Gabriel Jarry
email: gabriel.jarry@enac.fr

Daniel Delahaye
email: daniel.delahaye@enac.fr

Florence Nicol
email: florence.nicol@enac.fr

Eric Feron
email: feron@gatech.edu

Aircraft Atypical Approach Detection using Functional Principal Component Analysis

Keywords: Flight Path Safety Management, Atypical Flight Event, Non-Compliant Approach, Functional Principal Component Analysis, Unsupervised Learning, Anomaly Detection

Airports Terminal Maneuvering Areas (TMA) and Control Traffic Regions (CTR) are characterized by a dense air traffic flow with high complexity. In nominal operations, approach flight path safety management consists in procedures which guide the aircraft to intercept the final approach axis, and the runway slope with an expected configuration in order to land. Some abnormal flights are observed and considered as Non Compliant when the intermediate and the final leg intercepting conditions do not comply with the prescription of the operational documentation. This kind of trajectories generates difficulties for both crew and Air Traffic Control (ATC) and may induce undesirable events such as Non Stabilized Approaches or ultimate events like Control Flight Into Terrain (CFIT), in the worst cases. There is a real need for atypical flights detection tools in order to improve safety. In this paper, a post-operational detection method based on functional principal component analysis and unsupervised learning will be presented and compared to geometric features.

I. INTRODUCTION

A. Operational Motivations

Approach and landing accidents (i.e. accidents that occur during initial approach, intermediate approach and landing) represent every year 50% of total hull losses and 55% of fatalities. Moreover, a great majority of accidents presents significant atypical events from nominal approaches such as atypical speed or atypical altitude [START_REF] Tremaud | Getting To Grips With ALAR[END_REF], [START_REF] Jackman | Nearly Half of Commercial Jet Accidents Occur During Final Approach[END_REF]. In addition, Airports Terminal Maneuvering Areas (TMA) and Control Traffic Regions (CTR) are characterized by a dense air traffic flow with high complexity. This complexity will surely increase since IATA forecast a growth of air passenger worldwide from around 4 billion today, up to 7.8 billion in 2036 [START_REF] Iata | 2036 Forecast Reveals Air Passengers Will Nearly Double to 7.8 Billion[END_REF]. Consequently, there is a crucial need for aircraft atypical approach detection.

To respond to the International Civil Aviation Organization (ICAO) safety requirement, the French Civil Aviation Authority has launched since 2006 a national safety program, which for the time being, is divided into two State Safety Program (SSP) published for the period 2009-2013 [START_REF] Welterlin | Plan d'action Stratégique d'amélioration de La Sécurité, 2009-2013[END_REF] and 2013-2018 [START_REF] Welterlin | Plan d'action Stratégique d'amélioration de La Sécurité, Horizon 2018[END_REF]. A SSP for the period 2018-2023 is currently being published. The risk portfolio [START_REF] Welterlin | Cartographie Des Risques, PSE 2009-2013 -Version 2[END_REF] distinguishes undesirable events such as Non Stabilized Aproaches (NSA), from ultimate events such as Control Flights Into Terrain (CFIT) or mid-air collisions. Undesirable events may lead to final events and therefore jeopardize safety or reduce airfield capacity. Their identification and detection is an important issue.

In nominal operations, flight path safety management consists in procedures which guide the aircraft to intercept the final approach axis, and the runway slope with an expected configuration in order to land. A particular undesirable event called Non Compliant Approach (NCA) was defined in the second version of the 2008-2013 safety program risk portfolio [START_REF] Welterlin | Cartographie Des Risques, PSE 2009-2013 -Version 2[END_REF]. An approach is considered not compliant when the intermediate and the final leg intercepting conditions do not comply with the prescription of the operational documentation. It may occur during radar vectoring or not, and for visual or instrument approaches. A NCA is a potential precursor of NSA [START_REF] Vernay | Defining a Compliant Approach (CA): A joint response to enhance the safety level of approach and landing[END_REF]. A stabilized approach is one in which the pilot establishes and maintains a constant angle glide-path, an approach speed and an aircraft configuration towards a predetermined point on the landing runway.

Geometrical criteria with horizontal and lateral margins from the nominal path were defined to distinguish a compliant from a non-compliant approach. In particular, interception chevrons were created. They define a 45°maximum angle of procedure radial interception. This angle may be reduced to 30°in specific situations such as dependent parallel runways. Besides, a flight is expected to attend a 30 second level-off flight during the intermediate leg before descending on runway slope in order to reduce speed and to configure properly for landing. Figure 1 illustrates those criteria. For example, NCAs were found in different accidents such as the Air Nostrum flight 8313 on July, 30th 2011, where the aircraft suffered structural damages following hard landing at Barcelona Airport [START_REF]Accident Involving a Bombardier CL-600-2b19 (CRJ200), Registration EC-ITU, Operated by Air Nostrum[END_REF]. Peeks of descent rate above 3000ft/min were recorded and the aircraft flew over threshold at 315ft, where nominal Reference Datum Height (RDH), i.e. the nominal height above threshold on-glide is around 50ft. Another example is the crash of Asiana Airline flight 214 of July, 6th 2014 at San Francisco Airport, which counted 3 fatalities and 185 injuries [START_REF] Hart | Descent Below Visual Glidepath and Impact With Seawall Asiana Airlines Flight 214 Boeing 777-200er[END_REF]. The airplane was recorded with a very low speed in final approach, and finally stalled before crashing.

B. Previous Related Works in ATC

Compliance criteria were applied to flight operations to give a state of the current situation at Paris Charles-De-Gaulle (CDG) Airport between February and August 2014. The module NCA of the French Civil Aviation Authority tool called ELVIRA [START_REF] Lugardon | Manuel Utilisateur Elvira Qt[END_REF] was used. This module is a postoperational analysis tool that studies radar trajectories and describes their compliance. Over this period, 22% of flights were detected as non-compliant with approximately 2% being significant [START_REF] Avram | Non-Compliant Approach Study[END_REF]. It implies that the definition of compliance could be improved since a large majority of detected noncompliant flights, do not present significative safety issues. Too many false non-compliant alarms may occur, which is troublesome for Air Traffic Control (ATC) operations. Besides, there is a real lack of energetic features. This study led to the identification of different contributing factors and bias for NCAs such as extra energy owing to overspeed or downwind in final approach, or the influence of the QNH during the operations. Specific atypical situations called Glide Interception From Above (GIFA) were pointed out. These situations are particularly critical owing to the potential difficulties to manage the aircraft energy and because aircraft are nether designed nor certified to intercept glide slope from above [START_REF] Vincent | Compte Rendu Meeting -Glide Interception From Above[END_REF], [START_REF]EANPG Programme Coordinating Group, Sixty Second Meeting, Presentation, Facilitating Stabilized Approaches[END_REF].

To improve the safety and decrease the number of GIFA an online detection tool was set up at CDG Airport and used by ATCs on real time. It consists in four 3D-volumes using the Area Proximity Warning (APW) described in Figure 2. The first three volumes are warning volumes, the ATCs advise pilots that they are too high on glide. The final volume is a decision volume, where the ATC and pilots must take the decision to continue or to interrupt the approach.

The results of the experiments are positive since GIFAs are detected and an appropriate response is now taken. Approximately 5 flights over 700 per day rise an alarm and in about half cases, ATCs suggest a recovery slope as recommended [START_REF]EANPG Programme Coordinating Group, Sixty Second Meeting, Presentation, Facilitating Stabilized Approaches[END_REF].

