Abstract

The temporal behavior of stimulated Brillouin scattering (SBS) in single-mode optical fibers is investigated theoretically and experimentally. It is shown that if external feedback exists, the SBS and the transmitted pump intensities exhibit steady oscillations with a period of twice the transit time in the fiber. However, if the ratio between the SBS intensity and the input intensity is above a certain value, the oscillations decay.

© 1985 Optical Society of America

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References

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  1. D. Cotter, “Stimulated Brillouin scattering in monomode optical fibers,” Opt. Commun. 4, 10–19 (1983).
  2. K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “CW Brillouin laser,” Appl. Phys. Lett. 28, 608–609 (1976).
    [Crossref]
  3. D. Cotter, “Observation of stimulated Brillouin scattering in low loss silica fiber at 1.3 μ m,” Electron. Lett. 18, 495–496 (1983).
    [Crossref]
  4. L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-fiber stimulated Brillouin ring laser with submilliwatt pump threshold,” Opt. Lett. 7, 509–511 (1982).
    [Crossref] [PubMed]
  5. R. V. Johnson and J. H. Marburger, “Relaxation oscillation in stimulated Raman and Brillouin scattering,” Phys. Rev. A 4, 1175–1182 (1971).
    [Crossref]
  6. E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
    [Crossref]
  7. B. S. Kawasaki, D. C. Johnson, Y. Fujii, and K. O. Hill, “Bandwidth-limited operation of a mode-locked Brillouin parametric oscillator,” Appl. Phys. Lett. 32, 429–431 (1978).
    [Crossref]
  8. D. R. Ponikvar and S. Ezekiel, “Stabilized single-frequency stimulated Brillouin fiber ring laser,” Opt. Lett. 6, 398–400 (1981).
    [Crossref] [PubMed]
  9. W. E. Boyce and R. C. Diprima, Elementary Differential Equations and Boundary Value Problems, 2nd ed. (Wiley, New York, 1969).
  10. K. Boumgartel, U. Motschmann, and K. Sauer, “Self pulsing at stimulated scattering processes,” Opt. Commun. 51, 53–56 (1984).
    [Crossref]

1984 (1)

K. Boumgartel, U. Motschmann, and K. Sauer, “Self pulsing at stimulated scattering processes,” Opt. Commun. 51, 53–56 (1984).
[Crossref]

1983 (2)

D. Cotter, “Stimulated Brillouin scattering in monomode optical fibers,” Opt. Commun. 4, 10–19 (1983).

D. Cotter, “Observation of stimulated Brillouin scattering in low loss silica fiber at 1.3 μ m,” Electron. Lett. 18, 495–496 (1983).
[Crossref]

1982 (1)

1981 (1)

1978 (1)

B. S. Kawasaki, D. C. Johnson, Y. Fujii, and K. O. Hill, “Bandwidth-limited operation of a mode-locked Brillouin parametric oscillator,” Appl. Phys. Lett. 32, 429–431 (1978).
[Crossref]

1976 (1)

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “CW Brillouin laser,” Appl. Phys. Lett. 28, 608–609 (1976).
[Crossref]

1972 (1)

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

1971 (1)

R. V. Johnson and J. H. Marburger, “Relaxation oscillation in stimulated Raman and Brillouin scattering,” Phys. Rev. A 4, 1175–1182 (1971).
[Crossref]

Boumgartel, K.

K. Boumgartel, U. Motschmann, and K. Sauer, “Self pulsing at stimulated scattering processes,” Opt. Commun. 51, 53–56 (1984).
[Crossref]

Boyce, W. E.

W. E. Boyce and R. C. Diprima, Elementary Differential Equations and Boundary Value Problems, 2nd ed. (Wiley, New York, 1969).

Chodorow, M.

Cotter, D.

D. Cotter, “Stimulated Brillouin scattering in monomode optical fibers,” Opt. Commun. 4, 10–19 (1983).

D. Cotter, “Observation of stimulated Brillouin scattering in low loss silica fiber at 1.3 μ m,” Electron. Lett. 18, 495–496 (1983).
[Crossref]

Diprima, R. C.

W. E. Boyce and R. C. Diprima, Elementary Differential Equations and Boundary Value Problems, 2nd ed. (Wiley, New York, 1969).

