Design parameters of free-form color splitters for subwavelength pixelated image sensors

Summary Metasurface-based color splitters are emerging as next-generation optical components for image sensors, replacing classical color filters and microlens arrays. In this work, we report how the design parameters such as the device dimensions and refractive indices of the dielectrics affect the optical efficiency of the color splitters. Also, we report how the design grid resolution parameters affect the optical efficiency and discover that the fabrication of a color splitter is possible even in legacy fabrication facilities with low structure resolutions.


Highlights
The optimization of metasurface-based color splitters is conducted

The influence of design parameters on optimized optical efficiency is examined
There exist optimal ranges of structural parameters and the optical index contrast The number of layers plays a crucial role in determining device performance ll OPEN ACCESS

RESULTS
As schematically shown in Figures 1B and 1A, the color splitter deflects the incident light to its corresponding subpixel area.Instead of forming a lens-like structure, the design area is gridded into rectangular cells, and each cell is filled with a selection of two different dielectrics.The design parameters for the 2D color splitters can be classified into two categories: physical parameters and spatial resolution parameters.The physical parameters include color splitter period (P), thickness (t), the position of the focal plane (h), and the refractive indices of the two composing dielectrics (n 1 and n 2 ).The spatial resolution, determined by the number of grid layer N L and the grid elements in a layer N C , defines how the design area is gridded into cells of equal shape.Consequently, the design problem possesses N L 3 N C DoF and thus the number of possible structures is 2 NLNC .The default values of each design parameter are given in Table 1.As the transition from geometric optics to wave optics occurs for geometries with characteristic lengths comparable to or smaller than the wavelength, the color splitter configured with the default design parameters lies within the wave optics regime.
In this work, we define the optical efficiency h(l) using the electric field intensity at the focal plane (denoted by the dashed line in Figures 2A-2C).
h R;G;B ðlÞ = 1 2 3 Tðl; iÞ Here, E is the electric field at the focal plane, and T is the transmittance.Electric field distribution and the total transmittance are calculated with RETICOLO, a rigorous coupled-wave analysis package. 21x ˛ðx 1 ; x 2 Þ defines the area of the subpixel of interest.For simplicity, we assume that the wavelength range required for red (R), green (G), and blue (B) subpixels are 600 nm-700nm, 500-600 nm, and 400 nm-500 nm, respectively.Throughout the work, a normally incident light is assumed, and the optical efficiency is averaged between both transverse electric (TE) and transverse magnetic (TM) polarizations.Figures 2A-2C show the electric field intensity distribution inside an optimized color splitter with the default design parameters listed in Table 1. Figure 2D shows the electric field intensity distribution on the surface of the photodetectors.The optical efficiency h R;G;B ðlÞ of the same device is shown as red, green, and blue curves in Figure 2E.The peak optical efficiencies within red (600-700 nm), green (500-600 nm), and blue (400-500 nm) regions are 70.12%(669 nm), 57.15% (554 nm), and 77.39% (422 nm), respectively, which are comparable to those reported in other related works (see Table S1 for comparison). 5,7,9,10,16,18Both the field distributions and the optical efficiency plots clearly show that the intensity of light is concentrated at the corresponding subpixel area on the focal plane.The optimized device possesses an optical crosstalk of 16.8% (see Figure S1 for detailed analysis).The optical efficiency drops rapidly as the angle of incidence (q) deviates from normal to the surface.The color splitting effect still remains for q % 7 but the efficiency becomes as low as 45.6% at q = 5 as discussed in Figure S2.
In a conventional Bayer-type image sensor, a pixel consists of two green subpixels and one subpixel for red and blue, respectively.In order to account for such a subpixel ratio, we include two green subpixels in one period of a 1D image sensor.The default arrangement of the subpixels in this paper was set to RGBG as the design is periodic and the wavelength of the green light is in between red and blue (Figure 1B).In Figure S3, we compare the optical efficiency between the RGBG subpixel arrangement and the RGGB subpixel arrangement.As Figure S3 suggests, the arrangement of subpixels has a marginal effect on the device performance in terms of optical efficiency and crosstalk.
To understand how the design parameters of a color splitter affect its performance, we optimize the device geometry for various choices of design parameters.For given device design parameters, a conventional genetic algorithm with elitism is performed to obtain the optimal dielectric distribution in the grids. 22,23The optimization is configured with a population size of 200, and 100 epochs.The genotype of the individuals in the gene pool is represented by a binary array with array dimensions equal to N C and N L .The goal of the optimization is to maximize the average optical efficiency, h = ðh R + h G + h B Þ=3, where h R;G;B are the wavelength-averaged optical efficiencies obtained by averaging h R;G;B ðlÞ over the wavelength range corresponding to the subpixel type.During the optimization process, the optical efficiencies   were averaged over thirty wavelength points (405 nm, 415 nm, .695 nm) to reduce the computational cost, but the reported h were averaged over with much finer wavelength points (400 nm, 401 nm, .700 nm).As shown in Figure S4, the difference between 30 and 301 wavelengthpoint averaging is not significant (around 0.02 for (N L , N C ) = (8, 64) and 0.01 for (N L , N C ) = (4, 32)).
