## Analytical characterization of optical power and noise figure of forward pumped Raman amplifiers

Optics Express, Vol. 12, Issue 18, pp. 4235-4245 (2004)

http://dx.doi.org/10.1364/OPEX.12.004235

Acrobat PDF (230 KB)

### Abstract

We show that it is possible to find analytic expressions for characterizing the evolution of signal and noise photon numbers along the active fiber of a forward-pumped Raman amplifier with unequal signal and pump loss coefficients. We confirm the validity of the result by comparing the analytical solutions with numerical calculations and by analytically deriving the well-known 3 dB noise figure limit for high Raman gain. Apart from aiding the analysis and design of forward pumped Raman amplifiers, these results also enable one to find approximate analytical solutions for bidirectional Raman amplifiers and backward pumped Raman amplifiers with Rayleigh backscattering and Brillouin scattering.

© 2004 Optical Society of America

## 1. Introduction

2. M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Select. Topics in Quantum Electron. **8**, 548–559 (2002). [CrossRef]

2. M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Select. Topics in Quantum Electron. **8**, 548–559 (2002). [CrossRef]

4. V. E. Perlin and H. G. Winful, “Optimal design of flat-gain wide-band fiber Raman amplifiers,” IEEE J. Lightwave Technol. **20**, 250–254 (2002). [CrossRef]

5. P. C. Xiao, Q.J. Zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. **15**, 206–208 (2003). [CrossRef]

7. M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. **3**, 806–813 (1985). [CrossRef]

8. R. Chinn, “Analysis of counter-pumped small-signal fiber Raman amplifiers,” Electron Lett. **33**, 607–608 (1997). [CrossRef]

9. B. Bobbs and C. Warner, “Closed-form solution for parametric second Stokes generation in Raman amplifiers,” IEEE J. Quantum Electron. **24**, 660–664 (1988). [CrossRef]

7. M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. **3**, 806–813 (1985). [CrossRef]

8. R. Chinn, “Analysis of counter-pumped small-signal fiber Raman amplifiers,” Electron Lett. **33**, 607–608 (1997). [CrossRef]

10. Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. **13**, 948–950 (2001). [CrossRef]

14. P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. **10**, 159–161 (1998). [CrossRef]

15. P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, “Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,” IEEE J. Lightwave Technol. **21**, 2224–2228 (2003). [CrossRef]

16. A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. **20**, 1635–1643 (2002). [CrossRef]

## 2. Theoretical model

*z*, is measured from the end of the pump coupler to the right as shown in Fig. 1.

*n*(

*z*), and pump photon number,

*nP*(

*z*), along the fiber can be characterized by following differential equations [7

7. M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. **3**, 806–813 (1985). [CrossRef]

8. R. Chinn, “Analysis of counter-pumped small-signal fiber Raman amplifiers,” Electron Lett. **33**, 607–608 (1997). [CrossRef]

*γ*is the Raman gain coefficient,

*α*is the signal loss coefficient and

_{S}*α*is the pump loss coefficient. The experimental evidence for the adequacy and validity of Eq. (1) and Eq. (2) for describing Raman amplification in optical fibers can be found in [17

_{P}17. J. Auyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. **14**, 347–352 (1978). [CrossRef]

**3**, 806–813 (1985). [CrossRef]

**33**, 607–608 (1997). [CrossRef]

*n*(

*z*):

## 3. Approximate analytical solution using power series

**3**, 806–813 (1985). [CrossRef]

*α*=

_{P}*α*. To exploit this feature, we seek a solution for Eq. (4) as a power series of the parameter,

_{S}*α*= 1 -

*α*/

_{S}*α*,which represents the relative difference of signal and pump attenuation coefficients.

