Abstract

The nonlinear dynamics of cw-pumped Brillouin long-fiber ring lasers that contain a large number of longitudinal modes N beneath the Brillouin gain curve is controlled by a single parameter, namely, the Stokes feedback R. Below Rcrit, a stable train of dissipative solitonic pulses is spontaneously structured at the round-trip frequency fr without any additional intracavity mode locking. Experimental observations in cw-pumped fiber ring cavities, supported by numerical simulation in a coherent space–time three-wave model that includes the optical Kerr effect, prove the universality of the self-pulsing mechanism. Stability analysis shows that below Rcrit the steady Brillouin mirror regime is destabilized through a Hopf bifurcation. For R<Rcrit<R0 the bifurcation is supercritical and exhibits an asymptotically monostable oscillatory regime at twice fr for high enough N or at fr for lower N, in a finite transition region. For R0<R<Rcrit, the bifurcation is subcritical and exhibits dynamic bistability between the steady and the pulsed regimes in a finite hysteresis region whose width is proportional to the Kerr parameter. For R small enough, the cavity longitudinal modes merge into a dissipative solitonic Brillouin pulse: the dynamic three-wave model yields self-structured asymptotically stable trains of pulses for any initial conditions, in fair quantitative agreement (for pulse width, intensity, shape, and period) with the experiments in the entire self-pulsing domain. Amplification of spontaneous emission breaks down the stable-pulse regime in long devices (i.e., high N), so the fiber noise amplitude is higher than the coherent amplitude that separates two consecutive pulses.

© 1999 Optical Society of America

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    [CrossRef]
  47. The study of the detuned cavity in the bifurcation region was suggested to us by a referee.
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1998

L. Chen and X. Bao, “Analytic and numerical solutions for the steady state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
[CrossRef]

1997

A. Fellegara, A. Melloni, and M. Martinelli, “Measurement of the frequency response induced by electrostriction in optical fibers,” Opt. Lett. 22, 1615–1617 (1997).
[CrossRef]

V. Lecoeuche, B. Ségard, and J. Zemmouri, “Modes of destabilization of Brillouin fiber ring lasers,” Opt. Commun. 134, 547–558 (1997).
[CrossRef]

E. Picholle and A. Picozzi, “Guided-acoustic-wave resonances in the dynamics of stimulated Brillouin fiber ring laser,” Opt. Commun. 18, 327–330 (1997).
[CrossRef]

C. Montes, A. Picozzi, and D. Bahloul, “Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons,” Phys. Rev. E 55, 1092–1105 (1997).
[CrossRef]

1995

D. Yu, W. Lu, and R. G. Harrison, “Physical origin of dynamical stimulated Brillouin scattering in optical fibers with feedback,” Phys. Rev. A 51, 669–674 (1995).
[CrossRef] [PubMed]

J. Botineau, E. Picholle, and D. Bahloul, “Effective stimulated Brillouin gain in singlemode optical fibers,” Electron. Lett. 23, 2032–2033 (1995).
[CrossRef]

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 221–228 (1995).
[CrossRef]

Y. Takushima and K. Kikuchi, “Spectral gain hole burning and modulation instability in a Brillouin fiber amplifier,” Opt. Lett. 20, 34–36 (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]

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]

M. O. van Deventer and A. I. Root, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12, 585–590 (1994).
[CrossRef]

J. Botineau, C. Leycuras, C. Montes, and E. Picholle, “Co-herent modal analysis of a Brillouin fiber ring laser,” Opt. Commun. 15, 126–132 (1994).
[CrossRef]

1993

M. Dämmig, G. Zinner, F. Mitschke, and H. Welling, “Stimulated Brillouin scattering in fibers with and without external feedback,” Phys. Rev. A 48, 3301–3309 (1993).
[CrossRef] [PubMed]

D. J. Kaup, “The first-order perturbed SBS equations,” J. Nonlinear Sci. 3, 427–443 (1993).
[CrossRef]

1992

A. L. Gaeta and R. W. Boyd, “Stimulated Brillouin scattering in the presence of external feedback,” J. Nonlinear Opt. Phys. Mater. 1, 581–594 (1992).
[CrossRef]

1991

E. Picholle, C. Montes, C. Leycuras, O. Legrand, and J. Botineau, “Observation of dissipative superluminous solitons in a Brillouin fiber ring laser,” Phys. Rev. Lett. 66, 1454–1457 (1991).
[CrossRef] [PubMed]

S. P. Smith, F. Zarinetchi, and S. E. Ezekiel, “Narrow-linewidth stimulated Brillouin fiber laser and applications,” Opt. Lett. 16, 393–395 (1991).
[CrossRef] [PubMed]

N. Yoshisawa, T. Harigushi, and T. Kuroshima, “Proposal for stimulated Brillouin scattering suppression by fiber cabling,” Electron. Lett. 27, 1100–1101 (1991).
[CrossRef]

1990

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

A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990).
[CrossRef] [PubMed]

R. G. Harrison, J. S. Uppal, A. Johnstone, and J. V. Moloney, “Evidence of chaotic stimulated Brillouin scattering with weak feedback,” Phys. Rev. Lett. 65, 167–170 (1990).
[CrossRef] [PubMed]

1989

1988

B. Ya Zel’dovich and A. N. Pilipetskii, “Role of a soundguide and antisoundguide in stimulated Brillouin scattering in a single-mode waveguide,” Sov. J. Quantum Electron. 183, 818–822 (1988).
[CrossRef]

