volume | PIER Journals

Abstract

Most of the analytical work on general transparent dispersive media to date has been confined to second-order dispersion within the framework of the paraxial approximation. It is the aim in this article to lift this restriction. Specifically, a detailed discussion is provided of modulated (3+1)-dimensional nonparaxial spatiotemporally localized waves in second-order transparent dispersive media. Novel infinite-energy invariant wavepackets and finite-energy almost undistorted solutions are discussed in detail. Illustrative numerical examples of the latter are given for normal dispersion in fused silica and for anomalous dispersion in a Lorentz plasma.

1. Brittingham, James Neill, "Focus waves modes in homogeneous Maxwell's equations: Transverse electric mode," Journal of Applied Physics, Vol. 54, No. 3, 1179-1189, 1983. Google Scholar

2. Kiselev, A. P., "Modulated Gaussian beams," Radiophysics and Quantum Electronics, Vol. 26, No. 8, 755-761, 1983. Google Scholar

3. Ziolkowski, Richard W., "Localized transmission of electromagnetic energy," Physical Review A, Vol. 39, No. 4, 2005, 1989. Google Scholar

4. Besieris, Ioannis M., Amr M. Shaarawi, and Richard W. Ziolkowski, "A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation," Journal of Mathematical Physics, Vol. 30, No. 6, 1254-1269, 1989. Google Scholar

5. Lu, J.-Y. and James F. Greenleaf, "Nondiffracting X waves --- Exact solutions to free-space scalar wave equation and their finite aperture realizations," IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 39, No. 1, 19-31, 1992. Google Scholar

6. Ziolkowski, Richard W., Ioannis M. Besieris, and Amr M. Shaarawi, "Aperture realizations of exact solutions to homogeneous-wave equations," Journal of the Optical Society of America A, Vol. 10, No. 1, 75-87, 1993. Google Scholar

7. Saari, Peeter and Kaido Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Physical Review Letters, Vol. 79, No. 21, 4135, 1997. Google Scholar

8. Besieris, I., M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, "Two fundamental representations of localized pulse solutions to the scalar wave equation," Progress In Electromagnetics Research, Vol. 19, 1-48, 1998. Google Scholar

9. Salo, J., J. Fagerholm, Ari T. Friberg, and M. M. Salomaa, "Unified description of nondiffracting X and Y waves," Physical Review E, Vol. 62, No. 3, 4261, 2000. Google Scholar

10. Grunwald, R., V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, "Generation and characterization of spatially and temporally localized few-cycle optical wave packets," Physical Review A, Vol. 67, No. 6, 063820, 2003. Google Scholar

11. Saari, Peeter and Kaido Reivelt, "Generation and classification of localized waves by Lorentz transformations in Fourier space," Physical Review E, Vol. 69, No. 3, 036612, 2004. Google Scholar

12. Longhi, Stefano, "Spatial-temporal Gauss-Laguerre waves in dispersive media," Physical Review E, Vol. 68, No. 6, 066612, 2003. Google Scholar

13. Conti, Claudio, Stefano Trillo, Paolo Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X waves," Physical Review Letters, Vol. 90, No. 17, 170406, 2003. Google Scholar

14. Kiselev, A. P., "Localized light waves: Paraxial and exact solutions of the wave equation (a review)," Optics and Spectroscopy, Vol. 102, 603-622, 2007. Google Scholar

15. Hernández-Figueroa, Hugo E., Michel Zamboni-Rached, and Erasmo Recami, Localized Waves, John Wiley & Sons, Hoboken, NJ, 2008.
doi:10.1002/9780470168981

16. Hernández-Figueroa, Hugo E., Michel Zamboni-Rached, and Erasmo Recami, Non-Diffracting Waves, John Wiley & Sons, New York, 2013.
doi:10.1002/9783527671519

17. Yessenov, Murat, Layton A. Hall, Kenneth L. Schepler, and Ayman F. Abouraddy, "Space-time wave packets," Advances in Optics and Photonics, Vol. 14, No. 3, 455-570, 2022. Google Scholar

18. Lu, J.-Y. and J. F. Greenleaf, "Experimental verification of nondiffracting X waves," IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 39, No. 3, 441-446, 1992. Google Scholar

19. Reivelt, Kaido and Peeter Saari, "Experimental demonstration of realizability of optical focus wave modes," Physical Review E, Vol. 66, No. 5, 056611, 2002. Google Scholar

20. Bowlan, Pamela, Heli Valtna-Lukner, Madis Lõhmus, Peeter Piksarv, Peeter Saari, and Rick Trebino, "Measuring the spatiotemporal field of ultrashort Bessel-X pulses," Optics Letters, Vol. 34, No. 15, 2276-2278, 2009. Google Scholar

21. Saari, P., "X waves in ultrafast optics," Non-Diffracting Waves, 109-134, H. E. Hernandez-Figueroa, M. Zamboni- Rached, and E. Recami (eds.), Wiley, Hobokn, NJ, 2013.

