Sonia Boscolo

Christophe Finot

Nonlinear pulse shaping in fibres for pulse generation and optical processing

The development of new all-optical technologies for data processing and signal manipulation is a field of growing importance with a strong potential for numerous applications in diverse areas of modern science. Nonlinear phenomena occurring in optical fibres have many attractive features and great,

but not yet fully explored potential in signal processing. Here, we review recent progress on the use of fibre nonlinearities for the generation and shaping of optical pulses and on the applications of advanced pulse shapes in all-optical signal processing. Amongst other topics, we will discuss ultrahigh repetition rate pulse sources, the generation of parabolic shaped pulses in active and passive fibres, the generation of pulses with triangular temporal profiles, and coherent super-continuum sources. The signal processing applications will span optical regeneration, linear distortion compensation, optical decision at the receiver in optical communication systems, spectral and temporal signal doubling and frequency conversion.

Introduction

The use of photonic technologies for data processing in the all-optical domain has a strong potential for a variety of interesting applications in such diverse areas as optical telecommunications, metrology, optical sensing, microwave engineering, advanced microscopy image processing, optical computing and many others. Advantages of processing the information in the all-optical domain include the large available bandwidth and the (potential) parallelism intrinsic to the optical approach, which translate into high processing speeds. However, today electronic techniques of signal manipulation are advanced compared to all-optical processing devices, which are still at the research stage rather than under commercial development. A key feature of electronics, enabling many different applications, is the capability of generating electrical waveforms with arbitrary temporal profiles by use of simple integrated circuits. This capability is so far unmatched in the optical frequency range, where bulky and complex devices are required to shape the light fields. Hence, considerable knowledge still has to be accumulated and new methodologies need to be explored before a true breakthrough can be achieved in this field, allowing the range of functions and operations currently accomplished electronically, to be performed in the optical domain. Furthermore, by using optical rather than electronic processing additional functionality may be possible.

In order to realize such all-optical processing, nonlinear photonics is seen as a key technology. Optical fibre materials exhibit a nonlinear response to strong electric fields, such those of optical signals confined within the small fibre core. In high-speed optical communications such nonlinear effects generally degrade the integrity of the transmitted signal but the same effects can be used to realize a variety of optical functions that have practical applications in the field of light-wave technology. In this paper, we provide a snapshot of recent results and advances in the use of nonlinear effects in optical fibres for optical waveform generation and pulse shaping and in the applications of advanced pulse profiles in all-optical signal processing. We would like to note that the main attention in this paper will be focused on results obtained in our groups, and that it is not our intention here to comprehensively cover all the possible examples of fibre-based pulse shaping and signal processing.

Fibre nonlinearities

The third-order χ (3) optical nonlinearity in silica-based single-mode (SM) fibres is one of the most important effects that can be used for all-optical signal processing. This happens not only because the third-order nonlinearity provides ultra-fast response times in the femtosecond range, but also because it is responsible for a wide range of phenomena such as third-harmonic generation, nonlinear refraction (Kerr nonlinearity), and stimulated Raman and Brillouin scattering [START_REF] Shen | Principles of Nonlinear Optics[END_REF]. The rapid recent developments of micro-structured fibres with extremely small effective core areas and exhibiting enhanced nonlinear characteristics [START_REF] Knight | Photonic crystal fibers[END_REF], and of fibres using materials with refractive indexes higher than that of the silica glass [START_REF] Eggleton | Chalcogenide photonics[END_REF] have enabled dramatic reduction of the required fibre lengths for nonlinear processing compared to conventional fibres. Such progress has paved the way for the integration of fibre-based nonlinear processing functions on photonic chips.

The intensity dependence of the refractive index of a fibre gives rise to different effects depending on the shape of the input signal; the most widely studied effects are self-phase modulation (SPM) and cross-phase modulation (XPM). Considering the localized optical pulse evolution in a fibre medium and neglecting higher-order effects in the fibre, pulse propagation can be described by the nonlinear Schrödinger (NLS) equation [START_REF]Nonlinear Fiber Optics[END_REF]

ψ z = -i β 2 2 ψ tt + iγ|ψ| 2 ψ, (1)
where ψ(z, t) is the slowly varying pulse envelope in the co-moving system of coordinates, β 2 and γ = 2πn 2 /(λA eff ) are the respective group-velocity (second-order) dispersion (GVD) and Kerr nonlinearity coefficients of the fibre, n 2 is the nonlinear-index coefficient, λ the central wavelength of the pulse, and A eff the effective core area. As a result of the nonlinear term in (1), upon propagation in the fibre the pulse acquires an intensity-dependent nonlinear phase shift as φ(z, t) = γ|u(0, t)| 2 z, namely the frequency chirp -φ t . When the length scale associated with the pulse where the GVD effects take place: L D = T 2 0 /|β 2 | (with T 0 being some temporal characteristic value of the initial pulse) is much larger compared with both the fibre length and the nonlinear length L NL = 1/(γP 0 ) (with P 0 being the peak power of the initial pulse), the dispersive term in (1) can be neglected. In this case, the presence of a chirp causes a nonlinear broadening of the pulse spectrum. When two (or more) pulses co-propagate inside a fibre, one still obtains an NLS equation for each pulse but these equations are coupled through XPM [START_REF]Nonlinear Fiber Optics[END_REF]. In this case, the nonlinear phase evolution for each pulse depends also on the power of the other pulses according to the expression φ i (z, t) = γ i (|ψ i (0, t)| 2 + 2 j =i |ψ j (0, t)| 2 )z. XPM occurs only when the pulses overlap. Similar to the SPM case, XPM leads to additional nonlinear chirping and spectral shaping.

When the effects of chromatic dispersion are considered in combination with the Kerr nonlinearity, a rich pulse dynamics arises from the interplay between dispersive and nonlinear effects depending on the sign of the dispersion and the relative magnitudes of the associated length scales. A well-known and fascinating example is the formation of optical solitons [START_REF] Hasegawa | Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers I. Anomalous dispersion[END_REF][START_REF] Mollenauer | Solitons in optical fibers: fundamentals and applications[END_REF] in the anomalous dispersion regime of a fibre (β 2 < 0) as a result of a cooperation between GVD and SPM. On the contrary, in the regime of normal dispersion pulse dynamics is highly affected by the phenomenon of optical wave breaking [START_REF] Tomlinson | Optical wave-breaking of pulses in nonlinear optical fibers[END_REF][START_REF] Anderson | Wave-breaking in nonlinear optical fibers[END_REF][START_REF] Finot | Beneficial impact of wave-breaking for coherent continuum formation in normally dispersive nonlinear fibers[END_REF].

In the next Section we will discuss some newly emerged qualitative features of the optical pulse evolution in a fibre under the combined action of dispersion and nonlinearity.

Optical pulse generation and shaping using fibre nonlinearities

Techniques for generating, controlling, manipulating and measuring ultrashort optical pulses and specialized waveforms have become increasingly important in many scientific areas including, amongst others, ultrahigh-speed optical communications, optical signal processing and bio-photonics. To date, conventional picosecond and femtosecond pulse shaping in the optical domain has been implemented using devices, such as liquid crystal spatial light modulators [START_REF] Weiner | Femtosecond pulse shaping using spatial light modulators[END_REF][START_REF] Clarke | Reconfigurable optical pulse generator employing a Fourier-domain programmable optical processor[END_REF], wavelength-to-time mapping [START_REF] Nguyen | Dynamic parabolic pulse generation using temporal shaping of wavelength to time mapped pulses[END_REF][START_REF] Ye | Photonic generation of triangular-shaped pulses based on frequency-to-time conversion[END_REF], acousto-optic modulators and electro-optical phase arrays, that impart user-specified spectral amplitude and phase to the pulse. Though powerful and flexible, this approach has the drawback of requiring a rather bulky and expensive apparatus, entailing a level of cost and complexity that is not often commensurate to the used laser system and target application. In order to fulfill telecommunication requirements, more compact techniques for pulse shaping have been developed, which include the use of super-structured fibre Bragg gratings [START_REF] Petropoulos | Rectangular pulse generations based on pulse reshaping using a superstructured fiber Bragg grating[END_REF], long-period fibre grating filters [START_REF] Krcmarik | Nonlinear pulse compression of picosecond parabolic-like pulses synthesized with a long period fiber grating filter[END_REF], and arrayed waveguide gratings that are well suited for line by line processing of high-repetition periodic pulse shapes [START_REF] Hirooka | Bright and dark 40 GHz parabolic pulse generation using a picosecond optical pulse train and an arrayed waveguide grating[END_REF][START_REF] Kashiwagi | Fiber transmission characteristics of parabolic pulses generated by optical pulse synthesizer[END_REF]. However, in all these pulse shapers using linear techniques, the bandwidth of the input pulse determines the maximum bandwidth of the output. Indeed, a linear manipulation cannot increase the pulse bandwidth, and so to create shorter pulses nonlinear effects must be used. This limitation of the linear approaches becomes very stringent when synthesizing advanced pulse shapes such as rectangular pulses, for which a large optical bandwidth is required to reproduce accurately the spectral side-lobes linked to the temporal compactness of the pulse. Moreover, in those applications that require spectral narrowing, linear filtering introduces a power penalty that is at least proportional to the ratio of the target spectrum to the input one. The combination of third-order nonlinear processes and chromatic dispersion in optical fibres can provide efficient new solutions to overcome the aforementioned limitations. The pulse shaping examples discussed below highlight some of the advantages offered by a passive, all-optical nonlinear fibre-based approach to pulse shaping.

Ultrahigh repetition rate pulse train generation

As a first illustration of the possible nonlinear pulse shaping mechanisms in optical fibres, we review here the generation of ultrashort pulse trains with very high repetition rates resulting from the passive evolution of optical pulses in an anomalously dispersive fibre in the presence of Kerr nonlinearity. As mentioned in the previous Section, in the anomalous GVD regime of a fibre, the NLS equation (1) admits families of soliton solutions. While fundamental solitons are characterized by an exact balance between the GVD and SPM effects (N = L D /L NL = 1), in the case of higher-order solitons (N > 1) SPM dominates initially but GVD soon catches up and leads to pulse contraction. This stage of temporal compression can be used for generating pulses with ultrashort durations [START_REF] Kibler | Optimised one-step compression of femtosecond fibre laser soliton pulses around 1550 nm to below 30 fs in highly nonlinear fibre[END_REF][START_REF] Amorim | Sub-two-cycle pulses by soliton self-compression in highly nonlinear photonic crystal fibers[END_REF][START_REF] Mollenauer | Experimental observation of picosecond pulse narrowing and solitons in optical fibers[END_REF]. However, the compressed pulses exhibit a low amplitude structure outside the central lobe. An alternative technique that prevents sidelobe formation relies on the progressive evolution of the peak power and temporal duration of a fundamental soliton in a distributed amplifier or a dispersion decreasing (DD) fibre [START_REF] Chernikov | Femtosecond soliton propagation in fibers with slowly decreasing dispersion[END_REF][START_REF] Chernikov | Picosecond soliton pulse compressor based on dispersion decreasing fibre[END_REF].

