Abstract

We explore some general dynamical properties of a homogeneously broadened ring laser driven by an external coherent signal. By combining the information obtained from time-dependent solutions, their power spectra, Poincaré surface of sections, and Lyapunov exponents, we provide detailed evidence of the bifurcation structure of a laser with injected signal under high-gain conditions and show the existence of up to three basins of attraction for the same range of operating parameters. We also discuss the behavior of this system for experimentally accessible ranges of the gain and provide evidence for the persistence of a number of interesting pulsation phenomena.

© 1985 Optical Society of America

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References

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  1. A. N. Oraevskii, Radio. Electronikkalalo. Tek. Koreakoulu (Kertomus) 4, 718 (1959).
  2. See, for example, J. P. Gordon, Proc. IRE 50, 1898 (1962);A. S. Agabekyan, A. Z. Grazyuk, I. G. Zubarev, A. N. Oraevskii, V. I. Svergun, Radio. Electronikkalalo. Tek. Koreakoulu (Kertomus) 9, 2156 (1964).
    [CrossRef]
  3. For a brief review of the state of the art in the mid–late 1960’s, see N. G. Basov, A. Z. Grazyuk, I. G. Zubarev, L. V. Tevelev, in Quantum Electronics in Lasers and Masers, D. V. Skobel’tsyn, ed., Proceedings of the Lebedev Institute, Vol. 31 (Consultants Bureau, New York, 1968).
  4. M. B. Spencer, W. E. Lamb, Phys. Rev. A 5, 884 (1972).
    [CrossRef]
  5. See, for example, U. Ganiel, A. Hardy, D. Treves, IEEE J. Quantum Electron. QE-12, 704 (1976);R. Flamant, G. Megie, IEEE J. Quantum Electron. QE-16, 653 (1980).
    [CrossRef]
  6. S. Blit, U. Ganiel, D. Treves, Appl. Phys. 12, 69 (1977);Y. K. Park, G. Giuliani, R. L. Byer, Opt. Lett. 5, 96 (1980).
    [CrossRef]
  7. L. A. Lugiato, Lett. Nuovo Cimento 23, 609 (1978), and references therein.
    [CrossRef]
  8. T. Yamada, R. Graham, Phys. Lett. 53A, 77 (1975).
  9. M. J. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).
  10. L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
    [CrossRef]
  11. D. K. Bandy, L. M. Narducci, C. A. Pennise, L. A. Lugiato, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 585.
  12. F. T. Arecchi, G. Lippi, G. Puccioni, J. Tredicce, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984, p. 1227).
  13. K. Otsuka, H. Iwamura, Phys. Rev. A 28, 3153 (1983).
    [CrossRef]
  14. P. Mandel, Service de Chimie Physique II, Bruxelles, Belgium (personal communication).
  15. E. Brun, B. Derighetti, D. Meier, R. Holzner, M. Ravani, J. Opt. Soc. Am. B. 2, 156 (1985).
    [CrossRef]
  16. Y. Gu, D. K. Bandy, J. M. Yuan, L. M. Narducci, Phys. Rev. A (to be published), and references therein.
  17. All the Lyapunov exponent calculations were carried out in double precision on a Prime 850 computer.

1985 (1)

E. Brun, B. Derighetti, D. Meier, R. Holzner, M. Ravani, J. Opt. Soc. Am. B. 2, 156 (1985).
[CrossRef]

1983 (2)

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
[CrossRef]

K. Otsuka, H. Iwamura, Phys. Rev. A 28, 3153 (1983).
[CrossRef]

1981 (1)

M. J. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).

1978 (1)

L. A. Lugiato, Lett. Nuovo Cimento 23, 609 (1978), and references therein.
[CrossRef]

1977 (1)

S. Blit, U. Ganiel, D. Treves, Appl. Phys. 12, 69 (1977);Y. K. Park, G. Giuliani, R. L. Byer, Opt. Lett. 5, 96 (1980).
[CrossRef]

1976 (1)

See, for example, U. Ganiel, A. Hardy, D. Treves, IEEE J. Quantum Electron. QE-12, 704 (1976);R. Flamant, G. Megie, IEEE J. Quantum Electron. QE-16, 653 (1980).
[CrossRef]

1975 (1)

T. Yamada, R. Graham, Phys. Lett. 53A, 77 (1975).

1972 (1)

