A Diet

M Biancheri-Astier

Y Le Bihan

C Conessa

F Alves

M Grzeskowiak

M Benamara

G Lissorgues

Design of 1cm² coils for HF RFID instruments tracking with detection range improvement

Keywords: HF RFID, coil, magnetic coupling

This paper concerns an application of magnetic coupling RFID technology at 13.56 MHz (HF band) for tracking devices such as instruments. The tag size is defined to be ergonomically small compared to the hand, and fixed inside a maximum surface of 1 cm². The case of multiple detections is considered, and consequently the reader surface of control is considered wide enough to include several instruments at the same time during a logistic control process. The use of such a small RFID tag is almost impossible using a large reader loop of 15x30 cm², as chosen for the tests. The key idea of the paper is then the addition of a resonator that enables to create the mandatory physical link by means of magnetic coupling between the tag coil and the resonator coil and between the resonator coil and the reader loop. Finally the detection range is highly improved by the presence of this resonator and results demonstrate that it is possible to detect these small RFID "1 by 1 cm² tags" at a distance of 1.5 cm to 3 cm, depending on their orientations.

I. INSTRUMENT DETECTION IN A VOLUME

RFID (Radio Frequency Identification) is a fruitful technology that improves many applications of tracking in various contexts such as market or medicine [START_REF] Paret | Identification Radio-Fréquences et Cartes à puce sans contact[END_REF] [START_REF] Finkenzeller | RFID handbook, fundamentals and applications in contactless smart cards and IDs[END_REF]. The tracking process can concern persons, animals or devices in various scenarii in which some routing errors can be avoided by using RFID tags [START_REF] Talon | Traçabilité des instruments de chirurgie re-stérilisables: apport de la technologie RFID[END_REF][4] [START_REF] Al | Initial Clinical Evaluation of a Handheld Device for Detecting Retained Surgical Gauze Sponges Using RFID Technology[END_REF]. Additionally, the RFID tags can perform a history back-up memory that can be used for managing the life-cycle of the tagged device and optimize the logistic flux. Currently, technological locks, in RFID projects, are determined by specific cases constraints such as the electromagnetic environment (metal, water…), the problem geometry (e. g. limited volume) and the ability (and/or allowance) to use a radiating (in UHF) or a magnetic coupling (in LF and HF) mode of communication. To reduce routing errors by tracking the devices, the choice of the RFID technology must be customized regardless the whole system use and the ergonomic of the detection method. An example of project tackled in this paper concerns the tracking of surgical instruments with RFID tags [START_REF] Talon | Traçabilité des instruments de chirurgie re-stérilisables: apport de la technologie RFID[END_REF][4] [START_REF] Al | Initial Clinical Evaluation of a Handheld Device for Detecting Retained Surgical Gauze Sponges Using RFID Technology[END_REF] [START_REF] Diet | Improving LF Reader Antenna volume of detection for RFID token tag thanks to Identical Coaxial Loops (ICLs) and in/out-of phase multiple-loops structures[END_REF]. In such project, the goal is to fix some small tags, compared to the size of the instruments and the human hand, and to perform a detection of multiple tags for reconditioning surgical kits before the sterilization process. The management of each instrument lifecycle and its routing represents potential errors of (i) logistics flux, when surgical kits are incorrect, and (ii) nosocomial infection risks, because some specific sterilization process can be mandatory in function of the instrument use by the surgeon in the operating room. This tracking is fruitful if it can provide fast and reliable detection of several instruments, at the same time, and, if possible, inside a limited volume of control. Some optical solution, needing line of sight control process, by engraving Datamatrix or barcode were proposed [START_REF] Talon | Traçabilité des instruments de chirurgie re-stérilisables: apport de la technologie RFID[END_REF][4] [START_REF] Al | Initial Clinical Evaluation of a Handheld Device for Detecting Retained Surgical Gauze Sponges Using RFID Technology[END_REF]. Nevertheless, the use of RFID technology brings the possibility to detect several tags without manipulating the instrument or involving the constraint of visibility. Additionally, the chip memory in RFID tags is an interesting possibility to re-write and store data about the use of the instrument. In this paper, the project consists in tagging surgical instruments with HF RFID technology. The choice of HF (13.56 MHz) is due to the presence of metal (avoid UHF) and in the idea to be compliant with NFC equipment (HF band rather than LF). The constraints of the project are to provide a detection of multiple RFID tags by using anti-collision protocol and to define a volume of detection in which tens of instruments can be routed. In this paper, we focus on the ability to detect, by means of small RFID tags, these instruments put down on a large surface of control (detection), i. e. the reader loop surface. The key idea is to create a magnetic coupling between the tag and an additional resonator and a second magnetic coupling between this resonator and the reader loop, because the direct coupling between the small tag and the reader loop is too weak. In Figure 1, the theoretical maximization of the mutual coupling between the tag and the reader loop is proposed by means of a geometrical similarity (see Neumann formula and [START_REF] Diet | Improving LF Reader Antenna volume of detection for RFID token tag thanks to Identical Coaxial Loops (ICLs) and in/out-of phase multiple-loops structures[END_REF][7] [START_REF] Diet | Improvement of HF RFID Tag Detection with a Distributed Diameter Reader Coil[END_REF]). This solution evolves from a sub-loop to a separate loop and towards the use of a resonator by addition of a tuning capacitor, which increases the induced current in the resonator loop. Mainly, the resulting magnetic field is the sum of the magnetic field generated by the current in the reader loop and the one generated by the current in the resonator. As the magnetic field distribution is more optimized for the tag loop in the vicinity of the sub-loop, the spatial mutual coupling ability is increased. However, the feeding current of the reader is impacted by the presence of such a resonator and implies to tune the reader loop in consequence. A key factor is that the resonator tuning load (mainly the capacitor) defines the current induced in the resonator, especially its phase. In this paper, we present firstly the design of a small coil dedicated for the instruments tag. This coil is included inside a surface of 1 cm² and adapted to the RFID IC SL2002 ICODE SLI-X, from NXP. Secondly, the coil is measured in presence of a resonator, which is a tuned printed coil with an average diameter of 6 cm, chosen for a detection distance of 4 cm (see the part below and [START_REF] Diet | Improving LF Reader Antenna volume of detection for RFID token tag thanks to Identical Coaxial Loops (ICLs) and in/out-of phase multiple-loops structures[END_REF][7] [START_REF] Diet | Improvement of HF RFID Tag Detection with a Distributed Diameter Reader Coil[END_REF]). Finally, tests of detection are performed with a wide reader loop to demonstrate the improvement due to the presence of the resonator.