Our work consists in enhancing compliance criteria and developing methodologies to detect NCA and atypical flights post-operatively and online. This paper presents new criteria that extend those defined in the NCA module of ELVIRA and defines off-line methodologies based on the specific total energy of the aircraft.

C. Functional Data Analysis Approach

Functional Principal Component Analysis (FPCA), is a powerful mathematical tool from Functional Data Analysis (FDA). FDA consists in studying a sample of random functions generated from an underlying random variable [START_REF] Ramsay | Functional Data Analysis[END_REF]. They deeply evolved during the 2000s with Ramsay et Silverman [START_REF] Ramsay | Functional Data Analysis[END_REF]- [START_REF] Ramsay | Functional Data Analysis with R and MATLAB. Use R![END_REF]. Other theoretical and applied aspects like regression or clustering were published by Ferraty and Vieu [START_REF] Ferraty | Nonparametric Functional Data Analysis: Theory and Practice[END_REF], [START_REF] Ferraty | Recent Advances in Functional Data Analysis and Related Topics[END_REF].

The applications of FDA are numerous. In [START_REF] Ullah | Applications of functional data analysis: A systematic review[END_REF] Ullah et al. state many applications and underline its multidisciplinary purpose. In particular, FDA is used in various research fields such as medecine, biomedical, biology, finance, demography. In aeronautics, FDA is also widespread. Gregorutti [START_REF] Gregorutti | Forêts Aléatoires et Sélection de Variables: Analyse Des Données Des Enregistreurs de[END_REF] uses data from flight recorder to develop a prediction tool for long and hard landings. The tool mixes FDA and wavelets decomposition with machine learning like random forests. During his PhD a software was developed and is now commercialised by a company called SafetyLine. Suyundikov [START_REF] Suyundykov | Multivariate Functional Data Clusterization by PCA in Sobolev Space Using Wavelets[END_REF] presented a multivariate functional data clustering from trajectories using FPCA in Sobolev spaces. Hurter et al. [START_REF] Hurter | Functional Decomposition for Bundled Simplification of Trail Sets[END_REF] developed a bundling algorithm for radar trajectory visualization based on a smoothing splines decomposition and FPCA. Tastambekov [START_REF] Tastambekov | Aircraft Trajectory Prediction by Local Functional Regression[END_REF] developed an aircraft trajectory predictor based on local functional regression with wavelet decompostion and a k-mean clustering algorithm. Nicol [START_REF] Nicol | Statistical Analysis of Aircraft Trajectories: a Functional Data Analysis Approach[END_REF] applied FPCA to study the underlying mode and pattern of variation of aircraft trajectories. Barreyre et al. presented a novel outlier detection tool in functional data [START_REF] Barreyre | Multiple testing for outlier detection in functional data[END_REF], and a statistical outlier detection [START_REF] Barreyre | Statistical Methods for Outlier Detection in Space Telemetries[END_REF] for space telemetries, based on wavelet decomposition and principal component analysis. Finally, Yan et al. [START_REF] Yan | Functional principal components analysis on moving time windows of longitudinal data: dynamic prediction of times to event[END_REF] proposed to apply FPCA to a sliding window for dynamic prediction of longitudinal biomarker data, in order to enhance performance robustness.

In this paper we will propose a method to detect atypical approaches by applying recursively on a sliding window the following process. First, reduce the dimensions using a FPCA on total energy trajectories. Then, apply a hierarchical clustering and finally give a compliance score from outlier detection.

Our paper is divided in four parts. In the first part, mathe-matical backgrounds around functional data analysis and data clustering will be presented. In the second part, we will present different features that extend the actual geometric criteria to detect non-compliant approaches. Then, our atypical approach detection method will be presented. Finally, the method will be illustrated on real data and specific operational situations.

II. MATHEMATICAL BACKGROUNDS A. Functional Data and Functional Principal Component Analysis

In FDA, data are functions, and methoods focus on the statistical analysis of a sample of curves. In practice, we observed a discretization of these functions on a grid of time locations. For this reason, the first step of FDA consist in recovering the functional nature of curve data from discretised curve data by using a decomposition on a functional basis. There are different basis function systems in the literature like polynomial basis, Fourier basis, wavelet basis or smoothing spline basis [START_REF] Ramsay | Functional Data Analysis with R and MATLAB. Use R![END_REF]. Then, most of the multivariate statistical method may be extended to the functional setting, such as the Principal Component Analysis (PCA), in order to identify the most important source of variation in the sample of curves and cut down the complexity of the data.

In our context, approach trajectories are functional in nature, mapping a time interval to a state space R d . In this paper, such curve data are discretely recorded by radar every 4 seconds. Trajectories are observed on a time interval [0, T i ], which can be different for each trajectory.

PCA is a powerful statistical method that summarizes a significant amount of data information by creating new variables as the linear combination of existent variables. It is an orthogonal projection that concentrates the majority of the variance in the first components. It enables simpler representation and analysis of complex or even large dimension variables. In practice, data are projected over the eigen basis of the covariance matrix of the observations ordered by decreasing eigen values.

PCA was extended to the functional case called FPCA by Deville [START_REF] Deville | Méthodes Statistiques et Numériques de l'analyse Harmonique[END_REF] and Dauxois [START_REF] Dauxois | Les Analyses Factorielles En Calcul Des Probabiblités et En Statistique: Essai d'étude Synthétique[END_REF], [START_REF] Dauxois | Asymptotic Theory for the Principal Component Analysis of a Vector Random Function: Some Applications to Statistical Inference[END_REF]. When data are functions sampled from an underlying stochastic process, FPCA enables dimensionality reduction by estimating a truncated Karhunen-Loève decomposition. Therefore, when the principal components are determined, the trajectories can be represented by their decomposition coefficients on the principal component basis and considered as a small dimension vector.

In this paper, we will not use a grid of time location but a grid of curvilinear distance to the runway threshold location. The curvilinear distance is the distance flown by the aircraft along the trajectory. Indeed, to enable consistent comparisons between flights in the approach phase, flights must be compared regarding their curvilinear distance to the threshold rather than flight time since aircrafts do not operate at same speeds. Besides, in FPCA the entire interval is usually studied. However, in our methodology we propose to recursively apply the whole process on smaller intervals with a sliding window in order to give a local non-compliance score.

B. Introduction to Data Clustering

Data clustering, consists in grouping similar data samples together into subsets. The inputs are unlabeled data, and the idea is to find underlying information to classify those data [START_REF] Hastie | The Elements of Statistical Learning: Data Mining, Inference, and Prediction[END_REF].

A way to perform a data clustering is to solve an optimization problem that minimises the intra-class variance and maximises the inter-class variance over the possible clusters. In our paper, a clustering algorithm called Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDB-SCAN) will be used. This algorithm extends the DBSCAN clustering method [START_REF] Ester | A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise[END_REF] by converting it into a hierarchical clustering algorithm. It finally extracts a flat clustering based on the stability of clusters [START_REF] Campello | Density-Based Clustering Based on Hierarchical Density Estimates[END_REF]. The algorithm is divided into five steps. It first transforms the space according to the density. Secondly, it builds the minimum spanning tree of the distance weighted graph. Thirdly, a cluster hierarchy of the connected components is constructed. Then, the cluster hierarchy based on minimum cluster size is condensed. Finally, it extracts the stable clusters from the condensed tree.

