Refractive index/indices ellipsoid

Noctilucent cloud particles are now generally believed to be ice, although more by default—no serious competitor is still in the running—than because of direct evidence. The degree of linear polarization of visible light scattered by Rayleigh ellipsoids of ice is nearly independent of shape. This follows from (5.52) and (5.54) if the refractive index is 1.305, then P(90°) is 1.0 for spheres, 0.97 for prolate spheroids, and 0.94 for oblate spheroids. 

Figure 15.20. Schematic representation of refractive index ellipsoids of (a) polyimide prepared on isotropic substrates, and (b) uniaxially drawn polyimide.

For a crystal, the polarizability for a principal direction of the ellipsoid is calculated by using the same expression the refractive index n for that direction is taken to be related to the polarizability by the Lorenz-Lorentz expression,... 

As has been pointed out (63), this is a rather artificial model and, moreover, its application is quite unnecessary. In fact, (a> can be calculated from the refractive index increment (dnjdc), as has extensively been done in the field of light scattering. This procedure is applicable also to the form birefringence effect of coil molecules, as the mean excess polarizability of a coil molecule as a whole is not influenced by the form effect. It is still built up additively of the mean excess polarizabilities of the random links. This reasoning is justified by the low density of links within a coil. In fact, if the coil is replaced by an equivalent ellipsoid consisting of an isotropic material of a refractive index not very much different from that of the solvent, its mean excess polarizability is equal to that of a sphere of equal volume [cf. also Bullough (145)]. 

Figure 11. Schematic representation of the refractive index ellipsoid for a positive uniaxial material at frequency w. (Reprinted with permission from Williams, D. J. Atigew. Chem. Int. Ed. Engl 1984,23,690. Copyright VCH Publishers.)...

For the sake of illustration, the determination of the electrooptic coefficient for a uniaxial crystal is described below. Considering the nonlinear uniaxial medium of Figure 11, a D.C. electric field is applied in the z direction. The effect of the electric field is to modify the refractive index in the z direction by an amount proportional to the electric field, the modified ellipsoid is given as... 

If we try to understand the transmission of light waves in biaxial crystals, we start from the concept of the indicatrix, and to attempt to visualize what shape this must have to show the variation of refractive index with vibration direction for such crystals. From our previous knowledge of the indicatrix for uniaxial crystals, an ellipsoid of revolution with two principal refractive indices, n0 and ne, it is a simple step to see that the indicatrix for biaxial crystals will be a triaxial ellipsoid with three principal refractive indices, n7, np and na. 

Figure 2.7 The refractive index ellipsoid of a uniaxial liquid crystal phase with the optical axis parallel to ihe x-axis. The refractive index, no, of the ordinary ray is independent of the direction of propagation. The refractive index, ng, of the extraordinary ray is larger than n if the liquid crystalline phase is of positive birefringence. ...

Beam condensers, by using a pair of ellipsoid mirrors, produce very small images of the Jacquinot stop or the entrance slit at the sample position. The size of these images may be even further reduced by making use of a Weierstrass sphere. Weierstrass (1856) showed that a spherical lens has two aplanatic points . If a sphere of a glass with a refractive index n is introduced into an optical system which has a focus at a distance of r n from its center, then the beam is focused inside the sphere at a distance of r/n from the center (Fig. 3.5-9). In this case the angle O in Eq. 3.4-5 may approach 90°. Thus, a sample with a very small area can fully fit the optical conductance of the spectrometer (Fig. 3.4-2d). Microscopes usually have an optical conductance which is considerably lower than that of spectrometers. In this case, the sample and the objective are the elements limiting the optical conductance (Schrader, 1990 Sec. 3.5.3.3). 

The optical indicatrix is a useful construction for visualizing the variation in the refractive index of a crystal as a function of spatial direction.It shows directions (with respect to unit-cell edges) where the refractive index is greatest and where it is the smallest. It is a three-dimensional ellipsoid with a shape defined by the ends of vectors, each with the same fixed origin. The length of each vector is proportional to... 

Equation (22) describes an ellipsoid (see Fig. 2) called the index ellipsoid. The latter is very useful in deriving the refractive index of optical waves with different polarization and propagation direction. A wave traveling in a uniaxial polymer at an angle d with respect to the optic axis experiences two different index depending on its polarization if the wave is s-polarized (perpendicular to the plane of incidence) the refractive index is n and is independent of 0 for a p-polarized wave (polarization in the plane of incidence) the refractive index is given by... 

Fig. 3. Left The collinear phasematching condition in relation to refractive indexe ellipsoids. Right The...

When a crystal is subjected to a stress field, an electric field, or a magnetic field, the resulting optical effects are in general dependent on the orientation of these fields with respect to the crystal axes, it is useful, therefore, to express the optical properties in terms of the refractive index ellipsoid (or indicatrix) ... 

The electrooptic effect is defined through the optical indicatrix, or the refractive index ellipsoid, which can be written in its principal axes x = 1, y = 2, and z = 3 in the form... 

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