Abstract

Structural colors are common in nature. Generally single feathers or other integuments contain only one structural color, but those of the common bronzewing display a consistent color gradient from blue to red (462-647nm) over the proximo-distal length of individual barbs. We used optical microscopy and macro- and micro-spectrophotometry to characterize this color gradient, and transmission electron microscopy to investigate the nanostructure. Combining optical modeling and experimental results, we demonstrate that the rainbow-like iridescence is caused by multilayer interference from organized arrays of melanosome rods in a keratin matrix and that the color gradient results from subtle shifts in both diameter and spacing of melanosome rods. This result illustrates tight developmental control feathers and may provide inspiration for the design of multi-colored coatings or fibers.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
  37. D. G. Stavenga, H. L. Leertouwer, P. Pirih, and M. F. Wehling, “Imaging scatterometry of butterfly wing scales,”Opt. Express 17(1), 193–202 (2009).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2014

B. D. Wilts, K. Michielsen, H. De Raedt, and D. G. Stavenga, “Sparkling feather reflections of a bird-of-paradise explained by finite-difference time-domain modeling,”Proc. Natl. Acad. Sci. U.S.A. 111(12), 4363–4368 (2014).
[CrossRef] [PubMed]

2013

M. Kolle, A. Lethbridge, M. Kreysing, J. J. Baumberg, J. Aizenberg, and P. Vukusic, “Bio-Inspired Band-Gap Tunable Elastic Optical Multilayer Fibers,”Adv. Mater. 25(15), 2239–2245 (2013).
[CrossRef] [PubMed]

J. Sun, B. Bhushan, and J. Tong, “Structural coloration in nature,”RSC Advances 3(35), 14862–14889 (2013).
[CrossRef]

R. Maia, C. M. Eliason, P. P. Bitton, S. M. Doucet, and M. D. Shawkey, “Pavo: An R Package for the Analysis, Visualization and Organization of Spectral Data,”Methods. Ecol. Evol. 4, 906–913 (2013).

M. R. Nixon, A. G. Orr, and P. Vukusic, “Subtle design changes control the difference in colour reflection from the dorsal and ventral wing-membrane surfaces of the damselfly Matronoides cyaneipennis,”Opt. Express 21(2), 1479–1488 (2013).
[CrossRef] [PubMed]

C. M. Eliason, P.-P. Bitton, and M. D. Shawkey, “How hollow melanosomes affect iridescent colour production in birds,”Proc. Biol. Sci. 280(1767), 20131505 (2013).
[CrossRef] [PubMed]

2012

C. M. Eliason and M. D. Shawkey, “A photonic heterostructure produces diverse iridescent colours in duck wing patches,”J. R. Soc. Interface 9(74), 2279–2289 (2012).
[CrossRef] [PubMed]

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] [PubMed]

S. Vignolini, P. J. Rudall, A. V. Rowland, A. Reed, E. Moyroud, R. B. Faden, J. J. Baumberg, B. J. Glover, and U. Steiner, “Pointillist structural color in Pollia fruit,”Proc. Natl. Acad. Sci. U.S.A. 109(39), 15712–15715 (2012).
[CrossRef] [PubMed]

T. M. Jordan, J. C. Partridge, and N. W. Roberts, “Non-polarizing broadband multilayer reflectors in fish,”Nat. Photonics 6(11), 759–763 (2012).
[CrossRef] [PubMed]

H. K. Snyder, R. Maia, L. D’Alba, A. J. Shultz, K. M. C. Rowe, K. C. Rowe, and M. D. Shawkey, “Iridescent colour production in hairs of blind golden moles (Chrysochloridae),”Biol. Lett. 8(3), 393–396 (2012).
[CrossRef] [PubMed]

R. Maia, R. H. F. Macedo, and M. D. Shawkey, “Nanostructural self-assembly of iridescent feather barbules through depletion attraction of melanosomes during keratinization,”J. R. Soc. Interface 9(69), 734–743 (2012).
[CrossRef] [PubMed]

2011

M. G. Meadows, N. I. Morehouse, R. L. Rutowski, J. M. Douglas, and K. J. McGraw, “Quantifying iridescent coloration in animals: a method for improving repeatability,”Behav. Ecol. Sociobiol. 65(6), 1317–1327 (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] [PubMed]

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

D. G. Stavenga, B. D. Wilts, H. L. Leertouwer, and T. Hariyama, “Polarized iridescence of the multilayered elytra of the Japanese jewel beetle, Chrysochroa fulgidissima,”Philos. Trans. R. Soc. Lond. B 366(1565), 709–723 (2011).
[CrossRef] [PubMed]

D. G. Stavenga, H. L. Leertouwer, N. J. Marshall, and D. Osorio, “Dramatic colour changes in a bird of paradise caused by uniquely structured breast feather barbules,”Proc. Biol. Sci. 278(1715), 2098–2104 (2011).
[CrossRef] [PubMed]

2010

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(26-27), 2871–2880 (2010).
[CrossRef] [PubMed]

2009

A. E. Seago, P. Brady, J.-P. Vigneron, and T. D. Schultz, “Gold bugs and beyond: a review of iridescence and structural colour mechanisms in beetles (Coleoptera),”J. R. Soc. Interface 6(Suppl 2), S165–S184 (2009).
[CrossRef] [PubMed]

M. D. Shawkey, N. I. Morehouse, and P. Vukusic, “A protean palette: colour materials and mixing in birds and butterflies,”J. R. Soc. Interface 6(Suppl 2), S221–S231 (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(5910), 130–133 (2009).
[CrossRef] [PubMed]

