## Phase-sensitive amplification in a fiber

Optics Express, Vol. 12, Issue 20, pp. 4973-4979 (2004)

http://dx.doi.org/10.1364/OPEX.12.004973

Acrobat PDF (72 KB)

### Abstract

Phase-sensitive amplification (PSA) has the potential to improve significantly the performance of optical communication systems. PSA is known to occur in *χ*^{(2)} devices, and in a fiber interferometer, which is an example of a *χ*^{(3)} device. In this report some four-wave mixing processes are described, which produce PSA directly in fibers.

© 2004 Optical Society of America

## 1. Introduction

1. R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. **21**, 766–773 (1985). [CrossRef]

2. H. P. Yuen, “Reduction of quantum fluctuation and suppression of the Gordon-Haus effect with phase-sensitive linear amplifiers,” Opt. Lett. **17**, 73–75 (1992). [CrossRef] [PubMed]

3. Y. Mu and C. M. Savage, “Parametric amplifiers in phase-noise-limited optical communications,” J. Opt. Soc. Am. B **9**, 65–70 (1992). [CrossRef]

4. R. D. Li, P. Kumar, W. L. Kath, and J. N. Kutz, “Combating dispersion with parametric amplifers,” IEEE Photon. Technol. Lett. **5**, 669–672 (1993). [CrossRef]

5. W. Imajuku and A. Takada, “Reduction of fiber-nonlinearity-enhanced amplifier noise by means of phase-sensitive amplifiers,” Opt. Lett. **22**, 31–33 (1997). [CrossRef] [PubMed]

*χ*

^{(2)}medium [6]. PSA is also possible in a

*χ*

^{(3)}medium: Some schemes are based on degenerate backward [7

7. H. P. Yuen and J. H. Shapiro, “Generation and detection of two-photon coherent states in degenerate four-wave mixing,” Opt. Lett. **4**, 334–336 (1979). [CrossRef] [PubMed]

8. B. Yurke, “Use of cavities in squeezed-state-generation,” Phys. Rev. A **29**, 408–410 (1984). [CrossRef]

9. P. Kumar and J. H. Shapiro, “Squeezed-state generation via forward degenerate four-wave mixing,” Phys. Rev. A **30**, 1568–1571 (1984). [CrossRef]

7. H. P. Yuen and J. H. Shapiro, “Generation and detection of two-photon coherent states in degenerate four-wave mixing,” Opt. Lett. **4**, 334–336 (1979). [CrossRef] [PubMed]

9. P. Kumar and J. H. Shapiro, “Squeezed-state generation via forward degenerate four-wave mixing,” Phys. Rev. A **30**, 1568–1571 (1984). [CrossRef]

8. B. Yurke, “Use of cavities in squeezed-state-generation,” Phys. Rev. A **29**, 408–410 (1984). [CrossRef]

10. M. D. Levenson, R. M. Shelby, A. Aspect, M. Reid, and D. F. Walls, “Generation and detection of squeezed states of light by nondegenerate four-wave mixing in an optical fiber,” Phys. Rev. A **32**, 1550–1562 (1985). [CrossRef] [PubMed]

11. M. J. Potasek and B. Yurke, “Squeezed light generation in a medium governed by the nonlinear Schrödinger equation,” Phys. Rev. A **35**, 3974–3977 (1987). [CrossRef] [PubMed]

12. T. A. B. Kennedy and S. Wabnitz, “Quantum propagation: Squeezing via modulational polarization instabilities in a birefringent nonlinear medium,” Phys. Rev. A **38**, 563–566 (1988). [CrossRef] [PubMed]

13. M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B **7**, 30–34 (1990). [CrossRef]

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

## 2. Parametric amplification in a *χ*^{(2)} medium

*χ*

^{(2)}medium occurs when a pump wave interacts with signal and idler waves of lower frequency. If the signal and idler frequencies are identical, the process is said to be degenerate. (The backward and near-forward FWM schemes mentioned in Section 1 are even more degenerate, because the pump frequencies equal the common signal and idler frequency.) Degenerate PA is governed by the frequency-matching condition

*ω*

_{2}= 2

*ω*

_{1}, where

*ω*

_{2}and

*ω*

_{1}are the pump and signal frequencies, respectively, and the amplitude equations

*β*= 2

*β*

_{1}-

*β*

_{2}is the linear wavenumber mismatch and

*is the nonlinear coupling coefficient, which is proportional to*γ ¯

*χ*

^{(2)}. One can choose the amplitude units in such a way that |

*A*

_{j}|

^{2}is proportional to the photon flux

*P*

_{j}.

