Abstract

This paper solves the four coupled equations describing non-degenerate four-wave mixing, with the focus on amplifying a signal in a fiber optical parametric amplifier (FOPA). Based on the full analytic solution, a simple approximate solution describing the gain is developed. The advantage of this new approximation is that it includes the depletion of the pumps, which is lacking in the usual quasi-linearized approximation. With the proposed model it is thus simple to predict the gain of a FOPA, which we demonstrate with a highly nonlinear fiber to show that an undepleted FOPA can produce a flat gain spectrum with a bandwidth in the 100-nm range, centered on the zero-dispersion wavelength. When running the FOPA in depletion, this range can be slightly increased.

© 2011 Optical Society of America

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2010

R. Slavík, 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]

2009

C.-S. Brès, A. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Multicasting of 320-Gb/s Channel in Self-Seeded Parametric Amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009).
[CrossRef]

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, K. Rottwitt, and L. Grüner-Nielsen, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 21, 872–874 (2009).
[CrossRef]

M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, “Silica-based highly nonlinear fibers and their application,” IEEE J. Sel. Top. Quantum Electron. 15, 103–113 (2009).
[CrossRef]

2008

2007

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

2004

2003

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15, 957–959 (2003).
[CrossRef]

2002

M. N. Islam and O. Boyraz, “Fiber parametric amplifiers for wavelength band conversion,” IEEE J. Sel. Top. Quantum Electron. 8, 527–537 (2002).
[CrossRef]

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]

1993

1992

K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. 28, 883–894 (1992).
[CrossRef]

1990

1989

Alic, N.

C.-S. Brès, A. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Multicasting of 320-Gb/s Channel in Self-Seeded Parametric Amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009).
[CrossRef]

J. M. Chávez Boggio, J. R. Windmiller, M. Knutzen, R. Jiang, C. Brès, N. Alic, B. Stossel, K. Rottwitt, and S. Radic, “730-nm optical parametric conversion from near- to short-wave infrared band,” Opt. Express 16, 5435–5443 (2008).
[CrossRef] [PubMed]

Andrekson, P. A.

R. Slavík, 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]

P. Kylemark, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Semi-analytic saturation theory of fiber optical parametric amplifiers,” J. Lightwave Technol. 24, 3471–3479 (2006).
[CrossRef]

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]

Bogris, A.

R. Slavík, 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]

Boyraz, O.

M. N. Islam and O. Boyraz, “Fiber parametric amplifiers for wavelength band conversion,” IEEE J. Sel. Top. Quantum Electron. 8, 527–537 (2002).
[CrossRef]

Brès, C.

Brès, C.-S.

C.-S. Brès, A. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Multicasting of 320-Gb/s Channel in Self-Seeded Parametric Amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009).
[CrossRef]

Cao, X. D.

Cappelline, G.

Centanni, J. C.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15, 957–959 (2003).
[CrossRef]

Chávez Boggio, J. M.

Chen, Y.

Chraplyvy, A. R.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15, 957–959 (2003).
[CrossRef]

Croussore, K.

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]

Dasgupta, S.

R. Slavík, 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]

Ellis, A. D.

R. Slavík, 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]

Grüner-Nielsen, L.

R. Slavík, 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]

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, K. Rottwitt, and L. Grüner-Nielsen, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 21, 872–874 (2009).
[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]

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]

Herstrøm, S.

R. Slavík, 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]

Hirano, M.

M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, “Silica-based highly nonlinear fibers and their application,” IEEE J. Sel. Top. Quantum Electron. 15, 103–113 (2009).
[CrossRef]

Inoue, K.

K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. 28, 883–894 (1992).
[CrossRef]

Islam, M. N.

M. N. Islam and O. Boyraz, “Fiber parametric amplifiers for wavelength band conversion,” IEEE J. Sel. Top. Quantum Electron. 8, 527–537 (2002).
[CrossRef]

Jakobsen, D.

R. Slavík, 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]

Jiang, R.

Jopson, R. M.

C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004).
[CrossRef] [PubMed]

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15, 957–959 (2003).
[CrossRef]

Kakande, J.

R. Slavík, 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]

Kanaev, A. V.

Karlsson, M.

Knutzen, M.

Kogelnik, H.

Kuo, B. P.-P.

C.-S. Brès, A. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Multicasting of 320-Gb/s Channel in Self-Seeded Parametric Amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009).
[CrossRef]

Kylemark, P.

