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Abstract
High-performance angle-tolerant polarization tunable reflective color filters with an adopted one-dimensional silicon grating structure are proposed. The polarization-tunable color filtering properties and incident angle-insensitive properties are thoroughly investigated to achieve the angle-tolerant polarization controlled continuous color palettes. Benefiting from high refractive index of silicon, the proposed color filters evidently provide a high angular tolerance of 30° in conjunction with a large polarization-tunable color range. Moreover, full and vivid colors covering the entire visible range can be produced by simply altering the grating thickness. The combination between the all-dielectric nanostructure with a high angular tolerance and the polarization-tunable filtering properties, described in this paper, can be applied in anti-counterfeiting, imaging and filtering technologies, and so forth.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Whether in the real world or virtual reality, colors play extremely important roles for human beings in identifying their surroundings owing to their capability of carrying rich information [1,2]. In principle, colors can be produced by selectively projecting and collecting parts of white light (typically red, green, and blue), which is a characteristic of almost all color devices. This selective separation could be accomplished by various color filters. However, the most commonly used color filters, based on pigments and dye, have a susceptibility to moisture and high temperatures, resulting in their poor durability [3]. Due to the shortcomings of pigment and dye color filters, color filters based on nanostructures have recently attracted widespread attention owing to their potential characteristics of high resolution, high diffractive efficiency, small pixel size, and long-term stability [4–6]. One obvious example is a color filter based on plasmonic nanostructures. The interaction between light and metal nanostructures can achieve full-color light separation in either reflective systems or transmissive systems [7–9]. All-dielectric structural color filter is another notable example. The guided-mode resonance or electric/magnetic dipole modes resonance can enable all-dielectric nanostructures to selectively reflect or transmit particular wavelengths in the visible spectrum [10–12]. All-dielectric structural color filters are regarded as a better one than plasmonic color filters due to their low intrinsic losses and costs. Silicon (Si) nanostructures have been used to manufacture low-loss color filters due to the excellent properties of Si such as low intrinsic losses, high refractive index, and CMOS-compatible fabrication technique [13]. By altering the size and shape of Si nanostructures, vivid colors covering the whole visible spectrum have recently been produced [10–18]. Although the research on structural color filters have made rapid progress, the optical properties of most nanostructures are unalterable once they are manufactured with a fixed size, which hinders their applications in security tags and dynamic display.
In order to realize dynamic colors in nanostructures with a fixed size, several schemes such as the electro-optic effect, mechanical stretching, and the alteration of the surrounding environments were previously attempted [19–22]; however, all the above schemes are not straightforward enough, restricting their applications in biomedical sensors and anticounterfeiting [23–25]. Recently, a polarization-tunable color filtering scheme has attracted wide attention due to its relatively straightforward mechanism. The chromatic filter based on positive metallic nanoantennas proposed by Ellenbogen can realize dynamic color output by altering the incident polarization angle [26]. Similarly, a dynamic color output was also achieved with the aid of the interplay of light and the negative metallic nanoantennas [27], as proposed by Clark et al. However, the design and manufacture of color filters with cross-shape or two-dimensional nanostructures are costly, complex, and only suitable for small-area fabrication, which hinders their practical applications. Polarization-tunable color filters based on simple structures such as one-dimensional grating structures have also been reported [28–30]. As far as we know, most researches to date have focused on color tunable properties; little attention has been paid to the color tunable devices with high angular toleration.
Here, we present an angle-insensitive polarization tunable color filter based on a single-layered all-dielectric grating structure. The proposed tunable color filters not only readily achieve the continuous palettes of output colors by adaptively altering the polarization angle of incidence, but also produce full and vivid structural colors via a proper alteration of the single-layered Si grating thickness. Moreover, the reflective colors produced are insensitive to the incident angle of up to 30° for both the transverse magnetic (TM) polarized light and the transverse electric (TE) polarized light, which is mainly caused by the small propagation angle across the intermediate layers with high refractive index.
2. Structure and design
The schematic geometry of the proposed angle insensitive polarization-tunable all-dielectric color filter is shown in Fig. 1(a), which consists of a one-dimensional single-layer grating structure. The grating width, period, and thickness of the single-layer grating structure are denoted by a, p, and t, respectively. The duty ratio of the grating structure is defined as f = a/p. The materials of the grating layer and substrate are Si and silica (SiO2), whose optical constants are illustrated in Figs. 1(d) and 1(e). As shown in Fig. 1(d), the refractive index (n) of Si layer varies from 6.579 at 380 nm to 3.713 at 780 nm and its extinction coefficient (k) varies from 1.176 at 380 nm to 0.006 at 780 nm. The refractive index (n) of SiO2 substrate remains almost 1.46 and the extinction coefficient (k) keeps nearly zero in the entire visible spectral range, which is shown in Fig. 1(e). Here, Si is selected as the material of the grating layer, which mainly depends on high refractive index contrast between Si and air and its low absorption loss. The high refractive index contrast that exists in this structure is the physical origins of polarization-tunable properties and angle-insensitive properties. The surrounding media is air (nc = 1). Furthermore, a continuous palette between two different output colors could be obtained by varying the polarization angle φ of plane wave illuminated at an incident angle θ.
