Claire Mantel
email: claire.mantel@gipsa-lab.grenoble-inp.fr

Nathalie Guyader

Patricia Ladret

Gelu Ionescu

Thomas Kunlin

Characterizing eye movements during temporal and global quality assessment of h.264 compressed video sequences

published or not. The documents may come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Compressing a video sequence creates various kinds of artifacts. The impairments produced by each type of artifact combine to create the global impression of quality. This paper mainly focuses on a specific type of artifacts that we call 'temporally annoying artifacts'. We define these artifacts as defects induced by compression that are annoying because of their evolution over time, regrouping in this category variation of spatial defects ( 1 ) and specifically temporal defects (such as mosquito noise [START_REF] Segall | Pre-and post-processing algorithms for compressed video enhancement[END_REF][START_REF] Coudoux | A post-processor for reducing temporal busyness in low-bit-rate video applications[END_REF] ). Those 'temporally annoying artifacts' contain both spatial and temporal aspects but their transient nature is their main caracteristic and they cannot be isolated from other artifacts. Therefore those defects play a role complicated to evaluate in the overall quality of a video. We call 'temporal quality' the impression of quality produced by a video while considering only those 'temporally annoying artifacts'.

In the field of video quality, the first metrics that were developped were composed of image algorithms applied independently to frames one after another. The first reason to this approach is that the memory space and computing time required to work on a series of images is much larger than the one to work on one image at a time. Thanks to hardware performances progress, this limitation is less and less an issue. The second cause is the importance granted to the spatial aspects of a video sequence compared to the temporal ones concerning quality perception. Spatial defects are more simple to work on because it is possible to isolate the spatial aspects of compression defects by working on frames extracted from a video sequence. As a result, there have been few studies on the perception of temporal noises and, consequently, few studies have worked on how to measure them. Several studies however classify temporal defects among the main artifacts for h.264 video compression. [START_REF] Ninassi | Considering temporal variations of spatial visual distortions in video quality assessment[END_REF][START_REF] Keimel | No-reference video quality evaluation for high-definition video[END_REF][START_REF] Xia | Perceivable artifacts in compressed video and their relation to video quality[END_REF] It is then interesting to define the specificity of temporal quality assessment (QA-T) as a preliminary for designing a metric dedicated to evaluating 'temporally annoying artifacts' induced by h.264 video compression.

To better understand what parts of videos the observers use while they assess video quality, it is interesting to record their eye movements ( [START_REF] Larson | Can visual fixation patterns improve image fidelity assessment?[END_REF][START_REF] Ninassi | De la perception locale des distorsions de codage à l'appréciation globale de la qualité visuelle des images et vidéos[END_REF][START_REF] Le Meur | Do video coding impairments disturb the visual attention deployment?[END_REF][START_REF] Redi | Interactions of visual attention and quality perception[END_REF] ). Those recordings can then be compared with other recordings from different quality levels but also with the recordings of eye movements from observers freely exploring the same videos. Redi et al. showed that locations fixated during global quality assessment (QA-G) are different from those obtained during free-viewing exploration and that this deviation depends on degradations: the lower the quality is, the higher the deviation is. They found that the main difference is a shrinking of the area fixated: as quality level decreases, observers seem to concentrate their gaze on smaller regions ( 9 ) . LeMeur et al. found that QA-G task moderately modifies fixated locations for videos. However, this modification does not seem to depend on the quality level but rather on the fact that observers watch several times the same video under different quality level. [START_REF] Le Meur | Do video coding impairments disturb the visual attention deployment?[END_REF] The aim of this paper is to characterize the influence of temporal quality assessment (QA-T) over eye-movements. The results focus on two parameters extracted from eye movements, or more exactly from eye fixations: first, the fixated locations and second, the durations of the fixations. We precisely study the influence of task (global and temporal quality assessment) and of quality level on those two parameters. We also analyze the evolution of those parameters over time. To do so, we recorded eye-movements of observers while they were looking at video sequences in three different conditions: free exploration without instructions, exploration to assess global quality and exploration to assess temporal quality. The free-viewing exploration acts as a standard representing 'natural viewing' and the global quality assessment exploration acts as a standard representing 'global quality rating viewing'. To shorten notations, the three tasks are designated with the following abbreviations: QA-G for the Global Quality Assessment task, QA-T for the Temporal Quality Assessment task and FT for the free-viewing task.

The experiment we set up is detailed in section 2. To study the eye-movements recorded during the experiment, we computed three indicators: dispersion of locations fixated between observers, resemblance of locations fixated during QA-G and QA-T to the locations fixated during FT and, finally, fixation durations. These indicators are analyzed in section 3, before the conclusion of this paper presented in section 4.

