Polymer layer stack and methods of forming thereof

Abstract

A layer stack includes a polymer layer including a plurality of regions with different refractive indices. The plurality of regions includes a first region including a first polymer composition having a first refractive index; a second region including a second polymer composition having a second refractive index different from the first refractive index; and a third region including a third polymer composition having a third refractive index different from the first and second refractive indices. The first polymer composition is different from the second polymer composition, and the first and second polymer compositions are different from the third polymer composition. The first refractive index is between 1.3 and 1.8, the second refractive index is between 1.3 and 1.8, and the third refractive index is between 1.3 and 1.8.

Description

POLYMERLAYERSTACK ANDMETHODS OFFORMINGTHEREOFTECHNICAL FIELD

[0001] This application relates generally to the field of polymer optics, and in particular embodiments, to polymer layer stack and methods of forming them. BACKGROUND

[0002] In conventional optics, optical elements such as lenses, mirrors, and prisms manipulate light based on their shapes and material properties. The fundamental principles relating the structures of these elements to their functions are the refraction and reflection of light as it passes through them.

[0003] Conventional optical elements are made of materials with uniform refractive index, such that curvature of the optical surfaces is required to produce changes in the direction of light, such as focusing, collimation, or beam shaping. The technical demands associated with forming high-quality optical surfaces and precise element shapes are considerable.

[0004] Moreover, surface curvature may produce or exacerbate spherical aberration, chromatic aberration, coma, and other aberrations that degrade image quality. Correcting aberrations typically requires the incorporation of additional elements into an optical system and fine control over their spacing, increasing the complexity and bulk of the resulting device.

[0005] Constraints on the complexity, weight, and size (both cross-sectional area and thickness) of optical systems to be integrated into a given device may pose considerable design challenges for conventional optics. For example, mobile devices, medical imaging systems, and augmented or virtual reality (AR / VR) systems may require compact, lightweight, and high-performance optics.

[0006] AR / VR systems may present additional challenges because of the short distance between a viewer and an integrated display. For example, gaps between pixels or subpixel elements may be visible, giving rise to a screen-door effect. Simply reducing the distances between these elements without adjusting the size or curvature of the optics may result in an increased divergence angle, a greater sensitivity to dispersion, and a correspondingly reduced efficiency in colormixing. Micro- or nanoscale adjustments in curvature that might compensate for such changes may be impracticable for technical or economic reasons. SUMMARY OF THE INVENTION

[0007] Technical advantages are generally achieved by embodiments of this disclosure, which describes a Polymer Layer Stack and Methods of Forming Thereof.

[0008] According to embodiments, a layer stack includes a polymer layer including a plurality of regions with different refractive indices. The plurality of regions includes a first region including a first polymer composition having a first refractive index; a second region including a second polymer composition having a second refractive index different from the first refractive index; and a third region including a third polymer composition having a third refractive index different from the first and second refractive indices. The first polymer composition is different from the second polymer composition, and the first and second polymer compositions are different from the third polymer composition. The first refractive index is between 1.3 and 1.8, the second refractive index is between 1.3 and 1.8, and the third refractive index is between 1.3 and 1.8.

[0009] Embodiments enable light bending without the need for curved surfaces and simplify integration with other components. Embodiments further provide advantageous access to a broad design space of refractive indices with sub-micron spatial precision by including size-controlled regions with varying polymer compositions (and corresponding refractive indices).

[0010] In some embodiments, the layer stack has a first edge and an opposite second edge, and the refractive index of the polymer layer varies non- monotonically and asymmetrically from the first edge to the second edge. Flexibility in designing the refractive index within the polymer layer and throughout the stack is an advantage of embodiments.

[0011] In some embodiments, the first region includes a first plurality of layers, and one of the first plurality of layers includes a different polymer composition than another of the first plurality of layers.

[0012] In some embodiments, a thickness of the polymer layer is between 0.02 μm and 50 μm.

[0013] In some embodiments, the layer stack is part of a metalens.

[0014] In some embodiments, the first refractive index is between 1.3 and 1.4, and the first polymer composition includes poly(hexafluoropropylene oxide) or poly(1H,1H-perfluorobutyl acrylate). In certain embodiments, the first polymer composition further includes a first metal, and a weight percentage of the first metal in the first polymer composition is selected to produce the first refractive index.

[0015] In some embodiments, the second refractive index is between 1.4 and 1.5, and the second polymer composition includes poly(methyl methacrylate), poly(vinyl acetate), or poly(vinyl alcohol). In certain embodiments, the second polymer composition further includes a second metal, and a weight percentage of the second metal in the second polymer composition is selected to produce the second refractive index.

[0016] In some embodiments, the third refractive index is between 1.5 and 1.6, and the third polymer composition includes poly(vinyl chloride) or poly(benzyl methacrylate); the third refractive index is between 1.6 and 1.7, and the third polymer composition includes poly(2,4,6-tribromobenzyl methacrylate) or poly(2-chlorostyrene); or the third refractive index is between 1.7 and 1.8, and the third polymer composition includes poly(pentabromophenyl methacrylate). In certain embodiments, the third polymer composition further includes a third metal, and a weight percentage of the third metal in the third polymer composition is selected to produce the third refractive index.

[0017] In some embodiments, the minimum meaningful difference between any of the first, the second, and the third refractive indices is 0.01.

[0018] According to embodiments, a method of forming a layer stack includes forming a first layer over a substrate, the first layer including a first polymer composition and having a first refractive index; patterning the first layer to form a first plurality of wells; filling the first plurality of wells in the first layer with asecond polymer composition having a second refractive index different from the first refractive index, the first polymer composition being different from the second polymer composition; forming a second plurality of wells in the first layer; and filling the second plurality of wells with a third polymer composition having a third refractive index different from the first and the second refractive indices, the first and second polymer compositions being different from the third polymer composition. Embodiments advantageously enable a planar process of fabricating layer stacks that bend light without curved surfaces and spanning a broad design space of refractive indices.

[0019] In some embodiments, the first, second, and third refractive indices are separately selected from a set of refractive indices between 1.3 and 1.8 with a step size of 0.01.

[0020] In some embodiments, the method further includes forming a second layer over the first layer, the second layer including a fourth polymer composition having a fourth refractive index selected from a set of refractive indices between 1.3 and 1.8 with a step size of 0.01; forming a third plurality of wells in the second layer; and filling the third plurality of wells with a fifth polymer composition having a fifth refractive index selected from a set of refractive indices between 1.3 and 1.8 with a step size of 0.01. In certain embodiments, the method further includes forming a fourth plurality of wells in the second layer; and filling the fourth plurality of wells with a sixth polymer composition having a sixth refractive index selected from the set of refractive indices between 1.3 and 1.8 with a step size of 0.01.

[0021] In some embodiments, forming the first layer includes preparing a first solution that includes a first monomer and a first solvent, and coating the substrate with the first solution.

[0022] In certain embodiments, the first monomer includes an acrylate, a methacrylate, an epoxide, a vinyl ether, a thiol, an alkene, a silyldichloride, an amine, a benzocyclobutene, or an ormosil. Monomer chemistries covering a wide range of molecular functionalities advantageously allow for tuning of the refractive index. Two or more monomer functionalities may be parts of a single,multifunctional first monomer that endows a corresponding polymer with advantageous properties.

[0023] In certain embodiments, the first solvent includes acetic acid, acetone, acetonitrile, benzene, 2-butanone, n-butyl acetate, carbon tetrachloride, chlorobenzene, chloroform, 1,2-dichloroethane, diethyl ether, diethylene glycol, dimethylformamide, 1,2-dimethoxyethane, dimethyl sulfoxide, 1,4-dioxane, ethyl acetate, ethyl benzene, ethyl lactate, hexamethylphosphoramide, hexamethylphosphorous triamide, isopropanol, 1-methyl-2-pyrrolidone, methylene chloride, propylene glycol methyl ether acetate, pyridine, tetrahydrofuran, triethylamine, or water. Additional solvent may be added in certain embodiments in order to tailor the properties of the first solution or to advantageously modulate the properties of the resulting polymer.

[0024] In certain embodiments, the first solution further includes a metal, where a weight percentage of the metal in the first solution is selected to crosslink with the first monomer and produce the first refractive index. In some of these embodiments, the metal includes zinc, iron, silver, gold, aluminum, platinum, lead, or titanium. In some of these embodiments, interactions between the metal and nearby side chains may be designed to tune the refractive index or to endow the polymer with other advantageous functionalities.

[0025] In certain embodiments, the first solution further includes inorganic particles, where a weight percentage of the inorganic particles in the first solution is selected to produce the first refractive index. In some of these embodiments, the inorganic particles include SiO2 nanoparticles, CuO nanoparticles, ZnO, TiO2, silicon, α-quartz, silicon oxynitride, ZnS, CdS, or ferrocene.

[0026] In certain embodiments, the first solution further includes a second monomer, and a weight percentage of the second monomer is selected to copolymerize with the first monomer and produce the first refractive index.

[0027] In certain embodiments, the first solution further includes a plasticizer, where the plasticizer is selected to produce the first refractive index. In some of these embodiments, the plasticizer includes dimethyl phthalate, dimethyl terephthalate, or disodium terephthalate.

[0028] In certain embodiments, the first solution further includes a photoinitiator. In some of these embodiments, the photoinitiator includes azobisisobutyronitrile, 2,2-dimethoxy-2-phenylacetophenone, 2- isopropylthioxanthone, benzoyl peroxide, phenylbis(2,4,6- trimethylbenzoyl)phosphine oxide, or camphorquinone.

[0029] In certain embodiments, coating the substrate with the first solution includes spin coating, spray coating, thermal spray coating, flow coating, or dip coating.

[0030] In some embodiments, patterning the first layer includes disposing a patterned mask over the first layer; forming a plurality of cured regions in the first layer by exposure to a first actinic radiation through the patterned mask, a remainder of the first layer including a plurality of uncured regions; and rinsing the substrate to remove the plurality of uncured regions.

[0031] In some embodiments, patterning the first layer includes forming a plurality of cured regions in the first layer by a maskless lithography technique, a remainder of the first layer including a plurality of uncured regions; and rinsing the substrate to remove the plurality of uncured regions. In certain embodiments, the maskless lithography technique includes stereolithography 3D printing or direct laser writing.

[0032] In some embodiments, forming the first layer includes forming an alignment marker, and forming the second layer includes aligning a lithography tool with the alignment marker. Including an alignment marker may advantageously help to ensure that any overlap between layers is intentional.

[0033] In some embodiments, the method further includes selecting the first refractive index from the set of refractive indices between 1.3 and 1.8 with a step size of 0.01, and selecting the second refractive index from the set of refractive indices between 1.3 and 1.8 with a step size of 0.01.

[0034] In some embodiments, the method further includes, in response to selecting the first refractive index to be between 1.3 and 1.4, selecting the first polymer composition to include poly(hexafluoropropylene oxide) or poly(1H,1H-perfluorobutyl acrylate); in response to selecting the first refractive index between 1.4 and 1.5, selecting the first polymer composition to include poly(methyl methacrylate), poly(vinyl acetate), or poly(vinyl alcohol); in response to selecting the first refractive index between 1.5 and 1.6, selecting the first polymer composition to comprise poly(vinyl chloride) or poly(benzyl methacrylate); in response to selecting the first refractive index between 1.6 and 1.7, selecting the first polymer composition to comprise poly(2,4,6- tribromobenzyl methacrylate) or poly(2-chlorostyrene); and in response to selecting the first refractive index between 1.7 and 1.8, selecting the first polymer composition to comprise poly(pentabromophenyl methacrylate).

