## Stiffness analysis in the numerical solution of Raman amplifier propagation equations

Optics Express, Vol. 12, Issue 8, pp. 1656-1664 (2004)

http://dx.doi.org/10.1364/OPEX.12.001656

Acrobat PDF (326 KB)

### Abstract

For the first time, the stiffness of Raman amplifier propagation equations is analyzed. And based on this analysis, a novel method for propagation equations is proposed to enhance the stability of numerical simulation. To verify the reliability of this method, simulation experiments are employed by using our method and the existent predictor-corrector method with comparison. The results show that our backward differentiation formulae method behaves much better in stability with a comparative accuracy.

© 2004 Optical Society of America

## 1. Introduction

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

2. H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. **11**, 530–532 (1999). [CrossRef]

2. H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. **11**, 530–532 (1999). [CrossRef]

3. B. Min, W. J. Lee, and N. Park, “Efficient formulation of Raman amplifier propagation equations with average power analysis,” IEEE Photon. Technol. Lett. **12**, 1486–1488 (2000). [CrossRef]

4. S. Wang and C. Fan, “Generalised attenuation coefficients and a novel simulation model for Raman fibre amplifiers,” IEE Proc.-Optoelectron. **148**, 156–159 (2001). [CrossRef]

5. X. Liu, H. Zhang, and Y. Guo, “A novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. **15**, 392–394 (2003). [CrossRef]

6. X. Liu and B. Lee, “A fast stable method for Raman amplifier propagation equations,” Opt. Express **11**, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

6. X. Liu and B. Lee, “A fast stable method for Raman amplifier propagation equations,” Opt. Express **11**, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

## 2. Equation performance analysis

2. H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. **11**, 530–532 (1999). [CrossRef]

7. S. Namiki and Y. Emori, “Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes,” IEEE J. Select. Topics Quantum Electron. **7**, 3–16 (2001). [CrossRef]

*ζ*and

*ν*are optical frequencies; + and - denote the forward- and backward-propagating optical waves, respectively;

*α*

_{ν},

*T*,

*h*,

*k*and

*ε*

_{ν}are fiber loss, temperature, Planck’s constant, Boltzmann constant and Rayleigh-backscattering coefficient, respectively;

*g*

_{ζν}is Raman gain coefficient at frequency

*ν*due to pump at frequency

*ζ*(

*g*

_{ζν}defined here has been divided by the fiber effective area

*A*

_{eff}); Γ

*ζ*

_{ν}is the polarization factor between frequencies

*ζ*and

*ν*. Δ

*ν*and Δ

*ζ*are noise spectral bandwidths for simulation.

*n*forward propagating waves and

*m*backward propagating waves, can be modeled as

*n*denote the forward-propagating waves, while the ones from

*n*+1 to

*n*+

*m*denote the backward-propagating waves. In addition, to simplify the analysis, we ignore the Rayleigh-backscattering and thermal noise terms in Eq. (1). Hence, here

*F⃗*(

*z,P⃗*) at certain points (

*z*

_{c},

*P⃗*(

*z*

_{c})), the stiffness of Raman-amplified transmission system can be obtained. A detailed introduction of the corresponding knowledge about stiffness theory is contained in the appendix.

*P*

_{j}to be

*ε*

_{j}. In order to reduce the error, a valuable transformation, which has been used in the previous paper [5

5. X. Liu, H. Zhang, and Y. Guo, “A novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. **15**, 392–394 (2003). [CrossRef]

*ε*

_{j}becomes the rounding error of ln(

*P*

_{j}), and it is easy to conclude that the corresponding error of

*P*

_{j}is approximately

*P*

_{j}

*ε*

_{j}, which is remarkably less than

*ε*

_{j}because the power

*P*

_{j}is often in the range of [10

^{-4},5×10

^{-1}] (

*P*

_{j}is with the scale of international standard unit:

*W*).

*P⃗*‖ is calculated with the variation of step-size.

