708
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 39, NO. 6, JUNE 2003
Chaos Synchronization in Semiconductor Lasers
With Optoelectronic Feedback
S. Tang and J. M. Liu
Abstract—Chaos synchronization is experimentally investigated in semiconductor lasers with delayed optoelectronic
feedback. Depending on the coupling strength of the driving
signal to the receiver, the lasers fall into different regimes of
chaos synchronization or modulation. Synchronization happens
when the coupling strength to the receiver matches the feedback
strength to the transmitter, where the receiver output reproduces
the output of the transmitter regardless of the presence of channel
distortion or an encoded message. Modulation dominates when
the coupling strength is much less than the feedback strength,
where the receiver output follows the driving signal. In between,
there is no sharp transition between these two regimes but only a
mixture of the two phenomena. Driven oscillation is not observed
in this optoelectronic feedback system when the coupling strength
is increased to a level higher than the feedback strength.
Index Terms—Chaos synchronization, chaotic communication,
optoelectronic feedback.
I. INTRODUCTION
C
HAOS synchronization has been intensively investigated
in various nonlinear dynamical systems [1]–[5] for its potential applications in chaotic communications [6]–[10]. In a
synchronized chaotic communication, the quality of chaos synchronization between a transmitter dynamical system and a receiver dynamical system directly influences the reliability of
message recovery. Other phenomena such as modulation and
driven oscillation, which have also been observed in the studies
of chaos synchronization in coupled nonlinear dynamical systems [11], [12], are very different in their characteristics of reproducing a chaotic waveform from the transmitter even though
they resemble certain features of chaos synchronization. For the
applications of synchronized chaos to communications, it is of
great importance to investigate what is true chaos synchronization and how it can be distinguished from the other phenomena.
In true chaos synchronization, the receiver dynamical system
is required to be identical to the transmitter dynamical system
and the driving signal to the receiver is required to match that
to the transmitter. When the two systems are synchronized, the
output of the receiver reproduces that of the transmitter in both
waveform and magnitude. In modulation, the driving signal to
the receiver is small compared with that to the transmitter. The
output of the receiver is basically a direct linear response of
the receiver dynamical system to the input driving signal. The
Manuscript received August 21, 2002; revised October 28, 2002. This
work was supported by the U.S. Army Research Office under Contract
DAAG55-98-1-0269.
The authors are with the Department of Electrical Engineering, University of
California, Los Angeles, CA 90095-159410 USA.
Digital Object Identifier 10.1109/JQE.2003.810775
output of the receiver does not generally have the same magnitude as the output of the transmitter and the waveform may
be distorted by nonlinearity. In driven oscillation, the driving
signal to the receiver is high above the level which is required
for chaos synchronization. The output of the receiver is a direct,
but nonlinear, response of the receiver dynamical system to the
input driving signal. If the receiver dynamical system matches
the characteristics of the input driving signal, the receiver can
reproduce the waveform of the input driving signal even though
the receiver is now operated in a highly nonlinear regime.
In this paper, we experimentally investigate the synchronization of chaotic pulses in a system that consists of two
unidirectionally coupled semiconductor lasers with delayed
optoelectronic feedback. Chaos synchronization is observed
when the coupling strength from the transmitter to the receiver
matches the feedback strength to the transmitter. However,
when the coupling strength is lowered to much less than the
matched level for chaos synchronization, modulation phenomenon starts to dominate. Driven oscillation is not observed in
the optoelectronic feedback system when the coupling strength
is increased to a level higher than the feedback strength in the
transmitter. These results are observed when the receiver is
set in either an open- or closed-loop configuration. However,
the open-loop configuration shows better quality of chaos
synchronization and modulation [13]. Therefore, in this paper,
we focus on the demonstration of the characteristics of chaos
synchronization and its difference from modulation and driven
oscillation using the open-loop configuration.
The general concepts of the differences among chaos synchronization, modulation and driven oscillation are described in
Section II. Experimental demonstration of chaos synchronization and modulation are reported in Section III. Section IV concludes this paper.
