Christopher L Berhaut
email: christopher.berhaut@outlook.com

Rémi Dedryvère

Laure Timperman

Grégory Schmidt

Daniel Lemordant

Mérièm Anouti

Remi Dedryvère

Mériem Anouti

A new solvent mixture for use of LiTDI as electrolyte salt in Li-ion batteries

Keywords: Li-ion batteries, 4, 5-dicyano-2-(trifluoromethyl)imidazol-1-ide (LiTDI), electrolyte thermal stability, Solid Electrolyte Interphase, salt dissociation

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Introduction

To keep up with the ever-increasing number of renewable energy generating systems there is a need to improve the battery energy storage. Future batteries should present both a larger energy density and an improved security at lower cost. Lithium-ion batteries (LiB) remain the most suited candidate for this purpose owing to their low weight, low self-discharge and already good electrochemical performances. LiB technology has become popular over the years and today, LiBs are widely used to convert stored chemical energy into electrical energy in portable electronics [START_REF] Erickson | New Horizons for Conventional Lithium Ion Battery Technology[END_REF] as well as in larger scale applications such as electric vehicles. Over years, the success of this technology has been largely dependent on the choice of electrode materials and electrolyte formulations [START_REF] Duncan | Electrolyte Formulations Based on Dinitrile Solvents for High Voltage Li-Ion Batteries[END_REF][START_REF] Eshetu | LiPF6 electrolytes in contact with lithiated graphite: Comparing thermal stabilities and identification of specific SEI-reinforcing additives[END_REF][START_REF] Niedzicki | New type of imidazole based salts designed specifically for lithium ion batteries[END_REF]. Most of the commercial LiBs used in currently available electrical devices contain as electrolyte lithium hexafluorophosphate (LiPF6) dissolved in a mixture of alkylcarbonates. The main reasons for this [START_REF] Eshetu | LiPF6 electrolytes in contact with lithiated graphite: Comparing thermal stabilities and identification of specific SEI-reinforcing additives[END_REF][START_REF] Gnanaraj | LiPF3(CF2CF3)(3): A salt for rechargeable lithium ion batteries[END_REF][START_REF] Dahbi | Comparative study of EC/DMC LiTFSI and LiPF6 electrolytes for electrochemical storage[END_REF] are the high conductivity of LiPF6 solutions, the formation of a protective solid electrolyte interphase (SEI) on graphite negative electrodes and the ability of LiPF6 to passivate aluminum, a lowcost metal used as current collector for positive electrodes. However, the lack of thermodynamic stability and the moisture sensitivity of LiPF6 induce high risks of releasing HF and/or PF5 via its thermal decomposition or hydrolysis in the presence of water. These drawbacks have been highly criticized justifying the importance in finding a replacement. The most important function of the electrolyte is charge transportation from one electrode to the other and so the main requirements are: (i) a high ionic conductivity for battery power, (ii) a low viscosity for cell filling, (iii) high electrochemical and chemical stabilities to avoid solvent decomposition and gazing at the electrode interfaces, (iv) no phase change within the working temperature range and, (v) good wetting properties of the separator and the composite electrodes. Moreover, a large life-span without safety risks, especially when damaged, is highly considered by users and manufacturers.

Owing to its high thermal stability, low sensitivity to the presence of water and electrochemical stability [START_REF] Niedzicki | New covalent salts of the 4V class for Li batteries[END_REF], lithium 4,5-dicyano-2-(trifluoromethyl)imidazol-1-ide (LiTDI) is seen as a promising substitute for LiPF6 in LiBs. Recent studies [START_REF] Niedzicki | New type of imidazole based salts designed specifically for lithium ion batteries[END_REF][START_REF] Paillet | Power capability of LiTDI-based electrolytes for lithium-ion batteries[END_REF] demonstrate that LiTDI is able to protect aluminum from corrosion without releasing HF and can be used with standard cathode material due to its electrochemical stability (up to 4.5 V vs. Li + /Li) [START_REF] Paillet | Determination of the electrochemical performance and stability of the lithium-salt, lithium 4,5-dicyano-2-(trifluoromethyl) imidazolide, with various anodes in Li-ion cells[END_REF]. In alkylcarbonate mixtures like EC/DMC, LiTDI provides reasonable conductivities [START_REF] Paillet | Power capability of LiTDI-based electrolytes for lithium-ion batteries[END_REF][START_REF] Berhaut | LiTDI as electrolyte salt for Li-ion batteries: transport properties in EC/DMC[END_REF][START_REF] Niedzicki | LiTDI and solvent mixture based electrolytes for lithium-ion cells[END_REF][START_REF] Niedzicki | Optimization of the lithium-ion cell electrolyte composition through the use of the LiTDI salt[END_REF] which nevertheless remain lower than that obtained with LiPF6. As a matter of fact, LiTDI is not well dissociated in alkylcarbonates owing to the formation of ion pairs (IP) [START_REF] Berhaut | Ionic association analysis of LiTDI, LiFSI and LiPF6 in EC/DMC for better Li-ion battery performances[END_REF] and possibly larger aggregates. This is the main reason for the lower conductivity of electrolytes containing LiTDI as compared to LiPF6 or other lithium-salts such as lithium tris(pentafluoroethane)-trifluorophosphate (LiFAP), lithium bis(trifluoromethylsulfonyl)imide (LiTFSI) or lithium bis(fluorosulfonyl)imide (LiFSI) [START_REF] Berhaut | LiTDI as electrolyte salt for Li-ion batteries: transport properties in EC/DMC[END_REF].

In this work a new ternary mixture is proposed for using LiTDI as an electrolyte lithium salt. This mixture, called EGM in the following, is composed of ethylene carbonate (EC), γbutyrolactone (GBL) and methyl propionate (MP) in the weight proportions (1: 1: 1). The new LiTDI-EGM electrolyte, EGM in which LiTDI is added at a concentration of 1 mol.L -1 , will be compared to LiTDI-EC/DMC (1:1 by weight) and LiPF6-EC/DMC (1:1 by weight). LiPF6-EC/DMC has been used as an electrolyte for many purposes and its properties are well described in the literature [START_REF] Dahbi | Comparative study of EC/DMC LiTFSI and LiPF6 electrolytes for electrochemical storage[END_REF][START_REF] Sloop | Chemical Reactivity of PF 5 and LiPF6 in Ethylene Carbonate/Dimethyl Carbonate Solutions[END_REF][START_REF] Sun | Using a Boron-Based Anion Receptor Additive to Improve the Thermal Stability of LiPF6-Based Electrolyte for Lithium Batteries[END_REF][START_REF] Markevich | Reasons for capacity fading of LiCoPO4 cathodes in LiPF6 containing electrolyte solutions[END_REF]. The following properties will be considered: salt dissociation, electrolyte thermal stability, passivation of aluminum and lithiated graphite (SEI formation). Electrode material capacity retention will be studied within half-cells (graphite (Gr)/Li, LiNi1/3Mn1/3Co1/3O2 (NMC)/Li). Extended cycling will be performed using full NMC/Gr cells. The LiTDI-EGM electrolyte will also be tested under severe operating conditions, i.e. high (60 °C) and low (-20 °C) temperatures and under high discharge rates (up to 10D).

Experimental section

Battery preparation

Highly pure (GC grade, purity > 99 %) ethylene carbonate (EC), analytical g-butyrolactone (GBL, purity > 99 %), methyl propionate (MP, GC grade, purity > 99 %) and anhydrous dimethylcarbonate (DMC, purity > 99 %) purchased from Sigma-Aldrich, were used as received. The alkyl-carbonate mixture EC/DMC (1: 1, by weight) and the EGM mixture (EC/GBL/MP, 1: 1: 1, by weight) were prepared using a Sartorius 1602 MP balance with an accuracy of ±1 x 10 -4 g in an argon filled MBraun glove box at 25 °C containing less than 5 ppm of moister and oxygen. A GeneCust desiccant bag was placed in the solvent mixtures to reduce the water content to a minimal level (10 ppm).

Battery grade lithium hexafluorophosphate (LiPF6) purchased from Fluorochem was kept and used under the dry atmosphere of the glove box. Lithium 4,5-dicyano-2-(trifluoromethyl)imidazol-1-ide (LiTDI) was supplied by Arkema and kept in the glove box. LiTDI and LiPF6 based electrolytes were prepared in the glove box at concentrations ranging from 1 x 10 -4 mol.L -1 to 2 mol.L -1 and kept there before use.

Concentrations below 1 mol.L -1 were obtained by dilution. For cell testing, 2 % by weight of fluoroethylene carbonate (FEC) was added to the electrolytes.

Prior to any measurement, the water content of each electrolyte was measured using an 831 Karl-Fisher Coulometer (Metrohm). The water content of the LiPF6 and LiTDI based electrolytes was respectively lower than 20 ± 1 ppm and 100 ± 1 ppm (due to the high hygroscopic behaviour of LiTDI).

NMC and graphite electrodes are gifts from Arkema and were dried at 90 °C under vacuum for two hours before being stored in the glove box where the CR2032 coin-cells were then prepared.

Experimental methods

The relative permittivity εr of EGM and EC/DMC were obtained through impedance measurements. A two electrode Swagelok® system was filled with a small amount of solvent (≈ 0.1 mL) inside the glove box allowing a thin layer (e ≈ 0.8 mm) of liquid spreading between the two steel connectors.

The capacity C of the cell was determined by impedance measurements performed with an A.C. signal (10 mV) in the frequency range of 500 MHz to 100 mHz with 50 points per decade. The permittivity (ε) and relative permittivity (εr) of the mixture were then calculated using equation ( 1):

= = / Eq. 1
where e is the distance between the two disk electrodes, S the surface area of the disks, C the capacity of the cell and ε0 the vacuum permittivity (8.854187 x 10 -12 F.m -1 ).

