## Stabilization of chaotic spatiotemporal filamentation in large broad area lasers by spatially structured optical feedback

Optics Express, Vol. 5, Issue 3, pp. 48-54 (1999)

http://dx.doi.org/10.1364/OE.5.000048

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### 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.

© Optical Society of America

^{1;2}to a chaotic semiconductor laser displaying complex spatiotemporal chaos, however, is not straight-forward

^{3}. Due to the small timescales involved in semiconductor-laser dynamics an all-optical control scheme is required. A naive application of a delayed optical feedback (DOF) control or stabilization method to the semiconductor laser, however, even tends to increase spatiotemporal complexity in spatially distributed systems

^{3;4}. With careful choice of the feedback parameters obtained e.g. from a complex eigenmode analysis, DOF has, indeed, successfully been employed for a stabilization of the typical spatiotemporal chaos in multi-stripe semiconductor laser arrays

^{5}. In the broad-area laser, however, dynamic filamentation efects appear in the near-field spatiotemporal intensity trace as transversely migrating filaments and sub-ns pulsations

^{6}. Due to the continuous spectra of relevant spatial, spectral and temporal scales stabilization is even more involved in this high-power semiconductor laser system: Temporal, spatial, and spectral degrees of freedom have to be simultaneously stabilized by designing an appropriate control set-up.

^{7–9}. In an alternative approach based on effective Bloch equations for semiconductor lasers and amplifiers, the carrier-density dependence of the gain and refractive index and their respective dispersions are efficiently approximated by a superposition of several Lorentzians

^{10}. In microscopic simulations on the basis of Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers

^{11–13}, the full space and momentum dependence of the charge carrier distributions and the polarization has been included. Within the latter theory good quantitative agreement of simulation results with streak-camera measurements of the spatiotemporal near-field intensity dynamics of a broad-area semiconductor laser has been obtained

^{6}. The microscopic Wigner-function approach

^{12}provides next to the macroscopic spatiotemporal intensity dynamics, information on the complex internal interplay of the spatiotemporal light-field dynamics with the active semiconductor medium, demonstrating, in particular, for the case of high-power BAL the relevance of dynamic spatiospectral holeburning and spatiotemporal carrier-carrier as well as carrier-phonon scattering processes

^{13,14}.

^{15}we will considerably extend our study to the stabilization of BAL which support due to their geometry, material and waveguiding properties both, multiple transverse filaments and dynamic longitudinal structurs. Attempting to extend stabilization principles which have been successful in the case of a large discrete MSLA to the BAL one quickly realizes that next to a control of the temporal degrees of freedom by a temporal delay additionally, the spatiotemporal and spatiospectral dynamics have to be appropriately controlled in order to stabilize the whole system

^{15}. Thus, the spatiotemporal internal dynamics and the spatially inhomogeneous delayed optical feedback have to be included simultaneously in a theoretical description of the microscopic spatiotemporal efects which determine the interaction of the optical field with the active medium. To account for the microscopic processes which act in concert with the macroscopic spatiotemporal interactions we will base our investigation on the semiconductor laser model derived in

^{12}and applied to the description of free-running broad-area lasers

^{13}and tapered amplifiers

^{16}.

*semiconductor laser Maxwell-Bloch delay-equations*consist of Maxwell’s wave equations for the counterpropagating optical fields

*E*

^{±}=

*E*

^{±}(

**r**,

*t*) into which the efect of structured delayed optical feedback is included and an ambipolar transport equation for the charge carrier density

*N*=

*N*(

**r**,

*t*). This coupled system is-in turn-self-consistently coupled with spatially inhomogeneous semiconductor Bloch equations

^{17}for the Wigner distributions

*f*

^{e,h}(

*k*,

**r**,

*t*) of electrons (e) and holes (h) as well as the interband polarizations

*p*

^{±}(

*k*,

**r**,

*t*), where

**r**=(

*x, z*) indicates the longitudinal light field propagation direction

*z*, and the transverse direction

*x*, while

*k*refers to the dependence on the carrier-momentum wavenumber. The dynamics of the Wigner distributions is governed by the semiconductor laser Bloch equations

*gk*=-1/4

*ħd*

_{k}Im [

*E*

^{+}

*E*

^{-}

*p*

^{-}*

_{k}, where Im [·] indicates the imaginary part,

*d*

_{k}the interband dipole matrix element. The microscopic density-dependent scattering rates

^{12}and include carrier-carrier-scattering mechanisms and the interaction of carriers with optical (LO) phonons. Generally the frequency detuning

*ω̄*

_{k}=

*ω̄*

_{k}(

*T*

_{l}) and the spontaneous recombination coefficient

*T*

_{l}) depend on the lattice temperature

*T*

_{l}. The dynamic variation of the spatial distribution of

*T*

_{l}within the active layer is generally coupled with the carrier and light-field dynamics and may self-consistently included in the model

^{14}. However, we will in the following assume an approximately stationary temperature profile. The microscopic pump term

*V*

^{-1}∑

_{k}

*η*

_{eff}

*𝓣*/

*ed*includes the spatially dependent charge carrier density

*𝓣*, the injection efficiency

*η*

_{eff}=0:5, and the thickness

*d*=0.1

*µ*m of the active area. γ

_{nr}=5 ns is the rate due to nonradiative recombination. The coupling between the microscopic spatiospectral dynamics and the macroscopic propagation of the light field is mediated by the macroscopic nonlinear polarizations

*V*

^{-1}∑

_{k}

*d*

_{k}

*Γ*

_{r}and

*Γ*, respectively.

