Chaotic Modified Report

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Tutorials and ReviewsInternational Journal of Bifurcation and Chaos, Vol. 9, No. 11 (1999) 2129{2156c World Scienti_c Publishing Company

CHAOTIC COMMUNICATIONUSING TIME-DELAYED OPTICAL SYSTEMS

GREGORY D. VANWIGGEREN and RAJARSHI ROY_School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USAReceived August 20, 1998We discuss experimental demonstrations of chaotic communication in several optical systems. Ineach, an erbium-doped _ber ring laser (EDFRL) produces chaotic fluctuations of light intensityonto which is modulated a message consisting of a sequence of pseudorandom digital bits.This combination of chaos and message propagates at a wavelength of _ 1:5 microns throughstandard single-mode optical _ber from the transmitter to a receiver, where the message isrecovered from the chaos. We present evidence of the high-dimensional nature of the chaoticwaveforms and demonstrate chaotic communications through 35 km of single-mode optical

_ber at up to 250 Mbit/s, a rate that is, at present, limited only by the speed of our detectorelectronics.1. Introduction

A chaotic waveform that serves as a carrier ofinformation represents a generalization of the moretraditional sinusoidal carrier and o_ers the potentialfor enhanced privacy in communications. Inordinary radio communication, a speci_c frequencysine-wave carrier is modulated with a message andtransmitted. A radio receiver must be tuned tothe particular frequency of the carrier sine-wave inorder to recover the message. In conceptually thesame way, the experiments presented here demonstratethat information can be recovered from anoptical chaotic carrier using a receiver that is synchronized

or \tuned" to the chaotic dynamics of thetransmitter.The synchronization of chaotic systems plays animportant role in chaotic communications. The applicationof chaotic synchronization to secret communicationsystems was suggested in an earlierwork by Pecora and Carroll [1990, 1991]. They discoveredthat a chaotic transmitter could consist ofan electronic circuit that simulated the dynamics,for example, of the Lorenz model [Ditto & Pecora,1993]. The message to be concealed, assumed smallin magnitude, was added to the chaotic fluctuations,

assumed to be much larger, of one of the variables(let us choose the z variable for this purpose) andtransmitted to the receiver, while another chaoticvariable (let us choose x) was separately transmitted.The receiver consisted of a subsystem of thecircuits in the transmitter that generated the dynamicsof the y and z variables, and was drivenby the signal from the x variable of the transmitter.The receiver synchronized to the chaos of the


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transmitter if the conditional Lyapunov exponentsfor the systems were negative for the given operatingparameters. One could then recover the messagefrom the chaos through a subtraction at thereceiver.Cuomo and Oppenheim [1993] (see also

[Strogatz, 1994]) introduced an elegant variationof the method above that did not require theseparate transmission of a driving signal to thereceiver. They showed that the receiver couldactually synchronize to the chaotic dynamics of thetransmitter even when a message was added to the_E-mail: [email protected] G. D. VanWiggeren & R. Roychaotic driving signal from the transmitter. Thesynchronized output from the receiver was thenused to subtract out the information from the transmittedsignal. The synchronization was not perfect,and the message, treated as a perturbationof the chaotic signal, had to be small comparedto the chaos [Cuomo et al., 1993]. The developmentof techniques in which the message actuallydrives the chaotic transmitter system, in additionto being transmitted, was made by Wu and Chua[1993], Volkovskii and Rulkov [1993], and Parlitzet al. [1996]. The synchronization between receiverand transmitter can be exact, so message recoverycan be very accurate in principle. The experimentsreported in this paper are related in spiritto a method developed _rst in electronic systemsby Volkovskii and Rulkov [1993], who suggested theuse of an open-loop system in the receiver. A di_erent,adaptive approach to synchronization and securecommunications was introduced by Boccalettiet al. [1997].A proposal to use modulated unstable periodicorbits (UPOs) for secure communications and multiplexingwas made by Abarbanel and Linsay [1993].Multiplexing would be possible by using di_erentUPOs to carry di_erent messages. An alternate approachto chaotic communications with UPOs wasdeveloped by Hayes et al. [1993, 1994]. They symbolicallyencoded digital information into UPOs ofa chaotic system and used chaos control methodsto switch between di_erent orbits. This approachdoes not attempt to provide any privacy to theinformation being transmitted.The issue of privacy, however, arises naturallyin a discussion of chaotic communication andis an important motivation for chaotic communication


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research. In his pioneering paper, CommunicationTheory of Secrecy Systems, ClaudeShannon discussed three aspects of secret communicationsystems: concealment, privacy, and encryption([Shannon, 1949] see also [Hellman, 1977;Welsh, 1988]). These aspects apply to systems

that use chaotic waveforms for communication andcan be interpreted in that context. Concealmentof the information occurs because the chaotic carrieror masking waveform is irregular and aperiodic;it is not obvious to an eavesdropper thatan encoded message is being transmitted at all.Privacy in chaotic communication systems resultsfrom the fact that an eavesdropper must have theproper hardware and parameter settings in orderto decode and recover the message. In conventionalencryption techniques, a key is used to alterthe symbols used for conveying information. The

transmitter and receiver share the key so that theinformation can be recovered. In a chaotic communicationsystem, a transmitter that generates atime-evolving chaotic waveform acts as a \dynamicalkey" to transform the information symbols. Theinformation can be recovered with a receiver possessingthe same dynamical key, i.e. its con_gurationand parameter settings are matched to those ofthe transmitter. It is interesting to note that usinga chaotic carrier to dynamically encode informationdoes not preclude the use of more traditional digitalencryption schemes as well. Dynamical encoding

with a chaotic waveform can thus be considered anadditional layer of encryption.Two factors that are important to privacy considerationsin chaotic communication systems arethe dimensionality of the chaos and the e_ort requiredto obtain the necessary parameters for amatched receiver. Earlier work has shown that forcertain chaotic communication techniques, particularlythose involving additive masking of a messageby a chaotic carrier, the message can be recoveredfrom the transmitted signal by mathematically reconstructingthe transmitter's chaotic attractor if

the chaos is low-dimensional [Short, 1994a, 1994b;Perez & Cerdeira, 1995]. Higher-dimensional signals,especially those involving hyperchaotic dynamics,are likely to provide improved security. Thenumber of parameters that have to be matched forinformation recovery and the precision with whichthey must be matched are important aspects of receiverdesign. We will show how the con_gurationand operation of the receiver may be designed to


