J Melguizo-Gavilanes
email: josue.melguizo-gavilanes@cnrs.pprime.fr

V Rodriguez

P Vidal

R Zitoun

Dynamics of detonation transmission and propagation in a curved chamber: a numerical and experimental analysis

Keywords: Detonation dynamics, Mach reflection, re-initiation, detailed chemistry, high speed visualization, numerical simulation, OpenFOAM

The dynamics of detonation transmission from a straight channel into a curved chamber was investigated numerically and experimentally as a function of initial pressure (10 kPa ≤ p 0 ≤ 26 kPa) in an argon diluted stoichiometric H 2 -O 2 mixture. Numerical simulations considered the two-dimensional reactive Euler equations with detailed chemistry; hi-speed schlieren and OH* chemiluminescense were used for flow visualization. Results show a rotating Mach detonation along the outer wall of the chamber and the highly transient sequence of events (i.e. detonation diffraction, re-initiation attempts and wave reflections) that precedes its formation. An increase in pressure, from 15 kPa to 26 kPa, expectedly resulted in detonations that are less sensitive to diffraction. The decoupling location of the reaction zone and the leading shock along the inner wall determined where transition from regular reflection to a rather complex wave structure occurred along the outer wall. This complex wave structure includes a rotating Mach detonation (stem), an

Introduction

Understanding detonation interactions with curved geometries is necessary to propulsion and industrial and nuclear safety. Regarding propulsion, effort has been invested in developing prototypes of pulse and continuously rotating detonation engines (RDE), though many questions remain unanswered about design, integrity, control, vibration, noise, etc. [START_REF] Lu | Rotating detonation wave propulsion: experimental challenges, modeling, and engine concepts[END_REF][START_REF] Wolański | Detonative propulsion[END_REF][START_REF] Kailasanath | Review of propulsion applications of detonation waves[END_REF]. Regarding safety, most of the work has been devoted to prevention and mitigation of ignition, flame acceleration and detonation hazards. In mining industries, chemical processing and nuclear power plants, fuel leaks and accidental ignitions in congested or confined spaces can have devastating consequences such as losses of lives, infrastructure destruction, and high environmental impact [START_REF] Ciccarelli | Flame acceleration and transition to detonation in ducts[END_REF].

The simplest RDE chamber is the annular space between two coaxial cylinders in which one or several detonation waves propagate azimuthally consuming the reactive mixture. Promising RDE geometries are the hollow and semi-hollow configurations in which the internal wall of the combustor is fully or partially removed, resulting in larger specific impulse, pressure and rotation frequencies, e.g., [START_REF] Hansmetzger | A study of continuous rotation modes of detonation in an annular chamber with constant or increasing section[END_REF]. Literature includes experimental, numerical and theoretical studies on detonation propagation inside annular RDEs [START_REF] Sousa | Development of a fast evaluation tool for rotating detonation combustors[END_REF][START_REF] Schwer | Progress in efficient, high-fidelity, rotating detonation engine simulations[END_REF][START_REF] Pan | Wavelet pattern and self-sustained mechanism of gaseous detonation rotating in a coaxial cylinder[END_REF], curved channels/chambers of constant and varying widths [START_REF] Rodriguez | An experimental evidence of steadily-rotating overdriven detonation[END_REF][START_REF] Rodriguez | Experimental observations of semiconfined steadily-rotating detonation[END_REF][START_REF] Katta | Effect of increasing channel width on the structure of rotating detonation wave[END_REF][START_REF] Sugiyama | Numerical investigations on detonation propagation in a two-dimensional curved channel[END_REF][START_REF] Nakayama | Front shock behavior of stable curved detonation waves in rectangular-cross-section curved channel[END_REF][START_REF] Short | Steady detonation propagation in a circular arc: a detonation shock dynamics model[END_REF][START_REF] Short | Detonation propagation in a circular arc: reactive burn modelling[END_REF][START_REF] Short | Propagation of a stable gaseous detonation in a circular arc configuration[END_REF], wedges and bends [START_REF] Edward | Detonation interaction with wedges and bends[END_REF], and U-bend tubes [START_REF] Frolov | Shock wave and detonation propagation through u-bend tubes[END_REF]. Loss-of-coolant accidents in nuclear reactors may result in significant releases of hydrogen and the generation of flammable atmospheres that accumulate between the containment and the curved reactor building [START_REF] Council | Lessons learned from the Fukushima nuclear accident for improving safety of US nuclear plants[END_REF].

In the present study, a simplified hollow geometry amenable to fundamental analysis is used, namely a straight channel connected to a curved chamber with a small center body. The wave dynamics in this configuration is presumably representative of the early stages of detonation propagation inside a hollow RDE, and of that close to a curved wall of an industrial reactor.

Recently, this geometry was experimentally studied using a stoichiometric propane-oxygen mixture [START_REF] Rodriguez | An experimental evidence of steadily-rotating overdriven detonation[END_REF][START_REF] Rodriguez | Experimental observations of semiconfined steadily-rotating detonation[END_REF]. Its primary finding is a steadily rotating overdriven detonation wave along the outer wall of the chamber, whose distinguishing features are: (i) a Mach wave; (ii) a quasi-steady rotation velocity;

(iii) an almost constant Mach stem height, and (iv) very small detonation cells oriented parallel to the wall in the highly compressed regions behind the Mach front. Low-order modeling using Geometrical Shock Dynamics (GSD) [START_REF] Jourdain | Progress in understanding rotating detonation modes[END_REF] was capable of reproducing qualitatively the main global features in the chamber, i.e, wave diffraction and reflections, and a steadily rotating Mach wave. However, quantitatively capturing the details of the complex experimental transients requires unsteady, time accurate simulations.

The main objective of the present work is thus to numerically characterize the flow dynamics inside a curved chamber as a function of initial pressure and to interpret the results based on comparisons with our experimental data.

An argon-diluted stoichiometric H 2 -O 2 mixture is considered for which the implementation of a reliable detailed chemical kinetics scheme makes computations both tractable and representative, thereby reducing uncertainties on what modeling should be used for this rich phenomenological configuration.