Differential Scanning Calorimetry was performed with a PerkinElmer DSC 4000. Samples, initially at 30 °C, were first cooled down to -40 °C and then heated up to 70 °C before being cooled once more down to -60 °C for EC/GBL/MP based electrolytes or -40 °C for EC/DMC based electrolytes and finally heated up to 150 °C. Each scan was performed at 2 °C.min -1 and was followed by a 1 minute isothermal plateau.

Conductivity measurements were performed using a BioLogic Multichannel Conductivity

Meter based on frequency Response Analyser (MCM 10). Measurements where made using sealed cells with Pt parallel-plate-electrodes protecting the samples from air exposure. For temperature control, the cells were connected to a Peltier based temperature control unit with 10 slots (WTSH 10).

Density and viscosity measurements were carried out from 10 °C to 80 °C using respectively an Anton Parr digital vibrating tube densitometer (model 60/602, Anton Parr, France) and an Anton Parr rolling-ball viscometer (Lovis 2000M/ME, Anton Parr, France). In both cases the cell temperature was regulated within ± 0.02 °C. Dynamic viscosity values reported in this paper were calculated by taking into account the effect of the sample density and the buoyancy of the ball in each sample as a function of the temperature. The densitometer was firstly calibrated at all temperatures with degassed water and dehumidified air at atmospheric pressure as recommended by the constructor while ultra-pure water was used to calibrate the viscometer. The uncertainty of the density and viscosity measurements are better than 5 × 10 -5 g.cm -3 , and 1 %, respectively.

The salt dissociation coefficient (αD) was estimated herein using the Walden rule given by equation ( 2):

° °= °= Eq. 2
where Λ, Λ°, η and η° are respectively the molar conductivity, the high dilution limit molar conductivity, the medium viscosity and the solvent mixture viscosity. The limiting molar conductivity Λ° values are obtained by extrapolating at high dilution the Debye-Hückel-Onsager equation:

= ° -√ Eq. 3
where S is a constant.

Electrochemical measurements were performed on a MPG2 Biologic multichannel potentiostat at the exception of impedance measurements which were conducted on a VMP3 Biologic multichannel potentiostat.

Half and full-cells were tested using the same galvanostatic procedure with a constant charge rate of C/5 (Gr lithiation, NMC delithiation) and discharge rates ranging from D/24 to 10D.

NMC/Li half-cells and NMC/Gr cells were cycled between 3 V and 4.2 V vs. Li + /Li and Gr/Li half-cells between 0.01 V and 1 V vs. Li + /Li.

Electrochemical impedances were obtained using three electrode Swagelok® systems with lithium metal as reference electrode, NMC as counter electrode and Gr as working electrode.

Working and counter electrodes were separated by a Whatman® glass microfiber filter. An A.C. signal of 10 mV was applied in the frequency range of 500 kHz to 10 mHz with 20 points per decade.

The determination of the electrochemical stability window and aluminum passivation tests were carried out using three electrode Swagelok® systems with lithium metal as reference electrode and polished platinum disks, or aluminum disks, (10 mm ø) as working and counter electrodes. The system was run for five cycles at 1 mV.s -1 within the potential range of 0.01 V to 5 V for the determination of the electrochemical window and between 2 V and 4.7 V at 0.2 mV.s -1 in the case of aluminum passivation tests.

The Scanning Electron Microscope (SEM) images where obtained using a Zeiss ultra + equipped with a secondary electron sensor.

The X-ray photoelectron spectroscopy (XPS) measurements were carried out with a Kratos Axis Ultra spectrometer, using a focused monochromatized Al Kα radiation (hν = 1486.6 eV). The XPS spectrometer was directly connected to an argon dry box, through a transfer chamber, to avoid moisture/air exposure of the samples. The spectrometer was calibrated using the photoemission line Ag 3d5/2 (binding energy 368.3 eV). For the Ag 3d5/2 line, the full width at half-maximum was 0.58 eV under the recording conditions. The analyzed area of the samples was 300 µm x 700 µm. The analyzing depth of the XPS experiment was the first 5 nm of the surface of the electrode. Peaks were recorded with a constant pass energy of 20

eV. The pressure in the analysis chamber was around 1 x 10 -9 Pa. Short acquisition time spectra were recorded before and after each experiment and compared to make sure the samples did not suffer from degradation under X-ray beam during measurements. The binding energy scale was calibrated from the hydrocarbon contamination using C1s peak at 285.0 eV.

Core peaks were analyzed using a non-linear Shirley-type background [START_REF] Shirley | High-Resolution X-Ray Photoemission Spectrum of the Valence Bands of Gold[END_REF]. The peak positions and areas were optimized by a weighted least squares fitting method, using 70 % Gaussian, 30 % Lorentzian lineshapes. Quantification of the chemical species at the surface was performed on the basis of Scofield's relative sensitivity factors [START_REF] Scofield | Hartree-Slater subshell photoionization cross-sections at 1254 and 1487 eV[END_REF]. The coin-cells were opened in the argon-filled glove box and the electrodes were rinsed there with DMC. The DMC rinse was used to remove electrolyte salt residues that are not inherent to the SEI.

During rinsing, the electrode samples were soaked in 2 mL DMC for 30 min and then dried for 1 h (under reduced pressure) at ambient temperature. The electrode samples were then mounted on the XPS sample holder in the Ar glove box and sent to the IPREM laboratory under vacuum for XPS analysis. Baseline data for the anodes was achieved by analyzing a pristine sample.

Results and discussion

EGM as solvent mixture for a LiTDI based LiB electrolyte

The physical properties of the EGM mixture and each of its pure components [START_REF] Marcus | Ion solvation[END_REF][START_REF] Welton | Solvents and Solvent Effects in Organic Chemistry[END_REF] are reported in table 1 together with those [START_REF] Rodríguez | Viscosities of Dimethyl Carbonate or Diethyl Carbonate with Alkanes at Four Temperatures. New UNIFAC-VISCO Parameters[END_REF] of the EC/DMC mixture given for comparison.

Table 1.

GBL has been chosen as a component of the mixture for its high dipolar moment, its high permittivity and relatively low viscosity. Hence, lithium salts are readily dissolved in this solvent and their solutions exhibit high conductivity [START_REF] Huang | Study on γ-butyrolactone for LiBOB-based electrolytes[END_REF][START_REF] Xu | Nonaqueous Liquid Electrolytes for Lithium-Based Rechargeable Batteries[END_REF]. Another interesting property is the large temperature domain in which it remains liquid. However, GBL cannot be used in commercial cells with LiPF6 as lithium salt as it forms a highly resistive solid electrolyte interphase (SEI) on graphite which affects badly the battery performance [START_REF] Kinoshita | Effects of cyclic carbonates as additives to γ-butyrolactone electrolytes for rechargeable lithium cells[END_REF]. EC was chosen as a co-solvent mainly for its ability to form a stable SEI on graphite and also for its high polarity, high permittivity and high boiling point [START_REF] Fong | Studies of Lithium Intercalation into Carbons Using Nonaqueous Electrochemical Cells[END_REF]. Finally, MP was chosen as a third cosolvent for its low viscosity and low melting temperature with the aim of increasing the fluidity of the electrolyte and reducing the solidification point of the mixture. As a result, the EGM mixture exhibits a lower viscosity than EC/DMC and a higher permittivity which favours salt dissociation.

In the presence of 1 mol.L -1 LiTDI the conductivities range from 4.33 mS.cm -1 (LiTDI-EC) to 8.32 mS.cm -1 (LiTDI-EGM). Only by mixing EC, GBL and MP together could the optimum conductivity be obtained meaning that the contribution of the ternary mixture to the electrolyte conductivity is greater than the contributions offered by each of its separate pure solvents.

Electrochemical and thermal stability

The electrochemical stability windows of electrolytes containing LiTDI at 1 mol.L -1 in EC/DMC or EGM have been investigated on a platinum electrode in the presence of 2 % by weight of FEC. The latter is added as a SEI forming additive. Cyclic voltammograms presented in figure 1 show that the anodic limit of these electrolytes is limited to 4.7 V vs.

Li + /Li owing to the oxidation of TDI -anions [START_REF] Niedzicki | New covalent salts of the 4V class for Li batteries[END_REF]. Indeed, it has been reported that the solvent mixture itself is stable up to at least 5 V [START_REF] Tarascon | New electrolyte compositions stable over the 0 to 5 V voltage range and compatible with the Li1+xMn2O4/carbon Li-ion cells[END_REF]. It was also noticed that the use of EGM increased the potential at which the anion oxidation starts.

Figure 1.

The reduction peaks situated at 1.9 V vs. Li + /Li is attributed to water as the mean water content of LiTDI solutions are around 100 ppm. The following peaks at 1.5 V vs. Li + /Li and 0.8 V vs. Li + /Li correspond to the reduction of the SEI forming compounds: FEC and EC respectively.

Conductivity and ion-pair dissociation of LiTDI in EGM

Taking into account the thermal stability of LiTDI-EGM, the conductivity of this electrolyte was measured from -40 °C to 80 °C. Conductivity values are presented in figure 3 as a function of the salt concentration at several temperatures.

Figure 3.

The electrolyte conductivity increases with LiTDI concentration due to an increasing number of charged species before reaching a maximum that depends on the temperature (this phenomenon is the consequence of the increase in fluidity when the temperature is raised).

At higher salt concentrations, the electrolyte viscosity increase becomes predominant and as a result, the conductivity decreases. It should also be noted that the increasing salt concentration favors the formation of ion-pairs (IPs) and larger aggregates that also contribute to the drop in conductivity. From literature data, the conductivities of electrolytes containing LiTDI at 1 mol.L -1 can be compared: EGM allows the highest conductivity with 8.5 mS.cm -1 at 25 °C, followed by EC/DMC with 6.8 mS.cm -1 at 25 °C [START_REF] Berhaut | LiTDI as electrolyte salt for Li-ion batteries: transport properties in EC/DMC[END_REF], 8EC/16DMC/1DME (dimethoxyethane) with 6.13 mS.cm -1 at 25 °C [START_REF] Niedzicki | Optimization of the lithium-ion cell electrolyte composition through the use of the LiTDI salt[END_REF], and finally 3EC/7DEC(diethyl carbonate) with 3.39 mS.cm -1 at 20 °C [START_REF] Paillet | Power capability of LiTDI-based electrolytes for lithium-ion batteries[END_REF].