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make it more suitable for private communications.At this point we would like to emphasize that thesecurity of communication techniques is a complexand involved issue. In the work reported here, wedo not make any claims of secure communications.Indeed, we do not know of any systematic cryptographic

approach that has been taken to examinethe security of di_erent chaotic communication systems.We regard this as a very important openproblem for future analysis.Most realizations of chaotic communicationshave occurred in electronic circuits that simulatethe dynamics of simple model systems (Lorenz,Rossler, double scroll or Chua system, etc.), even forthe case of hyperchaotic systems. Peng et al. [1996]theoretically examined the question of synchronizationof hyperchaotic Rossler systems and showedChaotic Communication Using Time-Delayed Optical Systems 2131

that synchronization is successfully achieved over awide parameter range by using a transmitted signalthat is a linear combination of the original phasespace variables. Mensour and Longtin [1998] havestudied the synchronization of hyperchaotic systemsdescribed by Mackey{Glass delay-di_erential equations,with their use for private communications.Optical systems present a somewhat di_erentsituation; one often does not know a prioriwhat the equations are that should be used tomodel the system. Rather, insight into the formulationof appropriate models must be gained

through experimental observations of the systemdynamics. We follow this approach throughout thispaper. Our research into chaotic communicationsusing optical systems began when we experimentallyachieved chaotic synchronization of two mutuallycoupled Nd:YAG (neodymium doped yttriumaluminum garnet) lasers that were operated side byside in a single YAG crystal [Roy & Thornburg,1994; Sugawara et al., 1994]. We then proposed ascheme for digital communication with a transmitterlaser unidirectionally coupled to a distant laserused as a receiver [Colet & Roy, 1994]. The chaotic

dynamics for these systems is not high-dimensional[Alsing et al., 1997], and we showed that the messagecould be recovered fairly easily by time-delayembedding of the signals, unless one introducedcomplex modulations to increase the dimensionalityof the dynamics. The dynamics present in suchlasers is also rather slow, on the microsecond timescaleat best.Our next step was to examine a laser system


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that would provide not only a high-dimensional dynamics,but also a much greater bandwidth andhigh speed for signal transmission and recovery.This led us to the study of erbium doped _ber ringlasers (EDFRL); we developed a simpli_ed modelfor the operation of an EDFRL in [Williams et al.,

1997]. These lasers emit at 1.53{1.55 microns, thewavelength regime of choice for optical communicationsin _bers, which have minimum loss in thisrange. The emission is broadband (covering manynanometers), and we found the dynamics present inthis system to be extremely fast. We estimate thebandwidth of the light intensity fluctuations to bein the range of many gigahertz, but the bandwidthof our electronic detection equipment preventsdirect observation of the higher frequencies.Ring cavity optical systems with nonlinear elements,as pointed out by Ikeda and coworkers many

years ago [Ikeda et al., 1980; Ikeda & Matsumoto,1987], can possess very high-dimensional chaosresulting from the operation of the intra-cavitynonlinear elements and time-delayed feedback.Depending on the setting of the operating parametersof the erbium-doped _ber ring laser system, onecan observe both low- and high-dimensional dynamics[Ikeda et al., 1980]. Simulations of Ikeda typering systems by Abarbanel and Kennel reveal dimensionsof order twenty-_ve or higher [Abarbanel& Kennel, 1998]; they have also numerically demonstratedthe synchronization of two ring cavities via

unidirectional coupling. The dimensionality of thechaos in our experiments has been analyzed fromthe measured data using a false-nearest-neighbors(FNN) algorithm [Abarbanel, 1996] and found tobe of order 10 or higher. Erbium-doped _berring lasers therefore simultaneously o_er the advantagesof high-dimensional dynamics and high-speedcommunications.Very recently, we reported the _rst experimentson all-optical chaotic communication of 10 MHzsquare waves using erbium-doped _ber lasers andampli_ers [VanWiggeren & Roy, 1998]. A very

interesting experiment on chaotic communicationswas reported simultaneously by Goedgebuer andcolleagues [1998], who used a hybrid electro-opticsystem to encode a 2 kHz sine-wave in the chaoticwavelength fluctuations of a tunable semiconductorlaser and then decoded the information withan open-loop receiver. They also used a timedelaysystem [Larger et al., 1998] to generate highdimensionalchaotic dynamics.


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The experiments described in this paper takebetter advantage of the large bandwidth availablein an EDFRL, demonstrating communicationof return-to-zero (RZ) pseudorandom digital bitsat rates of 125 Mb/s. Results from an experimentshowing communication of a non-return-tozero

(NRZ) digital message at 250 Mbit/s are alsodiscussed. Currently, the bit-rate is limited by thebandwidth of our detection equipment. Though thehigh-dimensional dynamics of an EDFRL systemo_er the potential for privacy, we note that manyof the experiments and techniques discussed in thispaper are unlikely to be useful for private communications;rather, descriptions of those experimentsare included because they both established and furtheredour understanding. The more recent experimentsdescribed in this paper, however, do seemto o_er the possibility for enhanced privacy in communication.

A discussion of the privacy aspects ofeach method is included.2132 G. D. VanWiggeren & R. RoyAll of the communication experimentsdescribed in this paper use an EDFRL to generatechaotic waveforms and employ unidirectionalcoupling of the transmitter to the receiver. Eachtechnique described utilizes an open-ring receiverdesigned to mimic the dynamics of light propagatingonce through the transmitter ring. The messageto be communicated modi_es the chaotic dynamicsof the transmitter, and by comparing the output

dynamics from the transmitter to the open-ringreceiver dynamics, the message can be recovered.The message itself drives the chaotic dynamics ofthe transmitter. Thus, the manner in which themessage is dynamically encoded is at least partlydependent on earlier portions of the message.We took two distinct approaches in developingchaotic communication in EDFRLs. The _rstapproach involves coupling an optical message signalfrom an external laser directly into the EDFRL.In the second approach, an intensity modulator isinserted within the ring laser itself to encode the

message directly onto the chaotic carrier. A developmentof this technique incorporates two timedelaysinto the dynamics of the transmitter, makingthe information transmission more private.A feature of all of these communication systemsis that there is no encoding of symbols into unstableperiodic orbits of the chaotic system, as hasbeen proposed and demonstrated recently [Hayeset al., 1993, 1994]. Thus there is no loss of speed


