Abstract

Stimulated Brillouin scattering (SBS), excited by two pump waves in single-mode fibers, is investigated both theoretically and experimentally. Steady-state calculations, supported by experimental results, show that the SBS gain depends on the ratio of the pump coherence length to the characteristic gain length of the SBS as well as on the ratio of the frequency separation between the two pump waves to the SBS linewidth. These dependences are fully analyzed by following the evolution of the pressure wave that is generated by the SBS interaction. The competition between four-wave mixing and SBS is also considered.

© 1987 Optical Society of America

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Corrections

E. Lichtman, A. A. Friesem, R. G. Waarts, and H. H. Yaffe, "Stimulated Brillouin scattering excited by two pump waves in single-mode fibers: erratum," J. Opt. Soc. Am. B 5, 259-259 (1988)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-5-2-259

References

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  1. R. H. Stolen, “Nonlinear properties of optical fibers,” in Optical Fiber Telecommunications, S. E. Miller and A. G. Chynoweth, eds. (Academic, New York, 1979), p. 125.
    [CrossRef]
  2. D. Cotter, “Stimulated Brillouin scattering in monomode optical fiber,” J. Opt. Commun. 4, 10 (1983).
  3. N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34 (1965).
    [CrossRef]
  4. C. L. Tang, “Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process,” J. Appl. Phys. 37, 2945 (1966).
    [CrossRef]
  5. D. Cotter, “Observation of stimulated Brillouin scattering in low loss silica fiber at 1.3 μ m,” Electron. Lett. 18, 495 (1983).
    [CrossRef]
  6. L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-fiber stimulated Brillouin ring laser with submilliwatt pump threshold,” Opt. Lett. 7, 509 (1982).
    [CrossRef] [PubMed]
  7. G. C. Valley, “A review of stimulated Brillouin scattering excited with a broad-band pump laser,” IEEE J. Quantum Electron. QE-22, 704 (1986).
    [CrossRef]
  8. Yu. E. D’yakov, “Excitation of stimulated light scattering by broad-band pumping,” JETP Lett. 11, 243 (1970).
  9. G. P. Dzhotyan, Yu E. D’yakov, I. G. Zubarev, A. B. Mironov, and S. I. Mikhailov, “Influence of the spectral width and statistics of a Stokes signal on the efficiency of stimulated Raman scattering of nonmonochromatic pump radiation,” Sov. J. Quantum Electron 7, 783 (1977).
    [CrossRef]
  10. P. Narum, M. D. Skeldon, and R. W. Boyd, “Effect of laser mode structure on stimulated Brillouin scattering,” IEEE J. Quantum Electron. QE-22, 2161 (1986).
    [CrossRef]
  11. D. Cotter, “Suppression of stimulated Brillouin scattering during transmission of high-power narrowband laser light in mono-mode fibre,” Electron. Lett. 18, 638 (1982).
    [CrossRef]
  12. M. Tsubokawa, S. Seikai, T. Nakashima, and N. Shibata, “Suppression of stimulated Brillouin scattering in a single-mode fibre by an acousto-optic modulator,” Electron. Lett. 22, 473 (1986).
    [CrossRef]
  13. P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986 (1979).
    [CrossRef]
  14. B. Ya. Zel’dovich and V. V. Shkunov, “Influence of the group velocity mismatch on reproduction of the pump spectrum under stimulated scattering conditions,” Sov. J. Quantum Electron. 8, 1505 (1978).
    [CrossRef]
  15. J. Pelous and R. Vacher, “Thermal Brillouin scattering measurements of the attenuation of longitudinal hypersounds in fused quarts,” Solid State Commun. 16, 279 (1975).
    [CrossRef]
  16. K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098 (1978).
    [CrossRef]
  17. R. G. Smith, “Optical power handling of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489 (1972).
    [CrossRef] [PubMed]
  18. K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “Cw Brillouin laser,” Appl. Phys. Lett. 28, 608 (1976).
    [CrossRef]
  19. I. Bar-Joseph, A. A. Friesem, E. Lichtman, and R. G. Waarts, “Steady and relaxation oscillations of stimulated Brillouin scattering in single-mode fibers,” J. Opt. Soc. Am. B 2, 1606 (1985).
    [CrossRef]
  20. H. H. Yaffe, R. G. Waarts, E. Lichtman, and A. A. Friesem, “Multiple wave generation due to four wave mixing in a single-mode fibre,” Electron. Lett. 23, 42 (1987).
    [CrossRef]

