## Abstract

Nonlinear delay dynamics have found during the last 30 years a particularly prolific exploration area in the field of photonic systems. Besides the popular external cavity laser diode set-ups, we focus in this article on another experimental realization involving electro-optic (EO) feedback loops, with delay. This approach has strongly evolved with the important technological progress made on broadband photonic and optoelectronic devices dedicated to high-speed optical telecommunications. The complex dynamical systems performed by nonlinear delayed EO feedback loop architectures were designed and explored within a huge range of operating parameters. Thanks to the availability of high-performance photonic devices, these EO delay dynamics led also to many successful, efficient and diverse applications, beyond the many fundamental questions raised from the observation of experimental behaviours. Their chaotic motion allowed for a physical layer encryption method to secure optical data, with a demonstrated capability to operate at the typical speed of modern optical telecommunications. Microwave limit cycles generated in similar EO delay oscillators showed significantly improved spectral purity thanks to the use of a very long fibre delay line. Last but not least, a novel brain inspired computational principle has been recently implemented physically in photonics for the first time, again on the basis of an EO delay dynamical system. In this latter emerging application, the computed result is obtained by a proper ‘read-out’ of the complex nonlinear transients emerging from a fixed point, the transient being issued by the injection of the information signal to be processed.

## 1. Early electro-optic delay dynamics

The Lorenz butterfly effect (also known in a more academic way as ‘sensitivity to initial conditions’) originated from a 1963 publication, and it strongly contributed to the important revival of nonlinear dynamics theory. This occurred many years after the interrupted path drawn by Poincaré in the late nineteenth and early twentieth century. The field of nonlinear dynamics was later popularized by Yorke in the 1970s, while renaming this field ‘chaos theory’. Important efforts were then brought in this emerging scientific community to show evidence of chaotic motion in real world, whether in systems observable in Nature, or in artificial ones designed by humans. In the particular field of optics, a seminal idea was proposed in the late 1970s by a Japanese researcher, Ikeda, in order to observe chaotic motions in optics, the dynamical variable being the light intensity at the output of a ring cavity containing a Kerr medium (figure 1*a*) [1].

A rapid physical look at this ‘Gedanken’ experiment immediately reveals the central role of the delay induced by the time of flight of the light propagating inside the ring cavity. A simple dynamical modelling of the system can lead to the reduction of the number of dynamical variables to a single one, the laser output intensity. In this set-up, this light intensity level is also responsible for the optical phase variations induced by the Kerr effect, at a rate of change *γ* limited only by this ultrafast light–matter interaction phenomenon. One is then brought to a scalar linear first-order differential equation, driven by a nonlinear feedback term delayed in time. If no delay were present, it is known to always lead to a fixed point as the most complex possible solution. Bistable fixed points are however possible even without delay, because of the nonlinear feedback term occurring inside the cavity at the input of the Kerr medium. This nonlinear transformation is performed by an interference phenomena occurring between the input continuous-wave laser beam and the cavity feedback beam. The actual presence of the delay violently changes the dynamical perspectives on this simple system, increasing its number of degrees of freedom from 1 (the intensity at the time origin) to infinity. This delay-induced infinite number of degrees of freedom stems from the size of the initial conditions formed, when delay is present, by a functional describing the actual intensity variations in time, over a duration corresponding to the delay time *τ*_{D}. This delay interval can then be viewed as the ‘memory’ of the cavity. Again, a rapid physical analysis of the relative time scales would easily convince anyone that this delay potentially induces far from non-negligible ‘memory’ effects on the dynamics: the rate of change *γ* for a Kerr effect is of the order of ps^{−1} or even (sub-ps)^{−1} time scale, whereas the time of flight of the light in vacuum easily exceeds nanoseconds (three orders of magnitude greater!) even when only a few centimetre cavity length is concerned. This makes strong evidence of the importance of the large delay situation in which the set-up has to be considered.

In such an Ikeda ring cavity, high-complexity chaotic motions were indeed obtained numerically. This was also confirmed experimentally but on a modified set-up. This alternative set-up was however still following the typical ingredients described above, a delayed feedback loop with a modulated interference condition. In the study of [2], a bulk electro-optic (EO) birefringent interferometer was proposed instead of the optical cavity, and the delay was performed via a digital buffer after analogue-to-digital conversion of the photodetected interference intensity. After being transformed back to a continuous time signal by a digital-to-analogue converter and some amplification, this delayed electronic signal was finally used as the drive of the electro-optically induced phase shift in the interferometer. Instead of this bulk set-up, an integrated optics version was proposed right after in [3]. This technological solution is the one concerning most of the results reported in this article. It is indeed a very convenient experimental approach, both from the system integration perspectives and from that of the set-up stability and performances. As will be shown in this article, the extremely robust and reliable features of the related off-the-shelf optical telecommunication devices transformed the original Ikeda ring cavity into a flexible photonic laboratory tool for harnessing the complexity of delay dynamics.

## 2. Modelling and design

The Ikeda ring cavity dynamics principle, once translated into a signal-processing approach, appears as an obvious feedback loop chain. One of its main advantages is to allow for a clear separation between the linear and the nonlinear contributions in the dynamics. These issues will be detailed and analysed in the following subsections.

