R A B Zijlmans

S Welzel

O Gabriel

G Yagci

J H Van Helden

J Röpcke

D C Schram

R Engeln
email: r.engeln@tue.nl

Experimental study of surface contributions to molecule formation in a recombining N 2 /O 2 plasma

Keywords: numbers: 52.70.Kz, 82.33.Xj, 52.40.Hf, 52.50.Nr

A low pressure recombining Ar plasma to which mixtures of N 2 and O 2 were added has been studied to explore the relevance of surface related processes for the total chemistry. The abundances of the stable molecules N 2 , O 2 , NO, N 2 O and NO 2 have been measured by means of a combination of infrared tunable diode laser absorption spectroscopy and mass spectrometry.

A gas phase chemical kinetics model was developed in chemkin to investigate the contribution of homogeneous interactions to the conversion of the feedstock gases N 2 and O 2 . At a partial pressure of N 2 plus O 2 less than 8 Pa, significant discrepancies between measured and calculated concentrations of N 2 O and NO 2 were observed, indicating that heterogeneous processes are dominating the chemistry in this pressure range. At a partial pressure of N 2 plus O 2 higher than 40 Pa and a relatively high fraction of admixed O 2 we observed a fair agreement between measured and calculated concentrations of NO molecules, indicating that homogeneous processes (notably Natom induced) are more dominant than heterogeneous processes.

Introduction

The interaction of reactive species, like atomic or molecular radicals, with a surface is a very general phenomenon. In plasma processing it is this interaction that leads to the modification of surfaces, i.e. deposition or etching [START_REF] Van De Sanden | Plasma chemistry aspects of a-Si:H deposition using an expanding thermal plasma[END_REF][START_REF] Martinu | Plasma deposition of optical films and coatings: a review[END_REF][START_REF] Benedikt | Plasma chemistry during the deposition of a-C:H films and its influence on film properties[END_REF][START_REF] Förch | Soft plasma treated surfaces: Tailoring of structure and properties for biomaterial applications[END_REF]. In astro-chemistry it has been recognized that ice or dust particles in the interstellar space act as third body in association processes in which not only hydrogen molecules are formed, but also larger molecules [START_REF] Vidali | The formation of interstellar molecules via reactions on dust grain surfaces[END_REF]. In plasma assisted conversion of gases, the surfaces of a reactor also play an important role [START_REF] Tanaka | Synergistic effects of catalysts and plasmas on the synthesis of ammonia and hydrazine[END_REF][START_REF] Kim | Nonthermal plasma processing for air-pollution control: A historical review, current issues, and future prospects[END_REF][START_REF] Rousseau | Evidence of plasma-catalyst synergy in a lowpressure discharge[END_REF]. For example Gatilova et al [START_REF] Gatilova | NO formation mechanisms studied by infrared laser absorption in a single low-pressure plasma pulse[END_REF] studied the formation of NO in a low pressure N 2 /O 2 discharge. The authors focus mainly on the gas phase formation of NO, but recognize the importance of the surfaces of the plasma reactor. Castillo et al [START_REF] Castillo | Spectrometric and kinetic study of a modulated glow air discharge[END_REF][START_REF] Castillo | Low-pressure dc air plasmas. investigation of neutral and ion chemistry[END_REF] also conclude that mainly heterogeneous processes are responsible for the formation of NO.

In this paper the role of surface association reactions during the conversion of mixtures of N 2 and O 2 is in the centre of interest. Therefore a remote, recombining, low pressure argon plasma, produced by a cascaded arc (see also [START_REF] Brussaard | Stripping of photoresist using a remote thermal Ar/O 2 and Ar/N 2 /O 2 plasma[END_REF][START_REF] Van Hest | Analysis of the expanding thermal argon-oxygen plasma gas phase[END_REF]) is used. This type of plasma exhibits a high dissociation efficiency. Furthermore, the contribution of excited particles to the total kinetics is low, because there is only convective power input into the plasma from the moment it expands from the cascaded arc into the reactor volume. In combination with the low working pressure, this creates a condition in which radical surface interactions can readily be addressed.

N 2 and O 2 molecules are admixed to the argon expansion and are dissociated via charge exchange and subsequent dissociative recombination mainly into atomic radicals. The resulting steady state mole fractions of the stable molecules in the system (Ar, N 2 , O 2 , NO, N 2 O and NO 2 ) are measured by means of a combination of mass spectrometry and tunable diode laser absorption spectroscopy.

The relative importance of homogeneous interactions with respect to heterogeneous interactions is expected to vary significantly with the surface to volume ratio of the reactor and the pressure in the reactor. The effect of the pressure is investigated in this paper. Typical gas phase reaction rates between a stable molecule and a radical are reported to be in the range of 10 -18 m 3 /s (see the appendix). In general, the abundances of radicals in a plasma are orders of magnitude lower than the abundances of the stable molecules. When radical densities are a factor of 10 3 lower than the stable molecules, the gas phase reactions at a temperature of 1000 K can be expected to contribute to the chemistry at a partial pressure of N 2 and O 2 higher than 10 Pa. At this pressure a significant fraction of the stable molecules can be converted by these reactions within 1 s. The reaction rates of interactions between stable molecules are significantly lower, which implies that these types of interactions become dominant at even higher pressures. We will investigate the conversion of N 2 and O 2 at different partial pressures of N 2 and O 2 in the range of 1 -40 Pa to find indications for a difference in conversion between mainly homogeneous interactions at high pressure and mainly heterogeneous interactions at low pressure. Furthermore, the behavior of the formation of NO at low admixed O 2 fractions is scrutinised. For these conditions, a kind of threshold behavior in the formation of NO may be observed similar to the work of Nahorny et al [START_REF] Nahorny | Experimental and theoretical investigation of a N 2 -O 2 DC flowing glow discharge[END_REF] and Dilecce et al [START_REF] Dilecce | Experimental studies on elementary kinetics in N 2 -O 2 pulsed discharges[END_REF].

After a short introduction of the setup, the diagnostics and the expanding plasma in section 2, 3 and 4 respectively, the chemical model is presented in section 5. The measurements of the molecular densities for different plasma conditions are discussed in section 6 and summarized in the last section.

Experimental setup

The setup, which is sketched in figure 1, has a cascaded arc as remote plasma source, which consists of a stack of four copper plates with a central bore (∅ = 4 mm), through which the working gas (argon at a flow of 3000 standard cubic centimeters per minute, sccm) is directed. A discharge is maintained between cathode tip and anode plate, at a current setting between I = 45 A and I = 75 A, which results in ionization of a significant fraction of the working gas: in case argon is used around 10 -15 % of the gas is ionized [START_REF] Van De Sanden | The behaviour of heavy particles in the expanding plasma jet in argon[END_REF]. The copper plates are electrically insulated from each other, which stabilizes the discharge. During operation of the plasma source, the pressure in the arc channel is sub-atmospheric and in the relatively small volume of the arc channel (0.3 cm 3 ), the power input can reach values up to 10 kW. The mixture of atoms, ions and electrons flows through a nozzle and expands supersonically into the vessel. The stainless steel vessel has a diameter of 0.32 m and is 1.0 m long and is typically kept at a pressure of p = 20 Pa by a 3-stage pumping system. The pressure is monitored by three pressure sensors (a Baratron and two wide-range sensors), which are installed on top of the reactor.

Diagnostics

A compact and transportable InfraRed Multicomponent Acquisition system (irma), which is based on mid-infrared absorption spectroscopy has been installed [START_REF] Röpcke | IRMA: A tunable infrared multicomponent acquisition system for plasma diagnostics[END_REF]. It contains four helium cooled independently tunable lead-salt diode lasers which can be temporally multiplexed. Rapid scan software with real-time line shape analysis provides simultaneous measurements of the absolute concentrations of several molecular species.

In table 1 the types of molecules that are detected with this diagnostic are summarized, with the transition frequencies at which the analysis is performed.

The light beam of the diode lasers is directed through the plasma vessel twice via tubes that are directly connected to the vessel as sketched in figure 1. This ensures that the composition of the background gas is determined, and the absorption path length is 1.62 m. The laser light is directed to a mercury cadmium telluride detector (MCT, Graseby Infrared).

The instrumental broadening of the diode laser system is about 0.001 cm -1 , and by scanning the temperature and current of the laser, a spectral range of approximately 100 cm -1 can be achieved. The laser light output power is in the order of several hundred µW.

The hitran database [START_REF] Rothman | The HITRAN 2004 molecular spectroscopic database[END_REF] provides information on the line strengths, and by means of an advanced forward fitting implementation, the concentration is calculated real-time. The raw data of a typical absorption spectrum as a function of the laser current is shown in figure 2. Ramping the current through the laser diode at a constant operating temperature sweeps the laser frequency. A spectral region of approximately 1 cm -1 may be covered by means of an entire current ramp. One current ramp is performed in approximately 1 ms. The time resolution that is achieved in the measurements presented in this paper, including the fitting procedure and averaging is 10 ms.

In figure 2, in the range of one current ramp, absorption lines of both NO (1884.3 cm -1 ) and H 2 O (1884.6 cm -1 ) in an N 2 /O 2 /H 2 plasma can be recognized. The H 2 O absorption in a plasma reactor at low pressure is superimposed on a very broad absorption feature, caused by atmospheric H 2 O. The spectral feature attributed to NO consists of more than 20 different transitions, and the ratio between the absorption lines changes with temperature. By a careful analysis of this ratio, the gas temperature can be estimated.

The integrated absorption of the spectral features is directly related to the concentration of the type of molecules under investigation. The spectral line intensities of the transitions serve as the scaling factor for the concentrations and are tabulated in the hitran database (2004 edition [START_REF] Rothman | The HITRAN 2004 molecular spectroscopic database[END_REF]).

The accuracy of the absorption measurements is mainly limited by uncertainties in the temperature profile over the path length of the laser light through the plasma reactor, and is typically 10 %. The mimimum detectable mole fractions vary for the different types of species. The minimum detectable absorption was about 10 -3 which corresponds to a fractional detection limit down to 10 -5 for the experiments and species described in this paper.

Also a quadrupole mass spectrometer (QMA 430, Balzers) with a secondary electron multiplier was installed at the beginning of the pump line and is thus probing the stable species (Ar, N 2 , O 2 , NO, N 2 O and NO 2 ) present in the vessel [START_REF] Van De Sanden | Plasma chemistry aspects of a-Si:H deposition using an expanding thermal plasma[END_REF][START_REF] Van Hest | Use of in situ FTIR spectroscopy and mass spectrometry in an expanding hydrocarbon plasma[END_REF]. The minimum detectable mole fractions in the experiments described in this paper are in the order of 10 -5 -10 -4 .

