Timothy Parker

Umberto Grandi

Emiliano Lorini

Aurélie Clodic

Rachid Alami

Ethical Planning with Multiple Temporal Values 1

Keywords: . Ethical Planning, Temporal Values, Adjustable Morality

This paper describes an approach to ethical planning in robotics that combines classical planning, Linear Temporal Logic and lexicographically ordered value sets. We also discuss the features and limitations of the model, with particular attention to its philosophical implications.

Introduction

In this paper we present a framework for an ethical planner with potential applications to robotics. We will first introduce and explain the model, give some explanatory examples and then discuss various features and limitations of it.

The foundation for our model is classical planning, a well-studied problem in computer science [START_REF] Bylander | The Computational Complexity of Propositional STRIPS Planning[END_REF]. Given an initial state, a set of actions and a goal, we then attempt to find a plan (sequence of actions) that achieves the goal from the starting state. For simplicity, our current model assumes a single-agent setting with perfect knowledge and fully deterministic actions. Plans are then compared according to a lexicographically ordered value set, with the aim of finding a plan that satisfies as many as possible of our most important values, with less important value sets acting as tiebreakers.

Some of the most significant features of our model are (i) the inclusion of an adjustable morality level, allowing the robot to handle different goals with different levels of urgency, (ii) the ability for the robot to select a best plan even in cases where it cannot satisfy some or most of its values, (iii) the ability to ignore its given goal entirely in order to satisfy its most important values, (iv) the ease of generating simple explanations for the robot's behaviour.

In its current state, we do not consider our model to be a complete solution to the problem of ethical planning, as many of our simplifying assumptions make the model unsuitable for various applications. However, we nonetheless think that the concepts introduced by this model are worth discussing, and at the end of the paper we outline various ways of extending the model.

Related Work

For good overviews of the field of ethics in AI, see [START_REF] Müller | Ethics of Artificial Intelligence and Robotics[END_REF] and [START_REF] Tolmeijer | Implementations in Machine Ethics: A Survey[END_REF]. In this section we focus on papers that propose practical methods and applications for AI ethics, rather than more theoretical work. Note that many of these papers focus on single actions instead of entire plans. There are approaches utilising defeasible logic [START_REF] Dennis | A Defeasible Logic Implementation of Ethical Reasoning[END_REF], and weighted rules [START_REF] Alili | A Task Planner for an Autonomous Social Robot[END_REF] (though this paper focuses on social norms rather than ethical values). [START_REF] Arkin | Moral Decision Making in Autonomous Systems: Enforcement, Moral Emotions, Dignity, Trust, and Deception[END_REF] gives results on a variety of areas, including restricting robots to social norms and dealing with deception in robots. [START_REF] Vanderelst | An architecture for ethical robots inspired by the simulation theory of cognition[END_REF] uses a simulation-based approach to allow the robot to predict the consequences of its own and others actions, and evaluate these consequences according to various rules. [START_REF] Evans | Ethical Decision Making in Autonomous Vehicles: The AV Ethics Project[END_REF] focuses on a collision scenario involving an autonomous vehicle, proposing to prioritise the ethical claims depending on the situation, e.g., by giving more priorities to the claims of the more endangered agents. [START_REF] Anderson | GenEth: a general ethical dilemma analyzer[END_REF] describes an algorithm that attempts to learn ethical rules given a set of examples and responses from a professional ethicist. [START_REF] Lindner | Evaluation of the moral permissibility of action plans[END_REF] outlines an approach to the ethical evaluation of plans based on assigning utility scores to actions in various states. [START_REF] Yu | Building Ethics into Artificial Intelligence[END_REF] provides a recent survey of this area.

One approach that shares a lot of similarities with ours is [START_REF] Dennis | Formal verification of ethical choices in autonomous systems[END_REF]. This paper proposes a model for verifiable ethical decision making that uses a totally ordered set of values that functions very similarly to our lexicographically ordered sets of values. However, our model also differs from their approach in a number of ways. Firstly, we use Linear Temporal Logic to express our values instead of Propositional Logic. Secondly, our values can be evaluated directly over histories, whereas their values require contextual "ethical rules" that tie certain actions to the violation of certain values. This is an issue (discussed in the paper) if the violation of a value cannot be tied to a single action. One advantage of this model over ours is that it can easily accommodate plans that violate the same value multiple times, which our value struggles to handle correctly (see later discussion).

Model

One approach to ethical planning is to conceive of ethical values as restrictions on the space of available plans, in other words, plans are only considered if they satisfy all ethical values [START_REF] Arkin | An ethical governor for constraining lethal action in an autonomous system[END_REF][START_REF] Arkin | Moral Decision Making in Autonomous Systems: Enforcement, Moral Emotions, Dignity, Trust, and Deception[END_REF][START_REF] Brutzman | Run-time Ethics Checking for Autonomous Unmanned Vehicles: Developing a Practical Approach[END_REF] However, it is important that a robot is able to form a plan even when it is not possible to satisfy all of its values, so we decided to create a method for ranking plans according to the robot's values and goals so that the best-possible plan can always be found (this is a similar problem to non-dominance from [START_REF] Lang | Logical Preference Representation and Combinatorial Vote[END_REF]). We represent values in Linear Temporal Logic (LTL) [START_REF] Reynolds | Chapter 9 -Theorem-Proving for Discrete Temporal Logic[END_REF]. This uses the following syntax:

ϕ ::= p | ¬ϕ | ϕ 1 ∧ ϕ 2 | Xϕ | ϕ 1 U ϕ 2 ,
This is the standard syntax for propositional logic extended by the operators X (next) and U (until). We can also define the operators Gϕ def = ¬(⊤ U ϕ) (henceforth) and Fϕ def = ¬G¬ϕ (eventually). Our model of planning uses the notion of a finite history in LTL [START_REF] Giacomo | Linear Temporal Logic and Linear Dynamic Logic on Finite Traces[END_REF]. This is a finite, alternating sequence of states and actions. We can then evaluate formulas over histories according to the folowing syntax (we omit boolean cases which are defined as usual): Our approach is to order sets of values lexicographically, meaning that more important values are always prioritised over less important ones. The robot has a value set Ω = {Ω 1 , ..., Ω n } where Ω 1 are the most important values and Ω n are the least important. When comparing two histories, we first compare according to Ω 1 , and if they are equal according to Ω 1 we compare by Ω 2 and so on. To compare according to some Ω i set we have two approaches that were first explored in [START_REF] Lorini | A Logic of Evaluation[END_REF]. The first is qualitative. This means that if the values satisfied by history A are a strict subset of those satisfied by B then we prefer B to A. The second is quantitative, meaning we prefer B to A simply if B satisfies more values than A.

