Ingmar Steiner
email: ingmar.steiner|slim.ouni@loria.fr

Slim Ouni

Artimate: an articulatory animation framework for audiovisual speech synthesis

Keywords: articulatory modeling, 3D animation, electromagnetic articulography, audiovisual speech synthesis

We present a modular framework for articulatory animation synthesis using speech motion capture data obtained with electromagnetic articulography (EMA). Adapting a skeletal animation approach, the articulatory motion data is applied to a threedimensional (3D) model of the vocal tract, creating a portable resource that can be integrated in an audiovisual (AV) speech synthesis platform to provide realistic animation of the tongue and teeth for a virtual character. The framework also provides an interface to articulatory animation synthesis, as well as an example application to illustrate its use with a 3D game engine. We rely on cross-platform, open-source software and open standards to provide a lightweight, accessible, and portable workflow.

Background and Motivation

This paper presents a framework for creating portable kinematic articulatory models for audiovisual (AV) speech synthesis, driven by actual speech data. We refer to AV speech synthesis as the process of generating speech and displaying speechsynchronized animation for the face and articulators (viz. the tongue, lips, jaw) of a virtual character. While the interior of the vocal tract is not always visible during the synthesis of speech or speech-like motion, the moments during which it is, and does not appear as expected, can disrupt an otherwise convincing visual experience. The necessity of accounting for articulatory animation in realistic AV speech synthesis is widely acknowledged; in fact, the MPEG-4 (part 2) standard includes articulatory movements among the "facial action parameters" [START_REF]MPEG-4 Facial Animation: The Standard, Implementation and Applications[END_REF].

Nowadays, facial animation and full-body movements of virtual characters are commonly driven by motion data captured from human actors, using vertex and skeletal animation techniques, respectively [START_REF]Handbook of Virtual Humans[END_REF]. However, conventional motion capture approaches (which rely on optical tracking) cannot be directly applied to intraoral articulatory movements during speech, since the tongue and teeth are not fully visible. This practical restriction may account for the fact that many state-of-the-art AV synthesizers suffer from a lack of realistic animation for the tongue (and to a lesser extent, the teeth). Some systems use simple rules to animate the articulators, others omit them altogether [START_REF]Data-Driven 3D Facial Animation[END_REF].

Meanwhile, speech scientists have a number of medical imaging modalities at their disposal to capture hidden articulatory motion during speech, including realtime magnetic resonance imaging (MRI), ultrasound tongue imaging (UTI), and electromagnetic articulography (EMA). Such techniques are commonly used to visualize the articulatory motion of human speakers. Indeed, the resulting data has been applied to articulatory animation for AV speech synthesis [START_REF] Cohen | Modeling coarticulation in synthetic visual speech[END_REF][START_REF] Pelachaud | Modeling and animating the human tongue during speech production[END_REF][START_REF] King | A 3D parametric tongue model for animated speech[END_REF][START_REF] Engwall | Combining MRI, EMA & EPG measurements in a three-dimensional tongue model[END_REF]; using motioncapture data to animate such models can lead to significant improvements over rule-based animation [START_REF] Engwall | Real vs. rule-generated tongue movements as an audio-visual speech perception support[END_REF]. However, these synthesizers are generally focused towards clinical applications such as speech therapy or biomechanical simulation.

While the lips can be animated using optical tracking and the teeth and jaw are rigid bodies, the tongue is more complex to model, since its anatomical structure makes it highly flexible and deformable. With the exception of [START_REF] Pelachaud | Modeling and animating the human tongue during speech production[END_REF], the majority of previous work has modeled the tongue based on static shapes (obtained from MRI) and statistical parametric approaches to deforming them by vertex animation [START_REF] Engwall | Combining MRI, EMA & EPG measurements in a three-dimensional tongue model[END_REF] or computationally expensive finite element modeling (FEM) [START_REF] Stone | Modeling the motion of the internal tongue from tagged cine-MRI images[END_REF][START_REF] Dang | Construction and control of a physiological articulatory model[END_REF][START_REF] Vogt | Efficient 3D finite element modeling of a muscle-activated tongue[END_REF]. Moreover, the articulatory models used are generally specific to the synthesizer software, and cannot easily be separated for reuse in other AV synthesizers.

In this paper, we present a first glimpse at Artimate, a novel framework for three-dimensional (3D) vocal tract animation driven by articulatory data. This framework is designed to serve as a component ("middleware") for an AV speech synthesis platform, providing animation of the tongue and teeth of a computer-generated virtual character, synchronized with the character's facial animation. It differs from previous approaches to articulatory animation synthesis in that it combines motioncapture data from EMA with skeletal animation to generate a self-contained, animated model.

Instead of implementing its own low-level processing, Artimate leverages existing software for modeling and animation, focusing on kinematics for efficiency and open standards for portability. It provides a lightweight, cross-platform component that is specifically designed to be incorporated as a resource into a given AV synthesizer.

In this way, we hope to bridge the gap between research prototypes for articulatory animation synthesis and the wide range of potential applications which would benefit from more realistic articulatory animation for virtual characters.

Implementation

The Artimate framework has a modular design, which reflects the different aspects of its approach. The main modules are (a) the model compiler, (b) the synthesizer core, and (c) a demo application which illustrates how Artimate can be used. These modules are described in this section.

The basic design of Artimate is essentially comparable to that of [START_REF] Balcı | Xface open source project and SMIL-agent scripting language for creating and animating embodied conversational agents[END_REF], but while the latter provides only facial animation, Artimate focuses on the animation of the tongue and teeth, and features a platform-independent implementation.

The framework is managed using Apache Maven [START_REF] Sonatype | Maven: The Definitive Guide[END_REF],

Figure 1: Static model of the tongue and teeth, obtained from a stock 3D model website [START_REF] Bitmapworld | Free gums 3d model[END_REF].

which covers the dependencies, installation, and integration, as well as application programming interface (API) and user documentation and distribution.

Model compiler

The model compiler provides a template ("archetype" in Maven's terminology) which is used to generate an animated 3D model from appropriate source data. The user bootstraps the actual compiler from this template, provides speech data in the form of motion capture data obtained from a human speaker using EMA, and automatically compiles it into an animated model which can be used as a resource by downstream components. By default, a static 3D model of the tongue and teeth (Figure 1) is rigged, which was obtained from a stock 3D model website [START_REF] Bitmapworld | Free gums 3d model[END_REF] under a royalty-free license. However, the user can provide and configure a custom model to be rigged instead.

The actual processing in this module is performed by automatically generating and running a custom rigging and animation script, which is processed in Blender [START_REF] Blender | [END_REF], an open-source, 3D modeling and animation suite featuring a Python API. The resulting animated model is exported in the open, industrystandard interchange format COLLADA [START_REF] Barnes | COLLADA -Digital Asset Schema Release 1.5.0[END_REF] and bundled as a Java Archive (.jar file), which can be used as a dependency by an AV synthesizer.

EMA data

Artimate has been developed for use with EMA data obtained with a Carstens AG500 Articulograph [START_REF] Hoole | Five-dimensional articulography[END_REF], which provides up to 12 markers (referred to as receiver "coils" due to their technical composition) sampled at 200 Hz; each coil has 5 degrees of freedom (DOF): the location within the measurement volume (in Cartesian coordinates), and the coil axis orientation (two Euler angles).

Three reference coils are used to normalize for head movement of the speaker, and one is used to track jaw movements, which leaves up to eight coils available to track points (and tangent vectors) on the tongue surface. The three reference markers are constrained in Blender to lock the others into the scene, irrespective of any external normalization of measured marker positions (e.g., using the Carstens NormPos software).

For development, we used a small subset of an existing cor- pus of EMA data, featuring seven tongue coils; three along the mid-sagittal (tongue tip, mid, back center), and two on either side (tongue blade and mid left and right, respectively). This layout is shown in Figure 2.

Rigging

The static 3D model of the tongue and teeth [START_REF] Bitmapworld | Free gums 3d model[END_REF] is rigged with a pseudo-skeleton, or "armature" in Blender's terminology, which controls the deformation of the tongue mesh and jaw movements in a standard skeletal animation paradigm, with deformable bones. The model is configured with several points approximating the layout of the coils in the EMA data. These points act as seeds for the automatic rigging and animation targets; the relative movements of the EMA coils are transferred to the targets as animation data, without the need for registration. The tongue armature is then assembled using a simple directed graph (Figure 3) encoded by a GraphViz [START_REF] Gansner | An open graph visualization system and its applications to software engineering[END_REF] .dot file, which defines the structure of the tongue's armature; the components' rest position is determined by the animation targets' initial positions (Figure 4). The vertices of the tongue model mesh are then grouped and automatically assigned a weight map which determines the armature components' influence on the position of each vertex.

The armature is constrained to track the animation targets using inverse kinematics (IK) [START_REF] Schutter | Constraint-based task specification and estimation for sensor-based robot systems in the presence of geometric uncertainty[END_REF], while maintaining its volume during stretching and compression. Additional constraints can be added to the animation targets to smooth coil jitter and counter any measurement errors in the original EMA data; otherwise, any such errors will translate directly into noticeably unrealistic movements of the tongue model.

Finally, the result is exported as a self-contained, animated 3D model, which includes the articulatory movements of the human speaker, but is independent of the original EMA data. Several poses of the tongue model, taken directly from the animation data, are rendered in Figure 5.

Speech synthesis resources

The animation is currently generated in a single timeline, represented by the consecutive acquisition sweeps of the EMA data. If a phonetic annotation based on the acoustic segments spoken is available, corresponding temporal markers will be created in Blender, and it would be possible to restructure the animation as corresponding "actions" in a non-linear animation (NLA) paradigm. However, to maintain portability with external software, the linear animation timeline is used in the COL-LADA model. Nevertheless, the segmentation file is bundled into the generated resource, so that the animation can be split downstream for NLA synthesis.

The acoustic signals from the EMA recording session can be included as well, in case the audio is required externally.

Validation

In addition to generating the animated model in COLLADA format, the animated model is also saved as a .blend file for interactive exploration and debugging in Blender. Moreover, the positions of the EMA coils, IK targets, and vertices on the tongue mesh can be dumped to .pos files compatible with the Carstens EMA format, which can be used to externally visualize and evaluate the generated animation.

This permits validation of the generated articulatory trajectories by direct comparison with the source EMA data.

Core library

The resource generated by the model compiler includes the animation data derived from EMA, and optionally, segmentation and/or audio. This can be directly included into an AV synthesizer for articulatory animation of virtual characters. However, in most cases, unless a very low-level, direct control of the articulatory model is desired, it will be preferable to wrap the articulatory animation in a separate library, which exposes a public API for articulatory animation synthesis and handles the direct control of the model internally. This is the purpose of Artimate's core library module.

The core library, which is implemented in Java, serves as the interface between the external AV synthesizer and the articulatory model. Using a 3D engine wrapper, it handles loading the articulatory model and provides a lightweight, unit-selection synthesizer dedicated to articulatory animation synthesis. Animation units are selected from the database of available animation units provided by the animated model's segmentation, using an extensible cost function based on the requested segment durations and smoothness of concatenation.

This core library is simple enough to be extended or ported as needed, depending on external requirements.

