Phase-corrected metalens

Abstract

A computer-implemented method is provided for designing a metalens having a metasurface including nanostructures arranged on a substrate based on preliminary simulation data, the method includes dividing the nanostructures into overlapping d-unit supercells such that each nanostructure except for the periphery is at the center of one of the d-unit supercells, calculating a differentiable mapping function for the d-unit supercell that predicts the near field on a unit cell at the center of the supercell from design parameters of all the nanostructures in the supercell by fitting an interpolant to the preliminary simulation data of the d-unit supercells, adjusting the design parameters of all unit cells in the metalens collectively by using a Jacobian of the partial derivatives provided by the mapping function to solve for a locally optimal correction of all the design parameters to better approximate a target near field distribution throughout the metalens, and generating a manufacturable design of the metalens based on the optimized parameters.

Description

[Technical field]

[0001] The present disclosure relates generally to metalenses having metasurfaces, and more particularly to systems and methods for designing metalenses. [Background technology]

[0002] Metasurfaces are two-dimensional arrays of wavelength-scale structures that interact with electric and / or magnetic fields. Optical metasurfaces, also known as metalenses, interact with visible and near-visible wavelengths and provide fine control of the wavefront, potentially enabling substantial miniaturization in optoelectronics. In practice, metalenses have fallen well short of theoretically predicted efficiencies and so far offer no advantage over conventional bulk lenses and mirrors. The most promising design strategy is to decompose the metalens into a lattice of cells, calculate the desired optical phase retardation for each lattice cell, and select from a library of presimulated nanostructures a nanostructure that provides the appropriate phase retardation. For nanostructures that exhibit phase retardation that follows the predictions of Pancharatnam-Berry (PB) theory, the design problem is particularly simple, with the relative phase retardation expected to vary linearly with the orientation of rotation of the nanostructures. In practice, however, metalenses designed on the principle of unit cells suffer from unmodeled light-matter interactions and resonant coupling between neighboring nanostructures, especially at high numerical apertures (NA). Therefore, in order to solve the above problems, it is necessary to develop a new data processing (system). Summary of the Invention

[0003] This disclosure is based on the realization that rapid optimization of all unit cells simultaneously can improve the near-field distribution and performance of optical components, primarily by correcting interactions between neighboring cells, which improves the focusing efficiency of high NA metalens by 5-9%, as verified in high-resolution FDTD simulations.

[0004] Arrays of nanoscale waveguides have proven to be an attractive design motif for metalenses due to their high efficiency and simple design process in which the geometry of each waveguide is determined independently by Pancharatnam-Berry phase theory. We show that the phase delays are in fact not independent but provide corrections that lead to extremely high focusing efficiency.

[0005] Some embodiments of the present invention are based on the recognition that significant efforts have been made to discover classes of nanostructures that provide independent phase shifts as expected at the unit cell resolution. However, in practice, unexpected resonant coupling between neighboring nanostructures causes perturbations that move the realized phase retardation away from their intended values. While not fully understood, such coupling and the resulting perturbations have been identified as reducing the efficiency and overall performance of the metalens.

[0006] Some embodiments are based on the recognition that a computer-implemented method significantly enhances focusing efficiency by constructing a differentiable phase or electric field prediction function from simulations of a small neighborhood of the cell, and using this function to collectively optimize the realized phase delays of the overlapping neighborhood covering the entire metalens in Newton iterations. The main insights are that (1) a differentiable predictor of the electric field above the central cell in a small neighborhood can be constructed from simulation data, (2) this function can be roughly divided into functions for smaller overlapping neighborhoods, so that an exponential number of costly simulations are not required to accurately model the unknown physics, and (3) derivatives of this function can be used to obtain a (locally) optimal metalens design by solving a short series of band linear systems of equations for corrections to the nanostructure shape in the overlapping neighborhood covering the metalens.

[0007] Furthermore, because the data conversion system and computer-implemented method for data conversion according to embodiments of the present invention provide a special computational strategy for determining candidate tool locations, the data conversion system and computer-implemented method of the present invention can reduce central processing unit (CPU) usage, power consumption, and / or network bandwidth.