M. B. Spencer, W. E. Lamb, Phys. Rev. A 5, 884 (1972).
[CrossRef]

1962 (1)

See, for example, J. P. Gordon, Proc. IRE 50, 1898 (1962);A. S. Agabekyan, A. Z. Grazyuk, I. G. Zubarev, A. N. Oraevskii, V. I. Svergun, Radio. Electronikkalalo. Tek. Koreakoulu (Kertomus) 9, 2156 (1964).
[CrossRef]

1959 (1)

A. N. Oraevskii, Radio. Electronikkalalo. Tek. Koreakoulu (Kertomus) 4, 718 (1959).

Arecchi, F. T.

F. T. Arecchi, G. Lippi, G. Puccioni, J. Tredicce, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984, p. 1227).

Bandy, D. K.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
[CrossRef]

D. K. Bandy, L. M. Narducci, C. A. Pennise, L. A. Lugiato, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 585.

Y. Gu, D. K. Bandy, J. M. Yuan, L. M. Narducci, Phys. Rev. A (to be published), and references therein.

Basov, N. G.

For a brief review of the state of the art in the mid–late 1960’s, see N. G. Basov, A. Z. Grazyuk, I. G. Zubarev, L. V. Tevelev, in Quantum Electronics in Lasers and Masers, D. V. Skobel’tsyn, ed., Proceedings of the Lebedev Institute, Vol. 31 (Consultants Bureau, New York, 1968).

Blit, S.

S. Blit, U. Ganiel, D. Treves, Appl. Phys. 12, 69 (1977);Y. K. Park, G. Giuliani, R. L. Byer, Opt. Lett. 5, 96 (1980).
[CrossRef]

Brand, H.

M. J. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).

Brun, E.

E. Brun, B. Derighetti, D. Meier, R. Holzner, M. Ravani, J. Opt. Soc. Am. B. 2, 156 (1985).
[CrossRef]

Derighetti, B.

E. Brun, B. Derighetti, D. Meier, R. Holzner, M. Ravani, J. Opt. Soc. Am. B. 2, 156 (1985).
[CrossRef]

Ganiel, U.

S. Blit, U. Ganiel, D. Treves, Appl. Phys. 12, 69 (1977);Y. K. Park, G. Giuliani, R. L. Byer, Opt. Lett. 5, 96 (1980).
[CrossRef]

See, for example, U. Ganiel, A. Hardy, D. Treves, IEEE J. Quantum Electron. QE-12, 704 (1976);R. Flamant, G. Megie, IEEE J. Quantum Electron. QE-16, 653 (1980).
[CrossRef]

Gordon, J. P.

See, for example, J. P. Gordon, Proc. IRE 50, 1898 (1962);A. S. Agabekyan, A. Z. Grazyuk, I. G. Zubarev, A. N. Oraevskii, V. I. Svergun, Radio. Electronikkalalo. Tek. Koreakoulu (Kertomus) 9, 2156 (1964).
[CrossRef]

Graham, R.

M. J. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).

T. Yamada, R. Graham, Phys. Lett. 53A, 77 (1975).

Grazyuk, A. Z.

For a brief review of the state of the art in the mid–late 1960’s, see N. G. Basov, A. Z. Grazyuk, I. G. Zubarev, L. V. Tevelev, in Quantum Electronics in Lasers and Masers, D. V. Skobel’tsyn, ed., Proceedings of the Lebedev Institute, Vol. 31 (Consultants Bureau, New York, 1968).

Gu, Y.

Y. Gu, D. K. Bandy, J. M. Yuan, L. M. Narducci, Phys. Rev. A (to be published), and references therein.

Hardy, A.

See, for example, U. Ganiel, A. Hardy, D. Treves, IEEE J. Quantum Electron. QE-12, 704 (1976);R. Flamant, G. Megie, IEEE J. Quantum Electron. QE-16, 653 (1980).
[CrossRef]

Holzner, R.

E. Brun, B. Derighetti, D. Meier, R. Holzner, M. Ravani, J. Opt. Soc. Am. B. 2, 156 (1985).
[CrossRef]

Iwamura, H.

K. Otsuka, H. Iwamura, Phys. Rev. A 28, 3153 (1983).
[CrossRef]

Lamb, W. E.