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in communication that could occur through the useof higher-order orbits for symbolic coding or transientsto synchronization in closed-loop communicationsystems.This paper is organized as follows. Section 2contains a description of the experimental system

used for message injection from an external tunablesemiconductor laser system and dynamical encodingwith a chaotic EDFRL. The technique usedto recover the digital information is outlined. Theresults of experiments that use polarization orwavelength _lters in the EDFRL are given inSecs. 3.1.1{3.1.3. These experiments enabled usto reduce the interference between message andchaotic lightwaves and revealed many features of thewavelength and polarization dynamics of the lasersystem. In Secs. 3.2.1 and 3.2.2 the results of messageinjection experiments performed without any

wavelength or polarization _lters are given. Thepossibility of dynamical encoding and wavelengthmultiplexing are examined. Section 4 contains a discussionof the privacy and consistency of informationrecovery of the message injection techniques. Adi_erent approach for dynamical encoding with anintra-cavity modulator is described in Sec. 5, andthe results of these experiments are given in Sec. 6.No external laser for message injection is neededfor this technique, and the encoded information isrecovered by a division of signals in the receiver.A modi_ed intra-cavity modulation technique that

adds a second _ber loop (and hence a second timedelay)to signi_cantly enhance the privacy of theinformation is presented in Sec. 7 and the resultsare given in Sec. 8. We show that information isdynamically encoded and consistently and clearlyrecovered at 125 Mbits/s (return-to-zero, or RZformat) or 250 Mbits/s (non-return-to-zero, or NRZformat) and examine the result of mismatch of receivercon_guration or parameters. A discussion ofthe bit-error rate (< 105) and results for communicationthrough more than 35 km of _ber (includingan eye-diagram) is given. Section 9 contains a discussion

of the modi_ed intra-ring modulator techniqueincluding results of a false nearest-neighborsestimate of the dimensionality of the chaotic dynamicsfor this technique. Section 10 concludesthe paper with a brief recapitulation of goals andresults. We _nally mention some open questionsthat must be addressed before practical implementationof these ideas.2. Experimental Setup and


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Operation Using MessageInjectionSeveral variations of the message injection approachare described _rst, but their general form is shownin Fig. 1. To create the optical message, light producedby an external-cavity tunable diode laser isamplitude modulated by a lithium niobate Mach{Zehnder interferometer to form a sequence of pseudorandombits. After modulation, the lightwave isampli_ed in a controllable fashion by an erbiumdoped

_ber ampli_er (EDFA) with a 13 dBm maximumoutput power. This step governs the amplitudeof the message injected into the EDFRL. Theampli_ed message then passes through a polarizationcontroller consisting of a series of waveplates(_=4; _=2; _=4) arranged to permit complete controlover the polarization state of the message as itis coupled into the ring.

The message light is injected into the EDFRLthrough a 90/10 waveguide coupler, which allowsChaotic Communication Using Time-Delayed Optical Systems 2133

_ ____ _ _ _ _ _ _ _ __ ______ " $% & ' ) +, - & /1234 5 6789:; < = ?@< BPC DQ FRG SHT IQK GVV WM ITD OXY [ ]^ ' ac c f Yg hi j hmw nx pr s u vyz | z } ~ z . . . .. . . . . . . . . . . .. . . . . . .. . . . . .

Fig. 1. Experimental system for optical chaotic communicationconsisting of three parts. In the message modulationunit, cw laser light is intensity modulated to produce a messagesignal consisting of a series of digital bits. The messagesignal is injected into the transmitter where it is mixedwith the chaotic lightwaves produced by the erbium-doped_ber ring laser. The message, now masked by the chaoticlight, propagates through the communication channel to thereceiver where the message is recovered from the chaos.

10% of the message light to be injected into thering, while retaining 90% of the light already inthe ring. The light propagates to a 50/50 outputcoupler that sends half the light in the ringto the receiver unit, while the other half passesthrough EDFA. This EDFA has 17 dBm maximumoutput power and 30 dB small signal gain. Theactive (doped) _ber in the EDFA is 17 m long andis pumped by diode lasers with a 980 nm wavelength.After passing through the EDFA, the lightthen travels through a _lter consisting of a polarizationcontroller, again a series of waveplates,

and either a polarizer or bandpass _lter dependingon the variation of the method being used. Thetotal length of the active and passive _bers in thering totals approximately 40 m, which correspondsto a round trip time for light in the ring of about200 ns. Isolators in the EDFA ensure that lightpropagates unidirectionally in the ring, as indicatedin Fig. 1. Unless otherwise speci_ed in the descriptions


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of subsequent experiments, the transmitterEDFRL is operated at pump powers more than 10times threshold. Typical optical powers in the ringare between 10{40 mW depending on the losses inthe ring.Light exiting the ring through the 50/50

output coupler propagates in an optical _ber tothe receiver unit. Light entering the receiver unitis split at the 90/10 coupler. Ten percent passesthrough a variable attenuator that prevents photodiodeA (125 MHz bandwidth) from saturating.The other 90% of the light passes through EDFA 2and another _lter. EDFA 2 and the receiver _lterare intended to be replicas of EDFA 1 andthe _lter in the transmitter so that the receivercan synchronize to the dynamics of the transmitter.After passing through EDFA 2 and the receiver

_lter, the light passes through an attenuator and is

measured by photodiode B (125 MHz bandwidth).The signals from photodiodes A and B are recordedby a digital oscilloscope with a 1 GS/s sampling rateand 8-bit resolution.A model for the EDFRL without message injectionis given in [Williams et al., 1997]. It consists oftwo delay equations for the two polarizations of theelectric _eld and one di_erential equation for thepopulation inversion. The functional forms of theseequations, including message injection, areET (t) = f[ET(t_R); N(t); m(t_R)] (1)and

_N(t) = g[ET(t); N(t)] : (2)f and g are nonlinear functions of the populationinversion, N(t), and the complex slowly varying envelopeof the chaotic electric _eld in the transmitter,ET (t). The message is represented as m(t), andthe time it takes light to make a round-trip in theEDFRL is _R. Equation 1 shows that the electric

_eld ET (t) at any point in the cavity is functionallydependent on the electric _eld and message,ET (t _R) and m(t_R), at that point one roundtrip earlier.

