Thanh-An Pham

Emmanuel Soubies

Joowon Lim

Alexandre Goy

Ferréol Soulez

Demetri Psaltis

Michael Unser

PHASELESS DIFFRACTION TOMOGRAPHY WITH REGULARIZED BEAM PROPAGATION

Keywords: Beam propagation, intensity measurement, coherence tomography, image reconstruction

In recent years, researchers have obtained impressive reconstructions of the refractive index (RI) of biological objects through the combined use of advanced physics (nonlinear forward model) and regularization. Here, we propose an adaptation of these techniques for the more challenging problem of intensity-only measurements. It involves a difficult nonconvex optimization problem where phase and distribution of the RI must be jointly estimated. Using an adequate splitting, we leverage recent achievements in phase retrieval and RI reconstruction to perform this task. This yields an efficient reconstruction method with sparsity constraints.

INTRODUCTION

Having access to the map of the refractive index (RI) of biological samples has a broad range of applications [START_REF] Jin | Tomographic phase microscopy: Principles and applications in bioimaging[END_REF]. It can be obtained through optical diffraction tomography (ODT). There, the sample is illuminated by a set of tilted incident waves and holographic measurements of the resulting scattered fields are recorded (see Figure 1). The RI distribution is then recovered by solving an inverse scattering problem.

Pioneering works to solve the recovery problem were relying on direct linear inversion algorithms such as back-propagation [START_REF] Wolf | Three-dimensional structure determination of semi-transparent objects from holographic data[END_REF][START_REF] Devaney | Inverse-scattering theory within the Rytov approximation[END_REF]. Reconstructions were then dramatically improved using regularizationbased methods [4,[START_REF] Lim | Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography[END_REF]. However, the validity of linear models is restricted to weakly scattering samples. To overcome this limitation, the most recent reconstruction algorithms combine advanced physical models with modern regularization [START_REF] Kamilov | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF][START_REF] Kamilov | A recursive Born approach to nonlinear inverse scattering[END_REF]. These methods account for multiple scattering, which opens the door to the imaging of strongly scattering objects.

As a CCD camera can measure intensity only, holographic measurements must be acquired using an elaborate interferometric setup that needs a reference beam or multiple measurements per angle. Phaseless diffraction tomography allows one to simplify this setup by recording a single intensity measurement per angle. However, this comes at the price of a more challenging inverse problem. Existing methods tackle this difficulty by alternating between phase retrieval and RI estimation. The phase estimation step is generally performed using the popular Gerchberg-Saxton projection [START_REF] Gerchberg | A practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF]. The RI reconstruction step has known the same progression as for classical ODT, going from linear models [START_REF] Zheng | Wide-field, high-resolution Fourier ptychographic microscopy[END_REF][START_REF] Tian | Multiplexed coded illumination for Fourier ptychography with an LED array microscope[END_REF] to nonlinear ones with ad hoc regularization [START_REF] Maiden | Ptychographic transmission microscopy in three dimensions using a multi-slice approach[END_REF][START_REF] Tian | 3D intensity and phase imaging from light field measurements in an LED array microscope[END_REF][START_REF] Li | Separation of threedimensional scattering effects in tilt-series Fourier ptychography[END_REF].

Contributions In this paper, we propose phaseless diffraction tomography as an adaptation of the efficient regularized method [START_REF] Kamilov | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF] that solves inverse scattering. We thereby leverage the benefit of an advanced nonlinear physical model and sparse regularization. We first express the inverse problem within a variational framework that includes the total variation (TV) penalty together with a nonnegativity constraint. Then, using an adequate splitting strategy, we carry out the optimization by alternating between simpler steps. For each subproblem, we deploy an efficient numerical solution. Finally, we validate the proposed method on simulated and experimental data.

BEAM-PROPAGATION METHOD

We consider the 2D area Ω discretized in (Nx × Nz) points with steps δx and δz. We denote the RI distribution of the sample by n ∈ R Nx×Nz and the RI of the surrounding medium by n b ∈ R. Also, we introduce the RI variation δn = (n -n b 1) with 1 = ∑ NxNz k=1 e k . The incident plane wave of wavelength λ is referred to as u in ∈ C Nx×Nz . We represent the total field u tot (δn) ∈ C Nx×Nz (incident + scattered) as u tot q (δn) = aq(δn) e jk b q , (

where a(δn) ∈ C Nx×Nz is the complex envelope of the wave,

k b = 2πn b λ
is the background wavenumber, and the index q denotes the z slice of the corresponding matrix. The beam-propagation method (BPM) computes a(δn) slice-by-slice along the optical axis z using the recursive relation

aq(δn) = (aq-1(δn) * hprop) ⊙ pq(δn), (2)
a0(δn) = u in 0 , (3)
where ⊙ denotes the Hadamard product and * the convolution operation. In (2), aq-1(δn) is first propagated to the next slice by convolution with the propagation kernel hprop ∈ C Nx given by

hprop = F -1 { exp ( -jω 2 δz k b + √ k b 2 -ω 2 )} (diffraction step), ( 4
)
where F is the 1D discrete Fourier transform, ω ∈ R Nx is the frequency variable for the x direction, and all operations are component-wise. This convolution is followed by a point-wise multiplication with the qth slice of the phase mask p(δn) ∈ C N x×N z defined as

pq(δn) = exp (jk0δz(δn)q) (refraction step), (5)
where k0 = k b /n b is the wavenumber in free space. Finally, the BPM forward model is defined by the operator

B : R Nx×Nz → C Nx δn → aN z (δn) e jk b Nz , (6)
where aN z (δn) ∈ C Nx is computed using (2)-(3).

ADMM-BASED RECONSTRUCTION

We denote by P the number of the incident planes waves u in p ∀ p ∈ [1 . . . P ]. The forward model that links δn to the intensity measure-

ments yp ∈ R Nx is yp = |Bp(δn)| 2 + sp ∀ p ∈ [1 . . . P ], (7)
where sp ∈ R Nx is a vector of noise components, | • | denotes the component-wise magnitude, (•) 2 denotes the component-wise square operation, and Bp is the BPM model in ( 6) associated to u in p . To recover the RI variation δn, we minimize the TV-regularized negative log-likelihood of the noise distribution

δn ∈ { arg min δn∈χ ( 1 2 P ∑ p=1 ∥|Bp(δn)| 2 -yp∥ 2 Wp + τ ∥δn∥TV )} , ( 8
)
with τ a regularization parameter, χ ⊆ R Nx×Nz ≥0 a set that enforces the nonnegativity constraint, Wp = diag((w p 1 , . . . , w p Nx )) ∈ R Nx×Nx a diagonal matrix, and

∥ • ∥ W a weighted ℓ2-norm such that ∥v∥ 2 W = ∑ Nx m=1 w p m (vm) 2 .
To account for shot noise (Poisson), we set these weights to the inverse of the intensity of each measurement.

