Patent Grant
11385516
This invention relates to electronically controlling input light using a dielectrophoresis process (and more broadly generally phoretic processes from light, heat, chemical composition, magnetic fields, spin and orbital momentum in light fields, sound, etc.) on colloids or suspensions so that the direction and propagation of light in the colloid or suspension is controlled electronically to provide: fast light-beam steering, light focusing, light generation, and light wavefront control by particle placement within a control volume. These functions are very general and provide a means to address many disparate practical applications.
The material in this disclosure is a new sub-field of optics: the dynamic control of a refractive index distributions (and scattering in general) by controlling microscopic particle distributions to redirect light often along substantially curved trajectories with accuracy, precision, and speed in ways not possible before in order to replace or enhance traditional optics elements like lenses, mirrors, diffraction gratings, spatial light modulators etc. as well as to create new and hitherto unrealized capabilities.
Therefore, in this document I have taken the effort to develop this topic from first principles and to show many example embodiments for applications in a variety of fields. All the while I have focused on a small number of core principles and hardware elements. These core principles and hardware elements are the focus of what is actually claimed in this disclosure.
Fluidic Light Control
There are several fluidic methods discussed in the prior art for electronically redirecting light. Broadly speaking micro-optofluidic light guides and lenses are utilized for redirecting light and they are enabled by (1) changing the properties throughout a volume of a liquid (pressure, electric energy, orientation of suspended liquid crystals etc. . . . ) or (2) by changing the shape of a liquid's optical surface by using surface effects. Somewhat more specifically, electrostatic and quasi electrostatic fields in a liquid can provide a means for these changes and can provide: a net force on the bulk volume of a liquid and/or change the contact angle of a liquid's surface with a contacting solid by electronically changing the surface energy of liquids and solids.
In the case of substantially conductive liquids the surface shape is often changed electrically by means of an electrowetting precess, which requires that an insulator covers electrodes so that the electrodes at different potentials are not shorted through the liquid. A voltage across an electrode pair induces polarization charges in the insulators near the electrodes. These polarization charges interact with a polar liquid to change the liquid's surface energy and the wettability of the insulation contacting the liquid. This results in forces on a liquid in contact with the insulator that can move the liquid. It can also change the shape of the surface of a drop of a liquid as a function of applied electrode voltages.
However, in the case of substantially non-conductive liquids the surface shape is often changed electrically by means of dielectrophoresis, wherein the electrodes may now be in direct contact with a liquid that is surrounded by a gas or vacuum. A non-uniform electric field can then induce the liquid to move or to change shape by means of Kelvin forces.
The use of liquid drops in electrowetting and dielectrophoresis based light steering and focusing has disadvantages, including the need for: (1) small optics with limited light gathering ability, (2) multiple lenses to correct aberrations, (3) a limited range of refractive index due to the relatively low refractive index of most liquids so that the refractive power is small and light steering not appreciable.
There are numerous variations in the prior art for using the electrowetting approach including for example U.S. Pat. No. 7,898,740 entitled “Tunable Optical Array Device Comprising Liquid Cells,” issued to Jason Heikenfeld et. al. on 2011 Mar. 1. This prior-art patent shows an array of liquid-filled cells comprising at least two transparent and non-miscible fluids, each having a different refractive index. Voltages applied to the cells induce polarization charges that change the wetting angle of the boundary between the two non-miscible fluids, by a process called electrowetting, and this causes the optical boundary to change orientation so that light is refracted into different directions.
There are also numerous variations of the dielectrophoresis based approach of light control, including that discussed in a 2013 review article in the Journal of Physics D: Applied Physics, by Su Xu, et. al, entitled “Dielectrophoretically tunable optofluidic devices,” and having digital object identifier doi:10.1088/0022-3727/46/48/483001 J. Phys. D: Appl. Phys. 46 (2013) 483001 (14 pp). The article describes using dielectrophoresis for manipulating a quantity of liquid so that the shape of the liquid changes to provide lensing or guiding of light.
One particularly important point about the prior arts' use of dielectrophoresis for light control is that it is often the case that the focus of attention is a tiny liquid particle (e.g. a liquid drop), which is used in the dielectrophoresis process. The forces of the dielectrophoresis are directed on the liquid drops, which is immersed in a gas, another immiscible liquid or a vacuum medium and moved about by nonuniform electric fields.
In contradistinction, in this disclosure typically solid nanoparticles are immersed in a medium, typically a liquid and to a lesser extent potentially a gas. This may also include semi-solid particles like blood cells and other soft matter particles.
Alternately, in the prior-art the shape of liquid surfaces may be modified indirectly using soft-matter transparent elastomeric membranes, such as polydimethylsiloxane, in combination with actuators like piezoelectrics. Changing the fluid pressure causes liquid that is behind a membrane to push the membrane into a curved surface that mimics a lens shape that can subsequently redirect light.
The use of membrane-based light steering and focusing has disadvantages, including: (1) slow response time, (2) limited controllability of the membrane shape and (3) the need for actuators that may be complex and take up additional volume.
There are also numerous variations of a fluidic and membrane based approach of light control, including those discussed in the book “Introduction to Adaptive Lenses,” by Hongwen Ren and Shin-Tson Wu, printed in 2012 by John Wiley & Sons, Inc., ISBN 978-1-118-01899-6, which provides a review of light control techniques and documents optical surfaces formed by elastomeric membranes, piezoelectric actuators, electrowetting, dielectrophoretic optics comprising one or more liquids, liquid crystals, polymer dispersed liquid crystal lenses.
In yet another approach of the prior art two or more flowing liquids are used in guiding and/or focusing light. In particular, different flowing liquids that are moving relative to each other provide a refractive index gradient that is formed and controlled by the local geometry controlling the flow and speed of contacting liquids as a diffusion process works to mix some of one liquid having a low refractive index into another liquid having a higher refractive index. The gradient in the refractive index then provides an optical medium to modify the direction of light propagation.
The use of flowing liquids has disadvantages, including the need for: (1) an expendable feedstock of each liquid, (2) a pump to move each liquid in a flow, (3) the need for significant pump power to overcome friction and momentum and (4) the slow response time of flowing liquid system.
There are also numerous variations for liquid flow and induced refractive index gradient based approach of light control, including that discussed in an article published by the Optical Society of America by authors Richard S. Conroy, et. al., entitled “Optical waveguiding in suspensions of dielectric particles,” in Vol. 44, No. 36 of Applied Optics, 2005 Dec. 20. The article focuses on refractive index gradients formed by flowing liquids of differing refractive index.
Another example of controlling light by flowing fluids can be found in the journal review article “Micro-optofluidic Lenses: A review,” by Nam-Trung Nguyen and published by the American Institute of Physics in BioMicroFluidics 4, 031501 2010 and having digital object identifier doi:10.1063/1.3460392. The article notes various techniques of how flowing fluids, including those that have nanoparticle suspensions can redirect light by forming a refractive index gradient.
In yet another approach of conventional liquid crystals, including polymer dispersed liquid crystals, focus and redirect light not necessarily based on the shape of an optical surface, but rather on the orientation of liquid crystals within an optical volume to provide the needed phase shifts to allow the focusing and steering of light.
There are numerous variations on using liquid crystals for beam steering and focusing including: U.S. Pat. No. 6,958,868 entitled “Motion-Free Tracking Solar Concentrator,” issued to John George Pender on 2005 Oct. 25, which shows liquid crystal filled prism arrays that can steer light. Another example is U.S. Pat. No. 8,311,372 entitled “Liquid Crystal Waveguide Having Refractive Shapes For Dynamically Controlling Light,” issued to Michael H. Anderson et. al. on 2012 Nov. 13. This patent shows how to use evanescent field coupling to liquid crystals to steer light. Yet another example is U.S. patent application 2012/0188467 entitled “Beam steering devices including stacked liquid crystal polarization gratings and related methods of operation,” issued to Michael J. Escuti et. al. on 2012 Jul. 26. This patent shows stacks of polarization holograms (instead of the more common phase and amplitude holograms) and is used to steer light. The polarization holograms are formed from electronically controlled liquid crystals with cyclic variations in the liquid crystal orientations.
The use of liquid crystals has disadvantages, including the need for: (1) restrictions in the polarization of the incident and transmitted light and (2) the small difference in refractive index between the extreme values provided by the liquid crystal orientation, which limits the ability to redirect light over large angles.
A different approach to light control is found in U.S. patent application 2006/0050402 A1 entitled “Variable-Focus Lens and Image Taking Apparatus.” This patent application shows the use of electrophoresis, which is a strong polarity-dependent phenomenon related to double-layer charge formation at the surface some particles suspended in an aqueous media. The effect provides an effective particle charge and does not require electric non-uniform electric field to induce forces on particles, as is the case for dielectrophoresis.
In Ito's approach to light control electrophoresis is used to move particles in a liquid onto surfaces containing electrodes. The particles are used to build up new curved surfaces with different curvatures so that a variable focus lens is formed. In Ito's approach the rays of light move though the mixture of liquid and particles in piecewise straight line segments and not along curved trajectories. The technique is inherently a DC process and can have parasitic phenomena like electrolysis, it is also slow and very limited in its ability to move beyond light focusing as provided in Ito's disclosure.
Another approach to light control for displays is found in U.S. Pat. No. 7,106,297 B2 entitled “Color Display Device,” issued to John M. Shannon et. al. on 2006 Sep. 12. In this patent dielectrophoresis is used to move color absorbing particles into the path of input white light so that the output is given a color via selective absorption. The rays of light in this patent all move in piecewise straight line segments in the mixture of liquid and particles, and not along curved trajectories. Again, the only substantive modification to the input light is its remaining color in a subtractive color scheme. The ability to arbitrarily focus and steer light along curved trajectories is not provided by Shannon, as is required in the present invention.
Therefore, it is clear that there are many known methods for modifying light, however these known methods have an assortment of significant shortcomings that hinder applications. Consequently, there is a clear need for a method of controlling light that can overcome the shortcomings of the prior-art and provide new directions for a plurality of end-use applications.
The technical question addressed in this disclosure is how to actively take control of at least one of the refractive index and scattering properties of a particle-based colloid in such a way as to directly and electronically control the flow of light or indirectly and electronically control the manufacture of a device that controls the flow of light. The problem of how to provide fast, accurate, and precise light redirection of up to 2π radians in two dimensions and up to 4π steradians in three dimensions is addressed as are ray optics and modal optics where control of light fields in space and time are discussed. Particles of the colloid typically, but not necessarily, being nano-scale. The light controlled typically being at least one of narrowband, broadband, coherent, incoherent, low-power, high-power, polarization-specific (e.g. specific spin and orbital angular momentum), polarization-diverse, and “designer light fields” with temporal frequencies connected to spatial frequencies on a dispersion relation manifold.
This disclosure then addresses a plurality of optical problems and demonstrates methods and device embodiments to exploit the properties of particle-based optical colloids that allow electronic control of the colloid and the resulting control of the propagation direction of light for a number of applications, including: beam steering, focusing, concentration, wavefront-control, display of information, high-power laser beam combining, stable high-power fiber lasers, and other optical operations including those for optical device manufacture. Thus, the solution of a small number of engineering problems related to the control of an optical colloid may impact and enable a large number of applications and some of those application embodiments are demonstrated.
The application problems that are described herein typically require: low optical loss, broad spectral range, large angular steering range, high angular steering precision, high angular steering accuracy, polarization independence, voltage controllability, low power consumption, intensity-independent operation, fast responding, thin profile, configurable to steer, focus, concentrate and display information using light. Thus, part of the problem addressed is how to provide these properties, when needed, to the application embodiments through the demonstration of the small number of principles related to controlling a particle-based optical colloid.
The necessary problems of proper selection of materials for making devices functional, relatively safe, non-toxic and capable of working over temperature ranges that exceed the typical US Mil-spec temperature range (−55° C. to +125° C.) over extended periods of time is also addressed for systems comprising mixtures of solids and liquids.
More specific problem and application areas are addressed in
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To the best of the author's knowledge at the time of the writing of this document, the material in this document appears to describe a new subfield of optics. Additionally, recent review articles on the trajectory of the field of dielectrophoresis appear to neither teach or anticipate the material in this disclosure. An example of such a review article is given by Prof. Ronald Pethig of the School of Engineering, The University of Edinburgh, Edinburgh, Scotland EH9 3JL, United Kingdom, via the article “Review—Where Is Dielectrophoresis (DEP) Going?,” Journal of The Electrochemical Society, 164 (5) B3049-B3055 (2017).
In summary, this disclosure focuses on a small number of principles related to the electronic control of particle-based colloids that exploit dielectrophoresis (and other phoretic forces) for controlling the refractive index of the colloid and the resulting flow and properties of light. This has broad applicability and the many examples provided demonstrate the flexibility of the method.
The main focus of a solution to the above-stated problem is found in a process of Dielectrophoresis (DEP) in an optical colloid (or suspension), which modifies the properties of input light using small particles that are moved within a control volume to interact and scatter the input light under the influence of a non-uniform electromagnetic field. A secondary focus of a solution is found in the use of phoresis in general, i.e. where dielectrophoresis only being one form of phoresis and other phoretic forms are given later herein.
In particular, DEP provides ponderomotive forces on particles in a liquid (or gas) using particle interactions with a non-uniform electromagnetic field. Typically, a quasi-electrostatic field is the focus of attention, though this is not always the case in this disclosure. DEP-based ponderomotive forces do not require particles to be charged or to even be dielectric. In fact, all particles exhibit dielectrophoresis in the presence of nonuniform electric fields. However, a special case of particular interest is DEP for transparent dielectric particles driven by a harmonic non-uniform electric field, which can provide a (non-harmonic!) constant average ponderomotive force in one direction locally. This unidirectional ponderomotive force is a result of forces on a particle not being symmetric during each full oscillation of the electric field. The resulting average force can produce a constant drift velocity on solid nanoparticles because the friction of the viscous liquid on the particle resists the unbounded acceleration of the particle and the particle's velocity saturates to a constant. Moreover, the harmonic form of DEP avoids many parasitic phenomena, such as electrophoresis, electro-osmosis and electrolysis, which may make light control more difficult in practice.
Thus, while this disclosure has a strong focuses on harmonic DEP, the principles can also be applied to constant-field DEP processes if one is willing to address or potentially even utilize the parasitic phenomena in specific applications. Thus, this disclosure covers harmonic oscillating-field DEP, constant-field DEP processes, and even phoresis of other quantities (i.e. gradients in light intensity, temperature, magnetic field, etc. . . . ) to cause forces on particles to redirect light.
It should be pointed out that the word “dielectrophoresis” is typically associated with electrodes configured to provide a non-uniform harmonic electric field by means of impressed voltages. This is true in this disclosure too, however the concept of dielectrophoresis is also explicitly extended so that some embodiments may not have any electrodes at all. In this case it is typical that the electric fields are derived from light itself that compels the dielectrophoresis process. An example of this is a focused Gaussian laser beam where the focus region has non-uniform electric (and magnetic) fields. Thus, what is important is that there are non-uniform electromagnetic fields that are typically harmonic and not that there are electrodes. Also, the particles being controlled and coupled to light and optics can be dielectrics, metals, quantum dots, plasmonic dots, photonic bandgap dots, biological cells and many other complex and hybrid structures as will be discussed later for specific applications.
When we apply this ponderomotive force to large numbers of nanoparticles in a colloid we can control the distribution of those nanoparticles and the resulting refractive index as a function of space and time due to impressed non-uniform electric fields. Different size distributions of nanoparticles and different materials will separate differently in space and this allows for an optical element having many degrees of freedom so that various lens or beam steering aberrations are reduced or eliminated in one compact device. Therefore, instead of needing to use multiple lenses a single lens, having different electronically induced optical regions that are configured to eliminate aberrations, may be used. For example, by having some nanoparticles with positive material dispersion and other nanoparticles with negative material dispersion we can better manage chromatic aberrations in a single lens element.
A colloid is a homogeneous, noncrystalline substance comprising nanoparticles of one or more substances mixed and dispersed through a second substance. The nanoparticles do not settle or precipitate from a colloid and cannot be separated out by mechanical means like filtering or centrifuging the mixture. For example, under certain circumstances smoke is a colloid of particles in a gas medium. This is unlike nanoparticles in a suspension where such mechanical actions can separate the nanoparticles from a liquid eventually. Additionally, a colloid in a liquid host is often made stable against precipitation and flocculation by means of adding surfactants and other chemicals into the liquid or onto the nanoparticles. In this document the nanoparticles of interest typically have diameters of 1-100 nm, however this size range is not meant to limit the use of other particle sizes in this disclosure especially in the medical and three dimensional printing applications where larger particles (e.g. blood cells) are typical.
Consequently, as the size of the nanoparticles can be much smaller than the wavelength of light steered, e.g. visible light is from about 380-740 nm, we can approximately average the colloidal refractive index by using the volume fraction of the nanoparticles in the liquid. The direction of light passing through the colloid of nanoparticles is then controlled by using strategically located nonuniform electric fields to change the nanoparticle spatial distribution to redirect light. When used to steer light beams this is called Dielectrophoresis Beam Steering (DBS).
However, nanoparticles that are substantially smaller than the wavelength of light cost more to manufacture than larger nanoparticles and also have less surface area so that larger voltages are required to separate the nanoparticles and form a refractive index gradient. Therefore, the following disclosure also shows how to use larger particles by using a refractive index averaging process that includes diffraction effects of the light around the nanoparticles. This is accomplished by leveraging the theory of anomalous diffraction as applied to a colloid to properly describe the effective refractive index in a DEP driven colloid. It will also be seen that different probability distributions of nanoparticle size and material composition will interact in a complex way to further change the refractive index averaging process and that this can be leveraged to provide many additional degrees of freedom to control the redirection, concentration and focusing of light.
In this disclosure a distribution of dielectric nanoparticles in a liquid forms a colloid that is called an Index Gradient Liquid (IGL). This is different than an Index Matching Liquid (IML), also synonymously called an Index Matching Fluid (IMF), which has been used by this author in other disclosures for soft-matter beam steering. Thus, the refractive index of an IGL is electronically controlled to achieve a refractive index gradient for steering and focusing light, typically within a microfluidic Control Channel (pFCC), or a control volume in a solid and transparent host material. In contradistinction, an IML requires meso-scale advection of the optical liquid to achieve beam steering. IMLs and IGLs provide for different, but complimentary techniques to control light.
The foregoing discussion is only an introduction and other objects, features, aspects, and advantages will become apparent from the following detailed description and drawings of physical principles given by way of illustration. Note that figures are often drawn for improved clarity of the underlying physical principles and are not necessarily to scale and have certain idealizations introduced to show the essence of the method and embodiments and to make descriptions clear.
=50 and the nanosphere radius is b=100 nm at a wavelength of λ0=532 nm.
Introduction to Physical Principles
The following pages provide the details of the underlying theory and embodiments of DBS for both opto-electronic light steering and other optical operations from first principles where needed. The objective of providing the theory is so that there is no question as to exactly what it is that is being discussed in a very complex technical area. The detailed technical discussion also allows insights into the limits of the prior-art and ultimately allows those less familiar with the material to have easier access to the present material using one consistent and unified notation. This material is inherently multi-disciplinary in nature and draws from a number of diverse fields including, but not limited to: fluid flows, heat transfer, continuum mechanics, electromagnetics, electronics, chemistry, material science, optics, thermodynamics, computer science and mathematics.
The reader is advised that due to the large number of concepts needed to describe the material in this disclosure, the associated symbolic representations sometimes required symbol reuse or at least similar looking symbols. Such symbols are never mixed in the same discussion unless some form of symbolic distinction is provided. For example the identity matrix is I and the irradiance of light is I. The meaning of symbols is clear from the context of the discussion in the disclosure.
Dyadic Calculus for Dielectrophoresis
Consider 3-dimensional vector functions of space and time A and B, which are arbitrary and not necessarily field quantities from Maxwell's equations. The dyadic product of these three dimensional vectors is represented as AB and has nine components that may be written as a matrix having elements aij=AiBj and is given formally using the Einstein summation convention over two repeated indices as
AB=Ai{circumflex over (x)}iBj{circumflex over (x)}j=A1B1{circumflex over (x)}1{circumflex over (x)}1+A1B2{circumflex over (x)}1{circumflex over (x)}2+ . . . +A3B3{circumflex over (x)}3{circumflex over (x)}3, (1)
where double subscripts indicate summation over the three spatial dimensions and in general AB≠BA. No distinction is made here between covariant and contravariant vectors. In the following analysis all derivations are provided using cartesian dyadics, however the resulting equations are correct across any self-consistent system of coordinates.
We shall use the Levi-Civita tensor ϵijk, to make curl and other operations easy to compute. In particular, the Levi-Civita tensor takes the values
and we can show by direct computation that
ϵijkϵklm=ϵkijϵklm=δilδjm−δimδjl (3)
where δij is a Kronecker Delta function. The Levi-Civita tensor is used to write the cross product of A and B as
A×B={circumflex over (x)}kϵkijAiBj (4)
and the curl of A as
∇×A={circumflex over (x)}kϵkij∂iAj, (5)
where as usual we have applied the Einstein summation convention.
The first result of importance is the divergence of a dyadic product.
It may be expanded as follows
where in the future we may not always write obvious unit factors like ({circumflex over (x)}i·{circumflex over (x)}i) in intermediate results. Also note in the above equation we have used the fact that
(A·∇)B=A·(∇B)=A·∇B. (10)
Next, the cross product of the curl of a vector is expanded as
Similarly, its compliment is
B×(∇×A)={circumflex over (x)}i(Bj∂iAj−Aj∂jBi) (12)
Moreover,
(A·∇)B=Aj∂jBi{circumflex over (x)}i (13)
(14) (15)
so that
A special case of the above equation that is useful for analysis of Quasi Electro Static (QES) systems and DEP in particular, which specifies irrotational fields so that ∇×A=∇×B=0. Therefore, if we set A=B in Eq. 17 and assume irrotational fields then
where the last equality in the above equation was obtained by observing that for an arbitrary scalar U=A2/2 we can write the equation using the 3×3 identity matrix I. In particular,
so that
∇·U=∇·(UI)=∇U when U is a scalar. (20)
Next, by using Eq. 9 and its compliment, which is formed by swapping B for A and vice versa, we can solve separately for (B·∇)A and (B·∇)A
(A·∇)B=∇·(AB)−B(∇·A) (21)
(B·∇)A=∇·(BA)−A(∇·B) (22)
and now substitute these equations back into Eq. 17 so that
A special case for QES systems is where A and B are irrotational so that ∇×A=∇×B=0, then Eq. 23 can be written as
∇·(AB+BA)=∇(A·B)+A(∇·B)+B(∇·A) (24)
By setting A=B in the above equation we obtain
where A2=A·A. Now using Eq. 18 in Eq. 25 we obtain
so that
∇·(AA)=∇(A2)=∇·(A2I) when A is only irrotational. (27)
Additionally, from Eq. 25 and 27 we also have that
We summarize Eqs. 18 and 28 as
when A is only irrotational.
Notice that if A is solenoidal in Eq. 25 so that ∇·A=0 then we have from Eq. 25 that
when A is irrotational and solenoidal. Again, notice how different Eq. 30 is from Eq. 27.
The next relation that is needed is derived by using Eq. 2 and 3 so that the divergence of the ordered difference is
This result can be combined with Eq. 9 so that
which in the special case when A and B are solenoidal, i.e. with ∇·A=∇·B=0, then Eq. 33 reduces the above equation to
∇×(B×A)=A·∇B−B·∇A when A & B are solenoidal. (34)
Maxwell's Equations and Notation
We start with the space-time version of Maxwell's equations for macroscopic systems in SI units and the constitutive relations for a homogeneous, isotropic and linear materials. In particular, Maxwell's equations in SI units are
and the constitutive relations are(r,t)=ϵ(r,t)ε(r,t) (39)
(r,t)=μ(r,t)
(r,t) (40)
ƒ(r,t)=σ(r,t)ε(r,t). (41)
The electromagnetic quantities above include the electric field intensity ε [V/m], the electric field density [C/m2], the magnetic field intensity
[A/m], the magnetic field
[Wb/m2], the free electric current density
ƒ [A/m2], the free electric charge density ρƒ [C/m3], the permittivity ϵ [F/m], the permeability μ [H/m] and the conductivity σ [S/m]. If we assume that quantities are harmonic then we can decompose the space and time functions of fields and charges into functional products in phasor space and material properties into functions of space and frequency, which are to be calculated or assumed as the situation requires, so that in the frequency domain we have
(r,t)→D(r)eiωt (42)
ε(r,t)→E(r)eiωt (43)(r,t)→B(r)eiωt (44)
(r,t)→H(r)eiωt (45)
ρƒ(r,t)→ρƒ(r)eiωt (46)
σ(r,t)→σ(r,ω) (47)
ϵ(r,t)→ϵ(r,ω) (48)
μ(r,t)→μ(r,ω), (49)
where quantities like E(r) are in general complex valued quantities unless noted otherwise and i=√{square root over (−1)}. Alternately, we can write that ε(r, t)=Re[E(r)eiωt]. Either way we can rewrite Maxwell's equations as
∇×E(r)=−iω{tilde over (μ)}(r,ω)H(r) (50)
∇×H(r)=+iω{tilde over (ϵ)}(r,ω)E(r) (51)
∇·{{tilde over (ϵ)}(r,ω)E(r)}=0 (52)
∇·{{tilde over (μ)}(r,ω)H(r)}=0 (53)
where the third and fourth equations above follows from taking the divergence of the first and second equations and using the vector identity ∇·(∇×A)=0, where A is an arbitrary vector over space coordinates. Furthermore, we find that on using the constitutive relation ƒ=σƒε that Eq. 36 takes the form of Eq. 51 when
The above equations are provided to establish notation and sign conventions such as the phasor sign convention given by eiωt. Also, excitations may involve multiple frequencies so that in general ε(r, t)=Re[E1(r)eiω
Time-Averaged Maxwell Stress Tensor
In addition to Maxwell's equations Eqs. 35-38 and the constitutive equations Eqs. 39-41 there is the Lorentz force equation, which for a single charge is given by=qε+qv×
(59)
where q is the charge and v is the velocity vector of the charge in an inertial frame of reference. If we multiply both sides by N, which is the concentration of charged particles in units of number of charges per unit volume. Note that N is different than the concentration of nanoparticles per unit volume u. Therefore, then we can identify ρƒ=Nq as the free charge density, ƒ=Nqv as the free vector current density and the force per unit volume as
V=N
. Therefore, the force per unit volume, i.e. the force density, is
V=ρƒε+
×
. (60)
Also, because the speed of light squared is c2=1/(ϵμ) the Poynting vector divided by the speed of light squared in media is
Therefore,
Combining results by inserting Maxwell's equations Eq. 35-38 into the volume forces Eq. 60 and further exploiting Eq. 62 we obtain
and on rearranging terms and strategically adding a zero term using ∇·=0 then
Therefore, focusing initially on term-1 we find
where the first term in the last line is minus the gradient of the electric field energy density and the second term is the divergence of the dyadic product of field quantities.
Similarly, for T2 we can let →
, ε→
and ϵ→1/μ so that
Therefore, the Maxwell stress tensor, comprising field-induced mechanical stress matrix elements, is
The force per unit volume is given as
which has a first term due to radiation pressure and a second term due to field induced stresses.
For quasi electrostatic DEP systems the Maxwell stress tensor needs to be evaluated for the time average volume force, accordingly if we assume harmonic electric and magnetic fields with phase shifts then we may separate space and time. For example, for electric fields we can write
ε(r,t)=Re[{tilde over (E)}(r)eiωt]=E(r)cos(ωt) (70)(r,t)=Re[{tilde over (D)}(r)eiωt]=D(r)cos(ωt−ϕϵ) (71)
where ϕϵ=ϕϵ(r), E(r) and D(r) are real valued functions of position r. We can then write the time average Maxwell stress tensor as
Let us assume that electrodes, or nota bene a source of non-uniform electromagnetic radiation designed to replace electrodes, are introduced such that the electric field intensity being used to generate a ponderomotive force has an in-phase and a quadrature component so that by design
ε=ER cos(ωt)+EI sin(ωt) (73)=DR cos(ωt−ϕϵ)+DI sin(ωt−ϕϵ) (74)
where ER is not parallel to EI and the phase shift ϕϵ is due to the material properties. Then we consider the dyadic productε=DRER cos(ωt−ϕϵ)cos(ωt)+DREI cos(ωt−ϕϵ)sin(ωt)+DIER sin(ωt−ϕϵ)cos(ωt)+DIEI sin(ωt−ϕϵ)sin(ωt). (75)
The time average can be obtained by first using the trigonometric identities for arbitrary angles a and b
2 cos a cos b=cos(a+b)+cos(a−b) (76)
2 cos a sin b=sin(a+b)−sin(a−b) (77)
2 sin a cos b=sin(a+b)+sin(a−b) (78)
2 sin a sin b=−cos(a+b)+cos(a−b) (79)
so that the time average of ε over an integer number of cycles is formed. For example,
and applying this method to each term in Eq. 75 we obtain the time average dyadic product as
where the asterisk indicates the complex conjugate and Re[ ] is the real part operator. Also note that the complex permittivity is
{tilde over (ϵ)}={circumflex over (ϵ)}R+i{circumflex over (ϵ)}I. (82)
Additionally, we also find using essentially the same above procedure, that for an inner product the time average is
Therefore, the time average Maxwell stress tensor is
However, the electric component of the average ponderomotive force isE
=∇·
τE
(85)
and we immediately see a need to reduce the expressions for ∇·(EREI−EIER) and ∇·(ERER+EIEI). We shall assume that we are in the quasi electrostatic regime where both the real and imaginary parts of the electric field intensity are irrotational so from Eq. 32 we find that
∇·[EREI−EIER]=∇×(EI×ER) (86)
and from Eq. 27 with irrotational fields ∇×ER=V×EI=0 for quasi electrostatics we have
∇·[ERER+EIEI]=∇(ER2+EI2)=∇·(E2I) (87)
so, the quasi-electrostatic force, i.e. the Kelvin force density, is obtained from Eq. 84 as
This is the Kelvin force density applied to a spherical nanoparticle of radius b and volume
The value of {tilde over (ϵ)}={circumflex over (ϵ)}R+i{circumflex over (ϵ)}I associated with Eq. 88 is now derived heuristically by considering
We seek to find an expression for the local field in region Ω2 so that we can find an equation for the polarizability αS that relates the electric field intensity in the liquid EL to the polarization of the dielectric sphere. We start by considering the total local electric field in region Ω2 as the sum of the electric field intensity in the liquid plus the field due to the bound volume polarization and the field from the bound surface polarization on surface ω2. This results in a Lorentz relation between macroscopic effective fields (EL and Eloc) and polarization fields (EΩ
Eloc=EL+EΩ
where EL depends on the voltages applied to the electrodes, EΩ
p=αSEloc. (91)
This can be related to the potential of an average dipole moment. As we shall see this then provides the effective dielectric constant, which is the objective of our calculation.
To find the three field components of the Lorentz relation we need to find the various electric charges in the system first. In general, because the divergence of the macroscopic polarization is the volume charge density ∇·P=−ρb we have from Gauss's law ∇·E=(ρƒ+ρb)/ϵ0 that
P=D−ϵ0E, (92)
and the bound-surface-charge density per unit area is
σb=P·{circumflex over (n)}=(D−ϵ0E)·{circumflex over (n)}. (93)
Also, by integrating ∇·D=ρƒ over volume and using Gauss's integral theorem we obtain the free-surface-charge boundary condition
σƒ=(D2−D1)·{circumflex over (n)} (94)
where h is the normal unit vector pointing out of medium-1 into medium-2. At each of the boundaries we can write expressions for the bound and free charge densities.
For boundary ω1 the free, bound and total charge densities are
σƒ1=(D1−D0)·{circumflex over (z)}=ϵ0{tilde over (ϵ)}rLEL (95)
σb1=PL·{circumflex over (z)}=−ϵ0({tilde over (ϵ)}rL−1)EL (96)
σt1=σƒ1+σb1=ϵ0EL. (97)
where the complex relative permittivity {tilde over (ϵ)}rL was used to account for losses and phase shifts between the application of a harmonic electric field intensity and an induced polarization. Similarly, for the boundary ω3 we have
σƒ3=(D3−D2)·{circumflex over (z)}=−ϵ0{tilde over (ϵ)}rLEL (98)
σb3=PL·{circumflex over (z)}=ϵ0({tilde over (ϵ)}rL−1)EL (99)
σt3=σƒ3+σb3=−ϵ0EL, (100)
which shows that the total surface charge densities on the electrodes (i.e. σt1 and σt3) are of equal and opposite sign. It also shows that one component of the total electric field intensity within the dielectric is EL. However, this is not the only component because of the bound charges induced on the sphere. Therefore, the total local field must be EL plus a electric field component due to the sphere.
In particular, for the spherical boundary ω2 the charge densities are
σƒ2=ϵ0{tilde over (ϵ)}rLEL−ϵ0{tilde over (ϵ)}rSES=0 (101)
σb2L=−ϵ0({tilde over (ϵ)}rL−1)EL cos θ (102)
σb2S=+ϵ0({tilde over (ϵ)}rS−1)ES cos θ (103)
and therefore the total surface charge density on the sphere is
We can use the net charge distribution σt2 on the surface ω2 to obtain the total electric field at the origin of the nanosphere by integration from the source on the spherical surface to the observation point at the center of the sphere, i.e. in the (−{circumflex over (r)}) direction so that
Because we have taken the sum to ensure that we have the total electric field at the center of the sphere of radius b, we can identify that
so that we can change perspective from the sphere to the background liquid by using the electric field intensity in the vacuum as follows
Therefore, in the liquid we can finally write that relative to the local background medium of a liquid that
Next, we consider the bound polarization charges within the nanoparticle. We will assume that the atoms are on a cubic grid this is only a convenient assumption that can in general be relaxed. That is, the result is approximately correct for other geometries, including amorphous media like glass, but the calculation can become very complex in these other cases. In particular, the total electric field intensity is given by summing over source charges qj at positions rj so that
where p=qd and d is the displacement of the charges in the direction of the z-axis. Setting p=p{circumflex over (z)} and rm=xm{circumflex over (x)}+ymŷ+zm{circumflex over (z)} we obtain
But for a cubic lattice we are summing an antisymmetric function over a symmetric domain for the {circumflex over (x)} and ŷ components, which sums to zero for these two components. For the z-component we recognize that we can write rm2=xm2+ym2+zm2 and
therefore for a cubic lattice and as a good approximation for many other practical materials we have that
EΩ
From the above calculations we find that the Lorentz relation can be evaluated for the local electric field intensity with the spherical nanoparticle removed and the fields referenced to the liquid. The result in general is
Therefore, with a concentration of u particles per unit volume and a polarizability of as then using Eqs. 108 and 113
so that
If there is only one particle with a volume of =4/3πb3, where b is the particle radius, then the sphere's polarizability is
The effective polarization density of a collection of nanoparticles with concentration per unit volume u is then
so that
Note the real part of the {tilde over (ϵ)}rL is used to ensure consistency with the classical expression for electric dipole potential as a function of radius and angle—e.g. “Classical Electrodynamics 2nd Ed.” by J. D. Jackson.
Note that the above derivation is only exemplary and additional layers may be added to the dielectric nanoparticle to provide other functions for the complex dielectric constant that are consistent with multilayer spherical geometry.
For example, to ensure a stable colloid a surfactant may be added as a coating to the nanoparticle's outer layer. This would require that another layer of total bound charges at the boundary between the inner dielectric and the outer coating layer needing to be included into the analysis and a new expression for E.
Additionally, note that from Eq. 56 and assuming for convenience the imaginary part of the dielectric constant equal to zero we have
and we may place this in Debye relaxation form as follows
where we used the following definitions
and where τMW is the Maxwell-Wigner time constant. Therefore, after a little algebra the real and imaginary parts of K are
which is the complex Clausius-Mossotti (CM) factor for the solid sphere embedded in a dielectric liquid. The zero crossing frequency occurs when ω=ω0 so that KR(ω0)=0. Also, the peak frequency occurs when ω=ωp so that KI′(ωp)=0. This results in
Therefore, if we carefully select the material properties of the particles and liquid medium so that K0+K∞=0 then ω0=ωp=ωMW=1/τMW. This may not be the optimum configuration for controlling a particle distribution for a particular application.
Furthermore, note that KR and KI can take on both positive and negative values such that −½≤K0≤+1, −½≤K∞≤+1, −½≤KR≤+1 and 0≤|KI|≤¾. As we shall see this allows forces and drift velocities on nanoparticles to take on both positive or negative values depending on the frequency ω of the harmonic excitation of the voltages on electrodes. In the case of ϵS>ϵL we have that K∞>0. Therefore, to make K0<0 we would choose the liquid, which may be a mixture of several chemicals, to have σL>σS. This will provide a means to focus, defocus and collimate light using DEPS optics as needed for a particular application simply by changing the frequency and voltage amplitude on the electrodes, as is demonstrated later in specific embodiments.
Also, as an aside note that Eq. 121 is a modified bilinear transformation on the complex plane. Taking
z={tilde over (ϵ)}S/{tilde over (ϵ)}L (135)
we can then rewrite Eq. 121 as
On setting z=zR+izI and K=KR−iKI we can with some algebra obtain equations representing the CM factor on the complex plan parameterized by the real and imaginary parts of z. The curve equations are
which are orthonormal circles forming a modified Smith Chart that relates the dielectric constants of the liquid and the nanosphere to the CM factor.
Also, from Eq. 118 and Eq. 127 we can obtain the important results
{circumflex over (ϵ)}R=3ϵLKR (139)
{circumflex over (ϵ)}I=3ϵLKI, (140)
where KR and KI are a function of the constitutive parameters and the excitation frequency.
Finally, it should be noted that the conductivity of the particle σS is not always that of the bulk material making up the particle. In particular, as the diameter 2b of each particle decreases the impact of the surface electrical properties of the particle become more and more important. It can be shown that the conductivity is then given by
where σB is the bulk conductivity and kS is the surface conductance of the particle and is typically about 1-5 nS and may be dominated by the mobility of counterions in the surrounding electrical double-layer. The impact of counterions, which are associated with electric double layers, in increasing the effective polarizability of nano particles was first observed by Washizu et al., see M. Washizu, S. Suzuki, O. Kurosawa, T. Nishizaka, and T. Shinohara, IEEE Trans. Ind. Appl., 30, 835 (1994). Under certain conditions this can cause the magnitude (and even the sign) of the Clausius-Mossotti factors KR and KI to change from that anticipated from the bulk conductivity.
