See, for example, the special issue on optical parametric oscillation and amplification, J. Opt. Soc. Am. B 10, 1655–1791 (1993).

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, “Squeezing in detuned optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).

[CrossRef]

F. Arecchi, W. Gadomski, R. Meucci, and J. Roversi, “Delayed bifurcation at the threshold of a swept gain CO2laser,” Opt. Commun. 70, 155–160 (1989).

[CrossRef]

H. Zeghlache, P. Mandel, and C. Van der Broeck, “Influence of noise on delayed bifurcations,” Phys. Rev. A 40, 286–294 (1989).

[CrossRef]
[PubMed]

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Nuovo Cimento 10, 959–977 (1988).

[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of squeezed photon noise on twin laserlike beams,” Phys. Rev. Lett. 59, 2555–2558 (1987).

[CrossRef]
[PubMed]

W. Sharp, M. Squicciarini, D. Bromley, C. Green, J. Tredicce, and L. Narducci, “Experimental observation of a delayed bifurcation at the threshold of an argon laser,” Opt. Commun. 63, 344–348 (1987).

[CrossRef]

B. Morris and F. Moss, “Postponed bifurcations of a quadratic map with a swept parameter,” Phys. Lett. 118A, 117–120 (1986).

G. Broggi, A. Colombo, L. Lugiato, and P. Mandel, “Influence of white noise on delayed bifurcations,” Phys. Rev. A 33, 3635–3637 (1986).

[CrossRef]
[PubMed]

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys Rev. Lett. 57, 2520–2523 (1986).

[CrossRef]
[PubMed]

M. J. Collett and D. F. Walls, “Squeezing spectra for nonlinear optical systems,” Phys. Rev. A 32, 2887–2892 (1985).

[CrossRef]
[PubMed]

P. Mandel and T. Erneux, “Laser Lorenz equations with a time dependent parameter,” Phys. Rev. Lett. 53, 1818–1820 (1984).

[CrossRef]

P. Mandel and T. Erneux, “Temporal aspects of absorptive optical bistability,” Phys. Rev. A 28, 896–909 (1983).

[CrossRef]

P. Drummond, K. McNeil, and D. F. Walls, “Nonequilibrium transitions in subsecond harmonic generation,” Opt. Acta 28, 211–225 (1981).

[CrossRef]

F. Arecchi, W. Gadomski, R. Meucci, and J. Roversi, “Delayed bifurcation at the threshold of a swept gain CO2laser,” Opt. Commun. 70, 155–160 (1989).

[CrossRef]

G. Broggi, A. Colombo, L. Lugiato, and P. Mandel, “Influence of white noise on delayed bifurcations,” Phys. Rev. A 33, 3635–3637 (1986).

[CrossRef]
[PubMed]

W. Sharp, M. Squicciarini, D. Bromley, C. Green, J. Tredicce, and L. Narducci, “Experimental observation of a delayed bifurcation at the threshold of an argon laser,” Opt. Commun. 63, 344–348 (1987).

[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of squeezed photon noise on twin laserlike beams,” Phys. Rev. Lett. 59, 2555–2558 (1987).

[CrossRef]
[PubMed]

M. J. Collett and D. F. Walls, “Squeezing spectra for nonlinear optical systems,” Phys. Rev. A 32, 2887–2892 (1985).

[CrossRef]
[PubMed]

G. Broggi, A. Colombo, L. Lugiato, and P. Mandel, “Influence of white noise on delayed bifurcations,” Phys. Rev. A 33, 3635–3637 (1986).

[CrossRef]
[PubMed]

P. Drummond, K. McNeil, and D. F. Walls, “Nonequilibrium transitions in subsecond harmonic generation,” Opt. Acta 28, 211–225 (1981).

[CrossRef]

P. Mandel and T. Erneux, “Laser Lorenz equations with a time dependent parameter,” Phys. Rev. Lett. 53, 1818–1820 (1984).

[CrossRef]

P. Mandel and T. Erneux, “Temporal aspects of absorptive optical bistability,” Phys. Rev. A 28, 896–909 (1983).

[CrossRef]

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, “Squeezing in detuned optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).

[CrossRef]

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Nuovo Cimento 10, 959–977 (1988).

[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of squeezed photon noise on twin laserlike beams,” Phys. Rev. Lett. 59, 2555–2558 (1987).

[CrossRef]
[PubMed]

C. Fabre and S. Reynaud, “Quantum noise in optical systems: a semiclassical approach,” in Les Houches Session 53, J. Dalibard, J. M. Raimond, and J. Zinn-Justin, eds. (Elsevier, New York, 1992), pp. 675–711.

F. Arecchi, W. Gadomski, R. Meucci, and J. Roversi, “Delayed bifurcation at the threshold of a swept gain CO2laser,” Opt. Commun. 70, 155–160 (1989).

[CrossRef]

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, “Squeezing in detuned optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).

[CrossRef]

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Nuovo Cimento 10, 959–977 (1988).

