Abstract

The intensity noise properties of Brillouin fiber ring lasers are investigated both theoretically and experimentally. The fluctuating parameters that have the dominant influence on laser intensity noise are the pump rate and the cavity reinjection rate. The transfer functions that relate the laser intensity noise to the fluctuations of these parameters are determined in a theoretical study that is performed within the framework of the usual three-wave model of stimulated Brillouin scattering. The theoretical predictions are confirmed by experiments performed in a Brillouin fiber ring laser operating in a low-finesse cavity. Finally, the ability of all-fiber Brillouin lasers to reduce intensity noise of pump sources is discussed.

© 2002 Optical Society of America

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  5. Y. Cheng, J. T. Kringlebotn, W. H. Loh, R. I. Laming, and D. N. Payne, “Stable single-frequency traveling-wave fiber loop laser with integral saturable-absorber-based tracking narrow-band filter,” Opt. Lett. 20, 875–877 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  30. S. Randoux, V. Lecoeuche, and J. Zemmouri, “Polarization instabilities and antiphase dynamics in a Brillouin fiber ring laser,” Phys. Rev. A 56, R1717–R1720 (1997).
    [CrossRef]
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2001

2000

A. Bramati, J.-P. Hermier, V. Jost, and E. Giacobino, “Feedback control and nonlinear intensity noise of Nd:YVO4 microchip lasers,” Phys. Rev. A 62, 043806–1–043806–12 (2000).
[CrossRef]

A. Debut, S. Randoux, and J. Zemmouri, “Linewidth narrowing in Brillouin lasers: theoretical analysis,” Phys. Rev. A 62, 023803–1–023803–4 (2000).
[CrossRef]

1999

1998

S. Taccheo, P. Laporta, O. Svelto, and G. De Geronimo, “Theoretical and experimental analysis of intensity noise of a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66, 19–26 (1998).
[CrossRef]

D.-H. Lee, M. E. Klein, and K.-J. Boller, “Intensity noise of pump-enhanced continuous wave optical parametric oscillators,” Appl. Phys. B 66, 747–753 (1998).
[CrossRef]

B. C. Buchler, E. H. Huntington, C. C. Harb, and T. C. Ralph, “Feedback control of laser intensity noise,” Phys. Rev. A 57, 1286–1294 (1998).
[CrossRef]

1997

F. Marin, A. Bramati, V. Jost, and E. Giacobino, “Demonstration of high sensitivity spectroscopy with squeezed semiconductor lasers,” Opt. Commun. 140, 146–157 (1997).
[CrossRef]

S. Randoux, V. Lecoeuche, and J. Zemmouri, “Polarization instabilities and antiphase dynamics in a Brillouin fiber ring laser,” Phys. Rev. A 56, R1717–R1720 (1997).
[CrossRef]

1996

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H. A. Bachor, “Injection-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

1995

W. Muys, J. C. van der Plaats, F. W. Willems, H. J. van Dijk, J. S. Leong, and A. M. J. Koonen, “A 50-channel externally modulated AM-VSB video distribution system with three cascaded EDFA’s providing 50-dB power budget over 30 km of standard single-mode fiber,” IEEE Photonics Technol. Lett. 7, 691–693 (1995).
[CrossRef]

Y. Cheng, J. T. Kringlebotn, W. H. Loh, R. I. Laming, and D. N. Payne, “Stable single-frequency traveling-wave fiber loop laser with integral saturable-absorber-based tracking narrow-band filter,” Opt. Lett. 20, 875–877 (1995).
[CrossRef] [PubMed]

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical analysis of Brillouin fiber lasers: an experimental approach,” Phys. Rev. A 51, R4345–R4348 (1995).
[CrossRef] [PubMed]

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of a Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 2327–2334 (1995).
[CrossRef] [PubMed]

1994

C. Montes, A. Mamhoud, and E. Picholle, “Bifurcation in a cw-pumped Brillouin fiber-ring laser: coherent soliton morphogenesis,” Phys. Rev. A 49, 1344–1349 (1994).
[CrossRef] [PubMed]

