Notes from  “A Physically Based Anisotropic Iridescence Model for Rendering Morpho Butterflies Photo-realistically” paper by Iman Sadeghi

Iridescence is an optical phenomenon characterized as the property of surfaces in which hue changes according to the angle from which the surface is viewed.

Ordinary colors are caused by pigments inside the material and insensitive to directions

In contrast, iridescent colors are caused by micro-structures and therefore are sensitive to change of viewing directions. Iridescence is caused by multiple interaction of light with different micro-structures of the material in which phase shift and interference of the reflections modulates the incident light by amplifying or attenuating some frequencies more than the others.

According to studies, the physical origins for biological iridescences are light interference at multiple thin layers and also diffraction of light at gratings

In the case of Morpho butterflies iridescence is caused by the submicron structures on the scales of the wing. These structures cause an extraordinarily uniform color reflectance along a fixed viewing direction direction. The interaction of light with multiple layers of these microscopic structures are the main source of iridescence colors of the wings. Multi-layer nature of those structures plus their lamellar shapes with irregular heights are important aspects in producing those iridescence colors

Besides, it has been shown that the presence of pigment beneath the iridescent scales greatly

enhances the purity of the reflected colors by absorbing the undesirable background lights.

If the light intersect with the wings perpendicular to the direction of ridges the appropriate model would be Separate Lamellae. In contrast, if the light intersect with the wing along the ridges the appropriate model would be the multi-layer thin film described.

Multi-layer Thin Film Model

The motivation for choosing this model is that many studies with electron microscopy have showed that the main reason for iridescence reflectance of the light in many different species of insects is the light interference at multiple thin layers.

All of the film layers have exactly the same physical properties like index of refraction, width etc. Also, the model assumes that the absorption of light is zero at each interface. In this model

Y.Sun mathematically justifies that it is sufficient to consider only the rays that have exactly one reflection and ignore the ones with multiple reflections.

where N is the number of thin film layers, r is the Fresnel term and I in and I out are the incoming intensity and outgoing intensity of light respectively. Besides, σ a and σ b are defined as:

where θ is the viewing angle and θ is the refracted direction of ray after first interaction with the thin layers. This model is able to described some observed properties of the biological iridescence. For example it shows that when the viewing angle is zero the reflectance is dominated by a single peak around λ = 460.8nm which corresponds to cyan color. Also, it can

describe the movement of the reflectance peak toward the short-wavelengths when the viewing angle increase which matches the color change in wings of Morpho from cyan to blue and violet when the θ increases. When the viewing angle goes toward the π/2 the peak goes out side of the visible spectrum and the iridescent colors disappear.

The approximation formula is stated as:

Separate Lamellae Model

Similar to the multi-layer thin film model the separate lamellae model does not consider the anisotropic nature of the wings and basically provides a two dimensional reflectance model for these structures. This model assumes that the wing surface is composed of periodical tree-like structures. The derived analytical model of iridescence function of Morpho wings has the following assumptions:

1) the lamellae has finite length and separated by air layer with each adjacent lamellar

2) each tree is elevated randomly in vertical direction and the amount of elevation is not correlated with its neighbors.