## Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers

Optics Express, Vol. 12, Issue 4, pp. 545-550 (2004)

http://dx.doi.org/10.1364/OPEX.12.000545

Acrobat PDF (198 KB)

### Abstract

A mid-point shooting algorithm using the Newton–Raphson method is adopted for solving nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers. A series of novel backward-differentiation methods are constructed for the first time to our knowledge. Their combination can form a powerful solution for fiber amplifiers. Numerical results show that the approach can solve Raman amplifier propagation equations on various conditions including co-, counter-, and bidirectionally pumped cases. The computation speed of the present methods is about four times that of the backward-differentiation methods previously adopted.

© 2004 Optical Society of America

## 1. Introduction

1. M. Tang, P. Shum, and Y. D. Gong, “Design of double-pass discrete Raman amplifier and the impairments induced by Rayleigh backscattering,” Opt. Express **11**, 1887–1893 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-16-1887. [CrossRef] [PubMed]

6. A. A. B. Tio, P. Shum, and Y. D. Gong, “Wide bandwidth flat gain Raman amplifier by using polarization-independent interferometric filter,” Opt. Express **11**, 2991–2996 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-2991. [CrossRef] [PubMed]

3. X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. **230**, 425–431 (2004). [CrossRef]

9. Z. Tong, H. Wei, and S. S. Jian, “Theoretical investigation and optimization of bi-directionally pumped broadband fiber Raman amplifiers,” Opt. Commun. **217**, 401–413 (2003). [CrossRef]

10. X. M. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express **11**, 1452–1461 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452. [CrossRef] [PubMed]

3. X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. **230**, 425–431 (2004). [CrossRef]

11. X. M. Liu and B. Lee, “A fast and 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]

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

## 2. Physical model and mathematical algorithm

### 2.1. Physical model

11. X. M. Liu and B. Lee, “A fast and 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]

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

*v*, respectively;

*α*

_{v},

*η*

_{v},

*h*,

*K*, and

*T*are attenuation coefficient, Rayleigh-backscattering coefficient, Planck’s constant, Boltzmann constant, and temperature, respectively;

*A*

_{eff}is effective area of optical fiber;

*g*

_{µv}is Raman gain parameter at frequency

*v*resulting from the pump at frequency

*µ*; and the factor of Γ accounts for polarization randomization effects, whose value lies between 1 and 2.

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

14. X. M. Liu, “Corrections to: a novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. **15**, 1321–1321 (2003). [CrossRef]

10. X. M. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express **11**, 1452–1461 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452. [CrossRef] [PubMed]

11. X. M. Liu and B. Lee, “A fast and 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]

*z*=

*L*) to the starting point (

*z*=0), and we try to match boundary conditions at the input port of signals. Although this algorithm can solve the boundary value problems of distributed multipump Raman amplifiers on the general conditions, it is rather sensitive to starting conditions and even possibly fails for bidirectional multipump Raman amplifiers with higher power and longer span.

15. http://www.nr.com.

*z*

_{1}(i.e.,

*z*=0) and

*z*

_{2}(i.e.,

*z*=

*L*), the solutions are supposed to satisfy

*N*coupled differential equations of Eq. (1), satisfying

*n*

_{1}boundary conditions at the starting point

*z*

_{1}, and a remaining set of

*n*

_{2}=

*N*-

*n*

_{1}boundary conditions at the end point

*z*

_{2}.

### 2.2. Shooting algorithm

*z*

_{f}(e.g.,

*z*

_{f}=

*L*/2). At the same time, the multidimensional Newton–Raphson method is employed in the implementation of the new algorithm. The detailed procedure is as follows:

*n*

_{2}-vector

**V**

_{(1)}of starting parameters at

*z*

_{1}, a particular

*P*

_{i}(

*z*

_{1}) is generated from Eq. (2.b), i.e.,

*z*

_{1},

*z*

_{f}], and an

*N*-vector

**F**1 is constructed from the solution

*=(*

**P***P*

_{1},

*P*

_{2},…,

*P*

_{N})

^{T}of Eq. (2.a), i.e.,

**F**1=

**F**[

*(*

**P***z*

_{f},

**V**

_{(1)})].

*n*

_{1}-vector

**V**

_{(2)}of final parameters at

*z*

_{2}, a particular

*P*

_{i}(

*z*

_{2}) is produced from Eq. (2.c), i.e.,

*z*

_{2},

*z*

_{f}], and an

*N*-vector

**F2**is formed from the solution

*=(*

**P***P*

_{1},

*P*

_{2},…,

*P*

_{N})

^{T}of Eq. (2.a), i.e.,

**F2**=

**F**[

*(*

**P***z*

_{f},

**V**

_{(2)})].

**V**=(

**V**

_{(1)},

**V**

_{(2)}) that zeros the vector value of

**W**

_{(V)}=

**F1**-

**F2**, i.e.,

**W**

_{(V)}=

**F**[

*(*

**P***z*

_{f},

**V**

_{(1)})]-

**F**[

*(*

**P***z*

_{f},

**V**

_{(2)})]=0.

