Design of optical elements with metasurfaces

Abstract

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

Description

Technical Field

[0001] This disclosure relates to the design of optical elements having metasurfaces. Background Technology

[0002] Advanced optical elements can include metasurfaces, which are surfaces with distributed small structures (e.g., superatoms) arranged to interact with light in a specific way. For example, a superlens is composed of carefully arranged superatoms with subwavelength structures (e.g., distributed arrays of nanostructures). By adjusting the geometry of the superatoms, the phase above the element can be modified in response to a plane wave.

[0003] The initial steps in designing a superlens may include defining the target phase distribution of the superlens. In the case of a lens with a spherical or cylindrical shape, analytical formulas can be used to define the phase distribution. More generally, it is useful to represent the spatial phase data on a linear grid. For example, a rectangular lattice can be used to construct the entire superlens by using square units as building blocks. The radius of each superatom at each grid point can be calculated, and the corresponding structure can be added to each grid. That is, superatoms with the desired phase (e.g., nanorods or nanopillars) are placed at the center of each unit cell in the grid. In this way, a superlens with a specific phase distribution can be created. Summary of the Invention

[0004] This disclosure describes techniques for designing meta-optical elements (MOEs) such as superlenses, and the resulting designs and structures for optical elements.

[0005] For example, in one aspect, this disclosure describes a method for manufacturing an optical element having a metasurface composed of superatoms. The method includes (a) defining a lattice corresponding to the surface of the optical element to be formed; (b) providing an initial layout of superatoms in the lattice, wherein corresponding superatoms are assigned to each corresponding unit cell of the lattice; and (c) modifying at least one design parameter of the superatoms to achieve a target optical function of the optical element based on the modified layout of the superatoms. Modifying at least one design parameter includes allowing at least some of the superatoms to move freely within the lattice. The method further includes (d) repeating (c) until the modified layout achieves an optical function within a specified range of the target optical function; and (e) manufacturing the optical element having a layout for the superatoms determined by the most recent execution result of (d).

[0006] Some implementations include one or more of the following features. For example, in some implementations, allowing at least some superatoms to move freely within the lattice includes constraining at least some of the superatoms so that at least a portion of the superatoms remains within the corresponding unit cell to which the superatoms are allocated. In some implementations, allowing at least some superatoms to move freely within the lattice includes constraining at least a specified percentage of the superatoms so that at least a portion of the superatoms remains within the corresponding unit cell to which the superatoms are allocated. In some implementations, allowing at least some superatoms to move freely within the lattice includes constraining at least a specified percentage of the superatoms so that the center of the superatoms remains within the corresponding unit cell to which the superatoms are allocated. In some implementations, allowing at least some superatoms to move freely within the lattice includes constraining at least a specified percentage of the superatoms so that the center of the superatoms remains within the corresponding unit cell to which the superatoms are allocated.

[0007] In some embodiments, allowing at least some superatoms to move freely within the lattice includes constraining at least some of the superatoms so that the superatoms are completely contained within their respective unit cells. In some embodiments, allowing at least some superatoms to move freely within the lattice includes constraining at least a specified percentage of the superatoms so that the superatoms are completely contained within their respective unit cells.

[0008] In some embodiments, the optical element has superatoms arranged in a non-periodic pattern. In some embodiments, at least some of the superatoms are offset from the center of their respective unit cells. In some embodiments, the lattice has a square unit cell, a rectangular unit cell, or a hexagonal unit cell. In some embodiments, the optical element is a superlens.

[0009] This disclosure also describes an apparatus comprising an optical element having a metasurface composed of superatoms in a non-periodic arrangement. In some embodiments, the average density of superatoms (i.e., the number of superatoms per unit area) is non-uniform. In some embodiments, the superatoms are arranged in a lattice, and the average density of superatoms in the lattice is substantially uniform. For example, in some embodiments, the superatoms are arranged in a square lattice composed of square unit cells, each superatom occupying a corresponding square unit cell within the square unit cells. In some embodiments, the superatoms are arranged in a hexagonal lattice composed of hexagonal unit cells, each superatom occupying a corresponding hexagonal unit cell within the hexagonal unit cells. In some embodiments, the superatoms are arranged in a rectangular lattice composed of rectangular unit cells, each superatom occupying a corresponding rectangular unit cell within the rectangular unit cells.

