Émilie Samuel
email: emilie.samuel@univ-st-etienne.fr

Colin De La Higuera

Jean-Christophe Janodet
email: janodet@univ-st-etienne.fr

Extracting Plane Graphs from Images ⋆

Keywords: Plane graphs, images, interest pointels, segmentation, Delaunay triangulation

In order to use structural techniques from graph-based pattern recognition, a first necessary step consists in extracting a graph in an automatic way from an image. We propose to extract plane graphs, because of algorithmic properties these graphs have for isomorphism related problems. We also consider the problem of extracting semantically well-founded graphs as a compression issue: we get simple graphs from which can be rebuilt images similar to the initial image. The technique we introduce consists in segmenting the original image, extracting interest pixels on the segmented image, then converting these pixels into pointels, which in turn can be related by region-based triangulation. We show the feasibility and interest of this approach in a series of experiments.

Introduction

Representing images with graphs is an approach followed in pattern recognition in which it is hoped to benefit from the structural properties of the graphs, but also to be able to use the robustness associated with them [START_REF] Conte | Thirty years of graph matching in pattern recognition[END_REF][START_REF]Applied Graph Theory in Computer Vision and Pattern Recognition[END_REF]. A number of ideas have been analysed in order to use these graphs once built, involving edit distances [START_REF] Riesen | Approximate graph edit distance computation by means of bipartite graph matching[END_REF] or graph matching [START_REF] Lozano | Protein classification by matching and clustering surface graphs[END_REF]. Moreover, databases of synthetically generated graphs [START_REF] Santo | A large database of graphs and its use for benchmarking graph isomorphism algorithms[END_REF] or graphs related to pattern recognition benchmarks [START_REF] Riesen | IAM graph database repository for graph based pattern recognition and machine learning[END_REF] have been created.

Many different models of graphs extracted from images have already been proposed. A first alternative consists in segmenting the image, in representing each region by a vertex and putting an edge between two vertices whenever the regions have a common border. Further arguments can be added to indicate the size of the region, the length of the border, and so forth. The obtained graphs are called Region Adjacency Graphs [START_REF] Rosenfeld | Adjacency in digital pictures[END_REF] and have been improved, yielding dual graphs [START_REF] Kropatsch | Finding the structure of connected components using dual irregular pyramids[END_REF], ordered graphs [START_REF] Jiang | Optimal quadratic-time isomorphism of ordered graphs[END_REF], combinatorial maps [START_REF] Lienhardt | Topological models for boundary representation: a comparison with n-dimensional generalized maps[END_REF]. A second option consists in extracting interest points (they will now be the vertices) and in relating them according to some neighbourhood relationship (the Delaunay triangulation may be a way to do this). Again, numerical attributes can be added to the representation, either attached to the vertices or to the edges.

Yet neither of the two aforementioned techniques is satisfying: typical drawbacks in the first case are that the structure of the graph actually has little to do with the topological nature of the image (i.e. the spatial disposition of depicted objects, their borders and connexity), in particular when the segmentation is rough. In the second case, not only do many of the so-called interest points not seem to be that interesting (requiring to extract many more points than necessary in order to be sure to have all the important ones), but also, since these points are actually pixels (with non null thickness), the construction is hindered by discrete geometry issues (e.g. what is the frontier between two regions?).

What should be the features of a good graph extracted from an image? On one hand, it should be robust: if the image is slightly distorted, the graph should not be deeply modified. It should also have good algorithmic properties: typically the central isomorphism problem, the related problems of sub-graph isomorphism, maximum common subgraph and graph edit distance should be solvable with polynomial algorithms. Furthermore what we may call the semantic properties of the image should be present in the graph: this might mean that the graph, when drawn, actually is a symbolic representation of this image; for instance, the boundaries between the regions could coincide with the edges of the graph. Finally we would like to be able to rebuild an image from the graph and for this new image to be a good compression of the initial image. In other terms, the loss due to the extraction process should be minimal.

In this paper we explore the possibility of combining the advantages of both the techniques presented above: on the one hand, segmentation introduces robustness; on the other hand, extraction of interesting pixels is a good idea and much less pixels are going to be needed from a segmented image. The method we suggest is thus as follows: (1) segment the image into regions, (2) extract s interest pixels, (3) add s ′ intersection pixels (s ′ being generally less than s), (4) compute the corresponding interest pointels, (5) relate them according to the boundaries of the regions and (6) triangulate the pointels of each region.

Definitions

A (digital) image I is an array of n * m pixels. Each pixel p has a colour described by a 3 dimensional vector c I (p) in standard RBG colour system; that latter hypothesis corresponds to our experimental settings but any other colour system could be used. The coordinates of the pixels consist in pairs (i, j); their range goes from (0, 0) to (n -1, m -1). Each pixel is delimited from each of his four neighbours by a linel, and the four corners of the pixels are called pointels. Such pointels and linels are mathematical points and line respectively, with null thickness. This enables us to associate the coordinates of the pointels to those of the pixels; their range goes from (0, 0) to (n, m).

We now introduce the main criterion used in this paper to measure the loss between an original image I 1 and an image I 2 obtained after a series of trans-formations. We assume that both I 1 and I 2 have the same size (n * m). The loss is the L 1 distance between I 1 and I 2 divided by the number n * m of pixels. Formally, we first define the distance d p between any pixel p 1 in image I 1 and pixel p 2 in image I 2 as the normalised distance between corresponding colour vectors. That is, if (3 * 255). And we state:

c I1 (p 1 ) = (r 1 , g 1 , b 1 ) and c I2 (p 2 ) = (r 2 , g 2 , b 2 ), then d p (p 1 , p 2 ) = (|r 2 -r 1 | + |g 2 -g 1 | + |b 2 -b 1 |)/
loss(I 1 , I 2 ) = i=n-1 i=0 j=m-1 j=0 d p p(I 1 , i, j), p(I 2 , i, j) n * m , (1)
where p(I ℓ , i, j) denotes pixel with coordinates (i, j) in image I ℓ . We now introduce plane graphs. We recall that a graph G = (V, E) is planar if it can be embedded in the plane, in such a way that no two edges cross. Note that several non homeomorphic embeddings may exist for a given planar graph (i.e. several incomparable drawings of the same graph) [START_REF] Cori | Un code pour les graphes planaires et ses applications[END_REF]. Moreover, it is always possible to move the vertices so that the edges are drawn with straightline segments [START_REF] Fàry | On straight-line representation of planar graphs[END_REF]. Thus by a plane graph, we mean a planar graph for which the embedding is fixed and such that (1) every embedded edge is a straight-line and (2) no two embedded edges intersect, except at their endpoints.

Plane graphs are interesting to represent images for at least two reasons. On one hand, they can be nicely represented as combinatorial maps which also provide us with an elegant data structure [START_REF] Tutte | A census of planar maps[END_REF]. On the other hand, isomorphism and subisomorphism problems were recently studied for combinatorial maps [START_REF] Damiand | A polynomial algorithm for submap isomorphism: Application to searching patterns in images[END_REF], with the following interesting consequence:

Theorem 1 Let G = (V, E) and G = (V ′ , E ′ ) be two plane graphs. Deciding whether G and G ′ are isomorphic is solvable in time O(|V |×|V ′ |). Moreover, if G is face-connected, deciding if G is a pattern of G ′ is solvable in time O(|V |×|V ′ |).
A pattern of a graph can be seen as a subgraph obtained by erasing faces and then edges (called bridges) separating 2 erased faces and finally singular vertices (called hinges) separating 2 non-contiguous erased faces [START_REF] Damiand | A polynomial algorithm for subisomorphism of open plane graphs[END_REF]. Other positive algorithmic results for special classes of graphs are described in [START_REF] Jiang | Optimal quadratic-time isomorphism of ordered graphs[END_REF].

In this paper, we develop techniques to extract a plane graph G from an image. In order to rebuild an image, we also need to have (1) a vector P mapping each vertex with the coordinates of corresponding pointel and (2) a vector C mapping each face of G with the colour in which it should be drawn3 . Although G, P and C are known, a variety of images can still be drawn. Indeed, embedded edges will cross many pixels and divide them into several parts, so which colour should these pixels be given? We choose the following rule: for each split pixel, we consider the colours of its four corners (pointels) (inherited from the faces in which they stand -pointels that lie between regions are not taken into account), compute the average colour and assign it to the pixel.

We denote by Im(G, C, P ) the resulting image and can now measure the loss between Im(G, C, P ) and original image I using Eq.( 1). Obviously, the loss between an image obtained through some compression or representation by graphs could also be looked into w.r.t. the gain in terms of space. We claim that whereas the space needed to encode an image is n • m • K, where K is a constant corresponding to the size needed to encode one colour vector (here 24), to encode Im(G, C, P ) the space is about |V | • (log n + log m + 3 + K).

Construction

In this section, we propose a new way to represent an image as a graph, by taking advantage of both image preprocessing tasks introduced in Section 1: the segmentation task, combined to interest pixels extraction. This allows us to obtain graphs that preserve the semantics and the topology of the original images, while providing us with a valid approximation that minimises the loss. Remember that the goal is to extract, given an image I, a graph G such that, with correct P and C functions, we have loss(I, Im(G, P, C)) as small as possible.

Segmentation

Any segmentation process aims at defining disjoint regions made of homogeneous sets of pixels (in some predefined way). Many segmentation algorithms exist, and it is generally possible to target an approximate number of regions by tuning the parameters of any chosen algorithm (see Fig. 1). Aiming at minimising loss(I, Im(G, P, C)), it would be suitable to more or less preserve the regions of segmented image while building the graph: the boundaries between them should match, in as close a way as possible, the edges of the graph. We recall that two adjacent pixels are separated by a linel (see Section 2), so the regions are delimited by boundaries, sets of consecutive separating linels that form cycles. Of course, one given region may be delimited by more than one cycle, and the borders of the image need to be included. Now our thesis is that the vertices of the graph should be selected among the endpoints of the separating linels described above, that is, the pointels. Indeed, the vertices (and then the edges) must be chosen in such a way as to cover all the important elements structuring the boundaries. Obviously, too many vertices will make us deal with large graphs (which for combinatorial reasons is not recommended). Conversely, too few vertices will result in embedded edges (straight lines) which will be far apart from the boundaries. This means that we have to compress the information displayed in the segmented regions (which was already a compression of the original image): we are not interested in the exact shape of the regions, but in a rough although lifelike sketch (see Fig. 3 w.r.t. Fig. 1).

Extracting and Cleaning Up Interest Pixels

In order to select the vertices of the graph, we now make use of an interest pixels detector. No consensus exists on what an interest pixel is: each detector (e.g., [START_REF] Harris | A combined corner and edge detection[END_REF][START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF]) performs its own measures of local information (based on texture, colour, shape,. . . ) on the image and then extracts most stable and rich sets of pixels w.r.t. these features. While usually applied on original images, we run interest pixels detectors on segmented images. Indeed, segmented images are simplified, and detected pixels will thus be situated on, or close to, boundaries. This will prevent us from detecting pixels on textured areas, salient corners pixels situated inside a region, or pixels corresponding to noise. Note that if the number of extracted interest pixels is low, some regions (e.g. regions strictly included in others) may not be detected and thus may not be represented in the final graph.

Despite these precautions, the set of pixels obtained is generally not satisfying for two reasons. On the one hand, due to the principles of detectors, it is still possible for a pixel to be selected while it is far from any region boundary. We aim at eliminating them: for each interest pixel, we consider a mask containing this pixel and its eight neighbours, and then remove the interest pixel from the set if all the pixels in the mask belong to the same region.

On the other hand, the detector is often not able to come up with all the pixels that act as boundaries to three regions or more (i.e. pixels that belong to one region and have at least two other regions among their eight neighbours). Yet these pixels are necessary in order to have embedded edges of the graph close to the boundaries, so we aim at adding them: we consider 3x3 masks again, this time for all the pixels of the image, and add as interest pixels those whose mask intersect three different regions (or at least two, if the pixel is on image's border). We finally add the four corners of the image to the set.

We are clearly now in possession of a "clean" set of interest pixels that could be the vertices of the graph. However, we are still faced with a problem, wellknown to researchers in discrete geometry: when tracing the lines corresponding to the edges of such a graph, because of the discrete nature of the pixels, we can end up with zones that are not colourable, or even to edges which intersect. Thus the vertices of the graph should not be pixels but pointels (with null thickness).

From Pixels to Pointels

In order to define the vertices, we now draw a parallel between linels separating segmented regions and interest pixels. Since linels lie between pixels, we will analyse the four corners (pointels) of each interest pixel. Then the goal is to 00 00 00 [START_REF] Kropatsch | Finding the structure of connected components using dual irregular pyramids[END_REF] At this stage, the selected pointels become the vertices of the graph. Note that function P is also defined straightforwardly (we use coordinates of the pointels in the image). They are yet to be related: the edges are built by linking vertices, following the boundaries of each region (see Fig. 2 and3).