Li, G.

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]

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.

Lorenzen, M.

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, K. Rottwitt, and L. Grüner-Nielsen, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 21, 872–874 (2009).
[CrossRef]

Lundström, C.

R. Slavík, 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]

McKinstrie, C. J.

Nakanishi, T.

M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, “Silica-based highly nonlinear fibers and their application,” IEEE J. Sel. Top. Quantum Electron. 15, 103–113 (2009).
[CrossRef]

Nielsen, C. V.

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, K. Rottwitt, and L. Grüner-Nielsen, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 21, 872–874 (2009).
[CrossRef]

Noordegraaf, D.

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, K. Rottwitt, and L. Grüner-Nielsen, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 21, 872–874 (2009).
[CrossRef]

O’Gorman, J.

R. Slavík, 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]

Okuno, T.

M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, “Silica-based highly nonlinear fibers and their application,” IEEE J. Sel. Top. Quantum Electron. 15, 103–113 (2009).
[CrossRef]

Onishi, M.

M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, “Silica-based highly nonlinear fibers and their application,” IEEE J. Sel. Top. Quantum Electron. 15, 103–113 (2009).
[CrossRef]

Parmigiani, F.

R. Slavík, 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]

Petropoulos, P.

R. Slavík, 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]

Peucheret, C.

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, K. Rottwitt, and L. Grüner-Nielsen, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 21, 872–874 (2009).
[CrossRef]

Phelan, R.

R. Slavík, 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]

Radic, S.

C.-S. Brès, A. Wiberg, B. P.-P. Kuo, N. Alic, and S. Radic, “Multicasting of 320-Gb/s Channel in Self-Seeded Parametric Amplifier,” IEEE Photon. Technol. Lett. 21, 1002–1004 (2009).
[CrossRef]

J. M. Chávez Boggio, J. R. Windmiller, M. Knutzen, R. Jiang, C. Brès, N. Alic, B. Stossel, K. Rottwitt, and S. Radic, “730-nm optical parametric conversion from near- to short-wave infrared band,” Opt. Express 16, 5435–5443 (2008).
[CrossRef] [PubMed]

C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12, 2033–2055 (2004).
[CrossRef] [PubMed]

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15, 957–959 (2003).
[CrossRef]

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[CrossRef]

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C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, K. Rottwitt, and L. Grüner-Nielsen, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 21, 872–874 (2009).
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C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, K. Rottwitt, and L. Grüner-Nielsen, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 21, 872–874 (2009).
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R. Slavík, 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).
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R. Slavík, 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]

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R. Slavík, 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]

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R. Slavík, 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]

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R. Slavík, 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).
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[CrossRef]

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15, 957–959 (2003).
[CrossRef]

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, K. Rottwitt, and L. Grüner-Nielsen, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 21, 872–874 (2009).
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Figures (3)

Fig. 1
Fig. 1

Evolution of the power of the idler with Pp = 30 dBm and Ps = 0 dBm. The solid line shows the full analytic solution, the dash-dotted line is the usual quasi-linearized solution and the dashed line is the approximate solution found in this paper. It is evident that the setup in (c) results in complete power transfer, whereas the usual desired setup (a) results in the fastest growing idler. (b) shows that a zero wave-number mismatch results in an almost complete power transfer.

Fig. 2
Fig. 2

Gain calculated at γL = 3 W−1 with Pp = 30 dBm. The solid line is the analytically calculated gain, the dashed line is the gain calculated from Eq. (2) and the dash-dotted line is calculated using the approximations described in this paper. In (a) the pumps are not in saturation, so there is no significant difference between them. However, in (b) the pumps begin to be in saturation, which Eq. (2) does not take into account. In (c) the input power is so high that the pumps have depleted and for some values of Δβ/γ the power has started to couple back into the pumps resulting in lower gain. Only the full analytic solution accounts for this phenomenon.

Fig. 3
Fig. 3

Gain profiles for 30-dBm pumps. For cases in which the pumps are placed symmetrically around the ZDW, as in (a) and (b), it is seen that the gain is wide and flat over 113 and 123 nm for the undepleted (a) and depleted (b) case, respectively. It is possible to slightly misalign one of the pumps and continue to have wide and flat gain (c).

