Abstract

Localized waves (LW) are nondiffracting (“soliton-like”) solutions to the wave equations and are known to exist with subluminal, luminal, and superluminal peak velocities V. For mathematical and experimental reasons, those that have attracted more attention are the “X-shaped” superluminal waves. Such waves are associated with a cone, so that one may be tempted—let us confine ourselves to electromagnetism—to look [Phys. Rev. Lett. 99, 244802 (2007) ] for links between them and the Cherenkov radiation. However, the X-shaped waves belong to a very different realm: For instance, they can be shown to exist, independently of any media, even in vacuum, as localized non-diffracting pulses propagating rigidly with a peak-velocity V>c [ Hernández et al., eds., Localized Waves (Wiley, 2008 )]. We dissect the whole question on the basis of a rigorous formalism and clear physical considerations.

© 2010 Optical Society of America

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  1. See, e.g., S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
    [CrossRef]
  2. See H.E.Hernández-Figueroa, M.Zamboni-Rached, and E.Recami, eds., Localized Waves (Wiley, 2008) and references therein.
    [CrossRef]
  3. See E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal charge,” Phys. Rev. E 69, 027602 (2004) and references therein.
    [CrossRef]
  4. F. V. Hartmann, High-Field Electrodynamics (CRC Press, 2002).
  5. See also I. Ye. Tamm and I. M. Frank, “Coherent radiation of fast electrons in a medium,” Dokl. Akad. Nauk SSSR 14, 107 (1937).
  6. Also in papers like , and references therein, such a treatment is presented in a mathematically correct way, even if the language used in is sometimes ambiguous: For example, in the speed cn in the medium is just called c; furthermore, the point-charge associated with the Cherenkov radiation is called “superluminal,” despite the fact that its speed is lower than the light speed in vacuum: By contrast, in the existing theoretical and experimental literature on localized waves (see again, e.g., ) and in particular on X-shaped waves [cf. below], the word superluminal is reserved to group velocities actually larger than the speed of light in vacuum.
  7. A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509-533 (1982).
    [CrossRef]
  8. See E. Recami, “Classical tachyons and possible applications,” Riv. Nuovo Cimento 9(6), 1-178 (1986) and references therein.
    [CrossRef]
  9. J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X-waves: Exact solutions to the free-space scalar wave equation and their finite realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
    [CrossRef] [PubMed]
  10. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bi-directional traveling plane wave representation of exact solutions to the scalar wave equation,” J. Math. Phys. 30, 1254-1269 (1989).
    [CrossRef]
  11. E. Recami, “On localized X-shaped superluminal solutions to Maxwell equations,” Physica A 252, 586-610 (1998) and references therein.
    [CrossRef]
  12. J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441-446 (1992).
    [CrossRef] [PubMed]
  13. P. Saari and K. Reivelt, “Evidence of X-shaped propagation invariant localized light waves,” Phys. Rev. Lett. 79, 4135-4138 (1997).
    [CrossRef]
  14. P. Bowlan, H. Valtna-Lukner, M. Lohmus, P. Piksarv, P. Saari, and R. Trebino, “Pulses by frequency-resolved optical gating,” Opt. Lett. 34, 2276-2278 (2009).
    [CrossRef] [PubMed]
  15. See also F. Bonaretti, D. Faccio, M. Crerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” arXiv:0904.0952[physics.optics].
  16. See also P. Bowlan, M. Lohmus, P. Piksarv, H. Valtna-Lukner, P. Saari, and R. Trebino, “Measuring the spatio-temporal field of diffracting ultrashort pulses,” arXiv:0905.4381[physics.optics].
  17. For simplicity, we are here assuming the refractive index n to be constant.
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  19. M. Zamboni-Rached, “Analytic expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures,” J. Opt. Soc. Am. A 23, 2166-2176 (2006).
