domain: phys spi

text:
Shah Vladislav Popov

Nawaz Burokur

Fabrice Boust

Shah Nawaz Burokur

Conformal sparse metasurfaces for wavefront manipulation

come

I. INTRODUCTION

In the last two decades, metasurfaces that are thin two-dimensional metamaterials have proven themselves as a powerful tool to tailor wavefronts [START_REF] Yu | Light propagation with phase discontinuities: Generalized laws of reflection and refraction[END_REF][START_REF] Pfeiffer | Metamaterial huygens' surfaces: Tailoring wave fronts with reflectionless sheets[END_REF]. A myriad of metasurface-based planar devices has been proposed and validated. Nowadays there is an increasing interest in conformal metasurfaces to perfectly match curved shapes [START_REF] Germain | Phase-compensated metasurface for a conformal microwave antenna[END_REF][START_REF] Gregoire | 3-D conformal metasurfaces[END_REF][START_REF] Liang | Anomalous terahertz reflection and scattering by flexible and conformal coding metamaterials[END_REF][START_REF] Walia | Flexible metasurfaces and metamaterials: A review of materials and fabrication processes at micro-and nano-scales[END_REF][START_REF] Cheng | Alldielectric ultrathin conformal metasurfaces: lensing and cloaking applications at 532 nm wavelength[END_REF][START_REF] Kamali | Decoupling optical function and geometrical form using conformal flexible dielectric metasurfaces[END_REF][START_REF] Xu | High-efficiency broadband polarization-independent superscatterer using conformal metasurfaces[END_REF][START_REF] Wu | Modelling of free-form conformal metasurfaces[END_REF][START_REF] Li | Wide-angle beam steering based on an active conformal metasurface lens[END_REF]. As expected, the design and fabrication of conformal metasurfaces are more demanding compared to planar metasurfaces [START_REF] Walia | Flexible metasurfaces and metamaterials: A review of materials and fabrication processes at micro-and nano-scales[END_REF]. Adopting flexible substrates to implement metasurfaces opens a new way for integration with other elements and non-planar designs including lightweight wearable devices [START_REF] Walia | Flexible metasurfaces and metamaterials: A review of materials and fabrication processes at micro-and nano-scales[END_REF][START_REF] Kamali | Decoupling optical function and geometrical form using conformal flexible dielectric metasurfaces[END_REF][START_REF] Wu | Modelling of free-form conformal metasurfaces[END_REF]. In addition, for example, microwave antennas based on conformal metasurfaces do not only allow one to meet the aerodynamic specifications of aircrafts and satellites [START_REF] Germain | Phase-compensated metasurface for a conformal microwave antenna[END_REF][START_REF] Li | Wide-angle beam steering based on an active conformal metasurface lens[END_REF] but also break the fundamental constraint of their planar counterparts. For instance, the aperture of a flat metasurface antenna vanishes when the beam steering angle increases [START_REF] Nayeri | Reflectarray Antennas: Theory, Designs and Applications[END_REF].

The design of a conformal metasurface is generally based on geometrical optics approach [START_REF] Germain | Phase-compensated metasurface for a conformal microwave antenna[END_REF][START_REF] Gregoire | 3-D conformal metasurfaces[END_REF][START_REF] Liang | Anomalous terahertz reflection and scattering by flexible and conformal coding metamaterials[END_REF][START_REF] Walia | Flexible metasurfaces and metamaterials: A review of materials and fabrication processes at micro-and nano-scales[END_REF][START_REF] Cheng | Alldielectric ultrathin conformal metasurfaces: lensing and cloaking applications at 532 nm wavelength[END_REF][START_REF] Kamali | Decoupling optical function and geometrical form using conformal flexible dielectric metasurfaces[END_REF][START_REF] Xu | High-efficiency broadband polarization-independent superscatterer using conformal metasurfaces[END_REF][START_REF] Li | Wide-angle beam steering based on an active conformal metasurface lens[END_REF], where a proper spatial distribution of local reflection (transmission) coefficient should be established along the reflecting (transmitting) metasurface. It has been revealed that this heuristic approach has strong limitations in terms of efficiency and versatility [START_REF] Asadchy | Perfect control of reflection and refraction using spatially dispersive meta-surfaces[END_REF][START_REF] Estakhri | Wave-front transformation with gradient metasurfaces[END_REF]. On the other hand, more rigorously, metasurfaces can be described by means of continuous surface impedances [START_REF] Kuester | Averaged transition conditions for electromagnetic fields at a metafilm[END_REF][START_REF] Epstein | Huygens' metasurfaces via the equivalence principle: design and applications[END_REF]. Unfortunately, theoretical modelling of conformal metasurfaces appears to be exceptionally challenging, demanding accurate analysis of the metasurface geometry and dealing with curvilinear coordinates [START_REF] Wu | Modelling of free-form conformal metasurfaces[END_REF].

The contribution of this work is twofold. First, it is shown how numerical calculation of a Green's function * sburokur@parisnanterre.fr can be employed to design conformal sparse metasurfaces capable of creating arbitrary field patterns for arbitrary external excitations. The sparseness implies that the distance between neighboring meta-atoms is large enough to approximate polarization currents as a delta function and make the interaction between meta-atoms due to the macroscopic field (provided by a Green's function) dominant over strongly localized microscopic fields (appearing when geometrical parts constituting meta-atoms come very close to each other). The proposed approach does not use any complex local coordinate system matched to a particular geometry such that arbitrarily-shaped metasurfaces can be considered without any accommodation. Sparse metasurfaces possess strongly non-local response [START_REF] Ra'di | Metagratings: Beyond the limits of graded metasurfaces for wave front control[END_REF][START_REF] Epstein | Unveiling the properties of metagratings via a detailed analytical model for synthesis and analysis[END_REF][START_REF] Wong | Perfect anomalous reflection with a bipartite huygens' metasurface[END_REF][START_REF] Popov | Constructing the near field and far field with reactive metagratings: Study on the degrees of freedom[END_REF][START_REF] Epstein | Synthesis of passive lossless metasurfaces using auxiliary fields for reflectionless beam splitting and perfect reflection[END_REF][START_REF] Díaz-Rubio | From the generalized reflection law to the realization of perfect anomalous reflectors[END_REF][START_REF] Kwon | Lossless scalar metasurfaces for anomalous reflection based on efficient surface field optimization[END_REF] and can be described in terms of neither surface impedance nor local reflection and/or transmission coefficients being not subject to fundamental efficiency limitations as their "dense" counterparts [START_REF] Asadchy | Perfect control of reflection and refraction using spatially dispersive meta-surfaces[END_REF][START_REF] Estakhri | Wave-front transformation with gradient metasurfaces[END_REF][START_REF] Epstein | Synthesis of passive lossless metasurfaces using auxiliary fields for reflectionless beam splitting and perfect reflection[END_REF][START_REF] Epstein | Passive lossless huygens metasurfaces for conversion of arbitrary source field to directive radiation[END_REF][START_REF] Epstein | Arbitrary powerconserving field transformations with passive lossless omega-type bianisotropic metasurfaces[END_REF]. At the same time, the sparseness allows to establish a global theoretical model and get a microscopic insight into the theoretical analysis of conformal metasurfaces. Secondly, it is detailed how to realize these conformal sparse metasurfaces. To describe the design procedure, three semi-cylindrical sparse metasurfaces illuminated by an arbitrary complex wave configuration are experimentally demonstrated at microwave frequencies.

II. THEORY

Without loss of generality, we consider the case of TE polarization and a two-dimensional (2D) geometry. A translation symmetry is assumed along one of the three spatial dimensions. We compose a sparse metasurface by a finite set of N loaded wires distributed along the surface of an arbitrarily-shaped dielectric substrate. The wires are oriented along the translation-invariant direction. An example presented in Fig. 1 illustrates an array of loaded wires distributed along an arbitrarily-shaped substrate. Microscopically, a loaded wire represents itself as a chain of subwavelength meta-atoms. On the other hand, macroscopically, we model the loaded wire as uniform and having a deeply subwavelength effective radius r ef f .

We consider external sources radiating a background wave with the electric field directed along x-axis and exciting polarization currents in the loaded wires. However, here we do not impose any condition on the external sources unlike plane-wave excitation used for majority of metasurfaces presented in literature. The polarization currents excited in the wires can shape the field radiated by a metasurface. The total electric field E x (r) can be split into two terms: the field E (ext) x (r) due to external sources alone (in the absence of wires) and the field E (sct) x (r) re-radiated (scattered) by the polarization currents. The latter is found by means of a Green's function G xx (r, r ) of a system corresponding to a metasurfacebased device, which means that this Green's function is not the one of free space. It is calculated in the presence of a curved substrate (backed with a metal sheet in the case of a reflecting configuration) and other dielectric and/or metallic parts of the system. In the equation form, the Green's function is defined as

∂ 2 ∂y 2 + ∂ 2 ∂z 2 + k 2 0 ε r (r) G xx (r, r ) = iωµ 0 δ(r -r ), (1)
where r and r are 2D vectors in y0z plane, µ 0 is the permeability of free space, k 0 is the wavenumber of free space, ω is the angular frequency, ε r (r) is the relative permittivity corresponding to dielectric and metallic parts of the metasurface-based device in the absence of the loaded wires. As the considered system does not have a translational symmetry in the 2D plane, a Green's function G(r, r q ) is a function of r and r (not their difference). Then, the scattered electric field can be represented as the convolution and since the wires are distributed only along the top face of an arbitrarily-shaped substrate, the integration should be performed only over this face

E (sct) x (r) = G xx (r, r )J x (r )dr , (2)
where J x (r) is the electric current density corresponding to polarization currents in all wires. Following the assumption of infinitesimally thin wires, we approximate the current density as J x (r) = N q=1 I q δ(r -r q ), where I q and r q is the polarization line current excited in the q th wire and its position, respectively. Taking this into account, the total electric field can be written as

E x (r, φ) = E (ext) x (r, φ) + N q=1 G xx (r, φ; r q )I q , ( 3
)
where r and φ are polar coordinates: radius and polar angle, respectively. The total field E x (r, φ) is represented by the superposition of the wave radiated by external sources E

(ext) x (r, φ) and waves G(r, φ; r q )I q scattered by the wires.

