Amélie Noël

Jenny Faucheu
email: faucheu@emse.fr

Mathilde Rieu

Elodie Jean-Paul Viricelle

Elodie Bourgeat- Lami

Tunable architecture for flexible and highly conductive graphene-polymer composites

Keywords: A. Nanocomposites, A. Flexible composites, B. Electrical properties 1-Introduction

come

Tunable architecture for flexible and highly conductive graphene-polymer composites

In the past decades, flexible and textile substrates raised a strong interest in printed electronics applications. Wearable electronics would transform traditional textile and apparel products into lightweight, wireless and wearable intelligent devices [START_REF] Hu | Stretchable, Porous, and Conductive Energy Textiles[END_REF]. Currently, conductive inks, in particular ink-jet inks, are mostly based on dispersions of metallic particles in an aqueous or organic medium [START_REF] Khondoker | Synthesis and characterization of conductive silver ink for electrode printing on cellulose film[END_REF]. To reach the adequate particle sintering, an annealing step at high temperature is required, around 500-1000 °C [START_REF] Cauchois | Impact of variable frequency microwave and rapid thermal sintering on microstructure of inkjet-printed silver nanoparticles[END_REF]. Moreover, the resulting sintered material can undergo only very small deformations. For those two reasons, flexible and textile substrates cannot be used with these inks.

The aim of this paper is to present a simple processing route for producing conductive inks that form a continuous film at room temperature and that exhibit high deformability after drying. The strategy is based on architectured nanocomposites allying conductivity characteristics of Nanosize Multilayered Graphene (NMG) and deformability of a polymer binder. The nanocomposite architecture is tuned using a latex route [START_REF] Steward | An overview of polymer latex film formation and properties[END_REF]. The production of conductive nanocomposites by blending a polymer latex with graphene-based particles has been already presented by several authors [START_REF] Wu | Highly Conductive Nanocomposites with Three-Dimensional, Compactly Interconnected Graphene Networks via a Self-Assembly Process[END_REF], [START_REF] Li | Highly conducting graphene sheets and Langmuir-Blodgett films[END_REF] and [START_REF] Arzac | Comparison of the Emulsion Mixing and In Situ Polymerization Techniques for Synthesis of Water-Borne Reduced Graphene Oxide/Polymer Composites: Advantages and Drawbacks[END_REF]. However in all of these works, Graphene Oxide (GO) and reduced Graphene Oxide (rGO) was used, increasing the number of processing steps and chemical products needed. Moreover, in most of these works, the composite blends were not stable: the composite solids ended up flocculating and a subsequent hot-pressing step was needed to produce a continuous composite material with acceptable mechanical and electrical properties. For nanocomposite blend formulation, the latex route exhibits two major advantages as compared to the melt route or solution route.

First, this synthetic route is sustainable as a latex is made of polymer nanospheres in an aqueous suspension without using organic solvent. Second, the latex route favors the built-up of a tunable architecture of fillers. This specific architecture, in turn, favors the formation of a percolating network of fillers at lower filler content [START_REF] Tkalya | Latex-based concept for the preparation of graphene-based polymer nanocomposites[END_REF]. As a result, the final nanocomposite microstructure counts two interpenetrated networks, one made of the polymer matrix and the other one made of percolating fillers [START_REF] Grunlan | Water-Based Single-Walled-Nanotube-Filled Polymer Composite with an Exceptionally Low Percolation Threshold[END_REF], [START_REF] Grossiord | Strategies for dispersing carbon nanotubes in highly viscous polymers[END_REF]. The architecture of the polymer matrix is constrained by the spherical shape of the latex particles and the shape of the fillers can influence the percolation threshold. The use of 2D-fillers (platelets) would both favor filler contacts compared to 1D-fillers (nanotubes) and lower the percolation threshold compared to 3D-fillers (spheres). Thus, for geometrical considerations, platelet-like fillers [START_REF] Yang | The Prospective Two-Dimensional Graphene Nanosheets: Preparation, Functionalization, and Applications[END_REF] seems to be a good choice to build up an efficient network with polymer nanospheres.

The chosen size ratio between the polymer nanospheres and the fillers is a compromise. On the one hand, the platelets have to be large enough to limit the number of conductive platelets needed to cover the surface of the latex spheres and so the number of contacts. Indeed, intuitively, the contacts between fillers will have a lower conductivity than the intrinsic conductivity of the filler itself. On the other hand, the platelets have to be small enough in order not to destabilize the blend or hinder the film formation process of the latex nanospheres. Hereafter, the fabrication process and characterization of home-made conductive graphene-based nanoplatelets (NMG) are detailed. Secondly, the highly conductive behavior of NMG-based nanocomposites obtained through latex blends are discussed in terms of filler architecture through a percolation approach. Finally, it is demonstrated by mounting a simple electronic setup that this procedure provides a good candidate for conductive ink for printed electronics on flexible and textile substrates.

2-Experimental

2.1-Materials: Graphite flakes (7-10 µm with 99% purity) from Alfa Aesar and Sodium dodecyl benzene sulfonate (SDBS, Aldrich) were used as received. Potassium persulfate (KPS), methyl methacrylate and n-butyl acrylate monomers purchased from Aldrich were used without further purification.

2.2-Preparation of Nanosize Multilayered Graphene (NMG):

The preparation of the conductive particles, NMG, was performed using a bead milling procedure in wet media.

Prior to the milling procedure, the graphite flakes (40 g) and the surfactant (SDBS, 2 g) were dispersed in water (400 mL) under agitation during 12 hours. The surfactant favored the grinding process and the stability of both the graphite flakes and the NMG further produced.

The grinding experiments were carried out in a horizontal laboratory attritor (Minizeta, Netzsch Feinmahltechnik GmbH, Selb, Germany) during 4 hours at 1500 rpm, using Yttriumstabilized zirconium oxide grinding beads (ZetaBeads Plus, 0.4 mm, Netzsch) as grinding media. A sonicator was added in the attritor loop to improve graphite exfoliation and scission. A stable NMG suspension was isolated through decantation of the larger graphite flakes. It was used as-is or after increase of NMG-content by water evaporation.

2.3-Preparation of surfactant-free acrylate copolymer latexes via Free radical Emulsion

Polymerization: The polymer matrix used is a poly (methyl methacrylate-co-butyl acrylate) P(MMA-co-BA) latex spheres in a water suspension were synthesized using free-radical emulsion polymerization. The monomers (methyl methacrylate (12.5 g) and butyl acrylate (12.5 g) were poured in the reactor for degassing under nitrogen. Water (95 g) was added for further degassing and the reactor was pre-heated at 70 °C. After injection of the initiator (KPS (0.25 g)) dissolved in water (5.0 g), the polymerization was carried out during 5 hours at 70 °C. A final monomer conversion of 99 % was reached. The average particle size determined by DLS was 650 nm (0.3 Polydispersity Index).

2.4-Preparation of the nanocomposite films:

Prior to latex blending, the NMG platelets were introduced into the latex suspension drop-by-drop to increase the adsorption of NMG onto the latex nanospheres. The latex (1.0 g) was poured in a beaker and placed in an ultrasonic bath.

The adequate amount of NMG suspension was placed in a syringe. The NMG-content and the volume of suspension depended on NMG-content of the final composite that was aimed.

Finally, the composite latex blends were poured into silicone mold and dried at 40 °C.

2.5-Characterization:

Microstructural characterizations were performed using complementary techniques. Dynamic Light Scattering (DLS, Vasco) was performed on diluted suspensions. A drop of the NMG suspension was deposited on carbon-coated 300mesh copper grid and dried prior to Transmission Electron Microscopy observations (TEM Philips CM120) and High Resolution TEM (HR-TEM JEOL 2100 F) ) at the CTµ (Centre Technologique des Microstructures, Lyon 1 University). The NMG suspension was spincoated on Silicon wafers and washed using deionized water prior to Atomic Force Microscopy (AFM, JPK Nanowizard 3) characterizations. The washing step was performed to eliminate surfactant traces on the surface that might influence the AFM measurements. Thin foil TEM-specimens (< 100 nm) of the nanocomposite material were prepared using a diamond knife on a cryo-ultramicrotome equipment.

