Optical metasurfaces using sub-hogels

Abstract

To provide a sub-hogel configuration for a high-definition light field display.SOLUTION: Optical devices and three dimensional light field display technologies are also described, and more particularly, three dimensional holographic pixels (hogels) composed of monochromatic sub-hogels and engineered metasurfaces that act as directional optical elements for light field displays. The described sub-hogel structure designs and methods are suitable for achromatic metasurfaces to provide directional pixels for multi-view light field color displays.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] (CROSS-REFERENCE TO RELATED APPLICATIONS) This application claims priority to U.S. Patent Application No. 17 / 086,201, filed October 30, 2020, the entire contents of which are incorporated herein by reference.

[0002] This disclosure relates to three-dimensional light field display technology, and more particularly to three-dimensional holographic pixels (hogels) composed of monochromatic sub-hogels for light field displays. [Background technology]

[0003] Optical metasurfaces are artificial surfaces used to manipulate wavefronts. Optical metasurfaces generally consist of a two-dimensional lattice of pillar-shaped structures that interact with an impinging wavefront. The lattice constant and structure size are subwavelength thick relative to the wavelength range of the electromagnetic wave the structure is designed to interact with. The pillar dimensions and pillar spacing within the metasurface are altered to achieve desired optical properties. Optical metasurfaces can shape the amplitude, phase, and polarization of electromagnetic beams. The use of metasurfaces in light-field display technology can enable the fabrication of substantially flat optical devices, improve the performance of optical elements, and manipulate light to impart new properties to optical systems. In the development of light-field display technology, metasurfaces have shown promising potential as lightweight, thin optical components that can combine several functions into a single device.

[0004] In one example of an optical metasurface, U.S. Patent Application Publication No. 20170219739 to Lin describes a random spatially multiplexed metasurface, in which multiple optical elements are interleaved on a single metasurface, utilizing the full aperture of the metasurface for all optical elements, albeit individually because they occupy only a fraction of the total area. An achromatic metalens using this design is intended to resemble an interweaving of dedicated lenses for each color channel. Light from all color channels passes through all three lenses, such that for each lens, one-third of the light is focused at the intended achromatic focal point and two-thirds of the light is focused elsewhere.

[0005] In another example of optical metasurfaces, Lin, in U.S. Patent Application Publication No. 20170146806, describes an array of spatially multiplexed metalenses that can be used in light field displays without color channel separation. Different coded apertures can be created by sizing the subelements, and Lin describes an implementation of the aperture based on altering the phase of the wavefront. Summary of the Invention [Problem to be solved by the invention]

[0006] An object of the present disclosure is to provide a sub-hogel configuration for a high-definition light field display. Another object of the present invention is to provide a three-dimensional light field display, more specifically, an optical metasurface with three-dimensional holographic pixels (hogels) composed of monochromatic sub-hogels for a light field display. [Means for solving the problem]

[0007] In one aspect, an optical device is provided that includes a hogel array comprising a plurality of hogels, each hogel divided into a plurality of monochromatic sub-hogels, each of which comprises a plurality of monochromatic sub-pixels; and a directional optical element for directing light from the sub-pixels, the directional optical element being divided into a plurality of color regions, each color region designed to direct light of a particular color, and the monochromatic sub-hogels and the plurality of color regions configured such that the plurality of monochromatic sub-pixels are aligned with the color regions of the directional optical element designed to direct light of the particular color of the monochromatic sub-pixels.

[0008] In one embodiment, the directional optical element is a metasurface.

[0009] In another embodiment, the metasurface comprises nanostructures.

[0010] In another embodiment, the nanostructures comprise titanium dioxide.

[0011] In another embodiment, each of the monochrome sub-pixels is individually addressable.

[0012] In another embodiment, the plurality of monochromatic subhogels comprises at least one monochromatic red subhogel, at least one monochromatic green subhogel, and at least one monochromatic blue subhogel.

[0013] In another embodiment, each monochromatic sub-hogel comprises fewer monochromatic sub-pixels than can be individually distinguished by the human eye.

[0014] In another embodiment, each monochromatic sub-hogel has between 2 and 144 monochromatic sub-pixels.

[0015] In another embodiment, each subpixel is smaller than 10 μm 2 .

[0016] In another embodiment, the monochromatic subpixels in each monochromatic subhogel are arranged in a square configuration, a rectangular configuration, or a radial configuration.

[0017] In another embodiment, the directional optical element is a geometric metasurface, a Pancharatnam-Berry metasurface, an inversely designed metasurface, a dispersive phase compensation metasurface, or a combination thereof.

[0018] In another embodiment, the optical device is a light field display.

[0019] In another aspect, a method for designing a segmented optical metasurface is provided, the method including: defining a phase function for the metasurface; specifying materials for nanostructures in the metasurface; determining a fabrication configuration such that the metasurface is segmented into a plurality of color regions; determining nanostructure parameters for each color region; generating a transmission map for the metasurface based on the nanostructure parameters; designing each color region based on the nanostructure parameters and the transmission map to obtain a phase function, wherein each color region is designed to guide light of a particular optical bandwidth; calculating a figure of merit for the designed metasurface; and generating an output metasurface design for the metasurface.

[0020] In one embodiment, the nanostructure parameters of each color region are different.

[0021] In another embodiment, the material for the nanostructures is titanium dioxide.

[0022] In another embodiment, the metasurface is divided into red, green, and blue regions.

[0023] In another embodiment, the method further comprises, after calculating the figure of merit for the designed metasurface, adjusting the nanostructure parameters and recalculating the figure of merit.

[0024] In another embodiment, the parameters for the nanostructures in each color region include nanostructure height, nanostructure shape, unit cell spacing, resonance boundary parameters, or a combination thereof.

[0025] In another embodiment, the nanostructures have a consistent height across the color region.

[0026] In another embodiment, the metasurface is a geometric metasurface, a Pancharatnam-Berry metasurface, an inversely designed metasurface, a dispersion-phase-compensating metasurface, or a combination thereof.

[0027] In another aspect, a method for displaying a light field is provided, the method including: subdividing an integral image into a plurality of element images, each element image representing a two-dimensional array of angular descriptors associated with a pair of direction coordinates; decomposing each element image into a plurality of color channel-specific element images; sending each element image to a hogel, each hogel comprising a plurality of subpixels, the hogel being divided into monochrome sub-hogels comprising a plurality of monochrome subpixels, the color channel-specific element images being sent to monochrome sub-hogels of the same color; and creating a light field for display.

[0028] In one embodiment, the monochromatic subpixels are adjacent to each other in a monochromatic sub-hogel.

[0029] In another embodiment, each of the plurality of component images is of equal size.

[0030] In another embodiment, the color channel specific component images comprise a red channel, a green channel, and a blue channel.

[0031] In another embodiment, the method further comprises individually addressing the sub-pixels.

[0032] In another aspect, an optical display device is provided that includes a hogel array comprising a plurality of hogels, each hogel divided into a plurality of monochromatic sub-hogels, each monochromatic sub-hogel comprising a plurality of monochromatic sub-pixels. [Brief explanation of the drawings]

[0033] These and other features of the present invention will become more apparent in the following detailed description taken in conjunction with the accompanying drawings.

[0034] [Figure 1] Figure 1 illustrates how to design metasurfaces suitable for use in light field displays.

[0035] [Figure 2] FIG. 1 is a graphical representation of a refractive index plot for TiO2.

[0036] [Figure 3] FIG. 1 shows a graphical representation of subpixel full width at half maximum as a function of diffraction-limited pitch.

[0037] [Figure 4] FIG. 10 is a cross-sectional view of a 4×4 sub-hogel array along the y-axis.

[0038] [Figure 5] FIG. 1 illustrates an embodiment of the present disclosure showing an 8×8 array of sub-hogels.

[0039] [Figure 6] FIG. 10 illustrates the reduction in the number of metalenses required when eight-fold symmetry is applied to an 8×8 array of sub-hogels.

[0040] [Figure 7] FIG. 10 illustrates the conversion of a 6×6 pixel hogel element image into sub-element images and then into sub-hogel images.

[0041] [Figure 8] A simulated light field display with a further zoomed-in set of three (RGB) subhogels.

[0042] [Figure 9A] FIG. 10 shows a graphical representation of the intensity of the red channel of captured retinal images for nsh (subpixels per subhogel)=8, 16 compared to the intensity for the ideal nsh=1.

