Abstract

The space–time focusing of a (continuous) succession of localized X-shaped pulses is obtained by suitably integrating over their speed, i.e., over their axicon angle, thus generalizing a previous (discrete) approach. New superluminal wave pulses are first constructed and then tailored so that they become temporally focused at a chosen spatial point, where the wave field can reach very high intensities for a short time. Results of this kind may find applications in many fields, besides electromagnetism and optics, including acoustics, gravitation, and elementary particle physics.

© 2004 Optical Society of America

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  1. H. Bateman, Electrical and Optical Wave Motion (Cambridge U. Press, Cambridge, UK, 1915).
  2. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 356.
  3. R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1966), Vol. 2, p. 760.
  4. I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bi-directional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
    [CrossRef]
  5. R. Donnelly, R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. London, Ser. A 440, 541–565 (1993). See also Ref. 30 below.
    [CrossRef]
  6. J.-Y. Lu, J. F. Greenleaf, “Nondiffracting X-waves: exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
    [CrossRef]
  7. E. Recami, “On localized X-shaped superluminal solutions to Maxwell equations,” Phys. A 252, 586–610 (1998) and references therein.
    [CrossRef]
  8. R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
    [CrossRef]
  9. M. Zamboni-Rached, E. Recami, 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]
  10. For short review papers, see, for instance, E. Recami, “Superluminal motions? A bird’s-eye view of the experimental situation,” Found. Phys. 31, 1119–1135 (2001). Also see Ref. 11.
    [CrossRef]
  11. E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
    [CrossRef]
  12. See, e.g., J.-Y. Lu, H.-H. Zou, J. F. Greenleaf, “Biomedical ultrasound beam forming,” Ultrasound Med. Biol. 20, 403–428 (1994).
    [CrossRef] [PubMed]
  13. P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).
  14. M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” (e-print physics/0209101), Opt. Commun. 226, 15–23 (2003).
    [CrossRef]
  15. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003).
    [CrossRef]
  16. M. A. Porras, S. Trillo, C. Conti, P. Di Trapani, “Paraxial envelope X-waves,” Opt. Lett. 28, 1090–1092 (2003).
    [CrossRef] [PubMed]
  17. A. M. Attiya, “Transverse (TE) electromagnetic X-waves: propagation, scattering, diffraction and generation problems,” Ph.D. thesis (Cairo University, Cairo, 2001).
  18. See E. Recami, “Classical tachyons and possible applications,” Riv. Nuovo Cimento 9(6), 1–178 (1986) and references therein.
    [CrossRef]
  19. A. O. Barut, G. D. Maccarrone, E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
    [CrossRef]
  20. J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
    [CrossRef]
  21. J.-Y. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992). See also Ref. 22.
    [CrossRef]
  22. In the case of Ref. 21, the beam speed is larger than the sound (not of the light) speed in the considered medium.
  23. P. Saari, K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
    [CrossRef]
  24. D. Mugnai, A. Ranfagni, R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
    [CrossRef] [PubMed]
  25. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, S. Trillo, “Spontaneous formation of nonspreading X-shaped wavepackets” (e-print physics/0303083) in LANL Archives.
  26. M. Zamboni-Rached, E. Recami, F. Fontana, “Superluminal localized solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
    [CrossRef]
  27. M. Zamboni-Rached, F. Fontana, E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
    [CrossRef]
  28. M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
    [CrossRef]
  29. M. Zamboni-Rached, H. E. Hernández-Figueroa, “A rigorous analysis of localized wave propagation in optical fibers,” Opt. Commun. 191, 49–54 (2000).
    [CrossRef]
  30. I. M. Besieris, M. Abdel-Rahman, A. Shaarawi, A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Prog. Electromagn. Res. 19, 1–48 (1998).
    [CrossRef]
  31. S. He, J. Y. Lu, “Sidelobe reduction of limited-diffraction beams with Chebyshev aperture apodization,” J. Acoust. Soc. Am. 107, 3556–3559 (2000).
    [CrossRef] [PubMed]
  32. J.-Y. Lu, S. He, “High frame rate imaging with a small number of array elements,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 1416–1421 (1999).
    [CrossRef]
  33. J.-Y. Lu, “Experimental study of high frame rate imaging with limited-diffraction beams,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 84–97 (1998).
    [CrossRef]
  34. J.-Y. Lu, “Producing bowtie limited-diffraction beams with synthetic array experiments,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 893–900 (1996).
    [CrossRef]
  35. J.-Y. Lu, J. F. Greenleaf, “Producing deep depth of field and depth-independent resolution in NDE with limited-diffraction beams,” Ultrason. Imaging 15, 134–149 (1993).
    [CrossRef] [PubMed]
  36. A. A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2287–2295 (1998).
    [CrossRef]
  37. M. Abdel-Rahman, I. M. Besieris, A. M. Shaarawi, “A comparative study on the reconstruction of localized pulses,” in Proceedings of the IEEE Southeast Conference (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 113–117.
  38. A. M. Shaarawi, I. M. Besieris, T. M. Said, “Temporal focusing by use of composite X-waves,” J. Opt. Soc. Am. A 20, 1658–1665 (2003).
    [CrossRef]
  39. See, e.g., C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “A general formulation for the analysis of scalar limited-diffraction beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
    [CrossRef]
  40. H. Sõnajalg, M. Rätsep, P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. 22, 310–312 (1997).
    [CrossRef] [PubMed]
  41. J.-Y. Lu, “An X-wave transform,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47, 1472–1481 (2000).
    [CrossRef]
  42. P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).
  43. J. Salo, A. T. Friberg, M. Salomaa, “Orthogonal X-waves,” J. Phys. A. 34, 9319–9327 (2001).
    [CrossRef]
  44. M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernandez, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
    [CrossRef]
  45. I. S. Gradshteyn, I. M. Ryzhik, Integrals, Series and Products, 4th ed. (Academic, New York, 1965).
  46. A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
    [CrossRef]
  47. J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
    [CrossRef]
  48. P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in Time’s Arrows, Quantum Measurements and Superluminal Behavior, D. Mugnai, A. Ranfagni, L. S. Shulman, eds. (C.N.R., Rome, 2001), pp. 37–48.
  49. D. Mugnai, A. Ranfagni, R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A 311, 77–81 (2003).
    [CrossRef]
  50. G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426–435 (1952).
    [CrossRef]

