chaos synchronization uncorrected proofshome.iitk.ac.in/~siva/book/chap6.pdf · synchronized...
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6Chaos Synchronization
S. Sivaprakasam and C. Masoller Ottieri
6.1 INTRODUCTION
Chaos is one of the most appealing topics for researchers from various subject backgroundshaving non-linear dynamics as a common interest. This is due, primarily, to its significanceas a tool which one can use to make sense of a complex world. Examples of chaosinclude weather patterns, biological systems, fluid turbulence, mobile communications,traffic, population dynamics, and astrophysics – to name just a few. Synchronized chaoticsystems have received a lot of attention due to their potential use for secure communications.Synchronized semiconductor lasers are especially attractive due to their potential use inall-optical communication systems. In this book, chaos generation in semiconductor laserswith optical feedback is discussed in Chapter 2 as one of the regimes of operation ofthese systems. The use of chaotic semiconductor lasers in all-optical secure communicationsystems is discussed in Chapter 9.
Historically, synchronization in non-chaotic systems was realized before that of chaoticsystems. Christiaan Huygens first observed anti-phase synchronization of two pendulumclocks, with a common frame, in 1665. This was the subject of some of the earliestdeliberations of the Royal Society. Huygens found that the pendulum clocks swung at exactlythe same frequency and 180° out of phase. When he disturbed one pendulum the anti-phase state was restored within half an hour and pendulum clocks remained synchronizedindefinitely, thereafter, if left undisturbed. He found that synchronization did not occur whenthe clocks were separated beyond a certain distance, or oscillated in mutually perpendicularplanes. Huygens deduced that the crucial interaction came from very small movements ofthe common frame supporting the two clocks. He also provided a physical explanation forhow the frame motion set up the anti-phase motion [1].
The synchronization of chaotic systems is a subject with a long history that has been a ‘hottopic’ in the past ten years. Fujisaka and Yamada [2–5] did early work on synchronizationof chaotic systems, but it was not until the work of Pecora and Carroll [6, 7] that the subjectreceived a significant amount of attention. The term ‘chaotic synchronization’ refers to a
Unlocking Dynamical Diversity Edited by D.M. Kane and K.A. Shore© 2005 John Wiley & Sons, Ltd ISBN: 0-470-85619-X
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186 Chaos Synchronization
variety of phenomena in which chaotic systems adjust a given property of their motion toa common behaviour due to a coupling or to a driving force [8]. The systems might beidentical or different, the coupling might be unidirectional (master–slave or drive–responsecoupling) or bi-directional (mutual coupling) and the driving force might be deterministic orstochastic.
Intrinsic interest in synchronization phenomena from a dynamical systems and chaos theorypoint of view, and practical applications in secure communications [9], have spurred a widerange of research studies. Two systems are coupled unidirectionally if the dynamics of onesystem (called master or drive) affects the dynamics of the other (called slave or response),while the dynamics of the slave does not affect the dynamics of the master. The fact that twounidirectionally coupled chaotic systems can be used in a secure communication scheme wasfirst shown by Cuomo and Oppenheim [10, 11], who built a circuit version of the Lorenzequations and showed the possibility of using this system to transmit a small speech signal.While some years ago the word ‘chaos’ had a negative connotation in applied research, nowa days researchers are studying techniques for taking advantage of chaotic dynamics insteadof trying to avoid it. The use of chaos synchronization in secure communication systems isa good example of this modern point of view.
Chaos synchronization includes a number of different phenomena such as (1) completelyidentical oscillations in the coupled systems (complete synchronization); (2) frequencylocking ( frequency synchronization); (3) phase locking between the two systems while theiramplitudes remain uncorrelated ( phase synchronization); (4) the output of one system iscorrelated with the output of the other system but it lags in time (lag synchronization); and(5) the outputs of the two systems are functionally related (generalized synchronization).The latter usually occurs in real experiments of chaos synchronization, in which no completesynchronization is observed due to parameter mismatches. In each case, one can separatecases of full and partial synchronization and, thus, several measures have been proposedto determine the degree of synchronization. (these measures include correlation function,mutual information, synchronization diagrams, etc.) A recent and complete review of chaoticsynchronization phenomena can be found in [8].
In the field of lasers, the first experimental demonstrations of synchronization of twochaotic lasers were performed using Nd: YAG [12] and CO2 lasers [13, 14]. In the schemeused by Roy and Thornburg [12], one or both of the lasers were driven into chaos by periodicmodulation of their pump beams. Sugawara et al. [13] demonstrated synchronization of twochaotic passive Q-switched lasers by modulating the saturable absorber in the cavity of onelaser with the output of the other laser. Liu et al. [14] used two passive, optically-coupled,Q-switched lasers, and the amount of coupling used induced transitions from unsynchronizedperiodic oscillations to synchronized chaotic ones. Colet and Roy [15] were the first topropose a scheme for encoding data within a chaotic carrier from a loss-modulated solid-state laser, and the first experimental demonstration of chaotic communication with anoptical system (an erbium-doped fibre ring laser) was done in 1998 by Van Wiggerenand Roy [16–18]. In the past decade research from various groups, both theoretical andexperimental, has focused on understanding synchronization phenomena in unidirectionallycoupled lasers, its potential for use in secure communications, and its dependence on variouslaser parameters [19–93].
Synchronization phenomena in spatially extended systems (such as broad-area nonlinearoptical cavities, laser arrays, 1D chains and 2D lattices of coupled dynamical systems, neural
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Synchronization of Unidirectionally Coupled Semiconductor Lasers 187
networks, etc.), have also received a lot of attention. Local and global coupling havebeen considered and two different synchronization regimes have been found [8]. Eitherall elements of the ensemble display the same behaviour for any initial conditions (fullsynchronization) or the ensemble splits into groups of mutually synchronized elements(cluster synchronization). In the field of lasers, White and Moloney [94] proposed apractical application of synchronization of spatio-temporal chaos. They demonstrated thesuccessful multiplexing of random bit sequences between transmitter/receiver semiconductorlasers modelled by coupled nonlinear partial differential equations. They reported thesynchronization of spatiotemporal chaos through a single scalar complex variable (the totalelectric field). The separate messages were encoded into individual channels (the longitudinalmodes) by injecting weak signals at the relevant mode frequencies, thus enabling thetransmission of multiple messages through a single channel. Another application ofspatiotemporal chaos synchronization was proposed by García-Ojalvo and Roy [95, 96],who suggested a model system (a broad-area nonlinear optical ring cavity that exhibitsspatiotemporal chaos) that allows parallelism of information transfer using an optical chaoticcarrier waveform. The synchronization of laser arrays has also been the object of extensiveinvestigation [97–101].
