## The doctoral dissertations of the former Helsinki University of Technology (TKK) and Aalto University Schools of Technology (CHEM, ELEC, ENG, SCI) published in electronic format are available in the electronic publications archive of Aalto University - Aaltodoc. | |

Dissertation for the degree of Doctor of Science in Technology to be presented with due permission
of the Department of Engineering Physics and Mathematics for public examination and debate in
Auditorium C at Helsinki University of Technology (Espoo, Finland) on the 16^{th} of March, 2007, at 12
o'clock noon.

Overview in PDF format (ISBN 978-951-22-8673-7) [212 KB]

Dissertation is also available in print (ISBN 978-951-22-8672-0)

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.

This thesis consists of an overview and of the following 3 publications:

- M. F. Dahl, Electromagnetic Gaussian beams and Riemannian geometry, Progress In Electromagnetics Research, Vol. 60, pp. 265-291, 2006. © 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. © 2007 by author.
- M. F. Dahl, Contact geometry in electromagnetism, Progress In Electromagnetics Research, Vol. 46, pp. 77-104, 2004. © 2004 EMW Publishing. By permission.

**Keywords:**
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

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© 2007 Helsinki University of Technology

Last update 2011-05-26