Jonathan Roux
email: jonathan.roux@laas.fr

Éric Alata
email: eric.alata@laas.fr

Guillaume Auriol
email: guillaume.auriol@laas.fr

Vincent Nicomette
email: vincent.nicomette@laas.fr

Mohamed Kâaniche
email: mohamed.kaaniche@laas.fr

Toward an Intrusion Detection Approach for IoT based on Radio Communications Profiling

Keywords: Internet of Things, IoT, Security, RSSI, IDS, Smarthome, Detection

Nowadays, more and more Internet-of-Things (IoT) smart products, interconnected through various wireless communication technologies (Wifi, Bluetooth, Zigbee, Z-wave, etc.) are integrated in daily life, especially in homes, factories, cities, etc. Such IoT technologies have become very attractive with a large variety of new services offered to improve the quality of life of the endusers or to create new economic markets.

However, the security of such connected objects is a real concern due to weak or flawed security designs, configuration errors or imperfect maintenance. Moreover, the vulnerabilities discovered in IoT products are often difficult to eliminate because, most of the time, they cannot be patched easily. Therefore, protection mechanisms are needed to mitigate the potential risks induced by such objects in private and public connected areas.

In this paper, we propose a novel approach to detect potential attacks in smart places (e.g. smart homes) by detecting deviations from legitimate communication behavior, in particular at the physical layer. The proposed solution is based on the profiling and monitoring of the Radio Signal Strenght Indication (RSSI) associated to the wireless transmissions of the connected objects. A machine learning neural network algorithm is used to characterize legitimate communications and to identify suspiscious scenarios. We show the feasibility of this approach and discuss some possible application cases.

I. INTRODUCTION

The Internet of Things (IoT) has received increasing interest in the last few years. Indeed, the use of smart connected objects has become a reality in most of the activities of our daily life, at home as well as in public and professional spaces. It is estimated that the number of IoT objects will exceed 25 billions in the next 5 years [START_REF] Analysts | Internet of Things (IoT) Market Trends[END_REF]. Many of such objects (speakers, TVs, cameras, doors, shutters, lightbulbs, screwdrivers, etc.), that used to lack connectivity in the past, have nowadays the possibility to interact with other objects in the vicinity or remotely through the Internet, using heterogeneous communication protocols, such as Zigbee, Bluetooth or Z-Wave. While such evolution enables the development of new attractive services for the users, serious concerns can be raised with respect to the new opportunities offered by IoT objects to attackers to threaten the security and privacy of the users [START_REF] Kolias | Learning IoT Security "Hands-on[END_REF]. Indeed, IoT devices are relevant attack targets because they often collect critical information about the network (such as usage, environmental data, location, other connected devices), which can endanger the privacy of users in case of compromission. Some of these devices also have the ability to control objects used to ensure physical security, like locks and windows. Consequently, seriously considering the security of such connected objects is a crucial objective.

Nevertheless, new vulnerabilities involving connected IoT devices are daily reported. This is related to the fact that several IoT objects are designed without thoroughly addressing security concerns due to short lifecycles and economic pressure, or simply because of a lack of security expertise. The user motivation is rather related to the service offered by the device, its communication capabilities or its ergonomy. This does not motivate the IoT companies to invest on the security issues. Finally, specific constraints inherent to IoT devices prevent the integration of traditional security solutions in their design and implementation due to their limited resources [START_REF] Keranen | RFC 7228 Terminology for Constrained-Node Networks[END_REF]. Also, the lack of standardisation in this area led to a profusion of new IoT-specific and heterogeneous communication protocols which may have not been thoroughly tested from the security perspective [START_REF] Al-Fuqaha | Internet of Things: A Survey on Enabling Technologies, Protocols, and Applications[END_REF].

Implementing security practices such as those recommended by the NIST [START_REF] Ross | Systems Security Engineering: Considerations for a Multidisciplinary Approach in the Engineering of Trustworthy Secure Systems[END_REF] and the US Department of Homeland Security [START_REF]Strategic Principles for Securing the Internet of Things (IoT)[END_REF] contribute both to securely produce and securely use connected objects. However, such preventive measures are not sufficient and should be complemented with intrusion detection or intrusion tolerance solutions to cope with residual security threats.

In this paper, we investigate a novel intrusion detection approach aimed at addressing the challenges raised by the use of IoT devices in smart places such as smart homes or smart factories. The objective is: i) to automatically detect illegitimate behaviors and communications through the local network that might be initiated from compromised devices of the corresponding network, or from external devices inside or outside the network, and ii) to provide a solution that can be easily used by the end-users with the lowest impact on the delivered services. State-of-the art network intrusion detection techniques generally focus on TCP/IP communications. Some solutions have been also proposed to monitor specific IoT and short range communication protocols. However, these techniques cover partially the large set of protocols and communication links used in such environments. To address this problem, our technique is based on the monitoring and profiling of radio communication signals in the smart area to be protected (smart home, smart factory, etc.). The classification of legitimate and illegitimate communications is based on machine learning algorithms using neural networks. This approach is independent of the IoT protocols used and can be deployed in various environments using heterogeneous wireless communication technologies.

This paper is organized as follows. Section II first presents a state of the art of IoT attacks and network solutions to detect them. Section III presents the adversary model and the main concepts behind our proposed intrusion detection approach and preliminary experiments. Finally, Section IV discusses future work and the experiments planned to validate our approach.

II. RELATED WORK

In this section we discuss related work, focussing first on IoT related attacks and then on the security protection mechanisms proposed to cope with such attacks. Some classifications of the vulnerabilities of IoT devices already exist. The Open Web Application Security Project (OWASP) lists the potential attack surface that an attacker can exploit: memory leak, network, web interface, etc. [START_REF] Owasp | OWASP Internet of Things Project -OWASP[END_REF]. Most of these vulnerabilities concern the objects design and production. Some examples are discussed in [START_REF] Dhanjani | Abusing the Internet of Things[END_REF]. The massive DDoS IoT botnet attack based on the Mirai malware is another relevant example [START_REF] Bertino | Botnets and Internet of Things Security[END_REF]. However, this classification does not show any particularity compared to vulnerabilities that can be found in classical IT, except those related to low resources of IoT devices.

The classification proposed in [START_REF] Ronen | Extended functionality attacks on IoT devices: The case of smart lights[END_REF] categorizes the attacks according to their impact on the functionality provided by the IoT device. Four different categories are distinguished:

• Ignoring the functionality • Reducing the functionality • Misusing the functionality • Extending the functionality. This classification enables a direct interpretation of the consequences of attacks as perceived by the end-users (e.g., open a door, increase the temperature, shutdown a device, etc.).

Other papers focus on the vulnerabilities and attacks targeting the short distance IoT communication protocols. Such attacks on the local network open new possible attack surfaces not yet well explored by the scientific community. For example, M. Ryan [START_REF] Ryan | Bluetooth With Low Energy Comes Low Security[END_REF] presents a vulnerability in Bluetooth Low Energy allowing attackers to brute-force encryption keys just by listening to the association phase. T. Zillner [START_REF] Zillner | ZigBee exploited: The good the bad and the ugly[END_REF], presents a Zigbee security testing tool that is aimed at identifying potential vulnerabilities in IoT devices that implement this protocol. E. Ronen and al. [START_REF] Ronen | IoT Goes Nuclear -Creating a ZigBee Chain Reaction[END_REF] investigate the possibility to create an IoT worm in Philips Hue lightbulbs.

In respect of possible protection solutions against attacks involving IoT devices, the IoT specific characteristics such as ad-hoc communication, low resources and frequent design weaknesses require to reconsider the traditional security mechanisms. For example, antivirus solutions may not be practically feasible in IoT context, due to the impossibility to fix or install new software on most of connected devices. Moreover, the low power and limited resources (CPU, memory,...) available on these devices is a significant barrier to implement these solutions. Other mechanisms such as blacklisting or firewalls suffer from the same problems. Furthermore, they are not adapted to IoT communications. Instead of implementing the security mechanisms in the devices, a more practical solution would be to deploy them on dedicated components such as on a gateway connected to the IoT network. This approach has been investigated in some research works aimed at developing intrusion detection systems for the IoT.

The IoT SENTINEL proposed in [START_REF] Miettinen | IoT Sentinel: Automated Device-Type Identification for Security Enforcement in IoT[END_REF] is designed to probe the WiFi and Ethernet traffic flows in an IoT network. The connected devices are identified by their MAC address. By analyzing the traffic flows, the tool can identify the type of the device as well as its potential vulnerabilities. All vulnerable objects are isolated from the other ones using filtering rules. These objects may also be prevented from connecting to the Internet. However, this solution does not cover the communication flows through other IoT protocols such as Bluetooth or ZigBee. Moreover, the proposed filtering mechanism is problematic due to the impacts on the device functionalities. Indeed, as IoT devices can not be easily patched, a vulnerable object could become useless if the vulnerabilities are not fixed.

Another intrusion detection system able to detecting attacks on IPv6 and 6LoWPAN networks was proposed in [START_REF] Raza | SVELTE Real-time intrusion detection in the Internet of Things[END_REF] which is interesting from the WAN point of view. Nevertheless, since most of connected devices only communicate up to the link level (e.g. 802.15.4), potential attacks on these communication links may not be detected.

To conclude, current solutions to monitor the activities and to detect attacks on the IoT networks cover partially the large set of protocols and communication links that are typically used in such environments. These solutions also require the development of specific probes for each relevant protocol. In the following section we propose a complementary solution that is aimed at filling this gap. It consists in monitoring and profiling the radio communication signals at the physical layer, especially by observing the Received Radio Signal Indication (RSSI). This solution can cover all the communication protocols available in the target IoT network. In the literature, the RSSI is mostly used for indoor localization [START_REF] Goldoni | Experimental analysis of RSSI-based indoor localization with IEEE 802[END_REF], [START_REF] Dang | An improved indoor localization based on RSSI and feedback correction of anchor node for WSN[END_REF]. Yet, the possibility to use it to detect a physical intrusion within a Bluetooth connected area was explored in [START_REF] Sung | Intelligent Security IT System for Detecting Intruders Based on Received Signal Strength Indicators[END_REF]. In our case, we extend this approach to detect not only physical intrusions, but also logical ones such as those discussed in the beginning of this section.