In this chapter we review synchronization of chaotic semiconductor lasers and discussvarious types of synchronization in both, unidirectionally and mutually coupled lasers.In Section 6.2 we present numerical and experimental results of synchronization onunidirectional or master–slave configuration. In Section 6.3 we discuss results of thesynchronization of mutually coupled semiconductor lasers. In Section 6.4 we present ourconclusions and personal perspectives on the research field, at present and in the nearfuture.
6.2 SYNCHRONIZATION OF UNIDIRECTIONALLY COUPLEDSEMICONDUCTOR LASERS
Two lasers are coupled unidirectionally if the dynamics of one laser (called master ortransmitter) influences the dynamics of the other (called slave or receiver), and the masterlaser is isolated from the slave laser, so that the dynamics of the slave does not affect thedynamics of the master. A typical experimental setup for unidirectionally coupled external-cavity lasers is shown in Figure 6.1.
Mirasso, Colet and García-Fernández [20] were the first to show theoretically that chaoticsemiconductor lasers with optical feedback, unidirectionally coupled, can be synchronizedand used in encoded communications systems. In their scheme a message was encoded inthe chaotic output of the master laser, and it was transmitted to the slave laser using anoptical fibre. The slave laser was assumed to operate under similar conditions as the masterlaser. The message could be decoded by comparing the chaotic input and output of the slavelaser.
Since then, a lot of theoretical and experimental studies have focused on understandingthe synchronization properties of unidirectionally coupled chaotic semiconductor lasers[21, 24, 25, 30–32, 34, 36–38, 40–48, 50, 52, 53, 57–61, 63–70, 72, 74–80, 82–86,88–93]. Synchronization of power dropouts in the low-frequency fluctuations regime
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188 Chaos Synchronization
ML
SL
BS1
BS2
BS4
BS3
NDF
NDF
OI1
OI2
PD1
PD2
CA
M1
M2
CRO
CRO
Figure 6.1 Experimental set-up used in [36]: ML – Master Laser; SL – Slave Laser; BS1, BS2,BS3, BS4 – Beam Splitters; PD1, PD2 – Photo Detectors; OI – Optical Isolators; M1, M2 – Mirrors;NDF – Neutral Density Filter; CA – Coupling Attenuator; CRO – Digital Oscilloscope.
(LFF, see Chapter 2) was first shown numerically by Ahlers et al. [34]. The theoretical studieswere based on simple rate equations for the complex electric fields and carrier densities inthe lasers, which are extensions of the well-known Lang–Kobayashi equations [102] (seeChapter 2). These equations are:
•Em = 1+ i�
2
[Gm�t�− 1
�p�m
]Em�t�+�mEm�t − �� exp �−i�m��+√Dm�m�t� (6.1)
•Nm = Jm
e− Nm�t�
�n�m
−Gm�t� �Em�t��2 (6.2)
•Es = 1+ i�
2
[Gs�t�− 1
�p�s
]Es�t�+�sEs�t − �� exp �−i�s��
+Em�t − �c� exp −i ��m�c +��t��+√Ds�s�t� (6.3)
•Ns = Js
e− Ns�t�
�n�s
−Gs�t� �Es�t��2 (6.4)
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Synchronization of Unidirectionally Coupled Semiconductor Lasers 189
Here, the indices m and s refer to the master and slave lasers respectively. Em�Es are theslowly varying complex fields, and Nm�Ns are the normalized carrier densities. The equationsare written in the reference frame where the complex optical fields of the lasers are given byEm�t� exp �i�mt�, Es�t� exp �i�st�, where �m, �s are the optical frequencies of the solitarylasers. The external optical feedback in each laser is described by the parameters �m and�s, which are the feedback levels of the master and slave lasers respectively and � isthe delay time in the external cavity. The term Em�t − �c� exp −i ��m�c +��t�� in theright-hand-side of Equation (6.3) accounts for the light injected from the master laser tothe slave laser, is the injection rate, �c is the time for light to travel from the masterlaser to the slave laser and �� = �m − �s is the frequency detuning between the lasers.�m�t� and �s�t� are independent complex Gaussian white noises that represent the effects ofspontaneous emission and spontaneous recombination. D measures the noise intensity. Theother parameters are as follows: �p is the photon lifetime, � is the linewidth enhancement
factor, G = N/(
1+ �E�2)
is the optical gain, is the gain saturation coefficient, �� isthe phase accumulation after one round-trip in the external cavity, J is the injection currentdensity, e is the electronic charge and �n is the carrier lifetime.
This model does not include multiple reflections in the external cavity, and therefore it isvalid for weak feedback levels. It is assumed that the mirrors are positioned such that theexternal cavity length is the same for both lasers. It is also assumed that the optical fielddoes not experience any distortion during its propagation from the master to the slave laser.
In experiments of synchronization of chaotic external-cavity lasers, two configurations forthe slave laser have been considered. In the first configuration, the slave laser is subjected toboth, optical feedback from an external reflector and optical injection from the master laser(we will refer to this as closed-loop configuration). In the second configuration, the slavelaser is subjected only to optical injection from the master laser (we will refer to this asopen-loop configuration). In the theoretical model, the open-loop configuration correspondsto setting �s = 0.
Sivaprakasam and Shore [37] were the first to demonstrate experimentally thesynchronization of two chaotic external-cavity semiconductor lasers. The experimentalarrangement is shown schematically in Figure 6.1. Two laser diodes (APL-830 of linewidth200 MHZ) were driven by ultra-low noise current sources (ILX-Lightwave LDX-3620) andtheir temperature was controlled using thermo-electric controllers (ILX-Lightwave LDT-5412) to a precision of 0.01 K. Both lasers were subjected to optical feedback from externalmirrors (M1 and M2) and the feedback strength was controlled using a continuously variableneutral density filter (NDF1). The length of the external cavity is 25 cm in both the cases.The optical isolators (OFR-IO-5-NIR-HP) ensure the lasers were free from unwanted backreflection. The typical isolation is –41 dB. Isolator (OI1) ensured the master laser was isolatedfrom the slave laser. The coupling attenuator (CA) enables the percentage of master powerfed into slave laser to be controlled. PD1 and PD2 were two identical fast photodetectors(EG&G - FFD040B) with a response time of 2.5 ns. The output of the master laser is coupledto the photodetector (PD1) by the beamsplitters BS1 and BS2. Beamsplitter (BS3) acts asthe coupling element between the master and slave. Beamsplitter (BS4) couples the slaveoutput to the photodetector (PD2). The photodetector output signals are stored in a digitalstorage oscilloscope (Fluke Combiscope PM3394B, 200MHz) and then acquired by a PC.
