Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings

Abstract

The Fourier modal method (FMM), often also referred to as rigorous coupled-wave analysis (RCWA), is known to suffer from numerical instabilities when applied to low-loss metallic gratings under TM incidence. This problem has so far been attributed to the imperfect conditioning of the matrices to be diagonalized. The present analysis based on a modal vision reveals that the so-called instabilities are true features of the solution of the mathematical problem of a binary metal grating dealt with by truncated Fourier representation of Maxwell's equations. The extreme sensitivity of this solution to the optogeometrical parameters is the result of the excitation, propagation, coupling, interference, and resonance of a finite number of very slow propagating spurious modes. An astute management of these modes permits a complete and safe removal of the numerical instabilities at the price of an arbitrarily small and controllable reduction in accuracy as compared with the referenced true-mode method.

High order integral equation method for diffraction gratings

“…Efficient numerical methods are needed to analyze the diffraction and scattering of light by these periodic structures. Existing numerical methods for diffraction gratings include general-purpose methods such as the finitedifference time-domain (FDTD) method and the finite element method (FEM) [3], and more special methods such as the analytic modal method [4][5][6][7], numerical modal methods [8][9][10][11][12][13][14][15][16][17][18], the boundary integral equation (BIE) methods [19][20][21][22][23][24][25][26][27][28], etc. Although FDTD and FEM are extremely versatile, they are typically less efficient than the special methods.…”

“…Although FDTD and FEM are extremely versatile, they are typically less efficient than the special methods. Numerical modal methods, especially the Fourier modal method (FMM) , are simple to implement and very popular, but they also have a number of limitations [11,29]. Conventional BIE methods are applicable to gratings with piecewise constant but otherwise arbitrary refractive index profiles, but they are relatively complicated to implement since the quasiperiodic Green's function appearing in the integral operators requires sophisticated lattice sum techniques to evaluate.…”

Modal Method Based on Subsectional Gegenbauer Polynomial Expansion for Lamellar Gratings: Weighting Function, Convergence and Stability

“…All we can Finally, we test the stability of our approach over a grating configuration that posed numerical problems to the FMM [13] and for which solutions have been proposed by some authors [14,15]. It consists of a grating with a relative dielectric permittivity equal to −100 and all the other parameters are given in Figure 4.…”

The application and improvement of Fourier transform spectrometer experiment

“…FTS is very important for the study of atomic and molecular physics, astronomy and physics, spectroscopy, atmospheric remote sensing, analytical chemistry and other fields [4][5][6] . And it is also essential equipment for industrial inspection, customs inspection and so on [7] .…”