[0008] According to some embodiments of the invention, there is provided a computer-implemented method for designing a metalens. The metalens has a metasurface including nanostructures disposed on a substrate based on preliminary simulation data. The method employs a processor coupled to a memory, the memory storing a target near field profile, preliminary simulation data for the metalens, and instructions for implementing the method. The preliminary simulation data represents first design parameters including shapes and orientation angles for the nanostructures on the metalens. The instructions, when executed by the processor, perform the steps of the method. The method steps include dividing the nanostructures into overlapping d-unit supercells such that each nanostructure other than the periphery is at the center of one d-unit supercell; calculating a differentiable mapping function for the d-unit supercell that predicts the near field on a unit cell at the center of the supercell from design parameters of all the nanostructures in the supercell by fitting an interpolator to preliminary simulation data for the d-unit supercell; adjusting the design parameters of all unit cells in the metalens collectively by using a Jacobian of the partial derivatives provided by the mapping function to solve for a locally optimal correction of all design parameters to better approximate the target near field distribution throughout the metalens; and generating a manufacturable design of the metalens based on the optimized parameters.

[0009] The presently disclosed embodiments will be further described with reference to the accompanying drawings, in which: The drawings shown are not necessarily to scale, with emphasis instead generally being placed upon illustrating the principles of the presently disclosed embodiments. [Brief description of the drawings]

[0010] [Figure 1A] FIG. 1 shows a fragment of a metalens consisting of four oriented nanopillars on a substrate. [Figure 1B] FIG. 1 shows a top-down view of a metalens fragment divided into four equal-sized unit cells, three of which contain nanopillars, each nanopillar having a specified width, length, and orientation angle. [Figure 1C] FIG. 1 shows a side view of a metalens fragment, with the lower arrow indicating the incident planar electromagnetic wavefront and the upper straight lines representing the scattering pattern imparted by the nanopillars. [Figure 2A] FIG. 13 shows a top down view of a metalens section with two overlapping 3×3 neighborhoods of unit cells highlighted with grey squares, in accordance with an embodiment of the present invention. [Figure 2B] FIG. 13 illustrates how the average retardation provided by one unit cell changes as the nanopillar orientation angles of two of its neighboring cells are changed. [Figure 2C] FIG. 13 is a diagram of the near field phase provided by a high numerical aperture focusing metalens in accordance with some embodiments of the present invention. [Diagram 3] FIG. 2 illustrates a process for obtaining simulation data used to train a predictor according to some embodiments of the present invention. [Figure 4] FIG. 1 illustrates a metalens design system according to an embodiment of the present invention. [Diagram 5] 1 is a flowchart of a metalens design algorithm according to an embodiment of the present invention. [Figure 6A] FIG. 1 illustrates the error introduced when predicting the near-field phase delay across a unit cell by Pancharatnam-Berry phase theory (PB). [Figure 6B] FIG. 13 illustrates the error introduced when predicting the near-field phase delay across a unit cell by selecting the nearest three-cell supercell in a simulation data set. [Figure 6C] FIG. 13 illustrates the error introduced when predicting the near-field phase delay across a unit cell by averaging the two nearest 3-cell supercells in a simulation data set. [Figure 7A] FIG. 13 shows the error in predicting the near-field versus the central error of the 3×3 supercell near-field distribution of the PB theory. [Figure 7B] FIG. 13 illustrates the error introduced when using an interpolant trained on a 1×3 supercell. [Figure 7C] FIG. 13 illustrates the error introduced when averaging interpolations from 1×3 and 3×1 supercells. [Figure 7D] 4 shows the results of linear regression for these interpolations. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0011] While the above drawings illustrate embodiments disclosed herein, other embodiments are contemplated as set forth in the description. This disclosure presents illustrative embodiments by way of representation and not by way of limitation. Those skilled in the art can devise numerous other variations and embodiments which fall within the scope and spirit of the principles of the embodiments disclosed herein. [Detailed Description]

[0012] The following description provides exemplary embodiments only and is not intended to limit the scope, applicability, or configuration of the present disclosure. Rather, the following description of exemplary embodiments provides those skilled in the art with an enabling description for implementing one or more exemplary embodiments. It is contemplated that various changes may be made in the function and arrangement of elements without departing from the spirit and scope of the disclosed subject matter, as set forth in the appended claims.