### (a) Modelling: a signal-processing approach

The simplified physical equation, written in the Kerr medium box shown in figure 1*a*, can be re-written with a splitting of the linear terms in the left-hand side and the nonlinear term in the right-hand side. One can then analyse the dynamics as follows: in the time domain, it consists of a linear first-order differential process driven by a nonlinear delayed feedback term; in the Fourier domain, the left-hand side plays the role of a linear first-order filter (actually a low pass filter) driven at its input by a delayed nonlinear transformation of its output. This results in a delay differential equation (DDE), obtained from a straightforward calculation as described in equation (2.1), using conversion rules between the Fourier and the time domain (according to Fourier transformations (FT), and )
2.1where *Z*(*ω*)=FT[*z*(*t*)] and *Δφ*(*ω*)=FT[*δφ*(*t*)] are the Fourier transforms of the corresponding time variations. The Fourier transform *H*(*ω*) represents the linear filtering performed on the input signal and thus leading to the output signal. This filter is actually involved in the feedback loop architecture, that is, it is the first-order low pass filter in the particular case of equation (2.1). It is typically characterized by a −3 dB cut-off frequency of (2*πτ*)^{−1}=*γ*/(2*π*). One could note that any other filter could be in principle considered with the same modelling approach. This results in a change of the Fourier filtering function *H*(*ω*), according to the actual filter to be considered. The consequence in the time domain would consist of the presence of higher order differential terms and also eventually of additional integral terms. One might recall that *H*(*ω*) is defined as FT[*h*(*t*)], where *h*(*t*) is the so-called temporal impulse response of the corresponding linear filter, that is, the filter output signal when the input is a Dirac function *δ*(*t*). With this modelling approach allowing for different kinds of filters, the right-hand side of the differential equation remains unchanged, because it consists of the nonlinear delayed feedback driving the input of the linear filter. In our particular experimental scheme, the nonlinear transformation is ruled by a two-wave interference phenomena, i.e. . *A* and *B* are constant parameters defined by the feedback gain and the contrast of the interference, respectively. *B*=1 for a unity contrast, that is, when balanced wave amplitudes are concerned in the interference, and *A*∝*n*_{2}*kI*_{0}/2, as illustrated in figure 1*a*.

In practical situations where broadband or high-frequency delayed optoelectronic feedback is involved, the low pass filtering has typically to be replaced by a bandpass filter to reflect the actual filtering performed by the electronic branch of the loop. In such a situation, the most simple linear filter model corresponding to a bandpass filtering is a two pole (second-order) Fourier transfer function *H*(*ω*). More physically and ‘time domain’ speaking, this filtering situation is the one obtained in a damped oscillator, with the well-known differential model of the following form:
2.2where the overdot and double overdot indicate first and second derivatives, respectively, with respect to time, *m* is the damping factor and 2*π*/*ω*_{0} the pseudo-oscillation period. In this case of a second-order bandpass filter, the discussion with respect to the damped oscillator analogy typically involves the nature of the eigenvalues. On the one hand, one can have complex eigenvalues when *m*≪1, in the case of a weakly damped oscillator and thus a narrow filtering around a central resonance frequency *ω*_{0}. This is the typical situation met in high spectral purity delay optoelectronic microwave oscillators (*ω*_{0}/(2*π*)≃10 GHz, and −3 dB bandwidth *mω*_{0}≃10 s MHz), as proposed in [4]. On the other hand, the eigenvalues can be purely real and negative, which is met in a strongly damped oscillator with *m*≫1, *ω*_{0} being then the geometric mean of the characteristic cut-off angular frequencies . In this case, the filtering features are as follows: (i) a −3 dB high cut-off frequency *ω*_{h}/(2*π*)=*f*_{h}≃10 GHz corresponding to a low pass filter as in equation (2.1) with *τ*=*ω*^{−1}_{h}=(2*mω*_{0})^{−1}≃10 ps and (ii) a low cut-off frequency *ω*_{l}/(2*π*)=*f*_{l}≃10 s kHz corresponding to a high pass filter, and introducing a slow integration characteristic time as in equation (2.2) *θ*=*ω*^{−1}_{l}=2*m*/*ω*_{0} of a few microseconds. The case of negative real eigenvalues characterizes the situation typically met in broadband optoelectronic chaos communication applications, as proposed in [5–7]. The modified Ikeda dynamics from (2.1) into (2.2) has been named integro-differential delay equation (iDDE), as is written as follows in a normalized amplitude form, keeping the same nonlinear delayed feedback term in the right-hand side:
2.3where *β* is the normalized weight of the nonlinear function *f*_{NL} in the dynamics and *Φ*_{0} is a static phase shift defining the interference condition around which the dynamical modulation of the Mach–Zehnder (MZ) occurs. The normalized dynamical variable *x*(*t*) is determined so that it appears as a scale-free argument of the nonlinear function. It is physically proportional to the original Kerr phase shift *δφ* in the Ikeda model, or in the MZ case, it is proportional to the electro-optically induced phase shift or also to the radiofrequency (RF) voltage driving the MZ. One would notice that a constant term was added to the right-hand side (*f*_{NL}[0]) only for convenience, because this writing allows one to highlight the fact that *x*≡0 is a natural steady state in integro-differential delay dynamics. The model in equation (2.3), eventually with minor modifications depending on particular experimental conditions, was successfully used to investigate whether numerically or analytically many specific dynamical motions observed with the bandpass version of the generic EO intensity set-up (figure 1*b*) [8–12]. It is sometimes more convenient to re-write the dynamics in a vectorial form, with a time *s*=*t*/*τ*_{D} normalized to the delay. The equations of motion read then as follows:
2.4The previous formulation of the dynamics might be particularly useful in the broadband case, in order to highlight a few parameters revealing the important underlying multiple time-scales problem: *ε*=*τ*/*τ*_{D}≪1 and *δ*=*τ*_{D}/*θ*≪1 as in [10,13], where the various relevant time scales are spanned over more than six orders of magnitude.

The actual dynamical features, which can be reached experimentally, strongly depend on the devices and system organization. To clarify this from the practical point of view, we will address a few experimental details in the next subsection, trying to connect the technical characteristics of the devices used to build the EO delay oscillator, together with some expected dynamical features.