An extension to data clustering is called anomaly or outliers detection. After the clustering process, it is possible to consider as outlier the elements that fall outside the clusters, i.e. the elements that are far from any cluster. An algorithm called Global-Local Outlier Score from Hierarchies (GLOSH) gives a score between 0 and 1 for outliers [START_REF] Campello | Hierarchical Density Estimates for Data Clustering, Visualization, and Outlier Detection[END_REF]. It compares the density of a point to the density of any points in the associated current and child cluster. Points with substantially lower density than the cluster density are likely to be considered outliers.

In our method, hierarchical clustering and outlier scoring will be applied to the decomposition coefficients on the principal component basis of the FPCA to compute the local compliance coefficient.

III. COMPLIANCE CRITERIA EXTENSION

In this section we introduce the extension and the new compliance features we have created. On all the figures, colors are defined as the following. Green is compliant, orange is a warning, red is critical, blue corresponds to the study interval and gray means that it does not belong to the study interval. The current situation and the conformity limits for trajectories from ELVIRA NCA module [START_REF] Lugardon | Paramétrages Elvira Qt[END_REF] are the baseline for our features. Indeed, the chevrons, the 30 seconds level-off flight and the glide path define geometric limits.

In order to make the notion of compliance more restrictive, reducing the number of false alarms, we now consider two limits. A warning limit and a critical limit. For both altitude and lateral feature, the limits defined in the ELVIRA module become the warning limit. Besides, we introduce the critical limit for the altitude feature as the low altitude of the GIFA's 3D volumes. For the horizontal feature, the critical limit, is defined as twice the warning limit. Both limits are represented in Figure 3 The false alarms are possibly due to the lack of features like energetic features for example. In the following, we present complementary features. In operations, an aircraft is Fig. 3: Actual compliance criteria of NCA Module: horizontal limit (top) and vertical limit (bottom) supposed to attend a level-off flight before intercepting glide slope. Consequently, we introduce a new feature Glide Angle (GA), which corresponds to the slope to join the touchdown point from current position. Considering the earth as a sphere implies an altitude correction to compute the feature. This is illustrated in figure 4. To give an example, with 11NM distance from the runway threshold, there is an altitude difference of 107ft. The feature is illustrated for a trajectory in figure 5. Fig. 4: Illustration of the altitude correction [START_REF] Perez | Conception de procédures aux instruments -Cours IPD3f[END_REF] To define a warning limit and a critical limit, we computed the slope angle of intercepting Glide Slope at 4350ft and 4700ft instead of 4000ft at FAP. The warning limit (resp. critical limit) for the Glide Angle feature is consequently set to 0.4°(resp. 0.7°) up to the published glide angle and increases linearly to 0.9°(resp 1.2°) approaching the touchdown point to consider the latent fluctuation of the tangent function used to compute the feature nearby the touchdown point. Fig. 5: New criteria : Glide Angle compliance Finally, we introduced two other features, the Ground Speed (GS) and the Vertical Speed (VS) to complete the energy analysis of the trajectories. The nominal, warning and critical limits are based on operational on-glide deceleration issues. The nominal operational on-glide deceleration in the literature is between 10 and 20 kts/NM [START_REF] Lemozit | Aircraft Energy Management during Approach[END_REF]. Consequently, we consider nominal a constant ground speed deceleration of 15 kts/NM from the Chevrons to the stabilization height at 1000ft with the aircraft computed approach speed v app (average speed over the last 3NM). For the warning limit, we consider a 17.5 kts/NM on-glide deceleration from Chevrons to v app + 15kts at 1000 ft and for the critical limit a 20 kts/NM deceleration from Chevrons to v app +30kts at 1000 ft. Those limits are illustrated in Figure 6.

Regarding the vertical speed, while on-glide, the aircraft vertical speed is directly linked to the ground speed. Besides, we consider that before the FAP, the vertical speed is not supposed to change since the aircraft has to operate a level-off flight. We finally consider as warning (resp. critical) a vertical speed 50% (resp. 100%) greater than the nominal vertical speed after the stabilization height.

IV. ATYPICAL FLIGHTS DETECTION METHODS

In this paper, the data we used is composed of 20746 landing radar records at CDG Airport during December 2011. The radar records are composed with the longitude, the latitude, the altitude, the ground speed, the time, the vertical speed, the heading, and the aircraft type. Radar information are recorded every 4 seconds.

A. Energy Motivations

The real problem for aircraft to land is an excess of energy. Excess of energy corresponds to situations where an aircraft is Fig. 6: New criteria : Ground Speed (top) and Vertical Speed (bottom) function of the curvilinear distance for example too high on glide owing to GIFA resulting in high potential energy, or with an overspeed owing to down wind in final approach or late power reduction resulting in high kinetic energy. By using the total energy we have a tool that is able to detect both cases of non-compliance.

The main idea is to use the total energy of the aircraft in the runway coordinate system. Radar records do not contain aircraft mass. Since, our study only concerns last phases of flight before landing, we assume that the mass is constant. Therefore, we compute the total specific energy E T (energy per unit mass) as:

E T = E c + E p ; E c = 1 2 • (G 2 s + V 2 z ); E p = g • h
Where Ep is the specific potential energy, Ec the specific kinetic energy, G s is the ground speed, V z the vertical speed, h the height and g the gravity constant.

Our input is the total energy function of the curvilinear distance.

B. Algorithm

Our method consists of applying the following process on a sliding window (defined by its width ν and its shift δ) recursively. First, apply a smoothing spline decomposition and a FPCA over the pieces of trajectories. Then, project over the k first principal components as representation of each piece of trajectory. Finally, apply a clustering to detect outliers far from every cluster. The HDBSCAN algorithm was used to perform the clustering. The Global-Local Outlier Score from Hierarchies (GLOSH) is used to give an outlier scoring. The value given is between 0 for a nominal sample and 1 for an outlier.

With this algorithm, each shift of the sliding window is attributed a coefficient. To represent and give a smooth representation of the coefficient, we use an averaging process to give the discrete score. The local compliance coefficient at a fix point for example 10NM is computed by averaging all the sliding window shift coefficients containing the 10NM point.

Finally, the detection phase is done by computing the length of the maximum interval for which compliance coefficients are over a threshold τ . If the maximum interval length is greater than a reference length λ, the trajectory is considered as atypical.

C. Why a sliding window is used ?

The first question is, why do we apply a sliding window and not using the whole trajectory as usually done in FPCA ? To answer this question we applied the process on the whole trajectory and discuss the pros and cons in the following.

We used radar records at CDG Airport during December 2011. We focus on the last 15 nautical miles (18NM to 3NM from the threshold) before stabilization of A320 landing on runway 26L. First, using the whole trajectory implies only to use the threshold τ to seperate nominal from atypical trajectories. In this illustration, we fixed τ such that we detect the most distant percentile of trajectories.

The detected atypical flights are analysed using the geometric limits defined in Section III. Over the 20 flights detected, there are 7 flights that were too high and did not respect level flight, 9 flights with an overspeed. Besides, we notice a huge Glide Interception From Above shown in Figure 7. This method also detects interesting cases. First, a flight with very strong speed reduction at 9NM and with a big lateral deviation. Second, a flight that intercepts the glide at 3000ft instead of 4000ft. Then, a flight with a low speed at 5NM which speed up back to the approach speed. Finally, a landing after a go around, which therefore started with a low speed and had a small overspeed at the beginning of the glide slope.

