Mathematical properties of functions characterizing symmetries of spatial distribution of second harmonic–sum frequency radiation are determined. The conditions at which there occurs no generation and the conditions at which the polarization of the generated radiation is linear are found. The revealed properties are systematized by their manifestations in the directivity patterns of the generated radiation. Methods based on these properties for determining the components of the nonlinear dielectric susceptibility tensor are proposed. The relationship is described between the revealed properties as well as conditions and previously found similar properties and conditions for the phenomena of second–harmonic generation and sum–frequency generation.

The solution to the problem of second harmonic–sum frequency generation by two coherent plane electromagnetic waves with elliptical polarizations and equal frequencies in a thin spherical layer is analyzed graphically. Asymmetries are introduced that quantitatively describe the shape of three-dimensional directivity patterns (spatial distribution of the power density of second harmonic–sum frequency radiation). Three-dimensional directivity patterns and asymmetries are analyzed for various combinations of the parameters: ratio of the complex amplitudes of the incident waves, angle between the wave vectors of the incident waves (the opening angle), ellipticities, orientations of polarization ellipses, spherical particle size. It is found that, at small particle sizes, each anisotropy type corresponds to its own individual directivity pattern. It is revealed that, for one of the anisotropy types, the shape of the directivity pattern almost does not change for nearly all possible ranges of the above parameters.