III. INTRUSION DETECTION APPROACH

The proposed intrusion detection system (IDS) essentially targets smart places connected to IoT devices. Its main goal is to detect potential attacks that can occur through wireless communications. In order to design an efficient IDS, we must consider together the behavior of the attackers, the behavior of the users themselves and the dynamic evolution of the smart places. This design is discussed in the first subsections. The last subsection is dedicated to a preliminary experimentation.

A. Attacker model and hypothesis

We consider that the attacker can proceed remotely or near the smart place, without physical interaction with the devices of the smart place. Its objective may be to modify the information system to prepare a future physical intrusion, to collect confidential information, or to control a device of the smart place in order to bounce elsewhere. To achieve these objectives, he can either use his own device (in the case of a proximity attack), or he can use an already compromised device of the smart place (via the Internet wired connection for instance). His behavior can follow one of the following scenarios, considering the location where the attack is initiated, the time of the attack or the protocol used. The attack can be initiated:

1) from a location that is never occupied by legitimate users during legitimate communications (e.g.,from the garbage chute); 2) from a so-called "legitimate" location using a wireless technology never used by legitimate users; 3) from a legitimate location using an usual wireless technology (from the legitimate users point of view), but at an unusual period of the day (e.g., at night). 4) from a legitimate location using an usual wireless technology at an usual period of the day, though following an unusual scenario (e.g., by targeting shutters with XBee without disabling the alarm first with BlueTooth); 5) in accordance with legitimate users behaviors. This paper focuses on the four first malicious behaviors. This requires the characterization of the behavior of legitimate users as well as the dynamic evolution of the smart place (with the addition of future wireless technologies).

B. IDS Design

The proposed approach is based on the monitoring of wireless activities using probes and on the identification of misbehaviors. The probes are deployed in strategic locations of the smart place, by a security expert (as for a classic alarm). We do not need to consider the content of the messages themselves. However, the probes are used to characterize the radio communication activities on the bandwidths of interest (those that are used by wireless technologies). Our goal is not to geo-locate the emitter devices. The identification of attacks is based on a comparison of the observations by the different probes of wireless bandwidth usage with a reference model of the behavior of legitimate users. This approach leads to three main challenges:

1) It is not easy for the legitimate users to specify their usage behavior; 2) Their usage behavior can evolve depending on the time of the day or on specific events or periods (weekend, meeting, holidays, etc.); 3) New legitimate devices, possibly with new wireless technologies, may be connected to the smart place. The first challenge corresponds to the difficulty of defining a generic behavioral model for legitimate users of a smart place. In addition, the model may differ from one smart place to another. Machine learning techniques are well suited to address this challenge. In our study, we use neural networks. As for CSS Figure 1. Architecture of the approach anomaly-based intrusion detection systems, these techniques require a learning phase.

To address the second challenge, different usage modes can be defined. Each mode corresponds to a model that can be different. Switching from one mode to another can be done automatically (for instance, at the beginning of the weekend) or activated by a legitimate user during a specific event or period.

To address the last challenge, three kinds of probes are considered for the design of the IDS:

• For the standard wireless technologies, we can use widely available probes (wifi, etc.). • For other wireless technologies that are open and available today, ideally, a probe should be designed to retrieve the RSSI signal. • To anticipate future wireless protocols without changing the deployed IDS, a Software-Defined Radio (SDR) solution should be used to observe the corresponding bandwidth.

C. IDS architecture and deployment

The architecture of the IDS is presented in figure 1. It consists of two main components: a Central Security System (CSS), and the Radio Probes.

The CSS processes the observations collected from the probes and implements the intrusion detection algorithm to detect illegitimate behaviors. All the probes in the architecture are connected to this device via a secure channel (wired or not). The main goal of the CSS is to aggregate all the information gathered by the probes, and to process them with the neural network, that has to be initially trained to recognize what is a normal behaviour and what is not.

The Radio Probes are small sensors used to listen to all the wireless communications inside and around the smart place. They are configured to collect specific data representative of the communications behaviour, such as the RSSI (Received Signal Strength Indication), which is a power measure of the received signal of an antenna. These probes are deployed to ensure the coverage of the smart place, then their different RSSI measures are correlated to identify the location of the broadcast emission. The neural network can be trained to recognize the legitimate areas in which devices usually communicate within the smart place. In case a transmission is observed from an illegitimate place, outside of the home for instance, where no devices are normally used, the neural network will detect an attack.

However, to avoid too many false positives, the reception timestamp and the radio activity during an interval of time are added to the RSSI information received from the probes. They are correlated with the specified detection mode to allow the neural network to detect unusual communication durations or unusual peaks of activity.

D. Preliminary experiment

To evaluate the feasibility of our solution, a first preliminary experiment has been carried out in our laboratory. The considered smart place, as described in figure 1, is representative of an appartment. At this stage, only one IoT protocol, Zigbee, is considered. Our experimental setup is composed of the following elements: 1) Five Radio Probes (with ZigBee tranceivers and Raspberry Pi3 boards) monitor the Zigbee traffic and extract RSSIs from the gathered communication before sending them to the CSS; 2) One Central Security System, a laptop which gathers the information received from the Raspberry Pi and sends them as inputs to a neural network software; 3) Some IoT devices with one Zigbee transceiver on each, to mimic the legitimate objects in the smart place; 4) A Laptop with Zigbee transmitter to communicate with the IoT devices, either to generate attacks or to perform legitimate transmissions.

Currently, a feasibility test has been carried out to check whether the RSSI perceived by the different probes is significantly modified, when communications are initiated from different places. For this purpose, one probe is moved in the environment and performs several transmissions from different places. The results show that the RSSI is significantly modified according to the position of the devices from the probes and that it is possible to correlate the information gathered by multiple probes. The table I shows the average RSSI (in -dBm) for six different transmissions collected by 2 probes. The results show that it is possible to correlate the RSSI of a transmission on two probes to learn information about the transmitter position (here Tx3 is closer to the two probes than Tx6). This first experiment, which is a fundamental prerequisite for our solution, provides some preliminary promising insights about the relevance of our approach to detect intrusions based on the modification of the radio communication patterns. More significant and comprehensive experiments are planned to validate our approach.

IV. CONCLUSION & FUTURE WORK

This paper presents a new intrusion detection solution to protect connected areas like smart homes and smart factories. It is based on the monitoring and anomaly detection of wireless radio communications through neural networks. As future work, we plan to implement the whole proposed solution and to evaluate experimentally its detection efficiency (false positives, false negatives, etc.). The planned experimentation protocol will require the definition of realistic IoT network configurations, including legitimate and malicious communication scenarios. The training period of the neural network also needs to be ajusted according to the scenarios. Finally, several heterogenous IoT communication protocols will also be investigated.


Manuel Buisson

Yann Gueguen

Romain Laniel

Christopher Cantoni

Patrick Houizot

Bruno Bureau

Jean-Christophe Sangleboeuf

Pierre Lucas
email: pierre@u.arizona.edu

Mechanical model of giant photoexpansion in a chalcogenide glass and the role of photofluidity

Keywords: Photoexpansion, photofluidity, photoinduced fluidity, chalcogenide glasses, eigenstrain

An analytical model is developed to describe the phenomenon of giant photoexpansion in chalcogenide glasses. The proposed micro-mechanical model is based on the description of photoexpansion as a new type of eigenstrain, i.e. a deformation analogous to thermal expansion induced without external forces. In this framework, it is the viscoelastic flow induced by photofluidity which enable the conversion of the self-equilibrated stress into giant photoexpansion. This simple approach yields good fits to experimental data and demonstrates, for the first time, that the photoinduced viscous flow actually enhances the giant photoexpansion or the giant photocontraction as it has been suggested in the literature. Moreover, it highlights that the shear relaxation time due to photofluidity controls the expansion kinetic. This model is the first step towards describing giant photoexpansion from the point of view of mechanics and it provides the framework for investigating this phenomenon via numerical simulations.

INTRODUCTION

Chalcogenide glasses cover a wide range of applications [START_REF] Adam | Chalcogenide Glasses[END_REF][START_REF] Eggleton | Chalcogenide photonics[END_REF]; first and foremost because of their transparency in the infrared range. This includes thermal imaging cameras [START_REF] Zhang | Production of complex chalcogenide glass optics by molding for thermal imaging[END_REF], generation of new infrared sources [START_REF] Troles | Low loss microstructured chalcogenide fibers for large non linear effects at 1995 nm[END_REF], chemical and biological sensors [START_REF] Lucas | Advances in chalcogenide fiber evanescent wave biochemical sensing[END_REF], optical data storage [START_REF] Wuttig | Phase-change materials for rewriteable data storage[END_REF], etc. Chalcogenide glasses are also photosensitive materials. They exhibit a wide range of photoinduced phenomena [START_REF] Kolobov | Photoinduced phenomena in amorphous chalcogenides: from phenomenology to nanoscale[END_REF][START_REF] Shimakawa | Photoinduced effects and metastability in amorphous semiconductors and insulators[END_REF] including: photodarkening/bleaching [START_REF] Ren | In-situ measurement of reversible photodarkening in ion-conducting chalcohalide glass[END_REF], photoinduced fluidity [START_REF] Hisakuni | Optical microfabrication of chalcogenide glasses[END_REF][START_REF] Tanaka | Photoinduced fluidity in chalcogenide glasses[END_REF], photoinduced structural relaxation [START_REF] Golovchak | Physical ageing in glassy As-Se induced by above-band gap photoexposure[END_REF][START_REF] Lucas | Photoinduced structural relaxation in chalcogenide glasses[END_REF], photoinduced expansion [START_REF] Hisakuni | Giant photoexpansion in As2S3 glass[END_REF] or contraction [START_REF] Calvez | Reversible giant photocontraction in chalcogenide glass[END_REF], etc. Some of these effects can be scalar or vectorial, i.e.: isotropic or anisotropic respectively, depending on the light polarization [START_REF] Krecmer | Reversible nanocontraction and dilatation in a solid induced by polarized light[END_REF].

