All-Dielectric Color Filter with Ultra-Narrowed Linewidth

In this paper, a transmissive color filter with an ultra-narrow full width at half of the maximum is proposed. Exploiting a material with a high index of refraction and an extremely low extinction coefficient in the visible range allows the quality factor of the filter to be improved. Three groups of GaP/SiO2 pairs are used to form a Distributed Brag reflector in a symmetrical Fabry-Pérot cavity. A band-pass filter which is composed of ZnS/SiO2 pairs is also introduced to further promote the purity of the transmissive spectrum. The investigation manifests that a series of tuned spectrum with an ultra-narrow full width at half of the maximum in the full visible range can be obtained by adjusting the thickness of the SiO2 interlayer. The full width at half of the maximum of the transmissive spectrum can reach 2.35 nm. Simultaneously, the transmissive efficiency in the full visible range can keep as high as 0.75. Our research provides a feasible and cost-effective way for realizing filters with ultra-narrowed linewidth.


Introduction
Color filter is a kind of optical device which plays a significant role in many industrial fields, such as spectroscopy instruments, imaging sensors, and displays [1][2][3][4]. The precision and sensibility of many optical systems used in those fields are mainly decided by such an optical device. Generally, there are two types of optical filters: reflective filter and transmissive filter [5]. Usually, the reflective filter is realized by a perfect absorber, of which the filtered wavelengths are fully absorbed. As a result, its function is embodied in the reflective characteristic. On the contrary, the filtering function of a transmissive filter is obtained by permitting the required wavelengths to pass through. In comparison with the reflective filter, the transmissive filter is proper to be used in displays, charge-coupled devices, and hyperspectral imaging. With respect to the transmissive filter, the quality factor or the full width at half of the maximum (FWHM) of the transmissive spectrum is a principal characteristic if we want a filter with high accuracy and resolution [3]. In addition, the efficiency of the filter is also an important characteristic that should be considered. That is because a higher transmittance means a better energy use of incident light, which will be beneficial to reduce the dissipation of power and obtain a high intensity of the signal in optical systems using these filters.

Simulation Models
The simulated structure in this paper is shown in Figure 1. The light incidents from the top of the device and outputs from a glass substrate. As shown in Figure 1, the device's structure can be divided into two components-a symmetrical FP cavity and a band-pass filter. The FP cavity is realized by two DBRs and a SiO 2 interlayer. The DBR consists of several groups of GaP/SiO 2 pairs. The thicknesses of GaP and SiO 2 in each group and the number of groups have been optimized for obtaining a transmissive spectrum with an ultra-narrow FWHM and high intensity. The ZnS/SiO 2 multilayer is set on top of the cavity as a band-pass filter for the purpose of improving the purity of the transmissive spectrum. The refractive indices and extinction coefficients of various materials used in the simulation are shown in Figure 2. All the data are quoted from other reports [43][44][45].
Micromachines 2021, 12, x FOR PEER REVIEW 2 of 12 array metal films [10,36], metal grating [23,37], and nanoparticle arrays. In comparison, the FP cavity based on multilayered thin films presents many advantages, such as easy to be fabricated, facilitating large-scale production. In recent years, there have been a large number of reports on the design of filters based on the FP cavity [16,20,21,[38][39][40][41]. Using a dielectric layer sandwiched by two metallic layers to form an FP cavity is a convenient way, which is also named as metal-dielectric-metal (MDM) structure. However, the strong absorption of metallic layers limits the quality factor [39]. To further improve the quality factor of such structure, a distributed Bragg reflection (DBR) is a preferable candidate to substitute the metallic reflective mirror. Since the DBR structure is fabricated by various dielectric layers, the absorption in metallic layers can be avoided. So far, there are still few reports on filters with extremely narrow FWHM (<10 nm) in the visible range with a relatively simple DBR structure.
