Hadi Fayad
email: <hadi.fayad@univ-brest.fr

Keywords: Fiducial Markers, Four-Dimensional Computed Tomography, standards, Humans, Movement, Respiration respiratory motion, patient external surface, 4D CT

Purpose

Current respiratory motion monitoring devices used for motion synchronization in medical imaging and radiotherapy provide either 1D respiratory signals over a specific region or 3D information based on few external or internal markers. On the other hand, newer technology may offer the potential to monitor the entire patient external surface in real time. The main objective of this study was to assess the motion correlation between such an external patient surface and internal anatomical landmarks motion. Methods Four dimensional Computed Tomography (4D CT) volumes for ten patients were used in this study. Anatomical landmarks were manually selected in the thoracic region across the 4D CT datasets by two experts. The landmarks included normal structures as well as the tumour location. In addition, a distance map representing the entire external patient surface, which corresponds to surfaces acquired by a Time of Flight (ToF) camera or similar devices, was created by segmenting the skin of all 4D CT volumes using a thresholding algorithm. Finally, the correlation between the internal landmarks and external surface motion was evaluated for different regions (placement and size) throughout a patient s surface. ' Results Significant variability was observed in the motion of the different parts of the external patient surface. The larger motion magnitude was consistently measured in the central regions of the abdominal and the thoracic areas for the different patient datasets considered. The highest correlation coefficients were observed between the motion of these external surface areas and internal landmarks such as the diaphragm and mediastinum structures as well as the tumour location landmarks (0.8 0.18 and 0.72 0.12 for the abdominal ± ± and the thoracic regions respectively). Worse correlation was observed when one considered landmarks not significantly influenced by respiratory motion such as the apex and the sternum. Discussion and conclusions There were large differences in the motion correlation observed considering different regions of interest placed over a patients' external surface and internal anatomical landmarks. The positioning of current devices used for respiratory motion synchronization may reduce such correlation by averaging the motion over correlated and poorly correlated external regions. The potential of capturing in real-time the motion of the complete external patient surface as well as choosing the area of the surface that correlates best with the internal motion should allow reducing such variability and associated errors in both respiratory motion synchronization and subsequent motion modeling processes.

Technical note: Correlation of respiratory motion between external patient surface and internal anatomical landmarks

INTRODUCTION

Respiratory motion reduces the qualitative and quantitative accuracy of multimodality imaging as well as the precision in the dose delivered to tumor and healthy tissues during radiation therapy (RT). To minimize the impact of respiratory induced organ and tumor motion different methods have been developed. The AAPM Task Group 76 report lists five different approaches to deal with [START_REF] Keall | The management of respiratory motion in radiation oncology report of AAPM Task Group 76 AAPM Report 91[END_REF] respiratory motion in treatment planning: (1) motion-encompassing methods ; (2) forced shallow breathing ; (3) breath hold ; (4) [START_REF] Van Herk | Errors and margins in radiotherapy[END_REF] [3 ] [START_REF] Nakamura | Reproducibility of the abdominal and chest wall position by voluntary breath-hold technique using a laser-based monitoring and visual feedback system[END_REF] gating ; and (5) respiratory-synchronized radiotherapy offered currently by the CyberKnife system (Accuray Inc, Sunnyvale, US) or

[5 ] [START_REF] Schweikard | Robotic motion compensation for respiratory movement during radiosurgery[END_REF] in the future through the use of Dynamic Multileaf collimators .

[7 ]

The most advanced methods are based on modeling the respiratory motion and incorporating the information extracted from these models to improve dose calculation and delivery during RT. For example in the case of the CyberKnife Synchrony system a set of three ™ optical fiducial markers are attached to a snugly fitting vest the patient wears during treatment, to provide a breathing signal. The only current recommendation for the placement of the external markers is either on the chest or the abdomen remaining visible by the optical tracking system used to monitor their position , . In order to ensure continuous correspondence between the external and internal [6 8 ] motion , , small gold markers are also implanted prior to treatment in the vicinity of the target organ or lesion, allowing a model to be [6 8 ] defined between the motion of the external and internal markers. This model is updated based on frequent x-ray snapshots of the position of both marker sets simultaneously . A technology that can potentially eliminate the need for an internal external motion model [START_REF] Schweikard | Respiration tracking in radiosurgery[END_REF] involves the use of the Calypso system (Calypso Medical Technologies, Seattle, US) which uses implanted transponders with an associated wireless tracking with a frequency of up to 10Hz , . However, the issue remains the use of implanted markers which involves an [9 10 ] invasive procedure. A potential alternative to the use of implanted markers will be modeling the internal structure motion using respiratory phase and amplitude information in association with displacement fields derived from 4D CT acquisitions using deformable registration [11 , . However, any such model depends on the accuracy of the phase and amplitude motion information which is currently derived by 12 ] external markers.

In the literature, several studies have concentrated in the past on investigating the external-internal motion correlation. Mageras et al.[ and Vedam et al. have shown a strong correlation between external markers and the diaphragm but their study did not include 13 ] [START_REF] Vedam | Quantifying the predictability of diaphragm motion during respiration with a noninvasive external marker[END_REF] tumor motion. Gierga et al. used fluoroscopy to gather tumor respiratory motion data, recording this information using standard video [START_REF] Gierga | The correlation between external and internal markers for abdominal tumors: implications for respiratory gating Int[END_REF] recorder interfaced with the fluoroscopy monitor. Results of this study indicated that abdominal tumor motion correlates well with the external markers placed on the patient s abdominal skin. A good correlation was found between the motion frequency of the external and ' internal markers. However, in the same study, a substantial variability was observed in the tumor location considering a given position in the respiratory cycle. This variability was dependent on the external marker location, subsequently affecting the tumor to external marker motion ratio. Finally, this study did not include an assessment of the reproducibility of the observed correlation, neither a study of the potential correlation of phase between internal and external motion.

Tsunashima et al.

studied the correlation between the 1D respiratory waveform measured by an external sensor and the 3D tumor [START_REF] Tsunashima | Correlation between the respiratory waveform measured using A respiratory sensor and 3D tumor motion in gated radiotherapy Int[END_REF] motion. Different tumor types were considered which included lung, liver and esophagus. The correlation between the respiratory waveform and tumor motion showed the difficulty to obtain information from the respiratory waveform inhale phase because of the short inspiration period. In addition, a phase shift was observed between the 3D tumor motion and the respiratory signal. This shift was variable and in the range of 0.0 to 0.3s regardless of the organ being measured, which means that the respiratory waveform may not always express the 3D tumor motion with fidelity. Within the same context, Beddar et al. studied the correlation between the respiratory signal and [START_REF] Beddar | Correlation between internal fiducial tumor motion and external marker motion for liver tumors imaged with 4D-CT Int[END_REF] liver tumors. They also noted that, in some cases, internal anatomical landmarks do not move in synchrony with the 1D respiratory signal also implying that such an 1D signal may not adequately describe tumor respiratory motion.

All these studies indicate that external 1D surrogate measures of respiratory motion may correlate with tumor motion but with non-negligible amplitude and phase differences. These two parameters obtained through regular monitoring need to be taken into consideration in any respiratory motion model in order to ensure that external surrogate measures predict well internal tumor motion. On the other hand, the existing studies also suggest that the position of the external surrogates and the acquisition protocol used can influence the observed internal-external motion correlation in an arbitrary fashion. In order to overcome the observed variability associated with the location of the external markers we have previously proposed the use of patient surface monitoring using a novel external detection [START_REF] Keall | The management of respiratory motion in radiation oncology report of AAPM Task Group 76 AAPM Report 91[END_REF] system , without the need for invasive internal or external markers. This system consists of a Time of Flight (ToF) camera which [18 ] [19 ] actively illuminates the patient with an incoherent light signal and captures the reflected light allowing 3D distance maps of the patient surface to be acquired with rates of up to 30Hz. Alternative techniques that may also facilitate a complete 3D patient surface observation include the use of speckle projection as in the case of the VisionRT system , (Vision RT Ltd, London, UK). As our previous study ™ [ 20 21 ] has demonstrated considering such information allows an improved accuracy in modeling patient specific respiratory motion [START_REF] Fayad | A Patient Specific Model Based On 4D CT Data and a Time of Flight Camera (ToF)[END_REF] compared to using 1D respiratory signal information. However, in order to optimize the utilization of the information provided by the patient dynamic external surface maps there is first a need to understand the correlation between the motion of the external patient surface, to be used as a surrogate measure of respiratory motion, and the internal organ/structures motion.

