Jonathan M Histon

John Hansman

Blake Gottlieb

Howard Kleinwaks

MIT, Cambridge, MA Sarah Yenson

Daniel Delahaye

Stéphane Puechmorel

R John Hansman

Stephane Puechmorel

Structural considerations and cognitive complexity in air traffic control

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Cognitive Complexity of an air traffic situation is an important factor in determining airspace capacity. Cognitive Complexity is generally assumed to correlate with the cognitive difficulty of controlling the air traffic situation under normal and abnormal conditions. The objective of the work discussed in this paper is to understand the factors that influence complexity, particularly those factors which relate to the underlying structural elements in air traffic control.

A better understanding of all of the factors that impact Cognitive Complexity is important for advancing both safety and efficiency in the ATC system. A controller can become overloaded if the Cognitive Complexity of a situation exceeds their capabilities. In current operations, traffic management procedures restrict capacity to prevent such situations from developing.

Because structure can be imposed by external factors, understanding the role of structure can help assess the potential impact of proposed changes to the ATC system such as airspace redesign, modifications of operational procedures or the introduction of new technologies.

Previous metrics of ATC complexity have concentrated on geometrical factors [START_REF] Delahaye | Air Traffic Complexity: Towards Intrinsic Metrics[END_REF] [2] [3] [START_REF] Inc | An Evaluation of Air Traffic Control Complexity[END_REF]. A deeper understanding of how structure impacts Cognitive Complexity can extend these previous efforts and help provide a more complete understanding of the critical system properties for operational and design applications.

Structure and Cognitive Complexity

In order to understand how structure influences a controllers Cognitive Complexity a simple model is presented to provide a framework linking structure based abstraction with the controllers mental model and the air traffic control task.

Generalized ATC Process Model

Figure 1 presents a simple model which depicts hypothesized relationships between structure, Cognitive Complexity, and the basic tasks of air traffic control. In the model the controller receives surveillance information about the air traffic situation and processes it to produce a set of output commands [START_REF] Histon | Introducing Structural Considerations into Complexity Metrics[END_REF]. The processing of the information is represented in terms of an adapted version of Endsley's model of Situation Awareness [START_REF] Endsley | Situational Awareness Requirements for En-route Air Traffic Control[END_REF].

Figure 1. Generalized ATC Process Model

Key high-level ATC processes of planning, evaluating, and monitoring, identified by Pawlak et al., are included as parts of the decision making process [START_REF] Pawlak | A Framework for the Evaulation of Air Traffic Control Complexity[END_REF]. The planning process produces a distinct cognitive element identified in the model as the controller's "Current Plan" [START_REF] Histon | Introducing Structural Considerations into Complexity Metrics[END_REF] which is an internal representation of the schedule of events and commands to be implemented as well as the resulting aircraft trajectories that will ensure that the air traffic situation evolves in an efficient and conflict-free manner.

In Figure 1, Cognitive Complexity is associated with the working mental model of the controller. It is hypothesized that the needs of the controller's current task will determine the working mental used to accomplish that task. The working mental model is an internal representation of the dynamics of the real system and will operate at a level of abstraction appropriate for the task being performed. Cognitive Complexity is an attribute of the controllers working mental model and is related to the scope and details of the factors being

SITUATION AWARENESS LEVEL 2 Comprehension

LEVEL 3 Projection

LEVEL 1 Perception

DECISION PROCESS

Monitoring

Evaluating

Implementing

PERFORMANCE OF ACTIONS

STRUCTURE AIR TRAFFIC CONTROLLER

Planning

SURVEILLANCE

AIR TRAFFIC SITUATION

WORKING MENTAL MODEL

STRUCTURE BASED ABSTRACTIONS COGNITIVE COMPLEXITY

CURRENT PLAN

COMMANDS

considered as well as any simplifying abstractions employed by the controller.

In Figure 1, structure is presumed to be a basis for complexity reducing abstractions. Structure is composed of elements that shape and influence the evolution of the air traffic situation including physical objects, such as radio beacons (e.g. VORs, NDBs etc…), and non-physical elements such as rules, procedures, and constraints. Structure can also be an internal construct such as techniques or standard procedures used in a given situation. Such internalised structures are difficult to identify and capture but appear to be important mechanisms by which controllers can control and limit the Cognitive Complexity of situations.

Mental Models

Mental models have been defined by Rouse and Morris (as cited in [START_REF] Endsley | Situation Models: An Avenue to the Modeling of Mental Models[END_REF]) as "mechanisms whereby humans are able to generate descriptions of system purpose and form, explanations of system functioning and observed system states, and predictions of future states." Mogford has suggested there are two components of a controller's mental model: 1) a domain model of the airspace, aircraft, and required procedures for that airspace; and 2) a model of the devices used for surveillance and command implementation (i.e. the equipment used to perform the ATC task) [START_REF] Mogford | Mental Models and Situation Awareness in Air Traffic Control[END_REF].

Controllers use mental models as mechanisms to generate information to support their Situation Awareness [START_REF] Mogford | Mental Models and Situation Awareness in Air Traffic Control[END_REF]. A "working" mental model is a task-specific representation of the overall mental model. It is a mechanism used to support decision making processes at a level of abstraction appropriate for the current task.

Abstractions

Abstractions are a means of representing the essential characteristics of a mental model in a more cognitively compact form that is manageable within the constraints of human memory and processing limitations. Rasmussen states that the abstraction process is "not merely removal of details of information on physical or material properties. More fundamentally, information is added on higher level principles governing the co-function of the various functions or elements at the lower levels [START_REF] Rasmussen | Information Processing and human-Machine Interaction: An Approach to Cognitive Engineering[END_REF]."

Abstractions are powerful because they capture the most important and relevant aspects of a system without including the overhead of less critical details. This is hypothesized to reduce the Cognitive Complexity of performing that task, allowing the controller to attend to other aspects of the system.

Structure as a Basis for Simplifying Abstractions

Structure appears to form the basis for abstractions used in the working mental model in many ATC situations [START_REF] Histon | Introducing Structural Considerations into Complexity Metrics[END_REF]. Structural elements can limit the potential future states of an air traffic situation. For example, standard operating procedures define standard ingress and egress points from a sector. The effect of these constraints is to reduce the number of likely aircraft trajectories entering and leaving a sector. Experience of the recurring patterns of behavior created by the presence of the underlying structure allows the creation of abstractions that simplify the level of detail required in the working mental model used to generate projections or perform the key ATC tasks. By simplifying the working mental model, those abstractions reduce the Cognitive Complexity of managing the evolution of the air traffic situation.

Observability of Cognitive Complexity

In Figure 1, Cognitive Complexity and the concept of complexity reducing abstractions are shown as internal to the controller working mental model. Mogford has previously identified complexity as an internal construct that cannot be directly measured or observed [START_REF] Mogford | The Complexity Construct in Air Traffic Control: A Review and Synthesis of the Literature[END_REF]. However, Cognitive Complexity can be investigated indirectly through observations of the controller and the ATC system. A second observation channel is direct measurement of system states. These states can range from simple measures such as radar data to full aircraft states and trajectories as well as aggregate measures such as the number of aircraft in a sector or more complicated geometric properties.

A third observation channel is the commands and communication activity of the controller. Since this is a direct measurement of the controllers output it can be used to provide insight into the controllers' internal processes.

Finally, in ATC much of the structure is codified and can be observed through formal documentation, such as Standard Operating Procedures (SOPs), regulations, airways and airspace boundaries.

Because Cognitive Complexity is internal to the controller and is not directly measurable it is necessary to create a model which approximates the internal Cognitive Complexity. As shown in Figure 2, the model is a conceptual framework that interprets observables and maps them into an external construct labeled as complexity. Importantly, because no direct measurement of Cognitive Complexity can be made, complexity models can only be validated against other observables or through hypothesis testing.

Complexity models can generate metrics used to predict internal Cognitive Complexity in operational settings or can be used to understand how system changes may impact Cognitive Complexity. An overly simple example of the application of such model is the number of aircraft in a sector, which forms the basis for the Monitor Alert function in the Enhanced Traffic Management System (ETMS) used to limit system capacity.

Initial Observations

Each of the four observable channels in Figure 2 can be used to probe how structure may act to reduce cognitive complexity. The following sections present initial examples of observations from each channel.

Subjective Responses

In order to determine the perceived complexity of an ATC situation, subjective responses can be elicited from controllers by a variety of mechanisms such as direct query, interviews, expert elicitation, etc (see [START_REF] Laudeman | Dynamic Density: An Air Traffic Management Metric[END_REF] [4] [START_REF] Manning | The Relationship Between Air Traffic Control Communication Events and Measures of Controller Taskload and Workload[END_REF] [13] [START_REF] Mogford | Research Techniques for Documenting Cognitive processes in Air Traffic Control: Sector Complexity and Decision Making[END_REF]). However, as noted above, significant divergence between the perceived and Cognitive Complexity can occur. Therefore, the subjective data should be considered exploratory and should be correlated with objective measures whenever possible.

Method

In order to gather observations of controllers' perception of key Cognitive Complexity factors, a series of site visits to ATC facilities in the United States, Canada and France were conducted. The site visits included both en-route and terminal area control centers. The site visits consisted of focused interviews with current controllers, observations of live operations, as well as discussions with Traffic Management Unit and training personnel.

During the site visits, questions were posed that were designed to elicit a controller's perceptions regarding the use of structure. For example, controllers were asked to identify key factors which influence complexity. They were also asked to identify specific sectors or situations which they considered "easy" and "hard".

