Abstract

The possibility of controlling chaos in a model of an external-cavity laser diode with optimized impulsive delayed feedback is demonstrated. An examination is made of the application of such feedback by means of modulation of three different parameters, which determine the laser operation: (i) laser-drive current, (ii) laser-field phase, and (iii) laser-cavity losses. Account is taken of practical constraints arising from a technical delay in application of the control signal. The roles of the phase, width, and shape of the feedback pulses are elucidated for optimizing the control process. The effectiveness of the various methods of applying the control is analyzed by computing domains of control and finding the optimal conditions for control. It is demonstrated also that the preliminary targeting of dynamics to an unstable orbit by means of the present method is applicable to dynamical systems with many degrees of freedom and thus facilitates the process of controlling chaos.

© 1998 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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1997

S. I. Turovets, A. Valle, and K. A. Shore, “Effects of noise on the turn-on dynamics of a modulated class-B laser in the generalized multistability domain,” Phys. Rev. A 55, 2426–2434 (1997).
[CrossRef]

1996

C. R. Mirasso, P. Colet, and P. Garcia-Fernandez, “Synchronization of chaotic semiconductor lasers: application to encoded communications,” IEEE Photon. Tech. Lett. 8, 299–301 (1996).
[CrossRef]

1995

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
[CrossRef]

K. Petermann, “External Optical Feedback Phenomena in Semiconductor Lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 480–489 (1995).
[CrossRef]

N. Watanabe and K. Karaki, “Inducing periodic oscillations from chaotic oscillations of a compound-cavity laser diode with sinusoidally modulated drive,” Opt. Lett. 20, 725–727 (1995).
[CrossRef]

L. N. Langley, S. Turovets, and K. A. Shore, “Targeting periodic dynamics of external cavity semiconductor laser,” Opt. Lett. 20, 725–727 (1995).
[CrossRef] [PubMed]

G. Li, R. K. Boncek, X. Wang, and D. H. Sackett, “Transient and optoelectronic feedback-sustained pulsation of laser diodes at 1300 nm,” IEEE Photon. Technol. Lett. 7, 854–856 (1995).
[CrossRef]

1994

L. N. Langley and K. A. Shore, “Intensity noise and linewidth characteristics of laser diodes subject to phase conjugate optical feedback,” IEE Proc.: Optoelectron. 141, 103–108 (1994).

Y. Liu and J. Ohtsubo, “Experimental control of chaos in a laser diode interferometer with delayed feedback,” Opt. Lett. 19, 448–450 (1994).
[CrossRef] [PubMed]

A. T. Ryan, G. P. Agrawal, G. R. Gray, and E. C. Gage, “Optical-feedback-induced chaos and its control in semiconductor lasers,” IEEE J. Quantum Electron. 30, 668–679 (1994).
[CrossRef]

S. Bielawski, M. Bouazaoui, D. Derosier, and P. Glorieux, “Controlling unstable periodic orbits by delayed continuous feedback,” Phys. Rev. E 49, 971–973 (1994).
[CrossRef]

W. M. Yee and K. A. Shore, “Nearly degenerate four-wave mixing in laser diodes with nonuniform longitudinal gain distribution,” J. Opt. Soc. Am. B 11, 1211–1218 (1994).
[CrossRef]

K. A. Shore and D. T. Wright, “Improved synchronisation algorithm for secure communications systems using chaotic encryption,” Electron. Lett. 30, 1203–1204 (1994).
[CrossRef]

1993

S. Bielawski, M. Bouazaoui, D. Derosier, and P. Glorieux, “Stabilization and characterization of unstable steady states in a laser,” Phys. Rev. A 47, 3276–3279 (1993).
[CrossRef] [PubMed]

L. N. Langley and K. A. Shore, “The effect of phase conjugate optical feedback on the intensity noise in laser diodes,” Opt. Lett. 18, 1432–1434 (1993).
[CrossRef] [PubMed]

S. Bielawski, D. Derosier, and P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

G. R. Gray, A. T. Ryan, G. P. Agrawal, and E. C. Gage, “Optical-feedback-induced chaos and its control in semiconductor lasers,” in Chaos in Optics, R. Roy, ed., Proc. SPIE 2039, 45–57 (1993).
[CrossRef]

S. Hayes, C. Greboggi, and E. Ott, “Communicating with chaos,” Phys. Rev. Lett. 70, 3031–3034 (1993).
[CrossRef] [PubMed]