II. SMALL COIL DESIGN FOR HF RFID TAGS

In order to tag surgical instruments for the ARTIC project, a coil is designed on a PCB and solder with the SL2002 ICODE SLI-X IC from NXP which is a HF RFID chip. The maximum size of the coil is limited to 1 cm² for ergonomic consideration [START_REF] Talon | Traçabilité des instruments de chirurgie re-stérilisables: apport de la technologie RFID[END_REF][6] as the tag should not disturb the manipulation of the instruments. The RFID chip includes HF harvesting and has an equivalent capacitance of 23.5 pF, which must be tuned with a coil of 5.86 μH. We designed, with CST electromagnetic calculator, and realized a two faces spiral coil on FR4 substrate with dedicated corner plots for soldering the IC without matching capacitor. The coil parameters are given in Figure 2. The coil was realized by micro-etching and its impedance was measured with an impedance analyzer at the RFID operating frequency (13.56 MHz). Comparison between simulations and measurements are in good agreement, as shown in Table 1. It can be noticed that the quality factor is in the range of 35-40 because of the numerous number of turns (20 on each of the two faces) and consequently the total length of the spiral line. Moreover, reducing the number of turns would have increased this quality factor but this implies the addition of an external capacitor which has not been considered inside the 1 cm² surface at the moment. The tag coil is magnetically coupled with the resonator at a distance in the range of 3 to 5 cm [3][6]. To design the resonator as a tuned coil on PCB, we evaluate the optimum size thanks to a simple computation under MATLAB. The tag coil is supposed to be an equivalent single turn of 1 cm diameter and we calculate the magnetic induction thanks to [START_REF] Paret | Identification Radio-Fréquences et Cartes à puce sans contact[END_REF].

y r R Rr k k K k E y r R y r R r y r R π Iμ B B k K k E y r R y r R r y r R π y Iμ B y θ r 2 2 2 2 2 2 2 2 2 1 2 2 0 2 2 2 2 2 2 1 2 2 0 4 2 0 2 (1)
The magnetic induction expressed in (1) corresponds to the schematic in Figure 3, in which the mutual inductance value is evaluated by numerically integrating (sum) the orthonormal component of the magnetic field, given by ( 1), over a delimited surface (length parameterized), at different distance. Results in Figure 3 show an optimum reader loop length (and consequently surface size) in function of the distance for a given surface size of the tag. This drives us to define the resonator loop diameter to be in the range of 6 cm for a distance between 3 and 4 cm.

III. USING A RESONATOR FOR IMPROVING THE DETECTION

A resonator is designed by means of a hexagonal spiral coil of value 1.3 μH (by measurements). This value enables us to adapt the tuning with a variable capacitor (hundred of pF) in order to be able to adapt the resonator tuning in future tests if the environment is disturbed by metal, ferrite, or by another resonator. The resonator impedance is measured in Figure 4 and its quality factor is in the range of 30, which is high enough for powering the tag by magnetic coupling and low enough for the mandatory bandwidth of such RFID systems (several kHz) [START_REF] Paret | Identification Radio-Fréquences et Cartes à puce sans contact[END_REF][2]. To model the resonator influence, we define the equivalent system in Figure 5, in which the magnetic coupling between the tag coil and the reader loop is considered to be insufficient and negligible (Mant/tag = 0). The impedance seen by the RFID tag IC is function of the spiral coil designed coupled with the resonator. We expressed this impedance as Z, as given by ( 2) In ( 2), we expressed Z under the hypothesis of a resonator perfectly tuned at the operating frequency (f0 = 13.56 MHz) and with a sufficiently high quality factor value that enables to use the equivalent series resistance rs while keeping the same inductance Lres in serie or parallel. Theoretically, the impedance seen by the IC is supposed to stay tuned at the operating frequency and its resistive part is increasing with the (square of the) mutual inductance value. This is emphasized when the resonator has a high quality factor (i.e. a low rs value). In practice, there is a modification of the reactive part of Z if the resonator is mistuned and/or if the resonator is disturbed by its environment, including the presence of another resonator [START_REF] Kiani | Design and Optimization of a 3-Coil Inductive Link for Efficient Wireless Power Transmission[END_REF] [START_REF] Benamara | Equivalent Input and Output Impedances in HF RFID System Including Resonator[END_REF]. We measure the variation of Z with a VNA, in order to be able to move the spiral coil, as reported in Figure 6. We chose two positions for the spiral coil to illustrate the horizontal mode (position 2) and the vertical mode (position 1) [START_REF] Diet | Improving LF Reader Antenna volume of detection for RFID token tag thanks to Identical Coaxial Loops (ICLs) and in/out-of phase multiple-loops structures[END_REF][8] [START_REF] Benamara | Equivalent Input and Output Impedances in HF RFID System Including Resonator[END_REF], at a distance of 2 cm, as seen in Figure 6. It can be noticed that the inductance value measured, in Figure 6 (right), are slightly higher with the VNA than with the impedance-meter because the measurements are influenced by the soldered connector, cable, and by the fact that loads measured on VNA ports have a high reflection coefficient magnitude. Moreover, the measurements show the increase of the real part of Z, emphasized for the position 1 in which the coupling is higher, as predicted by [START_REF] Finkenzeller | RFID handbook, fundamentals and applications in contactless smart cards and IDs[END_REF]. The variation of the reactive part of Z (inductance) is noticeable, as mentioned above to be due to the realization.

r M ω L jω 1 ω L 1 L L 1 L jω r M ω L jω L jω //C R M ω L jω I V ω Z
The spiral coil is impacted by the presence of the resonator but the variation of its equivalent inductance value is low enough to test the detection of a tag. This tag is composed by the RFID IC soldered on the designed PCB of the spiral coil.

IV. TESTS OF DETECTION

The RFID detection test is performed thanks to a RFID reader from Ib technology (RFID-UNI-EVAL with 13.56 MHz ICODE SLI-X chip module) that can be connected toward an external tuned reader loop. We choose a loop with dimension 32x16 cm², as represented in Figure 7, and arbitrarily positioned the resonator center at coordinates (X=26 cm,Y=5 cm), as shown in the Figure 7. The reader loop is realized with thin adhesive copper tapes. As it can be seen in the figure, the different spiral coils used for the measurements and for the tag are considered small compared to the large reader loop. This is confirmed by the fact that it is impossible to detect the tag without the resonator, except at the edges of the reader loop where the maximum detection range are, respectively, 1 cm and 0.5 cm for the horizontal and vertical modes. These modes correspond to the orientation of the reader loop surface in relation to the tag coil surface.

The detection tests (and range), with the resonator, are reported in Figure 8 for the horizontal and vertical modes. Maximum distances of tag identification show that the resonator highly improves the detection performances in horizontal mode (above the resonator itself) towards 3 cm. In vertical mode, the tag is rotated in the plane and the detection distance can reach 1.5 cm above the edges of the resonator. This is due to the magnetic field distribution (field lines). However, it is important to notice that the detection range is also improved in the space in-between the loop and the resonator (for example into the delimited area from X = 28 to 30 cm and from Y = 0 to 7 cm). This is due to the coupling of the resonator with the reader loop which modifies nonuniformly the magnetic field distribution. Moreover, a too wide resonator would drives to a too strong coupling with the reader coil which can affect the tuning of the reader loop and penalize the detection performances.

V. CONCLUSION

In this paper, we presented the possibility to simulate and design a small tag coil, inside a surface of 1cm², for tracking multiple devices with RFID. The detection ability is possible over a wide surface if using additional tuned resonators.

This key idea is demonstrated in the case of using a single resonator, and based on the assumptions that both (i) the tag coil is weakly coupled with the resonator coil, and (ii) the resonator coil is also weakly coupled with the reader loop. If one of these coupling factors is too strong, additional works are necessary to overcome the tuning frequency shifting. This is the perspectives of this work because the context drives us to use several resonators over the wide reader loop surface, consequently generating multiple and additional magnetic coupling factors.

Figure 1 :

 1 Figure 1 : principle of increasing magnetic coupling by geometrical similarity, and evolution of the solution using a resonator. Physical principle is to feed (left) or to induce (middle or right, thanks to resonance) a current in the small (sub-)loop whose size is in the range of the tag loop, and optimized for a targeted distance of detection

Figure 2 :

 2 Figure 2 : designed coil (left) and meshing under CST (right)

Figure 3 :

 3 Figure 3 : MATLAB simulation for a single turn coil of 1 cm diameter, in function of the reader loop size at different distance

Figure 4 :

 4 Figure 4 : Impedance measurements of the resonator tuned at 13.56 MHz (magnitude in dB at left/blue, phase in radians at right/green)

Figure 5 :

 5 Figure 5 : Equivalent model of the tag coil coupled with the resonator coil, and the resonator coil coupled with the reader loop antenna

Figure 6 :

 6 Figure 6 : Measurements of Z (real part /imaginary part) for two positions of the small spiral tag coil (schematic on top).

