Abstract

The male Rajah Brooke’s birdwing butterfly, Trogonoptera brookiana, has black wings with bright green stripes, and the unique microstructure in the wing scales causes wavelength-selective reflection. It has been reported that the reflectance spectrum has several peaks in the visible wavelength range. However, there has been little progress in the interpretation of the spectral shape, and questions remain unanswered. For example, what are the physical origins of the observed reflectance peaks, and how are their wavelengths determined? To answer these questions, we performed a detailed analysis of the photonic structure of the wing scale of Trogonoptera brookiana. The reflectance spectrum also shows strong polarization dependence. This paper describes the analysis for TM polarization, which is perpendicular to the longitudinal ridges on the scale. We first constructed a realistic structural model that reproduced the experimentally determined reflectance spectrum. We then simplified the model and calculated the reflectance spectrum while varying several structural parameters. For three of the four observed spectral peaks, our calculations revealed the reflection paths for constructive interference to explain the peak wavelengths. A possible origin of the fourth peak is discussed. Such detailed understanding of natural photonic structures can inspire optical component design.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (1)

B. D. Wilts, B. Apeleo Zubiri, M. A. Klatt, B. Butz, M. G. Fischer, S. T. Kelly, E. Spiecker, U. Steiner, and G. E. Schröder-Turk, “Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development,” Sci. Adv. 3(4), e1603119 (2017).
[Crossref]

2016 (4)

A. Singer, L. Boucheron, S. H. Dietze, K. E. Jensen, D. Vine, I. McNulty, E. R. Dufresne, R. O. Prum, S. G. J. Mochrie, and O. G. Shpyrko, “Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales,” Sci. Adv. 2(6), e1600149 (2016).
[Crossref]

B. D. Wilts, M. A. Giraldo, and D. G. Stavenga, “Unique wing scale photonics of male rajah brooke’s birdwing butterflies,” Front. Zool. 13(1), 36 (2016).
[Crossref]

M. Giraldo and D. Stavenga, “Brilliant iridescence of morpho butterfly wing scales is due to both a thin film lower lamina and a multilayered upper lamina,” J. Comp. Physiol. A 202(5), 381–388 (2016).
[Crossref]

M. Giraldo, S. Yoshioka, C. Liu, and D. Stavenga, “Coloration mechanisms and phylogeny of morpho butterflies,” J. Exp. Biol. 219(24), 3936–3944 (2016).
[Crossref]

2015 (1)

B. D. Wilts, A. Matsushita, K. Arikawa, and D. G. Stavenga, “Spectrally tuned structural and pigmentary coloration of birdwing butterfly wing scales,” J. R. Soc., Interface 12(111), 20150717 (2015).
[Crossref]

2014 (2)

K. Zhang, S. Zhou, Y. Tang, G. Wang, H. Zhou, T. Fan, and D. Zhang, “Polarization-sensitive color in iridescent scales of butterfly ornithoptera,” RSC Adv. 4(94), 51865–51871 (2014).
[Crossref]

S. Yoshioka, H. Fujita, S. Kinoshita, and B. Matsuhana, “Alignment of crystal orientations of the multi-domain photonic crystals in parides sesostris wing scales,” J. R. Soc., Interface 11(92), 20131029 (2014).
[Crossref]

2013 (1)

S. Yoshioka, Y. Shimizu, S. Kinoshita, and B. Matsuhana, “Structural color of a lycaenid butterfly: analysis of an aperiodic multilayer structure,” Bioinspiration Biomimetics 8(4), 045001 (2013).
[Crossref]

2012 (1)

D. G. Stavenga, H. L. Leertouwer, T. Hariyama, H. A. De Raedt, and B. D. Wilts, “Sexual dichromatism of the damselfly calopteryx japonica caused by a melanin-chitin multilayer in the male wing veins,” PLoS One 7(11), e49743 (2012).
[Crossref]

2011 (5)

S. Yoshioka and S. Kinoshita, “Direct determination of the refractive index of natural multilayer systems,” Phys. Rev. E 83(5), 051917 (2011).
[Crossref]

S. Yoshioka, B. Matsuhana, S. Tanaka, Y. Inouye, N. Oshima, and S. Kinoshita, “Mechanism of variable structural colour in the neon tetra: quantitative evaluation of the venetian blind model,” J. R. Soc., Interface 8(54), 56–66 (2011).
[Crossref]

H. L. Leertouwer, B. D. Wilts, and D. G. Stavenga, “Refractive index and dispersion of butterfly chitin and bird keratin measured by polarizing interference microscopy,” Opt. Express 19(24), 24061–24066 (2011).
[Crossref]

M. Kambe, D. Zhu, and S. Kinoshita, “Origin of retroreflection from a wing of the morpho butterfly,” J. Phys. Soc. Jpn. 80(5), 054801 (2011).
[Crossref]

L. P. Biró and J. P. Vigneron, “Photonic nanoarchitectures in butterflies and beetles: valuable sources for bioinspiration,” Laser Photonics Rev. 5(1), 27–51 (2011).
[Crossref]

2009 (3)

B. D. Wilts, H. L. Leertouwer, and D. G. Stavenga, “Imaging scatterometry and microspectrophotometry of lycaenid butterfly wing scales with perforated multilayers,” J. R. Soc., Interface 6(suppl_2), S185–S192 (2009).
[Crossref]

R. B. Morris, “Iridescence from diffraction structures in the wing scales of callophrys rubi, the green hairstreak,” J. Entomol., Ser. A: Gen. Entomol. 49(2), 149–154 (2009).
[Crossref]

D. Zhu, S. Kinoshita, D. Cai, and J. Cole, “Investigation of structural colors in morpho butterflies using the nonstandard-finite-difference time-domain method: Effects of alternately stacked shelves and ridge density,” Phys. Rev. E 80(5), 051924 (2009).
[Crossref]

2008 (3)

S. Yoshioka, T. Nakano, Y. Nozue, and S. Kinoshita, “Coloration using higher order optical interference in the wing pattern of the madagascan sunset moth,” J. R. Soc., Interface 5(21), 457–464 (2008).
[Crossref]