THE SUBJECTIVE EXPERIMENT

The purpose of this experiment is to collect data on the observers' eye movements while they watch video sequences under three different tasks: FT, QA-G and QA-T. For the two quality assessment tasks (QA-G and QA-T) video sequences are displayed at different quality levels.

Stimuli

We used 10 different video sequences from the CDVL database ( 10 ). A frame extracted from each sequence is shown in figure 1. Those sequences are originally 1080p videos with a 4:2:2 color sampling but for practical reasons we subsampled the color space to 4:2:0 and we downscaled the videos to 720p using Lanczos filter.

Each video sequence was then compressed at 4 bitrates in the h.264 norm ( 11 ) using two different coding configurations: constant bitrate (CBR) and variable bitrate (VBR). The chosen configurations correspond to the two main categories of Figure 1: A frame extracted from each sequence used for the experiment encoding: CBR for broadcasting TV and VBR for storage (Blu-Ray). We used two different coding configurations because they play an important role in the resulting quality ( 4 ). Each sequence was cut to 10s, taking care of avoiding its first and last 2s during which the coding quality can be unstable. The stimuli therefore consisted in 9 different versions of the 10 video sequences: the original (perfect) quality and the compressed 8 versions (2 configurations x 4 bitrates).

The 4 bitrates are different for each sequence as can be seen in Figure 2. The bitrates were chosen fitly for each video to produce 4 different quality levels that are, in our opinion, similar from one sequence to another.

Methodology

The progress of an experiment session is schematized in figure 3. Participants began by watching each of the 10 sequences in perfect quality in a random order without any instruction: this part of the experiment is later referred to as the freeviewing task (FT).

The experimenter then explained to the participant that the rest of the test session consisted in a quality assessment task: observers had to watch videos and to rate their quality. One half of observers participated in the Global Quality rating task (QA-G) and was asked to assess the quality of videos without further explanations.

Conversely, the other half of observers participated in the Temporal Quality rating task (QA-T) and were asked to rate the temporal quality of the video sequences. The explanation was done in three phases:

• First the observers were given a semantic explanation of the concept of temporal quality and 'temporally annoying artifacts'. Temporal quality was defined as the impression of quality produced by a video while considering only 'defects that are annoying because they vary from one frame to another'. It was specified that the 'temporally annoying artifacts' can be 'thin as a pixel or thick as groups of pixels'. • Then observers were asked to put in practice this definition: they watched compressed videos and had to tell where they saw 'temporally annoying artifacts'

• Finally, their identifying of 'temporally annoying artifacts' was validated by the experimenter who pointed out the locations containing 'temporally annoying artifacts' in the above-mentionned videos

We used 5 different video sequences for observers to grasp the concept of temporal quality. Those videos are different from those used during the experiment. Examples of the videos used and are visible in Figure 4. During this explanation, we also took care of displaying a video presenting static blocking defects to observers. It allowed to emphasize that even though it was a very noticeable defect, it should not be taken into account for rating tempoal quality as it did not belong in the 'temporally annoying artifacts'. During the learning phase, observers performed a few tryouts to make sure they took correctly in hand the rating pro-cess. Then, they performed the QA-G or QA-T two times: they watched and rated all of the 90 stimuli displayed in random order two times (following recommendation from ITU [START_REF]Subjective video quality assessment methods for multimedia applications[END_REF] to check the rating of a video).

To avoid fatigue and weariness among participants, we split each of the two quality ratings performed by one observer in two 15min parts separated by a short break. The average overall duration of a test session was 1h15 for observers participating in the QA-G task and 1h20 for those participating in the QA-T task (the difference is due to the explaining of what we call the temporal quality).

Observers

36 observers participated in the experiment: 18 of them performed the QA-G task and the other 18 the QA-T task. The participants were 18 men and 18 women and their age ranged from 23 to 35. They all had a normal or corrected-to-normal visual acuity and correct color vision (tested with Snellen chart and Ishihara plates). All observers were naive regarding the purpose of the experiment and were no expert in video compression.

Set-up

Video sequences were displayed without upscaling on a Sony Bravia KDL40EX500-AEP TV placed at 1,5m from the observers which corresponds to 3 times the height of the videos, as advised by VQEG in. [START_REF] Vqeg | Tutorial on the objective perceptual assessment of video quality: Full reference television[END_REF] The experimental room has dark blue walls, it is isolated from exterior lighting and was lit by two fluorescent daylight-colored (6500K) light bulbs. We used an EyeLink II eye-tracker from SR Research. We recorded eye movements at a frequency of 500Hz in a monocular pupil tracking mode.

Observers were asked to rate the quality of videos on a 9-point scale, shown in Figure 5. Only the upper and lower grades of this scale were defined, respectively with excellent and really bad, to avoid influencing the observers answers (as advised in [START_REF] Jones | Graphic scaling of qualitative terms[END_REF][START_REF] Reiter | Comparing apples and oranges: Subjective quality assessment of streamed video with different types of distortion[END_REF] ). We used the same scale but different questions for the two quality assessment tasks (QA-G and QA-T).