1993

P.-A. Nicati, K. Toyama, S. Huang, and H. J. Shaw, “Temperature effects in a Brillouin fiber ring laser,” Opt. Lett. 18, 2123–2125 (1993).
[CrossRef] [PubMed]

G. A. Ball, G. Hull-Allen, C. Holton, and W. W. Morey, “Low noise single frequency linear fibre laser,” Electron. Lett. 29, 1623–1625 (1993).
[CrossRef]

1992

S. Sanders, J. W. Dawson, N. Park, and K. J. Vahala, “Measurements of the intensity noise of a broadly tunable, erbium-doped fiber ring laser, relative to the standard quantum limit,” Appl. Phys. Lett. 60, 2583–2585 (1992).
[CrossRef]

W. Lu, A. Johnstone, and R. G. Harrison, “Deterministic dynamics of stimulated scattering phenomena with external feedback,” Phys. Rev. A 46, 4114–4122 (1992).
[CrossRef] [PubMed]

1991

A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44, 3205–3209 (1991).
[CrossRef] [PubMed]

F. Zarinetchi, S. P. Smith, and S. Ezekiel, “Stimulated Brillouin fiber-optic gyroscope,” Opt. Lett. 16, 229–231 (1991).
[CrossRef] [PubMed]

1990

R. W. Boyd, K. Rzażewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef] [PubMed]

1987

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

1986

H. P. Huen, “Generation, detection, and application of high-intensity photon number eigenstate fields,” Phys. Rev. Lett. 56, 2176–2179 (1986).
[CrossRef]

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

1981

1965

J. A. Armstrong and A. W. Smith, “Intensity fluctuations in a GaAs laser,” Phys. Rev. Lett. 14, 68–70 (1965).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong and A. W. Smith, “Intensity fluctuations in a GaAs laser,” Phys. Rev. Lett. 14, 68–70 (1965).
[CrossRef]

Bachor, H. A.

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H. A. Bachor, “Injection-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

Bahloul, D.

Ball, G. A.

G. A. Ball, G. Hull-Allen, C. Holton, and W. W. Morey, “Low noise single frequency linear fibre laser,” Electron. Lett. 29, 1623–1625 (1993).
[CrossRef]

Becher, C.

Boller, K. J.

Boller, K.-J.

D.-H. Lee, M. E. Klein, and K.-J. Boller, “Intensity noise of pump-enhanced continuous wave optical parametric oscillators,” Appl. Phys. B 66, 747–753 (1998).
[CrossRef]

Bongrand, I.

Botineau, J.

Boyd, R. W.

A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44, 3205–3209 (1991).
[CrossRef] [PubMed]

R. W. Boyd, K. Rzażewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef] [PubMed]

Bramati, A.

A. Bramati, J.-P. Hermier, V. Jost, and E. Giacobino, “Feedback control and nonlinear intensity noise of Nd:YVO4 microchip lasers,” Phys. Rev. A 62, 043806–1–043806–12 (2000).
[CrossRef]

F. Marin, A. Bramati, V. Jost, and E. Giacobino, “Demonstration of high sensitivity spectroscopy with squeezed semiconductor lasers,” Opt. Commun. 140, 146–157 (1997).
[CrossRef]

Buchler, B. C.

B. C. Buchler, E. H. Huntington, C. C. Harb, and T. C. Ralph, “Feedback control of laser intensity noise,” Phys. Rev. A 57, 1286–1294 (1998).
[CrossRef]

Cheng, Y.

Cheval, G.

Dawson, J. W.

S. Sanders, J. W. Dawson, N. Park, and K. J. Vahala, “Measurements of the intensity noise of a broadly tunable, erbium-doped fiber ring laser, relative to the standard quantum limit,” Appl. Phys. Lett. 60, 2583–2585 (1992).
[CrossRef]

De Geronimo, G.

