Optical analysis of spatially periodic patterns in nematic liquid crystals: Diffraction and shadowgraphy

Abstract

Optical methods are most convenient for analyzing spatially periodic patterns with wave vector q in a thin layer of a nematic liquid crystal. In the standard experimental setup a beam of parallel light with a "short" wavelength λ<<2π/q passes the nematic layer. Recording the transmitted light the patterns are either directly visualized by shadowgraphy or characterized more indirectly by the diffraction fringes due to the optical-grating effects of the pattern. In this work we present a systematic short-wavelength analysis of these methods for the commonly used planar orientation of the optical axis of liquid crystal at the confining surfaces. Our approach covers general three-dimensional experimental geometries with respect to the relative orientation of q and of the wave vector k of the incident light. In particular, we emphasize the importance of phase-grating effects, which are not accessible in a pure geometric optics approach. Finally, as a by-product we present also an optical analysis of convection rolls in Rayleigh-Bénard convection, where the refraction index of the fluid is isotropic in contrast to its uniaxial symmetry in nematic liquid crystals. Our analysis is in excellent agreement with an earlier physical optics approach by Trainoff and Cannell [Phys. Fluids 14, 1340 (2002)], which is restricted to a two-dimensional geometry and technically much more demanding.