Triangulation

To complete our method, we finally make use of a kind of Delaunay triangulation. The reason is threefold. (1) Such a technique will improve the robustness of the graph: if the positions of the vertices are slightly modified by transformations applied to the original image, the Delaunay triangulation will remain stable.

(2) In a problem such as the plane graph isomorphism discussed in Section 2, triangulation edges will prevent one from matching faces with similar boundaries but different shapes. Note that the benefit of Delaunay triangulation has been investigated for other problems in Pattern Recognition [START_REF] Finch | Matching delaunay graphs[END_REF]. [START_REF] Cori | Un code pour les graphes planaires et ses applications[END_REF] The first stage of our technique is a segmentation. Yet the resulting regions may be strictly included one into another (see Fig. 1). On the other hand, in the last stage, the edges are built following the boundaries of the regions. So the graph we get is generally not connected (as shown in Fig. 3), which is not combinatorially suitable; the use of the Delaunay triangulation will settle this issue. Now, in order to preserve existing edges, a Delaunay triangulation takes place in each face of the embedded graph independently of the others. As a region may not be convex and often contains other included regions, the triangulation edges that are not totally included in the region are eliminated. Finally, we merge together the triangulated faces, and obtain the target plane graph G.

Experiments

The public benchmark of the Berkeley Segmentation Dataset [START_REF] Martin | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics[END_REF] contains 300 images (481x321 RGB) of natural scenes, coming from the Corel dataset (widely used in computer vision). Each image contains at least one discernible object, and was segmented at varying levels by several individuals without any particular instruction on the type of criterion to use (see Fig. 1 for an example). The number of regions is at least 2, for an average of 20. As for the interest pixels detector, we have used the one from [START_REF] Bres | Detection of interest points for image indexation[END_REF], based on multi-resolution contrast information. Below, I denotes an image, I k its segmentation into k regions, and Bg(I k , s) the graph obtained by extracting s interest pixels from the segmented image and following all the steps described in Section 3 (Bg=Build Graph).

Losses Due to the Graph Extraction Process

Firstly, we have evaluated loss(I, I k ), that is the loss due to the segmentation of I into k regions. For the 1633 available segmentations, the average loss is 10.0%. We also wanted to study its correlation with the number of regions. So, for regions ranging from 3 to 30, we selected 10 images (for a total of 280 segmentations) and computed the corresponding loss, together with the standard deviation. The results are shown on the left of Fig. 4: no correlation can be found. As the individuals who divided the images did not have any particular instructions, the segmentation was mostly semantic. Regions thus depict objects or parts of objects, and they are not necessarily homogeneous in terms of colours, whatever the level of segmentation.

Next, we have studied how the number of interest pixels affected the loss from I k (the image segmented into k regions) to Bg(I k , s) (the graph obtained by extracting s interest pixels from I k ). To this extent, we have extracted 10 to 500 interest pixels and built corresponding graphs. The results are shown on the right of Fig. 4. The loss ranges from 3.1% for 10 interest pixels to 0.2% for 500. Basically, the larger the extracted graph is, the closer the boundaries of the regions are followed by the edges, thus the lower loss(I k , Bg(I k , s)) is.

Finally note that the processing of interest pixels induces widely less loss than the segmentation stage (< 3% w.r.t. 10%). To confirm this, we have considered one particular image at different levels of segmentation k and extracted a fixed number of interest pixels. See Table 1.

Comparison with Another Plane Graph Extraction Method

In order to compare our graph extraction method, we use another more classic one: interest pixels are directly extracted from original images (without presegmentation), then converted into interest pointels as described in Sect. (with, added, the 4 corners pointels in order to fit whole image) and related by the Delaunay triangulation. We denote Bg(I, t s ) this graph and compare loss(I, Bg(I, t s )) with loss(I, Bg(I k , s)) (see Fig. 5): our method induces slightly less loss, while providing a graph that preserves the semantics of original image.

Size of the Graph

Given an image, in order to obtain a graph of a given size |V |, we can work on the segmentation level and the number of extracted pixels. To assess the trade-off between both, we compare loss(I, Bg(I k , s)) for different values of k and s. Of particular interest is the case where Bg(I k , s) and Bg(I k ′ , s ′ ) have similar sizes. The situation for 2 images A and B is depicted in Table 2. loss(I, Bg(I k , s)) is quite similar whether we are considering an important number of interest pixels and a small number of regions, or conversely. Thus any of parameters k and s can be tuned to obtain of graph of a given size, with a slight preference for the first due to the decrease of loss(I k , Bg(I k , s)) when s becomes larger. More precisely, to assess the influence of the number s of detected interest pixels on |V |, we extracted 10 to 500 interest pixels from 1633 segmented images, added the intersection pixels and computed the corresponding pointels; left-hand curve of Fig. 6 shows that |V | is a quasi-linear function of s.

Finally, concerning the influence of the level of segmentation k on |V |, we selected 10 segmentations for a number of regions varying from 3 to 30 (yielding 280 segmentation files); for each of them, we extracted 50, 100, 150 and 200 interest pixels and computed the corresponding pointels (see right of Fig. 6): Even if the curves do not increase steadily, for some fixed s, the more regions there are, the more selected pointels there are; this is due to the intersection pixels, that grows up with the number of regions. Finally, even though the method is not intended as a compression algorithm, it can be noted that the original images have a theoretical size of 450kB (encoded as JPEG, they are about 70kB). The segmented files have an average size of 28kB, and a graph with 150 vertices requires less than 1kB.

Discussion, Open Questions and Conclusion

The bottleneck problem to using graphs in pattern recognition for image recognition tasks is to extract good graphs, where good means both reasonably small and semantically stable. We have introduced a new method for extracting graphs from images, which compresses the original information while preserving the semantics. Experiments show that the loss due to this process is limited. Moreover, these graphs have nice properties that are interesting for isomorphism-related problems: they are planar and connected, and their size is low. Moreover, this method is general enough to be applied to any kind of images (while usual methods are often adapted to particular classes of images (e.g see [START_REF] Riesen | IAM graph database repository for graph based pattern recognition and machine learning[END_REF]).

It would be interesting to study alternative loss functions, to experiment with different segmentation algorithms and other ways of creating connexity and structural rigidity. Finally, we aim at proving the validity of the extracted graphs in pattern recognition tasks, and contributing to the field with a free distribution of the described software.

Fig. 1 .

 1 Fig. 1. An image and two segmentations at different levels, coming from the Berkeley Segmentation Dataset [15].

Fig. 2 .

 2 Fig.2. On the left, the border between 3 regions: 9 (hatched) interest pixels + 1 (striped) intersection pixel yield 12 interest pointels (big dots). On the right, the extracted plane graph: the edges are built following regions boundaries . select among these pointels, those which are the most relevant. The rule is as follows: (1) Consider each pointel π of each interest pixel separately; (2) Let p 0 , p 1 , p 2 and p 3 be the 4 pixels which share pointel π; (3) If ∃p i such that c I (p i-1 ) = c I (p i ) and c I (p i ) = c I (p i+1 ) (modulo 4) then the pointel is selected.At this stage, the selected pointels become the vertices of the graph. Note that function P is also defined straightforwardly (we use coordinates of the pointels in the image). They are yet to be related: the edges are built by linking vertices, following the boundaries of each region (see Fig.2 and 3).

Fig. 3 .

 3 Fig.3. Several plane graphs obtained from rightmost segmentation in Fig.1. For the leftmost (resp. middle, rightmost) graph, 25 (resp. 50, 100) interest pixels were detected (and none of them deleted using the uniform mask rule), then 37 (resp. 36, 36) intersection pixels were added, generating 68 (resp. 108, 194) vertices (pointels).

Fig. 4 .

 4 Fig. 4. loss(I, I k ) w.r.t. the number of regions (on the left) and loss(I k , Bg(I k , s)) w.r.t. the number of interest pixels (on the right).

Fig. 5 .

 5 Fig. 5. loss(I, Bg(I, ts)) and loss(I, Bg(I k , s)) w.r.t. the number of interest pixels.

Fig. 6 .

 6 Fig. 6. Number of pointels w.r.t. the number of interest pixels (on the left) and to the number of regions (on the right).

Table 1 .

 1 For one image, losses corresponding to different levels of segmentation.

	s k |V | loss(I, I k )) loss(I k , Bg(I k , s)) loss(I, Bg(I k , s))
	50 3 104	0.084	0.006	0.089
	50 7 123	0.084	0.005	0.089
	50 9 175	0.084	0.006	0.090
	50 13 149	0.084	0.007	0.091
	50 20 207	0.083	0.007	0.090

Table 2 .

 2 Influence of k and s on the losses for extracting a graph of size |V |. s k |V | loss(I, I k ) loss(I k , Bg(I k , s)) loss(I, Bg(I k , s))

	Image A 80 7 182	0.084	0.004	0.088
	20 18 188	0.076	0.023	0.098
	B 80 11 146	0.166	0.009	0.175
	20 25 145	0.166	0.022	0.188

Faces of plane graphs can be denoted and saved using directed edges, and a convention stating that such or such face is on the right of such or such directed edge.

⋆ This work was partially supported by an ANR grant BLANC 07-1-184534 (project Sattic), and the IST Programme of the European Community, under the PASCAL 2 Network of Excellence, IST-2006-216886.


David Paboeuf

Gaëlle Lucas-Leclin

Patrick Georges

Bernd Sumpf

Götz Erbert

Cyrille Varona

Pascal Loiseau

Gérard Aka

Bernard Ferrand

Diode pumping of Nd:ASL and its frequency doubling for blue emission around 450 nm

Keywords: diode pumped solid-state lasers, rare earth, second harmonic generation, neodymium, volume holographic grating, 10/10/ 13

We present the diode pumping of a Nd-doped strontium and lanthanum (Nd:ASL) crystal Sr1-xLax-yNdyMgxAl12-xO19 (0.05 ≤ x ≤ 0.5; y = 0.05) for second harmonic generation around 450 nm. In order to fulfill the pumping requirements of this crystal, we have developed a high-brightness pump source based on a tapered amplifier in an extended cavity with a volume Bragg grating for wavelength stabilization. A pump brightness of 110 MW.cm -2 sr -1 has been obtained with a linewidth lower than 80 pm at 798 nm. This laser source has been used to pump a Nd:ASL crystal to obtain 300 mW at 906 nm and 53 mW at 453 nm by intracavity doubling with a LBO crystal.

INTRODUCTION

Diode-pumped solid-state lasers operating in the blue spectral range have a large number of applications ranging from high-density optical data storage to phototherapy and medical diagnostic. One way to design such a laser is to perform non linear conversion of a near infrared laser line. The neodymium 4F3/2 -4I9/2 laser transition is one of the most used for the initial infrared source since it allows emission around 900 nm. For instance, efficient frequency doubling has been demonstrated with Nd:YAG (473 nm) [START_REF] Fan | Modeling cw operation of a quasi-three-level 946 nm Nd:YAG laser[END_REF], Nd:YVO4 (456 nm) [2] and Nd:GdVO4 (456 nm) [START_REF] Czeranowsky | Continuous wave diode pumped intracavity doubled Nd:GdVO 4 laser with 840 mW output power at 456 nm[END_REF]. In order to reach deeper blue wavelengths, two ways are possible. The first one is to design a cavity emitting on a deeper Stark sub-level with a conventional Nd-doped crystal; quasi-3-level laser emission at 899 nm in Nd:YAG has indeed been recently demonstrated [START_REF] Castaing | Diode-pumped Nd:YAG laser emitting at 899 nm and below[END_REF]. Another solution consists in developing new neodymium-doped laser materials. Among them, Nd-doped strontium and lanthanum (Nd:ASL) crystal Sr1-xLax-yNdyMgxAl12-x O19 is very promising since its quasi-three-level transition is located at 900 nm, one of the lowest existing in Nd-doped crystals [START_REF] Lupei | Spectroscopic and structural properties of Nd 3+ doped strontium lanthanum aluminate laser crystals[END_REF]. Unfortunately, the absorption transitions of Nd:ASL are narrow-band (< 3 nm) and located at 792 nm and 798 nm (see Fig. 7) where commercial high power laser diodes are difficult to find. Moreover the quasi-3-level operation of the laser implies that the lower level of the laser transition is thermally populated. A bright enough pump source is then required to exceed the transparency intensity (3 kW.cm-2) and get a positive gain. Under Ti:Sa pumping at 792 nm, 1.67 W of infrared laser at 900 nm have been demonstrated in [START_REF] Varona | CW blues laser emission by second harmonic generation of 900 nm oscillation of Nd-doped strontium and lanthanum aluminate (ASL)[END_REF]. Furthermore, intracavity doubling with a BiBO crystal produced 320 mW of blue laser power at 450 nm. In regard with these efficient results, the demonstration of the diode pumping of this crystal would be an interesting improvement in terms of compacity and wall-plug efficiency. With this purpose, we have developed our own high-brightness wavelength-stabilized pumping source at 798 nm based on a tapered amplifier in extended cavity with a volume Bragg grating. In this paper, we first present the design and characteristics of the pumping source and then the use of this source to perform infrared emission and intracavity second harmonic generation in Nd:ASL.

