Design of structurally colored surfaces based on scalar diffraction theory

Villads Egede Johansen; Jacob Andkjær; Ole Sigmund

doi:10.1364/JOSAB.31.000207

Design of structurally colored surfaces based on scalar diffraction theory

Villads Egede Johansen, Jacob Andkjær, and Ole Sigmund

Journal of the Optical Society of America B
Vol. 31,
Issue 2,
pp. 207-217
(2014)
•https://doi.org/10.1364/JOSAB.31.000207

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Villads Egede Johansen, Jacob Andkjær, and Ole Sigmund, "Design of structurally colored surfaces based on scalar diffraction theory," J. Opt. Soc. Am. B 31, 207-217 (2014)

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Abstract

In this paper we investigate the possibility of controlling the color and appearance of surfaces simply by modifying the height profile of the surface on a nanoscale level. The applications for such methods are numerous: new design possibilities for high-end products, color engraving on any highly reflective surface, paint-free text and coloration, UV-resistant coloring, etc. In this initial study, the main focus is on finding a systematic way to obtain these results. For now the simulation and optimization is based on a simple scalar diffraction theory model. From the results, several design issues are identified: some colors are harder to optimize for than others, and some can be produced by only a few height levels, whereas others require more complex structures. It is shown that a wide range of results can be obtained.

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[Crossref]
S. Kinoshita and S. Yoshioka, “Structural colors in nature: the role of regularity and irregularity in the structure,” ChemPhysChem 6, 1442–1459 (2005).
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[Crossref]
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[Crossref]
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[Crossref]
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O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).
[Crossref]
T. Borrvall, “Topology optimization of elastic continua using restriction,” Arch. Comput. Methods Eng. 8, 351–385 (2001).
[Crossref]

2014 (1)

J. Andkjær, V. E. Johansen, and O. Sigmund, “Inverse design of nano-structured surfaces for color effects,” J. Opt. Soc. Am. B 31, 164–174 (2014).

2013 (3)

Y. R. Wu, A. E. Hollowell, C. Zhang, and L. J. Guo, “Angle-insensitive structural colours based on metallic nanocavities and coloured pixels beyond the diffraction limit,” Sci. Rep. 3, 1194 (2013).

K. Yu, T. Fan, S. Lou, and D. Zhang, “Biomimetic optical materials: integration of natures design for manipulation of light,” Prog. Mater. Sci. 58, 825–873 (2013).
[Crossref]

M. J. Uddin and R. Magnusson, “Highly efficient color filter array using resonant Si3N4 gratings,” Opt. Express 21, 12495–12506 (2013).
[Crossref]

2012 (4)

G. Park, C. Lee, D. Seo, and H. Song, “Full-color tuning of surface plasmon resonance by compositional variation of Au@Ag core-shell nanocubes with sulfides,” Langmuir 28, 9003–9009 (2012).
[Crossref]

K. Kumar, H. Duan, R. S. Hegde, S. C. W. Koh, J. N. Wei, and J. K. W. Yang, “Printing colour at the optical diffraction limit,” Nat. Nanotechnol. 7, 557–561 (2012).
[Crossref]

J. E. Harvey, “Radiometry rocks,” Proc. SPIE 8483, 848304 (2012).
[Crossref]

K. S. Friis and O. Sigmund, “Robust topology design of periodic grating surfaces,” J. Opt. Soc. Am. B 29, 2935–2943 (2012).
[Crossref]

2011 (6)

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
[Crossref]

T. Xu, H. Shi, Y. Wu, A. F. Kaplan, J. G. Ok, and L. J. Guo, “Structural colors: from plasmonic to carbon nanostructures,” Small 7, 3128–3136 (2011).
[Crossref]

X. Li, Q. Tan, and G. Jin, “Surface profile optimization of antireflection gratings for solar cells,” Optik 122, 2078–2082 (2011).
[Crossref]

X. Sheng, S. G. Johnson, J. Michel, and L. C. Kimerling, “Optimization-based design of surface textures for thin-film Si solar cells,” Opt. Express 19, A841–A850 (2011).
[Crossref]