Ezekiel, S.

Fujii, Y.

B. S. Kawasaki, D. C. Johnson, Y. Fujii, and K. O. Hill, “Bandwidth-limited operation of a mode-locked Brillouin parametric oscillator,” Appl. Phys. Lett. 32, 429–431 (1978).
[Crossref]

Hill, K. O.

B. S. Kawasaki, D. C. Johnson, Y. Fujii, and K. O. Hill, “Bandwidth-limited operation of a mode-locked Brillouin parametric oscillator,” Appl. Phys. Lett. 32, 429–431 (1978).
[Crossref]

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “CW Brillouin laser,” Appl. Phys. Lett. 28, 608–609 (1976).
[Crossref]

Ippen, E. P.

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

Johnson, D. C.

B. S. Kawasaki, D. C. Johnson, Y. Fujii, and K. O. Hill, “Bandwidth-limited operation of a mode-locked Brillouin parametric oscillator,” Appl. Phys. Lett. 32, 429–431 (1978).
[Crossref]

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “CW Brillouin laser,” Appl. Phys. Lett. 28, 608–609 (1976).
[Crossref]

Johnson, R. V.

R. V. Johnson and J. H. Marburger, “Relaxation oscillation in stimulated Raman and Brillouin scattering,” Phys. Rev. A 4, 1175–1182 (1971).
[Crossref]

Kawasaki, B. S.

B. S. Kawasaki, D. C. Johnson, Y. Fujii, and K. O. Hill, “Bandwidth-limited operation of a mode-locked Brillouin parametric oscillator,” Appl. Phys. Lett. 32, 429–431 (1978).
[Crossref]

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “CW Brillouin laser,” Appl. Phys. Lett. 28, 608–609 (1976).
[Crossref]

Marburger, J. H.

R. V. Johnson and J. H. Marburger, “Relaxation oscillation in stimulated Raman and Brillouin scattering,” Phys. Rev. A 4, 1175–1182 (1971).
[Crossref]

Motschmann, U.

K. Boumgartel, U. Motschmann, and K. Sauer, “Self pulsing at stimulated scattering processes,” Opt. Commun. 51, 53–56 (1984).
[Crossref]

Ponikvar, D. R.

Sauer, K.

K. Boumgartel, U. Motschmann, and K. Sauer, “Self pulsing at stimulated scattering processes,” Opt. Commun. 51, 53–56 (1984).
[Crossref]

Shaw, H. J.

Stokes, L. F.

Stolen, R. H.

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

Appl. Phys. Lett. (3)

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “CW Brillouin laser,” Appl. Phys. Lett. 28, 608–609 (1976).
[Crossref]

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

B. S. Kawasaki, D. C. Johnson, Y. Fujii, and K. O. Hill, “Bandwidth-limited operation of a mode-locked Brillouin parametric oscillator,” Appl. Phys. Lett. 32, 429–431 (1978).
[Crossref]

Electron. Lett. (1)

D. Cotter, “Observation of stimulated Brillouin scattering in low loss silica fiber at 1.3 μ m,” Electron. Lett. 18, 495–496 (1983).
[Crossref]

Opt. Commun. (2)

D. Cotter, “Stimulated Brillouin scattering in monomode optical fibers,” Opt. Commun. 4, 10–19 (1983).

K. Boumgartel, U. Motschmann, and K. Sauer, “Self pulsing at stimulated scattering processes,” Opt. Commun. 51, 53–56 (1984).
[Crossref]

Opt. Lett. (2)

Phys. Rev. A (1)

R. V. Johnson and J. H. Marburger, “Relaxation oscillation in stimulated Raman and Brillouin scattering,” Phys. Rev. A 4, 1175–1182 (1971).
[Crossref]

Other (1)

W. E. Boyce and R. C. Diprima, Elementary Differential Equations and Boundary Value Problems, 2nd ed. (Wiley, New York, 1969).

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Figures (8)

Fig. 1
Fig. 1

A scheme of the system consisting of two mirrors, R1 and R2, and a single-mode fiber between them.

Fig. 2
Fig. 2

a, The normalized SBS intensity and b, the oscillation frequency at the transition from an unstable state (steady oscillations) to a stable state (relaxation oscillations), as a function of the gain factor β = glIs (0).

