Elucidating the stop bands of structurally colored systems through recursion
(Color online) Photographs of the beetles (a) Torynorhina flammea, (b) Chrysochroa raja, and (c) Gastrophysa viridula. Photographs are courtesy of (a) Richard Bartz, (b) Didier Descouens, and (c) James Lindsey at Ecology of Commanster. TEM cross-sections of the multilayers responsible for these colors are shown in (d) through (f) for T. flammea, C. raja, and G. viridula, respectively. Reflection spectra taken from the elytra of these three structurally colored green beetles, normalized with respect to each other, are shown in (g): dotted line, T. flammea; dashed line, C. raja; and solid line, G. viridula.
The one-dimensional, periodic optical structure that leads to the formation of optical stop bands. The dashed rectangle outlines a unit cell.
The unit cell of the structure of Fig. 2 .
An example of a path of a wave in a single-layer structure. In this case the wave reflects five times before exiting the layer and being reflected back. (A small incident angle is used in the figure to make the path clear, but throughout the paper we assume that the wave propagation is perpendicular to the surface). This path corresponds to the term in the equation for the reflectance, Eq. (4) .
Schematic depiction of the reflectance and transmittance properties of the system that lead to the recursion relations and enable representation of its optical band structure.
Propagation of light through a symmetric structure (a) and its time-reversed path (b).
Result for the reflectance from an infinite array of unit cells, as found from Eq. (15) . Here , and . The region of perfect reflectance is the optical stop band. The inset shows a plot of the reflectance over a broader range of wavelengths (plotted in the inset in terms of the wavenumber) and shows additional stop bands, outside the visible spectrum. Note that in the calculation we also added an additional air layer, which gives rise to the background reflectance.
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