In this paper, we propose a transmission filter structure based on an FP cavity. The reflective mirror is composed of GaP/SiO2 etalon. As a direct bandgap semiconductor material with a bandgap of 2.26 eV, GaP has a high refractive index in the visible range and can also be prepared by low-temperature atomic layer deposition technology or chemical vapor deposition technology, which is compatible with other thin-film deposition processing [42]. In addition, the absorption coefficient is close to zero when the wavelength is longer than 470 nm. A multilayer composed of ZnS/SiO2 pairs is used as a band-pass filter to improve the quality of the spectrum. Since the materials used have relatively small absorption in the visible range, it is expected to obtain a smaller FWHM through structural design. This research not only contributes to the preparation of large-area, high-resolution transmissive filters but also helps expand the application range of high-refractive-index semiconductor materials in the optical field.

Simulation Models
The simulated structure in this paper is shown in Figure 1. The light incidents from the top of the device and outputs from a glass substrate. As shown in Figure 1, the device's structure can be divided into two components-a symmetrical FP cavity and a band-pass filter. The FP cavity is realized by two DBRs and a SiO2 interlayer. The DBR consists of several groups of GaP/SiO2 pairs. The thicknesses of GaP and SiO2 in each group and the number of groups have been optimized for obtaining a transmissive spectrum with an ultra-narrow FWHM and high intensity. The ZnS/SiO2 multilayer is set on top of the cavity as a band-pass filter for the purpose of improving the purity of the transmissive spectrum. The refractive indices and extinction coefficients of various materials used in the simulation are shown in Figure 2. All the data are quoted from other reports [43][44][45].  A finite-difference time-domain (FDTD) simulation was performed by using commercial software (Version 8.24, Ansys Lumerical, Inc, Vancouver, BC, Canada). In the simulation, a plane electromagnetic (EM) wave was used as an excitation source. The intensity of the electric field was set to 1, and the incident direction was the z backward. The boundary condition in the z-direction and other directions were perfect match layer (PML) and periodic boundary conditions, respectively. In order to simulate the performance of the device in different polarization states, two polarization, i.e., transverse magnetic (TM), transverse electric (TE) modes, were used. In TM and TE modes, the polarization directions of the electric field were set to parallel to and perpendicular to the plane of incidence. A Broadband Fixed Angle Source Technique (BFAST) was used to simulate the transmissive spectra of the device as the incidence of the wide-band EM wave is oblique. A monitor of electric field power was located on the opposite side of the device to test the transmittance. In addition, a 2D electric field monitor was placed parallel to the incident plane to test the distribution of the electric field in each layer.

Optimization of the FP Cavity
As a principal component of the device, the FP cavity without a band-pass filter was optimized firstly. The main purpose of the optimization is to generate a kind of transmissive spectrum, which possesses an ultra-narrow FWHM and high transmittance simultaneously. It is well known that the spectral FWHM is determined by the reflectivity of the reflective ends, the refractive index, and the thickness of the interlayer [46]. Its expression is shown as follows: where R1 and R2 are the reflectivities of the reflective ends, c is the velocity of electromagnetic waves in vacuum, n is the refractive index of the interlay, and d is the physical thickness of the SiO2 interlayer. According to the equation, it is clear that the FWHM can be narrowed by increasing the reflectivities of the two DBRs in the proposed structure, the refractive index, and the thickness of the intermediate dielectric layer. Accordingly, the thicknesses of each layer and the number of GaP/SiO2 groups in the DBR will be the critical parameters, which should be optimized firstly.
In the beginning, the number of GaP/SiO2 groups is set to three. The thickness of each layer in the DBR is optimized by the Particle Swarm optimization method [47,48], which A finite-difference time-domain (FDTD) simulation was performed by using commercial software (Version 8.24, Ansys Lumerical, Inc, Vancouver, BC, Canada). In the simulation, a plane electromagnetic (EM) wave was used as an excitation source. The intensity of the electric field was set to 1, and the incident direction was the z backward. The boundary condition in the z-direction and other directions were perfect match layer (PML) and periodic boundary conditions, respectively. In order to simulate the performance of the device in different polarization states, two polarization, i.e., transverse magnetic (TM), transverse electric (TE) modes, were used. In TM and TE modes, the polarization directions of the electric field were set to parallel to and perpendicular to the plane of incidence. A Broadband Fixed Angle Source Technique (BFAST) was used to simulate the transmissive spectra of the device as the incidence of the wide-band EM wave is oblique. A monitor of electric field power was located on the opposite side of the device to test the transmittance. In addition, a 2D electric field monitor was placed parallel to the incident plane to test the distribution of the electric field in each layer.

