Abstract : To simulate the propagation of a monochromatic laser beam in a medium, we use the paraxial approximation of the Klein-Gordon (in the time-varying problem) and of the Maxwell (in the non time-depending case) equations.
In a first part, we make an asymptotic analysis of the Klein-Gordon equation.
We obtain approximated problems, either of Schrödinger or of transport-Schrödinger type. We prove existence and unicity of a solution for these problems, and estimate the difference between it and the exact solution of the Klein-Gordon equation.
In a second part, we study the boundary problem for the advection Schrödinger equation, and show what the boundary condition must be in order that the problem on our domain be the restriction of the problem in the whole space: such a condition is called a transparent or an absorbing boundary condition.
In a third part, we use the preceding results to build a numerical resolution method, for which we prove stability and show some simulations.