C. K. Jen, J. E. B. Oliveira, N. Goto, and K. Abe, “Role of guided acoustic wave properties in single-mode optical fibre design,” Electron. Lett. 24, 1419–1420 (1988).
[CrossRef]

R. M. Shelby, M. D. Levenson, and S. H. Perlmutter, “Bistability and other effects in a nonlinear fiber-optic ring resonator,” J. Opt. Soc. Am. B 5, 347–357 (1988).
[CrossRef]

1987

C. Montes and J. Coste, “Optical turbulence in multiple stimulated Brillouin backscattering,” Laser Part. Beams 5, 405–411 (1987).
[CrossRef]

1986

I. Bar-Joseph, A. A. Friesem, E. Lichtman, and R. G. Waarts, “Steady and relaxation oscillations of SBS in single-mode optical fibers,” J. Opt. Soc. Am. B 2, 1606–1611 (1986).
[CrossRef]

J. Coste and C. Montes, “Asymtotic evolution of stimulated Brillouin scattering: implications for optical fibers,” Phys. Rev. A 34, 3940–3949 (1986).
[CrossRef] [PubMed]

1983

D. Cotter, “Stimulated Brillouin scattering in monomode optical fiber,” J. Opt. Commun. 4, 10–19 (1983).
[CrossRef]

1982

1981

1980

1979

D. Heiman, D. S. Hamilton, and R. W. Hellwarth, “Brillouin scattering measurements on optical glasses,” Phys. Rev. B 19, 6583–6592 (1979).
[CrossRef]

1978

S. F. Morosov, L. V. Piskunova, M. M. Sushik, and G. I. Freidman, “Formation and amplification of quasisoliton pulses in head-on stimulated scattering,” Sov. J. Quantum Electron. 8, 576–580 (1978).
[CrossRef]

1976

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

1975

J. Pelous and R. Vacher, “Thermal Brillouin scattering measurements of the attenuation of longitudinal hypersounds in fused quartz from 77 to 300 K,” Solid State Commun. 16, 279–283 (1975).
[CrossRef]

Abe, K.

C. K. Jen, J. E. B. Oliveira, N. Goto, and K. Abe, “Role of guided acoustic wave properties in single-mode optical fibre design,” Electron. Lett. 24, 1419–1420 (1988).
[CrossRef]

Bahloul, D.

C. Montes, A. Picozzi, and D. Bahloul, “Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons,” Phys. Rev. E 55, 1092–1105 (1997).
[CrossRef]

J. Botineau, E. Picholle, and D. Bahloul, “Effective stimulated Brillouin gain in singlemode optical fibers,” Electron. Lett. 23, 2032–2033 (1995).
[CrossRef]

Bao, X.

L. Chen and X. Bao, “Analytic and numerical solutions for the steady state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
[CrossRef]

Bar-Joseph, I.

Bayvel, P.

Botineau, J.

J. Botineau, E. Picholle, and D. Bahloul, “Effective stimulated Brillouin gain in singlemode optical fibers,” Electron. Lett. 23, 2032–2033 (1995).
[CrossRef]

J. Botineau, C. Leycuras, C. Montes, and E. Picholle, “Co-herent modal analysis of a Brillouin fiber ring laser,” Opt. Commun. 15, 126–132 (1994).
[CrossRef]

E. Picholle, C. Montes, C. Leycuras, O. Legrand, and J. Botineau, “Observation of dissipative superluminous solitons in a Brillouin fiber ring laser,” Phys. Rev. Lett. 66, 1454–1457 (1991).
[CrossRef] [PubMed]

J. Botineau, C. Leycuras, C. Montes, and E. Picholle, “Stabilization of a stimulated Brillouin fiber ring laser by strong pump modulation,” J. Opt. Soc. Am. B 6, 300–312 (1989).
[CrossRef]

Boyd, R. W.

A. L. Gaeta and R. W. Boyd, “Stimulated Brillouin scattering in the presence of external feedback,” J. Nonlinear Opt. Phys. Mater. 1, 581–594 (1992).
[CrossRef]

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

Chen, L.

L. Chen and X. Bao, “Analytic and numerical solutions for the steady state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
[CrossRef]

Chodorow, M.

Coste, J.

C. Montes and J. Coste, “Optical turbulence in multiple stimulated Brillouin backscattering,” Laser Part. Beams 5, 405–411 (1987).
[CrossRef]

J. Coste and C. Montes, “Asymtotic evolution of stimulated Brillouin scattering: implications for optical fibers,” Phys. Rev. A 34, 3940–3949 (1986).
[CrossRef] [PubMed]

Cotter, D.

D. Cotter, “Stimulated Brillouin scattering in monomode optical fiber,” J. Opt. Commun. 4, 10–19 (1983).
[CrossRef]

Dämmig, M.

M. Dämmig, G. Zinner, F. Mitschke, and H. Welling, “Stimulated Brillouin scattering in fibers with and without external feedback,” Phys. Rev. A 48, 3301–3309 (1993).
[CrossRef] [PubMed]

Ezekiel, S.

Ezekiel, S. E.

Fellegara, A.

Freidman, G. I.

S. F. Morosov, L. V. Piskunova, M. M. Sushik, and G. I. Freidman, “Formation and amplification of quasisoliton pulses in head-on stimulated scattering,” Sov. J. Quantum Electron. 8, 576–580 (1978).
[CrossRef]

Friesem, A. A.

Gaeta, A. L.

A. L. Gaeta and R. W. Boyd, “Stimulated Brillouin scattering in the presence of external feedback,” J. Nonlinear Opt. Phys. Mater. 1, 581–594 (1992).
[CrossRef]

Giles, I. P.

