volume | PIER Journals

Abstract

Spatiotemporally localized luminal null electromagnetic fields are transverse with respect to the local flow of energy,whic h is equipartioned between the electric and magnetic fields,and the modulus of their local energy transport velocity equals the speed of light in vacuo. They have vortex structures on planes transverse to the direction of propagation,and,in general,are relatively simple so that explicit calculations can be made of the total energy and the total angular momentum they carry. A class of luminal null electromagnetic waves due originally to Robinson and Troutman is motivated by means of spherical Cunningham and Bateman transformations and their relationships to well-known scalar luminal localized waves are examined. This allows for the introduction of finite-energy localized null luminal electromagnetic waves with spatiotemporal spectra appropriate for applications in diverse physical areas.

1. Brittingham, J. N., "Focus wave modes in homogeneous Maxwell equations: Transverse electric mode," J. Appl. Phys., Vol. 54, 1179-1189, 1983.
doi:10.1063/1.332196

2. Kiselev, A. P., "Modulated Gaussian beams," Radio Phys. Quant. Electron., Vol. 26, 1014, 1983.
doi:10.1007/BF01034667

3. Belanger, P. A., "Packetlike solutions of the homogeneous wave equation," J. Opt. Soc. A, Vol. 1, 723, 1984.
doi:10.1364/JOSAA.1.000723

4. Ziolkowski, R. W., "Exact solutions of the wave equation with complex source locations," J. Math. Phys., Vol. 26, 861-863, 1984.
doi:10.1063/1.526579

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

6. Besieris, I. M., A. M. Shaarawi, and R. W. Ziolkowski, "A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys., Vol. 30, 1254-1269, 1989.
doi:10.1063/1.528301

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

8. Grunwald, R., V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. Nibbering, M. Piche, G. Rousseau, and M. Fortin, "Generation and characterization of spatially and temporally localized few-cycle optical wave packets," Phys. Rev. A, Vol. 67, 063820, 1-5, 2003.

9. Ziolkowski, R. W., D. K. Lewis, and B. D. Cook, "Experimental verification of the localized wave transmission effect," Phys. Rev. Lett., Vol. 62, 147-150, 1989.
doi:10.1103/PhysRevLett.62.147

10. Ziolkowski, R. W. and D. K. Lewis, "Verification of the localized wave transmission effect," J. Appl. Phys., Vol. 68, 6083-6086, 1990.
doi:10.1063/1.346896

11. Reivelt, K. and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E, Vol. 66, 056611-1-9, 2002.

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

13. Hillion, P., "Courant-Hilbert solutions of the wave equation," J. Math Phys., Vol. 33, 34-45, 1992.

14. Bateman, H., "The conformal transformations of four dimensions and their applications to geometrical optics," Proc. London Math. Soc., Vol. 7, 70-89, 1909.
doi:10.1112/plms/s2-7.1.70

15. Bateman, H., The Mathematical Analysis of Electrical and Optical Wave-Motion on the Basis of Maxwell's Equations, Dover, New York, 1955.

16. Borisov, V. V. and A. B. Utkin, "Generalization of Brittingham's localized solutions to the wave equation," Euro. Phys. J. B, Vol. 21, 477-480, 2001.
doi:10.1007/s100510170155

17. Kiselev, A. P., "Generalization of Bateman-Hillion progressive wave and Bessel-Gauss pulse solutions to the wave equation ," J. Phys. A: Math. Gen., Vol. 36, L345-L349, 2003.
doi:10.1088/0305-4470/36/23/103

18. Besieris, I. M. and A. M. Shaarawi, "Bateman conformal transformations within the framework of the bidirectional spectral representation," Progress In Electromagnetics Research, Vol. 48, 201-231, 2004.
doi:10.2528/PIER04021101

19. Ziolkowski, R. W., I. M. Besieris, and A. M. Shaarawi, "Aperture realizations of the exact solutions to homogeneous-wave equations," J. Opt. Soc. Am. A, Vol. 10, 75-87, 1993.
doi:10.1364/JOSAA.10.000075

20. Shaarawi, A. M., I. M. Besieris, R. W. Ziolkowski, and S. M. Sedky, "Generation of approximate focus wave mode pulses from wide-band dynamic Gaussian aperture ," J. Opt. Soc. Am. A, Vol. 12, 1954-1964, 1995.
doi:10.1364/JOSAA.12.001954