Time-Averaged Forces by Dielectrophoresis
From the force density Eq. 88, volume 89, and Eqs. 139-140 we can obtain the time-average force on a particle as Fdep=dep
whereby
In order to save some space it is convenient to write the above equation as
Fdep=∇[γ1E·E]+∇×(γ2Ei×ER) (143)
γ1=πb3KRϵR (144)
γ2=2πb3KIϵI. (145)
where KR and KI are now to be thought of in general as a function of space, time, constitutive parameters, and excitation frequency.
Moreover, a particle that is subject to a DEP-based ponderomotive force will feel the effects of its viscous fluid environment. In particular, Newton's second law at steady-state conditions becomes
where, the well known Stokes Law for spherical particle drag in a fluid was used to develop the second term and η is the dynamic viscosity of the fluid surrounding the nanoparticle. Therefore, the drift velocity is given by solving for dr/dt so that
which represents the forced motion of a particle due to a DEP process.
Next, let's consider the thermal motion of a particle under the influence of a DEP process. In particular, using Eq. 20 the average force can be written in terms of a conservative potential energy tensor −U so that the net flux of particles through the liquid under equilibrium and steady-state conditions is given by the drift current
jdrift(r)=u(r)vdrift(r)=−u(r)γ∇·U(r) (149)
where u(r) is the particle concentration and the diffusion of the particles due to thermal agitation and Brownian motion is given by
jdiff(r)=−D(r)∇·u(r), (150)
where u(r)=Iu(r), u is the particle concentration, i.e. the number particles per unit volume, and D is the to-be-determined diffusion coefficient of our analysis. Under equilibrium conditions the net vector flux of particles is zero so that jdrift+jdiff=0. Therefore,
jdrift+jdiff=−u(r)γ∇·U(r)−D(r)∇·u(r)=0. (151)
Observe that by the definition of the gradient if W is the energy then ∇W·dl=dW and ∇u·dl=du, where dl is the length element in the direction of the shortest distance between iso-value surfaces so that the gradient is a maximum directional derivative. Therefore,
so that
or on introducing additional terms that strategically have an inner product with l that is zero we find that
where the second term on the right sets the energy level to an arbitrary value and the third term has the property that the commutator of the outer products (GH−HG) is always a skew-symmetric matrix whereby Eq. 32 provides that ∇·(GH−HG)=V×(G×H). For this to occur we can imagine that the complex electric field intensity E is broken into an in-phase and a quadrature component that are directed along (for example) the x- and y-axis respectively and have no z-directed variations. In this way
where for example we might take
where functions g and h are real valued functions of the coordinates x and y only. Also, note that ER and EI are really not restricted to just the xy-plane. For example in cylindrical coordinates we might take a tangent plane to a unit cylinder at each polar azimuth angle as the plane wherein the in-phase and quadrature components of the electric field intensity are defined. This will have important consequences for beam steering from arbitrary formed hulls, e.g. the hull of an aircraft having complex form for proper aerodynamic properties.
Consequently, we can write the equations
However, in statistical mechanics the Maxwell Boltzmann statistics are used to describe the average distribution of non-interacting material particles over allowed energy states in thermal equilibrium. This distribution is applicable when the temperature is high enough or the particle density is low enough to render quantum effects negligible. Using this observation we write
where A is a constant related to the total number of particles in a unit volume and kB is Boltzmann's constant. Consequently,
Therefore,
and
hence
D=γkBT, (166)
which is an extension of the well known Einstein Diffusion Coefficient for the case where the Maxwell stress tensor has non-zero off-diagonal elements. Note that in general the diffusion coefficient can be a function of space, time and concentration.
The quasi-electrostatic approximation for the time averaged force associated with DEP assumes that the amplitude of the electric field intensity E is invariant in space. This can be relaxed if time variations are slow compared to the period of the harmonic excitation τ=2π/ω. Therefore, if t>>τ then approximately E=E(r, t)=ER(r, t)+iEI(r, t) so that the DEP particle flux is
By way of comparison, for the diffusion process the particle flux is
jdiff(r,t)=−D∇u(r,t), (168)
for advection in general
jadv(r,t)=u(r)v(r) (169)
and for electrophoresis (EP) for positive/negative charged particles
jep(r,t)=±u(r,t)μ(r,t)E. (170)
The total particle flux is then
and by continuity we can write
where Q(r, t) is a volume source in units of particle per unit volume.
For the special case where only DEP and diffusion are the dominant phenomena, i.e. gravity is ignored when surfactant coatings are used on nanoparticles to maintain a stable and separated colloid in a gravity field. We then obtain a total particle flux under quasi-electrostatic conditions of
j(r,t)=−D∇u(r,t)+u(r,t)γFdep(r,t) (173)
where the space and time dependence of Fdep(r, t) is introduced to account for slowly changing electric fields. In a particle-source-free region of space the continuity equation must hold whereby
and we obtain the equation
which is a Fokker-Planck type forced equation that is subject to initial, Neumann, and possibly periodic boundary conditions.
In particular, note that the solid boundaries of a control volume are subject to zero particle flux, normal to the surfaces, from the combined process of diffusion and electric-field-gradient forced advection. Therefore, the sum of the diffusive and advective particle flux is zero whereby jtot·{circumflex over (m)}=0, where {circumflex over (m)} is the unit normal vector
[−D∇u(rS,t)+vdep(rS,t)u(rS,t)]·{circumflex over (m)}=0, (178)
where rS represents points along the solid boundary ∂ΩS of the control volume Ω.
Moreover, when there are periodic boundary conditions that are enforced on the concentration u(r, t). e.g. because of a periodic potential inducing a periodic time-average electric field, then we must have that flows of particles leaving a periodic boundary surface ∂ΩL are matched by flows of particles entering the periodic boundary surface ∂ΩE. Then using subscripts E for particle “entering” and L for particles “leaving” we have for periodic regions typically that unit normals pointing out of the boundary satisfy {circumflex over (m)}L=−{circumflex over (m)}E={circumflex over (m)} and that
[−D∇u(rL,t)+vdep(rL,t)u(rL,t)]·{circumflex over (m)}L=[−D∇u(rE,t)+vdep(rE,t)u(rE,t)]·(−{circumflex over (m)}L). (179)
Next, observe that the volume integral of u must be a conserved quantity because the number of particles in the control volume is constant. Therefore, the time derivative of a constant (i.e. the total number of particles in a periodic unit cell) is zero.
As a check to confirm the model is compatible with this insight we can take the volume integral of Eq. 175 over the unit cell and use Gauss's integral theorem to obtain
This confirms that the boundary conditions and mathematical model are consistent with the intuitive sense of particle conservation and it means that the volume integral of u(r, t) over the control volume (and a unit cell in the case of a periodic system) is a conserved quantity so that
∫Ωu(r,0)d3r=∫Ωu(r,t)d3r. (181)
This provides a means to check a numerical calculation because over one (or more) complete units cells of the calculation we must find that
and any substantial deviations from this are an indication of problems with the numerical calculations. Thus, the above Eq. 182 is a necessary, but not a sufficient condition to show the validity of a calculation and may be used as a check of the fidelity of a numerical calculation.
The above analysis also shows that particle conservation, flow restrictions through the solid walls of the control volume, and periodic conditions insure that the particle concentration u should never be negative, i.e. u≥0 for all time and space.
Finally, the reader should note that strictly speaking ER and EI are not functions of time, however, because we are using a quasi electrostatic approximation they may vary slowly relative to the harmonic excitation. Therefore, the reader should be aware that in fact the following is valid:
by which ER and EI are now slow functions of time as well as functions of space.
An Alternative Formulation of Dielectrophoresis
For the avoidance of legal doubt please note that there is an alternative formulation of DEP-based ponderomotive forces that is formulated directly in terms of the excitation electric field phases so that even in the case of no gradients in the electric field magnitude there may still be a DEP process if there are gradients in the electric field phase. This alternative formulation is also included in this disclosure.
Recall from Eq. 142 that the time averaged DEP-based ponderomotive force on a particle is
where
|E(r)|2=ER(r)·ER(r)+EI(r)·EI(r) (185)
and when Eq. 73 is the electric field as expressed in phasor form. The electric field was written as
ε(r,t)=ER(r)cos(ωt)+EI(r)sin(ωt), (186)
to develop Eq. 184. However, we could have written the electric field equivalently as
ε(r,t)=εx(r,t),εy(r,t),εz(r,t)
(187)
where
εx(r,t)=Ex,0(r)cos[ωt+φx(r)] (188)
εy(r,t)=Ey,0(r)cos[ωt+φy(r)] (189)
εz(r,t)=Ez,0(r)cos[ωt+φz(r)], (190)
where φx(r), φy(r), φz(r) are the phase variation functions in the coordinate directions.
Next, we make the observation that an electrically neutral particle that is placed in an electric field will experience charge separation, i.e. it becomes polarized. Part of the particle at position r substantially acquires a net charge −q and part of the particle at position (r+d) acquires a net charge +q. Therefore, the total force on the particle is given by a Taylor expansion in three dimension so that
where is the polarization of the particle. Clearly, the polarization of the particle cannot happen instantaneously so that there must be a phase lag in the polarization from that of the driving electric field. This phase lag is accounted for by means of a complex-valued function K, which is called the Clausius-Mossotti factor.
The polarization field is(r,t)=
x(r,t),
y(r,t),
z(r,t)
(192)
Therefore, assuming that the particle is spherical and of radius b, then the induced polarization charge is jointly proportional to the volume 4πb3/3 and the quantity (3|K|ϵL), where ϵL is the dielectric constant of the liquid medium surrounding the particle and the number 3 comes from Eqs. 139-140. Therefore, on including equal phase lags φp in each coordinate direction we have that
where the polarization phase lag is φp. Whereby, on expanding Eq. 188 and Eq. 193 by trigonometric expansion so that, for examplex(r,t)=4πb3ϵLEx,0(r){(|K|cos φp)cos(ωt+φx(r))+(|K|sin φp)sin(ωt+φx(r))} (196)
then,x(r,t)=4πb3ϵLEx,0(r){KR cos[ωt+φx(r)]+KI sin[ωt+φx(r)]}. (197)
Using this relation, and its equivalent y and z component equivalents, it becomes possible to evaluate Eq. 191. The time average of the first term of this evaluation is
whereby on including the y and z components we obtain
where Ex,0=Ex,0(r), φx=φx(r), etc. and
|E(r)|2=Ex,02(r)+Ey,02(r)+Ez,02(r). (200)
Comparing Eq. 184 and Eq. 199 we can see the same form except that the second term in this alternative formulation of the DEP process does not necessarily need to have any spatial variations in the magnitude of the electric field. Instead, all that is needed are spatial variations in the electric field phase, which can also cause a DEP process. This demonstration explicitly extends the concept of a DEP process to also include phase gradients as well as magnitude gradients.
Noise & Impulse Based Low-Voltage Dielectrophoresis
This section extends the results of the previous section to show how we can exchange a large voltage amplitude of a small bandwidth signal with a small voltage amplitude of a large bandwidth signal. This allows will allow us to use at least one of a digitally derived voltage noise signal and voltage impulses in time.
To begin let's define the Fourier transform pair relating a time signal x(t) and its frequency representation as
where we have used a slightly unusual convention of placing the factor of 1/(2π) on the forward transform. This will help to keep formulas consistent with our phasor conventions for electromagnetism.
Generalizing Eqs. 188-190 we obtain
εc,m(r,t)=∫−∞+∞Em(r,ω)eiφ
where the c in the subscript of εc,m reminds us that this is a complex representation in the time domain, which requires taking the real part to obtain the actual function. The subscript m∈{1, 2, 3} corresponds to the x, y, and z coordinate axis. Similarly, we can generalize Eqs. 193-195 asc,m(r,t)=4πb3ϵL∫−∞+∞K(ω)Em(r,ω)eiφ
Notice that if Er(r, ω)=E0,m(r)δ(ω−ω0) then we recover the original equations Eqs. 188-190 and Eqs. 193-195 evaluated at ω=ω0 after taking the real part of the complex form. For example, Eqs. 193-195 are reproduced as follows
Starting with Eq. 191 we now calculate the time average of the complex valued force on a component by component basis. For example, if angle brackets operate on its contents to take the time average then Eq. 191 becomes
For the sake of simplicity let's work on just one general term and sum later. Therefore,
To evaluate the above expression note that time average is
where {circumflex over (δ)}(⋅) is the Kronecker δ-function—i.e. not the Dirac δ-function. To see that this is true note first that the time average of a sinusoidal signal is always over a single period of the sinusoidal signal, which is of course dependent of its frequency. Thus, for any pure sinusoidal signal the time average is zero.
That said, the special case of interest is when the sinusoidal signal has zero frequency. This occurs in the above average when α=−β. In this special case we have the time average of a complex constant exp i[φm(r, α)+φm(r, β)], which is just the constant itself.
Note that the factor of ½ in Eq. 208 arises due to the time averaging process. The easiest way to see this is to compare the time-averaging process of multiplied trigonometric functions compared to multiplied complex exponential functions. In particular,
therefore, when working with the complex form of the equation the averaging operator <⋅> must include a factor of ½ so that
Moreover, Eq. 208 is subject to constraints. In particular, if the phase is due to a time delay then phase is an odd function. To see this note consider cos[ω(t−t0)], where to is a pure time delay. Then it is clear that the phase is φ=−ωt0, which is an odd function of frequency ω. This intuitive sense of how the phase is distributed in frequency can be made more formal and we can also get the behavior of the amplitude as well.
In particular, let's consider a causal real valued signal, by which we will mean that the real signal exists after it is turned on, but never before it is turned on. While the reader may consider this to be the epitome of common sense, it also represents a kind of symmetry that we can exploit. In particular, we can write the causal real valued signal that is turned on at t=0 as
ƒcas(t)=ƒ(t)U(t), (211)
where U(t) is the unit step function
Then we can decompose the causal function into even and odd functions as follows
Let's define
Then the Fourier transform of the casual signal is
from which we see immediately that G(−ω)=G(ω) so that G(ω) is even and that H(−ω)=−H(ω) so that H(ω) is odd. Moreover, G(ω) is a purely real function and H(ω) is a purely imaginary number. For example,
Similarly, we find that
As we shall see this implies that both Em(r, ω) and φm(r, ω) are real functions. Also, note that we can instead treat the multiplication in the time-domain of ƒ(t)U(t) as convolution in the frequency domain. This alternate line of reasoning leads to the Kramers-Kronig relations that connects the real and imaginary parts of a complex function that is analytic in the upper half plane.
Now, recall that we had defined the Fourier transform of the causal electric field signal component as Em(r, ω)eiφ
Now there are four cases to consider corresponding to whether or not Em(r, ω) and φm(r, ω) are each even or odd real functions. Using the fact that an even times an even function is an even function; an odd function times an odd function is an even function; and an even function times an odd function is an odd function.
The first case occurs if Em(r, ω) is even in ω and φm(r, ω) is even in ω. Then we find that G(ω) is even and H(ω) is even. Thus, this case is inconsistent with a causal signal.
The second case occurs if Em(r, ω) is odd in ω and φm(r, ω) is odd in ω. Then we find that G(ω) is odd and H(ω) is even. Thus, this case is inconsistent with a causal signal.
The third case occurs if Em(r, ω) is odd in ω and φm(r, ω) is even in ω. Then we find that G(ω) is odd and H(ω) is even. Thus, this case is inconsistent with a causal signal.
The fourth case occurs if Em(r, ω) is even in ω and φm(r, ω) is odd in ω. Then we find that G(ω) is even and H(ω) is odd. Thus, this case is consistent with a causal signal.
Thus, we conclude that for a causal signal we must have that both Em(r, ω) and φm(r, ω) are real functions and that they are even and odd functions in frequency ω respectively so that
Em(r,−ω)=Em(r,ω) (220)
φm(r,−ω)=−φm(r,ω). (221)
We can use these results with Eq. 208 to evaluate Eq. 207 as
However, using Eq. 127 and taking the real part of the above expression we have that
now summing over m and q and changing back to an c dummy variable we get
which is equivalent to
which shows that even though the force is a nonlinear function of the electric field intensity, the principle of superposition of forces over different frequencies still holds. Thus, to overcome any high-voltage using one excitation frequency ω we instead use several discrete frequencies or even a continuous band of frequencies to drive a DEP process.
Note that from Eqs. 131-132 that KR(ω) is an even function and KI(ω) is an odd function. Moreover, for causal signals in Eq. 225 we have ε(r, ω)=ER(r, ω)+iEI(r, ω), which must have by Eqs. 216-218 that ER(r, ω) is an even function in ω and that EI(r, ω) is an odd function in ω. Thus, the first integral in Eq. 225 has all even functions and the second integral of Eq. 225 includes the product of two odd functions and an even function, which in total is an even function. So we have that both integrals of Eq. 225 have even kernels and integration over both positive and negative frequencies ω is justified.
Next, by way of example, consider the interesting case where the circulation of the electric fields is zero and only the first term survives in Eq. 225. Moreover, when the frequencies are much less than the zero crossing frequency (i.e. ω<<ω0) of the Clausius Mossotti factor then KR(ω)≈K0 is constant, and the force is
where the integral is the Power Spectral Density (PSD) of the signal and denoted by SEE(ω). This suggests that we may use at least one of a deterministic and a stochastic signal to generate the ponderomotive forces on particles in a colloid as both deterministic and a stochastic signals have a PSD. For the reader that is not familiar with stochastic processes they may imagine that a random signal is first recorded and then played back as a deterministic signal. Therefore,
F(r)=πb3ϵLK0∇SEE(r) (227)
≈ϕb3ϵLK0∇|E(r,ω)|2Δω (228)
where the last line assumed a constant gradient of the squared magnitude of the electric field. We can see that force is now proportional to the bandwidth Δω.
Equation 228 is an example of an amazing and general result, which is that we can trade signal amplitude for signal bandwidth to achieve the same ponderomotive force on particles in colloid. This can be done impulsively with impulse signals over one or more short periods of time. It can also be done with a noise signal over a sustained period of time. Moreover, these signals may be generated with low power digital electronics. Therefore, by increasing the conductivity of the liquid σL we can increase the noise bandwidth because the zero-crossing frequency of KR(ω) increases and the force on particles are driven by low-pass broadband noise. Alternately, frequencies greater than the zero crossing frequency may be obtained form high-pass filtered noise and the conductivity σL may be decreased if desired to increase the bandwidth above the zero-crossing frequency. One of the advantages of noise above the zero crossing frequency ω0 is that slower particle dynamics can evolve in time, which may be desired, and the fast dynamics is now associated with the high-pass noise-driven ponderomotive force process.
Finally, we may generalize the diffusion equation of Eq. 175 to be
where the Force Spectral Density (FSP) is
In summary, this section has shown that we can have a profound impact on reducing the needed voltage levels for a DEP process by using noise and other broadband signals. See for example
Summary of Nanoparticle Distributions in Colloids
The following equation summary assumes that only DEP and diffusion processes are at play and gravity effects are negligible in a stable colloid. Additionally, harmonic voltage excitations are provided to electrodes that are in the vicinity of a colloid of solid nano-particles in a liquid, both of which are typically (but not necessarily) optically transparent. The liquid and solid nanoparticles are typically not of the same bulk refractive index as a function of wavelength. Units are designated in terms of L for length, T for time, M for mass, I for current, θ for absolute temperature, and # for number of nanoparticles. In summary,
Volume-Averaged Refractive Index Distributions
The objective of this section is to show that DEP can be used to create refractive index gradients that allow a light beam to be directed along curved trajectories. In particular, a colloid comprising transparent sub-wavelength-scale particles in a transparent liquid (or transparent gas) host medium provides a volume averaged refractive index. By changing the local volume fraction of nanoparticles within a colloid the point-to-point effective refractive index may be changed dynamically in time.
If we initially assume that scattering processes have no impact on the average refractive index then the volume fractions for the liquid νL and solid νS components of a colloid must sum to unity so that
νL+νS=1, (232)
and we can derive a relation for the average refractive index. This is based on our intuition of how the refractive averaging process works and provides an average real refractive index of
The volume fraction of the solid nanoparticles νS has no units and is a pure real number between zero and one such that νS is obtained by multiplying the particle concentration by the volume per nanoparticle so that
νS≡u (234)
whereby it is possible to provide a direct linear relation between the nanoparticle concentration u and the average refractive index as
nA(r,t)=nL+u(r,t)(nS−nL). (235)
This is a direct linear relation between nanoparticle concentration we found in the previous sections and the average refractive index of the colloidal suspension.
Strictly speaking Eq. 235 is only correct when the radius b of the particle approaches zero, i.e. when b→0 and scattering processes are essentially zero. As the particle size approaches zero the required voltage for changing the spatial distribution of nanoparticles goes to infinity. So we are motivated to obtain a more realistic averaging equation between nL and nS so that voltages are in a practical range for engineered devices.
Additionally, note that when the particle size is much smaller than the free-space wavelength of light, typically b≤λ0/20, then we have Rayleigh scattering as a light beam passes through the colloid comprising the dispersed nanoparticles. As the particle size increases then typically we would use the full Mie scattering theory for nanoparticles of comparable size to the wavelength of light to describe the scattering process.
For the purposes of this discussion the scattering can be broken down into roughly two components. The first component is scattering into a direction that is far from the direction of wave propagation. For example, perpendicular to the direction of propagation of the incident light. This type of scattering is highly dependent on the size of the particle relative to the wavelength of the light. The second component is light scattering that is roughly in the direction of the incident light, which tends to be approximately independent of particle diameter. See for example “Light Scattering by Small Particles,” by H. C. Van de Hulst having ISBN 0-486-64228-3 and “Refraction by spherical particles in the intermediate scattering region,” by G. H. Meeten, 1996, Optics Communications 134 (1997) 233-240.
When the nanoparticle is made of a material that is transparent, but of a refractive index nS that is different than the surrounding liquid medium nL then it can be shown that particle size has a very weak dependence on the scattering process and we can simultaneously account for both diffraction and particle induced phase shifts. This means that even for particles that are large compared to the wavelength, an expression for the scattering cross-section close to the direction of propagation is roughly independent of the type of scattering theory that is used to derive it. That said, we are going to use the theory of Anomalous Diffraction (AD) to obtain analytic expressions for the scattering process and then use that information to develop an expression for the average refractive index that is similar to Eq. 233, but modified to account for scattering processes of practical larger nanoparticles. In this way we can avoid the complexities of Mie and other forms of scattering theory.
For DEP-based beam steering the colloid or suspension comprises a liquid and nanoparticle mixture having two operational frequencies. The first frequency is that associated with DEP control of the nanoparticle distribution in the liquid. This frequency is roughly on the order of 101 to 108 Hz for electrode-based DEP and at this frequency there are nonzero loss tangents so that there is a real and imaginary component to the dielectric constants.
In contradistinction, the optical frequency of light is approximately 1014-1015 Hz, where by convention we tend to specify free-space wavelength rather than frequency (i.e. roughly 1000-100 nm respectively). We will assume that the materials are lossless at this higher optical frequency. This is not a requirement, but it makes the calculations simpler and the interpretation clear.
So, for the sake of simplifying the description to its important points, in this document we assume a nonzero loss tangent only for the low frequency effects associated with DEP. In actual devices there will always be some losses at all frequencies.
As we shall come to see, the scattering of light by many lossless colloidal nanoparticles removes light from the direction of propagation. Thus, we will be able to introduce an effective complex refractive index to model the loss process. However, it must be stressed that the loss is a result of scattering by diffraction and not ohmic heating of the liquid or nanoparticles.
Consider an incident light field propagating in the z-direction given by its electric field phasor
E1=E0e−ikz (236)
where we have used a scalar to represent the incident electric field strength E0 instead of a vector quantity because spherical nanoparticles do not have a preferred orientation and we can therefore ignore the light's polarization in this special case. In the far field the scattered electric field intensity will take the form
where S(θ) is the normalized amplitude function of the scattering particle, r is the distance from the center of the nanoparticle to the far-field point (x, y, z) and θ is the angular direction of scattering relative to the direction of incidence, i.e. the z-direction.
The total field E of the incident and scattered field is then the superposition
but, in the far-field where z is large relative to x and y, we can use a Taylor expansion to obtain an approximation so that
and in the direction of incidence θ=0 therefore
where E1=E0e−ikz. This process can be extended for many identical spherical nanoparticles having a concentration of u [number of particles per m3] in a thin infinite planar slab of thickness z=1 according to
However, z>>√{square root over (x2+y2)}=ζ and the distance to the observation point is rα≈r so that for large numbers of particles observed from the far-field, after the light passes through the layer of nano particles, we have in the limit that
Moreover, the integral in ζ is easy to evaluate by observing that
where a=ik/(2r). So on substituting this result we find the expression reduces to
where in general we have S(0) is a complex quantity.
Let n=n′−in″ represent the effective complex refractive index of the colloidal suspension of nanoparticles that includes diffractive scattering losses from a large number of otherwise lossless dielectric nanospheres at the optical free-space wavelength λ0. That is, we say that in the direction of θ=0 we have losses and a non-zero imaginary component of the effective refractive index. Then the complex phase shift is
The effective complex refractive index of the colloid is then defined by writing
so that
where S′(0)=Re[S(0)] and S″(0)=Im[S(0)] so that on equating real and imaginary components on the left and right hand sides of the above equation we find a relation between the effective refractive index and the scattering amplitude as
Moreover, we can use the fact that the volume fraction of the nanoparticles is νS=u and the individual nanoparticle volume is
=4/3πb3 so that
and therefore
where the nanoparticle size parameter χ=kb was utilized. Then by using Eq. 250 the magnitude of the Poynting vector is
where Cext is defined to be the extinction cross-sectional area and
where G=πb2 is the geometric cross sectional area of a spherical nanoparticle. We further define the real part of the extinction efficiency in the direction of the incident light as
and the imaginary part of the extinction efficiency is
Therefore, using Eqs. 252-253
and similarly
So that we have developed the components of the complex effective refractive index in terms of the extinction scattering efficiency, the liquid's refractive index and the volume fraction of the spherical nanoparticles.
Next, consider a plane wave incident on a spherical particle as shown in
The diffraction process is accounted for by using Huygen's principle, which states that each point of a wavefront can be considered to be the origin and source of wavelets that radiate spherically. For example a plane wave
E=E0e−ikz (260)
can have an array of area elements, each having area dA across its surface, which serve as the sources for Huygen's wavelets. In this case we expect that in the far-field
where the proportionality constant (not to be confused with charge) q is a to-be-determined and
so that
where the last line is the original plane wave in the far field after free space propagation and where the integral on the second line is evaluated using Eq. 243 and the fact that r≈z in the far-field. On solving the last two lines for the proportionality constant we get that q=i/λ so that
This is the disturbance caused by a wavefront having area dA and amplitude E0 at a point having a distance r from the area element, where r is in a direction that is approximately in the direction of propagation of the incident wavefront. It is clear that when dA=rλ that there is no focusing or defocusing of the wave. Therefore, the width of the source must be approximately ω=√{square root over (rλ)}. The implication is that it is impossible to have rays traced from a particle having a width (diameter) of λ or smaller, and in general provides the Fresnel equation
which is an exact solution of the full time-domain wave equation under paraxial conditions, i.e. the paraxial wave equation, when the phase term in Eq. 266 uses expanding Huygens wavelets from each source point in a source plane with a Taylor expansion of the distance to second order
and the denominator in Eq 266 uses r≈z.
By way of example: the light scattering that gives rise to the apparent white color of a white paint is produced by large (1-10 micron sized) spherical micro-spheres of transparent titanium dioxide that are included in the paint by the manufacturer to scatter the light and provide the appearance of a white paint. However, the white color will typically not be present for small diameter nanospheres (e.g. <50 nm), which are sized significantly less than the wavelength of visible light. This idea may be generalized to any wavelength of interest and it is very important because we do not want the addition of nanoparticles to cloud and substantially scatter light from the IGL beam steering liquid.
With this analysis and example in mind we now return to
Eparticle+Ehole=Eincident (268)
This theorem can be modified slightly to account for a transparent dielectric particle that introduces optical phase shifts into the incident field. In particular, if from each area element of the hole we subtract off the complex amplitude of the incident wave and then add back in the complex phase-modified field we will have accounted for the phase shifts of the transparent bulk medium of the nanoparticle. In this way we can approximately account for the phase lag induced by the nanosphere of a refractive index different than its surrounding medium. Therefore, Babinet's principle in the far-field becomes
where from
Clearly, p is the phase lag directly through the central diameter of the nanosphere. Moreover, from Eq. 269 we have
where the identification of S(0) was obtained by comparison to Eq. 240. Moreover, from
Let s=sin τ then the complex scattering amplitude is
From Eqs. 256-257 we get the complex extinction efficiency as
Next, plug in the expression for Qext″ into Eq. 258 and simplifying using Eq. 271 we find
Thus, on setting nA=n′ for the average real refractive index we get
where the phase lag through the diameter of a nanosphere is
and where λ0 is the free-space wavelength of the light. These equations should be compared with the intuitively derived expression of Eq. 233, where we note that in the limit of small particles that the correction factor
so that Eq. 233 and Eq. 277 are identical. Notice also that no attempt to further developed the expression for n″ has been undertaken because the scattering losses are more strongly regulated by scattering processes roughly perpendicular to the direction of the incident light and are strongly size dependent. This requires the use of an appropriately chosen scattering formalism (e.g. Mie scattering) to properly account for the loss processes. Finally, the equation for the refractive index Eq. 277 can be connected to Eq. 231 by noting that
νS(r,t)=u(r,t) (280)
where =4/3πb3 and we consider the volume fraction a function of space and time under the control of voltages from electrodes. Then
By way of a numeric example, if b=25 nm, (nS−nL)=0.5 and λ0=500 nm then p=π/10 and the correction factor would be about 0.9902, so Eq. 233 is off by about 1%. However, the needed voltages for a 25 nm radius nanoparticle are high and will require high DEP voltages to control and potentially longer time periods to stabilize at the steady state. Therefore, if b=100 nm we find that the correction factor is 0.8507 and represents nearly a 15% error that does need to be taken into account.
In summary, light passing through a colloid or suspension will encounter liquid and nanospheres that are made of a material that in bulk is transparent. If at the optical frequencies the bulk materials are lossless then there will still be an extinction process that is governed by scattering of light from the nanoparticles. While scattering in directions substantially different than the direction of incidence is a strong function of nanoparticle size, this is not the case for light scattered into a narrow range of angles about the incident direction where the dependence on nanoparticle size is much weaker. In this case we can avoid using the exact Mie scattering theory or specific scattering approximations that are optimized for particle size relative the wavelength, e.g. the Rayleigh scattering approximation for particles small relative to wavelength, and use a scattering approximation called the Anomalous Diffraction approximation, which provides a compact analytic expression for the average refractive index of the mixture of liquid and nanoparticles.
The Anomalous Diffraction approximation, was originally developed by Dutch astronomer van de Hulst describing light scattering for optically soft spheres. It also goes by the name of the van de Hulst approximation, eikonal approximation, high energy approximation or soft particle approximation and it allows us to replace the lossless refractive indices of the liquid and nanoparticles with a complex, and therefore lossy dielectric constant n=n′−in″ of a homogenous material. The essence of the approximation was that n could be related to the scattering amplitude in the forward direction S(0) as the loss mechanism instead of a ohmic loss. The value of S(0) being derived from using a modified version of Babinet's principle to account for both diffraction and phase lags introduced in the incident wave from the transparent sphere. The technique can be generalized to account for particles that are not spherical and for intrinsically lossy liquids and nanoparticles, though the mathematics is more involved.
In practice, obtaining spherical nanoparticles of only one size b is often difficult or expensive. Typically there is a distribution of sizes that are available. With that in mind we can generalize Eq. 277 to
where the total volume fraction of the solid nanospheres is νS, which is divided into M distinct different sphere radii bm. The different radii then provide different phase diameter phase lags
where α is a convenient constant to be used shortly. Moreover, the sum over the individual number of nanoparticles in each size is
NT=N1+N2+N3+ . . . +NM (284)
where NT is the total number of nanoparticles. Therefore,
which can be rewritten in terms of the probability mass function ƒm
ƒ1+ƒ2+ƒ3+ . . . +ƒM=1. (286)
Therefore, the volume fraction of the mth sphere size is the ratio given by the volume of the nanospheres to the total volume as
where 0 is the maximum unit cell volume that allows one nanoparticle to fit therein and
0NT is the total volume of a colloid filled with nanoparticles. By using Eq. 287 and taking the limit of many nanoparticles we can move to a continuum of nanoparticle sizes so that
where ƒb(b) is the probability density function (PDF) of the random variable b. If we take
p=ab (289)
where
then db=dp/α and
Thus with different particle size distributions we will get different correction factors.
For example, if we assume that nanospheres are fabricated with a uniform distribution of sizes, perhaps by using a nano-sieve during fabrication
then Eq. 291 can be integrated directly and provides that
where for uniform probability distributions
This can be written in several ways, however let's observe that the average particle volume is
then for uniform distributions of nanospheres having radii ranging from 0 to bmax we find
where the average volume fraction of nanospheres in the colloid or suspension is
Also, note that in the limit of small particles that
so that we can see that very small particles with a range of sizes do not have a great impact on the overall performance if the particles are small. The impact of using such small particles is the need for high DEP voltages and perhaps high nanoparticle cost, which may not be desired.
Conversely, by moving to other probability distribution functions for particle sizes, and potentially a range of refractive indices, we can gain added degrees of freedom for forming optical elements having different layers and mitigating optical aberrations. The resulting average refractive index will take the form
nA=nL+δn1νS1
F1(p)+δn2
νS2
F2(p)+ . . . +δnM
νSM
FM(p) (299)
where δnj=nSj−nL, the refractive index of the jth material, the correction factors are Fj(p) and νSj
are the average volume fractions for the jth material used such that the total volume fraction is
νS=νS1
+
νS2
+ . . . +
νSM
. (300)
These different forms of averaging the refractive index provide the basis for correcting chromatic, spherical and other forms of aberrations in an optical system. It also provides the basis for solving inverse problems where the objective is to find the size and electrical properties of the particles and liquid via optical probing.
Finally, note that large particles in a colloid can have dipole moments interact, which may cause the particles to align and form a “pearl chain.” This periodic geometry may lead to diffraction effects. This is avoided by using small nanoparticles particles that are constantly randomized by the Brownian motion.
Generalized Refractive Index Averaging
In this section we discuss an experiment performed by the author. In particular, consider
In
In
The chemical species do not chemically react and can form a pure mixture. The atoms making up the individual molecules are about 0.1 nm in diameter. However, the MPPS and EA molecules are very roughly 1 nm in diameter with 10 atoms (in order of magnitude). In contradistinction, a 50 nm diameter solid nanoparticle (as considered is some embodiments in this disclosure) has 0.1 nm atoms and roughly 100 million atoms in it volume.
This condition is different than that which was discussed in the previous section because both are liquid and both are very small in diameter. Nonetheless, we find that at 22.5° C. the measured volume fraction of EA is 0.09584±0.00020 and the refractive indices are: nMPPS=1.498±0.0001, nEA=1.372±0.0001, and nMPPS+EA=1.494±0.001 (i.e. a less precise average measurement). This mixed refractive index is that of acrylic, i.e. poly methyl methacrylate (PMMA), which was used as a reference for calibration purposes. We may then assume that there is a volume fraction correction Φ so that
nMPPS+EA=nMPPS+(nEA−nMPPS)ΦνEA (301)
whereby
This means that if we use Eq. 277 so that
then we have that p=3.06±0.027 radians of phase shift. This appears to imply that there is π radians of phase shift associated with the interaction.
Note, that strictly speaking it is not justified to use Eq. 277 given the size of both molecular species is so small. It is still quite intriguing that we find a π radians of phase shift at the edge of the error bars. Clearly, more analysis is needed here to validate the result and extend the correction of Eq. 303 to this case.
Nonetheless, the value of Φ does provide an experimental volume fraction correction and it may be used in cases where the theoretical analysis is not sufficient to cover the case at hand. Thus, by this example we extend the volume averaging process of Eq. 277 into a general form
nA=nL+νS(nS−nL)Φ (304)
where Φ is empirically determined and is a function of the important parameters of the averaging process, which covers many other cases not covered so far in a detailed theoretical analysis herein.
Fermat's Principle, Atomic Transitions & the Eikonal
In this section we will develop the rules for light propagation both with and without light-emitting atomic transitions. Atomic transitions in a GRIN medium provide a loophole in the physics of light propagation that allows an optical system to bypass étendue conservation laws of ray optics and this principle is especially useful in some forms of DEP-based displays and elsewhere.
Additionally, in this section an exact solution to the well know Eikonal equation is provided and later applied to DEP-based lens design and other applications. The Eikonal equation is central to light propagation in a GRIN medium and it is well know to be extremely difficult if not impossible to solve exactly. Nonetheless, if the ray trajectories are known a-priori then the needed GRIN distribution can be found analytically. This exact solution to the Eikonal is both possible and very useful for DEP-based optics.
With these objectives in mind consider this: light that passes through a GRIN medium may interact with an active medium where nanoparticles can absorb and reemit light via both elastic and inelastic scattering precesses, therefore we consider the rules of combining probability amplitudes. From Eqs. 35-41 with regions of space with no macroscopic sources and a freely propagating electromagnetic wave we have
Take Eqs. 305-306 and multiply through by √{square root over (ϵ/2)}. Also, take Eqs. 307-308 and multiply through by i√{square root over (μ/2)}, where i=√{square root over (−1)}. Then we can add the curl and divergence equations together respectively to obtain
where ε and are real-valued functions of space and time. Let us also define the complex vector electromagnetic amplitude as
where the asterisk indicates the complex conjugate operation. As a consequence
However, because of the vector identity ∇·(V×ψ)=0 there is really only one equation as can be seen by taking the divergence of Eq. 314, therefore
where c=1/√{square root over (ϵμ)}. On rearranging and multiplying through by ℏ and using the fundamental quantum assumption Δε{tilde over (ω)} (where Δε is the change in energy in this context and is not to be confused with the electric field) and since k={tilde over (ω)}/c we obtain
which may be written in terms of an electromagnetic Hamiltonian operator =[ΔE/k∇x], which has units of energy, so that we obtain a vector form of Schrödinger's wave equation
and it can be shown that is Hermitian.