H,t
In agreement with contemporary theories of human motivation in philosophy [START_REF] Searle | Rationality in Action[END_REF], economics [START_REF] Harsanyi | Utilitarianism and Beyond[END_REF] and logic for AI [START_REF] Broersen | Goal generation in the BOID architecture[END_REF], we assume that preferences of a rational agent may originate either (i) from endogenous motivations such as goals or desires, or (ii) from ethical considerations based on moral values and norms. 2 To add goals to the model, we introduce the special set Ω D . This acts like one of the Ω i in the sequence Ω allowing us to determine which values are more or less important than the goal. The position of Ω D in the sequence is not fixed, but varies according to the "morality level" of the robot. For morality level m, Ω D is inserted into the set of values at position m, so the new sequence of value sets would be Ω 1 , ..., Ω m-1 , Ω D , Ω m , ... .

Examples

Figure 1. This is a visual representation of the first example described in this section.

Let us now show some examples of how this framework could be applied. Our first example is very simple and is intended to show how the logical machinery of the current model works, and how our model reacts to various kinds of value conflicts. The second is more of a blue-sky example and illustrates how we think this model (or a more advanced model based on ours) might behave in a real-word social setting.

Example 1 Imagine a robot in a childcare setting that is tasked (amongst other things) with distributing toys between children at playtime. For simplicity we will assume that there are only two children and at most four toys. We represent this example with three boxes, one box for each child and a central "toy pile". As this system is finite and has a fixed size we can describe it using propositional logic, but here we will use predicate logic as a shorthand. This means we only need one atomic formula: IN(a, b), meaning that toy a is in box b.

The robot has a single goal set and three value sets. They contain values as follows (we make use of the operators X, F and G, as described in section 3):

• Ω 1 = {FG(∃i IN(i, 1) ∧ ∃i IN(i, 2))}.
This is a subsistence value, ensuring that every child has a toy to play with.

• Ω 2 = {G(∀i (IN(i, 1) → X IN(i, 1)) ∧ ∀i (IN(i, 2) → X IN(i, 2)))}.
The second value set Ω 2 is a "respect for property" value, ensuring that the robot will not take toys out of a child's box.

• Ω D = {(∃x 1 IN(x 1 , 1) ↔ ∃y 1 IN(y 1 , 2)) ∧ ((∃x 1 , x 2 (IN(x 1 , 1) ∧ IN(x 2 , 1) ∧ x 1 ̸ = x 2 )) ↔ (∃y 1 , y 2 (IN(y 1 , 2) ∧ IN(y 2 , 2) ∧ y 1 ̸ = y 2 )))}.
The robot's goal Ω D is to ensure a fair distribution of toys, to reduce the likelihood of arguments or fights.

In this scenario the robot has a morality level of 3, meaning the effective ordering of values is Ω 1 , Ω 2 , Ω D . The only actions available to the robot are skip, which does nothing, and MOVE(a, b, c), which attempts to move toy a from box b to box c. If toy a is not in box b, then MOVE will do nothing. We can then consider various different "sharing problems" that the robot might face, and how it would react in each of them. For ease of comparison, all plans have length 2. 1. This table outlines six different variations on our example, and gives a preferred plan for each variation.

Initial state Box 1 Box 2 Pile Morality Level Preferred (Non-Dominated) Plan 1 / 0 / 0 1,2 3 MOVE(1, 3, 1), MOVE(2, 3, 2) 2 1,2 / 0 3,4 3 MOVE(3, 3, 2), MOVE(4, 3, 2) 3 1,2,3 4 / 0 3 skip, skip 4 1,2,3 4 / 0 2 MOVE(3, 1, 2), skip 5 1,2 / 0 / 0 3 MOVE(2, 1, 2), skip 6 1,2 / 0 3 3 MOVE(3, 3, 2), skip Table
Each initial case considered showcases a different aspect of our model. Variations one and two are straightforward cases where the robot can satisfy all of its values and achieve its goal. Variations three and four illustrate how the robot handles conflicts between its values and its goal, and how this is affected by the robot's morality level. In both variations, the robot cannot satisfy both Ω 2 and Ω D , so it will act to satisfy the more important set, as determined by its morality level. Variation five illustrates how a robot handles conflicts between its values. In this case the robot cannot satisfy both Ω 1 and Ω 2 , so it satisfies the more important set. Variation six illustrates how the robot may still act even if it cannot achieve its goal, in order to satisfy one of its value sets (in this case the robot fails to satisfy Ω D , but satisfies Ω 1 ).

Example 2 Consider a robotic shopkeeper. The most important values for this robot would be safety and legal values, such as not colliding with humans and not selling restricted products to underage customers. At a lower level the robot would need a number of social values, such as respecting the personal space of customers and staff, and prioritising assisting customers who have been waiting longer. Best practice values for the robot would include taking the shortest path between tasks and showing customers the way to a requested product instead of giving them directions. The robot would also need the capacity to update its goals on the fly, in order to correctly respond to requests by customers and staff. This would generally require a re-planning from the robot, in order to ensure that the new goal was taken into account.

We will now give an example of how this robot could behave in practice. Firstly the robot is asked by a customer where she can find flour, the robot would give directions to the customer instead of leading them to it, since (in this example) the robot can see several other customers that have requests, and its value to promptly listen to customer requests is more important than its value to be maximally helpful to the first customer. Then, while the robot is listening to a second customer, the store manager sends an urgent (and time-limited) request to the robot to go and clear a blocked aisle. This causes the robot to re-plan, whereupon it leaves the customer, since the morality level associated with the manager's command causes it to outweigh the value of listening to the customer. However, the robot would still pause to briefly apologise to the customer (in accordance with its social values) since it has time to do this and still complete the command.

This example also illustrates the importance of a capacity for ethical planning, not just ethical decision-making. For example, the robot needs to be able to calculate that it has time to apologise and clear the blocked aisle, but does not have time to listen to the customer's request and clear the blocked aisle. This illustrates that working out which sets of values are and are not satisfiable is often impossible without some capacity for planning. For this example this would likely require temporal planning (planning with real-time duration) instead of just classical planning, we discuss this extension further in the conclusion.

Discussion

In this section we will discuss various features of our model, both positive and negative and the practical and philosophical implications for trying to apply this model in a realworld setting.

