## Abstract

We propose three color filters (red, green, blue) based on a two-dimensional (2D) grating, which maintain the same perceived specular colors for a broad range of incident angles with the average polarization. Particle swarm optimization (PSO) method is employed to design these filters for the first time to our knowledge. Two merit functions involving the reflectance curves and color difference in CIEDE2000 formula are respectively constructed to adjust the structural parameters during the optimization procedure. Three primary color filters located at 637nm, 530nm and 446nm with high saturation are obtained with the peak reflectance of 89%, 83%, 66%. The reflectance curves at different incident angles are coincident and the color difference is less than 8 for the incident angle up to 45°. The electric field distribution of the structure is finally studied to analyze the optical property.

©2013 Optical Society of America

## 1. Introduction

Color filter, as a common optical element, selectively transmits or reflects a specific wavelength in the visible light region to display various colors. RGB (Red, Green and Blue) primary color filters which display red, green and blue colors respectively are extensively used in the fields of display, colorful decoration, anti-counterfeiting and so forth. The traditional chemical color filter comprising dye or pigment changes the color of the reflected light as the result of a wavelength-selective absorption of the particular functional groups which causes a material instability and a significant environmental burden. The optical thin film filters by multi-layer interference performs well with a high peak transmittance, a tailored bandwidth, and stable specifications [1,2]. However, its optical performance inevitably varies with angle of incidence, particularly a blue shift [3]. The guided mode resonance (GMR) filters have a sub nanometer bandwidth with very high efficiency and color filters in reflection with high purity can be obtained. But their optical properties are high angle sensitive and the angular tolerance is quite poor [4–6]. For recent years, color filters based on sub-wavelength structures have been widely studied owing to the development of numerical simulation algorithms and nano/micro fabrication techniques. Qin Chen proposed a color filter consisting of sub-wavelength triangular-lattice hole arrays in an aluminum film to excite surface plasmon resonance and found a few periods are sufficient to demonstrate filtering characteristic [7]. G. Y. Si fabricated a new color filter based on coaxial nanoring array structure which can support localized Fabry-Pérot plasmon modes so that extraordinary optical transmission can be observed from these sub-wavelength holes [8]. Yoshiaki Kanamori studied a transmission color filter based on silicon sub-wavelength gratings on quartz substrates [9]. However, all these color filters are sensitive to incident angles of light, which lead to a shift of spectrum with different incident angles. For many applications, e.g., special illumination, display and spectral analysis, the color filter is required to maintain the same perceived specular color for a broad range of incident angles. Therefore, it is of the essence to improve the incident angular tolerance of the color filters. Byoung-Ho Cheong etc. presented a reflective color filter with a relatively high angular tolerance [10]. Nonetheless, only a green filter was studied and the properties can be further improved.

In this paper, we propose three color filters (red, green, blue) based on a two-dimensional (2D) grating, which can maintain the same perceived specular colors for a broad range of incident angles with the average polarization. Particle swarm optimization (PSO) method is employed to design these filters for the first time to our knowledge. Two merit functions involving the reflectance curves and color difference respectively are adapted to adjust the structural parameters during the optimization procedure. Three primary color filters located at 637nm, 530nm and 446nm respectively are obtained with high reflectance. The reflectance curves at different incident angles are coincident and the color difference is less than 8 in CIEDE2000 formula for the incident angle up to 45°. The electric field distribution of the structure is finally studied to analyze the optical property.

## 2. Structure and design

2D sub-wavelength grating which is comprised by triangular-distributed cuboid units is adapted to construct the color filters. The schematic geometry is shown in Fig. 1 where *Λ _{x,y}* are grating periods along the

*x*and the

*y*axes, respectively;

*L*are lengths of grating ridge;

_{x,y}*d*are intervals of adjacent grating and

_{x,y}*t*is the thickness of grating layer. According to geometrical relationship, we could easily get

*Λ*,

_{x}=2L_{x}+2d_{x}*Λ*Considering the symmetry feature of the grating,

_{y}=2L_{y}+2d_{y}.*Λ*=

_{x}*Λ*and

_{y}*L*and

_{x}=L_{y}*d*are assumed. Thus, the duty ratio of the grating could be defined as