We then apply the popular alternating direction method of multipliers (ADMM) [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF] strategy to solve our inverse problem. The leading idea is to split the initial problem in a series of simpler subproblems for which we can deploy efficient algorithms. Starting from Algorithm 1 ADMM for minimizing [START_REF] Tian | Multiplexed coded illumination for Fourier ptychography with an LED array microscope[END_REF]

Require: {yp} p∈[1...P ] , δn (0) ∈ R N ≥0 , ρ > 0, τ > 0 1: w (0) p = 0 C Nx , ∀p ∈ [1 . . . P ] 2: k = 0 3: while (not converged) do 4: v (k+1) p = prox 1 2ρ ∥|•| 2 -yp∥ 2 Wp (Bp(δn (k) ) + w (k) p ρ ) 5: δn (k+1) = arg min δn∈χ ( 1 2 P ∑ p=1 ∥Bp(δn) -v (k+1) p + w (k) p ρ ∥ 2 2 + τ ρ ∥δn∥TV
This problem admits the augmented-Lagrangian form L(δn, v1, . . . , vP , w1, . . . , wP ) = 1 2

P ∑ p=1 ∥|vp| 2 -yp∥ 2 Wp + ρ 2 ∥Bp(δn) -vp + wp/ρ∥ 2 2 + τ ∥δn∥TV, ( 10
)
where wp and ρ are the Lagrangians and the penalty parameter [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF].

Algorithm 1 shows the steps to minimize (10) using ADMM.

Proximity Operator

At Step 4 of Algorithm 1, one has to compute the proximity operator of D(v) = 1 2ρ ∥|v| 2 -yp∥ 2 Wp defined as

prox D (x) = arg min v∈C Nx ( 1 2 ∥v -x∥ 2 2 + D(v) ) . ( 11
)
Here, we take advantage of the closed-form expressions that have been recently derived for both Gaussian and Poisson likelihoods in [START_REF] Soulez | Proximity operators for phase retrieval[END_REF]. Specifically, the proximity operator in ( 11) is computed component-wise according to

∀x ∈ C Nx , [prox D (x)] m = ϱm e jarg(xm) , ( 12
)
where ϱm is the positive root of the 3rd degree three polynomial

qG(ϱ) = 4w p m ρ ϱ 3 + ϱ ( 1 - 4w p m ρ [yp]m ) -|xm|, ( 13
)
which is found with Cardano's method.

Solving for δn

At Step 5 of Algorithm 1, we need to reconstruct the RI distribution from the complex "data" z

(k+1) p = v (k+1) p -w (k) p /ρ, taking into account that δn (k+1) = arg min δn∈χ ( 1 2 P ∑ p=1 ∥Bp(δn)-z (k+1) p ∥ 2 2 + τ ρ ∥δn∥TV ) . ( 14
)
This optimization problem is solved using the fast iterative shrinkagethresholding algorithm (FISTA) [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF], which has already been proven to be useful in this context [START_REF] Kamilov | Learning approach to optical tomography[END_REF][START_REF] Kamilov | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF]. Two quantities are required 1. The proximity operator of τ ρ ∥ • ∥TV which is computed efficiently using a standard iterative method [START_REF] Kamilov | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF].

The gradient of

F(δn) = 1 2 ∑ P p=1 ∥Bp(δn) -z (k+1) p ∥ 2 2
which is derived using classical differential rules.

Specifically, we have that

∇F(δn) = P ∑ p=1 Re ( J H Bp (δn)(Bp(δn) -z (k+1) p ) ) , ( 15
)
where JB p (δn) is the Jacobian matrix of the BPM forward model. It is computed efficiently by back-propagation as in [START_REF] Kamilov | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF]. Moreover, to reduce the computational cost, we compute the gradient only from a subset of angles L < P . We choose the angles such that they are equally spaced and increment them at each FISTA iteration. The computational complexity of ∇F(δn) for one angle corresponds to the cost of 6Nz FFTs of size Nx.

We implemented Algorithm 1 using the GlobalBioIm library [START_REF] Unser | GlobalBioIm: A unifying computational framework for solving inverse problems[END_REF].

NUMERICAL EXPERIMENTS

Simulated data

We simulated intensity measurements using a nonlinear accurate forward model [START_REF] Soubies | Efficient inversion of multiple-scattering model for optical diffraction tomography[END_REF]. The square area Ω = 33λ × 33λ includes the sample and the sensors. The medium has a RI n b = 1.33 (i.e., water). The setup is similar to the scheme in Figure 1. A cell-like phantom is included in a central area of side 16.5λ. As shown in Figure 2 (top left), the cell body and the two ellipses have a RI of 1.355, 1.432 and 1.457 respectively. We simulated on a very fine grid in order to reduce numerical errors (i.e., 1024 × 1024) and then down-sampled (512 × 512) to get the measurements used for the reconstruction (last column of this matrix). The plane waves have incident angles equally spaced between -π 4 and π 4 . Thirty one sets of measurements were acquired with a wavelength of 406 nm (i.e., P = 31). The reconstruction problem is challenging because of the limited-angle illuminations (missing cone). We computed the reconstruction error ∥δn-δn true ∥ F ∥δn true ∥ F with ∥ • ∥F the Frobenius norm. Our reference is the (linear) light field refocusing (LFR) method [START_REF] Zheng | Microscopy refocusing and dark-field imaging by using a simple LED array[END_REF] which is also used to initialize Algorithm 1. It provides a reasonably "good" start, which is crucial here since the optimization task is non-convex. The algorithm parameters were manually set to ρ = 10 -3 , L = 8 and the step size in FISTA to γ = 5 • 10 -4 .

Noiseless measurements For this experiment, we set the regularization parameter to τ = 1.5 • 10 -6 • ∥y P/2 ∥ 2 2 . We compare the proposed method with the BPM method in [START_REF] Kamilov | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF] that reconstructs the RI map from holographic measurements (BPMc). We initialize this algorithm with a filtered back-projection method (FBP) [START_REF] Kak | Principles of computerized tomographic imaging[END_REF]. As shown in Figure 2, the proposed method is able to recover the RI distribution. One can observe that the structures are slightly elongated along z, as a consequence of the missing cone. However, contrarily to the LFR solution, we can distinguish the two ellipses and the shape of the cell body. The reconstructed RI is also close to the true value. The reconstruction error is 6 • 10 -3 . The proposed method compares well against BPMc (error 5.4 • 10 -3 ) for which the phase was provided. the cell-like phantom and its associated noiseless intensity measurements;

(middle) the solutions from FBP [START_REF] Kak | Principles of computerized tomographic imaging[END_REF] and BPMc [START_REF] Kamilov | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF]; (bottom) the LFR solution [START_REF] Zheng | Microscopy refocusing and dark-field imaging by using a simple LED array[END_REF], and the RI distribution recovered by the proposed method. The elongated ellipses are due to missing informations along the optical axis.

