Refractive index complex tensor

This as been written in a reference frame coincident with the principal axes of n. The case of noncoaxial real and imaginary parts of the refractive index tensor requires a much more complex calculation and is treated in section 2.4.6. 

In an important series of papers [6,7], Jones established an approach for the treatment of materials where the refractive index tensor, n (z), varies along the propagation direction of the transmitted light. This procedure also lays the foundation for the analysis of complex systems possessing any combination of optical anisotropies. 

To see the relationship to the refractive index, we focus on the case where the tensor y reduces to a scalar. This is the case, for example, if fire electronic response of the medium is isotropic, in which case x(co) reduces to a scalar x(co), or where the field is, for example, along the laboratory-z axis and only the single yzz complex refractive index n(co) is given by... 

The 3x3 complex refractive index tensor N = n — ik is related to the dielectric tensor e and the magnetic tensor p by the Maxwell relation ... 

Birefringence and dichroism represent two optical methods which can be applied to materials under flow conditions, forming the basis of Optical Rheometry [3,4]. The aim of these two techniques is to measure the anisotropy of the complex refractive index tensor n = n - i n". Birefringence is related to the anisotropy of the real part, whereas dichroism deals with the imaginary part. Recent applications of birefringence measurements to polymer melts can be foimd in Chapter III.l of the present book. 

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 ... 

It should be mentioned here that the term is a second-rank tensor and has nine components because it is related to all the components of the polarization vectors and electric field vectors. Therefore, the dielectric constant is also a tensor of rank 2. The optical response of a medium at an optical frequency m can be represented equivalently by the complex refractive index as ... 

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... 

As shown above, there is a well-defined relationship between refractive index and electric susceptibility. According to Eq. (4.24), the refractive index is a measure of the degree of induced polarization in a given medium. We have to admit, however, in the case of an anisotropic medium that the derivation of the relationship between refractive index, dielectric constant, and permeability is very complex and a nuisance because eri and / n are tensors, and the refractive index is different for each direction of the propagated light. 

Anisotropy in the optical properties of a layer complicates the analytical expressions for reflectance since the complex dielectric function and the refractive index of the layer become tensors (1.1.2°). Determination of a film s anisotropy from its spectrum provides a wealth of information about the structure and molecular orientation in ultrathin films and therefore is of great importance in various areas of science and technology (Section 3.11). The theoretical approaches of Schopper [112] and Kuzmin et al. [113] (see the review in Ref. [114]) are... 

In these equations, eq and a are respectively the vacuum permittivity and the linear absorption coefficient. E a>) represents the apphed electric field at the frequeney co [ (complex conjugate of ( )], whilea>, —m) and/( >,< , —< ), which are fourth-order tensors, are respectively the third-order susceptibUity and hyperpolarisabUity. The real part of these tensors represents the induced refractive index change and is a function of the laser intensity (m) oc (imaginary part describes the 2PA process [40]. 

Thus, in principle, one can obtain the complex index of refraction from the upper left-hand quarter of the optical tensor M= [My, (i,j = 1,..., 3)]. We now explain how these optical constants can be derived from ellipsometry. 

The practical method to compute "backward"—that is, to calculate the dielectric constant tensor values, or the complex index of refraction—from a set of the observed polarizer and analyzer angles is not presented here. Instead, for a biaxial crystal, the technique indicated below is as follows ... 

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... 

Hamiltonian = matrix element of the Hamiltonian H I = nuclear spin I = nuclear spin operator /r( ), /m( ) = energy distributions of Mossbauer y-rays = Boltzmann constant k = wave vector L(E) = Lorentzian line M = mass of nucleus Ml = magnetic dipole transition m = spin projection onto the quantization axes = 1 — a — i/3 = the complex index of refraction p = vector of electric dipole moment P = probability of a nuclear transition = tensor of the electric quadrupole q = eZ = nuclear charge R = reflectivity = radius-vector of the pth proton = mean-square radi-S = electronic spin T = temperature v =... 

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