The electrical conductivity was measured using a four-probe setup (with gold contacts and 3.48 mm of distance between each probe) equipped with a conductimeter (Keithley 2400) [START_REF] Alwarappan | Probing the Electrochemical Properties of Graphene Nanosheets for Biosensing Applications[END_REF].

3-Results and Discussion

3.1-Processing steps of NMG/latex nanocomposites

In this work, the challenge relies on the production of adequate conductive fillers with specific dimensional characteristics that do not destabilize the latex during blending. In the literature the most popular route to synthesize NMG is based on the so-called Hummer's method [START_REF] Hummers | Preparation of Graphitic Oxide[END_REF]. Despite its popularity, this method presents noticeable disadvantages as it relies on a long and multistep synthesis using many chemical products. In Hummer's method, GO is produced from graphite flakes and reduced into rGO. After this reduction step, some functional groups containing oxygen remains on the rGO surface [START_REF] Gao | Hydrazine and Thermal Reduction of Graphene Oxide: Reaction Mechanisms, Product Structures, and Reaction Design[END_REF] that might affect both the interactions with the latex particles and the final electrical properties of the composite. As an alternative, Knieke et al. proposed a production of NMG based on the mechanical delamination of graphite in wet grinding media [START_REF] Knieke | Scalable production of graphene sheets by mechanical delamination[END_REF]. This mechanical method is not popular, yet it is cost-effective and avoids organic solvents. In our work, this procedure was chosen and the NMG water suspensions were conveniently used as-is in the subsequent processing steps of the nanocomposite material. The delamination process does require the use of a large amount of graphite flakes, but these are cheap and reusable raw materials. After 4 hours of wet grinding, 0.8 g of NMG in water suspension is obtained from 40 g of graphite flakes. The graphite suspensions are stabilized with addition of SDBS. This surfactant contains a benzene group that likely improves the NMG stabilization through π-π interactions, as compared to the most commonly used surfactant Sodium Dodecyl Sulfate (SDS) [START_REF] Lee | The Superior Dispersion of Easily Soluble Graphite[END_REF]. After the grinding step, the suspensions were settled overnight in order to remove the remaining graphite flakes through sedimentation. The concentration of the produced NMG suspensions was 2 mg mL -1 which is two times higher than common concentrations of graphene oxide suspensions obtained through Hummer's method [START_REF] Stankovich | Synthesis of graphene-based nanosheets via chemical reduction of exfoliated graphite oxide[END_REF].

To assess the NMG thickness h AFM measurements were carried out (Figure 1a). The NMG suspension was spin-coated on a silicon wafer and washed with deionized water in order to remove surfactant traces. This washing step might remove some of the NMG particles from the wafer [START_REF] Knieke | Scalable production of graphene sheets by mechanical delamination[END_REF]. The cumulative number distribution, N(h), given in Figure 1c, describes the amount of NMG sheets that have a thickness equal or smaller than a given thickness h and N(h) was assessed from 150 NMG sheets from various areas of the silicon wafer. The 50% percentile h 50 is 3 nm and the 90% percentile h 90 is around 3.5 nm, i.e. 90% of created NMG sheets are thinner than 4 nm. In addition, the sheet-like morphology is illustrated in Figure 1b showing profile of three different NMG particles.

To assess the NMG lateral size d, AFM and TEM observations were combined DLS measurements in order to ensure a more complete description of the NMG morphology. On the one hand, AFM gives the exact characterization of NMG morphology but this analysis remains very local. The lateral size over 500 NMG particles was assessed through AFM image analysis considering a disk-like equivalent particle. On the other hand, DLS measurements provide a statistical description of the size distribution; however, uncertainties might be introduced as the particles in suspension are considered spherical. Cumulative distributions of the lateral size assessed through AFM and DLS are plotted in Figure 2. Slight differences in general shape are noted but both distributions are consistent in that the 50% percentile d 50 are between 30 and 60 nm. The average lateral size of the NMG nanoplatelets produced was about ten times smaller than the latex diameter (650 nm), described hereafter.

On the TEM image shown on Figure 3a, two large NMG platelets (around 150 nm lateral size) are visible (arrows). The different grey shades are consistent different NMG thicknesses: this can come from NMG overlap or NMG folding. The HR-TEM image on Figure 3b shows a NMG platelet that counts 7 graphene layers.

In addition to adequate NMG size characteristics for latex blending, these NMG water suspensions exhibited good self-stability (no sedimentation nor flocculation after 2 weeks).

The suspensions were used as-is or after concentrated suspensions after water evaporation to produce NMG/polymer conductive inks through latex blending. The copolymer P(MMA-co-BA) has a glass transition temperature around 15°C. The latex was prepared through free radical polymerization without emulsifier. The latex nanospheres had an average diameter of The NMG suspensions of different solids contents were slowly added to the P(MMA-co-BA) latex in an ultrasonic bath and left standing for 10 min (Figure 4). The blends exhibited a homogeneous dark grey color and no sedimentation, phase separation nor flocculation was observed. They were poured into silicone molds and left to dry overnight at 40 °C. The latex film forming process is commonly described as a multistep process. As water evaporates, the latex nanospheres come into contact in a compact arrangement, and then deform to fill in the porosity and finally the polymer chains diffuse from one nanosphere to the other. Despite the presence of nanoplatelets in the water phase, flexible free-standing films (around 200 µm thick) were easily extracted from the silicon molds. Eight composite films (specimens 1 to 8) were produced and the NMG contents in the films are detailed in Table 1. A blank polymer film was also produced as a reference (specimen 0). Note that after the film forming process, surfactant molecules likely exude at the air-film and mold-film interfaces which might influence the conductivity measurements [START_REF] Kientz | Distribution of surfactants in latex films[END_REF]. Thus, all the composites films were washed several times in deionized water prior to further characterization.

3.2-Morphological characterization of NMG/latex nanocomposites

Thin foil TEM-specimens were prepared to assess the nanostructure of the final composite films (Figure 5 and6). On the TEM micrographs, light grey domains are holes in the thin cross section. The polymer matrix appears in medium grey as a homogeneous background.

Due to their very small thickness, the NMG platelets are visible when they are edge on, then they appear as dark sticks or dark aggregates. Compact arrangement of latex particles surrounded with NMG platelets are visible as domains distributed as a cellular structure.

These TEM observations are consistent with a geometrical description of the composite as a close-packed cubic arrangement of deformed spheres.

At lower magnification, NMG paths are visible throughout the nanocomposite material. The TEM micrograph shows a 2D-cross-section of the 3D-NMG network that is developed throughout the whole material. These cross-sections give insights of the network density. As expected, with an increase of NMG content from 0.5 % vol to 5% vol, the network density is clearly higher: an increasing number of paths are visible and an overall cellular architecture appears (Figure 6).

Thanks to the latex blending route, using fillers with adequate sizes, flexible NMG/polymer composite materials with an architectured microstructure were obtained. The influence of NMG content on the nanocomposite conductivity characteristics is explored hereafter.

3.3-Conductivity characteristics of NMG/latex nanocomposites

The electrical conductivity of the nanocomposite materials was measured using a four probe conductimeter. Ten measurements were performed on each side of the free-standing composite film. Figure 7a shows the average electrical conductivity as a function of the NMG content. Note that the conductivity of the blank sample (x NMG =0) was not measured but set to a value for common dielectric materials (10 -9 S.m -1 ) [START_REF] Mária Omastová | Chemical preparation and characterization of conductive poly(methyl methacrylate)/polypyrrole composites[END_REF]. The plot exhibits a steep increase at low NMG content and a slower increase for a NMG volume fraction above 2×10 -2 . A maximum value of 217 S.m -1 was reached at 6.1×10 -2 NMG volume fraction. To the best of our knowledge, this conductivity value is among the highest value available in the literature for graphene-latex composites [START_REF] Pham | Highly conductive poly(methyl methacrylate) (PMMA)-reduced graphene oxide composite prepared by self-assembly of PMMA latex and graphene oxide through electrostatic interaction[END_REF]. To highlight the influence of the NMG 3D-network on conductivity, the behavior of graphene-based nanocomposites is usually described through a probabilistic approach called percolation theory [START_REF] Stauffer | Introduction to percolation theory[END_REF]. In this approach, the probability of forming a percolating path of fillers, i.e. a path of fillers that crosses the whole sample, is described in terms of filler volume fraction and geometrical filler arrangement. In particular, the theoretical or statistical percolation threshold is defined as the minimum volume fraction of fillers needed to form the first percolating path. Experimentally, the addition of conductive fillers to a dielectric polymer can result in a conductive composite if enough paths are formed to enable electron transport. Intuitively, the minimum filler content needed, called experimental percolation threshold hereafter, is equal or higher than the value derived from the statistical description of the percolation threshold and depends on the sensitivity of the conductimeter equipment. In this work, the experimental percolation threshold was evaluated as shown in Figure 7a. As dielectric materials are commonly described as materials with conductivity lower than 10 -8 S.m -1 , the experimental percolation threshold lies between 1×10 -3 and 2×10 -3 . This value is also consistent with the steep increase observed for the conductivity in Figure 7.