[0043] [Figure 9B] FIG. 10 shows a graphical representation of the intensity and pixel shift of the red channel of the captured retinal image for nsh (subpixels per subhogel)=32, 64.

[0044] [Figure 9C] FIG. 10 shows a graphical representation of the intensity of the green channel of the captured retinal image for nsh=8, 16 compared to the intensity for the ideal nsh=1.

[0045] [Figure 9D] FIG. 10 shows a graphical representation of the intensity and pixel shift of the green channel of the captured retinal image for nsh (subpixels per subhogel)=32, 64.

[0046] [Figure 9E] FIG. 10 shows a graphical representation of the intensity of the blue channel of captured retinal images for nsh=8, 16 compared to the ideal intensity for nsh=1.

[0047] [Figure 9F] FIG. 10 shows a graphical representation of the intensity and pixel shift of the blue channel of the captured retinal image for nsh (subpixels per subhogel)=32, 64.

[0048] [Figure 10]10 is a plot of u versus v showing the intersection of the upper boundary conditions and the modal condition equations for the transverse electric (TE) symmetric and transverse electric asymmetric modes.

[0049] [Figure 11] FIG. 10 illustrates a linear relationship between β and V in accordance with one embodiment of the present disclosure.

[0050] [Figure 12] FIG. 10 shows a comparison of the calculated effective medium calculated using an approximation of the effective refractive index based on weighted indexes of pillars and voids in accordance with one embodiment of the present disclosure.

[0051] [Figure 13] FIG. 10 illustrates the smallest circle encompassing the top left sub-pixels of the R, G, and B sub-hogels in accordance with one embodiment of the present disclosure.

[0052] [Figure 14] FIG. 10 shows a graph of minimum display diagonal versus field of view for different sub-hogel sizes.

[0053] [Figure 15] Figure 1 shows triplets of monochromatic subhogels to form three color regions and the designed metasurface for each color region.

[0054] [Figure 16A] FIG. 10 is a top view of a metasurface design for a 3x3 3-sub hogel array arrangement in accordance with one embodiment of the present disclosure.

[0055] [Figure 16B] FIG. 10 is an isometric view of a metasurface design for a 3x3 3-sub hogel array arrangement in accordance with one embodiment of the present disclosure.

[0056] [Figure 17A]FIG. 10 is a top view of a metasurface design in one embodiment of the present disclosure comprising a radial array of 32 subhogels with three subhogel arrays.

[0057] [Figure 17B] FIG. 1B is an isometric view of the metasurface design according to an embodiment of the present disclosure, comprising a radial array of 32 sub-hogels. DETAILED DESCRIPTION OF THE INVENTION

[0058] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0059] The use of the word "a" or "an" when used herein in conjunction with the term "comprising" can mean "one," but is also consistent with the meanings of "one or more," "at least one," and "one or more."

[0060] As used herein, the terms "comprising," "having," "including," and "containing," and grammatical variations thereof, are inclusive or open-ended and do not exclude additional, unrecited elements and / or method steps. The term "consisting essentially of," when used herein in connection with a composition, device, article, system, use, or method, indicates that additional elements and / or method steps may be present, but that these additions do not materially affect the manner in which the recited composition, device, article, system, method, or use functions. The term "consisting of," when used herein in connection with a composition, device, article, system, use, or method, excludes the presence of additional elements and / or method steps. A composition, device, article, system, use, or method described herein as including certain elements and / or steps may also consist essentially of those elements and / or steps in some embodiments, and consist of those elements and / or steps in other embodiments, regardless of whether the embodiment is specifically recited.

[0061] As used herein, the term "about" refers to approximately a + / - 10% variation from a given value. It is to be understood that such a variation is always included in any given value provided herein, whether or not that variation is specifically referred to.

[0062] The description of ranges herein conveys both the range and the individual values ​​falling within that range, in the same place as the numbers used to denote the range, unless otherwise specified herein.

[0063] The use of any example or exemplary language, such as "such as," "exemplary embodiment," "illustrative embodiment," and "for example," is intended to illustrate or illustrate aspects, embodiments, variations, elements, or features related to the invention and does not limit the scope of the invention.

[0064] As used herein, the terms "connect" and "connected" refer to any direct or indirect physical association between elements or features of the present disclosure. Thus, these terms may be understood to refer to elements or functions that are partially or completely contained within, attached, coupled, disposed, joined, in communication, operably associated with, etc., even if other elements or functions are present between the elements or functions described as connected.

[0065] As used herein, the term "pixel" refers to the light source and light-emitting mechanism used to create a display. A pixel can comprise one or more subpixels, most commonly one red subpixel, one green subpixel, and one blue subpixel.

[0066] As used herein, the term "subpixel" refers to a structure comprised of a light-emitting element housed within an optical microcavity. The optical microcavity is operatively associated with a plurality of reflective surfaces to substantially collimate, manipulate, or condition the light. At least one of the reflective surfaces is a light-transmitting reflective surface connected to the optical microcavity to transmit light out of the microcavity. The present disclosure provides individually addressable red, green, and blue (RGB) subpixels. Subpixel sizes as currently described range from the nanoscale to several microns, which are significantly smaller than pixel sizes previously known in the art.

[0067] As used herein, the term "light field" at a basic level refers to a function that describes the amount of light flowing in all directions through a point in unoccluded space. A light field thus represents radiance as a function of light's position and direction in free space. Light fields can be synthetically generated by various rendering processes, or they can be captured from a light field camera or an array of light field cameras.

[0068] As used herein, the term "light field display" refers to a device that reconstructs a light field from a finite number of light field radiance samples input to the device. Typically, the radiance samples represent the color components red, green, and blue (RGB), although it should be understood that other combinations of colors may be possible. In a light field display, the light field may also be understood as a mapping from a four-dimensional space to a single RGB color. The four dimensions include the vertical and horizontal dimensions of the display and two dimensions that describe the directional components of the light field. The light field is defined as a function: LF:(x,y,u,v)→(r,g,b) where x,y are the Cartesian or position coordinates of the location in the light field, and u,v are the direction or angle descriptors. f , y f In the case of LF(x f ,y f ,u,v) represents a two-dimensional (2D) image called an "element image." An element image is a fixed x f , y f A light field is a directional image of a light field from a given point. When multiple elemental images are connected side-by-side, the resulting image is called an "integral image." The integral image can be understood as the entire light field required for a light field display.

[0069] As used herein, the term "metasurface" refers to an artificial surface used to manipulate wavefronts. The surface consists of a two-dimensional (2D) lattice of nanostructures that interact with an impinging wavefront, with lattice constants and structure sizes subwavelength. The properties of each subwavelength structure are selected to impart a specific local phase and amplitude to the wavefront. By controlling the phase and amplitude of the wavefront at each lattice site, the shape of the wavefront can be manipulated. Metasurfaces can be designed for various types of wavefronts, including but not limited to electromagnetic and acoustic wavefronts. Optical metasurfaces operate on light waves and can be used to planarize existing three-dimensional (3D) components, such as lenses. Optical metasurfaces can be fabricated using semiconductor techniques, thereby reducing production costs.

[0070] As used herein, the term "OLED" refers to an organic light-emitting diode, an optoelectronic device that emits light under the application of an external voltage. OLEDs can be divided into two major classes: those made with organic small molecules and those made with organic polymers. OLEDs are light-emitting diodes in which a light-emitting electroluminescent layer comprises a film of organic compounds that emits light in response to an electric current. Generally, OLEDs are solid-state semiconductor devices that include at least one conductive organic layer disposed between and electrically connected to an anode and a cathode. When an electric current is applied, the anode injects holes and the cathode injects electrons into the organic layer. The injected holes and electrons each migrate toward the oppositely charged electrode. When an electron and a hole localize on the same molecule, an exciton, a localized electron-hole pair with an excited energy state, is formed. When the exciton relaxes, light is emitted via a photoemissive mechanism. There are various types of OLEDs, including, but not limited to, active matrix OLEDs (AMOLEDs), top-emitting OLEDs, and bottom-emitting OLEDs. AMOLEDs have a total layer of cathode, organic molecules, and anode. The anode layer has a thin-film transistor (TFT) plane alongside it to form a matrix. This helps switch each pixel on or off as needed, thereby creating an image. The pixels are switched off whenever they are not needed or when a black image is displayed, thereby extending the device's battery life. This is the lowest power consumption type of OLED and also has a faster refresh rate, making it suitable for video. Applications for AMOLEDs include computer monitors, large-screen TVs, and digital signage or billboards. Top-emitting OLEDs have a substrate that can be either opaque or reflective. Top-emitting OLEDs are more suitable for active-matrix applications because they can be more easily integrated with non-transparent transistor backplanes. Manufacturers can use top-emitting OLED displays in smart cards. An OLED is bottom-emitting if the emitted light passes through a transparent or semi-transparent bottom electrode and substrate.

