Samuel Deslauriers-Gauthier
email: samuel.deslauriers-gauthier@inria.fr

Rachid Deriche
email: rachid.deriche@inria.fr

Estimation of Axonal Conduction Speed and the Inter Hemispheric Transfer Time using Connectivity Informed Maximum Entropy on the Mean

Keywords: Inter hemispheric transfer time, axon conduction velocity, diffusion MRI, EEG, MEG

The different lengths and conduction velocities of axons connecting cortical regions of the brain yield information transmission delays which are believed to be fundamental to brain dynamics. A critical step in the estimation of axon conduction speed in vivo is the estimation of the inter hemispheric transfer time (IHTT). The IHTT is estimated using electroencephalography (EEG) by measuring the latency between the peaks of specific electrodes or by computing the lag to maximum correlation on contra lateral electrodes. These approaches do not take the subject's anatomy into account and, due to the limited number of electrodes used, only partially leverage the information provided by EEG.

Using the previous published Connectivity Informed Maximum Entropy on the Mean (CIMEM) method, we propose a new approach to estimate the IHTT. In CIMEM, a Bayesian network is built using the structural connectivity information between cortical regions. EEG signals are then used as evidence into this network to compute the posterior probability of a connection being active at a particular time. Here, we propose a new quantity which measures how much of the EEG signals are supported by connections, which is maximized when the correct conduction delays are used.

Using simulations, we show that CIMEM provides a more accurate estimation of the IHTT compared to the peak latency and lag to maximum correlation methods.

INTRODUCTION

The different lengths and conduction velocities of axons connecting cortical regions of the brain yield information transmission delays which are believed to be fundamental to brain dynamics. [START_REF] Deco | Key role of coupling, delay, and noise in resting brain fluctuations[END_REF][START_REF] Caminiti | Diameter, length, speed, and conduction delay of callosal axons in macaque monkeys and humans: Comparing data from histology and magnetic resonance imaging diffusion tractography[END_REF] While early work on axon conduction velocity was based on ex vivo measurements, [START_REF] Hursh | Conduction velocity and diameter of nerve fibers[END_REF][START_REF] Ritchie | On the relation between fibre diameter and conduction velocity in myelinated nerve fibers[END_REF] more recent work makes use of a combination of diffusion Magnetic Resonance Imaging (MRI) tractography and electroencephalography (EEG) to estimate axon conduction velocity in vivo. [START_REF] Horowitz | In vivo correlation between axon diameter and conduction velocity in the human brain[END_REF][START_REF] Innocenti | Comments on the paper by horowitz et al[END_REF] An essential intermediary step in this later strategy is to estimate the inter hemispheric transfer time (IHTT) using EEG. The IHTT is estimated by measuring the latency between the peaks of specific electrodes or by computing the lag to maximum correlation on contra lateral electrodes. [START_REF] Saron | Visual evoked potential measures of interhemispheric transfer time in humans[END_REF] These approaches do not take the subject's anatomy into account and, due to the limited number of electrodes used, only partially leverage the information provided by EEG.

In previous work, [START_REF] Deslauriers-Gauthier | Inference and Visualization of Information Flow in the Visual Pathway using dMRI and EEG[END_REF][START_REF] Deslauriers-Gauthier | White Matter Information Flow Mapping from Diffusion MRI and EEG[END_REF] we proposed a method, named Connectivity Informed Maximum Entropy on the Mean (CIMEM), to estimate information flow in the white matter of the brain. CIMEM is built around a Bayesian network which represents the cortical regions of the brain and their connections, observed using diffusion MRI tractography. This Bayesian network is used to constrain the EEG inverse problem and estimate which white matter connections are used to transfer information between cortical regions. In our previous work, CIMEM was used to infer the information flow in the white matter by assuming a constant conduction velocity for all connections. In this context, the conduction speed, and thus the delays, were inputs used to help constrain the problem. Here, we interchange the variables and the parameters and instead assume that the connection used to transfer information across the hemispheres is known, due the design of the acquisition paradigm, but that its conduction velocity must be estimated. The result in a novel method to estimate the IHTT in addition to the axon conduction velocity that makes full use of the information provided by EEG and diffusion MRI.

METHODS

Our strategy to estimate the IHTT is built on a generative model of the EEG measurements which incorporates both the cortical surface and the underlying white matter connections. To make the current document self contained, we briefly summarize this generative model here. A more detailed presentation and discussion is available in our previous work. 8, 9

CIMEM Generative Model and Inverse Problem

The sources of brain activity are distributed on the cortical surface with their orientations fixed and normal to the surface. Let x = [x 1,1 , x 2,1 , ..., x N,1 , x 1,1 , ...x N,T ] ∈ R N T be the concatenated vector of source intensity where N is the number of sources and T is the number of time samples. These sources of brain activity are related to the EEG measurements through the forward model 10

m = Gx (1)
where m ∈ R M T is the concatenation of M measurements at each time sample. The matrix G ∈ R M T ×N T is the lead field matrix which projects sources onto sensors and is a known quantity. Because our formulation considers all time samples simultaneously, the lead field matrix is block diagonal with each block containing the usual lead field matrix.

In theory, any vector of source activity x can be used in the forward problem of Equation 1 to generate the measurements m. In practice however, certain configuration are more likely than others. For example, x can be expected to contain spatial correlation between neighboring sources and temporal correlation between sources connected by white matter axons. To capture the spatial correlation, we parcellate the brain into N S regions, assign a state S k to each, and group them in a region state vector S. The state of a cortical region provides information about the intensity of the individual sources within the region. For example, it can be expected that the sources of a region that is in the active state will have a higher average intensity than the sources in a region that is in an inactive state. To capture the temporal correlation, we connect the brain regions using N C white matter connections, assign a state C i to each of them, and group them in a connection state vector C. It is essential to note that, at the temporal resolution of EEG, white matter connections do not transfer information instantaneously. The callosal axons of the splenium, for example, are between 150 to 200 mm long and introduce a communication delay of 25 to 35 milliseconds between the left and right occipital lobes, assuming a conduction velocity of 6 m/s. [START_REF] Hursh | Conduction velocity and diameter of nerve fibers[END_REF] These delays are well above the millisecond temporal resolution of EEG. Connections in our model therefore introduce activation delays that reflect the length and conduction velocity of axons. As such, the state of a connection provides information on the cortical regions it connects across time and enforce a temporal regularization into the model. For example, if a connection is in the active state, it forces the regions it connects to also be in the active state with a specified delay. Here, both brain regions and connections are assigned two possible states: active or inactive. To link these states of cortical regions and connections to source intensity, we define a reference law dµ(x) = µ(x)dx which encodes our global prior knowledge of cortical source activity. Specifically, we define dµ(x) to have the form

dµ(x) = {C} N C i=1 ϕ(C i ) {S} N S k=1 π(S k |C γ(k) )dµ(x k |S k ) (2)
where the sum over state vectors indicates a sum over every possible state combinations, where x k contains the sources of the k th brain region, and where C γ(k) is the set of connections reaching the k th brain region. The density dµ(x k |S k ) encodes the likelihood of a configuration x k given the k th region's state, the density π(S k |C γ(k) ) encodes the likelihood of the region state given its connections, and ϕ(C i ) is the likelihood of the i th connection's state. A more derivation of Equation 2 is available in our prior work. [START_REF] Deslauriers-Gauthier | White Matter Information Flow Mapping from Diffusion MRI and EEG[END_REF] In summary, Equation 2 reflects that the activity of sources depends on the state of the region to which they belong and that the activity of a region depends on the activity on the regions to which it is connected. Because of the hierarchical relationship between sources, regions, and connections, Equation 2 defines a Bayesian network which can be used to compute the likelihood of any source configuration. In addition, the Bayesian network can also be used to infer the most likely brain configuration (source intensity, region states, connection states) given the observations. This can be achieved by finding the probability law p(x) whose average fits the measurements and that is closest to the reference law dµ(x). We therefore propose to solve the problem minimize

p(x),λ,λ0 p(x) ln p(x) µ(x) dx -λ(m -G xp(x)dx) -λ 0 (1 -dp(x)) (3)
where the first term is the Kullback-Leibler divergence between p(x) and µ(x), the second term is the data fit, and the third term is a scaling to ensure p(x) is a probability distribution. This can be shown [START_REF] Jaynes | Information theory and statistical mechanics[END_REF][START_REF] Amblard | Biomagnetic source detection by maximum entropy and graphical models[END_REF] to be equivalent to solving the problem

λ * = arg min λ ln exp (λ T Gx)dµ(x) -λ T m (4)
where λ * completely determines the optimal p(x), λ, and λ 0 of Equation 3. Because λ * captures the compromise between the EEG observation and the priors of the model, it can be used to compute a posterior probability for all variables of the model. Here, we are specifically interested in the posterior probability of active connections, defined as

Z(C i,n = 1) = exp (λ * T Gx)dµ(x|C i,n = 1) (5)
which is derived from the first term of Equation 4.

IHTT Estimation

The original purpose of CIMEM was to map information flow in the white matter from combined EEG and diffusion MRI measurements. In this context, the conduction velocity of white matter axons must be given as an input and serves to constrain the solution space. Our suggestion was to use diffusion MRI to select the available white matter connections and attempt to identify which of them are active at a given time. The output of our algorithm was then a subset of white matter connections whose delays and cortical endpoints explain the measurements. In contrast to this approach, here we assume that the white matter connections that support the data are known, but that their conduction velocity and induced delay must be estimated. A prime example is the estimation of the IHTT, where the communication between the left and the right hemisphere is assumed to go through callosal white matter axons. An approximation of the geometry of these axons can be obtained using diffusion MRI and added to the CIMEM model while ignoring all other brain connections. For each connection of the model and at every time instant, a variable is added to the model with two possible state: active (0) or inactive (1). EEG signals are then used as evidence into this network to compute the posterior probability of a connection being active at a particular time. Recall that Z(C i,n = 1) is the posterior probability that the i th connection is active at the n th time point obtained by solving the CIMEM problem of Equation 4. We define the connectivity power of the i th connection as

Γ d (C i = 1) = N -1 N -1 n=0 Z(C i,n = 1) 2 (6)
for a given delay d. The connectivity power is then computed for a series of delays and the estimated delay for a given connection is the one that maximizes Γ d (C i ). The rationale is that the CIMEM model will only be able to use the connection to explain the EEG measurements if the selected delay is correct.

