Abstract

It has long been known that random height variations of a repeated nanoscale structure can give rise to smooth angular color variations instead of the well-known diffraction pattern experienced if no randomization is present. However, until now there have been few publications trying to explain this and similar phenomena taking outset in electromagnetic theory. This paper presents a method for analyzing far-field reflection from a surface constructed by translated instances of a given structure. Several examples of the effect of random translations are given.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Hooke, Micrographia, http://www.gutenberg.org (1665).
  2. A. R. Parker, “515 million years of structural colour,” J. Opt. A 2, R15–R28 (2000).
    [CrossRef]
  3. P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single Morpho butterfly scales,” Proc. R. Soc. B 266, 1403–1411 (1999).
    [CrossRef]
  4. N. Okada, D. Zhu, D. Cai, J. B. Cole, M. Kambe, and S. Kinoshita, “Rendering Morpho butterflies based on high accuracy nano-optical simulation,” J. Opt. 42, 25–36 (2013).
    [CrossRef]
  5. C. W. Mason, “Structural colors in insects. II,” J. Phys. Chem., 31, 321–354 (1927).
  6. W. Lippert and K. Gentil, “Über Lamellare Feinstrukturen bei den Schillerschuppen der Schmetterlinge vom Urania- und Morpho-typ Z,” Morph. Ökol. Tiere 48, 115–122 (1959).
  7. A. R. Parker, T. Lenau, and A. Saito, “Biomimetics of optical nanostructures,” in Biomimetics in Photonics (CRC Press, 2012), pp. 55–115.
  8. S. Kinoshita, D. Zhu, and A. Saito, “Modeling and simulation of structural colors,” in Biomimetics in Photonics (CRC Press, 2012), pp. 191–242.
  9. P. Licinio, “Diffraction by disordered gratings and the DebyeWaller effect,” Am. J. Phys. 67, 1013–1016 (1999).
    [CrossRef]
  10. J. M. Rico-García and L. M. Sanchez-Brea, “Binary gratings with random heights,” Appl. Opt. 48, 3062–3069 (2009).
    [CrossRef]
  11. T. Buß, J. Teisseire, and S. Mazoyer, “Controlled angular redirection of light via nanoimprinted disordered gratings,” Appl. Opt. 52, 709–716 (2013).
    [CrossRef]
  12. F. Pratesi, M. Burresi, F. Riboli, K. Vynck, and D. S. Wiersma, “Disordered photonic structures for light harvesting in solar cells,” Opt. Express 21, A460–A468 (2013).
    [CrossRef]
  13. R. T. Lee and G. S. Smith, “Detailed electromagnetic simulation for the structural color of butterfly wings,” Appl. Opt. 48, 4177–4190 (2009).
    [CrossRef]
  14. A. Saito, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass-production and control of the Morpho-butterfly’s blue,” Ad. Fabric. Technol. Micro/Nano Optics and Photonics II 7205, 720506 (2009).
  15. A. Saito, M. Yonezawa, J. Murase, S. Juodkazis, V. Mizeikis, M. Akai-Kasaya, and Y. Kuwahara, “Numerical analysis on the optical role of nano-randomness on the Morpho butterfly’s scale,” J. Nanosci. Nanotech. 11, 2785–2792 (2011).
  16. M. A. Steindorfer and V. Schmidt, “Detailed simulation of structural color generation inspired by the Morpho butterfly,” Opt. Express 20, 21485–21494 (2012).
    [CrossRef]
  17. C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012).
  18. M. Zhou, S. B. Sørensen, E. Jørgensen, P. Meincke, O. S. Kim, and O. Breinbjerg, “An accurate technique for calculation of radiation from printed reflect arrays,” IEEE Antennas and Wireless Propagation Lett. 10, 1081–1084 (2011).
  19. 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, 051924 (2009).
    [CrossRef]
  20. P. Dutré, K. Bala, and P. Bekaert, Advanced Global Illumination (A K Peters, 2006).
  21. J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
    [CrossRef]
  22. T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).
  23. R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw Hill, 2000).
  24. S. Kinoshita, S. Yoshioka, Y. Fujii, and N. Okamoto, “Photophysics of structural color in the Morpho butterflies,” Forma, 17, 103–121 (2002).
  25. R. T. Lee, “A novel method for incorporating periodic boundaries into the FDTD method and the application to the study of structural color of insects,” Ph.D. thesis (Georgia Institute of Technology, 2009).
  26. R. S. Berns, F. W. Billmeyer, and M. Saltzman, Billmeyer and Saltzman's Principles of Color Technology (Wiley-Interscience Publication, 2000).
  27. J. Andkjær, V. E. Johansen, K. S. Friis, and O. Sigmund, “Inverse design of nanostructured surfaces for color effects,” J. Opt. Soc. Am. B 31, 164–174 (2014).
    [CrossRef]

