Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium

Expressions for the electric field at a sum frequency generated by a collinear elliptically polarized Gaussian beam and circularly polarized Laguerre-Gaussian beam in an isotropic chiral nonlinear medium are obtained in quadratures. The amount and locations of $C$ points in the cross section of a signal beam at a sum frequency are shown to be dependent on frequency and diffraction lengths ratios of fundamental beams and on the ellipticity degree of the Gaussian beam's polarization ellipse. Possible values of total topological charges of the emergent $C$ points are determined by the topological charge of the Laguerre-Gaussian beam and remain constant while the radiation propagates in nonlinear media. In case of nonzero total topological charge $C$ lines form helical structures, the parameters of which depend on the wave-vector mismatch. Otherwise, $C$ lines form a loop. As the wave-vector mismatch grows the loop undergoes deformation and breaks up, creating new $C$ lines.

Figure 1

The geometry of the fundamental beams interaction.

Figure 2

Left-hand $G$ lines (a, c) and transversal distributions of the nonlinear medium polarization field at the position of the waist of the fundamental beams (b, d) for the first (a, b) and the second (c, d) types of vector field ${\mathbf{P}}^{\text{(v)}}$ configurations. The charge $m=1$, and beams' parameters are $w_2/w_1=2,k_2/k_1=1.4,M_0=-0.7$ (a, b) and $w_2/w_1=0.3,k_2/k_1=2,M_0=0$ (c, d). The plane $z=0$ in the figure corresponds to the position of the fundamental beams' waists. Red (full) and blue (dashed) lines correspond to $C$ lines with positive and negative topological charges, respectively. $C$ points with positive and negative charges are marked by red (filled) and white circles. Blue (filled) ellipses are left-hand polarized and empty ellipses are right-hand polarized.

Figure 3

Left-hand $C$ lines (bold) and $G$ lines (thin) for $\nu =-5$ (a), $\nu =5$ (b), and $w_2/w_1=2,k_2/k_1=1.4,m=1,M_0=-0.7,l_0=0$. In figure (a) each $C$ line forms a left-hand screw, and in figure (b) it forms a right-hand one. Red (full) and blue (dashed) lines correspond to $C$ lines with positive and negative topological charges, respectively. $C$ points with positive and negative charges are marked by red (filled) and white circles.

Figure 4

Left-hand $C$ lines in the sum-frequency beam for $\nu =-4$ (a), $\nu =-8$ (b), and $w_2/w_1=0.3,k_2/k_1=2,m=1,M_0=0,l_0=l_3$. $C$ lines form three separate loops in figure (a) and two loops in figure (b). Red (full) and blue (dashed) lines correspond to $C$ lines with positive and negative topological charges, respectively.