22. Kondaksi, H. and A. F. Abouraddy, "Diffraction-free space-time light sheets," Nat. Photon., Vol. 11, No. 733-740, 2017. Google Scholar

23. Bhaduri, B., M. Yesserov, and A. F. Abouraddy, "Space-time wavepackets that travel at the speed of light in vacuum," Optica, Vol. 6, 139, 2019.
doi:10.1364/OPTICA.6.000139 Google Scholar

24. Papasimakis, Nikitas, Tim Raybould, Vassili A. Fedotov, Din Ping Tsai, Ian Youngs, and Nikolay I. Zheludev, "Pulse generation scheme for flying electromagnetic doughnuts," Physical Review B, Vol. 97, No. 20, 201409(R), 2018. Google Scholar

25. Comite, Davide, Walter Fuscaldo, S. K. Podilchak, V. Gómez-Guillamón Buendía, P. D. Hilario Re, Paolo Baccarelli, Paolo Burghignoli, and Alessandro Galli, "Microwave generation of X-waves by means of a planar leaky-wave antenna," Applied Physics Letters, Vol. 113, No. 14, 144102, 2018. Google Scholar

26. Fuscaldo, Walter, Davide Comite, Alessandro Boesso, Paolo Baccarelli, Paolo Burghignoli, and Alessandro Galli, "Focusing leaky waves: A class of electromagnetic localized waves with complex spectra," Physical Review Applied, Vol. 9, No. 5, 054005, 2018. Google Scholar

27. Mackinnon, L., "A nondispersive de Broglie wave packet," Foundations of Physics, Vol. 8, No. 3, 157-176, 1978. Google Scholar

28. Mackinnon, L., "Particle rest mass and the de Broglie wave packet," Lettere al Nuovo Cimento, Vol. 31, No. 2, 37-38, 1981.
doi:10.1007/BF02788163 Google Scholar

29. Shaarawi, Amr M., Ioannis M. Besieris, and Richard W. Ziolkowski, "A novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations," Journal of Mathematical Physics, Vol. 31, No. 10, 2511-2519, 1990. Google Scholar

30. Tippett, Michael K. and Richard W. Ziolkowski, "A bidirectional wave transformation of the cold plasma equations," Journal of Mathematical Physics, Vol. 32, No. 2, 488-492, 1991. Google Scholar

31. Hillion, Pierre, "Nondispersive solutions of the Klein-Gordon equation," Journal of Mathematical Physics, Vol. 33, No. 5, 1817-1821, 1992. Google Scholar

32. Borisov, Victor V. and Andrei B. Utkin, "Focus wave modes in noncollisional plasma," Canadian Journal of Physics, Vol. 72, No. 9-10, 647-650, 1994. Google Scholar

33. Rodrigues, W. A. and J. E. Maiorino, "A unified theory for construction of arbitrary speed (0 ≤ v < ∞) solutions to the relativistic equations," Random Operators and Stochastic Equations, Vol. 4, 355-400, 1996. Google Scholar

34. Abdel-Rahman, M., I. M. Besieris, and A. M. Shaarawi, "A comprehensive analysis of propagation of localized waves in collisionless plasma media," Journal of Electromagnetic Waves and Applications, Vol. 11, No. 12, 1649-1668, 1997. Google Scholar

35. Besieris, I. M., A. M. Shaarawi, and L. P. Ligthart, "A note on dimension reduction and finite energy localized wave solutions to the Klein-Gordon and scalar wave equations. Part I. FWM-type," Journal of Electromagnetic Waves and Applications, Vol. 14, No. 5, 593-610, 2000. Google Scholar

36. Besieris, I. M., A. M. Shaarawi, and L. P. Lighart, "A note on dimension reduction and finite energy localized wave solutions to the scalar wave and Klein-Gordon equations. II. X wave-type," Progress In Electromagnetics Research, Vol. 27, 357-365, 2000.
doi:10.2528/PIER99112301 Google Scholar

37. Perel, M. V. and I. V. Fialkovsky, "Exponentially localized solutions to the Klein-Gordon equation," Journal of Mathematical Sciences, Vol. 117, 3994-4000, 2003. Google Scholar

38. Sõnajalg, Heiki and Peeter Saari, "Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators," Optics Letters, Vol. 21, No. 15, 1162-1164, 1996. Google Scholar

39. Sõnajalg, Heiki, Margus Rätsep, and Peeter Saari, "Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium," Optics Letters, Vol. 22, No. 5, 310-312, 1997. Google Scholar

40. Porras, Miguel A., "Diffraction-free and dispersion-free pulsed beam propagation in dispersive media," Optics Letters, Vol. 26, No. 17, 1364-1366, 2001. Google Scholar

41. Orlov, S., A. Piskarskas, and A. Stabinis, "Localized optical subcycle pulses in dispersive media," Optics Letters, Vol. 27, No. 24, 2167-2169, 2002. Google Scholar

42. Zamboni-Rached, M., K. Z. Nóbrega, H. E. Hernández-Figueroa, and Erasmo Recami, "Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth," Optics Communications, Vol. 226, No. 1-6, 15-23, 2003. Google Scholar