DESIGN AND CHARACTERIZATION OF THE PUMPING SOURCE

Tapered laser diodes

Tapered laser diodes are one of the most promising designs proposed for high brightness emission in the 1-W power range. It consists of a single-mode ridge acting as a spatial filter and a large tapered amplifying section. Several achievements have been described in the literature, demonstrating high output powers in a nearly diffraction-limited beam [START_REF] Wenzel | High-brightness diode lasers[END_REF][START_REF] Kelemen | Tapered diode lasers at 976 nm with 8 W nearly diffraction limited output power[END_REF][START_REF] Dittmar | High-Power 808-nm Tapered Diode Lasers With Nearly Diffraction-Limited Beam Quality of M 2 = 1.9 at P = 4.4 W[END_REF]. A super-large optical-cavity structure (SLOC) for emission around 800 nm has been designed as in [START_REF] Dittmar | High-Power 808-nm Tapered Diode Lasers With Nearly Diffraction-Limited Beam Quality of M 2 = 1.9 at P = 4.4 W[END_REF]. The active region consists of a single GaAsP quantum well embedded in a 3-µm thick Al 0.45 Ga 0.55 As waveguide. The vertical far-field angle was as low as 18.3° (FWHM) and 95% of the power was included in an angle of 32.5°. The length L r of the index-guided straight ridge section was 2 mm, and the angle of the gain-guided tapered section was ϕ  = 4°. The full amplifier length was 4 mm. The front facet had a reflectivity R f of 0.5% whereas the rear facet reflectivity R r was < 5x10 - 3 . The devices were mounted p-side (epi-side) down on CuW submounts and, thereafter, on C-mounts. No laser emission was observed from these devices, but amplified spontaneous emission in a ~25 nm spectral range. With the same design (L r = 1 mm, ϕ  = 4°) but coatings R f ≅ 0.1% on the front facet and Rr ≅ 94% on the rear facet, a tapered laser diode has demonstrated an output power of about 4 W and a beam quality parameter M2 < 1.9 [START_REF] Dittmar | High-Power 808-nm Tapered Diode Lasers With Nearly Diffraction-Limited Beam Quality of M 2 = 1.9 at P = 4.4 W[END_REF]. Volume Bragg gratings have appeared to become essential optical components in the recent years, with a high spectral selectivity, low losses and reflectivity from a few % to 100 %. Is consists of a photo-refractive glass in which an index modulation has been recorded under UV illumination. Bragg gratings act as spectral and angular filter according to the Bragg condition: cos(θ)=λ/(2n 0 Λ). For a reflecting volume Bragg grating (VBG) with unslanted index fringes characterized by a thickness d, a grating period Λ, an average index n 0 and an index modulation Δn (see Figure 1), the diffraction efficiency η(θ,λ) using the same notations as in [10], is given by : ( )

Volume Bragg gratings

1 2 2 2 2 2 1 1 ! " " " " # $ % % % % & ' ! ! + = ( ) ) ( * sh (1)
ξ characterizes the mismatch to the Bragg condition. The computation of the diffraction efficiency in respect with the incident angle of the beam and wavelength for a 0.7-mm-thick VBG designed to reflect 20% at 798 nm is presented on Figure 2. From these calculations, the spectral selectivity of the grating is evaluated to 300pm (full-width at 1/e 2 ) and its angular selectivity to 2° ( full-width at 1/e 2 ) .

Tapered amplifier in extended cavity

The set-up of our tapered amplifier in extended cavity is presented on Figure 3. We used a VBG centered at 798 nm, with a reflectivity of 20 % and a spectral bandwidth of 200 pm, as the back mirror of the extended cavity. The Bragg grating acts as a spectrally selective end-mirror of the external cavity, and the useful output power is emitted from the tapered side. The emission from the ridge side of the tapered amplifier is focused into the Bragg grating with a high-NA aspheric lens pair (Thorlabs C230220P: f' coll =4.5 mm 0.55NA/ f' foc =15.4 mm 0.16NA) for diffraction-limited imaging. The major feature of our external cavity set-up is the focusing of the beam inside the Bragg grating, while it has been designed for a collimated beam. The resulting "cat's eye" effect improves the mechanical stability of the laser cavity as in self-aligned Littman cavities [START_REF] Fermigier | Self-aligned external-cavity semiconductor lasers for high resolution spectroscopy[END_REF] or interference-filter stabilized external cavities [START_REF] Zorabedian | Interference-filter-tuned, alignement stabilized, semiconductor external-cavity laser[END_REF]. Here the Bragg grating allows us to take benefit from this helpful effect in both transverse directions. Indeed, we evaluate the precision required on the lens pair position in X and Y directions to about ± 45 µm for a reduction of the power of 80% with our focused-beam configuration. In the longitudinal axis the misalignments tolerance is ∆Z = ± 12 µm. These values are much larger than the usual tolerances in collimated-beam configurations.

Whatever the operating current or the temperature, the laser emission was locked on the Bragg wavelength with a very low wavelength shift that remained included into the VBG spectral bandwidth (see Fig. 4). We measured a side-mode suppression ratio higher than 40 dB and a FWHM-linewidth lower than 80 pm (OSA-limited). The threshold of the extended cavity laser diode was as low as 1 A and the slope efficiency reached 0.9W.A -1 . A maximum output power of 2.5 W has been measured for an operating current of 3.5 A (see Fig. 5). The beam quality parameter (M 2 ) was as good as 1.2 in both directions for an operating current of 2.5 A (see Fig. 6). At higher operating currents we observed a degradation of the beam quality and measured a M 2 around 3 in the slow axis direction. This result in a pump brightness of 110 MW.cm -2 sr -1 . All these features make this pumping source fully suitable to the optical pumping of an Nd:ASL crystal and may result in a major improvement in terms of compactness and wall-plug efficiency compared to Ti:Sa pumping.

-

INFRARED OPERATION

With quasi-three level lasers, reabsorption at the oscillating wavelength is the main challenge to face. The available pump power being limited, the pump beam has to be strongly focused inside the crystal to reach a high intensity at the waist position. This results in a very diverging beam with reduced pump intensity at the crystal output. Reabsorption may then occur and limit the laser efficiency. Thus, simulations have been carried out to evaluate the gain inside the crystal and determine the theoretical adequate parameters. The small signal gain integrated on the transverse section of the crystal can be written as [13]:

( ) ( ) ( ) ( )dr z r A h z r I h z r I N z g c r p p aP aL p p aP eL ! " # # = 0 0 , 1 , , $ % & $ & % & & (2)
where N is the density of neodymium ions in the crystal, σ aP the absorption cross section at the pump wavelength, τ the fluorescence lifetime. I p (r,z) is the pump intensity taking into account the Gaussian propagation of the pump beam. In the model, we assume there is no change in the spatial profile of the pump due to absorption. σ eL and σ aL are, respectively, the emission and absorption cross sections at the laser wavelength. A(r,z) stands for the Gaussian profile of the laser mode. The double pass gain inside the crystal is given by:

( ) dz e G L z g ! = 0 2 0 (3)
The spectroscopic data used are listed in Table 1. The crystal is assumed to be cylindrical with a radius r c much larger than the beam size. Fig. 8 (left) shows the evolution of g 0 along the crystal for different pump waist radii and an incident pump power of 300 mW close to the experimental threshold values. The laser and pump waists are supposed to be equal and located at the input of the crystal. With a small pump waist (6 µm), the gain is very high at the beginning of the crystal. The beam being very divergent, the gain decreases very quickly and reabsorption occurs beyond a transparency length of ~1.4 mm. In contrast, with a too large pump waist, even at the beginning of the crystal, the gain is very low because of the strongly reduced incident pump intensity. We have also computed the evolution of the double pass gain G for different pump waist radii and crystal lengths on Figure 8 (right). From these simulations, we expect the laser gain will be the highest with a waist radius of 10 µm and a crystal length of 2-3 mm. σ eL @ 900 nm σ aL @ 900 nm σ aP τ 2.3 10 -20 cm 2 4 10 -21 cm 2 2.8 10 -20 cm 2 380 µs σ eL @ 906 nm σ aL @ 906 nm N (5% at.) We first investigated cw laser emission at 900 nm. The experimental setup is presented in Fig. 9. The strong astigmatism of the tapered amplifier was corrected by use of a cylindrical lens with a focal length of 300 mm to both correct and circularize the pump beam. An optical isolator (isolation factor > 20 dB) has been placed at the output of the tapered amplifier to avoid parasitic back-reflections. The pump waist radius could be varied by changing the focal length of the focusing lens. We used a 5%-doped Nd:ASL (N~1.7 10 20 cm -3 ) crystal grown by the Czochralski pulling technique. In order to minimize thermal population of the ground laser level, the copper mount of the Nd:ASL crystal was maintained at 14°C with a water cooling device. The resonator was a simple plane-concave cavity with the plane mirror (HR 900 nm -HT 800 nm) deposited on the input face of the crystal and a 100-mm-curvature concave output mirror transmitting 5% at 900 nm and highly transmitting at the pump wavelength. Both were anti-reflection coated at 1050 nm to prevent laser emission from the intense 4 F 3/2 -4 I 11/2 4-level-transition. In order to maximize the gain and avoid reabsorption the overlap between the laser beam and the pump beam has been optimized by setting up the cavity length and the focus position of the pump beam. As expected, the maximum output power has been obtained with a 3-mm long crystal and a pump waist radius of 10 µm corresponding to a f = 50 mm-focusing lens. We obtained 156 mW for 1.1 W incident pump power. The threshold was 200 mW. The optical-to-optical slope efficiency was 17%.

In order to evaluate the possibility of intracavity frequency doubling of the infrared beam we have tested a laser cavity with an output coupler highly reflecting at both the laser wavelength (R= 99.85%) and a pump waist increased to 20µm. The output mirror was also high reflective for the pump (R= 99.95%), allowing thus pump recycling for increased pump absorption. In these conditions, instead of the 900 nm line obtained previously, we observed a laser emission at 906 nm corresponding to the transition from the 4 F 3/2 level to the highest Stark sub-level of the 4 I 9/2 level. The output power at 906 nm through the low-transmission concave mirror reached 300 mW at the incident pump power of 1.5 W corresponding to an intracavity power of 200 W. This relatively high value is a consequence of the pump recycling which increases the inversion population in the crystal.

The red-shift of the NdASL laser wavelength with reduced intracavity losses has already been observed with Nd:GdVO 4 laser operating on the same 4 F 3/2 -4 I 9/2 transition [14] and with Er:Yb:glass [15]. It is a consequence of the lower absorption cross section at 906 nm as compared to 900 nm (see Table 1). This appears clearly on Fig. 10 which computes the double-pass gain of the two transitions in respect with the pump power taking into account the different experimental set-up. Indeed the threshold at 906 nm is lower than the one at 900 nm in the HR cavity (G=1.01). On the contrary with a higher output coupler (T=5%), the lowest threshold is for the 900-nm emission.

INTRACAVITY SECOND HARMONIC GENERATION

The major aim of this work is to perform intracavity second harmonic generation to reach the blue range. Promising results have already been obtained by intracavity doubling of the Nd:ASL emission under Ti:Sa pumping at 792 nm : 225 mW of blue laser power at 450 nm under 3 W of incident pump power resulting in an optical-to-optical efficiency of 7.5 %, and a power as high as 320 mW with a BiB 3 O 6 crystal in the same operating conditions [START_REF] Varona | CW blues laser emission by second harmonic generation of 900 nm oscillation of Nd-doped strontium and lanthanum aluminate (ASL)[END_REF]. Under diode pumping, second harmonic generation has then been achieved by inserting a LiB 3 O 5 (LBO) crystal into our previous laser cavity (Fig. 12). The LBO crystal was 15-mm long and cut for type-I phase-matching (see Table 2 for its main characteristics). We placed it as close as possible to the Nd:ASL crystal to maximize the intracavity laser intensity. In order to reduce the divergence of the laser beam and match the angular acceptance of the nonlinear crystal, the pump and laser waists have been increased to 20 µm. The temperature of the nonlinear crystal was controlled at 25° C by a Peltier module to adjust and stabilize the phase matching angle of the LBO. We obtained an output power of 53 mW at 453 nm for a maximum pump power of 1.5 W with the 3-mm long Nd:ASL crystal (see Fig. 11). Our results correspond to an optical-to-optical efficiency of 4%. We are mainly limited by the accessible pump power which is limited to 1.5 W. Moreover the results are limited by the degradation of the pump beam from the tapered amplifier at high operating current. The beam was close to the diffraction limit below 40 mW but degraded at maximum output powers with M 2 values around 3.

