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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    [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), <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]
  7. 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]
  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. <a href="http://www.mathworks.com/company/newsletter/clevescorner/may03_cleve.shtml">http://www.mathworks.com/company/newsletter/clevescorner/may03_cleve.shtml</a>
  11. J. D. Lambert, Computational methods in ordinary differential equations (John Wiley & Sons Ltd., London,1973).

APOC2003

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

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.

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.

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]

IEEE Photon. Technol. Lett

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]

IEEE Photon. Technol. Lett.

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

Other

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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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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