43. Porras, Miguel A., Gintaras Valiulis, and Paolo Di Trapani, "Unified description of Bessel X waves with cone dispersion and tilted pulses," Physical Review E, Vol. 68, No. 1, 016613, 2003. Google Scholar

44. Porras, Miguel A. and Isabel Gonzalo, "Control of temporal characteristics of Bessel-X pulses in dispersive media," Optics Communications, Vol. 217, No. 1-6, 257-264, 2003. Google Scholar

45. Porras, Miguel A. and Paolo Di Trapani, "Localized and stationary light wave modes in dispersive media," Physical Review E, Vol. 69, No. 6, 066606, 2004. Google Scholar

46. Longhi, Stefano, "Localized subluminal envelope pulses in dispersive media," Optics Letters, Vol. 29, No. 2, 147-149, 2004. Google Scholar

47. Ciattoni, Alessandro and Paolo Di Porto, "Electromagnetic nondiffracting pulses in lossless isotropic plasmalike media," Physical Review E, Vol. 70, No. 3, 035601, 2004. Google Scholar

48. Orlov, S. and A. Stabinis, "Angular dispersion of diffraction-free optical pulses in dispersive medium," Optics Communications, Vol. 240, No. 1-3, 1-8, 2004. Google Scholar

49. Malaguti, Stefania, Gaetano Bellanca, and Stefano Trillo, "Two-dimensional envelope localized waves in the anomalous dispersion regime," Optics Letters, Vol. 33, No. 10, 1117-1119, 2008. Google Scholar

50. Porras, M. A., P. Di Trapani, and W. Hu, "Optical modes: Localized and propagation-invariant wave packets in optically transparent dispersive media," Localized Waves, 217-241, H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami (eds.), Wiley-Interscience, Hoboken, NJ, 2008.

51. Malaguti, Stefania and Stefano Trillo, "Envelope localized waves of the conical type in linear normally dispersive media," Physical Review A, Vol. 79, No. 6, 063803, 2009. Google Scholar

52. Salem, Mohamed A. and Hakan Bağcı, "On the propagation of truncated localized waves in dispersive silica," Optics Express, Vol. 18, No. 25, 25482-25493, 2010. Google Scholar

53. Hall, Layton A. and Ayman F. Abouraddy, "Realizing normal group-velocity dispersion in free space via angular dispersion," Optics Letters, Vol. 46, No. 21, 5421-5424, 2021. Google Scholar

54. Hall, Layton A. and Ayman F. Abouraddy, "Spectral reorganization of space-time wave packets in presence of normal group-velocity dispersion," ArXiv Preprint ArXiv:2206.05387, 2022. Google Scholar

55. He, Hao, Cheng Guo, and Meng Xiao, "Nondispersive space-time wave packets propagating in dispersive media," Laser & Photonics Reviews, Vol. 16, No. 10, 2100634, 2022. Google Scholar

56. Yessenov, Murat, Sanaz Faryadras, Sepehr Benis, David J. Hagan, Eric W. Van Stryland, and Ayman F. Abouraddy, "Refraction of space-time wave packets in a dispersive medium," Optics Letters, Vol. 47, No. 7, 1630-1633, 2022. Google Scholar

57. Palastro, J. P., K. G. Miller, R. K. Follett, D. Ramsey, K. Weichman, A. V. Arefiev, and D. H. Froula, "Space-time structured plasma waves," Physical Review Letters, Vol. 132, No. 9, 095101, 2024. Google Scholar

58. Zhan, Qiwen, "Spatiotemporal sculpturing of light: A tutorial," Advances in Optics and Photonics, Vol. 16, No. 2, 163-228, 2024. Google Scholar

59. Brabec, Thomas and Ferenc Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Physical Review Letters, Vol. 78, No. 17, 3282, 1997. Google Scholar

60. Porras, Miguel A., "Propagation of single-cycle pulsed light beams in dispersive media," Physical Review A, Vol. 60, No. 6, 5069, 1999. Google Scholar

61. Bateman, Harry, Arthur Erdélyi, Wilhelm Magnus, and Fritz Oberhettinger, Tables of Integral Transforms, Vol. 1, McGraw-Hill, New York, 1954.

62. Gradshteyn, Izrail Solomonovich and Iosif Moiseevich Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 2014.

63. Fu, Xiquan, Liejia Qian, Shuangchun Wen, and Dianyuan Fan, "Propagation of the ultrashort pulsed beam with ultrabroad bandwidth in the dispersive medium," Physical Review A, Vol. 68, No. 6, 063818, 2003. Google Scholar

64. Kinsler, P. and G. H. C. New, "Few-cycle pulse propagation," Physical Review A, Vol. 67, No. 2, 023813, 2003. Google Scholar

65. Hillion, P., "Electromagnetic field propagation in neutral isotropic plasmas," The European Physical Journal D --- Atomic, Molecular, Optical and Plasma Physics, Vol. 29, No. 2, 273-277, 2004. Google Scholar

Professor Ioannis Besieris