Abstract

We propose an inverse design methodology for systematic design of nanostructured surfaces for color effects. The methodology is based on a 2D topology optimization formulation based on frequency-domain finite element simulations for E and/or H polarized waves. The goal of the optimization is to maximize color intensity in prescribed direction(s) for a prescribed color (RGB) vector. Results indicate that nanostructured surfaces with any desirable color vector can be generated; that complex structures can generate more intense colors than simple layerings; that angle independent colorings can be obtained at the cost of reduced intensity; and that performance and optimized surface topologies are relatively independent on light polarization.

© 2013 Optical Society of America

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    [CrossRef]
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  7. N. Okada, D. Zhu, D. Cai, J. B. Cole, M. Kambe, and S. Kinoshita, “Rendering Morpho butterflies based on high accuracy nano-optical simulation,” J. Opt. 42, 25–36 (2013).
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  9. 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).
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    [CrossRef]
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  26. O. Sigmund, “On the usefulness of non-gradient approaches in topology optimization,” Struct. Multidisc. Optim. 43, 589–596 (2011).
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    [CrossRef]
  31. M. B. Dühring and O. Sigmund, “Optimization of extraordinary optical absorption in plasmonic and dielectric structures,” J. Opt. Soc. Am. B 30, 1154–1160 (2013).
    [CrossRef]
  32. 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]

2013 (2)

N. Okada, D. Zhu, D. Cai, J. B. Cole, M. Kambe, and S. Kinoshita, “Rendering Morpho butterflies based on high accuracy nano-optical simulation,” J. Opt. 42, 25–36 (2013).
[CrossRef]

M. B. Dühring and O. Sigmund, “Optimization of extraordinary optical absorption in plasmonic and dielectric structures,” J. Opt. Soc. Am. B 30, 1154–1160 (2013).
[CrossRef]

2012 (4)

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]

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]

M. A. Steindorfer, V. Schmidt, M. Belegratis, B. Stadlober, and J. R. Krenn, “Detailed simulation of structural color generation inspired by the Morpho butterfly,” Opt. Express 20, 21485–21494 (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 (8)

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

A. Erentok and O. Sigmund, “Topology optimization of sub-wavelength antennas,” IEEE Trans. Anten. Propag. 59, 58–69 (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. Technol. Adv. Mat. 12, 064709 (2011).
[CrossRef]

M. Crne, V. Sharma, J. Blair, J. O. Park, C. J. Summers, and M. Srinivasarao, “Biomimicry of optical microstructures of Papilio palinurus,” Europhys. Lett. 93, 14001 (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]

O. Sigmund, “On the usefulness of non-gradient approaches in topology optimization,” Struct. Multidisc. Optim. 43, 589–596 (2011).
[CrossRef]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

2010 (3)

J. A. Andkjær, S. Nishiwaki, T. Nomura, and O. Sigmund, “Topology optimization of grating couplers for the efficient excitation of surface plasmons,” J. Opt. Soc. Am. B 27, 1828–1832 (2010).
[CrossRef]

A. R. Diaz and O. Sigmund, “A topology optimization method for design of negative permeability metamaterials,” Struct. Multidiscip. Optim. 41, 163–177 (2010).
[CrossRef]

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidiscip. Optim. 42, 483–493 (2010).
[CrossRef]

2009 (2)

2008 (1)

S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
[CrossRef]

2005 (2)

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

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005).
[CrossRef]

1993 (1)

D. C. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Euro. J. Appl. Math. 4, 321–339 (1993).
[CrossRef]

1988 (1)

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988).
[CrossRef]

1987 (1)

K. Svanberg, “The method of moving asymptotes—a new method for structural optimization,” Intern. J. Numer. Meth. Eng. 24, 359–373 (1987).
[CrossRef]

Akai-Kasaya, M.

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, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass production, and control of the Morpho butterfly’s blue,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics II, T. J. Suleski, W. V. Schoenfeld, and J. J. Wang, eds. (SPIE, 2009), Vol. 7205, p. 720506.

Andkjær, J. A.

Belegratis, M.

Bendsøe, M. P.

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988).
[CrossRef]

M. P. Bendsøe and O. Sigmund, Topology Optimization—Theory, Methods and Applications (Springer-Verlag, 2004).

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).