Demo application

To test the practicality of reusing the Artimate-compiled resource and core library, a demo application was developed, which also serves to illustrate the integration of the framework into an independent platform. This demo application is implemented in Java and consists of a simple graphical user interface, which displays an interactive 3D view of the animated model.

The demo application makes use of Ardor3D [20], one of a family of open-source, Java 3D game engines, which features mature COLLADA support. The core library is used to select animation units from the model, which are rendered by Ardor3D's internal animation system.

In addition to directly requesting specific units from the animation database, the demo application can also generate speech-synchronized animation from text via the multilingual text-to-speech (TTS) platform MARY [START_REF] Schröder | Multilingual MARY TTS participation in the Blizzard Challenge 2009[END_REF].

Conclusions

We have presented a modular framework for building and deploying a kinematic model for articulatory animation in AV speech synthesis. The animation is driven by motion capture in the form of EMA data and uses a skeletal animation approach with a deformable armature for the tongue. Models compiled in this manner can be reused as a self-contained resource by external applications, either with the core library as an interface, or directly accessing the model's animation data.

The Artimate framework will become publicly available under an open-source license in the first half of 2012, hosted at http://artimate.gforge.inria.fr/.

Future work includes extending model compiler support to other EMA data formats, such as those produced by alternative processing software (e.g., TAPADM [START_REF] Zierdt | Three-dimensional Artikulographic Position and Align Determination with MATLAB[END_REF]) or acquired with other Carstens or NDI device models.

Furthermore, we will investigate using Artimate to build an articulatory animation model from a much larger corpus, such as the mngu0 articulatory database, whose abundant EMA data is complemented by volumetric MRI scans that could be used to extract one or more static vocal tract models [START_REF] Richmond | Announcing the electromagnetic articulography (day 1) subset of the mngu0 articulatory corpus[END_REF][START_REF] Steiner | The magnetic resonance imaging subset of the mngu0 articulatory corpus[END_REF].

Acknowledgements

This publication was funded by the French National Research Agency (ANR -ViSAC -Project N. ANR-08-JCJC-0080-01).

Figure 2 :Figure 3 :

 23 Figure 2: EMA coil layout rendered in the transverse plane (major units in cm). The tongue coils are tip center (TTipC); blade left (TBladeL) and right (TBladeR); mid center (TMidC), left (TMidL) and right (TMidR); back center (TBackC). The absolute coil orientations (rendered as spikes) depend on their attachment in the EMA recording session.

Figure 4 :

 4 Figure 4: Rigged articulatory model with tongue in bind pose. The maxilla and upper teeth are displayed as wireframe meshes (in light and dark green, respectively), revealing the tongue, mandible, and lower teeth. The tongue armature is superimposed in gray, posed to meet the IK targets (not shown).

Figure 5 :

 5 Figure 5: The animated tongue model, posed according to several EMA data frames: a bunched configuration (left), a grooved tongue blade (center), and during an apical gesture (right). The asymmetry of the EMA data has been preserved. Note that the jaw opening has been exaggerated to improve visibility of the tongue surface.


Christelle L Douay

Thierry M Faure

Secondary

Secondary instability of centrifugal vortices in a cavity shear flow

Secondary instability of centrifugal vortices in a cavity shear flow

Ch. Douay * † , Th. Faure

ABSTRACT

We are interested in the development of centrifugal instabilities in an open cavity flow. Several works deal with cavity flows but very often these studies concern the shear layer and the acoustic of that two dimensional instability. We present here the development of a three dimensional instability, resulting in Taylor-Görtler-like vortices which are created by the curvature of the recirculation and its confinement by the walls. This primary instability manifests itself in the form of pairs of counter-rotating tori and the dynamic of these structures depends on experimental parameters -Reynolds number based on the cavity depth and the aspect ratio. For distinctive values of parameters, that primary instability exhibits a spanwise drift. A parametric study has been realized with flow visualization [START_REF] Faure | Three-dimensional centrifugal instabilities development inside a parallelepipedic open cavity of various shape[END_REF] and revealed, for different parameters, a secondary instability. In flow visualizations, that instability shows axial movement out of phase between vortices of the same pair. We want to characterize quantitatively this secondary instability. For that purpose, we study a case with the secondary instability and its spanwise drift.

In order to detect the vortices and to characterize their behaviors, Particle Image Velocimetry (PIV) measurements have been carried out. PIV fields are obtained with an optical flow algorithm [START_REF] Quénot | Particle image velocimetry with optical flow[END_REF]. This algorithm calculates continuous and differentiable fields so it is adapted for flows with strong velocity gradient. The study is realized in a plane parallel to the background of the cavity and located 30% below the upstream edge in order to visualize the vortices. A criterion for detecting structures in movEment, named Γ 2 , has been used and applied to the PIV fields and allows to track a vortex over time providing the vortices spanwise drift and size. The results show, for the configuration with secondary instability, a significant acceleration of the spanwise drift for the same Reynolds number. The size modifications and velocity modulation of these vortices are also identified. To understand the spanwise drift, the contribution of the Taylor-Görtler vortices on the velocity fields is studied. This work permits to characterize the modulation generated by the secondary instability on the velocity modulation provided by the vortices. This contribution is obtained from subtraction of the averaged velocity from instantaneous fields. The result highlights the link between that velocity modulation which is synchronized with the evolution of the spanwise drift but also with the fluctuation of the size of the vortex. We present the parameters which lead to the secondary instability and the features of this one.


Keywords: Rotating disk flows, cylindrical coordinate singularity, pseudo-spectral methods

The present work proposes a novel method to treat the cylindrical coordinate singularity which arises when dealing with three-dimensional confined rotating flows. In this work, we have developed a method which consists in discretizing the whole diameter -R ≤ r ≤ R with an even number of radial Gauss-Lobatto nodes. In the azimuthal direction, the overlap in the discretization and especially the use of parity conditions are avoided by introducing a shift equal to π/2K (K the number of mesh points in that direction) for θ > π in the Fourier transform. Spectral convergence of the method is illustrated on steady and unsteady analytical solutions. The ability of our numerical method to investigate complex unsteady flows is then illustrated for two rotating flow problems where other reliable experimental or numerical results are available in the literature.

Résumé :

Le présent travail propose une méthode pour traiter la singularité en coordonnées cylindriques qui se pose dans le traitement des écoulements 3D confinés en rotation. La présente méthode consiste à discrétiser sur tout le diamètre -R ≤ r ≤ R avec un nombre pair de noeuds dans la direction radiale. Dans la direction azimutale, le chevauchement des points de collocation et l'utilisation de conditions de parité sont évités en introduisant un décalage égal à π/2K (K, le nombre de points de maillage dans cette direction) pour θ > π dans la transformée de Fourier. La convergence spectrale de la méthode est illustrée sur les solutions analytiques stationnaires et instationnaires. La capacité de la méthode numérique pour étudier les écoulements instationnaires complexes est illustrée sur deux configurations pour lesquelles des résultats expérimentaux ou numériques fiables sont disponibles dans la littérature.

Introduction

When simulating flows in cylindrical configurations such as pipe flows or rotating cavity flows, the main difficulty arises from the singularity, which appears on the axis. More generally, the singularity at the centerline of a cylindrical coordinate system is due to the presence of terms 1/r n (n = 1, 2) in the Navier-Stokes equations governing the flow, where r is the radial distance. In the same time, the flow field itself does not have any singularity on the axis. Several numerical methods have been proposed in the literature [START_REF] Boyd | Chebyshev and Fourier Spectral Methods[END_REF] to overcome the singularities of the equations in cylindrical coordinates. Apparently these approaches depend greatly on whether a pseudo-spectral, finite-volume or finitedifference method is used, but in fact there are some similarities between them. Different methods have been developed in the past to avoid this singularity. Mercader et al. [START_REF] Mercader | An efficient spectral code for incompressible flows in cylindrical geometries[END_REF] proposed recently an efficient spectral code used to study the Rayleigh-Bénard convection in a vertical stationary or rotating cylinder. To overcome the singularity on the axis they considered radial expansions in the diameter of the cell in order to avoid clustering about the axis, with a selected number of points to ensure that the origin is not a collocation point. Moreover, they imposed implicitly some regularity conditions at r = 0 by forcing the proper parity of the Fourier expansions in the radial direction. This approach is named after Boyd [START_REF] Boyd | Chebyshev and Fourier Spectral Methods[END_REF] as unshifted Chebyshev polynomials of appropriate parity. The main advantages of their method are that it forces the minimal number of regularity conditions and that it reduces the size of the matrices. In this paper, we present an efficient and accurate pseudospectral method to solve the time-dependent three-dimensional Navier-Stokes equations for some rotating flow arrangements. The adopted approach is based on the previous work of Heinrichs [START_REF] Heinrichs | Spectral collocation schemes on the unit disc[END_REF], who proposed two spectral collocation schemes to solve the Poisson problem on an unit disk. It consists first of mapping the domain [0, 2π] × [0, R] into [0, 2π] × [-R, R] and using an even number of mesh points in the radial direction as also prescribed by Mohseni and Colonius [START_REF] Mohseni | Numerical treatment of polar coordinate singularities[END_REF]. It ensures that no extra pole conditions have to be specified at the axis. An angular shift equal to π/2K (K the number of mesh points in that direction) for θ > π in the discretization of angular direction is made to avoid an overlap of points and especially the use of parity conditions. To avoid the clustering of nodes around the rotation axis, a distribution of Gauss-Lobatto points is used. The flow is indeed laminar close to the rotation axis in most of the rotating flow applications, which does not require very thin meshes. The paper is organized as follows: the numerical approach is presented in Section 2 together with the validation of the method against analytical steady and unsteady solutions in Section 3. To show the capability of the present method to simulate axisymmetric and non axisymmetric flows, it is applied in Section 4 to the study of the vortex breakdown phenomenon in a cylindrical cavity considering different configurations such as the first bifurcation in a rotor-stator cavity. Finally some concluding remarks and future views are provided in Section 5.

Numerical approach

The motion is governed by the incompressible three dimensional Navier-Stokes equations written in the velocity pressure formulation, together with the continuity equation and appropriate boundary conditions. A cylindrical polar coordinate system (r, θ, z) is used. The components of the velocity vector V are denoted u, v and w in the (r, θ, z) directions respectively and P is the pressure. The scales for the dimensionless variables of space, time and velocity are h, Ω -1 and ΩR respectively, with h the interdisk spacing, R the outer radius of the rotating disk and Ω its rotation rate. The momentum equation, in dimensionless form, becomes:

∂V ∂t + (V.∇)V = -∇P + 1 Re ∆V + F in D (1)
where F represents a given body force. The continuity equation is given by

∇.V = 0 in D = D ∪ Γ.
The appropriate Dirichlet boundary conditions for the velocity vector write V = W on Γ = ∂D.

We have considered two geometrical configurations for which the main flow depends on two nondimensional parameters: the rotational Reynolds number Re and the aspect ratio G of the cavity defined as Re = ΩR 2 /ν and G = D/H, where D and H are respectively the diameter and the height of the cavity, Ω the rotation rate of the rotating disk and ν the kinematic viscosity of the fluid. The singularity at the junction of the stationary cylinder with the rotor is treated appropriately and the azimuthal velocity component has been regularized by employing a boundary layer function, v = e ±(z-1)/µ , with µ = 0.006 as described by Serre and Bontoux [START_REF] Serre | Vortex breakdown in a three-dimensional swirling flow[END_REF]. The numerical solution is based on a pseudo-spectral method with collocation Chebyshev polynomials in the radial r and axial z directions and Fourier collocation in the periodic azimuthal direction θ. The dimensionless space variables (r, z) have been normalized on [-1, 1], a requisite for the use of Chebyshev polynomials. The normalized variables (r, z) satisfy the following relations r = rG and z = z. The calculation domain is defined as

D = (r, θ, z) ∈ [-1, 1] × [0, 2π[×[-1, 1].
In the two non-periodic directions, a Gauss-Lobatto distribution for the collocation points is used.

