Color multispectral image sensor based on three-dimensional engineering material

Abstract

The invention relates to a color multispectral image sensor based on a three-dimensional engineering material, and relates to a method and equipment for constructing and using a multifunctional scattering structure. The disclosed methods and apparatus take into account multiple objective functions, and may be implemented using two-photon polymerization or multi-layer lithography based fabrication methods. Exemplary apparatuses for use as wave separators are also described. Results to validate the performance and benefits of the disclosed teachings are also described.

Description

[0001] This application is a divisional application of the invention patent application with application date of October 17, 2019, application number 201980067786X, and invention name “Color multispectral image sensor based on three-dimensional engineering materials”.

[0002] CROSS-REFERENCE TO RELATED APPLICATIONS

[0003] This application claims the benefit of U.S. patent application 62 / 748,677, filed on October 22, 2018, entitled “Color and Multi-Spectral Image Sensor Based On 3D Engineered Materials,” and U.S. patent application 62 / 776,685, filed on December 7, 2018, entitled “Color and Multi-Spectral Image Sensor Based On 3D Engineered Materials,” which are hereby incorporated by reference into this application.

[0004] Government funding statement

[0005] This invention was made with government support under DARPA Grant No. HR0011-17-2-0035. The government has certain rights in this invention. Technical Field

[0006] The present disclosure relates to multifunctional optical elements, and more particularly, to multifunctional metamaterial devices formed by structuring the refractive index in a single piece of material, such as three-dimensional (3D) scattering structures, methods for generating such structures, and methods for splitting an electromagnetic wave into multiple electromagnetic waves. Background Art

[0007] Optical systems are often designed through the modular combination of multiple elements to achieve complex functionality. For example, lenses and diffractive optics can be combined to perform hyperspectral imaging. This approach is both intuitive and flexible, providing access to a wide range of functionality from a limited set of components. However, the overall size and weight of the optical system can limit its range of applications. Recent advances in nanofabrication can alleviate this limitation by replacing bulky components with planar arrays of metasurfaces with resonant nanostructures of subwavelength thickness. By engineering the scattering of individual elements in the array, these devices can reproduce the versatility of complex optical systems in a single component. However, efforts to combine multiple metasurfaces to achieve more complex functionality have been hampered by reduced scattering efficiency, which is inversely proportional to the number of tasks performed simultaneously.

[0008] The inherent trade-off between versatility and efficiency in these systems is due to the limited degrees of freedom, which scale with the device's volume and maximum refractive index contrast. In particular, this limits the range of independent functions that can be implemented in any ultrathin system, such as sorting light according to frequency, polarization, and angle of incidence. Through contrast, three-dimensional scattering elements with thicknesses greater than the wavelength can often encode many simultaneous functions, although to date only with low efficiency due to weak scattering and refractive index contrast.

[0009] Historically, optical design has been modular, offering intuitive ways to build and reconfigure optical setups. Advances in nanofabrication have enabled the fabrication of structures with subwavelength feature sizes, enabling multifunctional optical components that can incorporate the functionality of more complex setups. Examples include metasurface lenses that can separate different polarization and spectral bands. However, the performance and functionality achievable using metasurfaces and other planar structures is inherently limited by the number of optical modes that can be controlled.

[0010] The ability to engineer high-contrast refractive indices on the subwavelength scale offers a vast optical design space for demonstrating multifunctional optical components. To date, this has primarily been achieved with two-dimensional structures or metasurfaces. However, their performance is limited by the available optical degrees of freedom.

[0011] To highlight the benefits of the teachings of the present disclosure in the following sections, an example of an image sensor is considered. Currently, most sensors record colors using absorption filters. Figure 1 A prior art image sensor is shown in which every four adjacent pixels have an absorptive color filter on top: two for green, one for blue, and one for red. The problem with this image sensor is that because most of the light is absorbed, the efficiency is limited to around 30%.

[0012] The disclosed methods and apparatus address the described problems and provide solutions to the aforementioned issues. Summary of the Invention

[0013] The resulting optical design space is a three-dimensional volume where the refractive index can be controlled at will with a spatial resolution less than the minimum relevant wavelength. In this case, the number of optical degrees of freedom is enormous and can be used to achieve completely non-intuitive multifunctional designs with high performance. The teachings of the present disclosure are based on this concept.