CONCLUSION

We have demonstrated the first diode pumping of a Nd:ASL crystal. We have developed our own high-brightness wavelength-stabilized pump source based on an extended cavity tapered amplifier with a volume Bragg grating suited to the optical pumping of this crystal (several watts, linewidth < 80pm). It should be noted that the set-up we have used to lock our tapered amplifier in extended cavity can be adapted to other wavelengths with the appropriate volume Bragg grating provided that tapered amplifier structures exist at that wavelength.

A maximum output power of 126 mW at 900 nm has been obtained with a 3-mm long crystal. With a HR output coupler and with pump recycling, we obtained 300 mW at 906 nm. Intracavity frequency doubling has also been performed with a LBO crystal to obtained 53 mW of blue laser at 453 nm. This performance was limited by the beam quality of the pump source at high power. We believe that these results are an important step on the way to a compact and efficient design of a Nd:ASL blue emitting laser. Given the recent results in reliability and power (10W) of tapered laser [16,17], a higher pumping power can be expected which should lead to higher laser powers. Moreover, our results could be improved with a more efficient doubling crystal such as BiBO (BiB 3 O 6 ).

Fig. 1

 1 Fig.1 Schematic of a reflecting volume Bragg grating.

Fig. 2

 2 Fig.2 Computation of the evolution of the diffraction efficiency of a VBG designed at 798 nm (d = 0.7 mm, Δn = 1.746x10 -4 , λ B = 798 nm)

Fig. 3

 3 Fig.3 Set-up of the tapered amplifier in extended cavity

Fig. 4 Fig. 5 Fig. 6

 456 Fig.4 Spectrum of the pumping source for various operating current

Fig. 7

 7 Fig.7 Absorption and emission spectrum of a Nd:ASL crystal

3 WFig. 9

 39 Photonics West -LASE 2008 published in Solid State Lasers XVII : Technology and Devices, Proc. of SPIE Vol. 6871, 68711P, (2008)

8/ 10 /Fig. 10

 1010 Photonics West -LASE 2008 published in Solid State Lasers XVII : Technology and Devices, Proc. of SPIE Vol. 6871, 68711P, (2008)

Fig. 11

 11 Fig. 11 SHG performance (dot -experimental values, line -parabolic fit) -Inset : experimental set-up for second harmonic generation

Table 1

 1 Spectroscopic data of Nd:ASL

Table 2

 2

LBO crystal characteristics length 10 mm deff 0.798 pm.V-1 (θ,ϕ) (90°, 22.5°) Δθ.L @ 906 nm 4.41 mrad.cm

ACKNOWLEDGMENTS

The authors thank Gilles Colas and Christian Beurthe for the polishing of the Nd:ASL crystals. This work has been partially supported by the European Community under the www.BRIGHT.eu integrated project (IP511722). D. Paboeuf acknowledges the funding of his PhD by the French Army (Délégation Générale de l'Armement).


E Frénod

A Mouton

Two-dimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates

In this paper, we present some new results about the approximation of the Vlasov-Poisson system with a strong external magnetic field by the 2D finite Larmor radius model. The proofs within the present work are built by using two-scale convergence tools, and can be viewed as a new slant on previous works of Frénod & Sonnendrücker and Bostan on the 2D finite Larmor Radius model. In a first part, we recall the physical and mathematical contexts. We also recall two main results from previous papers of Frénod & Sonnendrücker and Bostan. Then, we introduce a set of variables which are so-called canonical gyrokinetic coordinates, and we write the Vlasov equation in these new variables. Then, we establish some two-scale convergence and weak-* convergence results.

Introduction

Nowadays, domestic energy production by using magnetic confinement fusion (MCF) techniques is a huge technological and human challenge, as it is illustrated by the international scientific collaboration around ITER which is under construction in Cadarache (France). Since magnetic confinement, needed to reach nuclear fusion reaction, is a very complex physical phenomenon, the mathematical models which are linked with this plasma physics subject need to be rigorously studied from theoretical and numerical points of view. Such a work programme based on rigorous mathematical studies and high precision numerical simulations can bring some additional informations about the behavior of the studied plasma before the launch of real experiments.

The present paper can be viewed as a part of the recent work programme about the mathematical justification of the mathematical models which are used for numerical simulations of MCF experiments. Indeed, the first tokamak plasma models have been proposed by Littlejohn, Lee et al., Dubin et al. or Brizard et al. (see [START_REF] Littlejohn | A guiding center Hamiltonian : A new approach[END_REF], [START_REF] Lee | Gyrokinetic approach in particle simulation[END_REF][START_REF] Lee | Gyrokinetic particle simulation model[END_REF], [START_REF] Dubin | Nonlinear gyrokinetic equations[END_REF], [START_REF] Brizard | Nonlinear gyrokinetic tokamak physics[END_REF][START_REF] Brizard | Foundations of nonlinear gyrokinetic theory[END_REF]) nevertheless most of these models were established by using formal assumptions. For ten years, many mathematicians have been working on mathematical justification of these models, especially the gyrokinetic approaches like guiding-center approximations and finite Larmor radius approximations: many results in this research field are due to Frénod & Sonnendrücker et al. [START_REF] Frénod | Two-scale expansion of a singularly perturbed convection equation[END_REF][START_REF] Frénod | Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field[END_REF][START_REF] Frénod | Long time behavior of the Vlasov equation with a strong external magnetic field[END_REF][START_REF] Frénod | The finite Larmor radius approximation[END_REF], Golse & Saint-Raymond [START_REF] Golse | The Vlasov-Poisson system with strong magnetic field[END_REF][START_REF] Golse | The Vlasov-Poisson system with strong magnetic field in quasineutral regime[END_REF], Bostan [START_REF] Bostan | The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime[END_REF] or, more recently, Han-Kwan [START_REF] Han-Kwan | The three-dimensional finite Larmor radius approximation[END_REF]. These mathematical results mostly rely on two-scale convergence theory (see Allaire [START_REF] Allaire | Homogenization and two-scale convergence[END_REF], Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF]) or compactness arguments.

In this paper, we are focused on the 2D finite Larmor radius model and its mathematical justification: more precisely, the goal is to make a synthesis of previous mathematical proofs of the convergence of (f ǫ , Ẽǫ ), where

f ǫ (x, k, α, t) = fǫ x 1 - √ 2k sin α, x 2 + √ 2k cos α, √ 2k cos α, √ 2k sin α, t , (1.1)
and where ( fǫ , Ẽǫ ) is the solution of the following 2D Vlasov-Poisson system

             ∂ t fǫ + 1 ǫ ṽ • ∇ x fǫ + Ẽǫ + 1 ǫ ṽ2 -ṽ 1
• ∇ ṽ fǫ = 0 , fǫ (x, ṽ, 0) = f 0 (x, ṽ) ,

-∇ x φǫ = Ẽǫ , -∆ x φǫ = R 2 fǫ dṽ -ñe , (1.2)
towards the couple (f, Ẽ) which is the solution of the 2D finite Larmor radius model given in [START_REF] Bostan | The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime[END_REF] (see also Theorem 2 below). The main results on this model are due to Sonnendrücker, Frénod and Bostan, and indicate that (f ǫ , Ẽǫ ) somehow weak-* converges to the solution of the finite Larmor radius model. However the proofs within these articles are based on various assumptions and use various tools.

The first part of the present paper is devoted to a state-of-the-art about the twodimensional finite Larmor radius approximation. Firstly, we recall the procedure which allows us to obtain the dimensionless model (1.2) from the complete Vlasov-Poisson model by considering specific assumptions. Then we recall the two-scale convergence theorem of Frénod & Sonnendrücker [START_REF] Frénod | The finite Larmor radius approximation[END_REF] on the one hand, and the weak-* convergence theorem of Bostan [START_REF] Bostan | The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime[END_REF] on the other hand.

In a second part, we introduce a set of variables which are so-called canonical gyrokinetic coordinates and we reformulate the Vlasov-Poisson system (1.2) in these new variables. Then, we establish a two-scale convergence theorem which only relies on Frénod & Sonnendrücker's assumptions. Finally, we deduce directly from this result Bostan's Finite Larmor radius model (see [START_REF] Bostan | The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime[END_REF]) by adding a somehow non-physical assumption on the electric field which consists in considering a strong convergence of the sequence ( Ẽǫ ) ǫ > 0 .

2 State-of-the-art

Scaling of the Vlasov-Poisson model

This paragraph is devoted to the scaling of the following Vlasov-Poisson model:

         ∂ t f + ṽ • ∇ x f + e m i Ẽ + ṽ × B • ∇ ṽ f = 0 , f (x, ṽ, 0) = f 0 (x, ṽ) , -∇ x φ = Ẽ , -∆ x φ = e ε 0 R 3 ṽ f dṽ - e ε 0 ñe , (2.3)
where x = (x 1 , x2 , x3 ) ∈ R 3
x is the position variable, ṽ = (ṽ 1 , ṽ2 , ṽ3 ) ∈ R 3 ṽ is the velocity variable, t ∈ R + is the time variable, f = f (x, ṽ, t) is the ion distribution function, ñe is the electron density, Ẽ = Ẽ(x, t) is the self-consistent electric field generated by the ions and the electrons, B = B(x, t) is the magnetic field which is applied on the considered plasma, φ = φ(x, t) is the electric potential linked with Ẽ, e is the elementary charge and m i is the elementary mass of an ion.

In this model, the external magnetic field B is assumed to be uniform and carried by the unit vector e 3 . We also assume that the electron density ñe is given for any (x, t) ∈ R 3

x × R + . Following the same approach as in Bostan [START_REF] Bostan | The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime[END_REF], Frénod et al. [START_REF] Frénod | Two-scale expansion of a singularly perturbed convection equation[END_REF][START_REF] Frénod | The finite Larmor radius approximation[END_REF], Golse et al. [START_REF] Golse | The Vlasov-Poisson system with strong magnetic field[END_REF][START_REF] Golse | The Vlasov-Poisson system with strong magnetic field in quasineutral regime[END_REF] and Han-Kwan [START_REF] Han-Kwan | The three-dimensional finite Larmor radius approximation[END_REF], we add the following assumptions:

(i) The magnetic field is supposed to be strong, (ii) The finite Larmor radius effects are taken into account, (iii) The ion gyroperiod is supposed to be small.

We define the dimensionless variables and unknowns x′

= (x ′ 1 , x′ 2 , x′ 3 ), ṽ′ = (ṽ ′ 1 , ṽ′ 2 , ṽ′ 3 ), t ′ , f ′ , Ẽ′ and φ′ by x1 = L ⊥ x′ 1 , x2 = L ⊥ x′ 2 , x3 = L || x′ 3 , t = t t ′ , ṽ = v ṽ′ , f (L ⊥ x′ 1 , L ⊥ x′ 2 , L || x′ 3 , v ṽ′ 1 , v ṽ′ 2 , v ṽ′ 3 , t t ′ ) = f f ′ (x ′ 1 , x′ 2 , x′ 3 , ṽ′ 1 , ṽ′ 2 , ṽ′ 3 , t ′ ) , Ẽ(L ⊥ x′ 1 , L ⊥ x′ 2 , L || x′ 3 , t t ′ ) = E Ẽ′ (x ′ 1 , x′ 2 , x′ 3 , t ′ ) , φ(L ⊥ x′ 1 , L ⊥ x′ 2 , L || x′ 3 , t t ′ ) = φ φ′ (x ′ 1 , x′ 2 , x′ 3 , t ′ ) .
(2.4)

In these definitions, L ⊥ is the characteristic length in the direction perpendicular to the magnetic field, L || is the characteristic length in the direction of the magnetic field, v is the characteristic velocity and t is the characteristic time. We also rescale the electron density as follows:

ñe (L ⊥ x′ 1 , L ⊥ x′ 2 , L || x′ 3 , t t ′ ) = n ñ′ e (x ′ 1 , x′ 2 , x′ 3 , t ′ ) . (2.5)
Following the assumptions on the magnetic field B, we set B as being such that

B = B e 3 . (2.6)
Then, we set L || as the size of the physical device in game. We also link f , E and φ with the characteristic Debye length λ D by

f = n v 3 , E = λ D e n ε 0 , φ = λ D 2 e n ε 0 , (2.7)
and we take λ D as the characteristic length in the direction perpendicular to the magnetic field, i.e. L ⊥ = λ D .

