## Parametric amplification in presence of dispersion fluctuations

Optics Express, Vol. 12, Issue 1, pp. 136-142 (2004)

http://dx.doi.org/10.1364/OPEX.12.000136

Acrobat PDF (140 KB)

### Abstract

Parametric amplification in fibers with dispersion fluctuations is analyzed. The fluctuations are modelled as a stochastic process, with their size at a given position modelled as a Gaussian, and the autocorrelation decreasing exponentially. Two models are studied: in one the dispersion is piecewise constant, while in the other it is continuous. We find that the amplification does not depend on the models’ details and that only fluctuations with long correlation lengths affect the amplification significantly.

© 2004 Optical Society of America

## 1. Introduction

*ω*

_{p}co-propagates in an optical fiber with a weaker signal with frequency

*ω*

_{s}. By degenerate four-wave mixing (FWM), induced by the cubic nonlinearity of the glass, energy from the pump is transferred to the signal and, because of energy conservation, to an idler with frequency

*ω*

_{i}=2

*ω*

_{p}

*-ω*

_{s}as well [1, 2]. A key advantage of parametric amplification is its wide bandwidth, which is not limited by the properties of erbium, as in erbium doped fiber amplifiers, or by the properties of Raman phonons in glass, such as in Raman amplifiers [1, 2, 3

3. M-C Ho, K. Uesaka, M. Marhic, Y. Akasaka, and L.G. Kazovsky, “200-nm-bandwidth fiber optical amplifier combining parametric an Raman gain,” J. Lightwave Technol. **19**, 977–980 (2001). [CrossRef]

*β*

_{p,s,i}are the propagation constants of the pump, signal and idler.

4. M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am. B , **15**, 2269–2275 (1998). It is noted that in the Appendix in this paper the matrices **G** and **H** are implicitly, and incorrectly, assumed to commute. [CrossRef]

5. I. Brener, P.P. Mitra, D.D. Lee, and D.J. Thomson, “High-resolution zero-dispersion wavelength mapping in single-mode fiber,” Opt. Lett. **23**, 1520–1522 (1998). [CrossRef]

7. M. Eiselt, R.M. Jopson, and R.H. Stolen, “Non-destructive position-resolved measurement of the zero-dispersion wavelength in an optical fiber,” J. Lightwave Technol. **15**, 135–142 (1997). [CrossRef]

8. N. Kuwaki and M. Ohashi, “Evolution of longitudinal chromatic dispersion,” J. Lightwave Technol. **8**, 1476–1480 (1990). [CrossRef]

9. M. González-Herráez, P. Corredera, M.L. Hernanz, and J.A. Méndez, “Retrieval of the zero-dispersion wavelength map of an optical fiber from measurement of its continuous wave four-wave mixing efficiency,” Opt. Lett. **27**, 1546–1548 (2002). [CrossRef]

10. J.M. Chávez-Boggio, P. Dainese, and H.L. Fragnito, “Performance of a two-pump fiber optical parametric amplifier in a 10 Gb/s×64 channel dense wavelength division multiplexing system,” Opt. Commun. **218**, 303–310 (2003). [CrossRef]

*β*

_{p,s,i}to vary with position, and hence also the phase matching parameter Δ

*β*(1), and thus the gain. These variations in the fiber parameters can occur over long length scales, of the order of 100–1000 m [4

4. M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am. B , **15**, 2269–2275 (1998). It is noted that in the Appendix in this paper the matrices **G** and **H** are implicitly, and incorrectly, assumed to commute. [CrossRef]

5. I. Brener, P.P. Mitra, D.D. Lee, and D.J. Thomson, “High-resolution zero-dispersion wavelength mapping in single-mode fiber,” Opt. Lett. **23**, 1520–1522 (1998). [CrossRef]

7. M. Eiselt, R.M. Jopson, and R.H. Stolen, “Non-destructive position-resolved measurement of the zero-dispersion wavelength in an optical fiber,” J. Lightwave Technol. **15**, 135–142 (1997). [CrossRef]

9. M. González-Herráez, P. Corredera, M.L. Hernanz, and J.A. Méndez, “Retrieval of the zero-dispersion wavelength map of an optical fiber from measurement of its continuous wave four-wave mixing efficiency,” Opt. Lett. **27**, 1546–1548 (2002). [CrossRef]

