Abstract

We propose a semi-infinite 1-D photonic crystal approach for designing artificial reflectors which aim to reproduce color changes with the angle of incidence found in biological periodic multilayer templates. We show that both the dominant reflected wavelength and the photonic bandgap can be predicted and that these predictions agree with exact calculations of reflectance spectra for a finite multilayer structure. In order to help the designer, the concept of spectral richness of angle-tuned color-selecting reflectors is introduced and color changes with angle are displayed in a chromaticity diagram. The usefulness of the photonic crystal approach is demonstrated by modelling a biological template (found in the cuticle of Chrysochora vittata beetle) and by designing a bio-inspired artificial reflector which reproduces the visual aspect of the template. The bio-inspired novel aspect of the design relies on the strong unbalance between the thicknesses of the two layers forming the unit cell.

© 2006 Optical Society of America

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References

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  1. Serge Berthier, Les couleurs des Papillons, ou l’impérative beauté : propriétés optiques des ailes de papillons (Springer-Verlag, Paris, 2000).
    [PubMed]
  2. S. Kinoshita and S. Yoshioka (Eds), Structural Colors in Biological Systems: Principles and Applications (Osaka Univ. Press, Osaka, 2005).
  3. J.-P. Vigneron, M. Rassart, C. Vandenbem, V. Lousse, O. Deparis, L. P. Biró, D. Dedouaire, A. Cornet, P. Defrance, "Spectral filtering of visible light by the cuticle of metallic woodboring beetles and microfabrication of a matching bioinspired material," to be published in Phys. Rev. E, 2006.
  4. European NEST STREP BioPhot project (contract n°12915).
  5. P. Yeh, Optical waves in layered media, 2nd Ed. (John Wiley & Sons, Hoboken N. J., 2005).
  6. A. Dereux, J.-P. Vigneron, Ph. Lambin, A. A. Lucas, "Polaritons in semiconductor multilayered materials," Phys. Rev. B 38, 5438 (1988).
    [CrossRef]
  7. V. L. Welch, J.-P. Vigneron, A. R. Parker, "The cause of coloration in the ctenophore Beroë cucumis," Current Biology,  15, R985 (2005).
    [CrossRef]
  8. J.P. Vigneron, J.-F. Colomer, N. Vigneron and V. Lousse, "Natural layer-by-layer photonic structure in the squamae of Hoplia coerulea (Coleoptera)," Phys. Rev. E 72, 061904 (2005).
    [CrossRef]
  9. B. Gralak, G. Tayeb, S. Enoch, "Morpho butterflies wings color modeled with lamellar grating theory," Opt. Express 9, 567 (2001).
    [CrossRef] [PubMed]

2005

V. L. Welch, J.-P. Vigneron, A. R. Parker, "The cause of coloration in the ctenophore Beroë cucumis," Current Biology,  15, R985 (2005).
[CrossRef]

J.P. Vigneron, J.-F. Colomer, N. Vigneron and V. Lousse, "Natural layer-by-layer photonic structure in the squamae of Hoplia coerulea (Coleoptera)," Phys. Rev. E 72, 061904 (2005).
[CrossRef]

2001

1988

A. Dereux, J.-P. Vigneron, Ph. Lambin, A. A. Lucas, "Polaritons in semiconductor multilayered materials," Phys. Rev. B 38, 5438 (1988).
[CrossRef]

Colomer, J.-F.

J.P. Vigneron, J.-F. Colomer, N. Vigneron and V. Lousse, "Natural layer-by-layer photonic structure in the squamae of Hoplia coerulea (Coleoptera)," Phys. Rev. E 72, 061904 (2005).
[CrossRef]

Dereux, A.

A. Dereux, J.-P. Vigneron, Ph. Lambin, A. A. Lucas, "Polaritons in semiconductor multilayered materials," Phys. Rev. B 38, 5438 (1988).
[CrossRef]

Enoch, S.

Gralak, B.

Lambin, Ph.

A. Dereux, J.-P. Vigneron, Ph. Lambin, A. A. Lucas, "Polaritons in semiconductor multilayered materials," Phys. Rev. B 38, 5438 (1988).
[CrossRef]

Lousse, V.

J.P. Vigneron, J.-F. Colomer, N. Vigneron and V. Lousse, "Natural layer-by-layer photonic structure in the squamae of Hoplia coerulea (Coleoptera)," Phys. Rev. E 72, 061904 (2005).
[CrossRef]

Lucas, A. A.