With respect to the azimuthal direction θ, the distribution of collocation points consists in the discretization of the whole diameter. To avoid an overlap of collocation points, in the azimuthal direction, and especially the use of parity conditions, a shift equal to π/2K for θ > π is introduced in the Fourier transform. For the derivatives in the azimuthal direction θ, the Fourier basis writes:

The Navier-Stokes equations are discretized in time by using a semi-implicit scheme that combines an implicit second order Euler scheme for the diffusive terms and an explicit Adam-Bashforth scheme for the convective non-linear terms. The solution method is based on an efficient projection scheme and is a modification of the projection method initially proposed by Goda as described in [START_REF] Raspo | A spectral projection method for the simulation of complex three-dimensional rotating flows[END_REF]. For the computation of the non-linear terms, the derivatives in each space direction are calculated in the spectral space and the products are performed in the physical one. The connection between the two spaces is made through the use of a Fast Fourier Transform algorithm for the radial and axial directions and through a matrix multiplication in the azimuthal direction. For the diffusion terms the derivatives are performed in physical space using a simple matrix multiplication. Finally, all the resulting Helmholtz and Poisson problems are solved using a full diagonalization technique for each Fourier mode.

Validation against analytical solutions

The accuracy of the method is checked on the exact steady and time-dependent solutions defined in

D = [-1, 1] × [0, 2π[×[-1, 1]
. The results presented here have been obtained for G = 10 and Re = 250. The space accuracy of the method is evaluated by computing the discrete errors at the inner collocation points Er, see [START_REF] Raspo | A spectral projection method for the simulation of complex three-dimensional rotating flows[END_REF] for more details. In the steady case, the time step used is δt = 5 × 10 -3 and the values of the error for the velocity component u and for the pressure P are represented in figures 1(a) and 1(b) respectively. An exponential decay with the number of mesh points is obtained, which is characteristic of spectral methods. Figure 1(c) represents the values of the error for the radial velocity component u and for the pression P considering the time-dependent solution. The time step was decreased from δt = 5 × 10 -3 to δt = 5 × 10 -5 and the polynomial degrees are fixed to 40 for each space direction. The results are in line with the results obtained by Raspo et al [START_REF] Raspo | A spectral projection method for the simulation of complex three-dimensional rotating flows[END_REF], showing that the code is second-order accurate in time O(δt 2 ) for the radial velocity component and for the pressure inside the computational domain. The same temporal behavior was observed for the azimuthal and axial velocity components.

Results

The results obtained are compared to other reliable data available in the literature in two cases: the vortex breakdown phenomenon in a stationary cylinder driven by a rotating top disk considered by Serre & Bontoux [START_REF] Serre | Vortex breakdown in a three-dimensional swirling flow[END_REF] using DNS for different aspect ratios and the non axisymmetric flow in a rotor-stator cavity at a transitional Reynolds number [START_REF] Poncet | Revisiting the two first instabilities of the flow in an annular rotor-stator cavity[END_REF][START_REF] Schouveiler | Stability of a traveling roll system in a rotating disk flow[END_REF].

Vortex breakdown phenomenon in a cylinder with a rotating top disk

Vortex breakdown and transition to time-dependent regimes are initially investigated in a stationary cylinder with a rotating top disk. Two simulations are carried out. The first calculation for Re = 1850 and G = 1 is performed using a 126 × 8 × 126 mesh grid in the radial, azimuthal and axial directions respectively with a time step equal to 10 -2 . The second simulation was done using the same time step and grid arrangement but for G = 0.8 and Re = 2750.

In the first case, the base flow is laminar and axisymmetric. The iso-surface w = 0 of the axial velocity is presented in figure 2(a), showing that the flow is clearly axisymmetric. It highlights also the presence of an axisymmetric vortex breakdown along the rotation axis with two bubbles, which confirm previous experimental and numerical results (Fig. 2(b)). The present results are found in quite good agreement with the previous ones for the bubble characteristics defined in Fig. 2(c), even if the method is not optimized to simulate flow phenomenon close to the axis, where the grid is coarser. Concerning the case for Re = 2750 and G = 0.8, the flow gets unsteady as shown in Figure 3. The bubble is stretched by the shear along the axis and then is splitted into two bubbles, which finally merge, confirming the results described in [START_REF] Serre | Vortex breakdown in a three-dimensional swirling flow[END_REF]. We can deduce the time period of oscillations of the two bubbles. In the present case, the dimensionless time period is 36.2Ω -1 to be compared to the values 36Ω -1 of Stevens et al. [START_REF] Stevens | Oscillatory flow states in an enclosed cylinder with a rotating endwall[END_REF] for Re < 3500 and 55.6Ω -1 obtained by Serre & Bontoux [START_REF] Serre | Vortex breakdown in a three-dimensional swirling flow[END_REF]. Finally with the aim to show the capability of the method to compute non axisymmetric flows was made a third case for G = 0.5 and Re = 6500. A 126 × 48 × 65 mesh grid in the radial, azimuthal and axial directions respectively has been used together with a time step equal to 10 -2 . The results presented in Figure 4 confirms the expected three-dimensional nature of the flow. The surface identified as the breakdown zone (w = 0) (Fig. 4) has changed the bubble form having now a S-shaped form corresponding to the regime defined by Escudier [START_REF] Escudier | Observations of the flow produced in a cylindrical container by a rotating end wall[END_REF] as the precession regime of the lower breakdown structure. This behavior is due to the interaction of the vortex-breakdown with the structures from the vertical wall boundary layer as described in [START_REF] Serre | Three-dimensional swirling flow with a precessing vortex breakdown in a rotor-stator cylinder[END_REF]. In the present case, the cavity is cylindrical and the results obtained are in perfect agreement with Poncet et al. [START_REF] Poncet | Revisiting the two first instabilities of the flow in an annular rotor-stator cavity[END_REF] and in good agreement with the experimental results of Schouveiler et al. [START_REF] Schouveiler | Stability of a traveling roll system in a rotating disk flow[END_REF]. The circular rolls are here also observed only in a transient state with a dimensionless time period of 2.1Ω -1 after increasing the Reynolds number from 10000 to 20900 at t = 100 Ω -1 . The mode K = 17 is then perturbed with an amplitude of 5% at t = 160 Ω -1 to accelerate the transition to the spiral regime. 17 stationary spiral rolls are thus obtained with a dimensionless time period of 3Ω -1 . Schouveiler et al. [START_REF] Schouveiler | Stability of a traveling roll system in a rotating disk flow[END_REF] showed experimentally that the mode 18 is the most unstable. This weak discrepancy on the number of spiral arms can be explained by the fact that this instability is an Eckhaus instability. Thus, the number of arms is very sensitive to the time history of the flow and to the perturbations, which have been introduced experimentally or numerically. The present 17 spiral arms roll up in the sense of rotation of the rotor and form thus a positive angle with the tangential direction equal to ε = 25 • , which fully matches with [START_REF] Poncet | Revisiting the two first instabilities of the flow in an annular rotor-stator cavity[END_REF][START_REF] Schouveiler | Stability of a traveling roll system in a rotating disk flow[END_REF].

20 ème Congrès Français de Mécanique Besançon, 29 août au 2 septembre 2011

Conclusions

A novel method has been developed to take into account the axis singularity, which appears when using cylindrical coordinates within a rotating cavity. This method ensures that no extra pole conditions have to be specified at the axis. It has been first validated against analytical steady and unsteady solutions, while preserving the spectral accuracy. Then two different configurations have been used to test the method: the vortex breakdown phenomenon [START_REF] Serre | Vortex breakdown in a three-dimensional swirling flow[END_REF][START_REF] Serre | Three-dimensional swirling flow with a precessing vortex breakdown in a rotor-stator cylinder[END_REF] and the two first bifurcations in a rotor-stator cavity [START_REF] Poncet | Revisiting the two first instabilities of the flow in an annular rotor-stator cavity[END_REF][START_REF] Schouveiler | Stability of a traveling roll system in a rotating disk flow[END_REF]. For these two cases, the main characteristics of the mean flow and the instabilities have been favorably compared with the previous works, showing the capability of the method to predict with accuracy both axisymmetric and non-axisymmetric, steady and unsteady flows. The code has been recently extended for the simulation of turbulent flows using a "no-model" LES approach (Séverac & Serre [START_REF] Séverac | A spectral vanishing viscosity for the LES of turbulent flows within rotating cavities[END_REF]). Some calculations are in progress in the rotor-stator cavity considered by Craft et al. [START_REF] Craft | Some swirling flow challenges for turbulent CFD flow[END_REF] to highlight the presence of 3D large scale vortices embedded in the turbulent flow. It has been shown recently (private communication) that those patterns are very sensitive to the presence or not of a hub rotating with the rotor. They have been indeed observed only in the cylindrical case and not in the annular one. The new method is then absolutely necessary and appears moreover particularly well adapted to simulate such flows as the patterns are located far from the rotation axis, where the grid is coarser.

Figure 1 :

 1 Figure 1: Evolution of the error, log 10 (Er), for the velocity component u and for the pressure P versus the polynomial degrees N = M = K (a,b) and Er versus time step (c).

Figure 2 :

 2 Figure 2: Axial velocity w for Re = 1850 and G = 1: (a) iso-contours at different axial positions with the iso-values w = 0; (b) comparison of the bubble characteristic lengths; (c) schematic representation of the bubble characteristic lengths.

Figure 3 :

 3 Figure 3: Iso-surfaces of w = 0 and iso-contours of w in a (r, z) plane for Re = 2750 and G = 0.8.

Figure 4 :

 4 Figure 4: Iso-surface (w = 0) and iso-lines of the axial component of velocity w for G = 0.5 and Re = 6500: (a) t = 2100 Ω -1 , (b) t = 2292 Ω -1 and (c) t = 2484 Ω -1 .

