Complexity in electro-optic delay dynamics: modelling, design and applications

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design and applications

Laurent Larger

To cite this version:

Laurent Larger. Complexity in electro-optic delay dynamics: modelling, design and applications.

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences,

Royal Society, The, 2013, 371, pp.20120464. �10.1098/rsta.2012.0464�. �hal-00878742�

Laurent Larger

FEMTO-STInstitute, UMRCNRS6174, Opti s Dept. P.M. Dueux University of Fran he-Comté, 16 routede Gray, 25030 Besançon edex, Fran e Nonlinear delay dynami s have found during the last 30 years a parti ularly proli explorationareaintheeldofphotoni systems.Besidesthepopularexternal avitylaser diodesetups, wewill fo us inthis arti leon anotherexperimental realizationinvolving ele tro-opti (EO) feedba k loops,withdelay. Thisapproa hhasstronglyevolvedwith the important te hnologi al progress made on broadband photoni and optoele troni devi es dedi ated to high speed opti al tele ommuni ations. The omplex dynami al systemsperformedby nonlineardelayedele tro-opti feedba kloopar hite tures, were designed and explored within a huge range of operating parameters. Thanks to the availabilityofhighperforman ephotoni devi es,theseele tro-opti delaydynami sled alsotomanysu essful,e ient,anddiverseappli ations,beyondthemanyfundamental questions raisedfrom the observation of experimental behaviors. Their haoti motion allowedfor a physi al layeren ryption method to se ure opti aldata, and this in real timeat thetypi alspeedofmodernopti altele ommuni ations.Mi rowavelimit y les generated insimilar EOdelayos illators, showedsigni antlyimprovedspe tralpurity thanks to the use of a very long ber delay line. Last but not least, a novel brain inspired omputationalprin iplehasbeenre entlyimplementedphysi allyin photoni s for the rst time, again on the basis of an EO delay dynami al system. In this latter emerging appli ation, the omputed result is obtained by a proper Read-out of the omplexnonlineartransientsemergingfrom axedpoint,thetransientbeingissuedby theinje tionoftheinformationsignaltobepro essed.

Keywords:Nonlineardelaydynami s,ele tro-opti delayos illators, haos,mi rowaveoptoele troni os illators,photoni neuromorphi omputing,reservoir omputing.

1. Earlyele tro-opti delaydynami s

The Lorenz Buttery ee t (also known in a more a ademi way as sensitivity to initial onditions) is originating from a 1963 publi ation, and it strongly ontributedtotheimportantrevivalofnonlineardynami s theory.Thiso urred manyyearsaftertheinterruptedpathdrawnbyPoin aréinthelate

19

th

andearly

20

th

entury.Theeldofnonlineardynami swaslaterpopularizedbyYorkeinthe 70s,whilerenamingthiseld haostheory.Important eortswerethenbrought in thisemerging s ienti ommunity to showeviden e of haoti motion inreal world, whether insystems observableintheNature,or inarti ial onesdesigned byhumans.In the parti ular eldof Opti s,a seminal idea was proposedin the late 70sby a japanese resear her, K. Ikeda, inorder to observe haoti motions

Pro .R.So .A123;doi:10.1098/rspa.00000000 Thisjournalis

2011TheRoyalSo iety

inOpti s,the dynami alvariablebeingthe lightintensityat theoutputofaring avity ontaining aKerr medium (seeFig.1(left)) [15 ℄.

A rapid physi al look at this Gedanken experiment immediately reveals the entral role of the delay indu ed by the time of ight of the light propagating insidethe ring avity.Asimpledynami almodelingofthesystem anleadtothe redu tionof the numberof dynami al variables to a singleone, thelaser output intensity.Inthis setup, thislight intensitylevelisalso repsonsiblefor theopti al phase variations indu ed by the Kerr ee t, at a rate of hange

γ

limited only bythis ultra fastlight-matter intera tion phenomenon.Oneis thenbrought toa s alarlinearrstorderdierential equation, drivenbyanonlinearfeedba kterm delayed in time. If no delay were present, it is known to always lead to a xed point as the most omplex possible solution. Bistable xed points are however possible even without delay, be ause of the nonlinear feedba k term o urring insidethe avityattheinputoftheKerr medium.Thisnonlineartransformation isperformedbyaninterferen ephenomena o uring between theinputCW laser beam, and the avity feedba k beam. The a tual presen e of the delay violently hanges the dynami alperspe tivesonthis simple system,in reasing its number of degrees of freedom from 1 (theintensityat thetimeorigine), to innity. This delay-indu ed innite number of degrees of freedom stems from the size of the initial onditions formed, when delay is present, by a fun tional de ribing the a tual intensity variations in time, over a duration orresponding to the delay time

τ

D

. This delay interval an then be viewed as the memory of the avity. Again, arapidphysi alanalysisof therelativetimes ales, wouldeasily onvin e anyone that this delay potentially indu es a from far non negligeable memory ee ts on the dynami s: the rate of hange

γ

for a Kerr ee t is of the order of ps

−1

or even (sub-ps)

−1

time s ale, whereas the time of ight of thelight in va uum easily ex eeds ns (3 orders of magnitude greater!) even when only a few m avitylengthis on erned.Thismakesastrongeviden eoftheimportan eof thelarge delaysituation inwhi hthesetup hasto be onsidered.

Insu hanIkedaring avity,high omplexity haoti motionswereindeedobtained numeri ally. This was also onrmed experimentally, but on a modied setup. Thisalternativesetupwashoweverstillfollowingthetypi alingredientsdes ribed above,adelayedfeedba kloopwithamodulatedinterferen e ondition.In[13℄,a bulkele tro-opti birefringent interferometer wasproposedinsteadof theopti al avity,and the delaywasperformed viaa digital buer after analogue-to-digital onversion of the photodete ted interferen e intensity. After being transformed ba k to a ontinuous time signal by a digital-to-analogue onverter and some ampli ation, this delayed ele troni signal was nally used as the drive of the ele tro-opti ally indu ed phase shift in the interferometer. Instead of this bulk setup, an integrated opti s version was proposed right after in [33 ℄. This te hnologi al solution is the one on erned by most of the results reported in this arti le.It isindeed avery onvenient experimental approa h, both fromthe system integration perspe tives, as well as from the ones of the setup stability and performan es. As it will be shown in this arti le, the extremely robust and reliable features of the related o-the-shelf opti al tele ommuni ation devi es, transformed the original Ikeda ring avity into a exible photoni lab tool for harnessingthe omplexityof delaydynami s.

2. Modelinganddesign

TheIkedaring avitydynami sprin iple,on etranslatedinto asignalpro essing approa h,appearsasanobviousfeedba kloop hain.Oneofitsmainadvantageis to allowfor a learseparation between thelinearandthenonlinear ontributions in the dynami s. These issues will be detailed and analysed in the following subse tions.

Figure1.Ikedadynami s:fromtheGedankenexperimenttotheele tro-opti implementation. Left:TheIkedaring avityingredients:anonlinearmodulationoftheintensityisperformedvia aninterferen e onditionmodulatedbythephaseshift

∆ϕ

o urredatoneroundtripearlierin aKerrmedium(length

L

,Kerr oe ient

n

2

).Theinputlaserbeamhasa onstantintensity

I

0

andawaveve tor

k = 2π/λ = 2πν/c

inva uum.Thedynami softhephaseshiftisruledbythe Kerrresponsetime

τ = γ

−1

.Thedynami s anbeobservedthroughthe intensityu tuations atanyoutputofthetwopartiallyree tingmirrorsofthe avity.Right:Theintensity ele tro-opti nonlinear delay os illator: alaser diodeis seeding anele tro-opti Ma h-Zehnder(MZ) modulatorwith a ontinuouswave light beamof power

P

0

;the MZ outputinterferen eis (i) dynami allymodulatedbytheele tri alinput

G x(t)

,and(ii)stati allysetbytheosetphase

Φ

0

(setbya onstantvoltageonthed ele trode);theresultingnonlineartransformation

f

NL

(x)

isdelayedintimeby

τ

D

(ighttimethroughaber);themodulatedanddelayedopti alintensity isdete tedbyaphotodiode(sensitivity

S

);theresultingele tri alsignalislteredintheFourier domaina ording to

H(ω)

,andamplied(gain

G

),thenservingastherfinputoftheMZ.

