Abstract

The effects of stimulated Raman scattering on femtosecond pulse generation using a parabolic amplifier and a grating-pair compressor are presented. We derive an explicit analytical form for the Stokes pulse evolution. We find that the evolution of the Stokes pulse can be divided into four regimes; small Gaussian Stokes pulse, small asymmetric Stokes pulse, signal depletion, and parabolic Raman pulse. To achieve efficient pulse compression, one should operate the parabolic amplifier in the small Stokes pulse regime where the signal pulse is not seriously distorted. We also derive an analytical expression to obtain a critical fiber length for the small Stokes pulse regime. The derived theory is applied to a realistic high-power femtosecond pulse generation process through a split-step Fourier numerical simulation. The pulse compression results confirm that our derived critical fiber length leads to the highest peak power and shortest width of compressed pulse.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
    [CrossRef] [PubMed]
  5. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19, 461-469 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2004

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

G. Chang, A. Galvanauskas, H. G. Winful, and T. B. Norris, "Dependence of parabolic pulse amplification on stimulated Raman scattering and gain bandwidth," Opt. Lett. 29, 2647-2649 (2004).
[CrossRef] [PubMed]

2003

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

2002

2000

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

1997

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, "Optical pulse compression in fiber Bragg gratings," Phys. Rev. Lett. 79, 4566-4569 (1997).
[CrossRef]

1996

1992

1989

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, "Raman response function of silica-core fibers," J. Opt. Soc. Am. B 6, 1159-1166 (1989).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, "Theory of stimulated Raman scattering in optical fibers in the pulse walk-off regime," IEEE J. Quantum Electron. 25, 273-279 (1989).
[CrossRef]

1988

J. Herrmann and J. Mondry, "Stimulated Raman scattering and self-phase modulation of ultrashort light pulses in optical fibres," J. Mod. Opt. 35, 1919-1932 (1988).
[CrossRef]

1987

1986

1985

1969

E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. QE-5, 454-458 (1969).
[CrossRef]

Afanasjev, V. V.

Agrawal, G. P.

Anderson, D.

Anderson, P.

Billet, C.

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

Broderick, N. G.

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, "Optical pulse compression in fiber Bragg gratings," Phys. Rev. Lett. 79, 4566-4569 (1997).
[CrossRef]

Chang, G.

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, "Theory of stimulated Raman scattering in optical fibers in the pulse walk-off regime," IEEE J. Quantum Electron. 25, 273-279 (1989).
[CrossRef]

Clausnitzer, T.

Cockings, O.

Dudley, J. M.

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19, 461-469 (2002).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Fermann, M. E.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Finot, C.

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

Fuchs, H.-J.

Galvanauskas, A.

Gordon, J. P.

Grudinin, A. B.

Harde, H.

M. Kuckartz, R. Schultz, and H. Harde, "Theoretical and experimental studies of the combined self-phase modulation and stimulated Raman-scattering in single-mode fibres," Opt. Quantum Electron. 19, 237-246 (1987).
[CrossRef]

Harvey, J. D.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19, 461-469 (2002).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Haus, H. A.

Headley, C.

Herrmann, J.

J. Herrmann and J. Mondry, "Stimulated Raman scattering and self-phase modulation of ultrashort light pulses in optical fibres," J. Mod. Opt. 35, 1919-1932 (1988).
[CrossRef]

Ibsen, M.

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, "Optical pulse compression in fiber Bragg gratings," Phys. Rev. Lett. 79, 4566-4569 (1997).
[CrossRef]

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, "Theory of stimulated Raman scattering in optical fibers in the pulse walk-off regime," IEEE J. Quantum Electron. 25, 273-279 (1989).
[CrossRef]

Kley, E.-B.

Knox, W. H.

Kruglov, V. I.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19, 461-469 (2002).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Kuckartz, M.

M. Kuckartz, R. Schultz, and H. Harde, "Theoretical and experimental studies of the combined self-phase modulation and stimulated Raman-scattering in single-mode fibres," Opt. Quantum Electron. 19, 237-246 (1987).
[CrossRef]

Laming, R. I.

