Abstract

We show that the phases of waves coupled by four-wave mixing (FWM) can be calculated in closed form, for either degenerate or nondegenerate FWM. The phase of each wave is the sum of a linear function of distance z, and elliptic integrals of the third kind. A variety of behaviors as a function of z are possible. The results can be used for obtaining the phase transfer function between any input wave and any output wave, a subject of interest for phase regeneration of communication signals.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  5. Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
    [CrossRef]
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    [CrossRef]
  7. K. Inoue, “Optical level equalisation based on gain saturation in optical parametric amplifiers,” Electron. Lett. 36, 1016–1017 (2000).
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    [CrossRef]
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2012 (1)

2011 (1)

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[CrossRef]

2010 (1)

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

2007 (1)

K. Croussore and G. Li, “Phase regeneration of NRZ-DPSK signals based on symmetric-pump phase-sensitive amplification,” IEEE Photon. Technol. Lett. 19, 864–866 (2007).
[CrossRef]

2006 (2)

2005 (1)

2000 (1)

K. Inoue, “Optical level equalisation based on gain saturation in optical parametric amplifiers,” Electron. Lett. 36, 1016–1017 (2000).
[CrossRef]

1991 (3)

1989 (1)

1982 (1)

C. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Andrekson, P. A.

C. Lundström, B. Corcoran, M. Karlsson, and P. A. Andrekson, “Phase and amplitude characteristics of a phase-sensitive amplifier operating in gain saturation,” Opt. Express 20, 21400–21412 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[CrossRef]

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bergman, K.

Blessing, D. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bogris, A.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Byrd, P. F.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer, 1971).

Cappellini, G.

Caves, C.

C. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

Chen, Y.

Corcoran, B.

Croussore, K.

Dasgupta, S.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Ellis, A. D.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Essiambre, R.-J.

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94, 952–985 (2006).
[CrossRef]

Friedman, M. D.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer, 1971).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, A. Jeffrey, ed., 5th ed. (Academic, 1994).

Grüner-Nielsen, L.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[CrossRef]

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Han, Y.

Haus, H. A.

Herstrøm, S.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Hsia, C. H.

M. E. Marhic, C. H. Hsia, and J. M. Jeong, “Optical amplification in a nonlinear fiber interferometer,” IEEE Electron. Lett. 27, 210–211 (1991).
[CrossRef]

Inoue, K.

K. Inoue, “Optical level equalisation based on gain saturation in optical parametric amplifiers,” Electron. Lett. 36, 1016–1017 (2000).
[CrossRef]

Jakobsen, D.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Jeong, J. M.

M. E. Marhic, C. H. Hsia, and J. M. Jeong, “Optical amplification in a nonlinear fiber interferometer,” IEEE Electron. Lett. 27, 210–211 (1991).
[CrossRef]

Kakande, J.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Karlsson, M.

C. Lundström, B. Corcoran, M. Karlsson, and P. A. Andrekson, “Phase and amplitude characteristics of a phase-sensitive amplifier operating in gain saturation,” Opt. Express 20, 21400–21412 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[CrossRef]

Kim, C.

Kim, I.

Li, G.

Lundström, C.

C. Lundström, B. Corcoran, M. Karlsson, and P. A. Andrekson, “Phase and amplitude characteristics of a phase-sensitive amplifier operating in gain saturation,” Opt. Express 20, 21400–21412 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[CrossRef]

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Marhic, M. E.

M. E. Marhic, C. H. Hsia, and J. M. Jeong, “Optical amplification in a nonlinear fiber interferometer,” IEEE Electron. Lett. 27, 210–211 (1991).
[CrossRef]

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices, (Cambridge, 2007).

McKinstrie, C. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[CrossRef]

O’Gorman, J.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Parmigiani, F.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Petropoulos, P.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Phelan, R.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Puttnam, B. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[CrossRef]

Richardson, D. J.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, A. Jeffrey, ed., 5th ed. (Academic, 1994).

Sjödin, M.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Slavik, R.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Snyder, A. W.

Sygletos, S.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Syvridis, D.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Tipsuwannakul, E.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[CrossRef]

Toda, H.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[CrossRef]

Tong, Z.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[CrossRef]

Trillo, S.

Weerasuriya, R.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Winzer, P. J.

