## 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.

©1999 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-control^{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.

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 proposed^{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}.

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
discussed^{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}.

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 equations^{17} for the Wigner distributions
${f}_{k}^{e\mathit{,}h}$
=*f*^{e,h}
(*k*,
**r**, *t*) of electrons (e) and holes (h) as well as
the interband polarizations ${p}_{k}^{\pm}$=*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

where the microscopic nonlinear carrier generation rate is given by
*gk*=-1/4*ħd*_{k}
Im
[*E*
^{+}
${p}_{k}^{+*}$+*E*
^{-}
*p*
^{-}*_{k}, where Im [·] indicates the imaginary part,
${f}_{k\mathit{,}\mathit{\text{eq}}}^{e\mathit{,}h}$
are the quasi-equilibrium
carrier distributions, and *d*_{k}
the interband dipole
matrix element. The microscopic density-dependent scattering rates
${\gamma}_{k}^{e\mathit{,}h}$
and
${\gamma}_{k}^{p}$
are microscopically
determined^{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 ${\mathrm{\Gamma}}_{k}^{\mathit{\text{sp}}}$
=${\mathrm{\Gamma}}_{k}^{\mathit{\text{sp}}}$
(*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 ${\mathrm{\Lambda}}_{k}^{e\mathit{,}h}$
=Λ${f}_{k\mathit{,}\mathit{\text{eq}}}^{e\mathit{,}h}$
(1-${f}_{k}^{e\mathit{,}h}$
)/(*V*
^{-1}∑_{k}
${f}_{k\mathit{,}\mathit{\text{eq}}}^{e\mathit{,}h}$
(1-${f}_{k}^{e\mathit{,}h}$
))
represents the pump-blocking effect, where
Λ=*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
${P}_{\mathit{\text{nl}}}^{\pm}$=*V*
^{-1}∑_{k}
*d*_{k}${p}_{k}^{\pm}$, which in Maxwell’s wave equation

are the source of the optical fields. Note that the nonlinear polarizations
${P}_{\mathit{\text{nl}}}^{\pm}$ contain all spatiotemporal gain- and refractive index variations. In the
feedback term $\frac{\kappa}{{\tau}_{r}}{E}^{\pm}\left(x\sigma ,z=L,t-\tau \right)$ the resonator round trip times of the internal and external
resonator are *Γ*_{r}
and
*Γ*, respectively.

The back-coupling-strength is denoted by $\kappa =\left(1-{R}_{0}\right)\sqrt{\frac{{R}_{1}}{{R}_{0}}}$. 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

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
*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

with the ambipolar difusion coeficient ${D}_{f}^{12}$,
the macroscopic gain
*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}${\mathrm{\Gamma}}_{k}^{\mathit{\text{sp}}}$
${f}_{k}^{e}$${f}_{k}^{h}$
. For the
numerical integration of the system of stif nonlinear partial diferential equations
the *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*.

We model the spatiotemporal dynamics of a typical broad-area laser. The transverse
width w=100 *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.

In the free-running state, a steady state is never reached. Nevertheless one can
observe a typical width of the filaments
*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}.

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 BAL^{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.

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.

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