K. Michielsen and D. Stavenga, “Gyroid cuticular structures in butterfly wing scales: biological photonic crystals,” J. R. Soc., Interface 5(18), 85–94 (2008).
[Crossref]

A. Ingram and A. Parker, “A review of the diversity and evolution of photonic structures in butterflies, incorporating the work of john huxley (the natural history museum, london from 1961 to 1990),” Philos. Trans. R. Soc., B 363(1502), 2465–2480 (2008).
[Crossref]

2007 (1)

2006 (2)

D. G. Stavenga, S. Foletti, G. Palasantzas, and K. Arikawa, “Light on the moth-eye corneal nipple array of butterflies,” Proc. R. Soc. London, Ser. B 273(1587), 661–667 (2006).
[Crossref]

R. O. Prum, T. Quinn, and R. H. Torres, “Anatomically diverse butterfly scales all produce structural colours by coherent scattering,” J. Exp. Biol. 209(4), 748–765 (2006).
[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]

2004 (1)

S. Yoshioka and S. Kinoshita, “Wavelength-selective and anisotropic light-diffusing scale on the wing of the morpho butterfly,” Proc. R. Soc. London, Ser. B 271(1539), 581–587 (2004).
[Crossref]

2003 (1)

P. Vukusic and J. Sambles, “Photonic structures in biology,” Nature 424(6950), 852–855 (2003).
[Crossref]

2001 (1)

1999 (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. London, Ser. B 266(1427), 1403–1411 (1999).
[Crossref]

1995 (2)

1993 (1)

1985 (1)

H. Ghiradella, “Structure and Development of Iridescent Lepidopteran Scales: the Papilionidae as a Showcase Family,” Ann. Entomol. Soc. Am. 78(2), 252–264 (1985).
[Crossref]

Apeleo Zubiri, B.

B. D. Wilts, B. Apeleo Zubiri, M. A. Klatt, B. Butz, M. G. Fischer, S. T. Kelly, E. Spiecker, U. Steiner, and G. E. Schröder-Turk, “Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development,” Sci. Adv. 3(4), e1603119 (2017).
[Crossref]

Arikawa, K.

B. D. Wilts, A. Matsushita, K. Arikawa, and D. G. Stavenga, “Spectrally tuned structural and pigmentary coloration of birdwing butterfly wing scales,” J. R. Soc., Interface 12(111), 20150717 (2015).
[Crossref]

D. G. Stavenga, S. Foletti, G. Palasantzas, and K. Arikawa, “Light on the moth-eye corneal nipple array of butterflies,” Proc. R. Soc. London, Ser. B 273(1587), 661–667 (2006).
[Crossref]

Biró, L. P.

L. P. Biró and J. P. Vigneron, “Photonic nanoarchitectures in butterflies and beetles: valuable sources for bioinspiration,” Laser Photonics Rev. 5(1), 27–51 (2011).
[Crossref]

Boucheron, L.

A. Singer, L. Boucheron, S. H. Dietze, K. E. Jensen, D. Vine, I. McNulty, E. R. Dufresne, R. O. Prum, S. G. J. Mochrie, and O. G. Shpyrko, “Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales,” Sci. Adv. 2(6), e1600149 (2016).
[Crossref]

Butz, B.

B. D. Wilts, B. Apeleo Zubiri, M. A. Klatt, B. Butz, M. G. Fischer, S. T. Kelly, E. Spiecker, U. Steiner, and G. E. Schröder-Turk, “Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development,” Sci. Adv. 3(4), e1603119 (2017).
[Crossref]

Cai, D.

D. Zhu, S. Kinoshita, D. Cai, and J. Cole, “Investigation of structural colors in morpho butterflies using the nonstandard-finite-difference time-domain method: Effects of alternately stacked shelves and ridge density,” Phys. Rev. E 80(5), 051924 (2009).
[Crossref]

Cole, J.

D. Zhu, S. Kinoshita, D. Cai, and J. Cole, “Investigation of structural colors in morpho butterflies using the nonstandard-finite-difference time-domain method: Effects of alternately stacked shelves and ridge density,” Phys. Rev. E 80(5), 051924 (2009).
[Crossref]

De Raedt, H. A.

D. G. Stavenga, H. L. Leertouwer, T. Hariyama, H. A. De Raedt, and B. D. Wilts, “Sexual dichromatism of the damselfly calopteryx japonica caused by a melanin-chitin multilayer in the male wing veins,” PLoS One 7(11), e49743 (2012).
[Crossref]

Dietze, S. H.

A. Singer, L. Boucheron, S. H. Dietze, K. E. Jensen, D. Vine, I. McNulty, E. R. Dufresne, R. O. Prum, S. G. J. Mochrie, and O. G. Shpyrko, “Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales,” Sci. Adv. 2(6), e1600149 (2016).
[Crossref]

Dufresne, E. R.

A. Singer, L. Boucheron, S. H. Dietze, K. E. Jensen, D. Vine, I. McNulty, E. R. Dufresne, R. O. Prum, S. G. J. Mochrie, and O. G. Shpyrko, “Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales,” Sci. Adv. 2(6), e1600149 (2016).
[Crossref]

Fan, T.

K. Zhang, S. Zhou, Y. Tang, G. Wang, H. Zhou, T. Fan, and D. Zhang, “Polarization-sensitive color in iridescent scales of butterfly ornithoptera,” RSC Adv. 4(94), 51865–51871 (2014).
[Crossref]

Fischer, M. G.

B. D. Wilts, B. Apeleo Zubiri, M. A. Klatt, B. Butz, M. G. Fischer, S. T. Kelly, E. Spiecker, U. Steiner, and G. E. Schröder-Turk, “Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development,” Sci. Adv. 3(4), e1603119 (2017).
[Crossref]

Foletti, S.

D. G. Stavenga, S. Foletti, G. Palasantzas, and K. Arikawa, “Light on the moth-eye corneal nipple array of butterflies,” Proc. R. Soc. London, Ser. B 273(1587), 661–667 (2006).
[Crossref]

Fujita, H.

