Abstract

We analyze parallel-to-serial transmitters and serial-to-parallel receivers that use ultrashort optical pulses to increase the bandwidth of a fiber-optic communication link. This method relies on real-time holographic material for conversion of information between spatial and temporal frequencies. The analysis reveals that the temporal output of the pulses will consist of chirped pulses, which has been verified experimentally. When the signal pulses are transmitted along with a reference pulse, the distortions of the received signal, caused by dispersion and other factors in the fiber, are canceled because of the phase-conjugation property of the receiver. This self-referencing scheme simplifies the receiver structure and ensures perfect timing for the serial-to-parallel conversion.

© 1998 Optical Society of America

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References

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  1. C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. XX, pp. 65–153.
  2. A. M. Weiner, J. P. Heritage, E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5, 1563–1572 (1988).
    [CrossRef]
  3. Y. T. Mazurenko, “Holography of wave packets,” Appl. Phys. B 50, 101–113 (1990).
    [CrossRef]
  4. A. M. Weiner, D. E. Leaird, D. H. Reitze, E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
    [CrossRef]
  5. M. C. Nuss, R. L. Morrison, “Time-domain images,” Opt. Lett. 20, 740–742 (1995).
    [CrossRef] [PubMed]
  6. M. C. Nuss, M. Li, T. H. Chiu, A. M. Weiner, A. Patrovi, “Time-to-space mapping of femtosecond pulses,” Opt. Lett. 19, 664–666 (1994).
    [CrossRef] [PubMed]
  7. P. C. Sun, Y. Mazurenko, W. S. C. Chang, P. K. L. Yu, Y. Fainman, “All optical parallel-to-serial conversion by holographic spatial-to-temporal frequency encoding,” Opt. Lett. 20, 1728–1730 (1995).
    [CrossRef] [PubMed]
  8. K. Ema, M. Kuwata-Gonokami, F. Shimizu, “All-optical sub-Tbits/s serial-to-parallel conversion using excitonic giant nonlinearity,” Appl. Phys. Lett. 59, 2799–2801 (1990).
    [CrossRef]
  9. Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Belyaev, “Ultrafast space-time transformation of signals using spectral nonlinear optics,” Opt. Spectrosc. (USSR) 78, 122–128 (1995).
  10. Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Beliaev, E. B. Verkhovskij, “Time-to-space conversion of fast signals by the method of spectral nonlinear optics,” Opt. Commun. 118, 594–600 (1995).
    [CrossRef]
  11. P. C. Sun, Yu. T. Mazurenko, Y. Fainman, “Femtosecond pulse imaging: ultrafast optical oscilloscope,” J. Opt. Soc. Am. A 14, 1159–1170 (1997).
    [CrossRef]
  12. H. A. Haus, K. Tamura, L. E. Nelson, E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
    [CrossRef]
  13. R. M. Brubaker, Q. N. Wang, D. D. Nolte, E. S. Harmon, M. R. Melloch, “Steady-state four-wave mixing in photorefractive quantum wells with femtosecond pulses,” J. Opt. Soc. Am. B 11, 1038–1044 (1994).
    [CrossRef]
  14. A. Patrovi, A. M. Glass, D. H. Olsen, G. J. Zydzik, H. M. O’Bryan, T. H. Chiu, W. H. Knox, “Cr-doped GaAs/AlGaAs semi-insulating multiple quantum well photorefractive devices,” Appl. Phys. Lett. 62, 464–466 (1993).
    [CrossRef]

1997 (1)

1995 (5)

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Belyaev, “Ultrafast space-time transformation of signals using spectral nonlinear optics,” Opt. Spectrosc. (USSR) 78, 122–128 (1995).

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Beliaev, E. B. Verkhovskij, “Time-to-space conversion of fast signals by the method of spectral nonlinear optics,” Opt. Commun. 118, 594–600 (1995).
[CrossRef]

H. A. Haus, K. Tamura, L. E. Nelson, E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

M. C. Nuss, R. L. Morrison, “Time-domain images,” Opt. Lett. 20, 740–742 (1995).
[CrossRef] [PubMed]

P. C. Sun, Y. Mazurenko, W. S. C. Chang, P. K. L. Yu, Y. Fainman, “All optical parallel-to-serial conversion by holographic spatial-to-temporal frequency encoding,” Opt. Lett. 20, 1728–1730 (1995).
[CrossRef] [PubMed]

1994 (2)

1993 (1)