10. J.M. Chávez-Boggio, P. Dainese, and H.L. Fragnito, “Performance of a two-pump fiber optical parametric amplifier in a 10 Gb/s×64 channel dense wavelength division multiplexing system,” Opt. Commun. **218**, 303–310 (2003). [CrossRef]

4. M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am. B , **15**, 2269–2275 (1998). It is noted that in the Appendix in this paper the matrices **G** and **H** are implicitly, and incorrectly, assumed to commute. [CrossRef]

8. N. Kuwaki and M. Ohashi, “Evolution of longitudinal chromatic dispersion,” J. Lightwave Technol. **8**, 1476–1480 (1990). [CrossRef]

11. J.M. Chávez Boggia, S. Tenenbaum, and H.L. Fragnito, “Amplification of broadband noise pumped by two lasers in optical fibers,” J. Opt. Soc. Am. B **18**, 1428–1435 (2001). [CrossRef]

**15**, 2269–2275 (1998). It is noted that in the Appendix in this paper the matrices **G** and **H** are implicitly, and incorrectly, assumed to commute. [CrossRef]

*et al*. [12

12. F. Kh. Abdullaev, S.A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A **220**, 213–218 (1996). [CrossRef]

*et al*. restrict their work to fibers with periodically varying quadratic dispersion, and treat this case perturbatively. Karlsson considers the fiber dispersion to be derived from a stochastic process, but his subsequent analysis is flawed [4

**15**, 2269–2275 (1998). It is noted that in the Appendix in this paper the matrices **G** and **H** are implicitly, and incorrectly, assumed to commute. [CrossRef]

## 2. Propagation in the presence of FWM

**15**, 2269–2275 (1998). It is noted that in the Appendix in this paper the matrices **G** and **H** are implicitly, and incorrectly, assumed to commute. [CrossRef]

*ω*

_{p,s,i}separately, ignoring weaker fields at other frequencies. In the undepleted pump approximation, we then find for the pump amplitude

*A*

_{p},

*P*

_{0}=|

*A*

_{p}|

^{2}. The signal and idler amplitudes,

*A*

_{s}

*, A*

_{i}satisfy the set of linear differential equations [4

**15**, 2269–2275 (1998). It is noted that in the Appendix in this paper the matrices **G** and **H** are implicitly, and incorrectly, assumed to commute. [CrossRef]

*γP*

_{0}-Δ

*β*/2, A is the column vector with elements

*A*

_{s}and

*A**

_{i}, and

*γ=ωn*

^{(2)}/(

*cA*

_{eff}) is the effective nonlinearity of the fiber. Here

*n*

^{(2)}is the nonlinear refractive index,

*A*

_{eff}the effective area of the modes, and

*ω*the frequency of the light. In the treatment below we take

*γ*to be the same for the three frequencies

*ω*

_{p,s,i}.

*matrix*

**M**that describes how the amplitudes of the signal and idler evolve with position

**M**is written in terms of hyperbolic functions, which is appropriate for

*α*

^{2}>0, and the gain therefore grows exponentially. Otherwise, when

*α*

^{2}<0, sinusoidal functions must be used, and the gain is expected to be small. In this work we take the initial conditions

*A*

_{i}(0)=0 and

*A*

_{s}(0)=1. In the absence of dispersion fluctuations, and when

*α*

^{2}>0, the signal gain for a system of length

*L*is thus

*γP*

_{0}.

## 3. Stochastic models for dispersion variations

*β*, is taken to be a stochastic variable and therefore so is Δk. We write Δk as Δk=Δ+

*δ*k where Δ is the average value of Δk, but

*δ*k varies randomly. Following Karlsson [4

**15**, 2269–2275 (1998). It is noted that in the Appendix in this paper the matrices **G** and **H** are implicitly, and incorrectly, assumed to commute. [CrossRef]

**15**, 2269–2275 (1998). It is noted that in the Appendix in this paper the matrices **G** and **H** are implicitly, and incorrectly, assumed to commute. [CrossRef]

*L*

_{c}is the correlation length, the typical length scale over which the fluctuations occur.

*piecewise constant*, whereas in Model II, which is Gaussian to all orders, the

*fluctuations vary continuously*. Figure 1 shows the examples of Model I and II for dispersion variations.

