Thibaut De Terris

Olivier Andreau

Patrice Peyre

Frédéric Adamski

Imade Koutiri

Cyril Gorny

Corinne Dupuy

Optimization and comparison of porosity rate measurement methods of Selective Laser Melted metallic parts

Keywords:

The systematic occurrence of porosities inside selective laser melted (SLM) parts is a well-known phenomenon. In order to improve the density of SLM parts, it is important not only to assess the physical origin of the different types of porosities, but also to be able to measure as precisely as possible the porosity rate so that one may select the optimum manufacturing parameters.

Considering 316 L steel parts built with different input energies, the current paper aims to (1) present the different types of porosities generated by SLM and their origins, (2) compare different methods for measuring parts density and (3) propose optimal procedures. After a preliminary optimization step, three methods were used for quantifying porosity rate: the Archimedes method, the helium pycnometry and micrographic observations.

The Archimedes method shows that results depend on the nature and temperature of the fluid, but also on the sample volume and its surface roughness.

During the micrographic observations, it has been shown that the results depend on the magnification used and the number of micrographs considered.

A comparison of the three methods showed that the optimized Archimedes method and the helium pycnometry technique gave similar results, whereas optimized micrographic observations systematically underestimated the porosity rate.

In a second step, samples were analyzed to illustrate the physical phenomena involved in the generation of porosities. It was confirmed that: (1) low Volume Energy Density (VED) causes non-spherical porosities due to insufficient fusion, (2) in intermediary VED the small amount of remaining blowhole porosities come from gas occlusion in the melt-pool and (3) in excessive VED, cavities are formed due to the key-hole welding mode.

Introduction

The development of Metallic Additive Manufacturing (AM) has greatly expanded since the early 2000s. AM enables the production of components with complex geometries which are impossible to fabricate with conventional processes (forge, foundry,…). Metallic AM processes can be classified in two main categories: (1) direct deposition techniques and (2) powder bed fusion techniques. The first category consists in depositing matter via projected powder or a feeding wire, melted by either a laser or an electric arc. In the second category, the powder-bed can be melted using a laser beam, such as Selective Laser Melting (SLM), or using an electron beam as in the Electron Beam Melting process (EBM). A third technology, the Metal binder jet printing, is also in full development [START_REF] Nandwana | Powder bed binder jet 3D printing of Inconel 718: densification, microstructural evolution and challenges[END_REF]. There is growing interest in AM because of the wide range of metallic materials that can be used to produce increasingly complex parts. Different applications in various fields such as aeronautics and space, transport, energy, medical [START_REF] Herderick | Additive manufacturing of metals: a review[END_REF][START_REF] Vayre | Metallic additive manufacturing: state-of-the-art review and prospects[END_REF][START_REF]Introduction to Additive Manufacturing Technology, A Guide for Designers and Engineers[END_REF][START_REF] Herzog | Additive manufacturing of metals[END_REF][START_REF] Sames | The metallurgy and processing science of metal additive manufacturing[END_REF], already exist. Moreover, AM can provide mechanical properties that are similar or better than conventional processes (such as casting or forging) [START_REF] Herzog | Additive manufacturing of metals[END_REF][START_REF] Sames | The metallurgy and processing science of metal additive manufacturing[END_REF].

There are two main common defects in SLM which directly influence the mechanical properties of the parts: a deleterious surface roughness (with Ra usually superior to 7 μm) and the systematic occurrence of porosities inside the final parts. These defects and their origin are well described by [START_REF] Debroy | Additive manufacturing of metallic componentsprocess, structure and properties[END_REF] for Ti-6Al-4 V and 316 L stainless steel [START_REF] Fatemi | Multiaxial fatigue behavior of wrought and additive manufactured Ti-6Al-4V including surface finish effect[END_REF]. also indicates that compared to wrought parts, SLM parts have a significantly lower fatigue life behavior in multiaxial loading conditions, due to internal defects which cause stress concentration.

There are three 1st order parameters in SLM which directly impact ⁎ Corresponding author.

E-mail address: thibaut.de_terris@ensam.eu (T. de Terris).

T the quality of the fusion: the laser power P (W), the scanning speed V (mm.s -1 ) and the laser beam diameter D (mm). These parameters can be united in a single parameter called Volume Energy Density (VED) [START_REF] Mancisidor | Reduction of the residual porosity in parts manufactured by selective laser melting using skywriting and high focus offset strategies[END_REF][START_REF] Wang | Mechanisms and characteristics of spatter generation in SLM processing and its effect on the properties[END_REF][START_REF] Gunenthiram | Experimental analysis of spatter generation and melt-pool behavior during the powder bed laser beam melting process[END_REF] (Eq. ( 1)). It is also possible to consider other parameters such as the powder layer thickness Δh (μm) or the hatch spacing H (μm) between adjacent fusion lines.

= VED P V π D 4* * * 2 (1)
With: VED : Volume Energy Density (J. mm -3 ), P : laser power (W), V : scan velocity (mm.s -1 ), D : laser beam diameter (mm) Several recent works [START_REF] Kaufmann | Influence of process parameters on the quality of aluminium alloy en AW 7075 using Selective Laser Melting (SLM)[END_REF][START_REF] Gu | Influences of energy density on porosity and microstructure of selective laser melted 17-4PH stainless steel[END_REF][START_REF] Campanelli | Capabilities and performances of the selective laser melting process[END_REF] have shown that increased VED favors densification. Below a given VED threshold, SLM conditions do not allow for sufficient re-melting of the prior layer, leading to a lack of fusion (LOF) which results in residual pores forming in-betweens layers. Studies also show that small (< 10 μm) blowhole pores (coming from gas entrapment in powder particles and in the powder layer or from powder humidity) are formed throughout the VED range [START_REF] Debroy | Additive manufacturing of metallic componentsprocess, structure and properties[END_REF]. At high VED, and depending on powder material, the porosity rate may once again increase as shown by [START_REF] Choi | Densiication behavior of 316L stainless steel parts fabricated by selective laser melting by variation in laser energy density[END_REF] and [START_REF] Kasperovich | Correlation between porosity and processing parameters in TiAl6V4 produced by selective laser melting[END_REF].

Various physical phenomena (Fig. 1) can explain the presence of local unmelted zones inside a part. These are listed here below:

-Inappropriate manufacturing parameters, mostly corresponding to insufficient VED, can lead to poor overlap between SLM fusion beads (= fusion lines) and thereby increase the amount of unmelted zones. Too much hatching distance and excessive powder layer thickness can also generate a lack of cohesion between SLM beads [START_REF] Li | Densification behavior of gas and water atomized 316L stainless steel powder during selective laser melting[END_REF]. -A high VED leads to the formation of a vapor plume. The interaction between this vapor plume (generated above the impact zone) and the laser can lead to the formation of porosities [START_REF] Andreau | Texture control of 316L parts by modulation of the melt pool morphology in selective laser melting[END_REF]. As shown by [START_REF] Gunenthiram | Analysis of laser-melt pool-powder bed interaction during the selective laser melting of a stainless steel[END_REF] this vapor plume is composed of macro droplets and nanocondensates which interfere with the laser by absorbing the radiation, hence limiting the bead penetration and possibly promoting LOF. -The presence of spatter ejections on the powder bed can also generate porosities. The size of the spatters, usually much larger than the powder grains (< 50 μm), can prevent the powder from spreading homogeneously around the spatter, thus increasing the risk of LOF [START_REF] Wang | Mechanisms and characteristics of spatter generation in SLM processing and its effect on the properties[END_REF][START_REF] Taheri Andani | A study on the effect of energy input on spatter particles creation during selective laser melting process[END_REF]. According to [START_REF] Wang | Mechanisms and characteristics of spatter generation in SLM processing and its effect on the properties[END_REF] there are three types of spatter ejections : (1) metal jet caused by the recoil pressure of the vapor plume associated with the extreme expansion of the powder phase gas, (2) droplet spatters ejected from the turbulent melt-pool flow, (3) powder spatters generated at the powder layer/melt-pool front interface. -The phenomenon of powder denudation investigated in [START_REF] Matthews | Denudation of metal powder layers in laser powder bed fusion processes[END_REF] where the powder surrounding the bead is attracted towards it and causes a lack of powder on both sides of SLM beads. This has two effects: (1) the local powder thickness for the next layer is higher, (2) a nonoptimal lateral overlap between beads, due to an excessive hatching distance, which can be a source of LOF [START_REF] Li | Densification behavior of gas and water atomized 316L stainless steel powder during selective laser melting[END_REF]. -Melt-pool hydrodynamic instabilities like balling (favored by low VED) or humping (which appears at high laser powers and velocity) [START_REF] Kasperovich | Correlation between porosity and processing parameters in TiAl6V4 produced by selective laser melting[END_REF][START_REF] Gunenthiram | Analysis of laser-melt pool-powder bed interaction during the selective laser melting of a stainless steel[END_REF] and vaporization affect bead geometry.

Noting the importance of bead dimensions combined with the hatching distance H (μm) and the layer thickness Δh (μm), Tang [START_REF] Tang | Prediction of lack-of-fusion porosity for powder bed fusion[END_REF] suggested a formula (Eq. ( 2)) to geometrically estimate if the overlap is sufficient to avoid LOF. In the Eq. ( 2), W is the SLM bead width and e is the SLM bead depth (in μm) measured in a cross-section which is perpendicular to the scan direction as described in Fig. 2.

It can be clearly seen in Fig. 2 that to avoid LOF between each of the SLM beads, W must be higher than H in order to guarantee a sufficient lateral overlap, and e must be higher than Δh to provide a sufficient remelting (vertical overlap) of the previous layers and avoid LOF between each of the SLM layers.

⎛ ⎝ ⎞ ⎠ + ⎛ ⎝ ⎞ ⎠ ≤ H W h e Δ 1 2 2 (2)
With : H : hatching distance (μm), W : SLM bead width (μm), Δh : layer thickness (μm), e : SLM bead depth (μm) As indicated before, LOF porosities exhibit angular and random morphologies (Fig. 3a), of relatively large sizes (< < 100 μm) and contain unmelted powder. They are located mainly in-between SLM beads and come either from a poor choice of parameters or from process instabilities.

A number of near-spherical porosities will always remain, even with a correctly optimized SLM process. They can be divided in two types: blowhole pores and key-hole pores.

