Abstract

In a series of recent experiments investigating stimulated Raman scattering (SRS) in a multimode fiber, the evolution of the Stokes wave in the low-order modes was studied. In this paper we investigate the theory of SRS in a two-mode optical fiber in order to analyze the behavior of the Stokes modes under different pump and excitation conditions. We find that when the participating waves are phase matched a subtle nonlinear interaction occurs between the pump and Stokes modes. This interaction does not involve the stimulated four-photon mixing process. We also suggest a potential loss mechanism that is due to the scattering of Stokes light into higher-order leaky modes.

© 1993 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).
  2. P. L. Baldeck, F. Raccah, R. R. Alfano, “Observation of self-focusing in optical fibers with picosecond pulses,” Opt. Lett. 12, 588–589 (1987).
    [CrossRef] [PubMed]
  3. K. S. Chiang, “Stimulated Raman scattering in a multimode optical fiber: evolution of modes in Stokes waves,” Opt. Lett. 17, 352–354 (1992).
    [CrossRef] [PubMed]
  4. R. H. Stolen, E. P. Ippen, A. R. Tynes, “Raman oscillation in glass optical fibers,” Appl. Phys. Lett. 20, 62–64 (1972).
    [CrossRef]
  5. R. H. Stolen, E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
    [CrossRef]
  6. J. AuYeung, A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
    [CrossRef]
  7. F. Capasso, P. Di Porto, “Coupled mode theory of Raman amplification in lossless optical fibers,” J. Appl. Phys. 47, 1472–1476 (1976).
    [CrossRef]
  8. T. P. McLean, “Linear and nonlinear optics of condensed matter,” in Interaction of Radiation with Condensed Matter, lectures presented at Winter College Trieste, January 14–March 26, 1976 (International Atomic Energy Agency, Vienna, 1977), pp. 3–92.
  9. R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” in Progress in Quantum Electronics, J. H. Sanders, S. Stenholm, eds. (Pergamon, New York, 1979), Vol. 5.
    [CrossRef]
  10. R. H. Stolen, J. P. Gordon, W. J. Tomlinson, H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
    [CrossRef]
  11. S. J. Garth, “Small frequency shift stimulated four-photon mixing in optical fibers: optimum phase-matching conditions,” Appl. Opt. 31, 742–745 (1992).
    [CrossRef] [PubMed]
  12. E. A. Golovchenko, A. N. Pilipetskii, Sov. Lightwave Commun. 1, 271–275 (1991).
  13. S. Trillo, S. Wabnitz, “Parametric and Raman amplification in birefringent fibers,” J. Opt. Soc. Am. B 9, 1061–1082 (1992).
    [CrossRef]

1992

1991

E. A. Golovchenko, A. N. Pilipetskii, Sov. Lightwave Commun. 1, 271–275 (1991).

1989

1987

1978

J. AuYeung, A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
[CrossRef]

1976

F. Capasso, P. Di Porto, “Coupled mode theory of Raman amplification in lossless optical fibers,” J. Appl. Phys. 47, 1472–1476 (1976).
[CrossRef]

1973

R. H. Stolen, E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

1972

R. H. Stolen, E. P. Ippen, A. R. Tynes, “Raman oscillation in glass optical fibers,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

Alfano, R. R.

AuYeung, J.

J. AuYeung, A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
[CrossRef]

Baldeck, P. L.

Capasso, F.

F. Capasso, P. Di Porto, “Coupled mode theory of Raman amplification in lossless optical fibers,” J. Appl. Phys. 47, 1472–1476 (1976).
[CrossRef]

Chiang, K. S.

Di Porto, P.

F. Capasso, P. Di Porto, “Coupled mode theory of Raman amplification in lossless optical fibers,” J. Appl. Phys. 47, 1472–1476 (1976).
[CrossRef]

Garth, S. J.

Golovchenko, E. A.

E. A. Golovchenko, A. N. Pilipetskii, Sov. Lightwave Commun. 1, 271–275 (1991).

Gordon, J. P.

Haus, H. A.

Hellwarth, R. W.

