Abstract

We analyze the properties of chirped optical X-shaped pulses propagating in material media without boundaries. We show that such (“superluminal”) pulses may recover their transverse and longitudinal shapes after some propagation distance, whereas the ordinary chirped Gaussian pulses can recover their longitudinal width only (since Gaussian pulses suffer a progressive transverse spreading during their propagation). We therefore propose the use of chirped optical X-type pulses to overcome the problems of both dispersion and diffraction during pulse propagation.

© 2004 Optical Society of America

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References

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  1. 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]
  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]
  3. E. Recami, “On localized ‘X-shaped’ superluminal solutions to Maxwell equations,” Physica A 252, 586–610 (1998), and references therein.
    [CrossRef]
  4. 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]
  5. 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. Phys. J. D 21, 217–228 (2002).
    [CrossRef]
  6. See, e.g., 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]
  7. H. Sõnajalg, P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators,” Opt. Lett. 21, 1162–1164 (1996).
    [CrossRef] [PubMed]
  8. H. Sõnajalg, M. Ratsep, P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. 22, 310–312 (1997).
    [CrossRef] [PubMed]
  9. 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]
  10. C. Conti, S. Trillo, “Paraxial envelope X-waves,” Opt. Lett. 28, 1090–1093 (2003).
    [CrossRef]
  11. M. A. Porras, G. Valiulis, P. Di Trapani, “Unified description of Bessel X-waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003).
    [CrossRef]
  12. M. A. Porras, I. Gonzalo, “Control of temporal characteristics of Bessel-X pulses in dispersive media,” Opt. Commun. 217, 257–264 (2003).
    [CrossRef]
  13. 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]
  14. S. Longhi, “Spatial-temporal Gauss–Laguerre waves in dispersive media,” Phys. Rev. E 68, 066612 (2003).
    [CrossRef]
  15. M. Zamboni-Rached, A. M. Shaarawi, E. Recami, “Focused X-shaped pulses,” J. Opt. Soc. Am. A 21, 1564–1574 (2004).
    [CrossRef]
  16. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).
  17. The chirped Gaussian pulse can recover its longitudinal width, but with a diminished intensity, owing to progressive transverse spreading.
  18. This problem could be overcome, in principle, by using a lens. But it would not be a good solution, because a different lens would be necessary (besides a different chirp parameter C) for each different value of ZT1=T0.
  19. I. S. Gradshteyn, I. M. Ryzhik, Integrals, Series and Products, 4th ed. (Academic, New York, 1965).
  20. Obviously, in the case of finite apertures, one must take into account the finite field depth of the X-shaped pulses. We shall see this in Section 4.
  21. Again, if we consider finite-aperture generation, one must take into account the finite field depth of the X-shaped pulses, as we shall see in Section 4.
  22. We call spot size the transverse width, which for Gaussian pulses is the transverse distance from the pulse center to the position at which the intensity decreases by a factor 1/eand, in the case of the considered X-shaped pulses, the transverse distance from the pulse center to where the first zero of the intensity occurs.
  23. Remembering that (Vg)X=(Vg)Gauss/cos θand (β2)X= cos θ(β2)Gauss.
  24. One can easily verify that in this case the distance Ldispcoincides with zdiff=3πw02/λ0,which is the distance where a Gaussian pulsedoubles its transverse width.

2004

2003

C. Conti, S. Trillo, “Paraxial envelope X-waves,” Opt. Lett. 28, 1090–1093 (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]

M. A. Porras, G. Valiulis, P. Di Trapani, “Unified description of Bessel X-waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (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]

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

2002

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. Phys. J. D 21, 217–228 (2002).
[CrossRef]

2001

See, e.g., 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]

1998

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

1997

1996

1992

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]

1989

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]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

Besieris, I. M.

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]

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]

Di Trapani, P.

M. A. Porras, G. Valiulis, P. Di Trapani, “Unified description of Bessel X-waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003).
[CrossRef]

Fontana, F.

See, e.g., 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]

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]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Integrals, Series and Products, 4th ed. (Academic, New York, 1965).

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]

Hernández-Figueroa, H. E.

