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Optics Express

  • Editor: J. H. Eberly
  • Vol. 5, Iss. 3 — Aug. 2, 1999
  • pp: 48–54

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

Christian Simmendinger, Dietmar Preiber, and Ortwin Hess  »View Author Affiliations


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

In semiconductor laser dynamics there are mainly two reasons for instabilities: (1) Due to its very high gain and outcoupling rate, the semiconductor laser is very sensitive to delayed optical feedback (DOF) caused by distant reflecting surfaces such as e.g. an optical fiber. (2) In high-power semiconductor lasers the nonlinear interaction of spatial with temporal degrees of freedom leads to chaotic spatiotemporal instabilities. Clearly, for practical reasons it is highly desired to suppress these delay-induced and spatiotemporal instabilities. The strong nonlinearities which one encounters, in particular, in the high-power coupled multi-stripe laser arrays (MSLA) or broad area laser (BAL) structures are usually circumvented by resorting to small and low power lasers; laser arrays only emit stable laser radiation by arranging the lasers such that they are uncoupled, i.e. sufficiently far separated and isolated. Clearly, for coupled and high power semiconductor laser structures alternative schemes for controlling the complex temporal and spatiotemporal dynamics are desired. The application of schemes from the field of chaos-control1;2 to a chaotic semiconductor laser displaying complex spatiotemporal chaos, however, is not straight-forward3. 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 systems3;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 arrays5. In the broad-area laser, however, dynamic filamentation efects appear in the near-field spatiotemporal intensity trace as transversely migrating filaments and sub-ns pulsations6. 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.

Next to empirical experimental investigations it is, in particular, by the help of realistic and therefore as much as possible microscopically founded simulations of the spatiotemporal dynamics of the BAL with which one hopes to understand the internal processes and to obtain the vital quantities required for a successful stabilization. Various approaches have been made in order to include in a numerical modeling of BAL both, spatial and temporal variations as well as characteristic semiconductor laser properties. For that purpose, approximate Maxwell-Bloch equations have been proposed7–9. 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 Lorentzians10. In microscopic simulations on the basis of Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers11–13, 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 obtained6. The microscopic Wigner-function approach12 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 processes13,14.

It is the purpose of this contribution to shed light on the internal complex spatiotemporal dynamics of the light field in BAL and, in particular, on the stabilizing influence of (appropriately tailored) spatially structured delayed optical feedback. The spatiotemporal dynamics are vividly visualized in the form of animations of the results of microscopic numerical simulations. In extension of a previous study where the stabilization of MSLA and single-mode BAL has been discussed15 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 system15. 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 in12 and applied to the description of free-running broad-area lasers13 and tapered amplifiers16.

The 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 equations17 for the Wigner distributions f k e,h =fe,h (k, r, t) of electrons (e) and holes (h) as well as the interband polarizations p k ±=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

∂ ∂t fk e,h= gk− γk e,h ( fk e,h− f k,eq e,h)+ Λk e,h− Γksp fke fkh− γnr fk e,h
(1a)
∂ ∂t pk±= − ( i ω¯k+ γkp) pk±+ 1 iℏ dk E± ( fke+ fkh−1),
(1b)

where the microscopic nonlinear carrier generation rate is given by gk=-1/4ħdk Im [E + p k +*+E - p -* k , where Im [·] indicates the imaginary part, f k,eq e,h are the quasi-equilibrium carrier distributions, and dk the interband dipole matrix element. The microscopic density-dependent scattering rates γ k e,h and γ k p are microscopically determined12 and include carrier-carrier-scattering mechanisms and the interaction of carriers with optical (LO) phonons. Generally the frequency detuning ω̄k =ω̄k (Tl ) and the spontaneous recombination coefficient Γ k sp = Γ k sp (Tl ) depend on the lattice temperature Tl . The dynamic variation of the spatial distribution of Tl within the active layer is generally coupled with the carrier and light-field dynamics and may self-consistently included in the model14. However, we will in the following assume an approximately stationary temperature profile. The microscopic pump term Λ k e,h =Λ f k,eq e,h (1- f k e,h )/(V -1∑k f k,eq e,h (1- f k e,h )) represents the pump-blocking effect, where Λ=ηeff𝓣/ed includes the spatially dependent charge carrier density 𝓣, the injection efficiency ηeff =0:5, and the thickness d=0.1µm 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 P nl ±=V -1∑k dk p k ±, which in Maxwell’s wave equation