Consequently, Maxwell's equations are equivalent to a vector Schrödinger wave equation for wave propagation within a homogenous, isotropic and linear material and we can heuristically identify ψ as a vector quantum probability amplitude for photons that captures the polarization properties. As such ψ is subject to being constructed by similar rules as were used to construct the original Schrödinger equations. In particular, we can leverage the rules for combining probability amplitudes for photons and electrons, as developed by the physicist Richard P. Feynman (e.g. see “Quantum Mechanics and Integrals,” by Feynman and Hibbs, ISBN: 64-25171) wherein we distill that:
With this in mind consider light that has transitioned from a first medium into a colloidal medium at an optical boundary and assume that this light has a phase of
ϕ=k·r−{tilde over (ω)}t (319)
where k is the vector wavenumber, r is a point in space along the path of the light, {tilde over (ω)} is the radian frequency of the light (i.e. not the frequency of the harmonic excitation of electrodes) and t is the time. By Eq. 318 the phase of the electromagnetic wave is also the phase of the corresponding probability amplitude. Therefore, multiply Eq. 319 through by the reduced Planck constant so that
ϕ=
k·r−
{tilde over (ω)}t (320)
and use the fundamental quantum assumptions whereby the energy transition in an atom (or quantum dot etc. . . . ) is Δε={tilde over (ω)} and the vector momentum of the light is p=
k. On taking the time derivative we then obtain the Legendre transformation
The propagator of the probability amplitude to go from point-1 in space-time to point-2 in space-time along an mth arbitrary path is then
where i=√{square root over (−1)} and C(m) is a vector of proportionally to account for polarization effects on the mth path. To go from point-1 to point-2 to point-3 and so on to point-N we have by the rule for non-interfering paths that
where C(m) is the vector proportionally constant that is different for each path and polarization. Additionally, we must also consider all possible separate interfering paths along which the photons may also travel, therefore let m take on different values for different spatial paths so that we may write a Feynman path integral for the light path in
Note that the exponential in the above equation takes on a broad range of complex values as the phasor rotates in response to the evolving wave. It can be shown that on a statistical average the phasor sum of the path integral is therefore zero. To avoid this and obtain a non-trivial solution we must require that the exponent is stationary about a specific path in space-time so that variations 6 of the exponential integral must obey
δ∫t
or on recognizing that p=p0nτ and vdt=dr=τds where τ is the tangent vector to the path of the ray, n is the spatial distribution of the refractive index, ds is the element of arc length and p0=ε0/c is the quantum of photon momentum with a photon initially having energy ε0 so that
which we identify as a modified form of Fermat's Principle that can account for the effects of quantum transitions from atoms and colloidal quantum dots.
There is much that can be said about obtaining expressions for energy transitions in atoms, molecules and quantum dots. However, for now we will take Δε=0 and recover the usual Fermat's Principle from Eq. 328 so that
δ∫r
where {dot over (r)}=dr/ds and we can identify the Lagrangian as =n(r)√{square root over ({dot over (r)}·{dot over (r)})} and the variational problem of Eq. 328 is solved by the Euler-Lagrange equations
However,
because √{square root over ({dot over (r)}·{dot over (r)})}=1. Additionally,
where I is the identity matrix. We may therefor write Eq. 330 as
This equation is extremely useful for deriving the trajectories of rays in a refractive index distribution. For example, this is done in a subsequent sections on LiDAR beam steering.
Also note that the derivative of a position vector r with respect to arc length s has unit magnitude and direction. It is the unit tangent vector to a ray's trajectory and we can write
so that the linear optical momentum of a ray is
where p0 is the free-space quantum of linear momentum of a photon associated with the free-space wavelength λ0. The factor of p0 usually has no impact on ray optics and divides out on both sides of equations where it might be used. In particular, we see immediately that the Optical Path Length (OPL) function S must be given as
p=p0∇S(r) (337)
so that
p·p=p02n2=p02∇S(r)·∇S(r) (338)
or in rectangular coordinates we get the nonlinear partial differential equation
where S=S(x, y, z) and n=n(x, y, z). Equation 339 is called the Eikonal equation in terms of the OPL function S. This function is notoriously difficult to solve and it has been studied extensively as it has applications in such diverse fields as oil exploration (i.e. for acoustic waves probing for oil) and optics.
The Eikonal equation of Eq. 339 is equivalent to the vector equation Eq. 334. So a solution of one can be converted into the solution of the other. Therefore, in this sense both are Eikonal equations, but expressed with different dependent variables.
Equation 334 is now rewritten as
which can be rewritten again as
Now let's define
m(r)=ln n(r) (342)
so if m(r) can be found then
n(r)=em(r). (343)
On rewriting Eq. 341 we get
which can be expanded by the chain rule to give
{umlaut over (r)}+[{dot over (r)}·∇m(r)]{dot over (r)}=∇m(r). (345)
Let's further take
q(r)=∇m(r) (346)
then we have
{umlaut over (r)}+[{dot over (r)}·q(r)]{dot over (r)}=q(r). (347)
This equation can be written as three separate equations
{umlaut over (x)}(s)+[{dot over (x)}(s)qx(r)+{dot over (y)}(s)qy(r)+ż(s)qz(r)]{dot over (x)}(s)=qx(r) (348)
ÿ(s)+[{dot over (x)}(s)qx(r)+{dot over (y)}(s)qy(r)+ż(s)qz(r)]{dot over (y)}(s)=qy(r) (349)
{umlaut over (z)}(s)+[{dot over (x)}(s)qx(r)+{dot over (y)}(s)qy(r)+ż(s)qz(r)]ż(s)=qz(r) (350)
or by inspection of these three equations we get
Therefore, we can write
{umlaut over (r)}+[{dot over (r)}(s){dot over (r)}T(s)]q(r)=q(r), (352)
where the T superscript indicates a transpose operation and r(s) is now considered to be a column vector.
Next, it is desirable to be able to solve for q(r) by matrix operations, as we shall see shortly it is not possible, nonetheless the algebraic manipulations are performed anyway to emphasize an important point that is described in more detail below. Consequently, we obtain
q(r)=∇(r)=[I−{dot over (r)}(s){dot over (r)}T(s)]−1{umlaut over (r)}, (353)
where I is the identity matrix and the (−1) exponent indicates matrix inverse. Moreover, we then write that
so we can integrate to get the expression
m(s)=m(0)+∫0s{dot over (r)}T(ξ)[I−{dot over (r)}(ξ){dot over (r)}T(ξ)]−1{umlaut over (r)}(ξ)dξ( (355)
and then use Eq. 343 to get the refractive index along the trajectory of a ray by
n(s)=em(0)e∫
and
n(s)=n0e∫
where n0=n(0). Thus, it appears that the refractive index along a single ray trajectory depends on the direction of the trajectory {dot over (r)}(s) and the principle normal to the trajectory {umlaut over (r)}(s), which has a magnitude equal to the curvature. In general, it appears that each refractive index value along a ray trajectory is based on having a unique arc length s, tangent vector, principle normal vector and curvature.
At first look Eq. 357 appears both reasonable and compelling, however it is wrong. The problem is that the matrix [I−{dot over (r)}(s){dot over (r)}T(s)] always has a determinant of zero and therefore its inverse does not exist. This is a result of the fact that {dot over (r)} is a unit tangent vector to the ray trajectory and the sum of the squares of its components always equals unity. Consequently, there is no general procedure to just specify the ray trajectories and obtain the needed refractive index distribution—more information is needed or a symmetry needs to be exploited in specific cases.
In summary, by starting with Maxwell's equations this disclosure develops a general theory of ray propagation that includes scattering by atoms, molecules and quantum dots as a means to overcome some of the limitations imposed by étendue conservation. It was then shown that in general it not possible to specify the ray trajectories to determine the refractive index. Instead one must specify the refractive index and then determine the ray trajectories. This approach is taken later in this disclosure to describe a number of detailed embodiments for DEP-based devices including DEP-based mixed reality displays and DEP-based variable focus lenses.
Redirecting Light Using Colloidal Shock Waves
Next, we will consider redirecting light using electric fields from electrodes that are all typically placed on the same surface, even if the surface is curved. It will be shown how this electrode configuration can produce colloidal shockwaves that redistribute nanoparticles into a desired spatial distribution for switching and graded refractive index optics.
Equation 231 shows that ponderomotive forces on nanoparticles can be achieved from the first term on the right hand side. However, for in-phase and quadrature excitation the first two terms of Eq. 231 may be required. This can be seen when equation Eq. 231 is rewritten as
where ER is the in-phase electric field (i.e. the real component of the field) and EI is the quadrature electric field (i.e. the imaginary component of the field), which is not in the same direction as ER.
One way to induce the in-phase and quadrature excitations in the above equation is to use a poly-phase harmonic traveling-voltage-wave on the electrode array in
but more or less phases are also possible typically with equal phase separation between adjacent electrodes. The voltage sources are connected to electrodes, such as electrode 7a. The electrodes may be placed on or imbedded in a solid dielectric material 7b, which is typically transparent at the optical wavelengths of interest. In contact with the solid dielectric material is a colloidal nanoparticle fluid 7c. As discussed before the nanoparticles are dispersed in the liquid and the particular spatial distribution of the nanoparticles provides an optical gradient to control light. Although it is not shown in
Added degrees of freedom to the nanoparticle distribution are applied to the electrodes by energizing them with different voltage amplitudes, timing, phases and frequencies in order to better control the forces on transparent dielectric nanoparticles and the resulting distribution of the refractive index as a function of space and time. This is shown in
In practice it is usual for a 2-phase electrode array to be configured to provide pure DEP forces normal to the electrode array, i.e in the y-direction. It is also usual for a 4-phase DEP system to create traveling wave forces to allow movement of nanoparticles parallel the plane of the electrode array, i.e. in the x-direction. This second kind of DEP is called traveling wave DEP or twDEP.
The electric field may be represented by a harmonic wave traveling from left-to-right. A convenient model for the phasor form for the electric field to first order is given by
E=E0[i{circumflex over (x)}(x)+ŷ]e−iκx−βy (362)
where ƒ(x) is a real valued function that models the driving electronic's ability to change the amplitude of the harmonic voltage excitation as a function of electrode position in the plane of the electrodes, κ is the spatial radian frequency of the fields, β is the electric field decay rate, and {circumflex over (x)} and ŷ are unit vectors in the x- and y-directions. Note that when the amplitude has a change from +V0 to −V0 this represents a π radian phase shift, which is often used in practice because the electrodes can be interdigitated. Also, there is an exponential-like decay of the electric field strength as the point of observation moves away from the plane of the electrodes in the positive y-direction.
By expanding the last equation using the well-known Euler's formula eiθ=cos θ+i sin θ we obtain E=ER+iEI such that
ER=E0e−βy[{circumflex over (x)}ƒ(x)sin(κx)+ŷ cos(κx)] (363)
EI=E0e−βy[{circumflex over (x)}ƒ(x)cos(κx)−ŷ sin(κx)]. (364)
from which we find
ER·ER=E02e−2βy[ƒ2(x)sin2(κx)+cos2(κx)] (365)
EI·EI=E02e−2βy[ƒ2(x)cos2(κx)+sin2(κx)] (366)
ER·EI=E02e−2βy[ƒ2(x)−1]sin(κz)cos(κz) (367)
E2=E02e−2βy[ƒ2(x)+1] (368)
EI×ER=E02e−2βyƒ(x){circumflex over (z)} (369)
where E2=ER·ER+EI·EI. Notice that κ does not play a role in the last two equations. After some algebra Eq. 231 becomes
This is immediately identified as a Fokker-Planck Equation (FPE) in two dimensions. In the context of advection of particles the FPE is sometimes also known as a Smoluchowski equation. The elements of the FPE include a transverse gradient operator given by
a characteristic nanoparticle advection velocity vector is given by
where the primes indicate derivatives with respect to x and where
Vx0=4γγ2/βE02 (373)
Vy0=4γγ1βE02. (374)
Both Vx0 and Vy0 have units of velocity. The value for the electric field E0 magnitude and the electric field decay rate β is given by solving Maxwell's equations for the electric fields for a specific electrode geometry and voltage amplitude. For example see, “Closed-form solutions in the electrical field analysis for dielectrophoretic and travelling wave inter-digitated electrode arrays,” by Dong Eui Chang et. al., J. Phys. D: Appl. Phys. 36 (2003) 30733078. Where we find (approximately) both a closed form and infinite sum expression for the electric field over the electrodes given by
where
Therefore, to first order we see that Eqs. 373-374 suggest taking for E0 and κ=β the expressions
where β was obtained by considering ζ.
Note, the field expression in Eq. 375 is most accurate when there is a 50% duty cycle in the electrode geometry, i.e. so that d1=d2, and should probably be used only near this condition. More elaborate and accurate expressions are provided for the electric field in terms of elliptic and Legendre polynomial functions in the above mentioned journal paper by Dong Eui Chang et. al., this should be used if a large deviation from a 50% duty cycle is needed or if more phases are used to support the voltage traveling wave so as to make the fields more independent of x-directed variations.
Also, as a reminder to the reader, the voltage across the electrodes is what is being maintained and not the number free-charges in vacuum. Therefore, the electric field provided in Eq. 375 is only a function of the voltage and electrode geometry and not the dielectric constant in the control volume where the electric fields interact with the colloid. If instead there were a constant number of free charges on the electrodes then the introduction of a colloid would reduce the free charge density from σƒ to σƒ/ϵr as free charges combined with bound charges in the colloid.
By inspection of
where the factor Vy0e−2βy represents the strength of an advective velocity field, in the y-direction, which is induced by a DEP process over the electrodes where
Vy0=4γγ1βE02, (388)
which may take on either positive or negative values depending on the harmonic excitation frequency used to drive the electrodes as has been noted before with respect to the Clausius-Mossotti factor. Equation 387 for u=u(y, t) is now going to be solved for the control volume region 0<y<L assuming a constant and uniform concentration u0 as an initial condition. The concentration is therefore subject to the initial condition and boundary conditions given by
The last two equations are Neumann boundary conditions and are chosen to ensure that the net nanoparticle flux is zero at the walls of the containment volume holding the nanoparticle colloid. The boundary conditions also ensure that the area under the integral is a constant so that the number of nanoparticles in the containment volume is conserved, therefore
∫0Lu(y,t)dy=u0L=const. ∀t≥0. (392)
See the similar derivation leading to Eq. 181.
As an example and in order to explore the behavior of colloid kinetics we next numerically solve the FPE for the one-dimensional system of equations Eq. 387-391 using the appropriate definitions under Eq. 231 and the electric field amplitude and decay rate from Eqs. 385-386.
In particular, it is assumed that the liquid forming the colloid is a silicone fluid called dimethyl siloxane Trimethylsiloxy (terminated) [CAS No: 63148-62-9], having a specific gravity of 0.913, relative dielectric constant of 2.6, loss tangent of 0.0004 at 100 Hz (to obtain the liquid's conductivity), and kinematic viscosity of 5 cSt (i.e. centiStokes, which needs to be converted into the dynamic viscosity in the calculation). It is also assumed that ionic compounds are dissolved in the dimethyl siloxane to increase the conductivity of the liquid dimethyl siloxane by a factor of 2.0×107, which takes the conductivity of the liquid from that comparable to a piece of highly insulating glass (i.e. a near perfect insulator) to the conductivity of typical drinking water and sets the Maxwell-Wigner frequency 1/τMW of the Clausius-Mossotti factor to about 500 kHz. The nanoparticles are assumed to be spherical diamond or lanthanum glass beads with a diameter of 50 nm, relative dielectric constant of 5.7, and loss tangent of 0.0006 at 100 Hz.
The electrodes are assumed to be long strips in the z-direction that are parallel to each other to form a periodic sheet in the plane y=0. The electrodes have a 50% duty cycle in the x-direction so that d=d1=d2=50 μm, like that in
A DEP-induced colloidal shock wave is a sharp change in concentration u, nanoparticle volume fraction νS, and average refractive index nA, in a narrow spatial region that travels as a wave through the colloid. The wave is created by the rapid onset of a strong advective flow field caused by ponderomotive forces from non-uniform harmonic electromagnetic fields.
The parameters listed above are now used as the basis of a simple example for the formation of a colloidal shock wave. Starting in
Notice in
Therefore, it may be advantageous in some applications to have an initial voltage spike to quickly move the nanoparticles away from the wall at y/L=0 and then use very small voltage amplitudes to maintain the nanoparticles away from the wall at y/L=0. This is the basis of an optical switch that can redirect light by total internal reflection. Alternatively, using a noise signal that is high-pass or low-pass filtered with (for example) the zero crossing frequency ω0 of the Clausius-Mossotti factor KR(ω) as one edge of the filter response provides ponderomotive forces using a smaller voltage levels at the expense of using a noise circuit, which is often low cost and even controllable using digital techniques.
It should be noted that light propagation through the optical boundaries formed by the containment walls for the colloid and the colloid itself is not necessary. An amazing alternative is that light can be launched directly into the colloid, parallel or near parallel to the xz-plane and in the region 0<y<L. In this case the light can interact with the colloid over an extended range. For example, by using different voltages on different long-thin electrodes so that ƒ((x)≠1 then the we can gain control over a two dimensional concentration and refractive index u(x, y, t). If the electrodes are not long thin electrodes, but instead pixelated electrodes, then control over the concentration u(x, y, z, t) for 0<y<L becomes possible by introducing a function g(z) that varies the electrode-to-electrode voltage in the z direction. So for example the electric fields approximately become
E=E0[i{circumflex over (x)}(x)+ŷ+i{circumflex over (z)}g(z)]e−iκx−iκze−βy (393)
This opens up the possibility of creating an electronically controllable GRIN environment for creating lenses, gratings, holograms, beam steering, switches, shutters, and other optical functions all on demand and under the control of a computer. Thus, the elements of a general purpose broadband, polarization independent opto-electronic computer are provided and in the more narrow case reconfigurable opto-electronic instrumentation and prototype optical breadboards. The use of cartesian coordinates for electrodes is also only an example and electrodes following general curvilinear are possible. This is discussed at length in the section on augmented reality lenses where hyperbolic shaped electrodes and coordinates are introduced to combine the function of a lens and a beam steering device into one device. See for example the discussion associated with
The big picture here is that:
1. A spatial distribution of nanoparticles in a colloid can be controlled by DEP.
2. Rapid onset DEP induces a shock wave if the field-strength is strong enough.
3. At steady state the ponderomotive forces balance diffusive forces.
4. The direction of ponderomotive forces depends on excitation frequency.
5. Voltage changes between electrodes allows arbitrary GRIN in the colloid.
6. A first embodiment has light passing through the control volume.
7. A second embodiment has light remaining within the control volume.
8. Computer controlled and reconfigurable optics becomes possible.
9. Controllable GRIN distributions converts optics design to software design.
It is especially important to note that this technology makes it possible to create an optical “breadboard” where arbitrary optical components are programmed into existence. Such a programmable optical system can be used for making arbitrary optics like lenses, beam steering devices, and may even provide important capabilities for a general purpose classical or quantum optical computer.
Optical Traps and Dielectrophoresis
Dielectrophoresis occurs when a force is exerted on a particle subjected to a non-uniform electric field. This force does not require the particle to be charged, the particle can be metallic, dielectric, anisotropic, etc. All particles exhibit a dielectrophoretic force in the presence of non-uniform electric fields. Also, in this disclosure we are usually interested in harmonic non-uniform fields, though this is not a requirement.
The derivations described in previous sections for ponderomotive forces on particles are by means of harmonic electric fields provided by specific electrode configurations. However, in this section the technique is shown to actually be more general. In particular, the field of trapping atoms, molecules and large-particles (e.g. 100 Trillion atoms are in a blood cell) by intense lasers is described by essentially the same mathematics.
For historical reasons the name “particle trapping” or “optical tweezers” are commonly used. However, laser-based particle trapping is essentially just the application of ponderomotive forces on particles located in a non-uniform electromagnetic field found at the focus of a laser beam. In contradistinction, dielectrophoresis is typically considered to be the use of a dedicated set of conductive electrodes supporting time harmonic voltages from a lumped-parameter voltage source. In the case of dielectrophoresis the harmonic excitation of electrodes is in a frequency ranged very roughly from 1 Hz to 106 Hz, while for optical trapping the excitation frequency is on the order of 1015 Hz. There is really no restriction on the excitation frequency and other spectral bands are also possible.
The electric fields of a focusing laser approach a maximum magnitude at the location of the focus where the beam is at a minimum cross-sectional area. The symmetry of a focused laser beam can then trap a particle with forces of very roughly 1 pico newton in vacuum at the beam constriction (focus) region. Given the density of elements of the periodic table 1 pico newton corresponds to spheres of very roughly 1-10 microns diameter i.e. about the size a human blood cell. Note that in the case of optical trapping a first order correction to the physics provided by the theory of dielectrophoresis is a scattering force that tends to provide a very slight displacement of the position of the trapped particle equilibrium point, this is not enough of a difference to justify considering optical trapping and dielectrophoresis different phenomena even if the scattering correction is useful to know in some applications. Of course under suitable conditions the light scattering forces can be made very large, i.e. on the order of 100,000 greater than the gravitational force on a nano particle. Therefore, for particle trapping we can choose the parameters of the system so that the trapping forces dominate significantly.
So for this disclosure we shall consider dielectrophoresis, optical particle trapping, and optical tweezers as deriving from the same physical principles. This is made explicit from Eq. 88 and the magnetic analog to Eq. 88. By taking the electric fields from a laser as a Transverse Electromagnetic (TEM) so that both electric and magnetic terms are needed. Taking EI=0 and BI=0 then the force density is split between electric and magnetic energy density.
but in magnitude we can rewrite the second term in terms of electric fields by noting that for TEM waves ER=cBR, {circumflex over (ϵ)}{circumflex over (μ)}=1/c2 and from Eqs. 118-129 we see that
Therefore, on multiplying through by the volume of a spherical particle from Eq. 89 and using equal phases for electric and magnetic fields we obtain the force from the force density on a single particle as
or on plugging in the expression for K∞ we get on further multiplying and dividing by c
where c is the speed of light in the medium ϵL. However, c=co/nL, where co is the free space speed of light, nL=√{square root over (ϵL)}, nS=√{square root over (ϵS)}, and by Poynting's theorem we find that the irradiance is I=cϵoϵLrER2 [W/m2] so that
where m=nS/nL. Equation 399 is the classical equation for the confining force formed by an optical trap and it was derived directly from the equations for dielectrophoresis. The only real difference in the derivation to that of electrode-based DEP is the inclusion of the magnetic energy density term BR2/(2{circumflex over (μ)}) to account for TEM wave propagation instead of having physical electrodes. The subscript L is now interpreted as liquid or as any other medium within which the particle is being trapped—for example a vacuum or a gas.
To understand the scattering forces we first note that the energy density U [energy per unit volume] has the same base units as a pressure , i.e. a force per unit area, and it can be shown that
The energy flux is then
Moreover, the scattering force will be related to the time rate of change of the momentum density pv as light passes through a volume A Δt of the medium so that
which can be time averaged whereby S
=I is the irradiance and the average scattering force on a particle is then
where Ascat is the scattering cross section. If we assume that the refractive indices of the medium is nL and the solid particle nS are not complex then the irradiance is
Iscat=Iƒ(,r,λ,nL,nS) (405)
where I is the incident irradiance, is the particle volume, r is the distance between the particle and the observer, λ is the wavelength in the medium and ƒ is a function to be determined. Clearly we must have that ƒ is dimensionless because Iscat and I have the same units. Therefore, based on unit analysis and our expectation that the intensity falls off as the square of the distance from the particle we expect that
where g is a to-be-determined function of the refractive indices. It is interesting to note that this λ−4 factor causes the blue end of the visible spectrum to scatter more strongly and causes a blue sky on earth.
A more detailed analysis, for example see chapter 3 of “The Scattering of Light and Other Electromagnetic Radiation,” by Milton Kerker, published in 1969, which provides
where k=2πnL/λ0 so on using c=co/nL we get
where m=nS/nL, nS is the refractive index of the sphere and nL is the refractive index of the surrounding medium and {circumflex over (τ)} is a unit vector in the direction of the incident radiation onto the particle. The surrounding medium may be vacuum, gas or liquid in general. To have a stable optical trap, the gradient force must be much lager than the scattering force. Alternately, to have particle acceleration we must have the scattering force is greater than the trap forces.
The main point of this section is that ponderomotive forces from electrodes or from non-uniform light fields are nearly identical in the underlying phenomena and can be considered to be variations on the same idea of position and orientation control of material particles by non-uniform and harmonic electromagnetic fields. In the case of DEP with electrodes the magnetic field is often ignored as part the quasi-electrostatic approximation used to simplify the math, but in reality magnetic fields are still there, but smaller in effect. In the case of DEP with a focused laser (or other spatial variations in the light field) there are both electric and magnetic fields working together to propagate the light in free space and the quasi electrostatic approximation is not used.
Generalized Phoresis and Self-Propelled Particles
Much of this disclosure focuses on the specific example of dielectrophoresis as a means to control the distribution of particles in a colloid or suspension. However, this is only one form of phoresis. In general, phoresis is a process whereby a non-uniform quantity (e.g. an electric field, a magnetic field, a light intensity field, etc.) provides a force on a particle. Other forms of phoresis are possible and these phoretic processes may also control scattering and the propagation of light.
For example, a previous section showed how fast acting electronics can cause a colloidal shock wave for fast optical switching. However, under certain conditions, even faster motion of particles is possible using smaller electric field amplitudes and smaller voltages on electrodes. To achieve even faster refractive index changes that already described carefully designed self-propelled particles are introduced into the colloid that derive their energy from the environment e.g. from light.
The self-propelled particles have a definite orientation that is defined by the direction that the particles move when activated. Moreover, the self-propelled particles may take specific orientations relative to the external environment when they are under the influence of impressed torques from at least one of electric and magnetic fields. Alternately, the particles may have random orientations even in the presence of at least one of electric and magnetic fields if the impressed torques are applied by oscillating fields that act too fast for the mechanical rotation of the self-propelled particles to response.
Thus, a new advective flow exists when all self-propelled particles are in coordinated motion and all particles are oriented in the same direction locally. Alternately, a new effective diffusion constant of the particles exists when the orientation of the particles is random. This new effective diffusion constant is typically much greater than the diffusion constant when the particles are not self-propelled.
Nano and micro-scale particles in a liquid move in a typically low Reynolds number environment where viscous dissipation of the particle motion overcomes the effects of fluid momentum. Therefore, if a micro or nano-scale self-propelled particle design is based on a “swimming machine” that tries to mechanically swim in a colloidal environment (e.g. perhaps using a propeller) then the swimming will not occur and the particle will fail to move. An analogy is that of a human trying to swim in a pool of thick viscous honey so that the mechanical effort of swimming is substantially nullified by viscous dissipation. Therefore, to provide self-propelled particles in colloid requires a different approach than mechanical propulsion.
One such non-mechanical approach to self-propelled particles is based on self-induced phoretic processes. In particular, phoresis (or a phoretic process) is the motion of particles due to a gradient of a specific quantity. There are actually many kinds of phoresis that are based on different gradient quantities and they can provide particle forces and motion. A partial list of phoretic processes includes gradients by:
Of particular interest to us here is the observation that self-propelled particles may be formed by processes including: self-electrophoresis, self-diffusiophoresis, and self-thermophoresis. These processes are often created by means of particles having bilateral symmetry with opposite-facing sides with different material and chemical properties. These particles are called Janus particles, in reference to the two-faced Roman god of beginnings and endings.
Therefore, externally applied dielectrophoresis, as has been discussed throughout this disclosure, may be augmented by internally impressed forms of phoresis. Alternately, dielectrophoresis may be replaced by any suitable external phoretic process, which may then be augmented by internally impressed forms of phoresis.
An example Janus particle is a dielectric nano-sphere of titanium dioxide with half of the particle's exterior coated in a thin layer of gold. Such a particle, when mixed with pure water (i.e. having low conductivity) and subjected to ultraviolet (UV) light, will self-propel through the liquid with the gold side of the Janus particle in the direction of the velocity of motion. Such a particle is capable of self-propulsion under very low light conditions As the particles are in random orientations so too are the self-propelled directions unless external fields of suitable oscillating frequency are applied. See for example the article entitled “Highly Efficient Light-Driven TiO2—Au Janus Micromotors,” by Renfeng Dong, et. al. ACS Nano 10, 839 (2016) having DOI:10.1021/acsnano.5b05940.
In this case the gradient that is formed is based on electron-hole pairs. The UV light causes electron-hole separation and an electron current to flow form the TiO2 to the gold, while chemical reactions occur on the gold and TiO2 surfaces so that a self generated electric field is generated about the particle and the field interacts with the charges to produce a self-induced propulsion force based on electrophoresis. This is called self-electrophoresis and it requires that the liquid medium have very low conductivity so that the electric fields are as large as possible and localized. Therefore, if a plurality of half-gold-coated TiO2 microspheres are located in water and exposed to sunlight then the UV portion of the sunlight will cause the particles to self-propel. Alternatively, if the particles are in a stronger UV light field then the self-propulsion will be even more pronounced.
The enhancement in the diffusion coefficient of randomly oriented self-propelled particles compared to a normal unstructured nano-sphere of the same size can be as much as 30,000 or possibly more. The above cited article by Moran and Posner notes that phoretic self-propelled particles can move so fast that they cover a distance of 100 body lengths per second. By way of comparison, a Ferrari-488 sports car moving at its top speed of 205 mph (330 km/hr) only moves about car lengths per second. Moreover, the direction of the self-propulsion can be controlled by the addition of an external directional bias, like that provided by a DEP-process. As has been discussed herein this my be utilized for controlling light scattering and bending light.
In addition to self-propelled particles based on phoretic processes, it also possible to have self-propelled particles based on non-phoretic processes. An example is a chemical reaction on one side of the particle that causes the release of particles that provide asymmetric moment change.
From conservation of mass we have that the net particle flux j=j(r, t) flowing through the surface ∂Ω of a volume Ω must obey the relation
so that the particle flow out of the volume is equal to the time rate of decrease in the particle concentration u=u(r, t) in the volume plus the particles generated per unit volume Q=Q(r, t). This gives the well-known continuity equation
The particle flux is an aggregation of many potential types including: diffusion (diff), external phoretic (xp), self-phoretic (sp), and other non-phoretic (np) fluxes so that
where for example the terms that would survive for phoretic light control by DEP and self-electrophoretic self-propelled particles are
jdiff=−D∇u (412)
jdep=uγ/{∇(γ1E·E)+V×(γ2EI×ER)} (413)
jEP≈γγ3uIuvÊ (414)
where Iuv is the UV irradiance, E is the unit vector of the electric field that sets the local direction of the coordinated motion of the self-propelled TiO2—Au Janus particles, γ is the intrinsic mobility, γ3 is the relative mobility.
Therefore, at least one suitable external phoretic process, such as but not limited to dielectrophoresis, may work alone or in combination with at least one internal phoretic process to modify the scattering properties of a colloid or suspension to allow substantial control over light scattering and the direction of propagation of light. Self-propelled particles are one manifestation of this process and can provide additional degrees of freedom for controlling light from all bands of the electromagnetic spectrum.
Materials Considerations for Optical Dielectrophoresis
For an optical colloid containing transparent solid nanoparticles a midrange of refractive indices is obtained by using different kinds of well-known glass, which can provide a refractive index ranging from roughly 1.5 to 1.9.
However, for nanoparticles with extremely high refractive index the range of potential materials is more restrictive. A first example is diamond (dust), which has a refractive index of 2.417 at 589.29 nm wavelength. A second example is gem quality silicon carbide, which has a refractive index of 2.650 at 589.29 nm wavelength. A third example is the common titanium dioxide (rutile), which has an ordinary refractive index of 2.621 and an extraordinary refractive index of 2.919 at about 589.29 nm wavelength, this provides an average refractive index of 2.770 for random oriented nanoparticles at about 589.29 nm wavelength. A fourth example is gem quality strontium titanate, which has a refractive index of 2.410 at 589.29 nm wavelength. A fifth example is lonsdaleite also called hexagonal diamond, which has been made in the lab and is 58 percent harder than natural diamond, with a refractive index of about 2.404. A sixth example is Barium Titanate having a refractive index of 2.426 at 589.29 nm wavelength. A seventh example is Corundum, which is a transparent crystalline form of aluminum oxide (Al2O3), having a hardness of 9.0 on the Mohs scale, it also has an ordinary and extraordinary refractive index of 1.77 and 1.76 respectively and a melting point of 2044° C. Corundum may also fluoresce or phosphoresce under UV light.
An eighth example are garnets, which come in many colors including colorless. Most garnets are colored, however colorless Leuco Garnets have a refractive index roughly between 1.715 and 1.735, which is a cubic crystal so that it has a single isotropic refractive index. A generic chemical formula of garnet is A3B2(SiO4)3. The A and B sites can be partially filled by calcium, magnesium or aluminum ions, none of which cause colorations. If no chromophores are in the A and B sites, e.g. only calcium and aluminum, then garnets exhibits diamagnetism.
The sites A or B can also be iron (providing purplish red, brown, yellow crystals), manganese (orange, pink), chromium (red, green), and vanadium (green). In principle, the diamagnetism of colorless gannet can be used in the presence of a strong magnetic field to provide additional forces on nanoparticles in a liquid medium. Alternately, colored iron garnets are some of the most magnetic gemstones and are ferrimagnetic, for example Yttrium Iron Garnet (YIG) and is transparent for light that is above 600 nm wavelength. It is anticipated that the force on a YIG nanoparticle from an external magnetic field would be more pronounced than for diamagnetic systems (and oppositely directed) and could be used instead of a DEP related force in certain light control applications.
The low-frequency (i.e. very roughly less than 1 MHz) dielectric properties of solids varies considerably from material to material. Most of the materials listed above have a high dielectric constant. For example, rutile has a dielectric constant in its principle crystallographic directions of 100 and 175, so that on average a nanosphere would have a dielectric constant of about 138. See for example “Static Dielectric Constant of Rutile,” by R. A. Parker, Physical Review Vol. 124, Number 6, December 1961. Similarly Barium Titanate has a dielectric constant of about 2000 at room temperature. A high value for the low-frequency dielectric constant tends to restrict the Clausius-Mossotti factor KR of Eqs. 131-132 to take on only positive (or only negative) values so that besides zDEP only pDEP (or nDEP) are available for controlling light.
In comparison to many other materials, diamond nanoparticles prepared by chemical vapor deposition and other means has a dielectric constant of about 5.7 and is transparent from the Ultraviolet (about 230 nm) to the far infrared. This is especially attractive for automotive LiDAR work where 1550 nm is desired for an eye-safe region and also for defense applications. Moreover, a wide band gap means that there is no thermal generation of charge carriers at elevated temperatures. This eliminates the phenomena of thermal run away under intense laser irradiation as might be found in LiDAR applications.
Additionally, diamond does not become nonlinear at high radiation intensities. The above properties, coupled with high thermal conductivity, chemical stability, mechanical stability, a loss tangent of about 0.00005 and a dielectric strength of 106V/cm give excellent life expectancy properties to the material. Diamond may however require coatings, such as surfactants, added to maintain colloidal stability to avoid precipitation out of the liquid-solid mixture. The low dielectric constant and the low loss tangent at low-frequencies makes a single colloid providing nDEP, pDEP and zDEP through KR possible at electrode excitations less than about 1 MHz. Diamond nanoparticles are available with sizes less than 10 nm.
Nanoparticles may also be structured. One form of structured nanoparticle is one with a void therein to lower the effective refractive index of the surrounding medium. An example of a particle that can be used for this purpose is a fullerene like C60, C200 and nanotubes, which are all cage molecules. Another example of a cage molecule are borospherenes. Additionally, derivatives of cage molecules formed by chemical structures attached externally to the cage molecule as well as endohedral forms of cage molecules, which enclose other atoms like nitrogen therein may be useful in applications like DEP-enhanced quantum computing.
Nanoparticles may also be structured so that they are of the Janus type, with different material properties on at least two sides. This provides a means for self-phoretic processes that self-propel the particles with energy derived from the environment. This can enhance the diffusion process and lower the required fields for distributing the nanoparticles.
For an optical colloid the liquid is desired to be transparent, colorless, low toxicity and low cost. Such liquids having a low refractive index are more common than the extremely high refractive index solids. For example silicone fluids are extremely stable against thermal oxidation. In particular, the dimethyl silicone fluid remains essentially unchanged when exposed to temperatures up to 150° C. in air. Methylphenyl silicone fluid exhibits even better heat resistance than dimethyl silicone fluid and can be used for long durations in environments reaching 250° C. Silicone fluid also has excellent low-temperature resistance to changes. For example dimethyl silicone fluid remains a liquid even at −50° C. Methylphenyl silicone fluid remains a liquid even in environments of −65° C. Moreover, the ambient temperature has little effect on the fluid's viscosity. The refractive index of the methylphenyl silicone fluid called methylphenylpolysiloxane is about 1.498 at 589.29 nm wavelength. This refractive index can be further decreased in the liquid state with additives like ethyl acetate (refractive index of about 1.370) and ethanol (refractive index of about 1.372) if needed. Ethanol also has a large usable temperature range and may be used by itself. The chemical stability of the liquid is further enhanced by hermetic containment, and the elimination of dissolved oxygen gas by exposure to a vacuum. Isopar, a form of synthetic isoparaffins, is also a transparent liquid that has: a low freeze point, good chemically stability, low electrical conductivity, low surface tension for good wetting and spreading properties. Finally, even water with a dissolved salt like potassium chloride to set the conductivity can be used in some applications.
The external solid materials used to define the μFCC in the devices are glass, transparent plastics (e.g. acrylic) that has a refractive index of 1.4905 at 589.29 nm wavelength. Additionally, materials like semiconductors often can have transparent spectral windows outside of the visible spectrum, where some applications may require transparent materials e.g in the infrared and ultraviolet.
The above mentioned materials are only representative and other materials, particle shapes, particle structures, etc. are possible.
Light-Beam Steering by Dielectrophoresis
Introduction to Beam Steering
There is almost no end to the number of potential applications of broadband, polarization diverse, low-to-high power light beam steering. Some of the applications include: automotive LiDAR, electronic steering of illumination lights, robotic vision, power beaming to keep drones fling over extended periods, dynamic private and/or covert point-to-point communications, bar-code readers, high-power fiber lasers, additive 3D-printing, subtractive laser machining, free-space network links for a local and private covert moving computer network, laser light shows for entertainment, optimized hydroponic lighting, remote chemical & bio-hazard monitoring, bridge inspection, sensor drones, active lightning suppression (by draining clouds of excess charge with UV lasers), wind-sheer monitoring for aircraft, topography mapping, bathymetry mapping, surveying, GIS for natural resource management, augmented reality sensors, drought monitoring, drone collision avoidance, drone based structural inspection, construction site monitoring, security, and many others.
The beam steering described below provides a large field-of-view, including spherical, hemispherical, and circular thereby providing flexibility in LiDAR coverage. This is very useful for LiDAR for autonomous cars and drones.
