L Bergantin

C Coquet

J Dumon

A Negre

T Raharijaona

N Marchand

F Ruffier

Indoor and outdoor in-flight odometry based solely on optic flows with oscillatory trajectories

Estimating distance traveled is a frequently arising problem in robotic applications designed for use in environments where GPS is only intermittently or not at all available. In UAVs, the presence of weight and computational power constraints makes it necessary to develop odometric strategies based on minimilastic equipment. In this study, a hexarotor was made to perform up-and-down oscillatory movements while flying forward in order to test a self-scaled optic flow based odometer. The resulting self-oscillatory trajectory generated series of contractions and expansions in the optic flow vector field, from which the flight height of the hexarotor could be estimated using an Extended Kalman Filter. For the odometry, the downward translational optic flow was scaled by this current visually estimated flight height before being mathematically integrated to obtain the distance traveled. Here we present three strategies based on sensor fusion requiring no, precise or rough prior knowledge of the optic flow variations generated by the sinusoidal trajectory. The "rough prior knowledge" strategy is based on the shape and timing of the variations in the optic flow. Tests were performed first in a flight arena, where the hexarotor followed a circular trajectory while oscillating up and down over a distance of about 50m under illuminances of 117lux and 1518lux. Preliminary field tests were then performed, in which the hexarotor followed a longitudinal bouncing 20m-long trajectory over an irregular pattern of grass.

I. Introduction

Estimating distance traveled by an aerial robot is a problem which frequently arises when designing applications for use in situations where GPS is available only intermittently or not at all. In UAVs (Unmanned Aerial Vehicles), reducing the Size, Weight and Power (SWaP) of the perceptual equipment is often of great importance in order to ensure that the robot's task will be performed successfully. Several visual odometric approaches involving the use of either optic flow [START_REF] Strydom | Visual odometry: autonomous uav navigation using optic flow and stereo[END_REF][START_REF] More | Visual odometry using optic flow for Unmanned Aerial Vehicles[END_REF], events, images & IMU (Inertial Measurement Unit) combinations [START_REF] Vidal | Ultimate SLAM? Combining events, images, and IMU for robust visual SLAM in HDR and high-speed scenarios[END_REF] or the sparse-snapshot method [START_REF] Denuelle | A sparse snapshotbased navigation strategy for UAS guidance in natural environments[END_REF] have been successfully tested on flying robots. All these approaches require ground height information providing the factor used to scale the visual information. This scaling factor can be determined separately using a static pressure sensor [START_REF] Kendoul | Optic flowbased vision system for autonomous 3D localization and control of small aerial vehicles[END_REF] or stereovision [START_REF] Strydom | Visual odometry: autonomous uav navigation using optic flow and stereo[END_REF][START_REF] Denuelle | A sparse snapshotbased navigation strategy for UAS guidance in natural environments[END_REF], or it can be integrated when using the hybrid approach [START_REF] Vidal | Ultimate SLAM? Combining events, images, and IMU for robust visual SLAM in HDR and high-speed scenarios[END_REF], for example. One of the approaches used to estimate the 2D position of a drone is a combination of onboard odometry and visual mapping, known as SLAM (Simultaneous Localisation and Mapping) [START_REF] Faessler | Autonomous, vision-based flight and live dense 3D mapping with a quadrotor micro aerial vehicle[END_REF][START_REF] Lee | Stability-based scale estimation for monocular SLAM[END_REF][START_REF] Milijas | A comparison of lidar-based slam systems for control of unmanned aerial vehicles[END_REF]. Most of these approaches require the use of computationally intensive algorithms and feedback from the environment (such as the detection of a beacon or feedback from a map). A minimalistic alternative is IMU based dead reckoning -i.e. inertial integration [START_REF] Shurin | QDR: A Quadrotor Dead Reckoning Framework[END_REF]. A dead reckoning signal could be used by a UAV to return to the close proximity of its base station before reaching it a second time using other means of perception. In this case, the landing of the UAV on its base station can provide a new known starting point. Another minimalistic alternative consists in using optic flow cues, such as translational optic flow and optic flow divergence cues. Translational optic flow has been used on UAVs to control landing visually [START_REF] Ruffier | Optic flow regulation: the key to aircraft automatic guidance[END_REF], to follow uneven terrain [START_REF] Expert | Flying over uneven moving terrain on optic-flow cues without any need for reference frames or accelerometers[END_REF] and to attempt visual odometry and localisation [START_REF] Iida | Biologically inspired visual odometer for navigation of a flying robot[END_REF][START_REF] Kendoul | Optic flowbased vision system for autonomous 3D localization and control of small aerial vehicles[END_REF] (see [START_REF] Serres | Optic flow-based collisionfree strategies: From insects to robots[END_REF] for a review). Self-oscillations have been observed in honeybees flying forward in horizontal [START_REF] Kirchner | Freely flying honeybees use image motion to estimate object distance[END_REF][START_REF] Julien R Serres | A bee in the corridor: centering and wall-following[END_REF], doubly tapered [START_REF] Portelli | Honeybees' speed depends on dorsal as well as lateral, ventral and frontal optic flows[END_REF] and highroofed [START_REF] Portelli | Honeybees change their height to restore their optic flow[END_REF][START_REF] Portelli | Altitude control in honeybees: joint vision-based learning and guidance[END_REF] tunnels. The self-oscillatory motion generates a series of expansions and contractions in the optic flow vector field, providing the optic flow divergence cue. Visually controlled landing has been achieved based on the optic flow divergence cue [START_REF] Herissé | Landing a VTOL unmanned aerial vehicle on a moving platform using optical flow[END_REF][START_REF] Van Breugel | Monocular distance estimation from optic flow during active landing maneuvers[END_REF][START_REF] De Croon | Monocular distance estimation with optical flow maneuvers and efference copies: a stabilitybased strategy[END_REF][START_REF] Ho | Distance and velocity estimation using optical flow from a monocular camera[END_REF]. The instabilities due to oscillatory movements have been used to determine the flight height of a micro-flyer based on the linear relationship between the oscillation and the fixed control gain [START_REF] De Croon | Monocular distance estimation with optical flow maneuvers and efference copies: a stabilitybased strategy[END_REF]. The instabilities due to depth variations have been used to assess the optic flow scale factor of the scene observed to perform visual odometry onboard an underwater vehicle [START_REF] Creuze | Monocular odometry for underwater vehicles with online estimation of the scale factor[END_REF]. The local optic flow divergence was measured by means of two optic flow magnitudes perceived by two basic optic flow sensors placed on a chariot performing back-and-forth oscillatory movements in front of a moving panorama [START_REF] Bergantin | Estimation of the distance from a surface based on local optic flow divergence[END_REF]. The local optic flow divergence was then used to estimate the local distance between the chariot and the moving panorama by means of an Extended Kalman Filter (EKF) [START_REF] Bergantin | Estimation of the distance from a surface based on local optic flow divergence[END_REF]. A SOFIa (Self-scaled Optic Flow time-based Integration) model has been previously tested as a means of modeling the visual odometer of honeybees with simulations [START_REF] Bergantin | Oscillations make a self-scaled model for honeybees' visual odometer reliable regardless of flight trajectory[END_REF] as well as with preliminary indoor flights [START_REF] Bergantin | Minimalistic in-flight odometry based on two optic flow sensors along a bouncing trajectory[END_REF]. The SOFIa method to estimate the distance traveled is based on the integration of the local translational optic flow scaled by the drone's flight height, determined by means of an EKF taking the local optic flow divergence as measurement [START_REF] Bergantin | Oscillations make a self-scaled model for honeybees' visual odometer reliable regardless of flight trajectory[END_REF]. The SOFIa model was found to be about 10 times more accurate than the values obtained in simulations based on the raw mathematical integration of the optic flow [START_REF] Bergantin | Oscillations make a self-scaled model for honeybees' visual odometer reliable regardless of flight trajectory[END_REF]. Using an integration scheme of this kind can therefore be regarded as a minimalistic dead reckoning method based on the optic flow. Here we investigated how to include some knowledge about the oscillations occurring during the trajectory in an odomet-ric strategy based on optic flow cues alone. For this purpose, the optic flow based odometric scheme called SOFIa was tested both indoors and outdoors on a hexarotor equipped with optic flow sensors (see Figure 1). First we applied the SOFIa method using only 2 optic flow measurements perceived along the longitudinal axis of the drone, with no prior knowledge of the optic flow variations. In order to improve the odometric accuracy, a sensor fusion strategy based on the parameters of the self-oscillation using 4 optic flow sensors embedded in the hexarotor was then tested. The idea was to use some prior knowledge about the oscillations imposed on the drone in order to measure the optic flow divergence and the translational optic flow cues more accurately. Two different sensor fusion strategies, based on precise and on rough prior knowledge of the optic flow variations, respectively, were tested. The sensor fusion strategy based on rough prior knowledge consisted solely in using the shape and timing of the variations in the optic flow. All three optic flow based odometric processing methods were tested first indoors on bouncing circular trajectories about 50m-long under illuminances of 117lux and 1518lux and then outdoors on bouncing longitudinal trajectories about 20m-long in the presence of various wind and irregular grass conditions. In section 2, the hexarotor used to perform both indoor and outdoor experiments is described. In section 3, the measurement of the local translational and divergence optic flow cues is discussed. In section 4, the minimalistic visual odometric method is discussed. In section 5, the odometric processing method based on raw measurements of 2 optic flow sensors without any prior knowledge of the optic flow variations is discussed. In section 6, the sensor fusion odometric process-Fig. 1: Hexarotor oscillating up and down while flying forward over the ground at the flight height h. a) The hexarotor's velocity V can be decomposed into the components V x and V h . Along the hexarotor's longitudinal axis x, the optic flow sensors are set at angles ˘φ with respect to the hexarotor's vertical axis, at the distance D with respect to the ground. They perceive the optic flow magnitudes ωpφq and ωp´φq, respectively. This configuration is also present along the hexarotor's lateral axis y. b) If V h is positive, the optic flow divergence component is a contraction (in blue) (i); if it is negative, the optic flow divergence component is an expansion (in blue) (ii). The contraction or expansion of the optic flow is superimposed in the ventral optic flow vector field on the translational optic flow (in red).

ing method based on 4 optic flow sensors is discussed, both with precise and with rough prior knowledge of the optic flow variations. In section 7, the indoor experimental setup is first described, and experiments are then presented showing that the two sensor fusion strategies based on the knowledge of optic flow variations increased the measurement quality of the local optic flow cues. Lastly, the performances of the three minimalistic in-flight optic flow based odometric processing methods are compared. In section 8, we first discuss the outdoor experimental setup and then present experiments showing that the same considerations also apply to preliminary flight tests performed outdoors. In section 9, conclusions are drawn and projects for future studies are discussed.