S. Taccheo, P. Laporta, O. Svelto, and G. De Geronimo, “Theoretical and experimental analysis of intensity noise of a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66, 19–26 (1998).
[CrossRef]

Debut, A.

A. Debut, S. Randoux, and J. Zemmouri, “Experimental and theoretical study of linewidth narrowing in Brillouin fiber ring lasers,” J. Opt. Soc. Am. B 18, 556–567 (2001).
[CrossRef]

A. Debut, S. Randoux, and J. Zemmouri, “Linewidth narrowing in Brillouin lasers: theoretical analysis,” Phys. Rev. A 62, 023803–1–023803–4 (2000).
[CrossRef]

Ezekiel, S.

Freitag, I.

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H. A. Bachor, “Injection-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

Gaeta, A. L.

A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44, 3205–3209 (1991).
[CrossRef] [PubMed]

Giacobino, E.

A. Bramati, J.-P. Hermier, V. Jost, and E. Giacobino, “Feedback control and nonlinear intensity noise of Nd:YVO4 microchip lasers,” Phys. Rev. A 62, 043806–1–043806–12 (2000).
[CrossRef]

F. Marin, A. Bramati, V. Jost, and E. Giacobino, “Demonstration of high sensitivity spectroscopy with squeezed semiconductor lasers,” Opt. Commun. 140, 146–157 (1997).
[CrossRef]

Harb, C. C.

B. C. Buchler, E. H. Huntington, C. C. Harb, and T. C. Ralph, “Feedback control of laser intensity noise,” Phys. Rev. A 57, 1286–1294 (1998).
[CrossRef]

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H. A. Bachor, “Injection-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

Harrison, R. G.

W. Lu, A. Johnstone, and R. G. Harrison, “Deterministic dynamics of stimulated scattering phenomena with external feedback,” Phys. Rev. A 46, 4114–4122 (1992).
[CrossRef] [PubMed]

Hermier, J.-P.

A. Bramati, J.-P. Hermier, V. Jost, and E. Giacobino, “Feedback control and nonlinear intensity noise of Nd:YVO4 microchip lasers,” Phys. Rev. A 62, 043806–1–043806–12 (2000).
[CrossRef]

Holton, C.

G. A. Ball, G. Hull-Allen, C. Holton, and W. W. Morey, “Low noise single frequency linear fibre laser,” Electron. Lett. 29, 1623–1625 (1993).
[CrossRef]

Huang, S.

Huen, H. P.

H. P. Huen, “Generation, detection, and application of high-intensity photon number eigenstate fields,” Phys. Rev. Lett. 56, 2176–2179 (1986).
[CrossRef]

Hull-Allen, G.

G. A. Ball, G. Hull-Allen, C. Holton, and W. W. Morey, “Low noise single frequency linear fibre laser,” Electron. Lett. 29, 1623–1625 (1993).
[CrossRef]

Huntington, E. H.

B. C. Buchler, E. H. Huntington, C. C. Harb, and T. C. Ralph, “Feedback control of laser intensity noise,” Phys. Rev. A 57, 1286–1294 (1998).
[CrossRef]

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H. A. Bachor, “Injection-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

Itaya, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

Johnstone, A.

W. Lu, A. Johnstone, and R. G. Harrison, “Deterministic dynamics of stimulated scattering phenomena with external feedback,” Phys. Rev. A 46, 4114–4122 (1992).
[CrossRef] [PubMed]

Jost, V.

A. Bramati, J.-P. Hermier, V. Jost, and E. Giacobino, “Feedback control and nonlinear intensity noise of Nd:YVO4 microchip lasers,” Phys. Rev. A 62, 043806–1–043806–12 (2000).
[CrossRef]

F. Marin, A. Bramati, V. Jost, and E. Giacobino, “Demonstration of high sensitivity spectroscopy with squeezed semiconductor lasers,” Opt. Commun. 140, 146–157 (1997).
[CrossRef]

Klein, M. E.

