Gerard Granet - Academia.edu (original) (raw)
Papers by Gerard Granet
Journal of the Optical Society of America A
The problem of diffraction by snake gratings is presented and formulated as an eigenvalue eigenve... more The problem of diffraction by snake gratings is presented and formulated as an eigenvalue eigenvector problem. A numerical solution is obtained thanks to the method of moments where a tensor product of pseudo-periodic functions and Legendre polynomials is used as expansion and test functions. The method is validated by comparison with the usual Fourier modal method (FMM) as applied to crossed gratings. Our method is shown to be more efficient than the FMM in the case of metallic gratings.
We present a new semi-analytical formulation for diffraction by structured cylinders. A pseudo-sp... more We present a new semi-analytical formulation for diffraction by structured cylinders. A pseudo-spectral modal method is used to solve the Maxwell equations written in curvilinear coordinates. The program is compared with the numerical results obtained with finite element method using Comsol Multiphysics
Journal of the Optical Society of America A, 2017
The problem of diraction by slanted lamellar dielectric and metallic gratings in classical mounti... more The problem of diraction by slanted lamellar dielectric and metallic gratings in classical mounting is formulated as an eigenvalue eigenvector problem. The numerical solution is obtained by using the moment method with Legendre polynomials as expansion and test functions which allows to enforce in an exact manner the boundary conditions which determine the eigen-solutions. Our method is successfully validated by comparison with other methods including in the case of highly slanted gratings. Index terms Diraction and gratings, Computational electromagnetic methods, Electromagnetic optics.
Apres une presentation sommaire des enjeux et motivations relatifs a la generation de jets photon... more Apres une presentation sommaire des enjeux et motivations relatifs a la generation de jets photoniques, ce papier se propose de presenter une nouvelle structure guidee permettant d'obtenir de fortes concentrations energetiques. Afin d'en preciser l'interet, nous nous proposons de relater les resultats actuellement disponibles dans la litterature [1-2]. La modelisation numerique a ete realisee par l'utilisation de plusieurs methodes numeriques, en particulier la Methode Modale de Fourier (FMM) [3] et la Methode modale basee sur les polynomes de Gegenbauer (MMGE) [4]. Nous souhaitons apres avoir valide les resultats lors d'une confrontation avec un prototype experimental, utiliser ce modele dans le cadre d'une detection en champs proches a travers des structures opaques.
Journal of the Optical Society of America A, 2020
We report on the derivation of a spectral element method whose originality comes from the use of ... more We report on the derivation of a spectral element method whose originality comes from the use of a hierarchical basis built with modified Legendre polynomials. We restrict our work to TM polarization, which is the most challenging. The validation and convergence are carefully checked for metallic dielectric gratings. The method is shown to be highly efficient and remains stable for huge truncation numbers. All the necessary information is given so that non-specialists can implement the method.
International Journal of Numerical Modelling: Electronic Networks, Devices and Fields
SummaryThis paper deals with the development of a semi‐analytical model for the fast computation ... more SummaryThis paper deals with the development of a semi‐analytical model for the fast computation of the quasi‐static field induced by any eddy current probe in a 3D conductor of complex shape. The workpiece is considered as a planar half‐space mock‐up with an interface air conductor characterized by an arbitrary 2D surface a(x, y). The curvilinear coordinate method consists in introducing a change of coordinates to be able to write analytically and easily the boundary conditions at the interface air‐metal. Due to the novel generalized metric space, the covariant form of Maxwell equations must be considered. The curvilinear coordinate method is widely used in the optical community for the analysis the diffraction phenomenon on gratings. The method has been applied recently for eddy current calculations in the planar case for 2.5D configurations characterized by a 3D eddy current probe and a 2D layered stratified conducting media. By an extension to 3D problems, this work constitutes ...
