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, providing precise control of wavefronts and potentially enabling substantial miniaturization in optoelectronics. In practice, metalenses have fallen far short of theoretically predicted efficiencies and currently offer no advantages 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 nanostructures that provide the appropriate phase retardation from a library of pre-simulated nanostructures. For nanostructures that exhibit retardation that follows the predictions of Pancharatnam-Berry (PB) theory, the design problem is particularly simple: the relative phase retardation is expected to vary linearly with the orientation of the nanostructure's rotation. However, in practice, metalenses designed based on the unit cell principle 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 recognizes 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. This improves the focusing efficiency of high-NA metalenses 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 not in fact independent but provide corrections that result in extremely high focusing efficiency.

[0005] Some embodiments of the present invention are based on the recognition that considerable effort has been expended to discover classes of nanostructures that provide predicted independent phase shifts at the unit cell resolution. However, in practice, unexpected resonant coupling between neighboring nanostructures causes perturbations that shift the realized phase retardation away from their intended values. While not fully understood, such coupling and the resulting perturbations have been observed to reduce the efficiency and overall performance of metalens.

[0006] Some embodiments are based on the recognition that a computer-implemented method significantly improves 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 an overlapping neighborhood covering the entire metalens in Newton iterations. The key 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, thereby avoiding the need for an exponential number of costly simulations 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 present invention, there is provided a computer-implemented method for designing a metalens. The metalens has a metasurface including nanostructures arranged on a substrate based on preliminary simulation data. The method uses 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. When executed by the processor, the instructions 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 the design parameters of all of 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 local optimum 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 are not necessarily to scale, emphasis instead generally being placed upon illustrating the principles of the presently disclosed embodiments. [Brief explanation of the drawings]

[0010] [Figure 1A] FIG. 1 shows a fragment of a metalens consisting of four oriented nanopillars on a substrate. [Figure 1B] Shown is a top-down view of a metalens fragment divided into four equal-sized unit cells, three of which contain nanopillars, each with a specified width, length, and orientation angle. [Figure 1C] Shown is a side view of a metalens fragment, with the lower arrow indicating the incident planar electromagnetic wavefront and the upper straight line representing the scattering pattern imparted by the nanopillars. [Figure 2A] FIG. 1 shows a top view of a metalens fragment with two overlapping 3×3 neighborhoods of unit cells highlighted with gray squares, in accordance with embodiments of the present invention. [Figure 2B] FIG. 10 illustrates how the average retardation provided by one unit cell varies as the nanopillar orientation angles of two of its neighboring cells are changed. [Figure 2C] FIG. 10 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. [Figure 3] FIG. 1 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. [Figure 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. 10 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. 10 illustrates the error introduced when predicting the near-field phase delay across a unit cell by averaging the two nearest three-cell supercells in a simulation data set. [Figure 7A] FIG. 10 shows the error in predicting the near field relative to the central error of the 3×3 supercell near-field distribution of the PB theory. [Figure 7B] FIG. 10 illustrates the error introduced when using an interpolator trained on a 1×3 supercell. [Figure 7C] FIG. 10 illustrates the error introduced when averaging interpolations from 1×3 and 3×1 supercells. [Figure 7D] 10A and 10B show the results of linear regression for these interpolations. DETAILED DESCRIPTION OF THE INVENTION

[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 explanation]

[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 will provide those skilled in the art with an enabling description for implementing one or more exemplary embodiments. Various changes are contemplated that 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 to provide a thorough understanding of the embodiments. However, those skilled in the art will understand that the embodiments may be practiced without these specific details. For example, systems, processes, and other elements of the disclosed subject matter may be shown as components in block diagram form in order to avoid obscuring the embodiments with unnecessary detail. In other instances, well-known processes, structures, and techniques may be shown without unnecessary detail in order to avoid obscuring the embodiments. Furthermore, like reference numbers and names in the various drawings indicate 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. While 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 specifically described process may occur in all embodiments. A process may correspond to a method, a function, a procedure, a subroutine, a subprogram, etc. 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 the main function.

