Abstract

Structural coloration produces some of the most brilliant colors in nature and has many applications. Motivated by the recently proposed transparent displays that are based on wavelength-selective scattering, here we consider the new problem of transparent structural color, where objects are transparent under omnidirectional broad-band illumination but scatter strongly with a directional narrow-band light source. Transparent structural color requires two competing properties, narrow bandwidth and broad viewing angle, that have not been demonstrated simultaneously previously. Here, we use numerical optimization to discover geometries where a sharp 7% bandwidth in scattering is achieved, yet the peak wavelength varies less than 1%, and the peak height and peak width vary less than 6% over broad viewing angles (0–90°) under a directional illumination. Our model system consists of dipole scatterers arranged into several rings; interference among the scattered waves is optimized to yield the wavelength-selective and angle-insensitive response.

© 2015 Optical Society of America

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References

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2015 (1)

Y. Shen, V. Rinnerbauer, I. Wang, V. Stelmakh, J. D. Joannopoulos, and M. Soljačić, “Structural colors from Fano resonances,” ACS Photon. 2, 27–32 (2015).
[Crossref]

2014 (11)

O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Phys. Rev. Lett. 112, 123903 (2014).
[Crossref] [PubMed]

S. Magkiriadou, J.-G. Park, Y.-S. Kim, and V. N. Manoharan, “Absence of red structural color in photonic glasses, bird feathers, and certain beetles,” Phys. Rev. E 90, 062302 (2014).
[Crossref]

C. W. Hsu, B. Zhen, W. Qiu, O. Shapira, B. G. DeLacy, J. D. Joannopoulos, and M. Soljačić, “Transparent displays enabled by resonant nanoparticle scattering,” Nat. Commun. 5, 3152 (2014).
[Crossref] [PubMed]

J. Bauer, S. Hengsbach, I. Tesari, R. Schwaiger, and O. Kraft, “High-strength cellular ceramic composites with 3d microarchitecture,” Proc. Natl. Acad. Sci. U.S.A. 111, 2453–2458 (2014).
[Crossref] [PubMed]

C. W. Hsu, B. G. DeLacy, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Theoretical criteria for scattering dark states in nanostructured particles,” Nano Lett. 14, 2783–2788 (2014).
[Crossref] [PubMed]

V. R. Shrestha, S.-S. Lee, E.-S. Kim, and D.-Y. Choi, “Non-iridescent transmissive structural color filter featuring highly efficient transmission and high excitation purity,” Sci. Rep. 4, 4921 (2014).
[Crossref] [PubMed]

H. Men, R. M. Freund, N. C. Nguyen, J. Saa-Seoane, and J. Peraire, “Fabrication-adaptive optimization with an application to photonic crystal design,” Oper. Res. 62, 418–434 (2014).
[Crossref]

J. Andkjær, V. E. Johansen, K. S. Friis, and O. Sigmund, “Inverse design of nanostructured surfaces for color effects,” J. Opt. Soc. Am. B 31, 164–174 (2014).
[Crossref]

V. E. Johansen, J. Andkjær, and O. Sigmund, “Design of structurally colored surfaces based on scalar diffraction theory,” J. Opt. Soc. Am. B 31, 207–217 (2014).
[Crossref]

V. E. Johansen, “Optical role of randomness for structured surfaces,” Appl. Opt. 53, 2405–2415 (2014).
[Crossref] [PubMed]

H. Men, K. Y. K. Lee, R. M. Freund, J. Peraire, and S. G. Johnson, “Robust topology optimization of three-dimensional photonic-crystal band-gap structures,” Opt. Express 22, 22632–22648 (2014).
[Crossref] [PubMed]

2013 (2)

C. Yang, L. Hong, W. Shen, Y. Zhang, X. Liu, and H. Zhen, “Design of reflective color filters with high angular tolerance by particle swarm optimization method,” Opt. Express 21, 9315–9323 (2013).
[Crossref] [PubMed]

M. A. Kats, R. Blanchard, P. Genevet, and F. Capasso, “Nanometre optical coatings based on strong interference effects in highly absorbing media,” Nature Mater. 12, 20–24 (2013).
[Crossref]

2012 (7)

