Abstract

Recently, a method for obtaining diffraction–attenuation resistant beams in absorbing media has been developed in terms of suitable superposition of ideal zero-order Bessel beams. In this work, we show that such beams keep their resistance to diffraction and absorption even when generated by finite apertures. Moreover, we shall extend the original method to allow a higher control over the transverse intensity profile of the beams. Although the method is developed for scalar fields, it can be applied to paraxial vector wave fields, as well. These new beams have many potential applications, such as in free-space optics, medical apparatus, remote sensing, and optical tweezers.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. 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]
  3. M. Zamboni-Rached, “Diffraction–attenuation resistant beams in absorbing media,” Opt. Express 14, 1804–1809 (2006).
    [CrossRef] [PubMed]
  4. H.E.Hernández-Figueroa, M.Zamboni-Rached, and E.Recami, eds., Localized Waves: Theory and Applications, (Wiley, 2008), and references therein.
    [CrossRef]
  5. 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]
  6. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  7. 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]
  8. 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]
  9. 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]
  10. E. Recami, “On localized x-shaped superluminal solutions to Maxwell equations,” Physica A (Amsterdam) 252, 586–610 (1998).
    [CrossRef]
  11. P. Saari and K. Reivelt, “Evidence of x-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
    [CrossRef]
  12. 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]
  13. 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]
  14. 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]
  15. 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]
  16. 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]
  17. 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]
  18. 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]
  19. 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]
  20. S. V. Kukhlevsky and M. Mechler, “Diffraction-free sub-wavelength beam optics at nanometer scale,” Opt. Commun. 231, 35–43 (2004).
    [CrossRef]
  21. 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]
  22. S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B 70, 235123 (2004).
    [CrossRef]
  23. M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
    [CrossRef]
  24. 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]
  25. 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]
  26. Z. Bouchal and J. Wagner, “Self-reconstruction effect in free propagation wavefield,” Opt. Commun. 176, 299–307(2000).
    [CrossRef]
  27. 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.
  28. 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]
  29. C. J. R. Sheppard and P. Saari, “Lommel pulses: an analytic form for localized waves of the focus wave mode type with bandlimited spectrum,” Opt. Express 16, 150–160(2008).
    [CrossRef] [PubMed]
  30. M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).
    [CrossRef]
  31. 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/tan⁡θ≡Z and to ρ<(1−z/Z)R.
  32. In this paper we use cylindrical coordinates (ρ,ϕ,z).
  33. Fortunately, these conditions are satisfied for a great number of situations.
  34. The same that was given in .
  35. In an absorbing medium like this, at a distance of 25 cm, these beams would have their initial field intensity attenuated 148 times.
  36. 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.
  37. 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. .
  38. In this case, we can consider both ϵb(ω) and σ(ω) real quantities.
  39. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  40. Notice that, according to Section , the absorption coefficient of a Bessel beam is αθ=αcos⁡θ=2kIcos⁡θ. When θ→0, the Bessel beam tends to a plane wave and αθ→α.
  41. The idea developed in this section generalizes that exposed in Section 5 of , which was addressed to nonabsorbing media.
  42. The same is valid for a truncated higher-order Bessel beam.
  43. 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.
  44. Here, θm is the axicon angle of the mth Bessel beam in Eq. .
  45. That is, the shortest diffractionless distance is larger than the distance L.
  46. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

2009 (1)

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

2008 (2)

C. J. R. Sheppard and P. Saari, “Lommel pulses: an analytic form for localized waves of the focus wave mode type with bandlimited spectrum,” Opt. Express 16, 150–160(2008).
[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]

2006 (1)

2005 (1)

2004 (5)

2003 (4)

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]

2002 (3)

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]

2001 (2)

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]

2000 (2)

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]

1998 (1)

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

1997 (1)

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

1993 (1)

1992 (1)

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]

1989 (1)

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]

1987 (1)

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

1978 (1)

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]

Besieris, I. M.

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]

Borghi, R.

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]

Bouchal, Z.

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

Conti, C.

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]

Dartora, C. A.

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]

Di Trapani, P.

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]

Durnin, J.

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

Eberly, J. H.

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

Fontana, F.

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]

Goodman, J. W.

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

Greenleaf, J. F.

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]

Griebner, U.

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.

Grunwald, R.

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.

Hernandez, F. H. E.

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]

Hernández-Figueroa, H. E.

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]

Jackson, J. D.

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

Janner, D.

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

Jedrkiewicz, O.

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]

Kebbel, V.

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.

Kukhlevsky, S. V.

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

Longhi, S.

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

Lu, J.-Y.

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]

Mechler, M.

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

Miceli, J. J.

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

Mugnai, D.

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]

Neumann, U.

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.

Nóbrega, K. Z.

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, 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]

Piskarskas, A.

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]

Porras, M. A.

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]

Ranfagni, A.

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]

Recami, E.

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, 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, 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, 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, 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, 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]

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

Reivelt, K.

P. Saari and 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, 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]

Saari, P.

Santarsiero, M.

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]

Shaarawi, A.

Shaarawi, A. M.

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]

Sheppard, C. J. R.

C. J. R. Sheppard and P. Saari, “Lommel pulses: an analytic form for localized waves of the focus wave mode type with bandlimited spectrum,” Opt. Express 16, 150–160(2008).
[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]

Trillo, S.

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]

Trull, J.

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]

Valiulis, G.

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]

Wagner, J.

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

Wilson, T.

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]

Zamboni-Rached, M.

