Abstract

We have investigated wavelength-dependent light scattering in biomimetic structures with short-range order. Coherent backscattering experiments are performed to measure the transport mean free path over a wide wavelength range. Overall scattering strength is reduced significantly due to short-range order and near-field effects. Our analysis explains why single scattering of light is dominant over multiple scattering in similar biological structures and is responsible for color generation.

© 2011 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. E. V. Petrova, V.P. Tishkovets, and K. Jockers, “Interaction of particles in the near field and opposition effects in regolith-like surfaces,” Solar Syst. Res. 43, 100–115 (2009).
    [CrossRef]
  31. A. F. Koenderink, M. Megens, G. van Soest, W. L. Vos, and A. Lagendijk, “Enhanced backscattering from photonic crystals,” Phys. Lett. A 268, 104–111 (2000).
    [CrossRef]
  32. J. Huang, N. Eradat, M. E. Raikh, and Z. V. Vardeny, “Anomalous coherent backscattering of light from optical photonic crystals,” Phys. Rev. Lett. 86, 4815–4818 (2001).
    [CrossRef] [PubMed]
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    [CrossRef]
  34. P. D. García, R. Sapienza, L. S. Froufe-Pérez, and C. López, “Strong dispersive effects in the light-scattering mean free path in photonic gaps,” Phys. Rev. B 79, 241109 (2009).
    [CrossRef]

2010 (7)

B. Q. Dong, X. H. Liu, T. R. Zhan, L. P. Jiang, H. W. Yin, F. Liu, and J. Zi, “Structural coloration and photonic pseudogap in natural random close-packing photonic structures,” Opt. Express 18, 14430–14438 (2010).
[CrossRef] [PubMed]

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

M. H. Ur-Rashid, A. B. Imran, T. Seki, M. Ishii, H. Nakamura, and Y. Takeoka, “Angle-independent structural color in colloidal amorphous arrays,” Chemphyschem 11, 579–583 (2010).
[CrossRef]

I. Lee, D. Kim, J. Kal, H. Baek, D. Kwak, D. Go, E. Kim, C. Kang, J. Chung, Y. Jang, S. Ji, J. Joo, and Y. Kang, “Quasi-amorphous colloidal structures for electrically tunable full-color photonic pixels with angle-independency,” Adv. Mater. 22, 4973–4977 (2010).
[CrossRef] [PubMed]

H. Noh, S. F. Liew, V. Saranathan, R. O. Prum, S. G. J. Mochrie, E. R. Dufresne, and H. Cao, “Double scattering of light from biophotonic nanostructures with short-range order,” Opt. Express 18, 11942–11948 (2010).
[CrossRef] [PubMed]

H. Noh, S. F. Liew, V. Saranathan, R. O. Prum, S. G. J. Mochrie, E. R. Dufresne, and H. Cao, “Contribution of double scattering to structural coloration in quasi-ordered nanostructures of bird feathers,” Phys. Rev. E 81, 051923 (2010).
[CrossRef]

2009 (5)

Y. Takeoka, M. Honda, T. Seki, M. Ishii, and H. Nakamura, “Structural colored liquid membrane without angle dependence,” ACS Appl. Mater. Interfaces 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 homogenous, non-brilliant and angle-independent structural colors,” Chem. Commun. 3603–3605 (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]

E. V. Petrova, V.P. Tishkovets, and K. Jockers, “Interaction of particles in the near field and opposition effects in regolith-like surfaces,” Solar Syst. Res. 43, 100–115 (2009).
[CrossRef]

P. D. García, R. Sapienza, L. S. Froufe-Pérez, and C. López, “Strong dispersive effects in the light-scattering mean free path in photonic gaps,” Phys. Rev. B 79, 241109 (2009).
[CrossRef]

2008 (1)

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

2007 (1)

X. T. Peng and A. D. Dinsmore, “Light propagation in strongly scattering, random colloidal films: the role of the packing geometry,” Phys. Rev. Lett. 99, 143902 (2007).
[CrossRef] [PubMed]

2006 (1)

G.-J. Gao, J. Blawzdziewicz, and C. S. O’Hern, “Frequency distribution of mechanically stable disk packings,” Phys. Rev. E 74, 061304 (2006).
[CrossRef]

2005 (1)

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B 72, 153102 (2005).
[CrossRef]

2004 (1)