[0071] As used herein, the term "hogel" is an alternative term for holographic pixel, which is a cluster of traditional pixels with directional control. An array of hogels can generate a light field. "Hogel pitch" would then be defined as the distance from the center of one hogel to the center of an adjacent hogel.

[0072] As used herein, the term "subhogel" (or subhogel) is a cluster of conventional sub-pixels with directional control. An array of subhogels can comprise a hogel.

[0073] As used herein, the term "monochromatic" refers to a narrow color channel and refers to an emission having a narrow optical bandwidth.

[0074] As used herein, the term "element image" refers to a fixed x f , y f , LF(x f ,y f ,u,v), the two-dimensional (2D) image LF(x f ,y f ,u,v). The element image is a fixed x f , y f 1 is a directional image of the light field from a position.

[0075] As used herein, the acronym "FWHM" refers to "full width half maximum," which is an expression of the spread of a function given by the difference between the two extreme values ​​of the independent variable where the dependent variable is equal to half of its maximum value.

[0076] As used herein, the acronym "FRED" refers to Fred Optical Engineering Software. FRED is a commercially available 3D computer-aided design (CAD) computer program for optical engineering used to simulate the propagation of light through optical systems. FRED can process both incoherent and coherent light using Gaussian beam propagation.

[0077] As used herein, the term "transmittance" refers to the percentage of light transmitted per incident light.

[0078] As used herein, the term "wavelength" is a measure of the distance between two equivalent peaks (high points) or valleys (low points) of a wave, which is a repeating pattern of moving energy such as light or sound.

[0079] As used herein, the term "simulation" refers to the creation of a computer model of something, particularly for research purposes or to develop and refine manufacturing specifications. Various simulation methods may be used, including but not limited to the following: The finite-difference time-domain (FDTD) method is used to solve problems in electromagnetics and optical engineering, solving Maxwell's equations for complex geometries. FDTD is a versatile finite difference method in the time domain that handles nonlinear material properties in a natural way and allows users to measure system response over a wide range of frequencies. An equivalent technique is Rigorous Coupled Wave Analysis (RCWA), a semi-analytical method commonly employed to solve field diffraction problems for periodic structures. RCWA decomposes the field into a set of plane waves and represents the field by a sum of spatial harmonics in Fourier space. RCWA benefits from reduced simulation complexity and time, but becomes inaccurate for more complex geometries. Ray tracing simulations, such as those performed by FRED, are used to prototype opto-mechanical systems. Given an initial set of rays, ray tracing simulates the resulting light field by propagating the rays through space and calculating their interactions with any surfaces they strike.

[0080] It is contemplated that the various embodiments of the compositions, devices, articles, methods, structures, apparatus, and uses disclosed herein may be implemented by those skilled in the art either as is or by making such modifications or equivalents without departing from the scope of the invention.

[0081] This specification describes a sub-hogel configuration for high-definition light field displays. Optical devices and three-dimensional light field display technologies, more specifically, three-dimensional holographic pixels (hogels) composed of monochromatic sub-hogels, and engineered metasurfaces that act as directional optical elements for light field displays, are also provided. The described sub-hogel structure design and methods are suitable for achromatic metasurfaces to provide directional pixels for multi-view light field color displays. To date, the metasurface research community has not discovered an efficient broadband achromatic metalens. To simplify the design of metasurfaces for organic light-emitting diode (OLED) or projector-based displays, we describe a hogel containing an array of monochromatic sub-hogels, where each sub-hogel contains a unique monochromatic metalens.

[0082] A hogel is a directional light-emitting structure composed of multiple subpixels that emits light of different colors and intensities in different directions. While this disclosure illustrates a hogel with multiple RGB subpixels, it should be understood that a hogel can include different combinations of subpixel numbers and colors. A light field display is composed of an array of hogels. An observer sees a spot of light emitted from each hogel in the array. The collection of each spot of light from the hogel array produces an image seen by the observer. A second observer in a different location sees a spot of light from each hogel in the array, but because they are viewing the light field display from a different location and therefore a different direction, they see a different image than the first observer. In the case of an n × m array of hogels, both observers see an image produced by the n × m array of light spots. A hogel consists of a 2D pixel array (or subpixel array) and a directional optical element, such as a lens or metasurface. Light emitted by each pixel or subpixel travels perpendicular to the pixel array. Light from each pixel passes through a directional optical element and is directed in a predetermined direction. A hogel with a p×q pixel array sends light in p×q different directions. A light field display consists of an (n*p)×(m*q) pixel array and an n×m array of directional optical elements, such that there are p×q pixels per hogel in the n×m hogel array. A hogel is the product of combining a pixel array with an array of directional optical elements. Each pixel consists of subpixels, and typically, three adjacent RGB subpixels form a pixel. Thus, a pixel array is also a subpixel array. In a sub-hogel light field display, the subpixel array that makes up each hogel is reorganized to accommodate the directional optical element (in this case, a metasurface) so that similar-colored subpixels are grouped into clusters instead of grouping RGB subpixels of the same pixel together.

[0083] According to the present disclosure, metasurfaces can be considered regular spatially multiplexed metasurfaces, with metalenses for each color channel interleaved on a single metasurface. While metasurfaces are described herein as an example of directional optical elements, it is contemplated that other directional optical elements may be used. The aperture is determined by the subhogel size, with each subhogel interacting with a single color channel. The metasurfaces described herein are segmented to accommodate color regions, with each such segment having a corresponding cluster of similarly colored subpixels, allowing the metasurface color region segment to be tuned to the wavelength of the corresponding subpixel cluster. A subhogel is a similarly colored cluster of subpixels coupled with a corresponding metasurface color region segment.

[0084] One major hurdle faced in designing directional optical elements, particularly metasurfaces for light field display technology, has been achieving nanoscale pixel sizes to provide the pixel densities required for high-definition light field displays on the order of billions of pixels. The design described here achieves sub-10 micron pixel sizes while providing adequate subhogel sizes, enabling metasurface designs that can be fabricated using known fabrication tools and methods. To tailor and realize achromatic metasurfaces with directional pixel functionality, it is proposed to cluster similarly colored subpixels (R subpixels, G subpixels, and B subpixels) and stack them with regions of metasurface tuned to the spectrum of the light emitted by the similarly colored (monochromatic) subpixel clusters that direct the emitted light, referred to herein as monochromatic subhogels, monochromatic subhogel arrays, and / or monochromatic subhogel clusters.

[0085] This disclosure provides design considerations and methods necessary to design high-definition multi-view light field displays with monochromatic subhogel arrays combined with achromatic metasurfaces to provide directional pixels. Metasurfaces are therefore strong candidates for augmenting traditional refractive or diffractive optics. The metasurface concept utilizing monochromatic subhogels overcomes current limitations in achieving the high angular resolution required for compelling light field displays.

[0086] Described herein are monochromatic subhogel designs in which clusters of similarly colored subpixels can be combined with optical surfaces such as geometric metasurfaces to achieve the required directional pixel functionality. The techniques described throughout this disclosure offer advantages over other achromatic metasurfaces. These advantages include, but are not limited to, expected improved efficiency, the elimination of the need for a polarized light source, and the fact that subpixel size is not limited by the metasurface's ability to cover the entire visible spectrum. When used in displays, reducing pixel size improves upon pixels previously known in the art by enabling the system to output more light beams in more different directions, thereby enabling the creation of displays with higher angular resolution that improve the effective resolution of multidimensional objects. Increasing the number of light field display views allows a viewer located at any viewing position to receive multiple views simultaneously. This is known as a super multiview (SMV) display. SMV displays achieve improved angular resolution, eliminating the accommodation-convergence conflict and producing displays with higher-quality depth of field.