A. Saito, M. Yonezawa, J. Murase, S. Juodkazis, V. Mizeikis, M. Akai-Kasaya, and Y. Kuwahara, “Numerical analysis on the optical role of nano-randomness on the Morpho butterfly’s scale,” J. Nanosci. Nanotechnol. 11, 2785–2792 (2011).
[Crossref]

A. Saito, “Material design and structural color inspired by biomimetic approach,” Sci. Tech. Adv. Mater. 12, 064709 (2011).
[Crossref]

2009 (2)

A. Saito, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass-production and control of the Morpho-butterfly’s blue,” Proc. SPIE 7205, 720506 (2009).
[Crossref]

R. T. Lee and G. S. Smith, “Detailed electromagnetic simulation for the structural color of butterfly wings,” Appl. Opt. 48, 4177–4190 (2009).
[Crossref]

2007 (1)

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).
[Crossref]

2005 (1)

S. Kinoshita and S. Yoshioka, “Structural colors in nature: the role of regularity and irregularity in the structure,” ChemPhysChem 6, 1442–1459 (2005).
[Crossref]

2003 (1)

P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature 424, 852–855 (2003).
[Crossref]

2002 (1)

S. Kinoshita, S. Yoshioka, Y. Fujii, and N. Okamoto, “Photophysics of structural color in the Morpho butterflies,” Forma 17, 103–121 (2002).

2001 (1)

T. Borrvall, “Topology optimization of elastic continua using restriction,” Arch. Comput. Methods Eng. 8, 351–385 (2001).
[Crossref]

2000 (1)

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory: errata,” Appl. Opt. 39, 6374–6375 (2000).
[Crossref]

1999 (1)

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
[Crossref]

1987 (1)

K. Svanberg, “The method of moving asymptotes—a new method for structural optimization,” Internat. J. Numer. Methods Engrg. 24, 359–373 (1987).
[Crossref]

Akai-Kasaya, M.

Andkjær, J.

J. Andkjær, V. E. Johansen, and O. Sigmund, “Inverse design of nano-structured surfaces for color effects,” J. Opt. Soc. Am. B 31, 164–174 (2014).

Bala, K.

P. Dutré, K. Bala, and P. Bekaert, Advanced Global Illumination (A K Peters/CRC Press, 2006).

Bekaert, P.

P. Dutré, K. Bala, and P. Bekaert, Advanced Global Illumination (A K Peters/CRC Press, 2006).

Berns, R. S.

R. S. Berns, F. W. Billmeyer, and M. Saltzman, Billmeyer and Saltzman’s Principles of Color Technology (Wiley-Interscience, 2000).

Billmeyer, F. W.

R. S. Berns, F. W. Billmeyer, and M. Saltzman, Billmeyer and Saltzman’s Principles of Color Technology (Wiley-Interscience, 2000).

Borrvall, T.

T. Borrvall, “Topology optimization of elastic continua using restriction,” Arch. Comput. Methods Eng. 8, 351–385 (2001).
[Crossref]

Duan, H.

K. Kumar, H. Duan, R. S. Hegde, S. C. W. Koh, J. N. Wei, and J. K. W. Yang, “Printing colour at the optical diffraction limit,” Nat. Nanotechnol. 7, 557–561 (2012).
[Crossref]

Dutré, P.

P. Dutré, K. Bala, and P. Bekaert, Advanced Global Illumination (A K Peters/CRC Press, 2006).

Fan, T.

K. Yu, T. Fan, S. Lou, and D. Zhang, “Biomimetic optical materials: integration of natures design for manipulation of light,” Prog. Mater. Sci. 58, 825–873 (2013).
[Crossref]

Friis, K. S.

K. S. Friis and O. Sigmund, “Robust topology design of periodic grating surfaces,” J. Opt. Soc. Am. B 29, 2935–2943 (2012).
[Crossref]

Fujii, Y.

S. Kinoshita, S. Yoshioka, Y. Fujii, and N. Okamoto, “Photophysics of structural color in the Morpho butterflies,” Forma 17, 103–121 (2002).

Gao, Y.

