volume | PIER Journals

Abstract

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.

1. Brittingham, J. N., "Packetlike solutions of the homogeneous-wave equation," J. Appl. Phys., Vol. 54, 1179-1189, 1983.
doi:10.1063/1.332196 Google Scholar

2. Kiselev, A. P., "Modulated Gaussian beams," Radiophys. Quantum Electron., Vol. 26, 755-761, 1983.
doi:10.1007/BF01034890 Google Scholar

3. Belanger, P. A., "Packetlike solutions of the homogeneous-wave equation," JOSA A, Vol. 1, 723-724, 1984.
doi:10.1364/JOSAA.1.000723 Google Scholar

4. Sezginer, A., "A general formulation of focus wave modes," J. Appl. Phys., Vol. 57, 678-683, 1985.
doi:10.1063/1.334712 Google Scholar

5. Ziolkowski, R. W., "Exact solutions of the wave equation with complex source locations," J. Math. Phys., Vol. 26, 861-863, 1985.
doi:10.1063/1.526579 Google Scholar

6. Ziolkowski, R. W., "Localized transmission of electromagnetic energy," Phys. Rev. A, Vol. 39, 2005-2033, 1989.
doi:10.1103/PhysRevA.39.2005 Google Scholar

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.
doi:10.1109/5.104212 Google Scholar

8. Bandres, M. A. and J. C. Gutierrez-Vega, "Cartesian beams," Opt. Lett., Vol. 32, 3459-3461, 2007.
doi:10.1364/OL.32.003459 Google Scholar

9. Bandres, M. A. and J. C. Gutierrez-Vega, "Circular beams," Opt. Lett., Vol. 33, 177-179, 2008.
doi:10.1364/OL.33.000177 Google Scholar

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.
doi:10.1088/1464-4258/10/5/055006 Google Scholar

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.
doi:10.1063/1.528301 Google Scholar

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

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

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

15. Hillion, P., "Nondispersive waves: Interpretation and causality," IEEE Trans. Ant. Propag., Vol. 40, 1031-1035, 1992.
doi:10.1109/8.166527 Google Scholar

16. Hillion, P., "Generalized phases and nondispersive waves," Acta Appl. Math., Vol. 30, 35-45, 1993.
doi:10.1007/BF00993341 Google Scholar

17. Kiselev, A. P. and M. V. Perel, "Highly localized solutions of the wave equations," J. Math. Phys., Vol. 41, 1934-1955, 2000.
doi:10.1063/1.533219 Google Scholar

18. Kiselev, A. P., "Relatively undistorted waves. New examples," J. Math. Sci., Vol. 117, 3945-3946, 2003.
doi:10.1023/A:1024666808547 Google Scholar

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.
doi:10.1088/0305-4470/36/23/103 Google Scholar

20. Kiselev, A. P., "Relatively undistorted cylindrical waves, depending on three spatial variables," Math. Notes, Vol. 79, 587-588, 2006.
doi:10.1007/s11006-006-0066-y Google Scholar

21. Kiselev, A. P., "Localized light waves: Paraxial and exact solutions of the wave equation (a review)," Opt. & Spectr., Vol. 102, 603-622, 2007.
doi:10.1134/S0030400X07040200 Google Scholar

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

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.
doi:10.2528/PIER97072900 Google Scholar

24. Hillion, P., "Spinor focus wave modes," J. Math. Phys., Vol. 28, 1743-1748, 1987.
doi:10.1063/1.527484 Google Scholar

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.
doi:10.1364/JOSAA.23.002039 Google Scholar

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

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.
doi:10.1016/0375-9601(91)90711-G Google Scholar

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.
doi:10.1088/1464-4258/10/11/115007 Google Scholar

29. Mourad, E. and H. Ismail, "Relativistic orthogonal polynomials are Jacobi polynomials," J. Phys. A: Math. Gen., Vol. 29, 3199-3202, 1996.
doi:10.1088/0305-4470/29/12/023 Google Scholar

30. Volterra, V., "Sur les vibrations des corps elastiques isotropes," Acta Math., Vol. 18, 161-232, 1894.
doi:10.1007/BF02418279 Google Scholar

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

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

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

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

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

36. Essex, E. A., "Hertz vector potentials of electromagnetic theory," Amer. J. Phys., Vol. 54, 1099-1101, 1977.
doi:10.1119/1.10955 Google Scholar

37. Hillion, P., "More on focus wave modes in Maxwell equations," J. Appl. Phys., Vol. 60, 2981-2982, 1986.
doi:10.1063/1.337773 Google Scholar

38. Hillion, P., "The Bateman solutions of the spinor wave equation," Mod. Phys. Lett. A, Vol. 8, 2111-2115, 1993.
doi:10.1142/S0217732393001823 Google Scholar

39. Wu, T. T., "Electromagnetic missiles," J. Appl. Phys., Vol. 57, 2370-2373, 1985.
doi:10.1063/1.335465 Google Scholar

40. Shen, H. M and T. T.Wu, "The properties of the electromagnetic missile," J. Appl. Phys., Vol. 66, 4025-4034, 1989.
doi:10.1063/1.344011 Google Scholar

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.
doi:10.1364/JOSAA.10.000075 Google Scholar

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. Google Scholar

Dr. Amalia Torre