Abstract

The effects of mode coupling on the static and dynamic responses of a nonlinear interferometer that is pumped by two beams are investigated. Novel forms of bistable operation are identified along with self-pulsing.

© 1986 Optical Society of America

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  1. For a recent review see Optical Bistability, Dynamic Nonlinearity, and Photonic Logic, B. S. Wherrett and S. D. Smith eds. (Cambridge U. Press, Cambridge, 1985).
  2. R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A 18, 1129–1144 (1978); R. Bonifacio and P. Meystre, “Critical slowing down in optical bistability,” Opt. Commun. 29, 131–134 (1979).
    [CrossRef]
  3. S. L. McCall, “Instability and regenerative pulsation phenomena in Fabry–Perot nonlinear optic media,” Appl. Phys. Lett. 32, 284–286 (1978); R. Bonifacio, M. Gronchi, and L. A. Lugiato, “Self-pulsing in bistable operation,” Opt. Commun. 30, 129–133 (1979).
    [CrossRef]
  4. K. Ikeda, H. Diado, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980); H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, “Observation of chaos in optical bistability,” Phys. Rev. Lett. 46, 474–477 (1981).
    [CrossRef]
  5. G. P. Agrawal, “Effect of mode coupling on optical bistability in a bidirectional ring cavity,” Appl. Phys. Lett. 38, 505–507 (1981).
    [CrossRef]
  6. G. P. Agrawal, “Use of a bidirectional ring cavity for optical bistable devices,” IEEE J. Quantum Electron. QE-18, 214–218 (1982).
    [CrossRef]
  7. M. L. Asquini and F. Casagrande, “Optical bistability in a bidirectional ring cavity,” Z. Phys. B 44, 233–239 (1981).
    [CrossRef]
  8. A. E. Kaplan and P. Meystre, “Enhancement of the Sagnac effect due to nonlinearly induced nonreciprocity,” Opt. Lett. 6, 590–592 (1981).
    [CrossRef] [PubMed]
  9. A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40, 229–232 (1982).
    [CrossRef]
  10. A. E. Kaplan, “Optoelectronic enhancement of the Sagnac effect in a ring resonator and related effect of directional bistability,” Appl. Phys. Lett. 42, 479–481 (1983).
    [CrossRef]
  11. M.-M. Cheung, S. D. Durbin, and Y. R. Shen, “Optical bistability and self-oscillation of a nonlinear Fabry-Perot interferometer filled with a nematic-liquid-crystal film,” Opt. Lett. 8, 39–41 (1983); S. M. Arakelyan, A. S. Karayan, and Yu. S. Chilingaryan, “Dynamics of a nonlinear Fabry–Perot cavity with a nematic liquid crystal,” Opt. Spectrosc. (USSR) 55, 298–301 (1983); Y. R. Shen, “Optical nonlinearity and bistability in liquid crystals,” Phil. Trans. R. Soc. Lond. A 313, 327–332 (1984).
    [CrossRef] [PubMed]
  12. J. V. Moloney, F. A. Hopf, and H. M. Gibbs, “Effects of transverse beam variations on bifurcatons in an intrinsic bistable ring cavity,” Phys. Rev. A 25, 3442–3445 (1982); J. V. Moloney, “Self-focusing-induced optical turbulence,” Phys. Rev. Lett. 53, 556–559 (1984).
    [CrossRef]
  13. T. Bischofberger and Y. R. Shen, “Transient behavior of a nonlinear Fabry–Perot,” Appl. Phys. Lett. 32, 156–158 (1978); J. A. Goldstone and E. M. Garmire, “On the dynamic response of nonlinear Fabry-Perot interferometers,” IEEE J. Quantum Electron. QE-17, 366–373 (1981).
    [CrossRef]
  14. A. E. Kaplan, “Light-induced nonreciprocity, field invariants, and nonlinear eigenpolarizations,” Opt. Lett. 8, 560 (1983).
    [CrossRef] [PubMed]
  15. E. M. Wright, W. J. Firth, and I. Galbraith, “Beam propagation in a medium with a diffusive Kerr-type nonlinearity,” J. Opt. Soc. Am. B 2, 383–386 (1985).
    [CrossRef]
  16. D. J. Hagan, H. A. Mackenzie, H. A. Al Attar, and W. J. Firth, “Carrier diffusion measurement in InSb by the angular dependence of degenerate four-wave mixing,” Opt. Lett. 10, 187–189 (1985).
    [CrossRef] [PubMed]
  17. Y. Silberberg and I. Bar-Joseph, “Optical instabilities in a nonlinear Kerr medium,” J. Opt. Soc. Am. B 1, 662–670 (1984).
    [CrossRef]
  18. A. E. Kaplan and C. T. Law, “Isolas in four-wave mixing optical bistability,” IEEE J. Quantum Electron. (to be published).
  19. K. Ujihara, “Mean field theory of optical bistability in a phase-conjugate resonator using nearly degenerate four-wave mixing,” IEEE J. Quantum Electron. (to be published).
  20. L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, “Breathing, spiking, and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
    [CrossRef]

