IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 45, NO. 7, JULY 2009
879
Time-Delay Identification in a Chaotic Semiconductor
Laser With Optical Feedback: A Dynamical
Point of View
Damien Rontani, Alexandre Locquet, Marc Sciamanna, David S. Citrin, and Silvia Ortin
Abstract—A critical issue in optical chaos-based communications is the possibility to identify the parameters of the chaotic
emitter and, hence, to break the security. In this paper, we study
theoretically the identification of a chaotic emitter that consists of
a semiconductor laser with an optical feedback. The identification
of a critical security parameter, the external-cavity round-trip
time (the time delay in the laser dynamics), is performed using
both the auto-correlation function and delayed mutual information methods applied to the chaotic time-series. The influence
on the time-delay identification of the experimentally tunable
parameters, i.e., the feedback rate, the pumping current, and
the time-delay value, is carefully studied. We show that difficult time-delay-identification scenarios strongly depend on the
time-scales of the system dynamics as it undergoes a route to
chaos, in particular on how close the relaxation oscillation period
is from the external-cavity round-trip time.
Index Terms—Nonlinear dynamics, optical feedback, semiconductor laser, time-delay identification.
I. INTRODUCTION
S
EMICONDUCTOR lasers with an external cavity (ECSLs)
have been considered as rich sources of optical chaos, with
well-known chaotic regimes such as the so-called coherence
collapse [1] and low-frequency fluctuation (LFF) regimes [2].
In these regimes, ECSLs have received considerable attention
Manuscript received July 24, 2008; revised October 29, 2008. Current version
published June 10, 2009. The work of D. S. Citrin was supported in part by the
National Science Foundation under Grant ECCS 0523923.
D. Rontani is with the Ecole Supérieure d’Electricité (Supélec), Laboratoire
Matériaux Optiques, Photonique et Systèmes (LMOPS), Centre National de
la Recherche Scientifique (CNRS), F-57070 Metz, France and with the Unité
Mixte Internationale UMI 2958 GeorgiaTech-CNRS, F-57070 Metz, France.
He is also with the School of Electrical and Computer Engineering, Georgia
Institute of Technology, Atlanta, GA 30332 USA (e-mail: damien.rontani@supelec.fr).
A. Locquet is with the Unité Mixte Internationale UMI 2958 GeorgiaTechCNRS, F-57070 Metz, France (e-mail: alocquet@georgiatech-metz.fr).
M. Sciamanna is with the Ecole Supérieure d’Electricité (Supélec), Laboratoire Matériaux Optiques, Photonique et Systèmes (LMOPS), Centre National
de la Recherche Scientifique (CNRS), F-57070 Metz, France and with the
Unité Mixte Internationale UMI 2958 GeorgiaTech-CNRS, F-57070 Metz,
France (e-mail: marc.sciamanna@supelec.fr).
D. S. Citrin is with the School of Electrical and Computer Engineering,
Georgia Institute of Technology, Atlanta, GA 30332 USA and with the Unité
Mixte Internationale UMI 2958 GeorgiaTech-CNRS, F-57070 Metz (e-mail:
david.citrin@ece.gatech.edu).
S. Ortin is with the Instituto de Física de Cantabria, Consejo Superior de Investigaciones Científicas (CSIC)-Universidad de Cantabria, Santander E-39005,
Spain (e-mail: ortin@ifca.unican.es).
Digital Object Identifier 10.1109/JQE.2009.2013116
since they constitute key elements of optical secure chaotic communications [3]. Indeed, the addition of an external cavity introduces an infinite number of degrees of freedom through the
time delay in the dynamical representation of the semiconductor
laser, leading to the generation of high-dimensional chaos [4].
The unpredictable and noisy appearance of the signals emitted
by hyper-chaotic optical systems has been used to enhance security and privacy in communications. A chaotic carrier conceals and transmits an information bearing message, which is
decrypted by synchronization at the receiver end (a physical
copy of the chaotic emitter) [3], [5]–[8]. The security of such
a chaos-based communication scheme relies mainly on the difficulty to identify the emitter parameters and on the sensitivity
of synchronization to parameter mismatch [9]. Therefore, it is
of key importance to study the feasibility of an eavesdropper to
retrieve information on the parameters of a given chaotic generator from its transmitted time series.
In delayed hyper-chaotic systems, the security assumption relies on the computational complexity to reconstruct
a high-dimensional attractor from the time series. However,
the knowledge of the time delay allows for the projection
of the high-dimensional attractor onto a reduced-dimensional phase space, which makes the system vulnerable to
low-computational-complexity identification techniques [10].
It is thus crucial that the delay not be easily identifiable in
a cryptosystem. Until recently, ECSLs with a single optical
feedback were considered as weakly secure systems in terms
of time-delay identification, such that the use of several external cavities has been suggested [12]. However, we have
shown recently that a simple ECSL with a single optical
feedback could, with a careful choice of parameters, hide its
time-delay signature when standard estimators are employed
[13]. Interestingly, the region of laser and feedback parameters
where such a difficult time-delay identification occurs does
not necessarily correspond to the situation where the chaos
complexity (dimension and entropy) is higher [4]. Indeed, in
laser diodes with optical feedback, high-dimensional chaos is
typically found where the optical feedback strength is large, but
then the time-delay value is easily retrieved from the analysis
of the chaotic output using straightforward techniques [13].
Time-delay identification should therefore be considered as
an additional argument to appreciate the level of security of
chaos-based communications, besides chaos complexity and
robustness of chaos synchronization.