Conceptually, the method we use is similar toone described by Volkovskii and Rulkov [1993], but2134 G. D. VanWiggeren & R. Roy0 100 200 3000.70.80.911.11.2


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1.3Time (ns)Photodiode A (V)Transmitted signal without messageA0.71

1.30.711.30.711.3BI(t)Timedelay embedding of transmitted signalI(t+T)I(t+2T)0 100 200 3000.7

0.80.911.11.21.3Time (ns)Photodiode B (V)Receiver outputC0.7 0.9 1.1 1.30.70.91.1

1.3Synchronization plotTransmitted intensityReceived intensityD4 6 8 10012345DimensionPercentage of FNNs

Falsenearestneighbor analysis of attractor dimensionEFig. 2. (A) shows the transmitted signal measured by photodiode A when no message is injected into the transmitter.(B) shows a time-delay-embedding plot of the data in (A); the lack of structure indicates that the data is not low-dimensional.(C) shows the signal simultaneously detected by photodiode B. The signals recorded by the photodiodes are clearlysynchronized,as shown in (D). (E) gives the results of a false-nearest-neighbors analysis. It indicates a dimension of _8 for thedynamics of the transmitted signal.it has been modi_ed to incorporate time-delays, opticalphase, and polarization e_ects. The light that


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is transmitted from the EDFRL to the receiver unitis s(t) = ET(t) + m(t). If the injected messagepower is not too large (typically on the order of afew milliwatts or less), the message is \masked" bythe larger (10{40 mW unless otherwise speci_ed)chaotic intensity fluctuations. For simplicity, imagine

that photodiode A detects its fraction of s(t)at the same moment that the remaining fraction ofChaotic Communication Using Time-Delayed Optical Systems 2135s(t) is incident at EDFA 2. The ampli_er and _lterthen operate on the remaining fraction (_90%) ofs(t) to produce the waveform ER(t + _R), whicharrives at photodiode B with a time-delay equalto one round-trip time in the transmitter, _R. Atprecisely this moment, photodiode A is detectings(t + _R) = ET(t +_R) +m(t + _R). Note thatET (t+_R) is produced when EDFA 1 and the _lter(in the EDFRL) operate on s(t) = ET(t) +m(t).

Because EDFA 1 and the _lter in the transmitteroperate on light in the same way as EDFA 2 and thereceiver _lter, ER(t+_R) = ET(t+_R). Mathematically,this occurs because f and g are the same inboth systems, the systems have negative conditionalLyapunov exponents, and they display synchronizationin a global sense [Abarbanel & Kennel, 1998],The two photodiodes, therefore, measure the intensities

jET (t + _R) +m(t+_R)j2 and jER(t + _R)j2respectively. The di_erence of these two measurementsat any time is 2RefE_T (t+_R) _m(t+_R)g+

jm(t + _R)j2:

When no message is transmitted, it is clear fromthe preceding discussion that the transmitter andreceiver should be synchronized. In other words,the two photodiodes should measure the same intensities,

jER(t+_R)j2 = jET (t+_R)j2: The resultsof an experiment in which no message was transmittedare shown in Fig. 2. Figure 2A shows thesignal from the transmitter with the laser operatedfar above threshold (greater than ten times thepump power required for threshold). A time-delayembedding of the data in Fig. 2A is given in Fig. 2B;it shows that the chaos is not low-dimensional.

Figure 2C shows the signal recorded by photodiodeB. The straight line shown in Fig. 2D demonstratesthe synchronization of the light intensitiesas measured by photodiodes A and B. We used anumerical false-nearest-neighbors (FNN) algorithm[Abarbanel, 1996] in an attempt to estimate thedimensionality of the signal. The results are presentedin Fig. 2E. The dimensionality of a timeseriescan be estimated by observing the dimension


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at which the percentage of false-nearest-neighborsgoes to zero. Interestingly, we observe that for theanalysis of 250 000 points acquired at 1 GS/s, theFNN algorithm gives a dimension of 8. The limitedbandwidth of our photodiodes does not allowmeasurement of the much higher frequencies that

may be present in the real signal. Consequently,there may be additional attractor dimensions thatdo not manifest themselves in the data we obtain.Figure 2E shows that the natural dynamics of thering laser are not low-dimensional for the operatingparameters used, even without the presence of aninjected message to influence the dynamics of thering laser.3. Results for ExperimentsUsing Message InjectionTwo basic techniques using an injected opticalmessage have been investigated. The _rst technique

follows from the description above, but a generalized\_lter" is used in both the transmitter and receiverunits. Several variations of this techniqueare presented. The second technique is conceptuallysimilar to the _rst, but no generalized _lter isnecessary.3.1. Message injection techniquewith \_lter"3.1.1. Polarization discriminationwith in-ring polarizerIn our _rst variant of this method, m(t) is injectedinto the EDFRL with a polarization orthogonal toET (t) but with the same wavelength. The polarizationof the injected m(t) is determined using thepolarization controller in the message modulationunit. The _lter in the EDFRL transmitter is apolarizer. The polarization controller and polarizerin the EDFRL are aligned such that the m(t)in the EDFRL is \blocked" upon reaching the _lter,while still allowing an orthogonally polarizedET (t) to pass through relatively unchanged. In thismanner, m(t) is prevented from circulating in theEDFRL and mixing with ET (t) on more than onepass through the EDFA, though it is transmittedto the receiver unit. The _lter in the receiver unitis also aligned to prevent light polarized in thesame direction as m(t) from reaching photodiode B.Subtracting the signals at the photodiodes, as mentionedearlier, gives 2 RefE_T (t) _ m(t)g + jm(t)j2:The cross-term is eliminated in this techniquebecause ET (t) andm(t) are orthogonally polarized.This leaves just jm(t)j2 after subtraction.