Example Results

Table 1 shows a list of the key factors controllers reported as influencing Cognitive Complexity. No attempt has been made to rank the factors. However, they have been found to fall into three categories: Airspace Factors, Traffic Factors, and Operational Constraints.

Airspace Factors are those related to properties of the airspace. Represented are both internal properties, such as the distribution of navigational aids, and external properties, such as sector shape and coordination activities. In general, these factors are quasi-static, characterizing the underlying context within which a traffic load exists.

Traffic Factors, are those dependent on the instantaneous distribution of traffic. They represent more dynamic and transient effects than Airspace Factors. Most previous efforts focused on measures associated with Traffic Factors.

Finally, Operational Constraints are additional operational requirements that placed restrictions on possible control actions. These factors tend to represent short-term or temporary variations in operational conditions.

During the site visits to Boston Center controllers were asked to identify "easy" and "hard" sectors. The Albany sector was consistently identified as being harder than the Utica sector. Further analysis of the implications of these results is discussed in the following section.

System State Measurements

Direct observation of system states can be used to investigate the role and use of structure. Many of the previously proposed metrics of complexity have been based on such measurements including as the number of aircraft in a sector, the number of aircraft in transition (in altitude, speed, heading etc…), relative distances between aircraft, and sector transit time (see [START_REF] Delahaye | Air Traffic Complexity: Towards Intrinsic Metrics[END_REF] [2] [4] [START_REF] Mogford | The Complexity Construct in Air Traffic Control: A Review and Synthesis of the Literature[END_REF]). Often these metrics are calibrated against observations of subjective data.

Observations of system states are appealing as they tend to be non-intrusive and require less data collection effort than subjective observations. However, they can also create a focus on only the geometrical relationships between aircraft which may miss some important structural factors that modify the Cognitive Complexity of a situation.

Method

System states can be determined from radar data and other state histories of aircraft together with logs of status states of other pertinent equipment such as radars, radios, weather etc… System state observations may also include flight plan data to capture presumed aircraft intent. In some cases observations of system states might also include aggregate or computed system properties [START_REF] Delahaye | Air Traffic Complexity: Towards Intrinsic Metrics[END_REF].

To obtain insight into the existence of structure within the current ATC system, observations of traffic patterns have been made using data obtained from the Enhanced Traffic Management System (ETMS). Guided by findings from the subjective observations, this data was analyzed for evidence of key structural elements that could form the basis for structure based abstractions.

Example Results

Examples of observations using system state analysis of the "easier" (Utica) and "harder" (Albany) sectors as identified by Boston controllers are shown in Figure 3 Analysis of the state data showed that there is more crossing traffic and a larger number of aircraft in vertical transition in the "hard" Albany sector than in the "easy" Utica sector (shown in Figure 3) which contained traffic flying predominantly in a uniform westbound direction.

In the subjective data controllers indicated the importance of the standard flows through a sector. The trajectories of aircraft that appeared to be on standard flows were identified and are highlighted by thick lines in Figures 3 and4. As shown in Figure 4 the "hard" Albany sector has more standard ingress and egress points, more standard flows and a greater amount of interaction between those flows than the Utica sector.

It is interesting to note that a higher fraction of the traffic is on identified standard routes in the "hard" Albany Sector (~ 43%) than in the Utica Sector (~ 20%). It appears that the controllers rely more heavily on the structured routes in the sectors they perceive as "hard" but are willing to allow more flexibility in situations which they consider "easy" This is consistent with subjective observations that the standard route abstraction is useful for reducing Cognitive Complexity. It also points out the fact that the controllers are active participants in managing their own cognitive complexity and will manage the system to maintain a safe buffer between their perceived complexity and their internally defined complexity limits.

System state data can also be used to identify important structural patterns at larger scales. Figure 5 shows an example of aircraft trajectories into Chicago O'Hare airport. The aircraft are clearly consolidated into flows that merge through a branching structure. Similar patterns have been observed at all major U.S. hubs. In order to generate the high density flows into the key airports it is necessary to merge the traffic into the 3 or 4 major flows which are typical of a hub airport. As can be seen in Figure 5 these mergers take place as defined "critical points" which are typically limited to a single critical point per flow per sector. It is hypothesized that there is a critical point abstraction which allows controllers to focus on critical merging and crossing points within their sector. In the case of a merge point, a critical point abstraction can reduce a multi-dimensional interaction problem to a one dimensional one of "time-of-arrival" at the merge point as the key control parameter.

Codified Structure

In ATC much of the structure is formally codified and documented. The explicitly documented structural elements used in conjunction with other channels to identify possible sources of structure based abstractions.

Codified elements of structure are not necessarily static objects reflecting airspace design. They include procedures and dynamic elements such as traffic management restrictions (e.g. Miles-In-Trail).

Method

Codified structural elements are identified through system documentation including navigational and airspace charts, standard operating procedures, letters of agreement, the controllers handbook and training material.

Example Results:

Communication Analysis

An additional source of observational data is the direct outputs from a controller in the form of commands and coordination activities which can be observed through the various forms of ATC communication. Previous investigations of ATC communications have generally focused on the duration of and frequency of verbal communication events [START_REF] Manning | The Relationship Between Air Traffic Control Communication Events and Measures of Controller Taskload and Workload[END_REF]. However, additional insight into the controllers processes and the use of structure can be gained by more detailed analysis of the content of the controllers command output.

Method

In order to investigate the use of detailed communication analysis to gain insight into controllers decision processes and Cognitive Complexity a sector focused study was conducted of the Logen Sector of the Atlanta ARTCC. Controller-Pilot verbal communications were recorded [15] and correlated with ETMS data and weather data. Figure 7 depicts an example of the tracks through the sector over a 24 hour period. During this period 61% of all aircraft through the sector were destined for Atlanta-Hartsfield airport. The detailed analysis consisted of classifying communication events using a software application developed in Microsoft Visual Basic for this purpose which allowed the verbal data to be categorized and classified while the analysts could simultaneously visualize the traffic and weather situation. A screenshot of the communications analysis tool is shown in Figure 8.

Figure 8. Screenshot of communications analysis tool (left) and visual display of air traffic situation based on ETMS data (right).

Based on a preliminary sample of the audio data, a classification scheme was developed which focused on controller-pilot communications and in particular the controllers commands which are considered the output of the controllers control process. It should be noted that pilot-controller communications were not analyzed. The command classification scheme reflected the commands observed in this sector, but is easily adapted for other sectors or applications.

Example Results

In order to illustrate how observations of commands could be used to investigate cognitive processes, a small pilot study was performed using 2.25 hours of audio and flight path data collected on Monday March 11, 2002. Initial analysis of the controller-pilot communication events showed that 47% of the events were commands.

The detailed analysis shown in Figure 9 indicates that three basic command types accounted for 61% of all issued commands over the observed period. When the commands were analyzed in the context of the codified structure of this sector (represented by the MACEY TWO STAR in Figure 6) it was found that a high percentage of the issued commands were consistent with the controllers executing the codified procedure. For example the standard flows through the sector were observed to converge and merge at the LOGEN navigation fix and 89 % of the altitude crossing restrictions were issued for LOGEN which is the defined procedure for Atlanta arrivals for eastbound landings which was the condition during the observed period. When the key LOGEN fix was blocked, the number of crossing restrictions was substantially reduced, suggesting the controllers were no longer using the structure provided by the MACEY TWO STAR. In addition, and perhaps in reflection of the loss of capacity which occurred when the underlying structure was compromised, a strategic response of the system was observed. The traffic rates through the sector were reduced by approximately 1/3 during the disrupted operations compared with the baseline operations. The mix of traffic was also shifted with the air carrier traffic being reduced from 71% in the nominal condition to 34% in the disrupted condition.

Experimental Probes of Structure Based Abstractions

One approach to validate "structure based abstractions" is to subject the abstractions to experimental validation. If a structure based abstraction is hypothesized and the structural basis is manipulated as an independent variable then, according to the model in Figure 2, the subjective data (perceived complexity), objective data (system states and performance) and the command data (ATC commands) should reflect the change in perceived complexity. This approach was tested using the "critical points" abstraction discussed in the section System State measurements. The experiment was conducted in order to evaluate the hypothesis that a single co-located merge point was an effective structural abstraction.

Method

A simplified Air Traffic Control Task was simulated where subjects attempted to merge streams of traffic using only speed commands to achieve a minimum in trail separation upon leaving the sector. An example of the simulation display is shown in Figure 11. The independent variables in the experiment were the number of incoming traffic streams and if the streams merged at a single "colocated" merge point or if each stream merged separately. The test matrix is shown in Table 2.

The dependant variables were the number of separation violations, number and magnitude of command and subjective ratings of the difficulty (using a modified Cooper-Harper rating scale). The incoming traffic streams were constructed of modular segments which were rotated for each test case to provide identical load statistics with an unpredictable pattern to the controller.

Results

In the initial study, 12 university students participated as controllers. Initial results support the hypothesis that a co-located merge point simplifies the merging task. The average number of separation violations is shown in Figure 12. The average number of violations was less than 1 for the colocated merge points and approximately 2.3 for the non-collocated merge points. Due to the relatively small number of violations these results are only significant at the 90% confidence level but are consistent with the underlying hypothesis and the other observational data. It is interesting to note that close to 90% of all separation violations which occurred in the non co-located scenarios involved an aircraft joining the main horizontal flow at the second or third merge point. As seen in Figure 13 the amount of commanded speed change was higher for the non co-located merge points. This result is statically significant and is consistent with the hypothesis that the abstractions based on the co-located merge points simplified the controller tasks presumably through a less complex mental model.