The master laser and slave lasers were driven into chaos by application of appropriateoptical feedback from the external cavity mirrors. The optical feedback for the master and
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190 Chaos Synchronization
slave led to a reduction in their threshold currents of 7.0% and 9.7% respectively. (The free-running threshold current �Ith� values were 57 mA and 53 mA respectively for the master andslave laser.) The temperature and current were adjusted to ensure that the master (26�66 �C,1�136Ith) and slave (27�92 �C, 1�035Ith) operate at the same wavelength. A small percentage(3%) of the master optical output is injected into the slave and this led to synchronization ofthe slave to the master. In Figure 6.2(a), the time evolution of the master and slave lasers isshown. The upper trace (master laser) is shifted vertically for clarity. The slave laser actuallylags behind the master laser by a time equal to the time for light to travel between the twolasers �c, but this cannot be noticed in Figure 6.2(a) due to the different time scales: �c isof the order of nanoseconds, and the time-resolution is of the order of microseconds. Thesynchronization can conveniently be illustrated by using a synchronization diagram, whichconsists of a plot of the master intensity vs. the slave intensity. If the two lasers were perfectlysynchronized, the synchronization diagram will be a straight line with a positive gradient.
0.0 0.1 0.2 0.3 0.4
0.040
0.045
0.050
0.055
0.060
0.065
0.070
0.056 0.060 0.064 0.068 0.072
0.040
0.042
0.044
0.046
0.048
0.050
(a)
Out
put P
ower
(ar
b. u
nits
)
Time (ms)
Slav
e O
utpu
t (ar
b. u
nits
)
Master output (arb. units)
(b)
Figure 6.2 (a) Time traces of the master (upper) and slave (lower) laser output, the time traces areshifted vertically for clarity, (b) the corresponding synchronization diagram [37], (c) SynchronizedLFFs for master (ML) and slave (SL) lasers, (d) one shot of enlarged LFF waveforms correspondingto (c) [41].
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Synchronization of Unidirectionally Coupled Semiconductor Lasers 191
0 1000 2000
Time (ns)
Out
put P
ower
(a.
u.)
ML
0 1000 2000Time (ns)
Out
put P
ower
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u.)
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(c)
0 100 200 300
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0 100 200 300
Time (ns)
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Figure 6.2 (continued )
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192 Chaos Synchronization
Less than perfect synchronization leads to a broadening of the diagram. Figure 6.2(b) showsthe synchronization diagram corresponding to Figure 6.2(a).
The frequency detuning between the master and slave lasers is a critical parameteraffecting synchronization. The frequency detuning is defined as the master laser frequencyminus the slave laser frequency, �� = �m −�s. In the experiments of [104] the detuningwas varied by modifying the slave laser bias current (a change in the bias current by1 mA shifts the frequency by 1 GHz, and positive or negative detuning arise dependingupon whether the slave laser bias current was increased or decreased). Synchronizationplots for four different values of detuning are shown in Figure 6.11. Figure 6.11(a) showsdegradation in synchronization for a positive detuning (�� = 6 GHz). Good synchronizationis shown in Figure 6.11(b), where the gradient is positive. As the detuning is made negative,the synchronization plot starts to branch with a portion of negative gradient as shown inFigure 6.11(c) (�� = −3 GHz). As the detuning is further increased to −6 GHz the newlydeveloped branch with a negative gradient dominates and the branch with a positive gradientdisappears. This is shown in Figure 6.11(d). The appearance of a negative gradient in thesynchronization diagram was termed ‘inverse synchronization’. The output intensities of thelasers, for two different detunings (corresponding to normal and inverse synchronization),are shown in Figure 6.12. These time traces serve as direct time-domain evidence of theinverse behaviour seen in synchronization diagrams of Figure 6.12.
Inverse synchronization was modelled theoretically in [104] based on non-resonantcoupling between the master and slave lasers. It was argued that when the injected light isdetuned from the slave laser mode wavelength, the coupling into the slave lasing mode isvery weak and, therefore, the effects of non-resonant amplification dominate. Even thoughthe injected light is non-resonant with the slave laser cavity mode, it is still amplified throughthe stimulated emission process. The effect of the non-resonant amplification of light (thatdoes not couple into the slave laser mode) is a reduction of the carriers in the device. Thus,fewer carriers take part in the gain process for the slave laser, reducing its optical output.Wedekind and Parlitz [105] also observed the regime of inverse synchronization (they referto it as anti-synchronization) and proposed a different theoretical explanation, based onpolarization effects due to the Faraday isolator used for the unidirectional coupling. Thus,more theoretical and experimental work is needed in order to clarify the physical mechanismsunderlying inverse synchronization phenomena.
The experimental synchronization of LFF power dropouts was later done by Ohtsubo andco-workers [41] and the time traces are shown in Figure 6.2. In Figure 6.2(c) the upper andlower time trace correspond to the master and slave laser output respectively. The enlargedwaveforms of laser outputs are shown in Figure 6.2(d).
From a theoretical point of view the synchronization of two identical lasers in a closed-loopconfiguration leads to two qualitatively different synchronization regimes [66, 74, 77, 86].One regime occurs when the master and slave lasers are subjected to the same opticalfeedback strength:
�m = �s� (6.5)
and the slave laser is subjected to strong enough optical injection (roughly speaking,an injection strength corresponding to the injection locking region of a laser under CW
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Synchronization of Unidirectionally Coupled Semiconductor Lasers 193
optical injection). In this regime the slave laser output at time t synchronizes with the injectedfield at time t, which, taking into account the travel time from the master to the slave, �c, isthe field of the master laser at time t−�c. The intensities are related by:
Is�t� = a Im�t −�t� (6.6)
where �t = �c and a is a proportionality constant. Analytic conditions for the occurrence ofthis type of synchronization were given by Revuelta et al. [75].