L. F. Rojas-Ochoa, J. M. Mendez-Alcaraz, J. J. Sáenz, P. Schurtenberger, and F. Scheffold, “Photonic properties of strongly correlated colloidal liquids,” Phys. Rev. Lett. 93, 073903 (2004).
[CrossRef] [PubMed]

2003 (2)

C. S. O’Hern, L. E. Silbert, A. J. Liu, and S. R. Nagel, “Jamming at zero temperature and zero applied stress: the epitome of disorder,” Phys. Rev. E 68, 011306 (2003).
[CrossRef]

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

2002 (1)

Y. Huang, Z. Sun, and E. M. Sevick-Muraca, “Assessment of electrostatic interactions in dense colloidal suspensions with multiply scattered light,” Langmuir 18, 2048–2053 (2002).
[CrossRef]

2001 (1)

J. Huang, N. Eradat, M. E. Raikh, and Z. V. Vardeny, “Anomalous coherent backscattering of light from optical photonic crystals,” Phys. Rev. Lett. 86, 4815–4818 (2001).
[CrossRef] [PubMed]

2000 (3)

A. F. Koenderink, M. Megens, G. van Soest, W. L. Vos, and A. Lagendijk, “Enhanced backscattering from photonic crystals,” Phys. Lett. A 268, 104–111 (2000).
[CrossRef]

L. E. McNeil and R. H. French, “Multiple scattering from rutile TiO2 particles,” Acta Mater. 48, 4571–4576 (2000).
[CrossRef]

S. Torquato, T. M. Truskett, and P. G. Debenedetti, “Is random close packing of spheres well defined?,”Phys. Rev. Lett. 84, 2064–2067 (2000).
[CrossRef] [PubMed]

1998 (1)

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

1990 (1)

S. Fraden and G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[CrossRef] [PubMed]

1987 (1)

W. L. Griffith, R. Triolo, and A. L. Compere, “Analytical scattering function of a polydisperse Percus-Yevick fluid with Schulz -(Γ-) distributed diameters,” Phys. Rev. A 35, 2200–2206 (1987).
[CrossRef] [PubMed]

1983 (1)

J. G. Berryman, “Random close packing of hard spheres and disks,” Phys. Rev. A 27, 1053–1061 (1983).
[CrossRef]

1981 (1)

Y. Chonde and I. M. Krieger, “Emulsion polymerization of styrene with ionic comonomer in the presence of methanol,” J. Appl. Polymer Sci. 26, 1819–1827 (1981).
[CrossRef]

1979 (1)

L. Blum and G. J. Stell, “Scattering function for polydisperse fluids of hard or permeable spheres,” J. Chem. Phys. 71, 42–46 (1979).
[CrossRef]

1969 (1)

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007).
[CrossRef]

Baek, H.

I. Lee, D. Kim, J. Kal, H. Baek, D. Kwak, D. Go, E. Kim, C. Kang, J. Chung, Y. Jang, S. Ji, J. Joo, and Y. Kang, “Quasi-amorphous colloidal structures for electrically tunable full-color photonic pixels with angle-independency,” Adv. Mater. 22, 4973–4977 (2010).
[CrossRef] [PubMed]

Berryman, J. G.

J. G. Berryman, “Random close packing of hard spheres and disks,” Phys. Rev. A 27, 1053–1061 (1983).
[CrossRef]

Blawzdziewicz, J.

G.-J. Gao, J. Blawzdziewicz, and C. S. O’Hern, “Frequency distribution of mechanically stable disk packings,” Phys. Rev. E 74, 061304 (2006).
[CrossRef]

Blum, L.

L. Blum and G. J. Stell, “Scattering function for polydisperse fluids of hard or permeable spheres,” J. Chem. Phys. 71, 42–46 (1979).
[CrossRef]

Cao, H.

H. Noh, S. F. Liew, V. Saranathan, R. O. Prum, S. G. J. Mochrie, E. R. Dufresne, and H. Cao, “Contribution of double scattering to structural coloration in quasi-ordered nanostructures of bird feathers,” Phys. Rev. E 81, 051923 (2010).
[CrossRef]

H. Noh, S. F. Liew, V. Saranathan, R. O. Prum, S. G. J. Mochrie, E. R. Dufresne, and H. Cao, “Double scattering of light from biophotonic nanostructures with short-range order,” Opt. Express 18, 11942–11948 (2010).
[CrossRef] [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, R. O. Prum, S. G. J. Mochrie, E. R. Dufresne, and H. Cao, “How non-iridescent colors are generated by quasi-ordered structures of bird feathers,” Adv. Mater. 22, 2871–2880 (2010).
[CrossRef] [PubMed]

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]

Chonde, Y.