The subjective Cooper-Harper ratings of difficulty in controlling the situation were also consistent with the hypothesis. As seen in Figure 14 subjects perceived the co-located scenarios as less difficult than the non co-located scenarios.

A more detailed description of the experiment can be found in [START_REF] Gottlieb | Testing Air Traffic Controllers' Cognitive Complexity Limits[END_REF] and [START_REF] Kleinwaks | Testing Air Traffic Controllers' Cognitive Complexity Limits[END_REF] but the results suggest that 'structure based abstractions" can be successfully probed through indirect experimental techniques.

Conclusions / Summary

In order to develop an understanding of the impact of structure on the controllers internal Cognitive Complexity, a broad observational and holistic approach has been proposed to overcome the lack of direct observability of controllers internal processes. Based on this approach, some initial examples of structure-based abstractions (e.g. standard flows and critical points) have been identified and supported by experimental and observational probes. The underlying structural abstractions which have been investigated clearly act to reduce the Cognitive Complexity of the controllers working mental model and allow the capacity of the ATC system to be higher than when the structure based abstractions are not available. The structure may sufficiently reduce controllers cognitive load, to allow more flexibility for aircraft in the system. While the results are preliminary the multi-channel approach appears to be a promising method to investigate the impact of structure on controllers Cognitive Complexity and it's impact on ATC operations.

Figure 2 .

 2 Figure 2. Observability Channels of the ATC system.

Figure 2

 2 Figure 2 shows four observation channels that may be used to investigate the role of structure in reducing Cognitive Complexity. The subjective channel includes controllers' subjective responses to queries posed through interviews, questionnaires, or other subjective complexity assessment

  and Figure 4. The figures show images of aircraft trajectories through the Utica and Albany sectors respectively. Approximately the same number of aircraft passed through each sector during the twenty four hour traffic sample (Utica: 268 aircraft; Albany: 231 aircraft).

Figure 3 .

 3 Figure 3. Image of 24 hours of traffic through Utica sector ("easier").

Figure 4 .

 4 Figure 4. Image of 24 hours of traffic through Albany sector ("harder").

Figure 5 .

 5 Figure 5. Standard Flows of Aircraft destined for O'Hare airport in Chicago.As noted above controllers often rely on an abstraction based on an aircraft's membership in a standard flow. This abstraction is a key element in managing high density traffic. By classifying

Figure 6

 6 shows an example of structure in the MACEY TWO Standard Terminal Arrival Route (STAR) for Atlanta Hartsfield which delineates a common path and sequence of transitions that structure the flow of aircraft into the North-East corner post into Atlanta. When conjoined with other observation channels, the explicitly codified structure can provide valuable insight into the influence of structure on Cognitive Complexity.

Figure 6 .

 6 Figure 6. MACEY TWO Standard Terminal Arrival Route (STAR) (Courtesy [15]).

Figure 7 .

 7 Figure 7. 24 Exposure of traffic flows through the Logen sector (Oct. 18, 2001).

Figure 9 .

 9 Figure 9. Relative occurrences of commands.Additional insight into the importance of this codified structure for a controller's internal

Figure 10 .

 10 Figure 10. Commands per aircraft with the LOGEN fix blocked and clear of convective weather.

Figure 11 .

 11 Figure 11. Screenshot of ATC simulation environment. Numbers indicate the speed of the aircraft.

Figure 12 .

 12 Figure 12. Separation Violations.

Figure 14 .

 14 Figure 13. Average Speed ChangeCommanded per Aircraft.

Table 1 . Key factors reported by controllers as influencing complexity.

 1

	AIRSPACE FACTORS
	Sector dimensions
	Shape
	Physical size
	Number of Flight Levels
	Effective "Area of regard"
	Letters of Agreement / Standardized Procedures
	Number and position of standard ingress / egress points
	Spatial distribution of airways / Navigational aids
	Standard flows
	Number of
	Orientation relative to sector shape
	Trajectory complexity
	Interactions between flows (crossing points, merges)
	Coordination with other controllers
	Hand-offs
	Point-outs
	TRAFFIC FACTORS
	Density of traffic
	Clustering
	Sector-wide
	Aircraft encounters
	Number of
	Distance between aircraft
	Relative speed between aircraft
	Location of point of closest approach (near airspace
	boundary, merge points etc…)
	Difficulty in identifying
	Sensitivity to controller's actions
	Ranges of aircraft performance
	Aircraft types (Boeing 747 vs Cessna 172)
	Pilot abilities
	Control services required (IFR vs VFR)
	Number of aircraft in transition
	Altitude / Heading / Speed
	Sector transit time
	OPERATIONAL CONSTRAINTS
	Restrictions on available airspace
	Presence of convective weather
	Activation of special use airspace
	Aircraft in holding patterns
	Buffering capacity
	Procedural restrictions
	Noise abatement procedures
	Traffic management restrictions (e.g. miles-in-trail
	requirements)
	Communication limitations
	Wind Effects
	Direction, strength, changes

Table 2 . Merge point arrangements used in experimental study of critical point abstraction.

 2

		Number of
		Initial Flight Paths
	1	2	3	4
				Co -
				Located
				Non
				Co-located

Acknowledgements

The authors would like to thank the air traffic controllers who shared their experiences and provided opportunities for observation. The MIT work was supported by FAA grant # 96-C-001 through the National Center of Excellence for Aviation Operations Research (NEXTOR), Agreement # SA 1603JB. The CENA work was supported by CENA, ENAC, and LAAS. The image in Figure 5 was captured using "Flight Explorer," a commercial product providing an ETMS data feed. The authors also acknowledge Hayley Davison for her contributions and perspectives on the concept of abstractions and mental models.


Daniel Delahaye

Philippe Paimblanc

Stéphane Puechmorel

Jonathan Histon

John Hansman

A new air traffic complexity metric based on dynamical system modelization

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Ruth Charney

Luis Paris

Convexity of parabolic subgroups in Artin groups

Introduction

Let S be a finite set. A Coxeter matrix over S is a square matrix M = (m s,t ) s,t∈S indexed by the elements of S satisfying m s,s = 1 for all s ∈ S, and m s,t = m t,s ∈ {2, 3, 4, . . . } ∪ {∞} for all s, t ∈ S, s = t. The Coxeter graph which represents the above Coxeter matrix is the labelled graph Γ = Γ(M ) defined as follows. (1) S is the set of vertices of Γ. (2) Two vertices s, t ∈ S are joined by an edge if m s,t ≥ 3. (3) This edge is labelled by m s,t if m s,t ≥ 4.

The Coxeter system of Γ is the pair (W, S) = (W Γ , S), where S is the set of vertices of Γ, and W is the group defined by the following presentation. W = S s 2 = 1 for all s ∈ S (st) ms,t = 1 for all s, t ∈ S, s = t and m s,t = ∞ . if m is even a(ba) m- 1 2 if m is odd

In other words, Π(a, b : m) denotes the word aba • • • of length m. Let Σ = {σ s | s ∈ S} be an abstract set in one-to-one correspondence with S. The Artin system of Γ is the pair (A, Σ) = (A Γ , Σ), where A is the group defined by the following presentation.

A = Σ | Π(σ s , σ t : m s,t ) = Π(σ t , σ s : m s,t ) for all s, t ∈ S, s = t and m s,t = ∞ .

The group A itself is called the Artin group of Γ. Observe that the map Σ → S, σ s → s, induces an epimorphism θ : A Γ → W Γ . The kernel of θ is called the colored Artin group of Γ, and it is denoted by CA Γ . The Coxeter groups were introduced by Tits in his manuscript [START_REF] Tits | Groupes et géométries de Coxeter[END_REF]. The latter is one of the main sources for the celebrated Bourbaki book "Groupes et algèbres de Lie, Chapitres IV, V et VI" [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines[END_REF]. They play an important role in many areas such as Lie theory, hyperbolic geometry, and, of course, group theory. Furthermore, there is a quite extensive literature on Coxeter groups. We recommend [START_REF] Davis | The geometry and topology of Coxeter groups[END_REF] for a detailed study of the subject. The Artin groups were also introduced by Tits, as extensions of Coxeter groups [START_REF] Tits | Normalisateurs de tores. I. Groupes de Coxeter tendus[END_REF]. They are involved in several fields (singularities, knot theory, mapping class groups, and so on), and they have been the object of many papers in the last three decades. Most results, however, involve only special classes of Artin groups, such as spherical type Artin groups (where the corresponding Coxeter group is finite) or right-angled Artin groups (where all m s,t = 2, ∞). Artin groups as a whole are still poorly understood. In particular, it is not known if they are torsion free, or if they have solvable word problem (see [START_REF] Godelle | Basic questions on Artin-Tits groups. Configuration Spaces[END_REF]). The present paper is an exception to that rule since it concerns all Artin groups.

The flagship example of an Artin group is the braid group B n on n strands. It is associated to the Coxeter graph A n-1 depicted in Figure 1, and its associated Coxeter group is the symmetric group S n .

s 1 s 2 s 3 s n-2 s n-1 Figure 1. The Coxeter graph A n-1 .
For T ⊂ S, we denote by W T the subgroup of W generated by T , we denote by Γ T the full subgraph of Γ spanned by T , we set Σ T = {σ s | s ∈ T }, and we denote by A T the subgroup of A generated by Σ T . By [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines[END_REF], the pair (W T , T ) is the Coxeter system of Γ T , and by [START_REF] Van Der Lek | The homotopy type of complex hyperplane complements[END_REF], (A T , Σ T ) is the Artin system of Γ T . The group A T (resp. W T ) is called a standard parabolic subgroup of A (resp. of W ).

