A new formulation of the Fourier modal method (FMM) that applies the correct
rules of Fourier factorization for crossed surface-relief gratings is presented.
The new formulation adopts a general nonrectangular Cartesian coordinate system,
which gives the FMM greater generality and in some cases the ability to save
computer memory and computation time. By numerical examples, the new FMM is
shown to converge much faster than the old FMM. In particular, the FMM is
used to produce well-converged numerical results for metallic crossed gratings.
In addition, two matrix truncation schemes, the parallelogramic truncation
and a new circular truncation, are considered. Numerical experiments show
that the former is superior.

New formulation of the Fourier modal method for crossed surface-relief gratings

Two-dimensional symmetric and asymmetric subwavelength binary gratings are investigated. A method for determining the three effective indices of a two-dimensional (2-D) subwavelength grating is presented, as well as a theoretical formalization for the effective index parallel with the normal to the surface. It is shown that a 2-D asymmetric binary grating on the surface of a dielectric substrate is analogous to a biaxial thin film. If the grating is symmetric, then the two effective indices perpendicular to the normal are equal, and the grating is analogous to a uniaxial thin film. Using these effective indices and the quarter-wave Tschebyscheff synthesis technique, we designed two- and three-level binary gratings to suppress reflections over a broad band. It is shown that for a substrate index of ns = 3.0 a three-level 2-D binary grating reduced reflections below 0.1% from 8 μm to 12 μm.

Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings

Using theorems of Fourier factorization, a recent paper [J. Opt. Soc. Am. A 13, 1870 (1996)] has shown that the truncated Fourier series of products of discontinuous functions that were used in the differential theory of gratings during the past 30 years are not converging everywhere in TM polarization. They turn out to be converging everywhere only at the limit of infinitely low modulated gratings. We derive new truncated equations and implement them numerically. The computed efficiencies turn out to converge about as fast as in the TE-polarization case with respect to the number of Fourier harmonics used to represent the field. The fast convergence is observed on both metallic and dielectric gratings with sinusoidal, triangular, and lamellar profiles as well as with cylindrical and rectangular rods, and examples are shown on gratings with 100% modulation. The new formulation opens a new wide range of applications of the method, concerning not only gratings used in TM polarization but also conical diffraction, crossed gratings, three-dimensional problems, nonperiodic objects, rough surfaces, photonic band gaps, nonlinear optics, etc. The formulation also concerns the TE polarization case for a grating ruled on a magnetic material as well as gratings ruled on anisotropic materials. The method developed is applicable to any theory that requires the Fourier analysis of continuous products of discontinuous periodic functions; we propose to call it the fast Fourier factorization method.

https://mcgrating.com/references/ref21.pdf

Grating theory: new equations in Fourier space leading to fast converging results for TM polarization

We extend the rigorous eigenmode theory of binary surface-relief gratings to accommodate three-dimensional-modulation profiles, to formulate a synthesis problem for doubly periodic resonance-domain diffractive elements, and to demonstrate some of the problem’s symmetry properties. Several solutions for multiple beam splitters with ~90% transmission-mode diffraction efficiency are obtained by nonlinear parametric optimization. The polarization sensitivity and the required fabrication accuracy are analyzed for some solutions.

Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles

We apply the rigorous electromagnetic grating theory and an electromagnetic wave-propagation model with Kirchhoff boundary conditions to analyze and optimize the performance of arrays of cylindrical dielectric multilevel diffractive lenses. We show that the diffraction efficiency of a high-numerical-aperture lens array can be improved if the surface-relief profile is constructed with the use of rigorously optimized grating groove structures, which differ significantly from a multilevel approximation of a triangular profile.

Electromagnetic theory and design of diffractive-lens arrays

A finite-difference method for dielectric and conducting crossed gratings

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A finite-difference method for dielectric and conducting crossed gratings