II. GENERAL THEORY
A block diagram of two unidirectionally coupled nonlinear
dynamical systems is shown in Fig. 1. The transmitter consists
of a nonlinear dynamical system with a delayed feedback. The
, is driven
dynamical output of the transmitter, denoted as
by and deterministically dependant on the delayed feedback
, where is the feedback delay time. Theresignal
fore, the output of the transmitter can be written as
, where denotes the nonlinear system funcis the strength
tion of the transmitter dynamical system and
of the feedback. The receiver consists of a similar nonlinear dynamical system, whose system function is denoted by also for
simplicity because the two nonlinear dynamical systems are required to be the same for chaos synchronization. Driven by the
0018-9197/03$17.00 © 2003 IEEE
TANG AND LIU: CHAOS SYNCHRONIZATION IN SEMICONDUCTOR LASERS WITH OPTOELECTRONIC FEEDBACK
709
coupled signal from the transmitter, the receiver output can be
, where is the transmiswritten as
is
sion time for the driving signal to reach the receiver and
the coupling strength of the driving signal to the receiver.
A. Chaos Synchronization
When the receiver is an identical dynamical system to the
transmitter, which are both described by the function and the
coupling strength to the receiver matches the feedback strength
, the receiver can synto the transmitter so that
chronize with the transmitter as
. Indeed, the transmitter and the receiver syn[14]–[16], even though
chronize with a time shift
. Depending on
the receiver is driven by the signal
, the receiver can fall into two regimes of
the difference
retarded or anticipated synchronization [17]–[19], where the receiver output lags behind or leads ahead the transmitter output,
respectively. When the two nonlinear dynamical systems are not
is not exactly the same as ,
completely identical, or when
synchronization quality deteriorates.
The quality of chaos synchronization and its time shift can be
quantified by calculating the shifted correlation coefficient
Fig. 1. Block diagram of two unidirectionally coupled nonlinear dynamical
systems with delayed feedback for chaos synchronization. Here, G denotes the
nonlinear system function of the dynamical systems and F denotes the function
of distortion.
where
denotes the time average. In this calculation, the
chaotic waveform from the receiver is continously time shifted
with respect to the waveform of the transmitter.
by a value
The correlation coefficient is calculated correspondingly for
each time shift . For the coupled systems shown in Fig. 1, the
with
correlation coefficient should peak at
due to chaos synchronization. When there exists parameter
drops.
mismatch, the value of
receiver is required to match the characteristics of the input
signal. Different from the case of chaos synchronization, the
output of the receiver in driven oscillation directly follows the
. Driven oscillation
input driving signal as
has been observed both experimentally and numerically in optical feedback systems [11], [12]. Therefore, when we examine
the shifted correlation coefficient, a correlation peak should also
with
due to the direct response of
appear at
the receiver dynamical system to the input driving signal. The
indicates the quality of this direct response,
value of
which is either modulation or driven oscillation depending on
the strength of .
As discussed above, in the calculation of the shifted correlation coefficient, two correlation peaks should appear. One is
, which is associated with chaos synchronization, and the
, which is caused by the direct response of the reother is
, chaos synceiver dynamical system. When
,
chronization dominates. Otherwise, when
varies
direct response dominates. The time shift
with the variation in the feedback delay time . In contrast, the
does not vary with .
time shift
B. Modulation and Driven Oscillation
C. Synchronization With Channel Distortion
Other than chaos synchronization, the output of the receiver
is also a direct response of the receiver dynamical system to the
input driving signal. Consequently, there is always some correand the signal
lation between the output signal
that is the input to the receiver dynamical system. When the
, the receiver is
coupling strength is small such that
operated in a linear operating regime and its output is a linear
response to the input driving signal. Small-signal modulation,
amplification and attenuation all fall within this category. In this
regime, the output of the receiver is a linear response to the input
. Distortion exists within this catesignal as
gory if the linear response is not perfect because of nonlinearity
is very strong
in the system. When the coupling strength
, the response of the receiver dynamical
such that
system becomes highly nonlinear. Nevertheless, it is still possible that the output of the receiver well reproduces the input
driving signal if the dynamics of the receiver matches the characteristics of the input signal. This phenomenon is called driven
oscillation, nonlinear amplification, or type II synchronization
[11], [12]. In driven oscillation, the nonlinear dynamics of the
In reality, there is always channel distortion in transmission.
