Abstract

We exactly solve the initial-boundary value problem of interaction of three waves in the limit when one of these waves is strongly damped. The solution is applied to the characterization of transient effects in Raman amplifiers, with a special emphasis on the possibility of generating Stokes pulses with peak powers that are orders of magnitude higher than the input power of the pump beam.

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References

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  1. C. Headley and G. P. Agrawal, Raman Amplification(Elsevier, 2005).
  2. P. M. Krummrich, E. Gottwald, A. Mayer, R. E. Neuhauser, and G. Fischer, in Optical Amplifiers and Their Applications, Vol.  44 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), paper OTuC6.
  3. C.-J. Chen and W. S. Wong, Electron. Lett. 37, 371 (2001).
    [CrossRef]
  4. A. Bononi and A. Papararo, in Optical Fiber Communications Conference, Vol.  70 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), paper ThR1.
  5. V. M. Malkin, G. Shvets, and N. J. Fisch, Phys. Rev. Lett. 82, 4448 (1999).
    [CrossRef]
  6. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2002).
  7. M. Maier, W. Kaiser, and J. A. Giordmaine, Phys. Rev. 177, 580 (1969).
    [CrossRef]
  8. A. Hook, D. Anderson, and M. Lisak, J. Opt. Soc. Am. B 6, 1851 (1989).
    [CrossRef]

2001

C.-J. Chen and W. S. Wong, Electron. Lett. 37, 371 (2001).
[CrossRef]

1999

V. M. Malkin, G. Shvets, and N. J. Fisch, Phys. Rev. Lett. 82, 4448 (1999).
[CrossRef]

1989

1969

M. Maier, W. Kaiser, and J. A. Giordmaine, Phys. Rev. 177, 580 (1969).
[CrossRef]

Agrawal, G. P.

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

Anderson, D.

Bononi, A.

A. Bononi and A. Papararo, in Optical Fiber Communications Conference, Vol.  70 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), paper ThR1.

Chen, C.-J.

C.-J. Chen and W. S. Wong, Electron. Lett. 37, 371 (2001).
[CrossRef]

Fisch, N. J.

V. M. Malkin, G. Shvets, and N. J. Fisch, Phys. Rev. Lett. 82, 4448 (1999).
[CrossRef]

Fischer, G.

P. M. Krummrich, E. Gottwald, A. Mayer, R. E. Neuhauser, and G. Fischer, in Optical Amplifiers and Their Applications, Vol.  44 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), paper OTuC6.

Giordmaine, J. A.

M. Maier, W. Kaiser, and J. A. Giordmaine, Phys. Rev. 177, 580 (1969).
[CrossRef]

Gottwald, E.

P. M. Krummrich, E. Gottwald, A. Mayer, R. E. Neuhauser, and G. Fischer, in Optical Amplifiers and Their Applications, Vol.  44 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), paper OTuC6.

Headley, C.

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

Hook, A.

Kaiser, W.

M. Maier, W. Kaiser, and J. A. Giordmaine, Phys. Rev. 177, 580 (1969).
[CrossRef]

Krummrich, P. M.

P. M. Krummrich, E. Gottwald, A. Mayer, R. E. Neuhauser, and G. Fischer, in Optical Amplifiers and Their Applications, Vol.  44 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), paper OTuC6.

Lisak, M.

Maier, M.

M. Maier, W. Kaiser, and J. A. Giordmaine, Phys. Rev. 177, 580 (1969).
[CrossRef]

Malkin, V. M.

V. M. Malkin, G. Shvets, and N. J. Fisch, Phys. Rev. Lett. 82, 4448 (1999).
[CrossRef]

Mayer, A.

P. M. Krummrich, E. Gottwald, A. Mayer, R. E. Neuhauser, and G. Fischer, in Optical Amplifiers and Their Applications, Vol.  44 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), paper OTuC6.

Neuhauser, R. E.

P. M. Krummrich, E. Gottwald, A. Mayer, R. E. Neuhauser, and G. Fischer, in Optical Amplifiers and Their Applications, Vol.  44 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), paper OTuC6.

Papararo, A.