D. G. Stavenga, H. L. Leertouwer, P. Pirih, and M. F. Wehling, “Imaging scatterometry of butterfly wing scales,”Opt. Express 17(1), 193–202 (2009).
[CrossRef] [PubMed]

2007

A. Garahan, L. Pilon, J. Yin, and I. Saxena, “Effective optical properties of absorbing nanoporous and nanocomposite thin films,”J. Appl. Phys. 101(1), 014320 (2007).
[CrossRef]

2006

S. M. Doucet, M. D. Shawkey, G. E. Hill, and R. Montgomerie, “Iridescent plumage in satin bowerbirds: structure, mechanisms and nanostructural predictors of individual variation in colour,”J. Exp. Biol. 209(2), 380–390 (2006).
[CrossRef] [PubMed]

H. Yin, L. Shi, J. Sha, Y. Li, Y. Qin, B. Dong, S. Meyer, X. Liu, L. Zhao, and J. Zi, “Iridescence in the neck feathers of domestic pigeons,”Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 051916 (2006).
[CrossRef] [PubMed]

2003

J. Zi, X. D. Yu, Y. Z. Li, X. H. Hu, C. Xu, X. J. Wang, X. H. Liu, and R. T. Fu, “Coloration strategies in peacock feathers,”Proc. Natl. Acad. Sci. U.S.A. 100(22), 12576–12578 (2003).
[CrossRef] [PubMed]

M. D. Shawkey, A. M. Estes, L. M. Siefferman, and G. E. Hill, “Nanostructure predicts intraspecific variation in ultraviolet-blue plumage colour,”Proc. Biol. Sci. 270(1523), 1455–1460 (2003).
[CrossRef] [PubMed]

1999

R. O. Prum, “Development and evolutionary origin of feathers,”J. Exp. Zool. 285(4), 291–306 (1999).
[CrossRef] [PubMed]

1993

G. Jellison., “Data analysis for spectroscopic ellipsometry,”Thin Solid Films 234(1-2), 416–422 (1993).
[CrossRef]

1991

1981

1972

M. F. Land, “The physics and biology of animal reflectors,”Prog. Biophys. Mol. Biol. 24, 75–106 (1972).
[CrossRef] [PubMed]

1970

H. Durrer and W. Villiger, “Schillerradien des Goldkuckucks (Chrysococcyx cupreus (Shaw)) im Elektronenmikroskop,”Z. Zellforsch. Mikrosk. Anat. 109(3), 407–413 (1970).
[CrossRef] [PubMed]

1962

H. Durrer and W. Villiger, “Schillerfarben der Nektarvogel (Nectarinnidae),”Rev. Suisse Zool. 69, 801–804 (1962).

1957

C. Grégoire, “Topography of the organic components in mother-of pearl,”J. Biophys. Biochem. Cytol. 3(5), 797–808 (1957).
[CrossRef] [PubMed]

Aizenberg, J.

M. Kolle, A. Lethbridge, M. Kreysing, J. J. Baumberg, J. Aizenberg, and P. Vukusic, “Bio-Inspired Band-Gap Tunable Elastic Optical Multilayer Fibers,”Adv. Mater. 25(15), 2239–2245 (2013).
[CrossRef] [PubMed]

Andrew, P.

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(5910), 130–133 (2009).
[CrossRef] [PubMed]

Baumberg, J. J.

M. Kolle, A. Lethbridge, M. Kreysing, J. J. Baumberg, J. Aizenberg, and P. Vukusic, “Bio-Inspired Band-Gap Tunable Elastic Optical Multilayer Fibers,”Adv. Mater. 25(15), 2239–2245 (2013).
[CrossRef] [PubMed]

S. Vignolini, P. J. Rudall, A. V. Rowland, A. Reed, E. Moyroud, R. B. Faden, J. J. Baumberg, B. J. Glover, and U. Steiner, “Pointillist structural color in Pollia fruit,”Proc. Natl. Acad. Sci. U.S.A. 109(39), 15712–15715 (2012).
[CrossRef] [PubMed]

Bhushan, B.

J. Sun, B. Bhushan, and J. Tong, “Structural coloration in nature,”RSC Advances 3(35), 14862–14889 (2013).
[CrossRef]

Bitton, P. P.

R. Maia, C. M. Eliason, P. P. Bitton, S. M. Doucet, and M. D. Shawkey, “Pavo: An R Package for the Analysis, Visualization and Organization of Spectral Data,”Methods. Ecol. Evol. 4, 906–913 (2013).

Bitton, P.-P.

C. M. Eliason, P.-P. Bitton, and M. D. Shawkey, “How hollow melanosomes affect iridescent colour production in birds,”Proc. Biol. Sci. 280(1767), 20131505 (2013).
[CrossRef] [PubMed]

Brady, P.

A. E. Seago, P. Brady, J.-P. Vigneron, and T. D. Schultz, “Gold bugs and beyond: a review of iridescence and structural colour mechanisms in beetles (Coleoptera),”J. R. Soc. Interface 6(Suppl 2), S165–S184 (2009).
[CrossRef] [PubMed]

Cao, H.

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(26-27), 2871–2880 (2010).
[CrossRef] [PubMed]

Chittka, L.

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(5910), 130–133 (2009).
[CrossRef] [PubMed]

D’Alba, L.