Therefore the objective of this work was to study the correlation between a patient external surface and internal structures' displacement due to respiratory motion. This correlation was studied using the segmented external patient surfaces from 4D CT patient images and internal landmark identification by experts.

MATERIALS AND METHODS

CT datasets

The clinical datasets used in this study were acquired on a GE Lightspeed 8-slice CT with cine CT scan capability . Each cine CT [START_REF] Pan | Improvement of the cine-CT based 4D-CT imaging[END_REF] scan covered 2cm (8 2.5mm) with 19 to 23 images acquired per slice location. A 0.5 s gantry rotation was used to ensure a high temporal * resolution. The interval between each cine image was 0.45 sec. The total axial length covered in the cine CT acquisition for each patient was 28cm. The real-time position management (RPM, Varian Medical Systems) system was used to obtain the patient s respiration signal

' [
. No specific indication on how to breathe was given to the patients, but they were asked to breathe normally and regularly.

]

The acquired 4D CT images were sorted using an improved phase binning approach by rejecting parts of the respiratory cycle and corresponding CT images associated with irregular respiration as described by Pan et al . Binned 4D CT series were composed of ten [START_REF] Pan | Improvement of the cine-CT based 4D-CT imaging[END_REF] phases (0 , 10 up to 90 of a mean respiratory cycle). The software used searches for each slice location the CT image whose phase is

% %… %
closest to the one of the ten phases. Each binned thoracic CT volume had 512 512 112 voxels, with dimensions 0.97 0.97 2.5mm * * × × 3 corresponding to the x, y and z directions respectively). 4D CT datasets from 10 patients were used in this study. For all of these patient datasets no motion or streak artifacts were visually detectable by an experienced radiologist in any of the ten binned CT volumes.

Anatomical Landmark Motion Measurements

Two clinical oncologists were asked to select some easily identified anatomical landmarks in all of the 4D CT images for the 10 patient datasets. These points were placed in regions with high and low expected displacements during respiration , . As a result, 13 [24 25 ] anatomical landmarks, covering all the regions of the thorax were used in this study. These included the right and left apex, the aortic cross, the sternum, the carina, the right and left nipples, the right and left diaphragm highest position and the upper, low, left and right

" "
boundaries of the tumor (see ). The motion of these points was subsequently extracted by calculating the 3D distance between Figure 1 these landmarks considering for each patient all respiratory synchronized 4D CT image series. An example of this distance in the case of the right diaphragm higher position landmark is shown in and is given by the following equation: figure 2 where, and are respectively the x, y and z coordinates of the k-th landmark in the reference CT corresponding to the end expiration position (x, y, and z represent the components in the three dimensions) and and are respectively the x, y and z coordinates of the k-th landmark in the other 4D CT frames corresponding to the remaining nine phases. The calculated distance was recorded in mm.

The two clinical oncologists were asked to repeat once the same task of identifying the anatomical landmarks on the 4D CT images of all 10 patients considered in the study. The repeatability of the anatomical landmarks identification from the two experts were compared separately (intra-observer variability) and to each other (interobserver variability) .

[26 ]

External Surface Motion Measurements

4D CT volumes were segmented using a threshold based segmentation algorithm and patient surfaces were extracted. Each [START_REF] Zhou | Automated Segmentations of Skin, Soft-tissue and Skeleton from Torso CT images[END_REF] extracted surface corresponds to a binary 3D volume, with the voxels constituting the patient s surface set to 1 and all other voxels set to 0.

'

In addition, a distance map representing the distance of the patient skin from a given plane was calculated. This latter map corresponds to the distance map that can be acquired by a ToF camera , capable of yielding 3D surface maps (acquisition rate >30Hz) in which each [START_REF] Schaller | Time-of-Flight Sensor for Respiratory Motion Gating[END_REF] point corresponds to the measured Euclidean distance between the camera and the object.

A number of regions of interest (ROI) were subsequently placed on the calculated distance map in order to simulate the specific location of external markers. Given the external surface pixel s dimension of 0.97 2.5mm (corresponding to the x and z directions of a CT

RESULTS

The repeatability results in identifying the internal anatomical landmarks by the two clinical oncologists revealed an intra-operator variability of 1.7 0.7mm and 1.9 0.6mm for the first and the second expert respectively. In terms of inter-operator reproducibility a ± ± difference of 1.1 0.4mm and 1.7 0.6mm were measured considering the two repeated anatomical landmark identifications from the two ± ± oncologists. Therefore both intra-and inter-operator reproducibility results showed average differences smaller than the largest distance between two neighboring voxels (2.5mm). Variable motion amplitude was measured for the different internal anatomical landmarks considered. For example, the mean motion amplitude and corresponding standard deviation measured on the 10 patients included in this study was 18.1 6.6mm, 3.0 1.2mm and 12.3 8.1mm for the left diaphragm, the apex and the lower tumour landmark (see )

± ± ± figure 1
respectively.

Considering the correlation between the external regions motion and that of the internal anatomical landmarks, shows the In addition, a much lower correlation is obtained when one is considering the more peripherally placed external regions (i.e. ROIs 5 and 6 in

). figure 3(a) shows the 13 horizontal rectangular external regions considered (see also section II.C). For all patients considered, the Figure 4(a) mean and standard deviation of the motion correlation between these external rectangular regions and all of the internal landmarks are shown in . A higher correlation in the measured motion is shown for the ROIs placed over the abdominal and thoracic areas figure 4(b) rather than the regions located in-between. The amplitude of motion in the abdominal ROIs (regions 12 and 13, see

) was on figure 4(a) average a factor of two higher than the thoracic ROIs (regions 1 to 4, see ), while there was very little motion seen in the figure 4(a) in-between ROIs (0.2 0.8mm). On the other hand, no phase shifts were observed between the motion of the centrally arranged abdominal ± and thoracic regions. As noticed before, the correlation between external ROIs and the aortic cross, the carina, the diaphragm and the tumor location is clearly better than the correlation with the motion of the apex or the sternum (

). figures 4(c) and 4(d)

In addition as shown in , small square vertical regions were placed over the abdominal area, which has consistently across figure 5(a ). The worse correlation measured for these peripherally placed regions was associated with a phase shift in the measured motion 5(a) relative to the more centrally located ROIs or the internal anatomical landmark motion. As demonstrates this correlation is figure 5(c) stronger when the less correlated internal landmarks identified before (apex and sternum) are excluded. The assumption behind the removal of these internal landmarks from the correlation analysis is that regions like the apex and the sternum are, as observed, influenced very little by respiratory motion.

Finally, describes the variation observed in the external-internal landmarks motion correlation amongst four different patients figure 6 included in the study.

shows the patient respiratory traces demonstrating the variability in both the rate and amplitude of the Figure 6(a) motion. Despite this variability there is a high correlation between the motion associated with the external surface regions in the abdominal area and the internal anatomical landmarks. This is true for the landmarks associated with the normal structures (diaphragm, mediastinum structures) as well as those describing the tumour motion as shown in respectively. figures 6(b) and 6(c)

DISCUSSION AND CONCLUSIONS

Respiratory motion synchronization in medical imaging and external beam radiotherapy is currently based on the use of devices which are providing a respiratory signal trace used for synchronization purposes. This process is based on the assumption that a good correlation exists between the respiration induced motion of external surrogate measures and internal structures. Within this context different studies have already been performed demonstrating variable correlation between external markers and internal anatomical structures such as the diaphragm or tumour motion. In all of these studies external markers were placed on specific positions of the patient s skin. On the other ' hand, the correlation with the respiration induced motion of internal structures has been studied considering either normal anatomical landmarks or the tumour motion but rarely both.