Step-like or comb-like [START_REF] Chernikov | Comblike dispersion-profiled fiber for soliton pulse train generation[END_REF] dispersion profiled structures based on carefully chosen sets of fibres have been successfully used to emulate continuous DD fibre profiles.

The generation of high repetition rate pulse sources has largely benefited from the nonlinear compression effect taking place in the anomalous dispersion region, and several works have demonstrated the nonlinear reshaping of a sinusoidal beat-signal into well-separated pulses, the pulse repetition rate being simply determined by the frequency separation between the two continuous-wave (CW) laser sources. This technique has been successfully demonstrated with various experimental setups including DD fibres, adiabatic Raman compression in standard fibres and step-like and comb-like dispersion profiled fibres. More recently, this nonlinear compression effect has been observed through a multiple four-wave mixing (FWM) process in a single anomalously dispersive fibre, and repetition rates ranging from a few tens GHz to 1 THz have been achieved [START_REF] Fatome | 20-GHz-to-1-THz repetition rate pulse sources based on multiple four-wave mixing in optical fibers[END_REF]. In the frequency domain, such nonlinear compression can be viewed as the result of a modulation instability [START_REF] Tai | Observation of modulational instability in optical fibers[END_REF] experienced by the initial beat-signal during propagation in the fibre, with the generation of new sidebands via FWM: two pump photons at ω 1 and ω 2 interact to generate two first-order sideband photons at ω 1 -Ω and ω 2 + Ω, Ω = ω 2 -ω 1 being the pump frequency detuning. High-order sidebands are then generated due to multiple FWM interactions.

A typical experimental setup for the generation of high repetition rate pulse trains at telecommunication wavelengths is sketched in Fig. 1a. The initial beat-signal is easily obtained via temporal superposition of two CWs with slightly different wavelengths delivered by external cavity lasers or a single CW directly modulated by an intensity modulator. A phase modulator driven at a frequency of a few hundreds of megahertz is used to avoid the deleterious effects of Brillouin back-scattering. The beatsignal is then amplified up to the Watt level by an erbium-doped fibre amplifier (EDFA), and propagated through a suitable optical fibre with anomalous dispersion to realize the nonlinear reshaping. According to the pulse source parameters that are chosen (pulse repetition rate, initial peak power, nonlinearity and dispersion coefficients of the compression fibre, and fibre length), various output pulse shapes can be obtained. Intensity and phase measurements based on the frequency resolved optical gating (FROG) technique [START_REF] Trebino | Frequency-Resolved Optical Gating: The measurement of Ultrashort Laser Pulses[END_REF] show that high-quality compressed pulses with a Gaussian intensity profile and nearly constant phase can be achieved at a typical duty cycle of 1/5 [START_REF] Fatome | 20-GHz-to-1-THz repetition rate pulse sources based on multiple four-wave mixing in optical fibers[END_REF][START_REF] Fatome | Multiple four-wave mixing in optical fibers: 1.5-3.4-THz femtosecond pulse sources and real-time monitoring of a 20-GHz picosecond source[END_REF] (Fig. 1b). Lower duty cycles can still be obtained with a single fibre, but at the expenses of increased sidelobes in the wings of the pulses [START_REF] Pitois | Generation of 20-Ghz picosecond pulse trains in the normal and anomalous dispersion regimes of optical fibers[END_REF]. Such sidelobes can be partly avoided by using specially designed arrangements of segments of fibres [START_REF] Finot | All-fibered high-quality low duty-cycle 20-GHz and 40-GHz picosecond pulse sources[END_REF][START_REF] Fortier | All-fibered high-quality low duty-cycle 160-GHz femtosecond pulse source[END_REF][START_REF] Mansouri | All-fibered high-quality stable 20-GHz and 40-GHz picosecond pulse generators[END_REF] or other comb-like structures. for the optimum output shape and for the best compression, respectively. The temporal intensity profiles at different stages as predicted by numerical simulations (subplots 1) are compared to the experimental results (subplots 2) obtained by the FROG technique (b2 -results adapted from [START_REF] Fatome | 20-GHz-to-1-THz repetition rate pulse sources based on multiple four-wave mixing in optical fibers[END_REF]) or by an optical sampling oscilloscope (c2 -results adapted from [START_REF] Hammani | Peregrine soliton generation and breakup in standard telecommunications fiber[END_REF]).

We would like to note that the reshaping of a sinusoidal beat-signal into ultrashort structures can be understood in terms of the nonlinear dynamics of "Akhmediev breathers" [START_REF] Akhmediev | Modulation instability and periodic solutions of the nonlinear Schrödinger equation[END_REF][START_REF] Hammani | Spectral dynamics of modulation instability described using Akhmediev breather theory[END_REF]. In this context, the extreme spatio-temporal localization that can be observed (see, e.g., Fig. 1c) is explained as a result of the formation of Peregrine solitons [START_REF] Kibler | The Peregrine soliton in nonlinear fibre optics[END_REF][START_REF] Hammani | Peregrine soliton generation and breakup in standard telecommunications fiber[END_REF], mathematical solutions of the standard NLS equation that have been first derived in 1983 in the hydrodynamics context [START_REF] Peregrine | Water waves, nonlinear Schrödinger equations and their solutions[END_REF] but have never been demonstrated experimentally before 2010 and using experiments in optical fibres. These observations offer a promising way to better understand the problem of the occurrence of rogue waves in optics and hydrodynamics [START_REF] Dudley | Extreme events in optics: Challenges of the MANUREVA project[END_REF][START_REF] Solli | Optical rogue waves[END_REF].

Generation of parabolic pulses in active and passive fibres

Strong pulse reshaping can also occur in the normal regime of dispersion of a fibre. An important example that has generated a great deal of attention in the last decade is the occurrence of self-similar dynamical effects in the nonlinear propagation of ultrashort pulses in normally dispersive (ND) optical fibre amplifying media [START_REF] Dudley | Self-similarity in ultrafast nonlinear optics[END_REF]. Results have demonstrated a fundamentally new operating regime where the interaction of Kerr nonlinearity, dispersion and optical gain is exploited to generate a particular class of pulses with a parabolic intensity profile and a strictly linear frequency chirp. These pulses then propagate in a self-similar manner, holding certain relations (scaling) between pulse power, duration and chirp parameter. The results for the self-similar parabolic pulses are obtained by assuming that a fibre amplifier can be described by the NLS equation with gain

ψ z = -i β 2 2 ψ tt + iγ|ψ| 2 ψ + g 2 ψ, (2)
where g is the distributed gain coefficient. Using a self-similar ansatz and standard amplitude/phase decomposition, one can find self-similar parabolic pulse solutions in the quasi-classical limit of Eq. ( 2)

( β 2 |(|ψ|) tt |/(2γ|ψ| 3 ) ≪ 1 )
, whose dynamic evolution is summarized as [START_REF] Fermann | Self-similar propagation and amplification of parabolic pulses in optical fibers[END_REF][START_REF] Boscolo | Self-similar parabolic optical solitary waves[END_REF]

|ψ(z, t)| = a(z) 1 -t 2 /τ (z) 2 θ(τ (z) -|t|), arg ψ(z, t) = ib(z)t 2 + φ 0 (z), (3a)
a(z) = a 0 e gz/3 , τ (z) = τ 0 e gz/3 , b(z) = -g/(6β 2 ), ( 3b
)
in the case of constant gain. Here, θ(x) is the Heaviside function. The amplitude term indeed reveals a parabolic intensity profile, and the phase term a parabolic phase modulation or linear chirp with chirp rate b/2. The peak amplitude and width of the pulse scale from their initial values, and the initial values depend only on the initial energy and amplifier parameters. These solutions possess the remarkable property of being a global attractor to the system for arbitrary initial conditions [START_REF] Kruglov | Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters[END_REF][START_REF] Finot | Influence of the initial phase profile on the asymptotic self-similar parabolic dynamics[END_REF].

By analogy with the well-known stable dynamics of solitary waves -solitons, these self-similar parabolic pulses have come to be known as "similaritons". In contrast to solitons, similaritons can tolerate strong nonlinearity without wave breaking. The normal GVD effectively linearizes the accumulated phase of the pulse allowing for the spectral bandwidth to increase without destabilizing the pulse [START_REF] Anderson | Wave-breaking-free pulses in nonlinear-optical fibers[END_REF]. The unique properties of similariton pulses have motivated many theoretical and experimental studies (see, e.g., [START_REF] Finot | Influence of the initial phase profile on the asymptotic self-similar parabolic dynamics[END_REF][START_REF] Boscolo | Self-similar parabolic optical solitary waves[END_REF][START_REF] Finot | Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers[END_REF][START_REF] Billet | Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm[END_REF][START_REF] Bale | Impact of third-order fibre dispersion on the evolution of parabolic optical pulses[END_REF][START_REF] Bale | Effects of fourth-order fiber dispersion on ultrashort parabolic optical pulses in the normal dispersion regime[END_REF]). Experimental demonstrations of similariton generation relying on amplification from either rare-earth doping (with ytterbium [START_REF] Fermann | Self-similar propagation and amplification of parabolic pulses in optical fibers[END_REF][START_REF] Limpert | High-power femtosecond Yb-doped fiber amplifier[END_REF][START_REF] Malinowski | Ultrashort-pulse Yb3+ fiber based laser and amplifier system producing ¿ 25 W average power[END_REF][START_REF] Dupriez | High-power, high repetition rate picosecond and femtosecond sources based on Yb-doped fiber amplification of VECSELS[END_REF][START_REF] Papadopoulos | Generation of 63 fs 4.1 MW peak power pulses from a parabolic fiber amplifier operated beyond the gain bandwith limit[END_REF] or erbium [START_REF] Billet | Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm[END_REF] dopants) or Raman scattering [START_REF] Finot | Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers[END_REF][START_REF] Finot | Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers[END_REF][START_REF] Hammani | Real time measurement of long parabolic optical similaritons[END_REF] have been achieved, confirming the potential of this method especially when dealing with the generation of high-power ultrashort pulses. Thanks to the high level of linearity of the chirp developed during parabolic amplification and as long as the impact of third-order fibre dispersion is moderate [START_REF] Bale | Impact of third-order fibre dispersion on the evolution of parabolic optical pulses[END_REF][START_REF] Zhang | Propagation of high-power parabolic pulses in cubicon fiber amplifiers[END_REF][START_REF] Hammani | Propagation and breakup of pulses in fiber amplifiers and dispersion-decreasing fibers with third-order dispersion[END_REF], it is possible to efficiently compensate for the chirp slope, so that compressed pulses with very low substructures and temporal durations much shorter than the initial seed pulse can be obtained [START_REF] Fermann | Self-similar propagation and amplification of parabolic pulses in optical fibers[END_REF][START_REF] Limpert | High-power femtosecond Yb-doped fiber amplifier[END_REF][START_REF] Malinowski | Ultrashort-pulse Yb3+ fiber based laser and amplifier system producing ¿ 25 W average power[END_REF][START_REF] Dupriez | High-power, high repetition rate picosecond and femtosecond sources based on Yb-doped fiber amplification of VECSELS[END_REF][START_REF] Papadopoulos | Generation of 63 fs 4.1 MW peak power pulses from a parabolic fiber amplifier operated beyond the gain bandwith limit[END_REF][START_REF] Billet | Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm[END_REF]. As an illustration, Figs. 2a and2b show the setup and results of an experiment where similaritons are generated through Raman amplification at telecommunication wavelengths [START_REF] Hammani | Real time measurement of long parabolic optical similaritons[END_REF]. Moreover, recent fibre lasers that use self-similar pulse shaping in the normal dispersion regime have been demonstrated to achieve high-energy pulses [START_REF] Ilday | Self-similar evolution of parabolic pulses in a laser[END_REF][START_REF] Wise | High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion[END_REF][START_REF] Aguergaray | Experimental realization of a modelocked parabolic Raman fiber oscillator[END_REF][START_REF] Oktem | Soliton-similariton fibre laser[END_REF].