-Blowhole pores (Fig. 3b) reflect the presence of gas bubbles trapped in the material upon solidification. They are mostly dependent on the properties of the powder bed (compactness, powder contamination) and on the work atmosphere (shielding gas, humidity) as explained by [START_REF] Debroy | Additive manufacturing of metallic componentsprocess, structure and properties[END_REF]. A powder layer usually has a compactness of around 50%. During the melting of a layer, a fraction of the gas contained between the powder particles cannot escape before the solidification and gets entrapped as bubbles in the SLM bead. These spherical porosities are located in the heart of a bead and are much Fig. 2. Illustration of the key dimensions to estimate the formation of LOF, according to [START_REF] Tang | Prediction of lack-of-fusion porosity for powder bed fusion[END_REF].

smaller than those related to the key-hole. -At high laser intensity (for I > 10 6 W.cm -² according to [START_REF] William | Laser Material Processing[END_REF]) the SLM welding regime turns from a conduction mode into a key-hole mode [START_REF] Qi | Selective laser melting of Al7050 powder: Melting mode transition and comparison of the characteristics between the keyhole and conduction mode[END_REF]. Recoil pressure on the melt-pool generated by intense metal vaporization can create a vapor capillary (key-hole). When the laser beam is absorbed by the key-hole walls rather than the layer surface, this significantly increases the penetration depth and generates thin and deep fusion beads (Fig. 3c) as shown by [START_REF] Fabbro | Melt pool and keyhole behaviour analysis for deep penetration laser welding[END_REF]. In this case, porosities come from the collapse of key-hole walls under the action of surface tensions. The destabilization and resulting closure forms near-spherical large cavities with entrapped gas, located near the SLM bead root. Previous works [START_REF] Qi | Selective laser melting of Al7050 powder: Melting mode transition and comparison of the characteristics between the keyhole and conduction mode[END_REF][START_REF] Fabbro | Melt pool and keyhole behaviour analysis for deep penetration laser welding[END_REF] considered the Width-to-Depth Ratio of fusion beads (WDR = W/e) to distinguish conduction mode welds (for WDR > 1) from key-hole mode welds (WDR < < 1).

Examples of typical porosities obtained during our SLM experiments on 316 L steel are shown in Fig. 3.

The generation of porosities is a function of manufacturing conditions. It is important to know their physical origins to be able to limit their formation. It is also important to develop experimental methods that permit the analysis of porosity amounts as quickly and as efficiently as possible. Using either micrographic analysis of cross sections [START_REF] Kasperovich | Correlation between porosity and processing parameters in TiAl6V4 produced by selective laser melting[END_REF][START_REF] Parimi | Microstructural and texture development in direct laser fabricated IN718[END_REF], Archimedes method [START_REF] Spierings | Comparison of density of stainless steel 316L parts produced with selective laser melting using different powder grades, Solid Free, Solid Freeform Fabr[END_REF][START_REF] Spierings | Comparison of density measurement techniques for additive manufactured metallic parts[END_REF] or computed tomography [START_REF] Kasperovich | Correlation between porosity and processing parameters in TiAl6V4 produced by selective laser melting[END_REF][START_REF] Spierings | Comparison of density measurement techniques for additive manufactured metallic parts[END_REF][START_REF] Wits | Porosity testing methods for the quality assessment of selective laser melted parts[END_REF], a number of recent works have considered how porosity amounts vary according to SLM conditions. However, there are very few studies that compare experimental methods to one another [START_REF] Spierings | Comparison of density measurement techniques for additive manufactured metallic parts[END_REF][START_REF] Wits | Porosity testing methods for the quality assessment of selective laser melted parts[END_REF][START_REF] Wang | Scanning optical microscopy for porosity quantification of additively manufactured components[END_REF], and there is a lack of details about which experimental procedures are the most reliable for estimating porosity amounts. Another problem is the lack of uniformity in the results of many recent studies. For example [START_REF] Wits | Porosity testing methods for the quality assessment of selective laser melted parts[END_REF] shows a high degree of correlation between Archimedes method and micrographic analyses, while [START_REF] Spierings | Comparison of density measurement techniques for additive manufactured metallic parts[END_REF] shows these methods yield different results.

The objective of the present paper is: (1) to compare and validate several experimental methods for evaluating closed porosities in SLM parts, (closed porosities are those that are completely isolated inside of the metal, whereas open porosities are pores near the part surface that are not completely enclosed, as shown in Fig. 4), and (2) to apply the methods to the study of porosity formation in 316 L stainless steel samples.

Experimental methods

Material

The material used for SLM experiments was a gas-atomized 316 L stainless steel powder. Particles had a spherical morphology, with a few agglomerated satellites. Their size was between 25 μm (D10) and 60 μm (D90) (Fig. 5). The theoretical density of 316 L stainless steel is 7.99 g.cm -3 according to the supplier material data sheet. The chemical composition is given in Table 1.

SLM conditions

The test samples used for this study were manufactured using an SLM125 HL machine, from SLM Solutions GmbH, under an argon atmosphere and less than 500 ppm O 2 . They were rectangular cuboids of 10 × 10 x 25 mm.

The manufacturing parameters were kept similar, except for the scanning speed V, as summarized in the Table 2. The manufacturing strategy also remained the same (number of contours, lasing strategy and scheme, hatch distance).

Experimental methods and procedures for density measurements 2.3.1. The archimedes method

The Archimedes measurements were performed according to the method described by [START_REF]ASTM International, Standard Test Method for Density Of Powder Metallurgy (PM) Materials Containing Less Than Two Percent Porosity[END_REF]. The device used was a Mettler Toledo MS104TS hydrostatic weighing system equipped with a MS-DNY-43 solid sample weighing module. This technique is based on the principle of Archimedes: "A body wholly or partly immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced" [START_REF] York- | Encyclopedic dictionary of polymers[END_REF] (the volume of the displaced fluid is therefore equal to the immersed volume of the body), which enables to determine the volume of the samples, and therefore their density. Samples are weighed in the air and then in a fluid as shown in (Fig. 6).

The measurements make it possible to know the mass in the air (M air ) and the mass in the fluid (M fluid ) of each sample. And, knowing the density of the fluid (ρ fluid , which varies with the temperature) and of the air (ρ air ), it becomes possible to calculate the density of the samples according to the Eq. (3) [START_REF] Spierings | Comparison of density measurement techniques for additive manufactured metallic parts[END_REF]: In this study, each sample was measured 10 times, and different fluids (demineralized water, pure ethanol and acetone) were tested. Corresponding fluid properties are shown in Table 3. Between each measurement the samples were oven-dried (110 °C, 30 min) to evaporate the water trapped in the open pores, in order not to influence the mass in the air during the next measurement. During the measurements it was noticed that air bubbles tend to settle on the surface of the samples because of the surface asperities. In order to improve the density results accuracy, air bubbles shall be removed. Hence, the samples were previously soaked and stirred in an auxiliary beaker containing the same fluid as the one used for the measurements, and the bubbles were evacuated with a brush. The measurements in the fluid were performed with no visible bubbles remaining on the surface of the sample. With this method, a simple measurement takes about twenty seconds, not accounting for the sample preparation time and the analysis of the results, as well as the 10 necessary measurements.

= - - + ρ M M M ρ ρ ρ ( ) *

Helium pycnometry

The Helium pycnometry measurements were performed on a Micromeritics AccuPyc 1330 device. This technique makes it possible to determine the absolute density of a sample (solid or powder) after the measurement of its volume. The measurement process consists in placing a sample in a tank whose volume V 1 is calibrated and perfectly known, at atmospheric pressure P a . This tank is then filled with helium at a pressure P 1 . The gas is then expanded to a pressure P 2 in another atmospheric pressured empty vessel (called expansion), of a volume V 2 (Fig. 7).

The pycnometer measures the volume of a displaced gas from a primary tank (V tank ) at a pression P 1 to an expansion tank (V expansion ) at a pression P 2 . The pressure variation permits to determine the volume of the sample (V sample ) and the application of the Mariotte's law allows calculating this volume (Eq. ( 5)):

= - - - - V V V 1 sample tank expansion P P P P a a 1 2 (5)
With V sample : sample volume measured (cm 3 ), V tank : volume of the sample tank (cm 3 ), V expansion : volume of the expansion tank (cm 3 ), P 1 : gas pressure in the sample tank (kPaG), P 2 : gas pressure in the expansion tank (kPaG), P a : atmospheric pressure (kPa) Since the mass has been measured accurately beforehand, and knowing the volume of the sample, it is possible to calculate its density. In the case of helium pycnometry, several purge cycles are previously performed to remove all the air present in the tank and in the open pores of the sample. Then 10 successive measurements are made while keeping the tank closed and airtight. Samples were oven-dried at 110 °C during 30 min, and air cooled before helium pycnometry measurements. This method, composed of 10 cycles, takes about 45 min per sample.

Micrographic cross section

SLM Samples were cut, resin mounted and polished to realize micrographs. These micrographs were acquired on the XY (perpendicular to the manufacturing direction) and YZ (parallel to the manufacturing direction) planes (Fig. 8) using a ZEISS Axio Imager optical microscope. Three different magnifications were compared (x50, x100 and x200) as well as various number of micrographs. The images were acquired in similar light conditions and were binarized using the same threshold. They were analyzed by image processing (Python script) to know the porosity rate of the sample (ratio between the number of black pixels and the number total of pixels). It has to be noticed that, unlike the first two methods, micrographic analysis provides a porosity rate and not a density measurement.

According to [START_REF] Gu | Influences of energy density on porosity and microstructure of selective laser melted 17-4PH stainless steel[END_REF], micrographs can also be used to calculate pore circularity (PC) and determine their origins. The closer (PC) is to 1 (which is the maximum), the more circular is a pore. A low PC corresponds to LOF porosities whereas higher PC provides an estimation of entrapped gas porosities. This shape factor can be calculated with the following formula (Eq. ( 6)) where PC is the pore circularity, A the pore area (pix²) and Pp the pore perimeter (pix) :

= PC π A Pp 4 * * 2 (6)
With: A : the pore area (pix²), Pp : the pore perimeter (pix) This analysis is complementary to the Archimedes method or helium pycnometry because it allows to locate the pores in the sample, in order to estimate their size and shape. However it should be reminded that the closed pores related to a lack of fusion are full of powder whereas retained powder is removed during the polishing of the surface. This can induce a difference between the results of the various methods, especially for samples containing many LOF porosities. On the other hand, the metallographic method does not take into account the entire sample, but is only representative of the observed zones. This method last at least three minutes, considering only the micrographs acquisition of 4 different analyzed surfaces.