Optimization of the FP Cavity
As a principal component of the device, the FP cavity without a band-pass filter was optimized firstly. The main purpose of the optimization is to generate a kind of transmissive spectrum, which possesses an ultra-narrow FWHM and high transmittance simultaneously. It is well known that the spectral FWHM is determined by the reflectivity of the reflective ends, the refractive index, and the thickness of the interlayer [46]. Its expression is shown as follows: where R 1 and R 2 are the reflectivities of the reflective ends, c is the velocity of electromagnetic waves in vacuum, n is the refractive index of the interlay, and d is the physical thickness of the SiO 2 interlayer. According to the equation, it is clear that the FWHM can be narrowed by increasing the reflectivities of the two DBRs in the proposed structure, the refractive index, and the thickness of the intermediate dielectric layer. Accordingly, the thicknesses of each layer and the number of GaP/SiO 2 groups in the DBR will be the critical parameters, which should be optimized firstly.
In the beginning, the number of GaP/SiO 2 groups is set to three. The thickness of each layer in the DBR is optimized by the Particle Swarm optimization method [47,48], which has been a mature module of the software. The aim of optimization is that the mean reflectance at the interface between the SiO 2 interlayer and the DBR reaches the highest value. According to the result of optimization, the thicknesses of GaP and SiO 2 in a single etalon are fixed to 38 nm and 100 nm, respectively. In order to illustrate the result of optimization clearly, the reflective spectra of the DBRs with different thicknesses of GaP and SiO 2 are plotted in Figure 3. It can be seen from Figure 3a that the DBR with 38 nm-thick GaP and 100 nm-thick SiO 2 shows a better reflection in the range from 450 nm to 750 nm, which covers the visible range exactly, than other DBRs. The reflectance of the optimized DBR reflector is higher than 0.8 in the full range. Especially in the range from 460 nm to 680 nm, the mean reflectance surpasses 0.9. Once the thicknesses of GaP and SiO 2 are fixed, the number of GaP/SiO 2 pairs can be determined facilely. It is obvious in Figure 3a that the reflectance of the DBR reflector becomes higher and higher, even close to 1.0 in the range from 450 nm to 750 nm, as the number of Gap/SiO 2 pairs increases. According to such a result, it seems that the more numbers of Gap/SiO 2 pairs the DBR has, the better performance the FP cavity possesses. However, the mean reflectance of the DBR is not the only factor that should be considered. Subsequently, the transmissive spectrum of the FP cavity with 170 nm-thick SiO 2 interlayer is simulated and plotted in Figure 3b. It is straightforward that the intensity of the transmissive spectrum at the center wavelength, i.e., the resonant wavelength of the FP cavity, decreases dramatically as the number of groups increases. As shown in the inset in Figure 3b, the transmission at 520 nm almost vanishes when the number of GaP/SiO 2 pairs reaches five. In another aspect, the more GaP/SiO 2 pairs will lead to a more complicated fabrication processing and higher cost undoubtedly. Accordingly, three groups of GaP/SiO 2 in the DBR are the best choice for the device. has been a mature module of the software. The aim of optimization is that the mean reflectance at the interface between the SiO2 interlayer and the DBR reaches the highest value. According to the result of optimization, the thicknesses of GaP and SiO2 in a single etalon are fixed to 38 nm and 100 nm, respectively. In order to illustrate the result of optimization clearly, the reflective spectra of the DBRs with different thicknesses of GaP and SiO2 are plotted in Figure 3. It can be seen from Figure 3a that the DBR with 38 nm-thick GaP and 100 nm-thick SiO2 shows a better reflection in the range from 450 nm to 750 nm, which covers the visible range exactly, than other DBRs. The reflectance of the optimized DBR reflector is higher than 0.8 in the full range. Especially in the range from 460 nm to 680 nm, the mean reflectance surpasses 0.9. Once the thicknesses of GaP and SiO2 are fixed, the number of GaP/SiO2 pairs can be determined facilely. It is obvious in Figure 3a that the reflectance of the DBR reflector becomes higher and higher, even close to 1.0 in the range from 450 nm to 750 nm, as the number of Gap/SiO2 pairs increases. According to such a result, it seems that the more numbers of Gap/SiO2 pairs the DBR has, the better performance the FP cavity possesses. However, the mean