Vol. 13

Front:[PDF file] Back:[PDF file]
Latest Volume
All Volumes
All Issues

Relativistic Laguerre Polynomials and Splash Pulses

By Amalia Torre
Progress In Electromagnetics Research B, Vol. 13, 329-356, 2009


New solutions of the homogeneous wave equation of the type usually referred to as relatively undistorted waves are presented. Such solutions relate to the so-called "splash modes", from which indeed they can be generated by applying the Laguerre polynomial operator. Accordingly, the solutions here presented resort to the relativistic Laguerre polynomials --- introduced about one decade ago within a purely mathematical context --- which in fact appear as modulating factor of the basic "splash mode" waveform. Similar solutions of the homogeneous spinor wave equation are also suggested.


Amalia Torre, "Relativistic Laguerre Polynomials and Splash Pulses," Progress In Electromagnetics Research B, Vol. 13, 329-356, 2009.


    1. Brittingham, J. N., "Packetlike solutions of the homogeneous-wave equation," J. Appl. Phys., Vol. 54, 1179-1189, 1983.

    2. Kiselev, A. P., "Modulated Gaussian beams," Radiophys. Quantum Electron., Vol. 26, 755-761, 1983.

    3. Belanger, P. A., "Packetlike solutions of the homogeneous-wave equation," JOSA A, Vol. 1, 723-724, 1984.

    4. Sezginer, A., "A general formulation of focus wave modes," J. Appl. Phys., Vol. 57, 678-683, 1985.

    5. Ziolkowski, R. W., "Exact solutions of the wave equation with complex source locations," J. Math. Phys., Vol. 26, 861-863, 1985.

    6. Ziolkowski, R. W., "Localized transmission of electromagnetic energy," Phys. Rev. A, Vol. 39, 2005-2033, 1989.

    7. Ziolkowski, R. W., I. M. Besieris, and A. M. Shaarawi, "Localized wave representations of acoustic and electromagnetic radiation," Proc. IEEE, Vol. 79, 1371-1378, 1991.

    8. Bandres, M. A. and J. C. Gutierrez-Vega, "Cartesian beams," Opt. Lett., Vol. 32, 3459-3461, 2007.

    9. Bandres, M. A. and J. C. Gutierrez-Vega, "Circular beams," Opt. Lett., Vol. 33, 177-179, 2008.

    10. Torre, A., "A note on the general solution of the paraxial wave equation: A Lie algebra view ," J. Opt. A: Pure Appl. Opt., Vol. 10, 055006, 2008.

    11. Besieris, I. M., A. M. Shaarawi, and R. W. Ziolkowski, "A bidirectional traveling plane wave representation of exact solutions of the wave equation," J. Math. Phys., Vol. 30, 1254-1269, 1989.

    12. Bateman, H., The Mathematical Analysis of Electrical and Optical Wave-motion on the Basis of Maxwell’s Equations, Dover, New York, 1955.

    13. Courant, R. and D. Hilbert, Methods of Mathematical Physics, Vol. 2, Interscience, New York, 1962.

    14. Hillion, P., "Electromagnetic inhomogeneous pulses," Journal of Electromagnetic Waves and Applications, Vol. 5, 959-969, 1991.

    15. Hillion, P., "Nondispersive waves: Interpretation and causality," IEEE Trans. Ant. Propag., Vol. 40, 1031-1035, 1992.

    16. Hillion, P., "Generalized phases and nondispersive waves," Acta Appl. Math., Vol. 30, 35-45, 1993.

    17. Kiselev, A. P. and M. V. Perel, "Highly localized solutions of the wave equations," J. Math. Phys., Vol. 41, 1934-1955, 2000.

    18. Kiselev, A. P., "Relatively undistorted waves. New examples," J. Math. Sci., Vol. 117, 3945-3946, 2003.

    19. Kiselev, A. P., "Generalization of Bateman-Hillion progressive wave and Bessel-Gauss pulse solutions of the wave equation via a separation of variables," J. Phys. A: Math. Gen., Vol. 36, L345-L349, 2003.