A different regime (found numerically by Ahlers et al. [34]) occurs when: (1) the masterand slave lasers have the same amount of total external injection, and (2) the master injectionstrength is strong enough. Condition (1) implies that the master feedback strength is equalto the sum of the slave feedback strength and the optical coupling strength:
�m = �s + (6.7)
Condition (2) implies that the dynamics induced by the optical injection from the masterlaser dominates the dynamics induced by the external cavity of the slave laser (roughlyspeaking, > �s). In this regime the output of the slave laser synchronizes with the outputof the master laser as:
Is�t� = Im�t −�t� (6.8)
where:
�t = �c − � (6.9)
A qualitative experimental verification of the condition (6.7) was done in [50]. The outputpower levels of the master and slave lasers at different current levels were measured whenthe lasers were synchronized in their output intensity. The feedback power to the transmitter(Pfm) and receiver (Pfs) lasers and the transmitter power coupled to the receiver laser (Pc)were measured. The quantities Pfm, Pfs, and Pc are proportional to �m, �s, and respectively.A plot of Pfm vs. (Pfs + Pc) is shown in Figure 6.3. The linear dependence between Pfm
and (Pfs + Pc) provides a qualitative the verification of Equation (6.7). The experimentwas done using very slow photodetectors, hence it was not possible to verify the lag timeof Equation (6.9).
When �c < �, �t = �c − � < 0 and the slave laser intensity anticipates the masterlaser intensity. Masoller [55] has shown that this is a particular case of the anticipatingsynchronization regime discovered by Voss [103].
Figure 6.4 displays numerical solutions corresponding to anticipated synchronization [55].In Figures 6.4(a)–(c), the master laser operates in the LFF regime: the intensity suddenlydrops toward zero and then recovers gradually, only to drop out again after a random timeinterval. The intensity dropouts are actually the envelope of a series of fast, picosecondpulses. In Figure 6.4(a) the laser parameters are identical, and the noise level is zero. Forlarge enough coupling strength, , and for an adequate slave feedback level (�s = �m −),the slave laser anticipates by � − �c (5 ns) the output of the master laser. The dotted line
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194 Chaos Synchronization
0 1 2 3 40
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14 (d)
Slav
e la
ser
outp
ut (
arb.
uni
ts)
Master laser output (arb. units)
(a) (b)
(c)
Figure 6.3 Synchronization plots for transmitter-receiver detuning of (a) +6 GHz, (b) 0 GHz,(c) −3 GHz and (d) −6 GHz and with the transmitter laser operating at a low-frequency fluctuationregime [62].
plots Im�t +� −�c�− Is�t�, and proves that after a transient time, the slave laser is perfectlysynchronized to the future chaotic output of the master laser.
The physical origin of this behaviour can be understood by looking at the simultaneousturn-on of the master and slave lasers, Figure 6.4(b). The lasers emit the first intensity pulseat approximately the same time (because they are identical, and the initial conditions arethat both lasers are off at t = 0). The master laser emits a train of pulses at the relaxation-oscillation period, before relaxing to the solitary steady state. This train of pulses interfereswith the steady state, when it returns from the external mirror, at time � after the emissionof the first pulse. A fraction of the master intensity is transmitted to the slave laser, andthe train of pulses interferes with the slave laser emission, at time �c after the emission ofthe first pulse. Therefore, if the coupling is strong enough, the slave laser will respond in
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Synchronization of Unidirectionally Coupled Semiconductor Lasers 195
0 200 400 600 800 10000 200 400 600 800 1000
(b)
Out
put (
arb.
uni
ts)
Time (ns)
(a)
Figure 6.4 Time traces of the master (upper traces) and slave (lower traces) lasers for detunings(a) −6 GHz and (b) 0 GHz. The master laser operates in the low-frequency fluctuation regime [62].
a similar manner as the master laser, only that it will do it at time �c, while the masterlaser will do it at time �. The simultaneous turn-on of the lasers allows an understandingof the mechanism of anticipated synchronization, but the lasers also synchronize if they areturned-on independently. In Figure 6.4(c) there are small parameter mismatches between thelasers, and a small amount of noise. The lasers are not perfectly synchronized, and because Js
is slightly lower than Jm� Im�t + � − �c�− Is�t�, fluctuates about a mean value different fromzero. Bursts of desynchronization are observed when Im drops to zero. Figure 6.4(d) displayssolutions corresponding to anticipated synchronization when the master laser operates in thecoherence collapse regime (which occurs for larger injection current, see Chapter 2).
In [55] the degree of synchronization and the lag time between the lasers were quantifiedwith the similarity function, defined as:
S2��0� =⟨Im�t + �o�− Is�t��
2⟩√�I2
m� �I2s �
(6.10)
If Im(t) and Is(t) are independent time series, S��0� �= 0 for all �0. If the lasers are synchronizedsuch that Is(t) = Im�t + � − �c), S�� − �0� = 0. Figure 6.5(a) shows the similarity functionwhen there is perfect anticipated synchronization. S(�0) presents a sharp minimum at �0 =� −�c. There are also additional minima at �0 = n(� −�c) (with n integer), which arise from
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196 Chaos Synchronization
0.0 0.5 1.0 1.5 2.0 2.50.00
0.05
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Pfm
(mW
)
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Figure 6.5 Pfm vs (Pfs +Pc� under conditions of synchronization [50]. Pfm is the feedback powerto the master laser, Pfs is the feedback power to the slave laser and Pc is the power coupled to theslave laser. The size of the squares correspond to a power variation of about 0.01 mW. Maintenanceof the synchronization requires current fluctuations less than 0.25 mA, which would produce a changeof approximately 0.01mW in the feedback power.
time correlations of the master intensity. When there is noise and parameter mismatches,Figure 6.5(b), S(�0) has a similar shape, but the minima are shallower.