Figure 7 :

 7 Figure 7 : RFID reader, loop, and coordinates axis for the detection tests

Figure 8 :

 8 Figure 8 : Detection distance corresponding to Figure 7 configuration for Hm (top) and Vm (bottom) orientation of the RFID tag. Vm corresponds to any position orthonormal with XY plane.

Table 1 : simulation and measurements of the small coil at 13.56 MHz L (μH) Rs (Ω) / Q 5.93 12.7 / 39.78 5.84 14.3 / 34.8 CST measurements

 1

ACKNOWLEDGMENT


J.-E Franzkowiak

G Prudhomme

P Mercier

S Lauriot

E Dubreuil

L Berthe

PDV-based estimation of ejecta particles' mass-velocity function from shock-loaded tin experiment

A metallic tin plate with a given surface finish of wavelength λ 6 0 µm and amplitude h 8 µm is explosively driven by an electro-detonator with a shock-induced breakout pressure P SB = 28 GPa (unsupported). The resulting dynamic fragmentation process, the so-called "micro-jetting," is the creation of high-speed jets of matter moving faster than the bulk metallic surface. Hydrodynamic instabilities result in the fragmentation of these jets into micron-sized metallic particles constituting a self-expanding cloud of droplets, whose areal mass, velocity, and particle size distributions are unknown. Lithium-niobate-piezoelectric sensor measured areal mass and Photonic Doppler Velocimetry (PDV) was used to get a time-velocity spectrogram of the cloud. In this article, we present both experimental mass and velocity results and we relate the integrated areal mass of the cloud to the PDV power spectral density with the assumption of a power law particle size distribution. Two models of PDV spectrograms are described. The first one accounts for the speckle statistics of the spectrum and the second one describes an average spectrum for which speckle fluctuations are removed. Finally, the second model is used for a maximum likelihood estimation of the cloud's parameters from PDV data. The estimated integrated areal mass from PDV data is found to agree well with piezoelectric results. We highlight the relevance of analyzing PDV data and correlating different diagnostics to retrieve the physical properties of ejecta particles.

I. INTRODUCTION

The fragmentation of a roughened metallic plate under shock-loading has been extensively investigated in the last decade, theoretically [START_REF] Zellner | Effects of shock-breakout pressure on ejection of micron-scale material from shocked tin surfaces[END_REF][START_REF] Zellner | Probing the underlying physics of ejecta production from shocked Sn samples[END_REF][START_REF] Buttler | Unstable Richtmyer-Meshkov growth of solid and liquid metals in vacuum[END_REF][START_REF] Durand | Large-scale molecular dynamics study of jet breakup and ejecta production from shock-loaded copper with a hybrid method[END_REF][START_REF] Durand | Power law and exponential ejecta size distributions from the dynamic fragmentation of shock-loaded Cu and Sn metals under melt conditions[END_REF][START_REF] An-Min | Molecular dynamics simulations of jet breakup and ejecta production from a grooved Cu surface under shock loading[END_REF] and experimentally. [START_REF] Mercier | Photonic Doppler velocimetry in shock physics experiments[END_REF][START_REF] Mercier | Heterodyne velocimetry and detonics experiments[END_REF][START_REF] Bizeuil | Experimental study of shock waves driven matter ejection on the free surface of a metal[END_REF][START_REF] Rigg | Investigation of ejecta production in tin using plate impact experiments[END_REF][START_REF] De Rességuier | Dynamic fragmentation of laser shock-melted tin: Experiment and modelling[END_REF][START_REF] De Rességuier | Microjetting from grooved surfaces in metallic samples subjected to laser driven shocks[END_REF][START_REF] Monfared | Experimental observations on the links between surface perturbation parameters and shock-induced mass ejection[END_REF] Many physical mechanisms have to be taken into account: spalling, microspalling, melting, and micro-jetting being the main ones. The increasing interest to characterize these surface destruction products has been leading to many developments, such as X-ray shadowgraphy, piezoelectric pins, Asay foils, [START_REF] Asay | Ejection of material from shocked surfaces[END_REF] Miesizing diagnostic, Photonic Doppler Velocimetry (PDV), [START_REF] Strand | Velocimetry using heterodyne techniques[END_REF] and holography. The purpose is to determine the areal mass of ejecta, the particle sizes, and their velocities. In this article, we focus on the fastest ejecta particles obtained from shockwave loading at a metal vacuum tin interface with a particular surface finish (P SB = 28 GPa). We describe a PDV simulation model to analyze time-velocity spectrograms of ejecta clouds. Finally, we show that experimental PDV results can be related to simultaneous piezoelectric measurements through numerical investigations of PDV spectra and mass-velocity distributions, in line with experimental results.

A standard PDV setup, [START_REF] Strand | Velocimetry using heterodyne techniques[END_REF] presented in Fig. 1, has been implemented in our experiment. It consists in mixing two different frequencies to get a beat frequency ∆f related to the velocity V of the object. Moving particles are illuminated with a single mode continuous-wave fiber laser at λ = 1.55 µm using a fiber collimator (PDV probe). Doppler-shifted wavelets are collected in the same collimator and mixed with a reference a) Email: jean-eloi.franzkowiak@cea.fr beam (λ = 1.55 µm, i.e., no frequency up-shifting). For a single moving object, the interference results in a time-beating intensity I(t) on the photodetector whose beat frequency ∆f can be related to V using the formula V = λ 2 ∆f , I(t) = α s I s + α r I r + 2 α r α s I s I r cos(2π∆ft + φ).

(1)

I s and I r are, respectively, the beam intensities of the backscattered signal and of the reference signal. α r and α s are the coupling parameters of the PDV setup and φ is the phase difference between reference and backscattered waves.

For a large number of particles with different velocities, the backscattered electric field E s (t) is the sum of each contribution with its given amplitude E j , frequency ω j , and phase φ j ,

E s (t) = j E j exp i(ω j t + φ j ) . ( 2
)
Assuming a random particle arrangement (i.e., uncorrelated positions and phases φ j uniformly distributed between 0 and 2π), the resulting intensity I s (t) ∝ |E s (t)| 2 has a speckle structure [START_REF] Moro | Defining parametric dependencies for the correct interpretation of speckle dynamics in photon Doppler velocimetry[END_REF] in both time and frequency (the amplitude and the phase are treated as random variables). The Power Spectral Density (PSD), or the so-called PDV spectrogram, is the squared modulus of the Short-Term Fourier Transform (STFT) of I(t) given by Eq. (1). Recently, Andriyash et al. [START_REF] Andriyash | Optoheterodyne Doppler measurements of the ballistic expansion of the products of the shock wave-induced surface destruction: Experiment and theory[END_REF] In this article, we study the properties of ejecta clouds with typical particle sizes between 1 and 10 µm and velocity gradients around 1500 m/s. We develop two complementary models of PDV spectra for sparse ejecta cloud using an assumption on the ejecta size distribution (based on holographic measurements [START_REF] Sorenson | Ejecta particle size distributions for shock loaded Sn and Al metals[END_REF] ). We show that the integrated areal mass-velocity distribution estimated from PDV data is in line with piezoelectric experimental results, making possible to solve the inverse problem of determining the areal mass of ejecta with a non-invasive optical diagnostic.

II. SIMULATION OF PDV SPECTRA

In Secs. II A and II B, two models of PDV spectra for high-speed ejecta particles are presented.