Experimental Methodology

Apparatus and procedure

The experimental apparatus is shown in Fig. 1, it includes a detonation tube, a curved chamber and an expansion vessel. After vacuuming, the entire system is filled with a premixed 2H 2 :O 2 :2Ar mixture at the desired initial pressure p 0 .

A deflagration is ignited at one end of the tube and accelerates to detonation by means of a Shchelkin spiral before entering the curved chamber. In all experiments, the steady Chapman-Jouguet (CJ) regime was reached. This was verified by several dynamic pressure transducers (P1-P4) that recorded the pressure evolution behind the wave front to obtain an average detonation velocity (D CJ ). The detonation tube is 2-m long and has a 51-mm inner diameter. The detonation enters the curved chamber through a 200-mm long straight section with a 30-mm × 20-mm cross-section. The detonation then diffracts in a larger volume bounded by a 80-mm radius outer wall and a small 30-mm radius inner wall. The combustion products finally expand inside a large vessel which is separated from the curved chamber by a 18-µm Mylar foil (Mylar 1). More details on the experimental apparatus and procedure are given in [START_REF] Rodriguez | An experimental evidence of steadily-rotating overdriven detonation[END_REF][START_REF] Rodriguez | Experimental observations of semiconfined steadily-rotating detonation[END_REF].

Flow visualization

The curved chamber allows for 217-mm × 275-mm optical access with one UV-silica window on one face and a borosilicate-crown glass window on the other face, both 27-mm thick. Two visualization techniques were used to investigate the dynamics: (i) chemiluminescence based on the hydroxyl radical (OH*) to resolve the overall evolution, and (ii) schlieren to focus on details of interest. Both were coupled to hi-speed cameras.

OH* visualizations were performed using an intensified Princeton PI-MAX4 camera operated in dual frame imaging mode with a OH filter centered on 311 ± 5 nm. Two pictures were recorded per shot separated by a 10µs delay. The full-frame resolution was 1024 × 1024 pixels with a 100-ns exposure time. The experimental sequences shown in Figs. 2 and3 are thus a composite of three shots per initial pressure. This visualization technique has been extensively used as a typical combustion marker [START_REF] Zhao | Experimental and numerical study of OH* chemiluminescence in hydrogen diffusion flames[END_REF][START_REF] Hidaka | Shock-tube measurement of the rate constant for excited hydroxyl (A2.SIGMA.+) formation in the hydrogen-oxygen reaction[END_REF][START_REF] Kathrotia | Study of the H+O+M reaction forming oh: Kinetics of OH chemiluminescence in hydrogen combustion systems[END_REF][START_REF] Gutman | Shock-tube study of oh chemiluminescence in the hydrogen-oxygen reaction[END_REF]. Its main limitation is that excited OH molecules have relatively long life times (compared to detonation time scales) and remain active in burnt gases. This can make the interpretation of the images difficult, especially, in the center of the chamber where different combustion processes interact, i.e., decoupled leading shock-reaction zone, and detonations.

Schlieren visualizations were performed using a mercury-vapor lamp and a Shimadzu HPV-X2 camera. Videos were recorded at 500 kHz with an exposure time of 200 ns.

Computational methodology

Governing equations and numerical methods

The flow field in the curved chamber was modeled using the reactive Euler equations with detailed chemistry and temperature dependent thermodynamics (i.e. JANAF tables to compute species/mixture heat capacities, enthalpies and entropies) [START_REF] Burcat | Third millenium ideal gas and condensed phase thermochemical database for combustion (with update from active thermochemical tables)[END_REF]. The governing equations were solved in a twodimensional planar geometry using the OpenFOAM Toolbox [START_REF] Weller | A tensorial approach to computational continuum mechanics using object-oriented techniques[END_REF]; chemical kinetics was modeled with the 9-species, 21-reactions Mével's mechanism for hydrogen-oxidation [START_REF] Mével | Hydrogen-nitrous oxide delay time: shock tube experimental study and kinetic modelling[END_REF], whose ignition delay time predictions (metric of interest in the current study) are neither the fastest nor the slowest among the available alternatives. The spatial discretization of the solution domain is performed using finite volumes (FV) which allows for body-fitted hexagonal meshes that avoid the staircasing present when numerically integrating curved geometries with finite differences. The solver includes a hybrid implementation for the numerical fluxes that make it suitable for a wide range of Mach numbers; this is achieved by combining PISO methods (i.e. pressure-velocity coupling) in regions with low flow velocities (M ≤ 0.3), together with standard techniques for high speed flows (i.e. flux-splitting and limiters). Here, due to the nature of the flow at hand, only the latter scheme is active. The numerical fluxes at cell-faces are thus computed using a central-upwind Kurganov-Tadmor scheme [START_REF] Kurganov | New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations[END_REF][START_REF] Kurganov | Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations[END_REF]. The flow variables on cell faces are reconstructed through a second-order interpolation with a symmetric van Albada flux limiter which results in non-oscillatory bounded transitions across shocks and contact discontinuities. The chemical source terms are computed using a stiff ODE solver (i.e. fourth-order Rosenbrock). The PBiCGStab (Preconditioned Biconjugate Gradient Stabilized) method is used for all the linear systems resulting from the discretization of the governing equations, preconditioned through the DILU (Diagonal Incomplete-LU) technique. Finally, the time integration uses a second order Crank-Nicolson method, and the time step is dynamically adapted during the computation with an acoustic Courant number of 0.5 to ensure stability of the numerical scheme. The implementation, verification and validation of this solver follows the methodology presented in [START_REF] Kraposhin | Adaptation of Kurganov-Tadmor numerical scheme for applying in combination with the PISO method in numerical simulation of flows in a wide range of Mach numbers[END_REF]; the verification stage used standard methodologies in which the hydrodynamic and chemical parts of the solver are independently tested. For the combined implementation, unsteady 1-D simulations of detonations were carried out to reproduce their well-documented non-linear dynamics [START_REF] Romick | The effect of diffusion on the dynamics of unsteady detonations[END_REF].