A. Dereux, J.-P. Vigneron, Ph. Lambin, A. A. Lucas, "Polaritons in semiconductor multilayered materials," Phys. Rev. B 38, 5438 (1988).
[CrossRef]

Parker, A. R.

V. L. Welch, J.-P. Vigneron, A. R. Parker, "The cause of coloration in the ctenophore Beroë cucumis," Current Biology,  15, R985 (2005).
[CrossRef]

Tayeb, G.

Vigneron, J.P.

J.P. Vigneron, J.-F. Colomer, N. Vigneron and V. Lousse, "Natural layer-by-layer photonic structure in the squamae of Hoplia coerulea (Coleoptera)," Phys. Rev. E 72, 061904 (2005).
[CrossRef]

Vigneron, J.-P.

V. L. Welch, J.-P. Vigneron, A. R. Parker, "The cause of coloration in the ctenophore Beroë cucumis," Current Biology,  15, R985 (2005).
[CrossRef]

A. Dereux, J.-P. Vigneron, Ph. Lambin, A. A. Lucas, "Polaritons in semiconductor multilayered materials," Phys. Rev. B 38, 5438 (1988).
[CrossRef]

Vigneron, N.

J.P. Vigneron, J.-F. Colomer, N. Vigneron and V. Lousse, "Natural layer-by-layer photonic structure in the squamae of Hoplia coerulea (Coleoptera)," Phys. Rev. E 72, 061904 (2005).
[CrossRef]

Welch, V. L.

V. L. Welch, J.-P. Vigneron, A. R. Parker, "The cause of coloration in the ctenophore Beroë cucumis," Current Biology,  15, R985 (2005).
[CrossRef]

Current Biology

V. L. Welch, J.-P. Vigneron, A. R. Parker, "The cause of coloration in the ctenophore Beroë cucumis," Current Biology,  15, R985 (2005).
[CrossRef]

Opt. Express

Phys. Rev. B

A. Dereux, J.-P. Vigneron, Ph. Lambin, A. A. Lucas, "Polaritons in semiconductor multilayered materials," Phys. Rev. B 38, 5438 (1988).
[CrossRef]

Phys. Rev. E

J.P. Vigneron, J.-F. Colomer, N. Vigneron and V. Lousse, "Natural layer-by-layer photonic structure in the squamae of Hoplia coerulea (Coleoptera)," Phys. Rev. E 72, 061904 (2005).
[CrossRef]

Other

Serge Berthier, Les couleurs des Papillons, ou l’impérative beauté : propriétés optiques des ailes de papillons (Springer-Verlag, Paris, 2000).
[PubMed]

S. Kinoshita and S. Yoshioka (Eds), Structural Colors in Biological Systems: Principles and Applications (Osaka Univ. Press, Osaka, 2005).

J.-P. Vigneron, M. Rassart, C. Vandenbem, V. Lousse, O. Deparis, L. P. Biró, D. Dedouaire, A. Cornet, P. Defrance, "Spectral filtering of visible light by the cuticle of metallic woodboring beetles and microfabrication of a matching bioinspired material," to be published in Phys. Rev. E, 2006.

European NEST STREP BioPhot project (contract n°12915).

P. Yeh, Optical waves in layered media, 2nd Ed. (John Wiley & Sons, Hoboken N. J., 2005).

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Figures (8)

Fig. 1
Fig. 1

Reflectance spectra measured on a biological sample (cuticle of Chrysochroa vittata beetle - ventral side) at angles of incidence equal to 20° (magenta line), 30° (red line) and 45° (green line) using TE-polarized light. A metallic mirror was used as reference in a variable-angle specular reflectance measurement setup. Note the shift of the dominant reflected wavelength towards the shorter wavelengths as the angle of incidence was increased.

Fig. 2.
Fig. 2.

Layer thicknesses (top graph: solid and dotted lines correspond to low-index layer thickness d 1 and high-index layer thickness d 2, respectively), spectral richness (middle graph) and bandgap normalized width at 0° incidence (bottom graph) as functions of layer thickness ratio in the case of an air/chitin photonic crystal (n 1 = 1.0, n 2 = 1.56). The spectral richness was calculated at angles of incidence increasing from 0° to 80° by steps of 10°. Vertical lines correspond to a structure made of 10-nm thick air layers and 194-nm thick chitin layers. All calculations were done for TE polarization.

Fig. 3.
Fig. 3.