Z. Shi and Y. Gao, “Design of Dammann gratings by particle swarm optimization,” in Symposium on Photonics and Optoelectronics, June 2010.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed., McGraw-Hill Physical and Quantum Electronics Series (McGraw-Hill, 1996).

Guo, L. J.

T. Xu, H. Shi, Y. Wu, A. F. Kaplan, J. G. Ok, and L. J. Guo, “Structural colors: from plasmonic to carbon nanostructures,” Small 7, 3128–3136 (2011).
[Crossref]

Harvey, J. E.

J. E. Harvey, “Radiometry rocks,” Proc. SPIE 8483, 848304 (2012).
[Crossref]

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
[Crossref]

Hegde, R. S.

K. Kumar, H. Duan, R. S. Hegde, S. C. W. Koh, J. N. Wei, and J. K. W. Yang, “Printing colour at the optical diffraction limit,” Nat. Nanotechnol. 7, 557–561 (2012).
[Crossref]

Hollowell, A. E.

Ishikawa, Y.

Jensen, J. S.

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
[Crossref]

Jin, G.

X. Li, Q. Tan, and G. Jin, “Surface profile optimization of antireflection gratings for solar cells,” Optik 122, 2078–2082 (2011).
[Crossref]

Johansen, V. E.

J. Andkjær, V. E. Johansen, and O. Sigmund, “Inverse design of nano-structured surfaces for color effects,” J. Opt. Soc. Am. B 31, 164–174 (2014).

Johnson, S. G.

X. Sheng, S. G. Johnson, J. Michel, and L. C. Kimerling, “Optimization-based design of surface textures for thin-film Si solar cells,” Opt. Express 19, A841–A850 (2011).
[Crossref]

Juodkazis, S.

Kaplan, A. F.

T. Xu, H. Shi, Y. Wu, A. F. Kaplan, J. G. Ok, and L. J. Guo, “Structural colors: from plasmonic to carbon nanostructures,” Small 7, 3128–3136 (2011).
[Crossref]

Karthaus, O.

O. Karthaus, Biomimetics in Photonics (CRC Press, 2012).

Kimerling, L. C.

X. Sheng, S. G. Johnson, J. Michel, and L. C. Kimerling, “Optimization-based design of surface textures for thin-film Si solar cells,” Opt. Express 19, A841–A850 (2011).
[Crossref]

Kinoshita, S.

S. Kinoshita and S. Yoshioka, “Structural colors in nature: the role of regularity and irregularity in the structure,” ChemPhysChem 6, 1442–1459 (2005).
[Crossref]

S. Kinoshita, S. Yoshioka, Y. Fujii, and N. Okamoto, “Photophysics of structural color in the Morpho butterflies,” Forma 17, 103–121 (2002).

S. Kinoshita, Structural Colors in the Realm of Nature (World Scientific, 2008).

Koh, S. C. W.

K. Kumar, H. Duan, R. S. Hegde, S. C. W. Koh, J. N. Wei, and J. K. W. Yang, “Printing colour at the optical diffraction limit,” Nat. Nanotechnol. 7, 557–561 (2012).
[Crossref]

Krywonos, A.

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
[Crossref]

Kumar, K.

K. Kumar, H. Duan, R. S. Hegde, S. C. W. Koh, J. N. Wei, and J. K. W. Yang, “Printing colour at the optical diffraction limit,” Nat. Nanotechnol. 7, 557–561 (2012).
[Crossref]

Kuwahara, Y.

Lee, C.

Lee, R. T.

R. T. Lee and G. S. Smith, “Detailed electromagnetic simulation for the structural color of butterfly wings,” Appl. Opt. 48, 4177–4190 (2009).
[Crossref]

Li, X.

X. Li, Q. Tan, and G. Jin, “Surface profile optimization of antireflection gratings for solar cells,” Optik 122, 2078–2082 (2011).
[Crossref]

Lohmann, A. W.

A. W. Lohmann, “About the philosophies of diffraction,” in International Trends in Optics, J. W. Goodman, ed. (Academic, 1991), Chap. 11, pp. 155–164.

Lou, S.