2014

2013

2012

2011

M. Zhou, S. B. Sørensen, E. Jørgensen, P. Meincke, O. S. Kim, and O. Breinbjerg, “An accurate technique for calculation of radiation from printed reflect arrays,” IEEE Antennas and Wireless Propagation Lett. 10, 1081–1084 (2011).

A. Saito, M. Yonezawa, J. Murase, S. Juodkazis, V. Mizeikis, M. Akai-Kasaya, and Y. Kuwahara, “Numerical analysis on the optical role of nano-randomness on the Morpho butterfly’s scale,” J. Nanosci. Nanotech. 11, 2785–2792 (2011).

2009

J. M. Rico-García and L. M. Sanchez-Brea, “Binary gratings with random heights,” Appl. Opt. 48, 3062–3069 (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, 051924 (2009).
[CrossRef]

R. T. Lee and G. S. Smith, “Detailed electromagnetic simulation for the structural color of butterfly wings,” Appl. Opt. 48, 4177–4190 (2009).
[CrossRef]

A. Saito, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass-production and control of the Morpho-butterfly’s blue,” Ad. Fabric. Technol. Micro/Nano Optics and Photonics II 7205, 720506 (2009).

2002

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

2000

A. R. Parker, “515 million years of structural colour,” J. Opt. A 2, R15–R28 (2000).
[CrossRef]

1999

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

J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, “Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory,” Appl. Opt. 38, 6469–6481 (1999).
[CrossRef]

P. Licinio, “Diffraction by disordered gratings and the DebyeWaller effect,” Am. J. Phys. 67, 1013–1016 (1999).
[CrossRef]

1959

W. Lippert and K. Gentil, “Über Lamellare Feinstrukturen bei den Schillerschuppen der Schmetterlinge vom Urania- und Morpho-typ Z,” Morph. Ökol. Tiere 48, 115–122 (1959).

1927

C. W. Mason, “Structural colors in insects. II,” J. Phys. Chem., 31, 321–354 (1927).

Akai-Kasaya, M.

A. Saito, M. Yonezawa, J. Murase, S. Juodkazis, V. Mizeikis, M. Akai-Kasaya, and Y. Kuwahara, “Numerical analysis on the optical role of nano-randomness on the Morpho butterfly’s scale,” J. Nanosci. Nanotech. 11, 2785–2792 (2011).

A. Saito, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass-production and control of the Morpho-butterfly’s blue,” Ad. Fabric. Technol. Micro/Nano Optics and Photonics II 7205, 720506 (2009).

Andkjær, J.

Antonakakis, T.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Bada, F.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Bala, K.

P. Dutré, K. Bala, and P. Bekaert, Advanced Global Illumination (A K Peters, 2006).

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012).

Bekaert, P.

P. Dutré, K. Bala, and P. Bekaert, Advanced Global Illumination (A K Peters, 2006).

Belkhir, A.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Berns, R. S.

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

Billmeyer, F. W.

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

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw Hill, 2000).

Breinbjerg, O.

M. Zhou, S. B. Sørensen, E. Jørgensen, P. Meincke, O. S. Kim, and O. Breinbjerg, “An accurate technique for calculation of radiation from printed reflect arrays,” IEEE Antennas and Wireless Propagation Lett. 10, 1081–1084 (2011).

Burresi, M.

Buß, T.

Cai, D.

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

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, 051924 (2009).
[CrossRef]

Cherednichenko, K.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

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, 051924 (2009).
[CrossRef]

Cole, J. B.