[0014] The disclosed method is based on designing three-dimensional scattering elements through iterative gradient-based optimization, while considering multiple objective functions. The present method and device improve upon existing optical devices by encoding various functions into complex multi-scattering within a volume, rather than on a single surface. The disclosed method does not rely on local effective medium assumptions or higher refractive index contrasts that are typical for subsurfaces, allowing for efficient devices with rough features above the diffraction limit. Embodiments according to the present disclosure can also be manufactured using standard multilayer fabrication with modest requirements on feature size and number of layers.

[0015] According to a first aspect of the present disclosure, there is provided a 3D scattering structure formed in a set of three-dimensional (3D) patterns based on one or more sets of target functions, wherein the 3D scattering structure is configured to: receive electromagnetic waves; and scatter the electromagnetic waves to provide the one or more sets of target functions.

[0016] According to a second aspect of the present disclosure, a method for separating an electromagnetic wave into multiple waves with different wavelengths is disclosed, providing: applying an electromagnetic wave to a first side of a three-dimensional (3D) scattering structure, the 3D scattering structure being formed into a set of 3D patterns; and scattering the electromagnetic wave to generate multiple electromagnetic waves with different wavelengths, the multiple electromagnetic waves exiting the 3D scattering structure at its output second side. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Other aspects of the disclosure are provided in the description, drawings, and claims of the present application.

[0018] Figure 1 A prior art image sensor is shown.

[0019] Figure 2A-2A' An exemplary three-dimensional (3D) scattering structure according to an embodiment of the present disclosure is shown.

[0020] Figures 2B-2C Shown Figure 2A-2A' Wavelength separation functionality of an embodiment.

[0021] Figures 3A-3C An exemplary three-dimensional (3D) scattering structure according to another embodiment of the present disclosure is shown.

[0022] Figures 4A-4E Performance results according to exemplary embodiments of the present disclosure are shown.

[0023] Figure 5 An exemplary arrangement including a microwave 3D scattering structure according to an embodiment of the present disclosure is shown.

[0024] Figure 6 Shown with Figure 5 Example performance results associated with the arrangement of .

[0025] Figure 7 Several steps of an exemplary optimization algorithm according to an embodiment of the present disclosure are shown.

[0026] Figure 8 An exemplary flow chart illustrating various implementation steps of an optimization algorithm based on this concept and in accordance with further embodiments of the present disclosure is shown. DETAILED DESCRIPTION

[0027] Figure 2A An image sensor (200) according to an embodiment of the present disclosure is shown. The image sensor (200) includes a three-dimensional (3D) scattering structure (201) that functions as a spectrum separator. The 3D scattering structure (201) includes a plurality of dielectric pillars (205) that are formed to scatter light in a predetermined pattern. Incident light (202) passing through the 3D scattering structure (201) is scattered by the dielectric pillars. By arranging the dielectric pillars (205) according to one or more target functions, the scattering pattern is customized to perform a desired function. As an example, the 3D scattering structure (201) can be designed as a spectrum separator to simultaneously sort and focus the incident light (202) into any number of wavelengths ( ), each wavelength is directed to a single pixel on the focal plane (203) placed below the 3D scattering structure (201), as Figure 2A According to an embodiment of the present disclosure, the 3D scattering structure (201) may be a porous polymer cube or a dielectric cluster or a semiconductor (eg silicon) particle cluster embedded in a silicon oxide matrix.

[0028] Those skilled in the art will understand that Figure 1 Compared with the prior art image sensor (100), Figure 2A The image sensor (200) does not function based on absorption and, therefore, provides a significant improvement in efficiency compared to existing solutions. This will be quantified later using exemplary embodiments of the present teachings. As described in more detail throughout the disclosure, the disclosed apparatus and method provide the following additional benefits over existing solutions:

[0029] · Figure 2A The 3D scattering structure (201) can be manufactured by known photolithography processes.

[0030] · Figure 2A The 3D scattering structure (201) can be designed to function as a spectrum separator for any spectral band of infrared, mid-infrared, etc. In other words, in addition to hyperspectral imaging, thermal imaging is another potential application of the disclosed teachings.