II. The SOFIa hexarotor

The hexarotor was developed together with Hexadrone TM and equipped with 4 Pixart PAW3903 optic flow sensors (see Figure 2 and Table I). The Pixart PAW3903 optic flow sensors were embedded on printed circuits to be set on the drone. The hexarotor's onboard low-level flight controller was the PX4 autopilot system [START_REF] Meier | PX4: A nodebased multithreaded open source robotics framework for deeply embedded platforms[END_REF], using a trajectory tracking algorithm 1 . Based on the intrinsic attitude stability of the hexarotor, we can assume that no rotational component is measured by the optic flow sensors. In addition, the pitch and roll components were taken to be negligible. The downward translational optic flow can therefore be measured along the x axis of the optic flow sensors.

III. Measurement of the local optic flow cues

The translational optic flow is the angular speed magnitude of the optic flow vector field generated by the translational motion of a drone flying above the ground [START_REF] Gibson | The perception of the visual world[END_REF]. The theoretical local translational optic flow ω th T can be expressed as the ratio between the V x component of the drone's velocity and its flight height h (see Figure 1):

ω th T " V x h (1)
The local translational optic flow can be measured on a hexarotor as the sum of two optic flow magnitudes ωpφq and ωp´φq perceived by two optic flow sensors oriented at angles ˘φ with respect to the hexarotor's vertical axis, divided by a known factor of 2 ¨cospφq 2 (see mathematical proof in Appendix C):

ω meas T " ωpφq `ωp´φq 2 ¨cospφq 2 " V x h (2)
In the case of a hexarotor equipped with 4 optic flow sensors as illustrated in Figure 2.b, three translational optic flow cues can be measured as follows:

' the sum of the two optic flow magnitudes perceived by the two optic flow sensors set along the longitudinal axis x, namely ω meas T 1 , ' the sum of the two optic flow magnitudes perceived on the x axis by the two optic flow sensors set along the lateral axis y, namely ω meas T 2 , ' the median value of the four optic flow magnitudes sensed along the hexarotor's longitudinal axis by the 4 optic flow sensors, scaled by a 1{ cospφq factor, namely ω meas T 3

.

The series of contractions and expansions generated in the optic flow vector field by up-and-down oscillatory movements is known as the optic flow divergence. When a drone flies forward while oscillating up and down above the ground, the optic flow divergence is superimposed on the translational optic flow in the optic flow vector field. Due to the oscillatory movements, the state vector X " rh, V h s T is locally observable [START_REF] Ho | Distance and velocity estimation using optical flow from a monocular camera[END_REF]. The theoretical local optic flow divergence ω th div can be expressed as the ratio between the V h component of the drone's velocity and h (see Figure 1):

ω th div " V h h (3)
We have previously proved mathematically that the local optic flow divergence can be measured on a micro-flyer as the difference between two optic flow magnitudes ωpφq and ωp´φq perceived by two optic flow sensors oriented at angles ˘φ with respect to the normal to a surface, divided by a known factor of sinp2φq (see mathematical proof in Appendix C) [START_REF] Bergantin | Estimation of the distance from a surface based on local optic flow divergence[END_REF]:

ω meas div " ωpφq ´ωp´φq sinp2φq " V h h (4)
In the case of a hexarotor equipped with 4 optic flow sensors, two optic flow divergence cues can be measured as follows:

' the difference between the two optic flow magnitudes perceived by the two optic flow sensors set along the longitudinal axis x, namely ω meas div x , ' the difference between the two optic flow magnitudes perceived by the two optic flow sensors set along the lateral axis y, namely ω meas div y .

IV. The SOFIa visual odometer method A model for the honeybee's visual odometer called SOFIa (Self-scaled Optic Flow time-based Integration model) was tested in simulations [START_REF] Bergantin | Oscillations make a self-scaled model for honeybees' visual odometer reliable regardless of flight trajectory[END_REF]. The SOFIa model is based on the integration of the local translational optic flow ω T scaled by the estimated distance with respect to the ground ĥ:

XSOFIa " ż ω T ¨ĥ dt (5)
ĥ was estimated by means of an EKF. The use of an EKF was necessary due to the non-linearity of the local optic flow divergence, as the measurement depends on the ratio between the two states V h and h (see equation ( 3)). a) State space representation of the hexarotor along the vertical axis:: The hexarotor's system was modeled in the form of a double integrator receiving as its input the acceleration a z on the vertical axis z given by the drone's IMU. The hexarotor's state space representation can therefore be expressed as follows:

$ & % 9 X " f pX, a z q " A ¨X `B ¨az " " 0 1 0 0 ȷ ¨X `" 0 1 ȷ ¨az Y " gpXq " rXp2q{Xp1qs " V h {h " ω div (6)
where X " " h, V h ‰ T is the hexarotor's state vector.

V. Odometric method based on 2 optic flow sensors with No Prior Knowledge (NPK) of the optic flow variations

The local optic flow divergence ω 2S div was measured by taking the difference between the two raw optic flow magnitudes perceived by the 2 optic flow sensors set along the x axis, while the local translational optic flow ω 2S T was measured in the form of their sum. To estimate the flight height ĥ, the EKF received the following:

' input: the acceleration of the drone a z , ' measurement: the local optic flow divergence ω 2S div . See Appendix B for the EKF calculations. ĥ was then used to scale the integration of the local translational optic flow ω 2S

T in order to perform the odometry. This odometric method based on 2 raw optic flow measurements does not require any prior knowledge about any parameters to assess the distance traveled.

VI. Fusion strategies based on 4 optic flow sensors A. Fusion strategy using Precise Prior Knowledge (PPK) of the optic flow variations

Here we investigated how to use prior knowledge about the self-oscillations to further improve the accuracy of the Fig. 3: a) The sensor fusion based on 2 Optic Flow (OF) sensors is achieved using an Extended Kalman Filter (EKF). The embedded computer handles the outputs of the optic flow sensors set on the hexarotor, whose outputs are used to measure the local optic flow divergence ω 2S div and the local translational optic flow ω 2S T . The EKF receives as its input the hexarotor's acceleration a z and as its measurement ω 2S div to estimate the current flight height ĥ. The EKF output ĥ scales ω 2S T , which is then integrated in order to perform the odometry. b) The sensor fusion based on 4 optic flow sensors is achieved by inserting additional Kalman Filters (KF). The EKF receives as its input the hexarotor's acceleration a z and as its measurement ω KF div to estimate the current flight height ĥ. The EKF output ĥ scales ω KF T , which is then integrated in order to perform the odometry.

distance traveled estimates with 4 optic flow sensors.

The optic flow divergence induced by the self-oscillation serving as an input to a Kalman Filter (KF) was expressed as follows (see Figure 4.a):

ω div " 9 h h ÝÑ U div pkq "
A osc 2πf osc cosp2πf osc kδtq h 0 `Aosc sinp2πf osc kδtq [START_REF] Expert | Flying over uneven moving terrain on optic-flow cues without any need for reference frames or accelerometers[END_REF] where f osc , the oscillation frequency, was equal to 0.28Hz, A osc , the oscillation amplitude, was equal to 0.25m, and h 0 , the average flight height, was equal to 0.55m. To fuse ω meas div x

and ω meas div y , a KF was used (see Figure 3). At each k th step, the KF received as input the current value of the model in the equation ( 7) and as measurements ω meas div x and ω meas div y . See Appendix A for the KF calculations. The translational optic flow induced by the forward motion serving as the input to a KF was expressed as follows (see Figure 4.b):

ω T " V x h ÝÑ U T pkq " ω KF T pk ´1q ¨ĥpk ´1q h 0 `Aosc sinp2πf osc kδtq (8)
V x p0q « ω KF T pk " 0q ¨ĥpk " 0q was initialized at 0.45m{s. To fuse the three translational optic flow cues ω meas , a KF was used (see Figure 3). At each k th step, the KF received as input the current value of the model in the equation ( 8) and as measurements ω meas

B. Fusion strategy using Rough Prior Knowledge (RPK) of the optic flow variations

Here we investigated how to implement the sensor fusion strategy based on 4 optic flow sensors without any knowledge Fig. 4: Inputs U div (a) and U T (b) to the Kalman Filters (KF) used to fuse optic flow divergence cues and translational optic flow cues with both the Precise Prior Knowledge (PPK) fusion strategy (in purple) and the Rough Prior Knowledge (RPK) fusion strategy (in green). In the RPK fusion strategy, a sinus helps the KFs to keep the timing of the oscillations and the shapes of the inputs. This is only a rough approximation of the complex optic flow cues variations that are used for the inputs of the PPK fusion strategy. of the oscillation amplitude A osc or the average flight height h 0 , just using the information about the general shape and timing of the oscillations during the trajectory. with the RPK strategy (d.ii). The flight height estimates ĥ converged within 4s with the ground truth values h given by the MoCap system in all 3 cases (b, c and d.iii). The average percentage error of ĥ with respect to h after convergence was ´9.77% with the NPK method ( range: r´61.5%, 65.34%s) (b.iv), ´2.16% with the PPK strategy ( range: r´36.89%, 34.13%s) (c.iv) and ´2.55% with the RPK strategy ( range: r´36.88%, 34.25%s) (d.iv). The final percentage error of the distance traveled estimates XSOFIa with respect to the ground truth X gt was ´8.57% with the NPK method (b.v), ´1.22% with the PPK strategy (c.v) and ´2.80% with the RPK strategy (d.v).

For this purpose, we approximated very roughly both the optic flow divergence and the translational optic flow cues in the form of a sinusoidal signal serving as the input to both KFs as follows (see Figure 4): U div pkq " ´UT pkq " sinp2πf osc kδtq [START_REF] Gibson | The perception of the visual world[END_REF] where the oscillation frequency f osc was taken to be equal to 0.28Hz. At each k th step, the two KFs received as input the current value of the model in the equation ( 9) and as measurements the optic flow divergence measurements (ω meas div x

and ω meas div y

) and the translational optic flow measurements

(ω meas T 1 , ω meas T 2
and ω meas T 3

), respectively. See Appendix A for the KF calculations. By using as KF input U T pkq " ´sinp2πf osc kδtq, the fusion strategy takes into account the fact that the variation of the translational optic flow is inversely proportional to the flight height h, since it depends only on the ratio V x {h (see equation ( 1)). As shown in Figure 4, only a rough approximation of the actual shape and timing is taken into account.

