1. Introduction

A complementary metal oxide semiconductor (CMOS) image sensor, which converts optical signals into electrical ones, is an optical system that relays light from environment to handy camera sensor without degradation [1]. Following development of high-resolution display from large-area to near-eye display, high resolution cameras are on demand for videos or pictures [2]. To this end, size reduction of pixel has been highly required since spatial limitation of mobile device disrupts getting whole sensor size larger. However, it might deteriorate image quality due to interference among neighboring pixels, quantum efficiency, and wavelength purity. Hence the low signal-to-noise ratio has pushed re-increased pixel size [3,4]. In this regard, to enhance signal-to-noise ratio, advanced image sensor technologies such as backside illumination, deep-trench isolation, more pixel binning or novel electric circuit design have allowed pixel size to shrink into 0.7 µm [5–9].

Metasurfaces, which consist of subwavelength antennas, have been developed to replace conventional optical elements including color filters in CMOS image sensor thanks to their high degree of freedom and ultra-compact form factor [10–13]. Even if reflection-mode color filters have been reported with high resolution, saturated color purity, and high reflection efficiency with dielectrics by group resonance and second Kerker effect, the architecture of image sensors requires transmission-mode color filters [14–16]. Among a number of approaches to polarization independent transmissive metasurface color filters (MCFs), there are two representative design methods based on resonance effect. One is extraordinary optical transmission (EOT), which enables high color purity at plasmonic resonance through subwavelength aperture arrays on metal film [17–20]. The other design method is Mie-resonance that reflects light and partially absorbs light at multipolar resonances of subwavelength antennas based on dielectric metasurface [21–23]. It showed apparent advantages in transmission efficiency but has relatively low color purity compared with plasmonic MCFs [21–23]. For example, the peak transmissions of plasmonic MCFs are (R, G, B) = (< 20%) in Ref. [18] and (R, G, B) = (> 80%, > 70%, > 60%) in Ref. [19] but the peak transmissions of dielectric MCFs are (C, M, Y) = (90%, 90%, 90%) in Ref. [21] and (R, G, B) = (90%, 60%, 50%) in Ref. [22], where R, G, B, C, M, and Y mean red, green, blue, cyan, magenta, and yellow, respectively. Furthermore, hybrid MCFs with dielectric nanowire encapsulated by metal have been studied to improve color purity of dielectric MCFs and efficiency of metallic MCFs [24,25]. However, it is unpractical that most of transmissive MCFs have not considered reflections that might lower image quality, giving rise to flare effect by multiple reflection in CMOS image sensor [26]. Although the light is absorbed at Mie-resonance of amorphous silicon rods, it is not enough to suppress reflection because of low extinction coefficient at green and red wavelength bands [14]. Thus, absorption-based MCF with suppressed reflection is indisputably required that operates like conventional pigment or dye color filters.

Hyperbolic metamaterials (HMMs) are defined by the shape of their isofrequency curve that is caused by opposite sign of electric permittivity or magnetic permeability along primary axis on orthogonal coordinates [27]. They consist of metal-dielectric composition in form of alternative stacks with subwavelength layers or subwavelength wire arrays. By their diverging isofrequency curves that support high-k wave, HMMs have exhibited several advances not only as substrates that enhance local density of states coupled with emitters but also as lenses that enable deep subwavelength resolutions [28,29]. Especially, HMM waveguides or resonators presented wavelength-independent mode propagation along negative direction and ultracompact light trapping in arbitrary object size [30–32].

Herein, we propose absorptive MCF that functions like conventional color filters based on truncated-cone HMM waveguide absorbers that operate as a bandstop filter at undesired wavelengths. The truncated-cone HMM waveguide consists of several layers of metal-dielectric films with tapered angle. In a design of spectral response of the truncated-cone HMM waveguide, it looks elusive to decide not only the number of layers, metal filling ratio, and thickness that relate quality of effective medium but also widths of bottom and top of truncated-cone that relate resonance wavelengths. Thereby, we investigate geometry parameters for high figure of merit using particle swarm optimization (PSO) method which is stochastic optimization method in suggested range. The optimized waveguides at blue, green, and red colors are examined in color details, incident angle dependency, and expected color acceptance of virtual CMOS image sensor. Furthermore, amorphous silicon substrate is considered to emulate photodiode of CMOS image sensors. In the last section, the feasibility of MCF array in 255 nm of sub-pixel pitch is demonstrated with potential fabrication process.

