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J. Opt. Soc. Am. B / Vol. 14, No. 1 / January 1997
Turovets et al.
Nonlinear dynamics of a laser diode subjected to
both optical and electronic feedback
S. I. Turovets,* J. Dellunde,† and K. A. Shore
School of Electronic Engineering and Computer Systems, University of Wales, Bangor, Bangor LL57 1UT, Wales, UK
Received May 28, 1996
It is demonstrated that tailored optoelectronic feedback can be used selectively to excite periodic dynamical
output from external cavity semiconductor lasers undergoing a period-doubling bifurcation cascade on the
route to the chaotic coherence-collapse regime. The optoelectronic feedback can effectively suppress or invert
the bifurcation sequence so that low-order periodic motion can be resonantly excited from high-order periodic
or chaotic dynamics. The robustness of coherence-collapse control to intrinsic laser-diode noise is investigated. The application of the technique in chaotic communications and its role in chaos control are discussed.
© 1997 Optical Society of America. [S0740-3224(97)00401-3]
1. INTRODUCTION
The phenomenon of coherence collapse relates to a sudden increase of single-mode line width and amplitude intensity fluctuations that may occur in semiconductor lasers when they are subjected to external optical feedback.
The phenomenon is known to depend on the strength of
the optical feedback and the distances to external reflectors. In particular, it has been found that very low external reflectivities such as occur from the connector on
the fiber pigtail of a laser diode module (of the order of
0.01%) can cause the laser to enter the noisy coherencecollapse regime. To ensure immunity to this phenomenon in practical devices rather expensive optical isolators must be used to avoid the relevant optical feedback
regimes. Recent reviews of the effect and further references thereto can be found in Refs. 1 and 2.
At the same time, it is well established that the
coherence-collapse phenomenon belongs to the class of deterministic chaotic processes.3 Thus it is relevant to consider techniques for avoidance of coherence collapse by
taking advantage of recently developed techniques for
chaos control (suppression) that use relatively small external signals. The first such technique was suggested
by Ott et al.4 and is based on the idea of preparing and
maintaining the system at one of the unstable states by
applying a special feedback mechanism. This method
and its modifications, such as occasionally proportional
impulsive feedback,5 (OPIF) have already been applied to
control laser chaos in experiments6,7 and numerical
simulations.8 In particular, Bracikowski and Roy have
resolved the problem of multimode instability (the socalled green problem) in the Nd:YAG laser with intracavity frequency doubling on the KTP crystal,9 and Bielawski
et al. have stabilized a chaotic output in fiber lasers.10
Another possible way to tame chaos involves its synchronization by an external periodic signal.11,12 The frequency of the modulation signal should be chosen near a
maximum in the chaotic spectrum, whereas the amplitude is proportional to the largest Liapunov exponent. In
the optical frequency domain this approach has much in
0740-3224/97/010200-09$10.00
common with the well-known injection-locking stabilizing
technique,13 although to the best of our knowledge their
direct relationship has not been investigated.
Simple modifications of both of these methods of chaos
control have already been applied by Agrawal and
co-workers14,15 with moderate success to resolve the
coherence-collapse problem in single-mode and multilongitudinal-mode laser diodes. Quite recently an external microwave modulation approach was studied numerically in the context of chaos control and synchronization
of laser diodes subject to optical feedback.16,17 However,
as noted by the authors of Refs. 16 and 17, there were a
number of drawbacks in the methods, including the need
for extremely high sampling frequencies (;1 GHz), an inability to stabilize the system in the presence of noise,
and a priori unknown parameters of control (as many as
four are needed in the OPIF scheme), so the adjustments
to correct values become quite tedious in practice.
As an alternative control technique it has also been
proposed to optimize various system parameters so the laser is less susceptible to optical feedback.2,15 In particular, it was noted that quantum-well laser diodes with
small values of the linewidth enhancement factor a and
large nonlinear gain satisfy these requirements. In the
limit a → 0 the amplitude–phase coupling (which is specific to semiconductor lasers) is broken, and the dynamics
then has much in common with incoherent delayed optical feedback. The regime a → 0 is still able to produce
instability in the form of regular and chaotic spiking,18–20
but the instability, being purely amplitude in nature,
takes place for feedback strengths far beyond those typical of coherence collapse. In general, as was pointed out
by Petermann,2 ideas for laser designs with high immunity to optical feedback are based on increasing the damping rate of relaxation oscillation of the solitary laser.