Fig. 1. (a) Schematic geometry of the polarization-mediated color filter with the structural parameters denoted. The filter can generate two different colors under TM polarization and TE polarization, respectively. (b) Reflectance spectra for linearly polarized illumination of the proposed polarization-tunable color filter with polarization angle ranging from φ = 0° to φ = 90° in steps of 10°. The grating structure with dimensions of a = 100 nm, p = 210 nm, and t = 70 nm where a, p, and t represent the grating width, period, and thickness, respectively. (c) Mapping of transmittance spectra of (b) to black stars in the CIE 1931 chromaticity diagram, showing active polarization-dependent blue to yellow color filtering. The reflectance spectra for the color filters at different polarization angles are further presented with white solid lines and the arrow indicates the direction. (d) and (e) Optical constants of Si and SiO2, respectively. The refractive index (n) and extinction coefficient (k) are shown as black solid line and red dashed line, respectively.
In this work, a finite-difference-time-domain (FDTD) solutions commercial software is performed to analyze the reflectance spectra as well as the electric and magnetic field density profiles of the angle-insensitive polarization-tunable color filters. It is worth noting that the Bloch boundary condition is a suitable alternative to the periodic boundary condition because the angular tolerance of the proposed color filters with periodicity will be discussed in this paper. Hence, we choose the Bloch boundary condition to analyze and calculate the period section of the structure on the x-axis and y-axis and choose the perfectly matched layer on the z-axis.
3. Results and analysis
The polarization-tunable property of the proposed color filter can be observed from Figs. 1(b) and 1(c). Figure 1(b) depicts the reflectance spectra of the proposed color tunable filter for different polarizations at normal incidence. The polarization-tunable color filter has dimensions of a = 100 nm, p = 210 nm, and t = 70 nm. Since the electric field of an arbitrarily linearly polarized incident light can be represented as the sum of two linearly orthogonal polarizations, the reflectance for an arbitrarily linear polarization can be regarded as the linear superposition of the reflectance for two linearly orthogonal polarizations. Thereby the reflectance of the proposed filter can be characterized by the following formula [30]:
Meanwhile, to objectively recognize and classify the colors produced by the proposed color filters, the CIE standard illuminant D65 is chosen as the incoming source in this work because the spectral energy distribution of D65 is closest to that of the natural light. According to the illuminant D65 and the reflectance spectra shown in Fig. 1(b), the corresponding chromaticity coordinates are calculated and are plotted in the CIE 1931 chromaticity diagram illustrated in Fig. 1(c). As can be observed from Fig. 1(c), a continuous palette between blue and yellow colors can be produced by rotating the incident polarization angle.
The polarization-tunable color filters with different grating thicknesses are designed to show the ability of Si nanostructures to produce full and vivid structural colors. As the thickness of the single-layer grating increases from 60 nm to 200 nm with an interval of 20 nm, the reflectance spectra of eight nanostructures for TM and TE polarizations at normal incidence are shown in Figs. 2(a) and 2(b), respectively. Obviously, the reflectance peaks in Fig. 2(a) and the reflectance valleys in Fig. 2(b) shift from short wavelengths to long wavelengths, which is attributed to the increase in the thickness of the single-layer grating. The redshift phenomena will be more intuitive to show in Figs. 3(a) and 3(b).
Fig. 2. (a) and (b) Reflectance spectra of eight nanostructures with different grating thicknesses for TM and TE polarizations. The grating thickness increases from 60 nm to 200 nm in steps of 20 nm. Here, the grating width and period keep constant. (c) and (d) Reflectance spectra of (a) and (b) to black stars in the CIE 1931 chromaticity diagram. The reflectance spectra of the color filters with different grating thicknesses for TM and TE polarizations are further presented with white solid lines and the arrows indicate the direction. (e) Colors produced by eight nanostructures at different polarization angles. The polarization angle ranges from φ = 0° to φ = 90° in steps of 30°.
Fig. 3. (a) Reflectance spectra of the proposed color filters with different grating thicknesses for TM polarization at normal incidence. (b) Reflectance spectra of the proposed color filters with different grating thicknesses for TE polarization at normal incidence.