[0013] In the following description, specific details are given for a thorough understanding of the embodiments. However, those skilled in the art can understand that the embodiments can be realized without these specific details. For example, systems, processes, and other elements in the disclosed subject matter may be shown as components in block diagram form to avoid obscuring the embodiments with unnecessary detail. In other examples, well-known processes, structures, and techniques may be shown without unnecessary detail to avoid obscuring the embodiments. Additionally, like reference numbers and names in the various drawings refer to like elements.

[0014] Also, particular embodiments may be described as a process that is depicted as a flowchart, a flow diagram, a data flow diagram, a structure diagram, or a block diagram. Although a flowchart may describe operations as a sequential process, many of the operations may be performed in parallel or simultaneously. Additionally, the order of operations may be rearranged. A process may terminate when its operations are completed, or may have additional steps not described or included in the diagram. Moreover, not all operations in any process specifically described may occur in all embodiments. A process may correspond to a method, a function, a procedure, a subroutine, a subprogram, or the like. When a process corresponds to a function, the end of the function may correspond to a return of the function to the calling function or to the main function.

[0015] Additionally, embodiments of the disclosed subject matter may be implemented at least partially either manually or automatically. Manual or automated implementations may be performed, or at least assisted, by use of machines, hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware, or microcode, the program code or code segments to perform the necessary tasks may be stored on a machine-readable medium. A processor may perform the necessary tasks.

[0016] Thus, some embodiments of the present disclosure can reduce the power consumption of a computer (processor) and improve the functionality of a computing system.

[0017] In some embodiments of the present disclosure, the system can be performed efficiently and accurately in less time and with less computing power, thus reducing central processing unit usage and power consumption through use of the methods or systems described in the present disclosure.

[0018] We present a fast bulk optimization of unit cells across an entire metasurface that exploits the interactions between neighboring cells to generate a target (desired) near-field profile and improve the overall device efficiency. Bulk optimization means simultaneously tuning all design parameters of the entire metasurface to maximize the overall performance of the device, taking into account the interactions between different parts of the device. This is in contrast to local or greedy optimization, which adjusts only one or a small subset of parameters to improve performance in one part of the metasurface, often at the expense of degrading performance elsewhere.

[0019] Unit cell decomposition (UCD) has become the most promising design strategy for optical metalenses. UCD considers metalenses as lattices of nanostructures that can be independently selected to control the near field. Lenses designed with UCD have shown steadily improving focusing efficiency, but theoretical predictions fall short (potentially replaced by freeform designs once simulation and fabrication challenges are resolved). One reason for the efficiency gap is that the electric field effects associated with individual nanostructures are often based on periodic assumptions (i.e., nanostructures tile a plane in the same way), but unmodeled coupling effects can substantially alter this response when placed in heterogeneous neighborhoods of diverse nanostructures. Attempts to correct this typically use extended supercell simulations in a generate-and-test loop to find alternative nanostructures that provide the target near field in the context of their neighborhood. These one-at-a-time search procedures have limited effectiveness and are computationally very expensive, even when pre-trained neural networks are used as simulator proxies.

[0020] Of note, most simulator proxies are distinguishable with respect to their inputs as well as their parameters, thus providing an as yet unexploited derivative for directly tuning unit cell parameters. More usefully, this allows for the optimization of all cells in a metalens collectively to generate a target (desired) near-field distribution by solving a short set of linear equations, each with a special structure that allows for linear time and space complexity. In this way, metalenses consisting of millions of cells can be quickly optimized with modest computing resources. We apply this to the design of a focusing TiO2 nanofin metalens previously designed by UCD following Pancharatnam-Berry (PB) phase theory to significantly improve the focusing efficiency.

[0021] Figure 1A shows a fragment of a metalens consisting of four oriented nanopillars on a substrate. Figure 1B shows a top view of the metalens fragment divided into four equal-sized unit cells, three of which contain nanopillars, each nanopillar having a specified width, length, and orientation angle. Figure 1C shows a side view of the metalens fragment, with the arrows below indicating the incident planar electromagnetic wavefront and the straight lines above representing the scattering pattern imparted by the nanopillars.

[0022] The nanofins 110 are typically a vast number of oriented rectangular pillars fabricated on a substrate 120 ( FIG. 1A ) arranged in a regular lattice 130 ( FIG. 1B ), where each pillar introduces a local phase delay such that a planar wavefront 140 ( FIG. 1C ) passing through the metalens is reshaped in an optically useful way, for example into a spherical wavefront that diverges (150) or converges to a point on the far-field focal plane.