reflectance of the DBR is not the only factor that should be considered. Subsequently, the transmissive spectrum of the FP cavity with 170 nm-thick SiO2 interlayer is simulated and plotted in Figure 3b. It is straightforward that the intensity of the transmissive spectrum at the center wavelength, i.e., the resonant wavelength of the FP cavity, decreases dramatically as the number of groups increases. As shown in the inset in Figure 3b, the transmission at 520 nm almost vanishes when the number of GaP/SiO2 pairs reaches five. In another aspect, the more GaP/SiO2 pairs will lead to a more complicated fabrication processing and higher cost undoubtedly. Accordingly, three groups of GaP/SiO2 in the DBR are the best choice for the device. Once the structural parameters of the DBR are fixed, the resonant wavelength of the FP cavity is determined only by the physical thickness of the SiO2 interlayer. The resonant wavelength of the FP cavity can be expressed as follows [46]: where d is the thickness of SiO2, n( ) λ is the refractive index of SiO2, m is the eigenmode order, and m λ stands for the resonant wavelength of m th eigenmode. Figure  4 shows the evolution of the transmissive spectrum as the thickness of the intermediate dielectric layer SiO2 changes. It is obvious that three ultra-narrow bright bands appear in sequence, which represent three different orders of resonance mode, as the thickness of SiO2 increases from 50 nm to 300 nm. The peak of the spectrum at the resonant wavelength Once the structural parameters of the DBR are fixed, the resonant wavelength of the FP cavity is determined only by the physical thickness of the SiO 2 interlayer. The resonant wavelength of the FP cavity can be expressed as follows [46]: where d is the thickness of SiO 2 , n(λ) is the refractive index of SiO 2 , m is the eigenmode order, and λ m stands for the resonant wavelength of mth eigenmode. Figure 4 shows the evolution of the transmissive spectrum as the thickness of the intermediate dielectric layer SiO 2 changes. It is obvious that three ultra-narrow bright bands appear in sequence, which represent three different orders of resonance mode, as the thickness of SiO 2 increases from 50 nm to 300 nm. The peak of the spectrum at the resonant wavelength presents a redshift ranging from 420 nm to 700 nm as the thickness of SiO 2 increases. That means three primary colors for displays can be obtained easily via such a structure.
the interlayer of the FP cavity is fixed to a certain thickness. However, the high order indicates that the photons will travel in multiple rounds in the cavity before exiting from the other side. That will lead to a decay of the transmissive spectrum once loss exists in the dielectric layers [46]. As a result, using the resonant wavelength of an eigenmode in a higher-order to pursuit a narrower FWHM is not a suitable scheme for realizing a filter with high efficiency. In this paper, we choose the 1st order eigenmode to realize the function of the FP cavity. In order to analyze the characteristics of the FP cavity further, we extracted the transmissive spectra of the cavity with various thicknesses of the SiO2 interlayer (see Figure 5). It is straightforward that the center wavelength shifts from blue range to red range sequentially along with the increment of the thickness of SiO2. When the thickness increases from 128 nm to 248 nm, the center wavelength shifts from 465 nm to 636 nm. The variation range of the transmissive spectrum almost covers the visible range. The detailed data of the transmissive spectra are listed in Table 1. As shown in the table, the FWHMs of all spectra are less than 10 nm. In particular, the FWHMs of spectra corresponding to the center wavelengths at 465 nm, 520 nm, and 620 nm reach 6.46 nm, 2.05 nm, and 5.35 nm, respectively. Meanwhile, the transmittances can reach 0.756, 0.822, and 0.902, respectively.  According to Equation (1), it can be known that the resonant wavelength of the eigenmode in a high order will present a narrower FWHM than that in low order when the interlayer of the FP cavity is fixed to a certain thickness. However, the high order indicates that the photons will travel in multiple rounds in the cavity before exiting from the other side. That will lead to a decay of the transmissive spectrum once loss exists in the dielectric layers [46]. As a result, using the resonant wavelength of an eigenmode in a higher-order to pursuit a narrower FWHM is not a suitable scheme for realizing a filter with high efficiency. In this paper, we choose the 1st order eigenmode to realize the function of the FP cavity.