Expressivity of LTL An advantage of LTL is its ability to express complex temporal values. Many values are temporally very simple, as they either state that a certain state of affairs must always hold (Gϕ), must hold at some point (Fϕ) or must hold at the end of the plan (FGϕ) (where ϕ is some propositional formula). However some goals or values might be more temporally complex. For example, during cold weather, a robot should close doors shortly after opening them, this would be naturally written as G(C → (O → XX¬O)) (we use XX rather than X so that the robot has time to go through the door before having to close it). We can also have values that prohibit/require certain actions based on what will happen during the plan. For example, a house robot might have the goal to water the plants if it will not rain today, but also have the values to not water if it will rain, and not water more than once. These would be represented as G¬R → FW , FR → G¬W and ¬F(W ∧ XFW ), respectively (W is a proposition that is true only while the robot is watering);

One downside of LTL values is that LTL formulas are evaluated over single histories, meaning that the satisfaction of a value can only ever depend on the features of that history. This is fine for a lot of values, but means we cannot directly express values like "minimise harm" since the satisfaction of this value depends on all possible histories. However, it is possible to indirectly represent these values, as plans that do less harm will satisfy more of our safety values, meaning the least harmful plan should always be at least non-dominated.

As LTL formulas are always either true or false over any given history, we cannot neatly express values that are violated or satisfied by degree (for example, "do not speed"). One way to represent these values is by a series of "breakpoints". For example, we might have one value that says "do not exceed 105% of the speed limit" and another, more significant value which says "do not exceed 120% of the speed limit" (we can add as many of these breakpoints as we need).

Fitting values to settings Our model will be most effective when using a value set that is specialised for the situation in which the robot operates. This is because by specialising our values we can ensure that we have a value set that is relatively small and easy to work with but is effective at deciding between plans.

For example, any robot capable of harming a human should have a value or values that prioritise human safety. In many settings, we can safely assume that the best plan will never involve harm to humans, meaning we only need the value "humans must not be harmed". This helps keep the value set small, and is hopefully reassuring for humans to work with, as they know that the robot will never choose to harm people if it can avoid it. However, in other settings we might regularly expect a robot to have to decide between plans based on the degree of harm done to humans (such as an emergency rescue robot). This will require a more complex and nuanced value-set to allow the robot to identify the ethically ideal outcome.

Furthermore, an advantage of the lexicographic ordering of our model is that it can simultaneously handle many different kinds of values, including ethical values, social norms and codes of best practice. We would expect most robots to have some values from each of these categories, but how many and of what kind will depend on the setting of the robot. In particular, we would expect that the majority of the robot's values will be in the sets that it most frequently uses to discriminate between plans.

For example, we have already discussed how an emergency rescue robot would probably need a large and complex "safety" value set, to enable it to make correct decisions when large numbers of human lives are at stake, and perhaps not all can be saved. On the other hand, it would be unlikely to need a very complex set of social norms, since these would rarely be called upon to decide between plans.

For contrast, consider a more "social" setting for a robot, such as the robot shopkeeper from section 4. As mentioned this robot would need a few safety values, and these values would cause some plans to be abandoned, but we would almost always expect there to be many plans that fulfill all safety values, and that a relatively small and simple set of safety values would be sufficient for any (plausible) scenario that the robot might encounter. On the other hand, this robot would likely need a quite extensive and detailed set of values pertaining to social norms (see section 4). Furthermore, a lot of these values would likely be different for interacting with other robots, staff, and customers.

Explainability and Simplicity

We feel that one of the main attractions of our model is not necessarily its ability to handle certain scenarios "better" than other models, but its simplicity and the ease of generating explanations. For example, another approach to ethical planning would be to assign weights to violating values and then always select the plan with the lowest weight [START_REF] Alili | A Task Planner for an Autonomous Social Robot[END_REF]. Provided that there are finitely many values, and each value can only be violated finitely many times in any given plan, it is possible to calculate weights such that this system would produce identical decisions to a lexicographically ordered system. Therefore, weighted value systems are at least as flexible as ours.

However, we feel that our model has a significant advantage over weighted values when it comes to verification of the model, particularly "informal verification" (i.e how easy is it for a layperson to understand and trust this model). This is because it should be very easy to work out which sets of values the robot would prefer to satisfy, which can be quite difficult to work out for weighted values, particularly without a calculator.

In terms of explanation, the simplest kind of explanation to give is to explain why one specific plan is preferred to another. Our model is able to do this very efficiently, as whenever plan π 1 is preferred to π 2 , there will be some value set Ω k according to which π 1 is better than π 2 (and according to every more important value set, they are equal). We know that π 1 will satisfy either a strict superset of the values satisfied by π 2 , or just more values (using either qualitative or quantitative comparison), and provided that the value sets are relatively small and the values have some intuitive appeal, this should make it easy to generate appealing explanations. A related, but much harder form of explanation is to explain why the robot violated a particular value. For example, in variation 5 of our example, we might ask why the robot violated its "respect property" value. To generate an explanation we can consider all plans that do respect property, and see that they all violate the robot's primary value (every child gets at least one toy). This also provides a further motivation for keeping value sets as small as practically possible, as it will make explanations easier to understand.

Specificity of Values

Most human ethical theories were not designed as ethical decision procedures, meaning that the values they contain may be highly unsuitable for direct use in a robot planner. Consider Kant's universalisability principle "act only in accordance with that maxim through which you can at the same time will that it become a universal law". We can argue that this value requires us not to waste electricity, as if everyone wasted electricity, global warming would be significantly worsened. However, reaching this conclusion is likely to be far beyond the capacities of most robots, so it will be more expedient simply to give it a value like "minimise electricity usage" or even "turn off the lights in empty rooms". Indecision An issue that occurs specifically with qualitative comparison is that certain pairs of histories will be incomparable for some value sets, if both satisfy values that are not satisfied by the other history (at the same priority level). For instance, the robot must choose between interrupting someone's conversation and asking them to move, or invading their personal space. While this will be undesirable whenever a decision has to be made quickly (such as an autonomous vehicle) it may sometimes be desirable. For example, in some scenarios we may want the robot to only make a decision when the preferred option is clearly better (such as by satisfying strictly more values), otherwise, it should seek a human opinion.

Ignoring the Goal In classical planning, the aim is simply to find some plan that satisfies the goal. We can therefore assume that the robot will only act in order to achieve its goal. However, in our model it is entirely possible that the robot may carry out a plan that is entirely unrelated to its goal. For example, I may ask my personal assistant robot to buy me a pizza, but at the same time it spots a child about to be run over by a speeding car. If the robot calculates that there is no way to save the child besides sacrificing itself to stop the car then it would immediately do so as long as it has a value like "humans must not be harmed". The extent to which this behaviour is desirable is debatable, as it may lead to "morally superior robot villains" [START_REF] Leben | Ethics for Robots: How to Design a Moral Algorithm[END_REF].