    [CrossRef]
  20. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A 10, 75-87 (1993).
    [CrossRef]
  21. See also, e.g., C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A 18, 2594-2600 (2001).
    [CrossRef]
  22. S. He and J.-Y. Lu, “Sidelobes reduction of limited-diffraction beams with Chebyshev aperture apodization,” J. Acoust. Soc. Am. 107, 3556-3559 (2000).
    [CrossRef] [PubMed]
  23. J.-Y. Lu, H.-H. Zou, and J. F. Greenleaf, “Biomedical ultrasound beam forming,” Ultrasound Med. Biol. 20, 403-428 (1994).
    [CrossRef] [PubMed]
  24. I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1-48 (1998).
    [CrossRef]
  25. A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, and M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2289-2295 (1998).
    [CrossRef]
  26. I. M. Besieris and A. M. Shaarawi, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A 33, 7227-7254 (2000).
    [CrossRef]
  27. A. M. Shaarawi, I. M. Besieris, and T. M. Said, “Temporal focusing by use of composite X-waves,” J. Opt. Soc. Am. A 20, 1658-1665 (2003).
    [CrossRef]
  28. M. Zamboni-Rached, A. M. Shaarawi, and E. Recami, “Focused X-shaped pulses,” J. Opt. Soc. Am. A 21, 1564-1574 (2004).
    [CrossRef]
  29. M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217-228 (2002).
    [CrossRef]
  30. E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9(1), 59-73 (2003) [special issue on “Nontraditional Forms of Light”].
    [CrossRef]
  31. E. Recami, M. Zamboni-Rached, and H. E. Hernández-Figueroa, “Localized waves: a historical and scientific introduction,” Chap. 1 in , pp. 1-41.
  32. M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “Structure of nondiffracting waves, and some interesting applications,” Chap. 2 in , pp. 43-77.
  33. M. Zamboni-Rached, “Localized solutions: structure and applications,” M.Sc. thesis (Campinas State University, 1999).
  34. M. Zamboni-Rached, “Localized waves in diffractive/dispersive media,” Ph.D. thesis (Campinas State University, Aug. 2004) [can be download at http://libdigi.unicamp.br/document/?code=vtls000337794] and references therein.
  35. R. Mignani and E. Recami, “Tachyons do not emit Cherenkov radiation in vacuum,” Lett. Nuovo Cimento 7, 388-390 (1973).
    [CrossRef]
  36. Cf. also R. Folman and E. Recami, “On the phenomenology of tachyon radiation,” Found. Phys. Lett. 8, 127-134 (1995).
    [CrossRef]
  37. E. Recami and R. Mignani, “Classical theory of tachyons (extending special relativity to superluminal frames and objects),” Riv. Nuovo Cimento 4, 209-290 (1974).
    [CrossRef]
  38. E. Recami and R. Mignani, “Classical theory of tachyons (extending special relativity to superluminal frames and objects): Erratum,” Riv. Nuovo Cimento 4, 398 (1974).
    [CrossRef]
  39. A. Sommerfeld, K. Ned. Akad. Wet. Amsterdam 7, 346-367 (1904).
  40. A. Sommerfeld, Nachr. Ges. Wiss. Göttingen 5, 201-236 (1905).
  41. E. C. G. Sudarshan, personal communications (1971).
  42. See also J. J. Thomson, Philos. Mag. 28, 13 (1889).
  43. P. Saari “Superluminal localized waves of electromagnetic field in vacuo,” in D.Mugnai, A.Ranfagni, and L.S.Shulman, eds. Time Arrows, Quantum Measurement and Superluminal Behaviour (CNR, 2001), pp. 37-48. arXxiv:physics/01030541 [physics.optics]. In this paper the author uses, however, G+(r,t,r′,t′)−G−(r,t,r′,t′) instead of G+(r,t,r′,t′)/2+G−(r,t,r′,t′)/2, a choice that does not apply to a non-homogeneous problem such as ours, in which we deal with a (superluminal) charge.
  44. E. Recami and R. Mignani, “Magnetic monopoles and tachyons in special relativity,” Phys. Lett. B 62, 41-43 (1976).
    [CrossRef]
  45. R. Mignani and E. Recami, “Complex electromagnetic four-potential and the Cabibbo-Ferrari relation for magnetic monopoles,” Nuovo Cimento A 30, 533-540 (1975).
    [CrossRef]
  46. The same variables were adopted in , in the paraxial approximation context, while we are addressing the general exact case.
  47. I. M. Besieris and A. M. Shaarawi, “Paraxial localized waves in free space,” Opt. Express 12, 3848-3864 (2004).
    [CrossRef] [PubMed]