Figure 5 :

 5 Figure 5: Axial velocity w: space-time diagram (z = -0.773 and r = 0.809); (r, θ) planes in the stator boundary(z = -0.773) layer for G = 17.45: (t = 100 Ω -1 , Re = 10000), (t = 110 Ω -1 , Re = 20900), (t = 200 Ω -1 , Re = 20900).Schouveiler et al.[START_REF] Schouveiler | Stability of a traveling roll system in a rotating disk flow[END_REF] have shown experimentally that the flow remains laminar until the appearance of circular rolls (CR) in the stator boundary layers for Re ≥ 10500. These patterns coexist in the experiments with spiral rolls, denoted SR1, for Re ≥ 20000. These spirals are located at the periphery of the cavity, also along the stator side. Poncet et al.[START_REF] Poncet | Revisiting the two first instabilities of the flow in an annular rotor-stator cavity[END_REF] performed DNS results in an annular cavity and obtained only the spiral patterns for this range of parameters, the circular rolls being observed only in a transient state. In the present case, the cavity is cylindrical and the results obtained are in perfect agreement with Poncet et al.[START_REF] Poncet | Revisiting the two first instabilities of the flow in an annular rotor-stator cavity[END_REF] and in good agreement with the experimental results of Schouveiler et al.[START_REF] Schouveiler | Stability of a traveling roll system in a rotating disk flow[END_REF]. The circular rolls are here also observed only in a transient state with a dimensionless time period of 2.1Ω -1 after increasing the Reynolds number from 10000 to 20900 at t = 100 Ω -1 . The mode K = 17 is then perturbed with an amplitude of 5% at t = 160 Ω -1 to accelerate the transition to the spiral regime. 17 stationary spiral rolls are thus obtained with a dimensionless time period of 3Ω -1 . Schouveiler et al.[START_REF] Schouveiler | Stability of a traveling roll system in a rotating disk flow[END_REF] showed experimentally that the mode 18 is the most unstable. This weak discrepancy on the number of spiral arms can be explained by the fact that this instability is an Eckhaus instability. Thus, the number of arms is very sensitive to the time history of the flow and to the perturbations, which have been introduced experimentally or numerically. The present 17 spiral arms roll up in the sense of rotation of the rotor and form thus a positive angle with the tangential direction equal to ε = 25 • , which fully matches with[START_REF] Poncet | Revisiting the two first instabilities of the flow in an annular rotor-stator cavity[END_REF][START_REF] Schouveiler | Stability of a traveling roll system in a rotating disk flow[END_REF].

  Traitement numérique de la singularité à l'axe pour les écoulements 3D en cavités tournantes N. Peres, S. Poncet et E. Serre . Laboratoire de Mécanique, Modélisation et Procédés Propres (M2P2) IMT La Jetée, Technopôle de Château Gombert, 38 rue F. Joliot Curie 13451 Marseille, France


Mathieu Beau

Feynman Integral and one/two slits electrons diffraction : an analytic study

In this article we present an analytic solution of the famous problem of diffraction and interference of electrons through one and two slits (for simplicity, only the one-dimensional case is considered).

In addition to exact formulas, we exhibit various approximations of the electron distribution which facilitate the interpretation of the results. Our derivation is based on the Feynman path integral formula and this work could therefore also serve as an interesting pedagogical introduction to Feynman's formulation of quantum mechanics for university students dealing with the foundations of quantum mechanics.

I. INTRODUCTION

The quantum mechanical problem of diffraction and interference of massive particles is discussed, though without detailed formulas, by Feynman in his famous lecture notes. [START_REF] Feynman | The Feynman Lectures on Physics[END_REF] A more exact treatment, though still lacking in detail, is in his book with Hibbs. [START_REF] Feynman | Quantum Mechanics and Path Integrals[END_REF] It was first observed experimentally by Jönsson in 1961. [START_REF] Jönsson | Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten[END_REF] Moreover, there are also experiments for neutrons diffraction for single and double slit (see [START_REF] Zeilinger | Single-and double-slit diffraction of neutrons[END_REF] and the references therein) and in quantum optics about interference between photons, (e.g., [START_REF] Pfleegor | Interference of Independent Photon Beams[END_REF] ). The crucial point of this paper is to deal with different optical regimes: the usual Fraunhofer regime, and (less commonly taught to students) the Fresnel regime and intermediate regimes. Recall that these regimes depend on the distance between the slits and the screen, where the Fraunhofer regime corresponds to the case when the distance between the slits and the screen is infinite and the other regimes appear when this distance is finite, the intermediate and Fresnel regimes being distinguished by the value of the Fresnel number N F ≡ 2a 2 /λL, where 2a is the width of the slit, L is the distance between the screen and the slit and λ is the wave lenght of the electron. Thus, the purpose of this article is firstly to present the theory of the slit experiment using the Feynman path integral formulation of quantum mechanics, which may be of pedagogical interest as compared with the optical Young experiment (Section II, III, IV), secondly to give an analytical derivation of the final formulas for the intensity of the electron on the screen, and to analyse these formulas using some approximations based on the asymptotic behavior of the Fresnel functions [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] occurring in these expressions (Section V).

In particular we show how the physical parameters, especially the Fresnel number, affect the form of the diffraction and interference images. There exists some pedagogical papers about the multiple slit experiments (see e.g. [START_REF] Frabboni | Two and three slit electron interference and diffraction experiments[END_REF] for experimental and numerical discussions and, [START_REF] Barut | Path integrals and quantum interference[END_REF] for theoretical discussions using path integral formulation), but surprisingly the approximations obtained in the Section V has never been published.

II. FEYNMAN FORMULATION OF QUANTUM MECHANICS AND WHY IT COULD BE OF INTEREST FOR STUDENTS

Students are often surprised to learn that under certain physical conditions the experimental behavior of matter is wave-like. In fact, this can present a didactical obstacle in teaching the first course of quantum mechanics where the principle of wave-particle duality can appear mysterious, especially with electron diffraction experiments for one-and two slits. Despite the interesting historical ramifications, students often have many metaphysical questions which are not answered satisfactorily in introductory quantum mechanics courses.

A formal and complete solution of the electron diffraction problem based on Feynman's path integral approach may be helpful in this regard. It could demystify the diffraction experiment and clarify the principle of duality between wave and particle by analogy with wave optics. In addition, it could to be an interesting introduction to quantum mechanics via the Feynman approach based on the notion of path integral rather than the Schrödinger approach, highlighting the analogy between quantum mechanics and optics. Moreover, there is an interesting article that has recently been published in the European Journal of Physics Education that discusses using the Feynman path integral approach in secondary schools to teach the double slit experiment. [START_REF] Fanaro | A Didactic Proposed for Teaching the Concepts of Electrons and Light in Secondary School Using Feynmans Path Sum Method[END_REF] We begin by outlining Feynman's formulation of quantum mechanics. We recall some of the fundamental equations before studying the electron diffraction problem. For more detail and for historical remarks about this theory, see Ref. [START_REF] Feynman | Quantum Mechanics and Path Integrals[END_REF] , 10 , 11 . We will give an equivalent formulation to that based on Schrödinger's equation, but which is related to the Lagrangian formulation of classical mechanics rather than the Hamiltonian formulation. Consider a particle of mass m under the influence of an external potential V [x(t), t] where x(t) = (x [START_REF] Feynman | The Feynman Lectures on Physics[END_REF] (t), x [START_REF] Feynman | Quantum Mechanics and Path Integrals[END_REF] (t), .., x (d) (t)) is the (d-dimensional, d ≥ 1) coordinate of the particle at time t.

The Lagrangian of the particle has the simple form

L[x(t), ẋ(t), t] = m 2 ẋ(t) 2 -V [x(t), t],
where ẋ(t) = dx(t)/dt denotes the velocity of the particle at time t. If the particle is at position x i at time t i and at x f at time t f > t i then the action is given by

S[x f , t f ; x i , t i ] ≡ t f t i L[x(t), ẋ(t), t] dt.
In classical mechanics, the total variation of the action δS for small variations of paths at each point of a trajectory δx(t) is zero, which leads to the Euler-Lagrange equations well known to students of physics:

d dt ∂L[x(t), ẋ(t), t] ∂ ẋ(α) (t) - ∂L[x(t), ẋ(t), t] ∂x (α) (t) = 0, α = 1, .., d .
However, in quantum mechanics the Least Action Principle as described above is generally not true. Thus, the concept of classical trajectory is also no longer valid: the position as a function of time is no longer determined in a precise way. Instead, in quantum mechanics the dynamics only determines the probability for a particle to arrive at a position x f at time t f knowing it was at a given position x i at time t i < t f . In other words, knowing the state of the particle at time t i , denoted by |x i , t i , the question is: what is the probability of transition from the state |x i , t i to the final state |x f , t f ? This probability is given by the square of the modulus of the so-called amplitude denoted

x f , t f |x i , t i , i.e. | x f , t f |x i , t i | 2 .
The expression for the amplitude in Feynman's formulation, though equivalent, differs from Schrödinger's firstly because the Lagrangian formulation is more general (there is not necessarily a Hamiltonian) and secondly because it does not refer to waves. In Feynman's approach the amplitude of transition is given by an 'integral' over all possible trajectories of a phase whose argument is the action divided by Planck's constant h, symbolically,

K(x f , t f ; x i , t i ) = Dx(t)e iS[x(t)]/h , where K(x f , t f ; x i , t i ) ≡ x f , t f |x i , t i is the amplitude, where the action functional is S[x(t)] ≡ t f t i L[x(t), ẋ(t), t]dt,
and where Dx(t) represent the path measure to be interpreted as follows: the integral formula is a short-hand for a limit of multiple integrals:

K(x f , t f ; x i , t i ) ≡ lim n→∞ 1 (2iπhǫ/m) d/2 R d dx 1 (2iπhǫ/m) d/2 . . . R d dx n-1 (2iπhǫ/m) d/2 exp i hǫ n k=1 ( m(x k -x k-1 ) 2 2ǫ 2 -V (x k )) , (1)
where

ǫ = (t f -t i )/n, x 0 = x i et x n = x f .
To understand this formula, we recall some ideas from Feynman's thesis FIG. 1. Representation of paths between (x i , t i ) and (x f , t f ), for the continuous case at the left (integral over continuum paths, see (1)) and for the discrete case at the right (integral over discrete paths before the continuum limit, see ( 4)). In the left figure, the continuous line is the classical path, the dashed lines are two paths obtained by variations around the classical paths and the dotted line is an arbitrary path. In the picture at the right the discrete path for fixed n is shown together with the classical discrete path.

Then, if we divide the time interval [t i , t f ] into a sequence of small intervals t i = t 0 < t 1 < t 2 < . . . < t n-1 < t f = t n , where t k = t i + k(t f -t i )/n, the transition amplitudes are multiplied (because successive events are independent):

x f , t f |x i , t i = x f , t f |x n-1 , t n-1 . . . x 2 , t 2 |x 1 , t 1 x 1 , t 1 |x i , t i .
Integrating over all possible values of the intermediate positions x k leads to the formula (1), see Fig. 1. The intuition associated with this formula is as follows: we integrate the phase over all possible trajectories of the particle; unlike the classical case, the particle can follow paths that differ from the classical trajectory which minimizes the action. (Paths far away from the classical path usually make negligible contributions.) This image, though based on the classical image of a trajectory, illustrates the change in the mathematical description of the particle (wave-like behaviour) which can no longer be represented as a material point as its trajectory is not clearly defined with certainty.

The integral equation describing the evolution of the wave function Ψ(x, t) (i.e. the state |x, t ) over time, is given by

Ψ(x ′ , t ′ ) = R d K(x ′ , t ′ ; x, t)Ψ(x, t)dx . (2)
It can be shown that the 'wave function' Ψ(x, t) is also the solution of Schrödinger's equation:

- h2 2m ∆ x Ψ(x, t) + V (x, t)Ψ(x, t) = ih∂ t Ψ(x, t)
, where ∆ x is the d-dimensional Laplacian and ∂ t ≡ ∂/∂t. For a proof of the equivalence between both formulations, see 2 , 10 .

