Abstract

In this work, starting by suitable superpositions of equal-frequency Bessel beams, we develop a theoretical and experimental methodology to obtain localized stationary wave fields (with high transverse localization) whose longitudinal intensity pattern can approximately assume any desired shape within a chosen interval 0zL of the propagation axis z. Their intensity envelope remains static, i.e., with velocity v=0, so we have named “frozen waves” (FWs) these new solutions to the wave equations (and, in particular, to the Maxwell equation). Inside the envelope of a FW, only the carrier wave propagates. The longitudinal shape, within the interval 0zL, can be chosen in such a way that no nonnegligible field exists outside the predetermined region (consisting, e.g., in one or more high-intensity peaks). Our solutions are notable also for the different and interesting applications they can have—especially in electromagnetism and acoustics—such as optical tweezers, atom guides, optical or acoustic bistouries, and various important medical apparatuses.

© 2005 Optical Society of America

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  1. J.-Y. Lu, J. F. Greenleaf, “Nondiffracting X-waves: exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
    [CrossRef]
  2. A. O. Barut, G. D. Maccarrone, E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
    [CrossRef]
  3. E. Recami, “On localized X-shaped superluminal solutions to Maxwell equations,” Physica A 252, 586–610 (1998).
    [CrossRef]
  4. See, e.g., M. Zamboni-Rached, E. Recami, H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. J. Phys. 21, 217–228 (2002).
  5. P. Saari, K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
    [CrossRef]
  6. D. Mugnai, A. Ranfagni, R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000). This paper aroused some criticisms, to which those authors replied.
    [CrossRef] [PubMed]
  7. J.-Y. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
    [CrossRef]
  8. For a review, see E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, 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]
  9. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  10. R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
    [CrossRef]
  11. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
    [CrossRef] [PubMed]
  12. I. M. Besieris, A. M. Shaarawi, 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]
  13. R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
    [CrossRef]
  14. J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
    [CrossRef]
  15. A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
    [CrossRef]
  16. S. Esposito, “Classical vgr≠c solutions of Maxwell’s equations and the photon tunneling effect,” Phys. Lett. A 225, 203–209 (1997).
    [CrossRef]
  17. A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
    [CrossRef]
  18. P. Saari, in Time’s Arrows, Quantum Measurements and Superluminal Behavior, D. Mugnai, A. Ranfagni, and L. S. Shulman, eds. (C.N.R., Rome, 2001), pp. 37–48.
  19. M. Zamboni-Rached, H. E. Hernández-Figueroa, “A rigorous analysis of localized wave propagation in optical fibers,” Opt. Commun. 191, 49–54 (2001).
    [CrossRef]
  20. M. Zamboni-Rached, E. Recami, F. Fontana, “Localized superluminal solutions to Maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
    [CrossRef]
  21. M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernandez F., “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
    [CrossRef]
  22. M. Zamboni-Rached, F. Fontana, E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
    [CrossRef]
  23. M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, 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]
  24. S. Longhi, “Spatial-temporal Gauss–Laguerre waves in dispersive media,” Phys. Rev. E 68, 066612 (2003).
    [CrossRef]
  25. M. A. Porras, S. Trillo, C. Conti, P. Di Trapani, “Paraxial envelope X waves,” Opt. Lett. 28, 1090–1093 (2003).
    [CrossRef] [PubMed]
  26. M. A. Porras, I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217, 257–264 (2003).
    [CrossRef]
  27. M. A. Porras, R. Borghi, M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003).
    [CrossRef]
  28. M. Zamboni-Rached, H. E. Hernández-Figueroa, E. Recami, “Chirped optical X-shaped pulses in material media,” J. Opt. Soc. Am. A 21, 2455–2463 (2004).
    [CrossRef]
  29. A. M. Shaarawi, I. M. Besieris, T. M. Said, “Temporal focusing by use of composite X-waves,” J. Opt. Soc. Am. A 20, 1658–1665 (2003).
    [CrossRef]
  30. M. Zamboni-Rached, A. Shaarawi, E. Recami, “Focused X-shaped pulses,” J. Opt. Soc. Am. A 21, 1564–1574 (2004) and references therein.
    [CrossRef]
  31. Z. Bouchal, J. Wagner, “Self-reconstruction effect in free propagation wavefield,” Opt. Commun. 176, 299–307 (2000).
    [CrossRef]
  32. Z. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. 27, 1376–1378 (2002).
    [CrossRef]
  33. J. Rosen, A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 843–845 (2002).
    [CrossRef]
  34. R. Piestun, B. Spektor, J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
    [CrossRef]
  35. “Method and apparatus for producing stationary (intense) wave fields of arbitrary shape,” patent application EPO-4424387.0 (Italy), May 27, 2004.
  36. However, when we get complete control over the longitudinal shape, we cannot have total control also over the transverse localization, since our stationary fields are of course constrained to obey the wave equation.
  37. 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]
  38. Such a choice of the longitudinal intensity pattern does imply an interesting freedom, since we can consider more generally any expansion ∑m=−∞∞Bmexp(i2πmz∕L)=F(z)exp[iϕ(z)], quantity ϕ(z) being an arbitrary function of the coordinate z.
  39. The same apparatus could also be used to generate higher-order FWs, when the zero-order Bessel beams in superposition (23) are replaced with higher-order Bessel functions. Experimentally, higher-order Bessel beams can be produced by angular modulation of the slits.
  40. C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
    [CrossRef]