C. Extended Kalman Filter for the fusion strategy with 4 optic flow sensors

To estimate the drone's flight height ĥ, the EKF used received the following: Fig. 6: a) Comparison of the position of the hexarotor on the vertical plane (x,z) estimated with the Rough Prior Knowledge (RPK) fusion strategy (dashed line) with the ground truth given by the MoCap system (continuous line). The flight height estimates ĥ were plotted on the distance traveled estimates XSOFIa , while the ground truth h was plotted on X gt . This flight test was performed at an illuminance of 1518lux. b) The final percentage error in the distance traveled estimates XSOFIa with respect to the ground truth X gt was ´2.63%. Fig. 7: Distributions of the final percentage errors in the distance traveled estimates XSOFIa with respect to the ground truth X gt (traveled along the x axis) for 7 datasets recorded at 1518lux and 7 datasets recorded at 117lux. a.i) At an illuminance of 1518lux, the final percentage error ranged between ´8.57% and 5.52% (with a median value of ´1.14%) in the case of the No Prior Knowledge (NPK) method (in black), between ´1.65% and 1.08% (with a median value of ´0.8%) in that of the Precise Prior Knowledge (PPK) strategy (in purple) and between ´3.95% and 0.63% (with a median of ´1.55%) in that of the Rough Prior Knowledge (RPK) strategy (in green). a.ii) At an illuminance of 117lux, the final percentage error ranged between ´0.72% and 8.4% (with a median value of 4.73%) with the NPK method, between ´4.02% and 2.38% (with a median of ´0.27%) with the PPK strategy and between ´4.65%% and 2% (with a median value of ´1.14%) with the RPK strategy. b) Upon combining all 14 datasets recorded, the final percentage error ranged between ´8.57% and 8.4% (median value: 0.47%) with the NPK method, between ´4.02% and 2.38% (median value: ´0.53%) with the PPK strategy and between ´4.65% and 2% (median value: ´1.34%) with the RPK strategy.

' input: the acceleration of the drone a z , ' measurement: the local optic flow divergence ω KF div filtered by the KF based on the optic flow divergence measurements. See Appendix B for the EKF calculations. ĥ was used to scale the local translational optic flow ω KF T , which was then integrated to perform the odometry as follows:

XSOFIa " ż ω KF T ¨ĥ dt (10)

VII. Indoor experimental flight tests A. Indoor experimental setup

Indoor flight tests were performed in the Mediterranean Flight Arena (see Figure 5.a). The position and orientation used in the hexarotor's control system were taken from the motion-capture (MoCap) system installed in the flight arena, consisting of 17 motion-capture cameras covering a 6 × 8 × 4 m (lxLxH) volume using a VICON TM system. Datasets including the optic flow measurements were recorded via the Robot Operating System (ROS) and processed with the Matlab/Simulink 2022 software.

B. Indoor experimental results

The sensor fusion strategies based on Precise Prior Knowledge (PPK) and Rough Prior Knowledge (RPK) of the optic flow variations (using 4 optic flow sensors) were compared with the strategy based on No Prior Knowledge (NPK) of the optic flow variations (using 2 optic flow sensors). 7 bouncing circular flight tests over a distance of about 50m were performed with the hexarotor under an illuminance of 117lux (5.36 ¨10 ´6W {cm 2 ) and an illuminance of 1518lux (2.71 ¨10 ´4W {cm 2 ), amounting to a total number of 14 flight tests. First, the 14 datasets were processed with the NPK method (see Section (V)). The 14 datasets were then processed using the PPK strategy (see Section (VI-A)) and the RPK strategy (see Section (VI-B)). Supp. video n°1 shows a synchronised video of the odometry results obtained with the RPK fusion strategy for the same flight test processed in Figure 5 and6. The KF parameters discussed in Appendix A were defined experimentally as Φ " 10, Γ " 10 and H k " 10, based on the first dataset recorded under an illuminance of 1518lux and used to process all 14 datasets for both PPK and RPK fusion strategies. As shown in Figure 5, the optic flow measurements were processed with the three strategies (NPK, RPK and PPK), taking the same dataset recorded under an illuminance of 1518lux. The increase in the Signal-to-noise Ratio (SnR, computed as the square ratio of the root mean square of the signal and the root mean square of its noise) for the local optic flow divergence and the local translational optic flow in the case of the PPK and the RPK strategies in comparison with the NPK method affected the average percentage error of the flight height estimates after convergence (at 4s). The average percentage error of the flight height estimates was ´9.77% with the NPK method, ´2.16% with the PPK strategy and ´2.8% with the RPK strategy. Similar results were obtained with all 14 datasets. The SnR of the local translational optic flow measured with the NPK method ranged between 18.08dB and 24.79dB, between 24.84dB and 29.93dB with the PPK strategy and between 24.84dB and 31.39dB with the RPK strategy. The SnR of the local optic flow divergence measured with the NPK method ranged between 5.41dB and 5.71dB, and between 6.47dB and 7.11dB with both the PPK and RPK strategies. The flight height estimates ĥ and distance traveled estimates XSOFIa were used to assess the position of the hexarotor on the vertical plane (x,z). An example is shown in Figure 6, where the flight height estimates ĥ were plotted on the distance traveled estimates XSOFIa (which are given directly in meters) and compared with the ground truth values given by the MoCap system. Since the 2D position estimates were based on the optic flow based odometry, they were subject to an accumulated error increasing with the distance covered. Overall, the final percentage error in the distance traveled estimates XSOFIa with respect to the ground truth X gt (traveled along the x axis) ranged between ´8.57% and 8.4% with the NPK method, between ´4.02% and 2.38% with the PPK strategy and between ´4.65% and 2% with the RPK strategy (see Figure 7.b). Similar results were obtained taking the two different illuminances separately (see Figure 7.a).

VIII. Preliminary outdoor experimental flight tests A. Outdoor experimental setup

Outdoors, for trajectory tracking purposes, the hexarotor was equipped with a TeraRanger Evo 3m distance sensor in order to measure the flight height of the drone and with a Pixhawk GPS sensor (from Holybro) in order to measure the horizontal position. These 2 sensors were connected directly to the PX4 flight controller. In order to validate the precision of the TeraRanger Evo 3m distance sensor with the help of the MoCap system, a test was performed in the flight arena, in which we observed that TeraRanger Evo 3m was very reliable. This reliability was confirmed by the quality of the TeraRanger Evo 3m sensor's output, which was devoid of high frequency noise when measured on the field (see Section (2) of Supp. Information). According to the PX4 documentation, the standard deviation of the horizontal position error is 0.8 m using GPS outdoors. We observed that the hexarotor flying in the horizontal plane had a maximum deviation of about 1 m with respect to the desired trajectory (see Figure 8). Besides, the physical distance on the hexarotor of 5cm between the TeraRanger Evo 3m sensor and the optic flow sensors has been subtracted to the flight height estimates ĥ to be compared to the ground truth h (measured by the TeraRanger Evo 3m sensor) in Figures 9, 10 and 11. The outdoor experiments were performed using the same set of Pixart PAW3903 optic flow sensors, but adding a neutral density filter of 2 in front of the lenses to attenuate the solar luminosity. Without these filters, the optic flow sensors would have been saturated.

B. Results of preliminary outdoor experiments

4 bouncing longitudinal flight tests over a distance of about 20m were performed with the hexarotor outdoors over a field irregularly covered with grass (see Supp. video n°2 and 3). In Figure 9, the optic flow measurements of the outdoor flight number 2 were processed with the NPK method, the PPK strategy and the RPK strategy (see the odometry results obtained with the RPK fusion strategy in Supp. video n°2 for the outdoor flight number 2). The values of the KF parameters defined in Appendix A were those previously used in the indoor flight tests. As with the indoor flight tests, the SnRs of the local optic flow divergence and those of the local translational optic flow were greater with the PPK and RPK strategies than with the NPK method (see Section (1) of Supp. Information). Figure 10 gives the 2D position estimation on the vertical plane (x,z) for the outdoor flight number 2. Due to the presence of wind disturbances and the greater convergence time required by the EKF in outdoor visual setting, the distance traveled XSOFIa was estimated only after a convergence time of 5s. Overall, the final percentage Fig. 9: Hexarotor flying over a field irregularly covered with grass (outdoor flight n ˝2) (a). The same dataset recorded outdoors was processed with the No Prior Knowledge (NPK) method (b), the Precise Prior Knowledge (PPK) strategy (c) and the Rough Prior Knowledge (RPK) strategy (d). The local Optic Flow (OF) divergence (in blue) measured had a Signal-to-noise Ratio (SnR) of 5.74dB with the NPK method (b.i), 6.15dB with the PPK strategy (c.i) and 6.14dB with the RPK strategy (d.i). The local translational optic flow (in red) measured had a SnR of 5.9dB with the NPK method (b.ii), 7.41dB with the PPK strategy (c.ii) and 7.4dB with the RPK strategy (d.ii). The flight height estimates ĥ converged within 5s with the ground truth values h given by the distance sensor in all 3 cases (b, c and d.iii). The average percentage error of ĥ with respect to h after convergence was 1.1% with the NPK method ( range: r´56.77%, 85.65%s) (b.iv), 7.12% with the PPK strategy ( range: r´33.91%, 79.94%s) (c.iv) and 7.19% with the RPK strategy ( range: r´33.91%, 83.34%s) (d.iv). The distance traveled estimates XSOFIa with respect to the ground truth X gt were computed only after convergence. The final percentage error was 6.83% with the NPK method (b.v), 3.68% with the PPK strategy (c.v) and 4.08% with the RPK strategy (d.v).

error in the distance traveled estimates XSOFIa with respect to the ground truth values X gt ranged between 1.77% and 17.23% with the NPK method, between ´8.92% and 9.4% with the PPK strategy and between ´8.37% and 9.7% with the RPK strategy (see Figure 11). As with the indoor flights (see Figure 7), Figure 11 shows comparable results between the PPK and the RPK fusion strategies. The RPK fusion strategy seems to be very interesting because it only uses knowledge of the timing and general shape of the translational and divergence optic flow cues. This knowledge can be considered reasonable since it is available onboard the UAV: it is in fact the drone itself that shapes the optic flow cues by creating these oscillations.

IX. Conclusion

In this study, we investigated how to use information about the oscillating trajectory to improve a minimalistic odometry based on optic flow cues. The experiments were performed onboard a hexarotor first indoors, following circular bouncing trajectories at a frequency of 0.28Hz over distances of about 50m under illuminances of 117lux and 1518lux. The results were not affected by the illuminance conditions. A few tests were then performed outdoors, where the hexarotor followed bouncing longitudinal trajectories over a distance of about 20m over a field irregularly covered with grass in the presence of various wind conditions. Fig. 10: a) Comparison between the position of the hexarotor on the vertical plane (x,z) estimated with the Rough Prior Knowledge (RPK) fusion strategy (dashed line) and the ground truth values given by the distance sensor (continuous line) (outdoor flight n ˝2). The flight height estimates ĥ were plotted on the distance traveled estimates XSOFIa , while h was plotted on X gt (the ground truth values). b) The final percentage error in the distance traveled estimates XSOFIa with respect to the ground truth values X gt was 4.08%. Fig. 11: In the 4 datasets recorded outdoors, the final percentage error in the distance traveled estimates XSOFIa with respect to the ground truth values X gt ranged between 1.77% and 17.23% (median value: 4.67%) with the No Prior Knowledge (NPK) method (in black), between ´8.92% and 9.4% (median value: ´2.49%) with the Precise Prior Knowledge (PPK) strategy (in purple) and between ´8.37% and 9.7% (median value: ´2.1%) for Rough Prior Knowledge (RPK) (in green).

