Abstract

We report the observation of spectral oscillations at the output of an optical fiber after the propagation of a reshaped optical soliton over several soliton periods. The experimental results are in good agreement with the results of an analytical study that permits the calculation of the radiated field associated with a hyperbolic-secant-shaped input pulse, after a few soliton periods of propagation. For a nonsoliton input we show that the spectra of the asymptotic soliton and of the radiated field, which is stripped from the input pulse, remain overlapped during propagation, and the phase difference between the two fields is both distance and frequency dependent. This causes an interference that gives rise to the observed oscillations. Furthermore, a numerical simulation that uses a split-step Fourier-transform technique gives results that agree well with the experimental observations.

© 1993 Optical Society of America

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  1. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].
  2. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
    [CrossRef]
  3. V. E. Zakharov and A. V. Mikhailov, “Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method,” Zh. Eksp. Teor. Fiz. 74, 1953–1973 (1978) [Sov. Phys. JETP 47, 1017–1027 (1978)].
  4. H. Segur, “Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. II,” J. Math. Phys. 17, 714–716 (1976).
    [CrossRef]
  5. J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
    [CrossRef]
  6. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
    [CrossRef]
  7. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
    [CrossRef] [PubMed]
  8. M. N. Islam, E. R. Sunderman, C. E. Soccolich, I. Bar-Joseph, N. Sauer, T. Y. Chang, and B. I. Miller, “Color center lasers passively mode-locked by quantum wells,” IEEE J. Quantum Electron. 25, 2454–2463 (1989).
    [CrossRef]
  9. C. E. Soccolich, M. W. Chbat, M. N. Islam, and P. R. Prucnal, “Cascade of ultrafast soliton-dragging and -trapping logic gates,” IEEE Photon. Technol. Lett. 4, 1043–1046 (1992).
    [CrossRef]
  10. M. N. Islam, C. E. Soccolich, and D. A. B. Miller, “Low-energy ultrafast fiber soliton logic gates,” Opt. Lett. 15, 909–911 (1990).
    [CrossRef] [PubMed]
  11. M. Nakazawa, H. Kubota, K. Kurokawa, and E. Yamada, “Femtosecond optical soliton transmission over long distances using adiabatic trapping and soliton standardization,” J. Opt. Soc. Am. B 8, 1811–1817 (1991).
    [CrossRef]
  12. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986); J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [CrossRef] [PubMed]
  13. D. J. Richardson, R. I. Laming, D. N. Payne, V. I. Matsas, and M. W. Phillips, “Pulse repetition rates in passive, selfstarting, femtosecond soliton fibre laser,” Electron. Lett. 27, 1451–1453 (1991).
    [CrossRef]
  14. S. M. J. Kelley, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
    [CrossRef]

1992 (3)

J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
[CrossRef]

C. E. Soccolich, M. W. Chbat, M. N. Islam, and P. R. Prucnal, “Cascade of ultrafast soliton-dragging and -trapping logic gates,” IEEE Photon. Technol. Lett. 4, 1043–1046 (1992).
[CrossRef]

S. M. J. Kelley, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
[CrossRef]

1991 (2)

M. Nakazawa, H. Kubota, K. Kurokawa, and E. Yamada, “Femtosecond optical soliton transmission over long distances using adiabatic trapping and soliton standardization,” J. Opt. Soc. Am. B 8, 1811–1817 (1991).
[CrossRef]

D. J. Richardson, R. I. Laming, D. N. Payne, V. I. Matsas, and M. W. Phillips, “Pulse repetition rates in passive, selfstarting, femtosecond soliton fibre laser,” Electron. Lett. 27, 1451–1453 (1991).
[CrossRef]

1990 (1)

1989 (1)

M. N. Islam, E. R. Sunderman, C. E. Soccolich, I. Bar-Joseph, N. Sauer, T. Y. Chang, and B. I. Miller, “Color center lasers passively mode-locked by quantum wells,” IEEE J. Quantum Electron. 25, 2454–2463 (1989).
[CrossRef]

1986 (2)

1978 (1)

V. E. Zakharov and A. V. Mikhailov, “Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method,” Zh. Eksp. Teor. Fiz. 74, 1953–1973 (1978) [Sov. Phys. JETP 47, 1017–1027 (1978)].