K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronised chaos with applications in communications,” Phys. Rev. Lett. 71, 65–68 (1993).
[CrossRef] [PubMed]

L. N. Langley and K. A. Shore, “The effect of external optical feedback on timing jitter in modulated laser diodes,” J. Lightwave Technol. 11, 434–441 (1993).
[CrossRef]

1992

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilisation of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1261 (1992).
[CrossRef] [PubMed]

1990

E. Ott, C. Greboggi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

T. Shinbrot, E. Ott, C. Greboggi, and J. A. Yorke, “Using chaos to direct trajectories to targets,” Phys. Rev. Lett. 65, 3215–3218 (1990).
[CrossRef] [PubMed]

Agrawal, G. P.

A. T. Ryan, G. P. Agrawal, G. R. Gray, and E. C. Gage, “Optical-feedback-induced chaos and its control in semiconductor lasers,” IEEE J. Quantum Electron. 30, 668–679 (1994).
[CrossRef]

G. R. Gray, A. T. Ryan, G. P. Agrawal, and E. C. Gage, “Optical-feedback-induced chaos and its control in semiconductor lasers,” in Chaos in Optics, R. Roy, ed., Proc. SPIE 2039, 45–57 (1993).
[CrossRef]

Bielawski, S.

S. Bielawski, M. Bouazaoui, D. Derosier, and P. Glorieux, “Controlling unstable periodic orbits by delayed continuous feedback,” Phys. Rev. E 49, 971–973 (1994).
[CrossRef]

S. Bielawski, M. Bouazaoui, D. Derosier, and P. Glorieux, “Stabilization and characterization of unstable steady states in a laser,” Phys. Rev. A 47, 3276–3279 (1993).
[CrossRef] [PubMed]

S. Bielawski, D. Derosier, and P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

Boncek, R. K.

G. Li, R. K. Boncek, X. Wang, and D. H. Sackett, “Transient and optoelectronic feedback-sustained pulsation of laser diodes at 1300 nm,” IEEE Photon. Technol. Lett. 7, 854–856 (1995).
[CrossRef]

Bouazaoui, M.

S. Bielawski, M. Bouazaoui, D. Derosier, and P. Glorieux, “Controlling unstable periodic orbits by delayed continuous feedback,” Phys. Rev. E 49, 971–973 (1994).
[CrossRef]

S. Bielawski, M. Bouazaoui, D. Derosier, and P. Glorieux, “Stabilization and characterization of unstable steady states in a laser,” Phys. Rev. A 47, 3276–3279 (1993).
[CrossRef] [PubMed]

Colet, P.

C. R. Mirasso, P. Colet, and P. Garcia-Fernandez, “Synchronization of chaotic semiconductor lasers: application to encoded communications,” IEEE Photon. Tech. Lett. 8, 299–301 (1996).
[CrossRef]

Cuomo, K. M.

K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronised chaos with applications in communications,” Phys. Rev. Lett. 71, 65–68 (1993).
[CrossRef] [PubMed]

Derosier, D.

S. Bielawski, M. Bouazaoui, D. Derosier, and P. Glorieux, “Controlling unstable periodic orbits by delayed continuous feedback,” Phys. Rev. E 49, 971–973 (1994).
[CrossRef]

S. Bielawski, M. Bouazaoui, D. Derosier, and P. Glorieux, “Stabilization and characterization of unstable steady states in a laser,” Phys. Rev. A 47, 3276–3279 (1993).
[CrossRef] [PubMed]

S. Bielawski, D. Derosier, and P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

Gage, E. C.

A. T. Ryan, G. P. Agrawal, G. R. Gray, and E. C. Gage, “Optical-feedback-induced chaos and its control in semiconductor lasers,” IEEE J. Quantum Electron. 30, 668–679 (1994).
[CrossRef]

G. R. Gray, A. T. Ryan, G. P. Agrawal, and E. C. Gage, “Optical-feedback-induced chaos and its control in semiconductor lasers,” in Chaos in Optics, R. Roy, ed., Proc. SPIE 2039, 45–57 (1993).
[CrossRef]

Garcia-Fernandez, P.

C. R. Mirasso, P. Colet, and P. Garcia-Fernandez, “Synchronization of chaotic semiconductor lasers: application to encoded communications,” IEEE Photon. Tech. Lett. 8, 299–301 (1996).
[CrossRef]

Gills, Z.