From a nonlinear dynamics point of view this is quite understandable because thresholds of possible bifurcations
in a nonlinear system normally increase with increasing
friction or viscosity coefficients under otherwise equal
conditions.21 In a similar manner, the feasibility of improving noise performance of lasers by optical seeding
© 1997 Optical Society of America
Turovets et al.
Vol. 14, No. 1 / January 1997 / J. Opt. Soc. Am. B
from a stable master laser and negative electronic feedback can be explained. Here we investigate the possibility of using continuous optoelectronic feedback based on
the chaos-control algorithm proposed by Pyragas.22 The
use of negative electronic feedback for suppressing spiky
behavior of ruby lasers is not at all new23 and has become
a routine procedure for stabilizing a number of laser systems (especially against long-term drifts and 1/f noise),
including laser diodes.24 Nevertheless, there has been
little research on this approach in the context of chaos
control, in part because unavoidable technical delay in
the electronic feedback loop of such an arrangement, in
its traditional form, can change negative feedback into
positive feedback and thus serve to destabilize the laser.
For a recent review of related instabilities see Ref. 18.
Furthermore, it is generally believed that this scheme
suffers from limited loop bandwidth and is thus unable to
reduce short-term fluctuations and also that it changes
the laser’s bias and hence its output. However, the elegant and simple solution proposed by Pyragas is free
from the majority of these drawbacks as far as it concerns
deterministic chaotic behavior because the difference
feedback (with a time delay chosen to be commensurate
with a selected unstable periodic orbit inside a strange attractor) acts selectively in changing stability and in not
biasing the target itself. In addition, the influence of
parasitic technical delay has been shown to be periodically self-canceling as a technical delay time,8 which
makes implementation of this scheme feasible even in
fast systems with relatively long feedback loops. The
earlier implementations of this algorithm in lowdimensional dynamic systems seemed to be promising
with respect to robustness to noise and engineering flexibility and hence in benefits for possible device
applications.25,26 To the best of our knowledge, this kind
of control has never been investigated with respect to
chaos in an infinite-dimensional system with phase-type
instabilities such as a laser diode in the coherencecollapse regime. Such an investigation is the aim of the
present paper.
201
df ~ t !
5 1/2a G n @ n ~ t ! 2 n th# G
dt
2
k ext
tL
As ~ t 2 t ext!
As ~ t !
sin@ u ~ t !# 1 F f ~ t ! ,
(3)
where
u ~ t ! 5 f ~ t ! 2 f ~ t 2 t ext! 1 v tht ext .
(4)
In the rate equations n(t) is the carrier density, s(t) is
the photon density, f (t) is the electric field phase, I(t) is
the injection current, and e is the electronic charge.
Typical laser parameters for a distributed feedback laser
are used, where V is the active region volume (1.5
3 10216 m3), tsp is the carrier lifetime (2 ns), G n is the
gain slope (2.125 3 10212 m3 s21 ), n th is the threshold
carrier density (9.9 3 1023 m23 ), e is the saturation parameter (3 3 10223 m3), g is the spontaneous-emission
factor (1 3 1025 ), tph is the photon lifetime (2 ps), G is the
confinement factor (0.4), a is the linewidth enhancement
factor (5.5), n 0 is the transparency carrier density (4
3 1023 m23 ), vth is the operating frequency (l 5 1.55
mm), R 2 is the laser facet reflectivity (0.309), h is the
laser-to-fiber coupling efficiency (0.4), and tL is the laser
cavity round-trip delay (9 ps). In optical feedback terms,
R ext is the external reflector reflectivity and text is the external cavity round-trip delay (0.23 ns, which corresponds
approximately to the length of external cavity, 10
cm). k ext is the feedback coupling parameter, given by
k ext 5
1 2 R2
AR 2
AR exth .
The model can, in general, account for the effects of
Langevin noise terms, F n (t), F s (t), and F f (t), but the calculations described below consider mainly noise-free deterministic dynamics unless specified otherwise. Rate
equations (1)–(3) are solved numerically with a first-order
Euler algorithm with an integration time step of 0.1 ps.
3. OPTOELECTRONIC FEEDBACK
2. MODEL
It is assumed that the laser operates in a single longitudinal mode and is subject to weak optical feedback from
an external mirror. A rate-equation treatment of this
configuration was recently developed on the basis of the
Lang–Kobayashi model to take into account feedback effects in modulated external cavity laser diodes.27 The
equations are as follows:
F
dn ~ t !