*R*

_{0}=0.01 (

*R*

_{0}=0.33) and

*R*

_{1}=0.7 being the reflectivity of the front (back) laser facet and the external mirror, respectively.

*R*=0.5 mm and

*L*

_{e}=2370

*µ*m are the radius of curvature and the length of the external resonator, respectively, and

*ω*=100

*µ*m is the transverse width of the emitter. In (2)

*K*

_{z}denotes the wavenumber of the propagating fields,

*n*

_{l}is the refractive index of the active layer,

*L*the length of the structure, and

*α*=

*α*(

*T*

_{l}) the linear absorption coefficient. The parameter

*η*includes transverse (x) and vertical (y) variations of the refractive index due to the waveguide structure and the waveguiding properties are described by the confinement factor Γ

^{17}. The optical properties additionally depend on the local density of charge carriers, whose dynamics is governed by the carrier transport equation

*G*=

*X*

^{″}

*∊*

_{0}/2

*ħ*(|

*E*

^{+}|

^{2}+|

*E*

^{-}|

^{2})-1/4

*ħ*Im [

*E*

^{+}

*P*

^{+}*

_{nl}+

*E*

^{-}

*P*

^{-}*

_{nl}], and the spontaneous emission

*W*=

*V*

^{-1}∑

_{k}

*Hopscotch*method

^{18}is used as a general scheme and the operators are discretized by the

*Lax-Wendroff*

^{19}method. Details on the numerical method may be found in

^{20}. Here, the spatial resolution of the grid lies in the

*µ*m - regime at integration time steps of about 0.5

*fsec*.

*µ*m and its longitudinal length L=800

*µ*m are typical values of commercially available devices. In the simulations, the laser are electrically pumped at a two times its threshold current. The animations in Fig. 2 display the spatiotemporal evolvement of the intensity

*I*(

*x, z, t*) ~|

*E*

^{+}+

*E*

^{-}|

^{2}and charge carrier density

*N*(

*x, z, t*) within the active layer of the broad-area laser. The vertical extension displayed in the bottom frames corresponds to a transverse width of the current stripe and waveguide

*w*=100µm, and the horizontal axis to the longitudinal resonator with length

*L*=800

*µ*m. The electrical current is applied at

*t*=0. The initial 200 ps of the animation (reddish colors) visualize the free-running condition (i.e. without optical feedback). In the intensity distribution one can follow the formation, propagation, and vanishing of filaments appearing on the outcoupling facet as migrating filaments

^{6}. The processes which lead to this peculiar migrating behavior are a consequence of various processes acting in concert and highlighted by the direct correspondence of the spatiotemporal intensity and charge carrier density dynamics. As a result of the microscopic scattering processes the relaxation times of the carrier density are larger than that of the optical field and leading to a localization of filaments of high intensity in wave-guiding channels formed by the carrier density. As the animation on the right shows, the carrier density is dynamically locally depleted by a filament of high intensity. The optical filament is thus located in a region of low gain (low carrier-density) and relatively high refractive index. By the process of gain-guiding the filament thus provides itself with the dielectric waveguide which is necessary for its support during propagation in the laser cavity. However, due to the length of the (asymmetric) internal cavity of the BAL, the filaments longitudinally inhomogeneous, leading to a wave-like reflection. At the same time, with uniform injection of charge carriers the local carrier density outside the filament is not being depleted by stimulated emission, and consequently, rises quickly to levels above the threshold charge carrier density. A new optical filament is thus created. Moreover, the animation of the density shows that the induced waveguides persist considerably longer than the actual presence of the filaments which had initially been their origin, thus becoming a means of memory for other filaments to follow. Every new filament thus via the medium nonlinearly interacts with the previous filament, thereby destabilizing it. The result is a vividly irregular and fundamentally chaotic longitudinal and transverse interaction of optical filaments. Also, the gain-guiding processes inside the laser cavity ensure that by sustaining relatively stable high values of the density at the edge of the laser stripe the optical field has created its own optical waveguide, i.e. an effective waveguide is formed. Following this global focusing efect transverse modulations appear on a finer scale which promote finer scale filamentation instabilities through a transverse modulational instability. In direct comparison, the animations vividly show that the filamentation process in the BAL is a result of continuous competition between the anti-guiding efects, i.e. the carrier-induced refractive index

*δn*being negative, sub-sequent self-focusing, difraction, the tendency of the filaments to follow the gain (which is proportional to the density of charge carriers), and propagation efects.