For comparison purposes, the IHTT was also estimated using the peak latency and lag to maximum correlation methods. [START_REF] Saron | Visual evoked potential measures of interhemispheric transfer time in humans[END_REF] For the peak latency method, the IHTT is estimated by computing the temporal difference between the location of the maximum magnitude on the left and right lateral occipital electrodes. For the correlation method, the Pearson correlation is computed between the left and right lateral occipital electrodes for all time lags and the lag with maximal correlation is selected.

(b) (c) (d)

Data Simulation

To evaluate the performance of our algorithm, EEG signals were generated using a realistic head geometry and acquisition montage with 63 electrodes. The cortical surface used for simulation was reconstructed from a T1 weighted MRI of a healthy subject using FreeSurfer and decimated to 8000 vertices. EEG signals were generated by selecting the sources of the left and right lateral occipital cortical regions and assigning them intensities using a multivariate Gaussian with a mean of 1 and a covariance based on the cortical distance between sources. The left and right sources were then modulated by two temporal Gaussians with a variance of 80 ms whose difference in mean was adjusted to obtain the desired delay. The forward operator used to project cortical activity onto sensors was computed using OpenMEEG [START_REF] Gramfort | OpenMEEG: opensource software for quasistatic bioelectromagnetics[END_REF] with the source locations corresponding to the vertices of the cortical surface.

RESULTS

Examples of the simulated EEG signals are illustrated in Figure 1. As the delay between the left and right activation is reduced, both waveforms merge to produce a single peak. In addition, due to volume conduction, the peak does not occur at the same location on all electrodes. This effect will be more pronounced as the width of the activation, i.e. its duration, is increased. As both the peak latency and correlation methods rely only on the shape of the recorded signal, this will have an effect on their estimation of the IHTT.

A few representative examples of the connection power computed using CIMEM as a function of the simulated delay are illustrated in Figure 2. The EEG measurements were simulated with a delay of 11, 8, 5, and 2, which corresponds to a conduction speed of 1.82, 2.5, 4, and 10 m/s, respectively, for a white matter connection of 100 mm and EEG signals sampled at 200 Hz. In all cases, we observe that the maximum of the connection power curve corresponds to the simulated delay illustrated by a vertical red line in the graphs. The mean estimated speed as a function of the simulated speed is illustrated in Figure 3. Both the peak and correlation method overestimate the conduction speed for simulated speed of 5 to 10 m/s. For the Gaussian waveform simulated in this study, these speeds correspond to the range where the left and right occipital activations overlap. As previously illustrated in 1, this distorts the shape of the waveform observed on the electrodes and shifts the position of the peak. On the other hand, the mean speed estimated with CIMEM more closely corresponds to the true simulated speed as the volume conduction, the anatomy of the subject, and all electrodes are taken into account.

CONCLUSION

The estimation of the inter hemispheric transfer time is a critical step in the estimation of in vivo axon conduction velocity. Current methods rely on sensor space delays and thus do take the subject's anatomy into account. In this work we have presented a new method to estimate the inter hemispheric transfer time and thus axon conduction velocity. Our new approach, based on our previously published CIMEM algorithm, takes the subjects anatomy into account and leverages all of the information provided by EEG. In simulations, CIMEM outperforms both the peak latency and lag to maximum correlation methods for the estimation of the IHTT.

A hypothesis that is common to the peak latency, lag to maximum correlation, and our method is that the inter hemispheric transfer time can be represented by a single value. In practice, due to the distribution of axonal diameters in the white matter, 14 using a distribution of values may be more realistic. We believe our method may be adapted to provide this information and modifying it is the topic of our current work.

In addition to estimating white matter delays and thus providing an insight into brain dynamics, our method may also be used to probe the microstructural information of specific white matter bundles. For example, it has been shown that axon diameters and myelin both affect axon conduction velocity. [START_REF] Hursh | Conduction velocity and diameter of nerve fibers[END_REF][START_REF] Drakesmith | Mapping axon conduction delays in vivo from microstructural MRI[END_REF] Therefore, by providing an estimate of conduction velocity, our could potentially be used to validate in vivo microstructure estimates provided by diffusion MRI.

Figure 1 .

 1 Figure 1. Examples of simulated EEG signals for a delay of 11, 8, 5, and 2 samples (a, b, c, d). The left and right lateral occipital sensors are displayed in green and red, respectively. All the other sensors are displayed in grey.

Figure 2 .

 2 Figure 2. connection power (Γ) as a function of the delay used for the reconstruction for simulated delays of 11, 8, 5, 2 samples (a, b, c, d). The maximum connection power corresponds to the simulated delay illustrated by the red vertical line.

ACKNOWLEDGEMENTS

This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665 : CoBCoM -Computational Brain Connectivity Mapping).


Y Li

V V Naletov

O Klein

J L Prieto

M Muñoz

V Cros

P Bortolotti

A Anane

C Serpico

G De Loubens

Nutation spectroscopy of a nanomagnet driven into deeply nonlinear ferromagnetic resonance

Strongly out-of-equilibrium regimes in magnetic nanostructures exhibit novel properties, linked to the nonlinear nature of magnetization dynamics, which are of great fundamental and practical interest. Here, we demonstrate that field-driven ferromagnetic resonance can occur with substantial spatial coherency at unprecedented large angle of magnetization precessions, which is normally prevented by the onset of spin-wave instabilities and magnetization turbulent dynamics. Our results show that this limitation can be overcome in nanomagnets, where the geometric confinement drastically reduces the density of spin-wave modes. The obtained deeply nonlinear ferromagnetic resonance regime is probed by a new spectroscopic technique based on the application of a second excitation field. This enables to resonantly drive slow coherent magnetization nutations around the large angle periodic trajectory. Our experimental findings are well accounted for by an analytical model derived for systems with uniaxial symmetry. They also provide new means for controlling highly nonlinear magnetization dynamics in nanostructures, which open interesting applicative opportunities in the context of magnetic nanotechnologies.

media has been of tremendous impact on the development of physics since the beginning of the 20th century and remains of crucial importance till nowadays in the study of nanotechnologies. In this area, a central role is played by magnetic resonance spectroscopy, which includes various techniques such as nuclear magnetic resonance (NMR), electron paramagnetic resonance (EPR), and ferromagnetic resonance (FMR), all based on the excitation of the Larmor precession of magnetic moments around their equilibrium position [START_REF] Kittel | Introduction to Solid State Physics[END_REF].

FMR differs from NMR and EPR by the fact that in ferromagnetic media, magnetic moments are coupled by strong exchange interactions which tend to align them, leading to a large macroscopic spontaneous magnetization. In these conditions, magneto-dipolar effects become important and determine large internal fields which enrich both the ground state, that can be spatially non-uniform, and the dynamics of magnetic moments. The complex interactions taking place in the media can be described by an appropriate effective field which sets the time scale of the magnetization precession, and which itself depends on the magnetization, making the dynamics, for sufficiently large deviations from the ground state, highly nonlinear [START_REF] Mayergoyz | Nonlinear magnetization dynamics in nanosystems[END_REF]. A special role in FMR is also played by the spin-waves (SWs), which are the collective eigenmodes associated to small magnetization oscillations around the equilibrium configuration [START_REF] Gurevich | Magnetization Oscillations and Waves[END_REF]. When pumping fields excite SWs well above their thermal amplitudes, a rich variety of phenomena emerges, such as the formation of dynamical solitons [START_REF] Wigen | Nonlinear Phenomena and Chaos in Magnetic Materials[END_REF][START_REF] Mohseni | Spin torque-generated magnetic droplet solitons[END_REF], SW turbulences and chaos [START_REF] Wigen | Nonlinear Phenomena and Chaos in Magnetic Materials[END_REF][START_REF] Rezende | Spin-wave instabilities, auto-oscillations, and chaos in yttrium-iron-garnet[END_REF][START_REF] Victor | Wave Turbulence Under Parametric Excitation: Applications to Magnets[END_REF][START_REF] Petit-Watelot | Commensurability and chaos in magnetic vortex oscillations[END_REF], and Bose-Einstein condensation of magnons [START_REF] Demokritov | Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping[END_REF], the quanta of SWs.

Recent developments in magnetic nanotechnologies have also demonstrated that FMR and SW dynamics can be excited either by microwave magnetic fields or by spin transfer torques, with the promise of innovative magnonic and spintronic devices for information and communication technologies [START_REF] Chumak | Magnon spintronics[END_REF]. In this area, spin torque nano-oscillators (STNOs) [START_REF] Kiselev | Microwave oscillations of a nanomagnet driven by a spin-polarized current[END_REF][START_REF] Houssameddine | Spintorque oscillator using a perpendicular polarizer and a planar free layer[END_REF][START_REF] Chen | Spin-torque driven ferromagnetic resonance in a nonlinear regime[END_REF][START_REF] Hamadeh | Autonomous and forced dynamics in a spin-transfer nano-oscillator: Quantitative magneticresonance force microscopy[END_REF][START_REF] Collet | Generation of coherent spin-wave modes in yttrium iron garnet microdiscs by spin-orbit torque[END_REF], which exhibit strong nonlinear properties [START_REF] Slavin | Nonlinear auto-oscillator theory of microwave generation by spin-polarized current[END_REF], have even been successfully implemented to perform neuromorphic tasks [START_REF] Torrejon | Neuromorphic computing with nanoscale spintronic oscillators[END_REF][START_REF] Romera | Vowel recognition with four coupled spintorque nano-oscillators[END_REF].

