T Boulier

M Bamba

A Amo

C Adrados

A Lemaitre

E Galopin

I Sagnes

J Bloch

C Ciuti

E Giacobino

A Bramati
email: bramati@spectro.jussieu.fr

Polariton-Generated Intensity Squeezing in Semiconductor Micropillars

The generation of squeezed and entangled light fields is a crucial ingredient for the implementation of quantum information protocols. In this context, semiconductor materials offer a strong potential for the implementation of on-chip devices operating at the quantum level. Here we demonstrate a novel source of continuous variable squeezed light in pillar-shaped semiconductor microcavities in the strong coupling regime. Degenerate polariton four-wave mixing is obtained by exciting the pillar at normal incidence. We observe a bistable behavior and we demonstrate the generation of squeezing near the turning point of the bistability curve. The confined pillar geometry allows for a larger amount of squeezing than planar microcavities due to the discrete energy levels protected from excess noise. By analyzing the noise of the emitted light we obtain a measured intensity squeezing of 20.3%, inferred to be 35.8% after corrections.

While very efficient sources of non classical light, essentially based on bulk non-linear crystals, have been developed in the last twenty years, the quest for integrated quantum sources is a major challenge for the cutting edge research activity in the field of quantum information technologies.

Successful examples in the discrete variable regime are represented by efficient single and entangled photons sources in semiconductor quantum dots [START_REF] Loo | Optical nonlinearity for few-photon pulses on a quantum dot-pillar cavity device[END_REF][START_REF] Claudon | A highly efficient single-photon source based on a quantum dot in a photonic nanowire[END_REF][START_REF] Dousse | Ultrabright source of entangled photon pairs[END_REF]. Photon antibunching and squeezing were theoretically predicted and experimentally demonstrated in pioneering resonance fluorescence experiments [START_REF] Kira | Quantum theory of secondary emission in optically excited semiconductor quantum wells[END_REF][START_REF] Carmichael | Proposal for the measurement of the resonant Stark effect by photon correlation techniques[END_REF][START_REF] Kimble | Photon antibunching in resonance fluorescence[END_REF][START_REF] Slusher | Observation of squeezed states generated by four-wave mixing in an optical cavity[END_REF]. In strong contrast, despite several theoretical proposals [START_REF] Liew | Multipartite polariton entanglement in semiconductor microcavities[END_REF][START_REF] Liew | Quantum entanglement in nanocavity arrays[END_REF] the continuous variables regime has seen so far only very few experimental achievements [START_REF] Fox | Squeezed light generation in semiconductors[END_REF][START_REF] Karr | Squeezing in semiconductor microcavities in the strong coupling regime[END_REF][START_REF] Baas | Optical bistability in semiconductor microcavities[END_REF] of squeezed states of light in semiconductor materials.

Here, for the first time in this type of systems, we achieve a high degree of intensity squeezing exploiting effective Kerr effect for microcavity polaritons. Let us note that, due to the giant polariton non-linearities [START_REF] Saba | High-temperature ultrafast polariton parametric amplification in semiconductor microcavities[END_REF], the power range used here (5-20mW) is two orders of magnitudes lower than in previous experiments [START_REF] Fox | Squeezed light generation in semiconductors[END_REF]. These results open the way to the implementation of ultra-low threshold quantum devices integrated in a semiconductor platform.

Exciton-polaritons are half-light half-matter bosonic particles arising from the strong coupling between exciton and photons in semiconductor microcavities [START_REF] Weisbuch | Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity[END_REF]. They exhibit strong non-linearities due to the Coulomb interactions of their excitonic part, responsible for the rich variety of quantum effects recently unveiled [START_REF] Carusotto | Quantum fluids of light[END_REF], ranging from Bose-Einstein condensation [START_REF] Kasprzak | Bose-Einstein condensation of exciton polaritons[END_REF] and superfluidity [START_REF] Amo | Collective fluid dynamics of a polariton condensate in a semiconductor microcavity[END_REF][START_REF] Amo | Superfluidity of polaritons in semiconductor microcavities[END_REF] to more complex phenomena such as dark and bright solitons [START_REF] Amo | Polariton superfluids reveal quantum hydrodynamic solitons[END_REF][START_REF] Sich | Observation of bright polariton solitons in a semiconductor microcavity[END_REF], magnetic monopole-like half-solitons [START_REF] Hivet | Half-solitons in a polariton quantum fluid behave like magnetic monopoles[END_REF], halfvortices [START_REF] Lagoudakis | Observation of half-quantum vortices in an exciton-polariton condensate[END_REF] and vortex lattices [START_REF] Tosi | Geometrically locked vortex lattices in semiconductor quantum fluids[END_REF].

The great potential of polaritons for non-classical fields generation was recognized very soon after the experimental demonstration of the coherent non-linear optical response of microcavity polaritons allowing for the observation of parametric amplification and oscillation in such systems [START_REF] Savvidis | Angle-resonant stimulated polariton amplifier[END_REF][START_REF] Stevenson | Continuous wave observation of massive polariton redistribution by stimulated scattering in semiconductor microcavities[END_REF][START_REF] Messin | Parametric polariton amplification in semiconductor microcavities[END_REF]. Especially interesting is the case of degenerate polariton four-wave mixing in which the strong nonlinearities originating in the polariton-polariton interactions allow describing the polariton system as an effective Kerr medium whose refractive index depends on the light intensity [START_REF] Baas | Optical bistability in semiconductor microcavities[END_REF][START_REF] Tassone | Lasing and squeezing of composite bosons in a semiconductor microcavity[END_REF][START_REF] Schwendimann | Statistics of polaritons in the nonlinear regime[END_REF]. Kerr media are well known to be very efficient sources of quadrature squeezed light. Indeed, quadrature squeezing of about 4% was observed in a planar semiconductor microcavity, by using degenerate polariton four-wave mixing in 2004 [START_REF] Karr | Squeezing in semiconductor microcavities in the strong coupling regime[END_REF]. Despite several attempts with specifically designed planar microcavity samples, significant improvements of this first result were prevented by the excess noise coming from the intrinsic multimode nature of the emission in a planar microcavity, which is characterized by a continuum of polariton modes and is favorable to the coupling with lattice phonons. Very recently, it has been predicted [START_REF] Bamba | Quantum squeezing generation versus photon localization in a disordered planar microcavity[END_REF] that a large amount of squeezing could be achieved by quantizing the in-plane polariton field modes via a geometric confinement (strong localization limit [START_REF] Bamba | Quantum squeezing generation versus photon localization in a disordered planar microcavity[END_REF]). This tight confinement separates the modes energetically, severely reducing any nonlinear multimode coupling responsible for increased excess noise and reduced quantum effects. In this paper we apply this method in a controlled way making use of semiconductor micropillars [START_REF] Bloch | Strong-coupling regime in pillar semiconductor microcavities[END_REF][START_REF] Bajoni | Polariton laser using single micropillar gaas-gaalas semiconductor cavities[END_REF] and we demonstrate up to 20.3% of directly measured squeezing in the amplitude quadrature of the emitted light.

Results

Experimental configuration. In our setup the pillars are fabricated by etching a λ GaAs microcavity containing two GaAs-InGaAs quantum wells, resulting in a polariton Rabi splitting of about 4meV. The cavity finesse is of the order of 700, which gives a 0, 1meV polariton linewidth. Fig. 1 shows the energy separation of the polariton modes due to the finite size of the pillar. As shown in Fig. 2(a) the cavity mirrors are asymmetric with the bottom Bragg mirror twice thicker than the top one (30 pairs of Al 0.1 Ga 0.9 As -Al 0.95 Ga 0.05 As versus 15 pairs for the top mirror), optimizing the cavity for experiments in reflection configuration. The cavity is wedged in one direction, providing a large choice of cavityexciton detuning by selecting pillars placed in different positions on the chip. We can choose the shape and size of the micropillars in order to engineer the spatial distribution of the confined modes and their energy separation. Figure 2 (b) shows the large square-shaped pillars (width of 20µm) and small square or round-shaped pillars (width from 6µm to 2µm) available on the studied sample.

The pillars are cooled down to 5K in a cryostat mounted in reflection configuration and excited with a CW single-mode Ti:Sa laser tuned to be quasi-resonant with the ground state level of the lower polariton branch. The laser is frequency locked to a Fabry-Perot cavity and its intensity is stabilized by mean of an electro-optical modulator. The laser was verified to be at the shot noise for frequencies above 1MHz. The spatial mode is filtered by a 5m long single-mode optical fiber shielded against thermal fluctuations. The laser beam is focused by a microscope objective treated with a near-infrared optimized coating and excites the sample at normal incidence. Upstream of the microscope objective, a motorized λ 2 plate coupled with a polarizing beam splitter allows us to finely control the pump power on the sample and a λ 4 plate placed between the beam splitter and the objective ensures that the light coming on the sample is circularly polarized. With the polarized beam splitter, the λ 4 plate forms an optical circulator since the light re-emitted from the sample was measured to be 95% circularly polarized, as expected from angular momentum conservation. The light reflected from the cavity is thus collected and sent to the detectors.

The shape of the laser spot on the sample is Gaussian with a waist of 5.5µm, well matched to the round pillars with a diameter of 6µm. In the studies presented here, we hence use a round micropillar of 6 microns, which ensures an optimal mode matching between its polaritonic ground state and the focalized excitation laser. The re-emitted light is collected by the same microscope objective and sent through a telescope and a spatial filter, allowing us to select only the light coming out of the pillar or some specific region if needed. As shown in Fig. 2(c), the sample emission is collected in a setup allowing performing both balanced and homodyne detection.

For the balanced detection the collected light is equally split in two parts by a λ 2 plate associated to a polarizing beam splitter (PBS 2) and each beam is collected by a photodiode. The high frequency (HF) outputs of the two identical photodiodes (81% quantum efficiency, with home-made low-noise electronics) are added or subtracted with a passive circuit, amplified by a low-noise, high frequency amplifier and sent to a spectrum analyzer giving us the spectral noise power (thereafter simply called "noise"). The choice of addition or subtraction allows us to quickly switch between the measurement of the intensity noise (addition mode) or the shot noise (difference mode). The common mode rejection of the balanced diodes was 45dB at 7MHz.

For the homodyne detection a portion of the pump laser is mixed on PBS 1 with the light emitted by the micropillar with an orthogonal polarization, and acts as local oscillator. In this case the HF output of the photodiodes are subtracted and the measured noise gives the noise in the quadrature of the reflected field that is in phase with the local oscillator. The relative phase between the local oscillator and the reflected field is controlled with a piezoelectric-mounted mirror, allowing measuring all quadratures. The shot noise is measured by blocking the light reflected from the microcavity.