2004 (3)

2003 (9)

C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, T. M. Said, “Temporal focusing by use of composite X-waves,” J. Opt. Soc. Am. A 20, 1658–1665 (2003).
[CrossRef]

M. Zamboni-Rached, F. Fontana, 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, 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]

S. Longhi, “Spatial-temporal Gauss–Laguerre waves in dispersive media,” Phys. Rev. E 68, 066612 (2003).
[CrossRef]

M. A. Porras, S. Trillo, C. Conti, P. Di Trapani, “Paraxial envelope X waves,” Opt. Lett. 28, 1090–1093 (2003).
[CrossRef] [PubMed]

M. A. Porras, I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217, 257–264 (2003).
[CrossRef]

M. A. Porras, R. Borghi, M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003).
[CrossRef]

For a review, see E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, 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]

2002 (4)

See, e.g., M. Zamboni-Rached, E. Recami, H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. J. Phys. 21, 217–228 (2002).

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

Z. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. 27, 1376–1378 (2002).
[CrossRef]

J. Rosen, A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 843–845 (2002).
[CrossRef]

2001 (2)

M. Zamboni-Rached, 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, 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, R. Ruggeri, “Observation of superluminal behaviors in wave propagation,” Phys. Rev. Lett. 84, 4830–4833 (2000). This paper aroused some criticisms, to which those authors replied.
[CrossRef] [PubMed]

Z. Bouchal, 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 252, 586–610 (1998).
[CrossRef]

1997 (3)

A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

S. Esposito, “Classical vgr≠c solutions of Maxwell’s equations and the photon tunneling effect,” Phys. Lett. A 225, 203–209 (1997).
[CrossRef]

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

1996 (2)

R. Piestun, B. Spektor, J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

1993 (1)

1992 (2)

J.-Y. Lu, 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]

J.-Y. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
[CrossRef]

1990 (1)

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

1989 (2)

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

I. M. Besieris, A. M. Shaarawi, 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, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

1985 (1)

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
[CrossRef]

1982 (1)

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

Barut, A. O.

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

Besieris, I. M.

A. M. Shaarawi, I. M. Besieris, T. M. Said, “Temporal focusing by use of composite X-waves,” J. Opt. Soc. Am. A 20, 1658–1665 (2003).
[CrossRef]

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

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, 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, M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003).
[CrossRef]

Bouchal, Z.

Z. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. 27, 1376–1378 (2002).
[CrossRef]

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

Conti, C.

Dartora, C. A.

For a review, see E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, 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. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

Di Trapani, P.

