Abstract

Operation of a degenerate dual-pump phase sensitive amplifier (PSA) is thoroughly numerically investigated using a multi-wave model, taking into account high-order waves associated with undesired four-wave mixing (FWM) processes. More accurate phase-sensitive signal gain characteristics are obtained compared to the conventional 3-wave model, leading to precise optimization of the pump configuration in a degenerate dual-pump PSA. The signal gain for different pump configurations, as well as the phase sensitivity, is obtained and interpreted by investigating the dominant FWM processes in terms of the corresponding phase matching. Moreover, the relation between dispersion slope and the width of the signal gain curve versus the pump-pump wavelength separation is revealed, permitting the application-oriented arbitrary tailoring of the signal gains by manipulating the dispersion profile and pump wavelength allocation.

© 2015 Optical Society of America

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References

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  1. Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2014 (3)

2013 (1)

Z. Tong and S. Radic, “Low-noise optical amplification and signal processing in parametric devices,” Adv. Opt. Photonics 5(3), 318 (2013).
[Crossref]

2012 (1)

2011 (2)

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
[Crossref]

2010 (1)

A. Agarwal, T. Banwell, and T. K. Woodward, “Low Distortion Multicasting of an Analog Signal by Self-Seeded Parametric Mixer,” IEEE Photonics Technol. Lett. 22(5), 332–334 (2010).
[Crossref]

2008 (2)

2006 (1)

A. Vedadi, A. Mussot, E. Lantz, H. Maillotte, and T. Sylvestre, “Theoretical study of gain distortions in dual-pump fiber optical parametric amplifers,” Opt. Commun. 267(1), 244–252 (2006).
[Crossref]

2002 (2)

J. Hansryd, P. A. Andrekson, M. Westlund, Jie Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

C. J. McKinstrie and S. Radic, “Parametric amplifiers driven by two pump waves with dissimilar frequencies,” Opt. Lett. 27(13), 1138–1140 (2002).
[Crossref] [PubMed]

1996 (2)

1982 (2)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields 26(8), 1817–1839 (1982).
[Crossref]

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982).
[Crossref]

Agarwal, A.

A. Agarwal, T. Banwell, and T. K. Woodward, “Low Distortion Multicasting of an Analog Signal by Self-Seeded Parametric Mixer,” IEEE Photonics Technol. Lett. 22(5), 332–334 (2010).
[Crossref]

Andrekson, P. A.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
[Crossref]

J. Hansryd, P. A. Andrekson, M. Westlund, Jie Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

Banwell, T.

A. Agarwal, T. Banwell, and T. K. Woodward, “Low Distortion Multicasting of an Analog Signal by Self-Seeded Parametric Mixer,” IEEE Photonics Technol. Lett. 22(5), 332–334 (2010).
[Crossref]

Bhatia, A.

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982).
[Crossref]

Blessing, D. J.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
[Crossref]

Bretenaker, F.

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields 26(8), 1817–1839 (1982).
[Crossref]

Chiang, T.-K.

Ferrini, G.

Foster, M. A.

Fragnito, H. L.

Fsaifes, I.

Fu, S.

Gao, M.

Goldfarb, F.

Gruner-Nielsen, L.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Grüner-Nielsen, L.

Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
[Crossref]

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westlund, Jie Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

Hedekvist, P.-O.

J. Hansryd, P. A. Andrekson, M. Westlund, Jie Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

Inoue, T.

Jie Li,

J. Hansryd, P. A. Andrekson, M. Westlund, Jie Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

Kagi, N.

Karlsson, M.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
[Crossref]

Kazovsky, L. G.

Kurosu, T.

Labidi, T.

Lantz, E.

A. Vedadi, M. E. Marhic, E. Lantz, H. Maillotte, and T. Sylvestre, “Investigation of gain ripple in two-pump fiber optical parametric amplifiers,” Opt. Lett. 33(19), 2203–2205 (2008).
[Crossref] [PubMed]

A. Vedadi, A. Mussot, E. Lantz, H. Maillotte, and T. Sylvestre, “Theoretical study of gain distortions in dual-pump fiber optical parametric amplifers,” Opt. Commun. 267(1), 244–252 (2006).
[Crossref]

Lian, J.

Liu, D.

Lundstrom, C.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
[Crossref]

Maillotte, H.

A. Vedadi, M. E. Marhic, E. Lantz, H. Maillotte, and T. Sylvestre, “Investigation of gain ripple in two-pump fiber optical parametric amplifiers,” Opt. Lett. 33(19), 2203–2205 (2008).
[Crossref] [PubMed]

A. Vedadi, A. Mussot, E. Lantz, H. Maillotte, and T. Sylvestre, “Theoretical study of gain distortions in dual-pump fiber optical parametric amplifers,” Opt. Commun. 267(1), 244–252 (2006).
[Crossref]

Marhic, M. E.

