ИСТИНА

Angular momentum (AM) of light beam and its connection with the polarization of the propagating radiation is the object of numerous studies. In a paraxial beam the AM can be represented as a sum of so-called spin and orbital contributions. Beams carrying AM are widely used in different applications. We analyze the sum-frequency generation (SFG) in isotropic chiral medium, and study the interconnection between the AM of elliptically polarized focused Gaussian beams of fundamental radiation and one of the originating signal beam at sum-frequency. Conventional example of such a medium is an isotropic solution of chiral molecules or a metamaterial consisting of randomly oriented chiral meta-elements. In such a medium the collinear SFG is possible only due to the interaction of non-collinear spatial Fourier-components of the fundamental radiation [1] (in a plane-wave approximation the sum-frequency signal is zero). The angular momenta of the elliptically polarized Gaussian fundamental beams are represented only by their spin parts. Their average values (per photon) are equal to the ellipticity degrees of the polarization ellipses of the beams taken with opposite signs, -M1 and -M2. The originating signal beam at sum-frequency is the superposition of two elliptically polarized Laguerre-Gaussian modes. Its AM has both spin and orbital parts. The transversal distribution of the flux density vector of the AM in signal beam has complex shape. Total AM at sum-frequency is proportional to the sum of the spin AM of the fundamental waves. The orbital part of the AM at sum-frequency is mainly determined by the lower frequency beam spin AM, while the spin contribution is mainly determined by the corresponding quantity of the higher frequency fundamental beam. The modulus of the AM per photon in the beam at sum-frequency is always less or equal to the modulus of a sum of spin AM per photon in the incident fundamental beams. The difference (the “defect” of the AM in sum-frequency beam) is either transferred to the medium, or retains in the propagating fundamental beams. Its value depends on the spin AM of the fundamental beams, their wavenumbers ratio and the angle between the axes of their polarization ellipses (fig. 1c). Variation of the polarization states of the fundamental beams allows for the control of the relative contributions of spin and orbital AM in signal beam, as well, as its total value.