[0023] FIG. 2A shows a top view of a metalens fragment with two overlapping 3×3 neighborhoods of unit cells highlighted with grey squares. FIG. 2B is a diagram illustrating how the average phase retardation provided by one unit cell changes as the nanopillar orientation angle of two of its neighboring cells is changed. FIG. 2C is a diagram of the near-field phase provided by a high numerical aperture focusing metalens, with dots 230 representing the desired ideal phase profile for efficient focusing. The grey jagged curve 240 shows the phase error incurred when using prior art techniques to design the metalens. The top straight line 250 shows the near-zero phase error incurred when using the method of the present invention.

[0024] FIG. 2A shows an example of a metalens design methodology, according to some embodiments of the present invention. The method is summarized as follows: (a) divide the metalens into closely overlapping neighborhoods 210 and 220, here 3×3 (FIG. 2A), each known as a supercell. (b) fit a differentiable function f to the supercell simulation data to predict the near-field response of the central cell in each neighborhood from the design parameters, e.g., nanofin orientation, of the neighborhood. As an example, FIG. 2B shows that the average right-hand circular polarization phase retardation in the central cell near field varies by more than 40° with the rotation of two adjacent nanofins. (c) Finally, the design parameters of all cells in the metalens are collectively optimized to produce the target near-field phase retardation pattern. FIG. 2C shows an example in which this process improves a poorly focused metalens to one that is perfectly focused (within physical limits) by adjusting the orientation of all nanofins so that the average phase shift of all cells matches the target value for a focusing lens, increasing the focusing efficiency by >5%.

[0025] PB theory predicts that the phase shift provided by a properly shaped waveguide is simply twice its planar orientation angle, i.e., f(θ)=2θ. As an example, Khorasaninejad et al. describe in Science an extended computational search in homogeneous locally periodic unit cell simulations to find nanofin shapes that provide a phase shift that follows PB theory, which also provides good transmission efficiency (Mohammadreza Khorasaninejad, Wei Ting Chen, Robert C. Devlin, Jaewon Oh, Alexander Y. Zhu, Federico Capasso, "Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging," Science 3 JUNE 2016, vol. 352 issue 6290).

[0026] Some embodiments of the present invention are based on the recognition that we have found that f has a nonlinear and dependence on neighboring nanofins that can result in phase errors of >±20° (see the jagged phase error curve in FIG. 2C), which is revealed in detailed finite-difference time-domain (FDTD) simulations of heterogeneous supercells. In addition, there are nanofin geometries that provide good transmission efficiency but have large deviations from ideal PB behavior. One such geometry is used here: 265×95×600 nm TiO2 nanofins on a SiO2 substrate in a 325 nm square grid. In general, we observe deviations from PB phase whenever the corners of neighboring nanofins are close together, exhibiting coupling effects, and when the phase shift varies substantially between adjacent fins, typically when a metalens provides nontrivial beam deflection. For example, in FIG. 2C, the PB phase error increases to >40° near the edges of the metalens, but the relative phase error between neighboring cells is not significant, resulting in modest beam deflection. This still leaves room for optimization.

[0027] 3 illustrates a process for obtaining simulation data used to train a predictor, according to some embodiments of the present invention. First, in step 321, general device parameters are selected, such as size, frequency band, and lattice spacing. In step 322, the size of the neighborhood is selected to be small enough to allow fast simulations, but large enough to capture the physical interactions between the central nanostructure and its neighboring nanostructures. In most cases, these interactions occur pairwise, and a 3×3 neighborhood is sufficient.

[0028] In step 323, nanopillar parameters to be modified are selected. Typically, these parameters govern the shape and / or placement of the nanopillar within its lattice cell. The process samples the space of possible near-field configurations in step 324. For the sampled near-field configurations of the metalens nanostructures, the process simulates electromagnetic field propagation by solving Maxwell's equations in step 325 and outputs electric field samples from the near-field scene beyond the metasurface in step 326.