RESULT ANALYSIS

We collected several type of data during this experiment: eye-movements recording and quality grades from observers.

The ratings obtained during the experiment seem coherent and consistant with a quality assessment experiment, as can be seen in Figure 6: the grades decrease with the decrease of quality.

However, the purpose of this paper is to specify the differences between the way an observer watches a video in a freeviewing task (FT), while assessing its global quality (QA-G) or while assessing its temporal quality (QA-T). Hence in this result analysis, we do not focus on the quality rating but rather on the eye movements recorded during the different tasks and for the different quality levels. From our data, we extracted three indicators: dispersion, that represents the variability between observer 'gazes, ROC, that compares the spatial positions of the locations fixated between the FT and the two QA tasks, and finally durations of the recorded fixations * . For the two first indicators, we used the raw data recorded by the eye tracker each 2 ms. The eye movements recordings contained 20 eye positions per frame for 25Hz vidos (or 16 eye positions per frame for videos at 30Hz) for each observer. We computed the median position of these 20 positions, hence we got one position per observer for each frame of a video sequence. We used these positions for all observers to compute the dispersion, which is the variability between observers. We also used these data to compare eye positions recorded during the FT and the two QA tasks. For the last indicator we used the fixation-events detected by the eyelink rather than using the raw eye positions. We extracted the duration of these detected fixations.

Protocol Validation

Before analyzing the results of this experiment, we tried to evaluate whether observers performing the QA-T task actually rated 'temporally annoying artifacts'. To do so, we studied the feedback of observers on the experiment.

At the end of each test session, we asked the participant to rate the difficulty he had to perform the task he was assigned for each video. Observers that participated in QA-T task had in addition to rate the difficulty they had for separating the temporal quality from the global quality. The difficulty to perform the QA-G task, QA-T task and separating temporal from global in the QA-T task are displayed in Figure 7 averaged over all participants and with the corresponding confidence intervals † . We used two statistical inference tests to assess the equality of the participants' answers: Mann-Whitney's U test for the means equality and Brown-Forsythe test for the variances. These tests showed that, for each video, the assumptions of equal mean and variance for the QA-G and the QA-T tasks were not rejected (with p -values above 0, 25 in both cases). This means that we did not find any significant difference on the perceived difficulty to rate global or temporal quality of videos. We also compared the difficulty observers had to separate temporal and global quality during QA-T to the difficulty observers to assess global quality during QA-G. Tests showed that we did not statistically rejected the equality of means for each video except for the video 'GoosePark' (for 'GoosePark' p = 0, 03, for all other videos p are above 0, 19). Moreover, we did not statistically rejected the equality of variances for each video except for the video 'Tulip' (for 'Tulip' p = 0, 01, for all other videos p are above 0, 3). It indicates that the perceived difficulty and the variations between observers of both questions do not present a statistically significant difference.

Figure 7: Evaluations of the difficulty observers had executing the task they were assigned for each video Those elements seem to indicate that our experiment protocol is valid and that observers performing QA-T managed to rate the temporal quality separately from the global quality. We therefore consider in the rest of this analysis that instructions have been correctly executed. However, like all data in subjective quality assessment, the measurements are based on the perception of the participants.

Measure of the resemblance between locations fixated

During the free-viewing task (FT), no instruction was given to observers and they ignored the focus of the experiment so FT represents the 'natural' way to watch a video and it therefore acts as a reference. To compare the locations fixated during each quality assessment (QA) task to those fixated during FT, we computed two indicators: the dispersion of locations fixated by all participants and the Area Under Curve (AUC) using a receiver operating characteristic (ROC) metric. AUC measures the differences between locations fixated during FT and locations fixated during the two quality assessment tasks.

Dispersion represents the variability between the different observers: the lower it is, the closer the locations gazed at by participants were. It is computed as the average distance between the positions gazed at by all participants.

AUC measures the resemblance of a set of fixated locations to a reference. As the free-viewing task represents our reference, we applied AUC to evaluate how much fixations from both QA tasks are similar to fixations from FT. Figure 8 illustrates the computation steps for the Area Under Curve (AUC) indicator. We first built a reference Visual Attention Density Map (V A F T Current ) from the locations fixated during FT. We applied isotropic Gaussian filters with a standard deviation of 1 • centered on each fixation to create the Visual Attention Density Map (V A F T Current ). The degree of similarity between location fixated from QA-G or QA-T and this reference V A F T Current is then assessed by computing a receiver operating characteristic (ROC) metric (as explained by Le Meur et al. [START_REF] Le Meur | Overt visual attention for free-viewing and quality assessment tasks: Impact of the regions of interest on a video quality metric[END_REF] ). The ROC curve represents the True Positive rate as a function of the False Positive rate and the AUC is the area under this curve. The true positives are the locations fixated during QA-G that are also fixated during FT. The false positives are locations fixated during QA-G for a frame of another video at another quality level (the frame number, video and quality level are randomly picked) that are also fixated during FT. The AUC values are between 0,5 and 1. The more similar the fixated locations are to the reference, the closer the AUC will be to 1; random locations on the contrary produce an AUC of 0,5.