2. Design principle

2.1 Hyperbolic metamaterial waveguide absorbers

To begin with, principle of cylindrical HMM waveguides operating as absorber is demonstrated. We consider anisotropic metamaterial waveguide composed of multiple metal-dielectric stack. From effective medium theory for the multiple metal-dielectric stack, electric permittivities along principal axes are effectively as follows [27]:

In Eq. (1), ɛ_p ɛ_v, and f denote parallel permittivity, vertical permittivity, and metal fraction composed of multiple metal-dielectric stack, respectively. Regarding pixel size and resonant wavelength for visible range, we calculate effective electric permittivity tensor composed by aluminum (Al) and silicon dioxide (SiO₂) of which optical constant is obtained from Refs. [33] and [34]. In visible wavelength range, electric permittivity along parallel axis is negative while one along vertical axis is positive as shown in Fig. 1(b). This opposite sign of electric permittivity results in hyperbolic isofrequency curves in Fig. 1(c) by Eq. (2):

 

Fig. 1. (a) Schematic diagram of effective cylindrical HMM waveguide. (b) Effective electric permittivity along parallel and vertical directions. (c) Isofrequency curve from effective electric permittivity for equal ratio of aluminum and silica at 400 nm (blue), 550 nm (green), and 700 nm (red). (d) Effective refractive index along reduced radius (k₀a) calculated by characteristic equation. (e) Power flow along propagation axis. Solid lines (light blue box) and dashed lines (light red box) denote forward and backward modes, respectively. (f) Absorption spectrum for wavelength and radius.

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This hyperbolic shape allows not only unlimited propagation modes along forward and backward directions but also trapped modes. For example, cylindrical HMM waveguide with abovementioned material composition is investigated to explain effective mode index and power flow along propagation direction at the wavelengths of 400, 550, and 700 nm. The effective refractive mode indices are calculated from characteristic equation in the work by Huang et al. [35]. As mentioned above, Fig. 1(d) verifies that propagation modes always exist in several bands unrelated to varying object size. Furthermore, normalized power flow in Fig. 1(e) shows light propagation direction in waveguide mode, and the light is trapped at the point of sign change. The results correspond to absorption spectrum for 500 nm length waveguide in Fig. 1(f), which is simulated from 3D electromagnetic finite-difference time-domain (FDTD) simulator [36]. As shown in Fig. 1(f), absorption spectrum shifts from 400 nm to 700 nm in increasing radius. This signifies that vertically-tapered cylindrical HMM waveguide can be broadband perfect absorber, which is a two-dimensional case of well-known perfect absorber based on tapered HMM gratings [31,32]. We utilize vertically-tapered cylindrical HMM waveguide as building block of absorptive MCFs shown in Fig. 2(a).

 

Fig. 2. Proposed metasurface and optimization process. (a) Unit-cell diagram composed of 12 Al-SiO₂ layers with 100 nm-thick silica spacer on silicon substrate and variable filling ratio. Light is normally incident with polarization-independency. (b) Gaussian lineshape for target wavelengths of 460, 530, and 650 nm with 50% efficiency. (c) A schematic of particle swarm optimization. Particle, M-dimensional coordinate, and particle position correspond to FDTD simulation, parameter space, and optimized parameters, respectively.