_{x}= d_{y}*f=2L*The unpolarized light is launched at an incident angle θ and an azimuth angle φ towards the color filter, shown in Fig. 1. θ is the angle from the incident light to the z axis and φ is the angle from the x axis to the orthogonal projection of the incident beam in the xy plane. The materials of incident medium and substrate are air and quartz, whose refractive indices are

_{x}L_{y}/Λ_{x}Λ_{y}.*n*and

_{0}= 1.0*n*respectively. Silicon is chosen as the material of the grating, whose refractive index and extinction coefficient comes from the data in the book [11] (the data between the nodes derived from linear interpolation).

_{s}= 1.46We calculated the optical properties of the structure including the reflectance at different incident angle and the electric field distribution for TE and TM polarized light with the wavelength ranged from 380nm to 780 nm by Finite-Difference Time-Domain (FDTD) method. FDTD method proposed by K. S. Yee in 1966 [12], is a powerful numerical analysis technique to compute the electromagnetic field and through it a wide frequency range can be covered with a single simulation run [12–15]. We study the effect of the azimuth angle on the optical property of the color filters. The reflectance of the optimized blue filter at the azimuth angle of 0°, 22.5° and 45° are calculated for different incident angles. A good incident angular tolerance is maintained as the azimuth angle varies. The azimuth angle has quite little effect on the optical property of the color filter. So, in this paper, the incident angle θ varies from 0° to 45° while the azimuth angle is fixed to 0° for all the calculations. Note that the reflectance in the paper is the total reflectance of all the possible diffraction order for the grating. Using the rigorous coupled wave analysis (RCWA), we find that the 0-th order diffraction intensity dominates in the reflection. Therefore, the reflective angle could be determined to equal to the incident angle, shown in Fig. 1.

To obtain high angular tolerance, PSO method is applied for the filter design owing to its advantage of fast convergence speed and less dependence on the initial parameters. It has been proved to be a very powerful and efficient method for designing a complicated thin film optical filter with layer number larger than 100 [16]. PSO which is first developed by Eberhart and Kennedy [17] roots in the social behavior of large number of birds or fish, with a simple but effective working schedule. For every iteration, each particle which represents a potential solution would adjust its movement according to its own experience as well as the experience of other particles. The underlying rules of cooperation and competition within social swarms give it good capability for global optimization with the help of memory rather than a simple random search. Therefore, it has a large possibility to fly into a better solution with a faster speed and finally generate better results [16]. In our case, flying particles search in a three-dimensional space built of three structural parameters mentioned above. The optimized variables are the length of grating ridge *L _{x,y}*, the unit interval

*d*and the grating thickness

_{x,y}*t*other than the grating period

*Λ*which can be received from

_{x,y}*L*and

_{x,y}*d*.

_{x,y}Merit function during the optimization procedure is the most important for the convergence and the final results. To design a color filter with a high angle tolerance, two merit functions are adopted for different steps. The first merit function is used to find out rough parameters of the structure with the most consistent reflectance spectrums at different incident angles. Two parts are included: one focuses on the reflectance curve at normal incidence to assure high intensity efficiency and a specified reflected color with good saturation; the other is concentrated on the spectral characteristic deviation at different incident angles. Upon the consideration above, the first merit function in the optimization is defined as:

*R*is the reflectance at the normal incidence, when

_{0}*R*and

_{θ,TM}*R*are the reflectance at TM and TE polarized oblique incidence respectively, and

_{θ,TE}*θ*represents the incident angle. Empirically, the spectral characteristic changes gradually as the incident angle increases, i.e., an approving spectral characteristic at a big incident angle indicates a good spectral characteristic at a small incident angle either. Hence, the largest incidence angle

*θ*= 45° is just used in our design to improve the efficiency because FDTD simulation is quite time consuming.

*R*is the target reflectance of the color filter specifying an ideal spectral response without any ripples. Taking an example of the blue filter,

_{target}*R*could be:

_{target}*R*=0@380nm-410nm,

_{target}*R*=100%@435nm-455nm and

_{target}*R*=0@480nm-780nm.