[0035] In some embodiments, the method further includes, in response to selecting the first refractive index between 1.3 and 1.4, selecting a monomer for forming the first layer, the monomer being hexafluoropropylene oxide or 1H,1H- perfluorobutyl acrylate; in response to selecting the first refractive index between 1.4 and 1.5, selecting a monomer for forming the first layer, the monomer being methyl methacrylate or vinyl acetate; in response to selecting the first refractive index between 1.5 and 1.6, selecting a monomer for forming the first layer, the monomer being vinyl chloride or benzyl methacrylate; in response to selecting the first refractive index between 1.6 and 1.7, selecting a monomer for forming the first layer, the monomer being 2,4,6-tribromobenzyl methacrylate or 2- chlorostyrene; and in response to selecting the first refractive index between 1.7 and 1.8, selecting a monomer for forming the first layer, the monomer being pentabromophenyl methacrylate.

[0036] According to embodiments, a method of forming a layer stack includes coating a first organic solution onto a substrate; forming a first structure by exposing portions of the first organic solution to a first actinic radiation, the first structure including a first polymer composition having a first refractive index selected from a set of refractive indices between 1.3 and 1.8 with a step size of 0.01; developing the first organic solution after forming the first structure; coating a second organic solution onto the substrate and onto the first structure; forming a second structure by exposing portions of the second organic solution to a second actinic radiation, the second structure including a second polymer composition having a second refractive index selected from a set of refractiveindices between 1.3 and 1.8 with a step size of 0.01; and developing the second organic solution after forming the second structure. Embodiments advantageously enable a process of fabricating layer stacks that bend light without curved surfaces and that span a broad design space of refractive indices.

[0037] In some embodiments, the method further includes coating a third organic solution onto the substrate and onto the first structure and the second structure ; forming a third structure by exposing portions of the third organic solution to a third actinic radiation, the third structure including a third polymer composition having a third refractive index selected from the set of refractive indices between 1.3 and 1.8 with a step size of 0.01; and developing the third organic solution after forming the third structure.

[0038] In some embodiments, the first structure includes an alignment marker, and the method further includes aligning the first structure to a lithography tool before exposing portions of the second organic solution to the second actinic radiation. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] For a more complete understanding of the present disclosure, and the advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which:

[0040] Figures 1A–1C illustrate an example polymer layer comprising a plurality of regions having a variety of polymer compositions and corresponding refractive index values, wherein Figure 1A depicts the plurality of regions, Figure 1B plots the refractive index values along an axis of the polymer layer, and Figure 1C presents a side view of the polymer layer, according to various embodiments;

[0041] Figures 2A and 2B illustrate an example layer stack comprising a plurality of layers, one layer of the plurality having a different polymer composition and refractive index value from another layer of the plurality , wherein Figure 2A depicts the plurality of layers and Figure 2B plots the refractive index values along a depth axis of the layer stack, according to various embodiments;

[0042] Figure 3 depicts a metalens comprising a layer stack, according to embodiments;

[0043] Figures 4A–4F illustrate cross-sectional views of the formation of a layer stack, wherein Figure 4A depicts a substrate, Figure 4B illustrates a first layer comprising a first polymer composition formed over the substrate, Figure 4C shows formation of a first plurality of wells in the first layer, Figure 4D depicts a second polymer composition filling the first plurality of wells, Figure 4E shows formation of a second plurality of wells in the first layer, and Figure 4F depicts a third polymer composition filling the second plurality of wells, according to various embodiments;

[0044] Figures 5A–5F illustrate cross-sectional views of the formation of a layer stack, wherein Figure 5A depicts a first layer (which may be formed by the process of Figures 4A–4F) disposed over a substrate, Figure 5B illustrates a second layer comprising a fourth polymer composition formed over the first layer, Figure 5C shows formation of a third plurality of wells in the second layer, Figure 5D depicts a fifth polymer composition filling the third plurality of wells, Figure 5E shows formation of a fourth plurality of wells in the second layer, and Figure 5F depicts a sixth polymer composition filling the fourth plurality of wells, according to various embodiments;

[0045] Figures 6A–6F illustrate cross-sectional views of the formation of a layer stack, wherein Figure 6A depicts a first organic solution comprising a first polymer composition coated onto a substrate, Figure 6B shows the result of forming a first structure over the substrate and developing the first organic solution, Figure 6C depicts a second organic solution comprising a second polymer composition coated onto the first structure and the substrate, Figure 6D shows the result of forming a second structure over the substrate and developing the second organic solution, Figure 6E depicts a third organic solution comprising a third polymer composition coated onto the first structure, the second structure, and the substrate, and Figure 6F shows the result of forming a third structure over the substrate and developing the third organic solution, according to various embodiments;

[0046] Figures 7A–7C illustrate monomers that may be part of a solution producing a polymer composition of a polymer layer, according to various embodiments, wherein Figure 7A presents generic monomers with R groups, Figure 7B provides example monomer–polymer pairs, and Figure 7C depicts a polymerizable metal salt and a polymerizable metal complex;

[0047] Figures 8A and 8B illustrates solvents that may be part of a solution producing a polymer composition of a polymer layer, according to various embodiments;

[0048] Figure 9 illustrates plasticizers that may be part of a solution producing a polymer composition of a polymer layer, according to embodiments;

[0049] Figure 10 illustrates photoinitiators that may be part of a solution producing a polymer composition of a polymer layer, according to various embodiments;

[0050] Figure 11 depicts monomers and corresponding polymers that may be selected to produce a polymer composition in response to selecting a refractive index in a range between 1.3 and 1.8, according to various embodiments;

[0051] Figure 12 is a flow chart for a method of forming a layer stack, according to various embodiments; and

[0052] Figure 13 is a flow chart for another method of forming a layer stack, according to various embodiments.

[0053] Corresponding numerals and symbols in the different figures generally refer to corresponding parts unless otherwise indicated. The figures are drawn to clearly illustrate the relevant aspects of the embodiments and are not necessarily drawn to scale.DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

[0054] The making and using of embodiments of this disclosure are discussed in detail below. It should be appreciated, however, that the concepts disclosed herein can be embodied in a wide variety of specific contexts, and that the specific embodiments discussed herein are merely illustrative and do not serve to limit the scope of the claims. Further, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of this disclosure as defined by the appended claims.

[0055] Gradient refractive index (GRIN) optics are an approach to reducing the complexity, weight, and size of optical systems. Unlike conventional optics that rely on refraction at the curved surfaces of single-index materials to manipulate light, GRIN optics use materials with a refractive-index profile that varies smoothly parallel to the surface normal. This approach enables light bending without the need for curved surfaces, offering new possibilities for compact and efficient optical designs. Integration with other components may also be simplified considerably; precisely spacing or cementing flat components may be easier in general than positioning and aligning curved shapes.

[0056] One of the challenges in GRIN optics is to tune the refractive index of the underlying material across a wide range of values (a large design space) and with spatial precision. Embodiments of the disclosure described herein enable the production of optical components with the advantages of conventional GRIN optics (such as flat surfaces) while accessing a broader design space and achieving sub-micron precision. Embodiments of the present disclosure achieve these advantages by forming a polymer stack comprising size-controlled regions with varying polymer compositions (and corresponding refractive indices).

[0057] Embodiments described herein enable the fabrication of GRIN optics with a wide range of refractive index values (from about 1.3 to about 1.8), with precise tuning of the refractive index (variations of about 0.01), and with high spatial resolution (as low as 0.1 μm). In doing so, embodiments further enable fabrication of GRIN optics within a large and flexible design space defined by a diverse pool of polymer and copolymer compositions further expanded by the use of additives such as metals, solvents, plasticizers, and photoinitiators.

[0058] The optics and techniques disclosed thus offer an attractive and advantageous approach toward building compact, lightweight, easily assembled optical systems for advanced devices such as compact camera lenses, anti- reflection layers for organic LED displays, portable or wearable displays for AR and VR systems, ocular implants, and imaging assemblies for space telescopes.

[0059] To provide perspective on the embodiments described in this application, an example polymer layer will be described with reference to Figures 1A–1C. The example polymer layer may be part of a layer stack with tunable refractive index, consistent with various embodiments. Figures 2A–2B present an example layer stack, and Figure 3 illustrates a metalens comprising a layer stack, according to embodiments.

[0060] Methods of fabricating a layer stack according to various embodiments of the present disclosure will be described with reference to Figures 4A–4F, 5A–5F, and 6A–6F, all of which provide cross-sectional views of a layer stack being formed. The process flows illustrated may represent embodiments of more general methods of forming a layer stack, as described with reference to Figures 12 and 13.

[0061] Some embodiment methods may comprise the polymerization of specified polymer compositions. Figures 7A–7C present structures of various monomers— including generic monomers with R groups, embodiment monomer–polymer pairs, and polymerizable metal salts and complexes—that may be part of polymer compositions consistent with embodiments of the present disclosure. Figures 8A / 8B, 9, and 10 further illustrate various solvents, plasticizers, and photoinitiators consistent with embodiments. Figure 11 illustrates specific monomers that may be selected in response to the choice of a given range of refractive indices, together with polymer structures corresponding to those monomers.

[0062] Various embodiments of the present disclosure may comprise a layer stack. An embodiment layer stack may in turn comprise a polymer layer, as illustrated by the example polymer layer 10 in Figures 1A–1C.

[0063] Figure 1A depicts a top-down view of the polymer layer 10. According to various embodiments, the polymer layer 10 comprises a plurality of regions with different refractive indices. In particular, the polymer layer 10 comprises a first region 200, a second region 210, and a third region 220, each region comprising a corresponding polymer composition having a corresponding refractive index different from those of the other regions.

[0064] For example, the first region 200 comprises a first polymer composition 102 (as indicated by the dotted fill in Figure 1A). The first polymer composition 102 is different from both a second polymer composition 104 (dashed fill) of the second region 210 and a third polymer composition 106 (chevron fill) of the third region 220; moreover, the second polymer composition 104 and the third polymer composition 106 are different from each other.

[0065] Embodiment layer stacks may comprise one or more polymer layers with any number of distinct regions comprising at least three different polymer compositions, as may be desirable for a given use of an embodiment layer stack. As illustrated in Figure 1A, the polymer layer 10 comprises 170 regions with 5 distinct polymer compositions, including a fourth polymer composition 108 (striped fill) and a fifth polymer composition 110 (brick fill).

[0066] Each polymer composition may have a corresponding refractive index value. In particular, the first polymer composition 102 may have a first refractive index, the second polymer composition 104 may have a second refractive index, and the third polymer composition 106 may have a third refractive index. In embodiments, each refractive index may be between about 1.3 and about 1.8.

[0067] According to embodiments, a polymer layer of the layer stack may have an arbitrary cross-sectional shape. In some embodiments, the polymer layer 10 has a first edge 12 and an opposite second edge 14, such that the first edge 12 and the second edge 14 are parallel and aligned. For example, as illustrated in Figure 1A, the polymer layer 10 may be rectangular and have two pairs of opposite edges.