Blair, J.

M. Crne, V. Sharma, J. Blair, J. O. Park, C. J. Summers, and M. Srinivasarao, “Biomimicry of optical microstructures of Papilio palinurus,” Europhys. Lett. 93, 14001 (2011).
[CrossRef]

Cai, D.

N. Okada, D. Zhu, D. Cai, J. B. Cole, M. Kambe, and S. Kinoshita, “Rendering Morpho butterflies based on high accuracy nano-optical simulation,” J. Opt. 42, 25–36 (2013).
[CrossRef]

Cole, J. B.

N. Okada, D. Zhu, D. Cai, J. B. Cole, M. Kambe, and S. Kinoshita, “Rendering Morpho butterflies based on high accuracy nano-optical simulation,” J. Opt. 42, 25–36 (2013).
[CrossRef]

Crne, M.

M. Crne, V. Sharma, J. Blair, J. O. Park, C. J. Summers, and M. Srinivasarao, “Biomimicry of optical microstructures of Papilio palinurus,” Europhys. Lett. 93, 14001 (2011).
[CrossRef]

Diaz, A. R.

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidiscip. Optim. 42, 483–493 (2010).
[CrossRef]

A. R. Diaz and O. Sigmund, “A topology optimization method for design of negative permeability metamaterials,” Struct. Multidiscip. Optim. 41, 163–177 (2010).
[CrossRef]

Dobson, D. C.

D. C. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Euro. J. Appl. Math. 4, 321–339 (1993).
[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]

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]

Dühring, M. B.

Elesin, Y.

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Time domain topology optimization of 3D nanophotonic devices,” Photon. Nanostruct. Fundam. Appl., doi: 10.1016/j.photonics.2013.07.008 (2013).
[CrossRef]

Erentok, A.

A. Erentok and O. Sigmund, “Topology optimization of sub-wavelength antennas,” IEEE Trans. Anten. Propag. 59, 58–69 (2011).
[CrossRef]

Friis, K. S.

Fuchi, K.

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidiscip. Optim. 42, 483–493 (2010).
[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]

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]

Ishikawa, Y.

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,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics II, T. J. Suleski, W. V. Schoenfeld, and J. J. Wang, eds. (SPIE, 2009), Vol. 7205, p. 720506.

Jensen, J. S.

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

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005).
[CrossRef]

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Time domain topology optimization of 3D nanophotonic devices,” Photon. Nanostruct. Fundam. Appl., doi: 10.1016/j.photonics.2013.07.008 (2013).
[CrossRef]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics2nd ed. (Wiley, 2002).

Johnson, S. G.

Juodkazis, S.

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]

Kambe, M.

N. Okada, D. Zhu, D. Cai, J. B. Cole, M. Kambe, and S. Kinoshita, “Rendering Morpho butterflies based on high accuracy nano-optical simulation,” J. Opt. 42, 25–36 (2013).
[CrossRef]

Kikuchi, N.

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988).
[CrossRef]

Kimerling, L. C.

Kinoshita, S.

N. Okada, D. Zhu, D. Cai, J. B. Cole, M. Kambe, and S. Kinoshita, “Rendering Morpho butterflies based on high accuracy nano-optical simulation,” J. Opt. 42, 25–36 (2013).
[CrossRef]

S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
[CrossRef]

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

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]

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]

Kolle, M.

M. Kolle, Photonic Structures Inspired by Nature (Springer, 2011).

Krenn, J. R.

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]

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.

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, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass production, and control of the Morpho butterfly’s blue,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics II, T. J. Suleski, W. V. Schoenfeld, and J. J. Wang, eds. (SPIE, 2009), Vol. 7205, p. 720506.

Lazarov, B. S.

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Time domain topology optimization of 3D nanophotonic devices,” Photon. Nanostruct. Fundam. Appl., doi: 10.1016/j.photonics.2013.07.008 (2013).
[CrossRef]

Lee, R. T.

Michel, J.

Miyamura, Y.

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,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics II, T. J. Suleski, W. V. Schoenfeld, and J. J. Wang, eds. (SPIE, 2009), Vol. 7205, p. 720506.

Miyazaki, J.

S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
[CrossRef]

Mizeikis, V.

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]

Murase, J.