[0015] Furthermore, embodiments of the disclosed subject matter may be implemented at least in part either manually or automatically. Manual or automated implementations may be performed, or at least assisted, by the 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, and therefore central processing unit usage and power consumption can be reduced by use of the methods or systems described in this disclosure.

[0018] We present a fast bulk optimization of unit cells across an entire metasurface that exploits interactions between neighboring cells to generate a target (desired) near-field profile and improve overall device efficiency. Bulk optimization refers to simultaneously tuning all design parameters across the entire metasurface to maximize the overall device performance, taking into account interactions between different parts of the device. This contrasts with 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 degraded 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 remain poor (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 a periodic assumption (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 discover alternative nanostructures that provide the desired near field in the context of that 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] Remarkably, most simulator proxies are distinguishable with respect to their inputs as well as their parameters, thus providing an unexploited derivative for directly tuning unit cell parameters. Even more powerfully, this allows us to generate a target (desired) near-field distribution by collectively optimizing all cells in a metalens 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 using modest computing resources. We apply this to the design of a focusing TiO nanofin metalens, previously designed by UCD according to Pancharatnam-Berry (PB) phase theory, to significantly improve focusing efficiency.

[0021] Figure 1A shows a metalens fragment consisting of four aligned nanopillars on a substrate. Figure 1B shows a top-down view of the metalens fragment divided into four equal-sized unit cells, three of which contain nanopillars, each with a specified width, length, and orientation angle. Figure 1C shows a side view of the metalens fragment, with the lower arrow indicating the incident planar electromagnetic wavefront and the upper line representing the scattering pattern imparted by the nanopillars.

[0022] Nanofins 110 are typically a large number of oriented rectangular pillars fabricated on a substrate 120 ( FIG. 1A ) arranged in a regular lattice 130 ( FIG. 1B ), with each pillar introducing a local phase delay so 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-down view of a metalens fragment, with two overlapping 3×3 neighborhoods of unit cells highlighted by gray squares. FIG. 2B illustrates how the average phase retardation provided by one unit cell varies 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 target (desired) ideal phase profile for efficient focusing. The jagged gray curve 240 represents the phase error resulting from using prior art techniques to design the metalens. The straight line at the top 250 represents the near-zero phase error resulting from using a method in accordance with the present invention.

[0024] FIG. 2A shows an example of a metalens design method 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 circularly polarized 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. In the example shown in FIG. 2C, this process refines a poorly focusing metalens to one that is perfectly focusing (within physical limits) by adjusting the orientation of all nanofins so that the average phase shift of all cells matches the target value for the focusing lens, increasing the focusing efficiency by >5%.

[0025] PB theory predicts that the phase shift introduced by a properly shaped waveguide is simply twice its planar orientation angle, i.e., f(θ) = 2θ. For example, in Science, Khorasaninejad et al. describe an extensive computational search in similar locally periodic unit cell simulations to find nanofin shapes that yield phase shifts that obey PB theory, while also providing 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 dependent relationship with neighboring nanofins that can result in phase errors of >±20° (see the jagged phase error curve in Figure 2C), as revealed in detailed finite-difference time-domain (FDTD) simulations of heterogeneous supercells. Additionally, nanofin geometries exist that offer excellent transmission efficiency but exhibit significant deviations from ideal PB behavior. One such geometry is used here: 265 × 95 × 600 nm TiO nanofins on a SiO substrate in a 325 nm square grid. Deviations from PB phase are typically observed whenever the corners of adjacent nanofins are close enough to exhibit coupling effects and when the phase shift varies substantially between adjacent fins, typically when a metalens provides nontrivial beam deflection. For example, in Figure 2C, the PB phase error increases to >40° near the edges of the metalens, while the relative phase error between neighboring cells is not significant enough to result 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 such as size, frequency band, and lattice spacing are selected. In step 322, a neighborhood size is selected that is small enough to allow fast simulation, but large enough to capture the physical interactions between a central nanostructure and its neighboring nanostructures. In most cases, these interactions occur pairwise, and a 3x3 neighborhood is sufficient.