Current technology used routinely for respiratory motion synchronization provides only an 1D respiratory signal over a limited area of the external patient s surface, such as the use of a pressure-sensitive belt over the chest or abdomen , and the use of passive infrared ' [START_REF] Kanoulas | Derivation of the tumor position from respiratory surrogates with periodical updating of internal/external correlation[END_REF] reflective markers placed on a patient s torso . Alternative approaches based on the use of implanted markers have been also proposed, ' [START_REF] Nehmeh | Effect of respiratory gating on reducing lung motion artifacts in PET imaging of lung cancer[END_REF] such as the use of gold seeds for real time tumour tracking , , or in the context of providing a real-time update for an internal [1 29 30 ] structure external markers motion model in the case of the CyberKnife Synchrony system , as well as the use of electromagnetic

- ™ [6 ]
transponders whose location is monitored using the Calypso system , .

[

On the other hand, technologies such as the VisionRT system , or a ToF camera , allow real-time monitoring of a patient

™ [20 21 ] [ 9 10
] s surface. Such an approach presents certain advantages over other devices, noticeably a marker-less fashion to monitor respiratory motion ' since no contact with the patient is needed as is the case with external (fiducial, optical, magnetic) or implanted markers, and pressure belts. The only prerequisite of these devices is that the patient should wear a snugly fitting vest during surface acquisition or alternatively being shirtless. In addition, a ToF camera allows monitoring the motion of the complete patient surface in a high frequency rate making available substantially more information for respiratory synchronised acquisitions or therapy compared to location specific 1D respiratory signals. Additional advantages of the ToF camera technology include its small size and the fact that in principle no prior calibration is required. In this study, equivalent information to that provided by systems capturing the motion of a patient s external surface was derived ' using 4D CT acquisitions of ten patients. The objective was to investigate the level of correlation between the motion of different regions of the external patient s surface and multiple internal anatomical landmarks corresponding to normal structures as well as the tumour.

'

The results of this study indicate that there exists a correlation between the motion induced by respiration in parts of the external patient surface and internal anatomical landmarks. The motion of chest and abdominal regions seems to correlate the best with different internal anatomical landmarks including those describing the tumour motion. This observed correlation seems to be consistent despite the variability in the motion amplitude observed for the 10 patients considered in this study. No phase shifts were observed between the motion of the chest and abdominal surface regions. Internal normal anatomical landmarks demonstrating high correlation in terms of respiration induced motion with regions of the external patient s surface include the diaphragm, and mediastinum structures such as the ' aortic cross and the carina. On the other hand, poor correlation associated with motion phase shifts was measured between these same internal anatomical landmarks and peripherally located surface regions in the abdomen and chest. Finally, anatomical landmarks such as the apex and the sternum, showing the smallest amplitude respiratory induced motion, were poorly correlated with the external patient s ' surface motion.

The difference in the motion correlation observed considering different regions of interest placed over a patients external surface hint ' to potential issues associated with the use and positioning of current devices for respiratory synchronization. More specifically such 1D respiratory traces are either derived from regions over which a variable correlation exists between external and internal motion (for example areas covered by pressure-sensitive belts) or through the specific placement of external markers (for example in the case of infrared reflective markers). Existing recommendations considering the positioning of such markers with the Varian RPM system ™ suggest their placement typically midway between the xyphoid process and the umbilicus , chosen in order to maximize the AP [START_REF] Keall | The management of respiratory motion in radiation oncology report of AAPM Task Group 76 AAPM Report 91[END_REF] respiratory induced motion. In our study, this position was amongst the surface regions representing a good correlation between external and internal structure motion including that of the tumour. Another example is the proposed placement on the chest or abdomen of the three external optical markers used with the CyberKnife Synchrony system. Our study demonstrates that the level of the placement of ™ such markers could have a substantial impact since there is little motion and hence a worse correlation between internal landmarks and external markers in the zone located between the chest and the abdominal region. In addition, within these regions (chest and abdomen) keeping the placement of the markers centrally rather than peripherally will also help in increasing the expected internal landmark and external fiducials motion correlation.

Clearly the ability to measure in real-time the motion of the complete external patient surface will allow reducing such variability and associated errors. This may be achieved by considering in the subsequent motion synchronization and modeling processes only those regions of interest demonstrating the best correlation with the internal structures respiratory motion induced displacements. Finally, ' ensuring a good correlation between internal and external motion could help in reducing the need for implanted devices which involve invasive procedures. However, given that our study was based on 4D CT acquisitions created using an average, although patient specific, respiratory motion, it will be always necessary to use an adaptative internal-external correlation which is able to dynamically capture any correlation changes in order to accurately predict the internal target position.

A limitation of the current study is that it has not considered the errors in terms of accuracy with which a device, such a ToF camera, can determine in a real-time fashion the motion amplitude due to respiration throughout a patient surface. Preliminary studies have suggested that this accuracy may be <2mm, with a minimum variability throughout the camera s field of view . These experimental ' [START_REF] Clement | Time of flight camera (ToF) for contact-less and marker-less 3D respiratory motion detection[END_REF] estimations have to be clearly verified on datasets comprising simultaneous patient 4D CT acquisitions and external surface measurements using such devices.

Hadi Fayad 1 *1 3 , Dimitris Visvikis 1 LATIM,

 131 , Tinsu Pan 2 , Jean-Fran ois Cl ment ç é Laboratoire de Traitement de l Information Medicale 1

' × 2 volume*Figure 3 Figure 4 Figure 5

 2345 Figure 3(a)

' figure 3

 3 figure 3(a) position of the six small ROIs placed on the external patient surface (see also section II.C). Considering all patients the mean and standard deviation of the correlation between all 13 internal landmarks and the 6 external ROIs of is shown in . A more figure 3(a) figure 3(b) detailed correlation between each of these external points and individual internal landmarks is shown in for the figures 3(c) and 3(d) normal anatomical landmarks and those concerning the tumor location respectively. As shown in this figure, internal anatomical landmarks such as the aortic cross, the carina, the diaphragm and those associated with the tumor location show higher correlation with the motion of the external ROIs than the apex or the sternum which exhibit the smaller motion amplitude of the different internal landmarks considered.

  figure 5(b) ) that the central region surrounding the umbilicus correlates the most with the internal landmark motion. The motion measured in this central abdominal region (3 to 8, see ) was larger by a factor of three relative to the side regions (1 to 2 and 9 11, see figure 5(a) -

Figure 1

 1 Figure 1Thirteen anatomical internal landmarks identified by radiation oncologists covering both normal and tumour structures.

Figure 2

 2 Figure 2Distance between the position of one of the considered anatomical landmarks (higher right diaphragm position) throughout the respiratory cycle as identified in the 4D CT image series.

Figure 3

 3 Figure 3 Analysis of the correlation between (a) six small regions of interest placed on the external surface (3 pixels diameter) and the internal landmarks (L1-L13) shown in . (b) Mean and standard deviation (error bars) of the measured motion correlation. Correlation between figure 1 the motion of each external circular ROI considered with (c) the normal structure landmarks and (d) tumor landmarks motion.

Figure 4

 4 Figure 4 Analysis of the correlation between (a) 14 rectangular regions covering the majority of the external surface and the internal landmarks (L1-L13) shown in . (b) Mean and standard deviation (error bars) of the measured motion correlation. Correlation between the motion figure 1 of each external rectangular ROI considered with (c) the normal structure landmarks and (d) tumor landmarks motion.

Figure 5

 5 Figure 5 Analysis of the correlation between (a) 11 square ROIs covering the abdominal region of the external patient surface and the internal landmarks (L1-L13) shown in . Mean and standard deviation (error bars) of the measured motion correlation (b) considering all of the figure 1 internal anatomical landmarks and (c) after removal of the anatomical landmarks demonstrating very small respiratory motion (apex, sternum).