The effect of channel distortion can be denoted by a function
, as is shown in Fig. 1. The receiver output is then driven by
.
the distorted signal such that
If the same distortion is applied to the feedback signal in the
transmitter, as is shown in Fig. 1, the transmitter and the receiver
. Thus, the output of the
can still synchronize with
receiver still reproduces the output of the transmitter without
in
distortion as
spite of the fact that the receiver is driven by the distorted signal
.
In modulation, the receiver output follows the distorted input
. In driven oscilladriving signal as
because
tion, the receiver cannot follow well
the dynamics of the receiver system does not match the characteristics of the distorted input signal. Therefore, when there
is channel distortion, which is the case in our experiment, the
and
correlation coefficient should be calculated between
around
for synchronization quality and it should
and
around the peak
be calculated between
710
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 39, NO. 6, JUNE 2003
Fig. 2. Schematic experimental setup for chaos synchronization using semiconductor lasers with delayed optoelectronic feedback. TLD: transmitter laser diode.
RLD: receiver laser diode. PDs: photodetectors. As: amplifiers. Atts: attenuators. BSs: beam splitters.
for direct response quality. The correct positions for the
,
and
are indicated by
measurements of
, and correspondingly in Fig. 1.
D. Synchronization With Encoded Message
Different message-encoding schemes have different effects
on the quality of chaos synchronization [20]. To maintain good
quality of chaos synchronization, it is important to keep the symmetry between the transmitter and the receiver in the process
of message encoding. If the message is fed back to drive the
transmitter when it is sent to the receiver, the receiver can still
synchronize with the transmitter regardless of the strength of
the message. Furthermore, in this situation, the output of the
receiver synchronizes with that of the transmitter even though
the receiver is driven by the total transmitted signal which is a
combination of the output of the transmitter and the message.
In the case of modulation, the receiver output follows the total
driving signal, which has the message included. In driven oscillation, the receiver output still follows the total driving signal,
but with deteriorated quality because the characteristics of the
input signal is changed by the message.
III. OPTOELECTRONIC FEEDBACK SYSTEM
Fig. 2 shows the schematic experimental setup for the investigation of chaos synchronization using two semiconductor lasers
with delayed optoelectronic feedback. The two nonlinear dynamical systems under investigation for chaos synchronization
are the two semiconductor lasers which are intrinsically nonlinear devices. In this setup, the transmitter laser has an optoelectronic feedback loop. The optical output from the transmitter laser is converted into an electronic signal by a photodetector PD1. After amplification, the electronic signal is fed back
to drive the transmitter laser again. Driven by the optoelectronic
feedback, the transmitter laser generates chaotic pulses on a subnanosecond time scale. To synchronize the chaotic pulses, an
identical receiver laser is unidirectionally coupled to the transmitter laser through an optoelectronic path. Through this path,
the optical signal from the transmitter is sent to the receiver
and detected by a photodetector PD3. After amplification, this
signal is used to drive the receiver laser. The receiver is configured in an open-loop configuration with no optoelectronic
feedback. The dynamics of the receiver laser is driven by the
coupled signal from the transmitter. Two attenuators are used to
adjust the feedback strength
to the transmitter and the couto the receiver, respectively.
pling strength
The lasers are InGaAsP–InP single-mode distributed feedback (DFB) lasers emitting at the same wavelength of 1.299 m.
These lasers are carefully chosen from the same batch with the
closest characteristics. The threshold of the transmitter laser is
34 mA and that of the receiver laser is 28 mA. The two lasers
are biased at carefully chosen levels to compensate for the difference in their thresholds so that their parameters are matched
at the operating conditions of the experiment. In the experiment,
both lasers are temperature stabilized at 21.00 C. The photodetectors are InGaAs photodetectors with a 6-GHz bandwidth.
The amplifiers are Avantek SSF86 amplifiers with a 3-dB passband of 0.4–3 GHz. The chaotic waveforms are measured with
a Tektronix TDS 694 C digitizing real-time oscilloscope with a
3-GHz bandwidth and a 1 10 samples/s sampling rate.