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2.2 Absorptive color filter design based on a particle swarm optimization method

Figure 2(a) explains that a schematic of the proposed metasurface is composed of 12 layers of Al-SiO₂, each of which has 50 nm thickness, sitting on 100 nm-thick SiO₂ spacer and Si substrate. It is a reasonable layer thickness in terms of thermal expansion by heat-up from absorption, where the linear thermal expansion along vertical direction is less than 3 nm for 600 nm-thick aluminum in varying the temperature from -20 °C to 200 °C [37]. Also, the spacer thickness is also used in Ref. [22] and contribution of multiple reflection to optical flare effect is investigated by scattering matrix method in Appendix A.1. Top and bottom diameters are manipulated in designing absorption range. However, since incident light is absorbed wavelength-continuously for uniform ratio of Al-SiO₂ building block, adjusting only D_top and D_bottom is insufficient to achieve high quality of color from identical meta-atom array. In addition, it is difficult to transmit target colors with suppressed reflection in few degrees of freedom. Taking this into account, we make use of different metal filling ratio ${f_n} = {{t_{\textrm{Metal}}^n} / {({t_{\textrm{Metal}}^n + t_{\textrm{Dielectric}}^n} )}}$ about every layer for qualitative design. Furthermore, period is minimized to reduce pixel size maintaining low reflection. Since in-plane coupling determined by the longest diameter at 700 nm resonance and the lattice resonance by the period contribute to reflection, the period reduction is limited [38]. Considering those limitations, the unit-cell structure is arranged in 250 nm period along x- and y-directions and exhibits no diffraction.

To maximize transmission efficiency and minimize reflection, we employ PSO algorithm. Although deep learning has been applied to achieve exceptional performance in designing complex shape of antennas, it requires huge dataset [39]. Meanwhile, gradient descent algorithm needs to differentiate loss function about all parameters in each iteration. Thus, stochastic algorithms might be more efficient alternatives for the case of large amount of parameters. In electromagnetic simulation based on PSO, each solution with randomly generated parameters in M-dimensional parameter space {x₁, x₂, …, x_M}, which is called particle, is solved and moves along better fitness [40]. In our metasurface optimization, parameter space consists of diameters (D_bottom, D_top) of tapered cylindrical HMM waveguide and metal filling ratio f_n of each layer described in Fig. 2(a).

In an iteration of optimization, randomly-distributed twenty particles are solved using FDTD simulation. From transmission and reflection spectra, we calculate loss function that needs to be minimized. The loss function is defined as sum of L₂ loss between transmission spectrum and Gaussian lineshape as shown in Fig. 2(b), and one between reflection and zero as shown in the following equation:

3. Results and discussion

3.1 Numerical analysis on metasurface color filters

Through 100 iterations for each color, we obtain three unit-cells, corresponding to each metasurface for RGB colors. Transmission and reflection spectra in Figs. 3(a)–3(c) are numerically calculated from optimized geometry in Figs. 3(d)–3(f). The details of geometry for each color are given in Table 1, where filling ratio 1 means there is only metal and 0 means only dielectric. Figures 3(a)–3(c) verify that transmission spectra reach the maximum near target wavelength and slightly suppressed-efficiency. Reflections are lower than 10% in total visible wavelength range but this is negligible even if multiple reflections between MCF and other optical elements in CMOS image sensor exist (< 1%).

 

Fig. 3. Transmission and reflection spectra from optimized metasurface for (a) blue, (b) green, and (c) red color. Insets in (a–c) depict perceptual colors from CIELAB. Dashed lines at 460 nm, 530 nm, and 650 nm indicate center wavelengths of target spectrum. (d)–(f) XZ cross-sections of enlarged optimized meta-atoms and normalized electric field at target wavelengths. White solid lines denote boundary of meta-atom and substrate. White arrows indicate the spatial position of electric field resonances excited by meta-atom and transmitted light.

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Table 1. Optimized geometry parameters for MCF.

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To demonstrate that the given metasurfaces operate as absorptive MCFs, normalized electric field distributions are examined under linear polarization illumination. At the wavelength of dashed lines in blue color, electric field at 460 nm propagates through substrate. Meanwhile, the light is confined at the respective resonant positions at 530 nm and 650 nm in Fig. 3(d). Likewise, light passes through meta-atom at each target wavelength of green and red color filters but is confined at unwanted wavelengths in Figs. 3(e) and 3(f). Then, light is absorbed because the confined light is dissipated by loss of metal.