Sparse configuration of the metasurface allows one to accurately take into account the interactions between the wires via Ohm's law

Z q I q = E (ext) x (r q ) - N p=1 Z (m) qp I p . (4)
A load-impedance density (or impedance per unit length of a wire) Z q is a characteristic of a loaded wire and can be engineered for instance by tuning the geometrical parameters of meta-atoms constituting a wire. The righthand side of Eq. (4) represents the total electric field at the position of the q th wire, where

Z (m) qp = -G xx (r q , r p ) (5)
is the mutual-impedance density (the electric field created by the p th wire at the position of the q th wire). The separation between two neighboring wires can be arbitrary, as long as polarization currents in the wires can be approximated by a 2D delta function, which limits the transverse size of a wire. On the other hand, the interaction between neighboring wires due to the macroscopic field G xx (r q , r p ) should be much stronger than the coupling via a strongly localized microscopic near-field. The latter appears when geometric parts constituting two neighboring wires come very close to each other. As it is demonstrated further, the simple model of δ function works surprisingly well even for complex designs. The self-action of the q th wire and its interaction with the environment is accounted via

Z (m) qq = - 1 2πr ef f G xx (r, r q )dr, (6)
where the integration is performed over the circumference of the wire of effective radius r ef f . Although being very simple, Eq. ( 4) has an important practical implication: it allows one to know in advance the impact of one polarization current on another and to accordingly adjust the load-impedance densities. Conceptually, it means that the developed approach is global, being in strong contrast with conventional theoretical models of metasurfaces which are essentially local. Interaction between wires described via the matrix of mutual-impedance densities indicates that sparse metasurfaces are intrinsically strongly non-local, i.e. the current in the q th wire depends on the one in the p th wire.

Meanwhile, by appropriately choosing Z q and the number of wires N , we are then able to tailor the field at will. Each term on the right-hand side of Eq. (3) at a given distance r (r = |r q | for all q) can be approximated by a partial Fourier sum over the polar angle φ: E

(ext) x (r, φ) = M n=-M C (ext) n e inφ and G xx (r, φ; r q ) = M n=-M C (q)
n e inφ . M is the maximum Fourier harmonic defined as the minimum number M such that

|C (ext,q) ±|M +r| |/max|C (ext,q) n | 1 (7)
for all r = 1, 2,... . The number M might affect many parameters (such as geometrical parameters and material properties of a sample, the distance r) but the most im-portant one is the physical aperture max[|r q -r p |]. The larger the aperture the greater is M . Evidently, the total electric field E x (r, φ) can also be represented by a partial Fourier sum M n=-M C m e inφ and the relation between the Fourier coefficients is as follows:

C n = C (ext) n + N q=1 C (q) n I q , (8)
where

C (ext) n and C (q)
n are known. When a sparse metasurface is composed of at least N = 2M + 1 loaded wires one can establish arbitrary azimuthal field distributions within the functional space of the 2M + 1 Fourier harmonics by adopting the Fourier coefficients C n .

Corresponding load-impedance densities can be found from Eq. ( 4) after solving Eq. ( 8) with respect to I q , which in this case has a single solution [START_REF]coefficients C (q) n is not degenerate since Gxx(r, φ; rq) related to different points rq are linearly independent[END_REF]. As a matter of fact, there is no guarantee that analytically found Z q would not require implementing active and/or lossy elements since [Z q ] = 0 in a general case. In order to additionally deal with only reactive load-impedance densities Z q = i [Z q ], one might need a number of wires N ≥ 2M +1 for constructing arbitrary radiation patterns, as recently discussed in Ref. [START_REF] Popov | Constructing the near field and far field with reactive metagratings: Study on the degrees of freedom[END_REF]. Essentially, the procedure described in this paragraph allows one to know in advance all possible configurations of the azimuthal field for a given geometry of metasurface and number of wires. It includes practical parameters such as beamwidth and sidelobes level.

To sum up, after establishing the geometry of a sparse metasurface (flat, cylindrical, or any other shape), excitation type and positions of N wires, we are able to calculate a Green's function G xx (r, φ; r q ) and the background field E 3) and ( 4), a relation between the radiated field and load-impedance densities is established and the procedure related to Eq. ( 8) is used to find the number of wires and to determine possible functional characteristics of a sparse metasurface such as beamwidth and sidelobes level in beam-forming applications. A crucial point of the above analysis is the Green's function which, being defined for an arbitrary finite-size system, does not have any applicable analytical form. Instead, it is suggested to compute it numerically with the help of full-wave numerical simulations as detailed in Appendix A. A design procedure to practically implement loaded wires is described in Appendix B. An optimization procedure is brought to solve the inverse scattering problem assuming only reactive load-impedance densities of wires, as detailed in Appendix C.

III. 2D SIMULATION EXAMPLE

Let us consider a simple example of a cylindrical sparse metasurface for far-field manipulation. The radius of the metasurface is fixed to 5λ 0 /6 (λ 0 is the vacuum operating wavelength) and it is excited by a point source placed in the center, as shown in Figs. 2(a) and (b). The source creates a cylindrical background wave, as illustrated in Fig. 3(a). Due to the symmetry, Green's functions G xx (r, φ; r q ) of different wires placed on the cylindrical substrate are simply shifted with respect to each other (as function of φ). It means that one has to analyse the Fourier decomposition of only one Green's function and of the background wave in order to find the parameter M . Figure 3(b) shows an example of the function G xx (r, φ; r q ) computed for a particular r q . Following the definition in Eq. ( 7), from Figs. 2(c) and (d), it can be seen that it is enough to have N = 2 × 10 + 1 = 21 wires in order to be able to construct all possible far-field radiation patterns within the Fourier space of 21 harmonics exp[inφ]. As an illustrative example, one can reconstruct in the far-field region the shape of a "heart" after the required corresponding load-impedance densities are found from Eqs. ( 8) and (4). The required load-impedance densities are plotted in Fig. 2(e) and correspond to passive elements ( [Z q ] > 0). The real part of Z q can be engineered in a similar fashion as proposed in in Ref. [START_REF] Wang | Systematic design of printable metasurfaces: Validation through reverse-offset printed millimeter-wave absorbers[END_REF]. The resulted far-field pattern is depicted in Fig. 2(f) and compared to the ideal shape of a "heart". Figure 2(a) shows the corresponding profile of the electric field in the proximity to the metasurface. In order to improve the ac-curacy of approximating some ideal curve, one needs to increase M which can be done by increasing the size of the metasurface (the radius in the considered case). Finally, as stated above, to best of our knowledge, there is no analytical formula for a Green's function of a system with a cylindrical substrate. Additional details on this example are given in Appendix A.

IV. 3D SIMULATION EXAMPLES

In this section, we provide several examples of designs of conformal sparse metasurfaces operating in the microwave frequency range, which perform different beamforming functionalities. The metasurfaces demonstrated in what follows represent a set of loaded wires uniformly distributed along the top face of a dielectric substrate. The bottom face of the substrate is metal-free. Operating in the transmission mode, the metasurfaces transform an incident wavefront from one side into a desired wavefront on the other side. The designs are developed with the optimization-aided procedure (see Appendix C), verified by means of 3D full-wave simulations and compared to predictions of the analytical model represented by Eqs. ( 3) and ( 4). The impact of using different number of wires composing a sparse metasurface is also analysed.

The design of loaded wires is performed within the local periodic approximation (LPA), developed in Ref. [START_REF] Popov | Designing metagratings via local periodic approximation: From microwaves to infrared[END_REF] and adopted to current examples in Appendix B. The wires are built up from printed capacitors (left design of Fig. 4(a)) and inductors (right design of Fig. 4(a)), which provide a wide range of accessible load-impedance densities as shown in Figs. 4(b) for 5 GHz operation. A particular importance of the LPA of Ref. [START_REF] Popov | Designing metagratings via local periodic approximation: From microwaves to infrared[END_REF] should be emphasized: Sparse metasurfaces cannot be designed as their dense counterparts. Indeed, the design procedure results in the load-impedance density of a loaded wire, which is its proper characteristic and depends on neither the substrate's thickness nor the inter-wire distance, see also Ref. [START_REF] Popov | Beamforming with metagratings at microwave frequencies: Design procedure and experimental demonstration[END_REF]. On the other hand, when designing conventional dense metasurfaces, one deals with such characteristics as surface impedances (or local reflection and/or transmission coefficients) which represent the integral response a unit cell and depend on the parameters of a substrate and the inter-element distance.

A. Semi-cylindrical sparse metasurface

In the first example, we demonstrate a sparse metasurface operating at 5 GHz (vacuum wavelength λ 0 ≈ 60 mm) and conformed to a semi-cylindrical shape of 1.67λ 0 = 100 mm radius. N = 29 loaded wires are uniformly distributed along the top face of a λ 0 /120 = 0.5 mm thick and 5.25λ 0 = 315 mm long F4BM220 substrate, which serve to control 2M + 1 = 29 independent Fourier harmonics of the far-field as follows from Eqs. [START_REF] Cheng | Alldielectric ultrathin conformal metasurfaces: lensing and cloaking applications at 532 nm wavelength[END_REF] and [START_REF] Kamali | Decoupling optical function and geometrical form using conformal flexible dielectric metasurfaces[END_REF]. The metasurface is illuminated by a point source placed at λ 0 /4 distance above a PEC wall joining the two ends of the semi-cylinder, as shown in the inset of Fig. 4(c). Distribution of the electric field created by the source and an example of function G xx (r, r q ) computed for this configuration are shown in Figs. 3(c) and (d), respectively. In order to emulate the 2D configuration of the considered theoretical model, a narrow strip of the metasurface (of λ 0 /6 width) is embedded in between two PEC plates which form a parallel-plate waveguide. The source is represented by a coaxial probe exciting TEM waveguide mode. A schematics of the system is shown in Fig. 4(c). directivity, which is calculated as follows

D(ϕ) = 2π|E f f (ϕ)| 2 2π 0 |E f f (ϕ)| 2 dϕ, (9)
where Because of the very narrow λ 0 /6 aperture in the transverse direction, the radiation pattern presents a wide beam in the E-plane to produce a 3D fan-shaped beam. However, if the distance between two PEC plates exceeds λ 0 /2, higher order waveguide modes can be excited, leading to a nonuniform field distribution along the x-direction that was assumed to be translation invariant. This problem can be overcome and a much narrower beam can be achieved in the E-plane by elaborating on the excitation source as discussed in the following Section.