Physical processes in the expanding plasma

The argon plasma, which is created in the cascaded arc channel, expands supersonically into the vessel. Due to the fast expansion into the reactor, the gas temperature and the electron temperature drop rapidly. The gas temperature drops from approximately 5000 K (0.4 eV) at the start of the expansion, down to 450 K (0.04 eV) in the background of the reactor. The electron temperature reduces from 1 eV at the start of the expansion down to 0.2 eV downstream after the stationary shock.

The feedstock gases (mixtures of N 2 and O 2 ) are injected directly into the reactor and they mix efficiently with the expansion [START_REF] Vankan | Inflow and shock formation in supersonic, rarefied plasma expansions[END_REF]. Due to the low electron temperature, direct electron-impact ionization or dissociation reactions are not efficient. In the expansion, the feedstock gases are dissociated by charge transfer and subsequent dissociative recombination mainly into atomic radicals. For example, the dissociation of N 2 molecules occurs via:

Ar + + N 2 → Ar + N + 2 , (1)
N + 2 + e → N + N * , ( 2
)
where N * represents one of the two excited states ( 2 P, 2 D) of the N atom (see also [START_REF] Schram | Plasma processing and chemistry[END_REF]). An extra pathway of dissociating molecules in the reactor is via reactions with oxygen and nitrogen atomic radicals, e.g. N + NO → N 2 + O and N+O 2 →NO+O. The radicals, produced in the expansion, flow through the reactor, and recombine before they are pumped away. By means of a Direct Simulation Monte Carlo (DSMC, [START_REF] Bird | Proc. 24th Int. Symp. on Rarefied Gas Dynamics, Monopoly[END_REF]), the flow pattern and temperature distribution are calculated. For a hot argon gas with an initial temperature of 5000 K, expanding from the cascaded arc, the resulting temperature distribution in the reactor is shown in figure 3. Furthermore, the calculated flow pattern shows a strong recirculation cell in the reactor, that extends from the plasma source until halfway the reactor. The velocity profile is also sketched in the inset in figure 3. Our results are in good agreement with the results of Selezneva et al [START_REF] Selezneva | Stationary supersonic plasma expansion: continuum fluid mechanics versus direct simulation Monte Carlo method[END_REF], who used a hybrid fluid-DSMC model.

In the results of the Monte Carlo simulation, five different parts can be discriminated, based on the calculated temperature distribution in the reactor. The rate of production of the various gas phase interactions that are possible between the species present in the reactor, depends strongly on the temperature. Therefore these parts of the reactor are considered separately in the analysis in this paper: I) the supersonic expansion, including the stationary shock, where the feedstock gases mix efficiently with the expansion [START_REF] Vankan | Inflow and shock formation in supersonic, rarefied plasma expansions[END_REF][START_REF] Gabriel | Two-dimensional flow characteristic of a hot expanding plasma[END_REF], II) the subsonic expansion, III) the hot part of the background, where the feedstock gases are injected into the reactor, IV) the cold part of the background and V) the part where the measurements are performed. In parts III and IV (part of the) recirculation cell is present. A fraction of the hot and cold background gases is directed towards part V. The division of the system into these five parts is sketched in figure 3.

CHEMKIN model

A model has been developed for this plasma system, which is based on a combination of five zero-dimensional modules. For each part, the temperature, pressure, input flow, output flow and recirculation ratios are specified, based on the DSMC calculations.

The solver of the commercial software package CHEMKIN [START_REF]CHEMKIN(R) Interface 4. Simulation software[END_REF] is used to perform the calculations involved.

Recycling of the species in the reactor between the different parts of the reactor influences the corresponding residence times. In figure 4, the different parts in the reactor are depicted, and the numbers denote the flow between them, normalized to the inlet flow of argon through the cascaded arc (3000 sccm). The residence time τ res , temperature T of the gas and the volume V of each part, are given in table 2 for the flow settings used in the experiments that will be described in detail in section 6.1. The shock position, and thus the length of the supersonic expansion, varies between 3 cm at 3000 sccm Ar and a background pressure of p = 100 Pa and 10 cm at 3000 sccm Ar and a background pressure of p = 20 Pa. Because the temperature in this region is high, and the flow of admixed gases in all studied cases is high, the residence time in part I is relatively low and the kinetics in this part are expected to play only a minor role compared to the total kinetics in the system. Variations in the size of the subsonic expansion are also believed to play only a minor role to the total kinetics in the system. Therefore the volumes of the different parts are kept equal for all studied pressures.

The set of chemical reactions involving molecules containing N and O atoms, is selected from Capitelli et al [START_REF] Capitelli | Plasma kinetics in atmospheric gases[END_REF] and is supplemented with charge exchange and dissociative recombination reactions [START_REF] Van Helden | The generation of molecules through plasma-surface interactions[END_REF]. In the simulation program, we identified 17 reactions contributing at least for a fraction of 10 -3 to the production or destruction of the stable molecules present in the plasma. The reactions with their rate constants are summarized in the appendix.

In the simulations we do not explicitly take into account excited N atoms, although they are known to be produced by dissociative recombination via reaction 2. These excited atoms are de-excited mainly by collisions with NO or O 2 molecules, and as a result their lifetime is in the order of 10 -4 s at a pressure of p = 100 Pa. Though, the rate of progress of N+O 2 →NO+O is known to be significantly faster with excited N atoms than with N atoms in the ground state [START_REF] Gonzalez | VTST kinetics study of the N( 2 D) + O 2 (X 3 Σg ) → NO(X 2 Π) + O( 3 P, 1 D) reactions based on CASSCF and CASPT2 ab initio calculations including excited potential energy surfaces[END_REF][START_REF] Kossyi | Kinetic scheme of the nonequilibrium discharge in nitrogen-oxygen mixtures[END_REF]. Therefore we implement a rate constant for this reaction which is a factor of five larger than reported by Capitelli et al [START_REF] Capitelli | Plasma kinetics in atmospheric gases[END_REF].

The starting conditions of the simulation can be summarized as:

(i) a fixed inflow of 2550 sccm Ar, 450 sccm Ar + and an equal flow of electrons from the cascaded arc into zone 1 (see also [START_REF] Van De Sanden | The behaviour of heavy particles in the expanding plasma jet in argon[END_REF]), (ii) the studied mixture of N 2 and O 2 is injected into zone 3, (iii) the number of zero-dimensional plasma zones is determined from the DSMC calculations as described in the previous section, together with the temperature in each plasma zone and the flows between the different plasma zones. The measurements are presented for two different pressures, to investigate the possible influence of heterogeneous processes.

Results and discussion

Mixtures of N 2 and O 2 admixed to an Ar expansion

A
In figure 5, the measured mole fractions of Ar, N 2 , O 2 , NO, N 2 O and NO 2 are plotted on a semi-logarithmic scale. All the species are measured with the quadrupole mass spectrometer. Furthermore, NO and N 2 O are measured by infrared absorption spectroscopy, which provides direct in-situ results. The results for NO and N 2 O from mass spectroscopy and tunable diode laser absorption spectroscopy measurements showed good agreement: within 10 %.

At both background pressures in which experiments have been performed the molecules N 2 and O 2 were dominantly present (cf. Figure 5 and6). The abundances of the other types of molecules (NO, NO 2 and N 2 O) are one to three orders of magnitude lower. Remarkable is the maximum abundance of NO: it is independent of pressure and reaches a fraction of 10 -2 of the total pressure (see also [START_REF] Van Helden | Resemblance in gas composition of Ar-N 2 -O 2 and Ar-NO plasmas[END_REF]). Contrary to NO, the abundance of N 2 O and NO 2 decreases significantly with an increase of pressure.

Simulated volume interactions

In figure 6, the measured mole fractions of Ar, N 2 , O 2 , NO, N 2 O and NO 2 are plotted for the same conditions as presented in figure 5(a) and 5(b). The results of the complementary Chemkin calculations, in which only gas phase (homogeneous) interactions are taken into account, are also shown in figure 6. The abundances of Ar and N 2 are reproduced by the model. The calculated mole fraction of O 2 is lower than the measured one at all flow ratios. The main reason is the absence of an effective volume loss channel for O atoms in the model. This leads to an unrealistically high calculated oxygen atom abundance in the reactor, up to a mole fraction of 6 × 10 -2 . The model also predicts that a small amount of NO is formed in the gas phase, especially at high fractions of admixed O 2 . However, the trend of the calculated density as function of the flow ratio does not agree with the measured trend. And for low oxygen flows the calculated NO mole fractions are orders of magnitude lower than the measured ones. The abundances of the molecules N 2 O and NO 2 are calculated to be smaller than 10 -8 , which is several orders of magnitude lower than measured.

It is clear that for both series of measurements presented in figure 6, significant discrepancies between calculated and measured mole fractions are observed. Since not only the trends but also the absolute values of the mole fractions are not in agreement with the results of the CHEMKIN model in which only homogeneous interactions are taken into account, we conclude that heterogeneous interactions play an important role in the conversion of N 2 and O 2 .

Threshold behavior of NO production

In figure 5(b), we observe a threshold behavior in the formation of NO for relatively low admixed O 2 fractions: the NO mole fraction becomes only significant for an admixed O 2 fraction higher than 10 %. This effect is also observed in for example the work of Nahorny et al [START_REF] Nahorny | Experimental and theoretical investigation of a N 2 -O 2 DC flowing glow discharge[END_REF] and Dilecce et al [START_REF] Dilecce | Experimental studies on elementary kinetics in N 2 -O 2 pulsed discharges[END_REF]. To investigate this threshold behavior in our system in more detail, the effective fraction of NO is presented as a function of the admixed fraction of O 2 . The effective fraction represents the mole fraction, based on all the types of species in the reactor, except argon. It is calculated by multiplying the measured mole fraction of NO by the ratio of the total injected flows and the admixed flows of N 2 and O 2 . In figure 7, the effective mole fractions are plotted for three different pressures: The threshold behavior becomes more pronounced for higher pressures. At higher pressures, gas phase interactions will influence the plasma chemistry more than heterogeneous interactions. The pressure dependence of this threshold suggests that heterogeneous interactions are responsible for cancelling the threshold effect.

The results of the previous calculations in which only homogeneous reactions are taken into account (see figure 6), seem to underpin this reasoning. A clear threshold behavior can be observed in figure 6. However, the admixed O 2 fraction at which the mole fraction of NO starts to increase drastically, is much higher than during the measurements (see figure 7).

For p = 20 Pa, the NO mole fraction is smaller than 10 -5 for admixed O 2 fractions up to 0.10, and for p = 100 Pa for admixed O 2 fractions up to 0.20. This shift to higher admixed O 2 fractions may be explained by an overestimated N atom density in the calculations (not plotted). In the calculations presented in figure 6, the N atom density for the conditions at low admixed O 2 fractions, is high compared to the O 2 density. In that case NO is produced through N + O 2 → NO + O, but the high abundance of N atoms results in the immediate destruction of NO via N + NO → N 2 + O. In the gas phase simulations, the destruction of NO proceeds too fast to explain the measured NO abundance. Since heterogeneous processes are not taken into account in the calculations, the loss of N atoms is highly underestimated. Consequently, the density of N atoms is considerably overestimated and so is the destruction rate for NO. Including heterogeneous processes in the calculations will reduce the pronounced threshold effect that appears in the gas phase model. The implementation of heterogeneous processes in the model will be treated in detail in our next paper.