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]

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, E. Recami, 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]

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]

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]

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]

Porras, M. A.

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, G. Valiulis, P. Di Trapani, “Unified description of Bessel X-waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (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]

Ratsep, M.

Recami, E.

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

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, E. Recami, 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]

See, e.g., 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), and references therein.
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Integrals, Series and Products, 4th ed. (Academic, New York, 1965).

Saari, P.

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. M.

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

Sõnajalg, H.

Trillo, S.

Valiulis, G.

M. A. Porras, G. Valiulis, P. Di Trapani, “Unified description of Bessel X-waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003).
[CrossRef]

Zamboni-Rached, M.

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

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, E. Recami, 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]

See, e.g., 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]

Ziolkowski, R. W.

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]

Eur. Phys. J. D

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. Phys. J. D 21, 217–228 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

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

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. Math. Phys.

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. Opt. Soc. Am. A

Opt. Commun.

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]

Opt. Lett.

Phys. Rev. E

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

See, e.g., 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. A. Porras, G. Valiulis, P. Di Trapani, “Unified description of Bessel X-waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003).
[CrossRef]

Physica A

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

Other

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

The chirped Gaussian pulse can recover its longitudinal width, but with a diminished intensity, owing to progressive transverse spreading.

This problem could be overcome, in principle, by using a lens. But it would not be a good solution, because a different lens would be necessary (besides a different chirp parameter C) for each different value of ZT1=T0.

I. S. Gradshteyn, I. M. Ryzhik, Integrals, Series and Products, 4th ed. (Academic, New York, 1965).

Obviously, in the case of finite apertures, one must take into account the finite field depth of the X-shaped pulses. We shall see this in Section 4.

Again, if we consider finite-aperture generation, one must take into account the finite field depth of the X-shaped pulses, as we shall see in Section 4.

We call spot size the transverse width, which for Gaussian pulses is the transverse distance from the pulse center to the position at which the intensity decreases by a factor 1/eand, in the case of the considered X-shaped pulses, the transverse distance from the pulse center to where the first zero of the intensity occurs.

Remembering that (Vg)X=(Vg)Gauss/cos θand (β2)X= cos θ(β2)Gauss.

One can easily verify that in this case the distance Ldispcoincides with zdiff=3πw02/λ0,which is the distance where a Gaussian pulsedoubles its transverse width.

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Figures (4)

Fig. 1
Fig. 1

Longitudinal-shape evolution of a chirped X-shaped pulse with C=-1. One can see that such a pulse recovers its full longitudinal shape at the position z=Ldisp=T02/β2.

Fig. 2
Fig. 2

Transverse-shape evolution of a chirped X-type pulse with C=-1, T0=0.4 ps, λ0=0.2 μm, and axicon angle θ=0.002 rad, which correspond to an initial transverse width of Δρ=24.63 μm. One can see that the pulse recovers its entire transverse shape at the distance z=Ldisp=T02/β2, which, in this case, is equal to 0.373 m.

Fig. 3
Fig. 3

(a) Longitudinal-shape evolution of a chirped Gaussian pulse propagating in fused silica, with λ0=0.2 μm, T0=0.4 ps, C=-1, and initial transverse width (spot size) Δρ0=0.117 mm. (b) Transverse-shape evolution for the same pulse.

Fig. 4
Fig. 4

(a) Longitudinal-shape evolution of a chirped X-type pulse propagating in fused silica with λ0=0.2 μm, T0=0.4 ps, C=-1, and axicon angle θ=0.00084 rad, which correspond to an initial transverse width of Δρ0=0.117 mm. (b) Transverse-shape evolution for the same pulse.    