(a)Modeling: a signal pro essing approa h

The simplied physi al equation as written in the Kerr medium box of Fig.1(left), an be re-written with a splitting of the linear terms in the left hand side,and thenonlinear term inthe right hand side. One an then analyse the dynami s as follows: in the time domain, it onsists of a linear rst order dierential pro ess driven by a nonlinear delayed feedba k term; in the Fourier domain, the left hand side plays the role of a linear rst order lter (a tually a low pass lter) driven at its input by a delayed nonlinear transformation of its output. Thisresults from a straightforward al ulation asdes ribed in Eq.(2.1), using onversion rules between the Fourier and the Time domain (a ording to Fourier Transformations (FT), d/d

t → iω·

FT,and

R

d

t → (iω)

−1

_·

FT):

τ

d

δϕ

d

t

(t) + δϕ(t) = f [δϕ(t − τ

D

)] = z(t)

FT

−→

∆ϕ(ω)

_Z(ω)

=

1

1 + iωτ

= H(ω),

(2.1) where

Z(ω) =

FT

[z(t)]

and

∆ϕ(ω) =

FT

[δϕ(t)]

aretheFourierTransformsof the orrespondingtimevariations.TheFourier Transform

H(ω)

representsthelinear

lteringperformedontheinputsignalandthusleading totheoutputsignal.This lterisa tuallyinvolvedinthefeedba kloopar hite ture,i.e.itistherstorder low pass lter in the parti ular ase of Eq.(2.1). It is typi ally hara terised by a

−3

dB ut-ofrequen y of

(2πτ )

−1

_{= γ/(2π)}

.One ouldnoti e that anyother lter ould be in prin iple onsidered with the same modeling approa h. This resultsina hangeof theFourierlteringfun tion

H(ω)

,a ordingtothea tual lter tobe onsidered. The onsequen e inthetimedomain would onsistinthe presen eofhigherorderdierentialterms,aswellasalsoeventually inadditional integralterms.Onemightre allthat

H(ω)

isdenedasFT

[h(t)]

,where

h(t)

isthe so- alledtemporalimpulseresponseofthe orrespondinglinearlter,i.e.thelter outputsignalwhentheinputisaDira fun tion

δ(t)

.Withthismodelingapproa h allowing for dierent kinds of lters, the right hand side of the dierential equation remains un hanged, sin e it onsists inthe nonlinear delayed feedba k driving the input of thelinear lter. In our parti ular experimental s heme, the nonlinear transformation is ruled by a two-wave interferen e phenomena, i.e.

z(t) = A[1 + B cos[δϕ(t − τ

D

) + Φ

0

]}

.

A

and

B

are onstant parameters dened by the feedba k gain, and the ontrast of the intereferen e respe tively.

B = 1

for a unity ontrast, i.e. when balan ed wave amplitudes are on erned in the interferen e, and

A ∝ n

2

kI

0

/2

,asillustrated inFig.1(left).

Inpra ti alsituationswhere broadbandor highfrequen ydelayed optoele troni feedba k is involved, the low pass ltering has typi ally to be repla ed by a bandpassltertoree tthea tuallteringperformedbytheele troni bran hof theloop.Insu hasituation,themostsimplelinearltermodel orrespondingto abandpass ltering, isatwo pole(se ond order) Fourier transferfun tion

H(ω)

. More physi ally and time domain speaking, this ltering situation is the one obtained in a damped os illator, with the well known dierential model of the following form:

H(ω) =

i2m ω/ω

0

1 + i2m ω/ω

0

− (ω/ω

0

)

2

,

FT

−1

−→

(2.2)

ω

0

2m

Z

δϕ

d

t + δϕ +

1

2mω

0

˙

δϕ = z

_{⇔ δϕ +}

2m

ω

0

˙

δϕ +

1

ω

2

0

¨

δϕ =

2m

ω

0

˙z

where

˙x

isthederivative of

x

withrespe tto time,

m

isthedampingfa tor,and

2π/ω

0

thepseudoos illationperiod.Inthis aseofase ondorderbandpasslter, the dis ussion with respe t to the damped os illator analogy typi ally involves thenatureoftheeigenvalues.Ontheonehand,one anhave omplexeigenvalues when

m ≪ 1

,inthe aseofaweaklydampedos illatorandthusanarrowltering arounda entralresonan efrequen y

ω

0

.Thisisthetypi alsituationsmetinhigh spe tralpuritydelayoptoele troni mi rowaveos illators(

ω

0

/(2π) ≃ 10

GHz,and

−3

dBbandwidth

mω

0

≃ 10

s MHz), asproposedin[43℄. Ontheother hand,the eigenvalues an be purelyreal and negative, whi h ismet ina strongly damped os illatorwith

m ≫ 1

,

ω

0

beingthenthegeometri meanof the ara teristi ut-opulsation

√

_ω

h

ω

l

.Inthis ase,thelteringfeaturesareasfollows:(i)a

−3

dB high ut-ofrequen y

ω

h

/(2π) = f

h

≃ 10

GHzis orrespondingtoalowpasslter asinEq.(2.1) with

τ = ω

−1

h

= (2m ω

0

)

−1

≃ 10

ps;(ii) andalow ut-ofrequen y

ω

l

/(2π) = f

l

≃ 10

s kHz orresponding to a high pass lter, and introdu ing a slow integration hara teristi time asin Eq.(2.2)

θ = ω

−1

The ase of negative real eigenvalues hara terises the situation typi ally met in broadband optoele troni haos ommuni ation appli ations, as proposed in [14 , 22, 3℄). The modied Ikeda dynami s from (2.1) into (2.2) has been named integro-dierential nonlinear delay equation, sin e it is written as follows in a normalized amplitude form, keeping the same

cos

2

nonlinear delayed feedba k term inthe right handside:

1

θ

Z

t

t

₀

x(ξ)

d

ξ + x(t) + τ

d

x

d

t

(t) = f

NL

[x(t − τ

D

)] − f

NL

[0]

(2.3)

= β · {cos

2

[x(t − τ

D

) + Φ

0

] − cos

2

Φ

0

},

where

β

is the normalisedweight of thenonlinearfun tion

f

NL

inthedynami s, and

Φ

0

isastati phaseshiftdeningtheinterferen e onditionaroundwhi hthe dynami almodulationoftheMZo urs.Thenormaliseddynami alvariable

x(t)

is determined so that it appears as a s ale free argument of the

cos

2

nonlinear fun tion. It is physi ally proportional to the original Kerr phase shift

δϕ

in the Ikedamodel,orintheMZ ase,itisproportionnalwhethertotheele tro-opti ally indu edphaseshiftoralsototherfvoltagedrivingtheMZ.Onewouldnoti ethat a onstant term wasadded to the right hand side (

f

NL

[0]

) only for onvenien e, be ause this writing allows to highlight the fa t that

x ≡ 0

is a natural steady state in integro-dierential delay dynami s. The model in Eq.(2.3), eventually with minor modi ations depending on parti ular experimental onditions, was su essfullyusedtoinvestigatewhethernumeri ally oranalyti allymanyspe i dynami almotionsobservedwiththebandpassversionofthegeneri ele tro-opti intensitysetup(seeFig.1(right),[17 ,8,36 ,6 ,32 ℄).Itissometimesmore onvenient to re-write thedynami s ina ve torial form, witha time

s = t/τ

D

normalized to thedelay.Theequations of motion readthenasfollows:

ε · ˙x(s) = −x(s) − δ · y(s) + f

NL

[x(s − 1)] − f

NL

[0],

(2.4)

˙y(s) = x(s).