In this paper, we extend our earlier work on time-delay identification in ECSL. Conventional techniques such as autocorre-
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 45, NO. 7, JULY 2009
lation function (ACF) and delayed mutual information (DMI)
are used to process the identification. We analyze further the
range of laser and feedback parameters that lead to difficult
time-delay identification and link these bad identification scenarios to the laser dynamics as it undergoes a bifurcation cascade to optical chaos. More specifically, we show that the timescales of the laser dynamics in its route to chaos influence the
time-delay identification and therefore also the security of such
a chaotic emitter. A careful tuning of the external cavity roundtrip time with respect to the intrinsic laser relaxation oscillation
frequency leads to situations where the time-delay signature is
lost in the laser chaotic output. Finally, we discuss the robustness
of our results using other signal processing techniques applied
to the laser chaotic time-series.
This paper is organized as follows. Section II presents the
model and specificities of the various identification techniques
used to retrieve the time-delay value. Section III details the influence of the different operational parameters (the feedback rate,
the time delay and the pumping current) on the identification.
Section IV gives a dynamical interpretation of the identification scenarios observed, based on the studies of the frequencies
that appear in the ECSL dynamics as it undergoes a route to
chaos. Section V discusses the influence of key internal parameters that modify the ECSL model and the possible application
of our results to chaos-based communications. Finally, we summarize our main conclusions in Section VI.
II. THEORETICAL FRAMEWORK
A. Rate Equation Model
In this study, we consider a single-mode semiconductor laser
with optical feedback modelled by the Lang–Kobayashi rate
equations [14]. This model has been successful in reproducing
dynamical behaviors experimentally observed in ECSLs. The
rate equations are
lated white Gaussian noises that satisfy the relations
and
and
[15]. The relaxation-oscillation period
is an intrinsic damping time of the free-running laser and is
with
defined by
. We consider the following pa,
rad,
ps,
ns,
rameter values:
m s ,
m ,
m s ,
m , and
s .
B. Time-Delay Estimator and Time-Delay Signature
The time delay is a key parameter for the security of nonlinear time-delay systems. Indeed, its estimation is of particular
importance with regard to computationally efficient identification of other parameters. Standard techniques, such as the ACF
and the DMI, can be used to identify the time delay [16].
1) Autocorrelation Function (ACF): If a random process
is ergodic and wide-sense-stationary, then the ACF is defined by
(3)
where
and
are sampled from the random process
,
, and
, where denotes time average. The ACF measures for a given value the
to be aligned
tendency of the cloud of points
along a straight line and thus measures a linear relationship beand
.
tween
2) Delayed Mutual Information (DMI): The mutual inforis a quantity originally used in information theory
mation
[17]. Given two continuous variables and with joint probaand marginal probability denbility density function
sity functions
and
, the mutual information of
and is defined as
(4)
(1)
(2)
where
is the slowly varying complex elecis the average carrier density in the active region,
tric field,
is the linewidth-enhancement factor that describes the ampliis
tude phase coupling,
is the
the optical gain where is the saturation coefficient,
is the angular frequency of
carrier density at transparency,
the solitary laser, is the feedback rate,
is the photon lifeis the carrier lifetime,
is the threshold current,
time,
is the pumping factor, and is the delay corresponding to the
round-trip time of light in the external cavity. The Langevin
models the spontaneous-emission noise. Its polar deforce
composition in amplitude and phase is given by the two terms
and
, where is the spontaneous-emission rate. The variables
and
represent uncorre-
where
is the expectancy operator, the two variables and
are obtained by sampling the random process
at two
, and such a process is assumed to be stationary
times and
, , and
and ergodic. The probability density functions
will be estimated by their respective histogram
, , and
computed from the time series and leading to the approximate
mutual information estimator, also called delayed mutual information (DMI)
(5)
The mutual information corresponds intuitively to the quantity of information that the two random variables
and
are sharing. In time-delay systems, the presence of a
delayed feedback term induces a nonlocal time dependence in
the time evolution of its state variables. The integral definition
of the estimators under consideration allows for the detection of
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RONTANI et al.: TIME-DELAY IDENTIFICATION IN A CHAOTIC SEMICONDUCTOR LASER WITH OPTICAL FEEDBACK: A DYNAMICAL POINT OF VIEW
881
=2
Fig. 1. ECSL intensity time series (first column), ACF (second column), and DMI (third column) for increasing value of the feedback rate
GHz (first row),
GHz (second row),
GHz (third row),
GHz (fourth row) with a time-delay value ns and : ns. The vertical dotted–dashed
and dashed lines give the time location of and , respectively.
=5
= 10
= 15
nonlocal time dependencies, linear for the ACF and nonlinear
for the DMI. Therefore, if a particular time scale, such as a
time delay, is present in a given time series, it should manifest
itself in the estimator through a local extremum. This signature
of the time scale, depending on its sign, will be referred to
as peak or valley later on in this article. The time location of
a peak (respectively, valley) will be considered as a possible
estimation of the time delay.
III. IDENTIFICATION OF TIME DELAY IN
LANG–KOBAYASHI EQUATIONS
Here, the security of a chaotic ECSL is investigated. A timedelay identification is performed on the intensity time-series
using ACF and DMI. The evolution of the resulting time-delay signatures are analyzed with the variation
of the following operational parameters: the feedback rate ,
the external-cavity round-trip time , and the pumping factor .
Throughout this study, the spontaneous-emission rate is taken
equal to zero.