[0010] Some embodiments of this disclosure can provide greater flexibility in the design of super-optical elements.

[0011] Other aspects, features, and advantages will become apparent from the following detailed description, drawings, and claims. Attached Figure Description

[0012] Figure 1 An example of a crystal lattice is shown.

[0013] Figure 2 An example of superatoms placed in a crystal lattice is shown.

[0014] Figure 3 Another example of superatoms placed in a crystal lattice is shown.

[0015] Figure 4 An example is shown where a single superatom is placed in a crystal lattice.

[0016] Figure 5 Another example of superatoms placed in a crystal lattice is shown.

[0017] Figure 6 This is a flowchart of an example method for designing optical elements with metasurfaces composed of superatoms.

[0018] Figure 7 and Figure 8 An example of a MOE with superatoms having an aperiodic layout is shown. Detailed Implementation

[0019] This disclosure describes techniques for designing super-optical elements such as superlenses. In some embodiments, these techniques provide greater freedom during the design phase, enabling a wider range of design options and, in some cases, improved super-optical elements (e.g., superlenses with improved optical performance). According to this disclosure, each superatom is not limited to being placed at the center of the unit cell in the grid. Instead, some or all of the superatoms can be offset from the center of the corresponding unit cell where the superatom is located.

[0020] Figure 1 An example of a lattice 20, such as a linear grid, composed of square unit cells 22, is shown. In other cases, the unit cells of the lattice may have other shapes. For example, the lattice may be composed of rectangular or hexagonal unit cells. Figure 2An example design of a super-optical element, such as a superlens, composed of superatoms 24, is illustrated, where each superatom is arranged within a corresponding unit cell 22 of a lattice 20. The lateral dimensions (e.g., diameters) of the superatoms 24 can differ from one another. For example, if the superatoms are realized as cylindrical prisms, some superatoms 24 can have relatively small diameters, while others can have relatively large diameters. Similarly, if the superatoms have square cross-sections, the lengths of the sides of the superatoms can differ from the lengths of the sides of other superatoms. Superatoms can thus encompass a range of sizes and shapes. Figure 2 In the example, the constrained superlens is designed such that the center of each superatom 24 is located at the center of the corresponding cell in the cell 22 of the lattice 20. Thus, the centers of the corresponding superatoms 24 are horizontally and vertically aligned, for example, as indicated by lines 26 and 28.

[0021] According to this disclosure, one or more constraints in the design of a superlens (or other superoptical element) can be relaxed. For example, the design process can allow at least some superatoms 24 to move freely within the lattice without requiring each superatom 24 to be located at the center of a corresponding unit cell in the unit cell 22 of the lattice 20. For a design process that allows at least some superatoms to move completely freely within the lattice, the density of superatoms in the resulting design can be non-uniform.

[0022] In some implementations, the movement of superatoms is subject to certain restrictions. For example, in some cases, due to the constraint that superatoms are completely contained within the unit cell, at least some superatoms are allowed to move freely within their respective unit cells 22. That is, the design rules may instruct that each superatom 24 is completely located within the unit cell to which the superatom is assigned in the unit cell 22 of the lattice 20, but do not necessarily instruct that the center of the superatom coincides with the center of the unit cell. As before, another degree of freedom in the design process is provided by the ability to vary the size and shape (e.g., diameter or edge length) of each superatom.

[0023] Figure 3 An example design is illustrated where each superatom 24 is entirely contained within its corresponding unit cell 22 of the lattice 20. However, the center of the superatom is not necessarily located at the center of the unit cell 22. Some or all of the superatoms 24 may be offset from the center of their respective unit cells 22. Thus, as indicated by lines 30 and 32, the centers of the corresponding superatoms 24 do not need to be horizontally or vertically aligned with each other. In some instances, the centers of the corresponding superatoms 24 may be vertically aligned but not horizontally aligned; while in other instances, the centers of the corresponding superatoms 24 may be horizontally aligned but not vertically aligned. In some instances, such as Figure 3As shown, at least some centers of the respective superatoms are not horizontally aligned with each other, nor are they vertically aligned with each other. For example, line 30 passing through superatoms in a particular row of the lattice does not pass through the center of each superatom in that row. Similarly, line 32 passing through superatoms in a particular column of the lattice does not pass through the center of each superatom in that column. The aforementioned technique is capable of producing superlenses or other super-optical elements with superatoms 24 arranged in a non-periodic pattern. However, if the movement of each superatom 24 is constrained such that the superatoms remain bonded to their assigned corresponding unit cells 22, the average density of superatoms in the lattice can be substantially uniform in some instances.