Using the whole interval presents different limitations. First, τ is a parameter that was fix based on the percentile the most distant and is dataset dependent. Besides, we considered as atypical, outliers with an important score. The distribution of the scores obtained with our process is shown on top of Figure 8. Score near 1 are in red, and score near 0 are in green. If we compare this distribution with the labels obtained by the energetic features that we develop previously, it shows that the outlier coefficient is not always appropriated since the warning and critical trajectories seem to be located on the right side of the distribution. A possible alternative could be to use a supervised learning model, using the resulting features as Fig. 7: A GIFA detected as outlier by the FPCA detection method labels. Then, to consider the outlier score to be a function of the probability given by the supervised learning method to be in the class. An illustration is shown on bottom of Figure 8. Besides, when the principal component analysis is made over the whole trajectory, local events could have been hidden by the process. All these reasons motivate the sliding window FPCA. In our context, it will give an outlier score for each interval, which can be interpreted as a local compliance score.

D. Hyperparameters and inherent effects

We must point out the inherent effects of the sliding window width and the coefficient computation. First, the sliding window width must be selected properly. If it is too short, we will only detect noise, and if it is too large, we will not detect local events. Then, the computation uses an average over all the intervals containing the point. This implies a smoothness of the compliance coefficient but also a decrease of the maximum value and a possible delay to change from atypical to nominal and inversely. It implies possible problems to generalize in a real-time situation. Finally, the detection rule also implies hyperparameters that must be fixed properly.

We underline that our method is made with different hyperparameters. Those parameters and their influence will be studied in future works. The motivation of this paper is to present a methodology and some first results.

V. CASE STUDY ANALYSES

In this subsection, we present the results we obtained for a fix configuration of our algorithm over specific situations. We selected a 2NM sliding window, which corresponds to a flight of around 30s, and a shift of 0.2NM (around the radar refresh time). The compliance coefficient of a point is obtained by averaging all the coefficients over the sliding windows that contain the point. The threshold τ was fixed at 0.6 and the reference length λ to 2NM, which corresponds to the sliding window width. For the FPCA, we used the 3 first principal components coefficients. Finally, we used 10 minimum samples per cluster in HDBSCAN.

a) Continuous Descent Approach (CDA): CDAs are situations where an aircraft operates a continuous descent and therefore does not attend the level-off flight. The geometric limits will always notify the situation with an altitude deviation warning since the flight overpass the altitude limits designed for the level-off flight. Nevertheless, it does not present any safety issue since it is a known procedure. The only possible issue with CDA is over-energy owing to over speed. We wanted to analyse how our methods deal with this kind of situations.

We consider 30 flights that intercepted the glide slope at an altitude above the published interception altitude, and then proceeded a CDA. For all the flights that presented a nominal speed none is considered as atypical. It means that our method does not detect abnormal energetic behaviour. Nevertheless, for those with a high ground speed like the flight illustrated in Figure 9, which has a ground speed around 250kts at FAP, the sliding window detect an overenergy. The energy was finally dissipated but before the FAP it shows that it was a potential dangerous situation.

Consequently, our methods seem to be relevant to study CDA. Indeed, only approach with high speed are considered as atypical. b) Glide Interception From Above: Other interesting events to analyse are GIFAs. In the dataset, there are 6 cases of GIFAs. The result obtained for the GIFA represented in Figure 7 is shown in Figure 10. The sliding window method is really efficient since the atypical behaviour is well localized before 6NM. The results are similar for the 6 cases.

Our method is efficient to detect important GIFAs. We need to underline an aspect of our method. Small GIFAs like bumpy profiles (flights which attend a level-off flight on glide to decelerate for example), which means a potential energy excess, might be counterbalanced by a low ground speed. Nevertheless, this is coherent since the non-compliance induced by the excess of altitude is averaged by the low speed in the energetic point of view. c) Ground Speed Warning: We now focus on flights which had a ground speed warning with the geometric features. A typical example is shown in Figure 11. The aircraft maintained a ground speed of 210 kts untill 6NM and finally reduced speed joining approach speed apparently after stabilization. For all these situations, the sliding window presents a large Non Compliance area for the last nautical miles and flight are detected as atypical.

VI. CONCLUSIONS

In this paper we presented an atypical flight detection method based on FPCA to enhance safety in flights approach and landing. The results of our method were compared to geometric features built with operational limits and analysed on typical flight approach patterns. Our method detects and localizes properly different type of abnormal energy situations. Nevertheless, we want to underline some limitations. An inappropriate sliding window size may induce changes in the results. Besides, local events such as brutal changes in the trajectory behaviour may be detected without being relevant safety issues.

Future works will focus on analysing our algorithm hyperparameters behaviour. Then, on developing a complete postoperational analysis tool based on our methodologies. Besides, we want to use those methods to develop a real-time detection tool using the result of our algorithms to label the data. In addition we are currently working on a novel data generation method in order to enhance our atypical flights database.

Fig. 1 :

 1 Fig. 1: Description of Compliant Approach Criteria and illustration of Stabilized Approach

Fig. 2 :

 2 Fig. 2: Illustration of the Area Proximity Warning set up at CDG

Fig. 8 :

 8 Fig. 8: Representation of the outlier coefficient (top) and of the supervised classification with a simple learning model (bottom)

Fig. 9 :

 9 Fig. 9: Ground speed (Top) and Sliding Window Energy Compliance Score (Bottom) of a Continuous Descent presenting an high ground speed around FAP

Fig. 10 :Fig. 11 :

 1011 Fig. 10: Sliding Window Energy Compliance Score over a GIFA

ACKNOWLEDGMENT

I would like to give a special thank to Mr. André Vernay, Mr. Yoni Malka and Mr. Paul-Emmanuel Thurat from the French Civil Aviation Safety Authority for their significant help to understand the operational problem. I would also like to show my gratitude to Mr. Gaël Vincent and Mr. Brice Panel from Paris Charles-De-Gaulle Airport ATC operations for their time and their explanation of the complex Paris northern airspace and Charles-De-Gaulle approach.


Optical probing of wave driven magneto-acoustic resonance P. Kuszewski, 1 J.-Y. Duquesne, 1 L. Becerra, 1 A. Lemaître, 2 S. Vincent, 1 S. Majrab, 1 F. Margaillan, 1 C. Gourdon, 1 and L. Thevenard The resonant interaction of electrically excited travelling surface acoustic waves and magnetization has hitherto been probed through the acoustic component. In this work it is investigated using time-resolved magneto-optical detection of magnetization dynamics. To that end, we develop an experimental scheme where laser pulses are used both to generate the acoustic wave frequency and to probe magnetization dynamics thus ensuring perfect phase locking. The light polarization dependence of the signal enables to disentangle elasto-optic and magneto-optic contributions and to obtain the in-plane and out-of-plane dynamic magnetization components. Magnetization precession is proven to be driven solely by the acoustic wave. Its amplitude is shown to resonate at the same field at which we detect piezo-electrically the resonant attenuation of the acoustic wave, clearly evidencing the magneto-acoustic resonance with high sensitivity.

