Abstract

In large high-power broad-area lasers the spatiotemporal filamentation processes and instabilities occur macroscopic as well as on microscopic scales. Numerical simulations on the basis of Maxwell-Bloch equations for large longitudinally and transversely extended semiconductor lasers reveal the internal spatial and temporal processes, providing the relevant scales on which control for stabilization consequently has to occur. It is demonstrated that the combined longitudinal instabilities, filamentation, and propagation effects may be controlled by suitable spatially structured delayed optical feedback allowing, in particular, the control of coherent regimes in originally temporally and spatially chaotic states.

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References

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  1. E. Ott, C. Grebogi, and J. A. Yorke, "Controlling chaos," Phys. Rev. Lett. 64, 1196-1199 (1990).
    [CrossRef] [PubMed]
  2. K. Pyragas, "Continuous control of chaos, by self-controlling feedback," Phys. Lett. A 170, 421-428 (1992).
    [CrossRef]
  3. C. Simmendinger and O. Hess, "Controlling delay-induced chaotic behavior of a semiconductor laser with optical feedback," Phys. Lett. A 216, 97-105 (1996).
    [CrossRef]
  4. M. Münkel, F. Kaiser and O. Hess, "Spatiotemporal dynamics of multi-stripe semiconductor lasers with delayed optical feedback," Phys. Lett. A 222, 67-75 (1996).
    [CrossRef]
  5. M. Münkel, F. Kaiser and O. Hess, "Suppression of instabilities leading to spatiotemporal chaos in semiconductor laser arrays by means of delayed optical feedback," Phys. Rev. E 56, 3868-3875 (1997).
    [CrossRef]
  6. I. Fischer, O. Hess, W. Elsaber and E. Gobel, "Complex Spatio-Temporal Dynamics in the Nearfield of a Broad-Area Semiconductor Laser," Europhys. Lett. 35, 579-584 (1996).
    [CrossRef]
  7. C. M. Bowden and G. P. Agrawal, "Maxwell-Bloch formulation for semiconductors: Effects of Coulomb exhange," Phys. Rev. A 51, 4132-4139 (1995).
    [CrossRef] [PubMed]
  8. J. Yao, G. P. Agrawal, P. Gallion and C. M. Bowden, "Semiconductor laser dynamics beyond the rate-equation approximation," Opt. Communic. 119, 246-255 (1995).
    [CrossRef]
  9. S. Balle, "Effective two-level-model with asymmetric gain for laser diodes," Opt. Commun. 119, 227-235 (1995).
    [CrossRef]
  10. C. Z. Ning, R. A. Indik, and J. V. Moloney, "Effective Bloch equations for semiconductor lasers and amplifiers," IEEE J. Quantum Electron. 33, 1543-1550 (1997).
    [CrossRef]
  11. O. Hess, S. W. Koch, and J. V. Moloney, "Filamentation and Beam Propagation in Broad-Area Semiconductor Lasers," IEEE J. Quantum Electron. QE-31, 35-43 (1995).
    [CrossRef]
  12. O. Hess and T. Kuhn, "Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers I: Theoretical Description," Phys. Rev. A 54, 3347-3359 (1996).
    [CrossRef] [PubMed]
  13. O. Hess and T. Kuhn, "Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers II: Spatio-temporal dynamics," Phys. Rev. A 54, 3360-3368 (1996).
    [CrossRef] [PubMed]
  14. E. Gehrig and O. Hess, "Nonequilibrium Spatiotemporal Dynamics of the Wigner-Distributions in Broad-Area Semiconductor Lasers," Phys. Rev. A 57, 2150-2163 (1998).
    [CrossRef]
  15. C. Simmendinger, M. Munkel, and O. Hess, "Controlling Complex Temporal and Spatio-Temporal Dynamics in Semiconductor Lasers," Chaos, Solitons & Fractals 10, 851-864 (1999).
  16. E. Gehrig, O. Hess, and W. Wallenstein, "Modeling of the Performance of High-Power Diode Amplifier Systems with an Opto-Thermal Microscopic Spatio-Temporal Theory," IEEE J. Quantum Electron. 35, 320-331 (1999).
    [CrossRef]
  17. O. Hess and T. Kuhn, "Spatio-Temporal Dynamics of Semiconductor Lasers: Theory, Modeling and Analysis," Prog. Quant. Electr. 20, 85-179 (1996).
    [CrossRef]
  18. I. S. Grieg and J. D. Morris, "A Hopscotch method for the Korteweg-de-Vries equation," J. Comp. Phys. 20, 60-84 (1976).
  19. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1989).
  20. O. Hess, Spatio-Temporal Dynamics of Semiconductor Lasers (Wissenschaft und Technik Verlag, Berlin, 1993).
  21. M. E. Bleich, D. Hochheiser, J. V. Moloney, and J. E. S. Socolar, "Controlling extended systems with spatially filtered, time-delayed feedback," Phys. Rev. E 55, 2119 (1997).
    [CrossRef]

Other (21)

E. Ott, C. Grebogi, and J. A. Yorke, "Controlling chaos," Phys. Rev. Lett. 64, 1196-1199 (1990).
[CrossRef] [PubMed]

K. Pyragas, "Continuous control of chaos, by self-controlling feedback," Phys. Lett. A 170, 421-428 (1992).
[CrossRef]

C. Simmendinger and O. Hess, "Controlling delay-induced chaotic behavior of a semiconductor laser with optical feedback," Phys. Lett. A 216, 97-105 (1996).
[CrossRef]

