Arizona Center for Mathematical Sciences,
Department of Mathematics, University of Arizona, Bldg # 89
Tucson, Arizona 85721
D. Hochheiser, J.V. Moloney and J. Lega
A robust control strategy, implemented as a spatial filter with delayed feedback, is shown to stabilize and steer the weakly turbulent output of a spatially-extended system. The latter is described by a generalized complex Swift-Hohenberg equation [1] which is used as a generic model for pattern formation in the transverse section of semiconductor lasers. Our technique is particularly adapted to optical systems and should provide convenient experimental control of filamentation in wide aperture lasers.
The aim of this letter is to present a robust control strategy for a class of partial differential equations exhibiting spatio-temporal disorder. The technique we propose takes advantage of the fact that the spatio-temporal spectrum of the turbulent output often satisfies, to a good approximation, the dispersion relation of the system. Control is then achieved by filtering this spectral output about a desired plane wave and feeding back the corresponding delayed signal into the system, together with a contribution proportional to the retarded field. This method is likely to be applicable to any partial differential equation which sustains traveling waves and is particularly suited for optical systems. In the latter, the farfield output is indeed a natural spatial Fourier transform of the complex envelope of the electromagnetic field and applying a temporal Fourier transform to successive readings of the farfield yields the desired spatio-temporal spectrum.
In the following, we demonstrate our method on the example of a generalized laser Swift-Hohenberg equation [1], which phenomenologically describes the dynamics of wide aperture semiconductor lasers. The Swift-Hohenberg [2] (SH) equation is a well-known generic model of pattern formation in extended systems, and it was shown in [1] that when coupled to an equation for the population inversion, a complex SH equation gives a good description of 2-level, class B lasers. Particular effects can then be included in this model to reproduce filamentation, a typical feature of wide aperture semiconductor lasers. The latter are interesting physical manifestations of spatially extended systems showing persistent weakly turbulent behavior. In contrast to wide aperture two-level lasers, they display strong dynamic filamentation instabilities immediately at threshold. Moreover, their large gain leads to strongly amplified spontaneous emission along the laser axis and hence, to a persistent noisy background behavior. Attempts to achieve high brightness (spatial and temporal coherence) of such lasers has prompted great engineering ingenuity in fabricating a great variety of complex constraining geometries [3]. Although stabilization is achieved, these techniques tend to be of limited utility, since laser operation is restricted to a limited range of physical parameters. Because of the current technological importance of such devices, our discussion will be centered on controlling weak turbulence in semiconductor lasers, and the experimental feasability of our method will be emphasized. The genericity of the SH equation nevertheless extends the scope of our technique to a wide range of nonlinear systems.
Control of chaos algorithms take advantage of the intrinsic nonlinear dynamics of the system and provide much greater diversity in manipulating the control to achieve a variety of stable operating scenarios. For instance, using occasional proportional feedback (OPF), Roy et al. [4], successfully controlled a solid state laser by stabilizing a variety of unstable periodic orbits of the chaotic attractor. This scheme, proposed as a practical alternative to the control algorithm originally devised by Ott et al. [5], has been successfully demonstrated for a variety of systems with few degrees of freedom [6, 7, 8, 9]. For higher dimensional systems, control duration has to be included as an extra parameter [10]. A limitation of the OPF is that control is applied as a small perturbation to a system parameter. In other words, one has to wait for the system to get near the desired state or in the vicinity of its stable manifold before control becomes possible.
Ideally, a robust control strategy for an infinite dimensional nonlinear dynamical system should require no a priori knowledge of the solution to be stabilized and should take into account the fact that the system may have no stable attracting state. Delay feedback has been successfully demonstrated as an alternative to the above control methods for few dimensional dynamical systems [11], and recently been proposed as a means of stabilizing the defect-mediated turbulent state of the complex Ginzburg-Landau (CGL) equation. In reference [12], control is achieved by feeding back a delayed, appropriately weighted spatial average of the unstable solution. In [13], the domain of stable traveling wave solutions of CGL is extended by feeding back past states of the system. The latter procedure relies on starting close to the traveling wave solution of interest. The former starts from a turbulent state but does not seem to allow steering among the stabilized solutions. A combined delay feedback in space and time [14] has also been discussed. However, time synchronization associated with a spatial shift does not discriminate between higher spatial harmonics, at least in the present situation where there are nowhere stable solutions. The method we propose here not only stabilizes unstable traveling waves in the turbulent regime but allows one to select and angle tune (steer) the system output starting from initial noise or a turbulent state.
