Abstract

We show that it is possible to find analytic expressions for characterizing the evolution of signal and noise photon numbers along the active fiber of a forward-pumped Raman amplifier with unequal signal and pump loss coefficients. We confirm the validity of the result by comparing the analytical solutions with numerical calculations and by analytically deriving the well-known 3 dB noise figure limit for high Raman gain. Apart from aiding the analysis and design of forward pumped Raman amplifiers, these results also enable one to find approximate analytical solutions for bidirectional Raman amplifiers and backward pumped Raman amplifiers with Rayleigh backscattering and Brillouin scattering.

© 2004 Optical Society of America

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References

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  1. G. P. Agrawal, Fiber-Optic Communications Systems, 2nd Edition, (Wiley InterScience, New York, 1997).
  2. M. N. Islam, �??Raman amplifiers for telecommunications,�?? IEEE J. Select. Topics in Quantum Electron. 8, 548 �?? 559 (2002).
    [CrossRef]
  3. M. N. Islam, Raman amplifiers for telecommunications: Physical Principles, (Springer-Verlag, New York, 2003).
  4. V. E. Perlin and H. G.Winful,�??Optimal design of flat-gain wide-band fiber Raman amplifiers,�?? IEEE J. Lightwave Technol.20, 250 �?? 254 (2002).
    [CrossRef]
  5. P. C. Xiao, Q. J. Zeng, J. Huang, and J. M. Liu, �??A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,�?? IEEE Photon. Technol. Lett. 15, 206 �?? 208 (2003).
    [CrossRef]
  6. M. N. Islam and R. W. Lucky, Raman amplifiers for telecommunications 2: Sub-systems and Systems, (Springer- Verlag, New York, 2003).
  7. M. L. Dakss, and P. Melman, �??Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,�?? J. Lightware Technol. 3, 806 �?? 813 (1985).
    [CrossRef]
  8. R. Chinn, �??Analysis of counter-pumped small-signal fiber Raman amplifiers,�?? Electron Lett. 33, 607 �?? 608 (1997).
    [CrossRef]
  9. B. Bobbs, C.Warner, �??Closed-form solution for parametric second Stokes generation in Raman amplifiers,�?? IEEE J. Quantum Electron. 24, 660 �?? 664 (1988).
    [CrossRef]
  10. Y. Yan, J. Chen; W. Jiang, J. Li, J. Chen, X. Li, �??Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,�?? IEEE Photon. Technol. Lett. 13, 948 �?? 950 (2001).
    [CrossRef]
  11. D. Zwillinger, Handbook of Differential Equations, (Academic Publishers, Boston, 1989).
  12. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).
  13. M. R. Spiegel, Shaums Outline Series Theory and Problems of Complex Variables with an Introduction to Conformal Mappings and Its Applications, (McGraw-Hill Inc., New York, 1991).
  14. P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, D. J. DiGiovanni �??Rayleigh scattering limitations in distributed Raman pre-amplifiers,�?? Photon. Technol. Lett. 10, 159 �?? 161 (1998).
    [CrossRef]
  15. P. Parolari, L. Marazzi, L. Bernardini, M. Martinelli, �??Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,�??IEEE J. Lightwave Technol. 21, 2224 �?? 2228 (2003).
    [CrossRef]
  16. A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, A. F. Evans, �??Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,�?? IEEE J. Lightwave Technol. 20, 1635 �?? 1643 (2002).
    [CrossRef]
  17. J. Auyeung and A. Yariv, �??Spontaneous and stimulated Raman scattering in long low loss fibers,�?? IEEE J. Quantum Electron. 14, 347 �?? 352 (1978).
    [CrossRef]
  18. A. Bononi, M. Papararo, A. Vannucci, �??Impulsive pump depletion in saturated Raman amplifiers,�?? Electron. Lett. 37, 886 �?? 887 (2001).
    [CrossRef]

Electron Lett. (1)

R. Chinn, �??Analysis of counter-pumped small-signal fiber Raman amplifiers,�?? Electron Lett. 33, 607 �?? 608 (1997).
[CrossRef]

Electron. Lett. (1)

A. Bononi, M. Papararo, A. Vannucci, �??Impulsive pump depletion in saturated Raman amplifiers,�?? Electron. Lett. 37, 886 �?? 887 (2001).
[CrossRef]

IEEE J. Lightwave Technol. (3)

P. Parolari, L. Marazzi, L. Bernardini, M. Martinelli, �??Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,�??IEEE J. Lightwave Technol. 21, 2224 �?? 2228 (2003).
[CrossRef]