The complexity of magnetization dynamics when strongly nonlinear regimes set in is usually detrimental to the reliable control of nanomagnetic devices, such as oscillators, memories, and logic gates. In this respect, it is important to establish how far from equilibrium magnetic nanostructures can be driven before the coherent magnetization dynamics becomes highly perturbed by the onset of SW instabilities [START_REF] Suhl | The theory of ferromagnetic resonance at high signal powers[END_REF]. In this article, we provide a crucial advancement in this problem. We demonstrate that FMR in a sufficiently confined nanostructure can exhibit unprecedented large angle magnetization precessions which are spatially quasi-uniform. The experimental evidence of the coherence of large precessions is brought by a new spectroscopic technique based on the application of a second probe excitation field, with frequency close to the one of the main time-harmonic field. This second excitation is used to drive small eigen oscillations of magnetization around the FMR large angle periodic oscillations, corresponding to coherent nutations of the magnetization. These nutation modes are substantially different from the usual SW modes around the ground state because they correspond to eigenmodes around a far-from-equilibrium state and their existence is connected to the one of a large coherent precession. Moreover, we show that the resonant excitation of these nutations can be used to control the nonlinear magnetization dynamics by affecting the switching fields associated with the bistability of the large angle FMR response pictured in Fig. 1a, which occurs beyond the foldover instability predicted by Anderson and Suhl [START_REF] Anderson | Instability in the motion of ferromagnets at high microwave power levels[END_REF].

The preservation of coherent magnetization dynamics, which we report on a thin disc with sub-micrometric diameter, is mainly due to the geometric confinement. It significantly reduces the density of normal modes and suppresses the nonlinear SW interactions present in bulk ferromagnets [START_REF] Kobljanskyj | Nano-structured magnetic metamaterial with enhanced nonlinear properties[END_REF][START_REF] Melkov | Nonlinear ferromagnetic resonance in nanostructures having discrete spectrum of spin-wave modes[END_REF]. In addition, in our experiments the ground state is with the magnetization perpendicular to the plane. In this case, the uniform mode in thin films lies at the bottom of the SW manifold, so that it has no degenerate mode to couple to [START_REF] Mcmichael | High-power ferromagnetic resonance without a degenerate spin-wave manifold[END_REF][START_REF] Yu | Principal mode of the nonlinear spin-wave resonance in perpendicular magnetized ferrite films[END_REF] and the angle of the purely circular precession driven by FMR can reach large values [START_REF] Gnatzig | Ferrimagnetic resonance in garnet films at large precession angles[END_REF]. These combined circumstances allow for the excitation of large-amplitude quasi-uniform precession of magnetization without simultaneous excitation of other SWs.

The experimental results of the present work also address the important issue of comparison between theory and experiments. In fact, by using the framework of dynamical systems [START_REF] Mayergoyz | Nonlinear magnetization dynamics in nanosystems[END_REF], exact analytical solutions of the Landau-Lifshitz-Gilbert (LLG) equation in the presence of an arbitrarily large time-harmonic excitation have been found in high symmetry cases [START_REF] Bertotti | Nonlinear magnetization dynamics under circularly polarized field[END_REF] and their stability analyzed [START_REF] Bertotti | Spin-wave instabilities in large-scale nonlinear magnetization dynamics[END_REF], but have not been verified experimentally yet. For instance, the hysteretic FMR illustrated in Fig. 1a has already been observed in measurements [START_REF] Fetisov | Nonlinear ferromagnetic resonance and foldover in yttrium iron garnet thin films -inadequacy of the classical model[END_REF][START_REF] Gui | Foldover ferromagnetic resonance and damping in permalloy microstrips[END_REF], but with much weaker bistable response characteristics than expected from theory.

In the following, we investigate the FMR of an individual nanodisc of yttrium iron garnet (YIG) in the perpendicular configuration. The choice of YIG is natural as it is the magnetic material with the smallest SW damping, making it attractive to study weakly dissipative magnetization dynamics in the linear and nonlinear regimes [START_REF] Serga | YIG magnonics[END_REF]. The nanodisc has a diameter of 700 nm and is patterned from a 20 nm thick YIG film of magnetization µ 0 M s = 0.21 T [START_REF] Kelly | Inverse spin hall effect in nanometer-thick yttrium iron garnet/Pt system[END_REF], µ 0 being the vacuum permeability. It is saturated out-of-plane by a magnetic field H 0 applied along its normal, z. A broadband antenna supplies a spatially uniform, linearly polarized microwave field of pulsation ω 1 oriented in the plane of the nanodisc. It can be decomposed into the left and right circularly polarized components, only the latter being efficiently coupled to the Larmor precession of the magnetization. In the following, h 1 will refer to the circular amplitude of the excitation field produced by the output power P 1 from the synthesizer. It drives the YIG nanodisc into FMR, thereby opening a precession angle θ of the magnetization M around H 0 and decreasing its longitudinal component

M z = M s cos θ.
This dynamics is characterized by magnetic resonance force microscopy (MRFM), which sensitively probes the variation ∆M z = M s -M z through the dipolar force between the YIG nanodisc and a magnetic nanosphere attached at the end of a soft cantilever [START_REF] Klein | Ferromagnetic resonance force spectroscopy of individual submicron-size samples[END_REF], as sketched in Fig. 1b. Further details about the sample, the MRFM set-up and the microwave calibration can be found in the Methods.

Linear spin-wave spectroscopy

In Fig. 1c, the SW spectroscopy of the YIG nanodisc is performed at ω 1 /(2π) = 10.5 GHz and low power P 1 = -30 dBm, which is pulse modulated at the frequency of the MRFM cantilever to improve the signal to noise ratio. Quantized radial SW modes are excited by the uniform pumping field [START_REF] Naletov | Identification and selection rules of the spin-wave eigenmodes in a normally magnetized nanopillar[END_REF]. Their spatial profiles indexed by the radial number are shown above the spectrum. The fundamental Kittel mode is the one excited at the largest field, µ 0 H K = 0.569 T, and corresponds to a uniform phase of the transverse magnetization in the disc. Due to the geometric confinement, it is well separated from other SW modes at lower field [START_REF] Hahn | Measurement of the intrinsic damping constant in individual nanodisks of Y 3 Fe 5 O 12 and Y 3 Fe 5 O 12 |Pt[END_REF]. Its full width at half maximum, µ 0 ∆H = 0.35 mT, is determined at even lower power (P 1 = -38 dBm) to avoid distortions of the resonance line due to the onset of foldover, which occurs when the change in effective field becomes comparable to the FMR linewidth, at P th 1 = -33 dBm or µ 0 h th 1 = 0.009 mT (see Methods). It corresponds to a Gilbert damping parameter α = γµ 0 ∆H/(2ω 1 ) = 4.7 • 10 -4 , where γ is the gyromagnetic ratio, in agreement with the value determined from broadband measurements (see Supplementary Fig. S1).

Deeply nonlinear FMR

The FMR spectrum of the YIG nanodisc radically changes at much stronger pumping fields. Fig. 1d shows the measurement with a continuous wave (cw) excitation at P 1 = +12 dBm, i.e., more than four orders of magnitude larger than the threshold of foldover instability. The cw excitation allows to reveal the bistable character of the nonlinear magnetization dynamics. By sweeping down the applied field (red curve) through the resonance of the Kittel mode, the precession angle substantially increases, which decreases the static demagnetizing field µ 0 M z and shifts the FMR condition ω 1 = γµ 0 (H 0 -M z ) to lower magnetic field by the same amount. This foldover shift to lower field continues until the pumping field cannot sustain anymore the large-amplitude magnetization dynamics, causing the sharp downward jump to the lower stable branch observed at µ 0 H down = 0.381 T. By sweeping up the applied field (blue curve), an upward jump to the higher stable branch is observed at µ 0 H up = 0.516 T. The extremely hysteretic foldover witnessed in this experiment is remarkable. Moreover, the maximal foldover shift µ 0 (H K -H down ) = 0.188 T corresponds to a reduction of nearly 90% of µ 0 M z induced by the microwave pumping, which translates into a mean precession angle of 84 • in the nanodisc. The evolution of the maximal normalized foldover shift as a function of the pumping field h 1 is plotted in Fig. 1e together with

∆M max z /M s = 1 -1 -4h 2
1 /∆H 2 calculated from the macrospin LLG equation [START_REF] Gurevich | Magnetization Oscillations and Waves[END_REF]. The measured foldover shift starts to deviate from the macrospin model beyond µ 0 h 1 0.1 mT, which is an order of magnitude larger than the threshold for foldover, when the angle of the uniform precession increases above 30 • , corresponding to ∆M z /M s 15%. This is the signature of the onset of SW instabilities [START_REF] Suhl | The theory of ferromagnetic resonance at high signal powers[END_REF], which is here significantly postponed compared to what is observed in larger YIG samples, where the Suhl threshold is reached even before the onset of foldover, for a uniform precession angle of only a couple of degrees [START_REF] De Loubens | Reduction of the spin-wave damping induced by nonlinear effects[END_REF][START_REF] Naletov | Quantitative measurement of the ferromagnetic resonance signal by force detection[END_REF]. In this respect, the experimental results presented in Fig. 1 demonstrate that the discretization of the excitation spectrum in nanostructures efficiently inhibits nonlinear interactions between Nutation spectroscopy in the rotating frame