The previous formulation of the dynami s might be parti ularly usefull in the broadband ase, in order to highlight a few parameters revealing the important underlying multiple time s ales problem:

ε = τ /τ

D

≪ 1

and

δ = τ

D

/θ ≪ 1

as in [36 , 42℄, where the various relevant time s ales are spanned over more than 6 orders of magnitudes.

The a tual dynami al features whi h an be rea hed experimentally, strongly dependon thedevi es andsystemorganisation. To larify thisfromthepra ti al point of view,we will addressa few experimental details inthenext subse tion, trying to onne t the te hni al hara teristi s of the devi es used to build the ele tro-opti delayos illator,togetherwithsome expe teddynami alfeatures.

(b)Devi e features, and delaysystem properties

In this subse tion, many numeri al values for the experimentally a hievable parameter range will be given and dis ussed in the ontext of the dynami al propertiesoftheele tro-opti delaydynami s.Ea hofthedevi esdepi tedinthe

(b.1)Amplitude parameters

As indi ated in Eq.(2.3), essentially two independent amplitude parameters anbedened forthe hara terization ofthenonlinearfun tion:(i)

β

isaweight for

f

NL

[x]

,a ting asa verti alstret hinginthegraph of

f

NL

);(ii)

Φ

0

isanoset phasedeningthemeanoperatingpoint, itisa tingasanhorizontal shiftinthe graphof

f

NL

(see Fig.2).

The physi al origine dening

β = πSGP

0

/(2V

π

rf

)

omes from the various ampli ation fa tors and onversion e ien ies of ea h devi e in the feedba k loop. The most important devi e is the ele tro-opti Ma h-Zehnder (MZ) modulator performing the nonlinear transformation in the dynami s, a

cos

2

-transformation. It is pra ti ally obtained from an ele tro-opti ally tuneable integrated opti s two wave interferometer. The onstru tive interferen e are leading to the maximum of the fun tion

f

NL

(plotted in Fig.2 Left), and the destru tive one orrespond to the minima of the fun tion. With an EO Ma h-Zehnder devi e, the interferen e ondition an be dynami ally ontrolled over a large bandwidth, i.e. the one typi ally involved in opti al tele ommuni ation systems for whi h they are ommonly designed and used. The ele tro-opti e ien yisquantied te hni ally bythehalfwavevoltage

V

π

rf

.It orrespondsto theinputvoltage amplituderequiredto indu ea

π

-phaseshiftintheinterferen e ondition,thus hangingforexamplea onstru tiveinterferen eintoadestru tive one.Thevoltageamplitude

∆V

applied totheMZele trodes,whenres aledwith respe t to

V

π

rf

, an be viewed as a measure of the nonlinear strength a tually involved in the dynami s. More mathemati ally speaking,

(1 + ∆V /V

π

rf

)

is an estimate of the equivalent degree for the polynomial that ould approximate the a tually used range of the nonlinear fun tion. This would be for example a parabola when

∆V ≃ V

π

rf

, or a ubi polynomial when

∆V ≃ 2V

π

rf

, et .... Highlynonlinear motions are thus subje t to the apability to provide a voltage large enough ompared to

V

π

rf

, over the full bandwidth of interest (

> 10

GHz for tele om devi es). Currently, the most e ient Tele om MZ have a

V

π

rf of about 3 Volts, but ommon values are usually loser to 5 Volts. The te hni al hallenge be omesthenobvious, for a ubi motion one wouldneed to drive the MZwith a.15Volts,whi his ommonlyslightly abovethelimitsof onventional MZ tele om drivers. Higher voltage spans are however possible, but usually at the ost of a strongly redu ed bandwidth. A proper signal ampli ation is thus required,whi histheroleoftheoptoele troni feedba k.Itisintendedtoprovide enough dete tion e ien y and ampli ation, up to the requested drive voltage amplitude to be applied to the MZ. When broadband operation is desired, this impliesa

50 Ω

terminationat theMZele trodes, whi h isimposinganadditional te hni al onstrain in terms of rf power drive apability (up to a few ele tri al Watts).Assumingonehasastandardtele ommaximumopti alpoweravailable atthephotodiodeinput(u tuations from0to10 mW),and takinginto a ount the typi al dete tion e ien y of 0.9 A/W for InGaAs tele om photodiodes loadedwith

50 Ω

transimpedan e amplier,the ele troni gainrequiredfromthe photodiodetothedriver,risestoafew30dBwhenquadrati nonlinearoperation is expe ted. Neverheless, broadband amplied photodiodes are ommer ially available with up to 2.4 k

Ω

transimpedan e instead of the previous 50

Ω

. This isrelaxing theremaining ele tri al gainto lessthan 20 dB. Thoughthis onsists alreadyina strong ampli ation, espe iallyfor bandwidth greater than 10 GHz,

itiste hni allyfeasible.Whenamplierdrivesaturationisexperien ed,one ould modify the model in Eq.(2.3) with an additional nonlinear transformation, a

tanh(x)

asthe argument of the

cos

2

mainnon linearityinsteadof

x

(see [17 , 6℄). The progress in novel photoni te hnologies (plasmoni and/or photoni rystal devi es) might further improve the a tually a hievable range for the nonlinear operation of ele tro-opti nonlinear delay os illators. This orresponds to the design apabilityof devi es withlower halfwave voltage.Alternative setups and physi al solutions an be found in the literature, however usually at the ost of strongly redu ed bandwidth. This is for example the ase for thewavelength haosgeneratorreportedin[21,31℄,whereapolynomialupto14th degree anbe a hievedduetothelarge,butslow,tuningrangeofatunableDBRsemi ondu tor laser. The a hievable wavelength deviation an drive up to several free spe tral range ofa strongly imbalan ed birefringent interferometer.

The parameter

Φ

0

= πV

bias

/(2V

π

d

)

isusually tunableovera large enough range, onsidering its

π

-periodi ity in the model. As indi ated in Fig.1(right), this parameter is simply adjusted via a onstant voltage applied to the d ele trode of the MZ. This allows to adjust theoperating point overmore than one period of the

cos

2

modulation transfer fun tion. With a

V

π

d

of about 4 to 7 Volts, with a purely apa itive load impedan e and only very low frequen y operation requirements,theonly pra ti alissueto he kisthepossibleslowandsmalldrift with external environment hanges. Slow time s ale relaxation phenomena, e.g. very slow dynami s indu ed by possible surfa e harge redistribution inside the ele tro-opti rystal, might o ur, thus indu ing a slow time dependen e of the a tual ele tro-opti e ien y.

(b.2)Time parameters

The key ondition typi ally required when high omplexity dynami s is desired, is the so- alled large delay onguration. In that ase, the delay

τ

D

is typi allymu hgreaterthan the hara teristi time

τ

.Thelinearmemory ofthe dynami s is thus s aled as the ratio

τ

D

/τ

. This ratio an be viewed as simply representingthe numberoftimesthatthefastesttemporalmotiflimitedby

τ

an be a umulated over a time interval orresponding to the delay

τ

D

.Complexity an be further in reased through nonlinear ee ts: it was shown in[27, 28℄that the attra tor dimension of the Ikeda dynami s in haoti regimes in reases as

βτ

D

/τ

.

