volume | PIER Journals

2006-02-25

Electromagnetic Gaussian Beams and Riemannian Geometry

Matias Dahl

Dr. Matias Dahl

Helsinki University of Technology

Finland

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Progress In Electromagnetics Research, Vol. 60, 265-291, 2006

doi:10.2528/PIER05122802

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{e^iPθ(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 ε_i, μ_j , the induced geometry is Riemannian if and only if ε_iμ_j = ε_jμ_i 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, g_ij = εμδ_ij and in more complicated media there are two Riemannian metrics due to different polarizations.

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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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