[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of squeezed photon noise on twin laserlike beams,” Phys. Rev. Lett. 59, 2555–2558 (1987).

[CrossRef]
[PubMed]

This method that treats quantum fluctuations as fluctuations of classical fields is equivalent to the Wigner representation, with truncation of higher-order terms.See R. Graham, “Statistical theory of instabilities in stationary nonequilibrium systems with applications to lasers and nonlinear optics,” in Quantum Statistics in Optics and Solid State Physics, G. Hohler, ed., Vol. 66 of Springer Tracts in Modern Physics (Springer-Verlag, New York, 1973);S. Reynaud and A. Heidmann, “A semiclassical linear input-output transformation for quantum fluctuations,” Opt. Commun. 71, 209–214 (1989).

[CrossRef]

W. Sharp, M. Squicciarini, D. Bromley, C. Green, J. Tredicce, and L. Narducci, “Experimental observation of a delayed bifurcation at the threshold of an argon laser,” Opt. Commun. 63, 344–348 (1987).

[CrossRef]

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys Rev. Lett. 57, 2520–2523 (1986).

[CrossRef]
[PubMed]

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, “Squeezing in detuned optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).

[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of squeezed photon noise on twin laserlike beams,” Phys. Rev. Lett. 59, 2555–2558 (1987).

[CrossRef]
[PubMed]

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Nuovo Cimento 10, 959–977 (1988).

[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of squeezed photon noise on twin laserlike beams,” Phys. Rev. Lett. 59, 2555–2558 (1987).

[CrossRef]
[PubMed]

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, “Squeezing in detuned optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).

[CrossRef]

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys Rev. Lett. 57, 2520–2523 (1986).

[CrossRef]
[PubMed]

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, “Squeezing in detuned optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).

[CrossRef]

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Nuovo Cimento 10, 959–977 (1988).

[CrossRef]

G. Broggi, A. Colombo, L. Lugiato, and P. Mandel, “Influence of white noise on delayed bifurcations,” Phys. Rev. A 33, 3635–3637 (1986).

[CrossRef]
[PubMed]

H. Zeghlache, P. Mandel, and C. Van der Broeck, “Influence of noise on delayed bifurcations,” Phys. Rev. A 40, 286–294 (1989).

[CrossRef]
[PubMed]

G. Broggi, A. Colombo, L. Lugiato, and P. Mandel, “Influence of white noise on delayed bifurcations,” Phys. Rev. A 33, 3635–3637 (1986).

[CrossRef]
[PubMed]

P. Mandel and T. Erneux, “Laser Lorenz equations with a time dependent parameter,” Phys. Rev. Lett. 53, 1818–1820 (1984).

[CrossRef]

P. Mandel and T. Erneux, “Temporal aspects of absorptive optical bistability,” Phys. Rev. A 28, 896–909 (1983).

[CrossRef]

P. Drummond, K. McNeil, and D. F. Walls, “Nonequilibrium transitions in subsecond harmonic generation,” Opt. Acta 28, 211–225 (1981).

[CrossRef]

F. Arecchi, W. Gadomski, R. Meucci, and J. Roversi, “Delayed bifurcation at the threshold of a swept gain CO2laser,” Opt. Commun. 70, 155–160 (1989).

[CrossRef]

B. Morris and F. Moss, “Postponed bifurcations of a quadratic map with a swept parameter,” Phys. Lett. 118A, 117–120 (1986).

B. Morris and F. Moss, “Postponed bifurcations of a quadratic map with a swept parameter,” Phys. Lett. 118A, 117–120 (1986).

W. Sharp, M. Squicciarini, D. Bromley, C. Green, J. Tredicce, and L. Narducci, “Experimental observation of a delayed bifurcation at the threshold of an argon laser,” Opt. Commun. 63, 344–348 (1987).

[CrossRef]

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Nuovo Cimento 10, 959–977 (1988).

[CrossRef]

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, “Squeezing in detuned optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).

[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of squeezed photon noise on twin laserlike beams,” Phys. Rev. Lett. 59, 2555–2558 (1987).

[CrossRef]
[PubMed]

C. Fabre and S. Reynaud, “Quantum noise in optical systems: a semiclassical approach,” in Les Houches Session 53, J. Dalibard, J. M. Raimond, and J. Zinn-Justin, eds. (Elsevier, New York, 1992), pp. 675–711.

F. Arecchi, W. Gadomski, R. Meucci, and J. Roversi, “Delayed bifurcation at the threshold of a swept gain CO2laser,” Opt. Commun. 70, 155–160 (1989).

[CrossRef]

W. Sharp, M. Squicciarini, D. Bromley, C. Green, J. Tredicce, and L. Narducci, “Experimental observation of a delayed bifurcation at the threshold of an argon laser,” Opt. Commun. 63, 344–348 (1987).

[CrossRef]

W. Sharp, M. Squicciarini, D. Bromley, C. Green, J. Tredicce, and L. Narducci, “Experimental observation of a delayed bifurcation at the threshold of an argon laser,” Opt. Commun. 63, 344–348 (1987).