[0028] In step 323, nanopillar parameters are selected to vary. Typically, these parameters govern the shape and / or placement of the nanopillar within its lattice cell. The process samples the space of possible nearby configurations in step 324. For the sampled nearby 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 processors 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 * is fitted to the simulation data 326 of the d-unit neighborhood supercell (d-unit supercell), d ) 510. The interpolator f is a function of the near-field prediction and the neighborhood design parameters θ1,...,θ d , θ ≈ {θ 1 ,…,θ 1 ,θ 2 ,θ 3 ,θ 4 ,θ 5 ,θ 6 ,θ 7 ,θ 8 ,θ 9 ,θ 10 ,θ 11 ,…,θ 12 ,θ 13 ,θ 14 ,θ 15 ,θ 16 ,θ 17 ,θ 18 ,θ 19 ,θ 20 ,θ 21 ,…,θ 22 ,θ 23 ,θ 24 ,θ 25 ,θ 26 ,θ 27 ,θ 28 ,θ 29 ,θ 30 ,θ 31 ,θ 32 ,θ 33 ,θ 34 ,θ 35 ,θ 36 ,θ 37 ,θ 38 ,θ 39 ,θ 40 ,θ 41 ,θ 42 ,θ 43 ,θ 44 ,θ 45 ,θ 46 ,θ 47 ,θ 48 ,θ 49 ,θ 50 ,θ 51 ,θ 52 ,θ 53 ,θ 54 ,θ 55 ,θ 56 ,θ 57 ,θ 58 ,θ 59 ,θ 60 ,θ 61 ,θ 62 ,θ 63 ,θ 64 ,θ 65 ,θ 66 ,θ 67 ,θ 68 ,θ 70 ,θ 71 ,θ 72 ,θ 73 ,θ 74 ,θ 75 ,θ 76 ,θ 77 ,θ 78 ,θ 79 ,θ 80 ,θ 81 ,θ 82 ,θ 83 ,θ 84 ,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 by a Jacobian matrix 540. During design, this Jacobian is used to minimize a penalty function through Newton iterations 550, which jointly optimize all design parameters 560. Assuming that the near field at any point depends on the 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 one second, the time required to simulate any of these metalenses could be hours or days. In other words, a step is performed in which a differentiable mapping function of the d-unit supercell that predicts the near field on a unit cell at the center of the supercell from the design parameters of all the nanostructures in the supercell is calculated 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 to 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, and in this 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 convention, 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 accommodate. Nevertheless, the far-field intensity distributions shown in Figure 6C show the improved focusing efficiency to 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 nanostructures 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 located on or within a substrate. The nanostructures are typically dielectrics such as SiO2 or waveguides such as TiO2. Typically, the planar dimensions of the cell are smaller than the operating wavelength of the device, and there are no such restrictions on the vertical dimensions of the nanostructure. The neighborhood centered 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- or I-beam-shaped nanopillar, a nanopillar with a cross-shaped cross section, or a more complex shape. Typically, nanostructures are formed from a single material, but they can also be layers of multiple 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," etc. in the claims to modify claim elements does not, by itself, imply any priority, precedence, or ordering of one claim element relative to another element, or the temporal order in which actions of a method are performed, but is merely used as a label to distinguish one claim element having a particular name from another element having the same name (except when order terms are used), thereby distinguishing the claim elements.

[0039] Although the present 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 employing 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, performing steps of the method, the method steps including: dividing the nanostructures into overlapping d-unit supercells such that each non-peripheral nanostructure is at the center of a single 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 the design parameters of all nanostructures within the supercell by fitting an interpolant to preliminary simulation data for the d-unit supercell; collectively adjusting the design parameters of all unit cells within the metalens by using a Jacobian of partial derivatives provided by the mapping function to solve for 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 formed in a double spiral 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 is below a manufacturing tolerance.

7. 10. The method of claim 1, wherein the calculating step uses an interpolation function to predict the 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 interpolation is performed on one-dimensional subsets of the d-unit supercell to obtain interpolated functions, and the resulting interpolated functions are averaged to provide a prediction and gradient for a two-dimensional d-unit supercell.

10. the d-unit supercells are different combinations of orientations of a set of nanofins; 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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