Synchronization in Multiple Time Delay Chaotic Laser Diodes Subject to Incoherent Optical Feedbacks and Incoherent Optical Injection

Abstract

We present the first report of the synchronization regimes in both unidirectionally and bidirectionally-coupled multiple time delay chaotic laser diodes subject to incoherent optical feedbacks and incoherent optical injection. We derive the existence conditions and numerically study the stability for lag, complete, and anticipating synchronization regimes and cascaded synchronization. We also study in detail the effect of parameter mismatches and noise on the synchronization quality. It is emphasized that sensitivity of the synchronization quality to parameter mismatches can lead to a high level of security due to the difficulty to replicate the receiver laser. We show that the injection current and feedback delay times are highly important parameters from this point of view.

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arXiv:0904.4359v1 [nlin.CD] 28 Apr 2009
SYNCHRONIZATION IN MULTIPLE TIME DELAY CHAOTIC LASER DIODES SUBJECT
TO INCOHERENT OPTICAL FEEDBACKS AND INCOHERENT OPTICAL INJECTION
E.M.Shahverdiev1,2and K.A.Shore1
1School of Electronic Engineering,Bangor University,Dean St.,Bangor, LL57 1UT, Wales, UK
2Institute of Physics, H.Javid Avenue,33, Baku, AZ1143, Azerbaijan
ABSTRACT
We present the first report of the synchronization regimes in both unidirectionally and bidirectionally-
coupled multiple time delay chaotic laser diodes subject to incoherent optical feedbacks and in-
coherent optical injection. We derive the existence conditions and numerically study the stability
for lag, complete, and anticipating synchronization regimes and cascaded synchronization. We
also study in detail the effect of parameter mismatches and noise on the synchronization quality.
It is emphasized that sensitivity of the synchronization quality to parameter mismatches can lead
to a high level of security due to the difficulty to replicate the receiver laser. We show that the
injection current and feedback delay times are highly important parameters from this point of view.
PACS number(s):05.45.Xt, 42.79.-e, 05.45.Vx, 42.55.Px, 42.65.Sf
INTRODUCTION
Chaos synchronization [1] is one of the main features in nonlinear science and plays an impor-
tant role in a variety of complex physical, chemical, and biological systems, see e.g.references in [2].
Particularly, chaos synchronization has been intensively studied in various nonlinear dynamical
systems for its potential applications in chaotic communications, information processing, nonlin-
ear systems performance optimization,etc.In a chaos based secure communications a message is
masked in the broadband chaotic output of the transmitter laser and synchronization between the
transmitter and receiver systems is used to recover a transmitted message [2-5].
Semiconductor lasers find very wide applications e.g. from CD and DVD players to optical com-
munications networks. In some of these applications light often re-enters the laser after reflection
at other optical elements.Due to their widespread availability and the ease with which they may
be operated in a chaotic regime ,for example, by using external-cavity feedback [2-5], semicon-
ductor lasers have also been extensively studied with a view to their utilisation in chaos-based
communication systems.
Delay differential equations(DDEs) are a large and important class of dynamical systems.
These equations arise in many scientific and engineering areas such as optics, economics, networks
where they model effects arising due to the finite propagation velocity of information, from the
latency of feedback loops and so on [6]. The behaviour of semiconductor lasers subject to optical
feedback(s) (whether delibrate or unwanted) can be described by a set of DDEs. In many practi-
cal applications it may occur that such lasers are subject to more than one optical reflection and
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thus may represent a system with multiple time delays. In addition to modelling such behaviour
in semconductor lasers, DDEs with multiple time-delays may provide more realistic models of
interacting complex systems. From the applications viewpoint additional time delays offer the
opportunity e.g. to stabilize the output of a nonlinear system [7]. It would be of great significance
also to investigate the influence of additional time delays on the synchronization quality between
multiple time delay systems.
When coherent optical feedback is applied to lasers for use in chaos-based communication sys-
tems, the chaos synchronization quality depends sensitively on the frequency detuning between
the transmitter and receiver lasers. In particular detuning by a few hundred megahertz between
the lasers would be expected to effect a significant degradation of the synchronization quality
and in turn will impact the efficiency with which message extraction at the receiver laser can
be accomplished. For practical deployment of such chaos-based communications systems it is of
great interest to utilize laser systems where fine tuning of the lasers frequencies is not required,
i.e. laser systems where the coherent effect of the phase may be neglected. Laser diodes subject
to incoherent optical feedback and incoherent injection offer such a facility. In this scheme, the
feedback and injected fields act on the carrier population in the laser diode active layer but do
not interact with the intracavity lasing fields. As a consequence, the phases of the feedback and
injection fields do not impact the laser dynamics and synchronization requires no fine tuning of
the laser optical frequencies [8-9]. The models used in [8] provide the platform for generalising to
the case of multiple incoherent feedbacks and injection considered in the present paper.
