Abstract

We propose a novel technique of doubling optical pulses in both frequency and time domains based on a combination of cross-phase modulation induced by a triangular pump pulse in a nonlinear Kerr medium and subsequent propagation in a dispersive medium.

© 2009 Optical Society of America

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References

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  1. S. Boscolo, A. I. Latkin, and S. K. Turitsyn, “Passive nonlinear pulse shaping in normally dispersive fiber systems,” IEEE J. Quantum Electron. 44, 1196-1203 (2008).
    [CrossRef]
  2. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).
  3. M. N. Islam, L. F. Mollenauer, R. H. Stolen, J. R. Simpson, and H.-T. Shang, “Cross-phase modulation in optical fibers,” Opt. Lett. 12, 625-627 (1987).
    [CrossRef] [PubMed]
  4. G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Temporal and spectral effects of cross-phase modulation on copropagating ultrashort pulses in optical fibers,” Phys. Rev. A 40, 5063-5072 (1989).
    [CrossRef] [PubMed]
  5. C. Yeh and L. Bergman, “Enhanced pulse compression in a nonlinear fiber by a wavelength division multiplexed optical pulse,” Phys. Rev. E 57, 2398-2404 (1998).
    [CrossRef]
  6. L. Helczynski, B. Hall, D. Anderson, M. Lisak, A. Berntson, and M. Desaix, “Cross-phase modulation induced pulse splitting--the optical axe,” Phys. Scr. T84, 81-84 (2000).
    [CrossRef]
  7. J. Li, B.-E. Olsson, M. Karlsson, and P. A. Andrekson, “OTDM demultiplexer based on XPM-induced wavelength shifting in highly nonlinear fiber,” IEEE Photonics Technol. Lett. 15, 1770-1772 (2003).
    [CrossRef]
  8. Y. Park, M. H. Asghari, T.-J. Ahn, and J. Azaña, “Transform-limited picosecond pulse shaping based on temporal coherence synthesization,” Opt. Express 15, 9584-9599 (2007).
    [CrossRef] [PubMed]
  9. P. Petropoulos, M. Ibsen, A. D. Ellis, and D. J. Richardson, “Rectangular pulse generation based on pulse reshaping using a superstructured fiber Bragg grating,” J. Lightwave Technol. 19, 746-752 (2001).
    [CrossRef]
  10. F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating,” IEEE Photonics Technol. Lett. 18, 829-831 (2006).
    [CrossRef]
  11. F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. F. Roelens, P. Petropoulos, and D. J. Richardson, “Ultra-flat SPM-broadened spectra in highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express 14, 7617-7622 (2006).
    [CrossRef] [PubMed]
  12. F. Parmigiani, M. Ibsen, T. T. Ng, L. Provost, P. Petropoulos, and D. J. Richardson, “An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper,” IEEE Photonics Technol. Lett. 20, 1461-1463 (2008).
    [CrossRef]
  13. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951-1963 (1994).
    [CrossRef]
  14. M. Nakazawa and T. Hirooka, “Distortion-free optical transmission using time-domain optical Fourier transformation and transform-limited optical pulses,” J. Opt. Soc. Am. B 22, 1842-1855 (2005).
    [CrossRef]
  15. R. S. Bhamber, S. Boscolo, A. I. Latkin, and S. K. Turitsyn, “All-optical TDM to WDM signal conversion and partial regeneration using XPM with triangular pulses,” in Proceedings of the 34th European Conference on Optical Communication (ECOC 2008), paper Th.1.B.2 (2008).
    [CrossRef]

2008

S. Boscolo, A. I. Latkin, and S. K. Turitsyn, “Passive nonlinear pulse shaping in normally dispersive fiber systems,” IEEE J. Quantum Electron. 44, 1196-1203 (2008).
[CrossRef]

F. Parmigiani, M. Ibsen, T. T. Ng, L. Provost, P. Petropoulos, and D. J. Richardson, “An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper,” IEEE Photonics Technol. Lett. 20, 1461-1463 (2008).
[CrossRef]

2007

2006

F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. F. Roelens, P. Petropoulos, and D. J. Richardson, “Ultra-flat SPM-broadened spectra in highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express 14, 7617-7622 (2006).
[CrossRef] [PubMed]

F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating,” IEEE Photonics Technol. Lett. 18, 829-831 (2006).
[CrossRef]

2005

2003

J. Li, B.-E. Olsson, M. Karlsson, and P. A. Andrekson, “OTDM demultiplexer based on XPM-induced wavelength shifting in highly nonlinear fiber,” IEEE Photonics Technol. Lett. 15, 1770-1772 (2003).
[CrossRef]