Based on the percolation theory, Kirkpatrick proposed an equation to define the conductivity of biphasic materials comprising a conductive phase and a dielectric phase [START_REF] Kirkpatrick | Percolation and Conduction[END_REF]:

= (1)
where σ c is the conductivity of the nanocomposite material, x NMG is the filler volume fraction,

x c is the percolation threshold, σ 0 is the filler's conductivity in the composite film and t is a critical exponent [START_REF] Mclachlan | AC and DC percolative conductivity of single wall carbon nanotube polymer composites[END_REF]. Note that this description is only valid for x NMG > x c . In literature, the statistical percolation threshold is said to depend on the lattice dimensionality (3 dimensions in our case) and lattice geometry (more likely close to faced-centered cubic in these latexbased materials) [START_REF] Sperry | Role of Water in Particle Deformation and Compaction in Latex Film Formation[END_REF]. The critical exponent t is a universal exponent that is said to mainly depend on the dimensionality of the lattice. Using Equation (1), a linear trend is obtained for the log-log plot showing log(x NMG -x c ) on the x-axis and log(σ c ) on the y-axis, with x c = 1.0×10 -3 , σ 0 = 10 4 S.m -1 , and t = 1.72 as illustrated in Figure 7 (inset). All three parameters (x c , σ 0 and t) are physically consistent. First, the percolation threshold for x c is consistent with the preliminary graphical evaluation mentioned earlier. The critical exponent for a statistical description of a 3D percolation network is documented in literature around 1.6-2.0 [START_REF] Efros | Critical Behaviour of Conductivity and Dielectric Constant near the Metal-Non-Metal Transition Threshold[END_REF], [START_REF] Clerc | The electrical conductivity of binary disordered systems, percolation clusters, fractals and related models[END_REF].

The filler's conductivity σ 0 describes the conductivity of graphene sheets assembled in the nanocomposite. Thus, both the intrinsic conductivity of the filler and the conductivity of the filler-filler contact have to be taken into account. Values for the intrinsic conductivity of graphene monolayers are largely documented in literature around 10 7 -10 8 S.m -1 [START_REF] Chen | Preparation of graphene by the rapid and mild thermal reduction of graphene oxide induced by microwaves[END_REF].

However, these extremely high values do not take into account the filler-filler contacts.

Marinho et al. conducted an experimental study on conductivity of powder compacts made of graphite, graphene and Carbon Nanotube (CNT) powders [START_REF] Marinho | Electrical conductivity of compacts of graphene, multi-wall carbon nanotubes, carbon black, and graphite powder[END_REF]. These authors measured 2.12×10 3 S.m -1 for the graphite compacts, 2.62×10 2 S.m -1 for graphene compacts and 5.43×10 2 S.m -1 for CNT compacts. The highest conductivity exhibited by the graphite compacts might be due to larger particle-particle contacts that overcome the lower intrinsic conductivity of graphite as compared to graphene and CNT. Based on these results, σ 0 = 10 4 S.m -1 is a consistent order of magnitude for the conductivity of NMG networks. Finally, the conductivity behavior observed for these nanocomposite materials is consistent with a threedimensional percolation behavior.

In order to demonstrate the potential application of our conductive films, a basic electronic setup based on a light emitting diode (LED) was mounted (Figure 8). In this setup, small strips (2 mm-wide, 20 mm-length and 200 µm-thick) were cut out of sample 8 (6.1×10 -2 NMG volume fraction). These strips were used to replace part of the copper wires that bond the LED to the batteries. An unexpected advantage of the nanocomposite material was obtained in that in a classical LED setup a current-limiting resistor is added to protect the LED. In our proposition, a built-in resistor was obtained and the resistance value could be tuned by changing the NMG content and the dimensional characteristics of the material strip.

Conclusions

To conclude, highly conductive graphene-based composite materials have been produced through a solvent-free procedure. This processing route based on latex blends and Nanosize Multilayer Graphene (NMG) provides a good candidate for conductive inks for printed electronics on flexible and textile substrates. The NMG was obtained after a few hours of mechanical mild grinding of graphite micro-flakes in water suspensions. Both the NMG suspensions and the composite latex blends exhibited good shelf-stability. The compositelatex blend is based on acrylate copolymers that are already mature in the ink and paint industry: they can form continuous and deformable films without neither high temperature curing nor additional hot-pressing, which is adequate for flexible and textile substrates.

Finally high conductivity values are reached (217 S.m -1 at 6.1×10 -2 NMG volume fraction) which is a basic requirement for electronic applications. To our knowledge, this value is among the highest value available in the literature for graphene-latex composites.

The final composite materials exhibit a micrometer-scale domain size with filler paths running throughout the material. The conductivity behavior is successfully described using a percolation approach consistent with the presence of a three dimensional filler network across the material. The potential interest for electronics was demonstrated by the use of the nanocomposite material in replacement of copper wires in a LED setup. These conductive nanocomposite suspensions could become a cheaper alternative to silver-based conductive inks for printed electronics and could open more versatile electronic applications due to the deformability of the polymer matrix, for instance, on textile substrates. xNMG [-] 0.0 1.6×10 -3 3.0×10 -3 5.0×10 -3 1.0×10 -2 2.0×10 -2 3.5×10 -2 5.0×10 -2 6.1×10 -2

Figure 1 .

 1 Figure 1.Thickness h of the NMG nanoplatelets. (a) AFM height image, (b) Height profiles of

Figure 2 .

 2 Figure 2. Cumulative distribution of the NMG lateral size d assessed using AFM (triangles)

Figure 3 .

 3 Figure 3. (a) TEM micrograph of NMG nanoplatelets (arrows), (b) HR-TEM micrograph of

Figure 4 .

 4 Figure 4. Processing steps of the nanocomposite materials, (a) blank latex, (b) NMG

Figure 5 :

 5 Figure 5: TEM micrograph of a thin foil extracted from sample 7 (x NMG = 5×10 -2 ).

Figure 6 .Figure 7 :

 67 Figure 6. TEM micrograph of a thin foil extracted from (a) sample 3 (x NMG = 5×10 -3 ) and (b) sample 7 (x NMG = 5×10 -2 )

Figure 8 :

 8 Figure 8: LED setup with conductive graphene-based nanocomposite wires (sample 8) in

Table 1 :

 1 NMG volume fraction (x NMG ) for each NMG/latex suspension

Acknowledged

The authors would like to thanks P-Y. Dugas (C2P2/CPE/CNRS) for his help concerning the TEM observations done at the CTµ (Centre Technologique des Microstructures, Université Lyon1).


Yi An

Dragana Vukovic

Abel Lorences Riesgo

Geert Morthier

Christophe Peucheret
email: christophe.peucheret@enssat.fr

Ultra-Fast All-Optical Self-Aware Protection Switching Based on a Bistable Laser Diode

We propose a novel concept of all-optical protection switching with link failure automatic awareness based on AOWFF. The scheme is experimentally demonstrated using a single MG-Y laser diode with a record switching time ~200 ps.