The reflectance spectra of the proposed color filters with different grating thicknesses are further presented with white solid lines in CIE 1931 chromaticity diagrams, as shown in Figs. 2(c) and 2(d). A smooth curve that represents the colors from blue to red is presented in Fig. 2(c) and (a) spiral shaped line is presented in Fig. 2(d). The difference between the two curves is caused by the different reflectance spectra of the proposed color filters at TM and TE polarization angles, which will be discussed in detail in a later section. As can be observed from Fig. 2(c), a large color gamut can be obtained for TM polarization, where the colors share a trait that is high pure. For TE polarization, the relatively small color gamut is presented in Fig. 2(d). The black stars plotted in Figs. 2(c) and 2(d) represent the calculated colors according to the illuminant D65 and the reflectance spectra shown in Figs. 2(a) and 2(b). Furthermore, the colors produced by eight nanostructures at different polarization angles have also been calculated and the results are recorded in Fig. 2(e). It can be observed from Fig. 2(e) that the continuous palettes between two different colors can be achieved by rotating the polarization angle. All the above results indicate that the color transition achieved by rotating the incident polarization angle is fairly robust.
The maps of the reflectance spectra of the proposed polarization-tunable color filters for TM and TE polarizations at normal incidence are shown in Figs. 3(a) and 3(b) as the grating thickness increases from 60 nm to 200 nm. The redshift phenomena of the reflectance peaks and the reflectance valleys are very intuitive. With the aid of the maps of the reflectance spectra, the reason for the difference between two white solid lines in Figs. 2(c) and 2(d) can be clearly analyzed. As can be observed from Fig. 3(a), the colors produced by the proposed color filters for TM polarization mainly depend on the highlighted areas and the highlighted areas have a red-shift tendency as the grating thickness increases from 60 nm to 200 nm. Hence, the color transition for TM polarization is presented with a smooth line in CIE 1931 chromaticity diagram. For TE polarization, the colors produced by the proposed color filters mainly depend on the highlighted areas I, II, and III depicted in Fig. 3(b). The width of highlighted area I gradually narrows with the grating thickness increasing from 60 nm to 80 nm and the width of red-shifted highlighted area II gradually widens with the grating thickness increasing from 80 nm to 130 nm, which makes the color produced transfer from yellow to green to red. As the grating thickness increases from 130 nm to 200 nm, the width of red-shift highlighted area II gradually narrows and the width of highlighted area III gradually widens. The phenomena make the color produced further transfer from red to magenta to green. It is noting that the proportion of highlighted areas gradually becomes higher and the purity and saturation of color become weaker as the grating thickness increases from 60 nm to 200 nm. Therefore, the color transition for TE polarization is presented with an inward spiral shaped line in CIE 1931 chromaticity diagram. To sum up, with the redshift of the reflectance peaks and valleys, full and vivid structural colors covering the whole visible range can be produced.
The angle-insensitive properties of the proposed polarization-tunable color filters for different polarizations have been punctiliously assessed with the aid of the relative resonance wavelength shift; here, the relative resonance wavelength shift is expressed by $\textrm{|}\Delta \lambda /{\lambda _0}\textrm{|}$, where ${\lambda _0}$ is the resonance wavelength at normal incidence and $\Delta \lambda$ is the deviation of resonance wavelength at an oblique incident from ${\lambda _0}$. As shown in Figs. 4(a) and 4(b), the reflectance spectra of the proposed color filters with the grating thicknesses of 60 nm, 130 nm, and 200 nm for TM and TE polarizations are studied with the incident angle increasing from 0° to 60°. The resonance wavelengths of the reflectance peaks and valleys for TM and TE polarizations keep almost constant with the incident angle increasing from 0° to 30°; moreover, the relative resonance wavelength shifts are no more than 2%. However, the resonance wavelengths of the reflectance peaks and valleys for TM and TE polarizations shift significantly when the incident angle is beyond of 30°; moreover, the relative resonance wavelength shifts are no more than 6%.
Fig. 4. (a) and (b) Reflectance spectra of the proposed color filters with different grating thicknesses for TM and TE polarizations with the incident angle increasing from 0° to 60°. The grating thickness increases from 60 nm to 200 nm in steps of 70 nm. (c) Chromaticity coordinates of the proposed color filters for TM polarization at the incident angles of 0°, 15°, 30°, 45°, and 60°. (d) Chromaticity coordinates of the proposed color filters for TE polarization at the incident angles of 0°, 15°, 30°, 45°, and 60°. (e) The left-wing colors produced by three nanostructures for TM polarizations at different incident angles. Likewise, the right-wing colors produced by three nanostructures for TE polarization at different incident angles.
