Abstract

Exact integral solutions of forward and backward Raman Stokes–pump interaction are presented. They enable one to determine the gain and depletion characteristics of optical Raman amplifiers, as well as a good approximation of the noise figure, without the need of numerically solving the nonlinear boundary value differential equations governing the Raman interaction.

© 2007 Optical Society of America

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References

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  1. C.Headley and G.P.Agrawal, eds., Raman Amplification (Elsevier, 2005).
  2. R. H. Enns and I. P. Batra, Phys. Lett. A 28, 591 (1969).
    [CrossRef]
  3. J. Auyeung and A. Yariv, IEEE J. Quantum Electron. 14, 347 (1978).
    [CrossRef]
  4. C. Yijiang and A. W. Snyder, J. Lightwave Technol. 7, 1109 (1989).
    [CrossRef]
  5. R. G. Smith, Appl. Opt. 11, 2489 (1972).
    [CrossRef] [PubMed]
  6. M. L. Dakss and P. Melman, J. Lightwave Technol. 3, 806 (1985).
    [CrossRef]
  7. Y. Aoki, J. Lightwave Technol. 6, 1225 (1988).
    [CrossRef]
  8. M. Premaratne, Opt. Express 12, 4235 (2004).
    [CrossRef] [PubMed]
  9. A. Bononi, M. Papararo, and A. Vannucci, Electron. Lett. 37, 886 (2001).
    [CrossRef]
  10. J. H. Lambert, Acta Helv. 3, 128 (1758).
  11. L. Euler, Oper. Math. 15, 268 (1927) L. Euler,[Leonhardi Euleri Opera Omnia Ser. 1, 1777)].
  12. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, Commun. Math. Phys. 5, 329 (1996).
  13. M. Santagiustina, in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference (Optical Society of America, 2007), paper CJ-20-Tue.
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    [CrossRef] [PubMed]
  15. R. W. Davies, P. Melman, W. H. Nelson, M. L. Dakss, and B. M. Foley, J. Lightwave Technol. 5, 1068 (1987).
    [CrossRef]

2004 (1)

2001 (1)

A. Bononi, M. Papararo, and A. Vannucci, Electron. Lett. 37, 886 (2001).
[CrossRef]

1996 (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, Commun. Math. Phys. 5, 329 (1996).

1990 (1)

1989 (1)

C. Yijiang and A. W. Snyder, J. Lightwave Technol. 7, 1109 (1989).
[CrossRef]

1988 (1)

Y. Aoki, J. Lightwave Technol. 6, 1225 (1988).
[CrossRef]

1987 (1)

R. W. Davies, P. Melman, W. H. Nelson, M. L. Dakss, and B. M. Foley, J. Lightwave Technol. 5, 1068 (1987).
[CrossRef]

1985 (1)

M. L. Dakss and P. Melman, J. Lightwave Technol. 3, 806 (1985).
[CrossRef]

1978 (1)

J. Auyeung and A. Yariv, IEEE J. Quantum Electron. 14, 347 (1978).
[CrossRef]

1972 (1)

1969 (1)

R. H. Enns and I. P. Batra, Phys. Lett. A 28, 591 (1969).
[CrossRef]

1927 (1)

L. Euler, Oper. Math. 15, 268 (1927) L. Euler,[Leonhardi Euleri Opera Omnia Ser. 1, 1777)].

1758 (1)

J. H. Lambert, Acta Helv. 3, 128 (1758).

Acta Helv. (1)

J. H. Lambert, Acta Helv. 3, 128 (1758).

Appl. Opt. (1)

Commun. Math. Phys. (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, Commun. Math. Phys. 5, 329 (1996).

Electron. Lett. (1)

A. Bononi, M. Papararo, and A. Vannucci, Electron. Lett. 37, 886 (2001).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Auyeung and A. Yariv, IEEE J. Quantum Electron. 14, 347 (1978).
[CrossRef]

J. Lightwave Technol. (4)

C. Yijiang and A. W. Snyder, J. Lightwave Technol. 7, 1109 (1989).
[CrossRef]

M. L. Dakss and P. Melman, J. Lightwave Technol. 3, 806 (1985).
[CrossRef]

Y. Aoki, J. Lightwave Technol. 6, 1225 (1988).
[CrossRef]

R. W. Davies, P. Melman, W. H. Nelson, M. L. Dakss, and B. M. Foley, J. Lightwave Technol. 5, 1068 (1987).
[CrossRef]

Oper. Math. (1)

L. Euler, Oper. Math. 15, 268 (1927) L. Euler,[Leonhardi Euleri Opera Omnia Ser. 1, 1777)].

Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. A (1)

R. H. Enns and I. P. Batra, Phys. Lett. A 28, 591 (1969).
[CrossRef]

Other (2)

C.Headley and G.P.Agrawal, eds., Raman Amplification (Elsevier, 2005).

M. Santagiustina, in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference (Optical Society of America, 2007), paper CJ-20-Tue.

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Figures (4)

Fig. 1
Fig. 1

Stokes ( P s ) and pump ( P p ) power evolution (BPA) along 80 km of standard fiber for the exact, depleted solution (solid curves) compared with the undepleted approximation (dashed curves) (photon to power conversion factors are calculated as in [6]).

Fig. 2
Fig. 2

Gain saturation for FPA (solid curves) and BPA (dashed curves) as a function of the input Stokes power ( P s ) for three different input pump powers P p ; a, 0.9 W ; b, 0.6 W ; c, 0.3 W . The circles (squares) are the numerical solutions of Raman interaction equations, including ASE and a second-order Stokes wave for FPA (BPA).

Fig. 3
Fig. 3

Pump depletion for FPA (solid curves) and BPA (dashed curves) (same fiber data and input pump powers as in Fig. 2). The circles (squares) are the numerical solutions of Raman interaction equations, including ASE and a second-order Stokes signal for FPA (BPA).

Fig. 4
Fig. 4

NF: for the FPA (BPA) the diamonds (circles) are the numerical solutions of Eqs. (1) for ϵ = 1 , n s ( 0 ) = 0 , while crosses (pluses) are the exact solution of Eqs. (1) for ϵ = 0 , n s ( 0 ) = 1 (same fiber data and input pump powers as in Fig. 2).

Equations (6)

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d n s ( ϵ ) d z = g ( n s ( ϵ ) + ϵ ) n p ( ϵ ) α n s ( ϵ ) ,
d n p ( ϵ ) d z = ± [ g ( n s ( ϵ ) + ϵ ) n p ( ϵ ) + n p ( ϵ ) ] .
n s ( z , C ) = 1 g W [ C exp [ ± g n p ( z ) ] n p α ( z ) ] ,
n p ( 0 ) n p ( z ) d n p ± n p [ W [ C exp ( ± g n p ) n p α ] + 1 ] = z .
F [ n p ( 0 ) ] = n p ( 0 ) n p ( L ) d n p n p [ W [ C [ n p ( 0 ) ] exp ( g n p ) n p α ] + 1 ] L ,
d F [ n p ( 0 ) ] d n p ( 0 ) = n p ( 0 ) n p ( L ) W [ C [ n p ( 0 ) ] exp ( g n p ) n p α ] n p [ W [ C [ n p ( 0 ) ] exp ( g n p ) n p α ] + 1 ] d n p { n p ( 0 ) [ W [ C [ n p ( 0 ) ] exp [ g n p ( 0 ) ] n p ( 0 ) α ] + 1 ] } 1 .

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