Design of optical elements having a metasurface

Abstract

Optical elements and methods of manufacturing optical elements having a metasurface composed of meta-atoms are disclosed. The methods can include defining a lattice corresponding to a surface on which the optical element is to be formed. In an example, the design process allows at least some of the meta-atoms to freely move within the lattice or within their respective cells. An example optical element can have a metasurface composed of a non-periodic layout of meta-atoms.

Description

DESIGN OF OPTICAL ELEMENTS HAVING A METASURFACEFIELD OF THE DISCLOSURE

[0001] The present disclosure relates to the design of optical elements having a metasurface.BACKGROUND

[0002] Advanced optical elements may include a metasurface, which refers to a surface with distributed small structures (e.g., meta-atoms) arranged to interact with light in a particular manner. Metalenses, for example, are composed of carefully arranged meta- atoms (e.g., a distributed array of nanostructures) with sub-wavelength structures. By adjusting the geometry of the meta-atoms, one can modify the phase above the elements in response to a plane wave.

[0003] An initial step in designing a metalens can include defining the target phase profile for the metalens. In the case of a lens having a spherical or cylindrical shape, one can use an analytical formula to define the phase profile. In a more general case, it can be useful to represent the spatial phase data on a rectilinear grid. For example, a rectangular lattice can be used to construct the entire metalens using a square cell as a building block. The radius of each meta-atom on each grid point can be calculated and a corresponding structure added to each grid. That is, a meta-atom (e.g., a nanorod or pillar) with a desired phase is placed at the center of each cell in the grid. In this way, it is possible to create a metalens with a particular phase profile.SUMMARY

[0004] The present disclosure describes techniques for designing meta-optical elements (MOEs) such as a metalens, as well as resulting designs and structures for the optical elements.

[0005] For example, in one aspect, the present disclosure describes a method of manufacturing an optical element having a metasurface composed of meta-atoms. The method includes (a) defining a lattice corresponding to a surface on which the optical element is to be formed; (b) providing an initial layout for the meta-atoms in the lattice, wherein a respective meta-atom is assigned to each respective cell of the lattice; and (c) modifying at least one design parameter for the meta-atoms in an attempt to achieve a target optical functionality for the optical element based on a modified layout for the meta-atoms. Modifying at least one design parameter includes allowing at least some of the meta-atoms to freely move freely within the lattice. The method further includes (d) repeating (c) until the modified layout achieves an optical functionality that is within a specified range of the target optical functionality; and (e) fabricating the optical element, wherein the optical element has a layout for the meta-atoms as determined by the most recent performance of (d).

[0006] Some implementations include one or more of the following features. For example, in some implementations, allowing at least some of the meta-atoms to move freely within the lattice includes subjecting at least some of the meta-atoms to a constraint that at least a portion of the meta-atom remains within the respective cell to which the meta-atom is assigned. In some implementations, allowing at least some of the meta-atoms to move freely within the lattice includes subjecting at least a specified percentage of the meta-atoms to a constraint that at least a portion of the meta-atom remains within the respective cell to which the meta-atom is assigned. In some implementations, allowing at least some of the meta-atoms to move freely move within the lattice includes subjecting at least some of the meta-atoms to a constraint that a center of the meta-atom remains within the respective cell to which the meta-atom is assigned. In some implementations, allowing at least some of the meta-atoms to move freely move within the lattice includes subjecting at least a specified percentage of the meta-atoms to a constraint that a center of the meta-atom remains within the respective cell to which the meta-atom is assigned.

[0007] In some implementations, allowing at least some of the meta-atoms to move freely within the lattice includes subjecting at least some of the meta-atoms to a constraint that the meta-atom remains entirely within its respective cell. In some implementations, allowing at least some of the meta-atoms to move freely within the lattice includes subjecting at least a specified percentage of the meta-atoms to a constraint that the meta-atom remains entirely within its respective cell.

[0008] In some implementations, the optical element has a non-periodic layout of meta- atoms. In some implementations, at least some of the meta-atoms are offset from a center of the respective cell in which the meta-atom is located. In some implementations, the lattice has square cells, rectangular cells, or hexagonal cells. In some implementations, the optical element is a metalens.

