Abstract

It is shown that highly focused pulses can be shaped by exciting a finite aperture with a spread-out pulse train of X waves. The basis of the proposed scheme is that the peaks of X waves, characterized by different apex angles, travel at different velocities. This property allows one to vary the temporal starting points of the initial excitations of a sequence of X waves so that all their peaks meet at a chosen focusing point. It is demonstrated that this simple criterion can be effective in producing a highly focused, composite X-wave pulse that exhibits a slower decay behavior than the individual X-wave components used in synthesizing it.

© 2003 Optical Society of America

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  1. J-Y. Lu, J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realization,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
    [CrossRef]
  2. J-Y. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
    [CrossRef] [PubMed]
  3. I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, A. A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Prog. Electromagn. Res. 19, 1–48 (1998).
    [CrossRef]
  4. R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
    [CrossRef]
  5. P. Saari, K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
    [CrossRef]
  6. E. Recami, “On the localized ‘X-shaped’ superluminal solutions to Maxwell’s equations,” Physica A 252, 586–610 (1998).
    [CrossRef]
  7. D. Mugnai, A. Ranfagni, R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
    [CrossRef] [PubMed]
  8. M. Zamboni-Rached, E. Recami, H. E. Hernandez-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]
  9. 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]
  10. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  11. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  12. A. A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, M. A. Abdel-Rahman, “Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,” J. Acoust. Soc. Am. 103, 2287–2295 (1998).
    [CrossRef]
  13. A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, “Generation of an Approximate TE Electromagnetic X-wave in free space using Huygens’s aperture,” Proceedings of the 16th National Radio Science Conference, NRSC’99 (Ain Shams University, Cairo, Egypt, 1999), pp. B3.1–11.
  14. A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Acoustic X-wave reflection and transmission at a planar interface: spectral analysis,” J. Acoust. Soc. Am. 107, 70–86 (2000).
    [CrossRef] [PubMed]
  15. J-Y. Lu, J. F. Greenleaf, “Diffraction-limited beams and their applications for ultrasonic imaging and tissue characterization,” in New Developments in Ultrasonic Transducers and Transducer Systems, F. L. Lizzi, ed., Proc. SPIE1733, 92–119 (1992).
    [CrossRef]
  16. J-Y. Lu, M. Fatemi, J. F. Greenleaf, “Pulsed-echo imaging with X wave,” in Acoustical Imaging, P. Tortoli, L. Masotti, eds., (Plenum, New York, 1996), Vol. 22, pp. 191–196.
  17. A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, I. M. Besieris, “Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,” Prog. Electromagn. Res. 30, 191–211 (2000).
    [CrossRef]
  18. A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air-dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–250 (2000).
    [CrossRef]
  19. A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” Proceedings of the 17th National Radio Science Conference, NRSC’2000 (Minufiya University, Minufiya, Egypt, 2000), pp. B11.1–12.
  20. A. M. Attiya, “Transverse (TE) Electromagnetic X-waves: propagation, scattering, diffraction and generation problems,” Ph.D. thesis (Cairo University, Cairo, Egypt, 2001).
  21. H. Sõnajalg, P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators,” Opt. Lett. 21, 1162–1164 (1996).
    [CrossRef] [PubMed]
  22. M. A. Porras, I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217, 257–64 (2002).
    [CrossRef]
  23. A. M. Shaarawi, I. M. Besieris, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A 33, 7227–7254 (2000).
    [CrossRef]
  24. J. Y. Lu, “An X-wave transform,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47, 1472–1481 (2000).
    [CrossRef]
  25. J. Salo, A. T. Friberg, M. Salomaa, “Orthogonal X-waves,” J. Phys. A 34, 9319–9327 (2001).
    [CrossRef]
  26. J. Salo, J. Fagerholm, A. T. Friberg, M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
    [CrossRef]
  27. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  28. T. K. Song, J-Y. Lu, J. F. Greenleaf, “Modified X waves with improved field properties,” Ultrason. Imaging 15, 36–47 (1993).
    [PubMed]

2002 (2)

M. Zamboni-Rached, E. Recami, H. E. Hernandez-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. A. Porras, I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217, 257–64 (2002).
[CrossRef]

2001 (1)

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

2000 (7)

J. Salo, J. Fagerholm, A. T. Friberg, M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[CrossRef]

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

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

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Acoustic X-wave reflection and transmission at a planar interface: spectral analysis,” J. Acoust. Soc. Am. 107, 70–86 (2000).
[CrossRef] [PubMed]

A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, I. M. Besieris, “Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,” Prog. Electromagn. Res. 30, 191–211 (2000).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air-dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–250 (2000).
[CrossRef]

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

1998 (3)

E. Recami, “On the localized ‘X-shaped’ superluminal solutions to Maxwell’s equations,” Physica A 252, 586–610 (1998).
[CrossRef]

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

1997 (1)

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

1996 (2)

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]

H. Sõnajalg, P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators,” Opt. Lett. 21, 1162–1164 (1996).
[CrossRef] [PubMed]

1993 (2)

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

T. K. Song, J-Y. Lu, J. F. Greenleaf, “Modified X waves with improved field properties,” Ultrason. Imaging 15, 36–47 (1993).
[PubMed]

1992 (2)

J-Y. Lu, J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realization,” 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).
[CrossRef] [PubMed]

1987 (2)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

Abdel-Rahman, M.