1
I~ t !
n~ t !
5
2
2 s ~ t !G n@ n ~ t ! 2 n 0 #
dt
eV
t sp
1 1 es~ t !
G
1 F n~ t ! ,
(1)
F
ds ~ t !
gG
1
s~ t !
5
2
1 s ~ t !G n@ n ~ t ! 2 n 0 # G
dt
t sp
t ph
1 1 es~ t !
1
k ext
tL
As ~ t ! As ~ t 2 t ext! cos@ u ~ t !# 1 F s ~ t ! ,
G
(2)
The laser is taken to be supplied with a dc injection current I dc 5 2I th , where I th is the threshold current, to ensure that the operation conditions are typical of those
classified as coherence-collapse regimes. No consideration is given here to low-frequency fluctuation processes
such as the Sisyphus effect1 that take place for biases just
above the threshold.2,24 The optoelectronic feedback is
used to modify the driving current slightly, proportionally
to the difference between the output powers that were
emitted at time t and a chosen previous time t 2 t el . We
take the value
H
I ~ t ! 5 I dc 1 2 x
J
@ s ~ t ! 2 s ~ t 2 t el!#
.
s0
(5)
We refer to x, which is attributed to amplification in the
feedback loop, and tel , which is assumed to be adjustable
time delay, as the control parameters unless specified otherwise. The photon density s 0 of the solitary laser under
the dc driving current is taken as a reference.
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J. Opt. Soc. Am. B / Vol. 14, No. 1 / January 1997
The motivation for choosing the feedback in the specified form is as follows: First, to effect stabilization the
feedback should be negative; second, the feedback is designed so that it has no effect on the state in which the
system has been initially prepared by preliminary
targeting28–34 (e.g., an unstable periodic orbit with period
tel). A similar kind of continuous delayed feedback for
purposes of chaos control was discussed following the paper by Pyragas22 but to the best of our knowledge has
never been used for controlling the coherence-collapse regime in laser diodes.
Equation (5) essentially assumes that feedback is instantaneous. In practice, of course, there are always
some unavoidable frequency cutoffs owing to finite bandwidth and also a technical phase shift attributed in part
to the finite speed of the electrical signal through the loop.
Turovets et al.
We take the latter into account simply by an additional
time delay ttex in both terms, representing output power
in Eq. (5). It is supposed, nevertheless, that the bandwidth is wide enough to permit microwave spectra of amplitude fluctuations centered at the relaxation frequency
(approximately 3 GHz for the parameters used here) to
pass without significant distortion. This is perfectly feasible with current electronic feedback circuit performance; see, e.g., Ref. 35.
4. RESULTS
A. Period-Doubling Bifurcation Sequence with Purely
Optical Feedback
In Fig. 1(a) the normalized photon-density maxima are
shown as a function of the reflectivity of the feedback mir-
Fig. 1. Bifurcation diagrams for the laser diode with optical feedback, showing photon density maxima versus control parameter. The
control parameter was linearly ramped during the 20-ms integration time in the dynamic range shown. The photon density is normalized with respect to that of the stable output of a solitary laser. The laser is biased at twice the solitary laser threshold current. (a)
Pure optical feedback, where the control parameter is external reflectivity R ext . The arrows indicate dynamic hysteresis at the postponed bifurcations owing to the finite speed of ramping. (b) Effect of additional optoelectronic feedback [Eq. (5)] under fixed external
mirror reflectivity R ext 5 0.0016 [corresponding to developed chaos in (a)]. The control parameter is the electronic feedback strength x;
the delay time is fixed (t el 5 360 ps). (c) Same as in (b) but with fixed electronic feedback strength (x 5 0.05) while the time delay tel
is varied. (d) Effect of varying the technical delay ttech on the stability of control (R ext 5 0.0016, x 5 0.05, t el 5 425 ps).
Turovets et al.