E f f (ϕ) is E x (r,

B. Cavity-excited sparse metasurface

In the second example, we consider planar sparse metasurfaces placed on the top of an open rectangular PEC cavity excited by a coaxial probe. Distribution of the excitation field created by the probe inside and outside the cavity are demonstrated in Fig. 3(e). A particular example of function G xx (r, r q ) calculated for this configuration is presented by Fig. 3(f). Results for two configurations with 10 and 20 wires are compared. For the given aperture size of 5λ 0 , the number 2M + 1 of independent Fourier harmonics forming the far-field equals to 42. While neither 10 nor 20 wires is enough to arbitrarily control 42 Fourier harmonics, it is however possible to perform efficient beam-forming. Schematics of the configurations are shown in Figs. 5(a) and(d). be particularly advantageous for reconfigurable designs. Fortunately, the presented analytical model allows one to optimize the number of elements by considering a beforehand computed Green's function and without involving time-consuming 3D full-wave simulations of real designs.

For the sake of comparison with literature, we would like to mention that cavity-excited metasurfaces have been considered in works done by A. Epstein et al. [START_REF] Epstein | Arbitrary powerconserving field transformations with passive lossless omega-type bianisotropic metasurfaces[END_REF][START_REF] Epstein | Cavity-excited huygens' metasurface antennas for nearunity aperture illumination efficiency from arbitrarily large apertures[END_REF]. In strong contrast to the single-layer design of sparse metasurfaces presented in Fig. 5, metasurfaces are designed as dense three-layer structures to emulate magnetic and/or bianisotropic response in Refs. [START_REF] Epstein | Arbitrary powerconserving field transformations with passive lossless omega-type bianisotropic metasurfaces[END_REF][START_REF] Epstein | Cavity-excited huygens' metasurface antennas for nearunity aperture illumination efficiency from arbitrarily large apertures[END_REF].

V. EXPERIMENTAL EXAMPLES

To validate the approach, we designed and fabricated three sparse metasurfaces using extremely thin (λ 0 /240 = 0.25 mm) flexible substrates F4BM220, as highlighted by the photographies in Figs. 6(a) and(b). The samples operating in the microwave frequency range with a central frequency fixed at 5 GHz were fabricated by means of conventional printed circuit board (PCB) technology. Each sample is composed of twenty nine wires equidistantly distributed along a 5.25λ 0 ≈ 315 mm long substrate, making the separation between two neighbouring wires approximately equal to λ 0 /5. The wires are built up from printed capacitors and inductors (see Fig. 4(a)), which provide a wide range of accessible loadimpedance densities as shown in Fig. 4(b).

The ultra-thin samples are then conformed to create semi-cylindrical surfaces of 100 mm radius. A photography of the assembled prototype is presented in Figs. 6(b) and(c). The external source exciting the samples is made of two microstrip dipole antennas printed on a metal-backed substrate [START_REF] Katehi | On the modeling of electromagnetically coupled microstrip antennas-the printed strip dipole[END_REF], whose design is detailed in Figs. 6(d)-(f). The distance between the source and sparse metasurface being 100 mm (≈ 1.7λ 0 ), the background field pattern is neither a plane wave nor a cylindrical wave, as highlighted by Fig. 7.

The three samples were designed to show different beam-forming examples: a single broadside beam at 0 • , a steered beam at 40 • and a multibeam configuration with two beams at ±30 • from broadside. The inverse scattering problem was solved by maximizing the power in the desired direction and minimizing the sidelobes level (with respect to the geometrical parameters A and B of the loaded wires) radiated by the sparse metasurface in desired directions, and the maximization procedure was implemented as particle swarm optimization [START_REF] Kennedy | Particle swarm optimization[END_REF] detailed in Appendix C. The different configurations were experimentally validated by radiation patterns measurements performed in an anechoic chamber. A horn antenna used a receiver is kept fixed and the assembled prototype is mounted on a rotating platform as shown in Fig. 7(a). Figures 7(b)-(d) present the experimental results obtained from the three samples. The level of spurious scattering (at the operating frequency) does not exceed -12 dB for the first sample, -9 dB for the second one and -13 dB for the third one. The radiation pattern of the source alone is shown in Fig. 7(e) to highlight the beam-forming capabilities of the designed metasurfaces.

VI. CONCLUDING REMARKS

It is important to note that in the demonstrated numerical and experimental examples, the sparse meta-surfaces are transmitting while possessing only electric response. On the other hand, conventional phase gradient approach to design transmitting metasurfaces demands implementing additionally to the electric response an effective magnetic one. It is necessary to suppress reflection and achieve 2π-range phase response to be able to establish a required phase gradient along the metasurface [START_REF] Yu | Light propagation with phase discontinuities: Generalized laws of reflection and refraction[END_REF][START_REF] Germain | Phase-compensated metasurface for a conformal microwave antenna[END_REF][START_REF] Cheng | Alldielectric ultrathin conformal metasurfaces: lensing and cloaking applications at 532 nm wavelength[END_REF][START_REF] Kamali | Decoupling optical function and geometrical form using conformal flexible dielectric metasurfaces[END_REF][START_REF] Li | Wide-angle beam steering based on an active conformal metasurface lens[END_REF]. A more rigorous approach is based on engineering of electric and magnetic surface impedances [START_REF] Pfeiffer | Metamaterial huygens' surfaces: Tailoring wave fronts with reflectionless sheets[END_REF][START_REF] Wu | Modelling of free-form conformal metasurfaces[END_REF][START_REF] Epstein | Huygens' metasurfaces via the equivalence principle: design and applications[END_REF][START_REF] Epstein | Passive lossless huygens metasurfaces for conversion of arbitrary source field to directive radiation[END_REF] (and sometimes electromagnetic coupling [START_REF] Epstein | Arbitrary powerconserving field transformations with passive lossless omega-type bianisotropic metasurfaces[END_REF][START_REF] Asadchy | Functional metamirrors using bianisotropic elements[END_REF][START_REF] Asadchy | Bianisotropic metasurfaces: physics and applications[END_REF]) to manipulate wavefronts according to the equivalence theorem [START_REF] Pfeiffer | Metamaterial huygens' surfaces: Tailoring wave fronts with reflectionless sheets[END_REF][START_REF] Schelkunoff | Some equivalence theorems of electromagnetics and their application to radiation problems[END_REF]. Following the theory presented in this study, realizing only electric response can be sufficient for an efficient control of wavefronts that might significantly simplify the design and fabrication of wavefront manipulation devices. Furthermore, intrinsic strongly non-local response of sparse metasurfaces overcomes the fundamental efficiency constraint of conventional metasurfaces imposed by the conservation of normal power flow density [START_REF] Ra'di | Metagratings: Beyond the limits of graded metasurfaces for wave front control[END_REF][START_REF] Epstein | Unveiling the properties of metagratings via a detailed analytical model for synthesis and analysis[END_REF][START_REF] Popov | Constructing the near field and far field with reactive metagratings: Study on the degrees of freedom[END_REF][START_REF] Epstein | Synthesis of passive lossless metasurfaces using auxiliary fields for reflectionless beam splitting and perfect reflection[END_REF][START_REF] Díaz-Rubio | From the generalized reflection law to the realization of perfect anomalous reflectors[END_REF][START_REF] Kwon | Lossless scalar metasurfaces for anomalous reflection based on efficient surface field optimization[END_REF].

To conclude, we have presented a theoretical approach that opens the way to consequently design conformal sparse metasurfaces without appealing to a complex theory. Due to the versatility of the approach, one can consider different metasurface geometries and arbitrary excitation sources within the same framework. The theoretical analysis represented by Eqs. ( 7) and ( 8) allows one to approach problems of superdirectivity [START_REF] Schelkunoff | A mathematical theory of linear arrays[END_REF] and subdiffraction focusing [START_REF] Huang | Planar diffractive lenses: Fundamentals, functionalities, and applications[END_REF]. Although the experiments have been performed at microwave frequencies, the theory is valid in any frequency range and may inspire research on novel applications of conformal metasurfaces. Particularly, sparse metasurfaces implemented on flexible substrates can be advantageous for realizing a reconfigurability mechanism based on mechanical deformations [START_REF] Walia | Flexible metasurfaces and metamaterials: A review of materials and fabrication processes at micro-and nano-scales[END_REF]. It can represent a fruitful approach to create an adaptive response without complicating a design with tunable elements (which also often bring additional ohmic losses) and bias networks.

(r, ϕ) radiated by external sources. In order to find G xx (r, r q ), we build a 2D simulation model using the commercially available finite-element-method software COMSOL Mul- tiphysics in this work. An elementary unit source is consequently placed at the different positions of loaded wires r q , q = 1, 2, ..., N . More precisely, this source is a hollow disk of radius r ef f with an electric surface current density used as a boundary condition and set equal to 1/(2πr ef f ) A/m. Figures 8(a) and (b) demonstrate a schematics of a simulation model used to calculate the Green's function for the example of the cylindrical spare metasurface discussed in Section III. The electric field created by the source is recorded at each position along the substrate. There is no need to know the Green's function over the whole 2D plane but only at certain points. Namely, we extract the electric field at the positions of the wires to construct the matrix of mutual-impedance densities Z (m) qp and to control azimuthal wavefront at the distance r. In beam-forming applications, far-field calculations should be performed and the electric field in the far-field as a function of the azimuthal angle is recorded. In Figs. 8(c)-(f), G xx (r, r q ) for the positions of the 2 nd and 5 th wires are shown. the corresponding uniform array of period d illuminated by a plane wave incident at angle θ. It allows one to retrieve the load-impedance density Z q from the complex amplitude of the specularly reflected TE-polarized plane wave. Indeed, a polarization current I excited in a wire by the incident plane wave is linked to the complex amplitude S 11 of the specularly reflected wave via the following formula [START_REF] Popov | Designing metagratings via local periodic approximation: From microwaves to infrared[END_REF]

I = - 2d kη (S 11 e 2iβ0(h+λ0) -R TE 0 e 2iβ0h )β 0 (1 + R TE 0 )e jβ0h , ( B1
)
where h is the thickness of the substrate, R TE m is the Fresnel's reflection coefficient from the substrate

R TE m = j 2 βm β s m - β s m βm tan(β s m h) 1 + j 2 βm β s m + β s m βm tan(β s m h) , ( B2
)
β m = k 2 0 -ξ 2 m and β s m = ε s k 2 0 -ξ 2
m are, respectively, normal components of the wavevector of the m th Floquet mode outside and inside the substrate, ξ m = k 0 sin(θ) + 2πm/d is the tangential component of the wavevector.