In this connection, one has to keep in mind that the use of chaotic carrier signals generated in
semiconductor lasers using coherent optical feedback also requires coherent optical injection into
the receiver laser system to achieve synchronization, however it is very difficult to guarantee such a
coherent coupling into the receiver laser after transmission of a chaotic carrier over a long distance
[10]. In fact, propagation of the transmitter laser output over distances exceeding the coherence
length of the laser will render the laser field incoherent. In this respect it is noted that in the
unidrectionally coupled time-delayed systems for complete synchronization to occur the distance
between the lasers should equal the feedback distance. Therefore, the realization of high-speed
synchronized chaos, which does not depend on coherent injection and feedback is highly desirable
for practical applications.
In that case the question of practical realization of synchronization schemes based on incoherent
feedback and incoherent injection can be addressed in several ways. In principle, a scheme de-
scribed in figure 1(a) could be one of the options. In fig. 1(a), which illustrates a double feedback
scheme the transmitter laser output propagates to two distant mirrors, and then one part of the
reflected beam is injected into the receiver laser and the other one is fed back into the transmitter
laser. As the propagation distance of the laser output can be made greater than a coherence
length of the laser, then a synchronization scheme based on the incoherent feedbacks and incoher-
ent injection is obtained.
As a another possibility, in the scheme proposed in [8] the linearly polarized output field of the
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laser first undergoes a 90◦polarization rotation through an external cavity formed by a Faraday
Rotator and mirror. It is then split by a non-polarizing beam splitter. One part is fed back into
the transmitter laser and the other part is injected into the receiver laser. As the polarization
directions of the feedback and injection fields are orthogonal to those of the transmitter and re-
ciever output, respectively, then the transmitter laser is subjected to incoherent optical feedback,
while the receiver laser is subjected to incoherent optical injection. However, as emphasized in
[11-12 ] the simple polarization rotation scheme usually gives rise to both incoherent and coherent
optical feedbacks. This fact is also confirmed in recent experimental and theoretical work [13],
which considers as an alternative a full two-polarization model. In the case of one short cavity and
one or more long cavities the use of a Faraday Rotator in the short cavity arm (fig.1(a)) will be
enough also to consider the scheme as one with incoherent optical feedbacks and optical injection.
A scheme based on the optoelectronic feedback and coupling is another phase-insensitive ap-
proach [14-16], see, fig. 1(b).The output from each laser is split by beamsplitters and directed
along different feedback loops and coupling loops. Each signal is converted into an electronic
signal by a photodetector and then amplified before being added to the injection current of a
laser. An optical isolator ensures unidirectionality of the coupling. It is noted that in this scheme
that in order to eliminate interference effects between laser fields their detuning should excced the
detection bandwidth. As emphasized in [16], apart from avoiding the complexity introduced by
the phase of the electric field, there is another advantage in using the optoelectronic feedback and
coupling scheme. Namely in this case there is no restriction to weak or moderate coupling and
feedback to avoid secondary round trips in the external cavity. At the same time one must take
into account that at very large coupling strengths saturation effects in photodetectors and am-
plifiers may become significant. Also, in the optoelectronic case the bandwidth of the electronics
may act as a low-pass filter on the full dynamics of the optical field [12].
It should be emphasized that mathematically in the cases of both incoherent feedback-incoherent
injection and optoelectronic feedback-optoelectronic injection one deals with similar rate equations
for the photon density and the carrier density. Moreover, an all-optical incoherent feedback and
incoherent injection system is dynamically equivalent to the optoelectronic system [16]. Equiv-
alence between the dynamics of laser diodes with incoherent optical feedback and injection and
dynamics of laser diodes with optoelectronic feedback and coupling is also underlined in recent
work [13].
As synchronization between the transmitter and receiver lasers is vital for message decoding in
chaos-based secure communications, it is of paramount importance to investigate chaos synchro-
nization regimes in multiple time delay laser systems. In this paper we present the first report
of chaos synchronization regimes between laser diodes with multiple incoherent feedbacks and
incoherent injection. We derive existence conditions for lag,complete, and anticipating synchro-
nization regimes and numerically study the stability of the synchronization regimes.