2001

2000

L. Helczynski, B. Hall, D. Anderson, M. Lisak, A. Berntson, and M. Desaix, “Cross-phase modulation induced pulse splitting--the optical axe,” Phys. Scr. T84, 81-84 (2000).
[CrossRef]

1998

C. Yeh and L. Bergman, “Enhanced pulse compression in a nonlinear fiber by a wavelength division multiplexed optical pulse,” Phys. Rev. E 57, 2398-2404 (1998).
[CrossRef]

1994

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951-1963 (1994).
[CrossRef]

1989

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Temporal and spectral effects of cross-phase modulation on copropagating ultrashort pulses in optical fibers,” Phys. Rev. A 40, 5063-5072 (1989).
[CrossRef] [PubMed]

1987

Agrawal, G. P.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Temporal and spectral effects of cross-phase modulation on copropagating ultrashort pulses in optical fibers,” Phys. Rev. A 40, 5063-5072 (1989).
[CrossRef] [PubMed]

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

Ahn, T.-J.

Alfano, R. R.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Temporal and spectral effects of cross-phase modulation on copropagating ultrashort pulses in optical fibers,” Phys. Rev. A 40, 5063-5072 (1989).
[CrossRef] [PubMed]

Anderson, D.

L. Helczynski, B. Hall, D. Anderson, M. Lisak, A. Berntson, and M. Desaix, “Cross-phase modulation induced pulse splitting--the optical axe,” Phys. Scr. T84, 81-84 (2000).
[CrossRef]

Andrekson, P. A.

J. Li, B.-E. Olsson, M. Karlsson, and P. A. Andrekson, “OTDM demultiplexer based on XPM-induced wavelength shifting in highly nonlinear fiber,” IEEE Photonics Technol. Lett. 15, 1770-1772 (2003).
[CrossRef]

Asghari, M. H.

Azaña, J.

Baldeck, P. L.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Temporal and spectral effects of cross-phase modulation on copropagating ultrashort pulses in optical fibers,” Phys. Rev. A 40, 5063-5072 (1989).
[CrossRef] [PubMed]

Bergman, L.

C. Yeh and L. Bergman, “Enhanced pulse compression in a nonlinear fiber by a wavelength division multiplexed optical pulse,” Phys. Rev. E 57, 2398-2404 (1998).
[CrossRef]

Berntson, A.

L. Helczynski, B. Hall, D. Anderson, M. Lisak, A. Berntson, and M. Desaix, “Cross-phase modulation induced pulse splitting--the optical axe,” Phys. Scr. T84, 81-84 (2000).
[CrossRef]

Bhamber, R. S.

R. S. Bhamber, S. Boscolo, A. I. Latkin, and S. K. Turitsyn, “All-optical TDM to WDM signal conversion and partial regeneration using XPM with triangular pulses,” in Proceedings of the 34th European Conference on Optical Communication (ECOC 2008), paper Th.1.B.2 (2008).
[CrossRef]

Boscolo, S.

S. Boscolo, A. I. Latkin, and S. K. Turitsyn, “Passive nonlinear pulse shaping in normally dispersive fiber systems,” IEEE J. Quantum Electron. 44, 1196-1203 (2008).
[CrossRef]

R. S. Bhamber, S. Boscolo, A. I. Latkin, and S. K. Turitsyn, “All-optical TDM to WDM signal conversion and partial regeneration using XPM with triangular pulses,” in Proceedings of the 34th European Conference on Optical Communication (ECOC 2008), paper Th.1.B.2 (2008).
[CrossRef]

Desaix, M.

L. Helczynski, B. Hall, D. Anderson, M. Lisak, A. Berntson, and M. Desaix, “Cross-phase modulation induced pulse splitting--the optical axe,” Phys. Scr. T84, 81-84 (2000).
[CrossRef]

Ellis, A. D.

Finot, C.

Hall, B.

L. Helczynski, B. Hall, D. Anderson, M. Lisak, A. Berntson, and M. Desaix, “Cross-phase modulation induced pulse splitting--the optical axe,” Phys. Scr. T84, 81-84 (2000).
[CrossRef]

Helczynski, L.

L. Helczynski, B. Hall, D. Anderson, M. Lisak, A. Berntson, and M. Desaix, “Cross-phase modulation induced pulse splitting--the optical axe,” Phys. Scr. T84, 81-84 (2000).
[CrossRef]

Hirooka, T.

Ibsen, M.

