Abstract

Nondegenerate four-wave mixing in fibers enables the tunable and low-noise frequency conversion of optical signals. This paper shows that four-wave mixing driven by pulsed pumps can also regenerate and reshape optical signal pulses arbitrarily.

© 2012 OSA

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References

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  1. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
    [CrossRef]
  2. S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron.  E88-C, 859–869 (2005).
    [CrossRef]
  3. K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994).
    [CrossRef]
  4. T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004).
    [CrossRef]
  5. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
    [CrossRef] [PubMed]
  6. A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express 14, 8989–8994 (2006).
    [CrossRef] [PubMed]
  7. M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
    [CrossRef]
  8. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
    [CrossRef] [PubMed]
  9. K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
    [CrossRef]
  10. D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express 14, 8995–8999 (2006).
    [CrossRef] [PubMed]
  11. R. Provo, S. G. Murdoch, J. D. Harvey, and D. Méchin, “Bragg scattering in a positive β4 fiber,” Opt. Lett. 35, 3730–3732 (2010).
    [CrossRef] [PubMed]
  12. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 23, 109–111 (2011).
    [CrossRef]
  13. M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B 20, 2425–2433 (2003).
    [CrossRef]
  14. C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14, 8516–8534 (2006).
    [CrossRef] [PubMed]
  15. C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993).
    [CrossRef]
  16. A. Hirano, T. Kataoka, S. Kuwahara, M. Asobe, and Y. Yamabayashi, “All-optical limiter circuit based on four-wave mixing in optical fibres,” Electron. Lett. 34, 1410–1411 (1998).
    [CrossRef]
  17. K. Inoue, “Suppression of level fluctuation without extinction ratio degradation based on output saturation in higher order optical parametric interaction in fiber,” IEEE Photon. Technol. Lett. 13, 338–340 (2001).
    [CrossRef]

2011 (1)

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 23, 109–111 (2011).
[CrossRef]

2010 (3)

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[CrossRef]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[CrossRef] [PubMed]

R. Provo, S. G. Murdoch, J. D. Harvey, and D. Méchin, “Bragg scattering in a positive β4 fiber,” Opt. Lett. 35, 3730–3732 (2010).
[CrossRef] [PubMed]

2006 (3)

2005 (2)

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
[CrossRef] [PubMed]

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron.  E88-C, 859–869 (2005).
[CrossRef]

2004 (1)

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004).
[CrossRef]

2003 (1)

2002 (2)

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[CrossRef]

2001 (1)

K. Inoue, “Suppression of level fluctuation without extinction ratio degradation based on output saturation in higher order optical parametric interaction in fiber,” IEEE Photon. Technol. Lett. 13, 338–340 (2001).
[CrossRef]

1998 (1)

A. Hirano, T. Kataoka, S. Kuwahara, M. Asobe, and Y. Yamabayashi, “All-optical limiter circuit based on four-wave mixing in optical fibres,” Electron. Lett. 34, 1410–1411 (1998).
[CrossRef]

1994 (1)

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994).
[CrossRef]

1993 (1)

Andrekson, P. A.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Asobe, M.

A. Hirano, T. Kataoka, S. Kuwahara, M. Asobe, and Y. Yamabayashi, “All-optical limiter circuit based on four-wave mixing in optical fibres,” Electron. Lett. 34, 1410–1411 (1998).
[CrossRef]

Cao, X. D.

Centanni, J. C.

Gnauck, A. H.

Goh, C. S.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004).
[CrossRef]

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Harvey, J. D.

Hedekvist, P. O.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Hirano, A.

A. Hirano, T. Kataoka, S. Kuwahara, M. Asobe, and Y. Yamabayashi, “All-optical limiter circuit based on four-wave mixing in optical fibres,” Electron. Lett. 34, 1410–1411 (1998).
[CrossRef]

Inoue, K.

K. Inoue, “Suppression of level fluctuation without extinction ratio degradation based on output saturation in higher order optical parametric interaction in fiber,” IEEE Photon. Technol. Lett. 13, 338–340 (2001).
[CrossRef]

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994).
[CrossRef]

Jopson, R. M.

Kataoka, T.

A. Hirano, T. Kataoka, S. Kuwahara, M. Asobe, and Y. Yamabayashi, “All-optical limiter circuit based on four-wave mixing in optical fibres,” Electron. Lett. 34, 1410–1411 (1998).
[CrossRef]

Kazovsky, L. G.

M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B 20, 2425–2433 (2003).
[CrossRef]

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[CrossRef]

Kikuchi, K.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004).
[CrossRef]

Kogelnik, H.

Kuwahara, S.

A. Hirano, T. Kataoka, S. Kuwahara, M. Asobe, and Y. Yamabayashi, “All-optical limiter circuit based on four-wave mixing in optical fibres,” Electron. Lett. 34, 1410–1411 (1998).
[CrossRef]

Li, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Li, J. S.

Marhic, M. E.

M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B 20, 2425–2433 (2003).
[CrossRef]

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[CrossRef]

McGuinness, H. J.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 23, 109–111 (2011).
[CrossRef]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[CrossRef] [PubMed]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[CrossRef]

McKinstrie, C. J.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 23, 109–111 (2011).
[CrossRef]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[CrossRef] [PubMed]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[CrossRef]

C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14, 8516–8534 (2006).
[CrossRef] [PubMed]

D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express 14, 8995–8999 (2006).
[CrossRef] [PubMed]

A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express 14, 8989–8994 (2006).
[CrossRef] [PubMed]

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron.  E88-C, 859–869 (2005).
[CrossRef]

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993).
[CrossRef]

Méchin, D.