1976 (1)

H. Segur, “Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. II,” J. Math. Phys. 17, 714–716 (1976).
[CrossRef]

1974 (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

1971 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

Bar-Joseph, I.

M. N. Islam, E. R. Sunderman, C. E. Soccolich, I. Bar-Joseph, N. Sauer, T. Y. Chang, and B. I. Miller, “Color center lasers passively mode-locked by quantum wells,” IEEE J. Quantum Electron. 25, 2454–2463 (1989).
[CrossRef]

Chang, T. Y.

M. N. Islam, E. R. Sunderman, C. E. Soccolich, I. Bar-Joseph, N. Sauer, T. Y. Chang, and B. I. Miller, “Color center lasers passively mode-locked by quantum wells,” IEEE J. Quantum Electron. 25, 2454–2463 (1989).
[CrossRef]

Chbat, M. W.

C. E. Soccolich, M. W. Chbat, M. N. Islam, and P. R. Prucnal, “Cascade of ultrafast soliton-dragging and -trapping logic gates,” IEEE Photon. Technol. Lett. 4, 1043–1046 (1992).
[CrossRef]

Gordon, J. P.

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Haus, H. A.

Islam, M. N.

C. E. Soccolich, M. W. Chbat, M. N. Islam, and P. R. Prucnal, “Cascade of ultrafast soliton-dragging and -trapping logic gates,” IEEE Photon. Technol. Lett. 4, 1043–1046 (1992).
[CrossRef]

M. N. Islam, C. E. Soccolich, and D. A. B. Miller, “Low-energy ultrafast fiber soliton logic gates,” Opt. Lett. 15, 909–911 (1990).
[CrossRef] [PubMed]

M. N. Islam, E. R. Sunderman, C. E. Soccolich, I. Bar-Joseph, N. Sauer, T. Y. Chang, and B. I. Miller, “Color center lasers passively mode-locked by quantum wells,” IEEE J. Quantum Electron. 25, 2454–2463 (1989).
[CrossRef]

Kelley, S. M. J.

S. M. J. Kelley, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
[CrossRef]

Kubota, H.

Kurokawa, K.

Laming, R. I.

D. J. Richardson, R. I. Laming, D. N. Payne, V. I. Matsas, and M. W. Phillips, “Pulse repetition rates in passive, selfstarting, femtosecond soliton fibre laser,” Electron. Lett. 27, 1451–1453 (1991).
[CrossRef]

Matsas, V. I.

D. J. Richardson, R. I. Laming, D. N. Payne, V. I. Matsas, and M. W. Phillips, “Pulse repetition rates in passive, selfstarting, femtosecond soliton fibre laser,” Electron. Lett. 27, 1451–1453 (1991).
[CrossRef]

Mikhailov, A. V.

V. E. Zakharov and A. V. Mikhailov, “Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method,” Zh. Eksp. Teor. Fiz. 74, 1953–1973 (1978) [Sov. Phys. JETP 47, 1017–1027 (1978)].

Miller, B. I.

M. N. Islam, E. R. Sunderman, C. E. Soccolich, I. Bar-Joseph, N. Sauer, T. Y. Chang, and B. I. Miller, “Color center lasers passively mode-locked by quantum wells,” IEEE J. Quantum Electron. 25, 2454–2463 (1989).
[CrossRef]

Miller, D. A. B.

Mitschke, F. M.

Mollenauer, L. F.

Nakazawa, M.

Payne, D. N.

D. J. Richardson, R. I. Laming, D. N. Payne, V. I. Matsas, and M. W. Phillips, “Pulse repetition rates in passive, selfstarting, femtosecond soliton fibre laser,” Electron. Lett. 27, 1451–1453 (1991).
[CrossRef]

Phillips, M. W.

D. J. Richardson, R. I. Laming, D. N. Payne, V. I. Matsas, and M. W. Phillips, “Pulse repetition rates in passive, selfstarting, femtosecond soliton fibre laser,” Electron. Lett. 27, 1451–1453 (1991).
[CrossRef]

Prucnal, P. R.

C. E. Soccolich, M. W. Chbat, M. N. Islam, and P. R. Prucnal, “Cascade of ultrafast soliton-dragging and -trapping logic gates,” IEEE Photon. Technol. Lett. 4, 1043–1046 (1992).
[CrossRef]

Richardson, D. J.