[0068] Figure 1B plots the refractive index n of the polymer layer 10 from the first edge 12 to the opposite second edge 14 along a representative axis 16. Accordingto embodiments, and as illustrated, the refractive index varies non-monotonically from the first edge 12 to the second edge 14, such that the refractive index may increase, decrease, or remain constant when crossing a boundary between adjacent regions of the polymer layer 10. In particular, in at least one such crossing, the refractive index decreases after having increased or increases after having decreased.

[0069] According to embodiments, and again as illustrated in Figure 1B, the refractive index also varies asymmetrically from the first edge 12 to the second edge 14. In particular, the plot in Figure 1B does not exhibit bilateral symmetry perpendicular to the representative axis 16. The combined non-monotonic and asymmetric profile of the refractive index in embodiment layer stacks represents a step beyond what may be achieved by conventional GRIN optics, which typically exhibit either a monotonic profile in refractive index (associated with a composition gradient from edge to edge along the reference axis) or a symmetric profile (associated with a composition gradient originating at the center of the reference axis).

[0070] Moreover, the refractive index may have a qualitatively different profile along axes parallel to the representative axis 16. Indeed, no reference axis chosen parallel to the representative axis 16 in the polymer layer and passing through the centers of a set of regions in Figure 1A would yield the same profile as in Figure 1B. In some embodiments, however, the patterning of the plurality of regions in Figure 1A may be chosen such that the refractive index profile is consistent across reference axes parallel to the representative axis 16.

[0071] As illustrated in Figure 1B, the distinct polymer compositions 102–110 have correspondingly distinct refractive index values. In some embodiments, the polymer layer 10 may instead comprise at least one pair of polymer compositions having the same refractive index. For example, it may be desirable to select a polymer composition for regions at the perimeter of the polymer layer 10 that may be more easily cut to shape, fitted into an optical housing or barrel, or otherwise mechanically processed than polymer compositions with the same refractive index value lying within the interior of the polymer layer 10. Suchchoices may enable more straightforward integration of the layer stack into larger devices.

[0072] To emphasize that the polymer layer 10 is three-dimensional and comprises a finite thickness as well as the cross-sectional area illustrated in Figure 1A, Figure 1C presents a side view of the polymer layer 10. A thickness of the polymer layer 10 may vary, and individual polymer layers within an embodiment layer stack need not have uniform thickness. According to various embodiments, a thickness of the polymer layer 10 may be between about 0.02 μm and about 50 μm.

[0073] As exemplified by the polymer layer 10 depicted in Figures 1A–1C, embodiments of the present disclosure enable the creation of complex, spatially varying refractive index profiles within a single polymer layer of a layer stack. In particular, the number of different patterns achievable in a single polymer layer comprising M regions and X compositions is XM; the corresponding number for a layer stack comprising N such layers is XMN; and when orderings of layers within the stack are also accounted for, the number rises even higher, to N!·XMN. Embodiments of the present disclosure advantageously enable precise control the refractive index profile within the polymer layer 10 through selection of any desired pattern from this vast design space.

[0074] Figure 2A provides a perspective view of an example layer stack 20 comprising a polymer layer comprising a plurality of regions, according to various embodiments. As illustrated, the example layer stack 20 comprises several additional polymer layers. A first region 200 may thus extend through the stack such that it comprises a first plurality of layers. According to embodiments, one of the first plurality of layers comprises a different polymer composition than another of the first plurality of layers. Indeed, in the particular example layer stack 20 illustrated in Figure 2A, only 2 of the 6 layers shown share a polymer composition.

[0075] Figure 2B plots the refractive index n of the example layer stack 20 from a lower layer with first polymer composition 102 to an upper layer with third polymer composition 106 along a representative direction 202. The refractive index values corresponding to the respective polymer compositions 102–110 areconsistent with those illustrated in Figures 1A / 1B. The refractive index profile again varies non-monotonically and asymmetrically, though neither property is required when passing from layer to layer of the layer stack 20. Flexibility in designing the refractive index of the layer stack 20 is a significant advantage enabled by embodiments.

[0076] Some advantages of embodiments described here may be illustrated by considering a region of a layer stack ρ = L1L2⋯LN with layers Li labeled by positive integer i ∈ [1, N] and having refractive indices ni. Given the layer thicknesses di, the total thickness of the region (and of the layer stack) may be d = Σi di.

[0077] When a ray moving through an ambient medium (refractive index n0) is incident on an outer surface of layer L1, it may subsequently propagate through the total thickness d and emerge on an opposite side of the layer stack, from the outer surface of layer LN. For purposes of this discussion, and without excluding the possibility that some embodiment layer stacks may be designed differently, the angle of incidence θ0 and the refractive indices n0 and {ni} may be chosen such that total internal reflection does not occur anywhere within the layer stack.

[0078] The ray may be incident on L1 at an angle θ0 relative to a surface normal. After propagating through the layers, the ray may exit the surface of LN at the same angle θ0 but shifted laterally by a displacement x = Σi xi. As long as the layers are large enough that the ray will not reach any side of the layer stack before reaching the interface between LN and the ambient medium, a single lateral coordinate with origin at the point of incidence may be used, without loss of generality.

[0079] That being the case, the ray’s path consists of N segments having lengths ℓi =^^^^+ ^^^and angles θi = arctan (xi / di). The total geometric path length is thus ℓcorresponding optical path length, which is the distance the ray would have to travel through vacuum to accumulate the same phase, is OPL = Σi ni ℓi.

[0080] An effective refractive index may help to characterize the region ρ. If the region were homogeneous, the ray would follow a straight-line path within it, forming a right triangle with legs d and x and hypotenuse (geometric path length)ℓh = √^^+ ^^≤ ℓ. The corresponding angle relative to the surface normal is θh = arctan (x / d). With that information, the refractive index of the homogeneous region nh could be determined directly from Snell’s law: nh = (n0 sin θ0 / sin θh) (Equation 1) But the corresponding optical path length, OPLh = nh ℓh, will only be equal to theexact OPL if the region is homogeneous.

[0081] A more general (and more accurate) effective refractive index neff may be obtained by requiring that the optical path lengths be equal: OPLeff = neff ℓh = Σi ni ℓi = OPL (Equation 2)

[0082] Useful forms of neff may be obtained that respectively depend on the segment lengths ℓi (and thus the thicknesses di and displacements xi), on the thicknesses and angles θi, and on the displacements and angles: neff = (Σi ni ℓi / ℓh) (Equation 3a) neff = ((Σi ni di / cos θi) / (d / cos θh)) (Equation 3b) neff = ((Σi ni xi / sin θi) / (x / sin θh)) (Equation 3c)

[0083] While these forms yield the same values for neff, the latter two forms also provide useful approximations. When all angles are relatively small (or, equivalently, when the layer refractive indices ni are relatively large), the cosines of Equation 3b may be approximated by 1, such that: neff ≈ (Σi ni di / d) (small angles) (Equation 4) In other words, the effective refractive index of the region may be approximated as a simple thickness-weighted average of the layer refractive indices.

[0084] Equation 3c may be rearranged slightly by noting that the individual layers may be homogeneous by construction, such that Snell’s law gives sin θi = (n0 sin θ0 / ni). Further invoking Equation 1 yields the (still exact): neff = ((Σi ni2xi / x) / ( n0 sin θ0 / sin θh)) = ((Σi ni2xi / x) / nh) (Equation 5) In other words, the effective refractive index may be found from the ratio between a displacement-weighted mean-square refractive index and the Snell’s law refractive index for the region.

[0085] When the region is very nearly homogeneous, nh ≈ neff allows further rearrangement:neff ≈^Σi ni2xi / x (nearly homogeneous) (Equation 6) Thus, in cases in which the region ρ is nearly homogeneous, the effective refractive index may be a displacement-weighted root-mean-square refractive index for the region. Some embodiments may be described well by this equation.

[0086] The expressions above apply equally well if dispersion is considered (i.e., ni → ni(λ), where λ is the wavelength of the ray under consideration) or in the limit of a large number of thin layers, when the sums may be converted into line integrals over the ray’s geometric path. Irrespective of whether a discrete or continuous form of the expressions is used, the effective refractive index will necessarily be in the range neff ∈ [nlow, nhigh], where nlow = min {ni} and nhigh = max {ni}. Essentially any value of neff within this range may be realized by a careful design of the region, and indeed may be realized by a variety of different designs, according to various embodiments.

[0087] An additional factor may enable embodiment layer stacks (such as the layer stack 20) to bend light (as well as displacing it) without themselves being curved: When regions in a layer stack have a spatial extent comparable to the total displacement x, a ray may encounter one or more internal interfaces between regions, thus being refracted in a plane perpendicular to that involved in the original refraction event.

[0088] As in conventional GRIN optics, the result may be that the ray bends, with an angle on exit into the ambient medium different from the angle on entrance, θout ≠ θ0. The ray may even (in some embodiments) exit from a surface of the layer stack perpendicular to the one on which it was originally incident. Embodiment layer stacks may thus allow enable focusing, collimation, shaping, and other modifications to an incident beam without the need for any surface curvature.

[0089] In some embodiments, layer stacks of the types depicted in Figures 1A–1C and 2A / 2B may be part of a metalens 300, as illustrated in Figure 3. Metalenses comprise a planar pattern of scattering elements (a metasurface) disposed over a base layer. The scattering elements may have sizes smaller than the wavelengths to be manipulated, such that (for example) a visible light metalens may comprisescattering elements with sizes on the order of tens of nanometers, according to embodiments.

[0090] As illustrated in Figure 3, in some embodiments the metasurface may be formed by patterning inorganic nanostructures 302 over a layer stack 301. The layer stack 301 may be the layer stack 10, the layer stack 20, or any other embodiment layer stack.

[0091] In conventional metalenses comprising metasurfaces fabricated by methods such as electron-beam lithography or nanoimprint lithography, the planar pattern may often be a binary pattern (such as that illustrated in Figure 3). A binary pattern may in turn produce a binary choice of phase shifts (δ+ or δ–) at a fixed number of building spots across the surface.

[0092] A phase shift δ± may have an equivalent effect to a change in refractive index. For example, a metalens of thickness d fabricated with a uniform base layer of refractive index n0 may have either of 2 effective refractive indices at a given building spot, neff = n0 + Δn±, where Δn± = (δ±λ / 2πd) (Equation 7) Such a metalens may therefore have 2Mpossible patterns.

[0093] In various embodiments, the metalens 300 comprising the layer stack 301 as its base layer may have regions of the layer stack 301 comparable in size to (and aligned with) the inorganic nanostructures 302, such that the number of building spots and the number of regions in a surface polymer layer may both be equal to M. For example, in an embodiment, regions in the layer stack 301 may be about 0.1 μm (about 100 nm) on a side, while the inorganic nanostructures 302 may be about 10 nm to about 50 nm long.

[0094] As a result, embodiment metalenses may have a multinary number of possible designs drawn from a space of size (2X)M. or (when the layer stack 301 comprises N layers) of size (2XN)M. This expanded range of possibilities exists even at comparable spatial resolutions as conventional metalenses, enabling more complex and efficient optical designs.

[0095] As discussed above with reference to Figure 1A, the refractive index of the first, second, and third polymer compositions 102–106 may be between about 1.3 and about 1.8, according to various embodiments. In some embodiments, the first refractive index of the first polymer composition 102 may be between about 1.3 and about 1.4, and the first polymer composition 102 may comprise (with reference to Figure 11) poly(hexafluoropropylene oxide) 1102 or poly(1H,1H- perfluorobutyl acrylate) 1106.