A. A general simulation of PDV spectrograms

The physical properties of ejecta cloud and their influence on PDV spectra will be detailed. If the particles are dragging in a gas, the PDV spectrum is time-dependent. [START_REF] Prudhomme | PDV experiments on shockloaded particles[END_REF][START_REF] Prudhomme | Frontal and tilted PDV probes for measuring velocity history of laser-shock induced calibrated particles[END_REF][START_REF] Prudhomme | Étude des nuages de particules éjectées sous choc: Apports de la vélocimétrie hétérodyne[END_REF] Metallic particles ejected in a vacuum will be considered in this article, leading to a statistical time-invariant spectrum if the PDV probe's efficiency does not depend on the viewing distance. In its most general description, the digitized signal U s (t) (in volts) supplied by a collection of N spherical particles depends on many parameters: the incoming gaussian irradiance E inc (r, z) (W/m 2 ), the probe's efficiency η(r, z), the system's response R sys (V/W), the coupling parameters and the reference intensity [respectively, α r , α s , I r (W)], the volume fractions inside the cloud (volume of matter divided by occupied volume), directly related to the size-velocity distribution, and finally the phase φ j and the velocity V j (m/s) of each particle. The noiseless expression of the raw PDV signal, corresponding to the "AC" part of the signal (digitizers are usually AC coupled so DC terms can be safely ignored), is, for a system of N particles,

U s (t) = 2R sys α s α r I r × Re        N j=1 η(r j , z j (t))E inc (r j , z j (t)) × exp -2 z j (t) z min (t) 1 l ext (t, z(t)) dz . exp 2ik 0 V j t + φ j , (3)
where k 0 (1/m) is the wave vector at λ = 1.55 µm and z min (t) corresponds to the head of the cloud at time t (position of the fastest particle). l ext (t, z(t)) is the extinction length of a slab of particles between z(t) and z(t) + δz(t), depending on the particle sizes in the slab and on the volume fraction in particles. Re is the real part. The efficiency η(r, z) depends on both particle's position and particle's characteristics [size, shape, complex refractive index, and backscattering efficiency Q bs (d p ) defined by the radar hypothesis in Ref . 22]. In the following, only spherical particles will be considered.

In Eq. ( 3), E inc (r j , z j (t)) is the gaussian distribution of the irradiance received by each particle at position (r j , z j (t)) (see Fig. 2). The negative exponential accounts for the attenuation by scattering (in the forward and backward directions). Some assumptions must be made to write Eq. ( 3):

• A first order solution of light multiple scattering in the ejecta cloud. A scatter-induced attenuation of the beam is taken into account and multiply scattered photons are neglected, which is valid for dilute systems. • Velocities and particle sizes are uncorrelated. This may only be true [START_REF] Durand | Mass-velocity and size-velocity distributions of ejecta cloud from shock-loaded tin surface using atomistic simulations[END_REF] for the high-velocity region of the spectrum (region visible to PDV). For the estimation of the optical extinction in each slab, a mean particle diameter d p and a mean extinction efficiency Q ext will be inferred from the size distribution. Based upon previous results obtained by Sorenson et al., [START_REF] Sorenson | Ejecta particle size distributions for shock loaded Sn and Al metals[END_REF] a power law dependence with a specific exponent γ d is assumed,

f d (d p ) ∝ d -γ d p . (4)
We define an effective volume V p and the corresponding diameter d p , determined from f d ,

V p = d max d min f d (d p ) π 6 d 3 p dd p • d max d min f d (d p )dd p -1 , (5)
d p =       (1 -γ d )(d 4-γ d max -d 4-γ d min ) (4 -γ d )(d 1-γ d max -d 1-γ d min )       -1 3 . ( 6
)
d min and d max (µm) are the minimum and maximum particle diameters. An effective particle extinction efficiency Q ext FIG. 2. Coordinates (r p , z p ) of a particle in 3D-space with respect to the PDV probe.

is defined similarly,

Q ext = d max d min Q ext (d p )f d (d p )dd p • d max d min f d (d p )dd p -1
. (7) In Eqs. ( 5) and (7), f d is the particle size distribution and Q ext (d p ) is the extinction efficiency of a spherical particle calculated using the Mie theory of scattering. The cloud being inhomogeneous along z due to the velocity gradient, we decide to split it into N s slabs of identical volume V s (t). In each particle slab, the volume fraction f vol (z(t), t) is estimated at each time t. We make the assumption that in each slab, the number of particles is large (N(z, z + δz) 1) and the volume of matter in a slab is approximated by N(z, z + δz)V p . As the cloud expands in a vacuum, the mesh evolves similarly such that each slab of particles keeps the same amount of matter,

f vol (z(t), t) = N(z(t), z(t) + δz(t))V p V s (t) . ( 8
)
The extinction length in Eq. ( 3), using the first order solution of light scattering, is written as

l ext (t, z(t)) = 2d p 3f vol (z(t), t)Q ext . ( 9
)
To get an estimation of the collection efficiency η(r, z), i.e., the optical power coupled in the system, we detail the calculation for a fixed single particle of coordinates (r p , z p ). Using the radar hypothesis, for a particle in the far field of the entrance pupil of the probe (lens), the scattered electric field E s impinging on the lens is given by the following relationship:

E s (r, z = 0) = d p 4r
E inc (r p , z p )Q bs (d p ) exp (-ik 0 r) (10) with r = (xx p ) 2 + (yy p ) 2 + z 2 p . The optical power coupling efficiency η(r p , z p ) is the scalar product of the field scattered by the particle in the entrance pupil with the back-propagated gaussian mode of the fiber G 0 (r) in the same plane,

η(r p , z p ) = pupil E * s (r, z = 0)G 0 (r)dr 2 × 1 E inc (r p , z p ) . ( 11
)
For each distance z between the probe and the particle, a mean coupling efficiency η(z) is obtained [η(r p , z p ) is averaged over a disk of radius Φ P /2 located at a distance z p from the probe with Φ P the width of the beam],

η(r, z)E inc (r, z) = η(z)E u (z), (12)
where E u (z) (W/m 2 ) is the equivalent uniform irradiance.

Given a discrete number of particles whose positions and velocities are known at t = 0 in the mesh, we can construct the raw PDV signal at each time step and obtain the PDV spectrum Φ(V, t) by computing a conventional STFT.

An example of such a PDV spectrogram Φ(V, t) is shown in Fig. 3. We observe the typical time-varying speckle fluctuations coming from the interferences between the Dopplershifted wavelets and the reference beam. For neighboring particles having close velocities, their velocity traces are not discriminated from the Fourier analysis and a single particle velocity cannot be extracted in the spectrogram. On the left, we can observe the evolution of the coherent irradiance inside the cloud versus the penetration distance z.