The complex amplitude of the specularly reflected wave can be obtained from 3D full-wave simulations. The principal geometry of a corresponding simulation model is sketched in Fig. 9 and consists of two parts: a structured wire, represented by a printed capacitor on top of a dielectric substrate, and an air region. Periodic boundary conditions are imposed on the side faces of the model. The latter is excited by a periodic Port 1 assigned to the upper face of the air region, which is highlighted with the red color in Fig. 9. The periodic Port 1 creates a plane wave incident at angle θ and is also used as a listening port to calculate the scattering parameter S 11 . Periodic Port 2 is set to the listening mode to accept the transmitted wave. The thickness of the air region above and below the substrate is set to the operating vacuum wavelength λ 0 to eliminate higher order evanescent Floquet modes.

By modeling the polarization current excited in a structured wire as Dirac delta function δ(y, z), the interaction with the substrate and between adjacent wires can be taken into consideration analytically via the mutualimpedance density Z m [START_REF] Popov | Designing metagratings via local periodic approximation: From microwaves to infrared[END_REF]

Z m = kη 2 +∞ n=1 cos[k sin[θ]nL]H (2) 0 [knd] + kη 2d +∞ m=-∞ R T E m β m .
(B3) It allows one to retrieve the load-impedance density Z q of a wire by subtracting the electric field induced by adjacent wires and the substrate where E 0 = (1 + R T E 0 ) exp[jβ 0 h] represents the external electric field at the location of the central wire and induced by the incident wave and its reflection from the substrate. It should be noted that the term corresponding to Z (m) qq in Ohm's law, Eq. ( 4), accounts for both the self-interaction and the interaction with a substrate and an environment, while in the LPA the mutualimpedance density subtracted on the right-hand side of Eq. (B4) corresponds only to the interaction with a substrate and other wires in the uniform array. Therefore, Eq. (B4) is accordingly modified by subtracting from the right-hand side the self-interaction term which, luckily, can be accounted by a simple analytical expression -k 0 ηH This Section provides a step-by-step guide to design sparse metasurfaces by means of the LPA and an optimization procedure. The design procedure can be decomposed in five main steps, as outlined in the flowchart depicted in Fig. 10. The design procedure begins by specifying the configuration of the metasurface-based design (reflection, transmission, cavity-excited, etc.), geometry of the metasurface and its substrate. Finally, external sources and their positions are selected to efficiently illuminate the metasurface. Once the basic parameters are fixed, the second step is to calculate the Green's function and the external field by means of the established simulation-based procedure detailed in Ap-pendix A. Then, the total electric field created by a sparse metasurface is given by Eq. ( 3).

Z q = E 0 I - k 0 η 4 H (2) 0 (k 0 r ef f ) -Z m , (B4)
The third step is independent from the previous two and requires to design a meta-atom (or meta-atoms) that will form the wires of a sparse metasurface. To that end, the local periodic approximation is used. A metaatom's design should provide a wide range of achievable impedance densities of a corresponding wire when changing its geometrical parameters (or an external bias if there are embedded tunable elements).

Each desired configuration of the diffraction pattern requires different set of impedance densities which obey Ohm's law represented by Eq. ( 4). In the fourth step of the procedure, one finds load-impedance densities of wires composing a sparse metasurface and required to establish a desired distribution of the electric field. In general case, an arbitrary electric field E x (r), as it is stressed in Section II, requires impedance densities that have Re[Z q ] = 0 and imply engineering of active and/or lossy elements. Including in the period additional wires and satisfying the power conservation conditions Re E (exc)

x (r q ) - N p=1 Z (m) qp I p I * q = kη 4 |I q | 2 , (C1)
makes it sufficient to use purely reactive loads. On the other hand, any realistic passive meta-atom possesses inevitable resistive response whether because of conduction and/or dielectric losses or embedded lossy tunable elements (such as varactor diodes, for instance). Although the LPA allows one to account for these practical aspects of a meta-atom's design, the capacity to judiciously engineer Re[Z q ] independently from Im[Z q ] can be very limited. Furthermore, the set of equations (C1) should be modified to the following set of inequalities Re E (exc)

x (r q ) - N p=1 Z (m) qp I p I * q > kη 4 |I q | 2 , (C2)
which accounts for the resistance of wires and indicates that the power received by a wire is greater than the power it radiates. In many situations, it is preferable to minimize the impact of Re[Z q ] resulting in increased ohmic losses. Instead of elaborating on the analytical procedure developed in Section II and justifying the number of wires, the inverse scattering problem can be efficiently solved by means of a numerical optimization procedure. It can be particularly useful for beam-forming applications representing a subset of all possible far-field configurations when the power is maximized only in certain directions. Therefore, it might not be required to control all 2M + 1 of the Fourier harmonics and a lesser number of wires than N = 2M + 1 can be used for beamforming. As optimization parameters, the optimization procedure uses geometrical parameters of meta-atoms (or external biases) and the total number of wires. The geometrical parameters of meta-atoms G q are related to the load-impedance density Z q of a wire by means of the LPA. In their turn, load-impedance densities determine polarization currents and consequently the total electric field found from Eqs. ( 4) and ( 3). The objective function may vary depending on the needs. The one that has been used in this work to equally split the radiated power between an arbitrary number of beams when minimizing the level of secondary lobes is

f (G q ) = t t n∈D |E f f (ϕ n )| 2 2π 0 |E f f (ϕ)| 2 dϕ t .
(C3) Here, D represents a discrete set of desired beams at angles ϕ n and t is the size of this set. As a concrete optimization algorithm, a particle swarm optimization has been employed in this work, which showed better performances than a real-valued genetic algorithm. A block diagram is displayed in Fig. 11. Once the optimal loadimpedance densities are found, the fourth step of the design procedure is completed.

The fifth and the final step of the design procedure is to compose the whole metasurface out of designed metaatoms with optimized geometrical parameters and validate the final design using 3D full-wave numerical simulations. It is important to note that the developed optimization procedure itself does not include full-wave simulations and utilizes only numerical arrays calculated beforehand. It makes the optimization and the development of a final design much faster than in the case when only direct optimization of a whole metasurface is performed.

TABLE I. Geometrical parameters of the samples. Schematics of capacitor and inductor elements is illustrated in Fig. 6(a). The parameter w is the same for all elements and equals to 0.25 mm, the thickness of the copper cladding is 35 µm.

Element number

FIG. 1 .

 1 FIG. 1. Illustration of an arbitrarily-shaped sparse metasurface transforming an arbitrary impinging wave into multiple beams.

FIG. 2 .

 2 FIG. 2. (a) Profile of the electric field created by a cylindrical sparse metasurface. The metasurface is represented by 21 loaded wires uniformly distributed along a cylindrical substrate of thickness λ0/120 and having the relative permittivity 2.2. The metasurface is excited by a point source placed in its center. (b) Zoom of a part of the sparse metasurface. (c) |C (q) n |/max[|C (q) n |] (d) |C (ext) n |/max[|C (ext) n |] vs. the number n of Fourier harmonic. (e) Load-impedance densities required to approximate the "heart" shaped far-field pattern. (f) Comparison of the far-field pattern created by the sparse metasurface (solid curve) and the ideal "heart" shape (dashed curve).

  φ) radiated by external sources. With Eqs. (

FIG. 3 .

 3 FIG. 3. Simulated distributions of the electric field created by an elementary source in different configurations: (a),(b) a cylindrical dielectric substrate in free space, (c),(d) a semi-cylindrical substrate over a PEC plane, (e), (f) an open Fabry-Perot cavity cover with a planar dielectric substrate. The panels (a),(c) and (e) demonstrate distributions of the excitation field, the external (elementary) source is shown by arrows. The panels (b),(d) and (f) demonstrate function Gxx(r, rq) obtained as the electric field radiated by an elementary source located at rq (shown by arrows).

FIG. 4 .

 4 FIG. 4. (a) Schematics of a printed capacitance (left) and inductance (right), w = 0.25 mm (r ef f = w/4, see Ref. [28]). (b) Retrieved load-impedance densities (only imaginary parts are shown) of capacitively-and inductively-loaded wires at 5 GHz. (c) 3D schematic of a semi-cylindrical sparse metasurfaces of 100 mm radius and incorporating 29 wires. The inset figure shows a 2D schematic. (d)-(f) Engineered radiation patterns :two-dimensional directivity vs. angle. Predictions of the theoretical model (solid red curves) are compared to the results of 3D full-wave simulations of three different designs of sparse metasurfaces (dashed blue curves). The used substrate is F4BM220 of 0.5 mm thickness. Operating frequency is 5 GHz, λ0 ≈ 60 mm.

Figures 4 (

 4 Figures 4(d)-(f) demonstrate three different configurations of the far-field corresponding to three different designs of semi-cylindrical sparse metasurfaces. The directivity plotted in Figs. 4(d)-(f) is the two-dimensional

FIG. 5 .