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94, 952–985 (2006).
[CrossRef]

Electron. Lett. (1)

K. Inoue, “Optical level equalisation based on gain saturation in optical parametric amplifiers,” Electron. Lett. 36, 1016–1017 (2000).
[CrossRef]

IEEE Electron. Lett. (1)

M. E. Marhic, C. H. Hsia, and J. M. Jeong, “Optical amplification in a nonlinear fiber interferometer,” IEEE Electron. Lett. 27, 210–211 (1991).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

K. Croussore and G. Li, “Phase regeneration of NRZ-DPSK signals based on symmetric-pump phase-sensitive amplification,” IEEE Photon. Technol. Lett. 19, 864–866 (2007).
[CrossRef]

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

Nat. Photonics (2)

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[CrossRef]

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. D (1)

C. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

Proc. IEEE (1)

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94, 952–985 (2006).
[CrossRef]

Other (6)

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices, (Cambridge, 2007).

M. Abramovitz and I. Stegun, eds., Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1964), Vol. 55 http://www.nrbook.com/abramowitz_and_stegun/ .

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer, 1971).

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http://www.wolfram.com .

www.mathworks.com .

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Figures (3)

Fig. 1.
Fig. 1.

Plot of the idler phase θ4 versus z, for Δβ=0; γ=20W1km1; P10=P20=P30=1W; P40=0.01W; θ40=0; θ0=π.

Fig. 2.
Fig. 2.

Plot of the idler phase θ4 verus z, for Δβ=0; γ=20W1km1; P10=P20=P30=1W; P40=109W; θ40=0. Solid line: θ0=0 (σ=1); dashed line: θ0=π (σ=1). The dotted line corresponds to the solution of Eq. (22).

Fig. 3.
Fig. 3.

Plot of signal output phase versus signal input phase, for γ=20W1km1; L=0.1km; P10=0.5W; P30=P40=0.001W; θ10=π/4rad; S=P30P40.

Equations (70)