Y. Takeoka, “Angle-independent structural coloured amorphous arrays,” J. Mater. Chem. 22, 23299–23309 (2012).
[Crossref]

V. Saranathan, J. D. Forster, H. Noh, S.-F. Liew, S. G. J. Mochrie, H. Cao, E. R. Dufresne, and R. O. Prum, “Structure and optical function of amorphous photonic nanostructures from avian feather barbs: a comparative small angle X-ray scattering (SAXS) analysis of 230 bird species,” J. R. Soc. Interface 9, 2563–2580 (2012).
[Crossref] [PubMed]

S. Magkiriadou, J.-G. Park, Y.-S. Kim, and V. N. Manoharan, “Disordered packings of core-shell particles with angle-independent structural colors,” Opt. Mater. Express 2, 1343–1352 (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] [PubMed]

A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express 20, 21558–21575 (2012).
[Crossref] [PubMed]

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by Morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[Crossref] [PubMed]

Y. Elesin, B. Lazarov, J. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photon. Nanostruct. 10, 153–165 (2012).
[Crossref]

2011 (2)

F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B 28, 387–397 (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] [PubMed]

2010 (6)

L. Cao, P. Fan, E. S. Barnard, A. M. Brown, and M. L. Brongersma, “Tuning the color of silicon nanostructures,” Nano. Lett. 10, 2649–2654 (2010).
[Crossref] [PubMed]

M. Harun-Ur-Rashid, A. BinImran, T. Seki, M. Ishii, H. Nakamura, and Y. Takeoka, “Angle-independent structural color in colloidal amorphous arrays,” Chem. Phys. Chem 11, 579–583 (2010).
[PubMed]

J. D. Forster, H. Noh, S. F. Liew, V. Saranathan, C. F. Schreck, L. Yang, J.-G. Park, R. O. Prum, S. G. J. Mochrie, C. S. O’Hern, H. Cao, and E. R. Dufresne, “Biomimetic isotropic nanostructures for structural coloration,” Adv. Mater. 22, 2939–2944 (2010).
[Crossref] [PubMed]

H. Noh, S. F. Liew, V. Saranathan, S. G. J. Mochrie, R. O. Prum, E. R. Dufresne, and H. Cao, “How noniridescent colors are generated by quasi-ordered structures of bird feathers,” Adv. Mater. 22, 2871–2880 (2010).
[Crossref] [PubMed]

A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B 43, 163001 (2010).
[Crossref]

Z. Ruan and S. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114, 7324–7329 (2010).
[Crossref]

2009 (11)

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, 213104 (2009).
[Crossref]

E. R. Dufresne, H. Noh, V. Saranathan, S. G. J. Mochrie, H. Cao, and R. O. Prum, “Self-assembly of amorphous biophotonic nanostructures by phase separation,” Soft Matter 5, 1792–1795 (2009).
[Crossref]

Y. Takeoka, M. Honda, T. Seki, M. Ishii, and H. Nakamura, “Structural colored liquid membrane without angle dependence,” ACS Appl. Mater. Inter. 1, 982–986 (2009).
[Crossref]

K. Ueno, A. Inaba, Y. Sano, M. Kondoh, and M. Watanabe, “A soft glassy colloidal array in ionic liquid, which exhibits homogeneous, non-brilliant and angle-independent structural colours,” Chem. Commun. 2009, 3603–3605 (2009).
[Crossref]

H. Kim, J. Ge, J. Kim, S.-e. Choi, H. Lee, H. Lee, W. Park, Y. Yin, and S. Kwon, “Structural colour printing using a magnetically tunable and lithographically fixable photonic crystal,” Nature Photon. 3, 534–540 (2009).
[Crossref]

L. Li, R. R. Gattass, E. Gershgoren, H. Hwang, and J. T. Fourkas, “Achieving λ/20 resolution by one-color initiation and deactivation of polymerization,” Science 324, 910–913 (2009).
[Crossref] [PubMed]

H. M. Whitney, M. Kolle, P. Andrew, L. Chittka, U. Steiner, and B. J. Glover, “Floral iridescence, produced by diffractive optics, acts as a cue for animal pollinators,” Science 323, 130–133 (2009).
[Crossref] [PubMed]

