The uniaxial indicatrix

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 

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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... 

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