Vol. 14, No. 1 / January 1997 / J. Opt. Soc. Am. B
ror and in the absence of additional optoelectronic feedback. As the level of optical feedback is increased the laser dynamics undergoes a sequence of period-doubling
bifurcations, culminating in chaotic behavior. The parameter controlling the bifurcation diagram has been linearly ramped during the 20-ms simulation time. The
lowest levels of feedback show that the laser has a stable
output, and then at R ext 5 0.0003 the laser enters a regime of self-pulsations through a Hopf bifurcation. At a
reflectivity of ;1023 the period of the oscillations is
doubled. This value of the reflectivity may be termed the
first period-doubling (PD) point. A second PD point is
also visible before the dynamics becomes complex. Such
a bifurcation diagram is typical for laser diodes in a short
external cavity (See, e.g., Ref. 15 and references therein)
as well as in other nonlinear dynamic systems. Some periodic windows are observed inside the chaotic region,
which corresponds to internal crises of a strange attractor
until finally at R ext 5 2.4 3 1023 the stable solution is reestablished through an external boundary crisis of the
strange attractor. The output power (in units of the solitary stable laser output) is smaller than 1.0 because the
laser is now relatively more strongly biased by feedback
and also because of interference effects at the chosen
phase of the reflected signal. The situation differs drastically from that of incoherent feedback when the average
number of photons in the cavity always exceeds that of a
solitary laser at the same pumping.
The laser that is subject to optical feedback shows frequency multistability and mode-hopping phenomena at
reflectivities above 0.2 3 1023 , which can be estimated
according to the formula2,24
k extt ext~ 1 1 a 2 ! 1/2/ t L . 1.
This implies that in the diagram there are some additional isolated branches that have arisen through saddlenode bifurcations between new-born modes and their unstable counterpart antimodes. It is also worth noting
here that, because of the finite rate of ramping, the effects
of postponed bifurcations such as the dynamic
hysteresis36 near thresholds of instabilities can be seen in
Fig. 1 (for the sake of clarity they are indicated by arrows
only for the Hopf and the first PD bifurcation). We believe that this effect, although it is generally treated as an
artifact, can be used in a positive manner as a possible
method for exploring and targeting unstable orbits that
are to be stabilized in the approach of Ott et al.4 Indeed,
the first step of that method requires identification of the
unstable state to be stabilized. In methods such as OPIF
the omission of such a procedure results in long time transients and parameter readjustments.
In a generalized multistability domain, short-lived stabilization of an unstable orbit can be achieved through a
delayed bifurcation. In such procedures a control parameter is scanned relatively rapidly so that a bifurcation is
suppressed (postponed) and the system stays for some
time in an unstable state, as is clearly seen in Fig. 1. It
is possible also to relate the rate of ramping to the width
of the dynamic hysteresis36 and ultimately to the positive
Liapunov exponent of the unstable orbit, which is one of
the key characteristics needed for implementation of
chaos control techniques.7
203
B. Reverse Bifurcation Sequence: Low-Order
Dynamics
The effect of applying additional optoelectronic feedback
in the form of Eq. (5) to a laser for a fixed external mirror
reflectivity of 1.6 3 1023 (in the chaotic coherencecollapse regime) is shown in Fig. 1(b). The strength of
the feedback (the dimensionless control parameter x) varied adiabatically from zero to 10% over a period of 20 ms.
The delay time tel was set to be equal to the basic period
of self-pulsation according to the general strategy described above.
Comparison of Figs. 1(a) and 1(b) shows a spectacular
difference that is due to the effects of optical and electronic feedback on the laser dynamics. It shows that the
electronic feedback can completely cancel the influence of
the optical feedback in a remarkably detailed inverse
period-doubling cascade. The bottom branch is a consequence of local nonlinear distortion of the signal form,
leading, in particular in the interval 0.02 , x , 0.04, to
chaotic trains of pulses that fall into only two clearly distinct classes (according to their form and height), resembling in this way a double scroll oscillator and thus having potential for use in chaotic optical communications.37
The second parameter that one can use to control the
dynamics is the delay time of the optoelectronic feedback.
Figure 1(c) shows the bifurcation diagram for a laser with
a fixed optoelectronic feedback strength (x 5 5%) but
with the feedback delay tel adiabatically varied. It is
seen that the most stable region appears to be centered
around 430 ps. Here the dynamics is of period 2 because
the optoelectronic feedback strength x is not large enough
to ensure period 1 dynamics [cf. Fig. 1(b)]. The system
enters this domain through inverse PD sequences. The
presence of bifurcation line crossings in the diagram is
due again to local nonlinear distortions, or, in other
words, to fingerprints of the redistribution of energy between harmonics in the amplitude spectra. In particular, the crossing point near t el 5 480 ps means that the
smaller peak in the time-dependent PD solution has a
fine structure consisting of two symmetrical peaks. This
richness of different periodic regimes demonstrates that
the laser diode dynamics can be tuned to chosen dynamical regimes, including effecting transitions from chaos to
periodic regimes and vice versa.
