## Effective shooting algorithm and its application to fiber amplifiers

Optics Express, Vol. 11, Issue 12, pp. 1452-1461 (2003)

http://dx.doi.org/10.1364/OE.11.001452

Acrobat PDF (208 KB)

### Abstract

A series of new methods based on the Runge-Kutta (RK) formula are proposed, which not only retain the merit of RK methods, in that the adaptive stepsize is easily implemented, but also dramatically decrease the error under the same conditions. Based on the new methods, an effective shooting algorithm is also proposed. A two-point boundary value problem for the Raman amplifier propagation equations is solved using the proposed algorithm. Our algorithm markedly increases the simulating speed for Raman amplifier propagation equations, as well as improves the accuracy, compared to the traditional algorithm.

© 2003 Optical Society of America

## 1. Introduction

1. T. Mizuochi, K. Kinjo, S. Kajiya, T. Tokura, and K. Motoshima, “Bidirectional unrepeatered 43 Gb/s WDM transmission with C/L band-separated Raman amplification,” J. Lightwave Technol. **20**, 2079–2085 (2002). [CrossRef]

8. A. Carena, V. Curri, and P. Poggiolini, “On the optimization of hybrid Raman/erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. **13**, 1170–1172 (2001). [CrossRef]

6. Q. H. Mao, J. S. Wang, X. H. Sun, and M. D. Zhang, “A theoretical analysis of amplification characteristics of bi-directional erbium-doped fiber amplifiers with single erbium-doped fiber,” Opt. Commun. **159**, 149–157 (1999). [CrossRef]

9. U. M. Ascher and L. R. Petzold, *Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations* (SIAM, Philadelphia, 1998). [CrossRef]

## 2. New methods

**represents a multi-dimensional vector containing elements**

*y***={**

*y**y*

_{1}(

*t*),

*y*

_{2}(

*t*),

*y*

_{3}(

*t*), …} and the initial conditions are given by

*c*,

_{i}*w*,

_{i}*d*} denote all parameters. The development of a specific case entails determining the best possible values for these constants by matching the expansion of this formula with a Taylor series expansion.

_{ij}**to ln(**

*z***) and**

*y***(**

*g**t,*) to

**y****(**

*f**t,*)/

**y****(Here we use bold scripts only for a shortened notation for a set of equations. Hence, hereafter the division or multiplication of bold scripts do not have any meaning of vector calculus.), Eq. (1) is changed to**

*y***=exp(**

*y***) into Eq. (5), one can obtain, respectively,**

*z***=exp(**

*y***) is advantageous in amplifier simulations because light amplification or attenuation resembles the exponential pattern with propagation distance. Thus it is intuitively obvious that such a change of variable may lead to faster convergence than some linear or polynomial approximation. Although such a technique was proposed recently [7**

*z*7. 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]

9. U. M. Ascher and L. R. Petzold, *Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations* (SIAM, Philadelphia, 1998). [CrossRef]

*C*and

*C*’ are constants. To compare error of fourth-order RK method with that of fourth-order ORK method, we specify one ordinary differential equation, i.e.,

*t*>~2, and the fourth-order RK method is then divergent for calculating Eq. (10). On the contrary, Fig. 1(b) shows that the error in calculating Eq. (10) based on the fourth-order ORK method is zero if the round-off error is ignored, and the result is independent of stepsize

*h*. All of the results can be explained from Eqs. (8) and (11), i.e., the local truncation error

*ε*=0 for each step.

## 3. Boundary value problems and the shooting algorithm

*t*∊[

*a, b*] with the general boundary conditions

*y*

_{2}(

*a*)=

*A*

_{1}, and the calculating process is from

*t*=

*a*to

*t*=

*b*. The other is assumed to be

*y*

_{1}(

*b*)=

*B*, and the calculating process is from

*t*=

*b*to

*t*=

*a*. However, the region of absolute stability for the fourth-order ORK/RK method is (-2.785, 0). For some specific cases of Eq. (12), therefore, the first type of initial guess is convergent, but the second is divergent,

*vice versa*. In the following, we provide a specific case of Eq. (12). That is,

*f*

_{1}=

*y*

_{2}and

*f*

_{2}=

*y*

_{1},

*a*=0.5 and

*b*=2, and

*y*

_{1}(

*a*)=exp(0.5)+exp(-0.5) and

*y*

_{2}(

*b*)=exp(2)-exp(-2). Their exact solutions are

*y*

_{1}(

*t*)=exp(

*t*)+exp(

*-t*) and

*y*

_{2}(

*t*)=exp(

*t*)-exp(

*-t*). We employ the first type of initial guess to numerically calculate this specific case.