    20. Kiselev, A. P., "Relatively undistorted cylindrical waves, depending on three spatial variables," Math. Notes, Vol. 79, 587-588, 2006.

    21. Kiselev, A. P., "Localized light waves: Paraxial and exact solutions of the wave equation (a review)," Opt. & Spectr., Vol. 102, 603-622, 2007.

    22. Hillion, P., "Splash wave modes in homogeneous Maxwell's equations," Journal of Electromagnetic Waves and Applications, Vol. 2, 725-739, 1988.

    23. Besieris, I. M., M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, "Two fundamental representations of localized pulse solutions to the scalar wave equation," Prog. Electrom. Res., Vol. 19, 1-48, 1998.

    24. Hillion, P., "Spinor focus wave modes," J. Math. Phys., Vol. 28, 1743-1748, 1987.

    25. Shaarawi, A. M., M. A. Maged, I. M. Besieris, and E. Hashish, "Localized pulses exhibiting a missilelike slow decay," JOSA, Vol. 23, 2039-2052, 2006.

    26. Natalini, P., "The relativistic Laguerre polynomials," Rend. Matematica, Ser. VII, Vol. 16, 299-313, 1996.

    27. Aldaya, V., J. Bisquert, and J. Navarro-Salas, "The quantum relativistic harmonic oscillator: generalized Hermite polynomials," Phys. Lett. A, Vol. 156, 381-385, 1991.

    28. Torre, A., W. A. B. Evans, O. El Gawhary, and S. Severini, "Relativisitic Hermite polynomials and Lorentz beams," J. Opt. A: Pure Appl. Opt., Vol. 10, 115007, 2008.

    29. Mourad, E. and H. Ismail, "Relativistic orthogonal polynomials are Jacobi polynomials," J. Phys. A: Math. Gen., Vol. 29, 3199-3202, 1996.

    30. Volterra, V., "Sur les vibrations des corps elastiques isotropes," Acta Math., Vol. 18, 161-232, 1894.

    31. Miller, W., Symmetry and Separation of Variables, Addison-Wesley, Reading, MA, 1977.

    32. Hillion, P., "The Courant-Hilbert solution of the wave equation," J. Math. Phys., Vol. 33, 2749-2753, 1992.

    33. Erdelyi, A., W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vols. 1 and 2, MacGraw-Hill, New York, London and Toronto, 1953.

    34. Gradshteyn, I. S. and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1965.

    35. Stratton, J. A., "Electromagnetic Theory," McGraw-Hill, 1941.

    36. Essex, E. A., "Hertz vector potentials of electromagnetic theory," Amer. J. Phys., Vol. 54, 1099-1101, 1977.

    37. Hillion, P., "More on focus wave modes in Maxwell equations," J. Appl. Phys., Vol. 60, 2981-2982, 1986.

    38. Hillion, P., "The Bateman solutions of the spinor wave equation," Mod. Phys. Lett. A, Vol. 8, 2111-2115, 1993.

    39. Wu, T. T., "Electromagnetic missiles," J. Appl. Phys., Vol. 57, 2370-2373, 1985.

    40. Shen, H. M and T. T.Wu, "The properties of the electromagnetic missile," J. Appl. Phys., Vol. 66, 4025-4034, 1989.

    41. Ziolkowski, R. W., I. M. Besieris, and A. M. Shaarawi, "Aperture realizations of exact solutions to homogeneous-wave equations," JOSA A, Vol. 10, 75-87, 1993.

    42. Abdel-Rahman, M., I. M. Besieris, and A. M. Shaarawi, "A comparative study on the reconstruction of localized pulses," Proc. IEEE Southeast Conf. (SOUTHEASTCON'97), 113-117, Blackburg, Virginia, April 1997. also at http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=00598622.