Mathieu Barbier

Christian Laugier

Olivier Simonin

Javier Ibañez-Guzmán

Probabilistic Decision-Making at Road Intersections: Formulation and Quantitative Evaluation

As drivers approach a road intersection, they must decide whether to cross it or to come to a stop. For this purpose, drivers make a situation assessment and adapt their behaviour accordingly. When this task is performed by a computer, the available information is partial and uncertain. Any decision requires the system to use this information as well as taking into account the behaviour of other drivers to avoid collisions. However, metrics such as collision rate can remain low in an interactive environment because of other driver's actions. Consequently, evaluation metrics must depend on other driving aspects. In this paper a decision-making mechanism and metrics to evaluate such a system at road intersection crossing are presented. For the former, a Partially Observable Markov Decision Process is used to model the system with respect to uncertainties in the behaviour of other drivers. For the latter, different key performance indicators are defined to evaluate the resulting behaviour of the system with different configurations and scenarios. The approach is demonstrated within an automotive grade simulator. It has showed at times, that whilst the vehicle can cross safely the intersection, it might not satisfy other key performance indicators related to highway code.

I. INTRODUCTION

Road intersections are likely the most complex segment in a road network. From an accidentology perspective, most major accidents occur at intersections, mainly due to human errors [START_REF]Bilan de l'accidentalite de l'annee 2014[END_REF]. The failure to fully understand the situations encountered are the main reasons, particularly by young and elderly drivers [START_REF] Ibañez-Guzmán | Vehicle to vehicle communications applied to road intersection safety, field results[END_REF].

Situation understanding is a major challenge, whether the vehicle is under manual or computer control. Thus, the understanding of situations and decisions taken are affected by the uncertainty associated to the digital representation of the environment. Further, interactions between drivers are done through a non-verbal exchange. For an autonomous vehicle this represents a major challenge. Given the uncertainties associated to the perception of the world by the vehicle onboard sensors and the unpredicted behaviour of other road network actors, approaches based on probabilistic methods are preferred to formulate this problem.

With non-deterministic methods, a particular attention should be paid to the evaluation process. It is not feasible to test every scenario on experimental grounds, hence ¹Renault S.A.S, 1 av. du Golf, 78288 Guyancourt, France. name.surmane@renault.com ²Inria Grenoble Rhône-Alpes , Chroma team, 655 Avenue de l'Europe, 38330 Montbonnot-Saint-Martin, France name.surmane@inria.fr ³INSA Lyon, CITI Lab., 6 avenue des Arts, 69680 Villeurbanne, France, France name.surmane@inria.fr simulators are used to test a larger variety of scenarios. In order to evaluate the system performance in a given scenario, a set of appropriate metrics has to be defined. Key performance indicators aim to evaluate performances and issues of decision-making for scenarios considering the highway code, navigation performances or passengers comfort. As the industry wants to deploy autonomous features in their vehicles, validation process is yet to be defined or updated in a rigorous manner.

This paper centres on the crossing of road intersections, these are considered complex but also where most accidents occur, in particular cross-cutting collisions. Fig. 1 represents such scenarios, it includes the subject vehicle (SV) and the other vehicle (OV). Three types of scenarios are examined: Scenario A, where the SV is required to yield and Scenario B and C, where the SV has the right of way.

The purpose of the paper is twofold: first, a decision system is formulated that will control the subject vehicle acceleration as it approaches the intersection according to the situation and associated uncertainties. A partially observable Markov decision process and an online solver are used to find the action to be performed by the vehicle to react to the situation and future evolution. The state and observation spaces includes behavioural variables that represent three possible manoeuvres at an intersections: Pass, Stop and Yield. The reward function considers problems related to driver's behaviour, comfort and risk. Secondly, the formulation of key performance indicators (KPIs) is proposed to get a better understanding of the experiment results given the road intersection scenario.

Section II presents related work on decision-making for road intersection crossing with POMDPs and metrics used in such a problem. A description of the proposed decision making model is presented in Section III. Section IV describes the different Key performance indicators , the testing system and an analysis of the obtained results.

II. RELATED WORK

Probabilistic and Bayesian approaches include uncertainty into their representation. Partially observable Markov decision process (POMDP) is a model that relies on a Bayesian network to represent dynamics of the system (with a special action variable) and a reward model to keep track of action values. Planning with POMDP models is well known and has been experimented for various robotic applications [START_REF] Kaelbling | Planning and acting in partially observable stochastic domains[END_REF]. The complexity of problems, like autonomous driving, with large number of possible states, observations and actions affect the possibility to find the optimal policy in a acceptable time. Efforts on methods to keep track of the belief [START_REF] Thrun | Monte carlo POMDPs[END_REF] and to solve POMDPs online [START_REF] Ross | Online planning algorithms for pomdps[END_REF][START_REF] Silver | Monte-carlo planning in large pomdps[END_REF][START_REF] Somani | Despot: Online pomdp planning with regularization[END_REF], have made their usage attractive and effective in the last couple of years. Applications with online solvers show that real life use is possible [START_REF] Hubmann | Automated driving in uncertain environments: Planning with interaction and uncertain maneuver prediction[END_REF][START_REF] Liu | Situation-aware decision making for autonomous driving on urban road using online pomdp[END_REF].

The state space of a POMDP represents all the variable required to represent the situation. At road intersection crossing, the state space is composed of variables for vehicle pose, speed and behaviour. The behavioural variables can represent, for example, the motion intention of drivers separated between stopping, hesitating, normal and aggressive [START_REF] Liu | Situation-aware decision making for autonomous driving on urban road using online pomdp[END_REF]. These were inferred from a previously learned context. The behaviour variable is modelled to either constant velocity or constant acceleration that are then used to update the pose of vehicles [START_REF] Bouton | Belief state planning for autonomously navigating urban intersections[END_REF]. However, these behavioural variables are not used to calculate the reward. Drivers intention enforced by regulation (traffic signs) in place is not taken into account in referred works. A coarse discretization is often used to represent the pose and velocity of vehicles as it simplifies the problem without a performance loss. However, in [START_REF] Bouton | Belief state planning for autonomously navigating urban intersections[END_REF][START_REF] Brechtel | Probabilistic decisionmaking under uncertainty for autonomous driving using continuous pomdps[END_REF] continuous representation is used but required an extadiscretization step learned or computed online.

The observation model, that represents the uncertainty of system to observe the situation, mostly takes into account measurement noises. It can also include a more complex analysis with, for example, occluded obstacles [START_REF] Brechtel | Probabilistic decisionmaking under uncertainty for autonomous driving using continuous pomdps[END_REF]. Therefore, the situation understanding is done within the POMDP, but in a more complex system this function can be done by other systems (especially for behaviour inference) such as [START_REF] Aoude | Driver behavior classification at intersections and validation on large naturalistic data set[END_REF][START_REF] Barbier | Classification of drivers manoeuvre for road intersection crossing with synthethic and real data[END_REF].

The POMDP reward model promotes states that are desirable for the system. Thus, states after the intersection are highly rewarded and penalties are given to collision states [START_REF] Hubmann | Automated driving in uncertain environments: Planning with interaction and uncertain maneuver prediction[END_REF][START_REF] Brechtel | Probabilistic decisionmaking under uncertainty for autonomous driving using continuous pomdps[END_REF]. However, these long-term reward systems are not sufficient for decision making. It can be enhanced by penalizing changes of acceleration to guarantee the comfort of passengers or deviations from a reference speed [START_REF] Hubmann | Automated driving in uncertain environments: Planning with interaction and uncertain maneuver prediction[END_REF][START_REF] Liu | Situation-aware decision making for autonomous driving on urban road using online pomdp[END_REF]. With recent improvement in deep reinforcement learning, such reward functions could be constructed from human demonstration or an annotated data-set [START_REF] Isele | Navigating Occluded Intersections with Autonomous Vehicles using Deep Reinforcement Learning[END_REF].

In the approaches previously described, the behaviour model for the other vehicle assumed that the other driver is moving as expected from the situation. It is different from the aggressive behaviour defined for vehicle driving above a reference speed [START_REF] Liu | Situation-aware decision making for autonomous driving on urban road using online pomdp[END_REF] but rather a driver that attempts a manoeuvre (for example to yield) when it did not concord with what is imposed by the situation. This idea to divide the behaviour between intention and expectation has been proposed in [START_REF] Lefevre | Risk assessment at road intersections: Comparing intention and expectation[END_REF], where vehicle to vehicle communications are used by the SV to observe the scene and to estimate the level of risk based this difference. For this purpose, a Bayesian network is used to estimate probabilities over the possible states of the system but only for stop and go type of behaviour.

Most of referred works used simulation tools to test their approach and even before experimentation on close roads a simulator has been used to obtain preliminary results [START_REF] Liu | Situation-aware decision making for autonomous driving on urban road using online pomdp[END_REF]. Simulation is convenient for that type of approach for two reasons: firstly, solving online a POMDP can be non-deterministic, thus the same scenario requires to be tested multiple times to ensure stability. Secondly, scenario variations are easier to test within a simulation and are also less costly to set up. To make sense of this large amount of results, some metrics are used in order to find successful tests. Selected metrics are often chosen arbitrarily and reflect the performances of the system and not its failures. Most of the times, authors looked at collision rate and travel time to gauge their approach. However, these do not take into account scenario specific metrics that could sometimes qualified the test a failure. With our proposed KPIs, we aim to have a more scenario driven approach and find what are the important performance indicators that could be used to judge a system.

III. POMDP MODEL APPLIED TO ROAD INTERSECTIONS

The underlying components of the model used in the decision making system are detailed in this section.