Introduction

Protection switching is a key technique to ensure survivability against outages in optical networks by providing an alternative path to carry the traffic in case of link failure, thus providing high reliability and availability of the communications [START_REF] Ramaswami | Optical Networks A Practical Perspective[END_REF]. Traditionally, some fault detection mechanisms are implemented, which report a failure to the network management system. The protection mechanism is then triggered from the control plane, possibly resulting in a relatively long delay to establish the protection path. For instance, in widely deployed SONET/SDH networks, the protection switching time is typically of the order of ~50 ms [START_REF] Ramaswami | Optical Networks A Practical Perspective[END_REF][START_REF]ITU-T Recommendation G.841[END_REF] plus 10 ms [START_REF] Ramaswami | Optical Networks A Practical Perspective[END_REF] for failure detection. For some critical applications, such a protection time is prohibitively long. The use of fast electronically tunable lasers has been suggested for protection with an improved switching time down to 8 ms [START_REF] Chowdhury | OFC'11[END_REF]. However, such a scheme still relies on electronic signaling in order to set the tunable laser currents after the fault has been detected. One way to reduce the protection time into the sub-ns scale is to enable the establishment of the protection path to be triggered directly in the optical domain.

For the very first time, this problem is addressed here. In this paper, we propose an all-optical protection scheme, with failure detection and protection trigger all implemented on the optical layer. Since there is no upper layer mechanism required, the protection can be performed ultra-fast. We also experimentally demonstrate the proposed scheme successfully, with reported protection switching time of about 200 ps. The widely opened eye diagrams show good performance at both 10 Gbit/s and 40 Gbit/s data modulation. Optical control pulses of 50 ps duration and less than 1 pJ/pulse energy are used for triggering the establishment of the protection path.

Operation principle

The proposed scheme relies on a wavelength-bistable laser (also known as all-optical wavelength flip-flop, AOWFF) used as a continuous wave (CW) light source with external data modulation at the transmitter, as shown in Fig. 1(a). In an AOWFF, light emission can be switched between two stable operation states on different wavelengths by two distinguishable optical control pulses (set and reset). The wavelength of the emitted signal will thus depend on the state of the AOWFF. A wavelength selective component such as an arrayed waveguide grating (AWG) can then be employed to route the signal on the predefined operation wavelength WL opt to the working fiber, or on the protection wavelength WL pro to the backup fiber in case an event triggers the protection switching mechanism. The fault detection and subsequent triggering of the AOWFF is implemented as follows. A periodic optical control pulses associated with WL opt is generated at the destination side and transmitted via the working path fiber back to the source for controlling the AOWFF. This control is denoted Ctrl R . The other control, associated with WL pro , is locally generated at the source side and denoted Ctrl L . The Ctrl R and Ctrl L pulses are generated periodically with the same repetition rate.

The timing diagram of Fig. 1(b) illustrates the operation of the scheme. Under normal operation (in the interval [t 0 , t 1 ]), Ctrl L appears first and turns on WL pro . Shortly after, the remote control pulse Ctrl R arrives and resets the operation back to WL opt . The time interval over which WL opt is switched on, until it is switched off by the next Ctrl L , is therefore a time-frame of fixed duration, which can be used for sending the data traffic. If Ctrl R does not reach the AOWFF due to link failure, as indicated in the interval [t 1 , t 2 ] in Fig. 1(b), the local control Ctrl L will enable WL pro as usual, however WL opt will not be reset. This will result in the data being now modulated on the protection wavelength and routed to the backup fiber by the AWG. Therefore the protection is realized. As soon as Ctrl R is back, the operation will be changed back to normal automatically, as illustrated from t 2 in Fig. 1(b).

In this way, the entire protection procedure, including link failure detection and protection trigger, which are all performed via the optical remote control pulses Ctrl R , is realized on the optical layer. The total protection time will then depend on the switching speed of the AOWFF device, which can be of the order of a few hundred picoseconds, as demonstrated further. It also depends on the propagation delay and the size of the frames, but these are common to all protection schemes, including those relying on the protocol. The inherent advantage of the scheme, suitable for critical applications, is that the establishment of the protection path is performed automatically without referring to the management system. This results in a substantial time advantage over traditional schemes where the fault detection is performed by time-averaged measurements followed by alarm trigger to the management system, which will then decide and take contingency actions, such as switching of fast tunable laser diodes after appropriately setting the driving currents. Even though the proposed optical layer protection switching scheme does not provide fault location, its main benefit is to allow an ultra-fast set up of the protection link, while the other functionalities such as fault monitoring can still be realized through the protocol. The scheme however requires synchronization of the local and remote optical clock sources as well as slotted operation. Fig. 2 shows the experimental setup used for the proof-of-concept demonstration of the proposed scheme. The AOWFF used in the experiment is a single modulated grating Y-branch (MG-Y) laser structure [4], in which flipflop operation over a large wavelength tunable range has been recently demonstrated, with switching time less than 200 ps and 0.16-0.34 pJ control pulse energy [START_REF] An | Fr-S18-O21[END_REF]. Two tunable CW lasers were externally modulated using two Mach-Zehnder modulators (MZMs) in order to generate the 50 ps long control pulses with, at first, a low repetition rate of 78.125 MHz. The control pulses were amplified in erbium-doped fiber amplifiers (EDFAs), filtered in optical band-pass filters (BPFs) and their relative delay was adjusted using an optical delay line (DL) before they were injected into the MG-Y laser via a circulator. The state of polarization of the control signals was beforehand adjusted using polarization controllers (PCs). The MG-Y laser operated at a bistable window with two states on 1554.37 nm or 1554.66 nm. Its output was then modulated with 10 Gbit/s or 40 Gbit/s non return-to-zero (NRZ) data with a 2 31 -1 pseudo-random binary sequence (PRBS). Two BPFs were used to emulate an AWG and route the signal to a different port depending on its wavelength. The signals at the operation wavelength and at the protection wavelength were then monitored on an optical sampling oscilloscope. To emulate the link failure, the remote control pulses were periodically suppressed using another MZM driven at 9.766 MHz, resulting in 3 out of 8 Ctrl R pulses being suppressed. The dynamic extinction ratio of the modulator was about 10 dB and the gating window generated by the MZM can be seen in Fig. 3(a). In order to illustrate the scheme, the waveforms on both wavelengths in both normal operation and protection regime are shown in Fig. 3(b)-(c). Under normal operation, as indicated in the figure, 5 frames of data traffic on WL opt can be seen. In Fig. 3, a short frame length of ~100 bits at 10 Gbit/s was chosen in the experiment to ease the visualization of the process on the oscilloscope traces, but the scheme is obviously highly scalable towards lower repetition rates of the control signals, allowing much longer data frames to be transmitted.

Experimental demonstration

The next 3 Ctrl R pulses were suppressed, emulating a cut in the working path. As shown in Fig. 3(c), the data was immediately switched to WL pro while no more data was sent on WL opt , as can be seen in Fig. 3(b), apart from some residual out-of-band crosstalk leaking through the 0.3 nm OBPF used for signal separation in our proof-ofconcept implementation. Fig. 4 shows the eye diagrams of both WL opt in normal operation and WL pro in protection operation with 10 Gbit/s and 40 Gbit/s data modulation. The eyes are both widely open, indicating good signal quality. BER measurements were also performed with 10 Gbit/s NRZ data modulation, as shown in Fig. 5(a). When modulating the MG-Y laser output without any control pulses, the sensitivity is about -34 dBm (at BER=10 -9 ). Unfortunately, due to the lack of a proper burst gating control, BER measurements for both normal and protection operations could only be performed in continuous gating mode. For error counting, the repetition rate of the control signals was set to 9.766 MHz, allowing 1024 bits of 10 Gbit/s signal in every period. As 10 bits were switched off from WL opt between two adjacent frames, as shown in Fig. 5(b), and assuming an equal distribution of marks and spaces when averaging over many frames, 5 bits on average would be erroneously detected even if no error occurs during normal operation, leading to a minimum achievable BER of 4.9×10 -3 . The measured BER of WL opt with sufficient receiver power was exactly lying on this level, indicating no performance degradation for the bits transmitted over the frame. On WL pro , some bits were affected by Ctrl L , as shown in Fig. 5(c), since it disturbed the carrier density of the MG-Y laser, causing a moderate power penalty compared to CW operation, as shown in Fig. 5(a).