In the experiment, the transmitter laser is biased close to its
mA and the feedback strength is adthreshold at
. The feedback current
is proportional
justed to be
to the feedback strength and the fluctuation of the photon density of the transmitter laser such that
, where is the photon density at the free-running conns,
dition. When the feedback delay time is tuned to
the transmitter laser enters a chaotic pulsing state. Meanwhile,
the receiver laser is biased at 34.0 mA, where it is found to have
the best parameter match with the transmitter laser. The transmission time from the output of the transmitter to the input of
ns. To investigate the
the receiver is measured to be
different phenomena of chaos synchronization, modulation, and
to the receiver is addriven oscillation, the coupling strength
justed with respect to the feedback strength to the transmitter.
The chaotic waveforms are measured for the output of the trans-
TANG AND LIU: CHAOS SYNCHRONIZATION IN SEMICONDUCTOR LASERS WITH OPTOELECTRONIC FEEDBACK
Fig. 3. Shifted correlation coefficient between the outputs of the transmitter
and the receiver, under the condition: (a) : ; (b) ;
and (c) : . The correlation peak t is associated with chaos
synchronization. The correlation peak t is caused by the direct response
of the receiver.
= 25
(1 )
= 0 25
(1 )
=
mitter
at point , the output of the receiver
at point
, and the distorted driving signal to the receiver
at point , respectively, as is shown in Fig. 2, corresponding
in Fig. 1. The waveform
to the three positions , , and
is a distorted version of the transmitter output
due to the distortion from the optoelectronic channel path. The
same distortion is applied to the feedback signal to the transmitter through the optoelectronic feedback loop.
Fig. 3 shows the shifted correlation coefficient between the
outputs of the transmitter and the receiver at three different cou, , and 2.5 , respectively. Two
pling strengths
with
correlation peaks are observed. One is
ns and the other is
with
ns.
is calculated between
and
, while
Note that
is calculated between
and
, when
is continously shifted with respect to
and
, as
is discussed above when the system has channel distortion.
Whether the system is dominated by chaos synchronization or
the direct response of the receiver depends on which correlation
,
peak is higher than the other. In Fig. 3(a), with
, which means that the receiver
we find that
correlates better with the driving signal than with the transmitter
output. Under this condition, the receiver is in the modulation
, we can see that
regime. In Fig. 3(b), with
is increased while
is decreased. Therefore, the relation
, which indicates that the receiver
becomes
output now correlates much better with the transmitter output
than with the driving signal. Consequently, the receiver is synchronized with the transmitter in this situation. Since the time
711
Fig. 4. Comparison of the chaotic waveforms and the correlation plots under
: as in Fig. 3(a). (a) Chaotic waveforms from the
the condition outputs of the transmitter (upper trace) and the receiver (lower trace). (b) Chaotic
waveforms of the driving signal (upper trace) and the output of the receiver
(lower trace). (c) Correlation plot between the two traces in (a). (d) Correlation
plot between the two traces in (b). The receiver waveform is shifted by t
T in Fig. 4(b) and (d).
T
in Figs. 4(a) and (c) and it is shifted by t
= 0 25
0
1 =
1 =
shift is
ns, the output of the receiver actually leads the output of the transmitter by 3.0 ns and the receiver
is anticipatedly synchronized with the transmitter. In Fig. 3(c),
, we observe that the correlation peak
with
drops as the synchronization quality deteriorates. Meanwhile,
is found to drop also. Therefore, the
the correlation peak
phenomenon of driven oscillation is not observed in this optois increased to 2.5 . One
electronic feedback system when
possible reason for the absence of such phenomenon in our experiment is the fact that we are not able to increase the coupling
strength to an even higher level as is done in an optical feedback
system [11] because both the amplifiers and the lasers saturate
when the driving signal gets sufficiently high.