A. A Preliminary on POMDP

A Partially Observable Markov Decision Process models the decision process of an agent acting in an uncertain environment. Formally it is composed of {X , A, Z, T, R, O}, with X the state space, A a set of actions that the agent can take, Z a set of observations that can be obtained by the agent. T is the transition function T (x , x, a) = P (x |x, a) describes how the system changes when the agent takes action a when in state x with x the state at the next time step. The reward function R(x, a) indicates the value obtained after performing an action in a given state. And O(z, x, a) = P (z|x, a) represent the probability to obtain an observation z ∈ Z given a state and an action. In an uncertain environment, the agent does not know the real state after an observation, it reasons with a belief b ∈ B with b : B → R ≥0 and x∈X b(x)dx = 1. Thus, the goal of the agent is to maximize the value V : B → R for an initial belief. In a POMDP there exists an optimal policy π * : B → A that maximize V π . The value of a policy can be estimated as the expected future sum of rewards for an initial belief b o is

V π (b 0 ) = ∞ t=0 γ t R(b t , a) = ∞ t=0 γ t E(R(x,
a)|b 0 , π). γ, the discount factor, penalized rewards more and more after each time step t. For the purpose of our approach the agent is our decision-making system that decides which acceleration is appropriate while approaching towards the intersection.

B. State Space

The state space contains all the information required to model the system. Our state vector is x = [d sv , d ov , s sv , s ov , i ov , e sv , e ov ]. With, the physical state of the subject vehicle is represented with its speed s sv and distance towards the intersection d sv . Similarly d ov , s ov represent the physical state the of the other vehicle. Since small variations on these variables are hard to perceive and do not add meaningful information, range is d sv , d ov , s sv , s ov ∈ N. Distances are taken from 0 m that correspond to the entrance to the intersection to 50m that represents the point at which drivers start to be aware of the approaching intersection. The considered vehicle velocity ranges from 0 m/s to 14 m/s (50 km/h), the maximum legal velocity in urban areas.

For the behavioural variables, a difference is made between intention of the i ov and expectation e ov as in [START_REF] Lefevre | Risk assessment at road intersections: Comparing intention and expectation[END_REF]. The longitudinal expectation e ov , represents what a driver should do according to priority rules at the intersection and the situation. In a perfect world, drivers should always follow the expected behaviour to avoid risky situations. However, there could be differences caused by misinterpretation of the scene or if the SV or road users are occluded. As a consequence, another variable, the longitudinal intention i ov represents what the driver is doing approaching the intersection. The SV, assumed driven by an autonomous system, will always comply with the current situation. Thus, the behaviour of the SV is only represented by its expectation e sv .

The behavioural variables represent high level manoeuvres that drivers can do when approaching an intersection and are defined on the same ensemble i ov , e ov , e sv ∈ {Stop, Y ield, P ass}. Stops means that a driver will brake to stop at the intersection. Y ield is a manoeuvre when drivers regulate their speed in order to let another vehicle to cross an intersection. And the P ass manoeuvre corresponds to a behaviour when a driver intends to cross first the intersection.Our State space differs from [START_REF] Lefevre | Risk assessment at road intersections: Comparing intention and expectation[END_REF] with the introduction of the Y ield manoeuvre, that will allow the selection of smaller deceleration while the SV approach the intersection, and to evaluate the reward on states with traffics laws (see section III-E).

C. Action Space

The action space contains a range of accelerations that the SV can do to change its state. Approaching an intersection, a vehicle is most likely to decelerate. Slow deceleration helps the vehicle to widen the gap with another vehicle. High deceleration helps to avoid collision or to adapt the speed approaching towards the intersection entrance. Small accelerations help to reduce the travel time. From these observations, actions were chosen as follow: a ∈ {-2, -1.5, -1, -0.5, 0, 1}ms -2 . A larger range of accelerations would increase model complexity. The proposed action range is sufficient to stop the vehicle to avoid collision or to come to a stop. The step between each action is also small enough to avoid jerk.

D. Transition Model

This model represents the probability of going to a state x given a previous state x and an action a. The structure of this model can be represented by the Bayesian network illustrated in Fig. 2. The action performed by the SV has the most impact on s sv and d sv . Kinematic equations are used to update the speed and the distance to the intersection. However, the vehicle might not be able to exactly perform the action. Thus, the transition probabilities are

P (s sv |a, s sv ) = N (µ s , σ a ) W ith µ s = s sv + a∆t (1)
P (d sv |a, d sv , s sv ) = N (µ d , σ a ) W ith µ d = d sv -(s sv ∆t + a0.5∆t 2 ) (2)
With ∆t = 0.5s the time when acceleration is applied. The execution of the action could be delayed or altered due to the control systems, thus N (µ s , σ a ) represents this uncertainty with σ a = 1. A gap acceptance model [START_REF] Pollatschek | A decision model for gap acceptance and capacity at intersections[END_REF] is used to find the transition of expectation for the OV and SV. It uses the time gap between the two vehicles, to calculate the probability for the vehicle to come to a stop. It requires the distance and speed from each vehicle. Consequently expectation of both vehicle will change with the subject vehicle actions and the other vehicle states.

Equation 3 is used for the transition of the OV intention, if the previous intention matches the current expectation, we assumed that the driver will not change its intention (thus the probability of 0.9 in eq.3). Otherwise, the intention of the OV is unknown, thus uniformly distributed over all possible intentions.

P (i ov |e ov , i ov ) P (i ov = i ov ) = 0.9 i ov = e ov P (i ov ) = U(I) otherwise (3)
Then the speed of the OV is inferred from the previous speed and previous intention. Intentions to stop or to yield will decelerate induce a deceleration, whereas the pass manoeuvre correspond correponds to a null acceleration. The distance from intersection entrance depends on the speed and the previous speed as for the SV.

E. Reward Function

The reward function is a weighted sum of 5 components:

• Comfort reward (r c ) aims to reduce strong deceleration.

Thus, r c (a = -2m/s²) = -R max for other actions no penalty is applied • Risk reward (r r in eq.4) penalizes small time gaps between two vehicles. As the distance has for origin an entrance of the intersection, this gap represents the remaining time before the two vehicles reach their respective entrance at the same time.

r r (s sv , d sv , s ov , d ov ) R max | dsv ssv -dov sov -5s| > 0 | dsv ssv -dov sov | otherwise (4)
• Speed reward (r s in eq.5) is calculated as the difference between the speed of the SV and a reference speed and if the action brings the speed toward the reference. This reference is obtained with a vehicle driving in the same intersection but without another vehicle.

r s (s sv , d sv , a)          R max |V ref (d sv ) -s sv | < 2 R max /2 V ref (d sv ) -s sv > 2 ∧ a > 0 R max /2 V ref (d sv ) -s sv < -2 ∧ a < 0 0 otherwise (5)
• Intention reward (r i ) penalizes the situations when the intention of vehicles is in contradiction regarding the traffic laws. The relation between the intention of each vehicle is shown in table I.

R max is the maximum obtainable reward and is set to 10. Consequently, any component has a higher importance than another. The table I represents the function r i that has been designed to reward states where the behaviour of OV and SV are compatible. For example, if one vehicle intends to stop/yield and the other to pass the situation is rewarded because collisions will be avoided and travel time optimized. Situations where both vehicles have the same intention are not rewarded since both vehicles is trying to let the other cross the intersection. If both of them intend to pass a collision could happen, thus these situations are penalized.

The last two components are important for the cooperation between two vehicles. If r i is low, the SV will try actions that modify its expectation in order to cooperate. If r e is low the SV can modify its speed to change the expectation of the OV.

The resulting reward function is:

R(x, a) = w c (d sv )r c (a) + w r (d sv )r r (s sv , d sv , s ov , d ov )
+ w e (d sv )r e (e ov , i ov ) + w s (d sv )r s (s sv , d sv a) (7)

+ w i (d sv )r i (e sv , i ov )
Functions w are used to weight each reward component. They can be changed to adapt the behaviour of the system with a scenario or the driving style of the SV. These functions are dependent of the distance towards the intersection of the SV. For example, the risk of collision could be less important far away from the intersection and one would prefer to prioritize the comfort of the passengers. However, close to the intersection it is important to avoid collisions even if it requires strong braking. Linear functions, like w(d sv ) = k 1 d sv + k 2 , are used to weight each reward. Different values of k 1 and k 2 will be tested in order to find appropriate values that guarantee safety of vehicles and possible ranges for personification. Compared with a more conventional approach (assign a high reward to terminal states and penalties for failures), this approach can take into account more subtle aspects of driving such as the one involving the behaviour of vehicles.

F. Observation Space and Model

States of the subject vehicle are observed with proprioceptive sensors less subject to uncertainty than the measurement of the OV that comes from perception systems. Thus, the noise level is smaller for the SV than for the OV. This information is added to observation model to take into account measurement uncertainties as Gaussian noises on the states of each vehicle N (0, 0.5) for the SV and N (0, 1) for the OV). For the intention of the OV, in our previous work [START_REF] Barbier | Classification of drivers manoeuvre for road intersection crossing with synthethic and real data[END_REF] we trained classifiers to recognize the intention of vehicle approaching an intersection. This framework is used to observe the intention of the other vehicle, whereas, [START_REF] Lefevre | Risk assessment at road intersections: Comparing intention and expectation[END_REF] assumed that driver's intentions were hidden. The observation model for this variable is deduced from classification matrices of the classifier. The resulting observation vector is z = [ŝ sv , ŝov , dsv , dov , îov ] and variables are defined on the same ensemble as the state space.

G. Solver

The large number of possible states and observations made the proposed model hard to solve with an offline solver. Thus, an online solver has been chosen that provide an approximation of the optimal policy at anytime. The advantage of such a solution is that an evaluation of the value of an action is always available but might not be perfectly estimated. The POMCP [START_REF] Silver | Monte-carlo planning in large pomdps[END_REF] has been chosen for its simplicity and its scalability. This solver combines a search tree with a particle filter. A belief is represented as a density of particles. The filter uses the transition model, in section III-D, to simulate the transition of a single particle when an action is chosen. The tree is used to keep track of the history and to direct the exploration which is controlled by an Upper confidence bound (UCB1) algorithm. It finds a trade-off between exploration and exploitation (focusing the search on the best performing branch). The next action a to be explored is selected by max a V (ha) = V (ha) + C log(N (h) N (ha) with N (.) being the number of times a node has been explored and h the history of observation and action that lead to the current node. Finally, C is used to control the exploration. With C = 0 the action that had the maximum value is always selected and with C >> 0 the action space is uniformly explored. With our implementation, γ=0.85 and a search time of 0.5s, an average of 1400 particles can be sampled from the current belief and simulated until the discounted reward becomes negligible. It is sufficient for studied scenarios, but to scale up the approach and include more vehicles or pedestrians, a better implementation using parallel processing or a learned estimation of V (ha) is required.