Comparison and optimization of experimental methods for porosity rate determination: experimental results and discussion

To achieve the most accurate measurements possible with the Archimedes and micrographs methods, various tests were conducted to determine the ideal measurement conditions for both methods.

Optimization of the archimedes method

Concerning the Archimedes method, different fluids (such as demineralized water, pure ethanol and acetone) were tested. Then the influence of surface roughness and sample volumes on measurement accuracy was addressed.

First, it appears clearly that demineralized water shows the lowest porosity rate among all the fluids (unlike [START_REF] Spierings | Comparison of density of stainless steel 316L parts produced with selective laser melting using different powder grades, Solid Free, Solid Freeform Fabr[END_REF] which does not show differences between water and acetone), but the highest standard deviation. Pure ethanol exhibits the highest porosity rate with the lowest standard deviation (Fig. 9). Fig. 9 also shows that the measured densities of the samples are extremely dependent on the value of the actual density of the fluid at the time of measurement. Since this true density value is not exactly known during the measurement, the theoretical density was considered and determined via tables (such as the International Alcoholometric Tables [START_REF]OIML, International Alcoholometric Tables[END_REF]), knowing the temperature of the fluid. The end result is therefore particularly dependent on the chemical composition and temperature of the fluid, and the accuracy with which the temperature is measured. Thus, a slight inaccuracy in the measurement of the fluid temperature combined with a large Δρ parameter (Table 3) has a clear influence on the final result. As an example, in the case of a sample with a low porosity rate (˜0.5%), a variation of ± 0,5 °C on the measured temperature leads to a difference of ± 9,2% on porosity rate with ethanol, whereas the same measurement using demineralized water would lead to a difference of ± 3,5%. This trend tends to decrease with higher porosities.

In addition, the differences obtained can be related to the difference in surface tensions between the fluids (Table 3). This difference affects the ability of the fluid to correctly wet the entire surface of the samples. Due to its low surface tension, ethanol is expected to better wet the samples, and thus yield lower porosity rates than those obtained with demineralized water. However, the results do not confirm this assumption. An explanation may be that low surface tensions stabilize occluded bubbles in a fluid, thus stabilizing air bubbles.

The comparison of experimental data provides similar evolution of porosity rate with experimental conditions for the three fluids. It was chosen to perform further optimization with water as the standard deviation obtained with demineralized water remained around 10% of the porosity rate value. Moreover, the choice of demineralized water is also validated by the fact that pure acetone and ethanol are hygroscopic. This property makes their purity deteriorate with time, as they absorb the water from the air.

In a second step, samples were polished to modify their surface roughness (Sa in μm) in order to address the influence of surface finish on the accuracy of density measurements, through possible variations of wetting. Indeed it was supposed that a sample with a high Sa is more likely to entrap air bubbles and to affect results. Three Sa conditions were compared: as SLM-built (˜12 +/-2 μm for the 7 experimental conditions), mirror polished (˜0.01 μm) and sand blasted (˜4 μm). It can be seen that for "as manufactured samples" (high Sa), the porosity rate and the standard deviation are lower than for the "mirror polished" (Fig. 10). A possible explanation may be that mirror polishing or sandblasting texturize the surface relatively homogeneously, thus making it hydrophobic according to a possible "Lotus Effect" [START_REF] Zhang | Lotus effect in wetting and self-cleaning[END_REF]. Indeed, lotus leaves are well known to be hydrophobic: water droplets falling onto the leaves will bead of and roll on, but they do not wet the surface. This is due to the surface micro-texture of the leaves. They are covered with micro-papilla (˜10 μm height), themselves covered with nano-papilla (˜100 μm). It provides air pocket formation, thus, water can only interact with the peaks of the roughness surface instead of wetting the entire surface, including peaks and valleys. First, the formation of micro air pocket at the sample surface can distort the results, and the wettability of the samples being greatly reduced. This would lead to an overestimation of the measured volume, and an underestimation of porosity rate. On the other hand, the "as manufactured" surface can absorb the fluid by capillarity and increase the wettability of the surface.

Finally, samples were cut to determine the influence of the analyzed volume. Initially they were around 2.5 cm 3 and they were cut to obtain samples of 1 cm 3 . Results show that the higher the samples volume, the lower the porosity rate as well as the standard deviation (Fig. 11). This means that minimizing the specific surface (S/V) is a key-point for stabilizing results.

To conclude, results presented here indicate that the Archimedes method should be preferentially used with demineralized water on "as manufactured" samples of at least 2 cm 3 in volume. Depending on experimental conditions, variations of up to +/-20% can occur for porosity rate values in the 1-5 % range. Other precautions are also recommended before starting measurements: (1) to clean samples in ultrasound bath to evacuate the powder stuck on the surface, [START_REF] Herderick | Additive manufacturing of metals: a review[END_REF] to dry them between each measurement and evacuate visible air bubbles by immersing samples in a beaker containing demineralized water and softly brushing them before measuring in the fluid.

It has also been observed that the larger the samples (at least 2 cm 3 ), the more accurate the measurements and the lower the errors. For this reason, samples of 2.5 cm 3 have been preferentially considered in this study. However, using larger volumes can become problematic for both Archimedes method and helium pycnometry for which the size of the measurement cell is limited.

Micrographic method

The micrographic method can be applied with different magnifications (x 50, x 100 and x 200) and the final results can vary a lot according to the number of pictures and the magnification considered.

The first step of the optimization was to determine which magnification is suitable for the porosity rate measurement. Indeed, a too low magnification could occult the small porosities, and a too high magnification could not be representative of the global sample. Microscope technical characteristics are summarized in the Table 4. Two areas of ˜9,5 mm 2 , corresponding to one x 50 micrograph, were first analyzed for different samples. Then, the same areas were analyzed with the x 100 and x 200 magnifications (respectively 4 micrographs/ area and 16 micrographs/area). It can be seen that the use of magnification increasingly important for the analysis tends to increase standard deviation (Fig. 12) and to modify the porosity rate: porosity rates higher than 1% (P6, P7) will tend to decrease, whereas under 1% they will increase. Those effects are more visible for samples with higher porosity levels, and are confirmed by [START_REF] Spierings | Comparison of density measurement techniques for additive manufactured metallic parts[END_REF]. In addition, the higher the porosity rate, the greater the standard deviations. Indeed, as porosities are not homogeneously distributed in the material, the porosity rate can vary between each micrograph leading to an increase in the standard deviation.

Micrographs obtained with a x100 magnification have pixel size of 0.69 μm which means first that porosities with an area upper to 0.5 μm² can be detected and taken into account (which is subject to a correct contrast between the metal and the porosity when acquiring the micrographs, as well as a good definition of the porosities contours). That is also why, this magnification was selected for porosity analysis.

In a second step, the number of pictures required to provide satisfactory results was determined with the x100 magnification. For the previous samples (Fig. 13) it appears that for the highest porosity rates (> 1%), at least 20 micrographs are needed to obtain a stabilized value (Fig. 13a) whereas Fig. 13b shows that the standard deviation is not stabilized before using 45 micrographs. For samples with a porosity rate below 1%, only 20 micrographs are shown to be sufficient to obtain stabilized porosity rates and standard deviations.

Finally, a possible procedure would be as follows: (a) a first micrograph has to be taken to estimate the porosity rate at x100 magnification, (b) if the value is higher than 1%, 45 micrographs have to be considered (analyzed area ˜107 mm²) and if the value is lower than 1%, 20 micrographs are enough (analyzed area ˜47 mm²).

For those reasons, the next micrographic analyses were carried out with the x100 magnification and with either 20 or 45 micrographs for each analyzed surface.

Comparison of the optimized measurement methods

Density measurements for Archimedes and helium pycnometry methods, and porosity calculations for the three methods are compared in Fig. 14 with error bars corresponding to the standard deviations.

First of all, it clearly appears that at low VED, the energy injected in the powder is not large enough to perfectly melt it. This results in very porous samples (P2 = 50 J/mm 3 ). An increase of the VED leads to a better densification of the samples (P3 = 75 J/mm 3 , P4 = 100 J/mm 3 ). Above a certain VED threshold (> 120 J/mm 3 ), the porosity rate increases in correlation with the formation of a key-hole regime (samples P6 = 200 J/mm 3 and P7 = 250 J/mm 3 ). This tendency is well known and several authors [START_REF] Kaufmann | Influence of process parameters on the quality of aluminium alloy en AW 7075 using Selective Laser Melting (SLM)[END_REF][START_REF] Gu | Influences of energy density on porosity and microstructure of selective laser melted 17-4PH stainless steel[END_REF][START_REF] Campanelli | Capabilities and performances of the selective laser melting process[END_REF] have already observed it on various materials (such as aluminum alloys, maraging steels, stainless steels) processed using various manufacturing parameters. The increase in porosity rate observed at high VED, and assumed to be due to the destabilization of the key-hole welding mode is in full agreement with recent simulation work by Tang [START_REF] Tang | A numerical investigation on the physical mechanisms of single track defects in selective laser melting[END_REF].

Moreover, it can be seen (Fig. 14a) that the two methods (Archimedes and Pycnometry) which give a quasi-direct access to the density Fig. 16. Micrographic cross-sections of samples P2, P4, P5 and P7 samples (50, 100, 125 and 250 J/mm 3 ) corresponding to respectively 1.8%, 1.1%, 1.7% and 5% porosity rates (Archimedes method). of each sample give rather similar results with less than 0.3% difference in porosity rate (Fig. 14b). This is in contradiction with [START_REF] Wits | Porosity testing methods for the quality assessment of selective laser melted parts[END_REF] who did not find a good accuracy in pycnometry measurements of samples having a low porosity rate (< 0.5%). The reason why lower porosity rates are obtained with helium pycnometry is possibly due to the fact that open pores are more likely to be filled with small helium atoms (0.31 Å) than with larger water molecules (2.75 Å). Thus fewer porosities will be considered with helium measurement. Another explanation is related to the difference of dynamic viscosities of helium and water, the first one being more than 40 time less viscous (Helium 25 °C, 1 bar : 19.68.10-6 Ns/m² ; water 25 °C, 1 bar : 891.10-6 Ns/m²). This is assumed to be the main reason why the density values obtained by helium pycnometry are greater than those obtained by the Archimedes method.