A. Kobyakov, M. Mehendale, M. Vasilyev, S. Tsuda, A. F. Evans, �??Stimulated Brillouin scattering in Raman-pumped fibers: a theoretical approach,�?? IEEE J. Lightwave Technol. 20, 1635 �?? 1643 (2002).
[CrossRef]

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

IEEE J. Quantum Electron. (2)

B. Bobbs, C.Warner, �??Closed-form solution for parametric second Stokes generation in Raman amplifiers,�?? IEEE J. Quantum Electron. 24, 660 �?? 664 (1988).
[CrossRef]

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 in Quantum Electr (1)

M. N. Islam, �??Raman amplifiers for telecommunications,�?? IEEE J. Select. Topics in Quantum Electron. 8, 548 �?? 559 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

P. C. Xiao, Q. J. Zeng, J. Huang, and J. M. Liu, �??A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,�?? IEEE Photon. Technol. Lett. 15, 206 �?? 208 (2003).
[CrossRef]

Y. Yan, J. Chen; W. Jiang, J. Li, J. Chen, X. Li, �??Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,�?? IEEE Photon. Technol. Lett. 13, 948 �?? 950 (2001).
[CrossRef]

J. Lightware Technol. (1)

M. L. Dakss, and P. Melman, �??Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,�?? J. Lightware Technol. 3, 806 �?? 813 (1985).
[CrossRef]

Photon. Technol. Lett. (1)

P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, D. J. DiGiovanni �??Rayleigh scattering limitations in distributed Raman pre-amplifiers,�?? Photon. Technol. Lett. 10, 159 �?? 161 (1998).
[CrossRef]

Other (6)

G. P. Agrawal, Fiber-Optic Communications Systems, 2nd Edition, (Wiley InterScience, New York, 1997).

D. Zwillinger, Handbook of Differential Equations, (Academic Publishers, Boston, 1989).

M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

M. R. Spiegel, Shaums Outline Series Theory and Problems of Complex Variables with an Introduction to Conformal Mappings and Its Applications, (McGraw-Hill Inc., New York, 1991).

M. N. Islam and R. W. Lucky, Raman amplifiers for telecommunications 2: Sub-systems and Systems, (Springer- Verlag, New York, 2003).

M. N. Islam, Raman amplifiers for telecommunications: Physical Principles, (Springer-Verlag, New York, 2003).

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Figures (4)

Fig. 1.
Fig. 1.

A schematic diagram of forward pumped Raman amplifier.

Fig. 2.
Fig. 2.

Photon number, n(z), against fiber length for two different pump powers PP and pump loss coefficients αP

Fig. 3.
Fig. 3.

I(z) against fiber length for two different pump loss coefficients,αP = 0.4 dB/km and αP = 0.3 dB/km.

Fig. 4.
Fig. 4.

Noise Figure against fiber length for two different pump loss coefficients: (a) αP = 0.4 dB/km (b) αP = 0.3 dB/km

Equations (40)