B. Analytical model of PDV spectrograms

We add new assumptions to get an analytical description of a PDV spectrum for an N-particle system expanding in a vacuum. In the case of linear Richtmyer-Meshkov instabilities (for kh < 1) and based on recent molecular dynamics results, [START_REF] Durand | Power law and exponential ejecta size distributions from the dynamic fragmentation of shock-loaded Cu and Sn metals under melt conditions[END_REF][START_REF] Durand | Mass-velocity and size-velocity distributions of ejecta cloud from shock-loaded tin surface using atomistic simulations[END_REF][START_REF] Dimonte | Simulations and model of the nonlinear Richtmyer-Meshkov instability[END_REF][START_REF] Dimonte | Ejecta source model based on the nonlinear Richtmyer-Meshkov instability[END_REF] the integrated mass-velocity distribution can be very well approximated by an exponential function. Consequently, an exponential distribution of velocities will be assumed, where J 0 is a multiplicative factor. The integration of f v between V min and V max is equal to the total number of particle N tot ,

f v (V ) = J 0 e -γ v (V -V min ) V max , (13)
N tot = J 0 V max γ v 1 -e -γv∆V V max . ( 14
)
∆V = V max V min . N tot is related to the total areal mass of the cloud M s (kg/m 2 ) through the relationship

4 N tot V p ρ πΦ 2 E = M s . ( 15
)
ρ is the density of tin (kg m 3 ) and πΦ 2 E /4 is the surface of ejection (m 2 ). Finally, f v is written as

f v (V ) = M s γ v πΦ 2 E 4 V p ρV max • 1 1 -exp -γ v ∆V V max • e -γ v (V -V min ) V max . ( 16
)
We add two points in order to precise the model:

• The first point deals with the self-expansion of the cloud that links the coordinates z of the particles to their velocities (no dragging), i.e.,

N(z)δz = N(V )δV . (17)
• The second point is focused on replacing the backscattering efficiency of each particle with an average efficiency Q bs ,

Q bs = d max d min Q bs (d p )f d (d p )dd p • d max d min f d (d p )dd p -1 . ( 18
)
As the cloud rapidly self-expands in a vacuum, the first point is meaningful. The second point is only valid if the statistical properties of the cloud are invariant by translation in the x and y directions and if the total number of particles in each slab is large (N 1). The average PSD, P(V ) = Φ(V , t) t ∞ (W), between velocities V and V + δV, is proportional to the average intensity scattered in the velocity range [V, V + δV ]. Position being related to velocity, the spatial integral in the exponential argument [Eq. ( 3)] can be replaced with an integral over velocities. For a collimated beam,

E u (z) = E u . Introducing the velocity distribution f v , the average PSD Φ(V , t) t ∞ is Φ(V , t) t ∞ = P(V ) 2 = +∞ -∞ f v (V ) • η • E u × exp       - 3Q ext V p d p πφ 2 P V max V f v (V )dV       dV 2 . ( 19
)
To take into account the Fourier window F w of finite width T w , the upper definition of Φ(V , t) t ∞ must be modified and a convolution between F w and √ P(V ) must be computed for each velocity V,

Φ(V , t) t ∞ = P(V ) ⊗ F w (V ) 2 . ( 20
)
For a rectangular window of width T w and an exponential velocity distribution f v (V ) [Eq. ( 13)], we finally get for each V

Φ(V , t) t ∞ = 2T w λ +∞ -∞ • βM s • e -γv( Ṽ -V min ) 2•V max • exp       - 3Q ext V p d p πφ 2 P • β • M s • V max γ v       e -γ v Ṽ -V min V max -e -γ v ∆V V max             • sinc 2T w λ (V -Ṽ ) d Ṽ 2 . ( 21
)
. t ∞ is the time average. If we average an infinite number of independent speckle realizations, the fluctuations are removed and the PSD is Φ(V , t) t ∞ , the parameter of the speckle statistics. When the sliding time step of the Fourier window is equal to its width, adjacent points in the PDV spectrogram are independent and time-averaging is equivalent to averaging over different particle cloud realizations (ensemble average). (W) is a multiplicative constant accounting for the collection efficiency of the probe ( = η • E u ) and ∆V = V max V min . The transverse size of the cloud φ E is equal to the width of the beam φ P . M s is the total areal mass of the cloud (kg/m 2 ), sin c(x) = sin(πx)/πx, and β (m s/kg) is given by the following relationship:

β = γ v πΦ 2 P 4 V p ρV max • 1 1 -exp -γ v ∆V V max . (22)
Φ(V , t) t ∞ can be summed to recover the total optical intensity (W). Equation (21) gives an estimate of the average PSD for a self-expanding particle cloud in a vacuum, whose size and velocity distributions have, respectively, a power law and an exponential dependence. Since the constant does not depend on the properties of the cloud, it will be held constant. This will not have any consequence for a relative comparison between different PDV spectra.

In Fig. 4, we can observe the time-averaged PSD (spectrogram of Fig. 3) and the PSD obtained from Eq. ( 21). The red curve would be obtained if an infinite number of independent realizations (see Fig. 3) were averaged.

With these two models, we get access to a fluctuating PDV spectrogram (model 1) and to what is known as an "average PDV spectrogram" for which the characteristic speckle fluctuations are removed (model 2). Before drawing a direct comparison with experimental results, the way in which the parametric dependencies of the ejecta cloud modify the shape of the spectrum has to be considered [Eq. ( 21)].

C. PDV spectrum and parametric dependencies

The influence of ejecta parametric dependencies on the PDV spectrum will be briefly discussed, based on the calculation of the average spectrogram Φ(V , t) t ∞ from Eq. ( 21) (model 2). We recall that this analysis is only valid for a self-expanding cloud in a vacuum. The integrated areal massvelocity M(V ) is the cumulative areal mass (mg cm 2 ) between V and V max with M(V min ) = M s . FIG. 4. Comparison between Φ(V, t) t averaged over 500 ns, obtained from model 1, and the PDV spectrum Φ(V , t) t∞ obtained from model 2 [Eq. ( 21)] for a polydisperse expanding cloud in a vacuum. The cloud's properties are described in the caption of Fig. 3. The red curves correspond to Eq. ( 21) and the blue ones correspond to Eq. ( 3). On the right are plotted the spatial evolutions of the volume fractions inside the cloud for the 2 models (total length of the cloud L cloud = 725 µm) at time t = 500 ns. z = 0 is the head of the cloud (fastest particles).

Influence of the areal mass M s

The properties of the cloud remain unchanged (size and velocity distributions), except its areal mass M s (mg/cm 2 ).

The impact of increasing the areal mass on the mean PDV spectrum is shown in Fig. 5. The maxima of the spectrum are shifted to higher velocities. It arises from the fact that particle volume fractions are increasingly densified at high velocities. Two phenomena must be taken into account: the first one is the increased attenuation by scattering due to increased particle densities and the second one is the increase of the mean collected intensity (more particles). The penetration of the beam is large for a small areal mass (see green curves in Fig. 5) and reduced for a larger one (red curves): the PSD drastically falls below V = 2500 m/s. Φ(V , t) t ∞ remains large at high velocities, where the beam has not yet been impacted by attenuation.

Influence of the velocity distribution

All properties are still unchanged except the characteristic exponent γ v of the velocity distribution. We recall the form of the velocity distribution f v ,

f v (V ) ∝ e -γ v (V -V min ) V max .
A change in velocity distribution drastically impacts both slopes and amplitude of the spectrum Φ(V , t) t ∞ . For large values of γ v (see green curves in Fig. 6), the velocity distribution is highly peaked at low velocities and the few high-speed particles (near 3650 m/s) are not detected. The location of the maximum always depends on a compromise between extinction and scattering of light. For small values of γ v , particle densities are increased in the high-velocity region and the trade-off between attenuation and scattering appears for higher velocities (around 2750 m/s for γ v = 18).

Influence of the size distribution

We study the influence of the exponent γ d on the PDV spectrum,

f d (d p ) = d -γ d p .
In Fig. 7, we can see the impact of increasing γ d . For γ d = 2, the average particle diameter is larger than for a highly peaked size distribution obtained with γ d = 5.5. As the areal mass density is kept constant (3 mg/cm 2 ), the number of particles contributing to the amplitude of the spectrum grows with increasing γ d . Regarding the smallest particles (γ d = 6), we can also notice that the optical attenuation of light is more severe and the position of the maximum is shifted to high velocities.

This parametric study is of great interest to understand the influence of the microphysical properties of the cloud on the PDV spectrum. We are aware that a real PDV spectrum is inherently noisy due to the speckle statistics (Fig. 3) and average PDV spectra shown in Figs. 5-7 will never be obtained in any ejecta experiment. However, in the case of a self-expanding ejecta in a vacuum, a relevant time-averaging can be done to mitigate speckle fluctuations and an average model of the PDV spectrum can be used to find the best fit and try to resolve the inverse problem of determining ejecta's properties with non-invasive PDV diagnostic.