[0087] Fan et al. described a metalens array that can be used in light field displays using a dispersive phase-compensating achromatic metasurface. (Fan, Zhi-Bin. A broadband achromatic metalens array for integral imaging in the visible. Light: Science and Applications. 2019) They achieved an average efficiency of 47% and a numerical aperture (NA) of 0.08, which is too small to realize the large field of view required for light field display applications. Lin, in US Patent Application Publication No. 20170146806, also described a metalens array that can be used in light field displays using a dispersive phase-compensating achromatic metasurface. They achieved an average efficiency of 39% and an NA of 0.21, which is still too small to realize the large field of view required for light field display applications. The NA of a phase-compensating achromatic metalens is limited by the maximum lens size that the nanostructure can accommodate, while still providing sufficient phase compensation to achieve achromaticity, which is directly related to the height of the nanostructure. Increasing the NA of phase-compensated achromatic metalenses will require advances in fabrication to achieve higher nanostructures.

[0088] The present invention utilizes subhogels to tune distinct regions of a metasurface to specific color channels, thereby enabling highly efficient and greatly simplified metasurface design. Because each region of the metasurface is monochromatic, the metasurface can be a geometric metasurface, which means that the phase is controlled by varying the size of the nanostructures that make up the metasurface.

[0089] previously reported a 90% efficient metalens system that eliminates the need for a polarized light source and eliminates extra losses and components (Khorasaninejad, Mohammadreza. Polarization-Insensitive Metalenses at Visible Wavelengths. American Chemical Society, Nano Letters. October 24, 2016). However, these devices suffer from chromatic aberration, which is less important for the present invention because each geometric metasurface is tuned to a narrow-bandwidth color channel. The present invention can also be used with Pancharatnam-Berry metasurfaces, which control the phase of the polarization plane by varying the orientation of fixed-size birefringent nanostructures. Other possible metasurface types include, but are not limited to, combinations of geometric metasurfaces (with varying nanostructure size and orientation) and Pancharatnam-Berry metasurfaces, inversely designed metasurfaces, and dispersion-phase-compensated metasurfaces.

[0090] Various features of the present invention will become apparent from the following detailed description, taken in conjunction with the drawings. The design parameters, design methods, configurations, and uses of the microcavity OLED design processes and structures disclosed herein are described with reference to various examples that represent non-limiting embodiments of the invention as described and claimed herein. Those skilled in the art to which the invention pertains will recognize that there may be other variations, examples, and embodiments of the invention not disclosed herein that can be implemented in accordance with the teachings of the present disclosure without departing from the scope of the invention.

[0091] Figure 1 illustrates a method for designing a metasurface suitable for use in a light field display. This method first requires the definition of a required phase function at 10. A phase function is selected for the metasurface under design based on the desired function of the metasurface. In one example, if the metasurface is to function as a lens, the phase function will focus light presented to the metasurface to a designed focal length, or focal spot. Following the selection of the phase function, a material type is specified at 12, which then allows fabrication constraints to be determined at 14. An ideal material for a metasurface for a light field display has a refractive index that ensures strong confinement to achieve a full 2π phase shift within the range imposed by fabrication and pixel size while maintaining high transmittance. While this disclosure describes the use of titanium dioxide (TiO2) to fabricate nanostructures in the metasurface, it should be understood that other materials and material combinations, optionally in combination with a surface mask, can be used to fabricate the nanostructures. Metasurface materials may include, but are not limited to, TiO2, SiO2, Si, GaN, AlO3, and Si3N4, or other materials with suitable properties. While the described metasurfaces comprise nanostructures, in this case nanopillars, it should be understood that the nanostructures of the described metasurfaces can have a variety of shapes, including, but not limited to, elliptical, square, rectangular, and square horizontal cross sections, and linear, angular, curved, pyramidal, and frustoconical vertical cross sections. Metasurfaces can also optionally be fabricated on silicon dioxide (SiO2), adding further flexibility to the TiO2 deposition method. Possible deposition methods include, but are not limited to, TiO2 deposited directly on top of the display or aligned in an additional step.

[0092] Once the fabrication constraints are determined, the array configuration is then determined at 16, and the unit cell spacing is defined at 18. The unit cell, denoted as U, specifies the center-to-center distance between adjacent pillars or nanostructures in the metasurface. The unit cell size is analogous to the lattice constant in a periodic crystal structure or the grating period in a diffraction grating, and the duty cycle is the feature dimension divided by the grating period. Approximations for the minimum and maximum unit cell sizes can be found using the equations described below. The optimal unit cell size can then be determined using FDTD simulations that compare transmission maps for different unit cell sizes. The optimal unit cell size is the size where the transmittance remains as close as possible to unity from the minimum and maximum diameters for single-mode resonance. Previous reports on metasurface design have used different unit cell sizes for different wavelengths. In one example, Khorasaninejad et al. reported U = 180 nm, 250 nm, and 350 nm for wavelengths of 405 nm, 532 nm, and 660 nm, respectively (Khorasaninejad, Mohammadreza. Visible Wavelength Planar Metalenses Based on Titanium Dioxide. IEEE Journal of Selected Topics in Quantum Electronics. Vol. 23, No. 3, May / June 2017).

[0093] Figure 2 shows a graph of the refractive index of titanium dioxide (TiO2) in the visible range versus the wavelength of light (nm). For visible-wavelength nanostructure-based metasurfaces for light field displays, TiO2 has been shown to have a negligible absorption coefficient, a refractive index in the range of 2.3–2.7, as shown, and to achieve high-aspect-ratio anisotropic structures with minimal surface roughness.

[0094] Returning to the method shown in Figure 1, for rectangular dielectric resonators, Aieta et al. reported using FDTD sweeps to find optimal unit cell parameters in silicon, taking into account function and resonance in rectangular dielectrics (Aieta, Francesco. Multiwavelength achromatic metasurfaces by dispersive phase compensation. Science Express. February 19, 2015). U.S. Patent No. 9,103,973 to Fattal et al. reports that the lattice constant should be selected so that the optical element does not scatter light in an undesirable manner, which can be prevented by selecting the lattice constant based on the non-scattering limit defined by: In the case of a square lattice, JPEG2026000955000002.jpg13153In the case of a hexagonal lattice, JPEG2026000955000003.jpg13153

[0095] Khorasaninejad et al. describe a method to optimize the height and unit cell size of nanostructures at the design wavelength, with the maximum diameter equal to the unit cell size, which must be small enough to satisfy the Nyquist Sampling Criterion (Khorasaninejad, Mohammadreza. Polarization-Insensitive Metalenses at Visible Wavelengths. American Chemical Society, Nano Letters. October 24, 2016). According to the Nyquist Criterion, if the sampling frequency is greater than twice the highest frequency to be sampled, the repetitive waveform can be correctly reconstructed. Therefore, JPEG2026000955000004.jpg11153Here, NA is defined as the numerical aperture of the metalens.

[0096] According to the present disclosure, the minimum unit cell limit can be defined based on the shape of the nanostructures. The minimum unit cell size is the maximum distance a between nanostructures. maxIn this case, the gap between adjacent nanostructures, which is synonymous with the distance between pillars, is smaller than the gap between nanopillars, and max This is defined as the distance between the pillars smaller than 1 / 4 wavelength at which light at the design wavelength does not resonate between the nanopillars. There is no resonance between the nanopillars so that the only contribution to the output is from the nanopillars themselves. Therefore, in the non-resonant state, the optical path length between the pillars must be smaller than a quarter wavelength, i.e., JPEG2026000955000005.jpg12153where n gap is the refractive index of the material surrounding the nanopillar. The minimum radius of single-mode resonance, r min When you use JPEG2026000955000006.jpg16153

[0097] In that case, the minimum unit cell size can be defined as follows: U min =a max +d min That is, JPEG2026000955000007.jpg13153Also, by this condition, for larger unit cells, d min is set. JPEG2026000955000008.jpg12153

[0098] Once the unit cell spacing is defined in 18, the boundaries of the resonant states for the nanopillars must also be defined in 20. Calculating the resonance boundary parameters, including the diameter and shape of the nanostructures in the metasurface, involves setting lower and upper limits on the cross-sectional area of ​​each nanostructure so that the nanostructures respond well and efficiently to the wavelength of the intended color of light. For the design of metasurfaces for 3D light field displays, a single-mode resonance per nanostructure is desirable. Therefore, the minimum diameter d of each pillar at which no resonance occurs is determined. min Similarly, the maximum diameter d at which single-mode resonance becomes multi-mode resonance max It is necessary to seek the following.