*A*

_{2}(

*z*) =

*A*

_{2}(0). Let

*κ*= (|

*γ*|

^{2}-

*δ*

^{2})

^{1/2}. The transfer functions satisfy the auxiliary equation |

*μ*|

^{2}- |

*ν*|

^{2}= 1. The input-output relation described by Eqs. (5)–(7) is the defining property of a PSA. In quantum optics this relation is called a squeezing transformation [6]. To illustrate the effects of PSA, consider the simplest case, in which

*δ*= 0, and suppose that

*A*

_{2}is real (measure the signal phase relative to the pump phase). Then, according to Eqs. (5)–(7), the in-phase signal quadrature (

*B*

_{1}+

*κz*), whereas the out-of-phase quadrature (

*B*

_{1}-

*κz*)]. This property is responsible for the phenomena described in Section 1.

## 3. Four-wave mixing in a *χ*^{(3)} medium

*χ*

^{(3)}medium. However, it is possible to produce, in a

*χ*

^{(3)}medium, an idler that is a non-frequency-shifted image of the signal. The bidirectional scheme [15

15. P. Narum and R. W. Boyd, “Nonfrequency-shifted phase conjugation by Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. **23**, 1211–1216 (1987). [CrossRef]

*ω*

_{3}and

*ω*

_{1}, and a forward signal with frequency

*ω*

_{2}= (

*ω*

_{3}+

*ω*

_{1})/2. The interaction of these waves produces a backward idler with frequency

*ω*

_{2}. In the unidirectional scheme described in this report, all the waves propagate in the forward direction. The scalar version of this degenerate FWM process (inverse MI) is illustrated in Fig. 1. It is governed by the frequency-matching condition

*ω*

_{3}+

*ω*

_{1}= 2

*ω*

_{2}and the amplitude equations

*β*= 2

*β*

_{2}-

*β*

_{3}-

*β*

_{1}is the linear wavenumber mismatch and

*is the nonlinear coupling coefficient, which is proportional to*γ ¯

*χ*

^{(3)}[16

16. C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. **8**, 538–547 and 956 (2002), and references therein. [CrossRef]

*and 2*γ ¯

*, respectively. However, the deviations from the stated values are qualitatively unimportant and, for typical frequencies, are quantitatively insignificant.*γ ¯

*χ*

^{(3)}medium the pumps are subject to SPM and CPM. Let

*A*

_{3}(0) =

*B*

_{3},

*A*

_{1}(0) =

*B*

_{1}and

*δ*=

*β*/2 +

*(*γ ¯

*P*

_{3}+

*P*

_{1})/2 and

*γ*= 2

γ ¯
B

_{3}

*B*

_{1}. Equation (14) has the same form as Eq. (4), so the input-output relation is described by Eqs. (5)–(7): Degenerate scalar FWM provides PSA in a fiber.

*χ*

^{(3)}medium involves pumps with frequencies

*ω*

_{4}and

*ω*

_{1}, a nominal signal with frequency

*ω*

_{2}and a nominal idler with frequency

*ω*

_{3}. If the pumps are co-polarized (scalar FWM), so are the signal and idler. If the pumps are cross-polarized (vector FWM), the signal is aligned with one pump and the idler is aligned with the other. Nondegenerate vector FWM is governed by the frequency-matching condition

*ω*

_{4}+

*ω*

_{1}=

*ω*

_{2}+

*ω*

_{3}and the amplitude equations

*β*=

*β*

_{2}+

*β*

_{3}-

*β*

_{4}-

*β*

_{1}is the linear wavenumber mismatch,

*is the nonlinear coupling coefficient for co-polarized waves and*γ ¯

*ε*is the ratio of the coupling coefficients for cross-polarized and co-polarized waves. For (polarization-maintaining) fibers with constant dispersion

*ε*= 2/3 [17

17. C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express **11**, 2619–2633 (2003) and references therein. [CrossRef] [PubMed]

*ε*= 1 [18

18. 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) and refrences therein. [CrossRef] [PubMed]

*ω*

_{3}=

*ω*

_{2}. This process is the vector version of the aforementioned unidirectional scheme. It is illustrated in Fig. 2.

*A*

_{4}(0) =

*B*

_{4},

*A*

_{1}(0) =

*B*

_{1},

*δ*=

*β*/2 +

*(*γ ¯

*P*

_{4}+

*P*

_{1})/2 and

*γ*=

γ ¯
εB

_{4}

*B*

_{1}. It follows from Eqs. (23) and (24) that

*μ*and

*ν*were defined in Eqs. (6) and (7). In quantum optics the input-output relation defined by Eqs. (25) and (26) is called a two-mode squeezing transformation [6]. If there is no input idler [

*B*

_{3}(0) = 0], the output idler is proportional to the complex conjugate of the input signal. Consequently, this FWM process is called phase conjugation (PC). If the input signal is split evenly between the two polarizations [|

*B*

_{3}(0)| = |

*B*

_{2}(0)|], Eq. (25) reduces to Eq. (5): Degenerate vector FWM produces PSA in a fiber.