_{target}*W*and

_{1}*W*are the weighting functions. Generally, the properties in high reflection region are more important. So the weighting factors in this band are larger than those of cut-off region. As an example of blue filter,

_{2}*W*and

_{1}*W*could be set for the optimization:

_{2}*W*=1@380nm-410nm&480nm-780nm,

_{1}*W*=4@435nm-455nm and

_{1}*W*=1@380nm-480nm&480nm-780nm and

_{2}*W*=12@435nm-455nm.

_{2}Although the optimal spectral characteristic could be obtained through the first merit, the color property is perhaps not the best one because of the complicated relation between the reflectance and the perceived color. A small spectral change may cause a large color variation. Therefore, color difference characterized by CIEDE2000 is under our consideration. As well, the second merit function emphasizes two aspects: one focuses on the specified reflectance curve at normal incidence while the other concentrates on color difference at different incident angles. Thus, the second merit function in the optimization is defined as:

*R*,

_{0}*W*and

_{1}*R*are the same as those in merit1.

_{target}*∆E*is the calculated color difference value in CIEDE2000 formula between the normal incident case and the oblique incident case. Similarly, it is optimized at the incident angle of 45°. CIEDE2000 formula, published by CIE in 2001, is the latest and the most sophisticated formula to characterize color difference [18]. The formula provides an improved vision uniformity for industrial color difference by introducing various factors to change weight of luminance difference, Chroma difference and hue difference [19,20].

_{00}*W*here is the same as that of Merit1, while

_{1}*W*is the weighting functions to adjust the effect of the color difference. To balance the two parts of merit2,

_{3}*W*=1 is set.

_{3}With the PSO method and two merit functions given, the optimal blue filter could be obtained by adjusting the weighting functions in merit1 and merit2. Then, according to the scaling properties of the Maxwell’s equations [21], we get the initial structure size of green filter and red filter which is derived from the blue one. On the basis of these initial structure sizes, green and red color filters are optimized in the same way. Figure 2 depicts the variation of the merit function Merit2 during the PSO optimization. The decrease of merit function is fast and monotonic and an optimal solution could be found in less than 30 iterations. It proves PSO has a good search ability and fast convergence. Consequently, all the three primary color filters with high angular tolerance are accomplished.

## 3. Results and analysis

Based on an initial set of the structural parameters, the optimization of the blue filter is undertaken using PSO method. By varying the structure parameter combinations of the grating period *Λ _{x,y}*, the length of grating ridge

*L*the unit interval

_{x,y}*d*and the grating thickness

_{x,y}*t*, the optimized design of blue color filter is obtained with the parameters

*Λ*,

_{x}=Λ_{y}=220nm*L*,

_{x}=L_{y}=90nm*d*, and

_{x}=d_{y}=20nm*t=80nm*. The reflectance curves of the unoptimized and optimized filters at unpolarized incidence are shown in Fig. 3. The coincidence of the reflectance spectrum at various incident angles with bandwidth

*Δλ= 55nm*can be observed. At the incident angle of

*0°, 15°, 30°, 45°*, the maximum reflectance reaches

*66%, 64%, 61%, 55%*, respectively. Gratifyingly, as the incident angle increases, the reflectance peak position does not change, i.e., the maximum reflectance is located at

*λ= 446 nm*for all the incident angles.

According to these reflectance curves, the chromaticity coordinates at various incident angles are calculated and marked in Fig. 4. The chromaticity coordinate shifts obviously to the white point when the incident light increases to *45°*, owing to the 12% reflectance decrease in the high reflection region as well as the slight reflectance increase in the cut-off region compared with those at normal incidence. However, as the incident angle increases gradually, the reflected specular color changes much less than the multilayer filter [3] in CIE1931 chromaticity diagram.

We also find that the spectral characteristic at TE polarized incidence is much better than that at TM polarized incidence. Figure 5 shows the reflectance of the optimized blue filter for TE and TM polarization at different incident angles. At TE polarized incidence, as the incident angles increasing gradually, the reflectance peak position does not change and the maximum reflectance has a little decrease. Yet, for TM polarization, at the incident angle of *0°* and *15°*, the reflectance spectrum remains consistent well; at the incident angle of *30°* and *45°*, the peak reflectance position has an obvious shift towards long wavelength and the maximum reflectance has an obvious decrease, either.

