1. Introduction

Imaging system in visible and infrared regime has found widespread use in both civilian and military applications. In visible imaging system, the color filters Green, Blue or Red are arranged regularly in specific format called Bayer's pattern. Each filter is dedicated to individual sensors or photo detectors which converts the transmitted light from filter into electrical signal. These electrical signals are then processed to generate colored digital images. The combination of filter and photo detector is called pixel and the regular array of filter and photo detector is known as pixel array. The advancement and scaling effect in Complementary metal oxide semiconductor (CMOS) and micro-electro-mechanical system (MEMS) technology has made it possible to make imaging system with higher pixel density [1–5]. This significantly improves the spatial resolution of the imaging system. The higher pixel density is generally achieved by shrinking the size of each pixel in an array. The pixel size will continue to decrease as the demand for imaging systems with higher pixel density increases.

However, with decrease in pixel size, the optically active area of pixel will also decrease. For example, if we consider imaging pixels to be square shaped, reducing the pixel size from 5µm to 1µm, as shown in Fig. 1(a), will reduce the optically active area by 25 fold. This decrease in optically active area will reduce the number of photons available for photo detector, hence decreasing optical efficiency of individual pixel, signal to noise ratio (SNR) and dynamic range (DR) of imaging system [4–9].

 

Fig. 1 (a) Reduction of dimension of square shaped pixel (or detector), (b) Incidence area considered for calculating NOE over two pixels.

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Different studies have been done and design optimization has been proposed to address these issues, by sorting photons by colors and not filtering them. The concept of compact single device to simultaneously generate three primary colors (red, green, blue) from white light has been proposed by Dammann et al in 1978. The compact device proposed by Dammann et al used the phase relief type grating to project primary colors in far field [10]. In other study Knop et al [11] propose binary phase gratings with deep rectangular grooves as transmission filter to project complementary colors, cyan, magenta and yellow in zero diffraction order. Recently, light splitting techniques using microscale plate like structure has been proposed [14]. Furthermore, photon sorter based on plasmonic structure has also been proposed [15] to increase optical performance of imaging pixels. However, there exists limitation in the proposed solutions. For example, in the phase relief type grating and binary phase grating proposed in [10–13], larger grating period has to be used for efficiently project white light to primary color. Furthermore, the buffer layer between grating structure and detector has to be more than 10µm, resulting in bulky device. Even in the micro scale plate like structure proposed in [14] the buffer layer is at least 5µm. The microscale plate structure used in [14] has aspect ratio (height of structure/width of structure) larger than 5, resulting in bulky device. In addition, the angular tolerance of such devices is very small ( ± 5 ^o). The sorting structure based on SPP proposed in [15] requires higher number of gratings covering larger area. Since it is based on SPP, the device is angle-sensitive. All of these proposed structures would cease to work with good efficiency if the pixel size decreases below 1µm. Therefore, as the pixel size continues to get smaller, different design methodologies have to be explored.

In this paper we investigate the dielectric based sorting structure for visible light. First, we will discuss the concept and necessary condition for spectral sorting using normalized optical efficiency as described in [16, 17]. We systematically explore the possibility of dielectric-only spectral sorting. We then present design steps and numerical studies on grating assisted sorting device for visible light.

2. Concept of spectral sorting

One of the measures to quantify the performance of optical pixels is normalized optical efficiency (NOE) [16, 17]. It is defined as the ratio of optical power available or absorbed by photo detector to the total power incident on the pixel area of interest. Normalized optical efficiency is limited by geometrical parameters like detector fill factor (FF), which is the ratio of actual detector size to pixel size, and area of imaging pixel (A). For visible imaging system, NOE also depends on transmission efficiency (η) of color filters used. Let’s consider two neighboring blue and green pixels in Bayer’s pattern as shown in Fig. 1(b).

If we consider the incident power over the sum of physical area of blue and green pixels A_inc = A₁ + A₂, as shown in Fig. 1(b), then NOE for given pixel can be approximated using following relation [16,17]:

where, $NO{E}_{B(or G)}$ is the normalized optical efficiency of blue (or green) pixel, $P_{DB(or G)}$ is the power absorbed by blue (or green) detector, $P_{inc}$ is the incident power, FF is the detector fill factor, A1 and A2 are the area of blue and green pixels respectively, ${\eta }_{B(or G)}$ is the transmission efficiency of blue (or green) transmission filter used.