For the homodyne setup it is important to note that the spatial mode of the sample emission can be strongly distorted by the non-linear interactions and it is no longer Gaussian-shaped at high polariton densities. For this reason it was found very difficult to achieve the mode matching with the local oscillator necessary to have efficient homodyne detection, except in the special case of very low detuning of the pump laser with respect to the ground state level of the lower polariton branch (since in this case we have a low density of polariton even above threshold). There, proper homodyne detection was achieved (mode matching of 90%). However, such a low detuning is not optimal and thus the squeezing measured with the homodyne detection is lower than at the optimum detuning.

Intensity squeezing measurements at the optimum pump-polariton detuning were performed with the balanced setup. Figure 3 (a,b) shows the intensity noise measured with a balanced detection: a scan at fixed pump power in Fig. 3(a) and a scan of the noise versus pump power in Fig. 3(b) for the polaritonlaser detuning ∆ = 0.19meV that we have identified as optimal for squeezing. In Fig. 3 and Fig. 4 the spectrum analyzer was set to zero frequency span and centered at 7MHz where the signal-to-noise ratio of the detection is maximum, with a sweep time ranging from 1s to 20s depending on the type of measurement. The resolution bandwidth (spectral bandwidth of the measurement) was 300kHz with a 30Hz video bandwidth (temporal integration over a 30Hz bandwidth). For the noise dependence on the pump power (Fig. 3(b)) the motorized plate controlling the excitation power on the pillar is scanned at the same time as the spectrum analyzer. We then compare the measured noise to the shot noise. The protocol involves recording the shot noise in difference mode, immediately switching to addition mode to record the intensity noise and then switching back to shot noise measurement (difference mode again) in order to check its stability. The dark noise was recorded after each measurement by measuring the electronic noise when the laser beam is blocked. All the presented noise measurements are obtained by normalizing the noise traces by the shot noise after subtraction of the dark noise from both the noise and the shot noise.

A slight energy detuning between the pump laser and the ground state of the lower polariton branch gives rise to optical bistability [START_REF] Baas | Optical bistability in semiconductor microcavities[END_REF] (see Fig. 3(c)), characteristic in Kerr nonlinear media. According to theoretical predictions the best squeezing rate is expected in the vicinity of the turning points of the bistability, as in other nonlinear systems [START_REF] Hilico | Linear input-output method for quantum fluctuations in optical bistability with two-level atoms[END_REF]. We therefore chose to sweep the pump power in the vicinity of the bistable region (noise profile measurement, Fig. 3(b)) to identify the best working points and we performed noise measurements with a fixed pump power corresponding to the maximum of intensity squeezing (steady state measurement, Fig. 3(a)).

Squeezing was achieved in a reproducible way in repeated experiments on several different neighboring pillars. According to previous calculations [START_REF] Romanelli | Two-mode squeezing in polariton four-wave mixing[END_REF] the best squeezing is expected for a slightly negative cavity-exciton detuning (0 > δ > -1meV) in planar microcavities. Here the spatial confinement of pillars tends to enhance the polariton-polariton non-linear interactions with respect to planar microcavities, due to the increase in polariton density. We hence chose to work at larger negative cavity-exciton detuning (δ ≈ -5meV) to have a larger photonic fraction, while preserving strong enough nonlinearities from the excitonic component. Experimental and theoretical results. The strongest noise reduction at fixed pump power was obtained with the laser energy blue-detuned by about ∆ = 0.19meV from the polariton lower energy level on a 6µm pillar with a cavity-exciton detuning δ ≈ -5meV. The measured intensity squeezing is 20.3 ± 1.5%, as shown in Fig. 3(a). After correction for detection efficiency (81%) and transmission losses (T = 70%), we infer a squeezing in the outgoing field of 35.8 ± 2.6%.

This result was established with a laser intensity lock reducing the low frequency (0-100kHz) intensity variations to 0.2%. The use of the spatial filter was found not to be crucial, as the laser spot was well matched to the size of the 6µm pillars. On the other hand, excessive spatial filtering led to a reduction of the squeezing, as expected from increasing losses. A coupled exciton-photon model with pump, losses and Kerr nonlinearity (described in Methods) predicts a squeezing value under these conditions of about 40%, in agreement with the experimental value. (a) -Squeezed intensity noise at fixed pump power near the lower turning point (linear scale), on a 6µm pillar at cavity-exciton detuning of -5meV and a polariton-laser detuning of 0.19meV. The noise data were normalized with a measurement of the shot noise and a measurement of the dark noise, with S(t) = noise(t)-<dark(t)> <shotnoise(t)>-<dark(t)> (<...> is the time average). The measured noise reduction is 20.3 ± 1.5% (-1dB) below the shot noise limit before corrections, 35.8% ± 2.6% after. The green line is the average noise and the red line is the shot noise. (b) -Intensity noise (spectral noise power, semi-logarithmic scale) as a function of the pump power for the same pillar under the same conditions as in (a). The noise increases smoothly toward a peak just before the bistability threshold, then suddenly drops to form a gap where intensity squeezing is observed, with about the same value as in (a). Here also the normalized spectral noise is obtained with 3 measurements (noise, shot noise and dark noise). In red is the curve obtained by increasing the pump power, in blue is the curve obtained by decreasing the pump power and the green line represents the shot noise (normalized to one). The thick black arrow points to where Fig. 3 (a) was recorded. (c) -Bistability curve obtained by plotting the reflectance versus the pump power, for the same pillar under the same conditions as in (a) and (b). The color code is the same as in (b). Above the thresholds the transmission is increased, hence a sharp reduction on the reflectance.

Effect of the pump-polariton detuning. The values of pump power at which the optimal working points occur as well as the specific shape of the bistability curve depend on ∆. This parameter is also expected to influence the intensity noise profile and the squeezing value. The noise dependence on the pump power for the same pillar and parameter values as mentioned above is shown in Fig. 3(b). Outside of the bistable region there is an excess noise due to thermal exciton reservoir populated by the pump laser. In the vicinity of the bistability thresholds this noise undergoes amplification or deamplification due to the nonlinear effects. The intensity noise strongly increases below the bistability thresholds then decreases, as observed in other nonlinear systems [START_REF] Carusotto | Quantum fluids of light[END_REF][START_REF] Hilico | Linear input-output method for quantum fluctuations in optical bistability with two-level atoms[END_REF][START_REF] Romanelli | Two-mode squeezing in polariton four-wave mixing[END_REF].

At higher values of ∆ the pump power required to reach the bistability threshold increases and the noise profiles are drastically altered. Indeed a bistable behavior appears above some critical value of ∆. For rather large ∆ only the lower turning point (reached by decreasing the pump power) is able to produce intensity squeezing. If ∆ is lower than the critical value, strong nonlinear modifications of the noise and reflectance can be observed despite the lack of bistable behavior and the noise curves are the same when increasing or decreasing the pump power. These effects were studied in our experiment by repeating the noise measurements at different polariton-laser detunings and are discussed in the Supplementary Note 1. Supplementary Fig. 1 and Supplementary Fig. 2 show, respectively, the reflectance and the noise plots obtained for ∆ = 0.10meV while Supplementary Fig. 3 and Supplementary Fig. 4 show, respectively, the reflectance and the noise plots obtained for ∆ = 0.21meV and Supplementary Fig. 5 and Supplementary Fig. 6 show the reflectance and the noise plots obtained for ∆ = 0.30meV . The optimal detuning for squeezing that we present here corresponds to the onset of the bistability behavior where the hysteresis cycle is very narrow (Fig. 3(b)). Homodyne detection. To confirm these findings we have measured the noise along all the field quadratures with the homodyne detection and checked that squeezing occurs only in the amplitude quadrature at the bistability turning points. As previously mentioned, for low ∆ the output of the pillar is nearly Gaussian. In such conditions we were able to achieve good mode matching between the local oscillator and the spatial mode emitted by the micropillar (about 90%) and implement an efficient homodyne detection. The path difference between the two beams, set to change linearly with time, determines the measured quadrature. The result is presented in Fig. 4 for a laser-polariton detuning of ∆ = 0.10meV. Squeezing is observed at the minima of the curve showing the characteristic arches. Let us note that the anti-squeezed quadrature (phase quadrature) exhibits a large excess noise of the order of 12dB due to thermal noise input. The fluctuations and irregularities of the noise arches are due to the fact that the interferometric homodyne setup is more sensitive to mechanical instabilities. A comparison with the intensity measurements setup allows checking that the squeezed quadrature indeed corresponds to the amplitude quadrature. The observed squeezing with this setup was of about 14% (27% inferred after corrections). Experimental measurement of squeezing with homodyne detection for a pump-polariton detuning of 0.10meV on the same pillar as in Fig. 3. The minima were verified to be in the amplitude quadrature. The squeezing obtained from the average between the values at 0, π and 2π is 14% before corrections and 27% after corrections. The reduced value of the squeezing here is mainly due to the non-optimal laser-polariton detuning.

Discussion

In conclusion we have reported an experimental demonstration of strong enhancement of squeezed light generation with confined polaritons in semiconductor micropillars. The best intensity squeezing was inferred to be 35.8% ± 2.6% below the shot noise limit and was confirmed by measurements with a homodyne detection setup. Several original optimizations allowed to reach these results. In particular, the special design of the pillars ensures a near-perfect mode matching between the pump and the polariton field; the discretization of the polariton energy modes allows for a strong reduction of the excess noise coming from phonon coupling. The reduced reflectivity of the upper Bragg mirror relative to that of the bottom mirror enhances the emission of light toward the top of the pillar, optimizing the light collection. Finally the relatively low quality factor of the cavity increases the radiative to non-radiative losses ratio which, together with the choice of a working point with a high photonic fraction, results in a larger squeezing in the emitted light. These results demonstrate the improvement of intensity-squeezed light and light-matter fields brought by reduced dimensionality in semiconductor microstructures and open the way to the realization of ultra-low threshold integrated sources for the quantum information-processing technology.