On the basis of the blue filter, according to the scaling properties of the Maxwell’s equations, the initial parameters for green and red filters could be obtained by tuning the scaling factor *w*. We find that when the scaling factor are set *w = 1.4* and *w = 1.8*, the initial green filter and red filter come out respectively. Compared with the blue filter, the reflectance curves of the green and red filters with the initial parameters deteriorate partly, shown in Fig. 6. For green and red filters, the central wavelength are *λ = 544nm* and *λ = 654nm* when the bandwidth are *Δλ = 94nm* and *Δλ = 130nm*, respectively. Apparently, as the scaling factor increases, the bandwidth of the reflectance curves increases accordingly and the ripples in the short wavelength region grow. As a result, the incident angular property of those two filters is not as good as that of the blue one. The further improvements can be obtained with similar design procedures by PSO method as described above.

Figure 7 shows the reflectance curves of green and red filters after optimization. For optimized green and red filters, the central wavelength are *λ = 530 nm* and *λ = 637nm* when the bandwidth are *Δλ = 60nm* and *Δλ = 92nm*, respectively. We could find the bandwidth of those two optimized filters are much narrower than initial ones with a decrease over *30nm*, and small ripples at short wavelength region are suppressed. Besides, the reflectance curves at large incident angles agree well with those at the normal incidence. Note that the reflectance maximum raises *17%* and *23%* for the optimized green and red filters respectively in comparison with that of the blue filter, which can be attributed to the sharp decline of the extinction coefficient of the grating material—silicon. The CIE 1931 chromaticity coordinates of the optimized green and red color filters are calculated and marked in Fig. 4. Intuitively, the chromaticity coordinates of each color for different angles are all located in a small region and the color difference is hardly observed for human eyes. It is undoubted that the chromaticity coordinates of two optimized filters are much farther from the central white point than those unoptimized ones, which means a better saturation and purity of the color they show. The fact is attributed to the less ripples at short wavelength and the narrower reflectance bandwidths.

To get the actual human eyes sense of the color difference emerged at different incident angles, we calculated CIEDE2000 color difference values to quantify the color difference seen from the human eyes. The results calculated are shown in Fig. 8. A remarkable improvement of the color difference can be seen from Fig. 8 after the optimization of the structural parameters, which implies a much slighter color variation for human visual sense.

To make clear the interaction between the incident light and those sub-wavelength structures, the electric field distribution is studied. Figure 9 and 10 illustrate the electric field profile of the optimized green filter at different incident angles for two polarizations. From the reflectance in Fig. 7(a), we could regard *λ=530 nm* as the peak reflectance wavelength, and *λ=610 nm* as the minimum reflectance wavelength in the same reflectance spectrum envelope. From Fig. 9(a) and Fig. 10(a), it can be seen that at the normal incidence for the peak reflectance wavelength (*λ=530nm*), there is an intense coupling from the low refractive index (n=n_{air}) region to the high refractive index (n=n_{Si}) region, exciting a drastic resonance in the silicon region, which results in most of the light reflecting and merely a little transmitting. Different from the behavior of the peak reflectance wavelength, nearly all the incident light for minimum reflectance wavelength (*λ=610nm*) transmitted through the grating layer without the coupling between the two different refractive index regions, shown in Fig. 9(b) and Fig. 10(b). Figures 9(c) and 9(d) present the electric field of the optimized green filter at the oblique incidence of *45°* for TM polarization. In contrast with the normal incidence in Figs. 9(a) and 9(b), we could find that the phenomena are identical for the corresponding incident wavelength, which is responsible for the high angular tolerance of the optimized color filter. It is clear that the explanation for the high angular tolerance for TM polarization reveals the same with that for TE polarization from Fig. 10.