LiTDI-EGM also exhibits a relatively high conductivity at low temperature (1.65 mS.cm -1 at -40 °C) making it a good candidate for low temperature applications. In comparison, LiPF6 in EC/DEC/DMC/MA (methyl acetate) has a conductivity of 2 mS.cm -1 at -40 °C [START_REF] Smart | Development of High Conductivity Lithium-ion Electrolytes for Low Temperature Cell Applications[END_REF]. The dissociation coefficient (αD) of LiTDI in EGM and EC/DMC was calculated using the Walden rule [START_REF] Berhaut | LiTDI as electrolyte salt for Li-ion batteries: transport properties in EC/DMC[END_REF][START_REF] Schreiner | Fractional Walden Rule for Ionic Liquids: Examples from Recent Measurements and a Critique of the So-Called Ideal KCl Line for the Walden Plot †[END_REF][START_REF] Longinotti | Fractional Walden Rule for Electrolytes in Supercooled Disaccharide Aqueous Solutions[END_REF], as reported in the experimental section, and values are plotted as a function of the temperature and LiTDI concentration in figures 4a and 4b. It is interesting to notice that the dissociation coefficient of LiTDI does not depend on the temperature. This is due to a coupling effect between the conductivity and viscosity of LiTDI [START_REF] Berhaut | LiTDI as electrolyte salt for Li-ion batteries: transport properties in EC/DMC[END_REF][START_REF] Berhaut | Ionic association analysis of LiTDI, LiFSI and LiPF6 in EC/DMC for better Li-ion battery performances[END_REF]. The dissociation coefficient of LiTDI is larger in EGM (37 %) than in EC/DMC (31 %) and is not temperature dependent. As seen in figure 4b, αD decreases sharply when the salt concentration is increased.

Passivation layers at interfaces

The ability of an electrolyte to prevent cathodic current collector corrosion or anodic graphite exfoliation by forming a stable passivation layer is essential to the cell life-span and its capacity retention.

Aluminum passivation

It has already been demonstrated that LiTDI in EC/DMC and EC/DEC does not continuously corrode aluminum [START_REF] Niedzicki | New covalent salts of the 4V class for Li batteries[END_REF][START_REF] Paillet | Power capability of LiTDI-based electrolytes for lithium-ion batteries[END_REF]. The cyclic voltammograms (CV) presented in figure 5 prove that, as expected, LiTDI-EGM also does not continuously corrode aluminum. Corrosion can be observed on the first CV cycle at 3.8 V vs. Li + /Li but stops as of the second CV cycle.

The inset x 500 SEM image in figure 5 represents the surface of an aluminum current collector after 20 cycles in the presence of LiTDI-EGM. The EDX analysis, not presented here, reveals the existence of Al and O elements only. corresponding to the LiTDI cyanide functions (C≡N) and at 402-403 eV (N3) which corresponds well to an azide or -(N -)-specie (402.1 eV) [START_REF] Zorn | X-ray Photoelectron Spectroscopy Investigation of the Nitrogen Species in Photoactive Perfluorophenylazide-Modified Surfaces[END_REF]. In conclusion, the F1s spectrum suggests that, like LiPF6, LiTDI helps form a AlF3 protective layer at the surface of the aluminum electrode.

Solid Electrolyte Interface

EIS study

The electrochemical impedance diagram of a lithiated graphite electrode dipped in the LiTDI-EGM+2%FEC electrolyte is displayed in figure 7 as a Nyquist plot. The necessity of an SEI forming additive such as FEC is demonstrated in figure 9 where are reported the capacity variations of the graphite electrode in LiTDI-EC/DMC and LiPF6-EC/DMC electrolytes, both additive-free, as a function of the cycling rate (discharge rates from D/24 to D/5).

Figure 9.

Without the use of a SEI forming additive, the capacity of the graphite electrode drops at about 50 % of the initial capacity when the discharge rate is increased from D/24 to D/5 in the presence of the LiTDI based electrolyte whereas the loss in capacity is only of 16 % with the LiPF6 based electrolyte. LiTDI is responsible for the formation of a more resistive SEI which limits the migration of Li + cations when the discharge rate is increased. As suggested by Shkrob et al. [START_REF] Shkrob | Chemical Stability of Lithium 2-Trifluoromethyl-4,5dicyanoimidazolide, an Electrolyte Salt for Li-Ion Cells[END_REF], in the absence of SEI forming additives LiTDI undergoes a peculiar decomposition leading to radicals that disfavor radical polymerization of the solvent over its anionic polymerization and, as a result, leads to the formation of a high resistance SEI.

XPS analysis and SEM observation

The F1s, C1s and N1s XPS spectra of the graphite surface cycled in LiTDI-EGM+2%FEC are displayed in figure 10 along with a SEM image showing the delithiated graphite electrode surface after two C/24 cycles. A granular layer corresponding to the extreme surface of the SEI covers the surface.

Figure 10.

A higher LiF concentration (18.0 % LiF) was found at the surface of the graphite electrode cycled in LiTDI-EGM+2%FEC against only 5.5 % in LiTDI-EC/DMC+2%FEC. This increase in LiF concentration is due to the decomposition of FEC and of TDI -anions which follows a different mechanism when EC/DMC is replaced by EGM. The increase in mineral compounds like LiF over organic species (lithium alkylcarbonates and polymers) thus results in a less resistive SEI. The F2 region in the same F1s spectrum corresponds to CF3 groups originating from the TDI -anion.

In the C1s spectrum, the CF3 peak is almost undetectable as compared to the peaks belonging to carbon species such as carbonates (C4) meaning that LiTDI is a minor compound at the surface of the electrode. Carbonates represent respectively 49 % and 43 % of the graphite surface after cycling in LiTDI-EC/DMC+2%FEC and LiTDI-EGM+2%FEC. The C1, C2 and C3 regions correspond respectively to the C-C, C-OH and C=O chemical bonds.

The N1s spectrum presents two different nitrogen regions. The N1 (398 eV) and N2 (399-400 eV) regions correspond respectively to nitride ions (N 3-as in Li3N) and to cyanide groups belonging to LiTDI. As TDI -anions are the sole source of nitrogen, the presence of lithium nitride suggests a possible reductive decomposition of TDI -.

Galvanostatic cycling performance

In this section, the electrolytes LiTDI-EGM+2%FEC (El-1), LiTDI-EC/DMC+2%FEC (El-2) and LiPF6-EC/DMC+2%FEC (El-3) were tested in NMC/Li and Gr/Li half-cells and in a NMC/Gr full cell at room temperature. As the LiTDI-EGM electrolyte has a sufficient conductivity at -20 °C (2.8 mS.cm -1 ), it was possible to test this electrolyte in the full cell configuration at -20 °C. Moreover, as the LiTDI-EGM mixture is thermally stable until at least 130 °C, unlike the LiPF6-based electrolyte which starts releasing HF and PF5 [START_REF] Paillet | Power capability of LiTDI-based electrolytes for lithium-ion batteries[END_REF],

experiments were also performed at 60 °C using the EL-1 electrolyte in the full cell configuration. Finally, extended cycling of NMC/Gr cells was performed.

Irreversible capacity

The percentage of the irreversible capacities (IR) was calculated according to the following equation:

= ( . . )
."

Eq. 4

where QC.i is the initial charge capacity and QC.n is the charge capacity at cycle n. Results are reported in table 2.

For the LiTDI based electrolytes similar results were obtained in EGM and in EC/DMC.

Gr/Li half-cells irreversible capacities are negligible as of the 4 th cycle and the coulombic efficiency reaches 99.99 %. For LiPF6 one more cycle is needed to reach these values. In NMC/Li half-cells the IR are higher and stabilization occurs only at the 5 th cycle. The IR in NMC/Gr cells are close to those of the NMC/Li half cells and also become negligible after 5

cycles. In all cases, the LiPF6 based electrolyte leads to higher irreversible capacities.

Table 2.

Power capability in half-cells

The discharge capacities of graphite and NMC electrodes were measured in half-cells at several discharge rates, the charge rate being kept constant at C/5 as of the 5 th cycle. In figure 11 are reported the graphite and NMC charge/discharge voltage profiles (a and b) using El-1 electrolyte along with the discharge capacities obtained for each electrolyte under increasing discharge rate (c and d).

El-1 and El-2 electrolytes exhibit similar performances in Gr/Li half-cells up to 6D with capacity retentions respectively equal to 92.5 % and 87.6 %. El-3 could not be used in Gr/Li half-cells at discharge rates over 2D.

El-1 displays better performances than El-2 or even El-3 in NMC/Li half-cells up to 6D with capacity retentions of respectively 66.7 %, 56.8 % and 64.4 %. The higher specific capacities obtained in NMC/Li and Gr/Li half-cells using El-1 are due to a combination of many factors such as the electrolyte conductivity, the interfacial resistances, the salt dissociation and lithium solvation.

Power capability of NMC/Gr full-cells

In NMC/Gr full cells, the limiting electrode is the NMC in order to avoid any lithium plating of the negative electrode. Hence, the reported discharge capacities are those of the NMC electrode, the charging rate being kept constant at C/5 from the 5 th cycle to the last. The cell voltage profile obtained using the El-1 electrolyte is displayed in figure 12 along with the NMC capacities using all three electrolytes as a function of the cycle number.

Figure 12.

The El-1, El-2 and El-3 electrolytes exhibit similar performances up to 7D with respectively 51.5 %, 53.8 % and 60.0 % capacity retention. At discharge rates over 7D, the cells using LiTDI-based electrolytes undergo a more important drop in capacity than those using LiPF6.