 5 FIG. 5. (a), (d) 3D schematics of cavity-excited flat sparse metasurfaces having 10 (a) and 20 (d) wires. The inset figures show 2D schematics of the corresponding configurations. (b),(c),(e),(f) Two-dimensional directivity vs. angle of engineered radiation patterns corresponding to analytical and 3D simulation results of four different designs of sparse metasurfaces. Predictions of the theoretical model (solid red curves) are compared to the results of 3D full-wave simulations of three different designs of sparse metasurfaces (dashed blue curves). The panels (a)-(c) and (d)-(f) correspond, respectively, to the designs incorporating 10 wires (the inter-wire distance is λ0/2) and 20 wires (the inter-wire distance is λ0/4). The used substrate is F4BM220 of 1.5 mm thickness, the cavity height is λ0 √ n/2 ≈ 33.54 mm, where n is the cavity length (5λ0) divided over the wavelength λ0. Operating frequency is 10 GHz, λ0 ≈ 30 mm.

  ϕ) from Eq. (3) calculated in the far-field region (r → ∞). The 3D simulation results (dashed blue curves) are compared to the corresponding far-field patterns predicted by the analytical model in Eq. (3) (solid red curves), where an almost perfect agreement is observed. It proves a high accuracy of the established design procedure based on the analytical model represented by Eqs. (3) and (4) on one hand and the LPA on the other hand.

  Figures 5(b),(c) and (e),(f) demonstrate engineered radiation patterns obtained by means of the developed analytical model for 10-and 20-element sparse metasurfaces, respectively. The verification is performed via 3D full-wave simulations (blue dashed curves) where an almost perfect agreement with the analytical curves (red solid) is observed. The comparison of Figs. 5(b) and (e) and Figs. 5(c) and (f) represents an important result: two times more wires lead to a maximum of only 2 dBi improvement of the directivity. Therefore, one should carefully choose the number of wires as comparable performances can be achieved with a lesser effort, which can

FIG. 6 .

 6 FIG. 6. (a) A photography of fabricated sparse metasurfaces on a thin flexible substrate. (b), (c) Photographies of an assembled experimental prototype: (b) the front side and (c) the back side. (d) A schematic of the side view of a PCB of the excitation source. (e), (f) Schematics of the top face (e) and the inner face (f) of the PCB.The schematics contain all the information necessary to design the source used to excite the samples in the experiment. The design is performed following Ref.[START_REF] Katehi | On the modeling of electromagnetically coupled microstrip antennas-the printed strip dipole[END_REF].

FIG. 7 .

 7 FIG. 7. (a) Schematic illustration of the experimental setup used to measure the far-field patterns. (b)-(d) Measured far-field radiation patterns set up by the three fabricated samples. (e) Measured far-field radiation pattern created by the source. Operating frequency is 5 GHz.

FIG. 9 .

 9 FIG. 9. Principal geometry of a 3D full-wave simulation model which is used to calculate the reflection coefficient S11 from an array of structured wires.

FIG. 10 .

 10 FIG. 10. A flowchart of the optimization-aided design procedure of a sparse metasurface.

FIG. 11 .

 11 FIG.11. A block diagram of a particle swarm optimization algorithm employed in this work to achieve a desirable radiation pattern.

  0 r ef f )/4.Appendix C: Optimization-aided design procedure

  Element type B (mm) A (mm) Element type B (mm) A (mm) Element type B (mm) A (mm)

		Sample 1		Sample 2		Sample 3
	1	capacitor	10	3,31	capacitor	10	2,58	capacitor	10	1,39
	2	capacitor	10	4,78	capacitor	10	5,68	capacitor	5	1,04
	3	capacitor	10	1,58	capacitor	10	4,46	capacitor	10	3,47
	4	inductor	10	5,45	capacitor	5	1,06	capacitor	5	1,30
	5	capacitor	10	4,81	capacitor	5	1,86	capacitor	10	0,53
	6	capacitor	5	0,88	capacitor	10	4,46	capacitor	10	4,72
	7	capacitor	5	1,99	capacitor	10	2,32	inductor	5	7,29
	8	capacitor	10	4,73	capacitor	10	4,73	capacitor	5	1,53
	9	capacitor	10	2,41	inductor	10	5,62	capacitor	5	1,18
	10	capacitor	10	1,16	capacitor	5	0,45	capacitor	10	0,56
	11	capacitor	10	4,31	capacitor	10	7,64	capacitor	10	1,33
	12	capacitor	10	3,30	capacitor	10	3,85	capacitor	5	1,07
	13	capacitor	10	0,60	capacitor	5	1,29	capacitor	5	1,04
	14	capacitor	10	3,52	capacitor	10	2,78	capacitor	10	6,89
	15	capacitor	10	3,42	capacitor	10	4,72	capacitor	5	0,97
	16	capacitor	10	3,52	capacitor	10	1,64	capacitor	10	6,89
	17	capacitor	10	0,60	inductor	10	4,20	capacitor	5	1,04
	18	capacitor	10	3,30	capacitor	10	0,84	capacitor	5	1,07
	19	capacitor	10	4,31	capacitor	10	0,74	capacitor	10	1,33
	20	capacitor	10	1,16	capacitor	10	3,57	capacitor	10	0,56
	21	capacitor	10	2,41	capacitor	5	0,82	capacitor	5	1,18
	22	capacitor	10	4,73	inductor	10	3,00	capacitor	5	1,53
	23	capacitor	5	1,99	capacitor	10	2,25	inductor	5	7,29
	24	capacitor	5	0,88	capacitor	10	3,34	capacitor	10	4,72
	25	capacitor	10	4,81	capacitor	5	1,29	capacitor	10	0,53
	26	inductor	10	5,45	capacitor	10	1,41	capacitor	5	1,30
	27	capacitor	10	1,58	capacitor	10	0,65	capacitor	10	3,47
	28	capacitor	10	4,78	capacitor	10	0,35	capacitor	5	1,04
	29	capacitor	10	3,31	capacitor	5	1,49	capacitor	10	1,39

(ext)x

ACKNOWLEDGMENT

The authors would like to thank Dr. Badreddine Ratni (Univ Paris Nanterre) in preparing the experimental measurement setups.

Appendix A: Numerical calculation of a Green's function

After establishing the geometry of a sparse metasurface (flat, cylindrical, or any other shape), excitation type and positions of N wires, one calculates a Green's function G xx (r, ϕ; r q ) and the background field E

Appendix B: Local periodic approximation

This Section provides details on the local periodic approximation developed in Ref. [START_REF] Popov | Designing metagratings via local periodic approximation: From microwaves to infrared[END_REF] for planar sparse metasurfaces in the reflection configuration. According to the LPA, in order to find the load-impedance density, a structured wire from a nonuniform array is placed in Table I provides geometrical parameters of elements constituting the fabricated experimental samples.


Giulia Marchello
email: giulia.marchello@inria.fr

Marco Corneli

Charles Bouveyron

The dynamic latent block model for the co-clustering of evolving binary matrices

Keywords: Co-clustering, matrices binaires dynamiques, modèle à blocs latents, algorithme SEM-Gibbs Co-clustering, dynamic binary matrices, latent block model, SEM-Gibbs algorithm

Nous considérons le problème du co-clustering des matrices binaires qui peuvent évoluer dans le temps et nous introduisons un modèle génératif pour le gérer. Le modèle proposé, appelé dynamic latent block model, étend le modèle des blocs latents binaire classique au cas dynamique. La modélisation de la dynamique en temps continu repose sur un processus de Poisson non homogène, avec une partition latente des intervalles de temps. Nous proposons d'utiliser l'algorithme SEM-Gibbs pour l'inférence du modèle.

Mots-clés.

1I n t r o d u c t i o n

In many applications, it is now frequent to have to summarize large binary matrices that may evolve along the time. For instance, e-commerce systems record in continuous time all purchases of products made by customers. It is of interest for those companies to cluster both customers and products to better understand the purchasing behaviors. The simultaneous clustering of rows and columns of a matrices is known as a co-clustering problem.

The Dynamic Latent Block Model introduced in the present work is a new and original way to consider, in a dynamic framework, the Latent Block Model [see e.g. [START_REF] Brault | Co-clustering through latent bloc model: A review[END_REF][START_REF] Keribin | Model selection for the binary latent block model[END_REF]] which is ap o p u l a rg e n e r a t i v em o d e lf o rc o -c l u s t e r i n g . W ea s s u m et h a tt h en u m b e ro fi n d i v i d u a l s 1 (rows) i =1 ,...,n and the number of objects (columns) j =1 ,...,p is fixed during the whole time period taken into account [0,T]. Moreover, we consider that the interaction occurring between the individual i and the object j at time u 2 [0,T], is represented by the triplet (i, j, u) with u  T ,a n di ti sd e n o t e db yx ij (u). Thus, we assume that the number of interactions between individuals and objects follows a non-homogenous Poisson process (NHPP) where the intensity function λ(t) only depends on the clusters they belong to [START_REF] Corneli | Block modelling in dynamic networks with non-homogeneous poisson processes and exact icl[END_REF].

In order to model the continuous time we consider the approach that has been used in Corneli et al. (2018) [START_REF] Corneli | The dynamic stochastic topic block model for dynamic networks with textual edges[END_REF]. In that paper the continuous time is handled by a time partition over [0,T] where the interactions are aggregated on the time intervals of such partition obtaining a sequence of static matrices that allows us to identify the time clusters. Therefore, the basic idea is to partition the time in small windows and to cluster every time period, in this way it is possible to assign every time interval to a time cluster. Intuitively, one time cluster can occur more than once in the temporal line when its peculiar features are repeated after some time. This behaviour, for instance, is typical when the phenomenon of seasonality is observed.

2T h e d y n a m i c l a t e n t b l o c k m o d e l

We consider a random matrix X(t) where the entry X ij (t), with i =1 ,...,n, j =1 ,...,p and t 2 [0,T],c o u n t st h en u m be ro fi n t e r a c t i o n sbe t w e e ni and j up to time t.

The latent structure of rows and columns of the matrix x(t) is identified by:

• z =(z ik ; i =1,...,n; k =1,...,K) : represents the clustering of rows into K groups: A 1 ,...,A K .R o wi belongs to cluster A k iff z ik =1and z i =(z ik ) k 2 (0 , 1 ) K is the group idicator of row i;

• w =( w j`; j =1 ,...,p; `=1 ,...,L):r e p r e s e n t st h ec l u s t e r i n go fc o l u m n si n t oL groups: B 1 ,...,B L . Coloumn j belongs to cluster B `iff w j`= 1and w j =(w j `)`2 (0 , 1 ) L is the group idicator of coloumn j;

Moreover, z and w are indipendent and they are distributed as follows:

p(z|γ)= K Y k=1 γ |A k | k (1)
where:

γ k =P{z i = k}; K P k=1 γ k =1and |A k | represents the number of rows in the cluster A k . p(w|⇢)= L Y `=1 ⇢ |B `| ` (2)
2 where:

⇢ `=P{w j = `}; L P `=1
⇢ `=1and |B `| represent the number of coloums in the cluster B `.