McKinstrie, C. J.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
[Crossref]

C. J. McKinstrie and S. Radic, “Parametric amplifiers driven by two pump waves with dissimilar frequencies,” Opt. Lett. 27(13), 1138–1140 (2002).
[Crossref] [PubMed]

Meng, Y.

Mussot, A.

A. Vedadi, A. Mussot, E. Lantz, H. Maillotte, and T. Sylvestre, “Theoretical study of gain distortions in dual-pump fiber optical parametric amplifers,” Opt. Commun. 267(1), 244–252 (2006).
[Crossref]

Namiki, S.

Park, Y.

Puttnam, B. J.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
[Crossref]

Radic, S.

Z. Tong and S. Radic, “Low-noise optical amplification and signal processing in parametric devices,” Adv. Opt. Photonics 5(3), 318 (2013).
[Crossref]

C. J. McKinstrie and S. Radic, “Parametric amplifiers driven by two pump waves with dissimilar frequencies,” Opt. Lett. 27(13), 1138–1140 (2002).
[Crossref] [PubMed]

Rieznik, A. A.

Shun, P.

Stolen, R. H.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982).
[Crossref]

Sylvestre, T.

A. Vedadi, M. E. Marhic, E. Lantz, H. Maillotte, and T. Sylvestre, “Investigation of gain ripple in two-pump fiber optical parametric amplifiers,” Opt. Lett. 33(19), 2203–2205 (2008).
[Crossref] [PubMed]

A. Vedadi, A. Mussot, E. Lantz, H. Maillotte, and T. Sylvestre, “Theoretical study of gain distortions in dual-pump fiber optical parametric amplifers,” Opt. Commun. 267(1), 244–252 (2006).
[Crossref]

Tang, M.

Ting, H.-F.

Tipsuwannakul, E.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
[Crossref]

Toda, H.

Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
[Crossref]

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Tong, Z.

Z. Tong and S. Radic, “Low-noise optical amplification and signal processing in parametric devices,” Adv. Opt. Photonics 5(3), 318 (2013).
[Crossref]

Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
[Crossref]

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Treps, N.

Vedadi, A.

A. Vedadi, M. E. Marhic, E. Lantz, H. Maillotte, and T. Sylvestre, “Investigation of gain ripple in two-pump fiber optical parametric amplifiers,” Opt. Lett. 33(19), 2203–2205 (2008).
[Crossref] [PubMed]

A. Vedadi, A. Mussot, E. Lantz, H. Maillotte, and T. Sylvestre, “Theoretical study of gain distortions in dual-pump fiber optical parametric amplifers,” Opt. Commun. 267(1), 244–252 (2006).
[Crossref]

Westlund, M.

J. Hansryd, P. A. Andrekson, M. Westlund, Jie Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

Woodward, T. K.

A. Agarwal, T. Banwell, and T. K. Woodward, “Low Distortion Multicasting of an Analog Signal by Self-Seeded Parametric Mixer,” IEEE Photonics Technol. Lett. 22(5), 332–334 (2010).
[Crossref]

Yang, F. S.

Adv. Opt. Photonics (1)

Z. Tong and S. Radic, “Low-noise optical amplification and signal processing in parametric devices,” Adv. Opt. Photonics 5(3), 318 (2013).
[Crossref]

IEEE J. Quantum Electron. (1)

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Hansryd, P. A. Andrekson, M. Westlund, Jie Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

IEEE Photonics Technol. Lett. (1)

A. Agarwal, T. Banwell, and T. K. Woodward, “Low Distortion Multicasting of an Analog Signal by Self-Seeded Parametric Mixer,” IEEE Photonics Technol. Lett. 22(5), 332–334 (2010).
[Crossref]

J. Lightwave Technol. (1)

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

Nat. Photonics (2)

Z. Tong, C. Lundstrom, 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(7), 430–436 (2011).
[Crossref]

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011).
[Crossref]

Opt. Commun. (1)

A. Vedadi, A. Mussot, E. Lantz, H. Maillotte, and T. Sylvestre, “Theoretical study of gain distortions in dual-pump fiber optical parametric amplifers,” Opt. Commun. 267(1), 244–252 (2006).
[Crossref]

Opt. Express (1)

Opt. Lett. (5)

Phys. Rev. D Part. Fields (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields 26(8), 1817–1839 (1982).
[Crossref]

Other (4)

C.-S. Brès, A. O. J. Wiberg, B. P.-P. Kuo, E. Myslivets, N. Alic, B. Stossel, and S. Radic, “RF Photonic Link Employing Optical Phase Sensitive Amplification,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2012), paper OM3B.5.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

I. Fsaifes, T. Labidi, F. Goldfarb, and F. Bretenaker, “Intermodulation distortion analysis of an analog photonic link employing parametric phase sensitive amplification,” in European Conference on Optical Communication, (ECOC, 2014), paper P.1.10.
[Crossref]

M. Baillot, T. Chartier, and M. Joindot, “Multiple four-wave mixing in optical fibers,” in European Conference on Optical Communication, (ECOC, 2014), paper We.3.7.2.

