Coordinate Transformation Methods

Gérard Granet (1)

1 LASMEA - Laboratoire des sciences et matériaux pour l'électronique et d'automatique

The C-method was born in the eighties in Clermont-Ferrand , France, from the need to solve rigorously diffraction problems at corrugated periodic surfaces in the resonance regime. The main difficulty of such problems is the matching of boundaries conditions. For that purpose, Chandezon et al introduced the so called translation coordinate system in which the boundary of the physical problem coincides with coordinate surfaces. The second ingredient of C-method is to write Maxwell's equation under the covariant form. This formulation comes from relativity where the use of curvilinear non orthogonal coordinate system is essential and natural. The main feature of this formalism is that Maxwell's equations remain invariant in any coordinate system, the geometry being shifted into the constitutive relations. The third ingredient of C-method is that it is a modal method. This nice property is linked with the translation coordinate system in which a diffraction problem may be expressed as an eigenvalue eigenvector problem with periodic boundary conditions. The key point of C-method is the joint use of curvilinear coordinates and covariant formulation of Maxwell's equations. All the new developments in the modeling of gratings like Adaptive Spatial Resolution, and Matched Coordinates derive from this fundamental observation.

Domaines: Optique [physics.optics]

Gérard Granet. Coordinate Transformation Methods. E. Popov, ed. Gratings: Theory and Numeric Applications, AMU (PUP), pp.8.1-8.30, 2012, 978-2-8539-9860-4. ⟨hal-00783940⟩