Periodic Fourier Modal Method (FMM) for 1D and 2D gratings) 

The Fourier modal method (FMM) was first introduced by Knop (1978) for the case of dielectric gratings, which involves approximating the eigenfunctions of the diffraction operator using a partial Fourier sum. Moharam and Gaylord (1981) later extended this method to analyze diffraction by planar and volume gratings bounded by two different media, coining the term rigorous coupled wave analysis/method (RCWA or RCWM). However, Li (1998) subsequently adopted the terminology FMM since the method involves expanding the electromagnetic field components and the permittivity function into Fourier series, making it essentially a modal method.
The name FMM is justified for at least two reasons. Firstly, it highlights the fact that RCWA is not fundamentally different from the method proposed by Knop. Secondly, it provides a general terminology that can be compared with the classical modal method (Botten et al., 1981a, 1981b), which is sometimes referred to as "Modal Method Modal Expansion."
The use of FMM for metallic gratings, especially in the case of transverse magnetic (TM) polarization, proved to be challenging for the scientific community until 1996. It was only then that Granet and Guizal, as well as P. Lalanne and G. M. Morris, independently reformulated the FMM for isotropic gratings under TM polarization. They suggested calculation rules that allow for a fast convergence of the series of the partial Fourier sums. Li (1996) later provided mathematical justification for their findings in the form of three Fourier factorization rules.

Polynomial Modal Method  for periodic and non periodic  strucutres (PMM/APMM)

In recent years, significant improvements have been made to modal methods based on polynomial decompositions of fields using Legendre, Chebychev, or Gegenbauer polynomials. These advancements have positioned polynomial modal methods (PMM) as serious competitors to the Fourier modal method (FMM), even when it is equipped with the adaptive spatial resolution (ASR) concept.
Two variants of the PMM have been suggested. In the first formulation, the structure is partitioned in the periodicity direction, and the restriction of the eigenvectors is approximated by orthogonal polynomials (such as Gegenbauer) on each subinterval corresponding to a homogeneous medium. These approximations are then used to solve the eigenvalue equation by introducing them into Maxwell's equations and obtaining an undersized matrix relation. The boundary conditions satisfied by the electromagnetic field components are then added to the matrix relation to deduce the complete eigenvalue equation. However, the connection of boundary conditions to the undersized matrix is technically complex and can be tedious to implement.
Therefore, a second formulation of the PMM has been introduced, which is conceptually synthetic and easier to implement. This version departs from an orthogonal set of polynomials and defines a set of modified polynomial bases that naturally and rigorously describe the transversal boundary conditions. This approach has a significant advantage over the first formulation, as it allows for easy generalization to crossed gratings, anisotropic and bi-anisotropic mediums, which would have been tedious with the former version.


PMM implementation in matched coordinates

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3D Aperiodic Fourier Modal Method (AFMM)

The aperiodic Fourier modal method (AFMM) is a hybrid method that combines a solver for periodic systems with perfectly matched layers (PMLs) to address the challenge of implementing PMLs in aperiodic structures. One of the key challenges in implementing PMLs in aperiodic structures is describing both the periodic incident plane wave (the input field) and the aperiodic scattered field (the output field) of the isolated device within the same formalism.
To address this challenge, we introduce a hybrid method that combines a Fourier basis with Maxwell's equations in complex coordinates and the Stratton and Chu formulation. This method provides a rigorous and efficient framework for the calculation of the scattering matrix of aperiodic structures with PMLs, allowing for accurate characterization of the device's electromagnetic response. By combining the advantages of both periodic and aperiodic methods, the AFMM represents a powerful tool for the design and analysis of complex electromagnetic devices in a variety of applications.


The scattering of plane waves by a grating was studied using the curvilinear coordinates method, which involves selecting a coordinate system such that the surface profile coincides with a surface of coordinates. When using a translation curvilinear coordinate system, Maxwell's equations can be reduced to an eigenvalue equation. To solve this equation numerically, the method of moments is used with periodic or pseudoperiodic functions as expansion and test functions.
The convergence of the method is closely related to the choice of expansion functions, which must fit the boundary conditions of the problem. For grating problems, it is natural to choose periodic functions. The effectiveness of the method relies on the appropriate selection of expansion functions, ensuring that the method converges rapidly and accurately. Overall, the curvilinear coordinates method offers a powerful and efficient tool for studying the scattering of plane waves by a grating.

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