1987 (1)

H. H. Yaffe, R. G. Waarts, E. Lichtman, and A. A. Friesem, “Multiple wave generation due to four wave mixing in a single-mode fibre,” Electron. Lett. 23, 42 (1987).
[CrossRef]

1986 (3)

G. C. Valley, “A review of stimulated Brillouin scattering excited with a broad-band pump laser,” IEEE J. Quantum Electron. QE-22, 704 (1986).
[CrossRef]

P. Narum, M. D. Skeldon, and R. W. Boyd, “Effect of laser mode structure on stimulated Brillouin scattering,” IEEE J. Quantum Electron. QE-22, 2161 (1986).
[CrossRef]

M. Tsubokawa, S. Seikai, T. Nakashima, and N. Shibata, “Suppression of stimulated Brillouin scattering in a single-mode fibre by an acousto-optic modulator,” Electron. Lett. 22, 473 (1986).
[CrossRef]

1985 (1)

1983 (2)

D. Cotter, “Stimulated Brillouin scattering in monomode optical fiber,” J. Opt. Commun. 4, 10 (1983).

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

1982 (2)

L. F. Stokes, M. Chodorow, and H. J. Shaw, “All-fiber stimulated Brillouin ring laser with submilliwatt pump threshold,” Opt. Lett. 7, 509 (1982).
[CrossRef] [PubMed]

D. Cotter, “Suppression of stimulated Brillouin scattering during transmission of high-power narrowband laser light in mono-mode fibre,” Electron. Lett. 18, 638 (1982).
[CrossRef]

1979 (1)

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986 (1979).
[CrossRef]

1978 (2)

B. Ya. Zel’dovich and V. V. Shkunov, “Influence of the group velocity mismatch on reproduction of the pump spectrum under stimulated scattering conditions,” Sov. J. Quantum Electron. 8, 1505 (1978).
[CrossRef]

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098 (1978).
[CrossRef]

1977 (1)

G. P. Dzhotyan, Yu E. D’yakov, I. G. Zubarev, A. B. Mironov, and S. I. Mikhailov, “Influence of the spectral width and statistics of a Stokes signal on the efficiency of stimulated Raman scattering of nonmonochromatic pump radiation,” Sov. J. Quantum Electron 7, 783 (1977).
[CrossRef]

1976 (1)

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

1975 (1)

J. Pelous and R. Vacher, “Thermal Brillouin scattering measurements of the attenuation of longitudinal hypersounds in fused quarts,” Solid State Commun. 16, 279 (1975).
[CrossRef]

1972 (1)

1970 (1)

Yu. E. D’yakov, “Excitation of stimulated light scattering by broad-band pumping,” JETP Lett. 11, 243 (1970).

1966 (1)

C. L. Tang, “Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process,” J. Appl. Phys. 37, 2945 (1966).
[CrossRef]

1965 (1)

N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34 (1965).
[CrossRef]

Bar-Joseph, I.

Boyd, R. W.

P. Narum, M. D. Skeldon, and R. W. Boyd, “Effect of laser mode structure on stimulated Brillouin scattering,” IEEE J. Quantum Electron. QE-22, 2161 (1986).
[CrossRef]

Chodorow, M.

Cotter, D.

D. Cotter, “Stimulated Brillouin scattering in monomode optical fiber,” J. Opt. Commun. 4, 10 (1983).

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

D. Cotter, “Suppression of stimulated Brillouin scattering during transmission of high-power narrowband laser light in mono-mode fibre,” Electron. Lett. 18, 638 (1982).
[CrossRef]

D’yakov, Yu E.

G. P. Dzhotyan, Yu E. D’yakov, I. G. Zubarev, A. B. Mironov, and S. I. Mikhailov, “Influence of the spectral width and statistics of a Stokes signal on the efficiency of stimulated Raman scattering of nonmonochromatic pump radiation,” Sov. J. Quantum Electron 7, 783 (1977).
[CrossRef]

D’yakov, Yu. E.

