Abstract

Coupled nonlinear equations that describe the nonlinear process of stimulated Raman scattering in optical fibers are derived. These equations account in a unified manner for the Raman amplification, the Stokes generation, the induced self-frequency shift, and the interpulse stimulated Raman-scattering-induced cross-frequency shift. The equations reduce to a well-known form for relatively wide picosecond pump pulses. Using these equations, we show theoretically that the effects of cross-phase modulation, self-frequency shift, and cross-frequency shift cause two optical pulses copropagating in the anomalous dispersion regime of the fiber to shed some of their energy and evolve into a narrower soliton, which has a higher frequency shift than a single propagating soliton. It is also shown that the self-frequency shift of femtosecond pulses is detrimental to Raman generation. As the input pulse width is reduced, the spectrum of the pulse shifts by an amount comparable with the Raman shift. The shift increases continuously with propagation at such a rapid rate that the Stokes pulse has no time to build up from noise to significant energy levels.

© 1996 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, California, 1995).
  2. K. X. Liu and E. Garmire, “Understanding the formation of the SRS Stokes spectrum in fused silica fibers,” IEEE J. Quantum Electron. 27, 1022–1030 (1991).
    [Crossref]
  3. V. A. Aleshkevich, G. D. Kozhoride, and M. V. Shamonin, “Generation of Stokes stimulated Raman scattering pulses from spontaneous noise in fiber lightguides,” J. Commun. Technol. Electron. 38, 104–109 (1993).
  4. C. Headley and G. P. Agrawal, “Noise characteristics of picosecond Stokes pulses generated in optical fibers through stimulated Raman scattering,” IEEE J. Quantum Electron. 31, 2058–2067 (1995).
    [Crossref]
  5. C. Headley and G. P. Agrawal, “Simultaneous amplification and compression of picosecond optical pulses during Raman amplification in optical fibers,” J. Opt. Soc. Am. B 10, 2383–2389 (1993).
    [Crossref]
  6. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [Crossref] [PubMed]
  7. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–3673 (1989).
    [Crossref]
  8. S. Kumar, A. Selvarajan, and G. V. Anand, “Influence of Raman scattering on the cross phase modulation in optical fibers,” Opt. Commun. 102, 329–335 (1993).
    [Crossref]
  9. 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]
  10. C. Headley, “Ultrafast stimulated Raman scattering in optical fibers,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1995).
  11. A. Höök, “Influence of stimulated Raman scattering on cross-phase modulation between waves in optical fibers,” Opt. Lett. 17, 115–117 (1992).
    [Crossref] [PubMed]
  12. G. P. Agrawal, “Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers,” Opt. Lett. 15, 224–226 (1990).
    [Crossref] [PubMed]

1995 (1)

C. Headley and G. P. Agrawal, “Noise characteristics of picosecond Stokes pulses generated in optical fibers through stimulated Raman scattering,” IEEE J. Quantum Electron. 31, 2058–2067 (1995).
[Crossref]

1993 (3)

C. Headley and G. P. Agrawal, “Simultaneous amplification and compression of picosecond optical pulses during Raman amplification in optical fibers,” J. Opt. Soc. Am. B 10, 2383–2389 (1993).
[Crossref]

V. A. Aleshkevich, G. D. Kozhoride, and M. V. Shamonin, “Generation of Stokes stimulated Raman scattering pulses from spontaneous noise in fiber lightguides,” J. Commun. Technol. Electron. 38, 104–109 (1993).

S. Kumar, A. Selvarajan, and G. V. Anand, “Influence of Raman scattering on the cross phase modulation in optical fibers,” Opt. Commun. 102, 329–335 (1993).
[Crossref]

1992 (1)

1991 (1)

K. X. Liu and E. Garmire, “Understanding the formation of the SRS Stokes spectrum in fused silica fibers,” IEEE J. Quantum Electron. 27, 1022–1030 (1991).
[Crossref]

1990 (1)

1989 (2)

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]

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–3673 (1989).
[Crossref]

1986 (1)

Agrawal, G. P.

C. Headley and G. P. Agrawal, “Noise characteristics of picosecond Stokes pulses generated in optical fibers through stimulated Raman scattering,” IEEE J. Quantum Electron. 31, 2058–2067 (1995).
[Crossref]

C. Headley and G. P. Agrawal, “Simultaneous amplification and compression of picosecond optical pulses during Raman amplification in optical fibers,” J. Opt. Soc. Am. B 10, 2383–2389 (1993).
[Crossref]

G. P. Agrawal, “Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers,” Opt. Lett. 15, 224–226 (1990).
[Crossref] [PubMed]

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, California, 1995).

