M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).

[CrossRef]

M. Zamboni-Rached, “Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves,” Opt. Express 12, 4001–4006(2004).

[CrossRef]
[PubMed]

M. Zamboni-Rached, H. E. Hernández-Figueroa, and E. Recami, “Chirped optical X-shaped pulses in material media,” J. Opt. Soc. Am. A 21, 2455–2463 (2004).

[CrossRef]

S. V. Kukhlevsky and M. Mechler, “Diffraction-free sub-wavelength beam optics at nanometer scale,” Opt. Commun. 231, 35–43 (2004).

[CrossRef]

S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B 70, 235123 (2004).

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and F. H. E. Hernandez, “Superluminal x-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617(2002).

[CrossRef]

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2002).

[CrossRef]

M. Zamboni-Rached and H. E. Hernández-Figueroa, “A rigorous analysis of localized wave propagation in optical fibers,” Opt. Commun. 191, 49–54 (2001).

[CrossRef]

M. Zamboni-Rached, E. Recami, and F. Fontana, “Localized superluminal solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603(2001).

[CrossRef]

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000). This paper aroused some criticisms, to which the authors replied.

[CrossRef]
[PubMed]

Z. Bouchal and J. Wagner, “Self-reconstruction effect in free propagation wavefield,” Opt. Commun. 176, 299–307(2000).

[CrossRef]

E. Recami, “On localized x-shaped superluminal solutions to Maxwell equations,” Physica A (Amsterdam) 252, 586–610 (1998).

[CrossRef]

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

[CrossRef]

J.-Y. Lu and 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]
[PubMed]

I. M. Besieris, A. M. Shaarawi, and 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. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).

[CrossRef]
[PubMed]

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves Opt. Acoust. 2, 105–112 (1978).

[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 of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).

[CrossRef]

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2002).

[CrossRef]

Z. Bouchal and J. Wagner, “Self-reconstruction effect in free propagation wavefield,” Opt. Commun. 176, 299–307(2000).

[CrossRef]

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

[CrossRef]
[PubMed]

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

[CrossRef]

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

[CrossRef]
[PubMed]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).

[CrossRef]
[PubMed]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).

[CrossRef]
[PubMed]

M. Zamboni-Rached, F. Fontana, and 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, and F. Fontana, “Localized superluminal solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603(2001).

[CrossRef]

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

J.-Y. Lu and 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]
[PubMed]

R. Grunwald, U. Griebner, U. Neumann, and V. Kebbel, “Self-reconstruction of ultrashort-pulse Bessel-like x-waves,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (Optical Society of America, 2004), paper CMQ7.

R. Grunwald, U. Griebner, U. Neumann, and V. Kebbel, “Self-reconstruction of ultrashort-pulse Bessel-like x-waves,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (Optical Society of America, 2004), paper CMQ7.

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and F. H. E. Hernandez, “Superluminal x-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617(2002).

[CrossRef]

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “Theory of frozen waves: modeling the shape of stationary wave fields,” J. Opt. Soc. Am. A 22, 2465–2475 (2005).

[CrossRef]

M. Zamboni-Rached, H. E. Hernández-Figueroa, and E. Recami, “Chirped optical X-shaped pulses in material media,” J. Opt. Soc. Am. A 21, 2455–2463 (2004).

[CrossRef]

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

[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and 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, 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]

M. Zamboni-Rached and H. E. Hernández-Figueroa, “A rigorous analysis of localized wave propagation in optical fibers,” Opt. Commun. 191, 49–54 (2001).

[CrossRef]

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B 70, 235123 (2004).

[CrossRef]

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

[CrossRef]
[PubMed]

R. Grunwald, U. Griebner, U. Neumann, and V. Kebbel, “Self-reconstruction of ultrashort-pulse Bessel-like x-waves,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (Optical Society of America, 2004), paper CMQ7.

S. V. Kukhlevsky and M. Mechler, “Diffraction-free sub-wavelength beam optics at nanometer scale,” Opt. Commun. 231, 35–43 (2004).

[CrossRef]

S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B 70, 235123 (2004).

[CrossRef]

J.-Y. Lu and 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]
[PubMed]

S. V. Kukhlevsky and M. Mechler, “Diffraction-free sub-wavelength beam optics at nanometer scale,” Opt. Commun. 231, 35–43 (2004).

[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).

[CrossRef]
[PubMed]

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000). This paper aroused some criticisms, to which the authors replied.

[CrossRef]
[PubMed]

R. Grunwald, U. Griebner, U. Neumann, and V. Kebbel, “Self-reconstruction of ultrashort-pulse Bessel-like x-waves,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (Optical Society of America, 2004), paper CMQ7.

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

[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and F. H. E. Hernandez, “Superluminal x-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617(2002).

[CrossRef]

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

[CrossRef]
[PubMed]

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2002).

[CrossRef]

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000). This paper aroused some criticisms, to which the authors replied.

[CrossRef]
[PubMed]

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).

[CrossRef]

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “Theory of frozen waves: modeling the shape of stationary wave fields,” J. Opt. Soc. Am. A 22, 2465–2475 (2005).

[CrossRef]

M. Zamboni-Rached, H. E. Hernández-Figueroa, and E. Recami, “Chirped optical X-shaped pulses in material media,” J. Opt. Soc. Am. A 21, 2455–2463 (2004).

[CrossRef]

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

[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and 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, and 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, and F. H. E. Hernandez, “Superluminal x-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617(2002).

[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]

M. Zamboni-Rached, E. Recami, and F. Fontana, “Localized superluminal solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603(2001).

[CrossRef]

E. Recami, “On localized x-shaped superluminal solutions to Maxwell equations,” Physica A (Amsterdam) 252, 586–610 (1998).