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, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass production, and control of the Morpho butterfly’s blue,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics II, T. J. Suleski, W. V. Schoenfeld, and J. J. Wang, eds. (SPIE, 2009), Vol. 7205, p. 720506.

Nishiwaki, S.

Nomura, T.

Okada, N.

N. Okada, D. Zhu, D. Cai, J. B. Cole, M. Kambe, and S. Kinoshita, “Rendering Morpho butterflies based on high accuracy nano-optical simulation,” J. Opt. 42, 25–36 (2013).
[CrossRef]

Ouedraogo, R.

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidiscip. Optim. 42, 483–493 (2010).
[CrossRef]

Park, J. O.

M. Crne, V. Sharma, J. Blair, J. O. Park, C. J. Summers, and M. Srinivasarao, “Biomimicry of optical microstructures of Papilio palinurus,” Europhys. Lett. 93, 14001 (2011).
[CrossRef]

Rothwell, E.

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidiscip. Optim. 42, 483–493 (2010).
[CrossRef]

Saito, A.

A. Saito, “Material design and structural color inspired by biomimetic approach,” Sci. Technol. Adv. Mat. 12, 064709 (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, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass production, and control of the Morpho butterfly’s blue,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics II, T. J. Suleski, W. V. Schoenfeld, and J. J. Wang, eds. (SPIE, 2009), Vol. 7205, p. 720506.

Saltzman, M.

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

Schmidt, V.

Sharma, V.

M. Crne, V. Sharma, J. Blair, J. O. Park, C. J. Summers, and M. Srinivasarao, “Biomimicry of optical microstructures of Papilio palinurus,” Europhys. Lett. 93, 14001 (2011).
[CrossRef]

Sheng, X.

Sigmund, O.

M. B. Dühring and O. Sigmund, “Optimization of extraordinary optical absorption in plasmonic and dielectric structures,” J. Opt. Soc. Am. B 30, 1154–1160 (2013).
[CrossRef]

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]

A. Erentok and O. Sigmund, “Topology optimization of sub-wavelength antennas,” IEEE Trans. Anten. Propag. 59, 58–69 (2011).
[CrossRef]

O. Sigmund, “On the usefulness of non-gradient approaches in topology optimization,” Struct. Multidisc. Optim. 43, 589–596 (2011).
[CrossRef]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

J. A. Andkjær, S. Nishiwaki, T. Nomura, and O. Sigmund, “Topology optimization of grating couplers for the efficient excitation of surface plasmons,” J. Opt. Soc. Am. B 27, 1828–1832 (2010).
[CrossRef]

A. R. Diaz and O. Sigmund, “A topology optimization method for design of negative permeability metamaterials,” Struct. Multidiscip. Optim. 41, 163–177 (2010).
[CrossRef]

O. Sigmund, “Manufacturing tolerant topology optimization,” Acta Mech. Sinica 25, 227–239 (2009).
[CrossRef]

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005).
[CrossRef]

M. P. Bendsøe and O. Sigmund, Topology Optimization—Theory, Methods and Applications (Springer-Verlag, 2004).

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Time domain topology optimization of 3D nanophotonic devices,” Photon. Nanostruct. Fundam. Appl., doi: 10.1016/j.photonics.2013.07.008 (2013).
[CrossRef]

Smith, G. S.

Srinivasarao, M.

M. Crne, V. Sharma, J. Blair, J. O. Park, C. J. Summers, and M. Srinivasarao, “Biomimicry of optical microstructures of Papilio palinurus,” Europhys. Lett. 93, 14001 (2011).
[CrossRef]

Stadlober, B.

Steindorfer, M. A.

Summers, C. J.

M. Crne, V. Sharma, J. Blair, J. O. Park, C. J. Summers, and M. Srinivasarao, “Biomimicry of optical microstructures of Papilio palinurus,” Europhys. Lett. 93, 14001 (2011).
[CrossRef]

Svanberg, K.

K. Svanberg, “The method of moving asymptotes—a new method for structural optimization,” Intern. J. Numer. Meth. Eng. 24, 359–373 (1987).
[CrossRef]

Temme, A.

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidiscip. Optim. 42, 483–493 (2010).
[CrossRef]

Wang, F.

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[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]

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]

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]

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.