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilisation of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1261 (1992).
[CrossRef] [PubMed]

Glorieux, P.

S. Bielawski, M. Bouazaoui, D. Derosier, and P. Glorieux, “Controlling unstable periodic orbits by delayed continuous feedback,” Phys. Rev. E 49, 971–973 (1994).
[CrossRef]

S. Bielawski, M. Bouazaoui, D. Derosier, and P. Glorieux, “Stabilization and characterization of unstable steady states in a laser,” Phys. Rev. A 47, 3276–3279 (1993).
[CrossRef] [PubMed]

S. Bielawski, D. Derosier, and P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

Gray, G. R.

A. T. Ryan, G. P. Agrawal, G. R. Gray, and E. C. Gage, “Optical-feedback-induced chaos and its control in semiconductor lasers,” IEEE J. Quantum Electron. 30, 668–679 (1994).
[CrossRef]

G. R. Gray, A. T. Ryan, G. P. Agrawal, and E. C. Gage, “Optical-feedback-induced chaos and its control in semiconductor lasers,” in Chaos in Optics, R. Roy, ed., Proc. SPIE 2039, 45–57 (1993).
[CrossRef]

Greboggi, C.

S. Hayes, C. Greboggi, and E. Ott, “Communicating with chaos,” Phys. Rev. Lett. 70, 3031–3034 (1993).
[CrossRef] [PubMed]

E. Ott, C. Greboggi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

T. Shinbrot, E. Ott, C. Greboggi, and J. A. Yorke, “Using chaos to direct trajectories to targets,” Phys. Rev. Lett. 65, 3215–3218 (1990).
[CrossRef] [PubMed]

Hayes, S.

S. Hayes, C. Greboggi, and E. Ott, “Communicating with chaos,” Phys. Rev. Lett. 70, 3031–3034 (1993).
[CrossRef] [PubMed]

Hunt, E. R.

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilisation of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1261 (1992).
[CrossRef] [PubMed]

Karaki, K.

Langley, L. N.

L. N. Langley, S. Turovets, and K. A. Shore, “Targeting periodic dynamics of external cavity semiconductor laser,” Opt. Lett. 20, 725–727 (1995).
[CrossRef] [PubMed]

L. N. Langley and K. A. Shore, “Intensity noise and linewidth characteristics of laser diodes subject to phase conjugate optical feedback,” IEE Proc.: Optoelectron. 141, 103–108 (1994).

L. N. Langley and K. A. Shore, “The effect of phase conjugate optical feedback on the intensity noise in laser diodes,” Opt. Lett. 18, 1432–1434 (1993).
[CrossRef] [PubMed]

L. N. Langley and K. A. Shore, “The effect of external optical feedback on timing jitter in modulated laser diodes,” J. Lightwave Technol. 11, 434–441 (1993).
[CrossRef]

Lenstra, D.

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
[CrossRef]

Li, G.

G. Li, R. K. Boncek, X. Wang, and D. H. Sackett, “Transient and optoelectronic feedback-sustained pulsation of laser diodes at 1300 nm,” IEEE Photon. Technol. Lett. 7, 854–856 (1995).
[CrossRef]

Liu, Y.

Maier, T. D.

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilisation of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1261 (1992).
[CrossRef] [PubMed]

Mirasso, C. R.

C. R. Mirasso, P. Colet, and P. Garcia-Fernandez, “Synchronization of chaotic semiconductor lasers: application to encoded communications,” IEEE Photon. Tech. Lett. 8, 299–301 (1996).
[CrossRef]

Murphy, T. W.

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilisation of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1261 (1992).
[CrossRef] [PubMed]

Ohtsubo, J.

Oppenheim, A. V.

K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronised chaos with applications in communications,” Phys. Rev. Lett. 71, 65–68 (1993).
[CrossRef] [PubMed]

Ott, E.

S. Hayes, C. Greboggi, and E. Ott, “Communicating with chaos,” Phys. Rev. Lett. 70, 3031–3034 (1993).
[CrossRef] [PubMed]

E. Ott, C. Greboggi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

T. Shinbrot, E. Ott, C. Greboggi, and J. A. Yorke, “Using chaos to direct trajectories to targets,” Phys. Rev. Lett. 65, 3215–3218 (1990).
[CrossRef] [PubMed]

Petermann, K.