IV. QUANTITATIVE EVALUATION

The following section presents the key performances indicators alongside with the testing system. Then, an evaluation of two configurations of the proposed decision-making and a baseline algorithm shows the importance to use the proposed metrics to understand system performances.

A. Evaluation approach: Key Performance Indicators

The evaluation of the proposed decision-making system requires that vehicles within the test environment react to each other actions, consequently such vehicles are also able to avoid collisions with the SV. Thus, few collisions were observed even if some undesired behaviour were sometimes observed. It motivated us to propose some key performance indicators to evaluate a decision-making system in relation with the tested scenario. As explained in [START_REF] Amodei | Concrete problems in AI safety[END_REF], it is a disputable designed choose to implement the metrics used to judge a system in its rewards function, as the system would only optimise the metrics. It is the reason why KPIs were not implemented in the reward function.

For an autonomous driving decision-making system, they can be divided in 4 categories: safety, navigation, trust and comfort. In this section, we propose some KPIs for each category with respect to the road intersection crossing context.

1) Safety: At cross road intersection crossing, it is unsafe and forbidden by the highway code to stop within the intersection. Thus, a first KPI can be formulated as the time the SV spent stopped within the intersection. Tests where this KPI is not null result in a failure.

2) Navigation: This category highlights the performances of the system to adapt its behaviours regarding the situation. First the travel time is looked at. A slow vehicle might be safe but will reduce the traffic flow. In the context of our experimentation, a vehicle driving at 8ms -1 covers the distance necessary to cross the intersection in 6.5s. Taking into account that deceleration is required at road intersection, a travel time lower than 15s is considered a maximum for scenario B (SV has the priority). For Scenario A, a travel time lower than 20s is a success because the SV vehicle is required to decelerate more. In both scenario the SV is allowed to come to a stop before the intersection if the situation required it. The time stopped before entering the intersection is used as a second KPI for the navigation category. The best possible outcome for both scenarios is that the SV has adapted its speed to negotiate and did not come to a stop. However, if the SV stopped, it should not wait for a long time (in scenario A the waiting time should be enough to let the OV cross). Thresholds for this category could be learned from data obtained while driving.

3) Trust: Drivers tend to maintain a certain time gap between them and other obstacles. From [START_REF] Fitzpatrick | Gaps accepted at stop-controlled intersections[END_REF], a gap of 4 seconds is a minimum not to frighten on-board passengers. With a smaller gap the trust of the driver in the system might be reduced and lead to dangerous system disengagements. At road intersections the gap between the two vehicles can be small during the approach, however when the SV enter the intersection this gap is required to be sufficient enough or non-existent (the OV has already crossed the intersection). Therefore, this KPI is formulated as the time gap when the SV enters the intersection. A successful test has the SV let the other vehicle pass first or the SV entered the intersection with a gap higher than 4 seconds. Fig. 3. Experimental setup, the bottom part represents elements of the testing system, and the top one the decision-making framework. The simulated environment is modified by selected actions and send back observations of the situation.

4) Comfort:

As the velocity of a vehicle is changed approaching towards the intersection, sudden changes of acceleration (jerk) are undesirable. From [START_REF] Yi | Nonlinear brake control for vehicle cw/ca systems[END_REF], passengers felt uncomfortable if the jerk is higher than 2ms -3 . The last KPI is the mean jerk and should not be higher than 2ms -3 .

Success conditions for each scenario and for each KPI are summarized in table II.

B. Testing System

The experimental setup, as shown in Fig. 3, works as follows: first a set of configurations and scenarios is created. A configuration is the set of k values used to compute weight functions. Next a scenario is loaded into a simulator and the decision-making system starts with a selected configuration. Each test last for 20 seconds, enough time for the SV to cross the intersection. KPIs defined in section IV-A are used at the end of the chain to find configurations that were most likely successful. To generate the environment of testing, we used an automotive grade simulator (Scaner) interfaced with a desktop computer that runs ROS and the decision-making system. Within the simulated environment, the observations of every vehicle could be precisely obtained. Thus, Gaussian noises were added on the speed and distance measurement to simulate the performances of a perception under real world conditions. For the SV a noise N (0, 0.5) is added for the measurement and a noise of N (0, 1) for the OV vehicle.

The layout in Fig. 1 is used for each test, it corresponds to an existing intersection at our test site. Regulation is changed regarding the requested scenario. The other two scenario changed the regulation and required the OV to come to a stop (Scenario B) or to yield (Scenario C). In order to change the encounter at the intersection, the initial speed for each vehicle is randomly assigned. Thus, the moment when the two vehicles reach the intersection is different.

Best configurations were found with an iterative process, with random configurations tested in different scenarios. Then, the search range was narrowed to best performing configurations and tested again with fewer configuration parameters free. As of now the process of selecting configuration range with successful outcomes is done manually, but in the future methods such as [START_REF] Legay | Statistical model checking: An overview[END_REF] or reinforcement learning could be used to improve this optimization.

For the POMCP solver, a discount of 0.85 (in our setup a horizon of 6 seconds) and C=30 gave the best results. In the Fig. 4, it can be observed that tests with these values are likely to be successful.

C. Results and Analysis

While optimising the success rate of the decision-making system, it appeared that some configurations were better used in a specific scenario. To demonstrate the usage of KPI to explain and compare the system performances , two configurations were chosen with different area of strength. The configuration 1 performed well across each scenario and III shows values selected for two configurations and the Fig. 5 the variation of these functions for the first configuration with respect to the distance to the intersection. For the baseline, we used the decision making within the simulator that is based upon used an Intelligent driver model to control the subject vehicle.

In the two configurations, none of the weighting function was equal to zero. Thus, each aspect of the driving implemented in the reward function is useful to the system. For the first configuration, the low weight value for the risk related w r reward can be explained by the effect of rewards related to the intention w i and expectation w e . If they prevented the vehicle to enter risky situations, the importance of the risk can be reduced. However, its importance is almost constant during the manoeuvre. The high importance for the speed reward w s shows that a key to successful executions is to follow a reference speed. However, it is balanced by a reduction of the comfort as can be seen with a decreasing importance of the comfort. Fig. 6a, as the SV slows down after three seconds it gave enough time for the OV to cross without changing its speed. The more cautious behaviour of the SV in scenario B is due to uncertainties whereas, in scenario A, the OV drivers are much faster as the simulation behaviour model is different.

In Fig. 6 that shows the velocity of the two vehicles and acceleration actions selected by the system in two different scenario. It can be observed that high deceleration actions are selected between distances of 10 and 0 meters. During this test, in Fig. 6b, it can be observed that from t=0s to t=4s the SV kept its speed because the OV is still in an acceleration phase. While being close to the intersection (<20m), the SV starts to adapt its speed and to be ready to come to a stop in case the gap is not sufficient if the OV does not come to a stop as it could be expected in the scenario B. In The table IV shows causes of failure found with the proposed KPIs. As for the baseline,the first configuration is that it is capable to cross the intersection at every test in every scenarios as for the baseline (no failure related to travel time). However, this is also the main causes of failure of the second configuration for scenario B and C. In the Scenario A, it appears that the baseline is most likely to stop at the giveaway intersection instead of slowing and pass as the proposed system is able to do. However, it is balanced with the gap related failure caused by our vehicle entering the intersection while the other vehicle is close. It can be observed that the failure related to the gap is less important for configuration 2 than configuration 1, but compare to the baseline in scenario B the POMDP is able to maintain a sufficient gap. In some of the test, with the POMDP the vehicle stopped within the intersection (Failure caused by unsafe stop critic), even if illegal, it might have helped the subject vehicle to react to the other vehicle unexpectedly crossing the intersection. From results obtained, it can be understood the area of improvement for each configuration are. The configuration 1 should reduce its tendency to stop within the intersection, whereas the second one should focus on reducing gap related failures in scenario A.

V. CONCLUSION AND PERSPECTIVE

In this paper, a POMDP based decision-making system has been proposed to solve road intersection problems taking into account the behaviour of another driver and associated uncertainties. Results of this approach have been evaluated with different key performances indicators that inform on system performances with respect to the road intersection crossing scenario. The evaluation demonstrated that the decision-making is able to cross the intersection, but the analysis reveals that the behaviour does not conform with certain of the proposed criteria. Performances are better or at least equal to the baseline. On-going works involve using the Statistical Model Checking [START_REF] Legay | Statistical model checking: An overview[END_REF] to automatically explore the configuration and scenario space. As such, KPIs will be formulated as Bounded Linear Temporal Logic (BLTL). From that, the bounded probability of occurrence of a KPI value could be found and helps with the validation of decision-making system.

For the proposed POMDP model, the scenario complexity needs to be increase to include pedestrians and more vehicles. Thus, the implementation of the online solver needs to be improve and use parallel computing and learning to be better estimate the value function. The model used for the dynamics of the vehicle and the observation of the physical could state can be improve to be closer to systems used in a vehicle.

The domain of validation of AI-based systems is an emerging topics that requires more attention [START_REF] Li | Artificial intelligence test: a case study of intelligent vehicles[END_REF] and the definition of key performance indicators is important to understand the behaviour of such system [START_REF] Amodei | Concrete problems in AI safety[END_REF].

Fig. 1 .

 1 Fig. 1. Crossroad intersection representation including possible scenarios and the main actors: Subject Vehicle (SV) on the right side, Other Vehicle (OV) on the left side.

Fig. 2 .

 2 Fig. 2. The transition function represented as Bayesian network. The square node represent the action chosen by the framework

Fig. 4 .