Considering these results and the theoretical density of the material (stainless steel 316 L: ρ theoretical = 7,99 g.cm -3 ), it becomes possible to calculate the porosity rate of each sample, and for each measurement method (Fig. 14b). However each theoretical density should be used with caution, and operators have to keep in mind that it probably does not correspond to the reality. Indeed, it is know that light elements used in alloys processed with SLM or EBM tend to evaporate [START_REF] Mukherjee | Printability of alloys for additive manufacturing[END_REF][START_REF] Klassen | Numerical simulation of multicomponent evaporation during selective electron beam melting of TiAl[END_REF], which modify the chemical composition and thus the material density.

According to Delesse's principle, which states that "the ratio of the area occupied by a component relative to the entire profile area is a consistent estimate of the volume fraction of the component in the object" [START_REF]Stereology Terms | Stereology Information Center[END_REF], we should measure similar porosity rates with the Archimedes' method, the gas pycnometry and the micrographic cross section.

However, it appears that the micrographic cross sections method tends to underestimate the porosity rate, even if several cross sections are considered (parallel and perpendicular to the building direction), unlike [START_REF] Wits | Porosity testing methods for the quality assessment of selective laser melted parts[END_REF] who shows similar results between the Archimedes method and the micrographic cross sections method. We can find that [START_REF] Mayhew | Caveat on the use of the Delesse principle of areal analysis forestimating component volume densities[END_REF] explains that the Delesse's principle should be used carefully, and can be applied only if equal reference areas are used. It means that the each micrograph must have the same size, which is our case in this study. This does not explains the underestimation of the porosity rates.

A possible explanation could be due to the porosities morphology, which is mainly random and irregular in additive manufacturing (especially for LOF at low VED, Fig. 3a). Depending on the depth at which the section is made, the porosity section is very different and the Delesse's principle is not checked (Fig. 15). That is why it can be necessary to repeat this method several times for a same sample (with a light polishing step between each observations), in order to stabilize the result by calculating an average porosity rate value over several sections.

Another explanation may be that the sample surface or near-surface porosity rate, which are usually higher due to contour-hatching transitions, are not considered in the micrographic observation.

It is interesting to notice that in the case of the sample P2 (50 J/ mm 3 ), LOF porosities are full of unmelted powder, which is taken into account during the Archimedes' Method or gas pycnometry. This gives the real volume of void for each porosity of the sample but, during the sample preparation for the micrographic cross sections analysis, this unmelted powder is evacuated from porosities and tends to distort the measured void volume.

Figs. [START_REF] Kasperovich | Correlation between porosity and processing parameters in TiAl6V4 produced by selective laser melting[END_REF] and 17 show micrographic cross-sections of each sample, before and after being etched with 92:5:3 reagent (92% HCl + 5% H 2 SO 4 + 3% HNO 3 , recommended in [START_REF] Vander | Metallographic techniques for tool steels[END_REF]) to reveal SLM beads. It can be seen that at low VED (sample P2, 50 J.mm -3 ), porosities are mainly LOF. The increase of the VED progressively leads to a better densification (sample P4). The remaining porosities are very localized melting defects (angular porosities) or entrapped gases (rounded porosities). These entrapped gas porosities can be promoted by the occurrence of a key-hole regime (larger pores), or by the presence of gas or humidity in the powder bed. Then, an excessive VED will melt the powder in keyhole mode and no longer in conduction mode, hence the presence of gas porosities (samples P5 to P7, respectively 125 and 250 J.mm -3 ).

Bead geometries were analyzed on micrographic cross-sections (Fig. 17), allowing to measure width to depth ratios (WDR). All the WDR are comprised between 0.48 and 0.8 (Table 5), which indicates that for all the experimental conditions e > > D, and that a full conduction regime was not achieved. For all SLM conditions, bead shapes suggest the occurrence of key-hole formation, even if key-hole porosities are not visible in each sample micrographic cross sections (samples P2 or P3 for example).

Moreover, [START_REF] Tang | Oxides, porosity and fatigue performance of AlSi10Mg parts produced by selective laser melting[END_REF] suggests that lack of fusion porosity can be predicted by the SLM bead overlap versus their dimensions (Eq. ( 2)). In the case where the bead overlap is inferior to 1, incomplete melting is avoided. It means that the overlap of SLM beads is sufficient between each bead of each layer, and between each layers of the sample to avoid LOF.

However, for all the conditions considered, the bead overlap is superior to 1 (Table 5), which means that Eq. ( 2) is not fully reliable to estimate the full consolidation limits.

Logically, the dimensional ratio calculated by Eq. ( 2) decrease as the densification increase, such as with sample P2 and sample P4 (respectively 1,78 ± 0,09 and 1,18 ± 0,08) (regardless gas porosities). This means that, even such ratios are superior to 1, they can be considered as an accurate geometrical method for estimating a-priori the degree of consolidation from the dimensions of single melt-pool geometries.

One explanation is that, at high VED, Eq. ( 2) is no longer relevant because porosities are not LOF anymore but entrapped gas due to the key-hole mode.

The repeatability of the three methods is related to the standard deviations of the samples compared to their average porosity rate. The Table 6 shows that Helium pycnometry and Archimedes methods exhibit an error ratio equal or lower than 10% for 10 measurements. This shows that several measurements (10 in our case) give quite similar results.

However, repeatability is probably not the best criterion for choosing one method over another. We advise instead to choose the method depending on the type of sample (shape, size, destructible or not), the results we want to obtain (type of porosity, location, average porosity rate, trend between samples), the material available and the time allotted.

Conclusion

Based on a detailed comparison of three experimental methods, it can be concluded that the Archimedes' method carried out in demineralized water is a reliable technique for estimating porosity rate in SLM samples.

Pycnometry measurements using helium and independent measurements of masses and volumes can be considered the most accurate technique for measuring porosity. However, because helium pycnometry is very time consuming, this method is ill-adapted as a systematic porosity rate control method. The results obtained with the Archimedes' method are very similar to those obtained by helium pycnometry. Because the Archimedes' method is both rapid and reliable, it is an efficient alternative to helium pycnometry. Another advantage of the Archimedes' method is that it is a non-destructive technique.

To ensure the reliability of the Archimedes' method, some preliminary precautions are necessary: air bubbles must be evacuated with the aid of a brush before weighing the samples in the fluid and sufficiently large samples (2 cm 3 minimum) must be used. However, open porosities and powder grains agglomerated on the surface are known to skew the measurements.

On the other hand, the position and the type of porosities present in the samples can only be determined by micrographic cross-section analysis. Because this method measures a 2D fraction area of a 3D porosity, the technique tends to underestimate the real porosity rate. Moreover, the volume of the pores cannot be accurately measured, especially in the case of irregular LOF pores.

Finally it has been shown that the VED input affects the overall porosity rate. Low input energies favor LOF pores, while excessive energies promote key-hole pores and spatters. Between these two extremes there is a wide range of VED in which it is fairly easy to densify material up to 99%, whereas if the SLM parameters are correctly optimized, densities up to 99,95% [START_REF] Mancisidor | Reduction of the residual porosity in parts manufactured by selective laser melting using skywriting and high focus offset strategies[END_REF] can be achieved.

Fig. 1 .

 1 Fig. 1. Main physical phenomena occurring during Selective Laser Melting.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Types of porosities obtained on 316 L steel: (a) lack of fusion, (b) blowhole, (c) key-hole cavity.

Fig. 5 .

 5 Fig. 5. 316 L powder, (a) particles morphology and (b) size distribution.

Fig. 6 .

 6 Fig. 6. Archimedes method density measurement.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Helium pycnometry density measurement.

Fig. 9 .

 9 Fig. 9. Comparison of the porosity rate of P2 to P7 samples according to the fluid used.

Fig. 10 .

 10 Fig. 10. Comparison of the porosity rate of P2 to P7 samples according to their surface finish (experiments in water).

Fig. 11 .

 11 Fig. 11. Porosity rate of P2 to P7 samples according to their analyzed volume.

Fig. 12 .

 12 Fig. 12. Porosity rate of 5 samples according to the micrographic magnification (analyzed surface = 19 mm²).

Fig. 13 .

 13 Fig. 13. Porosity rate (a) and standard deviation (b) of P2 to P7 samples according to the number of micrographs analyzed.

Fig. 14 .

 14 Fig. 14. (a) density measurements and (b) porosity rate estimation for P2 to P7 samples (50-250 J/mm 3 ).

Fig. 15 .

 15 Fig.[START_REF] Choi | Densiication behavior of 316L stainless steel parts fabricated by selective laser melting by variation in laser energy density[END_REF]. Porosity cross sections according to the depth of the micrographic section.

Fig. 17 .

 17 Fig. 17. SLM microstructures after chemical etching. Analysis of bead dimensions for samples P2, P4, P5 and P7 samples (50, 100, 125 and 250 J/mm 3 ).

Table 1

 1 Chemical composition of the powder (material data sheet).

	Fe	Cr	Ni	Mo	Mn	Si	C	N	P	S	O

Balance 17,7 12,7 2,31 0,89 0,61 0,018 0,09 0,008 0,004 0,028

Table 2

 2 Manufacturing parameters of each samples.

	P2	235	1221	0,07	50
	P3		814		75
	P4		611		100
	P5		489		125
	P6		305		200
	P7		244		250

Samples

P (W) V (mm/s) D (mm) Volume Energy Density (J. mm -3 )

Table 3

 3 Physical

	Fluids	Density (20 °C)	Surface tension (20 °C, 1 bar)	Dynamic Viscosity (20 °C)	Δρ (15 °C -25 °C)
	Demineralized Water	0,9982 g.cm -3	72,8.10 -3 N. m -1	0,913.10 -3 Pa s	0,20 %
	Pure Ethanol	0,7892 g.cm -3	22,3.10

properties of fluid used in Archimedes method (Δρ = variation of density in the 15-25 °C temperature range). -3 N. m -1 1,095.10 -3 Pa s 1,09 % Pure Acetone 0,7845 g.cm -3 23,7.10 -3 N. m -1 0,316.10 -3 Pa s 1,31 %

Table 4

 4 Main technical characteristics of the optical microscope.

	Magnification	Resolution	Micrograph size	Acquisition time of an area
				of 38 mm²
	x50	1.374 μm / pix	2452 * 2056 pix	6 seconds (4 micrographs)
	x100	0.689 μm / pix		16 seconds (16 micrographs)
	X200	0.343 μm / pix		53 seconds (64 micrographs)

Table 5

 5 Porosity measurements, bead dimensions and bead overlaps for P2, P4, P5, P7 samples.

		VED (J/mm3)	% porosity	WDR	Bead overlap
	P2	50	1.835 +/-0.188	0.8+/-0.06	1.78 +/-0.09
	P4	100	1.124 +/-0.156	0.48 +/-0.01	1.18 +/-0.08
	P5	125	1.723 +/-0.148	0.49 +/-0.04	1.34 +/-0.10
	P7	250	4.04 +/-0.172	0.5 +/-0.03	1.47 +/-0.03

Table 6

 6 Repeatability of the measurement for each method, where % P is the average porosity rate; σ is the standard deviation and ER the error ratio (ER=(σ*100)/%P).