2003

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” (e-print physics/0209101), Opt. Commun. 226, 15–23 (2003).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

M. A. Porras, S. Trillo, C. Conti, P. Di Trapani, “Paraxial envelope X-waves,” Opt. Lett. 28, 1090–1092 (2003).
[CrossRef] [PubMed]

M. Zamboni-Rached, F. Fontana, E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
[CrossRef]

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

See, e.g., C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “A general formulation for the analysis of scalar limited-diffraction beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernandez, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[CrossRef]

D. Mugnai, A. Ranfagni, R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A 311, 77–81 (2003).
[CrossRef]

2002

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

M. Zamboni-Rached, E. Recami, 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

For short review papers, see, for instance, E. Recami, “Superluminal motions? A bird’s-eye view of the experimental situation,” Found. Phys. 31, 1119–1135 (2001). Also see Ref. 11.
[CrossRef]

M. Zamboni-Rached, E. Recami, F. Fontana, “Superluminal localized solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
[CrossRef]

J. Salo, A. T. Friberg, M. Salomaa, “Orthogonal X-waves,” J. Phys. A. 34, 9319–9327 (2001).
[CrossRef]

2000

D. Mugnai, A. Ranfagni, R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
[CrossRef] [PubMed]

J.-Y. Lu, “An X-wave transform,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47, 1472–1481 (2000).
[CrossRef]

M. Zamboni-Rached, H. E. Hernández-Figueroa, “A rigorous analysis of localized wave propagation in optical fibers,” Opt. Commun. 191, 49–54 (2000).
[CrossRef]

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

1999

J.-Y. Lu, S. He, “High frame rate imaging with a small number of array elements,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 1416–1421 (1999).
[CrossRef]

1998

J.-Y. Lu, “Experimental study of high frame rate imaging with limited-diffraction beams,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 84–97 (1998).
[CrossRef]

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

A. A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2287–2295 (1998).
[CrossRef]

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

1997

P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

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

P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

H. Sõnajalg, M. Rätsep, P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. 22, 310–312 (1997).
[CrossRef] [PubMed]

A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

1996

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

J.-Y. Lu, “Producing bowtie limited-diffraction beams with synthetic array experiments,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 893–900 (1996).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

1994

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

1993

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

R. Donnelly, R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. London, Ser. A 440, 541–565 (1993). See also Ref. 30 below.
[CrossRef]

J.-Y. Lu, J. F. Greenleaf, “Producing deep depth of field and depth-independent resolution in NDE with limited-diffraction beams,” Ultrason. Imaging 15, 134–149 (1993).
[CrossRef] [PubMed]

1992

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

J.-Y. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992). See also Ref. 22.
[CrossRef]

1989

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

1986

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

1982

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

1952

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426–435 (1952).
[CrossRef]

Abdel-Rahman, M.

