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Progress In Electromagnetics Research | ISSN: 1070-4698, E-ISSN: 1559-8985 |

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## ELECTROMAGNETIC GAUSSIAN BEAMS AND RIEMANNIAN GEOMETRYBy M. Dahl
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)}E_{0}(x, t)E 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_{0}_{i}, μ_{j} , the induced geometry is Riemannian if and only if ε_{i}μ_{j} = ε_{j}μ_{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 = εμδ_{ij}_{ij} and in more complicated media there are two Riemannian metrics due to different polarizations.
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