H. K. Snyder, R. Maia, L. D’Alba, A. J. Shultz, K. M. C. Rowe, K. C. Rowe, and M. D. Shawkey, “Iridescent colour production in hairs of blind golden moles (Chrysochloridae),”Biol. Lett. 8(3), 393–396 (2012).
[CrossRef] [PubMed]

De Raedt, H.

B. D. Wilts, K. Michielsen, H. De Raedt, and D. G. Stavenga, “Sparkling feather reflections of a bird-of-paradise explained by finite-difference time-domain modeling,”Proc. Natl. Acad. Sci. U.S.A. 111(12), 4363–4368 (2014).
[CrossRef] [PubMed]

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] [PubMed]

Dong, B.

H. Yin, L. Shi, J. Sha, Y. Li, Y. Qin, B. Dong, S. Meyer, X. Liu, L. Zhao, and J. Zi, “Iridescence in the neck feathers of domestic pigeons,”Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 051916 (2006).
[CrossRef] [PubMed]

Doucet, S. M.

R. Maia, C. M. Eliason, P. P. Bitton, S. M. Doucet, and M. D. Shawkey, “Pavo: An R Package for the Analysis, Visualization and Organization of Spectral Data,”Methods. Ecol. Evol. 4, 906–913 (2013).

S. M. Doucet, M. D. Shawkey, G. E. Hill, and R. Montgomerie, “Iridescent plumage in satin bowerbirds: structure, mechanisms and nanostructural predictors of individual variation in colour,”J. Exp. Biol. 209(2), 380–390 (2006).
[CrossRef] [PubMed]

Douglas, J. M.

M. G. Meadows, N. I. Morehouse, R. L. Rutowski, J. M. Douglas, and K. J. McGraw, “Quantifying iridescent coloration in animals: a method for improving repeatability,”Behav. Ecol. Sociobiol. 65(6), 1317–1327 (2011).
[CrossRef]

Dufresne, E. R.

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(26-27), 2871–2880 (2010).
[CrossRef] [PubMed]

Durrer, H.

H. Durrer and W. Villiger, “Schillerradien des Goldkuckucks (Chrysococcyx cupreus (Shaw)) im Elektronenmikroskop,”Z. Zellforsch. Mikrosk. Anat. 109(3), 407–413 (1970).
[CrossRef] [PubMed]

H. Durrer and W. Villiger, “Schillerfarben der Nektarvogel (Nectarinnidae),”Rev. Suisse Zool. 69, 801–804 (1962).

Eliason, C. M.

C. M. Eliason, P.-P. Bitton, and M. D. Shawkey, “How hollow melanosomes affect iridescent colour production in birds,”Proc. Biol. Sci. 280(1767), 20131505 (2013).
[CrossRef] [PubMed]

R. Maia, C. M. Eliason, P. P. Bitton, S. M. Doucet, and M. D. Shawkey, “Pavo: An R Package for the Analysis, Visualization and Organization of Spectral Data,”Methods. Ecol. Evol. 4, 906–913 (2013).

C. M. Eliason and M. D. Shawkey, “A photonic heterostructure produces diverse iridescent colours in duck wing patches,”J. R. Soc. Interface 9(74), 2279–2289 (2012).
[CrossRef] [PubMed]

Estes, A. M.

M. D. Shawkey, A. M. Estes, L. M. Siefferman, and G. E. Hill, “Nanostructure predicts intraspecific variation in ultraviolet-blue plumage colour,”Proc. Biol. Sci. 270(1523), 1455–1460 (2003).
[CrossRef] [PubMed]

Faden, R. B.

S. Vignolini, P. J. Rudall, A. V. Rowland, A. Reed, E. Moyroud, R. B. Faden, J. J. Baumberg, B. J. Glover, and U. Steiner, “Pointillist structural color in Pollia fruit,”Proc. Natl. Acad. Sci. U.S.A. 109(39), 15712–15715 (2012).
[CrossRef] [PubMed]

Fu, R. T.

J. Zi, X. D. Yu, Y. Z. Li, X. H. Hu, C. Xu, X. J. Wang, X. H. Liu, and R. T. Fu, “Coloration strategies in peacock feathers,”Proc. Natl. Acad. Sci. U.S.A. 100(22), 12576–12578 (2003).
[CrossRef] [PubMed]

Garahan, A.

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Ghiradella, H.

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S. Vignolini, P. J. Rudall, A. V. Rowland, A. Reed, E. Moyroud, R. B. Faden, J. J. Baumberg, B. J. Glover, and U. Steiner, “Pointillist structural color in Pollia fruit,”Proc. Natl. Acad. Sci. U.S.A. 109(39), 15712–15715 (2012).
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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(5910), 130–133 (2009).
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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).
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D. G. Stavenga, B. D. Wilts, H. L. Leertouwer, and T. Hariyama, “Polarized iridescence of the multilayered elytra of the Japanese jewel beetle, Chrysochroa fulgidissima,”Philos. Trans. R. Soc. Lond. B 366(1565), 709–723 (2011).
[CrossRef] [PubMed]

Hill, G. E.

S. M. Doucet, M. D. Shawkey, G. E. Hill, and R. Montgomerie, “Iridescent plumage in satin bowerbirds: structure, mechanisms and nanostructural predictors of individual variation in colour,”J. Exp. Biol. 209(2), 380–390 (2006).
[CrossRef] [PubMed]

M. D. Shawkey, A. M. Estes, L. M. Siefferman, and G. E. Hill, “Nanostructure predicts intraspecific variation in ultraviolet-blue plumage colour,”Proc. Biol. Sci. 270(1523), 1455–1460 (2003).
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Hu, X. H.