Influence of quality on the eye movements: fixation locations and durations

To study the influence of quality over the chosen indicators, we present their variations as a function of the quality level. The Orgn quality level designates the original quality of the videos (the perfect quality). The terms V BR and CBR designates the two compression configurations, respectively Constant Bitrate and Variable Bitrate. The numbers associated to those configurations indicates the compression level, 1 being the lowest compression (highest quality) and 4 the highest compression (lowest quality).

Dispersion

To study how widely the fixated locations by the observers are spread during each task, we computed the dispersion between locations fixated. As previously said, for each video frame and for each observer, we computed the median position of the locations gazed at. We then computed the dispersion (the Euclidean distances between one point and all the others points) between those median positions of all observers for each frame. Then, the dispersion values are averaged over all the frames of the video. The dispersions for the three tasks are shown in Figure 9 as a function of quality level. The lower dispersion is, the closer the locations that observers fixated are.

Figure 9: Dispersion between the locations that participants fixated as a function of quality level for the three tasks (Global Quality, Temporal Quality and Free Task). Note that FT value exists only for the original quality so we drew the corresponding confidence interval for all quality levels Compared to the dispersion values for FT, the dispersion values for both quality tasks (QA-G and QA-T) are much lower when the quality of the displayed video are either perfect (Orgn) or low (CBR3, V BR3, CBR4, V BR4). It means that observers looked at closer locations when assessing the quality of perfect or low quality videos than when watching without instruction videos of perfect quality. The difference is smaller for the other quality levels.

AUC

To study the resemblance of the participants' fixations during each QA task and the free task, we computed the AUC indicator, shown in Figure 10 as a function of quality level. The AUC for Global Quality/Free Task measures the resemblance of the locations fixated during QA-G compared to the locations fixated during FT used as reference. The higher an AUC value is, the higher the resemblance is.

The AUC values for all quality levels are considerably higher for the FT/FT AUC than for QA-G/FT AUC or QA-T/FT AUC. Those results concord with the conclusions from Ninassi et al. or LeMeur et al. that observers performing a QA task do not gaze at the same locations as during a free task. [START_REF] Ninassi | Considering temporal variations of spatial visual distortions in video quality assessment[END_REF][START_REF] Le Meur | Do video coding impairments disturb the visual attention deployment?[END_REF] Figure 10: AUC indicator of resemblance of the locations fixated by the participants between QA-G and FT and between QA-T and FT as a function of quality level. Note that FT value exists only for the original quality so we drew the corresponding confidence interval for all quality levels. The resemblance of the positions fixated by one half of the participants in FT and the other half in FT acts as a reference for the natural eye movements This difference is smaller for QA-T than for QA-G, suggesting that participants look at locations closer to the FT ones when they rate temporal quality than when they rate global quality. The quality level of the video displayed has no visible influence on the resemblance of fixations from a quality rating task to the FT reference.

Fixation duration To further characterize the way participants look at videos, we computed the average fixation durations for the three tasks. The fixation duration, averaged over all participants and all sequences, is shown for each task in Figure 11 as a function of quality level.

Figure 11: Duration of visual fixations for each of the 3 tasks as a function of quality level, averaged over all participants and all sequences. Note that FT value exists only for the original quality so we drew the corresponding confidence interval for all quality levels.

The average fixation duration of observers performing a quality assessment strongly depend on the quality level of the displayed video. The lower the quality level of the videos displayed is, the longer observers stare at objects.

Observers rating temporal quality clearly stare at objects longer than during free-viewing or global quality rating.

Evolution of the parameters over time

To study the evolution of eye movements over time, we computed the evolution as a function of time of the three indicators presented above: dispersion of the participants' gazes, resemblance of locations fixated during QA tasks to the FT reference (through AUC) and fixation duration. The results for each task averaged over all sequences and all quality levels are visible in Figure 12 for the dispersion. The fixations duration for each task averaged over all sequences for the VB4 quality is visible in Figure 13. We are analyzing fixations duration for the VB4 quality level because it is the quality level for which the the average values for each tasks differ the most (cf Figure 11). Taking only into account the evolution of the curves over time and not their respective level, it is clearly visible that the three curves contain two parts: during approximately 1 to 2 seconds, the indicators are evolving a lot whereas they stabilize during the rest of the video sequences. Asd the results are similar for the evolution of AUC over time, we did not include the corresponding figure in this paper. This evolution is common to the free-viewing task as well as the quality rating ones, it suggests that the participants' attention is mainly attracted by the low-level features from the videos during the first 1 or 2 seconds and that the instructions only seem to influence the participants' eye movements afterwards.