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dn ( z ) dz = γ n p ( z ) ( n ( z ) + 1 ) α S n ( z )
d n P ( z ) dz = γ n p ( z ) ( n ( z ) + 1 ) α P n P ( z )
n P ( z ) = n P ( 0 ) exp ( α p z )
dn ( z ) dz + ( α S γ n P ( 0 ) exp ( α P z ) ) n ( z ) = γ n p ( 0 ) exp ( α P z )
n ( z ) = i = 0 n i ( z ) α i
d n 0 ( z ) dz + ( α P γ n P ( 0 ) exp ( α P z ) ) n 0 ( z ) = γ n p ( 0 ) exp ( α P z )
d n i ( z ) dz + ( α P γ n P ( 0 ) exp ( α P z ) ) n i ( z ) = α P n i 1 ( z ) , i 0
n 0 ( z ) = H P ( z ) exp ( u 0 ) n ( 0 ) + H P ( z ) ( Ei ( u 0 ) Ei ( u ( z ) ) ) u 0
Ei ( x ) = P . V . x u 1 exp ( u ) du , x > 0
n 1 ( z ) = α P H P ( z ) ( exp ( u 0 ) n ( 0 ) + u 0 Ei ) u 0 ) ) z α P H P ( z ) u 0 0 z Ei ( u ( z ) ) dz
0 z Ei ( u ( z ) ) dz ( C + ln ( u 0 ) ) z α P 2 z 2 + u 0 ln ( u 0 ) α P ( 1 u 2 ( z ) u 0 2 )
n 1 ( z ) = α α P × exp ( u 0 ) n ( 0 ) z + α P 2 u 0 z 2 + ( Ei ( u 0 ) C ln ( u 0 ) ) ) u 0 z u 0 1 + ln ( u 0 ) α P ( 1 u 2 ( z ) u 0 2 ) H P ( z )
n ( z ) H P ( z ) ( exp ( u 0 ) n ( 0 ) + u 0 ( Ei ( u 0 ) Ei ( u ( z ) ) )
+ α α P × exp ( u 0 ) n ( 0 ) z + α P 2 u 0 z 2 + ( Ei ( u 0 ) C ln ( u 0 ) ) ) u 0 z u 0 1 + ln ( u 0 ) α P ( 1 u 2 ( z ) u 0 2 )
+ i = 2 ( α P z ) i i ! ( exp ( u 0 ) n ( 0 ) + u 0 Ei ( u 0 ) ) α i
n ( z ) = G S ( z ) n ( 0 ) + ( α P α S ) H P ( z ) H S ( z ) ( z )
+ H S ( z ) u 0 Ei ( u 0 ) H P ( z ) u 0 Ei ( u ( z ) )
( z ) = α P 2 u 0 z 2 u 0 ( C + ln ( u 0 ) ) z u 0 1 + ln ( u 0 ) α P ( 1 u 2 ( z ) u 0 2 )
n ( z ) α = 0 = G S ( z ) n ( 0 ) + u 0 H S ( z ) × ( Ei ( u 0 ) Ei ( u ( z ) ) )
n ( z ) = G S ( z ) × n ( 0 )
n ( z ) = G S ( z ) × n ̂ ( z )
d n ̂ ( z ) dz = α P u 0 exp ( α P αz u 0 ( 1 exp ( α P z ) ) )
n ̂ ( 0 ) = n ( 0 ) + u 0 exp ( u 0 ) × I ( z )
I ( z ) = 1 u 0 α u ( z ) u 0 v α 1 exp ( v ) dv
I ( z ) = k = 0 ( α ) k u 0 k ( α + 1 ) k k ! ( u ( z ) u 0 ) α k = 0 ( α ) k u k ( z ) ( α + 1 ) k k !
Φ ( a ; b ; z ) = k = 0 ( a ) k z k ( b ) k k !
n ( z ) = G S ( z ) n ( 0 ) + u 0 H S ( z ) α ( Φ ( α ; α + 1 ; u 0 ) Φ ( α ; α + 1 ; u ( z ) ) ( u ( z ) u 0 ) α )
n noise ( 0 ) = u 0 exp ( u 0 ) α ( Φ ( α ; α + 1 ; u 0 ) Φ ( α ; α + 1 ; u ( z ) ) ( u ( z ) u 0 ) α )
n ( z ) α = 0 = G S ( z ) n ( 0 ) +
u 0 H S ( z ) × lim α 0 d ( Φ ( α ; α + 1 ; u 0 ) Φ ( α ; α ; + 1 ; u ( z ) ) ( u ( z ) u 0 ) α )
n ( z ) α = 0 = G S ( z ) n ( 0 ) + u 0 H S ( z ) × ( k = 1 u 0 k k · k ! + ln ( u 0 ) k = 1 u k ( z ) k · k ! ln ( u ( z ) ) )
Ei ( x ) = C + ln ( x ) + k = 1 x k k · k !
NF ( z ) = 1 + 2 ( n ( z ) G S ( z ) n ( 0 ) ) G S ( z )
NF a ( z ) = 1 + 2 ( α P α S ) H P ( z ) H S ( z ) ( z ) + 2 ( H S ( z ) u 0 Ei ( u 0 ) H P ( z ) u ( z ) Ei ( u ( z ) ) ) exp ( u 0 ) H S ( z )
NF e ( z ) = 1 G ( z ) + 2 u 0 exp ( u 0 ) α ( Φ ( α ; α + 1 , u 0 ) Φ ( α ; α + 1 , u ( z ) ) )
NF a ( z ) 2 ( u 0 Ei ( u 0 ) u ( z ) Ei ( u ( z ) ) ) exp ( u 0 )
Ei ( x ) = exp ( x ) x ( 1 + O ( 1 x ) )
dn ( z ) dz = ( k = 1 M γ k n Pk ( z ) ) ( n ( z ) + 1 ) α S n ( z )
dn P i ( z ) dz = γ i n P i ( z ) ( k = 1 k i M nPk ( z ) + n ( z ) + 1 ) α Pi n Pi ( z ) , i = 1,2 , , M
n ( z ) = i = 0 n i ( z ) ( 1 αM k = 1 M α k ) i

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