We now aim at probing the stability of the large-amplitude magnetization dynamics demonstrated above, which is periodic at ω 1 in the laboratory frame, hence referred to as P-mode [START_REF] Bertotti | Nonlinear magnetization dynamics under circularly polarized field[END_REF]. In the frame rotating with h 1 at ω 1 around the z-axis, the magnetization M 0 of a P-mode is fixed at a polar angle θ 0 and a phase lag ϕ 0 (see Fig. 2a and supplementary information). For this, we conduct two-tone measurements, where in addition to the strong cw excitation h 1 at ω 1 a second weak microwave field h 2 , pulse modulated at the cantilever frequency, is applied at ω 2 , as shown in Fig. 2b. MRFM is used to simultaneously detect ∆M 1 induced by the main cw pumping at ω 1 (by monitoring the cantilever frequency, as in Fig. 1d) and the additional change in longitudinal magnetization, ∆M 2 , induced by the second excitation at ω 2 (by monitoring the amplitude of the cantilever vibrations, as in Fig. 1c). The former informs us about the time-harmonic steady state regime driven by h 1 , whereas the latter allows us to spectroscopically probe the eigen excitations on top of this P-mode. The ∆M 2 spectrum measured at constant bias field µ 0 H 0 = 0.52 T by sweeping ω 2 /(2π) at low power P 2 = -16 dBm in the vicinity of the frequency ω 1 /(2π) = 10.5 GHz of the main pumping (P 1 = +8 dBm) is shown in Fig. 2d. It displays two narrow resonance peaks centered at 10.2 GHz and 10.8 GHz, i.e., symmetrically with respect to ω 1 /(2π). This means that in the frame rotating with h 1 at ω 1 , the magnetization is precessing at (ω 2 -ω 1 )/(2π) = ±ω P,0 /(2π) = ±0.3 GHz around its equilibrium position M 0 (cf. Fig. 2a).

In other words, it is submitted to a slow nutation motion in the laboratory frame. The dependence of the nutation frequency ω P,0 on the main pumping field h 1 at fixed µ 0 H 0 = 0.52 T is presented in Fig. 2e, whereas its evolution measured as a function of the down swept field H 0 at fixed P 1 = +8 dBm and P 1 = +1 dBm is shown in the 2D spectroscopy maps of Fig. 2f and2h, respectively.

Following the theoretical approach of ref. [START_REF] Bertotti | Spin-wave instabilities in large-scale nonlinear magnetization dynamics[END_REF], it is possible to calculate analytically the frequency ω P,0 of spatially uniform nutation around a given P-mode based on the macrospin LLG equation. Technical details are given in the Methods (the full derivation is presented in the supplementary information). In the limit of small damping, α 1, it can be expressed as a function of h 1 and the angles θ 0 and ϕ 0 of the P-mode as follows:

ω 2 P,0 γ 2 = µ 0 h 1 cos ϕ 0 sin θ 0 µ 0 h 1 cos ϕ 0 sin θ 0 + µ 0 M s sin 2 θ 0 . (1)
This analytical expression is plotted as green dashed lines in Figs. The dependence of the nutation frequency on h 1 observed in Fig. 2e can be explained as follows. There is a minimum amplitude of µ 0 h 1 0.15 mT to drive the P-mode at µ 0 H 0 = 0.52 T, which corresponds to a normalized foldover shift (H K -H 0 )/M s = 23%, i.e., an angle of precession θ 0 40 • . Above this amplitude, the nutation frequency is defined and increases with h 1 as predicted by Eq.1, which shows that there are two torques driving the nutation dynamics of the magnetization. The first one is linear, and directly provided by h 1 , which sets the Rabi frequency in a magnetic resonance experiment [START_REF] Capua | Ensemble-averaged rabi oscillations in a ferromagnetic CoFeB film[END_REF]. The second one is a demagnetizing torque specific to nonlinear FMR, which stiffens the nutation resonance.

We now interpret the nutation spectroscopy maps of Figs. 2f and2h, where ∆M 2 is measured at fixed h 1 by sweeping H 0 and ω 2 . When H 0 > H K , the magnetization dynamics is driven off resonantly by h 1 and has small amplitude. Hence, the weak microwave field The experimental results presented in Fig. 2 also demonstrate that the weak resonant excitation of the nutation mode can destabilize the strong foldover dynamics. Figs.2g and 2i display the evolution of ∆M 1 induced by the cw pumping h 1 at ω 1 , while exciting the nutation dynamics with h 2 as a function of H 0 and ω 2 , whose ∆M 2 spectroscopy is presented in Figs. 2f and2h, respectively. In these 2D maps, the foldover breakdown occurring at H down is easily identified thanks to the associated sharp change of ∆M 1 , and is anticipated as the nutation mode is excited. This is particularly clear in Fig. 2i, where the maximal foldover shift µ 0 (H K -H down ) is reduced by almost 0.05 T when |ω 2 -ω 1 |/(2π) 0.2 GHz, which corresponds to the nutation resonance. Moreover, these data suggest that higher order nutation modes can be excited by h 2 , since an anticipated foldover breakdown is also

observed at |ω 2 -ω 1 |/(2π) 0.35 GHz.
In order to investigate these other nutation modes, we perform the same measurements as in Figs. 2f-i, but for larger detunings ω 2 -ω 1 (P 2 is also increased from -19 to -13 dBm). The results obtained at P 1 = +1 dBm are reported in Figs. 3b-c (those obtained at P 1 = +8 dBm are presented in Supplementary Fig. S2). The spectroscopy map of the SW modes excited by h 2 in the absence of h 1 is shown in Fig. 3a. It displays the linear dispersion relation of the radial SW modes excited by the uniform field h 2 , already discussed in Fig. 1c. Due to the strong foldover regime driven by h 1 at H 0 < H K in Fig. 3b, each of these radial SW branches transforms into a pair of branches symmetric around ω 1 . Additionally, there is a pair of branches which appears at twice the main nutation frequency, which is due to the ellipticity of the trajectory, apparent on Fig. 2c. The macrospin approach used to derive Eq.1 cannot be used to account for these higher order nutation modes, although plane wave perturbations to the P-mode can also be analytically calculated [START_REF] Bertotti | Spin-wave instabilities in large-scale nonlinear magnetization dynamics[END_REF]. We therefore use micromagnetic simulations to calculate the SW nutation spectra shown in Fig. 3d, which are in good agreement with the experiments. The extracted SW nutation modes profiles (see Methods), shown as insets in Fig. 3d, indicate that there is some continuity between the radial SW modes excited in the linear regime on top of the equilibrium magnetization and the nutation modes excited on top of the P-mode driven in the nonlinear regime by h 1 .

Finally, Fig. 3c confirms that the excitation of the nutation resonances can destabilize the strong foldover dynamics.

DISCUSSION

As in the case of a spinning top, the nutation of magnetization demonstrated above is made possible by the specific properties of the dynamics on the unit sphere [START_REF] Mayergoyz | Nonlinear magnetization dynamics in nanosystems[END_REF]. Namely, it is topologically allowed for the magnetization to oscillate around its fixed point M 0 (θ 0 , ϕ 0 ) in the rotating frame, which is set by the drive h 1 . The nutation frequency results from the balance of torques acting on the angular momentum, and is given by Eq.1 in the case of a macrospin governed by the LLG equation. The accuracy of the latter to account for the experimental data means that the coherent precession of the magnetization vector is dominating the deeply nonlinear driven dynamics, despite the signatures of SW instabilities observed at very large pumping power. Their main effect is to slightly reduce the nutation frequency, which is well captured by full micromagnetic simulations. This can be ascribed to the shift of the phase between the pumping field and the average magnetization [START_REF] Hans | Nonlinear spin-wave excitations at low magnetic bias fields[END_REF] observed in our simulations, a key effect to explain the above threshold dynamics [START_REF] Victor | Wave Turbulence Under Parametric Excitation: Applications to Magnets[END_REF][START_REF] Gerrits | Direct Detection of Nonlinear Ferromagnetic Resonance in Thin Films by the Magneto-Optical Kerr Effect[END_REF].

The nutation spectroscopy of magnetization thus allows a more detailed investigation of the highly nonlinear regime, where auto-oscillation instabilities [START_REF] Rezende | Spin-wave instabilities, auto-oscillations, and chaos in yttrium-iron-garnet[END_REF][START_REF] Mcmichael | High-power ferromagnetic resonance without a degenerate spin-wave manifold[END_REF][START_REF] Lührmann | Nonlinearities in the ferrimagnetic resonance in epitaxial garnet films[END_REF][START_REF] Watanabe | Generation of megahertz-band spin currents using nonlinear spin pumping[END_REF] and instability patterns [START_REF] Bonin | Analysis of magnetization instability patterns in spin-transfer nano-oscillators[END_REF] have been evidenced. Moreover, its quantitative understanding, made possible thanks to the low density of SW modes in nanomagnets, should enable to test further the LLG equation governing the motion of magnetization against experimental measurements.

Our results also highlight that the dynamical states driven by a high power microwave signal can be controlled using a second signal with much lower power by the resonant excitation of the nutation modes. This could be applied in devices taking advantages of the bistable magnetization dynamics for microwave signal processing [START_REF] Fetisov | Nonlinear ferromagnetic resonance and foldover in yttrium iron garnet thin films -inadequacy of the classical model[END_REF], in analogy to microwave assisted magnetization switching [START_REF] Thirion | Switching of magnetization by nonlinear resonance studied in single nanoparticles[END_REF][START_REF] Pigeau | Optimal control of vortex-core polarity by resonant microwave pulses[END_REF][START_REF] Suto | Magnetization switching of a Co/Pt multilayered perpendicular nanomagnet assisted by a microwave[END_REF]. Furthermore, the frequency selectivity and energy efficiency of nutation excitations provide new potentials for the scheme of neuromorphic computing. Cognitive tasks have already been implemented using the nonlinear dynamics of nanomagnets, from the transient regime of a single STNO [START_REF] Torrejon | Neuromorphic computing with nanoscale spintronic oscillators[END_REF] to the collective behavior of mutually coupled STNOs controlled by external microwave signals [START_REF] Romera | Vowel recognition with four coupled spintorque nano-oscillators[END_REF][START_REF] Li | Probing phase coupling between two spin-torque nano-oscillators with an external source[END_REF]. An appropriate use of the nutation dynamics of magnetization would allow to gain additional control on nonlinear dynamics, which is highly desired in this field.