Circular Trajectory Beam Steering
The current discussion is about the steering of a light beam in a circular trajectory for achieving large angular turning of a light beam. For example, this large field of view is especially useful for automotive and drone LiDAR applications where the surroundings of the vehicle are critically important. Having built up the needed physics in the previous sections we will next consider light beam steering along a planar circle using only a harmonic excitation cos(ωt) between two electrodes configures into a V-shape that are not parallel and positioned along flat or curved surfaces.
As an introduction to DEP-based circular beam steering it will prove instructive to first consider the work of the current author as provided in
Two possible states of the refractive index as “seen” by the laser beam are shown in
The extent of the total angular divergence of the output rays in
when ξ0>>δξ where ξ0 is the radius of curvature and d is the diameter of the collimated input light source. When ξ0 is large or d is small the divergence can be made to approach zero. Another approach is to approximate the circle by straight μFCC segments (that have ξ0=∞) so that θT=0.
To better see how the apparatus of
To overcome the problem of light divergence and the need for liquid advection we consider
An IGL 14d completely fills the μFCC and it is controlled electronically to form a refractive index gradient on demand. The idea is to form a gradient to steer the light without divergence of the rays until just the correct position is encountered by the rays, at which point we electronically spoil the Graded Refractive INdex (GRIN) in the gradient spoiled region 14e and remove the gradient and launch the beam out of the μFCC into output rays 14f. In this case the channel is filled completely with liquid that typically does not flows, but responds to electric fields passing through it to form a GRIN channel. The electrodes and composition of the liquids used to form the electric fields are not shown in
Next, we develop the required refractive index gradient profile as seen looking towards the direction arrows 14i and 14j so that circular propagation is maintained. The required linearized refractive index 14k gradient is then shown in context in
In particular, from
where ncir is the refractive index needed for a circular trajectory of the beam and the constant of integration C was determined by assuming that at ξ=ξ0 the refractive index is n=n0. Next we take the Taylor expansion of the refractive index to first order to linearize the equation about ξ=ξ0, the result is
A μFCC having a width of (2δξ) symmetrically about ξ0 at the center of the μFCC would then provide a refractive index at the inner and outer channel boundary of
where δn=n0(δξ/ξ0), as shown in
One of the key points to be made in the above example is that the ratio of δξ/ξ0 sets the amount of change in the refractive index needed. Moreover, as the μFCC are small in width 2δξ the ratio δξ/ξ0 can be made very small, thereby keeping the needed refractive index change small even with circular beam steering of 2π radians and so practical colloids are available for the IGL.
In
The IGL comprises a mixture of one or more liquids with one or more nanoparticles types therein (with potentially many millions of individual particles) so that the bulk refractive index is a spatial average over the properties of the liquids and solid particles. In this way controlling the spatial distribution of solid particles in liquids also controls the volume average refractive index. This can be accomplished at fast electronic speeds, without the need for liquid advection.
Next, we will consider
n(r)=nL+v(r)(nS−nL). (420)
Near the radius of r=r0 we can use a Taylor approximation to rewrite this as
Moreover, we can now equate Eq. 421 to Eq. 418 after we first relate the coordinate system in
Note that in
From
ξ=R+r (422)
ξ0=R+r0 (423)
where R is the distance from the axis of rotation to the point where the electrodes would intersect and r0 is the position of the portion of the μFCC used to control the optical radiation. We can rewrite Eq. 418 as
Therefore, on setting the required ncirc equal the DEPS based n we obtain
which we can see by matching coefficients of (r−r0) that
which can be used to eliminate n0 so that
This is the high-level master equation that must be satisfied for a light beam to move in a circular trajectory at DEP steady-state. To go any further we must solve the quasi-electrostatic boundary value problem for the electric fields generated by the electrodes and then solve the DEP master equation Eq. 231 under steady-state conditions. Finally, note that the case of
With that objective in mind, consider an example arrangement of two electrodes as shown in cross-section in
Note that
A disk shaped and transparent containment vessel 18c is shown in a cutaway perspective view. The reader is to imagine that the disk also fills the region along the cut-away region 18d. The containment vessel contains a wedge-shaped profile that is revolved around the symmetry axis to form a μFCC 18e that contains within its volume an IGL based on a colloid comprising optically transparent nanoparticles in an optically transparent liquid, where each component of the colloid has a different bulk refractive index. Solid nanoparticles have a refractive index that is typically (but not necessarily) higher than the refractive index of the liquid and transparent containment vessel 18c. This allows the refractive index of the nanoparticles to be averaged with the liquid as a colloid so that the colloid's effective (volume averaged) refractive index is close to that of the solid containment vessel when there are no DEP based forces on nanoparticles, i.e. as shown in
On the top and bottom of the μFCC 18e are electrodes 18f and 18g. The electrodes may each comprise a number of sub-electrodes that are separate and identical in form, but which may be independently energized so that separate angular regions in the μFCC may be energized for light steering, as described below for
In
Next, the nanoparticle concentration in the μFCC 18e is developed for a the specific example case of the wedge shaped electrodes having harmonic voltage excitation. The voltage phasors across the top and bottom electrodes are
V(Top,t)=+V0eiωt (429)
V(Bottom,t)=−V0eiωt. (430)
In phasor form the voltage in the volume between the electrodes is the solution of
∇2V(r,θ)=0 (431)
subject to the boundary conditions of
V(r1≤r≤2,+θ0/2)=+V0 (432)
V(r1≤r≤r2,−θ0/2)=−V0. (433)
Thus, we can see that we can approximate the voltage between the electrodes at −θ0≤θ≤±θ0 as
Therefore, the electric field intensity in cylindrical coordinates is
so that
Therefore,
Also, we can identify the real and imaginary parts of the electric field intensity
Using the above analysis for ∇·(u∇E2) and EI×ER the master DEP equation Eq. 231 at a single spectral frequency is written as
therefore
Next, we will normalize this equation and make it unitless. This will simplify the equation somewhat and provide some insight into to the final functional form of the solution u. First note that the units of 3 are obtained by first expanding, whereby
Therefore, the units of β are
Given the differential equation Eq. 441 we seek a product solution of the form
u˜ratbDcβdu0e (444)
so that so that all of the quantities in Eq. 441 are included as well as the initial spatial condition u(r, 0)=u0 is a constant at concentration at time zero. In terms of the units
[u]˜[r]a[t]b[D]c[β]d[u0]e
(#L−3)=LaTb(L2T−1)c(L4T−1)d(#L−3)e (445)
so that
a+2c+4d−3e=−3 (446)
b−c−d=0 (447)
e=1 (448)
Therefore,
d=b−c (449)
and
Therefore,
or more generally on summing over different proportionality constants Aa,b and exponents we can provide a series solution of the form
or equivalently we can write
u=u0g(ρ,τ), (453)
where g is a function that needs to be determined and where we have introduced the dimensionless similarity variables
Consequently,
and at steady-state
where the functions g and G are yet to be found by rewriting
resulting in a normalized differential equation
An initial condition and boundary condition are taken respectively as
g(ρ,0)=1 (462)
g(∞,∞)=0. (463)
At steady state when t→∞ let us define G(μ)=g(ρ, ∞) where
0≤G(ρ1)<∞. (464)
The infinitely large wedge region is an approximation that allows to avoidance of calculating fringing fields and is reasonable given that most of the DEP forces are near the wedge apex anyway. The first condition above says that the electric field has no impact on the IGL far away from the wedge's apex. The second condition bounds the solution, however for the point particles being used in this model the concentration may go to infinity and we will have to manually restrict the concentration. This will be explained in detail later in the analysis.
Before proceeding note that there are other possible ways to model the situation. For example, by assuming that the number of nanoparticles, in a fixed region between r1≤r≤r2, is invariant due to the walls of a finite containment vessel, then
∫−θ
whereby another similarity analysis yields solutions (when β≠0) of the form
and the resulting differential equation Eq. 461 does not change, even though the expression for the solution is different. Thus, Eq. 456 transforms into Eq. 466 so that u0→ζD/β would be needed in subsequent equations. For the sake of retaining a simple model with features that are useful for this disclosure we now proceed using Eq. 456.
Therefore, at steady-state we expect that
The general solution to the steady-state equation may be found in several ways, however for the sake of expedience note that the solution is most likely an exponential. So by trial and error we can try trial-solutions of the form Exp[aρ−2], Exp[aρ−1], Exp[aρ1], Exp[aρ2], etc. From this we find that the exponential Exp[½ρ−2] solves the equation and that a second independent solution may be found for the second order differential equation by the method of reduction-of-order so that
where Ei[z] is the exponential integral function
The first term G1 to the steady-state solution is a pure exponential and is always greater than zero and decreasing as ρ increases. The second term G2 is not always positive and increasing without bound as ρ increases as shown in
In general we have the steady-state concentration v(ρ)=u(ρ, ∞)=u0G(ρ) where C1=1 as required for our normalized differential equation, so that
From Eqs. 89, 131, 148, 166, 442 and 454 we can write Eq. 470 as
and we can clearly see that KR can take on both positive and negative values depending on the values of K0, K∞, τMW and the frequency ω. Additionally, we can identify Re[{tilde over (ϵ)}] from Eq. 118, the arc length we identify as s=r(2θ0) and the total voltage drop across the electrodes as 2V0 so we can further identify the electric field E and electric field energy within a nanoparticle W as
Therefore, the steady-state nanoparticle concentration is a function of the ratio of (1) the energy in the volume of the nanoparticle and (2) the thermal energy. In particular
The factor of ½ in Eq. 475 is specific to the form of the impressed electric field used and may change for other electrode geometries. Clearly, the simple form of the above equation shows that an alternate variable to use instead of ρ is the ratio of the energies. Note that we can switch between the two forms of G1 shown in
From Eqs. 166, 442 and 454 we can write
where we call {tilde over (ρ)} the normalizing radius and r0 is the distance from the electrode apex to where the light is confined for steering as shown in
where, similar to the definition of ρ in Eq. 454
Therefore,
where the volume fraction of the solid spheres is νS=u0, occurs on completely mixing the IGL's liquid and nanoparticle components in a beaker outside if an energized circuit. Now solving for the ratio of refractive indices we obtain for the configuration of
Notice, that as the normalized radius of the μFCC increases towards infinity, →∞ that the ratio of refractive indices approaches unity because the trajectory of the light is a straight line and a GRIN profile is not needed. Additionally, for the case shown in
Finally, for the sake of completeness, when extending Eq. 483 from Rayleigh to Mie scattering with larger nanoparticle sizes, i.e. b>λ0/20, we have
where p is given by Eq. 278 and ρ0 can take on both positive and negative values to account for the different configurations in
Note that some care has to be taken in using the above equations because Eq. 231 essentially assumes that the nanoparticle spheres are points. This means that the points can get arbitrarily close to each other so that the concentration can tend to infinity as r→0. This cannot happen in practice as the solid spheres would touch each other and stop any further concentration from occurring. In particular, for uniformly sized spheres in three dimensions the densest packing is approximately 74%, i.e. the fraction π/(3√{square root over (2)}) of the volume, for the face centered cubic (FCC) packing and the hexagonal close packing (HCP). In contradistinction, a random packing of equal spheres generally has a density around 64%. Although the analysis in this document has focused on spherical nanoparticles, it is clear that other geometries for the nanoparticles can be utilized.
The maximum uniformly-sized spherical packing density provides a volume fraction of
and the random packing provides
νS(Irregular Packing)≈⅔. (487)
As a practical matter for spherical nanoparticles
0<νS<<⅔ (488)
and because νS=u
Thus far in the analysis of beam steering the electrodes 18f and 18g in
The DBS device just described is based on a wedge shaped μFCC that has electrodes with opposite polarity placed on the upper and lower surfaces of the μFCC to form a wedge geometry. An alternative method is shown in
In particular, it is now desired to create a GRIN liquid that confines and guides light much as a fiber optic confines and guides light: by having a core region with higher refractive index than the cladding region. Most fiber optics have a step change in the refractive index between the core and the cladding. Some fiber optics uses a GRIN refractive index profile so that a gradual change occurs between the core and the cladding refractive index. That said, fiber optics have static refractive index profiles while the DBS systems that are described below have dynamic GRIN profiles so that guided light beams are steered to a desired azimuth (or elevation) direction and then released, i.e. by removing the GRIN profile, so that the light is launched into straight-line propagation and out into free-space.
In
In
One of the applications of the embodiment of
In
Helical Trajectory Beam Steering
One of the top areas of interest for solid-state beam steering is for LiDAR, which was originally a portmanteau of light and radar, but is now considered an acronym for light detection and ranging. As a first example we will consider a system that can replace spinning mechanical beam steering systems.
A Dielectrophoresis Beam Steering (DBS) device with 2π radians of azimuth steering and over ±π/4 radians of elevation steering is shown in
Additionally, a nanoparticle colloid 24f located in a μFCC formed either between the outer transparent radome 24a and the optional inner solid 24c; or between the outer transparent radome 24a and the spatial extent of spatially varying electric fields used to contain the light by a GRIN profile.
An input light beam 24g (source not shown) is injected into the μFCC and it is reflected by means of an angle selective electronic mirror 24h into an upward spiraling light trajectory 24i. The upward spiraling light trajectory may optionally be converted into a downward spiraling light trajectory 24j by a reflection device 24k, which is typically just total internal reflection. The injection of the light into the μFCC may be accomplished by any means so that the angle selective electronic mirror 24h is only provided as an example.
A means to control the nanoparticle colloid concentration u(r, t) is also provided and described later. When the means to control the nanoparticle colloid is activated the light trajectory remains spiraling in the upward or downward directions of
There are three different modes for controlling the trajectory of the light beams. The first mode uses in-phase and quadrature electrode excitation from a traveling voltage wave on electrodes to induce a radial gradient in the refractive index that allows circular light propagation. This first mode (mode-1) has the same requirements as set forth in Eq. 419, sets ƒ(x)=1 and has the concentration u exponentially decaying away from the two dimensional electrode array into the IGL in the μFCC. In
The second mode (mode-2) sets ƒ(x) to a non-constant function of x, such that two separate traveling voltage waves are generated: either both propagating away from or towards each side of the upward spiraling light trajectory 24i. This allows nanoparticles to accumulate around the light's trajectory to induce a GRIN waveguide similar to a GRIN fiber optic. Therefore, while mode-1 creates a gradient in the radial direction, mode-2 creates a gradient predominately in the angular and axial directions, i.e. in a tangent plane to the outer surface of the inner solid 24d such that the gradient vector is always towards or away from the light's spiraling trajectory. This induces a greater refractive index about the desired trajectory and provides a light guide for the light. For the avoidance the doubt
A second magnification inset 24u shows a detail of transparent transistors 24v and transparent electrodes 24w that are printed on at least one of the surfaces of the μFCC as an array that can have each of the electrode pixels selected and energized to hold a charge by capacitive action between refresh signals to the transistors shown. A fast microprocessor could exploit a GRIN strategy like mode-1 or mode-2 and energize the electrode array so that a light beam would progress along a spiral until the light is launched out of the μFCC at (for example) point 24m.
While the above description was made for a cylindrical DBS device it is equally valid for other structural forms, for example a sphere or hemisphere. In such a case the μFCC would be formed between an outer spherical radome and an inner spherical material. Many variations of electrode geometry are also possible.
In the preceding discussion the electrodes have a geometry that allows their coverage to extend to full or nearly full coverage of the supporting manifold. For example in a flat plane electrodes can extend in one or two dimensions. One possible tessellation is a rectangular grid of square electrodes on a plane, another is a linear grid of electrode strips on a plane and other tessellations are possible. The planar grid many be changed to a cylinder, sphere or other two-dimensional surface.
In an alternative approach, the geometry of the electrodes may be chosen to allow a simple traveling wave to still accumulate nanoparticles in a local region. Consider the one dimensional electrode array in
Hybrid LiDAR, RADAR, & Imaging Sensors
One of the applications of the V-electrode beam steering device is to provide laser beam radiation into 2π radians of a polar angle with an additional capability to steer the beam roughly ±π/12 in the elevation direction. Additionally, one of the useful aspects of the V-electrode beam steering embodiment is that it easily integrates with a LiDAR receiver colocated with a optical focal plane array for optical image receipt and W-Band automotive frequencies (77-81 GHz) transceivers, typically from a circular array configuration. This represents a multisensor solution for autonomous vehicles and other applications.
In particular, the elements of a V-electrode hyper-spectral sensor comprising an optical camera, radar and LiDAR are shown in
A 2π radian image is obtained via a cone shaped mirror 25d embedded in the radome 25a. The cone shaped mirror reflects image field light 25e (LiDAR light and image light for the camera) and redirects it towards a lens system 25g for focusing and a pixelated imaging array, typically a charged couples device (CCD) 25g that senses optical radiation. The reflected and focused light is distorted, but this distortion is canceled in post processing of the image.
In
There are many possible configurations of a remote sensor for applications where LiDAR may be needed. However, for the purpose of this disclosure the main features incorporated in an embodiment are: a light beam, a curved control volume for containing a curved trajectory laser beam. Co-located with the LiDAR's DEP-based transmitter is a LiDAR receiver and may include other sensors like a RADAR, camera and acoustic sensor.
DEP-Based Generation of Thousands of LiDAR Beams
In this section we consider how to take a single input laser beam and spread it in such a way as to create many beams that can be used independently and controlled by a DEP process. By carefully choosing modes to excite in a control volume with a controllable refractive index distribution the light's energy is made to move along a new class of “accelerating” trajectories within a curved control volume forming a μFCC. This has substantial practical applications for building compact beam directors for applications like Light Detection and Ranging (LiDAR) where potentially millions of laser pulses per second need to be steered into many distinct angular directions.
In what follows a quick review of select historical results is followed by an extension of the prior art theory to the new situation of spatially varying refractive index in a colloidal IGL. The resulting modes are used to redirect light into specific angular directions over large solid angles.
Over thirty years ago in the paper “Nonspreading wave packet,” by M. V. Berry [Am. J. Phys. 47(3), March 1979] it was observed that a probability wave taking the form of an Airy function can propagate in free space without distortion and with constant acceleration without violating the Ehrenfest's theorem on nonaccelerating wave packets. In particular, one possible solution to the Shcrodinger equation in free space
is an Airy wave given by
which is demonstrated by means of direct substitution of Eq. 491 into the wave equation Eq. 490 and the Airy differential equation
It can be shown that just like plane waves of infinite extent this Airy wave packet can propagate without changing form and remain distortion free. No other functions other than the plane wave and the Airy wave have this property in a linear medium.
Thus, if one could find the electromagnetic counterpart to the above quantum wave functions then the unusual feature of the Airy packets to remain distortion-free over long distances by free acceleration during propagation might be realized. Again, it is worth emphasizing that this distortion-free property is realized without the need for a special non-linear medium and the property may be maintained in free-space or in a beam forming device.
The staring point for bringing this wave property into electromagnetics is the time-domain version of the wave equation, in particular from Maxwell's equations it is easy to show for free propagation that
If we substitute the following assumed field configuration (which temporarily breaks with the time sign convention eiωt in this disclosure so as to be consistent with the cited literature)
ε(r,t)=E(r)ei(kz−ωt) (494)
into Eq. 493 then we obtain
which has historically been difficult to solve unless a paraxial approximation is made. This requires that the field varies gradually along the z-axis so that
whereby Eq. 495 becomes
This can be written generically as a kind of Shcrodinger equation similar to that of Eq. 490. In particular, we obtain
where
and χ is the normalized transverse direction, x0 is a convenient transverse metric like the beam width, ζ is the normalized propagation distance and k is the wavenumber in the medium with refractive index n. Equation 498 has a solution
which clearly shows that the intensity profile of the wave is invariant in its propagation. This result does not require a nonlinear medium. Moreover, the excitation of the Airy beam can be accomplished through a truncation via an exponential aperture function at the light beam source so that
E(χ,0)=E0Ai[χ]eaχ (503)
where 0<a<<1, which ensures that for negative X-coordinate values that the exponential goes to zero and effectively truncates the Airy beam. The resulting extended beam is found by integrating Eq. 498 so that a practical beam takes the form
The Fourier transform of which is proportional to exp[ak2] exp[ik3/3] so that one way the Airy beam is generated is by means of a Gaussian beam and an optical Fourier transformation after a cubic phase is impressed on the Gaussian beam. The details of this are further discussed by G. A. Siviloglou et. al. in the paper entitled “Observation of Accelerating Airy Beams,” in Physical Review Letters [PRL 99, 213901 (2001)]. Unfortunately, the Airy beams formed in the paraxial approximation are limited to small angles.
More recent solutions to the full Maxwell's equations have been found that bear a striking resemblance to the paraxial Airy beam solutions and are capable of large angular deviations like those that are found in beam steering. For example, the journal paper entitled “Nondiffracting Accelerating Wave Packets of Maxwell's Equations,” by Ido Kaminer et. al. in Physical Review Letters [PRL 108, 163901 (2012)] demonstrates accelerating solutions to Maxwell's equations for circular propagating non-diffracting waves. Also, in the full three dimensions accelerating waves on spherical manifolds were also found and experimentally observed in the journal paper entitled “Observation of Accelerating Waves Packets in Curved Space,” by Anatoly Patsyk et. al. in [Physical Review X 8, 011001 (2018), DOI: 10.1103/PhysRevX.8.011001]. This last journal paper is of particular interest in this disclosure.
The main feature of accelerating electromagnetic light beams is that their center of momentum (similar to the center of mass) follows a geodesic formed by the μFCC, but the radiation's main lobes follows a curved path that is shape-preserving (non diffracting) over large angular extents while the main radiation lobes do not propagate on a geodesic. In contradistinction to the accelerating beams in flat space, accelerating beams in curved space change their acceleration trajectory due to an interaction that exists between the space curvature and interference effects so that the light's main lobe may follow non-geodesic trajectories while the beam focuses and defocuses periodically due to the spatial curvature of the medium that they traverse.
For the avoidance of doubt, a geodesic is the shortest possible curve between two points on a curved surface or manifold. In the case of a sphere in three spatial dimensions a geodesic is a circle that cuts the sphere such that the center of the cutting circle coincides with the center of the sphere. Examples of a geodesics on a sphere are circles of longitude, while examples of non-geodesics on a sphere are circles of latitude except for the equator circle.
Light that is launched in thin uniform colloidal spherical shell, with thickness much greater than the wavelength light and higher uniform refractive index than a surrounding transparent medium, will propagate along a geodesic trajectory due to total internal reflection in the spherical shell. Similar considerations are possible for other geometric shapes, like cylinders and ellipsoids. The fact that light obeys this principle can be derived in several ways, e.g. for ray optics it is possible to start with Fermat's principle and use the variational calculus. For example, a colloidal cylindrical shell provides a geodesic trajectory called a geodesic helix as is later derived in Eq. 814.
The key to creating thousands or even millions of separate beams for LiDAR is to exploit a “loop hole” of sorts in the physics of light beam propagation that allows a portion of a light field to propagate on non-geodesic and “accelerating” trajectories so long as there are counter balancing “accelerations”. More specifically, the loop-hole is that the peak intensity radiation lobe of a modal light field can follow a non-geodesic trajectory if other parts of the light field, e.g. the side-lobes of radiation, compensate so that on average the light's average momentum is on a geodesic. Thus, if one is willing to expend some extra optical energy for the extra modes and a means to generate those modes then it is possible to obtain a new ways to generate beams for LiDAR. In this disclosure the loop-hole in propagation physics is exploited using a DEP process.
The modal solutions that are developed herein allow a control volume comprising a thin spherical shell μFCC, which is simple and cost effective to manufacture, to support many separate laser pulses that are directed along different non-geodesic trajectories, e.g. latitude circles, instead of being restricted to just path-length-minimizing geodesics, where an index gradient is maintained in an optical shell—e.g. the spherical shell. The accelerating laser pulses are steered within the μFCC to the correct angular location and the IGL is relaxed from a gradient state into a uniform refractive index that is matched to the surrounding medium so as to release the light beam into a straight line trajectory that ultimately becomes a plurality of free-space light beams in rectilinear motion. The light may be continuous or pulsed.
Some of the main conceptual points being developed herein include:
The development herein of the beam steering of a plurality of beams uses a different assumptions from any of the prior art in accelerating beam theory. In particular, it shall be assumed herein that a beam steering and generation device has a liquid and nanoparticle mixture forming a colloidal shell forming a substantially two-dimensional shell in a three dimensional space. Further, that the colloidal shell has a graded refractive index formed therein.
In particular, consider as an example of the principles a thin spherical shell that has a graded refractive index that is only a function of the radius. Moreover, assume that refractive index monotonically decreases as the radius increases, which is a convenient but not a necessary assumption. An example, of this is obtained by generalizing Eq. 418 for the colloid's refractive index as
where R is the radius to the center of the thin dielectric shell that forms the μFCC, n0 is the refractive index at the center of the μFCC and α is a real refractive index slope adjustment that is obtained in practice by electronically setting harmonic excitation voltages, electrode-to-electrode phase shift and frequencies that influence the quasi-static ponderomotive forces on the nanoparticles in the IGL. Of course other linearized functions may be used, but Eq. 505 provides a reasonable example for this disclosure.
Therefore, for a spherical dielectric shell the quasi-electrostatic field we need to induce the suggested gradient takes the form
E=E0(i{circumflex over (ϕ)}+{circumflex over (r)})ƒ(ϕ)e−iκϕ−κr. (506)
There is clearly an exponentially decaying field as the point of observation moves from the inner surface at r1=R−δr towards the outer surface at r2=R+δr, where 0<δr<<R. This decaying field provides ponderomotive forces on nanoparticles via a gradient formed by a decaying electric field as r increases. Additionally, and for the sake of clarity, it should be noted that electrodes may be placed on or about the outer surface of the thin spherical surface or on the inside as well, however the sign of the real part of the complex Clausius-Mossotti factor would have to be reversed by choosing that quasi-static excitation frequency appropriately as can be seen in
With that in mind we consider an optical transverse magnetic (TM) electric field given by E=Er, Eθ, Eϕ
=
0, Eθ, 0
, where the conventional meaning of the spherical coordinates includes: r is the radius, θ as the polar angle and ϕ the azimuth angle. The wave equation Eq. 493 is then excited by a field ε=ε(r, θ, ϕ)eiωt which gives a Helmholtz wave equation
∇2Eθ(r,θ,ϕ)+k2(r)Eθ(r,θ,ϕ)=0. (507)
The ∇2Eθ term has a sub-term with r factors, however we can make the approximation that for factors that do not include derivatives with respect to the radius that r=R±δr is really just r≈R so that Eq. 507 becomes
This may be solved by the method of separation of variables by letting Eθ(r, θ, ϕ)=X(r)ψ(θ, ϕ) so that
Where we proceed in the usual way and assign constants +β and −β to the terms in X and ψ separately. Then the equations for X(r) and ψ(θ, ϕ) become
The equation in ψ can also be separated by setting ψ(θ, ϕ)=Y(θ)Z(ϕ) so that Eq. 511 becomes
which has terms in θ and ϕ only that are set equal to opposite signed constants ±m2. The resulting set of separated equations becomes
Note that by making the change of variables x=cos θ we find that Eq. 514 can be written as
which is the well-known standard form for the associated Legendre equation, which is solved for using a series solution as described by the well-known method of Frobenius—see for example “Special Functions for Scientists and Engineers,” by W. W. Bell, 1996, ISBN 0-486-43521-0. The resulting recursion relations from the Frobenius method require that the value of βR2 and |m| are constrained as
βR2=l(l+1)∀l=0,1,2,3, (517)
|m|<l. (518)
To solve the system of equations Eqs. 513-515 we need to make the observation that the graded refractive index within the relatively very thin μFCC is nearly linear so we can expand k2(r) about the center of the thin spherical μFCC region at r=R via a Taylor expansion and then retain the zeroth, first, second, etc. order terms as needed to accurately describe the situation.
The expansion of k2(r) to first order is
k2(r)≈k2(R)−2k(R)|k′(R)|(r−R) (519)
where by assumption k′(r)<0 for R−δr<r<R+δR in the μFCC is assumed in the analysis to draw out the negative sign in front of the second term so as to obtain a real valued solution to the radial modes. The k′(r)<0 assumption is not a requirement, but is done for convenience in the example being developed here. For example, in the case of the refractive index of Eq. 505 we find that k(r)=k0n(r) so that
Using Eq. 519 in Eq. 513 provides a general solution in terms of the Airy functions of the first kind (Ai) and the Airy of the second kind (Bi) so that
Alternately, the general expansion of k2(r) to second order is
k2(r)≈k(R)2+2k(R)k′(R)(r−R)+[k(R)k″(R)+k′(R)2](r−R)2 (522)
Using Eq. 522 in Eq. 513 provides a general solution in terms of the Parabolic Cylindrical D-function Dν(ξ) where
which is further given in terms of the confluent hypergeometric function—see for example Abramowitz and Stegun (1965). In particular, the solution takes the form
X(r)={tilde over (C)}1Dν
where {A1, A2, B1, B2, ν1, ν2} are substantially complicated functions of the parameters {k(R), k′(R), k″(R), β} and are not provided here to save space. A detailed analysis of this radial mode as well as higher order radial modes in (r−R)j for j≥2 are not provided here.
The solution to Eq. 514 is given in terms of the associated Legendre polynomials of the first kind (P polynomials) and the associated Legendre polynomials of the second kind (Q polynomials) so that
Y(θ)=C3Plm(cos θ)+C4Qlmi(cos θ) (525)
The solution to Eq. 515 is given in terms of the complex exponential functions
Z(ϕ)=C5e+imϕ+C6e−imϕ, (526)
where the requirement that Z(ϕ)=Z(ϕ+2π) requires that e±i2πm=1 and that m=0, ±1, ±2, . . . and therefore in general m is an integer. Note that the superposition of Eq. 526 represents propagation into ±ϕ directions.
It is clear that we can choose to have the electric field propagate in only one direction of ϕ so we choose a direction and set C6=0. It is also clear that associated Legendre polynomials of the second kind are not bounded and are unphysical for the device configuration being analyzed, therefore C4=0. Finally, the Airy function of the second kind is unbounded and also unphysical for the device configuration being analyzed, therefore C2=0. Additionally, the function ψ(θ, ϕ)=Y(θ)Z(ϕ) is a spherical harmonic and written formally as the normalized eigenmodes
where Nl,m is the normalization amplitude of the mode that forms an orthonormal set of spherical harmonics note the slightly different definitions given by “Physical Chemistry, A Molecular Approach,” by Donald A McQuarrie, ISBN 0-935702-99-7 and “Mathematical Methods for Physicists, 3rd ed,” by Arfken, ISBN 0-12-059820-5.
However, on using Eq. 517 in Eq. 521 with C2=0 for just a single mode having modal numbers l and m we obtain the simplest allowed modal structures, which in this disclosure are called the unstructured angular modes, which are
and in the specific case of the graded refractive index via Eq. 520 the electric field becomes
where a=k02 n02R2. The unstructured angular modes in Eq. 532 are configured with an unstructured radial mode, which in this case is an Airy beam having an electric field amplitude of E0.
Examples of the unstructured modes are plotted in
A different way of showing an example of an unstructured mode is provided in
Note, that typically all the radiation should be released at once. This is done so that the center of the light's momentum remains constant and we have multiple parallel beams leaving the system as LiDAR beams. Only one such linearly propagating light 28e beam is shown in
The reader should appreciate that the colloidal medium may also contain particles that are doped with atoms that can provide gain and lasing action so random light noise becomes an Airy beam in the thin colloidal shell having a GRIN distribution as described. Much more is discussed on the topic of colloids used for lasers in the section of this disclosure on DEP-based lasers.
Having just described the unstructured angular and transverse modal structures, we now look at the structured angular and radial modes, which are formed by the superposition of the basic unstructured modes. The structured modes provide additional useful features, like beam steering in elevation, for end-use applications like LiDAR and are now described.
Structured radiation modes within the beam steering device are formed as a superposition of basic unstructured eigenmodes. For example, we can write in general that a structured angular mode ψl(θ, ϕ, l0, m0) is given by
which can impact the trajectory of the light in the 2-dimensional (θ, ϕ) space. Different complex apodization weighting coefficients Al,m will impact the manifold trajectory differently. There are essentially an infinite number of potential candidates for Al,m however, as a specific example we consider the structured angular mode formed from a gaussian weighting, which is chosen only as an example that connects back with discussions of the prior art.
In particular, consider the superposition
where the gaussian weighting parameter is
M(l,m)=a0,0+a0,1m+a1,0l+a1,1lm+a0,2m2+a2,0l2+ . . . +aN,NlNmN. (535)
In general the trajectories of the radiation in the μFCC vary as a function of how the input light is modally structured.
If M(100, 90)=a0,0=1 then an example of a structured mode is shown in
However, as another example, if M(l0, m0)=l0−m0+1 then we have M(100, 90)=11 and a different type of structured mode is shown in
This allows light to be launched into any direction by controlling the modal structure of the radiation in the μFCC through the harmonic excitation voltages, frequency, and phase on the electrodes controlling nanoparticle ponderomotive forces within the control volume μFCC and potentially by adding selective absorbing nanoparticles or selective gain producing nanoparticles.
In summary, it has been demonstrated from first principles that an “accelerating light beam” can maintain its main and side-lobes on non-geodesic trajectory. This flexibility provides significant opportunities for LiDAR laser directors, where there is a need to have large numbers of laser pulses sent into potentially thousands or even millions of per second for mapping the environment.
Supplemental comments: This section has provided some insight into accelerated beams and the quest for diffraction-free beams. In particular, the material herein was focused on the spatial structuring of light. However, recent advances in beam formation technologies has pointed to way towards space-time beam synthesis. An example of this recent literature is the article entitled, “Weaving the Rainbow,” by Murat Yessenov, et. al. in Optics and Photonics News May 2019, pages 34-41. DEP-based technologies may also be ideal for the synthesis of diffraction-free space-time beams due to an ability to integrate together diffraction, lenses, and spatial light modulators all of which are needed to produce these new and emerging diffraction-free beam.
Precision Beam Steering
The ability to point a light beam (e.g. a laser beam) over both short and long distances with both accuracy and precision is a fundamental and enabling technology capability for many high-value applications.
End-use applications of accurate and precise beam pointing include, but are not limited to: LiDAR, 3D-Printing, power beaming for drones and aircraft, welding, cutting, drilling, point-to-point laser communications, remote laser spectroscopy for chemical sensing, micro-satellite propulsion, target tracking munitions, laser surgery, and entertainment via light shows. Additionally, a broadband and polarization independent beam steering solution also has the potential for controlling extremely narrow pulses in time as well as light from super continuum laser sources, which are multi-octave bandwidth, for creating atmospheric solitons, electromagnetic bullets and other exotic linear and non-linear waves in a medium like the atmosphere. This can help to control beam-spread.
For the avoidance of doubt a high-level of laser pointing accuracy is the ability to drive the laser beam to a desired direction with at most small systematic pointing errors so that a Gaussian probability distribution, which describes the beam's output direction statistically, is nearly perfectly centered on the desired output direction.
For the further avoidance of doubt a high-level of laser pointing precision is the ability to drive the laser beam to a desired direction while maintaining at most small random pointing errors so that a Gaussian probability distribution, which describes the beam's output direction, statistically has a small standard deviation centered in the desired direction.
Having both high accuracy and precision means that a laser beam can, in a statistical sense, hit a target at a large distance with high certainty, not withstanding time dependent changes to atmospheric and laser-platform conditions. For example, to have 1 mm of light steering accuracy at 1 km, 10 km, 100 km and 1000 km (under steady and uniform atmospheric conditions) requires 1,000 nrad, 100 nrad, 10 nrad and 1 nrad of light steering accuracy respectively.
The phenomena of Dielectrophoresis (DEP) provides a direct electronic way to achieve both high beam pointing accuracy and precision by utilizing either precision frequency generation, precision voltage levels, or a combination of both. It should be stressed that electronic timing signals, e.g. digital clock pulses, are among the most precise physical quantity that can be generated and measured by humans. Atomic clocks set one end of the time precision spectrum and cheap watch oscillators are at the other end of the spectrum in accuracy and precision. So anytime you can tie performance of a controlled physical process to electronically generated timing signals there is an incredible opportunity to achieve high-quality control. In this case the control is in steering a beam of light e.g. a laser beam.
Thus, by leveraging the digital control of the frequency of a harmonic, i.e. both traveling and non-traveling wave voltage signals, on DEP electrodes both very accurate and precise ponderomotive forces can be generated leading to accurate and precise beam steering. The Clausius-Mossotti factor effectively converts a harmonic clock oscillation at frequency ω into a non-uniform electric field to impress a force on nanoparticles that respond in aggregate to provide highly controllable spatial distributions of the nanoparticles, and by extension the effective refractive index. This can be seen for example in Eq. 281.
Note that as long as there is no significant variation in the harmonic voltage amplitude along the plane of a planar electrode array (i.e. each electrode has the same electric field amplitude) then we can expect that only the real part of the Clausius-Mossotti factor survives and it is the exponential decay of the electric field intensity away from the electrode array that gives the needed non-uniform electric field intensity for DEP. The real part of the Clausius-Mossotti factor can take on both positive an negative values leading to pDEP and nDEP processes as can be seen in
To begin an analysis of ray trajectories consider Eq. 334 and divide by n(r) so we can rewrite the equation in terms of the natural logarithm, whereby we obtain
where s is the arc length of a propagating ray and the dot notation represents derivative with respect to s. Let us also now consider an exponential GRIN medium over a restricted spatial extent where practical values of the refractive index may be generated, then
n(r)=n0eα cos θ
where α, β and γ are new variables that are all greater than zero and represent the rate of exponential change in the x, y and z directions; and cos θx, cos θy, and cos θz are the direction cosines of the desired and electronically formed gradient vector. To see this note that
∇n(r)=α cos θx,β cos θy,γ cos θz
n(r). (538)
The increasing exponential distribution of the refractive index in Eq. 537 is used here to model beam steering over a limited spatial extent of the control volume so that the range of the refractive index remains within practical values. The reason for using the exponential refractive index distribution is that nanoparticle distributions in a colloid under DEP processes also follow exponential spatial distributions over at least part of the colloid' control volume. So we anticipate that the refractive index will also follow an exponential distribution. Also, for small values of a the exponential is a good approximation to a linear curve, which is a nice intuitive feature.
