This invention relates to magneto-optical devices and in particular to magneto-optical uni-directional resonator systems and their applications.
Nonreciprocal optical devices are important components in large-scale optical system and networks. These devices regulate the signal propagation along a single direction, thereby preventing reflection between stages and allowing the separation of the forward and backward signal flow.
Two-way propagation of optical signal can destabilize the operation of an optical system and contribute to elevated noise level. From the reciprocity principle, regular dielectric optical systems allow signal propagation along both forward and the backward directions. Most optical devices, particularly in integrated circuits, reflect incident light to a certain extent, and thereby introduce coupling between cascaded stages. Optical resonances, commonly used in lasers, modulators and filters, would experience a resonance frequency pulling, depending on the amplitude and the phase of the inter-stage coupling. The resultant system response functions can vary substantially from the designed characteristics. To allow a practical tolerance for fabrication and assembly error, nonreciprocal devices are used to block the reflected beam, such as optical isolators used in most laser systems.
A common approach to obtain non-reciprocity at optical wavelength is based on a linear magneto-optical effect, known as gyrotropy. It has been used in most commercially-available bulk optical isolator or circulator structures, where large isolation and minimal insertion loss can be simultaneously achieved. The dimension of such devices, however, tends to be very large at a length scale on the order of millimeters. The strength of gyrotropy in existing magneto-optical material, measured by the Voigt parameter, is at most 10−2 and typically less than 10−3. Consequently the signal interaction length needs to be at least hundreds of wavelengths.
One device configuration to generate large nonreciprocal effect is known as the Voigt configuration, where the external bias (magnetic field) is applied perpendicular to the optical path. The magneto-optical effect manifests as a small difference in the propagation constant of the forward and backward waves. This difference can be converted into a large difference in forward and backward transmission coefficient, using various forms of interferometers, such as a Mach-Zehnder interferometer. The need for a long interaction length still applies to devices based on such nonreciprocal phase shift, including many integrated magneto-optical waveguide isolators. R. Wolfe, R. A. Lieberman, V. J. Fratello, R. E. Scotti, and N. Kopylov, “Etch-tuned ridged waveguide magneto-optic isolator,” Applied Physics Letters, vol. 56, pp. 426-428, 1990; M. Levy, I. Ilic, R. Scarmozzino, R. M. Osgood, Jr., R. Wolfe, C. J. Gutierrez, and G. A. Prinz, “Thin-film-magnet magnetooptic waveguide isolator,” IEEE Photonics Technology Letters, vol. 5, pp. 198-200, 1993; M. Levy, “The on-chip integration of magnetooptic waveguide isolators,” IEEE Journal of Selected Topics in Quantum Electronics, vol. 8, pp. 1300-6, 2002. In these devices, while the lateral dimensions have been reduced to several microns, the length remains comparable to bulk devices.
The Faraday configuration is another type of device design commonly used in bulk nonreciprocal devices. Here the external bias is aligned parallel to the propagation direction of the light beam. To reduce the length of the devices, magneto-optical resonators have been developed to trade the operational bandwidth for shorter optical path. Experimentally demonstrated, enhanced Faraday rotation in one-dimensional photonic crystal defect systems can significantly shorten the total device length necessary for a 45° polarization rotation by one order of magnitude. M. Inoue, K. Arai, T. Fujii, and M. Abe, “Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers,” Journal of Applied Physics, vol. 83, pp. 6768-6770, 1998; M. J. Steel, M. Levy, and R. M. Osgood, “High transmission enhanced Faraday rotation in one-dimensional photonic crystals with defects,” IEEE Photonics Technology Letters, vol. 12, pp. 1171-3, 2000. The application of such scheme in today's on-chip optical circuits, however, is fundamentally limited by the problem of weak light confinement in the transversal dimension, which results in large lateral component sizes, and the inconvenient co-linear magnetic biasing.
It is therefore desirable to provide magneto-optical uni-directional resonator systems which overcome the above shortcomings.
A resonator system comprises an optical resonator that supports one or more pairs of nearly degenerate defect states. One or more magnetic domains comprising at least one gyrotropic material in the optical resonator cause magneto-optical coupling between the two states so that the system lacks time-reversal symmetry. In one embodiment, a single magnetic domain is used that dominates induced magneto-optical coupling between the defect states.
