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

Full Article  |  PDF Article
More Like This
Pump scheme for gain-flattened Raman fiber amplifiers using improved particle swarm optimization and modified shooting algorithm

Hai-ming Jiang, Kang Xie, and Ya-fei Wang
Opt. Express 18(11) 11033-11045 (2010)

Efficient and robust shooting algorithm for numerical design of bidirectionally pumped Raman fiber amplifiers

Hai Ming Jiang and Kang Xie
J. Opt. Soc. Am. B 29(1) 8-14 (2012)

A fast and stable method for Raman amplifier propagation equations

Xueming Liu and Byoungho Lee
Opt. Express 11(18) 2163-2176 (2003)

View More...

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  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]
  2. E. M. Dianov, “Advances in Raman fibers,” J. Lightwave Technol. 20, 1457–1462 (2002).
    [Crossref]
  3. M. Karásek and M. Menif, “Channel addition/removal response in Raman fiber amplifiers: modeling and experimentation,” J. Lightwave Technol. 20, 1680–1687 (2002).
    [Crossref]
  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. 15, 27–29 (2003).
    [Crossref]
  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. 159, 149–157 (1999).
    [Crossref]
  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]
  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]
  9. U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations (SIAM, Philadelphia, 1998).
    [Crossref]
  10. A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer-Verlag, New York, 2000).
  11. B. F. Plybon, An Introduction to Applied Numerical Analysis (PWS-KENT Publishing Company, Boston, 1992), pp. 428–441.
  12. J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering (Second edition, Prentice Hall, New Jersey, 1992), pp. 464–475.
  13. 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]
  14. P. B. Bailey, L. F. Shampine, and P. E. Waltman, Nonlinear Two Point Boundary Value Problems (Academic Press, New York, 1968).
  15. L. F. Shampine, Numerical Solution of Ordinary Differential Equations (Chapman & Hall, New York, 1994).
  16. 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]
  17. R. Weiss, “The application of implicit Runge-Kutta and collocation methods to boundary value problems,” Math. Comp. 28, 449–464 (1974).
  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]
  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.)

2003 (2)

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

2002 (4)

2001 (1)

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]

1999 (1)

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]

1996 (1)

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]

1974 (1)

R. Weiss, “The application of implicit Runge-Kutta and collocation methods to boundary value problems,” Math. Comp. 28, 449–464 (1974).

Ascher, U. M.

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

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]

Bailey, P. B.

P. B. Bailey, L. F. Shampine, and P. E. Waltman, Nonlinear Two Point Boundary Value Problems (Academic Press, New York, 1968).

Carena, A.

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]

Curri, V.

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]

Dianov, E. M.

Enright, W. H.

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]

Guo, Y. L

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]

Hu, E. S.

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]

Kajiya, S.

Karásek, M.

Kazovsky, L. G.

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]

Kikuchi, N.

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]

Kinjo, K.

Lee, B.

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

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.)

Liu, X. M.

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

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.)

Mao, Q. H.

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]

Marhic, M.

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]

Mathews, J. H.

J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering (Second edition, Prentice Hall, New Jersey, 1992), pp. 464–475.

Mattheij, R. M. M.

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]

Menif, M.

Mizuochi, T.

Motoshima, K.

Muir, P. H.

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]

Perlin, V. E.

Petzold, L. R.

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

Plybon, B. F.

B. F. Plybon, An Introduction to Applied Numerical Analysis (PWS-KENT Publishing Company, Boston, 1992), pp. 428–441.

Poggiolini, P.

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]

Quarteroni, A.

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

Russell, R. D.

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]

Sacco, R.

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

Saleri, F.

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

Shampine, L. F.

L. F. Shampine, Numerical Solution of Ordinary Differential Equations (Chapman & Hall, New York, 1994).

P. B. Bailey, L. F. Shampine, and P. E. Waltman, Nonlinear Two Point Boundary Value Problems (Academic Press, New York, 1968).