[CrossRef]

W. Sharp, M. Squicciarini, D. Bromley, C. Green, J. Tredicce, and L. Narducci, “Experimental observation of a delayed bifurcation at the threshold of an argon laser,” Opt. Commun. 63, 344–348 (1987).

[CrossRef]

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, “Squeezing in detuned optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).

[CrossRef]

H. Zeghlache, P. Mandel, and C. Van der Broeck, “Influence of noise on delayed bifurcations,” Phys. Rev. A 40, 286–294 (1989).

[CrossRef]
[PubMed]

M. J. Collett and D. F. Walls, “Squeezing spectra for nonlinear optical systems,” Phys. Rev. A 32, 2887–2892 (1985).

[CrossRef]
[PubMed]

P. Drummond, K. McNeil, and D. F. Walls, “Nonequilibrium transitions in subsecond harmonic generation,” Opt. Acta 28, 211–225 (1981).

[CrossRef]

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys Rev. Lett. 57, 2520–2523 (1986).

[CrossRef]
[PubMed]

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys Rev. Lett. 57, 2520–2523 (1986).

[CrossRef]
[PubMed]

H. Zeghlache, P. Mandel, and C. Van der Broeck, “Influence of noise on delayed bifurcations,” Phys. Rev. A 40, 286–294 (1989).

[CrossRef]
[PubMed]

See, for example, the special issue on optical parametric oscillation and amplification, J. Opt. Soc. Am. B 10, 1655–1791 (1993).

L. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Nuovo Cimento 10, 959–977 (1988).

[CrossRef]

P. Drummond, K. McNeil, and D. F. Walls, “Nonequilibrium transitions in subsecond harmonic generation,” Opt. Acta 28, 211–225 (1981).

[CrossRef]

F. Arecchi, W. Gadomski, R. Meucci, and J. Roversi, “Delayed bifurcation at the threshold of a swept gain CO2laser,” Opt. Commun. 70, 155–160 (1989).

[CrossRef]

W. Sharp, M. Squicciarini, D. Bromley, C. Green, J. Tredicce, and L. Narducci, “Experimental observation of a delayed bifurcation at the threshold of an argon laser,” Opt. Commun. 63, 344–348 (1987).

[CrossRef]

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys Rev. Lett. 57, 2520–2523 (1986).

[CrossRef]
[PubMed]

B. Morris and F. Moss, “Postponed bifurcations of a quadratic map with a swept parameter,” Phys. Lett. 118A, 117–120 (1986).

G. Broggi, A. Colombo, L. Lugiato, and P. Mandel, “Influence of white noise on delayed bifurcations,” Phys. Rev. A 33, 3635–3637 (1986).

[CrossRef]
[PubMed]

H. Zeghlache, P. Mandel, and C. Van der Broeck, “Influence of noise on delayed bifurcations,” Phys. Rev. A 40, 286–294 (1989).

[CrossRef]
[PubMed]

P. Mandel and T. Erneux, “Temporal aspects of absorptive optical bistability,” Phys. Rev. A 28, 896–909 (1983).

[CrossRef]

M. J. Collett and D. F. Walls, “Squeezing spectra for nonlinear optical systems,” Phys. Rev. A 32, 2887–2892 (1985).

[CrossRef]
[PubMed]

P. Mandel and T. Erneux, “Laser Lorenz equations with a time dependent parameter,” Phys. Rev. Lett. 53, 1818–1820 (1984).

[CrossRef]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of squeezed photon noise on twin laserlike beams,” Phys. Rev. Lett. 59, 2555–2558 (1987).

[CrossRef]
[PubMed]

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and W. Kaige, “Squeezing in detuned optical parametric oscillators,” Quantum Opt. 2, 159–187 (1990).

[CrossRef]

C. Fabre and S. Reynaud, “Quantum noise in optical systems: a semiclassical approach,” in Les Houches Session 53, J. Dalibard, J. M. Raimond, and J. Zinn-Justin, eds. (Elsevier, New York, 1992), pp. 675–711.

In the case of a degenerate parametric oscillator a1is the common signal and idler field amplitude. In the nondegenerate case, α1is equal to the sum of the amplitude of the signal and idler fields.

This method that treats quantum fluctuations as fluctuations of classical fields is equivalent to the Wigner representation, with truncation of higher-order terms.See R. Graham, “Statistical theory of instabilities in stationary nonequilibrium systems with applications to lasers and nonlinear optics,” in Quantum Statistics in Optics and Solid State Physics, G. Hohler, ed., Vol. 66 of Springer Tracts in Modern Physics (Springer-Verlag, New York, 1973);S. Reynaud and A. Heidmann, “A semiclassical linear input-output transformation for quantum fluctuations,” Opt. Commun. 71, 209–214 (1989).

[CrossRef]

In Ref. 5, the dynamical behavior was studied for a degenerate OPO. Reference 8 demonstrates that all the results of Ref. 5 can be extended to the nondegenerate case provided that the signal and idler modes have the same transmission and loss coefficients.