Domain, initial and boundary conditions

The computational domain is shown in Figs. 2 and3 A hexagonal uniform mesh was used comprising 9,286,953 cells, which resulted in resolutions of 15, 10, 8, 5 points per ZND induction length, l ind , for p 0 = 10, 15, 20, 26 kPa, respectively. Simulations were run using 200 processors, and took approximately 10,000 CPU hours per case. The detonations were ignited by imposing three equally spaced hemi-spherical regions (r = 2.5 mm) of high pressure (p ign /p 0 ∼ 46) and temperature (T ign /T 0 ∼ 14) at the chamber inlet whose width is 30 mm. This initiation method resulted in steadily propagating cellular detonation fronts before the curved section of the chamber (see numerical soot foils in Fig. 4, and first speed data points along the outer wall in Figs. 5 and6), except for p 0 = 10 kPa.

Results and discussion

The numerically predicted and experimentally observed dynamics of the detonation as it propagated through the curved chamber is presented below as a function of initial pressure, p 0 . Only results for p 0 = 15 kPa and 26 kPa are discussed in subsections 4.1-4.3 because the intermediate pressure case was not found to provide additional insight. Subsection 4.4 describes the re-initiation dynamics for p 0 = 10 kPa. effects have been found to result in a λ decrease when compared to twodimensional simulations [START_REF] Mével | Detonation in hydrogen-nitrous oxide-diluent mixtures: An experimental and numerical study[END_REF]. average normalized propagation speed of D end /D CJ ∼ 1.16; this observation is in line with our previous findings using a stoichiometric propane-oxygen mixture [START_REF] Rodriguez | An experimental evidence of steadily-rotating overdriven detonation[END_REF]. For p 0 = 26 kPa, the velocity signal ends shortly after the deceleration period due to the arrival of the wave to the end of the curved section.

Overall dynamics

The evolution along the inner wall exhibits more interesting dynamics.

For p 0 = 15 kPa smooth decoupling of the LS and RZ as the detonation enters the curved chamber, subsequent re-coupling as the transverse detonation, shown in Fig. 2 at t = 131 µs, reaches the inner wall, followed, once again, by smooth decoupling as the detonation propagates into a diverging area.

The velocity profiles, for p 0 = 15 kPa and 26 kPa, show the signature of a single re-coupling event for the former, and the consecutive decoupling/recoupling events referred to earlier for the latter. A resolution study on the wave position, speed histories and particularly, on the location where the transition from regular-to-Mach detonation takes place along the outer wall, revealed that at least 10 pts per l ind are required to capture the experimentally observed structure at the end of the chamber for p 0 = 15 kPa. For higher pressures, p 0 = 26 kPa, in which the diffraction and re-initiation transients are less pronounced, a resolution of 5 pts per l ind was found to be sufficient. Along the outer wall, the detonation appears to be undisturbed initially but quickly transits to a Mach reflection with the leading shock normal to the wall (see t = 111 µs). Detonation and shock wave reflection types (i.e. regular or Mach) upon interactions with concave wedges/cylinders have been studied extensively. They are known to depend on the wedge angle/cylinder radius, front velocity and mixture characteristics [START_REF] Edward | Detonation interaction with wedges and bends[END_REF][START_REF] Hornung | Regular and Mach reflection of shock waves[END_REF][START_REF] Yuan | Numerical study of cellular detonation wave reflection over a cylindrical concave wedge[END_REF]. After the initial Mach stem is established, it grows, reaches a maximum and decreases.

Early stages of propagation

The increased pressure and temperature behind the stem results in induction lengths that can be of magnitude smaller than the reference steady ZND length, so the cell size significantly decreases. In these highly com-pressed regions the numerical resolution used in this work does not resolve the detonation structure, but is able to capture all the transient dynamics in the chamber. While the importance of the cellular structure of detonation fronts close to propagation limits is now widely accepted, the cellular pattern left behind this specific overdriven Mach detonation front is so small that a fair approximation is to neglect it, which provides supporting evidence to the applicability of GSD modeling for the analysis of the current configuration [START_REF] Jourdain | Progress in understanding rotating detonation modes[END_REF], and also suggests that the chemical kinetics may not be the main aspect driving this element of the phenomenon. These important features are further discussed in the last section. is analogous to detonations propagating under yielding confinement, namely a curved front, a transmitted shock and a shear layer [START_REF] Reynaud | A computational study of the interaction of gaseous detonations with a compressible layer[END_REF][START_REF] Taileb | Influence of the chemical modeling on the quenching limits of gaseous detonation waves confined by an inert layer[END_REF]. A closeup of this structure is shown in Fig. 8. Upon reflection from the inner wall, t = 143 µs, the early stages of a Mach detonation is shown whose typical features are a Mach stem, an incident curved detonation, and a shear layer. wall and propagates intermittently, since the extent of LS-RZ decoupling is minor along the inner wall (see t = 108 -116 µs). These regions will be consumed by transverse waves propagating along the detonation front in the same fashion as for p 0 = 15 kPa. Figure 8 also shows a closeup of this structure for p 0 = 26 kPa case. Evidence of this intermittent behavior can be seen in the numerical soot foils and velocity histories shown in Figs. 4 and6, respectively. Similarly to p 0 = 15 kPa, the early stages of a Mach reflection along the inner wall appears at t = 123 µs, while on the outer wall, a stronger and smaller Mach stem forms that decreases rapidly and transitions into the regular reflection described in Fig. 3.

Finally, as the Mach stem along the inner wall continues to propagate into an ever-increasing area, full decoupling occurs irrespective of the initial pressure considered. However Fig. 10 shows that, for p 0 = 26 kPa, this decoupling takes place earlier but further along the inner wall, which is simply a consequence of the increased initial density and associated differences in wave speeds. ence of a regular reflection, a Mach detonation (stem), a decoupled LS-RZ, a transverse detonation wave, reflected waves, and shear layers. However, differences exist in the reflection transition location (regular-to-Mach) and final wave structure. The normalized critical angle, θ crit /π, where the decoupled LS-RZ interacts with the detonation wave propagating along the outer wall, depends on the decoupling of the diffracting detonation along the inner wall (Fig. 10). When full decoupling occurs, the LS velocity decreases significantly allowing for enough time for the detonation propagating along the outer wall to catch up, and for a Mach reflection to redevelop (see Fig. 11).