2009 (1)

2007 (1)

See, e.g., S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
[CrossRef]

2006 (1)

2004 (3)

2003 (2)

A. M. Shaarawi, I. M. Besieris, and T. M. Said, “Temporal focusing by use of composite X-waves,” J. Opt. Soc. Am. A 20, 1658-1665 (2003).
[CrossRef]

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9(1), 59-73 (2003) [special issue on “Nontraditional Forms of Light”].
[CrossRef]

2002 (1)

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217-228 (2002).
[CrossRef]

2001 (1)

2000 (2)

S. He and J.-Y. Lu, “Sidelobes reduction of limited-diffraction beams with Chebyshev aperture apodization,” J. Acoust. Soc. Am. 107, 3556-3559 (2000).
[CrossRef] [PubMed]

I. M. Besieris and A. M. Shaarawi, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A 33, 7227-7254 (2000).
[CrossRef]

1998 (3)

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1-48 (1998).
[CrossRef]

A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, and M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2289-2295 (1998).
[CrossRef]

E. Recami, “On localized X-shaped superluminal solutions to Maxwell equations,” Physica A 252, 586-610 (1998) and references therein.
[CrossRef]

1997 (1)

P. Saari and K. Reivelt, “Evidence of X-shaped propagation invariant localized light waves,” Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

1995 (1)

Cf. also R. Folman and E. Recami, “On the phenomenology of tachyon radiation,” Found. Phys. Lett. 8, 127-134 (1995).
[CrossRef]

1994 (1)

J.-Y. Lu, H.-H. Zou, and J. F. Greenleaf, “Biomedical ultrasound beam forming,” Ultrasound Med. Biol. 20, 403-428 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (2)

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441-446 (1992).
[CrossRef] [PubMed]

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X-waves: Exact solutions to the free-space scalar wave equation and their finite realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
[CrossRef] [PubMed]

1989 (1)

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bi-directional traveling plane wave representation of exact solutions to the scalar wave equation,” J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

1986 (1)

See E. Recami, “Classical tachyons and possible applications,” Riv. Nuovo Cimento 9(6), 1-178 (1986) and references therein.
[CrossRef]

1982 (1)

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509-533 (1982).
[CrossRef]

1976 (1)

E. Recami and R. Mignani, “Magnetic monopoles and tachyons in special relativity,” Phys. Lett. B 62, 41-43 (1976).
[CrossRef]

1975 (1)

R. Mignani and E. Recami, “Complex electromagnetic four-potential and the Cabibbo-Ferrari relation for magnetic monopoles,” Nuovo Cimento A 30, 533-540 (1975).
[CrossRef]

1974 (2)

E. Recami and R. Mignani, “Classical theory of tachyons (extending special relativity to superluminal frames and objects),” Riv. Nuovo Cimento 4, 209-290 (1974).
[CrossRef]

E. Recami and R. Mignani, “Classical theory of tachyons (extending special relativity to superluminal frames and objects): Erratum,” Riv. Nuovo Cimento 4, 398 (1974).
[CrossRef]

1973 (1)

R. Mignani and E. Recami, “Tachyons do not emit Cherenkov radiation in vacuum,” Lett. Nuovo Cimento 7, 388-390 (1973).
[CrossRef]

1937 (1)

See also I. Ye. Tamm and I. M. Frank, “Coherent radiation of fast electrons in a medium,” Dokl. Akad. Nauk SSSR 14, 107 (1937).

1905 (1)

A. Sommerfeld, Nachr. Ges. Wiss. Göttingen 5, 201-236 (1905).

1904 (1)

A. Sommerfeld, K. Ned. Akad. Wet. Amsterdam 7, 346-367 (1904).

1889 (1)

See also J. J. Thomson, Philos. Mag. 28, 13 (1889).

Abdel-Rahman, M.

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1-48 (1998).
[CrossRef]

A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, and M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2289-2295 (1998).
[CrossRef]

Barut, A. O.

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509-533 (1982).
[CrossRef]

Besieris, I. M.

I. M. Besieris and A. M. Shaarawi, “Paraxial localized waves in free space,” Opt. Express 12, 3848-3864 (2004).
[CrossRef] [PubMed]

A. M. Shaarawi, I. M. Besieris, and T. M. Said, “Temporal focusing by use of composite X-waves,” J. Opt. Soc. Am. A 20, 1658-1665 (2003).
[CrossRef]

I. M. Besieris and A. M. Shaarawi, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A 33, 7227-7254 (2000).
[CrossRef]

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1-48 (1998).
[CrossRef]

A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, and M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2289-2295 (1998).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A 10, 75-87 (1993).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bi-directional traveling plane wave representation of exact solutions to the scalar wave equation,” J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

Biegert, J.

See also F. Bonaretti, D. Faccio, M. Crerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” arXiv:0904.0952[physics.optics].

Bonaretti, F.

See also F. Bonaretti, D. Faccio, M. Crerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” arXiv:0904.0952[physics.optics].

Bowlan, P.

P. Bowlan, H. Valtna-Lukner, M. Lohmus, P. Piksarv, P. Saari, and R. Trebino, “Pulses by frequency-resolved optical gating,” Opt. Lett. 34, 2276-2278 (2009).
[CrossRef] [PubMed]

See also P. Bowlan, M. Lohmus, P. Piksarv, H. Valtna-Lukner, P. Saari, and R. Trebino, “Measuring the spatio-temporal field of diffracting ultrashort pulses,” arXiv:0905.4381[physics.optics].

Chatzipetros, A.

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1-48 (1998).
[CrossRef]

A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, and M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2289-2295 (1998).
[CrossRef]

Crerici, M.

See also F. Bonaretti, D. Faccio, M. Crerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” arXiv:0904.0952[physics.optics].

Dartora, C. A.

See E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal charge,” Phys. Rev. E 69, 027602 (2004) and references therein.
[CrossRef]

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9(1), 59-73 (2003) [special issue on “Nontraditional Forms of Light”].
[CrossRef]

Di Trapani, P.