Provided that these color filters are used for CMOS image sensor, we can expect how colors are recognized by suggesting CIELAB color space (L*, a*, b*) [41]. The CIELAB expresses luminance of light spectrum and represents colors considering human’s non-uniform vision perception, whereas CIE 1931 coordinate describes only chromaticity from normalization of CIEXYZ [42]. The equations to calculate CIELAB is given in Appendix A.2. The optimized color filters in CIELAB space result in blue, green, and red colors for transmission spectra and almost dark brown and black colors for reflection spectra as rendered in inset of Figs. 3(a)–3(c) and Fig. 4. The CIELAB values (L*, a*, b*) of transmission are (31.4, 21.2, -56.4), (56.9, -51.5, 35.7), and (34.2, 34.5, 44.2) for blue, green, and red color filters, respectively. Meanwhile, CIELAB values of reflection colors are (22.0, -1.35, 30.0), (13.6, 18.1, -3.8), and (17.3, 0.622, -0.181) for blue, green, red color filters. It can be seen that RGB colors in transmission spectra achieve much higher chromaticity and larger lightness than reflection spectra. In particular, in Figs. 4(a) and 4(b), the reflection colors are concentrated on axis in L*a*, L*b*, and a*b* coordinates, which means their chromaticity approaches zero. Furthermore, the RGB colors are farther from center than reflection colors in Fig. 4(b). As perspective on figure of merits (FoMs) about transmission per reflection spectra, the ratio between summed-square-magnitude of transmission and one of reflection in visible range is examined in Fig. 4(c). It is noted that FoM is achieved higher than 14.63 dB for all color filters, which means high quality color filter with reflection suppressed.

 

Fig. 4. Colors calculated from CIELAB space. Transmission and reflection spectra correspond to each point and color on (a) L*a* and L*b* coordinates, and (b) a*b* coordinate. The points in gray ellipse represent reflection colors. (c) Transmission and reflection ratio.

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Here, we inspect the image quality captured by MCFs. Figure 5 assumes that our proposed MCFs are integrated with CMOS image sensor and Gretag-Macbeth chart is measured as reference colors. The image is reproduced by integration of transmission efficiency passing through each MCF. The transmission efficiency is measured by power monitor at the boundary of SiO₂ spacer and Si substrate for plane-wave incidence with elemental spectra of Gretag-Macbeth chart. To quantitatively analyze lightness and chromaticity variation from original image, we suggest advanced color difference metric CIEDE2000 (ΔE₀₀) [43]. The details of CIEDE2000 calculation method are explained in Appendix A.3 and Ref. [43]. The average and standard deviation of ΔE₀₀ are 15.53 and 4.217, respectively, where lightness accounts for most part of ΔE₀₀. Although there are the color differences, it seems that the color tolerance can be corrected by exploiting white balance, color correction matrix, and gamma correction in practical CMOS image sensors [44].

 

Fig. 5. Image observation from Gretag-Macbeth chart by optimized virtual MCFs.

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3.2 Angle tolerance for optimized metasurface color filters

We turn to illumination condition in Fig. 6. It is important to maintain performance about oblique incidence in CMOS image sensors. Unlike the conventional color filters composed of pigment or dye, MCFs depend on incident angle by nanostructure array. Illuminating light from 0 to 40 degrees of incidence angle in transverse electric (TE) and transverse magnetic (TM) waves as depicted in Figs. 6(a) and 6(d), chromaticity variation of transmitted light is shown along black arrow direction with inclement of 5 degrees on a*b* coordinate in Figs. 6(b) and 6(e). In addition, color modification is quantitatively analyzed by CIEDE2000 in Figs. 6(c) and 6(f). It is known that color difference is noticeable at condition of ΔE₀₀< 3 and appreciably noticeable at 3 < ΔE₀₀< 6 [45]. Thus, note that the optimized color filter is consistent in consideration of ΔE₀₀< 3 with respect to oblique incident angle.