The advantage of substituting microlens and color filters with metasurface-based color splitters becomes clear for sub-micron image sensors.Hence, we first investigate the effect of the physical dimensions of devices on the optical efficiency of the splitter.It should be noted that the subpixel size of the 2D color splitter is a quarter of the device period, P. In comparison to a Bayer-type image sensor array, a 2D color splitter extends infinitely in the y-direction so the pixel size is defined as the width of each subpixel in the x-direction.The pixel size of the color splitter with the default design parameter is 0.25 mm, which is less than half the size of the smallest commercially available image sensor of 0.56 mm. 3 Figure 3 shows how the optimized h varies depending on the period P and the thickness t while all the other design parameters including DoF and refractive indices are fixed to their default values.For the devices with a deep subwavelength period of p = 0.25 mm, the optimized average optical efficiencies are around the trivial value of 33%, which can be achieved with a simple antireflection layer.When p R 0.5 mm, the color splitters start to show meaningful performance.At a given P, the device performance monotonically increases and saturates as the thickness t increases.The saturation point of t for 0.75mm % p % 2 mm is around 1.5 mm, and thus we set t = 1.5 mm as the default value.We note that, however, the saturation point of t can vary as a function of the other design parameters.At a fixed t, the optimized h does not monotonically increase with P but has a specific optimal value.This result is reasonable since it becomes increasingly difficult to split incident light over a longer lateral distance within a given thickness.
The position of the focal plane from the color splitter, h, is a similar physical design parameter to P and t, which also defines the physical dimension of the device.The dependence of the focal plane position on the optical efficiency is shown in Figure S5.In a periodic grating, the modes with a high lateral wavenumber cannot be extracted in the far field.Hence, as the focal plane of the color splitter is located further from the meshed region, the device is expected to have a lower efficiency due to the loss of near field.The sharp drop in optical efficiency for h > 1 mm in Figure S5 agrees with this expectation.
The refractive indices of the composing dielectric materials are another critical factor determining optimal efficiency.In previous works, the selection of a color splitter was based on simple relations such as the Fabry-Perot resonance condition. 16Those relations only provide order-of-magnitude estimates.In this work, we tune the design parameters (t, n 1 , n 2 ) to find the global trend in optimized optical efficiency.For the sake of simplicity, we assume that the dielectrics filling each grid are dispersionless and have refractive indices of n 1 and n 2 , where n 1 % n 2 is assumed throughout the work.The default values of (n 1 , n 2 ) are (1.5, 2.0), which is similar to the refractive indices of silica and silicon nitride.Our analyses reveal that, unlike other nanophotonic devices such as metalens whose device performance monotonically increases with the refractive index contrast, [24][25][26] color splitters have a distinct relation between Figure 3. Dependency of design parameters: Period and thickness Effect of device period (P) and color splitter thickness (t) on the optical efficiency of a color splitter.N L and N C are fixed to (4, 32).The design parameters stated in Table 1 are used except for P, and t.  (A-C) Optimization based on a genetic algorithm was carried out for color splitters with thickness (a) t = 0.1 mm, (b) t = 0.5 mm, and (c) t = 1.5 mm.In each color plot, the lower refractive index n 1 is changed from 1 to 2 with a step size of 0.25, and n 2 is swept from 2 to 4 with the same step size, 0.25.Each square represents the optimized efficiency obtained with the genetic algorithm.The maximum efficiency in each case is (a) 46.86%,(b) 54.98%, (c) 58.25%.Except for t, n 1 , and n 2 , the design parameters given in Table 1 are used.
the optimal refractive index contrast and the thickness of the device.When all the other parameters are fixed to their default values, the optimal index contrast values, n 1n 2 , are found to be 2.25, 1, and 0.5 for t = 0.1, 0.5, and 1.5 mm, respectively, as illustrated in Figure 4. We speculate that the trend could be attributed to the fact that the maximum achievable vertical optical path length difference is determined by the product of optical index contrast and the thickness of the device.
The choice of DoF is important in both computational and experimental aspects.On the computational side, the design space grows exponentially with the DoF, and the computational load required for optimization grows accordingly.5][46][47] On the other hand, the DoF is directly related to the fabrication feasibility of the device.The number of layers, N L , determines the number of deposition steps, and the number of cells in a layer, N C , affects the minimum feature size.Despite its importance, previous works on metasurface-based color splitters mostly lack investigations on DoF.In this work, we fix the values of the other design parameters including the device thickness, and change N L and N C to isolate the effect arising from the device dimension change.N L and N C are chosen to be integer powers of 2. This implies the existence of trivial monotonicity.For example, a set of every possible combination with (N L = 1, N C = 8) is a subset of (N L = 4, N C = 16) so the optical efficiency of the latter must be equal to or greater than the previous one if the optimization converges to the global optimum.Since the number of possible combinations is sufficiently low for device designs with DoF %16, an exhaustive search was carried out for the corresponding conditions.For device designs with DoF R32, the previously-described genetic algorithm was carried out.
Figure 5 shows the optimized results for each N L and N C pair.In the figure, the trivial monotonic relation in the optimized efficiency is observed.Regardless of the number of layers, the optimal h almost saturates when N C R 32, which corresponds to the minimum feature size of 31 nm.The optimal h asymptotically approaches 60% for the default physical parameters.It is important to note that, the number of layers N L plays a pivotal role in determining the device performance.For example, even with N C = 4 (minimum feature size of 250 nm), it is possible to achieve the average optical efficiency of 54% (about 90% of the highest efficiency achieved in this work) by having 8 layers.A similar trend was found when N L is increased while keeping the thickness of each layer to be 375 nm as shown in Figure S6.The designs of color splitters for different DoF conditions are displayed in Figure S7.For low-efficiency devices, a line of reflection symmetry exists at the center of the red and blue subpixel.This line of reflection symmetry originates from RGBG subpixel arrangement which is also symmetric with respect to that line.However, such reflection symmetry isn't observed in the optimized devices.The lack of symmetry in the optimized devices implies that the enforcement of trivial symmetry conditions on the device design does not always lead to better performance.Finally, we note that the optical efficiency of a color splitting device can surpass what has been reported in this work and even reach near unity when the device is optimized with significantly higher DoF. 9