METHODS

Sample preparation.

A 20 nm thick Y 3 Fe 5 O 12 (YIG) film was grown by pulsed laser deposition on a (111) Gd 3 Ga 5 O 12 (GGG) substrate, as described in ref. [START_REF] Kelly | Inverse spin hall effect in nanometer-thick yttrium iron garnet/Pt system[END_REF]. It was used to pattern the studied YIG nanodisc by electron lithography and dry etching. After the insertion of a 50 nm thick SiO 2 insulating layer, a 150 nm thick and 5 µm wide gold antenna was defined on top of the nanodisc to provide the microwave excitation [START_REF] Hahn | Measurement of the intrinsic damping constant in individual nanodisks of Y 3 Fe 5 O 12 and Y 3 Fe 5 O 12 |Pt[END_REF].

MRFM set-up. The magnetic resonance force microscope is located between the poles of an electromagnet and operated under vacuum (10 -6 mbar) at a stabilized temperature of 288 K. The cantilever is an Olympus Biolever (spring constant 5 mN•m -1 ) with a 700nm-diameter sphere of an amorphous FeSi alloy (magnetic moment 0.28 pA•m 2 ) glued to its apex. In this study, MRFM spectroscopy is achieved by placing the center of this magnetic nanosphere at a distance of 1.5-1.8 µm above the center of the YIG nanodisc. The strayfield of the MRFM probe (10-16 mT) is subtracted from the corresponding spectra.

The displacement of the cantilever is monitored using optical techniques. Its mechanical frequency (f c ≈ 12.3 kHz) is tracked using a phase-locked loop and its vibration amplitude stabilized to 4 nm using a piezoelectric bimorph. When the cw microwave pumping excites the magnetization dynamics in the sample, its longitudinal component is reduced, so the static dipolar force with the magnetic probe diminishes. The associated variation of the cantilever frequency provides a quantitative magnetometry of the sample [START_REF] Lavenant | Mechanical magnetometry of cobalt nanospheres deposited by focused electron beam at the tip of ultra-soft cantilevers[END_REF]. In order to improve the signal to noise ratio, the microwave excitation is pulsed on and off at f c . In that case, the cantilever vibrations induced by the magnetization dynamics excited in the sample are enhanced by the quality factor Q ≈ 2000 of the mechanical detection [START_REF] Klein | Ferromagnetic resonance force spectroscopy of individual submicron-size samples[END_REF].

Microwave field calibration. We use the onset of foldover as a mean to calibrate the amplitude of the excitation field produced by the microwave antenna at the sample location [START_REF] Naletov | Identification and selection rules of the spin-wave eigenmodes in a normally magnetized nanopillar[END_REF]. At the threshold of foldover instability h th 1 = 0.62∆H 3/2 /M 1/2 s [START_REF] Anderson | Instability in the motion of ferromagnets at high microwave power levels[END_REF], and the slope of the FMR curve becomes infinite on the low field side of the resonance, which is experimentally observed at 10.5 GHz for an output power from the synthesizer P th 1 = -33 dBm. Using the FMR line width measured in the linear regime, one gets µ 0 h th 1 = 0.009 mT, i.e., a calibration factor a = 0.4 mT/ √ mW between microwave field and power. To get a better precision, we also fit the dependences on power of the critical fields H down and H up determined experimentally beyond the foldover onset [START_REF] Gui | Foldover ferromagnetic resonance and damping in permalloy microstrips[END_REF], which yields a = 0.41 ± 0.03 mT/ √ mW.

Analytical calculations. The nonlinear FMR excited in a uniaxial system by the superposition of two time-harmonic external fields, h ac (t) = h 1 (t) + h 2 (t) with |h 2 (t)| (e z × m) where h eff = κ eff m z e z + h 0 e z + h ac (t), ṁ is the time derivative of the normalized magnetization vector taken in the rotating frame, h 0 the normalized bias field, e z the unit vector along it, and κ eff the effective anisotropy constant. It is then written in spherical coordinates and considered in the case where only the right circularly polarized component of h 1 is applied. This allows to find its equilibrium points m 0 (P-modes), or equivalently (θ 0 , ϕ 0 ), and to analyze the foldover of FMR [START_REF] Bertotti | Nonlinear magnetization dynamics under circularly polarized field[END_REF]. The standard analysis of the stability of these P-modes allows to calculate the nutation frequency given in Eq.1 [START_REF] Bertotti | Spin-wave instabilities in large-scale nonlinear magnetization dynamics[END_REF]. Shortly, m is expanded around m 0 to linearize the LLG equation, and the complex amplitudes of magnetization perturbations are calculated in the rotating frame by projecting them in the plane (e 1 , e 2 ) orthogonal to m 0 , where sin θ 0 e 1 = (e z × m 0 ) × m 0 and sin θ 0 e 2 = (e z × m 0 ).

Finally, the P-mode linear response to the small additional microwave field h 2 is studied.

Micromagnetic simulations. The magnetization dynamics in the YIG nanodisc is calculated using the python module MicroMagnum, a micromagnetic finite difference simulator which can be runned on GPU [48]. The nominal geometry of the nanodisc (diameter 700 nm, thickness 20 nm) is discretized using a 128×128×1 rectangular mesh. The following magnetic parameters are used: M s = 1.67 (d) Micromagnetic simulations of the experimental data shown in (b). In all the panels, the perpendicular field H 0 is swept down.

• 10 5 A•m -1 , A ex = 4.3 • 10 -12 J•m -1 (exchange length 16

  2e, 2f and 2h, using the amplitude h 1 of the main driving field in the experiments, the angle of precession at the bias field H 0 determined from the normalized foldover shift, cos θ 0 = (H K -H 0 )/M s , and the phase lag which satisfies γµ 0 h 1 sin ϕ 0 = αω 1 sin θ 0 in the macrospin model. It reproduces rather well the experimental data, except in regions where the level of excitation is very large, due to the deviation from the macrospin behavior already observed in Fig.1e. In addition, we have conducted full micromagnetic simulations in the time domain (see Methods), which allow us to extract the nutation frequency from the relaxation of the magnetization towards the steady state regime driven by h 1 , shown in Fig.2c. The obtained results, plotted as red dotted lines in Figs.2e, 2f and 2h, quantitatively agrees with the data on the full range of parameters investigated.

h 2

 2 simply excites the linear Kittel resonance on top of it, which explains the bright linear dispersion ω 2 = γµ 0 (H 0 -M s ) observed in this region. The situation is quite different when H 0 < H K . In this case, h 1 drives the strong foldover regime demonstrated in Fig.1upon sweeping down H 0 , and h 2 excites the magnetization dynamics on top of the corresponding P-mode. The evolution of the two resonance branches symmetrically distributed around the main pumping frequency ω 1 as a function of H 0 is reproduced qualitatively by Eq.1, and quantitatively by micromagnetic simulations. The fact that the upper branch is continuous with the linear Kittel resonance branch observed above H K indicates that the perturbation of the P-mode driven by h 2 has a uniform phase, i.e., corresponds to a uniform nutation of the magnetization.

|h 1 (

 1 t)|, is calculated based on the macrospin LLG equation. The main stages of the analytical derivation presented in the supplementary information are the following. The LLG equation is first written in the frame of reference rotating around the z-axis at the angular frequency ω 1 of the dominant time-harmonic component: ṁ-αm× ṁ = -m×(h eff -ω 1 e z )+αω 1 m×

  nm), γ/(2π) = 28.5 GHz•T -1 •s -1 , and α = 5 • 10 -4 . The bias magnetic field is applied along the normal of the disc. The static equilibrium configuration of the magnetization is calculated at 0.59 T. Then a linearly polarized excitation field of constant amplitude is applied at 10.5 GHz in the plane of the disc, and for each value of the bias magnetic field, which is decreased by steps of 0.02 T, the resulting magnetization dynamics is calculated over 100 ns with a typical step of 3 ps. This allows to reproduce the foldover regime demonstrated in the experiments and to calculate the nutation frequencies. Those are obtained by fast Fourier transformation of the transient dynamics of the average magnetization simulated at each bias field. The nutation mode profiles are obtained by stroboscopically averaging the magnetization dynamics at the corresponding nutation frequencies in the rotating frame.

FIGURESFIG. 1 .FIG. 2 .

 12 FIGURES

FIG. 4 .

 4 FIG. 4. (Supplementary Fig.S1) Gilbert damping of the YIG nanodisc. (a) FMR peak measured at ω 1 /(2π) = 10.5 GHz and P 1 = -38 dBm. A Lorentzian fit to the data yields the full width at half maximum µ 0 ∆H together with the associated error bar. (b) Dependence on excitation frequency of µ 0 ∆H determined in the linear regime. A linear fit to the data yields the Gilbert damping parameter of the YIG nanodisc.

