Dr. Kofi EDEE was born in Lomé (Togo) in 1975. He received his Ph. D in computational physics for electromagnetic modeling, from University Clermont Auvergne. After postdoctoral training in the Microelectronics Technology Laboratory at Grenoble, he returns to Clermont-Ferrand where he currently is Associate professor and researcher of Pascal Institut. He also serves as Director of the Non Destructive Testing department. His research interests concern essentially mathematical and computational physics. More specifically, he develop advanced and accurate electromagnetic modal methods to treat optics and electromagnetic problems for modern nanophotonic devices as plasmonic lenses, waveguides, gratings, etc. He recently introduced a polynomial modal method based on Gegenbauer polynomial expansion and emphasized that the main advantage of this new method is that the continuity relations of the electromagnetic field can be written in an exact manner.
Computational Physics for Photonics – Plasmonics
- Method of moment
- Tensorial form of Maxwell’s equations and Modal Method
- Modal Method (Boundary value problem)
Theory and Numerical Methods
Fourier Modal Method FMM :
The Fourier modal method (FMM) introduced by Knop (1978) in the case of dielectric gratings, consists of approximating the eigenfunctions of the operator of diffraction by a partial Fourier sum. Moharam and Gaylord (1981) generalized this method to analyze diffraction by general planar gratings (or volume gratings) bounded by two different media and introduced the terminology: rigorous coupled wave analysis/method (RWCA or RWCM). Later on, Li (1998) adopted the terminology FMM, since the method is essentially a modal method that involved expansions of the electromagnetic field components and the permittivity function into Fourier series. The use of the name FMM is justified for at least two reasons:
- it means that the RCWA is not different from the method proposed by Knop
- It enters under a general terminology to be compared classical modal method (Botten et al., 1981a, 1981b), which is called in some cases “Modal Method Modal Expansion”
The case of metallic gratings, and especially the TM polarization case, challenged the com- munity until 1996, when Granet and Guizal and P. Lalanne and G. M. Morris6 (1996) simultaneously reformulated the FMM for isotropic gratings under transverse magnetic (TM) polarization by suggesting calculation rules allowing a fast convergence of the series of the partial Fourier sums. The mathematical justification of their findings was provided after by Li (1996) under the form of three Fourier factorization rules.
Polynomial Modal Method PMM :
In the very recent years, the new improvements made on the modal methods based on polynomial decompositions (Legendre, Chebychev, Gegenbauer) of the fields, position them as serious competitors to the FMM even when it is equipped with the ASR concept. Indeed, the Polynomial Modal Methods (PMMs) prove to be particularly suited for the simulation of metallic structures, precisely where the FMM finds its limits. The PMM based on Gegenbauer polynomials has been introduced in references and where it has been successfully applied to 1D periodic diffraction problems in cartesian coordinates. Then it has been efficiently extended to the case of non periodic structures and also implemented in cylindrical coordinates in order to compute the eigenvalues of a cylindrical waveguide with arbitrary cross section. Globally, two variants of the PMM have been suggested.
- First formulation of PMM
In the first formulation, a partition of the structure in the periodicity direction is defined. On each subinterval of the partition, corresponding to an homogeneous medium, the restriction of the eigenvectors is approximated by orthogonal polynomials (Gegenbauer ones for instance). These approximations are then introduced in Maxwell’s equations and an undersized matrix relation is obtained. The eigenvalue equation is deduced by adding to this undersized matrix relation the boundary conditions satisfied by the electromagnetic field components. We have shown throughout various examples that this method is outstanding in terms of convergence and stability.
- Second formulation of PMM
However the connection of the boundary conditions to the undersized matrix is very technical and this process can be quite tedious. That is why we have introduced a second formulation which is conceptually synthetic and easier to implement. It consists in departing from an orthogonal set of polynomials and defining a set of modified polynomials basis that naturally and rigorously describe the transversal boundary conditions. One of the strengths of this version of the PMM, is that it allowed for an easy generalization to crossed lamellar gratings which would have been tedious with the former version.
Curvilinear coordinates method (C-method) :
The curvilinear coordinates method was applied to study the scattering of plane waves by a grating. The main point of this method is the choice of a coordinate system such that the profile of the surface coincides with a surface of coordinates. Furthermore, in the case of a translation curvilinear coordinate system, Maxwell’s equations lead to an eigenvalue equation. Numerically, the eigenvalue equation is solved by using the method of moments with periodic or pseudoperiodic functions as expansion and test functions. The convergence of the method is closely related to the choice of the expansion functions that have to fit the boundary conditions of the problem. Hence, for grating problems, the choice of periodic functions is a natural one.
Perfectly matched layers and complex coordinates transform :
The concept of PMLs was first introduced in the frame of the finite difference time domain (FDTD) by Bérenger and is called the perfectly matched layers (PMLs) concept. Since then, the concept has been extended to other numerical methods, especially in the frequency domain. Sacks et al. showed how a plane diopter separating an isotropic medium and a uniaxial anisotropic medium can achieve a perfect impedance adaptation and thus suppress any reflection at this interface. The equivalence between the concept of PMLs and stretching coordinates applied to the Maxwell equations was established by Chew and Weedon in . Later on, Chew et al. interpreted the change of complex coordinates as a continuous extension of the electromagnetic field in the complex plane. After that, works based on the use of differential forms were proposed by Teixeira and Chew.