A. Bononi and A. Papararo, in Optical Fiber Communications Conference, Vol.  70 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), paper ThR1.

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2002).

Shvets, G.

V. M. Malkin, G. Shvets, and N. J. Fisch, Phys. Rev. Lett. 82, 4448 (1999).
[CrossRef]

Wong, W. S.

C.-J. Chen and W. S. Wong, Electron. Lett. 37, 371 (2001).
[CrossRef]

Electron. Lett.

C.-J. Chen and W. S. Wong, Electron. Lett. 37, 371 (2001).
[CrossRef]

J. Opt. Soc. Am. B

Phys. Rev.

M. Maier, W. Kaiser, and J. A. Giordmaine, Phys. Rev. 177, 580 (1969).
[CrossRef]

Phys. Rev. Lett.

V. M. Malkin, G. Shvets, and N. J. Fisch, Phys. Rev. Lett. 82, 4448 (1999).
[CrossRef]

Other

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2002).

A. Bononi and A. Papararo, in Optical Fiber Communications Conference, Vol.  70 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), paper ThR1.

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

P. M. Krummrich, E. Gottwald, A. Mayer, R. E. Neuhauser, and G. Fischer, in Optical Amplifiers and Their Applications, Vol.  44 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), paper OTuC6.

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Figures (3)

Fig. 1
Fig. 1

Step-like excitation. Output signal calculated with the exact formula (21) (black solid curve) and approximate formula (4.6) from [7] (red dashed curve). (left) A in = 10 4 , (right) A in = 10 2 , where A in is the size of the initial step. For both cases, P = 0.5 and L = 10 . When expressed in dimensional units, one dimensionless unit of power corresponds to 1 W , unit of time to 0.12 ms , and unit of length to 0.4 km . All are related to the example of a dispersion- compensating fiber considered in the body of the text.

Fig. 2
Fig. 2

Illustration of the Raman amplification of a sequence of short pulses. Parameters are P = 0.5 , L = 10 , the peak intensity of each input pulse 0.01: (a) input, (b) output.

Fig. 3
Fig. 3

(a) Output intensity of the leading pulse in the sequence as function of initial pulse duration T p , (b) output intensity versus input intensity for three values of the pulse duration T p : 0.002 (black solid curve), 0.001 (red dashed curve), 0.0005 (green dotted curve). Parameters are P = 0.5 , L = 10 .

Equations (21)

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X P s + ( n / c ) T P s = g R P s P p ,
X P p + ( n / c ) T P p = g R P s P p ,
t A x A = 2 A B ,
t B + x B = 2 A B ,
B ( 0 , t ) = P , 0 t < ,
B ( x , 0 ) = P , 0 x L ,
A ( L , t ) = A L ( t ) , 0 < t < ,
A ( x , 0 ) = 0 , 0 x L .
A ( x , t ) = ξ a ( ξ ) a ( ξ ) + b ( η ) ,
B ( x , t ) = η b ( η ) a ( ξ ) + b ( η ) .
A 0 ( t ) = t a ( t ) a ( t ) + b ( t ) ,
P = t b ( t ) a ( t ) + b ( t ) .
d d t [ a ( t ) + b ( t ) ] = [ A 0 ( t ) P ] [ a ( t ) + b ( t ) ] .
a ( t ) + b ( t ) = e P t , 0 t L ,
a ( t ) + b ( t ) = e α 0 ( t ) P t ,
a ( t ) = e α 0 ( t ) P t e P t
A 0 ( t + L ) = t a ( t + L ) exp [ α 0 ( t + L ) P ( t + L ) ] ,
A L ( t ) = t a ( t + L ) a ( t + L ) + exp [ P ( t L ) ] .
A 0 ( t + L ) e α 0 ( t + L ) = A L ( t ) [ e P ( t + L ) a ( t + L ) + e 2 P L ] .
a ( t + L ) = e α L ( t ) 0 t d t A L ( t ) e α L ( t ) P ( t L ) ,
A 0 ( t ) = A L ( t L ) exp ( 2 P L ) f ( t L ) 1 + exp ( 2 P L ) 0 t L d t A L ( t L ) f ( t L ) ,

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