Equations (33)

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d A 1 d z = i β 1 A 1 + i γ { [ | A 1 | 2 + 2 ( | A 2 | 2 + | A 3 | 2 + | A 4 | 2 ) ] A 1 + 2 A 2 A 3 A 4 * } ,
d A 2 d z = i β 2 A 2 + i γ { [ | A 2 | 2 + 2 ( | A 1 | 2 + | A 3 | 2 + | A 4 | 2 ) ] A 2 + 2 A 1 A 3 * A 4 } ,
d A 3 d z = i β 3 A 3 + i γ { [ | A 3 | 2 + 2 ( | A 1 | 2 + | A 2 | 2 + | A 4 | 2 ) ] A 3 + 2 A 1 A 2 * A 4 } ,
d A 4 d z = i β 4 A 4 + i γ { [ | A 4 | 2 + 2 ( | A 1 | 2 + | A 2 | 2 + | A 3 | 2 ) ] A 4 + 2 A 1 * A 2 A 3 } ,
P s ( z ) = P s ( 0 ) { 1 + [ 2 γ P p g sinh ( g z ) ] 2 } ,
P i ( z ) = P s ( 0 ) [ 2 γ P p g sinh ( g z ) ] 2 ,
d P α d z = s α 4 γ β P β sin θ ,
d ϕ α d z = β α + γ ( 2 β P β P α ) + 2 γ Π β P β P α cos θ ,
P 1 = P s + F ,
P 2 = P 3 = P p F ,
P 4 = F ,
G ( z ) = P s ( z ) P s ( 0 ) = 1 + F ( z ) P s .
d F d z = 4 γ ( P p F ) F ( P s + F ) sin θ ,
d θ d z = 2 γ [ ( P p F ) ( F P s + F + P s + F F ) 2 F ( P s + F ) ] cos θ + γ δ 4 γ F ,
δ = Δ β γ + 2 P p P s .
H = 4 γ ( P p F ) F ( P s + F ) cos θ + γ ( δ 2 F ) F .
d F d z = H θ , d θ d z = H F .
cos θ = ( δ 2 F ) F 4 ( P p F ) F ( P s + F ) .
( d F d z ) 2 = γ 2 [ 16 ( P p F ) 2 F ( P s + F ) ( δ 2 F ) 2 F 2 ] .
( d F d z ) 2 = 12 γ 2 F ( f 1 F ) ( f 2 F ) ( F f 3 ) ,
F ( z ) = f 3 f 2 sn 2 [ γ z 3 f 1 ( f 2 f 3 ) , m ] f 2 + f 3 + f 2 sn 2 [ γ z 3 f 1 ( f 2 f 3 ) , m ] ,
m 2 = f 2 ( f 1 f 3 ) f 1 ( f 2 f 3 ) .
Δ β = γ P s .
F ( z ) = 4 P p P s sinh 2 [ γ z P p ( 3 P p + 4 P s ) ] 3 P p + 4 P s + 4 P s sinh 2 [ γ z P p ( 3 P p + 4 P s ) ] .
r a = 16 P s P p 2 ( P p + 1 6 Δ β γ ) ( P p 1 2 Δ β γ ) ,
r b = P p + 1 6 Δ β γ 12 P s P s Δ β γ 2 P p Δ β γ ,
r c = P p 1 2 Δ β γ 4 P s 3 P p + 2 Δ β γ 6 P p + Δ β γ .
F ( z ) f 2 f 3 f 2 f 3 sinh 2 [ γ z 3 f 1 ( f 2 f 3 ) ] 1 + 1 f 2 f 2 f 3 f 2 f 3 sinh 2 [ γ z 3 f 1 ( f 2 f 3 ) ] .
3 f 1 ( f 2 f 2 ) g γ ,
f 2 f 3 f 2 f 3 P s ( 2 γ P p g ) 2 ,
F ( z ) F Q L ( z ) 1 + F Q L ( z ) P sat = P s [ 2 γ P p g sinh 2 ( g z ) ] 2 1 + P s P sat [ 2 γ P p g sinh 2 ( g z ) ] 2 ,
P sat = f 2 { 1 6 ( Δ β γ P s ) + P p , 6 P p < Δ β γ < P s , 1 2 ( Δ β γ P s ) + P p , P s < Δ β γ < 2 P p .
Δ β β 4 12 [ ( ω s ω ZD ) 4 ( ω p ω ZD ) 4 ] .

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