According to the reflectance spectra for TM and TE polarizations shown in Figs. 4(a) and 4(b) and the illuminant D65, the corresponding chromaticity coordinates are marked in the CIE 1931 chromaticity diagrams in Figs. 4(c) and 4(d). Obviously, the colors produced by the proposed color filters with the grating thicknesses of 60 nm, 130 nm, and 200 nm do not change significantly with the incident angle increasing from 0° to 30°; however, the colors produced change significantly when the incident angle exceeds 30°. This can be corroborated by the results presented in Fig. 4(e), both the wing colors have slight changes with the incident angle increasing from 0° to 30° and both the wing colors have significant changes as the incident angle is beyond of 30°. Furthermore, whether for TM polarization or TE polarization, the color produced by the proposed color filter with the grating thickness of 130 nm has relatively biggest changes as the incident angle increases because the reflectance spectra of the proposed color filter with the grating thickness of 130 nm at different incident angles appear relatively most distinct variations than those of the proposed color filters with the grating thickness of 60 nm and 200 nm at different incident angles. To accurately evaluate the color difference between two chromaticity coordinates, the CIE DE2000 formula [31] is considered to calculate the color difference between two chromaticity coordinates. The calculated results are listed in Table 1. Apparently, the color difference gradually increases as the incident angle increases. The color difference of the proposed color filters remains at a relatively low level that cannot easily detected by human eyes when the incident angle is below 30°. Therefore, the proposed color filters with different grating thicknesses can provide a high angular tolerance of 30°.
Table 1. CIEDE2000 Color-Difference in Reflection at Different Incident Angles
As shown in Fig. 5, the reflectance spectra of the proposed color filters with the grating thicknesses of 60 nm, 130 nm, and 200 nm for TM and TE polarizations at different incident angles are plotted to further study the incident angle properties of the proposed color filters. Apparently, the resonance peaks and valleys of the proposed color filters for TM and TE polarizations appear slight shifts and their reflectance spectra keep almost coincident when the incident angle increases from 0° to 30°. Unfortunately, when the incident angle is beyond of 30°, the resonance peaks and valleys of the proposed color filters for TM and TE polarizations shift significantly and the new excited peaks and valleys under the oblique incidence become so obvious that the reflectance spectra appear distinct variations, which is the reason why the colors produced by the proposed color filters at between large incident angle and the normal incidence exist large color difference. Moreover, the shifts of the resonance peaks and valleys of the proposed color filters for TM polarization at different incident angles are more obvious than those of the proposed color filters for TE polarization at different incident angles. These further confirm that the proposed color filters have good incident angle-insensitive properties when the incident angle is below 30°.
Fig. 5. Reflectance spectra of the proposed color filters with different grating thicknesses for TM and TE polarizations at different incident angles. The white dotted lines represent the shifts of the resonance peaks and valleys with the incident angle increasing from 0° to 70°. The black dotted lines represent the shifts of the new excited peaks and valleys under the oblique incidence with the incident angle increasing from 0° to 70°. (a) The proposed color filter with the grating thickness of 60 nm, TM polarization. (b) The proposed color filter with the grating thickness of 60 nm, TE polarization. (c) The proposed color filter with the grating thickness of 130 nm, TM polarization. (d) The proposed color filter with the grating thickness of 130 nm, TE polarization. (e) The proposed color filter with the grating thickness of 200 nm, TM polarization. (f) The proposed color filter with the grating thickness of 200 nm, TE polarization.
To make clear the interplay between light and the grating structures, the electric and magnetic density profiles of the proposed color filter with the grating thickness of 130 nm for TM and TE polarizations are studied. For TM polarization, Fig. 6 shows the electric and magnetic field density profiles at the incident angles of 0°, 30°, and 60° with different wavelengths. For TE polarization, Fig. 7 shows the electric and magnetic field density profiles at the incident angles of 0°, 30°, and 60° with different wavelengths. Obviously, the electric and magnetic density profiles at the resonance wavelengths appear slight distortions with the incident angle increasing from 0° to 30°, which is the reason why the resonance peaks and valleys do not shift significantly when the incidence angle is within 30°; however, the electric and magnetic density profiles at the resonance wavelengths appear dramatic distortions when the incident angle is beyond of 30°, which is the reason why the resonance peaks and valleys appear relatively large shifts when the incidence angle is beyond of 30° [32,33].