Molecule formation at low effective pressure

To be able to distinguish even more between the effects of heterogeneous and homegeneous processes, we also performed measurements on mixtures of N 2 and O 2 at a low effective pressure. In this case, the follow-up reactions in the volume of the reactor between surface produced species and radicals in the gas phase should be less significant with respect to the total kinetics in the plasma.

Mixtures of N 2 and O 2 were admixed to an argon/helium expansion at a pressure of p = 6 Pa. Since a relatively large amount of He was admixed, the diffusion of particles in the plasma towards the walls of the reactor proceeded faster because of the relatively low mass of He atoms. Furthermore, the partial pressure of the molecules of interest was significantly lower than in the experiments described in the previous section. This implies that the follow-up reactions between surface produced molecules and radicals is less important. A combination of 300 sccm Ar and 1500 sccm He is directed through the arc. A total flow of 200 sccm N 2 + O 2 is injected into the background of the reactor and mixes with the Ar/He expansion. The fraction of admixed O 2 of the total admixed O 2 and N 2 flow is varied from zero, i.e. pure N 2 flow, to one, i.e. pure O 2 flow. The pressure in the reactor is kept at p = 6 Pa and if the admixed gases mix perfectly with the expansion, the total flow of N 2 and O 2 can entirely be dissociated in the expansion by charge exchange and subsequent dissociative recombination. At these flow settings, p N 2 +O 2 = 1 Pa, which is a factor of 40 lower than in the experiments performed at p = 100 Pa as displayed in figure 6(b).

The mole fraction of the measured types of molecules is given in figure 8. Also in this experiment, mainly N 2 and O 2 are observed. The abundances of the other types of molecules (NO, N 2 O and NO 2 ) are two to three orders of magnitude lower than the abundances of N 2 and O 2 . Only for admixed O 2 fractions higher than 0.7, the observed ratio between the abundance of NO and N 2 is comparable to the measurements at higher pressures. For this range of high O 2 partial pressures homogeneous reactions can explain the observed behavior even at these low pressures.

The ratios between the abundances of NO 2 to NO and N 2 O to NO are higher than in figure 5. This indicates faster formation rates or slower destruction rates of N 2 O and NO 2 . At this point we cannot discriminate between these two effects. It is very likely that heterogeneous effects play a role, because in our gas phase simulations we do not observe significant production of NO 2 and N 2 O for low admixed fractions of O 2 , whereas the measured abundances of N 2 O and NO 2 seem to increase at lower pressures.

Furthermore, the threshold behavior in NO is absent at this low effective pressure. This is consistent with the results from the previous section, where the threshold behavior was most pronounced at the highest studied pressures.

Conclusions

To explore the contribution of surface related processes to the total kinetics in a low pressure recombining plasma, we measured the abundances of the stable molecules in N 2 /O 2 containing discharges for a variety of conditions by means of mass spectrometry and tunable diode laser absorption spectroscopy. A simulation has been developed in chemkin, that takes into account the gas phase chemistry.

The homogeneous interactions can explain the measured abundances of the stable molecules reasonably well for total partial pressures of N 2 and O 2 higher than 38 Pa and for a relatively high admixture of O 2 . For lower total partial pressures, significant deviations arise between measured and simulated gas composition. Extra loss channels for O and N atoms should be introduced, especially for admixed O 2 fractions lower than 0.2, most likely in the form of heterogeneous interactions. An extra loss channel for O atoms would increase the calculated O 2 density. An extra loss channel for N atoms reduces the predicted threshold effect for NO.

In our next paper we describe the development of a module in CHEMKIN that takes heterogeneous interactions (surface adsorbption, surface desorption, Ely-Rideal, Langmuir-Hinshelwood and recirculation of surface-produced species into the plasma volume) into account and we investigate, quantitatively, the contributions of these interactions to the overall plasma chemistry.

Figure 1 .

 1 Figure 1. A sketch of the expanding thermal plasma setup with on the left the cascaded arc.

Figure 2 .

 2 Figure 2. Absorption spectrum of a plasma containing a mixture of N 2 , O 2 and H 2 . Clearly visible are the absorptions of NO and H 2 O in the vessel (narrow peak at around 1884.6 cm-1) superimposed on the broad atmospheric water absorption.

Figure 3 .

 3 Figure 3. The calculated temperature distribution in the reactor and the division of the reactor into five different regions: I) the supersonic expansion, II) the subsonic expansion, III) the hot part of the background, IV) the cold background, including the walls of the reactor and V) the background downstream in the reactor, where the measurements are performed. The calculated velocity profile of the first half of the reactor is also sketched. Note that the IR measurements are performed at an axial position of 0.8 m.

F 6 FFigure 4 .

 64 Figure 4. Sketch of the different parts of the reactor, together with the flows from each part into one of the other parts, normalized to the input flow through the cascaded arc.

  plasma expansion is created from a flow of 3000 sccm argon in the cascaded arc. A total power of 5 kW (I arc = 75 A) is used to create an Ar plasma in the arc channel. This plasma expands from the exit of the arc channel into the reactor, which is kept at a background pressure of p bg = 20 Pa or p bg = 100 Pa. A total flow of 1800 sccm of mixtures of N 2 and O 2 is injected directly into the reactor. Based on the flow settings of the filling gases, the sum of the partial pressures of N 2 and O 2 amounts to p N 2 +O 2 = 38 Pa at a background pressure p bg = 100 Pa and p N 2 +O 2 = 7.6 Pa at p bg = 20 Pa. The molecular abundances of the species formed in the plasma vessel are investigated as function of the ratio of admixed O 2 flow over the total flow of O 2 and N 2 .

2 OFigure 5 .

 25 Figure 5. The measured mole fraction of the molecules in the plasma as a function of the flow ratio of O 2 over the total flow of the injected gas mixture of N 2 and O 2 . A flow of 3000 sccm argon is injected through the cascaded arc into the reactor, and a flow of 1800 sccm of the injected gas mixture of N 2 and O 2 is injected directly into the reactor. The results are given for a) p = 20 Pa and b) p = 100 Pa. The lines through the measured points serve only as a guide to the eye.

Figure 6 .

 6 Figure 6. The mole fraction of the measured molecules in the plasma (symbols) as function of the flow ratio of O 2 over the total flow of the injected gas mixture of N 2 and O 2 . A flow of 3000 sccm argon is injected through the cascaded arc into the reactor, and a flow of 1800 sccm of the injected gas mixture of N 2 and O 2 is injected directly into the reactor. The lines indicate the calculated mole fractions of Ar, N 2 , O 2 and NO based on purely homegeneous interactions. The results are given for a) p = 20 Pa and b) p = 100 Pa.

Figure 7 .

 7 Figure 7. Threshold behavior of NO for low admixed O 2 flows. For three pressures, the NO effective fraction is plotted as a function of the admixed O 2 . The effective fraction is the mole fraction, not taking into account argon, see the text for details. For p = 20 Pa and p = 100 Pa, a flow of 3000 sccm Ar and 1800 sccm of mixtures of N 2 and O 2 are admixed. For p = 9 Pa, the total flow was decreased by a factor of two.

2 Figure 8 .

 28 Figure 8. The measured molecular abundances in the plasma created from Ar/He: 300/1500 sccm, to which mixtures of N 2 + O 2 are admixed, where the total flow of N 2 and O 2 is 200 sccm. The pressure is set to p = 6 Pa and the current through the cascaded arc channel I = 75 A.

Table 1 .

 1 The frequencies of the used absorption lines for the detection of NO, NO 2 and N 2 O.

	Molecule Frequency (cm -1 )
	NO	1884.3
	NO 2	1600.9
	N 2 O	2203.7

Table 2 .

 2 The different regions in the reactor and the residence time in each region for the flow settings: 3000 sccm Ar injected into part I and 1800 sccm N 2 + O 2 injected into part III, at a pressure of p = 20 Pa.

	Region	Description	T (K) V (m 3 ) τ res (ms)
	I	supersonic expansion 3000	0.001	0.26
	II	subsonic expansion	2000	0.005	1.5
	III	hot background	1200	0.02	3.3
	IV	cold background	500	0.01	12
	V	measurement region	500	0.04	51

Appendix: gas phase reactions

Here, a summary of the implemented gas phase interactions is given. The reaction rate constants are obtained from Capitelli et al [START_REF] Capitelli | Plasma kinetics in atmospheric gases[END_REF]. The rate constants are presented by means of their Arrhenius parameters, as in:

where A is the prefactor, n the temperature exponent and Êact the activation energy, expressed in eV. reaction A (m 3 /s) n Êact (eV)


Matteo Santacesaria
email: santacesaria@cmap.polytechnique.fr

GLOBAL STABILITY FOR THE MULTI-CHANNEL GEL'FAND-CALDER ÓN INVERSE PROBLEM IN TWO DIMENSIONS

We prove a global logarithmic stability estimate for the multi-channel Gel'fand-Calderón inverse problem on a two-dimensional bounded domain, i.e. the inverse boundary value problem for the equation -∆ψ + v ψ = 0 on D, where v is a smooth matrix-valued potential defined on a bounded planar domain D.

Introduction

The Schrödinger equation at zero energy (1.1) -∆ψ + v(x)ψ = 0 on D ⊂ R 2 arises in quantum mechanics, acoustics and electrodynamics. The reconstruction of the complex-valued potential v in equation (1.1) through the Dirichlet-to-Neumann operator is one of the most studied inverse problems (see [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF], [START_REF] Mandache | Exponential instability in an inverse problem of the Schrödinger equation[END_REF], [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF], [START_REF] Novikov | New global stability estimates for the Gel'fand-Calderon inverse problem[END_REF], [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF], [START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions[END_REF] and references therein).

In this article we consider the multi-channel two-dimensional Schrödinger equation, i.e. equation (1.1) with matrix-valued potentials and solutions; this case was already studied in [START_REF] Xiaosheng | Inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials in two dimensions at fixed energy[END_REF][START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions[END_REF]. One of the motivations for studying the multi-channel equation is that it comes up as a 2D-approximation for the 3D equation (see [START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions[END_REF]Sec. 2]).

This paper is devoted to give a global stability estimate for this inverse problem in the multi-channel case, which is highly related to the reconstruction method of [START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions[END_REF]. The following inverse boundary value problem arises from this construction: given Φ, find v. This problem can be considered as the Gel'fand inverse boundary value problem for the multi-channel Schrödinger equation at zero energy (see [START_REF] Gel'fand | Some problems of functional analysis and algebra[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF]) and can also be seen as a generalization of the Calderón problem for the electrical impedance tomography (see [START_REF] Calderón | On an inverse boundary problem[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF]). Note also that we can think of this problem as a model for the monochromatic ocean tomography (e.g. see [START_REF] Baykov | Mode Tomography of Moving Ocean[END_REF] for similar problems arising in this tomography).