The findings obtained in this study show that the sensor fusion strategies based on the use of 4 optic flow sensors make it possible to measure the optic flow divergence and the translational optic flow cues more reliably thanks to the use of additional Kalman Filters. This was the case even when taking only rough prior knowledge about the optic flow variations into account, and more specifically, only the general shape and timing of the oscillations during the trajectory. This prior knowledge can be considered acceptable since the general shape and timing of the oscillations are imposed by the drone itself on its own forward trajectory. The sensor fusion strategies presented decreased the error in the flight height estimates, and thus decreased the percentage error in the distance traveled estimates in all the cases considered, improving the odometric performances. These considerations also applied in the case of the few outdoor flight tests performed in the presence of wind and an irregular pattern of grass. With all three odometric processing methods, the final distance traveled estimates were admittedly subject to small errors as the odometric strategy is a dead reckoning method involving no feedback from the environment. Nevertheless, we show here that the SOFIa model can be accurate and precise enough to move in close proximity to a target without GPS, indoors and outdoors. Likewise, this highly minimalistic optic flow based odometric strategy could also be used to enable a future drone to assess whether it is returning near its base station without any need for a GPS. So far, the present findings can be said to constitute the first experimental proofof-concept of the SOFIa model [START_REF] Bergantin | Oscillations make a self-scaled model for honeybees' visual odometer reliable regardless of flight trajectory[END_REF] before this optic flow based odometric strategy is implemented on a nanodrone requiring very little computational power [START_REF] Palossi | Ultra lowpower visual odometry for nano-scale unmanned aerial vehicles[END_REF]. We now intend to test the robustness of these strategies in a range of forward speeds, in cases where a large drone pitch occurs and in the presence of strong reliefs. Future studies will also include the implementation of an optic flow regulator keeping the translational optic flow around a given setpoint. each k th step, the current value of the corresponding model was computed and given to the corresponding KF as input (see Figure 3). In the following paragraph, the notation A ą 0 indicates a strictly positive definite matrix. The KF took the following iterative steps at each k th time: Prediction step (a) One-step ahead prediction

X k{k´1 " Φ ¨Xk´1{k´1 `Γ ¨Uk´1{k´1 (11)
with Φ ą 0, Γ ą 0. (b) Covariance matrix of the state prediction error vector

P k{k´1 " Φ ¨Pk´1{k´1 ¨ΦT `Q (12)
Correction step (c) Measurement update

X k{k " X k{k´1 `Kk ¨pY i k ´Hk ¨Xk{k´1 q (13)
with Y i k current value of the i th measurement, H k ą 0 and K k Kalman gain defined as:

K k " P k{k´1 ¨HT k ¨rH k ¨Pk{k´1 ¨HT k `Rk s ´1 (14)
The measurement update step was repeated for each i th measurement (2 times for the optic flow divergence and 3 times for the translational optic flow). (d) Covariance matrix of state estimation error vector

P k{k " P k{k´1 `Kk ¨rH k ¨Pk{k´1 ¨HT k `Rk s ¨KT k ( 15
) (e) Innovation Ỹk " Y k ´Hk ¨Xk{k (16)
Appendix B: Extended Kalman Filter calculations The discretized model for the hexarotor along the vertical axis (see equation ( 6)) can be expressed as follows:

" Xrk `1s " Φ ¨Xrks `Γ ¨U rks Y rks " C k ¨Xrks `Dk ¨U rks (17)
with Φ " e A¨dt (

Γ " p

ż dt 0 e A¨τ dτq ¨B " pA T ¨eA¨dt ´AT q ¨B (19) C k " gpX k q " " X 2 rks X 1 rks ȷ " " V h rks hrks ȷ ( 20
) D k " 0 ( 21
)
where dt is the discretization time. To estimate the flight height h, the EKF took the following iterative steps at each k th time: Prediction step (a) One-step ahead prediction

X k{k´1 " Φ ¨Xk´1{k´1 `Γ ¨Uk´1{k´1 (22)
(b) Covariance matrix of the state prediction error vector

P k{k´1 " Φ ¨Pk´1{k´1 ¨ΦT `Q (23)
Correction step (c) Measurement update X k{k " X k{k´1 `Kk ¨pY k ´Hk ¨Xk{k´1 q (24)

with K k Kalman gain defined as:

K k " P k{k´1 ¨HT k ¨rH k ¨Pk{k´1 ¨HT k `Rk s ´1 (25)
and H k Jacobian matrix for the non linear function defined as follows:

H k " Bg BX | X"X k{k´1 "
" ´Vh The local optic flow divergence can be measured as the difference between two optic flow magnitudes ωpφq and ωp´φq perceived by two optic flow sensors oriented at angles ˘φ with respect to the normal to a surface, divided by a known factor of sinp2φq:

ω meas div " ωpφq ´ωp´φq sinp2φq " V h h (29)
The local translational optic flow can be measured as the sum of ωpφq and ωp´φq, divided by a known factor of 2 ¨cospφq 2 :

ω meas T " ωpφq `ωp´φq 2 ¨cospφq 2 " V x h ( 30
)
Proof. We take a drone equipped with two optic flow sensors oriented toward the ground at angles φ and ´φ with respect to its vertical axis. The optic flow magnitudes perceived by each optic flow sensor can be expressed as follows:

ωpφq " || Ý Ñ V || D ¨sin { `Ý Ñ D , Ý Ñ V ˘" || Ý Ñ V || D ¨sinp π 2 ´φ `αq
The two components of the velocity vector Ý Ñ V of the drone flying above the ground can be expressed as follows:

V x " || Ý Ñ V || ¨cos α V h " || Ý Ñ V || ¨sin α with || Ý Ñ V || " b V 2 x `V 2 h
From which we obtain: ´φ"

cos α " V x b V 2 x `V 2 h sin α " V h b V 2 x `V 2 h Thus ωpφq " || Ý Ñ V || D ¨sin { ´Ý Ñ D , Ý Ñ V " b V 2 x `V 2
V x D ¨sin ˆπ 2 ´φ˙`V h D ¨sin φ " › › › Ý Ñ V x › › › D ¨sin { ´Ý Ñ D , Ý Ñ V x ¯`› › › Ý Ñ V h › › › D ¨sin { ´Ý Ñ D , Ý Ñ V h
We can then express the optic flow magnitudes ωpφq and ωp´φq perceived by the two optic flow sensors as follows:

ωpφq " V

Fig. 2 :

 2 Fig. 2: a) Hexarotor equipped with 4 optic flow sensors oriented towards the ground flying along a bouncing circular trajectory in the Mediterranean Flight Arena. b) 2 optic flow sensors were set along the longitudinal axis x at angles φ " ˘30 o with respect to the hexarotor's vertical axis z, while the other 2 optic flow sensors were set along the lateral axis y at angles φ " ˘30 o with respect to the axis z. c) Example of a flight test trajectory over a distance of 53m at an oscillation frequency of 0.28Hz.

  Fig.3: a) The sensor fusion based on 2 Optic Flow (OF) sensors is achieved using an Extended Kalman Filter (EKF). The embedded computer handles the outputs of the optic flow sensors set on the hexarotor, whose outputs are used to measure the local optic flow divergence ω 2S div and the local translational optic flow ω 2S T . The EKF receives as its input the hexarotor's acceleration a z and as its measurement ω 2S div to estimate the current flight height ĥ. The EKF output ĥ scales ω 2S T , which is then integrated in order to perform the odometry. b) The sensor fusion based on 4 optic flow sensors is achieved by inserting additional Kalman Filters (KF). ω meas div x and ω meas div y are taken as measurements by a KF (denoted KF div ) receiving as its input the current value U div of the model for the optic flow divergence. The output of the KF is the local optic flow divergence ω KF div . ω meas T 1 , ω meas T 2 and ω meas T 3 are taken as measurements by a KF (denoted KF T ) receiving as its input the current value U T of the model for the translational optic flow. The output of the KF is the local translational optic flow ω KF T .The EKF receives as its input the hexarotor's acceleration a z and as its measurement ω KF div to estimate the current flight height ĥ. The EKF output ĥ scales ω KF T , which is then integrated in order to perform the odometry.

3 .

 3 See Appendix A for the KF calculations.

Fig. 5 :

 5 Fig. 5: Hexarotor flying in the Mediterranean Flight Arena (a). The same dataset taken at 1518lux was processed with the No Prior Knowledge (NPK) method (b), the Precise Prior Knowledge (PPK) strategy (c) and the Rough Prior Knowledge (RPK) strategy (d). The local Optic Flow (OF) divergence (in blue) measured with the NPK method had a Signal-to-noise Ratio (SnR) of 5.62dB (b.i), while with both the PPK and the RPK strategies the SnR was 6.72dB (c and d.i). The local translational optic flow (in red) measured with the NPK method had a SnR of 19.12dB (b.ii), 25.74dB with the PPK strategy (c.ii) and 25.9dBwith the RPK strategy (d.ii). The flight height estimates ĥ converged within 4s with the ground truth values h given by the MoCap system in all 3 cases (b, c and d.iii). The average percentage error of ĥ with respect to h after convergence was ´9.77% with the NPK method ( range: r´61.5%, 65.34%s) (b.iv), ´2.16% with the PPK strategy ( range: r´36.89%, 34.13%s) (c.iv) and ´2.55% with the RPK strategy ( range: r´36.88%, 34.25%s) (d.iv). The final percentage error of the distance traveled estimates XSOFIa with respect to the ground truth X gt was ´8.57% with the NPK method (b.v), ´1.22% with the PPK strategy (c.v) and ´2.80% with the RPK strategy (d.v).

Fig. 8 :

 8 Fig. 8: Top view of the drone's horizontal trajectory during an outdoor flight test (outdoor flight n ˝2). The drone was equipped with a TeraRanger Evo 3m distance sensor to measure the flight height and a Pixhawk GPS to measure its position on the horizontal plane (x, y). The take-off point of the flight was taken to be r0, 0s. The optic flow based odometry is performed along the darker part of the trajectory.

Ỹk " Y k ´Hk ¨Xk{k ( 28 )

 28 Covariance matrix of state estimation error vector P k{k " P k{k´1 `Kk ¨rH k ¨Pk{k´1 ¨HT k `Rk s ¨KT k Appendix C: Computation of the local divergence and translational optic flow cues

TABLE I :

 I Characteristics of the optic flow sensors equipped on the hexarotor.