A. Patrovi, A. M. Glass, D. H. Olsen, G. J. Zydzik, H. M. O’Bryan, T. H. Chiu, W. H. Knox, “Cr-doped GaAs/AlGaAs semi-insulating multiple quantum well photorefractive devices,” Appl. Phys. Lett. 62, 464–466 (1993).
[CrossRef]

1992 (1)

A. M. Weiner, D. E. Leaird, D. H. Reitze, E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

1990 (2)

K. Ema, M. Kuwata-Gonokami, F. Shimizu, “All-optical sub-Tbits/s serial-to-parallel conversion using excitonic giant nonlinearity,” Appl. Phys. Lett. 59, 2799–2801 (1990).
[CrossRef]

Y. T. Mazurenko, “Holography of wave packets,” Appl. Phys. B 50, 101–113 (1990).
[CrossRef]

1988 (1)

Beliaev, A. G.

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Beliaev, E. B. Verkhovskij, “Time-to-space conversion of fast signals by the method of spectral nonlinear optics,” Opt. Commun. 118, 594–600 (1995).
[CrossRef]

Belyaev, A. G.

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Belyaev, “Ultrafast space-time transformation of signals using spectral nonlinear optics,” Opt. Spectrosc. (USSR) 78, 122–128 (1995).

Brubaker, R. M.

Chang, W. S. C.

Chiu, T. H.

M. C. Nuss, M. Li, T. H. Chiu, A. M. Weiner, A. Patrovi, “Time-to-space mapping of femtosecond pulses,” Opt. Lett. 19, 664–666 (1994).
[CrossRef] [PubMed]

A. Patrovi, A. M. Glass, D. H. Olsen, G. J. Zydzik, H. M. O’Bryan, T. H. Chiu, W. H. Knox, “Cr-doped GaAs/AlGaAs semi-insulating multiple quantum well photorefractive devices,” Appl. Phys. Lett. 62, 464–466 (1993).
[CrossRef]

Colombeau, B.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. XX, pp. 65–153.

Ema, K.

K. Ema, M. Kuwata-Gonokami, F. Shimizu, “All-optical sub-Tbits/s serial-to-parallel conversion using excitonic giant nonlinearity,” Appl. Phys. Lett. 59, 2799–2801 (1990).
[CrossRef]

Fainman, Y.

Froehly, C.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. XX, pp. 65–153.

Glass, A. M.

A. Patrovi, A. M. Glass, D. H. Olsen, G. J. Zydzik, H. M. O’Bryan, T. H. Chiu, W. H. Knox, “Cr-doped GaAs/AlGaAs semi-insulating multiple quantum well photorefractive devices,” Appl. Phys. Lett. 62, 464–466 (1993).
[CrossRef]

Harmon, E. S.

Haus, H. A.

H. A. Haus, K. Tamura, L. E. Nelson, E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

Heritage, J. P.

Ippen, E. P.

H. A. Haus, K. Tamura, L. E. Nelson, E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

Kirschner, E. M.

Knox, W. H.

A. Patrovi, A. M. Glass, D. H. Olsen, G. J. Zydzik, H. M. O’Bryan, T. H. Chiu, W. H. Knox, “Cr-doped GaAs/AlGaAs semi-insulating multiple quantum well photorefractive devices,” Appl. Phys. Lett. 62, 464–466 (1993).
[CrossRef]

Kuwata-Gonokami, M.

K. Ema, M. Kuwata-Gonokami, F. Shimizu, “All-optical sub-Tbits/s serial-to-parallel conversion using excitonic giant nonlinearity,” Appl. Phys. Lett. 59, 2799–2801 (1990).
[CrossRef]

Leaird, D. E.

A. M. Weiner, D. E. Leaird, D. H. Reitze, E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Li, M.

Mazurenko, Y.

Mazurenko, Y. T.

Y. T. Mazurenko, “Holography of wave packets,” Appl. Phys. B 50, 101–113 (1990).
[CrossRef]

Mazurenko, Yu. T.

P. C. Sun, Yu. T. Mazurenko, Y. Fainman, “Femtosecond pulse imaging: ultrafast optical oscilloscope,” J. Opt. Soc. Am. A 14, 1159–1170 (1997).
[CrossRef]

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Belyaev, “Ultrafast space-time transformation of signals using spectral nonlinear optics,” Opt. Spectrosc. (USSR) 78, 122–128 (1995).