As mentioned previously, a non-homogeneous Poisson process (NHPP) is used to count the interactions between the row i and the column j up to time t 2 [0,T],d e n o t e db y X ij (t):

X ij (t)|z ik w j`= 1∼P 0 @ t Z 0 λ k`( u)du 1 A (3)
where λ k`( t) represents the intensity function that only depends on the considered row cluster k and on the column cluster `. Moreover, λ k`( t) has to be positive and integrable on the time interval [0,T].

As in Corneli (2016) [START_REF] Corneli | Block modelling in dynamic networks with non-homogeneous poisson processes and exact icl[END_REF], we discretize the continous time interval [0,T] in U subintervals, where

I u =[t u-1 -t u [. , with: 0=t 0 <t 1 < •••<t U = T.
The number of interactions between i and j on the considered time partition I u is summarized by X iju and is defined as:

X iju := X ij (t u ) -X ij (t u-1 ), 8(i, j, u).
We intro duce a tensor X = {X iju } iju with dimensionality N ⇥ P ⇥ U .A sp r e v i o u s l y described in a more theoretical way, each time interval I 1 ,...,I U is assigned to a hidden time cluster C 1 ,..,C c .T om o d e lt h em e m b e r s h i pt ot i m ec l u s t e r s ,an e wl a t e n tv a r i a b l es has to be introduced, in particular s u = c if and only if the time interval I u belongs to the time cluster C c .F u r t h e r m o r e ,w ea s s u m et h a ts follows a multinomial distribution:

p(s|δ)= C Y c=1 δ |Cc| c , (4)
where:

δ c =P{s u = c}; C P c=1 δ c =1and |C c |
X iju |z ik w j`Suc =1⇠P(λ k`c ∆ u ) (5)
3 132 sciencesconf.org:jds2020:319968

%

where ∆ u indicates the length of the interval I u that is usually constant, ∆ u = ∆. Moreover, as pointed out in Corneli et al. (2018) [START_REF] Corneli | The dynamic stochastic topic block model for dynamic networks with textual edges[END_REF], one can set ∆ u =1without loss of generality, indeed if, for instance, we have a dataset divided in week, the time interval ∆ u could be set to one week. It is finally possible to introduce a tensor

⇤ = {λ k`c } k`c of dimension K ⇥ L ⇥ C.

3T h e l i k e l i h o o d

Since the increments of a Poisson process are independent it holds that:

p(X iju |z ik w jl s uc =1,λ k`) = U Y u=1 (λ klc ) X iju X iju ! exp (-λ klc ) ! (6)
Therefore, we can introduce the K ⇥ L ⇥ C tensor of order 3, λ,i d e n t i fi e db yt h e triplet (i, j, u) and whose elements are denoted as ⇡ k`c . At this point it is possible to write the complete data likelihood of the model, identified by the following equation:

p(X, z, w, s|γ, ⇢, δ, λ)=p(z|γ) • p(w|⇢) • p(s|δ) • p(X|z, w, s, λ) (7)
Looking at the right hand side of the Eq. ( 7) we notice that p(z|γ), p(w|⇢) and p(s|δ) have been defined in the previous section respectively by the Eqs. ( 1), ( 2) and (4). While the joint distribution of X,g i v e nz, w,a n ds,c a nb ee a s i l yo b t a i n e da sag e n e r a l i z a t i o n of the Eq. ( 6):

p(X|z, w, s, λ)= K Y k=1 L Y `=1 C Y c=1 ✓ (λ k`c ) R k`c P k`c exp (-|A k ||B `||C c |λ k`c ) ◆ (8)
where:

R k`c = n X i=1 p X j=1 U X u=1 z ik w j`suc X iju P k`c = n Y i=1 p Y j=1 U Y u=1 (z ik w j`suc X iju )!
Denoted as ✓ the set of the model parameters: ✓ =( γ, ⇢, δ, λ). The complete data log-likelihood can be defined as

`(✓|z, w, S, X)= X z X w X c log p(X, z, w, S|✓) (9)
4M o d e l i n f e r e n c e

As usual, we look for a way to maximize the log-likelihood in order to obtain the estimation of θ. In the co-clustering case, the EM algorithm is computationally infeasible [START_REF] Govaert | Co-clustering: models, algorithms and applications[END_REF], the idea is to use a stochastic version of it, known as SEM-Gibbs, proposed by Keribin C. (2010) [START_REF] Keribin | Estimation d'un modèle à blocs latents par l'algorithme sem[END_REF] and exploited, for instance, by Bouveyron et al. ( 2017) [START_REF] Bouveyron | The functional latent block model for the co-clustering of electricity consumption curves[END_REF] in the Functional Latent Block Model . Thanks to the Gibbs Sampling, in the SE step a partition for z, w and s is generated without computing the joint distribution. The algorithm starts with initial values for the parameter set θ (0) ,t h ec o l u m nc l u s t e r sw (0) and the time clusters s (0) .R e g a r d i n gt h eb u r n -i np e r i o d ,a f t e rac e r t a i nn u m b e ro fi t e r a t i o n so ft h ea l g o r i t h m , we can obtain the final parameters estimation by computing the mean of the sampled distribution. The optimal values for z, w and s are estimated by the mode of their sample distributions.

5C o n c l u s i o n a n d f u r t h e r w o r k

We intro duced in this short pap er a generative mo del for co-clustering dynamic binary matrices. We proposed to use the SEM-Gibbs algorithm for model inference. Regarding further work, we are currently working on an implementation of this algorithm and we plan to apply it on real-world data from Amazon recording fine food purchases over 8 years.

  represents the number of time intervals in the cluster C c . The originality of this work lies in the fact that a dependence is assumed between the two variables: X iju and S u .Once these additional assumpions have been made, we can rewrite Eq; (3) considering that the intensity functions are stepwise constant on each time cluster C c .H e n c e ,n o ww e are considering a non-homogeneus Poisson process (NHPP) with parmeter λ k`c and ∆ u :

sciencesconf.org:jds2020:319968

sciencesconf.org:jds2020:319968


Loïc Toraille

Antoine Hilberer

Thomas Plisson

Margarita Lesik

Mayeul Chipaux

Baptiste Vindolet

Charles Pépin

Florent Occelli

Martin Schmidt

Thierry Debuisschert

Nicolas Guignot

Jean-Paul Itié

Paul Loubeyre

Jean-François Roch
email: jean-francois.roch@ens-paris-saclay.fr

Combined synchrotron X-ray diffraction and NV diamond magnetic microscopy measurements at high pressures

Keywords: diamond, NV center, diamond anvil cell, magnetometry, synchrotron X-ray diffraction, iron

We report the possibility to simultaneously perform wide-field nitrogenvacancy (NV) diamond magnetic microscopy and synchrotron X-ray diffraction (XRD) measurements at high pressure. NV color centers are created on the culet of a diamond anvil which is integrated in a diamond anvil cell for static compression of the sample. The optically detected spin resonance of the NV centers is used to map the stray magnetic field produced by the sample magnetization. Using this combined scheme, the magnetic and structural behaviors can be simultaneously measured. As a proofof-principle, we record the correlated α-Fe to ε-Fe structural and magnetic transitions of iron that occur here between 15 and 20 GPa at 300 K.

Pressure can be used as a tuning parameter to disclose relations between magnetic, structural and electronic properties of a solid-state system. Much detailed measurements with far reaching applications have been made thanks to the development of the diamond anvil cell (DAC) [START_REF] Mao | Solids, liquids, and gases under high pressure[END_REF] which is an efficient and practical tool to generate hydrostatic static compression up to extreme pressures in the megabar range. The development of high-pressure physics is also tightly linked to the improvement of synchrotron radiation sources [START_REF] Shen | High-pressure studies with x-rays using diamond anvil cells[END_REF] that now feature focal spots in the micrometer range along with high brightness in the 20-40 keV range. Due to the transparency of diamond to X-rays, a broad panel of X-ray synchrotron-based methods can be applied for high-pressure studies as in situ probes of the materials inside the DAC. As an example, complex magnetic ordering can be resolved using X-ray magnetic circular dichroism (XMCD) which provides element-specific probing of magnetic moments [START_REF] Itié | High Pressure XAS, XMCD and IXS[END_REF]. Other magnetic measurement techniques have been developed at high pressure but remain challenging due to the constraints of their implementation in the DAC [START_REF] Eremets | High-pressure experimental methods[END_REF]. We report here that wide-field nitrogen-vacancy (NV) diamond magnetic microscopy based on the imaging of an ensemble of NV centers [START_REF] Levine | Principles and techniques of the quantum diamond microscope[END_REF] can be implemented on a synchrotron X-ray diffraction (XRD) beamline so as to address both the unit cell structure and volume of the compressed sample inside the DAC. The combination of these two diagnostics allowed us to measure the magnetic properties of the sample and to simultaneously determine its cristallographic structure by recording the XRD pattern. The technique uses a specifically engineered diamond anvil and a compact optical setup which is slid on and off the XRD bench.