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Figure 3 shows the results of an experimentusing this technique. Figure 3A is a time-traceof the transmitted signal, s(t), as measured byphotodiode A, and Fig. 3B is its power spectrum.No hint of the message can be discerned in the2136 G. D. VanWiggeren & R. Roy

0 100 200 3000.40.50.60.70.80.91Transmitted signalTime (ns)Photodetector A (V)A0 100 200 300105

100Frequency (MHz)dBPower spectrum of transmitted signalB0 100 200 3000.40.50.60.70.80.91Time (ns)

Photodetector B (V)Signal in receiver unitC0 100 200 3000.10.0500.050.10.15Recovered messageTime (ns)Difference of two photodetectors (V)E0 100 200 300105100Frequency (MHz)dBPower spectrum of received signalD1.52 1.53 1.54 1.55 1.56706050


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403020100Optical spectrum of transmitted signalWavelength (m)

dBmFFig. 3. (A) shows the transmitted signal measured by photodiode A when a message is injected into the transmitter.Itspower spectrum can be seen in (B). (C) shows the detected signal at photodiode B, and its power spectrum is alsogivenby (D). Subtracting the signals shown in (A) and (C) results in the recovered message shown in (E). An opticalspectrum isshown in (F) revealing a single peak. (F) demonstrates that both the message and chaotic light share the samewavelength.time-trace of the signal. The broadband nature ofthe transmitted signal is evident from the powerspectrum. The spectrum's gradual decline with increasingfrequency matches the spectral responseof our photodiodes (125 MHz bandwidth) and resultsfrom this limitation. Figure 3C is a time-traceof the signal measured by photodiode B, and itspower spectrum is shown in Fig. 3D. The powerspectrum of the received signal lacks the smaller,

_ne peaks of the transmitted signal. As can beseen in our next _gure, those _ne peaks are justthe power spectrum when repeating the sequenceChaotic Communication Using Time-Delayed Optical Systems 21370 50 100 150 2000.050

0.050.10.15Directly measured bitsno chaosPhotodiode A (V)Time (ns)A0 50 100 150 200 250105100Power spectrum of directly measured bitsdBFrequency (MHz)B

0 50 100 150 2000.0500.050.10.15Recovered messageDifference of two photodectors (V)Time (ns)C


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0 50 100 150 200 250105100Power spectrum of recovered messagedBFrequency (MHz)D

Fig. 4. A comparison showing the quality of the message recovery. (A) shows a portion of the message directlydetected atphotodiode A when the ring laser is turned o_, i.e. ET (t) = 0. No chaotic encryption is used. The power spectrum ofthedirectly detected message is shown in (B). The same message recovered from the chaotic transmitted signal, i.e. E T(t) 6= 0, isshown in (C). (D) shows the power spectrum of the recovered signal. The recovered message is somewhat degradedbut stilldecipherable.of bits. Figure 3E results from a subtraction ofthe waveform data in Fig. 3C from the data inFig. 3A. The \random" bits are clear and matchthe 125 Mbit/s pattern used for this experiment:

11001101110110100100101111101010. Figure 3F isthe optical spectrum of the transmitted signal s(t)showing that both m(t) and ET(t) have the samewavelength, _1.532 _m.Figure 4 is included as a measure of theaccuracy of this technique. For comparison,Fig. 4A shows a segment of the message sequenceas detected by photodiode A in the absence ofchaos (EDFA 1 is turned o_ and ET (t) = 0).Figure 4B is simply the power spectrum of the entirerecorded message sequence. Figure 4C showsa portion of the message obtained through thechaotic subtraction. Clearly, the recovered messageis degraded somewhat, probably due to imprecisionin matching the polarization-based _lters in boththe transmitter and receiver. The power spectrumof the recovered message, a segment of which isshown in Fig. 4C, is shown in Fig. 4D. The peakcorresponding to the 125 MHz bit-rate is evidentin Fig. 4B.3.1.2. Wavelength discrimination within-ring polarizerIn the preceding technique, the wavelengths of m(t)and ET (t) were the same, but their polarizations

were orthogonal. In this method, however, thewavelengths are di_erent, but their polarizationsare the same. The _lter in the EDFRL againconsists of the polarization controller followed bya polarizer. The polarization controller in themessage modulation unit is adjusted so that the2138 G. D. VanWiggeren & R. Roy0 100 200 3000.40.5


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0.60.70.80.91Transmitted signalTime (ns)

Photodetector A (V)A0 100 200 300105100Frequency (MHz)dBPower spectrum of transmitted signalB0 100 200 3000.40.50.60.7

0.80.91Time (ns)Photodetector B (V)Signal in receiver unitC0 100 200 3000.10.0500.050.10.15

Recovered messageTime (ns)Difference of two photodetectors (V)E0 100 200 300105100Frequency (MHz)dBPower spectrum of received signalD1.53 1.54 1.55 1.56706050403020100Optical spectrum of transmitted signalWavelength (m)dBmF


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Fig. 5. (A) shows the transmitted signal measure by photodiode A, while (C) shows the signal measuredsimultaneously byphotodiode B. Their power spectra are given in (B) and (D) respectively. Again, subtraction of the two time-tracesrevealsthe message, shown in (E). An optical spectrum, (F), shows that the message light and chaos have di_erentwavelengths.polarizations of m(t) and ET(t) are parallel as

they are transmitted from the EDFRL to thereceiver unit. Again, the _lter's purpose in theEDFRL is to \block" light associated with m(t)from continuing to circulate in the ring. A momentof explanation will be useful here.m(t) and ET(t) have the same polarization atthe coupler where the message is injected. As thelight propagates around the ring, the di_erence inwavelength, typically _20 nm, results in a di_erentoutput polarization state for light at the twowavelengths. We observe that this phenomenonChaotic Communication Using Time-Delayed Optical Systems 2139

0 100 200 3000.40.50.60.70.80.9Transmitted signalTime (ns)Photodetector A (V)A0 100 200 300105100

Frequency (MHz)dBPower spectrum of transmitted signalB0 100 200 3000.40.50.60.70.80.9Time (ns)Photodetector B (V)Signal in receiver unitC

0 100 200 3000.0500.050.10.15Recovered messageTime (ns)Difference of two photodetectors (V)E


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0 100 200 300105100Frequency (MHz)dBmPower spectrum of received signalD

1.550 1.555 1.560706050403020100Optical spectrum of transmitted signalWavelength (m)dBmFFig. 6. (A) and (C) show simultaneous signals measured by photodiodes A and B, respectively. (A)'s power spectrum

isshown in (B), and (C)'s power spectrum is given in (D). Subtracting the signal shown in (C) from the signal shown in(A)gives the recovered message shown in (E). (F) An optical spectrum shows that the wavelengths are not the same,but aremuch closer than they had been in the experiments performed in Fig. 5.of polarization dispersion occurs primarily in theerbium-doped _ber in the EDFA. At the _lter, m(t)is almost completely orthogonal to ET (t). Thus,m(t) can be \blocked" by the polarizer while ET (t)continues to circulate. The method works as describedabove, but this time the cross-term averagesto zero because m(t) and ET(t) have di_erent

optical frequencies; once again, only jm(t)j2remains after subtraction of the two photodiodesignals.2140 G. D. VanWiggeren & R. RoyFigure 5A shows the signal transmitted fromthe EDFRL to the receiver unit as detected byphotodiode A. Its power spectrum is shown inFig. 5B, and again, the spectrum is quite broadband.The signal recorded by photodiode B isshown in Fig. 5C, and its corresponding power spectrumis given in Fig. 5D. Subtracting the two timetracesresults in the trace displayed in Fig. 5E andshows excellent reproduction of the bits. Finally,Fig. 5F shows the optical spectrum of the transmittedsignal, revealing a narrow peak corresponding tom(t) at 1:532 _m and a broader peak correspondingto ET (t) at 1:555 _m.3.1.3. Wavelength discrimination within-ring band-pass _lterAnother method investigated was the use of bandpass