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

Abdel-Rahman, M. A.

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

Attiya, A. M.

A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, I. M. Besieris, “Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,” Prog. Electromagn. Res. 30, 191–211 (2000).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air-dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–250 (2000).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Acoustic X-wave reflection and transmission at a planar interface: spectral analysis,” J. Acoust. Soc. Am. 107, 70–86 (2000).
[CrossRef] [PubMed]

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

A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, “Generation of an Approximate TE Electromagnetic X-wave in free space using Huygens’s aperture,” Proceedings of the 16th National Radio Science Conference, NRSC’99 (Ain Shams University, Cairo, Egypt, 1999), pp. B3.1–11.

A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” Proceedings of the 17th National Radio Science Conference, NRSC’2000 (Minufiya University, Minufiya, Egypt, 2000), pp. B11.1–12.

Besieris, I. M.

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

A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, I. M. Besieris, “Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,” Prog. Electromagn. Res. 30, 191–211 (2000).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air-dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–250 (2000).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Acoustic X-wave reflection and transmission at a planar interface: spectral analysis,” J. Acoust. Soc. Am. 107, 70–86 (2000).
[CrossRef] [PubMed]

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, A. 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. A. 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 the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[CrossRef]

Chatzipetros, A. A.

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

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

El-Diwany, E. A.

A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, I. M. Besieris, “Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,” Prog. Electromagn. Res. 30, 191–211 (2000).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Acoustic X-wave reflection and transmission at a planar interface: spectral analysis,” J. Acoust. Soc. Am. 107, 70–86 (2000).
[CrossRef] [PubMed]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air-dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–250 (2000).
[CrossRef]

A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, “Generation of an Approximate TE Electromagnetic X-wave in free space using Huygens’s aperture,” Proceedings of the 16th National Radio Science Conference, NRSC’99 (Ain Shams University, Cairo, Egypt, 1999), pp. B3.1–11.

A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” Proceedings of the 17th National Radio Science Conference, NRSC’2000 (Minufiya University, Minufiya, Egypt, 2000), pp. B11.1–12.

Fagerholm, J.

J. Salo, J. Fagerholm, A. T. Friberg, M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[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]

Fatemi, M.

J-Y. Lu, M. Fatemi, J. F. Greenleaf, “Pulsed-echo imaging with X wave,” in Acoustical Imaging, P. Tortoli, L. Masotti, eds., (Plenum, New York, 1996), Vol. 22, pp. 191–196.

Friberg, A. T.

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

J. Salo, J. Fagerholm, A. T. Friberg, M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[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]

Gonzalo, I.

M. A. Porras, I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217, 257–64 (2002).
[CrossRef]

Goodman, J. W.

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

Greenleaf, J. F.

T. K. Song, J-Y. Lu, J. F. Greenleaf, “Modified X waves with improved field properties,” Ultrason. Imaging 15, 36–47 (1993).
[PubMed]

J-Y. Lu, J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realization,” 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).
[CrossRef] [PubMed]

J-Y. Lu, J. F. Greenleaf, “Diffraction-limited beams and their applications for ultrasonic imaging and tissue characterization,” in New Developments in Ultrasonic Transducers and Transducer Systems, F. L. Lizzi, ed., Proc. SPIE1733, 92–119 (1992).
[CrossRef]

J-Y. Lu, M. Fatemi, J. F. Greenleaf, “Pulsed-echo imaging with X wave,” in Acoustical Imaging, P. Tortoli, L. Masotti, eds., (Plenum, New York, 1996), Vol. 22, pp. 191–196.

Hernandez-Figueroa, H. E.

M. Zamboni-Rached, E. Recami, H. E. Hernandez-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]

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]

Lu, J. Y.

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

Lu, J-Y.

T. K. Song, J-Y. Lu, J. F. Greenleaf, “Modified X waves with improved field properties,” Ultrason. Imaging 15, 36–47 (1993).
[PubMed]

J-Y. Lu, J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realization,” 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).
[CrossRef] [PubMed]

J-Y. Lu, J. F. Greenleaf, “Diffraction-limited beams and their applications for ultrasonic imaging and tissue characterization,” in New Developments in Ultrasonic Transducers and Transducer Systems, F. L. Lizzi, ed., Proc. SPIE1733, 92–119 (1992).
[CrossRef]

J-Y. Lu, M. Fatemi, J. F. Greenleaf, “Pulsed-echo imaging with X wave,” in Acoustical Imaging, P. Tortoli, L. Masotti, eds., (Plenum, New York, 1996), Vol. 22, pp. 191–196.

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

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]

Mugnai, D.

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

Porras, M. A.

M. A. Porras, I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217, 257–64 (2002).
[CrossRef]

Ranfagni, A.

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

Recami, E.

M. Zamboni-Rached, E. Recami, H. E. Hernandez-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 the localized ‘X-shaped’ superluminal solutions to Maxwell’s equations,” Physica A 252, 586–610 (1998).
[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, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000).
[CrossRef] [PubMed]

Saari, P.