S. Yoshioka, H. Fujita, S. Kinoshita, and B. Matsuhana, “Alignment of crystal orientations of the multi-domain photonic crystals in parides sesostris wing scales,” J. R. Soc., Interface 11(92), 20131029 (2014).
[Crossref]

Gaylord, T. K.

Ghiradella, H.

H. Ghiradella, “Structure and Development of Iridescent Lepidopteran Scales: the Papilionidae as a Showcase Family,” Ann. Entomol. Soc. Am. 78(2), 252–264 (1985).
[Crossref]

Giraldo, M.

M. Giraldo and D. Stavenga, “Brilliant iridescence of morpho butterfly wing scales is due to both a thin film lower lamina and a multilayered upper lamina,” J. Comp. Physiol. A 202(5), 381–388 (2016).
[Crossref]

M. Giraldo, S. Yoshioka, C. Liu, and D. Stavenga, “Coloration mechanisms and phylogeny of morpho butterflies,” J. Exp. Biol. 219(24), 3936–3944 (2016).
[Crossref]

Giraldo, M. A.

B. D. Wilts, M. A. Giraldo, and D. G. Stavenga, “Unique wing scale photonics of male rajah brooke’s birdwing butterflies,” Front. Zool. 13(1), 36 (2016).
[Crossref]

Grann, E. B.

Hariyama, T.

D. G. Stavenga, H. L. Leertouwer, T. Hariyama, H. A. De Raedt, and B. D. Wilts, “Sexual dichromatism of the damselfly calopteryx japonica caused by a melanin-chitin multilayer in the male wing veins,” PLoS One 7(11), e49743 (2012).
[Crossref]

Ingram, A.

A. Ingram and A. Parker, “A review of the diversity and evolution of photonic structures in butterflies, incorporating the work of john huxley (the natural history museum, london from 1961 to 1990),” Philos. Trans. R. Soc., B 363(1502), 2465–2480 (2008).
[Crossref]

Inouye, Y.

S. Yoshioka, B. Matsuhana, S. Tanaka, Y. Inouye, N. Oshima, and S. Kinoshita, “Mechanism of variable structural colour in the neon tetra: quantitative evaluation of the venetian blind model,” J. R. Soc., Interface 8(54), 56–66 (2011).
[Crossref]

Jensen, K. E.

A. Singer, L. Boucheron, S. H. Dietze, K. E. Jensen, D. Vine, I. McNulty, E. R. Dufresne, R. O. Prum, S. G. J. Mochrie, and O. G. Shpyrko, “Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales,” Sci. Adv. 2(6), e1600149 (2016).
[Crossref]

Kambe, M.

M. Kambe, D. Zhu, and S. Kinoshita, “Origin of retroreflection from a wing of the morpho butterfly,” J. Phys. Soc. Jpn. 80(5), 054801 (2011).
[Crossref]

Kelly, S. T.

B. D. Wilts, B. Apeleo Zubiri, M. A. Klatt, B. Butz, M. G. Fischer, S. T. Kelly, E. Spiecker, U. Steiner, and G. E. Schröder-Turk, “Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development,” Sci. Adv. 3(4), e1603119 (2017).
[Crossref]

Kinoshita, S.

S. Yoshioka, H. Fujita, S. Kinoshita, and B. Matsuhana, “Alignment of crystal orientations of the multi-domain photonic crystals in parides sesostris wing scales,” J. R. Soc., Interface 11(92), 20131029 (2014).
[Crossref]

S. Yoshioka, Y. Shimizu, S. Kinoshita, and B. Matsuhana, “Structural color of a lycaenid butterfly: analysis of an aperiodic multilayer structure,” Bioinspiration Biomimetics 8(4), 045001 (2013).
[Crossref]

S. Yoshioka, B. Matsuhana, S. Tanaka, Y. Inouye, N. Oshima, and S. Kinoshita, “Mechanism of variable structural colour in the neon tetra: quantitative evaluation of the venetian blind model,” J. R. Soc., Interface 8(54), 56–66 (2011).
[Crossref]

S. Yoshioka and S. Kinoshita, “Direct determination of the refractive index of natural multilayer systems,” Phys. Rev. E 83(5), 051917 (2011).
[Crossref]

M. Kambe, D. Zhu, and S. Kinoshita, “Origin of retroreflection from a wing of the morpho butterfly,” J. Phys. Soc. Jpn. 80(5), 054801 (2011).
[Crossref]

D. Zhu, S. Kinoshita, D. Cai, and J. Cole, “Investigation of structural colors in morpho butterflies using the nonstandard-finite-difference time-domain method: Effects of alternately stacked shelves and ridge density,” Phys. Rev. E 80(5), 051924 (2009).
[Crossref]

S. Yoshioka, T. Nakano, Y. Nozue, and S. Kinoshita, “Coloration using higher order optical interference in the wing pattern of the madagascan sunset moth,” J. R. Soc., Interface 5(21), 457–464 (2008).
[Crossref]

S. Yoshioka and S. Kinoshita, “Polarization-sensitive color mixing in the wing of the madagascan sunset moth,” Opt. Express 15(5), 2691–2701 (2007).
[Crossref]

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. Yoshioka and S. Kinoshita, “Wavelength-selective and anisotropic light-diffusing scale on the wing of the morpho butterfly,” Proc. R. Soc. London, Ser. B 271(1539), 581–587 (2004).
[Crossref]

Klatt, M. A.

B. D. Wilts, B. Apeleo Zubiri, M. A. Klatt, B. Butz, M. G. Fischer, S. T. Kelly, E. Spiecker, U. Steiner, and G. E. Schröder-Turk, “Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development,” Sci. Adv. 3(4), e1603119 (2017).
[Crossref]

Lawrence, C.

Lawrence, C. R.

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single morpho butterfly scales,” Proc. R. Soc. London, Ser. B 266(1427), 1403–1411 (1999).
[Crossref]

Leertouwer, H. L.