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, "Optical pulse compression in fiber Bragg gratings," Phys. Rev. Lett. 79, 4566-4569 (1997).
[CrossRef]

Limpert, J.

Lisak, M.

Manassa, J. T.

Martinez, O. E.

Millot, G.

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

Mondry, J.

J. Herrmann and J. Mondry, "Stimulated Raman scattering and self-phase modulation of ultrashort light pulses in optical fibres," J. Mod. Opt. 35, 1919-1932 (1988).
[CrossRef]

Norris, T. B.

Payne, D. N.

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19, 461-469 (2002).
[CrossRef]

Pitois, S.

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

Richardson, D. J.

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, "Optical pulse compression in fiber Bragg gratings," Phys. Rev. Lett. 79, 4566-4569 (1997).
[CrossRef]

D. J. Richardson, V. V. Afanasjev, A. B. Grudinin, and D. N. Payne, "Amplification of femtosecond pulses in a passive, all-fiber soliton source," Opt. Lett. 17, 1596-1598 (1992).
[CrossRef] [PubMed]

Schreiber, T.

Schultz, R.

M. Kuckartz, R. Schultz, and H. Harde, "Theoretical and experimental studies of the combined self-phase modulation and stimulated Raman-scattering in single-mode fibres," Opt. Quantum Electron. 19, 237-246 (1987).
[CrossRef]

Stolen, R. H.

Taverner, D.

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, "Optical pulse compression in fiber Bragg gratings," Phys. Rev. Lett. 79, 4566-4569 (1997).
[CrossRef]

Thomsen, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Tomlinson, W. J.

Treacy, E. B.

E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. QE-5, 454-458 (1969).
[CrossRef]

Tünnermann, A.

Winful, H. G.

Zellmer, H.

Zollner, K.

Appl. Opt.

IEEE J. Quantum Electron.

D. N. Christodoulides and R. I. Joseph, "Theory of stimulated Raman scattering in optical fibers in the pulse walk-off regime," IEEE J. Quantum Electron. 25, 273-279 (1989).
[CrossRef]

E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. QE-5, 454-458 (1969).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

J. Mod. Opt.

J. Herrmann and J. Mondry, "Stimulated Raman scattering and self-phase modulation of ultrashort light pulses in optical fibres," J. Mod. Opt. 35, 1919-1932 (1988).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

M. Kuckartz, R. Schultz, and H. Harde, "Theoretical and experimental studies of the combined self-phase modulation and stimulated Raman-scattering in single-mode fibres," Opt. Quantum Electron. 19, 237-246 (1987).
[CrossRef]

Phys. Rev. Lett.

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, "Optical pulse compression in fiber Bragg gratings," Phys. Rev. Lett. 79, 4566-4569 (1997).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

Other

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

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Figures (6)

Fig. 1
Fig. 1

Numerical simulation of signal and Stokes pulse propagation in 8 m fiber. (a) Signal pulse, (b) Stokes pulse.

Fig. 2
Fig. 2

Peak power of the signal pulse (dashed curve), numerical simulation of the Stokes pulse (solid curve), analytical solution (open circles), and the approximation of Ref. [8]. Region I, the small Gaussian Stokes pulse regime; region II, the small asymmetric Stokes pulse regime; region III, the signal depletion regime; region IV, the parabolic Stokes pulse regime.

Fig. 3
Fig. 3

Time position in relative time coordinates in which the peak power of the Stokes pulse occurs. Numerical solution, solid curve; analytical solution, open circles. Dashed curves represent the signal pulse width T p ( z ) . Region I, the small Gaussian Stokes pulse regime; region II, the small asymmetric Stokes pulse regime.

Fig. 4
Fig. 4

Compressed pulse from the parabolic signal pulse at z = 5 m . (a) The compressed pulse output. (b) The peak power of compressed pulses versus grating distance.