Repeated Value Violations

As previously mentioned, the most natural way to represent most maintenace-type values is Gϕ. One downside of representing values like this is that our model does not differentiate between violating thse values once or multiple times, since both cause the value not to be satisfied by the history. For instance, consider the main example from section 4, and consider a variation where child 1 has all four toys and child 2 has none. Intuitively, the correct response according to the robot's values would be something like π 1 = MOVE(4, 1, 2), skip, as this satisfies the subsistence value while minimally violating the respect for property value. However, in our model π 1 is dominated by π 2 = MOVE(4, 1, 2), MOVE(3, 1, 2) since this also satisfies the robot's goal of equality and both π 1 and π 2 fail to satisfy the property value.

This behaviour could be corrected by replacing some or all values of the form Gϕ with values ϕ, Xϕ, XXϕ and so on, since this would track separate violations. However, this could cause issues for qualitative comparison since technically ϕ is a different value from Xϕ. Another solution would be to introduce the notion of "graded" G, G x , which operates similar to graded connectives in standard modal logic [START_REF] De Rijke | A Note on Graded Modal Logic[END_REF] and counts the number of times that the value is violated.

Adjustable Morality One significant feature of our method is that it allows us to adjust the morality level of the robot, by adjusting the position of the "goal set" Ω D in Ω. This determines which of the robot's values it is prepared to sacrifice (if necessary) in order to achieve its goal. This is illustrated in variations 3 & 4 of table 1. A similar concept of the morality level of an agent can be found in [START_REF] Lorini | A logic for reasoning about Moral Agents[END_REF], which considers morality level as the relative weighting of an agent's desires relative to their values.

The idea of an ethical robot that is able to "adjust" its own level of morality may be quite concerning, but we suggest two restrictions on this feature. Firstly, the morality level of a given goal should ideally be determined externally, rather than by the robot, so that the appropriate morality level is transmitted to the robot along with the goal. Secondly, there should be strict limits on the range of morality levels, so that the robots goals can never be prioritised over values of safety.

Our model is built on the assumption that the robot only ever has a single goal to work with, or that all of its goals are at a similar level of importance, and thus can be placed in a single value set. This could produce incorrect behaviour in a robot that can simultaneously hold goals of very different levels of importance (such as the robot shopkeeper in section 4), as either some goals will be treated as much more important than they actually are, or much less. Fortunately, this can be very easily remedied by extending the model to have multiple different goal sets (Ω D1 , Ω D2 , ...) each with their own morality level. This will allow the robot to correctly handle multiple goals.

Conclusions and Future Work

In this paper we have presented the current state of our framework, and discussed some of its features and limitations. There are also several directions in which we are planning to expand this model.

Our current model has perfect knowledge and deterministic actions, meaning that given a defined initial state, the same plan will always produce the same results. We could change this by making actions non-deterministic, or by moving from perfect to imperfect knowledge, meaning that instead of a single start state, the robot has a set of possible start states. Either change means that any given plan is now associated with a set of histories instead of a single history. This makes comparing plans much more complex, and there are many different approaches. Comparing sets of objects that represent possible outcomes is a well-studied topic [START_REF] Barberà | Ranking Sets of Objects[END_REF]. Two popular approaches are leximax (choose the best possible outcome) and leximin (avoid the worst possible outcome). Of the two, leximin seems more appealing as risk-averse robots seem preferable to risk-seeking robots, but it could produce some strange results, such as a robot attempting to prevent its owners from ever leaving the house, since they might be hit by a car.

Another generalization of our model would consist in moving from the single-robot to the multi-robot perspective. In a multi-robot setting robots may have different goals and value sets, leading to different preferences over histories. We can then apply solution concepts from game theory including Nash equilibrium and iterated deletion of weakly/strongly dominated strategies (IDWDS/IDSDS) to compute which joint plan or joint strategy the robots could select .

A similar extension would be to consider a multi-agent setting with both humans and robots. This would allow for and require a much more complex account of many values, particularly social values such as politeness. It also opens up many questions on risk and responsibility. How should a robot evaluate the plan where it gives a human a kitchen knife, knowing that this gives the human the ability to seriously hurt themselves and others?

If moving to a multi-agent setting we may want to model simultaneous or overlapping actions from different agents. This would require moving from classical planning to temporal planning [START_REF] Rintanen | Complexity of Concurrent Temporal Planning[END_REF] where actions have duration and can occur concurrently. This would also allow us to specify more meaningful deadlines for goals, or time affected social norms ("the fridge door must not be open for more than thirty seconds").

Another possible extension of our model is to replace some value sets with alternative methods of discriminating between plans. For example, instead of a final value set, we could have the robot distinguish between otherwise-equal plans based on expected energy consumption or some other scalar value. Similarly, if we had some more complex system for handling some type of values, such as moral claims in an emergency scenario or some highly complex system for following social norms, this could replace the corresponding value set. This vastly increases the potential flexibility of our framework, as well as provide an approach towards combining multiple decision-making systems specialised for different "levels of significance" but would come at a cost to simplicity.

  |= ϕ 1 U ϕ 2 ⇐⇒ ∃t ′ ≥ t : t ′ ≤ k AND H,t ′ |= ϕ 2 AND ∀t ′′ ≥ t : IF t ′′ < t ′ THEN H,t ′′ |= ϕ 1 .

	|= p	⇐⇒ p ∈ H st (t),
	H,t |= Xϕ	⇐⇒ t < k AND H,t + 1 |= ϕ,
	H,t

Here we take desire and goal as synonyms. An alternative conception presented in[START_REF] Lorini | Logics for Games, Emotions and Institutions[END_REF] consists in conceiving "goal" as an umbrella term covering both desire-based goals and value-based goals.

 [START_REF] Bylander | The Computational Complexity of Propositional STRIPS Planning[END_REF]The authors acknowledge the support of Labex CIMI (ANR-11-LABX-0040-CIMI) and Institut 3iA ANITI (ANR-19-PI3A-0004).