See also F. Bonaretti, D. Faccio, M. Crerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” arXiv:0904.0952[physics.optics].

Faccio, D.

See also F. Bonaretti, D. Faccio, M. Crerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” arXiv:0904.0952[physics.optics].

Folman, R.

Cf. also R. Folman and E. Recami, “On the phenomenology of tachyon radiation,” Found. Phys. Lett. 8, 127-134 (1995).
[CrossRef]

Frank, I. M.

See also I. Ye. Tamm and I. M. Frank, “Coherent radiation of fast electrons in a medium,” Dokl. Akad. Nauk SSSR 14, 107 (1937).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Greenleaf, J. F.

J.-Y. Lu, H.-H. Zou, and J. F. Greenleaf, “Biomedical ultrasound beam forming,” Ultrasound Med. Biol. 20, 403-428 (1994).
[CrossRef] [PubMed]

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X-waves: Exact solutions to the free-space scalar wave equation and their finite realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
[CrossRef] [PubMed]

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441-446 (1992).
[CrossRef] [PubMed]

Hartmann, F. V.

F. V. Hartmann, High-Field Electrodynamics (CRC Press, 2002).

He, S.

S. He and J.-Y. Lu, “Sidelobes reduction of limited-diffraction beams with Chebyshev aperture apodization,” J. Acoust. Soc. Am. 107, 3556-3559 (2000).
[CrossRef] [PubMed]

Hernández-Figueroa, H. E.

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9(1), 59-73 (2003) [special issue on “Nontraditional Forms of Light”].
[CrossRef]

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217-228 (2002).
[CrossRef]

E. Recami, M. Zamboni-Rached, and H. E. Hernández-Figueroa, “Localized waves: a historical and scientific introduction,” Chap. 1 in , pp. 1-41.

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “Structure of nondiffracting waves, and some interesting applications,” Chap. 2 in , pp. 43-77.

Kuperman, W. A.

See, e.g., S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
[CrossRef]

Lohmus, M.

P. Bowlan, H. Valtna-Lukner, M. Lohmus, P. Piksarv, P. Saari, and R. Trebino, “Pulses by frequency-resolved optical gating,” Opt. Lett. 34, 2276-2278 (2009).
[CrossRef] [PubMed]

See also P. Bowlan, M. Lohmus, P. Piksarv, H. Valtna-Lukner, P. Saari, and R. Trebino, “Measuring the spatio-temporal field of diffracting ultrashort pulses,” arXiv:0905.4381[physics.optics].

Lu, J.-Y.

S. He and J.-Y. Lu, “Sidelobes reduction of limited-diffraction beams with Chebyshev aperture apodization,” J. Acoust. Soc. Am. 107, 3556-3559 (2000).
[CrossRef] [PubMed]

J.-Y. Lu, H.-H. Zou, and J. F. Greenleaf, “Biomedical ultrasound beam forming,” Ultrasound Med. Biol. 20, 403-428 (1994).
[CrossRef] [PubMed]

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441-446 (1992).
[CrossRef] [PubMed]

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X-waves: Exact solutions to the free-space scalar wave equation and their finite realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
[CrossRef] [PubMed]

Maccarrone, G. D.

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509-533 (1982).
[CrossRef]

Mignani, R.

E. Recami and R. Mignani, “Magnetic monopoles and tachyons in special relativity,” Phys. Lett. B 62, 41-43 (1976).
[CrossRef]

R. Mignani and E. Recami, “Complex electromagnetic four-potential and the Cabibbo-Ferrari relation for magnetic monopoles,” Nuovo Cimento A 30, 533-540 (1975).
[CrossRef]

E. Recami and R. Mignani, “Classical theory of tachyons (extending special relativity to superluminal frames and objects),” Riv. Nuovo Cimento 4, 209-290 (1974).
[CrossRef]

E. Recami and R. Mignani, “Classical theory of tachyons (extending special relativity to superluminal frames and objects): Erratum,” Riv. Nuovo Cimento 4, 398 (1974).
[CrossRef]

R. Mignani and E. Recami, “Tachyons do not emit Cherenkov radiation in vacuum,” Lett. Nuovo Cimento 7, 388-390 (1973).
[CrossRef]

Nobrega, K. Z.

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9(1), 59-73 (2003) [special issue on “Nontraditional Forms of Light”].
[CrossRef]

Piksarv, P.

P. Bowlan, H. Valtna-Lukner, M. Lohmus, P. Piksarv, P. Saari, and R. Trebino, “Pulses by frequency-resolved optical gating,” Opt. Lett. 34, 2276-2278 (2009).
[CrossRef] [PubMed]

See also P. Bowlan, M. Lohmus, P. Piksarv, H. Valtna-Lukner, P. Saari, and R. Trebino, “Measuring the spatio-temporal field of diffracting ultrashort pulses,” arXiv:0905.4381[physics.optics].