Conclusion

A novel concept of all-optical protection switching was proposed. Link failure detection and protection trigger were both implemented on the optical layer. The concept was experimentally demonstrated using a wavelength tunable AOWFF based on a single MG-Y laser structure. The achieved protection switching time was less than 200 ps. The good quality of the signal following the establishment of the protection path was evaluated at 10 and 40 Gbit/s. The method provides a fast establishment of the protection path, resulting in a minimum loss of data for critical applications, while the failure can still be detected by conventional slower means (e.g. by monitoring the power at the output ports of the wavelength router) resulting in the network management being informed by the protocol and appropriate measures being taken.

Fig. 1

 1 Fig. 1 Proposed scheme for all-optical self-aware protection switching. (a) Physical configuration. (b) Timing diagram for normal operation in the interval [t 0 , t 1 ] and protection operation in [t 1 , t 2 ].

Fig. 2

 2 Fig. 2 Experimental setup for proof-of-concept demonstration of protection switching.

Fig. 3

 3 Fig. 3 All-optical protection switching demonstration: (a) Gating window for emulating the link failure. Waveform on (b) WLopt in normal mode and (c) WLpro in protection mode.

Fig. 4

 4 Fig. 4 Eye diagrams of the data at the transmitter output under normal operation on WLopt at (a) 10 Gbit/s and (b) 40 Gbit/s, as well as under protection operation on WLpro at (c) 10 Gbit/s and (d) 40 Gbit/s.

Fig. 5

 5 Fig. 5 Link performance with 10 Gbit/s NRZ data: (a) BER measurement. (b) Time interval between two adjacent frames on WL opt in normal operation and (c) Bits on WL pro affected by the local control pulses in protection operation.


Antonio Estrada

Denis Efimov
email: denis.efimov@inria.fr

Wilfrid Perruquetti
email: wilfrid.perruquetti@inria.fr

Position and velocity estimation through acceleration measurements

This paper proposes a solution to the problem of velocity and position estimation for a class of oscillating systems whose position, velocity and acceleration are zero mean signals.

The proposed scheme considers that the dynamic model of the system is unknown and only noisy acceleration measurements are available.

INTRODUCTION

The estimation of position/velocity through the integration of measurements from inertial sensors inherently causes errors to grow with time, commonly known as integration drift. In fact, the process of zero-mean noise integration leads to an output that increases with integration time, even in the case that the accelerometer is at rest [START_REF] Thong | Numerical double integration of acceleration measurements in noise[END_REF]. For this reason, velocity/position estimation based on inertial sensors requires some additional source of information providing for the corresponding correct signal value at some time instants. In general this information is updated with intervals significantly larger than the acceleration measurement rate. For instance, in inertial navigation systems (INS) aiding sensors are applied. An INS consists of a triad of orthogonal accelerometers and rate gyroscopes that are used to calculate the both the translational and rotational movements of an aircraft over time [START_REF] Rogers | Applied Mathematics in Integrated Navigation Systems[END_REF], a process know as dead reckoning. MEMS technology has made low-cost INS available. However, sensor biases coupled with the time integration relationships needed for dead reckoning cause that the solution quickly diverges. That is why externally referenced sensors, such as the global positioning system, are required to regulate the error growth [START_REF] Grewal | Global Positioning, Inertial Navigation and Integration[END_REF]; Kalman filters are commonly used in order to fuse the two sources of information. An alternative to the use of aiding sensors is the use of prior-knowledge of motion. In [START_REF] Liu | Development of a wearable sensor system for quantitative gait analysis[END_REF], for instance, the drift accumulation from gyroscope based orientation estimation, in human walking experiments, is compensated by detecting different gait phases with the aid of accelerometers and the knowledge of the human walking motion. In [START_REF] Schepers | Ambulatory estimation of foot placement during walking using inertial sensors[END_REF], foot placement for several strides is estimated. The drift is reduced applying the knowledge of the cyclic walking movement that leads to information on initial and final conditions.

In the present work we address the problem of velocity/position estimation based only in inertial information with no aiding sensor nor an explicit dynamical model. We propose a method that allows to adjust the estimated position for the case in which the acceleration measurements, which can be contaminated by zero mean noise, are unbiased and correspond to an oscillating motion with zero mean velocity and position. This proposal compensates the drift using the oscillating motion period. Then, a period estimation algorithm is fundamental considering that the oscillation period is in general not available. In [START_REF] Spence | A time-domain analysis of multiple periodic signals[END_REF], a method for estimating the period and amplitude profile of a signal with multiple periodicities is proposed based on the results reported in [START_REF] Feder | Parameter estimation and extraction of helicopter signals oberved with a wide-band interference[END_REF]. In [START_REF] Spence | A time-domain analysis of multiple periodic signals[END_REF], a likelihood computation is put in terms of a least mean square (LMS) residual power expression which is evaluated for different test periods. The estimated period is chosen as the one minimising the LMS residual power. In the field of music and speech analysis the pitch period (fundamental frequency) estimation is very important. For short-time estimation of pitch period, the algorithms that find the average fundamental frequency using autocorrelation or linear prediction techniques are the most commonly used [START_REF] Guangyu | Improving AMDF for pitch period detection[END_REF], [START_REF] Peeters | Music pitch representation by periodicity measures based on combined temporal and spectral representations[END_REF], [START_REF] Bernardin | Wavelet processing for pitch period estimation[END_REF], [START_REF] Min | A pitched-based rapid speech segmentation for speaker indexing[END_REF]. We propose a period estimation algorithm that takes advantage on the zero mean nature of the signals. This scheme is not restricted to sinusoidal dominant frequency and is computationally simple. Relying on the aforementioned oscillation period estimation, a solution for finite time tracking of position and velocity is proposed. The rest of the paper is organized as follows, in the next section the problem statement is presented. In section III the possibility of an asymptotic velociy/position estimation is discussed and the idea of using the oscillation period is introduced. Section IV is devoted to the period estimation proposal. In Section V the scheme for finite-time position/velocity estimation is presented. Finally, Section VI presents simulation results.

PROBLEM STATEMENT

Consider a system with the following acceleration dynamics ẍ = f (x, ẋ) (1) and measurement model Ψ(t) = ẍ(t) + η, (2) where η is a bounded zero-mean noise. The acceleration measurement is assumed unbiased. f (x, ẋ) is an unknown nonlinear function. It is assumed that the position x, and hence ẋ, ẍ, have a periodic or quasi-periodic behaviour, more precisely it is considered that position is oscillating between two bounds, (there exist two constants such that B m ≤ x (i) (t) ≤ B M , ∀t > 0, i = 0, 1, 2). The objective of this work is to present an alternative solution for velocity and position estimation in the case of only acceleration measurement for a class of periodical signals, i.e. x (i) (t) = x (i) (t + T f ) for some finite T f .

VELOCITY ESTIMATION

Velocity estimation through averaging

In the following the super index < i > is used to indicate the i-th time integral of a function y : R → R:

y <i> (t) = t 0 . . . τ2 0 y(τ 1 ) dτ 1 . . . dτ i . (3)
Consider (2), in order to obtain velocity estimation we proceed to integrate directly Ψ(t), then

Ψ <1> (t) = ẋ(t) -ẋ(0) + η <1> , (4) Ψ <2> (t) = x(t) -x(0) -ẋ(0) • t + η <2> ,
(5) dividing ( 5) by t we obtain

Ψ <2> (t) t = -ẋ(0) + x(t) -x(0) + η <2> t . (6)
Then, define the velocity estimation, x, as follows:

x =Ψ <1> (t) - Ψ <2> (t) t = ẋ(t) + η <1> - x(t) -x(0) + η <2> t . (7)
Let us recall that the position is bounded and that the noise is zero-mean and bounded. Then, the next limit can be obtained

lim t→∞ x(t) -x(0) + η <2> t = 0. (8)
Therefore, for t "sufficiently" large x ≃ ẋ(t) + η <1> .

(9) The amount of time, which is necessary to obtain a reliable estimation, will depend on the magnitude of the bounds of position and noise. Next, a similar averaging approach can be applied for position estimation. Such estimation would be asymptotic without a possibility to regulate the speed of convergence. In addition, integration on an infinite time interval can be technically difficult without a significant accumulation of computation error leading to large estimation error. Thus a finite time solution has to be found. In fact a possibility to find a solution in finite time is related with knowledge of the period. In next section, this statement about the necessity of period knowledge is further discussed.