Durnin, J.

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

Eberly, J. H.

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

Esposito, S.

S. Esposito, “Classical vgr≠c solutions of Maxwell’s equations and the photon tunneling effect,” Phys. Lett. A 225, 203–209 (1997).
[CrossRef]

Fagerholm, J.

A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Fontana, F.

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

Friberg, A. T.

A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Gonzalo, I.

M. A. Porras, I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217, 257–264 (2003).
[CrossRef]

Greenleaf, J. F.

J.-Y. Lu, 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]

J.-Y. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
[CrossRef]

Hernandez F., H. E.

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

Hernández Figueroa, H. E.

C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

Hernández-Figueroa, H. E.

M. Zamboni-Rached, H. E. Hernández-Figueroa, 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, H. E. Hernández-Figueroa, 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]

For a review, see E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, 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]

See, e.g., M. Zamboni-Rached, E. Recami, H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. J. Phys. 21, 217–228 (2002).

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

Huttunen, J.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Longhi, S.

S. Longhi, “Spatial-temporal Gauss–Laguerre waves in dispersive media,” Phys. Rev. E 68, 066612 (2003).
[CrossRef]

Lu, J.-Y.

J.-Y. Lu, 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]

J.-Y. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
[CrossRef]

Maccarrone, G. D.

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

Miceli, J. J.

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

Morgan, D. P.

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Mugnai, D.

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

Nóbrega, K. Z.

For a review, see E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, 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. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

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

Piestun, R.

R. Piestun, B. Spektor, J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
[CrossRef]

Porras, M. A.

M. A. Porras, S. Trillo, C. Conti, P. Di Trapani, “Paraxial envelope X waves,” Opt. Lett. 28, 1090–1093 (2003).
[CrossRef] [PubMed]

M. A. Porras, R. Borghi, M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003).
[CrossRef]

M. A. Porras, I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217, 257–264 (2003).
[CrossRef]

Ranfagni, A.

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

Recami, E.

M. Zamboni-Rached, H. E. Hernández-Figueroa, 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, E. Recami, “Focused X-shaped pulses,” J. Opt. Soc. Am. A 21, 1564–1574 (2004) and references therein.
[CrossRef]

C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, 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, E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
[CrossRef]

For a review, see E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, 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]

See, e.g., M. Zamboni-Rached, E. Recami, H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. J. Phys. 21, 217–228 (2002).

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

M. Zamboni-Rached, E. Recami, 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 252, 586–610 (1998).
[CrossRef]

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

Reivelt, K.

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

Rosen, J.

Ruggeri, R.

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

Saari, P.

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

P. Saari, in Time’s Arrows, Quantum Measurements and Superluminal Behavior, D. Mugnai, A. Ranfagni, and L. S. Shulman, eds. (C.N.R., Rome, 2001), pp. 37–48.

Said, T. M.

Salomaa, M. M.

A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Santarsiero, M.

M. A. Porras, R. Borghi, M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003).
[CrossRef]

Shaarawi, A.

Shaarawi, A. M.

A. M. Shaarawi, I. M. Besieris, T. M. Said, “Temporal focusing by use of composite X-waves,” J. Opt. Soc. Am. A 20, 1658–1665 (2003).
[CrossRef]

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

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

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

Shamir, J.

R. Piestun, B. Spektor, J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
[CrossRef]

Spektor, B.

R. Piestun, B. Spektor, J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
[CrossRef]

Trillo, S.

Wagner, J.

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

Yariv, A.

Zamboni-Rached, M.

M. Zamboni-Rached, A. Shaarawi, 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, 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, H. E. Hernández-Figueroa, 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, E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
[CrossRef]

For a review, see E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, 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. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

See, e.g., M. Zamboni-Rached, E. Recami, H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. J. Phys. 21, 217–228 (2002).

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

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

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

Ziolkowski, R. W.

