The present disclosure relates to polarization control devices which use cascaded subwavelength dielectric gratings.
Metasurfaces are optically-thin structures that can control the phase and polarization of electromagnetic waves through subwavelength patterning that tailors their electric, magnetic, and electro-magnetic/magneto-electric surface properties. While initial metasurfaces used metallic patterns, recent fabrication advances have enabled all-dielectric metasurfaces that can provide similar responses with significantly lower losses. Demonstrated dielectric elements include rods and fins that provide spatially varying phase and polarization shifts and silicon microdisks that use overlapping electric and magnetic resonances to provide reflectionless (Huygens') response.
Stacked or cascaded structures have also been proposed and demonstrated with strong bianisotropic responses, including circular dichroism and multifunction polarization conversion. Multilayer dielectric metasurfaces can also exhibit broadband or multichromatic operation. This disclosure presents polarization control devices comprised of cascaded subwavelength dielectric gratings to improve polarization control with varied spectral response.
This section provides background information related to the present disclosure which is not necessarily prior art.
This section provides a general summary of the disclosure, and is not a comprehensive disclosure of its full scope or all of its features.
A polarization control device is presented which operates on electromagnetic radiation at a given wavelength. The polarization control device is comprised of two or more metasurfaces stacked directly onto each other without intermediate layers interposed between the two or more metasurfaces. Each of the two or more metasurfaces has a grating structure formed by two dielectric materials, where a ratio of permittivity exhibited by the two dielectric materials is high and periodicity of the grating structure is less than the given wavelength. The orientation of the grating structure in each of the two or more metasurfaces also differs from each of the other grating structures in the two or more metasurfaces.
In some embodiments, the ratio of permittivity exhibited by the two dielectric materials is greater than four.
In some embodiments, the periodicity of the grating structure is less than the quotient of the given wavelength divided by five.
The filling fraction of the grating structure is preferably between twenty and one hundred percent.
In some embodiments, each of the two or more metasurfaces has a thickness in range of λ/20 and λ/4, where λ is the given wavelength.
The polarization control device may be design to perform different functions. In one instance, the polarization control device operates to rotate polarization state of light incident thereon. In another instance, the polarization control device operates to rotate polarization state of light incident thereon by a fixed angle independent of the angle of incidence. In yet another instance, the polarization control device operates to transmit light incident thereon as left-circular polarized in a first frequency band and to transmit the light incident thereon as right-circular polarized in a second frequency band, where the first frequency band does not overlap with the second frequency band.
The polarization control device is preferably fabricated using additive manufacturing.
Further areas of applicability will become apparent from the description provided herein. The description and specific examples in this summary are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure.
The drawings described herein are for illustrative purposes only of selected embodiments and not all possible implementations, and are not intended to limit the scope of the present disclosure.
Corresponding reference numerals indicate corresponding parts throughout the several views of the drawings.
Example embodiments will now be described more fully with reference to the accompanying drawings.
where ε⊥ and ε∥ are the effective permittivity along the grating's extraordinary and ordinary optic axes, and f is the filling ratio of medium 1. After homogenizing each layer in this way, the plane wave transmission and reflection performance for the cascaded structure can be rapidly calculated using analytic transfer matrix techniques for layered media. A wide range of desired responses can then be achieved by numerically optimizing three parameters per layer: fill factor fl, layer thickness dland optic axis rotation angle θl.
With reference to
Applying the homogenization procedure, each layer is treated as a nonmagnetic, uniaxial homogeneous medium with principal axes rotated by an angle θl in the xy-plane. The constitutive relation in each layer is then:
where I is the 3×3 identity matrix, and
Beginning with Faraday's Law and Ampere's Law in source-free media:
Considering monochromatic fields with exp(jωt) time evolution, in Cartesian coordinates, one can write these in a matrix form:
If the material properties depend only on z, plane wave fields have the form A(z)e−jk
Combining equations 2, 6 and 7 yields a system of six equations, of which the third and sixth are linear algebraic equations relating the six components of E and H. These can be solved for Ez and Hz in terms of the other four components, yielding the following 4×4 wave equation for the transverse field:
Noting that the structure is piecewise uniform and the material properties do not depend on z within each layer, equation 8 has four solutions for the total transverse field vector ψl=(Ex, Ey, Hx, Hy) of the form
which when substituted in equation 8 yields the eigenvalue equation:
Equation 10 can be solved numerically for each layer to find the four characteristic propagation constants qln and associated eigenmodes ψln. In general, the total transverse field ψl at a given position z within the structure can be decomposed into a weighted superposition of the eigenmodes with weights ϕl=(ϕ1, ϕ2, ϕ3, ϕ4)T. The total field and mode amplitudes are related by
where Al=(ψl1, ψl2, ψ13, ψl4) is a weighting matrix whose columns are the eigenmodes of Λl. The vector of mode amplitudes evolves within each layer according to a propagation matrix K
Finally, by combining equations 11 and 12 and enforcing that the transverse fields must match across each layer boundary, a wave matrix W can be constructed relating the mode amplitudes on either side of the cascaded structure. Ordering the modes in the incident and exit media according to their polarization and propagation direction as depicted in
The transmission and reflection coefficients for the cascaded structure are most conveniently represented by the scattering matrix S, which relates scattered to incident mode amplitudes. The scattering matrix can be obtained from the wave matrix as follows:
Given the grating parameters for each layer (grating materials, filling fraction fl, layer thickness d1 and optic axis rotation angle θl), the transfer matrix analysis method described above allows computing the scattering matrix extremely quickly by simply multiplying several 4×4 matrices. Thus, the computation can be included within the cost function for a numerical optimization to obtain a wide range of polarization and spectral responses, including broadband, multiband, and multifunctional devices.