Noisy measurements We simulated noisy measurements at three different noise levels. For each of them, we set the incident fields (u in p ) p∈[1...P ] such that |[u in p ]m| = A ∈ R>0 and simulated the resulting intensity measurements. We considered three scenarios with A = 1.75, 3 and 5. Then, these measurements were corrupted using a Poisson distribution. The resulting SNR are 5.32, 9.77 and 14.13 dB, respectively. Simulated measurements are shown in Figure 3 (top line). The regularizations were set to τ = 10 -6 • ∥y P/2 ∥ 2 2 for all noise levels. As shown in Figure 3 (bottom line), the proposed method is still able to recover the shape of the cell and the ellipses. The reconstruction errors are 9.47 • 10 -3 , 8.07 • 10 -3 and 6.17 • 10 -3 for A = 1.75, 3 and 5, respectively. Despite the noise, we can still distinguish the different elements of the phantom, which demonstrates the robustness of the method.

Experimental data

We validated our method on experimental data. Holographic measurements were collected using a standard Mach-Zehnder interferometer, which relies on off-axis digital holography (λ = 450 nm). The sample was the cross-section of two fibres immersed in a medium of RI n b = 1.525 (oil). We obtained P = 160 views ranging from -π 4 to π 4 . The RI variation is negative δn ∈ R ≤0 . The reconstructed area is Ω = 38λ × 97λ. We compare the performance of the proposed method with BPMc. The latter and Algorithm 1 were initialized with the solutions of the Rytov based-backpropagation [START_REF] Devaney | Inverse-scattering theory within the Rytov approximation[END_REF] and LFR respectively.

The FISTA step size was set at γ = 0.2/∥y P/2 ∥ 2 2 for BPMc and our method. We set the penalty parameter to ρ = 2.5 for Algorithm 1. The regularization parameter τ was tuned manually.

As shown in Figure 4, both BPMc and the proposed method are able to reconstruct the cross-section of the two fibres. Although the phase is missing, our method reaches performances similar to BPMc.

CONCLUSION

We have proposed a method to reconstruct a map of refractive index (RI) from intensity-only measurements. It is a non-trivial extension from complex to amplitude-only of a state-of-the-art method for RI reconstruction from holographic measurements. We have combined proximity operators for phase retrieval with an efficient RI reconstruction pipeline. Using an adequate splitting of the problem, our method can cope with different noise models and regularizers. We showed its robustness to noise and to the limited-angle acquisition settings that are the main difficulties for biological imaging.

Fig. 1 .

 1 Fig. 1. Diffraction-tomography setup. A sample of RI n = n b 1 + δn is immersed in a background medium of index n b and impinged by a set of incident waves (u in p ) p∈[1...P ]. The interaction of the wave with the object produces a total field u tot . Its squared magnitude is recorded by the detector.

9 :∥|vp| 2

 92 return δn (k) (8), we introduce the auxiliary variables vp ∈ C Nx ∀ p ∈ [1 . . . P ] to obtain the equivalent constrained problem δn ∈ arg min δn∈χ ( -yp∥ 2 Wp + τ ∥δn∥TV ) , s.t. vp = Bp(δn) ∀p ∈ [1 . . . P ].

Fig. 2 .

 2 Fig. 2. Simulated data and their reconstruction. From left to right: (top)

Fig. 3 .

 3 Fig. 3. Noisy measurements and their RI reconstructions. Each column corresponds to a noise level. From left to right: |[u in p ]m| = 1.75, |[u in p ]m| = 3, |[u in p ]m| = 5 ∀p ∈ [1 . . . P ]. Top to bottom: intensity measurements, LFR [20], and proposed method.

2 Fig. 4 .

 24 Fig. 4. RI reconstructions of two fibres. Top left: profile plots of reconstructions from BPMc and proposed method of the dashed lines on the right. Bottom left: intensity measurements experimentally acquired. Right: BPMc (complex measurements) and proposed method (intensity measurements).


Antoine Mérand
email: amerand@eso.org

Antxon Alberdid

Jacques Kluska

Isabelle Tallon-Bosc

Michel Tallon

Éric Thiébaut

Joel Sánchez-Bermudez

Rainer Schödel

Ferréol Soulez

John Young

Optical Interferometry Image Reconstruction Contest VIII

Keywords: Optical Interferometry, image reconstruction, contest

We report on the VIII edition of the imaging reconstruction contest held in the optical/infrared interferometry community. This edition sported a VLTI+CHARA dataset of a simulated image of a star surrounded by a proto-planetary disk hosting a planet. 5 teams responded to the challenge and produced images very close the original image, showing the maturity of image reconstruction techniques for a well sampled u,v coverage. The contest was organised

INTRODUCTION

The "Optical Interferometry Image Reconstruction Contest" is organised every two years [START_REF] Lawson | An interferometry imaging beauty contest[END_REF][START_REF] Lawson | interferometry imaging beauty contest[END_REF][START_REF] Cotton | 2008 imaging beauty contest[END_REF][START_REF] Malbet | The 2010 interferometric imaging beauty contest[END_REF][START_REF] Baron | The 2012 interferometric imaging beauty contest[END_REF][START_REF] Monnier | The 2014 interferometric imaging beauty contest[END_REF][START_REF] Sanchez-Bermudez | The 2016 interferometric imaging beauty contest[END_REF] on the same principles since its first edition in 2004: one or several datasets are produced in the OIFITS format and provided to teams which will attempt to reconstruct images. The contest aims at not only comparing different image reconstruction algorithms, but also to encourage the use of the OIFITS format for optical interferometric data. The contest was announced on March 13th, 2018, on the OLBIN email list * .

DATASET

This year, only one dataset was provided using the following method. First, OIFITS2 [START_REF] Duvert | OIFITS 2: the 2nd version of the data exchange standard for optical interferometry[END_REF] were generated using the Aspro software. [START_REF] Bourgès | ASPRO 2: Astronomical Software to PRepare Observations[END_REF] Since the contest intended to demonstrate the complementarity of the spatial frequency coverage (u,v coverage) of the VLTI and CHARA Array, we chose four different telescope configurations: CHARA 6T (that is, using all the available telescopes) as well as the 3 different 4 Auxiliary Telescopes configurations currently offered at VLTI. The declination was chosen to be +9 degrees and realistic observabilities were chosen. Aspro's simulator estimated we could have pointed the object 4 times with CHARA and 8 times with VLTI (on each of the AT configuration) within the observability window (limited by delay line range, object's elevation or domes' shadowing -for VLTI-). Figure 1 shows the resulting coverage.