D. G. Stavenga, H. L. Leertouwer, T. Hariyama, H. A. De Raedt, and B. D. Wilts, “Sexual dichromatism of the damselfly calopteryx japonica caused by a melanin-chitin multilayer in the male wing veins,” PLoS One 7(11), e49743 (2012).
[Crossref]

H. L. Leertouwer, B. D. Wilts, and D. G. Stavenga, “Refractive index and dispersion of butterfly chitin and bird keratin measured by polarizing interference microscopy,” Opt. Express 19(24), 24061–24066 (2011).
[Crossref]

B. D. Wilts, H. L. Leertouwer, and D. G. Stavenga, “Imaging scatterometry and microspectrophotometry of lycaenid butterfly wing scales with perforated multilayers,” J. R. Soc., Interface 6(suppl_2), S185–S192 (2009).
[Crossref]

Liu, C.

M. Giraldo, S. Yoshioka, C. Liu, and D. Stavenga, “Coloration mechanisms and phylogeny of morpho butterflies,” J. Exp. Biol. 219(24), 3936–3944 (2016).
[Crossref]

Magnusson, R.

Matsuhana, B.

S. Yoshioka, H. Fujita, S. Kinoshita, and B. Matsuhana, “Alignment of crystal orientations of the multi-domain photonic crystals in parides sesostris wing scales,” J. R. Soc., Interface 11(92), 20131029 (2014).
[Crossref]

S. Yoshioka, Y. Shimizu, S. Kinoshita, and B. Matsuhana, “Structural color of a lycaenid butterfly: analysis of an aperiodic multilayer structure,” Bioinspiration Biomimetics 8(4), 045001 (2013).
[Crossref]

S. Yoshioka, B. Matsuhana, S. Tanaka, Y. Inouye, N. Oshima, and S. Kinoshita, “Mechanism of variable structural colour in the neon tetra: quantitative evaluation of the venetian blind model,” J. R. Soc., Interface 8(54), 56–66 (2011).
[Crossref]

Matsushita, A.

B. D. Wilts, A. Matsushita, K. Arikawa, and D. G. Stavenga, “Spectrally tuned structural and pigmentary coloration of birdwing butterfly wing scales,” J. R. Soc., Interface 12(111), 20150717 (2015).
[Crossref]

McNulty, I.

A. Singer, L. Boucheron, S. H. Dietze, K. E. Jensen, D. Vine, I. McNulty, E. R. Dufresne, R. O. Prum, S. G. J. Mochrie, and O. G. Shpyrko, “Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales,” Sci. Adv. 2(6), e1600149 (2016).
[Crossref]

Michielsen, K.

K. Michielsen and D. Stavenga, “Gyroid cuticular structures in butterfly wing scales: biological photonic crystals,” J. R. Soc., Interface 5(18), 85–94 (2008).
[Crossref]

Mochrie, S. G. J.

A. Singer, L. Boucheron, S. H. Dietze, K. E. Jensen, D. Vine, I. McNulty, E. R. Dufresne, R. O. Prum, S. G. J. Mochrie, and O. G. Shpyrko, “Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales,” Sci. Adv. 2(6), e1600149 (2016).
[Crossref]

Moharam, M. G.

Morris, R. B.

R. B. Morris, “Iridescence from diffraction structures in the wing scales of callophrys rubi, the green hairstreak,” J. Entomol., Ser. A: Gen. Entomol. 49(2), 149–154 (2009).
[Crossref]

Nakano, T.

S. Yoshioka, T. Nakano, Y. Nozue, and S. Kinoshita, “Coloration using higher order optical interference in the wing pattern of the madagascan sunset moth,” J. R. Soc., Interface 5(21), 457–464 (2008).
[Crossref]

Nozue, Y.

S. Yoshioka, T. Nakano, Y. Nozue, and S. Kinoshita, “Coloration using higher order optical interference in the wing pattern of the madagascan sunset moth,” J. R. Soc., Interface 5(21), 457–464 (2008).
[Crossref]

Oshima, N.

S. Yoshioka, B. Matsuhana, S. Tanaka, Y. Inouye, N. Oshima, and S. Kinoshita, “Mechanism of variable structural colour in the neon tetra: quantitative evaluation of the venetian blind model,” J. R. Soc., Interface 8(54), 56–66 (2011).
[Crossref]

Palasantzas, G.

D. G. Stavenga, S. Foletti, G. Palasantzas, and K. Arikawa, “Light on the moth-eye corneal nipple array of butterflies,” Proc. R. Soc. London, Ser. B 273(1587), 661–667 (2006).
[Crossref]

Parker, A.

A. Ingram and A. Parker, “A review of the diversity and evolution of photonic structures in butterflies, incorporating the work of john huxley (the natural history museum, london from 1961 to 1990),” Philos. Trans. R. Soc., B 363(1502), 2465–2480 (2008).
[Crossref]

Pommet, D. A.

Prum, R. O.

A. Singer, L. Boucheron, S. H. Dietze, K. E. Jensen, D. Vine, I. McNulty, E. R. Dufresne, R. O. Prum, S. G. J. Mochrie, and O. G. Shpyrko, “Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales,” Sci. Adv. 2(6), e1600149 (2016).
[Crossref]

R. O. Prum, T. Quinn, and R. H. Torres, “Anatomically diverse butterfly scales all produce structural colours by coherent scattering,” J. Exp. Biol. 209(4), 748–765 (2006).
[Crossref]

Quinn, T.

R. O. Prum, T. Quinn, and R. H. Torres, “Anatomically diverse butterfly scales all produce structural colours by coherent scattering,” J. Exp. Biol. 209(4), 748–765 (2006).
[Crossref]

Sambles, J.

P. Vukusic and J. Sambles, “Photonic structures in biology,” Nature 424(6950), 852–855 (2003).
[Crossref]

Sambles, J. R.

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single morpho butterfly scales,” Proc. R. Soc. London, Ser. B 266(1427), 1403–1411 (1999).
[Crossref]

Sambles, R.

Schröder-Turk, G. E.