## 3. Numerical model

4. S. Wang and C. Fan, “Generalised attenuation coefficients and a novel simulation model for Raman fibre amplifiers,” IEE Proc.-Optoelectron. **148**, 156–159 (2001). [CrossRef]

*N*elemental amplifier sections for iteration [3

3. B. Min, W. J. Lee, and N. Park, “Efficient formulation of Raman amplifier propagation equations with average power analysis,” IEEE Photon. Technol. Lett. **12**, 1486–1488 (2000). [CrossRef]

*k*denotes the corresponding discrete point; subscripts

*p*and

*s*denote pump and signal, respectively;

*i*and

*j*denote the

*i*th pump and jth signal, respectively; Δ

*z*is the step-size between two adjacent discrete points;

*N*

_{nj}=

*hν*

_{sj}Δ

*ν*is the noise power within bandwidth Δ

*ν*at frequency

*ν*

_{sj}. It is necessary to point out that Rayleigh-scattering and the temperature dependence are not considered here. In addition, the polarization factor Γ is assumed to be 1.

*F**

_{pi}(

*z*

_{k±1},

*P⃗*) and

*F**

_{sj}(

*z*

_{k+1},

*P⃗*). These need a set of approximations of

*z*

_{k±1}) and

*z*

_{k+1}), which can be calculated by one-step method [4

4. S. Wang and C. Fan, “Generalised attenuation coefficients and a novel simulation model for Raman fibre amplifiers,” IEE Proc.-Optoelectron. **148**, 156–159 (2001). [CrossRef]

5. X. Liu, H. Zhang, and Y. Guo, “A novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. **15**, 392–394 (2003). [CrossRef]

*z*

_{2}),

*z*

_{3}),

*P*

^{-}

_{pi}(

*z*

_{N-1}),

*P*

^{-}

_{pi}(

*z*

_{N-2}),

*P*

^{-}

_{pi}(

*z*

_{N-3}),

*z*

_{1}),

*z*

_{2}) and

*z*

_{3}), which act as starting values, can be obtained by the semi-analytical method [4

**148**, 156–159 (2001). [CrossRef]

## 4. Simulation experiments and discussion on results

6. X. Liu and B. Lee, “A fast stable method for Raman amplifier propagation equations,” Opt. Express **11**, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

^{-3}

*W*, which is negligible, and the largest difference of gain is about 0.28dB, which is difficult to measure experimentally.

## 5. Conclusion

## Appendix

*y⃗*(

*t*)=(

*y*

_{1}(

*t*),

*y*

_{2}(

*t*), …,

*y*

_{n}(

*t*))

^{T}is the solution vector,

*ϕ⃗*(

*t*)=(

*ϕ*

_{1}(

*t*),

*ϕ*

_{2}(

*t*),…,

*ϕ*

_{n}(

*t*))

^{T}is a known portion,

*t*is the independent variable and

*A*∈

*R*

^{n×n}is the coefficient matrix. Now, assume that

*A*has

*n*distinct eigenvalues

*λ*

_{j}

*,j*=1, 2, …,

*n*. Then

*C*

_{1},…,

*C*

_{n}are constants, {

*v⃗*

_{j}} is a basis formed by the eigenvectors of

*A*. As an initial value problem, we assume that Re(

*λ*

_{j})<0 for all

*j*to make sure that the system is a stable one, which means when

*t*→∞, the solution tends to the particular solution

*φ⃗*(

*t*).

*φ*(

*t*)

*A*. On the other hand, the greater this module is, the shorter the corresponding time interval is needed to be. We are thus faced with a sort of paradox: the scheme is forced to employ a small integration step-size to track a component of the solution that is virtually flat for large values of

*t*. Generally, stiffness is presented to illustrate the difference of decreasing speed corresponding to each solution portion

*s*is called the stiffness ratio, and the system is regarded as a stiff one if

*s*reaches the magnitude of 10 or above.

*J*(

*t*)

_{c}is the Jacobi matrix of

*f*(

*t*,

*y⃗*(

*t*)) at a certain point (

*t*

_{c},

*y⃗*(

*t*

_{c})),

*λ*

_{j}=

*λ*

_{j}(

*t*

_{c}),

*j*=1, 2,…,

*n*is the eigenvalues of (

*J*(

*t*

_{c}). If a nonlinear system satisfies the condition of (6), then it is regarded as a stiff one and

*t*

_{c},

*y⃗*(

*t*

_{c})).

