P.-L Schaefer
email: pierre-loup.schaefer@univ-grenoble-alpes.fr

• G Barrier

• G Chagnon

• T Alonso

• A Moreau-Gaudry

Guillaume Barrier
email: barrier.guillaume@gmail.com

Grégory Chagnon
email: gregory.chagnon@univ-grenoble-alpes.fr

Thierry Alonso
email: thierry.alonso@univ-grenoble-alpes.fr

Alexandre Moreau-Gaudry
email: alexandre.moreau-gaudry@univ-grenoble-alpes.fr

Strain Gauges Based 3D Shape Monitoring of Beam Structures Using Finite Width Gauge Model

Keywords: Beam monitoring, 3D reconstruction, Shape reconstruction, Strain sensor, Strain measurement

This paper presents a new approach validated experimentally to reconstruct with strain gauges the deformed shape of a straight beam with circular cross section. It is based on a novel beam-specific strain gauge model that improves the strain measurement by taking into account the width of the gauges. These improved strain measurements are used by a 3D finite strain large displacement beam shape reconstruction method to recover the deformed shape iteratively. The whole reconstruction approach has been validated experimentally with 3D deformations of a beam instrumented with strain gauges. Results show that the strain gauge model developed improves reconstruction accuracy and that beam reconstruction can be achieved effectively.

Introduction

Real-time shape monitoring of a deformed material structure is an important problem with various domains of applications, such as in the construction [START_REF] Moreau-Gaudry | Analysis of the status of pre-release cracks in prestressed concrete structures using longgauge sensors[END_REF][START_REF] Lee | Real-time displacement measurement of a flexible bridge using digital image processing techniques[END_REF][START_REF] Xia | Deformation monitoring of a super-tall structure using real-time strain data[END_REF] or the medical field [START_REF] Park | Real-time estimation of 3-d needle shape and deflection for mri-guided interventions[END_REF][START_REF] Roesthuis | Three-dimensional needle shape reconstruction using an array of fiber bragg grating sensors[END_REF][START_REF] Lehmann | A real-time estimator for needle deflection during insertion into soft tissue based on adaptive modeling of needle[END_REF]. A simple and straightforward way to monitor shape consists in measuring directly the displacement of the material structure by using technologies based on electronics and optics, such as deflection gauge, CCD camera or laser scanner [START_REF] Logozzo | Recent advances in dental optics part ii: Experimental tests for a new intraoral scanner[END_REF][START_REF] Cali | An integrated approach to characterize the dynamic behaviour of a mechanical chain tensioner by functional tolerancing[END_REF]. Nevertheless, this type of direct monitoring can be challenging in some cases due to instrumentation constraints and hidden or unreachable areas [START_REF] Yan | A sensor for the direct measurement of curvature based on flexoelectricity[END_REF]. Structure shape monitoring can then be achieved indirectly by reconstructing the shape from the structure characteristics, such as strain.

The topic of using strain measures for beam shape monitoring has been widely covered in the litterature on both theoretical [START_REF] Glaser | Shape monitoring of a beam structure from measured strain or curvature[END_REF][START_REF] Cheng | 3d beam shape estimation based on distributed coaxial cable interferometric sensor[END_REF][START_REF] Todd | A local material basis solution approach to reconstructing the three-dimensional displacement of rod-like structures from strain measurements[END_REF][START_REF] Gherlone | An inverse finite element method for beam shape sensing: theoretical framework and experimental validation[END_REF][START_REF] Chadha | A generalized approach for reconstructing the three-dimensional shape of slender structures including the effects of curvature, shear, torsion, and elongation[END_REF] and applied aspects, such as needles deflections during medical intervention [START_REF] Park | Real-time estimation of 3-d needle shape and deflection for mri-guided interventions[END_REF][START_REF] Roesthuis | Three-dimensional needle shape reconstruction using an array of fiber bragg grating sensors[END_REF][START_REF] Henken | Error analysis of fbg-based shape sensors for medical needle tracking[END_REF] or bridge deformation over time [START_REF] Sigurdardottir | Error in the determination of the deformed shape of prismatic beams using the double integration of curvature[END_REF]. The strain measures are acquired through technologies such as strain gauges [START_REF] Glaser | Shape monitoring of a beam structure from measured strain or curvature[END_REF][START_REF] Gherlone | An inverse finite element method for beam shape sensing: theoretical framework and experimental validation[END_REF][START_REF] Wang | Strain modes based dynamic displacement estimation of beam structures with strain sensors[END_REF], or fiber Bragg gratings [START_REF] Sigurdardottir | Error in the determination of the deformed shape of prismatic beams using the double integration of curvature[END_REF][START_REF] Kim | Estimating deflection of a simple beam model using fiber optic bragg-grating sensors[END_REF][START_REF] Moon | Fiber-bragg-grating-based ultrathin shape sensors displaying single-channel sweeping for minimally invasive surgery[END_REF][START_REF] Wang | Shape detection algorithm for soft manipulator based on fiber bragg gratings[END_REF][START_REF] Childlers | Fiber optic position and shape sensing device and method relating thereto[END_REF] and are used to obtain deformations of the beam. Depending on the orientations of the sensors these measures can be used to recover the different deformations of the beam such as bending [START_REF] Todd | A local material basis solution approach to reconstructing the three-dimensional displacement of rod-like structures from strain measurements[END_REF], torsion [START_REF] Xu | Curvature, torsion, and force sensing in continuum robots using helically wrapped fbg sensors[END_REF] and shearing and elongation [START_REF] Chadha | A generalized approach for reconstructing the three-dimensional shape of slender structures including the effects of curvature, shear, torsion, and elongation[END_REF]. The strain measures are thus used as inputs in a reconstruction method to obtain the structure deformed shape. The reconstruction methods proposed in the literature are either two dimensional [START_REF] Glaser | Shape monitoring of a beam structure from measured strain or curvature[END_REF][START_REF] Cheng | 3d beam shape estimation based on distributed coaxial cable interferometric sensor[END_REF][START_REF] Sigurdardottir | Error in the determination of the deformed shape of prismatic beams using the double integration of curvature[END_REF][START_REF] Wang | Strain modes based dynamic displacement estimation of beam structures with strain sensors[END_REF][START_REF] Kim | Estimating deflection of a simple beam model using fiber optic bragg-grating sensors[END_REF][START_REF] Xu | Deflection estimation of bending beam structures using fiber bragg grating strain sensors[END_REF] or three dimensional [START_REF] Todd | A local material basis solution approach to reconstructing the three-dimensional displacement of rod-like structures from strain measurements[END_REF][START_REF] Chadha | A generalized approach for reconstructing the three-dimensional shape of slender structures including the effects of curvature, shear, torsion, and elongation[END_REF][START_REF] Moon | Fiber-bragg-grating-based ultrathin shape sensors displaying single-channel sweeping for minimally invasive surgery[END_REF][START_REF] Wang | Shape detection algorithm for soft manipulator based on fiber bragg gratings[END_REF][START_REF] Childlers | Fiber optic position and shape sensing device and method relating thereto[END_REF] depending on the deformation hypothesis made.

The goal of this paper is to propose a new approach to reconstruct the deformed shape of a circular straight beam undergoing 3D deformations and to qualify experimentally its accuracy. Section 2 introduces a novel beam-specific strain gauge model and a 3D beam shape reconstruction method based on beam theory. Section 3 presents a test bench composed of a beam instrumented with strain gauges for reconstruction purpose. The procedures and informations on the deformations experiments are described in Section 4. The reconstructed shapes are then compared with the real beam shapes. Section 5 and 6 present and discuss the results of the reconstruction experiments. Finally, Section 7 provides conclusion and plans for future work.

Models

This section introduces a novel model of strain gauge taking into account its width and a 3D beam shape reconstruction method based on Reissner's finite strain large displacement beam theory [START_REF] Reissner | On one-dimensional largedisplacement finite-strain beam theory[END_REF].

Characterization of Beam Deformation from Strain Gauges

Beam deformation

Let r 0 be the radius of the beam and O a point of the beam's neutral axis of curvilinear abscissa s. The deformation characteristics of the beam at point O are presented in figure 1.