 

Fig. 6. Incident angle dependence in terms of polarization of (a-c) TE modes and (d-f) TM modes. (a, d) Schematic diagrams of TE and TM incidence. (b, e) Chromaticity variation in a*b* coordinates. The dots are distributed in increase of 5 degrees along black arrows. (c, f) Color difference expressed by CIEDE2000 from CIELAB with respect to incident angle. The light gray boxes denote just-noticeable region.

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For TE wave, Fig. 6(b) indicates little chromaticity variation for blue and red color filters, but relatively larger shift for green color filter. The blue color shifts to deep blue and purple, the red color almost stays, and the green color moves toward dark green. From the color difference in Fig. 6(c), the colors are maintained until 20 degrees for the green, and 15 degrees for the blue color filters. And the ΔE₀₀ about the red color filter is not more than 3 until 40 degrees. In comparison with low chromaticity variation in Fig. 6(b), the major color difference is caused by lightness degradation for TE case.

On the other hand, it seems that chromaticity for TM wave decreases but lightness becomes brighter in Fig. 6(e). The blue and red colors are shifted toward bright cyan and bright red, respectively. Meanwhile, the green color is shifted toward bright green. The color difference in Fig. 6(f) shows that blue, green, and red color filters are consistent under 10 degrees, 15 degrees, and 13 degrees, respectively.

In comparison with TE and TM incidence, the color differences for TM wave are more affected than TE wave. This is because tangential electric field excites tangential electron oscillation at resonant wavelength but the effective wavelength of tangential electric field is reduced with incident angle of TM wave increasing. In both TE and TM waves, the color differences increase with the incident angle increase. This is because incident wave couples with meta-atom by effectively small cross-section. In addition, shorter wavelength gives rise to diffraction with incident angle increasing, where the 1st diffraction order exists from 5.06 degrees at 400 nm to 49.9 degrees at 643 nm.

3.3 Color filter arrays by proposed optimization method

Until now, optimized meta-atom is arranged in period about each color filter. However, we design absorptive metasurface color filter array (MCFA) to glimpse a feasibility of our proposed method on conventional RGB sub-pixel array. Since every color pixel shares vertical geometry parameters such as metal filling ratio, thickness of each layer, and number of total layers, optimization based on single meta-atom is restricted for three colors simultaneously. Accordingly, we design each color by spatially multiplexing meta-atoms in 2 × 2 Bayer arrangement as shown in Fig. 7(a). For normal incident of transverse electric (TE) and transverse magnetic (TM) waves, Bayer MCFA is optimized about whole geometry parameters of RGB sub-pixel simultaneously including top/bottom widths, sub-pixel pitch, thickness and filling ratio of each layer in each color pixel. The boundary conditions are set to periodic condition along x- and y-direction and perfectly matched layers (PMLs) along z-direction. To increase the degree of freedom, the number of layers is selected as 15.

 

Fig. 7. (a) Schematic diagram of MCFA and material distribution at the cross-section of each meta-atom. (b) Transmission and reflection spectra from optimized MCFA for entire RGB pixel. (c) Color distribution on CIELAB space. (d) Signal-to-noise ratio about color crosstalk for 435 nm, 530 nm, and 625 nm of wavelength.

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Since we assume that all transmitted power is absorbed by photodiode, transmission power is measured at the boundary of spacer and Si substrate and averaged for incident transverse electric and transverse magnetic waves. By particle swarm optimization, geometry parameters are given in Table 2 and described in Fig. 7(a). The order of layer thickness and metal filling ratio in Table 2 is bottom to top layer. In comparison with single color pixels, transmission spectra of each color filter and reflection spectrum of whole Bayer pattern are given in Fig. 7(b). As the purpose of our MCF, the reflection is low. From this spectra, Fig. 7(c) shows the color distribution of optimized MCFA. The luminance is higher than single MCF in Fig. 3 but the chromaticity is lower than single MCF due to spectral broadening caused by diffraction orders and in-plane coupling between meta-atoms. To quantitatively identify MCFA, we define signal-to-noise ratio (SNR) for color crosstalk from direct transmission from each color filter and diffraction from neighboring pixels as follows:

Table 2. Optimized geometry parameters for metasurface color filter array.