Yu. E. D’yakov, “Excitation of stimulated light scattering by broad-band pumping,” JETP Lett. 11, 243 (1970).

Dzhotyan, G. P.

G. P. Dzhotyan, Yu E. D’yakov, I. G. Zubarev, A. B. Mironov, and S. I. Mikhailov, “Influence of the spectral width and statistics of a Stokes signal on the efficiency of stimulated Raman scattering of nonmonochromatic pump radiation,” Sov. J. Quantum Electron 7, 783 (1977).
[CrossRef]

Friesem, A. A.

H. H. Yaffe, R. G. Waarts, E. Lichtman, and A. A. Friesem, “Multiple wave generation due to four wave mixing in a single-mode fibre,” Electron. Lett. 23, 42 (1987).
[CrossRef]

I. Bar-Joseph, A. A. Friesem, E. Lichtman, and R. G. Waarts, “Steady and relaxation oscillations of stimulated Brillouin scattering in single-mode fibers,” J. Opt. Soc. Am. B 2, 1606 (1985).
[CrossRef]

Hill, K. O.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098 (1978).
[CrossRef]

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

Johnson, D. C.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098 (1978).
[CrossRef]

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

Kawasaki, B. S.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098 (1978).
[CrossRef]

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

Kroll, N. M.

N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34 (1965).
[CrossRef]

Lichtman, E.

H. H. Yaffe, R. G. Waarts, E. Lichtman, and A. A. Friesem, “Multiple wave generation due to four wave mixing in a single-mode fibre,” Electron. Lett. 23, 42 (1987).
[CrossRef]

I. Bar-Joseph, A. A. Friesem, E. Lichtman, and R. G. Waarts, “Steady and relaxation oscillations of stimulated Brillouin scattering in single-mode fibers,” J. Opt. Soc. Am. B 2, 1606 (1985).
[CrossRef]

MacDonald, R. I.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098 (1978).
[CrossRef]

Mikhailov, S. I.

G. P. Dzhotyan, Yu E. D’yakov, I. G. Zubarev, A. B. Mironov, and S. I. Mikhailov, “Influence of the spectral width and statistics of a Stokes signal on the efficiency of stimulated Raman scattering of nonmonochromatic pump radiation,” Sov. J. Quantum Electron 7, 783 (1977).
[CrossRef]

Mironov, A. B.

G. P. Dzhotyan, Yu E. D’yakov, I. G. Zubarev, A. B. Mironov, and S. I. Mikhailov, “Influence of the spectral width and statistics of a Stokes signal on the efficiency of stimulated Raman scattering of nonmonochromatic pump radiation,” Sov. J. Quantum Electron 7, 783 (1977).
[CrossRef]

Nakashima, T.

M. Tsubokawa, S. Seikai, T. Nakashima, and N. Shibata, “Suppression of stimulated Brillouin scattering in a single-mode fibre by an acousto-optic modulator,” Electron. Lett. 22, 473 (1986).
[CrossRef]

Narum, P.

P. Narum, M. D. Skeldon, and R. W. Boyd, “Effect of laser mode structure on stimulated Brillouin scattering,” IEEE J. Quantum Electron. QE-22, 2161 (1986).
[CrossRef]

Pelous, J.

J. Pelous and R. Vacher, “Thermal Brillouin scattering measurements of the attenuation of longitudinal hypersounds in fused quarts,” Solid State Commun. 16, 279 (1975).
[CrossRef]

Rowell, N. L.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986 (1979).
[CrossRef]

Seikai, S.

M. Tsubokawa, S. Seikai, T. Nakashima, and N. Shibata, “Suppression of stimulated Brillouin scattering in a single-mode fibre by an acousto-optic modulator,” Electron. Lett. 22, 473 (1986).
[CrossRef]

Shaw, H. J.

Shibata, N.