[CrossRef]

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

[CrossRef]

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000). This paper aroused some criticisms, to which the authors replied.

[CrossRef]
[PubMed]

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2002).

[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 of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).

[CrossRef]

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

[CrossRef]
[PubMed]

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

[CrossRef]
[PubMed]

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

[CrossRef]
[PubMed]

Z. Bouchal and J. Wagner, “Self-reconstruction effect in free propagation wavefield,” Opt. Commun. 176, 299–307(2000).

[CrossRef]

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves Opt. Acoust. 2, 105–112 (1978).

[CrossRef]

M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).

[CrossRef]

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).

[CrossRef]

M. Zamboni-Rached, “Diffraction–attenuation resistant beams in absorbing media,” Opt. Express 14, 1804–1809 (2006).

[CrossRef]
[PubMed]

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “Theory of frozen waves: modeling the shape of stationary wave fields,” J. Opt. Soc. Am. A 22, 2465–2475 (2005).

[CrossRef]

M. Zamboni-Rached, “Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves,” Opt. Express 12, 4001–4006(2004).

[CrossRef]
[PubMed]

M. Zamboni-Rached, H. E. Hernández-Figueroa, and E. Recami, “Chirped optical X-shaped pulses in material media,” J. Opt. Soc. Am. A 21, 2455–2463 (2004).

[CrossRef]

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

[CrossRef]

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

[CrossRef]

M. Zamboni-Rached, F. Fontana, and 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, and F. H. E. Hernandez, “Superluminal x-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617(2002).

[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]

M. Zamboni-Rached and H. E. Hernández-Figueroa, “A rigorous analysis of localized wave propagation in optical fibers,” Opt. Commun. 191, 49–54 (2001).

[CrossRef]

M. Zamboni-Rached, E. Recami, and F. Fontana, “Localized superluminal solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603(2001).

[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 of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).

[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]

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves Opt. Acoust. 2, 105–112 (1978).

[CrossRef]

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

[CrossRef]

J.-Y. Lu and 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]
[PubMed]

I. M. Besieris, A. M. Shaarawi, and 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]

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]

M. Zamboni-Rached, H. E. Hernández-Figueroa, and E. Recami, “Chirped optical X-shaped pulses in material media,” J. Opt. Soc. Am. A 21, 2455–2463 (2004).

[CrossRef]

M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “Theory of frozen waves: modeling the shape of stationary wave fields,” J. Opt. Soc. Am. A 22, 2465–2475 (2005).

[CrossRef]

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

[CrossRef]

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2002).

[CrossRef]

S. V. Kukhlevsky and M. Mechler, “Diffraction-free sub-wavelength beam optics at nanometer scale,” Opt. Commun. 231, 35–43 (2004).

[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and 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 and H. E. Hernández-Figueroa, “A rigorous analysis of localized wave propagation in optical fibers,” Opt. Commun. 191, 49–54 (2001).

[CrossRef]

Z. Bouchal and J. Wagner, “Self-reconstruction effect in free propagation wavefield,” Opt. Commun. 176, 299–307(2000).

[CrossRef]

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).

[CrossRef]

M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).

[CrossRef]

S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B 70, 235123 (2004).

[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and F. H. E. Hernandez, “Superluminal x-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617(2002).

[CrossRef]

M. Zamboni-Rached, E. Recami, and F. Fontana, “Localized superluminal solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603(2001).

[CrossRef]

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

[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).

[CrossRef]
[PubMed]

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000). This paper aroused some criticisms, to which the authors replied.

[CrossRef]
[PubMed]

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

[CrossRef]

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

[CrossRef]
[PubMed]

E. Recami, “On localized x-shaped superluminal solutions to Maxwell equations,” Physica A (Amsterdam) 252, 586–610 (1998).

[CrossRef]

H.E.Hernández-Figueroa, M.Zamboni-Rached, and E.Recami, eds., Localized Waves: Theory and Applications, (Wiley, 2008), and references therein.

[CrossRef]

According to Eq. , the maximum value allowed for N is 158 and we choose to use N=20 just for simplicity. Of course, by using higher values of N we get better results.

The analytic calculation of these coefficients is quite simple in this case and their values are not listed here; we just use them in Eq. .

Notice that, according to Section , the absorption coefficient of a Bessel beam is αθ=αcosθ=2kIcosθ. When θ→0, the Bessel beam tends to a plane wave and αθ→α.

That is, the shortest diffractionless distance is larger than the distance L.

The idea developed in this section generalizes that exposed in Section 5 of , which was addressed to nonabsorbing media.

Fortunately, these conditions are satisfied for a great number of situations.

Here, θm is the axicon angle of the mth Bessel beam in Eq. .

The same is valid for a truncated higher-order Bessel beam.

In an absorbing medium like this, at a distance of 25 cm, these beams would have their initial field intensity attenuated 148 times.

In this case, we can consider both ϵb(ω) and σ(ω) real quantities.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

The same that was given in .

In this paper we use cylindrical coordinates (ρ,ϕ,z).

Notice that kρRm=N is the smallest value of all kρRm, therefore, if R≫2.4/kρRm=N→R≫2.4/kρRm for all m.

R. Grunwald, U. Griebner, U. Neumann, and V. Kebbel, “Self-reconstruction of ultrashort-pulse Bessel-like x-waves,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (Optical Society of America, 2004), paper CMQ7.

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

When generated by a finite aperture of radius R≫2.4/kρR situated on the plane z=0, the solution in Eq. becomes a valid approximation only in the spatial region 0<z<R/tanθ≡Z and to ρ<(1−z/Z)R.