_lters rather than polarizers as _lters. Once


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again, the purpose of the _lters was to preventlight associated with m(t) from continuing tocirculate around the ring. The wavelength of themessage light could be adjusted to _1 nm of thecentral wavelength of the _lter and still be successfully

\blocked". The _lter in the receiver was tuned

to match the _lter in the transmitter. The electric_eld polarization, because no polarizer was inthe ring, fluctuates very rapidly and e_ectively hasno polarization, and consequently, the polarizationstate of the message is not important. Because thewavelengths of the message, m(t), and the chaoticelectric _eld, ET (t), are still substantially di_erent,the cross-term in the subtraction of the photodiodesignals averages to zero. Again, only jm(t)j2remains.Figure 6A shows the transmitted signal, s(t),recorded at photodiode A. Its power spectrum is

shown in Fig. 6B. The signal detected at photodiodeB is shown in Fig. 6C, and its power spectrumis shown in Fig. 6D. Because the chaotic outputhas become almost periodic in this example, a comparisonof Figs. 6A and 6C clearly shows the e_ectof the bits on the transmitted signal. However, itis still not obvious at all from observing just thetransmitted signal that a message is included. Asubtraction of the two signals is shown in Fig. 6E.The optical spectrum of the transmitted signal issupplied in Fig. 6F. The wavelength separation inthis case is much smaller than in the previous case,

and is approximately 1 nm.0 50 100 150 200 250 30000.511.5Time (ns)Photodiode voltage (V)Transmitted and received signalA0 50 100 150 200 250 3000.20

0.20.40.6Time (ns)Difference of photodiode signals (V)Recovered messageB1.53 1.54 1.55 1.5610080


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6040200Wavelength (nm)dBm

Optical spectrumCFig. 7. (A) shows both the transmitted signal measured byphotodiode A (thin-line) and the signal measured by photodiodeB (thick-line). The thick-line in (B) is the message signaldirectly detected by photodiode A when the transmitter ringlaser is turned o_, and is included to verify that the messagerecovered from the chaos (thin-line) is indeed correct. Anoptical spectrum of the transmitted signal is shown in (C) toshow that both the message and chaotic light have the samewavelength.Chaotic Communication Using Time-Delayed Optical Systems 21413.2. Message injection techniquewithout \_lter"

3.2.1. Message wavelength attransmitter's resonantwavelengthThe techniques discussed in Sec. 3.1. required theuse of a _lter and that a distinction of polarizationor wavelength be made between the chaoticelectric _eld, ET (t), and the message signal, m(t).The techniques presented in this section will demonstratemessage communication without the use of

_lters and with the same wavelength for the messageand chaotic electric _eld.0 100 200 300 400

108

106104102100102dBPower spectrum of signal at photodiode AFrequency (MHz)A0 100 200 300 400108106104102100102dBPower spectrum of signal at photodiode BFrequency (MHz)BFig. 8. Power spectra corresponding to the signals measuredin Fig. 7A. (A) is the power spectrum for the transmitted signalrecorded at photodiode A. (B) shows the power spectrumfor the signal measured by photodiode B.


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The time-traces from such an experiment areshown in Fig. 7A. In this example, both EDFA 1and EDFA 2 are pumped by their diode pump lasersat 10 mW, a level slightly less than twice the thresholdpump power for the ring laser. This pumpingresults in_1 dBm optical power circulating in

the ring when no message is being injected. Theinjected message has a power of_4:5 dBm and a1553.01 nm wavelength. The _rst panel, Fig. 7A,shows data taken from photodiode A (the thin line)and from photodiode B (the thick line). As explainedearlier, a subtraction of the two signals isequal to 2 RefE_T (t) _ m(t)g + jm(t)j2. Figure 7Bshows that subtraction (thin line). For comparison,the thicker line in Fig. 7B is the message signalmeasured by photodiode A when the EDFRL isturned o_ to remove the chaotic masking. This isequivalent to measuring just jm(t)j2. The greater

amplitude of the \decoded" message is simply theresult of the cross term 2RefE_T (t) _m(t)g; in thiscase, the polarizations and phases of ET (t) andm(t) (relevant in the cross-term) have combined toimprove the message reception. The _delity is quitegood. An optical spectrum of the transmitted lightis shown in Fig. 7C. The message injection is strongenough that its optical frequency forces the EDFRLto have the same lasing frequency | 1553.01 nm.Figure 8 provides power spectra for thetransmitted signal (Fig. 8A) and for the signal atphotodiode B (Fig. 8B); they have a very close

resemblance. The narrowly-spaced discrete spikesresult from the repetitive 32-bit pattern used. Thebit-rate was 125 Mbit/s; the corresponding spike isevident in both power spectra.3.2.2. Message wavelength di_erentfrom transmitters resonantwavelengthWe also performed experiments with messageinjection at wavelengths which were not resonantwith the lasing wavelength of the EDFRL.Figure 9A shows signals measured by photodiodes Aand B when the wavelength of the injected message

is 1533.01 nm. In this case the EDFAs were pumpedat about 85 mW, many times threshold. This resultedin an optical power in the ring of _9:1 dBmwithout any message injection. The injected messagepower was_3:1 dBm. The subtraction ofthe traces in Fig. 9A is seen in Fig. 9B. Once again,the same pattern of bits is obtained. The opticalspectrum (Fig. 9C) shows two distinct peaks. The2142 G. D. VanWiggeren & R. Roy


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0 50 100 150 200 250 3000.40.60.81Time (ns)Photodiode voltage (V)Transmitted and received signalA0 50 100 150 200 250 3000.100.10.20.3Time (ns)Difference of photodiode signals (V)Recovered messageB

1.53 1.54 1.55 1.56100806040200Wavelength (nm)dBmOptical spectrumCFig. 9. (A) once again shows both the transmitted signalmeasured by photodiode A (thin-line) and the signal measuredby photodiode B (thick-line). (B) shows the resultsof subtracting the thickline from the thin-line. (C) givesan optical spectrum showing the lasing wavelengths of theEDFRL. The message injection is occurring at a wavelengthof _1:533 _m.0 50 100 150 200 250 3000.0100.010.020.030.040.050.06Time (ns)Photodiode A (V)Bandpassed transmitted signalFig. 10. The transmitted signal after passing through a

1 nm bandpass _lter at 1:533 _m The chaotic light at thiswavelength still masks the message.