Besides the correlation coefficient, comparison between the
chaotic waveforms and the correlation plots are also investigated
and demonstrated in Figs. 4–6 for the conditions corresponding
to Fig. 3(a)–(c), respectively. Fig. 4(a) and (b) shows the comparison of the waveforms between the transmitter output
and the receiver output
and between the driving signal
and the receiver output
, respectively, under the
as in Fig. 3(a). The corresponding
condition of
correlation plots are shown in Fig. 4(c) and (d). To better show
the correlation between these waveforms, the receiver waveform
ns in Fig. 4(a) and
is time shifted by
ns in Fig. 4(b) and
(c) and it is shifted by
(d). As can be seen, under this condition, the receiver output
correlates better with the driving signal with
than with the transmitter output with
. In this
712
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 39, NO. 6, JUNE 2003
Fig. 5. Comparison of the chaotic waveforms and the correlation plots under
the condition = as in Fig. 3(b). Each plot corresponds to the same plot
in Fig. 4(a)–(d).
Fig. 6. Comparison of the chaotic waveforms and the correlation plots under
the condition = 2:5 as in Fig. 3(c). Each plot corresponds to the same
plot in Figs. 4(a)–(d).
case, the system is dominated by the modulation phenomenon.
In this experiment, the output of the receiver is observed to be
smaller than that of the transmitter because the receiver is under
small-signal modulation. In Fig. 4, the output of the receiver is
and
, respecrescaled to be comparable with
tively. Because of the nonlinearity in the system, the best correfor the modulation phenomenon.
lation is only
Fig. 5 shows similar comparison between the waveforms and
as
the correlation plots but under the condition of
in Fig. 3(b). As can be seen, the receiver waveform now correlates much better with the transmitter output, with a correlation
, than with the driving signal, with
coefficient
. Thus, it is demonstrated that the receiver is
synchronized with the transmitter in this situation. The effect of
channel distortion is clearly observed if we compare the waveand
. This distortion is caused mainly by
forms
the amplifier after PD3 in the optoelectronic channel path. Nevertheless, we still observe good quality of chaos synchronization between the outputs of the transmitter and the receiver. The
in Fig. 5 is
quality of chaos synchronization with
even better than the quality of modulation with
in Fig. 4. This is because the quality of chaos synchronization is
not affected by the distortion in the driving signal to the receiver
when the same distortion is applied to the feedback signal to the
transmitter through the amplifier after PD1 in the optoelectronic
feedback loop.
Fig. 6 shows similar comparison between the waveforms and
as
the correlation plots but under the condition of
in Fig. 3(c). As can be seen, the correlation coefficient between
and
and that between
and
both
and
, redrop significantly to
is due to
spectively. The drop of the correlation peak
the mismatch in the strengths of the driving signals to the transmitter and the receiver, respectively. Meanwhile, the low value
indicates that the phenomenon of driven oscillation
of
is not established.
In order to study the transition between the two regimes of
is
modulation and synchronization, the coupling strength
gradually varied with respect to the feedback strength . Fig. 7
between
shows how the correlation coefficients, both
the transmitter output and the receiver output, indicated by the
between the driving signal and the receiver
circles and
. The correoutput, indicated by the squares, vary with
reaches its maximum when the coulation coefficient
for
pling strength matches the feedback strength at
, which agrees with the requirement for chaos synchronization that the receiver and the transmitter be driven by
is tuned away from , the synchrothe same force. When
nization quality deteriorates. In contrast, the correlation coeffiincreases monotonically as the coupling strength
cient
is decreased, which agrees with the expectation that linearity
of modulation is improved as the strength of the input signal is
, the receiver output starts to correduced. When
relates better with the driving signal than with the transmitter
output. Therefore, the system makes a transition from synchrois decreased from
to
nization to modulation when
. Note that the modulation quality at
is still worse than the synchronization quality at
. The
. Further demodulation quality peaks at around
does not improve the modulation quality because
creasing
the effect of noise starts to deteriorate the modulation quality
when the input signal gets sufficiently small. Perfect linearity
of modulation is not possible because of the inherent nonlinear
dynamical nature of this laser system. Driven oscillation, which
TANG AND LIU: CHAOS SYNCHRONIZATION IN SEMICONDUCTOR LASERS WITH OPTOELECTRONIC FEEDBACK
Correlation coefficients (1t ) associated with chaos synchronization
(1t ) caused by modulation, respectively, versus = .