I. INTRODUCTION

Recent years have witnessed a renewed and growing interest in the use of acoustic waves to excite spin waves as an alternative, fast, efficient, and heat-free means to generate and control (possibly coherently and remotely) information in magnetic materials [START_REF] Dreher | Surface acoustic wave driven ferromagnetic resonance in nickel thin films: Theory and experiment[END_REF][START_REF] Bombeck | Excitation of spin waves in ferromagnetic (Ga,Mn)As layers by picosecond strain pulses[END_REF][START_REF] Li | Acoustically assisted magnetic recording: A new paradigm in magnetic data Storage[END_REF][START_REF] Thevenard | Surface acoustic wave-driven ferromagnetic resonance in (Ga,Mn)(As,P) epilayers[END_REF][START_REF] Yahagi | Controlling nanomagnet magnetization dynamics via magnetoelastic coupling[END_REF][START_REF] Shen | Laser-Induced Spatiotemporal Dynamics of Magnetic Films[END_REF][START_REF] Davis | Fast strain wave induced magnetization changes in long cobalt bars: Domain motion versus coherent rotation[END_REF][START_REF] Gowtham | Traveling surface spin-wave resonance spectroscopy using surface acoustic waves[END_REF][START_REF] Chowdhury | Parametric Amplification of Spin Waves Using Acoustic Waves[END_REF][START_REF] Janušonis | Frequency tunable surface magneto-elastic waves[END_REF][START_REF] Thevenard | Strong reduction of the coercivity by a surface acoustic wave in an out-of-plane magnetized epilayer[END_REF][START_REF] Thevenard | Precessional magnetization switching by a surface acoustic wave[END_REF][START_REF] Chang | Parametric frequency mixing in a magnetoelastically driven linear ferromagnetic resonance oscillator[END_REF][START_REF] Hashimoto | All-optical observation and reconstruction of spin wave dispersion[END_REF][START_REF] Kim | Magnetization precession induced by picosecond acoustic pulses in a freestanding film acting as an acoustic cavity[END_REF][START_REF] Kuszewski | Resonant magneto-acoustic switching: influence of Rayleigh wave frequency and wavevector[END_REF] . Magnetization switching [START_REF] Li | Acoustically assisted magnetic recording: A new paradigm in magnetic data Storage[END_REF][START_REF] Davis | Fast strain wave induced magnetization changes in long cobalt bars: Domain motion versus coherent rotation[END_REF][START_REF] Thevenard | Strong reduction of the coercivity by a surface acoustic wave in an out-of-plane magnetized epilayer[END_REF][START_REF] Thevenard | Precessional magnetization switching by a surface acoustic wave[END_REF][START_REF] Kuszewski | Resonant magneto-acoustic switching: influence of Rayleigh wave frequency and wavevector[END_REF] , and parametric excitation of spin waves [START_REF] Chowdhury | Parametric Amplification of Spin Waves Using Acoustic Waves[END_REF][START_REF] Chang | Parametric frequency mixing in a magnetoelastically driven linear ferromagnetic resonance oscillator[END_REF] have for instance been demonstrated, thus opening new opportunities for applications. The efficiency of acoustic waves relies on magneto-elastic coupling that occurs in magnetostrictive materials. Electrically excited surface acoustic waves (SAWs), with typical frequencies up to the GHz range are particularly well suited to excite magnetization dynamics in ferromagnetic magnetostrictive layers. Their interaction was evidenced using piezo-electric detection of SAW-induced ferromagnetic resonance (SAW-FMR) [START_REF] Dreher | Surface acoustic wave driven ferromagnetic resonance in nickel thin films: Theory and experiment[END_REF][START_REF] Thevenard | Surface acoustic wave-driven ferromagnetic resonance in (Ga,Mn)(As,P) epilayers[END_REF] : the SAW amplitude and velocity resonantly decrease when the spin wave frequency is varied across the SAW frequency by an applied magnetic field. However, spin waves have not been concomitantly observed. Time-resolved detection of magneto-optical effects using probe laser pulses indeed could provide a direct and very sensitive method to access magnetization dynamics [START_REF] Shihab | Counter-rotating standing spin waves: A magneto-optical illusion[END_REF] . This requires a non-trivial synchronization of the electrical SAW generation with probe laser pulses. Until now this difficulty has been bypassed by utilizing fairly cumbersome synchrotron X-ray radiation [START_REF] Foerster | Direct imaging of delayed magnetodynamic modes induced by surface acoustic waves[END_REF] .However no resonant magnetization dynamics have been observed .

In this article, we demonstrate optically the forced magnetic precession induced by travelling SAWs. To that end, we implement a table-top set-up to generate SAW bursts electrically using interdigitated transducers (IDTs) with perfect phase locking to laser pulses, which enables the optical investigation of time-and space-resolved magnetization dynamics excited by the SAW. We illustrate the potentiality of this technique using a ferromagnetic semiconductor layer of GaMnAs on piezoelectric GaAs, taking advantage of its low FMR frequency which is easily tunable across the SAW frequency, and its sizable magneto-optical effects. Using the probe light polarization and the magnetic field dependencies of the time-resolved signal we separate the magneto-optical from the pure elasto-optical contribution. The in-plane and out-of plane magnetization dynamical components display the dependencies on the magnetic field expected from the SAW-induced torque acting on the magnetization [START_REF] Linnik | Theory of magnetization precession induced by a picosecond strain pulse in ferromagnetic semiconductor (Ga,Mn)As[END_REF][START_REF] Thevenard | Irreversible magnetization switching using surface acoustic waves[END_REF] .

Finally, we show that resonant magnetic excitation appears concomitantly with resonant SAW absorption, the detection of the former being more sensitive than the latter.

II. EXPERIMENTAL SETUP

SAWs such as Rayleigh waves are easy to implement and thus find applications in various fields of soft and condensed matter physics [START_REF] Yeo | Surface Acoustic Wave Microfluidics[END_REF][START_REF] Mamishev | Interdigital sensors and transducers[END_REF][START_REF] Poole | Acoustoelectric Current in Graphene Nanoribbons[END_REF][START_REF] De Lima | Modulation of photonic structures by surface acoustic waves[END_REF] . With frequencies up to the GHz, their dynamical strain components are confined within a few µm from the surface, and can thus efficiently interact with a magnetic layer on a substrate. SAWs can be generated either optically via thermoelasticity [START_REF] Yahagi | Controlling nanomagnet magnetization dynamics via magnetoelastic coupling[END_REF][START_REF] Janušonis | Frequency tunable surface magneto-elastic waves[END_REF][START_REF] Chang | Parametric frequency mixing in a magnetoelastically driven linear ferromagnetic resonance oscillator[END_REF][START_REF] Neubrand | Laser generation and detection of surface acoustic waves: Elastic properties of surface layers[END_REF] or electrically via the piezoelectric effect using IDTs [START_REF] Royer | Elastic Waves in Solids II. Generation, Acousto-optic Interaction, Application[END_REF][START_REF] Weiß | Multiharmonic Frequency-Chirped Transducers for Surface-Acoustic-Wave Optomechanics[END_REF] . The simplest IDT consists of two comb-shaped electrodes in zipper configuration that work as capacitors on a piezoelectric surface. When a radio-frequency (RF) voltage at f SAW is applied, SAWs are generated and propagate on either side of the IDT. The wavelength λ SAW of the excited SAW is determined by the period of the electrode teeth. Frequency and wavelength are related by λ SAW = v R f SAW where v R is the Rayleigh velocity. Typical λ SAW values are around a few microns. The strain components excited with the IDT are: longitudinal ε xx (x k SAW ), transverse ε zz (z perpendicular to the surface) and shear ε xz (negligibly small close to the surface) [START_REF] Royer | Elastic Waves in Solids II. Generation, Acousto-optic Interaction, Application[END_REF] . In more sophisticated IDT architectures it is possible to excite overtone harmonics of the fundamental frequency, such as the split-52 design [START_REF] Schülein | Fourier synthesis of radiofrequency nanomechanical pulses with different shapes[END_REF] , which is used in this work. Optical time-resolved measurement of magnetization dynamics requires a fixed phase between the laser pulses and the acoustic bursts. The most common solution is a phase-locked loop [START_REF] Holländer | Component selection in time-resolved magneto-optical wide-field imaging for the investigation of magnetic microstructures[END_REF] . The 'master' clock imposes the repetition rate of the 'slave' laser. The laser cavity length is continuously adjusted in order to meet the set frequency. We propose a simpler, elegant, and low-cost alternative approach where f SAW is built from the laser repetition rate thereby ensuring a stable phase lock. Two research groups have very recently reported independently similar synchronization methods [START_REF] Foerster | Direct imaging of delayed magnetodynamic modes induced by surface acoustic waves[END_REF][START_REF] Weiß | Multiharmonic Frequency-Chirped Transducers for Surface-Acoustic-Wave Optomechanics[END_REF] .