Goto, N.

C. K. Jen, J. E. B. Oliveira, N. Goto, and K. Abe, “Role of guided acoustic wave properties in single-mode optical fibre design,” Electron. Lett. 24, 1419–1420 (1988).
[CrossRef]

Hamilton, D. S.

D. Heiman, D. S. Hamilton, and R. W. Hellwarth, “Brillouin scattering measurements on optical glasses,” Phys. Rev. B 19, 6583–6592 (1979).
[CrossRef]

Harigushi, T.

N. Yoshisawa, T. Harigushi, and T. Kuroshima, “Proposal for stimulated Brillouin scattering suppression by fiber cabling,” Electron. Lett. 27, 1100–1101 (1991).
[CrossRef]

Harrison, R. G.

D. Yu, W. Lu, and R. G. Harrison, “Physical origin of dynamical stimulated Brillouin scattering in optical fibers with feedback,” Phys. Rev. A 51, 669–674 (1995).
[CrossRef] [PubMed]

R. G. Harrison, J. S. Uppal, A. Johnstone, and J. V. Moloney, “Evidence of chaotic stimulated Brillouin scattering with weak feedback,” Phys. Rev. Lett. 65, 167–170 (1990).
[CrossRef] [PubMed]

Hasegawa, A.

Heiman, D.

D. Heiman, D. S. Hamilton, and R. W. Hellwarth, “Brillouin scattering measurements on optical glasses,” Phys. Rev. B 19, 6583–6592 (1979).
[CrossRef]

Hellwarth, R. W.

D. Heiman, D. S. Hamilton, and R. W. Hellwarth, “Brillouin scattering measurements on optical glasses,” Phys. Rev. B 19, 6583–6592 (1979).
[CrossRef]

Hill, K. O.

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

Jen, C. K.

C. K. Jen, J. E. B. Oliveira, N. Goto, and K. Abe, “Role of guided acoustic wave properties in single-mode optical fibre design,” Electron. Lett. 24, 1419–1420 (1988).
[CrossRef]

Johnson, D. C.

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

Johnstone, A.

R. G. Harrison, J. S. Uppal, A. Johnstone, and J. V. Moloney, “Evidence of chaotic stimulated Brillouin scattering with weak feedback,” Phys. Rev. Lett. 65, 167–170 (1990).
[CrossRef] [PubMed]

Kaup, D. J.

D. J. Kaup, “The first-order perturbed SBS equations,” J. Nonlinear Sci. 3, 427–443 (1993).
[CrossRef]

Kawasaki, B. S.

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

Kikuchi, K.

Kodama, Y.

Kuroshima, T.

N. Yoshisawa, T. Harigushi, and T. Kuroshima, “Proposal for stimulated Brillouin scattering suppression by fiber cabling,” Electron. Lett. 27, 1100–1101 (1991).
[CrossRef]

Lecoeuche, V.

V. Lecoeuche, B. Ségard, and J. Zemmouri, “Modes of destabilization of Brillouin fiber ring lasers,” Opt. Commun. 134, 547–558 (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 Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 221–228 (1995).
[CrossRef]

Legrand, O.

E. Picholle, C. Montes, C. Leycuras, O. Legrand, and J. Botineau, “Observation of dissipative superluminous solitons in a Brillouin fiber ring laser,” Phys. Rev. Lett. 66, 1454–1457 (1991).
[CrossRef] [PubMed]

Levenson, M. D.

Leycuras, C.

J. Botineau, C. Leycuras, C. Montes, and E. Picholle, “Co-herent modal analysis of a Brillouin fiber ring laser,” Opt. Commun. 15, 126–132 (1994).
[CrossRef]

E. Picholle, C. Montes, C. Leycuras, O. Legrand, and J. Botineau, “Observation of dissipative superluminous solitons in a Brillouin fiber ring laser,” Phys. Rev. Lett. 66, 1454–1457 (1991).
[CrossRef] [PubMed]

J. Botineau, C. Leycuras, C. Montes, and E. Picholle, “Stabilization of a stimulated Brillouin fiber ring laser by strong pump modulation,” J. Opt. Soc. Am. B 6, 300–312 (1989).
[CrossRef]

Lichtman, E.

Lu, W.

D. Yu, W. Lu, and R. G. Harrison, “Physical origin of dynamical stimulated Brillouin scattering in optical fibers with feedback,” Phys. Rev. A 51, 669–674 (1995).
[CrossRef] [PubMed]

Mamhoud, A.

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]

Martinelli, M.

Melloni, A.

Mitschke, F.

M. Dämmig, G. Zinner, F. Mitschke, and H. Welling, “Stimulated Brillouin scattering in fibers with and without external feedback,” Phys. Rev. A 48, 3301–3309 (1993).
[CrossRef] [PubMed]

Moloney, J. V.

R. G. Harrison, J. S. Uppal, A. Johnstone, and J. V. Moloney, “Evidence of chaotic stimulated Brillouin scattering with weak feedback,” Phys. Rev. Lett. 65, 167–170 (1990).
[CrossRef] [PubMed]

Montes, C.