Figure 6

 6 Figure 6 Patient specific measurements (a) 1D respiratory traces used in the synchronization process of the 4D CT acquisitions. (b) Correlation between different external abdominal ROIs (shown in ) and internal anatomical landmarks corresponding to normal structures (c) figure 5(a) Correlation between different external abdominal ROIs (shown in ) and internal tumour landmarks. figure 5(a)

[START_REF] Mcclelland | A continuous 4D motion model from multiple respiratory cycles for use in lung radiotherapy[END_REF]

[START_REF] Mcclelland | A continuous 4D motion model from multiple respiratory cycles for use in lung radiotherapy[END_REF]

[START_REF] Mcclelland | A continuous 4D motion model from multiple respiratory cycles for use in lung radiotherapy[END_REF]

[START_REF] Mcclelland | A continuous 4D motion model from multiple respiratory cycles for use in lung radiotherapy[END_REF]

[START_REF] Mcclelland | A continuous 4D motion model from multiple respiratory cycles for use in lung radiotherapy[END_REF]

[START_REF] Mcclelland | A continuous 4D motion model from multiple respiratory cycles for use in lung radiotherapy[END_REF]

[START_REF] Mcclelland | A continuous 4D motion model from multiple respiratory cycles for use in lung radiotherapy[END_REF]

[START_REF] Mcclelland | A continuous 4D motion model from multiple respiratory cycles for use in lung radiotherapy[END_REF]

[START_REF] Mcclelland | A continuous 4D motion model from multiple respiratory cycles for use in lung radiotherapy[END_REF]

[START_REF] Mcclelland | A continuous 4D motion model from multiple respiratory cycles for use in lung radiotherapy[END_REF]

[START_REF] Mcclelland | A continuous 4D motion model from multiple respiratory cycles for use in lung radiotherapy[END_REF]


Pierre Seppecher
email: seppecher@imath.fr

Jean-Jacques Alibert

Francesco Dell Isola

We study the statics of some trusses, i.e. networks of nodes linked by linear springs. The trusses are designed in such a way that a few number of floppy modes are present and remain when considering the homogenized limit of the truss. We then obtain linear elastic materials with exotic mechanical interactions which cannot be described in the classical framework of Cauchy stress theory. For aim of simplicity, the structures described here are two-dimensional. The extension to the 3D case does not present any difficulty.

Linear elastic trusses leading to continua with exotic mechanical interactions

Introduction

The theory of Cauchy stress is so efficient in describing mechanical interactions in usual materials that its foundations have seldom been reconsidered (see [3,20,19,26,28,27,37,38,[START_REF] Dell'isola | [END_REF]17,21,22,23,24,25,[START_REF] Maugin | Mechanics of Generalized Continua, One Hundred Years After the Cosserats[END_REF][START_REF] Mindlin | [END_REF]31,32,33,[START_REF] Toupin | [END_REF]41]. Rapid progress in nanotechnologies makes now possible the design of new materials having exotic mechanical properties. One basic assumption of Cauchy stress theory is that the mechanical contact interaction between two parts of the material is only due to a surface density of contact forces concentrated on the "cut", i.e. on the dividing surface [START_REF] Truesdell | A First Course in Rational Continuum Mechanics[END_REF][START_REF] Noll | Proceeding of the Berkeley Symposium on the Axiomatic Method[END_REF][START_REF] Noll | [END_REF]. In this paper we describe some microscopic structures leading at the macroscopic level to materials for which this basic assumption is no longer valid. Microscopic (atomic) structures leading to high order gradient materials have already been described [1,6,42,43]. The advantage of the structures presented here is that the non classic properties are preponderant and are not limited to producing thin boundary layers.

The microscopic structures we consider are simple is the sense that they consist only of linear elastic springs joining nodes (what is usually called "trusses"). The idea is to design structures with a few number of floppy modes, i.e. global deformations which do not change the length of any spring of the structure and so do not need any energy.

In the different sections we consider structures with an increasing degree of complexity. We compute the deformation energy in terms of the displacement of a network of nodes. This energy is generally self-explanatory : writing the homogenized energy which corresponds to a very dense network of nodes is straightforward even if its rigorous mathematical derivation is not so easy [5,[START_REF] Bensoussan | Asymptotics Analysis for periodic Structures[END_REF]9,10,11,12,[START_REF] Cioranescu | Applied Mathematical Sciences[END_REF] (for a rigorous derivation of the homogenization of the structure presented in section 3.1 see [2]). It is not the aim of this paper to give such a rigorous mathematical analysis, we prefer to emphasize the mechanical features of the proposed structures. the stiffness of elastic bars is suitably scaled, the homogenized behavior of the structure is onedimensional and corresponds to a non-extensional flexion beam. This is not very difficult to check but the computation is much more easy when considering a slightly different structure (figure 2) in which we assume that the horizontal bars have infinite stiffness while all other bars have the same finite stiffness k. Indeed this alternative structure is made of a periodic superposition of basic elements described in figure 3. Thus its elastic energy E is easy to write in terms of the displacements

[i -1] -2u 2 [i] + u 2 [i + 1]) 2 .
The elastic energy of the beam then reads

E(u) = N -1 i=2 (k/16)(u 2 [i -1] -2u 2 [i] + u 2 [i + 1]) 2
Let us assume that the total length of the structure is ℓ and let us denote ǫ = ℓ/(N -1) the distance between two consecutive blue nodes. If the stiffness of the bars scales as ǫ -3 : k = 16Kǫ -3 , then

E(u) = ǫ N -1 i=2 K u 2 [i -1] -2u 2 [i] + u 2 [i + 1] ǫ 2 2 .
When the beam is made of a large number of small cells, i.e. when ǫ tends to zero, then one can recognize in the expression of E the finite difference approximation for the continuous energy

Ẽ(u) = ℓ 0 K ∂ 2 u 2 ∂x 2 1 2 .
This is the energy of the well known linear flexion beam. Nevertheless let us make some remarks before using this structure to build a new 2D material : First, we have assumed that all bars are linear elastic bars. They have spring-like behavior or correspond to long range linear interactions : the possibility of buckling is not considered here. Secondly, external forces are exerted only on the lowest (blue) nodes, the upper (red) nodes are considered as "internal". Finally note that, even when the first node is fixed (which corresponds to a Dirichlet condition u(0) = 0 in the continuous model) the truss is not isostatic. A floppy mode (rotation around node 1) remains. For instance, in order to counterbalance a vertical force applied in the middle of the beam, it is necessary to apply a suitable force on the second node (which corresponds in the continuous model to a torque applied at x = 0).

The flexion 2D material

Now let us consider many (M ) parallel beam structures similar to the one described in the previous section (each one with first node fixed) and let us link them by adding vertical bars which link each blue node with the blue node just above; the added vertical bars have stiffness k ′ ; see figure 4. Note that, for sake of clearnesss, figure 4 shows a slightly different structure than the one we just described.

This structure clearly has a floppy mode : the deformation shown in figure 4 needs no energy as no bar is extended. It is the only floppy mode. Indeed it is easy to check that fixing an extra node makes the structure isostatic. At a first insight, one could think that we have here designed a degenerated classical material with a vanishing shear stiffness. It is easy to reject this interpretation by considering the dimension of the space of floppy modes in both cases.

