Fourier description of the propagation and focusing of an extraordinary beam in a planar uniaxial medium

The scalar-angular-spectrum approach is used to examine a focused extraordinary wave propagating along an arbitrary direction in a homogeneous uniaxial planar medium, and the Fraunhofer diffraction formula is generalized for this case. The size of the focal spot is found to be inversely proportional to an effective index, depending on the refractive indices and the propagation direction. The validity of the paraxial model is checked by nonparaxial (but scalar) numerical calculations. They show that the paraxial formulas predict the spot size correctly, but if the beam propagates neither parallel with nor perpendicular to the optic axis, they do not reproduce the symmetries of the amplitude distribution in the focal line.