M. B. Spencer, W. E. Lamb, Phys. Rev. A 5, 884 (1972).
[CrossRef]

Lippi, G.

F. T. Arecchi, G. Lippi, G. Puccioni, J. Tredicce, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984, p. 1227).

Lugiato, L. A.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
[CrossRef]

L. A. Lugiato, Lett. Nuovo Cimento 23, 609 (1978), and references therein.
[CrossRef]

D. K. Bandy, L. M. Narducci, C. A. Pennise, L. A. Lugiato, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 585.

Mandel, P.

P. Mandel, Service de Chimie Physique II, Bruxelles, Belgium (personal communication).

Meier, D.

E. Brun, B. Derighetti, D. Meier, R. Holzner, M. Ravani, J. Opt. Soc. Am. B. 2, 156 (1985).
[CrossRef]

Narducci, L. M.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
[CrossRef]

D. K. Bandy, L. M. Narducci, C. A. Pennise, L. A. Lugiato, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 585.

Y. Gu, D. K. Bandy, J. M. Yuan, L. M. Narducci, Phys. Rev. A (to be published), and references therein.

Oraevskii, A. N.

A. N. Oraevskii, Radio. Electronikkalalo. Tek. Koreakoulu (Kertomus) 4, 718 (1959).

Otsuka, K.

K. Otsuka, H. Iwamura, Phys. Rev. A 28, 3153 (1983).
[CrossRef]

Pennise, C. A.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
[CrossRef]

D. K. Bandy, L. M. Narducci, C. A. Pennise, L. A. Lugiato, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 585.

Puccioni, G.

F. T. Arecchi, G. Lippi, G. Puccioni, J. Tredicce, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984, p. 1227).

Ravani, M.

E. Brun, B. Derighetti, D. Meier, R. Holzner, M. Ravani, J. Opt. Soc. Am. B. 2, 156 (1985).
[CrossRef]

Scholz, M. J.

M. J. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).

Spencer, M. B.

M. B. Spencer, W. E. Lamb, Phys. Rev. A 5, 884 (1972).
[CrossRef]

Tevelev, L. V.

For a brief review of the state of the art in the mid–late 1960’s, see N. G. Basov, A. Z. Grazyuk, I. G. Zubarev, L. V. Tevelev, in Quantum Electronics in Lasers and Masers, D. V. Skobel’tsyn, ed., Proceedings of the Lebedev Institute, Vol. 31 (Consultants Bureau, New York, 1968).

Tredicce, J.

F. T. Arecchi, G. Lippi, G. Puccioni, J. Tredicce, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984, p. 1227).

Treves, D.

S. Blit, U. Ganiel, D. Treves, Appl. Phys. 12, 69 (1977);Y. K. Park, G. Giuliani, R. L. Byer, Opt. Lett. 5, 96 (1980).
[CrossRef]

See, for example, U. Ganiel, A. Hardy, D. Treves, IEEE J. Quantum Electron. QE-12, 704 (1976);R. Flamant, G. Megie, IEEE J. Quantum Electron. QE-16, 653 (1980).
[CrossRef]

Yamada, T.

M. J. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).

T. Yamada, R. Graham, Phys. Lett. 53A, 77 (1975).

Yuan, J. M.

Y. Gu, D. K. Bandy, J. M. Yuan, L. M. Narducci, Phys. Rev. A (to be published), and references therein.

Zubarev, I. G.

For a brief review of the state of the art in the mid–late 1960’s, see N. G. Basov, A. Z. Grazyuk, I. G. Zubarev, L. V. Tevelev, in Quantum Electronics in Lasers and Masers, D. V. Skobel’tsyn, ed., Proceedings of the Lebedev Institute, Vol. 31 (Consultants Bureau, New York, 1968).