A. Influence of the Feedback Rate
The feedback rate controls the optical power reinjected in the
laser cavity and hence drives the contribution of the delayed intensity
in the time-evolution of
. We thus expect
that the stronger the feedback rate, the more important the inand
will be.
formation shared between
=5
= 0 75
Fig. 1 analyzes a first case of time-delay identification for increasing values of the feedback rate . Large values of produce
chaotic time series with a clear time-delay signature: a significant peak is present in both the ACF [Fig. 1(b4)] and the DMI
[Fig. 1(c4)], at a time corresponding to the time delay. Similar results have been obtained experimentally, from the analysis of the ACF [11]. The progressive decrease in feedback
rate produces the following: first, a decrease of the peak amplitude close to the time-delay location and its multiples, in
both the ACF and DMI [Fig. 1(b3) and c(3)] until it reaches
a minimum value [Fig. 1(b2) and (c2)]; second, a qualitative
change of behavior appears. For relatively weak feedback rates,
from each
numerous peaks and valleys, separated by
other, appear and are amplified in the vicinity of the origin
and the time delay
[Fig. 1(b1) and (c1)]. They
complicate the time-delay identification. A quantitative description of the feedback-rate influence in terms of time location
and amplitude of the time-delay signature is given in Fig. 2
for both the ACF and the DMI. Fig. 2 is obtained by considering the most significant peak (or valley) in a vicinity
of the time delay which is defined by
for the
for the DMI. In Fig. 2, the curves
ACF and
with triangle markers are drawn for the same parameters as the
ones of Fig. 1. The curves confirm the tendency already observed in Fig. 1: as the feedback rate is decreased, the amplitude starts to diminish until a global minimum value is reached,
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 45, NO. 7, JULY 2009
W : ;:
p : p :
p :
: :
:
Fig. 2. Impact of the feedback variation and pumping factor on the amplitude and time location of the most significant peak in the vicinity ( ) = [4 5 ns 5 5 ns]
of the time delay = 5 ns. (a1)–(b1) gives, respectively, the amplitude and time location of max
j0( )j (a2)–(b2) gives respectively the amplitude and
j
( )j. The solid lines with triangle (4), round (), and square ( ) markers stands for
= 1 05,
= 1 26 and
= 1 72,
time location of max
respectively. These three different values of , correspond to the following relaxation oscillation periods:
= 0 75 ns,
= 0 33 ns and
= 0 2 ns,
respectively. In (b1)–(b2), the dotted–dashed lines give the time location of the time delay .
H
p
and then it increases [Fig. 2(a1)–(a2)]. The decreasing region
of the curve corresponds to a weak chaotic regime where the
laser’s intrinsic nonlinearity and feedback terms have equivalent driving actions. In these conditions, the ACF and DMI may
still find structural relationships within the intensity time se. However, larger values of lead to well established
ries
chaotic regimes. In such case, most of the time-scale signatures
are weakened except for the time delay which is favored by the
in
linear driving action of the feedback term
the rate equations. The decrease in also influences the time
location of the largest peak at a time close to the time delay;
see Figs. 2(b1)–(b2). For small feedback rates, the largest autocorrelation or mutual information is obtained for a time close
, when analyzing a small time
to a high order multiple of
. Then, the idenwindow in the vicinity of the time delay
tification gives a value of time which is quite shifted from the
. However, as the feedback rate increases,
expected value
the greatest autocorrelation and mutual information values are
achieved for a time closer to the time delay. This explains the
monotonic decreasing behavior of the time shift identification
curve in Fig. 2(b1)–(b2).
In summary, this first analysis shows that, when the feedback rate is relatively weak, the chaotic output of the laser diode
shows only small evidence of the time-delay signature, both in
the ACF or the DMI. The largest contribution to the ACF or
DMI, in a time window around the time delay, is in fact closer
, and
to a high multiple of the relaxation oscillation period
this blurs the good identification of the expected value .
Fig. 2 also shows the influence of the pumping factor , which
as well as the value
controls the injection current
of the relaxation oscillation period
. When increasing ,
chaotic dynamics are observed for larger values of the feedback
rate but similar conclusions hold for the identification of the
time delay. The amplitude of the highest peak (or valley) in the
ACF and DMI in the vicinity of exhibits a minimum value
when the feedback rate increases, and the feedback rate corresponding to this minimum increases with the increase of .
Moreover, the time value obtained from the ACF or DMI idenwhen
tification techniques is shifted from the expected value
the feedback rate is small but gets closer to as the feedback rate
increases. However, the maximum time shift is smaller when
increases, hence progressively leading to a better identification
of .
B. Influence of the Time Delay Relatively to the
Relaxation-Oscillation Period
Here, we study how the value affects the time-delay identification based on the ACF and DMI. This influence is ana, the other significant ECSL
lyzed relatively to the value of
time scale in terms of identification. This line of reasoning is
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RONTANI et al.: TIME-DELAY IDENTIFICATION IN A CHAOTIC SEMICONDUCTOR LASER WITH OPTICAL FEEDBACK: A DYNAMICAL POINT OF VIEW
883
=2
Fig. 3. ECSL intensity time series (first column), ACF (second column), and DMI (third column) for increasing value of the feedback rate
GHz (first row),
GHz (second row),
GHz (third row),
GHz (fourth row) with a time-delay value : ns and : ns. The vertical dotted–dashed
and , respectively.
and dashed lines give the time location of =5
= 10
= 15
supported by the results of Section III-A, where the two time
scales coexist and
proves to have a blurring effect on the
time-delay signature [Fig. 1(b1)–(c1)]. Here, we will show that
may have an even stronger effect. If it is taken sufficiently
,
close to and provided that the pumping current is small (
which leads to relatively large values of
), then the timedelay signature will not be easily retrievable either in the ACF
ns and keeping
or the DMI. Fig. 3 illustrates this fact for
ns.