Most photostructural changes can be induced by irradiation with either above or below bandgap light. Some photoinduced effects are much more pronounced in the sub-bandgap range [START_REF] Kolobov | Photo-induced metastability in amorphous semiconductors[END_REF], including giant photoexpansion-contraction [START_REF] Hisakuni | Giant photoexpansion in As2S3 glass[END_REF][START_REF] Calvez | Reversible giant photocontraction in chalcogenide glass[END_REF], photofluidity [START_REF] Hisakuni | Optical microfabrication of chalcogenide glasses[END_REF][START_REF] Tanaka | Photoinduced fluidity in chalcogenide glasses[END_REF] and photoinduced structural relaxation [START_REF] Kozdras | Light-assisted physical aging in chalcogenide glasses: dependence on the wavelength of incident photons[END_REF], because, sub-bandgap light permits to illuminate a larger volume. In addition, above bandgap irradiation is associated with high absorption that is sufficient to induce a significant warming and even potential damage, while sub-bandgap irradiation has a negligible thermal contribution under standard irradiation power. Several proofs of the optical origin and athermal nature of photoinduced effects exist including the fact that photofluidity is more pronounced at low temperature [START_REF] Hisakuni | Optical microfabrication of chalcogenide glasses[END_REF] and exhibits polarization effects [START_REF] Asatryan | Optical polarization driven giant relief modulation in amorphous chalcogenide glasses[END_REF] or the formation of wavelength specific Bragg gratings [START_REF] Tanaka | Photoinduced Bragg gratings in As2S3 optical fibers[END_REF]. Nevertheless, it has been recently suggested that the thermal contribution to the photoexpansion itself may not be negligible even for weakly absorbed light [START_REF] Zhao | Role of photothermal effect in photoexpansion of chalcogenide glasses[END_REF].

Giant photoexpansion in chalcogenide glasses has many applications including the production of adaptive lenses [START_REF] Saitoh | Self-developing aspherical chalcogenide-glass microlenses for semiconductor lasers[END_REF], microlenses arrays [START_REF] Hisakuni | Optical fabrication of microlenses in chalcogenide glasses[END_REF], optical fiber microlenses [START_REF] Saitoh | Chalcogenide-glass microlenses for optical fibers[END_REF], structured concave/convex lenses [START_REF] Calvez | Reversible giant photocontraction in chalcogenide glass[END_REF] and relief patterning for the production of gratings [START_REF] Asatryan | Optical polarization driven giant relief modulation in amorphous chalcogenide glasses[END_REF][START_REF] Messaddeq | Light-induiced relief grating and a mechanism of metastable light-induced expansion in chalcogenide glasses[END_REF][START_REF] Yang | Simultaneous microscopic measurements of photodarkening and photoexpansion in chalcogenide films[END_REF]. This technique is promising, since complex optical elements can be optically created in a single step. Nevertheless, if the kinetic and the geometry of the expansion cannot be predicted, the development of this technique will be limited because of the large number of tests required to produce the desired lens geometry. Hence a mechanical model describing the photoexpansion would considerably improve the predictability of the expansion geometry and would generally ease the development of photo induced relief patterning techniques. However, the giant photoexpansion is a phenomenon which is very difficult to model since its physical origin is still debated, and since other photoinduced effects may contribute to the expansion. Usually, bulk glasses or thin films are irradiated with a laser beam having a diameter much lower than the sample size. This way, the irradiated volume fluidifies and expands while being confined by the quasi-rigid volume surrounding the laser path which remains mostly unchanged. If the irradiated volume can expand easily in the direction of free surfaces, it is stressed in the direction perpendicular to the bean axis: this is where the viscoelastic properties of the glass come into play. This can be easily understood thanks to the phenomenological model proposed by Tanaka et al. [START_REF] Tanaka | Giant photo-expansion in chalcogenide glass[END_REF]. Despite clear indications that the phenomenon is governed by mechanical stress, the mechanical aspect of this phenomenon has been largely ignored or poorly treated. Various models of photoexpansion have been proposed in the literature [START_REF] Kolobov | Photo-induced metastability in amorphous semiconductors[END_REF][START_REF] Lukacs | To the kinetics of photoinduced volume changes in chalcogenide glasses[END_REF], but they all treat the expansion as if it was stress-free. The objective of the present work is to treat the photoexpansion from the point of view of mechanics, by taking into account the combined effect of these stresses and the photoinduced fluidity. We show here, using some simple hypothesis, that the contribution of photofluidity can be treated in a systematic way and that the resulting model gives a good estimation of the kinetic and the strain field of the giant photoexpansion observed experimentally.

MICRO-MECHANICAL MODEL OF PHOTOEXPANSION VIA A SIMPLE THERMODYNAMICAL APPROACH

Eigen Strain

Taking into account the physics of the photoexpansion, we assume that viscosity and photoexpansion phenomena induce respectively two so called "eigenstrains": the viscous-strain, 𝜖 v , and the photoexpansion strain, 𝜖 p . As suggested by Mura [START_REF] Mura | Micromechanics of Defects in Solids[END_REF], eigenstrain refers to such a category of nonelastic strains such as thermal expansion or phase transitions which can induce a self-equilibrated stress field called "eigenstress fields" without any external force and surface constraint. We will not deal with the structural origin of the eigenstrain due to photoexpansion, but only with the resulting internal stress fields.

Model basis

Athermal nature

The photoexpansion is assumed to proceed continuously and quasi-statically under permanent irradiation; the kinetic of this evolution is described by a single time parameter (τ A0 ). The eigenstrain saturates at long time, and this saturation level is taken to be proportional to the local light intensity. In this work, we will neglect the thermal aspect of the photoexpansion and we will not sort out the question of the physical origin of the corresponding eigenstrain. In fact, we will show here quantitatively that thermo-viscoelastic coupling and heat conduction can be disregarded, as a first approximation. We also assume that photoexpansion occurs in a viscoelastic medium. Indeed, under irradiation, chalcogenide glasses undergo a large and athermal viscosity decrease named photofluidity or photoinduced fluidity [START_REF] Tanaka | Photoinduced fluidity in chalcogenide glasses[END_REF]. The viscosity under irradiation is inversely proportional to the number of photons absorbed per second per atom [START_REF] Gueguen | Photoinduced fluidity in chalcogenide glasses at low and high intensities: A model accounting for photon efficiency[END_REF][START_REF] Li | Direct calculation of light-induced structural change in diffusive motion in glassy As 2 S 3[END_REF] and does not depend on the stress [START_REF] Gueguen | Reologie de verres speciaux dans le domaine de transition et phenomenes photoinduits[END_REF]: this further justifies the assumption that irradiated glasses undergo a linear viscoelastic behavior. The glass structure is also assumed to be isotropic. For elasticity, the values of the bulk modulus k and of the Poisson's coefficient ν given from [START_REF] Guin | Indentation creep of Ge-Se chalcogenide glasses below Tg: elastic recovery and non-Newtonian flow[END_REF]:

k = 10.2  0.2 GPa ν = 0.307  0.01 (1)
will be adopted for this study.

Self-focusing

Since the glass undergoes photoinduced refractive index changes, the glass sample may focus the laser beam along the thickness through a photoinduced self-focusing process [START_REF] Hisakuni | Giant photoexpansion in As2S3 glass[END_REF]. This effect will be neglected in the present model. It was shown that the self-focusing is prominent when the ratio sample thickness/beam radius is much larger that 10 (ref. [START_REF] Hisakuni | Giant photoexpansion in As2S3 glass[END_REF]). In our experiment, this ratio is 9.3. The divergence of the laser beam due to the photoexpansion will also be neglected. Indeed, during the expansion, the average divergence of the beam is lower than 2°.

Model approach

A detailed mathematical description of the proposed model is presented in the Supplemental Material section. Here we describe the general approach adopted to quantify the mechanical response of the glass under irradiation. First the free energy per unit volume is expressed within the framework of the Generalized Standard Model [34] using two distinct tensors for photoexpansion and viscous flow which are treated as uncoupled. Dissipation due to viscous strain rate is also accounted for using a dissipation potential. These yield a state law for the stress which provides the stress-strain response under irradiation. Photoexpansion has only spherical contribution to straining and viscosity plays a deviatoric role, i.e. only shear viscosity comes into play. The problem is treated as rotationally symmetrical around the z-axis of the beam.

EXPERIMENTAL

Glass synthesis

The giant photoexpansion has been investigated on GeSe 9 glasses. The glass production procedure is detailed in ref. [START_REF] Lucas | Calorimetric characterization of photoinduced relaxation in GeSe9 glass[END_REF]. A glass rod was sliced and cut to the desired testing specimen geometry using a diamond saw. The surfaces of the specimens were mirror polished using SiC paper and alumina suspension with 0.25 μm particle size. The sample thickness was 1.5  0.02 mm. The glass transition temperature of GeSe 9 is 92°C (ref. [START_REF] Gueguen | Photoinduced fluidity in chalcogenide glasses at low and high intensities: A model accounting for photon efficiency[END_REF])

Irradiation

A tunable Ti-sapphire laser 3900S from Spectra Physics was used in continuous mode for all the experiments. The Ti-sapphire was pumped with a 5 W, 532 nm laser. The Ti-sapphire laser was tuned to 790 nm so that the excitation wavelength corresponds to the Urbach edge of the GeSe 9 sample. The photon energy is 1.57 eV, far below the bandgap Eg = 1.95 eV (ref. [START_REF] Gueguen | Photoinduced fluidity in chalcogenide glasses at low and high intensities: A model accounting for photon efficiency[END_REF]) measured as the photon energy corresponding to an absorption of 1000 cm -1 . At this photon energy, the absorption coefficient is lower than 10 cm -1 even if the photodarkening is taken into account [START_REF] Gueguen | Photoinduced fluidity in chalcogenide glasses at low and high intensities: A model accounting for photon efficiency[END_REF]. The collimated laser beam was perpendicular to the surface and was not focused on the sample. The samples were irradiated for various periods of time, and the profile of the permanent photoexpansion was collected using a stylus profilometer Dektak 6M with a 12.5 μm radius diamond tip a few minutes after the irradiation periods. The samples were glued on a metallic support with a hole in the middle, so that no reflection of the laser beam occurred on the metallic support. The power of the laser beam was set at P = 33.8 mW using 2.0 and 0.2 neutral density filters with a rotationally symmetrical intensity profile. The laser beam profile was characterized and is plotted on Figure 1. This experimental profile is modeled by a Gaussian fit according to equation [START_REF] Eggleton | Chalcogenide photonics[END_REF].