(2.8)

Since we want to take into account the smallness of the gyroperiod and the finite Larmor radius effects, we define the characteristic gyrofrequency ω i and the characteristic Larmor radius r L as

ω i = e B m i , r L = v ω i . (2.9)
With these notations, the Vlasov-Poisson system is rescaled as follows:

                                 ∂ t ′ f ′ + t v    rL λD ṽ′ 1 rL λD ṽ′ 2 rL L || ṽ′ 3    • ∇ x′ f ′ + t λ D e 2 n ε 0 m i v Ẽ′ + t ω i ṽ′ × e 3 • ∇ ṽ′ f ′ = 0 , f ′ (x ′ , ṽ′ , 0) = f 0 ′ (x ′ , ṽ′ ) , Ẽ′ =    -∂ x′ 1 φ′ -∂ x′ 2 φ′ -λD L || ∂ x′ 3 φ′    , -∆ (x ′ 1 ,x ′ 2 ) φ′ - λ D 2 L || 2 ∂ 2 x′ 3 φ′ = R 3 f ′ dṽ ′ -ñ′ e .
(2.10)

Taking into account the finite Larmor radius effects consists in considering a regime in which the Larmor radius is of the order of the Debye length. This implies

r L λ D = 1 . (2.11)
Since the magnetic field is assumed to be strong, the Larmor radius is small when compared with the size of the physical domain. Then it is natural to take

r L L || = ǫ , (2.12)
where ǫ > 0 is small. Assumption (iii) can be translated in terms of characteristic scales by

t ω i = 1 ǫ . (2.13)
Assumption (i) means that the magnetic force is much stronger than the electric force, so we consider

E e v m i ω i = ǫ . (2.14)
Then, removing the primes and adding ǫ in subscript, the rescaled Vlasov-Poisson model writes

               ∂ t fǫ + 1 ǫ ṽ1 ṽ2 • ∇ (x1,x2) fǫ + ṽ3 ∂ x3 fǫ + Ẽǫ + 1 ǫ ṽ × e 3 • ∇ ṽ fǫ = 0 , fǫ (x, ṽ, 0) = f 0 ǫ (x, ṽ) , -∇ (x1,x2) φǫ -ǫ ∂ x3 φǫ = Ẽǫ , -∆ (x1,x2) φǫ -ǫ 2 ∂ 2 x3 φǫ = R 3 fǫ dṽ -ñe , (2.15)
which is the model studied in previous works of Frénod & Sonnendrücker [START_REF] Frénod | The finite Larmor radius approximation[END_REF], Golse & Saint-Raymond [START_REF] Golse | The Vlasov-Poisson system with strong magnetic field[END_REF][START_REF] Golse | The Vlasov-Poisson system with strong magnetic field in quasineutral regime[END_REF], and Bostan [START_REF] Bostan | The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime[END_REF].

Previous results

In this paragraph, we recall two main results about the asymptotic behavior of the sequences ( fǫ ) ǫ > 0 and ( Ẽǫ ) ǫ > 0 when ǫ goes to 0. The first one is based on the use of two-scale convergence and homogenization techniques developed by Allaire [START_REF] Allaire | Homogenization and two-scale convergence[END_REF] and Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF], and was established by Frénod and Sonnendrücker in [START_REF] Frénod | The finite Larmor radius approximation[END_REF]. The second one relies on compactness arguments and was proved by Bostan in [START_REF] Bostan | The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime[END_REF]. After recalling these two results, we discuss the main differences between them. These differences are the source of the motivation of the present paper.

In order to simplify, we consider that the whole model (2.15) does not depend on x3 nor ṽ3 , and we assume that f 0 ǫ = f 0 for all ǫ. Then, it is reduced to a singularly perturbed 2D Vlasov-Poisson model of the form

             ∂ t fǫ + 1 ǫ ṽ • ∇ x fǫ + Ẽǫ + 1 ǫ ṽ2 -ṽ 1 • ∇ ṽ fǫ = 0 , fǫ (x, ṽ, 0) = f 0 (x, ṽ) , -∇ x φǫ = Ẽǫ , -∆ x φǫ = R 2 fǫ dṽ -ñe , (2.16)
where x = (x 1 , x2 ) ∈ R 2 and ṽ = (ṽ 1 , ṽ2 ) ∈ R 2 .

The following theorem can be attributed to Frénod and Sonnendrücker [START_REF] Frénod | The finite Larmor radius approximation[END_REF] (see Theorem 1.5) even if the setting of this paper is a charged particle beam Vlasov-Poisson model not involving any electron density. Nonetheless, the proof of [START_REF] Frénod | The finite Larmor radius approximation[END_REF] works again in the setting of model (2.16) which involves an electron density.

Theorem 1 (Frénod & Sonnendrücker [14]). We assume that, for a fixed p ≥ 2, f 0 is in L 1 (R 4 ) ∩ L p (R 4 ), is positive everywhere and such that R 4 |ṽ| 2 f 0 (x, ṽ) dx dṽ < +∞ .

(2.17)

We also assume that ñe does not depend on t, is in

L 1 (R 2 ) ∩ L 3/2 (R 2
) and satisfies

R 4 f 0 (x, ṽ) dx dṽ = R 2 ñe (x) dx . (2.18)
Then, the sequence ( fǫ , Ẽǫ )

ǫ > 0 is bounded L ∞ 0, T ; L p (R 4 ) × L ∞ 0, T ; W 1,3/2 (R 2 )
2 independently of ǫ. Furthermore, by extracting some subsequences,

fǫ -→ F = F (x, ṽ, τ, t) two-scale in L ∞ 0, T ; L ∞ # 0, 2π; L p (R 4 ) , Ẽǫ -→ Ẽ = Ẽ(x, τ, t) two-scale in L ∞ 0, T ; L ∞ # 0, 2π; W 1,3/2 (R 2 ) 2 .
(2. [START_REF] Lee | Gyrokinetic approach in particle simulation[END_REF])

Moreover, F is linked with G = G(ỹ, ũ, t) ∈ L ∞ 0, T ; L p (R 4
) by the relation

F (x, ṽ, τ, t) = G x + R(-τ ) ṽ, R(-τ ) ṽ, t , (2.20)
and ( G, Ẽ) is the solution of

                                     ∂ t G(ỹ, ũ, t) + 2π 0 R(-σ) Ẽ ỹ + R(σ) ũ, σ, t dσ • ∇ ỹ G(ỹ, ũ, t) + 2π 0 R(-σ) Ẽ ỹ + R(σ) ũ, σ, t dσ • ∇ ũ G(ỹ, ũ, t) = 0 , G(ỹ, ũ, 0) = 1 2π f 0 (ỹ, ũ) , Ẽ(x, τ, t) = -∇ x Φ(x, τ, t) , -∆ x Φ(x, τ, t) = R 2 G x + R(-τ ) ṽ, R(-τ ) ṽ, t dṽ - 1 2π ñe (x) , (2.21)
where

R(τ ) = sin τ 1 -cos τ cos τ -1 sin τ , R(τ ) = cos τ sin τ -sin τ cos τ . (2.22)
In this theorem, L ∞ # (0, 2π; L p (R 4 )) stands for the space of functions h = h(x, ṽ, τ ) being in L ∞ (0, 2π; L p (R 4 )) and 2π-periodic with respect to τ .

As a consequence of this theorem, extracting some subsequences, we have

fǫ * ⇀ f in L ∞ 0, T ; L p (R 4 ) , Ẽǫ * ⇀ Ẽ in L ∞ 0, T ; W 1,3/2 (R 2 ) 2 , ( 2.23)
where f (x, ṽ, t) = 2π 0 F (x, ṽ, τ, t) dτ , and Ẽ(x, t) = 2π 0 Ẽ(x, τ, t) dτ .

(2.24)

By using the relation between F and G, we can easily remark that ( f , Ẽ) is solution of

                               ∂ t f (x, ṽ, t) + 2π 0 2π 0 R(τ -σ) Ẽ x + R(σ -τ ) ṽ, σ, t dσ • ∇ x F (x, ṽ, τ, t) dτ + 2π 0 2π 0 R(τ -σ) Ẽ x + R(σ -τ ) ṽ, σ, t dσ • ∇ ṽ F (x, ṽ, τ, t) dτ = 0 , f (x, ṽ, 0) = 1 2π 2π 0 f 0 x + R(-τ ) ṽ, R(-τ ) ṽ dτ , -∇ x φ(x, t) = Ẽ(x, t) , -∆ x φ(x, t) = R 2
f (x, ṽ, t) dṽñe (x) .

(2.25) We notice that these equations still involve F and Ẽ. In order to make these dependencies disappear, Bostan has proposed in [START_REF] Bostan | The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime[END_REF] a reformulation of the Vlasov equation in guidingcenter coordinates. Before presenting it, we introduce the sequence ( fǫ

) ǫ > 0 defined by fǫ (x, ṽ, t) = fǫ x + ṽ2 -ṽ 1 , ṽ, t , (2.26)
and, in the same spirit, we define the initial guiding-center distribution f 0 by

f 0 (x, ṽ, t) = f 0 x + ṽ2 -ṽ 1 , ṽ, t . (2.27)
Theorem 2 (Bostan [START_REF] Bostan | The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime[END_REF]). We assume that ñe = 1, and that f 0 is 2π-periodic in x1 and x2 , is positive everywhere and satisfies

R 2 2π 0 2π 0 f 0 (x, ṽ) dx dṽ = 1 , R 2 2π 0 2π 0 |ṽ| 2 f 0 (x, ṽ) dx dṽ < +∞ . ( 2

.28)

We also assume that there exists

F 0 ∈ L ∞ (R + ) ∩ L 1 (R + ; r dr) such that ∀ (x, ṽ) ∈ [0, 2π] 2 × R 2 , f 0 (x, ṽ) ≤ F 0 |ṽ| . ( 2

.29)

We also assume that ( Ẽǫ ) ǫ > 0 admits a strong limit denoted with

Ẽ in L 2 0, T ; L 2 # [0, 2π] 2 2 . Then, up to a subsequence, fǫ weakly-* converges to a function f = f (x, v, t) in L ∞ [0, T ) × R 2 ; L ∞ # [0, 2π] 2 verifying f (x, v, t) = 1 2π g x, |v| 2 2 , t , (2.30
)

where g = g(x, k, t) is the solution of                              ∂ t g + E 2 ∂ x1 g -E 1 ∂ x2 g = 0 , g(x, k, 0) = 2π 0 f 0 (x, √ 2k cos α, √ 2k sin α) dα , E (x, k, t) = 1 2π 2π 0 Ẽ x 1 - √ 2k sin α, x 2 + √ 2k cos α, t dα , -∇ x φ(x, t) = Ẽ(x, t) , -∆ x φ(x, t) = 1 2π +∞ 0 2π 0 g x1 + √ 2k sin α, x2 - √ 2k cos α, k, t dα dk -1 .
(2.31)

In this theorem, L 2 # [0, 2π] 2 stands for the space of functions h = h(x) being in L 2 [0, 2π] 2 and 2π-periodic with respect to x1 and x2 . This last result introduces a mathematical justification of the approximation of the Vlasov-Poisson model (2.16) by the finite Larmor radius model which is exactly (2.30)-(2.31). However, in order to prove this convergence result, Bostan considered stronger assumptions on f 0 and ñe than needed to get existence of the weak-* limit ( f , Ẽ) from Theorem 1: the initial distribution f 0 is supposed to be 2π-periodic in x1 and x2 and ( Ẽǫ ) ǫ > 0 is supposed to admit a strong limit in some Banach space.

Convergence result in canonical gyrokinetic coordinates

This section is devoted to a two-scale convergence result for model (2.16) written in a new set of variables which are so-called canonical gyrokinetic coordinates. Then, by adding a nonphysical hypothesis for the electric field Ẽǫ , we obtain straightforwardly Bostan's weak-* convergence result.