Experimental focusing dam break problem in a non-local optical stochastic Kerr medium

Experimental focusing dam break problem in a non-local optical stochastic Kerr medium A. Aleksanyan, 1 H. Louis, 1 J.F. Henninot, 2 and E. Louvergneaux 1, * 1 Univ. Lille, CNRS, UMR 8523 -PhLAM -Physique des Lasers Atomes et Molcules, F-59000 Lille, France 2 Univ. Artois, CNRS, UMR 8181 -UCCS -Unite de Catalyse et de Chimie du Solide, F-62307 Lens Cedex, France (Dated: August 18, 2020)

We experimentally study the propagating of an optical intensity jump discontinuity in a non-local stochastic Kerr focusing nematic liquid crystal (LC) cell. We show both theoretically and experimentally that non-locality opens a novel route towards beam steering in our system. Indeed, the discontinuity trajectory follows a curve that bends with the injected power. Despite the stochastic nature of the medium and the constant presence of transverse instabilities, the development of a focusing dam-break shock dynamics is shown to survive. The distance Zs for the focusing shock to occur follows a power law with the beam power P according to Zs ∝ P χ , with χ = 4/3, as for shock dynamics in self-defocusing media.

Any physics student from wave optics class has heard about the phenomenon of diffraction, i.e. varying when wave envelope varies during propagation. They also know that this effect is embedded in Maxwell's equations and the necessary condition for this phenomenon to occur is for the field envelope to depend on the spatial coordinates. This is linear diffraction. However an additional term appears in the wave propagation equation if the wave propagates in nonlinear media. This term accounts for the medium polarization and contains the nonlinear part of the index of the medium. If the wave envelope injected into the medium has no transverse spatial dependence (ideal plane wave), no linear diffraction is observed, but diffraction can still occur providing that the nonlinearity varies with space. This is nonlinear diffraction. Notion that every physics student has not heard of unless they have pursued a course in nonlinear optics. Thus, there are two cases where an optical beam propagating through a nonlinear medium experiences diffraction, either (i) the initial beam has a structured profile (amplitude or/and phase varying with space) and the nonlinear medium is spatially homogeneous, then the beam linearly diffracts through the nonlinear medium, or (ii) the initial beam is assimilated to a plane wave but the nonlinear medium is spatially structured so that nonlinear diffraction is achieved during propagation (a combination of these two cases is also possible but it is out of the scopes of this study). The nonlinear diffraction was first introduced and evidenced by Isaac Freud in a seminal work using the spatially periodic modulation of the dielectric susceptibility of NH4CL [START_REF] Freund | Nonlinear Diffraction[END_REF]. Such material structuring will be later implemented for the realization of photonic lattices [START_REF] Joannopoulos | Photonic crystals putting a new twist on light[END_REF] and lead to novel nonlinear phenomena [START_REF] Gomila | Photonic bandgap inhibition of modulational instabilities[END_REF][START_REF] Peschel | Discrete cavity solitons[END_REF][START_REF] Koke | Stabilization of counterpropagating solitons by photonic lattices[END_REF][START_REF] Marsal | Experimental control of pattern formation by photonic lattices[END_REF]. The most documented studies on nonlinear optical propagation deal with the above-mentioned first situation, that is, an initial beam linearly diffracting along the propagation through a spatially homogeneous nonlinear medium. This is the main core of nonlinear optics * eric.louvergneaux@univ-lille.fr for beam propagation and this article belongs to this category.

Competition or rather "cohabitation" between linear diffraction and nonlinearity has been the subject of an extensive amount of publications. Localization through solitary waves [START_REF] Schapers | Interaction of localized structures in an optical patternforming system[END_REF][START_REF] Barland | Cavity solitons as pixels in semiconductor microcavities[END_REF][START_REF] Piccardi | Dark nematicons[END_REF] or self-similar structures [START_REF] Snyder | Accessible Solitons[END_REF][START_REF] Buccoliero | Quasi-periodic transformations of nonlocal spatial solitons[END_REF][START_REF] Alberucci | Breather solitons in highly nonlocal media[END_REF], modulational instability (also called filamentation) [START_REF] El | Modulational instability and formation of a nonlinear oscillatory structure in a "focusing" medium[END_REF][START_REF] Peccianti | Optical modulational instability in a nonlocal medium[END_REF][START_REF] Królikowski | Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media[END_REF], wave singularities such as vortices [START_REF] Izdebskaya | Stable vortex soliton in nonlocal media with orientational nonlinearity[END_REF], shock waves [START_REF] Hoefer | Dispersive and classical shock waves in bose-einstein condensates and gas dynamics[END_REF][START_REF] Wan | Dispersive superfluid-like shock waves in nonlinear optics[END_REF][START_REF] Ghofraniha | Shocks in Nonlocal Media[END_REF] or else wave collapse [START_REF] Conti | Observation of a gradient catastrophe generating solitons[END_REF][START_REF] Bertola | Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I[END_REF] are some of the manifestations of such dynamics. We are interested in the nonlinear propagation of an optical amplitude jump discontinuity (between two uniform values of the initial data) through nonlinear focusing media. This problem belongs to the class of Riemann problems [START_REF] El | Dispersive shock waves and modulation theory[END_REF][START_REF] Biondini | Riemann problems and dispersive shocks in self-focusing media[END_REF]. A Riemann problem [START_REF] Riemann | Über die fortpflanzung ebener luftwellen von endlicher schwingungsweite[END_REF] classically refers to the initial value problem for a transverse uni-dimensional system associated with hyperbolic equations consisting of two constant states with a step at the origin. When the initial wave amplitude is step-like, the terminology dam break (or dam break Riemann problem) is used, in analogy with dam break flows in hydrodynamics. This situation has been extensively studied in shallow water for the regularization of initial discontinuities via the emergence of dispersive shock waves (also known as undular bores) when dissipation is negligible compared to dispersion [START_REF] Peregrine | Calculations of the development of an undular bore[END_REF]. These behaviors were later evidenced in nonlinear optics, when the nonlinearity is of defocusing type, in temporal systems [START_REF] Fatome | Observation of optical undular bores in multiple fourwave mixing[END_REF][START_REF] Xu | Dispersive Dam-Break Flow of a Photon Fluid[END_REF] as well as in spatial ones [START_REF] Wan | Dispersive superfluid-like shock waves in nonlinear optics[END_REF]. However, that dynamics radically changes when the nonlinearity is of focusing nature. The equations become elliptic so a shock boundary condition is ill posed, associated with no longterm undular bore solution but modulational instability. In optics, it is predicted that, for a focusing nonlinearity, whose response is non-local, the shock prevails over modulational instability [START_REF] Ghofraniha | Shocks in Nonlocal Media[END_REF] and focusing dispersive dam break flows emerge in the form of an oscillatory structures endowed with characteristics similar to dispersive shock waves [START_REF] Assanto | Collisionless shock resolution in nematic liquid crystals[END_REF]. Wan et al. experimentally demonstrated in local focusing media that using a partially-coherent beam suppresses modulational instability but leads to modulations as spatial dispersive shock waves with negative pressure [START_REF] Wan | Diffraction from an edge in a self-focusing medium[END_REF]. Thus, there is scant publications and almost no experimental study of the focusing dam break problem for an optical beam propagating through focusing nonlinear nonlocal media. This is the motivation of this study.