In order to analyze the characteristics of the FP cavity further, we extracted the transmissive spectra of the cavity with various thicknesses of the SiO 2 interlayer (see Figure 5). It is straightforward that the center wavelength shifts from blue range to red range sequentially along with the increment of the thickness of SiO 2 . When the thickness increases from 128 nm to 248 nm, the center wavelength shifts from 465 nm to 636 nm. The variation range of the transmissive spectrum almost covers the visible range. The detailed data of the transmissive spectra are listed in Table 1. As shown in the table, the FWHMs of all spectra are less than 10 nm. In particular, the FWHMs of spectra corresponding to the center wavelengths at 465 nm, 520 nm, and 620 nm reach 6.46 nm, 2.05 nm, and 5.35 nm, respectively. Meanwhile, the transmittances can reach 0.756, 0.822, and 0.902, respectively. presents a redshift ranging from 420 nm to 700 nm as the thickness of SiO2 increases. That means three primary colors for displays can be obtained easily via such a structure. According to Equation (1), it can be known that the resonant wavelength of the eigenmode in a high order will present a narrower FWHM than that in low order when the interlayer of the FP cavity is fixed to a certain thickness. However, the high order indicates that the photons will travel in multiple rounds in the cavity before exiting from the other side. That will lead to a decay of the transmissive spectrum once loss exists in the dielectric layers [46]. As a result, using the resonant wavelength of an eigenmode in a higher-order to pursuit a narrower FWHM is not a suitable scheme for realizing a filter with high efficiency. In this paper, we choose the 1st order eigenmode to realize the function of the FP cavity. In order to analyze the characteristics of the FP cavity further, we extracted the transmissive spectra of the cavity with various thicknesses of the SiO2 interlayer (see Figure 5). It is straightforward that the center wavelength shifts from blue range to red range sequentially along with the increment of the thickness of SiO2. When the thickness increases from 128 nm to 248 nm, the center wavelength shifts from 465 nm to 636 nm. The variation range of the transmissive spectrum almost covers the visible range. The detailed data of the transmissive spectra are listed in Table 1. As shown in the table, the FWHMs of all spectra are less than 10 nm. In particular, the FWHMs of spectra corresponding to the center wavelengths at 465 nm, 520 nm, and 620 nm reach 6.46 nm, 2.05 nm, and 5.35 nm, respectively. Meanwhile, the transmittances can reach 0.756, 0.822, and 0.902, respectively.

Optimization of the Band-Pass Filter
To improve the quality of the spectrum further, we introduce a band-pass filter on top of the FP cavity to stop the light, of which the wavelength is right consistent with the superfluous resonant wavelength, pass through. The structural parameters of the band-pass filter, which is composed of ZnS/SiO 2 pairs, are optimized by utilizing a similar way to that used in the optimization of the FP cavity.
The transmissive spectra of ZnS/SiO 2 pairs with different thicknesses and number of pairs are compared in Figure 6. The main purpose for the optimization is depressing the spectral transmittance at the two ends of the visible range but allowing high spectral transmittance in the variation range of the resonant wavelength mentioned in Section 3.1. As shown in Figure 6a, the permitted band of the spectrum shifts to red as the thicknesses of ZnS and SiO 2 increase. In comparison with other spectra, the spectrum of the band-pass filter with 107 nm-thick ZnS and 107 nm-thick SiO 2 presents a permitted band that can cover the range from 465 nm to 636 nm. Though increasing the number of ZnS/SiO 2 pairs is helpful for widening the bandwidth and forming a steep edge (see Figure 6b), the performance of band-pass filter with four groups of ZnS/SiO 2 is sufficient to achieve the aim of optimization. In order to illustrate the results of optimization, the transmissive spectra of an integrated device are simulated.