Fig. 5
Fig. 5

Pulse compressor simulation results for the signal pulse versus propagation distance. The peak powers of the compressed pulses are plotted (dotted curve) with the corresponding pulse width (open circles).

Fig. 6
Fig. 6

Lambert W function.

Equations (55)

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ψ s z + i 2 β 2 s 2 ψ s T 2 = i γ s [ ψ s 2 + ( 2 f R ) ψ r 2 ] ψ s g s 2 ψ r 2 ψ s + α s 2 ψ s ,
ψ r z d ψ r T + i 2 β 2 r 2 ψ r T 2 = i γ r [ ψ r 2 + ( 2 f R ) ψ s 2 ] ψ r + g r 2 ψ s 2 ψ r + α r 2 ψ r ,
ψ s z + i 2 β 2 s 2 ψ s T 2 = i γ s ψ s 2 ψ s + α s 2 ψ s ,
ψ r z d ψ r T + i 2 β 2 r 2 ψ r T 2 = [ i γ r ( 2 f R ) + g r 2 ] ψ s 2 ψ r + α r 2 ψ r .
ψ s = A 0 exp ( α s 3 z ) 1 T 2 T p 2 ( z ) exp ( i { ϕ 0 + α s 12 β 2 [ T p 2 ( z ) 2 T 2 ] } ) ,
ψ r ( z , T ) = ψ r ( 0 , T + z d ) exp { α r 2 z + [ g r 2 + i γ r ( 2 f R ) ] ϕ ( z , T ) } ,
ϕ ( z , T ) = 0 z A 0 2 exp ( 2 α s 3 z ) f ( z , z , T ) d z ,
f ( z , z , T ) = { 1 ( T + z d z d ) 2 T p 2 ( z ) , T + z d z d < T p ( z ) 0 , otherwise } .
ψ r ( z , T ) = P r 0 exp ( α r 2 z + [ g r 2 + i γ r ( 2 f R ) ] { 3 A 0 2 2 α s B 2 [ T p 2 ( z ) T p 2 ( 0 ) ] d 2 z 3 A 0 2 12 B 2 + z A 0 2 B 2 ( T + 1 2 z d ) 2 } ) ,
P r peak ( z ) = P r 0 exp ( α r z + g r A 0 2 { 3 2 α s B 2 [ T p 2 ( z ) T p 2 ( 0 ) ] d 2 z 3 12 B 2 } ) ,
T 0 ( z ) = z d 2 .
ψ r ( z , T ) = P r 0 exp ( α r 2 z + [ g r 2 + i γ r ( 2 f R ) ] A 0 2 B 2 { 3 2 α s [ T p 2 ( z ) T p 2 ( z min ) ] 1 3 d [ T 3 T p 3 ( z min ) ] } ) ,
P r peak ( z ) = P r 0 exp ( α r z + 3 A 0 2 g r 2 α s B 2 { T p 2 ( z ) T 0 2 ( z ) [ 1 4 α s 9 d T 0 ( z ) ] } ) ,
T 0 ( z ) = 3 d 2 α s W [ 2 α s 3 d T p ( z ) ] .
z min ( z , T ) = 3 α s d W { α s d 3 d exp [ α s 3 d ( T + z d ) ] } + 1 d ( T + z d ) .
P r 0 exp ( α r z 0 + 3 A 0 2 g r 2 α s B 2 { T p 2 ( z 0 ) T 0 2 ( z 0 ) [ 1 4 α s 9 d T 0 ( z 0 ) ] } ) = 2 α s g s ,
ψ s z = i γ s [ ψ s 2 + ( 2 f R ) ψ r 2 ] ψ s g s 2 ψ r 2 ψ s ,
ψ r z d ψ r T = i γ r [ ψ r 2 + ( 2 f R ) ψ s 2 ] ψ r + g r 2 ψ s 2 ψ r ,
ψ s z + i β 2 2 ψ s T 2 = i γ s ψ s 2 ψ s + α s 2 ψ s .