At large feedback rates, the reduction of the time sepaand
does not produce a qualitative
ration between
change of the time-delay identification at strong feedback rates.
Both estimators still exhibit an easy-retrievable time-delay
signature: a sharp peak close to the time-delay location
[Fig. 3(b3)–(c3) and (b4)–(c4)]. Then, similar to Fig. 1, a
. This leads to a
diminution of , promotes the effect of
time-delay signature hardly retrievable for sufficiently close
[Fig. 3(b2)–(c2)], whereas it was still
values of and
possible to retrieve it when the two time scales were sufficiently
disparate [Fig. 1(b2)–(c2)]. We also notice that the ACF or
DMI evolve from a pulse-like shape at high feedback rates to
an oscillating shape at low feedback rate [Fig. 3(b1) and (c1)].
The transition between these two shapes is achieved for a
feedback rate close to the one that leads to a minimum in the
curves of Fig. 2(a1)–(a2), and corresponds to a particularly
=12
= 0 75
adapted situation to conceal the time-delay signature. Further
decrease of , leads to exponentially-damped oscillations of the
and
ACF, with peaks and valleys located at multiples of
, respectively. In such a case, no signature
multiples of
of the time delay remains, in contrast to the situation where the
and was larger [Fig. 1(b1)–(c1)].
difference between
IV. TIME-DELAY IDENTIFICATION: A DYNAMICAL
POINT OF VIEW
The previous section has shown that the identification of the
time delay strongly depends on the operational parameters of
, and
the ECSL: the feedback rate , the pumping current
the value of relatively to
. At high feedback rates, the estimators always possess a predictable behavior: a pulse-like shape
with a clear signature of . However, when the feedback rate is
weak, a large variety of behaviors has been reported and detailed
in the previous section. These behaviors are largely dependent
.
on the choice of and relatively to
These first results suggest a closer analysis of the time-scales
involved in the early stage of the laser chaotic dynamics at small
feedback rates. Different dynamical scenarios are expected de, which then also lead
pending on the ratio between and
to the different time-delay identification scenarios we have reported.
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 45, NO. 7, JULY 2009
Fig. 4. Bifurcation diagrams with the feedback rate taken as the bifurcation parameter. It shows various routes to chaos depending on the choice of and .
ns and : ns. (b) Period-doubling (PD) route to chaos for : ns. (c) Complex route to chaos
(a) QP route to chaos for : ns and : ns.
: ns and for = 0 85
=5
= 0 75
= 0 75
A. Influence of the Frequency Generation on the Time-Delay
Signature
1) Disparate Time-Scales Scenario: Here, the feedback rate
is taken as the bifurcation parameter, with all of the other parameters remaining identical to the ones of Fig. 1. In this scenario, the bifurcation diagram of the ECSL intensity shows a
quasi-periodic (QP) route to chaos [Fig. 4(a)]. The stationary
solution of the ECSL is first destabilized through a Hopf bifurcation (H1) which induces a time-periodic dynamics [Fig. 5(a1)],
GHz
GHz
at frequency
[Fig. 5(b1)]. The intensity being time-periodic, the ACF and the
DMI both exhibit an oscillating shape at the frequency of the
limit cycle oscillation [Fig. 5(c1) and (d1)]. Indeed, the ACF
and
presents alternately maximum correlation at
maximum anti-correlation at
. Similarly, the DMI
presents two series of peaks with different amplitudes: one loand the other at
.
cated at
An increase of feedback rate destabilizes the limit cycle
into a torus [Fig. 5(a2)] and produces new frequencies
in the power spectrum [Fig. 5(b2)]. Amongst them, one
by
strong frequency component appears separated from
GHz, which is a value close to the external-cavity
frequency
GHz. This induces a slow undamped periodic modulation of both time-delay estimators
[Fig. 5(c2) and (d2)]. A further increase of feedback rate is
followed by the appearance of numerous new frequencies
[Fig. 5(b3)] that increases the attractor complexity [Fig. 5(a3)].
However, the strong frequency components are still located
and
. This guarantees the
at the frequencies
persistence of a global order of the time series on a long
time-extension even if on a short time-extension the complexity
(disorder) is increased. As a consequence, the ACF and DMI
show a strong modulation [Fig. 5(c3) and (d3)].
Larger feedback rates destabilize the torus into a chaotic atis a ruin
tractor, whose structure in the projected space
of the torus geometry [Fig. 5(a4)]. This induces a progressive
loss of correlation in the intensity time series as time increases,
which is signed with a damp modulated shape in both the ACF
=12
= 0 75
and DMI [Fig. 5(c4) and (d4)]. Such signatures find their origin
in the power spectrum that retains significantly the trace of the
frequencies associated to the first time-scales appearing in the
bifurcation cascade [Fig. 5(b4)]. These are the undamped relaxborn from H1 and the time
ation-oscillation period
delay that appears when the torus bifurcation occurs.
are responIn conclusion, the fast oscillations at
sible for the oscillatory shape of the time-delay estimators and
disturb the identification of the time delay. At weak feedback
rates, not too far from the onset of chaos, the time-delay time
scale slowly modulates both the ACF and DMI. Only large feedback rates weaken sufficiently the relaxation-oscillation signature and offer a clear time-delay signature.
2) Close Time-Scales Scenario: Similarly, the influence of
the bifurcation cascade is investigated for close values of the
. In this case, the analysis is performed
two time scales and
using parameters identical to the ones in Fig. 3. The bifurcation
diagram in Fig. 4(b) shows a PD route to chaos. We observe in
this case that the frequencies generated by the ECSL nonlinear
dynamics also significantly influence the shape of the estimators.