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

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

I. M. Besieris, A. M. Shaarawi, 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, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
[CrossRef]

Eur. J. Phys. (1)

See, e.g., M. Zamboni-Rached, E. Recami, H. E. Hernández-Figueroa, “New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. J. Phys. 21, 217–228 (2002).

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

For a review, see E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, 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 (2)

J.-Y. Lu, 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]

J.-Y. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X-waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
[CrossRef]

J. Math. Phys. (3)

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, 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. Mod. Opt. (1)

R. Piestun, B. Spektor, J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996).
[CrossRef]

J. Opt. Soc. Am. A (4)

Nuovo Cimento A (1)

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

Opt. Commun. (7)

A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of non-diffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

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

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

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, 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. A. Porras, I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217, 257–264 (2003).
[CrossRef]

M. A. Porras, R. Borghi, M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003).
[CrossRef]

C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, H. E. Hernández Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222, 75–80 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Lett. A (1)

S. Esposito, “Classical vgr≠c solutions of Maxwell’s equations and the photon tunneling effect,” Phys. Lett. A 225, 203–209 (1997).
[CrossRef]

Phys. Rev. A (1)

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

Phys. Rev. E (5)

J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

M. Zamboni-Rached, E. Recami, 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, H. E. Hernandez F., “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

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

S. Longhi, “Spatial-temporal Gauss–Laguerre waves in dispersive media,” Phys. Rev. E 68, 066612 (2003).
[CrossRef]

Phys. Rev. Lett. (3)

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

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

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

Physica A (1)

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

Other (5)

P. Saari, in Time’s Arrows, Quantum Measurements and Superluminal Behavior, D. Mugnai, A. Ranfagni, and L. S. Shulman, eds. (C.N.R., Rome, 2001), pp. 37–48.

Such a choice of the longitudinal intensity pattern does imply an interesting freedom, since we can consider more generally any expansion ∑m=−∞∞Bmexp(i2πmz∕L)=F(z)exp[iϕ(z)], quantity ϕ(z) being an arbitrary function of the coordinate z.

The same apparatus could also be used to generate higher-order FWs, when the zero-order Bessel beams in superposition (23) are replaced with higher-order Bessel functions. Experimentally, higher-order Bessel beams can be produced by angular modulation of the slits.

“Method and apparatus for producing stationary (intense) wave fields of arbitrary shape,” patent application EPO-4424387.0 (Italy), May 27, 2004.

However, when we get complete control over the longitudinal shape, we cannot have total control also over the transverse localization, since our stationary fields are of course constrained to obey the wave equation.

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Figures (9)

Fig. 1
Fig. 1

(a) Comparison between the intensity of the desired longitudinal function F ( z ) and that of our FW, Ψ ( ρ = 0 , z , t ) , obtained from Eq. (9). The solid curve represents the function F ( z ) and the dotted one our FW. (b) 3D plot of the field intensity of the FW chosen in this case by us.

Fig. 2
Fig. 2

(a) Comparison between the intensity of the desired longitudinal function F ( z ) , given by Eq. (14), and that of our FW, Ψ ( ρ = 0 , z , t ) , obtained from Eq. (9). The solid curve represents the function F ( z ) and the dotted one our FW. (b) 3D plot of the field intensity of the FW chosen by us in this new case.

Fig. 3
Fig. 3

(a) FW with Q = 0.999 ω 0 c and N = 45 , approximately reproducing the chosen longitudinal pattern represented by Eq. (15). (b) Different FW, now with Q = 0.995 ω 0 c (but still with N = 45 ) yielding the same longitudinal pattern. We can observe that in this case (with a lower value for Q) a higher transverse localization is obtained.

Fig. 4
Fig. 4

Comparison of the desired longitudinal intensity pattern (solid curves) with those of the resulting FWs (dotted curves), with (a) N = 100 , (b) N = 250 , (c) N = 300 .

Fig. 5
Fig. 5

(a) Longitudinal residual intensity of the considered FW with N = 250 . (b) The same with N = 300 .

Fig. 6
Fig. 6

(a) Transverse intensity pattern for the peak of the considered FW with N = 250 . (b) Transverse residual intensity for this case.