The osculating plane to the beam's neutral axis at point O is noted P b and called the plane of bending. The distance between O and the center of the osculating circle C is the radius of curvature ρ(s) as shown in figure 1(c). The curvature of the neutral axis at point O is defined as the inverse of the radius of curvature and is noted κ(s) = 1/ρ(s). The plane orthogonal to P b containing O and C is noted P s and called the section plane. The intersection of the beam with the section plane P s then defines the cross section of the beam at point O as illustrated in figure 1(b). The rigid cross section thus remains orthogonal to the neutral axis, the shearing deformation is therefore not considered. The intersection between the surface of the cross section and the bending plane which does not belong to the segment OC is noted D. Let M be a reference material point Fig. 2 Scheme of a resistance strain gauge whose resistance value changes with the gauge deformation. The strain measured is the strain in the direction of the gauge length.

on the surface of the cross section. The angle M OD associated to M is noted θ(s) and called bending angle.

The axial strain at the surface of the beam at the angle position α related to M is then noted ε(s, α). Under the hypothesis of linear elastic deformations, the axial strain ε(s, α) can be expressed with the curvature κ(s), the bending angle θ(s) and δ(s) the bias due to other deformations than bending [START_REF] Roesthuis | Three-dimensional needle shape reconstruction using an array of fiber bragg grating sensors[END_REF][START_REF] Ryu | Fbg-based shape sensing tubes for continuum robots[END_REF]:

ε(s, α) = r 0 κ(s) cos(θ(s) -α) + δ(s) (1)
The torsion, tensile and compressive deformations of the beam are not taken into account. It should be noted that the value of strain ε(s, α) in ( 1) is independent from the choice of the material point M .

Finite Width Strain Gauge Model

Beam shape monitoring with strain gauges consists of reconstructing the shape of the beam from the acquired data of the strain gauges. A strain gauge is a sensor which provides a strain measure ε * of the strain ε of the surface where it is fixed. This device is composed of an electrical resistance, as illustrated in figure 2.

The deformation of a gauge has for effect to change its electrical resistance value which can be measured. The gauge strain measure ε * is the mean of the strain conditions existing under its surface [START_REF] Hoffmann | An introduction to measurements using strain gages[END_REF]. In the case of beam monitoring with strain gauges placed parallel to the beam, when the length of the strain gauge is small compared to the length of the beam, the variation of the axial strain along the length of the gauge is neglected. Moreover, when the plane of bending is orthogonal to the gauge the axial strain is constant along the width of the beam, such as in bridge deformation monitoring [START_REF] Sigurdardottir | Error in the determination of the deformed shape of prismatic beams using the double integration of curvature[END_REF]. Consequently, as the strain field is supposed uniform under the strain gauge, the averaging by the gauge is ignored [START_REF] Sigurdardottir | Error in the determination of the deformed shape of prismatic beams using the double integration of curvature[END_REF][START_REF] Wang | Strain modes based dynamic displacement estimation of beam structures with strain sensors[END_REF]. For some circular beam shape monitoring, such as needle shape reconstruction, the strain sensor width is not taken into account and the strain measurements are supposed to be point-related [START_REF] Abayazid | 3d flexible needle steering in soft-tissue phantoms using fiber bragg grating sensors[END_REF][START_REF] Henken | Error analysis of fbg-based shape sensors for medical needle tracking[END_REF][START_REF] Roesthuis | Three-dimensional needle shape reconstruction using an array of fiber bragg grating sensors[END_REF]. We name that model Infinitesimal Width Strain Gauge model (IW). Consequently, the expression of the strain measure ε * IW of a strain gauge located at angle α g using the IW model is:

ε * IW (s, α g ) =ε(s, α g ) (2) =r 0 κ(s) cos(θ(s) -α g ) + δ(s) (3)
Nevertheless, the dependance of ε to the variable α in [START_REF] Moreau-Gaudry | Analysis of the status of pre-release cracks in prestressed concrete structures using longgauge sensors[END_REF] shows that the axial strain on the surface of the beam is not constant along the width of the strain gauge. In order to obtain an accurate gauge model, we developed the Finite Width Strain Gauge model (FW), which takes into account the width of the gauges. The gauge width is then characterized by its semi-angle β defined in figure 3a. By using the strain field relation provided by Schajer [START_REF] Schajer | Use of displacement data to calculate strain gauge response in non-uniform strain fields[END_REF] with the hypothesis that the axial strain is constant along the length of the gauge, the strain measure ε * F W with the FW model corresponds to the averaging of the strain field on the width of the gauge:

ε * F W (s, α g ) = 1 2βr 0 αg+β αg-β ε(s, α)r 0 dα (4) = 1 2β αg+β αg-β [r 0 κ(s)cos(θ(s)-α)+δ(s)]dα (5) = r 0 κ(s) 2β αg+β αg-β cos(θ(s) -α)dα + δ(s) (6) = r 0 sin(β) β κ(s) cos(θ(s) -α g ) + δ(s) (7) = r 0 sinc(β)κ(s) cos(θ(s) -α g ) + δ(s) (8)
Consequently, the expression of the strain measure ε * with the IW model ( 3) and the FW model ( 8) differs by the term sinc(β) which results from the hypothesis of the strain averaging. The estimations of curvature, bending angle and bias are noted κ IW , θ IW and δ IW using IW model and κ F W , θ F W and δ F W using FW model. Equations ( 3) and ( 8) give the following relations:

κ F W = 1 sinc(β) κ IW , θ F W = θ IW , δ F W = δ IW (9)
Thus, the expressions of bending angle θ and bias δ are identical with both models. On the contrary, the expression of the curvature is different and its value is multiplied by a factor of 1/sinc(β) with the FW model. Consequently, the use of IW model leads to an underassessment of the curvature compared to the FW model. The percent error of the curvature estimation between FW and IW models is presented in figure 4.

It should be noted that when β → 0, sinc(β) → 1 and then ε * F W → ε * IW and κ F W → κ IW . Therefore, the FW model converges to the IW model when β tends to 0.

Beam Instrumentation with Strain Gauges

The strain ε on the surface of a cross section of the beam depends of the three unknowns which are the curvature κ, the bending angle θ and the bias δ as shown in [START_REF] Cali | An integrated approach to characterize the dynamic behaviour of a mechanical chain tensioner by functional tolerancing[END_REF]. In some case, the positions of the loads on a structure are restricted or can be anticipated. It is then possible to formulate hypothesis on the direction of deformations which allow to reduce the number of strain gauges necessary for shape reconstruction [START_REF] Sigurdardottir | Error in the determination of the deformed shape of prismatic beams using the double integration of curvature[END_REF]. As no restrictive hypothesis have been formulated on the direction of the deformation of the beam, it is thus necessary to place three strain gauges per instrumented cross section. The strain gauges are placed by groups of three with a 120 • between them [START_REF] Abayazid | 3d flexible needle steering in soft-tissue phantoms using fiber bragg grating sensors[END_REF][START_REF] Henken | Error analysis of fbg-based shape sensors for medical needle tracking[END_REF][START_REF] Roesthuis | Three-dimensional needle shape reconstruction using an array of fiber bragg grating sensors[END_REF], as shown in figure 3b. This configuration is named triplet in the rest of the paper. By orienting the strain gauges parallely to the beam, the axial surface strain can be measured. The sensor triplet located on the cross section is then composed of strain sensors G 1 , G 2 and G 3 whose respective local axial strain measures are ε * 1 , ε * 2 and ε * 3 . For the three strain gauges of a triplet, the FW model gives the following system:

ε * 1 = r 0 sinc(β)κ cos(θ) + δ ε * 2 = r 0 sinc(β)κ cos(θ - 2π 3 ) + δ ( 10
)
ε * 3 = r 0 sinc(β)κ cos(θ - 4π 3 ) + δ
Resolution of [START_REF] Glaser | Shape monitoring of a beam structure from measured strain or curvature[END_REF] gives the following values of curvature κ, bending angle θ and bias δ:

κ = 1 sinc(β) × C 3r 0 (11) cos(θ) = 2ε * 1 -ε * 2 -ε * 3 C (12) sin(θ) = √ 3(ε * 2 -ε * 3 ) C (13) δ = ε * 1 + ε * 2 + ε * 3 3 ( 14
)
where

C = √ 2 (ε * 1 -ε * 2 ) 2 + (ε * 2 -ε * 3 ) 2 + (ε * 1 -ε * 3 ) 2 1 2 (15)

Beam Shape Reconstruction

The values of the curvature and the bending angle can be obtained at each triplet location using [START_REF] Cheng | 3d beam shape estimation based on distributed coaxial cable interferometric sensor[END_REF], [START_REF] Todd | A local material basis solution approach to reconstructing the three-dimensional displacement of rod-like structures from strain measurements[END_REF] and [START_REF] Gherlone | An inverse finite element method for beam shape sensing: theoretical framework and experimental validation[END_REF]. The interpolation of those values gives the curvature estimate κ and the bending angle estimate θ along the whole length of the needle. Let (T, N 1 , N 2 ) be the convected material frame of the beam with T the row vector of the tangent of the beam neutral axis and N 1 and N 2 two material row vectors of the beam. The evolution of the frame along the length of the beam depends on the curvature κ and the bending angle θ [START_REF] Schaefer | Advanced sensors placement for accurate 3d needle shape reconstruction[END_REF]:

Y (s) = A(s)Y(s) (16)
Y(0) = Y 0 (17)
where:

Y(s) =   T(s) N 1 (s) N 2 (s)   (18)
A(s) = κ(s)   0 cos θ(s) -sin θ(s) -cos θ(s) 0 0 sin θ(s) 0 0   ( 19
)
The expression of A in [START_REF] Moon | Fiber-bragg-grating-based ultrathin shape sensors displaying single-channel sweeping for minimally invasive surgery[END_REF] shows that the frame (T, N 1 , N 2 ) satisfies parallel transport properties. Such frames are called rotation-minimizing frames and are characterized by their stability and their absence of singularities compared to other frames such as Serret-Frenet [START_REF] Wang | Computation of rotation minimizing frames[END_REF]. Equation ( 17) will be resolved with the iterative method called Local Coordinates Approach [START_REF] Hairer | Geometric Numerical Integration[END_REF]. The iterative step of the method is based on the Magnus expansion [START_REF]On the exponential solution of differential equations for a linear operator[END_REF] which gives an exponential representation of the solution of first order matrix differential equation such as [START_REF] Wang | Strain modes based dynamic displacement estimation of beam structures with strain sensors[END_REF]. The interval of resolution is noted [s i , s i+1 ] with s i+1 = s i + h, h being the step of the method. The approximation of Y(s i ) and Y(s i+1 ) are noted Y i and Y i+1 . The matrix of initial condition Y 0 contains the coordinates of the moving frame at length s 0 and is composed of the row vectors T(s 0 ), N 1 (s 0 ) and N 2 (s 0 ). Here the length s 0 corresponds to the clamped extremity of the beam.

Y 0 =   T(s 0 ) N 1 (s 0 ) N 2 (s 0 )   ( 20
)
The truncation of the Magnus expansion and the approximation by midpoint rule gives the following iterative step of order 2 [START_REF] Hairer | Geometric Numerical Integration[END_REF]:

Y i+1 = exp hA s i + h 2 Y i (21)
Equation ( 21) contains a matrix exponential. Using the formula about exponential map in SO(3) [START_REF] Simo | On a stress resultant geometrically exact shell model. part i: Formulation and optimal parametrization[END_REF] gives the following expression:

exp hA s i + h 2 =   c 1 s 1 c 2 -s 1 s 2 -s 1 c 2 c 1 + λs 2 2 λc 2 s 2 s 1 s 2 λc 2 s 2 c 1 + λc 2 2   (22)
where

c 1 = cos hκ s i + h 2 (23)
s 1 = sin hκ s i + h 2 (24)
c 2 = cos θ s i + h 2 (25)
s 2 = sin θ s i + h 2 (26)
λ = 1 -hκ s i + h 2 (27)
Thus the set (Y i ) i=0,..,n is obtained by using [START_REF] Childlers | Fiber optic position and shape sensing device and method relating thereto[END_REF] iteratively on the discretization of the beam length (s i ) i=0,..,n . It contains the coordinates of the tangents of the beam neutral axis (T i ) i=0,..,n . Let (M i ) i=0,..,n be the set of points of the neutral axis of the beam for the discretization (s i ) i=0,..,n with M i the approximation of the position of the neutral axis M(s i ) for i = 1, .., n and M 0 the coordinates of the beam extremity. This set can be calculated iteratively using the tangents obtained previously:

M i+1 = M i + hT i (28)
Finally (M i ) i=0,..,n is a set of 3D points representing the shape of the deformed beam. It should be noted that the resolution can be generalized to discrete set with non constant step by using h i = s i+1 -s i instead of h in ( 21) to [START_REF] Schajer | Use of displacement data to calculate strain gauge response in non-uniform strain fields[END_REF].

Materials

This section describes the beam instrumentations used and the test bench designed.

Beam Characteristics

The material chosen for beams is fiberglass. The main reasons for this choice is that fiberglass is flexible, so that deformations can be done easily, and that it is compatible with the imaging modalities used to recover the deformed shape of the beam. As a non-ferromagnetic material, fiberglass will not produce any effect during the computerized tomography (CT) acquisitions. Thus, the beams considered are fiberglass cylindrical hollow tubes of 450 mm length with an outer radius of 3 mm and an inner radius of 1 mm.

Beam Instrumentation

The strain gauges are fixed on the beam parallel to the beam direction, as shown in figure 5, in order to measure the beam axial surface strain. They have a Fig. 5 Strain gauge fixed on the beam.

length of 3 mm and a width of 2 mm. Thus, the value of the strain gauge semi-angle β, as defined in figure 3a, is:

β ≈ 19 • (29)
According to figure 4, this means that the curvature value is underestimated by approximately 2% by the IW model compared to the FW model. The strain gauges have a resistance value of 120 ohms and a gauge factor of 2. The bonding of the strain gauges on the beam was realized with a glue designed for fiberglass.

Test Bench

In order to evaluate the beam shape reconstruction accuracy, a test bench has been designed to compare the shape of a beam deformed experimentally and its reconstruction from strain gauges data. The bench is composed of a beam clamped to the side of the box, as shown in figure 7. The beam has been instrumented with two strain gauges triplets as illustrated in figure 3b. have shown that the shape reconstruction and sensor positioning are coupled problems which means that the best position does not only depend of the parameters chosen for the minimization but also of the reconstruction method. Consequently, the optimal position obtained with a certain reconstruction method can be not legitimate for an other reconstruction method. For these reasons, the optimal triplets locations used here are taken from Schaefer et Fig. 7 Beam deformation bench: Deformed position: A cable is attached to the beam near its extremity and is fixed to the box. The beam undergoes a deformation.

al. [START_REF] Schaefer | Advanced sensors placement for accurate 3d needle shape reconstruction[END_REF] as the reconstruction method in this work is identical to the method presented here. These locations refers to optimal locations for strain sensors on needles, but because it has been computed with the same reconstruction method from real experimental deformations we believe that it will give relevant triplets locations. Fitting the results provided in Schaefer et al. to the current beam length then gives triplets locations of 45 mm and 200 mm from the beam fixation.

Methods

This section describes the specifications of the beam experiment developed to validate the reconstruction process. In order to control the deformation applied on the beam of the test bench, cables are fixed to the beam and then hooked to one of the several attachment points disposed on the sides of the box, as shown in figure 7. For each of the deformation, a CT-scan of the bench had been realized and the strain gauges data have been acquired. Deformations 1 and 2 are the results of two orthogonal loads applied on the beams whereas a single load was applied in case of deformations 3 and 4. Thus, in case of deformations 1 and 2 the beam has a 3D shape whereas the deformations 3 and 4 are planar. The deformations characteristics of the beams are presented in table 1. Results show that the beam undergoes large displacements in case of deformations 2, 3 and 4 as the deflections are greater than 10% of the beam length.

Results

The deformed beam shapes have been retrieved from scanner images by segmentation and registration. The strain acquisitions from the two triplets were used to recover the curvature and bending angle values at the triplet locations. These values were then interpolated linearly on the whole length of the beam using null curvature constraint at the end of the beam. From the resulting curvature and bending angle estimates were reconstructed the beam shapes with the method presented in Section 2.2 using a step value h=1 mm. The beam shapes were reconstructed using both IW and FW gauge model for comparison purposes.

One of main criteria to evaluate the accuracy of the reconstruction is the distance between the tip of the beam shape and the tip of the reconstructed shape of the beam [START_REF] Gherlone | An inverse finite element method for beam shape sensing: theoretical framework and experimental validation[END_REF][START_REF] Moon | Fiber-bragg-grating-based ultrathin shape sensors displaying single-channel sweeping for minimally invasive surgery[END_REF][START_REF] Henken | Error analysis of fbg-based shape sensors for medical needle tracking[END_REF][START_REF] Park | Real-time estimation of 3-d needle shape and deflection for mri-guided interventions[END_REF][START_REF] Roesthuis | Three-dimensional needle shape reconstruction using an array of fiber bragg grating sensors[END_REF]. This distance is called tip error. The tip errors results of the reconstructions are presented in table 2. For the shape reconstruction of deformation 1 and 2, the tip errors are 4.8 mm and 12.9 mm with the IW model and 4.7 mm and 12.0 mm with the FW model. For the shape reconstruction of deformation 3 and 4, the tip errors are 7.9 mm and 8.3 mm with the IW model and 7.0 mm and 7.3 mm with the FW model. Thus, the tip errors of the four experimental deformations are smaller when the beam shape is reconstructed using the FW model instead of the IW model. The original beam shapes and their reconstructed shapes with the FW model are presented conjointly for each deformation in figure 9 and figure 10. For deformations 1 and 2, with respective beam deflections of 28.5 mm and 55.0 mm, the gain in accuracy with the FW model is 3% and 6% and for deformations 3 and 4 whose beam deflections are 62.9 mm and 98.1 mm the gain is 11% and 13%. Therefore the larger the beam deflection is, the bigger the gain in accuracy is.