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

A. A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2287–2295 (1998).
[CrossRef]

M. Abdel-Rahman, I. M. Besieris, A. M. Shaarawi, “A comparative study on the reconstruction of localized pulses,” in Proceedings of the IEEE Southeast Conference (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 113–117.

Attiya, A. M.

A. M. Attiya, “Transverse (TE) electromagnetic X-waves: propagation, scattering, diffraction and generation problems,” Ph.D. thesis (Cairo University, Cairo, 2001).

Barut, A. O.

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

Bateman, H.

H. Bateman, Electrical and Optical Wave Motion (Cambridge U. Press, Cambridge, UK, 1915).

Besieris, I. M.

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

A. A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2287–2295 (1998).
[CrossRef]

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

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

M. Abdel-Rahman, I. M. Besieris, A. M. Shaarawi, “A comparative study on the reconstruction of localized pulses,” in Proceedings of the IEEE Southeast Conference (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 113–117.

Chatzipetros, A.

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

Chatzipetros, A. A.

A. A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2287–2295 (1998).
[CrossRef]

Conti, C.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

M. A. Porras, S. Trillo, C. Conti, P. Di Trapani, “Paraxial envelope X-waves,” Opt. Lett. 28, 1090–1092 (2003).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, S. Trillo, “Spontaneous formation of nonspreading X-shaped wavepackets” (e-print physics/0303083) in LANL Archives.

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1966), Vol. 2, p. 760.

Dartora, C. A.

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

See, e.g., C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “A general formulation for the analysis of scalar limited-diffraction beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

Di Trapani, P.

M. A. Porras, S. Trillo, C. Conti, P. Di Trapani, “Paraxial envelope X-waves,” Opt. Lett. 28, 1090–1092 (2003).
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, S. Trillo, “Spontaneous formation of nonspreading X-shaped wavepackets” (e-print physics/0303083) in LANL Archives.

Donnelly, R.

R. Donnelly, R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. London, Ser. A 440, 541–565 (1993). See also Ref. 30 below.
[CrossRef]

Fagerholm, J.

A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Fontana, F.

M. Zamboni-Rached, F. Fontana, E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
[CrossRef]

M. Zamboni-Rached, E. Recami, F. Fontana, “Superluminal localized solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
[CrossRef]

Friberg, A. T.

J. Salo, A. T. Friberg, M. Salomaa, “Orthogonal X-waves,” J. Phys. A. 34, 9319–9327 (2001).
[CrossRef]

A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Integrals, Series and Products, 4th ed. (Academic, New York, 1965).

Greenleaf, J. F.

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

J.-Y. Lu, J. F. Greenleaf, “Producing deep depth of field and depth-independent resolution in NDE with limited-diffraction beams,” Ultrason. Imaging 15, 134–149 (1993).
[CrossRef] [PubMed]

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

J.-Y. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992). See also Ref. 22.
[CrossRef]

He, S.

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

J.-Y. Lu, S. He, “High frame rate imaging with a small number of array elements,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 1416–1421 (1999).
[CrossRef]

Hernandez, H. E.

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernandez, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[CrossRef]

Hernández-Figueroa, H. E.

See, e.g., C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “A general formulation for the analysis of scalar limited-diffraction beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” (e-print physics/0209101), Opt. Commun. 226, 15–23 (2003).
[CrossRef]

M. Zamboni-Rached, E. Recami, 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]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

M. Zamboni-Rached, H. E. Hernández-Figueroa, “A rigorous analysis of localized wave propagation in optical fibers,” Opt. Commun. 191, 49–54 (2000).
[CrossRef]

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1966), Vol. 2, p. 760.