International Journal of Applied Electromagnetics and Mechanics, 2018
The European Physical Journal Applied Physics, 2016
This paper addresses the problem of the development of a fast numerical model for the computation... more This paper addresses the problem of the development of a fast numerical model for the computation of the electromagnetic field in the quasi-static regime. A semi-analytical approach is achieved in order to obtain fast simulation tools dedicated to the simulation of complex configurations of eddy current (EC) inspection: an EC probe scans a conducting cylindrical tube of complex shape and with varying electromagnetic properties. In this paper, though the cylindrical tube to be inspected presents some angular symmetry, the complexity of the configuration lies in the arbitrary shape of the tube’s walls and the dependence of constitutive parameters of the material on the radial or axial coordinates. In this context, no modal approach can be used to expand the components of the fields. This problem is overcome by using a pseudospectral Fourier method. Maxwell’s equations are written in a covariant form in order to translate boundary conditions at each interface in an analytical form. Tangential components of the fields with respect to the boundary surface are expanded by using a high order Chebyshev polynomials. Spatial derivatives of the components of the fields with respect to the radial coordinate are thus approximated at some collocation points. Numerical validations are discussed in order to show the efficiency of the proposed numerical model.
Journal of the Optical Society of America A, 2016
An efficient numerical modal method for modeling a lamellar grating in conical mounting is presen... more An efficient numerical modal method for modeling a lamellar grating in conical mounting is presented. Within each region of the grating, the electromagnetic field is expanded onto Legendre polynomials, which allows us to enforce in an exact manner the boundary conditions that determine the eigensolutions. Our code is successfully validated by comparison with results obtained with the analytical modal method.
Journal of the Optical Society of America A
In this paper, the electromagnetic field scattered by a cylinder with an arbitrary cross section ... more In this paper, the electromagnetic field scattered by a cylinder with an arbitrary cross section is computed using a domain decomposition method in which the structure under consideration is enclosed with two fictitious circular cylinders. TE and TM polarizations are investigated. Our code is successfully validated by comparison with analytical results and with the finite element software COMSOL.
Journal of the Optical Society of America A, 2021
The Fourier modal method (FMM) is certainly one of the most popular and general methods for the m... more The Fourier modal method (FMM) is certainly one of the most popular and general methods for the modeling of diffraction gratings. However, for non-lamellar gratings it is associated with a staircase approximation of the profile, leading to poor convergence rate for metallic gratings in TM polarization. One way to overcome this weakness of the FMM is the use of the fast Fourier factorization (FFF) first derived for the differential method. That approach relies on the definition of normal and tangential vectors to the profile. Instead, we introduce a coordinate system that matches laterally the profile and solve the covariant Maxwell’s equations in the new coordinate system, hence the name matched coordinate method (MCM). Comparison of efficiencies computed with MCM with other data from the literature validates the method.
This paper deals with the computation of quasistatic fields induced in a conductor by a 3D Eddy C... more This paper deals with the computation of quasistatic fields induced in a conductor by a 3D Eddy Current (EC) probe. The shape of the slab is locally distorded and constituted by several homogeneous or non-homogeneous layers with nonparallel interfaces. The approach is based on writing Maxwell’s equations in a curvilinear system, leading to a simple analytical expression of boundary conditions. In homogeneous layers, the fields are expanded as sums of eigenfunctions. In the case of non-homogeneous layers, a pseudo-spectral method is introduced and combined with modal solutions to solve Maxwell’s equations. Some numerical experiments validate this innovative approach, resulting in a fast model, able to tackle a complex configuration never solved in the context of EC Nondestructive Testing.
Diffraction par des surfaces periodiques. Resolution en coordonnees non orthogonales. La diffract... more Diffraction par des surfaces periodiques. Resolution en coordonnees non orthogonales. La diffraction d'une onde electromagnetique par des reseaux metalliques de conductivite finie, ou dielectriques a perte, est etudiee dans le cas general de l'incidence oblique. On ecrit les equations de maxwell sous forme covariante apres avoir choisi un systeme de coordonnees non orthogonales adapte a la geometrie du probleme. On se ramene a un systeme de deux equations differentielles couplees du premier ordre, que l'on resout par une methode aux valeurs propres et vecteurs propres, apres l'avoir projete sur une base de fourier. Ce formalisme permet de calculer le champ electromagnetique sur la surface meme du reseau. Deux cas sont examines: celui des reseaux d'argent peu profonds sur lesquels porte une verification experimentale; celui d'une mer houleuse. Dans l'un et l'autre cas, des renforcements non negligeables du champ de surface sont mis en evidence. Enfin, la definition d'un operateur impedance de surface permet de reduire de facon notable le volume des calculs
Progress In Electromagnetics Research B, 2011
The propagation equation, written in a curvilinear coordinate system, is solved by using a pertur... more The propagation equation, written in a curvilinear coordinate system, is solved by using a perturbation method inspired from quantum physics and extended to imaginary eigenvalues and evanescent waves. The parameter of perturbation is the groove depth which is small compared to the period. The method is expanded up to second order for the non-degenerate problem. In this way the solutions have analytical form compared to a numerical method. They present the advantage to put in evidence the evolution of the energy distribution for different diffraction orders as a function of the magnitude of the perturbation. The efficiencies which are deduced from these analytical solutions are compared of those obtained by the curvilinear coordinate method. The good agreement between the two methods occurs for a groove depth with respect to the wavelength less than or equal to 0.16. Thus, this new approach opens a new range of applications for inverse problems.
