Huygens's principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface

Huygens's principle is used to derive equations for tracing the extraordinary ray in a uniaxial crystal when the crystal axis is normal to the refracting surface. Snell's law is used to trace ordinary rays that lead to an array of ordinary spherical wavelets centered on the refracting surface. Each spherical wavelet is then replaced by an extraordinary wavelet in the shape of a rotationally symmetric ellipsoid whose major axis is in the direction of the crystal axis. The envelope of these is the extraordinary wave front; the extraordinary ray is a vector from the center of the wavelet to its point of contact with the extraordinary wave front. The inverse problem is also solved, yielding expressions for the ordinary ray in terms of the extraordinary ray, making it possible to trace the extraordinary ray out of the system by using a fictive ordinary ray. We found the geometrical invariant for the uniaxial crystals.