FIG. 5 . 1 (P 1 =

 511 FIG. 5. (Supplementary Fig.S2) Spin-wave nutation modes at larger main pumping power.(a) Spectroscopy map of the SW modes excited by the low power excitation at ω 2 in the absence of the main pumping at ω 1 /(2π) = 10.5 GHz. (b) ∆M 2 spectroscopy map of the SW nutation modes excited by the low power excitation at ω 2 in the presence of the main excitation at ω 1 (P 1 = +8 dBm). (c) Simultaneous measurement of ∆M 1 induced by the main pumping at ω 1 .

ACKNOWLEDGEMENTS

Y.L. thanks the French National Research Agency for funding under contract No. ANR-14-CE26-0021 (MEMOS), V.V.N. the Russian Competitive Growth program of KFU, and C.S. the Labex NanoSaclay for support under the action "Rayonnement International". This research was partially funded by the ANR-18-CE24-0021 (MAESTRO) project.


M Sagnard

R Ecault

F Touchard

M Boustie

L Berthe

Development of the symmetrical laser shock test for weak bond inspection

Keywords:

Laser induced shockwave to certify bonded assembly.

• No NDT currently available to assess bonded CFRP structures integrity.

• Detection capabilities of the S-LASAT to assess bond integrity of contaminated lab samples.

• Behaviour of both healthy and contaminated bonded CFRP under laser shock.

Introduction

The reduction of carbon dioxide emissions is one of today's greatest challenges for the aeronautic industry. Today, numerous directives such as the European Emissions Trading Scheme or organisations such as the International Air Transportation Association (IATA), aim to neutralise the CO2 emission growth [1,[START_REF] Anger | Including aviation in the European emissions trading scheme: impacts on the industry, CO2 emissions and macroeconomic activity in the EU[END_REF].

Lee [START_REF] Lee | Can we accelerate the improvement of energy efficiency in aircraft systems?[END_REF] showed, through a short analysis of current and forthcoming technologies, that a plethora of solutions already exist (alternative fuels, changing cruises altitudes, …). Nowadays, reducing the weight of aeronautic structures is considered as one of the most promising lead for the reduction of CO2 emissions. One outcome of this weight reduction is the wide use of composite structures, because of their good strength-to-weight ratio [START_REF] Marsh | Aero engines lose weight thanks to composites[END_REF]. However, while the materials have evolved, the assembly process did not evolve along with them. Composite materials are currently assembled using rivets or bolts, which implies drilling holes in the panel. As experimentally demonstrated by Di Franco [START_REF] Franco | Influence of the distance between rivets in selfpiercing riveting bonded joints made of carbon fiber panels and AA2024 blanks[END_REF], this type of mechanical bonding creates high local stresses inside composite structures and their use should be limited. To overcome these problems, parts are usually oversized, thus neutralizing composite's advantages on the structure weight.

Assembling composite structures using epoxy bonds has been seen as a good option to avoid drilling holes. The mechanical strength of such assembly is comparable if not better to that achieved with bolting or riveting [START_REF]Handbook of Composites[END_REF] and removes their major drawbacks: oversized parts. If current aircrafts, such as the A350, were to be built only using this technique, the total weight of the plane could be reduced by 15% after structure optimisation [START_REF] Markatos | The effects of manufacturing-induced and in-service related bonding quality reduction on the mode-I fracture toughness of composite bonded joints for aeronautical use[END_REF].

However, it is known that after contamination of one of the bonded surfaces, the overall adherence of the joint can drop drastically. These faulty bonds are often referred to as "weak bonds" [START_REF] Davis | The importance of failure mode identification in adhesive bonded aircraft structures and repairs[END_REF]. There are currently no Non-Destructive Tests (NDT) capable of assessing the integrity of these assemblies, which renders the certification of these parts impossible.

Two ways of addressing this certification problem are currently under investigation. The first group of methods is solely focused on the ⁎ Corresponding author. E-mail address: maxime.sagnard@ensam.eu (M. Sagnard).

T pre-bond analysis of surfaces. They aim at detecting whether an area has been contaminated or not. For example, studies realised by De Vito [START_REF] Vito | Detection and quantification of composite surface contaminants with an enose for fast and reliable pre-bond quality assessment of aircraft components[END_REF] and Helwig [START_REF] Helwig | An optoelectronic monitoring system for aviation hydraulic fluids[END_REF] on opto-electronic systems showed promising results for the detection of surface contaminant such as hydraulic fluids, humidity or release agent residues [START_REF] Tornow | Quality assurance concepts for adhesive bonding of composite aircraft structurescharacterisation of adherent surfaces by extended NDT[END_REF]. The second group of methods is entirely centered on the post-bonded structure. By applying a mechanical force onto the joint, these techniques try to assess the adhesive strength of a bond and compare to a reference sample [START_REF] Malinowski | Mechanical and Non-Destructive Study of CFRP Adhesive Bonds Subjected to Pre-Bond Thermal Treatment and De-Icing Fluid Contamination[END_REF].

This study aims at assessing for the first time, the bond integrity of real contaminated aircraft parts using LAser Shock Adhesion Test (LASAT) in a symmetrical impacts configuration. Our recent papers [START_REF] Sagnard | Development of the symmetrical laser shock test for weak bond inspection[END_REF] has described the technique but no clear differentiation between contamination levels was achieved. Moreover, no comparison with conventional mechanical testing was done. Since, major improvements were realised on both the technique and the testing process.

The first part presents the laser shock, its application to adhesion testing and its symmetrical optimisation. Follows a study using contaminated samples. Its goal is to assess the capabilities of the technique at detecting weak bonds created using different contaminants. Eventually, this article describes an experiment realised on real aircraft parts that have been partially contaminated. This work has been realised thanks to the European project ComBoNDT [START_REF] Schlag | [END_REF], anchored in the H2020 framework.

The laser shock adhesion test (lasat)

The LASAT uses a laser pulse to generate shock waves within a test sample. Upon reaching the surface of the specimen, the laser creates a dense plasma through the vaporisation of a few micrometres of material. By expanding, the high-pressure plasma transmits part of its energy to the sample in the form of shock waves. Fig. 1(a) depicts this laser shock phenomenon on a bonded Carbon Fibre Reinforced Polymer (CFRP) sample. The sacrificial layer (in grey) is an aluminium adhesive tape (35 µm) which serves two main purposes. It primarily protects the sample from being partially vaporised, instead, it is the aluminium which is sublimated to generate the plasma. Secondly, the pressure generated by this technique highly depends on the laser/matter interaction. By always using the same sacrificial layer, one ensures the interactionand thus the loadingis the same regardless of the sample material.

The purpose of the confinement material is to increase the generated pressure within the material. It has been shown that such configuration can help reach up to four times the pressure created without any constraining medium [START_REF] Berthe | Study of plasma induced by laser in water confinement regime: Application to laser shock processing with and without thermal protective coating[END_REF]. For the coupon samples, water was used as a confinement medium.

The space/time diagram Fig. 1(b) describes the shock wave's behaviour within the material. The front of the shock wave is represented using solid lines. The release wave, that sets the material back to its initial state, is represented using dotted lines. The shock wave travels all the way to the back-face of the sample and turns into a release wave. Upon crossing the initial release wave, the reflected shock wave generates a high tensile stress (full circle). For a given material, the position of this high tensile stress simply depends on the shock duration which is defined by the laser's pulse duration ΔT.

The capability to test interfaces' adherence has already been shown for both aluminium [START_REF] Gilath | Material response at hypervelocity impact conditions using laser induced shock waves[END_REF] and composite structures [START_REF] Ecault | Experimental and numerical investigations of shock and shear wave propagation induced by femtosecond laser irradiation in epoxy resins[END_REF]. However, the technique in its current state lacks modularity. If the tested interface was not near the back of the sample, the maximum tensile stress would not be optimally located. A previous study on contaminated joints showed that for a strong enough bond, it was not possible to detect the weak bond without damaging the composite [START_REF] Ecault | Observation of the shock wave propagation induced by a high-power laser irradiation into an epoxy material[END_REF].

The Symmetrical LAser Shock Adhesion Test (S-LASAT [START_REF] Boustie | Method of non-destructive testing comprising the generation of a state of localised and controlled traction in a multi-material and/or multi-layer assembly[END_REF][START_REF] Ecault | Experimental and numerical investigations on the dynamic behaviour of aeronautic composites under laser shock -Optimization of a shock wave adhesion test for bonded composites[END_REF]) is proposed to overcome this limitation. This technique consists of generating shock waves from both sides of the tested specimen. As for the single shot setup, high tensile areas are generated on each opposite side of the sample (dotted circles in Fig. 2(a)). However, a higher tensile stress is created at the intersection of the two reflected shock waves (solid circle) and its location does not exclusively rely on the pulse duration.

The high tensile stress location can be shifted by applying a time delay between each laser pulse (Fig. 2(b)). The required time delay can be calculated using Eq. (1):

∑ ∑ = × - × T t l ρl Zl tr ρr Zr Δ i i i i i j j j (1)
where the indices r and l refer to the material respectively on the right and on the left of the area where the maximum tensile stress should be located, t li is the thickness of ply number i, ρ li the density of the ply, and Z li its shock impedance. Ghrib [START_REF] Ghrib | Generation of controlled delaminations in composites using symmetrical laser shock configuration[END_REF] has shown that by using this time delay, one can generate the maximum tensile stress in a specific location of the material, and for example generate a delamination in between two selected plies inside of a laminate structure. This technique has not yet been used to assess bonding quality on real aircraft parts.

However, if the LASAT can differentiate a weak bond from a healthy one, an additional NDT is required to reveal the outcome of the test to the user. Ultrasounds scanning was chosen for its ease of use and strong capability to spot defects such as bond openings.

A standard LASAT procedure starts with the definition of the threshold of a healthy sample. Once isolated, a fraction of the threshold energy is used to test the real panel. For example, in the case of bond assessment, anything below 80% of the specification's adhesion level will be considered as too weak. If a damage is spotted using ultrasounds, the part will have failed the test, the bond must be redone. Else, the adhesive capacities of the bond are validated, and the part can be used as is. The LASAT is only destructive for parts that do not meet the technical requirements.

