Mathematical Analysis of the Guided Modes of an Optical Fiber

Mathematical analysis of electromagnetic open Waveguides

“…With such a formulation we can work equivalently with the electric field or the magnetic field which allows us to take profit from the natural symmetry of Maxwell's équations with respect to E and H. This will lead us to the generalization of the results of A. Bamberger and A. S. Bonnet to the case ju(x) variable. The reader will easily check that our results coïncide with those of [2] when jn(x) =ju oo . The second major interest of this new formulation is to be preparatory for the dérivation of a numerical method for the computation of the guided modes.…”

“…For the sequel it is useful to introducé the local propagation velocity c(x) of the medium, defined by xe IR 2 xe K 2 …”

“…The theory for the gênerai case is of course much more complicated. Reeen tl y, A. Bamberger and A. S. Bonnet [2], with the help of the spectral theory of selfadjoint operators, made a major step in the understanding of the properties of electromagnetic waveguides. In fact they obtained very complete results in the spécifie case where the function jn(x) is constant everywhere.…”

“…This assumption is physically relevant in many applications and has a very important technical conséquence : the H^regularity of the magnetic field H as a function of x. This is the reason why the authors in [2] considered a formulation of the problem in which H was the unknown : adding an artificial term in the corresponding variational formulation, they got rid of the problem of the divergence-free condition and were able to develop the theory vol. 29, n° 5, 1995 in the H 1 functional framework.…”

“…The electromagnetic waveguides 506 P. JOLY, Ch. POIRIER have applications in various domains of physics (electronic components, optical fibers, integrated opties...) and have been already abundantly studied, for instance by D. Marcuse [22] who is the référence in the physical literature, or by A. Bamberger and A. S. Bonnet [2], A. S. Bonnet [7], R. Djellouli [13], N. Gmati [17], A. Bermudez and D. G. Pedreira [5], F. Kikuchi [20], concerning the mathematical or numerical studies.…”