Each of the two or more metasurfaces 32 has a grating structure formed by two dielectric materials, where the ratio of permittivity exhibited by the two dielectric materials is high and the periodicity of the grating structure is less than the given operating wavelength (λ). In example embodiments, the ratio of permittivity exhibited by the two dielectric materials is greater than four and the periodicity of the grating structure is less than a quotient of the given wavelength divided by five (i.e., periodicity<λ/5). By way of example, the two dielectric materials can be alumina and air. In this example, the ratio of permittivity is on the order of 9, where the permittivity of alumina is 9.7 and the permittivity of air is about one. For a polarization control device operating in the Ka band (26.5-40 GHz), the grating periodicity is less than 1500 microns. While particular reference is made to alumina and air, it is readily understood that different types of dielectric materials fall within the scope of this disclosure.
Additionally, each of the two or more metasurfaces 32 has a thickness in range of λ/20 and λ/4, where λ is the given operating wavelength. In the example embodiments, the filling fraction of the grating structure is preferably between twenty and one hundred percent. These particular parameters are merely illustrative and other values falling within the specified limits are contemplated by this disclosure. Different examples and implementations for such polarization control devices are further described below.
where φ is an arbitrary constant phase shift.
For simplicity, a filling fraction fl=0.5 was fixed for each layer, with grating period Λ=1000 μm to give Λ/λ0<0.13. The layer thicknesses dl and optic axis rotation angles θl were numerically optimized to minimize the difference between the desired (equation (15)) and analytically calculated reflection tensors over the operating band. In this example, four metasurfaces are stacked directly onto each other. Layer thicknesses are as follows: 1750 μm; 1050 μm; 1000 μm and 725 μm. Using more layers widens the bandwidth at the cost of more complexity. In the end, four layers were chosen as a reasonable trade-off to yield the design.
As proof of concept, the half-wave plate 40 was fabricated by Technology Assessment & Transfer, Inc. using a ceramic stereolithography process. A resin was prepared consisting of sinterable alumina powder, a monomer/initiator mixture, and dispersants. The resin was photocured layer-by-layer as in conventional stereolithography to produce a green state part, which was then thermally processed to remove the binder, and sintered. During sintering the part shrinks in a predictable manner by approximately 20%, which is compensated by scaling the design appropriately. The fabricated half-wave plate 40 is a disk approximately 9 cm in diameter.
where φ is an arbitrary constant phase shift. That is, linearly polarized incident light is transmitted without reflection, and the transmitted polarization is rotated counterclockwise by an angle α. In contrast to the half-wave plate 40, which can rotate only specific linearly polarizations, the isotropic polarization rotator 60 produces the same rotation angle regardless of the incident polarization. Isotropic rotation is an inherently chiral response and therefore is a more demanding design challenge than a half-wave plate.
In one example, the polarization rotator 60 was designed to provide α=90° rotation from 30-35 GHz within the Ka band using alumina (ϵ1=9.7, tan δ1=10−4) and air (ϵ2=1) subwavelength gratings. The grating period was fixed at Λ=1100 μm, while the filling fraction fl, layer thicknesses dl and optic axis rotation angles θl for each layer were numerically optimized to minimize the difference between the desired and analytically calculated transmission tensors. The optimization was repeated with an increasing number of layers until good results were achieved at all incident polarizations over the target frequency band.