Fig. 6. Power density profiles of the proposed color filter with the grating thickness of 130 nm at different wavelengths under TM polarization. (a) (e) (i) Electric field density profiles at the incident angles of 0°, 30°, and 60° with the resonance wavelength λ = 554 nm. (b) (f) (j) Magnetic field density profiles at the incident angles of 0°, 30°, and 60° with the resonance wavelength λ = 554 nm. (c) (g) (k) Electric field density profiles at the incident angles of 0°, 30°, and 60° with the non-resonance wavelength λ = 438 nm. (d) (h) (l) Magnetic field density profiles at the incident angles of 0°, 30°, and 60° with the non-resonance wavelength λ = 438 nm.
Fig. 7. Power density profiles of the proposed color filter with the grating thickness of 130 nm at different wavelengths under TE polarization. (a) (e) (i) Electric field density profiles at the incident angles of 0°, 30°, and 60° with the resonance wavelength λ = 498 nm. (b) (f) (j) Magnetic field density profiles at the incident angles of 0°, 30°, and 60° with the resonance wavelength λ = 498 nm. (c) (g) (k) Electric field density profiles at the incident angles of 0°, 30°, and 60° with the non-resonance wavelength λ = 559 nm. (d) (h) (l) Magnetic field density profiles at the incident angles of 0°, 30°, and 60° with the non-resonance wavelength λ = 559 nm.
Notably, compared with the differences between the power density profiles at the resonance wavelength λ = 498 nm for TE polarization at different incident angles, the electric and magnetic density profiles at the resonance wavelength λ = 554 nm for TM polarization appear more dramatic distortions at different incident angles. This confirms that the shifts of the resonance peaks and valleys of the proposed color filters for TM polarization at different incident angles are more obvious than those of the proposed color filters for TE polarization at different incident angles. Furthermore, the electric and magnetic density profiles at the non-resonance wavelengths appear dramatic distortions with the incident angle increasing from 0° to 60°, which is responsible for the emergence of new reflectance peaks and valleys.
To further explore the underlying mechanisms of angle-insensitive properties, different types of phase shift exist in the proposed structure are studied. The phase shift property and the reflection of this structure can be investigated with the aid of Smith method [34]. Moreover, the effective refractive indices of the proposed grating structure for TM and TE polarizations can be calculated by the effective media theory (EMT), as following [35,36]:
The total reflectance is as follow:
Fig. 8. (a) Simplified sandwich models with multiple reflections and transmissions for the analysis of the phase shift within the proposed structure. (b) and (c) Different types of the phase shift in the derivation of the magnitude of the reflectance for TM and TE polarizations.
For the proposed color filter with the thickness of 130 nm, Fig. 8(b) shows the above phase shift items at λ = 554 nm for TM polarization as a function of incidence angle; the function relationship between the incident angle and the above phase shift items at λ = 498 nm for TE polarization is shown in Fig. 8(c). Owing to the high refractive index contrast between the medium of air and the intermediate equivalent layer, the total reflection angle in the intermediate equivalent layer becomes very small, which implying a limited little propagation angle across the intermediate equivalent layer. Therefore, with such a little angle propagating in the intermediate equivalent layer, all the phase items will keep almost invariable up to 30° both for TM and TE polarizations, which is responsible for the principle of angle-insensitive properties. Different to the case where the incident angle is within 30°, the phase shift will change gradually with the incident angle over 30°, which will cause the resonance wavelength shift. It is worth noting that the total phase shift for TE-polarized light behaves better than that for TM-polarized light due to a higher refractive index contrast between the intermediate equivalent layer and air for TE polarization, which further confirms that the shifts of the resonance peaks and valleys of the proposed color filters for TM polarization at different incident angles are more obvious than those of the proposed color filters for TE polarization at different incident angles.
4. Conclusions
To conclude, the proposed angle robust polarization-tuned color filters adopt one-dimensional single-layer grating structure are constructed via a thorough investigation of their polarization tunable properties and incidence angle-insensitive properties. The proposed color filters not only readily achieve the continuous palettes of output colors by altering the polarization angles, but also produce full and vivid structural colors by adjusting the single-layer Si grating thickness. Simultaneously, the proposed color filters can maintain the almost invariable colors at an arbitrary polarization angle with the incidence angle increasing from 0° to 30°. The investigation of the different items of phase shift within the single-layer grating structure makes it clear that the reason for the angle-insensitive color filtering properties is mostly due to the small propagation angle across the intermediate layer with high refractive index. As a result, these angle robust polarization-tunable color filters could impact future applications such as anti-counterfeiting, display, imaging and filtering technologies, and so forth.
Funding
National Natural Science Foundation of China (61775140, 62005165); Shanghai Pujiang Program (21PJD048).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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