D. J. Richardson, R. I. Laming, D. N. Payne, V. I. Matsas, and M. W. Phillips, “Pulse repetition rates in passive, selfstarting, femtosecond soliton fibre laser,” Electron. Lett. 27, 1451–1453 (1991).
[CrossRef]

Satsuma, J.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Sauer, N.

M. N. Islam, E. R. Sunderman, C. E. Soccolich, I. Bar-Joseph, N. Sauer, T. Y. Chang, and B. I. Miller, “Color center lasers passively mode-locked by quantum wells,” IEEE J. Quantum Electron. 25, 2454–2463 (1989).
[CrossRef]

Segur, H.

H. Segur, “Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. II,” J. Math. Phys. 17, 714–716 (1976).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

Soccolich, C. E.

C. E. Soccolich, M. W. Chbat, M. N. Islam, and P. R. Prucnal, “Cascade of ultrafast soliton-dragging and -trapping logic gates,” IEEE Photon. Technol. Lett. 4, 1043–1046 (1992).
[CrossRef]

M. N. Islam, C. E. Soccolich, and D. A. B. Miller, “Low-energy ultrafast fiber soliton logic gates,” Opt. Lett. 15, 909–911 (1990).
[CrossRef] [PubMed]

M. N. Islam, E. R. Sunderman, C. E. Soccolich, I. Bar-Joseph, N. Sauer, T. Y. Chang, and B. I. Miller, “Color center lasers passively mode-locked by quantum wells,” IEEE J. Quantum Electron. 25, 2454–2463 (1989).
[CrossRef]

Sunderman, E. R.

M. N. Islam, E. R. Sunderman, C. E. Soccolich, I. Bar-Joseph, N. Sauer, T. Y. Chang, and B. I. Miller, “Color center lasers passively mode-locked by quantum wells,” IEEE J. Quantum Electron. 25, 2454–2463 (1989).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Yajima, N.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Yamada, E.

Zakharov, V. E.

V. E. Zakharov and A. V. Mikhailov, “Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method,” Zh. Eksp. Teor. Fiz. 74, 1953–1973 (1978) [Sov. Phys. JETP 47, 1017–1027 (1978)].

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Electron. Lett. (2)

D. J. Richardson, R. I. Laming, D. N. Payne, V. I. Matsas, and M. W. Phillips, “Pulse repetition rates in passive, selfstarting, femtosecond soliton fibre laser,” Electron. Lett. 27, 1451–1453 (1991).
[CrossRef]

S. M. J. Kelley, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. N. Islam, E. R. Sunderman, C. E. Soccolich, I. Bar-Joseph, N. Sauer, T. Y. Chang, and B. I. Miller, “Color center lasers passively mode-locked by quantum wells,” IEEE J. Quantum Electron. 25, 2454–2463 (1989).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

C. E. Soccolich, M. W. Chbat, M. N. Islam, and P. R. Prucnal, “Cascade of ultrafast soliton-dragging and -trapping logic gates,” IEEE Photon. Technol. Lett. 4, 1043–1046 (1992).
[CrossRef]

J. Math. Phys. (1)

H. Segur, “Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. II,” J. Math. Phys. 17, 714–716 (1976).
[CrossRef]

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

Opt. Lett. (3)

Suppl. Prog. Theor. Phys. (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Zh. Eksp. Teor. Fiz. (2)

V. E. Zakharov and A. V. Mikhailov, “Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method,” Zh. Eksp. Teor. Fiz. 74, 1953–1973 (1978) [Sov. Phys. JETP 47, 1017–1027 (1978)].

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

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Figures (8)

Fig. 1
Fig. 1

Total field amplitude |u(z, t = 0)| (normalized to the soliton power Pc) versus the distance of propagation (normalized to the soliton period Z0) for different values of a. From the bottom to the top trace the values of a are −0.2, −0.15, −0.1, −0.05, 0, 0.05, 0.1, 0.15, and 0.2.

Fig. 2
Fig. 2

Time variation of |f(z, t)| (solid curve), |f(z, t)|tanh2(t) (dashed curve), |f(z, t)|sech2(t) (dotted curve), and a2 sech2(t) (dashed–dotted curve), for z = 50 soliton periods, and a = 0.1.