L. D. Bonifacio, B. V. Lotsch, D. P. Puzzo, F. Scotognella, and G. A. Ozin, “Stacking the nanochemistry deck: Structural and compositional diversity in one-dimensional photonic crystals,” Adv. Mater. 21, 1641–1646 (2009).
[Crossref]

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

A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Eng. Optim. 41, 365–384 (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] [PubMed]

2008 (1)

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

2007 (7)

A. C. Arsenault, D. P. Puzzo, I. Manners, and G. A. Ozin, “Photonic-crystal full-colour displays,” Nature Photon. 1, 468–472 (2007).
[Crossref]

R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, K. Dovidenko, J. R. Cournoyer, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nature Photon. 1, 123–128 (2007).
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V. Saranathan, J. D. Forster, H. Noh, S.-F. Liew, S. G. J. Mochrie, H. Cao, E. R. Dufresne, and R. O. Prum, “Structure and optical function of amorphous photonic nanostructures from avian feather barbs: a comparative small angle X-ray scattering (SAXS) analysis of 230 bird species,” J. R. Soc. Interface 9, 2563–2580 (2012).
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S. Kinoshita, S. Yoshioka, Y. Fujii, and N. Okamoto, “Photophysics of structural color in the Morpho butterflies,” Forma 17, 103–121 (2002).

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L. Li, R. R. Gattass, E. Gershgoren, H. Hwang, and J. T. Fourkas, “Achieving λ/20 resolution by one-color initiation and deactivation of polymerization,” Science 324, 910–913 (2009).
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H. Kim, J. Ge, J. Kim, S.-e. Choi, H. Lee, H. Lee, W. Park, Y. Yin, and S. Kwon, “Structural colour printing using a magnetically tunable and lithographically fixable photonic crystal,” Nature Photon. 3, 534–540 (2009).
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M. A. Kats, R. Blanchard, P. Genevet, and F. Capasso, “Nanometre optical coatings based on strong interference effects in highly absorbing media,” Nature Mater. 12, 20–24 (2013).
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L. Li, R. R. Gattass, E. Gershgoren, H. Hwang, and J. T. Fourkas, “Achieving λ/20 resolution by one-color initiation and deactivation of polymerization,” Science 324, 910–913 (2009).
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R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, K. Dovidenko, J. R. Cournoyer, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nature Photon. 1, 123–128 (2007).
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H. M. Whitney, M. Kolle, P. Andrew, L. Chittka, U. Steiner, and B. J. Glover, “Floral iridescence, produced by diffractive optics, acts as a cue for animal pollinators,” Science 323, 130–133 (2009).
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Hamam, R. E.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
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K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by Morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
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M. Harun-Ur-Rashid, A. BinImran, T. Seki, M. Ishii, H. Nakamura, and Y. Takeoka, “Angle-independent structural color in colloidal amorphous arrays,” Chem. Phys. Chem 11, 579–583 (2010).
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Hengsbach, S.