We did not use a simulated image to generate the data, but rather analytical formulas to generate synthetic visibilities (squared amplitudes and closure phases). The instrument we simulated (MIRC 10 and PIONIER [START_REF] Berger | PIONIER: a visitor instrument for VLTI[END_REF] ) have low spectral resolution (∼50) and we did not introduce chromaticity in the simulated data. Noise was added as follow: each individual V 2 or CP had a gaussian noise, and an overall correlated noise was added for each baseline and each pointing. The model was composed of 1) a 0.4mas uniform disk star (65% of the total flux) 2) an inner ringed disk with a bright rim (29%) and projection angle of 44.2degrees 3) a planet as a 0.3mas UD at 3.75mas from the central star and projection angle of 27 degrees (2%) and 4) the outer disk extended up to 20mas. See Figure 1 for an overview of the model and the corresponding synthetic image.

The parameters of the model was adjusted to match qualitatively the visibility and closure phases which are observed for young stellar objects in near-infrared interferometry: [START_REF] Lazareff | Structure of Herbig AeBe disks at the milliarcsecond scale . A statistical survey in the H band using PIONIER-VLTI[END_REF] the visibilities drop rapidly as the baseline length increase because the disk is rapidly resolved. For longer baselines, the visibility's amplitude reaches a plateau, which height depends on flux ratio between the central (unresolved) star and the disk.

The goal of the contest was set to measure the orientation of the disk and estimate the position of the planet it it was detected, or estimate an upper limit of its flux otherwise.

IMAGE RECONSTRUCTION ENTRIES

We reproduce in the following sections the text accompanying each submission.

Submission from J. Kluska

We used the MiRA algorithm 13 together with the SPARCO approach. [START_REF] Kluska | SPARCO : a semi-parametric approach for image reconstruction of chromatic objects. Application to young stellar objects[END_REF] It minimises the following cost function: f = χ 2 + µf rgl , where f rgl is the cost function of the regularisation and µ is the regularisation weight. The SPARCO approach separates a geometrical model of one component of the observed object to reconstruct the image of the rest. We used the quadratic smoothness regularisation as it is retrieving the images well [START_REF] Renard | Image reconstruction in optical interferometry: benchmarking the regularization[END_REF] and helps to not fall into local minima. We divided our approach into 3 steps. At step 1 we selected the field-of-view of the image to be roughly 3 times larger than the reconstructed morphology and the pixel-size to be significantly smaller that the highest angular resolution reached by the longest baseline. The pixel size was therefore chosen to be 0.1 mas and the image is made of 256×256 pixels. For each step we took a different fixed model and the image reconstruction was therefore made on different component of the object. We tuned the regularisation weight at each step in order to retrieve an image of a good quality (10 8 , 10 9 and 10 4 for each step respectively).

Here we describe the 3 steps:

1. model = point source. Given the dataset and the instructions we assumed that there is a non-resolved star in the object. We modelled it as a point source and performed an image reconstruction of its environment. We fitted the flux of the point source bu performing a grid with different flux ratios and choosing the one that has the smallest value of the global cost function (f ) consisted in assuming that there is an unresolved star, in order to reconstruct the environment. The recovered image presented an elongated emission with a bright spot at coordinates x∼1.8 mas and y∼3.3 mas. There was also some flux present very close to the star itself as it was marginally resolved.

2. model = uniform disk + point source. In the second step we modelled the the central star as a uniform disk of a diameter of 0.45 mas (to account for the flux surrounding the star at step 1) and a point source for the planet at the coordinates mentioned above. We manually tuned the flux of the star and the planet to fit the data. This step allowed the regularisation to act on the image of the disk only as the presence of the star and the planet would break the effect of the regularisation (as they have a high surface brightness and a sharp morphology compared to the extended disk flux).It allowed an image of the disk of better quality.

3. model = image of the disk. In the last step we used the image of the disk obtained at step 2 to reconstruct the two point sources and relax the constraints we put by modelling them with geometrical models. The flux ratio was the complementary one to the fluxes of the star and planet at step 2.

The final image is the addition of the image of step 2 with the image of step 3 with the correct flux ratio.

The position of the planet was measured at step 1, with the error bars being the extension of the flux at this position (∆x = 1.8±0.3 mas; ∆y = 3.3±0.3 mas). The position angle of the disk was measured at step 2 by fitting a 2D-Gaussian the amplitude of the Fourier transform of the image of the disk. The result is 45.5 • ± 2 • .

Submission from É. Thiébaut

Reconstruction were carried out with MiRA 13 † , using the power spectrum (OI-VIS2) and phase closure data (OI-T3PHI) of the provided dataset. The image reconstruction settings were as follows: pixel size is δθ = 0.1 mas, synthesized field of view Ω = 20 mas (hence restored images are 200 × 200 pixels). The maximum sampled frequency is about 200 Mega cycles which corresponds to an angular resolution of 0.5mas; hence images have a fair amount of super-resolution. Positivity and normalisation constraints were applied (such that the sum of pixel values is equal to 1). The different reconstructions were all started with an initial image given by an unresolved source centered in the field of view. An hyperbolic edge preserving smoothness [START_REF] Thiébaut | Principles of image reconstruction in optical interferometry: tutorial[END_REF] was used as regularisation term with transition threshold parameter τ and relative weight µ. Whatever the combination of the regularisation parameters, the restored images (see Fig. 1) clearly show a bright central star surrounded by a disk (tilted with respect to the observer, so only a fraction of the inner edge is seen) and a faint unresolved point-like source (North-East from the star) believed to be an exo-planet. There are a few other point-like structures inside the void surrounding the star but much fainter (the brightest one is 3 times fainter than the supposed exo-planet). The planet is at position angle PA = 28.1±0.2 (from North toward East), and separation ρ = 3.71±0.01 mas. The integrated flux of the planet is 2.0±0.1% of the total flux while the integrated flux of the star is 65.6±0.2% of the total flux. Most of the remaining flux is in the disk-like structure. These parameters does not much depend on the regularization settings. The major axis of the proto-planetary disk has a position angle of about PA maj = 45±1 and a length of 6.5±0.5 mas. Regularization parameters µ= 3 4 and τ = 5 × 10 4 yield one of the best image and was chosen for the challenge.