In addition to fibre amplifiers and lasers, stable similariton pulses can be generated in passive fibres provided a suitable longitudinal variation of the dispersion is introduced [START_REF] Hirooka | Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group-velocity dispersion[END_REF][START_REF] Kibler | Parabolic pulse generation in comb-like profiled dispersion decreasing fibre[END_REF][START_REF] Plocky | Experiments on the generation of parabolic pulses in waveguides with length-varying normal chromatic dispersion[END_REF][START_REF] Finot | Parabolic pulse generation with active or passive dispersion decreasing optical fibers[END_REF] (Fig. 2c). This approach is based on the observation that a longitudinal decrease of the normal dispersion is formally equivalent to linear gain. Recently, a simple approach to the generation of parabolic pulses that uses progressive nonlinear pulse reshaping in a ND fibre with fixed dispersion has been demonstrated [START_REF] Finot | Parabolic pulse generation through passive nonlinear pulse reshaping in a normally dispersive two segment fiber device[END_REF].

However, in contrast with the asymptotic similariton solutions obtained in fibre amplifiers, the generated parabolic waveforms represent transient states of the nonlinear pulse evolution in the passive fibre medium [START_REF] Boscolo | Passive nonlinear pulse shaping in normally dispersive fiber systems[END_REF]. As such, they have a finite life distance that depends sensibly on the initial conditions (pulse shape, energy and chirp profile). Nevertheless, stabilization of the parabolic features is possible by use of a second propagation stage in a fibre with specially adjusted nonlinear and dispersive characteristics relative to the first fibre [START_REF] Finot | Parabolic pulse generation through passive nonlinear pulse reshaping in a normally dispersive two segment fiber device[END_REF][START_REF] Boscolo | Passive nonlinear pulse shaping in normally dispersive fiber systems[END_REF].

Generation of triangular pulses

Parabolic shapes are not the only pulse waveforms that can be generated in a passive ND fibre. Recently, the combination of pulse pre-chirping and nonlinear propagation in a section of ND fibre has been introduced as a method for passive nonlinear pulse shaping, which provides a simple way of generating various advanced field distributions, including flat-top-and triangular-profiled pulses with a linear chirp [START_REF] Boscolo | Passive nonlinear pulse shaping in normally dispersive fiber systems[END_REF]. In this scheme, Kerr nonlinearity and GVD lead to various reshaping processes of an initial conventional pulse (e.g., a Gaussian pulse) according to the chirping value and power level at the entrance of the fibre. In particular, triangular pulses can be generated for a positive initial chirp parameter (using the definition ibt 2 for the phase profile) and sufficiently high energies. These theoretical results have been confirmed experimentally by intensity and phase measurements of the generated pulses [START_REF] Wang | Generation of triangular-shaped optical pulses in normally dispersive fibre[END_REF] (Fig. 3b). In the experimental setup used (Fig. 3a), the control of the pulse prechirping value was realized by propagation through different lengths of standard SM fibre with anomalous GVD, which imposed a positive chirp parameter on the pulse. The pre-chirped pulses were amplified to different power levels using an EDFA, and then propagated through a ND fibre to realize the pulse reshaping.

Different nonlinear dynamics may lead to the generation of triangular pulses in a passive ND fibre. Indeed, it has been recently shown qualitatively and numerically that temporal triangular intensity profiles can result from the progressive reshaping of initially parabolic pulses driven by the fourth-order dispersion (FOD) of the fibre [START_REF] Bale | Effects of fourth-order fiber dispersion on ultrashort parabolic optical pulses in the normal dispersion regime[END_REF]. The overall temporal effect of FOD on parabolic pulse propagation is to stretch and enhance the power reduction in the pulse wings, leading to a triangular profile. Furthermore, the possibility of triangular pulse shaping in mode-locked ring-cavity fibre lasers has been first reported in [START_REF] Boscolo | Pulse shaping in mode-locked ring-cavity fibre lasers[END_REF]. It has been numerically demonstrated that for normal net dispersion, formation of two distinct steady-state solutions of stable single pulses can be obtained in a laser cavity in different regions of the system parameter space: the previously known similariton [START_REF] Ilday | Self-similar evolution of parabolic pulses in a laser[END_REF] and a triangular-profiled pulse with a linear frequency chirp.

Coherent continuums for optical telecommunications

One of the most well-known examples of pulse reshaping in the spectral domain is in super-continuum (SC) generation. SC generation in optical fibres and, more specifically, in photonic crystal fibres with enhanced nonlinearity, is an established technique for producing broadband light sources in a wide range of research fields including metrology, bio-photonics and optical telecommunications [START_REF] Alfano | The Supercontinuum Laser Source[END_REF][START_REF] Dudley | Supercontinuum Generation in Optical Fibers[END_REF]. In telecommunications, the broadened spectrum can be spectrally sliced to generate wavelength multiplexed pulse trains [START_REF] Boivin | Spectrum slicing of coherent sources in optical communications[END_REF][START_REF] Takushima | 10-GHz, over 20-channel multiwavelength pulse source by slicing super-continuum spectrum generated in normal-dispersion fiber[END_REF][START_REF] Yousoff | A 36-channel x 10-GHz spectrally sliced pulse source based on supercontinuum generation in normally dispersive highly nonlinear holey fiber[END_REF]. The principle of operation of a multi-wavelength source is illustrated in Fig. 4. An initial high-power mode-locked picosecond or femtosecond pulse train with a repetition rate of several GHz is spectrally broadened through SPM in a km-long highly nonlinear (HNL) fibre. At the fibre output, the spectrum is sliced by adequate bandpass filters (BPFs) to obtain pulse trains that have the same repetition rate as the initial train but new central frequencies. Typically the broadest spectra are generated in anomalously dispersive fibres, where the broadening mechanism is strongly influenced by soliton dynamics [START_REF] Dudley | Supercontinuum generation in photonic crystal fiber[END_REF]. However, solitonic effects and modulation instability lead to decreased temporal coherence and spectral flatness of the continuum generated in the anomalous dispersion regime [START_REF] Nakazawa | Coherence degradation in the process of supercontinuum generation in an optical fiber[END_REF]. By contrast, the nonlinear pulse evolution in ND fibres leads to flatter spectra and improved temporal coherence, but at the expenses of relatively narrower spectra [START_REF] Heidt | Pulse preserving flat-top supercontinuum generation in all-normal dispersion photonic crystal fibers[END_REF][START_REF] Hooper | Coherent supercontinuum generation in photonic crystal fiber with all-normal group velocity dispersion[END_REF]. Hence frequency broadening using a ND fibre is an attractive choice whenever a moderate amount of broadening alongside good spectral and temporal stability of the output pulses are targeted. In this context, the propagation of a parabolic pulse in a ND-HNL fibre, which exploits the parabolic pulse ability to resist the effects of optical wave-breaking, has been demonstrated to enhance the flatness of the resulting continuum over the whole C-band of optical telecommunications [START_REF] Parmigiani | Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating[END_REF][START_REF] Krcmarik | Nonlinear pulse compression of picosecond parabolic-like pulses synthesized with a long period fiber grating filter[END_REF][START_REF] Clarke | Reconfigurable optical pulse generator employing a Fourier-domain programmable optical processor[END_REF].

Applications of parabolic and triangular pulses in signal processing

Applications of parabolic pulses have not been limited to ultrashort high-power pulse generation and highly coherent continuum sources, but several optical processing techniques have taken advantage of their specific features. These include optical regeneration [START_REF] Finot | Regenerative 40 Gbit/s wavelength converter based on similariton generation[END_REF][START_REF] Boscolo | All-optical signal regeneration by temporal slicing of nonlinearly flattened optical waveform[END_REF] including pulse re-timing [START_REF] Parmigiani | Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating[END_REF], prereceiver nonlinear processing in the optical domain [START_REF] Boscolo | Time domain all-optical signal processing at a RZ optical receiver[END_REF][START_REF] Nasieva | Bit error rate improvement by nonlinear optical decision element[END_REF], and mitigation of linear waveform distortions [START_REF] Ng | Compensation of linear distortions by using XPM with parabolic pulses as a time lens[END_REF][START_REF] Hirooka | All-optical 40-GHz Time-Domain Fourier Transformation using XPM with a dark parabolic pulse[END_REF]. The simple intensity profile of triangular pulses is also highly desired for a range of processing applications, including time domain add-drop multiplexing [START_REF] Li | OTDM add-drop multiplexer based on XPMinduced wavelength shifting in highly nonlinear fiber[END_REF][START_REF] Parmigiani | Time domain add-drop multiplexing scheme enhanced using a saw-tooth pulse shaper[END_REF], wavelength conversion [START_REF] Parmigiani | An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper[END_REF][START_REF] Parmigiani | Efficient all-optical wavelengthconversion scheme based on a saw-tooth pulse shaper[END_REF][START_REF] Verscheure | Pulse doubling and wavelength conversion through triangular nonlinear pulse reshaping[END_REF], optical signal doubling [START_REF] Latkin | Doubling of optical signals using triangular pulses[END_REF], time-to-frequency mapping of multiplexed signals [START_REF] Bhamber | All-optical TDM to WDM signal conversion and partial regeneration using XPM with triangular pulses[END_REF] and enhanced spectral compression [START_REF] Andresen | Transform-limited spectral compression by self-phase modulation of amplitude-shaped pulses with negative chirp[END_REF][START_REF] Finot | Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime[END_REF]. In this Section, we will review some of these applications.

