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

References

1. Kurylev, Y. V., M. Lassas, and E. Somersalo, "Maxwell's equations with scalar impedance: Direct and inverse problems," Institute of Mathematics Research Reports, 2003.

2. Bossavit, A., "On the notion of anisotropy of constitutive laws. Some implications of the 'Hodge implies metric' result," COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 20, No. 1, 233-239, 2001.

3. Kravtsov, Y. A. and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media, Springer-Verlag, 1990.

4. Kachalov, A. and M. Lassas, "Gaussian beams and inverse boundary spectral problems," New Analytic and Geometric Methods in Inverse Problems, 127-163, 2004.

5. Kachalov, A., Y. Kurylev, and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC, 2001.

6. Ralston, J., "Gaussian beams and the propagation of singularities," Studies in Partial Differential Equations, Vol. 23, 206-248, 1982.

7. Kachalov, A. P., "Gaussian beams, Hamilton-Jacobi equations, and Finsler geometry," Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Vol. 297, 2003.

8. Kachalov, A. P., "Gaussian beams for Maxwell equations on a manifold," Journal of Mathematical Sciences, Vol. 122, No. 5, 2004.

9. Kachalov, A. P., "Nonstationary electromagnetic Gaussian beams in inhomogeneous anisotropic media," Journal of Mathematical Sciences, Vol. 111, No. 4, 2002.

10. Shen, Z., Lectures on Finsler Geometry, World Scientific, 2001.

11. Kozma, L. and L. Tamassy, "Finsler geometry without line elements faced to applications," Reports on Mathematical Physics, Vol. 51, 2003.
doi:10.1016/S0034-4877(03)80017-4

12. Antonelli, P. L., R. S. Insgarden, and M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic Publishers, 1993.

13. Miron, R. and M. Anastasiei, The Geometry of Lagrange Spaces: Theory and applications, Kluwer Academic Publishers, 1994.

14. Miron, R. and M. Radivoiovici-Tatoiu, Extended Lagrangian Theory of Electromagnetism, Vol. 27, No. 2, Reports on Mathematical Physics, 1989.

15. Asanov, G. S., Finsler Geometry, Relativity and Gauge Theories, 1985.

16. Bellman, R., Introduction to Matrix Analysis, McGraw-Hill book company, 1960.

17. Naulin, R. and C. Pabst, "The roots of a polynomial depend continuously on its coefficients," Revista Colombiana de Matematicas, Vol. 28, 35-37, 1994.

18. Guillemin, V. and S. Sternberg, "Geometric asymptotics," Mathematical Surveys, No. 14, 1977.

19. Dahl, M., "Propagation of electromagnetic Gaussian beams using Riemann-Finsler geometry," Licentiate thesis, 2006.

20. Abraham, R. and J. E. Mardsen, Foundations of Mechanics, 2nd ed..