[ ]

where I0 = 2P/πR2 = 82.45 W•cm -2 is the maximum intensity and R = 1.615 × 10 -2 cm the radius of the beam. The saturation level of the eigenstrain corresponding to photoexpansion is then taken to be proportional to the incident intensity and the photoexpansion kinetic is characterized by a single time parameter, independent of the incident intensity. For the sake of simplicity, we adopt the following uncoupled quasi-static photoexpansion evolution (see details in the Supplemental Information section):

𝜖 [ ] ( [ ]) (3)
With τ A0 the characteristic duration of photoexpansion and Λ the asymptotic value of the "linear photoexpansion coefficient" at I 0 . If a stress-free sample of thickness e is irradiated by a homogenous light of intensity I, it will expand at t→∞ by Δe, so that: (4)

Shear relaxation time

It has been previously demonstrated that the shear relaxation time τ 0 under bang gap irradiation is inversely proportional to the density of photons absorbed per second per atom [START_REF] Gueguen | Photoinduced fluidity in chalcogenide glasses at low and high intensities: A model accounting for photon efficiency[END_REF]. Under irradiation, photoinduced entropy changes occur and it can impact the viscosity η. The viscosity is proportional to the shear relaxation time according to the Maxwell's relationship: η = τ 0 μ. In order to simplify the model, we will only consider the average shear relaxation time of an irradiated volume, corresponding to a cylinder of thickness e and of laser beam radius R. The shear relaxation time, for r > R in the polar coordinate system (r; z) outside the laser beam is then set at τ 0 →∞, since for the GeSe 9 glass, in the dark at configurational equilibrium, it is supposed to be larger than centuries [START_REF] Gueguen | Photoinduced fluidity in chalcogenide glasses at low and high intensities: A model accounting for photon efficiency[END_REF]. Nevertheless, to insure the linearity of the proposed simple constitutive equations, the same relaxation time is assumed everywhere. This assumption implies that the glass is considered to be "photofluid" outside the irradiated volume, but this artifact has no practical consequence since the stress outside the laser beam is negligible, so that the impact of the relaxation time becomes negligible as well. The average shear relaxation time τ 0 , regarding the irradiated volume, is:

[ [ ] ] (5)
where N d is the number of atoms per unit volume, ζ the reflectance at the considered wavelength, P the power of the laser at the surface of the sample, α the absorption coefficient, the energy of a photon, and n a proportionality factor equal to 0.2 for the GeSe 9 glass [START_REF] Gueguen | Photoinduced fluidity in chalcogenide glasses at low and high intensities: A model accounting for photon efficiency[END_REF]. The absorption coefficient, at 790 nm, takes into account the photodarkening [START_REF] Gueguen | Photoinduced fluidity in chalcogenide glasses at low and high intensities: A model accounting for photon efficiency[END_REF]. The values of all the parameters used are detailed in Table I. The right term of Eq. ( 5), between parentheses, is the number of photons absorbed per second. The term left of the parenthesis is the number of atoms in the irradiated volume divided by an efficiency parameter n.

RESULTS

Analytical solution

Instead of computing a rather difficult analytical solution based on use of potentials functions [START_REF] Nowacki | Thermoelasticity[END_REF] (see for example an analogy with ref. [START_REF] Sottos | The influence of interphase regions on local thermal displacements in composites[END_REF] in the framework of thermoviscoelastic heated fibers embedded in a matrix), we have used the Laplace transform relative to the time (t→s) of our governing equations and we have constructed an approximated analytical solution with a form based on the separation of variables r and s. This approach permits to simply return to the rt original space by inverse Laplace transform. The following approximated analytical expression for the vertical displacement is then obtained:

[ ] ( ) (6)
where:

( ) [ ] [ ] (7)
As expected, the vertical displacement reaches an asymptotic value at t→∞, since . This displacement is both proportional to the light intensity and to z. The maximum of the vertical displacement is obtained at the sample surface corresponding to the center of the beam: u z (0, e/2, t) where the laser intensity is maximum.

Fitting parameters

From Eq. ( 6), we obtain the maximum displacement after extended irradiation time according to: [START_REF] Shimakawa | Photoinduced effects and metastability in amorphous semiconductors and insulators[END_REF] Eq. ( 8) can then be fitted to the plateau of the experimental points plotted on Figure 2 to yield Λ ~ 3.42 × 10 -3 . We implicitly assume here that the eigenstrain 𝜖 p does not disappear instantaneously (even partially) since the experimental points correspond to measurements postirradiation and not under irradiation. Using τ A0 ~ 257.1 s, as calculated from Eq. ( 5), we obtain, by fitting all the experimental points: τ A0 ~ 267.5 s. Compared to the vertical displacement and the expansion radius, the radial displacement is found to be rather negligible and has little influence on the widening of the expansion profile. The maximum radial displacement at t = 2400 s is located at the surface at r = 0.125 mm, and reaches only 0.125 × 10 -3 mm, whereas u z = 1.55 × 10 -3 mm at the same position and time. FIG. 2. Experimental vertical displacement of the surface, at r = 0: u z (0, e/2, t) (dots); and displacement predicted by Eq. ( 6) (lines) at the same position, when taking into account the contribution of photofluidty with a relaxation time of τ 0 = 257.1 s derived from Eq. ( 5): top curve; and when assuming no contribution of photofluidity τ 0 →∞,: bottom curve.

Comparison with experimental results

Figure 2 shows a comparison of the experimental maximum expansion with the theoretical one obtained from our model. We first observe that the experimental data are affected by a significant uncertainty illustrated by the oscillation around the saturation level of photoexpansion. We must recall that the measurements are not performed in situ but instead each data point corresponds to a separate expansion performed on distinct points on the sample. This range of experimental uncertainty is observed in the data of other authors [START_REF] Hisakuni | Giant photoexpansion in As2S3 glass[END_REF]. Ideally, measurements should be performed in-situ to optimally match model values. Nevertheless, the experimental data clearly show that the expansion takes place continuously, which supports the assumption that the eigenstrain 𝜖 p is time-dependent. Figure 2 also illustrates that if the photofluidity is not taken into account, i.e. if τ 0 →∞, the photoexpansion is considerably lower. This clearly illustrates the role of the photofluidity. Figure 3 shows a comparison of the experimental profiles with those predicted by our model. We can see that at r < R, the experimental profiles are nearly Gaussian, and that the radius of this Gaussian shape remains almost unchanged over time, and is close to R. This is consistent with the negligible radial displacement found in our calculation. Nevertheless, the experimental profiles deviate from the Gaussian shape as soon as r > R, even at short times, and this is not predicted by our calculations. This can be explained in part by the fact that the experimental beam is not perfectly Gaussian as shown in Fig. 1. This could also be associated with some contribution of the beam reflection on the convex surface within the sample. Both experimental data and calculation lead to a saturation of the photoexpansion at long times. The impact of the assumptions used in our model with regard to the small differences between experimental and theoretical profiles are discussed in the next section. FIG. 3. Expansion profiles at various times of exposure: experiments (dots) and model (lines).

DISCUSSION

Thermal contribution

Recently, Zhao et al. [START_REF] Zhao | Role of photothermal effect in photoexpansion of chalcogenide glasses[END_REF] have suggested that thermal contributions to the giant photoexpansion may not be negligible. They have treated the photoexpansion using a thermoelastic approach, in the framework of uncoupled thermoelasticity, using the Beer's law to estimate the heat. This approach clearly presumes that even with low absorption, there is a nonnegligible heat production leading to thermal expansion. However, it does not take into account the photofluidity, and, as noted by the authors, thermal expansion cannot be the sole contribution since it cannot explain the photocontraction phenomenon. Another interpretation proposed by Calvez et al. [START_REF] Calvez | Reversible giant photocontraction in chalcogenide glass[END_REF], suggests instead that the main contributors to photocontraction and photoexpansion are photorelaxation and photoexcitation respectively.

Our model does not take into account the thermal contribution described by Zhao et al. [START_REF] Zhao | Role of photothermal effect in photoexpansion of chalcogenide glasses[END_REF] since we assume negligible light absorption. However, we can calculate the maximum temperature increase during our experiment using the model proposed by Zhao et al. Using the data detailed in Table II [START_REF] Lucas | Photoinduced structural relaxation in chalcogenide glasses[END_REF][START_REF] Gueguen | Photoinduced fluidity in chalcogenide glasses at low and high intensities: A model accounting for photon efficiency[END_REF][START_REF] Guin | Indentation creep of Ge-Se chalcogenide glasses below Tg: elastic recovery and non-Newtonian flow[END_REF][START_REF] Mathew | Thermal diffusivity measurements of semiconducting amorphous GexSe100-x thin films by photothermal deflection technique[END_REF], we obtain ΔT ~ 33.7 K at t = 2400 s.