M. Tsubokawa, S. Seikai, T. Nakashima, and N. Shibata, “Suppression of stimulated Brillouin scattering in a single-mode fibre by an acousto-optic modulator,” Electron. Lett. 22, 473 (1986).
[CrossRef]

Shkunov, V. V.

B. Ya. Zel’dovich and V. V. Shkunov, “Influence of the group velocity mismatch on reproduction of the pump spectrum under stimulated scattering conditions,” Sov. J. Quantum Electron. 8, 1505 (1978).
[CrossRef]

Skeldon, M. D.

P. Narum, M. D. Skeldon, and R. W. Boyd, “Effect of laser mode structure on stimulated Brillouin scattering,” IEEE J. Quantum Electron. QE-22, 2161 (1986).
[CrossRef]

Smith, R. G.

Stegeman, G. I.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986 (1979).
[CrossRef]

Stokes, L. F.

Stolen, R. H.

R. H. Stolen, “Nonlinear properties of optical fibers,” in Optical Fiber Telecommunications, S. E. Miller and A. G. Chynoweth, eds. (Academic, New York, 1979), p. 125.
[CrossRef]

Tang, C. L.

C. L. Tang, “Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process,” J. Appl. Phys. 37, 2945 (1966).
[CrossRef]

Thomas, P. J.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986 (1979).
[CrossRef]

Tsubokawa, M.

M. Tsubokawa, S. Seikai, T. Nakashima, and N. Shibata, “Suppression of stimulated Brillouin scattering in a single-mode fibre by an acousto-optic modulator,” Electron. Lett. 22, 473 (1986).
[CrossRef]

Vacher, R.

J. Pelous and R. Vacher, “Thermal Brillouin scattering measurements of the attenuation of longitudinal hypersounds in fused quarts,” Solid State Commun. 16, 279 (1975).
[CrossRef]

Valley, G. C.

G. C. Valley, “A review of stimulated Brillouin scattering excited with a broad-band pump laser,” IEEE J. Quantum Electron. QE-22, 704 (1986).
[CrossRef]

van Driel, H. M.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986 (1979).
[CrossRef]

Waarts, R. G.

H. H. Yaffe, R. G. Waarts, E. Lichtman, and A. A. Friesem, “Multiple wave generation due to four wave mixing in a single-mode fibre,” Electron. Lett. 23, 42 (1987).
[CrossRef]

I. Bar-Joseph, A. A. Friesem, E. Lichtman, and R. G. Waarts, “Steady and relaxation oscillations of stimulated Brillouin scattering in single-mode fibers,” J. Opt. Soc. Am. B 2, 1606 (1985).
[CrossRef]

Yaffe, H. H.

H. H. Yaffe, R. G. Waarts, E. Lichtman, and A. A. Friesem, “Multiple wave generation due to four wave mixing in a single-mode fibre,” Electron. Lett. 23, 42 (1987).
[CrossRef]

Zel’dovich, B. Ya.

B. Ya. Zel’dovich and V. V. Shkunov, “Influence of the group velocity mismatch on reproduction of the pump spectrum under stimulated scattering conditions,” Sov. J. Quantum Electron. 8, 1505 (1978).
[CrossRef]

Zubarev, I. G.

G. P. Dzhotyan, Yu E. D’yakov, I. G. Zubarev, A. B. Mironov, and S. I. Mikhailov, “Influence of the spectral width and statistics of a Stokes signal on the efficiency of stimulated Raman scattering of nonmonochromatic pump radiation,” Sov. J. Quantum Electron 7, 783 (1977).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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

Electron. Lett. (4)

D. Cotter, “Suppression of stimulated Brillouin scattering during transmission of high-power narrowband laser light in mono-mode fibre,” Electron. Lett. 18, 638 (1982).
[CrossRef]

M. Tsubokawa, S. Seikai, T. Nakashima, and N. Shibata, “Suppression of stimulated Brillouin scattering in a single-mode fibre by an acousto-optic modulator,” Electron. Lett. 22, 473 (1986).
[CrossRef]

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

H. H. Yaffe, R. G. Waarts, E. Lichtman, and A. A. Friesem, “Multiple wave generation due to four wave mixing in a single-mode fibre,” Electron. Lett. 23, 42 (1987).
[CrossRef]