**V**=-[

**J**]

^{-1}

**W**, where

**V**

^{(1)}+δ

**V**

_{(1)}

**V**

_{(2)}+δ

**V**

_{(2)}.

**F1**-

**F2**)/(

**F1**+

**F2**)‖>

*ε*(

*ε*is the specified relative error), go to Step 1. Otherwise, output results.

### 2.3. Novel backward-differentiation methods

10. X. M. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express **11**, 1452–1461 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452. [CrossRef] [PubMed]

**11**, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

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

*and*

**P**t

**P**_{t-j}are the value of

*at the*

**P***t*th step and (

*t*-

*j*)th step, and the step size Δ

*z*=

*z*

_{t+1}-

*z*

_{t}. Here we use bold scripts only to shorten notation for a set of equations, i.e.,

*=(*

**P***P*

_{1},

*P*

_{2},…,

*P*

_{N}) and

*=(*

**f***f*

_{1},

*f*

_{2},…,

*f*

_{N}). Hence, hereafter the division or multiplication of bold scripts does not have any meaning of vector calculus. After the manipulation, Eq. (3.a) is simplified as

*k*=6 are demonstrated in Table 1. For an implicit method such as BDFs, a nonlinear system of equations must be solved at each step [15

15. http://www.nr.com.

*n*is the current iteration, and

*n*=0, 1, 2, ….

## 3. Simulation results

*L*=100 km,

*α*=0.2, and 0.35 dB/km for signals and pumps, respectively, and ignored ASE, Rayleigh scattering, and other noises. There are 19 signal channels spaced 200 GHz from 188.85 to 192.45 THz. The signal power of each channel is 1 mW. Four backward-propagating and three forward-propagating pumps are used; their powers are all 300 mW, and their wavelengths are 1425, 1445, 1465, 1485, 1435, 1455, and 1475 nm. The estimated values of each signal and each copropagating pump at

*z*=

*L*are 1 mW and 0.1 mW, and each counterpropagating pump at

*z*=0 is estimated as 0.1 mW, which is shown in Fig. 1(a).

^{-6}at the fourth iteration. Simulated results demonstrate that various cases of co-, counter-, and bidirectionally pumped Raman amplifiers can be solved by the proposed methods under normal conditions. However, some methods in the previous reports [10

**11**, 1452–1461 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452. [CrossRef] [PubMed]

**11**, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

*z*

_{f}affects the efficiency and stability of the proposed shooting algorithm. Usually, it can be specified as

*z*

_{f}=

*L*/2. Although the traditional BDFs can also obtain Fig. 1 on the basis of our shooting algorithm, its CPU time is increased about four times in comparison with the novel BDFs under the same conditions.

## 4. Discussions

17. Y. Zhu et al., “Experimental comparison of all-Raman and Raman/EDFA hybrid amplifications using 40 Gbit/s-based transmissions over 400 km TW-RS fibre,” Electron. Lett. **38**, 893–895 (2002). [CrossRef]

18. J. Ko, S. Kim, J. Lee, S. Won, Y. S. Kim, and J. Jeong, “Estimation of performance degradation of bidirectional WDM transmission systems due to Rayleigh backscattering and ASE noises using numerical and analytical models,” J. Lightwave Technol. **21**, 938–946 (2003). [CrossRef]

**11**, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

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

14. X. M. Liu, “Corrections to: a novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. **15**, 1321–1321 (2003). [CrossRef]

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

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

**11**, 1452–1461 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452. [CrossRef] [PubMed]

**11**, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

## 5. Conclusions

## References and links

1. | M. Tang, P. Shum, and Y. D. Gong, “Design of double-pass discrete Raman amplifier and the impairments induced by Rayleigh backscattering,” Opt. Express |

2. | H. H. Lee et al., “A gain-clamped-semiconductor-optical-amplifier combined with a distributed Raman-fiber-amplifier: a good candidate as an inline amplifier for WDM networks,” Opt. Commun. |

3. | X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. |

4. | H. Suzuki, N. Takachio, H. Masuda, and K. Iwatsuki, “Super-dense WDM transmission technology in the zero-dispersion region employing distributed Raman amplification,” J. Lightwave Technol. |

5. | A. Pizzinat, M. Santagiustina, and C. Schivo, “Impact of hybrid EDFA-distributed Raman amplification on a 4×40-Gb/s WDM optical communication system,” IEEE Photon. Technol. Lett. |

6. | A. A. B. Tio, P. Shum, and Y. D. Gong, “Wide bandwidth flat gain Raman amplifier by using polarization-independent interferometric filter,” Opt. Express |

7. | C. R. S. Fludger and V. Handerek, “Fundamental noise limits in broadband Raman amplifiers,” in Optical Fiber Communication Conference (OFC 2000), Vol. 37 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D. C., 2000), paper MA-5. |

8. | Y. Chen et al., “Bi-directionally pumped broadband Raman Amplifier,” ECOC’2001, Tu.L.3.4. |