B. D. Wilts, B. Apeleo Zubiri, M. A. Klatt, B. Butz, M. G. Fischer, S. T. Kelly, E. Spiecker, U. Steiner, and G. E. Schröder-Turk, “Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development,” Sci. Adv. 3(4), e1603119 (2017).
[Crossref]

Shimizu, Y.

S. Yoshioka, Y. Shimizu, S. Kinoshita, and B. Matsuhana, “Structural color of a lycaenid butterfly: analysis of an aperiodic multilayer structure,” Bioinspiration Biomimetics 8(4), 045001 (2013).
[Crossref]

Shpyrko, O. G.

A. Singer, L. Boucheron, S. H. Dietze, K. E. Jensen, D. Vine, I. McNulty, E. R. Dufresne, R. O. Prum, S. G. J. Mochrie, and O. G. Shpyrko, “Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales,” Sci. Adv. 2(6), e1600149 (2016).
[Crossref]

Singer, A.

A. Singer, L. Boucheron, S. H. Dietze, K. E. Jensen, D. Vine, I. McNulty, E. R. Dufresne, R. O. Prum, S. G. J. Mochrie, and O. G. Shpyrko, “Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales,” Sci. Adv. 2(6), e1600149 (2016).
[Crossref]

Spiecker, E.

B. D. Wilts, B. Apeleo Zubiri, M. A. Klatt, B. Butz, M. G. Fischer, S. T. Kelly, E. Spiecker, U. Steiner, and G. E. Schröder-Turk, “Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development,” Sci. Adv. 3(4), e1603119 (2017).
[Crossref]

Stavenga, D.

M. Giraldo, S. Yoshioka, C. Liu, and D. Stavenga, “Coloration mechanisms and phylogeny of morpho butterflies,” J. Exp. Biol. 219(24), 3936–3944 (2016).
[Crossref]

M. Giraldo and D. Stavenga, “Brilliant iridescence of morpho butterfly wing scales is due to both a thin film lower lamina and a multilayered upper lamina,” J. Comp. Physiol. A 202(5), 381–388 (2016).
[Crossref]

K. Michielsen and D. Stavenga, “Gyroid cuticular structures in butterfly wing scales: biological photonic crystals,” J. R. Soc., Interface 5(18), 85–94 (2008).
[Crossref]

Stavenga, D. G.

B. D. Wilts, M. A. Giraldo, and D. G. Stavenga, “Unique wing scale photonics of male rajah brooke’s birdwing butterflies,” Front. Zool. 13(1), 36 (2016).
[Crossref]

B. D. Wilts, A. Matsushita, K. Arikawa, and D. G. Stavenga, “Spectrally tuned structural and pigmentary coloration of birdwing butterfly wing scales,” J. R. Soc., Interface 12(111), 20150717 (2015).
[Crossref]

D. G. Stavenga, H. L. Leertouwer, T. Hariyama, H. A. De Raedt, and B. D. Wilts, “Sexual dichromatism of the damselfly calopteryx japonica caused by a melanin-chitin multilayer in the male wing veins,” PLoS One 7(11), e49743 (2012).
[Crossref]

H. L. Leertouwer, B. D. Wilts, and D. G. Stavenga, “Refractive index and dispersion of butterfly chitin and bird keratin measured by polarizing interference microscopy,” Opt. Express 19(24), 24061–24066 (2011).
[Crossref]

B. D. Wilts, H. L. Leertouwer, and D. G. Stavenga, “Imaging scatterometry and microspectrophotometry of lycaenid butterfly wing scales with perforated multilayers,” J. R. Soc., Interface 6(suppl_2), S185–S192 (2009).
[Crossref]

D. G. Stavenga, S. Foletti, G. Palasantzas, and K. Arikawa, “Light on the moth-eye corneal nipple array of butterflies,” Proc. R. Soc. London, Ser. B 273(1587), 661–667 (2006).
[Crossref]

Steiner, U.

B. D. Wilts, B. Apeleo Zubiri, M. A. Klatt, B. Butz, M. G. Fischer, S. T. Kelly, E. Spiecker, U. Steiner, and G. E. Schröder-Turk, “Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development,” Sci. Adv. 3(4), e1603119 (2017).
[Crossref]

Tanaka, S.

S. Yoshioka, B. Matsuhana, S. Tanaka, Y. Inouye, N. Oshima, and S. Kinoshita, “Mechanism of variable structural colour in the neon tetra: quantitative evaluation of the venetian blind model,” J. R. Soc., Interface 8(54), 56–66 (2011).
[Crossref]

Tang, Y.

K. Zhang, S. Zhou, Y. Tang, G. Wang, H. Zhou, T. Fan, and D. Zhang, “Polarization-sensitive color in iridescent scales of butterfly ornithoptera,” RSC Adv. 4(94), 51865–51871 (2014).
[Crossref]

Torres, R. H.

R. O. Prum, T. Quinn, and R. H. Torres, “Anatomically diverse butterfly scales all produce structural colours by coherent scattering,” J. Exp. Biol. 209(4), 748–765 (2006).
[Crossref]

Vigneron, J. P.

L. P. Biró and J. P. Vigneron, “Photonic nanoarchitectures in butterflies and beetles: valuable sources for bioinspiration,” Laser Photonics Rev. 5(1), 27–51 (2011).
[Crossref]

Vine, D.

A. Singer, L. Boucheron, S. H. Dietze, K. E. Jensen, D. Vine, I. McNulty, E. R. Dufresne, R. O. Prum, S. G. J. Mochrie, and O. G. Shpyrko, “Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales,” Sci. Adv. 2(6), e1600149 (2016).
[Crossref]

Vukusic, P.

P. Vukusic and J. Sambles, “Photonic structures in biology,” Nature 424(6950), 852–855 (2003).
[Crossref]

P. Vukusic, R. Sambles, C. Lawrence, and G. Wakely, “Sculpted-multilayer optical effects in two species of papilio butterfly,” Appl. Opt. 40(7), 1116–1125 (2001).
[Crossref]

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single morpho butterfly scales,” Proc. R. Soc. London, Ser. B 266(1427), 1403–1411 (1999).
[Crossref]

Wakely, G.

Wang, G.