[0029] 4 shows a block diagram of a metalens design system 400 according to some embodiments of the present disclosure. System 400 may include a human machine interface (HMI) 410 connectable to a keyboard 411 and a pointing device / medium 412, one or more processors 420, storage 430, memory 440, a network interface controller (NIC) 450 connectable to a network 490, including local area networks, wireless networks, and internet networks, a display interface 460 connected to a display device 465, and a printer interface 480 connectable to a printing device 485. The one or more processors 420 may hereinafter be referred to as processor 420 for convenience.

[0030] Memory 440 may be one or more memory units operating in conjunction with storage device 430 that stores computer executable programs (algorithm code) associated with processor 420. System 400 may receive input data, including preliminary simulation data, from a user 499 or from a previous simulation database 495 via a network 490 connected to NIC 450. NIC 450 includes a receiver and a transmitter for connecting to network 490 via wired and wireless networks (not shown). Once system 400 receives the input data, system 400 uses processor 420 and memory 440 to execute a metalens design algorithm 404 stored in storage device 430 using preliminary simulation data 402, target near field profile data 434, and a dynamic programming algorithm module to provide design parameters for the metalens. Storage device 430 may include algorithm module 408 as a computer implemented method configured to execute metalens design algorithm 404 using preliminary simulation data 402 and target near field profile data 434 including a target near field phase distribution.

[0031] 5 is a flowchart 500 illustrating the metalens design algorithm 404. The unknown true near field function f * A differentiable interpolant f(θ1, ..., θ d ) 510. The interpolant f is a function of the near-field prediction and the neighborhood design parameters θ1, ..., θ d ,θ, and the partial derivative of the local near-field value with respect to f. f is used to calculate the near-field error 520 of all overlapping neighborhoods on the metasurface. Choose a penalty function P() that is differentiable for the difference between the target near-field radiance distribution and the near-fields provided by the candidate metalens designs, and use the derivative of f to calculate the near-field error 520 of all design parameters Θ ≒ {θ, …, θ n},n>>d and its partial derivatives 530 with respect to n>>d. For example, P could be the squared error of the local phase delay integrated over the entire near field. The partial derivatives are represented as Jacobian matrix 540. During design, this Jacobian is used to minimize a penalty function through Newton iterations 550 that collectively optimize all design parameters 560. Assuming that the near field at any point depends on a limited neighborhood of the metalens, each Newton iteration reduces to solving a band system of linear equations, which can be done with O(n) time and space complexity. While all of the examples discussed herein, each containing thousands of cells, were fully optimized in less than a second, the time required to simulate any of these metalenses could be hours or days. In other words, a step is performed to calculate a differentiable mapping function of the d-unit supercell that predicts the near field on the unit cell at the center of the supercell from the design parameters of all the nanostructures in the supercell by fitting an interpolant to preliminary simulation data of the d-unit supercell. In this case, the calculating step continues until the scale of the corrections of the design parameters is below the manufacturing tolerances. Further, in this calculating step, an interpolation function may be used to predict the near field phase distribution across each of the unit cells, in which case the calculating step continues until the near field phase distribution matches a target near field phase distribution within a target error tolerance.

[0032] Furthermore, a d-unit supercell is a different combination of the orientation of a set of nanofins, and the interaction of each of the d-unit supercells with the incident wavefront is simulated by solving Maxwell's equations.

[0033] For example, FIG. 6A shows (from left to right) the far-field propagation intensity from a full FDTD simulation of a small NA=0.937 metalens designed with UCD using the PB phase rule, near-field phase optimization in accordance with the present invention (+7.8% focal efficiency), and optimization of both near-field phase and amplitude in accordance with the present invention (+8.1%). The optimization also results in a device with a better depth of focus. FIG. 6B shows the target phase distribution (thick black line 630) and the realized phase error of an NA=0.8 metalens optimized at near-field points every 10 nm. In this case, the initial phase error (high amplitude jagged line 640) is minimized (low amplitude thin black line 650) but is not zero because the target phase oscillates faster than the device physics can provide. Nevertheless, the far-field intensity distributions shown in FIG. 6C show improved focusing efficiency within 3% of the theoretical limit for the ideal case 660, the optimized case 670, and the PB case 680.

[0034] The metasurfaces described in this invention are arranged in a lattice of unit cells according to a regular tiling, which are usually rectangular or hexagonal, and less commonly triangular or phyllotactic patterns (e.g., the double helix pattern seen in sunflowers). In other words, the metasurfaces are arranged in a lattice of unit cells that divide a plane into rectangular, hexagonal, triangular, or phyllotactic shapes.