If we consider the detector size to be the same as pixel size (FF = 1), same pixel dimensions for blue and green pixels (A1 = A2) and transmission efficiency to be ${\eta }_{B(or G)}$, the maximum achievable NOE for each pixel would be $NOE=\frac{1}{2}$ . In general, if we consider the incident power in spectral region of interest from N – number of surrounding pixels to evaluate the efficiency of single pixel, then the maximum achievable optical efficiency for that pixel is $NOE=\frac{1}{N}$. However, the transmission efficiency of filters is generally less,${\eta }_{B(orG)}<1$, resulting in decrease in overall optical efficiency of the pixel. The normalized optical efficiency larger than $\frac{1}{N}\left(NOE>\frac{1}{N}\right)$ for a given pixel could therefore be defined as the necessary condition to quantify the spectral sorting, since it would require routing of incident light to that pixel.

3. Spectral sorting in the visible regime

Silicon is widely used as the photon detecting material in various visible applications. In imaging system, the color filters are used to filter fundamental colors, red, green and blue colored light, from white light. The separated colored light is then absorbed by silicon layer beneath, which generate electronic signal that are processed by signal processing unit in the imaging system. In this article we use silicon as the base layer on which the sorting structure is constructed. We first revisit the antireflection property of dielectric coating on silicon which could give insight to the sorting of visible light. We then systematically study and present design rules for designing sorting structure using dielectric materials on silicon base layer. For simplicity, the study is focused on a 2D scenario and constant refractive indices for dielectric layers are used. The refractive indices of silicon are interpolated from Palik’s handbook of Optical Constants of Solids [18].

Antireflection property of multilayer dielectric coating on silicon

Because of its high refractive index, reflection of visible light from bulk silicon is considerably high (~40%). Using dielectric layers on top of bulk silicon, selective antireflection (or in other words, selective absorption) of specific wavelength could be achieved. Let’s consider a simple two-layer system as shown in Fig. 2(a), where the thickness of both layer on top of silicon are considered to be d₁ = d₂ = 110nm. The layers are extending infinitely along x axis. The index of the top layer is denoted as n_gt and second layer as n_IL respectively.

 

Fig. 2 (a) Two layer dielectric structure on top of silicon, with individual thickness of 100nm. Reflection as a function of wavelength and refractive index of top layer $(n_{gt})$ with index of second layer (b) $n_{IL}=1.5$ and (c) $n_{IL}=2.0$, (d) reflection of two layer system with index of first layer $n_{gt}:2.1$.

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With the thickness fixed to specific values, we can vary the refractive index of dielectric layers to tune the wavelength at which anti-reflection occurs. Figures 2(b) and 2(c) shows the reflection as a function of wavelength and refractive index of first layer n_gt for TE polarized light. The index of second dielectric layer is kept constant at n_IL = 1.5 and n_IL = 2.0 respectively. As the refractive index of first layer is varied from n_gt = 1 to n_gt = 4, the reflection minima could be observed for the wavelengths satisfying anti reflection condition. With increase in refractive index of first layer, the optical path length increases which shifts the reflection minima towards wavelengths longer than visible in Figs. 2(b) and 2(c). Figure 2(d) shows the reflection as a function of wavelength with n_gt = 2.1 for n_IL = 1.5 and n_IL = 2.0.

With these parameters, n_gt = 2.1, n_IL = 1.5, n_IL = 2.0 and d₁ = d₂ = 110nm we arrange the antireflection structure periodically as shown in Fig. 3(a) where, Px is the size of individual silicon detector. For clarity, the second dielectric layers with refractive index of 1.5 and 2.0 are denoted as n_IL-1 and n_IL-2 respectively. Silicon base layer under n_IL-1 and n_IL-2 is denoted as Dt1 and Dt2 respectively. We then vary the value of Px and calculate the absorption using homemade RCWA code in each silicon detectors. Figures 3(b) and 3(c) shows the absorption in silicon with second dielectric layer having refractive index n_IL-1 = 1.5 and n_IL-2 = 2.0 respectively for different value of Px. We consider n_gt = 2.1 for the calculation. We see that the peak absorption wavelength corresponds to reflection minima obtained in Fig. 2(d) and peak absorption value is always below or around 0.5. This indicates that even in periodic arrangement, antireflection dielectric layers work as an individual unit, without affecting the performance of neighboring layer structure.

 

Fig. 3 (a) Unit cell of the periodic arrangement of layered structures designed to minimize reflection at different wavelengths. Absorption spectra of silicon under the layered system with first- and second- layer made of material with indices (b) $n_{gt}:2.1$ and $n_{IL-1}:2.0$, (c) $n_{gt}:2.1$ and $n_{IL-2}:1.5$ for different sizes $(Px)$ of the structure.