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Figures (7)

Fig. 1
Fig. 1

(a) Experimental result of dual-pump degenerate PSA: considerable high-order waves are generated by underlying high-order FWM processes. (b) Labeling of waves, separation, and offset for 7-wave model. The three waves in the dashed-line rectangle are those considered in the fundamental 3-wave model.

Fig. 2
Fig. 2

Maximum signal gain versus ΔλPP calculated with input signal phase φ0,max maximizing the gain when δλofs = 0 using 3- and 7-wave models (a) analytically and numerically without attenuation; (b) numerically with attenuation, respectively. The fiber parameters are given in the text.

Fig. 3
Fig. 3

Maximum signal gain vs. signal wavelength offset δλofs with respect to the zero dispersion wavelength λZDW and vs. pump-pump separation ΔλPP with input signal phase φ0,max to maximize the gain for (a) 3- and (b) 7-wave models. The right axis corresponds to the dispersion difference ΔDPP between the two pumps while the upper axis corresponds to the dispersion Dsig for the signal.

Fig. 4
Fig. 4

Case where δλofs = 0 nm. The input signal phase is set at the value φ0,max to maximize the gain for each pump configuration. (a) Maximum signal gain for 3- and 7-wave models versus pump-pump wavelength separation ΔλPP; the inset shows the wavelengths configuration relatively to the zero-dispersion wavelength (ZDW). (b) Corresponding evolution of the output powers of the seven waves. (c) Normalized phase mismatch angles of the relevant FWM processes. The input phases of the signal and pumps are chosen to correspond to the maximum gain of the PSA in the ordinary 3-wave regime. The regions featured by ellipses are discussed in the text, and the corresponding dominant processes are given in the insets.

Fig. 5
Fig. 5

(a-c) Same as Fig. 4 for δλofs = –10 nm (signal in normal dispersion region). (d-f) Same as Fig. 4 for δλofs = 10 nm (signal in anomalous dispersion region). Both cases are calculated with φ0,max adjusted to maximize the signal gain and with zero input pump phases.

Fig. 6
Fig. 6

Maximum signal gain versus ΔλPP. The input pump phases are equal to zero and the initial signal phase is taken at φ0,max to optimize the gain. The plots are calculated for different factors multiplying the dispersion slope Dλ when δλofs = 0. The initial wave powers remain the same for all the values Dλ.

Fig. 7
Fig. 7

(a) Same maximum signal gain versus pump-pump separation as in Fig. 5(d), obtained for δλofs = + 10 nm. The vertical arrows point at the values of ΔλPP corresponding to (b-f). (b-f) Gain (in dB) polar plot versus input signal phase with zero initial pump phases for the 3-wave (dashed line) and 7-wave (full line models) for (b) ΔλPP = 6.8 nm (first gain peak in (a)), (c) ΔλPP = 12.2 nm (first gain dip in (a)), (d) ΔλPP = 14.2 nm (second gain peak in (a)), and (e) ΔλPP = 20.0 nm (third gain peak in (a)), (f) ΔλPP = 41.6 nm (second gain dip in (a)).

Equations (4)

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d A 0 dz = α 2 A 0 +iγ{ [ | A 0 | 2 +2 i=0,i0 6 | A i | 2 ] A 0 + A 1 2 A 3 e iΔ β 0311 z + A 2 2 A 4 e iΔ β 0422 z +2 A 1 A 2 A 0 e iΔ β 0012 z +2 A 1 A 6 A 4 e iΔ β 0416 z +2 A 1 A 3 A 5 e iΔ β 0513 z +2 A 1 A 4 A 2 e iΔ β 0214 z +2 A 2 A 5 A 3 e iΔ β 0325 z +2 A 2 A 3 A 1 e iΔ β 2301 z +2 A 2 A 4 A 6 e iΔ β 2406 z +2 A 4 A 5 A 1 e iΔ β 4501 z +2 A 5 A 6 A 0 e iΔ β 5600 z +2 A 3 A 4 A 0 e iΔ β 3400 z +2 A 3 A 6 A 2 e iΔ β 3602 z }
Δ β mnkl = β m + β n β k β l
κ mnkl =Δ β mnkl +γ P mnkl ,
γ P mnkl =γ( P k + P l P m P n ),

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