	Samples references	Helium Pycnometry		Archimedes' Method		Micrographic cross section
		%P	σ	ER	%P	σ	ER	%P	σ	ER
	P2	1.539	0.168	10.9%	1.835	0.188	10.2%	1.009	0.43	42.6%
	P7	3.751	0.118	3.1%	4.040	0.172	4.3%	2.412	0.499	20.7%

Acknowledgments

Authors would like to deeply thank the French Aerospace Lab (ONERA), and particularly Marc THOMAS, Odile LAVIGNE and Stefan DRAWIN for allowing them to carry out helium pycnometry measurements.

They would also like to thank BPI France for financial support of the FAIR project and Air Liquide for helpful discussions and technical support.

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Maxime Pierre

come

Sensitivity of Xenon dual-phase time projection

chambers to rare event searches

Sensitivity of Xenon dual-phase time projection chambers to rare event searches Subatech, UMR 6457 CNRS, IMT Atlantique, Université de Nantes, 4 rue Alfred Kastler, Nantes maxime.pierre@etu.univ-nantes.fr

2 =

 2 Energy reconstruction • S1/S2 = NR vs ER discrimination • Dark Matter evidences: Astrophysics and cosmological evidences indicate a missing mass that counts for 25% of the total mass of the universe. • Best candidate: Weakly Interacting Massive Particle (WIMP) • Massive, neutral particle • Non-baryonic matter • Weakly interactive with baryonic matter and itself • Stable particle • Neutrinos nature: Dirac or Majorana particles ? • Experimental signature: Electronic recoil • Energy deposition of the two emitted electrons: Q-value of the process = 2457 keV • Observable: 0𝛖𝜷𝜷 Half-life • Cylindrical dual-phase TPC : 2.6 m diameter and 2.6 m height • 40 tons active volume of LXe • Two arrays of PMTs (top and bottom) • Muon and neutron veto systems • DARWIN will be sensitive to explore the WIMP parameter space down to the irreducible background from neutrinos (neutrino floor) [3]• If dark matter is observed DARWIN will be able to constrain its mass and its cross-experiments:136 Xe abundance ≥ 90%• DM dedicated experiments:natural abundance of 136 Xe = 8.9% • 136 Xe enrichement studied S 0ν = 3.0 × 10 27 y Nuclear recoil (NR) Electronic recoil (ER) | m ββ | 2 . | M 0ν | 2 . G 0ν (Q, Z) S 0ν = ln(2) × N a × ϵ × enr σ CL × M molar exposure bkg index × ΔE Phase-space factor Effective Majorana mass Nuclear matrix element (NME) • Direct detection: • Observable: WIMP-Nucleus cross-section and WIMP mass • Physical constants and confidence level factor • Studied isotope properties such as its abundance • Detector characteristics: detection efficiency, energy resolution, exposure • Backgrounds estimations


G Real

D Habault

X Cristol

J P Sessarego

D Fattaccioli

An Ultrasonic Testbench for Emulating the Degradation of Sonar Performance in Fluctuating Media

A scaled experimental protocol, in a water tank, is proposed to mimic the effects of medium heterogeneities on underwater acoustic propagation. The procedure consists in transmitting an ultrasonic wave (f = 2.25 MHz) through a RAndom Faced Acoustic Lens (RAFAL) in order to induce a spatially fluctuating sound pressure field. A scaling process based on a dimensional analysis ensures the representativeness of the experiment, when compared to the fluctuations of a few kHz continuous waves in the ocean. The regimes of saturation and unsaturation are explored by tuning the statistical parameters of the RAFAL's output face, the signal frequency and the propagation distance in the water tank. The control and reproducibility of the measurements is therefore ensured. The influence of volume effects is especially studied, since the experiments were conducted in free field conditions. Results in terms of mutual coherence function (second-order moment) are presented ; a good agreement is found between the values of radii of coherence calculated using experimental and simulation data, as well as theoretical results. The evaluation of the complex pressure distribution highlights the relevance of the experimental scheme, since behaviors typical of the regimes of fluctuations (saturation and unsaturation) are found.

Introduction

This paper is dedicated to underwater sound propagation in an uncertain fluctuating medium. The fluctuations of the characteristics of a medium can have a strong impact on wave propagation. This effect has been extensively studied over the last decades, in various domains, such as optics [START_REF] Tatarskii | Effects of the Turbulent Atmosphere on Wave Propagation[END_REF], electromagnetism [START_REF] Ishimaru | Wave propagation and scattering in random media[END_REF], seismology [START_REF] Sato | Seismic wave propagation and scattering in the heterogeneous earth[END_REF] and acoustics (in air and water) [START_REF] Dashen | Sound transmission through a fluctuating ocean[END_REF].

The general objective of our work is to develop and validate techniques to detect targets in perturbed oceans. It is well known that the perturbations on the wave propagation lead to a loss of efficiency of the detection techniques. Therefore there is a need for new techniques and processing which can overcome the effect of the fluctuations. These techniques must be validated under experimental conditions. That is why in our opinion it is essential to develop an experimental protocol in a water tank in which all environmental parameters are controlled. Furthermore the experiment must be as reproducible as possible.

Several types of studies of these phenomena in controlled environments such as laboratory water tanks can be found in the literature. An extensive review of the results obtained in these experiments is proposed by Dobbins [START_REF] Dobbins | Degradation of Coherence of Acoustic Signals Resulting from Inhomogeneities in the Sea[END_REF]. For instance, heating thermistors were used to generate thermal turbulence while an acoustic signal propagated through the perturbed medium (air or water). They provided very interesting results in terms of intensity moment [START_REF] Blanc-Benon | Intensity fluctuations of spherical acoustic waves propagating through thermal turbulence[END_REF], amplitude fluctuations [START_REF] Chotiros | Sound amplitude fluctuations due to a temperature microstructure[END_REF][START_REF] Stone | Transition regime for acoustic fluctuations in a randomly inhomogeneous medium[END_REF] or spatial [START_REF] Sederowitz | Covariance between acoustic signals traveling diverging paths in a random medium[END_REF] and time correlation [START_REF] Campanella | Time autocorrelation of sonic pulses propagated in a random medium[END_REF]. However, they are not fully reproducible since they only allow to recover the statistical behavior of the induced fluctuations. We propose here a fully deterministic protocol.

Our experimental study was motivated by results obtained in adaptive optics. Spectacular advances in terms of enhancement of the resolution of detection techniques have been observed in this field applied to observational astronomy [START_REF] Metchev | Adaptive optics observations of vega: eight detected sources and upper limits to planetary-mass companions[END_REF]. Experimental protocols have been developed in the optical wave propagation domain such as scaled experimental testbenches in order to reproduce the effect of the atmospheric turbulence under laboratory conditions. They provided helpful tools to develop, validate or discard adaptive signal processing techniques [START_REF] Hippler | Atmosphere-like turbulence generation with surface-etched phase-screens[END_REF][START_REF] Rampy | Production of phase screens for simulation of atmospheric turbulence[END_REF][START_REF] Wilcox | Atmospheric turbulence generator using a liquid crystal spatial light modulator[END_REF].

This idea partly followed some results presented by Booker [START_REF] Booker | Comparison between the extended-medium and the phase-screen scintillation theories[END_REF] which studied the non-intuitive representativeness of perturbations induced by an "almost-2D" rough phase screen, compared to those induced by an extended 3D fluctuating medium. This analysis was applied successfully to directive Gaussian beams in [START_REF] Andrews | Propagation of a gaussian-beam wave through a random phase screen[END_REF], with the conclusion that such protocols may be representative of extended medium configurations, under some specific conditions, such as the continuity in rms phase fluctuations in both cases.

The aim of this paper is to present such a protocol along with some experimental results which are compared with numerical results in order to validate the experimental design.

The focus here is on the volume effects. No interfaces (like the sea surface or the seabed) are taken into account. In this configuration, internal waves were proven to be responsible for the sound speed deviations from the mean value observed in shallow and coastal waters [START_REF] Garrett | Space-time scales of internal waves[END_REF]. Let us point out that a protocol allowing to mimic the induced typical statistical features of the received signals is sought out, rather than the reenactment of the phenomenon responsible for the signal distortions.

The experimental design described in this paper consists in transmitting an ultrasonic wave through an acoustic slab of small thickness placed in a water tank. The output face of the slab, that is on the hydrophone side, is one realization of a randomly rough surface. The statistical and geometrical parameters of the experiment are chosen in order that the perturbations induced in the medium can be similar to perturbations in a fluctuating ocean,provided correspondence rules involving statistics of the roughness, frequency and range of propagation.

A manufactured slab, characterized by a few statistical parameters, is used to reproduce the effects of 3D propagation in a fluctuating medium. One slab corresponds to one realization of the roughness parameter. The roughness of the output face induces refraction and diffraction of the acoustic wave, causing alternating focal points and shadow zones, which are typical of long-range propagation of sound waves through linear internal waves or phase shift and wavefront distortions, which usually occur at short ranges. The source is moved along horizontal and vertical axis in order to highlight various statistically independent regions of the slab ; independent realizations of the sound field propagated through the slab can hence be obtained. This leads to two advantages :

• for one configuration of source, receiver and slab, the experiment is deterministic and therefore can be controlled and reproduced ;

• for several positions of source, receiver and slab, the illuminated part of the slab varies and corresponds to several realizations of a same set of statistical parameters.