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4. Conclusion

In conclusion, we propose and design metasurface for ultracompact color pixels. Distinct from conventional transmissive MCFs that have principally inevitable reflection, absorbers substitute for transmission band manipulation. Theoretical and numerical calculations on cylindrical HMM waveguide provide insight into absorptive MCF. In design method, PSO enables optimization of meta-atom with respect to various parameters simultaneously. Optimized color filters achieve 14.75, 14.63, and 18.62 dB of transmission colors compared with negligible reflection for blue, green, and red color filters, respectively. In comparison with plasmonic MCFs and all-dielectric MCFs, our work produces little reflection and similar transmission with plasmonic MCFs. Using CIELAB, numerical investigation in visual perception of color at D65 environment is examined. It is notable that reflection is suppressed in terms of chromaticity. Moreover, virtual CMOS image sensor based on our proposed MCFs shows consistent image quality for every color of Gretag-Macbeth color chart. As a factor of image degradation, color difference calculated from CIEDE2000 is maintained under 15 degrees of oblique incident angle in both of TE and TM polarization. Finally, we design subwavelength color filters in arrangement with 255 nm sub-pixel pitch for practical application. And grayscale FIB milling can support feasibility of designed metasurface. Therefore, we expect that our proposed metasurface can be a candidate in application to miniaturized CMOS image sensor.

Appendix A

A.1 Multiple reflection inside SiO₂ spacer

Optical flare effect is caused by the multiple reflection in optical system of CMOS image sensor. One of potential sources of multiple reflection in the proposed MCFs is SiO₂ spacer (layer 2) between Si substrate (layer 3) and metasurface (layer 1) as described in Fig. 8(a). From the scattering matrix calculation, forward transmission (t₃₁) from metasurface to substrate and multiple reflection terms are as below:

(5)$$\begin{array}{l} t_{31}^{(j )} = t_{32}^{(j )}\{ t_{21}^{(j )}\exp ({i{n_2}{k_0}d} )+ t_{21}^{(j )}r_{32}^{(j )}r_{12}^{(j )}\exp ({3i{n_2}{k_0}d} )\,\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + t_{21}^{(j )}{({r_{32}^{(j )}r_{12}^{(j )}} )^2}\exp ({5i{n_2}{k_0}d} )+ H.O.\} \\ \,\,\,\,\,\,\, = t_{32}^{(j )}\left\{ {t_{21}^{(j )}\exp ({i{n_2}{k_0}d} )+ \frac{{t_{21}^{(j )}r_{32}^{(j )}r_{12}^{(j )}\exp ({i{n_2}{k_0}d} )}}{{1 - r_{32}^{(j )}r_{12}^{(j )}\exp ({2i{n_2}{k_0}d} )}}} \right\},\\ r_{multiple}^{(j )} = \frac{{t_{21}^{(j )}r_{32}^{(j )}r_{12}^{(j )}\exp ({i{n_2}{k_0}d} )}}{{1 - r_{32}^{(j )}r_{12}^{(j )}\exp ({2i{n_2}{k_0}d} )}},\,j \in \{{R,G,B} \}, \end{array}$$

where d denotes spacer thickness. To calculate multiple reflection in SiO₂ spacer, we calculate second and high order terms (H.O.) which is sum of multiple reflection terms r_multiple inside SiO₂ spacer.

 

Fig. 8. (a) Scattering process in proposed MCF consisting of metasurface (layer 1), SiO₂ spacer (layer 2), and Si substrate (layer 3). (b) Intensity of multiple reflection inside spacer for RGB color filters.

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From the FDTD simulation, we get the scattering matrix sets t₂₁, t₃₂, r₁₂ for red, green, and blue color filters. It is noted that intensity of multiple reflection by spacer is under 1.2% as shown in Fig. 8(b). Thus, multiple reflection at the boundary of SiO₂ spacer does not affect optical flare effect much.