From the more pra ti al point of view, the setting of the time parameters dependson the targeted issues.For fundamentalinvestigations of thedynami al properties of delay system, any setting an be adopted, depending on the studied onguration:a urately ontrolableandvariabletimeparametersusually involvesdis retetime([21 ℄)oranalogue-to-digital onversion([10,37,38 ,31 ℄).The delay fun tion is emulated by so- alled FIFO (st in rst out) memories in the digitalpro essing. ThedelayvalueisthenxedbytheFIFOmemory depth,and the digital lo kdening the speed at whi h the samples aretravelling through the FIFO. The delay value an be easily adjusted when tuning thedigital lo k frequen y.Whenthe FIFOisimplementedinprogrammable digitaldevi es(su h asFPGA),eventhelinearlter anbeimplementeddigitally,thusalsoproviding a largeexibility,stability,anda ura y,inthedenitionofthelter properties.

theorem),whi hisimposinglimitationsintermsofavailableanaloguebandwidth depending on the maximum sampling rate. One has also to be aware of some signal to noise ratio limitations depending on the level of quantization. The digital approa h hasmany advantages for more exibility and robustness of the experimentally explored omplex phenomena.Typi al valuesof theratio

τ

D

/τ

is ofthe orderof a few10s, and thereferen e time s ale

τ

is ofthe orderof one or afew 10sof

µ

s, resulting indelaysof the order ofms.

When high speed is targeted, thenatural hoi e is to perform thedelay via the ultrabroadband(10sTHz) apa ityofopti albers.Opti al bershave alsothe attra tive property to provide an ultra low absorption (typ. 0.2 dB/km) of the travelling light beamat the tele om wavelength (

1.5 µ

m). Thissolution however leadstoxeddelays,whi hvalueisdire tlydeterminedbythelengthoftheber. For example,a4km standardtele omgrade singlemodeberspoolprovides a. 20

µ

sdelay,withanattenuationoflessthan1dB.Theothertimes alesrulingthe derential dynami s through theele troni ltering feedba k, an be set to very fastdynami swhenTele omgradedevi esareused.Optoele troni ,ele tro-opti , andele troni devi es anprovidea essto hara teristi responsetimeasfastas 10ps.Thesedevi es aretypi allydesigned for ommuni ationbandwidthgreater than 10 GHz.The large delay onguration is then easily fulllled witha setup asinFig.1(right), sin e only afewmeters ofberisgenerating 10ns timedelay, thus performing a normalised linearmemory greater than 1000. Considering the berpigtailsof ommer ialdevi es,10snsdelaysaresimplyobtainedwithoutany additionalberspool.The memoryof thedelay anbein reasedby3additional ordersofmagnitude(1millionsize linearmemory)when ommer ial berspool of several 10s of km are used. Within this extreme situation, subtil dispersion phenomena might have to be further onsidered in the dynami s, sin e ultra-fastdynami s leadsto ontinuouslyspread timedelaysdependingon theFourier frequen y omponents, asdes ribedin[25 ℄.

( )Dynami al andar hite tural diversity ( .1)A few bifur ation issues

A basi and ommon interpretation of delay dierential equations (DDE, as inEq.(2.1)) is usually done asa rst approa h for theunderstanding of some of its numerous behavior. It is typi ally alledthe adiabati approximation, or the singular limit map, whi h onsists in taking the limit

ε = τ /τ

D

→ 0

. Under this asumption, thedynami s isredu edto amap

x

n+1

= f [x

n

] = β cos

2

[x + Φ

0

]

.The xedpoints

x

F

(β, Φ

0

)

areobtainedfrom thetrans endentalequation

x

F

= f [x

F

]

, and they are the same for the DDE model. Graphi ally, these xed points are foundasthe interse tionsbetweenthegraphof

y = f

NL

[x]

,andtherstbisse tor

y = x

. There exists at least one solution between 0 and the normalized weight

β

,due to the bounded hara ter of thetwo-wave intensity interferen e fun tion. The number of interse tions is in reasing linearly with

β

. The stability of these xed points under the map approximation, an be straightforwardly derived. It isruledbythe absolutevalue oftheslope ofthenonlinearfun tion,evaluated at thexedpoint,

|β sin[2(x

F

+ Φ

0

)]|

:it isstableifthevalue issmallerthan 1,and unstableotherwise (seeFig.2). Asa onsequen e, itis usually always possible to nd one stable xed points in the

(β, Φ

0

)

-plane, if one sele ts

Φ

0

su h that the xedpoint is loseenough to anextremumof

f (x)

(i.e. loseto a onstru tiveor

destru tiveinterferen e).Conversely,sin ethisvalueisproportionaltotheweight

β

, the stable xed point an always be destabilized by in reasing the feedba k gain

β

, ex ept for the exa t onstru tive or destru tive interferen e ondition. Pra ti al parameter values a essible experimentally, are overing easily the full

π

-periodi ityrangefor

Φ

0

.Howeverfor

β

,itisusuallylimitedtoafewunits(typ.

≃ 5

) inthe ase ofan ele tro-opti setupasinFig.1(right).

Bifur ations and route to haos aretypi ally explored asthefeedba k gain

β

is

Figure 2.Route to haosindelaydynami s. Left: graphof theIkedamap

x

n+1

= β cos

2

(x

n

+

Φ

0

)

, in whi h the pit hfork ase is represented along the positive slope of the map model, leading to whether a period doubling or a risis and a period doubling (see also Fig.3 for bifur ation diagrams). Center: unusual experimental time tra es obtained with an integro dierential(

ε ≈ 0.015

and

δ ≈ 0.628

) delay dynami salongthe positiveslope(up:

τ

D

-periodi limit y le onne ting thetwostable xedofthe pit hforkmap, withasymmetri duty y le; middle:perioddoubledofthepreviousregime;low:slow

θ

-periodi limit y le);thesesregimes are onlyobtainedwiththeintegro-dierentialmodel, theupperand lowerones beingbistable (hysteresisloopisobtainedwithrespe tto

Φ

0

,forthesame

β

).Right: ommonregimesofthe period doubling as ade, along the negative slope (up:period4 limit y le; middle:period-2 haos;low:fullydevelop haos).

in reased. Thisis experimentally onvenient sin e

β

an be dire tlyand linearly tuned by the input opti al power level

P

0

as in Fig.1(right)). Under the map model, this route to haos an be roughly divided into 2 s enarii depending on the sign of theslope around theoriginal stablexed point

x

F

(forsmall

β

s and dependingon

Φ

0

).

- Ifthe slopeispositive,apit hforkbifur ation o urs,leadingto theappearen e of two stable xed points (with also positive slopes) separated by an unstable xed point (see Fig.2): the dynami s exhibits bistability, the a tually observed stable xed point depends on the initial ondition. The upper xed point is the losesttoamaximumof

f (x)

,i.e.toa onstru tiveinterferen e.As

β

isin reased, it slides upward along

f

NL

, going through the onstru tive interferen e state, and then rea hing the negative slope region. Further bifur ation of this stable xed point along a negative slope region, is qualitatively the same as the one experien edbya stablexedpoint originally lo atedalongthenegative slope.If the a tual intial xedpoint if thelower one, it isthe losest to the minimum of

f (x)

,i.e.tothedestru tiveinterferen e. Onthe ontrarytotheupperxedpoint, it experien es a tangent bifur ation through a ollision with the unstable xed pointwhen

β

isin reased. Bothxedpointsthendisappearafterthebifur ation,

lo ated lose to the maximum of

f (x)

. The observed dynami s is then the one resultingfromthe bifur ationsequen eexperien edbythesame xedpoint from its stable steady state around a maximum of

f (x)

. If the parameter

β

is then de reased,thehysteresis y learound the risispoint anbevisited,highlighting the hara teristi bistability aroundthispoint (see Fig.3).