[0099] The transverse refractive index profile of many optical fibers is radially symmetric, and the index profile of almost all fibers exhibits only a small index contrast; therefore, the fiber can be assumed to be only weakly guided. This simplifies the calculation of fiber modes to obtain linearly polarized (LP) modes. For stronger guidance, transverse electric and transverse magnetic modes must be distinguished, where either the electric field or the magnetic field is exactly perpendicular to the fiber axis. There are also hybrid modes, which have non-zero longitudinal components of both the electric and magnetic fields. As used herein, HE and EH are combinations of symbols for the electric field (E) and the magnetic field (H). The dominant field along the propagation direction is represented by the first symbol. For example, HE has a relatively strong longitudinal magnetic field compared to the longitudinal electric field.

[0100] The wave equation for the complex electric field profile in cylindrical coordinates is: JPEG2026000955000009.jpg14153, where β is the imaginary part of the propagation constant. For discrete values ​​of β at a given wavelength, there is a solution to the radial equation describing the guided modes of the fiber. All guided modes have β values ​​that lie between the plane wave values ​​in the cladding and core.

[0101] The V number can be interpreted as a kind of normalized optical frequency and is a dimensionless parameter that is essential for many fiber properties. The V number is defined as follows: JPEG2026000955000010.jpg13153

[0102] For values ​​of V less than 2.405, the fiber supports only one mode, LP01, per polarization direction, known as single-mode or monomode fiber. For values ​​of V greater than 2.405, the number of supported modes can be estimated as follows: JPEG2026000955000011.jpg13153

[0103] Using V=2.405 in the above formula for V, the maximum radius of a single mode fiber can be found using: JPEG2026000955000012.jpg18153

[0104] Similarly, the minimum radius at which single-mode resonance begins is V=0.9, so using JPEG2026000955000013.jpg19153The minimum radius of the resonance can be determined.

[0105] Following the calculation of the diameter of the resonance boundary, the height of the nanostructure pillars in the metasurface must be specified in 22. The height of the nanostructures must be high enough to cover the 2π phase over the achievable diameter. Furthermore, due to fabrication constraints, it is desirable for the pillar heights to be equal, or at most, have a single height per design wavelength.

[0106] A collaborative methodology from the Capasso group at Harvard University relies on simulation results to ensure this range, stating h = 400 nm, 600 nm, and 600 nm for wavelengths of 405 nm, 532 nm, and 660 nm, respectively (Khorasaninejad, Mohammadreza. Visible Wavelength Planar Metalenses Based on Titanium Dioxide. IEEE Journal of Selected Topics in Quantum Electronics. Vol. 23, No. 3, May / June 2017). It has been suggested that for a 2π phase shift, the height can be estimated based on the phase accumulation along the length of the slab, such that the height is: (Khorasaninejad, Mohammadreza. Polarization-Insensitive Metalenses at Visible Wavelengths. American Chemical Society, Nano Letters. October 24, 2016). JPEG2026000955000014.jpg12153

[0107] However, the effective refractive index ranges from about 1 to the refractive index of the material, which can lead to unreliable estimates of the required height. Fattal et al., U.S. Patent No. 9,103,973, reports that the metasurface thickness should be no greater than the following height: JPEG2026000955000015.jpg16153

[0108] To ensure a large differential phase, the above formula is used for 540 nm, H<526 nm. This value is smaller than the currently known value determined here in simulations. However, one possible reason for the discrepancy in this formula may be that n-pillars is the effective refractive index of the metasurface, not simply the refractive index of the nanostructure (pillar) material. Aperiodic metasurfaces are expected to have an effective refractive index that is not constant at any particular point. Furthermore, if the effective refractive index value were calculated for the entire metasurface and then simplified to the ratio of filled to unfilled areas, the resulting effective refractive index estimate would be unreliable.

[0109] Through a series of steps, we derive an equation for the effective refractive index based on the pillar diameter and refractive index. We then use this equation to determine the minimum height required to achieve a 2π phase shift within the single-mode resonance region for each wavelength. Furthermore, we show that the phase accumulation per unit length, Δφ / H, is the slope of a linear fit to the data for pillar height versus total phase shift.

[0110] A set of equations is used to determine the effective refractive index of the nanopillars. As theoretical background to explain the development of this calculation, we first explain that optical modes in a dielectric slab are solutions to eigenvalue equations derived from Maxwell's equations, subject to boundary conditions imposed by the waveguide geometry. Maxwell's equations can be written as follows: JPEG2026000955000016.jpg14153Here, n is the value of the refractive index profile.

[0111] Since the structure is uniform along the z-axis, the solution to the wave equation is: JPEG2026000955000017.jpg13153Here, β is the propagation constant (z component of the wave vector), JPEG2026000955000018.jpg8153 and JPEG2026000955000019.jpg7153 is the wave function of the guided mode.

[0112] After removing JPEG2026000955000020.jpg7153, the wave equation becomes: JPEG2026000955000021.jpg12153

[0113] For each segment of the dielectric structure, the above equation is solved. It is known that for confined modes, the field amplitude decreases exponentially outside the guiding structure and varies sinusoidally within it. For TE (Transverse Electric) modes, the mode function is: E m If (x)=Asin hx+Bcoshx, then |x| <d / 2 Ce -qx If x>d / 2 De qx If x<-d / 2 where h and q are related to the propagation constants as follows: JPEG2026000955000022.jpg24153

[0114] Using the above boundary conditions and the modal equation for the TE symmetric mode (A=0, C=D), we obtain JPEG2026000955000023.jpg12153

[0115] In asymmetric mode, JPEG2026000955000024.jpg13153Therefore, the propagation constant is JPEG2026000955000025.jpg13153 and JPEG2026000955000026.jpg12153, in which case the above boundary conditions and mode condition equations for the TE symmetric mode become: utan u=v

[0116] The above boundary conditions and mode condition equations for the TE asymmetric mode are as follows: -ucot u=v

[0117] Finally, the definition of the V number can be sought to be: JPEG2026000955000027.jpg12153

[0118] Since u and v must be positive, the propagation constants can now be found by finding the intersection of the above boundary conditions and the mode condition equations for the TE symmetric and TE anti-symmetric modes, and the V number obtained from the definition of the V equation is defined as a circle of radius V as follows: u 2 +v 2 =V 2

[0119] the u value for the intersection of the confinement modes, and Using JPEG2026000955000028.jpg12153, the propagation constant can be calculated using: JPEG2026000955000029.jpg11153

[0120] To define the confined modes, it is useful to define a normalized propagation constant. JPEG2026000955000030.jpg12153

[0121] Once the pillar height is specified in 22, a transmission map is generated in 24. Once the nanostructure parameters are defined, the accessible nanostructure range is defined accordingly. The transmission map maps the complex transmission coefficient as a function of the nanostructure parameters. The magnitude of the complex transmission coefficient determines the optical efficiency of the nanostructure, and the phase of the complex transmission coefficient determines the phase imparted to the emitted light from the subpixel. These maps are generated using finite-difference time-domain (FDTD) analysis software. Due to the complexity of metasurface structures, analytical techniques often fail to provide valid solutions, so numerical modeling techniques are employed. FDTD is one of the most common methods for modeling electromagnetic structures because it can handle inhomogeneous, anisotropic, and frequency-dispersive materials. However, modeling metamaterials, which have a high contrast between the material properties of the structure and free space, makes numerical simulation difficult, and the accuracy of traditional FDTD methods is usually insufficient. Although very fine meshes can be used to improve simulation accuracy, increased computational resources are required. Therefore, traditional FDTD methods must be appropriately developed to accurately model metamaterials.

[0122] The phase and transmission maps are used to create a data set from which high transmission parameters with the desired phase can be selected. A series of sweeps are created that successively increase the diameter of nanopillars at a specific unit cell spacing and height. These simulations use periodic boundary conditions to emulate the field generated by an infinite array of equivalent nanopillars, from which phase and transmission parameters are extracted. The transition and phase maps are used to design aperiodic metasurfaces in which adjacent nanopillars have different diameters, and therefore the pillars have different nearest-neighbor interactions, which can alter their phase and / or transmission parameters. Ideally, the range of pillar diameters is small enough that the above design approximations are valid. However, this leaves room for further design optimization.