Appl. Phys. (1)

S. Blit, U. Ganiel, D. Treves, Appl. Phys. 12, 69 (1977);Y. K. Park, G. Giuliani, R. L. Byer, Opt. Lett. 5, 96 (1980).
[CrossRef]

IEEE J. Quantum Electron. (1)

See, for example, U. Ganiel, A. Hardy, D. Treves, IEEE J. Quantum Electron. QE-12, 704 (1976);R. Flamant, G. Megie, IEEE J. Quantum Electron. QE-16, 653 (1980).
[CrossRef]

J. Opt. Soc. Am. B. (1)

E. Brun, B. Derighetti, D. Meier, R. Holzner, M. Ravani, J. Opt. Soc. Am. B. 2, 156 (1985).
[CrossRef]

Lett. Nuovo Cimento (1)

L. A. Lugiato, Lett. Nuovo Cimento 23, 609 (1978), and references therein.
[CrossRef]

Opt. Commun. (1)

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
[CrossRef]

Phys. Lett. (2)

T. Yamada, R. Graham, Phys. Lett. 53A, 77 (1975).

M. J. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).

Phys. Rev. A (2)

M. B. Spencer, W. E. Lamb, Phys. Rev. A 5, 884 (1972).
[CrossRef]

K. Otsuka, H. Iwamura, Phys. Rev. A 28, 3153 (1983).
[CrossRef]

Proc. IRE (1)

See, for example, J. P. Gordon, Proc. IRE 50, 1898 (1962);A. S. Agabekyan, A. Z. Grazyuk, I. G. Zubarev, A. N. Oraevskii, V. I. Svergun, Radio. Electronikkalalo. Tek. Koreakoulu (Kertomus) 9, 2156 (1964).
[CrossRef]

Radio. Electronikkalalo. Tek. Koreakoulu (Kertomus) (1)

A. N. Oraevskii, Radio. Electronikkalalo. Tek. Koreakoulu (Kertomus) 4, 718 (1959).

Other (6)

For a brief review of the state of the art in the mid–late 1960’s, see N. G. Basov, A. Z. Grazyuk, I. G. Zubarev, L. V. Tevelev, in Quantum Electronics in Lasers and Masers, D. V. Skobel’tsyn, ed., Proceedings of the Lebedev Institute, Vol. 31 (Consultants Bureau, New York, 1968).

P. Mandel, Service de Chimie Physique II, Bruxelles, Belgium (personal communication).

D. K. Bandy, L. M. Narducci, C. A. Pennise, L. A. Lugiato, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 585.

F. T. Arecchi, G. Lippi, G. Puccioni, J. Tredicce, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984, p. 1227).

Y. Gu, D. K. Bandy, J. M. Yuan, L. M. Narducci, Phys. Rev. A (to be published), and references therein.

All the Lyapunov exponent calculations were carried out in double precision on a Prime 850 computer.

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Figures (13)

Fig. 1
Fig. 1

State equation |X| versus Y for a laser with injected signal corresponding to parameters C = 20, Δ = 1, and ① Φ / κ = 2, ② Φ / κ = 3, ③ Φ / κ = 5. For subsequent analyses, it is important to know the instability range for cases ① and ②. The ends of these ranges are marked by points A and B. The corresponding values of κ and γ are 0.5, 0.05 and 1/3,0.05, respectively.

Fig. 2
Fig. 2

Output oscillations of |X| corresponding to a slowly varying driving field of the type Y(τ) = Y0 + and to the parameters C = 20, Δ = 1, Φ / κ = 2, κ = 0.5, γ = 0.05 (a) On the forward sweep, Y0 = 0 and V = 2.5 × 10−3; (b) on the backward sweep, Y0 = 25 and V = −2.5 × 10−3. For each sweep, the total run time is 104 units of γ−1.

Fig. 3
Fig. 3

Output oscillations of |X| corresponding to a slowly varying driving field of the type Y(τ) = Y0 + and to the parameters C = 20, Δ = 1, Φ / κ = 3, κ = 1 / 3, γ = 0.05. (a) On the forward sweep, Y0 = 0 and V = 1.0 × 10−3; (b) on the backward sweep, Y0 = 30 and V =−1.0 × 10−3. For each sweep, the run time is 3 × 104 units of γ−1.

Fig. 4
Fig. 4

Output oscillations of |X| corresponding to a slowly varying driving field of the type Y(τ) = Y0 + and to the parameters C = 20, Δ = 0.4, Φ / κ = 0.8, κ = 0.5, γ = 0.05. (a) On the forward sweep, Y0 = 0 and V = 2 × 10−3, (b) on the backward sweep, Y0 = 10 and V = −2 × 10−3. For each sweep, the run time is 5000 units of γ−1.

