Abstract

Recently, the classical Talbot effect (self-imaging of optical wave fields) has attracted a renewed interest, as the concept has been generalized to the domain of pulsed wave fields by several authors. In this paper we discuss the self-imaging of three-dimensional images. We construct pulsed wave fields that can be used as self-imaging “pixels” of a three-dimensional image and show that their superpositions reproduce the spatial separated copies of its initial three-dimensional intensity distribution at specific time intervals. The derived wave fields will be shown to be directly related to the fundamental localized wave solutions of the homogeneous scalar wave equation – focus wave modes. Our discussion is illustrated by some spectacular numerical simulations. We also propose a general idea for the optical generation of the derived wave fields. The results will be compared to the work, published so far on the subject.

© 2002 Optical Society of America

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References

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  1. W. D. Montgomery, “Self-Imaging Objects of Infinite Aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [Crossref]
  2. J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
    [Crossref]
  3. Z. Bouchal and J. Wagner, “Self-reconstruction effect in free propagation of wavefield,” Opt. Commun. 176, 299–307 (2000).
    [Crossref]
  4. J. Wagner and Z. Bouchal, “Experimental realization of self-reconstruction of the 2D aperiodic objects,” Opt.Commun. 176, 309–311 (2000).
    [Crossref]
  5. R. Piestun, Y. Y. Schechner, and J. Shamir, “Self-imaging with finite energy,” Opt. Lett. 22(4), 200–202 (1997).
    [Crossref] [PubMed]
  6. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  7. R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
    [Crossref]
  8. Z. Bouchal and M. Bertolotti, “Self-reconstruction of wave packets due to spatio-temporal couplings,” J. Mod. Opt. 47, 1455 – 1467 (2000).
    [Crossref]
  9. Z. Bouchal, “Self-reconstruction ability of wave field” Proc. SPIE,  vol.4356, 217–224, (2001).
    [Crossref]
  10. J. Salo and M. M. Salomaa, “Diffraction-free pulses at arbitrary speeds,” J. Opt. A: Pure Appl. Opt. 3, 366–373 (2001).
    [Crossref]
  11. H. Wang, C. Zhou, L. Jianlang, and L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,”Microwave and Opt. Technol. Lett. 25, 184–187 (2000)
    [Crossref]
  12. J. Azaña and M. A. Muriel, “Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber grating,” Opt. Lett. 24, 1672–1674 (1999).
    [Crossref]
  13. P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Physics 7, 32–39 (1997).
  14. A. A. Maznev, T. F. Crimmins, and K.A. Nelson, “How to make femtosecond pulses overlap,” Opt. Lett. 23, 1378–1380 (1998).
    [Crossref]
  15. Zs. Bor and B. Rácz, “Group velocity dispersion in prisms and its application to pulse compression and traveling-wave excitation,” Opt. Commun. 54, 165–170 (1985).
    [Crossref]
  16. P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
    [Crossref]
  17. P. Saari, J. Aaviksoo, A. Freiberg, and K. Timpmann, “Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,” Opt. Commun. 39, 94–98 (1981).
    [Crossref]
  18. O. Svelto, Principles of Lasers (3rd ed. Plenum Press1989).
  19. I. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Progr. In Electromagn. Research 19, 1 (1998).
    [Crossref]
  20. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
    [Crossref]
  21. K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A 17, 1785–1790 (2000).
    [Crossref]
  22. K. Reivelt and P. Saari, “Optical generation of focus wave modes: errata,” J. Opt. Soc. Am. A 18, 2026–2026, (2001).
    [Crossref]
  23. K. Reivelt and P. Saari, “Optically realizable localized wave solutions of homogeneous scalar wave equation,” Phys. Rev. E (to be published) [accepted for publication].

2001 (3)

Z. Bouchal, “Self-reconstruction ability of wave field” Proc. SPIE,  vol.4356, 217–224, (2001).
[Crossref]

J. Salo and M. M. Salomaa, “Diffraction-free pulses at arbitrary speeds,” J. Opt. A: Pure Appl. Opt. 3, 366–373 (2001).
[Crossref]

K. Reivelt and P. Saari, “Optical generation of focus wave modes: errata,” J. Opt. Soc. Am. A 18, 2026–2026, (2001).
[Crossref]

2000 (5)

H. Wang, C. Zhou, L. Jianlang, and L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,”Microwave and Opt. Technol. Lett. 25, 184–187 (2000)
[Crossref]

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

J. Wagner and Z. Bouchal, “Experimental realization of self-reconstruction of the 2D aperiodic objects,” Opt.Commun. 176, 309–311 (2000).
[Crossref]

Z. Bouchal and M. Bertolotti, “Self-reconstruction of wave packets due to spatio-temporal couplings,” J. Mod. Opt. 47, 1455 – 1467 (2000).
[Crossref]

K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A 17, 1785–1790 (2000).
[Crossref]

1999 (1)

1998 (4)

R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
[Crossref]

A. A. Maznev, T. F. Crimmins, and K.A. Nelson, “How to make femtosecond pulses overlap,” Opt. Lett. 23, 1378–1380 (1998).
[Crossref]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

I. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Progr. In Electromagn. Research 19, 1 (1998).
[Crossref]

1997 (3)

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

R. Piestun, Y. Y. Schechner, and J. Shamir, “Self-imaging with finite energy,” Opt. Lett. 22(4), 200–202 (1997).
[Crossref] [PubMed]

P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Physics 7, 32–39 (1997).