C p heat capacity ~0.3 J•g -1 •K -1 (ref. 13) ρ density 4.34 g•cm -1 (ref. 36) αT thermal expansion coefficient 380 × 10 -7 K -1 (ref. 30)
Based on their thermoelastic model, we can evaluate the resulting maximum expansion: u z (t = 2400) ~ 2.9 × 10 -3 mm. By comparison to the maximum expansion of ~ 5 × 10 -3 mm observed experimentally, this would indicate that, the thermoelastic expansion appears to be the major contribution. But the model of Zhao et al. predicts a broadly distorted expansion profile which is not in agreement with experiment as depicted on Fig. 4. The model of Zhao et al. largely overestimates the expansion diameter in comparison to experimental observation. More specifically, the expansion diameter is notably larger than the beam diameter. This is not consistent with giant photoexpansion experiments where the diameter of the expansion is typically the same as the diameter of the laser beam according to various experimental reports [START_REF] Hisakuni | Giant photoexpansion in As2S3 glass[END_REF][START_REF] Saitoh | Self-developing aspherical chalcogenide-glass microlenses for semiconductor lasers[END_REF]. Similarly, Zhao et al. find theoretical expansion diameters at least 2 order of magnitude larger [START_REF] Zhao | Role of photothermal effect in photoexpansion of chalcogenide glasses[END_REF] when modeling the experiment of Hisakuni & Tanaka. Hence, the heat generated during photoexpansion seems to be largely overestimated in the model of Zhao et al., probably because photon energy is not mainly transferred into phonons, but rather contributes to the athermal eigenstrain of photoexpansion.

In order to investigate the extent of heat production during our photoexpansion experiment, we have performed a thermal imaging measurement derived from ref. [START_REF] Lepine | Optical microfabrication of tapers in low-loss chalcogenide fibers[END_REF]. Figure 5(a) shows a thermal image of a bulk GeSe 9 sample irradiated with a laser intensity of 4.24 W•cm -2 , at 785 nm, placed in front of a heat/cool plate stabilized at 40°C. The plate provides an upper temperature reference for the image. For comparison a GeSe 9 disk heated at 40°C is shown next to a similar disk at room temperature in Fig. 5(c). It is clear from these two figures that the irradiated sample does not even reach 40°C, since it still contrasts with the plate. Nevertheless, Fig. 5(b) clearly highlights that an expansion occurs, with a diameter of 1.5 mm, corresponding to the laser beam diameter. According to the model of Zhao et al., the temperature increase should exceed 100°C in the middle of the expansion and 40°C at its border, after less than 3 s. We have done this calculation using an absorption coefficient of 42 cm -1 at 785 nm, taking into account the photodarkening [START_REF] Gueguen | Photoinduced fluidity in chalcogenide glasses at low and high intensities: A model accounting for photon efficiency[END_REF]. This clearly illustrates that the heat is largely overestimated, or more precisely, that most of photon energy is not transferred into phonons but mainly converted in the creation of eigenstrain associated to photoexpansion.

Nevertheless, it must be pointed out that the experimental expansion profiles are not exactly Gaussian, but instead show a slight expansion at values larger than the beam diameter r > R. As shown above, this profile is not consistent with heat production due to absorption but may be partly explained by the fact that the experimental beam is not perfectly Gaussian (Fig. 1), as well as the possible contribution of reflections on the back surface of the glass sample. These back reflections could be significant due to the high refractive index of chalcogenide glasses and may not be coaxial with the incident beam due to self-focusing or deformation of the back surface as shown on Figure 5

5.2

The role of photofluidity

. Does photofluidity enhance photoexpansion?

It has been suggested, first by Tanaka et al. [START_REF] Tanaka | Giant photo-expansion in chalcogenide glass[END_REF], that the photofluidity enhances the giant photoexpansion. Nevertheless, up to now, no model existed to demonstrate the role of the photofluidity. We can compare here the maximum expansion, , produced without photofluidity when τ 0 →∞, and the expansion produced with photofluidity, when τ 0 is finite and positive.

( ) (9)
This ratio is larger than unity and is a decreasing function of ν whenever ν ∈ [0; 1/2[: the maximum relative enhancement of photo expansion due to photofluidity occurs when the Poisson's ratio is null and corresponds to a value of 100%. For the GeSe 9 glass, the photofluidity enhances the relative expansion by almost 30%. Eq. ( 9) shows that the contribution of photofluidity enhances the effect of an existing ϵ p ; however, due to other existing contributions and material properties, Eq. ( 9) is not sufficient to consider that the giant expansion is larger for materials with lower Poisson's ratio.

Simplifying assumption considering the viscous flow

In order to obtain an analytical solution, we have set τ 0 (r) = τ 0 . This corresponds to the assumption that the whole glass becomes photofluid, even outside the irradiated volume, at r > R.

We have underestimated the shear relaxation time at r > R, so we have overestimated the viscous strain and consequently the vertical displacement at r > R. Nevertheless, the vertical displacement is proportional to the local laser intensity, and thus is negligible at r > R, even if it is overestimated. Indeed, since the stress is very low far from the middle of the laser beam, the underestimation of the shear relaxation time does not really impact the viscous strain in this region.

Photofluidity and deviatoric stress

In our model, the viscous flow is only produced by deviatoric stress, and not affected by hydrostatic pressure. This hypothesis comes from the comparison of the tensile and shear relaxation functions of Ge-Se glass fibers under irradiation [START_REF] Gueguen | Reologie de verres speciaux dans le domaine de transition et phenomenes photoinduits[END_REF] which shows that the viscous flow tends to become volume-conservative. This is a common hypothesis, indeed the Trouton law which states that the "coefficient of viscosity is equal to one third of the coefficient of viscous traction", derives from the same hypothesis. Hence, we can assume that if the hydrostatic pressure plays a role in the viscous flow, it is minor. According to this hypothesis, the viscous flow does not allow the relaxation of the hydrostatic pressure, even at long time and even when the irradiation ceases. Thus, the expansion stays stressed until the "natural" structural relaxation produces a volume relaxation that decreases the residual stress.

Photocontraction

In our model, the eigenstrain due to photoexpansion is proportional to Λ. If we assume that the volume expansion is correlated to a photoinduced entropy increase, the opposite effect, the photoinduced relaxation, will promote a volume relaxation and a giant photocontraction [START_REF] Calvez | Reversible giant photocontraction in chalcogenide glass[END_REF]. We can take into account this phenomenon by changing the sign of Λ: Λ > 0 corresponds to photoexpansion, and Λ < 0 to photocontraction. In this latter situation, we can see from Eq. ( 6), that the photofluidity will also enhance the contraction.

CONCLUSION

It has been previously suggested that photofluidity plays a role in the giant photoexpansion effect, but so far no model has been proposed to quantify its contribution. The mechanical model in this study demonstrates for the first time that photofluidity indeed plays a major role in the giant photoexpansion process and establishes a numerical framework to quantify its contribution. In this model the photoexpansion is described as a new kind of athermal eigenstrain that induces local stress fields called egeinstress. The viscoelastic behavior generated by the photofluidity in turn enables these stresses to produce a giant photoexpansion. With some simple hypothesis, especially regarding the relationship between the eigenstrain and the light intensity, the problem of giant photoexpansion can be analytically solved and it is shown to provide a good prediction of experimental data both in term of magnitude and geometry of expansion. In addition, the model permits to emphasize the coherence between the viscosity under irradiation and the kinetic of giant photoexpansion. This model supports the idea that the photofluidity plays an important role in the giant photoexpansion in terms of amplitude and kinetic but not as important as expected by Tanaka [START_REF] Tanaka | Giant photo-expansion in chalcogenide glass[END_REF] concerning the amplitude. More advanced analytical solutions will be required to treat complex optical effects such refractive index changes or self-focusing where the relationship between the light intensity and the eigenstrain is refined or where the light intensity profile is more sophisticated (for specific relief patterning, as an example). Nevertheless, the proposed model can serve as a starting point to implement the proposed integration law into more advanced numerical simulation that take more complex physical effects into account.

FIG 1 :

 1 FIG 1: Intensity profiles of the laser beam, which is rotationally symmetrical. Red dots correspond to experimental laser beam profile and the blue line corresponds to Gaussian fit using Eq. (2).

FIG. 4 .

 4 FIG. 4. Experimental expansion pro_le at t = 2400 s and expansion predicted by the model of Zhao et al. (ref.21) at this time.

  (b).

FIG. 5 .

 5 FIG. 5. (a) Thermograph of a bulk GeSe 9 glass irradiated with a laser beam of intensity 150 mW and wavelength 785 nm. The thermogram was collected with a FLIR Thermocam E300 IR camera. The background behind the sample corresponds to a heat/cool plate Echotherm IC120 tuned at exactly 40C. (b) The same bulk sample after an irradiation period, with a giant photoexpansion of diameter 1.5 mm, on each side. (c) Thermograph of two bulk GeSe 9 , at 28°C on the left and 40°C on the right.

TABLE I .

 I Parameters used to calculate the shear relaxation time.

	Parameter	Description	Value
	N d	number of atoms per unit volume	3.34×10 22 cm -3 (ref. 30)
	ζ	reflectance	0.215 (ref. 30)
	e	sample thickness	1.5 mm
	λ	laser wavelength	790 nm
	hc/λ	energy of a photon	1.57 eV or 2.51×10 -19 J
	R	beam radius	1.615 mm
	P	power of the laser	33.8 mW
	α	absorption coefficient	9.89 cm -1 (ref. 30)

TABLE II .

 II Necessary parameters in the thermoelastic model ofZhao et al.(ref.21), at 20°C.

	Parameter	Description	Value
	D	Thermal diffusivity	3.22 × 10 -3 cm 2 •s -1 (ref. 42)

ACKNOWLEDGEMENT

The authors would like to thank the CNRS for supporting our international collaboration via the program PICS PhotoGlass (PICS 118254). P. L. acknowledges funding from the National Science Foundation (NSF Grant ECCS-1201865). M. B. acknowledges support from "La Fondation Rennes1, (semestre innovation)".


