title: 
A geometric interpretation of the complex tensor Riccati equation for Gaussian beams 

publication: 

volumeissue:  14  1  
pages:  95  111  
ISSN: 
14029251  
DOI: 
doi:10.2991/jnmp.2007.14.1.8 (how to use a DOI)  
author(s): 
M F DAHL 

publication date: 
February 2007 

abstract: 
We study the complex Riccati tensor equation DcG + GCG  R = 0 on a geodesic c on a Riemannian 3manifold. This nonlinear equation appears in
the study of Gaussian beams. Gaussian beams are asymptotic solutions to hyperbolic
equations that at each time instant are concentrated around one point in space. When
time moves forward, Gaussian beams move along geodesics, and the Riccati equation
determines the Hessian of the phase function for the Gaussian beam. The imaginary
part of a solution G describes how a Gaussian beam decays in different directions of
space. The main result of the present work is that the real part of G is the shape
operator of the phase front for the Gaussian beam. This result generalizes a known
result for the Riccati equation in R3
. The idea of the proof is to express the Riccati
equation in Fermi coordinates adapted to the underlying geodesic. In Euclidean geometry we also study when the phase front is contained in the area of influence, or
light cone. 

copyright: 
©
Atlantis Press. This article is distributed under the
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noncommercial use, distribution and reproduction in any medium,
provided the original work is properly cited. 

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