R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” in Progress in Quantum Electronics, J. H. Sanders, S. Stenholm, eds. (Pergamon, New York, 1979), Vol. 5.
[CrossRef]

Ippen, E. P.

R. H. Stolen, E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

R. H. Stolen, E. P. Ippen, A. R. Tynes, “Raman oscillation in glass optical fibers,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

McLean, T. P.

T. P. McLean, “Linear and nonlinear optics of condensed matter,” in Interaction of Radiation with Condensed Matter, lectures presented at Winter College Trieste, January 14–March 26, 1976 (International Atomic Energy Agency, Vienna, 1977), pp. 3–92.

Pilipetskii, A. N.

E. A. Golovchenko, A. N. Pilipetskii, Sov. Lightwave Commun. 1, 271–275 (1991).

Raccah, F.

Stolen, R. H.

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

R. H. Stolen, E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

R. H. Stolen, E. P. Ippen, A. R. Tynes, “Raman oscillation in glass optical fibers,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

Tomlinson, W. J.

Trillo, S.

Tynes, A. R.

R. H. Stolen, E. P. Ippen, A. R. Tynes, “Raman oscillation in glass optical fibers,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

Wabnitz, S.

Yariv, A.

J. AuYeung, A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

R. H. Stolen, E. P. Ippen, A. R. Tynes, “Raman oscillation in glass optical fibers,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

R. H. Stolen, E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

IEEE J. Quantum Electron.

J. AuYeung, A. Yariv, “Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
[CrossRef]

J. Appl. Phys.

F. Capasso, P. Di Porto, “Coupled mode theory of Raman amplification in lossless optical fibers,” J. Appl. Phys. 47, 1472–1476 (1976).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Sov. Lightwave Commun.

E. A. Golovchenko, A. N. Pilipetskii, Sov. Lightwave Commun. 1, 271–275 (1991).

Other

T. P. McLean, “Linear and nonlinear optics of condensed matter,” in Interaction of Radiation with Condensed Matter, lectures presented at Winter College Trieste, January 14–March 26, 1976 (International Atomic Energy Agency, Vienna, 1977), pp. 3–92.

R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” in Progress in Quantum Electronics, J. H. Sanders, S. Stenholm, eds. (Pergamon, New York, 1979), Vol. 5.
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

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Figures (7)

Fig. 1
Fig. 1

Schematic showing pump and Stokes modes in a two-mode step-index fiber. Normalized frequency is defined as V = ρk(nco2ncl2)1/2, where ρ is the fiber core radius, k = 2π/λ, and nco and ncl are the refractive index values in the core and cladding of the fiber, respectively. Note that the fiber is single mode for V < 2.4. The normalized propagation constant is defined as (β2/k2ncl2)/(nco2ncl2). The frequency shift between pump and Stokes waves is exaggerated for clarity.

Fig. 2
Fig. 2

Phase mismatch B [Eq. (8g)] versus normalized frequency for a two-mode step-index fiber. Fiber parameters are ρ = 4 μm and nco2ncl2 = 0.0147, so that Vλp = 3.07 μm. The fiber is designed so that it is single mode at λp = 1.32 μm (V = 2.32). Dispersion properties of the refractive index are calculated by assuming a GeO2-doped silica core and pure silica cladding. Note that B ≈ 0 for V = 3(λp = 1.02 μm). These fiber parameters are used for all the numerical examples in this paper.

Fig. 3
Fig. 3

Normalized overlap integrals ρ2cij versus normalized frequency for the fiber of Fig. 2. The overlap integrals are defined by Eq. (8f) and are evaluated numerically for the case of a step-profile fiber.

Fig. 4
Fig. 4

Normalized gain coefficients gl versus pump power ratio P2/P1 at normalized frequency V = 3 and V = 5. gl is defined by Eq. (10c) and is given by gl = (c1l + c2lP2/P1). The solid lines represent the gain of the fundamental Stokes mode, and the dotted lines represent the gain of the second-order modes.