As an initial analysis let's consider a 3-dimensional rectangular box forming a microfluidic control volume, which is surrounded by electrodes on each of the walls. The electrodes will impress traveling voltage waves, as has already been describes, to induce ponderomotive forces on nanoparticle contained within the control volume. We assume that for a desired pointing direction of the laser the refractive index is n0 at the boundary x=0 and that only x-directed ponderomotive forces are initially created. The origin of the box forming the control volume is at one corner of the box and the axis are along the edges of the box. An example of the coordinates and control volume are shown in
Note that n0 is the smallest refractive index in the x-direction and that it is different for different steering directions, as discussed below. If the refractive index gradient points only in the x-direction so that θx=0, θy=π/2 and θz=π/2 then
n(r)=n0eαx. (539)
Moreover, as the number of nanoparticles within the rectangular box forming the control volume are a conserved quantity we expect that the average refractive index nA is also conserved. This implies that for a control volume of length Lx in the x-direction and with only an x-directed gradient that the average refractive index is
so that
where α=α(V, ω) depends on the voltage and harmonic drive frequency of the electrodes and the average refractive index nA is given by Eq. 233. Assuming nanoparticles that are much smaller than the wavelength of light we can write an expression for the refractive index over a limited range of x-values where the refractive index is physically realizable so that when the voltages on the electrodes are set to zero and the distribution of the nanoparticles in the colloid within the rectangular box control volume is uniform then
nA=nL+(nS−nL)νS, (542)
where nL is the refractive index of the liquid, nS is the refractive index of the solid nanoparticles, and νS is the volume fraction of the solid nanoparticles relative to the total volume of both the liquids and the solids in the colloid—see Eq. 232.
As we shall soon see the value of n0, which changes with harmonic voltage amplitude and frequency, does not enter into the beam steering. Only the spatial rates of change α matters. To see this we use Eq. 539 so that Eq. 536 becomes
{dot over (u)}+αu2=α (543)
{dot over (v)}+αuv=0 (544)
{dot over (w)}+αuw=0 (545)
where
{dot over (x)}=u (546)
{dot over (y)}=v (547)
ż=w. (548)
These equations can be integrated directly from which we obtain
where {c1, c2, c3, c4, c5, c6} are constants of integration to be determined from the initial conditions of ray position and direction. If at s=0 the direction cosines of the propagation direction are u0=cos θu, v0=cos θv, and w0=cos θw then
Note that nowhere in the above equations Eqs. 549-560 does one find the value for the refractive index at a coordinate point as a parameter. Only the spatial rate of change of the refractive index, i.e. through α, is included in the equations. This is an amazing property that allows parallel rays of a laser beam, which have spatial extent via a beam diameter, to continue to track parallel to each other through the GRIN system in the control volume.
This is seen in
Additionally, note that the optical path length S is given by combining Eqs. 539, 541, and 549-560 according to the following
where the limit
corresponds to the case when there are no ponderomotive forces (α=0) and the laser beam propagates through the uniform colloid a distance s along a straight line.
Let's consider the special case, which may be generalized, where the initial ray is launched from a position x0, y0, z0
direction into the z-direction so that
u0, v0, w0
=
0, 0, 1
, then we have that
c1=0 (563)
c2=0 (564)
c3=1 (565)
c4=x0 (566)
c5=y0 (567)
c6=z0 (568)
and
In the following analysis we shall obtain a function z=z(x) that describes the ray trajectory. During these algebraic manipulations we will find that the following identities are useful
Therefore, we can write from Eq. 572 that
cosh(αs)=eα(x−x
sinh(αs)=√{square root over (e2α(x−x
and from Eq. 574 and using Eqs. 575-577 and Eqs. 584-585 we obtain
so that
and from Eqs. 576-577
Now on using Eq. 578 and Eq. 575 a second time we get
The next step is to use Eqs. 579-580 to obtain
1−2eiα(z−z
and now expanding the terms in (z−z0) by Euler's identity of Eq. 581, independently setting the real and imaginary parts of the resulting equation to zero to enforce the right hand side of Eq. 590 and using the trigonometric identity of Eq. 582 or Eq. 583, depending on if one chooses to work with the real or imaginary part of Eq. 590 respectively, we always obtain in either case a form of the equation describing the ray's path through the GRIN medium as
cos[α(z−z0)]=e−α(x−x
where the electronically controllable DEP parameter a describes the strength of the refractive index gradient for different ray trajectories and is a function of the electrode geometry, the harmonic voltage amplitude and the harmonic drive frequency ω, which excites ponderomotive forces via the Clausius-Mossotti factor.
Now that we have Eq. 591 describing parameterized ray trajectories we can obtain the equation for a lens surface. This is important because we desire to transition rays from a GRIN medium into homogeneous medium like glass or air. If we are not careful a laser beam will diverge at the boundary between the GRIN and homogeneous medium because on the GRIN side of the boundary provides different parts of the laser beam with different refractive indices. So while the beam propagation in the GRIN medium only depends on the gradient parameter a the Snell refraction across an optical boundary does have a refractive index at each point that needs compensation. To overcome this issue we will let the steered laser beam intersect normally at a curved boundary between the GRIN and homogeneous medium. Then at each point on the boundary Snell's law of refraction will always have the input angle set to zero independent of the changing refractive index in the GRIN medium. This is a simple and effective compensation for crossing the boundary without causing beam divergence.
To find the lens shape we first take the derivative of Eq. 591 with respect to x so that
A curve that is always perpendicular to the rays will have its slope given by the negative inverse of that in Eq. 593. Let us agree to call the surface of the lens by z={tilde over (z)}(x) then
While Eq. 591, which describes the ray, has a as a parameter, the equation for the lens surface cannot have α as a parameter—it needs its own independent parameter integration constant. So while in principle we could use Eq. 591 in Eq. 594 to eliminate α. Unfortunately, Eq. 591 cannot be inverted to find α=α(z, x) and we run the risk of not finding an analytic solution.
Therefore, to make some progress on the shape of the lens to minimize laser beam divergence for long-range accurate and precision beam steering observe that the function
ƒ(α)=cos[α(z−z0)]−e−α(x−x
has a Taylor series expansion about α=0 so that
Now take only the terms though second order in α as an approximation and complete the squares. This results in the equation
which is an equation of a circle having a center at (x0+1/α, z0) in the xz-plane and a radius 1/α. Thus by inspection we have a circle that is parameterized in a similar way as one might expect from a Möbius projective transformation over the complex plane. We can therefore write a second equation for the projective map that preserves a π/2 radian angle between the curves of the ray and the possible curves of the lens surface. In particular, the lens equation must be
where {tilde over (z)} was used instead of z to distinguish the ray's z-coordinate z from the lens surface z-coordinate {tilde over (z)} and (1/σ)=(zm/2) is the radius of the circle and zm is the max distance from the laser to the lens surface—i.e. the circle's diameter. If we set z(x)={tilde over (z)}(x) in Eqs. 597-598 then we find that
However, what is really amazing is that if we write the condition for the two curves being perpendicular then
which, provides on solving for x, that
This is the exact same value for x as in Eq. 601. This proves that the point of intersection is the same as the point of perpendicular crossing, i.e. as might have been expected from knowledge of the Möbius transformation. The second-order approximation to the rays and lens surfaces are provided in
In
Clearly only this limited range of the lens and ray circles are used in an actual design. Additionally, while the circular approximations to the rays and lens boundary are only approximate they give a good second order approximation to the physics and this lends itself to a better understanding. For actual manufacturing additional analysis or numerical optimization may be employed to get an optimized lens shape. Similarly, the above discussion on ray trajectories is only representative and other functional forms of the GRIN medium are possible instead of Eq. 537.
Note that the reason for all this effort in deriving a simple geometric interpretation of the ray trajectories and lens shape for transitioning to the external world is that it allows quick analysis of complex situations. Later in this disclosure more complex situations are introduced, for example in the application of spectral power combining for telecommunications, the geometric interpretation will then prove very useful to simplify the analysis.
In
Note that in principle only one of the first electrode array 35e or the second electrode arrays 35f are needed, however if only one array is used then there is need for a much higher voltage on the remaining electrode array to compensate.
The curving ray 35h partially follows the trajectory given by Eq. 591, which is approximately a circle as given by Eq. 597. The first lens surface 35j approximately follows the curve given by Eq. 598. The second lens surface 35k may also be taken from the family of curves given approximately by Eq. 598. In this way a first ray segment 35m within the colloid is curved and is described by Eq. 591, a second ray segment 35n is a straight trajectory within a solid transparent medium between the first lens surface 35j and the second lens surface 35k, and a third ray segment 35o is a straight trajectory within the air or vacuum of the external environment.
The beam steering solutions shown in the zx-plane in
The x-direction electrode arrays and the y-direction electrode arrays are energized with harmonic traveling wave voltage signals to induce forces. It is completely reasonable to use different voltage amplitudes at a fixed excitation frequency to adjust the gradient vector of the refractive index in the GRIN colloid. That said, if instead of using voltage amplitudes in quadrature the electrode array frequencies are chosen to achieve a particular ponderomotive force on nanoparticles then we may get the added benefits of (1) pDEP, nDEP and zDEP ponderomotive forces and (2) higher precision and accuracy for the impressed ponderomotive forces on nanoparticles and (3) lower voltage amplitude. Observe that
In
where ω0=1/τMW, ƒ(t)=F(t)/F0 is the normalized ponderomotive force and F0 is the magnitude of the maximum normalized ponderomotive force F(t) on a nanoparticle in a colloid. The result in a linearized force as shown in
As an example, consider that a single static refractive index gradient is desired. Then the digital voltage signals u (not to be confused with nanoparticle concentration) for pDEP and nDEP excitation are
where the ±corresponds to pDEP and nDEP; and sgn is the sign function. The above signals will swing the beam to plus or minus a specific beam steering angle ±ϕ with voltage waveforms as provided by
In another example, we might desire to have a light beam moving in time. In this case the harmonic voltage signal to the nanoparticle colloid must also be made to be changing in frequency over time via angle modulation. For this to be properly analyzed we must generalize the electrode drive frequency ω to be the instantaneous frequency so that
where θ is the instantaneous phase angle of the voltage signal applied to the electrodes. As a check on the concept we can see that on taking the derivative of the static-beam phase angle θ(t)=ω0t we obtain ω=ω0, which is a constant frequency as expected.
Also, for the sake of providing an example let's further assume that we choose a symmetric Clausius-Mossotti factor with K0=−1/a and K∞=+1/a, then we can use Eq. 602 with the concept of the instantaneous phase angle so that by integrating Eq. 604 and Taylor expanding about a=0 we find
and we see that if a=0 we recover the static beam pointing case.
For example, an electrode array at x=0 provides a force proportional to cos[θ(t)] and the electrode array at the x=Lx plane provides a force −cos[θ(t)], so that the combined force on a nanoparticle in the control volume is in the same direction via a superposition process. The digital voltages to the electrode arrays would then be
VXp(t)=V0 sgn[cos θXp(t)] (606)
VXn(t)=V0 sgn[cos θXn(t)] (607)
VYp(t)=V0 sgn[cos θYp(t)] (608)
VYn(t)=V0 sgn[cos θYn(t)] (609)
Thus, to steer a light beam around in a circular-cone pattern with the beam steering device at the cone apex a set of digital timing signals are sent to the electrode arrays. Representative forces and digital signals are shown in
Clearly, more complex scan patterns may be developed for a computer and applied to a laser scanning program. Also, a series of static forces may be applied in time to simulate a dynamic laser scanning process. Additionally, the particular waveforms used to control light are a function of the Clausius-Mossotti factors, which are further dependent on the low frequency dielectric constants, conductivities shapes and size distributions of the nanoparticles and liquids comprising the colloid. Therefore, the waveforms shown are simply to be taken as examples that need to be changed to accommodate the particular physical parameters of a given embodiment. Further still, not all physical material parameters will support nDEP, pDEP and zDEP simultaneously. As just one example, it is possible for the conductivities of the nanoparticles and liquid to be so small that the zDEP frequency is near zero so that only pDEP and zDEP are possible (i.e. nDEP is excluded from use).
In
The various parameters of the example are provided in
For example, we can replace diamond nanoparticles with a very low cost and common material like rutile, which is a form of titanium dioxide, and increase the steering angle by roughly the ratio of the refractive indices, i.e. by 2.770/2.410 about 15%. Alternately, increasing the value of the gradient a from 0.210 to 0.271 increases the external steering angle from ±55 degrees to nearly ±90 degrees, i.e. by about 60%, on entering the air environment from the beam steering device. Another important point is that the conductivity of the liquid is easily changed by adjusting the liquid's conductivity so that the zero crossing frequency ωz for zDEP is positioned at a desired design frequency. This is shown in
In summary, laser beam steering by electronically impressed ponderomotive forces on nanoparticles in a colloid will induce an exponential GRIN distribution, which is anticipated to provide:
The principles described can be extended in obvious ways by those skilled in the physical, mathematical and engineering arts, however the main point will always remain: A DEP process applied to colloids can be used to steer light by changing the effective refractive index spatially within a control volume containing a colloid (or suspension etc.) and that the steering direction may be tied to one or more accurate and precise electronic frequency standards that are applied as electronic signals to electrodes in addition to the one or more voltage amplitudes. Frequency control of the light direction provides a means for extreme accuracy and precision. Frequency and amplitude beam direction control can be combined. Voltage signals may be substituted with other signal types.
Additionally, while the exponential GRIN model provides the basis of this discussion it is not to be interpreted that this disclosure is restricted to only exponential GRIN distributions. Other refractive index distributions are also covered. For example, superpositions, perturbations and periodic tessellations of the exponential GRIN refractive index distribution are also covered in this disclosure.
Beam Steering Optics for DEP-Based Gratings
Today's holograms, diffraction gratings, and photonics crystals are typically static: once it has been created by incorporating the variations in dielectric constant in the host medium then its optical properties are set. It would therefore be nice to be able to overcome this limitation and create a completely configurable hologram, diffraction grating, and photonic crystal.
As an example of this type of general reconfigurability in this section we look at a specific type of diffraction hologram (essentially an optical grating) that can steer light selectively, i.e. by wavelength and input angle, into a user controllable direction. Extensions are provided that allow for spectrally broadband light, e.g. sunlight, as well. The devices ultimately shown may have as little as a few microns of thickness and are therefore extremely compact.
Ultimately, the objective here is to create very simple hardware comprising a thin colloid layer between two solid transparent sheets, printed transparent electrodes (or alternatively continuous opto- or electro-resistive planes supporting an arbitrary potential boundary function) on the solid sheets, and software in a controller that can reconfigure the colloid to steer light in a desired direction. This may have applications for lenses, augmented reality glasses, LiDAR, and many other areas. As an explicit example, later in this disclosure, a solar collector and concentrator that is based on this technology is explicitly shown.
However, before the details of how to do this using a DEP-based system are provided some background physics is needed first. In particular, we will need to better understand the spatial configuration that is needed for the holographic fringes (grating) elements, and by extension for nanoparticles in a colloid, in order to achieve the desired beam steering result.
With this in mind consider
The fringes in
A rigorous accounting of diffraction requires solving Maxwell's equations, nonetheless great insight can be obtained by first analyzing the conditions for constructive wave propagation in a VPH by accounting for the geometry-based phases. This analysis will ultimately provide design objectives for a distribution of particles to form a periodic GRIN media. The periodic GRIN then needs to be synthesized by generating a suitable field of ponderomotive forces through a DEP-process.
With this objective in mind consider
Interference of a first ray 41b with a second ray 41c from a neighboring fringe will provide constructive interference when the residual path lengths av and bv together are an integer multiple of the wavelength in the medium so that av+bv=mvλ, where mv∈{±1, ±2, ±3, . . . } and the “v” subscript indicates volume diffraction. Therefore, for the special case when the input homogenous medium is the same as the average refractive index of the VPH, so as to minimize unwanted reflections. Therefore,
mvλ=Λ(sin ψR+sin ψS), (610)
and by differentiation we get
Interference of a third ray 41d with a fourth ray 41e, which are on the same fringe, provides destructive interference when the change in the residual path length Δ≡λ because radiation from the first half of the fringe (from input edge to midpoint) will add destructively with the radiation diffracted from the second half of the fringe (from midpoint to output edge). Note that
=σ cos ψS so that by differentiation
Δ=σ sin ψSΔψS≡λ. (612)
Dividing Eq. 611 by Eq. 612 and using the relation d=σ cos ϕ we get the relative null-to-null spectral bandwidth
Also, using Eq. 611 and observing that at Bragg resonance ψR=ψS so that Eq. 610 provides
Finally, we note that the replay and scatter angles are θR=ψR+ϕ and θS=ψS−ϕ that ΔψR=ΔψS=ΔθR=ΔθS≡Δθ. Also, for volume diffraction we must have by Snell's law of reflection that ψR=ψS≡ψ so that
where
Together Eqs. 616 and 617 show that the spectral and angular bandwidths vary inversely with the VPH thickness d. Moreover, as we shall see later in this document in detail, the VPH thickness d may be chosen according to Eq. 676, so that the bandwidths depend inversely on another integer md and directly proportional to a coupling constant κ, which is further proportional to a modulation amplitude of the refractive index M.
Thus, the greater the thickness of the VPH the narrower the spectral and angular bandwidths and the greater the refractive index variation, through the modulation depth M, the wider the spectral and angular bandwidths.
For volume diffraction at Bragg resonance we must require ψR=ψS≡ψ so that the linear momentum of the photons is conserved along the fringe length and Eq. 610 may be written as
mvλ=2Λ sin ψ. (620)
In simple words, Eqs. 616-620 basically say that the fringes within the volume of the VPH act as a kind of distributed mirror that is only active when Bragg resonance occurs, i.e. at a specific set of angles and wavelengths where constructive interference occurs.
The reason it is a “mirror” is because the angle of incidence and the angle of refection are identical and equal to ψ relative to the along-fringe direction. When the radiation is far from the Bragg condition the light simply passes through the VPH with essentially no deviation or losses.
As an aside, the reader should also note that the analysis for the bandwidths made in the above paragraphs does not take into account the polarization of the electric field. We will find that the polarization can also impact the extent of the angular and special bandwidths, but the details of this dependence will have to wait for the solution of Maxwell's equations later in this document.
Next, let's rewrite Eq. 620 so that
and defining the grating vector (also called the reciprocal lattice vector) as
and the replay wavevector as
then
and Eq. 621 reduces to
where Ĝ is the unit vector in the direction of G. Also, similarly we can write
Adding Eqs. 625 and 626 gives
(kS−kR)·Ĝ=mvG. (627)
When mv=0 this is Snell's law of reflection along a fringe and more generally we have
kR+mvG=kS, (628)
which is a fundamental result governing volume diffraction in reciprocal space.
Next, we consider surface diffraction at “sharp” high-contrast discontinuous edges that form the boundary between the volume fringe region and an external isotropic region of a VPH. In
msλ=Λ sec ϕ(sin θR+sin θS), (629)
which may be written as
In terms of the reciprocal lattice vector this becomes
−msG·{circumflex over (τ)}=(kR−kS)·{circumflex over (τ)}, (631)
where {circumflex over (τ)} is a unit vector tangent to a VPH surface in the y-direction as shown in
(kS−kR)·{circumflex over (τ)}=msG·{circumflex over (τ)}, (632)
and a second fundamental equation is therefore
kR+msG=kS. (633)
So we see that if we take the inner product of G with kR+mcG=kS we get a description of volume diffraction and if we take the inner product of the unit surface tangent {circumflex over (τ)} into kR+msG=kS we get a description of surface diffraction from sharp VPH edges. Moreover, as these are completely separate and distinct physical processes that are modified and controlled independently by controlling the extent of the sharp high-contrast refractive index edges at the surface of the VPH.
In the real space of
For surface diffraction
It is often the case for surface diffraction that efficiently diffracted output modes are at a greater output angle than the input angle—both being measured relative to a surface normal. For example output surface mode 43ad is at a greater angle than the resonant input surface mode 43x. Also, there is little selectivity as surface modes are always resonate at some set of output angles for some set of input angles and wavelengths.
In contradistinction, volume diffraction is very selective and only redirects light that is substantially at certain incident angles and at certain wavelengths. This is a very selective process that is governed by the VPH thickness 43w such that all volume modes have nearly zero diffraction efficiency except for the first order modes, which have almost 100% efficiency.
Therefore, surface diffraction efficiently spreads light out over a large range of angles centered on the light's input direction, while volume diffraction is an extension of the concept of mirror refection from fringes where the “reflection” of light only occurs over specific discrete angles instead of a continuum of angles. Surface diffraction exists for all input angles and wavelengths. Volume diffraction only occurs at specific angles and wavelengths and otherwise light passes unimpeded. Volume diffraction is near 100% only for the first diffraction mode and all other modes have an efficiency near 0%. Surface and volume diffraction are separate phenomena and can be separated by appropriate apodization.
Thus, it is possible to choose the VPH thickness 43w, which is represented by d, as well as lattice constant Λ, fringe angle ϕ and modulation depth M of the relative permittivity (M is to be discussed in more detail later) to diffract a specific wavelength of light λ nearly perfectly into the normal output direction of the VPH. This is shown explicitly in the example of
Wavevector 43ak in
Wavevector 43an in
Additionally, the example of
Similarly, from the VPH second surface 43ai there is also diffraction, which includes output surface modes 43ay, 43az, 43ba and 43bb. These correspond to output surface modes 43ad, 43ae, 43f and 43ag respectively.
To overcome the parasitic diffraction of the surface features of a VPH while retaining the directional sensitivity of the volume of the VPH the surface features of the VPH need to be removed by gradually tapering the fringe contrast so that there are no sharp and periodic refractive index edges to the VPH fringe pattern. The result is a hologram that is suitable for use in intense broadband applications (e.g. for sunlight steering) and which can select both a narrow range of angles and a narrow range of wavelengths around a specific Bragg resonant angle and wavelength.
In the following disclosure on diffractive beam steering it is to be understood that unless otherwise stated that each VPH is formed with a holographic medium having varying sensitivity as a function of depth to eliminate all abrupt VPH edges. This substantially eliminates all parasitic surface diffraction so that only volume diffraction is present. Furthermore, in general we shall say that a VPH has surface passivation when its refractive index variations gradually taper to zero at the surface of the VPH so as to reduce or eliminate parasitic surface diffraction.
Next, we consider a more formal analysis of electromagnetic waves within a periodically varying medium in order to actually ascertain the efficiency of the redirection of light, which impacts what applications are possible. This may be approached in several general ways, including but not limited to, modal theory, coupled wave theory, translation matrix theory, the method of parallel stacked mirrors and purely numerical methods like the method of moments and the finite elements method. Each technique is helpful in gaining a slightly different insight into the underlying physics of periodically varying media.
For example, our already developing graphical representation of the physical properties is easily refined by the modal theory, which provides a detailed representation of photonics band gaps within a reciprocal space wavevector diagram. Simple expressions for the diffraction efficiency around Bragg resonance are easily derived by the approximate coupled wave theory. Properties of a periodic media away from Bragg resonance and with non-sinusoidal refractive index profiles are often more easily handled by means of a rigorous coupled wave analysis. Aperiodic fringes that are a result of fabrication errors and apodization are analyzed by the more general numerical translation matrix theory and method of parallel stacked mirrors as well as the method of moments. Many of these techniques can also be extended to non-isotropic host media. For a DEP-based system analysis can be complex comprising a hybrid analysis of diffraction and particle distribution.
Consider
To analyze this structure at optical wavelengths the starting point is Maxwell's curl equations in harmonic form
∇×E=−μiωH
∇×H=+ϵiωE. (634)
Taking the curl of the first relation and using the vector identity ∇×(∇×E)=∇(∇·E)−∇2E we find that
∇2E−∇(∇·E)+ω2μϵE=0. (635)
However, Maxwell's equations provide that ∇·(ϵE)=ρƒ, where ρƒ are the free volume charges. Let us assume that ρƒ=0, which is typical for non-metallic systems, then after expanding ∇·(ϵE) we obtain
∇·E=−E·∇ log ϵ, (636)
where log is the natural logarithm function. There are two interesting cases:
We start by analyzing the first case, so that the wave equation becomes
∇2E+ω2μϵE=0. (637)
Next, for the sake of simplicity let us assume that the relative permeability is unity so that μ=μ0μr=μ0, where μ0 is the free space permeability. Additionally, let the permittivity take a periodic form so that=ϵ0ϵavg(1+M cos[G·r]), (638)
where M<<1 is the modulation amplitude of the average relative permittivity ϵavg, ϵ0 is the free space permittivity, G=(2π/Λ) cos γ, sin γ
is the reciprocal lattice vector 45u (also called the grating vector, which is in the xy-plane with associated spatial period Λ) and r is the radius vector from the origin to a point in space within the VPH. The reciprocal lattice vector is always perpendicular to the fringes. Therefore,
∇2E+k2(1+M cos[G·r])E=0, (639)
where k2=ω2ϵ0μ0ϵavg=(ω/c)2ϵavg and c is the speed of light in vacuum. Now let the electric field be polarized in the unit z-direction {circumflex over (z)} so that Floquet-Bloch (FB) waves propagate in the x-y plane and E=E{circumflex over (z)}. Then from the well known FB theorem we can expand the solution for the electric field intensity in terms of periodic repetitions of the grating vector, which forms FB waves, so that
where vm is the amplitude of the mth mode in the expansion. Taking km=k0+mG, where m∈{0, ±1, ±2, . . . } then
Also, observe that
km−G=k0+mG−G=k0+(m−1)G=km−1, (643)
and similarly km+G=km+1, so that after reorganizing dummy summation variables
Clearly, the terms in square brackets must be zero for this to be true. As m∈{0, ±1, ±2, . . . } then we have an infinite number of equations
which may be put into a diagonally banded matrix format such that
where km2=km·km. Furthermore, the vector dispersion relation, which is the desired output of our efforts, is then given by
km=k0+mG. (647)
If we define the infinite square matrix above as A and the infinite element vector as v then Av=0 then we must require that unknown values of k0 satisfy det[A]=0, where det[⋅] is the determinant operator, so that in general vm≠0.
Let λ be the wavelength in a medium having average refractive index n0 so that λ=λ0/n0 with λ0 as the free space wavelength. Then, to compute the vector dispersion relation km for all m∈{0, ±1, ±2, . . . } it is of considerable importance to note that: [1] the value of k must be equal to 2π/λ to conserve energy and [2] k0 has the same y-component as the incident electromagnetic field to conserve photon linear momentum tangent to the VPH first surface 450.
Let the wavevector associated with m=0, i.e. the wavevector associated with the first Brillouin zone be given in component form by
k0=x0,ky0
, (648)
where ky0 taken from the range −∞<ky0<+∞ and is considered known, i.e. the independent variable. The vector k0, and by extension km, are therefore considered computed when the dependent variable kx0 is calculated from the independent variable ky0, which is the component of the wavevector tangent to the VPH boundary with the isotropic medium.
Note that when the computed value of k0 is a real-valued vector then it represents a traveling wave. When the computed value of k0 is complex-valued it represents an evanescent wave. There are two cases for which we expect evanescent waves. The first case is when |ky0|>2π/λ. This occurs to allow phase matching of waves in the input medium 45q to FB waves in the VPH at the VPH first surface 45o. Similar evanescent surface waves can occur at the VPH second surface 45p. The second case is when |ky0|≤2π/λ and there are Bragg resonances leading to at least one PBG. The PBG rejects the propagation of the incident wave through the VPH so that the wave is reflected from the fringes forming the VPH.
In the reciprocal space 45b a solution of Eq. 645 is calculated and graphically represented by a wavevector diagram for particular values of the modulation amplitude M, the wavelength λ and the reciprocal lattice vector G. In this particular case there are two PBGs that are shown. Shown are the wavevector diagrams for s-polarized (solid line) and p-polarized (thick dashed curves). The circles (thin dashed curves) are the shifted isotropic medium dispersion curves.
In the particular case shown we see that a construction line 45v cuts the wavevector diagram at periodic positions. Shown at these positions are the group velocity vectors given by the k-space gradient of the dispersion relation vg=∇kω(km), where ω=ckm, where c is the vacuum speed of light. An example of a group velocity vector for the s-polarized waves is 45w. The combination of an infinite number of waves, having the same group velocity vector, forms the FB excitation, which is the fundamental electromagnetic excitation within the VPH. The FB waves are analogous to plane waves in a non-periodic medium. The FB waves are the simplest waves that can exist in the periodic medium of the VPH. Simple isolated plane waves can not exist within the periodic media. The FB waves are formed by an infinite number of different km vectors having the same group velocity vector vg.
To obtain analytic expressions for the diffraction efficiencies near Bragg resonance, which will assist the reader in understanding the dependencies of for specifying the optimum VPH thickness and angular bandwidths associated with each VPH layer of the structure, it will prove instructive to look at an Approximate Coupled Wave Analysis (ACWA) of a VPH.
For the purpose of simplifying the analysis we will take the average refractive index of the VPH equal to the refractive index of the input medium, which is also equal to the refractive index of the output medium and is represented by n0. In general this is not a required restriction. Note that the maximum deviation of the refractive index from its average value may be given in terms of the modulation amplitude of the relative permittivity by the relation
Additionally, the VPH thickness is represented by d. The replay wave 45i is represented by R(x) and is at a replay angle θR measured relative to the x-axis. A resulting scattered wave is diffracted according to the contents of a memory cell, i.e. the structure of a VPH. Therefore we say that a scattered wave 45j results from a replay wave. The scattered wave is represented by S(x) and is at a scatter angle θS measured relative to the x-axis. The fringes of the VPH are separated by a normal lattice spacing 45t and form the reciprocal lattice vector 45u, which is represented by G. The reciprocal lattice vector is perpendicular to the fringes and at the reciprocal lattice angle γ as measured relative to the a-axis.
The electric field of an s-polarized replay wave in the input medium 45q points in the {circumflex over (z)} direction and is given by
ER={circumflex over (z)}e−ik
where kR=(2π/λ) cos θR, sin θR
. The electric field of an s-polarized scattered wave in the output medium 45r is
ES={circumflex over (z)}S(d)e−iK
where kS=(2π/λ) cos θS, sin θS
. Within the structure of the periodic structure there are VPH layers, which to first order are assumed to not interact with each other so that we can focus on the essential underlying physics. Within a VPH layer the s-polarized (TE) electric field is given by
ETE={circumflex over (z)}{R(x)e−ik
where the form of Eq. 652 is consistent with radiation that is at or near Bragg resonance because it assumed that kS=kR+G and the magnitudes kR≈kS, which can only occur at certain specific intersection points of the shifted isotropic dispersion diagrams forming the VPH wavevector diagram i.e. at or near Bragg resonance.
Next, by plugging in the above assumed form of the electric field into Eq. 639, writing cos[G·r] in terms of exponential functions and using a slow wave approximation wherein R″(x)=S″(x)≈0 (where each prime is a derivative with respect to x) we find that
The above equation can only be true when each term in large brackets is zero and when k=kR (i.e. by energy conservation), so that
where the coupling coefficient is
the detuning coefficient is
the replay obliquity coefficient is
CR=cos θR, (657)
and the scatter obliquity coefficient is
Additionally, the squared magnitude of the scatter wavevector is given by the dot product
kS2=(kR+G)·(kR+G) (659)
where the reciprocal lattice vector is
Furthermore, note that the reciprocal lattice angle γ, which is actually ambiguous by π radians in direction, is chosen by convention so that kS=kR+G (and not kS=kR−G). This is effectively setting all the km components of the prior modal analysis to zero except for one mode. Taking the derivative of Eqs. 654 and rearranging terms we find that
Where the Dirichlet boundary conditions for a transmission VPH are
S(0)=0
R(0)=1, (662)
and the corresponding Neumann boundary conditions are obtained from Eqs. 654 and the Dirichlet boundary conditions, whereby
On providing trial solutions R(x)˜eiα
where the bS and bR terms are associated with the corresponding terms with square roots. Therefore, after applying the Dirichlet and Neumann boundary conditions we find
The diffraction efficiency is calculated by first evaluating Faraday's equation H=∇×E/(−iωμ0) and then finding the time averaged Poynting vector =½ Re[E×H*] in the input and output media. The input media has an incident wave and the output media has its corresponding replay and scattered waves. The results are
from which we can take the ratio of the x-components to obtain the diffraction efficiencies. The s-polarized scattered diffraction efficiency is given as
Similarly, the s-polarized replay throughput efficiency is given by ηR=R,x/
0,x so that
Finally, it is possible to show that all of the results of the ACWA for s-polarization carry forward for the p-polarization with the understanding that κ is modified. The general expression for κ being
where {circumflex over (R)} and Ŝ are unit vectors pointing in the direction of the polarization for the replay and scattered waves. For s-polarized waves {circumflex over (R)} and Ŝ point in the same z-direction and {circumflex over (R)}·Ŝ=1. For p-polarized waves the replay unit polarization vector {circumflex over (R)} is shown as 45k and the scatter unit polarization vector Ŝ is shown as 45m.
At Bragg resonance we have the special cases
ηS=sin2[κd√{square root over (sec θR sec θS)}] (674)
and
ηR=cos2[κd√{square root over (sec θR sec θS)}] (675)
and energy is conserved because ηS+ηR=1, which is unlike the more general off Bragg resonance case described above. Moreover, an optimum VPH thickness d that will transmit all the replay wave into the scattered wave at 100% efficiency at Bragg resonance occurs when ηS≡1 (and ηR≡0), therefore
where md∈{0, 1, 2, . . . }, which controls both the special and angular bandwidths. Also observe that due to the transition symmetry associated with the reciprocal lattice vector G in Eq. 640 we can formally exchange the angles of θR and θS, i.e. θS↔θR, and still have perfect Bragg resonance. This is the underlying reason for a DAR.
In equation Eq. 676 the quantity 2md+1 is an odd number for any integer md. Let us define mTE=2md+1 for TE modes (s-polarized) waves and mTE=2md+1 for TM modes (p-polarized) waves. By combining Eqs. 676 and 673 for TE and TM polarized waves we find that to have 100% diffraction efficiency for both TE and TM polarizations requires that
As a consequence the ideal condition for no polarization dependent losses is that
where mTM and mTM are odd integers for transmission mode holograms and the steering angle 45n is give by (θR−θS). Note that the steering angle is generally not in the plane of
From a modal analysis perspective we can find the response of a VPH for p-polarized (TM) waves by generalizing Eq. 640 for the FB waves by writing
where vm are now in the x-y plane for p-polarization, instead of in the z-direction for s-polarization—i.e. see coordinates in
Also, after substituting Eqs. 681 and 638 into Eq. 636, forming the approximation
which is valid for M<<1, substituting this approximation into Eq. 636 and equating like exponentials in the resulting expression for Eq. 636 we obtain
which is substituted into Eq. 682. Next, take the inner product of the resulting equation with G and define um=vm·G so that
which may be put into a diagonally banded matrix format such that
where km2=km·km. If we identify the infinite square matrix above as A and the infinite element vector as u then Au=0 then we must require that unknown values of k0 satisfy det[A]=0, where det[⋅] is the determinant operator, so that in general um≠0. This again provides a wavevector diagram having a set of band gaps formed at the intersection of shifted circular dispersion curves. These p-polarized band gaps are different than in the case of the s-polarized waves analyzed before. This is shown in more detail in
In particular, within the reciprocal space we see the shifted circular dispersion curves, an example of which is 45c, the s-polarized dispersion curves (thick solid curves) and p-polarized dispersion curves (thick dashed curves) are shown. Examples of the p-polarized dispersion curves are 45d and 45e. Examples of the s-polarized dispersion curves are 45f and 45g. The orientation of the dispersion curves is made clear by the corresponding orientation of the fringes and the construction line 45h. The implication is that the geometry in reciprocal space is much the same as for p-polarization as it is for s-polarization, but there are nonetheless different optimal VPH thicknesses, angular bandwidths, spectral bandwidths, and diffraction efficiencies for the p-polarization.
For the avoidance of doubt, what is being shown in
In summary, a basic building block of a light steering system is a Surface Passivated Volume Phase Hologram (SP-VPH). The SP-VPH provides mirror-like diffraction wherein the diffracted light is predominately in the direction that a conventional mirror would provide reflection had it the same orientation as the SP-VPH fringes. The SP-VPH only provides this mirror-like response over a jointly narrow angular and spectral band about the Bragg resonance of the SP-VPH. Light that is not within the narrow angular and spectral bandwidths ideally is not diffracted and passes through the SP-VPH without redirection. The SP-VPH can be made more ideal in its diffractive response by fringe-to-fringe and along-fringe apodization techniques and by carefully choosing the thickness d, depth of modulation M, the steering angle (θR−θS) so as to minimize or eliminate polarization dependent losses.
Next, we consider the generation of the electric fields that can create the needed DEP-based ponderomotive forces on nanoparticles in a colloid to form a proper apodization on a VPH having an electronically controllable reciprocal lattice vector G. This will allow for a thin colloidal control volume to steer light having a specific wavelength λ, bandwidth Δλ, polarization, coherence length, and angular spread Δθ. Ideally, the steering direction would be variable and based on a potentially complex set of waveforms impressed on simple electrodes.
Beam Steering Fields for DEP-Based Gratings
In this section the electric potentials, electric fields, ponderomotive forces, and particle concentrations for simple gratings and VPH having an apodization are derived and demonstrated. Some methods for developing more elaborate gratings are then discussed. These derivations are useful for electrically small and large gratings. The electrically small gratings are typically used with dielectric particles and are dynamic VPHs. The electrically large gratings may be used with dielectric particles or metallic particles. This allows for extremely broadband reflective gratings are useful for applications like solar energy. Thus, the following discussion applies to both electrically small and large structures. The electrically small structures have a scale similar to the wavelength of light. The electrically large structures are much larger than the wavelength of light.
In particular, let us consider a sheet of a colloidal mixture that is located between two solid transparent sheets, such as glass or plastic. This forms a thin control volume that may be very large, even many square meters in area. It is to be assumed that if electrodes are used then the electrodes are directly printed on the two inner surfaces of the control volume.
For electrically large gratings, i.e. where the grating period is much larger than the wavelength of light, electrodes on each surface may be energized by hybrid analog-digital voltage source that provides an arbitrary voltage in space and time on the electrodes. The electrodes may be thought of as a means for spatial sampling of the desired potential waveform and must obey the Nyquest sampling theorem.
For electrically small gratings other means than electrodes are typically required to impress a desired electrical potential because it is difficult to fabricate such small electrodes. For example, light induced surface plasmon polaritons may be used to induce a potential on a thin continuous sheet of metal placed on a glass or plastic surface instead of patterned electrodes. So the objective of the analysis here is to see how harmonic potential fields may be used to reach inside the control volume and create a grating structure dynamically. Similar considerations are useful for photonics crystals.
Note that while the principle of superposition may be used to add together potentials in the control volume it may not be used to obtain the forces because a DEP process is nonlinear—e.g. requires calculating factors like |E|2. Therefore, the entire potential in the control volume needs to be specified at the beginning of the solution of the problem.