In the case of complex-valued potentials the global injectivity of the map v → Φ was firstly proved in [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF] for D ⊂ R d with d ≥ 3 and in [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF] for d = 2 with v ∈ L p : in particular, these results were obtained by the use of global reconstructions developed in the same papers. The first global uniqueness result (along with an exact reconstruction method) for matrix-valued potentials was given in [START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions[END_REF], which deals with C 1 matrix-valued potentials defined on a domain in R 2 . A global stability estimate for the Gel'fand-Calderón problem for d ≥ 3 was found for the first time by Alessandrini in [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF]; this result was recently improved in [START_REF] Novikov | New global stability estimates for the Gel'fand-Calderon inverse problem[END_REF]. In the two-dimensional case the first global stability estimate was given in [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF].

In this paper we extend the results of [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF] to the matrix-valued case; we do not discuss global results for special real-valued potentials arising from conductivities: for this case the reader is referred to the references given in [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF], [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF], [START_REF] Mandache | Exponential instability in an inverse problem of the Schrödinger equation[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF], [START_REF] Novikov | New global stability estimates for the Gel'fand-Calderon inverse problem[END_REF], [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF].

Our main result is the following:

Theorem 1.1. Let D ⊂ R 2 be an open bounded domain with C 2 bound- ary, let v 1 , v 2 ∈ C 2 ( D, M n (C
)) be two matrix-valued potentials which satisfy (1.4), with v j C 2 ( D) ≤ N for j = 1, 2, and Φ 1 , Φ 2 the corresponding Dirichlet-to-Neumann operators. For simplicity we assume also that v j | ∂D = 0 and ∂ ∂ν v j | ∂D = 0 for j = 1, 2. Then there exists a constant

C = C(D, N, n) such that (1.5) v 2 -v 1 L ∞ (D) ≤ C log(3 + Φ 2 -Φ 1 -1 ) -3 4 log(3 log(3 + Φ 2 -Φ 1 -1 )) 2 ,
where A denotes the norm of an operator A :

L ∞ (∂D, M n (C)) → L ∞ (∂D, M n (C)) and v L ∞ (D) = max 1≤i,j≤n v i,j L ∞ (D) (likewise for v C 2 ( D) ) for a matrix- valued potential v.
This is the first global stability result for the multi-channel (n ≥ 2) Gel'fand-Calderón inverse problem in two dimension. In addition, Theorem 1.1 is new also for the scalar case, as the estimate obtained in [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF] is weaker.

Instability estimates complementing the stability estimates of [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF], [START_REF] Novikov | New global stability estimates for the Gel'fand-Calderon inverse problem[END_REF], [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF] and of the present work are given in [START_REF] Mandache | Exponential instability in an inverse problem of the Schrödinger equation[END_REF], [START_REF] Isaev | Exponential instability in the Gel'fand inverse problem on the energy intervals[END_REF].

The proof of Theorem 1.1 is based on results obtained in [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF], [START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions[END_REF], which takes inspiration mostly from [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF] and [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF]. In particular, for z 0 ∈ D we use the existence and uniqueness of a family of solution ψ z 0 (z, λ) of equation (1.1) where in particular ψ z 0 → e λ(z-z 0 ) 2 I, for λ → ∞ (where I is the identity matrix). Then, using an appropriate matrix-valued version of Alessandrini's identity along with stationary phase techniques, we obtain the result. Note that this matrix-valued identity is one of the new results of this paper.

A generalization of Theorem 1.1 in the case where we do not assume that v j | ∂D = 0 and ∂ ∂ν v j | ∂D = 0 for j = 1, 2, is given in section 5.

This work was fulfilled in the framework of researches under the direction of R. G. Novikov.

Preliminaries

In this section we introduce and give details about the above-mentioned family of solutions of equation (1.1), which will be used throughout all the paper.

We identify R 2 with C and use the coordinates

z = x 1 + ix 2 , z = x 1 -ix 2 where (x 1 , x 2 ) ∈ R 2 . Let us define the function spaces C 1 z ( D) = {u : u, ∂u ∂ z ∈ C( D, M n (C))} with the norm u C 1 z ( D) = max( u C( D) , ∂u ∂ z C( D) ), where u C( D) = sup z∈ D |u| and |u| = max 1≤i,j≤n |u i,j |; we define also C 1 z ( D) = {u : u, ∂u ∂z ∈ C( D, M n (C))
} with an analogous norm. Following [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF], [START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions[END_REF], we consider the functions:

G z 0 (z, ζ, λ) = e λ(z-z 0 ) 2 g z 0 (z, ζ, λ)e -λ(ζ-z 0 ) 2 , (2.1)
g z 0 (z, ζ, λ) = e λ(ζ-z 0 ) 2 -λ( ζ-z 0 ) 2 4π 2 D e -λ(η-z 0 ) 2 + λ(η-z 0 ) 2 (z -η)(η -ζ) dReη dImη, (2.2)
ψ z 0 (z, λ) = e λ(z-z 0 ) 2 µ z 0 (z, λ), (2.3) µ z 0 (z, λ) = I + D g z 0 (z, ζ, λ)v(ζ)µ z 0 (ζ, λ)dReζ dImζ, (2.4) h z 0 (λ) = D e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 v(z)µ z 0 (z, λ)dRez dImz, (2.5)
where z, z 0 , ζ ∈ D and λ ∈ C and I is the identity matrix. In addition, equation (2.4) at fixed z 0 and λ, is considered as a linear integral equation for

µ z 0 (•, λ) ∈ C 1 z ( D). The functions G z 0 (z, ζ, λ), g z 0 (z, ζ, λ), ψ z 0 (z, λ
), µ z 0 (z, λ) defined above, satisfy the following equations (see [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF], [START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions[END_REF]):

4 ∂ 2 ∂z∂ z G z 0 (z, ζ, λ) = δ(z -ζ), (2.6) 4 ∂ 2 ∂ζ∂ ζ G z 0 (z, ζ, λ) = δ(ζ -z), (2.7) 4 ∂ ∂z + 2λ(z -z 0 ) ∂ ∂ z g z 0 (z, ζ, λ) = δ(z -ζ), (2.8) 4 ∂ ∂ ζ ∂ ∂ζ -2λ(ζ -z 0 ) g z 0 (z, ζ, λ) = δ(ζ -z), (2.9) -4 ∂ 2 ∂z∂ z ψ z 0 (z, λ) + v(z)ψ z 0 (z, λ) = 0, (2.10) -4 ∂ ∂z + 2λ(z -z 0 ) ∂ ∂ z µ z 0 (z, λ) + v(z)µ z 0 (z, λ) = 0, (2.11)
where z, z 0 , ζ ∈ D, λ ∈ C, δ is the Dirac's delta. (In addition, it is assumed that (2.4) is uniquely solvable for µ z 0 (•, λ) ∈ C 1 z ( D) at fixed z 0 and λ.) We say that the functions G z 0 , g z 0 , ψ z 0 , µ z 0 , h z 0 are the Bukhgeim-type analogues of the Faddeev functions (see [START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions[END_REF]). Now we state some fundamental lemmata. Let (2.12)

g z 0 ,λ u(z) = D g z 0 (z, ζ, λ)u(ζ)dReζ dImζ, z ∈ D, z 0 , λ ∈ C, where g z 0 (z, ζ, λ) is defined by (2.
2) and u is a test function.

Lemma 2.1 ([11]

). Let g z 0 ,λ u be defined by (2.12). Then, for z 0 , λ ∈ C, the following estimates hold:

g z 0 ,λ u ∈ C 1 z ( D), for u ∈ C( D), (2.13) g z 0 ,λ u C 1 ( D) ≤ c 1 (D, λ) u C( D) , for u ∈ C( D), (2.14) g z 0 ,λ u C 1 z ( D) ≤ c 2 (D) |λ| 1 2 u C 1 z ( D) , for u ∈ C 1 z ( D), |λ| ≥ 1. (2.15) Given a potential v ∈ C 1 z ( D) we define the operator g z 0 ,λ v simply as (g z 0 ,λ v)u(z) = g z 0 ,λ w(z), w = vu, for a test function u. If u ∈ C 1 z ( D), by Lemma 2.1 we have that g z 0 ,λ v : C 1 z ( D) → C 1 z ( D), (2.16) g z 0 ,λ v op C 1 z ( D) ≤ 2n g z 0 ,λ op C 1 z ( D) v C 1 z ( D) ,
where (2.16) and Lemma 2.1 imply existence and uniqueness of µ z 0 (z, λ) (and thus also [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF]). For v ∈ C 1 z ( D) such that v| ∂D = 0 the following formula holds:

• op C 1 z ( D) denotes the operator norm in C 1 z ( D), z 0 , λ ∈ C. In addition, g z 0 ,λ op C 1 z ( D) is estimated in Lemma 2.1. Inequality
ψ z 0 (z, λ)) for |λ| > ρ(D, K, n), where v C 1 z ( D) < K. Let µ (k) z 0 (z, λ) = k j=0 (g z 0 ,λ v) j I, h (k) z 0 (λ) = D e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 v(z)µ (k) z 0 (z, λ)dRez dImz, where z, z 0 ∈ D, λ ∈ C, k ∈ N ∪ {0}. Lemma 2.2 ([
(2.17) v(z 0 ) = 2 π lim λ→∞ |λ|h (0) z 0 (λ), z 0 ∈ D. In addition, if v ∈ C 2 ( D), v| ∂D = 0 and ∂v ∂ν | ∂D = 0 then (2.18) v(z 0 ) - 2 π |λ|h (0) z 0 (λ) ≤ c 3 (D, n) log(3|λ|) |λ| v C 2 ( D) , for z 0 ∈ D, λ ∈ C, |λ| ≥ 1.
Let

W z 0 (λ) = D e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 w(z)dRe zdIm z,
where z 0 ∈ D, λ ∈ C and w is some M n (C)-valued function on D. (One can see that [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF]). For w ∈ C 1 z ( D) the following estimate holds:

W z 0 = h (0) z 0 for w = v.) Lemma 2.3 ([
|W z 0 (λ)| ≤ c 4 (D) log (3|λ|) |λ| w C 1 z ( D) , z 0 ∈ D, |λ| ≥ 1. (2.19) Lemma 2.4 ([12]). For v ∈ C 1 z ( D) and for g z 0 ,λ v op C 1 z ( D) ≤ δ < 1 we have that µ z 0 (•, λ) -µ (k) z 0 (•, λ) C 1 z ( D) ≤ δ k+1 1 -δ , (2.20) |h z 0 (λ) -h (k) z 0 (λ)| ≤ c 5 (D, n) log(3|λ|) |λ| δ k+1 1 -δ v C 1 z ( D) , (2.21) where z 0 ∈ D, λ ∈ C, |λ| ≥ 1, k ∈ N ∪ {0}.
The proofs of Lemmata 2.1-2.4 can be found in the references given. We will also need the following two new lemmata. Lemma 2.5. Let g z 0 ,λ u be defined by (2.12), where u ∈ C 1 z ( D), z 0 , λ ∈ C. Then the following estimate holds:

g z 0 ,λ u C( D) ≤ c 6 (D) log(3|λ|) |λ| u C 1 z ( D) , |λ| ≥ 1. (2.22) Lemma 2.6. The expression (2.23) W (u, v)(λ) = D e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 u(z)(g z 0 ,λ v)(z)dRez dImz, defined for u, v ∈ C 1 z ( D) with u C 1 z ( D) , v C 1 z ( D) ≤ N 1 , λ ∈ C, z 0 ∈ D, satisfies the estimate |W (u, v)(λ)| ≤ c 7 (D, N 1 , n) (log(3|λ|)) 2 |λ| 1+3/4 , |λ| ≥ 1. (2.24)
The proofs of Lemmata 2.5, 2.6 are given in section 4.