The gain in accuracy according to the beam deflection is presented in figure 8. The tip error of the deformations increases with the deflection of the beam except for deformation 2. In fact the tip error of deformation 2 is bigger than the tip error of deformations 3 and 4 who have higher deflections. This can be explained by the nature of deformation 2 which combines large displacement and 3D aspect. The 3D aspect in particular seems to play an important role as the the tip errors of the 3D deformations 1 and 2 represent 16% and 22% of the beam deflections compared to 11% and 7% for the 2D deformations 3 and 4 as shown in table 3.

Discussion

In this study, a novel model taking into account the width of the gauge was presented. This model is useful when the strain field of the surface on which the gauge is fixed is not uniform. This is the case for the deformations of beam structures presented in this paper. Results show that using the FW model improves the reconstruction accuracy compared to the IW model. This reconstruction accuracy improvement brought by the FW model increases with the beam deflection. In fact, as beams with the largest deflection have also the bigger curvature value, the larger the deflection is, the bigger the curvature correction made by the FW model will be.

The number of sensors and their configurations on the beam constitute input parameters of the beam shape monitoring. The configurations of the sensors include the locations of the triplets on the beam and the angles where the sensors are positioned around the cross section. The accuracy with which the sensors are positioned on the beam can then be a potential source of error for these input parameters and is then a factor influencing the accuracy of the shape monitoring. A sensitivity analysis of circular beams instrumented with strain sensors has been performed by Henkel et al. [START_REF] Henken | Error analysis of fbg-based shape sensors for medical needle tracking[END_REF] in work concerning needles instrumented with fiber Bragg gratings. In this work, the instrumented needle was modelized by a beam with triplets of strain sensors positioned at 120 • . Besides the difference of sensor technology, the results presented are thus still relevant with our work. Using simulations Henken et al. showed that the error in the estimate of curvature and bending angle is linearly related to the angular position inaccuracy around the cross section. The maximal error for an innacurate positioning of 2 • is approximately 3% of the curvature and 2 • for the bending angle. Concern- ing the impact of triplets locations, simulation results show that for a number of triplet comprised between one to five, the maximum error due to triplets positioning inaccuracy of 2 mm for a 200 mm needle is 6% of the reconstruction error. These results were based on simulated shapes obtained from 2D distributed forces loads which were also used to evaluate the impact of the number of triplets on the reconstruction accuracy. It was shown that unsurprisingly the accuracy of shape sensing increases when a triplet is added, the effect becoming small when used in excess of five triplets. The combined effect of these inaccuracies in sensor configurations (locations and angular positions) gives a maximal error in shape estimation of 7.1% of the reconstruction error. The actual sensitivity of the shape sensing to the input parameters is hard to assess as it depends of multiple factors such as beam deformations and cannot be predicted beforehand. Nevertheless, the results of Henken et. al. show that the gain in accuracy provided by the use of the FW model (up to 13%) is not negligible compared to error caused by sensor positioning inaccuracy.

Papers in literature dealing with beam reconstruction from strain gauges present tip errors results of experimental reconstructions for beam structures with different length and deflections. These results can then be compared by expressing the tip errors as a percentage of the deflection. Thus, Gu et al. [START_REF] Gu | A flexible-arm as manipulator position and force detection unit[END_REF] presents the reconstruction of a 138 cm flexible arm with a tip error of 2.9 mm for a 42.7 mm deflection, which corresponds to an error of 7% of the deflection. In their work, Payo et al. [START_REF] Payo | Strain gauges based sensor system for measuring 3-d deflections of flexible beams[END_REF] reconstruct a 120 cm beam with a tip error of 3.8 mm for a 32 mm deflection, which corresponds to an error of 12% of the deflection. The results of deformations 3 and 4 which are 11% and 7% are consistent with these tip errors results. The tip errors of deformation 1 and 2 are higher with values of 16% and 22% of the beam deflection. The higher values of these errors can be explained by the 3D property of these deformations, implying that two strain triplets are not enough to capture all their complexities thus leading to less accurate reconstructions. Consequently the less good results obtained in those two cases come from the beam being instrumented with only six strain sensors. Nevertheless, to put these results into perspective it should be noted that the reconstruction results of Gu of the beam deflection being approximately 3% of the beam length whereas the deformations considered in our work include larger deflections from 6% to 22% of the beam length, as shown in table 3. Consequently, despite the large deflections of the beams considered, the beam shape reconstruction results presented in this paper obtained from a limited number of strain measures are consistent with the literature. These results demonstrate the ability of the novel techniques presented in this article to achieve the goal of monitoring beam deformed shapes correctly.

Differents improvements could be performed on the beam instrumentation in order to increase the beam shape reconstruction accuracy. In fact, as mentionned previously, changing the number of strain sensors by adding additional strain gauge triplets on the beam would be an effective way to improve the reconstruction accuracy in case of 3D deformations. Interestingly, the number of strain triplets is not the only change that could be considered to improve the reconstruction accuracy. The positioning of the strain sensors parallel to the beam thus allows to have access to the bending informations of the beam. By changing the direction of the strain sensors to another direction, not parallel to the beam, it would be possible to obtain informations related to the torsion of the beam. Future work will focus on the development of a model of beam instrumented with strain gauges able to capture both axial and shear strain and how to use these to improve the reconstruction accuracy.

Conclusion

A 3D shape monitoring system for beam with strain gauges has been presented. The strain gauges were fixed on the surface of the beam by groups of three on the same cross section. An acquisition system was developed to measure the strain at each gauge locations. A finite width strain gauge model was presented in order to recover the estimates of the curvature and the bending angle of the beam while taking into account physical characteristics of 3D deformations. These estimates were used as inputs of the 3D finite strain large displacement beam shape reconstruction method to approximate the deformed beam shape. Results showed that correct beam shape reconstructions were achieved with only a limited number of strain measures, demonstrating the effectiveness of beam instrumentation, the strain gauge model and the reconstruction method pro-posed. Moreover, the creation of a finite width gauge model is not restricted to beams with circular cross section as the technique of strain field averaging over gauge width can be applied to any cross section geometry.

The approach and model presented here are particularly suitable for applications monitoring beam undergoing 3D deformations with strain gauges of significant size compared to the structure. In the medical field for instance, needle shape monitoring from strain gauges, such as in Hammond et al. [START_REF] Hammond | Estimating surgical needle deflection with printed strain gauges[END_REF] and Bonvilain et al.

[39] is a potential domain of application for the work presented in this paper.

Fig. 1

 1 Fig. 1 Deformation of a beam element (a) The plane of bending P b is the osculating plane to the beam's neutral axis at point O and the section plane P s is the plane orthogonal to P b containing O and the center of the osculating circle C. (b) Section plane P s : The bending angle θ(s) related to a reference material point on the surface of the cross section is the angle between this point and the plane of bending P b . The maximum and minimum strains are noted ε max and -ε max . (c) Plane of bending P b : The radius of curvature ρ(s) is the distance between O and C.

Fig. 3

 3 Fig. 3 Strain gauge fixed on the surface of the beam. (a) The segment OC is the radius between the center of the cross section of the beam and the center of the gauge. The segment OM i is the radius between the center of the cross-section and the side of the gauge. The angle between OC and OM i is called the semi-angle of the gauge and is noted β. (b) Sensor triplets composed of strain gauges G 1 , G 2 , G 3 placed on a beam cross section with a 120 • angle between them. The angular position α i of the gauge G i on the cross section is expressed relatively to the reference position of the gauge G 1 . Thus α 1 = 0, α 2 = 2π 3 and α 3 = 4π 3 .