1993 (1)

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[Crossref]

1987 (1)

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

1985 (1)

Zs. Bor and B. Rácz, “Group velocity dispersion in prisms and its application to pulse compression and traveling-wave excitation,” Opt. Commun. 54, 165–170 (1985).
[Crossref]

1981 (1)

P. Saari, J. Aaviksoo, A. Freiberg, and K. Timpmann, “Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,” Opt. Commun. 39, 94–98 (1981).
[Crossref]

1967 (1)

Aaviksoo, J.

P. Saari, J. Aaviksoo, A. Freiberg, and K. Timpmann, “Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,” Opt. Commun. 39, 94–98 (1981).
[Crossref]

Abdel-Rahman, M.

I. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Progr. In Electromagn. Research 19, 1 (1998).
[Crossref]

Azaña, J.

Bertolotti, M.

Z. Bouchal and M. Bertolotti, “Self-reconstruction of wave packets due to spatio-temporal couplings,” J. Mod. Opt. 47, 1455 – 1467 (2000).
[Crossref]

Besieris, I.

I. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Progr. In Electromagn. Research 19, 1 (1998).
[Crossref]

Bor, Zs.

Zs. Bor and B. Rácz, “Group velocity dispersion in prisms and its application to pulse compression and traveling-wave excitation,” Opt. Commun. 54, 165–170 (1985).
[Crossref]

Bouchal, Z.

Z. Bouchal, “Self-reconstruction ability of wave field” Proc. SPIE,  vol.4356, 217–224, (2001).
[Crossref]

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

J. Wagner and Z. Bouchal, “Experimental realization of self-reconstruction of the 2D aperiodic objects,” Opt.Commun. 176, 309–311 (2000).
[Crossref]

Z. Bouchal and M. Bertolotti, “Self-reconstruction of wave packets due to spatio-temporal couplings,” J. Mod. Opt. 47, 1455 – 1467 (2000).
[Crossref]

Caironi, D.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

Chatzipetros, A.

I. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Progr. In Electromagn. Research 19, 1 (1998).
[Crossref]

Crimmins, T. F.

Danielius, R.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

Di Trapani, P.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

Dubietis, A.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

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]

Freiberg, A.

P. Saari, J. Aaviksoo, A. Freiberg, and K. Timpmann, “Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,” Opt. Commun. 39, 94–98 (1981).
[Crossref]

Friberg, A. T.

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[Crossref]

Jianlang, L.

H. Wang, C. Zhou, L. Jianlang, and L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,”Microwave and Opt. Technol. Lett. 25, 184–187 (2000)
[Crossref]

Liu, L.

H. Wang, C. Zhou, L. Jianlang, and L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,”Microwave and Opt. Technol. Lett. 25, 184–187 (2000)
[Crossref]

Maznev, A. A.

Miceli, J. J.

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

Montgomery, W. D.

Muriel, M. A.

Nelson, K.A.

Piestun, R.

Piskarskas, A.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

Rácz, B.

Zs. Bor and B. Rácz, “Group velocity dispersion in prisms and its application to pulse compression and traveling-wave excitation,” Opt. Commun. 54, 165–170 (1985).
[Crossref]

Reivelt, K.

K. Reivelt and P. Saari, “Optical generation of focus wave modes: errata,” J. Opt. Soc. Am. A 18, 2026–2026, (2001).
[Crossref]

K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A 17, 1785–1790 (2000).
[Crossref]

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

K. Reivelt and P. Saari, “Optically realizable localized wave solutions of homogeneous scalar wave equation,” Phys. Rev. E (to be published) [accepted for publication].

Saari, P.

K. Reivelt and P. Saari, “Optical generation of focus wave modes: errata,” J. Opt. Soc. Am. A 18, 2026–2026, (2001).
[Crossref]

K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A 17, 1785–1790 (2000).
[Crossref]

P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Physics 7, 32–39 (1997).

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

P. Saari, J. Aaviksoo, A. Freiberg, and K. Timpmann, “Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,” Opt. Commun. 39, 94–98 (1981).
[Crossref]

K. Reivelt and P. Saari, “Optically realizable localized wave solutions of homogeneous scalar wave equation,” Phys. Rev. E (to be published) [accepted for publication].