The better capacity retention observed in the case of the LiPF6-based electrolyte at high discharge rate is likely the consequence of a lower battery internal resistance. Indeed, El-3 has a higher conductivity, which contributes to a lower internal resistance, as well as lower interfacial resistances than the LiTDI-based electrolytes. At a 10D rate only 38.5%, 26.9% and 52.0% of the initial capacity is recovered respectively with El-1, El-2 and El-3. Both LiTDI-based electrolytes present similar behavior even if El-1 leads to better capacities up to 4D but not at higher rates. As a conclusion, LiTDI is a very promising lithium salt for LiB as it could sustain up to 7D discharge rates with more than 50 % capacity retention in NMC/Gr cells.

In the following two sections we investigate how NMC/Gr cells containing LiTDI are affected by low (-20 °C) and high (60 °C) functioning temperatures and compare the more thermally stable El-1 to El-2.

Power capability in NMC/Gr cells at low temperature (-20 °C)

In order to study the impact of low temperatures on the capacities of NMC/Gr cells using the LiTDI based El-1 and El-2 electrolytes, two different tests were carried out. In the first test, the SEI formation was done at room temperature and the cell was afterwards discharged According to the results presented in figure 13, the temperature at which is formed the SEI has only a limited impact on the cycling capacities measured at -20 °C. However, the latter are lower than those obtained at room temperature and the gap between the two increases with the charge/discharge rate. At C rate, only 31 % of the capacity measured at room temperature is obtained at -20 °C. Still, it can be noticed that the EGM solvent mixture allows better results than EC/DMC. For each test, the coulombic efficiency was found to be close to 100 %.

The charge/discharge profiles of NMC/Gr cells cycled at -20 °C (Fig. 14.a) and at room temperature (Fig. 14.b) are reported Figure 14. At -20 °C, the charge capacity at the first cycle is always higher than those of the following cycles. This phenomenon does not occurs at room temperature and means that part of the lithium inserted in the graphite electrode remains trapped between the layers owing to a low kinetic. The positive NMC electrode is likely responsible for this phenomenon as this material is less electron-conducting at low temperatures [START_REF] Amin | Characterization of Electronic and Ionic Transport in Li1-xNi0.33Mn0.33Co0.33O2 (NMC333) and Li1-xNi0.50Mn0.20Co0.30O2 (NMC523) as a Function of Li Content[END_REF] inducing an ohmic voltage drop.

Figure 14.

When the cell temperature is set back to room temperature after cycling at -20 °C, full capacity recovery is achieved for the El-1 electrolyte with a coulombic efficiency of 99.9 % and for El-2, a value close to the full capacity is obtained (Figure 15). Hence, the capacity loss observed at -20 °C when the charge/discharge rate is increased is reversible.

Power capability in NMC/Gr cells at high temperature (60 °C)

Similar power capability tests were carried out at 60 °C. Because of the increase in electrolyte fluidity and conductivity and decrease in the internal resistance of the cells, higher capacities and coulombic efficiencies are expected. However, as presented in figure 15, a decrease in NMC/Gr cell capacity is observed at 60 °C. Moreover, even if the SEI layer is formed at room temperature instead of 60 °C, only a little increase in the discharge capacities is noted. This means also that the SEI is not responsible for the lower capacities measured at 60 °C as compared to room temperature. It was also noticed that instead of being close to 100 %, the coulombic efficiency is less than 99 %: 90 % at a C/5 rate and 95 % at a C rate in both electrolytes. This means that parasitic reactions occur at the surface of the electrodes like solvent oxidation at the positive electrode or/and SEI dissolution at the negative electrode.

Figure 16.

When the temperature is set back to room temperature, the discharge capacities stayed unchanged as indicated by the graph reported in Figure 16, proving that the capacity loss observed at 60 °C is irreversible.

Figure 17.

The observed capacity loss at 60°C could be related to the decomposition of EC as reported by Kerner et al [START_REF] Kerner | Towards more thermally stable Li-ion battery electrolytes with salts and solvents sharing nitrile functionality[END_REF]. At 70°C, the authors found that EC, in the presence of LiTDI, underwent a decomposition process leading to the production of CO2 as it does when in the presence of thermal decomposition products of PF6 - [START_REF] Sloop | Chemical Reactivity of PF 5 and LiPF6 in Ethylene Carbonate/Dimethyl Carbonate Solutions[END_REF].

Durability tests in NMC/Gr full-cell configuration

Extended cycling was performed at room temperature using the Gr/NMC cell with LiTDIbased electrolytes (El-1 and El-2) and LiPF6-based electrolyte (El-3). The evolution of the discharge capacity with the cycle number is displayed in figure 17. The charge/discharge rate was set at 1C/1D after forming the SEI at C/24 (one cycle only).

Figure 18.

According to the graphs presented in figure 17, both LiTDI-based electrolytes permitted higher discharge capacities than the LiPF6-based electrolyte. The LiTDI-EGM+2%FEC electrolyte leads to the best results as 850 cycles could be performed before the cell lost 30 % of its initial capacity.

Conclusion

A new electrolyte using the LiTDI lithium salt has been formulated. The solvent used is a ternary EC/GBL/MP mixture which allows a better dissociation of the lithium salt than the For comparison a NMC/Gr cell cycled at 25 °C using El-1 (green triangles) was added. Tables Table 1. Melting temperature (T m ), boiling temperature (T b ), density at 25 °C (d 25 ), viscosity at 25 °C ( 25 ), dipolar moment (), relative permittivity ( r ) and conductivity at 25 °C in the presence of LiTDI at 1 mol.L -1 ( LiTDI ) of single solvents EC, GBL, MP [START_REF] Marcus | Ion solvation[END_REF] , DMC [START_REF] Rodríguez | Viscosities of Dimethyl Carbonate or Diethyl Carbonate with Alkanes at Four Temperatures. New UNIFAC-VISCO Parameters[END_REF] and

EGM

  The thermal stability of LiTDI in EC/DMC and in EGM was investigated through DSC experiments. The DSC traces of the EGM and EC/DMC based electrolytes, obtained at 2 °C.min -1 , are reported in figure2. Both electrolytes can withstand temperatures up to ≈ 130 °C, LiTDI-EGM being more stable owing to the lower amount of volatile solvents such as MP and DMC (Tb < 100 °C) in the mixture. As expected, the LiTDI-EGM electrolyte undergoes no phase change between -60 °C and 130 °C whereas LiTDI-EC/DMC solidifies at -40 °C and presents, on the reverse scan, a characteristic eutectic peak followed by the fusion of the remaining crystals. No solidification peak is observed in the case of LiTDI-EGM suggesting a stability of this electrolyte at even lower temperatures.

Figure 2 .

 2 Figure 2.

Figure 4 .

 4 Figure 4.

Figure 5 .

 5 Figure 5.

Figure 6 .

 6 Figure 6.

Figure 7 . 14 Ω

 714 Figure 7.

Figure 8 .

 8 Figure 8.

Figure 11 .

 11 Figure 11.

(Figure 13 .

 13 Figure 13.

Figure 15 .

 15 Figure 15.

FiguresFigure 1 .Figure 2 .

 12 Figures

Figure 3 .

 3 Figure 3. Conductivity of LiTDI in EGM as a function of the salt concentration and

Figure 4 .

 4 Figure 4. LiTDI dissociation coefficient in EC/DMC and EGM measured using the Walden

4 Figure 5 .

 45 Figure 5. First and second voltammograms of an aluminum electrode in LiTDI-EGM between 2 V and 4.7 V at a 0.2 mV.s -1 scan rate. The first CV started with an open circuit potential of

Figure 6 .

 6 Figure 6. XPS spectra of the surface of an aluminum electrode after 20 CV cycles in the

Figure 7 .

 7 Figure 7. Nyquist plot of a lithiated graphite electrode dipped in LiTDI-EGM+2%FEC.

Figure 8 .

 8 Figure 8. Variations Rel, RSEI and Rct of a graphite electrode dipped in (a) LiPF6-

Figure 9 .

 9 Figure 9. Discharge capacities of a graphite electrode cycled in a Gr/Li half-cell using LiTDI-

Figure 10 .

 10 Figure 10. XPS spectra of the elemental composition of the surface of a delithiated graphite

Figure 11 .Figure 12 .

 1112 Figure 11. Charge/discharge profiles for (a) Gr/Li, and (b) NMC/Li half-cells containing El-1

Figure 13 .

 13 Figure 13. Discharge capacities at -20 °C of a NMC/Gr cell after SEI formation at (a) room

1 Figure 14 .

 114 Figure 14. Charge/discharge plots obtained at (a) -20 °C, and at (b) room temperature (Troom).

Figure 15 .

 15 Figure 15. Room temperature discharge capacities of the NMC electrode previously cycled at

Figure 16 .

 16 Figure 16. Discharge capacities of the NMC electrode at several charge/discharge rates

Figure 17 .Figure 18 .

 1718 Figure 17. Room temperature discharge capacities of the NMC electrode previously cycled at

  22classical EC/DMC binary mixture. The LiTDI-EGM electrolyte is capable of withstanding temperatures going from -60 °C to 130 °C without undergoing phase transitions. Its conductivity is higher than 1 mS.cm -1 at -40 °C making it an interesting electrolyte for low temperature applications. This electrolyte formulation is stable up to 4.7 V vs. Li + /Li, does not corrode aluminum and allows the formation of a stable SEI on graphite in the presence of FEC. In terms of power capability, it is able to withstand high discharge rates (up to 7D) in full NMC/graphite cells. LiTDI-EGM, also being more fluid and conductive than LiTDI-EC/DMC, allows better capacities at low temperatures (-20 °C). However, at high temperature (≥ 60 °C), an irreversible capacity loss occurs for full Gr/NMC cells. EC is suspected to decompose in the presence of LiTDI at 60 °C or more. Extended cycling of a Gr/NMC cell containing the LiTDI-EGM electrolyte show that 850 cycles could be performed before losing 30 % of the initial capacity.