Let m and n be two positive integers such that m ≤ n. A fundamental example of a standard parabolic subgroup is the braid group B m embedded in B n via the homomorphism which sends the standard generators of B m to the first m -1 standard generators of B n .

Let G be a group, and let S be a generating set for G. An expression for an element α ∈ G is a word α = s ε 1 1 • • • s ε on S S -1 which represents α. The length of α (with respect to S), denoted by lg S (α), is the minimal length of an expression for α. A geodesic for α is an expression of length lg S (α). Geometrically, this corresponds to a geodesic path from 1 to α in the Cayley graph of (G, S).

Let T be a subset of S, and let H be the subgroup of G generated by T . We say that H is convex in G with respect to S if for all α ∈ H and all geodesic α = s ε 1 1 • • • s ε of α, we have s 1 , . . . , s ∈ T . Or equivalently, the Cayley graph of (H, T ) is a convex subspace of the Cayley graph of (G, S). In particular, if H is convex, then lg S (α) = lg T (α) for all α ∈ H, that is, H is isometrically embedded in G.

The following is of importance in the study of Coxeter groups.

Theorem 1.1 (Bourbaki [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines[END_REF]). Let Γ be a Coxeter graph, let (W, S) be its Coxeter system, and let T be a subset of S. Then W T is a convex subgroup of W with respect to S.

In the present paper we prove that the same result holds for Artin groups. That is:

Theorem 1.2. Let Γ be a Coxeter graph, let (A, Σ) be its Artin system, and let T be a subset of S. Then A T is a convex subgroup of A with respect to Σ.

Our original motivation for studying the convexity of parabolic subgroups in Artin groups comes from a question asked us by Arye Juhász. In a work in preparation [START_REF] Juhász | On relatively extra large Artin groups and their relative asphericity[END_REF], he provides a solution to the word problem for a certain class of Artin groups, and he proves that these groups are torsion free. His proof uses a condition on the groups that he calls the A-S condition, a form of convexity for a certain type of word. It follows immediately from Theorem 1.2 that this condition is satisfied by all Artin groups. More generally, as for Coxeter groups, the study of Artin groups often goes through the study of their (standard) parabolic subgroups. So any result on these subgroups is likely to be useful in further developments in the theory of Artin groups.

Although Theorem 1.2 may seem natural, it is a surprise for the experts. As far as we know, it was not even known for the braid group B m embedded in B n , although the proof in this case is easy (see Proposition 2.1 below). Theorem 1.2 also comes as a surprise because, in general, the family of standard generators (that is, Σ) is not the best for studying combinatorial questions on Artin groups. In the most well understood case, when A is of spherical type, a larger generating set is generally used. This is called the set of simple elements and we denote it by S. It follows from [START_REF] Charney | Geodesic automation and growth functions for Artin groups of finite type[END_REF] that (A T , S T ) is isometrically embedded in (A, S), but the image is not convex since there are S-geodesics for elements of A T whose terms do not belong to S T S -1 T .

The proof

As promised, we start with the proof of Theorem 1.2 in the particular case of the braid group B m embedded in B n . We treat this case separately for two reasons. Firstly, because some readers may want to use Theorem 1.2 in that case without necessarily learning all the background on Artin groups needed to prove our theorem. Secondly, because the proof of Theorem 1.2 is, in a sense, a (non trivial) extension of the proof of Proposition 2.1. So, the reader may want to keep in mind the proof of Proposition 2.1 when reading the proof of Theorem 1.2.

Recall that a braid on n strands is an

n-tuple β = (b 1 , . . . , b n ) of paths, b i : [0, 1] → R 3 , such that
(1) b i (0) = (i, 0, 0) for all i ∈ {1, . . . , n}, and there is a permutation w ∈ S n such that b i (1) = (w(i), 0, 1) for all i ∈ {1, . . . , n}; (2) (p 3 • b i )(t) = t for all i ∈ {1, . . . , n} and t ∈ [0, 1], where p 3 denotes the projection of R 3 on the third coordinate; (3) b i ∩ b j = ∅ for all i, j ∈ {1, . . . , n}, i = j. The isotopy classes of braids form a group, called the braid group on n strands and denoted by B n . By Artin [START_REF] Artin | Theorie der Zöpfe[END_REF][START_REF] Artin | Theory of braids[END_REF], this group has the following presentation.

B n = σ 1 , . . . , σ n-1 σ i σ i+1 σ i = σ i+1 σ i σ i+1 for 1 ≤ i ≤ n -2 σ i σ j = σ j σ i for |i -j| ≥ 2 .
In other words, B n is the Artin group associated to the Coxeter graph A n-1 depicted in Figure 1.

Proposition 2.1. Let m, n ∈ N, m ≤ n. Then B m is a convex subgroup of B n with respect to {σ 1 , . . . , σ n-1 }.
Proof. Let p 1,3 : R 3 → R 2 denote the projection on the first and third coordinates. Let β = (b 1 , . . . , b n ) be a braid. We project each b i on the plane R 2 via p 1,3 and, up to isotopy, we may suppose that these projections form only finitely many regular double crossings.

As usual, we indicate in each crossing which strand goes over the other. We obtain in this way a braid diagram which represents the isotopy class of β. Recall also that each generator σ i has a "canonical" diagram with precisely one crossing. We turn now to the proof of Theorem 1.2. We fix a Coxeter graph Γ, and denote by (W, S) its Coxeter system, and by (A, Σ) its Artin system.

Let α = σ ε 1 i 1 • • • σ ε i be
= σ ε 1 i 1 • • • σ ε i of length which represents α. Let α ∈ B m , and let α = σ ε 1 i 1 • • • σ ε i be
Let X, Y be two subsets of S, and let w be an element of W . We say that w is (X, Y )minimal if it is of minimal length among the elements of the double-coset W X wW Y . The following will be implicitly used throughout the paper. Proposition 2.2 (Bourbaki [START_REF] Bourbaki | Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines[END_REF]). Let X, Y be two subsets of S, and let w ∈ W .

(1) There exists a unique (X, Y )-minimal element lying in W X wW Y .

(

) 2
The following are equivalent.

• w is (∅, Y )-minimal,

• lg S (ws) > lg S (w) for all s ∈ Y ,

• lg S (wu) = lg S (w) + lg S (u) for all u ∈ W Y . (3) The following are equivalent.

• w is (X, ∅)-minimal,

• lg S (sw) > lg S (w) for all s ∈ X,

• lg S (uw) = lg S (u) + lg S (w) for all u ∈ W X .

Now, set S f = {X ⊂ S | W X is finite}.
Then the following can be easily proved using the previous proposition (see for instance [START_REF] Paris | K(π, 1) conjecture for Artin groups[END_REF]Lemma 3.2]). Lemma 2.3. Let be the relation on W × S f defined as follows. Set (u, X) (v, Y ) if the following three conditions hold,

(

) X ⊂ Y , (2) v -1 u ∈ W Y , and 1
3) v -1 u is (∅, X)-minimal. (
Then is a partial order relation on W × S f .

Recall that the derived complex of a partially ordered set (E, ≤) is defined to be the abstract simplicial complex made of the finite nonempty chains of E. The Salvetti complex of Γ, denoted by Sal(Γ), is defined to be the geometric realization of the derived complex of (W × S f , ). Note that the action of W on W × S f defined by w • (u, X) = (wu, X) leaves invariant the order relation , hence it induces a "natural" (free and properly discontinuous) action of W on Sal(Γ). The quotient space will be denoted by Sal(Γ) = Sal(Γ)/W .

The spaces Sal(Γ) and Sal(Γ) admit cellular decompositions. Recall that a finite Coxeter group W X can be realized as an orthogonal reflection group acting on R k , k = |X|. The Coxeter cell for W X is the convex hull of the orbit of a generic point p in R k . Viewed as a combinatorial object, it is independent of choice of the point p. Its 1-skeleton is the Cayley graph of (W X , X). The cell decompositions of Sal(Γ) and Sal(Γ), which we now describe, are made up of Coxeter cells for the subgroups W X , X ∈ S f . For (u, X) ∈ W × S f , we set

C(u, X) = {(v, Y ) ∈ W × S f | (v, Y ) (u, X)} ,
and we denote by B(u, X) the simplicial subcomplex of Sal(Γ) spanned by C(u, X). It is shown in [START_REF] Paris | K(π, 1) conjecture for Artin groups[END_REF] that B(u, X) is a closed disc of dimension |X| for all (u, X) ∈ W × S f ; in fact it is naturally isomorphic to the Coxeter cell for W X . The set {B(u, X) | (u, X) ∈ W × S f } is a regular cellular decomposition of Sal(Γ) (see also [START_REF] Salvetti | The homotopy type of Artin groups[END_REF][START_REF] Charney | Finite K(π, 1)'s for Artin groups[END_REF]).