 4 Fig. 4. Percentage of successful execution for different discount values and exploration coefficients. The area within the blue border represent area with high potential of success

Fig. 5 .

 5 Fig. 5. Reward coefficient as function of the distance to the intersection for the configuration 1

B with configuration 1 Fig. 6 .

 16 Fig. 6. Evolution of the speed in function of distance (above) and time (below) Changes of acceleration are shown by the color of each segment.

  Expectation reward (r e in eq.6) penalizes states where the expectation of the OV is different from its intention (similar to the notion of risk introduced in[START_REF] Lefevre | Risk assessment at road intersections: Comparing intention and expectation[END_REF]). r e (e ov , i ov ) R max /2 e ov = i ov -R max otherwise

		TABLE I
	REWARDS RELATED TO THE BEHAVIOUR OF DRIVERS WITH RESPECT TO
		TRAFFIC LAWS
	esv/iov	Stop	Yield	Pass
	Stop	0	0	Rmax/2
	Yield	Rmax/2	0	Rmax/2
	Pass	Rmax	Rmax/2 -Rmax

•


Gibbard-Satterthwaite Theorem

Pierre Bernhard
email: pierre.bernhard@inria.fr

Marc Deschamps
email: marc.deschamps@univ-fcomte.fr

One seminal question in social choice theory was: is it possible to find a social choice function such that each agent is always better off when telling the truth concerning his preferences no matter what the others report? In other words, can we find a strategy-proof voting rule? With at least three alternatives and two voters the answer is clearly no under a very general framework, as was proved independently by Allan Gibbard and Mark Satterthwaite. Since then, the Gibbard-Satterthwaite theorem is at the core of social choice theory, game theory and mechanism design.

Introduction

Since K. Arrow's 1951 analysis, which marks the revival of the theory of social choice, economists investigate from an axiomatic point of view the aggregation of individual preferences in order to obtain a social welfare function (i.e a complete and transitive ranking based on the individual preferences) or a social choice function (i.e one alternative from the individual preferences). Such questions concern huge domains of human beings, for example, the family's choice of the walls color 1 in the living room, the choice of the Palme d'Or winner by the jury of the Cannes International Film Festival, or the vote for the President of the European Commission. Thus, in addition to his (im)possibility theorem, K. Arrow has initiated a new domain in economic analysis.

Since then, the question of the strategic behavior of individuals during an election was raised again on several levels. Indeed for a very long time, since we find for example a letter from Pline the Younger in Roman Antiquity, the question of manipulation had attracted attention. Whether playing on who will be a voter, who can be a candidate, the choice of the voting rule, abstention, beliefs about preferences or preferences expressed during the vote, the idea that at least one individual can do a manipulation to his advantage has interested and concerned many thinkers.

Concerning this last form of manipulation, it was historically mentioned at least seven times before the Gibbard-Satterthwaite theorem (an expression first used by [START_REF] Schmeidler | Two proofs of the gibbard-satterthwaite theorem on the possibility of a strategy-proof social choice function[END_REF]). First, a story probably apocryphal tells that Borda responded to one of his critics who pointed out that his method was manipulable that it was intended for use by honest people. There is also a reference to this question in Ch. Dodgson (alias Lewis Carroll), who said in a specific voting system, "This voting principle makes an election more of a skill game than real test of voters' wishes" (see [Black, 1958]). In the modern period, [Black, 1948] discusses the link between unimodal preferences and strategic votes, while [Arrow, 1951, p. 7] explicitly states that he will not deal with this issue even though he later returns to it in a footnote [footnote 8, p. 80-81]. Arrow's general analysis, however, will lead [Vickrey, 1960, pp. 517-519] to conjecture that immunity to strategic manipulation is logically equivalent to the association of the axiom of independence of irrelevant alternatives and that of positive association. Finally, it will be R. Farquharson in his 1958doctoral dissertation, published in 1969, [Farquharson, 1969], who will introduce the distinction between "sophisticated strategy" and "sincere strategy", and then in an article with M. Dummett in 1961 when they make the conjecture: "It seems unlikely that there is any voting procedure in which it can never be advantageous for any voter to vote 'strategically', i.e non sincerely" [Dummett and Farquharson, 1961, p. 34]. In an interview given in 2006 to R. Fara and M. Salles, M. Dummett confesses that he felt at this time that proving this conjecture would be extremely difficult and that is why they did not try the demonstration. We refer the reader to [Barberà, 2010] for more details on this historical part.

Thus, for more than twenty years after [Arrow, 1951], all scholars of social choice theory seem to have been convinced that the question of the manipulation of preferences by an individual (that is, the fact that he does not express his true preferences in order to lead to a social choice that satisfies him better than would have been obtained if he had been honest) was an important question, easy to ex-plain, but very difficult to prove.

As an illustration of this question of the manipulation of preferences by an individual we can give the following example (from [Feldman, 1979, p. 459]):

Imagine a committee made up of 21 members each having one vote and whose actual preferences presented in descending order can be broken down into three groups. In a majority election where voters indicate their real preferences, A gets 10 votes, B gets 9 votes and C gets 2 votes; this leads to the election of A. However, if anticipating this result, the voters in group 3 manipulate their preferences and vote for B, this will lead to 10 voters for A and 11 voters for B and so B will be elected. This strategic choice of the voters of the group 3 allows them to obtain a more favorable result than they would have obtained if they had voted sincerely.

Therefore a question immediately comes to mind: is it possible to imagine a social choice function such that each individual, regardless of the others's choices, is always better off by expressing his true preferences? In other words, is there a social choice function such that always telling the truth is a strictly dominant strategy for each individual? The answer to this question was independently given by the philosopher Allan Gibbard [Gibbard, 1973] and the economist Mark Satterthwaite [Satterthwaite, 1975] (work summarizing part of his doctoral dissertation defended in 1973) and it is unfortunately negative. What we call since the Gibbard-Satterthwaite theorem can be broadly stated as follows: it is not possible to find a social choice function that is both non-manipulable and non-trivial. 1This last term deserves an explanation. By a trivial function we mean one of the following solutions (or a solution equivalent to it): 1/ a dictatorship (i.e all the power of decision resides with a single individual and he has no indifferent choice), 2/ the permanent choice of an alternative whatever the preferences expressed by the individuals are (which could, for example, correspond to a tradition which would be imposed in all circumstances), 3/ the perfect unanimity of all the voters (which in fact means having perfect clones and therefore no diversity in the preferences), or 4/ the majority between two alternatives only, whatever the number of other alternatives existing and the votes that are expressed for them. Thus, if all individuals know their preferences and those of others and that there are at least three possible alternatives and at least two voters, there is no social choice function that implies that each individual always has an interest in expressing his or her real preferences. We thus find here the conditions of Arrow's theorem and the same conclusion since when there are only two alternatives, majority voting is at the same time a non-dictatorial and non-manipulable social choice function.

Since then, the Gibbard-Satterthwaite theorem has been considered, along with Arrow's theorem, as one of the two most famous results of social choice theory and has led to a very large literature in this field. It also plays a crucial role in public economics and in the theory of incentive mechanisms, which can be broadly seen as a social engineering approach of finding the rules to achieve a specific outcome from agents interacting strategically and having private information (see [Börger, 2015]). Indeed, because of this theorem, the incentives of individuals must be considered as relevant constraints in the design of any mechanism.

Gibbard-Satterthwaite theorem

Many proofs of this theorem have been proposed and it is possible to consider that they take one of the following four paths: 1/ that used by A. Gibbard and which uses Arrow's theorem, 2/ that used by M. Satterthwaite thanks to a combinatorial argument and recurrences on the number of individuals and alternatives, 3/ that considering this theorem as the consequence of the fact that non-manipulability requires strong monotony (see [Moulin, 1988]), and 4/ that developed by S. Barberá and his coauthors using the concept of pivotal agents. For a first presentation of the question of manipulation we refer to [Feldman, 1979] and for a review of this literature we refer to [Sprumont, 1995] and [Barberà, 2010]. Following Gibbard's approach, we will prove the Gibbard-Satterthwaite theorem as a corollary of Arrow's (im)possibility theorem. The presentation we retain will distinguish the formal setup, the links between the two theorems, the proof of the Gibbard-Satterthwaite theorem, and that Arrow's theorem in turn can be seen as a corollary if we directly prove the Gibbard-Satterthwaite theorem.

Formal set-up

Let A = {a, b, . . .} be a set of three or more alternatives. Let P be the set of linear orders over A, and P = P n . An element Π = (P 1 , P 2 , . . . , P n ) ∈ P is called a profile, and the P i the individual preferences. The order in P i is denoted a i b for "player i prefers a to b". We further define

Definition 1 The domination set of a ∈ A in P i D(a, P i ) = {x ∈ A | a i x}
Moreover, for Π = (P 1 , . . . , P n ), D(a, Π) stands for the product set of the D(a, P i ). Thus, let Π = (P 1 , . . . , P n ) be another profile, the notation D(a, Π ) ⊇ D(a, Π) means: ∀i ∈ {1, . . . , n} , D(a, P i) ⊇ D(a, P i ).

Definition 2

• A social welfare function (or SWF) is an application f : P → P .

• A social choice function (or SCF) is an application F : P → A.

The order relation in f (Π) will be denoted a > b or, if needed a > f (Π) b. We need to define the following properties of SWF or SCF:

Definition 3 • The SWF f is Pareto efficient (or satisfies the unanimity rule) if [∀i ≤ n , a i b] ⇒ a > b .
• The SCF F is Pareto efficient, (or satisfies the unanimity rule) if whenever an alternative a ∈ A is the most preferred alternative in all individual preferences, it results that F (Π) = a.

• The SWF f is independent of irrelevant alternatives (IIA) if , for any a and b in A, the relative ranking of a and b in f (Π) only depends on their relative rankings in the P i , irrespective of the rankings of other alternatives.