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It is worth noting that we consider the same thickness for dielectric layers and the same refractive index of top layer in two antireflecting structure. Having same thickness and material on top layer for different antireflecting structure, we could exploit the effective medium theory (EMT) of lamellar grating to replace this layer with grating structure.

Subwavelength gratings have been widely studied for the application in antireflective surface, polarization elements and homogenous birefringent material using effective medium theory [19–22]. In EMT the subwavelength gratings are treated as homogenous medium as shown in Fig. 4(a).

 

Fig. 4 (a) Homogenization of grating with subwavelength period. The effective index of homogenous layer is calculated using effective medium theory. (b) Effective index as a function of wavelength for different grating widths with grating refractive index of $2.5$ and period $250nm$.

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The grating with subwavelength period P, and grating width W is generally represented as a homogenous layer with effective refractive index n_eff. The effective index of such grating can be estimated for TE polarization using second order EMT [23] as:

where,$f=\frac{W}{P}$, $n_0$ and $n_g$ are the filling factor of grating, refractive index of grating groove and grating material respectively. Figure 4(b) shows the effective index as a function of wavelength when the grating width $W$ is varied from $50nm-200nm$ in step of $25nm$ for grating with period $250nm$ and grating material with refractive index $2.5$. The refractive index of grating groove is considered to be $1.0$ for the calculation. It can be seen that the effective index could be changed just by changing the width of the grating. Therefore, for the known refractive index value of homogenous layer, corresponding grating parameters could also be calculated using effective medium theory.

Grating assisted sorting of visible light

We apply the effective medium theory to calculate appropriate grating width and period to replace the homogenous top layer of antireflective structure discussed in previous section. We study the absorption in silicon when two grating based antireflection structures are arranged periodically and illuminated by TE polarized normal incident light. The structures could be arranged in two different configurations, the unit cell of which are as shown in Figs. 5(a) and 5(d). For ease, we will name the configuration shown in Figs. 5(a) and 5(d) as $C1$ and $C2$ respectively. The size of the individual structures is chosen such that each structure contains at least one grating stub or a groove. This means that the individual sizes are multiple of grating period $Px=q{P}_g$, where $q=1,2,\dots$ and $P_g=250nm$ is the grating period. The grating width W of $125nm$ and grating material with refractive index of 2.5 are considered for the study. It can be seen that the configuration $C2$ is the complimentary of configuration $C1$ and has one grating stub in the junction of homogenous second layer. In the figure, the homogenous second layer with refractive index $1.5$ and $2.0$ are denoted as $n_{IL-1}$ and $n_{IL-2}$respectively and both have a thickness of $110nm$. The silicon under $n_{IL-1}$ and $n_{IL-2}$ are used to detect the light and are denoted as $Dt1$ and $Dt2$ respectively.

 

Fig. 5 (a) Unit cell of periodic arrangement of two layered structure with groove in junction of two antireflecting structures (configuration C1). Absorption as a function of wavelength in silicon under (b) $n_{IL-1}:1.5$ and (c) $n_{IL-2}:2.0$ for configuration C1. (d) Unit cell of periodic arrangement of two layered structure with grating stub in junction of two antireflecting structures (configuration C2). Absorption as a function of wavelength in silicon under, (e) $n_{IL-1}:1.5$ and (f) $n_{IL-2}:2.0$ for configuration C2.

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We calculate the absorption in $Dt1$ and $Dt2$ normalized to the total incident power in a unit cell for each configuration $C1$ and $C2$. Figures 5(b) and 5(c) show the absorption in $Dt1$ and $Dt2$ respectively for the configuration $C1$ when the size $(Px)$ of individual detectors or structures is varied from $5\mu m$ down to $0.5\mu m$. The dotted line indicates the $NOE=0.5$ which is the classical limit of absorption in each detector $Dt1$ and $Dt2$. We discussed in previous sections, that the sorting of light is the phenomenon that occurs when the absorption in each detector is higher than $0.5$, when normalized to the total incident power in the unit cell. It can be seen that the absorption in $Dt1$ is below $0.5$ even when the size is decreased down to $5\mu m$.While the absorption in $Dt2$ is higher than $0.5$ at 420nm for detector size as large as $2.5\mu m$. The absorption efficiency at $560nm$ in $Dt2$ remains near $0.5$ even when the size is increased from $0.5\mu m$ to $5\mu m$. It is visible from Fig. 5 that using configuration $C1$, only the blue light could be detected with higher efficiency, meaning we could only sort the blue light while no sorting could be achieved for green light.