### (b) Device features and delay system properties

In this section, many numerical values for the experimentally achievable parameter range will be given and discussed in the context of the dynamical properties of the EO delay dynamics. Each of the devices depicted in the set-up of figure 1*b* will be addressed.

#### (i) Amplitude parameters

As indicated in equation (2.3), essentially two independent amplitude parameters can be defined for the characterization of the nonlinear function: (i) *β* is a weight for *f*_{NL}[*x*], acting as a vertical stretching in the graph of *f*_{NL} and (ii) *Φ*_{0} is an offset phase defining the mean operating point, acting as a horizontal shift in the graph of *f*_{NL} (figure 2).

The physical origin defining *β*=*πSGP*_{0}/(2*V* _{πrf}) comes from various amplification factors and conversion efficiencies of each device in the feedback loop. The most important device is the EO Mach–Zehnder (EO-MZ) modulator performing the nonlinear transformation in the dynamics, a transformation. It is practically obtained from an electro-optically tuneable integrated optics two-wave interferometer. The constructive interference leads to the maximum of the function *f*_{NL} (plotted in figure 2*a*), and the destructive one corresponds to the minimum of the function. With an EO-MZ device, the interference condition can be dynamically controlled over a large bandwidth, that is, the one typically involved in optical telecommunication systems for which they are commonly designed and used. The EO efficiency is quantified technically by the half-wave voltage *V* _{πrf}. It corresponds to the input voltage amplitude required to induce a *π*-phase shift in the interference condition, thus changing, for example, a constructive interference into a destructive one. The voltage amplitude *ΔV* applied to the MZ electrodes, when rescaled with respect to *V* _{πrf}, can be viewed as a measure of the nonlinear strength actually involved in the dynamics. More mathematically speaking, (1+*ΔV*/*V* _{πrf}) is an estimate of the equivalent degree for the polynomial that could approximate the actually used range of the nonlinear function. This would be, for example, a parabola when *ΔV* ≃*V* _{πrf}, or a cubic polynomial when *ΔV* ≃2*V* _{πrf}, etc. Highly nonlinear motions are thus subject to the ability to provide a voltage large enough compared with *V* _{πrf}, over the full bandwidth of interest (greater than 10 GHz for telecom devices). Currently, the most efficient telecom MZs have a *V* _{πrf} of about 3 V, but common values are usually closer to 5 V. The technical challenge becomes then obvious: for a cubic motion one would need to drive the MZ with *ca* 15 V, which is commonly slightly above the limits of conventional MZ telecom drivers. Higher voltage spans are however possible but usually at the cost of a strongly reduced bandwidth. A proper signal amplification is thus required, which is the role of the optoelectronic feedback. It is intended to provide enough detection efficiency and amplification, up to the requested drive voltage amplitude to be applied to the MZ. When broadband operation is desired, this implies a 50 Ω termination at the MZ electrodes, which is imposing an additional technical constraint in terms of RF power drive capability (up to a few electrical watts). Assuming one has a standard ‘telecom’ maximum optical power available at the photodiode input (fluctuations from 0 to 10 mW), and taking into account the typical detection efficiency of 0.9 A W^{−1} for InGaAs telecom photodiodes loaded with 50 Ω transimpedance amplifier, the electronic gain required from the photodiode to the driver rises to approximately 30 dB when quadratic nonlinear operation is expected. Nevertheless, broadband amplified photodiodes are commercially available with up to 2.4 kΩ transimpedance instead of the previously mentioned 50 Ω. This relaxes the remaining electrical gain to less than 20 dB. Although this consists already of a strong amplification, especially for bandwidth greater than 10 GHz, it is technically feasible. When amplifier drive saturation is experienced, one could modify the model in equation (2.3) with an additional nonlinear transformation, a as the argument of the main nonlinearity instead of *x* [8,11]. The progress in novel photonic technologies (plasmonic and/or photonic crystal devices) might further improve the actually achievable range for the nonlinear operation of EO nonlinear delay oscillators. This corresponds to the design capability of devices with lower half-wave voltage. Alternative set-ups and physical solutions can be found in the literature, however usually at the cost of strongly reduced bandwidth. This is, for example, the case for the wavelength chaos generator reported in [14,15], where a polynomial up to fourteenth degree can be achieved owing to the large, but slow, tuning range of a tunable distributed Bragg reflector semiconductor laser. The achievable wavelength deviation can drive up to several free spectral range of a strongly imbalanced birefringent interferometer.

The parameter *Φ*_{0}=*πV* _{bias}/(2*V* _{πDC}) is usually tunable over a large enough range, considering its *π*-periodicity in the model. As indicated in figure 1*b*, this parameter is simply adjusted via a constant voltage applied to the DC electrode of the MZ. This allows one to adjust the operating point over more than one period of the modulation transfer function. With a *V* _{πDC} of about 4–7 V, with a purely capacitive load impedance and only very low-frequency operation requirements, the only practical issue to check is the possible slow and small drift with external environment changes. Slow time-scale relaxation phenomena, for example, very slow dynamics induced by possible surface charge redistribution inside the EO crystal, might occur, thus inducing a slow time dependence of the actual EO efficiency.

#### (ii) Time parameters

The key condition typically required when high-complexity dynamics is desired is the so-called large delay configuration. In that case, the delay *τ*_{D} is typically much greater than the characteristic time *τ*. The linear ‘memory’ of the dynamics is thus scaled as the ratio *τ*_{D}/*τ*. This ratio can be viewed as simply representing the number of times that the fastest temporal motif limited by *τ* can be accumulated over a time interval corresponding to the delay *τ*_{D}. Complexity can be further increased through nonlinear effects: it was shown in [16,17] that the attractor dimension of the Ikeda dynamics in chaotic regimes increases as *β*_{τD}/*τ*.

