Refractive index/indices tensor dependence

The value of the refractive index n of light in the anisotropic medium depends on the direction of propagation s and on the direction of the polarisation of the light. For the given relative permittivity tensor ji, the refractive index can be determined from the relation (Born and Wolf 1970 Landau et al. 1987)... 

Many complex fluids contain orientable molecules, particles, and microstmctures that rotate underflow, and under electric and magnetic fields. If these molecules or microstructures have anisotropic polarizabilities, then the index of refraction of the sample will be orientation-dependent, and thus the sample will be birefringent. In general, the anisotropic part of the index of refraction is a tensor n that is related to the polarizability a of the sample. The polarizability is the tendency of the sample to become polarized when an electric field is applied thus P = a E, where P is the polarization and E is the imposed electric field. When the anisotropic part of the index of refraction is much smaller than the isotropic part (the usual case), the index-of-refraction tensor n can be related to a by the Lorentz-Lorenz formula ... 

The permittivity of a vacuum Eq has SI units of (C /J m). The specific conductivity (Tc (l/( 2-m)) couples the electric field to the electric current density by J= OcE. From the relations described in (6b), it becomes evident that optically generated gratings correspond to spatial modulations of n, , or Xg. The parameters AA, , and Xg are tensorial. This means that the value of Xg depends on the material orientation to the electric field (anisotropic interactions). In general, P and E can be related by higher-rank susceptibility tensors, which describe anisotropic mediums. The refractive index n, and absorption coefficient K, can be joined to specify the complex susceptibility when K (Xp) 471/Xp such that... 

In this equation, A is the polarizability, B is the hyperpolarizability, and Xo1, %jui3) are the 1st, 2nd, and 3rd order susceptibilities, respectively. These are tensors, where i,j, k, l correspond to the space coordinates (x, y, z) and crystal axes. The refractivity is related to Xij"-The occurrence of birefringence in anisotropic media is a direct consequence of the fact that X,fu is a tensor. Susceptibilities of order higher than one are called nonlinear susceptibilities. The nonlinear refractive index, n2, and the nonlinear absorption coefficient, a2, both depend on the intensity of light, /. They are defined by Eqs. (5.10) and (5.11),... 

The tensors and 7 constitute the molecular origin of the second-and third-order nonlinear optical phenomena such as electro-optic Pock-els effect (EOPE), optical rectification (OR), third harmonic generation (THG), electric field induced second harmonic generation (EFI-SHG), intensity dependent refractive index (IDRI), optical Kerr effect (OKE), electric field induced optical rectification (EFI-OR). To save space we do not indicate the full expressions for and 7 related to the different second and third order processes but we introduce the notations —(Ajy,ui,cj2) and 7(—a , o i,W2,W3), where the frequency relations to be used for the various non-linear optical processes which can be obtained in the case of both static and oscillating monochromatic fields are reported in Table 1.7. 

The refractive index thus depends on the average value of a,-,-, which depends on the anisotropy of the polarisability and the degree of orientation of the sample. It is shown below that, for uniaxial orientation with respect to OZ3 (for which wj = 2 = t) and a cylindrical polarisability tensor with low asymmetry. 

Since polarisability is a tensor quantity, the resulting optical properties may also be directionally dependent unless this tensor is isotropic. A simple example is illustrated by Fig. 1 where one can envisage that the interaction of the bond electrons will be greater for the imposed field that is oscillating in a plane parallel to the direction of the bond than for a field oscillating in a plane perpendicular to the bond, i.e. where the polarisability is highest. This interaction leads to a decrease in the velocity of the incident wave by an amount defined by the refractive index, n. For a non-absorbing system, the polarisability is related to the refractive index by the Lorenz-Lorentz equation ... 

This condition can be fulfilled in unaxial birefringent crystals that have two different refractive indices no and n for the ordinary and the extraordinary waves. The ordinary wave is polarized in the x-y-plane perpendicular to the optical axis, while the extraordinary wave has its -vector in a plane defined by the optical axis and the incident beam. While the ordinary index no does not depend on the propagation direction, the extraordinary index n depends on the directions of both E and k. The refractive indices Uo, and their dependence on the propagation direction in uniaxial birefringent crystals can be illustrated by the index ellipsoid defined by the three principal axes of the dielectric tensor. If these axes are aligned with the jc-, y-, z-axes, we obtain with n = the index ellipsoid. 

Next we will introduce the optical dielectric impermeability tensor of a crystal. The coefficients (17, ) of this tensor depend on the distribution of bond charges in the material [15,71]. The 17, are found by taking the reciprocal of the relative permittivity or dielectric constant [71]. The 17, have been defined in terms of the refractive index of the crystal as [71]... 

Optical Kerr Effect. Another important method used to characterize polymers is the optical Kerr effect (OKE). The optical Kerr effect differs from the quadratic electrooptic effect in that the birefringence effects are induced solely by an optical field (37). In this measurement, an intense linearly polarized pump pulse induces birefringence in the nonlinear sample through an intensity-dependent refractive index change. The sample is placed between crossed polarizers and a weak, typically tunable, continuous wave (cw) probe laser (usually at a different wavelength and polarized at 45° to the pump pulse) overlaps the pumped region. The increased transmission of the probe beam when the pump pulse arrives is proportional to (Xeff), a combination of elements of the tensor. Many... 

When an organic monolayer or thin film is present on the water surface, its effect is to introduce a thin layer of different refractive index, typically with n 1.45 this change in the refractive index results in violation of the Brewster angle condition and an enhancement in reflectivity. The increase in reflectivity due to the presence of an organic layer depends upon its thickness, refractive index, and whether it is isotropic and is described by a single value of the refractive index or whether the film is anisotropically ordered and must be described by a 4 X 4 dielectric tensor. The optical theory behind BAM is closely associated with that used to describe the ellipsometry of multilayered thin films on nonmetallic subsflates. ... 

After passing through a glass specimen, the light polarization is modified because glass becomes anisotropic when submitted to stresses (residual or transient Appendix G). The refractive index is not imique (Chapter 4, Appendix A) but depends on the stress tensor. The indices along principal directions are... 

For the analysis of the reflection spectra, we consider as usual the frequency-dependent complex dielectric tensor s (co). We restrict ourselves here to the case that the direction of polarisation is parallel to the stacking axis, and denote the real and imaginary parts of the complex dielectric function as usual s (jo) = i(co) -i- is2(co). It is related to the complex index of refraction, n = ni -i-in2, via n = The real part of... 

The refractive indices of the propagating eigenmodes in nematic LCEs depend on the principal values of the relative dielectric tensor as in ordinary nematics, that is, the index of the ordinary mode is given by... 

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