713
(1 )
(1t ),
Fig. 7.
and Fig. 8. Synchronization quality t and modulation quality respectively, versus parameter mismatch I =I .
should take place at
if it exists, is not observed in
this optoelectronic feedback system due to the reason discussed
above. When the transmitter laser is operated at a different value
is basically not
of the feedback strength , the curve
. Meanwhile,
changed under the same relative value of
can shift to the left or the right depending on
the curve
is increased or decreased. Nevwhether the absolute value of
ertheless, the existence of two regimes of synchronization and
modulation and the characteristic of the transition between the
two regimes does not change with .
The effect of parameter mismatch in chaos synchronization
of the receiver laser.
is studied by changing the bias current
The optimum bias current of the receiver laser is found to be
mA where synchronization shows the best quality.
mA
Therefore, the bias current of the receiver at
mA
matches the bias current of the transmitter at
after the difference in the thresholds of these two lasers are taken
into account.
In Fig. 8, the circles show the correlation coefficient
between the synchronized outputs of the transmitter and the reis varied with respect to the matching point of
ceiver when
under the condition
. When the bias current of the
receiver laser matches that of the transmitter laser at
, the synchronization quality is the best. When the bias current
is tuned away from this matching point, the synchronization
quality starts to drop precipitately. In the modulation regime,
parameter mismatch does not have such an effect because the
correlation between the driving signal and the receiver output
does not require the parameters of the receiver to match those
of the transmitter for linear modulation. The modulation quality
actually increases monotonically as the bias current of the reincreases, as is shown in Fig. 8 by the squares under
ceiver
. The melioration of the modulathe condition
is caused by the fact that
tion quality with the increase in
the system is operated in a small-signal modulation regime with
is increased. Due to the
decreased modulation index when
inherent nonlinear dynamical nature of this laser system, the
Fig. 9. Time shift t associated with chaos synchronization and the time
shift t associated with modulation, respectively, versus T
, under the
variation of the feedback delay time while T is fixed at 17.2 ns.
1
1
0
quality of modulation at
is still worse than that of
.
the chaos synchronization at
As we have observed in Fig. 3, the correlation peak
associated with chaos synchronization appears at
ns and the correlation peak
associated with
ns.
direct response of the receiver appears at
and
vary with the feedback
In order to investigate how
delay time , the length of the feedback loop in the transmitter is
, 1.5 and 0.0 ns, respectively,
adjusted such that
while is fixed at 17.2 ns. Fig. 9 shows the result of the time
and
, respectively, versus
. It is clear that
shifts
varies accordingly with the variation in . All the data points of
fall within one straight line with a slop of 1.0, which further
. In contrast,
is not a function
confirms that
of and it remains a constant at 17.2 ns when is changed
is only related to . Therefore, it is proven that
because
714
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 39, NO. 6, JUNE 2003
TABLE I
REGIMES IN THE OPTOELECTRONIC FEEDBACK SYSTEM
Fig. 10. Quality of chaos synchronization under different strengths of
message encoding with additive chaos modulation. Circles: Correlation
coefficient between the outputs of the transmitter and the receiver. Squares:
Correlation coefficient between the total transmitted signal and the output of
the receiver.
chaos synchronization has a time shift of
, which is tightly
related to the dynamical parameter of the feedback delay time
in the transmitter, while modulation as the direct response of
the receiver is only related to .
Another characteristic of chaos synchronization is that the
synchronization quality remains the same when there is an
encoded message, if the message-encoding process does not
break the symmetry between the transmitter and the receiver.
In Fig. 10, the circles represent the correlation coefficient
between the outputs of the transmitter and the receiver when
a message is encoded through additive chaos modulation [20].
The message signal has the same wavelength as that of the
chaotic transmitter. The message is added to the chaotic carrier
waveform at the position of the beam splitter BS2. Because
the encoded message is also fed back to drive the transmitter
laser while it is sent to the receiver, the symmetry between the
transmitter and the receiver is maintained. As a consequence,
chaos synchronization is maintained. It can be clearly seen in
Fig. 10 that the synchronization quality between the outputs of
the transmitter and the receiver remains unchanged when the
amplitude of the message is increased. The correlation between
the total transmitted signal, which is the combined signal of the
transmitter output and the message and the receiver output, is
also shown in Fig. 10 by the squares. It is seen that the value of
this correlation decreases monotonically when the amplitude
of the message is increased. These two correlations shown in
Fig. 10 clearly demonstrate that the receiver is synchronized
to the transmitter output but not to the total transmitted signal.