In the ase of a DDE model, novel solutions arises (see Fig.2- enter and -right) withrespe t to the above des ribed mapmodelsituation (Fig.2-left). A re ently observed and analyzed one on erns a

τ

D

-periodi motion onne ting plateaus. Thevaluesoftheplateausdonot orrespondtothe ommonlimit y leofthemap witha period of twi e the delay, but they are orresponding to ea h of the two stablexedpointsinFig.2.Whereassu hasolutionwasproventobeunstableor metastableforstandardDDEwithlowpassfeedba klter,thebandpassfeedba k wasre entlyfoundtoleadtoastableversionofthisparti ular

τ

D

-periodi solution [41 ,40℄.

-Iftheslopeisnegative,thebifur ations enarioisverysimilartothewellknown perioddoubling as aderouteto haos,popularizedintheliteraturebythelogisti map.Thenonlinearfun tionaround amaximumofintereferen eisindeedlo ally a on ave parabola, exa tly as for the logisti map. In the real world of DDEs however, only the beginning of the period doubling as ade is observed up to period 8 or 16, depending on the noise level in the experiment. The amplitude separation in the limit y le is smaller and smaller as the period doubling in reases:whenthisseparationissmallerthanthenoiseexperiment,they annot bedistinguished.Thelimit y lesoftheDDEtaketheformofalternatingplateaus of duration orresponding to the delay

τ

D

. The amplitudes of the plateaus are mat hingtheones al ulatedfor themap.Thetransitionsbetweenthesu essive plateausarehowevernotinstantaneousasinthemap,buttheyo urwith

ε

-small transition layers. These layers are thus s aled in time withthe response time

τ

, or equivalently, they are s aled with the inverse of the bandwidth limiting the feedba k ltering. These jumps are also hara terised by stronger and stronger overshoots or weakly damped os illations, whi h are underlining the in reasing roleofthe ontinuoustimedynami sintheglobalsolution.Afterwhatis alledthe a umulationpointforthemap(valueof

β

leadingtoaninniteperiodlimit y le in

2

n

as

n → ∞

),areverseperioddoubling as adeisobserved(

n

nowde reasing within reasing

β

).The a umulation point in

β

islo atedat thepositionof the verti al dashedline inFig.3. A hara teristi dynami s inthis reverse as ade is qualitatively hara terisedby

2

n

plateaus onsistingof small haoti u tuations withamplitudes

β/2

n

andwith

ε

-smalltime s ales, ea h plateaubeingseparated by

2

n

− 1

forbidden amplitude bandgaps. As

β

passes thefully developed haos limit found for the map (for whi h

n = 0

, i.e. when the full interval

[0, β]

is denselyvisited by the orresponding dynami s), thea tually observed dynami s arestronglydominated bythenumerous ontinuoustimemodesofDDEs,whi h are orresponding to the large amplitude and high frequen ies eigenmodes of theDDE hara teristi equation:

1 + ελ = f

′

_(x

F

)e

−λ

.Withinthis

β

-range,many qualitative featuresof the observed dynami s an be usedto illustrate the mu h strongerinuen eofthe ontinuoustimedynami s omparedto thedis retetime mapapproximation.Thehigh omplexityregimesarethus stronglydieringfrom themap situation ( ompare left and right diagrams for high

β

in Figs.3): (i)the amplitude probability density distribution be omes very smooth (whereas it is

dis ontinuous for the map); (ii)instead of the periodi ity windows of the map, one anobserveso- alled higherharmoni syn hronisation[16 ℄ revealing omplex resonan es withrational numbers between the short times ale

τ

, and the large delay

τ

D

; the haoti attra tor, instead of being in luded in a 1D phase spa e, rea hes dimensions of the order of

βτ

D

/τ

. In a tual experiments, the haoti attra tor dimensions an easily rea h more than several hundreds up to several thousands.

Figure 3.Bifur ation diagrams. Left: for the Ikeda map (numeri s). Many values of the asymptoti solutionsare al ulated forea hofthe 800

β

-valuesalongthehorizontalaxis.The valuesare usedto al ulateaprobabilitydensityfun tion(PDF)forea h

β

.ThePDFis olor en oded.Right:forthe ontinuoustimeIkedamodel(experimental,lowpasslterasinEq.(2.1)). Thebifur ationparameter

β

wasslowlys annedwithatriangularsignal ontrollingtheopti al powerseedingtheMZ ofthesetupinFig.1Right.Thisallowed toin reaseandthende rease

β

intime,veryslowly(0.5s) omparedtothe hara teristi times alesofthedynami s.Many points (

10

7

) are olle ted all along the s an, thus allowing to al ulate an approa hed PDF with

10

4

values at ea h of the 1024 horizontal positions for

β

. The PDF is en oded ingrey s ale.

Φ

0

isadjustedtoa omparablevaluewithrespe ttotheleftbifur ationdiagram.Noti e thatsolutionsrepresentedinFig.2Centeraretypi alofthenonlineardelayintegro-dierential model,and annotbeobservedwiththemap,orthedelaydierentialmodelofthisgure.The solutionsofFig.2Rightare ommontoea hofthedynami almodels.

( .2)Setupdesign variations: even more motion omplexity

ThoughtheinitialIkedaDDEmodelalreadyfeaturesri handhighly omplex dynami s, the experimental investigation of su h systems largely ontributed to the emergen e of even rea her dynami s. New experiments performing EO delay dynami s in the framework of several pra ti al appli ations, gave rise to various modeling modi ations (see the se tions below). This istypi ally one of the histori al reason motivating fundamental studies on the integro-dieren ial nonlinear delay dynami s des ribed in Eq.(2.3) or (2.4) instead of the DDE in Eq.(2.1). The presen e of the integral term was originating from the design of broadband haosgenerators for highspeed opti al haos ommuni ations.In the aseofsu hawidebandwidthspanningover6ordersofmagnitudeintheFourier domain, from a few 10s of kHz to more than 10 GHz, it is te hnologi ally too di ult tohave aDCpreservingfeedba k.Thebroadbandrf ele troni amplier of on ern behave ne essarily as bandpass lters. Su h a situation required the introdu tion of an additional slow time s ale

θ

atta hed to the low ut-o frequen y ofafew10sofkHz.Thisledto anotyet exhaustive zoology ofmany

[36 ℄,novel risis transisitionfromxedpoint to haos[6℄,

τ

D

-periodi limit y le alongthe positiveslope[41 ℄,Neimark-Sa ker(torus)bifur ationofthelimit y le inveryweaklydamped (

m ≪ 1

)mi rowave delayos illators[8℄,et ....

Among other te hnologi al modi ations of the original Ikeda dynami s, novel ele tro-opti ar hite tures alsoledtoasurprisingmodelingofthisdynami sba k to a map, however preserving the high-dimensional feature of thesolutions [23 ℄. Comparedto the setupinFig.1(right), themodi ationleading heretoa orre t mapmodeling isrelatedtotheuseof ahighrepetitionrate(>GHz) modelo ked pulsed laser sour e.The ondition leading to a mapmodel is related to the fast optoele troni devi es apable to resolve the nonlinear feedba k of the pulses within a time shorter than the repetition period of the laser. Dimensionality of the solutions is then ruled by the number of independent pulses stored in the feedba kdelay line,ea h pulsebeingruled bythenonlinear map.