Y. Chonde and I. M. Krieger, “Emulsion polymerization of styrene with ionic comonomer in the presence of methanol,” J. Appl. Polymer Sci. 26, 1819–1827 (1981).
[CrossRef]

Chung, J.

I. Lee, D. Kim, J. Kal, H. Baek, D. Kwak, D. Go, E. Kim, C. Kang, J. Chung, Y. Jang, S. Ji, J. Joo, and Y. Kang, “Quasi-amorphous colloidal structures for electrically tunable full-color photonic pixels with angle-independency,” Adv. Mater. 22, 4973–4977 (2010).
[CrossRef] [PubMed]

Compere, A. L.

W. L. Griffith, R. Triolo, and A. L. Compere, “Analytical scattering function of a polydisperse Percus-Yevick fluid with Schulz -(Γ-) distributed diameters,” Phys. Rev. A 35, 2200–2206 (1987).
[CrossRef] [PubMed]

Debenedetti, P. G.

S. Torquato, T. M. Truskett, and P. G. Debenedetti, “Is random close packing of spheres well defined?,”Phys. Rev. Lett. 84, 2064–2067 (2000).
[CrossRef] [PubMed]

Dinsmore, A. D.

X. T. Peng and A. D. Dinsmore, “Light propagation in strongly scattering, random colloidal films: the role of the packing geometry,” Phys. Rev. Lett. 99, 143902 (2007).
[CrossRef] [PubMed]

Dong, B. Q.

Dufresne, E. R.

H. Noh, S. F. Liew, V. Saranathan, R. O. Prum, S. G. J. Mochrie, E. R. Dufresne, and H. Cao, “How non-iridescent colors are generated by quasi-ordered structures of bird feathers,” Adv. Mater. 22, 2871–2880 (2010).
[CrossRef] [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, R. O. Prum, S. G. J. Mochrie, E. R. Dufresne, and H. Cao, “Double scattering of light from biophotonic nanostructures with short-range order,” Opt. Express 18, 11942–11948 (2010).
[CrossRef] [PubMed]

H. Noh, S. F. Liew, V. Saranathan, R. O. Prum, S. G. J. Mochrie, E. R. Dufresne, and H. Cao, “Contribution of double scattering to structural coloration in quasi-ordered nanostructures of bird feathers,” Phys. Rev. E 81, 051923 (2010).
[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]

Dyck, J.

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

Eradat, N.

J. Huang, N. Eradat, M. E. Raikh, and Z. V. Vardeny, “Anomalous coherent backscattering of light from optical photonic crystals,” Phys. Rev. Lett. 86, 4815–4818 (2001).
[CrossRef] [PubMed]

Farrell, R. A.

Forster, J. D.

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]

Fraden, S.

S. Fraden and G. Maret, “Multiple light scattering from concentrated, interacting suspensions,” Phys. Rev. Lett. 65, 512–515 (1990).
[CrossRef] [PubMed]

French, R. H.

L. E. McNeil and R. H. French, “Multiple scattering from rutile TiO2 particles,” Acta Mater. 48, 4571–4576 (2000).
[CrossRef]

Froufe-Pérez, L. S.

P. D. García, R. Sapienza, L. S. Froufe-Pérez, and C. López, “Strong dispersive effects in the light-scattering mean free path in photonic gaps,” Phys. Rev. B 79, 241109 (2009).
[CrossRef]

Gao, G.-J.

G.-J. Gao, J. Blawzdziewicz, and C. S. O’Hern, “Frequency distribution of mechanically stable disk packings,” Phys. Rev. E 74, 061304 (2006).
[CrossRef]

García, P. D.

P. D. García, R. Sapienza, L. S. Froufe-Pérez, and C. López, “Strong dispersive effects in the light-scattering mean free path in photonic gaps,” Phys. Rev. B 79, 241109 (2009).
[CrossRef]

Go, D.

I. Lee, D. Kim, J. Kal, H. Baek, D. Kwak, D. Go, E. Kim, C. Kang, J. Chung, Y. Jang, S. Ji, J. Joo, and Y. Kang, “Quasi-amorphous colloidal structures for electrically tunable full-color photonic pixels with angle-independency,” Adv. Mater. 22, 4973–4977 (2010).
[CrossRef] [PubMed]

Griffith, W. L.

