Abstract

A multiple interferometer is proposed and stimulated Raman radiation, stimulated Mandel’stam–Brillouin radiation, and Cerenkov-type optical parametric oscillation are considered in this interferometer, which is excited by an external light beam. An extremely low threshold intensity of the exciting beam is noted. Optical bistability and hysteresis, self-pulsing of the radiation intensities, amplification of amplitude-phase deviations in the incident pumping beam, limitation of the intensity of the beam transmitted by the interferometer, and nonlinear reflection of the beam from the multiple interferometer are predicted and discussed. It is also noted that the threshold for generation of the second Stokes component of stimulated Raman radiation and stimulated Mandel’stam–Brillouin radiation under the conditions discussed is usually not achieved, even for high excess of the exciting beam intensity over the threshold for generation of the first Stokes component.

© 1983 Optical Society of America

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References

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  1. V. N. Lugovoi, “Bistability and self-pulsing in Cerenkov-type optical parametric interactions,” IEEE J. Quantum Electron. QE-17, 384–386 (1981).
    [CrossRef]
  2. V. N. Lugovoi, “Cerenkov-type optical parametric oscillation in a double Fabry–Perot interferometer,” Kvant. Elektron. (Moscow) 9, 1653–1658 (1982).
  3. V. N. Lugovoi, “On stimulated Raman emission in an optical resonator,” Zh. Eksp. Teor. Fiz. 56, 683–693 (1969).
  4. V. N. Lugovoi, “On the theory of a non-linear optical resonator,” Opt. Acta 24, 743–756 (1977).
    [CrossRef]
  5. V. N. Lugovoi, “Nonlinear optical resonators (excited by external radiation),” Kvant. Elektron. 6, 2053–2077 (1979).
  6. T. Bischofberger and Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferometer,” Phys. Rev. A 19, 1169–1176 (1979).
    [CrossRef]
  7. H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
    [CrossRef]
  8. D. A. B. Miller, S. D. Smith, and A. Johnston, “Optical bistability and signal amplification in a semiconductor crystal: applications of new low-power nonlinear effects in InSb,” Appl. Phys. Lett. 35, 658–663 (1979).
    [CrossRef]
  9. D. F. Walls, P. D. Drummond, and K. J. McNeil, “Bistable systems in nonlinear optics,” in Optical Bistability, C. M. Bowden and et al., eds. (Plenum, New York, 1981), pp. 51–83.
    [CrossRef]
  10. A. Schenzle and H. Brand, “Multiplicative stochastic processes in statistical physics,” Phys. Rev. A 20, 1628–1647 (1979).
    [CrossRef]
  11. H. Brand, R. Graham, and A. Schenzle, “Wigner distribution of bistable steady states in optical subharmonic generation,” Opt. Commun. 32, 359–364 (1980).
    [CrossRef]
  12. L. M. Narducci, R. Gilmore, D. H. Feng, and G. S. Agarwal, “Absorption spectrum of optically bistable systems,” Phys. Rev. A 20, 545–549 (1979).
    [CrossRef]
  13. M. Reid, K. J. McNeil, and D. F. Walls, “Unified approach to multiphoton lasers and multiphoton bistability,” Phys. Rev. A 24, 2029–2043 (1981).
    [CrossRef]
  14. V. N. Lugovoi, Introduction to the Theory of Stimulated Raman Scattering(Nauka, Moscow, 1968).
  15. V. N. Lugovoi, “New type of light parametric oscillator and amplifier,” Pis’ma Zh. Eksp. Teor. Fiz. 25, 563–566 (1977).
  16. V. N. Lugovoi, “Inverse Cerenkov effect and new phenomena in non-linear optics,” Opt. Acta 25, 337–349 (1978).
    [CrossRef]
  17. V. N. Lugovoi, “Bistability and hysteresis phenomena in an optical parametric oscillator,” Phys. Status Solidi B 94, 79–86 (1979).
    [CrossRef]
  18. V. N. Lugovoi, “On the saturation in a Cerenkov-type optical parametric oscillator,” Phys. Lett. A 78, 338–340 (1980).
    [CrossRef]
  19. V. N. Lugovoi, “Theory of a Cerenkov-type optical parametric oscillation,” Kvant. Elektron. 7, 2093–2104 (1980).
  20. W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, 1429–1450 (1964).
    [CrossRef]
  21. V. N. Lugovoi, “On molecular oscillator with two natural frequencies of the resonator in the limits of the emission linewidth,” Radioteckh. Elektron. 6, 1700–1706 (1961).
  22. V. N. Lugovoi, “On mutual quenching of axial modes in two-mode lasers,” Kvant. Elektron. 5, 344–348 (1978).
  23. V. N. Lugovoi, “On the mutual quenching of modes in stimulated Raman emission,” Phys. Lett. A 69, 402–404 (1979).
    [CrossRef]
  24. V. N. Lugovoi, “Mutual quenching of modes at second harmonic generation,” Zh. Eksp. Teor. Fiz. 76, 1943–1949 (1979).
  25. V. N. Lugovoi, “On the theory of stimulated Raman radiation in an optical resonator,” Opt. Acta 27, 1551–1565 (1980).
    [CrossRef]