For an assumed potential function to be consistent with the allowed field configurations in the control volume it must be constrained by Maxwell's equations, therefore consider Ampere's law
where =ϵε,
ƒ=σε so that on taking the divergence of the above equation, using the identity ∇·(∇×
)=0, and using the quasi-electrostatic approximation ε=−∇ϕ we get
which is a differential equation for ∇2ϕ(r, t) whose solution is
∇2ϕ(r,t)=∇2ϕ(r,0)e−t/t
where t0=ϵ/σ the characteristic time of the system. Therefore, for t>>t0 we have to a good approximation that
∇2ϕ(r,t)=0 (690)
which is true for all quasi electrostatic processes. Then potential functions on the plane y=±Ly are assumed to have the form of a traveling wave potential
ϕ(x,y,t)=V cos(ωt±kxx)e−k
where V is the harmonic voltage amplitude, and kx, ky are wave numbers. This equation is self-consistent with Maxwell's equations, and will only satisfy Laplace's equation for the potential when we restrict kx=ky in the control region −Ly<y<+Ly, which may be demonstrated by direct substitution into Eq. 690. Note that two oppositely traveling waves potentials will form a standing wave potential which is also allowed by Maxwell's equations. Thus, the simple forms allowed are
ϕ1(r,t)=V cos(ωt±βx)e−β(L
ϕ2(r,t)=V cos(ωt)cos(βx)e−(L
where we have let β now represent the wavenumber.
The objective is to provide potentials on the electrode planes that are configured to provide podermotive forces normal to the electrode arrays in the y-direction and tangent to the electrode array in the x-direction in such a way as to construct a grating with a desired grating vector and apodization within the control volume. Based on Eq. 231 at a single spectral component we assume the functional forms
ϕ=ϕ(r,t) u=u(r,t) ε=ε(r,t) F=F(r,ω) (694)
ER=ER(r) EI=EI(r) (695)
KR=KR(ω) KI=Ki(ω) (696)
where it is understood that KR and KI are also functions of the constitutive parameters. Then the governing equations, in the order of the calculation flow, are as follows
the last equation is a Fokker-Planck type of equation that is expressing DEP-based particle advection and it must be coupled with boundary conditions and an initial condition.
Note that the solid boundaries of the control volume at y=±Ly are subject to zero particle flux normal to the surfaces from the combined process of diffusion and electric-field-gradient forced advection, so that the sum of the diffusive and advective particle flux is zero whereby
[−D∇u(x,±Ly,t)+γu(x,±Ly,t)Fdep(x,±Ly)]·ŷ=0 (702)
Moreover, there is also a periodic boundary condition on the concentration u(z, y, t) because there is a periodic force on the particles in the liquid. We shall assume that the potential h is given a periodicity of length 4Lx. Therefore, the electric field will have periodicity of 4Lx because the quasi-electrostatic calculation uses ε=−∇ϕ. This then implies that the force and concentration have a spatial periodicity of 2Lx because the force is proportional to terms like ∇|E|2 which doubles the spatial frequency and therefore halves the spatial period.
Therefore, when considering the potential periodicity we require that sin(βx)=sin(β(x+4Lx)), however, when we consider force periodicity we require that sin(2βx)=sin(2β(x+2Lx)). In both cases we have that
This means that the particle flux at x=−Lx must be the negative of the particle flux at x=+Lx, which is due to the periodicity of system. Therefore,
[−D∇u(−Lx,y,t)+γu(−Lx,y,t)Fdep(−Lx,y)]·{circumflex over (x)}+[−D∇u(Lx,y,t)+γu(Lx,y,t)Fdep(Lx,y)]·{circumflex over (x)}=0. (704)
The result of Eqs. 702 and 704 is that the volume integral of u over a periodic unit cell is conserved, as was shown in Eq. 180. Note, for this example the unit cell is defined to be
−Lx<y<+Lx −Ly<x<+Ly −Lz<z<+Lz. (705)
Next, consider a first approach to making the forces on particles periodic in the x-direction is to exploit the fact that balanced traveling waves in opposite directions will form a periodic standing wave. There are two obvious cases. In the first case oppositely traveling waves are separated by the control volume thickness of 2Ly. In the second case oppositely traveling wave are induced separately on y=−Ly and y=+Ly. This gives rise to two general forms for the potential in the control volume that are consistent with Maxwell's equations under quasi-electrostatic conditions.
The first general form is given by
where y=Ly is the location of the top potential plane, VT,m is mth voltage amplitude on the top potential plane, and βT,m is the mth wavenumber on the top potential plane. Similarly, y=−Ly is the location of the bottom potential plane, VB,m is mth voltage amplitude on the bottom potential plane, and βB,m is the mth wavenumber on the bottom potential plane. Notice that the top and bottom are configured to have oppositely traveling potential waves.
The second general form is given by
where y=Ly is the location of the top potential plane, VT,m is mth voltage amplitude on the top potential plane, and βT,m is the mth wavenumber on the top potential plane. Similarly, y=−Ly is the location of the bottom potential plane, VB,m is mth voltage amplitude on the bottom potential plane, and βB,m is the mth wavenumber on the bottom potential plane. The phase ψ is introduced as a means to vary the relative standing wave positions. Notice that the top and bottom are configured to have oppositely traveling potential waves. Also notice that the ψ term is not included in the first general form because we can always find a new start time when the traveling potential waves are aligned.
The first and second general forms will create a ponderomotive force field in the control volume. This force can quickly become algebraically complex due to the many cross terms that result from forming terms like |E|2. Therefore, initially we will look at only a single term in each of Eqs. 706-707.
Assuming that Eq. 706 only has one term with m=1, we use Eqs. 697-698 to obtain the quasi-electrostatic fields and then use Eqs. 699-700 to find the force field on a particle of radius b. Then we can use Eq. 701 plus initial and boundary conditions to generate the particle concentration u(z, y, t).
Therefore, using one term of Eq. 706 we get in the control volume gap −L<y<+L counter propagating potential waves
and using Eq. 698 we get an expression for ε(r, t), which can be written using trigonometric identities in the form of Eq. 698 whereby the x and y components of ER are give by
ER,x=VTβTe−T(L
ER,y=−VTβTe−β
and the x and y components of EI are give by
EI,x=−VTβTe−β
EI,y=−VTβTe−β
Then from Eq. 700 we get the time average force on a particle as
F3=2πb3ϵLGKI(ω)+HKR(ω),GKR(ω)−HKI(ω)0
(713)
G=2(−VB2βB3e−2β
H=2VTVBβTβB(βT+βB)e(β
In the special case of β=βT=βB this reduces to
F2=4πb3ϵLβ3e−2βLKI(ω)(VT2e2βy−VB2e−2βy)+2VTVBKR(ω)sin(2βx),KR(ω)(VT2e2βy−VB2e−2βy)−2VTVBKI(ω)sin(2βx),0
(716)
If we also take V=VT=VB then the force on a particle in the colloid is
F1=8πb3ϵLV2β3e−2βLKI(ω)sinh(2βy)+KR(ω)sin(2βx),KR(ω)sinh(2βy)−KI(ω)sin(2βx),0
(717)
In
Beam Steering Sensor Drones
In this section some applications of accurate and precise GRIN light control are described. In particular, we will look at some representative LiDAR and 3D-printing systems that utilize devices previously described by the exponential GRIN model.
In particular, in
The upper section 51c further comprises a transparent first surface 51f, a solid and transparent optical radome 51g. The transparent first surface 51f may be flat or curved as has already been described in this disclosure. Internal to the transmitter is a nanoparticle colloid 51h that fills a control volume having a second surface 51i approximately in the shape of a segment of a circle as has also already been described in detail.
The LiDAR transceiver has an outbound laser beam 51j sourced from the transmitter beam 51k. The transmitter beam is curved along an approximately circular trajectory by the application of a harmonic traveling wave voltage to one or more electrode arrays 51m.
An outbound laser pulse strikes a target (not shown), which becomes the source of backscattered light 51n. Some of the back-scatted light refracts through the transparent first surface 51f and then refracts through the second surface 51i to traverse an uninterrupted circular light trajectory 51o back to the plane of the laser source 51b at the beginning of the upper section 51c. This light then enterers a radiation collector comprising a non-imaging optical concentrator formed by optical reflectors 51p and 51q. In this embodiment the optical collector is in the form of a well-known compound parabolic concentrator, which may be formed by a mirrored surface or by TIR. The optical collector provides gain that can improve the signal-to-noise ratio of the receiver. As can be seen from
Note, while
In the 2012 science fiction movie “Prometheus,” which was directed by Ridley Scott, the human crew of a spacecraft used several small spherically shaped drones to explore a subterranean-like network of tunnels on the planet LV-223. The creators of the movie called the automated drones “Spectagraphs,” and they were used to provide near real-time maps of potentially dangerous and remote caves and spacecraft corridors. This type of laser scanning sensor drone can be deployed today using a solid-state (or “wet-state” if you prefer due to the nanoparticle colloid) beam steering LiDAR as discussed herein.
In particular, in
The sensor drone has ducted fans 54e and 54f for propulsion. The fans are located along a centerline of the drone and typically operate in opposite directions to cancel out fan induced torques. The ducted fans can operate at adjusted differential angular velocities speeds to allow a net torque to exist so that the drone can rotate around the rotational axis of the ducted fans. The drone may have may well-known methods for thrust vectoring. In
Around the sensor are a plurality of sensor ports 54g, 54h and 54i. Inside of each sensor port is a beam-steering LiDAR transceiver such as 54j, 54k, and 54m. Each steered laser beam 54p may be based on a different wavelength or time-domain pulse position code so that the onboard electronics can discern between backscatter from different LiDAR transceivers. Optionally some of the sensor ports may contain other types of sensors, such as but not limited to imaging sensors.
Lenses & Optics by Dielectrophoresis
Adjustable Focus Lenses for Smart Phones
In this section embodiments light control devices are provided that use DEP processes on nanoparticles in colloids to make variable focus lenses for diverse applications like smart-phone cameras, and robotic vision. The lenses are generally provided in a flat configuration, which is unlike that of a conventional lens that has a one or more curved surfaces. Furthermore, unlike a conventional lens the DEP-based lenses have variable focus based on electrode drive frequency, voltage amplitude, phase between voltages on adjacent electrodes, the (electrical, mechanical, and thermal) properties of the colloid liquid and nanoparticles as well as the distribution of nanoparticle sizes and composition. In the case of nanoparticle composition it is possible to design the distribution of nanoparticles so that the optical dispersion properties of the nanoparticles and liquid cancel to make the lens achromatic.
A first lens example is shown in
The lens has a transparent containment vessel to hold an IGL, which comprises a first containment structure 55b and a second containment structure 55c. These containment structures provide a volume within which both the refractive IGL 55d and the electric fields used in DEP are located.
Circular annular electrodes comprising a first annular electrode 55e and a second annular electrode 55f are separated by an electrical insulator 55g. The first and second annular electrodes support phasor voltages +V0 and −V0 and produce electric fields 55h, which are indicated in
Input light 55a that enters the DEP optoelectronic lens along the symmetry axis 55i is focused as output light 55j. The solutions shown in
Note that a conventional lens uses a constant refractive index and a variable thickness as a function of lens radius to form the lens. In contradistinction the lens in
A perspective view of the annular electrodes is shown in
A perspective and cut-away view of the annular electrodes and containment vessel of a electrophoresis-based lens are shown in
By using a cylindrical geometry the same thing can be accomplished to form a lens. Therefore, consider the perspective and cutaway view of an electronically controllable lens in
The colloid containment volume is a μFCC that contains the IGL and electrodes used to control GRIN profile in the liquid. A traveling wave on the concentric electrodes forces nanoparticles towards or away from the optical axis 58c so that at steady-state the constant thickness containment volume has a GRIN profile in the radial direction. A conventional circular lens, i.e. either convex and converging or concave and diverging, has an optical path length that varies radially from the input surface to the output surface of the lens. This same radial function is made using this DBS device, but now the shaped input and output lenses surfaces are replaced with a GRIN volume. The function of a converging or diverging lens is now obtained by the sign of the real part of the complex Clausius-Mossotti factor, which is controlled by the harmonic frequency of excitation of the electrodes. In this way input light 58d becomes focused output light 58f after it passes through the colloid containment volume 58b and its corresponding IGL containment structure 58e. It also allows for a dynamic zoom-in and zoom-out function as the magnification is changed by changing the harmonic electrode excitation frequency ω, the voltage amplitude or both. In practice the perspective cutaway view in
It should also be noted that if the transparent nanoparticles are replaced by opaque nanoparticles it is possible to have an electronic shutter function or neutral density filter implemented. By this example we can also see that while the main focus of this disclosure has been for transparent nanoparticle there are also obvious extensions to nanoparticles that are opaque or even absorbing in certain optical bands to make a spectral lens filter.
The lens embodiments of
The lens used in a DEP-process typically control the refractive index n instead of the surface shape. In the present example it is assumed that n only varies with the radius r of the thin lens. This lens has the z-axis as the optical axis of symmetry and the location of a focus 59b. Let us take the origin of the coordinate system 59c as the center of the lens circle on the output side of the lens. Therefore, the distance from the origin of coordinate system 59c to the focus 59b is the distance ƒ.
Furthermore, it is assumed that light rays are incident on the input side of the lens from infinity, so that all the input rays to the lens are parallel. Next, take n=n(r) and n0=n(0), then the Optical Path Length (OPL) of the rays from the source at infinity are all the same and we shall call it L∞, which is assumed to be in vacuum having a refractive index of unity. Then we can consider the ray path at a radial distance r and another ray at r=0. As the actual output ray 59d is very close to the thin lens approximating ray 59e, we shall take the angle θ to be the same in both cases below.
The OPL must be the same for all ray trajectories at the focus by Fermat's principle, therefore
where the angle θ is formed between the output ray that is focusing to a focus point at a distance ƒ from the output surface of the lens. This can be rewritten as
Moreover, by trigonometry we find that
Additionally, let us assume initially that the solid transparent nanoparticles in the colloid are smaller than approximately λ/10, this constraint can easily be relaxed later. Then Eq. 233 is valid and we have
n=nL+(nS−nL)ν (721)
with ν as the volume fraction of the solid nanoparticles (the subscript of S was dropped for notational convenience) so that Eq. 719 becomes
Where ν0 is the volume fraction at r=0. Then
For a thin lens with a focal length that is much longer than the radius of the lens we have
where the factor of 2 is to account for a diameter. By applying the Taylor expansion for an exponential twice we get
Therefore, on plugging this approximation back into Eq. 723 we get
where
rm2=2ν0fw(nS−nL) (727)
and rm is the maximum radius of the volume fraction distribution so that at r=rm the volume fraction of nanoparticles is zero. As the diameter dm is twice the radius rm we can write
dm2=8ν0fw(nS−nL). (728)
The volume fraction at r=0 is ν0 and it is anticipated to be a maximum of the volume fraction distribution. Also, it would be more convenient to use the average volume fraction νA over the lens control volume instead of the maximum volume fraction of nanoparticles. Therefore, the average over the lens, which has a radius R is
Again, because a diameter is twice a radius we can write
It is also interesting to note that when dm=D then from Eq. 730 we find
ν0=3/2νA (731)
Plugging Eq. 730 back into Eq. 728 we obtain the optical power of the lens in diopters (when ƒ is in meters) as
which is valid when dm≥D and where it has been taken that the ratio of the lens diameter to the lens thickness is
This ratio is typically ten or greater. Note that if dm<D then an annular region is formed around the lens that is devoid of nanoparticles. This means that the lens produces images with distortions. Therefore, this case has been eliminated from analysis here even though voltages can still be increased to further concentrate the nanoparticles and increase the lens magnification.
When the diameter of the nanoparticle distribution approaches infinity, ie. dm→∞, then there is a uniform distribution of nanoparticles within the lens diameter D, νA=ν0 and the lens has zero diopters of optical power. When dm=D, then there are zero nanoparticles at the edge of the lens and there is maximum optical power. Thus,
One of the amazing things about this type of lens is that it provides a fluid lens with the optical power of a solid lens. This is seen in
Please note that especially for the smaller diameter performance projections in
Spectral & Non-Spectral Light Combiners
The ability to combine laser beams is an enabling technology for telecommunications and high power laser applications. Power combiners come in two general varieties: (1) single wavelength signal and power combiners and (2) multi-wavelength signal and power combiners (also called spectral signal and power combiners).
The advantage of a single wavelength combiner for M-beams is that the electric fields add and the power is proportional M2. The disadvantage of a such a configuration is that the frequency of each beam must be exactly the same and phase-locked. Failure to achieve this will cause destructive cancellation of the laser beam. The cost and complexity of this configuration is therefore often not compatible with cost constraints and physical scaling to ever larger systems.
In contradistinction, a M-beam spectral power combiner has total combined power proportional to M. However, there is no need to keep the beams at the same frequency or phase locked. The only real requirement is that the frequencies are sufficiently far apart and that no beam frequency is a linear combination of other beam frequencies via a diophantine equation ƒi=miƒi+mkƒk for integers {mj, mk} and frequencies {ƒi, ƒj, ƒk} where i, j, k∈1, 2, 3, 4, . . . , M to avoid potential nonlinearities from generating overlapping harmonics.
Power and signal combiners can also be used as part of a telecommunications system. The combiner takes all the beam and combines them to a common output that is useful for wavelength division multiplexing of a communications signal.
Therefore, consider
Based on this insight into focusing light in an exponential GRIN medium we consider
Light carrying fiber optic 61c and 61d have output ends symmetrically provided with ball lenses 61e and 61f that provide rays that are on trajectory solution circles, as discussed for
The output of the ball lenses are the ray bundles 61i and 61j, which pass through the GRIN medium to a focal point 61k on the optical axis 61a. An optional mirror 61m is provided having partial reflectivity. When the mirror is present it forms the basis for lasing feedback as light energy is reflected back into the optical fibers along ray bundles 61i and 61j. The direction of propagation is shown by propagation direction arrows 61n and 61o. Note that when the optional mirror 61m is present energy propagates in two directions and this is helpful in applications such as but not limited to interferometry.
Also, it is critical to appreciate that some energy from Light carrying fiber optic 61c is received by Light carrying fiber optic 61d and vice-versa. For example, in a three dimensional geometry with multiple fibers, see for example
Thus, with the optional mirror 61m included a single wavelength power combiner is possible. If the optional mirror 61m is not included then each fiber laser may be of a different wavelength and the fiber lasers need not be phased locked. This is called spectral power combining as the different wavelengths of light do not interfere destructively. The beam combiner output 61p is diverging due to the geometry of the trajectories through the GRIN medium so a lens 61s is used to refocus the combined laser beam onto a target 61t. The lens 61s is a DEP-based lens, like that shown in
It is for this reason that
In particular,
Additionally,
Bistable Optical Switches
In what follows we shall examine the properties of a reflectance switch and then show how an optical switch based on DEP is both similar to a reflectance switch and different from a reflectance switch. In particular, we shall see that a DEP-based optical switch can redirect light by a process of refraction through the volume of a GRIN medium that effectively replaces reflection from a surface. Nonetheless, it is quite helpful to first look at the properties of an optical switch that is based on reflection because many of the properties are effectively the same, they just occur over different spatial extents.
Consider
n2(y,t)=nL+(nS−nL)νS(y,t) (735)
where ν(y, t) is the electronically modulated volume fraction of the solid nanoparticles as a function of time t and the distance y away from the optical boundary at y=0, where increasing y moves deeper into the colloidal medium away from the solid-colloid boundary. By appropriate choice of electrodes, material conductivities, material dielectric constants of the liquids/solids forming the colloid and a DEP drive frequency ω the ponderomotive forces on the nanoparticles in the vicinity of the boundary between the first and second medium can be made to be: (1) towards the boundary, (2) away from the boundary or (3) a zero force that is neither towards or away form the boundary.
These ponderomotive forces have a magnitude and direction that are due to the magnitude and sign of the real and/or imaginary parts of the Clausius-Mossotti factor—i.e. KR and KI respectively. The specific electrode configuration used and the phasing of harmonic voltage signals between the electrodes dictates if the ponderomotive forces on nanoparticles is related to KR, KI or both KR and KI. See for example, Eqs. 131-129, Eq. 231, the listed expressions for the relative mobilities {γ1, γ2} and
Continuing the example associated with
n2(max)=1.40+(1.55−1.40)(⅔)=1.50 (736)
n2(avg)=1.40+(1.55−1.40)(⅓)=1.45 (737)
n2(min)=1.40+(1.55−1.40)(0)=1.40. (738)
These three example refractive index values correspond to: (1) a DEP process piling up nanoparticles at the boundary so that a maximum volume fraction of about νS=⅔ is achieved, (2) a relaxed state where the Clausius-Mossotti factor is zero and particles diffuse into an unforced homogeneous distribution in the colloid where the volume fraction of nanoparticles is about νS=⅓ and (3) a DEP process pushing nanoparticles away from the boundary so that approximately νS=0.
These three refractive index cases are shown in
Next, we relax the initial assumption of a flat optical boundary surface between the solid first medium and a colloidal second medium and consider in detail how the DEP-based optical switch is actually based on refraction and not reflection. This assumption is even less appropriate in a situation where there are two regions of a colloid having nanoparticles. At issue is the diffusive process that occurs on the nanoparticles in the liquid that creates a GRIN volume medium instead of a well defined optical boundary.
To see how this works out consider
While we will often continue to talk about this effectively as a “TIR” reflection as a short-hand way to describe this volume refraction process the reader is to appreciate that the process is as has just been described. For the avoidance of doubt, the distinction here is between conventional-TIR from an optical surface compared to a refraction from a GRIN volume region is important because many prior-art display patents, including those for VR, AR, and MR systems have explicitly focused on systems that reflect and/or defract either actively via opto-electronics or passively. Consequently, companies like MagicLeap, Microsoft, Oculus, and Google, which are developing virtual and/or augmented reality glasses generally neither teach or anticipate the possibility of optical elements that are based on a different set of physical principles as have been described herein.
To help the reader a new abbreviation for total internal reflection is adopted herein to help the reader make a distinction: In particular let's adopt the acronyms Total Internal Reflection (TIR) and Total Internal Refraction (TIR2).
As just one specific example, the U.S. Pat. No. 9,804,379 from Magic Leap Inc. and inventors Brian T. Schowengerdt and Mathew D. Watson issued 2017 Oct. 31 shows dozens of disparate alternate embodiments for virtual reality glasses to apparently protect the augmented reality U.S. Pat. No. 9,753,286 and its derivatives. However, the embodiments in U.S. Pat. No. 9,804,379 neither teach or anticipate: (1) volume refraction in a GRIN medium to replace reflecting surfaces by volume refractions, (2) electro-active materials being a mixture of separate materials of different phases like a solid and liquid forming a colloid, (3) that a reflector (or its equivalent) does not have to be a discrete unit, but instead may be self-assembled on demand by an electric signal and (4) how to achieve about 120 degrees field of view for a human observer.
On GRIN Lens Fabrication Using DEP Electrodes
It is often desired to have a Graded Refractive Index (GRIN) lens that is permanent and solid instead of dynamic and a colloid. However, up to this point in this disclosure the material has been focused on dynamic systems. Therefore, in this section we focus on the manufacturing of passive structures, including precision GRIN lenses using electrodes that provide the GRIN in a colloid, which is then cured by at least one of passive drying, heat, photochemical curing.
One method to achieve this is by using a mixture that comprises a colloid or suspension that has liquid phase chemical components that are at least one of thermally cured, chemically cured and photochemically cured. This means that an initially liquid medium, having solid particles therein, is made solid by the application of a drying heat, light that induces a chemical hardening, or by the inclusion of a chemical hardening agent. Any particles in the initial suspension or colloid would be trapped in the resulting hardened material. So that in the case of a transparent colloid or suspension the resulting hardened structure is also transparent and has a GRIN distribution.
However, before the colloid or suspension is hardened by curing it can be manipulated by non-uniform and osculating electromagnetic fields. The electromagnetic field is sourced by electrodes.
This opens up many unique and advantageous possibilities for manufacturing. Because DEP can work on any type of particle it is quite possible to use manufactured three dimensional (3D) structures that have may different types of materials therein.
Broadband and Broad-Angle Anti-Reflection Coatings
Ultra broadband and broad angle Anti-Reflection Coatings are of critical importance in many applications. For example, in solar energy applications, where the solar energy has a wavelength from about 390 nm to 1,800 nm and may be incident from a wide range of directions.
To accommodate such broadband multi-directional light we can turn to nature. In particular, investigations on insect eye efficiency in the 1967 proved interesting because electron microscope images showed that nano-scale bumps and inclusions on the eyes of butterflies and moths increased the efficiency of light transmission into the eyes of these insects substantially near perfect 0% reflectance. The physics that nature exploited was to provide a very gradual change in the refractive index between two optical media. The more gradual the change the better the broadband control of reflected light. See for example the article “Light Conversions,” by American Scientist, Vol. 107, p-140, May-June 2019.
By using a DEP process it is possible to create a GRIN region between two homogenous media that is very gradual in changing its effective refractive index value so that reflection is minimized. For example, when using a liquid of refractive index 1.48 and glass nanoparticles of say 1.95 then it is possible to slowly transition between intermediate values of refractive index. For example, from a refractive index of 1.60 to 1.75, by varying the solid nanoparticle volume fraction from about 15% to 36%. This transition can be made permanent by using a curable colloid. This can provide a means for extremely broadband antireflection coatings that are between optical elements. However, to extend broadband anti-reflection coatings to an interface between air and an optical medium, e.g. glass or acrylic, something more needs to be done.
In particular, if we consider the solid nanoparticles to now be structured solid nanoparticles having a void therein then the average refractive index of the structured nanoparticle is substantially lower than if it were solid. The void being contained by a thin shell of solid material that makes up a solid structured particle. If the shell has substantially the same refractive index as the liquid (which is not a requirement) then it is as if there is a controllable void that can be positioned in space by the ponderomotive forces of a DEP process. A plurality of such voids may be given a spatial distribution by suitable electric fields. The volume fraction of the voids can then range from nearly 0% to 100% when a plurality of void diameters are deployed—see
However, for the case of an antireflection region the basic strategy is to use structured particles (micro and nano particles) with voids therein to create a gradual change in the refractive index over a large optical distance, typically from hundreds of nanometers to tens of microns, by DEP forces. This is many wavelengths of the usable radiation, but still small by human scale. Then, if desired, the voids having a specific spatial distribution are locked into place via a curing process of the liquid host medium and afterward the electromagnetic fields are permanently removed to complete the manufacturing. With a refractive index of near 1 close to the surface and a refractive index matched to the inner optical medium away form the surface the light will have a minimum of reflectance over a broad range of wavelengths and angles.
Thus, by using both solid particles and particles with voids a very great range of refractive indices may be obtained. On interesting example of natural thin shelled structured nanoparticles are fullerenes, such as C60 and C200, which can be transparent.
Finally, other approaches may also be deployed. For example, the solid nanoparticles can be made of a material that may be selectively etched away by chemical, photochemical, or other means to create a sponge-like material with varying degrees of porosity. So, in this second approach a plurality of solid particles, of potentially different diameters, are selectively distributed throughout a curable liquid to optimize the pore locations after an etch.
Displays by Dielectrophoresis
Display Pixels Based on Total Internal Refraction
Total Internal Reflection and Total Internal Refraction are different phenomena. Before describing the DEP-based display technology we first consider the difference.
Total Internal Reflection (TIR) is the well-known process that occurs when light traveling in a medium of high refractive index is reflected completely from a surface at the boundary to a lower refractive index back into the region of higher refractive index. The redirection of light obey's Snell's law of reflection and the critical angle from the normal to the surface is θH=sin−1[(nL/nH)], where nL is the lower refractive index, nH is the higher refractive index, OH is the reflection angle in the medium of higher refractive index as measured from the surface normal direction.
In contradistinction, Total Internal Refraction (TIR2) occurs when light traveling in a medium of high refractive index is refracted along a continuous trajectory from the region of higher refractive index back into the region of lower refractive index. The redirection of light obey's Snell's law of refraction as applied along infinitesimally small portions of the trajectory.
Generally, if the same input ray was a used starting from a medium of homogeneous refractive index and the outputs compared between Total Internal Reflection and Total Internal Refraction we would note different internal processes and outcomes. The reflected ray trajectory for Total Internal Reflection is: (1) piecewise continuous, (2) not smooth and differentiable. The refracted ray trajectory for Total Internal Refraction is: (1) continuous, (2) smooth and differentiable through a volume of space. Moreover, the output rays for both process would be spatially displaced, however the directions of the rays would eventually be the same direction and parallel because of the principle of conservation of linear photon momentum.
So for the purpose of this disclosure we often talk about Total Internal Reflection with the common acronym of (TIR). However, based on the context of the discussion the phenomenon of Total Internal Refraction (TIR2) may be more appropriate to use and it will be clear from the context of the discussion what applies. That is to say the directions of the output rays from Total Internal Reflection and Total Internal Refraction are the same even though the rays may be displaced relative to each other. In many applications the displacement is not critical from a physical point-of-view and we can think of the refractive process as a distributed form of a reflected process. However, within the present context Total Internal Reflection and Total Internal Refraction are different physical processes.
In this section the one notable advantage of Total Internal Refraction compared to Total Internal Reflection is that the ray trajectory can pass through a medium optimized for a scattering process that can redirect light out of a pixel region towards a human's eye. The use of Total Internal Refraction appears to be new and not taught or anticipated by the prior art in the context of display technology.
Therefore, we now consider electrode configurations optimized for total internal refraction and for asserting ponderomotive forces on nanoparticles so that a dynamic pixel can be formed. In particular, given a traveling wave on an electrode array that is on one side of an μFCC such that the electric field intensity is
E=E0(i{circumflex over (x)}+ŷ)ƒ(t)e−κ
where ƒ(t) is a switching pulse in time then we can show that Eq. 231 at a single spectral line reduces to
which is subject to an initial condition
u(y,0)=u0, (741)
where u0 is a constant homogenous nanoparticle concentration. Also, boundary conditions that stop nanoparticle diffusion flow at the boundaries of a μFCC at y=0 and y=Ly are
and a conserved number of nanoparticles in a hermetically sealed environment require that
∫0L
Eqs. 739-744 are difficult to solve analytically, however it is easy to see that we can turn the second term on the right in Eq. 740 OFF or ON on demand, which is to be done electronically. When ƒ(t)=0 we are left with an unforced diffusion equation. When ƒ(t)=1 we have a forced diffusion equation and the forcing function is proportional to γ1, which is in turn proportional to the real part of the Clausius-Mossotti factor KR. Recall that KR can take on both positive and negative values depending on the dielectric constants, conductivities and the electrode excitation frequency ω.
This further leads to three types of electrode configurations: pDEP only, nDEP only and hybrid nDEP-pDEP. Thus, depending on the specific graph of the Clausius-Mossotti factor over frequency we can choose to use one or two arrays of electrodes. One array on one side of the two primary sides of a μFCC and the other array on the other side of the μFCC. The two configurations may differ in cost or switching speed performance and any design may give preference to one or the other configuration for a specific application.
Therefore, in the discussion that follows it is often the case that one of the electrode arrays shown in the specification is “optional,” or that if only one electrode array is shown that it is to be understood that two electrode arrays may be substituted. For example, when there are two adjacent arrays then changing the direction of the diffusion of nanoparticles in the μFCC may be based on turning on one array and turning off the other array or alternately using complimentary pDEP and nDEP process on opposite arrays. However, when there is only one array then changing the direction of the diffusion of nanoparticles in the μFCC must be based on switching between different values of ω, which are called ωp and ωn for pDEP and nDEP processes respectively. Additionally, zero forces on the nanoparticles at certain frequencies ωz is possible in zDEP processes.
Consider
These electrode arrays are similar to that shown in
By changing the active electrodes to the second array of electrodes 68dd and deactivating the first array of electrodes 68d the nanoparticles are swept to the opposite side of the μFCC, as show in
With this in mind consider light that has transitioned from a first medium 68k into a colloidal medium 68m at the optical boundary 68h so that Eq. 328 applies. More specifically, if the nanoparticles are lossless dielectrics then Δε=0. The result is shown as curved scattering ray 68n. Ultimately the scattering ray emerges as a linearly propagating output ray 680. The radius of curvature of the ray is controlled by the voltage amplitude on the electrode array 68p.
The light in the curved scattering ray 68n can be scattered by dielectric nanoparticles, semiconductor nanoparticles, metallic nanoparticles and hybrid nanoparticle structures that may include spherical dielectrics covered with metal shells, e.g. gold or silver, as well as structured nanoparticles that have various kinds of periodicity introduced, an example of which are photonic crystals. It is even possible to have micro- and nano-particles that are biological in origin such as blood cells and viruses. Each class of particles can interact with light differently to produce a desired color and spectral response and avoid the restrictions of etendue conservation (resulting in TIR) by means of light interacting with atomic processes. An example of the scattered light is shown as scattered light 68q, which may include fluorescing processes.
As another example consider a solid having a first medium of n1=1.50 and a colloidal medium n2 that can take on a range of refractive index values, such as 1.55, 1.50 and 1.45. This can be accomplished with a colloidal liquid having nL=1.45 and solid nanoparticles with nS=1.60. If the volume fraction νS of the solid nanoparticles in the colloid can take on values {⅔, ⅓, 0} then we find that
n2(max)=1.45+(1.60−1.45)(⅔)=1.55 (745)
and
n2(avg)=1.45+(1.60−1.45)(⅓)=1.50 (746)
and
n2(min)=1.45+(1.60−1.45)(0)=1.45 (747)
and the average homogenized refractive index is 1.50 as shown.
With this in mind consider
Next, in
Pixel Color By Diffused Quantum & Plasmonic Dots
Next, we consider nanoscale particles that can energy-shift and/or scatter light. Quantum dots (QDs) are semiconductor nano-particles that have a size-dependent emission wavelength, narrow emission peaks and a broad excitation range. The QDs are semiconductor nano-crystals in which excitons are confined in three spatial dimensions to effectively make an artificial atom. The confinement of electrons is obtained by forming a tiny semiconductor in very small size, e.g. several hundred to thousands of atoms per particle. Due to quantum confinement effects these QD artificial atoms show controllable and discrete energy levels. The QDs provide a process of optical excitation, relaxation of an excited state to a lower energy level, recombination of electrons and holes and fluorescent emission at a wavelength that is typically different than the wavelength of the optical excitation. These set of processes taken together are called photoluminescence (PL).
For example, in
Similarly, in
The PDs are based on a surface plasmon spectral response from metal coated nanoparticles. Different metallic shell thicknesses set the resonance color, which can be set to the Cyan, Magenta, and Yellow (CMY) CMY subtractive color model or to the RGB additive color model. When the PDs are illuminated by a broad-band white light source, such as that often found as a backlight in a display system or from the sun, the resonance color will appear. In this way a backlight may be best coupled with an RGB color model and a front light may be best coupled with a CMY color model. Therefore, CMY can be produced for displays that use ambient white light to create a display image based on a subtractive color model and this provides a means for electronic paper.
Additive color systems that start with darkness (i.e. an absorbing black background plane) and that provide a PL response at different wavelengths (e.g. RGB) can combine RGB light in specific proportions to produce a range of colors. In contradistinction, subtractive color systems start with white light (i.e. light that is ambient or from a display's back plane) and has nanoparticles that subtract wavelengths from the incident light in specific proportions can also produce a range of colors. Both additive and subtractive color models are possible in using DEP based displays.
In particular, in
Separate and apart from the μFCC 71e there is an array of transparent DEP electrodes 71f (shown as the dashed line below the μFCC to represent being in the vicinity of the boundary), comprising individual electrodes such as electrode 71g. The transparent electrodes are made from indium tin oxide, carbon nanotubes or other forms of transparent conductors. Note, the electrode drive circuits are not shown here to reduce clutter in the figure, however the array of electrodes is similar in construction and operation to that shown in
The external context of the magnified cross-section 71a is shown in the zoomed out view 71m, where a sheet of transparent material, e.g. clear glass or acrylic plastic, forms the waveguide 71n for the (typically but not necessarily) narrow-band guided light. This radiation is edge-launched light 71o. Specifically, the TIR confined light is shown as guided rays 71p and 71q. An example of TIR is provided at TIR point 71r.
The waveguide 71n is effectively a controllable leaky waveguide. For example edge-launched ray 71s can leak into the μFCC 71t where it interacts with QDs that PL into output display light 71u propagating through a transparent solid 71v to a human eye 71w. The display light is added to scene light 71x, which has as its source a scene object 71y. In this way an AR image and display is formed.
In particular, the second layer is the transparent waveguide layer 72b, which traps edge-launched light within a high refractive index medium. This waveguide is a controllable leaky waveguide. The control is obtained by a DEP process layer 72c forming a μFCC. This DEP layer comprises a colloid that has a transparent liquid and solid nanoparticles. The nanoparticles may be simple dielectrics, layered dielectrics, quantum dots (for PL response), metal-coated dielectrics (for plasmon resonances), photonic crystals and combinations of these structures. The nanoparticles are often chemically treated to ensure that the colloid is stable and the nanoparticles do not precipitate under gravity or accelerations.
The next layer comprises as many sublayers (not shown) as needed to integrate electrodes and Transparent Thin Film Transistors (TTFTs). The TTFT layer 72d has TTFTs like 72e in the form of an array. On this layer there are transparent electrode-select wires such as 72f. There are also transparent signal wires such as 72g. The signal wires provide the harmonic drive signal at frequency ωp or ωn for pDEP and nDEP processes respectively. This provides the needed ponderomotive forces on nanoparticles in the DEP process layer 72c. The harmonic signal may be a sine wave or a digitally derived squarewave or other signal with a strong sinusoidal harmonic. Additionally, there are electrodes, such as 72h, that are energized only when their corresponding select wire is energized. The electrodes provide the electric fields that provide the DEP-based ponderomotive forces on nanoparticles.
The transparent waveguide layer 72b in combination with a fluidic containment layer 72i provides the surfaces between which is formed the DEP process layer and μFCC, which is used to contain the colloidal fluid. Note that
The TTFT layer 72d functions as follows. There are at least three potential harmonic signals types that can be applied to electrodes. These include a pDEP harmonic signal having frequency cc, a nDEP harmonic signal having frequency ωn and a zero amplitude signal, which we will call zDEP. These types of DEP are derived from the Clausius-Mossotti Factors, examples of which are provided in
For the purpose of this description it is assumed that the relaxation time of nanoparticles from a steady-state spatial distribution to a homogeneous unforced distribution is long compared to the time it takes for nanoparticles to proceed from a homogeneous unforced distribution to a steady state spatial distribution when harmonic signals are present. Said another way, the forced response time is shorter than the unforced response time. The unforced response time may be due to a voltage amplitude of zero during zDEP or it may be due to Clausius-Mossotti factor that is zero at specific electrode drive frequency. We will call this the time division multiplexing assumption (TDMA). It is not a requirement for operation, but it simplifies the problem somewhat and allows a straight forward description of signals used to form an impressed graphic.