_{P}*n*(

_{i}*z*) :

*i*= 0,1,⋯,∞ are some auxiliary functions that need to be determined recursively by using lower order solutions. It is interesting note that the parameter

*α*tends to zero as

*α*tends to

_{P}*α*. Thefore, when signal and pump loss coefficients are equal, the above power series reduces to a single term

_{S}*n*

_{0}(

*z*) with known analytical result. Substitution of Eq. (5) to Eq. (4) results in following set of coupled system of differential equations:

*u*

_{0}=

*γn*(0)/

_{P}*α*,

_{P}*u*(

*z*) =

*u*

_{0}exp(-

*α*) and

_{P}z*H*(

_{P}*z*) = exp(-

*α*-

_{P}z*u*(

*z*)) . In Eq. (8), Ei(

*x*) denotes the exponential integral [12]

*n*

_{0}(

*z*) given in Eq. (8) to Eq.(7) when

*i*= 1 gives the following analytic expression for

*n*

_{1}(

*z*) :

*C*= 0.5772156649⋯ is called the Euler’s constant [12]. Substitution of this expression to Eq. (10) results in following approximate expression:

*n*(

_{i}*z*) :

*i*= 2,3,⋯,∞ . Substituting these obtained values to Eq. (5) gives the following expression for signal and noise power evolution along the Raman amplifier:

*H*(

_{P}*z*) = exp(-

*α*-

_{P}z*u*(

*z*)),

*H*(

_{S}*z*) = exp(-

*α*-

_{S}z*u*(

*z*)) and Raman signal gain at distance

*z*from left end of fiber (c.f., Fig. 1) is given by

*G*(

_{S}*z*) = exp(

*u*

_{0})

*H*(

_{S}*z*). The term ℘(

*z*) represent a scattering noise contribution when

*α*=

_{P}*α*and is given by

_{S}*α*→ 0 (i.e.

*α*→

_{P}*α*. Then

_{S}*H*(

_{P}*z*) →

*H*(

_{S}*z*) and we get the following result:

**3**, 806–813 (1985). [CrossRef]

## 4. Exact analytical solution

*n̂*(

*z*) is an unknown function which assumes input photon number (i.e.

*n*(0)) when Raman amplifier is free of noise. Substitution of Eq. (18) to Eq. (4) gives a differential equation for

*n̂*(

*z*):

*u*

_{0}=

*γn*(0)/

_{P}*α*,

_{P}*u*(

*z*) =

*u*

_{0}exp(-

*α*) and

_{P}z*α*= 1-

*α*/

_{S}*α*. Integrating along the fiber and using

_{P}*n̂*(0) =

*n*(0) results in

*I*(

*z*)is defined as

*v*) and integrating term by term (note: this operation is valid given that the resulting series is uniformly convergent for positive real numbers),

*I*(

*z*) can be written in the following form:

*α*)

_{k}=

*α*(

*α*+ 1) ⋯ (

*α*+

*k*- 1), [12]. But noting that the confluent hypergeometric function is defined by [12]

*n*(

*z*) along the fiber

*n*(0) at input with following value:

_{noise}*α*→

_{P}*α*because exact analytical solution for this case has already been reported in literature [3, 5

_{S}5. P. C. Xiao, Q.J. Zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. **15**, 206–208 (2003). [CrossRef]

*α*→

_{P}*α*,α → 0 and hence second bracketed term in Eq.(24) assumes 0/0 indeterminate form. Therefore, limit of Eq. (24) as

_{S}*α*→ 0 needs to be calculated. Applying LHospitals rule [12], Eq. (24) can be written in the limit

*α*→ 0 as:

*α*→ 0 results in

*x*) is defined using Euler’s constant

*C*= 0.5772156649⋯ as [12]

## 5. Noise Figure

*z*), at distance

*z*from input can then be written as [8

**33**, 607–608 (1997). [CrossRef]

_{a}(

*z*) for power series solution Eq. (14)

_{e}(

*z*) for exact analytical solution using Eq. (24)

_{a}(

*z*) and NF

_{e}(

*z*) approach 2.0 (ie. 3dB) for large Raman gain,

*G*(

_{S}*z*) Large Raman gain amounts to large pump powers (ie. large

*u*

_{0}) and long amplifiers (ie.

*z*≈ 1/

*α*). Under these conditions, first term of Eq. (30) approaches zero rapidly and hence, Eq. (30) can be written in following approximate form

_{P}*u*

_{0}can be calculated by using an asymptotic expansion of exponential integral for large

*x*[12].