Fig. 4

 4 Fig. 4 Percent error between the curvature values obtained with IW and FW models.

  An acquisition system was developed to retrieve strain gauges values in real-time. As this process involves sensing extremely small resistance variations, Wheatstone bridges are used in order to measure accurately the resistance changes. The output signals of the Wheatstone bridges are then amplified by a signal conditioner (model Seneca Z-SG Strain Gauge Converter) before being dispatched to the data acquisition card of a PC. The experimental set-up of the acquisition system is presented in figure 6a. The resistance signals are then processed by the PC which computes the strain of each gauges in real-time, as shown in figure 6b.

( a )

 a Acquisition process of the strain gauges signals. (b) Real-time monitoring of the strain gauges resistance values.

Fig. 6

 6 Fig. 6 Strain gauges acquisition system

Fig. 8

 8 Fig. 8 Gain in reconstruction accuracy using the FW model.

  Deformation 1 : the beam deflection is 28.5 mm which corresponds to 6% of the beam length.

  (xOy) plane (xOz) plane (b) Deformation 2 : the beam deflection is 55.0 mm which corresponds to 12% of the beam length.

Fig. 9

 9 Fig.9Original shape (plain) and reconstructed shape (dotted) of a 3D deformed beam. Loads are located at 250 mm and 400 mm directed towards the (Oz) and (Oy) axis. Units are in millimeters.

  et al. have been obtained with 16 strain measures and the results of Payo et al. with 10 strain measures. Furthermore, in both works of Payo et al. and Gu et al. only small displacement deformations were considered, the value

  Deformation 3 : the beam deflection is 62.9 mm which corresponds to 14% of the beam length.

  (xOy) plane (xOz) plane (b) Deformation 4 : the beam deflection is 98.1 mm which corresponds to 22% of the beam length.

Fig. 10

 10 Fig.10Original shape (plain) and reconstructed shape (dotted) of a 2D deformed beam. Loads are located at 400 mm directed towards the (Oy) axis. Units are in millimeters.

Table 1 :

 1 Beam characteristics for deformations 1 to 4.

	Deformation Beam length (mm)	Beam loads position	Deflection (mm)
	1	450	250 mm (Oz) / 400 mm (Oy)	28.5
	2	450	250 mm (Oz) / 400 mm (Oy)	55.0
	3	450	400 mm (Oy)	62.9
	4	450	400 mm (Oy)	98.1

Table 2 :

 2 Beam reconstruction characteristics. E IW and E F W are the reconstruction tip errors for the strain gauge models IW and FW. The gain in reconstruction accuracy with the FW strain gauge model is expressed in absolute terms and relatively to E IW .

	Deformation E IW (mm) E F W (mm) Gain in accuracy (mm) Gain in accuracy (% of E IW )
	1	4.8	4.7	0.1	3%
	2	12.9	12.0	0.9	6%
	3	7.9	7.0	0.9	11%
	4	8.3	7.3	1.0	13%

Table 3 :

 3 Comparison of beam reconstruction characteristics of the experimental deformations with the results of Gu et al.[START_REF] Gu | A flexible-arm as manipulator position and force detection unit[END_REF] and Payo et al.[START_REF] Payo | Strain gauges based sensor system for measuring 3-d deflections of flexible beams[END_REF]. Deflection and tip error E are given as relative values for comparison purposes.

	Experiment	Deflection (% of Beam length) Cross sections instrumented Strain measures E (% of Deflection)
	Gu et al.	3%	4	16	7%
	Payo et al.	3%	4	10	12%
	Deformation 1	6%	2	6	16%
	Deformation 2	12%	2	6	22%
	Deformation 3	14%	2	6	11%
	Deformation 4	22%	2	6	7%

Acknowledgment This work is part of the project GAME-D, financed by the French National Agency for Research (ref: ANR-12-TECS-0019) and supported by Laboratory of Excellence CAMI (ref: ANR-11-LABX-0004-01).

The authors would like to thank Cecilia Hughes for English corrections and P. A. Barraud for providing valuable advice concerning electronic instrumentation.

Conflicts of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


Sébastien Dubos
email: sebastien.dubos@cea.fr

Olivier Limousin

Claire Blondel

Rémi Chipaux

Youri Dolgorouky

Olivier Gevin

Yves Ménesguen

Aline Meuris

Thierry Orduna

Thierry Tourette

Aymeric Sauvageon

Transactions on Nuclear Science -Copy for Review Low Energy characterization of Caliste HD, a fine pitch CdTe-based Imaging Spectrometer

Caliste HD is a recently developed micro-camera designed for X and gamma-ray astronomy, based on a 1×1 cm 2 CdTe Schottky pixelated detector. Its entire surface is composed of 256 pixels, disposed on a 16×16 pixel matrix. This spectrometer is buttable on its 4 sides and can be used to create a large focal plane. Caliste HD is designed for space environment. Its IDeF-X front-end electronics has a low power consumption, excellent noise performance and a wide dynamic range, from 2 keV to 1 MeV. Moreover, electronic noise performances of this device were optimized to set the low level-energy threshold lower than 2 keV.

This paper focuses on the Caliste HD performance near the low energy limit. For this purpose, we have exposed the Caliste HD module to a mono-energetic X-ray beam, and set energies between 2 and 12 keV. We measured accurately the detection efficiency in this energy range and found it to be ranging from 39% to 75% for energies from 2.2 keV to 11.6 keV, considering only particles detected in the single-event photopeak and ignoring events impinging between two adjacent pixels. This efficiency detection profile thereby highlights crucial effects of the Pt electrode opacity on Caliste HD low energy response, and suggests the presence of absorption zones at the interface between CdTe crystal and platinum. Respective thickness of each layer were estimated by simulation and confirmed by RBS (Rutherford Backscattering Spectroscopy).

Besides, using a mono-energetic beam allows fine energy resolution measurement, which was found to be ranging from 560 to 760 eV FWHM between 2 and 12 keV. In addition, the linearity of this spectrometer and the issue of charge sharing between adjacent pixels were studied. This study revealed that spectroscopic performances remain excellent for such boundary operating conditions.

I. INTRODUCTION

HE Caliste HD (HD is standing for "High Dynamic") module is the third generation of a development series of CdTe-based imaging spectrometers at CEA Saclay [START_REF] Meuris | Caliste 64, an Innovative CdTe Hard X-Ray Micro-Camera[END_REF][START_REF] Limousin | A CdTe imaging spectrometer for space science with a 580µm pixel pitch[END_REF][START_REF] Meuris | Caliste HD: A new fine pitch Cd(Zn)Te imaging spectrometer from 2 keV up to 1 MeV[END_REF]. This advanced device is designed for future X or gamma-ray astronomy missions and Solar physics [START_REF] Takahashi | The ASTRO-H Mission[END_REF][START_REF] Harrison | The Nuclear Spectroscopic Telescope Array (NuSTAR)[END_REF][START_REF] Tagliaferri | NHXM: a new hard X-ray imaging and polarimetric mission[END_REF][START_REF] Ferrando | The COSPIX mission: focusing on the energetic and obscured Universe[END_REF][START_REF] Ferrando | Simbol-X: mission overview[END_REF][START_REF] Meuris | Caliste-SO X-ray micro-camera for the STIX instrument on-board Solar Orbiter space mission[END_REF]. It operates typically in the energy range between 2 keV and 1 MeV, with excellent spatial resolution and spectroscopic performances.

The modular design of this small imaging-spectrometer which covers a 1 cm² active surface equipped with 256 pixels, can be used to form large detection areas, composed of sets of modules arranged side by side and disposed at the focal plane of a space telescope to cover a large field of view, typically tens of arcmin for X-ray astronomy [START_REF] Limousin | MACSI: an 8 cm 2 CdTe Detection Plane for Time-Resolved Hard X-Ray Imaging with High Spectroscopic Capability[END_REF]. In addition, the small pitch of the pixelated detector of 625 micrometers provides an appropriate spatial resolution required for this kind of instrument.

As well as good spatial resolution is required, spectral performance is a crucial parameter for X and gamma-ray astronomy. Previous generations of imaging-spectrometers developed at CEA -named Caliste 64 [START_REF] Meuris | Caliste 64, an Innovative CdTe Hard X-Ray Micro-Camera[END_REF] and Caliste 256 [START_REF] Limousin | A CdTe imaging spectrometer for space science with a 580µm pixel pitch[END_REF], with respectively 64 pixels of 1 mm pitch and 256 pixels of 580 µm pitch -already showed excellent spectral performances and a low energy threshold estimated around 2 keV. New improvements on the Caliste HD module results in even better energy resolutions and the low energy threshold can be set below 2 keV [START_REF] Meuris | Caliste HD: A new fine pitch Cd(Zn)Te imaging spectrometer from 2 keV up to 1 MeV[END_REF].