1982 (1)

V. N. Lugovoi, “Cerenkov-type optical parametric oscillation in a double Fabry–Perot interferometer,” Kvant. Elektron. (Moscow) 9, 1653–1658 (1982).

1981 (2)

V. N. Lugovoi, “Bistability and self-pulsing in Cerenkov-type optical parametric interactions,” IEEE J. Quantum Electron. QE-17, 384–386 (1981).
[CrossRef]

M. Reid, K. J. McNeil, and D. F. Walls, “Unified approach to multiphoton lasers and multiphoton bistability,” Phys. Rev. A 24, 2029–2043 (1981).
[CrossRef]

1980 (4)

H. Brand, R. Graham, and A. Schenzle, “Wigner distribution of bistable steady states in optical subharmonic generation,” Opt. Commun. 32, 359–364 (1980).
[CrossRef]

V. N. Lugovoi, “On the saturation in a Cerenkov-type optical parametric oscillator,” Phys. Lett. A 78, 338–340 (1980).
[CrossRef]

V. N. Lugovoi, “Theory of a Cerenkov-type optical parametric oscillation,” Kvant. Elektron. 7, 2093–2104 (1980).

V. N. Lugovoi, “On the theory of stimulated Raman radiation in an optical resonator,” Opt. Acta 27, 1551–1565 (1980).
[CrossRef]

1979 (9)

V. N. Lugovoi, “Bistability and hysteresis phenomena in an optical parametric oscillator,” Phys. Status Solidi B 94, 79–86 (1979).
[CrossRef]

V. N. Lugovoi, “On the mutual quenching of modes in stimulated Raman emission,” Phys. Lett. A 69, 402–404 (1979).
[CrossRef]

V. N. Lugovoi, “Mutual quenching of modes at second harmonic generation,” Zh. Eksp. Teor. Fiz. 76, 1943–1949 (1979).

L. M. Narducci, R. Gilmore, D. H. Feng, and G. S. Agarwal, “Absorption spectrum of optically bistable systems,” Phys. Rev. A 20, 545–549 (1979).
[CrossRef]

V. N. Lugovoi, “Nonlinear optical resonators (excited by external radiation),” Kvant. Elektron. 6, 2053–2077 (1979).

T. Bischofberger and Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferometer,” Phys. Rev. A 19, 1169–1176 (1979).
[CrossRef]

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

D. A. B. Miller, S. D. Smith, and A. Johnston, “Optical bistability and signal amplification in a semiconductor crystal: applications of new low-power nonlinear effects in InSb,” Appl. Phys. Lett. 35, 658–663 (1979).
[CrossRef]

A. Schenzle and H. Brand, “Multiplicative stochastic processes in statistical physics,” Phys. Rev. A 20, 1628–1647 (1979).
[CrossRef]

1978 (2)

V. N. Lugovoi, “Inverse Cerenkov effect and new phenomena in non-linear optics,” Opt. Acta 25, 337–349 (1978).
[CrossRef]

V. N. Lugovoi, “On mutual quenching of axial modes in two-mode lasers,” Kvant. Elektron. 5, 344–348 (1978).

1977 (2)

V. N. Lugovoi, “On the theory of a non-linear optical resonator,” Opt. Acta 24, 743–756 (1977).
[CrossRef]

V. N. Lugovoi, “New type of light parametric oscillator and amplifier,” Pis’ma Zh. Eksp. Teor. Fiz. 25, 563–566 (1977).

1969 (1)

V. N. Lugovoi, “On stimulated Raman emission in an optical resonator,” Zh. Eksp. Teor. Fiz. 56, 683–693 (1969).

1964 (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, 1429–1450 (1964).
[CrossRef]

1961 (1)

V. N. Lugovoi, “On molecular oscillator with two natural frequencies of the resonator in the limits of the emission linewidth,” Radioteckh. Elektron. 6, 1700–1706 (1961).