Our experimental setup follows the same strategy as presented in [START_REF] Peccianti | Optical modulational instability in a nonlocal medium[END_REF]. A step-like intensity beam profile is injected into a transverse unidimensional LC cell (Fig. 1). Such a Heaviside-like profile is prepared by focusing the Gaussian beam by a cylindrical lens (CYL) onto a thin metallic blade with a sharp edge. A 4f imaging system is constructed via two f = 100 mm lenses (L) to image the light profile from the plane Σ of the cutter (C) to the entrance plane Σ of the LC cell. The cell is mounted on a 3D stage and additional rotating stages allowing for precise adjustments via translations and rotations in any direction. To ensure that there is no pre-propagation in air at the cell entrance, a series of videos of the injected probe beam are recorded for different locations z of the cell. These videos are averaged over 20 s acquisition time to reveal the possible air diffraction pattern and adjust the entrance of the cell Σ with the plane Σ of the cutter. The single-mode frequency doubled N d +3 : Y V O 4 (λ 0 = 532 nm) laser with electric field A is linearly polarized along the x-axis initially perpendicular to the extraordinary axis of LC molecules. The input laser beam radii at the entrance of the cell, measured via a camera beam profiler (Thorlabs BC106N-VIS/M), are ω x ≈ 11.5 µm and ω y ≈ 960 µm (in the cutter plane). Thus, no more than one filament can form in the vertical x direction. The nonlinear medium is a E7 nematic LC of 75µm thickness sandwiched between two glass substrates with planar (parallel to the walls) anchoring conditions. The entrance of the cell is closed by a glass substrate with a planar anchoring condition to avoid depolarizing effects during beam injection. A white light imaging system (WL+IS1 on Fig. 1) is constructed to image the plane Σ along with the injected one-dimensional beam profile insuring that the latter is parallel to the walls of the LC cell. The propagation evolution of the beam is tracked via imaging system IS2 using the scattered light in the x direction. The propagating optical field is scattered by LC molecule fluctuations depending on director axis orientation, input field polarization and observation direction [START_REF] De Gennes | The physics of liquid crystal[END_REF][START_REF] Khoo | Liquid crystals, Physical Properties and Nonlinear Optical Phenomena[END_REF]. It is not a direct image of the optical field but of its scattering by the refractive index profile. However, as in all studies on optical beam propagation through nematic LCs, it allows for qualitative analysis of the dynamics.

The model describing the nonlinear propagation dynamics in such complex media is governed by a system of coupled equations for the optical reorientation angle θ of the LC molecules and the light envelope amplitude A [START_REF] Hutsebaut | Measurement of the self-induced waveguide of a solitonlike optical beam in a nematic liquid crystal[END_REF], such as

γ ∂θ ∂t = K 2 ∂ 2 θ ∂y 2 + C 1T A 2 sin(2θ) + √ ξ (1) 0 = -2ik 0 n ⊥ ∂A ∂z + ∂ 2 A ∂y 2 +k 2 0 n 2 a sin(θ) 2 A+ik 0 n ⊥ βA, ( 2
)
where γ is viscosity, K 2 is the Frank's twist elastic constant, n a = n -n ⊥ is the optical anisotropy with n /n ⊥ being the extraordinary/ordinary indices, C 1T = 0 n 2 a /4, k 0 is the laser wavenumber and β are losses. is the intensity of the thermal noise source term ξ which is Gaussian and delta correlated [START_REF] Agez | Noisy precursors in one-dimensional patterns[END_REF]. At third order in θ, dimensionless form of Eqs. [START_REF] Freund | Nonlinear Diffraction[END_REF][START_REF] Joannopoulos | Photonic crystals putting a new twist on light[END_REF]

is ∂ψ ∂T = σ 2 0 ∂ 2 ψ ∂Y 2 + 2 |a| 2 ψ + √ ηξ i ∂a ∂Z = 1 2 ∂ 2 a ∂Y 2 + 1 C 2 a ψ 2 a + i α 2 a (3)
where following scalings are introduced: optical field A with respect to the optical Fredericksz threshold is scaled as A F r = π/L K 3 /C 1T , such that a = A/A F r , L being the cell thickness and K 3 the bend elastic constant.

Y = y/ √ 2w y , Z = z/ 2k 0 n ⊥ w 2 y , T = tK 3 π 2 / γL 2 , ψ = θw 2 y k 2 0 n 2 a . α = 2k 0 n ⊥ w 2 y β, C a = w 2 y k 2 0 n 2 a , √ η = √ εL 2 C a / π 2 K 2 . σ 0 = L πwy K2
2K3 accounts for the degree of non-locality.

It is worth to mention that such a system of equations Eq. ( 3), even for stationary, deterministic and lossless conditions, is fundamentally different from the local nonlinear Schrdinger equation usually referred to study dam break problems in optics [START_REF] Biondini | Riemann problems and dispersive shocks in self-focusing media[END_REF][START_REF] Wan | Diffraction from an edge in a self-focusing medium[END_REF] as well as in hydrodynamics [START_REF] El | Dispersive shock waves and modulation theory[END_REF], hence, different dynamics is anticipated.

Fig. 2(b) depicts the spatio-temporal evolution of the y-transverse profile of the optical intensity |A| 2 after z = 1.2 mm propagation for Eqs.(1,2) in the local limit, that is K 2 = 0, without noise nor losses ( = 0 and β = 0) and for an initial amplitude value of the optical field at z = 0 leading to focusing dam break shock. An oscillatory wave regularizing the discontinuity is clearly evidenced. Its evolution with time stays stationary even at the early stage of the dam break (as also visible in [START_REF] Wan | Diffraction from an edge in a self-focusing medium[END_REF]). On the contrary, taking into account for the nonlocal response of the nonlinearity (K 2 = 6.57 10 -12 N ) gives a radically different dynamics [Fig. 2(c)]. The oscillatory wave regularizing the discontinuity still remains but its leading edge drifts with time to the highest intensity side. The whole transverse structure is steered during the propagation to the highest intensity value by more than 50 µm in the example of Fig. 2(c). Thus, alloptical beam-steering is observed owing to the non-local nature of our system.