[0009] The present disclosure also describes an apparatus including an optical element having a metasurface composed of a non-periodic layout of meta-atoms. In some implementations, an average density of the meta-atoms (i.e., number of meta-atoms per area) is non-uniform. In some implementations, the meta-atoms are arranged in a lattice, and an average density of the meta-atoms in the lattice is substantially uniform. For example, in some implementations, the meta-atoms are arranged in a square lattice composed of square cells, each of the meta-atoms being in a respective one of the square cells. In some implementations, the meta-atoms are arranged in a hexagonal lattice composed of hexagonal cells, each of the meta-atoms being in a respective one of the hexagonal cells. In some implementations, the meta-atoms are arranged in a rectangular lattice composed of rectangular cells, each of the meta-atoms being in a respective one of the rectangular cells.

[0010] Some implementations of the present disclosure can provide greater flexibility in the design of meta-optical elements.

[0011] Other aspects, features, and advantages will be readily apparent from the following detailed description, the accompanying drawings, and the claims.BRIEF DESCRIPTION OF THE DRAWINGS

[0012] FIG. 1 illustrates an example of lattice.

[0013] FIG. 2 illustrates an example of placement of meta-atoms in the lattice.

[0014] FIG. 3 illustrates another example of placement of meta-atoms in the lattice.

[0015] FIG. 4 illustrates placement of an individual meta-atom in the lattice.

[0016] FIG. 5 illustrates another example of placement of meta-atoms in the lattice.

[0017] FIG. 6 is a flow chart of an example method of designing an optical element having a metasurface composed of meta-atoms.

[0018] FIGS. 7 and 8 illustrate examples of MOEs having a non-periodic layout of meta- atoms.DETAILED DESCRIPTION

[0019] The present disclosure describes techniques for designing meta-optical elements such as metalenses. In some implementations, the techniques can provide a greater degree of freedom in the design phase, which can lead to a greater range of design options and, in some cases, to an improved meta-optical element (e.g., a metalens with improved optical performance). In accordance with the present disclosure, each meta- atom is not limited to being placed at the center of a cell in the grid. Instead, some or all of the meta-atoms may be offset from the center of the respective cell in which the meta- atom is located.

[0020] FIG. 1 illustrates an example of a lattice 20, such as a rectilinear grid, composed of square cells 22. In other cases, the cells of the lattice may have other shapes. For example, the lattice may be composed of rectangular or hexagonal cells. FIG. 2 illustrates an example design of a meta-optical element, such as a metalens, composed of meta-atoms 24, where each meta-atom is disposed within a respective cell 22 of the lattice 20. A lateral dimension (e.g., diameter) of the meta-atoms 24 may differ from one another. For example, if the meta-atoms are implemented as cylindrical pillars, some of the meta-atoms 24 may have a relatively small diameter, whereas other meta-atoms may have a relatively large diameter. Likewise, if the meta-atoms have a square cross-section, the length of a side of the meta-atoms may differ from the length of the side of other meta-atoms. The meta-atoms may, therefore, span a range of sizes and shapes. In the example of FIG. 2, the design of the metalens is constrained such that the center of each meta-atom 24 is located at the center of a respective one of the cells 22 of the lattice 20. Thus, the centers of the respective meta-atoms 24 are aligned, both horizontally and vertically, as indicated, for example, by the lines 26 and 28.

[0021] In accordance with the present disclosure, one or more constraints in the design of the metalens (or other meta-optical element) can be relaxed. For example, instead of requiring that each meta-atom 24 be located at the center of a respective one of the cells 22 of the lattice 20, the design process can allow at least some of the meta-atoms 24 to move freely within the lattice. For design processes in which at least some of the meta- atoms are permitted to move completely freely within the lattice, the density of meta- atoms in the resulting design may be non-uniform.

[0022] In some implementations, movement of the meta-atoms may be restricted to some extent. For example, in some cases, at least some of the meta-atoms are allowed to move freely within a respective cell 22 subject to the constraint that the meta-atom remains entirely within the cell. That is, the design rules may indicate that each meta-atom 24 be located entirely within a respective one of the cells 22 of the lattice 20 to which the meta- atom is allocated, but not necessarily that the center of the meta-atom coincide with the center of the cell. As before, another degree of freedom in the design process is providedby the ability to vary the size and shape (e.g., diameter or length of a side) of each metaatom.