  Since h " D ¨cospφq is the distance of the drone from the ground, equation (33) can be written as follows:

		¨sin	ˆπ 2	´φ˙`V h D	¨sin φ	(31)
	ωp´φq "	V x D	¨sin	ˆπ 2	´φ˙´V h D	¨sin φ	(32)
	Subtracting equations (31) and (32), we obtain:
	ωpφq ´ωp´φq " 2 ¨Vh D	¨sinpφq	(33)
	ωpφq ´ωp´φq " 2 ¨Vh h	¨sinpφq ¨cospφq	(34)
	Using the trigonometric formula sinpφq ¨cospφq " 1 2 we can rewrite equation (34) as follows:	sinp2φq,
	ω meas div "	ωpφq ´ωp´φq sinp2φq	h h
	Summing equations (31) and (32), we obtain:
	ωpφq `ωp´φq " 2 ¨Vx h		¨sinp	π 2	´φq ¨cospφq	(35)
	Using the trigonometric formula sinp π 2 ´φq " cospφq, we can rewrite equation (35) as follows:
	ω meas T	"	ωpφq `ωp´φq 2 ¨cospφq 2 "	V x h

x D

https://github.com/gipsa-lab-uav/trajectory control

Acknowledgements

We thank Dr. J. Blanc for improving the English manuscript and J.M. Ingargiola for his help with designing the printed circuit boards for the optic flow sensors. Financial support was provided via a ProxiLearn project grant (ANR-19-ASTR-0009) to F.R. from the ANR (Astrid Program). The participation of L.B. in this research project was supported by a joint PhD grant from the Délégation Générale de l'Armement (DGA) and Aix Marseille University. The participation of C.C. was supported by the CNRS Innovation via a Prematuration project. L.B, C.C., and F.R. were also supported by Aix Marseille University and the CNRS (Life Science, Information Science Institute as well as Engineering Science & Technology Institute). This work has been partially supported by ROBOTEX 2.0 (Grants ROBOTEX ANR-10-EQPX-44-01 and TIRREX ANR-21-ESRE-0015). We are grateful to the three anonymous Referees, whose suggestions have helped us to greatly improve the manuscript.

Appendix

Appendix A: Kalman Filter calculations In the PPK strategy, the optic flow divergence and the translational optic flow cues were expressed as in the equations ( 7) and ( 8), respectively. In the RPK strategy, the optic flow divergence and the translational optic flow cues were both expressed as in the equation [START_REF] Gibson | The perception of the visual world[END_REF]. With each optic flow cue, at


Y I Qiang

Qiushi Huang
email: huangqs@tongji.edu.cn

Jinshuai Zhang

Xiangmei Wang
email: wangzs@tongji.edu.cn

Runze Qi

Zhong Zhang

Rongkun Xu

Taiping Peng

Philippe Jonnard

Angelo Giglia

Thermal and temporal stability of V/Sc multilayer mirror for the soft X-ray water window region

V/Sc multilayer has been developed as a promising candidate for the water window region working near both the Sc-L edge (3.11 nm) and the V-L edge (λ=2.42 nm). To meet the application requirements, its thermal and temporal stability of it are studied. A series of V/Sc multilayers with a d-spacing ranging from 1.59 to 1.62 nm and a number of bilayers of 100 and 400 was fabricated. Different samples were annealed at temperatures from 100 to 300 °C. Grazing incidence X-ray reflectivity and soft X-ray reflectivity (SXR) measurements showed that the multilayer structure and reflectivity remain mostly unchanged at 100 °C. An obvious deterioration started at 200 °C, while the SXR reflectivity drops close to zero at 250 °C. The multilayer structure completely broke down at 300 °C. Both the X-ray diffraction and transmission electron microscopy measurements demonstrated a severe crystallization occurring at 250 °C which can be mainly identified as Sc(002) and V(110). Energy dispersive X-ray spectroscopy measurements have further confirmed a significant elemental mixture occurring after 250 °C annealing. The severe intermixing and polycrystallization resulted in the degradation of the multilayer structure. On the other hand, the V/Sc multilayer is relatively stable after half year storage in air environment showing a slight decrease of SXR reflectivity from 18.4% to 16.5%.

Introduction

Soft X-ray imaging at the water window wavelength region (λ=2.3 nm-4.4 nm) is of great importance for biology science due to the natural imaging contrast between carbon and water [13]. Spatial resolution and exposure time are the two critical parameters to evaluate the performance of a microscope. Presently a high spatial resolution of 10 nm [4,5] has been demonstrated while the exposure time and the system efficiency still needs to be improved particularly for the compact microscopes based on the plasma source and high harmonic generation (HHG) source [6]. On the other hand, the rapid advancement of free-electron laser (FEL) facilities with the wavelength ranging into water window region is also motivating the development of the related microscopy techniques [7]. The performance of the microscope is affected by each optical component, like the condenser, objective mirror, polarizer, and so on [5,8,9]. Multilayer mirrors are widely used in the water window microscopes. They have the advantages of providing relatively high collection efficiency, with large numerical aperture and well-defined spectral selectivity.

Due to the short wavelengths of the water window region, the near normal incidence multilayers require a d-spacing of less than 2 nm and each layer thickness is below 1 nm. This imposes a big challenge to the development of related multilayer optics. Sc-, Ti-, and V-based multilayers are usually selected in the water window range given to their absorption edges at λ=3.109 nm, 2.730 nm, and 2.420 nm, respectively [4]. Among which, Cr/Sc multilayer is the most widely studied and used near the Sc L-edge [1012]. The highest reflectance of Cr/Sc achieved was 32.1% at =3.11 nm, in which the structure is optimized by the introduction of diffusion barrier layers of B4C [12]. Despite the improved experimental reflectance, this value is still much lower than the theoretical prediction of around 60%. To further enhance the reflectance, Loch et al. proposed a new material combination of V/Sc. It has the same theoretical reflectance as Cr/Sc at near the Sc-L edge with even a little higher single-shot damage threshold in theory [13]. Moreover, the same V/Sc multilayer can also work at near V-L edge at grazing incidence which is a unique advantage for application. Inspired by this result, we have recently studied this multilayer experimentally which showed smaller interface widths than the Cr/Sc multilayer with no interface engineering methods applied. A high reflectance of 18.4% was measured at 3.129 nm under a near normal incidence of 11º based on the first experiment on V/Sc [14], demonstrating a new and competitive candidate for the multilayer mirror working in the water window region. The detailed results can be found in reference [14].

Despite the promising optical performance, stability is another critical factor for the application of multilayer mirrors with harsh working environment, e.g. under high irradiation power [15,16]. Thermal stability of the conventional Cr/Sc multilayer has been widely studied. Majkova et al [15] has found that Cr/Sc multilayer shows a reduction of 1 st Bragg peak intensity by 25% after annealing for 33 h under 280 °C and a structure collapse after annealing under 650 °C for 3 h. Eriksson et al. have also observed a continuous degradation starting at 100 °C and a complete destruction at 510 °C of the pure Cr/Sc multilayer [16]. Prasciolu et al. have found that Cr/Sc multilayer shows a reduction of 1 st Bragg peak intensity by 15% after annealing at 150°C for 1 h and a structure collapse after annealing at 450°C for 1 h [7]. The small difference of thermal stability was attributed to the different layer thickness and fabrication methods. The thermal stability of Cr/Sc can be enhanced by introducing nitrogen in the working gas or by using B4C as barrier layers. For instance, the nitridated Cr/Sc multilayers display an improved structural quality upon annealing to 250 °C and a final breakdown above 510 °C [16]. Prasciolu et al. have reported that the thermal stability of Sc/B4C/Cr multilayers is much improved than Sc/Cr and the structure collapsed after annealing for over 12 h under 600 °C [7].

The thermal and temporal stabilities of the new V/Sc multilayer candidate were not reported before. In order to use the V/Sc multilayer in various applications, the layer structure and reflectivity changes of V/Sc multilayers under elevated temperatures and long-term storage are investigated in this paper. A moderate thermal stability and relatively good temporal stability are found based on the different characterization results. The stability of V/Sc is compared to the one of Cr/Sc and the related physical mechanisms for the changes are discussed.

Experiments

The multilayers were deposited by direct current magnetron sputtering technique. The base pressure used in the deposition chamber is 6.0×10 -5 Pa. High purity argon (99.999%) is used as the sputtering gas with a sputtering pressure of 0.13 Pa. The depositions rates of V and Sc were 0.04 nm/s and 0.07 nm/s, respectively. The deposition parameters were designed with a d-spacing of 1.60 nm and a thickness ratio of γ~0.5. Five (after you write "four" samples. But perhaps are you adding a non-anneled sample for comparison purpose. At the moment it is not clear.) samples with 100 bilayers for thermal stability study and a sample with 400 bilayers for temporal stability study were deposited. A B4C layer was added on the top of all multilayers as capping layer with 2-3 nm thickness to prevent oxidation. It will only induce about 1% (absolute) reduction of the theoretical reflectance of the multilayer [Erreur ! Signet non défini.]. The annealing experiments were performed on the four samples with 100 bilayers in a high vacuum environment with a pressure lower than 5×10 -4 Pa [17]. The four samples were annealed for one hour at 100 °C, 200 °C, 250 °C, and 300 °C, respectively. Each annealing process was conducted with a heating rate of 5 °C/min. (New paragraph) All multilayers were measured by the grazing incidence X-ray reflectometry (GIXR) at the Cu-Kα emission line (=0.154 nm) before and after annealing. The soft X-ray (SXR) reflectivity of the as-deposited and annealed samples was measured using the reflectometer at the Bending magnet for Emission Absorption and Reflectivity (BEAR) beamline at the ELETTRA synchrotron facility. The X-ray diffraction (XRD) scans between 2θ = 30 ° ~ 50 ° were collected to study the crystallization change of the annealed samples. The morphology and internal structure of the multilayer sample with 400 bilayers as-deposited and the sample with 100 bilayers after annealing at 250 °C were further characterized by transmission electron microscopy (TEM). The TEM measurements were performed by using the FEI Tecnai G2 F20 equipment by Materials Analysis Technology Inc. High resolution bright field images, selected area electron diffraction (SAED) patterns were taken to study the layer structure changes and energy dispersive X-ray spectroscopy (EDX) were used to analyze the elemental distribution.

The V/Sc multilayer with 400 bilayers is further stored in air environment with ~20% humidity at room temperature for 6 months in order to study its temporal stability. The layer structure of the sample was monitored by measuring the GIXR during the storage period. The SXR reflectivity of the sample was measured at the ELETTRA synchrotron both before and after the 6 months storage.

Results and discussion

Thermal stability

GIXR measurements

(there was a mix of present and past tense in this paragraph; I put it at the present tense but after I have seen that the following paragraph is at the past tense) The measured GIXR curves of the four annealed samples are shown in Fig. 1. For the sample annealed at 100 °C as shown in Fig. 1(a), the reflectivity and bandwidth of the 1 st and 2 nd order Bragg peaks are almost the same as that of the as-deposited one, which indicates a similar periodic multilayer structure. The position of the small oscillation before the 1 st Bragg peak slightly shifts that may be caused by the change of the surface B4C layer or the first few bilayers of V and Sc. For the sample annealed at 200 °C as shown in Fig. 1(b), the 1 st Bragg peak reflectivity decreases by a small amount of 15% (relatively) while the 2 nd Peak reflectivity drops significantly by 38% (relatively). It is also found that the two Bragg peaks shift slightly to smaller angles, indicating a slight expansion of the d-spacing by 0.02 nm. This means the V/Sc multilayer structure started to change obviously at 200 °C. As the annealing temperature increases to 250 °C, Fig. 1(c), the 1 st Bragg peak reflectivity drops dramatically by one order of magnitude and the 2 nd Bragg peak cannot be observed.