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idAldZ=|Al|2Al+2jl=14|Aj|2Al+2AmAnAk*exp(iεlΔβZ/γ),l=14,
ifl=1or2,k=3l,m=3,n=4;εl=1,ifl=3or4,k=7l,m=1,n=2;εl=1.
idAldZAl*=Pl(2P0Pl)+2P1P2P3P4exp(iεlθ),l=14,
dAldZAl*+dAldZAl*=dPldZ=2iP1P2P3P4[exp(iεlθ)exp(iεlθ)]=4εlP1P2P3P4sinθ
dP1dZ=dP2dZ=dP3dZ=dP4dZ=4P1P2P3P4sinθ.
dθldZ=2P0Pl+2PlP1P2P3P4cosθl=1,2,3,4.
4P1P2P3P4cosθ=K2Sx+2x2.
dθldZ=2P0Pl+12Pl(K2Sx+2x2)=2P0(Pl0εlx)+12(Pl0εlx)(K2Sx+2x2).
dZ=±dx2h(x),
h(x)=3(xη1)(xη2)(xη3)(xη4),
dθl=±dxh(x)(ul+vlflx),
ul=P0Pl0+εlS4,
vl=εl2(K22+Pl02εlPl0S2),
fl=εlPl0.
θl=0x(ul+vlflx)dx3(xη1)(xη2)(xη3)(xη4).
θl,u=ul30xdx(xη1)(xη2)(xη3)(xη4)=2ul0xdZ=2ulZ(x).
θl,v=vl30xdx(flx)(xη1)(xη2)(xη3)(xη4)=vl3[η2xdx(flx)(xη1)(xη2)(xη3)(xη4)η20dx(flx)(xη1)(xη2)(xη3)(xη4)]
θl,v=vl32(flη2)(flη1)(η3η1)(η4η2)×((η2η1){Π[δ,(η3η2)(flη1)(η3η1)(flη2),k]Π[sin1(η2ηη1),(η3η2)(flη1)(η3η1)(flη2),k]}+(flη2)[F(δ,k)F(sin1η2ηη1,k)])=2vl(flη2)(flη1){Zc(η2η1)[Π(δ,nl,k)Π(sin1η2ηη1,nl,k)]+(flη2)Z},
δ=sin1[xη2η(xη1)]=sin1[sn(ZZ0Zc,k)].
θl=θl0+θl,u+θl,v=θl0+2(ul+vlflη1)Z+2Zcvl(η2η1)(flη2)(flη1)[Π(δ,nl,k)Π(sin1η2ηη1),nl,k],
θl=θl0+2(ul+vlflη1)Z+2Zcvl(η2η1)(flη2)(flη1)×[2m+12Π(π2,nl,k)+(1)mΠ(δ,nl,k)sΠ(sin1η2ηη1,nl,k)].
2P0P4<2P1P2P3P4cosθ.
K2+2P402+SP40=0,
P402+P40(Δβγ+P10+P20P30)+4P10P20P30cosθ0=0.
P40=2(2P10P20P30cos2θ0)1/3.
dθ4dZ=2P0S2.
θ4,Π=2Zcv4(η2η1)(f4η2)(f4η1)Π(δ,n4,k)=2Zcv4(η2η1)(P40η2)(P40η1)Π[δ,η(P40η1)(P40η2),k].
θ4,Π=σπ2,
Pl0=Pl0εlx1(Zd)andθ0=sin1[dx1(Zd)dZ/(4Pl0P20P30P40)].
Δβγ=3(P10+P20)+5P30.
h(x)=(P102x)2(P30+x)(P40+x)14[K1Sx+3x2]2,
S=Δβγ+ΔP0,
C02=7/4,
Sx3x2+2P1P3P4cosθ=K1,
K1=2P10P30P40cos(θ30+θ402θ10),
ΔP0=2P10P30P40,
dθ1dZ=2P0P1+2P3P4cosθ,
dθldZ=2P0Pl+P1PlP3P4cosθ,l=3,4.
θ1=0x(u1+v1f1x+w1x)dx(7/4)(xη1)(xη2)(xη3)(xη4)=A1+w1B1
u1=2P07P104+S2v1=12w1=12(K1+3P1024SP102)f1=P102.
u3=2P05P302S2v3=52w3=12(K1+3P302+SP30)f3=P30.
dθ1,3dZ=a+bx+cdx,
x(Z)=η1+(η2η1)1η·sn2(u,k),
θ1,3=θ10,30+(a+bη1+1dη1)γL+b(η2η1)z=0Ldz1ηsn2(u,k)c(η2η1)(dη1)(dη2)z=0Ldz1η(dη1dη2)sn2(u,k).
0udu1gsn2(u,k)=Π[g,am(u,k),k],
θ1,3=θ10,30+(a+bη1+cdη1)γL+A+B,
A=bZc(η2η1){Π[δ,am(uL,k),k]Π[δ,am(u0,k),k]},
B=cZcη2η1(dη1)(dη2){Π[δdη1dη2,am(uL,k),k]Π[δdη1dη2,am(u0,k),k]},
i(dAl*dZAldAldZAl*)=2Pl(2P0Pl)+4P1P2P3P4cosθ,l=14,
dθldZ=2P0Pl+2PlP1P2P3P4cosθ.
dθdZ=Δβγ+P1+P2P3P4+2P1P2P3P4(P31+P41P11P21)cosθ.
dθdZ=dθdxdxdZ=dθdx4P1P2P3P4sinθ,
dθdZ=4P1P2P3P4d(cosθ)dx.
Δβγ+P1+P2P3P4+4ddx(P1P2P3P4cosθ)=0.
Sx2x2+4P1P2P3P4cosθ=K2,
K2=4P10P20P30P40cosθ0,
dxdZ=4P1P2P3P4sinθ=±4P1P2P3P4(1cos2θ)=±2h(x),
h(x)=4(P10x)(P20x)(P30+x)(P40+x)14(K2Sx+2x2)2.
dZ=±dx2h(x).
s=sign(sinθ0).
dZ=dx2h(x)=dx23(xη1)(xη2)(xη3)(xη4),
Z=0xdx2h(x)=η2xdx2h(x)η20dx2h(x)=123η2xdx(xη1)(xη2)(xη3)(xη4)123η20dx(xη1)(xη2)(xη3)(xη4)=Zc[F(sin1xη2η(xη1),k)F(sin1η2ηη1,k)]=Z0+ZcF(sin1xη2η(xη1),k),
η=(η3η2)/(η3η1),
k=[(η3η2)(η4η1)/(η4η2)(η3η1)]1/2=[η(η4η1)/(η4η2)]1/2,
Zc=[3(η3η1)(η4η2)]1/2,
Z0=ZcF(sin1η2ηη1,k).
xη2η(xη1)=sn(ZZ0Zc,k),
x(Z)=η1+(η2η1)[1η·sn2(ZZ0Zc,k)]1.
Z=0xdx2h(x)=η2xdx2h(x)+η20dx2h(x)=Z0ZcF(sin1xη2η(xη1),k),
Z0=sZcF(sin1η2ηη1,k).

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