Therefore, chaos synchronization is further proven by the fact
that the output of the receiver synchronizes with the output
of the transmitter even when the receiver is driven by a total
transmitted signal that is encoded with an arbitrary message.
With this additive chaos modulation, the beating noise from the
interference of the message signal and the transmitter output
needs to be considered. However, in this optoelectronic feedback system, the effect of the beating noise is found to be small,
which is due to the fact that the power of the beating noise is
distributed over a broad spectrum when both the intensity and
the phase of the transmitter output fluctuate chaotically.
In the case of modulation with an encoded message, the receiver output follows the total driving signal which is also the
total transmitted signal including both the transmitter output
and the message. The receiver cannot recover the transmitter
output alone since it is mixed with the message. Because there
is channel distortion in the experiment, the receiver output actually follows the total driving signal that is already distorted. Detailed discussions on the different effects of message encoding
in chaos synchronization and modulation under different encoding schemes deserve further investigation and will be reported in a separate paper.
The experimental results are summarized in Table I for the
different regimes in this optoelectronic feedback system. Compared with the general theory in Section II, the experimental
results match well with the theory. A global picture of the two
regimes of synchronization and modulation are observed in the
experiment with this optoelectronic feedback system and the
characteristics of each regime match well with the theory. The
regime of driven oscillation is not observed in this optoelectronic feedback system.
IV. CONCLUSIONS
Chaos synchronization is experimentally investigated using
two semiconductor lasers with delayed optoelectronic feedback.
and
are observed between
Two correlation peaks
the outputs of the transmitter and the receiver when the chaotic
waveform of the receiver is continously shifted with respect to
the chaotic waveform of the transmitter. The correlation peak
characterizes chaos synchronization between the transis asmitter and the receiver and the correlation peak
sociated with the direct modulation response of the receiver to
the driving signal. The time shifts are found to be
and
, respectively. The system is in chaos synchronizawhen the coupling strength to the
tion with
receiver matches the feedback strength to the transmitter. The
system is dominated by modulation with
when the coupling strength to the receiver is sufficiently less
than the feedback strength to the transmitter. In between, there
TANG AND LIU: CHAOS SYNCHRONIZATION IN SEMICONDUCTOR LASERS WITH OPTOELECTRONIC FEEDBACK
is no sharp transition between these two regimes but only a mixture of the two phenomena. Driven oscillation is not observed in
this optoelectronic feedback system when the coupling strength
is increased to a level higher than the feedback strength in the
transmitter. In chaos synchronization, the receiver output reproduces the output of the transmitter regardless of the presence of
channel distortion or encoded message. In modulation, the receiver output follows the driving signal, which is the total transmitted signal including any encoded message and distortion.
REFERENCES
[1] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,”
Phys. Rev. Lett., vol. 64, no. 8, pp. 821–824, Feb. 1990.
[2] L. Kocarev and U. Parlitz, “General approach for chaotic synchronization with applications to communication,” Phys. Rev. Lett., vol. 74, pp.
5028–5031, June 1995.
[3] H. D. I. Abarbanel and M. B. Kennel, “Synchronizing high-dimensional
chaotic optical ring dynamics,” Phys. Rev. Lett., vol. 80, no. 14, pp.
3153–3156, Apr. 1998.
[4] A. Uchida, M. Shinozuka, T. Ogawa, and F. Kannari, “Experiments on
chaos synchronization in two separate microchip lasers,” Opt. Lett., vol.
24, no. 13, July 1999.
[5] Y. Liu, P. Davis, and T. Aida, “Synchronized chaotic mode hopping
in DBR lasers with delayed opto-electric feedback,” IEEE J. Quantum
Electron., vol. 37, pp. 337–352, Mar. 2001.
[6] P. Colet and R. Roy, “Digital communication with synchronized chaotic
lasers,” Opt. Lett., vol. 19, no. 24, pp. 2056–2058, Dec. 1994.
[7] J. P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based
on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett., vol. 80, no. 10, pp. 2249–2252, Mar.
1998.