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]

Yoshioka, S.

S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
[CrossRef]

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

Zhu, D.

N. Okada, D. Zhu, D. Cai, J. B. Cole, M. Kambe, and S. Kinoshita, “Rendering Morpho butterflies based on high accuracy nano-optical simulation,” J. Opt. 42, 25–36 (2013).
[CrossRef]

Acta Mech. Sinica (1)

O. Sigmund, “Manufacturing tolerant topology optimization,” Acta Mech. Sinica 25, 227–239 (2009).
[CrossRef]

Appl. Opt. (1)

Chem. Phys. Chem. (1)

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

Comput. Methods Appl. Mech. Eng. (1)

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988).
[CrossRef]

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D. C. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Euro. J. Appl. Math. 4, 321–339 (1993).
[CrossRef]

Europhys. Lett. (1)

M. Crne, V. Sharma, J. Blair, J. O. Park, C. J. Summers, and M. Srinivasarao, “Biomimicry of optical microstructures of Papilio palinurus,” Europhys. Lett. 93, 14001 (2011).
[CrossRef]

IEEE Trans. Anten. Propag. (1)

A. Erentok and O. Sigmund, “Topology optimization of sub-wavelength antennas,” IEEE Trans. Anten. Propag. 59, 58–69 (2011).
[CrossRef]

Intern. J. Numer. Meth. Eng. (1)

K. Svanberg, “The method of moving asymptotes—a new method for structural optimization,” Intern. J. Numer. Meth. Eng. 24, 359–373 (1987).
[CrossRef]

J. Nanosci. Nanotechnol. (1)

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]

J. Opt. (1)

N. Okada, D. Zhu, D. Cai, J. B. Cole, M. Kambe, and S. Kinoshita, “Rendering Morpho butterflies based on high accuracy nano-optical simulation,” J. Opt. 42, 25–36 (2013).
[CrossRef]

J. Opt. Soc. Am. B (4)

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. (2)

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]

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]

Opt. Express (2)

Rep. Prog. Phys. (1)

S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
[CrossRef]

Sci. Technol. Adv. Mat. (1)

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

Struct. Multidisc. Optim. (2)

O. Sigmund, “On the usefulness of non-gradient approaches in topology optimization,” Struct. Multidisc. Optim. 43, 589–596 (2011).
[CrossRef]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

Struct. Multidiscip. Optim. (2)

A. R. Diaz and O. Sigmund, “A topology optimization method for design of negative permeability metamaterials,” Struct. Multidiscip. Optim. 41, 163–177 (2010).
[CrossRef]

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidiscip. Optim. 42, 483–493 (2010).
[CrossRef]

Other (7)

M. P. Bendsøe and O. Sigmund, Topology Optimization—Theory, Methods and Applications (Springer-Verlag, 2004).

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Time domain topology optimization of 3D nanophotonic devices,” Photon. Nanostruct. Fundam. Appl., doi: 10.1016/j.photonics.2013.07.008 (2013).
[CrossRef]

M. Kolle, Photonic Structures Inspired by Nature (Springer, 2011).

J. Jin, The Finite Element Method in Electromagnetics2nd ed. (Wiley, 2002).

CIE, “Selected colorimetric tables,” http://www.cie.co.at .

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

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,” in Advanced Fabrication Technologies for Micro/Nano Optics and Photonics II, T. J. Suleski, W. V. Schoenfeld, and J. J. Wang, eds. (SPIE, 2009), Vol. 7205, p. 720506.

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Figures (8)

Fig. 1.
Fig. 1.

Computational domain composed of an air region ΩA, a TiO2 region ΩB, and a design domain ΩD, where SiO2 and TiO2 is distributed to form the nanostructure. The periodicity of the structure is modeled with Bloch–Floquet boundary conditions at Γp.

Fig. 2.
Fig. 2.

Color-matching functions for finding the RGB channels of a light spectrum. The functions are derived using the 1931 CIE 2° standard observer weighting function [21] to obtain the XYZ tristimulus values and afterward converted to the RGB working space using a D65 reference while following ISO 22028-1:2004.

Fig. 3.
Fig. 3.

Perfectly periodic nanostructure (a) with corresponding diffraction pattern (b) and a height randomized structure (c) with the diffraction effect eliminated (d). The display colors are exaggerated for the purpose of illustration. The area surrounded by the dashed line corresponds to one “unit cell.”