## 4. Conclusion

To conclude, new color filters of 2D sub-wavelength grating structure which is comprised by triangular-distributed cuboid units are proposed. By applying PSO method to optimize the structure dimensions of the filters, three primary color filters with good incident angular property are obtained, whose central wavelength are *λ= 446 nm*, *λ = 530 nm* and *λ = 637 nm* respectively. The simulation reveals that the obtained color filters offer a high incident angular tolerance, keeping the same color at the increasing incident angles, up to *45°*. In addition, the reflective intensity, the reflective purity of the color filters and color difference at different incident angles has approving performance. It is the coupling from the low refractive index to the high refractive index and the resonance excited in the high refractive index cell that bringing out the good incident angular property. Consequently, the three primary color filters of good incident angular property have potential applications in display, colorful decoration, anti-counterfeiting and so forth.

## Acknowledgments

It is a pleasure for authors to acknowledge the funding support from National High Technology Research and Development Program 863 (2012AA040401) and National Natural Science Foundation of China (No. 61275161, 61007056).

## References and links

**1. **M. Born and E. Wolf, *Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light,* 7th Ed. (Cambridge University, 1999).

**2. **H. A. Macleod, *Thin Film Optical Filters.* (Institute of Physics Pub, 2001).

**3. **A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and S. A. Yanshin, “Design of multilayer coatings with specific angular dependencies of color properties,” in Conference on Optical Interference Coatings (Optical Society of America, 2007), paperWB2.

**4. **S. S. Wang and R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. **19**(12), 919–921 (1994). [CrossRef] [PubMed]

**5. **S. S. Wang and R. Magnusson, “Multilayer waveguide-grating filters,” Appl. Opt. **34**(14), 2414–2420 (1995). [CrossRef] [PubMed]

**6. **S. Tibuleac and R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A **14**(7), 1617–1626 (1997). [CrossRef]

**7. **Q. Chen and D. R. S. Cumming, “High transmission and low color cross-talk plasmonic color filters using triangular-lattice hole arrays in aluminum films,” Opt. Express **18**(13), 14056–14062 (2010). [CrossRef] [PubMed]

**8. **G. Y. Si, E. S. P. Leong, A. J. Danner, and J. H. Teng, “Plasmonic coaxial fabry-pérot nanocavity color filter,” Proc. SPIE **7757**, 7757 (2010).

**9. **Y. Kanamori, M. Shimono, and K. Hane, “Fabrication of transmission color filters using silicon subwavelength gratings on quartz substrates,” IEEE Photon. Technol. Lett. **18**(20), 2126–2128 (2006). [CrossRef]

**10. **B. H. Cheong, O. N. Prudnikov, E. Cho, H. S. Kim, J. Yu, Y. S. Cho, H. Y. Choi, and S. T. Shin, “High angular tolerant color filter using subwavelength grating,” Appl. Phys. Lett. **94**(21), 213104 (2009). [CrossRef]

**11. **E. D. Palik, *Handbook of Optical Constants of Solids.* (Academic, 1985).

**12. **K. Yee, “Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media,” IEEE Trans. Antenn. Propag. **14**(3), 302–307 (1966). [CrossRef]

**13. **A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans. Electromagn. Compat. EMC **22**(3), 191–202 (1980). [CrossRef]

**14. **K. S. Kunz and R. J. Luebbers, *The Finite Difference Time Domain Method for Electromagnetics.* (CRC, 1993).

**15. **T. Allen and C. H. Susan, *Computational Electrodynamics: the Finite-Difference Time-Domain Method.* (Artech House, 2005).

**16. **Z. Luo, W. Shen, X. Liu, P. Gu, and C. Xia, “Design of dispersive multilayer with particle swarm optimization method,” Chin. Opt. Lett. **8**, 342–344 (2010).

**17. **R. C. Eberhart, J. Kennedy, and Y. Shi, *Swarm Intelligence.* (Morgan Kaufmann, 2001).

**18. **CIE, *Improvement to Industrial Colour Difference Evaluation.* (CIE, 2001).

**19. **M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl. **26**(5), 340–350 (2001). [CrossRef]

**20. **G. Sharma, W. Wu, and E. N. Dalal, “The CIEDE2000 color-difference formula: implementation notes, supplementary test data, and mathematical observations,” Color Res. Appl. **30**(1), 21–30 (2005). [CrossRef]

**21. **J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, *Photonic Crystals: Molding the Flow of Light,* 2nd Ed. (Princeton University, 2008).