Fig. 3
Fig. 3

Numerical solutions for the normalized SBS intensity as a function of time for various feedback conditions, a, No feedback, β = 30 (relaxation oscillations); b, R = 5 × 10−6, β = 13 (steady oscillations); and c, R = 2 × 10−2, = 13 (relaxation oscillations). An absorption of αl = 0.15 was assumed.

Fig. 4
Fig. 4

Numerical solutions for the transmitted pump intensity as a function of time. a, b, and c correspond to Figs. 3a, 3b, and 3c, respectively.

Fig. 5
Fig. 5

A scheme of the experimental setup. AO, acousto-optic modulator, D1 and D2, detectors, BS1, BS2, and BS3, beam splitters; R1 and R2, mirrors; OSC, oscilloscope; RFSA, rf spectrum analyzer; SFP, scanning Fabry–Perot.

Fig. 6
Fig. 6

Oscilloscope traces of the SBS intensity as a function of time for various feedback conditions. (a) 2-MHz relaxation oscillations: R = 0, the input intensity is 400 mW, and b0 = 0.04. (b) 2-MHz steady oscillations: the reflectivity is very small, the input intensity is 350 mW, and b0 = 0.1. (c) relaxation oscillations: R = 2 × 10−2, the input intensity is 420 mW, and b0 = 0.65. The horizontal scale in (a) and (b) is 1 μsec/division, and in (c) it is 2 μsec/division.

Fig. 7
Fig. 7

Oscilloscope traces of the transmitted pump intensity as a function of time for the same conditions as in Fig. 6. (a) and (b) correspond to Figs. 6(b) and 6(c), respectively. The horizontal scale is 1 μsec/division.

Fig. 8
Fig. 8

Typical rf spectrum of the SBS intensity when feedback is present. The vertical scale is logarithmic (10 dB/division), and the horizontal scale is 2 MHz/division.

Equations (25)

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- B ζ + B τ = + g l I B - α l B ,
I ζ + I τ = - g l I B - α l I ,
B ( 1 , τ ) = R B ( 0 , τ - 1 ) ,
I ( 0 , τ ) = I in + R I ( 1 , τ - 1 ) ,
I s ζ - B s ζ = 0 ,
b s ( ζ ) = b 0 ( 1 - b 0 ) e - γ ζ 1 - b 0 e - γ ζ ,
i s ( ζ ) = 1 - b 0 + b s ( ζ ) .
b 0 = 1 - R e γ 1 - R ,
I s ( 0 ) = I in 1 - R 2 e γ .
b ( ζ , τ ) = b s ( ζ ) [ 1 + b ( ζ ) e λ τ + c . c . ] ,
i ( ζ , τ ) = i s ( ζ ) [ 1 + i ( ζ ) e λ τ + c . c . ] ,
b = b e λ ζ
i = i e λ ζ .
b ζ = 2 λ b - β i s i ,
i ζ = - β b s b .
2 i ζ 2 + ( β i s - 2 λ ) i ζ - β 2 b s i s i = 0.
u 2 2 i u 2 + u ( 1 + 2 λ γ - 1 1 - b 0 u ) i u - b 0 u ( 1 - b 0 u ) 2 i = 0.
i = n = 0 a n u n + r ,
i = C 1 ( δ - 1 + 1 1 - b 0 u ) 1 δ + C 2 u 1 - δ 1 - b 0 u ,
b = [ C 1 γ b 0 u δ ( 1 - b 0 u ) 2 + C 2 γ ( 1 - δ + b 0 δ u ) u 1 - δ ( 1 - b 0 u ) 2 ] 1 β b s ,
i ( 0 ) = R 1 - b 0 1 - b 0 e - γ i ( 1 ) ,
b ( 0 ) = b ( 1 ) .
R = 1 - b 0 e γ - b 0 .
[ t 3 ( δ s - s + 1 ) - s 3 e - γ ( δ t - t + 1 ) ] C 1 + δ ( t 3 - s 3 e - 2 ( γ - λ ) ) C 2 = 0 ,
b 0 ( t - s ) C 1 + [ t ( 1 - δ s ) - s ( 1 - δ t ) e 2 λ ] C 2 = 0 ,

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