Methods

Theoretical model. Due to the lateral confinement of the micropillar, the polariton modes are well identified and we can selectively pump a polariton mode by tuning the spatial and spectral profile of the pumping light. For the theoretical analysis, we simply consider a single cavity mode with frequency ω c and an exciton state with frequency ω x . The Hamiltonian of the coupled system (polariton) in the microcavity is:

Ĥ0 = ω c â † â + ω x b † b + Ω R 2 â † b + b † â + g 2 b † b † bb (1)
Here, â and b are the annihilation operators of a photon and an exciton, respectively, and Ω R is the coupling strength (vacuum Rabi splitting) between the cavity photon and the exciton. The last term represents a third-order nonlinearity for the excitons with strength g. The polaritons are subject to an effective Kerr effect through it. Additionally, considering the photonic and excitonic reservoirs, the quantum Langevin equations for the cavity photons and the excitons are, respectively, derived as:

i ∂â ∂t = ω c - iκ 2 â + Ω R 2 b -i √ κ âin + F p e -iωpt (2)
i ∂ b ∂t = ω x - iγ 2 b + Ω R 2 â + g b † bb -i √ γ bin (3)
Here, κ and γ are the dissipation rates of the cavity photon and exciton, respectively. âin and bin are the input operators from photonic and excitonic reservoirs, respectively, and we assume they have a zero average âin = bin = 0 (incoherent input). Instead, we explicitly consider an applied coherent pumping with amplitude F p and frequency ω p . Then, the input-output relation for the photon operators is expressed as:

âout = âin + F p e -iωpt + √ κâ (4)
From the above Langevin equations, we first calculate the mean fields of the cavity photon and exciton induced by the coherent pump. Supposing â = αe -iωpt and b = βe -iωpt , the steady-state mean fields are:

i ∂ ∂t α = ω c -ω p - iκ 2 α + Ω R 2 β -i √ κF p = 0 (5) i ∂ ∂t β = ω x -ω p - iγ 2 β + Ω R 2 α + g |β| 2 β = 0 (6)
Once the mean fields are determined, the reflectance is calculated via the input-output relation as:

âout = α out e -iωpt = F p + √ κα e -iωpt (7)
R = | âout | 2 | âin + F p e -iωpt | 2 = |F p + √ κα| 2 |F p | 2 (8)
The intensity noise is calculated by the fluctuation around the mean field value. Defining the fluctuation operators as δâ = âe iωpt -α and δ b = be iωpt -β, and linearizing the corresponding equations, the quantum Langevin equations are rewritten as:

i ∂ ∂t δâ = ω c -ω p - iκ 2 δâ + Ω R 2 δ b -i √ κâ in (9) i ∂ ∂t δ b = ω x + 2g |β| 2 -ω p - iγ 2 δ b + Ω R 2 δâ + gβ 2 δ b † -i √ γ bin (10)
From these equations and the input-output relation, we can infer the fluctuation δâ out = âout e iωpt -α out of the output photons. The variance of the output is expressed as:

(∆N ) 2 = â † out â † out âout âout -â † out âout 2 = 2 |α out | 2 δâ † out δâ out + 2ℜ (α * out ) 2 δâ out δâ out (11)
Then, the normalized intensity noise is expressed as:

(∆N )

The correlations of the output are calculated from the ones of input operators. Here, we assume that the correlation of the photonic input as â † in (t)â in (t ′ ) = âin (t)â in (t ′ ) = 0 (extracavity photon vacuum) while the excitonic reservoir has an incoherent population, namely b † in (t) bin (t ′ ) = N th δ(tt ′ ) and bin (t) bin (t ′ ) = 0. We also assume that there is no correlation between the photonic and excitonic inputs.

N th represents the thermal noise and is chosen to be linear with the pump power (N th = σ * W p (mW), with σ the proportionality constant and W p the pump power). Parameters. In the calculations, we have used the following values: δ = ω cω x = -5meV, Ω R = 4meV and κ = 0.11meV. The other parameters have been estimated by fitting experimental data for several values of ∆, the pump-polariton detuning and are shown in Table I. From the theoretical results of this simple model, we observe that the excitonic dissipation rate γ tends to increase with ∆. This can be explained by an increased population in the excitonic reservoir. ∆ = ω pω polariton γ σ 0.14meV 0.43meV 0.2mW -1 0.15meV 0.40meV 0.3mW -1 0.19meV 0.89meV 0.12mW -1 0.26meV 0.63meV 0.06mW -1 This model is able to reproduce correctly the reflectance traces obtained in the experiment, and the intensity noise for pump intensities above the bistability threshold. Note that the prediction of the intensity noise for low pump intensities does not reproduce well the experimental data. This is probably due to the fact that our model does not take disorder effects into account. For very low densities, the disorder becomes the dominant effect, so we cannot capture the corresponding noise behavior adequately. Supplementary Fig. 7 shows the comparison between the model and the experiment for the intensity noise at ∆ = 0.15meV. The bistability curve of the reflectance and the corresponding intensity noise dependence on the pump power for a pump-polariton detuning of 0.26meV are shown in Supplementary Fig. 8 and in Supplementary Fig. 9, respectively.

Figure 1 :

 1 Figure 1: Energy modes separation. Dispersion curve for the 6µm circular pillar used in the experiment (-5meV cavity detuning), obtained with a non-resonant pumping at 1550meV. The lower polariton branch with several spatial modes and the weakly excited upper branch are visible. The modes are energetically separated and squeezing is protected against multimode nonlinear coupling.

Figure 2 :

 2 Figure 2: Sample and setup. (a) -Principle of squeezed light production with a pillar-shaped microcavity. The microcavity is made of a pair of quantum wells (white-red-white layers) sandwiched by two Bragg mirrors (alternation of light blue and dark blue layers); the green layer is the substrate. The orange beam represents the pump laser arriving on the sample at normal incidence, slightly blueshifted with respect to the energy of the lower polariton branch. The red beam represents the squeezed light emitted by the squeezed polariton field present in the microcavity. (b) -Image of the sample obtained with scanning electron microscopy. The different shapes and sizes of the pillars are visible. The scale bar corresponds to 100µm. (c) -Scheme of the experimental setup for the noise detection. The higher squeezing values were measured with the balanced detection setup, realized by blocking the local oscillator with the beam blocker. The choice of adding (+) or subtracting (-) the photodiodes signals determines the measurement: intensity noise (+) or shot noise (-). To measure the noise quadratures of the field one unblocks the local oscillator and set the mixer to (-); in this case the shot noise is measured by blocking the light reflected from the microcavity. PBS: polarizing beam splitter; λ 2 , λ 4 : half-wave, quarter-wave plate.

Figure 3 :

 3 Figure 3: Intensity squeezing and bistability. (a) -Squeezed intensity noise at fixed pump power near the lower turning point (linear scale), on a 6µm pillar at cavity-exciton detuning of -5meV and a polariton-laser detuning of 0.19meV. The noise data were normalized with a measurement of the shot noise and a measurement of the dark noise, with S(t) =

Figure 4 :

 4 Figure4: Homodyne detection. Experimental measurement of squeezing with homodyne detection for a pump-polariton detuning of 0.10meV on the same pillar as in Fig.3. The minima were verified to be in the amplitude quadrature. The squeezing obtained from the average between the values at 0, π and 2π is 14% before corrections and 27% after corrections. The reduced value of the squeezing here is mainly due to the non-optimal laser-polariton detuning.

Table I :

 I Parameters.Table of parameters used in the theory, deduced by fitting the experimental data of the reflectance

Acknowledgements -This work was partly supported by the FP7 ITN "Clermont4" (235114), by the ANR projects PNANO-07-005 "GEMINI" and Quandyde (ANR-11-BS10-001), by the French RENATECH network and the KAKENHI 24-632 grant. The authors are grateful to J. Ph. Karr and Q. Glorieux for useful discussions.

Author contribution

All authors contributed to all aspects of this work.


Xavier Devroey

Gilles Perrouin

Maxime Cordy

Pierre-Yves Schobbens

Axel Legay
email: axel.legay@inria.fr

Patrick Heymans

Towards Statistical Prioritization for Software Product Lines Testing

Keywords: Software Product Line, Software Testing, Test Prioritization, Statistical Testing

Software Product Lines (SPLs) are inherently difficult to test due to the combinatorial explosion of the number of products to consider. To reduce the number of products to test, sampling techniques such as combinatorial interaction testing have been proposed. They usually start from a feature model and apply a coverage criterion (e.g. pairwise feature interaction or dissimilarity) to generate tractable, fault-finding, lists of configurations to be tested. Prioritization can also be used to sort/generate such lists, optimizing coverage criteria or weights assigned to features. However, current sampling/prioritization techniques barely take product behaviour into account. We explore how ideas of statistical testing, based on a usage model (a Markov chain), can be used to extract configurations of interest according to the likelihood of their executions. These executions are gathered in featured transition systems, compact representation of SPL behaviour. We discuss possible scenarios and give a prioritization procedure validated on a web-based learning management software.

INTRODUCTION

Software Product Line (SPL) engineering is based on the idea that products of the same family can be built by systematically reusing assets, some of them being common to all members whereas others are only shared by a subset of the family. Such variability is commonly captured by the notion of feature. Individual features can be specified using languages such as UML, while their inter-relationships are organized in a Feature Diagram (FD) [START_REF] Kang | Feature-Oriented domain analysis (FODA) feasibility study[END_REF].

SPL testing the most common quality assurance technique in SPL engineering. As opposed to single-system testing where the testing process considers only one software product, SPL testing is concerned about minimizing the test effort for all the SPL products. Testing these products separately is clearly infeasible in real-world SPLs, which typically consist of thousands of products. Automated model-based testing [START_REF] Utting | Practical model-based testing: a tools approach[END_REF] and shared execution [START_REF] Kim | Shared execution for efficiently testing product lines[END_REF] are established testing methods that allows test reuse across a set of software. They can thus be used to reduce the SPL testing effort. Even so, the problem remains entire as these methods still need to cover all the products.

Other approaches consist in testing only a representative sample of the products. Typical methods select these products according to some coverage criterion on the FD (e.g. all the valid couples of features must occur in at least one tested product [START_REF] Cohen | Interaction testing of highly-configurable systems in the presence of constraints[END_REF][START_REF] Perrouin | Pairwise testing for software product lines: comparison of two approaches[END_REF]). An alternative method is to associate each feature with a weight and prioritize the products with the highest weight [START_REF] Henard | Multi-objective test generation for software product lines[END_REF][START_REF] Johansen | Generating better partial covering arrays by modeling weights on sub-product lines[END_REF]. This actually helps testers to scope more finely and flexibly relevant products to test than a covering criteria alone. Yet, assigning meaningful weights is cumbersome in the absence of additional information regarding their behaviour.