*A*

_{1}, which satisfies the original problem. The basic algorithm is as follows:

- Solve the differential equation using Eq. (7) with the initial conditions
*y*_{1}(0.5)=exp(0.5)+exp(-0.5) and*y*_{2}(0.5)=*A*_{1}. - Evaluate the solution
*y*_{2}(*t*) at*t*=2, compare this value with the target value of*y*_{2}(2)=exp(2)-exp(-2), and obtain the relative error. - Adjust the value of
*A*_{1}by an interpolating method until a desired level of tolerance and accuracy is achieved. - Once the specified accuracy has been achieved, the numerical solution is complete.

*A*

_{j+1}of the initial guess of the (

*j*+1)-th iteration, in step (3), the interpolating method employs the following expression

*A*and

_{j}*β*are the initial guess and the solution

_{j}*y*

_{2}(

*b*) in the

*j*-th iteration,

*A*

_{j-1}and

*β*

_{j-1}are the corresponding values in the (

*j*-1)-th iteration.

*y*

_{1}(

*t*), (b) for

*y*

_{2}(

*t*), and (c) for the relative error of

*y*

_{1}(

*t*) and

*y*

_{2}(

*t*) in the last iteration. In the legends of Fig. 2, the 1st-, 2nd-, and 3rd-iteration represent the first-, second-, and third-iteration in the shooting algorithm, respectively, and

*exact*indicates the exact solution of

*y*

_{1}(

*t*) and

*y*

_{2}(

*t*).

^{-5}. The numerical results also show that it is

*divergent*if the fourth-order RK method is adopted.

*t*=

*a*to

*t*=

*b*, however, some cases are divergent. But, it is convergent when calculating from

*t*=

*b*to

*t*=

*a*. The next section will describe this in detail.

## 4. Applications for Raman amplifier propagation equations

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

*P*,

_{i}*v*and

_{i}*α*are the power, frequency and attenuation coefficient for the

_{i}*i*th wave, respectively;

*A*is the effective area of optical fiber; the factor of Γ accounts for randomization polarization effects, the value of which lies between 1 and 2; and

_{eff}*g*(

_{R}*v*) is the Raman gain coefficient from wave

_{j}-v_{i}*j*to

*i*. The frequency ratio

*v*/

_{i}*v*describes vibrational losses. The minus and plus signs on the left hand side describe the backward-propagating pump waves and forward-propagating signal waves, respectively. The frequencies

_{j}*v*are numerated in the decreasing order of frequency (

_{i}*i*=1, 2, …,

*m*). The terms from

*j*=1 to

*j*=

*i*-1 and from

*j*=

*i*+1 to

*j*=

*m*cause amplification and attenuation of the channel at frequency

*v*.

_{i}*z*=

*L*instead of

*z*=0, the stability of the shooting algorithm is remarkably improved.

*A*=80×10

_{eff}^{-12}, Γ=2, the length of fiber

*L*=80 km,

*α*=0.2 and 0.35 dB/km for signals and pumps, respectively. There are 19 signal channels spaced at 200 GHz/channel, from 188.85 to 192.45 THz. The power of each channel is 1 mW, and powers of the two backward-propagating pumps are 500 mW, corresponding to wavelengths of 1455 and 1475 nm. The initial guesses of all signals at

*z*=

*L*are assumed to be 1 mW [see Fig. 4(b)].

*g*is illustrated in Fig. 3 [2

_{R}2. E. M. Dianov, “Advances in Raman fibers,” J. Lightwave Technol. **20**, 1457–1462 (2002). [CrossRef]

*z*=

*L*, where (a) and (b) for the procedure of the pump and signal, respectively, and (c) for all of the pumps and signals in the fourth iteration. In the legends, the 1st-, 2nd-, 3rd-, and 4th-iteration represent the first-, second-, third-, and fourth-iteration in shooting algorithm, respectively. From Fig. 4(a) and (b), it can be seen that only four iterations of the shooting algorithm can reach the constraint conditions with a relative error of <10

^{-3}. Figure 4(c) shows the strong stimulated Raman scattering interactions of pump-to-pump, signal-to-signal and pump-to-signal. The numerical results also show that, when calculating from

*z*=0 to

*z*=

*L*, it is divergent when each pump power exceeds 400 mW under our simulation parameters. To the contrary, it is convergent if starting from

*z*=

*L*. However, when each pump power exceeds 1 W, our proposed shooting algorithm is also divergent. The reason for this is that it is beyond the absolute stability region of the fourth-order ORK method, i.e., (-2.785, 0). Although this situation for each pump power >1 W in a Raman amplifier is infrequent, the divergent problem can still be solved by the multi-step method [19].

