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We study the convergence properties of the two-dimensional Rigorous Coupled Wave Approach (RCWA) for p-polarized monochromatic incident light. The RCWA is a semi-analytical numerical method that is widely used to solve the boundary-value problem of scattering by a grating. The approach requires the expansion of all electromagnetic field phasors and the relative permittivity as Fourier series in the spatial variable along the direction of the periodicity of the grating. In the direction perpendicular to the grating periodicity, the domain is discretized into thin slices and the actual relative permittivity is replaced by an approximation. The approximate relative permittivity is chosen so that the solution of the Maxwell equations in each slice can be computed without further approximation. Thus, there is error due to the approximate relative permittivity as well as the trucation of the Fourier series. We show that the RCWA embodies a Galerkin scheme for a perturbed problem, and then we use tools from the Finite Element Method to show that the method converges with increasing number of retained Fourier modes and finer approximations of the relative permittivity. Numerical examples illustrate our analysis, and suggest further work.

A design tool was formulated for optimizing the efficiency of inorganic, thin-film, photovoltaic solar cells. The solar cell can have multiple semiconductor layers in addition to antireflection coatings, passivation layers, and buffer layers. The solar cell is backed by a metallic grating which is periodic along a fixed direction. The rigorous coupled-wave approach is used to calculate the electron-hole-pair generation rate. The hybridizable discontinuous Galerkin method is used to solve the drift-diffusion equations that govern charge-carrier transport in the semiconductor layers. The chief output is the solar-cell efficiency which is maximized using the differential evolution algorithm to determine the optimal dimensions and bandgaps of the semiconductor layers.

We study the convergence properties of the two-dimensional Rigorous Coupled Wave Approach (RCWA) for s-polarized monochromatic incident light. The RCWA is widely used to solve electromagnetic boundary-value problems where the relative permittivity varies periodically in one direction, i.e., scattering by a grating. This semi-analytical approach expands all the electromagnetic field phasors as well as the relative permittivity as Fourier series in the spatial variable along the direction of periodicity, and also replaces the relative permittivity with a stairstep approximation along the direction normal to the direction of periodicity. Thus, there is error due to Fourier truncation and also due to the approximation of grating permittivity. We prove that the RCWA is a Galerkin scheme, which allows us to employ techniques borrowed from the Finite Element Method to analyze the error. An essential tool is a Rellich identity that shows that certain continuous problems have unique solutions that depend continuously on the data with a continuity constant having explicit dependence on the relative permittivity. We prove that the RCWA converges with an increasing number of retained Fourier modes and with a finer approximation of the grating interfaces. Numerical results show that our convergence results for increasing the number of retained Fourier modes are seen in practice, while our estimates of convergence in slice thickness are pessimistic.

The rigorous coupled-wave approach (RCWA) and the differential evolution algorithm (DEA) were coupled in a practicable approach to maximize absorption in optical structures with three-dimensional morphology. As a model problem, optimal values of four geometric parameters and the bandgaps of three i-layers were found for an amorphous-silicon, multi-terminal, thin-film tandem solar cell comprising three p-i-n junctions with a metallic hexagonally corrugated back-reflector. When the optical short-circuit current density was chosen as the figure of merit to be maximized, only the bandgap of the topmost i-layer was significant and the remaining six parameters played minor roles. While this configuration would absorb light very well, it would have poor electrical performance. This is because the optimization problem allows for the thicknesses and bandgaps of the semiconductor layers to change. We therefore devised another figure of merit that takes into account bandgap changes by estimating the open-circuit voltage. The resulting configuration was found to be optimal with respect to all seven variable parameters. The RCWA+DEA optimization approach is applicable to other types of photovoltaic solar cells as well as optical absorbers, with the choice of the figure of merit being vital to a successful outcome., Comment: 14 pages, 5 figures

The rigorous coupled wave approach (RCWA) was implemented to investigate optical absorption in a triple-p-i-n-junction amorphous-silicon solar cell with a 2D metallic periodically corrugated backreflector (PCBR). Both total and useful absorptances were computed against the free-space wavelength $\lambda_o$ for both s- and p-polarized polarization states. The useful absorptance in each of the three p-i-n junctions was also computed for normal as well as oblique incidence. Furthermore, two canonical boundary-value problems were solved for the prediction of guided-wave modes (GWMs): surface-plasmon-polariton waves and waveguide modes. Use of the doubly periodic PCBR enhanced both useful and total absorptances in comparison to a planar backreflector. The predicted GWMs were correlated with the peaks of the total and useful absorptances. The excitation of GWMs was mostly confined to $\lambda_o < 700$ nm and enhanced absorption. As excitation of certain GWMs could be correlated with the total absorptance but not with the useful absorptance, the useful absorptance should be studied while devising light-trapping strategies., Comment: 17 pages; 8 figures