Late stages of propagation

A plot summarizing the normalized transition angles for all pressures considered is shown in Fig. 12; θ crit /π decreases with increasing pressure and seems to have an asymptote at about θ crit /π ∼ -0.3.

The differences in wave structure at the very late stages of propagation, available arc length for propagation after transition, ∆s = r e (θ crit /π-θ end /π) -where r e is the radius of the outer wall and θ end /π = -0.5 is the angular location at which the outer wall meets the straight section of the chamberis larger for p 0 = 15 kPa (and p 0 = 10 kPa). This allows for the competition between the transverse detonation that propagates in shocked mixture and the rotating Mach stem propagating along the outer wall to last longer, and an "S-shape" arises as a result. Experimental evidence showing this wave structure is included in Fig. 13. As mentioned in [START_REF] Rodriguez | An experimental evidence of steadily-rotating overdriven detonation[END_REF][START_REF] Rodriguez | Experimental observations of semiconfined steadily-rotating detonation[END_REF], it is worth emphasizing that the evolution described above is a universal typical feature of re-initiation of diffracting detonations, and is not specific to the H 2 -O 2 -Ar mixture considered. Indeed, many experiments on detonation transmission from a straight channel to an open volume using different mixtures (see [START_REF] Pintgen | Detonation diffraction in mixtures with various degrees of instability[END_REF], for example), have identi-

Re-initiation dynamics

A further decrease in pressure to p 0 = 10 kPa resulted in the decoupling of the LS and RZ in the straight section of the chamber. This simply means that the amount of energy deposited in the gas through our numerical ignition scheme (described in subsection 3.2) was not sufficient to initiate a detonation directly. In contrast to the detonation transmission cases analyzed above, this case represents the re-initiation of a detonation by the transmission of an initially planar shock into the curved chamber. It should be noted that the numerical simulations first revealed the existence of this regime, and the accompanying experiments were run afterwards to validate the numerical predictions. To do so, the experimental setup was slightly modified. An additional 18 -µm Mylar foil was used to separate the detonation tube from the curved chamber (see Mylar 2 in Fig. 1). The interaction of the detonation with the Mylar foil resulted in a repeatable, normal transmitted shock into the straight section of the curved chamber (see schlieren image at t = 128 µs in Fig. 15). The late stages of the process in the curved chamber are shown in Fig. 16;

the intermediate dynamics are the same as described above for p 0 = 15 kPa, except that the transition from regular to Mach detonation occurs earlier at θ crit /π ∼ 0.1 (see Fig. 10). The experimental observations are in remarkable agreement with the numerical predictions regarding the wave dynamics, curvatures, separation distances between reaction zones and decoupled leading diffracted shocks, transverse detonation wave angles, and final wave structures at all time instances shown. The success in capturing the experimentally observed dynamics as a function of pressure using an inviscid formulation, that does not deliberately resolve the detonation cellular structure in the highly compressed regions of the chamber, demonstrates that the main drivers dictating the global features in the chamber are the geometry and the initial pressure.

To substantiate this hypothesis we used the latter experimental methodology but filled the chamber with air, whose ratio of specific heats, γ, is close to that of the argon-diluted stoichiometric H 2 -O 2 mixture studied (5% lower). The results are shown in Fig. 17; the similarities with the reactive cases discussed throughout this manuscript are evident. A more comprehensive experimental and numerical study of the inert wave dynamics as a function of γ is currently being carried out and will be the object of a future submission.

Conclusion

Experiments and two-dimensional Euler simulations with detailed kinetics were performed to characterize, as a function of initial pressure, the deto- We chose a numerical resolution that does not resolve the cellular structure in the highly compressed regions behind the detonation Mach stem and transverse detonation. The agreement obtained with the experiments sup-ports our previous conclusion [START_REF] Rodriguez | An experimental evidence of steadily-rotating overdriven detonation[END_REF][START_REF] Rodriguez | Experimental observations of semiconfined steadily-rotating detonation[END_REF] that resolving all the details of the detonation structure are not essential to capture the intricate transients experimentally observed. Finally, our results also support the applicability of GSD modeling [START_REF] Jourdain | Progress in understanding rotating detonation modes[END_REF] or variations thereof [START_REF] Yuan | A model for the trajectory of the transverse detonation resulting from re-initiation of a diffracted detonation[END_REF][START_REF] Yuan | Diffraction of cellular detonation wave over a cylindrical convex wall[END_REF] for the analysis of the current configuration, if one is only interested in capturing the most salient features of the dynamics.

Figure 1 :

 1 Figure 1: Schematic of experimental setup showing the detonation tube, the curved chamber and the expansion vessel. The angular position, θ, is measured counterclockwise from the positive x-axis. Note that Mylar 1 is used for all experiments whereas Mylar 2 is put in place for inert shots only. Adapted from Ref. [9].

  , and reproduces the curved chamber shown in Fig. 1. Four initial pressures were considered p 0 = 10, 15, 20 and 26 kPa to investigate the detonation dynamics in the chamber as a function of increasing pressure. Initial conditions were quiescent mixture at T 0 = 288 K, and mass fractions Y H 2 = 0.034779, Y O 2 = 0.276021, Y Ar = 0.6892, corresponding to a stoichiometric H 2 -O 2 mixture with %Ar ,vol = 40. A slip boundary condition for velocity, and zero gradient for temperature, species and pressure were imposed on the chamber walls.

Figure 2

 2 Figure2shows a time sequence of pressure contours superimposed on H atom numerical fields together with experimental OH* chemiluminescense for p 0 = 15 kPa. H atom is a useful chemical activity marker because of its short life time; it peaks inside the reaction zone and quickly decays thereafter, as seen from one-dimensional ZND profiles. The times reported elapse from the initiation of the detonation at the chamber inlet in the numerical simulations.