Observe that every w ∈ W \ {1} sends the closed disc B(u, X) homeomorphically onto B(wu, X), and that int(B(u, X)) ∩ int(B(wu, X)) = ∅, for all (u, X) ∈ W × S f . It follows that the cellular decomposition of Sal(Γ) induces a cellular decomposition of the quotient Sal(Γ) = Sal(Γ)/W . This decomposition has one cell for each X ∈ S f , corresponding to the orbit of B(1, X) under W . The closure of this cell will be denoted by B(X). Note that this cellular decomposition is not regular in general.

The k-skeletons of Sal(Γ) and Sal(Γ) for k = 0, 1, 2 can be described as follows.

The 0-skeleton. Let u ∈ W . Then C(u, ∅) = {(u, ∅)}, and B(u, ∅) is a vertex of Sal(Γ). We denote it by x(u). So, the 0-skeleton of Sal(Γ) is the set {x(u) | u ∈ W }, which is in one-to-one correspondence with W . The 0-skeleton of Sal(Γ) is reduced to a single point, B(∅), that we denote by x 0 .

The 1-skeleton. Let u ∈ W and s ∈ S. Then C(u, {s}) = {(u, ∅), (us, ∅), (u, {s})}, and B(u, {s}) is an edge of Sal(Γ) connecting x(u) to x(us). We denote it by a(u, s) and orient it from x(u) to x(us). Note that there is another edge connecting x(u) to x(us), namely a(us, s), but this edge is oriented in the other direction (see Figure 2). Observe that the action of W preserves orientations of edges. Since all edges are of this form, we see that the 1-skeleton of Sal(Γ) is just the Cayley graph of (W, S). Descending to Sal(Γ), the 1-skeleton consists of a loop at x 0 formed by the edge ās = B({s}), for each s ∈ S (see Figure 2).

The 2-skeleton. Let s, t ∈ S, s = t. Set X = {s, t}. First, notice that X ∈ S f if and only if m s,t = ∞. Now, assume m s,t = ∞, set m = m s,t , and take u ∈ W . Then

x(u) a (u, s) a(v, s) x(v)
x(us) x 0 ās B(u, X) is isomorphic to the Coxeter cell for W X . Namely, it is a 2m-gon with vertices

{x(uw) | w ∈ W X }. The boundary of B(u, X) is the loop a(u, s)a(us, t) • • • a(uΠ(s, t : m -1), r)a(uΠ(t, s : m -1), r ) -1 • • • a(ut, s) -1 a(u, t) -1
where r = t and r = s if m is even, and r = s and r = t if m is odd (see Figure 3). Hence, B(X) is a 2-cell whose boundary is From the above it follows that the 2-skeleton of Sal(Γ) is precisely the Cayley 2-complex associated with the standard presentation of A.

ās āt • • • ār ā-1 r • • • ā-1 s ā-1 t = Π(ā s , āt : m)Π(ā t , ās : m) - 1 . a(u, t) a(u, s) a(ut, s) a(us, t) a(uts, t) a(ust, s) āt ās
Proposition 2.4. We have π 1 (Sal(Γ), x 0 ) = A, π 1 (Sal(Γ), x(1)) = CA, and the exact sequence associated to the regular covering Sal(Γ) → Sal(Γ) is

1 -→ CA -→ A -→ W -→ 1.
Now, assume we are given a subset T of S, and set

S f T = {X ∈ S f | X ⊂ T }. Observe that the embedding W T × S f T → W × S f induces an embedding ι T : Sal(Γ T ) → Sal(Γ).
The following provides our main tool for proving Theorem 1.2.

Theorem 2.5 (Godelle, Paris [START_REF] Godelle | K(π, 1) and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups[END_REF]). The embedding ι T : Sal(Γ T ) → Sal(Γ) admits a retraction π T : Sal(Γ) → Sal(Γ T ).

In the proof of Theorem 1.2, we will need the following explicit description of the map π T . Let (u, X) ∈ W × S f . Write u = u 0 u 1 , where u 0 ∈ W T and u 1 is (T, ∅)-minimal. Set X 0 = T ∩ u 1 Xu -1

1 . Note that, since u 1 W X u -1 1 is finite, W X 0 is finite, hence X 0 ∈ S f T . Then we set π T (u, X) = (u 0 , X 0 ). It is proved in [START_REF] Godelle | K(π, 1) and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups[END_REF] that the map π T : W × S f → W T × S f T induces a continuous map π T : Sal(Γ) → Sal(Γ T ) which is a retraction of ι T . Now, the following lemma follows from the above description.

Lemma 2.6. Let u ∈ W . Write u = u 0 u 1 , where u 0 ∈ W T and u 1 is (T, ∅)-minimal. Then (1) π T (x(u)) = x(u 0 ).

(2) Let s ∈ S and set t = u 1 su -1

1 . If t ∈ T , then π T (a(u, s)) = a(u 0 , t). If t / ∈ T , then π T (a(u, s)) = x(u 0 ).
Proof. We leave Part (1) to the reader and turn to Part [START_REF] Artin | Theory of braids[END_REF]. We take u ∈ W and s ∈ S, and define u 0 , u 1 and t as in the Lemma. Recall that C(u, {s}) = {(u, ∅), (us, ∅), (u, {s})}, and that a(u, s) is the edge of Sal(Γ) spanned by C(u, {s}).

Assume first that t ∈ T . We have us = u 0 u 1 s = u 0 tu 1 , u 0 t ∈ W T and u 1 is (T, ∅)minimal, hence π T ((u, ∅)) = (u 0 , ∅), π T ((us, ∅)) = (u 0 t, ∅), and π T ((u, {s})) = (u 0 , {t}). Therefore, π T (a(u, s)) = a(u 0 , t).

Assume now that t ∈ T . We claim that u 1 s is (T, ∅)-minimal. If lg S (u 1 s) < lg S (u 1 ), then this is clear since u 1 is (T, ∅)-minimal. Suppose that lg S (u 1 s) > lg S (u 1 ). If u 1 s were not (T, ∅)-minimal, then there would exist s 0 ∈ T such that lg S (s 0 u 1 s) < lg S (u 1 s) and since u 1 is (T, ∅)-minimal, lg S (s 0 u 1 ) > lg S (u 1 ). By the "folding condition" (see [START_REF] Brown | Buildings[END_REF]Chap. II,Sec. 3]), this would imply that s 0 u 1 s = u 1 , that is, t = u 1 su -1 1 = s 0 ∈ T , a contradiction. Thus, if t ∈ T , then us = u 0 u 1 s, u 0 ∈ W T , and u 1 s is (T, ∅)-minimal. It follows that π

T ((u, ∅)) = π T ((us, ∅)) = π T ((u, {s})) = (u 0 , ∅), hence π T (a(u, s)) = x(u 0 ). Proof of Theorem 1.2. Let α = σ ε 1 s 1 • • • σ ε s ∈ (Σ Σ -1 ) * be a word in the alphabet Σ Σ -1 . Set γ( α) = āε 1 s 1 • • • āε s .
This is a loop in Sal(Γ) based at x 0 . Note that, if α is the element of A represented by the word α, then α is the element of A = π 1 (Sal(Γ), x 0 ) represented by the loop γ( α).

Denote by γ( α) the lift of γ( α) in Sal(Γ) with initial point x(1). The path γ( α) can be described as follows. For i ∈ {0, 1, . . . , } we set

u i = s ε 1 1 s ε 2 2 • • • s ε i i = s 1 s 2 • • • s i ∈ W . For i ∈ {1, . . . , }, we set a i = a(u i-1 , s i ) if ε i = 1 and a i = a(u i , s i ) if ε i = -1. Then γ( α) = a ε 1 1 • • • a ε . Let i ∈ {1, . . . , }. Write u i = v i w i , where v i ∈ W T and w i is (T, ∅)-minimal. If ε i = 1, let t i = w i-1 s i w -1 i-1 . If ε i = -1, let t i = w i s i w -1 i . Define τ i = σ ε i t i if t i ∈ T 1 if t i / ∈ T τ = τ 1 τ 2 • • • τ ∈ (Σ T Σ -1 T ) * . Note that lg( τ ) ≤ lg( α).
By Lemma 2.6, the projection of π T (a i ) in Sal(Γ T ) is the edge āt i if t i ∈ T , and the vertex x 0 otherwise. Thus the image of π T (γ( α)) in Sal(Γ T ) is the loop corresponding to τ . In other words, π T (γ( α)) = γ( τ ).

Claim 1. If α ∈ (Σ T Σ -1 T ) * , then τ = α. Proof of Claim 1. In this case we have u i ∈ W T , hence v i = u i and w i = 1 for all i ∈ {0, 1, . . . , }. It follows that t

i = s i ∈ T , so τ i = σ ε i s i for all i. Hence, τ = τ 1 τ 2 • • • τ = σ ε 1 s 1 σ ε 2 s 2 • • • σ ε s = α .
Claim 2. Suppose α ∈ (Σ Σ -1 ) * represents the element α ∈ A T . Then τ ∈ (Σ T Σ -1 T ) * also represents α.

Proof of Claim 2. Choose a word α ∈ (Σ T Σ -1 T ) * which represents α. By construction, γ( α ) and γ( α) represent α, hence they are homotopic in Sal(Γ) relative to x 0 . So, γ( α ) is homotopic to γ( α) relative to endpoints. It follows that π T (γ( α )) = γ( α ) is homotopic to π T (γ( α)) = γ( τ ) in Sal(Γ T ). We conclude that τ and α represent the same element of A T , namely, α.