[0031] • The spectrum separation function can be combined with other desired functions such as polarization separation.

[0032] • The embodiments according to the present disclosure can also be designed to perform optical image processing, such as Gabor filtering for edge detection.

[0033] Figure 2A' An image sensor (200') according to an embodiment of the present disclosure is shown, which includes an exemplary three-dimensional (3D) scattering structure (21) used as a spectral filter. Incident light (22) entering from above is scattered when passing through the 3D scattering structure (21) and is classified in a focal plane (23) consisting of four sub-pixels displayed as red, blue, green (x polarization) and green (y polarization). Figure 2A' As shown, the red (600–700 nm) and blue (400–500 nm) spectral bands are sorted into opposite quadrants. Furthermore, the green (500–600 nm) spectral band is further separated based on linear polarization. The red and blue quadrants can be polarization-independent.

[0034] According to an embodiment of the present disclosure, a 3D scattering structure (21) can be designed using an adjoint variable method, which generates a structure that optimizes a specific objective function. As an example, and with reference to Figure 2A' The objective function can be selected based on the focusing efficiency of incident light into one of four target regions, depending on frequency and polarization. Starting with an empty volume, full-wave finite-difference time-domain (FDTD) simulations are performed to calculate the sensitivity of this figure of merit to refractive index perturbations. The prescribed scattering structure is iteratively formed and updated. In other words, the optimized design is generated by iteratively updating the initial geometry, improving performance with each step. Sensitivity can be calculated in just two simulations, allowing for efficient optimization of three-dimensional devices with modest resources. Sensitivity can be calculated for multiple incident wavelengths across the visible spectrum, assigning each spectral band to a different quadrant: red (600–700 nm), green (500–600 nm), and blue (400–500 nm). The spectrally averaged sensitivity is then used to update the device's refractive index.

[0035] Figures 2B-2C Shown Figure 2A' Simulated intensity of incident light within the 3D scattering structure (21). Figure 2A' Diagonal cross-section analysis of the intensity in the red and blue quadrants of the image. Each wavelength undergoes multiple scattering before converging to its respective target area. Figure 2CThe intensity distribution of incident light passing through a green pixel in a diagonal cross section is shown for two orthogonal input polarizations. In both cases, a plane wave incident from above (λ = 550 nm) is preferentially routed to the pixel corresponding to its polarization. Simultaneously, for the red and blue spectral bands, both polarizations are assigned to the same region, preserving the mirror symmetry of the objective function.

[0036] According to an embodiment of the present disclosure, Figure 2A' The 3D scattering structure (21) sorts red, green, and blue light with efficiencies of 84%, 60%, and 87%, respectively. Throughout this disclosure, efficiency is defined as the fraction of the total power incident on the device that reaches the target quadrant within the spectrum for which the device is designed (i.e., Figure 2A' averaged over the visible spectrum).

[0037] With reference to Figures 2A and 2A', those skilled in the art will appreciate that the disclosed concepts offer considerable flexibility in defining the target scattering function, with independent control for any incident polarization, angle, or frequency. However, the complex three-dimensional structures present significant manufacturing challenges. Large-scale implementation of these devices in image sensors at visible wavelengths will require high manufacturing yields with sub-100 nm resolution. This can be achieved through multilayer lithography, in which the three-dimensional device is built through repeated material deposition and patterning. Here, each layer comprises a series of patterned mesas composed of a high-refractive-index dielectric. The interstitial spaces are filled with a low-refractive-index dielectric, forming a flat surface that serves as a substrate for subsequent layers.