Figure 2 :

 2 Figure 2: Overall dynamics for p 0 = 15 kPa: numerical pressure contours and H atom field vs. experimental OH* chemiluminescence.

Figure 3 Fig. 3 .Figure 3 :

 333 Figure3shows the same numerical and experimental fields for p 0 = 26 kPa. The dynamics of the detonation front is strongly affected by the pressure increase. Notably, the detonation front is less sensitive to the initial area change as it encounters the curved region of the chamber. For t ≥ 143 µs full decoupling of the LS and RZ occurs along the inner wall, whereas a regular reflection establishes itself along the outer wall until the last time shown in

Figure 4 :

 4 Figure 4: Numerical soot-foils.

Figures 5 and 6

 6 Figures 5 and 6 show the position and speed of the LS and RZ are reported as a function of time along the outer and inner walls. The angular normalized position, θ/π, is measured counterclockwise from the positive x-axis (i.e.θ/π = 0); see Fig.1. Following this convention, the detonation propagates from the II (θ/π = 1.0) to the IV (θ/π = -0.5) quadrant during its transit through the chamber. Samples were collected at a frequency of 1 MHz (1 µs).The evolution along the outer wall is qualitatively similar for 15 kPa and 26 kPa, showing unnoticeable decoupling between the LS and RZ, followed by significant acceleration/deceleration of the front. The velocity peaks shown in Figs.5 and 6are attained during propagation as a regular reflection, whereas the transition from regular-to-Mach detonation is accompanied by a decrease in the front propagation speed. For p 0 = 15 kPa, the final wave structure ("S-shape") is overdriven and accelerates linearly for t ≥ 200 µs reaching an

Figure 5 :

 5 Figure 5: Wave positions and speeds histories along outer (top) and inner (bottom) walls for p o = 15 kPa. The steady detonation propagation velocity, D CJ = 1971 m/s; D avg and u avg are the leading shock and reaction zone average propagation velocity over the entire sampling period. D end /D CJ is the average normalized propagation velocity measured after the final wave complex is established, i.e., t ≥ 200 µs.

Figure 7

 7 Figure7shows a close-up of the early stages of detonation diffraction for p 0 = 15 kPa. A planar detonation front first enters the curved chamber and, close to the inner wall, its RZ begins to decouple from the LS due to the propagation of expansion waves originating from the region where the front first encounters an area increase (see 96 µs ≤ t ≤ 101 µs). The high sensitivity of chemical reaction rates to decreasing temperature results in significantly longer induction times behind the leading shock at lower temperature. These waves propagate perpendicular to the detonation front towards the outer wall making it curved. An increased detonation front curvature results in a faster local temperature and velocity decrease, so the reaction zone further decouples from the leading shock (111 µs ≤ t ≤ 131 µs).

Figure 6 :

 6 Figure 6: Wave positions and speeds histories along outer (top) and inner (bottom) walls for p o = 26 kPa. The steady detonation propagation velocity, D CJ = 1993 m/s; D avg and u avg are the leading shock and reaction zone average propagation velocity over the entire sampling period. D end /D CJ is the average normalized propagation velocity measured after the final wave complex is established, i.e., t ≥ 180 µs.

p 0 = 15

 15 kPa t = 96 µs t = 101 µs t = 111 µs t = 121 µs t = 131 µs t = 143 µs

Figure 7 :

 7 Figure 7: Early stages of detonation propagation for p 0 = 15 kPa: numerical pressure contours and H atom field.

Figure 7 (

 7 Figure 7 (111 µs ≤ t ≤ 131 µs) also indicates that the first re-initiation mechanism appears to be driven by the competition among reflected expansions from the outer wall, compression waves from downward traveling triple points, and the progress of the decoupled LS-RZ. A re-initiation center, and the birth of a combustion front that subsequently consumes shocked mixture are visible at t = 121 µs. The structure of this combustion front (t = 131 µs)

Figure 9

 9 Figure 9 clearly displays the effect of increasing initial pressure on the early dynamics. As mentioned above, the sensitivity to area changes is significantly decreased for p 0 = 26 kPa. The detonation front becomes curved in the inner region due to the numerous expansions emanating from the inner

Figure 8 :

 8 Figure 8: Close up to detonation diffraction structure close to inner wall for increasing pressure; solid lines are temperature contours extracted from computed flow fields.

p 0 = 26

 26 kPa t = 86 µs t = 91 µs t = 96 µs t = 108 µs t = 116 µs t = 123 µs

Figure 9 :

 9 Figure 9: Early stages of detonation propagation for p 0 = 26 kPa: numerical pressure contours and H atom field.

Figure 11

 11 Figure 11 shows a close-up of the late stages of detonation propagation in the curved chamber for p 0 = 15 and 26 kPa.The global dynamics for both pressures bear similarities with the pres-

Figure 10 :

 10 Figure 10: Location of full decoupling of leading shock and reaction zone along inner wall for increasing pressure.

t

  = 236 and 196 µs for p 0 = 15 and 26 kPa, respectively, seem to be simply a consequence of the location where transition occurs. In other words, the t = 191 µs t = 216 µs t = 236 µs p 0 = 15 kPa t = 186 µs t = 191 µs t = 196 µs p 0 = 26 kPa

Figure 11 :

 11 Figure 11: Late stages of detonation propagation for increasing pressure: numerical pressure contours and H atom field.

Figure 12 :

 12 Figure 12: Critical normalized angle (θ crit /π) as a function of pressure for transition from regular to Mach detonation along outer wall -numerical results.

Figure 13 :

 13 Figure 13: schlieren images showing "S-shape" wave structure during late stages of propagation in chamber -p 0 = 15 kPa.

Figure 14 : 5 O 2

 1452 Figure 14: Experimental evidence of transverse detonation waves in different mixtures and geometries: 0.5 H 2 + 0.5 N 2 O for p 0 = 47.5 kPa adapted from [40] (left) and C 3 H 8 + 5 O 2 for p 0 = 12 kPa adapted from [9, 10] (right).