Fig. 3
Fig. 3

Spectra of the total field for a = 0.1, as given by Eqs. (50), for different propagation distances. Clockwise from top left: z = 10, 20, 50, and 100 soliton periods. The frequency represents the separation from the carrier frequency and is normalized to 1/tc.

Fig. 4
Fig. 4

a, Location of the first nine pairs of spectral peaks for a = 0.1 and z = 10 soliton periods. The peaks are located at the intersections of the parabola P z and the curves C p. The frequency normalization is the same as in Fig. 3.b, Evolution of the positive-frequency spectral peaks 1 (bottom trace), 2, 5, 10, and 20, with the distance of propagation, for a = 0.1. The frequency normalization is the same as in a, and the distance is normalized to a soliton period.

Fig. 5
Fig. 5

Experimental spectra, plotted versus wavelength, after 68 soliton periods of propagation, a, a = −0.356; b, a = −0.307.

Fig. 6
Fig. 6

Spectra obtained by a numerical simulation corresponding to the parameters of Figures 5a and 5b.

Fig. 7
Fig. 7

Spectra of the total field for a = −0.0898. The solid curves show the spectra obtained by the analytical result of Eqs. (50), and the dashed curves show the result of a numerical simulation. The frequency normalization is the same as in Fig. 3.a, z = 34 soliton periods; b, z = 68 soliton periods.

Fig. 8
Fig. 8

Solid curve: spectrum obtained for a = −0.0898 and z = 1000 soliton periods after truncation of the output field by the function ρ(t) = 1/2{tanh[5(t + 10)] + tanh[−5(t − 10)]}. This spectrum is indistinguishable, at this scale, from the spectrum of the asymptotic soliton. Dashed curve: total output spectrum obtained with the same parameters. The frequency normalization is the same as in Fig. 3.

Equations (64)

