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

[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, 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. 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, 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. 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]

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]

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

[CrossRef]

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

[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]

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 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, 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, 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, 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]

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]

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]

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

[CrossRef]

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.

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.

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

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

The same that was given in .

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

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. .

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

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

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.

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

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

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.

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

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

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

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

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