2R regeneration

As a first example, here we discuss the usage of parabolic pulses in 2R (re-amplification, re-timing) signal regeneration in an extension of the technique proposed by Mamyshev [START_REF] Mamyshev | All-optical data regeneration based on self-phase modulation effect[END_REF]. The method relies on SPM in a nonlinear medium (fibre) and subsequent offset spectral filtering to suppress the noise in the zero-bit slots and the amplitude fluctuations in the one-bit pulses of return-to-zero (RZ) optical data streams.

In the schematic of the Mamyshev regenerator depicted in Fig. 5a, the signal to be regenerated is first amplified to the optimum power level, and an optical BPF is also used to reduce the amplified spontaneous emission noise power. Then spectral broadening is achieved in a HNL fibre, and the broadened spectrum is partially sliced by a second BPF, which is shifted with respect to the input signal frequency in order to transmit SPM-generated frequencies and reject the original spectrum. The central frequency and output pulse shape of the regenerated signal are determined by the central frequency and spectral profile of the offset filter, respectively. The SPM phase shift experienced in the HNL fibre depends on the instantaneous power (cf. Section 2). Hence, a high-intensity pulse induces more SPM phase shift and spectrally broadens more than the low-intensity noise [START_REF] Mamyshev | All-optical data regeneration based on self-phase modulation effect[END_REF]. Filtering the new SPM-generated frequencies with the offset filter leads to a power transfer function for the pulses and the noise which is close to a binary one [START_REF] Provost | Design scaling rules for 2R-optical self-phase modulation-based regenerators[END_REF]. Spectral filtering of similariton pulses can improve the ability of the Mamyshev regenerator to eliminate spurious noise pulses as well as to simultaneously reduce any fluctuations in the signal one-bit level [START_REF] Finot | Regenerative 40 Gbit/s wavelength converter based on similariton generation[END_REF]. This ability relies on the specific features of self-similar amplification. Indeed, a spectrum broadened through self-similar amplification has a smooth profile (cf. Section 3.4), whereas a SPM-broadened spectrum is generally accompanied by oscillatory structures covering the entire frequency range. Moreover, the spectrum and chirp obtained by SPM both depend on the details of the initial pulse (shape, chirp and energy), leading to some amplitude and time fluctuations of the regenerated pulses. In contrast, the similariton-based spectral broadening and similariton-induced chirp depend only on the initial pulse energy and/or amplifier parameters, respectively (cf. Section 3.2), so that both amplitude and timing jitter are reduced. In particular, the energy-dependent spectral broadening is responsible for efficient discrimination of high-energy pulses (ones) from low-energy pulses (zeros) with the offset BPF. An im-plementation of the similariton-based regeneration technique at 40 Gbit/s data rate where similariton pulses are generated through distributed Raman amplification in a ND fibre is illustrated in Figs. 5b,c. It is worth noting here that a distributed amplification is helpful in increasing the energy yield of the regenerator which is a highly dissipative element [START_REF] Finot | Active Mamyshev regenerator[END_REF].

Linear-distortion compensation

As a second example of the application of parabolic pulses in signal processing, we review here the realization of a time-domain all-optical Fourier transformation (OFT) technique. Time-domain OFT in which the spectral profile is converted into a waveform in the time domain, makes it possible to eliminate the waveform distortion caused by linear perturbations in optical fibres, such as higher-order dispersion, jitter, and polarization-mode dispersion [START_REF] Romagnoli | Time-domain Fourier-optics for polarization-mode dispersion[END_REF][START_REF] Jiang | Timing jitter eater for optical pulse trains[END_REF][START_REF] Nakazawa | Distortion-free optical transmission using time-domain optical Fourier transformation and transform-limited optical pulses[END_REF][START_REF] Hirooka | Optical adaptive equalization of high-speed signals using timedomain optical Fourier transformation[END_REF]. Time-domain OFT is realized ideally by applying a quadratic temporal phase modulation (time lens) to a distorted pulse: ψ chirp (t) = ψ(t) exp(ibt 2 ), and passing the linearly chirped pulse through a dispersive medium such as a short piece of fibre. When the accumulated dispersion of the medium D = β 2 L equates the inverse of the chirp coefficient b (i.e., when the chirp is fully compensated), the output waveform is proportional to the spectral amplitude before OFT: χ(t) = i/(2πD) exp(-ibt 2 ) ψ(t/D), where ψ(ω) is the Fourier transform of ψ(t), and the frequency ω is converted into time t through the relationship ω = t/D. Since generally the spectral envelope profile has no distortion even if its time-domain waveform is distorted by linear perturbations, this frequency-to-time mapping indicates that a distortion-free pulse waveform can be obtained in the time domain after OFT. In [START_REF] Romagnoli | Time-domain Fourier-optics for polarization-mode dispersion[END_REF][START_REF] Jiang | Timing jitter eater for optical pulse trains[END_REF][START_REF] Nakazawa | Distortion-free optical transmission using time-domain optical Fourier transformation and transform-limited optical pulses[END_REF][START_REF] Hirooka | Optical adaptive equalization of high-speed signals using timedomain optical Fourier transformation[END_REF], this compensation technique was investigated by using electro-optic phase modulators. But this way the quadratic modulation region was restricted to around the center of the sinusoidal modulation imparted by the phase modulator. Thus, any parts of the pulse extending beyond the quadratic region could not be properly transformed, and only the section of the target pulse sitting within the quadratic region could be compensated. The use of XPM with parabolic pulses as a time lens has been recently proposed as a promising way to overcome the fore-mentioned limitation [START_REF] Ng | Compensation of linear distortions by using XPM with parabolic pulses as a time lens[END_REF][START_REF] Hirooka | All-optical 40-GHz Time-Domain Fourier Transformation using XPM with a dark parabolic pulse[END_REF]. Indeed, the chirp induced by XPM on the target pulse is proportional to the gradient of the intensity profile of the parabolic control pulse: -φ t = -2bt, b = ∓2γa 2 L/τ 2 , so it is exactly linear and thus ideal OFT can be achieved by complete chirp compensation. Here, a and τ are the respective peak amplitude and characteristic width of the parabolic pulse (cf. Section 3.2), and the sign ∓ is for either a standard "bright" or a dark parabolic pulse [START_REF] Hirooka | Bright and dark 40 GHz parabolic pulse generation using a picosecond optical pulse train and an arrayed waveguide grating[END_REF]. The parabolic XPM-based OFT technique has been successfully experimentally demonstrated to achieve cancellation of the third-order dispersion-induced distortion of short optical pulses [START_REF] Ng | Compensation of linear distortions by using XPM with parabolic pulses as a time lens[END_REF][START_REF] Hirooka | All-optical 40-GHz Time-Domain Fourier Transformation using XPM with a dark parabolic pulse[END_REF].

Optical decision at the receiver

In this section, we review a possible application of parabolic/flat-top pulses to the improvement of the signal bit-error rate (BER) in an optical communication system by using optical decision before the receiver. An attractive approach to the design of the optical receiver for high-speed fibre communication systems using RZ data formats is based on employing an additional all-optical decision element (ODE) just before the conventional receiver [START_REF] Yoshikane | Benefit of SPMbased all-optical reshaper for long-haul DWDM transmission systems[END_REF][START_REF] Matsuura | Performance improvement of optical RZ-receiver by utilizing an all-optical waveform converter[END_REF][START_REF] Ghelfi | Impact of an additional all-optical decision element in band-limited receivers for RZ systems[END_REF][START_REF] Boscolo | Time domain all-optical signal processing at a RZ optical receiver[END_REF]. In this way, the first decision is carried out in the optical domain, thus improving the quality of the received signal. The attractive possibility of improving the BER by using optical devices has been studied in some recent works [START_REF] Rochette | Bit error ratio improvement with 2R optical regenerators[END_REF][START_REF] Rochette | 3R optical regeneration: An all-optical solution with BER improvement[END_REF][START_REF] Nasieva | Bit error rate improvement by nonlinear optical decision element[END_REF]. The conceptual basis of the BER improvement by optical devices comes from the information theory: an intrinsic BER improvement is possible whenever the decision performed by the device in the optical domain is based on information that is different from the information used by the decision circuit of the (sub-optimal) receiver in the electrical domain. In [START_REF] Nasieva | Bit error rate improvement by nonlinear optical decision element[END_REF] an application of this general concept was demonstrated, based on using an advanced optical receiver enhanced by a nonlinear ODE. The example of an ODE scheme used in [START_REF] Nasieva | Bit error rate improvement by nonlinear optical decision element[END_REF] exploits the nonlinear mechanism that leads to the formation of parabolic pulses and the resultant flattening of the signal waveform (cf. Section 3.2). The method is based on the nonlinear reshaping of a pulse towards a broad and flat temporal waveform that can occur upon propagation in a ND fibre [START_REF] Finot | Beneficial impact of wave-breaking for coherent continuum formation in normally dispersive nonlinear fibers[END_REF], and subsequent slicing of the pulse temporal profile by an optical temporal gate (e.g., an amplitude modulator with a sharp nonlinear transfer function or a nonlinear optical loop mirror provided with a clock) [START_REF] Boscolo | All-optical signal regeneration by temporal slicing of nonlinearly flattened optical waveform[END_REF] or XPM-based techniques. The basic mechanism that is responsible for BER improvement in this scheme can be understood in the following way. The pulse temporal broadening and flattening in the ND fibre permits to recover the center of the bit slot for those pulses that are significantly shifted from the bit slot center in the input pulse train because of timing jitter (Fig. 7a). Such strongly time-shifted pulses, once they are converted into electrical pulses and arrive at the decision point, might be missed by the threshold detector in those conventional receivers that use a relatively small fraction of the bit period. In contrast, the bit slot recovery in the ND fibre enables such pulses to be recognized as marks in the ODE-enhanced receiver. The slicing of the central portion of the broadened pulse temporal profile by the temporal gate enables efficient suppression of the timing jitter of the pulses which, in turn, improves the receiver sensitivity [START_REF] Boscolo | Time domain all-optical signal processing at a RZ optical receiver[END_REF]. This approach is illustrated in Fig. 7b, which shows that BER improvement over the conventional scheme, indeed, can be achieved using nonlinear signal processing in the optical domain before the conventional receiver. It is worth noting that the same processing technique based on temporal slicing of nonlinearly flattened pulse waveforms can also be used in all-optical 3R (2R + re-timing) signal regeneration [START_REF] Boscolo | All-optical signal regeneration by temporal slicing of nonlinearly flattened optical waveform[END_REF].