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i u z = 1 2 2 u t 2 + | u | 2 u ,
t c 2 z c = λ 2 D ( λ ) 2 π c ,
P c z c = λ A eff 2 π n 2 ,
u s ( z , t ) = A sech ( A t q ) exp ( i Ω t + i Φ ) ,
u i ( z = 0 , t ) = ( N + a ) sech ( t ) ,
u s j 2 = + | u s j ( z , t ) | 2 d t = 4 ( N + a j + 1 / 2 ) ,
u s 2 = j = 1 N u s j 2 = 2 N ( N + 2 a ) .
u n s 2 = 2 a 2 .
u ( z = 0 , t ) = ( 1 + a ) sech ( t ) ,
u s ( z , t ) = ( 1 + 2 a ) sech [ ( 1 + 2 a ) t ] exp ( i ( 1 + 2 a ) 2 2 z ) .
u s ( 1 + 2 a ) 1 u s , t ( 1 + 2 a ) 1 t , z ( 1 + 2 a ) 2 z .
u ( 0 , t ) = 1 + a 1 + 2 a sech ( t 1 + 2 a ) ,
u s ( z , t ) = sech ( t ) exp ( i z 2 ) .
u ( 0 , t ) ( 1 a ) sech [ ( 1 2 a ) t ] .
u ( z , t ) = u s ( z , t ) + u p ( z , t ) .
i f z = 1 2 2 f t 2 ,
f ( z , t ) = 1 2 π + f ˜ ( ω ) exp ( i ω t i ω 2 z 2 ) d ω ,
f ˜ ( ω ) = 1 2 π + f ( 0 , t ) exp ( i ω t ) d t .
2 [ u s * ( 0 , t ) f ( 0 , t ) ] t 2 = u s * ( 0 , t ) u p ( 0 , t ) .
u p ( z , t ) = 2 f ( z , t ) t 2 + 2 tanh ( t ) f ( z , t ) t tanh 2 ( t ) f ( z , t ) + u s 2 ( z , t ) f * ( z , t ) .
u p ( 0 , t ) = u ( 0 , t ) u s ( 0 , t ) = a [ 1 2 t tanh ( t ) ] sech ( t ) .
f ( 0 , t ) = a 2 { exp ( t ) ln [ 1 + exp ( 2 t ) ] + exp ( t ) ln [ 1 + exp ( 2 t ) ] } .
0 + x α 1 ln ( 1 + x ) d x = π α sin ( π α ) ,
f ˜ ( ω ) = a ( π 2 ) 1 / 2 sech [ ( π / 2 ) ω ] 1 + ω 2 .
ω = t z + i ζ ,
f ( z , t ) = ( i 2 π ) 1 / 2 exp ( i t 2 2 z ) + f ˜ ( t z + i ζ ) × exp ( z 2 ζ 2 ) d ζ .
+ f ˜ ( t z + i ζ ) exp ( z 2 ζ 2 ) d ζ = p = 0 ( i ) p ( 2 p ) ! [ + ζ 2 p exp ( z 2 ζ 2 ) d ζ ] [ 2 p f ˜ ( ω ) ω 2 p ] ω = t / z .
+ ζ 2 p exp ( z 2 ζ 2 ) d ζ = 2 π ( 2 p 1 ) ! ! z p + ( 1 / 2 ) ,
f ( z , t ) = exp ( i t 2 2 z ) ( i z ) 1 / 2 p = 0 1 p ! ( i 2 z ) p ( 2 p f ˜ ( ω ) ω 2 p ) ω = t / z .
u p ( z , t ) a ( i π 2 z ) 1 / 2 { tanh 2 ( t ) i sech 2 ( t ) exp ( i z ) + 1 z [ i ( 1 2 t tanh ( t ) + ( t 2 2 + σ ) tanh 2 ( t ) ) ( t 2 2 + σ ) sech 2 ( t ) exp ( i z ) ] } ,
| u ( z , t ) | 2 = | u s ( z , t ) | 2 + | u p ( z , t ) | 2 + 2 Re [ u s * ( z , t ) u p ( z , t ) ] ,
2 Re [ u s * ( z , t ) u p ( z , t ) ] = a ( 2 π z ) 1 / 2 sech ( t ) { [ 1 2 sech 2 ( t ) ] cos ( z 2 + π 4 ) + 1 z [ 1 2 t tanh ( t ) + ( t 2 2 + σ ) [ 1 2 sech 2 ( t ) ] ] × sin ( z 2 + π 4 ) } .
u p ( z , 0 ) = a ( i π 2 z ) 1 / 2 p = 0 ( 2 p ) ! p ! ( i 2 z ) p c p × [ exp ( i z ) + ( 1 ) p + 1 ( 2 p + 1 ) 1 z ] ,
c p = q = 0 p ( 1 ) p q ( π 2 ) 2 q E 2 q ( 2 q ) ! .
( δ A ) ( z ) = Re [ + u s * ( z , t ) u p ( z , t ) d t ] ,
[ δ ( A Ω ) ] ( z ) = Im [ + u s * ( z , t ) u p ( z , t ) t d t ] ,
( δ q ) ( z ) = Re [ + t u s * ( z , t ) u p ( z , t ) d t ] ,