Progress In Electromagnetics Research M, 2011
A rigorous modal theory of conical diffraction from curved strip gratings is presented. In this a... more A rigorous modal theory of conical diffraction from curved strip gratings is presented. In this approach, the C-method with adaptive spatial resolution is used in conjunction with the combined boundary conditions. The method is successfully validated by comparison with a case where the solution can also be obtained in the Cartesian coordinate system.
The paper relates to the scattering of a plane wave by a 1-dimensional local perturbation. Above ... more The paper relates to the scattering of a plane wave by a 1-dimensional local perturbation. Above a given deformation, the scattered field can be represented by a superposition of a continuous spectrum of propagating or evanescent plane waves, the so-called Rayleigh - Fourier integral. We demonstrate that this integral decreases as 1/√r in the far-field zone with an angular dependence given by the scattering amplitudes associated with the propagating waves. Applying the Green's second theorem to the asymptotic field, we obtain 2 reciprocity relations and a power balance criterion. Comments are proposed.
Waves in Random Media, 2004
We present a method giving the bi-static scattering coefficient of a one-dimensional dielectric r... more We present a method giving the bi-static scattering coefficient of a one-dimensional dielectric random rough surface illuminated by a plane wave. The theory is based on Maxwell's equations written in a nonorthogonal coordinate system. For each medium, this method leads to an eigenvalue system. The scattered field is expanded as a linear combination of eigensolutions satisfying the outgoing wave condition.
Progress In Electromagnetics Research, 2002
Journal of the Optical Society of America A
The problem of diffraction by snake gratings is presented and formulated as an eigenvalue eigenve... more The problem of diffraction by snake gratings is presented and formulated as an eigenvalue eigenvector problem. A numerical solution is obtained thanks to the method of moments where a tensor product of pseudo-periodic functions and Legendre polynomials is used as expansion and test functions. The method is validated by comparison with the usual Fourier modal method (FMM) as applied to crossed gratings. Our method is shown to be more efficient than the FMM in the case of metallic gratings.
We present a new semi-analytical formulation for diffraction by structured cylinders. A pseudo-sp... more We present a new semi-analytical formulation for diffraction by structured cylinders. A pseudo-spectral modal method is used to solve the Maxwell equations written in curvilinear coordinates. The program is compared with the numerical results obtained with finite element method using Comsol Multiphysics
Journal of the Optical Society of America A, 2017
The problem of diraction by slanted lamellar dielectric and metallic gratings in classical mounti... more The problem of diraction by slanted lamellar dielectric and metallic gratings in classical mounting is formulated as an eigenvalue eigenvector problem. The numerical solution is obtained by using the moment method with Legendre polynomials as expansion and test functions which allows to enforce in an exact manner the boundary conditions which determine the eigen-solutions. Our method is successfully validated by comparison with other methods including in the case of highly slanted gratings. Index terms Diraction and gratings, Computational electromagnetic methods, Electromagnetic optics.
Apres une presentation sommaire des enjeux et motivations relatifs a la generation de jets photon... more Apres une presentation sommaire des enjeux et motivations relatifs a la generation de jets photoniques, ce papier se propose de presenter une nouvelle structure guidee permettant d'obtenir de fortes concentrations energetiques. Afin d'en preciser l'interet, nous nous proposons de relater les resultats actuellement disponibles dans la litterature [1-2]. La modelisation numerique a ete realisee par l'utilisation de plusieurs methodes numeriques, en particulier la Methode Modale de Fourier (FMM) [3] et la Methode modale basee sur les polynomes de Gegenbauer (MMGE) [4]. Nous souhaitons apres avoir valide les resultats lors d'une confrontation avec un prototype experimental, utiliser ce modele dans le cadre d'une detection en champs proches a travers des structures opaques.