To obtain relevant comparisons between configurations studied in a water tank and real scale oceanic cases, it was necessary to find a way to tightly relate these two kinds of configurations. This is achieved through a dimensional analysis. This analysis is based on the use of two specific parameters defined by Flatté [START_REF] Dashen | Sound transmission through a fluctuating ocean[END_REF] the strength parameter(Φ) and the diffraction parameter (Λ). The values of these parameters may be used to define three kinds of regime : unsaturated, partially saturated and fully saturated. The scaling analysis includes a third parameter, the ratio of the vertical and horizontal acoustic correlation lengths of the acoustic perturbations induced in the medium to the wavelength (L z /λ). With these 3 parameters a correspondence can be obtained between the water tank experiment and a perturbed ocean. The studied oceanic configurations presented here correspond to the propagation of acoustic waves in the mid-frequency band (1-15 kHz) over distances of the order 1-10 km. They correspond to short times of propagation compared with the daily period of medium fluctuations so that the studied phenomena are considered spatially random but frozen in time.

This paper is organized as follows.

In Section 2, we present a short summary of the dimensional analysis. Section 3 describes the numerical computations developed to chose the parameters of the experiment and to validate the experimental results. The experimental procedure is presented in Section 4. The experimental and numerical results are presented and analyzed in Section 5. Comparisons are made for the second-order moments of the sound pressure as well as for the complex pressure distributions. Conclusions and perspectives are given in Section 6.

Scaling Procedure

The experimental scheme consists in transmitting an ultrasonic wave from a transducer through a vertical slab named RAndom Faced Acoustic Lens (called RAFAL in what follows) characterized by a plane input face and a randomly rough output face. This random roughness is chosen as a Gaussian normal process with a zero mean and a standard deviation ξ 0 . The acoustic field is recorded on a hydrophone which is moved along a vertical axis. It is expected that the experimental configuration would induce perturbations similar to the perturbations obtained in an oceanic case [START_REF] Real | Rafal: RAndom Faced Acoustic Lens used to model internal waves effect on underwater acoustic propagation[END_REF]. As explained previously, in order to make qualitative and quantitative comparisons of the statistical characteristics of the sound field obtained with the experiment in the water tank and for some realistic oceanic cases, a scaling analysis is developed. The main points are summarized in this section along with the details on the choice of the parameters of both slab and oceanic cases.

The approach is based on the use of the Λ and Φ parameters and the evaluation of the ratio of the correlation length of the acoustic field to the acoustic wavelength L z /λ [START_REF] Tatarskii | Effects of the Turbulent Atmosphere on Wave Propagation[END_REF]. The main assumption is that it is possible with these three parameters to qualify and quantify the induced fluctuations.

In the oceanic case, these parameters depend on the signal frequency f s , the propagation range R, the standard deviation of the sound speed fluctuations δc 0 as well as on the vertical and horizontal correlation lengths of these fluctuations (L Vs and L Hs ) (see Table 1).

For the RAFAL, each configuration is described by geometrical and physical parameters, namely the standard deviation of the roughness amplitude ξ 0 , the vertical and horizontal correlation lengths of the roughness, respectively L V and L H , the signal frequency f and finally the distance x dist between the RAFAL's average output face and the receiver (see Figure 1). Therefore the main step is to express the three parameters (Λ, Φ and L z ) in terms of these quantities. This is done by first computing an analytical approximation of the sound pressure p based on several approximations, mainly the parabolic approximation and the Integral Small Table 1: Expression of the dimensional parameters for the oceanic case and the scaled experiment case.

Ocean RAFAL

Φ 2 = k 2 0 √ 2π δc 0 c 0 2 L Hs R Φ 2 ℓ = k 2 1 ξ 2 0 1 -c 1 c 2 2 Λ = R ak 0 L 2 Vs Λ ℓ = 1 2π R F L V 2 L z = α λ s c 0 δc 0 L 2 Vs RL Hs 1/2 L ℓ z = ρ 2 2q + x k 1 + b q 2 R
Slope Approximation (ISSA) [START_REF] Voronovich | Wave scattering from rough surfaces[END_REF]. The evaluation of Λ ℓ is based on the calculation of the Fresnel radius (obtained using its classical definition [START_REF] Brühl | Fresnel zones for broadband data[END_REF][START_REF] Sears | Optics[END_REF]) and Φ ℓ is obtained after analytical calculation of the average sound field < p >. Finally, the vertical correlation length L ℓ z is obtained from the vertical intercorrelation function of p. The details of these calculations can be found in [START_REF]An ultrasonic testbench for reproducing the degradation of sonar performance in random ocean[END_REF].

Table 1 summarizes the expressions of Λ, Φ and L z /λ in typical oceanic configurations [START_REF] Dashen | Sound transmission through a fluctuating ocean[END_REF][START_REF] Tatarskii | Effects of the Turbulent Atmosphere on Wave Propagation[END_REF] and in the experimental configuration. The acoustic wavenumber is defined as k i = 2πf /c i , where f is the signal frequency and c i the sound velocity in medium i. The parameters which appear in this Table are defined as follows : a is a constant equal to either 2 in the case of a plane wave or 6 in the case of a point source, α = 0.24, ρ is the transducer radius

, b = x 1 k 1 + x 2 -x 1 k 2 , d 2 = k 2 1 ξ 2 0 1 -c 1 c 2 2 1 L 2 V , q = 1 + d 2 ρ 2 2 and R = ρ 2 /8 + b 2 d 2 1+d 2 ρ 2 /2 .
Hence, the comparison is based on the equality between the three parameters in the ocean configuration and for the slab case [START_REF]An ultrasonic testbench for reproducing the degradation of sonar performance in random ocean[END_REF][START_REF] Real | Rafal: RAndom Faced Acoustic Lens used to model internal waves effect on underwater acoustic propagation[END_REF]:

Λ = Λ ℓ , Φ = Φ ℓ , L z /λ s = L ℓ z /λ. (1)
The Λ -Φ plane provides a way to relate each value of these two parameters to a corresponding regime of fluctuations :

• the unsaturated regime, where phase fluctuations mostly arise from the medium inhomo-geneities;

• the partially saturated regime, where correlated multipath propagation may be observed ;

• the fully saturated regime, where uncorrelated eigenrays are observed.

Of course, this is a global way to present the various regimes and in particular the two boundaries must not be understood as a strict separation, the regimes overlap in the transition zones (see Figure 2 for example).

Numerical Simulations

Numerical computations were conducted with two aims:

• to determine the parameters of the experimental configuration in relation with the parameters chosen for the oceanic cases

• to make comparisons of some of the statistical characteristics of the sound propagation for the slab and the ocean cases.

Once the statistical parameters of the RAFAL's output face were chosen, the propagation of acoustic wave through the designed RAFAL was tested through computer simulations. If the original set of parameters was validated by the simulation results coupled with the scaling analysis, the RAFAL is manufactured in order to correspond as much as possible to the model used in the simulations.

Two codes were used to compute the sound propagation through a RAFAL and to choose the parameters for the experimental configuration. One is called RAYTAL (for RAY Tracing through an Acoustic Lens) and is based on a ray tracing method. The other one is called P3DTEx (for 3D Tank Experiment configuration) and is based on a 3D parabolic model.

In the computations, the directivity function of the source transducer was described as a plane circular piston ("Sombrero" function [START_REF] Gaskill | Linear systems, Fourier transforms, and optics[END_REF]) [START_REF] Kinsler | Fundamentals of acoustics[END_REF]. This provides an accurate description of the radiation of the transducer used in the experiment. The source position is (x S = -0.2 m; z S = 0 m), the RAFAL's input plane face is located at x = 0m and the average output face at x = 0.02 m.

As an example, Figure 2 presents a qualitative comparison of sound fields obtained with these two software, for one realization only (i.e. one profile and one source position) of RAFAL. The profile of the slab is shown on the left part of each figure. The calculation is carried out in 3D, but the propagation is observed on a plane orthogonal to the RAFAL which transverse section is drawn on the left of each figure. The sound levels are given following the color bar underneath. Figure 2 displays the acoustic pressure fields and ray beams for four configurations : the unperturbed case (a) that is the simple case where the acoustic slab presents two plane faces; the unsaturated (b), the partially saturated (c) and the fully saturated case (d). Figure 2 shows a good qualitative agreement between RAYTAL and P3DTEx: the ray beams and high acoustic levels are superimposed. The unperturbed case points out the directivity of the transducer. Because the sound speeds in water and in the slab are quite close, the rays are not significantly refracted when propagating through the RAFAL. In the unsaturated case, a refraction of the ray beam is noticed. This result is confirmed by P3DTEx. The resulting acoustic field is perturbed, but remains close to the result obtained in the unperturbed case. In the partially saturated case, there is a focalization of the acoustic field in the region (x = 0.2 m; z = 0.01 m), which is in agreement with the high acoustic amplitude computed with P3DTEx. The fully saturated case shows a large refraction of the rays and several focal points. These focal points lead to regions of high acoustic amplitude given by P3DTEx.

All these results show that the propagation through a RAFAL does not simply result in a phase screen process but can produce 3D pressure fields similar to those obtained with volume effects. These qualitative observations are in good agreement with the comments found in the literature about sound waves propagation through a field of linear internal waves. The third software P3DCOM (for Propagation in a 3D Corresponding Ocean) is also based on a 3D parabolic method and was used to compute the sound propagation in ocean [START_REF] Cristol | Alternative criteria for sonar array-gain limits from linear internal waves[END_REF]. It is used in the next sections to present more detailed comparisons between the slab case and the corresponding oceanic case. [START_REF] Brühl | Fresnel zones for broadband data[END_REF] The Experimental Procedure

The equipment

A diagram of the experimental set-up is given in Figure 3.

The measurements were conducted in a 3 m long, 1 m wide and 1 m deep water tank, filled with fresh water (see Figure 3). The temperature was continuously controlled by a probe so that the sound speed was accurately known at any time from the table in reference [START_REF] Del Grosso | Speed of sound in pure water[END_REF], assuming that its distribution is uniform inside the water tank. The dimensions of the slab are 15×15×0.2 cm 3 . The equipment was located in the middle of the tank depth and it was possible to ignore the influence of the bottom and the surface.

The transmitted signal was a CW chirp at f = 2.25 MHz with a duration of 22.2 µs (N p = 50 periods) and an amplitude of 5V. At this frequency and at a temperature of 20 • C, the wavelength λ is equal to 0.658 mm. The signal was generated with a HP 33120A Arbitrary Waveform Generator and sent through a Panametrics V306 transducer (Figure 4) after ×10 amplitude amplification by a NF 4005 High Speed Power Amplifier.

The receiver was an Acoustic Precision Needle hydrophone (Figure 4). The signal was received on the control computer after high-pass filtering and amplification by a Sofranel Pulse Receiver Model 5055PR.