The elastic energy of the whole structure is simply the sum of the elastic energies of the beams we just derived plus the energy concentrated in the additional vertical junctions. It writes

E(u) = M j=1 N -1 i=2 (k/16)(u 2 [i -1, j] -2u 2 [i, j] + u 2 [i + 1, j]) 2 + M -1 j=1 N i=1 k ′ (u 2 [i, j] -u 2 [i, j -1]) 2
where u[i, j] denotes the displacement of the i-th (blue) node of the j-th beam. With the previous scaling for k and a suitable scaling for k ′ : k ′ = K ′ ǫ we obtain by passing to the limit ǫ → 0 the following continuous energy :

Ẽ(u) = Ω K ∂ 2 u 2 ∂x 2 1 2 + K ′ ∂u 2 ∂x 2 2
subjected to the constraint u 1 = 0 in the whole domain Ω := (0, ℓ) × (0, ℓ ′ ) and to the Dirichlet boundary condition u = 0 on the side x 1 = 0. In order to understand the behavior of the continuum described by this elastic energy, let us consider the response of the system to some test load. We assume that one exerts a vertical force f concentrated on a vertical surface S in the middle of the domain (that is an academic example since applying such a concentrated force inside the material is not really easy). An equilibrium solution minimizes the total energy Ẽ -S f u 2 dx 2 . In a classical Cauchy material, an equilibrium state exists with (approximately) a constant shear deformation in the left hand part of the domain and no deformation in the right hand part (cf. figure 5). The situation is completely different for our new material. First, due to the aforementioned floppy mode, the Dirichlet condition is not sufficient for ensuring the existence of equilibrium. If the second node of all beams is also fixed (or alternatively if a suitable vertical force is applied on them) then a solution exists. This corresponds in the continuous model to adding an extra clamping constraint ∂u 2 ∂x 1 = 0 on the side S l := {x 1 = 0} (or alternatively to applying a suitable surface density of torques on it, that is adding in the energy a term of the form S l g ∂u 2 ∂x 1 dx 2 ). The equilibrium state is then very different from the classical case : the shear deformation varies linearly in the left hand part of the domain and remains constant but not vanishing in the right hand part (cf. figure 5).

Up to now the reader may be deceived. Indeed the interactions in the described material are limited to surface densities of forces and torques which are not very exotic actions. The point is that at the microscopic level the displacement of the considered truss is a "local rotation", the truss behaves like a Timochenko continuum [START_REF] Timoshenko | Strenght of Materials[END_REF] or a Cosserat continuum [START_REF] Cosserat | Note sur la thorie de l'action euclidienne Paris[END_REF][START_REF] Cosserat | Théorie des Corps Déformables[END_REF] in the sense cleared up by Bleustein [START_REF] Bleustein | [END_REF] (see also [18]). In the next section we build a structure endowed with local deformation instead of local rotation. Interactions will then be of a completely different nature.

3. The pantograph beam and the pantograph truss 3.1. The pantograph beam Now we build a structure based on the basic element described in figure 6 which possesses the same floppy mode as scissors. Indeed, for the clearness of the forthcoming figures we will simply draw a pair of scissors for representing this structure but remember that this is only a symbolic notation for a structure simply made of nodes joined by elastic bars. By linking many such cells we get the pantographic structure described in figure 7. The remarkable feature of this structure is its extensional floppy mode (see fig. 7). Note that fixing the deformation of the first cell tends to fix the deformation of all cells. In particular, fixing the two first (blue) nodes makes the structure isostatic.

Computing the elastic energy in terms of the displacement of the blue nodes is straightforward. We get

E(u) = N -1 i=2 k(u 2 [i -1] -2u 2 [i] + u 2 [i + 1]) 2 + k ′ (u 1 [i -1] -2u 1 [i] + u 1 [i + 1]) 2 We now recognize in ǫ -2 (u 1 [i -1] -2u 1 [i] + u 1 [i + 1]
) the finite difference approximation of the second derivative of u 1 with respect to x 1 . With a suitable scaling for k and k ′ , the continuous limit (N → ∞) model reads

Ẽ(u) = ℓ 0 K ∂ 2 u 2 ∂x 2 1 2 + K ′ ∂ 2 u 1 ∂x 2 1 2
The transverse displacement u 2 and the axial one u 1 decouple. With respect to the transverse displacement, the behavior of the beam is exactly the behavior of the flexion beam described in the previous section. Owing to the symmetry of the terms involving u 1 and u 2 in the expression of the energy, every result valid for the transverse displacement can be transposed to the axial displacement, only the mechanical interpretation differs. For instance, when the two first nodes are fixed, the displacement at the equilibrium under the action of a single axial force F at a middle point point x 1 = ℓ/2 minimizes Ẽ(u) -F u 1 (ℓ/2) under the constraints u 1 (0) = ∂u 1 ∂x 1 (0) = 0 : the deformation increases linearly from x 1 = 0 to x 1 = ℓ/2 and remains constant but non zero for x 1 > ℓ/2. Fixing the two first nodes corresponds in the continuous limit to imposing the constraint ∂u 1 ∂x 1 (0) = 0. This can be replaced by the dual condition : two opposite axial forces applied at the two first nodes (see figure 8) which corresponds in the continuous limit to applying a new type of action. Two opposite axial forces are often said to have no action on a body but it is clearly not the case here as two such forces tend to modify the deformation of the first cell and as this deformation tends to propagate in the whole beam. Such "double force" is a new type of interaction [21,22,36] which needs a new symbol ( see figure 8).

The pantographic truss

Consider now an array of many (M ) horizontal pantographic beams (see figure 9), and link them using vertical pantographic beams. We fix the first node of each horizontal beam. Then we get a structure with four floppy modes (see figure 10). The elastic energy of the structure is the sum of the energies of all pantotographic beams: This clearly leads to the continuum model

E(u) = M j=1 N -1 i=2 k(u 1 [i-1, j] -2u 1 [i, j] + u 1 [i+1, j]) 2 + k ′ (u 2 [i-1, j] -2u 2 [i, j] + u 2 [i+1, j]) 2 + M -1 j=2 N i=1 k(u 1 [i, j -1] -2u 1 [i, j] + u 1 [i, j +1]) 2 + k ′ (u 2 [i, j -1] -2u 2 [i, j] + u 2 [i, j +1]) 2
Ẽ(u) = Ω K ∂ 2 u 2 ∂x 2 1 2 + K ′ ∂ 2 u 2 ∂x 2 2 2 + K ′ ∂ 2 u 1 ∂x 2 1 2 + K ∂ 2 u 1 ∂x 2 2 2
The floppy modes are of the form u(x 1 , x 2 ) = (ax

1 x 2 + bx 1 + cx 2 + d, ex 1 x 2 + f x 1 + gx 2 + h)
which, taking into account the Dirichlet condition u = 0 on the boundary x 1 = 0, reduces to a four dimensional space of floppy modes:

u(x 1 , x 2 ) = ((ax 2 + b)x 1 , (ex 2 + f )x 1 ),
(see figure 10). Note that one can still act on the material at the fixed surface x 1 = 0 by applying a density G of torques (or fixing the "rotation ∂u 2 ∂x 1 ) or by applying a density F of "double-forces" (or fixing the "dilatation ∂u 1 ∂x 1 ). For instance, let us consider the extension to the 2D case of the situation described in figure 8. We apply a surface density of horizontal forces f on the middle surface S = {x 1 = ℓ/2} adding in the energy the term S -f u 1 dx 2 . The existence of an equilibrium is ensured only if we apply on the surface x 1 = 0 a surface density of horizontal forces -f plus a surface density of double forces adding in the energy the terms -S l f u 1 dx 2 and -S l F ∂u 1 ∂x 1 dx 2 with F = f ℓ/2).

Third gradient structure

Up to now the continuous materials we have designed are second gradient material (see [21])that is materials with an elastic energy depending only on the first and second gradient of the displacement. It is possible, to the expense of some extra complexity, to design trusses leading to higher gradient material and consequently supporting more exotic mechanical interactions.

Third gradient flexion beam

We first consider the basic element described in figure 11. Note that we build on the structure we already introduced (fig. 6) and remember that such a structure is only made of nodes and linear springs. The lowest horizontal bars are assumed to have infinite stiffness while all other ones have the same finite stiffness. One can easily check that the elastic energy of this element in terms of the displacements of the 4 lowest (red) nodes is proportional to

(u 2 [i] -3u 2 [i + 1] + 3u 2 [i + 2] -u 2 [i + 3]) 2 .
In order to simplify the following drawings, we use the symbol for representing this basic element. Then we construct a beam by link any four successive nodes of the beam by such basic elements (figure 12). It is non extensible (u 1 = 0) and its elastic energy reads When the two first nodes are fixed, a floppy mode remains which corresponds to deformations with constant curvature along the beam. This curvature is fixed for instance by fixing the displacement of the third node. One can also act on the beam by applying three vertical forces (f, -2f, f ) on three successive nodes. In the continuous limit we recognize the finite difference approximation of the third derivative of the transverse displacement and the energy reads

E(u) = N -3 i=1 k(u 2 [i] -3u 2 [i + 1] + 3u 2 [i + 2] -u 2 [i + 3]) 2 .
Ẽ(u) = ℓ 0 K ∂ 3 u 2 ∂x 3 1 3 .