[0038] To further illustrate the layered fabrication approach discussed above, refer to Figure 3A and 3C , showing Figure 3C hierarchical design of the 3D scattering structure (31). In other words, Figure 3C The 3D scattering structure (31) can be obtained by Figure 3A The 3D scattering structure (31) may be constructed by stacking a plurality of layers (301, ..., 305) on top of each other. The manufacturing process may be CMOS-compatible, where manufacturing constraints may be directly coupled to the design algorithm. Each layer (301, ..., 305) may be produced using photolithography. The 3D scattering structure (31) may be composed of titanium dioxide and silicon dioxide, which are materials that are transparent at visible frequencies. The layers (301, ..., 305) may be 2 μm x 2 μm layers, each 400 nm high. Those skilled in the art will appreciate that these are exemplary dimensions for descriptive purposes, and that embodiments of the present disclosure may conceivably have dimensions and numbers of layers other than those described above. As Figure 3B As shown, each layer can consist of a set of irregular titanium dioxide terraces surrounded by silicon dioxide. Figure 3B The photolithography process can begin by growing a thin layer of dielectric (e.g., titanium dioxide) on top of a substrate (e.g., silicon dioxide). The pattern is transferred to this layer via photolithography, and the unprotected material is etched away to reveal the two-dimensional dielectric structure. Finally, the surface is coated (deposited) with a low-refractive-index dielectric and mechanically polished (planarized). By repeating the same process for each layer and stacking the layers, the desired 3D structure is produced. As mentioned above, this photolithography process offers flexibility in material design and is compatible with industry-standard CMOS manufacturing processes.

[0039] In the following, some exemplary performance results associated with the disclosed apparatus are presented.

[0040] Similar to About Figure 2A' As described, Figure 3C An image sensor (300) is shown that includes a 3D scattering structure (31) arranged on top of a focal plane (33), the focal plane (33) including four sub-pixels corresponding to red, blue, green (x polarization), and green (y polarization), and arranged in separate quadrants. Incident light (32) entering from above is effectively scattered when passing through the 3D scattering structure (31) and is sorted into separate sub-pixels in the focal plane (33). The incident light (32) can be linearly polarized light, unpolarized light, or light with other polarization states. The incident light is sorted on the image sensor based on frequency and polarization of certain frequencies. When designing the 3D scattering structure (31), and to further demonstrate practical aspects of the present teachings, a minimum feature size requirement of 60 nanometers can be set so that the design can exclude some small features that are not suitable for moderate manufacturing processes.

[0041] Figure 4A Shown with Figure 3C Example transmission spectra of an image sensor (300) of FIG. 41C, 42C, 43C represent transmission plots as a function of wavelength for each color (blue, green, red), respectively. Figure 1 As discussed, the dashed line (44A) corresponds to a typical achievable result using a prior art absorption filter.

[0042] Figure 4B Shown with Figure 3C An exemplary transmission spectrum of an image sensor (300) when the incident plane of light is tilted 3 degrees. Curves (41C, 42C, 43C) represent transmission diagrams as a function of wavelength of color (blue, green, red), respectively. Figure 1 As discussed, the dashed line (44B) corresponds to the typically achievable results using prior art absorption filters.

[0043] Figure 4C Shown with Figure 3C An exemplary transmission spectrum of an image sensor (300) when the incident plane of light is tilted 6 degrees. Curves (41C, 42C, 43C) represent transmission graphs as a function of wavelength of color (blue, green, red), respectively. Figure 1 As discussed, the dashed line (44C) corresponds to a typical achievable result using a state-of-the-art absorption filter. Since the 3D scattering structure (31) is not optimized for any particular angle of incidence, it is expected that Figures 4B-4C The results shown are Figure 4A The ones shown are even worse.

[0044] Figure 4D Shown with Figure 3C The exemplary transmission spectra of the image sensor (300) are related to the image sensor (300) and the incident plane of the light is tilted by 20 degrees. In this case, the design has been optimized to take into account the 20 degree incident angle. The curves (41D, 42D, 43D) represent the transmission graph as a function of the wavelength of the color (blue, green, red), respectively. As shown in reference Figure 1 As discussed, the dashed line (44D) corresponds to a typical achievable result using a prior art absorption filter.

[0045] Throughout this document, for the purpose of describing the disclosed methods and devices, exemplary plane waves are used as inputs to structures fabricated in accordance with the teachings of this disclosure. However, one skilled in the art will appreciate that other devices in accordance with embodiments of the present disclosure may be fabricated in which the inputs may be waves other than plane waves. A Gaussian beam is one example. One may also envision structures that apply different functions to different kinds of input beam profiles. This is referred to throughout this document as "spatial distributions" or "optical modes". Mode profiles vary widely, and this is determined by the spatial distribution of amplitude and phase. Structures fabricated in accordance with the teachings of this disclosure may be linear devices, i.e., they may distinguish orthogonal modes.