The acoustical equipment was placed on motorized rails driven by a computer interface (see reference [START_REF] Papadakis | Scaled laboratory experiments of shallow water acoustic propagation: Calibration phase[END_REF] for more details). Automatic displacements of the source and the receiver allowed to provide several realizations for the same fluctuation regime and to carry out statistical studies. The measurements were repeated for several source depths in order to acoustically highlight several decorrelated regions of a RAFAL. In order for the realizations to be independent, the vertical displacement of the source was chosen to be equal to at least twice the vertical correlation length of the random roughness of the output face L V . The number of RAFALs that needed to be manufactured was determined by the number of independent realizations available on a specific slab. The displacement of the hydrophone on a vertical axis was used to simulate vertical arrays of 64 hydrophones separated by a distance of 0.3 mm. This value was chosen in order to satisfy the sampling criterion (s < λ/2).

Table 2 summarizes the depth shifts between two positions of the source and the number of slabs that were manufactured. In the first column, FS, PS and US stand for Full Saturation, Partial Saturation and UnSaturation, respectively.

The RAFAL manufacturing process

The material chosen for the slabs is referred to as Machinable Blue Wax (used in reference [START_REF] Calvo | Benchmarking of computational scattering models using underwater acoustic data from a corrugated wax slab[END_REF]). Its physical properties were measured and are listed in Table 3 . Its density is very close to the density in water so that they were assumed to be equal in the computations. Moreover, it was assumed that the propagation of shear waves could be disregarded since the shear wave attenuation was larger than 30dB/cm. The manufacturing of the slabs was carried out in four steps.

Step one, the profile defined from the computations was interpolated and edited using a Computer Aided Design (CAD) software. In step two, the output file of the CAD software was sent to a 3D printer, producing a first version of the sample. However because of the temperature limitations of the 3D printer material it was not possible to directly use this sample.

Therefore, step three consists in using a molding silicone (RTV 2-RTV 123) in order to obtain a "negative" mold of the original profile. This silicone maintains its shape at high temperature. In step four the melted Machinable Blue Wax was poured in the negative mold. This last step was realized under the dome of an air pumping system in order to avoid the presence of air bubbles at the surface of the slab.

Figure 5 displays the RAFAL at the different steps of the manufacturing process [START_REF] Real | De-coherence effects in underwater acoustics: scaled experiments[END_REF].

Note that the design of these slabs is quite time consuming. Consequently, the number of RAFAL that could be manufactured was limited and it was not possible to acquire more experimental data.

Experimental Results

The experimental results are presented in this section. Comparisons between the measured data and the simulations are provided, first of all in terms of second-order moment of the pressure field, or mutual coherence function (MCF). Then, the complex pressure distribution (CPD), or phasor, is computed. The latter is used to retrieve the a priori classification of the experimental configurations in the various regimes of fluctuations.

f = 2.25 MHz f s = 1kHz Config. # ξ 0 [mm] L V [mm] x dist [mm] δc 0 [m/s] L Vs [m] L Hs [m] R [km

Configurations studied

In order to be representative of realistic oceanic cases a set of 7 configurations was chosen from the procedure detailed in Section 2. They are shown in Table 4 where parameters for the tank experiment and the corresponding ocean configurations are listed. As said previously the experiment was conducted for a f = 2.25 MHz frequency in the water tank. A frequency of f s = 1 kHz was chosen for the corresponding oceanic configuration. This arbitrary choice was motivated by the fact that the P3DCOM software is definitely validated around this frequency.

The values of the parameters chosen for the oceanic cases are typical of what can be observed in the case of perturbations due to linear internal waves [START_REF] Dashen | Sound transmission through a fluctuating ocean[END_REF]. The standard deviation of the sound speed fluctuations is of the order of 1 m/s going from 0.77 m/s (FS1 configuration) to 2.24 m/s (US2 configuration). The values of the vertical correlation length of the sound speed fluctuations are also quite representative of what can be observed in oceanic media: from 12.4 m (US2) to 31.9 m (FS1). The ratio of vertical to horizontal correlation lengths of the sound speed fluctuations is a parameter that was tuned in order to obtain realistic results in terms of δc 0 . Nevertheless, this was performed taking into account the fact that the value of the horizontal correlation length of the sound speed fluctuations is typically of a few hundred meters. Here L Hs lies between 106.3 m (FS1) and 247.5 m (US2). Finally, the range of propagation is also a critical parameter in an oceanic medium, since the saturation increases with the propagation distance (if the environmental parameters are the same). Here the parameter R spans from a few hundred meters (R = 250 m for US2) to more than ten kilometers (R = 19.9 km for FS1). Figure 6 represents the locations of the seven configurations in the Λ -Φ plane. From these results, two configurations were classified as fully saturated (FS1, FS2 and FS3), three as partially saturated (PS1, PS2) and two as unsaturated (US1 and US2).

5.2

Second-order moment of the pressure field

In this section, we compare the second-order moment (or mutual coherence function, MCF) obtained in various configurations. This function, denoted Γ, provides information on the correlation of the signal along a vertical linear array. The cross-spectrum matrix was first calculated and averaged over the whole number of realizations. The obtained quantity was then averaged over the iso-spaced sensors, leading to a function of the sensor spacing. More precisely, Γ is defined as [START_REF] Carey | Sound transmission and spatial coherence in selected shallow-water areas: Measurements and theory[END_REF][START_REF] Collis | Observed limiting cases of horizontal field coherence and array performance in a time-varying internal wavefield[END_REF][START_REF] Wilson | Performance bounds for acoustic direction-of-arrival arrays operating in atmospheric turbulence[END_REF]: where l is the discrete sensor spacing, Π is the Fourier transform of the received signal, . * denotes the complex conjugate, < . > N is the ensemble average over the array (variable n) and < . > Nr is the ensemble average over the number of realizations.

Γ(l) = Π (n) Π * (n + l) |Π (n) ||Π (n + l) | Nr N , (2)
Comparisons are made between the experimental results and the P3DTEX and P3DCOM results. Furthermore, they also include the theoretical solution given for the MCF by Collis et al [START_REF] Collis | Observed limiting cases of horizontal field coherence and array performance in a time-varying internal wavefield[END_REF] Flatté [START_REF] Dashen | Sound transmission through a fluctuating ocean[END_REF][START_REF] Flatté | Calculations of wave propagation through statistical random media, with and without a waveguide[END_REF][START_REF] Flatté | Calculations of internal-wave-induced fluctuations in oceanacoustic propagation[END_REF] where the MCF is approximated by an exponential function parameterized by the radius of coherence, denoted ρ c in the continuous space domain and R c in the discrete sensor spacing domain. Hence, the radius of coherence corresponds to the width of the main lobe of the second-order moment, denoted L z in Section 2. The quadratic form of the phase-structure function is used [START_REF] Flatté | Calculations of wave propagation through statistical random media, with and without a waveguide[END_REF]. The results associated with this formula will be referred to as simplified theory (ST) :

Γ ST (l) = e -1 2 l Rc 2 .
(

) 3
Let us point out that this approximation of the MCF obviously provide the same value for the 3D fluctuating ocean and for the experiment because of the scaling procedure based on the same value of the normalized acoustic correlation length L z /λ.

The comparisons obtained from the computations provide a tool to test the results obtained in the water tank. The idea that the MCF should behave differently depending on a specific regime of fluctuation is counterintuitive: the radius of coherence can sometimes be larger in a fully saturated case than in an unsaturated case. Therefore, it is not relevant to compare results obtained for different regimes. We emphasize on the fact that a consistency in terms of the evolution of the MCF as a function of the regime is sought out, rather than a perfect agreement between the different calculations: for instance, an increase (or a decrease) of the radius of coherence between two different configurations should be noticed with the simulation, experimental and theoretical results.

In the next paragraphs, the MCF functions are first presented for three configurations. They are displayed as functions of the sensor spacing normalized by the wavelength, s/λ, so that they can be compared for the scaled experiment and for the oceanic configurations Then a comparison between radii of coherence is presented for the seven configurations.

MCF

The number of realizations used for each case is shown in Table 2. In Figure 7,Γ(s/λ) is shown for 3 cases (each one corresponding to a specific regime). The experimental results and the P3DTEx results are plotted with red circles and magenta diamonds respectively. When available, the P3DCOM results are plotted with cyan squares. The simplified theory is represented by a black solid line. Let us point out that P3DCOM could not be used for the unsaturated cases considered here because of the numerical spurious interference produced by the code at such very short distances (R = 250 m).

Figure 7a displays the MCF obtained for the US2 case. The MCF curves corresponding to measurements and to P3DTEx are in excellent agreement. Despite the fact that the number of realizations is much higher in the P3DTEx case (N r = 240, whereas in the measurements case, N r = 24), the agreement for the main lobe of the MCF is quite good. Furthermore, the curve corresponding to the simplified theory is quite close to the other two.

Figure 7b presents the results obtained for the PS1 configuration. The agreement between the results provided by the scaled measurements, the numerical code (P3DTEx and P3DCOM) and the simplified theory is still very good.

Figure 7c corresponds to the FS3 case. The number of realizations N r is equal to 22 and equal to 220 for the P3DCOM and P3DTEx computations. The agreement between the curves is still good overall, however some discrepancies appear. The experimental and the P3DTEx curves are quite close but are different from the other two (simplified theory and P3DCOM) which are in good accordance.

For the three configurations the experimental results and P3DTEX are very close, although the number of realizations in the experimental case is quite low. This means that the experiment was conducted very carefully.

We observe a rise of the MCF "late secondary lobes" for large normalized sensor spacing in all cases. This is due to the fact that the value of the MCF as the spacing goes to infinity is given by Γ(l → ∞) = Π(0) [START_REF] Dashen | Sound transmission through a fluctuating ocean[END_REF]. Although this value is statistically uncertain because of the small number of realization for the maximum sensor spacing, we can reasonably assume that is not 0, which is confirmed by the results in Figure 7. Nevertheless, the focus of our study is the estimation of the width of the first lobe and this secondary lobe could be easily suppressed by a windowing procedure.

|Π(0)| Nr Π(l) |Π(l)| Nr

Radius of coherence

Other quantitative features to be investigated are the radii of coherence estimated from the numerical and experimental data. The radius of coherence is defined as the sensor spacing

s/λ = ρ C such that Γ(ρ C ) = e -1 2 [7].
For each configuration located along the horizontal axis in Figure 8, 3 or 4 colored bars are presented. They correspond to the mean value of the radius of coherence obtained from each calculations (P3DTEx, P3DCOM and simplified theory) and the experimental data. The error bars correspond to the standard deviation of calculations of the radius of coherence. For the experimental data, the error bars are not significant because of the rather small number of realizations. For the simplified theory, there is obviously no reason for error bars. The analysis of Figure 8 confirms the conclusions already pointed out. The match between theoretical, simulated and measurements is quite good for most cases. As an example for the PS1 configuration the values for ρ C are in excellent agreement since they vary from 5.05 to 5.35. Another example is given with the FS3 configuration, for which ρ C is respectively equal to 3.31 for P3DCOM, to 3.27 for the simplified theory, and to 2.4 for the experimental case and for the P3DTEx calculation.