Proof of Theorem 1.1

We begin with a technical lemma, which will be useful to generalise Alessandrini's identity.

Lemma 3.1. Let v ∈ C 1 ( D, M n (C))
be a matrix-valued potential which satisfies condition (1.4) (i.e. 0 is not a Dirichlet eigeinvalue for the operator -∆ + v in D). Then t v, the transpose of v, also satisfies condition (1.4).

The proof of Lemma 3.1 is given in section 4. We can now state and prove a matrix-valued version of Alessandrini's identity (see [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF] for the scalar case).

(-∆ + v 1 )u 1 = 0, (-∆ + t v 2 )u 2 = 0 on D,
where t A stand for the transpose of A. Then we have the identity

(3.1) ∂D t u 2 (z)(Φ 2 -Φ 1 )u 1 (z)|dz| = D t u 2 (z)(v 2 (z) -v 1 (z))u 1 (z)dRez dImz. Proof. If v ∈ C 1 ( D, M n (C)) is any matrix-valued potential (which satisfies (1.4)) and f 1 , f 2 ∈ C 1 (∂D, M n (C)) then we have (3.2) ∂D t f 2 Φf 1 |dz| = ∂D t t f 1 Φ * f 2 |dz|,
where Φ and Φ * are the Dirichlet-to-Neumann operators associated to v and t v, respectively (these operators are well-defined thanks to Lemma 3.1). Indeed, it is sufficient to extend f 1 and f 2 in D as the solutions of the Dirichlet problems (-∆ + v) f1 = 0, (-∆ + t v) f2 = 0 on D and fj | ∂D = f j , for j = 1, 2, so that one obtains

∂D t f 2 Φf 1 -t t f 1 Φ * f 2 |dz| = ∂D t f 2 ∂ f1 ∂ν -t ∂ f2 ∂ν f 1 |dz| = D t f2 ∆ f1 -t ∆ f2 f1 dRez dImz = D t f2 v f1 -t t v f2 f1 dRez dImz = 0,
where for the second equality we used the following matrix-valued version of the classical scalar Green's formula:

(3.3) ∂D t ∂f ∂ν g -t f ∂g ∂ν |dz| = D t (∆f ) g -t f ∆g dRez dImz, for any f, g ∈ C 2 (D, M n (C)) ∩ C 1 ( D, M n (C)). Identities (3.2) and (3.3) imply ∂D t u 2 (z)(Φ 2 -Φ 1 )u 1 (z)|dz| = ∂D t t u 1 (z)Φ * 2 u 2 (z) -t u 2 (z)Φ 1 u 1 (z) |dz| = ∂D t ∂u 2 (z) ∂ν u 1 (z) -t u 2 (z) ∂u 1 (z) ∂ν |dz| = D t (∆u 2 (z)) u 1 (z) -t u 2 (z)∆u 1 (z) dRez dImz = D t t v 2 (z) u 2 (z) u 1 (z) -t u 2 (z) v 1 (z) u 1 (z) dRez dImz = D t u 2 (z)(v 2 (z) -v 1 (z))u 1 (z)dRez dImz.
Now let μz 0 denote the complex conjugated of µ z 0 (the solution of (2.4)) for a M n (R)-valued potential v and, more generally, the solution of (2.4) with g z 0 (z, ζ, λ) replaced by g z 0 (z, ζ, λ) for a M n (C)-valued potential v. In order to make use of (3.1) we define

u 1 (z) = ψ 1,z 0 (z, λ) = e λ(z-z 0 ) 2 µ 1 (z, λ), u 2 (z) = ψ 2,z 0 (z, -λ) = e -λ(z-z 0 ) 2 μ2 (z, -λ),
for z 0 ∈ D, λ ∈ C, |λ| > ρ (ρ is mentioned in section 2), where we called for simplicity µ 1 = µ 1,z 0 , µ 2 = µ 2,z 0 and µ 1,z 0 , µ 2,z 0 are the solutions of (2.4) with v replaced by v 1 , t v 2 , respectively. Equation (3.1), with the above-defined u 1 , u 2 , now reads

∂D ∂D e -λ(z-z 0 ) 2 t μ2 (z, -λ)(Φ 2 -Φ 1 )(z, ζ)e λ(ζ-z 0 ) 2 µ 1 (ζ, λ)|dζ||dz| (3.4) = D e λ,z 0 (z) t μ2 (z, -λ)(v 2 -v 1 )(z)µ 1 (z, λ)dRez dImz. with e λ,z 0 (z) = e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 and (Φ 2 -Φ 1 )(z, ζ) is the Schwartz kernel of the operator Φ 2 -Φ 1 .
The right side I(λ) of (3.4) can be written as the sum of four integrals, namely

I 1 (λ) = D e λ,z 0 (z)(v 2 -v 1 )(z)dRez dImz, I 2 (λ) = D e λ,z 0 (z) t (μ 2 -I)(v 2 -v 1 )(z)(µ 1 -I)dRez dImz, I 3 (λ) = D e λ,z 0 (z) t (μ 2 -I)(v 2 -v 1 )(z) dRez dImz, I 4 (λ) = D e λ,z 0 (z) (v 2 -v 1 )(z)(µ 1 -I)dRez dImz, for z 0 ∈ D.
The first term, I 1 , can be estimated using Lemma 2.2 as follows:

2 π |λ|I 1 -(v 2 (z 0 ) -v 1 (z 0 )) ≤ c 3 (D, n) log(3|λ|) |λ| v 2 -v 1 C 2 ( D) , (3.5)
for |λ| ≥ 1. The other terms, I 2 , I 3 , I 4 , satisfy, by Lemmata 2.1 and 2.4,

|I 2 | ≤ D e λ,z 0 (z) t (g z 0 ,λ t v 2 )(v 2 -v 1 )(z)(g z 0 ,λ v 1 )dRez dImz (3.6) + O log(3|λ|) |λ| 2 c 8 (D, N, n), |I 3 | ≤ D e λ,z 0 (z) t (g z 0 ,λ t v 2 )(v 2 -v 1 )(z)dRez dImz (3.7) + O log(3|λ|) |λ| 2 c 9 (D, N, n), |I 4 | ≤ D e λ,z 0 (z) (v 2 -v 1 )(z)(g z 0 ,λ v 1 )dRez dImz (3.8) + O log(3|λ|) |λ| 2 c 10 (D, N, n),
where N is the costant in the statement of Theorem

|I 2 | ≤ c 11 (D, N, n) (log(3|λ|)) 2 |λ| 2 , (3.10) |I 3 | ≤ c 12 (D, N, n) (log(3|λ|)) 2 |λ| 1+3/4 , (3.11) |I 4 | ≤ c 13 (D, N, n) (log(3|λ|)) 2 |λ| 1+3/4 . (3.12)
The left side J(λ) of (3.4) can be estimated as follows:

|λ||J(λ)| ≤ c 14 (D, n)e (2L 2 +1)|λ| Φ 2 -Φ 1 , (3.13)
for λ which satisfies (3.9), and L = max z∈∂D, z 0 ∈D |zz 0 |.

Putting together estimates (3.5)-(3.13) we obtain

|v 2 (z 0 ) -v 1 (z 0 )| ≤ c 15 (D, N, n) (log(3|λ|)) 2 |λ| 3/4 + 2 π c 14 (D, n)e (2L 2 +1)|λ| Φ 2 -Φ 1 (3.14)
for any z 0 ∈ D. We call ε = Φ 2 -Φ 1 and impose |λ| = γ log(3 + ε -1 ), where 0 < γ < (2L 2 + 1) -1 so that (3.14) reads

|v 2 (z 0 ) -v 1 (z 0 )| ≤ c 15 (D, N, n)(γ log(3 + ε -1 )) -3 4 log(3γ log(3 + ε -1 )) 2 (3.15) + 2 π c 14 (D, n)(3 + ε -1 ) (2L 2 +1)γ ε, for every z 0 ∈ D, with (3.16) 0 < ε ≤ ε 1 (D, N, γ, n),
where ε 1 is sufficiently small or, more precisely, where (3.16) implies that |λ| = γ log(3 + ε -1 ) satisfies (3.9). As (3 + ε -1 ) (2L 2 +1)γ ε → 0 for ε → 0 more rapidly then the other term, we obtain that

v 2 -v 1 L ∞ (D) ≤ c 16 (D, N, γ, n) log(3 log(3 + Φ 2 -Φ 1 -1 )) 2 (log(3 + Φ 2 -Φ 1 -1 )) 3 4
(3.17)

for any ε = Φ 2 -Φ 1 ≤ ε 1 (D, N, γ, n).
Estimate (3.17) for general ε (with modified c 16 ) follows from (3.17) for ε ≤ ε 1 (D, N, γ, n) and the assumption that v j L ∞ (D) ≤ N, j = 1, 2. This completes the proof of Theorem 1.1.

4.

Proofs of Lemmata 2.5, 2.6, 3.1.