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

Cooper, S.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Craster, R.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Demesy, G.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

DeSanto, J.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Dutré, P.

P. Dutré, K. Bala, and P. Bekaert, Advanced Global Illumination (A K Peters, 2006).

Friis, K. S.

Fujii, Y.

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

Gentil, K.

W. Lippert and K. Gentil, “Über Lamellare Feinstrukturen bei den Schillerschuppen der Schmetterlinge vom Urania- und Morpho-typ Z,” Morph. Ökol. Tiere 48, 115–122 (1959).

Gralak, B.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Granet, G.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Guenneau, S.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Harvey, J. E.

Ishikawa, Y.

A. Saito, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass-production and control of the Morpho-butterfly’s blue,” Ad. Fabric. Technol. Micro/Nano Optics and Photonics II 7205, 720506 (2009).

Johansen, V. E.

Jørgensen, E.

M. Zhou, S. B. Sørensen, E. Jørgensen, P. Meincke, O. S. Kim, and O. Breinbjerg, “An accurate technique for calculation of radiation from printed reflect arrays,” IEEE Antennas and Wireless Propagation Lett. 10, 1081–1084 (2011).

Juodkazis, S.

A. Saito, M. Yonezawa, J. Murase, S. Juodkazis, V. Mizeikis, M. Akai-Kasaya, and Y. Kuwahara, “Numerical analysis on the optical role of nano-randomness on the Morpho butterfly’s scale,” J. Nanosci. Nanotech. 11, 2785–2792 (2011).

Kambe, M.

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

Kim, O. S.

M. Zhou, S. B. Sørensen, E. Jørgensen, P. Meincke, O. S. Kim, and O. Breinbjerg, “An accurate technique for calculation of radiation from printed reflect arrays,” IEEE Antennas and Wireless Propagation Lett. 10, 1081–1084 (2011).

Kinoshita, S.

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

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, 051924 (2009).
[CrossRef]

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

S. Kinoshita, D. Zhu, and A. Saito, “Modeling and simulation of structural colors,” in Biomimetics in Photonics (CRC Press, 2012), pp. 191–242.

Krywonos, A.

Kuwahara, Y.

A. Saito, M. Yonezawa, J. Murase, S. Juodkazis, V. Mizeikis, M. Akai-Kasaya, and Y. Kuwahara, “Numerical analysis on the optical role of nano-randomness on the Morpho butterfly’s scale,” J. Nanosci. Nanotech. 11, 2785–2792 (2011).

A. Saito, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass-production and control of the Morpho-butterfly’s blue,” Ad. Fabric. Technol. Micro/Nano Optics and Photonics II 7205, 720506 (2009).

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. B 266, 1403–1411 (1999).
[CrossRef]

Lee, R. T.

R. T. Lee and G. S. Smith, “Detailed electromagnetic simulation for the structural color of butterfly wings,” Appl. Opt. 48, 4177–4190 (2009).
[CrossRef]

R. T. Lee, “A novel method for incorporating periodic boundaries into the FDTD method and the application to the study of structural color of insects,” Ph.D. thesis (Georgia Institute of Technology, 2009).

Lenau, T.

A. R. Parker, T. Lenau, and A. Saito, “Biomimetics of optical nanostructures,” in Biomimetics in Photonics (CRC Press, 2012), pp. 55–115.

Licinio, P.

P. Licinio, “Diffraction by disordered gratings and the DebyeWaller effect,” Am. J. Phys. 67, 1013–1016 (1999).
[CrossRef]

Lippert, W.

W. Lippert and K. Gentil, “Über Lamellare Feinstrukturen bei den Schillerschuppen der Schmetterlinge vom Urania- und Morpho-typ Z,” Morph. Ökol. Tiere 48, 115–122 (1959).

Mason, C. W.

C. W. Mason, “Structural colors in insects. II,” J. Phys. Chem., 31, 321–354 (1927).

Maystre, D.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Mazoyer, S.

Meincke, P.

M. Zhou, S. B. Sørensen, E. Jørgensen, P. Meincke, O. S. Kim, and O. Breinbjerg, “An accurate technique for calculation of radiation from printed reflect arrays,” IEEE Antennas and Wireless Propagation Lett. 10, 1081–1084 (2011).