Finite time velocity estimation

Assume that ẍ, ẋ and x are unbiased periodic functions with period T f . In fact, if ẋ or ẍ have a bias, then x would not be periodical and bounded. Moreover, a constant bias in position cannot be identified from acceleration or velocity measurements because none of them will reflect such a bias. Now, consider the equations ( 4) and ( 5)

evaluated at T f Ψ <1> (T f ) = ẋ(T f ) -ẋ(0) + η <1> , (10) Ψ <2> (T f ) =x(T f ) -x(0) -ẋ(0) • T f + η <2> , (11)
where x(T f ) = x(0) and ẋ(T f ) = ẋ(0). Hence, an estimation x(0) of ẋ(0) can be obtained at t = T f :

x(0) = - Ψ <2> (T f ) T f (12) = ẋ(0) - η <2> T f . ( 13
)
Then, if period is known, an estimation of velocity can be derived using equations ( 10) and ( 13):

x(t) = Ψ <1> (t) -x(0). ( 14) Next, the same approach can be repeated for position estimation.

Importance of period knowledge

Note that x(t) = x(0) may hold for different time instants, but the correct estimation of velocity can only be ensured for t = κT f with κ ∈ IN. It implies that estimation of the initial condition, ẋ(0), without knowledge of the period may lead to big errors. To illustrate this idea, consider the next expression for acceleration ẍ = sin(1.5t) + sin(4.2t), (15) which is plotted with its corresponding velocity and position signals in Fig. 1. Suppose that measurements start at t = A. Note that there is a subperiod, between times A and B, for both acceleration and velocity. Given that only the acceleration is available, this zero mean subperiod can be wrongly considered as the real period causing the error term (x(B) -x(A))/(B -A) to be present in the estimation x(0), see Eq. ( 13). In next section, a solution to the problem of period estimation is presented.

PERIOD IDENTIFICATION

The algorithm for period estimation proposed in this paper is based on the features of the phase portrait ( ẋ, ẍ). To introduce the idea, consider the phase portrait ( ẋ, ẍ) of (15) depicted in Fig. 2. It is clear that after one period the trajectories pass through each point at least once. Nevertheless, there are a number of points and furthermore trajectory segments, where the trajectory passes more than once over a cycle. If we can select a point which is reached only once over a whole period, then an estimation of the period can be obtained by detecting the time elapsed between two consecutive passings over such point. The question is how to properly select this point. Recall that the real phase portrait is not available because there is no velocity measurement. From Equation (4) it can be seen that the velocity estimation, ẋ, is shifted

Period Identification Algorithm

Recall that ẋM denotes the global extremum in velocity. Acceleration is in general a noisy measurement and the accuracy of integration is additionally limited by the sampling time. The algorithm for period estimation should perform the following tasks (1) To ensure the detection of every occurrence, once each period, of the extremum ẋM . Due to the integration error, δ η <1> , detected extremum cannot be separated from the real ẋM by a distance larger than δ η <1> . The algorithm should ensure that the above mentioned δ η <1> -vicinity is achieved at each new ẋM passage.

(2) To ensure that no local extremum is wrongly estimated as ẋM . In addition to possible extrema outside of the δ η <1> -vicinity, there may be local extrema inside such a vicinity. These local extrema should be disregarded. The above implies that it is necessary to wait until being outside of the δ η <1> -vicinity in order to decide that a new detected extremum will be considered as a new occurrence of ẋM .

(3) In order to avoid accumulation of drift, the integral of acceleration, on which the detection of ẋM is based, should be restarted once each period. This restart can be performed after a new ẋM occurrence has been confirmed.

If the above conditions are fulfilled then an estimation of the period can be made as the difference between two consecutive detected occurrences of ẋM . Some definitions are introduced prior to present the algorithm. α is a parameter to be selected as 1 or -1 in order to detect a global maximum, or a global minimum respectively. ẋM - is the extremum, different from ẋM , whose amplitude is the closest to ẋM . The notation Ψ <1> (t a , t b ) indicates the integral of Ψ from t = t a to t = t b . Next we present the proposed period detection algorithm:

Algorithm A1. Initialization of variables α = ±1, t 0 = 0, t 1 = 0, f l = 0, δ > δ η <1> Ψ <1> m = 0, C0. If αΨ <1> (t 0 , t) > αΨ <1> m -δf l then Ψ <1> m =Ψ <1> (t 0 , t), t 1 =t, f l =0. (16) C1. If αΨ <1> (t 1 , t) < -2δ and f l = 0 then Ψ <1> m =0, Tf =t 1 -t 0 , t 0 =t 1 , f l =1, ( 17
)
Remark 2. The only parameter to be tuned is δ whose value will be related to noise amplitude and is further discussed in the sequel.

We state the claims: Lemma 1. Each occurrence of ẋM is followed by only one fulfillment of condition C1 with a value for t 1 such that ẋ(t 1 ) > ẋM -αδ η <1> .

(18) Proposition 2. If assumptions S1-S3 are fulfilled then, an estimation of the real period, T f , is obtained by means of the algorithm A1 provided that

1 2 | ẋM -ẋM -| > δ > δ η <1> . ( 19
)
Assume that α = 1 is selected. Then, the algorithm A1 proceeds in the following manner:

• The condition C0 at the beginning is αΨ <1> (0, t) > 0, and it is fulfilled for any positive quantity of the integral αΨ <1> (0, t). • After the first fulfillment of C0 the assigment Ψ <1> m = Ψ <1> (0, t) is done in order to save the new maximum, and the time of its occurrence is saved on t 1 .

• The condition C1 is checked for the integral Ψ <1> (t 1 , t),

which is restarted at the instant t 1 of the last maximum detection of Ψ <1> (t 0 , t). This condition is satisfied if Ψ <1> (t 1 , t) becomes less than -2δ and f l = 0. • The condition C1 serves for the detection purpose of a passage of the global maximum ẋM (in order to distinguish it from ẋM-or other local extremums).

It is shown in the proof of Lemma 1, that after the first occurrence of ẋM the condition ( 18) is fulfilled. • Finally, note that when condition C1 is fulfilled, the integral Ψ <1> (t 0 , t) contains the integrated acceleration with the initial time corresponding to the last maximum. Additionally, the condition C0 takes the form Ψ <1> (t 0 , t) > -δ. This change in the form of condition C0 is needed in order to take into account a possible drift due to the integration error δ η <1> < δ.

Proofs.

Let the unique absolute extremum for ẋ be a global maximum denoted as ẋM (the proof in the case of a global minimum follows the same rationale). For this case α = 1 is selected and the conditions C0 and C1 are rewritten as

Ψ <1> (t 0 , t) > Ψ <1> m if f l = 0 (20a) Ψ <1> (t 0 , t) > -δ if f l = 1 (20b) and Ψ <1> (t 1 , t) < -2δ, f l = 0, (21)
respectively. Note that condition C0 can take two forms, (20a) for f l = 0 and (20b) for f l = 1.

Proof of Lemma 1. This proof is divided into two parts:

• First, let us show that the condition ẋ(t 1 ) ∈ [ ẋM -δ η <1> , ẋM ]. ( 22) is satisfied independently on the initial conditions ẍ• = ẍ(0), ẋ• = ẋ(0) and x • = x(0). Define T 1 M as the time of first appearance of ẋM and suppose that the initial condition ẋ• is such that ẋ

• < ẋM -δ η <1> , (23)
in the contrary case, (22) will be necessarily satisfied after the first fulfillment of C1. Then, considering (23), if ( 21) is not fulfilled between t = 0 and t

= T 1 M we have Ψ <1> (t 0 , T 1 M ) = Ψ <1> (0, T 1 M ) = ẋM -ẋ• + η <1> > ẋM -ẋM + δ η <1> -δ η <1> > 0.
Thereby, C0, that is (20a), will be necessarily true at least for ẋ(t) = ẋM , recall that the errors are comprised in δ η <1> . The above implies that the correspondent time t 1 captured due to C0 is such that ( 22) is verified. On the other hand, suppose that C1, see ( 21), is true before passing through ẋM . This implies that Ψ <1> (t 0 , t) has decreased 2δ in magnitude since its last largest value. The assignments Ψ <1> m = 0 and f l = 1 due to C1 implies that condition C0 takes the form (20b). Now, note that the latter must be fulfilled before a new true event of C1. Since t 0 is reset to the time of occurrence of the last largest value of Ψ <1> (t 0 , t), one obtains that Ψ <1> (t 0 , t) < 2δ holds, after the assignments in C1 with x M still to appear. Then, there exist a time t * such that ẋ(t * ) = ẋ• ; with t * < T f , and for t = t * one obtains Ψ <1> (t 0 , t * ) = ẋ(t * ) -ẋ(t 0 ) + η <1> ≥ -δ η <1> > -δ. Then, (20b) is fulfilled and C0 takes the form (20a) again. From the above, if ẋ(t * ) = ẋ(t 0 ) < ẋM -δ η <1> then condition (20) will be true at some time satisfying ẋ(t) ∈ [ ẋM -δ η <1> , ẋM ].