[0024] Figure 4 An example is shown where a single superatom 24 is placed within a unit cell 22 of a crystal lattice. Figure 4 In this context, P represents the lattice period, D represents the diameter of the superatom, (0,0) is the center of a specific unit cell 22 in the lattice, φ represents the offset of the unit cell center (0,0) in the x-direction (center to center), and φ represents the offset of the unit cell center (0,0) in the y-direction (center to center). Figure 2 In the example, the placement of superatom 24 in unit cell 22 is constrained such that 𝛥𝑥=0 and 𝛥𝑦=0. In contrast, in Figure 3 In the example, during the design of the superlens (or other super-optical element), the aforementioned constraints are relaxed so that the superatom 24 can be freely placed within the unit cell 22 as long as the following conditions are met:

[0025]

[0026] and

[0027]

[0028] In some embodiments, the design process for a superlens (or other superoptical element) may ensure that the placement of each superatom that is allowed to move freely satisfies the aforementioned constraints. In other embodiments, the design process may only instruct that at least a minimum specified percentage of the superatoms that are allowed to move freely should satisfy the aforementioned constraints.

[0029] In some implementations, further constraints on the design of the superlens (or other superoptical elements) can be relaxed. For example, the design can allow at least some superatoms to move freely within the lattice, provided that at least a portion of the superatoms remains within their respective assigned unit cells, without requiring each superatom 24 to be completely located within its assigned unit cell 22. In some instances, the design allows at least some superatoms 24 to move freely within their respective unit cells 22, provided that the center of the superatom remains within its assigned unit cell. That is, the positions of at least some superatoms 24 can be allowed to move freely, resulting in at least some superatoms extending into one or more adjacent unit cells 22, respectively. Figure 5 An example design is illustrated in which superatoms 24A are assigned to unit cell 22A. Superatoms 24A are primarily located within unit cell 22A, but partially extend into adjacent unit cells 22B. In some cases, superatoms (e.g., superatoms 24C) may extend into more than one adjacent unit cell. On the other hand, in some cases, superatoms (e.g., superatoms 24D) may be entirely located within a single unit cell. In each case, the maximum deviation of the superatom from the center point of the corresponding unit cell can depend on various factors, including, for example, the size of the superatom and the size of the lattice unit cell.

[0030] In some embodiments, the design process for a superlens (or other superoptical element) can ensure that the placement of each superatom satisfies the aforementioned constraints (e.g., the superatom can be placed anywhere within the corresponding unit cell, provided that at least a portion of the superatom remains within the unit cell, or as long as the center of the superatom remains within the unit cell). In other embodiments, the design process can only instruct that at least a minimum specified percentage of the superatoms should satisfy the aforementioned constraints. The aforementioned techniques are capable of producing superlenses or other superoptical elements with a non-periodic arrangement of superatoms 24. However, since each superatom 24 remains bonded to the corresponding unit cell 22, in some instances, the average density of superatoms in the lattice can still be substantially uniform.

[0031] Figure 6This is a flowchart illustrating an example of a process 100 for designing a superlens or other superoptical element comprising superatoms. As indicated by 102, a lattice (e.g., a linear grid or other grid) is defined for the substrate to which the superlens or other superoptical element is to be formed. Then, as indicated by 104, an initial layout of the superatoms in the lattice is established or provided. The initial layout may be based, for example, prior experience regarding reasonable starting points for achieving the target optical function of the superlens or other superoptical element. In the initial layout, corresponding superatoms are assigned to each unit cell of the lattice. The initial layout may include, for example, corresponding superatoms of a given size arranged at the center of each unit cell of the lattice. Next, as indicated by 106, an optimizer modifies one or more design parameters of the superatoms to achieve the target optical function. The design parameters that may be modified may include, for example, the size of each superatom (e.g., diameter or edge length) and its position within a given unit cell of the lattice. The optimizer may be implemented, for example, in software, firmware, hardware, or a combination thereof.