Claim 3. Let α ∈ (Σ Σ -1 ) * . If lg( τ ) = lg( α), then α ∈ (Σ T Σ -1 T ) * . Proof of Claim 3. In order to have lg( τ ) = lg( α), we must have t i ∈ T for all i ∈ {1, . . . , }. We will prove by induction on i that s i also lies in T for all i.

Let i = 1. Suppose first that ε 1 = 1. Then u 0 = w 0 = 1, so s 1 = w 0 s 1 w -1 0 = t 1 ∈ T . Suppose now that ε 1 = -1. If we had s 1 ∈ T , then s 1 would be (T, ∅)-minimal and we would have u 1 = w 1 = s 1 . In this case, s 1 = w 1 s 1 w -1 1 = t 1 ∈ T , a contradiction. So, s 1 ∈ T . Now assume that i ≥ 2. By induction, we have u i-1 = s 1 • • • s i-1 ∈ W T . Suppose first that ε i = 1. Since u i-1 ∈ W T , we have v i-1 = u i-1 and w i-1 = 1, hence s i = t i ∈ T . Suppose now that ε i = -1. If we had s i ∈ T , then we would have v i = u i-1 and w i = s i , which would imply that s i = t i ∈ T , a contradiction. So, s i ∈ T .

We can now complete the proof of the theorem. Let α ∈ A T , and let α ∈ (Σ Σ -1 ) * be a geodesic for α. By Claim 2, τ is also an expression for α. By construction, lg( τ ) ≤ lg( α), hence lg( τ ) = lg( α) since α is a geodesic. By Claim 3 we conclude that α ∈ (Σ T Σ -1 T ) * .

  The group W itself is called the Coxeter group of Γ. If a, b are two letters and m is an integer ≥ 2, we set Π(a, b : m) = (ab) m 2

R.

  Charney was partially supported by NSF grant DMS-1106726.

  a geodesic of α with respect to the generating set of B n . Let D be the diagram of α obtained from α. By removing the strands b m+1 , . . . , b n from D, we obtain another diagram D of α, but now viewed as an element of B m . Since no new crossings are introduced by this procedure, D has at most crossings. But α was geodesic, so the number of crossings cannot be less than . This means that the only strands involved in the crossings in D are b 1 , . . . , b m , and therefore σ i 1 , . . . , σ i ∈ {σ 1 , . . . , σ m-1 }.

Figure 2 . 1 -

 21 Figure 2. 1-skeletons of Sal(Γ) and Sal(Γ).

Figure 3 . 2 -

 32 Figure 3. 2-skeletons of Sal(Γ) and Sal(Γ).

  a word which represents an element α ∈ B n . By concatenating the canonical diagrams of the σ ε i i 's, we obtain a diagram of α with precisely crossings. Conversely, by applying standard methods, from a diagram of α with crossings, we can define a (non unique) word α


N A Shevchenko

N N Dadoenkova

I L Lyubchanskii

F F L Bentivegna

Y P Lee

Th Rasing

Nonlinear Optical Diffraction by Standing Acoustic Waves in a GaAs film

Keywords: Quadratic optical nonlinearity, Modulated structures, Nonlinear optical diffraction, Standing acoustic waves

Second-order nonlinear optical diffraction by standing acoustic waves in a crystalline plate is theoretically investigated. A detailed analysis of the polarization state of the second-harmonic light diffracted by both longitudinal and transversal acoustic waves is carried out. It is shown that longitudinal standing acoustic waves only allow p-polarized nonlinear optical diffraction, irrespective of the incoming state of polarization, whereas transversal standing acoustic waves allow all possible combinations of incoming and diffracted polarization states. Numerical estimates of the relative intensities of nonlinearly diffracted radiation peaks are made for a GaAs plate.

Introduction

The phenomenon of optical diffraction from periodically modulated structures (e.g., diffraction gratings) is well known in classical optics [START_REF] Born | Principles of Optics[END_REF] and is widely used for a broad range of optical devices [START_REF] Palmer | Diffraction Grating Handbook[END_REF]. If in addition such a medium exhibits nonlinear optical properties, observation of nonlinear optical diffraction (NOD), i.e., diffraction at higher harmonics of the incident frequency, is possible provided certain conditions are fulfilled. In media with a quadratic optical nonlinearity, NOD is thus expected to be observed at a frequency double that of the incident electromagnetic wave (EMW). The first theoretical and experimental investigation of quadratic NOD was made in a periodically poled ferroelectric NH [START_REF] Dadoenkova | Nonlinear magnetooptical diffraction from periodic domain structures in magnetic films[END_REF] Cl crystal, where the modulation period was comparable with the wavelengths of the second harmonic EMWs generated from the periodic structure [START_REF] Freund | Nonlinear diffraction[END_REF]. Further theoretical and experimental studies of NOD were carried out in such modulated structures as ordered magnetic domain arrays [START_REF] Dadoenkova | Nonlinear magnetooptical diffraction from periodic domain structures in magnetic films[END_REF][START_REF] Lyubchanskii | Nonlinear magneto-optical diffraction by twodimensional magnetic superstructures[END_REF][START_REF] Dadoenkova | Influence of magnetic field on nonlinear magneto-optical diffraction on two-dimensional hexagonal magnetic bubble lattice[END_REF][START_REF] Lazarenko | Linear and nonlinear magneto-optical diffraction from one-dimensional periodic structures[END_REF] and photonic crystals [START_REF] Berger | Nonlinear photonic crystals[END_REF][START_REF] Malvezzi | Second-harmonic generation in reflection and diffraction by GaAs photonic-crystal waveguide[END_REF][START_REF] Fedyanin | Nonlinear diffraction and second-harmonic generation enhancement in silicon-opal photonic crystals[END_REF][START_REF] Estephan | Sensing by means of nonlinear pptics with functionalized GaAs/AlGaAs photonic crystals[END_REF][START_REF] Saltiel | Generation of second-harmonic conical waves via nonlinear Bragg diffraction[END_REF][START_REF] Saltiel | Multiorder nonlinear diffraction in frequency doubling processes[END_REF]. One of the possible ways to modulate the nonlinear polarization in a crystalline medium is to excite a standing acoustic wave characterized by a periodically distributed displacement field in the sample with spatially localized nodes and antinodes. In linear optics, a similar phenomenon is well known as light diffraction by ultrasound waves [START_REF] Born | Principles of Optics[END_REF][START_REF] Tucker | Hypersound in Solid State Physics[END_REF]. In the case of NOD, the nonlinear polarization of the medium will be modulated by the deformations of the crystal through a nonlinear photoelastic interaction. Such a situation was discussed in the frame of acoustically-induced optical second harmonic generation (SHG) due to the interaction between optical waves and a travelling acoustic wave in a crystal [START_REF] Nelson | Theory of acoustically induced optical harmonic generation[END_REF]. Some theoretical aspects of nonlinear photoelastic interaction were discussed by several groups [16 -18]. The effect of acoustically-induced optical SHG was also observed in GaAs films in the infrared region where this material is transparent [START_REF] Boyd | Observation of acoustically induced phase-matched optical harmonic generation in GaAs[END_REF]. Later, strain-induced three-photon effects such as optical SHG in strained crystals and hyper-Rayleigh light scattering by dislocation deformation were theoretically investigated [START_REF] Jeong | Strain-induced three-photon effects[END_REF], as well as SHG due to misfit strain and misfit dislocations at filmsubstrate interfaces [START_REF] Lyubchanskii | Second-harmonic generation from realistic film-substrate interfaces: The effects of strain[END_REF]. Similarly, experimental observations of strain-induced SHG were reported in silicon wafers [START_REF] Reif | Nonlinear optical characterization of the surface of silicon wafers: In-situ detection of external stress[END_REF] and the influence of strain on SHG in Co 0.25 Pd 0.75 films on a PZT substrate [START_REF] Jeong | Direct observation of controlled strain-induced second harmonic generation in a Co 0[END_REF], in bismuth-substituted iron garnet films [START_REF] Kumar | Second harmonic generation study of internally generated strain in bismuth-substituted iron garnet films[END_REF], as well as in bulk silicon [START_REF] Schriever | Strain dependence of second harmonic generation in silicon[END_REF][START_REF] Cazzanelli | Second-harmonic generation in silicon waveguides strained by silicon nitride[END_REF] were discussed.

In this paper, we theoretically investigate NOD induced by standing longitudinal and transversal acoustical waves in a crystalline plate. This situation differs from the aforementioned works [START_REF] Nelson | Theory of acoustically induced optical harmonic generation[END_REF][START_REF] Barocchi | Multiphoton-phonon interactions and higher-order Brillouin scattering: (Steady-state gain amplification)[END_REF][START_REF] Levine | Calculation of the direct acoustically induced optical harmonic generation coefficient[END_REF][START_REF] Bokut | Optical second harmonic generation under collinear light diffraction on ultrasound[END_REF][START_REF] Boyd | Observation of acoustically induced phase-matched optical harmonic generation in GaAs[END_REF] insofar as it does not require the same phase matching conditions to be fulfilled.

The paper is organized as follows. In Section 2 we describe strain-induced nonlinear polarization on the basis of the photoelastic interaction, and Section 3 introduces the mechanical strain originating in standing acoustic waves in a crystalline film. Section 4 is devoted to the analysis of second-order NOD by both longitudinal and transversal standing acoustic waves in such a film and presents and discusses the results of numerical simulations carried out in the case of a GaAs film.

