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2006-02-25
Electromagnetic Gaussian Beams and Riemannian Geometry
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Progress In Electromagnetics Research, Vol. 60, 265-291, 2006
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
Matias Dahl, "Electromagnetic Gaussian Beams and Riemannian Geometry," Progress In Electromagnetics Research, Vol. 60, 265-291, 2006.
doi:10.2528/PIER05122802
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