The permutation symmetry relations of the nth order nonlocal nonlinear optical susceptibilities emerging in the absence of absorption are obtained from the basic statements of electrodynamics. The values of the nth order nonlocal nonlinear susceptibility tensor components with swapped frequencies and indices corresponding to the generated electric field and the electric field under the operator of differentiation in the constitutive equation only differ by their signs and signs of the interchanged frequencies. Additionally, there exist
n
2
−
n
relations between three components of this tensor with cyclically permuted indices and frequencies corresponding to the generated field, the field under the operator of differentiation in the constitutive equation and one of the remaining fields involved into nonlinear process under consideration.
The components of electric field of the third harmonic beam, generated in isotropic medium with cubic nonlinearity by a monochromatic light beam carrying polarization singularity of an arbitrary type, are found analytically. The relation between C-points characteristics in the fundamental and signal beams are determined, as well as the impact of the phase mismatch on the shape of the C-lines.
In this work, we suggest a new algorithm for plotting the singularity lines of an electromagnetic field, which uses its derivatives in a certain area of space and the position of a single point belonging to an unknown singularity line. This algorithm is based on the common properties of rationally parameterizable curves. The implementation of this method is capable of plotting these lines significantly faster and with higher precision, as compared with ordinary methods. The advantages of the suggested method are illustrated by comparing the results of its application in problems of light-matter interaction, in which singularity lines are plotted using both the suggested algorithm, and ordinary algorithms.
The additional terms caused by the nonlocality of the nonlinear optical response of the medium in the expressions for the optical angular momentum density, the optical angular momentum flux density and the torque density on light, which are related to each other by the angular momentum transformation law, are obtained as a consequence of peculiarities of the momentum conservation law in such media. It is shown that the manifestation of the nonlocality of the optical response only changes the form of polarization of medium included in the expression for the angular momentum density, whereas the definition of the angular momentum flux density contains additional term depending on the nonlocal nth order nonlinear optical susceptibility.
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