*L*

_{c}between events. Thus, the distribution of the segment lengths

*L*

_{s}follows the exponentially decreasing function

5. I. Brener, P.P. Mitra, D.D. Lee, and D.J. Thomson, “High-resolution zero-dispersion wavelength mapping in single-mode fiber,” Opt. Lett. **23**, 1520–1522 (1998). [CrossRef]

7. M. Eiselt, R.M. Jopson, and R.H. Stolen, “Non-destructive position-resolved measurement of the zero-dispersion wavelength in an optical fiber,” J. Lightwave Technol. **15**, 135–142 (1997). [CrossRef]

9. M. González-Herráez, P. Corredera, M.L. Hernanz, and J.A. Méndez, “Retrieval of the zero-dispersion wavelength map of an optical fiber from measurement of its continuous wave four-wave mixing efficiency,” Opt. Lett. **27**, 1546–1548 (2002). [CrossRef]

10. J.M. Chávez-Boggio, P. Dainese, and H.L. Fragnito, “Performance of a two-pump fiber optical parametric amplifier in a 10 Gb/s×64 channel dense wavelength division multiplexing system,” Opt. Commun. **218**, 303–310 (2003). [CrossRef]

*δ*k(

*z*

_{0}) at the beginning of the fiber. Then

*δ*k(

*z*

_{1}), at

*z*

_{1}>

*z*

_{0}, is given by the conditional distribution

*δ*k(

*z*) from (9) by

*m*is the output of the normal distribution [0,1] of the same random number generator used for Model I and where

*r*=exp(-|

*z*

_{1}-

*z*

_{0}|/

*L*

_{c}). In our implementation, we sample Δk along the length with the step size of 0.004

*L*

_{c}or 0.001

*L*, whichever is smaller. Then we use the fourth order Runge Kutta integration method to integrate the coupled equations (3). Note that in Model I,

*δ*k is constant in each segment of average length

*L*

_{c}so that (4) can be used, whereas in Model II the calculation is done for at least 250 steps per correlation length. Therefore, the speed of computation for Model II is smaller than for Model I.

### 3.1. Numerical results

*N*=900,000 realizations for Model I, indicated in red, and

*N*=5,000 realizations for Model II, indicated in green. The ensemble is smaller for Model II than for Model I since, as discussed in Section 3, the speed of calculation is lower in this case. Since the gain is typically measured in decibells, the average gain can be computed in two ways. We can either calculate the gain in decibells for each realization and then average these, or first average over the realizations and then express this average in decibells. The first of these corresponds to the average that would be measured if the ensemble of fibers were connected in series; the second would be obtained if the ensemble were connected in parallel. Here we choose the former since it corresponds to situation we wish to describe. We also calculate the standard deviation σg of the gain distribution, which is given by the error bars in the graphs. The upward error corresponds to the width following from Model I, whereas the lower bar corresponds to that following from Model II. Note finally that the accuracy of the average gain is given by

*σ*

_{a}

*=σ*

_{g}/√

*N*, which is below 0.14 dB for Model II, and smaller for Model I.

*γP*

_{0}

*L*=4, corresponding by Eq. (4) to a maximum gain of cosh

^{2}

*γP*

_{0}

*L*=28.7 dB when Δ=0. However, other values of

*γP*

_{0}

*L*lead to the same conclusions. In Figs. 2 we show the average gain versus the dimensionless quantity

*γP*

_{0}

*L*

_{c}, for fixed values of Δ/

*γP*

_{0}and

*σ/γP*

_{0}. It is noted that from the governing Eqs. (3), the results only depend on these ratios. Therefore we immediately see that the effect of the fluctuations decreases when

*γP*

_{0}increases,

*i.e*., for increasing pump power or fiber nonlinearity. We also see that the effect of the correlation length of a fiber depends on how it compares with the gain length 1/(

*γP*

_{0}). We also show the amplification results for two extreme cases that are calculated easily: (1)

*σ*=0,

*i.e*., no fluctuations, and (2)

*L*

_{c}≫

*L*. When σ=0 then the “ideal gain” is simply given by Eq. (5). In Figs. 2 we show this result by a solid line. When

*L*

_{c}≫

*L*, then Δk does not vary over the length of fiber and so the average gain is given by the expectation value of (5) subject to distribution (6)

*α*

^{2}<0, the hyperbolic functions need to be replaced by sinusoidal functions. The result of Eq. (10) is given by the dotted line in Figs. 2.