[0023] FIG. 3 illustrates an example of a design in which each meta-atom 24 is located entirely within a respective cell 22 of the lattice 20. However, the center of the meta- atom is not necessarily located at the center of the cell 22. Some or all of the meta-atoms 24 may be offset from the center of the respective cell 22 in which the meta-atom is located. Thus, as indicated by the lines 30 and 32, the centers of the respective meta- atoms 24 need not be aligned with one another horizontally or vertically. In some instances, the centers of the respective meta-atoms 24 may be aligned vertically, but not horizontally, whereas in other instances, the centers of the respective meta-atoms 24 may be aligned horizontally, but not vertically. In some instances, as shown in FIG. 3, at least some of the centers of the respective meta-atoms are not aligned with one another horizontally and are not aligned with one another vertically. For example, the line 30 passing through meta-atoms in a particular row of the lattice does not pass through the center of each meta-atom in the row. Likewise, the line 32 passing through meta-atoms in a particular column of the lattice does not pass through the center of each meta-atom in the column. The foregoing techniques can result in a metalens or other meta-optical element having a non-periodic layout of meta-atoms 24. However, if movement of each meta-atom 24 is constrained such that the meta-atom remains bound to its respective assigned cell 22, the average density of meta-atoms in the lattice can, in some instances, be substantially uniform.

[0024] FIG. 4 illustrates an example of placement of an individual meta-atom 24 within a cell 22 of the lattice. In FIG. 4, P represents the lattice period, D represents the diameter of the meta-atom, (0,0) is the center of a particular cell 22 in the lattice, 4% represents the shift from the center (0,0) of the cell in the x direction (center-to-center), and 4y represents the shift from the center (0,0) of the cell in the y direction (center-to- center). In the example of FIG. 2, placement of a meta-atom 24 in the cell 22 is constrained such that 4% = 0 and 4y = 0. In contrast, in the example of FIG. 3, duringdesign of the metalens (or other meta-optical element), the foregoing constraint is loosened such that a meta-atom 24 can be freely placed within the cell 22 so long as:and

[0025] In some implementations, the design process for the metalens (or other meta- optical element) may be such that placement of each and every meta-atom that is allowed to move freely satisfies the foregoing constraints. In other implementations, the design process may simply indicate that at least a minimum specified percentage of the metaatoms that are allowed to move freely should satisfy the foregoing constraints.

[0026] In some implementations, a further constraint for designing the metalens (or other meta-optical element) can be relaxed. For example, instead of requiring that each meta- atom 24 be located entirely within a respective cell 22 of the lattice 20 to which the meta- atom is allocated, the design can allow at least some of the meta-atoms to move freely within the lattice so long as at least a portion of the meta-atom remains within the respective cell to which the meta-atom is allocated. In some instances, the design allows at least some of the meta-atoms 24 to move freely within their respective cells 22 so long as the center of the meta-atom remains within the cell to which the meta-atom is allocated. That is, the position of at least some of the meta-atoms 24 may be allowed to move freely, thereby resulting in at least some of the meta-atoms extending, respectively, into one or more adjacent cells 22. FIG. 5 illustrates an example of a design in which a meta-atom 24A is allocated to a cell 22A. The meta-atom 24A is located mostly in the cell 22A, but extends partially into an adjacent cell 22B. In some cases, a meta-atom (e.g., meta-atom 24C) may extend into more than one adjacent cell. On the other hand, in some cases, a meta-atom (e.g., meta-atom 24D) may be located entirely within a singlecell. In each case, the maximum deviation of the meta-atom from the center point of the corresponding cell can depend on various factors, including for example, the size of the meta-atom and the size of the lattice cell.

[0027] In some implementations, the design process for the metalens (or other meta- optical element) may be such that placement of each and every meta-atom satisfies the foregoing constraint (e.g., that placement of a meta-atom can be anywhere within the respective cell so long as at least a portion of the meta-atom remains within the cell or so long as the center of the meta-atom remains within the cell). In other implementations, the design process may simply indicate that at least a minimum specified percentage of the meta-atoms should satisfy the foregoing constraint. The foregoing techniques can result in a metalens or other meta-optical element having a non-periodic layout of meta- atoms 24. However, as each meta-atom 24 remains bound to a respective cell 22, the average density of meta-atoms in the lattice can, in some instances, still be substantially uniform.