The dependence of the lag time with the delay time given by Equation (6.9) wasverified experimentally by Liu et al. [82]. The experimental arrangement used is shownin Figure 6.6(a). The laser diodes are two similar single-mode DFB laser diodes (NEL-NLK1555) driven with a low-noise high precision injection current, and temperaturecontrolled. At the injection current Ib = 11�4 mA (1.5 Ith) the laser wavelength is 1537.17 nmwith a linewidth of about 4 MHz. The intensity variations of the laser outputs are detected by6 GHz bandwidth photodetectors (New Focus 1514LF) and observed on digital oscilloscope(Tektronix TDS694C) with a 3 GHz bandwidth and a 10 Gps (giga bits per second) samplingrate, as well as on a RF spectrum analyzer (Advantest R#267) with an 8 GHz bandwidth.The light output from the right facet (high-reflection (HR) coated, reflection >95%) of themaster laser is fed back to the left facet (antireflection (AR) coated, reflection <1%) of themaster laser. The light output from the AR-coated facet of the master laser output is injectedinto the slave laser on the AR-coated facet so that a strong injection can be achieved. Theeffect of multiple reflections is completely avoided in the current experiment due to theunidirectional ring configuration. The injection power of the slave laser is carefully tunedto match the feedback power of the master laser. The time lag �t between the master andslave laser outputs is calculated from two time series. When the delay time � is longerthan the propagation time, time lag becomes negative which means the slave laser outputanticipates the master laser. Figure 6.6(b) shows the variation of the lag time �t as a functionof the delay time �. Hence, Liu et al. demonstrated anticipated synchronization using theconfiguration shown in Figure 6.6(b).
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Synchronization of Unidirectionally Coupled Semiconductor Lasers 197
0 200 400 600–0.10
–0.05
0.00
0.05
0.10
I 1, I
2 (a
.u.)
(a)
0 10 20 30 40 50 60 70 80 90 100–0.10
–0.05
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I 1, I
2 (a
.u.)
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.u.)
(c)
0 50 100 150
Time (ns)
–4
–2
0
2
4
I 1, I
2 (a
.u.)
(d)
Figure 6.6 Numerical solutions of Equations (6.1)–(6.4) showing anticipated synchronization [55].The intensities of the master and slave lasers, I1 and I2, have been displaced vertically for clarity:I1 (I2) is the upper (lower) trace. The dotted line indicates the value of I1�t + � − �c) − I2�t�.� = 10 ns, and �c = 5 ns. (a) The lasers are identical and the noise level is zero. The parametersare: k = 500 ns−1, = 0�1, �n = 1 ns, � = 3, j = 1�01, �� = 3 rad, �m = 10 ns−1, �s = 5 ns−1, = 5 ns−1. (b) Simultaneous turn-on when the lasers are identical. The parameters are the same asin (a), (c) Same as (a), but the lasers have slightly different injection currents and optical frequencies�js = 1�0, � = 0�299GHz�, and the noise level is Dm = Ds = 1×10−4ns−1. �s = 0, = 10 ns−1, all otherparameters as (a). (d) Anticipated synchronization for larger injection current. The lasers are identicaland the noise level is zero. j = 2, �m = 2�5 ns−1, �s = 0�25ns−1, = 2�25ns−1, = 0�001, all otherparameters as (a).
Synchronization with a time lag �t = �c (Equation 6.6) corresponds to the synchronizationof the slave optical field with the injected field and has been referred to as isochronous orgeneralized synchronization [86]. Synchronization with a time lag �� = �c −� (Equation 6.8)corresponds to the case when the field of the slave laser anticipates the injected field byan anticipation time equal to � and it has been referred to as anticipated or completesynchronization [86].
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198 Chaos Synchronization
A numerical comparison of anticipated and isochronous synchronization was done in[74, 75]. It was shown that the two regimes exhibit different sensitivity to spontaneousemission noise, to parameter mismatches and they differ in the response of the slave laserto current modulation of the master laser. In the regime of isochronous synchronization aparameter region exists (for large coupling) so that there is good synchronization even whennoise and frequency detuning (of several gigahertz) are taken into account. In contrast, theregime of complete synchronization is easily destroyed by frequency detuning and noise.These results suggest that isochronous synchronization is a form of injection locking, wherethe slave intensity oscillations are phase locked to the master intensity oscillations, while theamplitudes of the oscillations remain functionally related.
The quality of the synchronization was measured with the correlation coefficients:
C1 =⟨[
Im�t + �1�−�Im�IIs�t�−�Is�]⟩
{⟨Im�t�−�Im��2
⟩ ⟨Is�t�−�Is��2
⟩}1/2 (6.11)
and
C2 =⟨[
Im�t + �2�−�Im�IIs�t�−�Is�]⟩
{⟨Im�t�−�Im��2
⟩ ⟨Is�t�−�Is��2
⟩}1/2 (6.12)
where �1 = −�c and �2 = � − �c. A large value of C1 implies good isochronoussynchronization, while a large value of C2 implies good anticipated synchronization.
The different sensitivity of the two regimes to frequency detuning is illustrated inFigures 6.7 and 6.8. Figure 6.7(a) displays the synchronization region when the lasers are
0 10 20 30
τo (ns)
0
1
2
3
S(τ o
)
0 10 20 30
τo (ns)
0
1
3
S(τ o
)
(a) (b)
2
Figure 6.7 Similarity function (a) for the same parameters as Figure 6.6(a); (b) for the sameparameters as Figure 6.6(c). S(�0) was calculated averaging over 100 time series with different noiserealizations [55].
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Synchronization of Unidirectionally Coupled Semiconductor Lasers 199
DigitalOscilloscope
PD1 BS
PD2 LD2
NDF
HWP O.I BS LD1 Mirror
NDF
MirrorO. IHWP
τ
τc
Pinj
|Es|2 |Em|2
Pext
(a)
3.0 3.2 3.4 3.61.4
1.6
1.8
2.0
time
lag
(ns)
delay time (ns)
(b)
Figure 6.8 (a) Experimental setup used in [84] to show complete synchronization. LD1 – masterlaser, LD2 – slave laser, PD – photodiode, OI – Optical isolator, BS – Beamsplitter, HWP – Half-wave plate and NDF – neutral density filter. Pinj (Pext) injection (feedback) power into the receiver(transmitter) laser. (b) Experimentally measured time lag as a function of time delay.
subject to the same feedback level (for large enough coupling isochronous synchronizationoccurs). The feedback level and bias current are such that both (identical) lasers operate inthe regime of coherence collapse. The horizontal axis is the frequency detuning between thelasers, the vertical axis is the optical injection rate, and the grey levels represent the value ofC1 (the dark grey represents large correlation). The synchronization region is asymmetric and
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200 Chaos Synchronization
broad, allowing for frequency detunings up to tens of gigahertz. Figure 6.7(b) displays thesame correlation coefficient as Figure 6.7(a) (and for the same parameter values), but whenthe slave laser is subjected to CW optical injection. The shapes of the chaotic synchronizationregion in Figure 6.7(a) and the CW injection-locking region in Figure 6.7(b) are very similar,thus suggesting that isochronous synchronization is an injection-locking-type phenomenon.It can also be observed that the chaotic synchronization region is broader than the CWinjection-locking region.