Miyamura, Y.

A. Saito, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass-production and control of the Morpho-butterfly’s blue,” Ad. Fabric. Technol. Micro/Nano Optics and Photonics II 7205, 720506 (2009).

Mizeikis, V.

A. Saito, M. Yonezawa, J. Murase, S. Juodkazis, V. Mizeikis, M. Akai-Kasaya, and Y. Kuwahara, “Numerical analysis on the optical role of nano-randomness on the Morpho butterfly’s scale,” J. Nanosci. Nanotech. 11, 2785–2792 (2011).

Murase, J.

A. Saito, M. Yonezawa, J. Murase, S. Juodkazis, V. Mizeikis, M. Akai-Kasaya, and Y. Kuwahara, “Numerical analysis on the optical role of nano-randomness on the Morpho butterfly’s scale,” J. Nanosci. Nanotech. 11, 2785–2792 (2011).

A. Saito, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass-production and control of the Morpho-butterfly’s blue,” Ad. Fabric. Technol. Micro/Nano Optics and Photonics II 7205, 720506 (2009).

Nicolet, A.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Okada, N.

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

Okamoto, N.

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

Parker, A. R.

A. R. Parker, “515 million years of structural colour,” J. Opt. A 2, R15–R28 (2000).
[CrossRef]

A. R. Parker, T. Lenau, and A. Saito, “Biomimetics of optical nanostructures,” in Biomimetics in Photonics (CRC Press, 2012), pp. 55–115.

Pratesi, F.

Riboli, F.

Rico-García, J. M.

Saito, A.

A. Saito, M. Yonezawa, J. Murase, S. Juodkazis, V. Mizeikis, M. Akai-Kasaya, and Y. Kuwahara, “Numerical analysis on the optical role of nano-randomness on the Morpho butterfly’s scale,” J. Nanosci. Nanotech. 11, 2785–2792 (2011).

A. Saito, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass-production and control of the Morpho-butterfly’s blue,” Ad. Fabric. Technol. Micro/Nano Optics and Photonics II 7205, 720506 (2009).

S. Kinoshita, D. Zhu, and A. Saito, “Modeling and simulation of structural colors,” in Biomimetics in Photonics (CRC Press, 2012), pp. 191–242.

A. R. Parker, T. Lenau, and A. Saito, “Biomimetics of optical nanostructures,” in Biomimetics in Photonics (CRC Press, 2012), pp. 55–115.

Saltzman, M.

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

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. B 266, 1403–1411 (1999).
[CrossRef]

Sanchez-Brea, L. M.

Schmidt, V.

Sigmund, O.

Smith, G. S.

Sørensen, S. B.

M. Zhou, S. B. Sørensen, E. Jørgensen, P. Meincke, O. S. Kim, and O. Breinbjerg, “An accurate technique for calculation of radiation from printed reflect arrays,” IEEE Antennas and Wireless Propagation Lett. 10, 1081–1084 (2011).

Steindorfer, M. A.

Stout, B.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Teisseire, J.

Thompson, P. L.

Vernold, C. L.

Vial, B.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Vukusic, P.

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

Vynck, K.

Wiersma, D. S.

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. B 266, 1403–1411 (1999).
[CrossRef]

Yonezawa, M.

A. Saito, M. Yonezawa, J. Murase, S. Juodkazis, V. Mizeikis, M. Akai-Kasaya, and Y. Kuwahara, “Numerical analysis on the optical role of nano-randomness on the Morpho butterfly’s scale,” J. Nanosci. Nanotech. 11, 2785–2792 (2011).

Yoshioka, S.

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

Zhou, M.

M. Zhou, S. B. Sørensen, E. Jørgensen, P. Meincke, O. S. Kim, and O. Breinbjerg, “An accurate technique for calculation of radiation from printed reflect arrays,” IEEE Antennas and Wireless Propagation Lett. 10, 1081–1084 (2011).

Zhu, D.

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

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, 051924 (2009).
[CrossRef]

S. Kinoshita, D. Zhu, and A. Saito, “Modeling and simulation of structural colors,” in Biomimetics in Photonics (CRC Press, 2012), pp. 191–242.