[0028] FIG. 6 is a flow chart that illustrates an example of a process 100 for designing a metalens or other meta-optical element that includes meta-atoms. As indicated by 102, a lattice (e.g., a rectilinear or other grid) is defined for the substrate on which the metalens or other meta-optical element is to be formed. Then, as indicated by 104, an initial layout of meta-atoms in the lattice is established or provided. The initial layout may be based, for example, on past experience as to a reasonable starting point for achieving a target optical functionality for the metalens or other meta-optical element. In the initial layout, a respective meta-atom is allocated to each cell of the lattice. The initial layout can include, for example, a respective meta-atom of a given size disposed at the center of each cell of the lattice. Next, as indicated by 106, an optimizer modifies one or more design parameters for the meta-atoms in an attempt to achieve the target optical functionality. The design parameters that can be modified may include, for example, the size (e.g., diameter or length of a side) of each meta-atom, as well as its location within a given cell of the lattice. The optimizer may be implemented, for example, in software, firmware, hardware, or a combination of the foregoing.

[0029] As described above in connection with FIG. 3, in some implementations, each respective meta-atom is located entirely within a respective cell of the lattice, but the center of the meta-atom need not necessarily be located at the center of the of the cell. Further, as described above in connection with FIG. 4, in some implementations, the center of each respective meta-atom is located within a respective cell of the lattice, but a portion of the meta-atom may expand into one or more adjacent cells.

[0030] In some implementations, instead of allowing the positions of all the meta-atoms to move during the design process, at least a first specified minimum percentage of the meta-atoms are allowed to move. Further, in some implementations, in the design process, at least a second specified minimum percentage of the meta-atoms that are allowed to move are required to satisfy a specified constraint (e.g., that the meta-atom remain entirety within its designated cell; or that the center of the meta-atom remain within its designated cell).

[0031] The process 100 can be iterative. That is, after performing the operation at 106, the process can determine (at 108) whether the current design is within a specified range of the target functionality. If the current design is within the specified range of the target functionality, the process 100 can end. On the other hand, if the current design is not within the specified range of the target functionality, the process 100 can repeat operations 106 and 108 until the design is within a specified range of the target functionality. In some implementations, the process may end if the design does not come within the specified range of the target functionality after a given amount of time or after a specified maximum number of iterations.

[0032] The target functionality may depend on the particular implementation. In some cases, the target functionality includes a specified field-of-view (FOV) efficiency. In some cases, the target functionality includes a specified reduction in the 0thdiffractive order. In some cases, the target functionality includes a specified maximum coupling of diffractive orders.

[0033] The resulting metastructure design then can be used to manufacture one or more meta-optical elements (MOEs). In some cases, the metastructure design can be transferred, for example, to an ultraviolet (UV)-curable resin by replication techniques. In general, replication refers to a technique by means of which a given structure is reproduced, e.g., etching, embossing or molding. In an example of a replication process, a structured surface is embossed into a liquid or plastically deformable material (a “replication material”), then the material is hardened, e.g., by curing using ultraviolet (UV) radiation or heating, and then the structured surface is removed. Thus, a negative of the structured surface (a replica) is obtained. The replication material can be disposed on a glass or other substrate.

[0034] In some cases, a master tool is provided for production of the MOEs and, in some cases, may be part of a “generation process” that includes fabrication of a master tool, the subsequent fabrication of at least one replication tool from the master, and the subsequent fabrication of the MOEs. For industrial production, typically second or third generation replicas may be produced. One reason to introduce a generation process is to protect the relatively expensive original master.

[0035] After the master tool is fabricated based on the layout design, the master tool can be used to manufacture one or more (negative) sub-masters or replicas, which in turn can be used directly or indirectly to replicate MOEs, for example, as part of a mass production manufacturing process. Manufacturing the MOEs may take place in some instances at a wafer-level in which tens, hundreds, or even thousands of MOEs are replicated in parallel using the same sub-master or other tool derived from the master.