Table 2 .

 2 and EC/DMC solvent mixtures. Irreversible capacities as a percentage of the initial charge capacity up to the 4 th cycle for Gr/Li, NMC/Li half-cells and NMC/Gr full cells.

			T m (°C)	T b (°C)	d 25	 25 (mPa.s)	 (D)	 r	 LiTDI (mS.cm -1 )
	EC		36	248	1.32	solid	4.87	90	4.33
	GBL		-44	204	1.12	1.7	4.12	39	7.66
	MP		-88	80	0.91	0.5	1.8 [20]	5.4	4.77
	DMC		5	95	1.06	0.58	0.9	3.1	1.93
	EC/GBL/MP	-48	130	1.16	1.08	/		58±5	8.32
	EC/DMC	-23	130	1.20	1.10	/		43±5	6.11
			Gr/Li			NMC/Li			NMC/Gr
	Cycle	2	3	4	2	3	4	2	3	4
	El-1	8.90	1.92	0	13.16	4.48	2.40	13.78	5.02	2.94
	El-2	9.06	0.22	0	13.29	4.93	2.35	13.36	6.21	2.87
	El-3	7.85	1.46	1.21	19.32	3.30	2.12	22.12	6.11	0.91


Pierre Lecomte

Quentin Leclere

Sébastien Ollivier

Quentin Leclère

Equivalent source model from acousto-optic measurements and application to an acoustic pulse characterization

Keywords: acousto-optic, inverse problem, refracto-vibrometry, acoustic pulse

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

Introduction

The characterization of acoustic sources is generally done with microphone arrays: from a set of discrete pressure measurements, the sources localization, their strengths or their radiation properties are characterized using multichannel signal processing. Popular approaches include beamforming [START_REF] Billingsley | The acoustic telescope[END_REF], near field holography [START_REF] Williams | Fourier Acoustics: Sound Radiation And Nearfield Acoustical Holography[END_REF], equivalent source methods and inverse problems [START_REF] Koopmann | A method for computing acoustic fields based on the principle of wave superposition[END_REF][START_REF] Nelson | Estimation of acoustic source strength by inverse methods: Part I, conditioning of the inverse problem[END_REF], time reversal [START_REF] Fink | Time reversal of ultrasonic fields. I. Basic principles[END_REF] or modal array signal processing [START_REF] Teutsch | Modal Array Signal Processing: Principles and Applications of Acoustic Wavefield Decomposition[END_REF].

In an experimental context, a main drawback of the measurements with microphone arrays is the acoustic influence of the array itself, especially at high frequencies. When the wavelength is of the same order of magnitude as the microphones or their support, the diffraction compromise the measurement validity. For instance, the spark sources characterization with conventional microphone suffer from those problems at high frequencies [START_REF] Yuldashev | Nonlinear propagation of sparkgenerated N-waves in air: Modeling and measurements using acoustical and optical methods[END_REF].

It would be of great interest to have alternative acoustic sources characterization method, which do not suffer from diffraction effects and limited microphone response bandwidth. Especially, if the source characterization is precise enough, then it could be used as a calibration source [START_REF] Torras-Rosell | An Acousto-Optic Method For Free-Field Microphone Calibration[END_REF]. This is the motivation of this work.

An approach of measuring sound in a non-invasive manner relies on the acousto-optic effect which describes the interaction of sound and light when a acoustic waves travels in a transparent medium. The resulting techniques are used in ultrasound acoustics [START_REF] Quate | Interaction of light and microwave sound[END_REF][START_REF] Adler | Interaction between light and sound[END_REF][START_REF] Pitts | Three-dimensional optical measurement of instantaneous pressure[END_REF] and more recently in audible domain [START_REF] Torras-Rosell | An acousto-optic beamformer[END_REF][START_REF] Torras-Rosell | Sound field reconstruction using acousto-optic tomography[END_REF].

The measurement of acousto-optic effect can be done with interferometric techniques [START_REF] Torras-Rosell | An acousto-optic beamformer[END_REF][START_REF] Yuldashev | Mach-Zehnder interferometry method for acoustic shock wave measurements in air and broadband calibration of microphones[END_REF]. However, by doing so, the optical measurement is proportional to the pressure spatial integration along the laser path. In order to solve this integral and find the acoustic pressure, hypothesis on the source radiation can be done. For instance, a radially symmetric radiation hypothesis allows to use the Abel transform [START_REF] Yuldashev | Mach-Zehnder interferometry method for acoustic shock wave measurements in air and broadband calibration of microphones[END_REF]. Otherwise, the acoustooptic tomography allows the characterization of the sound pressure field in a plane [START_REF] Torras-Rosell | Sound field reconstruction using acousto-optic tomography[END_REF]. Besides, for a qualitative characterization of the sound field, a Laser Doppler Vibrometer (LDV) can be used. In this case, the apparent output velocity given by the LDV is proportional to the acousto-optic effect. This latter technique is called refracto-vibrometry [START_REF] Zipser | Interferometric visualization of generation and propagation of sound[END_REF][START_REF] Zipser | Refracto-Vibrometry for Visualizing Ultrasound in Gases, Fluids and Condensed Matter[END_REF][START_REF] Zipser | Refracto-vibrometry -a novel method for visualizing sound waves in transparent media[END_REF].

In this context, one proposes to characterize acoustic sources by solving and inverse problem based on optical measurement. The measurements are obtained from a LDV whose laser beam passes through the sound pressure field with several angles of incidence and is reflected on a rigid surface. An acousto-optic Green function for monopole is highlighted and used to describe the acoustic sources as a combination of monopoles sources, which strengths are found by solving a regularized inverse problem.

The paper is organized as follows: The acousto-optic effect is recalled in and detailed for a monopole source in Sec. 2 and Appendix C. In particular, the effects of finite duration measurements are studied in Appendix B. Section 3 describes the Equivalent Source Method (ESM) inverse problem from acousto-optic measurements. Then, an experimental implementation of the proposed method is done for a spark source through Secs. 4 and 5. Particularly, the experimental method is described in Sec. 4 and the ESM and signal processing parameters are detailed in Sec. 5. Results are presented and discussed in Secs. 5.4 and 5.5. The paper concludes in Sec. 6.

Acousto-optic effect

This section recalls the acousto-optic effect which allows the measurement of sound with optical methods.

As an acoustic wave propagates in air, the variation of acoustic pressure induces a local variation of air refractive index. This phenomenon is called the acousto-optic effect and has been widely studied in ultrasound [START_REF] Quate | Interaction of light and microwave sound[END_REF][START_REF] Adler | Interaction between light and sound[END_REF][START_REF] Pitts | Three-dimensional optical measurement of instantaneous pressure[END_REF] and more recently for audible frequencies range [START_REF] Torras-Rosell | An acousto-optic beamformer[END_REF][START_REF] Torras-Rosell | Sound field reconstruction using acousto-optic tomography[END_REF][START_REF] Torras-Rosell | An Acousto-Optic Method For Free-Field Microphone Calibration[END_REF]. If the acoustic pressure p is much smaller than the static pressure p 0 , which is generally admitted in linear acoustics, the air refractive index can be approximated with the following linear equation:

n n 0 + n 0 -1 γp 0 p, (1)
where n is the air refractive index, p is the acoustic pressure, γ = 1.4 is the ratio of specific heats, n 0 = 1.0002772 and p 0 are the air refractive index and atmospheric pressure under static condition.

To measure the acousto-optic effect, an interferometer using a laser beam is commonly used. As the latter travels through the sound pressure field, a phase shift is induced and is expressed by [START_REF] Torras-Rosell | Sound field reconstruction using acousto-optic tomography[END_REF][START_REF] Yuldashev | Mach-Zehnder interferometry method for acoustic shock wave measurements in air and broadband calibration of microphones[END_REF]:

φ = k 0 L ndl, (2)
where k 0 = ω 0 /n 0 is the wave number of light in vacuum with ω 0 the laser beam pulsation, and L is the path followed by the laser beam.

Measuring the acousto-optic effect with a Laser Doppler Vibrometer

A Laser Doppler Vibrometer (LDV) is a device used to measure the velocity of vibrating structures. A laser beam is emitted by the LDV, travels through a sound pressure field and reflects back on a vibrating surface. The LDV output is a temporal signal which is proportional to the time derivative of the laser beam phase. From Eqs. (2) and Eq. ( 1) one has:

1 k 0 n 0 dφ dt = dL 0 dt + (n 0 -1) γn 0 p 0 d dt L p(l, t)dl, (3)
where L 0 is the laser beam length, that is to say the distance between the laser origin and the reflecting surface. The first term in the right-hand side of Eq. ( 3) is the velocity of the vibrating surface. The right term of the righthand side of Eq. ( 3) traduces the acousto-optic effect. In the case where the reflecting surface does not vibrate, dL 0 /dt 0. Thus, the acousto-optic effect is dominant and the LDV apparent velocity is given by:

v(t) = 1 k 0 n 0 dφ dt = n 0 -1 γp 0 n 0 d dt L p(l, t)dl, (4)
In the frequency domain, the apparent velocity of the LDV is given by [START_REF] Torras-Rosell | An acousto-optic beamformer[END_REF]:

v(ω) = n 0 -1 γp 0 n 0 iωF L p(l, t)dl , (5)
where F is the Fourier transform operator with temporal convention e +iωt , i = √ -1 is the imaginary unit. This indirect use of an LDV to measure sound is also referred as refracto-vibrometry [START_REF] Zipser | Interferometric visualization of generation and propagation of sound[END_REF][START_REF] Zipser | Refracto-Vibrometry for Visualizing Ultrasound in Gases, Fluids and Condensed Matter[END_REF][START_REF] Zipser | Refracto-vibrometry -a novel method for visualizing sound waves in transparent media[END_REF].

