ИСТИНА

A monochromatic light beam is generally pierced by imaginary lines, along which polarization ellipse is circular (C-lines). These lines intersect cross sections of the beam at the points of circular polarization singularity, or C-points. Polarization patterns in the vicinity of a C-point are divided into three morphological types: “lemon”, “star”, and “monstar”. The main quantity characterizing a polarization singularity is topological charge, which is the number of polarization ellipse rotation counted along one counter-clockwise loop around the C-point. “Lemon” and “monstar” patterns have the charge 1/2 (ellipse is rotated by 180 degrees counter-clockwise) and “star” pattern has the charge -1/2 (180 degrees clockwise rotation). C-points are vector analogs of phase singularities (optical vortices); topological charge of the latter shows the phase variation around the singularity. We found and study analytical expressions of the beam at double frequency reflected from the surface of the isotropic gyrotropic medium by normally incident fundamental beam with specially designed polarization pattern, containing a single C-point. This beam is a superposition of coaxial Gaussian mode with left circular polarization and two Laguerre-Gaussian vortex modes with opposite signs of topological charges and right circular polarization. Varying the amplitude and phase ratios of the modes one can realize all of three possible morphological types of the singularity. Since the medium does not have local quadratic response we take into account the effects of nonlinear optical activity. Also, the impact of heterogeneous surface layer is modeled by modified boundary conditions for the fields of the signal beam. In case of pure bulk nonlinearity the reflected beam contains two C-points with total topological charge equal to doubled charge of the initial singularity. However, their polarization handedness is not reversed by reflection if the C-point in the fundamental beam has negative charge. Media with surface response generate far more complicated polarization patterns of the signal beam, which in special cases inherit the symmetry of the fundamental beam. Total topological charges of left-handed and right-handed C-points depend only on the charge of initial C-point, but do not reflect the difference between its “lemon” and “monstar” patterns. For the media with both bulk and surface nonlinear response their values may not be equal to doubled charge of the initial C-point.