• The SCF F is monotonic if, given two profiles Π and Π ,

[F (Π) = a and D(a, Π ) ⊇ D(a, Π)] ⇒ F (Π ) = a .
• The SCF F is strategy-proof if, when Π and Π differ only in changing P i into P i , it results that either

F (Π) = F (Π ) or F (Π) i F (Π ) .
• The SWF f is called dictatorial if there exists a player k such that, ∀Π ∈ P, f (Π) = P k . (The social preferences are always player k' preferences.)

• The SCF F is called dictatorial if there exists a player k such that, ∀Π ∈ P, ∀x = F (Π), F (Π) k x. (F (Π) is player k's most preferred alternative.)

Arrow and Gibbard-Satterthwaite theorems

Theorem 1 (Arrow) If a SWF is Pareto efficient and IIA, then it is dictatorial.

Theorem 2 (Gibbard-Satterthwaite) If a SCF is strategy-proof and onto (i.e. its range is all of A: ∀a ∈ A, ∃Π ∈ P such that F (Π) = a), it is dictatorial.

Lemma 1 (Muller and Satterthwaite) If a SCF is strategy-proof and onto, it is monotonic and Pareto efficient.

Proof Let Π and Π be two profiles, F (Π) = a, D(a, Π ) ⊇ D(a, Π), but assume that F (Π ) = a. Make the change from Π to Π one player at a time in numeric order. Denote by Π k the profile obtained after changing P k to P k . (And Π 0 = Π).

At some point, we have that

F (Π i-1 ) = a = b = F (Π i )
. By strategy-proofness, it follows that a i b in P i while b i a in P i . But by hypothesis, if a i b in P i it is a fortiori true in Π . A contradiction. Therefore the SCF is monotonic.

Because F is assumed to be an onto function, for any given a ∈ A, there exists a profile Π such that F (Π) = a. Build Π by moving a at the top of the preferences of all players. By monotonicity, it still holds that F (Π ) = a. Now, get Π by shuffling at will the preferences of all players below a, leaving a at their top position. Because of the definition of monotonicity, and specifically its IIA-like character, it still holds that F (Π ) = a. Therefore the SCF is Pareto efficient.

Proof of the Gibbard-Satterthwaite theorem

This note is devoted to the proof of the Gibbard-Satterthwaite theorem viewed as a corollary of Arrow's theorem. We assume therefore that the latter is known. Given the above lemma 1, we need to prove Lemma 2 If a SCF is Pareto efficient and monotonic, it is dictatorial.

Or method of proof will be as follows: we assume that a SCF F is known which is Pareto efficient and monotonic. From it we construct a SWF f which we show to be Pareto efficient and IIA, therefore dictatorial, which will imply that F is dictatorial.

Let F be a SCF. Given a profile Π, our social ordering f (Π) is built via the following algorithm:

Algorithm • Let Π 1 = Π.
• For i ranging from 1 to n, do:

-define a i = F (Π i ),
define Π i+1 by moving a i at the bottom of all individual preferences in Π i .

• Define f (Π) as: for all i ∈ {1, . . . , n -1}, a i > a i+1 .

Proposition 1 If the SWF F is Pareto efficient and monotonic, the above algorithm produces a linear order on A.

Proof What we need to prove is that all elements of A will be ranked, i.e. that for all i ∈ {1, . . . , n}, and for all j < i, a i = a j . Let first i < n. Therefore, not all alternatives have been numbered as one of the a k . Define Π i as follows: Take an alternative b which has not yet been ranked. Raise it at the top of all individual preferences. This does not change the domination sets: D(a j , Π i ) = D(a j , Π i ) since in Π i , all the alternatives that have already been selected in the algorithm, including a j (remember that j < i), are stacked at the bottom of all individual preferences. Therefore, by monotonicity, if

F (Π i ) = a j , it also holds that F (Π i ) = a j . But by Pareto efficiency, F (Π i ) = b, a contradiction.
Finally, in Π n , n -1 different alternatives have been placed at the bottom of all individual preferences, thus the same and last alternative is alone at the top of all, and is therefore selected by F by Pareto efficiency.

The following propositions end our proof:

Proposition 2 If the SCF F is Pareto efficient and monotonic, the SWF defined by the algorithm is 1. Pareto efficient, 2. independent of irrelevant alternatives (IIA), and therefore dictatorial by Arrow's theorem. 2. Let i < j and therefore by our algorithm, a i > f (Π) a j . In Π, move another alternative b in some individual preferences, call the new profile Π . We claim that in the order created by our algorithm applied to Π , it still holds that a i > a j .

Assume that F (Π ) = b. Then at step 2, b is brought at the bottom of all individual preferences, and, as compared to profile Π, the domination sets of a 1 have been enlarged or kept unchanged for all individual preferences. Therefore F (Π 2 ) = a 1 . If b = a 2 , at step 2, for the same reason a 2 will be selected, and so forth until the end of the algorithm. Therefore, we will still select a i before a j .

Assume Repeat this argument at each step before b is selected, and once this happens, repeat the previous argument. It follows that, except for b, all other alternatives are chosen in the same order as for Π.

It follows that, by Arrow's theorem, f is dictatorial. There is a player k such that for all Π, f (Π) = P k , and in particular F (Π) is player k's preferred alternative. And this proves lemma 2, and consequently the Gibbard-Satterthwaite theorem.

Complement: Arrow's theorem as a corollary of Gibbard Satterthwaite

It can also be shown that, if the Gibbard-Satterthwaite theorem has been proved directly, Arrow's theorem is an easy corollary, thus establishing the equivalence between the two results. The process is symmetrical from the above one, deriving a SCF from a Pareto efficient and IIA SWF by simply picking the top alternative in the social preferences, and showing that it is onto and strategy-proof. We might also mention that Lemma 1 has an easy reciprocal.

Conclusion

Since its demonstration, the Gibbard-Satterthwaite theorem has generated a huge literature on the question of the manipulation of preferences, distinguishing, in par-ticular, the cases where the social choice function is manipulable by an individual from the case where it is manipulable by a coalition of individuals. It has also been extended to take into account set-valued ( [START_REF] Duggan | Strategic manipulability without resoluteness or shared beliefs: Gibbard-Satterthwaite generalized[END_REF]) and non-deterministic choice functions ( [Gibbard, 1977]), that is those where the result depends on the votes of individuals but also by chance. (Often both set-valued and non-deterministic.) In our opinion, three main ways of circumventing this theorem have been followed. The first is to restrict the preference domain (with functions defined on restricted sets of preference profiles, it is possible to find social choice functions that are both non-dictatorial and non-manipulable, e.g unimodal preferences à la Black). The second is to change the goal. Indeed, the framework in which the Gibbard-Satterthwaite theorem is situated is very strong since it seeks a social choice function for which telling the truth is a dominant strategy for each individual. The path followed by implementation theory is to simply ask that it be a Nash strategy, or a perfect subgame strategy, or a Bayesian Nash strategy, depending on the informational context of the individuals. Finally, more recently, a third way explain that this problem is real but may not be very important empirically. Indeed, besides the integrity, ignorance or stupidity of individuals that can prevent them from performing manipulations, the fact that a social choice function is manipulable does not imply that it will be manipulated. And since [START_REF] Bartholdi | The computational difficulty of manipulating an election[END_REF], economists consider that it may be empirically impossible for individuals to decide how to manipulate even when they have all the information to do so, as the problem may be NP-hard.

Cross References

Proof 1 .

 1 Let a, b ∈ A, and assume that in Π, ∀i ∈ {1, . . . , n}, a i b. Assume that at some step of our algorithm, F (Π i ) = b, but a has not yet been chosen. In Π i , raise alternative a at the top of all individual preferences. This does not change the domination sets of b in any individual preferences. Therefore, F should still select b. But by Pareto optimality, it should select a. A contradiction.

F

  (Π ) = c = b, and assume c = a 1 . This is possible only if in one individual preferences at least, b has been moved from below a 1 to above it. In these individual preferences only, bring b back below a 1 . Call this profile Π . D(a 1 , Π ) ⊇ D(a 1 , Π). Therefore F (Π ) = a 1 . But in going from Π to Π , b has only been moved down in some individual preferences. Therefore D(c, Π ) ⊇ D(c, Π ). Therefore F (Π ) = c. Hence a 1 = c contrary to the hypothesis. Hence F (Π ) = a 1 .

This statement implies that impossibility holds as soon as the social choice function has at least three alternatives in its range. We will only prove a slightly weaker version, namely under the stronger assumption that no alternative is out of its range.


Ngoc-Vu Hoang

Antonio Pereira

Hai Son Nguyen

Emmanuel Drouard

Bernard Moine

Thierry Deschamps

Regis Orobtchouk

Anne Pillonnet

Christian Seassal

Giant enhancement of luminescence down-shifting by a doubly-resonant rare-earth-doped photonic metastructure

Keywords: Luminescence enhancement, rare earth, down-shifting, photonic crystal, metastructure

Phosphors based on rare earth ions are used to convert photons to useful wavelengths for a wide range of applications, but the improvement of their emission efficiency remains a key challenge in thin-films.

In this paper, we demonstrate theoretically and experimentally that a photonic crystal (PC) structure can efficiently enhance the luminescence properties of a down-shifting layer (DSL). Our new metastructure combines a DSL (Y2O3:Eu 3+ ) and a two dimensional (2D) planar PC (SiNx), the latter being specifically engineered to control simultaneously the absorption in the DSL and the emission into free space. A global

emission enhancement of 77 times is obtained due to PC Bloch mode resonances. Additionally, this metastructure exhibits high transmission above 450 nm, and favours the emission of converted light at specific angles, with an enhancement factor of up to 690. We explain how the slow and fast Bloch modes of the PC structure induce these giant absorption and extraction enhancements of light.