The above resonator system may be used together with other components such as waveguides to form circulators, add drop filters, switches and memories.
a is diagram of a defect structure in a photonic crystal. The crystal consists of a triangular lattice of air holes with a radius of 0.35a, introduced in a dielectric of ε=6.25. The defect is created by reducing the radius of a single air hole.
b is a graphical plot of the frequency of the defect modes as a function of the radius of the center air hole in the defect region for the two-dimensional bismuth-iron-garnet photonic crystal. The calculation is performed using a plane-wave expansion method.
a and 2b are images representing the Hz field distribution of the two doubly-degenerate dipole modes in the BIG photonic crystal cavity as shown in
a is a diagram illustrating the cross-product between the E-fields of the two modes shown in
b is a diagram illustrating the corresponding domain pattern that maximizes the magneto-optical coupling constant of the two modes shown in
c is a diagram illustrating the spatial distribution of the modal cross-product of the dipole modes in an infiltrated BIG cavity in a silicon crystal. The Black dashed circle represents the position of the BIG rod.
d is a diagram illustrating the corresponding optimized domain pattern using only single domain. In
a is a graphical plot of the magneto-coupling strength in an infiltrated silicon photonic crystal cavity as a function of the radius of the BIG rod.
b is a graphical plot of the magneto-optical splitting between the dipole modes as a function of εa for the cavity in BIG crystal and the infiltrated silicon cavity with rBIG=0.4a.
a is a graphical plot of the frequency of the defect modes as a function of the radius of the center BIG rod in the defect region for the two-dimensional silicon photonic crystal.
b is a diagram illustrating an octapole mode in the infiltrated silicon cavity with rBIG=0.4a.
c is a diagram illustrating the modal cross product of the octapole mode of
d is a diagram illustrating the modal cross product of the octapole mode of
a is a schematic view of a three-port Y-junction circulator. The straight arrows indicate the incoming and outgoing waves. The curved arrows represent the two counter-rotating modes in the resonator.
b is a schematic view of a three-port Y-junction circulator constructed as a point defect coupled to three waveguides. Circles correspond to air holes in Bismuth Iron Garnet. The light and dark gray areas represent the magnetic domains with opposite out-of-plane magnetization direction.
c is a graphical plot of the transmission spectra at the output and isolated ports of a three-port junction circulator of
a-7d are diagrams illustrating the out-of-plane H field patterns of the three-port junction circulator shown in
a is a schematic view of a four-port channel add-drop filter (ADF). The straight arrows indicate the incoming and outgoing waves. The curved arrows represent the two counter-rotating modes in the resonator.
b is a graphical plot of the spectra of transfer efficiency of various four-port ADFs. The taller curve at the center represents an ideal ADF resonant at ω0 with a line-width γ. Such an ADF supports two resonant modes that are degenerate in both frequency and linewidth. The shorter dashed line curve at the center corresponds to a filter structure in which the two modes have the same width γ, and with a frequency split of 1.7γ. The two curves on the two sides of the center curve correspond to the transfer efficiency from ports 1 to 2 (2 to 1), in the presence of a strong magneto-optical coupling with a coupling strength κ.
a and 9b are diagrams illustrating a pair of degenerate even and odd resonant modes respectively of a 2D ring resonator consisting of a ring of high-index material. The structure can be seen as the intensity of the magnetic field along out-of-plane direction is represented as the “+” and “−” regions.
c is a diagram of the corresponding magnetic domain structure necessary to couple the modes plotted in
a is a top view of a micro-ring add/drop filter 150, where the high-index materials are outlined with black lines, in which a ring shaped structure 152 is identical to the structure in
b is a graphical plot of the field transmission spectra of the device excited with the left port of the top waveguide.
c is a graphical plot of the field transmission spectra of the device excited with the left port of the bottom waveguide.
For convenience in description, identical components are labeled by the same numbers in this application.
This disclosure covers optical devices employing planar optical resonances evanescently coupled to optical waveguides. Gyrotropic materials are introduced into the resonator to provide non-reciprocal effects to achieve optical isolation. Additionally, these non-reciprocal resonances are robust against many effects caused by disorders.
In this context, we specific demonstrate the concept of nonreciprocal resonances in two-dimensional photonic crystals by domain engineering. Photonic crystal slabs are promising candidates for large-scale optical integration that are necessary to address the increasing demand of optical information processing for broader communication bandwidth. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Physical Review Letters, vol. 58, pp. 2486-9, 1987; E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Physical Review Letters, vol. 58, pp. 2059-62, 1987; J. D. Joannopoulos, R. D. Meade, and J. N. Winn, “Photonic crystals: molding the flow of light,” Princeton, N.J.: Princeton University Press, 1995. Its advantages include the strong in-plane field confinement from the photonic band gap and the compatibility with planar on-chip circuits.