[0096] In some embodiments, the second refractive index of the second polymer composition 104 may be between about 1.4 and about 1.5, and the second polymer composition 104 may comprise poly(methyl methacrylate) 1110, poly(vinyl acetate) (1114A), or poly(vinyl alcohol) (1114B).

[0097] In some embodiments, the third refractive index of the third polymer composition 106 may be between about 1.5 and about 1.6, and the third polymer composition 106 may comprise poly(vinyl chloride) 1118 or poly(benzyl methacrylate) 1122. In other embodiments, the third refractive index may be between about 1.6 and about 1.7, and the third polymer composition 106 may comprise poly(2,4,6-tribromobenzyl methacrylate) 1126 or poly(2- chlorostyrene) 1130. In still other embodiments, the third refractive index may be between about 1.7 and about 1.8, and the third polymer composition 106 may comprise poly(pentabromophenyl methacrylate) 1134.

[0098] In various embodiments, any or all of the polymer compositions making up the plurality of regions may further comprise a metal, with each metal and each weight percentage thereof selected separately to produce the refractive index in the corresponding region. For example, the first polymer composition 102 may further comprise a first metal, with a weight percentage of the first metal selected to produce the first refractive index. As discussed in detail below, the first metal may be incorporated into the first polymer composition 102 in the form of metal counterions, polymerizable metal salts or complexes, metal fillers (in the form of inorganic particles), and the like. In respective embodiments, the second polymer composition 104 may further comprise a second metal; the third polymer composition 106 may further comprise a third metal; and so on, in any suitable combination.

[0099] According to various embodiments, the refractive indices of the polymer compositions making up the plurality of the regions may take any value between the lower value of about 1.3 and the upper value of about 1.8. Within this range, and according to embodiments, the minimum meaningful difference in refractive index may be 0.01, such that there are 51 distinct values of the refractive index . Thus, there may be on the order of 51Mpossible patterns of refractive index for a single polymer layer, according to embodiments.

[0100] Figures 4A–4F illustrate cross-sectional views of the formation of a layer stack according to embodiments of the present disclosure. In particular, these figures depict the formation of a polymer layer comprising several polymer compositions by a planar process.

[0101] Figure 4A depicts an incoming substrate 400. The substrate 400 may, in various embodiments, be or comprise another component in a larger optical system being fabricated that will include the layer stack. In some such embodiments, the substrate 400 may itself comprise a layer stack comprising one or more polymer layers and previously formed according to embodiments of the present disclosure; that is, the process flow depicted in Figures 4A–4F may add a polymer layer to an existing layer stack. In other embodiments, the substrate 400 may be or comprise a temporary support or sacrificial layer that may be removed during the process of fabrication (such as after the layer stack has sufficient mechanical strength to be manipulated without a support), during the process of integration (such as when the layer stack is cemented to or aligned with a component comprising a parallel surface), or at some other time after the process depicted in Figures 4A–4F. In still other embodiments, the substrate 400 may be a permanent support for the layer stack and may comprise any material suitable for that function.

[0102] In Figure 4B, a first layer 410 comprising a first polymer composition 402 is formed over the substrate 400. This layer formation may be achieved through any coating process compatible with the viscosity, flow characteristics, and other mechanical properties of the first polymer composition 402. In various embodiments, the coating method may be spin coating, spray coating, thermal spray coating, flow coating, or dip coating. In some embodiments, the firstlayer 410 may be allowed to dry for a period of time, with or without heating to accelerate the drying process.

[0103] Figure 4C illustrates the formation of a first plurality of wells 405 in the first layer 410. This patterning may be accomplished in several different ways.

[0104] According to various embodiments, patterning the first layer 410 may comprise three steps: First, a patterned mask is disposed over the first layer 410. Second, a plurality of cured regions 450 is formed in the first layer 410 by exposure to a first actinic radiation through the patterned mask. (A remainder of the first layer comprises a plurality of uncured regions.) Third, the substrate 400 is rinsed to remove the plurality of uncured regions, resulting in the workpiece as depicted in Figure 4C. disposing a patterned mask over the first layer 410.

[0105] In these embodiments, the first layer 410 may be coated onto the substrate 400 in the form of a solution comprising a solvent together with dissolved monomer molecules that may polymerize to form the first polymer composition 402. The formation of the plurality of cured regions 450 may thus be a photopolymerization, in which light is used to initiate and propagate the polymerization of monomers into a solid polymer. Because the plurality of cured regions 450 will be retained and the plurality of uncured regions will be removed, these embodiments comprise a negative-tone patterning technique.

[0106] In various embodiments, a critical dimension (smallest width) of the pattern to be formed in the first layer 410 may be about 0.1 μm. In some embodiments, the smallest area exposed through the mask may be a single region of the first layer 410. In some such embodiments, the patterning may form regions of any suitable cross-section, such as circles, ovals, squares, rectangles, other polygons, and the like, in any combination and arrangement within the first layer 410.

[0107] In certain embodiments, the patterning may form regions of a single cross- sectional shape capable of tiling the first layer 410, such as triangles, squares, rectangles, or hexagons. In other embodiments, the patterning may form regions with different cross sections (shape, size, or both) that are capable of tiling the first layer 410 in combination, such as hexagons and triangles (snub hexagontiling) or octagons and squares (truncated square tiling). In still other embodiments, the patterning may form regions capable of tiling the first layer 410 in an aperiodic fashion, such as in a Penrose tiling.

[0108] In various embodiments, the light used to effect the photopolymerization may be ultraviolet (UV) light. In some such embodiments, the patterning method may therefore be UV curing, with a UV light source shining through a mask disposed over the substrate 400. The wavelength (or, if not monochromatic, the spectrum) of the UV light source may comprise any wavelengths to which the monomers are photosensitive and that may initiate the polymerization, and the power of the UV light source may be set to any convenient wattage, according to various embodiments. In various embodiments, the UV light source may have a spectral maximum between about 100 nm and about 400 nm and may have a power above about 4 W. In some embodiments, the UV light source may have a spectral maximum at about 365 nm (the mercury i-line) and a power of about 35 W.

[0109] In other embodiments, the mask may be incorporated into a lithography tool. In these latter embodiments, the lithography tool may be configured to perform any suitable lithography technique, such as deep UV lithography (e.g., 193-nm deep UV or another wavelength between about 100 nm and about 300 nm), KrF lithography (e.g., 248-nm middle UV), i-line lithography (e.g., 365- nm near UV), or h-line lithography (e.g., 405-nm near UV or violet light).

[0110] In other embodiments, patterning the first layer 410 may comprise different steps: First, a plurality of cured regions 450 is formed in the first layer by a maskless lithography technique, such that a remainder of the first layer comprises a plurality of uncured regions. Second, the substrate 400 is rinsed to remove the plurality of uncured regions. In some embodiments, the rinsing liquid may be the same solvent as used to dissolve the monomers. In other embodiments, the rinsing liquid may be a different solvent miscible with the solvent used to dissolve the monomers.

[0111] In some embodiments, the maskless lithography technique used to form the first layer 410 may comprise stereolithography 3D printing at any of the above-mentioned wavelengths. In other embodiments, the maskless lithographytechnique may comprise direct laser writing to cure the polymer completely within the plurality of cured regions 450. In embodiments comprising direct laser writing, the wavelength of the light source (or the center of its spectrum) may be higher than for other patterning methods, reflecting the two-photon absorption mechanism that underpins the technique. For example, direct laser writing a material sensitive to 365-nm UV may be most effective using 730-nm red light.

[0112] In still other embodiments for forming the first layer 410, the first layer 410 may still be coated onto the substrate 400 in the form of a solution comprising a solvent together with dissolved monomer molecules. But rather than photopolymerizing or curing the solution according to a pattern, in these embodiments the solution may be polymerized to form a uniform film. The polymerization may be accomplished by any suitable means, such as by warming the substrate (if the solution is thermally polymerizable) or by flood exposure to an actinic radiation (with or without a photoinitiator), according to various embodiments. (In some embodiments, heating and photopolymerization may be combined.) The resulting first layer 410 may be patterned, such as by the deposition of a hard mask, and etched to form the first plurality of wells 405 by any suitable etching method, such as reactive ion etching. In some embodiments, the etching may be a timed etch.

[0113] In other embodiments not comprising a solution of monomers, the first layer 410 may have been coated onto the substrate 400 in the form of a solution comprising dissolved polymer strands with the first polymer composition 402. In such embodiments, the layer formation depicted in Figure 4B may amount to allowing solvent to evaporate from the coated solution (with or without heating the substrate). In such embodiments, the resulting first layer 410 may be a uniform film of the first polymer composition 402 that may be patterned, such as by deposition of a hard mask, and etched to form the first plurality of wells 405 by any suitable etching method, such as reactive ion etching.

[0114] As illustrated in Figure 4C, each well of the first plurality of wells 405 is 1.5 times a width (or critical dimension) of a remaining portion of the first layer 410. In various embodiments, the wells of the first plurality of wells 405 may have any width relative to the critical dimension, as long as such width is amenable tofabrication. In some embodiments comprising a plurality of regions with identical cross-section, each well of the first plurality of wells 405 may have width equal to the critical dimension.

[0115] Figure 4D depicts filling of the first plurality of wells 405 with a second polymer composition 406 that is different from the first polymer composition 402. The second polymer composition 406 further has a second refractive index different from the first refractive index of the first polymer composition 402. The filling process may be performed using coating methods similar to those mentioned above, according to various embodiments.

[0116] In Figure 4E, a second plurality of wells 407 is formed in the first layer 410. This step may use a similar patterning and development process as described in various embodiments with reference to Figure 4C, but according to a different pattern, such that the second plurality of wells 407 may be formed in previously exposed regions of the first layer 410 (as depicted), in previously unexposed regions of the first layer 410, or both.

[0117] Figure 4F depicts the result of filling the second plurality of wells 407 with a third polymer composition 412 that is different from both the first polymer composition 402 and the second polymer composition 406. Similarly, the third polymer composition 412 has a third refractive index different from both the first refractive index and the second refractive index. After filling, which may once again proceed by coating methods like those mentioned above, the third polymer composition 412 may (in some embodiments) be photopolymerized to set a final pattern. Alternatively, and according to various embodiments, additional rounds of patterning and filling of the first layer 410 may be performed until a desired pattern has been formed.

[0118] In some embodiments, the first, second, and third refractive indices may be separately selected from a set of refractive indices between about 1.3 and about 1.8 with a step size of about 0.01. There may therefore be about 51 distinct choices of refractive index in these embodiments. In embodiments drawing from the full design space of refractive index values, as many as 50 total rounds of patterning and filling may occur. The three-polymer design depicted in Figure 4F may have about 51·50·49 ≈ 125,000 possible three-index patterns.

[0119] In some embodiments, forming the first layer 410 may include forming an alignment marker (such as a crosshair or another registration symbol) that may be used to align tools used in further processing of the workpiece. In some embodiments, two or more alignment markers may be formed in the first layer 410. Irrespective of how many alignment markers may be present, they may be used to align a lithography tool as part of the process of forming a second layer 510 (with reference to Figure 5B) or subsequent layers, according to various embodiments.