W. L. Griffith, R. Triolo, and A. L. Compere, “Analytical scattering function of a polydisperse Percus-Yevick fluid with Schulz -(Γ-) distributed diameters,” Phys. Rev. A 35, 2200–2206 (1987).
[CrossRef] [PubMed]

Hart, R. W.

Honda, M.

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

Huang, J.

J. Huang, N. Eradat, M. E. Raikh, and Z. V. Vardeny, “Anomalous coherent backscattering of light from optical photonic crystals,” Phys. Rev. Lett. 86, 4815–4818 (2001).
[CrossRef] [PubMed]

Huang, Y.

Y. Huang, Z. Sun, and E. M. Sevick-Muraca, “Assessment of electrostatic interactions in dense colloidal suspensions with multiply scattered light,” Langmuir 18, 2048–2053 (2002).
[CrossRef]

Imran, A. B.

M. H. Ur-Rashid, A. B. Imran, T. Seki, M. Ishii, H. Nakamura, and Y. Takeoka, “Angle-independent structural color in colloidal amorphous arrays,” Chemphyschem 11, 579–583 (2010).
[CrossRef]

Inaba, A.

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

Ishii, M.

M. H. Ur-Rashid, A. B. Imran, T. Seki, M. Ishii, H. Nakamura, and Y. Takeoka, “Angle-independent structural color in colloidal amorphous arrays,” Chemphyschem 11, 579–583 (2010).
[CrossRef]

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

Jang, Y.

I. Lee, D. Kim, J. Kal, H. Baek, D. Kwak, D. Go, E. Kim, C. Kang, J. Chung, Y. Jang, S. Ji, J. Joo, and Y. Kang, “Quasi-amorphous colloidal structures for electrically tunable full-color photonic pixels with angle-independency,” Adv. Mater. 22, 4973–4977 (2010).
[CrossRef] [PubMed]

Ji, S.

I. Lee, D. Kim, J. Kal, H. Baek, D. Kwak, D. Go, E. Kim, C. Kang, J. Chung, Y. Jang, S. Ji, J. Joo, and Y. Kang, “Quasi-amorphous colloidal structures for electrically tunable full-color photonic pixels with angle-independency,” Adv. Mater. 22, 4973–4977 (2010).
[CrossRef] [PubMed]

Jiang, L. P.

Jockers, K.

E. V. Petrova, V.P. Tishkovets, and K. Jockers, “Interaction of particles in the near field and opposition effects in regolith-like surfaces,” Solar Syst. Res. 43, 100–115 (2009).
[CrossRef]

Joo, J.

I. Lee, D. Kim, J. Kal, H. Baek, D. Kwak, D. Go, E. Kim, C. Kang, J. Chung, Y. Jang, S. Ji, J. Joo, and Y. Kang, “Quasi-amorphous colloidal structures for electrically tunable full-color photonic pixels with angle-independency,” Adv. Mater. 22, 4973–4977 (2010).
[CrossRef] [PubMed]

Kal, J.

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H. Noh, S. F. Liew, V. Saranathan, R. O. Prum, S. G. J. Mochrie, E. R. Dufresne, and H. Cao, “Double scattering of light from biophotonic nanostructures with short-range order,” Opt. Express 18, 11942–11948 (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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H. Noh, S. F. Liew, V. Saranathan, R. O. Prum, S. G. J. Mochrie, E. R. Dufresne, and H. Cao, “Double scattering of light from biophotonic nanostructures with short-range order,” Opt. Express 18, 11942–11948 (2010).
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H. Noh, S. F. Liew, V. Saranathan, R. O. Prum, S. G. J. Mochrie, E. R. Dufresne, and H. Cao, “Contribution of double scattering to structural coloration in quasi-ordered nanostructures of bird feathers,” Phys. Rev. E 81, 051923 (2010).
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H. Noh, S. F. Liew, V. Saranathan, R. O. Prum, S. G. J. Mochrie, E. R. Dufresne, and H. Cao, “How non-iridescent colors are generated by quasi-ordered structures of bird feathers,” Adv. Mater. 22, 2871–2880 (2010).
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L. F. Rojas-Ochoa, J. M. Mendez-Alcaraz, J. J. Sáenz, P. Schurtenberger, and F. Scheffold, “Photonic properties of strongly correlated colloidal liquids,” Phys. Rev. Lett. 93, 073903 (2004).
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M. H. Ur-Rashid, A. B. Imran, T. Seki, M. Ishii, H. Nakamura, and Y. Takeoka, “Angle-independent structural color in colloidal amorphous arrays,” Chemphyschem 11, 579–583 (2010).
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M. H. Ur-Rashid, A. B. Imran, T. Seki, M. Ishii, H. Nakamura, and Y. Takeoka, “Angle-independent structural color in colloidal amorphous arrays,” Chemphyschem 11, 579–583 (2010).
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ACS Appl. Mater. Interfaces (1)