In order to generate the SAW RF frequency, a small fraction of a Ti:Sapphire laser at ∼75 MHz repetition rate (f L ) was directed to a Silicon photodiode (Thorlabs PDA10A-EC, bandwidth 150 MHz) to pick up the laser fundamental frequency (Fig. 1(a)). Using different RF multipliers, multiples of the laser frequency were then generated: n f L , where n = 2, 4, 6, 8. To ensure that n f L is free of any harmonics, the signal was filtered with 50 dB rejection band pass filters. The signal level was adjusted with a linear 40 dB amplifier and a set of attenuators.

The larger fraction of the 130 fs laser pulses (wavelength 722 nm) was sent to a pulse selector (Fig. 1(a)). The repetition rate was reduced by a factor 300 to f rep = 250 kHz. This beam was used to probe the magnetization and strain dynamics in a time window of 12 ns controlled with a motorized optical delay line stage.

As we shall see hereafter the IDT needs to be fed with a pulsed RF signal. In order to fabricate these pulses the n f L CW RF signal was mixed with a 400 ns-wide rectangular pulse train from a pulse generator (Keysight 81150A) at the pulse picker frequency f rep thus ensuring that both the envelope frequency and the RF frequency of the SAW are commensurate with f L . With the pulse generator we could modify the burst duration and its arrival time (electronic delay) to the transducer. The 'edge-to-reference' jitter 30 defined as the timing variation between the rising edge of the rectangular pulse and the optical pulse shows a standard deviation σ = 13.1 ps comfortably below 1% of the highest SAW period.

The linearly polarized laser pulses at f rep were focused on the sample surface with a long working distance objective lens with numerical aperture 0.4. The laser beam reflected off the sample traveled through the same objective. The polarization rotation induced by the dynamical magnetization and strain components was detected by a balanced optical bridge relying on magneto-optical effects [START_REF] Shihab | Counter-rotating standing spin waves: A magneto-optical illusion[END_REF][START_REF] Němec | Experimental observation of the optical spin transfer torque[END_REF] and the photo-elastic (PE) effect [START_REF] Royer | Elastic Waves in Solids II. Generation, Acousto-optic Interaction, Application[END_REF] , respectively. The bridge output was demodulated by a lockin amplifier at the frequency of a mechanical chopper f ch =541 Hz inserted on the laser beam before the slow photodiode (Fig. 1(a)), i.e. modulating the excitation SAW. Polarization rotations as small as 0.1 µrad can rou-tinely be measured.

The sample was a 45 nm thick, in-plane magnetized Ga 0.95 Mn 0.05 As layer on a (001) GaAs substrate, annealed for 16 h at 200 o C, with a Curie temperature T C = 120 K. It was placed in an Oxford MicroStat HiRes cryostat that ensures the required mechanical stability and optical access. The GaMnAs layer has a strong uniaxial magnetic anisotropy with the easy axis along [1-10] and exhibits a large magneto-elastic coupling [START_REF] Kuszewski | Resonant magneto-acoustic switching: influence of Rayleigh wave frequency and wavevector[END_REF] . An inplane magnetic field aligned with the hard axis [110] was applied to decrease the precession frequency and make it cross the SAW frequency. A 2x2 mm 2 square mesa of GaMnAs was prepared by chemical etching. At two opposite ends a set of 42 nm thick Aluminum IDTs with 1 mm aperture were deposited on the GaAs (Fig. 1(a)). In order to provide efficient operation of the IDTs even when varying the temperature (given the temperature dependence of v R (T )) and to achieve a compromise between excitation efficiency and bandwidth, the IDTs were designed as 25 pairs of split-52 electrodes. With a digit width and inter-digit spacing of 1.9 µm, the fundamental period was λ SAW =19 µm. This ensured that the wavelength of all excited SAW frequencies was substantially larger than the laser spot diameter (full-width at half maximum of about 1 µm). The RF voltage applied to the IDT triggers the propagation of a SAW along the [1-10] axis. The RF power was set to 25 dBm for most data presented here. We used the second IDT to detect the SAW electrically by the inverse piezoelectric effect after it had travelled d = 2.03 mm along the layer surface. Figure 2 presents an oscilloscope trace of the recorded signal (red line). The first 400 ns burst is the electromagnetic radiation. It travels with the speed of light and appears immediately at the receiver. After around t trans = d v R = 730 ns the acoustic echo arrives (v R ≈2780 m s -1 ). The pulsed RF allows to separate in time the electromagnetic radiation and the acoustic echo. The risetime of the acoustic echo is defined by the IDT geometry, the SAW velocity, and the burst duration [START_REF] Royer | Elastic Waves in Solids II. Generation, Acousto-optic Interaction, Application[END_REF] . The amplitude of the signal varies with the magnetic field up to T C , above which we observe only a field independent signal. This proves that the TRPR signal contains both a magnetic (field-dependent) contribution and a non-magnetic one arising from the elasto-optic effect.

III. RESULTS AND DISCUSSION

In order to prove that the optically detected dynamics is indeed triggered by the SAW we monitored the amplitude of the TRPR signal at various delays after the beginning of the the RF burst by setting an electronic delay ∆τ between the electric excitation and the optical pulse probe (Fig. 1(a)). First the probe beam was set at 100 µm from the receiving IDT (1.9 mm from the emitter, open circle in the sample scheme of Fig. 2). The open circle symbol curve represents the amplitude of the TRPR oscillations versus the electronic delay. It has the same shape as the acoustic echo and peaks slightly before the arrival of this echo on the receiving IDT, in agreement with the location of the detection spot. Then the probe beam was positioned 67 µm from the emitter IDT (full circle in the sample scheme of Fig. 2). The full circle curve represents the amplitude of the oscillations versus the electronic delay. It also has the same shape as the acoustic echo and is detected earlier, in agreement with the position of the detection spot. The 20% burst amplitude decrease from one burst to the other can be directly correlated to the loss of acoustic amplitude via magneto-elastic interaction, as will be seen further (Fig. 4(g)). For both spot positions, no signal is observed at the arrival time of the electromagnetic radiation. These results show that the TRPR signal is indeed generated by the SAW.