[0120] Figures 5A–5F illustrate cross-sectional views of the formation of a layer stack according to embodiments of the present disclosure. In these figures, a polymer layer comprising several polymer compositions is formed by a planar process over a preexisting layer. In some embodiments, Figure 5A may be the same layer stack as in Figure 4F, such that Figures 4A–4F and 5A–5F represent a single process flow for the formation of a bilayer stack. Taken together, this series of figures demonstrates how the technique described in Figures 4A-4F can be extended to create bi- and multilayer structures with tunable refractive index profiles. In other embodiments, an incoming workpiece depicted in Figure 5A may be fabricated by a different process, such as that described below with reference to Figures 6A–6F.

[0121] In Figure 5A, the incoming workpiece comprises a substrate 400 (which may be a substrate as described above) with a first layer 410 disposed over it. The first layer 410 comprises a first polymer composition 402, a second polymer composition 406, and a third polymer composition 412, each of which is different from the others and has a different refractive index. In some embodiments, the associated first, second, and third refractive indices may be separately selected from a set of refractive indices between about 1.3 and about 1.8 with a step size of about 0.01.

[0122] In Figure 5B, a second layer 510 is formed over the first layer 410. The second layer comprises a fourth polymer composition 502 having a fourth refractive index selected from a set of refractive indices between about 1.3 and about 1.8 with a step size of about 0.01.

[0123] Figure 5C illustrates the formation of a third plurality of wells 505 in the second layer 510. This patterning may be accomplished using any of the coating and patterning techniques described above, according to various embodiments.

[0124] As illustrated in Figure 5C, each well of the third plurality of wells 505 is 1.5 times a width (or critical dimension) of a remaining portion of the second layer 510. In various embodiments, the wells of the third plurality of wells 505 may have any width relative to the critical dimension, as long as such width is amenable to fabrication. In some embodiments comprising two or more polymer layers with a plurality of regions with identical cross-section, each well of the third plurality of wells 505 may have width equal to the critical dimension.

[0125] Figure 5D depicts filling of the third plurality of wells 505 with a fifth polymer composition 504 having a fifth refractive index selected from a set of refractive indices between about 1.3 and about 1.8 with a step size of about 0.01. After filling, which may proceed by coating methods like those mentioned above, the fifth polymer composition 504 may (in some embodiments) be photopolymerized to set a final pattern. Alternatively, additional rounds of patterning and filling of the second layer 510 may be performed until a desired pattern has been formed.

[0126] Accordingly, and in various embodiments, a fourth plurality of wells 515 may be formed in the second layer 510, as depicted in Figure 5E. This step may use similar patterning processes as described above, but according to a different pattern, such that the fourth plurality of wells 515 may be formed in previously exposed regions of the second layer 510 (as depicted), in previously unexposed regions of the second layer 510, or both.

[0127] Figure 5F depicts the result of filling the fourth plurality of wells 515 with a sixth polymer composition 508 having a sixth refractive index selected from a set of refractive indices between about 1.3 and about 1.8 with a step size of about 0.01. After filling, which may once again proceed by coating methods like those mentioned above, the sixth polymer composition 508 may (in some embodiments) be photopolymerized to set a final pattern. Alternatively, additional rounds of patterning and filling of the second layer 510 may be performed until a target pattern has been formed. As many additional layers maybe formed as desired, simply by repeating the steps already described with reference to Figures 5A–5F.

[0128] Figures 6A–6F illustrate cross-sectional views of another method for forming a layer stack according to embodiments of the present disclosure. In Figure 6A, a first organic solution 550 is coated onto a substrate 400, which may be a substrate in the sense described previously. The coating method may be any of those described previously, according to various embodiments.

[0129] Figure 6B depicts the result of a two-step process: A first structure 451 is formed over the substrate 400 by exposing portions of the first organic solution 550 to a first actinic radiation. (This exposure step may comprise a maskless lithographic step such as stereolithography 3D printing or direct laser writing, according to embodiments.) The first structure 451 comprises a first polymer composition 402 having a first refractive index selected from a set of refractive indices between about 1.3 and about 1.8 with a step size of about 0.01. Unexposed portions of the first organic solution 550 may be developed from the surface after the first structure 451 has formed, leaving the workpiece as depicted in Figure 6B, with a first plurality of wells 405 having been formed in the first layer 410.

[0130] Next, in Figure 6C, a second organic solution 552 is coated onto the first structure 451, such that it covers both the first layer 410 and the exposed portions of the substrate 400. The coating method may again be any of those described previously, according to various embodiments.

[0131] Figure 6D depicts the result of another two-step process: A second structure 452 is formed by exposing portions of the second organic solution 552 to a second actinic radiation. (This exposure step may again comprise a maskless lithographic step such as stereolithography 3D printing or direct laser writing, according to embodiments.) In some embodiments (and as illustrated), the second structure 452, once formed, may cover only the portions of the substrate 400 exposed in Figure 6B. In other embodiments, the second structure 452 may cover the first structure 451 partially or entirely, such that portions of a multilayer structure sharing a second polymer composition 406 may be formed all at once rather than layer by layer.

[0132] The second structure 452 comprises a second polymer composition 406 having a second refractive index selected from a set of refractive indices between about 1.3 and about 1.8 with a step size of about 0.01. Unexposed portions of the second organic solution 552 may be developed from the surface after the second structure 452 has formed, leaving the workpiece as depicted in Figure 6D, with a second plurality of wells 407 having been formed in the first layer 410. In some embodiments not depicted, patterning of the first layer 410 may be performed such that every region in the first layer 410 is filled with either the first polymer composition 402 or the second polymer composition 406, with no gaps.

[0133] According to various embodiments, a third organic solution 554 may be coated onto the second structure 452 as shown in Figure 6E, such that it covers both the first layer 410 (comprising the first structure 451 and the second structure 452) and any remaining exposed portions of the substrate 400. The coating method may yet again be any of those described previously, according to various embodiments.

[0134] Figure 6F depicts the result of yet another two-step process: A third structure 453 is formed by exposing portions of the third organic solution 554 to a third actinic radiation. (This exposure step may once again comprise a maskless lithographic step such as stereolithography 3D printing or direct laser writing, according to embodiments.) In some embodiments (and as illustrated), the third structure 453, once formed, may cover only the portions of the substrate 400 exposed in Figure 6D. In other embodiments, the third structure 453 may partially or entirely cover the first structure 451, the second structure 452, or both, such that portions of a multilayer structure sharing a third polymer composition 412 may be formed all at once rather than layer by layer.

[0135] The third structure 453 comprises the third polymer composition 412 having a third refractive index selected from a set of refractive indices between about 1.3 and about 1.8 with a step size of about 0.01. Unexposed portions of the third organic solution 554 may be developed from the surface after the third structure 453 has formed, resulting in the workpiece depicted in Figure 6F.

[0136] In some embodiments, the first structure 451 may comprise an alignment marker (such as a crosshair or another registration symbol). In someembodiments, two or more alignment markers may be part of the first structure 451. Irrespective of how many alignment markers may be present, they may be used to align the first structure 451 to a lithography tool before exposing portions of the second organic solution 552 to the second actinic radiation. Explicitly aligning the workpiece to the lithography tool may advantageously help to ensure that any overlap between layers is intentional rather than a product of manufacturing imprecision. One or more additional alignment markers may be part of the second structure 452 or the third structure 453 and used to align with the lithography tool before subsequent exposure steps.

[0137] As already mentioned with reference to various embodiments, forming the first layer 410 may comprise preparing a first solution comprising a first monomer and a first solvent, then coating the substrate 400 with the first solution. We now provide additional details of embodiment chemistries, which cover a wide range of molecular functionalities and advantageously allow for further tuning of the refractive index.

[0138] In various embodiments, the first monomer may be any molecule capable of forming a polymer, irrespective of whether polymerization occurs by chain growth (addition), with or without a photoinitiator; by step growth (condensation); or under other conditions ,such as by reaction with an ionic initiator, with catalytic assistance from an enzyme, or in the presence of a plasma.

[0139] Some condensation polymers form by reactions between pairs of precursor molecules that do not necessarily polymerize in isolation, such that both molecules, one molecule, or neither molecule may be a monomer in the sense of also being the minimal repeat unit of the resulting polymer. (See Scheme 5 of Figure 7B for an example.) In various embodiments comprising a condensation polymer, the first monomer may be a molecule that undergoes condensation reaction with itself to form the condensation polymer; a minimal repeat unit of the condensation polymer formed by the condensation reaction between two precursor molecules; or a precursor molecule comprising a structural unit of the minimal repeat unit of the condensation polymer.

[0140] Various choices of the first monomer consistent with embodiments of the present disclosure are presented in Figure 7A. Unless explicitly excluded withreference to a given monomer, the organic R groups appearing throughout Figure 7A (R1–R24) may be selected freely from hydrogen atoms (H) or lighter halogen atoms (F, Cl, Br); organyl groups (univalent organic groups of any kind with an open valence at a carbon atom); or organoheteryl groups (having an open valence at a heteroatom such as oxygen, nitrogen, sulfur, phosphorus, silicon, and the like).

[0141] That the various monomer functionalities in Figure 7A will be discussed separately should not be construed to mean that they cannot coexist. Indeed, in some embodiments, two or more of the monomer functionalities discussed here may be parts of a single, multifunctional first monomer that endows the corresponding polymer with advantageous properties.

[0142] In various embodiments, the first monomer may be an acrylate 700 or a methacrylate 702, both of which comprise a vinyl group (H2C=CH–) bonded to the carbon of a carbonyl (C=O). In the methacrylate group, a methyl group (CH3– ) replaces the hydrogen atom on the proximate vinyl carbon.

[0143] As shown for several representative examples in Figure 11, acrylates and methacrylates may polymerize to yield polymers with a saturated carbon backbone and pendant side groups comprising a carboxylic acid (–(C=O)OH) or an ester (–(C=O)OR). The former cases correspond to choosing R1 or R2 to be a hydrogen atom to obtain (in the respective embodiments) acrylic acid or methacrylic acid, which polymerize to poly(acrylic acid) (PAA, nPAA = 1.53) and poly(methacrylic acid) (PMAA, nPMAA = 1.40). PAA and PMAA are sufficiently acidic to carry a net negative charge from loss of hydrogen atoms; electrical neutrality is typically achieved by preparing the polymer as a neutral salt with alkali metal cations such as Na+or K+.

[0144] An acrylate 700 or methacrylate 702 may also, in some embodiments, have R1 or R2 comprising a carbamate group (–R(C=O)O–NH2), such that the monomer may form a poly(urethane acrylate) or poly(urethane methacrylate). R1 and R2 may not generally be a halogen atom, however.

[0145] In other embodiments, the first monomer may be an epoxide 704. Choosing R3–R6 to be hydrogen atoms corresponds to ethylene oxide (C2H5O),which polymerizes to yield poly(ethylene glycol) (n = 1.46). Some embodiments may comprise glycidol derivatives, in which one of the groups R3–R6 is a hydroxymethyl group (–CH2OH), as discussed further below with reference to Scheme 4 of Figure 7B. Some embodiments comprising an epoxide 704 may be epoxy resins.

[0146] In some embodiments, the first monomer may be a vinyl ether 706. Common choices of R7 may be alkyl groups such as methyl (–CH3), ethyl (– CH2CH3), n-propyl (–CH2CH2CH3), etc., with the corresponding polymers being poly(alkyl vinyl ether)s such as poly(vinyl methyl ether) (n = 1.47).