A useful example is the free particle amplitude calculation in one dimension. We simply replace the Lagrangian by the free Lagrangian, i.e. without potential, and use equation (1) to get :

K 0 (x ′ , t ′ ; x, t) = 1 2iπh(t ′ -t)/m e i m(x ′ -x) 2 2h(t ′ -t) . (3)
Let us derive formula (3) starting from the general formula (1) applied to a free particle in 1-dimension. We divide the integral, for fixed n, we have to integrate the phase e iS[x 0 ,..,xn]/h over the intermediate positions x j , j = 1, .., n -1 at times t j = jǫ, ǫ = (t f -t i )/n between

x 0 = x i and x n = x f , where S[x 0 , .., x n ] = m 2 n j=1 (x j -x j-1 ) 2 ǫ
, multiplied by the constant n j=1 m/2iπhǫ . We need to calculate, for a fixed number of subdivision n, the multiple integral:

K (n) 0 (x n , t n ; x 0 , t 0 ) ≡ 1 2iπhǫ/m ∞ -∞ dx 1 2iπhǫ/m . . . ∞ -∞ dx n-1 2iπhǫ/m exp i n j=1 ( m(x j -x j-1 ) 2 2hǫ ) . ( 4
)
We perform the integration one stage at a time (from j = 1 to j = n -1), using the well known formula for the convolution of two Gaussians. Given two Gaussians g σν (x)

≡ 1 √ 2πσ 2 ν e -x 2 2σ 2 ν , ν = 1, 2, we have: ∞ -∞ g σ 1 (x -x ′ )g σ 2 (x ′ ) = g √ σ 2 1 +σ 2 2 (x) . (5)
Hence, in (4), we can calculate the integral with respect to x 1 , changing variable to

x ′ 1 = x 1 -x 0 , obtaining: 1 2iπhǫ/m ∞ -∞ dx 1 2iπhǫ/m e i m(x 2 -x 1 ) 2 2hǫ e i m(x 1 -x 0 ) 2 2hǫ = 1 2iπh(2ǫ)/m e i m(x 2 -x 0 ) 2 2h(2ǫ)
.

Proceeding by recursion, we get (4) :

K (n) 0 (x n , t n ; x 0 , t 0 ) = 1 2iπh(nǫ)/m e i m(xn-x 0 ) 2 2h(nǫ)
, and hence (3) in the limit n → ∞.

Clearly the 3-dimensional propagator is a product of 1-dimensional propagators:

K (3D) 0 ( -→ x ′ , t ′ ; - → x , t) = K 0 (x ′ , t ′ ; x, t)K 0 (y ′ , t ′ ; y, t)K 0 (z ′ , t ′ ; z, t) = m 2iπh(t ′ -t) 3/2 e i m| - → x ′ -- → x | 2 2h(t ′ -t) , (6)
where -→ x = (x, y, z) is the position vector in 3-dimensions. Note that for reasons to be discussed below, we shall not make use of the 3-dimensional propagator in the example of election diffraction for one and two slits. We remark for completeness, that the propagator

(3) can be calculate by other methods, for example. [START_REF] Kleinert | Path Integrals in Quantum Mechanics[END_REF]

III. APPLICATION TO THE PROBLEM OF ELECTRON DIFFRACTION AND INTERFERENCE

Consider an electron source at (x, y, z) = (0, 0, 0), and two slits at z = D of width 2a and centered respectively at x = +b and x = -b, c.f. Fig. 2 . For the diffraction experiment with a single slit, just replace the system with a slit of size 2a centered at x = 0. At z = D + L is a screen on which electrons are recorded (or some other recording device). For more details about the experimental realization of the system, see Ref. [START_REF] Frabboni | Two and three slit electron interference and diffraction experiments[END_REF] Note that we neglect gravity for this problem. In addition, we assume that in the direction orthogonal to the plane in Fig. 2 (the x -z-plane), the slot is long enough to neglect diffraction effects; we consider only the horizontal (in-plane) deflection of the beam in order not to complicate the formulas.

Then we can reduce the dimension of the propagator ( 6) by integrating over y:

K (2D) 0 ( -→ r ′ , t ′ ; -→ r , t) = ∞ -∞ dy K (3D) 0 ( -→ x ′ , t ′ ; - → x , t) = m 2iπh(t ′ -t) e i m| - → r ′ -- → r | 2 2h(t ′ -t) , (7)
where -→ r = (x, z) is the position vector in the two dimensional plane orthogonal to the y-axis.

One can suggest different models for the slits given by distribution functions (e.g. Gaussian functions or 'door' functions). We focus on the more realistic model of 'door' functions (for the Gaussian function model, see Ref.

2 ):

χ [b-a,b+a] (w) =      0 w > b + a, w < b -a 1 b -a < w < b + a (8)
The question is: what is the probability of finding the electron at the point x on the screen knowing that it started at the point (x = 0, z = 0)? More precisely, suppose that the source emits electrons in large numbers, though small enough so that the distance between electrons is such that interactions can be neglected (no correlations). What is the intensity of electrons on the screen as a function of position x? The two questions are related since under these circumstances the motion of the electrons is independent (without mutual interaction,

given that the density of the beam is very low). Indeed, the intensity curve is obtained by simply multiplying the probability curve for an electron by the number of electrons emitted per unit time. As explained above formula (1), the probability is given by the square of the amplitude which we will compute using the propagator (6).

As explained in (7), we now take d = 2. In fact, we now argue that we can reduce the dimension further. For the z-direction, we really onght to use the two-dimensional propagator but we can reasonnably consider (as it is usually done 7 , 2 ) that the propagator is a product of two independant propagator in both directions x and z, where the last one is equal to a constant.

To see that, let us discuss the conceptual experiment for one slit. [START_REF] Feynman | Quantum Mechanics and Path Integrals[END_REF] We consider that the problem is divided in two separate motions, one starting from the source to the slit during the time T and the other one starting from the slit to the screen during the time τ .

Then, we would like to compute the probability amplitude for the electron from the source, at the intial position (x = 0, z = 0) at the time t = 0, to go to the screen, at the final position (x, z = D + L) at the time T + τ , knowing that it goes through the slit, at the intermediate position (w, z = D), b -a < w < b + a, at the time t = T . In fact, by the laws of quantum mechanics, there is no reasons to separate the motions in two independant parts, since we don't know where is the particle at the time T .In other words we don't know when the particle goes through the slit. Nevertheless, we can consider that this classical image is appropriated to study the problem. Indeed, the electron have in the z-direction the momentum p z = hk z (k z is the wave vector), which is related to the classical velocity

v z = D/T = L/τ
, where D is supposed to be very large compared to the dimensions in the x-direction, more precisely, x, a, b ≪ D, L. In addition we suppose that the wave lenght λ, which is approximatively equal to the z-direction wave lenght λ ≃ λ z = 2πh/(mv z ), is small compared to the distances λ ≪ D, L. Thus, the motion is approximatively classical in the z-direction and we can separate the problem as two independant motions. Notice that, quantum-mechanically, it is possible for the particle to go through the slit severals time before strike the screen, [START_REF] Yabuki | Feynman Path Integrals in the Young Double-Slit Experiment[END_REF] but that the probability of this event is relatively small. Now, let us compute the amplitude of the transition for the particle starting at the point (x = 0, z = 0) at the time t = 0, going through one slit at the position (w, z = D), b -a < w < b+a at the time t = T and arriving at the position (x, z = L+D) at the time t = T +τ :

K (2D) a,b ((x, L + D), T + τ ;(0, 0), 0) ≡ ∞ -∞ dw χ [b-a,b+a] (w) × K (2D) 0 ((x, L + D), T + τ ; (w, D), τ ) K (2D) 0 ((w, D), τ ; (0, 0), 0) .
Hence, the explicite formula is given by:

K (2D) a,b ((x, L + D),T + τ ; (0, 0), 0) = e i mD 2 2hT 2iπhT /m e i mL 2 2hτ 2iπhτ /m b+a b-a dw e i m(x-w) 2 2hτ 2iπhτ /m e i mw 2 2hT 2iπhT /m . (9)
Consequently, the two-dimensional propagator is the product of two independant onedimensional propagators in the x and in the z-directions, and since the propagator in zdirection is a constante (see the right hand sides of ( 9)):

K z (L + D, T + τ ; 0, 0) = e i mD 2 2hT 2iπhT /m e i mL 2 2hτ 2iπhτ /m , (10)
we can reduce in the one-dimension's x-direction the problem:

K x (x, T + τ ; 0, 0) = b+a b-a dw e i m(x-w) 2 2hτ 2iπhτ /m e i mw 2 2hT 2iπhT /m . ( 11
)
Then we can reduce the dimension of the problem keeping only the x-axis propagator, as you can see in the References 2 , 7 . In the Reference, 7 they consider the slits in the x-y-plane, in two-dimensions and not in one-dimension as we do in the present article.

If we take the limit a → ∞ (infinite slit), by (9) and by (5) we get: Then we find the amplitude corresponding to the transition between the position (x = 0, z = 0) at the time t = 0 and the position (x, z = L + D) at the time t = T + τ , knowing that the position in the z-direction is D at the time t = T : ((x, z), T + τ ; (0, 0), 0) , to get the result, we have used the convolution formula between two Gaussians (5).

K (2D) ∞ ((x, L + D), T + τ ; (0, 0), 0) = e i mD 2 2hT 2iπhT /m e i mL 2 2hτ 2iπhτ /m ∞ -∞ dw e i m(
K (2D) ∞ ((x, L + D), T + τ ;(0, 0), 0) = K 0,x (x, T + τ ; 0, 0) K 0,z (L + D, T + τ ; D, τ ) K 0,z (D, T ; 0, 0) , where K o,X (X ′ , t ′ ; X, t) = (m/2iπh(t ′ -t)) 1/2 exp (im(X ′ -X) 2 /(2h(t ′ -t))), X = x,

IV. THE EXACT RESULT IN TERMS OF A FRESNEL INTEGRAL

Now we want to compute the amplitudes A 1 (x; a, b), A 2 (x; a, b) at each point x on the screen, using the Feynman formulation, and then to add both amplitudes to get the total amplitude A(x; a, b) and finally to take the square of the modulus, obtaining the probability

P (x; a, b) ≡ |A(x; a, b)| 2 .
By (11), the formal expression for A 1 (x; a, b) is :

A 1 (x; a, b) = +∞ -∞ dw χ [b-a,b+a] (w)K 0 (x, T + τ ; w, T )K 0 (w, T ; 0, 0) , (12)
FIG. 2. Schematic depiction of the apparatus with the source, the two-slits and the screen. We have represented four differents paths going through the slits, from the source to the screen. The dashed lines represent three classical paths and the thin dashed line a curved path. The Feynman path Integral formulation allows one to compute the propagator summing over all different paths, see equations ( 1), ( 3), ( 12), ( 16), ( 17), ( 18).

where T is the travel time of the electron from the source to the slit and τ from the slit to the screen. Recall that to obtain formula (12), we used a similar argument to that which enabled us to write the formula (1), writing the integral over all possible paths as the product of independent amplitudes at successive times. However, in this case, at time T , we have to integrate over a finite interval (the slit), which results in the more complicated expression (12) rather than a Gaussian:

A 1 (x; a, b) = b+a b-a dw 1 2iπhτ /m e i m(x-w) 2 2hτ 1 2iπhT /m e i mw 2 2hT . ( 13
)
Notice that :

m(x -w) 2 2hτ + mw 2 2hT = ( m 2hτ + m 2hT )(y - x 1 + τ /T ) 2 + mx 2 2h(T + τ ) ,
and hence,

A 1 (x; a, b) = e i mx 2 2h(T +τ ) 2iπh(T + τ )/m b+a b-a dw T + τ 2iπhT τ /m exp i( T + τ 2hT τ /m )(w - x 1 + τ /T ) 2 . = e i mx 2 2h(T +τ ) (2i) 2 πh(T + τ )/m α (1) + (x) α (1) -(x) dw ′ exp iπ 2 w ′2 , (14)
where

α ± (x; a, b) ≡ (T + τ ) πhT τ /m (b ± a) - x πhτ /m T T + τ . (15)
In ( 14), we see that we have the integral of a Gaussian with complex argument. Decomposing the integral in real and imaginary parts, we get two integrals of cosine and sine functions respectively, with second degree polynomial arguments. These integrals are the well-known

Fresnel functions 6 :

C[u] ≡ u 0 dw cos ( πw 2 2 ) , S[u] ≡ u 0 dw sin ( πw 2 2
) .