Fig. 5
Fig. 5

Stokes power S1 and S2 versus distance in the nondepleted-pump approximation for different phase-matching conditions: (a) θ → ∞ [see Eq. (13)]; (b) θ = 0.02 m−1; (c) θ = 0.005; (d) θ = 0 [see Eqs. (14)]; (e) θ = −0.005 m−1; and (f) θ = −0.02 m−1. The fiber is operating at λp = 1.02 μm (V = 3), and phase mismatch θ is given by Eq. (11). Initial powers are P1(0) = P2(0) = 5 W and S1(0) = S2(0) = 0.1 W Labels 1 and 2 represent mode 1 and mode 2, respectively.

Fig. 6
Fig. 6

Stokes power S1 and S2 (solid curves) and pump power P1 and P2 (dotted curves) versus distance for the depleted pump case. The phase-matching conditions are the same as in the corresponding diagrams in Fig. 5, where B = θ − (α1α2) [see Eq. (11)]. The dashed curves shown in (c) and (e) are the nondepleted-pump approximations.

Fig. 7
Fig. 7

Phase mismatches defined by Eq. (A4) plotted against normalized frequency, V, for the case of Raman scattering and four-wave mixing in a two-mode step-index fiber. Fiber parameters are as in Fig. 2.

Equations (47)