[0147] Other embodiments may comprise a first monomer that is a thiol 708. While thiols will not typically polymerize in isolation, this monomer may undergo a thiol–ene reaction with an alkene 710 to form a thioether; for example, when groups R9–R11of the alkene 710 are chosen to be hydrogen atoms, the thiol–ene reaction may yield thioether R8S–CH2CH2R12. Because thiol–ene reactions often have near-quantitative yield (~100%), dithiols (HS–R–SH) may be advantageous first monomers for the formation of condensation polymers with alkenes, as discussed further below with reference to Scheme 5 of Figure 7B.

[0148] The first monomer may indeed be an alkene 710, in various embodiments. (Several of the other possible types of embodiment monomer correspond to very specific choices of R9–R12.) But no additional functionality need be present; for example, when R9–R12 are all chosen to be hydrogen atoms, yielding ethylene (C2H4), the corresponding polymer may be polyethylene (e.g., high-density polyethylene, n = 1.54).

[0149] Still other embodiments may comprise a first monomer that is a silyldichloride 712, which may be hydrolyzed to produce a dichlorosiloxane (ClRSi–O–SiR′Cl) that ultimately forms linear or cyclic silicone polymers. For example, choosing R13 and R14 to be methyl groups (–CH3) yields dimethyldichlorosilane, which hydrolyzes to dimethylsiloxane and may form polydimethylsiloxane (n = 1.40).

[0150] Some embodiments may comprise a first monomer that is an amine 714. Much like thiols, amines do not typically polymerize in isolation, but anamine 714 may react with molecules comprising an anhydride group (–(C=O)– O–(C=O)–) to yield an imide (–(C=O)–(N–R)–(C=O)–). For example, when R15 of the amine 714 is chosen to be a phenyl group, such that the amine 714 is aniline (C6H7N), the reaction may yield the corresponding phenylimide. Consequently, diamines (H2N–R–NH2) may be useful first monomers for the formation of condensation polymers with dianhydrides. In one such embodiment in which R15 is chosen to be a diphenylenyl ether (–C6H4–O–C6H4–), such that the amine 714 is 4,4’-oxydianiline, reaction with pyromellitic dianhydride (C6H2(C2O3)2) yields the well-known polyimide poly(4,4’-oxydiphenylene-pyromellitimide) (n = 1.70).

[0151] Other embodiments may comprise a first monomer that is a benzocyclobutene 716. One embodiment is 4-vinylbenzocyclobutene (VBCB), in which R17 or R18 is a vinyl group and all other R groups are hydrogen atoms. Poly(VBCB) is similar to polystyrene 728, except that the cyclobutene ring may undergo thermally or photochemically activated cycloaddition. In some instances, cycloaddition may form dicyclooctene crosslinks between polymer strands, advantageously stabilizing the polymer.

[0152] Still other embodiments may comprise a first monomer that is an ormosil (organically modified silicate) 718 derived from tetraethyl orthosilicate (TEOS). In some embodiments, R24 may be or comprise a vinyl group, yielding a vinyltriethylorthosilicate (VTES) that may be polymerized. In other embodiments, R24 may be an alkyl group or other nonpolymerizable unit; in these embodiments, ammonia may be used to crosslink ormosil molecules and form organically modified siloxanes.

[0153] Schemes 1–4 of Figure 7B illustrate several monomers that may be selected consistent with embodiments of the present application, together with the corresponding polymers. Each monomer may be selected for polymerization on its own to produce a target refractive index, in various embodiments, or may be selected for copolymerization with another monomer. In Scheme 5, two monomers that form a condensation polymer are presented together. Note that the schemes are read from top to bottom (as indicated by the “Monomer → Polymer” labeling 720) and numbered from left to right.

[0154] In Scheme 1, methyl acrylate 722 is shown polymerizing to poly(methyl acrylate) 724 (n = 1.48). Methyl acrylate 722 is an example of an acrylate 700, with R1 being a methyl group, in an embodiment.

[0155] In Scheme 2, styrene 726 is shown polymerizing to polystyrene 728 (n = 1.59). Styrene 726 is an example of an alkene 710 obtainable by several possible substitutions, such as by choosing R9 to be a phenyl group and R10–R12 to be hydrogen atoms, in an embodiment.

[0156] In Scheme 3, benzyl acrylate 730 is shown polymerizing to poly(benzyl acrylate) 732 (n = 1.57). Benzyl acrylate 730 is an example of an acrylate 700 with R1 being a benzyl group (–CH2–C6H5), in an embodiment.

[0157] In Scheme 4, phenyl glycidyl ether 734 is shown polymerizing to poly(phenyl glycidyl ether) 736 (n = 1.60). Phenyl glycidyl ether 734 is a glycidol derivative and an example of an epoxide 704 with R3–R5 being hydrogen atoms and R6 being a methylenyl phenyl ether (–CH2–O–C6H6), in an embodiment.

[0158] In Scheme 5, 1,3-propanedithiol 738 and diallyl phthalate 740 react by a thiol–ene reaction to form the minimal repeat unit of a thioether condensation polymer 742, in an embodiment.1,3-propanedithiol 738 is an example of a thiol 708, while diallyl phthalate 740 is an example of an alkene 710 (as well as a plasticizer related to those illustrated in Figure 9).

[0159] The various embodiments of the first monomer discussed above are all organic molecules, in the sense that they may comprise carbon–hydrogen bonds (C–H), carbon–halogen bonds (C–X, where X = F, Cl, Br, or I), or both. In many (but not all) embodiments, the first solution may correspondingly be an organic solution comprising an organic solvent, such as those depicted in Figures 8A and 8B. But in some embodiments comprising a first monomer with a dipole moment comparable to that of H2O (1.86 D)—or which may involve formation of the corresponding polymer or its minimal repeat unit by aqueous reactions such as hydrolysis—the first solvent may be water 856.

[0160] The solvents illustrated in Figures 8A and 8B are quite diverse, and their properties may not be listed exhaustively here. Even their dipole moments varywidely, from 0 (e.g., nonpolar molecules such as benzene 806 and carbon tetrachloride 812) to larger than that of water (e.g., dimethyl sulfoxide 828 at 4.1 D or hexamethylphosphoramide 838 at 5.38 D).

[0161] Figure 8A presents 16 possible embodiments of the first solvent: acetic acid 800, acetone 802, acetonitrile 804, benzene 806, 2-butanone 808, n-butyl acetate 810, carbon tetrachloride 812, chlorobenzene 814, chloroform 816, 1,2- dichloroethane 818, diethyl ether 820, diethylene glycol 822, dimethylformamide 824, 1,2-dimethoxyethane 826, dimethyl sulfoxide 828, and 1,4-dioxane 830.

[0162] In addition to water 856, Figure 8B presents another 12 possible embodiments of the first solvent: ethyl acetate 832, ethyl benzene 834, ethyl lactate 836, hexamethylphosphoramide 838, hexamethylphosphorous triamide 840, isopropanol 842, 1-methyl-2-pyrrolidone 844, methylene chloride 846, propylene glycol methyl ether acetate 848, pyridine 850, tetrahydrofuran 852, and triethylamine 854.

[0163] Whether alone (in some embodiments) or as part of a solvent system comprising one or more additional solvents (in others), the first solvent may be chosen for compatibility with the first monomer. In particular, the first solvent may be chosen such that the first monomer is straightforwardly soluble. Additional solvent may be added in some embodiments in order to tailor the properties of the first solution (such as polarity or viscosity) or to advantageously modulate the properties of the resulting polymer (such as the refractive index, tensile strength, glass transition temperature, and so on).

[0164] The first solution may comprise other components, in addition to the first monomer and the first solvent. According to various embodiments, these components may be one or more additional solvents (as just discussed); metals, in the form of metal counterions (as discussed above), polymerizable metal salts or complexes, or metal fillers (in the form of inorganic particles); one or more additional monomers that may copolymerize with the first monomer; plasticizers; and photoinitiators. Still other components may be included, in some embodiments.

[0165] As such, the first solution may further comprise a second monomer, wherein a weight percentage of the second monomer is selected to copolymerize with the first monomer and produce the first refractive index. In some such embodiments, any number of additional monomers may be selected, such that a resulting copolymer may comprise two, three, four, or more monomers.

[0166] Given a set of monomers labeled by integer k ≥ 1, the weight percentage of a specific monomer or polymerizable unit j may be written as: %wt of monomer j = (mj / Σk mk) (Equation 8) In other words, the weight percentage of monomer j is its mass fraction relative to the total mass of polymerizable units in the first solution. In some embodiments, the total mass may include the masses of polymerizable metal salts and complexes like those illustrated in Schemes 6 and 7 of Figure 7C.

[0167] By contrast, the mass of any solvent, metal fillers, plasticizers, photoinitiators, and any other component of the first solution will not contribute to the denominator of the weight percentage. The amounts of these additional components may nevertheless be expressed as a weight percentage so that definite weights need not be specified. For example, given a 100 g loading of various monomers and other polymerizable units, adding 0.1 wt% of another ingredient means adding 0.1 g of that ingredient to the first solution.

[0168] Copolymerization with judiciously chosen ratios of monomers may enable fine-tuning of the refractive index, transmittance, and other properties of a polymer composition used to form the first layer. In some embodiments, two monomers A and B comprising similar backbone structures and different side chains may yield a copolymer interpolating between the refractive indices nA and nB of the corresponding neat (single-monomer) polymers.

[0169] For example, copolymerizing methyl methacrylate (npoly(MMA) = 1.49) with benzyl acrylate 730 (nBpoly(BzA) = 1.57) may yield a copolymer with an intermediate refractive index, n ∈ [npoly(MMA), npoly(BzA)]. In one instance, the inventors mixed 50 wt% MMA with 50 wt% BzA and added 0.2 wt% of 2,2-dimethoxy-2- phenylacetophenone (DMPA) as a photoinitiator, yielding a polymer with intermediate refractive index 1.56. Copolymerizing methyl methacrylate with2,4,6-tribromophenyl acrylate (npoly(TBPA) = 1.70) may yield a copolymer with an intermediate refractive index from a correspondingly wider range.

[0170] In some embodiments, a metal salt or metal complex may be polymerized directly for the purpose of producing a polymer with a target refractive index, such that the salt or complex may be the first monomer. For example , Schemes 6 and 7 of Figure 7C respectively illustrate the polymerization of a metal salt and a metal complex to form crosslinked polymers.

[0171] In Scheme 6, zinc(II) acrylate 744 polymerizes to form poly(zinc(II) acrylate) 746. When zinc(II)acrylate 744 polymerizes on its own, it may produce highly branched and crosslinked polymers. In another embodiment, it may be used as a dopant (for example, in a first solution comprising poly(acrylic acid)) in order to achieve a targeted amount of crosslinking.

[0172] In Scheme 7, the vinyl groups of a tetrakis(2- vinylimidazole)dichloroiron(II) complex 748 polymerize to form a crosslinked polymer with a repeat unit 750. (The wavy lines on the majority of the imidazole rings in the repeat unit 750 represent additional polymer strands.) As with zinc(II) acrylate 744, the complex 748, when polymerized on its own, may produce highly branched and crosslinked polymers. In another embodiment, it may be used as a dopant (for example, in poly(2-vinylimidazole)) to achieve a targeted amount of crosslinking. In embodiments comprising a polymerizable metal complex, interactions between the metal and nearby side chains may be designed to tune the refractive index or to endow the polymer with other advantageous functionalities (such as self-healing).