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

H. Sõnajalg, P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators,” Opt. Lett. 21, 1162–1164 (1996).
[CrossRef] [PubMed]

Salo, J.

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

J. Salo, J. Fagerholm, A. T. Friberg, M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[CrossRef]

Salomaa, M.

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

Salomaa, M. M.

J. Salo, J. Fagerholm, A. T. Friberg, M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[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. M.

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

A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, I. M. Besieris, “Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,” Prog. Electromagn. Res. 30, 191–211 (2000).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Acoustic X-wave reflection and transmission at a planar interface: spectral analysis,” J. Acoust. Soc. Am. 107, 70–86 (2000).
[CrossRef] [PubMed]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air-dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–250 (2000).
[CrossRef]

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

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A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, “Generation of an Approximate TE Electromagnetic X-wave in free space using Huygens’s aperture,” Proceedings of the 16th National Radio Science Conference, NRSC’99 (Ain Shams University, Cairo, Egypt, 1999), pp. B3.1–11.

A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” Proceedings of the 17th National Radio Science Conference, NRSC’2000 (Minufiya University, Minufiya, Egypt, 2000), pp. B11.1–12.

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[CrossRef]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, E. A. El-Diwany, “Acoustic X-wave reflection and transmission at a planar interface: spectral analysis,” J. Acoust. Soc. Am. 107, 70–86 (2000).
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A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” Proceedings of the 17th National Radio Science Conference, NRSC’2000 (Minufiya University, Minufiya, Egypt, 2000), pp. B11.1–12.

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

J-Y. Lu, J. F. Greenleaf, “Diffraction-limited beams and their applications for ultrasonic imaging and tissue characterization,” in New Developments in Ultrasonic Transducers and Transducer Systems, F. L. Lizzi, ed., Proc. SPIE1733, 92–119 (1992).
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A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, “Generation of an Approximate TE Electromagnetic X-wave in free space using Huygens’s aperture,” Proceedings of the 16th National Radio Science Conference, NRSC’99 (Ain Shams University, Cairo, Egypt, 1999), pp. B3.1–11.

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Figures (7)

Fig. 1
Fig. 1

Schematic representation of the temporal focusing scheme. X waves traveling at different speeds arrive simultaneously and add up at the focusing point zf.

Fig. 2
Fig. 2

Behavior of a train of Nl=1001 source-free X waves designed to focus at zf=3.4374 m. The parameters are a0=10-6 m, z1=-1 m and 0.5°ξl1.5°. The plots show the axial profile of the CXW pulse at ct=1.0012, 2.3749, 4.4372, and 6.1559 m.  

Fig. 3
Fig. 3

Surface plots of the time dependence of (a) the initial source-free CXW pulse at z=0 and (b) the focused pulse at zf=3.4374 m.

Fig. 4
Fig. 4

Axial profile of the CXW pulse radiated from a circular aperture having diameter D=6 cm. The pulse is designed to focus at zf=3.4374 m. Plots show the axial profiles of the CXW pulse at ct=1.0012, 2.3749, 4.2654, and 6.1559 m. The parameters characterizing the X waves are the same as in Fig. 2.

Fig. 5
Fig. 5

Axial profile of the CXW pulse radiated from a circular aperture having diameter D=6 cm. The pulse is designed to focus at zf=2.0624 m. Plots show the axial profiles of the CXW pulse at ct=1.0012, 2.0311, 3.0211, and 4.0935 m. The parameters characterizing the X waves are the same as in Fig. 2.

Fig. 6
Fig. 6

Axial profile of the CXW pulse radiated from a circular aperture having diameter D=6 cm. The pulse is designed to focus at zf=2.0212 m. Plots show the axial profiles of the CXW pulse at ct=1.0012 and 2.0311 m. The parameters characterizing the X waves are the same as in Fig. 2.

Fig. 7
Fig. 7

Logarithmic plots of the normalized amplitudes of the peaks of the pulses shown in Figs. 4 and 5 versus the axial distance from the source. The decay patterns of the peaks of the two CXW pulses are compared to that of an individual X wave component exhibiting the longest diffraction range zdmax=3.4376 m corresponding to ξ1=0.5°. The pulses are normalized with respect to their peak amplitude at the aperture.

Equations (6)

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Ψ(ρ, z, ct)=Rel=1NlΦl(ρ, z, ct; zl, ξl),
Φl(ρ, z, ct; zl, ξl)
=-A(zl, ξl){a0+i[(z-zl)cosξl-ct]}((ρsinξl)2+{a0+i[(z-zl)cosξl-ct]}2)3/2.
(zf-zl)cosξl=ctf.
Ψ(ρ, z, ct)=Rel=1Nl-A[zf-(ctf/cosξl), ξl]{a0+i[(z-zf)cosξl-c(t-tf)]}((ρsinξl)2+{a0+i[(z-zf)cosξl-c(t-tf)]}2)3/2.
ΨRSI(0, z, t)=n=1NAn2πRn [-zΨ(ρ, z=z0, t)].

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