Tensor Dielectric

In the preceding sections the optical response of matter has been described by a scalar dielectric function e, which relates the electric field E to the displacement D. More generally, D and E are connected by the tensor constitutive relation (5.46), which we write compactly as D = e0e E. The dielectric tensor is often symmetric, so that a coordinate system can be found in which it is diagonal ... 

If j = c2, then the z axis—the direction of propagation—is called the optic axis or the c axis. In this analysis we have tacitly assumed that the coordinate transformation to principal axes diagonalizes both the real and imaginary parts of the dielectric tensor. 

The spring model suggests that the symmetry of the crystal lattice determines the different forms of the dielectric tensor that is, they are related to the seven types of crystalline solid (amorphous solids and most liquids are isotropic). This is summarized as follows ... 

For the low-symmetry triclinic and monoclinic crystals the principal axes for the real and imaginary parts of the dielectric tensor are different. This makes life very complicated, and we—along with most other authors—will avoid such complications. 

The wave equation for the electric field traveling through a medium whose properties are expressible by means of a static dielectric tensor e reads... 

It proves convenient to express the dielectric tensor in terms of values along principal axes, % and Cy, and the angle of rotation q> between these and the Cartesian axes used for Eq. (5.14). The connection between the two coordinate systems is by means of a rotation matrix R( ) ... 

Light scattering is due to fluctuations in the local dielectric tensor e of the medium. In fluids these fluctuations are dynamic and the scattered intensity will be a function of time and the frequency spectrum of the scattered light will differ from that of the incident light. The time dependence of the total scattered intensity is analyzed by measuring the intensity autocorrelation function... 

The dielectric tensor e in a viscoelastic medium is a function of the frequency at which it is measured. It can be represented in terms of a real and imaginary part e (co) = e (co) -ie"(a>). If the frequency dependence of e is determined by a single relaxation time, then the relationship between e and r is... 

The dielectric tensor describes the linear response of a material to an electric field. In many experiments, and particularly in optical rheometry, anisotropy in is the object of measurement. This anisotropy is manifested as birefringence and dichroism, two quantities that will be discussed in detail in Chapter 2. The nonlinear terms are responsible for such effects as second harmonic generation, electro-optic activity, and frequency tripling. These phenomena occur when certain criteria are met in the material properties, and at high values of field strength. 

This describes the x and y components of the electric vector of light propagating along the z axis and through an isotropic material of refractive index n. Evidently, the light has had a prior interaction with an anisotropic material and these two components have differing amplitudes and phases. For example, in section 1.2.1, a sample with a uniaxial dielectric tensor was observed to induce a phase difference, 8X - 8y = (2k/X) (n, - n2) d, where... 

The last two relations in equation (2.63) indicate that the vectors k, D, and B form a set of orthogonal vectors. For an anisotropic material, characterized by a dielectric tensor, the electric vector will not necessarily lie in tire plane perpendicular to k, and... 

If the sample possesses a biaxial dielectric tensor of the form,... 

Fluctuations in the optical properties of a material can induce spatial and temporal inhomogeneities that scatter light. In general, the dielectric tensor is taken as the following function of space and time,... 

Normally, the anisotropic part of the fluctuations will be much smaller than the isotropic contribution. Considering only concentration fluctuations, the isotropic fluctuation in the dielectric tensor is... 

In a fluctuating system, both the electric field, E, and the dielectric tensor, e, will undergo random oscillations described by equations (4.42) and (4.43). A solution is sought for this equation so that the mean electric field in the sample is of the form... 

The first term, (e), is the intrinsic part of the dielectric tensor, e., and will lead... 

If the fluctuations in the dielectric tensor are primarily due to isotropic concentration fluctuations, then using equation (4.49),... 

Among the first theories of form birefringence was the calculation of Peterlin and Stuart [58] who solved for the anisotropy in the dielectric tensor of a spheroidal particle with different dielectric constants e j and e2 parallel and perpendicular to its symmetry axis, respectively. If the spheroid is aligned along the z axis, and resides in a fluid of dielectric constant e, the contribution of a single particle to the difference between the principal values of the macroscopic dielectric tensor of the fluid is... 

Fourier transform of the dielectric tensor autocorrelation function, (4.89). 

Puschnig, P. and Ambrosch-Draxel, C. (1999) Density-functional study of the oligomers of poly-para-phenylene Band structures and dielectric tensors. Physical Review. E, Condensed Matter, 60, 7891-8. 

X. Gonze, D.C. Allan and M.P. Teter, "Dielectric tensor, effective charges and phonons in a-quartz by variational density-functional perturbation theory," Phys. Rev. Lett. 68 (1992), 3603-3606. 

The subscript i refers to the two polarization states parallel (i = ) and perpendicular [i = L) to the c-axis, which have to be distinguished for optically uniaxial samples, for instance wurtzite-structure ZnO or sapphire. Cubic crystals, for instance rocksalt-structure Mg Zni- O, are optically isotropic and have only one DF, because the dielectric tensor is reduced to a scalar. 

Rotation of B relative to A by angle Ob about the z axis creates dielectric tensors B(0b) a = 0) ... 

These materials can be rotated, with respect to each other, about the z axis perpendicular to the interfaces. Let the angle of zero rotation be that mutual orientation at which the principal axes of the materials are in the x, y, and z directions. Then the effects of rotating materials m and R by amounts and 0r with respect to dL (kept at 0l = 0) can be written in terms of dielectric tensors e" (0m) and eR(( ), i = m or R ... 

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) ... 

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. 

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