The TDMA allows electrodes to be temporarily energized to distribute nanoparticles within the colloid and then shutdown as other parts of the display are updated with new display information. Once an electrode is shut down, e.g. by setting c to the appropriate value to avoid QD interactions, the nanoparticles within its pixel domain will start to relax back to a homogeneous distribution, but long before the steady-state homogeneous distribution is reached all the other sections of the display are also provided an information update signal. In due course an updating signal is reapplied to each electrode again. The process continues over and over as the display images evolves in time.
An example of this is shown in
At Time-1 a signal 10000 is sent to row-select bus lines. This is read vertically from the top of the page to the bottom of the page of
At Time-2 a signal 01000 is sent to row-select bus lines and Row-2 electrodes are energized by the harmonic signals with frequencies {ωn, ωn, ωn, ωp, ωn}, while Row-1, Row-3, Row-4 and Row-5 are not receiving any harmonic signals and their associated nanoparticles are slowly relaxing back to a homogeneous distribution state.
At Time-3 a signal 00100 is sent to row-select bus lines and Row-3 electrodes are energized by the harmonic signals with frequencies {ωn, ωn, ωp, ωn, ωn}, while Row-1, Row-2, Row-4 and Row-5 are not receiving any harmonic signals and their associated nanoparticles are slowly relaxing back to a homogeneous distribution state.
At Time-4 a signal 00010 is sent to row-select bus lines and Row-4 electrodes are energized by the harmonic signals with frequencies {ωn, ωp, ωn, ωn, ωn}, while Row-1, Row-2, Row-3 and Row-5 are not receiving any harmonic signals and their associated nanoparticles are slowly relaxing back to a homogeneous distribution state.
At Time-5 a signal 00001 is sent to row-select bus lines and Row-5 electrodes are energized by the harmonic signals with frequencies {ωp, ωp, ωp, ωp, ωp}, while Row-1, Row-2, Row-3 and Row-4 are not receiving any harmonic signals and their associated nanoparticles are slowly relaxing back to a homogeneous distribution state.
The cycle then restarts at Time-1 and new data is provided to change the character from Z to a different character or graphic as needed. So as can be seen, the information is encoded in the frequency of a harmonic signal sent to the electrodes. Examples of harmonic signals that can may be used include sine waves or digital square wave signals with a zero constant voltage offset.
It is interesting to compare the data holding operation in a pixel that occurs between screen data refresh updates for a LCD and a QDD. In particular, the state of a LCD pixel is held by a voltage on a capacitor. However, in the case of a QDD the state of the pixel is held by the viscosity of the liquid used in the DEP process. It takes time for the nanoparticles to relax back to a homogenous state. Some colloids using silicon fluids have relatively stable viscosities over a large temperature range, thereby allowing stable relaxation times over temperature.
If the QDs in a colloid only have one diameter size then the display will be monochrome. For example a green letter Z in
Thus, designed Clausius-Mossotti characteristics over frequency ω are possible. For example, the general Clausius-Mossotti factor for a spherical nanoparticle with N layers can be built up from first principles and it will not in general be the expression Eq. 121.
The close proximity of the red, green and blue sub-pixels allows colors to be added together to produce a range of colors to the viewer under additive color mixing rules. As a general rule, higher voltages and electric fields are needed for smaller particles and larger gradients in the refractive index. The magnitude of each color is controlled by the number of QDs that are interacting with the evanescent field of the waveguide. This is accomplished by a combination of judicious choice of the harmonic excitation (for pDEP, nDEP and zDEP type responses) and voltage amplitude. In
It should be clear there are many different geometries and signaling schemes that may be used with the electrodes, some of which exploit in-phase ponderomotive forces, some of which exploit quadrature ponderomotive forces and some of which may be hybrid in-phase and quadrature ponderomotive forces. It should also be clear that specific optical bands such as the ultraviolet (UV) light considered in this disclosure for exciting QDs and the broad visible light used in PDs, may be considered only as examples and other optical bands may be used. It should also be clear that specific optical surfaces like the dark black background 74e may be replaced with mirror or other surfaces for some special applications. Additionally, while not shown here it should be obvious that touch displays may be included into the display design.
E-Paper by DEP-Diffusion of Large & Small Particles
Next, we discuss a way to use the basic designs and configurations disclosed as DEP-based displays as a form of electronic paper that only needs ambient light. First, note that while small dielectric nanoparticles in a liquid can interact with light as a modifier to the average refractive index, as provided for in Eqs. 235 and 277, larger particles tend to scatter light to make an effective white surface. If PDs (or other forms of color forming scatters colored particles) are mixed with larger dielectric spheres (white forming scatters) in a colloid then under a DEP process we can separate, by pDEP and nDEP, the white-background-forming dielectric spheres from the color-forming PDs. The user can see in each pixel either the white light scatters or the color scatters. This may be done with separate Cyan, Magenta, Yellow PDs in a subtractive color scheme to allow full color displays that are perfectly readable in bright sunlight and are more akin to paper than light emitting displays. This can be especially effective if the pDEP and nDEP occur at the same electrode drive frequency ω. If this is the case then at harmonic electrode drive frequency ω1 large dielectric spheres have a pDEP response while small PDs have a nDEP response so that that the same electromagnetic fields cause the dielectric spheres to move away from the electrodes and the PDs to move towards the electrodes. Conversely, at ω2 the large dielectric spheres have a nDEP response while small PDs have a pDEP response so that that the same electromagnetic fields cause the dielectric spheres to move towards the electrodes and the PDs to move away from the electrodes.
For the avoidance of doubt regarding scattering consider an analogy: white paint is often made by forming a colloid of microspheres of titanium dioxide, which scatters ambient light in such a way as to make it appear that the paint is white, even though the bulk titanium dioxide is essentially transparent. The same phenomena is also at work for table salt, which appears white in a pile of salt even though the individual grains of salt are substantially transparent when viewed under a microscope. Thus, numerous objects with transparent bulk properties can scatter ambient white light to produce a white scattering field, i.e. the white color needed to electronically form an electronic paper substitute.
These ideas are captured in
Similarly, in
Overview of Augmented & Virtual Reality Displays
In
A distillation and idealization of the geometry is shown in
The reader should note that the curvature of the wavefronts of the output rays controls the perception of the location of the data that is provided to the observer, i.e. it controls the distance from the observer where the data “hangs” or “hovers” in space in front of the observer. For example, the third ray segment 76j and the perturbation rays 76k and 76m are associated with flat or curved wavefronts to control depth precetion. If the wavefronts are flat then the data appears to be located between the first optical surface 76b and the second optical surface 76h. The two wavefront curvature control methods may also be combined to extend the range of curvature or to allow greater flexibility in how the control signals are provided to the display.
There are two options to achieve control over the curvature of the wavefronts. In the first method the data light source, which provides the first ray segment 76a is configured to provide the desired curvature of the wavefronts. In the second method, the effective mirror 76g is curved by DEP-based ponderomotive forces. One way to implement this second method is with wavefront curvature control DEP-electrodes on the first optical surface 76b and/or the second optical surface 76h.
Let the refractive index in the first internal environment 76d be called nA and the refractive index in the second internal environment 76f be called nB then if we ignore optical phase effects through each nanoparticle we can write
nA=ηL+(nS−nL)νA (748)
nB=nL+(nS−nL)νB (749)
where nL is the refractive index of the liquid forming the colloid and nS is the refractive index of the solid nanoparticles forming the colloid; also, νA and νB are the volume fractions of the nanoparticles in the first and second internal environments respectively. In general, at the effective mirror 76g
nA sin θA=nB sin θB (750)
and at the critical angle θB=π/2 with νB=0. Moreover, θA=θC when νA=νC (i.e. subscripts of C indicate critical angle) so that
where
Equation 752 can be modified for νA in general, by including all concentrations that are greater than νC, as TIR is still maintained, consequently
Therefore, in general we can have different critical angles θC such that
and we observe from
Therefore, on calculating the derivative we obtain
and the field-of-view angle θFOV can be calculated from Snell's law nA sin δϕ=(1) sin θo, where θ0 is the output angle after the ray leaves the second optical surface 76h. Therefore, approximately θo=nAδϕ and θFOV=2θ0 and after some algebra we arrive at
Clearly, a greater difference in the liquid and solid refractive index constituents of the colloid will yield a greater field-of-view as will a greater increase in the concentration δνA beyond that which is needed for TIR of the third ray segment 76j, which by assumption exits the device normal to the second optical surface 76h.
For example, if ethanol is used as the liquid with nL=1.37 and if titanium dioxide with nS≈2.6 are used then from Eq. 754 we find that νA≥0.12 when ϕ=35° and θM=45°, θFOV≈26° and δν=0.08. This requires a total volume fraction of the nanoparticles in the first internal environment of νC+δν=0.20.
It is also important to note that TIR is not a requirement at the effective mirror 76g boundary. Some light can leak through the effective mirror 76g boundary as it is substantially contained in the internal to the display. This allows for further increases in the FOV and relaxes some of the constraints on the refractive index values and volume fractions needed in colloid.
In
To form a pixel of light at the observer's eye 77k the drive frequency of the electrodes is changed to ωp on a line of electrodes, such as 77m, which are arrayed into the page of the image of
In this way a pulse of light in the waveguide, depicted here as light-guide rays 77f and 77g are redirected into pixel light 77p, which may be added to the light from the image source 77q, which is produced from a viewed physical object 77r in the field of view of the observer. The pulses of light in the light guide may come from any one of a number of sources, such as but not limited to: lasers or Light Emitting Diodes (LED).
In
The reduction of light passing through the transparent electrodes is helpful in some applications because the transparent electrodes are not perfectly transparent and reducing the interaction with the electrodes avoids optical losses. The avoidance the outer (environment facing) surfaces of the transparent slab 78b and a transparent corrugation slab 78c is helpful in some applications because dirt, water and other surface contaminants are avoided and the corresponding optical losses and distortions are avoided.
Therefore, edge-launched light 78a is directed into a pass through beam 78e or a pixel beam 78f. The pixel beam can then enter an observer's eye 78g along with external light 78h from a scene object 78i to form an augmented reality display.
The embodiments of
The embodiments of
The embodiments of
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The embodiments of
The embodiments of
Thus, the device of
In
The collimated light beam can now be selectively and electronically scattered as a function of position in the array by changing the harmonic drive frequency of one of the into-the-page arrays from ωp for pDEP to ωn for nDEP. For example, the into-the-page array associated with electrode 81f is shown operating at ωn so that the nanoparticles that are in the immediate vicinity experience a ponderomotive force that forces the particles away from the electrodes. This results in light scattering nanoparticles 81g that enter into the input light beam 81d to induce scattering and PL that can be seen by an observer's eye 81h. For example if QDs are used for the nanoparticles then a PL effect will be induced and a portion of the resulting light will be scattered out of the device. Because a symmetric QD is radiating into 47r steradians from a higher refractive index to a lower refractive index (air) it is possible to have a field of view of 2π radians in the plane of the page on one side of the display and 2π steradians of solid angle. The eye's entrance pupil further reduces the output solid angle 81i as shown.
In the case when the device is used as an AR display an external environment object 81j provides scene light 81k that is added to the internally generated light that provides the data overlay on the AR display 81m. Note that the linear array into-the-page is typically designed to be set to pDEP or nDEP operation along the entire linear array. This means that for the display to draw a line of image pixels along the into-the-page array there must be a source of pixilated light along the direction of the into-the-page array. This can be a linear array of lasers or other light sources that edge feed the AR device. In this way one or more pixilated light sources is energized to interact with the nanoparticles that are pushed into the light flux in the μFCC.
The AR display just described is monochrome if there is only one diameter of QDs used for the nanoparticles in the colloid. If non-spherical nanoparticles are used, e.g. with a jagged morphology, then it is possible to use passive particles, i.e. not QDs, and have the light scatter into the observer's eye. Thus the color range is then limited by the light sources and a full color display is possible. For the avoidance of doubt, note that spherical nanoparticles will only scatter light very weakly at about π/2 radians from the propagation direction of incident light. Therefore, a rough or even jagged nanoparticle morphology is better for light scattering, but typically needs to have its effective Clausius-Mossotti factor empirically determined due to the stochastic nature of the particle morphology.
The AR display just described can be converted to a regular display for computers or virtual reality applications with some minor modifications. In particular, an optional light absorbing backplane 81n may be used. Additionally the nanoparticles may be metallic to effect even better scattering properties.
The basic idea behind
Note that the lenses are near a table edge 86a, which helps to breakup the image and show when the optical elements are removed. The front lens element is 86b. The back lens element is 86c. The lens holder is 86d. The lens holder attaches to the front lens and the back lens and provides a plurality of electrical and potentially optical connections, which we call signal connections in general. An example of a signal connection is 86e.
Between the front lens element and the back lens element are the lens corrugations 86f. The lens corrugations may be affixed to the front lens element or the back lens element 86g. In the present example the lens corrugations are attached to the front lens element and may be integrated into the lens elements. Electrodes are located on the lens corrugations as are shown in
The lens-holder 86h has signal connections like 86e that connect to the electrodes on the lens corrugations 86f and controls the light source edge injectors 86i. The light source edge injectors may be lasers, LEDs, or it may be a light pipe with a means to distribute light from a common light source. In this last case the means to distribute light from a common light source may be a DEP-based light switch to redirect light. The light from the light source edge injectors is directed into the control volume gap 86j formed between the front and back lens elements, which is where the colloidal mixture of an optical liquid and nanoparticles are located.
Note that an augmented reality lens that is curved, as is the case of some single piece sunglasses, can provide upwards of 120 degrees field of view with data superimposed on the external scene.
Finally,
Augmented Reality Glasses Via Colloidal Shock Waves
In recent years a number of well funded and highly motivated companies have been developing augmented reality (AR), virtual reality (VR), and mixed reality (MR) headsets, which are here collectively called XR headsets. Billions of US dollars have been invested to create high-quality headsets as competing efforts push each other forward. Nonetheless, at the time of the writing of this document it is generally accepted that all of the efforts have fallen short in one or more ways, such as but limited to: cost, size, mass, optical performance, and curved stylish looks. The ultimate objective being a head set that looks exactly like a sunglasses, prescription glasses, or a stylish headset for a specific application like gaming or aerospace applications. This shortfall in performance has caused a search for ever more clever ways to make the AR and VR headsets cheap and high-performance to allow people to have general access to the new and emerging AR and VR markets for both enterprise and commercial applications.
However, one result of this prior and extensive effort is that the typical set of techniques that optical engineers use in optical design has now been tried in multiple ways and the desired outcome has not yet been achieved. Thus, new optical physics and techniques may be needed to overcome current limits and development a low-cost, high-performance, and attractive headset. One such new direction is provided in this disclosure using, in part, DEP-based colloidal shock wave optics.
One aspect of the low-cost nature of the XR headset discussed herein is based on the ability to use Commercial Off The Shelf (COTS) components for the lens, or at least a lens that is close to a COTS type lens. The COTS lenses may be, if desired, obtained from established sunglasses or from prescription lens providers or manufactured directly by established low-cost means in order to include minor modifications to optimize the design for XR applications.
A second aspect of the low-cost nature of the XR headset discussed herein is that, in principle, all the optical functions for data overlay on an external scene occur in or around a thin control volume, which is as small as about 10 μm over the entire lens area. Hence, a nanoparticle colloid is located between two stylishly curved lenses. The optical functions include: a laser source, laser beam scanning, focusing for wavefront curvature control, optical switching, and scattering into a large field-of-view for the user. This means that the optics of XR glasses are really just about the lenses instead of the entire headset. This is an improvement over today's designs where it is often the case that the designers hang various beam steering and light sources external to the lens. For example, DEP-based XR lenses would overcome the need to place fragile micro-electro-mechanical mirrors in or around the ear-piece of the XR glasses, which is found in some current designs.
To create an XR head set functionality several sequential steps are used to process light to provide graphical information to the user as an overlay on the external scene for AR. This same technique can be used in VR applications too, by simply blocking all or selective parts an external scene. In particular, the sequential light processing steps include:
1. Inject colored RGB light into a thin and stylishly-curved lens light-guide
2. DEP beam steering of injected light towards desired pixel location
3. DEP focusing of light for depth perception (may be simultaneous to steering)
4. DEP enabled beam propagation by TIR in solid light-guide lens
5. DEP optical-switch redirects light out of light-guide at the pixel location
6. DEP optical-switch enables light scattering at the pixel location
7. A meso-scale stochastic surface scatters broadband light into human eye
8. External scene-light may simultaneously pass through lens into an eye
9. An optional liquid crystal VR layer switches an external scene on and off
As can be see there are at least five DEP-based processing steps that may be used for injecting XR information into a users eye: beam steering, focusing, TIR light guiding, and two optical switching operations. These functions are all implemented in or around a thin layer of a nanoparticle colloid that is located between a first lens and a second lens. Each of these functions is based on manipulating the local refractive index of the colloid by electronic means. Therefore, what is about to be described in the disclosure herein below is a system of dynamic GRIN control for creating various optical devices in an optical system.
The DEP-based optical system is a radical departure from the current XR designs and comprises almost an entire self-contained system for doing optics. Moreover, the optics can even be implemented as dedicated hardware or in terms of a reconfigurable system that is capable of being modified by software updates. Such functionality may be useful in prototyping or in upgradeable hardware.
There are two areas where multiple processes are implemented into one device. The first hybrid device is the combined DEP-based beam steering and focusing device. The second hybrid device is the combined DEP-based optical switch, which is actually two switches comprising a switch that turns ON or OFF TIR-based light guiding on one surface of the optical light guide, while simultaneously turning OFF or ON the broadband stochastic scattering at another surface. Notice that as one side of the switch turns off the other side turns on and vice versa.
Some of the anticipated competitive advantages of a DEP-based XR lens design include:
1. Uses new physics to overcome current XR headset limitations
2. Combines multiple optical operations into one device to reduce size
3. Stylish looking curved lenses are now easy to implement
4. Compatible with low-cost commercial off-the-shelf (COTS) lenses
5. Replaces sunglass, prescription, and face-covering helmet lenses
6. Provides vivid RGB colors for XR headset displays
7. Provides wavefront curvature control for floating objects in front of user
8. Can simultaneously provide MR over different parts of a lens (peek mode)
9. Very large field of view (120 degree anticipated) and large eye-box
10. No ghost images and image shadows
11. Extremely low-cost easy to manufacture design
12. Can include an integrated eye-sensor to determine eye orientation
13. The entire optical package is in the lens, i.e. no external optics to lenses
14. Either dedicated optics or software updatable optics are possible
15. External observers cannot see graphics being displayed on the lenses
The reader is encouraged to review the underlying physics presented in the following sections: “Redirecting Light Using Colloidal Shock Waves” and “Hyper Accurate Precise Beam Steering.”
The description of an AR reality headset is now provided. In particular,
Note that the cross-sectional image referenced by the direction arrows 90a and 90b is similar to that provided in
A slightly more in-depth view of what is happening is provided by three magnified inset details. The first magnified detail 91h focuses on the roll that the colloidal layer provides. The nanoparticles in the colloid are moved into different regions by a fast acting shockwave, as has been described in the section “Redirecting Light Using Colloidal Shock Waves.”
The light guide core 91i allows light to propagate by TIR. An example guided ray 91j is shown reflecting by TIR off of the interface with the solid cladding layer 91k and the interface with low refractive (RI) cladding regions 91m and 91n of a colloidal cladding layer.
The colloidal cladding layer also comprises a medium refractive index region such as 91o that has its refractive index designed for a specific amount of light leakage from the refractive index of the light guide core 91i. The refractive index of the light guide core 91i can, if desired, be matched to the refractive index of the medium refractive index region such as 91o so that 100% of the intercepted light passes onto a subsequent scattering surface to form a pixel. A magnification of stochastic pixel region 91p is shown in the second magnified detail 91q and its geometry is discussed in a subsequent paragraph.
The pixel region formed by the medium (i.e. intermediate) refractive index region 91o allows light rays like 91v to leak out of the light guide and scatter off of a stochastic pixel region 91p into ray bundle 91r that is directed to the eye of an observer by propagating through the second lens 91s and air 91t. The high RI colloidal region 91u matches the refractive index of the solid medium used to form the stochastic scattering surface so that no light is scattered outside of an active pixel. Additionally, scene light from a scene region 91w also can propagate through the lens stack into the air 91t. Prescription lenses may be implemented by the specific shape of the lens back surface 91x.
A thin transparent electrode layer 91y comprises an array of electrodes, such as transparent electrode 91z. The transparent electrode layer 91y also has one surface with a patterned scattering surface. The thin transparent electrode layer 91y can be manufactured on a roll of thin plastic that has a nano-imprint pattern for the scattering surface.
The second magnified detail 91q shows the boundary between the colloid 91aa and a portion of the thin transparent electrode layer 91ab. A designer surface providing a light transformation with a specific angular dependence based on a randomly rough surface 91ac having slopes that accrue to provide a specific probability distribution function and angular extent of the scattered light. An example of a detailed analysis of such structures is provided in the text book “Designer Surfaces,” by Alexei A. Maradudin, et. al., Elsevier Publishers, 2008, ISBN 978-0-444-53048-6, which by reference is included in its entirety herein. One of the important features of a designer surface is that the feature sizes are meso-scale, which means the features are much bigger than the largest wavelength of light to be processed by the XR system, but small compared to physical structures. Therefore, many nanoparticles can easily fit between the peaks in the rough surface so that it becomes possible to have a colloidal refractive index that is matched to the refractive index of the material used to form the randomly rough surface. In this way the randomly rough surface can be turned on or off by changing the concentration u(x, y, z, t) of nanoparticles in the vicinity of the randomly rough surface by means of a DEP process.
The third magnified detail 91ad shows an alternative to the randomly rough surface of the second magnified detail 91q. In particular, a periodic surface 91ae provides a means for the diffraction of light in the light guide. As was the case for the randomly rough surface the nanoparticle concentration u(z, y, z, t) in the colloid 91f provides a means to turn the diffractive scattering on or off. However, the scale of the periodic structures in 91ad is much smaller than the stochastic structures in 91q and thus the ability to turn the surface ON or OFF may be less robust as there is less volume for the light to interact with a large number of nanoparticles.
In particular,
The next layer is the solid-phase external cladding layer 93b. This material has a refractive index of about 1.49. This is typical of polymethyl methacrylate (PMMA), i.e. acrylic. This layer provides a separation from the external environment and allows TIR to occur at its interface with the subsequent light guide layer 93c irrespective of the external environment, including dirt on the external surface of the external cladding layer 93b of the first lens.
The next layer is the solid-phase core light guide layer 93c having a RI of about 1.54, as is typical of cyclic olefin copolymer (COC). The light guide layer may be gently curved as is typical of stylish glasses and highly functional face masks. The light guide layer guides the light from the RGB laser to the general vicinity of a pixel.
The light guiding process is substantially in a non-colloidal medium. The only place that the guided light has any opportunity to interact with the colloid is at the boundary of the light guide layer with the colloidal cladding layer. However, the colloidal cladding layer is a liquid with a spatial distribution of high RI nanoparticles. Under ideal conditions for the light guiding process the colloidal cladding layer is configured so that there are no nanoparticles, i.e. just the colloidal liquid, at the boundary between the solid core and liquid cladding so that there is no opportunity for undesired low-level light scattering form light that is inside the light guide. This is shown as the segmented low RI cladding regions 93d and 93e, which occur on either side of a pixel region.
To get light to the eye of the user it is necessary to spoil the light guiding properties of the light guide over the desired pixel region by changing an electrical parameter such as one or more of the electrode-to-electrode voltage amplitude, phase, and excitation frequency so that the distribution of nanoparticles changes quickly and causes the average refractive index to ideally become the same as the light guide layer 93c. This is a process of RI matching so that there is no refraction or reflection at the solid-to-colloid interface. This is shown as the pixel RI matching (RIM) layer 93f, where the nanoparticle concentration νS=10% is a constant through the entire thickness of the layer.
Notice that a constant of νS=10% throughout the pixel RIM layer 93f implies that there must be a volume fraction νS=20% on adjacent sides, where the idealized distribution jumps from νS=0% to νS=20% at the halfway point across the control volume's gap width so that the average across the entire gap is νS=10% as expected by conservation of nanoparticles. The gap width is on the order of 10 μm.
The average RI of the colloid in a region Q, assuming that nanoparticle diameters are small enough to ignore, is given by
n(Ω)=nL+(nS−nL)νS. (759)
Thus, assuming a liquid phase polydimethyl siloxane having additives to increase conductivity and the Maxwell-Wigner frequency to about 1 MHz so that its RI is nL=1.49 and lanthanum high RI glass spheres of 50 nm diameter and with nS≈2.0. For this example the need for special coatings on the nanospheres to ensure colloidal properties will be ignored. Therefore, the RI in the layered regions are
n(93d)=n(93e)=1.49+(2.0−1.49)(0.0)=1.49 (760)
n(93c)=n(93f)=1.49+(2.0−1.49)(0.1)=1.54 (761)
n(93g)=n(93h)=n(93i)=n(93j)=1.49+(2.0−1.49)(0.2)=1.59 (762)
where the light guide layer 93c is index matched to the pixel RIM layer 93f; and the beam shaper RIM layers 93g and 93h are RI matched to the transparent electronics layer 93i, and the second lens 93j. The transparent electronics layer 93i has transparent electrodes, such as transparent electrode 93m, which may be fabricated by a thin layer of transparent indium tin oxide, conductive carbon nanotubes, graphene, borophene, and other transparent and conductive materials.
Notice that the pixel RIM layer 93f is RI matched to the light guide layer 93c, but is not RI matched to the transparent electronics layer 93i. This means that the stochastically designed random surface 93k in the pixel region is actively scattering light that is distributed from the light guide layer 93c. However, the stochastically designed random surface 93k is not scattering light outside of the pixel region because in these regions the RI of the colloid and the electronics layer, which includes the random surface, has a matched RI.
Ultimately, the light that is scattered off of the random stochastic surface passes through the second lens 93j and into the air region of the external eye environment 93n. The directional scattering properties of the random stochastic surface are chosen to deterministically send light into the user's eyes independent of the user's look direction.
The XR glasses of
The next objective of this section is to show how to implement a combined fast laser scanning and focusing device within the colloidal region between the external and internal lenses.
More specifically,
Each of the colloidal control volumes has a set of straight parallel electrodes parallel to the z-direction, such as electrode 94e. This electrode array has a spatially varying voltage from electrode-to-electrode to induce an exponential variation in the volume fraction of nanoparticles νS in the x-direction. Moreover, the variation in this volume fraction is fast because we have set up a nanoparticle shockwave in the y-direction where the colloid only needs to have DEP-based advection over a limited spatial range in the y-direction. Said another way, the speed of change of the refractive index gradient in the x-direction is fast because nanoparticles are being forced into or out of the path of the laser light in the y-direction. The y-direction is the thin gap holding the nanoparticle colloid.
The variation in the volume fraction is chosen so that the refractive index varies exponentially over a limited range of x-values. To see how this can be achieved we imagine that a DEP-based shock wave has pulled all of the nanoparticles towards one wall of the control volume in the y-direction e.g. see the steady state concentration 10a curve. Then as we move towards increasing values of x the electrode voltages are adjusted to allow the refractive index to vary as provided by Eq. 539 over a limited range of x. Said another way, the steady state concentration 10a curve is slightly adjusted from x to x+dx by variations in the electrode voltage so that each ray sees a corresponding exponentially varying refractive index. Then the solution to Eq. 536 are given by Eqs. 549-560 and the ray trajectory is approximately circular as provided in Eq. 597. Thus the curved laser radome, an example of which is 94f, takes the form of a section of an approximate circle as provided by Eq. 598. The curved laser radome is formed by one of the walls of the colloidal control volume, which is located in the transparent electrode layer 94g, which is further located on a portion of a lens section 94h forming the XR head set.
Thus, curved ray trajectories form a first and second scanned laser 94i and 94j, which independently redirect RGB laser light with different wavelength combinations into the large lenses in front of the eyes of the user. The two devices of
On the opposite side of the transparent electrode layer 94g layer is a second set of electrodes. This second set of electrodes is shown in
The focusing electrodes follow the light trajectory, therefore by varying the voltage from electrode-to-electrode to induce an additional δnFocus, as shown by the refractive index variation 94m about the trajectory, angularly over 0. It should be clear that in
In summary, an XR headset can be constructed by controlling a nanoparticle colloid between two solid lenses. In one portion of the XR lens the colloid provides a controllable cladding in a light guide. In another portion of the XR lens the colloid is used to create an exponential GRIN medium to both steer light towards the desired pixel and change the curvature of the wavefront to provide the illusion of depth to the user. The colloid response is a fast acting shock wave in the direction of the gap width between the lenses and this is adjusted to accommodate fast frame rates for RGB data overlays on the field of view of the user.
High-Power Lasers by Dielectrophoresis
Introduction to Prior-Art High-Power Fiber Lasers
In this section a new class of high-power fiber laser is provides enhanced capabilities based on dynamic control of its internal optical structure. However, before providing these methods let's review how a typical prior-art fiber laser works in this subsection. The interested reader may also get more details by seeing “High Power Fiber Lasers: A Review,” by Michalis N. Zervas and Christophe A. Codemard in IEEE Journal of Selected Topics in Quantum Electronics, Vol. 20, No. 5, September/October 2014.
One of the main problems with using a fiber optic to generate high-power laser light is in getting the optical pump power into the fiber's gain medium. A single-mode optical laser is very thin and also has a small numerical aperture. In contradistinction, a thick fiber optic is multi-modal and has large numerical aperture, but poor modal structure for the electromagnetic fields. To overcome this problem E. Snitzer proposed a hybrid solution in 1988 that uses a large numerical aperture for capturing pump light and a small numerical aperture for light amplification with a well defined modal structure.
In particular, while conventional low-power fiber optics have just a single core and a single cladding, a high-power fiber laser using Snitzer's approach typically has a core, an inner cladding and an outer cladding. Initially the core plus the inner cladding acts as the effective core and the outer cladding acts as the effective cladding, the pump energy is injected into one end of the fiber laser and the larger numerical aperture allows a large range of angular light directions to be injected into the fiber via solid-state lasers. The injected light then reflects by TIR within the effective core (core plus inner cladding). Thus, most of the light that is not TIR within the core is captured by TIR between the inner cladding and the outer cladding where there is no gain medium, moreover the overall modal structure of the electromagnetic fields is poor i.e. not a single well defined mode.
Nonetheless, by using Snitzer's approach light that is injected into one end of a high-power fiber laser can propagate to the other end, but with the energy in distinct regions. Clearly this is a good start as the light is in the fiber system, but not a sufficient condition for providing a high-power laser. In particular, the light energy that is propagating outside the core, i.e. in the inner cladding, now moves into the core. Thus, a means should exist for pump light to be removed from the inner cladding to the core with little ohmic loss. Moreover, the light that is trapped in the inner cladding has a well defined étendue which is a conserved physical quantity in an optical system. This etendue must be overcome if the light is to have enough room to “fit” into the fiber core. With this in mind we can turn to atomic processes for light absorption and radiation.
A first process, spontaneous emission typically occurs when an atom or molecule transitions from an excited energy state to an unexcited and lower energy state. If the excitation is due to absorption of a photon the spontaneous emission is called fluorescence when the time constant of the process is small and phosphorescence when the time constant is long.
A second process is stimulated emission, which occurs when an incoming photon of a specific energy interacts with an excited atom or molecule so that it emits light as it transitions to a lower energy state. Some atoms and molecules are better at this interaction than others. The input photon transfers it energy to the output photon(s) and can maintain phase, frequency, polarization, and direction of propagation that are identical to the input photon.
A third set of miscellaneous processes are also possible, for example a stimulating photon can in certain systems liberate two more equal but lower-energy photons into the same direction as the input photon. Also, two or more photons can combine within certain atoms and molecules and liberate a higher energy photon. Thus, both up conversion and down conversion (in photon energy) are possible. This means that the wavelength of the pump laser may be different than the wavelength of the laser's output. This allows the pump light to pass unhindered through a Bragg grating that is etched or doped in the ends of the fibers, after which atomic and molecular up or down conversion of pump light changes the wavelength of the pump light so that it is reflected back and forth in the fiber due to Bragg gratings on each end. One of the Bragg gratings is typically configured to transmit a small amount of the energy trapped in the fiber core to allow for a continuous beam output.
A lasing process is started by light interacting with an atom or molecule to produce a random spontaneous emission and potentially up or down conversion; and then stimulated emission becomes dominant in the steady-state. So the fiber's core can be doped with gain producing atoms and molecules that are predisposed to provide both spontaneous and stimulated emission. The gain producing atoms come from rare-earth elements such as dysprosium, erbium, holmium, neodymium, praseodymium, thulium and ytterbium. Currently, ytterbium (Yb) is the most common dopant for high-power fiber laser applications.
To transfer photons from the inner and outer cladding to the core where it can be amplified requires that we overcome two restrictions to light entering and staying in the core: (1) etendue conservation, which forces some of the light to move through both the core an inner cladding instead of just staying inside the core and (2) the circular symmetry of a fiber's cross-section, which allows some input rays to spiral along a three dimensional helix that never intersects with the core. These helix rays may be piecewise continuous or smooth curves depending on the refractive index spatial distribution.
The inclusion of a gain medium like ytterbium into the core breaks the etendue conservation as the ytterbium absorbs and reradiates some light that passes into the core into random directions by spontaneous emission—i.e. by fluorescence and phosphorescence. Some of the spontaneous emission goes into the needed direction along the fibers length and is captured by a TIR process and the amplitude of the light wave is then amplified by stimulated emission as an excited population inversion becomes pronounced. The light is then trapped in the single-mode core and is increasing in strength as it travels along the fiber. This light can also bounce back and forth along the fiber laser by including mirrors or Bragg gratings at the fiber ends.
So as light passing through both the core and inner cladding interacts with dopants like ytterbium in the core the some atoms get excited by fluorescence and phosphorescence from rays that are not properly aligned with the fiber core and radiate in random directions. Then, some of the fluorescence and phosphorescence light propagates into the desired direction along the core's optical axis and this provides a means to take pump light out of the inner core, where there is no gain medium, and transfer it to the inner core where there is a gain medium and population inversion.
To further increase the laser's efficiency we must take into account that there are still the previously mentioned helical rays, which never get a chance to pass into the gain producing core. To combat this situation various geometries are possible that can break the circular symmetry of the fiber optics. For example, offsetting the core from the center of the fiber or flattening one of the sides of the inner cladding. Many geometries have been proposed and implemented for symmetry breaking.
Additionally, many single fiber lasers can be combined into fiber bundles. If the fiber bundles are brought close enough together then evanescent light-field coupling allows the phases of the individual fiber lasers to synchronize. However, slight temperature variations between the fibers can cause tiny changes to the local refractive index of a single fiber. This leads to the loss of symmetry in the fiber bundle and increasing difficulty in keeping phase lock among the fiber lasers. Additionally, at the higher power levels a thermal run away effect is possible where local ohmic losses increase with increasing temperature, which further changes the refractive index and further increases the ohmic losses. Eventually the fiber can fail catastrophically. This process is even more problematic in fiber bundles where a balanced and symmetric system is needed for bundles of fiber lasers to phase lock and amplify successfully.
DEP-Based High-Power Fiber Lasers
Next, we consider how to use the optical properties of colloidal nanoparticles and the control of nanoparticle spatial distributions, via DEP-based ponderomotive forces, can be used to input, guide, amplify, and launch laser light, especially high-power laser light, without the use of the classical fiber optic structure previously discussed in the prior section and prior-art. Instead, only a single solid gain-medium doped fiber core (i.e. typically but not necessarily a single-mode fiber) is needed and that the solid inner cladding and outer cladding of the prior-art fiber lasers are not needed in certain embodiments. Furthermore, under certain special conditions even the need for a solid doped fiber core may not be needed. Instead a dynamic medium is introduced that can effectively act as the cladding or core and change its characteristics over time to improve power levels and efficiency.
Consider
This equation is the averaged RI of a colloid and its slope is adjusted electronically. Inside the linearized refractive index region a third RI segment 95c is shown. This third segment represents a solid fiber core. This fiber core may be doped with a gain medium like ytterbium to provide amplification of light and laser action. It is often the case that the RI of the core may have a RI distribution different from a constant to better control the modal structure of the electromagnetic fields within the core region, this non-constant case is also included by reference.
A fourth and fifth RI segment 95d-95e may take on any reasonable value or distribution of refractive index and has been shown as taking on the value of n0 for convenience, but may actually be a strong function of radius and voltages used in the DEP process. The critical issue in
The laser beam extent 95f is represented by dashed lines in
Therefore, the function of the solid prior-art fiber optic cladding to contain the pump beam is replaced by the refractive index distribution of the colloid in the circular control volume. The function of the prior-art fiber optics having non-circular cladding shapes, offset asymmetric cores and other asymmetries is replaced by the perturbation of the second order effects in the refractive index distribution in the colloid that disturbs the circular trajectory of the rays to non-circular so that all rays intersect the core.
Some prior figures can now be reused. In particular,
In summary, pump light is confined by the GRIN refractive index distribution 95a and 95a which are designed to provide a circular light trajectory and laser light is contained by the fiber core 95c. Advantages of this configuration include:
In another embodiment
In addition to the colloidal medium in the control volume there is also a solid doped fiber core 96f. This core is typically made of a highly transparent glass. As the pump beam 96a enters the control volume it comes under the influence of the non-uniform refractive index distribution detailed in
After the pump beam enters the control volume several process are at play. First, the wide pump beam, with initially parallel rays, begins to bend towards the region of higher refractive index so that an oscillation of the rays about ξ=ξ0 occurs. Second, these rays pass through the fiber core part of the time and are converted out of the pump beam into an amplified laser beam by spontaneous emission and stimulated emission respectively. Thus, we can see the schematic shows the oscillating rays of decreasing magnitude in a average circular trajectory as shown by the modified pump beam 96j. By properly setting the electrode voltages the pump beam can be made to bend in a circle electrodes are not shown in this figure to reduce image clutter, but examples are shown in other figures. Moreover, by setting different voltages at different positions along the control volume the confinement of the rays can be modified as needed, including release of the rays out of the control volume. As configured the pump beam decreases in oscillation magnitude as it moves in the propagation direction 96k. Thus, schematically the pump beam turns into the laser beam 96m, which is better described as modal energy in a single mode fiber, instead of ray energy.