Such a low energy threshold is a major improvement for this kind of detector, as it offers additional information for observations that require multiple wavelengths: it can be useful in astrophysics to observe thermal and non-thermal phenomena in a unique observation. Also, having a unique detector for both low and high energies is interesting to avoid cross-calibration issues and minimize systematic errors in reconstructed spectra.

Significant detection efficiency is therefore required, as well as high energy resolution, energy response linearity and fine pixel pitch.

A detailed study of the detector behavior at these boundary conditions was performed. In particular, a fine measurement of detection efficiency was done using a flux-calibrated X-ray source for low energies, and particularly in the 2 to 6 keV range. This energy band is interesting for the study of this type of CdTe imaging-spectrometer, as the photon collection efficiency that is supposed to be almost total for high energies, may sharply decrease below 6 keV. This behavior is suspected of being related to the absorption of the platinum entrance window, but some other effects can occur.

At first we present an overview of Caliste HD (Part II) and our experimental setup (Part III). Experimental results like integral non-linearity, imaging capabilities and energy resolution are considered. Later on, the data analysis process is detailed (Part IV). Then, we focus on results obtained for detection efficiencies at low energies (Part V). Finally, results T obtained are discussed and extracted parameters compared to RBS measurements. We also consider issues related to charge sharing between pixels (Part VI).

II. CALISTE HD OVERVIEW

Caliste HD [START_REF] Meuris | Caliste HD: A new fine pitch Cd(Zn)Te imaging spectrometer from 2 keV up to 1 MeV[END_REF] device can be divided into 3 parts: a pixelated Schottky CdTe detector subdivided into 256 pixels (first part) which is connected to a specially-developed frontend electronics, including 8 mixed analog-digital IDeF-X HD [START_REF] Michalowska | IDeF-X HD: A low power multi-gain CMOS ASIC for the readout of Cd(Zn)Te detectors[END_REF][START_REF] Gevin | Imaging X-ray detector front-end with high dynamic range: IDeF-X HD[END_REF] ASICs (Application Specified Integrated Circuit) each with 32 spectroscopic channels and disposed perpendicular to the detection surface (second part). This electrical body can communicate through a 4×4 pin grid array located at the opposite side of the detection surface (third part). The details of the assembly of the camera are presented Fig. 1. The IDeF-X HD is entirely designed at CEA Saclay (France) while the CdTe crystal is manufactured by Acrorad (Japan). The integration of these sub-parts into a detection module is realized by 3D Plus (France).

The CdTe detector is surrounded by a 1×1 cm 2 Al anode segmented into an array of 16×16 pixels and a Pt cathode. The latter constitutes the entrance window for incoming photons. The total thickness of the crystal in this work was set to 1 mm.

As shown on Fig. 2, each pixel is 525 µm large, the gap between two adjacent pixels is 100 µm and the matrix is surrounded by a 20-µm width guard ring, also separated from the pixels by a gap of 100 µm. The detection surface is 98 mm 2 , subtracting the area of the guard ring and its 100 µm gap to the outermost pixels. This detection surface is composed of a full metallized pixel surface of 70.56 mm 2 (72% of the total) and a metal free inter-pixel zone of 27.45 mm 2 (28% of the total). As a consequence, for a homogeneous impinging photon flux, 28% of photons are likely to hit an inter-pixel surface. This ratio can be compared to the charge-sharing ratio obtained by prediction and experimentally (Part VI). Each pixel is connected to a spectroscopic channel of an IDeF-X HD ASIC. A channel includes charge pre-amplifier, shaper, baseline holder and peak detector as well as analog memory. A large set of programmable parameters is available on the chip to optimize the signal to noise ratio such as peaking time, threshold or gain value. In addition, low level thresholds can be set individually in each channel.

The module readout is very flexible and the user can arbitrarily decide to use single or multiple events allowing charge sharing reconstruction or particle ionizing track rejection for instance.

Caliste HD is designed for space applications and complies with corresponding fabrication standards as well as radiation hardness, reliability and low power properties.

III. EXPERIMENTAL SETUP

In order to characterize Caliste HD near the low energy threshold, we need a calibrated low energy X-ray beam. This requirement is satisfied by the use of SOLEX (Source Of Low-Energy X-rays) [START_REF] Bonnelle | SOLEX: a tunable monochromatic X-ray source in the 1-20 keV energy range for metrology[END_REF], a facility located at CEA Saclay. In addition, a dedicated setup was developed to realize the interface (mechanical, vacuum, electrical and alignments) between SOLEX and Caliste HD.

A. SOLEX facility

The SOLEX facility is designed to produce a monoenergetic X-ray beam from 1 to 20 keV.

It is composed of an X-ray tube, which produces a polychromatic beam. We can make fine energy selection by directing the beam through a monochromator crystal. Indeed, by changing the incident angle of the beam on the monochromator, we can select a specific wavelength (Bragg law). Different monochromators can be used, depending on desired energies (Table I).

For our experiments, InSb was chosen for characterization runs between 2 and 5.5 keV while a quartz crystal was used at higher energies up to 15 keV The mono-energetic beam is directed to the Caliste characterization setup through an optional and adjustable collimation system to obtain the full spot shape on the detector surface.

Making an absolute efficiency measurement requires that the complete mono-chromator system can rotate and guide the beam to a calibrated proportional gas counter, which gives incident flux reference values [START_REF] Ménesguen | Efficiency calibration and surface mapping of an energy-dispersive detector with SOLEX: A compact tunable monochromatic X-ray source[END_REF].

The beam flux is adjustable between 100 and 1000 counts/s, for a typical acquisition time comprised between 10 and 15 minutes per run.

B. Caliste HD characterization setup

The Caliste characterization chamber is disposed at the output of the SOLEX beam. A bias voltage of 310 V was applied to the detector, and the peaking time set to 10.7 µs. These parameters provide optimal conditions for low energy measurements. A vacuum equal to 10 -5 mbar was established in the vessel, and an active thermal control maintains the sensor at -10°C. Caliste HD has its own internal set of thermal sensor which are used as house-keeping to monitor the operating temperature time to time.

Additionally, a 2-axis motion system is specially developed to allow centering the detector into the beam.

Communication with Caliste HD is established through a FPGA connected to the module. By the same way, we read out the signal amplitudes and time-tags corresponding to triggered pixels. The telemetry obtained is transferred to a computer by a spacewire link, acquired, stored and converted into the standardized FITS file format for off-line analysis. A real-time quick look facilitates alignments and proper Caliste HD operations.

IV. IMAGING AND SPECTROSCOPIC PROPERTIES

Two series of runs were performed: 11 runs between 2.1 and 3.1 keV and 39 runs between 3 and 12 keV, for a total of 50 runs. We emphasized the 2 -6 keV band with 40 runs in this energy range. Some additional points up to 16 keV were also acquired for energy calibration, including linearity check for the entire energy range of interest.

The ability of Caliste HD to detect and image the beam shape, the response linearity at low energies, gain distribution and energy resolution values below 12 keV were studied.

A. Energy calibration & linearity

At first, it was necessary to make an energy calibration of each individual pixel of Caliste HD for further analysis. Moreover, the linearity of this spectrometer for the lowest energies has to be clearly established.

For each pixel, the amplitude (in ADU -Analog/Digital Units) of the photopeak was plotted for all beam energies, and fitted by a linear function (Fig. 4). Calibration parameters obtained for each pixel were stored into a calibration file, systematically used in the following.

The mean value of the gain obtained is 68 ADU/keV, or 1895 ADU/fC. demonstrated here. Furthermore, the gain standard deviation for the 64 most active pixels during the runs is lower than 5%, which shows a good uniformity of the pixel matrix gain.

B. Imaging capabilities

The 16×16 fine pitch pixel array of Caliste HD was used to record the beam shape and to align the entire spot on the detector surface.

For almost all runs, the spot size was constrained to approximately 3×2 mm 2 . An example of image obtained for the lowest energy (2.137 keV) is displayed on Fig. 5. Due to the effect of the collimator used to reduce the beam shape, we obtained a non-symmetric beam on Caliste HD. Considering the X-axis for instance; a smooth increase of the count rate on the left side and a sharp cut on the right side are visible. Image obtained with a 2.137 keV spot. With a pitch of 625 micrometers, we can estimate the spot size to a maximum of 3.75 mm (X axis) and 2.5 mm (Y axis). The color scale represents the count rate of each pixel.