M. Münkel, F. Kaiser and O. Hess, "Spatiotemporal dynamics of multi-stripe semiconductor lasers with delayed optical feedback," Phys. Lett. A 222, 67-75 (1996).
[CrossRef]

M. Münkel, F. Kaiser and O. Hess, "Suppression of instabilities leading to spatiotemporal chaos in semiconductor laser arrays by means of delayed optical feedback," Phys. Rev. E 56, 3868-3875 (1997).
[CrossRef]

I. Fischer, O. Hess, W. Elsaber and E. Gobel, "Complex Spatio-Temporal Dynamics in the Nearfield of a Broad-Area Semiconductor Laser," Europhys. Lett. 35, 579-584 (1996).
[CrossRef]

C. M. Bowden and G. P. Agrawal, "Maxwell-Bloch formulation for semiconductors: Effects of Coulomb exhange," Phys. Rev. A 51, 4132-4139 (1995).
[CrossRef] [PubMed]

J. Yao, G. P. Agrawal, P. Gallion and C. M. Bowden, "Semiconductor laser dynamics beyond the rate-equation approximation," Opt. Communic. 119, 246-255 (1995).
[CrossRef]

S. Balle, "Effective two-level-model with asymmetric gain for laser diodes," Opt. Commun. 119, 227-235 (1995).
[CrossRef]

C. Z. Ning, R. A. Indik, and J. V. Moloney, "Effective Bloch equations for semiconductor lasers and amplifiers," IEEE J. Quantum Electron. 33, 1543-1550 (1997).
[CrossRef]

O. Hess, S. W. Koch, and J. V. Moloney, "Filamentation and Beam Propagation in Broad-Area Semiconductor Lasers," IEEE J. Quantum Electron. QE-31, 35-43 (1995).
[CrossRef]

O. Hess and T. Kuhn, "Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers I: Theoretical Description," Phys. Rev. A 54, 3347-3359 (1996).
[CrossRef] [PubMed]

O. Hess and T. Kuhn, "Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers II: Spatio-temporal dynamics," Phys. Rev. A 54, 3360-3368 (1996).
[CrossRef] [PubMed]

E. Gehrig and O. Hess, "Nonequilibrium Spatiotemporal Dynamics of the Wigner-Distributions in Broad-Area Semiconductor Lasers," Phys. Rev. A 57, 2150-2163 (1998).
[CrossRef]

C. Simmendinger, M. Munkel, and O. Hess, "Controlling Complex Temporal and Spatio-Temporal Dynamics in Semiconductor Lasers," Chaos, Solitons & Fractals 10, 851-864 (1999).

E. Gehrig, O. Hess, and W. Wallenstein, "Modeling of the Performance of High-Power Diode Amplifier Systems with an Opto-Thermal Microscopic Spatio-Temporal Theory," IEEE J. Quantum Electron. 35, 320-331 (1999).
[CrossRef]

O. Hess and T. Kuhn, "Spatio-Temporal Dynamics of Semiconductor Lasers: Theory, Modeling and Analysis," Prog. Quant. Electr. 20, 85-179 (1996).
[CrossRef]

I. S. Grieg and J. D. Morris, "A Hopscotch method for the Korteweg-de-Vries equation," J. Comp. Phys. 20, 60-84 (1976).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1989).

O. Hess, Spatio-Temporal Dynamics of Semiconductor Lasers (Wissenschaft und Technik Verlag, Berlin, 1993).

M. E. Bleich, D. Hochheiser, J. V. Moloney, and J. E. S. Socolar, "Controlling extended systems with spatially filtered, time-delayed feedback," Phys. Rev. E 55, 2119 (1997).
[CrossRef]

Supplementary Material (4)

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Figures (2)

Fig. 1.
Fig. 1.

Schematic geometry of the spatially structured delayed optical feedback stabilization scheme realized in the form of an unstable external resonator.

Fig. 2.
Fig. 2.

Initial frames from QuickTime movies of the spatiotemporal dynamics of the intensity (left, 2.2 MB) and charge carrier density (right, 1 MB) within the active layer of a broad area laser. Corresponding higher-quality movies have a size of 8.4 MB and 4.8 MB, respectively. The animations cover a time-period of 500 ps. Spatially structured delayed optical feedback is applied at 200 ps. Bottom figures: spatial intensity and charge carrier density distribution (vertical axis: transverse width w=100µm, horizontal axis: longitudinal resonator L=800µm); middle frames: the time-averaged (up to the time portrayed) intensity and density profiles at the out-coupling facet, and top frames: the intensity- and density-trace in the center of the laser stripe at (x=0; z=L). Note that the chance of colors from red to green in the left animation marks the application of feedback at 200 ps.

Equations (5)

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t f k e , h = g k γ k e , h ( f k e , h f k , eq e , h ) + Λ k e , h Γ k sp f k e f k h γ nr f k e , h
t p k ± = ( i ω ¯ k + γ k p ) p k ± + 1 i d k E ± ( f k e + f k h 1 ) ,
± z E ± + n l c t E ± = i 2 1 K z 2 x 2 E ± ( α 2 + i η ) E ± + i 2 Γ n l 2 0 L P nl ± + κ τ r E ± ( t τ )
σ = R ( w 2 ) 2 + ( L e + R ) 2 ,
t N = · ( D f N ) + Λ + G γ nr W ,

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