Figures 6.8(a) and 6.8(b) display the same as in Figures 6.7(a) and 6.7(b) but in the case ofan open-loop configuration (�s = 0). The horizontal axis is the frequency detuning betweenthe lasers, the vertical axis is the optical injection rate, and the grey levels represent the valueof C1. Again it can be observed that the chaotic synchronization region in Figure 6.8(a) andthe CW injection-locking-region in Figure 6.8(b) are very similar, suggesting that isochronoussynchronization is an injection-locking-type phenomenon. However, Fisher et al. [48]experimentally found that two semiconductor lasers in an open-loop unidirectional couplingscheme exhibit chaos-pass-filtering properties: a small perturbation superimposed on thechaotic output of the master laser is filtered out by the slave laser, which is selectivelysynchronized only to the chaotic dynamics. Thus, this contrasts with the interpretation ofisochronous synchronization as an injection-locking-type phenomenon, since in that case theslave laser would amplify both, the chaotic dynamics and the external perturbation in thesame way.
Comparing Figures 6.7(a) and 6.8(a) (that are done with the same grey-scale), it is clearthat the synchronization quality is in general lower when the slave laser does not have itsown external optical feedback. A possible explanation to this is based on the fact that, whenthe master and slave lasers are both external cavity lasers subjected to the same amount ofoptical feedback, under appropriate conditions an analytical synchronized solution exists inwhich the field of the slave laser lags by �c the field of the master laser [76]. When the slavelaser is a solitary laser, no such solution exists.
Thus, the synchronization with a close-loop scheme has advantages for applicationswhere a high degree of synchronization is required. However, it has the disadvantage thatadditional components have to be used in the experimental set-up. In particular, the externalmirror at the slave laser has to be very carefully positioned because small differences inthe external cavity feedback phases (�m� −�s�) can strongly degrade the synchronizationquality. Peil et al. [79] found experimentally and numerically a rich and complex set ofoutputs when varying the relative optical feedback phase. Their results demonstrate acharacteristic synchronization scenario in dependence on �m� −�s�, leading cyclically fromchaos synchronization to intermittent synchronization to almost uncorrelated oscillations,and back to chaos synchronization.
In the case of two identical lasers in an open-loop scheme, anticipating synchronizationoccurs for zero detuning and = �m. Thus, it is interesting to study the value of C2 in theparameter space (detuning, injection rate). This is displayed in Figure 6.8(c), which showsC2 on the same grey-scale as Figures 6.7(a) and 6.8(a). Comparing Figures 6.8(a) and 6.8(c)it is clear that the correlation coefficient C1 is usually larger than the correlation coefficientC2. However, in the region ∼�m and small detuning, C2 > C1. It can be observed thatthe dark region where good anticipating synchronization occurs is narrower than in theisochronous case.
The two regimes of synchronization also exhibit different sensitivity to the noise level. Thissensitivity is illustrated in Figure 6.9, which displays the synchronization diagrams: master
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Synchronization of Unidirectionally Coupled Semiconductor Lasers 201
output intensity, Im, vs. slave output intensity, Is. In Figure 6.9(a) the feedback levels are�s = �m and Im is shifted in time by −�c; In Figure 6.9(b) the feedback and injection levelsare related by +�s = �m and Im is shifted in time by � −�c. The noise level in both casesis the same, but the points in Figure 6.9(a) are clearly concentrated along a straight line,whereas in Figure 6.9(b) they are more scattered owing to noise-induced large bursts ofdesynchronization.
It has been shown numerically that by varying certain parameters a transition fromanticipated to isochronous synchronization can be observed [77, 86]. An example of suchtransition is illustrated in Figure 6.10, which shows the master laser and slave laser intensities(averaged in time to simulate the typical bandwidth of the detectors used in experiments).Figure 6.10(a) displays Im(t–�c), while Figures 6.10(b), 6.10(c) and 6.10(d) display Is(t)for different injection rates, . In Figure 6.10(b) = �m and Is�t� anticipates the injectedintensity Im�t − �c� by an anticipation time � (= 1 ns). In Figure 6.10(c) is slightly larger
C1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
–0.1
–0.2
–0.3
–0.4
Frequency Detuning fm – fs (GHz)
Inje
ctio
n R
ate
η (
ns–1
)
–30 –20 –10 0 10 20 300
5
10
15
20
25
30
35
40
45
50(a)
Figure 6.9 Comparison between isochronous synchronization and CW injection-locking when theslave laser is an external-cavity laser [88]. (a) Correlation coefficient C1 as a function of the frequencydetuning and the injection rate when the slave is an external cavity laser subjected to chaotic injectionfrom the master laser. �m = �s = 10 ns−1. (b) Correlation coefficient C1 as a function of the frequencydetuning and the optical coupling strength, when the slave is an external cavity laser subjected to CWinjection from the master laser. �s = 10 ns−1�
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202 Chaos Synchronization
C1
1.0
0.9
0.8
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–0.1
–0.2
–0.3
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Frequency Detuning fm – fs (GHz)
Inje
ctio
n R
ate
η (
ns–1
)
–30 –20 –10 0 10 20 300
5
10
15
20
25
30
35
40
45
50(b)
Figure 6.9 (continued )
than �m and isochronous synchronization occurs. The time traces shown in Figures 6.10(a)and 6.10(c) are most of the time equal, the main difference being a less pronounced dropoutin the intensity of the slave laser. In Figure 6.10(d) is slightly less than �m and synchroniza-tion disappears: the time traces shown in Figures 6.10(a) and 6.10(d) are completelydifferent.
Unidirectionally coupled external cavity semiconductor lasers operating on the LFF regimecan also exhibit intermittent synchronization. Wallace et al. [52] have shown that the outputsof master and slave lasers are synchronized in their sudden dropouts, but remain uncorrelatedin the process of power recovery. This is because during the recovery process the intensityof the master laser is very low and the optical injection from the master to the slave laser isnot strong enough to govern the dynamics of the slave laser.