From the more practical point of view, the setting of the time parameters depends on the targeted issues. For fundamental investigations of the dynamical properties of delay systems, any setting can be adopted, depending on the studied configuration: accurately controllable and variable time parameters usually involve discrete time [14] or analogue-to-digital conversion [15,18–20]. The delay function is emulated by the so-called first-in first-out (FIFO) memories in the digital processing. The delay value is then fixed by the FIFO memory depth, and the digital clock defining the speed at which the samples are travelling through the FIFO. The delay value can be easily adjusted when tuning the digital clock frequency. When the FIFO is implemented in programmable digital devices (such as field programmable gate arrays), even the linear filter can be implemented digitally, thus also providing a large flexibility, stability and accuracy in the definition of the filter properties. One has, however, to respect the basic rules of sampling theory (Shannon sampling theorem), which imposes limitations in terms of available analogue bandwidth 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 approach has many advantages for more flexibility and robustness of the experimentally explored complex phenomena. Typical values of the ratio *τ*_{D}/*τ* are of the order of a few tens, and the reference time scale *τ* is of the order of one or a few tens of microseconds, resulting in delays of the order of milliseconds.

When high speed is targeted, the natural choice is to perform the delay via the ultra broadband (tens of terahertz) capacity of optical fibres. Optical fibres have also the attractive property to provide an ultra-low absorption (typically 0.2 dB km^{−1}) of the travelling light beam at the telecom wavelength (1.5 μm). This solution however leads to fixed delays, which value is directly determined by the length of the fibre. For example, a 4 km standard telecom-grade single-mode fibre spool provides *ca* 20 μs delay, with an attenuation of less than 1 dB. The other time scales ruling the differential dynamics through the electronic filtering feedback can be set to very fast dynamics when telecom grade devices are used. Optoelectronic, EO and electronic devices can provide access to characteristic response time as fast as 10 ps. These devices are typically designed for communication bandwidth greater than 10 GHz. The large delay configuration is then easily fulfilled with a set-up as in figure 1*b*, because only a few metres of fibre generate 10 ns time delay, thus performing a normalized linear memory greater than 1000. Considering the fibre pigtails of commercial devices, tens of nanosecond delays are simply obtained without any additional fibre spool. The memory of the delay can be increased by three additional orders of magnitude (1 million size linear ‘memory’) when commercial fibre spools of several tens of kilometres are used. Within this extreme situation, subtle dispersion phenomena might have to be further considered in the dynamics, since ultra-fast dynamics leads to continuously spread time delays depending on the Fourier frequency components, as described in [21].

### (c) Dynamical and architectural diversity

#### (i) A few bifurcation issues

A basic and common interpretation of the DDE model (as in equation (2.1)) is usually done as a first approach for the understanding of some of its numerous behaviour. It is typically called the adiabatic approximation, or the singular limit map, which consists of taking the limit . Under this assumption, the dynamics is reduced to a map . The fixed points *x*_{F}(*β*,*Φ*_{0}) are obtained from the transcendental equation *x*_{F}=*f*_{NL}[*x*_{F}], and they are the same for the DDE model. Graphically, these fixed points are found as the intersections between the graph of *y*=*f*_{NL}[*x*], and the first bisector *y*=*x*. There exists at least one solution between 0 and the normalized weight *β*, owing to the bounded character of the two-wave intensity interference function. The number of intersections increases linearly with *β*. The stability of these fixed points under the map approximation can be straightforwardly derived. It is ruled by the absolute value of the slope of the nonlinear function, evaluated at the fixed point, : it is stable if the value is smaller than 1, and unstable otherwise (figure 2). As a consequence, it is usually always possible to find one stable fixed point in the (*β*,*Φ*_{0}) plane, if one selects *Φ*_{0} such that the fixed point is close enough to an extremum of *f*_{NL}(*x*) (i.e. close to a constructive or destructive interference). Conversely, since this value is proportional to the weight *β*, the stable fixed point can always be destabilized by increasing the feedback gain *β*, except for the exact constructive or destructive interference condition. Practical parameter values accessible experimentally cover easily the full *π*-periodicity range for *Φ*_{0}. However, for *β*, it is usually limited to a few units (typically ≃5) in the case of an EO set-up as in figure 1*b*.

Bifurcations and route to chaos are typically explored as the feedback gain *β* is increased. This is experimentally convenient because *β* can be directly and linearly tuned by the input optical power level *P*_{0} as in figure 1*b*. Under the map model, this route to chaos can be roughly divided into two scenarios depending on the sign of the slope around the original stable fixed point *x*_{F} (for small *β* and depending on *Φ*_{0}).

First, if the slope is positive, a pitchfork bifurcation occurs, leading to the appearance of two stable fixed points (with also positive slopes) separated by an unstable fixed point (figure 2): the dynamics exhibits bistability, the actually observed stable fixed point depends on the initial condition. The upper fixed point is the closest to a maximum of *f*_{NL}(*x*), that is, to a constructive interference. As *β* is increased, it slides upward along *f*_{NL}, going through the constructive interference state, and then reaching the negative slope region. Further bifurcation of this stable fixed point along a negative slope region is qualitatively the same as the one experienced by a stable fixed point originally located along the negative slope. If the actual initial fixed point is the lower one, it is the closest to the minimum of *f*_{NL}(*x*), that is, to the destructive interference. In contrast to the upper fixed point, it experiences a tangent bifurcation through a collision with the unstable fixed point when *β* is increased. Both fixed points then disappear after the bifurcation, leading to a crisis of the dynamics, with a jump onto the remaining fixed point located close to the maximum of *f*_{NL}(*x*). The observed dynamics is then the one resulting from the bifurcation sequence experienced by the same fixed point from its stable steady state around a maximum of *f*_{NL}(*x*). If the parameter *β* is then decreased, the hysteresis cycle around the crisis point can be visited, highlighting the characteristic bistability around this point (figure 3).

