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

[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]

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]

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]

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]

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]

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

[CrossRef]

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

[CrossRef]

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]

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]

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]

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

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

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

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

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]

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

[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]

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]

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 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].

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].

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]

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 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]

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 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].

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

[CrossRef]

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

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

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]

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

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]

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.

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

[CrossRef]

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].

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]

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]

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]

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].

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.

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

[CrossRef]

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.

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]

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

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

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

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

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

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].

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].

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

[CrossRef]

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.

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]

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

[CrossRef]
[PubMed]

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

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]

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

[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]

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).

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

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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).

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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).

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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).

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

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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).

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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).

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See also, e.g., C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A 18, 2594-2600 (2001).

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

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

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

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

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

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

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

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

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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.

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

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

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

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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).

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

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

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

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

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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).