A.2 CIEXYZ and CIELAB colorimetry

Since the development of CIE 1931, optical spectrum of scattered light and human perception have been connected corresponding to tristimulus values $\bar{x},\,\bar{y}$, and $\bar{z}$ from the observed data as shown in Fig. 9(a) [42]. Furthermore, standard illuminant, which is theoretical visible light source, was provided in representing daylight, incandescent light, or sunlight. In this paper, we use illuminant D65 to quantify color perception as shown in Fig. 9(b).

 

Fig. 9. (a) Tristimulus values of the CIE 1931 standard colorimetric observer. (b) Relative power spectrum of standard illuminant D65.

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Identifying colors, integral of multiplication among tristimulus values, spectrum from object, and illuminant spectrum is calculated:

(6)$$\begin{array}{l} X = \frac{{100}}{N}\int\limits_{380}^{780} {S(\lambda )I(\lambda )\bar{x}(\lambda )} \,d\lambda ,\\ Y = \frac{{100}}{N}\int\limits_{380}^{780} {S(\lambda )I(\lambda )\bar{y}(\lambda )} \,d\lambda ,\\ Z = \frac{{100}}{N}\int\limits_{380}^{780} {S(\lambda )I(\lambda )\bar{z}(\lambda )} \,d\lambda ,\\ N = \int\limits_{380}^{780} {I(\lambda )\bar{y}(\lambda )} \,d\lambda , \end{array}$$

where 100/N is normalizing constant, S and I are transmission/reflection spectra and illuminant D65 source spectrum, respectively. While CIE 1931 space considers only chromaticity and assumes linear human color perception, CIELAB space expresses lightness (L*) and non-linear response of human eye. In addition, the a* and b* denote chromaticity. To quantify color perception in CIELAB coordinate, CIEXYZ is used [41]:

(7)$$\begin{array}{l} {L^\ast } = 116f({{Y / {{Y_n}}}} )- 16,\\ {a^\ast } = 500({f({{X / {{X_n}}}} )- f({{Y / {{Y_n}}}} )} ),\\ {b^\ast } = 200({f({{Y / {{Y_n}}}} )- f({{Z / {{Z_n}}}} )} ),\\ {X_n} = 95.0489,\,\,{Y_n} = 100,\,\,{Z_n} = 108.8840,\\ f(t )= \left\{ \begin{array}{l} \sqrt[3]{t}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textrm{if}\,t > {\delta^3}\\ {t / {3{\delta^3} + {4 / {29}}}}\,\,\,\textrm{otherwise} \end{array} \right.,\,\,\delta = {6 / {29}}, \end{array}$$

where the normalized values X_n, Y_n, and Z_n are determined by illuminant. From Eqs. (6) and (7), the colors from scattered light are expected.

A.3 CIEDE2000 color difference

Color difference is a metric that quantifies how human perceives color difference noticeably. In general, the color difference has been defined as Euclidean distance in the presented coordinate such as CIERGB and CIELAB. However, there exists a non-uniformity of Euclidean color difference in CIELAB space in terms of colors due to human perceptive non-uniformities. Therefore, CIEDE2000 introduced several weighting factors and correction factors in regard to lightness, chroma, and hue in order to compare uniform difference among colors:

(8)$$\Delta {E_{00}} = \sqrt {{{\left( {\frac{{\Delta L^{\prime}}}{{{k_L}{S_L}}}} \right)}^2} + {{\left( {\frac{{\Delta C^{\prime}}}{{{k_C}{S_C}}}} \right)}^2} + {{\left( {\frac{{\Delta H^{\prime}}}{{{k_H}{S_H}}}} \right)}^2} + {R_T}\left( {\frac{{\Delta C^{\prime}}}{{{k_C}{S_C}}}} \right)\left( {\frac{{\Delta H^{\prime}}}{{{k_H}{S_H}}}} \right)} ,$$

where k_L, k_C, and k_H values are compensation terms regarding experimental condition, which are neutrally unity. In Eq. (8), the compensation terms for lightness, chroma, and hue are S_L, S_C, and S_H, respectively. Furthermore, the human color perception is corrected by R_T. The details of calculation procedures of compensation terms are given in Ref. [43].