Experimental material

Experimental setup

A Nd:Yag laser is used to generate a pulse of 7 ns which can deliver up to 2x7J. To create a symmetrical shot, the Gaïa laser used on the Hephaïstos platform is composed of two lasers which beams are polarised by 90°from one another. Thanks to a polariser, two distinct beams, referred to as Beam A and Beam B, are created and transported through optics to both sides of the tested sample (Fig. 3(a)).

Compared to the last work [START_REF] Sagnard | Development of the symmetrical laser shock test for weak bond inspection[END_REF], Diffractive Optical Elements (DOE) were added on both beam paths. This optics makes for a better control of the spatial repartition of the pressure on top of the sample. These serve to break down the initial laser beam, and reform it to assure a good homogeneity of the laser at the focal spot. Fig. 4(a) shows the laser repartition, at the focal spot, with and without DOE. One can a wide important intensity variation before the DOE ( ± 34% compared to the mean value). Moreover, this variation occurred over the whole spot area. After these optics were included in the experimental setup, a much smoother repartition can be observed (Fig. 4(b)), very close from a top-hat signal, with only a small and very localised intensity variation.

Prior to the shot, the tested area was protected using an aluminium adhesive tape, and the zone was confined using a water trickle to make sure at least 1 mm of water covered the sample during the test (Fig. 3(b)).

Coupon samples

The samples are made of two bonded Carbon Fibre Reinforced Polymers (CFRP) laminates. Each laminate is made of 8 layers [90/0/ 45/-45] S of Hexcel IM7/M21E for a total thickness of 1.5 mm. Epoxy resin is used for both the bond and the laminate. Two types of assemblies were created for this study: production type samples (P) bonded with 200 µm of FM300-K cured at 444°K and repair type samples (R) bonded with FM300-2M cured at 394°K. The FM300-K has an integrated knit carrier to facilitate its manipulation in the production facilities. This net is not present inside the FM300-2M bonds. Moreover, the samples have been prepared according to Airbus specifications: the repair samples were grinded all the way to the fibres before bonding. This step is not required when creating production type parts.

In this paper, two contaminations are studied: the finger print (FP) contamination for the production samples and the de-icing fluid (DI) contamination for the repair ones. Each contamination was realised at three given levels, low, medium, and high, which respectively represents a loss of 5%, 10% and 20% of adherence. For example, a repair sample contaminated with de-icing fluid will be referred to as R-DI-1, for the lowest level, R-DI-2 for the medium and R-DI-3 for the highest.

A standardised finger print solution (DIN ISO 9022-12) was used to mirror a FP contamination. It contains sodium chloride, urea, ammonium chloride, lactic acid, acetic acid, pyruvic acid, and butyric acid. The solution was diluted to obtain the desired adherence loss: low contamination was achieved by mixing 1 volume of solution with 9 volumes of demineralised water, the medium contamination was a 50/ 50 ratio, and the highest contamination was pure finger print solution.

The de-icer used was the Safeway KF developed by Clariant. The solution was diluted in demineralised water and applied, using a deepcoating method, with the following concentration (%vol): 2%, 7% and 10%. After pollution, the laminate was dried in a 40 °C oven for 2 h and let dry an addition 24 h at room temperature.

By definition, weak bonds cannot be spotted using standard NDTs. Thereby, to assure the joint is representative of a weak bond, ultrasound scans were realised after the bonding process. If initial flaws such as delamination or cracks were spotted, the sample would not be considered.

Each sample was also mechanically tested at Patras University [START_REF] Moutsompegka | Experimental study of the effect of pre-bond contamination with de-icing fluid and ageing on the fracture toughness of composite bonded joints[END_REF][START_REF] Moutsompegka | The effect of prebond contamination with fingerprint and ageing on the fracture toughness of composite bonded joints[END_REF] using a double cantilever beam (DCB) test to assess the mode-1 fracture toughness (GIC) and an end-notched flexure (ENF) test for the mode-2 fracture toughness (GIIC). Combined with ultrasound scanning, mechanical tests were a cross validation of the creation of weak bond within the coupon samples.

Fig. 5(a) summarizes the mechanical test results obtained for the production reference, P-RE, sample (blue) and the finger print contamination (orange), and Fig. 5(b) for the repair reference, R-RE, sample (blue) and de-icer contamination (orange). To ease later comparison with the LASAT, all values have been normalised by the reference sample value. Hence the de-icer results have been normalised using the sound repair sample threshold, and the finger print one by the reference production sample.

Both the GIC and GIIC tests show a general decrease of the bond strength after the contamination. However, because the range of GIC tests' uncertainty bars for a sound and a contaminated sample were overlapping, no direct conclusion could be drawn out. GIIC tests were more consistent and confirmed the loss of adherence/mechanical strength of the contaminated samples.

S-LASAT results on the coupon samples

Threshold determination

A specific threshold definition procedure was used to assess the bond strength of each coupon. Each coupon was first scanned to verify no damage was inherently present in the sample and an aluminium adhesive was applied onto the shot surface. The specimen was placed in a pre-design holder to assure the tested area would always be the same from a shot to another.

After each shot, the laser spot was moved to a different area and the intensity was increased, until a default was generated within the sample. However, this technique heavily relied on the good homogeneity of the sample, and this hypothesis could not be applied to certain local contamination such as the finger print one. Hence, to avoid heterogeneity problems and edge effects, a single area was tested in the middle of the sample. After each shot, the coupon was removed from the specimen holder, the adhesive tape discarded, and a second ultrasound scan was performed to assess the state of the sample. If no default was spotted, the laser intensity would be increased and the whole process started again, on the same location. The only variable input is the laser intensity, the final result being the sample opening threshold (Fig. 6). Water trickles were generated on both sides of the sample to properly confine the shot.

A laser was focused to a spot of 8 mm, allowing laser flux up to 1.3 GW/cm 2 per beam. Starting from 0.11 GW/cm 2 per beam, the laser intensity was increased by steps of 0.15 GW/cm 2 . The coupon samples where symmetrical, so no time delay had to be calculated in this case.

For ultrasonic testing, a 32 elements linear probe was used, with a frequency of 5 MHz. CFRPs act as a low-pass filter, limiting the probe scanning frequency. The sample is composed of three different layers: a CFRP laminate, the adhesive bond, and the second CFRP laminate. Each one of these layers has a specific material impedance that is detected by ultrasounds. Fig. 7(a) is a typical signal obtained for a sound sample. The upper signal is a A-Scan. It is a 1D local measurement and each spike (marked 1, 2 and 3) is due to an impedance modification in the scanned part. In this case, 1 is the signal obtained when the ultrasonic wave first encounters the sample, 2 is the signature signal of the adhesive bond, and 3 is the signal obtained when ultrasounds reach the back of the sample. The lower image, called B-Scan, is a 2D cross-section cartography of the sample.

To account for the material attenuation, and to achieve the same amount of signal signature for all three spikes, the signal was numerically compensated using a Time Correcting Gain (TCG) method.

Ultrasounds cannot go through empty area, typically here disbondings. The signal is blocked, and little to no information is available after the opening. With A-Scan, it can be spotted by a loss of amplitude of one of the echoes. For example, the A-Scan from Fig. 7(b) shows a loss of signal from the back-face echo (3) as well as a widening of the bond echo. The B-Scan confirms the damage location on the bond, and thus, the occultation of the signal below it. Each time the sample was scanned from both sides to assure no defaults were hidden by the damaged bond.

Results for the finger print and the di-icer contaminations are presented in Fig. 8. To confirm the ability of the S-LASAT to spot a specific contamination, the measured energy required to open the faulty bond must be lower than the one obtained for the reference sample. Since data has been normalised, a contamination correctly isolated must be below the 1.0 mark (represented by the red line entitled "detection threshold" in Fig. 8).

The experimental setup reduced the uncertainty of the measure to only the laser variations: there are no material uncertainties since the same spot is tested from one shot to another, the beam path is always the same and the aluminium adhesive as well as the water trickle are set up in a systematic way. For a given intensity, there is less than 0.1% of flux variation, which renders it invisible on the graph.

As the GIC and GIIC test showed, the contamination of the sample modified the adherence strength of the sample. The S-LASAT showed the same results: a contaminated bond requires less laser intensity loading to be opened than a healthy one. The same observation was made for both contaminations.

In the case of the finger print sample, the technique did not differentiate the three levels of contamination. However, unlike standard mechanical tests that apply a constant load increase, the LASAT was performed using a load step, limiting the results' accuracy. A smaller intensity step size could give more precise results. However, even with Fig. 6. Block scheme of the experimental procedure. such intervals, the technique was able to differentiate the second and third level of the de-icer contamination.

In comparison with the results obtained with the previous setup [START_REF] Sagnard | Development of the symmetrical laser shock test for weak bond inspection[END_REF], a clear difference is now made between a reference sample and the contaminated one. The new method to define threshold used to avoid heterogeneity greatly improved the results for the finger print contamination.

Put into perspective, these two studies show the level of control for the process to properly function. However, once reached, this degree of mastery allows for a good detection of weak bonds.

Damage analysis

Important differences have been spotted however when comparing the damage patterns of the repair and production reference samples. In the production reference sample, the disbonding seemed to propagate mainly between the epoxy from the pre-impregnated CRFP and the bond (Fig. 9(a)). However, cracks appeared primarily in the centre of the bond for a repair sample (Fig. 9(b)). The difference of bonding surfaces (CFRP fibres vs epoxy) could explain this variance as well as the presence of the knit carrier within the production bond.