The negatively-charged NV -center is a point defect in diamond which can be engineered as an atomic-like quantum system [START_REF] Lühmann | Screening and engineering of colour centres in diamond[END_REF]. It behaves as a magnetic quantum sensor due to the existence of an electron spin S = 1 associated with a bright photoluminescence (PL) [START_REF] Doherty | The nitrogen-vacancy colour centre in diamond[END_REF]. In absence of any magnetic field, this spin structure introduces discrete energy states inside the diamond bandgap. These states have different PL intensities due to their interaction with additional metastable levels. While optically pumping the NV centers with green laser light and simultaneously driving the electronic spin with microwave (MW) tuned at the resonance frequency of 2.87 GHz between the m S = 0 (high-PL) and m S = ±1 (low-PL) states, the optically detected magnetic resonance (ODMR) signal appears as a decrease of the PL intensity. When a magnetic field is applied to the NV center, the Zeeman effect removes the degeneracy between the m S = ±1 states and the resonance ODMR signal is thus split in two components. The difference between these two resonance frequencies gives access to the projection of the magnetic field on the quantization axis which is defined by the N-to-V direction of the NV center [START_REF] Taylor | High-sensitivity diamond magnetometer with nanoscale resolution[END_REF]. When collecting the PL from an ensemble of NV centers, the NV-ODMR spectrum consists in the cumulative response of many defects. Since single-crystal diamond only allows four different cristallographic orientations a, b, c or d for the NV centers, the signal associated to an ensemble of NV centers is the superposition of the PL emitted by these four families of NV centers [START_REF] Lai | Influence of a static magnetic field on the photoluminescence of an ensemble of nitrogen-vacancy color centers in a diamond single-crystal[END_REF]. The NV-ODMR signal of each family is then associated with a given projection of the magnetic field A c c e p t e d M a n u s c r i p t vector in the 111 set of equivalent symmetry directions. The three vector components of the magnetic field can then be retrieved from these four correlated measurements [START_REF] Chipaux | Magnetic imaging with an ensemble of nitrogen-vacancy centers in diamond[END_REF]. This technique can be extended to a layer of NV centers overlapping a magnetic structure in order to map the full vector magnetic field created by the sample magnetization.

The quantitative mapping of the magnetic field can then be modeled to determine the magnetization of micrometer sized magnetic structures [START_REF] Toraille | Optical magnetometry of single biocompatible micromagnets for quantitative magnetogenetic and magnetomechanical assays[END_REF][START_REF] Mccoey | Rapid, high-resolution magnetic microscopy of single magnetic microbeads[END_REF]. Ensemble-NV sensors have been applied in a number of applications, including the detection of the magnetic properties of samples in a diamond anvil cell at pressures up to 30 GPa [START_REF] Hsieh | Imaging stress and magnetism at high pressures using a nanoscale quantum sensor[END_REF][START_REF] Lesik | Magnetic measurements on micrometer-sized samples under high pressure using designed NV centers[END_REF][START_REF] Yip | Measuring magnetic field texture in correlated electron systems under extreme conditions[END_REF].

Elaborating on our work on NV magnetic measurement in a DAC relying on the activation of a layer of NV centers created in the culet of a diamond anvil [START_REF] Lesik | Magnetic measurements on micrometer-sized samples under high pressure using designed NV centers[END_REF], we built a compact and portable NV diamond magnetic microscope that ensured its compatibility with a XRD measurement on a synchrotron beamline (figure 1.a). This optical microscope, which is used to detect the magnetic field created by the sample, is constructed so that it can be slid on and off in front of the DAC which is settled on the XRD bench. The experiment was then performed at the PSICH É beamline of the SOLEIL synchrotron facility (Gif-sur-Yvette, France). Powder XRD patterns were collected using a monochromatic X-ray beam at 0.3738 Å wavelength and a Pilatus CdTe image plate detector. During the experiment performed at room temperature, we alternatively aligned the NV magnetometer with the DAC and removed it to free the way towards the X-ray detector used to record the XRD pattern. In the conditions of the experiment, we did not observe any modification of the NV centers luminescence properties that could be induced by the exposure to the X-ray beam, such as the potential photoionization of the negatively charged state NV -into the neutral charged state NV 0 [START_REF] Aslam | Photo-induced ionization dynamics of the nitrogen vacancy defect in diamond investigated by single-shot charge state detection[END_REF].

To demonstrate the complementarity between XRD and NV diamond magnetic microscopy methods we chose the α-Fe ↔ ε-Fe phase transition of iron [START_REF] Bassett | Mechanism of the body-centered cubic-hexagonal close-packed phase transition in iron[END_REF] as a testbed example. The structural transition between the body-centered-cubic (bcc) α phase and the hexagonal close-packed (hcp) ε phase is associated with a magnetic transition between the ferromagnetic state of α-Fe and the mainly non-magnetic state of ε-Fe. Both transitions have been extensively studied due to their implications in iron-based materials and their relation with magnetism and Earth sciences [START_REF] Mathon | Dynamics of the magnetic and structural α-phase transition in iron[END_REF][START_REF] Baudelet | Xray absorption spectroscopy and x-ray magnetic circular dichroism simultaneous measurements under high pressure: The iron bcc-hcp transition case[END_REF][START_REF] Monza | Iron under pressure: "Kohn tweezers" and remnant magnetism[END_REF][START_REF] Dewaele | Mechanism of the α -phase transformation in iron[END_REF][START_REF] Wei | High-pressure phase transition of iron: A combined magnetic remanence and Mössbauer study[END_REF][START_REF] Lebert | Epsilon iron as a spin-smectic state[END_REF].

As shown in figure 1.b, the DAC is made of non-magnetic material and a rhenium gasket is used to make the sample chamber. At the time of the experimental run, the spot size of the X-ray beam available at the PSICH É beamline had a diameter of about 80 μm. In order to maximize the XRD signal we then had to use a sample consisting of several iron beads of 99.5% purity which were packed together. As a consequence, the inhomogeneous magnetization of those irregularly packed samples of iron created a complex magnetic field distribution that could not be easily inverted to determine the magnetization texture [START_REF] Vervelidou | On the accuracy of palaeopole estimations from magnetic field measurements[END_REF]. Note that this experimental feature is a significant difference with our previous results of Ref. [START_REF] Lesik | Magnetic measurements on micrometer-sized samples under high pressure using designed NV centers[END_REF] where we measured the magnetic field created by an iron bead of 5 μm size. The magnetic response of this single iron bead could then A c c e p t e d M a n u s c r i p t be modeled as a simple magnetic dipole, leading to a quantitative measurement of the iron magnetization decrease along the α-Fe to ε-Fe transition.

The iron beads were embedded in NH 3 BH 3 for pressure transmission in the DAC. A gold bead of about 5 μm diameter was added in the DAC chamber to serve as a pressure gauge using the calibrated equation of state of gold [START_REF] Takemura | Isothermal equation of state for gold with a He-pressure medium[END_REF] to fit the gold unit cell volume. Although the NH 3 BH 3 salt induced a distortion of the samples optical microscopy images that could be recorded through the diamond anvil cell, it had no influence on the quality of the PL images. Moreover some properties of this solid-state pressure transmitting medium were useful: It is a weak scatterer with a large unit cell and, as shown below, it is not perturbing the iron XRD pattern. It also enables to minimize the gasket deformation which helps to preserve the integrity of the slit made in the metallic gasket for the propagation of the MW excitation to the NV ensemble [START_REF] Lesik | Magnetic measurements on micrometer-sized samples under high pressure using designed NV centers[END_REF]. A bias magnetic field of approximately 9 mT was applied using a permanent magnet oriented close to the DAC. This external magnetic field has two features: it magnetizes the iron samples and it creates a significant splitting of the eight NV-ODMR resonances associated to the four 111 axes of the NV center [START_REF] Toraille | Optical magnetometry of single biocompatible micromagnets for quantitative magnetogenetic and magnetomechanical assays[END_REF]. Using a membrane actuated DAC, we progressively increased the pressure up to 32.7 GPa. At each pressure step we successively recorded the XRD pattern and the NV-ODMR optical images. Figure 2 shows an example of results obtained at a pressure of 12.2 GPa and 23.5 GPa with the NV diamond magnetic microscope. Each pixel of the camera recorded the luminescence of an optically pumped ensemble of NV centers. By sweeping the MW frequency applied to the NV centers we then measured the full ODMR spectrum for each pixel of the image (see figure 2.a). Compared to the results of Ref. [START_REF] Lesik | Magnetic measurements on micrometer-sized samples under high pressure using designed NV centers[END_REF] significant noise of still unclear origin led to a smaller ODMR contrast which interfered with the processing of the data taken during this test run. To insure the validity of our results, we decided to only select a single set of resonance peaks associated to the orientation c of the NV centers which kept the highest ODMR contrast over the sample area and throughout the whole experiment run. Although we performed a least-square minimizing multi-Lorentzian fit of the full NV-ODMR spectrum for added robustness, we only retrieved the data which are associated with the splitting value ∆ C of the selected NV center family. We then reconstructed maps of this single splitting in the vicinity of the sample as shown in figure 2.b. Clear local variations of ∆ C can be observed with an increase close to the iron sample that reveals the α-Fe magnetization.

The XRD pattern acquired for each pressure step can be used to follow the α-Fe ↔ ε-Fe structural transition. Selected diffractograms, taken before, during, at the end and after this transition are shown in figure 3. Using the Fe and Au equations of state, we can assign each sharp Bragg peak to a specific crystallographic structure of each element. The diffraction peaks of α-Fe, ε-Fe and Au are clearly identified. Two weak peaks associated to NH 3 BH 3 are also observed but due to its large unit cell the most intense peaks of this compound appear at low diffraction angles, hence outside the region of interest for the transition in iron. The first fraction of the ε phase appears at 13.1 GPa while the last fraction of the α phase disappears at 20.0 GPa. At the end of A c c e p t e d M a n u s c r i p t the experiment run, the pressure was brought back down to its ambient value. We then observed the inverse transition with a hysteresis behavior; the α phase started to appear again at 12.9 GPa and the ε phase completely disappeared at 6 GPa. These values are consistent with those reported in the literature [START_REF] Monza | Iron under pressure: "Kohn tweezers" and remnant magnetism[END_REF][START_REF] Dewaele | Mechanism of the α -phase transformation in iron[END_REF]. Finally no chemical reaction between the iron sample and the NH 3 BH 3 salt could be detected.

