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Progress In Electromagnetics Research
ISSN: 1070-4698, E-ISSN: 1559-8985
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ELECTROMAGNETIC GAUSSIAN BEAMS AND RIEMANNIAN GEOMETRY

By M. Dahl

Full Article PDF (257 KB)

Abstract:
A Gaussian beam is an asymptotic solution to Maxwell's equations that propagate along a curve; at each time instant its energy is concentrated around one point on the curve. Such a solution is of the form

E = Re{eiPθ(x,t)E0(x, t)},

where E0 is a complex vector field, P >0 is a big constant, and θ is a complex second order polynomial in coordinates adapted to the curve. In recent work by A. P. Kachalov, electromagnetic Gaussian beams have been studied in a geometric setting. Under suitable conditions on the media, a Gaussian beam is determined by Riemann-Finsler geometry depending only on the media. For example, geodesics are admissible curves for Gaussian beams and a curvature equation determines the second order terms in θ. This work begins with a derivation of the geometric equations for Gaussian beams following the work of A. P. Kachalov. The novel feature of this work is that we characterize a class of inhomogeneous anisotropic media where the induced geometry is Riemannian. Namely, if ε, μ are simultaneously diagonalizable with eigenvalues εi, μj , the induced geometry is Riemannian if and only if εiμj = εjμi for some i ≠ j. What is more, if the latter condition is not met, the geometry is ill-behaved. It is neither smooth nor convex. We also calculate Riemannian metrics for different media. In isotropic media, gij = εμδij and in more complicated media there are two Riemannian metrics due to different polarizations.

Citation: (See works that cites this article)
M. Dahl, "Electromagnetic Gaussian Beams and Riemannian Geometry," Progress In Electromagnetics Research, Vol. 60, 265-291, 2006.
doi:10.2528/PIER05122802
http://www.jpier.org/PIER/pier.php?paper=0512282

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