Uniaxial indicatrix

Fig. 47. Left positive uniaxial indicatrix. Right negative uniaxial indicatrix.

Figure 4.6 A principal section of the negative uniaxial indicatrix...

Fortunately, but also evidently, many simplifications arise when considering the uniaxial indicatrix of crystals belonging to the trigonal, tetragonal, and hexagonal crystal systems. First, there is only one optic axis which, by convention, always lies along the z crystallographic axis hence, X, Y, and Z disappear, as z suffices to define this direction. The refractive index associated with this direction is called or n. The plane perpendicular to the optic axis is necessarily a circular section whose diameters all have the same refractive index denoted or n. Thus, and Hy disappear. Optically positive means... 

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 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. 

For a uniaxial crystal the indicatrix is symmetrical about the principal symmetry axis of the crystal - the optic axis. If v3 is the optic axis, the central section of the ellipsoid is a circle of radius n0 and the equation becomes... 

The direction of the principal axes of the index of refraction tensor n can be described by the indicatrix. For isotropic crystals the indicatrix is a sphere. For positive uniaxial crystals it is a prolate spheroid (ns > n0j) for negative uniaxial crystals it is an oblate spheroid (nol > n,). For orientations away from the principal axis orientations, the extraordinary ray will have a refractive index h - intermediate between nm and ne. 

FIGURE 5.10. Some crystals, each with their optical indicatrices drawn within them, (a) A uniaxial negative crystal (calcite) showing orientation of the indicatrix. (b) A uniaxial positive crystal (quartz) showing orientation of the indicatrix. 

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).

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... 

Fig. 3.11 The indicatrix of uniaxial birefringent materials, positive and negative. Positive materials have one principal refractive index greater than the other two. The optic axis is the axis of synunetry of the ellipsoid.

Figure 29. Media with a threefold, fourfold, or sixfold crystallographic axis must be optically uniaxial. Thus crystals belonging to the trigonal, tetragonal, or hexagonal system, including smectic B, must be uniaxial. The picture shows the indicatrix with its optic axis in the case of positive as well as negative birefringence.

In general the indicatrix is like a squashed (American) football. There are three principal refractive indices on perpendicular axes the maximum, the minimum, and an intermediate value perpendicular to both of these (A, C, and B in Fig. 3.13). For example, a nylon crystal has n(D parallel to chains) > n(D parallel to hydrogen-bonded sheet) > n(D parallel to intersheets). Such a material is biaxial. If two of these principal refractive indices are equal, the indicatrix is rotationally symmetric—an ellipsoid of revolution—and the material is uniaxial. For example, a polyethylene crystal has a larger refractive index for (D parallel to chains) and a smaller refractive index for any (D perpendicular to chains) (Fig. 3.14, left drawing). 

A uniaxial material has one optic axis and a biaxial material has two. A birefringent material appears to be isotropic when a plane light wave passes through it along an optic axis. In terms of the indicatrix, the cross section in the... 

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