Aurélien Thorette

Marco Romanelli

Marc Brunel

Marc Vallet

Frequency-locked chaotic opto-RF oscillator

come

Frequency-locked chaotic opto-RF oscillator Aurélien Thorette * , Marco Romanelli, Marc Brunel, and Marc Vallet Institut de Physique de Rennes, Université Rennes I -CNRS UMR 6251 Campus de Beaulieu, 35042 Rennes Cedex, France and * Corresponding author: aurelien.thorette@univ-rennes1.fr

A driven opto-RF oscillator, consisting of a dual-frequency laser (DFL) submitted to frequencyshifted feedback, is studied experimentally and numerically in a chaotic regime. Precise control of the reinjection strength and detuning permits to isolate a parameter region of bounded-phase chaos, where the opto-RF oscillator is frequency-locked to the master oscillator, in spite of chaotic phase and intensity oscillations. Robust experimental evidence of this synchronization regime is found and phase noise spectra allows to compare phase-locking and bounded-phase chaos regimes. In particular, it is found that the long-term phase stability of the master oscillator is well transferred to the opto-RF oscillator even in the chaotic regime.

Dual-frequency lasers (DFL) are convenient systems for the generation of continously tunable, high spectral purity, optically carried radiofrequency (RF) signals. Driven by applications such as microwave-photonic links, from radio-over-fibre to antenna feeds [1,2], or highly monochromatic THz generation [3], many solidstate DFLs have been investigated recently. This includes for instance diode-pumped solid-state lasers [4,5], laser diodes [6,7], vertical external-cavity surface-emitting lasers (VECSEL) [8], fiber lasers [9,10], or dual-axis cavities [11,12]. A general issue concerning DFL-based opto-RF oscillators is the stabilization of their beat frequency, and it has been shown that an optical feedback loop containing a frequency-shifting element can be successfully used to this end [13,14]. In particular, the DFL beat-note can be phase-locked to a master RF oscillator, so that the frequency stability of the master is transferred to the optically carried signal. In this context, an interesting synchronization regime called bounded-phase has been isolated [15,16], where the phase-locked state loses stability via a Hopf bifurcation. After the bifurcation, the system displays sustained amplitude and phase oscillations. The master and slave oscillators are not phase-locked anymore, yet they maintain frequency-locking [17,[START_REF] Brunel | Nonlinear optical cavity dynamics: from microresonators to fiber lasers[END_REF]. Being linked to the existence of a Hopf bifurcation, this behavior is rather generic [START_REF] Kelleher | [END_REF][20][21].

In the present work, we consider a DFL submitted to frequency-shifted feedback, i.e., an opto-RF oscillator, driven in a parameter region where it undergoes a subcritical bifurcation leading directly to a chaotic state, and investigate whether this situation is compatible with bounded-phase dynamics. Synchronization of chaotic lasers has already been studied, e.g. in spatially coupled [22,23] and injection locked [24] Nd:YAG, injectionlocked Nd:YVO 4 [25], injection-locked diode [26][27][28], as well as gas [29], lasers. In those systems each laser's output was chaotic per se, and the coupling had a regularizing effect, by inducing either phase or complete chaos synchronization. The case studied here is different in that our opto-RF oscillator is not, in itself, a chaotic oscillator. In the free-running regime, it has a regular behavior; it is the coupling that is responsible for the outbreak of chaos. Furthermore, in the cited previous studies, the authors were mostly interested in showing that synchronization occurred, without characterizing and discriminating phase-locking and bounded-phase regimes. In the present case, the heterodyne nature of our source readily offers experimental access to the phase difference, while in the previous works the phase had to be retrieved numerically from intensity time series [23,24,27]. By measuring phase noise spectra, in DFLs it is possible to obtain a precise experimental characterization and a quantitative comparison of the different synchronization regimes. The aim of this letter is to show a regime characterized by chaotic oscillations not only of the amplitude, but also of the phase, and to demonstrate experimentally that frequency-locking can be maintained, i.e., that boundedphase synchronization is possible also in the presence of chaos. Furthermore, we show that, in spite of the chaotic phase oscillations, the beat-note signal still keeps a good spectral purity at low offset frequencies.

FIG. 1. Experimental setup. The two polarization modes of a dual-frequency laser produce an optically-carried RF beatnote, which is locked to a master oscillator using frequencyshifted optical feedback. See text for details.

The experimental setup is similar to the one used in [17] and is described in Fig. 1. We study a solidstate Nd 3+ :YAG laser oscillating simultaneously on two orthogonally-polarized modes. An intra-cavity tunable birefringence, realized using two quarter-wave plates (QWP), generates a frequency difference δν = ν y -ν x . The interference of the two polarization modes after a polarizer P produces an optically-carried RF beatnote, which we choose to tune around 180 MHz. We then add an external coupling between the two modes, through a reinjection of the field E x on the other one E y , achieved using an external arm containing a QWP. Furthermore, the reinjected field is frequency-shifted using an acousto-optic modulator AO driven at the frequency f AO ≈ 90 MHz by a master RF oscillator. Provided that ν x + 2f AO ≈ ν y , injection from E y into E x can be neglected because the frequency difference between ν x and ν y + 2f AO is much larger than the cavity bandwith. The optical coupling between the two modes is characterized by its strength (related to diffraction efficiency of the modulator, spatial mode-matching of the external cavity and other experimental factors) and by the detuning δν -2f AO between the RF source and the laser beat-note. The laser exhibits relaxation oscillations at a frequency f R = 65 kHz, which allows to ignore delay effects in all optical paths, as this time scale is much longer than the round-trip time in the feedback arm, whose length is 70 cm. We describe the system using the following rate equa- tions for the electric fields and populations [30]:

de x ds = m x + βm y 1 + β e x 2 ( 1a
)
de y ds = m y + βm x 1 + β e y 2 + i∆e y + Γe x (1b) dm x,y ds = 1 -(|e x,y | 2 + β|e y,x | 2 ) -εm x,y [1 + (η -1)(|e x,y | 2 + β|e y,x | 2 )] (1c)
Here e x,y are the normalized amplitudes of the two complex fields E x = e x exp(2iπν x t) and E y = e y exp(2iπ(ν x +2f AO )t). According to these equations, e x has a constant phase, and can thus be considered without loss of generality as a real quantity, so that only the phase φ of e y corresponds to the relative phase between the two oscillators. The normalized population inversions are m x,y , and the normalized time s = 2πf R t is in units of the relaxation oscillations f R . β accounts for cross-saturation due to the spatial overlap in the gain medium, ε for the gain medium losses, η is the pump factor, Γ quantifies the self-injection strength, and ∆ = (ν y -ν x -2f AO )/f R is the detuning between the laser beat-note and the RF driver. Without loss of generality, we consider ∆ > 0.

Eqs. 1 are numerically integrated using an implicit Runge-Kutta method of order 5. Control parameters are ∆ and Γ while the others are fixed to β = 0.6, η = 1.2 and ε = 0.097. These values have been previously measured [30] and allow good model-experiment agreement. A mapping of the behavior of this dynamical system has been computed, unraveling chaotic, periodic, phasebounded and phase-unbounded attractors, sometimes exhibiting multistability and fractal attraction basins.

Figure 2 shows some of these features in a rich region of interest. Between the locking range ∆ < Γ, where the system reaches an equilibrium, and the unlocked region for ∆ Γ, where the system settles on a limit cycle, various attractors exist and co-exist. It can be noted from this mapping that the domains of bounded phase behavior and of stability of any attractor other than fixed point are neither simple rednor connected. It can also be noted that bounded-phase chaos, corresponding to the red points, is present only in small regions of the parameter space; nevertheless, as we will see, it can be clearly isolated experimentally.

Figure 3 shows a higher definition slice of the map at Γ = 0.91. On this phase bifurcation diagram, bounded phase regions are clearly seen, and different types of attractors are encountered as ∆ changes. Starting from the right, for ∆ > 1.02, the system settles on a stable periodic orbit, which corresponds to the bounded phase regime (B). As ∆ decreases and approaches the resonance with the relaxation oscillations, a Neimark-Sacker bifurcation leads to a torus, which is destabilized by other bifurcations in the multistable 0.97 < ∆ < 1.01 region, where periodic and chaotic (C) attractors can be found. When ∆ is further decreased away from resonance, the torus (T) is restored, before its chaotic break-up near the locking limit ∆ = Γ. In this particular region the phase remains bounded for a small range of detuning, where we witness a robust bounded-phase chaos regime (BC). Ultimately at the locking point, a subcritical Hopf bifurcation changes the stability of the fixed point, which becomes the only attractor in the phase space.

Knowing the parameters where it is supposed to arise, the previously unobserved bounded phase chaos could be investigated experimentally. Reinjection strength has been adjusted so that unlocking happens for δν -2f AO = 0.9f R , thus ensuring Γ = 0.9. Then detuning has been carefully set just after unlocking point, so that ∆ Γ. Time series of length 100 ms were recorded at a sampling rate of 10 MHz with a Rohde & Schwarz FSV sig- nal analyzer, performing IQ demodulation at 2f AO , thus allowing for a direct measurement of the quadratures of the beat-note signal, and of the relative phase. Results are shown in figure 4, where we see a complicated trajectory in the IQ plane, corresponding to the projection of a strange attractor. The important point is that the trajectory never makes a loop around the origin, so that the phase variations, albeit chaotic, are relatively small. Fig. 4 (b) presents a small portion of the measured time series, showing a spiking behavior, similar to the one expected from numerical simulations. The repetition rate looks quite regular, but the amplitude of the spikes is erratic. The phase, quickly increases during the spike then decreases before the next one, but as it remains bounded, the mean frequency of the opto-RF oscillator remains synchronized to the external RF source.