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

Acta Mater. (1)

L. E. McNeil and R. H. French, “Multiple scattering from rutile TiO2 particles,” Acta Mater. 48, 4571–4576 (2000).
[CrossRef]

Adv. Mater. (3)

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

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Figures (5)

Fig. 1
Fig. 1

(a) Transmission electron micrograph (TEM) showing the amorphous photonic structure (right part) in a feather barb of Cotinga maynana that produces blue color. Uniform spherical air cavities (white) are closely packed in β-keratin (grey). The structure is isotropic and has only short-range order. (b) Scanning electron micrograph (SEM) of our biomimetic sample made of random close-packed polystyrene spheres of two sizes. Inset is a photo image of the entire sample.

Fig. 2
Fig. 2

Small Angle X-ray Scattering (SAXS) measurement of the biomimetic sample. (a) SAXS pattern (inset) showing an isotropic ring pattern. The azimuthal-averaged SAXS intensity (black solid line in main panel) reveals the existence of a dominant spatial frequency in the structure. The red-dashed curve is obtained from the power Fourier spectrum of a computer-simulated structure shown in Fig. 4(a). (b) Log-linear plot of SAXS intensity (black solid line) at high q value featuring the oscillation and beating, caused by the form factors of bi-disperse spheres. Red dashed curve is from the calculation with form factors of two spheres with diameters 265 nm and 223 nm. The red dashed curve is shifted vertically for better comparison.

Fig. 3
Fig. 3

(a) Coherent backscattering (CBS) measurement (a) CBS intensity Is vs. scattering angle θB , measured at λ = 660 nm (red triangle), 580 nm (orange circle) and 473 nm (blue square). θB = 0 in the backscattering direction. Black solid lines represent the fitted curves. (b) Measured (black square) and estimated (lines) transport mean free path lt vs. wavelength λ. Green dash-dots curve represents lt estimated without short-range order and near-field effects, blue dashed line is with short-range order but no near-field effects, and red solid curve is with both.

Fig. 4
Fig. 4

(a) Computer-simulated structure of random close-packed spheres of diameters 265 nm (blue) and 223 nm (yellow). (b) Partial structure factors computed for the structure in (a). Blue solid curve is S 11, green dashed curve S 22, and red dash-dots curve S 12.

Fig. 5
Fig. 5

(a) Near-field effects on form factors can be included in an effective background refractive index nb , whose value is calculated from Eq. (6). It approaches the refractive index of air at short wavelength, and that of a homogenized medium at long-wavelength. The wavelength range of our CBS measurement is highlighted with color. For comparison, the value of nb obtained from Ref. [28] is plotted with blue dashed line. (b) Calculated scattering efficiency Qsca of a dielectric sphere in different backgrounds. The sphere has a diameter 244 nm and a refractive index of 1.58. Qsca = σsca /σgeo is the ratio of the scattering cross section σsca to the geometical cross section σgeo . The refractive index of the background is equal to nb from Eq. (6) (black solid line), nh = 1.39 of the homogenized medium (blue dashed line), and that of air 1 (red dash-dotted line).

Equations (6)

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σ t = π k 2 0 π B ( θ ) sin θ ( 1 cos θ ) d θ ,
B ( θ ) = x F 11 ( θ ) S 11 ( θ ) + ( 1 x ) F 22 ( θ ) S 22 ( θ ) + 2 x ( 1 x ) F 12 ( θ ) S 12 ( θ ) .
S 11 ( q ) = 1 N 1 N 1 n , m e i q ( r n ( 1 ) r m ( 1 ) ) N 1 N 1 δ ( q ) ,
S 22 ( q ) = 1 N 2 N 2 n , m e i q ( r n ( 2 ) r m ( 2 ) ) N 2 N 2 δ ( q ) ,
S 12 ( q ) = 1 N 1 N 2 n , m e i q ( r n ( 1 ) r m ( 2 ) ) N 1 N 2 δ ( q ) .
n b = 0 n ( r ) e α r / λ o r 2 d r 0 e α r / λ o r 2 d r

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