Phosphors based on rare earth (RE) ions play an increasingly important role in a variety of applications such as bio-imaging and labeling, [START_REF] Bouzigues | Biological Applications of Rare-Earth Based Nanoparticles[END_REF] displays, 2 lighting, 3 solar cells, [START_REF] Trupke | Improving Cell Efficiencies by Down-Conversion of High-Energy Photons[END_REF][START_REF] Richards | Luminescent Layers for enhanced Silicon Solar Cell Performance: Down-Conversion[END_REF][START_REF] Huang | Enhancing Solar Cell Efficiency: The Search for Luminescent Materials as Spectral Converters[END_REF][START_REF] Goldschmidt | Upconversion for Photovoltaics -A Review of Materials, Devices and Concepts for Performance Enhancement[END_REF] thermometry, [START_REF] Jaque | Luminescence Nanothermometry[END_REF] and X-ray imaging. [START_REF] Pereira | Low-absorption, Multi-Layered Scintillating Material for High Resolution Real-Time X-ray Beam Analysis[END_REF] Interest in these materials is related to their luminescent properties over a wide spectral range from ultraviolet (UV) to infra-red (IR), their high quantum efficiencies close to the theoretical maximum (100%), and their long emission lifetimes (up to a few milliseconds). Unfortunately, due to the forbidden nature of the 4f-4f transitions in trivalent rare earth ions, such materials suffer from a small absorption crosssection (around 10 -20 cm 2 ), [START_REF] Bunzli | New Opportunities for Lanthanide Luminescence[END_REF] making them unsuitable for thin film applications for example. In order to improve the photoluminescence (PL) efficiency, several approaches can be considered. The simplest one consists of using a large amount of material (up to several tens of microns thick), but this is not compatible with thin film applications. Another possibility is to increase the doping concentration.

However, the efficacy of this approach is limited because of emission quenching process. During the last few years, the maturity of nanophotonics, [START_REF] Koenderink | Nanophotonics: Shrinking Light-Based Technology[END_REF] together with the mastering of nanofabrication, has enabled novel solutions to tailor light-matter interaction at the subwavelength scale. These technologies can now be used to improve the efficiency of light absorption, emission, or extraction of RE doped thin films.

Thanks to plasmonic resonances, metallic structures precisely designed on the nanometer scale can give rise to a fluorescence enhancement in nearby emitters via processes occurring at the excitation and/or the emission wavelengths. By exploiting these properties, many studies have been devoted to plasmonicbased solutions using a handful of materials such as gold [START_REF] Fischer | Plasmon Enhanced Upconversion Luminescence Near Gold Nanoparticles -Simulation and Analysis of the Interactions: Errata[END_REF][START_REF] Eichelbaum | Plasmonic Enhancement or Energy Transfer? On the Luminescence of Gold-, Silver-, and Lanthanide-Doped Silicate Glasses and Its Potential for Light-Emitting Devices[END_REF][START_REF] Derom | Metal Enhanced Fluorescence in Rare Earth Doped Plasmonic Core-Shell Nanoparticles[END_REF][START_REF] Saboktakin | Plasmonic Enhancement of Nanophosphor Upconversion Luminescence in Au Nanohole Arrays[END_REF] ,silver [START_REF] Koenderink | Nanophotonics: Shrinking Light-Based Technology[END_REF][START_REF] Fischer | Plasmon Enhanced Upconversion Luminescence Near Gold Nanoparticles -Simulation and Analysis of the Interactions: Errata[END_REF][START_REF] Eichelbaum | Plasmonic Enhancement or Energy Transfer? On the Luminescence of Gold-, Silver-, and Lanthanide-Doped Silicate Glasses and Its Potential for Light-Emitting Devices[END_REF][START_REF] Pillonnet | Coupling Distance Between Eu 3+ Emitters and Ag Nanoparticles[END_REF] and aluminum. [START_REF] Abdellaoui | Plasmonic Enhancement of Eu:Y2O3 Luminescence by Al Percolated Layer[END_REF]18 However, in practice, and summarizing the current state of the art, the fluorescence enhancement obtained with such coupling is limited to factors of a few tens, [START_REF] Bunzli | New Opportunities for Lanthanide Luminescence[END_REF][START_REF] Fischer | Plasmon Enhanced Upconversion Luminescence Near Gold Nanoparticles -Simulation and Analysis of the Interactions: Errata[END_REF][START_REF] Eichelbaum | Plasmonic Enhancement or Energy Transfer? On the Luminescence of Gold-, Silver-, and Lanthanide-Doped Silicate Glasses and Its Potential for Light-Emitting Devices[END_REF][START_REF] Derom | Metal Enhanced Fluorescence in Rare Earth Doped Plasmonic Core-Shell Nanoparticles[END_REF][START_REF] Saboktakin | Plasmonic Enhancement of Nanophosphor Upconversion Luminescence in Au Nanohole Arrays[END_REF] the highest plasmon mediated enhancement being achieved when the resonance is tuned in the excitation wavelength range. [START_REF] Fischer | Plasmon Enhanced Upconversion Luminescence Near Gold Nanoparticles -Simulation and Analysis of the Interactions: Errata[END_REF] Recently, Lee et al. [START_REF] Lee | Simultaneous Enhancement of Upconversion and Downshifting Luminescence via Plasmonic Structure[END_REF] have reported improvements up to 29-and 174 times for down-shifted luminescence and up-shifted luminescence, respectively. However, due the complexity of the structures involved which combine simultaneously down-shifting and up-conversion layers with silver film and nanodisks, the origin of the enhancement factors is not fully understood. On the other hand, PC structures allow a careful engineering of the dispersion of photonic bands, and thereby make it possible to enhance both light extraction [START_REF] Wierer | III-nitride Photonic-Crystal Light-Emitting Diodes with High Extraction Efficiency[END_REF] and absorption. [START_REF] Park | Absorption Enhancement using Photonic Crystals for Silicon Thin Film Solar Cells[END_REF] Additionally, as these structure enable in-depth modification of the local density of optical states (LDOS), they may be used to control emission rates, [START_REF] Noda | Spontaneous-Emission Control by Photonic Crystals and Nanocavities[END_REF] and energy transfer mechanisms. [START_REF] Kedia | Influence of Photonic Crystal on Fluorescence Resonance Energy Transfer Efficiency between Laser Dyes[END_REF] These photonic approaches provide unique ways to improve the wavelength conversion in RE doped layers.

Most of the literature in this field is focused on enhancing the RE-emission extraction, by out-coupling guided light via Bloch mode resonances of a PC, which leads to an increase in luminescence by a factor of several times. [START_REF] Ko | Structural Effect of a Two-Dimensional SiO2 Photonic Crystal Layer on Extraction Efficiency in Sputter-Deposited Y2O3:Eu 3+ Thin-Film Phosphors[END_REF][START_REF] Ko | Fabrication of 2D Photonic Crystal Assisted Y2O 3: Eu 3+ Thin-Film Phosphors by Direct Nano-Imprinting[END_REF][START_REF] Zhang | Enhancing Extraction Efficiency of Mid-Infrared Fluorescence in Chalcogenide Glass via Photonic Crystal[END_REF] In a previous study using a Ce 3+ , Pr 3+ , Yb 3+ tri-doped CaYAlO4 layer, we theoretically demonstrated that Bloch mode resonances at excitation wavelengths can lead to a 10-fold increase of the UV light absorption. [START_REF] Deschamps | Nanostructured Down-Converter Module for Photovoltaic Application[END_REF] Most recently, Hofmann et al. have proposed a one dimensional (1D) PC design which combines absorption improvement and LDOS enhancement, yielding a 480-fold theoretical up-conversion enhancement of Er 3+ doped NaYF4. [START_REF] Hofmann | Increased Upconversion Quantum Yield in Photonic Structures due to Local Field Enhancement and Modification of the Local Density of States -A Simulation-Based Analysis[END_REF] Depending on the geometrical parameters of the photonic structure, it is therefore possible to optimize separately or simultaneously the RE ions absorption, emission rate and light extraction of the converting layer.

In this article, we present a new method to improve the down-shifting efficiency of light by combining a 2D planar PC slab with a luminescent DSL, as shown in Figure 1a. We choose the well-known phosphor Y2O3:Eu 3+ , now widely used in fluorescence lamps and field emission displays, [START_REF] Ronda | Rare Earth Phosphors Fundamentals and Applications[END_REF][START_REF] Justel | New Developments in the Field of Luminescent Materials for Lighting and Displays[END_REF][START_REF] Matsuura | Green and Blue Up-Conversion Luminescence of Trivalent Rare Earth Ion-Doped Y2O3 Nanocrystals[END_REF] as the DSL material.

The Eu 3+ ion is also well known for its intense red emission under excitation in the UV range, and for its quantum yield close to 100%. A detailed study on the spectroscopic properties of Eu 3+ ions can be found elsewhere. [START_REF] Binnemans | Interpretation of Europium (III) spectra[END_REF] For the PC layer located on the top of the DSL, a highly transparent and high optical refractive index material, namely SiNx, is used. As the luminescent down-shifting process involves an interplay between absorption, emission and extraction mechanisms, various strategies can be envisaged to enhance its efficiency. In this study, the PC structure is primarily designed in such a way as to enhance the absorption in the DSL. To reach this objective, the PC membrane should exhibit low Bloch modes at photonic band edges with almost-zero group velocity. We use the resonant coupling of these modes with the incident light to trap incoming photons within the PC structure and thus improve the absorption. [START_REF] Park | Absorption Enhancement using Photonic Crystals for Silicon Thin Film Solar Cells[END_REF] In order to specifically enhance the absorption in the DSL, the PC dispersion must therefore be engineered to exhibit Bloch resonances spatially and spectrally tuned to the absorbing medium. These photonic structures should also be chosen to exhibit dispersion characteristics that favour light emission and extraction in the longer wavelength range.