CONCLUSION

The aim of this paper was to characterize the influence of temporal quality assessment task (QA-T) on eye-movements of observers compared to the influence of Global Quality assessment task (QA-G) and compared to a Free Task (FT). To collect ground truth data, we set up an experiment combining quality assessment and recording of eye-movements with an eye-tracker. We analyzed the recorded eye-movements through three indicators: dispersion of fixated locations for each task, ROC/AUC between one task and the FT reference and fixation duration for each task.

Results showed that the quality level of the displayed video had no visible effect on the locations that observers fixated during both quality assessment tasks (QA-G and QA-T). However the quality highly influenced the fixations duration, i.e. the time during which observers kept their gaze fixated on one object. For both quality assessment tasks, the lower the quality of the displayed video were, the longer the fixations lasted.

Another interesting finding was that the instruction given to participants had no noticeable influence over the studied indicators (AUC, dispersion or fixation duration) during the first 1 to 2 seconds. It suggests that during those first diffusion instants, low-level (signal) properties of the videos might be the main factor driving the eye-movements. The assigned task only seemed to affect those indicators during the last 8 to 9 seconds of the diffusion, whichever the considered task (FT, QA-G, or QA-T).

We finally determined two elements that differed according to the type of quality assessed (global or temporal). The average fixation duration during QA-G was similar to that of FT whereas the average fixation duration during QA-T was much longer to that of FT. Conversely, the locations fixated during QA-T were resembling more closely to those fixated during FT than those fixated during QA-G.

Figure 2 :

 2 Figure 2: Frame per second and bitrates chosen for each sequence

Figure 3 :

 3 Figure 3: Organization of an experiment session

  (a) Frame extracted from Sprinkler (b) Difference between (a) and the consecutive frame (c) Frame extracted from RushHour (d) Difference between (c) and the consecutive frame

Figure 4 :

 4 Figure 4: (a) and (c) are frames from two of the videos used to explain what are 'temporally annoying artifacts'. Image (b) and (d) illustrate the variation from one frame to another. Those variations are due to the motion in the video and to the 'temporally annoying artifacts' present in the videos (the difference frame was not shown to observers, it only serves here as a way to show 'temporally annoying artifacts' on paper).

Figure 5 :

 5 Figure 5: The rating scale for the global quality evaluation task (QA-G) in French. The question asked to participants translates as 'What was the quality of the video?'

Figure 6 :

 6 Figure 6: MOS (Mean Opinion Scores) averaged over all observers and all sequences and their correspondng Confidence Interval for the QA-G and QA-T tasks

Figure 8 :

 8 Figure 8: Computation process of a similarity indicator: the Receiver Operating Characteristics (ROC) -Area Under Curve (AUC). In this example it evaluates the similarity of locations fixated during QA-G with locations fixated during FT (serving as a reference)

Figure 12 :Figure 13 :

 1213 Figure 12: Dispersion of the locations fixated by the participants as a function of time for each task (Global Quality, Temporal Quality and Free Task)

† The ratings attributed to the BlueSpruce video are quite lower than those for other sequences. We suppose this originates in the content of the sequence: a traveling on a spruce with snow falling on it that occupies the screen really homogeneously. We also believe we chose too high bitrates because we overestimated the impact produced by the present defects on the perceived quality


C Michel

S Tascu

V Doya

P Aschiéri

W Blanc

O Legrand

F Mortessagne

Experimental phase-space-based optical amplification of scar modes

Waves billiard which are chaotic in the geometrical limit are known to support non-generic spatially localized modes called scar modes. The interaction of the scar modes with gain has been recently investigated in optics in micro-cavity lasers and vertically-cavity surface-emitting lasers. Exploiting the localization properties of scar modes in their wave analogous phase space representation, we report experimental results of scar modes selection by gain in a doped D-shaped optical fiber.