The general conclusion is that the theoretical solution and P3DCOM are in excellent agreement. Similarly, the scaled experiment and P3DTEx results are in remarkable concordance, but for some configurations there is a discrepancy between the two groups of results. The largest differences are obtained for the full saturation regimes.

It is interesting to notice the slight differences between the results obtained using the tank measurements and P3DTEx: the results are very similar, but the radius of coherence calculated in the simulation framework exceeds the scaled experiment radius in all cases. This may be explained by the uncertainty on several experimental parameters, for instance:

• the position of the hydrophone was estimated with an error of 0.1mm

• the plane input face of RAFAL could be not perfectly normal to the main axis of propagation although the orthogonality was measured by comparing time travels of signals reflected from various locations of the input face. Furthermore, slight local errors in the randomly rough face amplitude of the RAFAL are induced, due to the manufacturing process.

• the physical characteristics of the material composing the RAFAL were measured with uncertainties (see Table 3 ).

This could lead to possible differences between the results obtained with the experimental data and P3DTEx. However these differences are quite small.

The global comparison between the experiment and P3DCOM is quite satisfying but some differences appear. They obviously could be the result of too rough approximations made in the scaling procedure. The choice of the 3 parameters (Λ, Φ and L V /λ) seems to be quite relevant. However a lot of approximations were used in order to evaluate the parameters for the slab case. For instance, the calculation of the sound pressure is based on the use of the ISSAwhich may not be valid for too high values of the RAFAL roughness. This is the case for the fully saturated configurations which correspond to a large roughness amplitude and a small vertical correlation length. Some further computations must be conducted to evaluate the influence of such approximations on the accuracy of the parameters. If necessary, different approaches may be considered in order to derive the acoustic pressure field in the fully saturated configuration.

Let us remark also that the MCF obtained in the experimental framework (with RAFAL or P3DTEX) is slightly narrower than the theoretical approximation of the MCF. Similarly to what was explained earlier, this result can be explained by the approximations used to derive an expression for the theoretical radius of coherence in Section 2.

Finally, despite some discrepancies between the radii of coherence estimated in the fully saturated regime, the experiment provides very satisfying results. The fact that the measurements were conducted with great care should be highlighted. Moreover, although the number of realizations is much smaller in the experimental case than in the simulations configurations, the results are in good concordance, meaning that, for the MCF analysis, the number of realizations provided here is sufficient. In particular our main objective is fulfilled that is to reproduce perturbations of the sound propagation similar to those observed in a perturbed ocean. This means that this experiment is validated as a powerful tool to test detection techniques.

One aspect of the scaling procedure was therefore validated throughout this section (the L V /λ parameter). Section 5.3 focuses on the validity of the other aspect of the procedure: the classification into regimes of fluctuations.

Complex pressure distribution

Another comparison between the experimental and numerical data may involve the behavior of the complex pressure distribution (CPD). Examples of CPD results can be found in [START_REF] Ehrhardt | Evaluating a linearized euler equations model for strong turbulence effects on sound propagation[END_REF][START_REF] Flatté | Wave propagation through random media: Contributions from ocean acoustics[END_REF]. In the fully saturated regime, large phase and amplitude variations are noticed, inducing a chaotic representation of the so-called "phasor". The CPD should therefore be concentrated in the zero area. In the unsaturated regime, the CPDs are distributed on the circle of mean complex pressure which corresponds to a variation of phase only. The partially saturated regime presents a transition between the other two.

Figure 9 displays CPD calculated on 16 hydrophones (corresponding to the 16 central sensors of the array) for 3 configurations. The real and imaginary parts of the complex pressure are displayed for each realization of the medium. They are normalized that is divided by the mean value of the pressure distribution for each configuration (p 0 = |p| 2

Nr

, plotted with the dashed line). Let us first point out that the comparisons are mostly qualitative. The experimental results cannot accurately be analyzed because of the small number of realizations available. But the good comparisons done for the MCF prove that the numerical computations obtained with a larger number of realizations provide a correct simulation of the experiment.

Figure 9 presents the CPD for three cases (US2, PS1 and FS3, from left to right). For each one, the three lines respectively correspond to the experimental data, the P3DTEx and the P3DCOM computations.

The main features are globally similar to what was expected. Figure 9a, which depicts the US2 case, shows a CPD that tends to follow the mean complex pressure circle. A very good correspondence with the mean complex pressure is observed in all cases (Figures 9a, 9d and9g). Figure 9c (FS3 case) shows that the CPD are relatively uniform but present numerous peaks inside the mean pressure circle. Nevertheless, the P3DTEx results enhances the analysis. The behavior of the phasor appears more clearly with an accumulation around the zero area in the FS3 configuration. As expected, the partially saturated case (PS1) appears as a transition between the FS3 and US1 configurations. This analysis also applies to the results obtained using P3DCOM (third line), where the behaviors are extremely similar to the P3DTEx values.

The CPD enhances the relevance of the scaling procedure in terms of regimes of fluctuations between oceanic environment and tank experiment setup.

Conclusions

This paper proposed an original approach to reproduce effects similar to those of internal waves on underwater acoustic propagation. Innovative corrective signal processing techniques could therefore be benchmarked using the data acquired in the water tank.

The experimental protocol consisted in propagating an ultrasonic wave through an acoustic slab. This procedure was carried out in a water tank in conditions such that scattering from the sea surface or the seabed were left aside. The manufacturing of the RAFAL, based on the use of a 3D printer and molding techniques, provide a tool to control the induced fluctuations in a highly reproducible manner, which was, to our knowledge, not proposed before. It allows indeed to reproduce a given result for a given source-slab-receiver configuration faithfully and in a deterministic manner. The reproducibility of the procedure is to be highlighted since it permits to validate or discard several potential corrective detection techniques in the same experimental conditions.

A scaling procedure based on a dimensionless analysis provides a direct link between the experiment and oceanic scale observations. The acoustic data acquired in the water tank can hence be related to realistic at-sea configurations.

We provide in this paper a twofold data analysis. First, the mutual coherence function (MCF) was calculated for several configurations: using experimental data, parabolic equation simulations for the scaled experiment case and for the corresponding ocean medium case, and simplified theory. It remained consistent throughout the explored regimes of saturation and unsaturation. The estimation of the radius of coherence was proven to be accurate, which enhances the relevance of the experimental scheme from a quantitative degradation point of view. The deviations observed in the fully saturated regimes were analyzed and should not be attributed to experimental uncertainties. Nonetheless, a different approach may be needed to tackle the fully saturated case, because of the important roughness of the slab in this case. Second, the evaluation of the distribution of the complex pressure showed remarkably good behavior when compared to the available literature references. This reflects the relevance of the experimental protocol. Especially, it validates the a priori classification into qualitative regimes of fluctuations.

Overall the aimed phenomena were observed under laboratory conditions. Some improving paths may be explored, such as the use of receiver arrays, which would decrease the time necessary to conduct an experiment, and hence would allow for more realization to be processed. The manufacturing process theoretically presents no limitations and the studies conducted here should be extended to other spectrum of fluctuations than Gaussian only. The representativeness of other phenomena (turbulence, small-scale fluctuations due to sea surface agitation,...) could therefore be studied.

Also, an extensive analysis of the intensity fluctuations (fourth-order moment) of the data collected during the experiments will soon be available. It would however be interesting to provide a comparison between the results associated with the scaled experiment and real-scale data. The authors plan on using the data acquired during the ALMA [START_REF] Fattaccioli | ALMA: a new experimental acoustic system to explore coastal waters[END_REF] experiment to do so.

Figure 1 :

 1 Figure 1: Experimental configuration diagram.

Figure 2 :

 2 Figure 2: Simulation results: ray tracing programs & P3DTEx -unperturbed case (a); unsaturated regime (b); partially saturated regime (c); fully saturated regime (d).

Figure 3 :

 3 Figure 3: Experimental configuration.

Figure 4 :

 4 Figure 4: Experimental equipment -top left: Panametrics V306 SU transducer; top right: AP Needle hydrophone; bottom: water tank.

Figure 5 :

 5 Figure 5: Manufacturing steps of the RAFAL -From left to right: CAD software (GLC) screenshot (step 1), 3D printer output (ABS) (step 2), negative mold (RTV2 -RTV123 Silicone) (step 3), final RAFAL (Machinable Blue Wax ) (step 4).

Figure 6 :

 6 Figure 6: Experimental configurations in the Λ -Φ plane.

Figure 7 :

 7 Figure 7: MCF -64-sensor vertical linear array -Experimental data(•), P3DTEx results (♦), P3DCOM results( ), Simplified theory (-).

Figure 8 :

 8 Figure 8: Comparison of radii of coherence ρ C obtained with P3DCOM, experimental data, P3DTEx and simplified theory from left to right in all configurations.

Figure 9 :

 9 Figure 9: Complex pressure distribution (CPD) and mean complex pressure distribution (dashed circle) -Configurations (from left to right): US2, PS1 and FS3.

Table 2 :

 2 Source depth shift ∆ Zs , in mm, as a function of the vertical correlation length L V , in mm. Corresponding number of realizations per RAFAL N r and number of manufactured RAFALs per configuration N RAF AL . Config. # L V [mm] ∆ Zs [mm] N r N RAF AL

	FS	4	10	22	2
	PS	7	15	21	3
	US	8	20	24	4

Table 3 :

 3 Measured physical properties -Machinable Blue Wax.

	Density [g/cm 3 ]

Table 4 :

 4 Scaled experiment and corresponding oceanic medium parameters.

Acknowledgements

The authors would like to thank Régine Guillermin and Guy Rabau, as well as the staff from the SERM (mechanical technical support) at the LMA, for their help during the experiment. This work was sponsored by the Mission pour la Recherche et l'Innovation Scientifique (MRIS, DGA) and Thales Underwater Systems.