Submission from J. Sanchez-Bermudez, A. Alberdi and R. Schödel

In order to recover the morphology from the delivered MIRC and PIONIER observables, we used two of the software available to the community: BSMEM and SQUEEZE. Our reconstruction process first consisted in combine all the data sets into a single OIFITS file. In order to have the same wavelength information for both MIRC and PIONIER, we performed a wavelength interpolation in the MIRC data set at PIONIER wavelengths. Since the source is considered a grey body (i.e., the morphology is the same at all the sampled wavelengths) we did not performed a chromatic reconstruction of the delivered data set. After the combination, we used SQUEEZE to perform the first reconstruction. This is a Monte Carlo -Markov Chain software that allows us to compute parallel reconstruction chains with a large variety of regularisers. For this exercise, we computed 10 different chains (with 250 iterations each one) to find the best image that suits the interferometric observables. We used a Laplacian regulariser with an hyper parameter value of ?=100 (other combinations of regularisers and hyper parameters were tested, but no significant morphological changes were observed in the images); the pixel scale was set to 0.1 mas/pixel (since the maximum angular resolution of the array was of θ=0.5 mas) over a pixel grid of 40 mas. SQUEEZE converged for all the ten different chains. The morphology of the object shows a circum-stellar disk/rim around a central star and a secondary source close to the north-east edge of the rim.

Additionally to these three morphological components, we observed several other point-like sources in the field-of-view. Since one of the objectives of the contest is to find the planetary component in the image, we had to refine our reconstruction to discard spurious components associated to systematics, noise, or to the secondary lobes of the synthesised beam. To refine the SQUEEZE solution, for the ten different images, we filter the three principal morphological components by masking all the other emission/noise in the field-of-view. With these masked maps, we then used BSMEM for a second iteration to find the best image. BSMEM uses a gradientdescent method to perform the minimisation process for the reconstruction. For this software, the regularisation uses mainly Gull-Skilling entropy and the hyper parameter value is selected automatically for the user. We performed 60 iterations for the reconstruction. With this setup, we managed to find images that fit, with highdegree of accuracy, the squared visibilities and closure phases. Finally a raw-mean image was created with the ten reconstructed frames. An additional convolved image with a point-spread-function of 0.5 mas (FWHM) was also created. Figure 1 displays the best-fit reconstructed images (raw and convolved), with the morphological elements labeled; Figure 2 shows the interferometric observables reconstructed from the raw image, together with the squared differences that result from the comparison with the data. Our results support the following conclusions over the reconstructed images:

• Three main components are detected: (a) a central source with ∼65.8% of the total flux in the image; (b) a circumstellar rim with ∼29.4% of the flux and a secondary companion, plausible a planet, with ∼2.2% of the flux. The rest of the flux in the image was spread over the field-of-view with noise peaks of the order of ∼ 5 × 10 -4 % of the total flux (∼ 10 3 times fainter than the image peak).

• The circum stellar rim is elongated with the projected position angle (PA) of the semi-major axis at ∼47 degrees (East of North). The rim also exhibits a non-uniform brightness distribution with the maximum located at PA∼309 degrees. The brightness distribution of the rim decreases quickly from the peak to the rest of the rim profile. Uncertainties in the observables, the sparse u,v coverage, and the limitations in the reconstruction make clumpy the emission along the rim profile. At a PA=139 degrees no emission is detected from the rim.

• The plausible planetary component is detected nearby the outer edge of the rim at a separation of ∼3.8 mas form the central source and at a PA∼28 degrees. During the first reconstruction, two additional point-like sources were found, one inside the rim cavity and a second one in the depleted emission area of the rim. At first sight, we thought that one of them could be the sought planetary companion. Nevertheless, after revising the secondary lobes of the synthesised beam, we noticed that these additional point-like sources lie at the loci of two of the secondary lobes. Therefore, we discarded them as real sources. This hypothesis was confirmed after the second reconstruction, when only a remanent of them appeared in the images.

• The peak of the squared differences between the data and the interferometric observables from the meanreconstructed images are 5 × 10 -5 for the V 2 and 0.1 deg for the closure phases of both, the MIRC and PIONIER data sets.

submission from J. Young

The BSMEM (BiSpectrum Maximum Entropy Method) software was first written in 1992 to demonstrate image reconstruction from optical aperture synthesis data. It has been extensively enhanced and tested since then. BSMEM was adapted during 2015-2016 to work with the "OImaging" graphical user interface developed at JMMC. The 2018 contest entry was produced with v2.1.1 of BSMEM ‡

The BSMEM algorithm applies a fully Bayesian approach to the inverse problem of finding the most probable image given the evidence, making use of the Maximum Entropy approach to maximize the posterior probability of an image. The algorithm uses a trust region method with non-linear conjugate gradient steps to minimise the sum of the log(likelihoodχ 2 of the data given the image and a regularization term expressed as the Gull-Skilling

entropy k I k -M k -I k log(I k /M k ).
The model image M k is usually chosen to be a Gaussian, a uniform disk, or a delta-function centered in the field of view, which conveniently fixes the location of the reconstructed object (the bispectra and powerspectra being invariant to translation). This type of starting model also acts as a support constraint by penalising the presence of flux far from the centre of the image.

The oifitslib utility programs were used to merge the CHARA and VLTI contest datasets into a single OIFITS file, and to remove the complex visibility and triple product amplitude data. An initial reconstruction was made from the combined dataset using a simple circular Gaussian prior model. The approximate orientation and axial ratio of the protoplanetary disk were inferred from the initial reconstruction, and a new model image consistent with these parameters was created. The model image was created in LITpro and comprised a compact circular Gaussian, representing the star, plus a flattened disk encompassing all of the believable circumstellar disk flux in the initial reconstruction (the model image can be found in the second HDU of the submitted FITS file). The submitted reconstruction was made using this second prior image.

Given the systematic errors deliberately introduced into the contest dataset, BSMEM's automatic hyperparameter estimation was not used. The optimum regularization weight was chosen using the "L-curve" approach. The final reduced chi-squared of the submitted image was 1.23.

From the submitted image, the position angle (N through E) of the protoplanetary disk is 43 degrees. The supposed planet is 3.76 mas from the central star, at position angle (N through E) 28.6 degrees.

3.5 submission from M. Tallon, I. Tallon-Bosc, É. Thiébaut and F. Soulez

We attempted to tackle the imaging contest 2018 by using model fitting with LITpro 17 and to get a normalised image from the model determined in the Fourier plane from fitting the provided OI-VIS2 and OI-T3PHI data. The best solution was built from these elements:

• a central star gathering 65.3±2.5% of the total flux;

• a component (planet) at separation 3.73±0.01 mas and position angle PA=27.3±0.3 degrees (i.e. North-East quadrant) gathering 1.9±0.1% of the total flux;

• an elongated proto-planetary disk, with a major axis at position angle PA=42.4±0.3 degrees, the length of the elongation found to be 6.05±0.05 mas;

• an elongated gaussian halo, with a major axis at position angle PA=47±2 degrees and a FWHM of 15.4±0.6 mas along this direction, gathering 4.2±0.25% of the total flux.