[0123] Using the FDTD-generated transmission map as a look-up table for nanostructure parameters, color regions in the metasurface can be designed in 26. The color regions in the metasurface are designed based on the nanostructure parameters and transmission map to achieve the phase specified by the phase function, with each color region designed to guide light of a specific optical bandwidth. The nanostructure parameters for each color region can include, but are not limited to, nanostructure height, nanostructure shape, unit cell spacing, and resonance boundary parameters. To create the desired phase profile / wavefront with uniform transmittance across the metasurface, the pillar diameters selected from the LUT minimize the following equation: JPEG2026000955000031.jpg10153where, T m is the average transmittance, and φ t is the desired phase, T(d) is the transmission parameter from the LUT, and φ(D) is the phase parameter from the LUT.

[0124] Desired phase φ t is given by the metasurface function: For a metalens with focal length f, the desired phase is given by: JPEG2026000955000032.jpg11153Here, x and y are position coordinates relative to the center of the metalens.

[0125] Finally, at 28, the parameters of the nanostructure array are optimized using FDTD simulation of the metasurface to maximize the calculated Figures of Merit (FOM), and at 30, the metasurface is arranged into the design. The Figures of Merit are performance metrics of a metasurface defined by the functionality of the metasurface. For example, the Figures of Merit of a metalens include, but are not limited to, its focal length, the full width at half maximum (FWHM) at the focal location, and the Strehl ratio (which is a comparison with an ideal intensity curve at the focal location). Given a design, the Figures of Merit can be calculated through the results of design simulation. After calculating the Figures of Merit of the designed metasurface, the parameters of the nanostructures can be adjusted, and then the Figures of Merit can be recalculated to see if the performance of the metasurface improves as a result of adjusting the nanostructure parameters. The metasurface optimization process can be iterative; nanostructure parameters can be adjusted and subsequent FOM calculations can be performed multiple times to optimize the metasurface design.

[0126] Sub-hogel metalens pitch Δx SH is given by: Δx SH =N SH Δx SP where N SH is the number of subpixels per subhogel, and Δx SP is the sub-pixel pitch. The required angular pitch φ of each sub-hogel SH is given by: φ SH =Φ*(N SH -1)+PS where P is the FWHM of the point spread function and Φ is the angular resolution of the display. It has been found that having the FHWM of P equal to twice the angular resolution provides a pleasant viewing experience.

[0127] Using the above equation, it is possible to determine the minimum number of subpixels per subhogel to overcome the diffraction limit Δθ. In Figure 3, Δx is plotted in a graph of FWHM [degrees] versus pitch [μm] for an angular resolution of 0.6 degrees and a PS of 1.2 degrees. SH For φ SH and Δθ are plotted. The diffraction limit is met for sub-hogels containing 4×4 or more sub-pixels (based on 4 μm×4 μm sub-pixels).

[0128] As subhogels become larger, the distance between the RGB subpixels that combine to form individual pixels increases, so the upper limit on the number of subpixels per subhogel is set by the eye's ability to distinguish individual subpixels. Figure 4 represents a cross-sectional view along the y-axis of a subhogel array with four subpixels per subhogel, highlighting three subpixels in adjacent subhogels that contribute to a single pixel viewed at a distance d perpendicular to the screen 44: a blue (B) subpixel 46, a green (G) subpixel 48, and a red (R) subpixel 50. Figure 4 illustrates the angle subtended by a single pixel as viewed by an observer 106 centered on the central subpixel, expressed as the subpixel pitch 100 (Δx SP ) and subhogel pitch 102 (Δx SH ) and the subpixels are (Δy SP ) in the y dimension. In this example, SP All RGB triplets of the subpixels that form each pixel are in the same row of subpixels, so that only contribute to the pixel size in the y dimension.

[0129] The angular spread of the RGB subpixels is greatest for an observer normal to the screen, and therefore this view limits the subhogel size. The light forming each pixel is spread over a viewing distance 104 given by: JPEG2026000955000033.jpg7153

[0130] If the angular resolution of the human eye is limited to β = 0.03°, then the minimum viewing distance of a pixel is JPEG2026000955000034.jpg35153

[0131] For a given minimum viewing distance, the maximum subhogel size can be set using the above formula, which may be set by the near point of the human eye or by the properties of the light field display. JPEG2026000955000035.jpg16153

[0132] The maximum subhogel pitch is found by solving the above quadratic equation: (2Δx SH +Δx SP ) 2 >>Δy SP , the maximum sub-hogel pitch can be approximated by: JPEG2026000955000036.jpg12153

[0133] 5 illustrates one embodiment of the present disclosure showing a hogel 60 comprising an array of 8×24 multicolor sub-hogels 52 for a total of 192 subpixels. The hogel shown has a full-color 64×64 view from an array of blue 46, green 48, and red 50 subpixels.

[0134] FIG. 6 illustrates a hogel 60 with multiple subpixels that reduces the number of metalenses required when eight-fold symmetry is applied to an 8×8 array of sub-hogels. As shown in FIG. 6, view symmetry reduces the number of unique metalenses 58 from 192 to 48 when four-fold symmetry is applied, and the number of unique metalenses 58 further reduces to 30 when eight-fold symmetry is applied. In this example, the number of unique metalenses would be one for each RGB subpixel. For larger screens, where the minimum viewing distance is greater, the number of sub-hogels can be reduced to 48 using a 16×16 array of subpixels 46, 48, and 50 per sub-hogel; applying four-fold symmetry reduces the number of unique metalenses to 12. Applying eight-fold symmetry would then reduce that number to nine.

[0135] The integral image for a sub-hogel display is formed by subdividing the hogel display's elemental image into equal-sized sub-elemental images with an integer number of pixels per sub-elemental image. For light-field displays, a fixed x f , y f , LF(x f ,y f , u,v), the element image is a two-dimensional (2D) image LF(x f ,y f ,u,v). The element image is a fixed x f , y f The sub-element image is a directional image of the light field from a position. Each sub-element image is then decomposed into three sub-element images, one for each color channel, such that each pixel in the sub-element image has a corresponding sub-pixel in each of the sub-element images. The sub-element images are placed adjacent to each other to create the element images of the sub-hogel display.

[0136] 7 illustrates the conversion of a 6×6 hogel 60 into a split hogel 62. As shown, split hogel 62 is divided into two multicolor sub-hogels 52a and 52b in the x-direction, although it should be understood that the division can occur in various directions. Hogel 60, which may be referred to as an elemental image, and split hogel 62, which may be referred to as a partial elemental image, have 6×6 pixels, each pixel comprising one red subpixel, one blue subpixel, and one green subpixel. The size of each subpixel is preferably 10 μm. 2 6. FIG. 7 further illustrates the conversion of 6×6 divided hogel 62 into sub-hogel elemental image 64. Note that the division of the elemental image into sub-elemental images only needs to occur along one direction, since like-colored subpixels are already adjacent to each other along the remaining axes, although it should be understood that, again, the division can occur in various orientations. The resulting sub-hogel elemental image 64 is composed of monochromatic sub-hogels 66 a, 66 b, and 66 c. While the pixels in monochromatic sub-hogels 66 a, 66 b, and 66 c are shown arranged in a rectangular 3×6 configuration, it should be understood that the sub-hogels can be arranged in other orientations, such as, for example, a rectangular, square, or radial orientation, with variations in the number of subpixels in each sub-hogel.

[0137] The initial hogel elemental image data is stored in a 6x6 matrix, while the sub-hogel elemental image data is stored in an 18x6 matrix. Using a MatLab (or equivalent) script, the hogel integral image can be converted to a sub-hogel integral image and written to a txt file. To display a light field, the entire image for display is commonly called an integral image, and the integral image is divided into multiple elemental images, which are sent to a light field display for display. Each elemental image is represented by an associated hogel in the hogel array and is composed of multiple hogels arranged in the hogel array. Each pixel in an elemental image has a corresponding view or orientation such that an observer facing the display at the normal to the display sees the pixel at the center of each hogel in the hogel array.

[0138] Ray-tracing software was used to simulate the subhogel display architecture to analyze how the display is perceived. Note that these simulations can be performed with any suitable software tool. One example of a ray-tracing software tool is FRED. As previously explained, FRED refers to Fred Optical Engineering Software (FRED), a commercially available 3D CAD computer program for optical engineering used to simulate the propagation of light through optical systems. Note that to simplify the simulation, a limited number of views were simulated.