C. Montes, A. Picozzi, and D. Bahloul, “Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons,” Phys. Rev. E 55, 1092–1105 (1997).
[CrossRef]

J. Botineau, C. Leycuras, C. Montes, and E. Picholle, “Co-herent modal analysis of a Brillouin fiber ring laser,” Opt. Commun. 15, 126–132 (1994).
[CrossRef]

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]

E. Picholle, C. Montes, C. Leycuras, O. Legrand, and J. Botineau, “Observation of dissipative superluminous solitons in a Brillouin fiber ring laser,” Phys. Rev. Lett. 66, 1454–1457 (1991).
[CrossRef] [PubMed]

J. Botineau, C. Leycuras, C. Montes, and E. Picholle, “Stabilization of a stimulated Brillouin fiber ring laser by strong pump modulation,” J. Opt. Soc. Am. B 6, 300–312 (1989).
[CrossRef]

C. Montes and J. Coste, “Optical turbulence in multiple stimulated Brillouin backscattering,” Laser Part. Beams 5, 405–411 (1987).
[CrossRef]

J. Coste and C. Montes, “Asymtotic evolution of stimulated Brillouin scattering: implications for optical fibers,” Phys. Rev. A 34, 3940–3949 (1986).
[CrossRef] [PubMed]

Morosov, S. F.

S. F. Morosov, L. V. Piskunova, M. M. Sushik, and G. I. Freidman, “Formation and amplification of quasisoliton pulses in head-on stimulated scattering,” Sov. J. Quantum Electron. 8, 576–580 (1978).
[CrossRef]

Oliveira, J. E. B.

C. K. Jen, J. E. B. Oliveira, N. Goto, and K. Abe, “Role of guided acoustic wave properties in single-mode optical fibre design,” Electron. Lett. 24, 1419–1420 (1988).
[CrossRef]

Pelous, J.

J. Pelous and R. Vacher, “Thermal Brillouin scattering measurements of the attenuation of longitudinal hypersounds in fused quartz from 77 to 300 K,” Solid State Commun. 16, 279–283 (1975).
[CrossRef]

Perlmutter, S. H.

Picholle, E.

E. Picholle and A. Picozzi, “Guided-acoustic-wave resonances in the dynamics of stimulated Brillouin fiber ring laser,” Opt. Commun. 18, 327–330 (1997).
[CrossRef]

J. Botineau, E. Picholle, and D. Bahloul, “Effective stimulated Brillouin gain in singlemode optical fibers,” Electron. Lett. 23, 2032–2033 (1995).
[CrossRef]

J. Botineau, C. Leycuras, C. Montes, and E. Picholle, “Co-herent modal analysis of a Brillouin fiber ring laser,” Opt. Commun. 15, 126–132 (1994).
[CrossRef]

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]

E. Picholle, C. Montes, C. Leycuras, O. Legrand, and J. Botineau, “Observation of dissipative superluminous solitons in a Brillouin fiber ring laser,” Phys. Rev. Lett. 66, 1454–1457 (1991).
[CrossRef] [PubMed]

J. Botineau, C. Leycuras, C. Montes, and E. Picholle, “Stabilization of a stimulated Brillouin fiber ring laser by strong pump modulation,” J. Opt. Soc. Am. B 6, 300–312 (1989).
[CrossRef]

Picozzi, A.

E. Picholle and A. Picozzi, “Guided-acoustic-wave resonances in the dynamics of stimulated Brillouin fiber ring laser,” Opt. Commun. 18, 327–330 (1997).
[CrossRef]

C. Montes, A. Picozzi, and D. Bahloul, “Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons,” Phys. Rev. E 55, 1092–1105 (1997).
[CrossRef]

Pilipetskii, A. N.

B. Ya Zel’dovich and A. N. Pilipetskii, “Role of a soundguide and antisoundguide in stimulated Brillouin scattering in a single-mode waveguide,” Sov. J. Quantum Electron. 183, 818–822 (1988).
[CrossRef]

Piskunova, L. V.

S. F. Morosov, L. V. Piskunova, M. M. Sushik, and G. I. Freidman, “Formation and amplification of quasisoliton pulses in head-on stimulated scattering,” Sov. J. Quantum Electron. 8, 576–580 (1978).
[CrossRef]

Ponikvar, D. R.

Randoux, S.

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 221–228 (1995).
[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]

Root, A. I.

M. O. van Deventer and A. I. Root, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12, 585–590 (1994).
[CrossRef]

Rzazewsky, K.

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

Ségard, B.

V. Lecoeuche, B. Ségard, and J. Zemmouri, “Modes of destabilization of Brillouin fiber ring lasers,” Opt. Commun. 134, 547–558 (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 Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 221–228 (1995).
[CrossRef]

Shaw, H. J.

Shelby, R. M.

Smith, S. P.

Stegeman, G. I. A.

Stokes, L. F.

Sushik, M. M.

S. F. Morosov, L. V. Piskunova, M. M. Sushik, and G. I. Freidman, “Formation and amplification of quasisoliton pulses in head-on stimulated scattering,” Sov. J. Quantum Electron. 8, 576–580 (1978).
[CrossRef]

Takushima, Y.

Thomas, P. J.

Uppal, J. S.

R. G. Harrison, J. S. Uppal, A. Johnstone, and J. V. Moloney, “Evidence of chaotic stimulated Brillouin scattering with weak feedback,” Phys. Rev. Lett. 65, 167–170 (1990).
[CrossRef] [PubMed]

Vacher, R.

J. Pelous and R. Vacher, “Thermal Brillouin scattering measurements of the attenuation of longitudinal hypersounds in fused quartz from 77 to 300 K,” Solid State Commun. 16, 279–283 (1975).
[CrossRef]

van Deventer, M. O.

M. O. van Deventer and A. I. Root, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12, 585–590 (1994).
[CrossRef]

van Driel, H. M.