Proof of Lemma 2.5. We decompose the operator g z 0 ,λ , defined in (2.12), as the product 1 4 T z 0 ,λ Tz 0 ,λ , where

T z 0 ,λ u(z) = 1 π D e -λ(ζ-z 0 ) 2 + λ( ζ-z 0 ) 2 z -ζ u(ζ)dReζ dImζ, (4.1) Tz 0 ,λ u(z) = 1 π D e λ(ζ-z 0 ) 2 -λ( ζ-z 0 ) 2 z - ζ u(ζ)dReζ dImζ, (4.2)
for z 0 , λ ∈ C. From the proof of [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF]Lemma 3.1] we have the estimate

Tz 0 ,λ u C( D) ≤ η 1 (D) |λ| 1/2 u C( D) + η 2 (D) log(3|λ|) |λ| ∂u ∂ z C( D) , (4.3) for u ∈ C 1 z ( D), z 0 ∈ D, |λ| ≥ 1.
As the kernel of T z 0 ,λ and Tz 0 ,λ are conjugated each other we deduce immediately

T z 0 ,λ u C( D) ≤ η 1 (D) |λ| 1/2 u C( D) + η 2 (D) log(3|λ|) |λ| ∂u ∂z C( D) , |λ| ≥ 1, (4.4) for u ∈ C 1 z ( D).
Combining the two estimates we obtain

g λ,z 0 u C( D) = 1 4 T z 0 ,λ Tz 0 ,λ u C( D) ≤ 1 4 η 1 (D) Tz 0 ,λ u C( D) |λ| 1/2 + η 2 (D) log(3|λ|) |λ| ∂ ∂z Tz 0 ,λ u C( D) ≤ η 3 (D) u C( D) |λ| + log(3|λ|) |λ| 3/2 ∂u ∂ z C( D) + log(3|λ|) |λ| u C( D) ≤ η 4 (D) log(3|λ|) |λ| u C 1 z ( D) , |λ| ≥ 1,
where we used the fact that

∂ ∂z Tz 0 ,λ u C(D) = u C(D) .
Proof of Lemma 2.6. For 0

< ε ≤ 1, z 0 ∈ D, let B z 0 ,ε = {z ∈ C : |z -z 0 | ≤ ε}. We write W (u, v)(λ) = W 1 (λ) + W 2 (λ),
where

W 1 (λ) = D∩Bz 0 ,ε e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 u(z)g z 0 ,λ v(z)dRez dImz, W 2 (λ) = D\Bz 0 ,ε e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 u(z)g z 0 ,λ v(z)dRez dImz.
The first term, W 1 , can be estimated as follows:

|W 1 (λ)| ≤ σ 1 (D, n) u C( D) v C 1 z ( D) ε 2 log(3|λ|) |λ| , |λ| ≥ 1, (4.5)
where we used estimates (2.16) and (2.22).

For the second term, W 2 , we proceed using integration by parts, in order to obtain

W 2 (λ) = 1 4i λ ∂(D\Bz 0 ,ε) e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 u(z)g z 0 ,λ v(z) z -z0 dz - 1 2 λ D\Bz 0 ,ε e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 ∂ ∂ z u(z)g z 0 ,λ v(z) z -z0 dRez dImz.
This imply

|W 2 (λ)| ≤ 1 4|λ| ∂(D\Bz 0 ,ε) u(z)g z 0 ,λ v(z) C( D) |z -z0 | |dz| (4.6) + 1 2|λ| D\Bz 0 ,ε e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 ∂ ∂ z u(z)g z 0 ,λ v(z) z -z0 dRez dImz ,
for λ = 0. Again by estimates (2.16) and (2.22) we obtain

|W 2 (λ)| ≤ σ 2 (D, n) u C 1 z ( D) v C 1 z ( D) log(3ε -1 ) log(3|λ|) |λ| 2 (4.7) + 1 8|λ| D\Bz 0 ,ε u(z) Tz 0 ,λ v(z) z -z0 dRez dImz , |λ| ≥ 1,
where we used the fact that ∂ ∂ z g z 0 ,λ v(z) = 1 4 e -λ(z-z 0 ) 2 + λ(z-z 0 ) 2 Tz 0 ,λ v(z), with Tz 0 ,λ defined in (4.2).

The last term in (4.7) can be estimated independently on ε by (4.8)

σ 3 (D, n) u C( D) v C 1 z ( D) log(3|λ|) |λ| 1+3/4 .
This is a consequence of (4.3) and of the estimate (4.9)

| Tz 0 ,λ u(z)| ≤ log(3|λ|)(1 + |z -z 0 |)τ 1 (D) |λ||z -z 0 | 2 u C 1 z ( D) , |λ| ≥ 1,
for u ∈ C 1 z ( D), z, z 0 ∈ D (a proof of (4.9) can be found in the proof of [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF]Lemma 3.1]). Indeed, for 0 < δ ≤ 1 2 we have

D u(z) Tz 0 ,λ v(z) z -z0 dRez dImz ≤ B z 0 ,δ ∩D |u(z)| | Tz 0 ,λ v(z)| |z -z 0 | dRez dImz + D\B z 0 ,δ |u(z)| | Tz 0 ,λ v(z)| |z -z 0 | dRez dImz ≤ u C( D) v C 1 z ( D) τ 2 (D, n) |λ| 1/2 B z 0 ,δ ∩D dRez dImz |z -z 0 | + u C( D) v C 1 z ( D) log(3|λ|) |λ| τ 3 (D, n) D\B z 0 ,δ dRez dImz |z -z 0 | 3 ≤ 2π u C( D) v C 1 z ( D) τ 2 (D, n) δ |λ| 1 2 + u C( D) v C 1 z ( D) τ 4 (D, n) log(3|λ|) |λ|δ ,
for |λ| ≥ 1. Putting δ = 1 2 |λ| -1/4 in the last inequality gives (4.8). Finally, defining ε = |λ| -1/2 in (4.7), (4.5) and using (4.8), we obtain the main estimate (2.24), which thus finishes the proof of Lemma 2.6.

Proof of Lemma 3.1. Take u ∈ H 1 (D, M n (C)) such that (-∆ + t v)u = 0 on D and u| ∂D = 0. We want to prove that u ≡ 0 on D.

By our hypothesis, for any f ∈ C 1 (∂D, M n (C)) there exists a unique f ∈ H 1 (D, M n (C)) such that (-∆ + v) f = 0 on D and f | ∂D = f . Thus we have, using Green's formula (3.3)

, ∂D t ∂u ∂ν f |dz| = D t (∆u) f -t u∆ f dRez dImz = D t t v u f -t u v f dRez dImz = 0
which yields ∂u ∂ν | ∂D = 0. Now consider the following straightforward generalization of Green's formula (3.3)

, ∂D t ∂f ∂ν g -t f ∂g ∂ν |dz| = D t (∆ -t v)f g -t f ((∆ -v)g) dRez dImz, (4.10

An extension of Theorem 1.1

As an extension of Theorem 1.1 for the case when we do not assume that v j | ∂D ≡ 0, ∂ ∂ν v j | ∂D ≡ 0, j = 1, 2, we give the following result.

(5.1) v 2 -v 1 L ∞ (D) ≤ C log(3 + Φ 2 -Φ 1 - 1
1 ) -α ,
where A 1 is the norm for an operator A :

L ∞ (∂D, M n (C)) → L ∞ (∂D, M n (C)), with kernel A(x, y), defined as A 1 = sup x,y∈∂D |A(x, y)|(log(3+|x-y| -1 )) -1 and |A(x, y)| = max 1≤i,j≤n |A i,j (x, y)|.
The only properties of 1 we will use are the following:

i) A L ∞ (∂D)→L ∞ (∂D) ≤ const(D, n) A 1 ;
ii) In a similar way as in formula (4.9) of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF] one can deduce

v L ∞ (∂D) ≤ const(n) Φ v -Φ 0 1 ,
for a matrix-valued potential v, Φ v its associated Dirichlet-to-Neumann operator and Φ 0 the Dirichlet-to-Neumann operator of the 0 potential.

We recall a lemma from [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF], which generalize Lemma 2.2 to the case of potentials without boundary conditions. We define (∂D) δ = {z ∈ C : dist(z, ∂D) < δ}. The proof of Lemma 5.2 for the scalar case can be found in [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF]: the generalization to the matrix-valued case is straightforward.

Proof of Proposition 5.1. Fix 0 < α < 1 5 and 0 < δ < 1. We have the following chain of inequalities

v 2 -v 1 L ∞ (D) = max( v 2 -v 1 L ∞ (D∩(∂D) δ ) , v 2 -v 1 L ∞ (D\(∂D) δ ) ) ≤ C 1 max 2N δ + Φ 2 -Φ 1 1 , log(3 log(3 + Φ 2 -Φ 1 - 1
))

δ 4 log(3 + Φ 2 -Φ 1 -1 ) + log(3 + 1 δ ) Φ 2 -Φ 1 1 + log(3 log(3 + Φ 2 -Φ 1 -1 )) 2 (log(3 + Φ 2 -Φ 1 -1 )) 3 4 ≤ C 2 max 2N δ + Φ 2 -Φ 1 1 , 1 δ 4 log(3 + Φ 2 -Φ 1 - 1 1 )
-5α

+ log(3 + 1 δ ) Φ 2 -Φ 1 1 + log(3 log(3 + Φ 2 -Φ 1 -1 1
))

2 (log(3 + Φ 2 -Φ 1 -1 1 )) 3 4
, where we followed the scheme of the proof of Theorem 1.1 with the following modifications: we made use of Lemma 5.2 instead of Lemma 2.2 and we also used i)-ii); note that C 1 = C 1 (D, N, n) and C 2 = C 2 (D, N, n, α).

Putting δ = log(3 + Φ 2 -Φ 1 - 1
1 ) -α we obtain the desired inequality

v 2 -v 1 L ∞ (D) ≤ C 3 log(3 + Φ 2 -Φ 1 -1 1 ) -α , (5.3)
with C 3 = C 3 (D, N, n, α), Φ 2 -Φ 1 1 = ε ≤ ε 1 (D, N, n, α) with ε 1 sufficiently small or, more precisely when δ 1 = log(3 + ε -1 1 ) -α satisfies:

δ 1 < 1, ε 1 ≤ 2N δ 1 , log(3 + 1 δ 1 )ε 1 ≤ δ 1 .
Estimate (5.3) for general ε (with modified C 3 ) follows from (5.3) for ε ≤ ε 1 (D, N, n, α) and the assumption that v j L ∞ ( D) ≤ N for j = 1, 2. This completes the proof of Proposition 5.1.

Let

  D be an open bounded domain in R 2 with C 2 boundary and v ∈ C 1 ( D, M n (C)), where M n (C) is the set of the n × n complex-valued matrices. The Dirichlet-to-Neumann map associated to v is the operator Φ : C 1 (∂D, M n (C)) → L p (∂D, M n (C)), p < ∞ defined by: (1.2) Φ(f ) = ∂ψ ∂ν ∂D where f ∈ C 1 (∂D, M n (C)), ν is the outer normal of ∂D and ψ is the H 1 ( D, M n (C))-solution of the Dirichlet problem (1.3) -∆ψ + v(x)ψ = 0 on D, ψ| ∂D = f ; here we assume that (1.4) 0 is not a Dirichlet eigenvalue for the operator -∆ + v in D.

Lemma 3 . 2 .