9. | Z. Tong, H. Wei, and S. S. Jian, “Theoretical investigation and optimization of bi-directionally pumped broadband fiber Raman amplifiers,” Opt. Commun. |

10. | X. M. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express |

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

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

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

14. | X. M. Liu, “Corrections to: a novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. |

15. | |

16. | H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Blaisdell, Waltham, Mass., 1968), pp. 192. |

17. | Y. Zhu et al., “Experimental comparison of all-Raman and Raman/EDFA hybrid amplifications using 40 Gbit/s-based transmissions over 400 km TW-RS fibre,” Electron. Lett. |

18. | J. Ko, S. Kim, J. Lee, S. Won, Y. S. Kim, and J. Jeong, “Estimation of performance degradation of bidirectional WDM transmission systems due to Rayleigh backscattering and ASE noises using numerical and analytical models,” J. Lightwave Technol. |

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

**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: December 22, 2003

Revised Manuscript: February 1, 2004

Published: February 23, 2004

**Citation**

Xueming Liu, "Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers," Opt. Express **12**, 545-550 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-4-545

Sort: Journal | Reset

### References

- M. Tang, P. Shum, and Y. D. Gong, �??Design of double-pass discrete Raman amplifier and the impairments induced by Rayleigh backscattering,�?? Opt. Express 11, 1887�??1893 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-16-1887">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-16-1887</a> [CrossRef] [PubMed]
- H. H. Lee et al., �??A gain-clamped-semiconductor-optical-amplifier combined with a distributed Raman-fiber-amplifier: a good candidate as an inline amplifier for WDM networks,�?? Opt. Commun. 229, 249�??252 (2004). [CrossRef]
- X. M. Liu and Y. H. Li, �??Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,�?? Opt. Commun. 230, 425�??431 (2004) [CrossRef]
- H. Suzuki, N. Takachio, H. Masuda, and K. Iwatsuki, �??Super-dense WDM transmission technology in the zero-dispersion region employing distributed Raman amplification,�?? J. Lightwave Technol. 21, 973�??981 (2003). [CrossRef]
- A. Pizzinat, M. Santagiustina, and C. Schivo, �??Impact of hybrid EDFA-distributed Raman amplification on a 4 x 40-Gb/s WDM optical communication system,�?? IEEE Photon. Technol. Lett. 15, 341�??343 (2003). [CrossRef]
- A. A. B. Tio, P. Shum, and Y. D. Gong, �??Wide bandwidth flat gain Raman amplifier by using polarization-independent interferometric filter,�?? Opt. Express 11, 2991�??2996 (2003),<a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-2991"> http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-2991</a> [CrossRef] [PubMed]
- C. R. S. Fludger and V. Handerek, �??Fundamental noise limits in broadband Raman amplifiers,�?? in Optical Fiber Communication Conference (OFC 2000), Vol. 37 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D. C., 2000), paper MA-5
- Y. Chen et al., �??Bi-directionally pumped broadband Raman Amplifier,�?? ECOC�??2001, Tu.L.3.4.
- Z. Tong, H. Wei, and S. S. Jian, �??Theoretical investigation and optimization of bi-directionally pumped broadband fiber Raman amplifiers,�?? Opt. Commun. 217, 401�??413 (2003). [CrossRef]
- X. M. Liu and B. Lee, �??Effective shooting algorithm and its application to fiber amplifiers,�?? Opt. Express 11, 1452�??1461 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452</a> [CrossRef] [PubMed]
- X. M. Liu and B. Lee, �??A fast and 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 wavelength-division-multiplexed high power laser diodes,�?? IEEE J. Sel. Top. Quantum Electron. 7, 3�??16 (2001). [CrossRef]
- X. M. Liu, H. Y. Zhang, and Y. L. Guo, �??A novel method for Raman amplifier propagation equations,�?? IEEE Photon. Technol. Lett. 15, 392�??394 (2003). [CrossRef]
- X. M. Liu, �??Corrections to: a novel method for Raman amplifier propagation equations,�?? IEEE Photon. Technol. Lett. 15, 1321�??1321 (2003). [CrossRef]
- <a href= "http://www.nr.com.">http://www.nr.com</a>
- H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Blaisdell, Waltham, Mass., 1968) pp. 192.
- Y. Zhu et al., �??Experimental comparison of all-Raman and Raman/EDFA hybrid amplifications using 40 Gbit/s-based transmissions over 400 km TW-RS fibre,�?? Electron. Lett. 38, 893�??895 (2002). [CrossRef]
- J. Ko, S. Kim, J. Lee, S. Won, Y. S. Kim, and J. Jeong, �??Estimation of performance degradation of bidirectional WDM transmission systems due to Rayleigh backscattering and ASE noises using numerical and analytical models,�?? J. Lightwave Technol. 21, 938�??946 (2003). [CrossRef]
- P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, �??Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,�?? J. Lightwave Technol. 21, 2224�??2228 (2003). [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.

### Figures

Fig. 1. |

« Previous Article | Next Article »

OSA is a member of CrossRef.