Firstly, we demonstrate that nonreciprocal transport among waveguides can be accomplished in a low-loss and large-bandwidth three-port junction circulator. The transmission between adjacent ports is uni-directional and can provide a 30 dB-isolation bandwidth well over 100 GHz. The highly reduced device dimension, on the order of a single wavelength, allows tight integration in functional systems. When absorbing materials are present at the isolated port, the circulator can serve as an optical isolator. In addition, channel add/drop can be performed with the wavelength-selective reflectors.
Similarly, a four-port circulator can be designed using a magnetized four-port optical add/drop filter. The magneto-optical coupling between the two degenerate resonances in the structure breaks the time reversal symmetry. The resonance tunneling therefore transmits light only along a single direction in certain wavelength range and such a resonator functions as an ultra-compact optical circulator.
More importantly, in the limit of a strong magneto-optical coupling, ideal channel add/drop characteristics can occur independent of small structural variations. The strong magneto-optical coupling dominates disorder-induced mode coupling and stabilize the ideal transmission lineshape in the presence of fabrication related roughness.
The suitable resonator structures that exhibit large nonreciprocal effects are not limited to photonic crystal systems. Micro-ring(disk) resonators, for example, can benefit from the same magneto-optical effects, as nonreciprocal devices and disorder-tolerant filters.
In addition, since the transport properties of these devices are strongly influenced by domain structures. We can exploit the nonvolatile magnetization (as permanent magnets) in some iron garnet films. These garnet films can be used directly in the photonic crystal cavity as the gyrotropic core or can be used to generate the bias field. By rewriting the magnetization direction of the magnetic domain, one can readily reprogram the switching property of the circulators and control the routing configuration in optical circuits. The required DC magnetic field can be created by an inductor, which has been routinely miniaturized to micron scale in integrated circuit and can be wafer bond to the optical chip. The direction of the inductor current dictates the polarity of the applied magnetic field, which in turn controls the magnetization of the permanent magnet and the magnetic core of the photonic crystal cavity, as shown in
Before we discuss specific magneto-optical resonances, let's briefly review some of the basic properties of magneto-optical materials. At optical wavelengths, the gyrotropy of a magneto-optical material is characterized by a dielectric tensor:
when the magnetization is along the z direction. Here, for simplicity, we ignore the absorption and assume ε⊥, ε∥and εα to be real. The εα in the off-diagonal elements has its sign dictated by the direction of magnetization. The strength of magneto-optical effects is measured by the Voigt parameter QM=εα/ε⊥.
Here we take photonic crystal cavities as an example, to solve the optical eigne-modes in the presence of magnetio-optical materials. To theoretically describe the modes in a magneto-optical defect structures in a photonic crystal, we use a Hamiltonian formalism, Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Physical Review E (Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics), vol. 62, pp. 7389-404, 2000, where the resonator mode
at a frequency ω is the solution of the eigenvalue equation
as defined by the Maxwell's equations.
This eigenvalue equation can be solved numerically. Here, however, to exhibit the general features of modal structures in a magneto-optical resonator, we exploit the fact that the Voigt parameter is typical less than 10−3 and use a perturbative approach where the Hamiltonian is split into a non-magneto part Θ0 and a gyrotropic perturbation V as Θ=Θ0+V, where
The effects of gyrotropy are entirely encapsulated by V, which induces magneto-optical coupling between the eigenmodes of the non-magnetic photonic crystal described by Θ0.
For concreteness, we consider the simple case of a system supporting two nearly-degenerate defect states, where the effect of magneto-optical coupling is particularly prominent. The structure shown in
These two modes can be categorized as an even mode
(
(
As shown in
However, in practical devices, the E fields may not be entirely in the same plane, but may deviate from such ideal, to the extent that the E fields no less than 20% of its peak value is aligned within 10 degrees from the plane (referred to below as “E fields being substantially confined to the plane”), where the plane may be defined in reference to certain physical characteristics of the devices. In practical devices, the H fields may not be entirely be normal to the same plane, but may deviate from such ideal, to the extent that the H fields no less than 20% of its peak value is aligned within by up to 10 degrees from the perpendicular direction to the plane. Where the H field deviates from the perpendicular direction to the plane by up to 10 degrees, the H field is said to be in a near normal direction.