The most important distinction between the semiconductor and other laser
systems is the marked asymmetry of its gain and refractive index spectra.
This causes a very strong nonlinear amplitude-phase coupling in the field
which leads to uncontrolled dynamical filamentation in the laser intensity
at and beyond threshold. In the semiconductor laser literature, this effect
is modeled by a nonlinear coupling between the electric field and the
carrier density through a coefficient known as the 
- factor
[15]. By introducing a similar term in the 2-level laser complex
Swift-Hohenberg (CSH) equation, we obtain the following system:
where is negative for semiconductor lasers.
Although we do not give details here, the use of this phenomenological
extension of the 2-level laser model is well justified, since these
coupled equations can in fact be derived from the microscopic many-body
semiconductor equations [16].
The complex order parameter 
is the scaled envelope of the electric
field and n is a scaled relative (total ?)
carrier density. The latter acts as a mean-flow
and has a profound influence in destabilizing the system, leading to a
very complicated linear growth behavior of the traveling wave solutions.
Here 
is the scaled cavity loss coefficient, a is proportional
to the inverse of the Fresnel number of the laser and measures the
characteristic length scale in the transverse dimension relative to the
wavelength of light, 
is the dimensionless detuning of the laser
frequency from the gain peak, 
is generally the two dimensional
Laplacian although we restrict our study here to one transverse dimension x,
and b is the dimensionless ratio of the carrier recombination to
polarization dephasing times in the Semiconductor Bloch equations [17].
The external pump parameter r(x) is the scaled external current applied
to the laser and the x-dependence is explicitly displayed in order to
emphasize that the pumping is only applied over a finite transverse section
of the laser. Outside the pumped region, the passive semiconductor acts as
a very strong absorber. In the discussion below we first assume that the
pump is infinitely extended in x, in order to take advantage of the known
properties of solutions to the CSH equation. We then show explicitly that many
of the properties of this idealized system carry over to the realistic
finitely pumped cross-section.
Equation (1) admits traveling wave solutions of the form 
where 
. The frequency of the
traveling wave of wavenumber k is given by 
. This latter expression shows
that the nonlinear amplitude-phase coupling
makes the frequency dependent on the distance above threshold for lasing.
Traveling waves of this form are also exact solutions of the two-level
Maxwell-Bloch [18] and the full Maxwell Semiconductor Bloch
[16] equations.
Figure 1.a shows
the analytic nonlinear dispersion curve ( 
plot) for the two-level
laser ( 
). In this case, if one starts with noisy initial data,
the system eventually stabilizes a traveling wave, which corresponds to a point
(indicated by the arrow in Fig.1a) on the anlytic dispersion curve. Also
shown on this figure is the spectrum of the transient
evolution towards the stable asymptotic state.
As soon as is non-zero and negative, all traveling wave solutions
are unstable and the output of system (1) is always turbulent
above threshold. More precisely, strong filamentation is observed,
independently of the initial condition. This state is referred to as
optical turbulence. Figures 1.b and 1.c
show the post-transient spectral distribution of energy in the
turbulent regime with an infinitely extended transverse (r=2) pump
at 
and 
, respectively. For a finite transverse
pump (Fig. 1.d), the spectral distribution of energy is shifted in frequency
( 
) from the analytic curve.
A striking observation in Figure 1.b-d
is that, although
the output is highly unstable, the dynamics involves random motion in the
vicinity of these curves. Full scale simulation of an unstable broad area
laser shows identical dispersion curves and these shapes are in agreement with
experimentally observed spectrally resolved far-field outputs
[19, 20]. In other words, the dynamic filamentation instability
in wide aperture
semiconductor lasers corresponds to random beam steering. This phenomenon
takes place on average time scales of hundreds of picoseconds.
The control strategy we now discuss is a direct consequence of these
observations.