[0032] As mentioned above Figure 3 In some embodiments, each corresponding superatom is entirely located within the corresponding unit cell of the lattice, but the center of the superatom is not necessarily located at the center of the unit cell. Furthermore, as described above... Figure 4 In some embodiments, the center of each corresponding superatom is located within the corresponding unit cell of the lattice, but a portion of the superatom may extend into one or more adjacent unit cells.

[0033] In some implementations, not all superatoms are allowed to move during the design process; instead, at least a first specified minimum percentage of superatoms are allowed to move. Further, in some implementations, during the design process, at least a second specified minimum percentage of the allowed-to-move superatoms are required to satisfy specified constraints (e.g., the superatoms remain entirely within their specified unit cells; or the centers of the superatoms remain within their specified unit cells).

[0034] 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 the specified range of the target function. If the current design is within the specified range of the target function, process 100 can terminate. On the other hand, if the current design is not within the specified range of the target function, process 100 can repeat operations 106 and 108 until the design is within the specified range of the target function. In some implementations, the process can terminate if the design does not reach the specified range of the target function after a given amount of time, or after a specified maximum number of iterations.

[0035] The target function can vary depending on the specific implementation. In some cases, the target function includes specified field-of-view (FOV) efficiency. In others, it includes specified reduction at the 0th diffraction order. In still others, it includes specified maximum coupling at the diffraction order.

[0036] The resulting metastructure design can then be used to fabricate one or more super-optical elements (MOEs). In some cases, the metastructure design can be transferred to a UV-cured resin using replication techniques. Generally, replication refers to a technique used to reproduce a given structure, such as etching, pressing, or molding. In an example of a replication process, a structured surface is pressed into a liquid or malleable material (“replication material”), which is then hardened, for example by using ultraviolet (UV) radiation or heat, and the structured surface is then removed. This yields a reversed form (replica) of the structured surface. The replication material can be placed on glass or other substrates.

[0037] In some cases, a master tool is provided for producing the MOE (Mean Exchange Entity), and in others, the master tool can be part of a "generation process" that includes the manufacture of the master tool, the subsequent manufacture of at least one copying tool derived from the master tool, and the subsequent manufacture of the MOE. For industrial production, second- or third-generation copies are typically produced. One reason for introducing the generation process is to protect the relatively expensive original master.

[0038] After a master tool is created based on a layout design, it can be used to manufacture one or more (reverse) master-daughter templates or replicas, which can then be used directly or indirectly to replicate MOEs, for example, as part of a mass production manufacturing process. In some instances, MOE manufacturing can be performed at the wafer level, where dozens, hundreds, or even thousands of MOEs are replicated in parallel using the same master-daughter templates or other tools derived from the master.

[0039] As an example, in some cases, fabricating a MOE includes providing a substrate (e.g., made of silicon) having a polymer layer on its surface; forming openings in the polymer layer; and depositing material in the openings to form superatoms based on a superstructure design. Adjacent superatoms can be separated from each other by the polymer material in the polymer layer. In some instances, the openings in the polymer layer are formed by an imprinting process. The imprinting process may include, for example, pressing an imprint into the polymer layer, and the method may include curing the polymer material before separating the imprint from the polymer layer. In some cases, the metamaterial is deposited in the openings by atomic layer deposition. In some instances, the material deposited in the openings to form superatoms is titanium dioxide, although in some embodiments, other materials may also be used for the superatoms.

[0040] In some cases, the fabrication of MOE involves etching superatoms into layers disposed on a substrate. The substrate may be made of, for example, glass or fused silica, and the layers may be made of, for example, polycrystalline silicon, amorphous silicon, crystalline silicon, silicon nitride, zinc oxide, titanium oxide, zinc aluminum oxide, or niobium oxide. In some cases, masks, such as organic (e.g., amorphous carbon) or inorganic (e.g., SiN, SiON, TiN) hard masks, are disposed on the layers to define the locations where the layers should be etched. The hard masks may be made of, for example, metals such as chromium, aluminum, or titanium.