We have also examined the influence that a technical
phase shift has on the performance of this configuration.
As is shown in Fig. 1(d), this phase shift strongly affects
the stabilizing effect of the electronic feedback. However, inasmuch as the technical phase shift enters symmetrically into both output power terms represented in
Eq. (5), it repeatedly cancels itself, giving a way to large
windows of stability.
The results indicate a recurrence of windows of controlled dynamics bounded by regimes of complex dynamics. In previous studies of modulated lasers8 it was
shown that optoelectronic feedback introduces a damping
that oscillates with the technical feedback delay and that
causes a periodic variation in the threshold of the perioddoubling bifurcation sequence. The appearance of the
windows of controlled dynamics consequently occurs periodically as a function of the technical feedback delay time
with a period determined by that of the laser modulation
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J. Opt. Soc. Am. B / Vol. 14, No. 1 / January 1997
Turovets et al.
frequency. This result has been established analytically
for the first period-doubling bifurcation8 and has also
been found numerically in simulations of optoelectronic
control of modulated semiconductor lasers.38 Such behavior arises from the use of the difference term @ s(t)
2 s(t 2 t )] in the optoelectronic feedback, where the effect of the technical delay that occurs in both terms has a
further advantage relative to traditional negative feedback approaches in which long technical delays can in fact
lead to further instabilities.
On the basis of such studies it is expected that using
the present approach will permit control of the dynamics
for longer technical delays, although, to avoid excessive
use of computing facilities, present simulations have been
limited to technical delays of no more than 0.5 ns. It will
be advantageous, nevertheless, to utilize optoelectronic
feedback schemes wherein technical delays of the order of
1 ns are obtained. This would seem to be feasible on the
basis of recent experimental work.35
C. Selective Excitation of Periodic Dynamics
Noting that an electronic feedback signal port to a laser
diode can readily be used as a port for an informationcarrying signal, we investigated the possibility of digital–
waveform conversion in the present configuration. Figures 2(a) and 2(b), respectively, demonstrate that when
the control parameter x or tel is increased by relatively
small values in a stepwise manner the chaotic dynamics
is successively replaced by period 4, period 2, and period 1
dynamics (after transients that last a few nanoseconds).
In this way one can make use of the signal waveform as a
means for representing information (rather than having
to use amplitude discrimination principles). It is also expected that this approach will be useful to steer chaotic
laser output in a prescribed sequence of double-scroll-like
pulses to implement a chaotic encryption algorithm (as
discussed further in Section 6).
Examination of corresponding results for the time dependence of the laser-light output power and control signal amplitudes (see Fig. 3) confirms that periodic dynamics can be extracted from chaos by the use of corrections to
the pump current that are typically less than 2% of the dc
injection current even when the control parameter is
changed suddenly (e.g., when the laser is modulated by
rectangular pulses). Such small current corrections imply that the relevant control electronics can be implemented with microelectronics circuitry.
D. Noise Effects
From a practical viewpoint it is important to establish the
robustness of the performance to noise perturbations.
This requires the retention in Eqs. (1)–(3) of the Langevin
sources that describe spontaneous emission and shotcarrier noise. First, we show in Fig. 4 the maxima of
emission spikes in the dynamic regime where selective excitation of periodic dynamics can be achieved. Although
the performance in the time domain is significantly affected, some ordering can still be readily recognized as the
feedback strength increases. Figure 5 shows the amplitude fluctuation spectra of the laser in the steady state
(before the first Hopf bifurcation) with purely optical feedback (top) and increasing strength of optoelectronic feed-
Fig. 2. Dynamics of the laser when optoelectronic feedback was
suddenly switched on in the coherence-collapse regime (R ext
5 0.0016) with the parameters of the optoelectronic loop
changed in a stepwise manner (t tech 5 0 in both cases): (a) Laser intensity versus time for four values of optoelectronic
strength x. By increasing x it is possible to select period-four
double-period and single-period dynamics. The time delay tel is
set at 360 ps. (b) Same as in (a), but now time delay tel is
changed from 360 to 425 ps in a stepwise manner.
