Wave interactions, nematics

Because the equilibrium order in heterophase systems is characterized by only one nonzero degree of freedom of the order parameter tensor, the fluctuation modes of all five degrees of freedom are uncoupled. Due to the uniaxial symmetry of the phase the two biaxial modes are degenerate and so are the two director modes. If a nematic layer is bounded by walls characterized by a strong surface interaction and a bulk-like value of the preferred degree of order, the fluctuation modes /3j s are sine waves, and their relaxation rates may be cast into... 

As a rule for nematics a is positive. Thus the interaction free-energy is minimum when the molecules are oriented along the polarization direction of the light beam. Should this condition not correspond to the initial alignment in the cell, optical reorientation occurs under the influence of the electromagnetic wave. 

The orientation interaction is considered for light waves and liquid crystals with a uniform initial director orientation (such crystals include twisted nematics and cholesterics, and crystals with hybrid or hyperhybrid structures). Conditions are found for critical behavior to occur for a nonthreshold interaction. A light-induced Frederiks transition is predicted for cells with a hybrid nematic orientation, in which radiation absorption may reorient the director. 

The magnitude of the response y depends in an extremely complicated way on the interaction geometry. In general, becomes anomalously large as A—>0. It is easy to check that in the one-constant case (A, = A2 = A,), for which instability develops when the director is twisted by an angle = tt, the behavior of y exhibits no anomalies atP = 0. We must have/0 7 0 for instability to be observable in the orientation interaction of a light wave with a nematic liquid crystal. 

The admixture of the chiral compound to a conventional nematic induces a macroscopic helical structure whose wave vector, that is, inversed pitch qo = 27t/Po depends on concentration (c), in general, nonmonotonically. Only in the case of the ideal mixture, without any short-range intermolec-ular interactions, we have the linear relationship... 

In this equation, y is the interaction strength, c(r) the crosslink concentration, the smectic order parameter, and Vz (r) the relative displacement of the rubber matrix. Witkowski and Terentjev [132] evaluated (15) for (r) = 1, which is valid deep in the smectic phase, i.e., far below the smectic-nematic transition. Using the so-called replica trick, they integrated out the rubbery matrix fluctuations and obtained an effective free-energy density that depends only on the layer displacements M(r). Under the restriction that wave vector components along the layer normal dominate over in-layer components, q q, and considering only long-... 

One optical feature of helicoidal structures is the ability to rotate the plane of incident polarized light. Since most of the characteristic optical properties of chiral liquid crystals result from the helicoidal structure, it is necessary to understand the origin of the chiral interactions responsible for the twisted structures. The continuum theory of liquid crystals is based on the Frank-Oseen approach to curvature elasticity in anisotropic fluids. It is assumed that the free energy is a quadratic function of curvature elastic strain, and for positive elastic constants the equilibrium state in the absence of surface or external forces is one of zero deformation with a uniform, parallel director. If a term linear in the twist strain is permitted, then spontaneously twisted structures can result, characterized by a pitch p, or wave-vector q=27tp i, where i is the axis of the helicoidal structure. For the simplest case of a nematic, the twist elastic free energy density can be written as ... 

Several theories aimed at explaining the phenomena have been proposed, each of which is founded on completely different concepts. Sripaipan et al. [21] proposed a nematic layer with free ends, in which the interaction between the longitudinal oscillations (induced by the motion of the free ends of the layer in compression) and the traverse oscillations establishes steady flow of the liquid and, as a result, rotation of the molecules. However, these authors used incorrect dispersion relations and their calculations are not consistent with observed layer compression patterns. Nagai and coworkers [26,27] hypothesized that with normal incidence of an ultrasound beam on the layer the rotation of molecules is attributable to radiation fluxes. Radiation fluxes are the steady acoustic flows caused by radiation forces in a traveling acoustic wave, the only provision being that the width of the ultrasound beam is smaller then the dimensions of the cell. In reality, radiation fluxes can only occur near the boundaries of the beam and produce a compression effect that is smaller than the one that is actually ob-... 

Figure 7. Equipment geometries for studying the wave and acousto-electrical interactions in nematics (1) substrate (y cut, x oriented quartz), (2) glass plate, (3) interdigital transducer, (4) shear transducer (y cut quartz), (5) compression transducer (x cut quartz), (6) nematics, (7) mirror coating, (8) optically transparent electrode, (9) generator, (10) waveguide (substrate), (11) phase meter.

In general, the classical Fredericks transition in nematics can be fairly well-explained using continuum theory of nematic liquid crystals developed by Frank, Ericksen and Leslie. Before we present a detailed analysis on the optical Fredericks transition, which couples the interaction between the applied electromagnetic field of a light wave and the orientation of hquid crystals, we would like to briefly review the classical results (de Geimes and Frost 1993 Virga 1994 Stewart 2004). Many of our ideas here are borrowed from Stewart (2004). 

Figure 5.9. Typical laser-nematic director axis interaction geometry in a wave mixing experiment to measure the nonlinear refractive index change induced in the nematic liquid crystal. Note that an external dc or ac voltage can be applied in conjunction with the optical field. Also the liquid crystal could be doped with photosensitive molecules or nanoparticulates to enhance their nonlinear scattering efficiency.

Consider the interaction of a linearly polarized extraordinary wave laser beam with a homeotropically aligned nematic liquid crystal as shown in Figure 9.14. The extraordinary refractive index as seen by a low-intensity laser is given by... 

Figure 9.14. Interaction of an extraordinary wave laser with a homeotropically aligned nematic liquid crystal at oblique incidence.

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