Optical indicatrix, equation

In order to obtain accurate values of the principal refractive indices in the plane of the sample, it is best to measure the refractive index as a fimction of the angle, ( ), around the film. By introducing a large number, 16 in our case, of angular measurements into the optical indicatrix equation, a precise identification is obtained of the symmetry axis direction and of both principal refractive indices in the plane of the film. The optical indicatrix equation in the plane of the film has the linear form ... 

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... 

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... 

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 refractive indices of anisotropic materials are conveniently represented in terms of the optical indicatrix, the surface of which maps the refractive indices of propagating waves as a function of angle. Solution of Maxwell s equations for an anisotropic medium leads to the result that for a particular wave-normal, two waves may propagate with orthogonal plane-polarisations and different refractive indices. An ellipsoid having as semi-axes the three principal refractive indices defines the optical indicatrix. In general for any wave-normal, the section of the indicatrix perpendicular to the wave-normal direction will be an ellipse, and the semi-axes of this ellipse are the refractive indices of the two propagating waves. 

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... 

Thus, the propagation of the electromagnetic waves into a crystal will have three main directions of refraction or the main optical axes (for each of them being characteristic one refraction index as n, n, n ), in a quadratic relation with the Cartesian coordinates associated to the crystalline system (x, y, z), through the form of a rotational ellipsoid which generates the so-called surface of the refraction index (indicatrix), with the equation ... 

---