## Design of reflective color filters with high angular tolerance by particle swarm optimization method |

Optics Express, Vol. 21, Issue 8, pp. 9315-9323 (2013)

http://dx.doi.org/10.1364/OE.21.009315

Acrobat PDF (3318 KB)

### 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 OSA

## 1. Introduction

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]

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]

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]

## 2. Structure and design

*Λ*are grating periods along the

_{x,y}*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}= 1.4612. 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]

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]

*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}*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

_{00}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]

*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}## 3. Results and analysis

*Λ*, the length of grating ridge

_{x,y}*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.

*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.

*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.

*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.

*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.

*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

*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

## References and links

1. | M. Born and E. Wolf, |

2. | H. A. Macleod, |

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. |

5. | S. S. Wang and R. Magnusson, “Multilayer waveguide-grating filters,” Appl. Opt. |

6. | S. Tibuleac and R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A |

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 |

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

9. | Y. Kanamori, M. Shimono, and K. Hane, “Fabrication of transmission color filters using silicon subwavelength gratings on quartz substrates,” IEEE Photon. Technol. Lett. |

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. |

11. | E. D. Palik, |

12. | K. Yee, “Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media,” IEEE Trans. Antenn. Propag. |

13. | A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans. Electromagn. Compat. EMC |

14. | K. S. Kunz and R. J. Luebbers, |

15. | T. Allen and C. H. Susan, |

16. | Z. Luo, W. Shen, X. Liu, P. Gu, and C. Xia, “Design of dispersive multilayer with particle swarm optimization method,” Chin. Opt. Lett. |

17. | R. C. Eberhart, J. Kennedy, and Y. Shi, |

18. | CIE, |

19. | M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE2000,” Color Res. Appl. |

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. |

21. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

**OCIS Codes**

(050.6624) Diffraction and gratings : Subwavelength structures

(230.7408) Optical devices : Wavelength filtering devices

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: February 20, 2013

Revised Manuscript: March 25, 2013

Manuscript Accepted: April 1, 2013

Published: April 8, 2013

**Citation**

Chenying Yang, Liang Hong, Weidong Shen, Yueguang Zhang, Xu Liu, and Hongyu Zhen, "Design of reflective color filters with high angular tolerance by particle swarm optimization method," Opt. Express **21**, 9315-9323 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-9315

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### References

- M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th Ed. (Cambridge University, 1999).
- H. A. Macleod, Thin Film Optical Filters. (Institute of Physics Pub, 2001).
- 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.
- 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]
- S. S. Wang and R. Magnusson, “Multilayer waveguide-grating filters,” Appl. Opt.34(14), 2414–2420 (1995). [CrossRef] [PubMed]
- S. Tibuleac and R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A14(7), 1617–1626 (1997). [CrossRef]
- 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. Express18(13), 14056–14062 (2010). [CrossRef] [PubMed]
- G. Y. Si, E. S. P. Leong, A. J. Danner, and J. H. Teng, “Plasmonic coaxial fabry-pérot nanocavity color filter,” Proc. SPIE7757, 7757 (2010).
- 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]
- 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]
- E. D. Palik, Handbook of Optical Constants of Solids. (Academic, 1985).
- 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]
- A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans. Electromagn. Compat. EMC22(3), 191–202 (1980). [CrossRef]
- K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics. (CRC, 1993).
- T. Allen and C. H. Susan, Computational Electrodynamics: the Finite-Difference Time-Domain Method. (Artech House, 2005).
- 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).
- R. C. Eberhart, J. Kennedy, and Y. Shi, Swarm Intelligence. (Morgan Kaufmann, 2001).
- CIE, Improvement to Industrial Colour Difference Evaluation. (CIE, 2001).
- 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]
- 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]
- 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).

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