 32 Let v 1 , v 2 ∈ C 1 ( D, M n (C)) be two matrix-valued potentials which satisfy (1.4), Φ 1 , Φ 2 their associated Dirichlet-to-Neumann operators, respectively, and u 1 , u 2 ∈ C 2 ( D, M n (C)) matrix-valued functions such that

  ) which holds (weakly) for any f, g ∈ H 1 (D, M n (C)). If we put f = u we obtain (4.11) D t u (-∆ + v)g dRez dImz = 0 for any g ∈ H 1 (D, M n (C)). By Fredholm alternative (see [5, Sec. 6.2]), for each h ∈ L 2 (D, M n (C)) there exists a unique g ∈ H 1 0 (D, M n (C)) = {g ∈ H 1 (D, M n (C)) : g| ∂D = 0} such that (-∆ + v)g = h: this yields u ≡ 0 on D. Thus Lemma 3.1 is proved.

Proposition 5 . 1 .

 51 Let D ⊂ R 2 be an open bounded domain with C 2 boundary, let v 1 , v 2 ∈ C 2 ( D, M n (C)) be two matrix-valued potentials which satisfy (1.4), with v j C 2 ( D) ≤ N for j = 1, 2, and Φ 1 , Φ 2 the corresponding Dirichlet-to-Neumann operators. Then, for any 0 < α < 1 5 , there exists a constant C = C(D, N, n, α) such that

Lemma 5 . 2 .

 52 For v ∈ C 2 ( D) we have that v(z 0 ) -2 π |λ|h (0) z 0 (λ) ≤ κ 1 (D, n)δ -4 log(3|λ|) |λ| v C 2 ( D) (5.2) + κ 2 (D, n) log(3 + δ -1 ) v C(∂D) , for z 0 ∈ D \ (∂D) δ , 0 < δ < 1, λ ∈ C, |λ| ≥ 1.


Trung Quang
email: quang-trung.le@enssat.fr

Laurent Le

Mathilde Bramerie

Sebastien Gay

Michel Lobo

Jean-Claude Joindot

Simon

Trung Hoang

Jean-Louis Nguyen

Oudar

Le

Laurent Bramerie

Mathilde Gay

Sebastien Lobo

Michel Joindot

Jean-Claude Simon

Hoang-Trung Nguyen

Jean-Louis Oudar
email: jean-louis.oudar@lpn.cnrs.fr

Cascadability assessment of a microcavity-saturable-absorber based phase-preserving amplitude regenerator in a DPSK transmission system

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

The differential phase-shift keying (DPSK) modulation format has recently attracted much attention for high-speed and ultra long-haul optical transmission. It offers, for instance, a 3-dB improvement in receiver sensitivity compared to OOK (with balanced detection) and an enhanced tolerance to dispersion and nonlinear effects 1 .

All-optical regeneration that aims to enhance the signal quality would be attractive to improve the performance of DPSK systems. In the literature, some interesting solutions for phase regeneration based on phase-sensitive interferometric techniques have been proposed [START_REF] Croussore | Proc. OFC'06[END_REF][START_REF] Devgan | Proc. OFC'05[END_REF] . However, those techniques are rather complex, and have instability issues. Another approach consists in phase-preserving amplitude regeneration that can prevent the accumulation of nonlinear phase noise (NLPN) during transmission [START_REF] Boscolo | [END_REF] . The cascadability of the phase-preserving amplitude regenerators based on four-wave mixing in fibre [START_REF] Matsumoto | Proc. ECOC'06[END_REF] and on nonlinear amplifier loop mirror [START_REF] Stephan | Proc. ECOC'09[END_REF] are experimentally demonstrated in the literature. Both use shot-pulse RZ DPSK signals at 10 Gbit/s.

Recently, we have shown that a novel microcavity saturable absorber (SA1) [START_REF] Nguyen | [END_REF] can provide phase-preserving amplitude noise reduction, and therefore, can prevent from the accumulation of NLPN in a RZ DPSK transmission system [START_REF] Le | Proc. OFC'10[END_REF] . The microcavity saturable absorber consists of a passive nonlinear mirror which promises a simple, compact and WDM compatible solution [START_REF] Bramerie | Proc. OFC'07[END_REF] .

In this paper, we report on the cascadability assessment of the SA1 based phase-preserving amplitude regenerator in a recirculating loop using conventional RZ33 DPSK at 42.7 Gbit/s. The numerical and experimental results demonstrate that the tolerance of phase-encoded signals to NLPN is considerably increased thanks to the re- generator. An improvement of bit error rate and of transmission distance can be achieved.

Numerical simulation

For a complete regeneration of DPSK signals, both phase and amplitude regenerations are needed. However, we illustrate in this section that when SA1 based phase-preserving amplitude regenerators are used in the transmission line, a reduction of SPM-induced NLPN at the receiver can be obtained. Indeed, by strongly reducing the amplitude noise, the SA1 prevents the accumulation of NLPN in the next transmission spans.

The transmission link is modelled as the concatenation of 100 km amplification spans composed of non-zero dispersion-shifted fibres (NZ DSF) followed by a dispersion-compensatingfiber (DCF) (Fig. 1). The NZ DSF has a dispersion of 4.5 ps.nm -1 .km -1 , a nonlinear coefficient of 2 W -1 .km -1 , and an attenuation of 0.23 dB/km. The total span loss is 26 dB, which is compensated for by an EDFA with noise figure of 4.5 dB. The chromatic dispersion is perfectly compensated, and the nonlinearities occur only in the transmission fibre. A phase-preserving amplitude regenerator based on SA1 is inserted every M spans.

The SA1 model is based on the same rate equation as for the classical saturable absorber [START_REF] Haus | [END_REF] but with an inverted sign of the last term (which is directly dependent on the input power) in order to obtain a decreasing reflectance in response to an increasing input power: dα(t, P ) dt = α 0 -α(t, P ) τ + αP τ P sat P is the input power, α is the device's absorption, α 0 is the small signal absorption, P sat is the saturation power, and τ is the recovery time. The parameters used in the simulation are P sat =7 mW, τ =3.5 ps and α 0 =5.3, where d is the total quantum-well section thickness. Those parameters are representative of the investigated device for the experimental studies.

The rms variations of the amplitude and differential phase (on the zero and π rails) for RZ DPSK signals transmitted through the system in the cases with and without SA1-based amplitude regenerators are plotted in Fig. 2. The fibre launched power is set at 4 dBm. The regenerators are located after every 200 km of transmission. As can be seen, in the absence of regenerators, both the amplitude and phase jitters increase continuously which can become critical for the system performance. In the case with regenerators, the amplitude noise is considerably reduced at each regenerator location. After the first regenerator, the slope of differential phase jitters with transmission distance is reduced. At 1000 km, the rms phase jitter is reduced by 40% compared to the non regenerated case. As a qualitative estimate of the transmission performance of the system, we use here the differential phase Q-factor (defined as π divided by the sum of the rms variations of the differential phase between adjacent bits on the zero and π rails) 11 , which provides a measure of the phase fluctuation in the signal. Fig. 3 shows the phase Q-factor in dB (20.log10(Q)) after 1000 km of transmission with and without SA1 versus fibre launched power. In the linear regime at low fibre launched power, the system performance is limited by OSNR degradation. No improvement by the regenerator is observed, as it preserves the signal phase. At high fibre launched power, Qfactor is improved. An increase of nonlinear tolerance is thus obtained thanks to the regenerator. This Q-factor margin can be used for increasing the transmission distance, which will be demonstrated experimentally in the next sections.

Experimental setup

The experimental setup is shown in Fig. 4. The transmitter based on Mach-Zehnder modulators generates an 8-ps-pulse-width RZ DPSK signal at 42.7 Gbit/s. The signal wavelength is centred at 1550 nm. An ASE source followed by a 3-nm band-pass optical filter, followed by a variable optical attenuator (ATT) is used in order to modify the OSNR at the transmitter output.

The signal is boosted by EDFA1 before being sent to SA1 via an optical circulator (OC). The transmission line consists of 100 km of nonzero dispersion-shifted fibre (NZ DSF) with chromatic dispersion of 4.5 ps/km/nm at 1550 nm, followed by a dispersion-compensating fibre (DCF). EDFA2 increases the launched power up to 13 dBm, while EDFA3 compensates for the residual loss. A gain equalizer is required to compensate the signal spectrum distortion due to the device's resonance. The preamplified RZ DPSK receiver consists of a fibre-based delay-line interferometer (DLI) and a balanced detector.

Experimental results

Figure 5 shows the measured bit error rate (BER) versus transmission distance with and without SA1 based amplitude regeneration. The OSNR at the transmitter (called OSNR T x ) is 22.7 dB (over 1 nm) and the fibre launched power is 13 dBm. This high value is unrealistic, but is required, in our experiment, to generate a high enough amount of NLPN in 100 km of transmission fibre. For longer distance between regenerators, the BER curves are shifted towards lower input power values since less launched power is needed for the same amount of total nonlinear phase noise in the system. In the case without SA1, the BER grows rapidly due to linear and nonlinear phase noise accumulation. When the phase-preserving amplitude regenerator is applied, the amplitude noise is reduced and the nonlinear phase noise is partly removed. As a consequence, the transmission distance is enhanced thanks to the regenerator. A transmission distance of 800 km is reached for a BER of 10 -4 compared to 500 km in the case without regeneration, a distance improvement ratio of 1.6 is thus obtained.

We define the distance improvement ratio (DIR) as the ratio of distances covered with and without regeneration for a given BER. Fig. 6 presents the evolution of DIR versus OSNR T x for a BER of 10 -4 and with fibre launched powers of 10 dBm and 13 dBm. At the first glance, the results show that the DIR generally exceeds 1.2 for all OSNR T x values. As can be seen, a difference in regeneration efficiency is shown when the fibre launched power varies from 10 dBm to 13 dBm. This can be explained by the fact that, at higher fibre launched power (13 dBm), the nonlinear effects that convert amplitude noise into phase noise are more efficient. Consequently, better regenerator efficiency is obtained (distance improvement ratio of 1.4 to 1.6 compared to 1.2 at 10 dBm). The greater the fibre launched power, the more efficient the nonlinear effects, and the better the distance improvement.

When OSNR T x increases the obtained DIR is slightly better. For a fibre launched power of 13 dBm, a DIR of 1.4 is obtained at low OSNR T x (13.8 dB), and this improvement ratio is 1.6 at high OSNR T x (22.7 dB). At a fibre launched power of 10 dBm, this evolution is less visible because the nonlinear effects are less efficient.

Conclusions

Cascaded performance of a phase-preserving amplitude regenerator based on a microcavity saturable absorber has been numerically and experimentally investigated in a RZ-DPSK transmission system at 42.7 Gbit/s via a recirculating loop setup. The obtained results show that the regenerator reduced the amplitude noise, which is the origin of nonlinear phase noise, thus improve the system performance. The best regenerator efficiency is achieved at high fibre launched power where the nonlinear effects are significant. A distance improvement ratio up to 1.6 is experimentally obtained at bit error rate of 10 -4 .