A.4 Potential fabrication method

We briefly discuss a potential fabrication method to manufacture the proposed MCFs. Although electron beam lithography has advantage in fabricating large-scale metasurface, the geometry condition of vertically tapered and multiple-layered Al-SiO­₂ meta-atom looks very challenging. In this regime, as the previous work of Ref. [32], focused ion beam can be attributed to control tapered angle and fabricate metal-dielectric simultaneously, which is manipulated by using grayscale exposure dose [46].

First, aluminum and silica films are alternatively deposited by e-beam evaporator and atomic layer deposition, respectively, in consideration of accuracy of layer thickness. And then meta-atom array is patterned by FIB based on predefined grayscale map as depicted in Fig. 10(a). To predict fabrication reliability, we investigate how geometric deviations influence color distortions, where the deviations include individual layer-thickness ${\widetilde t_n}$, bottom diameter ${\widetilde D_{bottom}}$, top diameter ${\widetilde D_{top}}$ in Eq. (9), and rounded-tip.

(9)$$\begin{array}{l} {\widetilde D_{bottom}} = {D_{bottom}} + {\delta _{bottom}},\\ {\widetilde D_{top}} = {D_{top}} + {\delta _{top}},\\ {\widetilde t_n} = {t_n} + {\delta _n},\,\,({n \le 12,n \in {\mathbb N}} )\end{array}$$

where δs are errors of geometric parameters and n denotes n-th layer of the proposed MCFs. The thickness error means variation of individual layer thickness, not total thickness of MCFs. In Fig. 10(b), calculated color distribution on CIELAB is given by 180 simulations of which geometric parameters are randomly selected in the range from -10 to 10 nm for ${\delta _{bottom}},\,{\delta _{top}}$ and ${\delta _n}$. To quantitatively evaluate fabrication error, standard deviations of (L*, a*, b*) are (3.44, 12.3, 6.15), (5.24, 7.44, 14.0), and (8.45, 6.58, 11.3) for blue, green, and red color filters, respectively. As examples of fabrication error, color table is illustrated in Fig. 10(c). It shows that colors present still their own colors in about 20 nm (-10 to 10 nm) of diameter tolerance and about 20 nm of thickness tolerance (-10 to 10 nm).

 

Fig. 10. (a) Fabrication process. The SiO₂ spacer is deposited and Al-SiO₂­ layers are alternatively deposited. And then FIB milling is performed by Ga⁺ ion. (b) Color tolerance by geometric fabrication error for randomly distributed error from -10 to 10 nm of individual layer-thickness and bottom/top diameters for Fig. 3(b). (c) Examples of color tolerance from -20 to 20 nm of bottom/top diameters simultaneously and from -10 nm to 10 nm of all layer thickness for Fig. 3(b). (d) Cross-sections of rounded-tip structure with its transmission and reflection spectra where solid-lines and dashed-lines denote spectra of optimized color filter and with rounding errors.

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Lateral errors can be manipulated by exposure dose calibration using commercial focused ion beams from Zeiss and FEI Co. that have 3 nm resolution. Meanwhile, vertical errors could be expected, determined by deposition-uniformity, which is less than 5% for commercial e-beam evaporators from Rocky Mountain Vacuum Tech Inc. and less than 1 nm-roughness for atomic layer deposition [47]. In addition, the influence from rounded-tip is not much critical as shown in Fig. 10(d). We set the geometry of top layer to paraboloid. There is little spectral variation for blue and red MCFs but enhanced transmission response for green MCF. Thus, the proposed fabrication method can be one fabrication method.

Funding

Samsung Electronics.

Acknowledgment

This work is supported by the Samsung Electronics’ University R&D program. [Research for nanostructure-based color router for high-sensitive miniaturized-pixel].

Disclosures

The authors declare no conflicts of interest.

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