Recami, E.

See E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal charge,” Phys. Rev. E 69, 027602 (2004) and references therein.
[CrossRef]

M. Zamboni-Rached, A. M. Shaarawi, and E. Recami, “Focused X-shaped pulses,” J. Opt. Soc. Am. A 21, 1564-1574 (2004).
[CrossRef]

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9(1), 59-73 (2003) [special issue on “Nontraditional Forms of Light”].
[CrossRef]

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217-228 (2002).
[CrossRef]

E. Recami, “On localized X-shaped superluminal solutions to Maxwell equations,” Physica A 252, 586-610 (1998) and references therein.
[CrossRef]

Cf. also R. Folman and E. Recami, “On the phenomenology of tachyon radiation,” Found. Phys. Lett. 8, 127-134 (1995).
[CrossRef]

See E. Recami, “Classical tachyons and possible applications,” Riv. Nuovo Cimento 9(6), 1-178 (1986) and references therein.
[CrossRef]

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509-533 (1982).
[CrossRef]

E. Recami and R. Mignani, “Magnetic monopoles and tachyons in special relativity,” Phys. Lett. B 62, 41-43 (1976).
[CrossRef]

R. Mignani and E. Recami, “Complex electromagnetic four-potential and the Cabibbo-Ferrari relation for magnetic monopoles,” Nuovo Cimento A 30, 533-540 (1975).
[CrossRef]

E. Recami and R. Mignani, “Classical theory of tachyons (extending special relativity to superluminal frames and objects),” Riv. Nuovo Cimento 4, 209-290 (1974).
[CrossRef]

E. Recami and R. Mignani, “Classical theory of tachyons (extending special relativity to superluminal frames and objects): Erratum,” Riv. Nuovo Cimento 4, 398 (1974).
[CrossRef]

R. Mignani and E. Recami, “Tachyons do not emit Cherenkov radiation in vacuum,” Lett. Nuovo Cimento 7, 388-390 (1973).
[CrossRef]

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “Structure of nondiffracting waves, and some interesting applications,” Chap. 2 in , pp. 43-77.

E. Recami, M. Zamboni-Rached, and H. E. Hernández-Figueroa, “Localized waves: a historical and scientific introduction,” Chap. 1 in , pp. 1-41.

Reivelt, K.

P. Saari and K. Reivelt, “Evidence of X-shaped propagation invariant localized light waves,” Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

Saari, P.

P. Bowlan, H. Valtna-Lukner, M. Lohmus, P. Piksarv, P. Saari, and R. Trebino, “Pulses by frequency-resolved optical gating,” Opt. Lett. 34, 2276-2278 (2009).
[CrossRef] [PubMed]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation invariant localized light waves,” Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

See also P. Bowlan, M. Lohmus, P. Piksarv, H. Valtna-Lukner, P. Saari, and R. Trebino, “Measuring the spatio-temporal field of diffracting ultrashort pulses,” arXiv:0905.4381[physics.optics].

P. Saari “Superluminal localized waves of electromagnetic field in vacuo,” in D.Mugnai, A.Ranfagni, and L.S.Shulman, eds. Time Arrows, Quantum Measurement and Superluminal Behaviour (CNR, 2001), pp. 37-48. arXxiv:physics/01030541 [physics.optics]. In this paper the author uses, however, G+(r,t,r′,t′)−G−(r,t,r′,t′) instead of G+(r,t,r′,t′)/2+G−(r,t,r′,t′)/2, a choice that does not apply to a non-homogeneous problem such as ours, in which we deal with a (superluminal) charge.

Said, T. M.

Shaarawi, A. M.

M. Zamboni-Rached, A. M. Shaarawi, and E. Recami, “Focused X-shaped pulses,” J. Opt. Soc. Am. A 21, 1564-1574 (2004).
[CrossRef]

I. M. Besieris and A. M. Shaarawi, “Paraxial localized waves in free space,” Opt. Express 12, 3848-3864 (2004).
[CrossRef] [PubMed]

A. M. Shaarawi, I. M. Besieris, and T. M. Said, “Temporal focusing by use of composite X-waves,” J. Opt. Soc. Am. A 20, 1658-1665 (2003).
[CrossRef]

I. M. Besieris and A. M. Shaarawi, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A 33, 7227-7254 (2000).
[CrossRef]

A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, and M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2289-2295 (1998).
[CrossRef]

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1-48 (1998).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A 10, 75-87 (1993).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bi-directional traveling plane wave representation of exact solutions to the scalar wave equation,” J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

Sheppard, C. J. R.

Sommerfeld, A.

A. Sommerfeld, Nachr. Ges. Wiss. Göttingen 5, 201-236 (1905).

A. Sommerfeld, K. Ned. Akad. Wet. Amsterdam 7, 346-367 (1904).

Sudarshan, E. C. G.