Optimization of the Band-Pass Filter
To improve the quality of the spectrum further, we introduce a band-pass filter on top of the FP cavity to stop the light, of which the wavelength is right consistent with the superfluous resonant wavelength, pass through. The structural parameters of the bandpass filter, which is composed of ZnS/SiO2 pairs, are optimized by utilizing a similar way to that used in the optimization of the FP cavity.
The transmissive spectra of ZnS/SiO2 pairs with different thicknesses and number of pairs are compared in Figure 6. The main purpose for the optimization is depressing the spectral transmittance at the two ends of the visible range but allowing high spectral transmittance in the variation range of the resonant wavelength mentioned in Section 3.1. As shown in Figure 6a, the permitted band of the spectrum shifts to red as the thicknesses of ZnS and SiO2 increase. In comparison with other spectra, the spectrum of the band-pass filter with 107 nm-thick ZnS and 107 nm-thick SiO2 presents a permitted band that can cover the range from 465 nm to 636 nm. Though increasing the number of ZnS/SiO2 pairs is helpful for widening the bandwidth and forming a steep edge (see Figure 6b), the performance of band-pass filter with four groups of ZnS/SiO2 is sufficient to achieve the aim of optimization. In order to illustrate the results of optimization, the transmissive spectra of an integrated device are simulated.    Figure 7 shows the transmissive spectra of the integrated device. It can be seen from the figure that the integrated device obtains an outstanding unimodal spectrum in the full visible range due to the suppression in the short wavelength range and longwavelength range. In addition, the integrated device keeps excellent performance. As a result, the FWHMs of the transmissive spectra in the visible range are still less than 10 nm. The FWHMs at center wavelengths of 465 nm, 520 nm, and 620 nm are 6.70 nm, 2.35 nm, and 6.91 nm, respectively. Although the introduction of ZnS/SiO 2 will weaken the transmittance slightly, the transmittance of the integrated device is still higher than 0.75 at various wavelengths. The transmittances reach 0.75, 0.82, and 0.90 at 465 nm, 520 nm, and 620 nm, respectively.
Micromachines 2021, 12, x FOR PEER REVIEW 7 of 12 visible range due to the suppression in the short wavelength range and long-wavelength range. In addition, the integrated device keeps excellent performance. As a result, the FWHMs of the transmissive spectra in the visible range are still less than 10 nm. The FWHMs at center wavelengths of 465 nm, 520 nm, and 620 nm are 6.70 nm, 2.35 nm, and 6.91 nm, respectively. Although the introduction of ZnS/SiO2 will weaken the transmittance slightly, the transmittance of the integrated device is still higher than 0.75 at various wavelengths. The transmittances reach 0.75, 0.82, and 0.90 at 465 nm, 520 nm, and 620 nm, respectively.

Evaluation of the Device's Performance
In order to evaluate the chromaticity of the transmissive spectrum, the color coordinate of the transmissive spectrum in the 1931 Commission Internationale de L'Eclairage (CIE) coordinate system is calculated according to the CIE-XYZ tristimulus value, which is obtained from the following expressions: where kis the adjustment factor, x, y and z are the optical efficiency functions, and R( ) λ is the calculated spectrum. The coordinate can be calculated as follows: The coordinates of spectra with different center wavelengths are shown in Figure 8. It can be seen from the figure that the chromaticities of various transmissive spectra are very close to the boundary of the CIE chart, especially for the spectrum with a center wavelength of 465 nm. It indicates that the spectrum has a high color purity. As the thick-

Evaluation of the Device's Performance
In order to evaluate the chromaticity of the transmissive spectrum, the color coordinate of the transmissive spectrum in the 1931 Commission Internationale de L'Eclairage (CIE) coordinate system is calculated according to the CIE-XYZ tristimulus value, which is obtained from the following expressions: where k is the adjustment factor, x, y and z are the optical efficiency functions, and R(λ) is the calculated spectrum. The coordinate can be calculated as follows: The coordinates of spectra with different center wavelengths are shown in Figure 8. It can be seen from the figure that the chromaticities of various transmissive spectra are very close to the boundary of the CIE chart, especially for the spectrum with a center wavelength of 465 nm. It indicates that the spectrum has a high color purity. As the thickness of the intermediate layer increases, the corresponding coordinate varies from the region of blue to the regions of green, yellow, and red successively. As shown in Figure 8, most of the coordinates locate outside the Adobe RGB and sRGB chromatic gamut. That means a larger scale of chromatic gamut can be obtained, and more vivid colors can be reproduced via such filter.