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K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by Morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
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K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by Morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
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H. Men, K. Y. K. Lee, R. M. Freund, J. Peraire, and S. G. Johnson, “Robust topology optimization of three-dimensional photonic-crystal band-gap structures,” Opt. Express 22, 22632–22648 (2014).
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J. D. Forster, H. Noh, S. F. Liew, V. Saranathan, C. F. Schreck, L. Yang, J.-G. Park, R. O. Prum, S. G. J. Mochrie, C. S. O’Hern, H. Cao, and E. R. Dufresne, “Biomimetic isotropic nanostructures for structural coloration,” Adv. Mater. 22, 2939–2944 (2010).
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E. R. Dufresne, H. Noh, V. Saranathan, S. G. J. Mochrie, H. Cao, and R. O. Prum, “Self-assembly of amorphous biophotonic nanostructures by phase separation,” Soft Matter 5, 1792–1795 (2009).
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J. Bauer, S. Hengsbach, I. Tesari, R. Schwaiger, and O. Kraft, “High-strength cellular ceramic composites with 3d microarchitecture,” Proc. Natl. Acad. Sci. U.S.A. 111, 2453–2458 (2014).
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L. D. Bonifacio, B. V. Lotsch, D. P. Puzzo, F. Scotognella, and G. A. Ozin, “Stacking the nanochemistry deck: Structural and compositional diversity in one-dimensional photonic crystals,” Adv. Mater. 21, 1641–1646 (2009).
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C. W. Hsu, B. Zhen, W. Qiu, O. Shapira, B. G. DeLacy, J. D. Joannopoulos, and M. Soljačić, “Transparent displays enabled by resonant nanoparticle scattering,” Nat. Commun. 5, 3152 (2014).
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Y. Shen, V. Rinnerbauer, I. Wang, V. Stelmakh, J. D. Joannopoulos, and M. Soljačić, “Structural colors from Fano resonances,” ACS Photon. 2, 27–32 (2015).
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K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by Morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
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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, 213104 (2009).
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V. R. Shrestha, S.-S. Lee, E.-S. Kim, and D.-Y. Choi, “Non-iridescent transmissive structural color filter featuring highly efficient transmission and high excitation purity,” Sci. Rep. 4, 4921 (2014).
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Y. Shen, V. Rinnerbauer, I. Wang, V. Stelmakh, J. D. Joannopoulos, and M. Soljačić, “Structural colors from Fano resonances,” ACS Photon. 2, 27–32 (2015).
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C. W. Hsu, B. Zhen, W. Qiu, O. Shapira, B. G. DeLacy, J. D. Joannopoulos, and M. Soljačić, “Transparent displays enabled by resonant nanoparticle scattering,” Nat. Commun. 5, 3152 (2014).
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C. W. Hsu, B. G. DeLacy, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Theoretical criteria for scattering dark states in nanostructured particles,” Nano Lett. 14, 2783–2788 (2014).
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M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
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H. M. Whitney, M. Kolle, P. Andrew, L. Chittka, U. Steiner, and B. J. Glover, “Floral iridescence, produced by diffractive optics, acts as a cue for animal pollinators,” Science 323, 130–133 (2009).
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Y. Shen, V. Rinnerbauer, I. Wang, V. Stelmakh, J. D. Joannopoulos, and M. Soljačić, “Structural colors from Fano resonances,” ACS Photon. 2, 27–32 (2015).
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K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optimiz. 12, 555–573 (2002).
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A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B 43, 163001 (2010).
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Y. Takeoka, “Angle-independent structural coloured amorphous arrays,” J. Mater. Chem. 22, 23299–23309 (2012).
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M. Harun-Ur-Rashid, A. BinImran, T. Seki, M. Ishii, H. Nakamura, and Y. Takeoka, “Angle-independent structural color in colloidal amorphous arrays,” Chem. Phys. Chem 11, 579–583 (2010).
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Y. Takeoka, M. Honda, T. Seki, M. Ishii, and H. Nakamura, “Structural colored liquid membrane without angle dependence,” ACS Appl. Mater. Inter. 1, 982–986 (2009).
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D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
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J. Bauer, S. Hengsbach, I. Tesari, R. Schwaiger, and O. Kraft, “High-strength cellular ceramic composites with 3d microarchitecture,” Proc. Natl. Acad. Sci. U.S.A. 111, 2453–2458 (2014).
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K. Ueno, A. Inaba, Y. Sano, M. Kondoh, and M. Watanabe, “A soft glassy colloidal array in ionic liquid, which exhibits homogeneous, non-brilliant and angle-independent structural colours,” Chem. Commun. 2009, 3603–3605 (2009).
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M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
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P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single Morpho butterfly scales,” Proc. R. Soc. Lond. B 266, 1403–1411 (1999).
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Y. Shen, V. Rinnerbauer, I. Wang, V. Stelmakh, J. D. Joannopoulos, and M. Soljačić, “Structural colors from Fano resonances,” ACS Photon. 2, 27–32 (2015).
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K. Ueno, A. Inaba, Y. Sano, M. Kondoh, and M. Watanabe, “A soft glassy colloidal array in ionic liquid, which exhibits homogeneous, non-brilliant and angle-independent structural colours,” Chem. Commun. 2009, 3603–3605 (2009).
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M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
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H. M. Whitney, M. Kolle, P. Andrew, L. Chittka, U. Steiner, and B. J. Glover, “Floral iridescence, produced by diffractive optics, acts as a cue for animal pollinators,” Science 323, 130–133 (2009).
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R. O. Prum, R. H. Torres, S. Williamson, and J. Dyck, “Coherent light scattering by blue feather barbs,” Nature 396, 28–29 (1998).
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J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2. (Princeton University, 2008).