The effect of gyrotrophy, described by the operator V, is to introduce magneto-optical coupling between the eigenmodes of Θ0. With the basis of the eigenmodes of Θ0, the Hamiltonian Θ can be rewritten as Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Physical Review E (Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics), vol. 62, pp. 7389-404, 2000.
To the first order of εα, the coupling strength between any two modes α and β can be derived as:
where the sign of εα is determined by the direction of the magnetization vector, and Ēα is the electrical field for the unperturbed mode |Ψα
Thus the magneto-optical coupling is closely related to the spatial arrangement of the magnetic domain as well as the vectorial field distribution of the defect states.
When we choose the standing-wave modes with real-valued electromagnetic fields as the eigenmode basis, the coupling constant (Veo) between them is purely imaginary. Therefore, for the structure as shown in
and |o
when ωe=ωo, the Hamiltonian of the system is
when magneto-optical materials are present in the cavity. The eigenstates for the Hamiltonian in Eq. (7) take the rotating wave form |e
±i|o
with a frequency splitting of 2|Veo|. We denote this pair of eigenstates in the magneto-optical cavity as |+
and |−
with their resonance frequencies being ωe+|Veo| and ωe−|Veo| respectively. Since the two counter rotating modes are related by a time-reversal operation, the frequency splitting between them clearly indicates the breaking of time-reversal symmetry and reciprocity. Also, importantly, even in the case where ωe deviates from ωo, for example due to fabrication related disorders that break the three-fold rotational symmetry, as long as the magneto-optical coupling is sufficiently strong, i.e. |Veo|>>|ωe-ωo|, |e
±i|o
remain the eigenstates of the system. Thus, in the limit of strong magneto-optical coupling, such modes assume a general waveform of circular hybridization independent of small structural disorders that would almost always occur in practical devices. For example, in application of optical isolators, |Veo|>3×|ωe-ωo| may suffice the condition of dominant magneto-optical coupling (i.e. for supporting at least one or more pairs of nearly degenerate defect states). On the other hand, |Veo|>10×|ωe-ωo| is preferred for good throughput in optical circulators. A device whose operation relies upon the presence of such rotating states will therefore be robust against small disorders.
Given the importance of obtaining large magneto-coupling strength|Veo|, we now proceed to maximize the spatial overlap between domain structure and the modal fields in Eq. (6). Due to the vectorial nature of defect states in photonic crystals, the cross product between the electric fields of the even mode |e
and the odd mode |o
changes sign rapidly in the cavity, as shown in
and |o
is shown in
Alternatively, since the cross product of the modal field is well localized in the defect due to the strong field confinement from the photonic band gap, we can employ a simpler domain configuration, while still maintaining a moderately strong magneto-coupling. As an example, consider a silicon/air cavity similar to
In reference to
In fabricated devices, one or more additional magnetic domain can exist, in which the magnetization direction can be oriented along direction different from the applied magnetic bias. These domains can be tolerated when the dominant domain achieves the optimal pattern described in the previous paragraph. These unwanted domains can be ultimately eliminated by the application of a strong external magnetic bias with the field strength greater than the saturation magnetic field.
In the Si/BIG hybrid cavity structure, in addition to the dipole modes, there exist other modes inside the photonic band gap (
As an application of the nonreciprocal states in the magneto-optical resonators, we now construct a three-port optical circulator by using these rotating modes to create direction-dependent constructive or destructive interference. The structure is schematically shown in
and |−
as described in section 1, which have separated resonance frequencies ω+ and ω− respectively under the strong magneto-optical coupling. For simplicity, we assume the entire structure have 120-degree rotational symmetry. Ideally, at the signal frequency the device shall allow complete transmission from ports 1 to 2, 2 to 3, and 3 to 1, while prohibiting transmission in the reversed directions. This transport characteristic can be accomplished with an appropriate choice of frequency splitting with respect to the decay rate of the resonance.