± ∂ ∂z E±+ nlc ∂ ∂t E±= i2 1 Kz ∂2 ∂ x2 E±− ( α2+iη) E±+ i2 Γ nl2 ∊0L Pnl±+ κ τr E± ( t−τ)
(2)

are the source of the optical fields. Note that the nonlinear polarizations P nl ± contain all spatiotemporal gain- and refractive index variations. In the feedback term κ τr E± ( xσ,z=L, t−τ) the resonator round trip times of the internal and external resonator are Γr and Γ, respectively.

Fig. 1. Schematic geometry of the spatially structured delayed optical feedback stabilization scheme realized in the form of an unstable external resonator.

The back-coupling-strength is denoted by κ= ( 1− R0) R1 R0. The spatially structured feedback, {realized in from of the external resonator configuration schematically displayed in Fig. 1-has in (2) been notationally suppressed. It is taken into account by

σ= R ( w2)2+ ( Le+R)2,
(3)

with 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 Le =2370µm are the radius of curvature and the length of the external resonator, respectively, and ω=100µm is the transverse width of the emitter. In (2) Kz denotes the wavenumber of the propagating fields, nl is the refractive index of the active layer, L the length of the structure, and α=α(Tl ) 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 Γ17. The optical properties additionally depend on the local density of charge carriers, whose dynamics is governed by the carrier transport equation

∂ ∂tN= ∇· ( Df∇N)+Λ+ G− γnr−W,
(4)

with the ambipolar difusion coeficient D f 12, the macroscopic gain G=X ″ ∊ 0/2ħ(|E +|2+|E -|2)-1/4ħ Im [E + P +* nl +E - P -* nl ], and the spontaneous emission W=V -1∑k Γ k sp f k e f k h . For the numerical integration of the system of stif nonlinear partial diferential equations the Hopscotch method18 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 in20. Here, the spatial resolution of the grid lies in the µm - regime at integration time steps of about 0.5fsec.

We model the spatiotemporal dynamics of a typical broad-area laser. The transverse width w=100 µm and its longitudinal length L=800 µm 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=800µm. 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 filaments6. 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.

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.

In the free-running state, a steady state is never reached. Nevertheless one can observe a typical width of the filaments wf ≈10–15µm13;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-field15.

The animations vividly can convey that next to a control of the temporal degrees of freedom spatial scales have to be simultaneously controlled in the BAL21. In the BAL system, this may be established if the time-delayed feedback is additionally structured in space and internal microscopic time scales13 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 pattern15 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 processes15 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.

The spatiotemporal intensity and charge-carrier density dynamics associated with a stabilization of a “large’ high-power broad-area laser displaying in the free-running condition transverse and longitudinal instabilities by spatially structured delayed optical feedback are discussed. Animations of numerical simulations on the basis of microscopic Maxwell-Bloch equations for spatially extended semiconductor lasers including the delayed optical feedback reveal the complex nonlinear spatiotemporal processes within a free-running multi-mode broad-area laser, providing the relevant parameters used in an unstable resonator external cavity set-up. With this appropriately tailored set-up the successful stabilization of a spatiotemporally chaotic BAL is demonstrated in direct visualization of the internal spatiotemporal intensity and density dynamics.

References

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, ” Physics 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, ” Physics Lett. A 216, 97–105 (1996). [CrossRef]

4.

M. MÄunkel, F. Kaiser, and O. Hess, “Spatiotemporal dynamics of multi-stripe semiconductor lasers with delayed optical feedback, ” Physics Lett. A 222, 67–75 (1996). [CrossRef]

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 56, 3868–3875 (1997). [CrossRef]

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. 35, 579–584 (1996). [CrossRef]

7.

C. M. Bowden and G. P. Agrawal, “Maxwell-Bloch formulation for semiconductors: Efects 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. MÄunkel, 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]

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

  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]

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