back (middle and bottom). It can be seen that the forms
of the spectra are determined mainly by the relaxation oscillation resonance, which is becoming more damped, as
expected, with increasing strength of optoelectronic feedback. Figures 6 and 7 present, respectively, the amplitude fluctuation spectra for a laser diode with only optical
feedback and with both forms of feedback for the several
typical values of the bifurcation parameter. The first
point to note from Figs. 6 and 7 is the larger broadening of
1/2 subharmonics in the period-doubling regime in comparison with the basic frequency. This is a manifestation
of effects of amplification near bifurcation points.31 Another important conclusion stems from a comparison of
Figs. 6 and 7: The final performance of the electronic
feedback with respect to amplitude noise is almost the
same as in the laser without control but out of the coher-
Turovets et al.
Vol. 14, No. 1 / January 1997 / J. Opt. Soc. Am. B
205
ence collapse zone. The general residual background
noise in coherence collapse is suppressed by almost 10 dB
and is expected to be suppressed much more under conditions of stabilizing the steady state in the coherencecollapse regime. The latter condition would, however, require higher values of the optoelectronic feedback [see
Fig. 1(b)]. In this respect the present control scheme
seems to possess advantages over the OPIF technique
studied by Gray et al.14
5. DISCUSSION
There are a number of common physical mechanisms
shared by almost all stabilizing techniques; higherinstability thresholds are associated with a higher effective viscosity in the system. In the present case optoelectronic feedback would cause a general shift to the right of
all the bifurcation thresholds (and consequently of the
chaotic domain) in Fig. 1(a). The stronger the optoelectronic feedback, the further the shift to the right, resulting in pushing the first instability point beyond external
Fig. 4. Selective excitation of dynamics including the effects of
Langevin noise sources describing spontaneous emission and
shot-carrier noise.
Fig. 5. Amplitude fluctuation spectra of the laser in the steady
state with R ext 5 0.0001, t el 5 360 ps, and x with the values
shown.
Fig. 3. Selective excitation of dynamics and related amplitudes
of the control signal: (a) same as in Fig. 2(a), but now only intensity maxima are plotted versus time; (b) amplitude of controlling corrections to the bias current I dc (in units of I dc) versus
time. The parameters are the same as in (a).
reflectivities that otherwise would correspond to welldeveloped chaos. This means that the coherence-collapse
regime can be eliminated for some parameter domain
that is practically accessible.
It should be emphasized that the use of traditional
negative feedback [with a feedback signal simply being
proportional to s(t 2 t tex)] inevitably leads to the lower
thresholds of instabilities under finite strength x and
time delay ttex (see, e.g., Ref. 18 and references therein)
and biases the steady-state output. The present tailored
form of delayed electronic feedback plays a positive role in
preventing conversion of a negative feedback coupling
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J. Opt. Soc. Am. B / Vol. 14, No. 1 / January 1997
Fig. 6. Amplitude fluctuation spectra of the laser with purely
optical feedback for several typical values of bifurcation parameter. R ext (Rext) 5 0.0008, 0.0011, 0.016 (just after the Hopf
bifurcation, near the first PD bifurcation, and in the developed
coherence-collapse regime, respectively).
Turovets et al.
It is noted that the present technique perturbs the system slightly even when transient processes have died out
and thus does not conform to the usual definition of chaos
control.4 In general, the latter case implies both the stabilization of orbits native to the strange attractor in hand
and the use of infinitesimal control signal corrections. In
many systems of practical interest, however, the first requirement appears to be purely academic. It is more important that chaos can be suppressed by small-amplitude
control signals. Sometimes one-to-one correspondence
between parameters of the control scheme and the final
point of destination after the moment when control has
been switched on i.e., the absence of multistability of controlled states, is desirable, although this problem can be
successfully resolved by preliminary targeting of the required regime.28–34
We wish to underline that in the present scheme the
penalty of nonvanishing control corrections is, in fact,
quite negligible: As one can see from Fig. 3(b), the amplitude of the control signal is just 2% of the bias current
and is compensated for by an opportunity to have a large
domain of continuously tuning control. In the Pyragas
case22 one requires extremely subtle tuning of the parameter to obtain the correct value for true control. This factor usually limits the possibility to observe pure chaos
control, especially in an autonomous system, such as the
one under consideration here, in which the basic unstable
limit cycles gradually change their periods with the system parameters.