The difficulty was to calculate the image from the model while controlling the artefacts brought by the FFT. The model is indeed perfectly determined in the Fourier plane whereas the parameters for the image calculation, like the pixel size or the resolution, are arbitrary: this is a major difference between model fitting and image reconstruction. The pixel size of the image has been chosen equal to 0.1 mas, small enough to prevent any pixelization effect. The image has then been convolved by a gaussian with a FWHM of 0.16 mas to reduce the truncation effect in the Fourier plane. That gives a resolution of about 0.3 mas, three times better than the diffraction limit of the measurements.

From this process, we can draw out some interesting remarks:

• Fitting the star and the planet by gaussians gives the same FWHM for both: 0.250.05 mas. This suggests that the data were computed from an image obtained by convolving a perfect object with a gaussian of this width. It is somehow confusing since it looks like we modelize the making of the image instead of the shape of the sources in the sky. • For modelling the proto-planetary disk, we have developed a new model: the intensity along a circle is represented by a set of points interpolated by cubic splines. This circle is then stretched, oriented and convolved by a kernel (here a gaussian). This yields a very flexible adjustment.

• The planet is obvious in the data. For example, if the planet is removed from the model, the fitter forces the disk to move and to produce a spot exactly at the location of the planet, by activating one cubic spline of the disk.

RESULTS AND CONCLUSIONS

All reconstructed images are shown on Figure 2. From the previous descriptions of the entries, it is clear that all resulting in a correct measurement of the disk orientation and planet position. We came up with two additional criteria:

• Was the gap between the inner and outer disks detected, at the correct position? Yes for Kluska, Thiébaut and Sanchez-Bermudez et al.

• Was the inner disk reconstructed in its entirety? Yes for Sanchez-Bermudez et al., Young and Tallon et al.

Considering that only one entry satisfies the two criteria (Sanchez-Bermudez et al.), this entry was deemed the winner of Optical Interferometry Image Reconstruction Contest VIII.

In Conclusions, all images managed well to reproduce the original model used for the data synthesis. We can also note that some images were produced using novel techniques. For example, the use of one algorithm to produce a first image which is then used to feed a second algorithm (J. Sanchez-Bermudez et al.). Additionally, we can see the advent of approaches not limited to "pixel-based" modeling of the image: J. Kluska used some analytical components, while M. Tallon et al. used a fully analytical modeling of the visibility. Interestingly, only the analytical modeling, using LITpro, was able to model the low surface brightness, extended outer disk.

Figure 1 .

 1 Figure 1. Leftmost panel: u,v coverage of the data set. CHARA/MIRC data points are in red, whereas VLTI/PIONIER data points are in blue. Central panels: Model used to generate the synthetic data showing each component described in the text, as well as a synthetic image (large upper right panel) in non linear colour scale. Rightmost panels: V 2 (upper) and phase closure (lower) as a function of spatial frequency. Same colour code as the leftmost panel.

Figure 2 .

 2 Figure 2. All reconstructed images at the same spatial and color scales. The color scale is a power law in order to show the lowest surface brightness details.

† MiRA version 2.0.0b available at https://github.com/emmt/MiRA

‡ BSMEM contains proprietary software, but may be downloaded for non-commercial use subject to a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License -see http://www.mrao.cam.ac.uk/research/optical-interferometry/bsmem-software/


Xiaoke Han

Nicolas Adnet

Isabelle Bruant

Frederic Pablo

Habiba Hafdallah Ouslimani
email: habiba.ouslimani@u-paris10.fr

Laurent Proslier

Alain C Priou

EXPERIMENTAL STUDY OF THE BEHAVIOR OF AN EBG-BASED PATCH ANTENNA SUBJECTED TO MECHANICAL DEFORMATIONS

This work deals with the behavior of a patch antenna equipped with squared electromagnetic bandgap (EBG) structures and subjected to various mechanical deformations (twisting and bending deformations). The EBG structures have a stop band frequency (rejection) feature, allowing the coupling and the undesired electromagnetic interferences to be reduced. The influences of the deformations on the mutual coupling and radiation patterns of an antenna equipped of those EBG elements are experimentally studied.

INTRODUCTION

Telecommunication and avionic needs are increasing, and as a consequence, the number of antennas structures keeps growing on aircrafts.

Most of the time, these antennas are prominent objects and lead up to manifold drawbacks such as: aerodynamic penalization, local weakening of the supporting region by the fixing holes of the antenna, prevalent structural health control and servicing, electromagnetic compatibility rules, etc. Patch antennas are excellent candidates to avoid these problems: they are thin, have a low profile and are conformable to planar and warped surfaces. Moreover, such a solution should allow the design of multifunctional antennas, achieved by gathering numerous functions embedded on the same substrate. In this kind of new multi-frequencies antennas, some radiating elements might be close to each other, and an unwanted coupling eAEects might arise if there is not a suitable distance separating two successive radiating elements. These undesired coupling can be reduced using metamaterial (MTM) structures, such as EBG structures [START_REF] Yang | Electromagnetic Bandgap Structures in Antenna Engineering[END_REF][START_REF] Sievenpiper | High impedance electromagnetic surfaces with a forbidden frequency band[END_REF][START_REF] Han | Understanding the coupling reduction eAEect in microtrip array antennas using high impedance surface (HIS)[END_REF][START_REF] Kawakami | Mutual coupling reduction eAEects of EBG structure located on cylinder surface[END_REF][START_REF] Liang | A novel broadband EBG using cascaded mushroom-like structure[END_REF][START_REF] Tran | High impedance surfaces based antennas for high data rate communications at 40 GHz[END_REF] and/or planar MTM surfaces such as Complementary Split Ring Resonator (CSRR) structures [START_REF] Han | CSRRS for e±cient reduction of the electromagnetic interferences and mutual coupling in microstrip circuits[END_REF][START_REF] Schurig | Electric-field-coupled resonators for negative permittivity metamaterials[END_REF][START_REF] Hafdallah Ouslimani | Analysis and reduction of electromagnetic coupling interferences in microstrip antenna arrays[END_REF]. Commonly, this kind of EBG (electromagnetic bandgap) structures is used for planar antennal structures but is not optimally designed when deformed due to mechanical loads (except the theoretically study of [START_REF] Kawakami | Mutual coupling reduction eAEects of EBG structure located on cylinder surface[END_REF]).

In this paper, the behavior of a patch antenna integrating EBG structures, subjected to prescribed mechanical deformations is studied. This work is a part of the MSIE † project, launched in October 2008 by the competitiveness French ASTech cluster and closed in December 2011. In Sections 2 and 3, the experimental setup and design of the EBG-based patch antenna are described. In Section 4, the experimental results are presented in planar and distorted configurations. They show that the twisting and bending loads do not aAEect the performance of the EBG-based patch antenna.