K. Zhang, S. Zhou, Y. Tang, G. Wang, H. Zhou, T. Fan, and D. Zhang, “Polarization-sensitive color in iridescent scales of butterfly ornithoptera,” RSC Adv. 4(94), 51865–51871 (2014).
[Crossref]

Wang, S. S.

Wilts, B. D.

B. D. Wilts, B. Apeleo Zubiri, M. A. Klatt, B. Butz, M. G. Fischer, S. T. Kelly, E. Spiecker, U. Steiner, and G. E. Schröder-Turk, “Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development,” Sci. Adv. 3(4), e1603119 (2017).
[Crossref]

B. D. Wilts, M. A. Giraldo, and D. G. Stavenga, “Unique wing scale photonics of male rajah brooke’s birdwing butterflies,” Front. Zool. 13(1), 36 (2016).
[Crossref]

B. D. Wilts, A. Matsushita, K. Arikawa, and D. G. Stavenga, “Spectrally tuned structural and pigmentary coloration of birdwing butterfly wing scales,” J. R. Soc., Interface 12(111), 20150717 (2015).
[Crossref]

D. G. Stavenga, H. L. Leertouwer, T. Hariyama, H. A. De Raedt, and B. D. Wilts, “Sexual dichromatism of the damselfly calopteryx japonica caused by a melanin-chitin multilayer in the male wing veins,” PLoS One 7(11), e49743 (2012).
[Crossref]

H. L. Leertouwer, B. D. Wilts, and D. G. Stavenga, “Refractive index and dispersion of butterfly chitin and bird keratin measured by polarizing interference microscopy,” Opt. Express 19(24), 24061–24066 (2011).
[Crossref]

B. D. Wilts, H. L. Leertouwer, and D. G. Stavenga, “Imaging scatterometry and microspectrophotometry of lycaenid butterfly wing scales with perforated multilayers,” J. R. Soc., Interface 6(suppl_2), S185–S192 (2009).
[Crossref]

Wootton, R. J.

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single morpho butterfly scales,” Proc. R. Soc. London, Ser. B 266(1427), 1403–1411 (1999).
[Crossref]

Yoshioka, S.

M. Giraldo, S. Yoshioka, C. Liu, and D. Stavenga, “Coloration mechanisms and phylogeny of morpho butterflies,” J. Exp. Biol. 219(24), 3936–3944 (2016).
[Crossref]

S. Yoshioka, H. Fujita, S. Kinoshita, and B. Matsuhana, “Alignment of crystal orientations of the multi-domain photonic crystals in parides sesostris wing scales,” J. R. Soc., Interface 11(92), 20131029 (2014).
[Crossref]

S. Yoshioka, Y. Shimizu, S. Kinoshita, and B. Matsuhana, “Structural color of a lycaenid butterfly: analysis of an aperiodic multilayer structure,” Bioinspiration Biomimetics 8(4), 045001 (2013).
[Crossref]

S. Yoshioka, B. Matsuhana, S. Tanaka, Y. Inouye, N. Oshima, and S. Kinoshita, “Mechanism of variable structural colour in the neon tetra: quantitative evaluation of the venetian blind model,” J. R. Soc., Interface 8(54), 56–66 (2011).
[Crossref]

S. Yoshioka and S. Kinoshita, “Direct determination of the refractive index of natural multilayer systems,” Phys. Rev. E 83(5), 051917 (2011).
[Crossref]

S. Yoshioka, T. Nakano, Y. Nozue, and S. Kinoshita, “Coloration using higher order optical interference in the wing pattern of the madagascan sunset moth,” J. R. Soc., Interface 5(21), 457–464 (2008).
[Crossref]

S. Yoshioka and S. Kinoshita, “Polarization-sensitive color mixing in the wing of the madagascan sunset moth,” Opt. Express 15(5), 2691–2701 (2007).
[Crossref]

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. Yoshioka and S. Kinoshita, “Wavelength-selective and anisotropic light-diffusing scale on the wing of the morpho butterfly,” Proc. R. Soc. London, Ser. B 271(1539), 581–587 (2004).
[Crossref]

Zhang, D.

K. Zhang, S. Zhou, Y. Tang, G. Wang, H. Zhou, T. Fan, and D. Zhang, “Polarization-sensitive color in iridescent scales of butterfly ornithoptera,” RSC Adv. 4(94), 51865–51871 (2014).
[Crossref]

Zhang, K.

K. Zhang, S. Zhou, Y. Tang, G. Wang, H. Zhou, T. Fan, and D. Zhang, “Polarization-sensitive color in iridescent scales of butterfly ornithoptera,” RSC Adv. 4(94), 51865–51871 (2014).
[Crossref]

Zhou, H.

K. Zhang, S. Zhou, Y. Tang, G. Wang, H. Zhou, T. Fan, and D. Zhang, “Polarization-sensitive color in iridescent scales of butterfly ornithoptera,” RSC Adv. 4(94), 51865–51871 (2014).
[Crossref]

Zhou, S.

K. Zhang, S. Zhou, Y. Tang, G. Wang, H. Zhou, T. Fan, and D. Zhang, “Polarization-sensitive color in iridescent scales of butterfly ornithoptera,” RSC Adv. 4(94), 51865–51871 (2014).
[Crossref]

Zhu, D.