Zolla, F.

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

Ad. Fabric. Technol. Micro/Nano Optics and Photonics II

A. Saito, Y. Miyamura, Y. Ishikawa, J. Murase, M. Akai-Kasaya, and Y. Kuwahara, “Reproduction, mass-production and control of the Morpho-butterfly’s blue,” Ad. Fabric. Technol. Micro/Nano Optics and Photonics II 7205, 720506 (2009).

Am. J. Phys.

P. Licinio, “Diffraction by disordered gratings and the DebyeWaller effect,” Am. J. Phys. 67, 1013–1016 (1999).
[CrossRef]

Appl. Opt.

Forma

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

IEEE Antennas and Wireless Propagation Lett.

M. Zhou, S. B. Sørensen, E. Jørgensen, P. Meincke, O. S. Kim, and O. Breinbjerg, “An accurate technique for calculation of radiation from printed reflect arrays,” IEEE Antennas and Wireless Propagation Lett. 10, 1081–1084 (2011).

J. Nanosci. Nanotech.

A. Saito, M. Yonezawa, J. Murase, S. Juodkazis, V. Mizeikis, M. Akai-Kasaya, and Y. Kuwahara, “Numerical analysis on the optical role of nano-randomness on the Morpho butterfly’s scale,” J. Nanosci. Nanotech. 11, 2785–2792 (2011).

J. Opt.

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

J. Opt. A

A. R. Parker, “515 million years of structural colour,” J. Opt. A 2, R15–R28 (2000).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Chem.

C. W. Mason, “Structural colors in insects. II,” J. Phys. Chem., 31, 321–354 (1927).

Morph. Ökol. Tiere

W. Lippert and K. Gentil, “Über Lamellare Feinstrukturen bei den Schillerschuppen der Schmetterlinge vom Urania- und Morpho-typ Z,” Morph. Ökol. Tiere 48, 115–122 (1959).

Opt. Express

Phys. Rev. E

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, 051924 (2009).
[CrossRef]

Proc. R. Soc. B

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

Other

R. Hooke, Micrographia, http://www.gutenberg.org (1665).

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012).

A. R. Parker, T. Lenau, and A. Saito, “Biomimetics of optical nanostructures,” in Biomimetics in Photonics (CRC Press, 2012), pp. 55–115.

S. Kinoshita, D. Zhu, and A. Saito, “Modeling and simulation of structural colors,” in Biomimetics in Photonics (CRC Press, 2012), pp. 191–242.

P. Dutré, K. Bala, and P. Bekaert, Advanced Global Illumination (A K Peters, 2006).

T. Antonakakis, F. Bada, A. Belkhir, K. Cherednichenko, S. Cooper, R. Craster, G. Demesy, J. DeSanto, G. Granet, B. Gralak, S. Guenneau, D. Maystre, A. Nicolet, B. Stout, F. Zolla, and B. Vial, Gratings: Theory and Numeric Applications, 1st ed. (Presses Eniversitaires de Provence, 2012).

R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw Hill, 2000).

R. T. Lee, “A novel method for incorporating periodic boundaries into the FDTD method and the application to the study of structural color of insects,” Ph.D. thesis (Georgia Institute of Technology, 2009).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1.
Fig. 1.

Same structure divided into cells with an air domain above.

Fig. 2.
Fig. 2.

Geometry of the translation of a structure lit by a plane electromagnetic wave.

Fig. 3.
Fig. 3.

Example of how a structure with some unit response (a) is influenced by different SAFs. In (b) the SAF for a strictly periodic structure is shown (the comb function), and the effect of combining the structure giving the unit response in (a) with the pattern giving rise to the SAF in (b) is shown in (c). The arrows indicate that all energy is emitted at discrete points. If the SAF is as it is in (d), then the response will end up as seen in (e).

Fig. 4.
Fig. 4.

(a) Strictly periodic structure. (b) The same structure but with per period height translations.

Fig. 5.
Fig. 5.

(a) Numerical calculation of Eq. (18) with z,n picked uniformly from [0,z,max] and N=100 repetitions and a periodicity in x of x=2λ. By changing x the SAF would either be dilated or constricted such that the grating modes still match Eq. (16). (b) 100 averages of the setup in (a) using Eq. (19).