The vast majority of chaos control studies have been reported so far for low-dimensional systems exhibiting deterministic chaos (as a rare exclusion see Ref. 39, in which
a Pyragas-like control scheme was applied to the delayed
Mackey–Glass equation). The infinite dimension of
phase space of the systems with optical feedback can be a
possible embarrassment for implementing control techniques. In principle, hyperchaos (i.e., more than one
positive Liapunov exponent) is possible in such a system,
so the control should be made in several directions in
phase space, whereas, we have only one readily adjustable parameter, an injection current, in the present system. Although some direct measurements of a correlation dimension from experimental signals in the
coherence-collapse regime point toward a relatively lowdimensional attractor, intuitively it might be the case
that in such systems control is applied to the wrong parameter or is not possible at all. The results of the
present treatment indicate that such concerns may be
misplaced and underline the justification for performing
further detailed investigations.
6. EXPLOITATION
Fig. 7. Amplitude fluctuation spectra of the laser with optoelectronic feedback for several typical values of the feedback
strength x 5 0.00, 0.05, and 0.1 [the developed coherence collapse: R ext 5 0.0016, the noisy period four, and the noisy period
one just near the PD bifurcation regimes, respectively; see also
Fig. 1(b)].
into a positive destabilizing feedback under finite ttex . In
addition, there is no influence on the steady state that is
being stabilized.
Practical applications for chaotic dynamics have been proposed in a number of disciplines including laser physics,
medicine, and, most recently, communications. The major advance that has facilitated these efforts is the development of techniques for controlling chaotic dynamics,
which have led, in particular, to the proposal of secure
communication systems that exploit the properties of chaotic dynamic systems.37 The idea here is to control chaotic dynamics in the manner that is prescribed by an
Turovets et al.
information-carrying signal. Techniques for the control
of chaos permit the selection and stabilization of one of
the periodic orbits that are available within the dynamics
of a chaotic system. Then the control signal is replaced
by an information-targeting signal that directs trajectories within a chaotic attractor in a prescribed sequence.
The transmitted data are then recovered from the transmitted perturbed chaos. One can also effect chaotic communications by mixing the message signal with the output from a chaotic transmitter and then recovering the
message from the received transmitted chaos, using a
rather different concept of chaos synchronization.40 An
efficient algorithm for improving the locking rate between
receiver and transmitter was reported previously.41 The
feasibility of optical injection-locking techniques for reciprocal synchronization of two distant chaotic laser diodes
was also considered recently.42
The scheme of chaos control proposed in the present
study permits the convenient manipulation of the regimes
of operation in such possible devices. The combination of
an efficient control algorithm and a practical chaotic optical transmitter offers an opportunity for a novel secure
communications system based on chaotic encryption. In
addition, chaos control techniques have potential for engineering immunity to coherence collapse caused by conventional residual feedback in the hybrid integration and
packaging of commercial laser diodes.
Vol. 14, No. 1 / January 1997 / J. Opt. Soc. Am. B
†
Present address, Department d’Estructura i Constituents de la Materia, Facultat de Fisica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain.
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7. CONCLUSION
In a conclusion, chaos control and selection of stable periodic outputs in an infinite-dimensional dynamic system
representing semiconductor lasers with weak optical feedback in the coherence-collapse regime has been demonstrated. In comparison with synchronization by external
modulation the present approach is seen to possess significant engineering advantages in that it does not require additional high-frequency sources, imposes a modest requirement on the bandwidth of the electronic
components in the feedback loop, is robust to noise, and is
amenable to optoelectronic integration. The potential
applications of chaos control in secure optical communications have been discussed. The approach also offers a
novel means for performing electronic-to-optical information conversion.
207
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ACKNOWLEDGMENTS
S. I. Turovets acknowledges financial support from the
Royal Society, London, which permitted a visit to University of Wales, Bangor, where most of this research was
carried out. He was also partially supported by the Belorussian Foundation for Fundamental Research and the
International Scientific Foundation. The research of J.
Dellunde at the University of Bath was supported by
Comisión Interministerial de Ciencia y Tecnologı́a (Spain)
project PB93-0769-C02-01.
18.
*Permanent address, Institute of Physics, Academy of
Sciences, 70 Skarina Avenue, Minsk 220072, Belarus.
22.
19.
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