Abstract

We examine the dynamics of a ring cavity containing a diffusive, monostable induced absorber. We find both temporal and spatial chaos over a range of input intensities and diffusion-length to spot-size ratios. The diffusive coupling enforces spatial correlations over a range of many diffusion lengths.

© 1987 Optical Society of America

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References

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  1. M. Lindberg, S. W. Koch, and H. Haug, “Oscillatory instability of an induced absorber in a ring cavity,” J. Opt. Soc. Am. B 3, 731–761 (1986); H. Haug, S. W. Koch, and M. Lindberg, Phys. Scr. (to be published).
    [Crossref]
  2. M. Wegner and C. Klingshirn, “Self-oscillations of an induced absorber (CdS) in a hybrid ring resonator,” submitted to Phys. Rev. A.
  3. M. Wegner and C. Klingshirn, “Coexisting oscillation modes and optical chaos in a hybrid ring cavity containing an induced absorber (CdS),” submitted to Phys. Rev. A.
  4. J. V. Moloney, F. A. Hopf, and H. M. Gibbs, “Effects of transverse beam variation on bifurcations in an intrinsic bistable ring cavity,” Phys. Rev. A 52, 3442–3445 (1982); J. V. Moloney, “Many parameter routes to optical turbulence,” Phys. Rev. A 33, 4061–4078 (1986).
    [Crossref] [PubMed]
  5. N. N. Rozanov, “Hysteresis phenomena in distributed optical systems,” Sov. Phys. JETP 53, 47–53 (1981); W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1009 (1985).
    [Crossref]
  6. H. Haug and S. Schmitt-Rink, “Basic mechanisms of the optical nonlinearities of semiconductors near the band edge,” J. Opt. Soc. Am. B 2, 1135–1142 (1985).
    [Crossref]
  7. M. Lambsdorff, C. Dörnfeld, and C. Klingshirn, “Optical bistability in semiconductors induced by thermal effects,” Z. Phys. B 64, 409–416 (1986).
    [Crossref]
  8. M. Le Berre, E. Ressayre, A. Tallet, and H. M. Gibbs, “High-dimensional chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
    [Crossref] [PubMed]
  9. J. D. Farmer, “Chaotic attractors of an infinite dimensional dynamical system,” Physica 4D, 366–393 (1982).

1986 (3)

M. Lindberg, S. W. Koch, and H. Haug, “Oscillatory instability of an induced absorber in a ring cavity,” J. Opt. Soc. Am. B 3, 731–761 (1986); H. Haug, S. W. Koch, and M. Lindberg, Phys. Scr. (to be published).
[Crossref]

M. Lambsdorff, C. Dörnfeld, and C. Klingshirn, “Optical bistability in semiconductors induced by thermal effects,” Z. Phys. B 64, 409–416 (1986).
[Crossref]

M. Le Berre, E. Ressayre, A. Tallet, and H. M. Gibbs, “High-dimensional chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
[Crossref] [PubMed]

1985 (1)

1982 (2)

J. D. Farmer, “Chaotic attractors of an infinite dimensional dynamical system,” Physica 4D, 366–393 (1982).

J. V. Moloney, F. A. Hopf, and H. M. Gibbs, “Effects of transverse beam variation on bifurcations in an intrinsic bistable ring cavity,” Phys. Rev. A 52, 3442–3445 (1982); J. V. Moloney, “Many parameter routes to optical turbulence,” Phys. Rev. A 33, 4061–4078 (1986).
[Crossref] [PubMed]

1981 (1)

N. N. Rozanov, “Hysteresis phenomena in distributed optical systems,” Sov. Phys. JETP 53, 47–53 (1981); W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1009 (1985).
[Crossref]

Dörnfeld, C.

M. Lambsdorff, C. Dörnfeld, and C. Klingshirn, “Optical bistability in semiconductors induced by thermal effects,” Z. Phys. B 64, 409–416 (1986).
[Crossref]

Farmer, J. D.

J. D. Farmer, “Chaotic attractors of an infinite dimensional dynamical system,” Physica 4D, 366–393 (1982).

Gibbs, H. M.

M. Le Berre, E. Ressayre, A. Tallet, and H. M. Gibbs, “High-dimensional chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
[Crossref] [PubMed]

J. V. Moloney, F. A. Hopf, and H. M. Gibbs, “Effects of transverse beam variation on bifurcations in an intrinsic bistable ring cavity,” Phys. Rev. A 52, 3442–3445 (1982); J. V. Moloney, “Many parameter routes to optical turbulence,” Phys. Rev. A 33, 4061–4078 (1986).
[Crossref] [PubMed]

Haug, H.