[0036] As an example, in some cases, manufacturing the MOEs includes providing a substrate (e.g., composed of silicon) having a polymeric layer on a surface of the substrate, forming openings in the polymeric layer, and depositing a material in the openings to form meta-atoms based on the metastructure design. Adjacent ones of the meta-atoms may be separated from one another by polymeric material of the polymericlayer. In some instances, the openings in the polymeric layer are formed by an imprinting process. The imprinting process can include, for example, pressing a stamp into the polymeric layer, and the method can include hardening the polymeric material before separating the stamp from the polymeric layer. In some cases, the meta-material is deposited in the openings by atomic layer deposition. In some instances, the material deposited in the openings to form the meta-atoms is titanium dioxide, although other materials may be used for the meta-atoms in some implementations.

[0037] In some cases, manufacture of the MOEs includes etching the meta-atoms into a stratum that is disposed on a substrate. The substrate can be composed, for example, of glass or fused silica, and the stratum can be composed, for example, of polysilicon, amorphous silicon, crystalline silicon, silicon nitride, zinc oxide, titanium oxide, aluminum zinc oxide, or a niobium oxide. In some cases, a mask, such as an organic (e.g., amorphous carbon) or inorganic (e.g., SiN, SiON, TiN) hardmask is disposed on the stratum to define where the stratum should be etched. The hardmask can be composed, for example, of metal, such as chrome, aluminum or titanium.

[0038] In some implementations, techniques other than, or in addition to, nano imprint lithography (NIL) or electronic beam lithography (EBL) can be used to manufacture the MOEs. Such other techniques can include, for example, deep ultraviolet (DUV) lithography or extreme ultraviolet (EUV) lithography techniques.

[0039] Each MOE, based on a metastructure design as described above (e.g., in connection with FIGS. 3-6), can include multiple meta-atoms forming a metasurface that has a specified optical phase profile. The metalens or other MOE can have a nonperiodic layout of meta-atoms. FIGS. 7 and 8 illustrate examples of MOEs having a nonperiodic layout of meta-atoms 124. The example MOE 120 of FIG. 7 is based on the design of FIG. 3, and the example MOE 122 of FIG. 8 is based on the design of FIG. 5. In some instances, the average density of the meta-atoms is non-uniform (e.g., where the meta-atoms are allowed to move completely freely during the design process). In some implementations, the meta-atoms are arranged in a lattice, and the average density ofmeta-atoms in the lattice is substantially uniform. The meta-atoms can be arranged, for example, in a square lattice composed of square cells, each of the meta-atoms being in a respective one of the square cells. In some implementations, the meta-atoms are arranged in a hexagonal lattice composed of hexagonal cells, each of the meta-atoms being in a respective one of the hexagonal cells. In some implementations, the meta-atoms are arranged in a rectangular lattice composed of rectangular cells, each of the meta-atoms being in a respective one of the rectangular cells.

[0040] Depending on the arrangement of the meta-atoms, the metastructure may function, for example, as a lens, a grating coupler, a fanout grating, a diffuser or other meta-optical element. In some implementations, the metasurfaces may perform other functions, including polarization control, negative refractive index transmission, beam deflection, vortex generation, polarization conversion, optical filtering, and / or plasmonic optical functions. Some implementations can help provide improved optical performance, such as higher efficiency, fewer unwanted signals, improved effectiveness of optical functionalities (e.g., polarization control), and / or the ability to employ wider field of views.

[0041] Various aspects of the subject matter and the functional operations described in this specification (e.g., operations described in connection with the process of FIG. 6) can be implemented in digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Thus, aspects of the subject matter described in this specification can be implemented as one or more computer program products, i.e., one or more modules of computer program instructions encoded on a computer readable medium for execution by, or to control the operation of, data processing apparatus. The computer readable medium can be a machine-readable storage device, a machine-readable storage substrate, a memory device, a composition of matter effecting a machine-readable propagated signal, or a combination of one or more of them. The apparatus can include, in addition to hardware, code that creates an executionenvironment for the computer program in question, e.g., code that constitutes processor firmware.

[0042] A computer program (also known as a program, software, software application, script, or code) can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program does not necessarily correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network.

[0043] The processes and logic flows described in this specification can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit).