ψ s ( z , T ) 2 = d ψ s ( z 0 , T ) 2 L ( T ) H ( z , T ) ,
ψ r ( z , T ) 2 = d ψ r [ z 0 , T + d ( z z 0 ) ] 2 L [ T + ( z z 0 ) d ] H ( z , T ) ,
H ( z , T ) = F ( T ) + G [ T + d ( z z 0 ) ] ,
L ( T ) = exp { 1 d 0 T [ g r ψ s ( z 0 , T ) 2 + g s ψ r ( z 0 , T ) 2 ] d T } ,
F ( T ) = g r 0 T ψ s ( z 0 , u ) 2 L ( u ) d u ,
G ( T ) = g s 0 T ψ r ( z 0 , u ) 2 L ( u ) d u .
ψ r z + i β 2 r 2 ψ r T 2 = i γ r ψ r 2 ψ r + α r 2 ψ r ,
ϕ c ( ω ) = a c ( ω ω 0 ) 2 ( 1 β c ω ω 0 ω 0 ) ,
z max d T p ( z max ) = T + z d ,
z min d + T p ( z min ) = T + z d .
z max d T p ( z max ) > z d T p ( z ) .
ϕ ( z , T ) = A 0 2 ( 3 2 α s [ exp ( 2 α s z 3 ) exp ( 2 α s a 3 ) ] + 1 3 d B 2 { T 3 [ T + ( z a ) d ] 3 } ) ,
z min ( z , T ) = z + T d 3 α s W { α s B 3 d exp [ α s ( T + z d ) 3 d ] } .
z min ( z , T 0 ) = z 2 3 α s W [ α s B 3 d exp ( α s z 6 ) ] ,
ϕ ( z , T ) = A 0 2 { 3 2 α s [ exp ( 2 α s z 3 ) 1 ] + 1 3 d B 2 [ T 3 ( T + z d ) 3 ] } ,
ϕ ( z , T ) = A 0 2 B 2 { 3 2 α s [ T p 2 ( z ) T p 2 ( z min ) ] + 1 3 d [ T 3 T p 3 ( z min ) ] } .
T 0 = T p [ z min ( z , T 0 ) ] = 1 2 d [ z z min ( z , T 0 ) ] .
T 0 = 3 d 2 α s W [ 2 α s 3 d T p ( z ) ] .
ϕ ( z , T 0 ) = 3 A 0 2 2 α s B 2 { T p 2 ( z ) T 0 2 ( z ) [ 1 4 α s 9 d T 0 ( z ) ] } .
ψ s y = i γ s d [ ψ s 2 + ( 2 f R ) ψ r 2 ] ψ s g s 2 d ψ r 2 ψ s ,
ψ r x = i γ r d [ ψ r 2 + ( 2 f R ) ψ s 2 ] ψ r g r 2 d ψ s 2 ψ r .
ψ s ( x , y ) = I s 1 2 ( x , y ) exp [ i θ s ( x , y ) ] ,
ψ r ( x , y ) = I r 1 2 ( x , y ) exp [ i θ r ( x , y ) ] ,
I s y + g s d I r I s = 0 ,
I r x + g r d I r I s = 0 ,
θ s y = γ s d [ I s + ( 2 f R ) I r ] ,
θ r x = γ r d ( I r + ( 2 f R ) I s ] .
I s ( x , y ) = I s 0 ( x ) L ( x ) d H ( x , y ) ,
I r ( x , y ) = I r 0 ( y ) L ( y ) d H ( x , y ) ,
H ( x , y ) = F ( x ) + G ( y ) ,
L ( u ) = exp { 1 d 0 u [ g r I s 0 ( v ) + g s I r 0 ( v ) ] d v } ,
F ( x ) = g r 0 x I s 0 ( u ) L ( u ) d u ,
G ( y ) = g s 0 y I r 0 ( u ) L ( u ) d u .
f ( W ) = W exp ( W ) .
W ( x ) exp [ W ( x ) ] = x .
W ( x ) = n = 1 ( n ) n 1 n ! x n .

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