F. Parmigiani, M. Ibsen, T. T. Ng, L. Provost, P. Petropoulos, and D. J. Richardson, “An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper,” IEEE Photonics Technol. Lett. 20, 1461-1463 (2008).
[CrossRef]

F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating,” IEEE Photonics Technol. Lett. 18, 829-831 (2006).
[CrossRef]

F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. F. Roelens, P. Petropoulos, and D. J. Richardson, “Ultra-flat SPM-broadened spectra in highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express 14, 7617-7622 (2006).
[CrossRef] [PubMed]

P. Petropoulos, M. Ibsen, A. D. Ellis, and D. J. Richardson, “Rectangular pulse generation based on pulse reshaping using a superstructured fiber Bragg grating,” J. Lightwave Technol. 19, 746-752 (2001).
[CrossRef]

Islam, M. N.

Karlsson, M.

J. Li, B.-E. Olsson, M. Karlsson, and P. A. Andrekson, “OTDM demultiplexer based on XPM-induced wavelength shifting in highly nonlinear fiber,” IEEE Photonics Technol. Lett. 15, 1770-1772 (2003).
[CrossRef]

Kolner, B. H.

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951-1963 (1994).
[CrossRef]

Latkin, A. I.

S. Boscolo, A. I. Latkin, and S. K. Turitsyn, “Passive nonlinear pulse shaping in normally dispersive fiber systems,” IEEE J. Quantum Electron. 44, 1196-1203 (2008).
[CrossRef]

R. S. Bhamber, S. Boscolo, A. I. Latkin, and S. K. Turitsyn, “All-optical TDM to WDM signal conversion and partial regeneration using XPM with triangular pulses,” in Proceedings of the 34th European Conference on Optical Communication (ECOC 2008), paper Th.1.B.2 (2008).
[CrossRef]

Li, J.

J. Li, B.-E. Olsson, M. Karlsson, and P. A. Andrekson, “OTDM demultiplexer based on XPM-induced wavelength shifting in highly nonlinear fiber,” IEEE Photonics Technol. Lett. 15, 1770-1772 (2003).
[CrossRef]

Lisak, M.

L. Helczynski, B. Hall, D. Anderson, M. Lisak, A. Berntson, and M. Desaix, “Cross-phase modulation induced pulse splitting--the optical axe,” Phys. Scr. T84, 81-84 (2000).
[CrossRef]

Mollenauer, L. F.

Mukasa, K.

Nakazawa, M.

Ng, T. T.

F. Parmigiani, M. Ibsen, T. T. Ng, L. Provost, P. Petropoulos, and D. J. Richardson, “An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper,” IEEE Photonics Technol. Lett. 20, 1461-1463 (2008).
[CrossRef]

Olsson, B.-E.

J. Li, B.-E. Olsson, M. Karlsson, and P. A. Andrekson, “OTDM demultiplexer based on XPM-induced wavelength shifting in highly nonlinear fiber,” IEEE Photonics Technol. Lett. 15, 1770-1772 (2003).
[CrossRef]

Park, Y.

Parmigiani, F.

F. Parmigiani, M. Ibsen, T. T. Ng, L. Provost, P. Petropoulos, and D. J. Richardson, “An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper,” IEEE Photonics Technol. Lett. 20, 1461-1463 (2008).
[CrossRef]

F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating,” IEEE Photonics Technol. Lett. 18, 829-831 (2006).
[CrossRef]

F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. F. Roelens, P. Petropoulos, and D. J. Richardson, “Ultra-flat SPM-broadened spectra in highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express 14, 7617-7622 (2006).
[CrossRef] [PubMed]

Petropoulos, P.

F. Parmigiani, M. Ibsen, T. T. Ng, L. Provost, P. Petropoulos, and D. J. Richardson, “An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper,” IEEE Photonics Technol. Lett. 20, 1461-1463 (2008).
[CrossRef]

F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating,” IEEE Photonics Technol. Lett. 18, 829-831 (2006).
[CrossRef]

F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. F. Roelens, P. Petropoulos, and D. J. Richardson, “Ultra-flat SPM-broadened spectra in highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express 14, 7617-7622 (2006).
[CrossRef] [PubMed]

P. Petropoulos, M. Ibsen, A. D. Ellis, and D. J. Richardson, “Rectangular pulse generation based on pulse reshaping using a superstructured fiber Bragg grating,” J. Lightwave Technol. 19, 746-752 (2001).
[CrossRef]

Provost, L.

F. Parmigiani, M. Ibsen, T. T. Ng, L. Provost, P. Petropoulos, and D. J. Richardson, “An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper,” IEEE Photonics Technol. Lett. 20, 1461-1463 (2008).
[CrossRef]

Richardson, D. J.