M. Lindberg, S. W. Koch, and H. Haug, “Oscillatory instability of an induced absorber in a ring cavity,” J. Opt. Soc. Am. B 3, 731–761 (1986); H. Haug, S. W. Koch, and M. Lindberg, Phys. Scr. (to be published).
[Crossref]

H. Haug and S. Schmitt-Rink, “Basic mechanisms of the optical nonlinearities of semiconductors near the band edge,” J. Opt. Soc. Am. B 2, 1135–1142 (1985).
[Crossref]

Hopf, F. A.

J. V. Moloney, F. A. Hopf, and H. M. Gibbs, “Effects of transverse beam variation on bifurcations in an intrinsic bistable ring cavity,” Phys. Rev. A 52, 3442–3445 (1982); J. V. Moloney, “Many parameter routes to optical turbulence,” Phys. Rev. A 33, 4061–4078 (1986).
[Crossref] [PubMed]

Klingshirn, C.

M. Lambsdorff, C. Dörnfeld, and C. Klingshirn, “Optical bistability in semiconductors induced by thermal effects,” Z. Phys. B 64, 409–416 (1986).
[Crossref]

M. Wegner and C. Klingshirn, “Self-oscillations of an induced absorber (CdS) in a hybrid ring resonator,” submitted to Phys. Rev. A.

M. Wegner and C. Klingshirn, “Coexisting oscillation modes and optical chaos in a hybrid ring cavity containing an induced absorber (CdS),” submitted to Phys. Rev. A.

Koch, S. W.

M. Lindberg, S. W. Koch, and H. Haug, “Oscillatory instability of an induced absorber in a ring cavity,” J. Opt. Soc. Am. B 3, 731–761 (1986); H. Haug, S. W. Koch, and M. Lindberg, Phys. Scr. (to be published).
[Crossref]

Lambsdorff, M.

M. Lambsdorff, C. Dörnfeld, and C. Klingshirn, “Optical bistability in semiconductors induced by thermal effects,” Z. Phys. B 64, 409–416 (1986).
[Crossref]

Le Berre, M.

M. Le Berre, E. Ressayre, A. Tallet, and H. M. Gibbs, “High-dimensional chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
[Crossref] [PubMed]

Lindberg, M.

M. Lindberg, S. W. Koch, and H. Haug, “Oscillatory instability of an induced absorber in a ring cavity,” J. Opt. Soc. Am. B 3, 731–761 (1986); H. Haug, S. W. Koch, and M. Lindberg, Phys. Scr. (to be published).
[Crossref]

Moloney, J. V.

J. V. Moloney, F. A. Hopf, and H. M. Gibbs, “Effects of transverse beam variation on bifurcations in an intrinsic bistable ring cavity,” Phys. Rev. A 52, 3442–3445 (1982); J. V. Moloney, “Many parameter routes to optical turbulence,” Phys. Rev. A 33, 4061–4078 (1986).
[Crossref] [PubMed]

Ressayre, E.

M. Le Berre, E. Ressayre, A. Tallet, and H. M. Gibbs, “High-dimensional chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
[Crossref] [PubMed]

Rozanov, N. N.

N. N. Rozanov, “Hysteresis phenomena in distributed optical systems,” Sov. Phys. JETP 53, 47–53 (1981); W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1009 (1985).
[Crossref]

Schmitt-Rink, S.

Tallet, A.

M. Le Berre, E. Ressayre, A. Tallet, and H. M. Gibbs, “High-dimensional chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
[Crossref] [PubMed]

Wegner, M.

M. Wegner and C. Klingshirn, “Coexisting oscillation modes and optical chaos in a hybrid ring cavity containing an induced absorber (CdS),” submitted to Phys. Rev. A.

M. Wegner and C. Klingshirn, “Self-oscillations of an induced absorber (CdS) in a hybrid ring resonator,” submitted to Phys. Rev. A.

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

M. Lindberg, S. W. Koch, and H. Haug, “Oscillatory instability of an induced absorber in a ring cavity,” J. Opt. Soc. Am. B 3, 731–761 (1986); H. Haug, S. W. Koch, and M. Lindberg, Phys. Scr. (to be published).
[Crossref]

H. Haug and S. Schmitt-Rink, “Basic mechanisms of the optical nonlinearities of semiconductors near the band edge,” J. Opt. Soc. Am. B 2, 1135–1142 (1985).
[Crossref]

Phys. Rev. A (1)

J. V. Moloney, F. A. Hopf, and H. M. Gibbs, “Effects of transverse beam variation on bifurcations in an intrinsic bistable ring cavity,” Phys. Rev. A 52, 3442–3445 (1982); J. V. Moloney, “Many parameter routes to optical turbulence,” Phys. Rev. A 33, 4061–4078 (1986).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