Optically, as the nonlinear dispersion curve (or spectrum) is an
experimentally accessible quantity [19], it is natural to
introduce an optical
feedback scheme which filters, in k and , the desired traveling
wave lying on the actual dispersion curve (attractor). The idea
is then to introduce a delayed feedback with a spatial filter (consisting of a
lens and aperture at the focal point of the lens). Ideally the delay
(proportional to 
) should be chosen to locate the desired
coordinate on the plot. If this scheme works, we simultaneously
have achieved stabilization and a beam steering capability by simply
tuning the filter along the experimental dispersion curve.
Our control technique is then the following: we add a feedback term of
the form 
to the above equation
where
is the Fourier transform operator and F is a suitable
aperture. This represents a time-delayed, spatially filtered feedback
of the original complex field 
at the output facet.
With this technique, the unstable region, which for 
corresponds
to the whole domain of existence of traveling waves, can be stabilized over
the full range of pump strengths and angular tuning (k-axis).
Figure 2.a and 2.b present a succinct overview of the control achieved
for different steering angles (proportional to k) of the idealized and
ramped pump systems respectively. For these plots, the external
pump is twice the lasing threshold value, the feedback strength
, and the feedback delay time 
is chosen to match the
selected angle (k-value) on the analytic dispersion curve.
Note that the controlled
far-field spectra are extremely sharp in k and for the infinitely
wide pump. For the ramped pump case in (b), there is some spectral broadening
in k and , indicating a modulated finite support traveling wave
(
Fig. 3.a) rather than an infinitely extended constant amplitude signal.
Besides, the controlled state is no
longer a traveling wave of the isolated system but instead is a solution
of the full partial differential equation with delay. Its wavenumber
(angle tuning) is precise but the frequency is off.
The traveling wave emanates from one side from a defect (source) and is
absorbed on the other (a sink), as shown in
Fig. 3.a for r=2 and k=4.
In this case the wave travels from right to left (since the corresponding
is positive). The situation is reversed if control is established
at -k: the wave then travels from left to right. Also shown on this
picture is beam steering from k=4 to k=7. When control is switched
from one wavenumber to the other (around t=0.27), the traveling wave at
k=4 is first damped out since it is no longer stabilized by the feedback
term. Then the mode at k=7 starts growing and is stabilized around
t=6.8 (arbitrary units).
Our observations indicate that the spatial filter is the critical component in enabling effective control. The time delay, while less critical, proves most effective when matched to the relevant frequency on the nonlinear dispersion curve. In the unramped case, we have seen locking to an intermediate controlled state which is stable but not the desired one when there is a significant frequency mismatch between the delay and the desired frequency on the dispersion curve. This resembles the asymptotic state of the ramped system.
For completion, we have checked that control can also be achieved when
is positive.We note in addition that a sharply absorbing boundary
has a tendency to select a particular transverse wavenumber from the family
of allowed traveling waves, which then invades the bulk, at least for a
two-level laser [21]. Here the control wins out. Finally, we chose
the value of b=0.01 in order to reduce the transient time between the
initial data and the controlled state. We have nevertheless confirmed
that control can be achieved when b=0.0001, which is more typical
of semiconductor lasers.
To summarize, stabilization of a turbulent filamentation regime can
be achieved by filtering the output signal at the desired wavenumber
and frequency and feeding back the resulting component into the system.
The non-desired components of the field are damped out by the feedback
term and the latter vanishes in the ideal case when the system has
reached the desired traveling wave.
Our procedure takes advantage of the fact that the spectrum
of the free-running laser is close to the analytic dispersion relation.
Beam steering is then achieved by tuning the filter along the experimental
dispersion curve. Such a technique can easily be implemented in an
optical system and is fast enough to allow convenient steering
and control. Besides, for , a typical value for semiconductor
lasers, control can be achieved with a feedback level less than 
, which is very small. Our
method is general enough to be applicable to any system displaying phase
or amplitude turbulence. Similar control should also work for a system where
the spectrum of the disordered output is not close to the
analytic dispersion curve, although one would then need to have some a-priori
knowledge of the latter.
The authors thank M. Bleich, N. Ercolani and R. Indik for stimulating discussions and wish to thank the Arizona Center for Mathematical Sciences (ACMS) for support. ACMS is sponsored by AFOSR contract F49620-94-1-0144DEF. J. V. Moloney and J. Lega also acknowledge support from the European Union Human Capital and Mobility Network ERB-CHRX-CT-940680.