1985 (2)

1984 (1)

1983 (4)

1982 (3)

J. V. Moloney, F. A. Hopf, and H. M. Gibbs, “Effects of transverse beam variations on bifurcatons in an intrinsic bistable ring cavity,” Phys. Rev. A 25, 3442–3445 (1982); J. V. Moloney, “Self-focusing-induced optical turbulence,” Phys. Rev. Lett. 53, 556–559 (1984).
[CrossRef]

A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40, 229–232 (1982).
[CrossRef]

G. P. Agrawal, “Use of a bidirectional ring cavity for optical bistable devices,” IEEE J. Quantum Electron. QE-18, 214–218 (1982).
[CrossRef]

1981 (3)

M. L. Asquini and F. Casagrande, “Optical bistability in a bidirectional ring cavity,” Z. Phys. B 44, 233–239 (1981).
[CrossRef]

A. E. Kaplan and P. Meystre, “Enhancement of the Sagnac effect due to nonlinearly induced nonreciprocity,” Opt. Lett. 6, 590–592 (1981).
[CrossRef] [PubMed]

G. P. Agrawal, “Effect of mode coupling on optical bistability in a bidirectional ring cavity,” Appl. Phys. Lett. 38, 505–507 (1981).
[CrossRef]

1980 (1)

K. Ikeda, H. Diado, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980); H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, “Observation of chaos in optical bistability,” Phys. Rev. Lett. 46, 474–477 (1981).
[CrossRef]

1978 (3)

R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A 18, 1129–1144 (1978); R. Bonifacio and P. Meystre, “Critical slowing down in optical bistability,” Opt. Commun. 29, 131–134 (1979).
[CrossRef]

S. L. McCall, “Instability and regenerative pulsation phenomena in Fabry–Perot nonlinear optic media,” Appl. Phys. Lett. 32, 284–286 (1978); R. Bonifacio, M. Gronchi, and L. A. Lugiato, “Self-pulsing in bistable operation,” Opt. Commun. 30, 129–133 (1979).
[CrossRef]

T. Bischofberger and Y. R. Shen, “Transient behavior of a nonlinear Fabry–Perot,” Appl. Phys. Lett. 32, 156–158 (1978); J. A. Goldstone and E. M. Garmire, “On the dynamic response of nonlinear Fabry-Perot interferometers,” IEEE J. Quantum Electron. QE-17, 366–373 (1981).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, “Use of a bidirectional ring cavity for optical bistable devices,” IEEE J. Quantum Electron. QE-18, 214–218 (1982).
[CrossRef]

G. P. Agrawal, “Effect of mode coupling on optical bistability in a bidirectional ring cavity,” Appl. Phys. Lett. 38, 505–507 (1981).
[CrossRef]

Akimoto, O.

K. Ikeda, H. Diado, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980); H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, “Observation of chaos in optical bistability,” Phys. Rev. Lett. 46, 474–477 (1981).
[CrossRef]

Al Attar, H. A.