Thus we obtain explicit analytical expressions for the amplitudes:

A 1 (x; a, b)= e i mx 2 2h(τ +T ) (2i) 2 πh(T + τ )/m × C[α + (x; a, b)] -C[α -(x; a, b)] + iS[α + (x; a, b)] -iS[α -(x; a, b)] , (16)
A 2 (x; a, b) = A 1 (x; a, -b) . (17)
For two slits, we can compute the total amplitude by summing the amplitudes for both slits:

A(x; a, b) = A 1 (x; a, b) + A 2 (x; a, b) . (18)

V. PHYSICAL PARAMETERS, APPROXIMATIONS AND INTERPRETATIONS

Recall that the length of the slits and the distance between them is small compared to the horizontal distances D and L, hence we can assume that the wave length of the electron λ = h/mv is approximatively given by h/mv z , where v z = L/τ = D/T . We will use this expression for the wave length in the following.

A. Diffraction by a single slit

We can write for the single-slit case (of size 2a), the analogue of the function defined by (15):

α(x; a) = N F (a) 1 + L/D 1 - x a 1 1 + L/D (19)
where N F (a) = 2a 2 /λL is the Fresnel number.

We can now easily compute the single-slit diffraction probability: esp. depending on whether it is greater or less than unity. To understand these differences explicitly, we will analyse the asymptotic behavior of these functions for different regimes of N F (a) using the known asymptotics of the Fresnel functions: (see [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] )

P (1Slit) (x; a) = |A 1 (x; a, b = 0)| 2 = 1 2λ(L + D) [C(α(x; a)) + C(α(x; -a))] 2 + [S(α(x; a)) + S(α(x; -a))] 2 . ( 20
C(±u) ≃ ± 1 2 + 1 πu sin πu 2 2 , u ≫ 1 , S(±u) ≃ ± 1 2 - 1 πu cos πu 2 2 , u ≫ 1 . (21)
If we have:

x aη -1 ≫ 1 N F (a)η ,
we get the asymptotic behavior for the functions defined by (19):

α(x; a) ≪ -1 and α(x; -a) ≫ 1 ⇔ ±α(x; ∓a) ≫ 1 , (22)
and so by ( 21) and ( 22), the asymptotics formulas of the Fresnel functions in (20) are given by:

C[α(x; ±a)] ≃ ± 1 2 + 1 πα(x; ±a) sin ( πα(x; ±a) 2 2 ), S[α(x; ±a)] ≃ ± 1 2 - 1 πα(x; ±a) sin ( πα(x; ±a) 2 2 ) . (23)
Then we get:

C[α(x; +a)] + C[α(x; -a)] ≃ 1 πα(x; a) sin ( πα(x; a) 2 2 ) + 1 πα(x; -a) sin ( πα(x; -a) 2 2 ), S[α(x; +a)] + S[α(x; -a)] ≃ -1 πα(x; a) cos ( πα(x; a) 2 2 ) - 1 πα(x; -a) cos ( πα(x; -a) 2 2 ) (24)
Applying the Fresnel function asymptotic forms (24) to (20) and using the definition (19), we deduce than if N F (a) ≪ 1 and if (x -aη)/aη ≫ 1/ N F (a)η ⇔ x -aη ≫ λL/2, we get the following asymptotic formula:

P (1Slit) (x; a) ≃ 2γ π 2 η 2 a 2 ( x 2 η 2 -a 2 ) 2 + 1 x 2 η 2 -a 2 sin 2 (πN F (a) x a ) , (25)
Moreover, since N F (a) ≪ 1 and so x aη ≫ 1, we have another asymptotic form (large distance on the screen):

P (1Slit) (x; a) ≃ 2γ π 2 x 2 sin 2 (πN F (a) x a ), (N F (a) ≪ 1, x/aη ≫ 1) , (26)
In this case we are in the so-called Fraunhofer regime analogous to plane wave diffraction in optics. [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF] In fact, notice that the distance between fringes is a/N F (a) = λ z L/2a, c.f. Fig 3a.

Both approximations ( 25) and ( 26) are also valid if N F (a) is of the order of unity (this is the intermediate regime) provided that x ≫ aη. This means that the pattern on the screen far from the position of the first lobe is well approximated by equations ( 25) and (26), see Fig. 3b.

On the contrary, if N F (a) ≫ 1, we get different asymptotics given by:

P (1Slit) (x) ≃ γ η N F (a) 2a + sin ( π 2 N F (a)η(1 -x aη ) 2 ) 2π √ η(a -x η ) + sin ( π 2 N F (a)η(1 + x aη ) 2 ) 2π √ η(a + x η ) 2 + γ η N F (a) 2a - cos ( π 2 N F (a)η(1 -x aη ) 2 ) 2π √ η(a -x η ) - cos ( π 2 N F (a)η(1 + x aη ) 2 ) 2π √ η(a + x η ) 2 , |x| < aη, ( 27
)
P (1Slit) (x)≃ 2γ π 2 η 2 a 2 ( x 2 η 2 -a 2 ) 2 + 1 x 2 η 2 -a 2 sin 2 (πN F (a) x a ) , |x| > aη , (28)
since the asynptotic behavior of the functions (19) are:

α(x; ±a) ≫ 1, if N F (a) ≫
then by ( 29) and (31), we have:

C[α(x; +a)] + C[α(x; -a)] ≃ 1 + 1 πα(x; a) sin ( πα(x; a) 2 2 ) + 1 πα(x; -a) sin ( πα(x; -a) 2 2 ), S[α(x; +a)] + S[α(x; -a)] ≃ 1 - 1 πα(x; a) cos ( πα(x; a) 2 2 ) - 1 πα(x; -a) cos ( πα(x; -a) 2 2
) , then we get (27).

If |x| > aη, by ( 30) and ( 21), we get the same approximations as ( 23) and ( 24), then we obtain (28).

B. Comment about the probability interpretation

We can see that (20) has the physical dimension of the inverse of a length squared and so it is neither a probability nor a probability density. This apparent problem can, however, be seen to be a matter of interpretation by looking at the formula (2). Indeed, the probability density for the diffraction problem is given by |Ψ(x, T + τ )| 2 , where Ψ(x, T + τ ) is a normalized wave function, i.e. R 1 dx |Ψ(x, T + τ )| 2 = 1. Therefore, we must choose an initial wave function (at time t = 0) which is also normalized so that its square modulus describes the probability distribution of the electron in the x 0 plane. Essentially, what we have done in (20) is to take the initial wave function to be a delta-function, whereas it should be the initial probability distribution which is a delta-function, i.e. Ψ(x, 0) should be 'the square root of a delta-function'.

To make this clearer, consider for example a wave function at time t = 0 given by the square root of a Gaussian φ σ (x 0 ) = normalized so as to get the probability of the presence of the electron at the point x on the screen by taking the square of the modulus. Indeed, the wave function at time t = T , i.e.

1 (2πσ 2 ) 1/4 e -x 2 0 4σ 2 , so that R 1 dx 0 |φ σ (x 0 )| 2 =
at the position of the slits, is given by:

φ σ (x, T ) = R 1 dx 0 K 0 (x, T ; x 0 , 0)φ σ (x 0 ) , (32)
where K 0 (x, T + τ ; x 0 , 0) is the free propagator defined by the equation (3). Using the identity

R 1 K 0 (x, T ; x 0 , 0) * K 0 (x, T ; x ′ 0 , 0)dx = δ(x 0 -x ′ 0 ) (33)
we have that φ σ (x, T ) remains normalized, i.e. |φ σ (x, T )| 2 dx = 1.

The wave function at time t = T + τ is given by:

Ψ σ (x, T + τ ) = +a -a dyK 0 (x, T + τ ; y, T )φ σ (y, T ). (34)
Now, the quantity of interest is the conditional probability (density) for the electron to be at the point x on the screen at the time T + τ given that it was in the interval [-a, +a] at time T , i.e. given that it passed through the slit:

P σ (x, T + τ | y ∈ [-a, +a], T ) ≡ |Ψ σ (x, T + τ )| 2 +a -a dy |φ σ (y, T )| 2 . ( 35
)
after which we wish to take the limit σ → 0. Using the relation (33) one can see that the condition probability (35) is normalized so that this procedure just amounts to division by a normalization factor. Thus

R 1 dx |Ψ σ (x, T + τ )| 2 = +a -a dy|φ σ (y, T )| 2 . ( 36
)
To take the limit σ → 0, note that

φ σ (x 0 ) = (8πσ 2 ) 1/4 g σ √ 2 (x 0 ) where g σ √ 2 (x 0 ) = e -x ′ 2 0 4σ 2 √ 4πσ 2 is a normalized Gaussian of variance σ √ 2, and hence g σ √ 2 (x 0 ) → δ(x 0 ), σ → 0.
Multiplying top and bottom of (34) by (8πσ 2 ) 1/2 we thus have

lim σ→0 P σ (x ∈ A, T + τ |y ∈ [-a, +a], T ) = = lim σ→0 (8πσ 2 ) -1/2 |Ψ σ (x, T + τ )| 2 (8πσ 2 ) -1/2 +a -a dy|φ σ (y, T )| 2 , = dx 0 K(x, T + τ ; x 0 , 0)g σ √ 2 (x 0 ) 2 a -a dy dx 0 K 0 (y, T ; x 0 , 0)g σ √ 2 (x 0 ) 2 = |K(x, T + τ ; 0, 0)| 2 +a -a dy|K 0 (y, T ; 0, 0)| 2 = λL 2a P 1Slit (x; a) , (37)
where

K(x, T + τ ; 0, 0) = +a -a dy K 0 (x, T + τ ; y, T )K 0 (y, T ; 0, 0) (38)
is the propagator through the slit and P (1Slit) (x; a) = |K(x, T + τ ; 0, 0)| 2 is given by (20).

Note that this now has the correct dimension of an inverse length.