Otherexperimental investigationsdedi atedto opti al haos ommuni ation[29, 26 , 25℄ proposed the introdu tion of a multiple delay ele tro-opti ar hite ture. Thespe i devi e introdu ing asupplementary delay wasa passive imbalan ed DPSK (dieren e phase shift keing) demodulator. This led to the design of a 4 time s ales delay dynami s, an additional short delay

δτ

D

being present due to the imbalan e interferometer. Novel bifur ation phenomena of the rst Hopf bifur ation were dis overed in su h a system, revealing a resonan e ondition between the smalldelay

δτ

D

andthelargedelay

τ

D

[42℄.Thebifur ation onsists in the destabilization of the Hopf limit y le ruled by the small delay, and the o urren e ofa stableamplitudemodulation oftheHopf y le,withan envelope periodruled by thelargedelay.

Lastbutnotleast,intheframeworkoftheinvestigationofanovel omputational prin iple referred to as Reservoir Computing and based on the omputational power provided by omplex transient motion of high-dimensional dynami al system, a many delayed feedba k (15 to 150 randomly distributed delays) dynami s was explored. Their motivation was relatedto theinvestigation of the onne tivity enhan ement of the equivalent virtual network of nodes, due to the multipledelayed feedba kar hite ture[31 ℄.Thea tual propertiesandfeaturesof thisnovelkind of omplexmultiple delay dynami sis still underinvestigations.

3. Delaydynami smeet appli ations

In the remaining se tions, we will fo us more on a few pra ti al appli ations that were explored in the re ent past years, spe i ally on the basis of various ele tro-opti nonlinear delay dynami s. We will onne t the spe i dynami al propertiesof the ele tro-opti delay os illatorsof on ern,to the a tualphysi al requirements expe ted for ea h of these appli ations. The sequen e of thethree proposed appli ations will moreover s an a bifur ation diagram in the reverse dire tion:rst,opti al haos ommuni ations will showhowone an makeuseof thehigh omplexity haoti regimestoperformdataen ryption;se ond,thelimit y le of an optoele troni mi rowave os illator will benet from high spe tral purity provided by a long delay; and last, even the stable xed point will be usefull as a stable starting point from whi h a omplex transient is generated. This nonlinear transient dynami s is the omplex response of the dynami s to

the on eptual basis for a novel universal omputing prin iple exploiting high dimensional phasespa e of omplex dynami s.

(a)Se ure haos ommuni ations

The idea to hide an information signal within a haoti arrier stems from the demonstration ofthe syn hronisation apability between two distant haoti os illators, provided a suitable oupling between the emitter and the re eiver an bedesigned [35℄. Indeed,whena waveform anbe syn hronised, this usually means it an be used as an information arrier. The most ommon arrier in lassi al ommuni ations systems is the sine waveform, for whi h a re eiver typi allyperformsfrequen ysyn hronisationinordertoa hievethedemodulation of the arried information. Repla ing the sine waveform by a broadband noise-like haoti waveform does hange the on ept of information transmission via a arrier signal, as long as syn hronisation of the haoti arrier is possible. Su h a syn hronisation apability for haoti signal, was orignally thought to beimpossiblebe auseofthewellknownsensitivitytoinitial onditionsof haoti dynami s. Chaos syn hronisation is thus the triggering s ienti result for the eld of haos ommuni ations. A rough explanation for the syn hronisation requirement in haos ommuni ations, an be given as follows (we refer the reader tothe proli literature on haossyn hronizationfor moredetails,seee.g. [5℄). Thetransmitted signal, whi h isavailablefor anyone listening to thepubli transmission hannel, is the result of the lear information signal superimposed ontothelargeamplitude haoti arrier.Re overyofthemaskedinformation into the haoti arrier, an be a hieved ifone is ableat thede oder side to repli ate thesame haoti arrierwaveform(usually alledidenti alsyn hronisation).Inan authorisedde oder,thelo allysyn hronised haoti arrierissubtra tedfromthe re eivedsignaltoresultinthere overyoftheoriginalinformation (seethemiddle plots in Fig.4, upper: masked signal, haos plus information; lower: unmasked information). Chaos, as a broadband signal mu h more omplex than a sine waveform, oers the possibility to prote t the transmitted information via the masking,andviathedi ultyfora hievingasu essfullsyn hronisation.Indeed, syn honisation usually requires the knowledge of numerous physi al parameters dened at the emitter for the generation of the haoti arrier. These physi al parameters and their pre ise setting at the emitter, usually form the se ret key required also at there eiverfor a properde odingvia haos syn hronisation.

Ele tro-opti delaydynami soeredinthat ontextseveralattra tiveadvantages: they in reased dramati ally the phase spa e dimension ompared to the rst experimental demonstration. The rst demonstrations were indeed making use of ele troni ir uits generating haoti arrier in a 3D phase spa e only [12 ℄, with easily re ognisablespe tra. Onthe ontrary,the haoti solutions ofhighly nonlinear delay dynami s exhibit a nearlyat and broadband Fourier spe trum, and a nearly Gaussian probability density fun tion. They are thus ressembling anywhite noise sour e, and they make mu h more hallenging theidenti ation of the se ret key parameters. The use of an opti al ar hite tures also provided a ess to standard opti al tele ommuni ation bandwidth, due to the use of o-the-shell tele om optoele troni , ele troni , and ele tro-opti devi es designed for more than 10 Gb/s opti al data transmissions. The single loop os illating

Figure4.Ele tro-opti phase haos ommuni ations:left:s hemati oftheemitterandre eiver setup;middle:10Gb/seyediagramsforthe haosen ryptedsignal(up)andre overedbitstream (low);right:satelliteviewoftheLumièrebrothersnetworkinBesançon,wheretherst10Gb/s eldexperiment was performed.TheLumièrebrothersare theinventorsof inema, theywere borninthe ityofBesançon.

onguration. Re ently, experimental investigations of haos ommuni ations su esfully a hieved eld experiment demonstration over installed ber opti network,atmorethan10Gb/sdatatransmission.Aspe i allydesignEOsetup was able to generate a haoti opti al phase used to arrythe digital data with aDPSK en oding [25℄.Thedemonstration wasrst performedon thesmallring network in our ity of Besançon, the so- alled Lumière brothers ring network (seeFig.4).Furtherexperimentsoverlargernetworksshowedthatourlab emitter-re eiversetup anbepluggedonaninstallednetwork,beingoperationalovermore than 100 km transmission link. The link quality of the en rypted transmission was found to be reasonnably good, with a net bit error rate at last of

10

−7

. Su hperforman e anbeeasilyimprovedtoerrorfreetransmissionwithstandard digitalerror orre tion en oding.