As previously, the progressive increase of feedback
rate leads to a time-periodic dynamics with a frequency
GHz
GHz [Fig. 6(b1)].
This induces an oscillating behavior of the ACF and DMI
[Fig. 6(c1) and (d1)]. A larger feedback rate induces a period-doubling bifurcation, which yields a new frequency at
[Fig. 6(b2)]. The coexistence of these two time scales
modulates the shape of the estimators [Fig. 6(c2)–(d2)],
and its multiples the time-locations of
making
the strongest contributions in the time-delay estimators. Further
increase of the feedback rate leads to the appearance of many
frequencies in the ECSL power spectrum, and has two effects
on the estimators: the global decrease of the amplitude of the
oscillations of the ACF and DMI, and a stronger modulation
[Fig. 6(b3)–(c3)]. Then, the chaotic regime exponentially
GHz in both the ACF
damps the fast oscillations at
and DMI [Fig. 6(c4)–(d4)]. In contrast to the previous case,
the time-delay time scale is not present either in the ECSL
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RONTANI et al.: TIME-DELAY IDENTIFICATION IN A CHAOTIC SEMICONDUCTOR LASER WITH OPTICAL FEEDBACK: A DYNAMICAL POINT OF VIEW
N; E
:
It
885
Fig. 5. Projection of the attractor in ( j j) plane (first column), power spectrum (jFFT( ( ))j ) (second column), ACF (third column), and DMI (fourth
column) for increasing value of the feedback rate = 0 35 GHz (first row), = 0 55 GHz (second row), = 0 85 GHz (third row), = 1 GHz (fourth row)
= 0 75 ns. The vertical dotted–dashed and dashed lines give the time location of
and , respectively.
with a time-delay value = 5 ns and
:
nonlinear dynamics after the occurrence of the first bifurcations
nor in the power spectrum which contains a single strong
. This prevents its clear signature
frequency component at
to appear in the estimators at low feedback rates. Both the ACF
and the DMI give only information concerning the value of the
undamped relaxation-oscillation period through an oscillating
behavior.
In conclusion, the cascade of bifurcations plays a significant
role in the behavior of the time-delay estimator. The first bifurcation appearing in the ECSL leads to oscillations at a “fundaof both the ACF and DMI, and subsemental frequency”
quent bifurcations shape the estimators. It appears that a necessary condition to conceal the time-delay signature is that it must
not appear early in the route to chaos [Fig. 5(b2)]. However,
the early appearance of is not the only aspect to consider: the
estimators starting to oscillate at the period of the undamped re. Thus, it is also important to study the
laxation oscillation
particular role of this frequency and its remaining presence as
the ECSL undergoes its cascade of bifurcations.
B. Influence of the First Frequencies From the Bifurcation
Cascade and Diversity of Time-Delay-Identification Scenarios
born from the first Hopf
The “fundamental frequency”
is the first time scale emerging in the ECSL dybifurcation
namics. Its presence seems to be systematically responsible for
the fast oscillating behavior of the time-delay estimators that
:
:
persists during the entire cascade of bifurcations and can blur
or even mask the time-delay signature [Figs. 1 and 3]. This secand see
tion aims at investigating deeper the properties of
how its characteristics can be used to increase even further the
time-delay concealment. Interestingly, a systematic study has
can be significantly shifted from
and
unveiled first that
that it will not always play the role of the fundamental frequency
before the chaotic regime is reached. In this latter case, the fundamental frequency associated to the last stable attractor plays
this role.
1) Influence of the First-Hopf Frequency Shift and Consequences on the Time-Delay Identification: In the previous secwas extremely close to the value
tions, it was found that
. This situation was true for a specific ratio
, but
of
may present significant shifts from
. Fig. 7 displays a
as a function of the time delay . Its
systematic analysis of
periodically oscilevolution is not monotonic. Frequency
(horizontal dashed line), with an oscillating
lates around
period close to
, and with a slowly damped amplitude of
oscillations. Similar conclusions have been obtained for other
sets of parameters in previous ECSL studies [18]–[21]. This interesting property can be used to increase the security of the
ECSL. Indeed the use of frequency-shifted undamped relaxation
oscillations could lead to fast oscillating ACF and DMI which
.
can prevent an eavesdropper to get insight on both or
ns or
ns
Our previous choice of parameters,
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N; E
It
Fig. 6. Projection of the attractor in ( j j) plane (first column), power spectrum (jFFT( ( ))j ) (second column), ACF (third column), and DMI (fourth
column) for increasing values of the feedback rate: = 0 6 GHz (first row), = 0 8 GHz (second row), = 1 2 GHz (third row), = 1 5 GHz (fourth row)
with a time-delay value = 1 2 ns and
= 0 75 ns. The vertical dotted-dashed and dashed lines give the time location of
and , respectively.
:
:
:
ns, has led each time to situations where
. But if we now consider a value of closer to
,
from
is maximum, and
where the frequency shift of
combine it with a low feedback rate, we may expect a damping
oscillating behavior of both the ACF and DMI at a time-scale
, hence preventing a
that corresponds neither to nor to
good time-delay identification.