[0041] In some implementations, other techniques besides nanoimprint lithography (NIL) or electronic beam lithography (EBL) can be used to fabricate MOEs. These other techniques may include, for example, deep ultraviolet (DUV) lithography or extreme ultraviolet (EUV) lithography.

[0042] Based on the above (for example, in combination) Figures 3 to 6 Each MOE in the superstructure design described herein may include multiple superatoms forming a metasurface with a specific optical phase distribution. Superlenses or other MOEs may have superatoms arranged in a non-periodic pattern. Figure 7 and Figure 8 An example of a MOE with a superatom 124 having an aperiodic layout is shown. Figure 7 Example MOE 120 based Figure 3 The design, and Figure 8 Example MOE 122 is based on Figure 5 The design involves various methods. In some cases, the average density of superatoms is non-uniform (e.g., where superatoms are allowed to move freely during the design process). In some embodiments, superatoms are arranged in a lattice, and the average density of superatoms within the lattice is substantially uniform. Superatoms can be arranged, for example, in a square lattice composed of square unit cells, with each superatom in a corresponding square unit cell. In some embodiments, superatoms are arranged in a hexagonal lattice composed of hexagonal unit cells, with each superatom in a corresponding hexagonal unit cell within a hexagonal unit cell. In some embodiments, superatoms are arranged in a rectangular lattice composed of rectangular unit cells, with each superatom in a corresponding rectangular unit cell within a rectangular unit cell.

[0043] Depending on the arrangement of the metaatoms, metastructures can be used as, for example, lenses, grating couplers, sector gratings, diffusers, or other meta-optical elements. In some embodiments, metasurfaces can perform other functions, including polarization control, negative refractive index transmission, beam deflection, vortex generation, polarization conversion, optical filtering, and / or plasmonic optics. Some embodiments can contribute to providing improved optical performance, such as higher efficiency, less unwanted signal, improved effectiveness of optical functions (e.g., polarization control), and / or the ability to employ a wider field of view.

[0044] The various aspects and functional operations of the subject matter described in this specification (e.g., in combination with...) Figure 6 The operations described in the process can be implemented in digital electronic circuits including the structures disclosed in this specification and their equivalents, or in computer software, firmware, or hardware including the structures disclosed in this specification and their equivalents, or in a combination of one or more of these. 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 operation by a data processing apparatus or for controlling the operation of a data processing apparatus. The computer-readable medium can be a machine-readable storage device, a machine-readable storage substrate, a memory device, a combination of substances that generate machine-readable propagating signals, or a combination of one or more of these. In addition to hardware, the apparatus may also include code that creates an operating environment for the associated computer program, such as code constituting processor firmware.

[0045] Computer programs (also known as programs, software, software applications, scripts, or code) can be written in any programming language, including compiled or interpreted languages, and can be deployed in any form, including as standalone programs or as modules, components, subroutines, or other units 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 as part of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), as a single file dedicated to the related program, or as multiple coordinating files (e.g., a file storing one or more modules, subroutines, or portions of code). Computer programs can be deployed to run on a single computer, or on multiple computers located in one location or distributed across multiple locations and interconnected via a communication network.

[0046] The processes and logic flows described in this specification can be executed by one or more programmable processors, which run one or more computer programs to perform functions by manipulating input data and generating output. These processes and logic flows can also be executed by special-purpose logic circuitry, such as field-programmable gate arrays (FPGAs) or application-specific integrated circuits (ASICs), and the device can also be implemented as special-purpose logic circuitry, such as an FPGA or an ASIC.

[0047] Processors suitable for running computer programs include, for example, general-purpose microprocessors and special-purpose microprocessors, as well as any one or more processors in any type of digital computer. Typically, a processor receives instructions and data from read-only memory or random access memory, or both. The basic components of a computer are a processor for executing 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, for example, semiconductor memory devices such as EPROM, EEPROM, and flash memory devices; magnetic disks such as internal hard disks or removable disks; magneto-optical disks; and CD-ROMs and DVD-ROMs. Processors and memory can be complemented by dedicated logic circuitry or integrated into dedicated logic circuitry.

