Many-wave approximations for light propagation in cholesteric liquid crystals.

Abstract

The eigensolutions of the Maxwell equations for light propagation in a perfect cholesteric liquid crystal are considered. They are Bloch waves having, in principle, infinitely many Fourier components. The relative amplitudes of these components are studied as a function of the angular frequency \ensuremath{\omega} and of the incidence angle ${\ensuremath{\theta}}_{i}$, in order to determine the conditions under which the actual eigensolutions can be suitably approximated by a superposition of only a few plane waves. When these conditions are satisfied, the wave vectors and the amplitudes of the components are given by simple analytical expressions. The main results are the following: (a) Far from the Bragg reflection bands, only one or two plane-wave components need be retained. Regions of the (\ensuremath{\omega},${\ensuremath{\theta}}_{i}$) plane are found, in which a very good approximation is represented by a linearly polarized wave (\ensuremath{\pi}- and \ensuremath{\sigma}- polarized-wave approximation) or by a nearly circular wave (quasinormal approximation). This allows selecting the optical conditions for which practically a single plane wave is excited in a cholesteric sample, a fact which is very important when dielectric inhomogeneities due to thermal fluctuations or defects and Brownian motions or fluorescence of guest molecules are studied. (b) Simple dispersion relations are obtained, which give the correct values of both real and imaginary parts of the wave vectors near the first-order reflection band. (c) It is shown that for the higher-order reflection bands, reasonably simple analytical expressions can be obtained, which give good qualitative results but which are not useful for a quantitative analysis. However, an exact dispersion relation simpler than the ones given in the literature is found. In all these cases, each approximation is discussed in detail. The approximate expressions found are in general simpler and more accurate than the ones previously given for similar cases. Their limits of validity for any practical purposes are found. The ensemble of the regions of validity of given approximations covers the largest portion of the (\ensuremath{\omega},${\ensuremath{\theta}}_{i}$) plane.