2) Influence of the Fundamental Frequency of the Last Stable
Attractor and Consequence on Time-Delay Identification: In
many investigated cases, the shape of the time-delay estimators
during the whole bifurcation casexhibit fast oscillations at
cade. However, we present here a different case where the ECSL
instead of
locks on a limit cycle with a frequency
being destabilized into a torus or a two-period limit cycle. Such
a situation is illustrated in the bifurcation diagram of Fig. 4(c)
ns and
ns. The two branches
taking
corresponding to the limit cycle H1 are replaced by two news
branches associated to a new limit cycle H1b. Then, an increase
of feedback rate destabilizes the limit cycle born from H1b at
GHz [Fig. 8(b2)] and makes the ECSL lock on a
PD limit cycle with fundamental frequency
GHz
[Fig. 8(b3)]. Finally, the system enters in a weakly developed
chaotic regime that inherits the spectral contents of this last
stable attractor [Fig. 8(b3)–(b4)]. In this scenario, the ECSL has
:
:
:
with
f
Fig. 7. Evolution of the first Hopf Frequency
as a function of the time
= 0 75 ns. The horizontal and vertical dotted-dashed lines
delay , for
represent, respectively, the relaxation oscillation frequency
=1
, and
.
multiples of the relaxation-oscillation period
:
f
=
known several different values for the frequency at the origin of
,
and
. These nonsmooth variations of
instabilities:
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RONTANI et al.: TIME-DELAY IDENTIFICATION IN A CHAOTIC SEMICONDUCTOR LASER WITH OPTICAL FEEDBACK: A DYNAMICAL POINT OF VIEW
N; E
It
887
Fig. 8. Projection of the attractor in ( j j) plane (first column), power spectrum (jFFT( ( ))j ) (second column), ACF (third column), and DMI (fourth
column) for increasing values of the feedback rate: = 1 GHz (first row), = 1 2 GHz (second row), = 1 4 GHz (third row), = 1 6 GHz (fourth row) with
a time-delay value = 0 85 ns and
= 0 75 ns. The vertical dotted-dashed and dashed lines give the time location of
and , respectively.
:
:
frequency seem to be linked to the discontinuous shape of the bifurcation diagram [Fig. 4(c)], where the system jumps and locks
on new stable attractors. These variations have visible consequences on the ACF and DMI [Fig. 8(c1)–(c3) and (d1)-d(3)]:
the oscillating shape is not related to any a priori known freor
. Moreover, the complex route to
quencies such as
chaos has shaped nontrivially the estimators such that it is virtually impossible to identify the time delay from either the ACF
or DMI [Fig. 8(c4) and (d4)].
3) Conclusion on the Diversity of Time-Delay-Identification
Scenarios: In conclusion, we have shown that the first Hopf frequency is not systematically responsible for the fast oscillating
shape of the ACF or DMI during the entire cascade of bifurcations. We have also illustrated the key role of the different frequencies born from the cascade of bifurcations occurring in the
ECSL. These frequencies can either shape or modify the fast
oscillating behaviors of both the ACF and the DMI. They are
in every case responsible for blurring the time-delay signature
when the feedback rate is taken relatively weak. These different
identification scenarios take their origin in the incredible diversity of routes to chaos existing in the ECSL.
:
:
:
V. DISCUSSION
Here, we discuss the robustness of the time-delay concealment with respect to other identification techniques and to the
influence of laser spontaneous emission noise and gain compression coefficient, as well as potential applications of our results for chaos-based communications.
A. Robustness of Time-Delay Concealment to Other
Identification Techniques
Numerous methods exist to identify the time delay, besides
the well known ACF and DMI. The most common ones are: the
filling factor analysis [22], the local linear models (LLMs) [10],
global nonlinear models such as those obtained with modular
neural networks (MNN) [23], and statistics of time-series extrema [24]. However, it is interesting to analyze whether these
time-delay-identification techniques would allow to retrieve any
information on the time-delay value.
We present here a time-delay extraction based on MNN. In
order to mimic the structure of the equation driving the evolution
, the MNN incorporates two modules,
of the light intensity
and a second one for
one for the non delayed part (ND) of
its delayed part (D). A feedforward neural network [23] is used
for each of the modules. The first module is fed with input data:
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 45, NO. 7, JULY 2009
ECSL are carefully chosen, the time-delay signature remains
hidden.
B. Influence of Internal Parameters
Fig. 9. Forecast error produced by the MNN on the intensity time series I (t)
generated with the following parameters: = 1 ns, = 0:75 ns and
= 2 GHz. The identification time-step is 5 ps. The vertical dotted-dashed
and , respectively.
and dashed lines give the time location of and the second module
is fed with
. The parameter
is the embedding time, is a
is the number of inputs of the ND part,
time-delay candidate,
is the number of required inputs for the delayed
and
part [10]. The output of the neural network is
(6)
where and are nonlinear functions associated with ND and
D, respectively. The forecast error of the MNN is defined by
and is a function of . If there exists a value
such that
is minimum, corresponding to a valley in
the forecast-error curve, then will be considered a time-delay
estimation. We apply the MNN technique to the intensity time
ns,
series corresponding to the operational parameters
ns, and
GHz. In previous sections, we
have shown that such a set of parameter leads to a chaotic laser
output from which it is not possible to retrieve the information
on the time delay using ACF or DMI. Fig. 9 presents the results
of a time-delay extraction based on a MNN composed of six
neurons in the first layer and three neurons in the second layer
for the delayed module and only one neuron for the nondelayed
module. The plot of the forecast error reveals only a minimum
for values close to 0, that corresponds to the linear correlation
time [23]. We do not observe, however, any other minimum in
the vicinity of the time delay . As a consequence, the time delay
cannot be inferred from the intensity time-series using MNN.
The other identification techniques described above have also
been tested and do not give more insight about the time-delay
value. However, it is interesting to note that the statistics of timeseries extrema is among these methods the one that identifies the
time delay of the system for the smallest value of the feedback
rate .