In order to disentangle the magneto-optic and the elasto-optic contributions in the TRPR signal we analyze the dependence of the signal on the probe light polarization and the applied magnetic field. The elasto-optic effect gives rise to a dynamic birefringence with axes parallel and perpendicular to the SAW wavevector k SAW . The resulting rotation of linear polarization is proportional to the P 44 component of the elasto-optic tensor of the cubic GaAs and the ε xx strain component [START_REF] Royer | Elastic Waves in Solid I: Free and Guided Propagation[END_REF][33][34] . It does not depend on the magnetic field. Besides we expect the contribution from two magneto-optical effects [START_REF] Shihab | Counter-rotating standing spin waves: A magneto-optical illusion[END_REF][START_REF] Němec | Experimental observation of the optical spin transfer torque[END_REF] : the polar magneto-optical Kerr effect (PMOKE) sensitive to the out-of-plane dynamic component of the magnetization δθ and independent of the incoming polarization and the (weaker) Voigt effect (magnetic linear dichroism (MLD)) sensitive to the in-plane dynamic component of 4(h) was taken. The surface amplitudes of the excited SAW strain components are εxx = 4.8 × 10 -5 and εzz = 2.2 × 10 -5 estimated following the procedure described in ref. [START_REF] Kuszewski | Resonant magneto-acoustic switching: influence of Rayleigh wave frequency and wavevector[END_REF] . The SAW frequency is 452 MHz, the applied field is µ0H =20 mT, the polarization angle is β=30 deg.

the magnetization δφ and to the field-dependent in-plane equilibrium orientation of the magnetization φ 0 (H). The TRPR signal can therefore be expressed as [START_REF] Shihab | Counter-rotating standing spin waves: A magneto-optical illusion[END_REF][START_REF] Royer | Elastic Waves in Solids II. Generation, Acousto-optic Interaction, Application[END_REF][START_REF] Němec | Experimental observation of the optical spin transfer torque[END_REF] :

δβ = Kδθ(H, t) + 2V δφ(H, t) cos (2 (β -φ 0 (H))) +P E ε xx (t) sin 2β (1) = Kδθ(H, t) + 2V δφ(H, t) cos 2φ 0 (H) cos 2β + (2V δφ(H, t) sin 2φ 0 (H) + P E ε xx (t)) sin 2β (2)
where β is the angle of the probe polarization with respect to k SAW , K and V are the Kerr and Voigt magneto-optical coefficients, respectively, and P E = Re n 3 P 44 /(n 2 -1) is the elasto-optic coefficient with n the refractive index. Indeed the experimental TRPR signal shows a clear dependence on the probe polarization (Fig. 3(a)) and on the magnetic field (Fig. 3(b)). For each time and field value, the signal is fitted as a function of β by δβ = F θ +G φ cos 2β +H φε sin 2β. The resulting fit is very good as seen in Fig. 3(b). The offset that appears under the application of a magnetic field µ 0 H =6 mT is related to the PMOKE signal F θ = Kδθ. The change of the phase is a good indication of the presence of MLD. The resulting F θ , G φ , H φε time-dependent functions are then fitted by sinusoidal fonctions at frequency f SAW in order to extract the field-dependence of their amplitudes f θ , g φ , h φε , respectively. We have f θ = Kδθ 0 (H) and g φ = 2V δφ 0 (H) |cos 2φ 0 (H)| where δθ 0 and δφ 0 are the amplitudes of the oscillating δθ(t) and δφ(t), respectively. In Fig. 3(c) we plot the amplitudes f θ , g φ , h φε as a function of the applied field. The amplitude of the PMOKE signal f θ (black curve) increases progressively to reach a maximum at 22.5 mT and then drops to zero. The MLD component g φ (red curve) shows a different behavior with two maxima and a zero at µ 0 H=16 mT. The h φε component reflecting a combination of PE and MLD (green curve) has a broad maximum and a clear 40 µrad offset owing to the field-independent elasto-optic effect.

To demonstrate that the physics of the magneto-elastic coupling is responsible for the experimental observations, the magnetization dynamics is modelled in the framework of the Landau-Lifshitz-Gilbert equation including the driving torque generated by the SAW [START_REF] Thevenard | Irreversible magnetization switching using surface acoustic waves[END_REF] . When the SAW is travelling along [1-10] the torque is mainly driven by the ε xx strain component [START_REF] Kuszewski | Resonant magneto-acoustic switching: influence of Rayleigh wave frequency and wavevector[END_REF] . We use the sample magnetic parameters (magnetization and magnetic anisotropy) obtained from vibrating sample magnetometer and cavity-FMR experiments. The calculated amplitudes 35 of the magneto-optical signals f θ and g φ (Fig. 3(d), black and red curves, respectively) show a peak around 25 mT, corresponding to the resonance condition of equal precession and SAW frequencies at f SAW = 301 MHz. The g φ amplitude (red curve) goes to zero at a field such that the static magnetization is at 45°of the easy axis (cos 2φ 0 = 0). The baseline for h φε is given by the PE effect that does not depend on the magnetic field. As seen from the comparison of Fig. 3(c) and (d) a good quantitative agreement between experimental and calculated f θ , g φ , h φε curves is obtained. To account for the amplitude and width of f θ , g φ , h φε we had to introduce a dispersion of the uniaxial in-plane magnetic anisotropy constant that is known to be crucial for SAW-induced magnetization dynamics [START_REF] Kuszewski | Resonant magneto-acoustic switching: influence of Rayleigh wave frequency and wavevector[END_REF] . These results show that we are able to disentangle the different contributions in the optical polarization signal and to accurately extract the dynamical magnetic contribution. They provide the first direct, time-domain detection of magneto-acoustic resonance induced by electrically generated propagating SAWs. Note that the resonant coupling of optically generated stationary SAWs with magnetization had been previously evidenced by Faraday effect in nickel [START_REF] Yahagi | Controlling nanomagnet magnetization dynamics via magnetoelastic coupling[END_REF][START_REF] Janušonis | Frequency tunable surface magneto-elastic waves[END_REF][START_REF] Chang | Parametric frequency mixing in a magnetoelastically driven linear ferromagnetic resonance oscillator[END_REF] . In our system, the presence of Kerr and Voigt effects moreover gives access to both the inplane and out-of-plane dynamical magnetization components.