IEEE J. Quantum Electron. (2)

G. C. Valley, “A review of stimulated Brillouin scattering excited with a broad-band pump laser,” IEEE J. Quantum Electron. QE-22, 704 (1986).
[CrossRef]

P. Narum, M. D. Skeldon, and R. W. Boyd, “Effect of laser mode structure on stimulated Brillouin scattering,” IEEE J. Quantum Electron. QE-22, 2161 (1986).
[CrossRef]

J. Appl. Phys. (3)

N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34 (1965).
[CrossRef]

C. L. Tang, “Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process,” J. Appl. Phys. 37, 2945 (1966).
[CrossRef]

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098 (1978).
[CrossRef]

J. Opt. Commun. (1)

D. Cotter, “Stimulated Brillouin scattering in monomode optical fiber,” J. Opt. Commun. 4, 10 (1983).

J. Opt. Soc. Am. B (1)

JETP Lett. (1)

Yu. E. D’yakov, “Excitation of stimulated light scattering by broad-band pumping,” JETP Lett. 11, 243 (1970).

Opt. Lett. (1)

Phys. Rev. B (1)

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986 (1979).
[CrossRef]

Solid State Commun. (1)

J. Pelous and R. Vacher, “Thermal Brillouin scattering measurements of the attenuation of longitudinal hypersounds in fused quarts,” Solid State Commun. 16, 279 (1975).
[CrossRef]

Sov. J. Quantum Electron (1)

G. P. Dzhotyan, Yu E. D’yakov, I. G. Zubarev, A. B. Mironov, and S. I. Mikhailov, “Influence of the spectral width and statistics of a Stokes signal on the efficiency of stimulated Raman scattering of nonmonochromatic pump radiation,” Sov. J. Quantum Electron 7, 783 (1977).
[CrossRef]

Sov. J. Quantum Electron. (1)

B. Ya. Zel’dovich and V. V. Shkunov, “Influence of the group velocity mismatch on reproduction of the pump spectrum under stimulated scattering conditions,” Sov. J. Quantum Electron. 8, 1505 (1978).
[CrossRef]

Other (1)

R. H. Stolen, “Nonlinear properties of optical fibers,” in Optical Fiber Telecommunications, S. E. Miller and A. G. Chynoweth, eds. (Academic, New York, 1979), p. 125.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

A scheme of the different waves involved in the two-pump-modes SBS.

Fig. 2
Fig. 2

The calculated normalized SBS gain as a function of the normalized frequency separation for the case when the coherence length of the pump wave is smaller than the fiber length.

Fig. 3
Fig. 3

The calculated normalized SBS gain as a function of the normalized coherence length. The solid curve shows the gain for the constructive interference case, while the dashed curve shows the gain for the destructive interference case.

Fig. 4
Fig. 4

A scheme of the experimental setup to measure phase-dependent gain. AO1 and AO2, acousto-optic modulators that modulate the light at frequencies Ω1 and Ω2, respectively; BS’s, beam splitters; M’s, mirrors; OBJ’s, 10× objectives; IS, optical isolator consisting of a polarizer and a quarter-wave plate; FP, Fabry–Perot filter; D, detector; SC, oscilloscope.

Fig. 5
Fig. 5

The intensity of the amplified Stokes wave. The time scale is 50 μsec/division. Ω1 is 1 MHz, and Ω2 is 6 kHz larger.

Fig. 6
Fig. 6

Experimental results of normalized SBS gain for different values of the normalized length. The crosses represent the results that were obtained for a fiber amplifier length of 3.5 m, while the diamonds represent the results for a 30-m fiber. Some of the normalized length values have two experimental points, which represent the maximum and the minimum SBS gain values. The solid and the dashed curves represent the theoretical prediction.

Fig. 7
Fig. 7

A scheme of the experimental setup to measure the dependence of the SBS’s gain on Ω/Γ, A.O., acousto-optic modulator; D1 and D2, detectors; BS’s, beam splitters; M, mirror; OBJ’s, 10× objectives.