K. Yu, T. Fan, S. Lou, and D. Zhang, “Biomimetic optical materials: integration of natures design for manipulation of light,” Prog. Mater. Sci. 58, 825–873 (2013).
[Crossref]

Magnusson, R.

M. J. Uddin and R. Magnusson, “Highly efficient color filter array using resonant Si3N4 gratings,” Opt. Express 21, 12495–12506 (2013).
[Crossref]

Michel, J.

X. Sheng, S. G. Johnson, J. Michel, and L. C. Kimerling, “Optimization-based design of surface textures for thin-film Si solar cells,” Opt. Express 19, A841–A850 (2011).
[Crossref]

Miyamura, Y.

Mizeikis, V.

Murase, J.

O’Shea, D. C.

D. C. O’Shea, “Scalar diffraction theory,” in Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003), pp. 17–35.

Ok, J. G.

T. Xu, H. Shi, Y. Wu, A. F. Kaplan, J. G. Ok, and L. J. Guo, “Structural colors: from plasmonic to carbon nanostructures,” Small 7, 3128–3136 (2011).
[Crossref]

Okamoto, N.

S. Kinoshita, S. Yoshioka, Y. Fujii, and N. Okamoto, “Photophysics of structural color in the Morpho butterflies,” Forma 17, 103–121 (2002).

Park, G.

Saito, A.

A. Saito, “Material design and structural color inspired by biomimetic approach,” Sci. Tech. Adv. Mater. 12, 064709 (2011).
[Crossref]

Saltzman, M.

R. S. Berns, F. W. Billmeyer, and M. Saltzman, Billmeyer and Saltzman’s Principles of Color Technology (Wiley-Interscience, 2000).

Sambles, J. R.

P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature 424, 852–855 (2003).
[Crossref]

Seo, D.

Sheng, X.

X. Sheng, S. G. Johnson, J. Michel, and L. C. Kimerling, “Optimization-based design of surface textures for thin-film Si solar cells,” Opt. Express 19, A841–A850 (2011).
[Crossref]

Shi, H.

T. Xu, H. Shi, Y. Wu, A. F. Kaplan, J. G. Ok, and L. J. Guo, “Structural colors: from plasmonic to carbon nanostructures,” Small 7, 3128–3136 (2011).
[Crossref]

Shi, Z.

Z. Shi and Y. Gao, “Design of Dammann gratings by particle swarm optimization,” in Symposium on Photonics and Optoelectronics, June 2010.

Sigmund, O.

J. Andkjær, V. E. Johansen, and O. Sigmund, “Inverse design of nano-structured surfaces for color effects,” J. Opt. Soc. Am. B 31, 164–174 (2014).

K. S. Friis and O. Sigmund, “Robust topology design of periodic grating surfaces,” J. Opt. Soc. Am. B 29, 2935–2943 (2012).
[Crossref]

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
[Crossref]

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).
[Crossref]

Smith, G. S.

R. T. Lee and G. S. Smith, “Detailed electromagnetic simulation for the structural color of butterfly wings,” Appl. Opt. 48, 4177–4190 (2009).
[Crossref]

Song, H.

Svanberg, K.

K. Svanberg, “The method of moving asymptotes—a new method for structural optimization,” Internat. J. Numer. Methods Engrg. 24, 359–373 (1987).
[Crossref]

Tan, Q.

X. Li, Q. Tan, and G. Jin, “Surface profile optimization of antireflection gratings for solar cells,” Optik 122, 2078–2082 (2011).
[Crossref]

Vukusic, P.

P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature 424, 852–855 (2003).
[Crossref]

Wei, J. N.

K. Kumar, H. Duan, R. S. Hegde, S. C. W. Koh, J. N. Wei, and J. K. W. Yang, “Printing colour at the optical diffraction limit,” Nat. Nanotechnol. 7, 557–561 (2012).
[Crossref]

Wu, Y.

T. Xu, H. Shi, Y. Wu, A. F. Kaplan, J. G. Ok, and L. J. Guo, “Structural colors: from plasmonic to carbon nanostructures,” Small 7, 3128–3136 (2011).
[Crossref]

Wu, Y. R.

Xu, T.