J. Zi, X. D. Yu, Y. Z. Li, X. H. Hu, C. Xu, X. J. Wang, X. H. Liu, and R. T. Fu, “Coloration strategies in peacock feathers,”Proc. Natl. Acad. Sci. U.S.A. 100(22), 12576–12578 (2003).
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T. M. Jordan, J. C. Partridge, and N. W. Roberts, “Non-polarizing broadband multilayer reflectors in fish,”Nat. Photonics 6(11), 759–763 (2012).
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M. Kolle, A. Lethbridge, M. Kreysing, J. J. Baumberg, J. Aizenberg, and P. Vukusic, “Bio-Inspired Band-Gap Tunable Elastic Optical Multilayer Fibers,”Adv. Mater. 25(15), 2239–2245 (2013).
[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(5910), 130–133 (2009).
[CrossRef] [PubMed]

Kreysing, M.

M. Kolle, A. Lethbridge, M. Kreysing, J. J. Baumberg, J. Aizenberg, and P. Vukusic, “Bio-Inspired Band-Gap Tunable Elastic Optical Multilayer Fibers,”Adv. Mater. 25(15), 2239–2245 (2013).
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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] [PubMed]

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] [PubMed]

D. G. Stavenga, H. L. Leertouwer, N. J. Marshall, and D. Osorio, “Dramatic colour changes in a bird of paradise caused by uniquely structured breast feather barbules,”Proc. Biol. Sci. 278(1715), 2098–2104 (2011).
[CrossRef] [PubMed]

D. G. Stavenga, B. D. Wilts, H. L. Leertouwer, and T. Hariyama, “Polarized iridescence of the multilayered elytra of the Japanese jewel beetle, Chrysochroa fulgidissima,”Philos. Trans. R. Soc. Lond. B 366(1565), 709–723 (2011).
[CrossRef] [PubMed]

D. G. Stavenga, H. L. Leertouwer, P. Pirih, and M. F. Wehling, “Imaging scatterometry of butterfly wing scales,”Opt. Express 17(1), 193–202 (2009).
[CrossRef] [PubMed]

Lethbridge, A.

M. Kolle, A. Lethbridge, M. Kreysing, J. J. Baumberg, J. Aizenberg, and P. Vukusic, “Bio-Inspired Band-Gap Tunable Elastic Optical Multilayer Fibers,”Adv. Mater. 25(15), 2239–2245 (2013).
[CrossRef] [PubMed]

Li, Y.

H. Yin, L. Shi, J. Sha, Y. Li, Y. Qin, B. Dong, S. Meyer, X. Liu, L. Zhao, and J. Zi, “Iridescence in the neck feathers of domestic pigeons,”Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 051916 (2006).
[CrossRef] [PubMed]

Li, Y. Z.

J. Zi, X. D. Yu, Y. Z. Li, X. H. Hu, C. Xu, X. J. Wang, X. H. Liu, and R. T. Fu, “Coloration strategies in peacock feathers,”Proc. Natl. Acad. Sci. U.S.A. 100(22), 12576–12578 (2003).
[CrossRef] [PubMed]

Liew, S. F.

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(26-27), 2871–2880 (2010).
[CrossRef] [PubMed]

Liu, X.

H. Yin, L. Shi, J. Sha, Y. Li, Y. Qin, B. Dong, S. Meyer, X. Liu, L. Zhao, and J. Zi, “Iridescence in the neck feathers of domestic pigeons,”Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 051916 (2006).
[CrossRef] [PubMed]

Liu, X. H.

J. Zi, X. D. Yu, Y. Z. Li, X. H. Hu, C. Xu, X. J. Wang, X. H. Liu, and R. T. Fu, “Coloration strategies in peacock feathers,”Proc. Natl. Acad. Sci. U.S.A. 100(22), 12576–12578 (2003).
[CrossRef] [PubMed]

Macedo, R. H. F.

R. Maia, R. H. F. Macedo, and M. D. Shawkey, “Nanostructural self-assembly of iridescent feather barbules through depletion attraction of melanosomes during keratinization,”J. R. Soc. Interface 9(69), 734–743 (2012).
[CrossRef] [PubMed]

Maia, R.

R. Maia, C. M. Eliason, P. P. Bitton, S. M. Doucet, and M. D. Shawkey, “Pavo: An R Package for the Analysis, Visualization and Organization of Spectral Data,”Methods. Ecol. Evol. 4, 906–913 (2013).

R. Maia, R. H. F. Macedo, and M. D. Shawkey, “Nanostructural self-assembly of iridescent feather barbules through depletion attraction of melanosomes during keratinization,”J. R. Soc. Interface 9(69), 734–743 (2012).
[CrossRef] [PubMed]

H. K. Snyder, R. Maia, L. D’Alba, A. J. Shultz, K. M. C. Rowe, K. C. Rowe, and M. D. Shawkey, “Iridescent colour production in hairs of blind golden moles (Chrysochloridae),”Biol. Lett. 8(3), 393–396 (2012).
[CrossRef] [PubMed]

Marshall, N. J.

D. G. Stavenga, H. L. Leertouwer, N. J. Marshall, and D. Osorio, “Dramatic colour changes in a bird of paradise caused by uniquely structured breast feather barbules,”Proc. Biol. Sci. 278(1715), 2098–2104 (2011).
[CrossRef] [PubMed]

McGraw, K. J.