But it is worth asking if we still have a good oscillator. For instance, it has been previously shown that even in the bounded phase regime, the quality and long-term stability of the master reference was transferred on the optical carrier [17]. This can be quantified by computing the power spectral density (PSD) of the relative phase. Experimental phase spectra extracted from the quadra- ture time series are presented in Fig. 5. It can be seen that the onset of chaos results in a broadening of the spectral peak around 50 kHz, corresponding to the value of ∆, and of its harmonic at 100 kHz. A broad peak at low frequency, around 5 kHz, also appears. Comparing with the time series in Fig. 4, it can be seen that this peak corresponds to the average repetition rate of the spikes. In this part of the phase spectrum, bounded chaos features a higher phase noise level than all other curves, but it still outperforms clearly the free running laser at low frequency. In particular, we see that the low frequency part of the spectrum is flat, leading to a phase noise at 100 Hz from the carrier which is comparable to the level of the phase-locking regime, and at least 30 dB lower than the unlocked regime. Comparing with the free-running case, it appears clearly that in bounded-phase chaos the longterm synchronization of the mean value of the phase is well preserved. For a qualitative comparison, the phase noise spectrum was computed from the numerical model, by introducing arbitrary gaussian white noise in the parameters η, ∆ and Γ. The results, shown in Fig. 6, are in good qualitative agreement with experiments. At 100 Hz from the carrier, the simulated phase noise is lower for the phase-locking than for bounded-phase chaos, contrary to the experiments. This suggests that we have reached the noise floor of the experimental detection system. In any case, the simulations confirm the main features of the phase noise spectrum in the bounded-phase chaotic regime: a broadening of the peak corresponding to the detuning frequency, and of its harmonics; the appearance of a large, continuous component in the region between around 1 kHz and 20 kHz, corresponding to the chaotic oscillations; and, most importantly, a flat spectrum in the 100 Hz -1 kHz region, with a noise level that is quite close to the phase-locking level, and in any case much lower than for the unlocked case. In short, it is confirmed that, when bounded-phase chaos occurs, synchronization is well preserved. A more quantitative agreement with the experiment would require a precise characterization of the noise sources of the experimental setup, beyond the scope of this work.

In summary, we have presented experimental evidence of bounded-phase chaos in a driven opto-RF oscillator destabilized by optical feedback. By measuring the phase noise spectra, we have shown that, in spite of this, the long-term phase stability of the master oscillator is still well transferred to the slave opto-RF oscillator. These features are well reproduced using a rate equation model. As it is the case for the non-chaotic bounded-phase regime [17], we can expect these features to be common to a large class of systems. In particular, the effects of the Henry factor and of the feedback delay on this regime deserve to be investigated in opto-RF oscillators using semiconductor lasers [31], for which chaotic sensing has been widely demonstrated [32]. The so-called chaotic lidar [START_REF] Lin | Quantum Electronics[END_REF] takes advantage of broadband chaos to achieve position sensing with a high resolution. Bounded-phase chaos, by imposing a good phase coherence to the chaotic waveform, will allow to combine this precise position sensing with the high resolution velocimetry allowed by phase coherence [34], i.e. to realize a chaotic lidar-radar.

FIG. 2 .

 2 FIG. 2. Map of different regimes of the system. Light gray: bounded phase (existence of at least a stable limit cycle for which φ is bounded), black: existence of at least one unbounded chaotic attractor, red: existence of bounded chaos. Dashed line Γ = 0.91 is the line along which the bifurcation diagram of Fig. 3 is computed.

  FIG. 3. (a) Simulated bifurcation diagram of phase extrema for Γ = 0.91. Final state at each point is taken as the initial conditions for the next point, as ∆ is increased. The largest Lyapunov exponent is computed for each point and plotted as the solid black line. (b) Projection of the phase space trajectory on the { (ey), (ey)} = {|ey| cos φ, |ey| sin φ} plane for different regimes encountered (dark: 50000 time units, light: 2000).

FIG. 4 .

 4 FIG. 4. (a) Experimental phasor plot in the chaotic bounded phase regime (light: 100ms, dark: 10ms). (b) Corresponding time series of the intensity ( √ I 2 + Q 2 , blue lower trace) and phase (φ, top red). I and Q are the two quadratures of the beat signal Ixy = |Ex + Ey| 2 = |ex| 2 + |ey| 2 + 2|exey| cos(2fAOt + φ), demodulated at the reference frequency 2fAO, so that I = 2|exey| cos φ and Q = 2|exey| sin φ. The phase φ has the same meaning as the one in Fig 3b.

FIG. 5 .

 5 FIG.5. Experimental phase noise for different regimes at Γ ≈ 0.9 (black: unlocked, blue: locked, green: bounded, red: bounded chaos).

FIG. 6 .

 6 FIG.6. Simulated phase noise spectrum for different regimes at Γ = 0.9 (black: drifting ∆ = 1.8, blue: locked ∆ = 0.8, green: bounded ∆ = 1.1, red: bounded chaos ∆ = 0.91). Computed by replacing η with η(1 + 0.03ξ(s)), ∆ by ∆(1 + 0.03ξ(s)) and Γ by Γ(1 + 0.03ξ(s)) where ξ(s) is the same stochastic Gaussian process.


Aurélien Thorette

Marco Romanelli

Marc Brunel

Marc Vallet

Bounded-phase chaotic dynamics in a vectorial laser


Kedong Yuan

Jean-François Soulé

Vincent Dorcet

Henri Doucet

Palladium-Catalyzed Cascade sp(2

Keywords: Palladium, C-H activation, catalysis, desulfitative coupling, cascade

published or not. The documents may come

Introduction

One current important area of modern synthetic chemistry is the development of methods minimizing both the requisite number of steps and formation of wastes. Among these methods, the metal-catalyzed functionalization of C-H bonds has emerged as a simpler and "greener" way for the access to useful molecules for biological or material applications, as such processes are capable of forming similar products while avoiding the use of stoichiometric organometallic reagents. Most of the examples of such C-H bond functionalizations concern the formation of C-C bonds via arylations, alkylations or alkenylations; 1 whereas, the formation of two C-C bonds via consecutive Heck type reaction followed by an sp 2 C-H bond activation has attracted less attention. [START_REF] Grigg | For examples of synthesis of heterocycles via Pd-catalyzed molecular queuing processes[END_REF]3 Among rare examples, Fagnou et al. described the formation of indolines from 2bromoaniline derivatives and heteroarenes via a domino palladium-catalyzed Heckintermolecular direct arylation. 3d In 2012, Wu et al. described the synthesis of 4-polyfluoroaryl pyrrolo [1,2-a]quinolines via intermolecular followed by intramolecular palladium-catalyzed sp [START_REF] Grigg | For examples of synthesis of heterocycles via Pd-catalyzed molecular queuing processes[END_REF] C-H bond activations. 3e Recently, Zhu and co-workers reported the synthesis of [3,4]-fused oxindoles via a double palladium-catalyzed sp [START_REF] Grigg | For examples of synthesis of heterocycles via Pd-catalyzed molecular queuing processes[END_REF] C-H bond activation. 3g On the other hand, the Heck reaction [START_REF]For reviews on Heck type reaction: (a) Heck[END_REF][START_REF] Miura | For Heck type reaction with benzenesulfonyl chlorides[END_REF] using N-allylbenzenesulfonamides and aryl halides as coupling partners has been described by Y. Dong (Scheme 1 top). [START_REF] Dong | For Heck reactions with N-allylsulfonamides and aryl halides[END_REF] In this article, we report on the unexpected one pot synthesis of 4-aryl-1,2,3,4tetrahydroquinolines, from N-allyl-N-benzenesulfonamides and ArSO 2 Cl through a palladiumcatalyzed cascade desulfitative addition-cyclization reaction (Scheme 1, bottom). [7][8][START_REF] Wang | For a review on transition-metal mediated C-S bond activation[END_REF] It should be mentioned that 1,2,3,4-tetrahydroquinoline is an important motif found in numerous compounds with important antitumoral, antibacterial or antioxidant activities. 10 Several methods are leading to the synthesis of 1,2,3,4-tetrahydroquinolines, including partial reduction of quinolines, condensation of anilines with two molecules of an aldehyde, or from a Schiff base and an alkene. 10,11 However, most of these methods involve a multi-steps synthesis and in several cases does not allow the introduction of specific functional groups at the desired positions.

Scheme 1.

Results and discussion

The N-allyl-N-arylsulfonamides 1-22 were first prepared in high yields from anilines, methyl-or benzene-sulfonyl chlorides and allyl bromides (Scheme 2).

Scheme 2. Structures of 1-22

We then examined the influence of the conditions on the products formation using N-allyl-N-ptolylbenzenesulfonamide 1 and 4-methylbenzenesulfonyl chloride as the coupling partners (Table 1). Using PdCl 2 catalyst and Li 2 CO 3 as base in dioxane as reaction conditions, which had been previously found operative for direct arylations of heterocycles, 12 no reaction occurred. On the other hand, under the same conditions, but in the presence of 1.1 equiv. of CuBr as additive, the unexpected 4-tolyl-1,2,3,4-tetrahydroquinoline 24 -resulting from a formal 6-endo-trig cyclization after desulfitative addition-was obtained in 75% yield with 89% conversions of 1 (Table 1,entry 2). It should be noted that no formation of the expected Heck type product 23 was observed under these conditions. The use of Pd(OAc) 2 , PdCl 2 (MeCN) 2 , PdCl(C 3 H 5 )(dppb), Pd(TFA) 2 or Pd 2 (dba) 3 catalysts did not increase the yield in the desired product 24 (Table 1, entries 3-7). With Pd 2 (dba) 3 catalyst in the absence of CuBr, deallylated 1 was the major product. With PdCl 2 catalyst, a decrease of the CuBr loading to 0.1 equiv. had almost no influence on the yield in 24; whereas using 1 mol% CuBr gave 24 in low yield (Table 1, entries 9-11). Then, the influence of several bases was examined using 5 mol% of PdCl 2 associated to 0.5 equiv. of CuBr. Lower yields in 24 were obtained with K 2 CO 3 and Na 2 CO 3 ; whereas, Cs 2 CO 3 or a reaction without base were completely ineffective (Table 1, entries 12-15). The decrease of the reaction temperature to 100 °C also affords 24 in good yield (Table 1, entry 16).