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E = E p 1 + E p 2 + E s 1 + E s 2 ,
E q l = 1 2 ( η N l ) 1 / 2 a q l ( z ) ψ l ( x , y ) exp [ i ( β l ( ω q ) z ω q t ) ] + c . c . ,
N l = ψ l 2 d A , η = 1 2 n co 0 c ,
Q l = | a q l | 2 .
l β l ( ω q ) d a q l d z ψ l exp [ i β l ( ω q ) z ] = i k q 2 P ( 3 ) ( ω q ) ,
P ( 3 ) ( ω p ) = 3 χ ( ω p , ω p , ω p ) ( | E p 1 | 2 E p 1 + | E p 2 | 2 E p 2 + 2 | E p 2 | 2 E p 1 + 2 | E p 1 | 2 E p 2 + E p 1 2 E p 2 * + E p 2 2 E p 1 * ) + 6 χ ( ω s , ω s , ω p ) ( | E s 1 | 2 + | E s 2 | 2 + E s 1 E s 2 * + E s 1 * E s 2 ) ( E p 1 + E p 2 ) .
P ( 3 ) ( ω s ) = [ P ( 3 ) ( ω p ) with p s ] .
χ ( ω p , ω p , ω p ) = χ ( ω s , ω s , ω s ) = χ , χ ( ω s , ω s , ω p ) = χ * ( ω p , ω p , ω s ) = χ + i χ R ,
exp { ± i [ β 1 ( ω p ) β 2 ( ω p ) ] z } , exp { ± i [ β 1 ( ω s ) β 2 ( ω s ) ] z } ,
exp { ± i [ β 1 ( ω p ) β 2 ( ω p ) + β 1 ( ω s ) β 2 ( ω s ) ] z } ,
d a p 1 d z = i k p n 2 ( c 11 | a p 1 | 2 a p 1 + 2 c 12 | a p 2 | 2 a p 1 ) + 2 i k p ( n 2 + i n 2 R ) [ c 11 | a s 1 | 2 a p 1 + c 12 | a s 2 | 2 a p 1 + c 12 a s 1 a s 2 * a p 2 exp ( i B z ) ] ,
d a p 2 d z = i k p n 2 ( c 22 | a p 2 | 2 a p 2 + 2 c 12 | a p 1 | 2 a p 2 ) + 2 i k p ( n 2 + i n 2 R ) [ c 12 | a s 1 | 2 a p 2 + c 22 | a s 2 | 2 a p 2 + c 12 a s 1 * a s 2 a p 1 exp ( i B z ) ] ,
d a s 1 d z = i k s n 2 ( c 11 | a s 1 | 2 a s 1 + 2 c 12 | a s 2 | 2 a s 1 ) + 2 i k s ( n 2 i n 2 R ) [ c 11 | a p 1 | 2 a s 1 + c 12 | a p 2 | 2 a s 1 + c 12 a p 1 a p 2 * a s 2 exp ( i B z ) ] ,
d a s 2 d z = i k s n 2 ( c 22 | a s 2 | 2 a s 2 + 2 c 12 | a s 1 | 2 a s 2 ) + 2 i k s ( n 2 i n 2 R ) [ c 12 | a p 1 | 2 a s 2 + c 22 | a p 2 | 2 a s 2 + c 12 a p 1 * a p 2 a s 1 exp ( i B z ) ] ,
n 2 = 3 4 χ c 0 n , n 2 R = 3 4 χ R c 0 n ,
c i j = ψ i 2 ψ j 2 d A N i N j ,
B = β 1 ( ω p ) β 2 ( ω p ) β 1 ( ω s ) + β 2 ( ω s ) .
d a p 1 d z = i k p n 2 ( c 11 P 1 + 2 c 12 P 2 ) a p 1 = i α 1 a p 1 ,
d a p 2 d z = i k p n 2 ( c 22 P 2 + 2 c 12 P 1 ) a p 2 = i α 2 a p 2 ,
d a s 1 d z = 2 i k s ( n 2 i n 2 R ) [ ( c 11 P 1 + c 12 P 2 ) a s 1 + c 12 a p 1 a p 2 * a s 2 exp ( i B z ) ] ,
d a s 2 d z = 2 i k s ( n 2 i n 2 R ) [ ( c 12 P 1 + c 22 P 2 ) a s 2 + c 12 a p 1 * a p 2 a s 1 exp ( i B z ) ] .
d a s 1 d z = 2 ( i + R ) [ γ 1 a s 1 + 1 2 P 12 a s 2 exp ( i θ z ) ] ,
d a s 2 d z = 2 ( i + R ) [ γ 2 a s 2 + 1 2 P 12 a s 1 exp ( i θ z ) ] ,
R = n 2 R n 2 , γ 1 = k s n 2 ( c 11 P 1 + c 12 P 2 ) , γ 2 = k s n 2 ( c 12 P 1 + c 22 P 2 ) , P 12 = 2 k s n 2 c 12 ( P 1 P 2 ) 1 / 2 ,
θ = B + α 1 α 2 ,
a s 1 ( z ) = a ˜ s 1 ( z ) exp ( i θ z / 2 ) , a s 2 ( z ) = a ˜ s 2 ( z ) exp ( i θ z / 2 ) .
a s l ( z ) = b l + e λ + z + b l e λ z , l = 1 , 2 ,