M. G. Meadows, N. I. Morehouse, R. L. Rutowski, J. M. Douglas, and K. J. McGraw, “Quantifying iridescent coloration in animals: a method for improving repeatability,”Behav. Ecol. Sociobiol. 65(6), 1317–1327 (2011).
[CrossRef]

Meadows, M. G.

M. G. Meadows, N. I. Morehouse, R. L. Rutowski, J. M. Douglas, and K. J. McGraw, “Quantifying iridescent coloration in animals: a method for improving repeatability,”Behav. Ecol. Sociobiol. 65(6), 1317–1327 (2011).
[CrossRef]

Meyer, S.

H. Yin, L. Shi, J. Sha, Y. Li, Y. Qin, B. Dong, S. Meyer, X. Liu, L. Zhao, and J. Zi, “Iridescence in the neck feathers of domestic pigeons,”Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 051916 (2006).
[CrossRef] [PubMed]

Michielsen, K.

B. D. Wilts, K. Michielsen, H. De Raedt, and D. G. Stavenga, “Sparkling feather reflections of a bird-of-paradise explained by finite-difference time-domain modeling,”Proc. Natl. Acad. Sci. U.S.A. 111(12), 4363–4368 (2014).
[CrossRef] [PubMed]

Mochrie, S. G. J.

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(26-27), 2871–2880 (2010).
[CrossRef] [PubMed]

Montgomerie, R.

S. M. Doucet, M. D. Shawkey, G. E. Hill, and R. Montgomerie, “Iridescent plumage in satin bowerbirds: structure, mechanisms and nanostructural predictors of individual variation in colour,”J. Exp. Biol. 209(2), 380–390 (2006).
[CrossRef] [PubMed]

Morehouse, N. I.

M. G. Meadows, N. I. Morehouse, R. L. Rutowski, J. M. Douglas, and K. J. McGraw, “Quantifying iridescent coloration in animals: a method for improving repeatability,”Behav. Ecol. Sociobiol. 65(6), 1317–1327 (2011).
[CrossRef]

M. D. Shawkey, N. I. Morehouse, and P. Vukusic, “A protean palette: colour materials and mixing in birds and butterflies,”J. R. Soc. Interface 6(Suppl 2), S221–S231 (2009).
[CrossRef] [PubMed]

Moyroud, E.

S. Vignolini, P. J. Rudall, A. V. Rowland, A. Reed, E. Moyroud, R. B. Faden, J. J. Baumberg, B. J. Glover, and U. Steiner, “Pointillist structural color in Pollia fruit,”Proc. Natl. Acad. Sci. U.S.A. 109(39), 15712–15715 (2012).
[CrossRef] [PubMed]

Nixon, M. R.

Noh, H.

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(26-27), 2871–2880 (2010).
[CrossRef] [PubMed]

Orr, A. G.

Osorio, D.

D. G. Stavenga, H. L. Leertouwer, N. J. Marshall, and D. Osorio, “Dramatic colour changes in a bird of paradise caused by uniquely structured breast feather barbules,”Proc. Biol. Sci. 278(1715), 2098–2104 (2011).
[CrossRef] [PubMed]

Partridge, J. C.

T. M. Jordan, J. C. Partridge, and N. W. Roberts, “Non-polarizing broadband multilayer reflectors in fish,”Nat. Photonics 6(11), 759–763 (2012).
[CrossRef] [PubMed]

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A. Garahan, L. Pilon, J. Yin, and I. Saxena, “Effective optical properties of absorbing nanoporous and nanocomposite thin films,”J. Appl. Phys. 101(1), 014320 (2007).
[CrossRef]

Pirih, P.

Prum, R. O.

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(26-27), 2871–2880 (2010).
[CrossRef] [PubMed]

R. O. Prum, “Development and evolutionary origin of feathers,”J. Exp. Zool. 285(4), 291–306 (1999).
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Qin, Y.

H. Yin, L. Shi, J. Sha, Y. Li, Y. Qin, B. Dong, S. Meyer, X. Liu, L. Zhao, and J. Zi, “Iridescence in the neck feathers of domestic pigeons,”Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 051916 (2006).
[CrossRef] [PubMed]

Reed, A.

S. Vignolini, P. J. Rudall, A. V. Rowland, A. Reed, E. Moyroud, R. B. Faden, J. J. Baumberg, B. J. Glover, and U. Steiner, “Pointillist structural color in Pollia fruit,”Proc. Natl. Acad. Sci. U.S.A. 109(39), 15712–15715 (2012).
[CrossRef] [PubMed]

Roberts, N. W.

T. M. Jordan, J. C. Partridge, and N. W. Roberts, “Non-polarizing broadband multilayer reflectors in fish,”Nat. Photonics 6(11), 759–763 (2012).
[CrossRef] [PubMed]

Rowe, K. C.

H. K. Snyder, R. Maia, L. D’Alba, A. J. Shultz, K. M. C. Rowe, K. C. Rowe, and M. D. Shawkey, “Iridescent colour production in hairs of blind golden moles (Chrysochloridae),”Biol. Lett. 8(3), 393–396 (2012).
[CrossRef] [PubMed]

Rowe, K. M. C.

H. K. Snyder, R. Maia, L. D’Alba, A. J. Shultz, K. M. C. Rowe, K. C. Rowe, and M. D. Shawkey, “Iridescent colour production in hairs of blind golden moles (Chrysochloridae),”Biol. Lett. 8(3), 393–396 (2012).
[CrossRef] [PubMed]

Rowland, A. V.