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P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single Morpho butterfly scales,” Proc. R. Soc. Lond. B 266, 1403–1411 (1999).
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J. D. Forster, H. Noh, S. F. Liew, V. Saranathan, C. F. Schreck, L. Yang, J.-G. Park, R. O. Prum, S. G. J. Mochrie, C. S. O’Hern, H. Cao, and E. R. Dufresne, “Biomimetic isotropic nanostructures for structural coloration,” Adv. Mater. 22, 2939–2944 (2010).
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Yang, S.-M.

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by Morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
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Yin, Y.

H. Kim, J. Ge, J. Kim, S.-e. Choi, H. Lee, H. Lee, W. Park, Y. Yin, and S. Kwon, “Structural colour printing using a magnetically tunable and lithographically fixable photonic crystal,” Nature Photon. 3, 534–540 (2009).
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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).
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S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
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S. Kinoshita, S. Yoshioka, Y. Fujii, and N. Okamoto, “Photophysics of structural color in the Morpho butterflies,” Forma 17, 103–121 (2002).

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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, 213104 (2009).
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K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by Morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
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Zhang, Y.

Zhen, B.

C. W. Hsu, B. Zhen, W. Qiu, O. Shapira, B. G. DeLacy, J. D. Joannopoulos, and M. Soljačić, “Transparent displays enabled by resonant nanoparticle scattering,” Nat. Commun. 5, 3152 (2014).
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Zhen, H.

ACS Appl. Mater. Inter. (1)

Y. Takeoka, M. Honda, T. Seki, M. Ishii, and H. Nakamura, “Structural colored liquid membrane without angle dependence,” ACS Appl. Mater. Inter. 1, 982–986 (2009).
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ACS Photon. (1)

Y. Shen, V. Rinnerbauer, I. Wang, V. Stelmakh, J. D. Joannopoulos, and M. Soljačić, “Structural colors from Fano resonances,” ACS Photon. 2, 27–32 (2015).
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Acta Mech. Sinica (1)

O. Sigmund, “Manufacturing tolerant topology optimization,” Acta Mech. Sinica 25, 227–239 (2009).
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Adv. Mater. (4)

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by Morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
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J. D. Forster, H. Noh, S. F. Liew, V. Saranathan, C. F. Schreck, L. Yang, J.-G. Park, R. O. Prum, S. G. J. Mochrie, C. S. O’Hern, H. Cao, and E. R. Dufresne, “Biomimetic isotropic nanostructures for structural coloration,” Adv. Mater. 22, 2939–2944 (2010).
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H. Noh, S. F. Liew, V. Saranathan, S. G. J. Mochrie, R. O. Prum, E. R. Dufresne, and H. Cao, “How noniridescent colors are generated by quasi-ordered structures of bird feathers,” Adv. Mater. 22, 2871–2880 (2010).
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L. D. Bonifacio, B. V. Lotsch, D. P. Puzzo, F. Scotognella, and G. A. Ozin, “Stacking the nanochemistry deck: Structural and compositional diversity in one-dimensional photonic crystals,” Adv. Mater. 21, 1641–1646 (2009).
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Appl. Opt. (2)

Appl. Phys. Lett. (1)

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, 213104 (2009).
[Crossref]

Chem. Commun. (1)

K. Ueno, A. Inaba, Y. Sano, M. Kondoh, and M. Watanabe, “A soft glassy colloidal array in ionic liquid, which exhibits homogeneous, non-brilliant and angle-independent structural colours,” Chem. Commun. 2009, 3603–3605 (2009).
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Chem. Phys. Chem (1)

M. Harun-Ur-Rashid, A. BinImran, T. Seki, M. Ishii, H. Nakamura, and Y. Takeoka, “Angle-independent structural color in colloidal amorphous arrays,” Chem. Phys. Chem 11, 579–583 (2010).
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Eng. Optim. (1)