14. P.K.A. Wai and C.R. Menyuk, “Anisotropic diffusion of the state of polarization in optical fibers with randomly varying birefringence,” Opt. Lett. **24**, 2493–2495 (1995). [CrossRef]

*L*

_{c}→0, so that the dispersion varies rapidly with position, we can use an argument that applies to both Models, but most directly to Model I. It is straightforward to see from Eqs. (3) that in the presence of dispersion fluctuations, the powers |

*A*

_{p,i}|

^{2}and its first derivative are both continuous, while the second derivative is discontinuous. Deviations between different realizations within a segment thus increase with

*L*

_{c}is the average length of the segments. The average number of segments is

*L/L*

_{c}, and the total deviation in the gain between different realizations thus scales as

*L/L*

_{c}∝

*L*

_{c}. Hence, when

*L*

_{c}→0, the effect of the fluctuations disappears. Thus in this limit result (5) applies (solid lines in Figs. 2) and the variations in the gain disappear, whereas when

*L*

_{c}→∞ we have result (10) (dotted line in Figs. 2).

*γP*

_{0}(Figs. 2(a) and (b)), |Δ|≳

*γP*

_{0}(Figs. 2(c) and (d)), and |Δ|⋍

*γP*

_{0}(Figs. 2(e) and (f)). These correspond to frequencies that are tuned close to the gain peak, far from the gain peak, and tuned close to the edges of the region with exponential gain, respectively. In all three regimes the gain behaves as required when

*L*

_{c}→0 and

*L*

_{c}→∞. Figures 2(a) and (b) show that when |Δ|≲

*γP*

_{0}, the gain decreases monotonically with

*L*

_{c}. We can understand this by considering the extreme case where Δ→0. Then the gain is maximum and fluctuations can only cause the gain to decrease. By increasing σ the gain decreases more strongly. In contrast, Figs. 2(c) and (d) show that when |Δ|≳

*γP*

_{0}, the gain increases monotonically with

*L*

_{c}. This can be understood in a similar way as before: when

*σ*=0,

*α*

^{2}<0, and no exponential significant gain arises. Fluctuations may cause some exponential gain to occur, thus increasing the average gain. Finally, Figs. 2(e) and (f) show results for |Δ|=

*γP*

_{0}. Now the fluctuation may cause the average gain either to increase, or to decrease, and the dependence on

*L*

_{c}is not monotonic.

## 4. Discussion and conclusions

*γP*

_{0}

*L*

_{c}≪1, do not affect the operation of the parametric amplifier, as can be understood using a simple scaling argument. This manifests itself in two ways. First, the gain in this limit approaches that in the absence of fluctuations. Secondly, in this limit the fluctuations in the gain rapidly vanish. Therefore, only dispersion fluctuations that vary on long length scales matter for the amplification, while the effect of short-length scale fluctuations decreases quickly with decreasing correlation length. As an example, consider a highly nonlinear fiber with

*γ*=20 W

^{-1}km

^{-1},

*L*=400 m and

*P*

_{0}=0.5 W, so that

*γP*

_{0}

*L*=4, and the maximum gain is 28.7 dB, as in Section 3. Any fluctuations with

*σ*≪

*γP*

_{0}=2 km

^{-1}is likely to be negligible. For fluctuations for which this inequality is not satisfied, those with

*L*

_{c}≈1m(so

*γP*

_{0}

*L*

_{c}=0.01) can likely still be ignored, while fluctuations with a correlation length of around 100 m (so

*γP*

_{0}

*L*

_{c}=1) need to be considered carefully.

## References and links

1. | G. P. Agrawal, |

2. | S. Kinoshita et al. Eds., |

3. | M-C Ho, K. Uesaka, M. Marhic, Y. Akasaka, and L.G. Kazovsky, “200-nm-bandwidth fiber optical amplifier combining parametric an Raman gain,” J. Lightwave Technol. |

4. | M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am. B , |

5. | I. Brener, P.P. Mitra, D.D. Lee, and D.J. Thomson, “High-resolution zero-dispersion wavelength mapping in single-mode fiber,” Opt. Lett. |

6. | J.S. Pereira |

7. | M. Eiselt, R.M. Jopson, and R.H. Stolen, “Non-destructive position-resolved measurement of the zero-dispersion wavelength in an optical fiber,” J. Lightwave Technol. |

8. | N. Kuwaki and M. Ohashi, “Evolution of longitudinal chromatic dispersion,” J. Lightwave Technol. |