S. Vignolini, P. J. Rudall, A. V. Rowland, A. Reed, E. Moyroud, R. B. Faden, J. J. Baumberg, B. J. Glover, and U. Steiner, “Pointillist structural color in Pollia fruit,”Proc. Natl. Acad. Sci. U.S.A. 109(39), 15712–15715 (2012).
[CrossRef] [PubMed]

Rudall, P. J.

S. Vignolini, P. J. Rudall, A. V. Rowland, A. Reed, E. Moyroud, R. B. Faden, J. J. Baumberg, B. J. Glover, and U. Steiner, “Pointillist structural color in Pollia fruit,”Proc. Natl. Acad. Sci. U.S.A. 109(39), 15712–15715 (2012).
[CrossRef] [PubMed]

Rutowski, R. L.

M. G. Meadows, N. I. Morehouse, R. L. Rutowski, J. M. Douglas, and K. J. McGraw, “Quantifying iridescent coloration in animals: a method for improving repeatability,”Behav. Ecol. Sociobiol. 65(6), 1317–1327 (2011).
[CrossRef]

Saranathan, V.

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(26-27), 2871–2880 (2010).
[CrossRef] [PubMed]

Saxena, I.

A. Garahan, L. Pilon, J. Yin, and I. Saxena, “Effective optical properties of absorbing nanoporous and nanocomposite thin films,”J. Appl. Phys. 101(1), 014320 (2007).
[CrossRef]

Schultz, T. D.

A. E. Seago, P. Brady, J.-P. Vigneron, and T. D. Schultz, “Gold bugs and beyond: a review of iridescence and structural colour mechanisms in beetles (Coleoptera),”J. R. Soc. Interface 6(Suppl 2), S165–S184 (2009).
[CrossRef] [PubMed]

Seago, A. E.

A. E. Seago, P. Brady, J.-P. Vigneron, and T. D. Schultz, “Gold bugs and beyond: a review of iridescence and structural colour mechanisms in beetles (Coleoptera),”J. R. Soc. Interface 6(Suppl 2), S165–S184 (2009).
[CrossRef] [PubMed]

Sha, J.

H. Yin, L. Shi, J. Sha, Y. Li, Y. Qin, B. Dong, S. Meyer, X. Liu, L. Zhao, and J. Zi, “Iridescence in the neck feathers of domestic pigeons,”Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 051916 (2006).
[CrossRef] [PubMed]

Shawkey, M. D.

C. M. Eliason, P.-P. Bitton, and M. D. Shawkey, “How hollow melanosomes affect iridescent colour production in birds,”Proc. Biol. Sci. 280(1767), 20131505 (2013).
[CrossRef] [PubMed]

R. Maia, C. M. Eliason, P. P. Bitton, S. M. Doucet, and M. D. Shawkey, “Pavo: An R Package for the Analysis, Visualization and Organization of Spectral Data,”Methods. Ecol. Evol. 4, 906–913 (2013).

R. Maia, R. H. F. Macedo, and M. D. Shawkey, “Nanostructural self-assembly of iridescent feather barbules through depletion attraction of melanosomes during keratinization,”J. R. Soc. Interface 9(69), 734–743 (2012).
[CrossRef] [PubMed]

C. M. Eliason and M. D. Shawkey, “A photonic heterostructure produces diverse iridescent colours in duck wing patches,”J. R. Soc. Interface 9(74), 2279–2289 (2012).
[CrossRef] [PubMed]

H. K. Snyder, R. Maia, L. D’Alba, A. J. Shultz, K. M. C. Rowe, K. C. Rowe, and M. D. Shawkey, “Iridescent colour production in hairs of blind golden moles (Chrysochloridae),”Biol. Lett. 8(3), 393–396 (2012).
[CrossRef] [PubMed]

M. D. Shawkey, N. I. Morehouse, and P. Vukusic, “A protean palette: colour materials and mixing in birds and butterflies,”J. R. Soc. Interface 6(Suppl 2), S221–S231 (2009).
[CrossRef] [PubMed]

S. M. Doucet, M. D. Shawkey, G. E. Hill, and R. Montgomerie, “Iridescent plumage in satin bowerbirds: structure, mechanisms and nanostructural predictors of individual variation in colour,”J. Exp. Biol. 209(2), 380–390 (2006).
[CrossRef] [PubMed]

M. D. Shawkey, A. M. Estes, L. M. Siefferman, and G. E. Hill, “Nanostructure predicts intraspecific variation in ultraviolet-blue plumage colour,”Proc. Biol. Sci. 270(1523), 1455–1460 (2003).
[CrossRef] [PubMed]

Shi, L.

H. Yin, L. Shi, J. Sha, Y. Li, Y. Qin, B. Dong, S. Meyer, X. Liu, L. Zhao, and J. Zi, “Iridescence in the neck feathers of domestic pigeons,”Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 051916 (2006).
[CrossRef] [PubMed]

Shultz, A. J.

H. K. Snyder, R. Maia, L. D’Alba, A. J. Shultz, K. M. C. Rowe, K. C. Rowe, and M. D. Shawkey, “Iridescent colour production in hairs of blind golden moles (Chrysochloridae),”Biol. Lett. 8(3), 393–396 (2012).
[CrossRef] [PubMed]

Siefferman, L. M.

M. D. Shawkey, A. M. Estes, L. M. Siefferman, and G. E. Hill, “Nanostructure predicts intraspecific variation in ultraviolet-blue plumage colour,”Proc. Biol. Sci. 270(1523), 1455–1460 (2003).
[CrossRef] [PubMed]

Snyder, H. K.