DISCUSSION
In conclusion, we systematically analyze the dependence of color splitter performance on various design parameters by leveraging numerical device optimization methods based on a genetic algorithm.We discover that the average optical efficiency of a color splitter with a micronscale form factor can be up to 60%, whereas the classical microlens and color filter configuration can have optical efficiency of up to 25% for red and blue and 50% for green.We show that it is not always beneficial to have a larger pixel if the thickness of the device is limited and there exist optimal refractive index pairs for composing dielectrics for a given device thickness.Unlike the case of metalens, the optical efficiency drops when the refractive index contrast becomes greater than the optimal value.We also report that the device performance can be greatly increased while maintaining a relatively large feature size by having multiple layers in the design scheme.We anticipate that the qualitative trend seen in the 2D color splitter design parameter tuning would be repeated for Bayer-type 3D color splitters, although the optimal values may differ due to the introduction of the additional dimension.Our results will serve as a design guideline for the future development of freeform metasurface-based color splitters for deep sub-micron image sensors.The optical efficiency saturates to 60%.Except for N L and N C , the design parameters given in Table 1 are used.

Figure 1 .
Figure 1.Schematic of the working principle (A) A simplified diagram of a conventional image sensor consisting of a microlens array and color filter.(B) A schematic diagram of a color splitter.The design area (P 3 t) is gridded into a grid of N C 3 N L , and refractive indices n 1 and n 2 are allocated to each cell for color splitting.Two and four arrows at the focal plane of the color splitter imply that an ideal color splitter can have a two-fold, four-fold increase in optical efficiency compared to the conventional design.

Figure 2 .
Figure 2. Performance of the optimized device (A-C) The electric field intensity profile inside the optimized device is given in Figure 1B for a normally incident light of (a) l = 650 nm, (b) l = 550 nm, (c) l = 450 nm.The depicted field distribution is the average of transverse electric and transverse magnetic polarized light.(D) The electric field intensity profile of red, green, and blue normally incident light on the surface of the photodetector.(E) Optical efficiency spectra of the same device.The default design parameters in Table1are used.The average optical efficiency is 58.29%.The device sorts red, green, and blue normal incident light with peak efficiencies of up to 70.12%, 57.15%, and 77.39% at 669 nm, 554 nm, and 422 nm wavelengths, respectively.

Figure 4 .
Figure 4. Dependency of design parameters: Refractive indices and thickness Effect of refractive indices on the optical efficiency for different device thicknesses, t. (A-C) Optimization based on a genetic algorithm was carried out for color splitters with thickness (a) t = 0.1 mm, (b) t = 0.5 mm, and (c) t = 1.5 mm.In each color plot, the lower refractive index n 1 is changed from 1 to 2 with a step size of 0.25, and n 2 is swept from 2 to 4 with the same step size, 0.25.Each square represents the optimized efficiency obtained with the genetic algorithm.The maximum efficiency in each case is (a) 46.86%,(b) 54.98%, (c) 58.25%.Except for t, n 1 , and n 2 , the design parameters given inTable 1 are used.

Figure 5 .
Figure 5. Dependency of design parameters: Degree of freedom Optimized optical efficiency calculated for multiple DoF configurations.The optical efficiency saturates to 60%.Except for N L and N C , the design parameters given in Table1are used.

Table 1 .
The default design parameters used in this work

Table 1
are used.The average optical efficiency is 58.29%.The device sorts red, green, and blue normal incident light with peak efficiencies of up to 70.12%, 57.15%, and 77.39% at 669 nm, 554 nm, and 422 nm wavelengths, respectively.