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 and 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, and 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, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. 10, 159–161 (1998).
    [Crossref]
  15. P. Parolari, L. Marazzi, L. Bernardini, and 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, and 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, and A. Vannucci, “Impulsive pump depletion in saturated Raman amplifiers,” Electron. Lett. 37, 886–887 (2001).
    [Crossref]

2003 (3)

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]

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

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

2002 (3)

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

M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Select. Topics in Quantum Electron. 8, 548–559 (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]

2001 (2)

Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and 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]

A. Bononi, M. Papararo, and A. Vannucci, “Impulsive pump depletion in saturated Raman amplifiers,” Electron. Lett. 37, 886–887 (2001).
[Crossref]

1998 (1)

P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. 10, 159–161 (1998).
[Crossref]

1997 (1)

R. Chinn, “Analysis of counter-pumped small-signal fiber Raman amplifiers,” Electron Lett. 33, 607–608 (1997).
[Crossref]

1988 (1)

B. Bobbs and C. Warner, “Closed-form solution for parametric second Stokes generation in Raman amplifiers,” IEEE J. Quantum Electron. 24, 660–664 (1988).
[Crossref]

1985 (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]

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]

Abramowitz, M.

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

Agrawal, G. P.

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

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]

Bernardini, L.

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

Bobbs, B.

B. Bobbs and C. Warner, “Closed-form solution for parametric second Stokes generation in Raman amplifiers,” IEEE J. Quantum Electron. 24, 660–664 (1988).
[Crossref]

Bononi, A.

A. Bononi, M. Papararo, and A. Vannucci, “Impulsive pump depletion in saturated Raman amplifiers,” Electron. Lett. 37, 886–887 (2001).
[Crossref]

Chen, J.

Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and 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]

Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and 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]

Chinn, R.

R. Chinn, “Analysis of counter-pumped small-signal fiber Raman amplifiers,” Electron Lett. 33, 607–608 (1997).
[Crossref]

Dakss, M. L.

M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. 3, 806–813 (1985).
[Crossref]

DeMarco, J. J.

P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. 10, 159–161 (1998).
[Crossref]

DiGio-vanni, D. J.

P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. 10, 159–161 (1998).
[Crossref]

Eskildsen, L.

P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. 10, 159–161 (1998).
[Crossref]

Evans, A. F.

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

Hansen, P. B.

P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. 10, 159–161 (1998).
[Crossref]

Huang, J.

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]

Islam, M. N.

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,” IEEE J. Select. Topics in Quantum Electron. 8, 548–559 (2002).
[Crossref]

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

Jiang, W.

Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and 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]

Judkins, J.

P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. 10, 159–161 (1998).
[Crossref]

Kobyakov, A.

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

Li, J.

Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and 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]

Li, X.

Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and 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]

Liu, J. M.

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]

Lucky, R. W.

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

Marazzi, L.

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

Martinelli, M.

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

Mehendale, M.

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

Melman, P.

M. L. Dakss and P. Melman, “Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,” J. Lightware Technol. 3, 806–813 (1985).
[Crossref]

Papararo, M.

A. Bononi, M. Papararo, and A. Vannucci, “Impulsive pump depletion in saturated Raman amplifiers,” Electron. Lett. 37, 886–887 (2001).
[Crossref]

Parolari, P.

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

Pedrazzani, R.

P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. 10, 159–161 (1998).
[Crossref]

Perlin, V. E.

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]

Spiegel, M. R.

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).

Stegun, A.

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

Stentz, J.

P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. 10, 159–161 (1998).
[Crossref]

Strasser, T. A.

P. B. Hansen, L. Eskildsen, J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGio-vanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” Photon. Technol. Lett. 10, 159–161 (1998).
[Crossref]

Tsuda, S.

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

Vannucci, A.

A. Bononi, M. Papararo, and A. Vannucci, “Impulsive pump depletion in saturated Raman amplifiers,” Electron. Lett. 37, 886–887 (2001).
[Crossref]

Vasilyev, M.

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

Warner, C.

B. Bobbs and C. Warner, “Closed-form solution for parametric second Stokes generation in Raman amplifiers,” IEEE J. Quantum Electron. 24, 660–664 (1988).
[Crossref]

Winful, H. G.

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]

Xiao, P. C.

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]

Yan, Y.

Y. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and 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]

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]

Zeng, Q.J.

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]

Zwillinger, D.

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

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, and 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, and 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, and 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 and 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 Electron. (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, and 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, and D. J. DiGio-vanni, “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)

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

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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