9. | M. González-Herráez, P. Corredera, M.L. Hernanz, and J.A. Méndez, “Retrieval of the zero-dispersion wavelength map of an optical fiber from measurement of its continuous wave four-wave mixing efficiency,” Opt. Lett. |

10. | J.M. Chávez-Boggio, P. Dainese, and H.L. Fragnito, “Performance of a two-pump fiber optical parametric amplifier in a 10 Gb/s×64 channel dense wavelength division multiplexing system,” Opt. Commun. |

11. | J.M. Chávez Boggia, S. Tenenbaum, and H.L. Fragnito, “Amplification of broadband noise pumped by two lasers in optical fibers,” J. Opt. Soc. Am. B |

12. | F. Kh. Abdullaev, S.A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A |

13. | A. Papoulis, |

14. | P.K.A. Wai and C.R. Menyuk, “Anisotropic diffusion of the state of polarization in optical fibers with randomly varying birefringence,” Opt. Lett. |

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

**ToC Category:**

Research Papers

**History**

Original Manuscript: October 21, 2003

Revised Manuscript: December 22, 2003

Published: January 12, 2004

**Citation**

Mitra Farahmand and Martijn de Sterke, "Parametric amplification in presence of dispersion fluctuations," Opt. Express **12**, 136-142 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-136

Sort: Journal | Reset

### References

- G. P.Agrawal, Nonlinear Fiber Optics (Academic Press, 1995)
- S. Kinoshita et al. Eds., Optical amplifiers and their application (Optical Society of America, Washington, DC 1999).
- M-C Ho, K. Uesaka, M. Marhic, Y. Akasaka, and L.G. Kazovsky, �??200-nm-bandwidth fiber optical amplifier combining parametric an Raman gain,�?? J. Lightwave Technol. 19, 977-980 (2001). [CrossRef]
- M. Karlsson, �??Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,�?? J. Opt. Soc. Am. B, 15, 2269-2275 (1998). It is noted that in the Appendix in this paper the matrices G and H are implicitly, and incorrectly, assumed to commute. [CrossRef]
- I. Brener, P.P. Mitra, D.D. Lee, and D.J. Thomson, �??High-resolution zero-dispersion wavelength mapping in single-mode fiber,�?? Opt. Lett. 23, 1520-1522 (1998). [CrossRef]
- J.S. Pereira et al, �??Measurement of zero-dispersion wavelength using a novel method based on four-wave mixing,�?? in Optical Fiber Communications Conference, Vol. 2, 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 345-346.
- M. Eiselt, R.M. Jopson, and R.H. Stolen, �??Non-destructive position-resolved measurement of the zero-dispersion wavelength in an optical fiber,�?? J. Lightwave Technol. 15, 135-142 (1997). [CrossRef]
- N. Kuwaki and M. Ohashi, �??Evolution of longitudinal chromatic dispersion,�?? J. Lightwave Technol. 8, 1476-1480 (1990). [CrossRef]
- M. González-Herráez, P. Corredera, M.L. Hernanz, and J.A. Méndez, �??Retrieval of the zero-dispersion wavelength map of an optical fiber from measurement of its continuous wave four-wave mixing efficiency,�?? Opt. Lett. 27, 1546-1548 (2002). [CrossRef]
- J.M. Chávez-Boggio, P. Dainese, and H.L. Fragnito, �??Performance of a two-pump fiber optical parametric amplifier in a 10 Gb/s�?64 channel dense wavelength division multiplexing system,�?? Opt. Commun. 218, 303-310 (2003). [CrossRef]
- J.M. Chávez Boggia, S. Tenenbaum, and H.L. Fragnito, �??Amplification of broadband noise pumped by two lasers in optical fibers,�?? J. Opt. Soc. Am. B 18, 1428-1435 (2001). [CrossRef]
- F. Kh. Abdullaev, S.A. Darmanyan, A. Kobyakov, and F. Lederer, �??Modulational instability in optical fibers with variable dispersion,�?? Phys. Lett. A 220, 213-218 (1996). [CrossRef]
- A. Papoulis, Probability, random variables, and stochastic processes(McGraw-Hill, New York, 1965).
- P.K.A. Wai and C.R. Menyuk, �??Anisotropic diffusion of the state of polarization in optical fibers with randomly varying birefringence,�?? Opt. Lett. 24, 2493-2495 (1995). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.