[0139] Figure 8 shows a simulated light field display with a further zoom in on a set of three monochrome (RGB) subhogels, specifically a red monochrome subhogel 66a, a green monochrome subhogel 66b, and a blue monochrome subhogel 66c. To conserve computational resources, nine views were simulated using ray tracing software. To test color mixing, the test image was a 4 x 8 pixel white image. Each monochrome (RGB) subhogel 66a, 66b, and 66c creates one pixel for a 3 x 3 view. Space 72 between the subhogels is reserved for additional views. Each monochrome subhogel can be composed of multiple subpixels, preferably between 2 and 144 monochrome subpixels.

[0140] In these simulations, a single wavelength served as the light source. A mixing ratio calculator was used to determine the color mixing ratios needed to mix white. The ratios obtained by the calculator are luminance ratios, which are photometric quantities. Therefore, the light source output ratios were set to the luminance ratios divided by the photopic luminosity function, a photopic function built into the ray tracing software tool.

[0141] To analyze the perception of sub-hogel displays, a human eye model, also known as the "Arizona eye," created by Photon Engineering, was incorporated into the model. The pupil diameter and eye accommodation can be set by the user. Accommodation can be defined as the process by which a vertebrate eye changes its optical power to maintain a sharp image or focus on an object as the object's distance changes. An analytical surface is placed behind the retina to capture the observed image. Because one degree of visual acuity corresponds to a length of 288 microns on the retina and the eye can resolve lines only 0.03 degrees apart, the retinal distance sampling should be 8 microns.

[0142] result The resulting setup produces a series of images that are evaluated qualitatively. Figures 9A-9F show the intensity of each color channel of the captured retinal image for different numbers of subpixels per subhogel. These ray tracing simulations do not consider diffraction. Therefore, the case with one subpixel per subhogel represents the case of combining an ideal metalens with a conventional subpixel array. The intensity peaks correspond to the retinal locations of the images formed by the pixels, or the perceived locations of the pixels, in arbitrary units. n sh For σ = 8, 16, the pixel locations in all color channels are closer to the ideal n sh Matches where n = 1, where n sh is defined as the number of subpixels per sub-hogel. As shown in Figure 9B, n sh For = 32, 64, the pixel position in the red channel is shifted in the positive x direction, and n sh =64 is n sh As shown in Figure 9D, the pixel positions in the green channel match the ideal case. As shown in Figure 9F, the pixel positions in the blue channel are shifted in the negative x direction, resulting in a shift of n sh =64 is n sh n = 32. sh As n increases, the distance between the subpixels of a pixel increases along the x-axis, so for a center pixel with this subpixel shape, the red subpixel is pushed in the positive x-direction, the green subpixel remains in the center, and the blue subpixel is pushed in the negative x-direction. sh is large enough, in this example, sh At = 32 and 64, a central observer perceives the white pixel as a separate RGB pixel. SP = 3.3 μm and Δy SP = 10 μm. sh For =16, the following equation describing the light forming each pixel is: JPEG2026000955000037.jpg9153 where Δx SHMAX = 63 μm, which corresponds to a maximum of 19 subpixels per subhogel, which is consistent with the results above. [Example]

[0143] An example of determining the effective refractive index of nanopillars for a metasurface designed for light field displays is described herein. Considering step-index guiding with n1 = 1.5, n2 = 1.6, d = 5 μm, and λ = 1.55 μm, we obtain JPEG2026000955000038.jpg10153V=5.64

[0144] It is expected that there are m-1=4 confined modes in the waveguide. Figure 10 shows a plot of u vs. v showing the intersections of the upper boundary conditions with the mode equations for the TE symmetric and TE asymmetric modes. From this figure, the u values ​​for the intersections are 1.33, 2.65, 3.94, and 5.14, respectively.

[0145] To find the value of the propagation constant and the effective refractive index, we extract the value for the dispersion relation in a symmetric waveguide, noting that the propagation constant scales from 0 to 1 using the following equation: JPEG2026000955000039.jpg16153This can be written as follows: The values ​​of the intersection points can then be used to find the values ​​of the propagation constant. The resulting β values ​​are 6.46, 6.40, 6.29, and 6.15.

[0146] Noting that the data is approximately linear within this range, the relationship between b and V is shown in FIG. 11 and can be explained as follows: b=0.349V-0.314

[0147] The relationship between the pillar diameter and the effective refractive index can be defined as follows: JPEG2026000955000041.jpg13153

[0148] To relate these variables, V is calculated from the pillar diameter, and b and β are calculated by defining a normalized propagation constant, so that the effective refractive index of each pillar can be approximated.

[0149] In general, the phase accumulation from a dielectric is defined as: JPEG2026000955000042.jpg11153

[0150] However, the effective refractive index value used in this equation assumes a constant effective refractive index (periodic structures with the same diameter and unit cell width). Because the effective refractive index as a function of diameter is approximately linear within the single-mode resonance limit defined above, the effective refractive index can be replaced with the effective refractive index difference from the maximum diameter to the minimum diameter. Δn rms = n rms (d max )-n effective (d min )

[0151] Therefore, the phase accumulation equation can be written as: JPEG2026000955000043.jpg12153

[0152] The phase accumulation per unit length can be written as: JPEG2026000955000044.jpg12153

[0153] If Δφ=2π, the minimum height required to achieve a 2π phase shift within the resonance limit can be found using: JPEG2026000955000045.jpg11153

[0154] Table 1 below calculates the phase accumulation and minimum height required for each wavelength. [Table 1]

[0155] In Figure 12, we compare the calculated effective medium calculated here with an approximation of the effective refractive index based on weighted exponents of the nanopillars and voids. The plot shows that this effective medium approximation (EMA) overestimates the effective refractive index. Further calculations also show that the total phase accumulation is lower within a defined range of diameters for each wavelength. [Example]

[0156] As described, an implementation of a monochromatic subhogel design for use as an achromatic metasurface for high-definition light field displays is provided. To determine the appropriate subhogel size for a minimum light field display with a 3.15-inch screen size and a 40° field of view (FOV), given that the angular resolution of the human eye is limited to β = 0.03°, the minimum viewing distance of the display is JPEG2026000955000047.jpg14153

[0157] The minimum viewing distance is JPEG2026000955000048.jpg15153

[0158] Near point d of the human eye np Since this (the closest point the eye can focus on, approximately 25 cm) is greater than the minimum viewing distance of the display, this has previously often led to the determination of the maximum subhogel pitch using the following relationship: JPEG2026000955000049.jpg12153

[0159] Therefore, assuming 4 μm x 4 μm subpixels, it is determined that an observer would only be able to distinguish individual subpixels for 16 x 16 subhogels at the near point. 8 x 8 subhogels would meet the angular resolution of the eye at the near point and at the smaller minimum viewing distance of the described embodiment of the light field display. [Example]

[0160] Detailed subhogel size calculations for light field displays are described below. The lower limit of the subhogel size is set by the diffraction limit and the Rayleigh criterion. The choice of subhogel size is not trivial. The size must be large enough to satisfy the Rayleigh criterion and the diffraction limit, but small enough so that the individual subpixels are indistinguishable at the smallest viewing distance, which is either the near point of the eye or the minimum viewing distance of the display. The diffraction limit is given by: JPEG2026000955000050.jpg16153 where Δθ is the subhogel angular pitch, λ is the source wavelength, Δx is the subhogel pitch, and φ out is the deflection angle. PS=2Φ pitch Then the expression for the angular pitch of the subhogels from

[0106] becomes: Δθ=(n sh +2)Φ pitch where n sh is the number of subpixels per subhogel along each direction (x and y), and Φ pitch h is the angular pitch of each view. The sub-hogel pitch should be chosen as follows: JPEG2026000955000051.jpg16153

[0161] The Rayleigh criterion for resolving two points through an aperture of diameter d (assuming a plane wave of light) is given by: JPEG2026000955000052.jpg12153 where φ is the smallest angle between two distinguishable points, and therefore, JPEG2026000955000053.jpg14153

[0162] By taking the ratio of the two limits, we find that the Rayleigh criterion dominates in setting the minimum subhogel pitch. shThe value of n indicates the number of rows and columns of the sub-pixel array within the sub-hogel. For example, sh = 8 corresponds to a sub-hogel with an 8x8 sub-pixel array. In this example, Figure 12 shows sh For the subpixel shapes shown in Figure 12, Table 2 shows the subpixel shapes for n = 4. sh It shows that the Rayleigh criterion is satisfied when >8. For a 64x64 view display, corresponding to 64x64 pixels per hogel, it is convenient to limit the number of subpixels per subhogel to a factor of 64. [Table 2]