Waarts, R. G.

Welling, H.

M. Dämmig, G. Zinner, F. Mitschke, and H. Welling, “Stimulated Brillouin scattering in fibers with and without external feedback,” Phys. Rev. A 48, 3301–3309 (1993).
[CrossRef] [PubMed]

Ya Zel’dovich, B.

B. Ya Zel’dovich and A. N. Pilipetskii, “Role of a soundguide and antisoundguide in stimulated Brillouin scattering in a single-mode waveguide,” Sov. J. Quantum Electron. 183, 818–822 (1988).
[CrossRef]

Yoshisawa, N.

N. Yoshisawa, T. Harigushi, and T. Kuroshima, “Proposal for stimulated Brillouin scattering suppression by fiber cabling,” Electron. Lett. 27, 1100–1101 (1991).
[CrossRef]

Yu, D.

D. Yu, W. Lu, and R. G. Harrison, “Physical origin of dynamical stimulated Brillouin scattering in optical fibers with feedback,” Phys. Rev. A 51, 669–674 (1995).
[CrossRef] [PubMed]

Zarinetchi, F.

Zemmouri, J.

V. Lecoeuche, B. Ségard, and J. Zemmouri, “Modes of destabilization of Brillouin fiber ring lasers,” Opt. Commun. 134, 547–558 (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 Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 221–228 (1995).
[CrossRef]

Zinner, G.

M. Dämmig, G. Zinner, F. Mitschke, and H. Welling, “Stimulated Brillouin scattering in fibers with and without external feedback,” Phys. Rev. A 48, 3301–3309 (1993).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys.

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

Electron. Lett.

C. K. Jen, J. E. B. Oliveira, N. Goto, and K. Abe, “Role of guided acoustic wave properties in single-mode optical fibre design,” Electron. Lett. 24, 1419–1420 (1988).
[CrossRef]

J. Botineau, E. Picholle, and D. Bahloul, “Effective stimulated Brillouin gain in singlemode optical fibers,” Electron. Lett. 23, 2032–2033 (1995).
[CrossRef]

N. Yoshisawa, T. Harigushi, and T. Kuroshima, “Proposal for stimulated Brillouin scattering suppression by fiber cabling,” Electron. Lett. 27, 1100–1101 (1991).
[CrossRef]

J. Lightwave Technol.

M. O. van Deventer and A. I. Root, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12, 585–590 (1994).
[CrossRef]

J. Nonlinear Opt. Phys. Mater.

A. L. Gaeta and R. W. Boyd, “Stimulated Brillouin scattering in the presence of external feedback,” J. Nonlinear Opt. Phys. Mater. 1, 581–594 (1992).
[CrossRef]

J. Nonlinear Sci.

D. J. Kaup, “The first-order perturbed SBS equations,” J. Nonlinear Sci. 3, 427–443 (1993).
[CrossRef]

J. Opt. Commun.

D. Cotter, “Stimulated Brillouin scattering in monomode optical fiber,” J. Opt. Commun. 4, 10–19 (1983).
[CrossRef]

J. Opt. Soc. Am. B

Laser Part. Beams

C. Montes and J. Coste, “Optical turbulence in multiple stimulated Brillouin backscattering,” Laser Part. Beams 5, 405–411 (1987).
[CrossRef]

Opt. Commun.

E. Picholle and A. Picozzi, “Guided-acoustic-wave resonances in the dynamics of stimulated Brillouin fiber ring laser,” Opt. Commun. 18, 327–330 (1997).
[CrossRef]

J. Botineau, C. Leycuras, C. Montes, and E. Picholle, “Co-herent modal analysis of a Brillouin fiber ring laser,” Opt. Commun. 15, 126–132 (1994).
[CrossRef]

V. Lecoeuche, B. Ségard, and J. Zemmouri, “Modes of destabilization of Brillouin fiber ring lasers,” Opt. Commun. 134, 547–558 (1997).
[CrossRef]

L. Chen and X. Bao, “Analytic and numerical solutions for the steady state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
[CrossRef]

Opt. Lett.

Phys. Rev. A

M. Dämmig, G. Zinner, F. Mitschke, and H. Welling, “Stimulated Brillouin scattering in fibers with and without external feedback,” Phys. Rev. A 48, 3301–3309 (1993).
[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]

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]

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

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 221–228 (1995).
[CrossRef]

D. Yu, W. Lu, and R. G. Harrison, “Physical origin of dynamical stimulated Brillouin scattering in optical fibers with feedback,” Phys. Rev. A 51, 669–674 (1995).
[CrossRef] [PubMed]

J. Coste and C. Montes, “Asymtotic evolution of stimulated Brillouin scattering: implications for optical fibers,” Phys. Rev. A 34, 3940–3949 (1986).
[CrossRef] [PubMed]

Phys. Rev. B

D. Heiman, D. S. Hamilton, and R. W. Hellwarth, “Brillouin scattering measurements on optical glasses,” Phys. Rev. B 19, 6583–6592 (1979).
[CrossRef]

Phys. Rev. E

C. Montes, A. Picozzi, and D. Bahloul, “Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons,” Phys. Rev. E 55, 1092–1105 (1997).
[CrossRef]

Phys. Rev. Lett.

E. Picholle, C. Montes, C. Leycuras, O. Legrand, and J. Botineau, “Observation of dissipative superluminous solitons in a Brillouin fiber ring laser,” Phys. Rev. Lett. 66, 1454–1457 (1991).
[CrossRef] [PubMed]

R. G. Harrison, J. S. Uppal, A. Johnstone, and J. V. Moloney, “Evidence of chaotic stimulated Brillouin scattering with weak feedback,” Phys. Rev. Lett. 65, 167–170 (1990).
[CrossRef] [PubMed]

Solid State Commun.