Optical signal doubling and frequency conversion

Here we discuss the use of triangular pulses in the context of optical signal doubling and frequency conversion. A technique of copying optical pulses in both the frequency and time domains based on a combination of either XPM with a triangular pump pulse or SPM of a triangular pulse in a nonlinear Kerr medium and subsequent propagation in a dispersive medium was introduced in [START_REF] Latkin | Doubling of optical signals using triangular pulses[END_REF] and experimentally demonstrated in [START_REF] Verscheure | Pulse doubling and wavelength conversion through triangular nonlinear pulse reshaping[END_REF]. In this scheme, spectral doubling of an optical signal is made possible by the XPM or SPM phase shift generated by a temporal intensity profile with a linear gradient: φ(t) = φ 0 (1 -|t/τ |), φ 0 ∝ a 2 γL (a and τ being the peak amplitude and characteristic width of a triangular pulse, respectively), which translates into a constant and distinct (opposite sign) frequency chirp induced onto the leading (down-shifted) and trailing (up-shifted) edges of the signal pulse. As a result of this constant and distinct chirp, under certain conditions the pulse spectrum develops a structure consisting of two equal peaks. The separation between the two spectral peaks increases with the maximum nonlinear phase shift φ 0 , while their form is almost preserved. Under subsequent propagation of the modulated signal pulse in a linear dispersive medium, the produced spectral separation of the pulse components leads to their subsequent separation in time, thus allowing temporal replication of the pulse. In the case of normal dispersion, the pulse develops into a structure consisting of two identical pulses where the original pulse splits down the middle. The two pulse parts separate in time while experiencing dispersive broadening. The relative velocity of separation is proportional to the separation between the two peaks of the nonlinearly modulated pulse spectrum. Pulse splitting and separation occurs also in the case of anomalous dispersion after an initial stage of pulse compression. Exact analytic expressions for the nonlinearly modulated pulse spectrum and the pulse temporal intensity distribution clearly show the features described above [START_REF] Latkin | Doubling of optical signals using triangular pulses[END_REF]. Applying spectral filtering to the XPM/SPM modulated signal is also possible to realize efficient and clean frequency conversion [START_REF] Parmigiani | An efficient wavelength converter exploiting a grating based saw-tooth pulse shaper[END_REF][START_REF] Parmigiani | Efficient all-optical wavelengthconversion scheme based on a saw-tooth pulse shaper[END_REF][START_REF] Verscheure | Pulse doubling and wavelength conversion through triangular nonlinear pulse reshaping[END_REF]. Figure 8 shows results of the experiment reported in [START_REF] Verscheure | Pulse doubling and wavelength conversion through triangular nonlinear pulse reshaping[END_REF]. Similarly to [START_REF] Wang | Generation of triangular-shaped optical pulses in normally dispersive fibre[END_REF], triangular pulses were generated by passive nonlinear reshaping in a ND fibre. Spectral pulse doubling was realized through the SPM experienced by the triangular pulses over their life distance in a HNL fibre. The output SPM-shaped spectrum from the HNL fibre was sent into either a linear standard SM fibre segment to realize temporal pulse doubling or a wavelength de-multiplexer filter for frequency conversion processing.

Conclusion

We have provided an overview of several recent examples of the use of nonlinear phenomena in optical fibres for the generation and shaping of optical pulses. We have discussed the generation of ultrashort pulse trains with very high repetition rates in the anomalous dispersion regime of a fibre, the generation of parabolic shaped pulses in active and passive ND fibres, the generation of pulses with triangular intensity profiles in passive ND fibres, and the generation of coherent continuums for optical telecommunications in ND fibres. Examples of new techniques or improvements in existing techniques of all-optical nonlinear fibre-based signal processing enabled by specialized pulse waveforms have been reviewed. Specifically, we have presented applications of parabolic pulses in optical signal regeneration, compensation of linear waveform distortions, and signal post-processing in optical communication systems. Applications of triangular pulses in optical signal doubling and frequency conversion have also been discussed.

Figure 1 :

 1 Figure 1: (a) Experimental setup for high repetition rate pulse train generation. (b), (c) Results obtainedfor the optimum output shape and for the best compression, respectively. The temporal intensity profiles at different stages as predicted by numerical simulations (subplots 1) are compared to the experimental results (subplots 2) obtained by the FROG technique (b2 -results adapted from[START_REF] Fatome | 20-GHz-to-1-THz repetition rate pulse sources based on multiple four-wave mixing in optical fibers[END_REF]) or by an optical sampling oscilloscope (c2 -results adapted from[START_REF] Hammani | Peregrine soliton generation and breakup in standard telecommunications fiber[END_REF]).

Figure 2 :

 2 Figure 2: (a) Experimental setup for parabolic pulse generation through Raman amplification at telecommunication wavelengths. (b) Results obtained in the Raman amplifier (adapted from [54]): Temporal intensity profile (subplot b1) and chirp profile (subplot b2) of the output pulse from the amplifier. The experimental results (solid black lines) are compared to a parabolic and a linear fit (open black circles), respectively. The intensity profile of the initial pulse is also shown (solid blue line). (c) Results obtained in a ND decreasing fibre (adapted from[START_REF] Finot | Parabolic pulse generation with active or passive dispersion decreasing optical fibers[END_REF]): Experimental temporal intensity and chirp profiles at the fibre output (black circles) compared to a parabolic and a linear fit (solid black lines), respectively. The initial intensity profile is also shown (blue diamonds).

Figure 3 :

 3 Figure 3: (a) Experimental setup for triangular pulse generation. (b) Experimental and numerical temporal intensity and chirp (φ t ) profiles retrieved from second-harmonic generation (SHG) FROG (left) and optical spectra (right) of the generated pulses for an example length of pre-chirping fibre. (c) SHG-FROG spectrogram of the pulse in (b). Results adapted from [68].

Figure 4 :

 4 Figure 4: Principle of operation of a multi-wavelength picosecond source.

Figure 5 :

 5 Figure 5: (a) Schematic of an optical regenerator based on SPM and filtering (after [96]). (b) Experimental setup for 40 Gbit/s similariton-based regeneration. (c) Top, autocorrelation traces of the pulse train at the Raman amplifier input (circles) and output for different values of the filter offset (dashed, dotted and solid curves). Bottom, output versus input proportion of the spurious noise pulse energy with respect to the ones energy for a fixed filter offset. Results adapted from [80].

Figure 6 :

 6 Figure 6: Schematic diagram of time-domain OFT (after [86]).

Figure 7 :

 7 Figure 7: (a) Schematic illustration of the optical decision at an ODE-enhanced receiver. (b) Numerically calculated probability density functions for the voltage of the marks and spaces in the conventional and ODE-enhanced receivers for a 40 Gbit/s RZ input signal with a random jitter of the pulse temporal position. Results adapted from [84].

Figure 8 :

 8 Figure 8: (a) Temporal (top) and spectral (bottom) intensity profiles after the ND reshaping fibre (left)and after the HNL fibre (right). (b) Top, evolution of the output spectrum from the HNL fibre with the input power (left), and dependence of the frequency spacing between the two spectral peaks on the input power (right). Bottom, evolution of the temporal intensity profile at the output of the linear SM fibre with the input power to the HNL fibre (left), and eye diagrams of a 10 Gbit/s pulse train obtained after spectral filtering for the ITU channels 34 and 42. Results adapted from[START_REF] Verscheure | Pulse doubling and wavelength conversion through triangular nonlinear pulse reshaping[END_REF].

Acknowledgements

We would like to acknowledge and thank our colleagues, viz., Prof. Sergei K. Turitsyn, Dr. Paul Harper, Dr. Anton I. Latkin, Prof. Keith J. Blow, Dr. Julien Fatome, Dr. Kamal Hammani, Dr. Bertrand Kibler, Prof. Guy Millot, Prof. John M. Dudley, and the Optoelectronics Research Center (Southampton, UK). We also acknowledge support from the British Council (Alliance grant 10.002), the Ministère des Affaires Étrangères et Européennes (Alliance grant 22836ZJ), the Agence Nationale de la Recherche, and the Conseil Regional de Bourgogne (PARI Photcom).


Daniel Weisz-Patrault
email: daniel.patrault@enpc.fr

Alain Ehrlacher
email: balain.ehrlacher@lami.enpc.fr

Nicolas Legrand
email: cnicolas.legrand@arcelormittal.com

Nathalie Labbe
email: nathalie.labbe@arcelormittal.com

Jaroslav Horský
email: horsky@fme.vutbr.cz

Tomáš Luks
email: fluks@lptap.fme.vutbr.cz

Experimental study of interfacial heat flux and surface temperature by inverse analysis with thermocouple (fully embedded) during hot steel strip rolling

Keywords: Inverse sensor, Thermocouple, Experimental study, Steel Strip Rolling

Knowledge of temperature distribution in the roll is fundamental aspect in cold rolling. An inverse analytical method has been previously developed to determine interfacial heat flux and surface temperature by measuring the temperature with a thermocouple (fully embedded) at only one point inside the roll. On this basis some pilot mill tests have been performed. The temperature sensor, the calibration procedure and rolling tests at different strip rolling conditions (5%, 10%, 15% and 20%) are described. Results show a good agreement with well-known theoretical models. Moreover the CPU times of the method (around 0.05 s by cycle) enable an online control of the rolling process.

Introduction

In steel strip rolling, two rolls are used as tools to reduce the thickness of a workpiece. For hot rolling conditions, knowledge of the heat fluxes entering the roll is fundamental for evaluation of thermal fatigue, which is one of the major factor of wear on rolls, as investigated by Coral et al. (2004) [START_REF] Corral | Modeling the thermal and thermoelastic responses of work rolls used for hot rolling steel strip[END_REF] or [START_REF] Li | Numerical simulation of temperature field and thermal stress field of work roll during hot strip rolling[END_REF] [START_REF] Li | Numerical simulation of temperature field and thermal stress field of work roll during hot strip rolling[END_REF]. Fatigue load is completely negligible for cold rolling conditions. However, in the roll gap the characterization of the thermal problem would lead to knowledge of viscosity and thickness of the lubricant. Moreover the roll expansion and thermal crown are important factors of the prediction of the flatness of products.