T. Xu, H. Shi, Y. Wu, A. F. Kaplan, J. G. Ok, and L. J. Guo, “Structural colors: from plasmonic to carbon nanostructures,” Small 7, 3128–3136 (2011).
[Crossref]

Yang, J. K. W.

K. Kumar, H. Duan, R. S. Hegde, S. C. W. Koh, J. N. Wei, and J. K. W. Yang, “Printing colour at the optical diffraction limit,” Nat. Nanotechnol. 7, 557–561 (2012).
[Crossref]

Yonezawa, M.

Yoshioka, S.

S. Kinoshita and S. Yoshioka, “Structural colors in nature: the role of regularity and irregularity in the structure,” ChemPhysChem 6, 1442–1459 (2005).
[Crossref]

S. Kinoshita, S. Yoshioka, Y. Fujii, and N. Okamoto, “Photophysics of structural color in the Morpho butterflies,” Forma 17, 103–121 (2002).

Yu, K.

K. Yu, T. Fan, S. Lou, and D. Zhang, “Biomimetic optical materials: integration of natures design for manipulation of light,” Prog. Mater. Sci. 58, 825–873 (2013).
[Crossref]

Zhang, C.

Zhang, D.

K. Yu, T. Fan, S. Lou, and D. Zhang, “Biomimetic optical materials: integration of natures design for manipulation of light,” Prog. Mater. Sci. 58, 825–873 (2013).
[Crossref]

Appl. Opt. (3)

R. T. Lee and G. S. Smith, “Detailed electromagnetic simulation for the structural color of butterfly wings,” Appl. Opt. 48, 4177–4190 (2009).
[Crossref]

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
[Crossref]

Arch. Comput. Methods Eng. (1)

T. Borrvall, “Topology optimization of elastic continua using restriction,” Arch. Comput. Methods Eng. 8, 351–385 (2001).
[Crossref]

ChemPhysChem (1)

S. Kinoshita and S. Yoshioka, “Structural colors in nature: the role of regularity and irregularity in the structure,” ChemPhysChem 6, 1442–1459 (2005).
[Crossref]

Forma (1)

S. Kinoshita, S. Yoshioka, Y. Fujii, and N. Okamoto, “Photophysics of structural color in the Morpho butterflies,” Forma 17, 103–121 (2002).

Internat. J. Numer. Methods Engrg. (1)

K. Svanberg, “The method of moving asymptotes—a new method for structural optimization,” Internat. J. Numer. Methods Engrg. 24, 359–373 (1987).
[Crossref]

J. Nanosci. Nanotechnol. (1)

J. Opt. Soc. Am. B (2)

K. S. Friis and O. Sigmund, “Robust topology design of periodic grating surfaces,” J. Opt. Soc. Am. B 29, 2935–2943 (2012).
[Crossref]

J. Andkjær, V. E. Johansen, and O. Sigmund, “Inverse design of nano-structured surfaces for color effects,” J. Opt. Soc. Am. B 31, 164–174 (2014).

Langmuir (1)

Laser Photon. Rev. (1)

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
[Crossref]

Nat. Nanotechnol. (1)

K. Kumar, H. Duan, R. S. Hegde, S. C. W. Koh, J. N. Wei, and J. K. W. Yang, “Printing colour at the optical diffraction limit,” Nat. Nanotechnol. 7, 557–561 (2012).
[Crossref]

Nature (1)

P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature 424, 852–855 (2003).
[Crossref]

Opt. Express (2)

M. J. Uddin and R. Magnusson, “Highly efficient color filter array using resonant Si3N4 gratings,” Opt. Express 21, 12495–12506 (2013).
[Crossref]

X. Sheng, S. G. Johnson, J. Michel, and L. C. Kimerling, “Optimization-based design of surface textures for thin-film Si solar cells,” Opt. Express 19, A841–A850 (2011).
[Crossref]

Optik (1)

X. Li, Q. Tan, and G. Jin, “Surface profile optimization of antireflection gratings for solar cells,” Optik 122, 2078–2082 (2011).
[Crossref]

Proc. SPIE (2)

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Fig. 1.

Geometry for the scalar diffraction setup.

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Fig. 2.