ness of the intermediate layer increases, the corresponding coordinate varies from the region of blue to the regions of green, yellow, and red successively. As shown in Figure 8, most of the coordinates locate outside the Adobe RGB and sRGB chromatic gamut. That means a larger scale of chromatic gamut can be obtained, and more vivid colors can be reproduced via such filter.  Figure 9 shows the distribution of electric fields in each layer in the integrated device at various wavelengths when the thickness of SiO2 is 170 nm. Correspondingly, the center wavelength of the transmissive spectrum is 520 nm. The intensity of the electric field at 520 nm in the intermediate medium gets to the maximum in comparison with the electric fields at other wavelengths, such as 460 nm, 600 nm, and 700 nm. The consistency between the wavelength where the electric field in cavity reaches the maximum and the center wavelength of the transmissive spectrum indicates that the FP microcavity plays a decisive role in the filter device.   Figure 9 shows the distribution of electric fields in each layer in the integrated device at various wavelengths when the thickness of SiO 2 is 170 nm. Correspondingly, the center wavelength of the transmissive spectrum is 520 nm. The intensity of the electric field at 520 nm in the intermediate medium gets to the maximum in comparison with the electric fields at other wavelengths, such as 460 nm, 600 nm, and 700 nm. The consistency between the wavelength where the electric field in cavity reaches the maximum and the center wavelength of the transmissive spectrum indicates that the FP microcavity plays a decisive role in the filter device.
ness of the intermediate layer increases, the corresponding coordinate varies from the region of blue to the regions of green, yellow, and red successively. As shown in Figure 8, most of the coordinates locate outside the Adobe RGB and sRGB chromatic gamut. That means a larger scale of chromatic gamut can be obtained, and more vivid colors can be reproduced via such filter.  Figure 9 shows the distribution of electric fields in each layer in the integrated device at various wavelengths when the thickness of SiO2 is 170 nm. Correspondingly, the center wavelength of the transmissive spectrum is 520 nm. The intensity of the electric field at 520 nm in the intermediate medium gets to the maximum in comparison with the electric fields at other wavelengths, such as 460 nm, 600 nm, and 700 nm. The consistency between the wavelength where the electric field in cavity reaches the maximum and the center wavelength of the transmissive spectrum indicates that the FP microcavity plays a decisive role in the filter device.  According to Equation (1), the ultra-narrow linewidth of the filter is ascribable to the high reflection on the GaP/SiO 2 DBR, while the high transmissive efficiency is attributable to the low absorption of the device. Figure 10b shows the absorbed power density in each layer at various wavelengths. It can be seen in Figure 10a that a slight absorption occurs in the GaP layer, which is adjacent to the intermediate layer. Accompany with the diminishing of reflectivity at the resonant wavelength, the transmissive efficiency at resonant wavelength gets to a very high value according to the formula A(λ) + T(λ) + R(λ) = 1. Figure 10d shows an overall absorption of the device in the full visible range. It can be seen from the figure that absorption in the full range is very low except for a strong peak near 460 nm and a small absorption peak near 700 nm. As shown in Figure 10a,c, the strong absorptions at these wavelengths occur in the top DBR of the FP cavity and the band-pass filter. That is because the GaP and SiO 2 layers in the DBR forms some submicrocavities as well as the ZnS and SiO 2 layers in the band-pass filter. When the wavelengths of incident light satisfy the resonant conditions in those submicrocavities, there will be an enhancement of the electric field in the interlayer. Thus, some discrete absorbing peaks at those resonant wavelengths will exist in the absorption curve. According to Equation (1), the ultra-narrow linewidth of the filter is ascribable to the high reflection on the GaP/SiO2 DBR, while the high transmissive efficiency is attributable to the low absorption of the device. Figure 10b shows the absorbed power density in each layer at various wavelengths. It can be seen in Figure 10a that a slight absorption occurs in the GaP layer, which is adjacent to the intermediate layer. Accompany with the diminishing of reflectivity at the resonant wavelength, the transmissive efficiency at resonant wavelength