λ ± = ( i + R ) ( γ 1 + γ 2 ) ± { ( i + R ) 2 [ ( γ 1 γ 2 ) 2 + P 12 2 ] i θ ( i + R ) ( γ 1 γ 2 ) θ 2 / 4 } 1 / 2
S l ( z ) = S l ( 0 ) exp ( 4 R γ l z ) ,
λ ± = ( i + R ) { γ 1 + γ 2 ± [ ( γ 1 γ 2 ) + P 12 2 ] 1 / 2 } ,
b 1 ± = 1 2 { 1 ± γ 1 γ 2 + P 12 [ ( γ 1 γ 2 ) 2 + P 12 2 ] 1 / 2 } , b 2 ± = 1 2 { 1 ± γ 2 γ 1 + P 12 [ ( γ 1 γ 2 ) 2 + P 12 2 ] 1 / 2 } ,
a s 1 ( 0 ) = a s 2 ( 0 ) = .
a q l = ( Q l ) 1 / 2 exp ( i ϕ q l ) ,
d P 1 d z = 4 k p n 2 R [ c 11 S 1 P 1 + c 12 S 2 P 1 + c 12 ( S 1 S 2 P 1 P 2 ) 1 / 2 cos θ ] + 4 k p n 2 c 12 ( S 1 S 2 P 1 P 2 ) 1 / 2 sin θ ,
d P 2 d z = 4 k p n 2 R [ c 12 S 1 P 2 + c 22 S 2 P 2 + c 12 ( S 1 S 2 P 1 P 2 ) 1 / 2 cos θ ] 4 k p n 2 c 12 ( S 1 S 2 P 1 P 2 ) 1 / 2 sin θ ,
d S 1 d z = 4 k s n 2 R [ c 11 P 1 S 1 + c 12 P 2 S 1 + c 12 ( S 1 S 2 P 1 P 2 ) 1 / 2 cos θ ] 4 k s n 2 c 12 ( S 1 S 2 P 1 P 2 ) 1 / 2 sin θ ,
d S 2 d z = 4 k s n 2 R [ c 12 P 1 S 2 + c 22 P 2 S 2 + c 12 ( S 1 S 2 P 1 P 2 ) 1 / 2 cos θ ] + 4 k s n 2 c 12 ( S 1 S 2 P 1 P 2 ) 1 / 2 sin θ .
d θ d z = B + d d z ( ϕ p 1 ϕ s 1 ϕ p 2 ϕ s 2 ) = B + k p n 2 { c 11 P 1 + 2 c 12 P 2 + 2 c 11 S 1 + 2 c 12 S 2 c 22 P 2 2 c 12 P 1 2 c 12 S 1 2 c 22 S 2 + 2 c 12 cos θ ( S 1 S 2 ) 1 / 2 [ ( P 2 P 1 ) 1 / 2 ( P 1 P 2 ) 1 / 2 ] } + k s n 2 { c 22 S 2 + 2 c 12 S 1 + 2 c 12 P 1 + 2 c 22 P 2 c 11 S 1 2 c 12 S 2 2 c 11 P 1 2 c 12 P 2 + 2 c 12 cos θ ( P 1 P 2 ) 1 / 2 [ ( S 1 S 2 ) 1 / 2 ( S 2 S 1 ) 1 / 2 ] } + 2 n 2 R c 12 sin θ { k p ( S 1 S 2 ) 1 / 2 [ ( P 2 P 1 ) 1 / 2 + ( P 1 P 2 ) 1 / 2 ] k s ( P 1 P 2 ) 1 / 2 [ ( S 1 S 2 ) 1 / 2 + ( S 2 S 1 ) 1 / 2 ] } .
P 1 k p + P 2 k p + S 1 k s + S 2 k s = 0 ,
P 1 = 4 k p n 2 R ( c 11 S 1 P 1 + c 12 S 2 P 1 ) , S 1 = 4 k s n 2 R c 11 P 1 S 1 , S 2 = 4 k s n 2 R c 12 P 1 S 2 α S 2 .
P 1 ( z ) k p + S 1 ( z ) k s + S 2 ( z ) k s = P 1 ( 0 ) k p α k s S 2 d z ,
S 1 ( z ) = S 1 ( 0 ) exp ( 4 k s n 2 R c 11 P 1 z ) , S 2 ( z ) = S 2 ( 0 ) exp ( 4 k s n 2 R c 12 P 1 α ) z ,
E = E p 1 + E p 2 + E s 1 + E s 2 + E a 1 + E a 2 ,
Ω = ω p ω s = ω a ω p ,
χ ( ω p , ω p , ω p ) = χ ( ω s , ω s , ω s ) = χ ( ω a , ω a , ω a ) = χ ( ω a , ω s , ω p ) = χ ( ω a , ω s , ω s ) = χ ( ω s , ω a , ω a ) = χ , χ ( ω p , ω s , ω s ) = χ * ( ω p , ω a , ω a ) = χ ( ω p , ω p , ω s ) = χ * ( ω p , ω p , ω a ) = χ ( ω a , ω p , ω p ) = χ * ( ω s , ω p , ω p ) = χ + i χ R .
Δ k 1 = β p 1 β p 2 β s 1 + β s 2 = B , Δ k 2 = β p 1 β p 2 + β s 1 + β s 2 , Δ k 3 = β p 1 β p 2 β a 1 + β a 2 , Δ k 4 = β p 1 β p 2 + β a 1 β a 2 , Δ k 5 = β p 1 + β p 2 β a 1 β s 2 , Δ k 6 = β p 1 + β p 2 β a 2 β s 1 , Δ k 7 = 2 β p 1 β a 1 β s 1 , Δ k 8 = 2 β p 1 β a 2 β s 2 ,
Δ k 1 Ω ( β 1 β 2 ) 1 2 Ω 2 ( β 1 β 2 ) , Δ k 6 Ω ( β 1 β 2 ) 1 2 Ω 2 ( β 1 + β 2 ) ,

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