Illustration of how a reflective grating can be turned equivalent to a complex distribution on an aperture plane. The phase lag is due to the extra distance traveled by the wave.

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Fig. 3.

Discretization of a surface into rectangular bumps.

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Fig. 4.

CIE 1931 2° color-matching functions converted to RGB weighting functions with a D65 illuminant as reference.

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Fig. 5.

Physics for the optimization problem.

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Fig. 6.

Reflection for a full reflective surface to illustrate how the color not scattered in the desired directions will be scattered in other directions. In this plot blue and green are scattered close to the specular direction, where red is suppressed, meaning that red will have to be scattered somewhere else.

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Fig. 7.

Analysis of possible colors produceable from the specular mode of a binary phase grating. The maximum height has been limited to 1500 nm. (a) Map of binary phase grating zero-order colors as a function of height and duty cycle (a few RGB values have been cut, as they were negative or exceeded 1) and (b) possible “color directions” for the colors in (a), where the color vector has been transformed into polar coordinates and the gray scale indicates the (largest obtained) value of the length, $r$ , of the color vector. The red, green, and blue points have been indicated, as well as white and black (which overlap, since they only vary in intensity), which are represented by a white circle, and the obtainable test reference is indicated with a yellow circle. (c) Example of how a structure can be realized for certain parameters of $h_{1}$ and $b$ .

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Fig. 8.

Optimization of ZOD for blue color without any regularization. (a) The obtained design, (b) the color response seen by the optimization algorithm, and (c) an analysis of the final structure using nonparaxial SDT. This shows that SDT does not capture the physics well, due to scattering from small feature sizes.

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Fig. 9.

Optimization of ZOD using regularization for different colors. All parameters except $C_{0}$ and $c_{1}$ are unchanged from the previous example. The color coordinates for the prescribed design colors are (1, 0, 0), (0, 1, 0), (0, 0, 1), and (0.89, 0.96, 0.65), respectively.

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Fig. 10.

Optimization for blue for $θ_{out} \in [- 10^{°}, 10^{°}]$ with no regularization (all other parameters are the same as for the ZOD example). (a) The design obtained, with the dashed line indicating one of the envelopes seen in the structure, (b) the color response seen by the optimization algorithm, and (c) an analysis of the final structure using nonparaxial SDT.

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Fig. 11.

Comparison of the envelope indicated by a dashed line in Fig. 10(a) and a flat surface having the same length. It is clearly seen how the shape scatters blue in an almost flat interval from $- 10^{°}$ to 10° compared to the flat response.

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Fig. 12.

Optimization for different colors for $θ_{out} \in [- 10^{°}, 10^{°}]$ using regularization. Except for $C_{0}$ and $c_{1}$ , all the parameters are unchanged from the previous example. The color coordinates for the prescribed design colors are (1, 0, 0), (0, 1, 0), (0, 0, 1), and (0.89, 0.96, 0.65), respectively.

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Equations (23)

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E (x^{'}, y^{'}) = \frac{1}{λ^{2} z^{2}} {| F {U (x, y)} (\frac{x^{'}}{λ z}, \frac{y^{'}}{λ z}) |}^{2},

F {U (x, y)} (ξ, η) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} U (x, y) e^{- 2 π i (x ξ + y η)} d x d y .

U (x, y) = A e^{2 π i 2 (h_{ref} - h (x, y)) / λ} = A e^{- 4 π i h (x, y) / λ} e^{4 π i h_{ref} / λ},

U (x, y) = {rect}_{A_{s}} (x, y) e^{- 4 π i h (x, y) / λ},

\hat{x} = x / λ, \hat{y} = y / λ, \hat{z} = z / λ

α = \hat{x} / \hat{r}, β = \hat{y} / \hat{r}, γ = \hat{z} / \hat{r},

α = \sin θ \cos ϕ,

β = \sin θ \sin ϕ,

γ = \cos ϕ .