A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Eng. Optim. 41, 365–384 (2009).
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Forma (1)

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

J. Appl. Phys. (1)

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
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J. Mater. Chem. (1)

Y. Takeoka, “Angle-independent structural coloured amorphous arrays,” J. Mater. Chem. 22, 23299–23309 (2012).
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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).
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J. Opt. Soc. Am. B (4)

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P. Kaelo and M. Ali, “Some variants of the controlled random search algorithm for global optimization,” J. Optimiz. Theory App. 130, 253–264 (2006).
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J. Phys. B (1)

A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B 43, 163001 (2010).
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J. Phys. Chem. C (1)

Z. Ruan and S. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114, 7324–7329 (2010).
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J. R. Soc. Interface (1)

V. Saranathan, J. D. Forster, H. Noh, S.-F. Liew, S. G. J. Mochrie, H. Cao, E. R. Dufresne, and R. O. Prum, “Structure and optical function of amorphous photonic nanostructures from avian feather barbs: a comparative small angle X-ray scattering (SAXS) analysis of 230 bird species,” J. R. Soc. Interface 9, 2563–2580 (2012).
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Nano Lett. (1)

C. W. Hsu, B. G. DeLacy, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Theoretical criteria for scattering dark states in nanostructured particles,” Nano Lett. 14, 2783–2788 (2014).
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Nano. Lett. (1)

L. Cao, P. Fan, E. S. Barnard, A. M. Brown, and M. L. Brongersma, “Tuning the color of silicon nanostructures,” Nano. Lett. 10, 2649–2654 (2010).
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Nat. Commun. (1)

C. W. Hsu, B. Zhen, W. Qiu, O. Shapira, B. G. DeLacy, J. D. Joannopoulos, and M. Soljačić, “Transparent displays enabled by resonant nanoparticle scattering,” Nat. Commun. 5, 3152 (2014).
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Nat. Mater. (1)

M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444–447 (2004).
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Nature (1)

R. O. Prum, R. H. Torres, S. Williamson, and J. Dyck, “Coherent light scattering by blue feather barbs,” Nature 396, 28–29 (1998).
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Nature Mater. (1)

M. A. Kats, R. Blanchard, P. Genevet, and F. Capasso, “Nanometre optical coatings based on strong interference effects in highly absorbing media,” Nature Mater. 12, 20–24 (2013).
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A. C. Arsenault, D. P. Puzzo, I. Manners, and G. A. Ozin, “Photonic-crystal full-colour displays,” Nature Photon. 1, 468–472 (2007).
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R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, K. Dovidenko, J. R. Cournoyer, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nature Photon. 1, 123–128 (2007).
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H. Kim, J. Ge, J. Kim, S.-e. Choi, H. Lee, H. Lee, W. Park, Y. Yin, and S. Kwon, “Structural colour printing using a magnetically tunable and lithographically fixable photonic crystal,” Nature Photon. 3, 534–540 (2009).
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Oper. Res. (1)

H. Men, R. M. Freund, N. C. Nguyen, J. Saa-Seoane, and J. Peraire, “Fabrication-adaptive optimization with an application to photonic crystal design,” Oper. Res. 62, 418–434 (2014).
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Opt. Mater. Express (1)

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Y. Elesin, B. Lazarov, J. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photon. Nanostruct. 10, 153–165 (2012).
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Phys. Rev. A (1)

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
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Phys. Rev. E (1)

S. Magkiriadou, J.-G. Park, Y.-S. Kim, and V. N. Manoharan, “Absence of red structural color in photonic glasses, bird feathers, and certain beetles,” Phys. Rev. E 90, 062302 (2014).
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D. A. B. Miller, “Fundamental limit to linear one-dimensional slow light structures,” Phys. Rev. Lett. 99, 203903 (2007).
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O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Phys. Rev. Lett. 112, 123903 (2014).
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Proc. Natl. Acad. Sci. U.S.A. (1)

J. Bauer, S. Hengsbach, I. Tesari, R. Schwaiger, and O. Kraft, “High-strength cellular ceramic composites with 3d microarchitecture,” Proc. Natl. Acad. Sci. U.S.A. 111, 2453–2458 (2014).
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Proc. R. Soc. Lond. B (1)