In the case of an iDDE model, novel solutions arise (figure 2*b,c*) with respect to the above-described map model situation (figure 2*a*). As recently observed and analysed, one concerns a *τ*_{D}-periodic motion connecting plateaus. The values of the plateaus do not correspond to the common limit cycle of the map with a period of twice the delay, but they correspond to each of the two stable fixed points in figure 2*a*. Whereas such a solution was proved to be unstable or metastable for standard DDE with low pass feedback filter, the bandpass feedback, leading to an iDDE model, was recently found to lead to a stable version of this particular *τ*_{D}-periodic solution [22,23].

Second, if the slope is negative, the bifurcation scenario is very similar to the well-known period-doubling cascade route to chaos, popularized in the literature by the logistic map. The nonlinear function around a maximum of interference is indeed locally a concave parabola, exactly as for the logistic map. In the real world of DDEs, however, only the beginning of the period-doubling cascade is observed up to period 8 or 16, depending on the noise level in the experiment. The amplitude separation in the limit cycle is smaller and smaller as the period doubling increases: when this separation is smaller than the noise experiment, they cannot be distinguished. The limit cycles of the DDE take the form of alternating plateaus of duration corresponding to the delay *τ*_{D}. The amplitudes of the plateaus match the ones calculated for the map. The transitions between the successive plateaus are however not instantaneous as in the map, but they occur with *ε*-small transition layers. These layers are thus scaled in time with the response time *τ*, or equivalently, they are scaled with the inverse of the bandwidth limiting the feedback filtering. These jumps are also characterized by stronger and stronger overshoots or weakly damped oscillations, which are underlining the increasing role of the continuous time dynamics in the global solution. After what is called the accumulation point for the map (value of *β* leading to an infinite period limit cycle in 2^{n} as ), a reverse period-doubling cascade is observed (*n* now decreasing with increasing *β*). The accumulation point in *β* is located at the position of the vertical dashed line in figure 3. A characteristic dynamics in this reverse cascade is qualitatively characterized by 2^{n} plateaus consisting of small chaotic fluctuations with amplitudes *β*/2^{n} and with *ε*-small time scales, each plateau being separated by 2^{n}−1 forbidden amplitude bandgaps. As *β* passes the fully developed chaos limit found for the map (for which *n*=0, i.e. when the full interval [0,*β*] is densely visited by the corresponding dynamics), the actually observed dynamics are strongly dominated by the numerous continuous time modes of DDEs, which correspond to the large amplitude and high frequency eigenmodes of the DDE characteristic equation: 1+*ελ*=*f*^{′}(*x*_{F}) e^{−λ}. Within this β-range, many qualitative features of the observed dynamics can be used to illustrate the much stronger influence of the continuous time dynamics compared with the discrete time map approximation. The high-complexity regimes thus strongly differ from the map situation (compare figure 3*a* and figure 3*b* for high *β*): (i) the amplitude probability density distribution becomes very smooth (whereas it is discontinuous for the map) and (ii) instead of the periodicity windows of the map, one can observe so-called higher harmonic synchronization [24] revealing complex resonances with rational numbers between the short time scale *τ*, and the large delay *τ*_{D}; the chaotic attractor, instead of being included in a one-dimensional phase space, reaches dimensions of the order of *βτ*_{D}/*τ*. In actual experiments, the chaotic attractor dimensions can easily reach more than several hundreds up to several thousands.

#### (ii) Set-up design variations: even more motion complexity

Although the initial Ikeda DDE model already features rich and highly complex dynamics, the experimental investigation of such systems largely contributed to the emergence of even richer dynamics. New experiments performing EO delay dynamics in the framework of several practical applications gave rise to various modelling modifications (see the sections below). This is typically one of the historical reasons motivating fundamental studies on the integro-differential nonlinear delay dynamics (iDDE model) described in equations (2.3) or (2.4) instead of the DDE in equation (2.1). The presence of the integral term originated from the design of broadband chaos generators for high-speed optical chaos communications. In the case of such a wide bandwidth spanning over six orders of magnitude in the Fourier domain, from a few tens of kilohertz to more than 10 GHz, it is technologically too difficult to have a DC preserving feedback. The broadband RF electronic amplifiers of concern behave necessarily as bandpass filters. Such a situation required the introduction of an additional slow time scale *θ* attached to the low cut-off frequency of a few tens of kilohertz. This led to a not yet exhaustive ‘zoology’ of many new dynamics typical of iDDE models, such as chaotic breathers [25], slow limit cycle, pulsating regime [10], novel crisis transition from fixed point to chaos [11], *τ*_{D}-periodic limit cycle along the positive slope [22], Neimark–Sacker (torus) bifurcation of the limit cycle in very weakly damped (*m*≪1) microwave delay oscillators [9], etc.

Among other technological modifications of the original Ikeda dynamics, novel EO architectures also led to a surprising modelling of this dynamics back to a map, however preserving the high-dimensional feature of the solutions [26]. Compared with the set-up in figure 1*b*, the modification leading here to a correct map modelling is related to the use of a high repetition rate (greater than gigahertz) mode-locked pulsed laser source. The condition leading to a map model is related to the fast optoelectronic devices capable of resolving the nonlinear feedback 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 feedback delay line, each pulse being ruled by the nonlinear map.