Wave chaos is investigating the wave behavior of bounded systems with chaotic classical dynamics. The common feature of chaotic modes in a linear (passive) medium is ergodicity [1]: modes of a chaotic cavity exhibit a generic statistically uniform field distribution [2]. Scar modes that result from constructive wave interferences along the least unstable classical periodic orbits break the ergodicity property [3] but are rather scarce events. Even though scar phenomenon has been intensively investigated since their numerical discovery by Mc Donald and Kaufman [4], their localization feature still arouses interest as in the profilic context of graphene investigations [5]. Interaction between spatially localized scar modes and nonlinear effects is of fundamental interest and constitutes a recent evolution of Wave Chaos. The subtle interplay of gain with wave chaos and specifically scar modes has been widely examined in optics in micro-cavity lasers [6][7][8] and in vertically-cavity surfaceemetting lasers [9]. The problem of the interaction between the lasing medium and the light field in a twodimensional (2D) laser cavity is still a vivid research area [10]. In the presence of a saturable nonlinear effect, which does not involve any phase change, scar modes take a crucial part on the improvement of the lasing efficiency. Thanks to their low-loss properties compared with generic ergodic modes, scar modes are commonly selected during the lasing process leading to enhanced emission directivity and lowered lasing threshold [11]. In recent studies, the use of gain has also proved to favor the selection of scar modes in guided optics: numerical investigations revealed that the interaction of the chaotic modes with a localized active medium in a D-shaped optical fiber amplifier led to a selective amplification of scar modes [12].

In this letter, we report the first, to our knowledge, experimental demonstration of a selective amplification of scar modes through gain in a highly multimode chaotic system. This amplification is based upon the realization of a D-shaped optical fiber doped with a spatially localized gain. The amplified scar modes are built on the 2-bounce periodic orbit (2-PO) (see inset of Fig. 1) of the associated D-shaped billard. Using the specific features of the scar modes in their wave-analogous phase space representation and a spatially coherent amplification process, we show that the selection mechanism of scars is strong enough to enhance scar modes and make them take over ergodic modes. Indeed, we can establish that certain non-scarred modes, displaying a good spatial overlap with the gain area, remain unamplified while the neighboring scar modes are clearly amplified, essentially because those non-scarred modes are not phase-space localized in the way the 2-PO scar modes are.

Our experimental system is a non standard Ytterbium doped multimode optical fiber with a core diameter of 121 µm, truncated at its half radius (see inset in Fig. 1). L1, lens (15-mm working distance), L2, lens (10-mm working distance), L3, lens (2.5-cm focal length f), DM, dichroic mirror (T=90%@1064 nm, R=90%@978 nm), P1, P2, 3-D positioning supports, F1, bandpass filter (3 nm @1064 nm), BC, beam collimator, BE, beam expander, M, mirror. Right inset, transverse cross section of the D-shaped fiber amplifier with the off-centered doped area (circle) and the 2-PO along the x-axis. The polymer cladding is not represented. Color online

We fabricate the fiber preform by a standard MCVD (modified chemical vapor deposition) method. The core is made of pure silica including a 15-µm-diameter doped area of Ytterbium, free from any other co-dopant. A low refractive index polymer surrounds the silica core and acts as an optical cladding. The doped area is off-centered and located in a specific position deduced from numerical investigations [12] which corresponds to the maximum of intensity for most of the 2-PO scar modes. The preform has been cut and shaped before being stretched for this purpose. The optical index mismatch between the doped area and the D-shaped core is induced by the ions concentration and takes the value ∆n = 5 × 10 -4 ([Yb 3+ ] = 900 ppm). This value of ∆n, coupled with a triangular index profile, prohibits any light guidance in the doped area. Both sides of the fiber are optically polished to avoid either at the input or at the output any diffraction induced by defects. The signal is given by a YAG laser (P s =400 mW) at λ s =1064 nm and expanded thanks to a couple of microscope objectives (×4 and ×10) that play the role of a beam expander (BE). The resulting beam, of 5 mm of diameter, is sent through a 15-mm working distance lens (L 1 ) and is transmitted by a dichroic mirror (DM, with 90 % of transmission at λ s ). The pump (P p =10 W) is provided by a cw laser diode (coupled in an angle-polished fiber of 105-µm diameter) at λ p =974 nm. The pump is focused thanks to a 10-mm working distance lens (L 2 ) on the input of the D-shaped optical fiber through the dichroic mirror (90 % of reflexion at λ p ). For an optimization of the pump absorption (excitation of a maximized number of modes), the beam is strongly focused into the multimode core [13,14]. The pump absorption coefficient has been measured under low pump power to prevent any population inversion and takes the value α p = 6.5 dB/m. The absorption and emission cross sections of the pump and the signal are respectively σ pa = 2.65 × 10 -24 m 2 , σ pe = 2.65 × 10 -24 m 2 , σ sa = 5.56 × 10 -26 m 2 and σ se = 6.00 × 10 -25 m 2 . The gain of the signal has been measured along a 20 mlength fiber, and the maximum value G = 4.5 dB (with an output signal to noise ratio of 27 dB) is obtained at the optimal length L opt = 15 m. This gain value, which is rather low as compared to standards of telecom optical amplifiers, serves our purpose in demonstrating that amplification is really at work in our highly multimode fiber. At the output of the fiber, the residual pump is filtered by a bandpass filter F 1 of 3 nm-width around λ s and a CCD camera is placed in the focal plane of a 2.5 cm-focal length lens to image the far-field intensity pattern. As we want to observe strong signatures of waves localization associated with scar modes, we choose to vizualize the signal intensity in the far-field configuration. Indeed, the square modulus of the spatial Fourier transform of the field distribution gives some information about the directions and modulus of the outgoing wave vectors.