Huttunen, J.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Jedrkiewicz, O.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, S. Trillo, “Spontaneous formation of nonspreading X-shaped wavepackets” (e-print physics/0303083) in LANL Archives.

Lu, J. Y.

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

Lu, J.-Y.

J.-Y. Lu, “An X-wave transform,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47, 1472–1481 (2000).
[CrossRef]

J.-Y. Lu, S. He, “High frame rate imaging with a small number of array elements,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 1416–1421 (1999).
[CrossRef]

J.-Y. Lu, “Experimental study of high frame rate imaging with limited-diffraction beams,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 84–97 (1998).
[CrossRef]

J.-Y. Lu, “Producing bowtie limited-diffraction beams with synthetic array experiments,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 893–900 (1996).
[CrossRef]

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

J.-Y. Lu, J. F. Greenleaf, “Producing deep depth of field and depth-independent resolution in NDE with limited-diffraction beams,” Ultrason. Imaging 15, 134–149 (1993).
[CrossRef] [PubMed]

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

J.-Y. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992). See also Ref. 22.
[CrossRef]

Maccarrone, G. D.

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

Morgan, D. P.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Mugnai, D.

D. Mugnai, A. Ranfagni, R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A 311, 77–81 (2003).
[CrossRef]

D. Mugnai, A. Ranfagni, R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
[CrossRef] [PubMed]

Nóbrega, K. Z.

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” (e-print physics/0209101), Opt. Commun. 226, 15–23 (2003).
[CrossRef]

See, e.g., C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “A general formulation for the analysis of scalar limited-diffraction beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernandez, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

Piskarskas, A.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, S. Trillo, “Spontaneous formation of nonspreading X-shaped wavepackets” (e-print physics/0303083) in LANL Archives.

Porras, M. A.

Ranfagni, A.

D. Mugnai, A. Ranfagni, R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A 311, 77–81 (2003).
[CrossRef]

D. Mugnai, A. Ranfagni, R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
[CrossRef] [PubMed]

Rätsep, M.

Recami, E.

See, e.g., C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “A general formulation for the analysis of scalar limited-diffraction beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernandez, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[CrossRef]

M. Zamboni-Rached, F. Fontana, E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
[CrossRef]

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” (e-print physics/0209101), Opt. Commun. 226, 15–23 (2003).
[CrossRef]

M. Zamboni-Rached, E. Recami, 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]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

M. Zamboni-Rached, E. Recami, F. Fontana, “Superluminal localized solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
[CrossRef]

For short review papers, see, for instance, E. Recami, “Superluminal motions? A bird’s-eye view of the experimental situation,” Found. Phys. 31, 1119–1135 (2001). Also see Ref. 11.
[CrossRef]

E. Recami, “On localized X-shaped superluminal solutions to Maxwell equations,” Phys. A 252, 586–610 (1998) and references therein.
[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, E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
[CrossRef]

Reivelt, K.

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

Ruggeri, R.

D. Mugnai, A. Ranfagni, R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A 311, 77–81 (2003).
[CrossRef]

D. Mugnai, A. Ranfagni, R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
[CrossRef] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Integrals, Series and Products, 4th ed. (Academic, New York, 1965).

Saari, P.

H. Sõnajalg, M. Rätsep, P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. 22, 310–312 (1997).
[CrossRef] [PubMed]

P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

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

P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in Time’s Arrows, Quantum Measurements and Superluminal Behavior, D. Mugnai, A. Ranfagni, L. S. Shulman, eds. (C.N.R., Rome, 2001), pp. 37–48.

Said, T. M.

Salo, J.

J. Salo, A. T. Friberg, M. Salomaa, “Orthogonal X-waves,” J. Phys. A. 34, 9319–9327 (2001).
[CrossRef]

Salomaa, M.

J. Salo, A. T. Friberg, M. Salomaa, “Orthogonal X-waves,” J. Phys. A. 34, 9319–9327 (2001).
[CrossRef]

Salomaa, M. M.

A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Shaarawi, A.

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

Shaarawi, A. M.

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

A. A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2287–2295 (1998).
[CrossRef]

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

M. Abdel-Rahman, I. M. Besieris, A. M. Shaarawi, “A comparative study on the reconstruction of localized pulses,” in Proceedings of the IEEE Southeast Conference (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 113–117.

Sõnajalg, H.