Similarly, Figs. 5(e) and 5(f) shows the absorption in $Dt1$ and $Dt2$ respectively for the configuration $C2$ when the size $(Px)$ of individual detectors is varied from $5\mu m$ to $0.5\mu m$. It can be seen that the absorption of green light $(520nm)$ in $Dt1$ remains lower than $0.5$ for detector size as small as $1.0\mu m$, however the absorption gets higher than $0.5$ when detector size is decreased down to $0.5\mu m$. For this size, the absorption efficiency is as high as $0.8$. The absorption of blue light $(420nm)$ in $Dt2$ for configuration $C2$ is much higher compared to that of configuration $C1$. It can be seen in Fig. 5(f) that the absorption at $(420nm)$ is higher than $0.8$ for the detector size of $0.5\mu m$, while it remains higher than 0.5 for the detector size as large as $2.5\mu m$. Moreover, the absorption in red also increases when detector size is decreased to $0.5\mu m$.

From Fig. 5 we learn that the sorting of visible light, with higher absorption efficiency, is possible for submicron detector size using configuration $C2$ which has at least one grating stub on top of second layer of each structure and one on top of junction of second layer. We assume the grating stub on top of the junction of homogenous second layer has a key role for sorting, which results in high absorption efficiency at wavelength of 520nm and 420nm.

We calculate the electromagnetic power dissipation in the silicon at 520nm and 420nm respectively for the configuration $C2$ calculated using COMSOL RF module. Figures 6(a) and 6(b) shows power dissipation in silicon as well as the power flow time average in the unit cell for wavelength of peak absorptions 520nm and 420nm respectively. The detector size of $0.5\mu m$ is used for the calculation. It can be seen from Fig. 6(a) that the grating stub on top of the junction of homogenous second layer directs the portion of incidence power at adjacent structure to the silicon under $n_{IL-1}$ at the wavelength of peak absorption of $Dt1(\lambda : 520nm)$. Similarly, at $420nm$, the grating stub on top of junction directs the portion of power incident at adjacent structure to silicon $(Dt2)$ under $n_{IL-2}$ as can be viewed from Fig. 6(b). This can be viewed as an increase in absorption cross section area of silicon detectors $Dt1$ and $C2$ facilitated by grating stub on top of the junction of second layer.

 

Fig. 6 Electromagnetic power dissipation in silicon as a function of spatial position for configuration $C2$ at wavelength of (a) $520nm$ and (b) $420nm$. The red arrow indicates the power flow time average in the structure.

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Using simple two-layer structure, we show that higher absorption efficiency for the detector size as small as $0.5\text{µ}m$ could be achieved. We found that for such layered structure, larger detector size acts as antireflection structure with maximum absorption limited to 50% of the total incident power in the unit cell. We also found that these layered systems are efficient for detector size lower than $1\text{ µ}m$. Such layered system can therefore be utilized to design the submicron color pixels which are difficult to achieve using conventional color filter technology. We can use simple steps to design submicron sized spectral sorter with high optical efficiency, which we summarize in the following steps:

With this design guideline, we further investigated three-layer structure for sorting of visible light. Figure 7(a) shows the three-layer structure on top of silicon considered for study. The thickness and refractive index of first, second and third layer coating are denoted as $d_1$ and $n_1$, $d_2$ and $n_2$, $d_3$ and $n_3$, respectively. First, we find the refractive index and thickness of each layers which could selectively antireflects blue and green light. The thickness of first, second and third layer coating are determined to be $d_1:60nm$, $d_2:100nm$ and $d_3:50nm$ respectively. The first and third homogenous layers of the structure antireflecting at blue and green light has same refractive index value of $n_1:2.1$ and $n_3:1.5$ respectively, whereas the second layer of the structure designed to antireflect blue light has refractive index value of $n_2:1.5$ and that for green light has refractive index value of $n_2:2.0$. Figure 7(b) shows the reflection as a function of wavelength for the multilayer structure designed to antireflect blue and green light. For convenience, we name the layered system designed to antireflect blue and green light with these parameters as B1 and G1 respectively.

 

Fig. 7 (a) Three layer structure for selective antireflection of colored light. (b) Reflection as a function of wavelength of the three layer antireflection structure (B1-G1) designed for blue-green light, with $d_1: 60nm$, $d_2: 100nm$ and $d_3:50nm$ and $n_1: 2.1$, $n_2: 1.5$ for blue, $n_2: 2.0$ for green light and $n_3: 1.5$ (c) Reflection as a function of wavelength of the three layer antireflection structure (G2-R1) designed for green-red light, with $d_1: 90nm$, $d_2: 100nm$ and $d_3:50nm$ and $n_1: 2.1$, $n_2: 1.5$ for green, $n_2: 2.0$ for red light and $n_3: 1.5$. (d) Unit cell of periodic arrangement of three layered structure with grating stub in junction between the two antireflecting structures, (e) Absorption spectra of B1-G1 structure, and (f) Absorption spectra of G2-R1 structure.