This concludes the part that ensures that t 1 will be such that (22) holds independently of the initial conditions.

• Once t 1 satisfies (22), condition C1 is fulfilled only once after each ẋM occurence. Consider that C1 is true after ( 22) is satisfied then,

ẋ(t 0 ) ∈ [ ẋM -δ η <1> , ẋM ], (24)
and note that C0 must be before C1 can be satisfied again. Next we show that C0, which is now in the form (20b), can only be true again in the vicinity ẋ(t) ∈ [ ẋM -δ η <1> , ẋM ]. Recall that the local extremum closest in magnitude to ẋM is ẋ-M . Then, for ẋ(t) = ẋ-M , after (24) holds, we obtain

Ψ <1> (t 0 , t) = ẋ- M -ẋ(t 0 ) + η <1> Ψ <1> (t 0 , t) ≤ ẋ- M -ẋM + η <1> . From (19) we have that ẋ- M -ẋM < -2δ, hence Ψ <1> (t 0 , t) < -2δ + δ η <1> Ψ <1> (t 0 , t) < -δ
thus, ẋ(t) must be larger than ẋ-M in order to fulfill again C0. This can only occur in vicinity of [ ẋMδ η <1> , ẋM ]. On the other hand, C0 will necessarily be true in the latter vicinity, to show this consider the case ẋ(t) = ẋM :

Ψ <1> (t 0 , t) = ẋ(t) -ẋ(t 0 ) + η <1> Ψ <1> (t 0 , t) ∈ ẋM -ẋM -[-δ η <1> , 0] + η <1> Ψ <1> (t 0 , t) > -δ η <1> > -δ.
Then, condition C0 is fulfilled and a new time t 1 will be captured such that ( 22) is satisfied. This concludes the proof of Lemma 1.

Proof of Proposition 2. Proposition 2 is a straightforward result of Lemma 1. Note that if ( 22) is satisfied, then for each subsequent C1 verification, the variables t 0 and t 1 will contain the time of detection of two consecutive global maxima of Ψ <1> , with an error which depends on the noise and the sampling, δ η <1> . The condition ( 22) cannot be ensured before a whole period has elapsed, then at least two periods are needed in order to ensure that a correct estimation of T f is obtained. Moreover, as the condition C1 is needed to compute the estimation, the first correct estimation may occur during the third period after launching the algorithm, when Ψ <1> exits the 2δ vicinity. It is worth to stress that the developed algorithm does not impose a restriction on possible values for the amplitude and the period of the signals. Let us remark that the accuracy does not depends on the parameter δ of the algorithm, the error in the estimation is directly related to δ η <1> which is inherent to the system.

VELOCITY AND POSITION ESTIMATION BASED ON PERIOD ESTIMATION

The period estimation, Tf , obtained in the previous section can be used to estimate ẋM and in turn to obtain a real time velocity estimation. The fact that ẋM is estimated at each period will be used in order to compensate the drift due to errors previously mentioned. Define T i M the time of i-th occurrence of the extremum ẋM , and denote T i M its estimate calculated after fulfillment of condition C1, then T i M = t 1 . Let T i f be the period estimation corresponding to the time T i M , which is obtained on the i-th iteration of Algorithm A1, then T i f = T i M -T i-1 M . Thus, based on (13) we obtain

x(T i M ) = - Ψ <2> ( T i-1 M , T i M ) ( T i f )
Finally, the estimation of velocity and position is obtained through the next expressions

x(t) =Ψ <1> ( T i M , t) -x( T i M ), (33)
x(t) =Ψ <2> ( T i M , t) -x( T i M ).

EXAMPLE

Expressions (33) and (34) (with the results obtained from algorithm A1) were applied for velocity and position estimation with acceleration measurements for a variant of Van der Pol oscillator ẍ + 0.1(x + 1)(x -1)(x -2.2) ẋ + x = 0 (35) Ψ(t) = ẍ(t) + η(t) (36)

The parameters in (35) were proposed in [START_REF] Karreman | Heart muscle contraction oscillation[END_REF] as a mathematical model for an isometric contraction of the mammalian cardiac muscle. The zero mean noise has been selected to satisfy |η(t)| < 0.2. The phase portrait (ẍ, ẋ) of ( 35), with initial conditions x 0 = 3 and ẋ0 = -2, is displayed in Fig. For a class of oscillatory motion, a scheme for velocity and position estimation based on noisy acceleration measurements is presented. The inherent error accumulation, due to integration drift, is compensated using information about the period of oscillation. An algorithm for estimation of the aforementioned period of oscillation is proposed. No model of the motion or a priori information on the period of the signals are required. As a result, the proposed scheme is suitable as a standalone solution for estimation of position and velocity using only inertial information without any aiding measurements. Further research is directed to the application of the proposed solution on a practical scenario with real measurement data.

  Fig. 1. Acceleration, Velocity and Position (sec).

  Fig. 3. Phase portrait (Ψ <1> , Ψ) for system (35).

  Fig. 4. Velocity and position estimation for (35).

This work was supported in part by the Government of Russian Federation (Grant 074-U01) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).


Mathilde Gay
email: gay@enssat.fr

Arthur O'hare

Laurent Bramerie

Zhenyu Hao

Schadrac Fresnel

Christophe Peucheret

Pascal Besnard

Siddharth Joshi

Sophie Barbet

François Lelarge
email: francois.lelarge@3-5lab.fr

Single quantum dash mode-locked laser as a comb-generator in four-channel 112 Gbit/s WDM transmission

Keywords:

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Various low cost transmitters have been proposed as solutions for wavelength division multiplexed (WDM) passive and broadband optical access networks. These solutions are mainly based on spectral slicing of broadband incoherent [START_REF] Kim | A low-cost WDM source with an ASE injected Fabry-Perot semiconductor laser[END_REF] or coherent [START_REF] Mun | A cost-effective WDM-PON using a multiple section Fabry-Pérot laser diode[END_REF][START_REF] Gubenko | Error-free 10 Gbit/s transmission using individual Fabry-Perot modes of low-noise quantum-dot laser[END_REF] sources. Quantum dash mode-locked lasers (QD MLLs) are attractive because they present very low amplitude and phase noise, high thermal stability and emit a broad optical spectrum [START_REF] Lelarge | Recent advances on InAs/InP quantum dash based semiconductor lasers and optical amplifiers operating at 1.55 µm[END_REF][START_REF] Liu | 1.6-m multiwavelength emission of an InAs-InGaAsP quantum-dot laser[END_REF]. A WDM source with nearly 30 channels on a 40 GHz grid could be realised using a single laser [START_REF] Akrout | Error-free transmission of 8 WDM channels at 10 Gbit/s using comb generation in a quantum dash based mode-locked laser[END_REF]. However, a major limitation in comparison to single-mode lasers when using the wavelength comb of a Fabry-Perot laser diode (FP LD) as a WDM source is the increase of relative intensity noise (RIN) of the selected mode due to mode partition noise [START_REF] Petermann | Laser diode modulation and noise[END_REF][START_REF] Akrout | Error-free transmission of 8 WDM channels at 10 Gbit/s using comb generation in a quantum dash based mode-locked laser[END_REF], particularly in the low frequency range. This limitation was mitigated at 2.5 Gbit/s in [START_REF] Nguyen | Bidirectional 2.5-Gb/s WDM-PON using FP-LDs wavelength-locked by a multiple-wavelength seeding source based on a mode-locked laser[END_REF] thanks to the injection-locking mechanism in a FP LD. However, this solution has never been shown at higher bit rate. A second solution consists in using a semiconductor optical amplifier (SOA) in the gain compression regime to reduce the mode partition noise. It was demonstrated in [START_REF] Sato | Reduction of mode partition noise by using semiconductor optical amplifiers[END_REF] with a spectrally filtered FP LD at 10 Gbit/s. This technique happens to be particularly attractive since it also provides gain to the selected line and is compatible with photonic integration. To the best of our knowledge, this method has never been assessed with a frequency comb generated by a QD-MLL, that could validate the concept at higher bit rates, for which the RIN contribution becomes more critical.