(b)Mi rowave optoele troni os illators

The limit y le solution of delay dynami s might oer also very attra tive high performan e features for another engineering appli ations. This on erns the generation of mi rowave os illations at ultra high spe tral purity level, e.g. for Radar sour es. The a tually os illating mi rowave frequen y

F

is roughly determined as

ω

0

/(2π)

ifwe adoptthemodelinEq.(2.3) for whi hthebandpass lter ishighlysele tive,i.e.with

m ≪ 1

.The longberdelaylineis hereusedto providealong( omparedtotheos illatingperiod)energystorageelement,whi h istypi allyrequiredwhenhighspe tralpurityisexpe ted.Inusualos illators,this energystoragefun tionistypi allyperformedbyresonatorswithveryhighquality fa tors. Quarz resonators are very well mature devi es for a hieving extreme stability via piezoele tri ee ts oupling ele tri al and me hani al vibrations. However, their Fourier frequen y range, when very high stability is on erned, is urrently limited up 100 MHz. In the mi rowave range, surfa e a ousti wave devi es are making impressive progress, but photoni te hnologies are oering for themoment themost e ient solutions,e.g. through so- alled optoele troni

ontrarytoos illatorsbasedonele tro-me hani al resonators,theirqualityfa tor is in reasing with the operating frequen y. This is easily explained due to the energy storage prin iple based on thedelay of a mi rowave signal arried by an opti al light beam: at a given linear loss oe ient (very low for bers, typ. 0.2 dB/km), the

1/e

attenuation is orresponding to a xed ber length

L

or delay

τ

D

,independently ofthemi rowavemodulationofthelight arrier. Quality fa tor an then be dened as thenumber of open loop os illating periods before their amplitude de ay at

1/e

. Consequently, this quality fa tor s ales linearly with the mi rowave frequen y

F

, i.e.

Q = (n.L/c) F = τ

D

F

. Su h a s aling is moreovervalidinprin ipleuptoveryhighfrequen ies,duetothehugebandwidth of Tele om grade

1.5 µ

m monomode sili a bers. Pra ti al limitations do not involved the ultra low loss feature, but other restri ting physi al phenomena at the origine of time delay u tuations. This omprises for example refra tive index variation due to temperature drifts, or laserphase noise transferonto the mi rowave due to ber dispersion. These limitations are urrently imposing in pra ti al OEO, a berlength limit a. of a few km (thus a few 10sof

µ

s), thus shorter than to the

1/e

attenuation limit. Figure 5 represents a typi al phase noisespe trumre orded aroundanOEOwitha entralfrequen y

F = 10

GHz.It wasdesigned withana uratelythermalized4kmberdelayline,anda50MHz bandwidthsele tiveele troni feedba klter.Thezoominsetisahighresolution measurement of the width and height of the rst side noise delay mode. This peak o urs at 50 kHz

= τ

−1

D

from the entral os illating mode. One an noti e theextremely thinwidth( a.40mHz) andtheextreme height (110dB,fromthe oorlevelat-140,tothetop levela uratelymeasuredintheinsetat -30),whi h aretypi al signatures ofthelarge delay hara ter ofsu h anOEO.

From the more theoreti al and nonlinear dynami s point of view, su h extreme

Figure5.Mi rowaveoptoele troni os illator.Left:lterfeaturesrelativetothedelaymodein theFourierdomain.Right:Phasenoisespe trummeasurementaroundthe entralfrequen yat

ω

0

/2π = 10

GHz;inset:spe tralfeatures(extremelythinandstrong)oftherstnoisysideband delay mode[9℄(thepeak at50Hzisonlyaparasiti peakdueto unavoidableele tri al power lineele tro-magneti pollution).

parameter onditions have brought also very interesting investigations. Indeed, withrespe ttothemodelgiveninEqs.(2.2)and(2.3),onehasaresonantltering feedba k orresponding to a very low damping

m ≃ 10

−3

_{≪ 1}

ontrast ompared to the very high damping of the previous se tion a where

m ≃ 10

5

≫ 1

.However, though the feedba k bandwidth is strongly redu ed, the potential omplexityorthe numberof degrees offreedom,isnot ne essarilyvery low, be ause the de rease of bandwidth is here ompensated by the in rease of thetime delay.The relatively short delay in the previous se tion was about 10s of ns, it is here extended to several 10s of

µ

s. The fondamental Fourier delay mode is now orresponding to amu h lowerfrequen y (typ. 50 kHz vs.10 MHz for haos ommuni ations), andthedelaymodesarebe omingmu h more dense intheFourier spa e. Thoughnarrowbandpass feedba klter is involved around 10GHz,thedensityofdelaymodesstillallowsformanytensto hundredsofthese modes to be potantially involved in the dynami s. The narrow lter width of a few 10sof MHz feeds ba kpotentially many of the50 kHz-spa ed delay modes. Highly omplex delaydynami s an be expe ted, and wasfoundto lead toother bifur ation s enarii when the feedba k gain is in reased: e.g. a Neimark-Sa ker bifur ation o urs from the

ω

0

-Hopflimit y le, leading to a torus hara terized bya

τ

D

-periodi envelope modulation ofthe

ω

0

mi rowave os illation [8 ℄.

( )Photoni nonlinear transient omputing

Thestablesteadystate isasolutionofapriori relativelypoorinterest.And indeed,ifone onsidersaxedpoint,onlyaverylow omplexitylevelisobtained. Thestablexedpointishowevernotaunique featureofthedynami alsystemit is generated from. One might at least asso iate to it the orresponding basin of attra tion.Dependingon thedynami s stru tureand dimension,thestablexed point anbe omemu hmoreattra tiveintermsofrelated omplexity.One ould thennotonly analyzethe lo alstru tureofaxedpoint,butalsoitsentirebasin ofattra tion, assoon asa omplex external signal an be designed to addressin prin ipleanypossibletraje torydriving to the xedpoint:thisis pre iselywhat is usedin an emerging appli ation, Reservoir Computing, whi h is making use ofnonlineartransients omplexityto perform omputation. Nonlinear Transient Computing was also suggested in [11℄ and [31℄ as a naming whi h might refer moretothephysi sandnonlineardynami sviewpoint.Thisnovel omputational prin iplewasoriginallyproposedindependentlyintheearly2000bythe omputer s ien e / neural network omputing ommunity [18℄, and the brain ognitive resear h ommunity [30 ℄.It was initiallyreferred to asE ho StateNetwork,and LiquidStateMa hine,respe tively.Morere ently,theunied nameofReservoir Computing [39 ℄wasproposed. The on eptis loselyinspiredand relatedtothe standard neural network omputing, as represented in Fig.6. One of the main dieren e is to assume that the internal stru ture of thenetwork doesnot need to be optimized during the learningphase. This internal stru ture typi ally only needstobeon erandomlydened,viaa onne tivitymatrix

W

I

.Thisisimposing a xed, but still omplex, dynami s for the global neural network. The learning phase then aims only at nding a suitable so- alled Read-Out output via a Read-Outmatrix

W

R

. TheRead-Out onsistssimplyin ndinga hyperplane in the phase spa e of the network dynami s, from whi h the orre t answer an be easily dedu ed. This Read-Out output onsists of a linear ombination of some of the network nodes that are animated by the omplex transient motion, onse utiveytotheinje tioninto thenetwork oftheinformation tobe pro essed.

dened byaninputlayer onne tiontothenodesofthenetwork.Thisintrodu es a data onne tivity matrix

W

D

, i.e. the way the input information is inje ted into the omplex phase spa e. One of the most important advantage ompared to lassi al neural network omputing, resides in the fa t that the learning is strongly simplied. It is usually performed su essfully via an always existing solution of a simple ridgeregression. This regressionresults inthe al ulation of

W

R

.Despite its mu h loweralgorithmi omplexity whi h is essentially redu ed to the learning pro edure, this approa h already showed very good results on various omplex tasks. Moreover, performan es at thelevel of the most e ient lassi al neural network approa hes have been a hieved, and sometimes they are evenoutperforming them[19 ℄.