K. Petermann, “External Optical Feedback Phenomena in Semiconductor Lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 480–489 (1995).
[CrossRef]

Roy, R.

R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilisation of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1261 (1992).
[CrossRef] [PubMed]

Ryan, A. T.

A. T. Ryan, G. P. Agrawal, G. R. Gray, and E. C. Gage, “Optical-feedback-induced chaos and its control in semiconductor lasers,” IEEE J. Quantum Electron. 30, 668–679 (1994).
[CrossRef]

G. R. Gray, A. T. Ryan, G. P. Agrawal, and E. C. Gage, “Optical-feedback-induced chaos and its control in semiconductor lasers,” in Chaos in Optics, R. Roy, ed., Proc. SPIE 2039, 45–57 (1993).
[CrossRef]

Sackett, D. H.

G. Li, R. K. Boncek, X. Wang, and D. H. Sackett, “Transient and optoelectronic feedback-sustained pulsation of laser diodes at 1300 nm,” IEEE Photon. Technol. Lett. 7, 854–856 (1995).
[CrossRef]

Shinbrot, T.

T. Shinbrot, E. Ott, C. Greboggi, and J. A. Yorke, “Using chaos to direct trajectories to targets,” Phys. Rev. Lett. 65, 3215–3218 (1990).
[CrossRef] [PubMed]

Shore, K. A.

S. I. Turovets, A. Valle, and K. A. Shore, “Effects of noise on the turn-on dynamics of a modulated class-B laser in the generalized multistability domain,” Phys. Rev. A 55, 2426–2434 (1997).
[CrossRef]

L. N. Langley, S. Turovets, and K. A. Shore, “Targeting periodic dynamics of external cavity semiconductor laser,” Opt. Lett. 20, 725–727 (1995).
[CrossRef] [PubMed]

W. M. Yee and K. A. Shore, “Nearly degenerate four-wave mixing in laser diodes with nonuniform longitudinal gain distribution,” J. Opt. Soc. Am. B 11, 1211–1218 (1994).
[CrossRef]

L. N. Langley and K. A. Shore, “Intensity noise and linewidth characteristics of laser diodes subject to phase conjugate optical feedback,” IEE Proc.: Optoelectron. 141, 103–108 (1994).

K. A. Shore and D. T. Wright, “Improved synchronisation algorithm for secure communications systems using chaotic encryption,” Electron. Lett. 30, 1203–1204 (1994).
[CrossRef]

L. N. Langley and K. A. Shore, “The effect of phase conjugate optical feedback on the intensity noise in laser diodes,” Opt. Lett. 18, 1432–1434 (1993).
[CrossRef] [PubMed]

L. N. Langley and K. A. Shore, “The effect of external optical feedback on timing jitter in modulated laser diodes,” J. Lightwave Technol. 11, 434–441 (1993).
[CrossRef]

Turovets, S.

Turovets, S. I.

S. I. Turovets, A. Valle, and K. A. Shore, “Effects of noise on the turn-on dynamics of a modulated class-B laser in the generalized multistability domain,” Phys. Rev. A 55, 2426–2434 (1997).
[CrossRef]

Valle, A.

S. I. Turovets, A. Valle, and K. A. Shore, “Effects of noise on the turn-on dynamics of a modulated class-B laser in the generalized multistability domain,” Phys. Rev. A 55, 2426–2434 (1997).
[CrossRef]

van Tartwijk, G. H. M.

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclass. Opt. 7, 87–143 (1995).
[CrossRef]

Wang, X.

G. Li, R. K. Boncek, X. Wang, and D. H. Sackett, “Transient and optoelectronic feedback-sustained pulsation of laser diodes at 1300 nm,” IEEE Photon. Technol. Lett. 7, 854–856 (1995).
[CrossRef]

Watanabe, N.

Wright, D. T.

K. A. Shore and D. T. Wright, “Improved synchronisation algorithm for secure communications systems using chaotic encryption,” Electron. Lett. 30, 1203–1204 (1994).
[CrossRef]

Yee, W. M.

Yorke, J. A.

T. Shinbrot, E. Ott, C. Greboggi, and J. A. Yorke, “Using chaos to direct trajectories to targets,” Phys. Rev. Lett. 65, 3215–3218 (1990).
[CrossRef] [PubMed]

E. Ott, C. Greboggi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

Electron. Lett.

