Indicatrix, optical

Wire models will make this and other features of the optical indicatrix clearer than plane diagrams can possibly do. 

Crystals, except those belonging to the cubic system, are anisotropic in this respect the force of repulsion varies with the orientation of the crystal with respect to the direction of the field. The graph representing vectorialiy the diamagnetic susceptibility in all directions in a crystal is an ellipsoid, whose orientation with respect to the unit cell is restricted by symmetry in exactly the same way as that of the optical indicatrix. Thus, for uniaxial crystals the magnetic ellipsoid an ellipsoid of revolution whose unique axis coincides with the threefold, fourfold, or sixfold axis of the crystal for orthorhombic crystals the ellipsoid has three unequal axes which necessarily coincide with the three axes of the crystal for monoclinie crystals the only restriction is that one of the principal axes of the magnetic ellipsoid must coincide with the b axis of the crystal while for triclinic crystals the orientation of the ellipsoid is not restricted in any way. 

In optics, the optical indicatrix (Figures 9 and 12) is a useful construct that ch aracterizes the birefringence of materials. The indicatrix is a surface that specifies the refractive indices of both the O and E rays traveling in any direction through the material. The indicatrix for a uniaxial material is defined by the equation... 

For optically uniaxial crystals we know that the refractive index values for extraordinary waves are variable, with that for ordinary waves fixed. We can link this observation with that concerning the vibration directions for the two waves travelling along a general wave normal direction the ordinary vibration direction is always perpendicular to the optic axis, while the extraordinary vibration is always in the plane containing the optic axis and wave normal direction. This suggests that we may connect the variation of the refractive index in the crystal with the vibration direction of the light. This concept allows a convenient representation of anisotropic optical properties in the form of a spatial plot of the variation of refractive index as a function of vibration direction. Such a surface is known as the optical indicatrix. 

For uniaxial crystals, the optical indicatrix is a single-surfaced ellipsoid of revolution similar in shape to the extraordinary ray velocity surface. To construct the optical indicatrix for a particular example, say calcite, we construct the ellipsoid of revolution so that the radius... 

Figure 4.4 Sections of the optical indicatrix for uniaxial crystals, (a) Optically negative crystal and (b) optically positive crystal...

If I is oriented along the principal-axis system of the optical indicatrix, then for each of the three components of the refractive index (nlr n2, n3) the anisotropic Lorentz-Lorenz equation is... 

Since the electro-optic coefficient rij is defined by the electric field dependence of the optical indicatrix, is related to the second order bulk susceptibility through i... 

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... 

If the crystal is biaxial, there are three principal refractive indices (a, /3, and 7, with a less than (3 less than 7). The optical indicatrix has... 

Optical indicatrix A three-dimensional ellipsoid whose surface is defined by vectors from an origin. These vectors have lengths proportional to the refractive indices of a crystal in those directions. 

Fig. 16. Combination of two optical uniaxial materials The figure shows the optical indicatrix of an uniaxially stretched plastic foil with horizontal direction of Ue and the indicatrix of a homeotropic oriented LC siloxane with horizontal direction of n (schematically).

Figure 1. The geometry relevant to deriving Equation 1. The elliptical outline represents the intersection between the optical indicatrix and the specimen plane. SS and FF respectively identify the slow and fast vibration directions in the specimen.

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... 

If the optical indicatrix is oriented at an angle 0 to the plane of the cell such as the plane of the glass walls and ITO electrodes, then the refractive index seen by light passing perpendicular to the cell waU is given by... 

Fig. 5.9. The optical indicatrix in the layer with a longitudinal polarization is rotated.

The optical indicatrix, averaged over three layers, should differ from that in systems where only perpendicular polarizations in the layer are present. 

Fig. 10.18 Illustration of (a) the lamellar crystal, (b) the optical indicatrix and (c) the Maltese-cross extinction in polymer spherulites...

Fig. 4.5 The nematic phase molecular orientation (a), optical indicatrix (b) and characteristic microscopic texture (c)...

Fig. 4.11 The optical indicatrix (a) and the microscopic texture (b) of the SmC biaxial phase...

At optical frequencies = and the same ellipsoid becomes the so-called optical indicatrix with its semi-axes exactly equal to refi action indices i, and n. ... 

Where does such a strong scattering anisotropy originate from It is evident that the optical anisotropy of nematic hquid crystals plays the crucial role. In fact, the scattering is caused by fluctuations of the director n, i.e. the local orientation of the order parameter tensor. The local changes in orientation of n imply local changes in orientation of the optical indicatrix. 

The optical properties of an anisotropic crystal can be conveniently described by the optical indicatrix (Fig. 11.2), which is derived from the optical surfaces by... 

For biaxial crystals, the optical indicatrix is a bilayer surface with four points of interlayer contact, which correspond to the directions of the two optical axes. In the simple case of light propagation in the principal planes XY, YZ, and XZ, the dependences of the refractive indices on the direction of light propagation are represented by a combination of an ellipse and a circle. Thus, in the principal planes, a biaxial crystal can be considered as a uniaxial crystal for example, a biaxial crystal with nz > ny > nx in the XY plane is similar to a negative uniaxial crystal with Ho=nz... 

The application of an external electric field deforms the optical indicatrix (Fresnel ellipsoid) of crystals lacking a center of symmetry in such a way that its birefringence is changed. The dependence of the birefringence on the electric field E is linear, and can be analytically described by a change of the impermeability tensor a = (e ) by the electric field E [see Eq. (8.4a)] and the polarization P, respectively ... 

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