Agarwal, G. S.

L. M. Narducci, R. Gilmore, D. H. Feng, and G. S. Agarwal, “Absorption spectrum of optically bistable systems,” Phys. Rev. A 20, 545–549 (1979).
[CrossRef]

Bischofberger, T.

T. Bischofberger and Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferometer,” Phys. Rev. A 19, 1169–1176 (1979).
[CrossRef]

Brand, H.

H. Brand, R. Graham, and A. Schenzle, “Wigner distribution of bistable steady states in optical subharmonic generation,” Opt. Commun. 32, 359–364 (1980).
[CrossRef]

A. Schenzle and H. Brand, “Multiplicative stochastic processes in statistical physics,” Phys. Rev. A 20, 1628–1647 (1979).
[CrossRef]

Drummond, P. D.

D. F. Walls, P. D. Drummond, and K. J. McNeil, “Bistable systems in nonlinear optics,” in Optical Bistability, C. M. Bowden and et al., eds. (Plenum, New York, 1981), pp. 51–83.
[CrossRef]

Feng, D. H.

L. M. Narducci, R. Gilmore, D. H. Feng, and G. S. Agarwal, “Absorption spectrum of optically bistable systems,” Phys. Rev. A 20, 545–549 (1979).
[CrossRef]

Gibbs, H. M.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

Gilmore, R.

L. M. Narducci, R. Gilmore, D. H. Feng, and G. S. Agarwal, “Absorption spectrum of optically bistable systems,” Phys. Rev. A 20, 545–549 (1979).
[CrossRef]

Gossard, A. C.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

Graham, R.

H. Brand, R. Graham, and A. Schenzle, “Wigner distribution of bistable steady states in optical subharmonic generation,” Opt. Commun. 32, 359–364 (1980).
[CrossRef]

Johnston, A.

D. A. B. Miller, S. D. Smith, and A. Johnston, “Optical bistability and signal amplification in a semiconductor crystal: applications of new low-power nonlinear effects in InSb,” Appl. Phys. Lett. 35, 658–663 (1979).
[CrossRef]

Lamb, W. E.

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, 1429–1450 (1964).
[CrossRef]

Lugovoi, V. N.

V. N. Lugovoi, “Cerenkov-type optical parametric oscillation in a double Fabry–Perot interferometer,” Kvant. Elektron. (Moscow) 9, 1653–1658 (1982).

V. N. Lugovoi, “Bistability and self-pulsing in Cerenkov-type optical parametric interactions,” IEEE J. Quantum Electron. QE-17, 384–386 (1981).
[CrossRef]

V. N. Lugovoi, “On the theory of stimulated Raman radiation in an optical resonator,” Opt. Acta 27, 1551–1565 (1980).
[CrossRef]

V. N. Lugovoi, “Theory of a Cerenkov-type optical parametric oscillation,” Kvant. Elektron. 7, 2093–2104 (1980).

V. N. Lugovoi, “On the saturation in a Cerenkov-type optical parametric oscillator,” Phys. Lett. A 78, 338–340 (1980).
[CrossRef]

V. N. Lugovoi, “Bistability and hysteresis phenomena in an optical parametric oscillator,” Phys. Status Solidi B 94, 79–86 (1979).
[CrossRef]

V. N. Lugovoi, “Nonlinear optical resonators (excited by external radiation),” Kvant. Elektron. 6, 2053–2077 (1979).

V. N. Lugovoi, “Mutual quenching of modes at second harmonic generation,” Zh. Eksp. Teor. Fiz. 76, 1943–1949 (1979).

V. N. Lugovoi, “On the mutual quenching of modes in stimulated Raman emission,” Phys. Lett. A 69, 402–404 (1979).
[CrossRef]

V. N. Lugovoi, “Inverse Cerenkov effect and new phenomena in non-linear optics,” Opt. Acta 25, 337–349 (1978).
[CrossRef]