Other experimental investigations dedicated to optical chaos communication [21,27,28] proposed the introduction of a multiple delay EO architecture. The specific device introducing a supplementary delay was a passive imbalanced difference phase-shift keying (DPSK) demodulator. This led to the design of a four time-scales delay dynamics, an additional short delay *δτ*_{D} being present owing to the imbalance interferometer. Novel bifurcation phenomena of the first Hopf bifurcation were discovered in such a system, revealing a resonance condition between the small delay *δτ*_{D} and the large delay *τ*_{D} [13]. The bifurcation consists of the destabilization of the Hopf limit cycle ruled by the small delay, and the occurrence of a stable amplitude modulation of the Hopf cycle, with an envelope period ruled by the large delay.

Last but not least, in the framework of the investigation of a novel computational principle referred to as ‘reservoir computing’ and based on the computational power provided by complex transient motion of high-dimensional dynamical systems, a many delayed feedback (15–150 randomly distributed delays) dynamics was explored. The motivation was related to the investigation of the connectivity enhancement of the equivalent virtual network of nodes, owing to the multiple delayed feedback architecture [15]. The actual properties and features of this novel kind of complex multiple delay dynamics are still under investigation.

## 3. Delay dynamics meet applications

In the remaining sections, we will focus more on a few practical applications that were explored in recent years, specifically on the basis of various EO nonlinear delay dynamics. We will connect the specific dynamical properties of the EO delay oscillators of concern to the actual physical requirements expected for each of these applications. The sequence of the three proposed applications will, moreover, scan a bifurcation diagram in the reverse direction: first, optical chaos communications will show how one can make use of the high-complexity chaotic regimes to perform data encryption; second, the limit cycle of an optoelectronic microwave oscillator will benefit from high spectral purity provided by a long delay; and last, even the stable fixed point will be useful as a stable starting point from which a complex transient is generated. This nonlinear transient dynamics is the complex response of the dynamics to an external signal encoding a problem to be solved. The nonlinear transient is the conceptual basis for a novel universal computing principle exploiting high-dimensional phase space of complex dynamics.

### (a) Secure chaos communications

The idea to hide an information signal within a chaotic carrier stems from the demonstration of the synchronization capability between two distant chaotic oscillators, provided a suitable coupling between the emitter and the receiver can be designed [29]. Indeed, when a waveform can be synchronized, this usually means that it can be used as an information carrier. The most common carrier in classical communications systems is the sine waveform, for which a receiver typically performs frequency synchronization in order to achieve the demodulation of the carried information. Replacing the sine waveform by a broadband noise-like chaotic waveform does not change the concept of information transmission via a carrier signal, as long as synchronization of the chaotic carrier is possible. Such a synchronization capability for a chaotic signal was originally thought to be impossible because of the well-known sensitivity to initial conditions of chaotic dynamics. Chaos synchronization is thus the triggering scientific result for the field of chaos communications. A rough explanation for the synchronization requirement in chaos communications can be given as follows (we refer the reader to the prolific literature on chaos synchronization for more details, e.g. [30]). The transmitted signal, which is available for anyone listening to the public transmission channel, is the result of the clear information signal superimposed onto the large amplitude chaotic carrier. Recovery of the masked information into the chaotic carrier can be achieved if one is able at the decoder side to replicate the same chaotic carrier waveform (usually called identical synchronization). In an authorized decoder, the locally synchronized chaotic carrier is subtracted from the received signal to result in the recovery of the original information (see figure 4*b*: (i) masked signal, chaos plus information and (ii) unmasked information). Chaos, as a broadband signal much more complex than a sine waveform, offers the possibility to protect the transmitted information via the masking and the difficulty of achieving a successful synchronization. Indeed, synchronization usually requires the knowledge of numerous physical parameters defined at the emitter for the generation of the chaotic carrier. These physical parameters and their precise setting at the emitter usually form the secret key required also at the receiver for a proper decoding via chaos synchronization. EO delay dynamics offered in that context several attractive advantages: they increased dramatically the phase space dimension compared with the first experimental demonstration. The first demonstrations were indeed making use of electronic circuits generating chaotic carriers in a three-dimensional phase space only [31], with easily recognizable spectra. On the contrary, the chaotic solutions of highly nonlinear delay dynamics exhibit a nearly flat and broadband Fourier spectrum and a nearly Gaussian probability density function. They thus resemble any white noise source, and they make much more challenging the identification of the secret key parameters. The use of optical architectures also provided access to standard optical telecommunication bandwidth, owing to the use of off-the-shelf telecom optoelectronic, electronic and EO devices designed for more than 10 Gb s^{−1} optical data transmissions. The single loop oscillating scheme also allowed one to implement unconditionally stable chaos synchronization configuration. Recently, experimental investigations of chaos communications successfully achieved field experiment demonstration over installed fibre-optic network, at more than 10 Gb s^{−1} data transmission. A specifically designed EO set-up was able to generate a chaotic optical phase used to carry the digital data with a DPSK encoding [21]. The demonstration was first performed on the small ring network in our city of Besançon, the so-called Lumière brothers ring network (figure 4). Further experiments over larger networks showed that our laboratory emitter–receiver set-up can be plugged on an installed network, being operational over a more than 100 km transmission link. The link quality of the encrypted transmission was found to be reasonably good, with a net bit error rate at last of 10^{−7}. Such performance can be easily improved to error-free transmission with standard digital error correction encoding.