M. Le Berre, E. Ressayre, A. Tallet, and H. M. Gibbs, “High-dimensional chaotic attractors of a nonlinear ring cavity,” Phys. Rev. Lett. 56, 274–277 (1986).
[Crossref] [PubMed]

Physica (1)

J. D. Farmer, “Chaotic attractors of an infinite dimensional dynamical system,” Physica 4D, 366–393 (1982).

Sov. Phys. JETP (1)

N. N. Rozanov, “Hysteresis phenomena in distributed optical systems,” Sov. Phys. JETP 53, 47–53 (1981); W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1009 (1985).
[Crossref]

Z. Phys. B (1)

M. Lambsdorff, C. Dörnfeld, and C. Klingshirn, “Optical bistability in semiconductors induced by thermal effects,” Z. Phys. B 64, 409–416 (1986).
[Crossref]

Other (2)

M. Wegner and C. Klingshirn, “Self-oscillations of an induced absorber (CdS) in a hybrid ring resonator,” submitted to Phys. Rev. A.

M. Wegner and C. Klingshirn, “Coexisting oscillation modes and optical chaos in a hybrid ring cavity containing an induced absorber (CdS),” submitted to Phys. Rev. A.

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Figures (7)

Fig. 1
Fig. 1

The fraction of intensity absorbed by the induced absorber A(N) for A0 = 0.3, B = 0.69, k = 8, and N0 = 0.37. The intersection of A(N) with the straight line gives the solution of the diffusion equation for Ld = 0. With the given parameters the absorber is in a monostable mode.

Fig. 2
Fig. 2

Lyaponov exponent as a function of input intensity for the local map. Period-n orbits exist in the ranges indicated as pn; s.s. indicates the steady state, and chaotic solutions exist for λ > 0. Parameters for A(N) are as in Fig. 1.

Fig. 3
Fig. 3

Attractor for the local map consisting of 5000 values of the excitation density plotted as a function of the previous value. The input intensity is Iin = 0.1.

Fig. 4
Fig. 4

Successive transmitted intensity profiles for Iin = 0.6 and various diffusion lengths. (a) Ld = 0, (b) Ld = 0.01, (c) Ld = 0.1, (d) Ld = 5.0, (e) Ld = 5.4, (f) Ld = 5.5, (g) Ld = 6.0, (h) Ld = 7.0. The dashed vertical lines delineate the range of input intensities that are chaotic in the local case.

Fig. 5
Fig. 5

Successive transmitted intensity profiles for Iin = 0.3 and various diffusion lengths. (a) Ld = 0, (b) Ld = 0.01, (c) Ld = 0.1, (d) Ld = 0.5. The dashed vertical lines delineate the range of input intensities that are chaotic in the local case.

Fig. 6
Fig. 6

Successive transmitted intensity profiles for Iin = 0.1 and various diffusion lengths. (a) Ld = 0, (b) Ld = 0.01, (c) Ld = 0.1. The dashed vertical lines delineate the range of input intensities that are chaotic in the local case.

Fig. 7
Fig. 7

Successive transmitted intensity profiles for Iin = 0.0635 and various diffusion lengths. (a) Ld = 0, (b) Ld = 0.02, (c) Ld = 0.05, (d) Ld = 0.1. The dashed vertical lines delineate the range of input intensities that are chaotic in the local case.

Equations (11)

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D a ( 2 N x 2 + 2 N z 2 ) - N τ + α ( N ) ω n 0 c 8 π E n ( x , z ) 2 = 0 ,
D a τ 2 N x 2 - N + τ ω L { 1 - exp [ - α ( N ) L ] } E n ( x , 0 ) 2 = 0.
A ( N ) = A 0 + B 1 + exp [ - k ( N - N 0 ) ] ,
L d 2 2 N x 2 - N + A ( N ) E n ( x ) 2 = 0
N ( x ) 0             as x ± ,
E n + 1 ( x ) = T I in e - x 2 + R { 1 - A [ N ( x ) ] } E n ( x ) ,
I n = [ 1 - A ( N ) ] E n ( x ) 2 .
λ = lim M [ 1 M n = 0 M ln f ( E n ) ]
N ( x ) = 1 2 L d { 0 x exp [ ( x - x ) / L d ] A [ N ( x ) ] E n ( x ) 2 d x + x exp [ ( x - x ) / L d ] A [ N ( x ) ] E n ( x ) 2 d x } .
N = A ( N ) L d 0 E n ( x ) 2 d x ,
C ( k ) = lim N 1 N i = 1 N [ I n ( x i ) - I n ] [ I n ( x i + k ) - I n ] ,

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