Shimizu, K.

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]

Sun, X. H.

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]

Tokura, T.

Uesaka, K.

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]

Waltman, P. E.

P. B. Bailey, L. F. Shampine, and P. E. Waltman, Nonlinear Two Point Boundary Value Problems (Academic Press, New York, 1968).

Wang, J. S.

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]

Weiss, R.

R. Weiss, “The application of implicit Runge-Kutta and collocation methods to boundary value problems,” Math. Comp. 28, 449–464 (1974).

Winful, H. G.

Wong, K. K.

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]

Yam, S.

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]

Zhang, H. Y

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]

Zhang, M. D.

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]

IEEE Photon. Technol. Lett. (3)

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]

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]

J. Lightwave Technol. (4)

Math. Comp. (1)

R. Weiss, “The application of implicit Runge-Kutta and collocation methods to boundary value problems,” Math. Comp. 28, 449–464 (1974).

Opt. Commun. (1)

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]

SIAM J. Sci. Comput. (1)

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]

Other (9)

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.)

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

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).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1. Relationship of relative error with t, for (a) the fourth-order RK method, and (b) the fourth-order ORK method.

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2. Calculation procedure for the shooting algorithm based on the fourth-order ORK method, where (a) for 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.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3. Raman gain spectrum gR (Δv) of the fiber, which was used in the simulation.

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4. Iteration procedure and numerical results for a distributed pump Raman amplifier, for (a) iteration procedures of a pump, (b) iteration procedures of a signal, and (c) results of all pumps and signals at the last iteration. The calculating procedure is from z=L to z=0.

Download Full Size | PPT Slide | PDF

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

d y d t = f ( t , y ) ,
y ( t = t 0 ) = y 0 .
y n + 1 = y n + ∑ i = 1 m w i k i ,
k i = h f ( t n + c i , y n + ∑ j = 1 m d ij k j ) , i = 1 , 2 , … m .
d z d t = g ( t , exp ( z ) ) .
z n + 1 = z n + ∑ i = 1 m w i q i ,
q i = h g ( t n + c i , exp ( z n ) ( 1 + ∑ j = 1 m d ij q j ) ) , i = 1 , 2 , … m .
y n + 1 = y n exp ( ∑ i = 1 m w i q i ) ,
q i = h g ( t n + c i , y n + y n ∑ j = 1 m d ij q j ) , i = 1 , 2 , … m .
y n + 1 = y n exp [ ( q 1 + 2 q 2 + 2 q 3 + q 4 ) h 6 ] ,
q 1 = g ( t n , y n ) ,
q 2 = g ( t n + h 2 , y n + y n q 1 h 2 ) ,
q 3 = g ( t n + h 2 , y n + y n q 2 h 2 ) ,
q 4 = g ( t n + h , y n + y n q 3 h ) ,
g ( t n ) = f ( t n ) y n .
ε = C h 5 [ In ( y ) ] ( 5 ) ( ξ ) .
ε = C ′ h 5 y ( 5 ) ( ξ ) .
dy dt = y ( t 3 200 + t 2 50 + t ) , y ( 0 ) = 1 , and t ∈ [ 0 , 3.6 ] .
y = exp ( t 4 800 + t 3 150 + t 2 2 ) .
d y 1 ( t ) d t = f 1 ( t , y 1 , y 2 ) ,
d y 2 ( t ) d t = f 2 ( t , y 1 , y 2 ) ,
y 1 ( a ) = α ,
y 2 ( b ) = β .
A j + 1 = { A j − 1 + ( β − β j − 1 ) ( A j − A j − 1 ) ( β j − β j − 1 ) , ( if j > 1 ) A j β β j , ( if j = 1 )
± d P i d z = [ − α i + ∑ j = 1 i − 1 g R ( v j − v i ) Γ A eff P j − ∑ j = i + 1 m v i v j g R ( v i − v j ) Γ A eff P j ] P i , ( i = 1 , 2 , … , m ) ,

Metrics