Dielectric tensor transverse

Outside of a small region around the center of the Brillouin zone, (the optical region), the retarded interactions are very small. Thus the concept of coulombic exciton may be used, as well the important notions of mixure of molecular states by the crystal field and of Davydov splitting when the unit cell contains many dipoles. On the basis of coulombic excitons, we studied retarded effects in the optical region K 0, introducing the polariton, the mixed exciton-photon quasi-particle, and the transverse dielectric tensor. This allows a quantitative study of the polariton from the properties of the coulombic exciton. 

We have indicated in Section I that the optical properties of the crystal are characterized by the transverse dielectric tensor ex(k, cu) (1.79). The real and imaginary parts of this tensor being related by the Kramers-Kronig relations resulting from the linearity, ex(k, co) is itself determined by its imaginary part. In what follows, we assume that an eigendirection of c1 is excited, and we consider t"(k, oj) and the optical conductivity cuc (k, a>) under the common denomination of optical absorption . In fact, it is the conductivity that determines the absorption by the crystal of the energy of the plane wave (see Appendix A). 

According to the theory of linear response the transverse dielectric tensor of a crystal at temperature T = 0 is determined by the relation... 

In determining the quantities Coulomb interaction has been completely taken into account. In this case, linear response theory determines only the so-called (see (44) and the next Ch. 7) transverse dielectric tensor ej y(w, k). This tensor relates the induction vector T> to the transverse part of the macrofield E Di(ui, k) = see also eqn (7.3). 

The transverse dielectric tensor and dissipation of light waves 7.4.1 The transverse dielectric tensor... 

The consideration of the strong dependence of dissipation of polaritons near exciton resonances will be performed below with the use of transverse dielectric tensor ej y(w, k). This tensor will be calculated assuming that the excitonic states, with complete account of the Coulomb interaction, are known. Since the derivation of the expression for k) is given in the monograph (3), see... 

This tensor is less general than the dielectric tensor of classical electrodynamics (1.69), since it contains the interaction with only the retarded transverse fields. For each wave vector K, (1.78) provides two solutions whose eigenpolarizations are orthogonal. The principal dielectric constants are obtained by the evaluation of the 2 x 2 determinants of (1.78) (i =1,2) ... 

The relations (7.27) and (7.33) express the dielectric tensor of the crystal in terms of polariton states. Below we apply it for the case of cubic crystals. In cubic crystals the normal waves p with small wavevectors are either transverse, or longitudinal. Therefore the product S m(k)S ]jin(k) in this medium always vanishes. In consequence (see also eqn 4.48)... 

It can be seen from this relation that, as could be expected, the dielectric tensors for transfer and longitudinal wave are different. The zeros of the value b(u>, k) determine the resonances of the dielectric tensor for transverse waves and the resonances of the value a(u>, k) determine the frequencies of longitudinal waves (i.e. zeros of dielectric tensor for longitudinal waves). 

Here, A is the vacuum-wavelength of the reflected light, riiso is the index of refraction of the liquid crystal in the isotropic phase, ngiass is the refractive index of the substrate, ez z) is the component of the dielectric tensor of the liquid crystal along the normal to the interface and (z) is the corresponding component in a transverse direction. The value of the pB is therefore connected to the thickness, optical anisotropy and 2 direction variation of refractive index of the interfacial layer. 

Thus the spectral function L(z) of an isotropic medium is represented as a linear combination of two spectral functions determined for an anisotropic medium pertinent to longitudinal ( ) ) and transverse (K ) orientations of the symmetry axis with respect to the a.c. field vector E. It is shown in GT, Section V, that these spectral functions are proportional to the main components of the dielectric-susceptibility tensor. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

In the uniaxially oriented case, the system is transversely isotropic and the dielectric constant tensor reduces to... 

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