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Quelques problemes de Reflexion-Transmission en optique
dispersive faiblement non linaire.

Abstract : This thesis is devoted to the mathematical study of the nonlinear
propagation of high frequency waves in homogeneous media made of two
materials with constant refractive index and split up by a straight interface.
The interface requires to describe the reflexion-transmission
phenomena at the crossing of the discontinuity. The approach is the
Geometric Optics.
The studied tyical example is the Maxwell-Lorentz system with added
various equations which decribe each media polarization.

The novelty in this work comes from the dispersive aspect of the
equations and the consideration of the exact boundary equations.
In this frame the dispersion relation, or the characteristic variety is
not homogeneous, which implies that the group velocities do depend on
the size of the frequencies, not only on their direction.
For boundary value problem, this has the consequence that new
characteristic phases coming from the nonlinear intercations between
the existing phases and the boundary are created.

The first chapter sets the problem in an historical and mathematical
context. There is also a summary of the three next chapters.
The second chapter which is central deals with the Geometric Optics
analysis for the transmission problem using assymptotic expansion of
any order. There is a precise analysis of the nonlinear generation of
phases at the boundary and a rigorous justification of the expansion
approximation. It also gives a new treatment of the evanescent modes.

In the same geometrical context the third chapter tackles the problem of
Dffractive Optics looking for approximate assymptotic solutions of any
order and considering larger times of propagation.

Finally, in a fourth chapter one constructs and tests numerical quasi
linearly exact 1D schemes for simulating long time propagation (in
space) from a boundary.
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Contributor : Vincent Lescarret <>
Submitted on : Friday, November 17, 2006 - 6:34:06 PM
Last modification on : Thursday, January 11, 2018 - 6:12:27 AM
Long-term archiving on: : Tuesday, April 6, 2010 - 11:01:38 PM


  • HAL Id : tel-00114849, version 1




Vincent Lescarret. Quelques problemes de Reflexion-Transmission en optique
dispersive faiblement non linaire.. Mathématiques [math]. Université Sciences et Technologies - Bordeaux I, 2006. Français. ⟨tel-00114849⟩



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