[0035] Each unit cell contains a photonic nanostructure on or in a substrate. Typically, the nanostructures are dielectrics such as SiO2 or waveguides such as TiO2. Typically, the planar dimensions of the cells are smaller than the operating wavelength of the device, and there are no such limitations on the vertical dimensions of the nanostructures. The neighborhood around the central unit cell can be defined as this cell plus any edge neighbors, optional corner neighbors, and possibly neighbors if the device physics supports longer range interactions. The figures show examples of TiO2 rectangular nanoprisms at various orientations within those cells.

[0036] The physical nanostructure specified inside each cell can be a rectangular nanoprism, an elliptical nanopillar, a split ring structure (e.g., a "c" shaped resonator), a hole in the substrate, an H-beam or I-beam shaped nanopillar, a nanopillar with a cross-shaped cross section, or a more complex shape. Typically, the nanostructures are formed of a single material, but they can also be layers of materials. Each nanostructure is described by a small number of design parameters that specify its shape and composition. The drawing shows an example where the design parameters are the orientation angle and two of the three rectangular prism dimensions.

[0037]

number

[0038] The above-described embodiments of the present disclosure can be implemented in any of numerous ways. For example, the embodiments may be implemented using hardware, software, or a combination thereof. The use of order terms such as "first," "second," and the like in the claims to modify claim elements does not, in itself, imply any priority, precedence, or ordering of a claim element relative to other elements, or the temporal order in which the actions of a method are performed, but is merely used as a label to distinguish a claim element having a particular name from other elements having the same name (except where order terms are used) to distinguish between the claim elements.

[0039] Although the disclosure has been described with reference to certain preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the disclosure. It is, therefore, the feature of the following claims to cover all such variations and modifications that come within the true spirit and scope of the disclosure.

Claims

1. 1. A computer-implemented method for designing a metalens having a metasurface including nanostructures disposed on a substrate based on preliminary simulation data, the method using a processor coupled to a memory, the memory storing a target near field profile, the preliminary simulation data for the metalens, and instructions for implementing the method, the preliminary simulation data representing design parameters including shapes and orientation angles for the nanostructures on the metalens, the instructions, when executed by the processor, perform steps of the method, the method steps including: Dividing the nanostructures into overlapping d-unit supercells such that each nanostructure other than the periphery is at the center of one d-unit supercell; calculating a differentiable mapping function for the d-unit supercell that predicts the near field on the unit cell at the center of the supercell from the design parameters of all nanostructures within the supercell by fitting an interpolant to preliminary simulation data of the d-unit supercell; adjusting the design parameters of all unit cells within the metalens collectively by using a Jacobian of partial derivatives provided by the mapping function to solve a locally optimal correction of all design parameters to better approximate a target near field distribution throughout the metalens; and generating a manufacturable design of the metalens based on the optimized parameters.

2. 10. The method of claim 1, wherein the metasurface is arranged in a lattice of unit cells that divide a plane into rectangular, hexagonal, triangular, or phyllotactic shapes.

3. 3. The method of claim 2, wherein the phyllotaxis is arranged in a double helix pattern.

4. The method of claim 1 , wherein the substrate is made of silicon dioxide.

5. The method of claim 1 , wherein the nanostructures are comprised of titanium dioxide.

6. The method of claim 1 , wherein the calculating step continues until the scale of the design parameter correction falls below a manufacturing tolerance.

7. 2. The method of claim 1, wherein the calculating step uses an interpolation function to predict a near field phase distribution across each of the unit cells.

8. The method of claim 7 , wherein the calculating step continues until the near field phase distribution matches a target near field phase distribution within a target error tolerance.

9. 2. The method of claim 1, wherein an interpolation is performed on a one-dimensional subset of the d-unit supercell to obtain an interpolated function, and the resulting interpolated function is averaged to provide a prediction and a gradient for a two-dimensional d-unit supercell.

10. The d-unit supercell is a set of nanofins with different combinations of orientations, The method of claim 1 , wherein the interaction of each of the d-unit supercells with an incident wavefront is simulated by solving Maxwell's equations.

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