E. C. G. Sudarshan, personal communications (1971).

Tamm, I. Ye.

See also I. Ye. Tamm and I. M. Frank, “Coherent radiation of fast electrons in a medium,” Dokl. Akad. Nauk SSSR 14, 107 (1937).

Thomson, J. J.

See also J. J. Thomson, Philos. Mag. 28, 13 (1889).

Trebino, R.

P. Bowlan, H. Valtna-Lukner, M. Lohmus, P. Piksarv, P. Saari, and R. Trebino, “Pulses by frequency-resolved optical gating,” Opt. Lett. 34, 2276-2278 (2009).
[CrossRef] [PubMed]

See also P. Bowlan, M. Lohmus, P. Piksarv, H. Valtna-Lukner, P. Saari, and R. Trebino, “Measuring the spatio-temporal field of diffracting ultrashort pulses,” arXiv:0905.4381[physics.optics].

Valtna-Lukner, H.

P. Bowlan, H. Valtna-Lukner, M. Lohmus, P. Piksarv, P. Saari, and R. Trebino, “Pulses by frequency-resolved optical gating,” Opt. Lett. 34, 2276-2278 (2009).
[CrossRef] [PubMed]

See also P. Bowlan, M. Lohmus, P. Piksarv, H. Valtna-Lukner, P. Saari, and R. Trebino, “Measuring the spatio-temporal field of diffracting ultrashort pulses,” arXiv:0905.4381[physics.optics].

Walker, S. C.

See, e.g., S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
[CrossRef]

Zamboni-Rached, M.

M. Zamboni-Rached, “Analytic expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures,” J. Opt. Soc. Am. A 23, 2166-2176 (2006).
[CrossRef]

M. Zamboni-Rached, A. M. Shaarawi, and E. Recami, “Focused X-shaped pulses,” J. Opt. Soc. Am. A 21, 1564-1574 (2004).
[CrossRef]

See E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal charge,” Phys. Rev. E 69, 027602 (2004) and references therein.
[CrossRef]

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9(1), 59-73 (2003) [special issue on “Nontraditional Forms of Light”].
[CrossRef]

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217-228 (2002).
[CrossRef]

E. Recami, M. Zamboni-Rached, and H. E. Hernández-Figueroa, “Localized waves: a historical and scientific introduction,” Chap. 1 in , pp. 1-41.

M. Zamboni-Rached, “Localized solutions: structure and applications,” M.Sc. thesis (Campinas State University, 1999).

M. Zamboni-Rached, “Localized waves in diffractive/dispersive media,” Ph.D. thesis (Campinas State University, Aug. 2004) [can be download at http://libdigi.unicamp.br/document/?code=vtls000337794] and references therein.

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “Structure of nondiffracting waves, and some interesting applications,” Chap. 2 in , pp. 43-77.

Ziolkowski, R. W.

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A 10, 75-87 (1993).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bi-directional traveling plane wave representation of exact solutions to the scalar wave equation,” J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

Zou, H.-H.

J.-Y. Lu, H.-H. Zou, and J. F. Greenleaf, “Biomedical ultrasound beam forming,” Ultrasound Med. Biol. 20, 403-428 (1994).
[CrossRef] [PubMed]

Dokl. Akad. Nauk SSSR (1)

See also I. Ye. Tamm and I. M. Frank, “Coherent radiation of fast electrons in a medium,” Dokl. Akad. Nauk SSSR 14, 107 (1937).

Eur. Phys. J. D (1)

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217-228 (2002).
[CrossRef]

Found. Phys. Lett. (1)

Cf. also R. Folman and E. Recami, “On the phenomenology of tachyon radiation,” Found. Phys. Lett. 8, 127-134 (1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9(1), 59-73 (2003) [special issue on “Nontraditional Forms of Light”].
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (2)

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X-waves: Exact solutions to the free-space scalar wave equation and their finite realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
[CrossRef] [PubMed]

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441-446 (1992).
[CrossRef] [PubMed]

J. Acoust. Soc. Am. (2)

S. He and J.-Y. Lu, “Sidelobes reduction of limited-diffraction beams with Chebyshev aperture apodization,” J. Acoust. Soc. Am. 107, 3556-3559 (2000).
[CrossRef] [PubMed]

A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, and M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2289-2295 (1998).
[CrossRef]

J. Math. Phys. (1)

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bi-directional traveling plane wave representation of exact solutions to the scalar wave equation,” J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

J. Opt. Soc. Am. A (5)

J. Phys. A (1)

I. M. Besieris and A. M. Shaarawi, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A 33, 7227-7254 (2000).
[CrossRef]

K. Ned. Akad. Wet. Amsterdam (1)

A. Sommerfeld, K. Ned. Akad. Wet. Amsterdam 7, 346-367 (1904).

Lett. Nuovo Cimento (1)