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We also design the multilayer structure which could selectively antireflect green and red light. Figure 7(c) shows the reflection spectra of multilayer structure designed to antireflect green and red light. Both structures have the same material with refractive index $n_1:2.1$ and $n_3:1.5$ in first and third layer respectively. Whereas the second layer of the structure designed to antireflect the green and red light has refractive index value of $n_2:1.5$ and $n_2:2.0$ respectively. The thickness of first, second and third layer coating are determined to be $d_1: 90nm$, $d_2: 100nm$ and $d_3:50nm$ respectively. We call the structure designed to antireflect green and red light as G2 and R1 respectively.

We then, apply the effective medium theory to convert the first homogenous layer of the multilayer coating into the grating structure. The first homogenous layer of B1 and G1 is replaced by the periodic grating with period $P_g=250nm$ and grating width $W=88nm$. The thickness of this first layer is unchanged at $d_1: 60nm$. Similarly, the first homogenous layer of G2 and R1 is replaced by the grating with period $P_g=250nm$ and grating width $W=125nm$, while the grating thickness $d_1: 90nm$ is unchanged. For all cases, the grating material with refractive index 2.5 is considered. The grating groove is considered to be air in the study.

The grating based antireflection structures are now arranged periodically as shown in Fig. 7(d) such that one grating stub is on the junction between two antireflecting structures. For ease of clarity, the second layer of B1 (or G2) and G1 (or R1) are represented as $n_{21}$ and $n_{22}$ respectively. The silicon used in B1(or G2) and G1(or R1) are represented as Dt1 and Dt2 . The size of $Px: 0.5\mu m$ is considered for individual structure in the study.

Figures 7(e) and 7(f) shows the absorption in $Dt1$ and $Dt2$ as a function of incidence wavelength for B1-G1 and G2-R1 systems respectively. It can be seen from Fig. 7(e) that the absorption peak in detectors $Dt1$ and $Dt2$ is higher than $50\% (or 0.5)$ at $480nm$ and $540nm$, which corresponds to the reflection minima of B1 and G1 respectively. Similarly, Fig. 7(f) shows the absorption peaks at $540nm$ and $620nm$ corresponding to reflection minima of G2 and R1 respectively higher than $50\% (or 0.5)$. The absorption is normalized to the total power incident in the unit cell and TE polarization of normal incident light has been considered for the study. With this specific lamellar grating design, one cannot achieve spectral sorting in TM polarization for the same wavelengths as in TE; a new dedicated design should be considered for this polarization.

Regarding the angular sensitivity, we show on Fig. 8 the behavior of the device as a function of incident angle. As it can be seen, the sorting effect is maintained within a significant range of incident angle of [-10°; + 10°], with no spectral shift occurring, which is consistent with the fact that the grating stubs behave as high scattering cross section localized antennas. As a comparison, the signature of a grating-enhanced guided resonance can be slightly observed in Fig. 8(b) in the blue spectral region, showing a high angular dispersion, compared to our resonance of interest.

 

Fig. 8 Absorption as a function of the incident angle θ, for configuration C2 with Px = 0.5µm, for (a) $n_{IL-1}:1.5$ and (b) $n_{IL-2}:2.0$; Region of spectral sorting is delimited by a dashed line.

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The presented results therefore show that spectral sorting of visible light using dielectric-only structure could be achieved while increasing optical efficiency of the detector as small as $0.5\mu m$ . This concept could be directly used for incident light coming from polarized sources, such as lasers for example. For unpolarized incident light, as in the scope of solid-state image sensors applications, one should consider extending the concept to 2D gratings structures, with lattice and stub feature showing 90° rotational symmetry.

4. Summary

We presented the concept of spectral sorting using normalized optical efficiency and study dielectric based multilayer structure for spectral sorting of visible light. We investigated the antireflection property of multilayer dielectric coating on silicon. We then applied effective medium theory and replaced one of the homogenous layers in multilayer coating by subwavelength grating structure and showed that spectral sorting with high optical efficiency could be achieved in visible regime. We found that grating assisted spectral sorter structures are more efficient when the detector size is less than $1\mu m$, enabling to shrink the detector size to the wavelength scale. The comprehensive design strategy derived could be used as design guideline to achieve sorting of visible light. Using those guidelines, we then studied a three-layer dielectric structure to achieve sorting of blue green and red colored light. We showed that even for pixel size as less as $0.5\mu m$ optical efficiency as high as 80% could be achieved.