H. K. Snyder, R. Maia, L. D’Alba, A. J. Shultz, K. M. C. Rowe, K. C. Rowe, and M. D. Shawkey, “Iridescent colour production in hairs of blind golden moles (Chrysochloridae),”Biol. Lett. 8(3), 393–396 (2012).
[CrossRef] [PubMed]

Stavenga, D. G.

B. D. Wilts, K. Michielsen, H. De Raedt, and D. G. Stavenga, “Sparkling feather reflections of a bird-of-paradise explained by finite-difference time-domain modeling,”Proc. Natl. Acad. Sci. U.S.A. 111(12), 4363–4368 (2014).
[CrossRef] [PubMed]

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] [PubMed]

D. G. Stavenga, H. L. Leertouwer, N. J. Marshall, and D. Osorio, “Dramatic colour changes in a bird of paradise caused by uniquely structured breast feather barbules,”Proc. Biol. Sci. 278(1715), 2098–2104 (2011).
[CrossRef] [PubMed]

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] [PubMed]

D. G. Stavenga, B. D. Wilts, H. L. Leertouwer, and T. Hariyama, “Polarized iridescence of the multilayered elytra of the Japanese jewel beetle, Chrysochroa fulgidissima,”Philos. Trans. R. Soc. Lond. B 366(1565), 709–723 (2011).
[CrossRef] [PubMed]

D. G. Stavenga, H. L. Leertouwer, P. Pirih, and M. F. Wehling, “Imaging scatterometry of butterfly wing scales,”Opt. Express 17(1), 193–202 (2009).
[CrossRef] [PubMed]

Steiner, U.

S. Vignolini, P. J. Rudall, A. V. Rowland, A. Reed, E. Moyroud, R. B. Faden, J. J. Baumberg, B. J. Glover, and U. Steiner, “Pointillist structural color in Pollia fruit,”Proc. Natl. Acad. Sci. U.S.A. 109(39), 15712–15715 (2012).
[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(5910), 130–133 (2009).
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Sun, J.

J. Sun, B. Bhushan, and J. Tong, “Structural coloration in nature,”RSC Advances 3(35), 14862–14889 (2013).
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Tong, J.

J. Sun, B. Bhushan, and J. Tong, “Structural coloration in nature,”RSC Advances 3(35), 14862–14889 (2013).
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Vigneron, J.-P.

A. E. Seago, P. Brady, J.-P. Vigneron, and T. D. Schultz, “Gold bugs and beyond: a review of iridescence and structural colour mechanisms in beetles (Coleoptera),”J. R. Soc. Interface 6(Suppl 2), S165–S184 (2009).
[CrossRef] [PubMed]

Vignolini, S.

S. Vignolini, P. J. Rudall, A. V. Rowland, A. Reed, E. Moyroud, R. B. Faden, J. J. Baumberg, B. J. Glover, and U. Steiner, “Pointillist structural color in Pollia fruit,”Proc. Natl. Acad. Sci. U.S.A. 109(39), 15712–15715 (2012).
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Villiger, W.

H. Durrer and W. Villiger, “Schillerradien des Goldkuckucks (Chrysococcyx cupreus (Shaw)) im Elektronenmikroskop,”Z. Zellforsch. Mikrosk. Anat. 109(3), 407–413 (1970).
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Vukusic, P.

M. R. Nixon, A. G. Orr, and P. Vukusic, “Subtle design changes control the difference in colour reflection from the dorsal and ventral wing-membrane surfaces of the damselfly Matronoides cyaneipennis,”Opt. Express 21(2), 1479–1488 (2013).
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M. Kolle, A. Lethbridge, M. Kreysing, J. J. Baumberg, J. Aizenberg, and P. Vukusic, “Bio-Inspired Band-Gap Tunable Elastic Optical Multilayer Fibers,”Adv. Mater. 25(15), 2239–2245 (2013).
[CrossRef] [PubMed]

M. D. Shawkey, N. I. Morehouse, and P. Vukusic, “A protean palette: colour materials and mixing in birds and butterflies,”J. R. Soc. Interface 6(Suppl 2), S221–S231 (2009).
[CrossRef] [PubMed]

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Wang, X. J.

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B. D. Wilts, K. Michielsen, H. De Raedt, and D. G. Stavenga, “Sparkling feather reflections of a bird-of-paradise explained by finite-difference time-domain modeling,”Proc. Natl. Acad. Sci. U.S.A. 111(12), 4363–4368 (2014).
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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).
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J. Zi, X. D. Yu, Y. Z. Li, X. H. Hu, C. Xu, X. J. Wang, X. H. Liu, and R. T. Fu, “Coloration strategies in peacock feathers,”Proc. Natl. Acad. Sci. U.S.A. 100(22), 12576–12578 (2003).
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J. Zi, X. D. Yu, Y. Z. Li, X. H. Hu, C. Xu, X. J. Wang, X. H. Liu, and R. T. Fu, “Coloration strategies in peacock feathers,”Proc. Natl. Acad. Sci. U.S.A. 100(22), 12576–12578 (2003).
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H. Yin, L. Shi, J. Sha, Y. Li, Y. Qin, B. Dong, S. Meyer, X. Liu, L. Zhao, and J. Zi, “Iridescence in the neck feathers of domestic pigeons,”Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 051916 (2006).
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J. Zi, X. D. Yu, Y. Z. Li, X. H. Hu, C. Xu, X. J. Wang, X. H. Liu, and R. T. Fu, “Coloration strategies in peacock feathers,”Proc. Natl. Acad. Sci. U.S.A. 100(22), 12576–12578 (2003).
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Adv. Mater.