F. Parmigiani, M. Ibsen, T. T. Ng, L. Provost, P. Petropoulos, and D. J. Richardson, “An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper,” IEEE Photonics Technol. Lett. 20, 1461-1463 (2008).
[CrossRef]

F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating,” IEEE Photonics Technol. Lett. 18, 829-831 (2006).
[CrossRef]

F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. F. Roelens, P. Petropoulos, and D. J. Richardson, “Ultra-flat SPM-broadened spectra in highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express 14, 7617-7622 (2006).
[CrossRef] [PubMed]

P. Petropoulos, M. Ibsen, A. D. Ellis, and D. J. Richardson, “Rectangular pulse generation based on pulse reshaping using a superstructured fiber Bragg grating,” J. Lightwave Technol. 19, 746-752 (2001).
[CrossRef]

Roelens, M. A. F.

Shang, H.-T.

Simpson, J. R.

Stolen, R. H.

Turitsyn, S. K.

S. Boscolo, A. I. Latkin, and S. K. Turitsyn, “Passive nonlinear pulse shaping in normally dispersive fiber systems,” IEEE J. Quantum Electron. 44, 1196-1203 (2008).
[CrossRef]

R. S. Bhamber, S. Boscolo, A. I. Latkin, and S. K. Turitsyn, “All-optical TDM to WDM signal conversion and partial regeneration using XPM with triangular pulses,” in Proceedings of the 34th European Conference on Optical Communication (ECOC 2008), paper Th.1.B.2 (2008).
[CrossRef]

Yeh, C.

C. Yeh and L. Bergman, “Enhanced pulse compression in a nonlinear fiber by a wavelength division multiplexed optical pulse,” Phys. Rev. E 57, 2398-2404 (1998).
[CrossRef]

IEEE J. Quantum Electron.

S. Boscolo, A. I. Latkin, and S. K. Turitsyn, “Passive nonlinear pulse shaping in normally dispersive fiber systems,” IEEE J. Quantum Electron. 44, 1196-1203 (2008).
[CrossRef]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951-1963 (1994).
[CrossRef]

IEEE Photonics Technol. Lett.

F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating,” IEEE Photonics Technol. Lett. 18, 829-831 (2006).
[CrossRef]

F. Parmigiani, M. Ibsen, T. T. Ng, L. Provost, P. Petropoulos, and D. J. Richardson, “An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper,” IEEE Photonics Technol. Lett. 20, 1461-1463 (2008).
[CrossRef]

J. Li, B.-E. Olsson, M. Karlsson, and P. A. Andrekson, “OTDM demultiplexer based on XPM-induced wavelength shifting in highly nonlinear fiber,” IEEE Photonics Technol. Lett. 15, 1770-1772 (2003).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. A

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Temporal and spectral effects of cross-phase modulation on copropagating ultrashort pulses in optical fibers,” Phys. Rev. A 40, 5063-5072 (1989).
[CrossRef] [PubMed]

Phys. Rev. E

C. Yeh and L. Bergman, “Enhanced pulse compression in a nonlinear fiber by a wavelength division multiplexed optical pulse,” Phys. Rev. E 57, 2398-2404 (1998).
[CrossRef]

Phys. Scr.

L. Helczynski, B. Hall, D. Anderson, M. Lisak, A. Berntson, and M. Desaix, “Cross-phase modulation induced pulse splitting--the optical axe,” Phys. Scr. T84, 81-84 (2000).
[CrossRef]

Other

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

R. S. Bhamber, S. Boscolo, A. I. Latkin, and S. K. Turitsyn, “All-optical TDM to WDM signal conversion and partial regeneration using XPM with triangular pulses,” in Proceedings of the 34th European Conference on Optical Communication (ECOC 2008), paper Th.1.B.2 (2008).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Initial pulse spectrum ( ϕ 0 = 0 ) and XPM-modulated spectra for different values of ϕ 0 . Top, Gaussian pulse with η = 0.35 ; bottom, sech pulse of the same FWHM temporal duration ( η = 0.33 ) .

Fig. 2
Fig. 2

Limits of spectral copying in the plane duty ratio parameter—maximum XPM phase shift. Light area, copying occurs; dark area, copying does not occur. Insets, typical pulse spectra.

Fig. 3
Fig. 3

Evolution of the temporal intensity profile of the modulated pulse in a medium with normal dispersion for ϕ 0 = 12 π and η = 0.35 .

Fig. 4
Fig. 4

Top, intensity and chirp profiles of the pulse of Fig. 3 at ξ = 0 and ξ = 1 . Bottom, pulse spectrogram at ξ = 1 .