In this paper, we propose an approach to prioritize the products to test according to criteria based on the actual behaviour of the products, thereby increasing the relevancy of the testing activities and decreasing the risk of leaving errors undetected in many products. Our work leans on statistical testing [START_REF] Whittaker | A markov chain model for statistical software testing[END_REF], which generates test cases from a usage model represented by a Discrete-Time Markov Chain (DTMC). This usage model represents the usage scenarios of the software under test as well as their respective probability. The latter information allows one to determine the likelihood of execution scenarios, and to rank these accordingly. We postulate that this ranking method can serve as a basis for prioritizing the products to test in an SPL. For instance, one may be interested only in products that can produce the executions that have the highest probability of occurrence. Usage models being extracted from running systems in a black-box manner, they are agnostic on vari- Figure 1: The soda vending machine example [START_REF] Classen | Featured Transition Systems : Foundations for Verifying Variability-Intensive Systems and their Application to LTL Model Checking[END_REF] ability and therefore cannot link a behaviour to the features needed to execute it. To overcome this, we assume the existence of design models describing the SPL behaviour. We then encode these models into a fundamental formalism -a Featured Transition System (FTS) -to obtain the missing link between executions and variability. The next step is to remove the behaviours of the FTS that were not selected during the analysis of the usage model. In our example, we would obtain a pruned FTS that exhibits only the most common behaviours, from which we can extract the products able to execute them. The remainder of this paper is organized as follows: Section 2 presents the theoretical background underlying our vision. Our approach is detailed in Section 3. The results of a preliminary experiment are presented in Section 4. Section 5 discusses related research and Section 6 concludes the paper with challenges and future research directions.

BACKGROUND

In this section, we present the foundations underlying our approach: SPL modeling and statistical testing.

SPL Modelling

A key concern in SPL modeling is how to represent variability. To achieve this purpose, SPL engineers usually reason in terms of features. Relations and constraints between features are usually represented in a Feature Diagram (FD) [START_REF] Kang | Feature-Oriented domain analysis (FODA) feasibility study[END_REF]. For example, Fig. 1a presents the FD of a soda vending machine [START_REF] Classen | Featured Transition Systems : Foundations for Verifying Variability-Intensive Systems and their Application to LTL Model Checking[END_REF]. A common semantics associated to a FD d (noted [[d]]) is the set of all the valid products allowed by d.

Different formalisms may be used to model the behaviour of a system. To allow the explicit mapping from feature to SPL behaviour, Featured Transition Systems (FTS) [START_REF] Classen | Featured Transition Systems : Foundations for Verifying Variability-Intensive Systems and their Application to LTL Model Checking[END_REF] were proposed. FTS are transition systems (TS) where each transition is labelled with a feature expression (i.e., a boolean expression over features of the SPL), specifying which products can execute the transition. Thus it is possible to determine products that are the cause of a violation or a failed test. Formally, an FTS is a tuple (S, Act, trans, i, d, γ) where S is a set of states; Act a set of actions; trans ⊆ S × Act × S is the transition relation (with (s1, α, s2) ∈ trans sometimes noted s1 α -→ s2); i ∈ S is the initial state; d is a FD; and γ : trans → [[d]] → {⊤, ⊥} is a total function labelling each transition with a boolean expression over the features, which specifies the products that can execute the transition. For instance: ¬f in Fig. 1b indicates that only products that have not the f ree feature may fire the pay, change, open, take and close transitions. A TS modelling the behaviour of a given product is obtained by removing the transition whose feature expression is not satisfied by the product.

We define the semantics of an FTS as a function that associates each valid product with its set of finite and infinite traces, i.e. all the alternating sequences of states and actions starting from the initial state available, satisfying the transition relation and such that its transitions are available to that product. According to this definition, an FTS is indeed a behavioural model of a whole SPL. Fig. 1b presents the FTS modeling a vending machine SPL. For instance, transition 3 pay/¬f -→ 4 is labelled with the feature expression c. This means that only the products that do have the feature Cancel (c) are able to execute the transition. This definition differs from the one presented in [START_REF] Classen | Featured Transition Systems : Foundations for Verifying Variability-Intensive Systems and their Application to LTL Model Checking[END_REF], where only infinite paths are considered. In a testing context, one may also be interested in finite paths.

Statistical Testing

Whittaker and Thomason introduce the notion of usage model in [START_REF] Whittaker | A markov chain model for statistical software testing[END_REF] and define it as a TS where transitions have probability values defining their occurrence likelihood. Formally, their usage model is actually equivalent to a DTMC where transitions are additionally decorated with actions, i.e. a tuple (S, Act, trans, P, τ ) where :

• (S, Act, trans) are defined as in FTS;

• P : S ×Act×S → [0, 1] is the probability function that associates each transition (si, α, sj) the probability for the system in state si to execute action α and reach state sj;

• τ : S → [0, 1] is the vector containing the probabilities to be in the initial state when the system starts, with the following constraint : ∃i : (τ (i) = 1 ∧ ∀j = i : τ (j) = 0);

• ∀si ∈ S : α∈Act,s j ∈S•s i α -→s j P (si, α, sj) = 1, that is, the total of the probabilities of the transitions leaving a state must be equal to 1.

Note that in their original definition, DTMCs have no action. We need them here to relate transitions in a DTMC with their counterpart in an FTS. Still, our alternate definition remains in accordance with the action-less variant. Fundamentally, in both cases, the probability function associates probability values to transitions. Thus, the fact that transitions are defined either as sequences of states and actions or as sequences of states is meaningless. Also, we consider that there is a single initial state i, that is, τ (i) = 1.

STATISTICAL PRIORITIZATION IN SPL TESTING

In our approach, we consider three models: a FD d to represent the features and their constraints (in Fig. 1a), an FTS f ts over d (in Fig. 1b) and a usage model represented by a DTMC dtmc (in Fig. 1c) whose states, actions, and transitions are subset of their counterpart in f ts: S dtmc ⊆ S f ts ∧ Act dtmc ⊆ Act f ts ∧ trans dtmc ⊆ trans f ts . Moreover, the initial state i of f ts is in S dtmc and has an initial probability of 1, that is, τ (i) = 1.

An alternative would have been to rely on a unified formalism able to represent both variability and stochasticity (see, e.g., [?]). However, we believe it is more appropriate to maintain a separation of concerns for several reasons. First, existing model-based testing tools relying on usage models (e.g., MaTeLO 1 ) do not support DTMCs with variability. Implementing our approach on top of such tools would be made harder should we use the aforementioned unified formalism. Second, the DTMC can be obtained from either users trying the software under test, extracted from logs, or from running code. These extractions methods are agnostic on the features of the system they are applied to. Third, since the DTMC is built from existing software executions, it may be incomplete (as in Fig. 1c). Because of that, there may exist FTS executions that are not exercised in the usage model, resulting in missing transitions in the DTMC. Keeping the FTS and usage models separate is helpful to identify and correct such issues.

Since the usage model does not consider features, there may exist paths in the DTMC that are inconsistent for the SPL. For example, in the usage model of Fig. 1c one can follow the path pay, change, tea, serveT ea, take. This path actually mixes "pay machine" (feature f not enabled) and "free machine" (feature f enabled). The combined use of DTMCs and FTS allows us to detect such inconsistencies.

We propose two testing scenarios: product-based test derivation and family-based test prioritization.

Product-Based Test Derivation

Product-based test derivation is straightforward: one selects one product (by selecting features in the FD), projects it onto the FTS, giving a TS with only the transitions of the product, prunes the DTMC to keep the following property true : S dtmc ⊆ Sts ∧ Act dtmc ⊆ Actts ∧ trans dtmc ⊆ transts. We assume that the probabilities of the removed transitions are uniformly distributed on adjacent transitions (since the probability axiom ∀si ∈ S : s j ∈S P (si, sj) = 1 holds). 1 see:

http://all4tec.net/index.php/en/ model-based-testing/20-markov-test-logic-matelo Finally, we generate test cases using statistical testing algorithms on the DTMC [START_REF] Feliachi | Generating transition probabilities for automatic model-based test generation[END_REF][START_REF] Whittaker | A markov chain model for statistical software testing[END_REF].

This scenario is proposed by Samih and Baudry [START_REF] Samih | Relating variability modelling and model-based testing for software product lines testing[END_REF]. Product selection is made on an orthogonal variability model (OVM) and mapping between the OVM and the DTMC (implemented using MaTeLo) is provided via explicit traceability links to functional requirements. This process thus requires to perform selection of products of interest on the variability model and does not exploit probabilities and traces of the DTMC during such selection. Additionally, they assume that tests for all products of the SPL are modeled in the DTMC. This assumption may be too strong in certain cases and delay actual testing since designing the complete DTMC for a large SPL may take time. We thus explore a scenario where the DTMC drives product selection and prioritization.

Family-Based Test Prioritization

Contrary to product-based test derivation, our approach (in Fig. 2) only assumes partial coverage of the SPL by the usage model. For instance, the DTMC represented in Fig. 1c does not cover serving soda behaviour because no user exercised it. The key idea is to generate traces from the DTMC according to their probability to happen (step 1). For example, one may be interested in analyzing the "serving teas for free" behaviour, which will correspond to the following actions (f ree, tea, serveT ea, take), since it is highly probable (p = 0.729). On the contrary, one may be interested in low probability because it can mean poorly tested products.

The generated traces are filtered using the FTS in order to keep only sequences that may be executed by at least one product of the SPL (step 2). The result will be a FTS', corresponding to a pruned FTS according to the extracted traces. Each valid trace is combined with the FTS' to generated a set of products that may effectively execute this behaviour. The probability of the trace to be executed allows us to prioritize products exercising the behaviour described in the FTS' (step 3).

Trace Selection in the DTMC

The first step is to extract traces from the DTMC according to desired parameters provided by the tester.

Formally, a finite trace t is a finite alternating sequence t = (i = s0, α0, ..., αn-1, sn) such as (sj, αj+1, sj+1) ∈ trans, ∀j ∈ [0, n -1].