Experimentally, (i) the intrinsic thermal fluctuations are known to affect the dynamics [START_REF] Agez | Noisy precursors in one-dimensional patterns[END_REF] and (ii) the scattering losses are relatively high (β ∼ 600m -1 ), so we expect competition between shock dynamics and filamentation (absent in the ideal case of simulations depicted in Fig. 2) as pointed out in [START_REF] Ghofraniha | Shocks in Nonlocal Media[END_REF][START_REF] Assanto | Collisionless shock resolution in nematic liquid crystals[END_REF] and wonder if shock dynamics can still survive in our stochastic system. Since diffraction and dissipation (scattering losses) are present, we also question on the type of regularization mechanism for shock dynamics that can occur.

Typical experimental propagation evolution of the initial optical intensity jump discontinuity with increasing nonlinearity is displayed on Fig. 3(a-c). The dynamics exhibits a continuous evolution of the fine structure of the propagating optical pattern due to the stochastic nature of the nonlinear medium as described in [START_REF] Louis | Experimental evidence of dynamical propagation for solitary waves in ultra slow stochastic non-local Kerr medium[END_REF]. However, its global shape remains quasi-stationary after a transient set up time. Thus, videos of the dynamics are recorded (20 s duration at 5 fps) with a 10 bit depth dynamic range to extract corresponding averaged images representative of the global shape of the propagating structure as presented on Fig. 3(a-c). The acquisition time of the camera is automatically adjusted by the software to get unsaturated videos taking into account eventual light pinning or saturated pixels due to cell impurities. The first 200 µm of propagation are not reported on experimental plots (i) to avoid scattered signals from the optical field injection at the entrance of the cell and mainly (ii) to allow for the input light to transfer its boundary condition discontinuity on the LC molecule distribution, that is to the medium refractive index. This latter point is crucial since our system is described by a set of two coupled equations [Eqs. [START_REF] Freund | Nonlinear Diffraction[END_REF][START_REF] Joannopoulos | Photonic crystals putting a new twist on light[END_REF]] and the initial boundary condition is only applied to the light envelope amplitude A, not to the optical reorientation angle θ of the LC molecules. In usual setups where the dam break Riemann problem is studied, only one variable accounts for the propagation dynamics of the system (e.g. nonlinear Schrdinger equation or Kortewegde Vries equation) and the initial condition is applied to this variable [START_REF] Xu | Dispersive Dam-Break Flow of a Photon Fluid[END_REF][START_REF] Zabusky | Interaction of Solitons in a Collisionless Plasma and the Recurrence of Initial States[END_REF][START_REF] Trillo | Observation of dispersive shock waves developing from initial depressions in shallow water[END_REF]. Also, the scattered light recorded by the camera IS2 is a result of the LC index distribution that has no initial jump discontinuity at z = 0 but will only acquire it over the early stage of propagation ( 100 -200 µm). Three values of the injected optical power P , representative of the dynamics of interest, are shown on Fig. 3(a-c), namely P = 17.5 mW , P = 193.5 mW , and P = 286.7 mW . For the lowest power [Fig. 3(a)] the regime is dominated by linear diffraction and observed small deviation experienced by the discontinuity is attributed to linear Fresnel diffraction. The dynamics drastically changes when increasing the laser power with a transverse shift of the sharp edge [Fig. 3(b,c)]. The edge steering is clearly evidenced on Fig. 3(d) through the transverse intensity profiles plotted at a fixed propagation distance, namely z = 354 µm, for the three powers of Fig. 3(a-c). A shift of ∼ 20 µm is obtained after only 354 µm propagation. The 0 reference of the transverse y-axis is fixed by the location of the discontinuity at z = 200 µm in the linear regime (that is P = 17.5 mW ). Fig. 3(d) also evidences the steepening of the edge with increasing the power. For the highest depicted power [Fig. 3(c)], the reached transverse steering at 1.2 mm propagation distance from the cell entrance is at least 40 µm, that is a larger than 33 mrad. The amount of steering obtained after 1.2mm propagation versus the input beam power P is plotted on Fig. 3(e) clearly demonstrating a power dependence of the sharp edge position shift. For a quantitative analysis of this nonlinear beam deviation, the transverse position Y s of the intensity discontinuity is tracked. As an example, Y s is marked by a black circle on profiles of Fig. 3(d). At a given propagating distance z, Y s corresponds to the abscissa of the maximum of the transverse intensity steepness calculated as S z (y) = (∂|A| 2 /∂y)| z . The measurement of this dependence brings out a power law for Y s as depicted on Fig. 4(d), namely Y s ∝ P 1 25 . Experimental results of Fig. 3 show that the beam steering predicted in Fig. 2 is a robust phenomenon with respect to noise and scattering losses. On the other hand, Fig. 3 does not allow for the evidence of focusing dispersive dam break flows in the form of an oscillatory structure as evidenced in Fig. 2(b,c). A numerical study would be necessary to identify the effects of noise, losses and e.g. nonlinearity saturation effects on the emergence of the focusing dispersive oscillating structure. This is out of the scope of this letter since the high transverse spatial resolution required to perform numerical simulations on a such wide beam profile requires many weeks of computing to reach the stationary state in the deterministic case. This is why in this letter we do not directly contrast our experimental results with numerical ones, based on Eqs. [START_REF] Freund | Nonlinear Diffraction[END_REF][START_REF] Joannopoulos | Photonic crystals putting a new twist on light[END_REF]. Transverse instabilities, in the form of modulational instabilities, starting from the early stage of propagation (so not the focusing dispersive dam break flows expected as a result of the regularization of the discontinuity) are always present even for the "linear regime" of Fig. 3(a) as mentioned in [START_REF] Wang | Observation of Optical Solitons and Abnormal Modulation Instability in Liquid Crystals with Negative Dielectric Anisotropy[END_REF]. The averaged pictures of Fig. 3(a-c) do not emphasize them since their transverse location and periodicity are wandering with time [START_REF] Louis | Experimental evidence of dynamical propagation for solitary waves in ultra slow stochastic non-local Kerr medium[END_REF] and are washed out by the averaging process. Further study would be necessary to identify these short wavelength instabilities with noise sustained modulational instability [START_REF] Peccianti | Optical modulational instability in a nonlocal medium[END_REF] or other stochastic resonance behavior [START_REF] Feng | Reconstruction of noisy images via stochastic resonance in nematic liquid crystals[END_REF]. Although no focusing dispersive dam break flows are experimentally observed in our stochastic nonlinear medium, shock-like dynamics is still observable. Focusing dam-break shock results from the underlying mechanism of the wave steepening driven by the nonlinearity which leads to a gradient catastrophe. The self-steepening dynamics of our experimental refractive index discontinuity is analyzed versus the propagation using the previously defined steepness S z (y) . For each z, the maximum value S maxz of S z (y) is extracted. The typical experimental evolution, during the propagation, of the maximum transverse steepening S maxz (z) for the discontinuity between the ground and high intensity regions, versus power injection P is illustrated in Fig. 4(a). The plotted S maxz (z) profiles correspond to the average of the 100 instantaneous steepness profiles extracted from each frame of the video. In order to avoid noise-induced hot spots and small scale transverse instabilities in the calculation of S, Savitzky-Golay algorithm is applied to smooth the signal. It is observed that above P ≈ 135 mW power, the evolution of S maxz (z) always reveals the occurrence of a maximum (Fig. 4(a)). It indicates that the jump distribution of the index profile self-steepens during the propagation distance z before "relaxing" or being "regularized" by the dispersive and dissipative medium. This is the characteristics of a shock-like dynamics. Such observation is also reported by Wan et al. [START_REF] Wan | Diffraction from an edge in a self-focusing medium[END_REF] for the diffraction from an edge but in a local self-focusing medium.