### (b) Microwave optoelectronic oscillators

The limit cycle solution of delay dynamics might also offer very attractive high-performance features for another engineering applications. This concerns the generation of microwave oscillations at ultra-high spectral purity level, e.g. for radar sources. The actually oscillating microwave frequency *F* is roughly determined as *ω*_{0}/(2*π*), if we adopt the model in equation (2.3) for which the bandpass filter is highly selective, i.e. with *m*≪1. The long fibre delay line is here used to provide a long (compared with the oscillating period) energy storage element, which is typically required when high spectral purity is expected. In usual oscillators, this energy storage function is typically performed by resonators with very high quality factors. Quartz resonators are very well mature devices for achieving extreme stability via piezoelectric effects coupling electrical and mechanical vibrations. However, their Fourier frequency range, when very high stability is concerned, is currently limited up to 100 MHz. In the microwave range, surface acoustic wave devices are making impressive progress, but photonic technologies are offering for the moment the most efficient solutions, for example, through so-called optoelectronic oscillators (OEOs). A basic, but very attractive feature is, moreover, that in contrast to oscillators based on electro-mechanical resonators, their quality factor increases with the operating frequency. This is easily explained owing to the energy storage principle based on the delay of a microwave signal carried by an optical light beam: at a given linear loss coefficient (very low for fibres, typically 0.2 dB km^{−1}), the 1/*e* attenuation corresponds to a fixed fibre length *L* or delay *τ*_{D}, independently of the microwave modulation of the light carrier. Quality factor can then be defined as the number of open loop oscillating periods before their amplitude decay at 1/*e*. Consequently, this quality factor scales linearly with the microwave frequency *F*, i.e. *Q*=(*n*.*L*/*c*) *F*=*τ*_{D} *F*. Such a scaling is moreover valid in principle up to very high frequencies, owing to the huge bandwidth of telecom grade 1.5 μm monomode silica fibres. Practical limitations do not involve the ultra-low loss feature, but other restricting physical phenomena at the origin of time delay fluctuations. This comprises, for example, refractive index variation owing to temperature drifts, or laser phase noise transfer onto the microwave owing to fibre dispersion. These limitations are currently imposing in practical OEOs a fibre length limit of a few kilometres (thus a few tens of microseconds), thus shorter than to the 1/*e* attenuation limit. Figure 5 represents a typical phase noise spectrum recorded from an OEO with a central frequency *F*=10 GHz. It was designed with an accurately thermalized 4 km fibre delay line and a 50 MHz bandwidth ‘selective’ electronic feedback filter. The zoom inset is a high-resolution measurement of the width and height of the first side noise delay mode. This peak occurs at from the central oscillating mode. One can note the extremely thin width (*ca* 40 mHz) and the extreme height (110 dB, from the floor level at −140 to the top level accurately measured in the inset at −30), which are typical signatures of the large delay character of such an OEO.

From the more theoretical and nonlinear dynamics point of view, such extreme parameter conditions have also brought very interesting investigations. Indeed, with respect to the model given in equations (2.2) and (2.3), one has a resonant filtering feedback corresponding to a very low damping *m*≃10^{−3}≪1. This is in strong contrast compared with the very high damping of the previous section, where *m*≃10^{5}≫1. However, though the feedback bandwidth is strongly reduced, the potential complexity or the number of degrees of freedom is not necessarily very low, because the decrease in bandwidth is here compensated by the increase in the time delay. The relatively short delay in the previous section was about tens of nanoseconds; it is here extended to several tens of microseconds. The fundamental Fourier delay mode now corresponds to a much lower frequency (typically 50 kHz versus 10 MHz for chaos communications), and the delay modes are becoming much more dense in the Fourier space. Although narrow bandpass feedback filter is involved around 10 GHz, the density of delay modes still allows for many tens to hundreds of these modes to be potentially involved in the dynamics. The narrow filter width of a few tens of megahertz feeds back potentially many of the 50 kHz spaced delay modes. Highly complex delay dynamics can be expected and was found to lead to other bifurcation scenarios when the feedback gain is increased: for example, a Neimark–Sacker bifurcation occurs from the *ω*_{0}-Hopf limit cycle, leading to a torus characterized by a 2*τ*_{D}-periodic envelope modulation of the *ω*_{0} microwave oscillation [9].