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

22. Heyman, E., "Pulsed beam propagation in an inhomogeneous medium," IEEE Trans. Antennas Propag., Vol. 42, 311-319, 1994.
doi:10.1109/8.280715

23. Cunningham, E., "The principle of relativity in electrodynamics and an extension thereof," Proc. London Math. Soc., Vol. 8, 77-98, 1909.
doi:10.1112/plms/s2-8.1.77

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

25. Bateman, H., "The transformations of coordinates which can be used to transform one physical problem into another," Proc. London Math. Soc., Vol. 8, 469-488, 1910.
doi:10.1112/plms/s2-8.1.469

26. Robinson, I., "Null electromagnetic fields," J. Math. Phys., Vol. 2, 290-291, 1961.
doi:10.1063/1.1703712

27. Robinson, I. and A. Troutman, "Some spherical gravitational waves in general relativity," Proc. R. Soc. London, Series A,265, 463, 1962.

28. Robinson, I. and A. Troutman, New Theories in Physics: Roc. Warsaw Symp. On Elementary Particle Physics, Z. Ajduk et al. (eds.), 454–497, Singapore World Scientific, 1989.

29. Troutman, A., "Robinson manifolds and Cauchy-Riemann spaces," Class. Quantum Grav., Vol. 19, R1-R10, 2002.
doi:10.1088/0264-9381/19/2/201

30. Penrose, R. and W. Rindler, Spinors and Space-Time, Vol. 2, Cambridge University Press, 1986.

31. Bialynicki-Birula, I., "Electromagnetic vortex lines riding atop null solutions of the Maxwell equations," J. Opt. A: Pure Appl. Opt., Vol. 6, S181-S183, 2004.
doi:10.1088/1464-4258/6/5/007

32. Weber, H., Die Partiellen Differential-Gleichungen der Mathematischen Physik nach Riemann's Vorlesungen, Friedrich Vieweg und Sohn, Brunschweig, 1901.

33. Silberstein, L., "Electromagnetische grundgleichungen in bivectorieller behandlung," Ann. d. Phys., Vol. 22, 579-586, 1907.
doi:10.1002/andp.19073270313

34. Feng, S., H. G. Winful, and R. W. Hellwarth, "Spatio-temporal evolution of single-cycle electromagnetic pulses," Phys. Rev. E, Vol. 59, 4630-4649, 1999.
doi:10.1103/PhysRevE.59.4630

35. Lekner, J., "Energy and momentum of electromagnetic pulses," J. Opt. A: Pure Appl. Opt., Vol. 6, 146-147, 2004.
doi:10.1088/1464-4258/6/1/026

36. Shaarawi, A. M., "Nondispersive wavepackets," Ph.D. Thesis, Virginia Polytechnic Inst. and State Univ., 1989.

37. Overfelt, P ., "Bessel-Gauss pulses," Phys. Rev. A, Vol. 44, 3941-3947, 1991.
doi:10.1103/PhysRevA.44.3941

38. Bialynicki-Birula, I. and Z. Bialynicki-Birula, "Vortex lines of the electromagnetic field," Phys. Rev. A, Vol. 67, 062114-1, 2003.
doi:10.1103/PhysRevA.67.062114

39. Berry, M., "Riemann-Silberstein vortices for paraxial waves," J. Opt. A: Pure Appl. Opt., Vol. 6, S175-7, 2003.

40. Kaiser, G., "Helicity, polarization and Riemann-Silberstein vortices," J. Opt. A: Pure Appl. Opt., Vol. 6, S243-5, 2003.
doi:10.1088/1464-4258/6/5/018

41. Lekner, J., "Helical light pulses," J. Opt. A: Pure Appl. Opt., Vol. 6, L29-L32, 2004.
doi:10.1088/1464-4258/6/10/L01

42. Desyatnikov, A. S., Y. S. Kivshar, and L. Torner, "Optical vortices and vortex solitons," Progress in Optics, Vol. 47, 291-391, E. Wolf (ed.), North-Holland, 2005.

43. Molina-Terriza, G., J. P. Torres, and L. Torner, "Management of the angular momentum of light; preparation of photons in multidimensional vector states of angular momentum," Phys. Rev. Lett., Vol. 88, 0136011-4, 2002.

Professor Ioannis Besieris

Dr. Amr Shaarawi