Nonlinear Optics with Elliptically Polarized Singular Beams and Short Pulses in Media with Spatial Dispersion

Intrinsic symmetry of nonlocal nonlinear optical susceptibility tensor in degenerate multi-wave mixing

“…In nonlinear optics, the relation between polarization of a medium in a given point of space at a given moment of time to electric field strengths in other moments of time in an area around this point is given by the well-known and rather complicated integral (see e.g. [1][2][3][4][5][6][7][8][9][10][11][12]). Along with the electric field strengths themselves, the number of which is determined by the order of nonlinearity of the medium, its integrand contains a tensor characterizing this nonlinearity and having rank dependent on the number of multipliers.…”

“…Along with the electric field strengths themselves, the number of which is determined by the order of nonlinearity of the medium, its integrand contains a tensor characterizing this nonlinearity and having rank dependent on the number of multipliers. If we assume that the components of electric field are strictly monochromatic and the characteristic size of area in which nonlocality of optical response is manifested is less than wavelengths of propagating waves, we can transform this integral into the expression for polarization of medium consisting of two groups of terms [11,12]. If we neglect the second group, this expression has the form of the well-known constitutive equation of nonlinear * Author to whom any correspondence should be addressed.…”

“…The second group of terms describes the contribution of both spectral components of electric field strengths and their spatial derivatives to the spectral component of medium polarization. If we only keep the terms which contain only one first-order spatial derivative of spectral components of electric field strengths in this group, then in a medium of nth order of nonlinearity this group will contain n terms [11,12]. Each of them contributes to the spectral component of polarization and contains n + 2-rank tensor of nonlocal nonlinear optical susceptibility dependent on n + 1 frequencies and each index of this tensor corresponds to exactly one frequency except for the last index which corresponds to the index of spatial derivative operator.…”

“…The symmetry of this tensor to permutations of indices and frequencies is determined by the reality of electric and magnetic fields, symmetry properties of the medium, and the possibility to perform certain indices permutations which keep the constitutive equation unchanged [1-3, 5, 6, 9]. In the absence of absorption in medium energy and momentum conservation laws impose additional restrictions on the intrinsic permutation symmetry of its components [11][12][13][14][15]. In some cases the combination of intrinsic symmetry of this tensor with spatial symmetry of the medium leads to zeroing of the tensor which forbids certain nonlinear processes.…”

Intrinsic symmetry of nonlocal nonlinear optical susceptibilities

“…The rank of the nonlocal nonlinear susceptibility tensors is by one greater than the rank of the local nonlinear susceptibility tensors of the same order of nonlinearity. Nonlinear optical effects caused by nonlocal nonlinear susceptibilities are well known (see [13,14] and references in [15,16]). If local nonlinear susceptibility tensor responsible for certain process is zero due to medium symmetry, the signal may be generated by non-zero components of nonlocal nonlinear susceptibility tensor of the same order (e.g.…”