Second harmonic generation of optical ellipse fields

Frequency doubling of polarization singularities is studied for elliptically polarized light. Phase matching requirements are briefly considered for birefringent and for periodic structures, while non-phase matched second harmonic generation (SHG) is discussed in detail. General expressions are presented for all the symmetry classes of uniaxial crystals for propagation parallel and perpendicular to the optic axis. For propagation parallel to the optic axis, the critical points of ellipse fields transform under SHG as follows: The singularity indices of all C-points (isolated points of circular polarization) are doubled in magnitude and reversed in sign. Also the handedness (right/left) of all C-points, as well as of all other regions of the ellipse field, are reversed. All azimuthal maxima become minima, and vice versa, and directions of steepest ascent and descent of all azimuthal saddle points are interchanged. All L-lines (lines of linear polarization) are preserved unchanged, and no new L-lines are produced. Similar results are found for linearly polarized vector fields, and the winding numbers (topological charges) of the point singularities of these fields are doubled in magnitude and reversed in sign. SHG for propagation perpendicular to the optic axis depends sensitively on dispersion, birefringence, and crystal thickness, and is found to give rise to a host of peculiar effects, including the generation of "dark" polarization singularities.