Acousto-optic effect for a monopole source

A monopole source with strength q(t) is located at a point x s in space. The pressure radiated at a point x is given by :

p(x, t) = q(t) * δ t -x-xs c 4π x -x s , ( 6
)
where c is the sound speed in air, δ is the delta Dirac distribution and * is the convolution operator. Inserting Eq. ( 6) in Eq. ( 4) one has:

1 k 0 n 0 dφ dt = dq(t) dt (n 0 -1) 4πγp 0 n 0 * L δ t -x-xs c x -x s dl I(t) , (7)
One considers the integral term in Eq. ( 7), denoted I(t). A laser beam emitted by the LDV is crossing the sound pressure field at a distance a from x s . From here, for sake of simplicity, one works in the plane containing the point x s and the laser beam. In this plane, the origin O is chosen at point x s . The local coordinate system is defined with unit vectors u = [1 0] T and v = [0 1] T such as u is parallel at the laser beam. The situation is displayed in Fig. 1: the acoustic pressure is null everywhere except at the radius r = ct.

The distance between the LDV and the rigid reflector is denoted L. If one restrains the signal acquisition duration for the LDV from time t = 0 to time t = √ a 2 + α 2 /c, the Dirac pulse travels up to the distance r = √ α 2 + a 2 from origin. Therefore, the integration path L for the term I(t) in Eq. ( 7) can be reduced to the interval [-α α] :

u v rigid reflector LDV a d = √ a 2 + x 2 r = ct O • -α • +α • •
I(t) = α -α δ t - √ a 2 +x 2 c √ a 2 + x 2 dx ( 8
)
The computation of I(t) in Eq. 8 is detailed in Appendix A. In the frequency domain, I(t) can be approximated by a Bessel function of second kind and degree 0, K 0 :

I(ω) 2K 0 (iωa/c) (9)
Note that this approximation is exact in the case of an infinitely long laser beam, see Appendix A. A study of the approximation error is conducted in Appendix B. Finally, by inserting Eqs. ( 9) and ( 6) into Eq. ( 5), the acousto-optic effect for a monopole gives the LDV apparent velocity approximated by:

v(ω, a) = iωq(ω)(n 0 -1) 4πγp 0 n 0 2K 0 (iωa/c). ( 10
)

Equivalent sources method from acousto-optic measurements

The equivalent source method (ESM) seeks to find a distribution of simple sources modeling the sound pressure field radiated by the acoustic source to be characterized [START_REF] Koopmann | A method for computing acoustic fields based on the principle of wave superposition[END_REF][START_REF] Song | Numerical errors associated with the method of superposition for computing acoustic fields[END_REF][START_REF] Jeans | The wave superposition method as a robust technique for computing acoustic fields[END_REF]. Usually based on microphone measurements, the ESM is adapted in this work for acousto-optic measurements using developments proposed in the Sec. 2. From Eq. 10, a distribution of N monopoles sources produces an apparent velocity at point x m given in the frequency domain by:

vm = N n=1 q n G(x m , x n ), (11)
where, q n and x n are respectively the strength and the location of the n-th monopole. G(x m , x n ) is the acousto-optic Green function given by:

G(x m , x n ) = iω(n 0 -1) 4πγp 0 n 0 2K 0 iω c a(x m , x n ) . (12)
The function a(x m , x n ) in Eq. ( 12) is the normal distance from the n-th monopole position x n to a laser beam originating from a LDV and passing by point x m . The analytical expression of a(x m , x n ) is given in Appendix C. A matrix formulation for M spatial locations is given by:

v = Gq, (13)
where v(M×1) is the apparent velocity vector, q (N ×1) is the source strengths vector and G (M ×N ) is the matrix of acousto-optic Green functions. The apparent velocity vector for the acoustic source to be modeled is denoted v (M ×1) . This vector is measured with a LDV at M spatial locations.

The complex error vector e (M ×1) between the v and v is defined as:

e = v -v. ( 14
)
In order to find the source strengths q which minimize e 2 in Eq. ( 14), while remaining a constrained energy solution, one defines the following regularized cost function J [START_REF] Gauthier | Beamforming regularization matrix and inverse problems applied to sound field measurement and extrapolation using microphone array[END_REF]:

J = v -Gq 2 +λ 2 Wq 2 , ( 15
)
where • denotes the 2-norm, λ is a Tikhonov regularization parameter [START_REF] Tikhonov | Methods for Solving Ill-Posed Problems[END_REF], and W (N ×N ) is a diagonal weighting matrix. In the reported cases, the choice of λ is done in a Bayesian framework [START_REF] Pereira | Empirical Bayesian regularization of the inverse acoustic problem[END_REF]. The solution q minimizing the cost function J, is given by [START_REF] Gauthier | Beamforming regularization matrix and inverse problems applied to sound field measurement and extrapolation using microphone array[END_REF]:

q = (G H G + λ 2 W H W) -1 G H v, (16)
where H is the Hermitian transpose conjugate operator.

Experimental Method

From this section, an experimental implementation of the method described in Sec. 3 is conducted for an acoustic pulse.

Apparatus

The experimental layout is shown in Fig. 2. A top view scheme is shown in Fig. 3. The acoustic pulse is generated with an electric spark between two metal electrodes spaced 10 mm apart. The electrodes diameter is 3 mm. The electrical circuit comes from a piezoelectric oven spark ignitor. The structure supporting the electrodes is built with 2-mm and 3-mm thin metal rods, in order to limit its acoustical influence at high frequencies. It can rotate around the vertical axis. This allows measurements with several source orientations θ, as shown in Fig. 3. The laser beam is emitted by a Polytech PSV 400 c LDV as shown on the left of Fig. 2. It crosses the sound pressure field and reflects back on the rigid reflector.

Geometrical layout

The origin O of the Cartesian coordinate system is chosen in the middle of the two electrodes. The x-axis is pointing towards the rigid reflector, the z-axis toward the ceiling and the y-axis is such that the coordinate system is direct. The LDV scanning head is at coordinate x L (-111.3 cm, 0, 0). A reference microphone, denoted Mic #1, is placed above the spark source at coordinates (0, 0, 11.5 cm) (green support in Fig. 2). An additional control microphone, denoted Mic #2, is placed at coordinates (0, -71 cm, 0) (red support in Fig. 2). Examples of waveforms measured by microphones 1 and 2 are shown in Fig. 4 in black circles. For these measurements, the sampling frequency is f s = 102.4 kHz. As a consequence, the acoustic signal of the pulse is strongly lowpass filtered by the anti-aliasing device of the Digital Acquisition System. In addition, the response of microphones Mic #1 and Mic #2 filters as well the waveforms. As a consequence, the pulse waveforms on acquired 3: Geometrical layout (top view). The measurement points are defined in the virtual scanning plan (Oyz). The electrodes are oriented such that their axis is at an angle θ from y-axis.

signals are not identical to the pressure waveforms, and the analysis of the temporal parameters, such as the rise time, the amplitude or the duration, is compromised [START_REF] Wright | Propagation in air of N waves produced by sparks[END_REF][START_REF] Yuldashev | Mach-Zehnder interferometry method for acoustic shock wave measurements in air and broadband calibration of microphones[END_REF]. However, the present study focuses on the "low" frequency characteristics of the source (< 40 kHz) which are not affected by these filtering effects. Note that for more readability purposes, the solid line signals in Fig. 4 are oversampled by a factor 10.

Measurement points

Measurement points are set in the plane (Oyz). They are organized following a rectangular mesh of dimensions (L y 20 cm × L z 23 cm). The y and z distances between scan points are identically set to ∆ P = 6.1 mm. In this situation, there are M = 1286 measurement points per spark source orientation. In the reported cases, measurements are operated for 10 different spark source orientations θ from θ min = 0 • to θ max = -90 • , with steps of 10 • . Note that the scanning grid is the same for the different source orientations, as shown in Fig. 3.

Measurement procedure and parameters

For each source orientation θ, the LDV scans the 1286-point grid. The acquisition is triggered by a threshold detection on the reference microphone signal (Mic # 1). More precisely, it is triggered a little before the shock wave arrives on the reference microphone, allowing the measurement of points closer to the origin. Acquisition duration is fixed at 625 µs. This short duration makes it possible to be freed from acoustic reflections, in particular that on the rigid reflector. At the same time, it is long enough to measure the furthest points in the scan grid. Rectangular and Hanning windows are operated on the optical and microphone signals respectively. Sampling frequency is set to f s = 102.4 kHz. The frequency range is [1.6 kHz, 40 kHz] with a step of ∆f = 1600 Hz.

The spark source generates a spark approximately every 0.1 s. For each measurement point, auto and cross spectra of the optical and microphone signals are averaged over 10 acquisitions. The LDV apparent velocity at a point of index m is computed from the following formula:

v m = S vm,p 1,m S p 1,m ,p 1,m S p 1,m ,p 1,m M . (17)
In Eq. ( 17), S vmp 1,m is the cross-spectrum between the LDV optical and reference microphone (Mic #1) signals for measurement of index m. S p 1,m ,p 1,m is the auto-spectrum of reference microphone signal (Mic #1) for the measurement m. The operator • M denotes an average over M measurements.

Using an average value helps reducing the estimation error on the spectrum v m [START_REF] Leclère | Multi-channel spectral analysis of multi-pass acquisition measurements[END_REF].

Equivalent source method parameters

Equivalent sources layout

The point sources are distributed around the spark electrodes axis, following a parallelepipedic layout. In the reported case, the distance along x, y and z axis between the equivalent point sources is identical and fixed at ∆ = 1.5 mm. This results in a maximum reproducible frequency of f max = c 2∆ 113 kHz [START_REF] Spors | Spatial aliasing artifacts produced by linear and circular loudspeaker arrays used for wave field synthesis[END_REF]. The parallelepiped is centered at the origin and its dimensions are (L x × L y × L z ) = ( 2 cm × 4 cm × 2 cm). In this situation, there are N = 5292 point sources. For the acousto-optic interaction described in Sec. 2.1, the distance between the laser beam and an equivalent point source is denoted a, and its expression is detailed in Appendix C.