C. Data Analysis protocol

Using our calibration file, we can reconstruct the sum spectrum, by summing all individual spectra obtained for each pixel. An example of sum spectrum obtained at 2.14 keV is shown on Fig. 6. Sum spectra are built using single events (1 hit pixel at a time) and multiple events (2, 3 or 4 simultaneously hit pixels, due to charge-sharing between neighboring pixels). According to the count rate at most equal to 1000 counts/s for the entire detector and the peaking time settled to 10.7 µs, it is very unlikely that spurious double events occur.

On the sum spectrum presented on Fig. 6, the low-level energy threshold is estimated to be as low as 1.3 keV. Due to the fact that this threshold is set close to the beam energy and regardless the energy distribution between neighbor pixels, almost no photon can be detected as a shared event with a 2.137 keV source.

With the increasing beam energy, the difference between impinging photons and low threshold becomes more significant to allow the detection of split events. These additional counts, resulting of charge-sharing between 2, 3 or 4 neighbor pixels degrade the energy resolution of the sum spectrum. Moreover, these reconstructed spectra show a significant energy loss (cf. Part VI), which implies that the beam energy cannot be accurately determined without an appropriate correction. By considering only spectra generated with single events between 2 and 12 keV, we can thereby suppress the contribution of shared events and consider a constant effective detection surface, for simplification. It is very useful for instance, to quantify the effects of the Pt entrance electrode and other opaque layers on the detection efficiency profile without other additional effects.

For all runs, a Gaussian function is fitted to the right side of the photopeak produced with single events. Derived parameters are used to determine the integral of the peak and energy resolution. On the other hand, a dead-time correction is applied to compute the count rate. The latter is compared to the incoming flux for detection efficiency extraction in the single event photopeak. Proceeding this way, we found the efficiency to be 42.71 ± 0.94 % at 2.137 keV.

Fitting the right side of the photopeak allows us to filter the contribution of partial events produced by asymmetric charge sharing where a fraction of the charge remains below the threshold. Such events are triggering one channel only and are consequently considered as singles, creating a peak tail towards the low energy.

Conversely, to derive the total detection efficiency, the entire sum spectrum was integrated and compared with the incoming photon flux at a given energy, after dead-time correction.

On Fig. 7 and8, we can identify absorption edges of Pt, Cd and Te. These lines are useful to identify the nature of different absorption layers.

VI. DISCUSSION

A. Charge-sharing effects

Experimental detection photopeak efficiencies for single events only are presented in Fig. 7 and8. This unusual presentation of singles is to facilitate the analysis of the entrance window absorption ignoring charge sharing and split event effects that would corrupt dead layer properties extraction.

Assuming a uniform illumination of the sensor entrance face, 72% of the photons interact to the vertical of a single pixel while 28% hit the inter-pixel gap, most likely causing a multiple event if the energy deposit is high enough. Actually, deriving this ratio of single event will lead to a slightly higher value due to the distortion of electric field lines near the pixels which gives the pixel a effective area larger than the metal surface. This effective area of a pixel is getting closer and closer to the physical pixel surface as the applied bias voltage gets higher.

Iniewski et al. [START_REF] Iniewski | Modeling charge-sharing effects in pixellated CZT detectors[END_REF] proposed a geometric-based model to estimate the probability of shared events in CdTe pixelated detectors. Meuris et al. [START_REF] Meuris | Charge sharing in CdTe pixilated detectors[END_REF] applied this model to the Caliste 64 geometry. This method describes very well chargesharing rates observed for different operating conditions, including voltage and detector geometries.

Considering V the applied bias voltage, d the thickness of the detector, k the Boltzmann's constant and T the temperature, the characteristic size of the electron cloud formed at the anode is, according to the diffusion equation and the Einstein's relationship:

! = ! !!" !"
Eq. [START_REF] Meuris | Caliste 64, an Innovative CdTe Hard X-Ray Micro-Camera[END_REF] Applying the model to our operating conditions (-10°C, 310 V bias voltage, 1 mm thick detector) one find a charge cloud spreading of approximately 12 µm when the cloud goes across the entire thickness, which is reasonable at such low energies.

On the other hand, Equation ( 2) gives the corresponding probability of a multiple event according to the geometry:

! = 1 - !!!!!!! !!! ! Eq. ( 2
)
Where a is the pixel size (525 µm), g the inter-pixel gap (100 µm), c the collection width (min(g/2, a/20)=a/20 here) and r the radius of the electron cloud (1.15 σ).

We find P = 23% (which can be compared to the geometric ratio of 28% probability to hit an inter-pixel surface).

According to this model, single events represents 77% of the incoming flux, regardless the energy. The effective ratio of shared events depends however on the value of the low energy threshold; if we consider double events for instance, in some cases the energy distribution between the two adjacent pixels is unbalanced and one of the two induced signals is below the threshold, and this event is marked as a single one.

This effect is even more pronounced as the beam energy is near the threshold. To overcome it and to have more representative values on the real charge-sharing ratio, we considered a whole 241 Am spectrum measured with comparable operating conditions by Caliste HD. The chargesharing rate was 23.5%, with 76.5% single events. These results nicely match the model.

For low energy measurements, when the threshold is not negligible compared to the deposited energy, the chargesharing rate effectively measured can be very different. On Table III, some experimental charge-sharing rates are displayed. Detected charge-sharing rate increases when energy is up to 3.1 -3.2 keV: below, almost all shared events are chance triggers. Energy repartition between two neighbor pixels touched simultaneously, with a 12 keV beam.

By increasing beam energies, these values tend to higher charge-sharing ratio as predicted before. Fig. 9 represents the correlation plot obtained considering double events for a 12 keV beam. We clearly see the charge-sharing profile of double events for the beam, plotting a pixel response against any neighbor pixel for split events. The repartition of shared events seems to be homogeneous along permitted energies.

Extrapolating the amount of shared events detected in the permitted range, we estimate a charge-sharing probability of 22.40%. This extrapolation allows computing the expected charge-sharing fraction for a virtually zeroed threshold level. This value is gain matching the expected split event rate.

In the following section, we keep this value of 22.7% for charge-sharing probability.

Moreover, we symmetrically see on Fig. 9 an energy loss for split double events: reconstructed spectrum gives 10% charge loss for the peak value (Fig. 10). This energy loss is relatively high compared to Caliste 64 [START_REF] Meuris | Charge sharing in CdTe pixilated detectors[END_REF] (1.2 keV at 12 keV here, and 0.3 keV at 60 keV for Caliste 64).

B. Detection efficiency simulation

As explained in section V, X-ray absorption edges are visible on the efficiency curve for Pt, Te and Cd elements. As the presence of a platinum layer that covers the entire entrance electrode, is already established, its exact thickness has to be determined accurately. Also, the characteristics of a dead layer composed of Cd and Te can be estimated.

It is possible to find an approximated value for the CdTe dead layer thickness by considering the L III (Te) edge at 4.345 keV. We measured an efficiency of 67.18 ± 2.84% for 4.300 keV, and 62.21 ± 2.52% for 4.375 keV. It implies 7.98 ± 0.45% efficiency loss at this edge.

Equation 3 gives an estimation of the CdTe dead layer thickness x, for A edge = Efficiency before edge / Efficiency after edge = 1.0798:

! = !" (! !"#!) ! !"#$% !"#!! ! !"#$%" !"#! × !

Eq. (3).

Where µ and ρ are the CdTe mass attenuation coefficients and density respectively. Applying to CdTe, we find a thickness of the CdTe dead layer equal to 475 ± 28 nm.

The experimental profile obtained Fig. 8 was fitted using NIST data for mass attenuation coefficients for Pt and CdTe [17]. A 1 mm thick CdTe detector with adjustable sensitive volume and dead zone disposed under a Platinum-made entrance window was simulated. We also considered a fraction of 77.34% single events.

The best fit was obtained by choosing a 34 nm thick Pt layer and a 440 nm thick CdTe dead layer (Fig. 8). With this simulation, only 8 experimental points out of 40 in the 2-8 keV range are not well fitted. The CdTe-like dead layer was slightly lower compared to predicted values. Iwase et al. [START_REF] Iwase | A 90 element CdTe array detector[END_REF] also identified a thin TeO x layer on CdTe crystals.

The model fitting is not accurate and systematically over estimated at very low energy, close to the threshold. We assume that this effect is due to the dispersion of the effective low threshold values from pixel to pixel, which results in a deficit on the count rate -as a matter of fact; energy calibration reveals a smooth standard deviation on the gain and offset values (2.11% and 2.71% respectively).