The period expansion is still very small, below 0.02 nm. A small peak appears at 1.5° which can be from the interference effect of a newly formed structure inside the stack. Evidently, the layer interfaces are much deteriorated after 250 °C annealing. For the sample annealed at 300 °C, Fig. 1(d), the 1 st Bragg peak almost vanishes and a small peak appears at around 1°. The multilayer structure completely breaks down after annealing at 300 °C. The detailed measured results and structural parameters of the annealed samples are listed in table 1. The reflectivity is normalized to the value of the as-deposited value.

XRD measurements

To study the structure changes occurring under annealing, the XRD measurements of the as-deposited and annealed V/Sc multilayers with 100 bilayers were performed and are displayed in Fig. 2. For the as-deposited sample, a broad diffraction peak was observed at about 35.6º which indicated a very weak polycrystallization. This can be mainly identified as the crystalline phase of hexagonal Sc (101). For the sample annealed under 100 °C, the diffraction peak was almost the same as the as-deposited one. As the temperature increased to 200 °C, the peak intensity was slightly enhanced and a new peak seems to appear at the lower angles of around 34.0º. After annealed at 250 °C, the diffraction peaks were significantly changed. The original broad peak become much narrower and centered at 34.0°, which was identified as hexagonal Sc (002). The crystallization of Sc (002) was also reported in the Cr/Sc multilayers under annealing [7]. Besides, a new diffraction peak also appeared at around 42.1°, that can be identified as the crystalline phase of cubic V (110). As the annealing temperature further increased to 300 °C, the two diffraction peaks were significantly enhanced while no other peaks appeared. Although a small amount of impurities of oxygen, nitrogen and carbon can exist in the layers as indicated by the EDX measurement in the next section, the crystallization of oxides and carbides were not considered as the content would be low even if existed.

Assuming that the diffraction peak corresponds to a single phase, it can be calculated by the Scherrer equation [18] that the grain sizes of Sc (002) and V (110) are 5.30 nm and 2.16 nm, respectively, after 250 °C annealing. For the sample annealed at 300 °C, the grain sizes increased to 9.80 nm for Sc (002) and 5.80 nm for V (110), respectively.

(New paragraph) According to the XRD results, The polycrystalline structure of the multilayer starts to change at 200 °C and new phases are forming at 250 °C with relatively large grain size as compared to the layer thicknesses. After 300 °C annealing, the crystallization further increased with much larger grain size. This is consistent with the deterioration process of the multilayer observed in the GIXR measurements. As the size of the crystallites are much larger than the single layer thickness of V and Sc (0.8 nm), a severe inter-diffusion probably occurred during the annealing process.

SXR reflectivity measurements

The SXR reflectivity results of V/Sc multilayers with 100 bilayers are shown in Fig. 3. Given to the small difference of the d-spacings among samples, they were measured under the incidence angle of 13º-11º to make the peak position appearing at around the same wavelength. The peak reflectance of the as-deposited V/Sc multilayer was 3.0% at =3.147 nm. After annealing at 100 °C, the reflectivity was 2.9% at 3.150 nm, which was almost the same as the as-deposited one. The peak reflectivity dropped significantly by 30% (relatively) to 2.1% at 3.155 nm after the sample was annealed at 200 °C. This deterioration was more pronounced than the change of hard X-ray reflectivity shown in the GIXR measurements. It can be explained by the effect from the oxidation and other structural changes of the surface layers that the hard X-ray reflectivity measurement is not very sensitive to. As the sample was annealed under 250 °C, the SXR reflectivity decreased down to 0.2% at 3.143 nm. With the annealing temperature rising up to 300 °C, the reflectance was too low to be measured and is not shown here. The SXR measurement results are also listed in table 1. Both the SXR and the GIXR results have indicated that the layer structure of the V/Sc multilayers started to deteriorate at around 200 °C, and it was mostly destructed at 250 °C. For the Cr/Sc multilayer, the deterioration starts at ~100 °C, while the relative drop of SXR reflectivity is around 30% at 300 °C and the structure breaks down at above 450 °C [7,15,16]. As a comparison, the thermal stability of pure V/Sc multilayer is not as good as Cr/Sc.

TEM measurements

The layer morphology and microstructure of the multilayers before and after annealing were further studied by TEM. The cross-sectional images of the as-deposited sample with 400 bilayers and the 250 °C annealed sample with 100 bilayers are shown in Fig. 4. The bottom part of the 400-bilayers stack represents the structure of the 100-bilayers multilayer before annealing. As depicted in Fig. 4, the bright layers are Sc and the dark layers are V. For the as-deposited sample (Fig. 4(a)), the multilayer exhibited a clear periodic layered structure with a d-spacing of about 1.62 nm. The SAED pattern in Fig. 4(b) showed a relatively weak polycrystallization of the layers and the diffraction spots were identified as Sc (101) [14]. This is consistent with the XRD results (Fig. 2(a)). For the sample annealed at 250 °C, the image of the whole multilayer structure is shown in Fig. 4(c). A major part of the layers especially at the middle area of the stack were degraded and barely observed.

Only a few layers at the bottom near the substrate and at the top part can still be seen. High resolution images of the middle and surface area of the multilayer are shown in Fig. 4(e) and (f), respectively. In the middle area of the stack (Fig. 4(e)), the layers are heavily crystallized with relatively large grains randomly distributed (is the size of the grains in agreement with the one deduced from the Scherrer formula?), while the layer interfaces completely disappeared. For the surface area (Fig. 4(f)), a few topmost layers were seen with slight polycrystallization. The crystallization became stronger for the layers located deeper inside the multilayer and the layer interfaces gradually blurred. The bottom layers near the substrate exhibited a similar phenomenon and it was not shown here. The SAED result also showed an enhanced diffraction pattern (Fig. 4(d)), comparing to the as-deposited one, and the main spots were identified as hexagonal Sc (002) and cubic V (110), respectively.

The increased crystallization with large grains and newly formed phases is consistent with the XRD result (Fig. 2(d)).

(New paragraph) The energy dispersive X-ray spectroscopy (EDX measurements were performed at the cross-section of the two samples as illustrated in Fig. 5. For the as-deposited sample (Fig. 5 ), no periodic oscillation can be seen. A small amount of oxygen, carbon and nitrogen were also detected in the multilayer while the absolute concentration cannot be determined due to the limited accuracy of EDX. It is clear that the V and Sc atoms significantly mixed at the interfaces after annealed at 250 °C. The increased interdiffusion contributed to the formation of V and Sc nano-grains and these two processes destroyed the original layer interfaces and resulted in the dramatic reduction of the reflectance for the annealed V/Sc multilayers. These processes first occurred at the middle part of the stack which may be caused by the larger interdiffusion of V and Sc atoms from neighboring layers (do you have an idea of the diffusion length to justify such a statement?) and thus an earlier crystallization of the layers, comparing to the surface and bottom part.

The thermal degradation of Cr/Sc multilayers were also caused by the formation of a mixture of Cr and Sc nanocrystallites that broadened the interfaces [16]. Nevertheless, the severe crystallization starts from the temperature of around 450 °C that is at higher temperature than for the V/Sc multilayer. The as-deposited Cr/Sc multilayer also exhibited an amorphous structure while the V/Sc multilayer already shows a polycrystalline structure before annealing. The formation of randomly distributed nano-grains at a lower temperature can be the main reason for the relatively poor thermal stability of V/Sc with respect to Cr/Sc.

It is worth to note that the thermal stability of the multilayer studied here is for pure V/Sc structure with no interface engineering methods applied. Although the thermal stability of V/Sc is a little worse than Cr/Sc, it may be improved by introducing diffusion barrier layers or nitridation, as have been done on the Cr/Sc multilayer. It is very likely that e.g. an atomic-scale B4C barrier layer can reduce the inter-diffusion of atoms under high temperature annealing, suppress the formation of large crystallites, and result in a better thermal stability [4]. Nevertheless, this will need to be investigated in the future.

Les paramètres requis sont manquants ou erronés. (perhaps you should indicate to which location corresponds the zero value. Is it the surface?)

Temporal stability

The GIXR measurements of the V/Sc multilayer with 400 bilayers stored for 2 months, 4 months, and 6 months are depicted as Fig. 6. The as-deposited result is also shown as a comparison. It can be found that the reflectance of the1 st and 2 nd order Bragg peaks of the sample stored after different times are essentially the same and the d-spacing remains as 1.59 nm. It means that the main part of the periodic layer structure were unchanged after 6 months storage. The oscillation of the reflectance curve at about 1°~2° was slightly shifted to the lower angle with time which indicates a gradual change of the surface B4C layer and possibly of the first few V/Sc bilayers of V and Sc. Similar phenomenon was also observed in other multilayer mirrors after a long time storage [19]. The SXR reflectance of the V/Sc sample with 400 bilayers was measured again at the BEAR beamline after a storage of 6 months and the result is displayed in Fig. 7. The reflectance curve of the as-deposited sample is also shown as a comparison.

They were both measured under a near normal incidence of 9 °. The peak reflectance dropped slightly from 18.4% to 16.5% at 3.128 nm. Considering that the aged sample was measured with a beam of higher energy resolution than the as-deposited one, 0.2 eV instead of 0.4 eV, there should be another ~0.6% drop of reflectivity of the aged sample (according to simulation), if using the same incidence beam. This also contributes to the narrower bandwidth of the aged sample. The reduction of the SXR reflectivity can originate from the surface oxidation and other chemical changes mostly in the surface structure after exposure to air for a long time [7,20]. Despite the minor reflectance decrease, the temporal stability of the V/Sc multilayer is relatively good for using as optical elements in X-ray microscopy applications.

3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17

Conclusions

V/Sc multilayer is a very promising candidate for high reflectivity mirror used in the water window region. To study its thermal and temporal stabilities, the structure evolution and the optical performance change of the multilayer under different temperatures annealing and a long-term storage were investigated. It is found from the GIXR and SXR measurements that the layer structure and the SXR reflectivity of V/Sc remain almost unchanged after 100 °C annealing. The SXR reflectivity deteriorates obviously after annealing at 200 °C, and the reflectivity drops almost to zero at 250 °C. According to the XRD and TEM measurements, a severe crystallization of V and Sc was found with randomly distributed nano-grains after 250 °C annealing. This was accompanied by an elemental mixture at the interfaces and these two processes ruined the ultra-thin layer structure. The multilayer was relative stable after 6 months storage in air with only ~2.5% decrease of the SXR reflectivity (absolutely). Although the V/Sc multilayer exhibits a little worse thermal stability than the widely used Cr/Sc multilayer, different interface engineering methods can be explored to improve it. Adding interface barrier layers or nitridation of the multilayer have both been demonstrated effective on enhancing the thermal stability [7,16].This can be further investigated in the future experiments.

Acknowledgement

_____________________________________________________________________

Fig. 1 .

 1 Fig.1. GIXR curves of the samples before and after annealing under different temperatures (a) 100°C, (b) 200°C, (c) 250°C, and (d) 300°C.

Fig. 2 .