Layer thicknesses (top graph: solid and dotted lines correspond to low-index layer thickness d 1 and high-index layer thickness d 2, respectively), spectral richness (middle graph) and normalized bandgap width at 0° incidence (bottom graph) as functions of layer thickness ratio in the case of a SiO2/TiO2 photonic crystal (n 1 = 1.5, n 2 = 2.7). The spectral richness was calculated at angles of incidence increasing from 0° to 80° by steps of 10°. Vertical lines correspond to a structure made of 10-nm thick TiO2 layers and 194-nm thick SiO2 layers. All calculations were done for TE polarization.

Fig. 4.
Fig. 4.

Spectral richness at an angle of incidence of 60° (top graph) and normalized bandgap width at 0° incidence (bottom graph) as functions of layer thickness ratio in the case of photonic crystal structures with two different values of the basic index (dotted lines: n 1 = 1.0, solid lines: n 1 = 1.5) and various values of index contrast: c = n 2/n 1 = 1.20 (blue lines), 1.56 (green lines), 1.80 (red lines), 2.20 (cyan lines). All calculations were done for TE polarization.

Fig. 5.
Fig. 5.

Dominant reflected wavelength as function of angle of incidence (top graph) and reflectance spectra (bottom graph) at angles of incidence equal to 0° (magenta line), 30° (red line) and 60° (green line) in the case of a periodic multilayer air/chitin structure (n 1 = 1.0, n 2 = 1.56, d 1 = 10 nm, d 2 = 194 nm). Values of the dominant reflected wavelengths were calculated by eq. 3 (crosses) or extracted from reflectance spectra (circles). The reflectance spectra were calculated for a structure of 20 periods by solving Maxwell’s equations exactly. All calculations were done for TE polarization.

Fig. 6.
Fig. 6.

Dominant reflected wavelength as function of angle of incidence (top graph) and reflectance spectra (bottom graph) at angles of incidence equal to 0° (magenta line), 30° (red line) and 60° (green line) in the case of a periodic multilayer SiO2/TiO2 structure (n 1 = 1.5, n 2 = 2.7, d 1 = 194 nm, d 2 = 10 nm). Values of the dominant reflected wavelengths were calculated by eq. 3 (crosses) or extracted from reflectance spectra (circles). The reflectance spectra were calculated for a structure of 20 periods by solving Maxwell’s equations exactly. All calculations were done for TE polarization.

Fig. 7.
Fig. 7.

Evolution of chromaticity coordinates as the angle of incidence is increased from 0° to 80° by steps of 5° in the case of 20-periods air/chitin multilayer structure (n 1 = 1.0, n 2 = 1.56, d 1 = 10 nm, d 2 = 194 nm).

Fig. 8.
Fig. 8.

Evolution of the chromaticity coordinates as the angle of incidence is increased from 0° to 80° by steps of 5° in the case of a 20-periods SiO2/TiO2 multilayer structure (n 1 = 1.5, n 2 = 2.7, d 1 = 194 nm, d 2 = 10 nm).

Equations (12)

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n ¯ = ( n 1 2 d 1 + n 2 2 d 2 d 1 + d 2 ) 1 / 2 = n 1 ( 1 + r c 2 1 + r ) 1 / 2
n ¯ = ( d 1 / n 1 2 + d 2 n 2 2 d 1 + d 2 ) 1 / 2 = n 1 ( 1 + r 1 + r / c 2 ) 1 / 2
k z = m π a = [ n ¯ ω c ] 2 k y 2
λ θ = 2 a n ¯ 2 sin 2 θ m .
d 1 = m λ θ = 0 2 n 1 [ ( 1 + r c 2 ) ( 1 + r ) ] 1 / 2
d 1 = m λ θ = 0 2 n 1 [ ( 1 + r / c 2 ) ( 1 + r ) 3 ] 1 / 2
Θ = λ θ λ θ = 0 = [ 1 1 + r 1 + r c 2 sin 2 θ n 1 2 ] 1 / 2 .
Θ = λ θ λ θ = 0 = [ 1 1 + r / c 2 1 + r sin 2 θ n 1 2 ] 1 / 2 .
Δ = Δ ω gap ω 0 = ε ( 1 ) + ε ( 1 ) ε ( 0 ) = 2 π ( 1 + r ) ( 1 c 2 ) sin ( π 1 + r ) 1 + r c 2 .
X = k R ( λ ) L ( λ ) x ( λ )
Y = k R ( λ ) L ( λ ) y ( λ )
Z = k R ( λ ) L ( λ ) z ( λ )

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