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single Morpho butterfly scales,” Proc. R. Soc. Lond. B 266, 1403–1411 (1999).
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Rep. Prog. Phys. (1)

S. Kinoshita, S. Yoshioka, and J. Miyazaki, “Physics of structural colors,” Rep. Prog. Phys. 71, 076401 (2008).
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Sci. Rep. (1)

V. R. Shrestha, S.-S. Lee, E.-S. Kim, and D.-Y. Choi, “Non-iridescent transmissive structural color filter featuring highly efficient transmission and high excitation purity,” Sci. Rep. 4, 4921 (2014).
[Crossref] [PubMed]

Science (2)

H. M. Whitney, M. Kolle, P. Andrew, L. Chittka, U. Steiner, and B. J. Glover, “Floral iridescence, produced by diffractive optics, acts as a cue for animal pollinators,” Science 323, 130–133 (2009).
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L. Li, R. R. Gattass, E. Gershgoren, H. Hwang, and J. T. Fourkas, “Achieving λ/20 resolution by one-color initiation and deactivation of polymerization,” Science 324, 910–913 (2009).
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SIAM J. Optimiz. (1)

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optimiz. 12, 555–573 (2002).
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E. R. Dufresne, H. Noh, V. Saranathan, S. G. J. Mochrie, H. Cao, and R. O. Prum, “Self-assembly of amorphous biophotonic nanostructures by phase separation,” Soft Matter 5, 1792–1795 (2009).
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M. T. H. Reid, Scuff-EM: Free, open-source boundary-element software, http://homerreid.com/scuff-EM .

Supplementary Material (1)

» Media 1: MP4 (2694 KB)     

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Figures (4)

Fig. 1
Fig. 1

Illustration of the desired scattering characteristics. (a) Light coming from the projector, which is monochromatic at wavelength λ0 and incident from a specific direction n ^ in, is scattered strongly into all viewing angles. (b) Ambient white light, which is broad band and incident from all directions (illustrated with different arrows), passes through with little scattering.

Fig. 2
Fig. 2

The evolution of the FOM for one instance of the step-wise optimization procedure. Starting from vacuum, rings are added one by one, with global optimization used to find parameters of the new ring and local optimization used to relax the whole structure. Insets show the intermediate and final configurations with 4, 15, and 40 rings, all drawn to the same length scale. Each ring is visualized as a torus with major radius ρj and volume proportional to weight wj, positioned at height zj; all insets are drawn to the same length scale. Media 1 animates this evolution of the configuration together with the corresponding scattering-angle-resolved spectrum of S(q).

Fig. 3
Fig. 3

Structure factor S(q) for the 40-ring configuration optimized for wavelength-selective scattering at λ0 = 600 nm and viewing-angle independence across 0 to 90 degrees. (a) Average value (solid line) and the 10th and the 90th percentiles (bars) with respect to the viewing angles. (b) Scattering-angle-resolved spectrum of S(q), for light incident along the z-axis. The y-axis scale accounts for the weight in the solid angle integration, |sinθdθ| = |d(cosθ)|.

Fig. 4
Fig. 4

Comparison between the S(q) model and the exact scattering response calculated using the boundary-element method (BEM) for a 10-ring configuration. (a) Scattering-angle-resolved spectrum of S(q) (in arbitrary units). (b) Normalized differential scattering cross section, (dσ/dΩ)/k4 (in arbitrary units), calculated using BEM for the corresponding system of dielectric rings (ε = 1.2) in air with incident light polarized perpendicular to the scattering plane. (c) The surface mesh used in the BEM calculation.

Equations (6)

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d σ d Ω = k 4 | e ^ in e ^ out * | 2 S ( q ) ,
S ( q ) = | j α j e i q r j | 2 .
q = k ( n ^ in n ^ out ) ,
FOM = A B + C ,
A = S ( λ 0 ) n ^ out , B = 1 λ 2 λ 1 λ 1 λ 2 d λ S ( λ ) n ^ out , C = max λ [ λ 1 , λ 2 ] S ( λ ) n ^ out S ( λ 0 ) n ^ out 2 ,
S ( q ) = | j w j e i q z z j J 0 ( q ρ ρ j ) | 2 ,

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