V. N. Lugovoi, “On mutual quenching of axial modes in two-mode lasers,” Kvant. Elektron. 5, 344–348 (1978).

V. N. Lugovoi, “New type of light parametric oscillator and amplifier,” Pis’ma Zh. Eksp. Teor. Fiz. 25, 563–566 (1977).

V. N. Lugovoi, “On the theory of a non-linear optical resonator,” Opt. Acta 24, 743–756 (1977).
[CrossRef]

V. N. Lugovoi, “On stimulated Raman emission in an optical resonator,” Zh. Eksp. Teor. Fiz. 56, 683–693 (1969).

V. N. Lugovoi, “On molecular oscillator with two natural frequencies of the resonator in the limits of the emission linewidth,” Radioteckh. Elektron. 6, 1700–1706 (1961).

V. N. Lugovoi, Introduction to the Theory of Stimulated Raman Scattering(Nauka, Moscow, 1968).

McCall, S. L.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

McNeil, K. J.

M. Reid, K. J. McNeil, and D. F. Walls, “Unified approach to multiphoton lasers and multiphoton bistability,” Phys. Rev. A 24, 2029–2043 (1981).
[CrossRef]

D. F. Walls, P. D. Drummond, and K. J. McNeil, “Bistable systems in nonlinear optics,” in Optical Bistability, C. M. Bowden and et al., eds. (Plenum, New York, 1981), pp. 51–83.
[CrossRef]

Miller, D. A. B.

D. A. B. Miller, S. D. Smith, and A. Johnston, “Optical bistability and signal amplification in a semiconductor crystal: applications of new low-power nonlinear effects in InSb,” Appl. Phys. Lett. 35, 658–663 (1979).
[CrossRef]

Narducci, L. M.

L. M. Narducci, R. Gilmore, D. H. Feng, and G. S. Agarwal, “Absorption spectrum of optically bistable systems,” Phys. Rev. A 20, 545–549 (1979).
[CrossRef]

Passner, A.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

Reid, M.

M. Reid, K. J. McNeil, and D. F. Walls, “Unified approach to multiphoton lasers and multiphoton bistability,” Phys. Rev. A 24, 2029–2043 (1981).
[CrossRef]

Schenzle, A.

H. Brand, R. Graham, and A. Schenzle, “Wigner distribution of bistable steady states in optical subharmonic generation,” Opt. Commun. 32, 359–364 (1980).
[CrossRef]

A. Schenzle and H. Brand, “Multiplicative stochastic processes in statistical physics,” Phys. Rev. A 20, 1628–1647 (1979).
[CrossRef]

Shen, Y. R.

T. Bischofberger and Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferometer,” Phys. Rev. A 19, 1169–1176 (1979).
[CrossRef]

Smith, S. D.

D. A. B. Miller, S. D. Smith, and A. Johnston, “Optical bistability and signal amplification in a semiconductor crystal: applications of new low-power nonlinear effects in InSb,” Appl. Phys. Lett. 35, 658–663 (1979).
[CrossRef]

Venkatesan, T. N. C.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

Walls, D. F.

M. Reid, K. J. McNeil, and D. F. Walls, “Unified approach to multiphoton lasers and multiphoton bistability,” Phys. Rev. A 24, 2029–2043 (1981).
[CrossRef]

D. F. Walls, P. D. Drummond, and K. J. McNeil, “Bistable systems in nonlinear optics,” in Optical Bistability, C. M. Bowden and et al., eds. (Plenum, New York, 1981), pp. 51–83.
[CrossRef]

Wiegmann, W.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

Appl. Phys. Lett. (2)

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, “Optical bistability in semiconductors,” Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

D. A. B. Miller, S. D. Smith, and A. Johnston, “Optical bistability and signal amplification in a semiconductor crystal: applications of new low-power nonlinear effects in InSb,” Appl. Phys. Lett. 35, 658–663 (1979).
[CrossRef]

IEEE J. Quantum Electron. (1)

V. N. Lugovoi, “Bistability and self-pulsing in Cerenkov-type optical parametric interactions,” IEEE J. Quantum Electron. QE-17, 384–386 (1981).
[CrossRef]

Kvant. Elektron. (3)

V. N. Lugovoi, “Nonlinear optical resonators (excited by external radiation),” Kvant. Elektron. 6, 2053–2077 (1979).

V. N. Lugovoi, “Theory of a Cerenkov-type optical parametric oscillation,” Kvant. Elektron. 7, 2093–2104 (1980).

V. N. Lugovoi, “On mutual quenching of axial modes in two-mode lasers,” Kvant. Elektron. 5, 344–348 (1978).

Kvant. Elektron. (Moscow) (1)

V. N. Lugovoi, “Cerenkov-type optical parametric oscillation in a double Fabry–Perot interferometer,” Kvant. Elektron. (Moscow) 9, 1653–1658 (1982).