Fig. 4.
Fig. 4.

Color optimization for 0° observation angle. The left column shows the material distribution; the center column the frequency response seen from the observer; and the right column shows the angular color spectrum including individual color distributions. Structures (a)–(h) are optimized for red (RGBr=[1,0,0]), green [0,1,0], blue [0,0,1], yellow [0,1,1], magenta [1,0,1], cyan [1,1,0], black [0,0,0], and white [1,1,1], respectively.

Fig. 5.
Fig. 5.

Color optimization for 0° observation angle but design freedom limited to simple layered Bragg-like structures. Other details can be read from the caption of Fig. 4.

Fig. 6.
Fig. 6.

Color optimization for 30° to 30° observation angle with full design freedom. Other details can be read from the caption of Fig. 4. The three curves in the reflection plots correspond to the reflection spectra for the three optimization angles (0°, 15°, and 30°).

Fig. 7.
Fig. 7.

Color optimization for combined Ez and Hz polarization. (a)–(c) Structures optimized for Hz polarization corresponding to those optimized for Ez in Figs. 4(c), 5(c), and 6(c), respectively. (d)–(f) Similar structures optimized for combined Ez and Hz polarization. Red curves in the reflection plots correspond to Hz polarization and black to Ez.

Fig. 8.
Fig. 8.

Optimization for prescribed iridescent colors. Red color [1,0,0] intensity is optimized for 43° observer direction, and blue color intensity [0,0,1] is optimized for 27° observer direction. Designs (a)–(c) correspond to optimized structures for Ez, Hz, and combined Ez to Hz polarization, respectively. Red curves correspond to the reflection in the 43° direction and blue to the reflection in the 27° direction.

Tables (1)

Tables Icon

Table 1. Objective Values for RGB Optimizations Corresponding to 0°, 0° Constrained to Simple Layerings and Wider Angular Spectrum [30°, 30°]a

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

·(μr1Ez)+k02ϵrEz=0,
x(sysxμr1Ezx)+y(sxsyμr1Ezy)+k02sxsyϵrEz=0,
Ezi=Ez0exp(jk0μrϵrk^·r),
Jsz=2cos(θin)ϵ0ϵrμrμ0Ezi,
Ez(x,d)=Ez(x,0)exp(jk0μrϵrdsin(θin)).
Ezsr(ρ,λ,θout)jk08πρexp(jk0ρ)0d[(n^ysin(θout)+n^xcos(θout))Ezs1jk0(n^xEzsx+n^yEzsy)]exp(jk0(xcos(θout)+ysin(θout)))dy,
L(λ,θout)=ρλcos(θinc)cos(θout)|Ezsr(ρ,λ,θout)|2|Ezi(λ)|2,
R(θout)=Kc380760D65(λ)r¯(λ)L(λ,θout)dλKcλD65(λ)r¯(λ)L(λ,θout)Δλ,
G(θout)=Kc380760D65(λ)g¯(λ)L(λ,θout)dλKcλD65(λ)g¯(λ)L(λ,θout)Δλ,
B(θout)=Kc380760D65(λ)b¯(λ)L(λ,θout)dλKcλD65(λ)b¯(λ)L(λ,θout)Δλ,
ϵr(γe)=ϵrSiO2+γe(ϵrTiO2ϵrSiO2).
maxγΦ:=mink=1,,N|RGB(θoutk)|2|RGBrk|2,s.t.|RGB(θoutk)×RGBrk|2|RGBrk|2τ2,k=1,,N,1VΩDΩDγdΩDβ0,0γ1,
n^×(H2H1)=J=z^Jsz,
×E=x^Ezyy^Ezx.
n^×H=1jωμrμ0(n^yEzy+n^xEzy)=1jωμrμ0n^·(Ez),
n^·(Ez2Ez1)=z^jωμrμ0Jsz
E1=z^Ez0exp(jk0μrϵrk^1·r),E2=z^Ez0exp(jk0μrϵrk^2·r+jϕ),
Jsz=jk0μrϵrjωμ0μrn^·(k^1Ez1k^2Ez2).
Jsz=2k^xϵ0ϵrμrμ0Ez,1.

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