Fig. 5
Fig. 5

Output field amplitude |X| as a function of time for C = 3, Δ = 0.5, Φ / κ = 0.5, κ = 0.1, γ = 0.01. Both (a) and (b) correspond to the same value of Y = 1.7 and represent solutions evolving under the action of coexisting attractors.

Fig. 6
Fig. 6

Output oscillations of |X| corresponding to a slowly varying driving field of the type Y(τ) = Y0 + and to the parameters C = 20, Δ = 5, Φ / κ = 0, κ = 0.5, γ = 2; Y0 = 0 and V = 2.5 × 10−3. The forward and backward sweeps are shown simultaneously.

Fig. 7
Fig. 7

Output oscillations of |X| corresponding to a slowly varying driving field of the Y(τ) = Y0 + and to the parameters C = 20, Δ = 0, Φ / κ = 15, κ = 0.5, γ = 2, Y0 =0 and V = 0.005. The backward sweep is practically identical with the forward sweep and is not shown.

Fig. 8
Fig. 8

Output field amplitude |X| as a function of time for C ≡ 20, Δ = 1, Φ / κ = 2, κ = 0.5, γ = 0.05. Both (a) and (b) correspond to the same value Y = 9.8 and represent solutions evolving under the action of coexisting attractors.

Fig. 9
Fig. 9

The behavior of the fundamental frequency Ω0 of the power spectrum as a function of the driving-field strength. Lines ① and ② correspond to the fundamental frequencies of solutions that evolve under the influence of distinct domains of attraction with a common range of existence (6 < Y < 11). Line ③ traces the behavior of Ω0 in the vicinity of the injection-locking threshold. The dashed lines correspond to the imaginary part of the unstable eigenvalue of the linearized equation.

Fig. 10
Fig. 10

Global behavior of the system in the range 5 < Y < 15. 1P denotes a simple periodic solution; 2P is a doubly periodic solution (i.e., the spectrum contains a fundamental frequency ω and its subharmonic ω/2, etc.); 2Q denotes quasi-periodic motion with two incommensurate frequencies (two-dimensional torus); W denotes a window in chaos (thus 3PW denotes a window with a triply periodic solution); the values of Y, e.g., Y = 11.12 at the end of domain I, denote the approximate thresholds.

Fig. 11
Fig. 11

State equation |X| versus Y for a laser with injected signal corresponding to the parameters C = 3, Δ = 1, ①, Φ / κ = 0.2, ②, Φ / κ = 0.5, ③. Φ / κ = 1.0. The points A, B, and C mark the right-hand boundaries of the instability ranges for the curves, corresponding to κ = 0.1, γ = 0.01.

Fig. 12
Fig. 12

Output oscillations of |X| corresponding to a slowly varying driving field of the type Y(τ) = Y0 + and to the parameters C = 3, Δ = 0.5, Φ / κ = 0.2, κ = 0.1, γ = 0.01. (a) On the forward sweep, Y0 = 0 and V = 1.5 × 10−4; (b) on the backward sweep, Y0 = 3 and V = −1.5 × 10−4. For each sweep, the run time is 2 × 104 units of γ−1

Fig. 13
Fig. 13

Output oscillations of |X| corresponding to a slowly varying driving field of the type Y(τ) = Y0 + and to the parameters C = 3, Δ = 0.5, Φ / κ = 0.5, κ = 0.1, γ = 0.01. (a) on the forward sweep, Y0 = 0 and V = 1.75 × 10−4; (b) on the backward sweep, Y0 = 3.5 and V = −1.75 × 10−4. For each sweep, the run time is 2 × 104 units of γ−1

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

d X d τ = κ [ ( 1 i Φ / κ ) X Y + 2 C P ] ,
d P d τ = ( 1 + i Δ ) P + X D ,
d D d τ = γ [ 1 2 ( X P * + X * P ) + D + 1 ] ,
Y = | X | [ ( 1 2 C 1 + Δ 2 + | X | 2 ) 2 + ( 2 C Δ 1 + Δ 2 + | X | 2 Φ κ ) 2 ] 1 / 2 ,
0 = LE 1 > LE 2 = LE 3 > LE 4 > LE 5 ,
| LE 1 | | LE 2 | | LE 3 | < 10 3 , LE 4 1.460 , | LE 5 | 1.590.
Ω = ( κ / 1 + κ ) | Δ Φ / κ | .

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