Asquini, M. L.

M. L. Asquini and F. Casagrande, “Optical bistability in a bidirectional ring cavity,” Z. Phys. B 44, 233–239 (1981).
[CrossRef]

Bandy, D. K.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, “Breathing, spiking, and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
[CrossRef]

Bar-Joseph, I.

Bischofberger, T.

T. Bischofberger and Y. R. Shen, “Transient behavior of a nonlinear Fabry–Perot,” Appl. Phys. Lett. 32, 156–158 (1978); J. A. Goldstone and E. M. Garmire, “On the dynamic response of nonlinear Fabry-Perot interferometers,” IEEE J. Quantum Electron. QE-17, 366–373 (1981).
[CrossRef]

Bonifacio, R.

R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A 18, 1129–1144 (1978); R. Bonifacio and P. Meystre, “Critical slowing down in optical bistability,” Opt. Commun. 29, 131–134 (1979).
[CrossRef]

Casagrande, F.

M. L. Asquini and F. Casagrande, “Optical bistability in a bidirectional ring cavity,” Z. Phys. B 44, 233–239 (1981).
[CrossRef]

Cheung, M.-M.

Diado, H.

K. Ikeda, H. Diado, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980); H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, “Observation of chaos in optical bistability,” Phys. Rev. Lett. 46, 474–477 (1981).
[CrossRef]

Durbin, S. D.

Firth, W. J.

Galbraith, I.

Gibbs, H. M.

J. V. Moloney, F. A. Hopf, and H. M. Gibbs, “Effects of transverse beam variations on bifurcatons in an intrinsic bistable ring cavity,” Phys. Rev. A 25, 3442–3445 (1982); J. V. Moloney, “Self-focusing-induced optical turbulence,” Phys. Rev. Lett. 53, 556–559 (1984).
[CrossRef]

Hagan, D. J.

Hopf, F. A.

J. V. Moloney, F. A. Hopf, and H. M. Gibbs, “Effects of transverse beam variations on bifurcatons in an intrinsic bistable ring cavity,” Phys. Rev. A 25, 3442–3445 (1982); J. V. Moloney, “Self-focusing-induced optical turbulence,” Phys. Rev. Lett. 53, 556–559 (1984).
[CrossRef]

Ikeda, K.

K. Ikeda, H. Diado, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980); H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, “Observation of chaos in optical bistability,” Phys. Rev. Lett. 46, 474–477 (1981).
[CrossRef]

Kaplan, A. E.

A. E. Kaplan, “Optoelectronic enhancement of the Sagnac effect in a ring resonator and related effect of directional bistability,” Appl. Phys. Lett. 42, 479–481 (1983).
[CrossRef]

A. E. Kaplan, “Light-induced nonreciprocity, field invariants, and nonlinear eigenpolarizations,” Opt. Lett. 8, 560 (1983).
[CrossRef] [PubMed]

A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40, 229–232 (1982).
[CrossRef]

A. E. Kaplan and P. Meystre, “Enhancement of the Sagnac effect due to nonlinearly induced nonreciprocity,” Opt. Lett. 6, 590–592 (1981).
[CrossRef] [PubMed]

A. E. Kaplan and C. T. Law, “Isolas in four-wave mixing optical bistability,” IEEE J. Quantum Electron. (to be published).

Law, C. T.

A. E. Kaplan and C. T. Law, “Isolas in four-wave mixing optical bistability,” IEEE J. Quantum Electron. (to be published).

Lugiato, L. A.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, “Breathing, spiking, and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
[CrossRef]

R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A 18, 1129–1144 (1978); R. Bonifacio and P. Meystre, “Critical slowing down in optical bistability,” Opt. Commun. 29, 131–134 (1979).
[CrossRef]

Mackenzie, H. A.

McCall, S. L.

