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

Abstract

Using energy and momentum conservation laws, we obtained the intrinsic symmetry relations for the nonlocal nonlinear optical susceptibility tensor in lossless nth order nonlinear medium of arbitrary symmetry class for the case when less than n + 1 electromagnetic waves with different frequencies interact. Particular attention is devoted to the relations of the components of this tensor, which cannot be obtained as limiting case from the symmetry relations for the nonlocal nonlinear susceptibility tensor describing interaction of exactly n + 1 waves with different frequencies. The examples of these symmetry relations for degenerate second- and third-order processes often considered are given.

Peculiarities of the intrinsic symmetry of linear and nonlinear optical susceptibility tensors in nonabsorbing inhomogeneous media with nonlocality of optical response

“…in equation ( 29) in first order to the spatial dispersion parameter obeys the same intrinsic symmetry relations as a counterpart tensor of nonlocal nth order nonlinear optical susceptibility of homogeneous media with the same sequence of frequency arguments [24].…”

“…These methods do not contradict each other, but, as mentioned in [5], the method based on the consideration of the energy conservation law allows both to obtain these symmetry relations and opens the possibility of determining the explicit form of the expressions for the energy density and the energy flux density of the electromagnetic field. Another advantage of the method we used is the possibility to use it to obtain intrinsic symmetry relations for the tensors of local [20][21][22] and nonlocal [23,24] nonlinear optical susceptibilities. The possibility of using reciprocity theorem to study symmetry properties of nonlinear optical susceptibility tensors is discussed in [25].…”

“…dr n of the products of the tensor χ(n) on all possible combinations of various degrees of vectors r − r 1 , r − r 2 ,… ,r − r n . We substitute the resulting expressions into the left-hand side of equation (24). The components of the vector G k can be represented similarly to the right-hand side of equation ( 8) in a most general form as sums of products of various combinations of spectral components of electric field strengths and their spatial derivatives of all orders for each allowed combination of frequencies in the components of some auxiliary tensor.…”

“…The components of the vector G k can be represented similarly to the right-hand side of equation ( 8) in a most general form as sums of products of various combinations of spectral components of electric field strengths and their spatial derivatives of all orders for each allowed combination of frequencies in the components of some auxiliary tensor. These components can be obtained by differentiating G k in equation (24) and subsequent equating of coefficients at the identical combinations of spectral components of electric field strength and their spatial derivatives in both left-hand and right-hand sides of the resulting equation. The solution of this system of linear equations makes it possible to write down rather cumbersome intrinsic symmetry relations for the components of tensors of local and nonlocal nth-order nonlinear optical susceptibility.…”