Reformulation of Vlasov equation

Following the ideas of Littlejohn [START_REF] Littlejohn | A guiding center Hamiltonian : A new approach[END_REF], Lee [START_REF] Lee | Gyrokinetic approach in particle simulation[END_REF][START_REF] Lee | Gyrokinetic particle simulation model[END_REF], and Brizard et al. [START_REF] Brizard | Nonlinear gyrokinetic tokamak physics[END_REF][START_REF] Brizard | Foundations of nonlinear gyrokinetic theory[END_REF], we define the variables (

x 1 , x 2 , k, α) ∈ R 2 × R + × [0, 2π] by linking them with (x 1 , x2 , ṽ1 , ṽ2 ) ∈ R 4 by x1 = x 1 -ṽ2 , x2 = x 2 + ṽ1 , ṽ1 = √ 2k cos α , ṽ2 = √ 2k sin α . (3.1)
This set of variables is so-called canonical gyrokinetic coordinates: indeed, if we define the characteristics X 1 , X 2 , K, A linked with x 1 , x 2 , k, α by

X1 = X 1 -Ṽ2 , X2 = X 2 + Ṽ1 , Ṽ1 = √ 2K cos A , Ṽ2 = √ 2K sin A , (3.2)
where X1 , X2 , Ṽ1 , Ṽ2 are the characteristics associated with the Vlasov equation (2.16.a), i.e. satisfying

       ∂ t X1 (t) = 1 ǫ Ṽ1 (t) , ∂ t X2 (t) = 1 ǫ Ṽ2 (t) , ∂ t Ṽ1 (t) = Ẽǫ,1 X1 (t), X2 (t), t + 1 ǫ Ṽ2 (t) , ∂ t Ṽ2 (t) = Ẽǫ,2 X1 (t), X2 (t), t - 1 ǫ Ṽ1 (t) , (3.3)
we have

       ∂ t X 1 (t) = -∂ x2 H ǫ X 1 (t), X 2 (t), K(t), A(t), t , ∂ t X 2 (t) = ∂ x1 H ǫ X 1 (t), X 2 (t), K(t), A(t), t , ∂ t K(t) = ∂ α H ǫ X 1 (t), X 2 (t), K(t), A(t), t , ∂ t A(t) = -∂ k H ǫ X 1 (t), X 2 (t), K(t), A(t), t , (3.4)
where the hamiltonian function H ǫ is defined by

H ǫ (x 1 , x 2 , k, α, t) = k ǫ + φ ǫ (x 1 , x 2 , k, α, t) , (3.5)
and φ ǫ is linked with φǫ by the relation

φ ǫ (x 1 , x 2 , k, α, t) = φǫ x 1 - √ 2k sin α, x 2 + √ 2k cos α, t . (3.6)
Then it is straightforward to see that, in the gyrokinetic canonical coordinates, the Vlasov-Poisson system (2.16) has the following shape:

                                           ∂ t f ǫ + E ǫ,2 ∂ x1 f ǫ -E ǫ,1 ∂ x2 f ǫ + √ 2k (E ǫ,1 cos α + E ǫ,2 sin α) ∂ k f ǫ + E ǫ,2 cos α -E ǫ,1 sin α √ 2k ∂ α f ǫ - 1 ǫ ∂ α f ǫ = 0 , f ǫ (x, k, α, 0) = f 0 x 1 - √ 2k sin α, x 2 + √ 2k cos α, √ 2k cos α, √ 2k sin α , E ǫ (x, k, α, t) = Ẽǫ x 1 - √ 2k sin α, x 2 + √ 2k cos α, t , Ẽǫ (x, t) = -∇ x φǫ (x, t) , -∆ x φǫ (x, t) = 2π 0 +∞ 0 f ǫ x1 + √ 2k sin α, x2 - √ 2k cos α, k, α, t dk dα -ñe (x, t) , (3.7)
where x = (x 1 , x2 ) and x = (x 1 , x 2 ), and where f ǫ = f ǫ (x, k, α, t) and E ǫ = E ǫ (x, k, α, t) are linked with f ǫ and Ẽǫ by

f ǫ (x, k, α, t) = fǫ x 1 - √ 2k sin α, x 2 + √ 2k cos α, √ 2k cos α, √ 2k sin α, t , E ǫ (x, k, α, t) = Ẽǫ x 1 - √ 2k sin α, x 2 + √ 2k cos α, t . (3.8)

Two-scale convergence

We set the following notations

Ω = R 2 × R + × [0, 2π] , Γ = R 2 × R + , S = R + × [0, 2π] , (3.9)
and we consider the following Banach spaces, involving periodicity with respect to α:

L p # 0, 2π; L p (Γ) = f ∈ L p (Ω) : f is periodic in α , W 1,p # 0, 2π; W 1,p (Γ) = f ∈ W 1,p (Ω), f (., ., 0) = f (., ., 2π) : f is periodic in α , W 2,p # 0, 2π; W 2,p (Γ) = f ∈ W 2,p (Ω), f (., ., 0) = f (., ., 2π), ∂ α f (., ., 0) = ∂ α f (., ., 2π) : f is periodic in α ,
and we can state the following theorem.

Theorem 3. We assume that, for a fixed p ≥ 2, f 0 and ñe satisfy the assumptions of Theorem 1. Then sequences

(f ǫ ) ǫ > 0 and ( Ẽǫ ) ǫ > 0 of system (3.7) are bounded independently of ǫ in L ∞ 0, T ; L p # (0, 2π; L p (Γ)) and L ∞ 0, T ; W 1,3/2 (R 2 )
2 respectively. As a consequence, there exist F = F (x, k, α, τ, t) and Ẽ = Ẽ(x, τ, t) such that, extracting some subsequences,

f ǫ -→ F two-scale in L ∞ 0, T ; L ∞ # 0, 2π; L p # (0, 2π; L p (Γ)) , Ẽǫ -→ Ẽ two-scale in L ∞ 0, T ; L ∞ # (0, 2π; W 1,3/2 (R 2 )) 2 . (3.10) Furthermore, there exist G = G(x, k, α, t) ∈ L ∞ 0, T ; L p # (0, 2π; L p (Γ)) and E = E(x, k, α, τ, t) ∈ L ∞ 0, T ; L ∞ # (0, 2π; W 1,3/2 # (0, 2π; W 1,3/2 (Γ))) 2 such that F (x, k, α, τ, t) = G(x, k, α + τ, t) , (3.11)
E(x, k, α, τ, t) = Ẽ x 1 - √ 2k sin α, x 2 + √ 2k cos α, τ, t , (3.12)
and verifying

                                                     ∂ t G + E 2 ∂ x1 G -E 1 ∂ x2 G + F k ∂ k G + F α ∂ α G = 0 , G(x, k, α, 0) = 1 2π f 0 x 1 - √ 2k sin α, x 2 + √ 2k cos α, √ 2k cos α, √ 2k sin α , -∇ x Φ(x, τ, t) = Ẽ(x, τ, t) , -∆ x Φ(x, τ, t) = S G x1 + √ 2k sin α, x2 - √ 2k cos α, k, α + τ, t dk dα - 1 2π ñe (x) , F k (x, k, α, τ, t) = √ 2k E 1 (x, k, α, τ, t) cos α + E 2 (x, k, α, τ, t) sin α , F α (x, k, α, τ, t) = E 2 (x, k, α, τ, t) cos α -E 1 (x, k, α, τ, t) sin α √ 2k , (3.13)
where the notation • stands for

u (x, k, α, t) = 2π 0 u(x, k, α -τ, τ, t) dτ . (3.14)
Proof of Theorem 3. Several parts of this proof are only sketched since they can be redundant with [START_REF] Frénod | The finite Larmor radius approximation[END_REF]. However, more details can be found in Mouton [START_REF] Mouton | Approximation multi-échelles de l'équation de Vlasov[END_REF].

Following the same way as in [START_REF] Frénod | The finite Larmor radius approximation[END_REF], we prove that, under the assumptions of Theorem 1, we have

f ǫ (•, t) L p # (0,2π;L p (Γ)) = f 0 L p (R 4 ) , ∀ t ≥ 0 , (3.15)
and, defining ρǫ as ρǫ (x, t)

= S f ǫ x1 + √ 2k sin α, x2 - √ 2k cos α, k, α, t dk dα , (3.16)
that the sequence (ρ ǫ ) ǫ > 0 is bounded in L ∞ 0, T ; L 3/2 (R 2 ) independently of ǫ. Then, we deduce that ( φǫ ) ǫ > 0 and ( Ẽǫ )

ǫ > 0 are bounded independently of ǫ in L ∞ 0, T ; W 2,3/2 (R 2 )
and L ∞ 0, T ; W 1,3/2 (R 2 ) 2 respectively. As a consequence, there exist F = F (x, k, α, τ, t), Φ = Φ(x, τ, t) and Ẽ = Ẽ(x, τ, t) such that

f ǫ -→ F two-scale in L ∞ 0, T ; L ∞ # 0, 2π; L p # (0, 2π; L p (Γ)) , φǫ -→ Φ two-scale in L ∞ 0, T ; L ∞ # 0, 2π; W 2,3/2 (R 2 ) , Ẽǫ -→ Ẽ two-scale in L ∞ 0, T ; L ∞ # 0, 2π; W 1,3/2 (R 2 ) 2 .
(3.17)

Considering a compact set K ⊂ Γ, we easily remark that the sequence (φ ǫ ) ǫ > 0 defined by

(3.6) is bounded in L ∞ 0, T ; W 1,3/2 # (0, 2π; W 1,3/2 (K)
) and that all its second order deriva-

tives except ∂ 2 k φ ǫ are bounded independently of ǫ in L ∞ 0, T ; L 3/2 # 0, 2π; L 3/2 (K))). The sequence (E ǫ ) ǫ > 0 defined by (3.8.b) is bounded in L ∞ 0, T ; W 1,3/2 # (0, 2π; W 1,3/2 (K))
2 independently of ǫ. As a consequence, we claim that there exist Φ = Φ(x, k, α, τ, t) and

E = E(x, k, α, τ, t) such that φ ǫ -→ Φ two-scale in L ∞ 0, T ; L ∞ # 0, 2π; W 1,3/2 # (0, 2π; W 1,3/2 (K)) , E ǫ -→ E two-scale in L ∞ 0, T ; L ∞ # 0, 2π; W 1,3/2 # (0, 2π; W 1,3/2 (K)) 2 . (3.18)
Furthermore, we remark that Φ and E are linked with Φ and Ẽ by the formula

E(x, k, α, τ, t) = Ẽ x 1 - √ 2k sin α, x 2 + √ 2k cos α, τ, t , Φ(x, k, α, τ, t) = Φ x 1 - √ 2k sin α, x 2 + √ 2k cos α, τ, t . (3.19)
Then the vector function A ǫ defined by

A ǫ =     -∂ x2 φ ǫ ∂ x1 φ ǫ ∂ α φ ǫ -∂ k φ ǫ     =       E ǫ,2 -E ǫ,1 √ 2k E ǫ,1 cos α + E ǫ,2 sin α E ǫ,2 cos α -E ǫ,1 sin α √ 2k       , (3.20)
has its three first components which are bounded in L ∞ 0, T ; W

1,3/2 # (0, 2π; W 1,3/2 (K)) , in-
dependently of ǫ and its fourth one in L ∞ 0, T ; L 3/2 # (0, 2π; L 3/2 (K)) and admits a two-scale limit denoted

A = A(x, k, α, τ, t) in L ∞ 0, T ; L ∞ # (0, 2π; W 1,3/2 # (0, 2π; W 1,3/2 (K))) 3 × L ∞ 0, T ; L ∞ # (0, 2π; L 3/2 # (0, 2π; L 3/2 (K))
) . The convergence of the three first components is the consequence of classical embedding of Sobolev spaces in L p spaces. Concerning the convergence of the fourth one we to use that E ǫ,1 and E ǫ,2 are bounded independently of ǫ in L ∞ 0, T ; W 1,3/2 # (0, 2π; W 1,3/2 (K)) and consequently in L ∞ 0, T ; L 6 # (0, 2π; L 6 (K)) and that (2k) -1/2 is, for any k max and any q < 2, in L q (0, k max ). The vector function A is linked with Φ and E as follows:

A =     -∂ x2 Φ ∂ x1 Φ ∂ α Φ -∂ k Φ     =       E 2 -E 1 √ 2k E 1 cos α + E 2 sin α E 2 cos α -E 1 sin α √ 2k       . (3.21)
In order to establish the two-scale limit model, we cannot simply apply Theorem 1.3 of [START_REF] Frénod | The finite Larmor radius approximation[END_REF]: indeed, the formulation (3.7.a) of Vlasov equation does not fit with the assumptions which are needed for applying this theorem since the differential operator f → -1 ǫ ∂ α f cannot be written under the form

f → 1 ǫ M     x 1 x 2 k α     + N • ∇f , (3.22)
where M is a constant square matrix satisfying T r(M) = 0, and N ∈ Im(M). However, the approach which is considered in [START_REF] Frénod | The finite Larmor radius approximation[END_REF] can be adapted to the present case.