At the fiber entrance, we control the direction of the transverse wave vectors propagating in the fiber. A selection of the injected transverse wave vectors is obtained by a (x, ŷ)-translation of the polished input of the optical fiber in the divergent part of the focused signal beam. The figure 2(a) presents the typical far field intensity of the signal obtained at the output of the multimode 15 mlength D-shaped fiber amplifier when pump is off. Even if the injection of the signal is optimized to excite a scar mode among few ergodic modes, the bending of the fiber tends to distribute the signal intensity among numerous other modes. The observed statistically uniform distribution of intensity reveals the superposition of a large number of modes in the fiber. Figure 2(b) represents a cut along the κ x -axis for κ y = 0. It confirms the broad distribution of transverse wavenumbers. For the same signal configuration, but with pump turned on, the figure 2(c) presents the far field intensity pattern of the outgoing signal. A pair of high intensity spots symmetrically distributed on both sides along the κ x -axis appears in the center of the far field. These two peaks demonstrate that the signal propagates along the fiber with two privileged transverse directions given by κ and -κ along the x-direction. Those directions are associated with those of the 2-PO. The observation of two symetrical peaks along the 2-PO direction is the fingerprint of a scar mode in the far-field intensity representation as can be seen in the κ x -cut of the far-field intensity pattern of a typical calculated scar mode (Fig. 2(f)). The transverse wavenumber corresponding to the observed experimental peaks can be estimated from figure 2(d). The calibration of the far field representation gives one pixel = (2.405±0.005)×10 -3 µm -1 and then, the transverse wavenumber associated to the peaks is deduced. We get κ = (0.279 ± 0.003) µm -1 = (17.0 ± 0.2)/R. This value has to be compared with the theoretical value deduced from a condition of phase coherence of waves that bounce back and forth along the 2-PO in the D-shaped billiard:

κ p L + ∆φ + π 2 = 2πp (1)
where L = 3R is the 2-PO-length, ∆φ is the phaseshift induced by the reflexion at the core/cladding interface, the additional π/2 phaseshift is a consequence of the single self-focal point of the 2-PO, and p is an integer that gives the order of the scar mode. For p = 7, we get a value κ p=7 = 17.08/R which is in perfect agreement with the measured value: the scar mode of order 7 is selected thanks to the localized gain. We change the signal launching condition by translating the fiber input in the (x, ŷ) plane (Fig. 1) in order to seed the amplification process on another scar mode. Figure 2(e) reports the enhancement of two symetrical peaks along the κ x -direction for this new input illumination. The measured value of the transverse wavenumber κ = (15.0 ± 0.2)/R has to be compared to the calculated value (Eq. 1) κ p=6 = 15.0/R. The agreement is again remarkable and one can deduce that the scar mode of order p = 6 is enhanced thanks to amplification. These two examples are the proof that, by controlling the initial illumination condition, one can select a given scar mode along the 2-PO thanks to the optical amplification process. These preliminary experimental results demonstrate our ability to selectively amplify scar modes in a highly multimode fiber thanks to a localized gain. In agreement with common fiber amplification processes, the spatial overlap of the modes with the active area is usually the main control parameter of the amplification efficiency. We propose a complementary interpretation of the scar modes enhancement by gain based on the physical nature of scars. According to E. J. Heller, "scars manifest themselves as enhanced probability in phase space, as measured by overlap with coherent states placed on the periodic orbits" [3]. Therefore, scar modes, viewed as fingerprints of classical orbits in the near-field intensity pattern, also show strongly privileged wave directions as highlighted through their far-field intensity. Both spatial and directional features of scar modes can be most efficiently combined in the so-called Husimi representation [15]. The Husimi representation is commonly used as a wave equivalent to the classical Poincaré surface of section. Thus, it establishes a strong correspondence between some particular modes and their associated classical trajectories in the semiclassical regime. In the case of 2D cavities, the Husimi function measures the normal derivative of the eigenfunction on the boundaries of the cavity [START_REF] Birkhoff | Dynamical systems[END_REF][START_REF] Crespi | [END_REF][18][19]. Fig. 3(b-d-f) present the Husimi representation of three different calculated modes of the chaotic optical fiber with metallic boundary in the 2dimensional space (θ, κ sin χ) with θ the curved abscissa, κ the transverse wavenumber of the modes and κ sin χ the projection of the transverse wavevector along the tangential direction to the boundary (Fig. 3(a)). The Husimi representation (Fig. 3(b)) of a generic ergodic mode (Fig. 3(a)) generally explores a large part of the available space: no specific transverse wavevector directions appear to be prevailing. This observation is consistent with the hypothesis of generic modes built upon an ergodic semiclassical behavior [2,[START_REF] Berry | Session XXXVI -Chaotic behavior of deterministic systems[END_REF]. Ergodic modes result from the superposition of a large number of plane waves with fixed transverse wavenumber κ but random directions χ. On the contrary, the Husimi representation (Fig. 3(d)) of a scar mode associated to the 2-PO (Fig. The spatial and directional extensions of the Husimi pattern define a range of angular abscissa ∆θ around θ 0 and of transverse wavevector directions ∆χ derived from ∆(κ sin χ) around χ 0 . Thus, a scar mode is built upon few highly directional plane waves, ensuring a strong spa-tial coherence. In a geometrical approach in terms of rays in the D-shaped billiard, this span of initial conditions extracted from the Husimi distribution determines an area in the billiard plane where rays are converging. This area is located in the vicinity of the so-called selffocal point of the 2-PO. This location of the active ions guarantees an efficient round-trip of rays through the active medium, as well as a spatially coherent amplification and also a good overlap with the field amplitude of the scar modes. We stress that all these conditions should be fulfilled to assure a scar mode amplification. Indeed, we observe that non-scarred (NS) modes can be insensitive to amplification despite a good spatial overlap with the gain medium. For instance, the field amplitude of the NS mode of Fig. 3(e) has a good spatial overlap with the doped area but is not selected during the amplification process [21]. Using a Beam Propagation Method algorithm [START_REF] Feit | [END_REF], we simulate the propagation of a gaussianlike initial signal injected into the fiber with a transverse wavenumber κ NS = 17.07/R that corresponds to the specific NS mode. The modulus of the transverse wavevector is fixed, but its direction is chosen so as to avoid the 2-PO direction and thus a 2-PO scar mode excitation. FIG. 4. Intensity spectrum at the input of the fiber amplifier (dashed) and after amplification at the output of the fiber (gray) associated to a gaussian signal launched into the fiber with κNS. Color online Fig. 4 presents the intensity spectrum at the output of the D-shaped fiber amplifier. The targeted NS mode is not amplified whereas the scar mode of order p = 7 is strongly amplified despite its reduced contribution in the initial spectrum: the spatial field overlap is not able to clarify this favored amplification of the scar mode. The explanation is supplied in the analysis of the Husimi representation (Fig. 3(f)). Indeed, the NS mode presents a broad distribution of transverse wavevectors directions in the Husimi representation that prohibits a coherent amplification mechanism. As a consequence, a strong localization in the Husimi representation around the signature of the 2-PO is essential to ensure the efficiency of modes amplification.