[0046] According to embodiments of the present disclosure, 3D structures may be fabricated wherein the classification of input electromagnetic waves may be based on 1) one or more wavelengths, 2) one or more polarizations, 3) the incident angle of the electromagnetic wave, 4) spatial distribution, or a combination thereof.

[0047] Further references Figures 3A-3C The trade-off between multifunctionality and device thickness is being investigated by designing a series of 3D scattering structures with different numbers of layers. Each structure follows the same design algorithm using 400 nm layers and as previously discussed Figures 3A-3C As described in the embodiments. Figure 4EThe classification efficiency, polarization contrast, and color contrast of each scattering structure are shown, averaged across the visible spectrum. While the performance of a single-layer metasurface is slightly better than that of empty space, the efficiency steadily increases with increasing device thickness. Furthermore, thicker structures show improved color and polarization contrast. Contrast is defined herein as the difference in normalized power between the two strongest quadrants and therefore reflects the ability to distinguish incident color and polarization. With five layers, the volume scattering element outperforms the absorptive filter in classification efficiency (58%), color contrast (28%), and polarization contrast (41%).

[0048] Microwave frequency

[0049] Maxwell's equations are known to be scale-invariant, meaning that the behavior of any physical system is preserved when the wavelength and dimensionality are scaled by a common factor. This fact has been used to demonstrate constrained design using large-scale analog operations at microwave frequencies. In other words, according to the teachings of this disclosure, scattering devices operating in the Ka-band (26-40 GHz) with centimeter-scale dimensions can be realized.

[0050] Figure 5 A microwave device (500) is shown that includes a 3D scattering structure (51) that acts as a microwave filter. The 3D scattering structure (51) is constructed from a stack of 20 patterned polypropylene sheets (refractive index = 1.5), each 1.6 mm thick, assembled into a cube. The minimum feature size can be limited to 1 mm. Metal boundaries can also be incorporated into the sides of the cube to limit interference from the measurement device. The microwave device (500) occupies an area of ​​35 mm x 35 mm, which is the same as its optical analog relative to the operating wavelength.

[0051] The 3D scattering structure (51) is characterized by measuring the complex microwave field scattered by the 3D scattering structure (51). In the example shown in the figure, the 3D scattering structure (51) is illuminated by a collimated Gaussian beam (full width half maximum, FWHM = 25 mm) generated by a vector network analyzer (not shown) coupled to free space via a microwave horn antenna (52) and a focusing mirror (56). As previously described, the input beam passes through the structure (51) and is scattered into the far field. The local electric field at the measurement plane (56) 62 mm outside the output aperture of the 3D scattering structure (51) is measured using a WR-28 waveguide flange in order to recover the complex scattering amplitude S21. The measurement is then deconvolved and back-propagated to obtain the result at the focal plane (55).

[0052] The analysis was repeated for the microwave frequency range within the Ka-band (26–40 GHz) and for two orthogonal polarizations of the input beam. To measure the scattering parameters for the orthogonal polarizations, the 3D scattering structure was rotated 90 degrees.

[0053] Figure 6 The simulated and measured intensities of the microwave field at the focal plane (55) and for a specific polarization are shown. The curve corresponding to the measured results is represented by a solid line, and the curve corresponding to the simulated results is represented by a dashed line. Figure 6 The graph shown in represents the combined intensity of all frequencies within the measurement bandwidth, normalized to the total measured power at the focal plane for each frequency. The pairs (61, 61'), (62, 62'), and (63, 63') correspond to green, blue, and red, respectively. The colors correspond to the observed hues of the simulated optical field when the wavelength is scaled by a factor of 1.75×10^4. The curve shown represents Figure 5 The relative classification efficiencies of the 3D scattering structure (51) across the measured spectra. These efficiencies are defined as the power projected through each target quadrant, normalized to Figure 5 The total power at the focal plane (55°) is obtained. Close agreement between experimental and simulated efficiencies is observed. Each band shows efficient classification, with low crosstalk of approximately 10% for out-of-band light. The sharp transitions between spectral bands highlight the improved color discrimination relative to typical dispersive scattering elements.