## 6. Comparison with simulations

*γn*(0) →

_{P}*g*where

_{Rm}P_{P}*g*is the modal Raman gain with and

_{Rm}*P*is the input pump power [8

_{P}**33**, 607–608 (1997). [CrossRef]

*g*= 0.67W

_{Rm}^{-1}km-1 ,

*α*= 0.22dB/km and

_{S}*P*= 400mW.

_{P}*n*(

*z*) of approximate power series solution in Eq. (13) against fiber length for two different pump loss coefficients

*α*= 0.30dB/km and

_{P}*α*= 0.40dB/km. The solid lines in Fig. 2 shows the approximate power series solution while dashed lines represent the numerical solution of Eq. (1) and Eq. (2). Figure 2 clearly shows that the analytical solution is in very good agreement with typical usage limits of Raman amplifiers. It also shows that the accuracy of the analytical solution increases as the Raman gain increases. This is expected because the mismatch of signal and pump loss coefficients influence mainly the noise added by the Raman amplifier

_{P}*I*(

*z*) (c.f. Eq. (21))against fiber length. The solid-lines in Fig. 3 represent the hyper-geometrical functions based solution, while triangles represent the results of direct numerical integration. For comparison,

*I*(

*z*) for

*α*=

_{S}*α*with

_{P}*α*= 0.30dB/km and

_{P}*α*= 0.40dB/km are shown using dashed lines in Fig. 3. These results clearly confirm the validity of the current derivation within the typical usage limits of Raman amplifiers. It also shows that commonly used approximation

_{P}*α*=

_{S}*α*is not accurate enough for characterizing forward-pumped Raman amplifiers.

_{P}*α*= 0.4

_{P}*dB*and (b)

*α*= 0.3

_{P}*dB*/

*km*. The solid-lines in Fig. 4 represent the analytical solution, Eq. (31), while triangles represent the numerical solution obtained by substituting the numerically integrated solution of Eq. (1) and Eq. (2) to Eq. (29). For comparison, widely used analytical expression of noise figure with equal pump and signal attenuation coefficients are shown using dashed lines. These results clearly confirm the validity of the current derivation within the typical usage limits of Raman amplifiers. It also shows that commonly used equal pump and loss coefficient approximation is not accurate enough for characterizing forward-pumped Raman amplifiers. In conclusion, the analytical results presented here can be used to get insight into the noise performance of forward pumped Raman amplifiers and hence to improve their design.

## 7. Applications

*M*different laser beams pumps the Raman fiber. Then, Eq. (1) and Eq. (2) need to be modified to

*γk*:

*k*= 1,⋯,

*M*are the Raman gain coefficients and

*α*:

_{Pk}*k*= 1,⋯,

*M*are the pump loss coefficient for

*M*pump wavelengths with photon numbers,

*n*(

_{Pk}*z*). All the other parameters assume same meaning as in Eq. (1) and Eq. (2). Using an approximation similar to impulsive pump depletion approximation [18

18. A. Bononi, M. Papararo, and A. Vannucci, “Impulsive pump depletion in saturated Raman amplifiers,” Electron. Lett. **37**, 886–887 (2001). [CrossRef]

*n*(

*z*) as

10. Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. **13**, 948–950 (2001). [CrossRef]

## 8. Conclusion

10. Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. **13**, 948–950 (2001). [CrossRef]

16. A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. **20**, 1635–1643 (2002). [CrossRef]

## References and links

1. | G. P. Agrawal, |

2. | M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Select. Topics in Quantum Electron. |

3. | M. N. Islam, |

4. | V. E. Perlin and H. G. Winful, “Optimal design of flat-gain wide-band fiber Raman amplifiers,” IEEE J. Lightwave Technol. |

5. | P. C. Xiao, Q.J. Zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. |

6. | M. N. Islam and R. W. Lucky, |

7. | M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. |

8. | R. Chinn, “Analysis of counter-pumped small-signal fiber Raman amplifiers,” Electron Lett. |