Several contributions use inverse methods to interpret experimental measurements. [START_REF] Tavares | Instantaneous interfacial heat fluxes during the 4 to 8 m/min casting of carbon steels in a twin-roll caster[END_REF] [START_REF] Tavares | Instantaneous interfacial heat fluxes during the 4 to 8 m/min casting of carbon steels in a twin-roll caster[END_REF] proposed an inverse method based on one-dimensional model (Lagrangian coordinate system moving at the same speed as the roll). The response time of the thermocouple is studied and a proper correction of the temperature data is introduced. [START_REF] Kotrbacek | Experimental study of heat transfer in hot rolling[END_REF] [START_REF] Kotrbacek | Experimental study of heat transfer in hot rolling[END_REF] proposed a comprehensive numerical inverse approach, where the thermocouple is modelled to interpret measurements to optimise the cooling of the roll.

In the present paper, an analytical inverse method (previously developed by Weisz-Patrault et al. [START_REF] Weisz-Patrault | Evaluation of temperature field and heat flux by inverse analysis during steel strip rolling[END_REF]) is used to interpret the measurements of a fully embedded thermocouple. The main improvement of this inverse method is the very short CPU times (around 0.05 s by cycle), which could be used for real time control of rolling processes.

Thermocouple and acquisition

The thermocouple is located in a cylindrical plug. The work roll is drilled then the plug is inserted and glued into the roll. The surface of the roll is then reground to avoid marks on the strip. The thermocouple is placed in a slot made by milling (at the surface of the plug) and brazed to the rest of body. A solder material covers the thermocouple. The thermocouple is placed along the axis of the roll in order to be parallel to the isotherms. Fig. 1 shows a schematic view of the plug inside the roll and a picture.

Fig.1 Plug with thermocouple

A high resolution (16 bits) data acquisition system has been specifically designed. The system is embarked with the rotating roll: the telemeters encoders and transmitters are fixed directly to the roll chock which enables to supply the embarked acquisition system and to collect the signal. The signal is then transferred to the main data acquisition system by wireless inductive transmission.

Material properties

The thermocouple is actually a multilayered component. The wire of the thermocouple is surrounded by an electrical insulation and a shield embedded in the solder material. The insulation is responsible for a delay in the sensor response. However considering the low rotation speed of the present study, these effects have not been modelled. The material considered for the inverse calculation is the solder material. The inverse solution used for interpreting the measurements requires a thermal diffusivity (and thermal conductivity for heat flux) independent on the temperature. The Fig. 2 shows these two quantities as functions of temperature. From these graphs, mean values are extracted for use in the inversion: D=4.2 mm 2 /s (thermal diffusivity) and λ=0.0173 W/mm/K (thermal conductivity).

Fig.2 Thermal diffusivity and thermal conductivity

Calibration

The depth of the thermocouple is unknown, because of the solder material and the grinding. Therefore a calibration is used by spraying warm water on the surface of the roll (no rotation). The roll being initially at the room temperature, the response of the thermocouple is a concave increasing curve as shown in Fig. 3. The Heat Transfer Coefficient (HTC) between water and steel (estimated to 19 750 W/m 2 /K) is used to determine the heat flux entering the roll. Even if the water does not surround the entire roll, it is almost equivalent for the signal delivered by the thermocouple that is heated by water. Therefore a well known analytical direct solution (based on zero-th Bessel function and time dependent exponentials) is used for this problem (constant surrounding temperature). The depth of the thermocouple is determined by matching the measurements and the calculated temperature. The thermocouple is estimated to be at 0.51 mm from the surface of the roll, which is close from the designed value (0.5 mm).

Fig.3 Calibration

Results

Rolling conditions are listed in Table .1. The work roll diameter is 469 mm. The cooling of the roll is done only by ambient air. The frequency of acquisition is 10 kHZ except for 20% reduction (5 kHZ). Fig. 5 presents the measured temperatures for all reductions. A small level of noise (+/-0.5 K) and a temperature increase between the beginning and the end of each cycle (transient state) are observed. The noise is not filtered because the inverse analysis developed by Weisz-Patrault et al. [START_REF] Weisz-Patrault | Evaluation of temperature field and heat flux by inverse analysis during steel strip rolling[END_REF] is already a filter (expansion into a Fourier series of the inputs multiplied for each harmonic by a factor). 6 and7 present (for the first cycle) the surface temperature and the interfacial heat flux reconstructed by inverse analysis. The higher the reduction is and the higher the interfacial heat flux and therefore the temperature peak at the surface of the roll are. This is due to the fact that the friction level and the plastic deformation increase. 7 shows the heating in the roll gap (sharp gradient) but also the cooling by air elsewhere. At the exit of the roll gap the cooling is higher than in the other parts because the difference of temperature between the roll and the air is much more significant at the exit of the roll gap.

Reduction

Summary

This paper presents an experimental study based on the interpretation of measured temperatures (thermocouple fully embedded inside the roll) by inverse analytical method. The surface temperature and the interfacial heat flux present good agreement with classical predictive models like the old analytical model of [START_REF] Tseng | Thermal behavior of aluminium rolling[END_REF] [START_REF] Tseng | Thermal behavior of aluminium rolling[END_REF]. This study emphasizes the heat due to friction and plastic deformation because several reductions have been tested. Moreover the very short computation times of the inverse analysis (around 0.05 s by cycle) enables its use in a close-loop control during rolling process.

Fig 5 .

 5 Fig 5. Temperature measurements for various reductions

Fig 7 .

 7 Fig 7. Interfacial heat fluxes (estimated by inverse analysis)


P Wang

J.-C Jurdic

Géminard

S Hu

T Biben

X Wang

P Jurdic

J.-C Géminard

Internal dynamics of actin structures involved in the cell motility and adhesion: Modeling of the podosomes at the molecular level

Keywords: Bone resorption, Osteoclast, Cellular attachment, Molecular turnover

Podosomes are involved in the spreading and motility of various cells to a solid substrate. These dynamical structures, which have been proven to consist of a dense actin core surrounded by an actin cloud, nucleate when the cell comes in the vicinity of a substrate. During the cell spreading or motion, the podosomes exhibit collective dynamical behaviors, forming clusters and rings. We design a simple model aiming at the description of internal molecular turnover in a single podosome: actin filaments form a brush which grows from the cellular membrane whereas their size is regulated by the action of a severing agent, the gelsolin. In this framework, the characteristic sizes of the core and of the cloud, as well as the associated characteristic times are expressed in terms of basic ingredients. Moreover, the collocation of the actin and gelsolin in the podosome is understood as a natural result of the internal dynamics.

Introduction

All along the adult life, two types of cell insure the permanent renewal of the bone material: the osteoclasts, which resorb the bone, and the osteoblasts, which secrete new material replacing the old one. When an osteoclast encounters a substrate, small structures, the podosomes, appear in the contact region. Podosomes are actin structures involved in the spreading and motility of various cells (dendritic cells, osteoclasts, macrophages) to a solid substrate [1,2,3,4]. They have been proven to consist of a dense actin core surrounded by an actin cloud [5,6].

In an initial stage, the podosomes form aggregates (clusters) in which they remain randomly distributed with a distance of about 1.4 μm between them. In a second stage, podosomes disappear at the center of the initial cluster, forming then an annulus (ring) that migrates toward the periphery of the contact region with a velocity of about 2 μm/min. During this process, podosomes which remain immobile, preferably disappear along the inner boundary of the annulus whereas others nucleate at the outer boundary. The whole process results in an increase of the surface area of the contact region between the cell and the substrate [5].

In the confocal microscope, the apparent shape of a podosome is a cone of typical height h 0.5 μm and base radius r p 0.15 μm. It is made of a dense assembly of actin filaments, the core, preferably oriented along the perpendicular to the cell membrane [7]. Interestingly, FRAP (Fluorescence Recovery After Photobleaching) experiments have proven that podosomes are dynamical structures in spite of their stationary shape during their lifespan which is about 2 min [5]: the podosomes grow during about 30 s before they reach an apparent steady-state during which the filaments continuously grow from the cellular membrane.

The mechanisms that regulate these structures are not known at present but probably involve actin regulators that are specifically found in podosomes, like cortactin and Wiskott-Aldrich syndrome protein (WASP), which localize directly underneath the podosome [6]. Among them, the gelsolin, an actin severing agent, has been proven to be essential for the podosome regulation [8,9]. Fluorescence experiments in which the actin and the gelsolin are marked by different fluorescent dyes show that the activity of the two molecules collocate: the concentration of the gelsolin is large in the podosome core as well as in the surrounding cloud (Fig. 1).

In a first approach [10], we focussed on the dynamics of the actin in the core and in the cloud. Reducing the complex biological system to a simplified model involving only the synthesis of actin filaments at the cell membrane and the severing, we accounted for the observed apparent shape of the single podosome and for experimental FRAP results, which proved that podosomes are dynamical structures. However, the previous model could not account for the colocalization of the gelsolin and actin fluorescence signals. In the present Letter, introducing equations governing the activity of the gelsolin molecule, we propose a far more complete model which explains why the concentration of the gelsolin is large in both the core and the cloud and we discuss the consequences for the dynamics of the podosomal structures.

Model and definitions

Following the same idea as developed in [10], we consider a podosome as a dense assembly of independent actin filaments, the actin core. Seeking for simplicity, and in accordance with recent experimental results [11], we will consider that the filaments are not branched. Each filament is supposed to grow, from a nucleation site located at the cell membrane, by addition of monomers whereas its size is limited by the action of the severing molecule, the gelsolin (Fig. 2). The filaments that are released from the core diffuse freely in the cytoplasm (the intra cellular medium) and form the actin cloud [5]. In the cloud, the free filaments are subsequently cut by the gelsolin.

In order to account for the dynamics of the actin in the core, we consider the probability b n (t), at time t, for a filament attached to the nucleation site to consist of (n + 1) monomers, linked by n bonds. The number of filaments in the core is M and the typical radius of the structure at the membrane, σ (Note that no additional assumption about the geometrical arrangement of the filaments in the core is made). For the freely-diffusing filaments, we consider c n ( r, t) the concentration, at time t and position r, of the filaments which consist of (n + 1) monomers linked by n bonds. In accordance, c 0 ( r, t) denotes the concentration of the freely-diffusing actin monomers.

In the core, the growth of the filaments is insured by the addition of actin monomers from a nucleation site with the frequency v. The growth velocity of the filaments is thus av, where a (∼ 2.7 nm) denotes the diameter of the actin monomer. Seeking for simplicity, we will assume that the typical size σ of the nucleation site is small compared to the typical diffusion length so that the local concentration of the actin monomer, at time t, is c 0 ( 0, t), constant in the podosome core and, thus, that the polymerization frequency v, which might depend on c 0 , is the same for all the filaments [12].