Complementary to the XRD measurement, we analyzed the maps of the splitting value ∆ C from the NV-ODMR spectrum which were determined for each pressure step. As previously described [START_REF] Barson | Nanomechanical sensing using spins in diamond[END_REF][START_REF] Broadway | Microscopic imaging of the stress tensor in diamond Using in situ quantum sensors[END_REF], hydrostatic pressure induces a shift of the center frequency of the spin resonance while non-hydrostatic stress components quadratically add to the frequency splitting induced by the magnetic field. Therefore the analysis of the NV-ODMR spectrum under pressure requires to discriminate between the influence of stress on the NV center spin states and their response to the applied magnetic field. In order to retrieve the relevant splitting associated to the magnetic field created by the iron magnetization, we used reference values of the splitting at a distance of 15 μm from the sample. On such length scale the external magnetic field can be considered as uniform, and stress induced variations measured were below 0.2 GPa (2 MHz ODMR frequency shift at most). Moving this reference zone, highlighted as a red square in figure 4.a, in the vicinity of the sample changed the magnetic field estimation results only by a factor smaller than other error sources. This procedure allowed us to remove both the contribution of the non-hydrostatic components of the stress tensor and the contribution of the applied magnetic field [START_REF] Lesik | Magnetic measurements on micrometer-sized samples under high pressure using designed NV centers[END_REF]. We could then extract the projection along the chosen c axis of the magnetic field that is only created by the magnetization of the samples located within the studied area of the pressure chamber (figure 4.a). Since the vector magnetization of the iron samples remains aligned along the applied magnetic field, the projection along the fixed c axis is a good estimator of the amplitude of the magnetic field that is created by the samples. At low pressure below 10 GPa a magnetic field of a few hundreds of μT appeared above the iron samples due to the α-Fe magnetization. As the pressure increased, this magnetic field decreased and finally disappeared. The data show that the magnetic transition starts between 12.2 GPa and 13.8 GPa, and ends at a pressure slightly above 20.0 GPa. Figure 4.b gives a comparison between the magnetic transition inferred from the NV-ODMR signal and the structural transition determined by the XRD pattern. From these results, we confirm that both transitions occur within a similar pressure range.

To conclude, this experiment demonstrates that NV diamond magnetic microscopy is a robust magnetic measurement technique which can be operated on a synchrotron beamline with the intrinsic constraints of this environment [START_REF] Fu | Sensitive magnetometry in challenging environments[END_REF]. As a proof-of-principle experiment, we followed simultaneously the evolution of the magnetization of an iron sample along the α-Fe to ε-Fe transition with the XRD pattern associated to these two phases. This combined set-up can be easily adapted depending on the investigated samples, for instance by optimizing the focusing of the X-ray beam on the sample inside the DAC. Our work demonstrates that NV diamond magnetic microscopy is well suited to complement fourth-generation synchrotrons whose improved brightness and A c c e p t e d M a n u s c r i p t focusing will enable the observation of the dynamic, electronic and structural properties of materials with upcoming spatial resolution of about 100 nm. Almax-Boehler anvils with 300 μm-diameter culets. A layer of NV centers was created on the tip of one of the anvils using nitrogen ion implantation. In the wide-field NV diamond magnetic microscope, a microscope objective (Obj, Nikon ×40 and numerical aperture of 0.4) focuses a continuous-wave laser beam at 532 nm wavelength on the NV layer for optical pumping. The red PL emitted by the NV center layer is collected by the same objective, then spectrally filtered from the laser excitation using a dichroic mirror (DM) and a long-pass optical filter (F), and finally imaged on a CMOS camera. When the NV microscope is removed, the DAC is illuminated by the X-ray beam whose size is limited by a pinhole (PH) of 50 μm diameter to avoid parasitic diffraction from the DAC gasket. The XRD pattern is then recorded on an image plate detector (IPD). The permanent magnet located near the DAC (M) is used to induce the magnetization of the iron samples and to discriminate between the four 111 orientations of the NV centers. (b) Implementation of ODMR in the DAC chamber delimited by the tips of the two anvils and a rhenium gasket (G). A single loop of insulated copper wire shown in orange is wrapped around the anvil hosting the NV centers as the antenna (A) which provides the MW excitation for the electron spin resonance. A slit is cut in the gasket to prevent MW screening by avoiding the onset of eddy currents induced in the rhenium metallic foil. The inset shows a close-up of the DAC chamber filled with NH 3 BH 3 as the pressure transmitting medium. Micrometer-size pieces of iron (in grey) and gold (in yellow) are deposited on the tip of the NV-doped anvil. The red line shows the implanted NV layer in the culet of the anvil with a concentration of about 10 4 defects/μm 2 and located at a depth of 20 nm below the surface. We observe no angular displacement of these parasitic peaks with pressure changes. Due to the size of the X-ray beam smaller than the diameter of the hole in the DAC gasket, no parasitic peaks associated to rhenium are observed. Filled markers indicate discernible peaks, while hollow markers indicate theoretical positions for the peaks which remain hidden in the noise.

Figure 1 .

 1 Figure 1. (a)Experimental setup used to sequentially perform NV-based magnetic measurement and XRD measurement in a DAC. The DAC is composed of type IIas Almax-Boehler anvils with 300 μm-diameter culets. A layer of NV centers was created on the tip of one of the anvils using nitrogen ion implantation. In the wide-field NV diamond magnetic microscope, a microscope objective (Obj, Nikon ×40 and numerical aperture of 0.4) focuses a continuous-wave laser beam at 532 nm wavelength on the NV layer for optical pumping. The red PL emitted by the NV center layer is collected by the same objective, then spectrally filtered from the laser excitation using a dichroic mirror (DM) and a long-pass optical filter (F), and finally imaged on a CMOS camera. When the NV microscope is removed, the DAC is illuminated by the X-ray beam whose size is limited by a pinhole (PH) of 50 μm diameter to avoid parasitic diffraction from the DAC gasket. The XRD pattern is then recorded on an image plate detector (IPD). The permanent magnet located near the DAC (M) is used to induce the magnetization of the iron samples and to discriminate between the four 111 orientations of the NV centers. (b) Implementation of ODMR in the DAC chamber delimited by the tips of the two anvils and a rhenium gasket (G). A single loop of insulated copper wire shown in orange is wrapped around the anvil hosting the NV centers as the antenna (A) which provides the MW excitation for the electron spin resonance. A slit is cut in the gasket to prevent MW screening by avoiding the onset of eddy currents induced in the rhenium metallic foil. The inset shows a close-up of the DAC chamber filled with NH 3 BH 3 as the pressure transmitting medium. Micrometer-size pieces of iron (in grey) and gold (in yellow) are deposited on the tip of the NV-doped anvil. The red line shows the implanted NV layer in the culet of the anvil with a concentration of about 10 4 defects/μm 2 and located at a depth of 20 nm below the surface.

Figure 2 .

 2 Figure 2. (a) Reconstruction of NV-ODMR spectra for the ensemble of NV centers corresponding to the pixel indicated by the red cross in panel (b), at pressures of P = 12.2 GPa (upper curve) and P = 23.5 GPa (lower curve). The PL intensity is normalized to its mean value taken outside the spin resonance. A bias magnetic field of around 9 mT is used to split the eight resonances associated with the four crystallographic families of NV centers, and to induce the sample magnetization. The splitting from the family exhibiting the highest ODMR contrast is ∆ C = 381.9 MHz at 12.2 GPa and decreases to ∆ C = 370.6 MHz at 23.5 GPa. The general shift of the NV-ODMR spectrum is due to the pressure increase. The processed data is shown in black while the red and blue solid lines are numerical fits using a set of Lorentzian functions. For increased robustness, only the data associated with the fitting of the family c is considered for the rest of the analysis. (b) Corresponding maps of ∆ C evaluated above the iron samples.

Figure 3 .

 3 Figure 3. (a) X-ray diffractograms recorded for increasing pressure steps. The data are displayed with a vertical offset for clarity. At 6.3 GPa, only Au and α-Fe diffraction peaks appear. At 13.8 GPa, the intensities of the peaks corresponding to α-Fe have decreased while ε-Fe peaks can be clearly observed. Disappearance of the last traces of α-Fe is observed between 19.0 and 20.0 GPa. From 20.0 up to 26.9 GPa, the diffraction signal only shows the presence of Au, NH 3 BH 3 and ε-Fe. The intensity of the Au diffraction peaks is evolving as the relative positions of the Au bead and the Fe sample change under the pressure induced deformation of the gasket. The diffraction peaks indicated by the stars are associated with the NH 3 BH 3 salt. The most intense NH 3 BH 3 diffraction peaks are found below 8 • . Additional peaks overlaid with white squares are caused by the diffraction of the permanent magnet used to split the spin resonance of the four NV families and to induce the magnetization of the iron sample. We observe no angular displacement of these parasitic peaks with pressure changes. Due to the size of the X-ray beam smaller than the diameter of the hole in the DAC gasket, no parasitic peaks associated to rhenium are observed. Filled markers indicate discernible peaks, while hollow markers indicate theoretical positions for the peaks which remain hidden in the noise.

Figure 4 .

 4 Figure 4. (a) Color maps of the magnetic field projection B sample,C of the magnetic field created by the iron magnetization along the c axis of the selected NV family. As the pressure increases, the local positive and negative variations of B sample,C progressively disappear, until the sample no longer exhibits any magnetic properties. Red squares (7 × 7 pixels) indicate the chosen reference area to extract sample magnetization effects from pressure and bias field ones. Black squares (11 × 11 pixels) indicate the area over the sample where signal is averaged for figure 2b. (b) Measurement of B sample,C averaged over the black square shown in panel (a) as the pressure in the DAC chamber increases. A greyscale pattern is overlaid to indicate the start and the end of the α-Fe to ε-Fe structural transition independently determined using the XRD pattern. The evolution of the magnetic field created by the iron magnetization is consistent with the XRD data.

Acknowledgments

We acknowledge the synchrotron SOLEIL for provision of beamtime under Proposal No 20181196. The experiment was completed on the PSICH É beamline. This project received funding from the European Union's research and innovation programme (H2020-FETFLAG-2018-2020) under grant agreement No 820394 (ASTERIQS), the Agence Nationale de la Recherche (project SADAHPT), the Investissements d'Avenir program through LabEx PALM (ANR-10-LABX-0039-PALM), and the DIM SIRTEQ of Région Île-de-France. B. Vindolet acknowledges support by a PhD research grant of Délégation Générale de l'Armement.