Geometric properties of electromagnetic waves

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Doctoral thesis (article-based)
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30, [75]
Research reports / Helsinki University of Technology, Institute of Mathematics. A, 520
This work studies geometrical properties of electromagnetic wave propagation. The work starts by studying geometrical properties of electromagnetic Gaussian beams in inhomogeneous anisotropic media. These are asymptotical solutions to Maxwell's equations that have a very characteristic feature. Namely, at each time instant the entire energy of the solution is concentrated around one point in space. When time moves forward, a Gaussian beam propagates along a curve. In recent work by A. P. Kachalov, Gaussian beams have been studied from a geometrical point of view. Under suitable conditions on the media, Gaussian beams propagate along geodesics. Furthermore, the shape of a Gaussian beam is determined by a complex tensor Riccati equation. The first paper of this dissertation provides a partial classification of media where Gaussian beams geometrize. The second paper shows that the real part of a solution to the aforementioned Riccati equation is essentially the shape operator for the phase front for the Gaussian beam. An important phenomena for electromagnetic Gaussian beams is that their propagation depend on their polarization. The last paper studies this phenomena from a very general point of view in arbitrary media. It also studies a connection between contact geometry and electromagnetism.
electromagnetism, Maxwell's equations, Riemann geometry, Finsler geometry, contact geometry, symplectic geometry, Hamilton-Jacobi equation, phase function, complex Riccati equation, Gaussian beams, propagation, polarization, helicity, Bohren decomposition, Moses decomposition
Other note
  • M. F. Dahl, Electromagnetic Gaussian beams and Riemannian geometry, Progress In Electromagnetics Research, Vol. 60, pp. 265-291, 2006. [article1.pdf] © 2006 EMW Publishing. By permission.
  • M. F. Dahl, A geometric interpretation of the complex tensor Riccati equation for Gaussian beams, Journal of Nonlinear Mathematical Physics, Vol. 14, No. 1, pp. 95-111, 2007. [article2.pdf] © 2007 by author.
  • M. F. Dahl, Contact geometry in electromagnetism, Progress In Electromagnetics Research, Vol. 46, pp. 77-104, 2004. [article3.pdf] © 2004 EMW Publishing. By permission.
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