Steepness oscillations S maxz along the propagation direction are linked to the corresponding transverse oscillations on the intensity profiles [Fig. 3(d)]. This feature is found in the numerical simulations carried out without noise nor losses ( = 0 and β = 0) but taking into account for the non-local response of the nonlinearity (K 2 = 6.57 10 -12 N ) [Fig. 4(b)]. These oscillations come from the transverse modulations previously discussed [Fig. 3(a-c)] and their interactions due to the non-locality. The location of S(z) maximum never occurs above approximately 600 µm propagating distance. The reason is certainly due to losses that overcome nonlinearity for this distance. Such a limitation in the nonlinear effects was already reported in [START_REF] Louis | Experimental evidence of dynamical propagation for solitary waves in ultra slow stochastic non-local Kerr medium[END_REF]. We extract the longitudinal coordinate Z s corresponding to the maximum value of S maxz (z) that measures the propagating distance needed for the focusing shock-like to occur. This distance Z s moves towards the cell entrance when increasing the initial beam power P as reported for shocks in optical defocusing media (see e.g. [START_REF] Ghofraniha | Measurement of scaling laws for shock waves in thermal nonlocal media[END_REF]) and in hydrodynamics [START_REF] Ghofraniha | Shocks in Nonlocal Media[END_REF][START_REF] Conti | Observation of a gradient catastrophe generating solitons[END_REF]). Indeed, it is known that the shock distance Z s scales with power P according to the law Z s ∝ P χ , with χ = 0.5 in the hydrodynamic limit. The evolution plot of Z s versus P for our experimental recordings is drawn on Fig. 4(c). It clearly evidences a power law Z s ∝ P 4/3 for powers larger than P ≈ 135 mW (ln(P ) ≈ -4.9). Thus, as for shocks in defocusing media a power law in the form of Z s ∝ P χ is found. Our χ = 4/3 value is related to the non-local nature of our system as shown by Conti et al. and Ghofrahina et al. in [20,[START_REF] Ghofraniha | Measurement of scaling laws for shock waves in thermal nonlocal media[END_REF]. A numerical study would state on the influence of the level of non-locality on the χ parameter in our system.

In conclusion, we experimentally show that the propagation of an optical intensity jump discontinuity (dam beak problem) in a non-local focusing Kerr medium follows a trajectory that bends with the injected beam power P due to the non-local nature of the nonlinearity. The transverse beam nonlinear deviation/shift Y s follows a power law with Y s ∝ P 1 25 . This opens a novel alloptical route towards beam steering. We also evidenced that the profile of the refractive index self-steepens along propagation and leads to a focusing shock dynamics. It is characterized by a power law, as for self-defocusing local Kerr media, for the shock distance to occur versus the injected beam power P , namely P -4/3 . Focusing dispersive dam break oscillatory structure, that results from the regularization of the shock, is not observed due to the intrinsic noise that sustains wandering small scale transverse instabilities. Further investigations are in progress to blackuce the stochasticity level in order to evidence experimentally focusing dispersive dam break flow or maybe dissipative one due to losses.

FIG. 1 .

 1 Figure 1

FIG. 2 .

 2 Figure 2

FIG. 3 .

 3 FIG. 3. Spatial evolution of the initial intensity jump discontinuity versus propagating distance z for increasing laser power measured before the sample entrance, namely (a) P = 17.5 mW , (b) P = 73.8 mW and (c) P = 193.5 mW . The intensity profiles are normalized to their maximum values. (d) Transverse intensity profiles at z = 354 µm propagating distance [dotted lines on (a), (b) and (c)] for the three power values of diagrams (a), (b) and (c). Black circles on (d) locate the scattered light intensity discontinuity YS (see text for definition). (e) Evolution of the intensity profile at z = 1200µm versus the injected light power P ; the transverse profiles collected for each value of P are normalized to 1 in order the render the image readable. All data are averaged over 20 seconds of dynamics.

FIG. 4 .

 4 FIG. 4. (a) Typical experimental maximum transverse steepening Smax z (z) (see text for definition) evolution with increasing power, P = 137.18 mW (black), 193.5 mW (blue), and 279.7 mW (red); the dot indicates the location ZS and refers to the maximum of Smax z (z). (b) Numerical maximum transverse steepening Smax z (z) evolution with increasing intensity values |A| 2 = 5.64 10 6 (black), |A| 2 = 7.98 10 6 (blue) and|A| 2 = 9.76 10 6 V 2 /m 2 (red) correspondingly. (c) Experimental evolution of the focusing shock-like distance Zs (see text for definition) versus injected power P in logarithmic scales showing a power law Zs ∝ P χ , with χ = -4/3 coefficient. (d) Experimental evolution of the transverse position Ys of the intensity discontinuity (see text for definition) at z = 1.2 mm versus injected power P in logarithmic scales showing a power law Ys ∝ P χ , with χ = 1/25 coefficient. All experimental values are averaged over 20 seconds of dynamics.


Claire Prada
email: claire.prada@espci.psl.eu

Sylvain Mezil

Daniel A Kiefer

Backward modes and local resonances in anisotropic elastic waveguides

come

Figure 1 :

 1 Figure 1: Zero-Group-Velocity Lamb modes measured in a 525µm thick silicon plate after excitation by a ~1 mm diameter and 8 ns duration laser pulse at point (0, 0). The horizontal axis corresponds to the [110] silicon crystallographic axis.