Salo, J.

J. Salo and M. M. Salomaa, “Diffraction-free pulses at arbitrary speeds,” J. Opt. A: Pure Appl. Opt. 3, 366–373 (2001).
[Crossref]

Salomaa, M. M.

J. Salo and M. M. Salomaa, “Diffraction-free pulses at arbitrary speeds,” J. Opt. A: Pure Appl. Opt. 3, 366–373 (2001).
[Crossref]

Schechner, Y. Y.

Shaarawi, A.

I. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Progr. In Electromagn. Research 19, 1 (1998).
[Crossref]

Shamir, J.

Sõnajalg, H.

P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Physics 7, 32–39 (1997).

Svelto, O.

O. Svelto, Principles of Lasers (3rd ed. Plenum Press1989).

Timpmann, K.

P. Saari, J. Aaviksoo, A. Freiberg, and K. Timpmann, “Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,” Opt. Commun. 39, 94–98 (1981).
[Crossref]

Turunen, J.

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[Crossref]

Valiulis, G.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

Wagner, J.

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

J. Wagner and Z. Bouchal, “Experimental realization of self-reconstruction of the 2D aperiodic objects,” Opt.Commun. 176, 309–311 (2000).
[Crossref]

Wang, H.

H. Wang, C. Zhou, L. Jianlang, and L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,”Microwave and Opt. Technol. Lett. 25, 184–187 (2000)
[Crossref]

Zhou, C.

H. Wang, C. Zhou, L. Jianlang, and L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,”Microwave and Opt. Technol. Lett. 25, 184–187 (2000)
[Crossref]

J. Mod. Opt. (1)

Z. Bouchal and M. Bertolotti, “Self-reconstruction of wave packets due to spatio-temporal couplings,” J. Mod. Opt. 47, 1455 – 1467 (2000).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

J. Salo and M. M. Salomaa, “Diffraction-free pulses at arbitrary speeds,” J. Opt. A: Pure Appl. Opt. 3, 366–373 (2001).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Laser Physics (1)

P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Physics 7, 32–39 (1997).

Microwave and Opt. Technol. Lett. (1)

H. Wang, C. Zhou, L. Jianlang, and L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,”Microwave and Opt. Technol. Lett. 25, 184–187 (2000)
[Crossref]

Opt. Commun. (3)

Zs. Bor and B. Rácz, “Group velocity dispersion in prisms and its application to pulse compression and traveling-wave excitation,” Opt. Commun. 54, 165–170 (1985).
[Crossref]

P. Saari, J. Aaviksoo, A. Freiberg, and K. Timpmann, “Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,” Opt. Commun. 39, 94–98 (1981).
[Crossref]

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

Opt. Lett. (3)

Opt.Commun. (1)

J. Wagner and Z. Bouchal, “Experimental realization of self-reconstruction of the 2D aperiodic objects,” Opt.Commun. 176, 309–311 (2000).
[Crossref]

Phys. Rev. Lett. (3)

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

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses,” Phys. Rev. Lett. 81, 570–573 (1998).
[Crossref]

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

Proc. SPIE, (1)

Z. Bouchal, “Self-reconstruction ability of wave field” Proc. SPIE,  vol.4356, 217–224, (2001).
[Crossref]

Progr. In Electromagn. Research (1)

I. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions of the scalar wave equation,” Progr. In Electromagn. Research 19, 1 (1998).
[Crossref]

Pure Appl. Opt. (1)

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[Crossref]

Other (2)

O. Svelto, Principles of Lasers (3rd ed. Plenum Press1989).

K. Reivelt and P. Saari, “Optically realizable localized wave solutions of homogeneous scalar wave equation,” Phys. Rev. E (to be published) [accepted for publication].

Supplementary Material (3)

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Figures (7)

Fig. 1.
Fig. 1.

(0.85 MB) The video clip of the temporal evolution of the superposition of two interfering optical pulses (red and cyanide line) that have equal carrier wavelengths and group velocities but different phase velocities. The blue line and the dotted line denote the amplitude and the envelope of the sum of the two waves respectively.

Fig. 2.
Fig. 2.

(a) The temporal evolution of the spatial intensity distributions of the pulsed self-imaging wave field (the 1.2 MB movie); (b) The spatial amplitude of a monochromatic self-imaging wave field (see also text in Sec. 4).

Fig. 3.
Fig. 3.

(a) An excerpt of the spatial amplitude distribution of a tilted pulse. vg and vp denote its group and phase velocities respectively, k0 and λ0 are the wave vector and the wavelength of the plane wave of the carrier wave number, t is an arbitrary time and ϕ is the tilt angle of the pulse; (b) On the phase and group velocities of the tilted pulses (see text).