Opt. Acta (3)

V. N. Lugovoi, “On the theory of a non-linear optical resonator,” Opt. Acta 24, 743–756 (1977).
[CrossRef]

V. N. Lugovoi, “Inverse Cerenkov effect and new phenomena in non-linear optics,” Opt. Acta 25, 337–349 (1978).
[CrossRef]

V. N. Lugovoi, “On the theory of stimulated Raman radiation in an optical resonator,” Opt. Acta 27, 1551–1565 (1980).
[CrossRef]

Opt. Commun. (1)

H. Brand, R. Graham, and A. Schenzle, “Wigner distribution of bistable steady states in optical subharmonic generation,” Opt. Commun. 32, 359–364 (1980).
[CrossRef]

Phys. Lett. A (2)

V. N. Lugovoi, “On the saturation in a Cerenkov-type optical parametric oscillator,” Phys. Lett. A 78, 338–340 (1980).
[CrossRef]

V. N. Lugovoi, “On the mutual quenching of modes in stimulated Raman emission,” Phys. Lett. A 69, 402–404 (1979).
[CrossRef]

Phys. Rev. (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, 1429–1450 (1964).
[CrossRef]

Phys. Rev. A (4)

L. M. Narducci, R. Gilmore, D. H. Feng, and G. S. Agarwal, “Absorption spectrum of optically bistable systems,” Phys. Rev. A 20, 545–549 (1979).
[CrossRef]

M. Reid, K. J. McNeil, and D. F. Walls, “Unified approach to multiphoton lasers and multiphoton bistability,” Phys. Rev. A 24, 2029–2043 (1981).
[CrossRef]

A. Schenzle and H. Brand, “Multiplicative stochastic processes in statistical physics,” Phys. Rev. A 20, 1628–1647 (1979).
[CrossRef]

T. Bischofberger and Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferometer,” Phys. Rev. A 19, 1169–1176 (1979).
[CrossRef]

Phys. Status Solidi B (1)

V. N. Lugovoi, “Bistability and hysteresis phenomena in an optical parametric oscillator,” Phys. Status Solidi B 94, 79–86 (1979).
[CrossRef]

Pis’ma Zh. Eksp. Teor. Fiz. (1)

V. N. Lugovoi, “New type of light parametric oscillator and amplifier,” Pis’ma Zh. Eksp. Teor. Fiz. 25, 563–566 (1977).

Radioteckh. Elektron. (1)

V. N. Lugovoi, “On molecular oscillator with two natural frequencies of the resonator in the limits of the emission linewidth,” Radioteckh. Elektron. 6, 1700–1706 (1961).

Zh. Eksp. Teor. Fiz. (2)

V. N. Lugovoi, “Mutual quenching of modes at second harmonic generation,” Zh. Eksp. Teor. Fiz. 76, 1943–1949 (1979).

V. N. Lugovoi, “On stimulated Raman emission in an optical resonator,” Zh. Eksp. Teor. Fiz. 56, 683–693 (1969).

Other (2)

D. F. Walls, P. D. Drummond, and K. J. McNeil, “Bistable systems in nonlinear optics,” in Optical Bistability, C. M. Bowden and et al., eds. (Plenum, New York, 1981), pp. 51–83.
[CrossRef]

V. N. Lugovoi, Introduction to the Theory of Stimulated Raman Scattering(Nauka, Moscow, 1968).

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Figures (5)

Fig. 1
Fig. 1

Plane pumping wave of intensity I+ is incident at an angle φ onto one of the mirrors of the double Fabry–Perot interferometer formed by external mirrors ① and ② and internal mirrors adjacent to a waveguide layer ⑤ on both sides. The mirrors are assumed to reflect at the frequency ωp of the incident wave, and the frequency ωp is assumed to be in the transmission bands of both external and internal Fabry–Perot interferometers. Mirrors ③ and ④ are assumed to be adjacent to the end surfaces of the layer ⑤, forming the Fabry–Perot resonator for the radiation generated.