[0048] While this specification contains numerous details, these details should not be construed as limiting the scope of this disclosure or any potentially claimed content, but rather as descriptions of features specific to particular embodiments. Certain features described in the context of separate embodiments may also be combined in the same embodiment. Conversely, various features described in the context of a single embodiment may be implemented individually or in any suitable sub-combination in multiple embodiments. Various modifications can be made to the foregoing examples. Accordingly, other embodiments are also within the scope of the claims.

Claims

1. A method for manufacturing an optical element having a metasurface composed of superatoms, the method comprising: (a) Defining a lattice corresponding to the surface to which the optical element is to be formed; (b) Providing an initial layout for the superatoms in the lattice, wherein the corresponding superatoms are assigned to each corresponding unit cell of the lattice; (c) Modify at least one design parameter of the superatoms to achieve a target optical function of the optical element based on a modified layout of the superatoms, wherein modifying at least one design parameter includes allowing at least some of the superatoms to move freely within the lattice; (d) Repeat (c) until the modified layout achieves the optical function within the specified range of the target optical function; (e) Manufacturing the optical element, wherein the optical element has a layout for the superatom determined by the latest execution result of (d).

2. The method according to claim 1, wherein, Allowing at least some of the superatoms to move freely within the lattice includes constraining at least some of the superatoms, wherein at least a portion of the superatoms is held within the corresponding unit cell to which the superatoms are assigned.

3. The method according to claim 2, wherein, Allowing at least some of the superatoms to move freely within the lattice includes constraining at least a specified percentage of the superatoms, the constraint being that at least a portion of the superatoms remain within the corresponding unit cell to which the superatoms are allocated.

4. The method according to claim 1, wherein, Allowing at least some of the superatoms to move freely within the lattice includes constraining at least some of the superatoms such that the center of the superatom remains within the corresponding unit cell to which the superatom is assigned.

5. The method according to claim 4, wherein, Allowing at least some of the superatoms to move freely within the lattice includes constraining at least a specified percentage of the superatoms, the constraint being that the center of the superatom remains within the corresponding unit cell to which the superatom is assigned.

6. The method according to claim 1, wherein, Allowing at least some of the superatoms to move freely within the lattice includes constraining at least some of the superatoms so that the superatoms are completely contained within the corresponding unit cell to which they are assigned.

7. The method according to claim 6, wherein, Allowing at least some of the superatoms to move freely within the lattice includes constraining at least a specified percentage of the superatoms, the constraint being that the superatoms remain completely within the corresponding unit cell to which the superatoms are allocated.

8. The method according to any one of claims 1 to 7, wherein, Manufacturing the optical element includes manufacturing an optical element with a superatomic arrangement of non-periodic structure.

9. The method according to any one of claims 1 to 8, wherein, At least some of the superatoms are offset from the center of the corresponding unit cell in which the superatom is located.

10. The method according to any one of claims 1 to 9, wherein, The lattice has a square unit cell.

11. The method according to any one of claims 1 to 9, wherein, The lattice has rectangular unit cells.

12. The method according to any one of claims 1 to 8, wherein, The lattice has hexagonal unit cells.

13. The method according to any one of claims 1 to 12, wherein, The optical element is a superlens.

14. An apparatus comprising: An optical element having a metasurface composed of superatoms arranged in a non-periodic pattern.

15. The apparatus according to claim 14, wherein, The average density of the superatoms is non-uniform.

16. The apparatus according to claim 14, wherein, The superatoms are arranged in a lattice, and the average density of the superatoms in the lattice is substantially uniform.

17. The apparatus according to claim 16, wherein, The superatoms are arranged in a square lattice composed of square unit cells, with each superatom residing in a corresponding square unit cell within the square unit cells.

18. The apparatus according to claim 16, wherein, The superatoms are arranged in a hexagonal lattice composed of hexagonal unit cells, with each superatom residing in a corresponding hexagonal unit cell within the hexagonal unit cells.

19. The apparatus according to claim 16, wherein, The superatoms are arranged in a rectangular lattice composed of rectangular unit cells, with each superatom residing in a corresponding rectangular unit cell within the rectangular unit cells.

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