Fig. 7
Fig. 7

(Color online) (a) Transverse intensity pattern at z = L 2 of the considered, higher-order FW. (b) Transverse section of the resulting stationary field for z = L 2 . We emphasize that Figs. 7 and 8 represent a cylindrical surface of stationary light.

Fig. 8
Fig. 8

(Color online) (a) Orthogonal projection of the three-dimensional intensity pattern of the higher-order FW depicted in Fig. 7. (b) The same field but under a different perspective.

Fig. 9
Fig. 9

Set of suitable, concentric annular slits, as a simple means for generating a FW.

Equations (34)

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ψ ( ρ , z , t ) = J 0 ( k ρ ρ ) e i β z e i ω t ,
k ρ 2 = ω 2 c 2 β 2 ,
ω β > 0 , k ρ 2 0
Ψ ( ρ , z , t ) = e i ω 0 t n = N N A n J 0 ( k ρ n ρ ) e i β n z ,
0 β n ω 0 c .
F ( z ) = m = B m e i ( 2 π L ) m z ,
B m = 1 L 0 L F ( z ) e i ( 2 π L ) m z d z .
n = N N A n e i β n z 2 F ( z ) 2 , with 0 z L .
β n = Q + 2 π L n ,
0 Q ± 2 π L N ω 0 c .
Ψ ( ρ = 0 , z , t ) = e i ω 0 t e i Q z n = N N A n e i ( 2 π L ) n z ,
A n = 1 L 0 L F ( z ) e i ( 2 π L ) n z d z .
Ψ ( ρ , z , t ) = e i ω 0 t e i Q z n = N N A n J 0 ( k ρ n ρ ) e i ( 2 π L ) n z ,
k ρ n 2 = ω 0 2 ( Q + 2 π n L ) 2 .
F ( z ) = { 4 ( z l 1 ) ( z l 2 ) ( l 2 l 1 ) 2 for l 1 z l 2 1 for l 3 z l 4 4 ( z l 5 ) ( z l 6 ) ( l 6 l 5 ) 2 for l 5 z l 6 0 elsewhere } ,
F ( z ) = { 4 ( z l 1 ) ( z l 2 ) ( l 2 l 1 ) 2 for l 1 z l 2 4 2 ( z l 3 ) ( z l 4 ) ( l 4 l 3 ) for l 3 z l 4 0 elsewhere } ,
F ( z ) = { 4 ( z l 1 ) ( z l 2 ) ( l 2 l 1 ) 2 for l 1 z l 2 0 elsewhere } ,
D = 0 L F ( z ) 2 d z L N N A n 2 ,
Δ ρ 2.4 k ρ n = 0 = 2.4 w 0 2 c 2 Q 2 ,
Ψ ( ρ , z , t ) = e i ω 0 t e i Q z n = N N A n J μ ( k ρ n ρ ) e i [ ( 2 π L ) n z ] ,
A n = 1 L 0 L F ( z ) e i [ ( 2 π L ) n z ] d z ,
Z = R tan θ ,
R L ω 0 2 c 2 β n = N 2 1 .
ψ ( ρ , z , t ) Λ J 0 ( k ρ ρ ) e i β z e i ω 0 t ,
k ρ = ω 0 c a f ,
β 2 = ω 0 2 c 2 k ρ 2 .
Ψ ( ρ , z , t ) = e i ω 0 t n = N N Λ n T n J 0 ( k ρ n ρ ) e i β n z ,
k ρ n = ω 0 c a n f ,
β n 2 = ω 0 2 c 2 k ρ 2 .
β n = Q + 2 π L n .
( Q + 2 π L n ) 2 = ω 0 2 c 2 ( ω 0 c a n f ) 2
a n = f 1 c 2 ω 0 2 ( Q + 2 π L n ) 2 .
T n = A n Λ n = 1 L Λ n 0 L F ( z ) e i [ ( 2 π L ) n z ] d z .
L Z min R f a max ,

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