 2 Fig. 2. XRD pattern of V/Sc multilayers under different annealing temperatures.

Fig. 3 .

 3 Fig. 3. SXR measurements of V/Sc multilayer as-deposited and annealed at different temperatures.

  (a)), periodic oscillations of the relative concentration (what do you call "relative concentration"? I would expect that the sum of V and Sc concentrations should give 100%, which is not the case from what I see on the figure.) of V and Sc were observed, despite the very small d-spacing of 1.59 nm. For the sample after annealing at 250 °C (Fig. 5(b)

Fig. 4 .Fig. 5 .

 45 Fig. 4. (a) TEM image, and (b) SAED pattern of the as-deposited 400-bilayer V/Sc multilayer; TEM image of (c) the whole stack, (e) the middle area and (f) the surface area, and (d) SAED pattern of the 100-bilayer V/Sc multilayer annealed at 250 °C.

Fig. 6 .

 6 Fig.6. Temporal evolution of GIXR in a half year for the V/Sc multilayer with 400 bilayers.

Fig. 7 .

 7 Fig. 7. SXR curves of the 400-bilayer V/Sc multilayer as-deposited and after six months storage.

  This work was supported by National Natural Science Foundation of China (No.11443007,No.11505129, No.11135007), National Key Scientific Instrument and Equipment Development Project (No.2012YQ13012505, 2012YQ24026402), Shanghai Pujiang Program (No. 15PJ1408000).


Antonin Cheymol

Gwendal Fouch

Lysa Gramoli

Yutaro Hirao

Emilie Hummel

Ma É Mavromatis

Yann Moullec

Ferran Argelaguet

Florian Nouviale

The Rubber Slider Metaphor: Visualisation of Temporal and Geolocated Data

Keywords: Human-centered computing-Human computer interaction (HCI)-Interaction paradigms-Graphical user interfaces, Human-centered computing-Human computer interaction (HCI)-Interaction paradigms-Virtual reality, Human-centered computing-Visualization-Visualization application domains-Information visualization

In the context of the IEEE VR 2022 3DUI Contest entitled "Arts, Science, Information, and Knowledge -Visualized and Interacted", this paper presents a VR application to highlight the usage of the rubber slider metaphor. The rubber slider is an augmentation of usual 2D slider controls where users can bend the slider axis in order to control an additional degree of freedom value in the application. This demonstration immerses users in a Virtual Environment where they will be able to explore a database of art pieces geolocated on a 3D earth model and their corresponding art movements displayed on a timeline interface.

INTRODUCTION

The visualization of temporal data is common for immersive analytics applications, in which data has a temporal dimension. In such applications, a main requirement is to enable the user to control the time dimension to filter and select data of interest. A number of solutions have been proposed in the literature in the 2D interaction domain, such as TimeZoom [START_REF] Dachselt | Timezoom: a flexible detail and context timeline[END_REF], which proposed an interactive zooming solution offering multiple levels of detail, or the work of Morawa [START_REF] Morawa | Combining timeline and graph visualization[END_REF], in which users can interact dynamically with the data visualization by using key points of a graph representing the timeline. In this paper, inspired by 2D slider interaction, we propose the Rubber Slider metaphor, a novel, enhanced, pseudo-haptic control interface. The Rubber Slider metaphor gives better control to users over the displayed time range thanks to the additional degree of freedom and the pseudo-haptic feedback [START_REF] Ujitoko | Survey of pseudo-haptics: Haptic feedback design and application proposals[END_REF].

The Rubber Slider metaphor is illustrated through a VR application, which allows for the exploration of geolocalized art pieces through different artistic and cultural periods. Art pieces are displayed on a 3D earth model, similarly to Orboot Earth AR 1 . Additionnaly, a side panel is used to show details about any selected art piece and a timeline enables the user to select a year that controls what art pieces are shown on the earth. The rubber slider metaphor is used to efficiently interact with the timeline.

VR APPLICATION STRUCTURE

In the proposed application, users are immersed in a virtual environment in which they can move freely and observe an earth model on which various previews of art pieces are displayed (see Fig. 1). The Earth manipulation Below the earth, a timeline is displayed, as well as artistic movements that took place in the displayed time range. Users can switch between three different interaction modes: (1) Earth manipulation and art piece selection, (2) Timeline control and (3) continent selection for data filtering. All the interactions of the application are performed using one hand-held controller (e.g. HTC Vive controller) and a dedicated button is used for the mode switch.

Earth manipulation and art piece selection

On the virtual earth model, previews of art pieces are displayed at their creation location if they belong to one of the art movements occurring during the year selected on the timeline (see Sect. 2.2). In order to interact with the virtual earth and previews, users use a virtual ray originating from their controller. The ray can be used to hover, highlight and select a preview or rotate the earth model if no preview intersects with the ray. Selection of a preview is performed by pressing the trigger button which displays details regarding the corresponding art piece on the side panel (Fig. 1), including title, author, creation date and description. Rotation of the earth model is performed by holding the trigger button, and dragging it with the controller. This allows to observe previews of art pieces that were not directly visible in the previous position.

Timeline control

By adjusting the timeline, users can visualize on the earth art pieces from the art movements taking place from paleolithic to contemporary eras. The bottom of the timeline displays a ruler of years while the top displays the art movements as a stack of colored rectangles, symbolising the time range of each movement. The art previews displayed on the earth model are surrounded with a halo of the same color as the movement they belong to. The central position of the ruler shows a cursor indicating the year currently selected. The timeline position and scale can be controlled using the rubber slider interaction, as explained in Sect. 3.

Continent selection for filtering

In this interaction mode, users can hover and highlight continents using a virtual ray. By pressing the trigger, the hovered continent gets selected which results in the filtering of the art movements on the timeline and art pieces on the earth model, so that only the art movement occurring on this continent are displayed.

SLIDER INTERACTION: THE RUBBER SLIDER

The rubber slider (Fig. 2) presented in this paper can be considered as a dual-value interaction interface. Similarly to a 2D slider, users can move the cursor along the main axis of the slider to select a bounded value V . The second bounded value, B, can be selected by pushing or pulling the cursor from the main axis, which will accordingly bend the slider main axis into a curve. When the rubber slider is bent, a color feedback is used. The more the slider is pushed or pulled, the more colored it gets, up to the maximum curvature.

Cursor and bending control

The rubber slider is used to control the selected year on the timeline and the time range to display. We propose two control modes to showcase different usages and control laws that can be used with the rubber slider.

Direct Control

In this mode, the cursor value V of the slider is directly converted to a year on the timeline. Meaning the leftmost position represents the earliest year while the rightmost position represents the latest year. The bending value B changes the scaling of the time range. Users can display a narrower time range on the timeline by bending the slider or a wider time range by unbending the slider.

Incremental Control

In this mode, the cursor value V of the slider is shifting the selected year on the timeline. When the cursor is on the left part of the slider, it decreases the selected year, when it is on the right part, it increases the selected year. The further from the center, the higher the increase/decrease value used, while around the center, a dead zone helps the user fix the selected year. Similarly to the direct control mode, the bending changes the scale of the time range displayed, but it also scales the increment number used to select value V . When users pull the slider, the time range displayed is scaled down, showing a narrower range of years, and the increment value used is decreased, allowing a more precise control when the users' arm are less extended.

While both values can be controlled continuously, a secondary optional mode is added to the rubber slider through a metaphor of hooks, enabling a discrete control of the bending value B.

Hooks metaphor

Manipulating two values by the use of a single limb can sometimes prove difficult. The hooks metaphor shown on Fig. 3 aims at tackling this issue by proposing a limited number of discrete bending values B symbolised by visual elements. Users can select a bending value by pulling the slider. When the slider reaches a hook, the value B is updated to that bending value. If users need to change the bending value B, either to increase or decrease it, they need to pull the cursor up. When the cursor exceeds the hook height, the rubber slider can be pushed or pulled to a new hook to select a new bending value B.

Pseudo-haptic feedback

A pseudo-haptic feedback was added to the bending interaction with the rubber slider. This not only improves the realism of the interaction, but allows users to manipulate it more precisely. To create this effect, we incrementally shift back the cursor position and slider curve when the user pulls it (see Fig. 3).

CONCLUSION

This paper presents a VR application to explore a database of art pieces. In this application, a rubber slider control interface can be used for a precise and adaptive control of time display and selection on a timeline. The slider can be pushed or pulled to up-or down-scale the timeline display range. During this manipulation, a pseudo-haptic feedback improves the feeling of effort on the slider. Furthermore, different control modes of the rubber slider are presented, including a hook metaphor, that showcase various possible usages of this control interface. Users can also interact with a 3D earth model to select specific geographical areas, rotate the earth model, or select art pieces, to get additional details.

The database of art pieces and movements is handmade from various data sources. While the size of the database is limited in this use case, more data and different data types (music, architecture, ...) could be added in the future. The rubber slider could be enhanced with other control laws, orientations and behaviors for different use cases, and tested with a user study. Other manipulation techniques, like PRISM [START_REF] Frees | Prism interaction for enhancing control in immersive virtual environments[END_REF] could be combined to the rubber slider manipulation to improve precision.

Figure 1 :

 1 Figure 1: Overview of the application with the earth model showing art pieces being selected with the user ray (left), the timeline showing art movements (bottom left) and the selected art piece description panel (right).

Figure 2 :

 2 Figure 2: The rubber slider, located under the timeline, being manipulated (and also pulled on the right) by the user.

Figure 3 :

 3 Figure 3: Hooks metaphor. The active hook is displayed in yellow. The purple cursor displays the position of the user controller (usually hidden) while the cyan one shows the pseudo-haptic position.


Jingtao Zhu
email: jtzhu@tongji.edu.cn

Jiayi Zhang

Haochuan Li

Yuchun Tu

Jinwen Chen

Hongchang Wang

Sarnjeet S Dhesi

Mingqi Cui

Jie Zhu

Philippe Jonnard

Improving the soft x-ray reflectivity of Cr/Ti multilayers by deposition of B4C

Keywords: Synchrotron radiation, multilayer, interface, Cr/Ti

Systematic analysis on interfaces of Cr/Ti multilayer has been performed in this work. The near-normal incidence reflectivity increases from 4.48% to 15.75% at wavelength of 2.73nm using co-deposition of B4C.

Introduction

Multilayer coatings for the soft x-ray region have drawn a great deal of attention due to their vital importance in high-resolution microscopes, deep-space telescopes and soft x-ray polarimetry. (Artyukov et al., 2009[START_REF] Windt | Optics for EUV, X-Ray, and Gamma-Ray Astronomy[END_REF], Kitamoto et al., 2010) In these applications, reflectivity is one of the most critical parameters of a multilayer. Cr/Ti multilayers are one of the suitable candidates in the "water window" wavelength range (2.4-4.4 nm) because the theoretical near-normal incidence reflectivity is as high as 64% at wavelengths just above the Ti 2p edge (2.73 nm). However, the reported reflectivity of real Cr/Ti multilayers at near-incidence angles is lower than 3%. (Ghafoor et al., 2006) The large disparity between the theoretical and measured reflectivity is caused by several physical limitations, among which the most important is the interfacial imperfections, especially for ultra-short periods (Λ~1.4 nm) used.