Following the same study process, micrographs where realised on both sets of contaminated samples. The damage inside the FP contaminated samples was similar to the one observed before: creation of a disbonding between the epoxy from the pre-impregnated CFRP and the bond. However, no clear crack pattern was found for DI samples, instead microcracks were observed. This type of defect is harder to detect using linear ultrasounds and may not be spotted in the case of a lower contamination. A closer study of the specimen also revealed particles probably from the Safeway KF (Fig. 9(c)).

For all samples, the damage location as well as the particles were mainly found towards the contaminated area, and right below the area where the laser shock was done. No delamination was found inside the CFRP laminate, confirming the observations made by ultrasounds. Fig. 9. Photomicrographs of the bond after damage created by S-LASAT: (a) production reference sample [START_REF] Sagnard | Development of the symmetrical laser shock test for weak bond inspection[END_REF], (b) repair reference sample [START_REF] Sagnard | Development of the symmetrical laser shock test for weak bond inspection[END_REF], and (c) highest degree of de-icing fluid contamination.

Application on real aircraft parts

Parts' geometry

A350 panels have been produced/recovered to test the technique using real parts. The production sample is an 800 × 800 × 1.5 mm CFRP plate on which two 700 × 50 × 1.7 CFRP stiffeners were bonded (Fig. 10). The plate is made of 14 prepreg layers with the following layup: [+45/90/0/90/+45/0/90] s . On one half, the stiffeners have been bonded without contamination, and on the other half of the plate both the de-icing fluid and the finger print solution have been applied on the stiffeners' surface before the bond. The goal will be to assess the quality of the bond holding the stiffeners to the CFRP plate. Only the upper stiffener has been tested.

The repair panel has been cut out of a A350 structure. The middle part of the sample has been grinded all the way to the fibres in a conic shape. Half of the surface was contaminated with both finger print and de-icing fluid solutions (Fig. 11(b)). The hole created in the panel was then patched, starting from the centre to the edge, with the missing CFRP ring (cf. Fig. 11(b)) while keeping the original lay-up of the part (ring diameter and orientation in Fig. 11(a)). The composite was then cured by GMI's tool ANITA EZ0901 hot bonder.

This part had hollow stiffeners on one side, making it impossible to test in this area using a symmetrical shot configuration. It is represented by the dark bar on Fig. 11(a): the darker section is the hollow part and the lighter sections on either side show the areas where the stiffener is assembled onto the panel. Thus, there is a total of three different areas on the patch. The first one only features the repair patch. The S-LASAT can be applied normally (Fig. 11(b). The second area above the hollow part was tested using the single shot setup since only one face was available. The third one features both the patch and the zone where the stiffeners have been bonded. These areas then have two bonds: the first one between the plate and the stiffener, and a second one between the plate and the repair patch. S-LASAT can be used, but the extra thickness coming from the stiffeners must be considered.

Because of the way the plate was grinded, the depth of the bond varied depending on the tested area. When the symmetrical shot was possible, the time delay between each beam had to be recalculated using Eq. ( 1), to assure a proper positioning of the maximum tensile strength. It varied from 0 ns to 1290 ns.

Ultrasound scans were performed by Airbus (Toulouse, France) prior to the test. The production panel did not show any default whereas the repair panel was already porous, regardless of the presence or not of contamination on the surface.

LASAT tests

These parts were too big to fit into the initial experimental setup. A robot has been installed to move the piece around, and the optics had to be rearranged (Fig. 12(a)). The water confinement was not feasible either. Instead, a solid constraining material in the form of a thick transparent adhesive layer was used. Its confinement capabilities are comparable to the water in the intensity range 0-3 GW/cm 2 . Tests on the coupon sample have also validated this choice. The sacrificial layer is the same as before. Fig. 12(b) is a picture of the production panel. Two bands of sacrificial aluminium tape were placed on the stiffener. They are covered by the transparent confinement tape. The "dots" are areas that have been test using S-LASAT. This panel is ready to undergo a full ultrasonic scan to find the opened areas.

The threshold for a healthy bond was measured on a not contaminated area. For the repair panel two thresholds were required, one for the symmetrical setup and one for the single shot setup. To simulate a real application, both panels were shot with an energy equivalent to 80% of that required to open a healthy bond. If an opening is detected using ultrasonic scans, the area will be defined as "faulty,", else the zone is validated. Results are summarised in Fig. 13. The results for the symmetrical shots on both the production and the repair panel are regrouped in the green section while the ones for the single shot setup are in the blue section. For each panel the reference bond threshold per setup is specified. For a given panel, results are divided into two "areas": the contaminated one, and the one where the bonding was done properly. The number of tests realised at 80% of the threshold is specified and should be compared to the number of opened bonds. For the technique to be validated, none of tests realised over the healthy area of a panel should have resulted in an opening, while all those done on a contaminated zone should have shown signs of opening.

Tests on the repair panel revealed that the single shot setup is not appropriate for this kind of structure. Only 20% of the shots on the contaminated surface revealed a loss of bond adherence. With an unoptimized technique, and such low contamination and adherence loss, the detection was not consistent enough to be used for certification purposes. However, 100% of the contaminated areas tested with S-LASAT were open and none of the correct bonds were damaged.

100% of detection was achieved on the production panel. None of the sound areas were open while ultrasounds revealed that the totality of the contaminated areas failed the test. These results showed the need of the symmetrical shot to properly test a sample.

Conclusion

In this paper, a Symmetrical Laser Shock Adhesion Test (S-LASAT) was applied on several contaminated CFRP assemblies to assess the weak bond detection capabilities of the technique. Two contaminations have been studied: de-icing fluid and finger print.

For coupon samples, two different reference thresholds were obtained: 0.99 GW/cm 2 per beam for production and 1.14 GW/cm 2 for repair. Each contaminated sample showed lower thresholds than their associated reference, validating the capacity of the technique to differentiate a healthy sample from a faulty coupon, given these contamination levels. Moreover, in the case of the de-icing fluid contamination, the technique also proved to be able to differentiate the first level of contamination for the other two.

When a bond was opened using S-LASAT, no damage was spotted within the laminated structures. This could not have previously been achieved using the non-optimised technique [START_REF] Ecault | Experimental and numerical investigations of shock and shear wave propagation induced by femtosecond laser irradiation in epoxy resins[END_REF]. Moreover, the addition of DOE in the experimental greatly increased the detection capabilities, and well as the robustness of the technique [START_REF] Sagnard | Development of the symmetrical laser shock test for weak bond inspection[END_REF].

Post-mortem observations showed different types of damage, depending on both the type of sample (production or repair) and the contamination.

A full-scale test on real aircraft parts was conducted. For the production panel, two stiffeners were bonded onto a panel. The repair part was a real A350 panel that was salvaged and fixed. Both parts were locally contaminated before the bonding.

All contaminated areas from the repair panel tested with S-LASAT have been spotted. However, due to geometric limitations, parts of the repair panel were tested with the single LASAT. Results with this technique were not good enough and further illustrated the high degree of precision required for the technique to be used efficiently to detect bond with very little adherence loss.

The production could be fully tested using only the symmetrical configuration. A 100% detection rate was achieved.

Further studies are now required to better understand the influence of chemical or physical bond contamination on its adherence properties. A better understanding of these phenomena could help to better master the technique and improve the results.

To overcome certain geometrical limitations such as the one faced when studying the repair panel, a double shock-based test is currently under study and has shown good results. It relies on two laser pulses, shot at the same side of the sample. This technique also allows a good localisation of the maximum tensile stress [START_REF] Ecault | Experimental and numerical investigations on the dynamic behaviour of aeronautic composites under laser shock -Optimization of a shock wave adhesion test for bonded composites[END_REF].

Fig. 1 .

 1 Fig. 1. (a) Representation of a typical LASAT experimental setup and (b) Space-time diagram of a single shot LASAT.

Fig. 2 .

 2 Fig. 2. Time/space diagram of S-LASAT shots: (a) without time delay and (b) with a time delay.

Fig. 3 .

 3 Fig. 3. (a) Schema of the experimental setup. (b) Close up on the sample setup.

Fig. 4 .

 4 Fig. 4. Spatial repartition of the laser intensity on beam A (a) before DOE were included in the experimental setup and (b) after DOE were included in the experimental setup.

Fig. 5 .

 5 Fig. 5. Mechanical tests for: (a) the production reference sample (P-RE) and the three levels of finger print contamination, low (P-FP-1), medium (P-FP-2) and high (P-FP-3) -(b) the repair reference sample (R-RE) and the three levels of de-icing fluid contamination, low (R-DI-1), medium (R-DI-2) and high (R-DI-3) [22,23].

Fig. 7 .

 7 Fig. 7. A-Scan and B-Scan of the bonded CFRP assembly: (a) no damage (b) with damage.

Fig. 8 .

 8 Fig. 8. Normalised results comparison between S-LASAT and the mechanical tests for: (a) the production reference sample (P-RE) and the three levels of finger print contamination, low (P-FP-1), medium (P-FP-2) and high (P-FP-3) -(b) the repair reference sample (R-RE) and the three levels of de-icing fluid.

Fig. 10 .

 10 Fig. 10. Production panel: (a) front view with laser shot position and (b) side view with specified thicknesses.

Fig. 11 .

 11 Fig. 11. (a) figure of the shot pattern realised on the repair panel and (b) cross-section of the repair patch geometry.

Fig. 12 .

 12 Fig. 12. (a) New experimental setup with the repair panel placed on the robot and (b) one of the stiffener of the production panel after laser shots.

Fig. 13 .

 13 Fig. 13. Results obtained for the production and the repair panels, for both S-LASAT (green) and single shot setups (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Acknowledgment

This project has received funding from the European Union's Horizon 2020 Research and Innovation Program under grant agreement N°636494.