*vice versa*. Because both interactions amplify the signal component and generate the idler component in the same way, the total output powers of the signal and idler are proportional to the total input power of the signal: The signal and idler gains produced by nondegenerate vector FWM do not depend on the signal polarization [18

18. 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) and refrences therein. [CrossRef] [PubMed]

## 4. Cascaded four-wave mixing processes in a *χ*^{(3)} medium

*ω*

_{2}+

*ω*

_{3}=

*ω*

_{4}+

*ω*

_{1}and the amplitude equations (15)–(18). Unlike the aforementioned FWM process, waves 1 and 3 are the pumps (rather than 1 and 4), wave 2 is the signal and wave 4 is the idler (rather than 3). Because power flows from the signal to the idler, the photon flux of the output idler cannot exceed the photon flux of the input signal: BS is intrinsically stable (does not provide gain).

*A*

_{1}(0) =

*B*

_{1},

*A*

_{3}(0) =

*B*

_{3},

*k*= (|

*γ*|

^{2}+

*δ*

^{2})

^{1/2}. The transfer functions satisfy the auxiliary equation |

*|*μ ¯

^{2}+|

*|*ν ¯

^{2}= 1. In quantum optics the input-output relation defined by Eqs. (33)–(36) is called a beam-splitter transformation [6]. In the current context

*B*

_{4}(0) = 0. For the ideal case in which

*δ*= 0,

*k*=

*π*/2 and

*γ*is real,

*B*

_{2}(

*z*′) =

*B*

_{2}(0)/2

^{1/2}and

*B*

_{4}(

*z*′) =

*iB*

_{2}(0)/2

^{1/2}: The output idler is a phase-shifted, but non-conjugated image of the signal. (The phase shift is required by photon-flux conservation.) Because the signal and idler frequencies are distinct, their relative phase can be modified between the BS and PC processes. (Because

*k*

_{4}≠

*k*

_{2}, propagation effects this change naturally.)

*ω*

_{5}+

*ω*

_{1}=

*ω*

_{2}+

*ω*

_{4}and the amplitude equations (15)–(18), with the subscripts 3 and 4 replaced by 4 and 5, respectively. The effects of SPM and CPM on the pumps are described by Eqs. (19) and (20). The signal and idler amplitudes obey the (linearized) equations

*δ*=

*β*/2 +

*(*γ ¯

*P*

_{5}+

*P*

_{1})/2,

*γ*=

γ ¯
εB

_{5}

*B*

_{1}and

*β*=

*β*

_{2}+

*β*

_{4}-

*β*

_{5}-

*β*

_{1}. It follows from Eqs. (37) and (38) that

*μ*and

*ν*were defined in Eqs. (6) and (7). Because

*B*

_{2}(

*z*′) is proportional to

*B*

_{2}(0) and

*z*′) is proportional to

*B*

_{4}(

*z*′)| = |

*B*

_{2}(

*z*′)|, Eq. (39) reduces to Eq. (5). It should be noted that the pump frequencies are not unique: Other combinations of pump frequencies provide PSA by cascaded BS and PC. Furthermore, BS can also be used to combine a signal with the frequency-shifted, and conjugated, idler produced by prior PC.

## 5. Summary

## Acknowledgments

## References and links

1. | R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. |

2. | H. P. Yuen, “Reduction of quantum fluctuation and suppression of the Gordon-Haus effect with phase-sensitive linear amplifiers,” Opt. Lett. |

3. | Y. Mu and C. M. Savage, “Parametric amplifiers in phase-noise-limited optical communications,” J. Opt. Soc. Am. B |

4. | R. D. Li, P. Kumar, W. L. Kath, and J. N. Kutz, “Combating dispersion with parametric amplifers,” IEEE Photon. Technol. Lett. |

5. | W. Imajuku and A. Takada, “Reduction of fiber-nonlinearity-enhanced amplifier noise by means of phase-sensitive amplifiers,” Opt. Lett. |

6. | R. Loudon, |

7. | H. P. Yuen and J. H. Shapiro, “Generation and detection of two-photon coherent states in degenerate four-wave mixing,” Opt. Lett. |

8. | B. Yurke, “Use of cavities in squeezed-state-generation,” Phys. Rev. A |

9. | P. Kumar and J. H. Shapiro, “Squeezed-state generation via forward degenerate four-wave mixing,” Phys. Rev. A |

