Conditions for Minimizing the Computational Complexity of the RCWA Calculation of the Diffraction Efficiency of Sawtooth Two-Layer Double-Relief Microstructures

Abstract

In this study, novel recommendations are presented and substantiated for selecting the number of modes and optical thicknesses of flat lattice slabs that make up microreliefs, which minimize the computational complexity of the rigorous coupled-wave analysis calculation of the diffraction efficiency (DE) of a sawtooth two-layer two-relief microstructure, while maintaining the specified reliability of the calculation results. The computational complexity can be controlled by allowing one or another level of oscillation of the DE curves, depending on the angle of incidence of the radiation incident on the microstructure. In particular, the complexity of the thousands of DE calculations in the optimization process can be reduced by using the proposed methodology as well as increased computational complexity to verify the accuracy of the solution obtained as a result of the implemented optimization.

Techniques for speeding up algorithms for implementing rigorous coupled-wave analysis

“…As a basic algorithm, the algorithm for determining for finding DE of the transmitted diffraction orders (see algorithm described in paper [3]) was used. The DE values obtained by each of the five ways are identical to each other and to the results of our work [4] (see Figures 4 and 5). As can be seen from the results presented in Table 1, the use of the properties of the Toeplitz matrix significantly speeds up the obtaining of results by the RCWA (compare the columns "Without the use of techniques" and "Technique № 2" with the columns "Technique № 1", "Joint techniques № 1 and 2" and "Joint techniques № 1, 2 and 3").…”

“…Moreover, the selection of optimal values of N m and N L requires multiple recalculations of the DE with different these values. The authors of this article encountered a similar problem both in the above-mentioned paper [4] and in their earlier works (see, for example, papers [5] and [6]). In connection with the foregoing, the issue of reducing the computational complexity in the tasks of reliably determining both individual values of the DE and entire dependences of the DE on such parameters as the wavelength λ and the angle of incidence of radiation θ is topical.…”

“…Magnetic permeability is equal to unity. The "Core Algorithm" block symbolizes the main computational part of the algorithm of the selected approach within the framework of RCWA, for example, described in articles [2] or [4]. Here it is assumed that dispersion formulas for permittivity of all M materials in the structure (ε 1 , ..., ε M ) and formulas for Fourier coefficients are included in the program text and are placed in rectangular blocks.…”

“…As a characteristic of the number of plates, it is convenient to take the ratio of the optical thickness of one plate to the wavelength of the incident radiation k = L/λ. This convenience is related to the solution of optimization problems (see our paper [4]). For example, in [4], the optimal number of modes N m for some tasks could reach 600, and k could drop to 0.05.…”

“…This convenience is related to the solution of optimization problems (see our paper [4]). For example, in [4], the optimal number of modes N m for some tasks could reach 600, and k could drop to 0.05. It is obvious that the smaller the parameter k, the greater the number of thin lattice plates N L .…”