The system as shown in
α+(−) is the normalized field amplitude of the counter-clockwise (clockwise) rotating mode. These modes resonate at frequencies ω+(−) and decay at rates γ+(−). Si+(−) is the normalized amplitude of the incoming (outgoing) wave at port i. The 3×2 matrices, K and D, represents the coupling between the resonances and the waves at the ports. Unique in magneto-optical system under DC magnetic bias, the full time-reversal operation should include the reversal of external DC magnetic field. J. D. Jackson, Classical electrodynamics, 3rd ed. New York: Wiley, 1999. Thus, a full time-reversal operation flips the rotation directions of the cavity modes, and hence transform K to K*. Taking into account energy conservation and time-reversal properties of the structure, we can arrive at the following relations:
These conditions, in combination of the 120-degree rotational symmetry of the structure, leads to:
In this derivation, we also assume that the main non-reciprocal effect of the magneto-optical materials is to introduce the frequency split between the counter rotating modes, while the coupling between these modes with the waveguides contain no non-reciprocal phase shift.
When wave at frequency ω is incident from port 1, the power transmission coefficients at ports 2 and 3 are solved from Eqs. (8)-(13) as:
The ideal circulator response with T1→3=1 and T1→2=0 can be obtained at an operational frequency ω0, when the resonant frequencies are chosen to satisfy the following conditions: A similar condition has been derived for a microwave ferrite circulator in D. M. Pozar, Microwave Engineering, (John Wiley, New York, 1998). Our derivation is more general since we do not assume the detailed modal field pattern in the resonator region.
ω+=ω0+γ30/√{square root over (3)} and ω−=ω0-γ−/√{square root over (3)} (15)
In such a case, ports 2 and 3 function as the isolated and the output ports, respectively. (The roles of ports 2 and 3 are switched with ω+<ω−.) From Eq. (14) and (15), the transfer function from the input port to the output port takes a symmetric form with respect to ω0 when γ+≈γ−. In this case, the structure possesses maximum bandwidth for given magneto-optical splitting |ω+-ω−|. The bandwidth for 30-dB isolation can be determined to be 0.0548|ω+-ω−|/π in the vicinity of ω0 using Eq. (14). Also, by rotational symmetry of the structure, we have T1→2=T2→3=T3→1=0, and T2→1=T3→2=T1→3=1. Thus, transmission at frequency ω0 is allowed only along the clockwise direction. Such a structure therefore behaves as an ideal circulator.
To validate the theoretical analysis, we compare the analytical coupled-mode theory conclusion with first-principles FDTD calculations with a gyrotropic material model. A. P. Zhao, J. Juntunen, and A. V. Raisanen, “An efficient FDTD algorithm for the analysis of microstrip patch antennas printed on a general anisotropic dielectric substrate,” IEEE Transactions On Microwave Theory and Techniques, vol. 47, pp. 1142-1146, 1999. Using the BIG resonator we discussed in Section 4, a three-port Y-junction circulator is created by coupling three waveguides to the cavity shown in
In the transmission calculation with FDTD method, we choose εa-0.02463. With the choice, the two rotating modes have frequencies 0.3465 (c/a) and 0.3471 (c/a), and quality factors 364 and 367, satisfying the conditions in Eq. (14). The FDTD calculations indeed demonstrate nearly ideal three-port circulator characteristics and agree nicely with the coupled-mode theory (
The steady-state field patterns at a frequency 0.3468 (c/a), where maximum isolation occurs, are shown in
The proposed device occupies only a small footprint of a few wavelength square. While the simulation in this paper is two-dimensional, the coupled-mode theory analysis, and hence the principles of the device, applies to three-dimensional cavity systems. For implementations in BIG thin films, the material exhibits strong gyrotropy with εa saturated at 0.06. T. Tepper and C. A. Ross, “Pulsed laser deposition and refractive index measurement of fully substituted bismuth iron garnet films,” Journal of Crystal Growth, vol. 255, pp. 324-31, 2003; N. Adachi, V. P. Denysenkov, S. I. Khartsev, A. M. Grishin, and T. Okuda, “Epitaxial Bi3Fe5O12(001) films grown by pulsed laser deposition and reactive ion beam sputtering techniques,” Journal of Applied Physics, vol. 88, pp. 2734-2739, 2000. From the coupled-mode theory, the bandwidth of the circulator scales linearly with the magneto-optical coupling strength. Hence the BIG device can provide a large bandwidth for 30 dB isolation up to 213 GHz, when operating at 633 nm. Since the quality factor of the resonator due to waveguide coupling can be as low as 140, the relative large material absorption in BIG can be still tolerated. At optical communication wavelength of 1550 nm, Ce:Yttrium Iron Garnet (Ce:YIG) has εa saturated at 0.009 with very low absorption. M. Huang and S. Y. Zhang, “Growth and characterization of cerium-substituted yttrium iron garnet single crystals for magneto-optical applications,” Applied Physics A (Materials Science Processing), vol. A74, pp. 177-180, 2002. For this material system, the bandwidth for 30 dB isolation at 1550 nm is estimated as 12.6 GHz.