J. Pelous and R. Vacher, “Thermal Brillouin scattering measurements of the attenuation of longitudinal hypersounds in fused quartz from 77 to 300 K,” Solid State Commun. 16, 279–283 (1975).
[CrossRef]

Sov. J. Quantum Electron.

B. Ya Zel’dovich and A. N. Pilipetskii, “Role of a soundguide and antisoundguide in stimulated Brillouin scattering in a single-mode waveguide,” Sov. J. Quantum Electron. 183, 818–822 (1988).
[CrossRef]

S. F. Morosov, L. V. Piskunova, M. M. Sushik, and G. I. Freidman, “Formation and amplification of quasisoliton pulses in head-on stimulated scattering,” Sov. J. Quantum Electron. 8, 576–580 (1978).
[CrossRef]

Other

The study of the detuned cavity in the bifurcation region was suggested to us by a referee.

H. M. Gibbs, Optical Bistability: Controlling Light by Light (Academic, New York, 1985).

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992), pp. 263–269.

A. Mamhoud, “Soliton morphogenesis in a cw-pumped Brillouin fiber-laser,” thèse de doctorat (Université de Nice—Sophia Antipolis, Nice, France, 1996).

A. Picozzi, C. Montes, J. Botineau, and E. Picholle, “Inertial model for stimulated Raman scattering inducing chaotic dynamics,” J. Opt. Soc. Am. B 15, 1309–1314 (1998); A. Picozzi, C. Montes, and E. Picholle, “Fast solitary waves against slow inertial instability in stimulated Raman scattering,” Phys. Rev. E 58, 2548–2557 (1998).
[CrossRef]

C. Montes and O. Legrand, “Nonstationary stimulated Brillouin backscattering,” in Electromagnetic and Acoustic Scattering: Detection and Inverse Problems, C. Bourrely, P. Chiappetta, and B. Torresani, eds. (World Scientific, Singapore, 1989), pp. 209–221; O. Legrand and C. Montes, “Apparent superluminous quasi-solitons in stimulated Brillouin backscattering,” J. Phys. Colloq. 50, C3–147–155 (1989).
[CrossRef]

C. Montes, A. Mamhoud, and E. Picholle, “Hopf bifurcation in cw-pumped fiber resonators: generation of stimulated Brillouin solitons,” in Nonlinear Coherent Structures in Physics and Biology, K. H. Spatschek and F. G. Mertens, eds. (Plenum, New York, 1994), pp. 357–363.

W. Lu and R. G. Harrison, “Nonlinear dynamical and chaotic features in stimulated scattering phenomena,” Europhys. Lett. 16, 655–660 (1991); W. Lu, A. Johnstone, and R. G. Harrison, “Deterministic dynamics of stimulated scattering phenomena with external feedback,” Phys. Rev. A 46, 4114–4122 (1992); R. G. Harrison, P. M. Ripley, and W. Lu, “Observation and characterization of deterministic chaos in stimulated Brillouin scattering with weak feedback,” Phys. Rev. A PLRAAN 49, R24–R27 (1994).
[CrossRef] [PubMed]

M. Niklès, “La diffusion Brillouin dans les fibres optiques: Etude et application aux capteurs distribués,” thèse de doctorat (Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 1997).

A comparison round-robin was organized in 1996 by L. Thévenaz (Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland) and F. Ravet (FPMs, Mons, Belgium) in the European program COST 241, to compare and evaluate the measurement procedures of several SBS parameters in fibers.

A. Küng, “L’émission laser par diffusion Brillouin stimulée dans les fibres optiques,” thèse de doctorat (Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 1997).

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Figures (20)

Fig. 1
Fig. 1

Basic lens-and-mirror Brillouin fiber ring laser design (Ar-ion setup).

Fig. 2
Fig. 2

Integrated Brillouin fiber ring laser (Nd:YAG setup).

Fig. 3
Fig. 3

SBS Marginal stability for modes M1 and M2 in the (G, ln R) plane. The curve Rcrit (Im ω=0) separates the stable (Im ω<0) and the unstable (Im ω>0) domains. Point (G0, R0) delineates the subcritical and supercritical bifurcations for N=60. At R=Rcrit, the first destabilized mode is M1 when G<G0 (or R>R0) but is M2 when G>G0 (or R<R0). For N=30 it is always mode M1 that is first destabilized at Rcrit. Below Rpulse and above Rthres a great number of longitudinal modes N˜ of the cavity, beneath the Brillouin gain curve, merge together into a solitonic pulse.

Fig. 4
Fig. 4

Three-wave dynamics [Eqs. (1)] for five feedback values R that correspond to the gain–length G=8>G06 of the Ar-ion experiment: transmitted pump |Ep(L, t)| and backscattered Stokes amplitude |EB(0, t)| versus round-trip time. Left, the first 40 round trips, starting from an initial acoustic noise. Right, the asymptotic behaviors: (a) steady, (b) oscillatory at M2, (c) beating between M2 and M1, (d) oscillatory at M1, and (e) solitonic pulsed. In this and figures that follow, SBSK indicates the SBS Kerr system.

Fig. 5
Fig. 5

Space–time dynamics in the ring cavity during one round trip, showing the periodic counterpropagating process in the asymptotic pulsed regime. Spatial distributions of the locally depleted pump Ep, the amplified Stokes EB, and the acoustic Ea wave envelopes inside the cavity at consecutive time intervals Δt/tr=1/8.