In this situation, the minimum and maximum distances a are a min = 0.02 cm and a max = 17.6 cm.

Physical parameters estimation

The physical parameters n 0 , γ, p 0 and c in Eq. 12 depend on the environmental conditions and should be calculated accordingly. In the reported case, the static pressure p 0 , temperature, and humidity were measured for each set of acquisition. From these measurements, the sound speed c and the specific heats ratio γ were calculated for each set of acquisition according to [START_REF] Cramer | The variation of the specific heat ratio and the speed of sound in air with temperature, pressure, humidity, and CO2 concentration[END_REF]. The refractive index is taken constant with n 0 = 1.0002772.

Measurement points selection

Before solving the inverse problem described in Sec. 3, a sub-set of optical measurements v are selected with two criteria:

Coherence. The first criterion is based on the coherence between the reference microphone and LDV optical signals [START_REF] Oppenheim | Digital Signal Processing[END_REF]. At low frequencies, the spark source radiates little energy. This is due to the spark source energy spectral density nature [START_REF] Lee | Energy spectral density of the sonic boom[END_REF] and it is related to the electrode gap width [START_REF] Wright | Acoustic Radiation from a Finite Line Source with N-Wave Excitation[END_REF]. Moreover, the velocity output of the LDV is proportional to the time derivative of the acoustic pressure (Eq. ( 5)). In the frequency domain, this is traduced by a factor iω which is low at low frequencies. Consequently, the signal to noise ratio between reference microphone and LDV optical signal is poor and the coherence as well. Besides, for some laser incidences and scan points, the laser path can meet an obstacle such as the electrodes support. The coherence criterion is also weak in these situation.

In summary, the selection based on coherence criterion ensures that measurements with low signal-to-noise ratio or wrong laser paths are discarded for the inverse problem. In the reported case, only the measurements with a coherence above 0.8 are retained.

Minimum radius. The second selection criterion is based on the radius between measurement points and origin. In the reported case, only the points at a minimum distance of 1.5 cm from origin are retained. This ensures that no measurement points are located inside the equivalent source volume and helps with the condition number of the matrix G. In fact, equivalent sources close to measurement points bring low distances a (See Appendix C), leading to high amplitude terms in the matrix G. Moreover, in the near field of the spark source, the non-linear propagation effects are predominant [START_REF] Yuldashev | Nonlinear spherically divergent shock waves propagating in a relaxing medium[END_REF] and could disturb the linear equivalent source model. This topic is left for further investigations.

After applying the sub-set selection on the LDV signals, the number of measurement points (i.e. the line number of matrix G) is noted M .

Beamforming regularization

The weighting matrix W of Eq. ( 16), is chosen from a delay-and-sum beamforming solution [START_REF] Gauthier | Beamforming regularization matrix and inverse problems applied to sound field measurement and extrapolation using microphone array[END_REF]. Thus, W is a diagonal matrix with its diagonal term W n,n given by:

W n,n = |G H n v| 2 G n 2 v 2 . ( 18
)
In Eq. ( 18), G n is the n-th column of the matrix G [START_REF] Gauthier | Beamforming regularization matrix and inverse problems applied to sound field measurement and extrapolation using microphone array[END_REF]. Finally, the diagonal term of W, W n,n , is a value between 0 and 1 which weights the equivalent source n for the inversion. In the reported cases, all the values such as W n,n < 0.02 are set to 0, and the corresponding equivalent sources are discarded for the inversion. After this selection, the number of equivalent sources (i.e. the column number of matrix G) is Ñ .

Additional Truncated Singular Value Decomposition

An additional Truncated Singular Value Decomposition (TSVD) [START_REF] Hansen | Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion[END_REF] is applied on the matrix G ( M × Ñ ) after beamforming regularization. Only the s highest singular values are kept with:

s = max s min , 2 λ 3 L x L y L z , (19)
Where • represents the rounding up to the nearest integer, s min = 10 in the reported case and λ is the wavelength. The term (2/λ) 3 L x L y L z is a criterion which gives the minimum required number of points in the equivalent source volume to avoid spatial aliasing. Therefore, this approach regularizes the matrix to be inverted from a physical criterion. For instance, at low frequencies, the spatial resolution of the equivalent sources layout is unnecessarily dense. It brings many singular values which are discarded with this approach.

Results

This section presents several results for the acoustic pulse equivalent source model.

Equivalent sources strengths

The normalized equivalent sources strengths |q n | 2 /|q n,max | 2 , are displayed at several frequencies in Fig. 5. Each equivalent source is represented as a dot whose size and color is proportional to its strength. The two electrodes are represented as thick lines.

In order to evaluate the reconstruction quality at the scan points, a global coherence criterion, ranging from 0 to 1, is defined as:

C = |v H v| 2 v 2 v 2 (20)
As well, a degree of over-determination for the inversion is given by:

S = M Ñ . ( 21
)
These two criteria are displayed for each frequency in Fig. 5. One observes that the highest equivalent source strengths are gathered in the vicinity of the gap in between the electrodes. This is in accordance with the spark source spatial location. As the frequency increases, the number of predominant equivalent sources decreases and a line-source shape appears. However, with a closer look, one observes that equivalent sources are located a few millimeters next to the electrode axis. This can be explained by the positioning precision of the experimental layout. As a matter of fact, there are slight misalignment between the LDV, the spark source and the equivalent sources respective coordinate systems. Consequently, the distances a (Eq. (C.4)) can reflect these misalignments and the inverse problem solution as well. In addition, the scans orientations into consideration and their number can influence the reconstruction results. This topic requires further investigations. Regarding the reconstruction quality highlighted by the criterion C, one observes that it is ranging from 0.97 at low frequencies (6400 Hz) to 0.81 at high frequencies (38.4 kHz).

Reconstructed pressure

From the equivalent monopole source strengths, the pressure at a point x can be synthesized with a free field Green function:

p(x, ω) = q(ω) e -i ω c x-xs 4π x -x s (22)
The sound pressure levels measured by the microphones Mic #1 are Mic #2 and shown on Fig. 6a and 6b respectively. The dashed line curves are the sound pressure levels including free field normal incidence corrections. The correction coefficients are taken from the B&K c datasheets. The solid line curves are the sound pressure levels reconstructed with the equivalent source.

From these figures, one notices that the equivalent sources reconstruct accurately the sound pressure level at both locations within the range of 2 dB for the considered frequencies. However, for the Mic #2 reconstructed pressure, there are ripples from 12 kHz to 20 kHz. By looking at the equivalent sources layout (Fig 5 ) and their directivity (Fig. 7) at these frequencies, one observes that they are rotated around the vertical axis with respect to the electrode axis. Thus, the pressure reconstructed at the Mic # 2 is slightly different from the situation where the equivalent sources would be aligned in the electrode axis.

Equivalent sources directivity

The normalized directivity at 1 m of the equivalent sources is shown in Fig. 7 at several frequencies: the equivalent source pressure, Eq. 22, is reconstructed at a radial distance of 1 m and various angles. Absolute value of this pressure is normalized by the maximum absolute value on the 4π steradian angle. One can see that the equivalent source directivities range from a quasi-omnidirectionnnal source at low frequencies to a line source directivity at higher frequencies. This is in accordance with the literature a spark source directivity generated between two electrodes [START_REF] Wright | Acoustic Radiation from a Finite Line Source with N-Wave Excitation[END_REF]. However, at mid frequencies, from 12 kHz to 20 kHz, the source orientation is shifted from the electrode axis. This is confirmed by the observation of equivalent source spatial distribution at these frequencies in Fig. 5.

Discussion

The results presented in an experimental context for an acoustic pulse show an equivalent sources model from acousto-optic measurements. The inverse problem solutions is regularized with a beamforming solution and an additional TSVD. The equivalent sources are found in the vicinity of the spark source gap. The direct problem reconstruction give high coherence values in comparison with the measurements. The reconstructed pressures at control microphones are accurate and the directivity of the equivalent source is in accordance with a line-source shape. However, several factors influence the results and are discussed here.

Positioning accuracy. In an experimental context, the positioning accuracy is limited and influences directly the localization/directivity of the equivalent sources, especially at high frequencies. In fact, for the experimental layout described in the paper, the positioning accuracy is around the half-centimeter range. Moreover, the spark source rotation axis is not exactly coincident with the equivalent sources rotation axis. Consequently, when computing the distances a for a theoretical configuration, a bias is introduced and compromises the results, as one can observes for the equivalent source location and orientation. The study of the positioning accuracy for the robustness of the proposed method is a topic for further investigations.

LDV laser beam focusing. Using a scanning LDV offers the possibility of measuring numerous points in an automated way. As the device is not moving during the scan, it uses lenses and mirrors to focus the laser on each measurement point location. As a result, the laser has a varying diameter along the integration path. When this diameter is of the same order of magnitude as the acoustic wavelength, an averaging on the laser cross section is done [32, p.45-46]. To minimize this effect, it is advised to focus the laser beam where the acousto-optic effect is pre-dominant, i.e. in the vicinity of the spark source for the current experiment. However, by doing so, the signal-to-noise ratio decreases as there will be less light reflected on the rigid surface. In the current experiment, the laser is focused on the rigid reflector in Fig. 2.