M. Kambe, D. Zhu, and S. Kinoshita, “Origin of retroreflection from a wing of the morpho butterfly,” J. Phys. Soc. Jpn. 80(5), 054801 (2011).
[Crossref]

D. Zhu, S. Kinoshita, D. Cai, and J. Cole, “Investigation of structural colors in morpho butterflies using the nonstandard-finite-difference time-domain method: Effects of alternately stacked shelves and ridge density,” Phys. Rev. E 80(5), 051924 (2009).
[Crossref]

Ann. Entomol. Soc. Am. (1)

H. Ghiradella, “Structure and Development of Iridescent Lepidopteran Scales: the Papilionidae as a Showcase Family,” Ann. Entomol. Soc. Am. 78(2), 252–264 (1985).
[Crossref]

Appl. Opt. (2)

Bioinspiration Biomimetics (1)

S. Yoshioka, Y. Shimizu, S. Kinoshita, and B. Matsuhana, “Structural color of a lycaenid butterfly: analysis of an aperiodic multilayer structure,” Bioinspiration Biomimetics 8(4), 045001 (2013).
[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]

Front. Zool. (1)

B. D. Wilts, M. A. Giraldo, and D. G. Stavenga, “Unique wing scale photonics of male rajah brooke’s birdwing butterflies,” Front. Zool. 13(1), 36 (2016).
[Crossref]

J. Comp. Physiol. A (1)

M. Giraldo and D. Stavenga, “Brilliant iridescence of morpho butterfly wing scales is due to both a thin film lower lamina and a multilayered upper lamina,” J. Comp. Physiol. A 202(5), 381–388 (2016).
[Crossref]

J. Entomol., Ser. A: Gen. Entomol. (1)

R. B. Morris, “Iridescence from diffraction structures in the wing scales of callophrys rubi, the green hairstreak,” J. Entomol., Ser. A: Gen. Entomol. 49(2), 149–154 (2009).
[Crossref]

J. Exp. Biol. (2)

M. Giraldo, S. Yoshioka, C. Liu, and D. Stavenga, “Coloration mechanisms and phylogeny of morpho butterflies,” J. Exp. Biol. 219(24), 3936–3944 (2016).
[Crossref]

R. O. Prum, T. Quinn, and R. H. Torres, “Anatomically diverse butterfly scales all produce structural colours by coherent scattering,” J. Exp. Biol. 209(4), 748–765 (2006).
[Crossref]

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

J. Phys. Soc. Jpn. (1)

M. Kambe, D. Zhu, and S. Kinoshita, “Origin of retroreflection from a wing of the morpho butterfly,” J. Phys. Soc. Jpn. 80(5), 054801 (2011).
[Crossref]

J. R. Soc., Interface (6)

S. Yoshioka, B. Matsuhana, S. Tanaka, Y. Inouye, N. Oshima, and S. Kinoshita, “Mechanism of variable structural colour in the neon tetra: quantitative evaluation of the venetian blind model,” J. R. Soc., Interface 8(54), 56–66 (2011).
[Crossref]

S. Yoshioka, T. Nakano, Y. Nozue, and S. Kinoshita, “Coloration using higher order optical interference in the wing pattern of the madagascan sunset moth,” J. R. Soc., Interface 5(21), 457–464 (2008).
[Crossref]

B. D. Wilts, H. L. Leertouwer, and D. G. Stavenga, “Imaging scatterometry and microspectrophotometry of lycaenid butterfly wing scales with perforated multilayers,” J. R. Soc., Interface 6(suppl_2), S185–S192 (2009).
[Crossref]

K. Michielsen and D. Stavenga, “Gyroid cuticular structures in butterfly wing scales: biological photonic crystals,” J. R. Soc., Interface 5(18), 85–94 (2008).
[Crossref]

S. Yoshioka, H. Fujita, S. Kinoshita, and B. Matsuhana, “Alignment of crystal orientations of the multi-domain photonic crystals in parides sesostris wing scales,” J. R. Soc., Interface 11(92), 20131029 (2014).
[Crossref]

B. D. Wilts, A. Matsushita, K. Arikawa, and D. G. Stavenga, “Spectrally tuned structural and pigmentary coloration of birdwing butterfly wing scales,” J. R. Soc., Interface 12(111), 20150717 (2015).
[Crossref]

Laser Photonics Rev. (1)

L. P. Biró and J. P. Vigneron, “Photonic nanoarchitectures in butterflies and beetles: valuable sources for bioinspiration,” Laser Photonics Rev. 5(1), 27–51 (2011).
[Crossref]

Nature (1)

P. Vukusic and J. Sambles, “Photonic structures in biology,” Nature 424(6950), 852–855 (2003).
[Crossref]

Opt. Express (2)

Philos. Trans. R. Soc., B (1)

A. Ingram and A. Parker, “A review of the diversity and evolution of photonic structures in butterflies, incorporating the work of john huxley (the natural history museum, london from 1961 to 1990),” Philos. Trans. R. Soc., B 363(1502), 2465–2480 (2008).
[Crossref]

Phys. Rev. E (2)

D. Zhu, S. Kinoshita, D. Cai, and J. Cole, “Investigation of structural colors in morpho butterflies using the nonstandard-finite-difference time-domain method: Effects of alternately stacked shelves and ridge density,” Phys. Rev. E 80(5), 051924 (2009).
[Crossref]

S. Yoshioka and S. Kinoshita, “Direct determination of the refractive index of natural multilayer systems,” Phys. Rev. E 83(5), 051917 (2011).
[Crossref]

PLoS One (1)

D. G. Stavenga, H. L. Leertouwer, T. Hariyama, H. A. De Raedt, and B. D. Wilts, “Sexual dichromatism of the damselfly calopteryx japonica caused by a melanin-chitin multilayer in the male wing veins,” PLoS One 7(11), e49743 (2012).
[Crossref]

Proc. R. Soc. London, Ser. B (3)

D. G. Stavenga, S. Foletti, G. Palasantzas, and K. Arikawa, “Light on the moth-eye corneal nipple array of butterflies,” Proc. R. Soc. London, Ser. B 273(1587), 661–667 (2006).
[Crossref]

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single morpho butterfly scales,” Proc. R. Soc. London, Ser. B 266(1427), 1403–1411 (1999).
[Crossref]

S. Yoshioka and S. Kinoshita, “Wavelength-selective and anisotropic light-diffusing scale on the wing of the morpho butterfly,” Proc. R. Soc. London, Ser. B 271(1539), 581–587 (2004).
[Crossref]

RSC Adv. (1)

K. Zhang, S. Zhou, Y. Tang, G. Wang, H. Zhou, T. Fan, and D. Zhang, “Polarization-sensitive color in iridescent scales of butterfly ornithoptera,” RSC Adv. 4(94), 51865–51871 (2014).
[Crossref]