Fig. 2
Fig. 2

Plane pumping wave of intensity I+ is incident onto one of the mirrors of the triple interferometer consisting of an external ring interferometer and of the internal double Fabry–Perot interferometer.

Fig. 3
Fig. 3

Plane pumping wave of intensity I+ is incident onto one of the mirrors of the triple interferometer consisting of an external ring interferometer and internal double interferometer, the double interferometer consisting, in turn, of an external ring interferometer and the internal Fabry–Perot interferometer.

Fig. 4
Fig. 4

Intensity of stationary oscillations of the exciting ( | Z | 2 ) and the generated ( w ) modes versus the external beam intensity (|F|2). Solid curves give stable regimes, and dashed curves correspond to unstable ones. Arrows show the hysteresis for a cyclic variation of the pumping intensity. Optical bistability is observed at | F | S 2 < | F | 2 < | F | thr 2.

Fig. 5
Fig. 5

Self-pulsing of the exciting (|Z|2) and the generated (w) modes in time.

Equations (18)

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Z ˙ = ( μ n + i Δ n ) Z ω n ρ Z w + i F , w ˙ = 2 μ 1 w + 2 ω 1 ρ | Z | 2 w ,
w = | Y 1 | 2 , ρ = 2 π μ μ + i Δ Γ | ( E n E 1 ) | 2 d r | E n | 2 d r ,
Γ = 3 λ 1 4 2 9 π 5 n 0 Q 0 μ .
N τ = 1 4 π | E τ | 2 d r
Δ 1 = ω 1 ρ | Z | 2 .
ω p ω s + ω ν ,
k s ± k ν + k p cos θ
w = | Y s | 2 , ρ = ω ν | a | 2 4 N n 2 ( μ ν + i Δ ν ) , a = χ ̂ ( ω s ) : E s E n * E ν d r , Δ ν ω s + ω ν ω p ,
w = | ρ | 2 { ρ μ n ρ Δ n ± [ ( | F | 2 | Z | 2 μ n 2 Δ n 2 ) | ρ | 2 + ( ρ μ n + ρ Δ n ) 2 ] 1 / 2 } | Z | 2 = μ 1 / ω 1 ρ .
D ( λ ) = | A λ E | = λ 3 + 2 q 1 λ 2 + ( q 1 2 + q 2 2 + 4 μ 1 ρ w ) λ + 4 μ 1 ( q 1 ρ + q 2 ρ ) w ,
f 1 , 2 > 0 ,
f 1 = μ n ρ + Δ n ρ + | ρ | 2 w , f 2 = b 0 w 3 + b 1 w 2 + b 2 w + b 3 , b 0 = ρ | ρ | 2 , b 1 = μ n ( 3 ρ 2 + ρ 2 ) + 2 Δ n ρ ρ + 2 μ 1 ( ρ 2 ρ 2 ) , b 2 = ρ ( 3 μ n 2 + Δ n 2 ) + 2 μ n Δ n ρ + 2 μ 1 ( μ n ρ Δ n ρ ) , b 3 = μ n ( μ n 2 + Δ n 2 ) .
| F | 2 > | F | thr 2 = ( μ n 2 + Δ n 2 ) μ 1 / ω 1 ρ
| F | S 2 = | F | thr 2 [ 1 ( μ n μ Δ n Δ ) 2 ( μ n 2 + Δ n 2 ) ( μ 2 + Δ 2 ) ] .
| E + | thr 2 K f | T 1 T 2 T n | 2 × res μ 1 ( μ n 2 + Δ n 2 ) ( μ 2 + Δ 2 ) L res 8 π ω 1 Γ μ 2 μ n 2 L res act ,
| E + | thr 2 K f | T 1 T 2 T n | 2 × res 2 μ s μ ν ( μ n 2 + Δ n 2 ) ( μ ν 2 + Δ ν 2 4 π 2 ω s ω ν | χ ̂ ( ω s ) | 2 μ n 2 μ ν 2 .
w = μ n ρ [ ( I + I + thr ) 1 / 2 1 ] .
ξ K f | T 1 T 2 T n | 2 S 2 | T 1 | 2 S 1 ,