To perform trace selection in a DTMC dtmc, we apply a depth-first Search (DFS) algorithm parametrized with a maximum length lmax for finite traces and an interval [P rmin, P rmax] specifying the minimal and maximal values for the probabilities of selected traces. Formally: DF S(lmax, P rmin, P rmax, dtmc) = {(i, α1, ..., αn, i)

∧ n < lmax ∧ ( ∃k : 0 < k < n • i = s k ) ∧ (P rmin ≤ P r(i, α1, ..., αn, i) ≤ P rmax)}
where τ dtmc (s0) = 1 and P r(i, α0, . . . , sn) = τ dtmc (i) × Π n-1 i=0 P dtmc (si, αi, si+1). We initially consider only finite traces starting from and ending in the initial state i (assimilate to an accepting state) without passing by i in between. Finite traces starting from and ending in i corresponds to a coherent execution scenario in the DTMC. With respect to partial finite traces (i.e., finite traces not ending in i), our trace definition involve a smaller state space to explore in the DTMC. This is due to the fact that the exploration of a part of the graph may be stopped, without further checks of the existence of partial finite traces, as long as the partial trace is higher than lmax. The DFS algorithm is hence easier to implement and may better scale to large DTMCs.

Practically, this algorithm will build a n-tree where a node represents a state with the probability to reach it and the branches are the α k labels of the transitions taken from the state associated to the node. The root node corresponds to the initial state i and has a probability of 1. Since we are only interested in finite traces ending in the initial state, the exploration of a branch of the tree is stopped when the depth is higher than then maximal path lmax. This parameter is provided to the algorithm by the test engineer and is only used to avoid infinite loops during the exploration of the DTMC. Its value will depend on the size of the DTMC and should be higher than the maximal "loop free" path in the DTMC in order to get coherent finite traces.

For instance, the execution of the algorithm on the soda vending machine (vm) example presented in Fig. 1b During the execution of the algorithm, the trace (f ree, tea, serveT ea, take) has been rejected since its probability (0.729) is is not between 0 and 0.1.

The downside is that the algorithm will possibly enumerate all the paths in the DTMC depending on the lmax value. This can be problematic and we plan in our future work to use symbolic executions techniques inspired by work in the probabilistic model checking area, especially automatabased representations [START_REF] Baier | Principles of model checking[END_REF] in order to avoid a complete state space exploration.

FTS-Based Trace Filtering and FTS Pruning

Generated finite traces from the DTMC may be illegal: behaviour that cannot be executed any valid product of the SPL. The set of generated finite traces has to be filtered using the FTS such that the following property holds: for a given FTS f ts and a usage model dtmc, a finite trace t generated from dtmc represents a valid behaviour for the product line pl modelled by f ts if there exists a product p in pl such as t ⊆ [[f ts |p ]]T S , where f ts |p represents the projection of f ts using product p and [[ts]]T S represents all the possible traces and their prefixes for a TS ts. The idea here Require: traces, f ts Ensure: traces, f ts ′ 1: Practically, we will build a second FTS' which will represent only the behaviour of the SPL appearing in the finite traces generated from the DTMC. Fig. 3 presents the algorithm used to build an f ts ′ from a set of traces (filtered during the algorithm) and a f ts. The initial state of f ts ′ corresponds to the initial state of the f ts (line 1) and d in f ts ′ is the same as for f ts (line 1). If a given trace is accepted by the f ts (line 3), then the states, actions and transitions visited in f ts when executing the trace t are added to f ts ′ (line 4 to 6). The accept(f ts, t) function on line 3 will return true if there exists at least one product in d f ts that has t as one of its behaviours. On line 7, the f Labels(f ts, t) function is used to enrich the γ f ts ′ function with the feature expressions of the transitions visited when executing t on the f ts. It has the following signature : f Labels : (F T S, trace) → γ → γ and f Labels(f ts, t)γ f ts ′ will return a new function γ ′ f ts ′ which will for a given transition tr = (si

S f ts ′ ← {i f ts } ; i f ts ′ ← i f ts ; d f ts ′ ← d f
α k -→ sj) return γ f ts tr if α k ∈ t or γ f ts ′ tr otherwise.
In our vm example, the set of finite traces generated from step 1 contains two illegal traces: (pay, change, tea, serveT ea, take) and (f ree, tea, serveT ea, open, take, close). Those 2 traces (mixing free and not free vending machines) cannot be executed on the f tsvm and will be rejected in step 2. The generated f ts ′ vm is presented in Fig. 4.

Product Prioritization

At the end of step 2 in Fig. 2, we have an FTS' and a set of finite traces in this FTS'. This set of finite traces (coming from the DTMC) covers all the valid behaviours of the FTS'. It is thus possible to order them according to their probability to happen. This probability corresponds to the the cumulated individual probabilities of the transitions fired when executing the finite trace in the DTMC. A valid finite trace t = (i, α1, . . . , αn, i) corresponding to a path (i α 1 -→ . . . αn -→ i) in the DTMC (and in the FTS') has a probability P r(t) (calculated as in step 1) to be executed. We may perform bookkeeping of P r(t).

The set of products able of executing a trace t may be calculated from the FTS' (and its associated FD). It corresponds to all the products (i.e., set of features) of the FD ([[d]]) that satisfy all the feature expressions associated to the transitions of t. Formally, for t and a FTS' f ts ′ , the set of products prod(t, f ts

′ ) = n k=1 {p | γ f ts ′ (s k α k -→ s k+1 )p = true}.
From a practical point of view, the set of products corresponds to the products satisfying the conjunction of the feature expressions γ f ts ′ (s k α k -→ s k+1 ) on the path of t and the FD d f ts ′ . As d f ts ′ may be transformed to a boolean formula where features become variables [START_REF] Czarnecki | Feature Diagrams and Logics: There and Back Again[END_REF], the following formula can be calculated using a SAT solver: n k=1 (γ f ts ′ (s k α k -→ s k+1 ))∧booleanF orm(d f ts ′ ). At this step, each valid finite trace t is associated to the set of products prod(t, f ts ′ ) that can actually execute t with a probability P r(t). Product priortization may be done by classifying the finite traces according to their probability to be executed, giving t-behaviourally equivalent classes of products for each finite trace t. For instance, for the trace tvm = (pay, change, tea, serveT ea, open, take, close) generated for our vm example the products will have to satisfy the ¬f ∧ t feature expression and dvm. This gives us a set of 8 products (amongst 32 possible): All of them executing tvm with a probability P r(tvm) = 0.009 which is the lowest probable behaviour of the soda vending machine.

{(v

CASE STUDY

To assess the feasibility of our proposed approach, we decided to apply it on Claroline [START_REF]Claroline[END_REF], an open-source web-based application dedicated to learning and on-line collaborative work. Analysing such systems using Markov Chains has already been investigated [START_REF] Sarukkai | Link prediction and path analysis using markov chains[END_REF] (e.g, for link prediction) suggesting that our approach apply to this domain. The instance of Claroline at University of Namur2 is the main communication channel between students and lecturers. Students may register to courses and download documents, receive announcements, submit their assignments, perform online exercises, etc. Due to space constraint, the complete models are not shown in this paper but may be downloaded at http://info.fundp.ac.be/~xde/fts-testing.

Inferring the Usage Model

To derive the DTMC usage model, we used the anonymized Apache access log provided by the IT support team of the university. This file [START_REF] Classen | Featured Transition Systems : Foundations for Verifying Variability-Intensive Systems and their Application to LTL Model Checking[END_REF][START_REF] Samih | Relating variability modelling and model-based testing for software product lines testing[END_REF] Go) contains all the HTTP requests made to access the Claroline instance from January 1st to October 1st 2013. We chose to consider only requests to access PHP pages accessible to users. It represents a total of 12.689.030 HTTP requests. We setup an automated way to infer the usage model from the log, as it would be intractable to do so manually. First, Claroline Log was translated into a MySQL database to ease processing of requests. Indeed, each entry of the table represents a single access to a page: we had to reconstruct the behaviour of an user by grouping requests. We followed a similar approach as Sampath [START_REF] Sampath | Prioritizing user-session-based test cases for web applications testing[END_REF]: we group visited PHP pages by the user (according to their line numbers in the log) to Claroline pages for a given IP and within a 45 minutes timeframe. Once these user traces are computed, we can learn an usage model from them. There are various machines learning approaches to learn such underlying DTMCs [START_REF] Verwer | Pautomac: a probabilistic automata and hidden markov models learning competition[END_REF]: we elected n-grams ( contiguous sequence of n elements) for their close relationship to DTMCs and their reported success in practice for real instances [START_REF] Verwer | Pautomac: a probabilistic automata and hidden markov models learning competition[END_REF]. Our implementation can be thought of a 2-gram one without smoothing (we do not consider unseen 2grams as we focus only on observed behaviour). Concretely, each page is mapped to a state in the usage model and each trace is completed so that it ends in the start state. In the obtained DTMC, the set Act contains actions that indicate the target state of the transition (since they correspond to request to a page). In that setting, they cannot be more than one transition for any two given states. We can therefore ignore actions in our probability computation method. Then, the probability of moving from state si to another sj is given as

P (si, sj) = occ(si, sj)
s∈S occ(si, s) where occ(si, sj) is the occurrence count of the related 2gram. We were able to process the entirety of our log database, ending up in an usage model formed of 96 states and 2149 transitions. The whole inference process took approximately 2 hours on a Ubuntu Linux machine with a Intel Core i3 (3.10GHz) processor and 4GB of memory.

Building Family Models

To obtain the Claroline FD and FTS we proceeded the following way. The FD was built manually by inspecting a locally installed Claroline instance (Claroline 1.11.73 ). The FD describes Claroline with 3 main features decomposed in sub-features : U ser, Course and Subscription.

Subscription may be open to everyone (opt Open Subscription) and may have a password recovery mechanism (opt Lost P assword). User corresponds to the different possible user types provided by default with a basic Claroline installation : unregistered users (U nregisteredU ser) who may access courses open to everyone and registered users (RegisteredU ser) who may access different functionalities of the courses according to their privilege level (Student, T eacher or Admin). The last main feature, Course, corresponds to the page dedicated to a course where students and teacher may interact. A course has a status (Available, AvailableF romT o or U navailable), may be publicly visible (P ublicV isibility)

or not (M embersV isibility), may authorize registration to identified users (AllowedRegistration) or not (Registration Denied) and may be accessed by everyone (F reeAccess), identified users (Identif iedAccess) or members of the course only (M embersAccess). Moreover, a course may have a list of tools (T ools) used to different teaching purposes, e.g., an agenda (opt CourseAgenda), an announcement panel (opt CourseAnnoucements), a document download section where teacher may post documents and students may download them (opt CourseDocument), an on-line exercise section (opt CourseExercise), etc. Since we are in a testing context, one instance of the FD does not represents a complete Claroline instance but rather only the minimal instance needed to play a test-set (i.e., test cases formed as sequences of HTTP requests). Basically, it will correspond to a Claroline instance with its subscription features, one particular user and one particular course. In order to represent a complete Claroline instance, we need to introduce cardinalities on the Course and U ser features [START_REF] Michel | A formal semantics for feature cardinalities in feature diagrams[END_REF]. Eventually we obtained a FD with 44 features.