S. L. McCall, “Instability and regenerative pulsation phenomena in Fabry–Perot nonlinear optic media,” Appl. Phys. Lett. 32, 284–286 (1978); R. Bonifacio, M. Gronchi, and L. A. Lugiato, “Self-pulsing in bistable operation,” Opt. Commun. 30, 129–133 (1979).
[CrossRef]

Meystre, P.

A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40, 229–232 (1982).
[CrossRef]

A. E. Kaplan and P. Meystre, “Enhancement of the Sagnac effect due to nonlinearly induced nonreciprocity,” Opt. Lett. 6, 590–592 (1981).
[CrossRef] [PubMed]

Moloney, J. V.

J. V. Moloney, F. A. Hopf, and H. M. Gibbs, “Effects of transverse beam variations on bifurcatons in an intrinsic bistable ring cavity,” Phys. Rev. A 25, 3442–3445 (1982); J. V. Moloney, “Self-focusing-induced optical turbulence,” Phys. Rev. Lett. 53, 556–559 (1984).
[CrossRef]

Narducci, L. M.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, “Breathing, spiking, and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
[CrossRef]

Pennise, C. A.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, “Breathing, spiking, and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
[CrossRef]

Shen, Y. R.

Silberberg, Y.

Ujihara, K.

K. Ujihara, “Mean field theory of optical bistability in a phase-conjugate resonator using nearly degenerate four-wave mixing,” IEEE J. Quantum Electron. (to be published).

Wright, E. M.

Appl. Phys. Lett. (4)

S. L. McCall, “Instability and regenerative pulsation phenomena in Fabry–Perot nonlinear optic media,” Appl. Phys. Lett. 32, 284–286 (1978); R. Bonifacio, M. Gronchi, and L. A. Lugiato, “Self-pulsing in bistable operation,” Opt. Commun. 30, 129–133 (1979).
[CrossRef]

G. P. Agrawal, “Effect of mode coupling on optical bistability in a bidirectional ring cavity,” Appl. Phys. Lett. 38, 505–507 (1981).
[CrossRef]

A. E. Kaplan, “Optoelectronic enhancement of the Sagnac effect in a ring resonator and related effect of directional bistability,” Appl. Phys. Lett. 42, 479–481 (1983).
[CrossRef]

T. Bischofberger and Y. R. Shen, “Transient behavior of a nonlinear Fabry–Perot,” Appl. Phys. Lett. 32, 156–158 (1978); J. A. Goldstone and E. M. Garmire, “On the dynamic response of nonlinear Fabry-Perot interferometers,” IEEE J. Quantum Electron. QE-17, 366–373 (1981).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. P. Agrawal, “Use of a bidirectional ring cavity for optical bistable devices,” IEEE J. Quantum Electron. QE-18, 214–218 (1982).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40, 229–232 (1982).
[CrossRef]

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, “Breathing, spiking, and chaos in a laser with injected signal,” Opt. Commun. 46, 64–68 (1983).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (2)

J. V. Moloney, F. A. Hopf, and H. M. Gibbs, “Effects of transverse beam variations on bifurcatons in an intrinsic bistable ring cavity,” Phys. Rev. A 25, 3442–3445 (1982); J. V. Moloney, “Self-focusing-induced optical turbulence,” Phys. Rev. Lett. 53, 556–559 (1984).
[CrossRef]

R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A 18, 1129–1144 (1978); R. Bonifacio and P. Meystre, “Critical slowing down in optical bistability,” Opt. Commun. 29, 131–134 (1979).
[CrossRef]

Phys. Rev. Lett. (1)

K. Ikeda, H. Diado, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980); H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, “Observation of chaos in optical bistability,” Phys. Rev. Lett. 46, 474–477 (1981).
[CrossRef]

Z. Phys. B (1)

M. L. Asquini and F. Casagrande, “Optical bistability in a bidirectional ring cavity,” Z. Phys. B 44, 233–239 (1981).
[CrossRef]

Other (3)

For a recent review see Optical Bistability, Dynamic Nonlinearity, and Photonic Logic, B. S. Wherrett and S. D. Smith eds. (Cambridge U. Press, Cambridge, 1985).