[0054] Referring again to Figures 3-5 , as previously mentioned, the disclosed device can be fabricated using multilayer photolithography. Embodiments according to the present disclosure can also be constructed using two-photon polymerization (TPP), where the desired structure is directly printed. This method is similar to 3D printing, but occurs at a microscopic scale. For example, a laser can be focused onto the center of a liquid polymer, causing the polymer to crosslink and harden at the laser's focal point. By moving the laser's focal point, three-dimensional structures with arbitrary geometries can be generated.

[0055] Optimization Algorithm

[0056] Gradient Descent

[0057] Reference again Figures 2A'-3C , and as previously described, a three-dimensional dielectric structure optimized to perform the target light scattering function is designed according to the teachings of the present disclosure. Figures 2A'-3C In the case of the exemplary embodiment shown, such a target scattering function includes focusing an incident plane wave to different locations depending on frequency and polarization. The exemplary three-dimensional (3D) scattering structure (21, 31) is composed of a spatially dependent refractive index distribution within a cubic design area. This represents a vast design space capable of expressing a wide variety of complex optical versatility. However, determining the optimal refractive index profile for a given objective function remains a challenging inverse design problem, especially for strongly scattering devices.

[0058] To overcome this challenge, and in accordance with the teachings of the present disclosure, an iterative approach guided by gradient descent can be implemented, where, starting with an initial refractive index profile, full-wave simulations (FDTD) are used to calculate the sensitivity of focusing efficiency to refractive index perturbations. Sensitivity can be calculated with only two simulations, allowing for efficient optimization of three-dimensional devices with modest resources. Based on the sensitivity, the initial design is modified to maximize performance while complying with manufacturing constraints. This updating process is repeated until the optimized device is able to effectively perform the target function.

[0059] To further clarify the above, refer to Figure 7 , Figure 7 The multiple steps of the gradient based optimization algorithm according to an embodiment of the present disclosure are shown. In step (81), the algorithm is initialized with a uniform refractive index distribution, ,in and The distribution is continuously updated to maximize the electromagnetic intensity at the target position in the focal plane. This objective function serves as a proxy for focusing efficiency while simplifying sensitivity calculations. In step 74 the electromagnetic fields from the two FDTD simulations (forward and adjoint) (steps (72, 73)) are calculated according to the following expressions:

[0060] (1)

[0061] in is the electric field inside the cube when illuminated from above with a plane wave, step (72); is the electric field inside the cube when illuminated from below by a point source at the target location, step (73). In the forward simulation, the phase and amplitude of the point source are given by the electric field at the target location. The sensitivity can be calculated for multiple incident wavelengths and polarizations across the visible spectrum, assigning each spectral band to a different quadrant: red (600 nm–700 nm), green (500 nm–600 nm), and blue (400 nm–500 nm). The spectrally averaged sensitivity is then used to update the refractive index of the device in step (74) using the following formula:

[0062] (2)

[0063] The step size α can be fixed to a small fraction (e.g., α = 0.001) to ensure that the change in the refractive index can be treated as a perturbation in the linear region. The sensitivity is recalculated after each update. After several iterations, the algorithm converges to the optimal design, step (75), where the resulting structure focuses the incident light with the desired efficiency.

[0064] Manufacturing limitations

[0065] A. Binary Refractive Index

[0066] During the optimization process, a set of constraints can be imposed on the refractive index profile required for the manufacturing process. According to embodiments of the present disclosure, a high contrast 3D scattering element can be constructed from two materials. While the gradient descent algorithm detailed above produces an optimized device with a gradient refractive index, the binary condition can be further improved by introducing an auxiliary density in the range [0, 1]. To implement. Figure 8 An exemplary flow chart illustrating various implementation steps of a gradient-based algorithm based on this concept and in accordance with further embodiments of the present disclosure is shown.