### (c) Photonic nonlinear transient computing

The stable steady state is a solution of *a priori* relatively poor interest. Indeed, if one considers a fixed point, only a very low complexity level is obtained. The stable fixed point is however not a unique feature of the dynamical system it is generated from. One might at least associate with it the corresponding basin of attraction. Depending on the dynamics structure and dimension, the stable fixed point can become much more attractive in terms of related complexity. One could then not only analyse the local structure of a fixed point, but also its entire basin of attraction, as soon as a complex external signal can be designed to address, in principle, any possible trajectory driving to the fixed point: this is precisely what is used in an emerging application, ‘reservoir computing’, which makes use of nonlinear transient complexity to perform computation. ‘Nonlinear transient computing’ was also suggested in [33] and [15] as a naming which might refer more to the physics and nonlinear dynamics viewpoint. This novel computational principle was originally proposed independently in the early 2000s by the computer science/neural network computing community [34] and the brain cognitive research community [35]. It was initially referred to as Echo State Network and Liquid State Machine, respectively. More recently, the unified name of ‘reservoir computing’ [36] was proposed. The concept is closely inspired by and related to standard neural network computing, as represented in figure 6. One of the main differences is to assume that the internal structure of the network does not need to be optimized during the learning phase. This internal structure typically only needs to be once randomly defined, via a connectivity matrix *W*^{I}. This imposes a fixed, but still complex, dynamics for the global neural network. The learning phase then aims only at finding a suitable so-called read-out output via a read-out matrix *W*^{R}. The read-out consists simply in finding a hyperplane in the phase space of the network dynamics, from which the correct answer can easily be deduced. This read-out output consists of a linear combination of some of the network nodes that are animated by the complex transient motion, consecutively to the injection into the network of the information to be processed. A suitable formatting of the input information is required, which is practically defined by an input layer connection to the nodes of the network. This introduces a data connectivity matrix *W*^{D}, that is, the way the input information is injected into the complex phase space. One of the most important advantages compared with classical neural network computing resides in the fact that the learning is strongly simplified. It is usually performed successfully via an always existing solution of a simple ridge regression. This regression results in the calculation of *W*^{R}. Despite its much lower algorithmic complexity which is essentially reduced to the learning procedure, this approach already showed very good results on various complex tasks. Moreover, performances at the level of the most efficient classical neural network approaches have been achieved, and sometimes they even outperform them [37]. Very recently, in the frame of the European project PHOCUS, this novel computational approach was addressed from the experimental point of view, with the use of a delay dynamics instead of a network of nodes. A first approach was tested with an electronic Mackey–Glass delay dynamics [38]. This first electronic demonstration was then also extended to a photonic design [15,39,40] directly inspired by the set-up in figure 1. These successful implementations of reservoir computing with delay dynamics can be explained by a known analogy between infinite-dimensional delay dynamics and spatio-temporal dynamics [41] (figure 7). Standard neural network computers are usually realized as a network of interconnected nodes, such as a network of neurons, thus forming a complex spatio-temporal dynamics. This dynamical analogy between spatio-temporal dynamics and delay dynamics allows *a priori* for the use of a delay dynamics for reservoir computing in the place of the common spatially extended networks. This analogy consists of a representation of a delay dynamics, as a virtual continuous-space and discrete-time dynamics. The discrete-time motion corresponds to a round trip in the delay oscillation loop, indexed by an integer *k* labelling the number of the current *τ*_{D}-interval. The virtual continuous-space consists of the fine temporal motion of the dynamics, of the order of *τ*, and occurring within one ‘coarse’ time delay interval. The nodes of the virtual network are then defined as *N* temporal positions within a time delay. Each node *x*_{n}(*k*)|*n*∈[1,*N*] has a discrete time motion from one delay interval to the next, as *k* is increased to *k*+1. The two-dimensional pattern *x*(*n*,*k*) forms a representation of the transient motion, on which one has to apply the read-out matrix *W*^{R} in order to find the expected answer (figure 6). Experiments based on nonlinear delay dynamics used as the virtual complex dynamical network involved in a nonlinear transient computer have been successfully conducted on benchmark tests, such as spoken digit recognition, time-series prediction and nonlinear channel equalization. Moreover, the achieved performances in these real-world physical set-ups are very close to the ones obtained from pure numerical simulations, opening very interesting perspectives on many practical problems that cannot be solved easily and/or fast enough with standard digital computers.

## 4. Conclusions and future issues

EO delay dynamical systems have triggered during the last 15 years an important scientific activity, nicely balanced and cross-fertilized between fundamental issues and novel applications. It took benefit from the intrinsic complexity and dimensionality of the numerous and various possible dynamics. Powerful applications have been at the origin of specific dynamical features owing to set-up modifications guided by applications. These specific dynamical features have been then also explored from the theoretical point of view. Examples in this review have covered applications concerned with three solutions of delay equations, from high-dimensional chaos to the complex transients leading to a stable fixed point, through the extreme regularity of a limit cycle solution. A typical illustration of the cross-fertilization can be the following one: broadband chaos was used in the frame of secure optical communication, which could only be designed with bandpass optoelectronic feedback, thus resulting in an integral term in the modelling equation. This term was then found to be at the origin of whether new periodic regimes or stable ones that were known to be unstable or metastable.

An important horizon is still open in this framework, and many issues remain to be addressed, again both from the application point of view and from the theoretical one. The nonlinear interactions between the limit cycle features and the phase noise performance in microwave OEOs require fundamental investigations of noisy delay equations. The phase space feature around the stable fixed point used for nonlinear transient computing needs to be more deeply understood, especially from the information theory point of view, in order to optimize the processing power of this novel computational principle.

Last but not least, we anticipate that novel fundamental dynamical properties will appear again via novel design approaches for improved applications involving delay dynamical systems. A probably important technological evolution in delay dynamics, which has already started in the case of external cavity lasers [42], concerns the use of integration possibilities and micro- and nanotechnologies for the realization of delay dynamical systems. This also includes optical disk resonators currently investigated for broadband comb generation and extreme stability time-frequency systems. These set-ups have clearly been identified as involving both the delay in the ring and also distributed nonlinear light–matter interactions between the numerous cavity/delay modes [43].

## Funding statement

This work was more recently essentially supported by the European PHOCUS project (FP7-2009-ICT-240763). L.L. also thanks the support of the Institut universitaire de France for the efficient and highly appreciated research conditions it offered him during the last 5 years, as a junior member of this institution. This project has been performed in cooperation with the Labex ACTION programme (contract ANR-11-LABX-01-01).

## Acknowledgements

Most of the results presented above were obtained in the framework of several projects and collaborations, in which L.L. has participated as a group leader of the FEMTO-ST institute. The many aspects of these results are thus definitely the consequence of numerous ‘highly constructive interferences’ with many co-workers (colleagues, collaborators and PhD students). I first apologize for the ones who might be missing in the following list, but I assure them also of my respectful recognition of their contribution. The main contributors can thus be found at least among the following persons: Jean-Pierre Goedgebuer, Jean-Marc Merolla, Maxime Jacquot, Yanne Chembo, John Dudley, Pierre Lacourt, Vladimir Udaltsov, Thomas Erneux, Pere Colet, Ingo Fischer, Claudio Mirasso, Jia-Ming Liu, Min Won Lee, Rajarshi Roy, Atsushi Uchida, Raymond Quéré, Luis Pesquera, Pierre Glorieux, Yves Pomeau, Maxi San Miguel, Daniel Gauthier, Lucas Illing, Dimitris Syvridis, Apostolos Argyris, Alan Shore, Valerio Annovazzi-Lodi, etc.

## Footnotes

One contribution of 15 to a Theme Issue ‘Dynamics, control and information in delay-coupled systems’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.