gets to a very high value according to the formula A(λ) + T(λ) + R(λ) = 1. Figure 10d shows an overall absorption of the device in the full visible range. It can be seen from the figure that absorption in the full range is very low except for a strong peak near 460 nm and a small absorption peak near 700 nm. As shown in Figure 10a,c, the strong absorptions at these wavelengths occur in the top DBR of the FP cavity and the band-pass filter. That is because the GaP and SiO2 layers in the DBR forms some submicrocavities as well as the ZnS and SiO2 layers in the band-pass filter. When the wavelengths of incident light satisfy the resonant conditions in those submicrocavities, there will be an enhancement of the electric field in the interlayer. Thus, some discrete absorbing peaks at those resonant wavelengths will exist in the absorption curve.  Figure 11 shows the evolution of the transmissive spectrum as the angle of incidence with different polarization modes increases from 0° to 50°. An evident shifting of the center wavelength occurs as the angle of incidence is greater than 10°. The variation almost reaches 50 nm for both of the modes, as shown in Figure 11, suggesting that the device is sensitive to the incident angle. Another important message from Figure 11 is that the evolution trends of the transmissive spectra in TM & TE modes are consistent. There is only a difference in the intensities of the transmissive spectra. The similar evolution trends in different modes are attributed to the isotropy of light response of the film in the x-y plane.  Figure 11 shows the evolution of the transmissive spectrum as the angle of incidence with different polarization modes increases from 0 • to 50 • . An evident shifting of the center wavelength occurs as the angle of incidence is greater than 10 • . The variation almost reaches 50 nm for both of the modes, as shown in Figure 11, suggesting that the device is sensitive to the incident angle. Another important message from Figure 11 is that the evolution trends of the transmissive spectra in TM & TE modes are consistent. There is only a difference in the intensities of the transmissive spectra. The similar evolution trends in different modes are attributed to the isotropy of light response of the film in the x-y plane.
The results of the simulation manifest that the filter based on the proposed structure in this paper presents an excellent performance. Moreover, Gudovskikh, A. S. et al. reported that GaP could be grown on a Si wafer by low-temperature plasma-enhanced atomic layer deposition [42]. As for ZnS and SiO 2 , all these materials can be deposited by atomic layer deposition according to other researchers' reports [49,50]. Accordingly, it provides us a feasible way to fabricate the device in the same chamber at one time without breaking down the vacuum. As a result, the device investigated in this paper also has a potential value for a practical application in the future. The results of the simulation manifest that the filter based on the proposed structure in this paper presents an excellent performance. Moreover, Gudovskikh, A. S. et al. reported that GaP could be grown on a Si wafer by low-temperature plasma-enhanced atomic layer deposition [42]. As for ZnS and SiO2, all these materials can be deposited by atomic layer deposition according to other researchers' reports [49,50]. Accordingly, it provides us a feasible way to fabricate the device in the same chamber at one time without breaking down the vacuum. As a result, the device investigated in this paper also has a potential value for a practical application in the future.

Conclusions
In summary, a transmissive filter composed of GaP/SiO2 DBR and ZnS/SiO2 is simulated. An ultra-narrow linewidth of spectrum in the visible range is obtained according to the results of the simulation. The FWHMs of transmission peaks corresponding to the center wavelengths at 465 nm, 520 nm, and 620 nm reach 6.70 nm, 2.35 nm, and 6.91 nm, respectively. Because the entire device is designed based on dielectric materials, a very low total absorption in the visible range is achieved. The transmittance of the device can reach 0.75, 0.82, and 0.90 at 465 nm, 520 nm, and 620 nm. In comparison with traditional MDM structure, the narrower linewidth and higher transmittance result in more saturated primary colors. Finally, a larger scale of the color gamut can be realized in comparison with that of Adobe RGB color gamut. Additionally, the device also shows polarizationindependent characteristics, which is very suitable for high-resolution imaging systems, detectors, and spectrometers.