EXPERIMENTAL SETUP

Due to increasing needs in navigation, communication and surveillance, aircrafts are equipped with more and more antennal structures. Then, the antenna radiating elements have to be closer from each other and thus, reducing their mutual coupling arises from such a design constraint. A solution can consist in using metamaterial structures to reduce electromagnetic interactions between the antenna radiating elements, such as EBG structures [START_REF] Yang | Electromagnetic Bandgap Structures in Antenna Engineering[END_REF][START_REF] Sievenpiper | High impedance electromagnetic surfaces with a forbidden frequency band[END_REF][START_REF] Han | Understanding the coupling reduction eAEect in microtrip array antennas using high impedance surface (HIS)[END_REF][START_REF] Kawakami | Mutual coupling reduction eAEects of EBG structure located on cylinder surface[END_REF][START_REF] Liang | A novel broadband EBG using cascaded mushroom-like structure[END_REF][START_REF] Tran | High impedance surfaces based antennas for high data rate communications at 40 GHz[END_REF], Complementary Split Ring Resonator (CSRR) structures [START_REF] Han | CSRRS for e±cient reduction of the electromagnetic interferences and mutual coupling in microstrip circuits[END_REF][START_REF] Hafdallah Ouslimani | Analysis and reduction of electromagnetic coupling interferences in microstrip antenna arrays[END_REF] and/or the planar MTM surfaces [START_REF] Schurig | Electric-field-coupled resonators for negative permittivity metamaterials[END_REF].

In flight conditions, planes are subjected to aerodynamic loads leading to the structure distortion. Flight cycles induce a combination of twist and bend modes to wings. In order to study the impact of this kind of modes on the electromagnetic behavior of EBG-based patch antenna, an experimental setup has been conceived. Fig. 1 presents this developed benchmark experimental setup [START_REF] Adnet | Modélisation numérique du couplage mécanique/électromagnétique pour l'étude de la sensibilité du comportement électromagnétique d'antennes patch aux déformations mécaniques[END_REF]. It allows twisted and bended mechanical deformations of the MTM antenna in flight range defined by the aeronautical project partners. The mechanical system is composed of two main parts: part A sets a twisted configuration (Fig. 1(b)), and part B (Fig. 1(c)) is dedicated to reaching the aimed bent state. When integrated with an electromagnetic measurement system (in an anechoic chamber), this set up is able to provide the antenna radiation characteristics under the considered mechanical deformations.

DESIGN OF THE ANTENNA AND EBG STRUCTURES

EBG Structure Features

The metamaterial EBG array structure, designed using CST MWS [11], is presented in Fig. 2(a). It consists of four EBG squared patches of 8.4 mm side length (w), spaced by a 0.275 mm gap (g) and fed by a cylindrical via of 0.5 mm diameter (d). This array structure is printed on a 0.787 mm height substrate with a permittivity equal to 2.25.

The EBG is characterized by the surface wave propagation measurement, using the setup presented Fig. 2(b), where two microwave ports are normally set to the y-axis. The first one (port 1) generates an electromagnetic wave ( Ẽ, H) travelling along the y-axis and received by the port 2.

The boundary conditions (Fig. 2(c)) are chosen in order to define a waveguide structure, for which the four-EBG pattern repeats to infinity along the negative and positive x-direction. This is achieved by setting PMC conditions at both of the x-locations. PEC conditions are moreover defined below and above the substrate, assuming that an infinite ground plane exists onto the rear substrate. Then, two open boundaries are set at the two port locations.

Figure 2(d) depicts the reflection coe±cient S 11 and transmission coe±cient S 21 associated with the computed HIS using the method described in [START_REF] Liang | A novel broadband EBG using cascaded mushroom-like structure[END_REF][START_REF] Tran | High impedance surfaces based antennas for high data rate communications at 40 GHz[END_REF] in the 4-8 GHz frequency range. It shows that the bandgap of the MTM matrix (where the entire surface waves are reflected) is included between 4.7 GHz and 6.3 GHz.

EXPERIMENTAL AND COMPUTED RESULTS

Experimental and computed S-parameters and radiation patterns around 6.0 GHz are compared in this section. Results are presented for patches 2, 4 and 7, located in regions of the antenna where 3 particular mechanical strain states are obtained (Fig. 3(a)). Indeed, close to the twisting system (patch 2) -clamping for the bending tests-maximum mechanical strains will be obtained. On the other hand, close to the bending system (patch 7) -clamping for the twisting tests-mechanical strains are minimal, but important vertical displacement are observed

S-parameters

First, Fig. 5 presents measured (blue) and computed (red) reflective (S ii ) and transmitting (S ij , i 6 = j) parameters in the planar state (non-deformed antenna). The solid and dotted curves respectively correspond to measured parameters for conventional and MTM antennas. Experimental and numerical results are in a good agreement.

It can be observed that the computed reflection parameters S ii have a very good adequacy for both the antennas. On the other hand, a tiny shift (about 20 MHz) can be noted for the experimental ones. The introduction of EBG patterns (MTM antenna) induces a maximal coupling reduction (S ij parameters) about 12 dB through computations and 30 dB through experimental measurements for a 6 GHz frequency. It can also be noted that the experimental bandgap is narrower than the numerical one.

The influence of twisting and bending loads on reflective and transmitting S-parameters, measured with (solid curves) and without (dotted curves) MTM, are respectively shown for a 30 ± twist angle in Fig. 6 and for a °70 mm tip displacement in Fig. 7.

It clearly appears that the S-parameters are non-sensitive to the mechanical distortions, whatever the considered antenna. Moreover, the metamaterial decoupling features are maintained meanwhile the MTM antenna is deformed.

Radiation Characteristics

This subsection presents the measured and computed antenna's radiation patterns. The previous mutual coupling parameters (S ij ) analysis has shown that the maximal coupling reduction (MTM antenna) is obtained for at 6 GHz. Thus experimental measurements were performed at this frequency.

The radiation pattern has then been measured, on the measurement StarGate (shown by Fig. 8) of the SATIMO Industries. It consists in an arch, installed in a 5 £5 £ 5 m 3 anechoic chamber. The antenna under test (AUT) mounted on the deformation device is located on the top of a mast such as to be at the center of the arch. The arch is equipped with several probes, regularly spaced around the perimeter, being able to work in emitting or receiving mode. Concerning our test measurement, the probes are set in receiving mode, in order to acquire the radiation patterns along the µ elevation angle, Figure 9 illustrates the bended and twisted configurations of the AUT, and assembling of the antenna on the support (Fig. 9(b)).

Each of the eight patches of the array antenna was characterized and separately fed while all the others were 50 ≠ adapted such as to avoid radiation interferences. The reflector plane is finite size metallic ground plane (or perfect electric plane, PEC).