It can be envisaged that in chaos-based communication systems there will be a need to broad-
cast a message to a number of receivers or else use may be made of repeater stations to extend
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the transmission distance. Both cases rely on the possibility of synchronization between a master
laser and a number of slave lasers. In other words the study of cascaded synchronization is of great
practical importance. In the light of this we also study cascaded synchronization of laser systems.
We also present the results of a detailed investigation on the effect of parameter mismatches and
noise on the synchronization quality.
SYSTEM MODELLING
Throughout the paper we deal with laser systems with double delay times. To model the
arrangement we generalise the approach of [8] to the case of multiple feedback lasers. Thus the
dynamics of the double delay time master laser is governed by the following system of equations:
dP1
dt
= (G1−
1
τp1)P1+ β1N1
(1)
dN1
dt
= I1−N1
τs1
− G1(P1+ k1P1(t − τ1) + k2P1(t − τ2))(2)
The receiver laser is described by the following set of equations
dP2
dt
= (G2−
1
τp2)P2+ β2N2
(3)
dN2
dt
= I2−N2
τs2
− G2(P2+ k3P2(t − τ1) + k4P2(t − τ2) + KP1(t − τ3))(4)
where Gj= GNj(1 − ǫjPj)(Nj− N0j), with j = 1 for the transmitter and j = 2 for the receiver.
In Eqs.(1-4), Pjand Njare the photon number and the electron-hole pair number, respectively,
in the active region of laser j.N0j is the value of Nj at transparency.τpj,τsj,Ij,GNj, and ǫj are
respectively the photon lifetime, the carrier lifetime,the injection current (in units of the electron
charge), the gain coefficient, and the gain saturation coefficient of laser j. β1 and β2 are the
spontaneous emission rates. k1,2and k3,4are the feedback rates for the transmitter and receiver
systems,respectively. τ1,2are the feedback delay times in the transmitter and receiver systems;K
is the coupling rate between the transmitter and the receiver;τ3is the time of flight between lasers.
Unless otherwise stated, the parameters of the lasers are chosen to be identical, except for the
feedback levels and coupling strengths. Throughout this paper xτ≡ x(t − τ).
Now comparing
dP1,τ3−τ1
dt
= (G1,τ3−τ1−
1
τp1)P1,τ3−τ1,(5)
dN1,τ3−τ1
dt
= I1−N1,τ3−τ1
τs1
− G1,τ3−τ1(P1,τ3−τ1+ k1P1,τ3+ k2P1,τ2+τ3−τ1) (6)
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with the receiver system, Eqs.(3-4) one finds that
P2= P1,τ3−τ1,N2= N1,τ3−τ1
(7)
is the synchronization manifold under the existence conditions
k1= k3+ K,k2= k4.(8)
Analogously we find that
P2= P1,τ3−τ2,N2= N1,τ3−τ2
(9)
is the synchronization manifold if
k2= k4+ K,k1= k3.(10)
We notice that depending on the relation between the coupling τ3and feedback τ1delay times,
manifold (7) is the retarded (τ3> τ1), complete (τ3= τ1),and anticipating (τ3< τ1) synchroniza-
tion manifold [17], respectively. It is also noted that with additional time delay the number of
possible synchronization manifolds is doubled. Note that existence conditions obtained here are
similar to those derived in [18] for semiconductor lasers with coherent feedbacks and injection,
i.e.these conditions are quite generic.
Most chaos based communication techniques use synchronization in unidirectional master-slave
system.Such a coupling scheme prevents the messages being exchanged between the sender and
receiver. A two way transmission of signals requires bidirectional coupling. With this in mind in
the paper we also consider chaos synchronization between bidirectionally coupled laser diodes with
double time delays. For the bidirectional coupling the systems to be synchronized are described
by the equations:
dP1
dt
dN1
dtτs1
and
dP2
dt
dN2
dtτs2
Now comparing the system of Eqs.(11-12) with the system of Eqs.(13-14) we establish that com-
plete synchronization P1= P2is possible under the conditions k1= k3,k2= k4,K1= K2.
We note that the existence conditions are necessary for synchronization, but these conditions indi-
cate nothing about the stability of the synchronization manifolds. Due to the high complexity of
the model the question of stability of the synchronization regimes cannot be studied by analytical
means. To identify stable synchronous regimes use is made of numerical modelling.
5
= (G1−
1
τp1)P1+ β1N1
(11)
= I1−N1
− G1(P1+ k1P1,τ1+ k2P1,τ2+ K1P2,τ3) (12)
= (G2−
1
τp2)P2+ β2N2
(13)
= I2−N2
− G2(P2+ k3P2,τ1+ k4P2,τ2+ K2P1,τ3) (14)