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

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References

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  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]
  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]
  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]
  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).

2003 (2)

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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163.
[Crossref] [PubMed]

2001 (2)

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]

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]

2000 (1)

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]

1999 (1)

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]

1978 (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]

Auyeung, J.

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

Emori, Y.

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]

Fan, C.

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]

Guo, Y.

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

Hu, S.

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.

Kidorf, H.

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]

Lambert, J. D.

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

Lee, B.

Lee, W. J.

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]

Liu, X.

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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163.
[Crossref] [PubMed]

Ma, M.

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]

Min, B.

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]

Namiki, S.

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]

Nissov, M.

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]

Park, N.

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]

Quarteroni, A.

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

Rabarijaona, E.

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]

Rottwitt, K.

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]

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).

Wang, S.

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]

Yariv, A.

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

Zhang, H.

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

IEE Proc.-Optoelectron. (1)

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]

IEEE J. Quantum Electron. (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]

IEEE J. Select. Topics Quantum Electron. (1)

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]

IEEE Photon. Technol. Lett. (3)

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]

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

Opt. Express (1)

Other (4)

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).

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

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

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Figures (5)

Fig. 1.
Fig. 1.

The stiffness ratio of PERA along the fiber distance.

Fig. 2.
Fig. 2.

The maximum error of ‖P⃗‖ varied with the iterating step based on model (2) and (3).

Fig. 3.
Fig. 3.

Pump power evolution along the fiber distance obtained from three methods.

Fig. 4.
Fig. 4.

Net gain of FRA obtained from three methods.

Fig. 5.
Fig. 5.

Signal power evolution along the fiber calculated by different methods, in which (a) for BDF method based on model (2), (b) for PC method, (c) for BDF method based on model (3) and (d) for VPI.

Tables (2)

Tables Icon

Table 1. Comparison of the largest difference of the values calculated by three methods

Tables Icon

Table 2. Coefficients of zero-stable BDF methods for p=0,1,…,5.

Equations (13)

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d P ν ± d z = F ν ± ( z , P )
F ν ± ( z , P ) = α ν P ν ± ± ε ν P ν ± ± P ν ± ς > ν g ς ν Γ ς υ · ( P ς + + P ς ) P ν ± ς < ν ν ς · g ν ς Γ ν ς · ( P ς + + P ς )
± 2 h ν ς > ν g ς ν · ( P ς + + P ς ) · [ 1 + 1 exp [ h ( ς ν ) k T ] 1 ] · Δ ν
4 h ν P ν ± ς < ν ν ς · g ν ς · [ 1 + 1 exp [ h ( ν ς ) k T ] 1 ] · Δ ς
{ d P d z = F ( z , P ) P = ( P 1 ( z ) , P 2 ( z ) , , P n + m ( z ) ) T F = ( F 1 ( z , P ) , F 2 ( z , P ) , , F n + m ( z , P ) ) T
{ d ln ( P ) d z = F * ( z , P ) P = ( P 1 ( z ) , P 2 ( z ) , , P n + m ( z ) ) T F * = ( F 1 ( z , P ) P 1 , F 2 ( z , P ) P 2 , , F n + m ( z , P ) P n + m ) T
P p i ± ( z k ± 1 ) = exp { 48 25 ln [ P p i ± ( z k ) ] 36 25 ln [ P p i ± ( z k 1 ) ] + 16 25 · ln [ P p i ± ( z k 2 ) ] 3 25 ln [ P p i ± ( z k 3 ) ] + 12 25 F p i * ( z k ± 1 , P ) }
P s j + ( z k + 1 ) = exp { 48 25 ln [ P s j + ( z k ) ] 36 25 ln [ P s j + ( z k 1 ) ] + 16 25 · ln [ P s j + ( z k 2 ) ] 3 25 ln [ P s j + ( z k 3 ) ] + 12 25 F s j * ( z k + 1 , P ) }
d y dt = A y ( t ) + ϕ ( t ) , t [ a , b ]
Re ( λ j ) < 0 , j = 1 , 2 , , n
s = max 1 j n Re ( λ j ) min 1 j n Re ( λ j ) 1
d y dt = f ( t , y ( t ) ) , t [ a , b ]
y ( z k + 1 ) = i = 0 p q i y ( z k i ) + h b 1 f k + 1

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