D.-H. Lee, M. E. Klein, and K.-J. Boller, “Intensity noise of pump-enhanced continuous wave optical parametric oscillators,” Appl. Phys. B 66, 747–753 (1998).
[CrossRef]

Koonen, A. M. J.

W. Muys, J. C. van der Plaats, F. W. Willems, H. J. van Dijk, J. S. Leong, and A. M. J. Koonen, “A 50-channel externally modulated AM-VSB video distribution system with three cascaded EDFA’s providing 50-dB power budget over 30 km of standard single-mode fiber,” IEEE Photonics Technol. Lett. 7, 691–693 (1995).
[CrossRef]

Kringlebotn, J. T.

Laming, R. I.

Laporta, P.

S. Taccheo, P. Laporta, O. Svelto, and G. De Geronimo, “Theoretical and experimental analysis of intensity noise of a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66, 19–26 (1998).
[CrossRef]

Lecoeuche, V.

S. Randoux, V. Lecoeuche, and J. Zemmouri, “Polarization instabilities and antiphase dynamics in a Brillouin fiber ring laser,” Phys. Rev. A 56, R1717–R1720 (1997).
[CrossRef]

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical analysis of Brillouin fiber lasers: an experimental approach,” Phys. Rev. A 51, R4345–R4348 (1995).
[CrossRef] [PubMed]

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of a Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 2327–2334 (1995).
[CrossRef] [PubMed]

Lee, D.-H.

D.-H. Lee, M. E. Klein, and K.-J. Boller, “Intensity noise of pump-enhanced continuous wave optical parametric oscillators,” Appl. Phys. B 66, 747–753 (1998).
[CrossRef]

Leong, J. S.

W. Muys, J. C. van der Plaats, F. W. Willems, H. J. van Dijk, J. S. Leong, and A. M. J. Koonen, “A 50-channel externally modulated AM-VSB video distribution system with three cascaded EDFA’s providing 50-dB power budget over 30 km of standard single-mode fiber,” IEEE Photonics Technol. Lett. 7, 691–693 (1995).
[CrossRef]

Loh, W. H.

Lu, W.

W. Lu, A. Johnstone, and R. G. Harrison, “Deterministic dynamics of stimulated scattering phenomena with external feedback,” Phys. Rev. A 46, 4114–4122 (1992).
[CrossRef] [PubMed]

Machida, S.

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Mamhoud, A.

Marin, F.

F. Marin, A. Bramati, V. Jost, and E. Giacobino, “Demonstration of high sensitivity spectroscopy with squeezed semiconductor lasers,” Opt. Commun. 140, 146–157 (1997).
[CrossRef]

McClelland, D. E.

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H. A. Bachor, “Injection-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

Montes, C.

Morey, W. W.

G. A. Ball, G. Hull-Allen, C. Holton, and W. W. Morey, “Low noise single frequency linear fibre laser,” Electron. Lett. 29, 1623–1625 (1993).
[CrossRef]

Muys, W.

W. Muys, J. C. van der Plaats, F. W. Willems, H. J. van Dijk, J. S. Leong, and A. M. J. Koonen, “A 50-channel externally modulated AM-VSB video distribution system with three cascaded EDFA’s providing 50-dB power budget over 30 km of standard single-mode fiber,” IEEE Photonics Technol. Lett. 7, 691–693 (1995).
[CrossRef]

Narum, P.

R. W. Boyd, K. Rzażewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef] [PubMed]

Nicati, P.-A.

Nilsson, O.

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Park, N.

S. Sanders, J. W. Dawson, N. Park, and K. J. Vahala, “Measurements of the intensity noise of a broadly tunable, erbium-doped fiber ring laser, relative to the standard quantum limit,” Appl. Phys. Lett. 60, 2583–2585 (1992).
[CrossRef]

Payne, D. N.

Picholle, E.

Picozzi, A.

Ponikvar, D. R.

Ralph, T. C.