Strain-induced nonlinear polarization

In the dipole approximation the quadratic nonlinear optical polarization vector P NL (2ω) can be written in a Cartesian referential as [START_REF] Shen | The Principles of Nonlinear Optics[END_REF]:

(2) 0 [START_REF] Palmer | Diffraction Grating Handbook[END_REF] ( 2 : , ) ( ) ( )

NL i ijk j k P E E ω = ε χ -ω ω ω ω ω , (1)
where ( )

ω E
is the electric field of the incident EMW at frequency ω, (2) ijk χ is the quadratic nonlinear optical susceptibility (NOS) tensor of the medium, and 0 ε is the vacuum permittivity.

When a spatial modulation of the optical properties of the medium takes place -e.g., due to the presence of a standing acoustic wave -the NOS tensor can be expressed as:

(2) (2) (2,0) (2,0) 0 ( ) ( ) ( ) ( ) ( ) =   ∂χ χ = χ + = χ +     ∂   lm ijk ijk ijk ijklm lm ijk lm lm u p u u u r r r r r , (2)
where (2,0) ijk χ is the nonlinear optical susceptibility tensor in the absence of modulation, ijklm p is the nonlinear photoelastic tensor and lm u is the strain tensor. In the linear approximation, the latter is symmetric, i.e. lm ml u u = . The tensor p ijklm was first introduced in Ref. [START_REF] Nelson | Theory of acoustically induced optical harmonic generation[END_REF] for the description of acoustically-induced optical SHG. From Eq. (2) it follows that the symmetry properties of the nonlinear photoelastic tensor ijklm p derive from those of both the NOS tensor

(2) ijk χ and the strain tensor lm u . Thus the symmetry of the strain tensor imply that

ijklm ikjlm ijkml ikjml p p p p = = =
, which strongly reduces the number of independent components of the NOS tensor (2) ijk χ . In addition, that tensor has non-zero components in non-centrosymmetric crystals only, i.e., in media without inversion center [START_REF] Shen | The Principles of Nonlinear Optics[END_REF].

The second-harmonic electric field E 2ω (r,t) emitted from a non-magnetic medium is a solution of the quadratic nonlinear wave equation which, in SI units, writes [START_REF] Shen | The Principles of Nonlinear Optics[END_REF]:

2 2 2 NL 2 2 2 2 0 2 2 2 ( , ) ( , ) ( , ) n t t t c t t ω ω ω ω ∂ ∂ ∇ ×∇ × + = -µ ∂ ∂ E r P r E r , (3)
where 2 n ω is the refractive index of the medium at the second harmonic frequency, c is the velocity of light in vacuum, and 0 µ is the vacuum permeability. Taking the Fourier transform of Eq. ( 3) with respect to time and using the slowly varying envelope approximation, where

2 (2 , ) (2 , ) ω ⋅∇ ω ∆ ω ≫ k E r E r
, leads to the following reduced form [START_REF] Shen | The Principles of Nonlinear Optics[END_REF][START_REF] Mills | Nonlinear Optics: Basic Concepts[END_REF]:

2 2 NL 2 (2 , ) 2 (2 , ) exp( ) i i c ω ω ⋅∇ ω = - ω ⋅ k E r P r q r . (4)
Here (2 , ) ω E r and NL (2 , ) ω P r are the temporal Fourier transforms of the electric field at the second harmonic frequency and the corresponding nonlinear polarization, and

2 2 ω ω = - q
k k is the phase-mismatch vector, while ω k and 2ω k are the wave vectors of the fundamental and second harmonic EMWs, respectively. The solution of wave equation ( 4) in the plane-wave approximation can then be expressed as [START_REF] Shen | The Principles of Nonlinear Optics[END_REF][START_REF] Mills | Nonlinear Optics: Basic Concepts[END_REF]:

NL 2 1 (2 , ) (2 ,
) exp( )

V i i d cn V ω ω ω = ω ⋅ ∫ E q P r q r r , ( 5
)
The integral is taken over the interaction volume V, i.e., in the irradiated region where the electric field of the incident EMW is non-zero.

Strain induced by standing acoustic waves

We now consider a non-centrosymmetric thin crystalline film whose surfaces are perpendicular to the y-axis of a (xyz) Cartesian referential (Fig. 1). In what follows, calculations will be made for a GaAs film, whose zinc blende crystal structure (with a 43m or T d point group symmetry) is indeed non-centrosymmetric. We suppose the system to be continuous with respect to its elastic properties. The form of standing acoustic waves in a bulk elastic medium can be deduced from the following system of equations of motion [START_REF] Kittel | Introduction to Solid State Physics[END_REF]:

2 2 2 ( , ) ( , ) i l ijkl j k u t u t c t r r ∂ ∂ ρ = ∂ ∂ ∂ r r , (6)
where ρ and ijkl c are the density of the crystal and the elastic stiffness tensor, respectively, and the summation convention over repeated indices is assumed. For cubic crystals without inversion center, only three components of the ijkl c tensor are independent [START_REF] Kittel | Introduction to Solid State Physics[END_REF]:

11 xxxx yyyy zzzz c c c c ≡ = = , 12
xxyy yyzz zzxx c c c c ≡ = = , 44 xyxy yzyz zxzx c c c c ≡ = = . ( 7
)
The same symmetry rules hold for the zinc blende structure. Consequently, Eq. ( 6) reduces to the following system:

( )

2 2 2 2 2 2 11 44 12 44 2 2 2 2 y x x x x z u u u u u u c c c c t x y z x y x z   ∂   ∂ ∂ ∂ ∂ ∂ ρ = + + + + +       ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂     , ( )
2
  ∂ ∂ ∂ ∂   ∂ ∂ ρ = + + + + +       ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂     , ( 8
) ( ) 2 2 2 2 2 2 11 44 12 44 2 2 2 2 y x z z z z u u u u u u c c c c t z y x z x y z   ∂   ∂ ∂ ∂ ∂ ∂ ρ = + + + + +       ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂    
.

The acoustic waves generated within the crystal are assumed to propagate along the z-axis of the referential (Fig. 1), which is also the four-fold [001] axis of the crystal. Thus the mechanical displacement vector u is function of the sole z spatial coordinate and of time t. Hence, Eqs. [START_REF] Berger | Nonlinear photonic crystals[END_REF] simplify to the following set of wave equations:

2 2 2 2 2 2 44 44 11 2 2 2 2 2 2
, ,

y y x x z z u u u u u u c c c t z t z t z ∂ ∂ ∂ ∂ ∂ ∂ ρ = ρ = ρ = ∂ ∂ ∂ ∂ ∂ ∂ . (9)
It can be seen from Eqs. ( 9) that in such a crystal an acoustic wave excited along the z-axis actually consists of two components propagating independently with different phase velocities.

The displacement ( , )

L T z t z t z t = + u u u ( , ) ( , )
, writes as the sum of a longitudinal and a transverse part, each of which satisfies the wave equation with the appropriate velocity.

One possible solution of Eqs. [START_REF] Malvezzi | Second-harmonic generation in reflection and diffraction by GaAs photonic-crystal waveguide[END_REF] in the bulk of the crystal is a standing acoustic wave that results from the interference between two traveling waves propagating in opposite directions -as is bound to happen in a crystal plate whose extension along the z-axis is finite. The displacement vector for the bulk standing acoustic wave can then be written as:

( ) ( ) ( , ) cos exp cos exp L T L L T T L T z z z t i t i t v v     Ω Ω = Ω + Ω         u A A . ( 10
)
Here [0, 0, ]

L z A = A and [ , , 0] T x y A A = A
are the amplitudes of the longitudinal and transverse acoustic standing waves, respectively, and , ,

L T L T z v L π Ω =
, are their frequencies, where L z is the length of the crystal along the z-axis. For standing waves, there is no propagation of the mechanical energy along that axis, i.e., the amplitudes of the displacement vector vary with position z and time t but their extrema are stationary. Values of z obeying the conditions ( )

, , ( ) 1 2 ,

L T L T z v m m

π Ω = + ∈ ℕ , denote the nodes of the vibration, i.e., points in the crystal where the vibration is zero for all values of t, whereas the amplitudes of the vibration are maximal at any time t at its antinodes, for which , , ( ) ,

L T L T z v m m π Ω = ∈ ℕ .
As usual with standing waves, the distance between successive nodes or antinodes is equal to half the wavelength of the exciting acoustic wave (Fig. 1).

In practice, such transversal and longitudinal acoustical waves can be excited in thin films and plates with piezoelectric transducers, as thoroughly described for instance in Ref. [START_REF] Tucker | Hypersound in Solid State Physics[END_REF]. It is also possible to use optical methods to excite both travelling and standing acoustical waves in bulk materials as well as in thin films and multilayers [START_REF] Rogers | Optical generation and characterization of acoustic waves in thin films: Fundamentals and applications[END_REF].