The influence of a few solvents was also investigated. No reaction occurred in xylene or DMF; whereas 1 was obtained in 65% yield in diethyl carbonate (Table 1, entries 17-19). The addition of 1.2 equiv. of TEMPO was found to quench almost completely the reaction without its incorporation in 1 or 2. This result suggests that no radical species is involved in this 6-endo-trig cyclization (Table 1, entry 20). From 1 using 5 mol% PdCl 2 catalyst in the presence of CuBr and Li 2 CO 3 at 140 °C, but without benzenesulfonyl chloride, no cyclization to afford a 1,2,3,4tetrahydroquinoline occurred and 1 was recovered (Table 1, entry 21). When CuBr 2 was used as additive instead of CuBr, a low conversion of 1 was observed, and 24 was formed in very low yield together with unidentified side-products (Table 1, entry 22). Finally, the replacement of 1 by tert-butyl allyl(phenyl)carbamate led to several unidentified side-products; whereas, the desired cyclized product was not detected by GC/MS analysis of the crude mixture (Table 1, entry 23). The use of benzenesulfonyl chlorides as coupling partner is crucial for this reaction, as 4-

ACS Catalysis

The influence of substituent on the N-allyl moiety was also investigated. From (E)-N-(but-2enyl)-N-phenylmethanesulfonamide 8 and p-TolSO 2 Cl, 41 was obtained in 81% as a single diastereomer. Similar results were obtained for the coupling of 8 and 9 with 4-fluoro-and 4methoxybenzenesulfonyl chlorides affording 42 and 43 in 78% and 88% yields, respectively.

The trans-stereochemistry was unambiguously assigned by X-ray analysis of 43 (see SI). The regioselectivity of the reaction with aniline derivatives bearing meta-bromo or meta-methoxy substituents was then studied.

In both cases, the formation of mixture of 1,2,3,4tetrahydroquinolines was obtained. The major products 44a and 45a arises from coupling at less hindered position. Similar regioselectivity was observed using 3,4-dimethylaniline, as 46a and of the use of a deuterated aniline derivative on the reaction rate (Scheme 8, middle). This result suggests that the C-H bond activation step is probably not rate limiting for the catalytic cycle.

From the N-allyl-N-phenylmethanesulfonamide 63 containing two deuterium atoms on the allyl moiety (Scheme 8, bottom), the formation of 64 as a mixture of two diastereomers was observed.

The deuterium migration seems to confirm a mechanism involving a β-D elimination followed by a reinsertion in the Pd-D species (See scheme 9).

Conclusions

In summary, we report here the first palladium-catalyzed synthesis of 4-aryl-1,2,3,4tetrahydroquinoline derivatives via successive insertion of an alkene into a Pd-Ar bond followed by an intramolecular sp 2 C-H functionalization. The reaction is chemo-and diastereo-selective and proceeds with an easily accessible phosphine-free air stable palladium catalyst, and Li 2 CO 3 as inexpensive base associated to CuBr. The reaction can be successful for a variety of substituents both on aniline, allyl and on the benzenesulfonyl moieties. Due to the described usefulness of 1,2,3,4-tetrahydroquinoline derivatives, such simple reaction conditions offer a new attractive method for access to such structures.

Experimental Section

General procedure for the synthesis of N-allyl-N-phenylmethanesulfonamides 1-22, 54, 63:

A mixture of the aniline derivative (5 mmol) and triethylamine (0.505 g, 5 mmol) was stirred in CH 2 Cl 2 (15 mL) at 22 °C, then, R 2 SO 2 Cl (5 mmol) was slowly added to the mixture and the reaction was monitored by TLC. When a complete conversion of the starting material was observed, the mixture was concentrated then filtrated through a short silica column and evaporated to afford the desired aryl sulfamide. A mixture of this aryl sulfamide, allyl bromide derivative (7.5 mmol) and K 2 CO 3 (1.725 g, 12.5 mmol) in acetone (30 mL) was refluxed overnight. After reaction cooling down, the mixture was concentrated and purified by silica gel chromatography (ethyl acetate/pentane) to afford the desired products 1-22, 54 and 63.

General procedure for the synthesis of 24-53, 55-62 and 64: Page 14 of 18

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To an oven dried 25 mL Schlenk tube under argon, N-allyl-N-phenylsulfonamides (0.5 mmol), arenesulfonyl chlorides (1.0 mmol), PdCl 2 (4.4 mg, 0.0025 mmol), CuBr (0.035 g, 0.25 mmol), Li 2 CO 3 (0.110 g, 1.5 mmol) and 1,4-dioxane (1.5 mL) were successively added. Then, the reaction mixture was settled in a preheated (140 o C) oil bath for 20 h under stirring. Upon the reaction finished, the crude mixture was purified on silica chromatography (ethyl ether/ pentane, ethyl acetate/ pentane). ). [START_REF]For reviews on Pd(II)/Pd(IV) catalysis: (a) Muniz[END_REF] C NMR (100 MHz, CDCl 3 ): δ 143. 6, 142.0, 137.0, 136.8, 136.0, 132.9, 130.3, 129.7, 129.1, 128.1, 127.4, 126.8, 125.0, 124.6, 45.5, 43.0, 30.3, 21.6, 21.0 Reaction procedures and 1 H and [START_REF]For reviews on Pd(II)/Pd(IV) catalysis: (a) Muniz[END_REF] C NMR of all new compounds. This material is available free of charge via the Internet at http://pubs.acs.org."

4-p-Tolyl

Scheme 4 .

 4 Scheme 3. Control Reaction with an Aryl Bromide

46bScheme 5 .Scheme 6 .

 56 Scheme 5. Scope of the Substituents on the Aniline and Allyl moieties.

Scheme 9 .

 9 Scheme 8. Experiments using Deuterated Derivatives

tolylbenzenesulfonamide 1

 1 (0.144 g, 0.5 mmol) and 4-methylbenzenesulfonyl chloride (0.191 g, 1 mmol), 24 was obtained in 83% (0.156 g) yield as a white solid: mp 107-109 °C.[START_REF] Alberico | For reviews on Pd-catalyzed C-H bond activation[END_REF] H NMR (400 MHz, CDCl 3 ): δ 7.97 (d, J = 8.2 Hz, 1H), 7.56 (d, J = 8.3 Hz, 2H), 7.29-7.20 (m, 3H), 7.02 (t, J = 7.6 Hz, 1H), 6.98 (d, J = 7.9 Hz, 2H), 6.78 (d, J = 7.7 Hz, 1H), 6.55 (d, J = 8.3 Hz, 2H), 4.15-4.07 (m, 1H), 3.83 (dd, J = 8.5, 7.0 Hz, 1H), 3.80-3.70 (m, 1H), 2.45 (s, 3H), 2.31 (s, 3H), 2.00-1.88 (m, 1H), 1.77-1.65 (m, 1H

Table 1 .

 1 Influence of the Reaction Conditions for the Po-Catalyzed Functionalization of 1. TolSO 2 Cl (2 equiv.), base (3 equiv), 1,4-dioxane. a in xylene, b in DMF, c in diethyl carbonate, d p-TolSO 2 Cl (1 equiv.) and TEMPO as additive (1.2 equiv.), e without p-TolSO 2 Cl. f CuBr 2 instead of CuBr, g Using tert-butyl allyl(phenyl)carbamate instead of 1.

	Entry	Catalyst (mol%)	Base	CuBr (equiv.)Temp (°C)	Conv.	Yield in 24
						of 1 (%)	(%)
	1	PdCl 2	Li 2 CO 3	-	140	<10	0
	2	PdCl 2	Li 2 CO 3	1.1	140	89	75
	3	Pd(OAc) 2	Li 2 CO 3	1.1	140	81	75
	4	PdCl 2 (CH 3 CN) 2	Li 2 CO 3	1.1	140	83	74
	5	PdCl(C 3 H 5 )(dppb)	Li 2 CO 3	1.1	140	76	68
	6	Pd(TFA) 2	Li 2 CO 3	1.1	140	70	67
	7	Pd 2 (dba) 3	Li 2 CO 3	1.1	140	74	53
	8	Pd 2 (dba) 3	Li 2 CO 3	-	140	100	trace
	9	PdCl 2	Li 2 CO 3	0.5	140	88	83
	10	PdCl 2	Li 2 CO 3	0.1	140	80	73
	11	PdCl 2	Li 2 CO 3	0.01	140	23	20
	12	PdCl 2	K 2 CO 3	0.5	140	73	31
	13	PdCl 2	Na 2 CO 3	0.5	140	85	42
	14	PdCl 2	Cs 2 CO 3	0.5	140	0	0
	15	PdCl 2	-	0.5	140	40	0
	16	PdCl 2	Li 2 CO 3	0.5	100	83	74
	17	PdCl 2	Li 2 CO 3	0.5	120	0	0 a
	18	PdCl 2	Li 2 CO 3	0.5	120	0	0 b
	19	PdCl 2	Li 2 CO 3	0.5	120	72	65 c
	20	PdCl 2	Li 2 CO 3	0.5	140	<10	<5 d
	21	PdCl 2	Li 2 CO 3	0.5	140	0	0 e
	22	PdCl 2	Li 2 CO 3	0.5 f	140	30	<10
	23	PdCl 2	Li 2 CO 3	0.5	140	<20 g	-

Conditions: 1 (1 equiv.), p-

  . Elemental analysis: calcd (%) for C 23 H 23 NO 2 S (377.50): C 73.18, H 6.14; found: C 73.30,

	H 6.00.
	ASSOCIATED CONTENT
	Supporting Information.

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ACKNOWLEDGMENT

We thank the ministère de la recherche for a fellowship to K. Y. and the Centre National de la Recherche Scientifique and "Rennes Metropole" for providing financial support.

AUTHOR INFORMATION

Corresponding Author E-mail: jean-francois-soule@univ-rennes1.fr; henri.doucet@univ-rennes1.fr

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.