[0044] Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive instructions and data from a read only memory or a random access memory or both. The essential elements of a computer are a processor for performing instructions and one or more memory devices for storing instructions and data. Computer readable media suitable for storing computer program instructions and data include all forms of non volatile memory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto optical disks; and CD ROM and DVD-ROM disks. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry.

[0045] While this specification contains many specifics, these should not be construed as limitations on the scope of the disclosure or of what may be claimed, but rather as descriptions of features specific to particular implementations. Certain features described in this specification in the context of separate implementations also may be combined in the same implementation. Conversely, various features described in the context of a single implementation also may be implemented in multiple implementations separately or in any suitable sub-combination. Various modifications can be made to the foregoing examples. Accordingly, other implementations also are within the scope of the claims.

Claims

What is claimed is:

1. A method of manufacturing an optical element having a metasurface composed of meta-atoms, the method comprising:(a) defining a lattice corresponding to a surface on which the optical element is to be formed;(b) providing an initial layout for the meta-atoms in the lattice, wherein a respective meta-atom is assigned to each respective cell of the lattice;(c) modifying at least one design parameter for the meta-atoms in an attempt to achieve a target optical functionality for the optical element based on a modified layout for the meta-atoms, wherein modifying at least one design parameter includes allowing at least some of the meta-atoms to move freely within the lattice;(d) repeating (c) until the modified layout achieves an optical functionality that is within a specified range of the target optical functionality; and(e) fabricating the optical element, wherein the optical element has a layout for the meta-atoms as determined by the most recent performance of (d).

2. The method of claim 1 wherein allowing at least some of the meta-atoms to move freely within the lattice includes subjecting at least some of the meta-atoms to a constraint that at least a portion of the meta-atom remains within the respective cell to which the meta-atom is assigned.

3. The method of claim 2 wherein allowing at least some of the meta-atoms to move freely within the lattice includes subjecting at least a specified percentage of the meta- atoms to a constraint that at least a portion of the meta-atom remains within the respective cell to which the meta-atom is assigned.

4. The method of claim 1 wherein allowing at least some of the meta-atoms to move freely within the lattice includes subjecting at least some of the meta-atoms to a constraint that a center of the meta-atom remains within the respective cell to which the meta-atom is assigned.

5. The method of claim 4 wherein allowing at least some of the meta-atoms to move freely within the lattice includes subjecting at least a specified percentage of the meta- atoms to a constraint that a center of the meta-atom remains within the respective cell to which the meta-atom is assigned.

6. The method of claim 1 wherein allowing at least some of the meta-atoms to move freely within the lattice includes subjecting at least some of the meta-atoms to a constraint that the meta-atom remains entirely within the respective cell to which the meta-atom is assigned.

7. The method of claim 6 wherein allowing at least some of the meta-atoms to move freely within the lattice includes subjecting at least a specified percentage of the meta- atoms to a constraint that the meta-atom remains entirely within the respective cell to which meta-atom is assigned.

8. The method of any one of claims 1-7 wherein fabricating the optical element includes fabricating an optical element having a non-periodic layout of meta-atoms.

9. The method of any one of claims 1-8 wherein at least some of the meta-atoms are offset from a center of the respective cell in which the meta-atom is located.

10. The method of any one of claims 1-9 wherein the lattice has square cells.

11. The method of any one of claims 1-9 wherein the lattice has rectangular cells.

12. The method of any one of claims 1-8 wherein the lattice has hexagonal cells.

13. The method of any one of claims 1-12 wherein the optical element is a metalens.

14. An apparatus comprising: an optical element having a metasurface composed of a non-periodic layout of meta-atoms.

15. The apparatus of claim 14 wherein an average density of the meta-atoms is non- uniform.

16. The apparatus of claim 14 wherein the meta-atoms are arranged in a lattice, and wherein an average density of the meta-atoms in the lattice is substantially uniform.

17. The apparatus of claim 16 wherein the meta-atoms are arranged in a square lattice composed of square cells, each of the meta-atoms being in a respective one of the square cells.

18. The apparatus of claim 16 wherein the meta-atoms are arranged in a hexagonal lattice composed of hexagonal cells, each of the meta-atoms being in a respective one of the hexagonal cells.

19. The apparatus of claim 16 wherein the meta-atoms are arranged in a rectangular lattice composed of rectangular cells, each of the meta-atoms being in a respective one of the rectangular cells.

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