Regarding the FTS, we employed a web crawler (i.e., a Python bot that systematically browse and record information about a website) [START_REF]Scrapy[END_REF] on our local Claroline instance to discover states of the FTS which as for the DTMC represent visited page on the website. Transitions have been added in such a way that every state may be accessed from anywhere. This simplification, used only to ease the FTS building, is consistent with the Web nature of the application and satisfy the inclusion property: S dtmc ⊆ S f ts ∧ Act dtmc ⊆ Act f ts ∧ trans dtmc ⊆ trans f ts . Moreover, some user traces show that the navigation in Claroline is not always as obvious as it seems (e.g., if the users access the website from an external URL sent by e-mail). Finally, transitions have been tagged manually with feature expressions based on the knowledge of the system (via the documentation and the local Claroline instance). To simulate a web browser access, we added a "0" initial state connected to and accessible from all states in the FTS. The final FTS consists of 107 states and 11236 transitions.

Setup and Results

The DF S algorithm has been applied 4 times to the Claroline DTMC with a maximal length of 98, which corresponds to the number of states, a maximal probability of 1 and 4 different minimal probabilities: 10 -4 , 10 -5 , 10 -6 and 10 -7 to see if a pattern emerge. The execution time took from less 1 presents the parameters provided to the DF S and the resulting traces before (# DTMC tr.) and after checking with the FTS (# Valid tr.). Surprisingly, all traces generated from the DTMC are valid. This is caused by the nature of the Claroline FD. Most of the features are independent from each other and few of them have exclusive constraints. As expected, the average size of the traces increase as the P rmin decrease to tend to 9, 88, the average size of the traces used to generate the DTMC.

As explained in section 3, it is possible to prune the original FTS using the valid traces in order to consider only the valid products capable of executing those traces. In this case, it eventually reduce the number of states and transitions from 107 and 11236 (resp.) to 16 and 66 (resp.) in run 1 and to 69 and 844 (resp.) in run 4. As expected, by controlling the interval size we can reduce the number of traces to be considered and yield easily analysable FTS'.

Table 2 presents the 5 most frequents features appearing in the feature expressions associated to the transitions of the valid traces. In all the runs, CourseDocument appears the most frequently, from 47% in run 1 to 53% in run 4.

Discussion

After running our approach on the Claroline case study, we made the following observations :

First, the independence of the features in the FD and the low size of valid FTS traces does not allow to reduce significantly the number of products associate to each trace. This is caused by the web nature of the application which allows a very free way of travelling in the FTS with lots of independent feature expressions (i.e., most feature expressions concern optional features with few dependencies to other features). In order to get a more concrete product, one could generate longer traces in the DTMC by coupling probabilistic approach to more classical state/transition coverage criteria [START_REF] Mathur | Foundations of software testing[END_REF]. This is left for future work.

Although it is possible to generate the list of products for each trace by taking the conjunction of the feature expressions of the transitions of the trace and the FD, this seems of little interest in this case. The valid FTS traces are short and the FD has too much variability to get a reasonable list of products to test. However, we can tackle this problem by taking only needed features to ensure the execution of a valid trace on the FTS. For instance, if we take the less probable trace in run 1 (probability=1.0149 × 10 -4 ) and its associated feature expression RegisteredU ser∧CourseDocument, the minimal feature configuration to execute the trace on the FTS is done by conjunction the feature expression with the FD and negation all optional features except Course Document and RegisteredU ser. This gives us a set of 260 possible products. This set is reduced to 20 products if we consider only courses available to student with an id, which is the most classical scenario in the UNamur Claroline instance. The tester, with knowledge of the application domain, may also use the feature frequency (presented in table 2) in order to reduce the number of products associated to each trace.

Thirdly, Claroline has a fine grained access control system. In its basic setup, it comes with three user roles: Administrator, Teacher and User. According to its role, a user may or may not perform different actions and access different functionalities. Claroline also allows public access to some pages and functionalities (e.g., a course description) to anonymous users. We think that since those 4 user roles have very different usage profiles, a better approach would be to create 4 different usage models, one for each user role. This confirm our design choice to keep usage model and FTS separated. Unfortunately, it was not possible in our case with the provided Apache access log since the user roles have been erased in the anonymisation process.

Finally, we also tried to implement other selection criteria to apply the DTMC. One may want to get the most probable traces in the DTMC (max). On the Claroline case, this gives us a lot of small traces, most of them with 2 or 3 transitions implying each time the download.php page. After a few investigations, we discovered that this page is used to download documents, text, images, etc. from the platform and that it is the most frequent HTTP request in the Apache access log that we processed. This is coherent with the fact that all the states in the DTMC may return to the initial state, allowing very small traces. On the contrary, getting the less probable traces (min) is hard since it requires a full exploration of the DTMC. Other abstraction/simulation techniques will be investigate in future work in order to allow such criteria and to improve the actual DF S algorithm.

The main threat to our case study is the nature of the considered application. This first case study has been performed on a very particular kind of application: a web application accessible trough PHP pages in a web browser. This kind of application allows a very flexible navigation from page to page either by clicking on the links in the different pages or by a direct access with a link in a bookmark or an e-mail for instance. In our future work, we will apply our approach on other cases from various domains (e.g., system engineering) where the transitions from state to state are more constrained.

RELATED WORK

To the best of our knowledge, there is no approach prioritizing behaviours statistically for testing SPLs in a familybased manner. The most related proposal (outlined in section 3) has been devised by Samih and Baudry [26]. This is a product-based approach and therefore requires selecting one or more products to test at the beginning of the method. One also needs that the DTMC covers all products of the SPL, which is not our assumption here.

There have been SPL test efforts to sample products for testing such as t-wise approaches (e.g. [START_REF] Cohen | Interaction testing of highly-configurable systems in the presence of constraints[END_REF][START_REF] Cohen | Coverage and adequacy in software product line testing[END_REF][START_REF] Perrouin | Pairwise testing for software product lines: comparison of two approaches[END_REF]). More recently sampling was combined with prioritization thanks to the addition of weights on feature models and the definition of multiple objectives [START_REF] Henard | Multi-objective test generation for software product lines[END_REF][START_REF] Johansen | Generating better partial covering arrays by modeling weights on sub-product lines[END_REF]. However, these approaches do not consider SPL behaviour in their analyses.

To consider behaviour in an abstract way, a full-fledged MBT approach [START_REF] Utting | Practical model-based testing: a tools approach[END_REF] is required. Although behavioural MBT is well established for single-system testing [START_REF] Tretmans | Model based testing with labelled transition systems[END_REF], a survey [START_REF] Oster | Model-based software product lines testing survey[END_REF] shows insufficient support of SPL-based MBT. However, there have been efforts to combine sampling techniques with modelling ones (e.g. [START_REF] Lochau | Model-based pairwise testing for feature interaction coverage in software product line engineering[END_REF]). These approaches are also product-based, meaning that may miss opportunities for test reuse amongst sampled products [START_REF] Von Rhein | The PLA model: on the combination of product-line analyses[END_REF]. We believe that benefiting from the recent advances in behavioural modelling provided by the model checking community [START_REF] Asirelli | Design and validation of variability in product lines[END_REF][START_REF] Asirelli | Formal description of variability in product families[END_REF][START_REF] Baier | Principles of model checking[END_REF][START_REF] Classen | Symbolic model checking of software product lines[END_REF][START_REF] Classen | Model checking lots of systems: efficient verification of temporal properties in software product lines[END_REF][START_REF] Fischbein | A foundation for behavioural conformance in software product line architectures[END_REF][START_REF] Lauenroth | Model checking of domain artifacts in product line engineering[END_REF], sound MBT approaches for SPL can be derived and interesting family-based scenarios combining verification and testing can be devised [START_REF] Devroey | A Vision for Behavioural Model-Driven Validation of Software Product Lines[END_REF].

Our will is to apply ideas stemming from statistical testing and adapt them in an SPL context. For example, combining structural criteria with statistical testing has been discussed in [START_REF] Gouraud | A new way of automating statistical testing methods[END_REF][START_REF] Thévenod-Fosse | An investigation of statistical software testing[END_REF]. We do not make any assumption on the way the DTMC is obtained: via an operational profile [START_REF] Musa | The operational profile[END_REF] or by analyzing the source code or the specification [START_REF] Thévenod-Fosse | An investigation of statistical software testing[END_REF]. However, an uniform distribution of probabilities over the DTMC would probably be less interesting. As noted by Witthaker [START_REF] Whittaker | A markov chain model for statistical software testing[END_REF], in such case only the structure of traces would be considered and therefore basing their selection on their probabilities would just be a means to limit their number in a mainly random-testing approach. In such cases, structural test generation has to be employed [START_REF] Feliachi | Generating transition probabilities for automatic model-based test generation[END_REF].

CONCLUSION

In this paper, we combine concepts stemming from statistical testing with SPL sampling to extract products of interest according to the probability of their execution traces gathered in a discrete-time markov chain representing their usages. As opposed to product-based sampling approaches, we select a subset of the full SPL behaviour given as Featured Transition Systems (FTS). This allows us to construct a new FTS representing only the executions of relevant products. This such pruned FTS can be analyzed all at once, to enable allow test reuse amongst products and/or to scale model-checking techniques for testing and verification activities.

Future work will proceed to the improvement of the first implementation and its validation on other cases from various domains. There remains a number of challenges, the main one being an efficient trace extraction algorithm using abstraction and/or simulation techniques. We will also associate statistical prioritization to more classical test selection criteria like transition coverage and state coverage in order to improve the quality of the extracted traces.

