Index ellipsoid

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

Figure 9. Top An isotropic atom/molecule and crystal with isotropic polarizabilities will give rise to a spherical wave surface and index ellipsoid. Middle Another isotropic atom/molecule and crystal with larger isotropic polarizabilities will give rise to a smaller spherical wave surface and a larger spherical index ellipsoid. Bottom An anisotropic atom/molecule and crystal with anisotropic polarizabilities will give rise to an ordinary spherical wave surface and an ellipsoidal extraordinary wave surface. The index ellipsoid will have major and minor axes.

Figure 12. (a) Index ellipsoid defined by ne and no with a ray of light propagating in an arbitrary direction OP (b) an ellipse that is formed by the intersection of the plane normal to OP and the index ellipsoid. The principal axes of this ellipse are the angularly dependent index ne (9) and the angularly independent index nQ. 

Figure 16. Index ellipsoid of an anisotropic material (a) in the absence of an applied field, (b) in a medium field, and (c) in a strong field.

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.)...

If all three principal values are positive, the quadric surface is an ellipsoid with semiaxes a, = TfiVz, but if one or two of the principal values are negative the quadric surface is a hyperboloid. For example, the (relative) impermeability tensor 3 is defined by nfn, where k is the permittivity and n0 is the permittivity of free space. As for any symmetric 7(2) the components of 3 define the representation quadric I3ijxixj= 1, which here is called the indicatrix or optical index ellipsoid. Referred to principal axes the indicatrix is... 

With respect to the crystal, this is visualized by the index ellipsoid or indicatrix. (In a homogeneous medium this would be a sphere.) Without external field, the indicatrix is oriented along the main crystallographic directions. 

Figure 4.8 Relation between the crystal and the index ellipsoid (right). The application of an electric field rotates and deforms the ellipsoid...

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. ...

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... 

As discussed in Sect. 2.1.2, the index of an anisotropic medium is described by the index ellipsoid (Eq. 22). If the coordinate system is chosen such that the axes do not match with the principal symmetry axes of the crystal, the index ellipsoid is described by the more general expression [11]... 

The index ellipsoid can also be expressed in a more convenient way as... 

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... 

When an electric field is applied, the index ellipsoid will change in power series expansion to the field, thus. 

Now consider shear flow along x with a velocity gradient du/dz. The flow induces a birefringence proportional to the velocity gradient with the principal axes of the index ellipsoid inclined at 45° to the x, z axes. In the steady state = dr/dz. Therefore... 

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... 

The tilt angle profile G z) can be used to calculate the average refractive index, n, of the liquid crystal at different applied voltages. For a given tilt angle, the refractive index is given by the index ellipsoid equation [54] ... 

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... 

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