*k*denotes the discrete point;

*h*is the step-size;

*a*

_{i}and

*b*

_{-1}are constants, whose values are shown in the following table corresponding to the order of

*p*. The numerical scheme derived in this paper is a three-order (

*p*=3) BDF method and the corresponding coefficients come from the following table.

## References and Links

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

2. | H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. |

3. | B. Min, W. J. Lee, and N. Park, “Efficient formulation of Raman amplifier propagation equations with average power analysis,” IEEE Photon. Technol. Lett. |

4. | S. Wang and C. Fan, “Generalised attenuation coefficients and a novel simulation model for Raman fibre amplifiers,” IEE Proc.-Optoelectron. |

5. | X. Liu, H. Zhang, and Y. Guo, “A novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. |

6. | X. Liu and B. Lee, “A fast stable method for Raman amplifier propagation equations,” Opt. Express |

7. | S. Namiki and Y. Emori, “Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes,” IEEE J. Select. Topics Quantum Electron. |

8. | S. Hu, H. Zhang, and Y. Guo, “Simulation model for high-speed wide-bandwidth Raman-amplified WDM system,” in Proc. APOC2003, Wuhan, China, Paper No. 5281–06. |

9. | A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer-Verlag, New York, 2000). |

10. | http://www.mathworks.com/company/newsletter/clevescorner/may03_cleve.shtml. |

11. | J. D. Lambert, Computational methods in ordinary differential equations (John Wiley & Sons Ltd., London, 1973). |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(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: February 23, 2004

Revised Manuscript: March 27, 2004

Published: April 19, 2004

**Citation**

Song Hu, Hanyi Zhang, and Yili Guo, "Stiffness analysis in the numerical solution of Raman amplifier propagation equations," Opt. Express **12**, 1656-1664 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1656

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### References

- J. Auyeung and A. Yariv, �??Spontaneous and stimulated Raman scattering in long low loss fibers,�?? IEEE J. Quantum Electron. 14, 347-352 (1978). [CrossRef]
- H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, �??Pump interactions in a 100-nm bandwidth Raman amplifier,�?? IEEE Photon. Technol. Lett. 11, 530-532 (1999). [CrossRef]
- B. Min, W. J. Lee, and N. Park, �??Efficient formulation of Raman amplifier propagation equations with average power analysis,�?? IEEE Photon. Technol. Lett. 12, 1486-1488 (2000). [CrossRef]
- S. Wang, and C. Fan, �??Generalised attenuation coefficients and a novel simulation model for Raman fibre amplifiers,�?? IEE Proc.-Optoelectron. 148, 156-159 (2001). [CrossRef]
- X. Liu, H. Zhang, and Y. Guo, �??A novel method for Raman amplifier propagation equations,�?? IEEE Photon. Technol. Lett. 15, 392-394 (2003). [CrossRef]
- X. Liu, and B. Lee, �??A fast stable method for Raman amplifier propagation equations,�?? Opt. Express 11, 2163-2176 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163</a> [CrossRef] [PubMed]
- S. Namiki, and Y. Emori, �??Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelengthdivision-multiplexed high-power laser diodes,�?? IEEE J. Select. Topics Quantum Electron. 7, 3-16 (2001). [CrossRef]
- S. Hu, H. Zhang, and Y. Guo, �??Simulation model for high-speed wide-bandwidth Raman-amplified WDM system,�?? in Proc. APOC2003, Wuhan, China, Paper No. 5281-06.
- A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer-Verlag, New York, 2000).
- <a href="http://www.mathworks.com/company/newsletter/clevescorner/may03_cleve.shtml">http://www.mathworks.com/company/newsletter/clevescorner/may03_cleve.shtml</a>
- J. D. Lambert, Computational methods in ordinary differential equations (John Wiley & Sons Ltd., London,1973).

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