[0163] The smallest resolvable circle in the plane of the display has diameter d given by pixMAX It has. d pixMAX =d min tantanβ≒d min β where d min is the minimum viewing distance of the display, and β = 0.03°, which is the angular resolution of the eye. For humans to perceive the intended pixel color rather than the individual sub-pixels, the sub-pixels that make up a pixel must be of diameter d pixMAX At the near point of the eye (25 cm), d pixMAX =131μm

[0164] 13 shows the smallest diameter 78 encompassing the top left subpixel of the R sub-hogel, the G sub-hogel, and the B sub-hogel. These three sub-pixels make up one pixel. The diameter is given by: JPEG2026000955000055.jpg11154where Δx *sh (Δy *sh ) is the sub-hogel pitch in the X (Y) direction of the R sub-hogel, G sub-hogel, or B sub-hogel, and Δx *sp (Δy *sp) is the subpixel pitch in the X (Y) direction of the R subpixel, G subpixel, or B subpixel. pix =d pixMAX Set n sh By defining ∑ to be a positive real number, we find that the maximum number of subpixels per subhogel is 16 for a minimum viewing distance of 25 cm (the near point of the human eye). However, the minimum viewing distance of many displays is greater than the near point of the human eye, and is given by: JPEG2026000955000056.jpg16153Where, W disp is the display diagonal and FOV is the field of view. Figure 14 plots the minimum display diagonal versus FOV for different sub-hogel sizes.

[0165] FIG. 15 shows a triplet of three monochromatic subhogels 66a, 66b, and 66c. Each monochromatic subhogel 66a, 66b, and 66c is composed of a single-color 4×4 subpixel. Clustering similarly colored subpixels allows the metasurface 82 to be designed with unique properties tailored for each color region. The advantage of this configuration is that the metasurface 82, which acts as a directional optical element, can be designed to have unique properties for specific wavelengths or colors of light. Creating color regions in the metasurface 82 that are larger than a single subpixel enables practically manufacturable metasurface designs. The metasurface color region 84 for red is aligned directly on top of the red subhogel 66a. The metasurface color region 86 for green is aligned directly on top of the green subhogel 66b. Similarly, the metasurface color region 88 for blue is aligned directly on top of the blue subhogel 66c.

[0166] FIG. 16A shows a top view of a metasurface 82 design according to one embodiment of the present disclosure, designed for a 3×3 array of three monochromatic subhogels using the metasurface design method disclosed by FIG. 1.

[0167] FIG. 16B shows an isometric view of the metasurface 82 design according to one embodiment of the present disclosure, comprising a 3×3 array of three sub-hogels, designed using the metasurface design method disclosed by FIG. 1.

[0168] 17A shows a top view of a metasurface 82 design according to one embodiment of the present disclosure, designed for a radial array of 32 monochromatic subhogels. The metasurface 82 shown has different color regions for each subpixel color: color region 84 is designed for red subhogels, metasurface color region 86 is designed for green subhogels, and metasurface color region 88 is designed for blue subhogels. Metasurface color regions 84, 86, and 88 are composed of nanopillars 90 designed using the metasurface design method disclosed in FIG. 1.

[0169] FIG. 17B shows an isometric view of the metasurface 82 design according to one embodiment of the present disclosure, comprising a metasurface tailored for a radial array of 32 subhogels, and further shows an isometric view of a nanostructure, in this case, a nanopillar, comprising a metasurface designed using the disclosed metasurface design method.

[0170] All publications, patents, and patent applications mentioned in this specification are indicative of the level of skill of those skilled in the art to which this invention pertains, and are hereby incorporated by reference. The reference to any prior art in this specification is not, and should not be construed as, an acknowledgment or any form of suggestion that such prior art forms part of the common general knowledge.

[0171] The invention being thus described, it will be obvious that the same may be varied in many ways. Such variations are not to be regarded as a departure from the scope of the invention, and all such modifications as would be obvious to one skilled in the art are intended to be included within the scope of the following claims.

Claims

1. 1. An optical device, comprising: a hogel array comprising a plurality of hogels, each hogel divided into a plurality of monochromatic sub-hogels each comprising a plurality of monochromatic sub-pixels; a directional optical element for directing light from the subpixels, the directional optical element being divided into a plurality of color regions, each color region being designed to direct light of a particular color, the monochromatic subpixels and the plurality of color regions being configured such that the plurality of monochromatic subpixels are aligned with the color regions of the directional optical element designed to direct light of the particular color of the monochromatic subpixels; An optical device comprising:

2. The optical device of claim 1 , wherein the directional optical element is a metasurface.

3. The optical device of claim 2 , wherein the metasurface comprises nanostructures.

4. The optical device of claim 3 , wherein the nanostructures comprise titanium dioxide.

5. 5. The optical device of claim 1, wherein each of the monochromatic sub-pixels is individually addressable.

6. 6. The optical device of claim 1, wherein the plurality of monochromatic sub-hogels comprises at least one monochromatic red sub-hogel, at least one monochromatic green sub-hogel, and at least one monochromatic blue sub-hogel.

7. 7. The optical device of claim 1, wherein each monochromatic sub-hogel comprises fewer monochromatic sub-pixels than can be individually distinguished by the human eye.

8. 8. The optical device of claim 1, wherein each monochromatic sub-hogel has between 2 and 144 monochromatic sub-pixels.

9. Each subpixel is 10 μm 2 The optical device according to claim 1 , wherein the optical device is smaller than

10. The optical device of claim 1 , wherein the monochromatic sub-pixels in each monochromatic sub-hogel are arranged in a square configuration, a rectangular configuration, or a radial configuration.

11. 11. The optical device of claim 1, wherein the directional optical element is a geometric metasurface, a Pancharatnam-Berry metasurface, an inversely designed metasurface, a dispersive phase compensation metasurface, or a combination thereof.

12. 12. The optical device according to claim 1, wherein the optical device is a light field display.

13. A method for designing a segmented optical metasurface, comprising: defining a phase function for the metasurface; and specifying a material for nanostructures in the metasurface; and determining a fabrication configuration such that the metasurface is divided into a plurality of color regions; determining nanostructure parameters for each color region; generating a transmission map for the metasurface based on the nanostructure parameters; and designing each color region based on the nanostructure parameters and the transmission map to obtain the phase function, wherein each color region is designed to guide light of a specific optical bandwidth; Calculating a figure of merit for the designed metasurface; and generating an output metasurface design for the metasurface; and A method comprising:

14. The method of claim 13 , wherein the nanostructure parameters of each color region are different.

15. 15. The method of claim 13 or 14, wherein the material for the nanostructures is titanium dioxide.

16. 16. The method of any one of claims 13 to 15, wherein the metasurface is divided into red, green, and blue regions.

17. 17. The method of claim 13, further comprising, after calculating the figure of merit for the designed metasurface, adjusting the nanostructure parameters and recalculating the figure of merit.

18. 18. The method of claim 13, wherein the parameters for the nanostructures in each color region include nanostructure height, nanostructure shape, unit cell spacing, resonance boundary parameters, or a combination thereof.

19. 19. The method of any one of claims 13 to 18, wherein the nanostructures have a consistent height across the color region.

20. 20. The method of claim 13, wherein the metasurface is a geometric metasurface, a Pancharatnam Berry metasurface, an inversely designed metasurface, a dispersion-phase-compensated metasurface, or a combination thereof.

21. 1. A method for displaying a light field, comprising: subdividing the integral image into a plurality of component images, each component image representing a two-dimensional array of angular descriptors associated with a pair of orientation coordinates; Decomposing each element image into a plurality of color channel-specific element images; Sending each element image to a hogel, each hogel comprising a plurality of subpixels, divided into monochromatic sub-hogels comprising a plurality of monochromatic sub-pixels, and each element image specific to a color channel being sent to a monochromatic sub-hogel of the same color; Creating a light field for display A method comprising:

22. 22. The method of claim 21, wherein the monochromatic subpixels are adjacent to each other in the monochromatic sub-hogel.

23. 23. The method of claim 21 or 22, wherein each of the plurality of elemental images is of equal size.

24. 24. The method of any one of claims 21 to 23, wherein the color channel specific component images comprise a red channel, a green channel, and a blue channel.

25. 25. The method of any one of claims 21 to 24, further comprising individually addressing the sub-pixels.

Images