Figure 15 showsFigure 15 :

 1515 Figure 15 shows the numerical and experimental re-initiation dynamics. The interaction of the normal shock with the curved section results in the development of an inert Mach stem along the outer wall, and a diffracted curved shock along the inner wall at t = 155 µs. The thermodynamic state, i.e., pressure and temperature ratios, behind the Mach stem are p/p 0 ∼ 23 and T /T 0 ∼ 4.7, whose associated delay time is less than 1 µs. Consequently, an ignition center forms close to the wall (see fields at t = 158 µs). Although the inviscid formulation used in this work is only capable of capturing purely chemical ignitions, since thermal and hydrodynamic boundary layers are not part of the computed flow field, the agreement observed with the experimental sequences indicates that chemistry is dominant over diffusive effects in this regime. The interaction of the curved shock, that develops from the explosion, with the previously shocked mixture and Mach stem initiates a detonation. The early stages of this transient are shown at t = 166 µs. The initially straight Mach stem has now become a curved detonation moving upstream, and the ignition center has similarly developed into an additional detonation propagating downstream. A larger transverse detonation, whose structure was already depicted in Fig. 8, consumes the mixture behind the curved diffracted shock (see t = 180 µs) to become a slightly curved detonation that takes the full extent of the chamber at t = 196 µs.

Figure 16 :

 16 Figure 16: Late stages of detonation propagation for p 0 = 10 kPa: numerical pressure contours and H atom field vs. experimental hi-speed schlieren.

p 2 /p 1 = 11 tFigure 17 :

 21117 Figure 17: Inert case: transmission of a shock wave (measured wave speed: u s = 1067 m/s; computed pressure ratio: p 2 /p 1 = 11) into the curved chamber filled with air; magnitude of density gradient, |∇ρ| (numerical) vs. experimental schlieren.

Acknowledgements

Computations were carried out on the supercomputer facilities of the Mésocentre de Calcul de Poitou Charentes. Experiments were conducted in the Detonation Team of the Institut Pprime UPR 3346 CNRS. This work was supported by a CPER FEDER Project of Région Nouvelle Aquitaine. Special thanks to Alain Claverie CNRS Research Engineer for his invaluable technical assistance.


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Self-assembled, 10 nm-tailored, near infrared plasmonic metasurface acting as broadband omnidirectional polarizing mirror

Keywords: plasmonics, metasurfaces, self-assembly, metal nanorods, polarization

Plasmonic metasurfaces tailored at the few-nm scale and fabricated by highthroughput lithography-free methods are needed to efficiently produce optical components with optimal performance and small footprint. Herein, we report the design of such metasurfaces and their fabrication over cm 2 areas by self-assembly. They present an optical cavity structure, in which a 2D array of parallel Au nanorods is coupled to a metal reflector through a thin spacer. The nanorods present a quasi-monodisperse ~10 nm diameter and separation, and a broad polydisperse length distribution. These outstanding nanostructural features, combining periodicity in one direction with randomness in another, endow the nanorod array with unconventional plasmonic and anisotropic effective optical properties. For light polarized along the nanorods, the metasurface displays strong destructive optical interference and thus near perfect absorption over a broad range of angles of incidence and of wavelengths in the near infrared. In contrast, for light polarized perpendicular to the nanorods, no interference occurs and thus the metasurface behaves as a metal mirror. In sum, the metasurface can be used as broadband omnidirectional polarizing mirror, an original thin and monolithic optical component merging the functionalities of a metal mirror and of a wire-grid polarizer.

Introduction

Over the past years, plasmonic metasurfaces have arisen as powerful material platforms for manipulating the spectrum, polarization and topology of optical beams. [1] Accordingly, they have enabled flat and subwavelength-thin optical components such as lenses, beam steerers, phase shifters, coatings with structural colors, polarization controllers or mirrors, opening the way to photonic devices with new functionalities and a smaller footprint. [1][2][3][4][5][6][7][8] To present optimal optical functionalities, the structure of plasmonic metasurfaces must be tailored at a scale much smaller than their operation wavelength. Therefore, for operation in the ultraviolet, visible and near infrared regions, such tailoring must be achieved at the scale of few nm. Such accuracy lies at the limit of the techniques currently employed for plasmonic metasurface fabrication, for instance electron beam lithography or focused ion beam lithography, techniques that in addition suffer from a low fabrication throughput.

In this context, there is a strong demand for developing alternative lithography-free approaches to fabricate plasmonic metasurfaces tailored at the scale of few nm for applications in the ultraviolet, visible and near infrared regions. Particularly appealing are self-assembly methods, which enable the accurate fabrication and organization of nanostructures and are excellent for upscaling. Despite of these positive aspects, the potential of these methods for the fabrication of plasmonic metasurfaces tailored at the scale of few nm has been seldom considered. [9][10][11] Indeed, in most of the works reporting self-assembled plasmonic metasurfaces, the characteristic scale for tailoring lies in the range of few tens of nm. [11][12][13][14][15][16][17] Herein, we report the design and fabrication by a high-throughput self-assembly approach, of plasmonic metasurfaces tailored at the scale of a few nm. They are based on a twodimensional (2D) array of parallel Au nanorods presenting a quasi-monodisperse few-nm diameter and separation, and a broad polydisperse length distribution in the range from few tens of nm to 1 m. These combined nanostructural features endow the nanorod array with unconventional plasmonic and anisotropic effective optical properties in the near infrared. In this spectral region, no transverse plasmon is excited in the nanorods because of their few-nm diameter. Therefore, the array is transparent to incoming near infrared light when it is polarized perpendicular to the nanorods. In contrast, the array partially absorbs incoming near infrared light when it is polarized parallel to the nanorods. This partially absorbing character is achieved over a broad spectral range thanks to the excitation of longitudinal plasmon resonances at different wavelengths in the nanorods with different lengths, and to the few-nm separation between them.