Very re ently, in the frame of the European proje t PHOCUS, this novel

Figure6.Photoni NonlinearTransientComputing,orReservoirComputing,withele tro-opti delaydynami s.Left:generalar hite tureofneuralnetwork omputing.Middle:pi tureofthe EOsetup,ands hemati ofthesetupsimilartoFig.1.Right:experimentalresultwithanalready learntReadOutmatrix

W

R

,multipliedbythe2Dpatternrepresentingthetransientresponseof thedelaydynami s.Thehorizontalaxisrepresentsthedis retetimeindex

k

labellingthetime delayintervaloverthepro essingduration.Theverti alaxisrepresentsthenodeindex

n

(from 1to 150),labelling thesampled amplitudeswithinea hofthesu essive timedelay intervals. Thetransientisgeneratedbyanexternalsignal orrespondingtotheinputinformation,herean a ousti spee h(spokendigit)tobere ognised. Thematrixprodu tgivesaReadOutoutput wheredigit7 anbeeasilyidentiedasthemostred- oloredlineamongthe10possibledigits (0to9).

omputationalapproa hwasaddressedfromtheexperimentalpointofview,with the useof a delay dynami s insteadof a network of nodes. A rst approa h was tested with an ele troni Ma key-Glass delay dynami s [1 ℄. This rst ele troni demonstration was thenalso extended to a photoni design [24, 34, 31 ℄ dire tly inspired by the setup in Fig.1. These su essful implementations of Reservoir Computing withdelay dynami s, an be explained by a knownanalogy between innitedimensionaldelaydynami s,andspatio-temporaldynami s[2℄(seeFig.7). Standard neural network omputer are usually materialized as a network of inter onne tednodes,su hasnetworkofneurons,thusforminga omplex spatio-temporal dynami s. This dynami al analogy between spatio-temporal dynami s and delaydynami s, allows apriorifor theuseofa delaydynami s for Reservoir Computing,inthepla eofthe ommonspatiallyextendednetworks.Thisanalogy

Figure7.Graphi alinterpretationofthespa e-timeanalogyforadelaydynami alsystemused asa NonlinearTransient Computer.Left: theinput dataare multiplexedintime throughthe useofamask.Timemultiplexingplaystheroleofaddressingthevirtualspatialnodeswiththe inputdata,withdierentweights.Thevirtualspatialnodesarenetemporalpositionswithin atimedelay ofthe nonlineardelayedfeedba k dynami s. Theyare separatedbythe quantity

δτ

.Thesamenodesare usedtoperformthe Read-Out,asalinear ombinationofthesenode amplitudes during the transient response of the delay dynami s. Middle: the delay dynami s is interpreted intermsof nodes inter onne tions.The impulseresponse

h(t)

provides a short term onne tivity,orequivalentlyashortdistan espatial onne tivitybetweenthenodes.The nonlineardelayedtermappearsasalongterm onne tivity,oratemporalupdatefromonetime delayto thenext,for ea hnode.Right:spatio-temporalgraphofanonlineardelay dynami al transient. Thene temporalstru ture is the virtual spatial oordinate represented verti ally. Thetimestepfromonetimedelaytothenext,representsthedis retetimeiteration.

dis rete-timedynami s.Thedis rete-timemotionis orrespondingtoaroundtrip inthedelayos illation loop,indexedbyaninteger

k

labellingthenumberof the urrent

τ

D

-interval. The virtual ontinuous-spa e onsists of the ne temporal motionofthe dynami s, ofthe orderof

τ

,ando uring withinone  oarse time delay interval. The nodes of the virtual network arethen dened as

N

temporal positions within a time delay. Ea h node

x

n

(k)|n ∈ [1, N]

has a dis rete time motion from one delay interval to the next, as

k

is in reased to

k + 1

. The 2D pattern

x(n, k)

forms a representation of the transient motion, on whi h one has to apply the Read-Out matrix

W

R

in order to nd the expe ted answer (see Fig.6). Experiments based on nonlinear delay dynami s used asthe virtual omplex dynami al network involved in a nonlinear transient omputer, have been su essfully ondu tedonben hmarktestssu hasspokendigitre ognition, timeseriespredi tionand nonlinear hannelequalization.Moreover,thea hieved performan es in these real-world physi al setups are very lose to the ones obtained from pure numeri al simulations, opening very interesting perspe tives onmanypra ti alproblemsthat annotbesolvedeasilyand/orfastenoughwith standard digital omputers.

4. Con lusionsand futureissues

Ele tro-opti delay dynami al systems have triggered during the last 15 years an important s ienti a tivity, ni ely balan ed and ross-fertilized between fundamental issues and novel appli ations. It took benet from the intrinsi

Powerfulappli ationshavebeenattheorigineofspe i dynami alfeaturesdueto setupmodi ationsguidedbyappli ations.Thesespe i dynami alfeatureshave beenthenalsoexploredfromthetheoreti alpointofview.Examplesinthisreview have overed appli ations on erned by three solutions of delay equations, from highdimensional haosto the omplex transients leading toa stablexed point, through the extreme regularity of a limit y le solution. A typi al illustration of the ross-fertilisation an be the following one: broadband haos was used in the frame of se ure opti al ommuni ation, whi h ould only be designed with bandpass optoele troni feedba k, thus resulting in an integral term in the modeling equation.Thistermwasthenfoundtobeat theorigineofwhethernew periodi regimes,or stableoneswhi h wereknownto be unstableor metastable. Aimportanthorizonisstillopeninthisframework,andmanyissuesremaintobe addressed,againbothfromthe appli ationpointofviewandfromthetheoreti al one. The nonlinear intera tions between the limit y le features, and the phase noise performan e in mi rowave OEO, requires fundamental investigations of noisy delay equations. The phase spa e feature around the stable xed point usedfornonlineartransient omputingneedsto be deeperunderstood,espe ially from the information theory point of view, in order to optimize the pro essing powerofthis novel omputational prin iple.

Last but not least, we anti ipate that novel fundamental dynami al properties willappearagainvianoveldesignapproa hesforimprovedappli ationsinvolving delaydynami alsystems.A probably important te hnologi al evolution indelay dynami s,whi h alreadystartedinthe aseofexternal avitylasers[4℄, on erns theuseofintegrationpossibilitiesandmi ro-nano-te hnologiesfortherealisation of delay dynami al systems. This also in ludes opti al disk resonators urrently investigatedforbroadband ombgenerationandextremestabilitytime-frequen y systems.Thesesetups have been learly identied asinvolvingboth thedelay in the ring, and also a distributed nonlinear light-matter intera tions between the numerous avity/delaymodes[7 ℄.

A knowledgment

Most of the results presented above were obtained in the frameworkof several proje ts and ollaborations, inwhi hL.L. has parti ipatedas a groupleaderof the FEMTO-ST institute. The many aspe ts of these results are thus denitely the onsequen e of numerous highly onstru tive interferen es with many o-workers ( olleagues, ollaborators, PhD students). I rstapologise for theones whomight bemissinginthefollowing list, butIensurethemalso for my respe tful re ognitionof their ontribution.Themain ontributors an thus be found at least among the following persons: Jean-Pierre Goedgebuer, Jean-Mar Merolla, Maxime Ja quot, Yanne Chembo, John Dudley,Pierre La ourt, Vladimir Udaltsov, Thomas Erneux, PereColet,IngoFis her,ClaudioMirasso,Jia-MingLiu,MinWonLee,RajarshiRoy,Atsushi U hida,RaymondQuéré,LuisPesquera,PierreGlorieux,YvesPomeau,MaxiSanMiguel,Daniel Gauthier, Lu as Illing, Dimitris Syvridis, Apostolos Argyris, Alan Shore,Valerio Annovazzi-Lodi,et ....Thisworkwasalsomorere entlyessentiallysupportedbytheEuropeanPHOCUS proje t(FP7-2009-ICT-240763)andbytheLabexACTIONprogram( ontra t

ANR-11-LABX-highlyappre iatedresear h onditionsitoeredhimduringthelast5years,asaJuniormember ofthisinstitution.

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