Figure 2 :

 2 Figure 2: Family-based test prioritization approach

  gives 5 finite traces: DF S(7; 0; 0.1; DT M Cvm) = { (pay, change, cancel, return); (f ree, cancel, return); (pay, change, tea, serveT ea, open, take, close); (pay, change, tea, serveT ea, take); (f ree, tea, serveT ea, open, take, close)}

Figure 3 Figure 4 :

 34 Figure 3: FTS' building algorithm

  , b, cur, t, eur); (v, b, cur, t, usd); (v, b, cur, t, c, eur); (v, b, cur, t, c, usd); (v, b, cur, t, s, eur); (v, b, cur, t, s, usd); (v, b, cur, t, s, c, eur); (v, b, cur, t, s, c, usd)}

  Act f ts ′ ← Act f ts ′ ∪ t 6:trans f ts ′ ← trans f ts ′ ∪ transitions(f ts, t) 7:γ f ts ′ ← f Labels(f ts, t)γ f ts ′

		ts
	2: for all t ∈ traces do
	3:	if accept(f ts, t) then
	4:	S f ts ′ ← S f ts ′ ∪ states(f ts, t)
	5:
	8:	else
	9:	traces ← traces \ {t}
	10:	end if
	11: end for 12: return f ts ′

Table 1 :

 1 Traces generation

		Run 1	Run 2	Run 3	Run 4
	lmax	98	98	98	98
	P rmin	1E -4	1E -5	1E -6	1E -7
	P rmax	1	1	1	1
	#DTMC tr.	211	1389	9287	62112
	#Valid tr.	211	1389	9287	62112
	Avg. size	4, 82	5, 51	6, 35	7, 17
	σ size	1, 54	1, 54	1, 62	1, 66
	Avg. proba. 2, 06E -3 3, 36E -4 5, 26E -5 8, 10E -6
	σ proba	1, 39E -2 5, 46E -3 2, 12E -3 8, 18E -4
	#FTS' st.	16	36	50	69
	#FTS' tr.	66	224	442	844

Table 2 :

 2 Most frequent features in valid traces

	Run 1 CourseDocument, CourseForum, CourseAnnouce-
	ments, CourseExercise, RegisteredUser
	Run 2 CourseDocument, CourseAnnoucements, Course-
	Forum, RegisteredUser, CourseExercise
	Run 3 CourseDocument, RegisteredUser, CourseForum,
	CourseAnnoucements, CourseWork
	Run 4 CourseDocument, RegisteredUser, CourseForum,
	CourseAnnoucements, CourseWork
	than a minute for the first run to ±8 hours for run 4. Addi-
	tionally to those values, the algorithm has been parametrized
	to consider each transition only once. This modification has
	been made since we discovered after a few tests that the algo-
	rithm produced lot of traces with repeated actions which is
	of little interest in the product prioritization context. Those
	repetitions were due to the huge number of loops in the
	Claroline DTMC.
	Table

http://webcampus.unamur.be

http://sourceforge.net/projects/claroline/files/ Claroline/Claroline_1.11.7/

Acknowledgments

Thanks to Jean-Roch Meurisse and Didier Belhomme for providing the Webcampus' Apache access log.


Anaïs Gouin

Anthony Bretaudeau

Claire Lemaitre

Fabrice Legeai

Identification of mis-assemblies

Remove scaffolds (entire or parts)

"inner" "end"

Analysis of chains matching known false duplications

Identification of mis-assemblies

  Harris RS: Improved pairwise alignment of genomic DNA. Ann Arbor: ProQuest; 2007:84 [2] Kajitani R. et al, Efficient de novo assembly of highly heterozygous genomes from whole-genome shotgun short reads, Genome Res. 2014; 24(8):1384-95 [3] Nguyen V.H., Lavenier D., PLAST: parallel local alignment search tool for database comparison, BMC Bioinformatics, vol 10, no 329, 2009 [4] Waterhouse et al, OrthoDB: a hierarchical catalog of animal, fungal and bacterial orthologs, Nucleic Acids Research, 2013, PMID:23180791 here : potential problems of allelism within a scaffold, at contig level? Other parameters to take into account : SNPs count (to select regions) / mate pairs (to validate corrected assembly)

Our strategy allows a good improvement of the initial assembly. Performing a new assembly with a tool handling heterozygosity (such as Platanus) is still more efficient.

									Number of scaffolds
												Coverage of scaffolds
									dist1
								dist2		lim1 ≤	added coverage	≤ lim2
									Expected size : ~ 375 Mb
			"inners"			"ends"		Initial assembly	Platanus [2]	Corrected
										Allpaths	assembly	assembly
	Added coverage	lim2 lim1	Potential bad annotations : duplications? "included" Expected coverage % query covered	B	lim2 lim1 Added coverage	A	"borders" Potential bad annotations : duplications? Expected coverage max(dist1,dist2)	BUSCO* stats	Total size (Mb) Nb. scaffolds N50 (bp) No hit Single hit Multi hit	526,0 48 272 39 593 14 1497 782	470,1 41 633 75 578 9 2047 237	434,9 41 536 52 733 15 1904 374
									*BUSCO : Benchmarking sets of Universal Single-Copy Orthologs (Arthropoda
									species) [4]


Gaëtan Benoit

Dominique Lavenier

Claire Lemaitre
email: claire.lemaitre@inria.fr

Guillaume Rizk
email: guillaume.rizk@inria.fr

C Deltel

P Lemaitre

D Peterlongo

Lavenier

Bloocoo: a memory efficient read corrector

Next generaPon sequencing technologies generate a high amount of short DNA sequences, but may contain imperfecPons. Some applicaPons, such as assembly, yield beSer results with high quality data, triggering the need for short-read correcPon soVware.

Error correcPon -common methods • Exploits high depth of coverage • MulPple sequence alignment • Suffix tree • Kmer frequency spetrum Most methods use large amount of memory And therefore limited to small datasets No tool able to handle large whole genome re-sequencing data files Mo&va&ons

Inser&on in Bloom filter 3. Error Correc&on

Bloom filter of solid kmers

Kmer spectrum crea&on

Kmer counPng and inserPon of solid kmers in Bloom filter : GATB library DSK: disk streaming of kmers [START_REF] Rizk | DSK: k-mer counPng with very low memory usage[END_REF] Bloom filter

Methods

IdenPfy and correct errors by querying the bloom filter for solid kmers

Correc&on

Corrected read file

Results

Easy integraPon in GATB assembly pipeline

  ..... as in DSK[START_REF] Rizk | DSK: k-mer counPng with very low memory usage[END_REF]



  Simulated read sets on C.elegans  1% error rate, 40 x, 100 bp, 40 M reads Comparison with Musket[START_REF] Liu | Musket: a mulPstage k-mer spectrum-based error corrector for Illumina sequence data[END_REF], Racer[START_REF] Ilie | RACER: Rapid and accurate correcPon of errors in reads[END_REF], SOAPec[START_REF] Li | De novo assembly of human genomes with massively parallel short read sequencing[END_REF], Bless[START_REF] Heo | BLESS: Bloom filter-based error correcPon soluPon for high-throughput sequencing reads[END_REF] Real «big» data :  Human NA12878, 2.8 G reads ~ 100x  440 GB fasta file  ~ 2 billion errors corrected  Requiring only 3.8GB of memory, executed in ~ 30 h Bloocoo: Able to correct large datasets Easy integraPon in the GATB assembly pipeline Time/memory scalable on big datasets Works also on indel errors (ion torrent) the real C. elegans dataset SRX026594 (70x) with and without error-correcPon.• Keep solid kmers ( with abundance above threshold)• Insert them in a bloom filter• Fast cache-coherent implementaPon• 11 bits /kmer  0.5 % false posPves a 56x D.Melanogaster dataset (98M reads) on state-of-the-art methods . From A survey of error-correc)on methods, 2013[START_REF] Yang | A survey of error-correcPon methods for next-generaPon sequencing[END_REF] -indel correction (ion torrent homopolymers)


P.-L Etchepare
email: apierreluc.etchepare@ensiacet.fr

H Vergnes

D Samélor

D Sadowski
email: ddaniel.sadowski@ensiacet.fr

C Brasme

B Caussat

C Vahlas

Amorphous alumina coatings on glass bottles using direct liquid injection MOCVD for packaging applications

Keywords: MOCVD, direct liquid injection, amorphous alumina, aluminum tri-isopropoxide, complex geometry, process modeling, hydrolytic resistance

In the field of packaging, coatings are commonly applied on containers to avoid interactions between them and their content. For glass bottles, application of a thin film prevents interactions with the phase in contact and consequently the alteration of surface properties of the latter. In this article, we propose an innovative way to apply amorphous alumina coatings on glass bottles by metalorganic chemical vapor deposition from aluminum tri-isopropoxide. A numerical model, using the Computational Fluid Dynamics code FLUENT, has been developed to calculate local profiles of gas flow, temperature, concentration and deposition rates into the reactor. The submicrometric alumina films have been deposited at reduced pressure between 480°C and 670°C. Uniform thickness profiles have been determined on cross sections over the length of the bottle and have been successfully simulated. Strongly improved hydrolytic resistance with regard to the uncoated bottles reveals the excellent performance of the films.

Introduction

Aluminum oxide is a promising material with an increasing technological and industrial interest regarding its numerous properties for a wide panel of applications: refractive index and optical transmittance for the optical field [1], insulating properties for microelectronic components [2,3], chemical inertness and high hot-hardness for wear resistance [4], catalyst support [5], protection against corrosion [6,7]. Alumina thin films also provide enhanced barrier properties [8][9][10]. Different techniques exist to form alumina coatings on metal or polymer substrates such as electron beam evaporation [2], magnetron sputtering [3], atomic layer deposition [9], sol-gel [6] or metal-organic chemical vapor deposition (MOCVD) [1]. Among them, MOCVD is one of the most attractive for the preparation of such coatings on complex-in-shape geometries with conformal coverage i.e. uniform thickness along the surface and controlled nature and structure of the material. Alumina MOCVD requires the decomposition of a precursor, such as aluminum tri-isopropoxide (ATI) [6,[11][12][13][14][15], aluminum tri-sec-butoxide [1] or aluminum dimethyl-isopropoxide [4]. ATI is able to produce amorphous and stoichiometric alumina coatings with a smooth and dense microstructure in the temperature range 450°C to 700°C [11][12][13][14]. Deposition of thin films inside cavities or near-closed geometry has only been scarcely investigated [10]. The objective of this study is to apply alumina coating on an almost closed geometry substrate such as a glass bottle, to understand the phenomena occurring during the elaborated MOCVD process by implementing a numerical model, and to test the barrier properties of the film. In that which follows, the MOCVD reactor and processing conditions will be exposed in parallel with the design and assumptions of the process modeling. Results of local profiles of gas velocity and precursor concentration will be presented, followed by experimental and calculated thickness profiles. Coating morphologies and performances will be investigated using Scanning Electron Microscopy (SEM) and hydrolytic resistance respectively.