RESULTS AND DISCUSSION

In this work, the geometrical parameters of the PC have been carefully chosen to optimize the absorption of incident light in a 100-nm thick Y2O3:Eu 3+ layer, and considering the specific case of the solar spectrum as the incident light. In our design, a square symmetry of air holes was chosen. The PC structural parameters, including the SiNx thickness t, the lattice period p, and the aspect ratio D/p (D being the holes diameter), were varied to ensure high absorption in the DSL. The influence of the PC structural parameters on the absorption of the DSL is summarized with the figure of merit (FOM), which is calculated as

𝐹𝑂𝑀 = ∫ 𝐴(𝜆). 𝑃𝐿𝐸(𝜆). 𝑆(𝜆). 𝑑𝜆

where 𝐴(𝜆) is the absorption spectrum of the device which is numerically calculated by the Rigorous Coupled Wave Analysis method (RCWA, see Materials and Methods section for more details), 𝑃𝐿𝐸(𝜆) is the experimental photoluminescence excitation spectrum of Eu 3+ (see Figure 2a), and 𝑆(𝜆) is the AM 1.5G solar irradiance. According to this optimization, the set of parameters selected for the fabrication is: 100-nm-thick SiNx square lattice PC with period p of 250 ± 6 nm, holes diameter D of 175 ± 25 nm (i.e. an aspect ratio D/p of 0.7 ± 0.1), and etching depth h of 90 ± 10 nm. The FOM of our new downshifting structure was calculated by scanning the PC geometrical parameters and it is shown in Figure The effect of the PC structure on the transmission of the metastructure over the solar spectrum was first investigated. Transmittance spectra of the metastructure (sample with PC -blue curve) and of the reference (full stack with unpatterned SiNx -red curve) measured at normal incidence are shown in Figure 2a. These spectra clearly reveal that the PC structure generates two resonant modes in the DSL at 400 and 419 nm. As expected, these two modes are perfectly coupled with the excitation bands of Eu 3+ between 350 and 430 nm (grey curve). Another important result is that the transparency of the DSL is globally not affected by the presence of the SiNx PC. More precisely, the transmittance remains within 80% to 93% in the visible and near infrared domains. However, the maximum transmission is blueshifted, which can be attributed to a decrease of the effective refractive index due to the presence of the air holes in the structure. Figure 2b, showing a macroscopic image of the whole sample, illustrates this high transparency. To demonstrate the performance of the proposed device, PL measurements were first performed in an integrating sphere with an optical excitation of 400 nm at normal incidence (with a power density of 0.82 mW/mm 2 and a spot size of 1 mm diameter, more details can be found in Materials and Methods section).

The emission spectra of the metastructure and reference samples are shown in Figure 3. The observed emission spectra exhibit peaks corresponding to the transitions 𝐷 0 → 𝐹 𝐽 7 5 (with J = 0-3) of Eu 3+ ions in the cubic phase. As expected, the Y2O3:Eu 3+ film exhibits very low PL intensity when the SiNx layer is unpatterned (red curve), and only the luminescence of the main transition 𝐷 0 → 𝐹 2 7 5 at 611 nm is observed. The light trapping effect is clearly evidenced on the patterned sample (blue curve). Indeed, thanks to the SiNx PC a 77 ± 3 fold enhancement of the emission is measured and all the emission peaks corresponding to the transitions 𝐷 0 → 𝐹 𝐽 7 5 (with J = 0-3) are now clearly distinguishable. This PL enhancement is about 10 times higher than the best results reported in literature for DSLs such as Y2O3:Eu 3+ thin films. [START_REF] Ko | Structural Effect of a Two-Dimensional SiO2 Photonic Crystal Layer on Extraction Efficiency in Sputter-Deposited Y2O3:Eu 3+ Thin-Film Phosphors[END_REF][START_REF] Ko | The variation of the enhanced photoluminescence efficiency of Y2O3:Eu 3+ films with the thickness to the photonic crystal layer[END_REF] We will now analyse the contributions of the different mechanisms to this remarkable conversion increase, in both the absorption and emission wavelength ranges. Angularly resolved PL experiments were also performed on the reference (Figure 5a) and on the metastructure (Figure 5b) under excitation at 400 nm and at normal incidence (with a power density of 0.82 mW/mm 2 and a spot size of 1 mm diameter). The PL was measured under a conic angle 𝜙 = 0 o for various emission angles 𝜃 ranging from 10 to 170°. The angular distribution of the light emission corresponding to the main transition 𝐷 0 → 𝐹 2 7 5 is shown in Figure 5c. The corresponding configuration is shown in Figure 5d. As expected, the emission pattern of the reference (Figure 5a) does not exhibit an optical dispersion characteristic; light emitted around 611 nm is extracted directly into free-space over a large angular range. On the contrary, the emission pattern of the metastructure (Figure 5b) clearly exhibits two photonic bands corresponding to Bloch resonances in a background of free-space modes. Thus, the metastructure preferentially extracts emitted photons at angles selected by the Bloch mode dispersion characteristics, and they overlap with the main transition of the emitter at 611 nm (Supporting Information for details, Figures SI.8 and SI.9). ). (d) Schematic of the angularly resolved PL experiment.

These results clearly show that our metastructure not only controls the absorption at 400 nm with slow Bloch modes, but also influences the extraction in the Eu 3+ emission range due to fast Bloch modes, i.e. with a significant slope leading to a high group velocity. Such dispersions provide efficient out-coupling channels via Bloch resonances, and thus enhance extraction of emission. [START_REF] Boroditsky | Spontaneous Emission Extraction and Purcell Enhancement from Thin-Film 2-D Photonic Crystals[END_REF] The angular distribution of emission at 611 nm, depicted in Figure 5c, shows a high degree of anisotropy -mostly unidirectional with an oblique emission angle: 𝜃 = 58 𝑜 (back emission) and 𝜃 = 119 𝑜 (front emission). In addition to that, the metastructure favours the transmitted emission since the back emission, i.e. below the metastructure, into the substrate, is about twice as intense as the front emission (Figure 5c,d). We attribute this property to the vertical asymmetry of the stack, which includes a higher refractive index medium below the PC structure compared to the ambient medium above it. Figure 6 presents the total PL enhancement 𝜂 𝑡𝑜𝑡 (𝜃, 𝜙) extracted from the angularly resolved PL measurements performed on the metastructure and on the reference. This total enhancement 𝜂 𝑡𝑜𝑡 (𝜃, 𝜙), which is the average of the enhancement factor with emission angle 𝜃 ∈ [0, 𝜋], and emission plan making an angle 𝜙 ∈ [0,2𝜋] with the PC axis (see Figure 5d) is given by 𝜂 𝑡𝑜𝑡 (𝜃, 𝜙) = 1 4𝜋 ∬ 𝜂(𝜃, 𝜙)𝑠𝑖𝑛𝜃. 𝑑𝜃. 𝑑𝜙.

The measurements were therefore performed according to both planes of incidence 𝛤-X (i.e. 𝜙 = 0) and 𝛤-M (i.e. 𝜙 = 𝜋 4 ⁄ ) to determine how the emission is enhanced in different directions and thereby to infer the total enhancement. In the case of the 𝛤-M direction (red dots), the enhancement is almost constant and equal to 𝜂 𝑡𝑜𝑡 ~ 26 ± 7 times for all angles. Such enhancement is consistent with the slow Bloch mode observed previously at 400 nm following the 𝛤-M direction, and is responsible for the absorption enhancement (Figure 4c). In the case of the 𝛤-X direction (blue dots), a similar trend is observed, namely a constant background enhancement of 𝜂 𝑡𝑜𝑡 ~ 33 ± 7, except for four highly intense emission angles located at 59 o , 70 o , 109 o and 120 o . These are attributed to preferential out-coupling to fast Bloch modes displayed in Figure 5b. Moreover, we highlight that the highest global wavelength conversion enhancement is obtained at 𝜃 = 59°, with 𝜂 𝑡𝑜𝑡𝑎𝑙 ~ 690. These results show a strong angular dependence of the enhanced emission, suggesting that the increased absorption is not the only mechanism responsible for such luminescence enhancement. Indeed, for each (𝜃, 𝜙), the enhancement factor is also given by 𝜂 𝑡𝑜𝑡 (𝜃, 𝜙) = 𝜂 𝑎𝑏𝑠 . 𝜂 𝑒𝑥𝑡 (𝜃, 𝜙). 𝜂 𝜏 where 𝜂 𝑎𝑏𝑠 is the absorption enhancement factor due to slow Bloch modes at 400 nm (see Figure 4), 𝜂 𝑒𝑥𝑡 (𝜃, 𝜙) is the extraction enhancement factor due to fast Bloch modes at 611 nm (see Figure 5)

and 𝜂 𝜏 the emission rate enhancement factor of the metastructure with respect to the reference. Decay time measurements (Supplemental Information, Figure SI.10) show that the luminescence lifetime of Eu 3+ does not change when the SiNx structure is patterned as a PC ( 1.1 ms), indicating that the Purcell effect is not a relevant mechanism in our case (i.e. 𝜂 𝜏 = 1). In summary, the observed emission enhancement is related to an improvement in the absorption efficiency, extraction efficiency, or a combination of both. A detailed analysis of Figures 4 and 6 reveals that, when there is no fast mode at play, i.e. in the 𝛤-M direction and for most angles in the 𝛤-X direction (except in the case of the intense emission peaks), the total enhancement arises only from an increase in the absorption. In these case, the absorption is on average enhanced 29-fold (the variations observed are within experimental uncertainties), and fast Bloch modes do not enhance light extraction (i.e. 𝜂 𝑒𝑥𝑡 = 1). Using the same approach, we can therefore determine the extraction enhancement factor for each highly intense emission angles in the 𝛤-X direction. For example at 𝜃 = 59 𝑜 , with 𝜂 𝑡𝑜𝑡 ~690 and 𝜂 𝑎𝑏𝑠 ~ 29, we found that the extraction was enhanced by a factor of ~24. It is worth noting that this giant enhancement occurs only in one direction (i.e. an oblique emission which is favoured in the transmitted direction). This is an important property which can open new avenues for the implementation of DSL in photovoltaic applications. Indeed, such obliquely emitted light is expected to be efficiently trapped in a solar cell that could be positioned underneath the metastructure. Integrating these metastructures with silicon solar cell is currently under investigation. More generally, using this concept, tight adjustments of the geometrical parameters of the PC will enable one to tune the wavelength and emission angle of the converted light.