Third gradient 2D structure

Consider now an array of many (M ) horizontal 3rd gradient beams (see figure 13), and link them using vertical elastic springs. We fix the two first nodes of each beam. Then we get a structure with one floppy mode. The elastic energy of the whole structure the sum of the elastic energies of the beams we just derived plus the energy concentrated in the additional vertical junctions.

It takes the form

E(u) = M j=1 N -3 i=1 k(u 2 [i, j]-3u 2 [i+1, j]+3u 2 [i+2, j]-u 2 [i+3, j]) 2 + M -1 j=1 N i=1 k ′ (u 2 [i, j]-u 2 [i, j-1]) 2
and with a suitable scaling for k and k ′ we obtain the following continuous energy :

Ẽ(u) = Ω K ∂ 3 u 2 ∂x 3 1 2 + K ′ ∂u 2 ∂x 2 2
with the constraints u 1 = 0 on the domain Ω and u 2 = 0, ∂u 2 ∂x 1 = 0 on the boundary x 1 = 0. Let us again consider the problem defined in section 2 (figure 5). The equilibrium solution is still different. First such a solution exists only if the third node of all beams is also fixed (or alternatively if a suitable vertical force is applied on them). This corresponds in the continuous model to adding an extra clamping constraint ∂ 2 u 2 ∂x 2 1 = 0 on the side S l := {x 1 = 0} (or alternatively to applying a suitable surface density of "triple-forces" on it, that is adding in the energy a term of the form S l h ∂ 2 u 2 ∂x 2 1 dx 2 ). This type of triple-forces can also be interpreted as "double-torques" and we use the symbol for them. In the equilibrium state the curvature increases in the left hand part of the domain and remains constant but not vanishing in the right hand part (cf. figure 14).

Conclusion

We could carry on this enumeration of more and more intricate structures. For instance, building on the structural element introduced and symbolized in the previous section, we could consider the element described in figure 15. We let the reader check that it has a floppy mode which mimics the third derivative of u 1 with respect of x 1 (that is the second derivative of the dilatation) and that, introducing a new symbol to denote it, the element described in figure 16 has a floppy mode which mimics the fourth derivative of u 2 with respect of x 1 . This enumeration would not be of great interest : the reader should yet be convinced by the examples we have presented that homogenization of trusses can lead to any high gradient materials.

In these materials, contact interactions between parts of the domain are far belong those described by the Cauchy stress theory. In our examples we have illustrated the possibility of surfaces densities of torques, "double-forces" or "triple-forces". Our trusses give a clear microscopic interpretation of all these different actions : they are different distributional limits of highly oscillating forces. The denominations that P. Germain [21] gave to them are illustrated.

Let us recall that it has been rigorously proved in [4] that all types of high gradient materials can be achieved by homogenizing trusses.

The highest gradient terms are clearly active when the lower gradient terms do not play any role, that is when the structures possess floppy modes. The reader who prefer to work with non degenerated materials and thus to consider only trusses with no floppy mode (apart those corresponding to rigid motions) can add linear elastic links between each node and its neighbors. This will simply add a classical (first gradient) elastic term in the elastic energy. These supplementary links have to be much weaker than the links we have used for building our structures (the scaling with respect to ǫ is different). That explains why, for natural materials, these new types of interactions scarcely matter : to obtain a material in which a new type of action is preponderant, one must carefully build with very stiff links a structure having a floppy mode and then, possibly add some weaker links.

2 .Figure 1 .

 21 Figure 1. A beam truss.

Figure 2 .

 2 Figure 2. A alternative beam truss. The horizontal bars have infinite stiffness. All other bars have the same finite stiffness. The diagonal bars do not interact : this can be achieved either by using a 3D structure or by adding an extra node at the junction.

Figure 3 .

 3 Figure 3. An element of the beam of fig.2.

Figure 4 .

 4 Figure 4. A 2D flexion structure. For the clearness of the figure, any red node has been joined with the blue one just above, but the computation is straightforward only when assuming that the blue nodes actually join the blue nodes just above. For the same reason we draw the first version of the flexion beam while the alternative version is used in the computation. The truss admits a floppy mode.

Figure 5 .

 5 Figure 5. Comparison of the equilibrium states of classical (left hand drawing) and flexion (right hand drawing)materials subjected to a Dirichlet condition on the left hand side S l and a vertical force on a vertical middle surface S. In the classical case, the balance of forces is ensured by a suitable surface density of reaction forces on S l while, in the flexion case, a suitable density of torque has to be applied on S l .

Figure 6 .

 6 Figure 6. Basic element of the pantographic structures. We assume that the bars linking blue and red nodes have very high (infinite) stiffness corresponding to pure flexion (no extension) for the scissors. The structure is symbolized by two crossing rods. Its floppy mode plays an essential role.

Figure 7 .

 7 Figure 7. The pantographic structure and its extensional floppy mode.

Figure 8 .

 8 Figure 8. An axial force is applied at the middle point of a pantographic beam. On the left hand side of the pantograph, in addition to the reaction force, two opposite forces on the right hand side of the pantograph must be applied in order to enable the equilibrium of the structure. In the continuous limit (below) such additional action is a point-wise "double force".

Figure 9 .

 9 Figure 9. The pantographic 2D structure.

Figure 10 .

 10 Figure 10. When imposing a Dirichlet condition on one side, the pantographic 2D structure has four floppy modes.

Figure 11 .

 11 Figure 11. The basic element of the 3rd gradient beam and its major floppy mode.

Figure 12 .

 12 Figure 12. A 3rd gradient beam made by superposition of basic elements of figure 11.

Figure 13 .

 13 Figure 13. A 3rd gradient 2D structure.

Figure 14 .

 14 Figure 14. The equilibrium state of a flexion third gradient material clamped on the left hand side S l and submitted to a vertical force on the vertical middle surface.

Figure 15 .

 15 Figure 15. A structural element for a material sensitive to the second derivative of the dilatation.

Figure 16 .

 16 Figure 16. A structural element for a material sensitive to the fourth derivative of the transverse displacement.


V Ménoret

R Geiger

G Stern

P Cheinet

B Battelier

N Zahzam

F Pereira

Dos Santos

A Bresson

A Landragin

P Bouyer

A TRANSPORTABLE COLD ATOM INERTIAL SENSOR FOR SPACE APPLICATIONS

I. INTRODUCTION

Atom interferometry has hugely benefitted from advances made in cold atom physics over the past twenty years, and ultra-precise quantum sensors are now available for a wide range of applications [START_REF] Cronin | Optics and interferometry with atoms and molecules[END_REF]. In particular, cold atom interferometers have shown excellent performances in the field of acceleration and rotation measurements [START_REF] Canuel | Six-axis inertial sensor using cold-atom interferometry[END_REF][START_REF] Peters | High-precision gravity measurements using atom interferometry[END_REF], and are foreseen as promising candidates for navigation, geophysics, geo-prospecting and tests of fundamental physics such as the Universality of Free Fall (UFF). In order to carry out a test of the UFF with atoms as test masses, one needs to compare precisely the accelerations of two atoms with different masses as they fall in the Earth's gravitational field. The sensitivity of atom interferometers scales like the square of the time during which the atoms are in free fall, and on ground this interrogation time is limited by the size of the experimental setup to a fraction of a second. Sending an atom interferometer in space would allow for several seconds of excellent free-fall conditions, and tests of the UFF could be carried out with precisions as low as 10 -15 [START_REF] Varoquaux | How to estimate the differential acceleration in a two-species atom interferometer to test the Equivalence Principle[END_REF].

However, cold atoms experiments rely on complex laser systems, which are needed to cool down and manipulate the atoms, and these systems are usually very sensitive to temperature fluctuations and vibrations. In addition, when operating an inertial sensor, vibrations are a major issue, as they deteriorate the performances of the instrument. This is why cold atom interferometers are usually used in ground based facilities, which provide stable enough environments. In order to carry out airborne or space-borne measurements, one has to design an instrument which is both compact and stable, and such that vibrations induced by the platform will not deteriorate the sensitivity of the sensor.