Aleshkevich, V. A.

V. A. Aleshkevich, G. D. Kozhoride, and M. V. Shamonin, “Generation of Stokes stimulated Raman scattering pulses from spontaneous noise in fiber lightguides,” J. Commun. Technol. Electron. 38, 104–109 (1993).

Anand, G. V.

S. Kumar, A. Selvarajan, and G. V. Anand, “Influence of Raman scattering on the cross phase modulation in optical fibers,” Opt. Commun. 102, 329–335 (1993).
[Crossref]

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–3673 (1989).
[Crossref]

Garmire, E.

K. X. Liu and E. Garmire, “Understanding the formation of the SRS Stokes spectrum in fused silica fibers,” IEEE J. Quantum Electron. 27, 1022–1030 (1991).
[Crossref]

Gordon, J. P.

Haus, H. A.

Headley, C.

C. Headley and G. P. Agrawal, “Noise characteristics of picosecond Stokes pulses generated in optical fibers through stimulated Raman scattering,” IEEE J. Quantum Electron. 31, 2058–2067 (1995).
[Crossref]

C. Headley and G. P. Agrawal, “Simultaneous amplification and compression of picosecond optical pulses during Raman amplification in optical fibers,” J. Opt. Soc. Am. B 10, 2383–2389 (1993).
[Crossref]

C. Headley, “Ultrafast stimulated Raman scattering in optical fibers,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1995).

Höök, A.

Kozhoride, G. D.

V. A. Aleshkevich, G. D. Kozhoride, and M. V. Shamonin, “Generation of Stokes stimulated Raman scattering pulses from spontaneous noise in fiber lightguides,” J. Commun. Technol. Electron. 38, 104–109 (1993).

Kumar, S.

S. Kumar, A. Selvarajan, and G. V. Anand, “Influence of Raman scattering on the cross phase modulation in optical fibers,” Opt. Commun. 102, 329–335 (1993).
[Crossref]

Liu, K. X.

K. X. Liu and E. Garmire, “Understanding the formation of the SRS Stokes spectrum in fused silica fibers,” IEEE J. Quantum Electron. 27, 1022–1030 (1991).
[Crossref]

Selvarajan, A.

S. Kumar, A. Selvarajan, and G. V. Anand, “Influence of Raman scattering on the cross phase modulation in optical fibers,” Opt. Commun. 102, 329–335 (1993).
[Crossref]

Shamonin, M. V.

V. A. Aleshkevich, G. D. Kozhoride, and M. V. Shamonin, “Generation of Stokes stimulated Raman scattering pulses from spontaneous noise in fiber lightguides,” J. Commun. Technol. Electron. 38, 104–109 (1993).

Stolen, R. H.

Tomlinson, W. J.

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–3673 (1989).
[Crossref]

IEEE J. Quantum Electron. (3)

K. X. Liu and E. Garmire, “Understanding the formation of the SRS Stokes spectrum in fused silica fibers,” IEEE J. Quantum Electron. 27, 1022–1030 (1991).
[Crossref]

C. Headley and G. P. Agrawal, “Noise characteristics of picosecond Stokes pulses generated in optical fibers through stimulated Raman scattering,” IEEE J. Quantum Electron. 31, 2058–2067 (1995).
[Crossref]

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–3673 (1989).
[Crossref]

J. Commun. Technol. Electron. (1)

V. A. Aleshkevich, G. D. Kozhoride, and M. V. Shamonin, “Generation of Stokes stimulated Raman scattering pulses from spontaneous noise in fiber lightguides,” J. Commun. Technol. Electron. 38, 104–109 (1993).

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

S. Kumar, A. Selvarajan, and G. V. Anand, “Influence of Raman scattering on the cross phase modulation in optical fibers,” Opt. Commun. 102, 329–335 (1993).
[Crossref]

Opt. Lett. (3)

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, California, 1995).

C. Headley, “Ultrafast stimulated Raman scattering in optical fibers,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1995).

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Figures (5)

Fig. 1
Fig. 1

Imaginary (solid curve) and real (dashed curve) parts of the Raman response function. The imaginary portion h r ( ω ) is proportional to Raman gain, and the real part h r ( ω ) is proportional to Raman-induced index changes.