The severing of the actin filaments is due to the action of the gelsolin [13]. Gelsolin is a potent actin-filament severing-protein. In the presence of micromolar concentration of calcium, gelsolin rapidly binds to the side of actin filaments, which is followed by a relatively slow severing process. After severing, the gelsolin caps the barbed end. We mention that the activity of the severing agent is also regulated by other species, especially phospholipids (PIP, PIP 2 ,. . . [13]).

In order to account for the severing, in the framework of our simplistic model, we shall keep only the basic ingredients of the gelsolin severing activity. We will assume that the concentration of calcium ions is large and that their diffusion is fast. In this limit, the activity of the gelsolin will not be limited by any kinetic effect associated with that of the calcium ions. In the same way, seeking for simplicity, we will neglect any effect associated with the phospholipids. Thus, the gelsolin diffuses in the cytoplasm with the diffusion coefficient D g and attaches to any available bond between two actin monomers with the kinetic constant k on . Once a bond is occupied by a gelsolin molecule, it is cut after the typical time τ ≡ 1/β. Even if the gelsolin molecule has been recognized to cap the barbed end, we will assume that the molecule is released in the cytoplasm right after severing. The main reason for this choice is that the orientation of the actin filaments in the podosome core is not known at present. Thus, considering the internal dynamics of the core, we are unable to decide if the gelsolin molecule remains attached to the filament inside the core or diffuses away with the released part. Both situations can be considered in the model, leading to slightly different systems of equations. However, capping does not change the qualitative behavior of these equations and we shall ignore it in this first approach.

The dynamics of the gelsolin will be accounted for by considering its local concentration in the cytoplasm g( r) and the probabilities f n (t), respectively h n ( r, t), for one bond between two actin monomers in the core, respectively in the cytoplasm, to be occupied by a gelsolin molecule. In order to make these definitions clear, we consider a first example and write the total number of gelsolin molecules bound to filaments having n bonds in the core. In average, nf n gelsolin molecules are attached to one filament having n bonds in the core. In addition, the number of filaments having n bonds is Mb n . As a result, the total number of gelsolin molecules bound to all the filaments having n bonds in the core is

(Mb n ) × (nf n ) = M(nf n b n ). Second example,
locally in the cloud, at time t and position r, the total concentration of the gelsolin, free or bound to filaments, is c g ≡ g( r, t) + ∞ n=1 nh n ( r, t)c n ( r, t). The actin monomers and the filaments diffuse in the cytoplasm when not attached to the core. We denote D 0 the diffusion coefficient of the actin monomers. Following Einstein's prescription [14], we can assume that the diffusion coefficient of an actin filament made of (n + 1) monomers, free of gelsolin molecule (h n = 0), decreases with its molecular weight according to D n = D 0 /(n + 1). When gelsolin molecules are attached to the filament, we can accordingly assume that:

D n ≡ D 0 1 + n(1 + αh n ) , ( 1
)
where α ≡ m g /m a 2.1 is the ratio of the mass m g 90 kDa of the gelsolin molecule to the mass m a 43 kDa of the actin monomer [12].

Set of equations governing the dynamics

In the present section, we shall establish the set of equations governing the dynamics of the distribution b n , diffusion fields, g and c n , as well as of the probabilities, f n and h n , which account for the dynamics of the podosome core and surrounding cloud.

The core

The internal dynamics of the actin core is accounted for by the distribution b n (t) of the filament length and the probability f n which describes the occupation of the bonds by the gelsolin molecule.

Taking into account the polymerization at the nucleation site and the severing, the equation governing the temporal evolution of the distribution b n (t) is given by:

db n dt = v(b n-1 -b n ) -βnf n b n + β ∞ k=n+1 f k b k . ( 2
)
The first term accounts for the growth of the filaments due to the polymerization which results in an advection of the distribution toward larger n with the frequency v. The two additional terms account for the severing: First, b n (t) is decreased because filaments of size n are cut at any of the nf n occupied bonds with the characteristic frequency β; second, b n (t) increases when any filament of size k, larger than n, whose (n + 1) th bond (counted from the membrane) is occupied by a gelsolin molecule (which occurs with the probability f k ) is cut, at that specific position, after the time 1/β. The dynamics of the fraction f n is more difficult to account for and, in order to make the result understandable, we shall detail the contributions of the various mechanisms. Consider the variation of the number, nf n b n , of gelsolin molecules attached to filaments having the size n. The addition of a monomer at the base does not change the number of gelsolin molecules so that the contribution of the polymerization is:

d dt (nf n b n ) pol. = v(n -1)f n-1 b n-1 -vnf n b n . ( 3
)
The severing of a filament having the size n, at any of its occupied bonds, leads to the loss of the number nf n of gelsolin molecules for the considered population, so that the first contribution of the severing is:

d dt (nf n b n ) sev.1 = -(βnf n )nf n b n (4)
where the prefactor (βnf n ) takes into account that the filament is likely to be cut at nf n bonds with the characteristic time 1/β. The severing of a filament having the size k, larger than n, at the (n + 1) th bond, leads to a filament of size n whose bonds are occupied by a gelsolin molecule with the probability f k , so that the second contribution of the severing is:

d dt (nf n b n ) sev.2 = ∞ k=n+1 (βf k )nf k b k (5)
where the prefactor (βf k ) accounts for the fact that the (n + 1) th bond is occupied with the probability f k and cut with the characteristic time 1/β if so. Finally, the gelsolin molecules that are in solution, in the vicinity of the core, attach to the available bonds [number n(1f n )] with the characteristic time k on g 0 , where g 0 is the local concentration of the gelsolin, which leads to:

d dt (nf n b n ) att. = (k on g)n(1 -f n )b n (6)
Summing all the contributions listed in the Eqs. ( 3) to ( 6), we get the equation governing the dynamics of

f n b n d dt (nf n b n ) = v[(n -1)f n-1 b n-1 -nf n b n ] -βn 2 f 2 n b n + βn ∞ k=n+1 f 2 k b k (7) +k on g 0 n(1 -f n )b n .
The coupled equations ( 2) and ( 7) govern the dynamics of the distribution b n and of the fraction f n . We point out that Eqs. ( 2) and ( 7) couple the internal dynamics of the core with that of the diffusion fields in the cloud through the local concentration of the gelsolin g 0 and the velocity v which might depend on the local concentration of the actin monomer c 0 .

The cloud

The dynamics of the cloud can be accounted for by considering, in addition to the severing, the diffusion of the various species in solution. Before we establish the boundary conditions, we first consider the concentration of the gelsolin, g, the concentration of the actin monomers and filaments, c n , and then the fraction of occupied bonds, h n .

The gelsolin molecules, on the one hand, diffuse with the diffusion coefficient D g , detach from the bond when a filament is cut and attach to the available bonds of the filaments in solution. Thus, because of the severing process, the diffusing filaments constitute volume sources and sinks and the field g( r, t) is governed by:

∂g ∂t = D g Δg + β ∞ k=1 h k kc k -k on g ∞ k=1 [1 -h k ]kc k ( 8
)
where Δ denotes the Laplacian operator. At the same time, the filaments diffuse in the cytoplasm, are cut with the characteristic time 1/β at any of the nh n ( r, t) occupied bonds, so that:

∂c n ∂t = ∇.[D n ∇c n ] -βnh n c n + 2β ∞ k=n+1 h k c k . ( 9
)
Note that the diffusion coefficient depends on space. Indeed, D n depends on the fraction of bonds occupied by the gelsolin (Eq. 1). The factor 2 in the last term is due to the fact that there are two possibilities for getting a filament of size n when cutting a larger filament. Finally, we mention that the equation ( 9) holds true for the monomer and that we get for n = 0:

∂c 0 ∂t = D 0 Δc 0 + 2β ∞ k=1 h k c k . ( 10
)
The dynamics of the fraction h n is again more difficult to account for and we shall detail the contributions of the various mechanisms. We consider the contributions of all the mechanisms that induce a variation of the number nh n c n , in the unit volume, of gelsolin molecules attached to the filaments of size n. The filaments diffuse and carry, each, a number nh n ( r, t) molecules:

∂ ∂t (nh n c n ) dif f. = ∇.(nh n D n ∇c n ) (11)
Again, the diffusion does not reduce to a simple Laplacian equation, not only because the diffusion coefficient depends on space and time but also because the number of gelsolin molecules on the filaments depends on space and time.

In addition, the severing of the filaments having the length n leads, on the one hand, to the decrease of the local concentration of the gelsolin attached to the filaments of size n according to:

∂ ∂t (nh n c n ) sev.1 = -βn 2 h 2 n c n . ( 12
)
Indeed, nh n ( r, t) gelsolin molecules are attached to the filament which is thus cut with the frequency βnh n ( r, t), nh n ( r, t) gelsolin molecules being transfered to smaller filaments or released in solution. On the other hand, the severing of larger filaments (k > n), at two specific positions occupied with the probability h k ( r, t), leads to a filament of size n and provides nh k ( r, t) gelsolin molecules:

∂ ∂t (nh n c n ) sev.2 = ∞ k=n+1 β(nh k )(2h k )c k (13)
Finally, free gelsolin molecules attach to the filaments at the available bonds:

∂ ∂t (nh n c n ) att. = k on gn(1 -h n )c n ( 14
)
From the contributions Eqs. ( 11) to ( 14), we get the equation governing the product h n c n :

∂ ∂t (nh n c n ) = ∇.[nh n D n ∇c n ] -βn 2 h 2 n c n (15) +2βn ∞ k=n+1 h 2 k c k + k on gn[1 -h n ]c n
The equations ( 2) and ( 7) govern the dynamics of the distribution b n (t) and of the fraction f n (t) in the core whereas the equations ( 8), ( 9) and (15) govern the dynamics of c n ( r, t) and g( r, t), the actin and gelsolin concentration fields, and the fraction h n ( r, t) in the cloud.

The boundary conditions

Due to the polymerization, the core acts as a sink of monomeric actin whereas, due to the severing, it acts as a source of both actin monomers and filaments. In the same way, due to the attachment to the available bonds, the core acts as a sink of gelsolin molecules but, due to the severing process, is a source of gelsolin, free or attached to diffusing filaments. In order to account for these sources and sinks, we shall write the boundary conditions at the core, supposed to be centered in r = 0.