Signal to noise ratio. Using an LDV for measuring the acousto-optic effect is a misuse of such a device. In fact, a LDV is sensitive to the mechanical vibrations of a moving surface (first right-hand term in Eq. ( 3)). As the acousto-optic effect (second right-hand term in Eq. ( 3)) is weak in comparison with the mechanical vibration, a high signal-to-noise ratio is needed to observe it with an LDV. This implies several limitations for the acoustic sources in an experimental context. A first solution is to drive the acoustic source in harmonic mode with a single frequency, to increase the signal-tonoise ratio [START_REF] Torras-Rosell | Sound field reconstruction using acousto-optic tomography[END_REF]. In the present case and transient spark source is used, and a limited duration acquisition is done to maximize the signal-to-noise ratio. However, the acousto-optic effect decreases rapidly as the laser distance to the source increases. This limits the measurement to near field of the spark source. A solution would be to increase the electrode gap, but this will increases the non-linear propagations effects as mentioned in the following paragraph. An other lead would be to use more sensible interferometers, such as Mach-Zender interferometers [START_REF] Yuldashev | Mach-Zehnder interferometry method for acoustic shock wave measurements in air and broadband calibration of microphones[END_REF].

Spark source propagation. The spark source presents several advantages in an experimental context. It is a transient source which allows the characteri-zation in a non-anechoic environment. The finite duration acquisition allows to discard the wave reflections on the rooms walls, LDV scanning head of rigid reflector. Moreover, it is an transparent source, i.e. the laser beam has no obstacle on its path, except the electrodes and their support in the case of and electric spark source. This allows to measure the acousto-optic effect with various incidence and enhance its characterization. However, a spark source typically propagates with nonlinearities [START_REF] Yuldashev | Nonlinear spherically divergent shock waves propagating in a relaxing medium[END_REF]. These effects are predominant in the near field of the spark source gap and increase as the electrode gap increases [START_REF] Wright | Acoustic Radiation from a Finite Line Source with N-Wave Excitation[END_REF]. Besides, in the proposed method, a linear superposition of linear monopole sources is used to construct and equivalent sources model. Consequently, the equivalent source model is valid in a leastsquare sense but it may not be the most physically valid solution. To improve this point, it would be possible to describe the acouto-optic effect for an elementary point source propagating with nonlinearites. This topic is left for further investigations.

Conclusion and Future Works

This paper presented a method for acoustic sources characterization using optical measurements: the acousto-optic effect is measured with a LDV at various distance and incidence from the sources. From these measurements, an inverse problem is solved in order to construct and equivalent monopole sources model. The model uses an acousto-optical Green function.

Effects of finite length laser beam and acquisition duration were studied. The method was validated in an experimental context for an acoustic pulse generated with a spark source. Results showed good agreement for the equivalent sources model in term of spatial location, reconstructed pressures and directivity.

The main drawbacks of the experimental method were highlighted. In particular, it would be advised to use more sensible optical interferometers.

Future works include the comparison of the proposed method with acoustooptic tomography [START_REF] Torras-Rosell | Sound field reconstruction using acousto-optic tomography[END_REF] and studies of the method robustness to positioning accuracy, number and orientation of measurements, and nonlinear effects. For this latter point, a nonlinear propagation code for spark source will be used [START_REF] Lechat | Simulation du champ proche généré par une source impulsionnelle à arc électriques (Simulation of the near field generated by an electric arc pulse source)[END_REF].

du = -x c √ a 2 + x 2 dx, (A.7) du = -c 2 (t -u) 2 -a 2 c √ a 2 + x 2 dx, (A.8) dx √ a 2 + x 2 = -cdu c 2 (t -u) 2 -a 2 . (A.9)
By doing the variable change with u, one has:

I(t) = -2 β -β
-cδ(u) c 2 (tu) 2a 2 du, (A.10)

Finally: In order to work in the frequency domain, the Fourier transform of I(t) is computed as follows: One can split the previous integral in two parts:

I(t) = 2c a
I(ω) =
I(ω) = 2 +∞ 1 1 √ u 2 -1 e -iω a c u du -2 +∞ √ a 2 +α 2 a 1 √ u 2 -1 e -iω a c u du (A.17)
The first term of the right-hand side in Eq (A.17) is an integral form of a modified Bessel functions of second kind of degree 0, K 0 [34, Eq. 9. this case, one observes that the residual R decreases as the frequency and point-source-distance increase. For instance, in the measurement points and equivalent source layouts described in Sec. 4.3 and Sec 5.1 respectively, the distance a spans from a min = 0.02 cm to a max = 17.4 cm, and the minimum measurement frequency is 1600 Hz. In this situation, R is always inferior to -20 dB.

Appendix C. Analytic expression of function a

Let S(x n ) be a monopole at spatial location x n . Let P (x m ) be a LDV measurement point at spatial location x m . Finally, let L(x L ) be the LDV laser beam origin. By considering the plane containing the points (S, P, L), the geometrical layout is schematized in From law of cosines one has:

γ = arccos x n -x L 2 + x m -x L 2 -x m -x n 2 2 x n -x L x m -x L (C.1)
As well, from Pythagorean theorem, one has:

sin(γ) = a x n -x L (C.2)
Thus,

a = x n -x L sin arccos x n -x L 2 + x m -x L 2 -x m -x n 2 2 x n -x L x m -x L (C.3) Finally, a = x n -x L 1 - x n -x L 2 + x m -x L 2 -x m -x n 2 2 x n -x L x m -x L 2 (C.4)

Figure 1 :

 1 Figure 1: Geometrical layout for the measurement of the acousto-optic effect induced by a Dirac pulse.

Figure 2 :

 2 Figure2: Layout of the experiment: A LDV (left on the picture) emits a laser beam which crosses the sound pressure field and reflects back on the rigid reflector (right of the picture). The acoustic field is radiated by a spark source emitted between two electrodes (middle of the picture). Two quarter-inch microphones are placed at coordinates Mic # 1: (0, 0, 11.5 cm) and Mic # 2: (0, -71 cm, 0), corresponding to green and red support respectively.

Figure

  Figure3: Geometrical layout (top view). The measurement points are defined in the virtual scanning plan (Oyz). The electrodes are oriented such that their axis is at an angle θ from y-axis.

Figure 4 :

 4 Figure 4: Example of waveform measured at Mic #1 (0, 0, 11.5 cm) and Mic #2 (0, -71 cm, 0) (a) Mic #1 signal. (b) Mic # 2 signal. The time origin is set arbitrarily. The acquisition sampling frequency is f s = 102.4 kHz. The original samples are plotted as black circles. For better readability, the waveforms are also plotted in solid lines after being oversampled by a factor of 10.

13

 13

Figure 5 :

 5 Figure 5: Normalized equivalent sources strengths at several frequencies. The electrodes are represented as black thick lines. The size and color of the sources are proportional to their strength. (a) f = 6400 Hz, C = 0.97052, S = 2.2355. (b) f = 12800 Hz, C = 0.95703, S = 2.3427. (c) f = 19200 Hz, C = 0.93532, S = 2.3745. (d) f = 25600 Hz, C = 0.90293, S = 2.3998. (e) f = 32000 Hz, C = 0.86254, S = 1.9696. (f) f = 38400 Hz, C = 0.81213, S = 1.3220.

Figure 6 :

 6 Figure 6: Measured and reconstructed sound pressure level in dB SPL (reference 20 × 10 -6 Pa): (a) Microphone #1 (0, 0, 11.5 cm). (b) Microphone #2 (0, -71 cm, 0). Dashed line curves: measured pressure including free field normal incidence corrections. Solid line curves: equivalent source reconstruction.

Figure 7 :

 7 Figure 7: Equivalent source 3D directivity at 1 m for several frequencies. Amplitude and color are normalized by the maximum value in the 4π steradian angle. (a) f = 6400 Hz. (b) f = 12800 Hz. (a) f = 19200 Hz. (b) f = 25600 Hz. (a) f = 32000 Hz. (b) f = 38400 Hz.

1 c 2 a 2 t 2 - 1 . (A. 11 ) 2 a 2 t 2 - 1 .

 1211121 The two aforementioned situations can be grouped with an indicator func-

2 a 2 t 2 - 1 e

 21 -iωt dt (A.15) With variable change u = ct a , du = c a dt, one has:

Figure B. 8 :

 8 Figure B.8: Contour plots for the magnitude of term R (Eq. (B.1)) versus the ratios a/λ and α/λ.

Figure B. 9 :

 9 Figure B.9: Contour plots for the magnitude of term R (Eq. (B.1)) versus the distance laser-point-source, a, and the frequency f . In Eq. (B.1), the sound speed is set to c = 340 m/s and the laser length to 1 m.

Figure C. 10 :

 10 Figure C.10: Geometrical layout in the plane (S, P, L). The laser beam, in red, crosses the scan point P at a distance a from the source S.

Acknowledgments

This work was supported by the Labex CeLyA of Université de Lyon, operated by the French National Research Agency (ANR-10-LABX-0060/ ANR-11-IDEX-0007).

Appendix A. Computation of integral I in Eq. [START_REF] Torras-Rosell | An Acousto-Optic Method For Free-Field Microphone Calibration[END_REF] The integral I(t) of Eq. ( 8) is recalled here for readability:

The function under the integral in I is pair with respect to variable x, thus:

In order to compute integral I in Eq. ( 8), two situations are considered with respect to variable t.

. The first situation is when t / ∈ ] a c ,

], that is to say when the spatial Dirac pulse is outside the laser beam in spatial interval [-α, α]. In this situation,

], that is to say, when the spatial Dirac pulse is crossed by the laser beam in spatial interval

, t -a c ] be the variable such that:

, and from Eq.(A.4), one has:

Appendix B. Study of the residual in Eq. (A.18)

The integral I(ω) can be approximated by K 0 (iωa/c) if the second term of the right-hand side in Eq. (A.18) is negligible when compared to K 0 (iωa/c). In order to study this residual, one expresses the following term:

where λ = c/f is the acoustical wavelength. The magnitude R is shown in Fig. B.8. One observes that it decreases as the ratio a/λ decreases and the ratio α/λ increases. Physically, this means that the error of approximation decreases when the laser length is big in comparison to the acoustical wavelength and when the distance a is small in comparison to the acoustical wavelength. In other words, a long laser, close to the point source, relatively to the acoustical wavelength, decreases the approximation error. For a practical application where the laser length is not changing but the distance to the source is, R can be plotted versus frequency, as shown in