K. A. Shore and D. T. Wright, “Improved synchronisation algorithm for secure communications systems using chaotic encryption,” Electron. Lett. 30, 1203–1204 (1994).
[CrossRef]

IEE Proc.: Optoelectron.

L. N. Langley and K. A. Shore, “Intensity noise and linewidth characteristics of laser diodes subject to phase conjugate optical feedback,” IEE Proc.: Optoelectron. 141, 103–108 (1994).

IEEE J. Quantum Electron.

A. T. Ryan, G. P. Agrawal, G. R. Gray, and E. C. Gage, “Optical-feedback-induced chaos and its control in semiconductor lasers,” IEEE J. Quantum Electron. 30, 668–679 (1994).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

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Figures (10)

Fig. 1
Fig. 1

Laser intensity u versus time t: a single event of the targeting by means of a perturbation of the driving current. Time of the correction kick (marked by a circle) is equal to φtarg/2T=0.0179 (2T=0.326τsp=652 ps). The targeting kicks have a super-Gaussian form with the form factor m=2, pulse strength βtarg=0.1, and pulse width w/2T=0.4. At the bottom a 2T-periodic signal is given for reference. The normalized parameters are as follows: v=1000, Γ=0.4, r=4×10-3, n0=1.7, nth=4.2075, α=2500, λ0=8.2, ωτ=4.256, ε=7.059×10-3, τ=0.1, k=4.6.

Fig. 2
Fig. 2

Targeting the unstable T cycle by short-kick perturbations of the driving current. The parameters of the targeting pulse are the same as in Fig. 1. The time scale is given here in units of a self-pulsation period, i.e., 2T, which is 652 ps. (a) Relaxation time Trel versus the time needed to apply a targeting kick. (b) Final 2T-periodic regime phase taken with respect to the phase of initial 2T-periodic oscillations and normalized to 2π versus time needed to apply a targeting kick. Sharp attractor phase switchings by π (or 0.5 in normalized units) correspond to the location of peaks in the Fig. 2(a).

Fig. 3
Fig. 3

Time-domain demonstration of the processes of targeting and subsequent activation of the control scheme to stabilize the unstable T cycle by means of short-kick perturbations of losses (w0). The targeting-kick parameters are φtarg /2T=0.113829, βtarg=0.2. Time is normalized to τsp; other parameters are as in Fig. 1. (a) Laser intensity u versus time t. After having delivered the laser to the T cycle by a targeting pulse [marked by the circle in Fig. 3(b)] the control scheme is activated [the first peak in Fig. 3(b)]; β1=0.8; φ1=0.6 (in units of T=0.163). (b) Controlling-kick strength β1[u(φ1+ti)-u(φ1+ti-Ti)] versus time t, measured in units of intensity and tending to zero when control is achieved.

Fig. 4
Fig. 4

Time-domain demonstration of the processes of targeting and subsequent activation of the control scheme to stabilize the unstable T cycle using the finite-width-kick perturbations of a driving current. The parameters of the control scheme are w/T=0.4; β2=0.2; φ2/T=0.58, m=3. The parameters of the targeting pulse are φtarg /2T=0.030825, βtarg=0.1. Time is normalized to τsp; the rest of parameters are as in Fig. 1. (a) Laser intensity u versus time t. After having delivered the laser to the T cycle by a targeting pulse [marked by the circle in Fig. 4(b)], the control scheme is activated [the first peak in Fig. 4(b)]. (b) Controlling kicks signal K2(t) versus time; measured in the same units as the pump current and tending to zero when control is achieved.

Fig. 5
Fig. 5

Domains of control in the control-parameter space for the stabilizing technique using the short-pulse (w0) perturbation of losses. The solitary laser parameters are the same as in Fig. 1. (a) Contour plot of the relaxation time to the T-periodic unstable orbit in the phase space: feedback strength β1 versus feedback phase φ1. The relaxation time trel is measured from the moment the laser is prepared near the T cycle to the moment the system relaxes to T cycle with an accuracy of 10-4. The time is normalized to the oscillation period T. The external curve corresponds to trel=300 and gives the practical external boundaries of the control domain. The inner curves correspond to trel=45, 20, 15, 10, and 8. (b) Same as in Fig. 5(a), but in the kφ1 parameter space (the optical-feedback strength versus the phase of the additional electronic feedback). The isolines correspond to the relaxation-time values: 300, 45, 20, 10, 8 (in units of T). β1=0.8.

