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Based on the scattering matrix formalism, the method of quantization of the electromagnetic field in photonic nanostructures with three-dimensional modulation of the dielectric constant (S-quantization in the three-dimensional case) is developed. Quantization is based on equating the eigenvalues of the scattering matrix to unity, which is equivalent to equating each other the set of fields, which are expanded into Fourier series, of the waves incidenting on the structure to the set of waves which travel away from the structure (outgoing waves). In the photon nanostructure, the spatial variation of the electromagnetic field of the modes is calculated on the basis of the R̃ and T̂ matrices describing the reflection and transmission of the Fourier components through the structure. To calculate the reflection and the transmission coefficients of individual coordinateand Fourier-components, the structure is divided into parallel layers, in which the dielectric constant varies in a two-dimensional coordinates. Using the Fourier transform, Maxwell’s equations are written in the form of a matrix connecting the Fourier components of the electric field at the boundaries of neighboring layers. Based on the calculated vectors of reflection and transmission for all polarizations and Fourier components, a scattering matrix is formed for the entire structure and quantization is performed by equating the eigenvalues of the scattering matrix to unity. The developed method makes it possible to obtain spatial profiles of eigenmodes without solving a system of nonlinear integro-differential equations, and significantly reduces the computational resources necessary for calculating the probability of spontaneous emission in three-dimensional systems. 4 Физика и техника полупроводников, 2018, том 52, вып. 9

Квантование электромагнитного поля в трехмерных фотонных структурах на основе формализма матрицы рассеяния (S-квантование)