Firstly, we prove that there exists a function G such that F (x, k, α, τ, t) = G(x, k, α+τ, t). To reach such a result, we consider a test function ψ = ψ(x, k, α, τ, t) on Ω × [0, 2π] × [0, T ] which is 2π-periodic in α and τ . If we multiply (3.7.a) by ψ(x, k, α, t ǫ , t) and integrate over Ω × [0, T ], we obtain

T 0 Ω f ǫ (x, k, α, t) ∂ t ψ x, k, α, t ǫ , t + 1 ǫ ∂ τ ψ x, k, α, t ǫ , t + A ǫ (x, k, α, t) • ∇f ǫ (x, k, α, t) - 1 ǫ ∂ α ψ x, k, α, t ǫ , t dx dk dα dt = - Ω f 0 (x 1 - √ 2k sin α, x 2 + √ 2k cos α, √ 2k cos α, √ 2k sin α) × ψ(x 1 , x 2 , k, α, 0, 0) dx 1 dx 2 dk dα . (3.23)
Multiplying (3.23) by ǫ and letting ǫ → 0, we obtain the weak formulation of ∂ τ F -∂ α F = 0, which indicates that there exists a function G ∈ L ∞ 0, T ; L p # (0, 2π; L p (Γ)) such that

F (x, k, α, τ, t) = G(x, k, α + τ, t) . (3.24)
Secondly, we introduce the sequence (g ǫ ) ǫ > 0 defined by

g ǫ (x, k, α, t) = f ǫ x, k, α - t ǫ , t . (3.25)
In the spirit of [START_REF] Frénod | The finite Larmor radius approximation[END_REF], we prove that g ǫ strongly converges to 2π G in a given Banach space. For that, we notice that, up to a subsequence, g ǫ two-scale converges to G in L ∞ 0, T ; L ∞ # 0, 2π; L p # (0, 2π; L p (Γ)) since we have

T 0 Ω g ǫ (x, k, α, t) ψ x, k, α, t ǫ , t dx dk dα dt = T 0 Ω f ǫ (x, k, α, t) ψ x, k, α + t ǫ , t ǫ , t dx dk dα dt → 2π 0 T 0 Ω F (x, k, α, τ, t) ψ(x, k, α + τ, τ, t) dx dk dα dt dτ = 2π 0 T 0 Ω G(x, k, α, t) ψ(x, k, α, τ, t) dx dk dα dt dτ , (3.26)
for any test function ψ on Ω×[0, 2π]×[0, T ] which is 2π-periodic in α and τ . As a consequence, g ǫ weakly-* converges to 2π G in L ∞ 0, T ; L p # (0, 2π; L p (Γ)) . Let us prove that this weak-* convergence is a strong convergence in a given Banach space. This is the aim of the following lemma: Proof of Lemma 4. For any compact subset K of Γ, (g ǫ ) ǫ > 0 and (A ǫ ) ǫ > 0 are bounded independently of ǫ in L ∞ 0, T ; L p # (0, 2π; L p (K)) and L ∞ 0, T ; W

1,3/2 # (0, 2π; W 1,3/2 (K)) 3 ×L ∞ 0, T ; L 3/2
# (0, 2π; L 3/2 (K)) respectively. Then, remarking that g ǫ is solution of

     ∂ t g ǫ (x, k, α, t) + A ǫ x, k, α - t ǫ , t • ∇g ǫ (x, k, α, t) = 0 , g ǫ (x, k, α, 0) = f 0 (x 1 - √ 2k sin α, x 2 + √ 2k cos α, √ 2k cos α, √ 2k sin α) , (3.27)
we use similar arguments as the ones given after equation (3.20) to deduce

1. (A ǫ ) ǫ > 0 is bounded independently of ǫ in L ∞ 0, T ; L 3/2 # (0, 2π; L 3/2 (K)) 4 , 2. g ǫ (x, k, α, t) A ǫ (x, k, α -t ǫ , t) ǫ > 0 is bounded in L ∞ 0, T ; L r # (0, 2π; L r (K)) 4 inde- pendently of ǫ with r defined by 1 r = 1 p + 1 q (r ∈ ]1, 3 2 [), 3. (∂ t g ǫ ) ǫ > 0 is bounded in L ∞ 0, T ; W 1,r * # (0, 2π; W 1,r * 0 (K)) * independently of ǫ with 1 r * + 1 r = 1. If r < 3/2, the embedding W 1,r * # (0, 2π; W 1,r * 0 (K)) ⊂ W 1,3/2 # (0, 2π; W 1,3/2 0 (K)
) is compact with density. Furthermore, Rellich-Kondrakov's theorem (see [START_REF] Adams | Sobolev Spaces[END_REF]) gives the compact embedding L p # (0, 2π; L p (K)) ⊂ (W

1,3/2 # (0, 2π; W 1,3/2 0 (K))) * since p ≥ 2.
Then, we apply Aubin-Lions' lemma (see [START_REF] Lions | Quelques méthodes de résolution de problèmes aux limites non linéaires[END_REF]) and we prove that the functional space U defined by

U = u ∈ L ∞ 0, T ; L p # (0, 2π; L p (K)) : ∂ t u ∈ L ∞ 0, T ; W 1,r * # (0, 2π; W 1,r * 0 (K)) * , (3.28) is compactly embedded in L ∞ 0, T ; (W 1,3/2 # (0, 2π; W 1,3/2 0 (K))) * . Since g ǫ ∈ U
for all ǫ, we deduce that the weak-* convergence of

g ǫ to 2π G in L ∞ 0, T ; L p # (0, 2π; L p (K)) is a strong convergence in L ∞ 0, T ; (W 1,3/2 # (0, 2π; W 3/2 0 (K))) * . If r ≥ 3/2, the compact embedding W 1,r * # (0, 2π; W 1,r * 0 (K)) * ⊂ W 1,3/2 # (0, 2π; W 1,3/2 0 (K)) *
is gotten directly. If we introduce the functional space U ′ defined by

U ′ = u ∈ L ∞ 0, T ; L p # (0, 2π; L p (K)) : ∂ t u ∈ L ∞ 0, T ; W 1,3/2 # (0, 2π; W 1,3/2 0 (K)) * , (3.29)
we remark that the sequence (g ǫ ) ǫ > 0 is bounded in U ′ independently of ǫ. By using Aubin-Lions' lemma, we prove that

U ′ ⊂ L ∞ 0, T ; (W 1,3/2 # (0, 2π; W 1,3/2 0 (K))) * is a compact em- bedding. Then, g ǫ strongly converges to 2π G in L ∞ 0, T ; (W 1,3/2 # (0, 2π; W 1,3/2 0 (K))) * .
Let us finish the proof of Theorem 3 by establishing a transport equation satisfied by G. Let us consider a test function ψ = ψ(x, k, α, t) on Ω. We denote its compact support in Γ by K and we assume that ψ and its first order derivatives are 2π-periodic in the α direction. Then we have

T 0 2π 0 K g ǫ (x, k, α, t) ∂ t ψ(x, k, α, t)dx dk dα dt + T 0 2π 0 K g ǫ (x, k, α, t) A ǫ x, k, α - t ǫ , t • ∇ψ(x, k, α, t) dx dk dα dt + 2π 0 K f 0 (x 1 - √ 2k sin α, x 2 + √ 2k cos α, √ 2k cos α, √ 2k sin α) × ψ(x, k, α, 0) dx dk dα = 0 . (3.30) Since the vector function Z ǫ (x, k, α, t) = g ǫ (x, k, α, t) A ǫ (x, k, α -t ǫ , t) is bounded indepen- dently of ǫ in L ∞ 0, T ; L r # (0, 2π; L r (K)) 4 with r ∈ ]1, 3 2 [, there exists a function Z = Z(x, k, α, τ, t) in L ∞ 0, T ; L ∞ # (0, 2π; L r # (0, 2π; L r (K))) 4 such that Z ǫ two-scale converges to Z in L ∞ 0, T ; L ∞ # (0, 2π; L r # (0, 2π; L r (K)))
4 and weakly-* converges to

2π 0 Z(•, •, •, τ, •) dτ in L ∞ 0, T ; L r # (0, 2π; L r (K)) 4 .
If we now restrict our choice of test functions ψ to the one having their support satisfying

K ∩ R 2 × {0} = ∅ , (3.31) then, A ǫ (x, k, α -t ǫ , t) is bounded in L ∞ 0, T ; W 1,3/2 # (0, 2π; W 1,3/2 (K)) 4 independently
of ǫ and admits a two-scale limit in space

L ∞ 0, T ; L ∞ # (0, 2π; W 1,3/2 # (0, 2π; W 1,3/2 (K))) 4 which is exactly A. Since sequence (g ǫ ) ǫ > 0 converges to 2π G in space L ∞ 0, T ; (W 1,3/2 # (0, 2π; W 1,3/2 0 (K))
) * strongly, we can deduce that

Z = 2π G A , (3.32)
almost everywhere on Ω×[0, T ]\{(x, k, α, t), k = 0} and then almost everywhere on Ω×[0, T ], and we obtain the weak formulation of

       ∂ t G(x, k, α, t) + 2π 0 A(x, k, α -τ, τ, t) dτ • ∇G(x, k, α, t) = 0 , G(x, k, α, 0) = 1 2π f 0 (x 1 - √ 2k sin α, x 2 + √ 2k cos α, √ 2k cos α, √ 2k sin α) , (3.33)
when ǫ → 0 in (3.30). Finally, we extend this conclusion to every compact subsets K of Γ by remarking that the measure of R 2 × {0} is null.

In order to obtain Poisson type equations (3.13.c) and (3.13.d), we consider a test function ψ = ψ(x, τ, t) on R 2 ×[0, 2π]×[0, T ] which is 2π-periodic in τ , we multiply Ẽǫ (x, t), ∇ x φǫ (x, t) and ∆ x φǫ (x, t) by ψ(x, t ǫ , t) and we integrate in x and t. We obtain

T 0 R 2 Ẽǫ (x, t) ψ x, t ǫ , t dx dt → 2π 0 T 0 R 2 Ẽ(x, τ, t) ψ(x, τ, t) dx dt dτ , (3.34)
T 0 R 2 ∇ x φǫ (x, t) ψ x, t ǫ , t dx dt → 2π 0 T 0 R 2 ∇ x Φ(x, τ, t) ψ(x, τ, t) dx dt dτ , (3.35)
T 0 R 2 ∆ x φǫ (x, t) ψ x, t ǫ , t dx dt → 2π 0 T 0 R 2 ∆ x Φ(x, τ, t) ψ(x, τ, t) dx dt dτ , (3.36)
when ǫ converges to 0. Since F is the two-scale limit of (f ǫ ) ǫ > 0 , we also have

T 0 R 2 S f ǫ x1 + √ 2k sin α, x2 - √ 2k cos α, k, α, t) dk dα ψ x, t ǫ , t dx dt → 2π 0 T 0 R 2 S F x1 + √ 2k sin α, x2 - √ 2k cos α, k, α, τ, t) × ψ(x, τ, t) dk dα dx dt dτ , (3.37)
when ǫ → 0. Then, gathering convergence results (3.34)-(3.37), we obtain the weak formulation of (3.13.c) and (3.13.d).

Weak-* convergence under non-physical hypothesis

Firstly, we have a direct corollary of Theorem 3:

Corollary 5. Up to some subsequences,

• f ǫ weakly-* converges to f ∈ L ∞ 0, T ; L p # (0, 2π; L p (Γ)) with f defined by f (x, k, α, t) = 2π 0 F (x, k, α, τ, t) dτ , (3.38)
• Ẽǫ weakly-* converges to Ẽ ∈ L ∞ 0, T ; W 1,3/2 (R 2 ) 2 with Ẽ defined by Ẽ(x, t) = 2π 0 Ẽ(x, τ, t) dτ , (3.39)
• φǫ weakly-* converges to φ ∈ L ∞ 0, T ; W 2,3/2 (R 2 ) with φ defined by

φ(x, t) = 2π 0 Φ(x, τ, t) dτ . (3.40)
From now, we add the following non-physical assumption:

Ẽǫ → Ẽ strongly.

(3.41) Lemma 6. Under the hypotheses of Theorem 3 and (3.41), we have

∀ (x, k, α, t) ∈ Ω × [0, T ] , ∂ α 2π 0 Φ(x, k, α -τ, τ, t) dτ = 0 . (3.42)
Proof of Lemma 6. We consider a compact subset K of Γ and a test function

ψ = ψ(x, k, α, t) on Ω×[0, T ] which support in (x, k) is included in K. Since (φ ǫ ) ǫ > 0 is bounded independently of ǫ is L ∞ 0, T ; W 1,3/2 # (0, 2π; W 1,3/2 (K)
) and that all its second order derivatives, except

∂ 2 k φ ǫ , are also bounded in L ∞ 0, T ; L 3/2 # (0, 2π; L 3/2 (K)) , we have T 0 2π 0 Γ ∇ x ∂ α φ ǫ (x, k, α - t ǫ , t) ψ(x, k, α, t) dx dk dα dt = - T 0 2π 0 Γ ∇ x φ ǫ (x, k, α, t) (∂ α ψ)(x, k, α + t ǫ , t) dx dk dα dt = T 0 2π 0 Γ E ǫ (x, k, α, t) (∂ α ψ)(x, k, α + t ǫ , t) dx dk dα dt = T 0 2π 0 Γ Ẽǫ (x, t) × (∂ α ψ) x1 + √ 2k sin α, x2 - √ 2k cos α, k, α + t ǫ
, t dx dk dα dt . Following the same approach, we use the fact that the sequences (∂ 2 k,α φ ǫ ) ǫ > 0 and (∂ 2

α φ ǫ ) ǫ > 0 are bounded in L ∞ 0, T ; L 3/2 # (0, 2π; L 3/2 (K)) independently of ǫ to obtain ∂ 2 k,α φ ǫ (x, k, α - t ǫ , t) * ⇀ ∂ 2 k,α 2π 0 Φ(x, k, α -τ, τ, t) dτ , ∂ 2 α φ ǫ (x, k, α - t ǫ , t) * ⇀ ∂ 2 α 2π 0 Φ(x, k, α -τ, τ, t) dτ , (3.47) in L ∞ 0, T ; L 3/2
# (0, 2π; L 3/2 (K)) . On the other hand, we have