In order to compare the sensitivity of the magnetooptical signal detection and the SAW amplitude detection after propagation along the layer, we show in Fig. 4 the PMOKE component f θ (out-of-plane magnetization dynamics) and the variation of the electrically detected SAW amplitude (through the receiving IDT) as a function of the applied magnetic field for the four SAW frequencies. The PMOKE data (top panel of Fig. 4) were normalized by the measured PE level (field-independent baseline of h φε ) in order to correct for slightly different SAW amplitudes. For each frequency we observe a clear and strong peak of the PMOKE curve at the resonance field (which has close values for the four frequencies because of the steep variation of the FMR frequency with the field [START_REF] Kuszewski | Resonant magneto-acoustic switching: influence of Rayleigh wave frequency and wavevector[END_REF] ). The electrically detected SAW amplitude (bottom panel of Fig. 4) is a rather noisy signal at 151, 301 and 452 MHz, with a much smaller dynamical range. At f SAW = 301, 452, and 602 MHz, it shows a dip at the same field as the peak of the corresponding PMOKE signal. The dip is more pronounced at the higher frequency of 602 MHz as expected from the increase of the absorbed SAW power with the SAW frequency [START_REF] Dreher | Surface acoustic wave driven ferromagnetic resonance in nickel thin films: Theory and experiment[END_REF][START_REF] Kuszewski | Resonant magneto-acoustic switching: influence of Rayleigh wave frequency and wavevector[END_REF] . The dip is however not detectable at f SAW = 151 MHz whereas the PMOKE peak is clearly detected. We also plot in Fig. 4(h) the amplitude of the photo-elastic signal detected on GaAs, in between the GaMnAs mesa and the receiving IDT (crossed-circle symbol in the scheme of Fig. 2), normalized by its low-field value. The data was taken for an incoming beam polarisation of 45 deg, which maximizes the strain-induced birefringence. Its field-dependence is similar to that of the electrical signal (Fig. 4(h)) in shape and amplitude. These results show the much larger dynamical range and sensitivity of the magneto-optical SAW-FMR signal, because it is detected on a zero background, with respect to electrical or optical signal of the SAW amplitude variation, detected on a non-zero background.

IV. CONCLUSION

We have developed a sensitive time-domain optical technique to investigate the magneto-elastic coupling be-tween piezo-electrically generated travelling SAWs and magnetization in ferromagnetic layers at variable temperature. The time-resolved magnetization precession was clearly proved to originate from the magneto-elastic coupling with the SAW and shows a resonant behavior at equal SAW and precession frequencies. Compared to the detection of the SAW attenuation, the magnetooptical time-resolved SAW-FMR signal is more sensitive and provides a better signal-to-noise ratio. The detection threshold of the time-resolved signal is mainly governed by the convolution of the SAW wavelength and the laser spot size w. As such, magnetization precession could still be observed at f SAW =900 MHz, for which for λ SAW ∼ 3w. Working at higher SAW frequencies (above 1 GHz) will require a tighter focusing of the laser spot, either by working at higher photon energy (keeping a decent Kerr signal) and/or by increasing the numerical aperture of the focusing objective (keeping a good linear polarization). More generally, we believe that this approach, combining high space/time sensitivity and access to the two components of magnetization will find a wider use in any experiment requiring the synchronization of a radio-frequency electrical stimulus with the ultra-fast optical detection of the magnetic effects it induces, and should benefit broadly the magnetization dynamics community. Applied to magneto-strictive materials, this technique opens the way for a deeper insight into the magnon-phonon coupling and the exploration of the nonlinear regime of acoustic-wave induced magnetization dynamics. Since electrically generated SAWs are a wellmastered technology in microelectronics, SAW-induced magnetization control could be readily implemented in logic or memory devices, with the unprecedented possibility to use wave physics tools such as focusing, diffrac-tion, and wave-guiding to address a magnetic bit.

1 1

 1 Sorbonne Université, CNRS, Institut des Nanosciences de Paris, 4 place Jussieu,75252 Paris France 2 Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91460 Marcoussis, France (Dated: August 2, 2018)

Figure 1 (

 1 Figure1(b) shows the time-resolved polarization rotation (TRPR) at 60 K for excitation SAW frequencies 151, 301, 452 and 602 MHz as a function of the optical delay. The frequency of the detected signals matches exactly the excitation frequency. The amplitude of the signal varies with the magnetic field up to T C , above which we observe only a field independent signal. This proves that the TRPR signal contains both a magnetic (field-dependent) contribution and a non-magnetic one arising from the elasto-optic effect.In order to prove that the optically detected dynamics is indeed triggered by the SAW we monitored the amplitude of the TRPR signal at various delays after the beginning of the the RF burst by setting an electronic delay ∆τ between the electric excitation and the optical

FIG. 1 .

 1 FIG. 1. (a) Schematics of the experimental set-up, (b) Time-resolved polarization rotation signal for four SAW frequencies (vertically off-set for clarity). The probe polarization was set at 60°with respect to kSAW . The applied magnetic field was 19.5 mT. The temperature was T=60K.

FIG. 2 .

 2 FIG. 2. Time-domain signal detected electrically (red solid line) by an oscilloscope, and amplitude of the optically detected TRPR oscillations (circles) as a function of the electronic delay. TRPR was measured in two places on the sample (open and closed symbols on the scheme). The crossedcircle symbol indicates where the optical SAW-FMR data shown in Fig.4(h) was taken. The surface amplitudes of the excited SAW strain components are εxx = 4.8 × 10 -5 and εzz = 2.2 × 10 -5 estimated following the procedure described in ref.[START_REF] Kuszewski | Resonant magneto-acoustic switching: influence of Rayleigh wave frequency and wavevector[END_REF] . The SAW frequency is 452 MHz, the applied field is µ0H =20 mT, the polarization angle is β=30 deg.

FIG. 3 .

 3 FIG. 3. Polarization and field dependence of the magnetization and strain dynamics for fSAW = 301 MHz at T = 60 K. (a) TRPR signal for six polarization orientations as a function of the optical delay for µ0H= 6 mT. (b) TRPR at fixed time delay (cross section of Fig. 3(a), green dashed line) versus probe polarization for two field values. (c) Field variation of the three components f θ , g φ , h φε related to polar Kerr effect (β-independent), MLD (cos2β term) and mixed PE/MLD (sin2β term), respectively, at fSAW =301 MHz. (d) Calculated amplitudes of f θ , g φ , h φε ; the parameters are K=18 mrad, V =0.8 mrad, PE=0.77 rad with εxx = 5.3 10 -5 . The Gaussian distribution on the in-plane anisotropy constant has a standard deviation equal to 12 % of its mean value.

FIG. 4 .

 4 FIG.4. Top panels: Field dependence of the PMOKE signal (out-of-plane magnetization dynamics) normalized by the fieldindependent photo-elastic signal for four SAW frequencies at T =60 K. Bottom panels: variation of the SAW amplitude with the magnetic field measured by acoustic-to-electrical conversion at the receiving IDT and in (h) normalized PE signal detected on GaAs in between the GaMnAs mesa and the receiving IDT (see crossed-circle symbol in the schematics of Fig.2), with a polarization angle β=45 deg. The amplitudes of excited SAW strain components are estimated as εxx = 5.3 × 10 -5 and εzz = 1.8 × 10 -5 at fSAW =301 MHz (they are slightly different at other frequencies but low enough to stay in the linear regime).

  berg, 2000). 33 Saito, T., Matsuda, O., Tomoda, M. & Wright, O. B. Imaging gigahertz surface acoustic waves through the photoelastic effect. J. Opt. Soc. Am. B 27, 2632 (2010).

Santos, P. V. Acoustic field mapping on GaAs using microscopic reflectance and reflectance anisotropy. Appl. Phys. Lett. 74, 4002 (1999).

 35 Because we are only considering the uniform FMR mode, it is unecessary to take into account the absorption and optical phase shift of the light in the layer, as in Ref. 17..

ACKNOWLEDGMENTS

This work has been supported by the French Agence Nationale de la Recherche (ANR13-JS04-0001-01). The authors also acknowledge D. Hrabovsky (MPBT-Physical Properties Low Temperature facility of Sorbonne Université), H. J. von Bardeleben (INSP) for magnetometry measurements, and C. Testelin (INSP) for providing scientific equipment.