Murdoch, S. G.

Provo, R.

Radic, S.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 23, 109–111 (2011).
[CrossRef]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[CrossRef] [PubMed]

A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express 14, 8989–8994 (2006).
[CrossRef] [PubMed]

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron.  E88-C, 859–869 (2005).
[CrossRef]

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
[CrossRef] [PubMed]

Raymer, M. G.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 23, 109–111 (2011).
[CrossRef]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[CrossRef] [PubMed]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[CrossRef]

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
[CrossRef] [PubMed]

Schenato, L.

Set, S. Y.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004).
[CrossRef]

Tanemura, T.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004).
[CrossRef]

Uesaka, K.

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[CrossRef]

van Enk, S. J.

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[CrossRef]

Westlund, M.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Wong, K. K. Y.

M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Am. B 20, 2425–2433 (2003).
[CrossRef]

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[CrossRef]

Yamabayashi, Y.

A. Hirano, T. Kataoka, S. Kuwahara, M. Asobe, and Y. Yamabayashi, “All-optical limiter circuit based on four-wave mixing in optical fibres,” Electron. Lett. 34, 1410–1411 (1998).
[CrossRef]

Electron. Lett. (1)

A. Hirano, T. Kataoka, S. Kuwahara, M. Asobe, and Y. Yamabayashi, “All-optical limiter circuit based on four-wave mixing in optical fibres,” Electron. Lett. 34, 1410–1411 (1998).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Wavelength translation across 210 nm in the visible using vector Bragg scattering in a birefringent photonic crystal fiber,” IEEE Photon. Technol. Lett. 23, 109–111 (2011).
[CrossRef]

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994).
[CrossRef]

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004).
[CrossRef]

K. Inoue, “Suppression of level fluctuation without extinction ratio degradation based on output saturation in higher order optical parametric interaction in fiber,” IEEE Photon. Technol. Lett. 13, 338–340 (2001).
[CrossRef]

IEICE Trans. Electron (1)

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron.  E88-C, 859–869 (2005).
[CrossRef]

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

Opt. Commun. (1)

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Frequency diagram for nearby (left) and distant (right) Bragg scattering. Long arrows denote pumps (p and q), whereas short arrows denote idler and signal sidebands (r and s). Downward arrows denote modes that lose photons, whereas upward arrows denote modes that gain photons. The directions of the arrows are reversible.

Fig. 2
Fig. 2

(a) Predicted amplitude of the idler generated by a Gaussian signal and Gaussian pumps. The dotted, dashed, dot-dashed and solid curves denote interaction lengths of 0.5, 1.0, 2.0 and 4.0, respectively. (b) Comparison of the analytical predictions (dashed curve) and numerical results for a length of 4.0. The dot-dashed (solid) curve was obtained by solving Eqs. (4) and (5) numerically, without (with) the phase-modulation terms.

Fig. 3
Fig. 3

(a) Predicted amplitude of the idler generated by a Gaussian signal, a Gaussian pump and a super-Gaussian pump. The dotted, dashed, dot-dashed and solid curves denote interaction lengths of 0.5, 1.0, 2.0 and 4.0, respectively. (b) Comparison of the analytical predictions (dashed curve) and numerical results for a length of 4.0. The dot-dashed (solid) curve was obtained by solving Eqs. (4) and (5) numerically, without (with) the phase-modulation terms.

Fig. 4
Fig. 4

Amplitudes of a noisy Gaussian signal (top) and the regenerated idler (middle and bottom) produced by Gaussian pumps (left), and a Gaussian pump and a super-Gaussian pump (right). Solid curves denote the idler, whereas dashed curves denote the (rescaled) pumps. The top, middle and bottom figures correspond to distances of 0.0, 0.5 and 4.0, respectively. The curves were obtained by solving Eqs. (4) and (5) numerically.

Equations (10)

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

ω p + ω s = ω q + ω r , k p + k s = k q + k r ,
k ( ω ) β 0 ( ω a ) + β 1 ( ω a ) ω + β 2 ( ω a ) ω 2 / 2 + β 3 ( ω a ) ω 3 / 6 + β 4 ( ω a ) ω 4 / 24 ,
δ ( ω p 2 ω q 2 ) [ β 2 + β 4 ( ω p 2 + ω q 2 ) / 12 ] .
( z + β r t ) A r = i 2 γ ( | A p | 2 + | A q | 2 ) A r + i 2 γ A p A q * A s ,
( z + β s t ) A s = i 2 γ ( | A p | 2 + | A q | 2 ) A s + i 2 γ A p * A q A r ,
A j ( t , z ) = a j f j ( t β j z ) ,
( z + β r t ) A r ( t , z ) = i 2 γ A p ( t β s z ) A q * ( t β r z ) A s ( t β s z ) ,
z A r ( τ , z ) = i 2 γ A p ( τ + β r z z ) A s ( τ + β r s z ) A q * ( τ ) ,
A r ( τ , l ) = i 2 γ 0 l A p ( τ + β r s z ) A s ( τ + β r z z ) d z A q * ( τ ) .
A r ( t , l ) = i γ ¯ a s f q * ( t β r l ) ,

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