There are a number of significant variations to the configuration of
In particular,
A harmonic voltage source 97f between the first and second electrodes provides the nonuniform electric field needed for controlling a helical light trajectory outside the fiber laser core 97a, which is where the majority of the light from a light laser pump beam is located before being converted into the desired output laser light by spontaneous and stimulated emission. This provides a non-uniform electric field for the needed ponderomotive forces of a DEP-process to create the refractive index distribution for guiding pump light and breaking propagation symmetries to improve pump conversion efficiency into laser light.
The shaped channel 97b can be any suitable shape for the purpose of providing the desired electric fields. The shaped channel may be part of at least one of the first electrode 97c and the second electrode 97d, however in this example only the first electrode 97c is shown with a V-shaped groove.
The shaping of the V-groove can easily be shown to provide a highly non-uniform field in the vicinity of the gain-medium-doped fiber core by means of the Schwartz-Christoffel transformation. In particular, for the electroquasistatic excitation of the electrodes we anticipate that the voltage potential V=V(x, y) will obey Laplace's equation
∇2V(x,y)=0. (764)
Let us strategically investigate two-dimensional electric fields and potentials as cross sectional cuts at potentially different orientations into our high-power DEP-based laser system. Let the complex number z=x+iy (note that z is not a coordinate direction here) and let us map this to a complex number w=U+iV by
w=ƒ(z) (765)
which is analytic on the complex w-plane if it admits derivatives that are independent of direction then
which are the Cauchy-Riemann equations. On eliminating U we obtain
and a similar equation exists for U. Additionally, if we assume that z=z(t) and w=w(t), where t is an arbitrary parameter (note t is not the time here) then the derivative of w=ƒ(z) becomes
and this has a complex argument angle given by
where
are the angles of the tangent vectors to the parametric curves z(t) and w(t) to their corresponding real axes. Therefore,
Then at two separate values of t we have that
Moreover, if the parametric trajectories z(t) and w(t) are simple straight lines then the values of θz(t) and θw(t) are easy to determine. If in the z-plane the parametric curve z(t) is initially taken as the real axis so that
and in the w-plane the parametric curve is taken as the negative real axis and a line at an angle of α with the real axis of the w-plane then
where ρ>0 is the radius. Then if t1≤0≤t2 we have that
θz(t1)=π (777)
θ2(t2)=0 (778)
and
θw(t1)=π (779)
θw(t2)=α (780)
so that the change in the argument of the derivative of the mapping function is the steering angle through which the parametric curve in the w-plane changes as it passes through the w-plane origin, therefore
This is exactly the case for the function
where A is a complex number, which is chosen for scale and rotation. The argument of Eq. 782 is easy to determine if we use the phasor form of the complex number z=ρeiϕ so that
Therefore, the change in the argument of Eq. 782 is
This is the same as required by Eq. 781. Therefore, we may generalize Eq. 782 for the derivative of the mapping function to allow for a piecewise parametric curve in the w-plane, then in terms of a product expansion
where B is a complex integration constant.
For the case considered in the simple example of
z=−1, z2=0, z3=1, z4=2 (787)
where in the w-plane the corresponding points are
w1=−1, w2=−i√{square root over (3)}, w3=1, z4∞ (788)
and the corresponding steering angles are
so the mapping function is
where 2F1 is the hypergeometric function
and (a)n is the rising Pochhammer symbol defined by
The approximation in the second line for ƒ(z) is in anticipation of obtaining a closed form analytic solution corresponding to having the gap between the first electrode 97c and the second electrode 97d large enough that potential isovcurves are nearly parallel to the inner surface of the second electrode 97d when measured near the second electrode. The constants A and B are then found from solving the following two equations simultaneously to ensure the correspondence of z and w points. Accordingly,
so we then obtain
where Γ(⋅) is the Gamma function. The solution is plotted in
Thus, input pump light 97i is contained in the close vicinity of the coiled fiber core, which is doped with a gain medium like ytterbium, as the fiber coils around the first electrode 97c the pump light is both guided to travel in a circle with minimal interactions with the metallic electrodes and maximally interact with the fiber laser so as to convert pump light into laser light. The geometry of the electrodes, and the resulting refractive index distribution, are chosen to break symmetries like cylindrical symmetries to maximize pump and fiber interaction. The geometry of the electrodes can also provide control of the polarization (spin angular momentum) and orbital angular momentum of the light passing through the fiber core.
There are a number of different embodiments that are possible including: using fiber bundles, having more electrodes so that there is more control over symmetry breaking, using periodic standoffs to hold up the fibers in the V-groves, different shape grooves, and having multiple layers of wound fiber. These are included herein by reference.
A completely different embodiment is shown in cross section for a colloidal fiber laser in
There are a couple of ways to break the cylindrical symmetry to avoid helical rays propagating into the page of the image. One way is to move the fiber core to a non-symmetrical position and out of the center of the cylindrical symmetry. An alternative is to change the voltage amplitude or excitation frequency as a function of the electrode angle. This can pull the nanoparticles into asymmetric nanoparticle distribution 98e inside the control volume in the containment vessel.
Moreover, if the nanoparticle distribution and refractive index distribution are changed in time this helps to avoid local heating, thermal runaway and to allows better conversion of pump light into the desired laser light output. The colloidal nanoparticles are less prone to laser radiation damage and the colloid can be made into a medium that flows to carry away unwanted heat.
It is worth noting that the nanoparticle colloid may also by photo reactive so that is cures on being illuminated with (typically) UV light. This would allow for a cladding to be manufactured comprising a complex refractive index distribution as a solid cladding. The DEP process only used during manufacturing to set up the desired GRIN distribution. Also, the same idea allows for regions of complex shaped gain medium where the colloid used in manufacturing contains nanoparticles doped with a gain medium like ytterbium atoms.
In summary, a DEP process in a nanoparticle colloid allows the optical properties of a fiber laser to use either dynamic or static GRIN environments for the cladding and core. Core made from conventional gain-medium doped glass may be augmented by colloidal claddings. Cores made from gain-medium doped nanoparticles are also possible. A fiber laser can now have cores that are very close together so that strong coupling allows phase lock across a fiber bundle. This stronger synchronizing signal among the fiber cores can allow a larger array of fiber lasers to be bundled and greater power to be delivered than current fiber lasers. Localized heating can be managed by dynamic reconfiguration of the refractive index along the fiber laser to mix the pump rays and ensure that symmetries are broken and that the pump rays pass through the gain medium in the core to allow stimulated emission to convert poorly directed pump rays into rays that are consistent with the modal structure of the core.
Fiber-Bundle Laser with Beam Combiner & Steering
In this section several components derived in the disclosure are combined to provide a high-power laser. This section therefore discusses a system of DEP-based components that come together for laser applications like high-power metal cutting, welding, sintering for 3D-printers, and additive manufacturing in general.
In particular,
The fiber-bundle may comprise one or more bundled fiber cores that are doped with a gain medium like ytterbium atoms. The fiber cores are close packed and surrounded by a colloid that will provide beam confinement, symmetry breaking containment of pump energy and cooling. The colloid refractive index distribution may be dynamically changed to extend power levels and reduce localized heating and thermal runaway effects.
There are many ways that the fiber bundles can be incorporated into the system. In the embodiment shown in the inset image of
Thus, dielectrophoresis-based light-control is used in high-power laser applications like metal welding, cutting, and hole drilling. Variations of the same technology used in these industrial process may also be used for power beaming to drones to keep them aloft for extended periods (e.g. for communications, or package delivery), provide the energy for unconventional forms of aerospace propulsion and many other applications.
Underlying these applications are physical process that are controlled by DEP-based optics, including: beam steering, beam combining, wavefront control for conventional focusing, wavefront control as part of an adaptive-optics system to overcome atmospheric thermal and non-linear effects, laser gain-medium control, bundled fiber laser arrays, and overall thermal control of the high-power laser.
For example, in
Solar Collectors by Dielectrophoresis
Introduction to High-Temperature Solar Processing
The direct use of the sun's energy has many advantages for certain high-temperature applications. Moreover, photovoltaics has significant disadvantages for energy intensive applications that require very high temperature heating. This is because even at the theoretical limits of solar cell performance, i.e. the 33.7% at 1.34 eV band gap Shockley-Queisser limit, roughly 67% of the solar energy is lost in its conversion to electricity and therefore it really makes no sense to subsequently convert the remaining 33% of sunlight's energy back into heat from electricity—all the while throwing away 67% of the sunlight as heat anyway. A better use of sunlight for heat intensive applications is to use 100% of the sunlight radiant heat directly.
This has strong implications for lowering costs on industrial processes like smelting of steel, water purification by desalination, and utility-scale high temperature materials processing. This is especially true of applications that can also benefit from electricity generation, e.g. for sale to the public, by solar thermal processes when the materials processing is not active. Thus, allowing the down time between materials processing runs to also generate revenues by electricity sale to help offset operational costs.
However, high-temperature materials processing (and electricity generation) by the direct use of the sun's radiant energy requires sun trackers. However, today's prior-art sun tracking technology wastes as much as 90% of the solar energy collection area, is open to the environment, and is based on high-cost low reliability mechanical parts. This is not economical and it is not environmentally friendly.
The underlying problem is more easily seen with the aid of
Current solar tracking systems have the greatest efficiency at harvesting the available solar energy falling on a fixed area of land when the sun is least intense, in the early morning and late in the evenings. This is completely contrary and incompatible to what is needed to maximize performance and allow the solar concentration process to be contained in a compact building to protect the environment instead of spread out over huge fields. Ideally, the efficiency in capturing solar energy across a fixed area should be 100% independent of the time of day and the dangerous temperatures and light flux of concentrated solar energy should be contained and separate and from the external environment (e.g. birds, etc.).
How the sun interacts with current solar array geometries is more easily seen with the aid of
A first array 107a shows sixteen solar receivers that have 0-DOF, are zenith facing and have a footprint bounded by square a1a2a3a4. Note that there is no panel-to-panel shadowing in this particular configuration, but the sun does not usually point directly at the collectors either.
A second array 107b shows sixteen solar receivers that have 0-DOF, are statically tilted up at an angle equal to the latitude angle of the installation and have a footprint bounded by the rectangle b1b2b3b4. Additionally, the spacing and tilting of each solar panel is chosen so that there is no panel-to-panel shadowing at any time of the year. Thus, the sun is more closely directed to the normal of each collector panel, but the interstitial area between panels is increasing compared to the first array 107a.
A third array 107c shows sixteen solar receivers that have 1-DOF that are capable of dynamically tracking the sun east to west, have a rotational axis that is parallel the ground 107g and have a footprint bounded by rectangle c1c2c3c4. The receivers in this configuration are typically solar thermal parabolic troughs—the receivers shown only represent the input area of the parabolic troughs. Again, the spacing and tilting of the solar receiver is usually chosen so that there is no receiver-to-receiver shadowing during the majority of the day. Thus, the interstitial area between panels is increasing compared to the second array 107b.
A fourth array 107d shows sixteen solar receivers that have 1-DOF with a rotational axis that is tilted up at an angle equal to the latitude angle of the installation and have a footprint bounded by rectangle d1d2d3d4. Additionally, the receivers dynamically track the sun from east to west and also have a more direct average insolation in the north-south direction than array 107c. Again, the spacing and tilting of the solar collector is usually chosen so that there is no panel-to-panel shadowing during the majority of the day. Thus, the interstitial area between panels is increasing compared to the third array 107c.
A fifth array 107e shows sixteen solar receivers that have 2-DOF, are capable of dynamically tracking the sun daily from east to west as well as seasonally from north to south and have a footprint bounded by rectangle e1e2e3e4. Again, the spacing and tilting of the solar mirrors is usually chosen so that there is no panel-to-panel shadowing during the majority of the day. Thus, the interstitial area between panels is increasing compared to the fourth array 107d.
A sixth array 107f shows sixteen solar receivers that have 2-DOF, are capable of dynamically tracking the sun daily from east to west as well as seasonally from north to south and have a footprint bounded by rectangle f1f2f3f4. As in the previous cases the receivers in this configuration are typically mirrors (heliostats) that reflect the sunlight 107x into a solar tower 107w for processing. Again, the spacing and tilting of the solar panel is usually chosen so that there is no panel-to-panel shadowing during the majority of the day. Thus, the interstitial area between panels is increasing compared to the fifth array 107e.
A linearized mathematical model for the energy captured by a collector configuration is provided by the present author (Leo D. DiDomenico) in the article “Towards doubling solar harvests using wide-angle, broad-band microfluidic beam steering arrays,” Opt. Express 23, A1398-A1417 (2015).
In summary, the model states that the time averaged total cumulative energy E [kwh] received by a solar power system, such as those in
E=PATηaηrηoηc (799)
ηc=cos ψ
, (800)
where cos ψ
is the average cosine of the angle ψ between the collector panel normal and the direction of the sun when the sun is above the local horizon for the solar array configurations of
Moreover, from orbital mechanics it can be shown that the annual average above-horizon cosine of the angle between the collector normal and the sun is the product of the average East-To-West (ETW) & North-To-South (NTS) area projections whereby
Additionally,
where Jo is the Bessel function of the first kind of order zero, ϵ≈0.409 radians is the Earth's tilt angel (obliquity angle) to the ecliptic in radians, {tilde over (ϵ)}≈0.0167 is the orbital eccentricity of Earth, θLAT is the latitude angle of the solar receiver, and a is the tilt angle of the solar collector towards due south. Therefore,
where the case for 107f is typically determined numerically due to the non-uniform placement of heliostats.
From the above results we can determine that the energy collected for a fixed number of solar collectors is greatest for the fifth array 107e. However, if the we instead require that an industrial high-temperature solar plant operates in a containment building then the constrained parameter is not the number of solar collector heliostats, but rather the area of the collector, which is related to the building size and large capital expenditures. Said in another way, for a fixed containment building area some solar collector array configurations will be able to have more collector panels than for other solar array configurations. Therefore, some configurations will collect more energy than others.
With that insight let's compare the fifth array 107e with the first array 107e. In particular, from Eqs. 799 and 804 and assuming that insolation P, time T, plant area A, receiver efficiency ηr and optical efficiency ηo are the same and θLAT=35° we obtain the amazing result
where the fact that in practice as little as 10% of sun-tracker area is used for 2-DOF sun trackers so that
Therefore, when there is a compelling reason to have a small fixed area, such as the need for a safety containment building, then there is roughly 20× energy advantage to using a area filling solar collector. This is exactly the type of solar collector that can be provided by a DEP process.
Additionally, a microfluidic solar concentrator array has the potential to provide between 10,000-45,000 suns of concentration with temperatures in excess of 3,500° C.—there are few materials that can withstand such a heat flux. Most materials, e.g. steel, will melt or vaporize almost immediately. Such an intense energy flux is useful for smelting steel, copper, glass, and other materials. The heat produced is essentially instant, free, and non-polluting.
There is no need for coal or natural gas to operate a high-temperature solar heat plant. All that is needed is a good insulator to ensure that heat is not lost form its containment vessel. From making better silicon ingots for electronics, to steel, copper, gold, glass, and even chemical reactors there is a real opportunity to directly use focused sunlight for large-scale industrial processes. Even the area of solar desalination has new opportunities with a DEP light steering technology.
Thus, the ability to steer light passing through large physical areas by fluidic techniques is an enabling technology for heat intensive industrial processes that occur in containment buildings. This in combination with recent advances in solar desalination and smelting offer real hope for water resources and clean industrial steel manufacturing.
For example, recent advances in smelting technology allow for carbon-free smelting as noted in “STEP—A Solar Chemical Process to End Anthropogenic Global Warming. II: Experimental Results,” by Stuart Licht et. al., The Journal of Physical Chem. C 2011 115 (23), 11803-11821 DOI: 10.1021/jp111781a.
As another example, recent advances in the desalination of water as noted in “Highly efficient solar vapour generation via hierarchically nanostructured gels,” by Fei Zhao, et. al. in Nature Nanotechnology 13, 489-495 (2018) having DOI: 10.1038/s41565-018-0097-z; and “Architecting highly hydratable polymer networks to tune the water state for solar water purification,” by Xingyi Zhou, et. al. in Science Advances 2019 Jun. 28, Vol. 5 no. 6, eaaw5484, and having DOI: 10.1126/sciadv.aaw5484. In these articles we see that material scientists have found a way to use hydrogels comprising water binding and light absorbing materials to increase the rate of evaporation from the previous theoretical limit of about 1.6 L/h/m2 to a now measured 3.2 L/h/m2. This is the highest solar evaporation rate ever reported and it is about 12 times the amount produced by todays commercially available solar stills. This technology can in principle be combined with the intense light and heat produced by the solar concentrators of this disclosure for commercial scale water desalination.
Thus, there is a significant motivation to provide large area (i.e. at least 1 m2) digitally controlled pieces of glass that can be used in collector arrays to steer and concentrate sunlight over large areas in a cost effective way. Such a glass panel could be arrayed into large solar power and material processing plants. The principles of operation of such a transparent collector and concentrating panel is the subject of the next subsection.
DEP Solar Collectors, Concentrators, and Directors
Consider the prior-art of
This solar collector can be reconfigured to eliminate the need for advective flows along the long length of the microfluidic control channels (μFCCs), the need for vacuum regions in the μFCCs, and the need for an index matching liquid (IML) reservoir. Instead it is possible to use a colloidal system of nanoparticles under the influence of DEP-based ponderomotive forces to control the direction of sunlight for large area solar collectors, concentrators, and redirectors of sunlight. Applications include: concentrating solar power, concentrating solar desalination, concentrating solar smelting of steel, and others.
More specifically, it is possible to provide a DEP-based solar collector having curved (or piecewise flat) μFCCs that act as controllable mirrors and that is based on the GRIN optical switch principles. This has the distinct advantage of allowing electrodes to be easily manufactured on or in proximity to the surface of the transparent window-like sheet instead of in the proximity of the μFCC inside the solar collector panel.
In
The sunlight enters the input surface 108c and exits an output surface 108d. As the output direction is typically taken to have a fixed direction the output surface is optionally provided with a corrugated surface that allows sunlight to exit normal the solid transparent medium 108e to assist in reducing reflection losses.
There are a plurality of μFCC that form an array of control volumes 108f. These control volumes are thin and extend into the direction into the page of
A first inset drawing 108h and a second inset drawing 108i provide descriptions of the functionality of the device when sunlight is redirected or allowed to pass respectively.
In the first inset drawing 108h intermediate light ray 108j is directed to output light ray 108k by a process of TIR at TIR point 108m. This process occurs between the solid transparent medium 108e and a substantially liquid side and low refractive index side 108p of μFCC 108n. The opposite side of the μFCC has a high density of nanoparticles. In the first inset drawing 108h the right hand side of the μFCC has a low density of nanoparticles (and a low refractive index) and the left hand side of the μFCC has a high nanoparticle density (and high refractive index). Thus, if the solid transparent medium 108o has a high refractive index relative to the low refractive index side 108p of the colloid in the μFCC then losses TIR can occur.
Alternatively, if the nanoparticle concentration is uniform throughout the μFCC 108q, as is the case in the second inset drawing 108i, then the intermediate light ray 108r passes directly through the μFCC 108q into output light ray 108s. This occurs because the refractive index of the colloid in the μFCC is uniform and substantially equal to the refractive index of the solid transparent medium 108t. Thus the μFCC becomes invisible instead of effectively mirrored by TIR.
The change in the refractive index inside each μFCC is controlled by the voltage excitation on the electrode array 108g. Also, the extent of the refractive index separation along the μFCC is also controlled by the excitation on electrode arrays. The electrode array comprises long thin electrodes that are long in length into the page of
In the solar collector of
To control the direction of the ponderomotive forces relative to the electrode array requires that different types of phase excitation are used. Consider
Other types of excitation, i.e. different amplitudes, phase and number of phases used may produce different mixtures of x-directed and y-directed podermotive forces on the nanoparticles. Therefore, we can use a 4-phase excitation in
There are many variations on this approach to light control, for example, making a stack of similar cross sections, but of thinner composition and with only smaller portions of the curved μFCC for segmented light control allows smaller voltage amplitudes to be used in the system.
From a system's perspective a strategy may be employed that is based on tracking the sun from north-to-south (NTS), which requires essentially half the angular tracking capability compared to tracking the sun east-to-west (ETW). For a 1-axis (1-degree-of-freedom) tracker NTS tracking historically has not made any sense for prior-art-systems because the sun moves predominantly in the ETW direction. However, when the curvature of the fluidic channels is directed NTS then by Fermat's principle and the variational calculus it can be shown that light propagating along curved microfluidic mirrors traverse a geodesic helix.
In particular, by Fermat's principle we know that a light ray in going from point A to point B must traverse an optical path length that is stationary with respect to variations of that path. Consequently, the variation 6 of the action integral must be zero
δ∫ABds=0, (808)
where ds is a differential element of optical path length. The trajectory is a path in three dimensional space that is restricted to the manifold provided by a control channel. In cylindrical coordinates the line element ds is
where n is the constant refractive index of the rigid transparent solid forming the control channels, r is the radius of the cylinder, ϕ is the cylinder's polar angle, z is the cylinders z-coordinate which is directed from ETW, σ is the parameterization of the trajectory and L is the Lagrangian. Moreover, as the Lagrangian L is not an explicit function of the parameterization σ we may consider the more convenient L′=L2 as the effective Lagrangian with a constant r (i.e. for the spacial case of cylindrical control channels i=0) so that the trajectory of the light rays must in general satisfy the two Euler-Lagrange equations
the solution of which is easily shown to be a geodesic helix
ϕ=c1z+c2, (814)
where c1 and c2 are constants determined from the incident conditions of the ray onto the control channel manifold. Thus light corkscrews on the μFCC for a small distance as it changes some of its photon momentum from the ETW direction to the NTS direction.
The effects of the helical trajectory are especially pronounced in the early morning or late afternoon, however the direction of rotation in the early morning is opposite to that of late afternoon. This rotation or “twisting of light's trajectory” has the wonderful property of twisting predominantly ETW propagating sunlight into the NTS direction, thereby allowing a measure of control in the collection of sunlight that moves ETW.
In particular,
In particular, a hybrid solar collector-concentrator 109a comprises a beam steering system 109b, an optional reflector 109c and a solar receiver 109d. The solar receiver may, for example, be a photovoltaic receiver, a solar thermal receiver or a combination of electricity generation and water heating so as to provide a means to both cool photovoltaic cells and use any remaining waste heat for practical purposes. In the north-south cross sectional view shown (assuming the system location is in the northern latitudes of earth) the beam steering system 109b intercepts light over an annual angular range 109e, which is about 48° and is due to the earth's tilt angle and fixed rotation direction with respect to the plane of the ecliptic. The sunlight that enters into the beam steering system 109b comprises rays that are substantially parallel, having about 0.275° angular radius (to form a light cone) about the mean direction of energy propagation. The hybrid solar collector-concentrator 109a is configured electronically to accept this light independent of the time of day and redirect it substantially towards the solar receiver 109d.
The purpose of the optional mirror reflector 109c is to allow either a horizontal bottom surface, as might be required on a flat roof, or a larger output angular extent of light from the beam steering system so as to reduce complexity and cost. Consequently the optional reflector 109c, when utilized, may or may not be configured to be horizontal as is shown in the figure. An example of a mirror reflection 109f is shown for a horizontal mirror surface and a large beam steering angular output.
The magnified region 109g is shown for two example cases: the summer solstice and the winter solstice. The winter solstice magnification 109h shows an input light bundle 109i refracting into a transparent medium at a first surface 109j and light refracting out of the transparent medium at a second surface 109k. The output light bundle 109m typically has a wider angular extent than the input light bundle 109i due to the processing of the light by the beam steering system. Additionally, the input and output surfaces typically are provided with anti-reflection coatings, such as multi-layer dielectrics or a graded refractive index surface, which allow refraction without appreciable reflection losses.
In between the first surface 109j and the second surface 109k fluidic control channels have IMF injected or extracted so that a curved TIR based mirror is dynamically reconfigured. The winter solstice magnification 109h shows schematically that the upper portions of a fluidic control channel array are index matched by a uniform nanoparticle distribution. This is depicted, for example by the example deactivated mirror 109n, which has dashed curved contours. The lower portions of the control channels are not index matched so that activated mirrors, such as 109o, are available to redirect the sunlight.
The summer solstice magnification 109p shows an input light bundle 109q from a different input angle, however, now each fluidic control channel is reconfigured by changing the distribution of nanoparticles so that the curved mirrors of the mirror array allow a greater steering angle. An example of a fully activated mirror is shown as activated mirror 109r. Subsequently, sunlight that is refracted from the second surface forms an output light bundle 109s that is substantially the same as output light bundle 109m as projected onto the plane of the page independent of the time of day and the position of the sun.
Next,
Specifically, a concentrating solar collector 110a, comprising a wedge shaped section of a compound parabolic concentrator is typically situated on the ground 110b and positioned with the output aperture 110c pointing substantially due north when the solar power plant is on the northern hemisphere of earth. The concentrating solar collector 110a comprises an active input slab 110d that redirects sunlight, while allowing the light to pass through the device. As already mentioned this is similar to a beam steering system 109b described in
The concentrating solar collector 110a is shown inside a compass rose that is situated on the ground 110b. North (N), South (S), East (E), and West (W) are shown. Additionally, the in the sky above the concentrating solar collector 110a is shown a band of “figure eights,” which are plots of equations of time 110aa separated by fifteen minute intervals. This band shows graphically in some detail the various positions of the sun over the course of a year and over each day of the year. The summer solstice side 110bb and the winter solstice side 110cc depict the angular extent of the sun in the sky, which is about π/4 radians. In contradistinction the horizon-to-horizon angular extent is about π radians. The particular, design shown in
A typical input ray 110h, which is incident from the sun, is shown intercepting the active input slab 110d and being redirected thereby into an internal ray 110i, which may (as shown) reflect off of the reflective bottom 110g into internal ray 110j. This ray is further redirected by the reflective side 110e into an internal ray 110k that intercepts the output aperture 110c. Different rays passing through the system may not need to utilize all three redirections shown here. The only required redirection is that provided by the active input slab 110d which provides a first stage of concentration. This concentration is caused simply by redirecting the input light towards the output at an angle close to the planar surface of the active input slab—see
Optionally, a safety region 110m of the input slab may be made opaque and of low thermal conductivity. This region being formed to protect wildlife, such as birds, from intense light and heat. Thus, the concentrating solar collector 110a, has the unique capability of also being a containment vessel for intense concentrated sunlight and heat generated by concentrated sunlight. This is a particularly important capability because currently deployed conventional solar towers have been known to spontaneously ignite or even flash vaporize many hundreds of birds in the course of an hour or two, forming what those in the concentrating solar power industry call “streamers”, which are dead or dying birds that are literally on fire and falling out of the sky after flying into the intense solar radiation field of unprotected and exposed solar tower receivers.
The concentrating solar collector 110a is also nearly 100% area covering. This is unlike todays heliostats, which can waste up to 90% of the land area and solar energy. The concentrating solar collector 110a therefore has much lower impact on the environment and is especially useful for solar applications where there is an intrinsic need for a containment building. Note that the containment building may be rigid or even inflated, such as large sports complexes. The roof of the inflated building would hold an array of concentrator panels with colloidal and DEP-based light control. Inflated building are low-cost and can support huge weights, for example even the weight of large amounts of snow on an inflated sports building will not collapse the building.
The next example of a solar tracker and concentrator is based on trading hardware complexity for software complexity by using reflective particles, e.g. metallic particles in a simple gap between two transparent sheets. These particles may be nanoparticles or they many be micron scale particles. In this solar tracker electrically large gratings are formed by controlling the distribution of reflecting particles within the colloid. This is related to the material already discussed in Section 7.8 “Beam Steering Fields For DEP-Based Gratings. As the sun moves the grating reconfigures to redirect sunlight towards whatever type of solar receiver is desired: solar cells, concentrating solar power molten salts, a solar smelter, a solar desalination receiver and others. The great advantage of an electrically large grating formed now by reflecting particles, see for example
To give the reader an idea of how the electrically large grating sun tracker is used in a high-concentrating photovoltaic solar panel a very simple design is provided as an example. In particular,
The emitted rays take the form of a cone of light 112g from each point along the electrically large grating. Note that only a finite number of these cones are shown in
Next, we reconsider periodic potentials to generate particle distributions like that shown in
In the development of DEP based gratings in the section “Beam Steering Fields for DEP-Based Gratings,” we initially assumed that the potential plane supported a continuous potential. This is not a necessary condition and we can sample the potential plane with electrodes. This is shown in
In
Therefore, by the used of simple printed electrodes on flat panels we can impress non-uniform electric fields on particles in a thin colloidal layer to steer sunlight into a desired direction. If each panel has slightly different direction then we can focus large fields of sunlight to a common focus for the generation of electricity, desalination of water, smelting of steel, and many other applications.
Additional Applications
DEP-Based 2D-Reconfigurable Optics
The ability to control the position of many different types of particles in time by means of harmonic voltages, frequencies, and phases opens up the possibility of creating optical elements under control of software. Some possible parameters that may be controlled include refractive index, scattering cross section, focal length, beam scan angle, optical switch state (pass vs. reflect), polarization (spin angular momentum) state, orbital angular momentum state, wavelength, and potentially many other quantities.
As particles in a colloid get smaller and smaller the voltages needed to move the particles increase as there are fewer induced charges on the particles to provide electrical forces. To have fast acting optics at modest voltages therefore requires that the distance that the particles have to travel is small. This is a motivation for keeping all the optical processing in a thin control volume.
Therefore, consider
Between the electrode arrays and the slabs is located a colloid 116e (or suspension) having particles comprising at least one of a dielectric, a metal, a quantum dot, a plasmonic dot, a photonic crystal dot (photonic band gap dot), and a biological cell. Complex particles comprising different hybrid combinations and structures are also possible.
Connected to the electrodes are a first harmonic excitation 116f and a second harmonic excitation 116g. The excitations may be grouped into sets. For example, if every other electrode in the first electrode array 116c is energized with the same voltage excitation then there is a set of two repeating excitations for the first electrode array {A1ei(ω
The parameters that can be placed under software control include: voltage amplitude Vm, frequency ω, phase θm, and the number M of electrodes in a group. This is available on each electrode array and provides a large number of degrees of freedom that may be exploited to change the optical properties of the colloid. Light can pass through and reaming the thin layer of colloid or it can pass through the slabs and the colloid. Various functions have already been described in the prior sections on beam steering and optical switching.
Clearly, the more electrodes (pixels electrodes for example) will give greater control at the expense of more complexity, however the result is greater control over light in a way that controllable by software.
Quantum Computing
The ability to control one or more nano-scale particles in space and time by means of harmonic voltages, frequencies, and phases opens up the possibility of positioning and interacting with atomic and molecular species for the purposes of quantum computing by means of a DEP process. Of particular interest here are room temperature quantum computers based on endohedral fullerenes, endohedral borospherenes, and other cage molecules.
A fullerene is a structural form of carbon, i.e. an allotrope of carbon, and can take many forms such as C60, C200 and large carbon nanotubes. The C60 molecule has 60 carbon atoms in a cage-like ball structure, i.e. a Buckyball, that resembles a soccer ball (football) with geometry of a truncated icosahedron having a diameter of about 1 nm. These cage-like molecules are called cage molecules.
Endohedral C60 fullerenes molecules have additional materials inside the Buckball cage. Those additional materials may include at least one of atoms, ions, and multi-atom particle clusters. Endohedral borospherenes, such as B40, are also cage molecules like similar to the Buckball, but with an antiprismatic symmetry D2d symmetry group like that of a baseball with a unique axis of symmetry. Endohedral borospherenes also have additional materials inside the molecular cage. Endohedral nanotubes are also possible.
The most stable forms of fullerenes have been calculated from first principles to be C60, C240, C540, C960, C1500, and C2160 by the US Naval Research Labs. These fullerenes have diameters ranging from about 1 nm to about 50 nm. Thus, the largest fullerenes are comparable to the size of nanoparticles typically considered for modification of the refractive index of a liquid or gas host medium.
A carbon Buckyball C60 made only of isotopic C13 atoms can, in principle, serve as a 60-qubit quantum computer through the nuclear spin of ½ for each atom therein. However, it is fragile and typically requires cryogenic temperatures. In contradistinction, a room temperature atomic cage molecule can hold atoms to form a spin quantum register. For example, free nitrogen and phosphorus atoms possess a half-filled p-shell and a total electron spin of S=3/2. If these atoms are caged in fullerene molecules C60 and C70 respectively they retain their atomic character, which is useful for quantum computing.
Therefore, these cage molecules provide a way to protect the fragile quantum states of atoms, ions, and atomic clusters that are inside the cages from the external environment. One type of external environment is that of a colloid. That is to say that cage molecules can form a colloid (or suspension) in either a liquid or gas—even a very dilute gas.
The fullerenes (and borospherenes) can have covalently bonded chemical groups that extend the size of the molecules and give them modified shapes, chemistry, and physical properties. These derivatives are included here as well. Thus, another class of particles that is included in this disclosure are the cage molecules, their derivatives and endohedral forms. These particles, may be used to provide at least one of DEP-based assembly and a means for quantum interactions for quantum computing in a colloid or suspension.
The cage molecules are not the only systems of interest for DEP-based arrangement of quantum computing systems. Other potential systems for DEP-based arrangement and interaction with electromagnetic radiation include: solid-state NMR of a nitrogen vacancy in diamond particles used in a DEP process, quantum dots used for charge qubits, and quantum dots used for spin quibits.
While the above descriptions in each of the sections contains many specific details for dielectrophoresis based light control, these details should not be construed as limiting the scope of the invention, but merely providing illustrations of some of the possible methods, physical embodiments and applications. In particular, the present invention is thus not limited to the above theoretical modeling and physical embodiments, but can be changed or modified in various ways on the basis of the general principles of the invention, including but not limited to forces on particles that are induced by other forms of phoresis where gradients in a physical quantity induce ponderomotive forces to curve a light beam's path.
Note that the theoretical discussion provided in this disclosure may reuse some mathematical symbols to mean different things in different locations of the text for historical and pragmatic reasons. The meaning is readily discernible by those skilled in the art when taken in context of the associated descriptions.
Many of the figures in the disclosure are not to scale, but are instead provided to maximize understanding of the underlying concepts.
Many potential applications follow from a few physical principles and a few generic embodiments. This disclosure provides many example embodiments in different applications to establish the flexibility of the underlying principles and to teach by example.
Every effort was made to provide accurate analysis of the physics. Nonetheless, typographical and other errors in equations sometimes make it through reviews. Therefore, derivations, individual equations, textural descriptions, and figures should be taken together so that clarity of meaning is ascertained from a body of information even in the event of an unintended error in an equation.
Finally, the scope of the invention should in general be determined by the appended claims and their equivalents and not only by the examples, embodiments, and theoretical analysis provided in this disclosure.
This invention has applicability for controlling light by controlling scattering from small particles. The control includes optical operations as beam steering and focusing as well as general wavefront modification without significant restrictions due to polarization, direction of input light and direction output light. Specific applications include, but are not limited to: Light Detection and Ranging (LiDAR), electronically focused camera lenses, robotic visions system, adaptive automotive headlights, light-art, free-space photonic network configurations for computing, laser machining, laser power beaming to remotely power drones, 3D-printing, topographic mapping, automated inspection, remote sensing, point-to-point communications, computer displays, augmented reality displays, virtual reality displays, mixed reality displays, electronic paper, sensor drones, topography mapping, surveying, augmented reality sensors, drought monitoring sensors, aircraft collision avoidance, drone based structural inspection, construction site monitoring, security, laser scanning for bar code readers, optical reflectance switches for telecommunications, solar concentrators, solar power plants, solar desalination plants, solar smelting plants, solar radiation control windows, light-beam power combiner, laser systems, laser gyroscopes, a laser machining, a manufacturing machine for making graded refractive index devices, and software reconfigurable optics.
First, and foremost I would like to express sincere thanks and gratitude to my wife Susan and daughter Anora for tolerating the countless days and nights I spent away from them theorizing, experimenting in the lab, and writing for more than a decade of entrepreneurial R&D as I sought a way to contribute to overcoming global warming and make the world a better place through invention.
Second, I would like to thank my parents Maria and Leo DiDomenico, whose sacrificed so much for my life and scientific training and asked for so little in return. Their time of life was entirely too short and they will be forever missed.
This invention claims the benefit of patent applications: [1] PCT application number PCT/US18/016930, entitled “Light Steering and Focusing by Dielectrophoresis,” which was filed by patent attorney Christopher Peil for the inventor Leo D. DiDomenico on 2018 Feb. 5, the entirety of which is incorporated herein by this reference; [2] U.S. 62/558,829 filed by patent attorney Christopher Peil for the inventor Leo D. DiDomenico on 2017 Sep. 14 and entitled, “Method and Apparatus for Precision, Non-Mechanical Digital Control of Light,” is hereby incorporated in its entirety; [3] U.S. 62/693,894 filed by patent attorney Christopher Peil for the inventor Leo D. DiDomenico on 2018 Jul. 3 and entitled, “Light Steering, Focusing and Fluorescing by Dielectrophoresis,” is hereby incorporated in its entirety; [4] U.S. 62/794,505 filed by patent attorney Christopher Peil for the inventor Leo D. DiDomenico on 2019 Jan. 8 and entitled, “Controlling Light by Dielectrophoresis For LiDAR, Beam Steering, Augmented Reality, Electronic Lens Focusing, Sensor Drones, and Other Applications,” is hereby incorporated in its entirety.
| Number | Name | Date | Kind |
|---|---|---|---|
| 7106297 | Shannon | Sep 2006 | B2 |
| 20060050402 | Ito | Mar 2006 | A1 |
| 20070115550 | Nakayama | May 2007 | A1 |
| 20110042215 | Hou | Feb 2011 | A1 |
| 20120188295 | Joo | Jul 2012 | A1 |
| 20140080040 | Fontecchio | Mar 2014 | A1 |
| 20150268529 | Lin | Sep 2015 | A1 |
| 20150369220 | Takatori | Dec 2015 | A1 |
| Number | Date | Country | |
|---|---|---|---|
| 20190353975 A1 | Nov 2019 | US |
| Number | Date | Country | |
|---|---|---|---|
| 62460080 | Feb 2017 | US | |
| 62456614 | Feb 2017 | US |
| Number | Date | Country | |
|---|---|---|---|
| Parent | PCT/US2018/016930 | Feb 2018 | US |
| Child | 16530889 | US |