Planar Configuration

Figure 10 presents the measured and computed E-plane (Fig. 10(a)) and H-plane (Fig. 10(b)) Cartesian radiation diagrams obtained for patch 4, located in the middle of the patch network, in the planar case (non-deformed structure), for the conventional and MTM antennas. It appears that the computed results are in good agreement with the measured radiation pattern in the [°60 ± , 60 ± ] elevation angle range (µ), but "edges eAEects" can be observed out of this range. This point could be explained as follows: the measured antennas were manufactured with a finite metallic ground plane, and consequently, the radiation characteristics are available in the whole free space, i.e., the angular range goes from °180 ± to 180 ± . On the other hand, as the patch antenna model corresponds to a cavity recessed in an infinite metallic ground plane model, the radiation patterns computed with CST software were post-processed in the upper half-space, i.e., an angular range from °90 ± to 90 ± in the elevation view. Then, the contribution of lower half-space electromagnetic fields can not be taken into account via this numerical model.

Finally, it can be noticed in Fig. 10 that the main lobe (ML) directions are respectively 0-degree (vertical) in the H-plane and ±18-degrees oriented from the vertical axis in the E-plane diagrams, for conventional and MTM antennas. The EBG structures do not aAEect the radiation patterns. Moreover, it reduces the back radiation especially in the E-plane.

Figure 11 presents the total gain of the conventional and MTM antennas in the planar configuration. It is obtained by summing the individual gain of each of the 8 patches. It can be noticed that the measured data are well estimated by CST EM numerical model. Moreover, the unitary radiation patterns combination leads to a main lobe and some side secondary lobes, due to the patch interspaces corresponding to D patches = ∏ 6 GHz .

Twisting EAEects

Figure 12 presents the impact of a 30-degrees twisting angle on measured H-plane polar radiation patterns for patch 2, located close to the maximal mechanical strains zone (close to the twisting systems) for the conventional (red curves) and MTM (blue curves) antennas.

It can be observed that the impact of the twisting (dotted curves) lies in a tilt about 28 degrees of the H-plane radiation diagram in comparison with the planar configuration (solid curves), whatever the considered antenna. On the other hand, the shapes of polar radiation diagram seem to be non-sensitive to this mechanical distortion whatever the considered antenna. Finally, the EBG structures do not aAEect the radiation patterns.

Bending Loads

CONCLUSIONS

The influence of the prescribed mechanical deformations to an EBG metamaterial-based antenna is investigated in this paper. Two antennas have been manufactured: the first one is a conventional antenna, composed of eight rectangular patches, whereas the other one is equipped with squared EBG metamaterial structures, located between the seven couples of patches. The EBG structures are used here to reduce the mutual coupling. A 30 dB reduction has been achieved according to some measurements performed around the resonant frequency. A mechanical experimental setup has been conceived to study the electromagnetic behavior of the twisted and bended antennas. Results show that the metamaterial performances of the EBG-based antenna are not modified. Nevertheless, it is only observed that the main lobe direction is sensitive to the structural deformation of the antenna.

In future works, it would be interesting to study other designs of EBG-based patch antennas, so as to answer whether these conclusions could be generalized.

Figure 1 .

 1 Figure 1. Experimental set up, (a) general view, (b) twisting system and (c) bending system.

Figure 2 .

 2 Figure 2. Bandgap computation of the studied EBG matrix, (a) elementary pattern, (b) EBG characterization principle, (c) waveguide boundary conditions and (d) computed reflection (S 11 (dB)) and transmission (S 21 (dB)) coe±cients.

Figure 3 .

 3 Figure 3. Manufactured antennas, (a) conventional and MTM full antennas, (b) conventional antenna zoom in and, (c) MTM antenna zoom in.

3. 2 .

 2 Antenna Coupling Reduction Using EBG StructuresTwo 8-patch antenna arrays (Fig.3(a)) were designed to run in the MTM bandgap previously pointed out, around 6 GHz. The first antenna is a conventional one, composed of 8 patches of 15.64 mm length and 20 mm width regularly spaced with a 50 mm gap printed on a 503.84 mm £ 67 mm £ 0.787 mm dielectric substrate (RT/Duroid 5880) having a relative permittivity of 2.2 ± 0.02. The second one, hereafter named MTM antenna, consists in the conventional antenna, with 4£6 squared EBG patterns integrated between patches. Detailed structures of these two antennas are presented in Fig.3(b) and Fig. 3(c) near patches 4 and 5. Specific dimensions of EBG patterns are given Fig. 4.

Figure 4 .

 4 Figure 4. Inserted EBG component (D = 50 mm, L p = 15.64 mm, W p = 20 mm, W sub = 67 mm, x feed = 4.7 mm).

Figure 5 .

 5 Figure 5. Measured and simulated S-parameters in the planar case.

Figure 6 .

 6 Figure 6. Measured S-parameters in the planar (blue) and twisted (red) configurations.

Figure 7 .

 7 Figure 7. Measured S-parameters in the planar (blue) and bended (red) configurations.

Figure 8 .

 8 Figure 8. The measurement StarGate of the SATIMO industries.

Figure 9 .

 9 Figure 9. The AUT and its supporting plate, (a) mechanical boundary conditions, (b) components of the assembly, (c) +30 ± Twisted AUT et (d) °70 mm bended AUT.

Figure 10 .

 10 Figure 10. Cartesian radiation diagrams of patch 4, in (a) E-plane and (b) H-plane.

Figure 11 . 8 patches

 118 Figure 11. 8 patches Cartesian radiation diagrams, (a) E-plane for the conventional antenna, (b) E-plane for the MTM antenna, (c) Hplane for the conventional antenna, (d) H-plane for the MTM antenna.

Figure 13 Figure 12 .

 1312 Figure 13 presents the impact of a 70 mm tip bending displacement load on measured E-plane polar radiation patterns for patch 4, located in the middle of the patch network, close to the maximal mechanical strains, for the conventional (red curves) and MTM (blue curves) antennas. It can be observed that the impact of the bending (dotted

Figure 13 .

 13 Figure 13. Impact of bending on the E-plane polar patterns for patch 4.

Figure 14 .

 14 Figure 14. Combined Cartesian pattern, (a) "Planar-Twisted" measurement -H-plane, and (b) "Planar-Bended" measurement -E-plane.

Figure 15 .

 15 Figure 15. Recombined 3D polar pattern, (a) planar, y-z view, (b) twisted, y-z view, (c) planar, x-z view and (d) bended, x-z view.

† MSIE (Matériaux et Structures Intelligences pour l' Éectromagnétisme/Smart Material and Structures for Electromagnetism).

ACKNOWLEDGMENT

The authors would like to thank INEO Defense for having manufactured the antennas, and also thank SATIMO Industries for the use of their measurement facilities.