Synchronizing several lasers becomes important when the lasers are used in practicalcommunication systems. Sivaprakasam and Shore experimentally demonstrated thepossibility of synchronizing three chaotic external cavity diode lasers in a cascadescheme [72]. The experimental set-up is shown in Figure 6.13. The master laser and boththe slave lasers are driven to LFF by application of appropriate feedback from the externalcavity. The effective external reflectivities for the master, slave-1 and slave-2 are 1�6×10−3,5�0 × 10−4, and 4�0 × 10−3 respectively. The free running threshold current (Ith) valuesare 56 mA, 49 mA and 55 mA respectively for master, slave-1 and slave-2 laser. The
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Synchronization of Unidirectionally Coupled Semiconductor Lasers 203
temperature and current are adjusted to ensure that the master (25�66 �C, 1�17Ith) and slave-1(26�35 �C, 1�20Ith) operate at the same wavelength. A small percentage (15%) of the masteroptical output power is fed to the slave-1, which leads to the synchronization of slave-1to the master. 6.5 percentage of the slave-1 output power is fed to the slave-2 laser. Thetemperature and operating current of the slave-2 (26�96 �C, 1�15Ith) are adjusted so as toobtain synchronization between the slave-1 and slave-2 lasers. The process of synchronizingslave-2 and slave-1 laser does not affect the synchronization between the slave-1 and masterlaser. Hence, all the three lasers are synchronized.
The time evolution of the intensities of the three lasers is shown in Figure 6.14: trace(a), (b) and (c) are the output intensities of master, slave (1) and slave (2) lasers respectively.It is noticeable from this figure that the output of slave laser (1) lags behind master laser,and slave laser (2) lags behind the slave laser (1).
C11.0
0.9
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0.0
–0.1
–0.2
–0.3
–0.4
Frequency Detuning fm – fs (GHz)
Inje
ctio
n R
ate
η (
ns–1
)
–30 –20 –10 0 10 20 300
5
10
15
20
25
30
35
40
45
50(a)
Figure 6.10 Comparison between isochronous synchronization, CW injection-locking, andanticipated synchronization when the slave laser is a solitary laser ��s = 0� [88]. (a) Correlationcoefficient C1 as a function of the frequency detuning and the injection rate, when the slave laser issubjected to chaotic injection from the master laser. �m = 10 ns−1. (b) Correlation coefficient C1 as afunction of the frequency detuning and the optical coupling strength, when the slave laser is subjectedto CW injection from the master laser. (c) Correlation coefficient C2 as a function of the frequencydetuning and the optical coupling strength, when the slave laser is subjected to chaotic injection fromthe master laser.
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204 Chaos Synchronization
C11.0
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–0.1
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Inje
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ate
η (ns
–1)
(b)
C21.0
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–0.4
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Frequency Detuning fm – fs (GHz)
Inje
ctio
n R
ate
η (ns
–1)
(c)
Figure 6.10 (continued )
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Synchronization of Unidirectionally Coupled Semiconductor Lasers 205
Figure 6.11 Effect of noise in the isochronous and anticipated synchronization regimes [76].(a) Im�t − �c) is plotted against Is�t�. The parameters are �m = �s = 20 ns−1, = 50 ns−1, �� = 0,D = 0�02 ns−1. C1 = 0�99 (b) Im�t +� −�c� is plotted against Is�t�. The parameters are �m = 20 ns−1,�s = 0, = 20 ns−1, �� = 0, D = 0�02 ns−1. C2 = 0�94�
0.06
0.08
1(a)
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1.00(b)
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1.00(c)
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0.06
0.08
1.00
t (ns)
(d)
I s(t
)I s
(t)
I s(t
)I m
(t –
τc)
Figure 6.12 Transitions between anticipated and isochronous synchronization are illustrated byplotting the time traces of the time-averaged master and slave intensities [88]. (a) Intensity of the masterlaser (lagged �c in time) for �m = 10 ns−1. Intensity of the slave laser for (b) = �m; (c) = 12 ns−1;(d) = 9 ns−1.
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206 Chaos Synchronization
M1
M2
M3
PD1
PD2
PD3
ML
SL1
SL2
OI1
OI2
OI3
OI4
OI5
BS1
BS2
BS3
BS4
BS5
BS6
BS7
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NDF2
NDF3
CA1
CA2
Figure 6.13 Schematic diagram of the experiment of cascade synchronization in Ref. [74]:ML – Master laser; SL1 – Slave laser-1; SL2 – Slave laser-2; M1– M3 – external cavity mirrors;NDF’s – neutral density filters, BS1–BS7 – Beam splitters, PD1–PD3 – Photodetectors, OI1–OI5 –Optical isolators, CA1 and CA2 – Coupling attenuator.
6.3 SYNCHRONIZATION OF MUTUALLY COUPLEDSEMICONDUCTOR LASERS
A lot of research has focused on mutually coupled lasers, either subjected only to mutualinjection, or subjected to mutual injection and optical feedback from an external mirror[106–117]. When the lasers are subjected only to their mutual optical injection andhave dissimilar intensities, their coupling strength may be asymmetric, and in this case,Hohl et al. [106] found that the coupled lasers exhibit a form of synchronization whichis characterized by small oscillations in one laser, and large oscillations in the other.Heil et al. [109] have found that two mutually coupled lasers may exhibit subnanosecondsynchronized chaotic dynamics, in which a leading laser synchronizes its lagging counterpart.
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Synchronization of Mutually Coupled Semiconductor Lasers 207
0 100 200 300 400 500
Time (nS)
(a)
(c)
(b)
Opt
ical
out
put (
arb.
uni
ts)
Figure 6.14 Cascade synchronization [74]. Time-series plots of (a) master laser, (b) slave laser-1and (c) slave laser-2. The time traces have been shifted in y-axis for clarity.
The time lag between the intensities is equal to the time for light to travel from one laserto the other. This effect was interpreted as a spontaneous symmetry-breaking phenomenon.Fujino and Ohtsubo also observed the synchronization of the chaotic outputs from mutuallyinjected lasers [111]. It was shown that the synchronization mechanism was not based oncomplete chaos synchronization but on injection-locking phenomena (which the authorsrefer to as generalized chaos synchronization). A recent theoretical study on synchronizationof mutually coupled multimode semiconductor lasers shows that the two lasers can besynchronized within each laser-cavity mode, while the synchronization across different cavitymodes is significantly weaker [112].
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208 Chaos Synchronization
The rate equations commonly used in the literature to model mutually coupled single-modelasers are
•E1�2 = 1+ j�
2
[G1�2�t�− 1
�p1�2
]E1�2�t�+�1�2E2�1�t − �c� (6.13)
•N1�2 = J1�2
e− N1�2�t�
�n�1�2
−G1�2�t�∣∣E1�2�t�
∣∣2 (6.14)
where E1 and E2 are the complex optical fields of the two lasers, N1 and N2 are the carrierdensities in the two lasers, and the other parameters have the same meaning as in the previoussection. These equations were derived in [116], in the limit of small mutual coupling, startingfrom Maxwell equations, supplemented with adequate boundary conditions.