Fig. 6
Fig. 6

Mean reflectivities computed from Eqs. (1) obtained after 1800 and 7200 round trips, showing the long transient evolution in the transition region. For low R the small value of the square mean amplitude |EB(0, t)|2 (dashed curves) with respect to the mean instantaneous intensity |EB(0, t)|2 (solid curve) accounts for the solitonic localization.

Fig. 7
Fig. 7

Mean reflectivity |EB(0, t)|2 computed from the dynamic three-wave model [Eqs. (4)] for γe=0 (asterisks) compared with the analytical stationary output IB(0)st=|EB(0)|2 given by Eq. (6) (solid curve), showing the discrepancy, in the pulsed region (R<0.01), of the mean intensity and the stationary intensity. Dashed curve, |EB(0, t)|2 for the model [Eqs. (4)].

Fig. 8
Fig. 8

Experimental results for the Ar-ion experiment at λp=514.5 nm for R1% and input pump powers P from 200 to 40 mW: (a) stationary Brillouin mirror (P200 mW), (b) oscillating at M8 in the transition region, and (c) stable-pulsed regime (P40 mW).

Fig. 9
Fig. 9

Experimental results for the Nd:YAG experiment at λp=1319 nm for R8%: (a) stationary Brillouin mirror (P100 mW), (b) oscillatory regime at M2 (P7080 mW), (c) oscillatory regime at M1 (P7080 mW), and (d) stable-pulsed regime (P30 mW). The spectrum is included in (b) and (c).

Fig. 10
Fig. 10

Spatial distribution of the three-wave envelopes inside the cavity at t/tr=5120, corresponding to both stable attractors in the middle of bistable domain (b) of Fig. 11 [Ln/(cτ)=12.8, R=0.16]: (a) stationary Brillouin mirror, (b) three-wave solitary structure.

Fig. 11
Fig. 11

Hysteresis curve in the transition region of the subcritical bifurcation corresponding to the Nd:YAG parameters of Table 2 [Ln/(cτ)=12.8] (a) for the actual Kerr parameter κ=1.6×10-3 and (b) for the enhanced Kerr parameter κ=3.9×10-3. Mean Stokes reflectivity |EB(0, t)|2 versus feedback rate ρ=R. Slowly varying ρ in the directions of the arrows causes both branches to bifurcate at different critical values: from the stationary to the pulsed regime at ρ=0.390 for (a) and at ρ=0.395 for case (b) and from the pulsed to the stationary regime at (a) ρ=0.394 and (b) ρ=0.406. Inside the respective hysteresis domains both solutions are asymptotically stable.

Fig. 12
Fig. 12

Long oscillatory transients at fr corresponding to the temporal evolution that separates the two attractor states of Fig. 11 at the boundary of the hysteresis domains. Extrema (|EB(0, t)|max, |EB(0, t)|min) of the Stokes oscillations, the Brillouin solitonic pulses, or both encompass as many as 43,000 round trips for the mirror-to-pulse transition at (a) ρ=0.390 and at (b) ρ=0.395 and as many as 18,000 round trips for the pulse-to-mirror transition at (a) ρ=0.394 and at (b) ρ=0.406.

Fig. 13
Fig. 13

Pulse period Δt for the entire self-pulsing domain ΔG=gLΔP/S of the Nd:YAG experiment (at R=0.08): experimental (points between asterisks) and numerical (solid curve). Their values correspond to subluminous behavior (Δt/tr>1) until P60 mW and to superluminous behavior (Δt/tr<1) at P>60 mW.

Fig. 14
Fig. 14

Mean reflectivity IB(0)/Ip(0) for the entire self-pulsing domain ΔG of Fig. 13: experimental (points between asterisks) and numerical (solid curve).

Fig. 15
Fig. 15

Peak pulse intensity IB(0)/Ip(0) for the entire self-pulsing domain ΔG of Fig. 13: experimental (points between asterisks) and numerical (solid curve).

Fig. 16
Fig. 16

Temporal pulse width δt/tr for the entire self-pulsing domain ΔG of Fig. 16: experimental (points between asterisks) and numerical (solid curve).

Fig. 17
Fig. 17

Experimental soliton pairs corresponding to the Nd:YAG experiment at R=0.8 generated for (a) pump power P=48 mW in the subluminous (Δt/tr=1.0088) self-pulsing domain exhibiting the thinnest δt/tr=0.058 and an amplification rate IB/Ip0=4.83, (b) P=61 mW in the luminous (Δt/tr=1) self-pulsing state with δt/tr=0.094 and IB/Ip0=4.82, (c) P=64 mW in the superluminous (Δt/tr=0.9908) self-pulsing domain with δt/tr=0.107 and IB/Ip0=4.71.

Fig. 18
Fig. 18

Numerical soliton pairs [from Eqs. (1)] corresponding to the Nd:YAG experiment at R=0.0784 (cf. Table 6) generated for (a) a pump power P=61 mW in the subluminous self-pulsing domain exhibiting the thinnest δt/tr=0.075 and an amplification rate IB/Ip0=7.60, (b) P=63 mW in the luminous self-pulsing state with δt/tr=0.076 and IB/Ip0=7.90, (c) P=73 mW in the superluminous self-pulsing domain with δt/tr=0.119 and IB/Ip0=5.47.

