A combination of up- and down-going plane waves used to describe the field inside grooves of a deep grating

Abstract

INTRODUCTIONThe purpose of the present research is to extend the range of application of Yasuura's method [1,2] in solving the problem of diffraction by a grating. Although alternative terminology for the method (e.g., a least-squares boundary residual method or a modified Rayleigh method) exists, we employ the name throughout this paper. It is an accepted knowledge [3,4] that Yasuura's method, in particular, the conventional Yasuura method with Floquet modes as basis functions does not have a wide range of application. Although the convergence of the sequence of solutions obtained by the method is proven, the rate of convergence is often so slow for deep gratings that we cannot find solutions with accuracy. Let D and 2H be the period and the depth of a sinusoidal grating made of a perfectly conducting metal. The period is assumed to be comparable to the wavelength, i.e., we are working in the resonance region. For an E-wave (s polarization) problem where 2H/D = 0.5, taking 71 Floquet modal functions, we can obtain a solution with 1 percent error in both energy conservation and boundary condition. Employment of additional Floquet modal functions easily causes numerical trouble in making least-squares approximation on the boundary. Hence, a practical limit in 2H/D in the E-wave case is 0.5 so long as we use conventional doubleprecision arithmetic. Similarly, the limit in the H-wave (p polarization) case seems to be a little less than 0.4. To accelerate the convergence of solutions, Yasuura's method is equipped with a smoothing procedure [5,6]. It has been shown that: in the above problem, we can obtain a solution with I percent error using 17-41 modal functions (the number depends on the order of the smoothing procedure and on the polarization). Hence, Yasuura's method with the smoothing procedure is effective in making a systematic research that needs to handle problems with complicated boundaries, e.g., Fourier gratings. Although Yasuura's method with the smoothing procedure solves most of the problems for commonly used gratings, the limit in 2H/D has as yet been scarcely dealt with. There still is a limit at 211/D = 0.7 or 0.8 even if we employ