Diffusing filaments are released in solution when filaments of the core are cut at the appropriate bond whereas, the total number of filaments in the core being M and v monomers being added per unit time, Mv monomers are consumed:

S D n ( ∇c n ).d S = Mv δ(n) -Mβ ∞ k=n+1 f k b k ( 16
)
where S is a surface enclosing the core and, d S the surface element oriented outwards. The diffusion coefficient, which depends on h n ( r, t), is evaluated at the boundary. The first term on the right-hand side accounts for the consumption of monomers (δ stands for the Kronecker delta function) due to the polymerization. The severing of a longer filament (k > n) in the core, after a time 1/β if the bond is occupied (which occurs with the probability f k ) releases in solution nf k gelsolin molecules which then diffuse attached to the filament:

S nh n D n ( ∇c n ).d S = -nMβ ∞ k=n+1 f 2 k b k . ( 17
)
Finally, per unit time, k on g 0 gelsolin molecules attach to any available bond in the core [g 0 ≡ g(0, t) stands for the gelsolin concentration at the core] whereas one molecule is released in solution each time a bond is cut:

D g S ∇g.d S = -Mβ ∞ n=1 nf 2 n b n (18) +Mk on g 0 ∞ n=1 n(1 -f n )b n
Far away from the core, the concentration of any filament vanishes, whereas the concentrations of the monomers and of the gelsolin tend respectively to the overall concentrations c ∞ 0 and g ∞ :

lim r→∞ c n ( r, t) = c ∞ 0 δ(n) (19) lim r→∞ g( r, t) = g ∞ (20)
We mention also that, accordingly, the concentration of gelsolin molecules attached to filaments of size n also vanishes so that lim r→∞ (nh n c n ) = 0 (∀n).

Parameters of the problem

From now on, it is particularly interesting to consider the parameters of the problem. First, the equation governing the internal dynamics of the core [Eqs. ( 2) and ( 7)] suggest that the pertinent timescale is τ = 1/β whereas the equation governing the actin diffusion field [Eqs. ( 9) and ( 15)] suggest that the pertinent length-scale is D 0 /β. We shall thus report results expressed in terms of the dimensionless variables t ≡ βt and r ≡ r β/D 0 . Second, one can consider that the pertinent concentration scale is the concentration of the actin far away from the core so that one can express all the concentrations relative to c ∞ 0 . We thus define, for instance, cn = c n /c ∞ 0 and g = g/c ∞ 0 . The remaining independent parameters of the problem are α, D g , g ∞ , M, σ, k on and the function v. The parameter α is the ratio of the gelsolin and actin molecular weights which plays a role in the dependency of D n on h n . We define d ≡ D g /D 0 , γ ≡ g ∞ /c ∞ 0 and ζ ≡ k on g ∞ /β in accordance with the choice of the dimensionless variables. The parameters M and σ account for the surface density of the actin filaments inside the podosome core. Indeed, assuming that σ D 0 /β one can consider that the surface in the boundary conditions [Eqs. ( 16) to (18)] is a sphere of radius σ/2. Assuming then radial diffusion fields, one exhibits the dimensionless parameter Φ ≡ (2M/πσ 2 )/( D 0 /β c ∞ 0 ) which compares the density of the filaments in the core with the surface density of the actin monomer in a layer of thickness D 0 /β.

Results

We shall first study the steady-state solution to the problem and then discuss the characteristic time associated with the dynamics of the system. The solutions of the problem are calculated numerically.

The core

The internal dynamics of the core, governed by the equations ( 2) and ( 7), is coupled to the cloud only by the velocity v which might depend on the local concentration of the actin monomer and by the local concentration of the gelsolin, g 0 . Thus, it is pertinent to consider the steady-state solution for given v * ≡ v/β and ζ 0 ≡ k on g 0 /β. In the figure 3a, we report typical solutions for b n and f n b n as functions of n in the steady state. We observe that the distribution b n presents the same typical shape already described in [10]. From the similarity of the shapes of b n and f n b n , we can conclude that f n does not drastically depend on n and, thus, that the gelsolin bound to the filaments is rather homogeneously distributed among the filaments having different lengths. We define and report the podosome size, n ≡ ∞ n=0 (n + 1)b n , and the average fraction of gelsolin on the filaments in the core, 3b). We find numerically that f g and n/ πv * /2 are functions of ζ 0 / √ v * which can be understood as the product of the attachment characteristic frequency k on g 0 and of the characteristic time of the core growth

f g ≡ ( ∞ n=0 nf n b n )/( ∞ n=0 nb n ) (Fig.
τ core ≡ 1/ √ βv (∝ n/v * ). For ζ 0 / √ v *
1, the gelsolin tends to occupy all the bonds so that f g 1 and n πv * /2 as we already demonstrated analytically [10]. For smaller ζ 0 / √ v * , the concentration f g < 1, which leads to a less effective severing process and, accordingly, to a larger typical core size n.

Thus, the characteristics of the podosome core do not depend drastically on the local concentration of the gelsolin, g 0 . Indeed, changing ζ 0 by a factor 10 4 (from 100 to 0.01) changes n by a factor 5 only. By contrast, n ∝ v * 1/2 and, thus, the characteristic time of the podosome growth, τcore ∼ v * -1/2 drastically depend on the polymerization velocity which might depend on the local concentration of the actin monomer.

The cloud

The dynamics of the actin and of the gelsolin in the surrounding core is governed by the equations ( 8), ( 9) and (15). The podosome core, assumed to be centered in r = 0, is accounted for by the boundary conditions ( 16) to (20).

Considering again the steady state, we can report the typical concentration c n of the actin filaments of size n and the concentration nh n c n of gelsolin bound to the filaments of size n in the cloud as functions of the distance to the core r. In the figure 4a, one observes that the diffusion length associated with the longer filaments (larger n) is shorter. This can be understood easily considering the fact that the gelsolin can cut the longer filaments in a larger number of locations. Thus, the longer filaments have a smaller probability to reach a large distance from the core. By contrast to c n , each concentration nh n c n has a non-monotonic dependency on the distance r and reaches a maximum at a finite distance from the core (Fig. 4b). The increase near the core is due to the continuous binding of free gelsolin molecules whereas, far from the core, the decrease results from the decrease of the filament concentration c n . We define and report the total concentration of the actin monomer, c a ≡ ∞ n=0 (n + 1)c n , and of the gelsolin, c g ≡ g + ∞ n=0 nh n c n , as functions of the distance r from the podosome core (Fig. 4c). We note that both the actin and the gelsolin accumulate around the core in a region having a dimensionless radius, denoted rc , which is governed by the competition between the diffusion and the severing: The typical diffusion-coefficient is of the order of D 0 /n whereas the typical severing frequency is about nβ. Considering that r c is of the order of the associated diffusion lengh D 0 /β/n, we get r c ∼ D 0 /v and, thus, rc ∼ 1/ √ v * . Interestingly, considering the dynamics, one can estimate the time necessary to establish the steady-state diffusion field to be of the order of τ cloud ∼ r 2 c /(D 0 /n) so that τcloud ∼ τcore . Thus, provided that the polymerization frequency v is given, the dynamics of the whole system involves a single characteristic time τ = 1/ √ βv. We point out that, the typical number n being a slowly varying function of the gelsolin concentration at the core, the typical size and, thus, the characteristic time associated with the cloud dynamics depend only slowly on the local concentration of the gelsolin.

Discussion

In a former publication [10], two of us and a co-worker presented the onlyavailable model of the turnover of the actin within the podosomal structure. They showed that such models can explain, for instance, that the typical height of the core is auto-regulated by the interplay between the polymerization and severing processes or why the steady-state core might have a finite lifespan. However, the model was too simplistic to account for the collocal-ization of the actin and gelsolin molecules or to envisage a modeling of the podosomes collective-dynamics in a so simple framework.

Here, we presented a model of the podosome dynamics including not only the actin turnover but also the gelsolin dynamics. We show that the polymerization of actin filaments at the cell membrane and the diffusion and attachment kinetics of the gelsolin account for the observation of large concentrations of the actin and of the gelsolin in the podosome region, in both the core and the cloud. The model is thus compatible with the collocalization of the molecules observed in the fluorescence microscope. In addition, the model predicts that the characteristics of the podosome in the steady-state, typical size and characteristic time for instance, are not sensitive to the local concentration of the gelsolin. The main conclusions of the former publication still hold true when the dynamics of the gelsolin is taken into account. The present model might easily be modified in order to include the capping.

The interplay between the gelsolin and actin concentration fields is now accounted for. We then think that the present model includes the ingredients that make it possible to envisage a modeling of the podosomes collectivedynamics based on a system of equations similar to that presented here. The dependency of the polymerization frequency on the local concentration of actin at the core and the interaction between cores through the diffusion fields in the cloud might explain the collective dynamics of the system. Especially, the formation and the migration of podosome rings [5] in the framework of the present model shall be investigated in the near future.

Figure captions

Figure 1 :

 1 Figure 1: Fluorescence images of an osteoclast -The podosomes correspond to the bright dots. The use of two dyes exhibiting different fluorescence wavelengths makes it possible to image separately the actin (A) and the gelsolin (G). The experiment clearly demonstrates the colocalization of the actin and gelsolin fluorescence signals in the podosome core [Fixed osteoclast observed in an Axioplan 2 Imaging from Zeiss: actin marked with Phalloidin (wavelength 488 nm) and gelsolin with Anti-Gelsolin bound with fluorescent second antibody (wavelength 562 nm)]

Figure 2 :

 2 Figure 2: Sketch of the podosomal structure.

Figure 3 :

 3 Figure 3: Typical solution for the podosome core -(a) Typical b n and f n b n (v * = 1000, ζ 0 = 10). (b) Fraction f g (open squares) and normalized size n/ πv * /2 (full diamonds) vs. ζ 0 / √ v * .

Figure 4 :

 4 Figure 4: Typical solution for the cloud -(a) Concentration of the actin filaments cn in the cloud. (b) Concentration nh n cn of gelsolin bound to the filaments of size n in the cloud. (c) Total concentrations of the gelsolin cg and of the actin ca in the cloud around the core. One can notice that, in spite of the small concentration of gelsolin far away from the podosome (γ = 5.855 10 -2 ), the concentrations of the actin and of the gelsolin in the core region are of the same order (v * = 1000, ζ 0 = 10, α = 0, d = 1, φ = 2.516 10 -3 , ζ = 1.171 10 2 and γ = 5.855 10 -2. These parameters insure ζ 0 = 10 which corresponds to the core described in Fig.3).

  Figure 3

Acknowledgments

J.-C.G. and X.W. acknowledge the support from the CNRS and the NSFC (contract #020041FSD). S.H. aknowledges support from the "Région Rhône-Alpes" (Programme MIRA).