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2015-04-24

Relativistic Bateman-Hillion Solutions for the Electromagnetic 4-Potential in Hermite-Gaussian Beams

By Robert Ducharme
Progress In Electromagnetics Research M, Vol. 42, 39-47, 2015
doi:10.2528/PIERM15030104

Abstract

The electromagnetic field equations are solved to give the 4-potential in Hermite-Gaussian beams as a function of both the 4-positions of the beam waist and each point in the field. These solutions are the sums of products of position-dependent complex 4-vectors and modified Bateman-Hillion functions. It is assumed that the time difference between the beam waist and each other point is equal to the distance between the points divided by the speed of light. This method is shown to generate solutions that preserve their forms under Lorentz transformations that also correspond to the well known paraxial solutions for the case of nearly parallel beams.

Citation


Robert Ducharme, "Relativistic Bateman-Hillion Solutions for the Electromagnetic 4-Potential in Hermite-Gaussian Beams," Progress In Electromagnetics Research M, Vol. 42, 39-47, 2015.
doi:10.2528/PIERM15030104
http://jpier.org/PIERM/pier.php?paper=15030104

References


    1. Bateman, H., "The transformation of the electrodynamical equations," London Math. Soc., Vol. 8, 223-264, 1910.
    doi:10.1112/plms/s2-8.1.223

    2. Hillion, P., "The Courant-Hilbert solution of the wave equation," J. Math. Phys., Vol. 33, 2749-2753, 1992.
    doi:10.1063/1.529595

    3. Kiselev, A. P., A. B. Plachenov, and P. Chamorro-Posada, "Nonparaxial wave beams and packets with general astigmatism," Phys. Rev. A, Vol. 85, 043835-1-043835-11, 2012.
    doi:10.1103/PhysRevA.85.043835

    4. Siegman, A. E., Lasers, University Science Books, Mill Valley, California, 1986.

    5. Svelto, O., Principles of Lasers, Springer, New York, 2010.
    doi:10.1007/978-1-4419-1302-9

    6. Brabec, T. and F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys., Vol. 18, No. 2, 545-591, 2000.
    doi:10.1103/RevModPhys.72.545

    7. Kiselev, A. P., "Localized light waves: Paraxial and exact solutions of the wave equation (a review)," Optics and Spectroscopy, Vol. 102, No. 4, 603-622, 2007.
    doi:10.1134/S0030400X07040200

    8. Overfelt, P. L., "Bessel-Gauss pulses," Phys. Rev. A, Vol. 44, No. 6, 3941-3947, 1991.
    doi:10.1103/PhysRevA.44.3941

    9. Moses, H. E. and R. T. Prosser, "Acoustic and electromagnetic bullets: Derivation of new exact solutions of the acoustic and Maxwell’s equations," SIAM J. Math. Phys., Vol. 50, No. 5, 1325-1340, 1990.
    doi:10.1137/0150079

    10. Konar, S., M. Mishra, and S. Jana, "Nonlinear evolution of cosh-Gaussian laser beams and generation of flat top spatial solitons in cubic quintic nonlinear media," Physics Letts. A, Vol. 362, 505-510, 2007.
    doi:10.1016/j.physleta.2006.11.025

    11. Konar, S. and S. Jana, "Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity," Optics Communication, Vol. 236, No. 1, 7-20, 2004.
    doi:10.1016/j.optcom.2004.03.012

    12. Komar, A., "Interacting relativistic particles," Phys. Rev. D, Vol. 18, 1887-1893, 1978.
    doi:10.1103/PhysRevD.18.1887

    13. Crater, H. W. and P. Van Alstine, "Two-body dirac equations for particles interacting through world scalar and vector potentials," Phys. Rev. D, Vol. 36, No. 10, 3007-3036, 1987.
    doi:10.1103/PhysRevD.36.3007

    14. Saleem, M. and M. Rafique, Special Relativity Applications to Particle Physics and the Classical Theory of Fields, Ellis Horwood, New York, 1992.

    15. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.