  Interference phenomena are the source of some of the spectacular colors of
animals and plants in nature. In some of these systems, the physical structure
consists of an ordered array of layers with alternating high and low refractive
indices. This periodicity leads to an optical band structure that is analogous
to the electronic band structure encountered in semiconductor physics; namely,
specific bands of wavelengths (the stop bands) are perfectly reflected. Here,
we present a minimal model for optical band structure in a periodic multilayer
and solve it using recursion relations. We present experimental data for
various beetles, whose optical structure resembles the proposed model. The stop
bands emerge in the limit of an infinite number of layers by finding the fixed
point of the recursive relations. In order for these to converge, an
infinitesimal amount of absorption needs to be present, reminiscent of the
regularization procedures commonly used in physics calculations. Thus, using
only the phenomenon of interference and the idea of recursion, we are able to
elucidate the concepts of band structure and regularization in the context of
experimentally observed phenomena, such as the high reflectance and the
iridescent color appearance of structurally colored beetles.