In conclusion, we have shown that independently from the
method of identification, if the operational parameters of the
1) Influence of the Spontaneous-Emission Noise: The results
of the previous sections are based on a fully deterministic model
of the ECSL. However, the stochastic processes modeled by the
Langevin force appearing in (1) may affect our results. Indeed,
this additional stochastic part of the Lang–Kobayashi equations
blurs the cascade of bifurcations that has already proven to
be directly responsible for the blurred time-delay signatures
observed in both the ACF and DMI. When the feedback rate
is weak, the noise acts as a significant driving force for the
dynamics that weakly excites the intrinsic nonlinearity of the
ECSL. However, this stochastic excitation, only visible at low
feedback rate, does not influence the time-delay identification:
the signature is still blurred by the time-scales related to the
cascade of bifurcations. Larger feedback-rate values make the
noise effect negligible in comparison with the delayed-feedback
term. Fig. 10 shows this result by comparing the evolution for
increasing feedback rate strength of a model of ECSL with
and without spontaneous-emission noise. In conclusion, the
presence of noise in the equation does not affect the time-delay
identification, and the results of Section III remain valid.
2) Influence of Gain Saturation: Gain saturation is phenomenologically introduced in the rate equations by considering
an explicit intensity dependence of the gain. It has been reported
that the saturation gain has a stabilizing effect on the ECSL
dynamics [25]. As a consequence, reducing the value of will
relatively to the one of
favor the driving action of
. The consequences are first, a decrease of the
extremum located around , and, second, the appearance of fast
oscillations in its vicinity. Fig. 11 presents these results: three
different identification scenarios based on the ACF are considand for increasing values
ered for various choices of ,
of . Interestingly, the first row shows, at a fixed feedback rate,
, where a
that for a large time separation between and
clear delay signature were expected [Fig. 11(d4)], a progressive
tendency to the disappearance of the time-delay signature
is observed [Fig. 11(d1)]. These results hold for the various
parameters used in Fig. 11. In conclusion, decreasing values
of the gain saturation favor the fast time scales emerging from
the laser intrinsic nonlinearities with respect to the delay time
scale. This has also proven to enhance the range of feedback
rate and separation of time-scales to conceal the time delay.
C. Application to Chaos-Based Communications
Time-delay concealment is a requirement for the security of
chaos-based communication setups involving ECSLs. But it
is crucial to check that chaos synchronization is also possible
for conditions leading to time-delay concealment. For this, we
consider a typical communication scheme constituted of two
and a slave
unidirectionally coupled ECSLs: a master
[3]. In this configuration, it is known that two synchronization regimes exist depending on the coupling conditions: the
injection-locking-based synchronization (ILBS) and complete
synchronization (CS) [26]. A good level of synchronization is
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RONTANI et al.: TIME-DELAY IDENTIFICATION IN A CHAOTIC SEMICONDUCTOR LASER WITH OPTICAL FEEDBACK: A DYNAMICAL POINT OF VIEW
889
Fig. 10. Influence of the rate of spontaneous emission on the time delay identification. The first row corresponds to the case with the spontaneous emission
s , the second row corresponds to the case
s . Each column is associated to a given feedback rate. From left to right, the feedback rates
noise
: GHz,
: ns. They are
ns and GHz,
GHz and
are:
GHz. The time delay and the relaxation period are equal to represented by the dotted and dotted-dashed lines, respectively.
= 10
=25
=5
= 10
= 15
=0
=5
= 0 75
=1
= 0 75
Fig. 11. Influence of the gain saturation " on the time delay signature for three different cases of difficult identification. The first row corresponds to ns,
: ns p
: ,
: ns p
: ns
GHz, the second row to : ,
GHz and the third row to : ns, ns, p
: ,
: GHz. The vertical dotted and dotted–dashed lines represent the time delay and the relaxation oscillation period, respectively. Each column
corresponds to a different value of the saturation gain, as indicated on the figure.
= 0 2 ( = 1 72) = 10
( = 1 05) = 2 5
=05
= 0 33 ( = 1 26) = 5
essential for chaos-based communication. The synchronization
quality is measured by the following cross-correlation function:
=5
is
GHz. These results thus
injection into the slave
demonstrate the potential of chaotic regimes with a high level
of time-delay concealment for chaos based communication.
VI. CONCLUSION
with
and
. We
find that the different sets of parameters that lead to good timedelay concealment allow for weak quality CS, whereas very
good quality ILBS is achieved. Indeed, the maximum value of
is larger than 0.95 as soon as the master
optical
In this paper, we have analyzed the security of an ECSL in
terms of time-delay identification using standard time-delay estimators such as the ACF and the DMI. The key role of the feedback rate and the pumping factor, as well as the choice of the
time-delay
value in comparison with the relaxation-oscilla, has been underlined. It appears that the diffition period
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 45, NO. 7, JULY 2009
cult identification scenarios occur for relatively weak feedback
rates and weak pumping factors with close values of the two
. These difficult identifications find their
time-scales and
origin in the specific nonlinear dynamics and time-scales appearing in the ECSL in its bifurcation cascade leading to chaos.
Indeed, at weak feedback rates, the chaos is reminiscent of the
time-scales involved in the early stage of the laser dynamics,
such as the undamped relaxation oscillation time-scale and possibly PD and QP dynamics. The time-delay estimators then exhibit complex modulated shapes showing these different laser
are close to each other,
dynamics time-scales. When and
the true value is efficiently concealed thanks to a shift of the
with respect to the relaxation oscilfirst Hopf frequency
lation frequency. The laser output at the Hopf frequency starts
.
pulsating at a frequency which is neither close to nor to
The impact of additional laser internal parameters is also investigated. It appears that the decrease in gain saturation coefficient
as well as to inallows to use more distant values of and
crease the pumping factor while maintaining time-delay concealment. The robustness of our results has been checked with
other signal processing techniques such as neural networks and
filling-factor methods. We expect our results to be of interest for
a proper design of a laser chaotic emitter that would allow for
the best concealment of its system parameters, hence also improving security in optical chaos-based communications.