C. Interference and diffraction for two slits

Similarly, one can find the two-slit diffraction probability formula using ( 16), (17) :

P (2Slit) (x; a, b) = P 1 (x; a, b) + P 2 (x; a, b) + I 12 (x; a, b) , (39)
with the diffraction terms :

P 1 (x; a, b) = |A 1 (x; a, b)| 2 = γ 2λLη [C(α + (x; a, b)) -C(α -(x; a, b))] 2 + [S(α + (x; a, b)) -S(α -(x; a, b))] 2 , P 2 (x; a, b) = |A 2 (x; a, b)| 2 = P 1 (x; a, -b) , (40)
and the interference term :

I 12 (x; a, b) = A 1 (x; a, b)A 2 (x; a, b) * + A 2 (x; a, b)A 1 (x; a, b) * = γ λLη ([C(α + (x; a, b)) -C(α -(x; a, b))][C(α + (x; a, -b)) -C(α -(x; a, -b))] + [S(α + (x; a, b)) -S(α -(x; a, b))][S(α + (x; a, -b)) -S(α -(x; a, -b))]) . ( 41
)
Notice that there is an additional term compared to the single-slit case called the interference term, which is of course quite similar to that in optics. [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF] This results in a modulation effect of the curve given by (39) by the sum of the diffraction terms (40) (modulo a multiplicative factor), see Fig. 4.

Let us define the Fresnel numbers

N F (a) ≡ 2a 2 /λ z L, N F (b) ≡ 2b 2 /λ z L and N F ≡ 2ab/λ z L = N F (a)N F (b)/2.
Assume that the distance between the slits is large compared to the size of the slits b ≫ a.

In the experiment considered one fixes both parameters a and b and varies the distance between the screen and the slits (keeping the same value for η). Notice that because b ≫ a, N F ≫ 1 does not necessarily imply that N F (a) ≫ 1. Thus we will see that both parameters play different roles.

First, we establish the asymptotics of (39) for different asymptotic value of N F (a). Under the condition N F (a) ≪ 1 and at large scales |x -bη| ≫ aη et |x + bη| ≫ aη, we get similar expressions for P 1 (x; a, b) and P 2 (x; a, b) as (26). We have to compute the asymptotic expression for the interference term. This yields

P (2Slit) (x; a, b) ≃ 2γ π 2 (x -bη) 2 sin 2 πN F η(1 - x bη ) + 2γ π 2 (x + bη) 2 sin 2 πN F η(1 + x bη ) - γ π 2 (x 2 -b 2 η 2 ) cos (2π(N F + N F (a)) x a ) -cos (2πN F η(1 + x aη )) - γ π 2 (x 2 -b 2 η 2 ) cos (2π(N F -N F (a)) x a ) -cos (2πN F η(1 - x aη ))) . ( 42
)
One can observe that there are two phases: the separated phase (N F ≫ 1) and the mixed phase (N F ≪ 1). At the same time there are the Fresnel and Fraunhofer regimes depending on the values of N F (a) as explained above. The distinction between two phases is purely geometric and characterizes the separation respectively mixture of diffraction curves. Indeed, similar to optics, to observe the two diffraction curves separately, the fringe modulation λ z L/2a must be less than the distance between the origins of the two curves (being centered in ±bη) because otherwise both curves are mixed. Thus the criterion is written λ z L/2a < bη and therefore N F η > 1. This obviously unlike the case N F ≪ 1 where the two curves are combined (one added to the other) and where we observe modulation interference.

To see this more formally, consider firstly the case N F η ≪ 1. If |x| > λL/2a then |x| ≫ bη and we can give an approximation of (42). Indeed, the first two terms are approximately equal and contribute 4γ π 2 x 2 sin 2 ( 2πa λL x).

In the last two terms we develop the cosine functions and get

2 cos ( 2πb λL x) cos ( 2πa λL x) -cos ( 2πabη λL ) cos ( 2πb λL x) ≈ -4 cos( 2πb λL x) sin 2 ( 2πa λL x),
where we used cos ( 2πabη λL ) ≃ 1 because N F η ≪ 1. Adding the terms we obtain

P (2slit) (x; a, b) ≃ 8γ π 2 x 2 sin 2 ( 2πa λL x) cos 2 ( 2πb λL x) . (43)
This is the familiar optical formula: see Reference, 14 formula (10) Chap. VIII-6. It shows that the diffraction curves are modulated by interference fringes. The distance between two interference fringes is of the order of λ z L/2b whereas that between minima of the diffraction curves is of the order of λ z L/2a ≫ λ z L/2b, see Fig. 4a, Fig. 4b (far from the first lobe) and Fig. 4c.

Secondly, consider the case that N F η ≫ 1 (while still assuming N F (a) ≪ 1). If x -bη > -λL/2a (or x + bη < λL/2a) then (x -bη) -2 ≫ (x + bη) -2 (respectively (x -bη) -2 ≪ (x + bη) -2 ) so one of the two terms is negligible in the respective domain. Moreover, in both cases, the interference term is small compared to the diffraction term since (x -bη) -2 ≫ (x 2 -b 2 η 2 ) -1 . The total probability is therefore approximatively equal to a sum of the two diffraction curves centered at ±bη modulated by an interference term which oscillates rapidly with a relatively small amplitude, c.f. Fig 4c :

P (2slit) (x; a, b) ≃ 2γ π 2 (x ∓ bη) 2 sin 2 πN F η(1 ∓ x bη ) + O( 1 x 2 -b 2 η 2 ), |x ± bη| > λL 2a . (44)
Now, consider N F (a) ≫ 1. Since, unlike the previous cases, we do not need special conditions for the position x on the screen, we find similar formulas to ( 27) and (28) for the direct terms P 1 and P 2 , except that x is replaced by x -bη for the slit centered at +b and by x + bη for the slit centered at -b.

For the interference term, inserting the asymptotics (21) into (41) above results in the sum of two terms, one being the product of differences of cosin-functions, the other the product of differences of sin-functions. The problem is obviously symmetric about x = 0, so we need only consider the case x > 0. Then there are again two cases:

(i) |x -b| > a; and (ii) |x -b| < a.

In the first case, both terms decrease like 1/(x + b)(x -b) with various fluctuating factors as in (42). In the second case, P 1 behaves as in (27) but centred around x = b, and the other terms are negligible. We do not write the asymptotic formulas explicitly because the result is simply the observation that in this case we obtain a sum of two separated diffraction curves in the Fresnel regimes, i.e. curves that tend to the door functions in the limit, see Fig. 4d.

Notice an interesting behavior of the interference pattern in Fig. 4c, where we see that the interference amplitudes are very small compared to the diffraction amplitude inside a band 25 < ∼ |x/a| < ∼ 75, so that there are no interference fringes. This is also discernible in Fig. 4b of the Reference 7 which corresponds to the calculated two-slit diffraction images, where one can observe the absence of fringes in a band. However, this phenomenon is not apparent on the corresponding experimental image. This is probably due to the difference in defocussing between the calculated and experimental images, see Fig. 3b and Fig. 4b of the Reference. [START_REF] Frabboni | Two and three slit electron interference and diffraction experiments[END_REF] Indeed, the existence of such a band is quite sensitive to the value of the parameter N F (a).

VI. CONCLUSION AND REMARKS

-We have briefly presented the Feynman approach to quantum mechanics, based on the Lagrangian formulation of classical mechanics, and the associated change in paradigm in the transition from classical to quantum mechanics. We note that in this approach, the transition from classical to quantum mechanics is quite natural because it relies mostly on concepts well known to students of analytical mechanics and does not confuse particle and wave behavior. It thus avoids some metaphysical questions and leads directly to the solution of the diffraction and interference problems above, and hence to a better understanding of the quantum mechanics of such quintessential phenomena. This justifies introducing the Feynman formulation at an early stage especially as the semi-classical approach that is often used in a first course relies on the idea of quantification of the action. A parallel introduction of the Feynman integral thus makes sense as it clarifies the passage classical to quantum.

-Secondly, it seemed of interest to derive explicit formulas for the problem of diffraction / interference by one or two slits, and to discuss the results based on the physical parameters of the system, notably the Fresnel numbers and the distance scale at which we observe on the screen. The properties of the diffraction and interference patterns are not apparent from the exact formulas (20) (39), so it is useful to establish asymptotic forms (25) (26) (27) for the case of one slit, and (42) for the case of two slits.

We summarize the various conclusions. In case of a single slit:

-If N F (a) ≪ 1, this is the Fraunhofer regime for which the distribution curve is similar to the plane wave case, c.f. equations ( 25), (26) and Fig. 3a.

-If N F (a) ≫ 1, this is the Fresnel regime for which the diffraction curve approximates the form of the slit, c.f. equations ( 27), (28) and Fig. 3c.

-If N F (a) ∼ 1 one is in the intermediate regime for which there is a spreading around the center of the electronic distribution and we find the case of Fraunhofer distances on the screen, c.f. equation (20) and Fig. 3b.

In the case of two slits of width 2a, and separated by a distance 2b with b ≫ a, we can make similar distinctions as in the one-slit diffraction case but there is also a transition between two phases dependent on the optical resolution:

-If N F ≪ 1, one is in the mixed phase, i.e. we observe an interference curve modulated by a diffraction curve for a slit of size a in this case N F (a) ≪ 1, then we are in the regime of Fresnel, c.f. equations (42), (43) and Fig. 4a.

-If N F ≫ 1, one is in the separated phase, and there are two interference curves (the interference amplitudes are lower) modulated by the diffraction curves corresponding to both slits, each curve being centered respectively at ±bη; the shapes of the diffraction curves modulating the signals of each of the slits depends on N F (a) and are similar to the case of a single slit as summarized above (with three regimes: Fresnel, Fraunhofer and intermediate), -If N F ∼ 1 we observe a separation between two interference curves, modulated by the diffraction curve corresponding to one slit at the intermediate regime. see Fig. 4b and Eq.

(42).

Note that the fringes corresponding to the diffraction are at a distance λL/2a and those for interference at about λL/2b. The analytical properties of our asymptotics of two slits do not permit us to estimate these distances more exactly, but by analogy with optics they may be considered adequate.

In perspective, I suggest to take into account the quantum-mechanical way in the zdirection to solve the problem completely. Indeed, recall that as we discussed in the Section III, we consider in this article that the problem is separated in two motions, one between the source and the slits and the other one between the slits and the screen, which is rigorously not true. This is a challenging task since we have also to compute the loop path corrections, [START_REF] Yabuki | Feynman Path Integrals in the Young Double-Slit Experiment[END_REF] but it could be an interesting contribution to the European Journal of Physics.

  z, are the 1-dimensional free propagators. Then, integrating over the intermediate positions D between -∞ and +∞, we find the two dimensional free propagator(7) between (x = 0, z = 0) at t = 0 and (x,z = L + D) at the time t = τ + T : ∞ -∞ dD K (2D)∞ ((x, z = L + D), T + τ ; (0, 0)

)

  Let us introduce the following parameters η ≡ 1 + L/D and γ = η -1. We can then plot the functions (20) for various values of these parameters, see Fig 3. Note that the Fresnel functions in (20) behave differently depending on the value of the Fresnel number N F (a),

  Note that the function (27) oscillates rapidly in the interval [-a, +a] (esp. near the edges) around a constant value N F (a)γ/2a 2 η = 1/(λ z (L + D)), whereas for |x| > aη, the function (28) decreases rapidly to 0. Hence P (1Slit) tends at large N F (a) to the 'door' function defined by (8), as might have been expected, see Fig.3c.

FIG. 3 .