A. E. Kaplan and C. T. Law, “Isolas in four-wave mixing optical bistability,” IEEE J. Quantum Electron. (to be published).

K. Ujihara, “Mean field theory of optical bistability in a phase-conjugate resonator using nearly degenerate four-wave mixing,” IEEE J. Quantum Electron. (to be published).

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Figures (7)

Fig. 1
Fig. 1

Two possible setups to observe mode coupling. (a) Ring resonator with Kerr-type nonlinear medium pumped by two coherent input beams 1 and 2. (b) The nonlinear susceptibility couples two mutually incoherent beams of orthogonal polarizations.

Fig. 2
Fig. 2

Intracavity intensities I1, I2 in the case of symmetric pumping, as a function of the input intensity I01 = I02 = I0. (a) Corresponds to a linear phase shift ϕ = −1.65 and (b) to ϕ = −1.8. The dashed parts of the curves indicate the regions of instability.

Fig. 3
Fig. 3

Intracavity intensities I1 and I2 for nearly symmetric pumping as a function of the input intensity I01. The linear phase shift is ϕ = −1.8. In (a) the asymmetry in the pump intensities is I01/I02 = (1 − 0.001) and in (b) I01/I02 = (1 − 0.003).

Fig. 4
Fig. 4

Intracavity intensities I1 and I2 as function of I01 for asymmetric pumping and fixed linear phase shifts ϕ1 = −5 and ϕ2 = −2.5. The different graphs correspond to different values of I02: (a) I02 = 1, (b) I02 = 6, (c) I02 = 12, (d) I02 = 15, (e) I02 = 16.

Fig. 5
Fig. 5

Total phase shifts ψ1 and ψ2 for the same cases as in Fig. 4.

Fig. 6
Fig. 6

Dynamic evolution of I1. The time is in units of the response time τ of the medium. (a) I01 = 3.7, (b) I01 = 3.8, (c) I01 = 4.3, (d) I01 = 4.35, (e) I01 = 4.3551. Other parameters as in Fig. 4(c).

Fig. 7
Fig. 7

Period of the oscillation as function of I01. The asymptote corresponds to the point of reappearance of multiple-valued solutions.

Equations (15)

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τ 1 d θ 1 d t + θ 1 = a I 1 ( t ) + b I 2 ( t ) , τ 2 d θ 2 d t + θ 2 = c I 1 ( t ) + d I 2 ( t ) .
I 1 ( t ) = I 01 1 + [ ϕ 1 + θ 1 ( t ) ] 2 , I 2 ( t ) = I 02 1 + [ ϕ 2 + θ 2 ( t ) ] 2 ,
I i = I 0 i 1 + ( ϕ i + θ ) 2 ,
S = S 0 1 + ( ϕ + a S ) 2 ,
R = I 01 I 02 ,
I 1 = I 0 1 + ( ϕ + 3 a I 1 ) 2 = I 2 ,
I 0 = ( 2 / 3 ) [ - ϕ ( 3 ϕ 2 - 5 ) ± ( 3 ϕ 2 - 1 ) ϕ 2 - 3 ] ,
I 0 = P ( 2 ϕ + 3 S ) ,
I 0 = [ 1 + ( S + ϕ ) 2 ] ( 2 ϕ + 3 S ) .
λ i = - 1 + ( a / 2 ) D 1 + ( d / 2 ) D 2 + ( - 1 ) i { [ ( a / 2 ) D 1 ] 2 + [ ( d / 2 ) D 2 ] 2 + ( 1 / 2 ) ( 2 b c - a d ) D 1 D 2 } ,
D i = I 0 i d T i d θ i ,             i = 1 , 2
T i = I i I 0 i = 1 1 + ( ϕ i + θ i ) 2 .
2 b c a d .
2 b c - a d = 1.
2 ( 2 b c - a d ) D 1 D 2 < - ( a D 1 ) 2 - ( d D 2 ) 2 .

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