Equations (36)

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ψ(ρ, z, t)=J0(kρρ)exp(iβz)exp(-iωt),
Ψ(ρ, z, t)=-S(ω)J0n(ω)ωc ρ sin θ×exp[iβ(ω)z]exp(-iωt)dω,
S(ω)=T02π(1+iC)exp[-q2(ω-ω0)2],
q2=T022(1+iC),
βω=cos θcn(ω)+ω nω;
2βω2=cos θc2 nω+ω 2nω2.
Ψ(ρ, z, t)=T0exp[iβ(ω0)z]exp(-iω0t)2π(1+iC)×-dωJ0n(ω)ωcsin θρ×expi (ω-ω0)Vg (z-Vgt)×exp(ω-ω0)2iβ22 z-q2.
Ψ(ρ=0, z, t)=exp[iβ(ω0)z]exp(-iω0t)×T0T02-iβ2(1+iC)z×exp-(z-Vgt)2(1+iC)2Vg2[T02-iβ2(1+iC)z].
T1T0=1+Cβ2zT022+β2zT0221/2.
ZT1=T0=-2CT02β2(C2+1).
z=zc+Δz,
t=tczc/Vg,
Ψ(ρ, zc=0, Δz)=T0exp(iβ0Δz)[2π(1+iC)]1/2-dωJ0(kρ(ω)ρ)×exp-T022(1+C2) (ω-ω0)2×expi(ω-ω0)ΔzVg+(ω-ω0)2β2Δz2+(ω-ω0)2T02C2(1+C2),
Ψ(ρ, zc=ZT1=T0, Δz)=T0expiβ0zc-Δz-czccos θn(ω0)Vg2π(1+iC)×-dωJ0(kρ(ω)ρ)×exp-T022(1+C2) (ω-ω0)2×exp-i(ω-ω0)ΔzVg+(ω-ω0)2β2Δz2+(ω-ω0)2T02C2(1+C2),
|Ψ(ρ, zc=0, Δz)|2=|Ψ(ρ, zc=ZT1=T0, -Δz)|2.
kρ=n2(ω)ω2c2-n2(ω)ω2c2cos θ1/2.
n(ω)ωAu+B,
A=ω0nωω0+n(ω0)=cVgcos θ,B=n(ω0)ω0,
kρsin θcA2u2+2ABu+B2
J0(kρ(ω)ρ)=J0ρ sin θcA2u2+2ABu+B2.
J0(mR)=J0(mx)J0(my)+2p=1Jp(mx)Jp(my)cos(pϕ),
J0ρ sin θcA2u2+2ABu+B2=J0ρ sin θc AuJ0ρ sin θc B+2p=1Jpρ sin θc AuJpρ sin θc B(-1)p.
Ψ(ρ, z=zc, t=zc/Vg)=T0exp[iβ(ω0)z]exp(-iω0t)2π(1+iC)×exp-tan2 θρ28Vg2(-iβ2zc/2+q2)-iβ2zc/2+q2×Γ(1/2)J0n(ω0)ω0sin θρc×I0tan2 θρ28Vg2(-iβ2zc/2+q2)+2p=12pΓ(p+1/2)Γ(p+1)Γ(2p+1)×J2pn(ω0)ω0sin θρc×I2ptan2 θρ28Vg2(-iβ2zc/2+q2),
exp-tan2 θρ28Vg2(-iβ2zc/2+q2)
J0n(ω0)ω0sin θρc,
ΔρG(zc)=2c(T02+β2Czc)2+β22zc2T0sin θ[n(ω0)+ω0nω|ω0]
ΔρB=2.4cn(ω0)ω0sin θ,
ΔρG(zc=ZT1=T0)ΔρB=2n(ω0)2.4(n(ω0)+ω0nω|ω0) T0ω0,
T0ω0>2.4(n(ω0)+ω0nω|ω0)2n(ω0),
T0ω0<2.4(n(ω0)+ω0nω|ω0)2n(ω0),
zc=T02|C|+T02n(ω0)ω02.42(1+C2)n(ω0)+ω0nωω02-4T02n2(ω0)ω021/2|β2|(1+C2),
n2(ω)=1+j=1NBjωj2ωj2-ω2,
Ldiff=Rtan θ,
ZT1=T0Ldiff-2CT02β2(C2+1)Rtan θ.
-2CT02β2(C2+1)=Rtan θ.
ΨGauss(ρ, z, t)=exp-ρ2w02+2iz/k0exp(z-Vgt)2(1+iC)2Vg2T02[1-i(1+iC)β2z/T02]1+2izw02k01-i(1+iC) β2zT021/2,

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