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Morpho butterflies wings color modeled with lamellar grating theory

Boris Gralak, Gérard Tayeb, and Stefan Enoch

Open Access Open Access

Abstract

We describe an approach for converting reflection coefficients of any structure into colors, taking into account human color perception. This procedure is applied to the study of the colors reflected by Morpho rhetenor butterflies wings. The scales of these wings have a tree-like periodic structure which is modeled with the help of a rigorous lamellar grating electromagnetic theory. In this way, we are able to determine the colors reflected by the wing under various illumination conditions.

©2001 Optical Society of America

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Figures (13)
Fig. 1.
Fig. 1. a: Relative energy distribution of the D65 illuminant (blue) and its 5th order polynomial approximation (red) versus the wavelength λ. b: Spectral tristimulus values.

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Fig. 2.
Fig. 2. a: Transmission electron microscope image showing the cross-section through a single Morpho rhetenor scale. Image reprinted from [1], with permission from P. Vukusic and the Royal Society. b: Modeled structure; the two red lines define a grating layer, the optical index of black regions is n, and that of white regions is 1.

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Fig. 3.
Fig. 3. a: k i is the incident wave vector, with angles θ i and φ i . s i and p i are respectively normal and parallel to the incidence plane Π i . The polarization of the incident electric field E i is defined by the angle δi . b: Schematic representation of two conical subregions of the upper hemisphere.

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Fig. 4.
Fig. 4. Characteristic dimensions of the modeled scale (in nanometers). The white regions are filled with air (or with a liquid solvent), and the gray regions are filled with a material with optical index n. In this figure, and using the tree image, we will say that each tree has four branches. The lower layer models the bottom membrane of the scale. Apart from the bottom air layer, all the dimensions are scaled.

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Fig. 5.
Fig. 5. Diffracted efficiencies by a scale in air and in IPA. The reflected angles are between 0 and 180° and the transmitted angles are between 180 and 360°. The normals to the scale correspond to 90° and 270°.

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Fig. 6.
Fig. 6. Total reflection and transmission for both polarizations.

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Fig. 7.
Fig. 7. Reflected colors by a scale with 16 branches in air. The horizontal axis gives the θ r value, from 0 to 80°, and the vertical axis gives the φ r value, from 0 to 180°. The angular shift between two neighboring squares is equal to 10°.

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Fig. 8.
Fig. 8. Same as Fig.7, but the scale is immersed in IPA.

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Fig. 9.
Fig. 9. Characteristic dimensions of the second modeled scale (in nanometers). The nine lower lamellae have the same dimensions as those described in Fig.4, and the upper lamellae have decreasing widths. Apart from the bottom air layer, all the dimensions are scaled.

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Fig. 10.
Fig. 10. Reflected colors by the structure of Fig.9 lying in air.

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Fig. 11.
Fig. 11. Reflected colors by the structure of Fig.9 lying in IPA.

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Fig. 12.
Fig. 12. Reflected energy near the normal when the structure with 16 branches in air is illuminated from the left (φ i =0) and from the right (φ i =180°).

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Fig. 13.
Fig. 13. Field maps of the squared modulus of the total electric field for the structure of Fig.9 illuminated by a E//polarized plane wave in normal incidence.

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Equations (5)

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

X = 1 k ∫ D ( λ ) R ( λ ) x ¯ ( λ ) d λ
Y = 1 k ∫ D ( λ ) R ( λ ) y ¯ ( λ ) d λ
Z = 1 k ∫ D ( λ ) R ( λ ) z ¯ ( λ ) d λ
k = ∫ D ( λ ) y ¯ ( λ ) d λ
[ R G B ] = [ 3.240479 − 1.537150 − 0.498535 − 0.969256 1.875992 0.041556 0.055648 − 0.204043 1.057311 ] [ X Y Z ]
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