Fig. 6
Fig. 6

Domains of control for the stabilizing technique using the short-pulse (w0) perturbation of pump current, feedback-monitored by the difference of intensities. The solitary laser-parameter values and meaning of the plots are the same as in Fig. 5. Contour plot of the relaxation time to the T-periodic unstable orbit in the β2φ2 space (feedback strength versus feedback phase). The isolines correspond to the relaxation-time values 300, 40, 20, and 15. In small islands of control, only the external curve trel=300 is shown.

Fig. 7
Fig. 7

Domain of control for the stabilizing technique using stepwise perturbations of the laser field phase increment in the external resonator, feedback monitored by the difference of intensities. The parameter values of the laser diode and meaning of the plots are the same as in Figs. 5 and 6. The parameters of the control-scheme pulses are w/T=1; m=> (a square form). (a) Contour plot of the relaxation time to the T-periodic unstable orbit in the space; feedback strength β3 and feedback phase φ3. The isolines correspond to the relaxation-time values 300, 45, 30, 15, and 10. (b) Contour plot of the relaxation time in the kφ3 phase space (optical-feedback strength versus the phase of the additional electronic feedback) for β3=-2. The isolines correspond to the relaxation-time values 300, 45, and 20.

Fig. 8
Fig. 8

Effects of the finite width of correcting pulses and the finite technical delay on the domains of control for the stabilizing technique using the pump current. Contour plots of the relaxation time to the T-periodic unstable orbit in the τel/Tβ2 space (feedback technical delay versus feedback strength) for φ2=0. The parameters of the super-Gaussian pulses are the following: w/T=0.2, m=2. The actual scale of the technical delay is shifted from zero by a half-width of the controlling pulse. Solid curves correspond to trel=300 and essentially give the external boundaries of the control domain. Control is possible inside this domain and not achievable outside of it. The dotted curves correspond to the negative relaxation time trel=-10 (the scheme is unstable). The patterns demonstrate two tendencies in the behavior of the boundaries of control: their recurrency with a period 2T and narrowing with increasing technical delay.

Fig. 9
Fig. 9

Contour plots of the relaxation time to the T-periodic unstable orbit in the β2τel/T space (feedback strength versus feedback technical delay) for the different super-Gaussian pulses parameters. Here m=3 and (a) w/T=0.4 and (b) w/T =0.8. The actual scale of the technical delay does not include a half-width of the controlling pulse. The pictures demonstrate broadening of the boundaries of control with increasing control-pulse width.

Fig. 10
Fig. 10

Noise performance of the control scheme using the driving current in three-dimensional phase space representation. The parameters are the same as in Fig. 4; the Langevin forces are given by Eqs. (12).

Equations (18)

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dS(t)dt=S(t)Gn[N(t)-N0]Γ11+εS(t)-1τph+γΓN(t)τsp+2 kextτL S(t)S(t-τext)×cos[θ(t)]+Fs(t),
dN(t)dt=I(t)eV-N(t)τsp-S(t)Gn[N(t)-N0]×11+εS(t)+Fn(t),
dΦ(t)dt=12 αGn[N(t)-Nth]Γ-kextτL S(t-τext)S(t) sin[θ(t)]+Fϕ(t),
θ(t)=Φ(t)-Φ(t-τext)+ωthτext.
kext=1-R2R2 Rextη,
u˙=vu[g(n, u)Γ-1-K1(t)]+rn+2kuuτ cos θ,
n˙=P0+K2(t)-n-ug(n, u),
Φ˙=α(n-nth)Γ-kuτ/u sin θ,
θ=Φ-Φτ+ωτ+K3(t).
Kj(t)=βji[u(φj+ti)-u(φj+ti-Ti)]×f[t-(φj+ti+τel)].
f(t)=M exp(-at2m),a=[Γ(1/2m)/mw]2m,
M=1/w,
f(t)=s2m-1/Qt2m-s2m,Q=π/m sin(π/2m),
s=w/Q.
Kj(targ)(t)=βtargf(t-φtarg),
Fn(t)=-2S(ti)γNΓτspΔt xs+2N(ti)τspΔtV xn,
Fs(t)=2S(ti)γNΓτspΔt xs,
FΦ(t)=1S(t) S(ti)γNΓ2τspΔt xΦ.

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