Fig. 1 :

 1 Fig. 1: Scheme of a regenerated transmission.

3 Fig. 2 :

 32 Fig. 2: Amplitude and phase jitter vs. transmission distance with and without SA1.

Fig. 3 :

 3 Fig. 3: Phase Q-factor after 1000 km of transmission vs. fibre launched power.

Fig. 4 :

 4 Fig. 4: Recirculating loop experiment of cascaded microcavity-saturable-absorber-based regenerator.

Fig. 5 :

 5 Fig. 5: BER vs. transmission distance.

Fig. 6 :

 6 Fig. 6: Distance improvement ratio at BER of 10 -4 with different fibre launched powers and transmitter OSNR.

Mo.2.F.3


Yousra Ben

M ' Sallem

Trung Quang
email: quang-trung.le@enssat.fr

Laurent Le

Bramerie

Thai Quoc

Eric Nguyen

Pascal Borgne

Sophie La Besnard

Leslie A Rochelle

Jean-Claude Rusch
email: rusch@gel.ulaval.ca

Simon

Quantum-dash mode-locked laser source for wavelength-tunable 56 Gbit/s DQPSK

Quantum-Dash Mode-Locked Laser Source for

Wavelength-Tunable 56 Gbit/s DQPSK Yousra Ben M'Sallem [START_REF] Healy | Proc. ECOC'07[END_REF][START_REF] Zeller | Proc. CLEO'[END_REF] , Quang Trung Le [START_REF] Healy | Proc. ECOC'07[END_REF] , Laurent Bramerie [START_REF] Healy | Proc. ECOC'07[END_REF] , Quoc-Thai Nguyen [START_REF] Healy | Proc. ECOC'07[END_REF] , Eric Borgne [START_REF] Healy | Proc. ECOC'07[END_REF] , Pascal Besnard [START_REF] Healy | Proc. ECOC'07[END_REF] , Sophie LaRochelle [START_REF] Zeller | Proc. CLEO'[END_REF] , Leslie A. Rusch [START_REF] Zeller | Proc. CLEO'[END_REF] , Jean-Claude Simon

Introduction

Since the phenomenal traffic growth of the Internet, data traffic has rapidly overtaken voice as the dominant information type transported by today's high speed communication networks. The next generation of optical networks requires efficient densely packed WDM channels to transmit massive amounts of information. An important challenge is the cost and the wavelength versatility of WDM transmitters.

WDM transmission using comb generation in a mode-locked laser (MLL) has been proposed as a cost-effective solution. The multiple frequencies (modes) of the MLLs offer several advantages compared to sets of single mode distributed feedback (DFB) lasers: fixed channel spacing set by the MLL mode spacing, reduced channel crosstalk due to the coherence of the MLL modes, etc. 1 In addition, no individual channel frequency control is needed.

Optical frequency combs have been demonstrated using different configurations based on mode locked lasers. Examples include amplitude or frequency modulated mode-locked lasers, e.g., Er:Yb:glass laser oscillators 2 , fibre ring lasers 3 or mode-locked semiconductor lasers 4 . Semiconductor lasers consume significantly less power as no pump diode laser is required. Recently, semiconductor quantum-dot or quantum-dash mode-locked lasers have been heavily investigated owing to remarkable properties, including low-noise, high thermal stability, and a broad gain spectrum 5 . They have been examined as particularly attractive sources for wavelength division multiplexing (WDM) using on-off keying (OOK) 6 . Phase-shift keying (PSK) has recently emerged as an alternative to conventional OOK modulation allowing for much higher spectral efficiencies. Despite their relatively complex transmitter and receiver setups, PSK is a strong candidate for high data-rate and spectrally efficient DWDM systems 7 . In this paper, we demonstrate for the first time that a quantumdash mode-locked laser (QD-MLL) can be used as multi-wavelength laser for 56 Gbit/s DQPSK transmission. We demonstrate that relative intensity noise (RIN) and phase noise over all modes are compatible with PSK modulation. We quantify DQPSK performance via bit error rate (BER) measurements and compare performance to that of an external cavity laser (ECL) source.

QD-MLL characterisation

The laser is fabricated with an active core consisting of 6 InAs-based QD layers. A cavity length of 420 µm yields a mode spacing of 100 GHz; the device structure was reported in [START_REF] Akrout | Proc. ECOC'08[END_REF]. The optical spectrum of the laser is given in Fig. 1. Due to mode-locking, a wide spectrum is observed. The 3 dB optical bandwidth is about 8 nm, covering ten optical modes.

Fig. 2 shows the RIN measurement results of 1) a commercial ECL, 2) all QD-MLL modes taken together, and 3) each filtered QD-MLL mode (from 1548.3 nm to 1555.6 nm). Generally, the QD lasers have low intensity noise thanks to low amplified spontaneous emission (ASE)

5

. The RIN of the total of all modes is even lower than that of a commercial ECL. However, -70 when using a QD-MLL wavelength comb as a WDM source, the RIN for each filtered mode due to mode partition noise 8 . As shown in Fig. 2, the RIN floor of one filtered mode can be as high as -100 dBc/Hz for frequencies below 500 MHz, decreasing to -130 dBc/Hz at 10 GHz for the worst case observed. RIN is much higher than that of a commercial ECL (see Fig. 2). Fig. 3 summarizes the RIN (at 100 MHz) and power level of each QD-MLL mode examined. Not surprisingly, the RIN of the non-dominant modes are strong compared to the QD-MLL dominant modes. In the next section we assess the impact of this higher RIN level on BER performance for DQPSK.

Experimental setup

The experimental setup (Fig. 4) consists of a 56 Gbit/s DQPSK transmitter and receiver. A 100 GHz quantum-dash Fabry-Perot modelocked laser source is used at the transmitter. The output of the laser passes through an optical isolator and then an arrayed waveguide grating (AWG) wavelength demultiplexer (for this experiment we use a tunable filter).

The selected lasing mode is amplified using an EDFA and externally modulated using a DQPSK, LiNbO3 Mach-Zehnder interferometer (MZI). With the test equipment available, we generate two 2 9 -1 pseudo random binary sequences (PRBS) delayed by 256 bits to decorrelate them. At the receiver the signal is preamplified, filtered and demodulated with delay interferometers (DLI). The DIs and balanced detectors convert the optical phase modulation into intensity modulation. The 28 Gbit/s bit stream passes through a 1:4 electronic demultiplexing stage; the 7 Gbit/s streams are sent to the error detector. The experimental test bed was implemented on the Persyst platform at Foton laboratory.

Experimental results

Results of DQPSK back-to-back measurements are shown in Fig. 5. For reference, the curve with square markers indicates BER vs. receiver input power for a DFB; BER is 10 -9 at -26 dBm. BER for an ECL source is reported with circle markers. Using one mode of the QD-MLL as a source yields BER reported with triangle mark-

Fig. 1 :

 1 Fig. 1: Optical spectrum of the QD-MLL.

Fig. 2 :

 2 Fig. 2: Relative intensity noise (RIN).

Fig. 3 :

 3 Fig. 3: Power level and RIN of each DQPSK QD-MLL channel.

Fig. 4 :

 4 Fig. 4: Experimental setup.

Fig. 7 :

 7 Fig. 7: Power level and penalty (BER=10 -9 ) of each DQPSK QD-MLL channel.

  Bit error rate vs. receiver input power.

		-90 -100		1548.3 1549.1 1549.9 1550.6		1.E-05 1.E-03 1.E-04			DFB 1549 nm ECL 1549.1 nm QD 1549.1 nm
	RIN (dBc/Hz)	-130 -120 -110	ECL	1551.5 1552.3 1553.1 1553.9 1556.5 1554.8 1555.6	BER	1.E-10 1.E-09 1.E-08 1.E-07 1.E-06
		-140				1.E-11	-33	-31	-29	-27	-25	-23
		10 Fig. 5: -150 100 1000 10000 Total mode of QD-MLL		Receiver input power (dBm)
			Frequency (MHz)

  Bit error rate vs. receiver input power for 11 ECL and QD-MLL channels.

			ECL 1548.3	QD 1548.3
			ECL 1549.1	QD 1549.1
	1.E-03		ECL 1549.9 ECL 1550.7	QD 1549.9 QD 1550.7
			ECL 1551.5	QD 1551.5
	1.E-04		ECL 1552.3 ECL 1553.1	QD 1552.3 QD 1553.1
	1.E-05		ECL 1553.9 ECL 1554.8	QD 1553.9 QD 1554.8
			ECL 1555.6	QD 1555.6
	1.E-06		ECL 1556.5	QD 1556.5
	1.E-07 BER
	1.E-08		QD
	1.E-09	ECL		Error floors for QD at 1550.7 & 1556.5
	1.E-10
	1.E-11
	-35	-30	-25	-20	-15
		Receiver input power (dBm)
	Fig. 6:

Mo.1.F.4

Acknowledgements

This work was supported by the French National projects TELDOT and PERSYST II and by Québec's Regroupement Stratégique Centre for Advanced Systems and Technologies in Communications (SYTACom). The authors would like to thank A. Shen and G.-H. Duan from Alcatel-Thales III-V Lab for the laser fabrication, and P. Chanclou from Orange Lab for lending the RIN measurement equipment.

ers. All sources were centred at 1549 nm. The ECL and DFB give nearly identical BER. The QD-MLL has a power penalty of 2.5 dB at 10 -9 compared to the DFB and ECL, yet nonetheless is error-free (BER<10 -9

). This penalty is due to high RIN when filtering the QD-MLL to isolate one mode as discussed in the previous section.

The performance of our system is evaluated for each mode of the QD-MLL. For each wavelength, the DQPSK system is first optimised using the wavelength-adjustable ECL source. Next each mode of the QD-MLL is isolated in turn, and BER measurements are compared for ECL and QD-MLL, see Fig. 6. Results show that the DQPSK performance for the QD-MLL is as stable as the tuned ECL over the entire 8 nm range, from 1548.3 nm to 1556.4 nm, covering 11 WDM channels with 100 GHz spacing. The receiver sensitivity of the system using ECL varies between -26 dBm and -26.5 dBm in this wavelength range. When the QD-MLL is used, error-free performance is obtained over nine channels. Error floors at BER of 4×10 -8 occur with two channels (1550.7 nm and 1556.5 nm). This is to be expected given the RIN and power level of those modes.

Finally, Fig. 7 shows the power penalty (BER of 10 -9

) of DQPSK signals using each QD-MLL mode. Power penalty from 2 dB to 4 dB is obtained over nine WDM channels. For the weak modes, the power penalty rapidly increases. Not surprisingly, the fluctuations in non-dominant modes are strong compared to that of the dominant modes.

Conclusions

We proposed and experimentally demonstrated for the first time the feasibility of a cost-effective, multi-wavelength seeding source based on QD-MLL for high speed and spectral efficient WDM systems using DQPSK modulation at 56 Gbit/s. We reported error-free signal operation over nine WDM channels with 100 GHz spacing. Therefore, this strategy is viable solution for future high bit rate optical communications networks.