Journal of the Optical Society of America A, 2020
We report on the derivation of a spectral element method whose originality comes from the use of ... more We report on the derivation of a spectral element method whose originality comes from the use of a hierarchical basis built with modified Legendre polynomials. We restrict our work to TM polarization, which is the most challenging. The validation and convergence are carefully checked for metallic dielectric gratings. The method is shown to be highly efficient and remains stable for huge truncation numbers. All the necessary information is given so that non-specialists can implement the method.
International Journal of Numerical Modelling: Electronic Networks, Devices and Fields
SummaryThis paper deals with the development of a semi‐analytical model for the fast computation ... more SummaryThis paper deals with the development of a semi‐analytical model for the fast computation of the quasi‐static field induced by any eddy current probe in a 3D conductor of complex shape. The workpiece is considered as a planar half‐space mock‐up with an interface air conductor characterized by an arbitrary 2D surface a(x, y). The curvilinear coordinate method consists in introducing a change of coordinates to be able to write analytically and easily the boundary conditions at the interface air‐metal. Due to the novel generalized metric space, the covariant form of Maxwell equations must be considered. The curvilinear coordinate method is widely used in the optical community for the analysis the diffraction phenomenon on gratings. The method has been applied recently for eddy current calculations in the planar case for 2.5D configurations characterized by a 3D eddy current probe and a 2D layered stratified conducting media. By an extension to 3D problems, this work constitutes ...
International Journal of Applied Electromagnetics and Mechanics, 2018
The European Physical Journal Applied Physics, 2016
This paper addresses the problem of the development of a fast numerical model for the computation... more This paper addresses the problem of the development of a fast numerical model for the computation of the electromagnetic field in the quasi-static regime. A semi-analytical approach is achieved in order to obtain fast simulation tools dedicated to the simulation of complex configurations of eddy current (EC) inspection: an EC probe scans a conducting cylindrical tube of complex shape and with varying electromagnetic properties. In this paper, though the cylindrical tube to be inspected presents some angular symmetry, the complexity of the configuration lies in the arbitrary shape of the tube’s walls and the dependence of constitutive parameters of the material on the radial or axial coordinates. In this context, no modal approach can be used to expand the components of the fields. This problem is overcome by using a pseudospectral Fourier method. Maxwell’s equations are written in a covariant form in order to translate boundary conditions at each interface in an analytical form. Tangential components of the fields with respect to the boundary surface are expanded by using a high order Chebyshev polynomials. Spatial derivatives of the components of the fields with respect to the radial coordinate are thus approximated at some collocation points. Numerical validations are discussed in order to show the efficiency of the proposed numerical model.
Journal of the Optical Society of America A, 2016
An efficient numerical modal method for modeling a lamellar grating in conical mounting is presen... more An efficient numerical modal method for modeling a lamellar grating in conical mounting is presented. Within each region of the grating, the electromagnetic field is expanded onto Legendre polynomials, which allows us to enforce in an exact manner the boundary conditions that determine the eigensolutions. Our code is successfully validated by comparison with results obtained with the analytical modal method.
Journal of the Optical Society of America A
In this paper, the electromagnetic field scattered by a cylinder with an arbitrary cross section ... more In this paper, the electromagnetic field scattered by a cylinder with an arbitrary cross section is computed using a domain decomposition method in which the structure under consideration is enclosed with two fictitious circular cylinders. TE and TM polarizations are investigated. Our code is successfully validated by comparison with analytical results and with the finite element software COMSOL.
Journal of the Optical Society of America A, 2021
The Fourier modal method (FMM) is certainly one of the most popular and general methods for the m... more The Fourier modal method (FMM) is certainly one of the most popular and general methods for the modeling of diffraction gratings. However, for non-lamellar gratings it is associated with a staircase approximation of the profile, leading to poor convergence rate for metallic gratings in TM polarization. One way to overcome this weakness of the FMM is the use of the fast Fourier factorization (FFF) first derived for the differential method. That approach relies on the definition of normal and tangential vectors to the profile. Instead, we introduce a coordinate system that matches laterally the profile and solve the covariant Maxwell’s equations in the new coordinate system, hence the name matched coordinate method (MCM). Comparison of efficiencies computed with MCM with other data from the literature validates the method.