Using the same rotating states that enable the construction of a three-port optical circulator, we can design another important optical device that shows nonreciprocal transmission characteristics as well as significant suppression of disorders. This type of device is based on a four-port channel add/drop filter.
Optical channel add/drop filters (ADFs) have been intensely researched since they are essential for wavelength-division multiplexed (WDM) systems. ADFs allow an optical signal at a specific wavelength to be injected into or/and extracted from a bus waveguide while leaving channels at other wavelengths intact. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Physical Review Letters, vol. 80, pp. 960-963, 1998; S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel drop filters in photonic crystals,” Optics Express, vol. 3, pp. 4-11, 1998. Using resonance tunneling through photonic crystals defects, photonic crystal optical ADFs with a device dimension approaching micron scales have been successfully proposed or demonstrated by many research teams. S. Noda, T. Baba, and Optoelectronic Industry and Technology Development Association (Japan), Roadmap on photonic crystals. Dordrecht; Boston: Kluwer Academic Publishers, 2003; H. Takano, B. S. Song, T. Asano, and S. Noda, “Highly efficient in-plane channel drop filter in a two-dimensional heterophotonic crystal,” Appl. Phys. Lett. (USA), vol. 86, pp. 241101-241102, 2005.
We start by briefly reviewing the operating principle of the original channel ADF structure in photonic crystals, highlighting only those features that are relevant for the discussions of magneto-optical effects. Sketched in
The structure can be analyzed with a coupled-mode approach. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE Journal of Quantum Electronics, vol. 40, pp. 1511-1518, 2004; H. A. Haus and W. P. Huang, “Coupled-mode theory,” Proceedings of IEEE, vol. 79, pp. 1505-1518, 1991; C. Manolatou, M. J. Khan, S. H. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE Journal of Quantum Electronics, vol. 35, pp. 1322-31, 1999. With a mirror symmetry, a properly designed defect supports an even modes |e
and an odd mode |o
oscillating at complex frequencies of ωe,o+i2γe,o respectively. The time evolution of the cavity mode amplitudes αe,o in the presence of the incoming (outgoing) waves at the port i with amplitude Si+(−) are described by the following equations:
By observing the constraints from energy conservation and time-reversal symmetry, one can show that: W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE Journal of Quantum Electronics, vol. 40, pp. 1511-1518, 2004.
When an accidental degeneracy of the complex frequencies is maintained, i.e. ωe=ωo and γe=γo, an input wave from port 1 excites a circularly hybridized state|+
=(|e +i|o
)/√{square root over (2)}. Such a state then decays only into port 2 and port 4. S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel drop filters in photonic crystals,” Optics Express, vol. 3, pp. 4-11, 1998. The destructive interference between the decayed wave from the cavity and direct transmission from port 1 to 4 leads to zero transmission at port 4, while the decayed wave at port 2 creates a complete transfer on resonance. The time-reversed transfer from port 2 to port 1 occurs at an identical frequency through the other resonant mode |−
=(|e −i|o
)/√{square root over (2)}.
Mathematically, when the degeneracy condition is satisfied, the operation of the device can be more easily described with Eqs. (4)-(6), which can be shown to be equivalent to Eqs. (1)-(3) via a unitary transformation:
Using Eqs. (4) -(6), the spectrum for transmission, transfer and reflection can be determined as
Thus, complete transfer between the bus and drop waveguides and zero reflection can be achieved on resonance ωo with a bandwidth of 2γ. (ideal curve 52 in
In fabricated devices, the dielectric function εr(r) would unavoidably deviate from the designed dielectric function εd(r). The effects of small perturbations, i.e. Δεr=εr-εd, can be determined by introducing an off-diagonal element into the frequency matrices in Eqs. (1) and (4). The disorders in the vicinity of the cavity affect mainly the real part of the frequencies. Their effects can be expressed as a coupling strength between the even and odd modes: Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Physical Review E (Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics), vol. 62, pp. 7389-404, 2000.