III. EXPERIMENT AND COMPARISONS

The experimental setup is described below. An electrodetonator is used to generate a shockwave in a 1 mm-thick tin plate. In the first experiment, the free surface of the plate is flat (e.g., diamond turned). For the second one, the surface was finished to get 2D periodic triangular patterns of width 60 µm and height 8 µm. The distance between the probes and the plate is set to 7 mm. The lateral size of the plate is 4 mm. For the second experiment, the shockwave interaction with the perturbed surface creates metal sheets which fragment into high-speed micron-sized metallic particles in a vacuum.

The unsupported peak pressure at the free surface is approximately 28 GPa. The signals were recorded with a sampling frequency of 50 GS/s (20 ps time step) during several µs. We do not focus on the ejection mechanism which has been extensively studied, and this way experimental conditions will be kept constant (shockwave pressure, plate thickness, and surface finish). PDV probes are gradient index (GRIN) lenses with an output beam diameter of 100 µm (purchased from IDIL Fibres Optiques, Inc., reference COCOM02472). Despite the fact that the coupling efficiency of this probe has not been precisely defined, our analysis will not be limited since we are interested in relative comparisons between PDV spectrograms. The optical power delivered by the PDV probe is set to 300 mW at λ = 1.55 µm. The use of piezoelectric probes for areal mass measurements has been demonstrated [START_REF] Mears | Impact stress measurement using piezoelectric probes with PZT and lithium niobate elements[END_REF][START_REF] Vogan | Piezoelectric characterization of ejecta from shocked tin surfaces[END_REF][START_REF] Buttler | Dynamic comparisons of piezoelectric ejecta diagnostics[END_REF] and we used a standard lithium-niobate (LN) sensor (purchased from Dynasen, Inc.).

A. PDV spectrogram results

To get an estimate of the free-surface velocity, a first experiment was performed on a flat tin surface of width 1 mm (the setup is described in Fig. 8). Two frontal PDV probes are located 1.75 mm off center at 7 mm from the surface. The shock is not perfectly plane at 1.75 mm and a correction must be made since PDV measures the line-of-sight velocity vector V z . From a hydrodynamic simulation using CEA Hesione code, an angle of θ 6 • is found between the freesurface velocity vector V fs and the z-axis such that V fs V z / cos θ = 2013 m/s (see Fig. 9). This free-surface velocity will be our reference for the following analysis on piezoelectric and PDV data.

In a second experiment, simultaneous frontal PDV and LN piezoelectric probes have been implemented (Fig. 8). The average background noise B Φ exp is estimated in a region of the spectrum Φ exp (Fig. 10) without signal (V ≥ 5000 m/s). Since the probe's collection efficiency is not constant with z (diverging beam), both the fastest and the slowest particles appear at later time, the high optical energy density near the probe increasing the penetration of the beam in the cloud. Around FIG. 9. First experiment: Velocity of an explosively driven 1 mm-thick flat tin surface measured by two frontal PDV probes located at 1.75 mm off center. The mean velocity is V fs 2013 m/s (with an angle correction of 6 • ). FIG. 10. Second experiment: Experimental PDV spectrogram Φ exp (V, t) for a shockwave loaded 1 mm-thick tin plate experiment (60 × 8 µm grooves) at P SB 28 GPa, the configuration being described in Fig. 8. Fourier window: Rectangular, T w = 50 ns, δT w = 100%, no zero-padding. The black dashed box defines the limits of our analysis: time ranges from t = 1.95 µs to t = 2.9 µs and velocities from V min = 2281 m/s to V max = 3676 m/s. B Φexp is the average background noise of the PDV spectrogram. t 3.8 µs, the fastest particles hit the probe, corresponding to a time of flight of approximately 2 µs for a particle moving at 3500 m/s. Within the first microsecond, we assume that the statistics of the cloud is unchanged (the collection efficiency of the probe is assumed to be constant) and the data Φ exp (V, t) are averaged on a finite interval, between t = 1.95 µs and t = 2.9 µs, to get Φ exp (V ) t .

B. Piezoelectric pin results

The piezoelectric pin used in this experiment was a Lithium-Niobate (LN) y + 36 • -cut crystal of diameter 1.27 mm. The compression of the crystal, induced by accumulation of particles and in the absence of any applied electric field, leads to the creation of a charge density D i given by

D i = d ij σ j , ( 23
)
where d ij is the piezoelectric sensitivity and σ j is the applied stress. It will be assumed that the stress is applied in the zdirection (i = j for uniaxial compression), that all particles are ejected instantaneously at shock breakout (t 0 ), and that particle collisions are inelastic. The sensitivity of the LN-pin is assumed to be d ii = 24 pC/N and the fail pressure is around 0.6 GPa. The time-dependent voltage response U(t) across an impedance R (50 Ω) shown in Fig. 11 is linked to the timedependent applied stress σ(t) (GPa),

σ(t) 1 d ii A t t 0 U(t) R dt. ( 24
)
The integrated areal mass M a (t) is calculated as follows: where V p (t) (m/s) are particle velocities. The expression of M a (t) as a function of velocity V is called M(V ). At V = V fs , M(V = V fs ) = M s , the total areal mass of ejecta. In Fig. 12, both pressure on the pin and integrated areal mass versus velocity are depicted. With V fs 2013 m/s (first experiment), the total areal mass is M(V fs ) = M s 7.3 mg/cm 2 .

M a (t) = t t 0 σ(t) V p (t) dt , (25)

C. Maximum likelihood estimation

To make comparison between experiment and simulations, we make the assumption of a power-law size distribution with exponent 18,29 γ d = 5.6. Model 2 described by Eq. ( 21) is parametric and will be used for simulating average PDV spectra Φ(V , t) t ∞ with the given cloud's characteristics (γ d , γ v , etc.). To the extent that a simple visual comparison is not relevant, we perform a Maximum Likelihood (ML) estimation of the parameters with our model. The purpose is to estimate the mass-velocity function M(V ) from PDV data. The model has been reduced to four parameters, two of whom with a real interest: the integrated areal mass density M(V min ) and the coefficient from the velocity distribution γ v . The third parameter is a multiplicative coefficient κ and the fourth parameter is the background noise amplitude B Φ . For a given point of the PDV spectrogram (V j , t i ) in the absence of signal, the probability distribution of background noise B Φ exp (V j , t i ) (experimental results) is well approximated by a negative exponential,

p(B Φ exp (V j , t i )) = 1 B Φ exp exp - B Φ exp (V j , t i ) B Φ exp . ( 26
)
If the background noise were absent ( B Φ exp = 0), the distribution of independent values Φ exp (V, t i ) (experimental PDV spectrogram) at different times t i (i = 1, . . ., N) (N being the number of independent realizations from the PDV spectrogram) would follow a speckle statistics. The probability density function (PDF) would be

p(Φ exp (V , t i )) = 1 Φ(V , t) t ∞ exp - Φ exp (V , t i ) Φ(V , t) t ∞ . ( 27
)
To ensure the independence between Φ(V, t i ) and Φ(V, t j ) at constant velocity, the Fourier analysis on the raw PDV signal is made with a rectangular window (width T = 50 ns, sliding time step δT = T , and no zero-padding). However, as background noise is present, we must correct Eq. ( 27) to take into account both the signal Φ exp (V, t i ) and average noise B Φ exp . When background noise is present, the PDF p(Φ exp (V, t i )) is the PDF of the sum of three random variables:

• X = Φ exp (V, t i ) whose PDF is given by Eq. ( 27),

• Y = B Φ exp (V j , t i ) whose PDF is given by Eq. ( 26),

• and Z = 2 • √ X • √ Y cos(Θ)
, where Θ follows an uniform distribution between 0 and 2π,

√

X and √ Y being described by two independent Rayleigh distributions. We get,

p(Φ exp (V , t i )) = 1 ( Φ(V , t) t ∞ + B Φ exp ) × exp - Φ exp (V , t i ) ( Φ(V , t) t ∞ + B Φ exp ) . ( 28
)
For the experimental time-averaged PDV spectrum Φ exp (V , t) t , the PDF becomes Gamma distributed [START_REF] Goodman | Speckle Phenomena in Optics: Theory and Applications[END_REF] (PDF of the mean of N independent speckle realizations) and Eq. ( 28) FIG. 12. Cumulative areal mass M(V ) (mg/cm 2 ) and applied stress σ (GPa) on the piezoelectric probe versus reduced velocity V /V fs with V fs = 2013 m/s, determined from the first experiment. becomes

p( Φ exp (V , t) t ) = N N Φ exp (V , t) N-1 t Γ(N)( Φ(V , t) t ∞ + B Φ exp ) N × exp -N Φ exp (V , t) t ( Φ(V , t) t ∞ + B Φ exp ) . ( 29
)
The first three parameters of

ζ = [M(V min ), γ v , κ, B Φ ] are hidden in Φ(V , t ∞ = Φ(V , t, M(V min ), γ v , κ) t ∞ .
Γ is the Gamma function. M(V min ) will be denoted as M Vmin . The goal is to find the maximum of the likelihood function defined by 31) is verified (joint confidence region at 95%). M(V min ) has units of mg/cm 2 . Scatter plot: ML estimation. Bottom right: Comparison between experimental results and ML estimation of the mass-velocity distribution M(V ) (mg/cm 2 ) for V /V fs between 1.11 and 1.8. The error bar at ±2σ and V min /V fs = 1.13 is calculated using the likelihood ratio method (boundaries of L).

L : (κ, γ v , M Vmin , B Φ ) → N V k=1 N N Φ exp (V k , t) N-1 t Γ(N)( Φ(V k , t, κ, γ v , M Vmin ) t ∞ + B Φ ) N × exp - N Φ exp (V k , t) t (Φ(V k , t, κ, γ v , M Vmin ) t ∞ + B Φ ) . ( 30
)
N V =
density function of Eq. ( 29). The joint confidence region for the set of parameters [M(V ), γ v , κ, B Φ ] is a four-dimensional volume in the space of parameters and is estimated with the Likelihood Ratio (LR) method, 2 log L(ζ o ) L(ζ ) being approximated as a χ 2 4 (1-α) with α = 0.05 (confidence region of 95%),

ζ : 2 log L(ζ o ) L(ζ) ≤ χ 2 4 (1 -α) . (31)
In Fig. 14, three projections of a constant likelihood function are provided to illustrate the joint confidence region for the set of parameters ζ, such that

L(ζ) = L(ζ o ) exp -1 2 χ 2 4
(1 -α) . κ and B Φ have a small impact on the estimation. On the contrary, the main uncertainty on M V min comes from the coupling of crucial importance between M V min and γ v . The integrated mass-velocity function M(V ) (bottom right) is deduced from the two parameters M Vmin and γ v using

M(V ) = M Vmin exp -γ v (V -V min ) V max . ( 32
)
A good agreement is obtained between the ML estimation and the experimental results [M V min = 2.05 ± 0.36 mg/cm 2 for the ML estimation and M(V min 2.48 mg/cm 2 ) for experimental LN-pin data]. It appears that for V /V fs > 1.3, the LN-pin does not detect the presence of high-speed ejecta particles. In this case, an estimate from PDV data is relevant. Further comparisons between model 2 [Eq. ( 21)] and experimental PDV data, for which a time-averaging of the spectrum can be done over a larger interval, are of great interest. If the number of points used to average the spectrum is large, the probability density function of the averaged spectrum will be well approximated by a normal distribution and the ML estimate will converge with which it is obtained from a non-linear least square minimization. Furthermore, since the joint confidence intervals will be reduced, the estimation will be improved. Our FIG. 15. Comparison between the ML estimate M(V ) (mg cm 2 ) performed on experimental PDV data and experimental M(V ) measured with a piezoelectric LN-pin, for V /V fs between 1 and 1.8. The error bar at ±2σ and V /V fs = 1 is estimated from the uncertainty determined at V /V fs = 1.13 using the likelihood ratio method (boundaries of L).

estimation from PDV data being limited to V min = 2281 m/s, an extrapolation of M(V ) to smaller velocities is possible but uncertain (no information gain). In our case, kh = 0.42 (k = 2π/60 µm 1 and h = 4 µm) and an extrapolation using the exponential model of the mass-velocity seems therefore relevant. A total areal mass of M s = 11.2 ± 3.4 mg/cm 2 is obtained from our PDV-based analysis, compared to LN-pin result M s = 7.3 mg/cm 2 (see Fig. 15).

IV. DISCUSSION

Section IV discusses how the estimate is modified when various assumptions are violated and how combining different diagnostics may improve the analysis.

A. Random particle arrangement

In this article, a random particle arrangement is assumed and fully justified by the speckle nature of the PDV spectrogram, Φ(V , t) t ∞ being described by the negative exponential distribution of a random phasor. The following relationship can be used to test whether or not experimental results follow this distribution:

Φ N exp (V , t) t = Φ exp (V , t) N t N!. ( 33
)
This relation has been verified up to N = 4 for the experimental results shown in Fig. 13.

B. Particle shapes

The average extinction efficiency Q ext is determined using the Mie theory only valid for spherical particles. In practice, particles are not perfectly spherical and ellipsoidal shapes may be present in the ejecta. An extended Mie scattering theory or any electromagnetic solver can be used to study how the analysis might be affected. For randomly oriented ellipsoidal particles with respect to the PDV beam, the extinction efficiency will not be drastically modified for aspect ratios (the ratio between the length and the width of the particle) between 1 (sphere) and 2. We believe that assuming particles' sphericity is reasonable and introducing ellipsoidal shapes will unnecessarily increase the complexity of the problem for a first order theory of scattering in the ejecta cloud.

C. Particle sizes

In this article, a power law dependence with exponent γ d = 5.6 has been assumed, between d min = 1 µm and d max = 10 µm. Since most of the mass is present in the region near d min , a precise knowledge of the lower bound of the distribution is of paramount importance. Since it is likely that submicron particles are present in the ejecta, the impact of lowering d min down to 0.3 µm must be studied. We also analyze how the areal mass estimate is modified when the critical exponent is lowered down to γ d = 3.4.

The impact of lowering d min is shown in Fig. 16. Down to 0.5 µm, the estimated areal mass M s decreases almost linearly. For d min = 0.3 µm, M s increases up to 8 mg/cm 2 . In this region, extinction cross sections are so small that their contribution to the extinction of light is reduced. On the contrary, their contribution to the total mass of the cloud is not negligible. The evolution of the estimated areal mass M s versus γ d is shown in Fig. 17. The decrease is mostly attributed to an increased average particle diameter in the ejecta cloud.

The analysis will drastically be improved by a precise knowledge of the size distribution. Indeed, the inversion from PDV data is severely under-constrained and the data will never be inverted if a particle size distribution is not assumed (the consistency of the ML estimator would be lost). Further experimental investigations are needed to relax this strong assumption. A log-normal distribution may also be assumed for the size distribution: for σ, we get 21.2, and 22.2 and 28.4 mg/cm 2 . The increased estimate is justified by the reduced number of small particles (tail of the log-normal distribution) in the cloud.