10. | M. D. Levenson, R. M. Shelby, A. Aspect, M. Reid, and D. F. Walls, “Generation and detection of squeezed states of light by nondegenerate four-wave mixing in an optical fiber,” Phys. Rev. A |

11. | M. J. Potasek and B. Yurke, “Squeezed light generation in a medium governed by the nonlinear Schrödinger equation,” Phys. Rev. A |

12. | T. A. B. Kennedy and S. Wabnitz, “Quantum propagation: Squeezing via modulational polarization instabilities in a birefringent nonlinear medium,” Phys. Rev. A |

13. | M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B |

14. | M. E. Marhic and C. H. Hsia, “Optical amplification in a nonlinear interferometer,” Electron. Lett. |

15. | P. Narum and R. W. Boyd, “Nonfrequency-shifted phase conjugation by Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. |

16. | C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. |

17. | C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express |

18. | C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.5040) Nonlinear optics : Phase conjugation

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 9, 2004

Revised Manuscript: September 27, 2004

Published: October 4, 2004

**Citation**

C. McKinstrie and S. Radic, "Phase-sensitive amplification in a fiber," Opt. Express **12**, 4973-4979 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4973

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### References

- R. Loudon, �??Theory of noise accumulation in linear optical-amplifier chains,�?? IEEE J. Quantum Electron. 21, 766�??773 (1985). [CrossRef]
- H. P. Yuen, �??Reduction of quantum fluctuation and suppression of the Gordon�??Haus effect with phase-sensitive linear amplifiers,�?? Opt. Lett. 17, 73�??75 (1992). [CrossRef] [PubMed]
- Y. Mu and C. M. Savage, �??Parametric amplifiers in phase-noise-limited optical communications,�?? J. Opt. Soc. Am. B 9, 65�??70 (1992) [CrossRef]
- R. D. Li, P. Kumar, W. L. Kath and J. N. Kutz, �??Combating dispersion with parametric amplifers,�?? IEEE Photon. Technol. Lett. 5, 669�??672 (1993). [CrossRef]
- W. Imajuku and A. Takada, �??Reduction of fiber-nonlinearity-enhanced amplifier noise by means of phase-sensitive amplifiers,�?? Opt. Lett. 22, 31�??33 (1997) [CrossRef] [PubMed]
- R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford University Press, Oxford, 2000)
- H. P. Yuen and J. H. Shapiro, �??Generation and detection of two-photon coherent states in degenerate four-wave mixing,�?? Opt. Lett. 4, 334�??336 (1979). [CrossRef] [PubMed]
- B. Yurke, �??Use of cavities in squeezed-state-generation,�?? Phys. Rev. A 29, 408�??410 (1984). [CrossRef]
- P. Kumar and J. H. Shapiro, �??Squeezed-state generation via forward degenerate four-wave mixing,�?? Phys. Rev. A 30, 1568�??1571 (1984). [CrossRef]
- M. D. Levenson, R. M. Shelby, A. Aspect, M. Reid and D. F.Walls, �??Generation and detection of squeezed states of light by nondegenerate four-wave mixing in an optical fiber,�?? Phys. Rev. A 32, 1550�??1562 (1985). [CrossRef] [PubMed]
- M. J. Potasek and B. Yurke, �??Squeezed light generation in a medium governed by the nonlinear Schrödinger equation,�?? Phys. Rev. A 35, 3974�??3977 (1987). [CrossRef] [PubMed]
- T. A. B. Kennedy and S. Wabnitz, �??Quantum propagation: Squeezing via modulational polarization instabilities in a birefringent nonlinear medium,�?? Phys. Rev. A 38, 563�??566 (1988). [CrossRef] [PubMed]
- M. Shirasaki and H. A. Haus, �??Squeezing of pulses in a nonlinear interferometer,�?? J. Opt. Soc. Am. B 7, 30�??34 (1990). [CrossRef]
- M. E. Marhic and C. H. Hsia, �??Optical amplification in a nonlinear interferometer,�?? Electron. Lett. 27, 210�??211 (1991). [CrossRef]
- P. Narum and R. W. Boyd, �??Nonfrequency-shifted phase conjugation by Brillouin-enhanced four-wave mixing,�?? IEEE J. Quantum Electron. 23, 1211�??1216 (1987). [CrossRef]
- C. J. McKinstrie, S. Radic and A. R. Chraplyvy, �??Parametric amplifiers driven by two pump waves,�?? IEEE J. Sel. Top. Quantum Electron. 8, 538�??547 and 956 (2002), and references therein. [CrossRef]
- C. J. McKinstrie, S. Radic and C. Xie, �??Parametric instabilities driven by orthogonal pump waves in birefringent fibers,�?? Opt. Express 11, 2619�??2633 (2003) and references therein. [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) and refrences therein. [CrossRef] [PubMed]

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