Fig. 6.
Fig. 6.

Results for the same setup as in Fig. 5(a), but now with z,n taken from a triangular distribution.

Fig. 7.
Fig. 7.

100 averages of the same setup as Fig. 6(a), but for double the interval width of the distribution used in Fig. 6(a).

Fig. 8.
Fig. 8.

Results for the same setup as in Fig. 5(a), but now with z,n only taking the values 0 and z,max.

Fig. 9.
Fig. 9.

Results for the same setup as in Fig. 5(a), but with in-plane movement as specified in Eq. (20).

Fig. 10.
Fig. 10.

Color representation of different SAFs. All unit structures are repeated with a period of 2 μm, and incoherence has been taken into account by averaging over 200 samples. The oscillatory effect seen at large angles is due to the fact that the angular spacing between wavelengths gets larger. Notice how the response for blue is flat for all plots except the uniform random distribution only going to 110 nm.

Fig. 11.
Fig. 11.

Verification of the proposed method by comparison to a full-wave simulation for the Ez polarized case (c.f. [26]). (a) Unit structure with black representing material having εr=3.5 and white being air. (b) Ten random height translations of this structure with the heights chosen as described in the text. Both structures have periodic boundary conditions. The far-field transformation for (a) and (b) are calculated on the lines y=700nm, for x[0,2000]nm and y=1500nm, for x[0,20000]nm, respectively. (c) Unit response for (a) simulated at λ=750nm as well as the SAF calculated from Eq. (22). (d) Comparison of the approximation and full simulation reference.

Equations (22)

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

Hf(θ,ϕ)=nNHnf(θ,ϕ),
Hnf(θ,ϕ)=jkejkr4πrr^×SnJeejkr^(θ,ϕ)·rodSn,
Ef=ηHf×r^,
E=12|Ef×Hf¯|=12η|Hf|2=12η|nNHnf|2,
Δp=k^·Δr/λ.
r0new=r0+Δr,
  Hnf,t(θ,ϕ)=ej2πk^·Δr/λjkejkr4πrr^SnJeejkr^(θ,ϕ)·(ro+Δr)dSn=ejk(k^·Δrr^(θ,ϕ)·Δr)jkejkr4πrr^SnJeejkr^(θ,ϕ)·rodSn=ejk(k^r^(θ,ϕ))·ΔrHnf(θ,ϕ),
E=12η|nejk(k^r^)·ΔrnH0f|2=12η|H0f|2|nejk(k^r^)·Δrn|2=AF.
I(θ,ϕ)=r2E(θ,ϕ),
L(θ,ϕ)=I(θ,ϕ)Ascosθ=r2AscosθE(θ,ϕ),
L(θ,ϕ)=r2η2Ascosθ|H0f(θ,ϕ)|2|nejk(k^r^(θ,ϕ))·Δrn|2=r2η2A0cosθ|H0f(θ,ϕ)|2=unit responseN|1Nnejk(k^r^(θ,ϕ))·Δrn|2=SAF,
L(θ)=rη2d0cosθ|H0f(θ,ϕ=90°)|2N×|1Nnejk(k^r^(θ,ϕ=90°))·Δrn|2,
k^=(0,0,1),Δrn=(nx,0,0),nZ,
SAF(θ)=1NnZejk(k^r^(θ))·Δrn=1NnZejknxsinθ.
nZej2πnxλsinθ=nZδ(xλsinθn)comb(xλsinθ),
xλsinθ=mZ
k^=(0,0,1),Δrn=(nx,0,z,n),nZ,
SAF(θ)=1NnZejk(nxsinθ(cosθ+1)z,n)=1NnZejknxsinθejk(cosθ+1)z,n,
|SAF(θ)|2=|1NnZejknxsinθejk(cosθ+1)z,n|2,
k^=(0,0,1),Δrn=(nx+Δxn,0,0),nZ,
SAF(θ)=1NnZejk(nx+Δxn)sinθ=1NnZejknxsinθejkΔxnsinθ.
k^=(sinθi,0,cosθi),Δrn=(nx,0,z,n),

Metrics