Fig. 2
Fig. 2

Change in the mean frequency as a function of distance for (i) a fundamental soliton, (ii) two copropagating fundamental solitons with no interpulse SRS, (iii) two copropagating fundamental solitons with no interpulse SRS and no XPM, and (iv) two copropagating fundamental solitons. Parameters are tp = ts = 100 fs, β2p = −5 ps2 km−1, β2s = −5.35 ps2 km−1, γp = 3.50 km−1 W−1, γs = 3.27 km−1 W−1, P0p = 143 W, and P0s = 164 W.

Fig. 3
Fig. 3

(a) Shapes and (b) spectra of the Stokes pulse, showing the effect of intrapulse stimulated Raman scattering when the Stokes pulse copropagates with a pump pulse. Input powers for both pulses correspond to a fundamental soliton.

Fig. 4
Fig. 4

(a) Shapes and (b) spectra showing the growth of the Stokes pulse from noise during the propagation of a sixth-order soliton with parameters tp = ts = 1 ps, β2p = −5 ps2 km−1, β2s = −5.35 ps2 km−1, γp = 3.50 km−1 W−1, γs = 3.27 km−1 W−1, gp = 1.75 km−1 W−1, and gs = 1.63 km−1 W−1.

Fig. 5
Fig. 5

Frequency shift plotted as a function of the pulse width experienced by a sixth-order soliton over one dispersion length.

Tables (1)

Tables Icon

Table 1 Pulse Width, Dispersion Length, and Input Pump Power for N = 6 Soliton with gp = 1.75 (km W)−1 and β2p = −5 ps2/km

Equations (30)