We report on the operation of an atom interferometer on board a plane carrying out parabolic flights (Airbus A300 Zero-G, operated by Novespace). We have constructed a compact and stable laser setup, which is well suited for onboard applications. Our goal is to implement a dual-species Rb-K atom interferometer in order to carry out a test of the UFF in the plane. In this perspective, we are designing a dual-wavelength laser source, which will enable us to cool down and coherently manipulate the quantum states of both atoms. We have successfully tested a preliminary version of the source and obtained a double species magneto-optical trap (MOT).

II. ATOM INTERFEROMETRY

Atom interferometers have been demonstrated almost twenty years ago [START_REF] Kasevich | Atomic interferometry using stimulated Raman transitions[END_REF] and are now thoroughly understood [START_REF] Bordé | Atomic clocks and inertial sensors[END_REF]. A simple way to understand our atom interferometer is as follows. Before the measurement itself, the atoms are cooled and prepared to the correct quantum state, before being released. This means that during the interferometric sequence, they are in pure free-fall. At three precise instants, separated by a time T, the atoms are illuminated with a laser, retroreflected on a reference mirror attached to the plane (Fig. 1). This can be seen as creating a ruler, the graduation being the phase of the retroreflected light wave. As they are illuminated, the internal state of the atoms can change, with a probability which depends on the phase of the laser and therefore on their position in the laser ruler. At the end of the interferometric sequence, we measure the proportion of atoms which have undergone the transition, and this measurements gives us a precise information on their acceleration with respect to the reference mirror, in the direction of the laser. In other words, this measurement consists in a precise readout of the position of the center of mass of the atomic cloud with respect to the reference mirror. The probability P for an atom to undergo the transition scales as [START_REF] Bordé | Atomic interferometry with internal state labeling[END_REF] : One can see from (2) that the sensitivity of the device to acceleration scales like the square of the interrogation time T. To carry out precise measurements, it is therefore of interest to increase T as much as possible. A way to do this is to be in weightlessness conditions. The experiment will then fall along with the atoms, allowing for long interrogation times. On the ground, T is limited to a fraction of a second by the size of the vacuum cell. In microgravity conditions, free-fall times up to one second have been demonstrated [START_REF] Van Zoest | Bose-Einstein Condensation in Microgravity[END_REF], and in space this could be increased even more.

€ P ∝ cos Φ ( ).
In case the acceleration is not constant, one has to take into account the frequency response of the atom interferometer [START_REF] Cheinet | Measurement of the sensitivity function in a time-domain atomic interferometer[END_REF]. The phase shift Φ is the sum of the contribution of all the frequency components. Vibrations are a crucial issue when carrying out acceleration measurements with an atom interferometer, as they will contribute to the acceleration a(t), and therefore to the phase Φ. In the plane, vibrations are up to four orders of magnitude higher than in laboratory-based atom interferometry experiments, and are a real challenge one must face in order to carry out measurements with a high sensitivity.

III. SINGLE SPECIES EXPERIMENTAL SETUP

A. General design

The experiment is made of three sub-units, and comprises a ultra-high vacuum cell in which we cool and manipulate the atoms, a laser source and a microwave frequency reference based on an ultra-stable quartz oscillator. All the components have been tested in microgravity onboard the plane. Altogether, the setup weighs less than 700 kg, and has an electrical power consumption of the order of 1 kW (Fig. 2).

Fig. 2.

Experimental apparatus in the plane. The science chamber is protected by a magnetic shield (orange straps, in the front). Behind the science chamber is the rack containing the electrical power supplies and the free-space laser. The bigger unit, on the right, holds the telecom laser source, the 6.8 GHz frequency reference, all the control electronics and the computer.

ICSO 2010

Rhodes, Greece International Conference on Space Optics 4 -8 October 2010

During the flight, the plane carries out a series of 31 parabolas, each providing about 22 seconds of microgravity. Before each parabola, there is a pull-up phase, which lasts for about 20 seconds, and during which the acceleration is about 1.8g. There is also a 1.8g pull-out phase after the parabola, and about 2 minutes of 1g flight before the next parabola. The experiment is able to operate both during the 0g and 1g phases of the flight.

B. Laser source

Having a very compact and stable laser source is a major challenge one has to face before thinking of sending an atom interferometer in space. In laboratories, the laser systems used to cool down and interrogate the atoms usually rely on Extended Cavity Diode Lasers (ECDLs), and free-space optics. It is possible to miniaturize such systems, and very stable sources have been demonstrated, for example in the Quantus droptower experiment, or for the Pharao mission [START_REF] Van Zoest | Bose-Einstein Condensation in Microgravity[END_REF]10,[START_REF] Schmidt | A portable laser system for high precision atom interferometry experiments[END_REF]. In order to cool and manipulate 87 Rb, we need a laser source emitting at 780 nm. We have chosen to take advantage of existing fiber optics components in the telecom C-band around 1.5 µm, and generate the final optical frequency by Second Harmonic Generation (SHG) [START_REF] Lienhart | Compact and robust laser system for rubidium laser cooling based on the frequency doubling of a fiber bench at 1560 nm[END_REF][START_REF] Stern | Light-pulse atom interferometry in microgravity[END_REF]. This way, we have a setup which is almost entirely fibered and therefore immune to vibration-induced misalignments. The variety of available components in the C-band allows us to have a versatile setup for the experiment.

Starting from a 1560 nm fibered DFB laser diode, we use a phase modulator to generate optical sidebands at 6.8 GHz in order to address the hyperfine transition of 87 Rb [START_REF] Steck | Alkali D Line Data[END_REF]. The light is amplified to 5W with an Erbium-Doped Fiber Amplifier (EDFA), and outcoupled from the fiber. It is then sent through a PPLN crystal in a double-pass configuration, which generates the light at 780 nm that is sent into the science chamber via some fibers (Fig. 3). This Second Harmonic Generation (SHG) stage is the only free-space part of the laser system, and has been designed to be as compact and robust as possible.

During the parabolic flight campaigns, the experiment undergoes very strong vibrations during transportation and in the plane, as well as temperature and humidity changes. The laser system is stable enough to operate without any adjustment. Another key feature of our laser source is that we have a single laser, which we use to both cool and interrogate the atoms. The sidebands generated by phase modulation are also intrinsically phase-locked to one another, and since they are carried by the same beam, the system is insensitive to fluctuations induced by path differences. Fig. 3. Compact laser source [START_REF] Stern | Light-pulse atom interferometry in microgravity[END_REF]. A Distributed feedback (DFB) fibered laser diode at 1560 nm is frequencydoubled in pigtailed PPLN waveguide, and the resulting 780 nm light is used to stabilize the laser on a Rb transition. The slave laser is also a 1560 nm DFB, frequency-locked on the master diode via a beatnote. Light from the slave DFB is phase-modulated at 6.8 GHz in order to generate the sidebands needed to cool and address the atoms. It is then amplified to 5W in the EDFA, and frequency-doubled in a PPLN crystal in a double-pass configuration. An acousto-optic modulator (AOM) then sends the light towards the MOT or Raman fibers, to respectively cool and interrogate the atoms.

ICSO 2010

Rhodes, Greece International Conference on Space Optics 4 -8 October 2010

IV. TOWARDS A TEST OF THE UFF : A DUAL-WAVELENGTH LASER SOURCE

In order to carry out a test of the UFF, one needs to compare the accelerations of two test masses as they fall freely in the Earth's gravitational field. This has been done with macroscopic test masses using torsion balances, and Lunar Laser Ranging (LLR) [START_REF] Schlamminger | Test of Equivalence Principle using a rotating torsion balance[END_REF][START_REF] Williams | Progress in Lunar Laser Ranging tests of relativistic gravity[END_REF]. With cold atoms, a preliminary test has been carried out with 85 Rb and 87 Rb [START_REF] Fray | Atomic interferometer with amplitude gratings of light and its application to atom based tests of the Equivalence Principle[END_REF]. However, the two Rubidium isotopes have very similar masses and internal compositions, and it would be interesting to test the UFF at the quantum level with two very different objects.