B. C. Buchler, E. H. Huntington, C. C. Harb, and T. C. Ralph, “Feedback control of laser intensity noise,” Phys. Rev. A 57, 1286–1294 (1998).
[CrossRef]

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H. A. Bachor, “Injection-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

Randoux, S.

A. Debut, S. Randoux, and J. Zemmouri, “Experimental and theoretical study of linewidth narrowing in Brillouin fiber ring lasers,” J. Opt. Soc. Am. B 18, 556–567 (2001).
[CrossRef]

A. Debut, S. Randoux, and J. Zemmouri, “Linewidth narrowing in Brillouin lasers: theoretical analysis,” Phys. Rev. A 62, 023803–1–023803–4 (2000).
[CrossRef]

S. Randoux, V. Lecoeuche, and J. Zemmouri, “Polarization instabilities and antiphase dynamics in a Brillouin fiber ring laser,” Phys. Rev. A 56, R1717–R1720 (1997).
[CrossRef]

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of a Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 2327–2334 (1995).
[CrossRef] [PubMed]

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical analysis of Brillouin fiber lasers: an experimental approach,” Phys. Rev. A 51, R4345–R4348 (1995).
[CrossRef] [PubMed]

Rzazewski, K.

R. W. Boyd, K. Rzażewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef] [PubMed]

Sanders, S.

S. Sanders, J. W. Dawson, N. Park, and K. J. Vahala, “Measurements of the intensity noise of a broadly tunable, erbium-doped fiber ring laser, relative to the standard quantum limit,” Appl. Phys. Lett. 60, 2583–2585 (1992).
[CrossRef]

Ségard, B.

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[CrossRef] [PubMed]

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S. Taccheo, P. Laporta, O. Svelto, and G. De Geronimo, “Theoretical and experimental analysis of intensity noise of a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66, 19–26 (1998).
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W. Muys, J. C. van der Plaats, F. W. Willems, H. J. van Dijk, J. S. Leong, and A. M. J. Koonen, “A 50-channel externally modulated AM-VSB video distribution system with three cascaded EDFA’s providing 50-dB power budget over 30 km of standard single-mode fiber,” IEEE Photonics Technol. Lett. 7, 691–693 (1995).
[CrossRef]

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W. Muys, J. C. van der Plaats, F. W. Willems, H. J. van Dijk, J. S. Leong, and A. M. J. Koonen, “A 50-channel externally modulated AM-VSB video distribution system with three cascaded EDFA’s providing 50-dB power budget over 30 km of standard single-mode fiber,” IEEE Photonics Technol. Lett. 7, 691–693 (1995).
[CrossRef]

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S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

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[CrossRef]

A. Debut, S. Randoux, and J. Zemmouri, “Linewidth narrowing in Brillouin lasers: theoretical analysis,” Phys. Rev. A 62, 023803–1–023803–4 (2000).
[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of a Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 2327–2334 (1995).
[CrossRef] [PubMed]

Appl. Phys. B

S. Taccheo, P. Laporta, O. Svelto, and G. De Geronimo, “Theoretical and experimental analysis of intensity noise of a codoped erbium-ytterbium glass laser,” Appl. Phys. B 66, 19–26 (1998).
[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

A. Debut, S. Randoux, and J. Zemmouri, “Linewidth narrowing in Brillouin lasers: theoretical analysis,” Phys. Rev. A 62, 023803–1–023803–4 (2000).
[CrossRef]

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[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic representation of a Brillouin fiber ring laser and of the noise sources that contribute largely to its intensity noise. The optical isolator is inserted within the cavity to prevent pump recoupling.

Fig. 2
Fig. 2

Transfer functions for fluctuations of reinjection rate R. (a) In a low-finesse cavity (g=6.04, βA=10.93, R=0.36), curve i is obtained at threshold from the analytical calculation, whereas the curves ii, iii, and iv are obtained from numerical simulations performed, respectively, with μ/μthr=1.01, μ/μthr=1.2, and μ/μthr=2.0. (b) In a high-finesse cavity (g=6.04, βA=10.93, R=0.95), curve i is obtained at threshold from the analytical calculation, and curves ii and iii are obtained from numerical simulations performed, respectively, with μ/μthr=1.2 and μ/μthr=5.0. Frequency ν is expressed in units of cavity FSR.