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Nat. Photonics

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Philos. Trans. R. Soc. Lond. B

D. G. Stavenga, B. D. Wilts, H. L. Leertouwer, and T. Hariyama, “Polarized iridescence of the multilayered elytra of the Japanese jewel beetle, Chrysochroa fulgidissima,”Philos. Trans. R. Soc. Lond. B 366(1565), 709–723 (2011).
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S. Yoshioka and S. Kinoshita, “Direct determination of the refractive index of natural multilayer systems,”Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 051917 (2011).
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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).
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B. D. Wilts, K. Michielsen, H. De Raedt, and D. G. Stavenga, “Sparkling feather reflections of a bird-of-paradise explained by finite-difference time-domain modeling,”Proc. Natl. Acad. Sci. U.S.A. 111(12), 4363–4368 (2014).
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Figures (13)

Fig. 1
Fig. 1

a) Stereo light microscope image. Panels b-d) show higher magnification images of barbules in distal region, middle region, and proximal regions, respectively. Scale bars a) 2 mm, b-d) 200 μ m .

Fig. 2
Fig. 2

Schematic for geometry of spectrometer for specular reflectance measurement. α is the angle between the incident beam and the surface normal; similarly, β is the angle between the detector and the surface normal; and θ is the angle between feather rachis (bold black line with an arrow) and the plane-of-incidence (yellow plane). The values of α and β can be read directly from the spectrometer. The arrow of the feather stands for the distal end of the feather. γ is the angle between the stage plane on which the feather lies and the horizontal plane.

Fig. 3
Fig. 3

Specular reflectance spectra for the green region of barbules at incident angles from 10 ° to 45 °

Fig. 4
Fig. 4

Panels a-c) microscopic images of blue, green and red barbules from MSP, respectively and small black squares in the middle indicate the sampled spot (the length of the black square is 4 μm ). d) Color variation of the barbules measured by MSP, the vertical axis is the normalized reflectance intensity with arbitrary unit. The color for each curve is based on the human visual perception according to the standard CIE1931 [25].

Fig. 5
Fig. 5

TEM images of a) cross section of a red barbule, b) longitudinal section of a green barbule and c) cross section of a non-iridescent brown barbule. Scale bars: a-c) 500 nm.

Fig. 6
Fig. 6

Schematic of multilayer structure in barbules. d 1 is the keratin cortex thickness; d m , the diameter of melanosomes; a , the spacing between melanosomes layers and d 0 , the spacing between neighboring melanosomes

Fig. 7
Fig. 7

a) Measured (green line) and modeled spectra (black line) on the average layer thickness for barbules in green region. b-c) The modeled bluest and reddest spectra (black lines) based on largest and smallest thickness of melanosome and keratin layers in blue and red barbules; and the blue and red curves are the bluest and reddest spectra measured by MSP, respectively.

Fig. 8
Fig. 8

Modeled results for angle-resolved reflectance spectrum for middle barbules (green solid line). The black square data points are obtained from experimental spectra in Fig. 3.

Fig. 9
Fig. 9

Reflectance of green barbule measured by MSP using differently polarized input beams. The blue curve is the unpolarized reflectance, and the red, green are the s- and p- polarization, respectively.

Fig. 10
Fig. 10

The hue dependence on the outmost keratin layer (cortex layer) thickness based on multilayer modeling result.

Fig. 11
Fig. 11

Green solid line is the decrement rate of full width at half maximum (FWHM), which is the derivative of FWHM with respect to layer number and blue dash line is the increment rate of intensity which is the derivative of intensity with respect to layer number.

Fig. 12
Fig. 12

TEM images of cross-sections of barbules with different colors under the same magnification: a) red, b) green and c) blue. Scale bars 100nm

Fig. 13
Fig. 13

The modeled color range for barbules in green zone and the experimentally measrued hue for green barbules at different incident angles.

Tables (1)

Tables Icon

Table 1 Spacing and diameter of melanosomes in the barbule nanostructure measured using TEM results (Errors are standard deviation from 100 measurements)

Equations (14)

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n ker = 1.532 + 5890 / λ 2
n m e l = 1.56 + 36000 / λ 2
k m e l = 1.62 e λ 142
n ˜ m e l , a v g = n e f f i κ e f f
n e f f 2 = [ A + A 2 + B 2 ] / 2
κ e f f 2 = [ A + A 2 + B 2 ] / 2
A = V m e l ( n m e l 2 κ m e l 2 ) + ( 1 V m e l ) ( n ker 2 κ ker 2 )
B = 2 V m e l n m e l κ m e l + 2 ( 1 V m e l ) n ker κ ker
I ( j 1 ) j = 1 t j [ 1 r j r j 1 ]
r j = n j cos φ j 1 n j 1 cos φ j n j cos φ j 1 + n j 1 cos φ j
r j = n j 1 cos φ j 1 n j cos φ j n j cos φ j + n j 1 cos φ j 1
n 0 cos φ 0 = n 1 cos φ 1 = = n j cos φ j n 0 cos φ 0 = n 1 cos φ 1 = = n j cos φ j
L j = [ e i b j 0 0 e i b j ]
S = [ S 11 S 12 S 21 S 22 ] = I 01 L 1 I 12 L 2 · · · I ( m 1 ) m L m I m ( m + 1 )

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