In recent years there has been a huge effort to develop techniques that can reduce the interface width of multilayers. One recent attempt involved the intentional incorporation of light-element impurities such as N into multilayers. It has been shown that the incorporation of N in Cr/Sc (Ghafoor et al., 2008) or B4C/La (Tsarfati et al., 2010) multilayers can form nonstoichiometric nitride multilayers with reduced interdiffusion or reaction between the constituent layers and leads to major improvement of soft x-ray reflectivity. Another scheme is the introduction of B4C, which have been found effective in Cr/Sc [START_REF] Ghafoor | [END_REF] and Cr/V (Huang et al., 2016) system. Concerning Cr/Ti multilayers, B or C is a better choice than N because of a stronger absorption of N in the application wavelength range of Cr/Ti multilayers, which is at a slightly lower wavelength than the N K edge (3.02 nm). In this work, we intentionally incorporated B and C during deposition and investigated the microstructure and optical performance of the multilayers.

Fabrication

The Cr/Ti multilayers were deposited by DC magnetron sputtering system with B and C impurities incorporated by the deposition of ultra-thin B4C at the interfaces of Cr/Ti multilayers. All the multilayers were deposited onto natively oxidized Si (001) wafer with a roughness of 0.3nm. The pressure of residual gas before deposition was 5.0×10 -5 Pa and that of Ar sputtering gas was 0.13 Pa with a purity of 99.999%. Sub-nm thin layers were obtained by optimizing target-substrate distances and magnetron power. Target-substrate distances were all 8 cm. The magnetron power of B4C, Cr and Ti were 40 W, 20 W and 40 W respectively.

Experimental details can be found in the PhD thesis by Yunchun Tu (Tu, 2015).

Characterization for the interfaces of Cr/Ti multilayers

Firstly, we studied the effect at interfaces of different amounts of the incorporated B4C. We deposited a series of Cr/Ti multilayers, which can be divided into four groups. All the multilayers were designed with period number N=50. B4C was incorporated at both Cr-on-Ti and Ti-on-Cr interfaces for the first group, only at Cr-on-Ti interfaces for the second group, and only at Ti-on-Cr interfaces for the third group. The multilayers in those three groups were deposited with the same thicknesses of Cr layers and Ti layers (dTi=dCr=0.7 nm). Each of those three groups consisted of six multilayers with different amounts of B4C, where one multilayer had no B4C, and the nominal amounts (in thickness) of B4C for the other five multilayers were 0.08, 0.13, 0.18, 0.22 and 0.27 nm, respectively. For comparison, the last group contained four pure Cr/Ti multilayers without B4C. The period thicknesses of the four groups of multilayers ranged from 1.4 nm to 1.7 nm and their thickness ratio (Γ=dCr/Λ) was ~0.5.

All the multilayers were characterized by low angle x-ray reflectometry (XRR).

Measurements were performed on a Bede D1 diffractometer with a Cu x-ray source (λ=0.154 nm). Only the first order Bragg peak could be observed for each multilayer because of the ultra-short period. We present the peak reflectivity as a function of the grazing incident angle in Figure 1, where thicker multilayers correspond to the smaller grazing angle (left side). It can be noticed that incorporating B4C at either or both interfaces leads to higher reflectivity, namely, improves multilayer quality. In particular, it is better to incorporate B4C at the Cr-on-Ti interfaces than at the Ti-on-Cr interfaces, and incorporating B4C at both types of interfaces is the best of all. Thickness of 0.13 nm B4C is chosen because the fitted roughness is lowest. In order to study the interface stoichiometry, we performed x-ray photoelectron spectroscopy (XPS) on two samples. One had no B4C incorporated and the other had 0.13 nm thick B4C at both interfaces. The measurements were performed on a Thermo Scientific K-Alpha system with an Al Kα source. The binding energy scale was calibrated from hydrocarbon contamination using the C 1s peak at 285.0 eV. A non-linear Shirley-type background was subtracted from the raw spectra. The peak positions and areas of core level peaks were optimized by a weighted least-squares fitting method using 70% Gaussian, 30% Lorentzian line shapes. Atomic concentrations were calculated on the basis of Scofield's relative sensitivity factors. To eliminate the contamination of the multilayer such as the oxidation at the surface, the XPS spectra were recorded after ~10 nm were removed at the surface by argon etching. Considering the ultra-short period and the estimated analysed depth it is safe to regard the recorded spectra as averaged over the whole multilayer period. It should be, for some material combinations, that the chemical interaction of materials in the multilayer may change after energetic Ar + ion bombardment. However, in a previous research on XPS study of Pb/B4C multilayer (Supruangnet et al., 2016), C1s core level remained the same as pure carbon after Ar + etching. B1s core level shifts only because of the change from boride phase to oxide compound. Pb 3d3/2 and Pb 3d5/2 core levels for pure plumbum remained the same while the only change occurs in the oxidation of Pb compound. It can be confirmed that Ar + etching will not introduce extra chemical interaction in B4C-based multilayers.

(new paragraph)The Cr 2p and Ti 2p spectra of the two multilayers show no difference in peak position, which is probably due to the small chemical shifts of possible compounds (<0.2eV for CrB2 or TiB2). [START_REF] Wanger | Handbook of x-ray photoelectron spectroscopy Eden Prairie[END_REF] The B 1s and C 1s spectra of the multilayer with B4C are shown in Figure 2 along with the spectra of a B4C monolayer reference sample which was deposited and measured by the same method described above. The B 1s spectrum of the monolayer consists of two peaks at 188.4 eV, characteristic of B-B or B-C bond from B atoms in the B12 icosahedra, and 190.6 eV, characteristic of B-C bond from B atoms in the C-B-C chain. The B 1s spectrum of the Cr/Ti multilayer is dominated by a peak at 187.4 eV which is attributed to TiB2 (187.5 eV) (Mavel et al., 1973) rather than CrB2 (188.0 eV). (Mavel et al., 1973) The C 1s spectrum of the monolayer consists of two peaks at 282.8 eV, characteristic of B-C bond in the icosahedra, and 284.1 eV, characteristic of B-C bond in the C-B-C chain. The C 1s spectra of the Cr/Ti multilayer is dominated by a peak at 281.9 eV which is attributed to TiC (281.7 eV) [START_REF] Galuska | [END_REF] rather than Cr3C2 (282.9 eV). (Goretzki et al., 1989) For the pure Cr/Ti multilayer, the atomic concentrations of Cr, Ti, and O are 58.9%, 27.8%, and 13.4%, respectively. For the Cr/Ti multilayer with B4C, the atomic concentrations of Cr, Ti, B, C, and O are 47.8%, 22.9%, 17.1%, 4.4% and 7.9%, respectively. There are two reasons why there is less oxygen in Cr/B4C/Ti/B4C multilayer: firstly, B4C can hinder the diffusion of oxygen to inner layers; secondly, with the deposition of B4C, imperfection of grain boundary can be decreased and thus there occur less ways of oxygen diffusing into inner layers.

According to the calculated results, different concentrations of oxygen will not influence the reflectivity. Encouraged by the preliminary results, we deposited two Cr/Ti multilayers, one without B4C and the other with 0.13 nm thick B4C at both interfaces, which can work as high reflection mirror for wavelength λ=2.73 nm (453.5 eV photon energy). The period number is N=600 and the designed period thickness is Λ=1.38 nm. Figure 3 shows the cross-section transmission electron microscopy (TEM) images of the two samples measured using an FEI microscope (Tecnai G2 F20). The multilayer with B4C shows much better contrast between Cr layers and Ti layers although the flatness is similar. Figure 4

Reflectivity measurements in water window range

The soft x-ray reflectivity measurements were performed by using a high precision polarimeter on Beamline I06 at the Diamond Light Source, United Kingdom. (Wang et al., 2011) The Cr filter was selected to suppress the high harmonic contamination from the collimated plane grating monochromator. Linear horizontal polarization synchrotron radiation with energy of 453.5 eV was used in the measurements. The reflectivity curves are plotted in Figure 4 as a function of the glancing angle, close to the near-normal incidence. The reflectivity of the pure Cr/Ti multilayer is 4.48% (at 79.95°) and the interface width is determined to be 0.41 nm by simulation using the IMD code. [START_REF] Windt | [END_REF] The reflectivity of the Cr/Ti multilayer with B4C is 15.75% (at 82.75°) and the interface width is determined to be 0.30 nm.

The required optical constants were generated following the method in the reference (Berkeley center for x-ray optics). The large improvement of the reflectivity, by a factor 3, is due to the reduced diffusion between Cr and Ti layer and the suppression of crystallization.

Summary

In this paper, we have considered Cr/Ti multilayers with ultra-short period (Λ~1.4 nm). We intentionally incorporated B and C into Cr/Ti multilayers by depositing ultra-thin B4C at the interfaces. We found that incorporating 0.13 nm thick B4C at both interfaces dramatically improves the structure and composition of the Cr/Ti multilayers. B and C incorporated during deposition is mainly bonded to Ti. The formation of non-stoichiometric Ti compound hinders the diffusion and improves the interface quality of the multilayer. crystallization is also supressed after introduction of B4C. As a result, Cr/B4C/Ti/B4C multilayers (with period number N=600) exhibits a reflectivity 3.5 higher near-normal incidence soft x-ray reflectivity (15.75% absolute reflectivity) than pure Cr/Ti multilayers.

Figure 1

 1 Figure 1 (color online) Peak reflectivity versus peak angle of the first order Bragg peaks in the XRR curves of the Cr/Ti multilayers. Symbols represent the data and solid lines are only guides for the eyes.

Figure 2

 2 Figure 2 (color online) (a) B 1s and (b) C 1s XPS spectra of the B4C monolayer and Cr/Ti multilayer with B4C. Open symbols represent the measured data, solid lines the fitted spectra, short dashed lines the background, and long dashed lines the peaks used in the fitting if more than one peak is involved.

  presents the selected area electron diffraction (SAED) results of Cr/Ti multilayers with and without B4C. The diffraction ring and spots weakened after co-deposition of B4C. The crystallization in Cr/Ti multilayer has been suppressed. These results confirm show that the TiBxCy formation prevents diffusion of Cr.

Figure 3

 3 Figure 3 Cross-section TEM graphs of Cr/Ti multilayers (a) with and (b) without B4C.

Figure 4

 4 Figure 4 SAED (selected area electron diffraction) results of (a) Cr/Ti multilayer and (b) Cr/B4C/Ti/B4C multilayer.

Figure 5

 5 Figure 5 (color online) Soft x-ray reflectivity of the Cr/Ti multilayers with (black) and without (blue) B4C. Solid lines represent the measured data and dashed lines show the calculated reflectivities.

Acknowledgements

The authors acknowledge Francesco Maccherozzi for the support during the reflectivity measurements, Kawal Sawhney for helpful discussion and Yuchun Tu for the experimental suggestions.