Optical tensors

This ellipsoid is tilted with respect to the x, y, z-axes and the parameters By are a function of the electric field E. In order to couple the six constants By to the three components of E, 18 coefficients are needed. They are arranged in the form of a 3 x 6 matrix, which is sometimes called the electro-optic tensor ... 

The form of the electro-optic tensor for 6mm symmetry is identical with that for the 4mm symmetry dealt with above. Therefore the induced birefringence for... 

In the case of polycrystalline ceramic (6mm) the form of the electro-optic tensor is the same as that for m3m symmetry except that R66 = (Rn - RX2). Therefore, when a field is applied along the x3 axis, the induced birefringence is again... 

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. 

An alternative quantity is also traditionally used to characterize the Pockels non-linearity. Here an electro-optical tensor r is defined which is related to the Pockels susceptibility j (-a) a),0) by equation (31) (Singer et al, 1987),... 

Qualitative information concerning the distribution and zones of concentration of stresses in the various areas of flow and the different dies considered was obtained by birefidngence. Like many other authors [37-39], we shall assume that the photo-elastic law, which expresses the relation between the optical tensor and stress tensor, is linear for the experiments performed here. Thus, at least in principle, this relation may be used to obtain an experimental measurement of stresses at all points of the flow [37, 38,40, see also chapter III.l in this book]. 

In this case, the electro-optic tensor elements are defined by the following expression ... 

Since the dielectric tensor is real and symmetric, its inverse is also real and symmetric, and the tensor f must be symmetric in its hrst two indices. The electro-optic tensor elements as dehned in Eqs. (40) and (41) are expressed in the contracted notation ... 

From the two descriptions of the Pockels effect in the frameworks of electro-op-tics and nonlinear optics, one can show that the electro-optic tensor elements and the second-order nonlinear susceptibility elements are related by... 

In deriving Eq. (48) one ignores the anisotropy of the linear part of the dielectric constant and one assumes that Kleinman s symmetry is valid. Under these conditions, using group theory to reduce the number of non-vanishing tensor elements, the electro-optic tensor for poled polymers is given by... 

Fig. 11. Real (Qi) and imaginary (Q2) part of the off-diagonal magneto-optical tensor Q = Q2 —iQ2 measured for various Co-Fe alloys at a photon energy of 2.0 eV (after Prinz et al. 1981). The alloys were annealed during vapour-deposition at 250 ° C. Note that the tensor element Q used by Prinz et al. differs in sign from the tensor element S considered in eq. (3), Qx = Re(S/er) Q-, = Im(S/er).

Many important and interesting systems and properties could not be considered in this presentation either for conciseness, as they would require some preliminary long explanation, or because they are not yet available in CRYSTAL, although they have been implemented in other codes. The calculation of NMR tensors, Raman intensity tensors, and electro-optic tensors are only a few examples from this long list of omissions. 

Individual molecules will contribute to macroscopic electro-optic activity in an additive manner hence, the linear dependence on chromophore number density, N, in the master equation for electrooptic activity, rss = 2N j8 /(principle element of the electro-optic tensor,... 

The first term in Equation (14.6) is related to initial refractive indices of the medium at three primary directions, n, Uy, n. The second term refers to the linear electro-optic effect, which is known as the Pockels effect, and the third term refers to the quadratic electro-optic effect, known as the Kerr effect. Here, and Sjj are electro-optic tensors for the linear and quadratic electro-optic effects, respectively. The second-order Kerr effect is small as compared to the first-order linear effect, so it is usually neglected in the presence of linear effect. However, in crystals with centro-symmetric point groups, the linear effect vanishes and then the Kerr effect becomes dominant. 

Symmetry point group Dielectric tensor Elastic tensor ors , Piezoelectric tensor dimn Elasto-optic tensor Pmn Electrooptic tensor rmk Nonlinear susceptibility tensors x > 1 X< > Table number... 

Going back to contracted notation, the electro-optic tensor (r, ) for crystals belonging to the tetragonal 4 group can be represented by the well-known matrix [15]... 

Hillar Aben, Andrei Errapeirt, and Leo Ainola, On real and imaginary algorithms of optical tensor field tomography, Proc. Estonian Acad. Sci. Phys. Math., 2006,55,2, 112-127. 

Since the electro-optic tensor has the same symmetry as the tensor of the inverse piezoelectric effect, the linear electro-optic (Pockels) effect is confined to the symmetry groups in which piezoelectricity occurs (see Table 8.3). The electro-optic coefficients of most dielectric materials are small (of the order of 10 m V ), with the notable exception of ferroelectrics such as potassium dihydrogen phosphate (KDP KH2PO4), lithium niobate (liNbOs), lithium tantalate (LiTaOs), barium sodium niobate (Ba2NaNb50i5), or strontium barium niobate (Sro.75Bao.25Nb206) (Zheludev, 1990). For example, the tensorial matrix of KDP with symmetry group 42m has the form... 

There is a further geometry of practical interest for light incident on chiral nematic films, related to the pitch of the helix in which we consider light propagating in a direction normal to the helix axis, i.e., as in the fingerprint texture, but with a pitchp less than A. In this short pitch chiral nematic case, the chiral optical tensor is averaged in space and the macroscopic optic axis is col-linear with the helix axis [27]. The macro-... 

Zuber, G., 8c Hug, W. (2004). Rarefied basis sets for the calculation of optical tensors. 1. The importance of gradients on hydrogen atoms for the Raman scattering tensor. Journal of Physical Chemistry A, 108, 2108. 

The measured elements of the optical tensor in the laboratory XYZ system are related to its elements in molecular x yV system by ensemble averaging ... 

A typical incident light wavelength of 532 nm was adopted in all optical tensor calculations. The Maxwell-Boltzmann (1/[1 — factor with T = 298.15... 

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