P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

H. Sõnajalg, M. Rätsep, P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. 22, 310–312 (1997).
[CrossRef] [PubMed]

P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 356.

Toraldo di Francia, G.

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426–435 (1952).
[CrossRef]

Trillo, S.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

M. A. Porras, S. Trillo, C. Conti, P. Di Trapani, “Paraxial envelope X-waves,” Opt. Lett. 28, 1090–1092 (2003).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, S. Trillo, “Spontaneous formation of nonspreading X-shaped wavepackets” (e-print physics/0303083) in LANL Archives.

Trull, J.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, S. Trillo, “Spontaneous formation of nonspreading X-shaped wavepackets” (e-print physics/0303083) in LANL Archives.

Valiulis, G.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, S. Trillo, “Spontaneous formation of nonspreading X-shaped wavepackets” (e-print physics/0303083) in LANL Archives.

Zamboni-Rached, M.

M. Zamboni-Rached, F. Fontana, E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
[CrossRef]

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” (e-print physics/0209101), Opt. Commun. 226, 15–23 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernandez, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[CrossRef]

See, e.g., C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “A general formulation for the analysis of scalar limited-diffraction beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

M. Zamboni-Rached, E. Recami, 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]

M. Zamboni-Rached, E. Recami, F. Fontana, “Superluminal localized solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
[CrossRef]

M. Zamboni-Rached, H. E. Hernández-Figueroa, “A rigorous analysis of localized wave propagation in optical fibers,” Opt. Commun. 191, 49–54 (2000).
[CrossRef]

Ziolkowski, R. W.

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

R. Donnelly, R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. London, Ser. A 440, 541–565 (1993). See also Ref. 30 below.
[CrossRef]

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

Zou, H.-H.

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

Eur. Phys. J. D

M. Zamboni-Rached, E. Recami, 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.

For short review papers, see, for instance, E. Recami, “Superluminal motions? A bird’s-eye view of the experimental situation,” Found. Phys. 31, 1119–1135 (2001). Also see Ref. 11.
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control

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

J.-Y. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992). See also Ref. 22.
[CrossRef]

J.-Y. Lu, S. He, “High frame rate imaging with a small number of array elements,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 1416–1421 (1999).
[CrossRef]

J.-Y. Lu, “Experimental study of high frame rate imaging with limited-diffraction beams,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 84–97 (1998).
[CrossRef]

J.-Y. Lu, “Producing bowtie limited-diffraction beams with synthetic array experiments,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 893–900 (1996).
[CrossRef]

J.-Y. Lu, “An X-wave transform,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47, 1472–1481 (2000).
[CrossRef]

J. Acoust. Soc. Am.

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

A. A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, M. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2287–2295 (1998).
[CrossRef]

J. Math. Phys.

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

J. Opt. Soc. Am. A

J. Phys. A.

J. Salo, A. T. Friberg, M. Salomaa, “Orthogonal X-waves,” J. Phys. A. 34, 9319–9327 (2001).
[CrossRef]

Laser Phys.

P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

Nuovo Cimento A

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

Nuovo Cimento, Suppl.

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426–435 (1952).
[CrossRef]

Opt. Commun.

See, e.g., C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “A general formulation for the analysis of scalar limited-diffraction beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernandez, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[CrossRef]

A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

M. Zamboni-Rached, H. E. Hernández-Figueroa, “A rigorous analysis of localized wave propagation in optical fibers,” Opt. Commun. 191, 49–54 (2000).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” (e-print physics/0209101), Opt. Commun. 226, 15–23 (2003).
[CrossRef]

Opt. Lett.

Phys. A

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

Phys. Lett. A

D. Mugnai, A. Ranfagni, R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A 311, 77–81 (2003).
[CrossRef]

Phys. Rev. E

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

M. Zamboni-Rached, E. Recami, F. Fontana, “Superluminal localized solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
[CrossRef]

M. Zamboni-Rached, F. Fontana, E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

Phys. Rev. Lett.

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

D. Mugnai, A. Ranfagni, R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

Proc. R. Soc. London, Ser. A

R. Donnelly, R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. London, Ser. A 440, 541–565 (1993). See also Ref. 30 below.
[CrossRef]

Prog. Electromagn. Res.