[0173] In some embodiments, metal salts or complexes may instead be mixed with a neat (non–metal-containing) polymer or a partially polymerized first solution. Solid metal salt, melted metal salt, metal salt solution, or any combination or admixture thereof may be used to obtain a desired amount of crosslinking and to tune the refractive index.

[0174] In various embodiments, a weight percentage of the metal (and thus of a polymerizable metal salt or complex) may be selected to copolymerize with the first monomer and produce the first refractive index. In various embodiments,the metal may comprise zinc, iron, silver, gold, aluminum, platinum, lead, or titanium.

[0175] For example, poly(acrylic acid) has a refractive index of 1.53, and poly(methyl methacrylate) has a refractive index of 1.49. The inventors mixed 10 wt% zinc(II) acrylate 744 with 90 wt% methyl methacrylate and added 0.4 wt% of DMPA as a photoinitiator, yielding a polymer with a refractive index of 1.52.

[0176] In some embodiments, the first solution may comprise a metal filler. That is to say, in these embodiments, the first solution may comprise inorganic particles, wherein a weight percentage of the inorganic particles is selected to produce the first refractive index. In particular, the inorganic particles may be suspended or dissolved in the first solution, according to respective embodiments, such that the inorganic particles may be distributed evenly within the resulting polymer matrix.

[0177] In various embodiments, the inorganic particles may comprise silicon dioxide (SiO2) nanoparticles (n = 1.44), copper(II) oxide (CuO) nanoparticles (n = 2.63), zinc(II) oxide (ZnO) (n = 2.0), titanium dioxide (TiO2) (n = 2.61), silicon (n = 3.98), α-quartz (n = 1.6), silicon oxynitride (n ~ 1.73 when oxygen and nitrogen are in a 1:1 ratio), zinc(II) sulfide (ZnS) (n = 2.38), cadmium(II) sulfide (CdS) (n = 2.38), or ferrocene (Fe(C5H5)2) (n = 1.45). According to various embodiments, a higher-index filler may be used to raise the refractive index of a lower-index polymer (and vice versa).

[0178] According to various embodiments, the first solution may further comprise a plasticizer. During and after polymerization, plasticizers may be disposed between polymer chains, increasing the chains’ free volume and making the polymer more flexible, with a corresponding reduction in the glass-transition temperature. Figure 9 illustrates several plasticizers that may be part of the first solution, namely, dimethyl phthalate 900, dimethyl terephthalate 902, and disodium terephthalate 904, according to embodiments. Phthalates and terephthalates may enable improved flexibility in a polymer without significantly affecting its optical clarity. Other embodiments may employ plasticizers such as diallyl phthalate 740, dibutyl phthalate, dioctyl sebacate, triethyl citrate, or epoxidized soybean oil.

[0179] Some embodiment monomers may form condensation polymers, and others may form radicals (i.e., initiate a polymerization chain reaction) with a relatively low activation energy, such that thermal initiation may be an option. But in various embodiments, the first solution may further comprise a photoinitiator. For some embodiment choices of first monomer, polymerization may be slow or even negligible over timescales compatible with manufacturing throughput unless a photoinitiator is added. Including a photoinitiator in an embodiment may therefore allow a wider selection of polymer chemistries.

[0180] In various embodiments, a photoinitiator may be included in the first solution in response to the selection of a first monomer that undergoes radical polymerization, such as 2-chlorostyrene 1128. The photoinitiator may be a Norrish type I photoinitiator forming radicals by cleavage (such as an α- hydroxyketone or a phosphine oxide); a Norrish type II photoinitiator forming radicals by proton abstraction (such as a benzophenone or thioxanthone); a cationic photoinitiator sharing features of both Norrish types (such as an iodonium or sulfonium salt); or any other suitable photoinitiator. In embodiments comprising Norrish type II or cationic photoinitiators, the first solution may additionally comprise a co-initiator (proton donor) such as an ether, amine, alcohol, or thiol.

[0181] Figure 10 illustrates various photoinitiators consistent with embodiments of the present disclosure: azobisisobutyronitrile (AIBN) 1000, 2,2-dimethoxy-2- phenylacetophenone (DMPA) 1002, 2-isopropylthioxanthone (ITX) 1004, benzoyl peroxide 1006, phenylbis(2,4,6-trimethylbenzoyl)phosphine oxide 1008 (a bis(acyl) phosphine oxide or BAPO), and camphorquinone 1010. The BAPO 1008 and camphorquinone 1010 are type-II photoinitiators; the others are type I. ITX 1004 and camphorquinone 1010 may be selected in embodiments comprising thicker polymer layers, because they may absorb longer wavelength light (nearer to or even within the visible range).

[0182] In various embodiments, choosing a different photoinitiator for polymerization of a given monomer may yield polymers with slightly different properties owing to the formation of different polymer end groups. For example, polymers formed by reaction with AIBN 1000 may have cyano (–C≡N) andisopropyl (–CH(–CH3)–CH3) end groups, whereas those formed by reaction with benzoyl peroxide 1006 may have phenyl end groups. The presence of these different end groups enables further fine-tuning of the properties of the resulting material.

[0183] In various embodiments, selecting the first refractive index from a set of refractive indices between about 1.3 and about 1.8 with a step size of about 0.01 may comprise selecting a refractive index within one of several narrower ranges, namely: between about 1.3 and about 1.4; between about 1.4 and about 1.5; between about 1.5 and about 1.6; between about 1.6 and about 1.7; and between about 1.7 and about 1.8.

[0184] Figure 11 depicts monomers (and the corresponding polymers) that may be selected in response to a choice of refractive index within one of these narrower ranges, according to various embodiments. A labeled refractive index scale at the top of Figure 11 and vertical dividers within the figure may serve to illustrate the correspondence.

[0185] In response to selecting the first refractive index to be between about 1.3 and about 1.4, the first polymer composition may be chosen that is enriched in fluorine atoms, which tend to produce a lower refractive index. According to embodiments, the first polymer composition may specifically comprise poly(hexafluoropropylene oxide) 1102 or poly(1H,1H-perfluorobutyl acrylate) 1106. In other embodiments, a monomer may be selected for forming the first layer, the monomer being hexafluoropropylene oxide 1100 or 1H,1H- perfluorobutyl acrylate 1104. Fluorine-enriched polymers often have low refractive indices

[0186] In response to selecting the first refractive index to be between about 1.4 and about 1.5, the first polymer composition may be chosen to comprise poly(methyl methacrylate) 1110 or poly(vinyl acetate) 1114A, according to embodiments. In some embodiments, the first polymer composition may be chosen to comprise poly(vinyl alcohol) 1114B, which may be obtained by hydrolysis of poly(vinyl acetate) 1114A. In other embodiments, a monomer may be selected for forming the first layer, the monomer being methyl methacrylate 1108 or vinyl acetate 1112.

[0187] In response to selecting the first refractive index to be between about 1.5 and about 1.6, the first polymer composition may be chosen to comprise poly(vinyl chloride) 1118 or poly(benzyl methacrylate) 1122, according to embodiments. In other embodiments, a monomer may be selected for forming the first layer, the monomer being vinyl chloride 1116 or benzyl methacrylate 1120. Heavier halogen atoms (Cl, Br, I), aromatic rings, and heavier metals (Fe, Ag, Cr, etc.) often appear in molecules with higher refractive index.

[0188] In response to selecting the first refractive index to be between about 1.6 and about 1.7, the first polymer composition may be chosen to comprise poly(2,4,6-tribromobenzyl methacrylate) 1126 or poly(2-chlorostyrene) 1130, according to embodiments. (Note that these compositions combine heavier halogens with an aromatic ring.) In other embodiments, a monomer may be selected for forming the first layer, the monomer being 2,4,6-tribromobenzyl methacrylate 1124 or 2-chlorostyrene 1128.

[0189] In response to selecting the first refractive index to be between about 1.7 and about 1.8, the first polymer composition may be chosen to comprise poly(pentabromophenyl methacrylate) 1134, according to embodiments. (Note the perhalogenated aromatic ring.) In other embodiments, a monomer may be selected for forming the first layer, the monomer being pentabromophenyl methacrylate 1132.

[0190] The process flow described above with reference to Figures 4A–4F may represent various embodiments of a more general method, as illustrated by a flow chart in Figure 12.

[0191] In box 1201, a first layer is formed over a substrate, the first layer comprising a polymer composition and having a first refractive index. Next, in box 1202, the first layer is patterned to form a first plurality of wells.

[0192] In box 1203, the first plurality of wells in the first layer is filled with a second polymer composition having a second refractive index different from the first refractive index. The first polymer composition is also different from the second polymer composition.

[0193] In box 1204, a second plurality of wells is formed in the first layer. Then, in box 1205, the second plurality of wells is filled with a third polymer composition having a third refractive index different from the first and second refractive indices. The first and second polymer compositions are different from the third polymer composition.

[0194] The process flow described above with reference to Figures 6A–6F may represent various embodiments of another more general method, as illustrated by a flow chart in Figure 13.

[0195] In box 1301, a first organic solution is coated onto a substrate. Next, in box 1302, a first structure is formed by exposing portions of the first organic solution to a first actinic radiation. The first structure comprises a first polymer composition having a first refractive index selected from a set of refractive indices between 1.3 and 1.8 with a step size of 0.01. Then, in box 1303, after the first structure has been formed, the first organic solution is developed.

[0196] In box 1304, a second organic solution is coated onto the substrate and onto the first structure. Next, in box 1305, a second structure is formed by exposing portions of the second organic solution to a second actinic radiation. The second structure comprises a second polymer composition having a second refractive index selected from a set of refractive indices between 1.3 and 1.8 with a step size of 0.01. Then, in box 1306, after the second structure has been formed, the second organic solution is developed.

[0197] Although the description has been described in detail, it should be understood that various changes, substitutions and alterations can be made without departing from the spirit and scope of this disclosure as defined by the appended claims. Moreover, the scope of the disclosure is not intended to be limited to the particular embodiments described herein, as one of ordinary skill in the art will readily appreciate from this disclosure that processes, machines, manufacture, compositions of matter, means, methods, or steps, presently existing or later to be developed, may perform substantially the same function or achieve substantially the same result as the corresponding embodiments described herein. Accordingly, the appended claims are intended to includewithin their scope such processes, machines, manufacture, compositions of matter, means, methods, or steps.

Claims

WHAT IS CLAIMED:

1. A layer stack comprising: a polymer layer (10) comprising a plurality of regions with different refractive indices, the plurality of regions comprising a first region (200) comprising a first polymer composition (102) and having a first refractive index, a second region (210) comprising a second polymer composition (104) having a second refractive index different from the first refractive index, a third region (220) comprising a third polymer composition (106) having a third refractive index different from the first and the second refractive indices, the first polymer composition (102) being different from the second polymer composition (104), the first (102) and second (104) polymer compositions being different from the third polymer composition (106), wherein the first refractive index is between 1.3 and 1.8, the second refractive index is between 1.3 and 1.8, and the third refractive index is between 1.3 and 1.

8.