( A δ Φ + Ω δ q ) ( z ) = Im [ + t u s * ( z , t ) u p ( z , t ) t d t ] .
Im [ u s * ( z , t ) u p ( z , t ) t ] = a ( π 2 z 3 ) 1 / 2 × sech ( t ) [ 2 t sech 2 ( t ) + t 2 tanh ( t ) ] cos ( z 2 + π 4 ) .
( δ A ) ( z ) = ( δ Ω ) ( z ) = ( δ q ) ( z ) = ( δ Φ ) ( z ) = 0.
Δ φ ( ω , z ) = 0 z ( k s k p ) ( z ) d z 1 2 ( 1 + ω 2 ) z .
u ˜ s ( z , ω ) = ( π 2 ) 1 / 2 sech ( π 2 ω ) exp ( i z 2 ) .
u ˜ p ( z , ω ) = ω 2 f ˜ ( z , ω ) 2 i ω F [ f ( z , t ) tanh ( t ) ] ( ω ) F [ f ( z , t ) tanh 2 ( t ) ] ( ω ) + F [ f * ( z , t ) sech 2 ( t ) ] ( ω ) ,
u ˜ p ( z , ω ) = [ ω 2 1 + 2 i ω sgn ( ω ) ] f ˜ ( z , ω ) .
u ˜ p ( z , ω ) = a ( π 2 ) 1 / 2 sech ( π 2 ω ) × exp { i [ ω 2 2 z + 2 arctan ( | ω | ) ] } .
u ˜ p ( z , ω ) = a u ˜ s ( z , ω ) exp { i [ 1 + ω 2 2 z + 2 arctan ( | ω | ) ] } .
Δ φ ( ω , z ) = 1 + ω 2 2 z + 2 arctan ( | ω | ) .
Δ φ ( ω , z ) = 2 p π ,
| u ˜ | 2 = | u ˜ s | 2 + | u ˜ p | 2 + 2 Re ( u ˜ s * u ˜ p ) .
2 Re ( u ˜ s * u ˜ p ) = π a sech 2 ( π 2 ω ) × cos ( 1 + ω 2 2 z + 2 arctan ( | ω | ) ) = π a sech 2 [ ( π / 2 ) ω ] 1 + ω 2 [ ( 1 ω 2 ) cos ( 1 + ω 2 2 z ) 2 | ω | sin ( 1 + ω 2 2 z ) ] .
ω ( 1 + 2 a ) 1 ω .
u ˜ s ( z , ω ) = ( 1 + 2 a ) ( π 2 ) 1 / 2 sech [ π 2 ( ω 1 + 2 a ) ] × exp [ i ( 1 + 2 a ) 2 z 2 ] , u ˜ p ( z , ω ) = a u ˜ s ( z , ω ) exp { i [ ω 2 + ( 1 + 2 a ) 2 2 z + 2 arctan ( | ω | 1 + 2 a ) ] } , u ˜ ( z , ω ) = u ˜ s ( z , ω ) + u ˜ p ( z , ω ) .
ρ ( t ) = 1 / 2 { tanh [ 5 ( t + 10 ) ] + tanh [ 5 ( t 10 ) ] } .
u p ( z , 0 ) = ( 2 f ( z , t ) t 2 ) t = 0 + exp ( i z ) f * ( z , 0 ) ,
f ( z , 0 ) = ( i z ) 1 / 2 p = 0 1 p ! ( i 2 z ) p ( 2 p f ˜ ( ω ) ω 2 p ) ω = 0 .
[ 2 f ( z , t ) t 2 ] t = 0 = ( i z ) 1 / 2 p = 0 1 p ! ( i 2 z ) p × { i z [ 2 p f ˜ ( ω ) ω 2 p ] ω = 0 + 1 z 2 [ 2 p + 2 f ˜ ( ω ) ω 2 p + 2 ] ω = 0 } .
u p ( z , 0 ) = 1 z p = 0 1 p ! ( 1 2 z ) p [ ( exp { i [ z + ( 2 p + 1 ) π 4 ] } 1 z exp [ i ( 2 p + 1 ) π 4 ] ) [ 2 p f ˜ ( ω ) ω 2 p ] ω = 0 1 z 2 exp [ i ( 2 p + 1 ) π 4 ] [ 2 p + 2 f ˜ ( ω ) ω 2 p + 2 ] ω = 0 ] = a ( i π 2 z ) 1 / 2 p = 0 1 p ! ( i 2 z ) p 2 p × [ exp ( i z ) + ( 1 ) p + 1 ( 2 p + 1 ) 1 z ] ,
2 p = 1 a ( 2 π ) 1 / 2 [ 2 p f ˜ ( ω ) ω 2 p ] ω = 0 .
sech ( π 2 ω ) = 1 + p = 1 ( π 2 ) 2 p E 2 p ( 2 p ) ! ω 2 p , | ω | < 1 ,
1 1 + ω 2 = 1 + p = 1 ( 1 ) p ω 2 p .
E 2 q = ( 1 ) q 2 ( 2 π ) 2 q + 1 ( 2 q ) ! r = 0 ( 1 ) r ( 2 r + 1 ) 2 q + 1 .
E 0 = 1 , E 2 = 1 , E 4 = 5 , E 6 = 61 , E 8 = 1385 , E 10 = 50521 ,
f ˜ ( ω ) = a ( π 2 ) 1 / 2 { 1 + p = 1 [ q = 0 p ( 1 ) p q ( π 2 ) 2 q E 2 q ( 2 q ) ! ] ω 2 p } .
[ 2 p f ˜ ( ω ) ω 2 p ] ω = 0 = a ( π 2 ) 1 / 2 ( 2 p ) ! q = 0 p ( 1 ) p q ( π 2 ) 2 q E 2 q ( 2 q ) ! .

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