D. Multiple scattering

The first order solution presented in this paper does not take into account multiply scattered photons. For the highvelocity region of the spectrum, they may contribute [START_REF] Andriyash | Optoheterodyne Doppler measurements of the ballistic expansion of the products of the shock wave-induced surface destruction: Experiment and theory[END_REF] at second order in the formation of the Doppler signal. The discrepancy between our model and experimental results observed around 2600 m/s in Fig. 13 may also be explained by the multiple scattering of light in the ejecta cloud. After the first scattering event, the photon can be preferentially scattered in the forward direction and will not acquire any Doppler shift since incoming and scattered directions will be identical. If this event takes place in the high-velocity region of the cloud above 2600 m/s, a second scattering may eventually occur in the back direction around 2600 m/s, increasing the PSD in this region with respect to the amplitude obtained using a first order theory of scattering. Further investigations are deemed necessary to validate this assumption. The general method described in this paper can be applied to other models of average PDV spectrum Φ(V , t) t ∞ for which more physics is taken into account.

E. Combining different diagnostics

Additional diagnostics are needed to improve the inversion. An advantage of PDV inversion is that the retrieved areal mass depends neither on the time of shock breakout nor on the distance between the PDV probe and the cloud (for a collimated beam). Asay foil, X-ray absorption and piezoeletric pins diagnostics can be used to measure the massvelocity distribution in the ejecta cloud. In this case, the massvelocity function can be related to the velocity distribution M(V ) ∝ ∫ V max V f v (V )dV . Different particle size distributions can accordingly be tested to find out which one is likely to explain experimental PDV results. Further experiments using Asay foil diagnostic will be very relevant and will certainly give more information on the shape of the mass-velocity distribution (the assumption of an exponential function will be relaxed). Small Asay foils 32 may be implemented nearest to a PDV probe such that the two diagnostics will look at the same region of the ejecta. As already discussed in Sec. IV C, any particle sizing diagnostic will also drastically improve the analysis.

V. CONCLUSION

In this article, we have developed two models to study the way how the properties of an ejecta cloud expanding in a vacuum influence the shape of the time-velocity PDV spectrogram. The first one accounts for the speckle fluctuations observed on PDV data and the second one describes an average spectrum which would be obtained from averaging an infinite number of realizations. This model [Eq. (21)] is used to study the influence of the areal mass, the size and the velocity distributions. A comparison between the time-averaged PDV spectrum measured on a shock-loaded tin plate experiment and our model is presented, showing promising comparisons between experiment and simulation. A maximum likelihood analysis was done to retrieve the properties of the cloud, and an integrated areal mass of M(V /V fs = 1.13) = 2.05 ± 0.36 mg/cm 2 is deduced, in line with piezoelectric pin result M(V /V fs = 1.13) 2.48 mg/cm 2 , showing that correlations between piezoelectric and PDV diagnostics are possible. The total areal mass estimated from PDV is M s = 11.2 ± 3.4 mg/cm 2 . This analysis also highlights the limited depth probed by the PDV beam in the particle cloud: below a given velocity (V min 2281 m/s), the optical intensity is too low and the signal to noise ratio falls drastically, limiting the range of the analysis. A future study will focus on both improving the depth probed by the beam and relaxing the strong hypothesis on particle sizes. Finding the role of particle cloud's properties on PDV spectra is a first step to solve the inversion of the problem, and correlations between different diagnostics will be necessary to reduce the number of solutions and improve the analysis. We believe that PDV may generate valuable data for an areal mass estimation of ejecta particles.

  studied the influence of the multiple scattering of light on PDV spectra, based on a two-flux solution of the Radiative Transfer Equation (RTE) in the ejecta cloud. They applied this model to analyze the properties of densely packed debris (∼30 µm size) with relatively slow velocity gradients (between 100 and 500 m/s).

FIG. 1 .

 1 FIG. 1. Standard PDV setup. 1: Laser (beam intensity I 0 ), 2: optical circulator, 3: optical collimator, 4: moving particles, 5: reference (beam intensity I r ), 6: optical coupler, 7: photodetector and digitizer.

FIG. 3 . 2 P 4 =

 324 FIG. 3. Noiseless simulated PDV spectrogram Φ(V, t) (right) and evolution of the irradiance E inc (z) inside the cloud (left) from Eq. (3) for a polydisperse expanding cloud in a vacuum at time t = 500 ns. z = 0 is the head of the cloud (fastest particles) and L cloud = 725 µm. Velocities are exponentially distributed with γ v = 18 [cf. Eq. (13)], and particle sizes follow a power law distribution with characteristic exponent γ d = 4. d min = 1 µm, d max = 10 µm, V min = 2200 m/s, V max = 3650 m/s, and M s = 2 mg/cm 2 . Collimated optical probe (ω 0 = 150 µm, z f = 0), φ P = 300 µm, I 0 = E inc (r, 0)π Φ 2 P 4 = 400 mW. [Short-Time Fourier Transform (STFT) parameters: sliding window F w : Hanning, width T w = 50 ns, time step δT w = 10%, zero-padding Z p = 4.]

FIG. 5 .

 5 FIG. 5. Influence of the areal mass M s (mg/cm 2 ) on the average PDV spectrum Φ(V , t) t∞ (W) and on M(V ) (mg/cm 2 ) inside the cloud. γ d = 4, γ v = 18, V min = 2200 m/s, V max = 3650 m/s, d min = 1 µm, and d max = 10 µm.

FIG. 6 .

 6 FIG. 6. Influence of velocity distribution f v on the average PDV spectrum Φ(V , t) t∞ (W) and on M(V ) (mg/cm 2 ) inside the cloud. γ d = 4, M s = 6 mg/cm 2 , V min = 2200 m/s, V max = 3650 m/s, d min = 1 µm, and d max = 10 µm.

FIG. 8 .

 8 FIG. 8. Experimental setup.

FIG. 11 .

 11 FIG. 11. LN-pin voltage U(t) (V), located at 7 mm from the sample.

FIG. 13 .

 13 FIG. 13. Comparison between the normalized experimental time-averaged PDV data Φ exp (V , t) t (blue) and the PDV spectrum Φ(V ) t∞ obtained from the ML estimation: ζ o = [M(V = 2281 m/s) = 2.05 mg/cm 2 , γ v = 23.28, κ = 6.6 × 10 3 , B Φ = 2.8 × 10 3 ] (red), for V between 2281 m/s and 3676 m/s. Fourier parameters: Rectangular window (T w = 50 ns, δT w = T w without zeropadding). 20 PDV independent slices between 1.95 and 2.9 µs were used to average the spectrum. The black dashed line is the estimated background noise amplitude B Φ . The error bars (± 2 σ(V ), standard deviation) are estimated from the experimental PDV spectrogram.

  FIG. 14. Top left, top right, and bottom left: Projections of the constant likelihood function L(ζ ) such that the equality of Eq. (31) is verified (joint confidence region at 95%). M(V min ) has units of mg/cm 2 . Scatter plot: ML estimation. Bottom right: Comparison between experimental results and ML estimation of the mass-velocity distribution M(V ) (mg/cm 2 ) for V /V fs between 1.11 and 1.8. The error bar at ±2σ and V min /V fs = 1.13 is calculated using the likelihood ratio method (boundaries of L).

FIG. 16 .

 16 FIG. 16. Influence of lowering d min on the estimated areal mass M s from PDV data (power law particle size distribution with exponent γ d = 5.6).

31

 31

  FIG. 17. Influence of lowering γ d on the estimated areal mass M s from PDV data (power law particle size distribution with exponent γ d ). d min = 0.5 µm and d max = 10 µm.