[8] G. D. VanWiggeren and R. Roy, “Optical communication with chaotic
waveforms,” Phys. Rev. Lett., vol. 81, no. 16, pp. 3547–3550, Oct. 1998.
[9] S. Sivaprakasam and K. A. Shore, “Message encoding and decoding
using chaotic external-cavity diode lasers,” IEEE J. Quantum Electron.,
vol. 36, pp. 35–39, Jan. 2000.
[10] “Special issue on applications of chaos in modern communication systems,” IEEE Trans. Circuits Syst. I, vol. 48, Dec. 2001.
[11] Y. Liu, H. F. Chen, J. M. Liu, P. Davis, and T. Aida, “Synchronization
of optical-feedback-induced chaos in semiconductor lasers by optical
injection,” Phys. Rev. A, vol. 63, no. 031 802(R), pp. 1–4, Feb. 2001.
[12] A. Murakami and J. Ohtsubo, “Synchronization of feedback-induced
chaos in semiconductor lasers by optical injection,” Phys. Rev. A, vol.
65, no. 033 826, pp. 1–7, Feb. 2002.
[13] S. Tang and J. M. Liu, “Synchronization of high-frequency chaotic optical pulses,” Opt. Lett., vol. 26, no. 9, pp. 596–598, May 2001.
715
[14] V. Ahlers, U. Parlitz, and W. Lauterborn, “Hyperchaotic dynamics and
synchronization of external-cavity semiconductor lasers,” Phys. Rev. E,
vol. 58, no. 6, pp. 7208–7213, Dec. 1998.
[15] H. Fujino and J. Ohtsubo, “Experimental synchronization of chaotic oscillations in external-cavity semiconductor lasers,” Opt. Lett., vol. 25,
no. 9, pp. 625–627, May 2000.
[16] I. Fischer, Y. Liu, and P. Davis, “Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential
for chaos communication,” Phys. Rev. A, vol. 62, no. 011 801(R), pp.
1–4, June 2000.
[17] H. U. Voss, “Anticipating chaotic synchronization,” Phys. Rev. E, vol.
61, no. 5, pp. 5115–5119, May 2000.
[18] C. Masoller, “Anticipation in the synchronization of chaotic semiconductor lasers with optical feedback,” Phys. Rev. Lett., vol. 86, no. 13,
pp. 2782–2785, Mar. 2001.
[19] H. U. Voss, “Dynamic long-term anticipation of chaotic states,” Phys.
Rev. Lett., vol. 87, no. 1, pp. 1–4, July 2001.
[20] S. Tang, H. F. Chen, S. K. Hwang, and J. M. Liu, “Message encoding and
decoding through chaos modulation in chaotic optical communications,”
IEEE Trans. Circuits Syst. I, vol. 49, pp. 163–169, Feb. 2002.
S. Tang received the B.S. and M.S. degrees in electronics from Peking University, Beijing, China, in 1992 and 1997, respectively. She is currently working
toward the Ph.D. degree in electrical engineering at University of California,
Los Angeles.
Her current research interests include optical communication systems, semiconductor lasers, and nonlinear optics.
J. M. Liu received the B.S. degree in electro-physics from National Chiao Tung
University, Taiwan, R.O.C., in 1975 and the M.S. and Ph.D. degrees in applied
physics from Harvard University, Cambridge, MA, in 1979 and 1982, respectively. He also became a licensed professional electrical engineer in 1977.
He was an Assistant Professor with the Department of Electrical and Computer Engineering, State University of New York at Buffalo, from 1982 to 1983,
and a Senior Member of Technical Staff with GTE Laboratories, Inc., from 1983
to 1986. He is currently a Professor with the Electrical Engineering Department,
University of California, Los Angeles. His research interests include the development and application of picosecond and femtosecond wavelength-tunable infrared laser pulses, nonlinear and dynamic processes in semiconductor materials
and lasers, wave propagation in optical structures, and chaotic synchronization
in optical systems.
Dr. Liu is a Fellow of the Optical Society of America, a Senior Member of
the IEE Laser and Electro-Optics Society, and a Member of American Physical
Society, Phi Tau Scholastic Honor Society, and Sigma Xi. He is also a founding
member of the Photonics Society of Chinese-Americans.