In the experiments of [110] a different mutual coupling configuration was used. Theexperimental set-up is shown in Figure 6.15. Two single-mode (side mode suppression ratioequal to −20 dB), Fabry–Perot laser diodes emitting at 830 nm were used, but only one issubjected to optical feedback from an external reflector (this laser will be called external-cavity laser, or laser 1, while the laser without external feedback will be called solitary laser,or laser 2). The laser operating temperatures were stabilized using thermo-electric elements
ML
SL
BS1
BS2
BS3
NDF
OI1PD1
CA
M1
M2
PD2
OSAOI2
OI3
Figure 6.15 Experimental setup used in Ref. [110]. ML – external-cavity laser, SL – solitary laser,BS1–3, Beam Splitters, NDF – Neutral density filter, OI1–3, Optical Isolators, M1–2, Mirrors,CA – Coupling Attenuator, PD1–2, Photodetectors, OSA – Optical spectrum analyzer.
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Synchronization of Mutually Coupled Semiconductor Lasers 209
and controllers to a precision of 0.01 K. The output of each laser was coupled to a fastphotodetector (Newport - AD-70xr) and monitored using a digital oscilloscope (LeCroy-LC564A). The optical isolators ensure that no feedback from the photodetectors reaches thelaser diodes. Laser 1 (laser‘2) was biased at 1.08 (1.04) times the free-running threshold.The time of flight between the two lasers was 3.5 ns. Laser 1 was operated in an externalcavity configuration with external reflectivity 1�75 × 10−3, which drives the laser into theLFF regime (when the external mirror was removed, no power dropouts were observed).Beamsplitters BS1 and BS3 coupled the transmitter laser output to the receiver laser. Thecoupling attenuator (CA) was used to control the amount of light coupled between the lasers(the percentage of laser 1 output power reaching laser 2 and vice versa is 0.14% throughoutthe experiment). It is noted that, when the optical feedback from the external cavity mirroris not applied, the lasers did not show any LFF dynamics.
Figure 6.16 shows the time traces of the intensities of laser 1 and laser 2. It can beseen that close to 250 ns and 450 ns the output of laser 2 drops and recovers ahead of theoutput of laser 1. Hence, laser 2 is leading laser 1 by an ‘anticipation time’, �A = 3�5 ns,which was measured with the digital oscilloscope. The measurement of anticipation timewas confirmed by studying the quality of synchronization between the laser 1 and 2 outputs.The laser 2 output was plotted against laser 1 output so as to obtain the synchronizationplot. The synchronization plot was then least square fitted to a straight line and the slope (m)and its variation (�m) was calculated. The inverse of the variation (1/�m) represents thequality of the synchronization (SQ). Good synchronization would be indicated by m = 1and low variation (�m) implying high synchronization quality. On the other hand, poorsynchronization would give a relatively large variation (�m) and hence a low synchronization
200 300 400 500
Time (ns)
Out
put (
arb.
uni
ts)
Figure 6.16 Time traces of the external-cavity laser (upper) and solitary laser (lower) outputs [110].The vertical lines identify that the solitary laser is ahead of the external-cavity laser. The externalcavity round trip time is 13.5 ns.
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210 Chaos Synchronization
0.65
0.70
0.75
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0.85
0.90
0.95
1.00 τ = 13.5 nsτ = 6.7 ns
Sync
hron
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(SQ
)(N
orm
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ed u
nits
)
Transmitter laser time shift (τs), (ns)
0 1 2 3 4 5
Figure 6.17 Synchronization quality as a function of the time-shift in external-cavity laser output fortwo external cavity round trip times, � = 13�5 ns (�) and 6.7 ns (×) [110].
quality. Figure 6.17 shows the dependence of the (normalized) synchronization quality (SQ)on the laser 1 time shift �S , where �S is the time by which the laser 1 output is shifted relativeto the laser 2 output. It is shown for two external-cavity delay times. It can be seen that thesynchronization quality, in both cases, shows a sharp maximum at �S = 3�5 ns, which is theflight time from one laser to the other. This indicates that laser 2 is leading laser 1 by 3.5 nsirrespective of the external-cavity round-trip time. Thus, in mutual coupling a regime ofanticipating synchronization was found in which the anticipation time does not depend on theexternal-cavity round-trip time of the laser 1. Anticipating synchronization depends on theoperating wavelength of both lasers. Once the lasers are synchronized, both the lasers emit ata common wavelength denoted by �sys and it is found that �sys < �1 (where �1 is the centraloperating wavelength of laser 1). The operating temperature of one of the lasers is modified bya fraction of a degree such �sys < �1, and hence rendering laser 2 to lag behind laser 1 [114].
6.4 CONCLUSION
The aim of this chapter was to give an overview of the field of synchronization ofchaotic semiconductor lasers, from the theoretical and experimental points of view. InSection 6.2 we discussed synchronization of unidirectionally coupled lasers (in master–slaveor driver-response configuration). We considered the case in which the slave laser has itsown feedback (closed-loop scheme) and the case in which it is a solitary laser (open-loop scheme). We discussed the two well-known synchronization regimes: isochronous,which has been interpreted as an injection-locking-type phenomenon that requires thefrequency detuning between the lasers to be within a certain injection-locking range, andanticipated synchronization, which requires perfect matching of the optical frequencies andall the internal laser parameters. In terms of the well-known synchronization regimes ofchaotic systems, anticipated synchronization is identified with complete synchronization, andisochronous synchronization is identified with generalized synchronization. We reviewed
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recent, exciting developments such as inverse synchronization, which is a phenomenon notyet fully understood. In Section 6.3 we discussed synchronization of mutually coupled lasers,in a face-to-face configuration, and in the case when one of the lasers is also subjected to itsown feedback from an external mirror. We have chosen to discuss only the synchronizationof edge-emitting lasers and not of vertical-cavity surface-emitting lasers, because, in spite ofthe fact that there are several theoretical studies that indicate the possibility of synchronizingVCSELs, no experimental demonstration has been done to the best of our knowledge. Wehope that future developments of technology towards secure all-optical communicationsbased on chaotic synchronized systems will clarify relevant issues such as the physicalmechanisms underlying complete and generalized synchronizations.
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