_rst of these peaks (1533 nm) corresponds tothe message injection, whereas the second peak(1558 nm) corresponds to the natural lasing wavelengthof the EDFRL. The very broad linewidthis characteristic of the EDFRL. The message lightat 1533 nm stimulates the EDFRL to emit at thesame wavelength and the fraction of the light that


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remains in the ring continues to circulate stimulatingadditional emission. Consequently, the light detectedat 1533 nm consists of a combination of themessage itself plus chaotic light produced by theEDFRL.It was important to determine whether it would

be possible simply to isolate the message wavelength(1533.01 nm) using a bandpass _lter andobserve the message directly. We performed thatexperiment and observed that the message was wellobscured by the chaotic laser light. Figure 10 showsone of these measurements. The sequence of bitsis not visible even after isolating just the messagewavelength. This experiment indicates that wavelengthdivision multiplexing may be possible, whilestill using chaos to hide the information. In summary,the message wavelength can be varied aroundthe natural lasing wavelength of the EDFRL and

chaotic communication can still occur.4. Discussion of MessageInjection ApproachAll of the variations demonstrated in Sec. 3.1 wereChaotic Communication Using Time-Delayed Optical Systems 2143able to consistently recover a 125 Mbit/s digitalmessage. Each variation uses a matched chaoticreceiver to extract the message but the variationsall have limitations in their ability to mask information.In Sec. 2.1.1, the polarizations of ET (t) andm(t) are orthogonal. At any point in the transmissionchannel they are both, in general, elliptically

polarized. But by using a polarization controller, itis possible to change their polarization from ellipticalto linear. Once that is accomplished, a properlyoriented polarizer could remove the maskingET (t) leaving only m(t). Varying the polarizationsof ET (t) andm(t) in time could make this unmaskingmore di_cult.In the experiment described in Sec. 3.1.2, thewavelengths of the message and chaotic _eld are different,though they are polarized identically. Manyoptical devices exist to isolate particular wavelengths.Those devices could be used to obtain the

message without the masking. The technique describedin Sec. 3.1.3 has the limitation that m(t)and ET (t) are separated by _1 nm in wavelength.The same wavelength dependent _lters could possiblyremove ET (t) without distorting m(t) whenthe technique in Sec. 3.1.3 is used. This might bemore di_cult in the one used in Sec. 3.1.3 thanin Sec. 3.1.2 as the wavelength separation is muchsmaller. To make more di_cult isolating just the


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message wavelength, one could imagine a _lter inthe EDFRL itself which allows many wavelengthsof ET (t) to be transmitted, thereby making it dif-

_cult to remove all of them without removing m(t)as well. One could also imagine varying the wavelengthof m(t) or transmitting m(t) as two or more

wavelengths.As stated earlier, chaotic communication doeslend itself naturally to issues of encryption andprivacy. From that perspective, the techniques ofSec. 3.2 are probably to be preferred. But thismethod, too, has some limitations. Principally, wewere unable to control the phase relationship of theelectric _eld and the message _eld. Consequently,the cross-term, 2 RefE_T (t) _m(t)g, had to be minimizedfor more consistent message recovery. Withouta _lter, this meant that ET (t) had to be keptas small as possible. The low pump powers used

in Sec. 3.2.1 were chosen for that reason. Even so,the message recovery was not consistent. This inconsistencybecame a greater problem as the pumppower in EDFA 1 was increased or as the wavelengthof the injected message light came closerto the resonant lasing wavelength of the EDFRL.In Sec. 3.2.2, the message wavelength was separatedfrom the EDFRL's natural lasing line. In thisregime, larger pump powers in EDFA 1 could beused before the interference term mentioned earlierbecomes too large for reasonably consistent messagerecovery.

5. Experimental Setup andOperation Using anIntra-ring Intensity ModulatorIn the experiments described earlier, consistentmessage recovery is di_cult to achieve because ofinterference e_ects between the chaotic light in thering and the injected optical message. Without a

_lter, the interference term, 2RefE_T (t)_m(t)g, fluctuatesdue to changing relative phase and polarizationbetween ET (t) and m(t). A new approach istaken to overcome this di_culty. In this approach,the message is applied directly to the chaotic light

in the EDFRL using an electro-optic modulatorlocated within the EDFRL. The message, in thisapproach, is a modulation rather than an injectedlightwave; consequently, the message does not interferewith the chaotic electric _eld in the EDFRL.Figure 11 shows a transmitter consisting ofan EDFRL and a LiNBO3 intensity modulator.The erbium-doped _ber ampli_er (EDFA) has a


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small signal gain of _30 dB and a maximum outputpower of 17 dBm. Its erbium-doped _ber isthe active medium for the ring laser. As before,the intensity modulator uses a waveguide Mach{Zehnder interferometer as the basis for its operation.The modulator also acts as a polarizer because

its waveguides are polarizing. A 90/10 output couplerdirects 10% of the light out of the ring and intothe communication channel. The remaining 90% ofthe light continues to circulate around the ring.In the receiver, the light is split again. Thetransmitted signal is directly measured at photodiodeA. The remaining light passes through avariable time-delay device. A precision time-delayis achieved using a GRIN (graded index) lens tocouple light from a _ber into free-space. The lighttraverses a distance, the source of the variable timedelay,before it is incident on another GRIN lens

coupler which couples the light from free- space into_ber. By controlling the separation between the twoGRIN lens couplers, the time-delay between photodiodesA and B can be precisely controlled. For thisexperiment, photodiode B measures the same signal2144 G. D. VanWiggeren & R. Roy_

______ ___ ___ _____ _ _" #$& ' ( ) #+, - . / -1234 5 6 78 9 : 8 9;< > ?> @ A > @ B CEF H J H KLH KM NPQST UVW XYZ[\]_a 'bcdef eg hjk l mno o p l mpq rs tuv wyz { { | } z~ .