Equations (16)

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ψ ̃ ( Ω ) = ψ ̃ 0 ( Ω ) exp ( Ω 2 2 ) + ψ ̃ ( Ω ) exp ( ( Ω Δ Ω ) 2 2 ) + ψ ̃ + ( Ω ) exp ( ( Ω + Δ Ω ) 2 2 ) ,
ψ ̃ 0 ( Ω ) = 2 π u 0 T [ 1 + 1 2 erf ( ( i Ω η 1 ) 2 ) 1 2 erf ( ( i Ω + η 1 ) 2 ) ] ,
ψ ̃ ( Ω ) = 1 2 2 π u 0 T exp ( i ϕ 0 ) [ erf ( i ( Ω Δ Ω ) 2 ) erf ( ( i ( Ω Δ Ω ) η 1 ) 2 ) ] ,
ψ ̃ + ( Ω ) = ψ ̃ ( Ω ) , η = T T T , Ω = ω T , Δ Ω = ϕ 0 η .
ψ ̃ 0 ( Ω ) u 0 T η exp ( 1 ( 2 η 2 ) ) cos ( Ω η ) ,
ψ ̃ ( Ω ) 2 exp ( ϵ 2 ) π u 0 2 T 2 2 exp ( ϵ 2 ) ( 1 + 2 ϵ 2 π + 2 ϵ 4 3 π + 7 ϵ 6 45 π + ) .
ψ ̃ ( Ω ) = ψ ̃ 0 ( Ω ) + ψ ̃ ( Ω ) + ψ ̃ + ( Ω ) ,
ψ ̃ 0 ( Ω ) = 2 u 0 T [ π sech ( π Ω 2 ) exp ( ( 1 i Ω ) η ) 1 i Ω F ( 1 , ( 1 i Ω ) 2 ; ( 3 i Ω ) 2 ; exp ( 2 η ) ) exp ( ( 1 + i Ω ) η ) 1 + i Ω F ( 1 , ( 1 + i Ω ) 2 ; ( 3 + i Ω ) 2 ; exp ( 2 η ) ) ] ,
ψ ̃ ( Ω ) = u 0 T exp ( i ϕ 0 ) [ B ( exp ( 2 η ) ; ( 1 + i ( Ω Δ Ω ) ) 2 , 0 ) + 1 2 Ψ ( ( 1 + i ( Ω Δ Ω ) ) 4 ) 1 2 Ψ ( ( 3 + i ( Ω Δ Ω ) ) 4 ) ] , ψ ̃ + ( Ω ) = ψ ̃ ( Ω ) .
ψ ( z , t ) = u 0 i ( 2 π D ) d t exp ( t 2 ( 2 T 2 ) ) exp ( i ϕ ( t ) i ( t t ) 2 ( 2 D ) ) .
ψ ( ξ , τ ) = ψ 0 ( ξ , τ ) exp ( τ 2 ( 1 + i sgn ( D ) ξ ) 2 ( 1 + ξ 2 ) ) + ψ ( ξ , τ ) exp ( ( τ Δ τ ) 2 2 ( 1 + ξ 2 ) + i Φ ( ξ , τ ) ) + ψ + ( ξ , τ ) exp ( ( τ + Δ τ ) 2 2 ( 1 + ξ 2 ) + i Φ + ( ξ , τ ) ) ,
ψ 0 ( ξ , τ ) = u 0 1 i sgn ( D ) ξ [ 1 + 1 2 erf ( X ( ξ , τ ) ) 1 2 erf ( X + ( ξ , τ ) ) ] ,
ψ ( ξ , τ ) = 1 2 u 0 1 i sgn ( D ) ξ [ erf ( X ( ξ , τ Δ τ ) ) + erf ( X 0 ( τ Δ τ ) ) ] ,
ψ + ( ξ , τ ) = ψ ( ξ , τ ) , X 0 ( ξ , τ ) = i 2 τ sgn ( D ) ξ ( 1 i sgn ( D ) ξ ) ,
X ± ( ξ , τ ) = i 2 ( τ sgn ( D ) ξ ( 1 i sgn ( D ) ξ ) ± η 1 1 i sgn ( D ) ξ sgn ( D ) ξ ) ,
Φ ± ( ξ , τ ) = ϕ 0 τ 2 ξ 2 2 τ Δ τ Δ τ 2 2 sgn ( D ) ξ ( 1 + ξ 2 ) , ξ = D T 2 , τ = t T , Δ τ = sgn ( D ) Δ Ω ξ .

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