ACKNOWLEDGMENT
The authors would like to thank L. Pesquera for useful discussions and suggestions. The authors acknowledge the support
of Conseil Régional de Lorraine and Fondation Supélec.
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[13] D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of timedelay signature in the chaotic output of a semiconductor laser,” Opt.
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Damien Rontani received the M.S. degree in electrical and computer engineering both from the Ecole
Supérieure d’Electricité (Supélec), Metz, France, and
the Georgia Institute of Technology (Georgia Tech),
Atlanta, in 2005. He is currently working toward the
Ph.D. degree jointly at Supélec and Georgia Tech.
His research interests include nonlinear dynamics of semiconductor lasers, security analysis
of time-delay systems, chaos synchronization and
cryptography.
Alexandre Locquet (M’99) received the M.S.
degree in electrical engineering from Faculté Polytechnique de Mons, Belgium, in 2000, the Ph.D.
degree in engineering science from Université de
Franche-Comté, France, in 2004, and the Ph.D.
degree in electrical and computer engineering from
the Georgia Institute of Technology (Georgia Tech),
Atlanta, in 2005. His doctoral work was related to
optical chaos-based communications.
He is currently a Postdoctoral Researcher with
the Unité Mixte Internationale Georgia Tech-CNRS,
Metz, France. His interests are in physical-layer security, semiconductor laser
dynamics, and time series analysis.
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RONTANI et al.: TIME-DELAY IDENTIFICATION IN A CHAOTIC SEMICONDUCTOR LASER WITH OPTICAL FEEDBACK: A DYNAMICAL POINT OF VIEW
Marc Sciamanna graduated with a degree in electrical engineering from the Faculté Polytechnique de
Mons, Belgium, in 2000. He received the Ph.D. degree from Faculté Polytechnique de Mons, as a Research Fellow of the “Fonds National de la Recherche
Scientifique” (FNRS). His Ph.D. dissertation was entitled “Nonlinear dynamics and polarization properties of externally driven semiconductor lasers.”
Currently, he is a permanent Researcher, Lecturer and faculty staff member of Supélec (Ecole
Supérieure d’Electricité), Metz, France, and LMOPS
CNRS UMR-7132 Laboratory, Metz, France. His research interests include
the physics and nonlinear dynamics of semiconductor lasers, the polarization
properties of VCSELs, the study of dynamical instabilities related to optical
feedback, optical injection, or large current modulation, synchronization and
chaotic encryption using laser diodes. He is author of about 50 research
papers in international journals and conference proceedings. He has given
several invited talks in international conferences and has worked as a scientific
program committee member in SPIE Photonics Europe 2006. He organized
and cochaired the PHASE international workshop (PHysics and Applications
of SEmiconductor LASERs), Supélec, Metz, France, in 2005 and acted as a
Guest Editor of a special issue of Optical and Quantum Electronics related to
the PHASE workshop.
Dr. Sciamanna acted as a Board Member of the IEEE Photonics [formerly the
Lasers and Electro-Optics Society (LEOS)] Benelux Chapter, co-founded the
IEEE/LEOS Benelux Student Chapter and co-chaired the IEEE/LEOS Workshop on Low-Cost Photonics (Mons, Belgium, June 13, 2003). He is ViceChairman of working group 2 “Physics of Devices” in European Action COST
288. He was the recipient of the Prize for Best Engineer in Electricity from
Faculté Polytechnique de Mons (2000), the OSA-Newfocus student travel grant
award (2002), the IEEE/LEOS Graduate Student Fellowship Award (2002), and
the SPIE F-MADE Scholarship Award in Optical Engineering (2003).
891
David S. Citrin (M’93–SM’03) received the
B.A. degree is in physics from Williams College,
Williamstown, MA, in 1985, and the M.S. and
Ph.D. degrees in physics from the University of
Illinois at Urbana-Champaign, Urbana, in 1987 and
1991, respectively, where he worked on the optical
properties of quantum wires.
After receiving the Ph.D. degree, he was a Postdoctoral Research Fellow with the Max Planck Institute for Solid State Research, Stuttgart, Germany
(1992–1993), where he worked on exciton radiative
decay in low-dimensional semiconductor structures. From 1993 to 1995, he was
a Center Fellow with the Center for Ultrafast Optical Science at the University of Michigan, Ann Arbor, where his work addressed ultrafast phenomena in
quantum wells. He was an Assistant Professor of Physics with Washington State
University from 1995 to 2001, at which time he joined the faculty at the Georgia
Institute of Technology, Atlanta, where he is currently a Professor in the School
of Electrical and Computer Engineering. His interests are in ultrahigh-speed
devices, nanoscale engineering, and terahertz technology. The present focus of
his group is the identification and evaluation of novel nonlinear and ultrafast
optical processes in various materials and nanostructures for applications in optoelectronics and photonics. His group also works on photonic crystals and on
terahertz sources and sensors.
Dr. Citrin has served as an Associate Editor for the IEEE JOURNAL OF
QUANTUM ELECTRONICS.
Silvia Ortin graduated in electrical engineering
from the University of Cantabria, Spain, in 2002. She
is currently working toward the Ph.D. degree at the
Institute of Physics of Cantabria (IFCA), Santander,
Spain.
Her research interests include optical chaos communications, chaotic time-series analysis, and nonlinear dynamics.
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