This paper deals with the computation of quasistatic fields induced in a conductor by a 3D Eddy C... more This paper deals with the computation of quasistatic fields induced in a conductor by a 3D Eddy Current (EC) probe. The shape of the slab is locally distorded and constituted by several homogeneous or non-homogeneous layers with nonparallel interfaces. The approach is based on writing Maxwell’s equations in a curvilinear system, leading to a simple analytical expression of boundary conditions. In homogeneous layers, the fields are expanded as sums of eigenfunctions. In the case of non-homogeneous layers, a pseudo-spectral method is introduced and combined with modal solutions to solve Maxwell’s equations. Some numerical experiments validate this innovative approach, resulting in a fast model, able to tackle a complex configuration never solved in the context of EC Nondestructive Testing.
Diffraction par des surfaces periodiques. Resolution en coordonnees non orthogonales. La diffract... more Diffraction par des surfaces periodiques. Resolution en coordonnees non orthogonales. La diffraction d'une onde electromagnetique par des reseaux metalliques de conductivite finie, ou dielectriques a perte, est etudiee dans le cas general de l'incidence oblique. On ecrit les equations de maxwell sous forme covariante apres avoir choisi un systeme de coordonnees non orthogonales adapte a la geometrie du probleme. On se ramene a un systeme de deux equations differentielles couplees du premier ordre, que l'on resout par une methode aux valeurs propres et vecteurs propres, apres l'avoir projete sur une base de fourier. Ce formalisme permet de calculer le champ electromagnetique sur la surface meme du reseau. Deux cas sont examines: celui des reseaux d'argent peu profonds sur lesquels porte une verification experimentale; celui d'une mer houleuse. Dans l'un et l'autre cas, des renforcements non negligeables du champ de surface sont mis en evidence. Enfin, la definition d'un operateur impedance de surface permet de reduire de facon notable le volume des calculs
Progress In Electromagnetics Research B, 2011
The propagation equation, written in a curvilinear coordinate system, is solved by using a pertur... more The propagation equation, written in a curvilinear coordinate system, is solved by using a perturbation method inspired from quantum physics and extended to imaginary eigenvalues and evanescent waves. The parameter of perturbation is the groove depth which is small compared to the period. The method is expanded up to second order for the non-degenerate problem. In this way the solutions have analytical form compared to a numerical method. They present the advantage to put in evidence the evolution of the energy distribution for different diffraction orders as a function of the magnitude of the perturbation. The efficiencies which are deduced from these analytical solutions are compared of those obtained by the curvilinear coordinate method. The good agreement between the two methods occurs for a groove depth with respect to the wavelength less than or equal to 0.16. Thus, this new approach opens a new range of applications for inverse problems.
Progress In Electromagnetics Research M, 2011
A rigorous modal theory of conical diffraction from curved strip gratings is presented. In this a... more A rigorous modal theory of conical diffraction from curved strip gratings is presented. In this approach, the C-method with adaptive spatial resolution is used in conjunction with the combined boundary conditions. The method is successfully validated by comparison with a case where the solution can also be obtained in the Cartesian coordinate system.
The paper relates to the scattering of a plane wave by a 1-dimensional local perturbation. Above ... more The paper relates to the scattering of a plane wave by a 1-dimensional local perturbation. Above a given deformation, the scattered field can be represented by a superposition of a continuous spectrum of propagating or evanescent plane waves, the so-called Rayleigh - Fourier integral. We demonstrate that this integral decreases as 1/√r in the far-field zone with an angular dependence given by the scattering amplitudes associated with the propagating waves. Applying the Green's second theorem to the asymptotic field, we obtain 2 reciprocity relations and a power balance criterion. Comments are proposed.
Waves in Random Media, 2004
We present a method giving the bi-static scattering coefficient of a one-dimensional dielectric r... more We present a method giving the bi-static scattering coefficient of a one-dimensional dielectric random rough surface illuminated by a plane wave. The theory is based on Maxwell's equations written in a nonorthogonal coordinate system. For each medium, this method leads to an eigenvalue system. The scattered field is expanded as a linear combination of eigensolutions satisfying the outgoing wave condition.
Progress In Electromagnetics Research, 2002