 3 FIG. 3. Diffraction curve for a single-slit, with η = 2. The abscissae are the distances in units of a and the ordinates are the relative populations. We have N F (a) = 0.01 for the Figure (3a), 0.5 for (3b), and 100 for (3c).

FIG. 4 .

 4 FIG.4. Interference curves (39) from two slits, with b/a = 13, η = 2. The abscissae are the distances in units of a and the ordinates are the relative populations. We have N F (a) = 0.001 for the Figure (4a) , 0.015 for (4b), 0.12 for (4c) and 6 for (4d).

  c.f. (44), Fig.4cfor N F (a) ≪ 1 and Fig.4dfor N F (a) ≫ 1.

ACKNOWLEDGMENTS

I would like to thank the Professor Tony Dorlas for discussions, encouragements and English corrections of the manuscript.


Di Loreto
email: diloreto@lirmm.fr

Gouaich Abdelkader
email: gouaich@lirmm.fr

Post stroke social gaming

Introduction

One of the most challenging aspects for post stroke rehabilitation is to include other people in the rehabilitation process. We can identify at least three levels of inclusion in social rehabilitation:

1. Patient to Patient (Quite-symmetric collaboration or competition) This first level presents challenges related to the patients' physical abilities.

In fact, the specific abilities that will be affected by stroke depend on the location, type and size of the lesion. Each patient is then characterized by a specific combination of deficits (National Stroke Association, 2010). Therefore, stroke rehabilitation programs are strongly personalized: they are adapted to a particular patient, to regain as much function as possible. Because each stroke is different from another it is possible to treat patients in independently but it's very difficult to treat them together. For example, two patients who have different recovery degrees would not be able to collaborate on the same exercise exactly because of their specific impairments.

Patient to Therapist (collaboration or competition)

This second level presents challenges related to the different physical abilities (as in the first case) but it's typically used during rehabilitation in care centers. In fact, therapists and patients could complete tasks together as a complement to personal rehabilitation.

Patient to Family (Asymmetric collaboration or competition)

This last level presents challenges linked to the different physical abilities (as in the first case) and it's not very diffused in real life (i.e. the family does not do rehabilitation exercises with the patient's at least in care centers).

Proposition

The above described scenario could be complicated when the post stroke therapy used is a 'virtual' therapy such as with serious games [START_REF] Loreto | Mixed reality as a means to strengthen post-stroke rehabilitation -International Conference on Human-Computer Interaction HCI International[END_REF]). In fact besides described challenges, the acceptance of virtual therapy has to be taken into account. For example, while therapists physically interact with patients during the rehabilitation process (e.g., they correct the patient's position) they could feel that playing with the patient 'is not part of their job' (and then susceptible to degrade the therapy value). On the other hand, we believe in the importance to add the social aspect to elders' therapythe average age of post stroke patients is 65 years old -even if technology mediated. In this assumption we are supported for example by [START_REF] Gajadhar | Out of sight out of mind: coplayer effects on seniors' player experience[END_REF]. Gajadhar et al. underlined the importance of social interaction in technology mediated exergames. They found that online co-play was deemed less rewarding than physical co-playing or when elders thought they were playing against a computer.

For these reasons we have decided to exploit social gaming for post stroke rehabilitation. The concept we propose is to insert rehabilitation serious games inside a more complex virtual world with a virtual economy to overcome the problem of heterogeneity on abilities. In fact, most on-line rpg-like games present a virtual economy based on items (virtual resources) that could be crafted by a player or an NPC (non player character). The following characteristics may be found in virtual resources in mimicry of tangible property (see Castronova,2001).

1. Rivalry: Possession of a resource is limited to one person or a small number of persons within the virtual world's game mechanics. 2. Persistence: Virtual resources persist across user sessions. In some cases, the resource exists for public view even when its owner is not logged into the virtual world. 3. Interconnectivity: Resources may affect or be affected by other people and other objects. The value of a resource varies according to a person's ability to use it for creating or experiencing some effect. 4. Secondary markets: Virtual resources may be created, traded, bought, and sold. Realworld assets (typically money) may be at stake. 5. Value added by users: Users may enhance the value of virtual resources by customizing and improving upon the resource.

These conditions create an economical system with properties similar to those seen in contemporary economies. Our idea is to exploit this already existing economy to assign to the patient a role typically taken by an NPC (non playing character) in RPG games: the crafter. In this way we can 1) use motion based games (the main task in post stroke rehabilitation is to reach an object) while creating the sense of social presence, 2) insert the rehabilitation game in a more structured world consisting of different games, each one appealing to a different age range, and not only to elder people.

Games general interaction

Each crafting games for post stroke is then coupled with another (RPG) game, involving crafting. The full game consist then of two (or more) different games (A+B). Game A is a movement based game, conceived to be played by a stroke affected person, and based on stroke rehabilitation principles. Obviously different instances of movement based games could be played by different patients. Game B is thought to be played by a younger generation, used to RPG games and to their dynamic.

A practical example

As an instantiation of the above described framework we developed the 'RehabCraft' game for post stroke rehabilitation, which is conceived to be played as a couple with the 'Legendary eagles' game.

The legendary eagles' game is an rpg game based on the following history. Sometime in the future mankind has managed to practically destroy the world via magnetic weapons. The world was knocked off its axis and continents have sunk into the ocean. Only small islets remain. Several decades later, the survivors have managed to remodel their lives. They have built homes on the islets, as well as shops and a school. The top, elite students of this school are recognized worldwide as the best people in the world: the Legendary Eagles. The game addresses a target of 20 years old people.

In this game is enclosed an arena where the players participate in tournaments. To participate to these tournaments they ask a forger to build for them swords and shields. Obviously the use of swords and shields is a gameplay choice linked to the kind of game described above. Whatever object could be crafted in the movement based game. In order to ask to the forger to create an item the player has to collect different materials in the main RPG game. Once the player has gathered the right amount of materials he can ask the forger to make him a new weapon.

On the other side the forger role is then taken by a patient playing a crafting game. The patient's movement quality is stocked during the game and influences the 'quality label' assigned to the crafted object. This quality label is an attribute that will stick to the object once retransmitted to the RPG game, in combination with the crafter nickname. In this way we are able to create a sort of classification list, giving feedbacks to the patient that can help to enhance his motivation (for example, how many fights were won with his crafted sword). The crafting games were conceived after most than a year of study and participatory design with therapists. This participatory work allowed us to extract from the therapists' expertise general rules to apply in movement based post stroke rehabilitation games (see for example the level design section description.

Hereafter the detail of the construction of a rocking horse (a fun element added to the game) and its relationship with the RPG game.

Assets:

Tools are used for the Crafting Game (i.e., the patient game). Materials are used for both, the Crafting Game and the RPG Game.

Tools:

Materials :

• Scissors • Saw • Sandpaper • Paintbrushes • Hammer • Wood • Cutted wood • Scrubbed wood • Paint • Painted Wood • Paper • Nails
• Leather strips and scraps

Level design

Level 0: animated intro to put the player in media res.

Co-Player Level 1-A-Find the rocking horse schema:

The user playing GAME B has to help an NPC character. Once the quest is finished he receives the rocking horse schema B-Collecting Woods: The user playing GAME B has to defeat a set of enemies in order to collect wood.

Patient Level 1 -Cutting the pieces: Using the schema retrieved by the Co-Player character, the patient has to cut each piece of wood in the right shape. Game Mechanics based on therapists' expertise: Templates for the shapes are traced onto the wood and the patient player has to cut the pieces according to the outlines (following a path). Shapes have to be proposed one by one and the shape to cut has not to be too complex.

Tool used: Saw Material used: Wood NB: When the GAME A is played independently, the rocking horse schema and the wood are given to the patient player from the start.

Co-Player Level 2-Find the paint: The user playing GAME B has to help an NPC character.

Once the quest is finished he receives the paint for the rocking horse.

Patient Level 2 -Scrub the pieces:

Game Mechanics based on therapists' expertise: Use sandpaper the patient scrubs the wood to remove any unevenness in the wood (free movement). Tool used: Sandpaper Material used: Cutted Wood Co-Player Level 3-Retrieve the nails: The user playing GAME B has to help an NPC character. Once the quest is finished he receives the nails for assembling the rocking horse.

Patient Level 3 -Paint the pieces: Using the paint retrieved by the Co-Player character, the patient player has to paint each pieces of wood.

Game Mechanics based on therapists' expertise:

Step 1: the patient has to blend the colours with a circular movement (following a path).

Step 2: Once colours are blended he can paint the wood pieces using whichever movement he wants (free movement).Woods pieces have to be proposed one by one.

Tool used: Paintbrushes Material used: Scrubbed Wood NB: When the GAME A is played independently, the paint is given to the patient player from the start.

Co-Player Level 4-Find paper and other materials (details): The user playing GAME B has to help an NPC character. Once the quest is finished he receives the materials for decorating the rocking horse.

Patient Level 4 -Decorate the pieces: the patient creates decorative items for the rocking horse.

Decorative items are ears, reins, tail (all the elements that could be cut down).

Game Mechanics based on therapists' expertise: To make the ears, the patient cut two equal triangles in the same way he cut wood (following a path). For reins and tail he has to cut leather strips (free movement). Tool used: Scissors Material used: paper and leather strips Patient Level 5 -Assemble the pieces: the patient assembles the pieces of the rocking horse according to a plan. Game Mechanics based on therapists' expertise: A template for the shapes is show to the player.

Step 1: The patient player has to slide the proposed shape in the right spot (free movement).

Step 2: Then, using the hammer the patient player has to fix nails on the template shape (free movement) Shapes have to be proposed one by one.

Tool used: Hammer Material used: Nail, Painted Wood

Figure 1 -

 1 Figure 1-The relationship between players in our idea

Figure 2 -

 2 Figure 2-A scene from the Legendary Eagles prototype

Figure 3 -

 3 Figure 3 -A level from the RehabCraft prototype

Game Workflow

Importance of this approach

The above described approach to social gaming for post stroke rehabilitation allow us to address all the inclusion levels described in the introduction:

1. Patient to patient

The different patients' physical abilities could be addressed inside the above described approach in two ways.

a) In a collaborative approach players can work on the same object (at the same time or not). b) In a competitive approach players can compete on the crafting of the same object, because of the quality label described in section 3.

Patient to therapist

With the above described approach we can potentially cope with this aspect. On the other hand therapist's acceptance of the playing approach is strongly subjective.

Patient to family

This last level presented difficulties linked to the different physical abilities (an healthy person playing with a sick person). With the above described approach we can create social gaming, leaving the healthy person playing a more difficult game while the patient plays a game conceived for his rehabilitation. In fact it's the game play level which gives meaning to the relationship between the different games/players.

Experimentation:

A first experiment could be conducted assessing as:

Experimental hypothesis:

Motivation and duration of rehabilitation sessions are increased for group B when compared with group A.

Experimental scheme:

To experiment with the presented approach for social therapeutic games, the following experimental scheme is proposed:

-Two groups (A,B) of N patients are selected. For both groups, the therapeutic game is proposed in addition to classical therapy. Patient are free to stop each rehabilitation session whenever they want. -Group A (control group) members are asked to play to only rehabCraft -Group B members are asked to play with RehabCraft with a connection to another RPG game (to be defined) played by family member.

Independent variables:

-Motivation level assessed through a questionnaire.

-Time of rehabilitation (measured in seconds for each session)