In the considered metasurfaces, where the nanorod array is incorporated into an optical cavity structure inspired by the "optical Salisbury screen" design, [18] these unconventional plasmonic and anisotropic effective optical properties lead to (i) no destructive optical interference and thus a high, mirror-like reflectance, for the polarization perpendicular to the nanorods; (ii) strong destructive optical interference and thus a near-perfect absorption of light for the polarization parallel to the nanorods. [18][19][20][21][22][23][24][25] The effective optical properties of the nanorod array present a suitable wavelength-dependence to enable such well-contrasted interference and reflection/absorption properties to occur in the near infrared in a broad range of wavelengths and angles of incidence. This makes the metasurface behave as a broadband omnidirectional polarizing mirror in this spectral region. Such optical component merges, within a thin film, the functionalities of a metal mirror and of a wire-grid polarizer. It reflects the incoming light as efficiently as a metal mirror when it is polarized perpendicular to the nanorods, and near-perfectly absorbs it when it is polarized parallel to them. Such original self-assembled component is useful for lowering the footprint of photonic devices by merging two components into a monolithic one with a subwavelength thickness.

Concept

To introduce the key structural elements and working principle of such metasurfaces, we first consider the simplified metasurface design depicted in figure 1a. It consists of a periodic 2D array of parallel and monodisperse cylindrical Au nanorods with a few-nm diameter (D) and separation ( = D, the same from tip-to-tip and side-to-side), supported on a thin transparent spacer (amorphous aluminum oxide, noted hereafter Al2O3) coating a metal reflector (Al).

This optical cavity design is inspired by the "optical Salisbury screen" configuration, [18] in which we replaced the top continuous metal film by a nanorod array. We consider the cases in which the metasurface is illuminated at normal incidence, with light polarized either perpendicular to the nanorods (⊥) or parallel to them (//). Figure 1b depicts in a simplified way the key optical features of this metasurface in the near infrared. For the ⊥ polarization, the nanorod array is transparent to the incoming light, which is thus mainly reflected at the Al reflector (r ~ rAl). Therefore, for this polarization the metasurface reflects light as efficiently as a metal mirror. In contrast, for the // polarization, the nanorod array absorbs a fraction of the incoming light. Therefore, the amplitude of the directly reflected wave cannot be neglected versus that of the waves escaping the cavity after multiple internal reflections. If the nanorod array and cavity are properly designed, suitable values of the amplitudes (a) and phases () of these waves can be achieved to enable strong destructive optical interference and thus a near-zero reflectance/near-perfect absorption (ae j ~ 0) at selected wavelengths. [20,21] According to this picture, the key to the polarization-dependent near infrared response of the metasurface is the optical anisotropy of the nanorod array in this spectral region. This anisotropy arises from the excitation of a longitudinal plasmon for the // polarization in the near infrared, while no transverse plasmon can be excited for the ⊥ polarization in this spectral region. [27][28][29] To demonstrate this fact, finite difference time domain (FDTD) simulations were performed. Details about these simulations are given in ref. 26 and Supporting Information S1. The FDTD field maps of the nanorod array displayed in figure 1c demonstrate the excitation of a longitudinal plasmon and no transverse plasmon in the near infrared, for a nanorod length L = 50 nm and D =  = 10 nm. A similar trend is observed in the FDTD transmittance spectra of the nanorod array, displayed in figure 1d, which were obtained for different L values (from 50 nm to ∞). For L = 50 nm, the longitudinal plasmon peaks at ~ 950 nm and then it shifts deeper in the infrared as L increases.

These plasmonic properties together with the deep-subwavelength values of D and , make the nanorod array behave as an anisotropic effective layer in the near infrared. The anisotropy is uniaxial with the optical axis oriented parallel to the nanorods. For the ⊥ polarization, because no transverse plasmon is excited in the near infrared, the effective layer is transparent in this spectral region. In contrast, for the // polarization, the excitation of the longitudinal plasmon makes the effective layer present a near infrared optical absorption that depends on the wavelength and nanorod length.

This anisotropy leads to very different optical interference properties and thus very different reflectance spectra of the metasurface in the near infrared for the // and ⊥ polarizations, as shown in figure 1e. For the ⊥ polarization, no interference occurs due to the transparency of the effective layer. Therefore, the metasurface presents a high reflectance that is independent of the wavelength and nanorod length. For the // polarization, the wavelength-dependent and nanorod length-dependent absorption of the effective layer yields interference properties that depend on these two quantities. These trends are seen in figure 1e where the FDTD reflectance of the metasurface is represented for different L values (from 50 nm to ∞). For selected nanorod lengths (50, 100 nm) a band with very low reflectance appears as a result of strong destructive interference for the // polarization. The position of this band depends on L.

However, note that it is not necessarily close to the peak wavelength of the longitudinal plasmon, which may yield to a too strong absorption to enable interference at this wavelength.

In Supporting Information S1, it is also shown how the interference properties of the metasurface depend on  and on the nanorod cross-section morphology.

These results suggest that a metasurface based on nanorods presenting monodisperse few-nm D and  and broadly polydisperse lengths (and thus an inhomogeneously broadened effective longitudinal plasmon absorption [30] ) will present, over a broadened range in the near infrared, a high reflectance for the ⊥ polarization and a low reflectance for the // polarization. Let us stress that, to ensure a strong reflectance contrast between the two polarizations, in such metasurfaces the separation between nanorods should remain small. Indeed, increasing much this separation would make the absorption of the effective layer under // polarized light become too low to achieve the balance needed between directly and multiply reflected waves for destructive interference. In that case the reflectance would thus become similar for both polarizations. Keeping the nanorod separation at a small value is also important to ensure that the nanorod array displays an effective behavior. Finally, it is expected that the observed interference phenomena will enable the reflectance of the metasurface to be independent of the angle of incidence, in the same way as in conventional optical Salisbury screens. [21] Summarizing, metasurfaces with a 2D nanorod array/spacer/reflector structure, where the parallel Au nanorods present a quasi-monodisperse few-nm diameter and separation together with broadly polydisperse lengths, are appealing to achieve broadband omnidirectional polarizing mirrors operating in the near infrared.