Fig. 3
Fig. 3

Low-frequency value |GR(ν0)| of the transfer function versus ratio μ/μthr for (a) a low-finesse cavity and (b) a high-finesse cavity. Solid curves, analytical results; points were obtained from numerical simulations performed with the parameters g=6.04, βA=10.93, and (a) R=0.36 and (b) R=0.95.

Fig. 4
Fig. 4

Transfer functions for the fluctuations of the pump rate for (a) a low-finesse cavity and (b) a high-finesse cavity. All the curves were obtained in the same conditions and with the same parameters as in Fig. 2.

Fig. 5
Fig. 5

Longitudinal profiles of the relative amplitudes of the pump and Stokes fields recorded at two different times and for modulation frequencies of (a), (b) ν0=0.35 and (c), (d) ν0=0.5. The parameters used for the numerical simulations are g=6.04, βA=10.93, R=0.36, and μ/μthr=1.02.

Fig. 6
Fig. 6

Low-frequency value |GP(ν0)| of the transfer function versus ratio μ/μthr for (a) a low-finesse cavity and (b) a high-finesse cavity. Solid curves, analytical results; the points were obtained from numerical simulations performed with the parameters g=6.04, βA=10.93, and (a) R=0.36 and (b) R=0.95.

Fig. 7
Fig. 7

Schematic representation of the experimental setup used to determine the transfer functions for the fluctuations of the reinjection rate. The reflectivity of the beam splitters inserted within the ring cavity is 30%. The pump wave circulates in the clockwise direction, and the Stokes wave circulates in the counterclockwise direction.

Fig. 8
Fig. 8

Measured values of the transfer function for cavity loss modulation at (a) low pump power and (b) high pump power. Dashed curves, numerical simulations performed with the parameters g=6.04, βA=10.93, R=0.36, and (a) μ/μthr=1.093 and (b) μ/μthr=1.165.

Fig. 9
Fig. 9

Schematic representation of the experimental setup used to determine the transfer functions for the fluctuations of the pump rate.

Fig. 10
Fig. 10

Measured values of the transfer function for pump modulation at (a) low pump power and (b) high pump power. Dashed curves, numerical simulations performed with the parameters g=6.04, βA=10.93, R=0.36, and (a) μ/μthr=1.115 and (b) μ/μthr=1.361.

Equations (58)