Experimental

The classical MOCVD of ATI requires conditioning the liquid compound inside a bubbler and vaporizing it with N 2 flow. However, with this technique it is difficult to get constant vapor pressure of the precursor. Taking in account the low activation energy for this process in the entire temperature range of interest [14] which results in mass transport limited operation, it is tedious to establish a robust and reproducible process. Direct liquid injection (DLI) allows overcoming the previously mentioned drawbacks [16,17]. However, the implementation of a DLI technology in a CVD process involving ATI requires its dissolution in a solvent to allow its injection, vaporization and transport to the deposition area. Cyclohexane (C 6 H 12 ) was selected as a convenient solvent of ATI.

ATI is sensitive to humidity and is subjected to ageing upon exposure to ambient atmosphere [11]. Water vapor undergoes partial and total hydrolysis of the ATI molecule Al(O i Pr) 3 in a few volatile compounds Al(O i Pr) 3-n (OH) n (n = 1, 2) and Al(OH) 3 respectively [11,18]. ATI (98% pure, Acros Organics CAS N° 555-31-7) was used as received. It was stored in a glove box and manipulated under dry argon atmosphere: It was sealed into a glass ampoule inside the glove box and then cyclohexane (99.5% extra dry, Acros Organics CAS N° 110-82-7) was injected with a syringe using Schlenk technique under argon flow.

Depositions were performed in a vertical hot-wall reactor, composed of a quartz tube and an inductively heated, external coaxial stainless steel (SS) tube for thermoregulation. Figure 1a presents a photograph of the MOCVD setup. The glass bottle to be coated was positioned on a metallic support. Temperature was monitored by a thermocouple grazing its external bottom surface on the central axis of the bottle. All experiments were performed at 480°C at that point, corresponding to an experimentally determined temperature profile at the internal walls of the bottle shown in Figure 1b. The DLI of the solution is carried out with a Kesmtream Vapbox 500 ® instrument. The reactive gas phase was introduced from the upper part of the reactor through a SS nozzle. It was composed of the vaporized solution and of dilution gas. 99.9992% pure N 2 (Air Products) was used and was fed through lines equipped with MKS mass flow rate controllers. The total flow rate (dilution N 2 , cyclohexane, ATI) was maintained unchanged at 585 standard cubic centimeters per minute (sccm) for all experiments. The liquid flow rate was 0.7 g/min corresponding to an inlet ATI molar fraction of 1.7 10 -3 . The regulated total pressure was set at 666 Pa. At the reactor outlet, products of the reactions were trapped upstream to the pump. Deposition experiments duration was 30 min. After treatment, the coated bottles were fractured following their height. Film thickness and surface morphology along the internal surface of each bottle was measured on cross sections by SEM performed on a LEO 435VP instrument.

Coatings performance was determined by the hydrolytic stability, which is the resistance to the release of soluble mineral substances from the inner surface of the container in contact with water. The measurement conditions come from the European pharmacopoeia taken as reference [START_REF][END_REF]. Glass bottles were carefully cleaned and then filled with distilled water. They were placed into an autoclave and heated during 1 h at 121°C. The hydrolytic resistance was measured by titrating released alkali with 0,01M hydrochloric acid.

Process modeling

A numerical model using the ANSYS suit and the Computational Fluid Dynamics (CFD) code FLUENT 12.1 was used to simulate the deposition of alumina from ATI into the glass bottle. An axisymmetric, 2D configuration was considered, composed of 100,620 cells. The mesh was refined in regions close to the bottle surface and in those with high gradients in particular at the aperture of the bottle, using the velocity gradient as indicator. Grid independence of the results was verified. Conservation equations of mass, momentum, energy and gas species concentration including kinetics of alumina deposition were solved to calculate local profiles of gas flow, temperature, mass fraction and deposition rate inside the reactor. Steady-state, laminar conditions and compressibility effects were considered. Heat of reaction was not taken into account due to the low ATI molar concentration. Properties of the gaseous species and the multicomponent diffusion coefficients were calculated using the kinetic theory. Lennard-Jones parameters for N 2 and H 2 O were taken from the FLUENT database whereas those for C 6 H 12 and C 3 H 6 were found in the literature [START_REF] Bird | Transport Phenomena[END_REF]. These parameters were unknown for ATI and taken from the molecule of n-decane which has almost the same number of carbon atoms. The following apparent chemical reaction was used to describe the deposition of alumina from ATI [14]:

[Al(OC 3 H 7 ) 3 ] 4 → 2 Al 2 O 3 + 12 C 3 H 6 + 6 H 2 O (1)
The corresponding apparent heterogeneous kinetic law is:

R si = k 0 * exp(-E a /RT)*[ATI] n (kg/(m2s)) (2)
where k 0 is a pre-exponential constant, E a (J/mol) an activation energy, T (K) the temperature, [ATI] the ATI concentration and n the apparent reaction order. The kinetic parameters have been taken from Vergnes et al. [18] as k 0 = 5.81 10 7 kg.m 2.5 /(mol 1.5 s), E a = 78.1 kJ/mol and n = 1.5. The inlet ATI and C 6 H 12 mass fractions were taken equal to those provided by the DLI system at the upper section of the nozzle inlet. On the walls, a classical no-slip condition was used for gas velocity and the mass flux density of each species was assumed to be equal to the corresponding heterogeneous reaction rate. The experimentally measured thermal profile was fixed as thermal boundary conditions on the bottle walls.

Results and Discussion

Figure 2 presents calculated local profiles in large and fine scales of gas velocity (Fig. 2a,2b) and of ATI molar fraction (Fig. 2c,2d). Gas flow presents a parabolic profile inside the nozzle and near to the bottle inlet, characteristic of laminar flow. The gas velocity at the entrance of the bottle reaches 97 m/s, and 12 m/s at half-height on the central axis which reveals the existence of high convective transfer rates. These values correspond to a residence time inside the bottle shorter than 100 ms. Moreover, a high velocity (50 m/s) can be observed close to the inlet walls of the bottle contrary to lateral and bottom walls. So, complex combinations of convective and diffusive phenomena exist in the bottle during deposition. ATI is rapidly consumed at the inlet of the bottle. The ATI molar fractions at the bottle inlet decrease to 50% of the initial molar fraction due to the deposition of alumina. However, the ATI concentration close to the internal walls is quite homogeneous even if a higher concentration can be noticed at half-height due to the higher convective flux in this zone. Figure 3 presents the experimentally measured thickness of the coating along the bottle (squares). The thickness profile is almost constant on the lateral walls between 0 and 50 mm, ranging between 0.25 and 0.48 µm with a local maximum at 30 mm. The thickness highly increases at the bottle entrance to exceed 3 µm. Similarly, the calculated profiles present a smooth curved evolution along the bottle from 0 to 50 mm and a high increase at the inlet. A satisfactory agreement between experimental results and simulation is obtained which validates in a first approach the model and in particular the adopted kinetic scheme. At the bottle inlet, the calculated high gas velocities which involve a high supply of ATI, lead to the high thicknesses. On the lateral walls between 0 and 50 mm, the curved thickness profile can be related to the experimental thermal profile (Fig. 1a) and also to the ATI molar fraction field in this area (Fig. 2b). The experimental conversion rate of ATI, calculated by the ratio between the inlet ATI molar fraction and the deposited alumina molar fraction, is 14%. A macroscopic observation of the coated bottle reveals colorful iridescences which indicate effective deposition on the entire internal surface, but with thickness variations. Figure 4 presents a photograph of a coated glass bottle with iridescences at the bottom part and a macroscopic white stain at the entrance and the corresponding SEM cross-section micrographs. The coating morphology at the bottom on the lateral walls looks dense and smooth, i.e. it matches exactly the glass surface, in agreement with previous studies [13]. At the bottle inlet, a smooth and glassy layer also exists but with spherical sub-micrometric nodules at the film surface. Krumdieck et al. [15] also noticed this phenomenon during deposition of Al 2 O 3 from ATI on stainless steel using pulsedpressure MOCVD. They correlated the surface morphology with the process i.e. the generation of pulsed concentrated precursor droplets heated and pyrolysed close to the surface. This phenomenon could also be related to gas phase nucleation or homogeneous reactions of ATI, high growth rate or misalignment of the nozzle with the glass bottle. EDX measurements indicate that no carbon is present in the coating. Alumina thin films deposited in the equivalent operating conditions on Si substrate do not diffract X-rays [11][12][13]15]. The hydrolytic resistance was measured on three bottles, coated in the same processing conditions. Because of the low capacity (50 ml) of the bottles, results are extrapolated to 100 ml with respect to the European pharmacopoeia [START_REF][END_REF]. The mean hydrolytic stability of the coated bottles corresponds to a volume of 0.27 ± 0.01 ml of 0.01M HCl. This value favorably compares with that of uncoated glass bottle, which is 3.3 ml of 0.01M HCl. It thus appears that alumina coatings on soda-lime glass bottles present excellent performances improving the hydrolytic resistance of uncoated bottles.

Conclusions

Alumina coatings were processed on glass bottles by MOCVD from ATI at 666 Pa in a vertical hot-wall reactor. A numerical model based on an apparent heterogeneous kinetic law was established from operating conditions to simulate the deposition of alumina. The good agreement between the experimental and the calculated thickness profiles validates the process model for the tested conditions. The results point up high film thicknesses at the bottle entrance due to high ATI molar fractions and high gas convective flux. The model allows better understanding the chemical and physical phenomena occurring inside the bottle. The coating morphology is dense and smooth on the internal walls except at the bottle entrance where spherical nodules are present at the surface. It is expected that the model will help optimizing the process in order to obtain uniform thickness profiles of smooth alumina films. The first results obtained for hydrolytic resistance reveal excellent performances of the amorphous alumina films.

Fig. 1 :

 1 Fig. 1 : (a) Photograph of the MOCVD reactor for deposition of alumina coating inside glass bottles -(b) Temperature profile on the internal wall (sketched) of the bottle

Fig. 2 :

 2 Fig. 2 : Local profiles at two different scales of gas velocity (a,b) and ATI molar fraction (c,d).

Fig. 3 :

 3 Fig. 3 : Experimental and simulated profiles of alumina thickness along the height of the internal bottle wall

Fig. 4 :

 4 Fig. 4 : (a) Photograph of a coated glass bottle and (b) SEM cross-section micrographs of alumina coatings

Acknowledgments

We are indebted to Hervé Guillon, Kemstream Montpellier for advice on the operation of the DLI facility.