Such perturbation lifts the degeneracy and creates eigenstates that are drastically different from the preferred states |e ±i|o
resulting in significant reflection and reduction in transfer efficiency. As an example, we show in
6. Channel Add/Drop Filters with Magneto-Optical Materials in the Cavity
Here we seek to fundamentally suppress the effects from resonant-frequency splitting originated from fabrication inaccuracy by breaking time-reversal symmetry. It was recently shown that when magneto-optical material is introduced into the cavity region, the resulting eigenstates can assume a circularly “hybridized” waveform |e ±i|0
Z. Wang and S. F. Fan, “Optical circulators in two-dimensional magneto-optical photonic crystals,” Optics Letters, vol. 30, pp. 1989-1991, 2005. In this case, the time-reversal symmetry is broken, and the time-reversed pair |e
±i|o
oscillates at different frequencies. When the frequency separation induced by magneto-optics is much larger than the splitting caused by fabrication disorders, the eigenstates of the systems are largely immune from fabrication disorders. Below, we will show that magneto-optical effects can be very beneficial for channel ADF functions.
Analytically, the effect due to the presence of magneto-optical materials can be described with imaginary and anti-symmetric off-diagonal elements in the frequency matrix. Z. Wang and S. Fan, “Magneto-optical defects in two-dimensional photonic crystals,” Applied Physics B: Lasers and Optics, vol. 81, pp. 369-375, 2005. In the case of two modes, if we start with a structure that satisfies the degeneracy condition, after the introduction of magneto-optical effects, Eq. (1) is modified as:
Diagonalizing the frequency matrix yields equations that are the same as Eqs. (4)-(5), except with ω+=ω0+κ and ω−=ω0-κ. The transmission, reflection, and transfer spectra can be calculated from Eqs. (5) and (9) as
Thus, ideal channel add/drop characteristics are maintained, while the transport properties become direction-dependent and nonreciprocal.
The presence of disorders introduces additional real off-diagonal elements (i.e. Ve,o as defined in Eq. (8)) into the frequency matrix. However, |e
±i|o
remain the eigenstates for Eq. (9), as long as magneto-optical coupling dominates, i.e. |κ|>>|Veo|. Consequently, the ideal operation of the ADF is protected against disorders when significant magneto-optical coupling is present.
The nonreciprocal operation can be also applied to other integrated optical device, such as micro-ring (micro-disk), micro-toroid or micro-sphere resonators. As an example, the eigenmodes in a micro-ring resonator can also be categorized as even and odd modes (
a is a top view of a micro-ring add/drop filter 150, where the high-index materials are outlined with black lines, in which a ring shaped structure 152 is identical to the structure in
The magneto-optical effect again renders the eigenstates to be two counter propagating modes at disparate frequencies, which can be probed when the resonator 152 is coupled to two parallel waveguides 156 and 158 shown in
While the simulations here are for two-dimensional structures, the operating principles, as described by coupled-mode theories, can be readily applied to three-dimensional structures including of photonic crystal slabs (
Additionally, the role of the output port and the isolated port for the three-port and the four-port circulators are determined by the magnetization direction of the magneto-optical material. When combined with integrated inductor co-fabricated with the optical device, the circulator structures can be used as electrically reprogrammable optical switches. By changing the direction of the current flow in the inductor 204, the bias magneto-optical material can be inverted, so that the output port and the isolated port are interchanged. This can be achieved with or without the permanent magnet 202, as shown in
For magneto-optical materials 198 with large coercity, namely a permanent magnet, the resonant cavity can serve dual purposes as both the optical resonator and the permanent magnet. In such cases, external magnetic bias may not be necessary and the switching characteristic of the device are stable against small changes of the external field. Such devices can serve as optical read-only memory cell, as the binary information is stored as the switching direction of the input signal.
The gyrotropic material used for achieving the magneto-optical coupling between nearly degenerate defect states may comprise bismuth iron garnet, or other iron garnet, or diluted magnetic semiconductors.
Embodiments of the resonator system of this invention may comprise a two-dimensional photonic crystal resonator, a photonic crystal slab, or a micro-ring (or micro-disk/micro-toroid/micro-sphere) resonator.
While the invention has been described above by reference to various embodiments, it will be understood that changes and modifications may be made without departing from the scope of the invention, which is to be defined only by the appended claims and their equivalent. All references referred to herein are incorporated by reference in their entireties.
This non-provisional application claims the benefit of provisional application No. 60/755,274, filed Dec. 29, 2005, which application is incorporated herein in its entirety by this reference.
Number | Date | Country | |
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60755274 | Dec 2005 | US |