T 0 2π 0 Γ ∂ 2 k,α φ ǫ (x, k, α - t ǫ , t) ψ(x, k, α, t) dx dk dα dt = T 0 2π 0 Γ E ǫ,2 (x, k, α, t) cos α √ 2k (∂ α ψ)(x, k, α + t ǫ , t) dx dk dα dt - T 0 2π 0 Γ E ǫ,1 (x, k, α, t) sin α √ 2k (∂ α ψ)(x, k, α + t ǫ , t) dx dk dα dt = T 0 2π 0 Γ Ẽǫ,2 (x, t) cos α √ 2k × (∂ α ψ)(x 1 + √ 2k sin α, x2 - √ 2k cos α, k, α + t ǫ , t) dx dk dα dt - T 0 2π 0 Γ Ẽǫ,1 (x, t) sin α √ 2k × (∂ α ψ)(x 1 + √ 2k sin α, x2 - √ 2k cos α, k, α + t ǫ , t) dx dk dα dt , (3.48)
and

T 0 2π 0 Γ ∂ 2 α φ ǫ (x, k, α - t ǫ , t) ψ(x, k, α, t) dx dk dα dt = T 0 2π 0 Γ √ 2k E ǫ,1 (x, k, α, t) cos α (∂ α ψ)(x, k, α + t ǫ , t) dx dk dα dt + T 0 2π 0 Γ √ 2k E ǫ,2 (x, k, α, t) sin α (∂ α ψ)(x, k, α + t ǫ , t) dx dk dα dt = T 0 2π 0 Γ Ẽǫ,1 (x, t) √ 2k cos α × (∂ α ψ)(x 1 + √ 2k sin α,
= T 0 2π 0 Γ Ẽ2 (x, t) cos α √ 2k × 2π 0 (∂ α ψ)(x 1 + √ 2k sin α, x2 - √ 2k cos α, k, α + τ, t) dτ dx dk dα dt - T 0 2π 0 Γ Ẽ1 (x, t) sin α √ 2k × 2π 0 (∂ α ψ)(x 1 + √ 2k sin α, x2 - √ 2k cos α, k, α + τ, t) dτ dx dk dα dt = T 0 2π 0 Γ Ẽ2 (x, t) cos α √ 2k × 2π 0 (∂ τ ψ)(x 1 + √ 2k sin α, x2 - √ 2k cos α, k, α + τ, t) dτ dx dk dα dt - T 0 2π 0 Γ Ẽ1 (x, t) sin α √ 2k × 2π 0 (∂ τ ψ)(x 1 + √ 2k sin α, x2 - √ 2k cos α, k, α + τ, t)
+ T 0 2π 0 Γ Ẽ2 (x, t) √ 2k sin α × 2π 0 (∂ α ψ)(x 1 + √ 2k sin α, x2 - √ 2k cos α, k, α + τ, t) dτ dx dk dα dt = T 0 2π 0 Γ Ẽ1 (x, t) √ 2k cos α × 2π 0 (∂ τ ψ)(x 1 + √ 2k sin α, x2 - √ 2k cos α, k, α + τ, t) dτ dx dk dα dt + T 0 2π 0 Γ Ẽ2 (x, t) √ 2k sin α × 2π 0 (∂ τ ψ)(x 1 + √ 2k sin α, x2 - √ 2k cos α, k, α + τ, t) dτ dx dk dα dt = 0 . (3.51)
Then we deduce that (3.54)

∇ (k,α) ∂ α 2π 0 Φ(x, k, α -τ, τ, t) dτ = 0 , ( 3
Then, the two-scale limit model (3.13) can be reduced to

                                             ∂ t G + E 2 ∂ x1 G -E 1 ∂ x2 G + F α ∂ α G = 0 , G(x, k, α, 0) = 1 2π f 0 x 1 - √ 2k sin α, x 2 + √ 2k cos α, √ 2k cos α, √ 2k sin α , -∇ x Φ(x, τ, t) = Ẽ(x, τ, t) , -∆ x Φ(x, τ, t) = S G x1 + √ 2k sin α, x2 - √ 2k cos α, k, α + τ, t dk dα - 1 2π ñe (x) , F α (x, k, τ, t) = E 2 (x, k, α, τ, t) cos α -E 1 (x, k, α, τ, t) sin α √ 2k , (3.55
Γ × [0, T ] such that f (x, k, α, t) = 1 2π g(x, k, t) , ∀ (x, k, α, t) ∈ Ω × [0, T ] , (3.57)
and verifying

                                       ∂ t g + E 2 ∂ x1 g -E 1 ∂ x2 g = 0 , g(x, k, 0) = 2π 0 f 0 (x 1 - √ 2k sin α, x 2 + √ 2k cos α, √ 2k cos α, √ 2k sin α) dα , E (x, k, t) = 1 2π 2π 0 Ẽ x 1 - √ 2k sin α, x 2 + √ 2k cos α, t) dα , -∇ x φ(x, t) = Ẽ(x, t) , -∆ x φ(x, t) = 1 2π S g x1 + √ 2k sin α, x2 - √ 2k cos α, k, t) dk dα -ñe (x) . ( 3
∂ α f (x, k, α, t) = ∂ α 2π 0 F (x, k, α, τ, t) dτ = ∂ α 2π 0 G(x, k, α + τ, t) dτ = ∂ α 2π 0 G(x, k, τ, t) dτ = 0 . (3.59)
Then, if we integrate (3.55.a) in α, we obtain Since E does not depend on α, we can integrate it in α and divide it by 2π: then, we have We can remark that, even if they are based on the non-physical hypothesis (3.41) for ( Ẽǫ ) ǫ > 0 , Bostan's results on the mathematical justification of the 2D finite Larmor radius approximation have been improved: indeed, we have proved that the hypotheses of Theorem 3 coupled with (3.41) are sufficient to prove Theorem 7. Furthermore, this result has been generalized to a non-periodic initial distribution f0 and to a non-uniform electron density ñe .

∂ t f + E 2 ∂ x1 f -E 1 ∂ x2 f = 0 , ( 3
E (x, k, t) = 1 2π 2π 0 2π 0 E(x, k, α -τ, τ, t) dτ dα = 1 2π 2π 0 2π 0 E(x, k, α, τ, t) dτ dα = 1 2π 2π 0 2π 0 Ẽ(x 1 - √ 2k sin α, x 2 + √ 2k cos α, t) dα . ( 3

Conclusions and perspectives

In a first part, we recalled a two-scale convergence result for a 2D Vlasov-Poisson model for a charged particle beam which is due to Frénod and Sonnendrücker. After adapting it to plasma modeling by adding an electron density ñe and some reasonable compatibility conditions, we recalled that such a result trivially implies, up to some subsequences, the weak-* convergence of ( fǫ , Ẽǫ ) to a couple ( f , Ẽ). Then we compared this conclusion to Bostan's results which was established in [START_REF] Bostan | The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime[END_REF] under stronger assumptions.

In a second part, we introduced a new set of variables involving the guiding-center position coordinates and the transverse part of the kinetic energy which is denoted with k. After rewriting the Vlasov equation in these new coordinates, we established a two-scale convergence result by using Frenod & Sonnendrücker's assumptions and the compatibility conditions for ñe which were added in Theorem 1. Then, we proved that (f ǫ , Ẽǫ ) weakly-* converges to a couple (f, Ẽ) and, under an additional non-physical property for ( Ẽǫ ) ǫ > 0 , we established a system of constraints satisfied by (f, Ẽ) through a few computations lines, remarking that this limit model is exactly the 2D finite Larmor radius model which was studied in [START_REF] Bostan | The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime[END_REF].

The first main remark we can do about the present work concerns the formulation of the two-scale limit model in canonical gyrokinetic coordinates under hypothesis (3.41): indeed, the transport equations (3.55.a) and (3.58.a) indicate that k is an adiabatic invariant not only for the weak-* limit system but also for the two-scale limit system. Since k is the dimensionless transverse kinetic energy, this remark confirms previous results about the adiabatic invariant property of the magnetic momentum (see Littlejohn [START_REF] Littlejohn | A guiding center Hamiltonian : A new approach[END_REF], Lee [START_REF] Lee | Gyrokinetic approach in particle simulation[END_REF][START_REF] Lee | Gyrokinetic particle simulation model[END_REF], Brizard et al. [START_REF] Brizard | Nonlinear gyrokinetic tokamak physics[END_REF][START_REF] Brizard | Foundations of nonlinear gyrokinetic theory[END_REF], Grandgirard et al. [START_REF] Grandgirard | A drift-kinetic semi-lagrangian 4D code for ion turbulence simulation[END_REF]).

The second remark which can be done is about the mathematical results which have been established in this paper: by linking previous two-scale and weak-* convergence under common assumptions, i.e. the assumptions from Theorem 3 coupled with (3.41), we have improved Bostan's results by generalizing them to some cases involving a non-uniform electron density or a non-periodic initial distribution f0 .

Since the canonical gyrokinetic variables allow us to simplify the formulation of the twoscale limit model and, under (3.41), allow us to highlight the adiabatic invariant property of the magnetic momentum, they may be a useful tool for a mathematical justification of the full 3D finite Larmor radius model.

From a numerical point of view, the model (3.13) can be used to build a two-scale numerical method in order to simulate the high frequency oscillations of the solution ( fǫ , Ẽǫ ) of the Vlasov-Poisson model (2.16), such as it has been done before for low Mach number problems (see [START_REF] Frénod | Two-scale numerical simulation of the weakly compressible 1D isentropic Euler equations[END_REF]), charged particle beams problems (see [START_REF] Frénod | Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method[END_REF] and [START_REF] Mouton | Two-scale semi-lagrangian simulation of a charged particle beam in a periodic focusing channel[END_REF]) or drift problems in the ocean (see [START_REF] Ailliot | Long term object drift in the ocean with tide and wind[END_REF]). As an example, a numerical method based on the computation of the characteristics associated with the limit transport equation will be simpler to be developed on the formulation (3.13.a) than the formulation (2.21.a): indeed, since we have

∂ x1 E 2 -∂ x2 E 1 = ∂ k F k + ∂ α F α = 0 , (4.63)
the transport equation (3.13.a) can support a time splitting without losing its conservation property (see [START_REF] Besse | Semi-Lagrangian schemes for the Vlasov equation on an unstructured mesh of phase space[END_REF]).

Lemma 4 .

 4 For any compact subset K of Γ, and up to a subsequence, g ǫ strongly converges to 2πG in L ∞ 0, T ; (W

∇∂∇∇ x ∂ α 2π 0 Φ

 0 x ∂ α φ ǫ (x, k, α -t ǫ , t) ψ(x, k, α, t) dx dk dα dt = ψ) x1 + √ 2k sin α, x2 -√ 2k cos α, k, α + τ, t dτ dx dk dα dt = τ ψ x1 + √ 2k sin α, x2 -√ 2k cos α, k, α + τ, tdτ dx dk dα dt = 0 . (3.44) On another hand, we deduce from (3.18) that we also have x ∂ α φ ǫ (x, k, α -t ǫ , t) ψ(x, k, α, t) dx dk dα dt → (x, k, ατ, τ, t) dτ ψ(x, k, α, t) dx dk dα dt .

2 #

 2 , k, ατ, τ, t) dτ = 0 ,(3.46) in L ∞ 0, T ; L 3/(0, 2π; L 3/2 (K)) and that (x, k, α, t) → ∂ α 2π 0 Φ(x, k, α-τ, τ, t) dτ does not depend on x.

  φ ǫ (x, k, α -t ǫ , t) ψ(x, k, α, t) dx dk dα dt = ψ)(x 1 + √ 2k sin α, x2 -√ 2k cos α, k, α + τ, t) dτ dx dk dα dt

α 2π 0 Φ

 0 .52) and that the function (x, k, α) → ∂ α 2π 0 Φ(x, k, ατ, τ, t) dτ is constant on Ω. Then, for any (x, k, t), the function α → 2π 0 Φ(x, k, ατ, τ, t) dτ is monotonic and periodic on [0, 2π] and then must be constant. As a consequence, we finally obtain ∂ (x, k, ατ, τ, t) dτ = 0 .(3.53)Consequently, if we consider the hypothesis (3.41) alongwith those from Theorem 3, we can write 2π 0 Φ(x, k, ατ, τ, t) dτ = 2π 0 Φ(x, k, -τ, τ, t) dτ .

Theorem 7 .

 7 ) and the notation • stands foru (x, k, t) = 2π 0 u(x, k, -τ, τ, t) dτ .(3.56) Under the hypotheses ofTheorem 3 and (3.41), there exists a function g defined on

  .60) and the equation (3.58.a). By integrating the initial condition (3.55.b) in α and dividing it by 2π, we obtain the initial condition (3.58.b).

  .58) Proof of Theorem 7. Since F and G are linked by the relation (3.11), f does not depend on α. Indeed, we have

Acknowledgments: We would like to thank Daniel Han-Kwan for a helpful remark he did and which allowed us to correct a mistake.