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

Riv. Nuovo Cimento

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

Ultrason. Imaging

J.-Y. Lu, J. F. Greenleaf, “Producing deep depth of field and depth-independent resolution in NDE with limited-diffraction beams,” Ultrason. Imaging 15, 134–149 (1993).
[CrossRef] [PubMed]

Ultrasound Med. Biol.

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

Other

H. Bateman, Electrical and Optical Wave Motion (Cambridge U. Press, Cambridge, UK, 1915).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 356.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1966), Vol. 2, p. 760.

A. M. Attiya, “Transverse (TE) electromagnetic X-waves: propagation, scattering, diffraction and generation problems,” Ph.D. thesis (Cairo University, Cairo, 2001).

M. Abdel-Rahman, I. M. Besieris, A. M. Shaarawi, “A comparative study on the reconstruction of localized pulses,” in Proceedings of the IEEE Southeast Conference (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 113–117.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, S. Trillo, “Spontaneous formation of nonspreading X-shaped wavepackets” (e-print physics/0303083) in LANL Archives.

In the case of Ref. 21, the beam speed is larger than the sound (not of the light) speed in the considered medium.

P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in Time’s Arrows, Quantum Measurements and Superluminal Behavior, D. Mugnai, A. Ranfagni, L. S. Shulman, eds. (C.N.R., Rome, 2001), pp. 37–48.

I. S. Gradshteyn, I. M. Ryzhik, Integrals, Series and Products, 4th ed. (Academic, New York, 1965).

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Figures (7)

Fig. 1
Fig. 1

Space–time evolution of the superluminal pulse represented by Eq. (14); the chosen parameter values are a=10-12 s, Vmin=1.001c, and Vmax=1.005c, and the focusing point is at zf=200 cm. One can see that this solution is associated with a rather good spatiotemporal focusing: The field amplitude at z=zf is 40.82 times larger than the initial one. The field amplitude is normalized at the space–time point ρ=0, z=zf, t=tf.

Fig. 2
Fig. 2

Space–time evolution of the superluminal pulse represented by Eq. (20), for the same values of the parameters as in Fig. 1 (namely, a=10-12 s, Vmin=1.001c, and Vmax=1.005c); the focusing point is again at zf=200 cm. This solution, too, is associated with a rather good spatiotemporal focusing: The field amplitude at z=zf is 40.65 times greater than the initial one. The field amplitude is normalized at the space–time point ρ=0, z=zf, t=tf.

Fig. 3
Fig. 3

Space–time evolution of the superluminal pulse represented by Eq. (32). Now the parameters have the values m=1, a=10-12 s (and therefore ωc=1 GHz), Vmin=1.001c, Vmax=1.005c, and the focusing point is again at zf=200 cm. This solution, too, is associated with a very good spatiotemporal focusing: The field amplitude at z=zf is 1000 times higher than the initial one. Once more, the field amplitude is normalized at the space–time point ρ=0, z=zf, t=tf.

Fig. 4
Fig. 4

Surface plots of (a) the initial excitation |Ψ|2 on the aperture plane z=0 and (b) the source-free pulse at the focusing point z=zf=200 cm. The spatiotemporally focused pulse corresponds to a=10-12 s, Vmin=1.001c, and Vmax=1.005c. The radius of the aperture is chosen equal to 20 cm.

Fig. 5
Fig. 5

Axial profiles of the field |Ψ|2 radiated from a finite aperture at distances z=(a) 50, (b) 150, (c) 200, (d) 250 cm. All the other parameters are chosen as in Fig. 4.

Fig. 6
Fig. 6

Surface plots of (a) the initial excitation |Ψ|2 on the aperture plane z=0 and (b) the source-free pulse at the focusing point z=zf=300 cm. The spatiotemporally focused pulse has a=10-12 s, Vmin=1.001c, and Vmax=1.005c. The radius of the aperture is chosen equal to 20 cm.

Fig. 7
Fig. 7

Axial profiles of the field |Ψ|2 radiated from a finite aperture at distances z=(a) 75, (b) 225, (c) 285, (d) 375 cm. All the other parameters are chosen as in Fig. 6.