Fig. 19
Fig. 19

Numerical train of correlated pulses [from Eqs. (1)] for a ring cavity of N=58 longitudinal modes beneath the Brillouin gain curve in the presence of thermal noise fn/KEp2=10-5. (a) Train of 32 pulses for Stokes amplitude |EB(0, t)|, also showing saw-toothed behavior for the transmitted pump |Ep(L, t)|. (b) Eye diagram for the 32 superimposed pulses, showing its stability. [Ln/(cτ)=27.03, μa=6.75, mue=1μa×10-4, κ=3.7×10-3, N=58 may correspond to L=75 m and P=108 mW for an Ar-ion laser at λp=514.5 nm or to L=490 m and P=50 mW for a Nd:YAG laser at λp=1319 nm.]

Fig. 20
Fig. 20

Numerical train of decorrelated pulses [from Eqs. (1)] for a ring cavity of N=209 longitudinal modes beneath the Brillouin gain curve in the presence of thermal noise fn/KEp2=10-5. (a) Train of 32 pulses for the Stokes amplitude |EB(0, t)| and for the saw-toothed transmitted pump |Ep(L, t)|. (b) Eye diagram for the 32 superimposed pulses showing its decorrelation [G=8, R=0.0196, Ln/(cτ)=51.2, μa=12.8, mue=1μa×10-4, κ=2.62×10-3, N=209 may correspond to L=270 m and P=30 mW for an Ar-ion laser at λp=514.5 nm or to L=1800 m and P=15 mW for a Nd:YAG laser at λp=1319 nm.]

Tables (6)

Tables Icon

Table 1 Computation Parameters for the cw-Pumped Ar-Ion Brillouin Fiber Ring Lasera

Tables Icon

Table 2 Computation Parameters for the cw-Pumped Nd:YAG Brillouin Fiber Ring Lasera

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Table 3 Solution of Dispersion Equation (10): Critical Feedback Rcrit and Oscillation Frequency ω/ωr for Modes M1 and M2 Corresponding to N=60 and N=30a

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Table 4 Pulse Characteristics for the Ar-Ion cw-Pumped Brillouin Fiber Ring Lasers from Eqs. (1)a

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Table 5 Pulse Characteristics for the Nd:YAG cw-Pumped Brillouin Fiber Ring Laser from Eqs. (1)a

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Table 6 Pulse Characteristics for the Nd:YAG cw-Pumped Brillouin Fiber Ring Laser from Eqs. (1)a

Equations (45)

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t+cnx+γeEp=-KEBEa+iKr[|Ep|2+2|EB|2]Ep,
t-cnx+γeEB=KEpEa*+iKr[2|Ep|2+|EB|2]EB,
(t+γa)Ea=KEpEB*+fn,
K=1σ0cn78ρ0cca1/2ωpπp12,
N=nLΔνBc=nLγaπcL[m]5λp2[µm].
Ep(0, t)=Ecw+ρpEp(L, t),EB(L, t)=ρBEB(0, t),
Ep(0, t)=Ecw,EB(L, t)=REB(0, t).
t+cnx+γeEp=-KEBEa,
t-cnx+γeEB=KEpEa*,
(t+γa)Ea=KEpEB*.
[x+(n/c)(t+2γe)]Ip=-gIpIB,
[x-(n/c)(t+2γe)]IB=-gIpIB,
Ip(0, t)=Icw,IB(L, t)=RIB(0, t).
Ipst(x)=2DIp0(2D-Ip0)exp(-2Dx)+Ip0,
IBst(x)=Ipst(x)-2D,
R exp(2DG)=R+2(1-R)D.
d2Ydx2+(ω-ifD)2+f(1-f)D2sinh2[D(x0+x)]Y=0,
f(ω)=1-i(ω/μa2)1-2i(ω/μa2),
Y=½[(1-D)sinh Dx+D cosh Dx)]f(ω)(δIp-δIB),
d(δIp-δIB)dx=iω(δIp+δIB).
Y(x)=z1/4[C1Ai(-z)+C2Bi(-z)],
z=[3Ω(x+1)/2]2/3,Ω=ω-ifD.
A(ω)+B(ω)sin(ΩG)+C(ω)cos(ΩG)=0,
Dcrit=1/6,IBst/Ip0=2/3,
Rcritint=13 exp(G/3)-2.
γeffcnLln1R,
δζ=1p=1μS-μa-1,
δσ=(μS/μa)1/2[1-(μSμa)1/2](1-μS/μa),
EB/Ep0=[1-(μSμa)1/2](2μS/μa-1)1/2,
IB(0, δω)=IB(L)expγa2γa2+(δω)2Geff,
IB(0, δω)IB(L)exp(Geff)exp[-Geff(δω)2/γa2],
Δω=2γaln 2/Geff.
N˜=ΔωδωFSR=γanLπcln 2Geff=Nln 2Geff.
δt0=4γaGeff ln 2,
δδt0tr=4πN˜Geff=4πNGeff ln 2,
Gcritint=3 ln1+2R3R,
IB,minIB,max exp(-1/δ).
N>Nmax84G ln 2π22G,
Y(x)=C1+C2Ω(x+1)sin[Ω(x+1)]+C2-C1Ω(x+1)cos[Ω(x+1)],
A(ω)=2iωR1-f exp(-DfL)a1Ω-a2,
B(ω)=a1b2+a2b1,C(ω)=a2b1-a1b2,
a1=iω-fS0+,
a2=Ω(iω-fS0+1)-Ω,
b1=Ω(L+1)iω+fS0+1L+1+Ω,
b2=iω+fSL-(/L+1).

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