## 5. Conclusions

## Appendix

## Acknowledgment

## References and links

1. | T. Mizuochi, K. Kinjo, S. Kajiya, T. Tokura, and K. Motoshima, “Bidirectional unrepeatered 43 Gb/s WDM transmission with C/L band-separated Raman amplification,” J. Lightwave Technol. |

2. | E. M. Dianov, “Advances in Raman fibers,” J. Lightwave Technol. |

3. | M. Karásek and M. Menif, “Channel addition/removal response in Raman fiber amplifiers: modeling and experimentation,” J. Lightwave Technol. |

4. | N. Kikuchi, K. K. Wong, K. Uesaka, K. Shimizu, S. Yam, E. S. Hu, M. Marhic, and L. G. Kazovsky, “Novel in-service wavelength-band upgrade scheme for fiber Raman amplifier,” IEEE Photon. Technol. Lett. |

5. | X. M. Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” (to be submitted) |

6. | Q. H. Mao, J. S. Wang, X. H. Sun, and M. D. Zhang, “A theoretical analysis of amplification characteristics of bi-directional erbium-doped fiber amplifiers with single erbium-doped fiber,” Opt. Commun. |

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

8. | A. Carena, V. Curri, and P. Poggiolini, “On the optimization of hybrid Raman/erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. |

9. | U. M. Ascher and L. R. Petzold, |

10. | A. Quarteroni, R. Sacco, and F. Saleri, |

11. | B. F. Plybon, |

12. | J. H. Mathews, |

13. | U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, |

14. | P. B. Bailey, L. F. Shampine, and P. E. Waltman, |

15. | L. F. Shampine, |

16. | W. H. Enright and P. H. Muir, “Runge-Kutta software with defect control for boundary value ODEs,” SIAM J. Sci. Comput. |

17. | R. Weiss, “The application of implicit Runge-Kutta and collocation methods to boundary value problems,” Math. Comp. |

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

19. | X. M. Liu and B. Lee, “A series of fast and accurate algorithms and their applications in fiber transmission systems,” (submitted to IEEE J. Quantum Electron.) |

**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: May 5, 2003

Revised Manuscript: May 29, 2003

Published: June 16, 2003

**Citation**

Xueming Liu and Byoungho Lee, "Effective shooting algorithm and its application to fiber amplifiers," Opt. Express **11**, 1452-1461 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-12-1452

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### References

- T. Mizuochi, K. Kinjo, S. Kajiya, T. Tokura, and K. Motoshima, ???Bidirectional unrepeatered 43 Gb/s WDM transmission with C/L band-separated Raman amplification,??? J. Lightwave Technol. 20, 2079???2085 (2002). [CrossRef]
- E. M. Dianov, ???Advances in Raman fibers,??? J. Lightwave Technol. 20, 1457???1462 (2002). [CrossRef]
- M. Karásek and M. Menif, ???Channel addition/removal response in Raman fiber amplifiers: modeling and experimentation,??? J. Lightwave Technol. 20, 1680???1687 (2002). [CrossRef]
- N. Kikuchi, K. K. Wong, K. Uesaka, K. Shimizu, S. Yam, E. S. Hu, M. Marhic, and L. G. Kazovsky, ???Novel in-service wavelength-band upgrade scheme for fiber Raman amplifier,??? IEEE Photon. Technol. Lett. 15, 27???29 (2003). [CrossRef]
- X. M. Liu and B. Lee, ???A fast and stable method for Raman amplifier propagation equations,??? (to be submitted)
- Q. H. Mao, J. S. Wang, X. H. Sun, and M. D. Zhang, ???A theoretical analysis of amplification characteristics of bi-directional erbium-doped fiber amplifiers with single erbium-doped fiber,??? Opt. Commun. 159, 149-157 (1999). [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]
- A. Carena, V. Curri, and P. Poggiolini, ???On the optimization of hybrid Raman/erbium-doped fiber amplifiers,??? IEEE Photon. Technol. Lett. 13, 1170-1172 (2001). [CrossRef]
- U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations (SIAM, Philadelphia, 1998). [CrossRef]
- A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer-Verlag, New York, 2000).
- B. F. Plybon, An Introduction to Applied Numerical Analysis (PWS-KENT Publishing Company, Boston, 1992), pp. 428-441
- J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering (Second edition, Prentice Hall, New Jersey, 1992), pp. 464-475.
- U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (SIAM, Philadelphia, 1995). [CrossRef]
- P. B. Bailey, L. F. Shampine, and P. E. Waltman, Nonlinear Two Point Boundary Value Problems (Academic Press, New York, 1968).
- L. F. Shampine, Numerical Solution of Ordinary Differential Equations (Chapman & Hall, New York, 1994).
- W. H. Enright and P. H. Muir, ???Runge-Kutta software with defect control for boundary value ODEs,??? SIAM J. Sci. Comput. 17, 479???497 (1996). [CrossRef]
- R. Weiss, ???The application of implicit Runge-Kutta and collocation methods to boundary value problems,??? Math. Comp. 28, 449???464 (1974).
- V. E. Perlin and H. G. Winful, ???Optimal design of flat-gain wide-band fiber Raman amplifiers,??? J. Lightwave Technol. 20, 250???254 (2002). [CrossRef]
- X. M. Liu and B. Lee, ???A series of fast and accurate algorithms and their applications in fiber transmission systems,??? (submitted to IEEE J. Quantum Electron.)

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