Sci. Adv. (2)

A. Singer, L. Boucheron, S. H. Dietze, K. E. Jensen, D. Vine, I. McNulty, E. R. Dufresne, R. O. Prum, S. G. J. Mochrie, and O. G. Shpyrko, “Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales,” Sci. Adv. 2(6), e1600149 (2016).
[Crossref]

B. D. Wilts, B. Apeleo Zubiri, M. A. Klatt, B. Butz, M. G. Fischer, S. T. Kelly, E. Spiecker, U. Steiner, and G. E. Schröder-Turk, “Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development,” Sci. Adv. 3(4), e1603119 (2017).
[Crossref]

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Figures (9)

Fig. 1.
Fig. 1. Wing scale structure of $Trogonoptera$ $brookiana$. (a) Photograph of the butterfly. (b) SEM image of the scale surface. (c) TEM image of a cross section of the scale. (d) Structural model with dimensions indicated. Nine lamellae are assumed in this model. Scale bar: (a) 1 cm, (b) 10 $\mu$m, and (c) 1 $\mu$m.
Fig. 2.
Fig. 2. Polarization-dependent reflectance spectrum. (a) Experimental results determined by the microspectrophotometer. (b) Zeroth-order diffraction efficiency calculated using the structural model shown in Fig. 1(d). Black and red curves correspond to TM and TE polarizations, respectively. Refractive indices of cuticle and ridge center are assumed to be $n_{\mbox {c}}=1.575$ and $n_{\mbox {rc}}=1.575+0.067i$, respectively.
Fig. 3.
Fig. 3. Structural models and reflectance spectra. (a), (c), and (e): structural models named M0, M1, and M2, respectively. M0 corresponds to the cross section of the model shown in Figs. 1(d). (b), (d), and (f): reflectance spectra for TM polarization corresponding to the models shown in (a), (c), and (e), respectively. In (b) and (d), the zeroth-order diffraction efficiency calculated using RCWA is shown as reflectance. In (f), the red curve is the reflectance calculated using a method for multilayer systems [31], while the black curve is the same spectrum shown in (d). In these calculations, we assume nine lamellae with refractive index $n_{\mbox {c}}=1.575$. The ridge center refractive index $n_{\mbox {rc}}$ is $1.575+0.067i$ in (b) and $1.575+0.01i$ in (d). In (e) (model M2), the two layer types, with thicknesses 125 nm and 96 nm, have refractive indices 1.575 and 1.184, respectively.
Fig. 4.
Fig. 4. Reflectance spectra of model structures with different numbers of layers. (a) A model structure based on M1. (b) Reflectance spectrum (zeroth-order diffraction efficiency) with 9 (black) and 20 (red) layer pairs in the structure shown in (a). Refractive index values $n_{\mbox {c}}=1.575$ and $n_{\mbox {rc}}=1.575+0.01$i are assumed. (c) A more realistic model including triangular structures on the tops of the ridge centers, where $n_{\mbox {c}}=1.575$ and $n_{\mbox {rc}}=1.575+0.067i$ are assumed. (d) Reflectance spectrum (zeroth-order diffraction efficiency) with 9 (black) and 20 (red) layer pairs in the model structure shown in (c).
Fig. 5.
Fig. 5. Reflectance spectra of M1-type structures with different ridge separations. (a) Model structure. The separation $\Lambda$ and the width of the pillar (ridge) $w$ are varied, keeping the ratio $w / \Lambda$ at a constant value of 0.32, obtained using the values $w=277$ nm and $\Lambda =866$ nm (see Fig. 1(d)). The number of layer pairs is assumed to be 20. Assumed refractive index values are $n_{\mbox {c}}=1.575$ and $n_{\mbox {rc}}=1.575+0.01i$. (b) Calculated spectra (zeroth-order diffraction efficiency) for, from top to bottom, $\Lambda$ = 400 nm, 500 nm, 600 nm, and 866 nm. The value of $\Lambda$ is displayed in the top right corner of each plot. The gray vertical lines show the wavelengths calculated by applying the Bragg condition, Eq. (1).
Fig. 6.
Fig. 6. Bragg diffraction from tilted planes. (a) Schematic illustration of reflection. $\theta$ is the angle of incidence on the planes, whose separation distance is $\delta$. (b) Reciprocal representation. The incident and scattered vectors are denoted by $\boldsymbol {k}_{\mbox {i}}$ and $\boldsymbol {k}_{\mbox {s}}$, respectively, and $\boldsymbol {G}$ is the reciprocal vector associated with the reflection. The origin of reciprocal space is denoted by $\Gamma$. (c) Schematic illustration explaining normal reflection.
Fig. 7.
Fig. 7. Diffraction efficiencies for (a) $\Lambda$=500 nm and (b) 866 nm calculated for the model structure shown in Fig. 5(a). The blue and red curves show the zeroth- and first-order diffraction efficiencies, respectively. The black curve shows the sum of the zeroth- and $\pm$1st-order diffraction efficiencies.
Fig. 8.
Fig. 8. (a) Model structure for photonic band diagram. The black rectangle shows the unit cell. The assumed refractive index of the cuticle (grey region) is 1.575. (b) Reciprocal space. The rectangle represents the first Brillouin zone. Point X’ denotes the zone boundary along the $k_y$ direction. (c) Photonic band diagram. The lowest four electromagnetic modes that can symmetrically couple to the external plane wave are shown in color (red or blue), while black curves show the modes that cannot couple. For comparison, the reflectance spectrum is shown on the right. This is the spectrum for 20 layer pairs shown in Fig. 4(d).
Fig. 9.
Fig. 9. Effects of the triangular structure on top of the ridge center. Red and black curves show the zeroth-order diffraction efficiency for the model structure with and without the triangular structure, respectively. The structural models are similar to those shown in Fig. 4(a) and (c), but for 9 layer pairs. Refractive indices $n_{\mbox {c}}=1.575$ and $n_{\mbox {rc}} = 1.575+0.067i$ are assumed in these calculations.

Equations (1)

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m λ B = 2 n ¯ δ cos θ ,