L (α, β) = \frac{λ^{2}}{A_{s}} {| F {U (\hat{x}, \hat{y}, 0)} (α, β) |}^{2} = \frac{1}{A_{s}} {| F {U (x, y, 0)} (α / λ, β / λ) |}^{2},

L (α) = \frac{1}{L_{0}} {| F {U (x)} (α / λ) |}^{2},

h (x) \approx \sum_{n = 0}^{N - 1} rect (x / d - n) h_{n}, where rect (x) = {\begin{matrix} 1 & for 0 < x \leq 1, \\ 0 & otherwise . \end{matrix}

L (α) = \frac{1}{L_{0}} {| F {{rect}_{L_{0}} (x) e^{- 4 π i h (x) / λ)}} (α / λ) |}^{2} = \frac{1}{N d} {| F {\sum_{n = 0}^{N - 1} {rect}_{[0, 1]} (x / d - n) e^{- 4 π i h_{n} / λ}} (α / λ) |}^{2} = \frac{1}{N d} {| \sum_{n = 0}^{N - 1} e^{- 4 π i h_{n} / λ} F {{rect}_{[0, 1]} (x / d - n)} (α / λ) |}^{2} = \frac{1}{N d} {| \sum_{n = 0}^{N - 1} e^{- 4 π i h_{n} / λ} d e^{- π i d α / λ} e^{2 π in d α / λ} sinc (d α / λ) |}^{2} = {| \binom{\underset{︸}{e^{- π i d α / λ} sinc (d α / λ)}}{const.shape} \frac{1}{N} \sum_{n = 0}^{N - 1} \binom{\underset{︸}{e^{- 4 π i h_{n} / λ}}}{height} \binom{\underset{︸}{{(e^{2 π i d α / λ})}^{n}}}{translation} |}^{2} .

L (α) = {| C (α) \frac{1}{N} \sum_{n = 0}^{N - 1} H_{n} (α) T_{n} (α) |}^{2} .

P = C \frac{1}{N} {\hat{P}}_{N}, where {\hat{P}}_{n} = {\begin{matrix} T_{1} {\hat{P}}_{n - 1} + H_{N - 1 - n} & for n > 0, \\ H_{N - 1} & for n = 0, \end{matrix}

\frac{\partial P}{\partial h_{j}} = C \frac{1}{N} \sum_{n = 0}^{N - 1} T_{n} \frac{\partial}{\partial h_{j}} e^{- 4 π i h_{n} / λ} = - \frac{4 π i}{N λ} C T_{j} e^{- 4 π i h_{j} / λ} = - \frac{4 π i}{N λ} C T_{j} H_{j} .

R = \int_{0}^{\infty} I (λ) \bar{r} (λ) d λ,

G = \int_{0}^{\infty} I (λ) \bar{g} (λ) d λ,

B = \int_{0}^{\infty} I (λ) \bar{b} (λ) d λ,

\min_{h} f (h) = \max_{i = 1, 2, \dots, M} \frac{- {‖ C (h, θ_{i}) ‖}^{2}}{{‖ C_{0} (θ_{i}) ‖}^{2}} + c_{1} \frac{{‖ \nabla h ‖}^{2}}{{(N d)}^{2}}, s.t. h_{\min} \leq h \leq h_{\max}, f_{i} (h) = \frac{‖ C (h, θ_{i}) \times C_{0} (θ_{i}) ‖^{2}}{ε^{2} {‖ C_{0} (θ_{i}) ‖}^{2}} - 1 \leq 0,

{‖ C \times C_{0} ‖}^{2} = {‖ n \sin ϕ ‖ C ‖ \cdot ‖ C_{0} ‖ ‖}^{2} = {‖ C ‖}^{2} \cdot {‖ C_{0} ‖}^{2} \cdot \sin^{2} ϕ,

L (0) = {| \frac{1}{N} \sum_{n = 0}^{N - 1} e^{- 4 π i h_{n} / λ} |}^{2} .

L (b, h_{1}) = {| \frac{\sum_{n \in H_{0}} e^{- 4 π i h_{0} / λ}}{\sum_{n \in H_{0}} 1} + \frac{\sum_{n \in H_{1}} e^{- 4 π i h_{1} / λ}}{\sum_{n \in H_{1}} 1} |}^{2} = {| (1 - b) + b e^{- 4 π i h_{1} / λ} |}^{2},

Abstract

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