[0067] Further references Figure 8 ,density First, it is initialized. Through the S-type projection filter, this density is related to the refractive index distribution, steps (82, 83):

[0068] (3)

[0069] where the parameter β controls the filter strength. For small β, the refractive index profile is equal to the density scaled by the available refractive index range. For large β, the S-type filter approximates a Heaviside function, and the refractive index profile is pushed to either extreme. Importantly, the filter function is continuously differentiable, so the sensitivity can be expressed in terms of density: , as shown in steps (85, 86). Similar to what is described with respect to equation (2), the sensitivity can be calculated based on averaging across the desired spectral range, as shown in step (87). During optimization, step (84), the density The design is parameterized by β, gradually increasing the strength of the filter. In the early stages of this iterative process, β is small, which is equivalent to the unfiltered case. Over time, as the number of iterations increases, the optimized refractive index profile is gradually pushed towards the binary design as the strength increases, even if the density remains continuous. In step (88), the density is updated using the calculated sensitivity. The convergence criteria are then checked in step (88). If such criteria are not met, the parameter β is increased in step (89) to update the density, and the algorithm proceeds to the next iteration. If the convergence criteria are met in the current iteration, the optimized design is achieved, as shown in step (850).

[0070] B. Minimum feature size

[0071] In addition to the material limitations described above, other embodiments according to the present disclosure and consistent with the resolution limitations imposed by the manufacturing process can be envisioned. For example, diffraction and proximity dose effects limit electron beam lithography to features of approximately 10 nanometers. The minimum feature size of the device design can be increased by introducing "expansion" density To achieve this, the density is expressed at each point The maximum density in the neighborhood Ω :

[0072] (4)

[0073] For sufficiently large exponents M, this operation approximates morphological dilation. However, it is continuously differentiable with respect to the parameters. Therefore, the sensitivity can be expressed in terms of the undilated density: During the optimization process, the device is divided into two groups: parameterized, while the refractive index is given by the expanded density Define the neighborhood Ω as a circle, where the radius represents the minimum feature size.

[0074] C. Connection layer design

[0075] As about Figures 2A'-3C As discussed in the illustrated embodiments, some device designs may be intended to be fabricated using multilayer 2D lithography, consisting of several vertically invariant patterned plates. In this case, optimization can be constrained by averaging the calculated sensitivity in the vertical direction within each layer. In effect, the pixels within each layer depend on a shared 2D profile.

[0076] As another example, refer to Figure 5 3D scattering structures (51) where the design is further constrained so that each layer is fully connected without floating blocks. Connectivity can be imposed directly by periodically adding bridges between disconnected islands within each layer. This intervention does not take sensitivity into account and typically results in a small degradation of device performance. Therefore, connectivity constraints can be applied, for example, every 40 iterations, allowing performance to recover thereafter.

Claims

1. A 3D scattering structure formed in a set of three-dimensional (3D) patterns based on one or more sets of objective functions, wherein the 3D scattering structure is configured to: Receiving electromagnetic waves; and Electromagnetic waves are scattered to provide one or more sets of objective functions.

2. The 3D scattering structure of claim 1 , formed by stacking layers, wherein the set of 3D patterns is generated using a combination of two-dimensional (2D) dielectric structures, and Each stacked layer includes: a dielectric layer above the substrate; and A 2D pattern on a dielectric layer, the 2D pattern comprising etched 2D structures according to one or more sets of target functions.

3. The 3D scattering structure according to claim 1 or 2, wherein the one or more target functions are based on classifying electromagnetic waves into one or more target regions, and wherein the classification is performed according to 1) one or more wavelengths, 2) one or more polarizations, 3) the angle of incidence of the electromagnetic waves, 4) spatial distribution, or a combination thereof.

4. The 3D scattering structure according to claim 3, wherein: The one or more wavelengths include wavelengths corresponding to red, green, and blue; and The one or more polarizations include one or more polarization directions.

5. An image sensor comprising a 3D structure according to claim 3 or 4, wherein the one or more target areas comprise one or more pixels.

6. An image sensor comprising the 3D scattering structure according to claim 4, wherein the one or more target areas include a first sub-pixel corresponding to red, a second sub-pixel corresponding to blue, a third sub-pixel corresponding to green having a first polarization direction, and a fourth sub-pixel corresponding to green having a second polarization direction, and wherein the first, second, third, and fourth sub-pixels are adjacent sub-pixels.

7. A camera comprising a plurality of image sensors according to claim 5. The 3D scattering structure according to claim 2 , wherein each of the plurality of layers is 2 μm×2 μm and has a thickness of 400 nm.

9. The 3D scattering structure according to claim 3, operating in a wavelength range of 400 nm to 700 nm.

10. The 3D scattering structure of claim 2, wherein the dielectric layer comprises titanium dioxide, and the matrix comprises silicon dioxide.

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