R. Mignani and E. Recami, “Tachyons do not emit Cherenkov radiation in vacuum,” Lett. Nuovo Cimento 7, 388-390 (1973).
[CrossRef]

Nachr. Ges. Wiss. Göttingen (1)

A. Sommerfeld, Nachr. Ges. Wiss. Göttingen 5, 201-236 (1905).

Nuovo Cimento A (2)

R. Mignani and E. Recami, “Complex electromagnetic four-potential and the Cabibbo-Ferrari relation for magnetic monopoles,” Nuovo Cimento A 30, 533-540 (1975).
[CrossRef]

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509-533 (1982).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Philos. Mag. (1)

See also J. J. Thomson, Philos. Mag. 28, 13 (1889).

Phys. Lett. B (1)

E. Recami and R. Mignani, “Magnetic monopoles and tachyons in special relativity,” Phys. Lett. B 62, 41-43 (1976).
[CrossRef]

Phys. Rev. E (1)

See E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal charge,” Phys. Rev. E 69, 027602 (2004) and references therein.
[CrossRef]

Phys. Rev. Lett. (2)

See, e.g., S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
[CrossRef]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation invariant localized light waves,” Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

Physica A (1)

E. Recami, “On localized X-shaped superluminal solutions to Maxwell equations,” Physica A 252, 586-610 (1998) and references therein.
[CrossRef]

Prog. Electromagn. Res. (1)

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1-48 (1998).
[CrossRef]

Riv. Nuovo Cimento (3)

E. Recami and R. Mignani, “Classical theory of tachyons (extending special relativity to superluminal frames and objects),” Riv. Nuovo Cimento 4, 209-290 (1974).
[CrossRef]

E. Recami and R. Mignani, “Classical theory of tachyons (extending special relativity to superluminal frames and objects): Erratum,” Riv. Nuovo Cimento 4, 398 (1974).
[CrossRef]

See E. Recami, “Classical tachyons and possible applications,” Riv. Nuovo Cimento 9(6), 1-178 (1986) and references therein.
[CrossRef]

Ultrasound Med. Biol. (1)

J.-Y. Lu, H.-H. Zou, and J. F. Greenleaf, “Biomedical ultrasound beam forming,” Ultrasound Med. Biol. 20, 403-428 (1994).
[CrossRef] [PubMed]

Other (14)

E. Recami, M. Zamboni-Rached, and H. E. Hernández-Figueroa, “Localized waves: a historical and scientific introduction,” Chap. 1 in , pp. 1-41.

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “Structure of nondiffracting waves, and some interesting applications,” Chap. 2 in , pp. 43-77.

M. Zamboni-Rached, “Localized solutions: structure and applications,” M.Sc. thesis (Campinas State University, 1999).

M. Zamboni-Rached, “Localized waves in diffractive/dispersive media,” Ph.D. thesis (Campinas State University, Aug. 2004) [can be download at http://libdigi.unicamp.br/document/?code=vtls000337794] and references therein.

Also in papers like , and references therein, such a treatment is presented in a mathematically correct way, even if the language used in is sometimes ambiguous: For example, in the speed cn in the medium is just called c; furthermore, the point-charge associated with the Cherenkov radiation is called “superluminal,” despite the fact that its speed is lower than the light speed in vacuum: By contrast, in the existing theoretical and experimental literature on localized waves (see again, e.g., ) and in particular on X-shaped waves [cf. below], the word superluminal is reserved to group velocities actually larger than the speed of light in vacuum.

See H.E.Hernández-Figueroa, M.Zamboni-Rached, and E.Recami, eds., Localized Waves (Wiley, 2008) and references therein.
[CrossRef]

F. V. Hartmann, High-Field Electrodynamics (CRC Press, 2002).

See also F. Bonaretti, D. Faccio, M. Crerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” arXiv:0904.0952[physics.optics].

See also P. Bowlan, M. Lohmus, P. Piksarv, H. Valtna-Lukner, P. Saari, and R. Trebino, “Measuring the spatio-temporal field of diffracting ultrashort pulses,” arXiv:0905.4381[physics.optics].

For simplicity, we are here assuming the refractive index n to be constant.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

The same variables were adopted in , in the paraxial approximation context, while we are addressing the general exact case.

P. Saari “Superluminal localized waves of electromagnetic field in vacuo,” in D.Mugnai, A.Ranfagni, and L.S.Shulman, eds. Time Arrows, Quantum Measurement and Superluminal Behaviour (CNR, 2001), pp. 37-48. arXxiv:physics/01030541 [physics.optics]. In this paper the author uses, however, G+(r,t,r′,t′)−G−(r,t,r′,t′) instead of G+(r,t,r′,t′)/2+G−(r,t,r′,t′)/2, a choice that does not apply to a non-homogeneous problem such as ours, in which we deal with a (superluminal) charge.