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Beliaev, E. B. Verkhovskij, “Time-to-space conversion of fast signals by the method of spectral nonlinear optics,” Opt. Commun. 118, 594–600 (1995).
[CrossRef]

Melloch, M. R.

Morrison, R. L.

Nelson, L. E.

H. A. Haus, K. Tamura, L. E. Nelson, E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

Nolte, D. D.

Nuss, M. C.

O’Bryan, H. M.

A. Patrovi, A. M. Glass, D. H. Olsen, G. J. Zydzik, H. M. O’Bryan, T. H. Chiu, W. H. Knox, “Cr-doped GaAs/AlGaAs semi-insulating multiple quantum well photorefractive devices,” Appl. Phys. Lett. 62, 464–466 (1993).
[CrossRef]

Olsen, D. H.

A. Patrovi, A. M. Glass, D. H. Olsen, G. J. Zydzik, H. M. O’Bryan, T. H. Chiu, W. H. Knox, “Cr-doped GaAs/AlGaAs semi-insulating multiple quantum well photorefractive devices,” Appl. Phys. Lett. 62, 464–466 (1993).
[CrossRef]

Paek, E. G.

A. M. Weiner, D. E. Leaird, D. H. Reitze, E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Patrovi, A.

M. C. Nuss, M. Li, T. H. Chiu, A. M. Weiner, A. Patrovi, “Time-to-space mapping of femtosecond pulses,” Opt. Lett. 19, 664–666 (1994).
[CrossRef] [PubMed]

A. Patrovi, A. M. Glass, D. H. Olsen, G. J. Zydzik, H. M. O’Bryan, T. H. Chiu, W. H. Knox, “Cr-doped GaAs/AlGaAs semi-insulating multiple quantum well photorefractive devices,” Appl. Phys. Lett. 62, 464–466 (1993).
[CrossRef]

Putilin, S. E.

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Beliaev, E. B. Verkhovskij, “Time-to-space conversion of fast signals by the method of spectral nonlinear optics,” Opt. Commun. 118, 594–600 (1995).
[CrossRef]

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Belyaev, “Ultrafast space-time transformation of signals using spectral nonlinear optics,” Opt. Spectrosc. (USSR) 78, 122–128 (1995).

Reitze, D. H.

A. M. Weiner, D. E. Leaird, D. H. Reitze, E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Shimizu, F.

K. Ema, M. Kuwata-Gonokami, F. Shimizu, “All-optical sub-Tbits/s serial-to-parallel conversion using excitonic giant nonlinearity,” Appl. Phys. Lett. 59, 2799–2801 (1990).
[CrossRef]

Spiro, A. G.

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Beliaev, E. B. Verkhovskij, “Time-to-space conversion of fast signals by the method of spectral nonlinear optics,” Opt. Commun. 118, 594–600 (1995).
[CrossRef]

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Belyaev, “Ultrafast space-time transformation of signals using spectral nonlinear optics,” Opt. Spectrosc. (USSR) 78, 122–128 (1995).

Sun, P. C.

Tamura, K.

H. A. Haus, K. Tamura, L. E. Nelson, E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

Vampouille, M.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. XX, pp. 65–153.

Verkhovskij, E. B.

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Beliaev, E. B. Verkhovskij, “Time-to-space conversion of fast signals by the method of spectral nonlinear optics,” Opt. Commun. 118, 594–600 (1995).
[CrossRef]

Wang, Q. N.

Weiner, A. M.

Yu, P. K. L.

Zydzik, G. J.

A. Patrovi, A. M. Glass, D. H. Olsen, G. J. Zydzik, H. M. O’Bryan, T. H. Chiu, W. H. Knox, “Cr-doped GaAs/AlGaAs semi-insulating multiple quantum well photorefractive devices,” Appl. Phys. Lett. 62, 464–466 (1993).
[CrossRef]

Appl. Phys. B (1)

Y. T. Mazurenko, “Holography of wave packets,” Appl. Phys. B 50, 101–113 (1990).
[CrossRef]

Appl. Phys. Lett. (2)

K. Ema, M. Kuwata-Gonokami, F. Shimizu, “All-optical sub-Tbits/s serial-to-parallel conversion using excitonic giant nonlinearity,” Appl. Phys. Lett. 59, 2799–2801 (1990).
[CrossRef]

