Ellipsoid, refractive index

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

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.)...

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. ...

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... 

Fig. 2.19 The indicatrix, or refractive-index ellipsoid, for a general anisotropic medium. OxqX2X3 are the axes of the ellipsoid and PO represents the direction of propagation (wave-normal) of light through the medium. OA and OB are the principal axes of the section of the ellipsoid normal to OP, shown shaded. The possible D vectors for the light are parallel to these axes and their lengths represent the corresponding values of the refractive indices if the ellipsoid is drawn correctly to scale. (Reproduced by permission of Oxford University Press.)...

The calculation of the principal refractive indices for non-orthorhombic crystals is a little more complicated because the axes of the indicatrix, or refractive-index ellipsoid (see section 2.8.1), carmot be predicted in advance of the calculation. It is therefore necessary to calculate the values of all six independent components of the polarisability tensor of the crystal with respect to arbitrarily chosen axes and then to find the principal axes of the resulting tensor. 

Figure 15.20 shows schematically refractive index ellipsoids of polyimide prepared on isotropic substrates and imiaxially drawn polyimide. The films prepared on an isotropic substrate have no refractive index anisotropy in the... 

It may be easier to visualize the eigen refractive indices and the eigen electric field vectors using the refractive index ellipsoid [5]. The major axes of the refractive index ellipsoid arc parallel to the X, y, and z axes of the principal frame and have the lengths 2nx, 2riy, and 2n, respectively, as shown in Figure 2.3. The ellipsoid is described by the equation... 

Second, we consider the optical anisotropy of the blue phases. Generally speaking, the refractive indices of a crystal form an ellipsoid, as discussed in Chapter 2. Now the blue phases have cubic symmetries. On a macroscopic scale, the refractive index ellipsoid must have the same cubic symmetries. Cubic symmetries contain four-fold rotational symmetry around three orthogonal axes. Therefore the refractive index ellipsoid must be a sphere, that is, the refractive index in any direction is the same at macroscopic scale. Due to this optical isotropy, when a blue phase sample is sandwiched between two crossed polarizers, the transmittance is zero. This is the dark state of the blue phase display based on field induced birefringence. 

A coordinate showing the refractive index ellipsoid and the direction of applied... 

Substituting matrices (14.7) and (14.8) into Equation (14.6), we obtain the refractive index ellipsoid of blue phase LC under electric field as... 

The optic axis of the induced refractive-index ellipsoid is along the electric field direction. From Equation (14.15) and (14.16), we can rewrite the induced birefringence as [37]... 

Donald et al. [2] reported banded structures formed by several thermotropic polymers oriented by shear at temperatures above their softening points. Similar structures were also noted in fibers drawn from polymers with rigid backbones above the softening points. Viney et al. [3] point out that the banded structures observed in shear are due to the variation in the direction of the long molecular axis with respect to the direction of shear. Evidence obtained by both optical microscopy and electron diffraction measurements supports this view. Donald and Windle [4] studied the banded structure by electron microscopy and commented that The near sinusoidal variation in the direction of the principal axis of the refractive index ellipsoid is indeed reflecting the variations in the molecular orientation. Their transmission electron microscopy indicates that the transition from... 

The deformed helical ferroelectric (DHF) effect. If the voltage applied to the smectic C phase is lower than the untwisting field value, the helix is not completely suppressed but only distorted (Fig. 14). For a square voltage, there will be an alternation between two deformed helical states, and optically it appears as switching of the refractive index ellipsoid [6,121). In contrast to ferroelectric switching, the response time for the DHF effect is independent of... 

Figure 7.35 Orientation of refractive index ellipsoids in a banded spherulite.

It is customary to define a refractive index ellipsoid which becomes an ellipse in the plane. The lengths of the long axes of the refractive index ellipse are proportional to the maximum and the minimum... 

The optical properties of liquid crystals determine their response to high frequency electromagnetic radiation, and encompass the properties of reflection, refraction, optical absorption, optical activity, nonlinear response (harmonic generation), optical waveguiding, and light scattering [1], Most applications of thermotropic liquid crystals rely on their optical properties and how they respond to changes of the electric field, temperature or pressure. The optical properties can be described in terms of refractive indices, and anisotropic materials have up to three independent principal refractive indices defined by a refractive index ellipsoid. 

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