C. Experimental verification of layers thickness

We used an additional technique to explore respective thicknesses and composition of the different material layers in the detector that have been derived from efficiency curve fitting. A representative CdTe sample was analyzed by RBS (Rutherford Backscattering Spectroscopy) at the ARAMIS facility, CSNSM, Orsay, France.

ARAMIS is a Tandem-type accelerator working with a minimal high-voltage of 2 MV. It delivers He 2+ beam when used in Van de Graaf mode [START_REF] Cottereau | ARAMIS: An ambidextrous 2 MV accelerator for IBA and MeV implantation[END_REF]. This beam is used to probe superficial layers analyzing the backscattered spectrum.

A CdTe detector, disposed into a vacuum chamber, was exposed to a monoenergetic flux of 3.2 MeV He 2+ ions. Energies of backscattered ions were measured with a Ge spectrometer, for a scattering angle of 165°.

From the energy spectrum obtained in Fig. 11 two main structures are visible: single peaks which represent the energy lost in scattered events with sample nuclei (depending on the sample atomic numbers) and a large continuum for low energies which depends on the stopping-power of the electrons in the sample (depending on the electron density and distances traversed across the sample material). Due to this second effect, energy peaks are progressively shifted towards lower energies. Analyzing the resulting RBS spectrum, respective thickness and composition of individual layers are estimated using peaks shifts and width as well as relative peak heights.

We used the SIMNRA software [20] to fit this spectrum. We constrained the fit with three successive layers: o a Pt layer with an adjustable thickness, o another adjustable layer composed with an adjustable ratio of Cd, Te and O, o a stoichiometric CdTe layer with an infinite thickness. The best fit was obtained and over plotted in Fig. 9 with the following parameters:

-a platinum layer of 29 nm, -A 392 nm thick layer composed of cadmium (1.52%), tellurium (30.35%) and oxygen (68.13%).

The layer thickness of the Pt electrode is very well matching the 34 nm thick Pt layer derived from efficiency curve fitting on a different sample, bur same production batch. The second layer is found to be a TeO 2 oxidized dead layer comparable to the suspected 440 nm dead layer. RBS allows the identification of the dead layer nature.

VII. CONCLUSION

In this paper, we described the characterization of the Caliste HD imaging-spectrometer at boundary conditions, as close as possible to the low energy threshold. With a threshold estimated to be 1.3 keV, a 2.1 keV spot is easily detected and the detection efficiency is estimated equal to 43% for single events. This is a very useful information for application in space astrophysics where Caliste broad energy range is demonstrated. Moreover, precise measurement allows accurate sensitivity estimation of imaging spectrometers.

The linearity for low energies was also clearly verified, as well as energy resolution found between 560 and 760 eV.

Identification of successive X-ray absorption edges on the increasing efficiency curve allowed us to estimate the nature and the thickness of different absorption layers: the Pt entrance electrode and also a probable TeO 2 dead zone. These layers limit the efficiency for the lowest energy ranges, typically below 8 keV. For higher energies, the detection efficiency rapidly increases and tends to very high values.

Respective thicknesses and compositions of superficial layers were also confirmed by RBS measurement made using the ARAMIS platform. We found a 29 nm thick Pt layer surrounding a 392 nm thick TeO 2 dead layer.

By using SOLEX and ARAMIS, we acquired complete and accurate knowledge of Caliste HD at low energy down to 2 keV.

Fig. 1 .

 1 Fig. 1. Detailed components of a complete Caliste HD module.

Fig. 2 .

 2 Fig. 2. Scheme of the right top corner of the subdivided anode with pixel size and inter-pixel zone, plus the 20 µm guard ring.

Fig. 3 .

 3 Fig. 3. Caliste HD characterization setup (opened).

Fig. 4 .

 4 Fig. 4. Incident energy (keV) as a function of peak amplitude (ADU), for the 64 most active pixels -First campaign.

Fig. 5 .

 5 Fig. 5.Image obtained with a 2.137 keV spot. With a pitch of 625 micrometers, we can estimate the spot size to a maximum of 3.75 mm (X axis) and 2.5 mm (Y axis). The color scale represents the count rate of each pixel.

Fig. 6 .

 6 Fig. 6. Example of sum-spectrum obtained with the 2.137 keV beam. The photopeak can be fitted by a Gaussian function.

Fig. 9 .

 9 Fig. 9.Energy repartition between two neighbor pixels touched simultaneously, with a 12 keV beam.

Fig. 10 .

 10 Fig. 10. Reconstructed spectrum for double events, with a 12 keV beam.

Fig. 11 .

 11 Fig. 11. RBS spectrum obtained with a 4 He beam of 3.2 MeV. Experimental points are in red, and the simulated curve is blue.

TABLE I .

 I EXAMPLES OF MONOCHROMATORS AND ASSOCIED ENERGY RANGES

	Crystal	Minimal energy (eV)	Maximal energy (eV)
	InSb (111)	1964	6809
	Quartz (R)	4086	14957

TABLE III .

 III EXAMPLES OF CHARGE-SHARING RATES MEASURED FOR

		DIFFERENT BEAM ENERGIES
	Beam energy	Single-events rate	Multiple-events rate
	2.137 keV	99.33%	0.67%
	2.172 keV	99.25%	0.75%
	2.272 keV	99.46%	0.54%
	3.072 keV	99.03%	0.97%
	3.672 keV	98.17%	1.83%
	6 keV	93.93%	6.07%
	7 keV	93.40%	6.60%
	10 keV	89.11%	10.09%
	12 keV	83.57%	16.43%

Transactions on Nuclear Science -Copy for Review

ACKNOWLEDGMENTS

We thank Cyril Bachelet (CSNSM, Orsay, France) who gave us the possibility to make RBS measurements on the ARAMIS facility. His help for data analysis was also precious.

This work is performed in a frame of a PhD Grant supported by CEA and CNES.

D. Energy resolution

Energy resolutions (FWHM) were derived for our different series of runs. Values were found between 560 and 758 eV in the studied energy range, considering the sum spectrum of all pixels, or between 441 and 655 eV if we consider only one pixel centered on the beam. We observe a slight degradation of the resolution between 2.1 and 2.5 keV, but no clear evolution is visible at higher energies (Table II). These results are close, and better compared to energy resolution measurements performed on other Caliste HD samples, where we found 820 eV FWHM at 14 keV on a sum spectrum of a 241 Am source [START_REF] Meuris | Caliste HD: A new fine pitch Cd(Zn)Te imaging spectrometer from 2 keV up to 1 MeV[END_REF][START_REF] Michalowska | IDeF-X HD: A low power multi-gain CMOS ASIC for the readout of Cd(Zn)Te detectors[END_REF].

V. DETECTION EFFICIENCY

Experimental values of detection efficiencies obtained by considering single and multiple events for beam energies between 2 and 12 keV (50 runs) are displayed Fig. 7. We emphasized the energy range between 2 and 6 keV (40 runs) on Fig. 8.

Data analysis for each run is similar, and different measurement series are presented: the first campaign with the InSb monochromator used in SOLEX facility, and the second campaign with both InSb and quartz crystals.

Considering single events, uncertainties on detection efficiencies remain below 1% for a majority of runs from the first campaign. They are approximately between 2 and 4% for the second campaign due to less favorable operating conditions. Uncertainties on total detection efficiencies are below 1% for almost all runs.

Determined values for detection efficiency considering single events are all comprised between 39.49 ± 0.81 % (2.172 keV beam) and 74.23 ± 1.86 % (11.525 keV beam).

Regarding all events, total detection efficiency values are between 40.43 ± 0.38 % (2.172 keV) and 98.82 ± 1.54 % (12 keV). For the lowest energy, the detection efficiency measured for single events is also the total detection efficiency of the entire detector since no split event could trigger Caliste HD in these conditions.

X-ray absorption edges of platinum (M IV edge at 2.204 keV, M III edge at 2.649 keV and eventually L III edge at 11.568 keV) and also tellurium absorption edges (L III edge at 4.345 keV, L II edge at 4.616 keV) appear on our efficiency curves and particularly on the efficiency curve measured for single events, as explained Part. IV-C.

We can also detect an effect of cadmium absorption edges (L III (Cd)=3.554 keV, L II (Cd)=3.742 keV), but less visible considering error bars. Variations of experimental detection efficiency for these characteristic energies imply an effect of the Pt entrance electrode, as well as the presence of a CdTe-like thin dead layer that absorb a significant fraction of low-energy photons. Attenuation due to these elements progressively decreases as the beam energy increases. At higher energies, the efficiency curve saturates after 8 keV, with values that tend toward 75% for single events. Taking into account the total efficiency, including multiple events, the curve saturates at 98%.