In this paper, we demonstrate the first WDM transmission of four 100 GHz spaced channels, each modulated at 28 Gbit/s, generated using a single QD MLL followed by an SOA limiting the RIN of each individual filtered mode. Transmission over 100 km of a capacity of 112 Gbit/s with less than 0.5 dB penalty proves the concept.

Laser structure and characterisations

The active region of the QD MLL is composed of quantum dash layers with InGaAsP quantum wells in a Fabry-Perot (FP) structure [START_REF] Lelarge | Recent advances on InAs/InP quantum dash based semiconductor lasers and optical amplifiers operating at 1.55 µm[END_REF]. The geometric structure is a buried hetero-structure with an FP cavity length of 420 µm, resulting in a free spectral range of 100 GHz. The chip is placed in a module in order to improve the coupling stability. The optical coupling is realized using bulk lenses. The total coupled power for a bias current of 150 mA is +6 dBm at 22°C, which are the operational conditions used in the following system experiment. Fig. 1(a) shows the optical spectrum at the output of the laser under these conditions. The spectral width is 7.3 nm at -3 dB, covering 10 modes (numbered from 1 to 10 in Fig. 1(a)).

A commercially available SOA, consisting of a 1 mm long InP buried heterostructure, was used for RIN reduction (CIP). It was biased at 180 mA during all the experiments, leading to 20 dB small signal gain and 6 dBm saturated output power.

The RIN of the laser was deduced from measurements of the spectral density of optical noise and electrical power [START_REF] Poette | Highly sensitive measurement technique of relative intensity noise and laser characterization[END_REF]. The results are presented in Fig. 1(b). When all modes are jointly detected, the RIN (green) is below -135 dBc/Hz thanks to the small amplified spontaneous emission noise of this kind of laser [START_REF] Lelarge | Recent advances on InAs/InP quantum dash based semiconductor lasers and optical amplifiers operating at 1.55 µm[END_REF]. When a single mode is selected, the RIN increases due to mode partition noise. The RIN spectrum of the noisiest line of the frequency comb (channel #8) is shown in purple in Fig. 1(b). The RIN measured when this same selected mode is amplified by the SOA is represented by the blue line in Fig. 1(b). As expected, the RIN decreases in the low frequency range (below 4 GHz) thanks to the SOA gain compression. A similar behavior is observed for all modes, validating the RIN limiting effect of the SOA with our QD-MLL comb-generator. Next, the benefits of this technique will be highlighted in a high-speed WDM transmission experiment.

4× × × ×28 Gbit/s WDM transmission setup

The experimental setup is presented in Fig. 2(a). At the transmitter, the QD-MLL is protected by an isolator and its output is amplified by an erbium doped fiber amplifier (EDFA), which could, in principle, be replaced by an SOA for integration purposes. Four adjacent modes are selected with a multiport filter (corresponding to channels #7 to 10 in Fig. 1(a)). These channels include the spectral line that shows the highest intensity noise (# 8). The programmable filter transfer function consists of four flat-top passband regions spaced by 100 GHz and with 3 dB bandwidths of 60 GHz. Each channel will, in turn, have its RIN reduced by the saturated SOA and be analyzed at the end of the link. The SOA input power varied between -6 dBm and -3 dBm depending on the channel (due to lines power variations of the laser spectrum), corresponding to a gain compression comprised between 5.5 dB and 8 dB.

The four channels are then combined in odd-even pairs with two 3 dB couplers and modulated in the on-off keying (OOK) format by two Mach-Zehnder modulators (MZs), each driven by a 2 31 -1 pseudo random binary sequence (PRBS) at 28 Gbit/s. Adjacent channels are modulated by a different MZ modulator and are uncorrelated. The two pairs of modulated channels are then combined together through a third 3 dB coupler. Their power is equalized using optical attenuators. The channels are then amplified before being injected into the 100 km link made of non-zero dispersion shifted fiber (NZDSF) with a chromatic dispersion of +5 ps/nm/km at 1550 nm. The power per channel is 0 dBm at the link input. Chromatic dispersion is compensated by a dispersion compensating fiber (DCF). The propagation losses are compensated by another EDFA. The channel being investigated is then selected with a 100 GHz 3 dB-bandwidth flat top filter and analyzed at the receiver.

- Fig. 2 (a) Setup for system experiments. (b) Bit error rate as a function of receiver input power in back-to-back (B2B), after 100 km WDM transmission for channels #7, 8, 9 and 10 without the use of the SOA and with SOA.

Results

Fig. 2(b) presents the bit error rate (BER) evolution as a function of the receiver input power for different configurations. The open triangle curve is a reference curve obtained using an external cavity single mode laser (ECL) in a back-to-back (B2B) experiment (transmitter directly in front of the receiver). The full triangle curve is obtained with the ECL after 100 km transmission over the same link, showing no significant degradation. The circle curves are obtained for each investigated channel after 100 km transmission without the use of the SOA. Error floors appear between 10 -5 and 10 -8 , depending on the channel, because of the increased RIN when a single mode is selected. The square curves are those obtained after the introduction of the SOA. These curves are close to the case of the single mode laser curve, showing less than 0.5 dB penalty at a BER of 10 -9 . This shows that the RIN reduction in the 0-4 GHz range produced by the SOA leads to penalty removal and error floor suppression in the transmission experiment, allowing the transmission of 4 WDM channels at 28 Gbit/s using a single laser.

In order to further investigate the condition of operation, the impact of the input power to the SOA on the transmission quality of channel 9 is studied next. For that, the setup presented in Fig. 2(a) is used with the SOA placed at the output of channel 9 and with a variable power in front of the SOA. The BER as a function of receiver input power is measured at different SOA input powers and the power penalty at a BER of 10 -9 is calculated with respect to the single mode laser reference after 100 km (Fig. 3). It can be observed that, at power levels higher than -7 dBm, corresponding to a gain compression larger than 5.5 dB, the transmission penalty is smaller than 1 dB. This should therefore be the SOA minimum input power condition for the RIN to be suitably reduced. As a consequence, considering a QD-MLL laser generating 10 channels and 5 dB losses in the channel demultiplexer, the required power level in front of the SOA remains compatible with the use of a single SOA as a booster stage before the demultiplexer. This makes a potential integration possible. Fig. 3 Penalty at a BER of 10 -9 after 100 km transmission as a function of input power to the SOA.

Conclusion

In conclusion, we have achieved for the first time the transmission of four WDM channels with 100 GHz spacing using a single QD-MLL laser followed by an SOA as a WDM transmitter. The total capacity of 112 Gbit/s was reached with 28 Gbit/s per channel modulation over a distance of 100 km. The SOA was used as a limiting amplifier for mode partition noise reduction, leading to no appreciable penalty after 100 km for all four WDM channels compared to a single mode external cavity laser. It is worth noting that the scheme could be extended to the ten modes available from this laser, thus offering a total capacity of 280 Gbit/s with a single laser. This would require using a wavelength demultiplexer instead of the 4-channel filter. This work demonstrates that the use of a single QD-MLL followed by an SOA as a WDM transmitter is a cost-effective solution for WDM passive optical networks.

  Fig.1 (a) Optical spectrum of the QD-MLL. (b) Relative intensity noise spectra of all jointly-detected modes of the QD-MLL (green), of line #8 alone (purple) and of line #8 alone after amplification by the SOA (blue).

Acknowledgements

This work was supported in part by the French government through the ANR project Teldot (ANR 2009-VERS-12) and by the European Commission through the ITN project Prophet.