Our idea is to use 87 Rb and 39 K. These two atoms have transitions at wavelengths which are close to one another (780 nm and 767 nm respectively). This makes them ideal candidates for testing the UFF. Both wavelengths can be generated from frequency-doubled telecom lasers, and we have shown that the fact that they are quite close allows for very efficient common mode noise rejection between the two interferometers [START_REF] Varoquaux | How to estimate the differential acceleration in a two-species atom interferometer to test the Equivalence Principle[END_REF].

To manipulate simultaneously the two atoms, we are constructing a dual-wavelength laser source. We will start from two DFB lasers at 1560 and 1534 nm, and frequency-double them to 780 and 767 nm. In order to have as little differential-mode noise between the two atom interferometers as possible, we will have the two lasers locked on the same frequency reference, by the means of an optical frequency comb specially developed by MenloSystems for the parabolic flights.

A. Frequency-stabilization of the lasers

An optical frequency comb is a femtosecond laser, which emits short pulses at a given repetition rate, typically of the order of 100 MHz. In the Fourier domain, this laser has a broad spectrum with very narrow lines separated from each other by the repetition rate [START_REF] Udem | Optical frequency metrology[END_REF]. The frequency of each spectral line can be written as

€ f n = f 0 + nf rep (3)
where n is the index of the considered line, f rep the repetition rate of the laser, and f 0 the so-called Carrier-Envelope Offset (CEO). It is possible to stabilize both f 0 and f rep , either on an optical or RF reference. This way, the comb is an absolute reference for optical frequencies, covering a very broad spectrum.

Our optical comb has been designed to be as compact as possible. It has a repetition rate of 250 MHz, and covers a spectrum of about 100 nm, between 1500 and 1600 nm. We have two Beat Detection Units (BDUs), which enable us to select comb lines around given frequencies (1560 and 1534 nm), and to make a beatnote between the comb and our continuous wave DFB lasers.

The 1560 nm part of the dual-frequency source will be quite similar to what we have described above for the single-wavelength version. Part of the light from the master laser is sent to the 1560 nm BDU. The resulting beatnote will be used to lock the repetition rate of the comb. This way, the comb will be stabilized on the master laser, which in turn is locked on 85 Rb transition. The 1534 nm DFB will then be locked on the comb via the second BDU. This way, the two lasers can be stabilized on the same frequency reference (a Rubidium transition), and there will be almost no relative drift.

B. Serrodyne modulation for the 1534 nm Potassium laser

Instead of using a master-slave configuration as we have on the Rubidium laser, we have tested a singlelaser configuration for the Potassium part of the source. An important challenge is the fact that between the phase of the experiment where the atoms are cooled and the phase where they are interrogated, the laser has to be detuned of several hundreds of MHz. In the Rubidium source, this is done by changing the value of the frequency-lock between the master and slave DFB diodes. For the 1534 nm source, we have implemented a serrodyne modulation technique [START_REF] Johnson | Broadband optical serrodyne frequency shifting[END_REF][START_REF] Houtz | Wideband, efficient optical serrodyne frequency shifting with a phase modulator and a nonlinear transmission line[END_REF].

Shifting the frequency of the electric field can be seen as applying a linear increase to the phase of the electric field. Although it is not possible to do this with a phase modulator, one can carry out an analogous operation by applying a sawtooth waveform, with an amplitude corresponding to a phase of 2π. The modulated electric field E can be written

€ E(t) = E 0 cos ωt + ϕ(t) ( ) (4)
where E 0 is the amplitude of the electric field, ω the pulsation of the un-modulated light, and φ(t) the phase modulation. In the case of a sawtooth waveform,

€ ϕ(t) = αt [mod] 2π (5)
ICSO 2010

Rhodes, Greece International Conference on Space Optics 4 -8 October 2010

The efficiency of serrodyne modulation depends on the steepness of the falling edges of the sawtooth waveform. If the edge is not steep enough, spurious sidebands will be created, impairing the quality of the frequency-shifting. The generation of the sawtooth can be easily achieved by means of a passive Non Lienar Transmission Line (NLTL). To achieve a frequency shift of α/2π, one applies a sine wave of pulsation α on the input of the NLTL, and the output is connected to the input of the optical phase modulator.

Serrodyne modulation is a convenient way to shift the frequency of our laser between the cooling and interrogating phases of the experiment. The frequency shift is very fast compared to a usual frequency lock, and only one laser is needed.

C. Free-space bench

As for the Rubidium system, the Second Harmonic Generation takes place in free-space. Beforehand, the 1560 and 1534 nm lasers are combined, and amplified to 5 W each in a dual-frequency EDFA. They are then outcoupled, and sent through two crystals in a double-pass cascade configuration. The first crystal is phasematch for the frequency-doubling of 1560 nm light. The second crystal is tuned to the 1534 nm phase-matching condition. The 780 nm light is then sent to fibers in a configuration which is similar to the single-wavelength setup. The 767 nm light is first sent through a double-pass AOM, in order to generate a sideband 462 MHz away from the carrier, corresponding to the hyperfine splitting of 39 Rb. The reason why do not use phase modulation like we did for the Rubidium laser is that the splitting is much smaller (462 MHz vs 6.8 GHz), meaning that the spurious sidebands created by the modulation and the frequency-doubling would still be close to atomic resonance, and contribute to important parasitic signal.

The dual-wavelength laser source is shown in Fig. 4. As we did for the 780 nm source, we have favored fiber components in the design of the system. The free-space bench will be larger (approx. 40 x 70 cm 2 ), but we have designed it to be stable enough, using aeronautic-grade aluminium for the optical flat.

V. PRELIMINARY RESULTS

The single-wavelength Rubidium setup has been thoroughly tested, and has enabled us to carry out precision measurements in the zero-G aircraft. We have also tested part of the frequency-comb, and obtained a satisfying behaviour, the femtosecond laser remaining mode-locked throughout the parabolic flight.

We are now assembling and testing the dual-wavelength source. We have shown that it was possible to trap and cool down atoms with a serrodyne-modulated laser, even after the SHG stage, which tends to enhance spurious sidebands. We have also tested the EDFA and the SHG module. Using the 767 nm light and a direct modulation of the DFB diode [START_REF] Stern | Frequency doubled 1534 nm laser system for potassium laser cooling[END_REF], we have obtained a dual-species K-Rb magneto-optical trap. This is the first step towards a dual-species atom interferometer and a test of the UFF with cold atoms. We have constructed a very robust laser source, and shown that it was possible to carry out atom interferometry experiments in a hostile environment like that of an aircraft carrying out parabolic flights. A new laser source is under design, which will give us the possibility to address simultaneously two atomic species in order to test the Universality of Free Fall with cold atoms as test masses. This work also opens the way for terrain measurements with cold atom sensors, in fields such as gravity mapping or inertial navigation.

Furthermore, cold atoms are seen as promising candidates for space-based measurements, in particular for tests of fundamental physics [START_REF] Ertmer | Matter wave explorer of gravity (MWXG)[END_REF]. The design of the laser system for a space mission will be a key point. With our frequency-doubled telecom laser source, we propose an interesting alternative to extended cavity diode lasers. The use of fiber components allows for the implementation of a wide variety of functions in an intrinsically stable and compact setup.

( 1 )Fig. 1 .

 11 Fig. 1. Schematic view of the atom interferometer. The atoms are cooled down to 10 µK. The cooling beams arethen turned off for the interferometric measurement, so that the cold atom cloud is in free fall. The retroreflecting mirror defines the phase reference of the laser ruler.

Fig. 4 .

 4 Fig. 4. Dual-wavelength laser source

The I.C.E. collaboration is funded by the Centre National d'Etudes Spatiales, as are RG, and GS. VM is funded by Délégation Générale de l'Armement. This work is also supported by the European Space Agency under the SAI Program. Laboratoire Charles Fabry, ONERA and SYRTE are all members of IFRAF.