*w*

_{f}≈10–15

*µ*m

^{13;14}. The formation of multiple filaments is also reflected in the far-field: the emitted far-field widens and becomes more structured. Increasing instabilities which lead to the observed characteristic migration of the filaments in the near-field across the laser facet then cause an even stronger widening of the far-field

^{15}.

^{21}. In the BAL system, this may be established if the time-delayed feedback is additionally structured in space and internal microscopic time scales

^{13}are taken into account. Indeed, simulations have shown that spatially un-structured (flat) optical feedback causes a stabilization of the migrating filaments leading to a chevron-like spatiotemporal pattern

^{15}and spatially and temporally appropriately tailored optical feedback is necessary for stabilization of a BAL. In the animation of Fig. 2 spatially structured delayed optical feedback pertaining to the unstable resonator configuration is applied at the time

*t*=200 ps. The animation shows that the application of the tailored optical feedback (marked by a change in color from red shading to green shading) is highly effective: it takes the BAL only a few ps for an autocatalytic spatiospectral mode coupling processes

^{15}to induce a spatially stable light field. Note that due to the relaxation times of the charge carrier density (~ 5 ns) being about two orders of magnitude larger than the propagation and feedback times the approach to cw laser emission is only gradual (

*t*>5 ns)

^{15}and has — for reasons of display and file size — not been included in the animations.

## References

1. | E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos, ” Phys. Rev. Lett. |

2. | K. Pyragas, “Continuous control of chaos, by self-controlling feedback, ” Physics Lett. A |

3. | C. Simmendinger and O. Hess, “Controlling delay-induced chaotic behavior of a semiconductor laser with optical feedback, ” Physics Lett. A |

4. | M. MÄunkel, F. Kaiser, and O. Hess, “Spatiotemporal dynamics of multi-stripe semiconductor lasers with delayed optical feedback, ” Physics Lett. A |

5. | M. MÄunkel, 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 |

6. | I. Fischer, O. Hess, W. ElsÄaßer, and E. GÄobel, “Complex Spatio-Temporal Dynamics in the Nearfield of a Broad-Area Semiconductor Laser, ” Europhys. Lett. |

7. | C. M. Bowden and G. P. Agrawal, “Maxwell-Bloch formulation for semiconductors: Efects of Coulomb exhange, ” Phys. Rev. A |

8. | J. Yao, G. P. Agrawal, P. Gallion, and C. M. Bowden, “Semiconductor laser dynamics beyond the rate-equation approximation, ” Opt. Communic. |

9. | S. Balle, “Effective two-level-model with asymmetric gain for laser diodes, ” Opt. Commun. |

10. | C. Z. Ning, R. A. Indik, and J. V. Moloney, “Effective Bloch equations for semiconductor lasers and amplifiers, ” IEEE J. Quantum Electron. |

11. | O. Hess, S. W. Koch, and J. V. Moloney, “Filamentation and Beam Propagation in Broad-Area Semiconductor Lasers, ” IEEE J. Quantum Electron. |

12. | O. Hess and T. Kuhn, “Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers I: Theoretical Description, ” Phys. Rev. A |

13. | O. Hess and T. Kuhn, “Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers II: Spatio-temporal dynamics, ” Phys. Rev. A |

14. | E. Gehrig and O. Hess, “Nonequilibrium Spatiotemporal Dynamics of the Wigner-Distributions in Broad-Area Semiconductor Lasers, ” Phys. Rev. A |

15. | C. Simmendinger, M. MÄunkel, and O. Hess, “Controlling Complex Temporal and Spatio-Temporal Dynamics in Semiconductor Lasers, ” Chaos, Solitons & Fractals |

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. |

17. | O. Hess and T. Kuhn, “Spatio-Temporal Dynamics of Semiconductor Lasers: Theory, Modeling and Analysis, ” Prog. Quant. Electr. |

18. | I. S. Grieg and J. D. Morris, “A Hopscotch method for the Korteweg-de-Vries equation, ” J. Comp. Phys. |

19. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, |

20. | O. Hess, |

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 |

**OCIS Codes**

(140.5960) Lasers and laser optics : Semiconductor lasers

(270.3100) Quantum optics : Instabilities and chaos

**ToC Category:**

Focus Issue: Spatial and Polarization Dynamics of Semiconductor Lasers

**History**

Original Manuscript: April 8, 1999

Published: August 2, 1999

**Citation**

Christian Simmendinger, Dietmar Preiber, and Ortwin Hess, "Stabilization of chaotic spatiotemporal filamentation in large broad area lasers by spatially structured optical feedback," Opt. Express **5**, 48-54 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-5-3-48

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### References

- 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]

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