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εpτ+εpζ=-gBεs,
εsτ-εsζ=gB*εp,
1βA Bτ+B=εpεs*+f(ζ, τ).
εp(ζ=0, τ)=μ exp[iϕ0(τ)],
εs(ζ=1, τ)=R exp(iδs)εs(ζ=0, τ).
Apτ+Apζ=-gAaAs,
Asτ-Asζ=gAaAp,
1βA Aaτ+Aa=ApAs.
Ai(ζ, τ)=Ai0(ζ)+δAi(ζ, τ)(i=p, s, a).
δAp˜τ+δAp˜ζ=gIs(ζ)(δAp˜-δAa˜-δAs˜),
δAs˜τ-δAs˜ζ=gIp(ζ)(δAp˜+δAa˜-δAs˜),
1βA δAa˜τ=δAp˜+δAs˜-δAa˜,
Ap(ζ=0, τ)=μ,
As(ζ=1, τ)=[R+δR(τ)]As(ζ=0, τ),
δAp˜(ζ=0, τ)=0,
δAs˜(ζ=1, τ)=δAs˜(ζ=0, τ)+δR˜(τ),
GR(ν)=δAs˜(ζ=0, ν)δR˜(ν).
δAs˜(ζ, ν)ζ=i2πν1-ln RβA+i2πνδAs˜(ζ, ν),
δAs˜(ζ=1, ν)=δAs˜(ζ=0, ν)+δR˜(ν).
GR(ν)=δAs˜(ζ=0, ν)δR˜(ν)=1exp{i2πν[1-ln R/(βA+i2πν)]}-1.
|GR(ν)|=12πν(1+Γc/γA).
δAs˜(ζ=0, τ)=11+Γc/γA -τ δR˜(τ)dτ.
Ap(ζ=0, τ)=μ+δμ(τ),
As(ζ=1, τ)=RAs(ζ=0, τ),
δAp˜(ζ=0, τ)=δμ˜(τ),
δAs˜(ζ=1, τ)=δAs˜(ζ=0, τ),
GP(ν)=δAs˜(ζ=0, ν)δμ˜(ν),
δAs˜(ζ, ν)ζ=C1δA˜(ζ, ν)-C2δμ˜(ν)exp(-i2πνζ),
C1=i2πν1-ln RβA+i2πν,
C2=-ln(R)2βA+i2πνβA+i2πν.
GP(ν)=δAs˜(ζ=0, ν)δμ˜(ν)=C2C1+i2πν exp(C1)-exp(-i2πν)exp(C1)-1.
δAs˜(ζ, τ)=as(ν0)sin[2πν0τ+ϕ(ν0)]×s0(ν0)+n=1+ sn(ν0)sin(nπζ).
|GP(ν)|=|ln(R2)|2πν(1+Γc/γA).
δAs˜(ζ=0, τ)=|ln(R2)|1+Γc/γA -τ δμ˜(τ)dτ.
dδAs˜(ζ)dζ=-2gIs(ζ)δAs˜(ζ)+C,
Is(t)=Is¯[1+ms(νm)sin(2πνmt)],
Ws(ν)=|GR(ν)|2WR(ν)+|GP(ν)|2WP(ν),
δBs˜(ζ, τ)=δAs˜(ζ, τ)-ζδR˜(τ)
δBs˜(ζ=1, τ)=δBs˜(ζ=0, τ),
δAp˜τ+δAp˜ζ=gIs(ζ)(δAp˜-δAa˜-δBs˜-ζδR˜),
δBs˜τ-δBs˜ζ=gIp(ζ)(δAp˜+δAa˜-δBs˜-ζδR˜)-ζ δR˜τ+δR˜,
1βA δAa˜τ=δAp˜-δAa˜+δBs˜+ζδR˜.
dδBs˜(ζ)dζ=-2gIs(ζ)[δBs˜(ζ)+ζδR˜]+K,
|GP(ν0)|=δAs˜(ζ=0)δR˜=g/2+M/DIs(0)-N/D,
M=m0+m1Is(0)+m2Is2(0)+m3Is3(0).
δAs0˜(ζ)=δAs˜(0)+a1ζ+a22ζ2+a36ζ3+a424ζ4,
δAs1˜(ζ)=δAs˜(1)+b1(ζ-1)+b22(ζ-1)2+b36(ζ-1)3+b424(ζ-1)4.
Is0(ζ)=Is(0)+α1ζ+α22ζ2+α36ζ3,
Is1(ζ)=Is(0)+β1(ζ-1)+β22(ζ-1)2+β36(ζ-1)3,
δAs0˜(ζ=1/2)=δAs1˜(ζ=1/2),
dδAs0˜(ζ)dζζ=1/2=dδAs1˜(ζ)dζζ=1/2,
d2δAs0˜(ζ)dζ2ζ=1/2=d2δAs1˜(ζ)dζ2ζ=1/2.
C=2g ND δAs˜(ζ=0),
N=n0+n1Is(0)+n2Is2(0)+n3Is3(0),
D=d0+d1Is(0)+d2Is2(0).
dδAs˜(ζ)dζ=-2gIp(ζ)δAp˜(ζ),
dδAp˜(ζ)dζ=-2gIs(ζ)δAs˜(ζ).
|GP(ν0)|=δAs˜(ζ=0)δAp˜(ζ=0)=μ2Is(0)-N/D.

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