Fig. 8
Fig. 8

Experimental results of the normalized SBS gain for different values of the normalized frequency separation. The solid curve represents the theoretical prediction.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

E l ( z , t ) = { E l 1 ( z ) exp [ - i Ω 2 ( t - z c ) ] + E l 2 ( z ) × exp [ i Ω 2 ( t - z c ) ] } exp [ i ( ω l t - k l z ) ] x ^ + c . c . ,
E s ( z , t ) = { E s l ( z ) exp [ - i Ω 2 ( t + z c ) ] + E s 2 ( z ) × exp [ i Ω 2 ( t + z c ) ] } exp [ i ( w s t + k s z ) ] x ^ + c . c . ,
P ( z , t ) = { P - 1 ( z ) exp [ - i Ω ( t - z v ) ] + P 0 ( z ) + P + 1 ( z ) × exp [ i Ω ( t - z v ) ] } exp [ i ( ω 0 t - k 0 z ) ] z ^ + c . c .
d E s 1 d z = - i g 1 [ E l 1 P * 0 exp ( i Ω c z ) + E l 2 P * + 1 exp ( i Ω v z ) ] ,
d E s 2 d z = - i g 1 [ E l 1 P * - 1 exp ( - i Ω v z ) + E l 2 P * 0 exp ( - i Ω v z ) ] ,
d P - 1 d z + η P - 1 = i g 2 v E l 1 E s 2 * exp ( - i Ω v z ) ,
d P 0 d z + η P 0 = i g 2 v [ E l 1 E s 1 * exp ( i Ω c z ) + E l 2 E s 2 * exp ( - i Ω c z ) ] ,
d P + 1 d z + η P + 1 = i g 2 v E l 2 E s 1 * exp ( i Ω v z ) .
P - 1 ( z ) = i g 2 E l 1 E s 2 * Γ - i Ω exp ( - i Ω v z ) ,
P 0 ( z ) P 01 + P 02 = i g 2 [ E l 1 E s 1 * Γ + i Ω v c exp ( - i Ω v z ) + E l 2 E s 2 * Γ - i Ω v c exp ( - i Ω v z ) ] ,
P + 1 ( z ) = i g 2 E l 2 E s 1 * Γ + i Ω exp ( - i Ω v z ) ,
P - 1 ( z ) = g 3 Γ 2 + Ω 2 I l 1 I s 2 ,
P 0 ( z ) = g 3 Γ 2 [ I l 1 I s 1 + I l 2 I s 2 + 2 I l 1 I l 2 I s 1 I s 2 cos ( 2 Ω c z + Φ ) ] ,
P + 1 ( z ) = g 3 Γ 2 + Ω 2 I l 2 I s 1 ,
d d z ( E s 1 E s 2 ) = - g 1 g 2 [ E l 1 2 Γ - i Ω v / c + E l 2 2 Γ - i Ω E l 1 E l 2 * Γ + i Ω v / c exp ( i 2 Ω c z ) E l 2 E l 1 * Γ - i Ω v / c exp ( - i 2 Ω c c ) E l 2 2 Γ + i Ω v / c + E l 1 2 Γ + i Ω ] ( E s 1 E s 2 ) .
d I s 1 d z - [ g ( 0 ) I l 1 + g ( Ω ) I l 2 ] I s 1 ,
d I s 2 d z - [ g ( 0 ) I l 2 + g ( Ω ) I l 1 ] I s 2 ,
g ( δ ω ) = ( 16 π g 1 g 2 c ) Γ Γ 2 + ( δ ω ) 2
( E s 1 E s 2 ) = C 1 ( E l 1 E l 2 ) exp [ - g ( 0 ) ( I l 1 + I l 2 ) z ] + C 2 ( E l 2 * - E l 1 * ) ,
( E s 1 E s 2 ) = C 1 ( u 1 ) exp ( γ 1 z ) + C 2 ( u 2 ) exp ( γ 2 z ) ,
γ 1 , 2 = 1 2 { a 11 + a 22 + i 2 Ω c ± [ ( a 11 - a 22 - i 2 Ω c ) 2 + 4 a 12 2 ] 1 / 2 } ,
L c h 1 g ( 0 ) ( I l 1 + I l 2 ) .

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