Nonlinear diffraction by standing acoustic waves

The plane of incidence of light is taken to be the (yz)-plane. The wave vectors of the secondharmonic EMWs diffracted from the crystal must satisfy the nonlinear Bragg condition Alternatively, that vectorial equality can be decomposed in two scalar conditions for in-plane and out-of-plane wave vector components, respectively:

( ) 2 2 , sin 2 sin( )
N L T k k N Q ω ω ω ω   ϕ = ϕ +   , (11a) ( ) 2 2
cos 2 cos( )

N k k ω ω ω ω   ϕ = ϕ   . ( 11b
)
Here angles φ ω and ( )

L T L T N n n ω ω ω 2ω Λ = - ϕ λ . ( 12
)
In Eq. ( 12) , L T Λ are the wavelengths of L-and T-standing acoustic waves, respectively, 2ω λ is the wavelength in vacuum of the diffracted EMW at frequency 2 ω, and n ω and n 2ω are the refractive indexes of the crystal at the fundamental and second harmonic frequencies. Figure 2 shows the dependence of The efficiency of NOD induced by standing acoustic waves strongly depends on the polarization state of light. For a given phase-mismatch vector in the crystal, the electric fields , (2 , ) s p ω E q of second-harmonic s-and p-polarized EMWs are phenomenologically related to those at the fundamental frequency through the second-order NOS tensor as follows:

(

i cn ω ω     χ χ ω   ω = - ⋅       ω χ χ ω           E q q q E E q q q E . ( 13
)
The components of that tensor depend on those of both the nonlinear photoelastic tensor and the deformation tensor of the crystal in the spatial frequency domain, hence they differ depending on the nature of the standing acoustic wave responsible for nonlinear diffraction.

Taking into account the symmetries of GaAs (T d point group), they are, for L-acoustic waves:

(2, ) ;

( ) 0

N s ss χ = q , (14a) ( ) ( ) (2, ) ; 2 ( ) ( ) cos
N zxxzz p ss zz N p u ω χ = ϕ q q , (14b) (2, ) ;
( ) 0

N s pp χ = q , (14c) ( ) ( )
(2, ) 2 2 ; 2 2
( ) [ sin(2 ) sin( ) ( sin ( ) cos ( )) cos( )] ( )

N N N p pp yyzzz zyyzz zzzzz zz p p p u ω ω ω ω ω χ = ϕ ϕ + ϕ + ϕ ϕ q q ; (14d)
and for T-acoustic waves:

(

N s ss xxxxz xz p u χ = q q , 2, ) ; ( ) ( )
sin yxxyz N p ss yz N p u ω χ = ϕ q q , ( (15a) ( ) ( ) (2, ) ; 2 ( ) ( )
) (2, ) 15b
( ) [ sin ( ) cos ( )] ( )

N s pp xyyxz xzzxz xz p p u ω ω χ = ϕ + ϕ q q , (15c) ( ) ( ) ( )
(2, ) 2 2 ; 2 2 2 ( ) [( sin ( ) cos ( )) sin( ) sin(2 ) cos( )] ( )
N N N N p pp yyyyz yzzyz zyzyz yz p p p u ω ω ω ω ω χ = ϕ + ϕ ϕ + ϕ ϕ q q . (15d)
The components of the strain tensor in the frequency domain that appear in Eqs. ( 14) -( 15) are obtained by Fourier transform from their counterparts in real space, with:

( )

1 ( ) ( ) exp ( ) ( ) ⊥ = ⋅ = ∫ lm lm lm V u u i d I q I q V q r q r r , (16a)
where integration in Eq. ( 16a) is taken over the irradiated volume, q and q ⊥ are the longitudinal and transversal components of wave vector q , and I( q ) and I lm ( q ⊥ ) are integrals defined as:

( ) ( ) 2 2 1 exp sinc 2 -   = =     ∫ H H q H I q iq z dz H , (16b) ( ) (
)

2 2 0 0 , , 1 1 , , , ; , ; , , , , 1 ( )
= ( ) exp cos [( ) ] [( ) ] = 2 ( ) ( ) ρ ρ ρ ρ ϕ ϕ π ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ π Ω Ω   + -     + - Ω Ω   + -     ∫ ∫ R lm lm L T L T L T L T l L T m m L T l L T L T L T L T I q u d iq d R J q N R J q N R v v i A Q A Q q N R q N R v v . (16c)
In Eq. (16b), integral I( q ) is evaluated over the film thickness H, sinc( ) sin( )

x
q n n ⊥ ω ω ω ω ω = π λ ϕ - ϕ .
For the sake of clarity, the order of diffraction N, upon which the angle of diffraction at 2ω depends, was not explicitly mentioned in Eqs. ( 16) and in the expressions of q and q ⊥ .

From Eqs. ( 14) -( 15), it appears that L-acoustic waves can only give rise to p-polarized nonlinear diffraction, whatever the state of polarization of the incoming beam, whereas T-standing acoustic waves for the first order of nonlinear diffraction (N = 1), calculated using Eqs. ( 14) -( 15) and the numerical values of the nonlinear photoelastic tensor components of GaAs taken from Ref. [START_REF] Bokut | Optical second harmonic generation under collinear light diffraction on ultrasound[END_REF], for the following combinations of polarization state of incident and diffracted second-harmonic EMWs: ( )

{ } { } { } { } { } ( ) 2 (2) , , 2 (2 )
) s p ω → ω , ( ) (2 ) p p ω → ω , ( ) (2 ) s s ω → ω and ( ) (2
) p s ω → ω . In accordance with Eqs. ( 14), ( 15) and ( 17), only the first two combinations yield any NOD for L-acoustic waves, whereas all four are possible with T-acoustic waves. As can be seen on Fig. 3, the maximal efficiency of the NOD process should be observed for a ppolarized incident EMW in the case of diffraction by L-acoustic waves. Similarly, the diffraction of s-polarized EMWs by T-acoustic waves is less efficient than that of p-polarized EMWs.

Conversely, the results shown on the example of GaAs at 10.6 µm suggest a potential use of NOD for the determination of the components of the nonlinear photoelastic tensor of that crystal and of any crystal with similar symmetries -as a function of wavelength. With that application in mind, the approach developed in this paper can thus serve as a guide for the choice of the best combination of polarization states and angles of both the incident and the diffracted fields.

Conclusions

In this paper, we proposed a full theoretical analysis of second-order nonlinear optical diffraction due to the interaction of electromagnetic waves with both longitudinal and transverse standing acoustic waves in a GaAs crystal. We showed that this effect strongly depends on the polarization state of light. In particular, longitudinal standing acoustic waves only allow ppolarized nonlinear optical diffraction, whatever the incoming state of polarization, whereas transversal standing acoustic waves allow all possible polarization combinations of incoming and diffracted polarization states. A potential application of nonlinear optical diffraction from standing acoustic waves in crystals is the determination of the non-zero components of their nonlinear photoelastic tensor. In all cases, the standing wave frequency is F L,T = 250 MHz, the intensity of the acoustic wave is equal to 30 W/cm 2 , the diameter of the laser spot is 400 µm, and the thickness of the crystalline film is 0.5 mm.

z u z t associated

  to the first component is parallel to the direction of excitation and propagates with the velocity is a so-called longitudinal wave (L-wave). The displacement ( , ), ( , )x y u z t u z t    associated to the second component is perpendicular to the direction of excitation and propagates with a phase velocity is a transverse wave (T-wave). Thus the overall general expression for the displacement vector, ( , )

  three-wave interaction, where L Q and T Q are the wave vectors of the longitudinal and transverse standing acoustic waves, respectively[3, 4 -6, 8, 9].

2 φNω

 2 refer to the directions of the incident and N-th order diffracted EMWs, respectively (Fig.1of the L-and T-standing acoustic waves in the crystal. From Eq. (11a) one can derive the maximum number of diffraction orders that can be observed in each case:

  the incidence angle φ ω for various values (250 MHz, 500 MHz, and 1 GHz) of the acoustic wave frequencies made for GaAs at the CO 2 laser wavelength 10.6 ω λ = µm.

  I ω and ( ) I ω are the intensities of the diffracted and incident beams. Again, the order of diffraction N has been omitted in Eq.[START_REF] Levine | Calculation of the direct acoustically induced optical harmonic generation coefficient[END_REF].

Figure 3

 3 Figure 3 depicts the angular dependencies of the diffraction efficiencies

Figure

  Figure captions for the manuscript "Nonlinear Optical Diffraction by Standing Acoustic Waves in a GaAs film" by N.A. Shevchenko et al.

Fig. 1 .

 1 Fig. 1. Schematic of nonlinear acousto-optical diffraction by standing acoustic waves in the transmission geometry.

Fig. 2 .

 2 Fig. 2. Dependence of the maximum number of observable diffraction orders upon incidence angle ω ϕ for NOD due to standing L-acoustic waves (a) and T-acoustic waves (b) at various acoustic frequencies.

Fig. 3 .

 3 Fig. 3. Angular dependencies of the diffraction efficiency of the first order of NOD induced by a standing acoustic wave in GaAs for the following geometries: (a) and (b) represent NOD by a standing L-acoustic wave for the ( ) (2 ) s p ω → ω and ( ) (2 ) p p ω → ω diffractions, respectively; (c), (d), (e) and (f) correspond to NOD by a standing T-acoustic wave for the ( ) (2 ) s s ω → ω , ( ) (2 ) s p ω → ω , ( ) (2 ) p s ω → ω , and ( ) (2 ) p p ω → ω diffractions, respectively.

  Tacoustic waves allow all four types (s → s, s → p, p → s, and p → p ) of nonlinear diffraction processes. In order to estimate the relative intensities of the second-harmonic diffracted beams in

	each configuration, one can introduce nonlinear diffraction efficiencies	, NOD D α ββ defined as

Acknowledgments

This work was supported by the FP7-People-2009-IRSES Project "NOWAPHEN" No. 247556 (N. A. Shevchenko, N. N. Dadoenkova, and I. L. Lyubchanskii), the Ministry of Education, Science and Technology (MEST) and the National Research Foundation (NRF) of South Korea through q-Psi (Y. P. Lee), and the Netherlands Organization for Scientific Research (NWO, Th. Rasing).