Equations (46)

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ψ(ρ, z-Vt)=-S(ω)J0ωc ρsinθ×expi ωccosθz-ccosθ tdω,
z=Vn(t-tn).
tf=zf/V1.
tn=(1/V1-1/Vn)zf.
Ψ(ρ, z, t)=n=1NAnψn(ρ, z-Vn(t-tn)),
Ψ(ρ, z, t)=VminVmaxdVA(V)×ψρ, z-Vt-1Vmin-1Vzf,
V_VminVmaxA(V)VdVVminVmaxA(V)dV>c.
X(ρ, z-Vt)=V[aV-i(z-Vt)]2+V2c2-1ρ21/2.
ψ(ρ, z, t)X=VaV-iz-Vt-1Vmin-1Vzf2+(V2/c2-1)ρ21/2.
Ψ(ρ, z, t)=VminVmaxVA(V)PV2+QV+RdV,
P=a+it-zfVmin2+ρ2c2,
Q=2t-zfVmin-ai(z-zf),
R=[-(z-zf)2-ρ2].
A(V)=1.
Ψ(ρ, z, t)=VminVmaxVPV2+QV+RdV.
Ψ(ρ, z, t)
=PVmax2+QVmax+R-PVmin2+QVmin+RP+Q2P3/2ln 2P(PVmin2+QVmin+R)+2PVmin+Q2P(PVmax2+QVmax+R)+2PVmax+Q,
A(V)=1/V,
Ψ(ρ, z, t)=VminVmax1PV2+QV+RdV.
Ψ(ρ, z, t)
=1Pln2P(PVmax2+QVmax+R)+2PVmax+Q2P(PVmin2+QVmin+R)+2PVmin+Q.
A(V)=1/V2
Ψ(ρ, z, t)=VminVmax1VPV2+QV+RdV.
Ψ(ρ, z, t)=1RlnVmax[2R+QVmin+2R(PVmin2+QVmin+R)]Vmin[2R+QVmax+2R(PVmax2+QVmax+R)].
A(V)=1/V3,
Ψ(ρ, z, t)=VminVmax1V2PV2+QV+RdV.
Ψ(ρ, z, t)=PVmin2+QVmin+RRVmin-PVmax2+QVmax+RRVmax+Q2R3/2lnVmin[2R(PVmax2+QVmax+R)+2R+QVmax]Vmax[2R(PVmin2+QVmin+R)+2R+QVmin].
ωc=m/a,
m=1±Δω±/ωc-ln(1±Δω±/ωc),
X(m)(ρ, z-Vt)0ωmexp(-aω)J0ωc ρsinθ×expi ωccosθz-ccosθ tdω=immXtm.
Ψ(ρ, z, t)=immtmVminVmaxVA(V)PV2+QV+RdV,
A(V)=1.
Ψ(ρ, z, t)
=immtmPVmax2+QVmax+R-PVmin2+QVmin+RP+Q2P3/2ln2P(PVmin2+QVmin+R)+2PVmin+Q2P(PVmax2+QVmax+R)+2PVmax+Q.
A(V)=1/V
Ψ(ρ, z, t)
=immtm1P×ln2P(PVmax2+QVmax+R)+2PVmax+Q2P(PVmin2+QVmin+R)+2PVmin+Q.
A(V)=1/V2,
Ψ(ρ, z, t)=immtm1RlnVmax[2R+QVmin+2R(PVmin2+QVmin+R)]Vmin[2R+QVmax+2R(PVmax2+QVmax+R)],
A(V)=1/V3
Ψ(ρ, z, t)=immtmPVmin2+QVmin+RRVmin-PVmax2+QVmax+RRVmax+Q2R3/2lnVmin[2R(PVmax2+QVmax+R)+2R+QVmax]Vmax[2R(PVmin2+QVmin+R)+2R+QVmin].
ψ(ρ, z-Vt)=-S(ω)J0ωV ρV2c2-11/2×expi ωV (z-Vt)dω,
Ψ(ρ, z, t)=VminVmaxdVA(V)-S(ω)J0ωV ρV2c2-11/2×expi ωVz-Vt-1Vmin-1Vzfdω,
Ψ(ρ, z, t)=VminVmaxdV-B(ω, V)J0ωV ρV2c2-11/2×expi ωV (z-Vt)×expiω1Vmin-1Vzfdω.
S¯(ω, V)B(ω, V)expiω1Vmin-1Vzf
ΨRS(II)(ρ, z, t)=02πdϕ0D/2dρρ×12πR[Ψ]+[ctΨ] z-zR.

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