Fig. 4.
Fig. 4.

The examples of the supports of the angular spectrums of plane waves of (a) the specified tilted pulse for β= 40 rad m-1 and γ= 1 (k0 is the wave vector of the carrier wavelength), (b) the self-imaging superposition of the tilted pulses (see text). The “rainbow” on the pictures denotes the visible spectral region.

Fig. 5.
Fig. 5.

(a) The Fourier spectrum and (b) the spatial amplitude of a train of sinusoidal waves.

Fig. 6.
Fig. 6.

The instantaneous intensity distribution of a FWM if β= 40 rad m-1, θ (T)(k0 ,β) = 0.22 deg, γ= 1 and the frequency spectrum A(k) has a rectangular shape and extends over the wavelengths 400–800nm.

Fig. 7.
Fig. 7.

(1.4 MB movie) A numerical example of the evolution of a self-imaging spatial image (smiling human face) consisting of eight self-imaging pixels, each consisting of five FWM’s of different constant β(see text).

Equations (38)

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

2 Ψ 1 c 2 2 Ψ t 2 = 0 ,
Ψ x y z t = exp [ iωt ] 0 π sin θ 0 2 π dϕA k θ ϕ
× exp [ ik ( x cos ϕ si n θ + y sin ϕ sin θ + z cos θ ) ]
Ψ x y z t = Ψ ( x , y , z + d , t ) ,
kd cos θ = ψ + 2 πq ,
k z = k cos θ = ψ + 2 πq d ,
Ψ ρ z t = exp [ iωt ] n c n exp [ inϕ ] 0 π sin θ A k θ ,
× J n ( sin θ ) exp [ ikz cos θ ]
Ψ ρ z t = exp [ iωt ] q a q J 0 [ 1 ( 2 πq d ) 2 ] exp [ i 2 πq d z ] .
Ψ 0 z t = exp [ iωt ] q a q exp [ i 2 πq d z ] ,
Ψ ( T ) x z t = 0 dkA ( k ) exp [ ik ( x sin θ ( k ) + z cos θ ( k ) ct ) ] .
θ ( T ) ( k , β ) = arccos [ γ ( k 2 β ) k ] ,
v g ( k ) = ( dk z ) 1 ,
k z = k cos θ ( T ) k β = γk 2 βγ
v g ( k ) = c γ ,
v p ( β ) = c cos θ ( T ) ( k 0 , β ) = ck 0 γ ( k 0 2 β ) ,
Ψ ( SI ) x z t = q a q 0 dkA ( k ) exp [ ik ( x sin θ ( T ) k β q + z cos θ ( T ) k β q ct ) ] ,
k x k β = k x k 0 β + dk x k β dk k 0 ( k k 0 )
k z k β = k z k 0 β + dk z k β dk k 0 ( k k 0 )
= k z k 0 β + γ ( k k 0 )
Ψ ( SI ) x z t q a q C q x ct exp ik 0 ( x sin θ ( T ) k 0 β q + z cos θ ( T ) k 0 β q ct ) ,
C q ( x , ct ) = k 0 dkA ( k + k 0 ) exp [ ik ( x dk x k β q dk k 0 + ct ) ]
Ψ ( SI ) x z t C x ct q a q exp ik 0 ( x sin θ ( T ) k 0 β q + z cos θ ( T ) ( k 0 , β 0 ) ct ) .
kd cos θ ( T ) ( k 0 , β q ) = ψ + 2 πq .
2 πq d = γk 0 2 γβ ,
β q = k 0 2 πq γd
θ ( T ) k β q = arccos ( γ ( k k 0 ) k + 2 π kd q ) ,
θ ( T ) k 0 β q = arccos ( 2 π k 0 d q ) .
0 < q < k 0 d 2 π
Ψ ( z ) = A 0 sin [ ½ ( 2 n + 1 ) Δ k z z ] sin ( ½ Δ k z z ) ,
Ψ ρ z t = 0 dkA ( k ) J 0 [ sin θ ( k ) ] exp [ ik ( z cos θ ( k ) ct ) ] .
Ψ ( SI ) ρ z t = q a q 0 dkA ( k ) ,
× J 0 [ sin θ ( T ) k β q ] exp [ ik ( z cos θ ( T ) k β q ct ) ]
Ψ ( SI ) ρ z t = q a q exp [ i 2 γ β q z ] 0 dkA ( k ) .
× J 0 [ 1 ( γ ( k 2 β q ) k ) 2 ] exp [ ik ( ct ) ]
k z , q = γ ( k k 0 ) + 2 π d q ,
k z , q = γk + 2 π d q ,
U ' ν ψ ω = U ν ψ ω m = M M n = 0 N δ ( ω ω m ) δ ( ν ν mn ) ,

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