2. The layer stack of claim 1, wherein the layer stack has a first edge (12) and an opposite second edge (14), and wherein the refractive index of the polymer layer varies non-monotonically and asymmetrically from the first edge (12) to the second edge (14).

3. The layer stack of claim 1 or claim 2, wherein the first region (200) comprises a first plurality of layers (102–110), one of the first plurality of layers comprising a different polymer composition than another of the first plurality of layers.

4. The layer stack of any one of claims 1–3, wherein a thickness of the polymer layer is between 0.02 μm and 50 μm.

5. The layer stack of any one of claims 1–4, wherein the layer stack is part of a metalens (300).

6. The layer stack of any one of claims 1–5, wherein the first refractive index is between 1.3 and 1.4, and the first polymer composition (102) comprises poly(hexafluoropropylene oxide) (1102) or poly(1H,1H-perfluorobutyl acrylate) (1106).

7. The layer stack of claim 6, wherein the first polymer composition (102) further comprises a first metal, wherein a weight percentage of the first metal in the first polymer composition (102) is selected to produce the first refractive index.

8. The layer stack of any one of claims 1–7, wherein the second refractive index is between 1.4 and 1.5, and the second polymer composition (104) comprises poly(methyl methacrylate) (1110), poly(vinyl acetate) (1114A), or poly(vinyl alcohol) (1114B).

9. The layer stack of claim 8, wherein the second polymer composition (104) further comprises a second metal, wherein a weight percentage of the second metal in the second polymer composition (104) is selected to produce the second refractive index.

10. The layer stack of any one of claims 1–9, wherein the third refractive index is between 1.5 and 1.6, and the third polymer composition (106) comprises poly(vinyl chloride) (1118) or poly(benzyl methacrylate) (1122); wherein the third refractive index is between 1.6 and 1.7, and the third polymer composition (106) comprises poly(2,4,6-tribromobenzyl methacrylate) (1126) or poly(2-chlorostyrene) (1130); or wherein the third refractive index is between 1.7 and 1.8, and the third polymer composition (106) comprises poly(pentabromophenyl methacrylate) (1134).

11. The layer stack of claim 10, wherein the third polymer composition (106) further comprises a third metal, wherein a weight percentage of the third metal in the third polymer composition (106) is selected to produce the third refractive index.

12. The layer stack of any one of claims 1–11, wherein the minimum meaningful difference between any of the first, the second, and the third refractive indices is 0.

01.

13. A method of forming a layer stack, the method comprising: forming a first layer (410) over a substrate (400), the first layer comprising a first polymer composition (402) and having a first refractive index; patterning the first layer (410) to form a first plurality of wells (405);filling the first plurality of wells (405) in the first layer (402) with a second polymer composition (406) having a second refractive index different from the first refractive index, the first polymer composition (402) being different from the second polymer composition (406); forming a second plurality of wells (407) in the first layer (410); and filling the second plurality of wells (407) with a third polymer composition (412) having a third refractive index different from the first and the second refractive indices, the first (402) and second (406) polymer compositions being different from the third polymer composition (412).

14. The method of claim 13, wherein the first, second, and third refractive indices are separately selected from a set of refractive indices between 1.3 and 1.8 with a step size of 0.

01.

15. The method of claim 13, further comprising: forming a second layer (510) over the first layer (410), the second layer (510) comprising a fourth polymer composition (502) having a fourth refractive index selected from a set of refractive indices between 1.3 and 1.8 with a step size of 0.01; forming a third plurality of wells (505) in the second layer (510); and filling the third plurality of wells (505) with a fifth polymer composition (504) having a fifth refractive index selected from a set of refractive indices between 1.3 and 1.8 with a step size of 0.

01.

16. The method of claim 15, further comprising: forming a fourth plurality of wells (515) in the second layer (510); and filling the fourth plurality of wells (515) with a sixth polymer composition (508) having a sixth refractive index selected from the set of refractive indices between 1.3 and 1.8 with a step size of 0.

01.

17. The method of any one of claims 13–16, wherein forming the first layer (410) comprises: preparing a first solution comprising a first monomer and a first solvent; and coating the substrate (400) with the first solution.

18. The method of claim 17, wherein the first monomer comprises an acrylate (700), a methacrylate (702), an epoxide (704), a vinyl ether (706), a thiol (708), analkene (710), a silyldichloride (712), an amine (714), a benzocyclobutene (716), or an ormosil (718).

19. The method of claim 17 or claim 18, wherein the first solvent comprises acetic acid (800), acetone (802), acetonitrile (804), benzene (806), 2-butanone (808), n- butyl acetate (810), carbon tetrachloride (812), chlorobenzene (814), chloroform (816), 1,2-dichloroethane (818), diethyl ether (820), diethylene glycol (822), dimethylformamide (824), 1,2-dimethoxyethane (826), dimethyl sulfoxide (828), 1,4-dioxane (830), ethyl acetate (832), ethyl benzene (834), ethyl lactate (836), hexamethylphosphoramide (838), hexamethylphosphorous triamide (840), isopropanol (842), 1-methyl-2-pyrrolidone (844), methylene chloride (846), propylene glycol methyl ether acetate (848), pyridine (850), tetrahydrofuran (852), triethylamine (854), or water (856).

20. The method of any one of claims 17–19, wherein the first solution further comprises a metal, wherein a weight percentage of the metal in the first solution is selected to crosslink with the first monomer and produce the first refractive index.

21. The method of claim 20, wherein the metal comprises zinc, iron, silver, gold, aluminum, platinum, lead, or titanium.

22. The method of any one of claims 17–21, wherein the first solution further comprises inorganic particles, wherein a weight percentage of the inorganic particles in the first solution is selected to produce the first refractive index.

23. The method of claim 22, wherein the inorganic particles comprise SiO2 nanoparticles, CuO nanoparticles, ZnO, TiO2, silicon, α-quartz, silicon oxynitride, ZnS, CdS, or ferrocene.

24. The method of any one of claims 17–21, wherein the first solution further comprises a second monomer, wherein a weight percentage of the second monomer is selected to copolymerize with the first monomer and produce the first refractive index.

25. The method of any one of claims 17–21, wherein the first solution further comprises a plasticizer, wherein the plasticizer is selected to produce the first refractive index.

26. The method of claim 25, wherein the plasticizer comprises dimethyl phthalate (900), dimethyl terephthalate (902) or disodium terephthalate (904).

27. The method of any one of claims 17–26, wherein the first solution further comprises a photoinitiator.

28. The method of claim 27, wherein the photoinitiator comprises azobisisobutyronitrile (1000), 2,2-dimethoxy-2-phenylacetophenone (1002), 2- isopropylthioxanthone (1004), benzoyl peroxide (1006), phenylbis(2,4,6- trimethylbenzoyl)phosphine oxide (1008), or camphorquinone (1010).

29. The method of any one of claims 17–28, wherein coating the substrate with the first solution comprises spin coating, spray coating, thermal spray coating, flow coating, or dip coating.

30. The method of any one of claims 13–29, wherein patterning the first layer (410) comprises: disposing a patterned mask over the first layer (410); forming a plurality of cured regions (450) in the first layer by exposure to a first actinic radiation through the patterned mask, a remainder of the first layer comprising a plurality of uncured regions; and rinsing the substrate to remove the plurality of uncured regions.

31. The method of any one of claims 13–29, wherein patterning the first layer (410) comprises: forming a plurality of cured regions (450) in the first layer by a maskless lithography technique, a remainder of the first layer comprising a plurality of uncured regions; and rinsing the substrate to remove the plurality of uncured regions.

32. The method of claim 31, wherein the maskless lithography technique comprises stereolithography 3D printing or direct laser writing.

33. The method of any one of claims 15–32, wherein forming the first layer (410) includes forming an alignment marker, wherein forming the second layer (510) comprises aligning a lithography tool with the alignment marker.

34. The method of any one of claims 13–33, further comprising selecting the first refractive index from the set of refractive indices between 1.3 and 1.8 with a step size of 0.01, and selecting the second refractive index from the set of refractive indices between 1.3 and 1.8 with a step size of 0.

01.

35. The method of claim 34, further comprising in response to selecting the first refractive index to be between 1.3 and 1.4, selecting the first polymer composition to comprise poly(hexafluoropropylene oxide) (1102) or poly(1H,1H-perfluorobutyl acrylate) (1106); in response to selecting the first refractive index between 1.4 and 1.5, selecting the first polymer composition to comprise poly(methyl methacrylate) (1110), poly(vinyl acetate) (1114A), or poly(vinyl alcohol) (1114B); in response to selecting the first refractive index between 1.5 and 1.6, selecting the first polymer composition to comprise poly(vinyl chloride) (1118) or poly(benzyl methacrylate) (1122); in response to selecting the first refractive index between 1.6 and 1.7, selecting the first polymer composition to comprise poly(2,4,6-tribromobenzyl methacrylate) (1126) or poly(2-chlorostyrene) (1130); and in response to selecting the first refractive index between 1.7 and 1.8, selecting the first polymer composition to comprise poly(pentabromophenyl methacrylate) (1134).

36. The method of claim 34 or claim 35, further comprising: in response to selecting the first refractive index between 1.3 and 1.4, selecting a monomer for forming the first layer, the monomer being hexafluoropropylene oxide (1100) or 1H,1H-perfluorobutyl acrylate (1104); in response to selecting the first refractive index between 1.4 and 1.5, selecting a monomer for forming the first layer, the monomer being methyl methacrylate (1108) or vinyl acetate (1112); in response to selecting the first refractive index between 1.5 and 1.6, selecting a monomer for forming the first layer, the monomer being vinyl chloride (1116) or benzyl methacrylate (1120);in response to selecting the first refractive index between 1.6 and 1.7, selecting a monomer for forming the first layer, the monomer being 2,4,6-tribromobenzyl methacrylate (1124) or 2-chlorostyrene (1128); and in response to selecting the first refractive index between 1.7 and 1.8, selecting a monomer for forming the first layer, the monomer being pentabromophenyl methacrylate (1132).

37. A method of forming a layer stack, the method comprising: coating a first organic solution (550) onto a substrate; forming a first structure (451) by exposing portions of the first organic solution (550) to a first actinic radiation, the first structure (451) comprising a first polymer composition (402) having a first refractive index selected from a set of refractive indices between 1.3 and 1.8 with a step size of 0.01; developing the first organic solution (550) after forming the first structure (451); coating a second organic solution (552) onto the substrate (400) and onto the first structure (451); forming a second structure (452) by exposing portions of the second organic solution (552) to a second actinic radiation, the second structure (452) comprising a second polymer composition (406) having a second refractive index selected from a set of refractive indices between 1.3 and 1.8 with a step size of 0.01; and developing the second organic solution (552) after forming the second structure (452).

38. The method of claim 37, further comprising: coating a third organic solution (554) onto the substrate (400) and onto the first structure (451) and the second structure (452); forming a third structure (453) by exposing portions of the third organic solution (554) to a third actinic radiation, the third structure (453) comprising a third polymer composition (412) having a third refractive index selected from the set of refractive indices between 1.3 and 1.8 with a step size of 0.01; and developing the third organic solution (554) after forming the third structure (453).

39. The method of claim 37 or claim 38, wherein the first structure (451) comprises an alignment marker, further comprising aligning the first structure (451) to a lithography tool before exposing portions of the second organic solution (552) to the second actinic radiation.

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