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P ( r , t ) = P L ( r , t ) + P N ( r , t ) + P K ( r , t ) + P R ( r , t ) = 0 χ L ( t t ) E ( r , t ) d t + 0 E ( r , t ) × R N ( t t ) F N ( t ) d t + χ K E ( r , t ) · E ( r , t ) + χ R ( t t ) × E ( r , t ) · E ( r , t ) d t .
χ N ( t ) = R N ( t t ) F N ( t ) d t ,
E ( r , t ) = ½ x ˆ [ E p ( r , t ) exp ( i ω p t ) + E s ( r , t ) × exp ( i ω s t ) + c .c . ] ,
P R ( r , t ) = ½ x ˆ [ P R p ( r , t ) exp ( i ω p t ) + P R s ( r , t ) × exp ( i ω s t ) + c .c . ] .
F N ( z , t ) = ½ x ˆ [ f N ( z , t ) exp ( i Ω R t ) + c .c . ] ,
E · E = ½ | E p | 2 + ½ | E s | 2 + ½ E p E s * × exp [ i ( ω p ω s ) t ] + ½ E p * E s × exp [ i ( ω p ω s ) t ] + ½ E p E s × exp [ i ( ω p ω s ) t ] + ½ E p * E s * × exp [ i ( ω p ω s ) t ] + ¼ E p 2   exp ( i 2 ω p t ) + ¼ E p * 2   exp ( i 2 ω p t ) + ¼ E s 2   exp ( i 2 ω s t ) + ¼ E s * 2   exp ( i 2 ω s t ) .
P R j ( r , t ) = 1 / 4 0 E j ( t ) χ R ( t t ) [ | E j ( t ) | 2 + | E k ( t ) | 2 ] d t + 1 / 2 0 E k ( t ) × χ R ( t t ) E j ( t ) E k * ( t ) × exp [ i ( ω j ω k ) ( t t ) ] d t ,
2 E 1 c 2 2 E t 2 = μ 0 2 P t 2 ,
E j ( r , t ) = T j ( x , y ) A j ( z , t ) exp ( i β 0 j z ) ( j = p , s ) ,
u p z + 1 v g p u p t + i β 2 p 2 2 u p t 2 β 3 p 6 3 u p t 3 + α p 2 u p = i γ p ( 1 f R ) u p ( | u p | 2 + 2 | u s | 2 ) + i γ p f R u p × h r ( t t ) [ | u p ( t ) | 2 + | u s ( t ) | 2 ] d t + i γ p f R u s × h r ( t t ) u p ( t ) u s * ( t ) exp [ i Ω R ( t t ) ] d t + i u s h p ( t t ) f N ( z , t ) exp [ i Ω R ( t t ) ] d t
u s z + 1 v g s u s t + i β 2 s 2 2 u s t 2 β 3 s 6 3 u s t 3 + α s 2 u s = i γ s ( 1 f R ) u s ( | u s | 2 + 2 | u p | 2 ) + i γ s f R u s × h r ( t t ) [ | u s ( t ) | 2 + | u p ( t ) | 2 ] d t + i γ s f R u p × h r ( t t ) u s ( t ) u p * ( t ) × exp [ i Ω R ( t t ) ] d t + i u p h s ( t t ) f N * ( z , t ) × exp [ i Ω R ( t t ) ] d t .
u j = κ A j T 2 ( x , y ) d x d y 1 / 2 , κ 2 = ½ n j ( 0 / μ 0 ) 1 / 2 .
γ j ω j n 2 c A eff   κ 2 , n 2 = 3 8 n j χ K 1 + 2 χ 0 3 χ K , χ R ( t ) = χ 0 h r ( t ) , A eff = T 2 d x d y 2 T 4 d x d y ,
f R = 1 + 3 χ K 2 χ 0 1 .
h j ( t ) = ω j R N ( t ) 4 c n j ( j = p , s ) .
j ( t ) = i γ j f R u j h r ( t t ) [ | u j ( t ) | 2 + | u k ( t ) | 2 ] d t + i γ j f R u k h r ( t t ) u j ( t ) u k * ( t ) × exp [ i Ω R ( t t ) ] d t ,
j ( t ) = i γ j f R u j ( t ) [ | u j ( t ) | 2 + | u k ( t ) | 2 ] h r ( t t ) d t + i γ j f R u j ( t ) | u k ( t ) | 2 h r ( t t ) × exp [ i Ω R ( t t ) ] d t .
( t ) = i γ j f R u j ( t ) [ | u j ( t ) | 2 + | u k ( t ) | 2 ] γ j f ˜ R h r ( Ω R ) u j ( t ) | u k ( t ) | 2 ,
g j = 2 f R γ j | h ˜ r ( Ω R ) | .
u p z + 1 v g p u p t + i β 2 p 2 2 u p t 2 + α p 2 u p = i γ p u p [ | u p | 2 + ( 2 f R ) | u s | 2 ] g p 2 u p | u s | 2 ,
u s z + 1 v g s u s t + i β 2 s 2 2 u s t 2 + α s 2 u s = i γ s u s [ | u s | 2 + ( 2 f R ) | u p | 2 ] g s 2 u s | u p | 2 ,
q = g p γ p = 2 f R h ˜ r ( Ω R ) = 0.50 ,
u p z + 1 v g p u p t + i β 2 p 2 2 u p t 2 = i γ p u p [ | u p | 2 + ( 2 f R ) | u s | 2 + i 1 / 2 u p u s g p ( t t ) × u s * ( t ) exp [ i Ω R ( t t ) ] d t + i u s h p × ( t t ) f N ( z , t ) exp [ i Ω R ( t t ) ] d t ,
u s z + 1 v g s u s t + i β 2 s 2 2 u s t 2 = i γ s u s [ | u s | 2 + ( 2 f R ) | u p | 2 ] + i 1 / 2 | u p | 2 × g s ( t t ) u s ( t ) exp [ i Ω R ( t t ) ] d t + i u p × h s ( t t ) f N * ( z , t ) exp [ i Ω R ( t t ) ] d t .
˜ j ( ω ) = i γ j f R h ˜ r ( ω 1 ) u ˜ j ( ω ω 1 ) × u ˜ j ( ω 1 ω 2 ) u ˜ j * ( ω 2 ) d ω 2 d ω 1 × i γ j f R h ˜ r ( ω 1 ) u ˜ j ( ω ω 1 ) × u ˜ k ( ω 1 ω 2 ) u ˜ k * ( ω 2 ) d ω 2 d ω 1 × i γ j f R h ˜ r ( ω 1 Ω R ) u ˜ j ( ω 1 ω 2 ) × u ˜ k ( ω 1 ω 2 ) u ˜ k * ( ω 2 ) d ω 2 d ω 1 ,
ν j , avg = ν | u ˜ j ( z , ν ) | 2 d ν | u ˜ j ( z , ν ) | 2 d ν ( j = p , s ) ,
N 2 = γ p P 0 p t p 2 | β 2 p | = γ s P 0 s t s 2 | β 2 s | .
Z 0 = π 2 t p 2 | β 2 p | .
L D = t p 2 | β 2 p | , L W = t p | d | ,
g p P th L w 16.

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