9. | B. Bobbs and C. Warner, “Closed-form solution for parametric second Stokes generation in Raman amplifiers,” IEEE J. Quantum Electron. |

10. | Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. |

11. | D. Zwillinger, |

12. | M. Abramowitz and A. Stegun, |

13. | M. R. Spiegel, |

14. | P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. |

15. | P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, “Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,” IEEE J. Lightwave Technol. |

16. | A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,” IEEE J. Lightwave Technol. |

17. | J. Auyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. |

18. | A. Bononi, M. Papararo, and A. Vannucci, “Impulsive pump depletion in saturated Raman amplifiers,” Electron. Lett. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

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

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 27, 2004

Revised Manuscript: August 25, 2004

Published: September 6, 2004

**Citation**

Malin Premaratne, "Analytical characterization of optical power and noise figure of forward pumped Raman amplifiers," Opt. Express **12**, 4235-4245 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-18-4235

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### References

- G. P. Agrawal, Fiber-Optic Communications Systems, 2nd Edition, (Wiley InterScience, New York, 1997).
- M. N. Islam, �??Raman amplifiers for telecommunications,�?? IEEE J. Select. Topics in Quantum Electron. 8, 548 �?? 559 (2002). [CrossRef]
- M. N. Islam, Raman amplifiers for telecommunications: Physical Principles, (Springer-Verlag, New York, 2003).
- V. E. Perlin and H. G.Winful,�??Optimal design of flat-gain wide-band fiber Raman amplifiers,�?? IEEE J. Lightwave Technol.20, 250 �?? 254 (2002). [CrossRef]
- P. C. Xiao, Q. J. Zeng, J. Huang, and J. M. Liu, �??A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,�?? IEEE Photon. Technol. Lett. 15, 206 �?? 208 (2003). [CrossRef]
- M. N. Islam and R. W. Lucky, Raman amplifiers for telecommunications 2: Sub-systems and Systems, (Springer- Verlag, New York, 2003).
- M. L. Dakss, and P. Melman, �??Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,�?? J. Lightware Technol. 3, 806 �?? 813 (1985). [CrossRef]
- R. Chinn, �??Analysis of counter-pumped small-signal fiber Raman amplifiers,�?? Electron Lett. 33, 607 �?? 608 (1997). [CrossRef]
- B. Bobbs, C.Warner, �??Closed-form solution for parametric second Stokes generation in Raman amplifiers,�?? IEEE J. Quantum Electron. 24, 660 �?? 664 (1988). [CrossRef]
- Y. Yan, J. Chen; W. Jiang, J. Li, J. Chen, X. Li, �??Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,�?? IEEE Photon. Technol. Lett. 13, 948 �?? 950 (2001). [CrossRef]
- D. Zwillinger, Handbook of Differential Equations, (Academic Publishers, Boston, 1989).
- M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).
- M. R. Spiegel, Shaums Outline Series Theory and Problems of Complex Variables with an Introduction to Conformal Mappings and Its Applications, (McGraw-Hill Inc., New York, 1991).
- P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, D. J. DiGiovanni �??Rayleigh scattering limitations in distributed Raman pre-amplifiers,�?? Photon. Technol. Lett. 10, 159 �?? 161 (1998). [CrossRef]
- P. Parolari, L. Marazzi, L. Bernardini, M. Martinelli, �??Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,�??IEEE J. Lightwave Technol. 21, 2224 �?? 2228 (2003). [CrossRef]
- A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, A. F. Evans, �??Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,�?? IEEE J. Lightwave Technol. 20, 1635 �?? 1643 (2002). [CrossRef]
- J. Auyeung and A. Yariv, �??Spontaneous and stimulated Raman scattering in long low loss fibers,�?? IEEE J. Quantum Electron. 14, 347 �?? 352 (1978). [CrossRef]
- A. Bononi, M. Papararo, A. Vannucci, �??Impulsive pump depletion in saturated Raman amplifiers,�?? Electron. Lett. 37, 886 �?? 887 (2001). [CrossRef]

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