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, 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]

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, 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, 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]

Ziolkowski, R. W.

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]

Eur. Phys. J. D (1)

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]

IEE J. Microwaves Opt. Acoust. (1)

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]

IEEE J. Sel. Top. Quantum Electron. (1)

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]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

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]

J. Math. Phys. (1)

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. Opt. Soc. Am. A (4)

Opt. Commun. (5)

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]

Opt. Express (3)

Phys. Rev. A (2)

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]

Phys. Rev. B (1)

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

Phys. Rev. E (3)

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]

Phys. Rev. Lett. (4)

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]

Physica A (Amsterdam) (1)

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

Other (18)

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/tan⁡θ≡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=αcos⁡θ=2kIcos⁡θ. 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.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

(a) Three-dimensional field intensity of the resulting beam. (b) Orthogonal projection of the resulting beam intensity (normalized with respect to its maximum value, | Ψ | max 2 ) in logarithmic scale.

Fig. 2
Fig. 2

(a) Three-dimensional field intensity of the resulting beam and (b) its orthogonal projection.

Fig. 3
Fig. 3

Beam’s transverse intensity pattern at z = L / 2 .

Fig. 4
Fig. 4

Truncated version of the ideal diffraction–attenuation resistant beam obtained in the example of Section 2.

Fig. 5
Fig. 5

Ideal beam presenting a moderate exponential growth in an absorbing medium.

Fig. 6
Fig. 6

Truncated version of the beam with exponential growth.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

n ( ω ) = n R ( ω ) + i n I ( ω ) ,
ψ = J 0 [ ( k ρ R + k ρ I ) ρ ] exp ( i β R z ) exp ( β I z ) exp ( i ω t ) ,
β R = n R ω c cos θ , β I = n I ω c cos θ ,
k ρ R = n R ω c sin θ , k ρ I = n I ω c sin θ ,
Ψ ( ρ , z , t ) = m = N N A m J 0 ( k ρ m ρ ) e i β m z e i ω t = e i ω t m = N N A m J 0 ( ( k ρ R m + i k ρ I m ) ρ ) e i β R m z e β I m z ,
k ρ m 2 = n 2 ω 2 c 2 β m 2 ,
β R m β I m = k ρ R m k ρ I m = n R n I ,
β R m = Q + 2 π m L ,
0 Q + 2 π m L n R ω c ,
Ψ ( ρ , z , t ) = e i ω t e i Q z m = N N A m J 0 ( ( k ρ R m + i k ρ I m ) ρ ) e i 2 π m L z e β I m z ,
β I m = ( Q + 2 π m L ) n I n R ,
Δ = ( β I ) max ( β I ) min β ¯ I = 4 π N L Q .
A m = 1 L 0 L F ( z ) e β ¯ I z e i 2 π m L z d z .
Δ ρ 2.4 k ρ R m = 0 = 2.4 n R 2 ω 2 c 2 Q 2 .
F ( z ) = { 1 for     0 z Z 0 elsewhere ,
2 E + k 2 E = 0 ,
k = ω c n ( ω ) = ω μ ( ω ) ϵ ( ω ) = ω μ ( ϵ b + i σ ω ) ,
k = k R + i k I ,
k R ω μ 0 ϵ b 2 [ 1 + ( σ ϵ b ω ) 2 + 1 ] 1 / 2 ,
k I ω μ 0 ϵ b 2 [ 1 + ( σ ϵ b ω ) 2 1 ] 1 / 2 ,
E = E x e x + E z e z ,
E x ( ρ , z , t ) = e i ω t m = N N A m J 0 ( k ρ m ρ ) e i β m z = e i ω t e i Q z m = N N A m J 0 ( ( k ρ R m + i k ρ I m ) ρ ) e β I m z e i 2 π L m z ,
E z = E x x d z .
E z ( ρ , ϕ , z , t ) = e i ω t m = N N A m k ρ m β m J 1 ( k ρ m ρ ) cos ϕ e i β m z .
E E x e x = ( e i ω t m = N N A m J 0 ( k ρ m ρ ) e i β m z ) e x ( p a r a x i a l a p p r o x i m a t i o n ) ,
B = i ω × E .
B i ω E x z e y = ( e i ω t m = N N A m β m ω J 0 ( k ρ m ρ ) e i β m z ) e y ( p a r a x i a l a p p r o x i m a t i o n ) .
B n c ( e i ω t m = N N A m J 0 ( k ρ m ρ ) e i β m z ) e y ,
B n c e z × E = n c E x e y ( p a r a x i a l a p p r o x i m a t i o n ) .
u 1 4 Re ( ϵ * + | n | 2 μ * c 2 ) | E x | 2 | E x | 2 ( p a r a x i a l a p p r o x i m a t i o n ) .
Ψ ( ρ , ϕ , z , t ) = e i ω t e i Q z e i μ ϕ m = N N A m J μ ( k ρ m ρ ) e β I m z e i 2 π L m z ,
F ( z ) = { 1 for     0 z Z 0 elsewhere ,
R 2.4 k ρ R m = N ,
Z m = N > L R tan θ m = N > L R > L n R 2 c 2 ω 2 β R m = N 2 1 .
Ψ T ( ρ , ϕ , z , t ) = 1 2 π 0 2 π d ϕ 0 R d ρ ρ e i k D D ( z Ψ ( ρ , ϕ , z , t ) ) z = 0 ,
F ( z ) = { exp ( z / Z ) for     0 z Z 0 elsewhere ,

Metrics