A. Patrovi, A. M. Glass, D. H. Olsen, G. J. Zydzik, H. M. O’Bryan, T. H. Chiu, W. H. Knox, “Cr-doped GaAs/AlGaAs semi-insulating multiple quantum well photorefractive devices,” Appl. Phys. Lett. 62, 464–466 (1993).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. M. Weiner, D. E. Leaird, D. H. Reitze, E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

H. A. Haus, K. Tamura, L. E. Nelson, E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Beliaev, E. B. Verkhovskij, “Time-to-space conversion of fast signals by the method of spectral nonlinear optics,” Opt. Commun. 118, 594–600 (1995).
[CrossRef]

Opt. Lett. (3)

Opt. Spectrosc. (USSR) (1)

Yu. T. Mazurenko, A. G. Spiro, S. E. Putilin, A. G. Belyaev, “Ultrafast space-time transformation of signals using spectral nonlinear optics,” Opt. Spectrosc. (USSR) 78, 122–128 (1995).

Other (1)

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. XX, pp. 65–153.

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Figures (7)

Fig. 1
Fig. 1

Structure of the parallel-to-serial transmitter. cw writing beams emanating from point sources separated by Δ propagate from right to left, coupled in by a beam splitter (dashed line). An incident short pulse propagates from left to right, with the spectrum spatially dispersed at plane 2 (by a grating in plane 1), where a real-time 4WM process mixes the cw interference (with reference source not shown) and the pulsed spectrum.

Fig. 2
Fig. 2

Structure of the serial-to-parallel receiver. The incoming data pulses are shown in gray, the reference pulse in black. Real-time hologram at plane 2 transfers interference information of the pulses to a readout cw beam. The output cw signal is coupled out by a beam splitter (dashed line) and is focused onto a detector array at output plane 3.

Fig. 3
Fig. 3

Time-domain output simulation of parallel-to-serial transmitter. The simulation is shown for a pulse duration of 90 fs, 1.55-μm wavelength, and Δ = 0.3 mm.

Fig. 4
Fig. 4

Coupling output pulses of transmitter into an optical fiber. The output pulses, of the size of w″(x), are delayed by Δ″/c and laterally shifted by Δ″. Lens F′ couples the pulses into a fiber, where the lateral shift information is lost.

Fig. 5
Fig. 5

Intensity output of diffracted data points. Spots are displaced by x 0, a distance that guarantees a signal-to-cross-talk ratio of 20 dB. The shaded area corresponds to light intensity gathered by a detector of width D.

Fig. 6
Fig. 6

Schematic of experiment setup. G’s, dispersion gratings; f g , Ronchi grating.

Fig. 7
Fig. 7

Plot of the cross-correlation traces. The solid curves are the measured curves, and the dashed curves are the theoretical curves.

Tables (1)

Tables Icon

Table 1 Calculated and Experimental Results of the FWHM for 20-, 28-, and 39-lp/mm Gratings

Equations (41)