CONCLUSION

In summary, we have designed and fabricated a new metastructure that combines a luminescent downshifting layer with a photonic crystal, in order to control and enhance the down-shifting process. We have experimentally demonstrated efficiency enhancement of luminescence of the active layer using optical resonances of the PC structure. Our metastructure results in a 77 fold-enhancement of luminescent downshifting efficiency, which is the highest reported PL enhancement for DSLs such as Y2O3:Eu 3+ thin films. and mixing Y2O3 (99.99% purity, Alfa Aesar) and Eu2O3 (99.999% purity, Alfa Aesar). Thin films were grown at room temperature in an oxygen-gas atmosphere (PO2 = 10 -3 mbar). The laser fluence was held constant 3.2 J/cm 2 . Depositions were performed on high purity fused silica glass (Suprasil grade, Heraeus), which was placed parallel to the target surface at a target-to-substrate distance of 5 cm. The number of laser pulses was adjusted to achieve a 100 nm film thickness. The deposition rate (in nm/pulse) was first determined by means of a profilometer (Tencor Alpha Step 100). The deposited films were then subject to thermal annealing under air at 800 °C at a rate of 0.5 °/min for 6 h, and then cooled down to 20 °C at a rate of 0.5 °C/min. Such thermal treatment allows for improvement of the crystallization of Y2O3, and the low ramping rate was used to avoid the formation of cracks on the film surface.

A 100-nm thick SiNx layer was deposited on top of the Y2O3: Eu 3+ layer by low-frequency plasmaenhanced chemical vapor deposition, using a 440 kHz SEMCO-Engineering reactor. The SiNx stoichiometry was controlled by adjusting the SiH4/NH3 gas ratio. The optical properties of SiNx were determined by ellipsometry measurements to ensure a low extinction coefficient k (< 10 -6 for λ  350 nm) and the proper refractive index n in the working spectrum (Supporting Information, Figure SI.5).

Laser interference lithography (LIL) [START_REF] Meng | Absorbing photonic crystals for silicon thinfilm solar cells: Design, fabrication and experimental investigation[END_REF] was used to create a square lattice of air holes in the SiNx layer.

HMDS (hexamethyldisilizanean adhesion promoter) and then Ma-N2403 photoresist were first spincoated on the layer at 6000 rpm for 30 s, followed by baking on a contact hotplate at 90 o C for 60 s.

Exposure was performed with a 266 nm laser (P = 10 mW) on a sample holder connected to a mirror at 90°. Under illumination fringes are generated on the resist with an inter-fringe period p. This is directly related to the angle , which corresponds to the angle between the laser beam and the normal of the sample surface (p = λlaser/(2sinθ)). The hole radius is controlled by the exposure time (336 s). After tilting the sample at 90°, a second exposure was performed in order to create a 2D pattern with square symmetry.

Finally, the resin was developed using a MIF726 solution for 20 s. After LIL, a reactive ion etching process was used to transfer the pattern from the photoresist on to the SiNx layer. The etching was performed using a gas mixture of SF6, Ar and CHF3 (flow rate of 15, 10 and 40 sccm respectively), a total pressure of 30 mTorr and 100 W of power. The etching time was set to 70 s. At the end, an additional oxygen cleaning plasma was performed for 120 s to completely remove the photoresist (flow rate of 20 sccm, pressure of 20 mTorr and 60 W of power).

Photoluminescence measurements. PL enhancement measurements were performed using a tunable pulsed laser ( = 5 ns, Ekspla NT340) as the excitation source. The fluorescence was collected through an optical fiber (1.1 mm diameter, NA = 0.2) connected to an integrating sphere (Labsphere), and coupled with a Triax190 spectrometer (Jobin Yvon) equipped with a CCD camera. All measurements were performed inside the integrating sphere under normal incidence with a 400 nm excitation wavelength (with a power density of 0.82 mW/mm 2 and a spot size of 1 mm diameter), six accumulations and 10 s of integration time. Angularly-resolved PL experiments were performed outside the integrating sphere using the same experimental setup. All spectra were corrected by the response of the experimental setup.

Angularly resolved transmission measurements. Incident broad light (halogen lamp) was focused onto the metastructure via a microscope objective (x20, NA = 0.42), corresponding to a 50 µm spot size.

The transmitted light was collected via a second microscope objective (x5, NA = 0.2) whose the back focal plan was projected onto the entrance slit of a spectrometer. Finally, the output of the spectrometer was coupled to the sensor of a CCD camera (1024 x 256). This experiment allowed to measure in single shot the angularly resolved transmission (within the numerical aperture of the collecting objective): the horizontal axis of the CCD sensor is the spectral axis and the vertical axis is the angular one.

ASSOCIATED CONTENT Supporting Information

The Supporting Information is available free of charge on the ACS Publications website.

This includes detailed information on the design of the 2D photonic crystal (figure of Merit); details on the fabrication and characterization of the new metastructure; SEM image of the fabricated PC structure

SI. 1

 1 in the Supporting Information.

Figure 1 .

 1 Figure 1. (a) Schematic view of the new down-shifting metastructure combining a 100-nm thick Y2O3:Eu 3+ as the DSL and a 2D PC layer. (b) SEM top view of the 2D PC covering the Y2O3:Eu 3+ layer. (c) SEM cross-sectional

Figure 2 .

 2 Figure 2. (a) Transmittance spectrum of the metastructure (sample with PC -blue curve) and of the reference (full stack with unpatterned SiNx -red curve). The photoluminescence excitation spectrum of Eu 3+ is also given by the

Figure 3 .

 3 Figure 3. Luminescence properties of the metastructure (sample with PC -blue curve) and of the reference (full

Figure 4 .

 4 Figure 4. (a) Experimental measurements and (b) theoretical simulations of the absorption spectra at normal

Figure

  Figure 4a depicts absorption spectra of the metastructure and reference samples. The absorption was

Figure 5 .

 5 Figure 5. Angularly-resolved PL experiments on the reference (a) and on the metastructure sample (b) under

Figure 6 .

 6 Figure 6. Total PL enhancement 𝜂 𝑡𝑜𝑡 (𝜃, 𝜙) extracted from the angularly-resolved PL measurements for the incident plans 𝛤-X (i.e. 𝜙 = 0) and 𝛤-M (i.e. 𝜙 = 𝜋 4 ⁄ ). The enhancement factor is given by 𝜂 𝑡𝑜𝑡 (𝜃, 𝜙) = 𝜂 𝑎𝑏𝑠 . 𝜂 𝑒𝑥𝑡 (𝜃, 𝜙). 𝜂 𝜏 , where 𝜂 𝑎𝑏𝑠 is the absorption enhancement factor due to slow Bloch modes at 400 nm, 𝜂 𝑒𝑥𝑡 (𝜃, 𝜙) is the extraction enhancement factor due to fast Bloch modes at 611 nm and 𝜂 𝜏 is the emission rate enhancement factor. The values given in the figure correspond to the 𝛤-X direction.

6 .

 6 Optical simulations and measurements demonstrate that the fabricated metastructure based on our design leads to slow Bloch resonances spectrally tuned with the absorption peaks of the DSL in the UV range. This coupling allows enhancement of the absorption in the DSL by up to 29 times without disturbing the transmission at other wavelengths. Due to fast Bloch modes in resonance with emission of Eu 3+ at 611 nm, which strongly improve the extraction process, giant luminescence enhancement can be achieved, up to a factor of 690 at specific angles in the transmitted direction. Such oblique emission is an important property which can open the door for the implementation of DSL in photovoltaic applications. One can expect a boost in efficiency if effective solar spectrum harvesting is realized, especially in the UV and near-UV range. More generally, this light management approach, combining DSLs and PCs, is relevant to improve the PL efficiency of other devices, such as LEDs and photodetectors where efficiency and directionality of the emitted light needs to be controlled. Specific metastructures can then be designed using our approach, and fabricated over wide areas using laser interference lithography or nanoimprinting, which is expected to open new routes for the implementation of a wide variety of emitting devices based on luminescent materials.MATERIALS AND METHODSOptical simulations. All the optical simulations use RCWA[START_REF] Moharam | Rigorous Coupled-Wave Analysis of Planar-Grating Diffraction[END_REF] method. In our RCWA code developed in-house, analytical Bessel functions are used to calculate the Fourier coefficients of the permittivity for each cylindrical hole, rather than the Fast Fourier transform (FFT) techniques. This enables a faster computation. Moreover, in contrast to most other methods, the computation time does not depend on the thickness of the stack for a given spectral resolution. The number of orders used in the simulation, from -5 to +5 in each direction for the square lattice of holes, is sufficient to ensure accurate results. The absorption and angularly resolved transmission of the metastructure are simulated using a period of 250 nm, an etching depth of 80 nm, and the radius distribution shown in Figure SI.Metastructure fabrication. Y2O3:Eu 3+ (10 at.%) was first deposited by pulsed laser deposition using an ArF excimer laser beam ( = 193 nm,  = 17 ns, Coherent CompexPro) operating at 5 Hz. The Y2O3:Eu 3+ target was previously synthesized by solid-state reaction (8 h at 1400 °C in air) after grinding

and calculation of the diameter distribution of the holes; theoretical calculation of the dispersion diagram of the Bloch modes of the metastructure; details on extraction enhancement for different in-plane directions; lifetime measurements performed on the metastructure and the reference sample.

ACKNOWLEDGMENTS

We thank Dr Thomas David for SEM-FIB measurements. This work was supported by ARC 4 -Energies programme of the Auvergne-Rhône-Alpes Region.

AUTHOR INFORMATION

Corresponding Authors

* E-mail: ngoc-vu.hoang@univ-lyon1.fr * E-mail: antonio.pereira@univ-lyon1.fr * E-mail: christian.seassal@ec-lyon.fr

Author contributions

The manuscript was written with contributions from all authors. All authors have given approval to the final version of the manuscript.

Notes

The authors declare no competing financial interest.