  An artificial opal is a compact arrangement of transparent spheres, and is an
archetype of a three-dimensional photonic crystal. Here, we describe the optics
of an opal using a flexible model based upon a stratified medium whose
(effective) index is governed by the opal density in a small planar slice of
the opal. We take into account the effect of the substrate and assume a well-
controlled number of layers, as it occurs for an opal fabricated by
Langmuir-Blodgett deposition. The calculations are performed with transfer
matrices, and an absorptive component in the effective index is introduced to
account for the light scattering. This one-dimensional formalism allows
quantitative predictions for reflection and transmission, notably as a function
of the ratio between the irradiation wavelength and the sphere diameter, or as
a function of the incidence angle or of the polarization. It can be used for an
irradiation from the substrate side or from the vacuum side and can account for
defect layers. The interface region between the opal and the substrate (or
vacuum) is shown to have a strong influence, regardless of the exact opal
structure. This break in the periodicity at the interface is a general, but
often ignored feature, of any external coupling to a photonic crystal. Our
calculations provide also the main features of the Bragg peak for reflection,
including its width and strength. Comparisons of this versatile model with
experiments show that despite its simplicity, it is powerful enough to explain
numerous observations.