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s t = p t - t 0 exp j ω 0 t ,
S ω = P ω - ω 0 exp - j ω - ω 0 t 0 ,
s 1 x ;   ω = exp - j   ω - ω 0 c   α x w x ,
s 2 f x ;   ω = W f x + ω - ω 0 2 π c   α ,
t f w = n   A n exp j 2 π n Δ f w ,
t f x = n   A n exp j 2 π n Δ   ω w ω   f x .
s 3 x ;   ω = n   A n w - x + n ω w ω   Δ × exp j   ω - ω 0 c   α x - n ω w ω   Δ ,
s 3 x ;   ω = n   A n w - x + n ω w ω   Δ α × exp - j   n ω - ω 0 ω w Δ c ω   α ,
s 0 x ;   t = F ω - 1 P ω - ω 0 exp - j ω - ω 0 t 0 × n   A n w - x + n ω w ω   Δ α × exp - j   n ω - ω 0 ω w Δ c ω   α .
1 ω = 1 ω 0 + δ ω = 1 ω 0 1 - δ ω ω 0 + O δ ω 2 ω 0 2 1 ω 0 - δ ω ω 0 2 .
s 0 x ;   t = F ω - 1 P δ ω exp - j δ ω t 0 × n   A n w - x + n Δ - n Δ   δ ω ω 0 × exp - j   n δ ω Δ c + j   n δ ω 2 Δ c ω 0 ,
p t = exp - t 2 2 τ 2 FT   P ω = 2 π τ   exp - τ 2 ω 2 2 .
s 0 x ;   t = n   A n 2 π exp j ω 0 t -   τ   exp - τ 2 δ ω 2 2 × exp j   n Δ δ ω 2 c ω 0 w - x + n Δ - n Δ   δ ω ω 0 exp j δ ω t n d δ ω ,
s 0 x ;   t = n   A n exp j ω 0 t w - x + n Δ 1 4 1 + ξ 2 n 2 × exp - j   tan - 1 ξ n 2 exp - t n 2 2 τ 2 1 + j ξ n 1 + ξ 2 n 2 ,
s n t = F ω - 1 P δ ω exp - j δ ω t 0 A n × exp - j   n δ ω Δ c + j   n δ ω 2 Δ c ω 0 ,
r t = p t exp j ω 0 t ,
S n ω = P δ ω exp - j δ ω t 0 A n × exp - j   n δ ω Δ c + j   n δ ω 2 Δ c ω 0 ,
R ω = P δ ω ,
W f x ;   ω = W f x + δ ω 2 π c   α ,
f x = ω 0 + δ ω x 2 π cF .
R ω S n * ω | W f x ;   ω | 2 = A n | P δ ω | 2 exp j t 0 + n Δ c δ ω - j   n Δ c ω 0 δ ω 2 | W f x ;   ω | 2 .
H n x = -   R ω S n * ω | W f x ;   ω | 2 d ω
H n x = A n | P δ ω | 2 exp j t 0 + n Δ c   δ ω - j   n Δ c ω 0 δ ω 2 ,
δ ω = - x ω 0 x + α F ,
δ ω = - x ω 0 α F 1 1 + x α F = - x ω 0 α F 1 - x α F + O   x α F 2 = - x ω 0 α F + ω 0 x α F 2 + O x α F 3 ,
δ ω 2 = ω 0 x α F 2 + O x α F 3 ,
H n x = A n P - x ω 0 α F + ω 0 x α F 2 2 exp - j   n Δ c x ω 0 α F × exp j ω 0 t 0 - x α F + x α F 2 .
H n x ;   t = A n P - x ω 0 α F + ω 0 x α F 2 2 × exp - j   n Δ c x ω 0 α F × exp j ω 0 t 0 - x α F + x α F 2 K n t .
K n t w ct α w c α t - t 0 - n Δ c .
H n x ;   t = A n exp - τ 2 ω 0 2 x α F 2 exp - j   n Δ c x ω 0 α F × exp jt 0 - x ω 0 α F + ω 0 x α F 2 K n t .
h x ;   t = n   h n x ;   t = n   A n   exp - α 2 x λ r + n Δ α λ 0 2 4 τ 2 ω 0 2 K n t ,
R ω a ω exp j Φ ω W x ;   ω × S ω a ω exp j Φ ω W x ;   ω * = R ω S * ω | a ω | 2 | W x ;   ω | 2 .
M = F N . A . Δ ,
SXR = erf α 2 τ ω 0 λ r D 2 erf α 2 τ ω 0 Δ λ w + D 2 λ r - erf α 2 τ ω 0 Δ λ w - D 2 λ r + 2   exp - 1 2 α Δ λ w ω 0 τ 2 erf α 2 τ ω 0 λ r D 2 ,
U r f x ;   ω = P ω - ω 0 W f x + ω - ω 0 2 π c   α ,
U r x ;   t = 1 2 π -   P ω - ω 0 W ω x 2 π cF + ω - ω 0 2 π c   α × exp j ω t d ω ,
U r x ;   t = 1 2 π   P - ω 0 x x + α F -   W ω x 2 π cF + ω - ω 0 2 π c   α exp j ω t d ω .
U r x ;   t = P - ω 0 x α F + ω 0 x α F 2 c / α 1 + x α F × exp j ω 0 t 1 - x α F + x α F 2 w ct / α 1 + x α F .
U s x ;   t = A n P - ω 0 x α F + ω 0 x α F 2 c / α 1 + x α F × exp j ω 0 t 1 - x α F + x α F 2 × exp j ω 0 t 0 x α F - x α F 2 × exp j ω 0 n Δ c x α F w c t - t 0 - n Δ c α 1 + x α F .
U s * x ;   t U r x ;   t = A n P - ω 0 x α F + ω 0 x α F 2 2 × c / α 1 + x α F 2 exp - j ω 0 n Δ c x α F × exp j ω 0 t 0 - x α F + x α F 2 × K n x ;   t ,
K n x ;   t = w ct / α 1 + x α F w c t - t 0 - n Δ c α 1 + x α F w ct α w c t - t 0 - n Δ c α K n t .

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