Specifically, the design resulted in a polarization control device comprising nine (9) subwavelength grating layers with total thickness of 10.1 mm (about 0.85 λ0). Starting at front layer, layer thickness in microns (μm) is 1160, 1120, 1200, 920, 1240, 920, 1200, 1120, 1160; whereas, starting with the front layer, the grating angle in degrees is 0, 30, 60, 30, 62, 93, 64, 93, 124. Starting again with the front layer, the filling fraction for each layer is 0.30, 0.65, 0.36, 0.38, 0.65, 0.38, 0.36, 0.65, 0.30. While an exemplary embodiment for a polarization rotator has been described above with specific values and arranged in a specific configuration, it will be appreciated that these devices may be constructed with many different configurations and/or values as necessary or desired for a particular application. The above configurations, components and values are presented only to describe one particular embodiment that has proven effective and should be viewed as illustrating, rather than limiting, the present disclosure.
Similar to the polarization rotator, each grating layer's thickness, filling fraction and orientation were allowed to vary as design parameters. In one embodiment, the dual band circular polarizer is comprised of sixteen (16) subwavelength grating layers with total thickness of 15.7 mm. Starting at front layer, layer thickness in microns (μm) is 400, 1100, 1200, 1000, 650, 1200, 1200, 500, 1200, 1200, 950, 600, 1200, 900, 1200 and 1200; whereas, starting with the front layer, the grating angle in degrees is 121, 81, 22, 52, 99, 28, 69, 39, 9, 53, 23, 141, 111, 30, 93, and 63. Starting again with the front layer, the filling fraction for each layer is 0.55, 0.59, 0.30, 0.70, 0.46, 0.40, 0.30, 0.70, 0.30, 0.52, 0.70, 0.46, 0.70, 0.30 and 0.30. While an exemplary embodiment for a circular polarizer has been described above with specific values and arranged in a specific configuration, it will be appreciated that these devices may be constructed with many different configurations and/or values as necessary or desired for a particular application. The above configurations, components and values are presented only to describe one particular embodiment that has proven effective and should be viewed as illustrating, rather than limiting, the present disclosure.
Stereolithography was developed by 3D Systems, Inc. and is a widely used 3D printing process that builds parts using a liquid photocurable resin and a scanned UV laser or projected UV image.
The foregoing description of the embodiments has been provided for purposes of illustration and description. It is not intended to be exhaustive or to limit the disclosure. Individual elements or features of a particular embodiment are generally not limited to that particular embodiment, but, where applicable, are interchangeable and can be used in a selected embodiment, even if not specifically shown or described. The same may also be varied in many ways. Such variations are not to be regarded as a departure from the disclosure, and all such modifications are intended to be included within the scope of the disclosure.
When an element or layer is referred to as being “on,” “engaged to,” “connected to,” or “coupled to” another element or layer, it may be directly on, engaged, connected or coupled to the other element or layer, or intervening elements or layers may be present. In contrast, when an element is referred to as being “directly on,” “directly engaged to,” “directly connected to,” or “directly coupled to” another element or layer, there may be no intervening elements or layers present. Other words used to describe the relationship between elements should be interpreted in a like fashion (e.g., “between” versus “directly between,” “adjacent” versus “directly adjacent,” etc.). As used herein, the term “and/or” includes any and all combinations of one or more of the associated listed items.
Although the terms first, second, third, etc. may be used herein to describe various elements, components, regions, layers and/or sections, these elements, components, regions, layers and/or sections should not be limited by these terms. These terms may be only used to distinguish one element, component, region, layer or section from another region, layer or section. Terms such as “first,” “second,” and other numerical terms when used herein do not imply a sequence or order unless clearly indicated by the context. Thus, a first element, component, region, layer or section discussed below could be termed a second element, component, region, layer or section without departing from the teachings of the example embodiments.
Spatially relative terms, such as “inner,” “outer,” “beneath,” “below,” “lower,” “above,” “upper,” and the like, may be used herein for ease of description to describe one element or feature's relationship to another element(s) or feature(s) as illustrated in the figures. Spatially relative terms may be intended to encompass different orientations of the device in use or operation in addition to the orientation depicted in the figures. For example, if the device in the figures is turned over, elements described as “below” or “beneath” other elements or features would then be oriented “above” the other elements or features. Thus, the example term “below” can encompass both an orientation of above and below. The device may be otherwise oriented (rotated 90 degrees or at other orientations) and the spatially relative descriptors used herein interpreted accordingly.
This application claims the benefit of U.S. Provisional Application No. 63/073,997, filed on Sep. 3, 2020. The entire disclosure of the above application is incorporated herein by reference.
This invention was made with government support under N00014-15-1-2390 awarded by the Office of Naval Research and FA9550-18-1-0466 awarded by the U.S. Air Force. The government has certain rights in the invention.
Number | Date | Country | |
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63073997 | Sep 2020 | US |