E. C. G. Sudarshan, personal communications (1971).

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Figures (3)

Fig. 1
Fig. 1

Example of an X-type LW endowed with finite energy (even without truncation), which consequently gets deformed while propagating: (b) represents the pulse depicted in (a) after it has traveled 50 km . See [32] and the text.

Fig. 2
Fig. 2

This figure, which appeared in [3], but is taken from [8], intuitively shows, among other things, that a superluminal charge [35, 36, 37, 38, 7, 8, 30, 31] traveling at constant speed in vacuum would not lose energy: see [3] and the text. Reprinted with kind permission of Società Italiana di Fisica.

Fig. 3
Fig. 3

Example of a finite-energy X-type LW, corresponding to an exact, analytic solution of Eq. (18), totally free of backward components. This figure represents the real part of the field, normalized at ρ = z = 0 for t = 0 , with the choices a = 3.99 × 10 6 m , d = 20 m , V = 1.005 c , and α 0 = 1.26 × 10 7 m 1 . In this case the frequency spectrum starts at ω ω min 3.77 × 10 15 Hz and afterward decays exponentially with the bandwidth Δ ω 7.54 × 10 13 Hz . The value ω min can be regarded as the pulse central frequency; since Δ ω ω min 1 , there exists a well-defined carrier wave, which clearly shows up in the plots. Any finite-energy LW gets deformed while propagating: (b) represents the pulse depicted in (a) after it has traveled 2.78 km .

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

( 2 1 c n 2 2 t 2 ) ψ ( r , t ) = 4 π c n j ( r , t ) ,
G ( r , t , r , t ) = l G + ( r , t , r , t ) + ( 1 l ) G ( r , t , r , t ) ,
G ± ( r , t , r , t ) = δ ( t ( t R c n ) ) c n R
ψ ( r , t ) = d x 3 d t G + ( r , t , r , t ) j ( r , t ) .
ψ ( ρ , ζ ) = 1 2 π c n [ ( 0 d ω ( i π ) q H 0 2 ( ρ γ n 1 | ω | v ) e i ω ζ v ) + ( 0 d ω ( i π ) q H 0 1 ( ρ γ n 1 ω v ) e i ω ζ v ) ] ,
ψ ( ρ , ζ ) = 1 2 c n d ω i q H 0 1 ( ρ γ n 1 ω v ) e i ω ζ v
ψ ( ρ , z , t ) = { 2 q β n ζ 2 γ n 2 ρ 2 for ζ < γ n 1 ρ 0 elsewhere } ,
ψ X ( ρ , ζ ) = 0 d ω S ( ω ) J 0 ( ρ ω V V 2 c n 2 1 ) e i ω ζ V ,
X X ( ρ , ζ ) = V ( a V i ζ ) 2 + ( V 2 c n 2 1 ) ρ 2 .
X ̃ ( ρ , ζ ) = V ζ 2 ( V 2 c n 2 1 ) ρ 2 ,
ψ RS ( II ) ( ρ , z , t ) = 0 2 π d ϕ 0 D 2 d ρ ρ 1 2 π R { [ X ] ( z z ) R 2 + [ c n t X ] ( z z ) R } .
ψ ( ρ , ϕ , z , t ) = ν = [ 0 d ω ω c ω c d k z A ν ( k z , ω ) × J ν ( ρ ω 2 c 2 k z 2 ) e i k z z e i ω t e i ν ϕ ]
A ν ( k z , ω ) = μ = S ν μ ( ω ) δ [ ω ( V k z + b μ ) ] ,
ψ ( ρ , ζ , t ) = q c 0 d ω N 0 ( ρ ω V γ 1 ) cos ( ω V ζ ) ,
ψ ( ρ , z , t ) = { q β [ ζ 2 ρ 2 γ 2 ] 1 2 when 0 < ρ γ 1 < | ζ | 0 elsewhere } .
E ( ρ , ζ ) = q γ 2 Y ( ρ e ̂ ρ + ζ e ̂ z ) , B = q β γ 2 Y ρ e ̂ θ ,
Y [ ζ 2 ρ 2 γ 2 ] 3 2
S = c 4 π q 2 β γ 4 Y 2 ρ ( ρ e ̂ z ζ e ̂ ρ ) .
ζ z V t ; η z c t .
ψ ( ρ , ζ , η ) = ( V c ) 0 d σ σ d α J 0 ( ρ γ 2 σ 2 2 ( β 1 ) σ α ) exp [ i α η ] exp [ i σ ζ ] A ( α , σ ) ,
A ( α , σ ) = ϴ ( α α 0 ) V c e d α e a σ ,
ψ ( ρ , ζ , η ) = X V Z e α 0 Z ,
Z ( d i η ) c V + c ( a i ζ V X 1 ) .

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