Fig. 11. Experimental setup for the intra-ring modulatorapproach. An intensity modulator is used to encode a digitalmessage onto chaotic lightwaves produced by the erbiumdoped_ber ring laser. Chaotic light from the ring travels to areceiver where a precise time-delay between the photodiodes

allows for the message to be recovered from the chaos.as photodiode A, but with a time-delay matchedto within 0.1 ns of the round trip time of theEDFRL, _R.The slowly varying envelope of the chaoticlightwave after the EDFA in the transmitter canbe represented as E. The lightwave propagatesthrough the ring and is amplitude modulated asit passes through the modulator to create m(t)E,where m(t) is the message signal. Note that inthis case, the message, m(t) is a scalar modulationrather than a vector lightwave as in the

previous experiments. Ninety percent of this lightcontinues until it is ampli_ed in the EDFA to createE0_= m(t)E, while the remaining fraction isoutput to the receiver. The lightwave exiting theampli_er has a signi_cantly greater intensity, butits waveform is very similar to the input wave. Forsimplicity, we write for the lightwave after the ampli

_er E0_= m(t)E because the relative amplitudesof E0 and m(t)E are not as important to the operation


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as is the shape of the waveform. The shapeof the waveform is only slightly distorted in passingthrough the ampli_er due to noise and nonlinearitiesin the ampli_cation. E0 also circulates throughthe ring and is modulated to produce m(t + _R)E0,and a fraction is output to the receiver.

In the receiver, photodiode B is delayed relativeto photodiode A by one round trip time, _R,to within an accuracy of _0.1 ns. Consequently,when photodiode A is measuring m(t+_R)E0, photodiodeB is measuring m(t)E. Since E0_=m(t)E,a division of the signal recorded at photodiode Aby the signal recorded at photodiode B should givem(t + _ ), thereby recovering the message from thechaotic carrier.6. Results of Intra-ring ApproachFigure 12 gives the results of such an experiment.A repeating digital message with a

32 bit length was applied to the modulatorin the ring. The message had the pattern:01111101010110011011101101001001. The recoveredmessage shown in Fig. 12E replicates thispattern. Figures 12A and 12C, however, show nosign of the message. The power spectra of thetwo measured signals (Figs. 12B and 12D, respectively)show many peaks; the peak at 125 MHz correspondsto the bit-rate. That peak is also visible inthe power spectrum of the recovered message bits,Fig. 12F.Only a small message modulation was used to

obtain Fig. 12; the depth of message modulationthat could be used is limited by the dynamics ofthe ring laser. If the message modulation is toosmall, bit recovery is impaired because the noisemay be larger in amplitude than the communicatedmessage. If the message modulation amplitude istoo large, it drives the laser into an unstable spikingregime. In that spiking regime, the transmittedintensities are near zero much of the time. Becauseour message is recovered through a division process,any noise present in the signal detected byphotodiode B when the signal is near zero has very

detrimental e_ects. Longer sequences of pseudorandombits tended to require even smaller modulationChaotic Communication Using Time-Delayed Optical Systems 2145amplitudes to prevent the spiking behavior thanemployed in Fig. 12.Once an appropriate adjustment has beenmade, the message recovery is very consistentbecause, unlike in the earlier experiments, no interferenceterm exists to distort message recovery. The


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intra-ring modulator method also has additionalprivacy bene_ts when compared with the techniquesdescribed in Sec. 3.1. In this method, the messageis incorporated as a part of the chaotic carrier0 100 200 300 4000.2

0.40.60.81Signal measured by photodiode ATime (ns)Voltage (V)A0 100 200 300 4000.20.40.60.81

Signal measured by photodiode BTime (ns)Voltage (V)C0 100 200 300 4000.911.11.21.3Recovered messageTime (ns)Voltage (V)E0 100 200 300 400105100dBPower spectrum of signal at photodiode AFrequency (MHz)B0 100 200 300 400105100dBPower spectrum of signal at photodiode BFrequency (MHz)D0 100 200 300 400

105100dBPower spectrum of recovered messageFrequency (MHz)FFig. 12. (A) shows the signal from the transmitter as recorded by photodiode A. (C) is the signal recorded byphotodiode Bin the receiver. A division of the two signals recovers the message as shown in (E). Panels B, D, and F show thepower spectra


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of the signals shown in A, C, and E respectively. Note that the 125 MHz peak in (B) and (D) is not the most prominentpeak.(F) A peak at the bit-rate of 125 MHz is clearly visible.2146 G. D. VanWiggeren & R. Roylightwave, m(t)E. Consequently, the message cannotbe separated from the chaotic electric _eld usingoptical devices such as polarizers or bandpass

_lters.As mentioned earlier, two additional factorsthat are important to privacy considerations arethe dimensionality of the transmitted chaotic lightwave,and the e_ort required to obtain the necessaryparameters for message recovery. The signalfrom the transmitter EDFRL has been analyzed usinga false nearest-neighbors algorithm [Abarbanel,1996]. Using 100 000 data points, the analysis indicatesthat the dimensionality is high, of order 10 orgreater.We have observed that only one parametermust be known in order to construct a receivercapable of recovering the message. Precise knowledgeof the round-trip time, _R, of the EDFRLis su_cient for message recovery. Though thenonlinear dynamics of the coupled light _eld andpopulation inversion in the EDFA of the transmitterresult in chaotic fluctuations of light intensity,the EDFA does not signi_cantly alter awaveform's shape after just one pass through theampli_er. Consequently, an EDFA in the receivermatched to the EDFA in the transmitteris not actually necessary to recover the message.

Having only one parameter, _R, to be matchedlimits the potential privacy of the communicationmethod. The same reasoning can be applied tothe methods described in Sec. 3. There, too,the matched EDFA is unnecessary, though it wasincluded in the experiments.7. Modi_ed Intra-ring MethodA new con_guration that requires multiple-matchedparameters in the receiver and allows consistent andclear message communication is shown in Fig. 13.An additional outer-loop is added to the solitaryEDFRL in the previous system. The outer-loop extracts

a portion of the light in the inner-ring, delaysit relative to the light in the inner-ring, andreinjects it. With just a small amount of reinjectedlight, the spiking behavior observed withlarger modulation amplitudes in the previous experimentis eliminated. The transmitter also becomesmore stable. Unlike the consistently chaoticoutput of the previous experiment, the transmitter


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is now only intermittently chaotic without messagemodulation._

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