In this paper, we have presented an experimental observation of scar modes amplification in the D-shaped optical fiber with localized gain. An active medium with a spatial location controlled by the phase space signature of scars is a propitious system to selectively amplify scar modes. Investigations about scar modes in nonlinear systems are opening new attractive issues. For instance, having a scar amplifier may offer new perspectives for optical communications. The intensity spectrum of a scar modes amplifier has a response similar to the spectral response of a Fabry-Perot cavity. Each scar mode appears well-separated with a spacing between adjacent transverse wavenumbers equal to 2π/L ′ with κL ′ = 2πp. These well-resolved modes injected in a multimode fiber would be less sensitive to crosstalk and could be employed as independent transmission channels in a process of mode division multiplexing [23]. We express grateful thanks to S. Trzesien and M. Ude for their priceless involvement in the preform and fiber manufacture.

FIG. 1 .

 1 FIG.1. Experimental setup for the amplification of scar modes in the double clad Ytterbium doped fiber amplifier. L1, lens (15-mm working distance), L2, lens (10-mm working distance), L3, lens (2.5-cm focal length f), DM, dichroic mirror (T=90%@1064 nm, R=90%@978 nm), P1, P2, 3-D positioning supports, F1, bandpass filter (3 nm @1064 nm), BC, beam collimator, BE, beam expander, M, mirror. Right inset, transverse cross section of the D-shaped fiber amplifier with the off-centered doped area (circle) and the 2-PO along the x-axis. The polymer cladding is not represented. Color online

FIG. 2 .

 2 FIG. 2. Far field intensity observations (a), (c) and their transverse horizontal cut at κy = 0 (b), (d) for the unpumped and pumped configuration respectively for κp = 7. Far field intensity pattern and transverse horizontal cut superimposed for the pumped configuration with κp = 6 (e). Horizontal cut of the far field intensity of a calculated scar mode (f). Color online

3

 3

FIG. 3 .

 3 FIG. 3. Field intensity of specific modes and their respective Husimi representations. For a generic speckle mode (a),(b); for a scar mode along the 2-PO (c), (d); and for the mode with κNS (e),(f). Color online