Results and discussion

Fabrication of 2D arrays of Au nanorods

As a first step for the fabrication of such metasurfaces, we focused on growing 2D arrays of parallel Au nanorods with a quasi-monodisperse few-nm diameter and separation together with broadly polydisperse lengths. At such aim, Au was deposited at glancing incidence onto periodic nanorippled amorphous aluminum oxide (Al2O3) films supported on transparent fused silica substrates, [31] as shown in figure 2a. By such means, the Au deposit grows selectively on the ripple sides that face the incoming metal flux, to form parallel Au nanorods whose diameter and separation are driven by the dimensions of the nanoripples. [32] In contrast, there is no geometrical restriction on the nanorod length, which is thus driven by stochastic nucleation-diffusion-growth events. Therefore, while the nanorods grow with a very narrow distribution of diameter and separation distance, their length distribution can be very broad.

The experimental pulsed laser deposition setup used to grow the nanorod arrays is schematized in figure 2b and related experimental details are given in Supporting Information S2. Figure 2b also shows a topographic image typical of the nanorippled Al2O3 films used.

Herein, we chose nanoripples with a 27 nm period to grow nanorods with a near 10 nm diameter and separation. Parallel nanorods with such small and quasi-monodisperse diameter and separation were successfully grown, as seen in the transmission electron microscopy image shown in figure 2c. In this image, the average values of D, side-to-side separation // and tip-to-tip separation ⊥ appear to be 12.6 nm, 15.7 nm, and 8.6 nm, respectively. Figure 2c also confirms the broad length polydispersity of the nanorods. The corresponding length distribution, characterized by digital image analysis (Supporting Information S3), is shown in figure 2d. This distribution spans from a few tens of nm to near one m, with a maximum near 200 nm.

Note that all these structural features are representative of the whole surface of the grown material, which was deposited homogeneously over a cm 2 area. This homogeneity is seen in the inset of figure 2c, which shows a macroscopic image of a nanorod array sample. The nanorod array absorbs partially visible light and displays a homogeneous grey color over the cm 2 sample area. Such cm 2 scale homogeneity, achieved by a single step process, shows the suitability of our approach to fabricate nanostructure arrays tailored down to the few-nm scale, with a much higher throughput than lithography techniques. Furthermore, to our knowledge of the literature, [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49] this is one of the first times that 2D arrays of metal nanorods with such structural features (parallel orientation, few-nm quasimonodisperse D and , very broad L distribution) are fabricated by self-assembly on a transparent substrate. These previous works relied mainly on approaches such as physical vapor deposition on nanorippled or facetted surfaces, [33][34][35][36][37][38][39][40][41][42][44][45][46][47] or by self-ordering of colloids onto wrinkled surfaces. [48,49] As shown in Supporting Information S4, similar features were reported only in refs. 26, 42 and 47.

Effective optical properties of the 2D arrays of Au nanorods

The outstanding structural features of the fabricated nanorod arrays enable them to present effective optical properties with strong uniaxial anisotropy. To unveil such properties, the transmittance and reflectance of a nanorod array on nanorippled Al2O3 film/fused silica were measured in the visible and near infrared, for several angles of incidence and polarizations of the incoming light. The obtained spectra are shown in Supporting Information S5. These spectra were fitted using a layered model depicted in figure 3a. In this model, the nanorod array was represented by a uniaxial effective layer presenting a complex refractive index N// = n// + ik// along the nanorods, and a complex refractive index N⊥ = n⊥ + ik⊥ perpendicular to them. These four wavelength-dependent quantities were left free during the fit, which was realized using the transfer matrix method with mean-square error minimization algorithms.

By such means, an excellent fit quality was achieved (Supporting Information S5).

The best fit spectra of n//, k//, n⊥ and k⊥ are displayed in figure 3b. They confirm the strong anisotropy and show that, in the near infrared, the effective layer is transparent (resp. partially absorbing) for the ⊥ polarization (resp. // polarization). In addition, it is worth noting that the extinction coefficient k// presents similar spectral features to those of a continuous Au film, yet with lower values and a weaker increase with the wavelength. Correspondingly, when the incoming light is polarized along the nanorods, the nanorod array behaves as a partially absorbing metal film in a broad spectral range. Moreover, as explained hereafter, the wavelength-dependence of N// is suitable for achieving strong destructive interference effects in the cavity structure for a broad range of wavelengths in the near infrared. In contrast, no such strong interference would occur for the incoming light polarized perpendicular to the nanorods in the near infrared.

Metasurfaces based on the fabricated 2D nanorod arrays

To evaluate the optical properties of metasurfaces based on the fabricated nanorod arrays, we simulated the visible-near infrared reflectance spectra of the design shown in figure 3c, where the wavelength-dependent n//, k//, n⊥ and k⊥ of the effective layer were those shown in figure 3b. Simulations were done with the transfer matrix method (necessary details are given in Supporting Information S6). The simulated spectra obtained for different values of the spacer thickness t are shown in figure 3d. For the ⊥ polarization, whereas some oscillations occur in the visible as a result of destructive interference, such interference is absent in the near infrared so that a high reflectance is observed in this spectral region. Remarkably, the reflectance becomes as high as that of an Al mirror (0.93) for wavelengths above 1000 nm if the spacer is thick enough (t = 200 nm). For the // polarization, a low reflectance resulting from strong destructive interference can be achieved in a broad spectral range in the near infrared if the spacer is thick enough. For t = 200 nm, a reflectance close to 0.3 can be achieved for wavelengths above 1000 nm. Note that these values are lower than expected from the simulations of figure 1. One first reason to that might be that the actual nanorod dimensions and morphology slightly depart from those used in these simulations. This is partly supported by the FDTD simulations shown in Supporting Information S1. A second reason might be that, because each nanorod has a polydisperse environment, it is excited by a near field with different spectral and spatial features compared with that of periodic FDTD models. Such different features would lead to a different excitation of each nanorod, which might support multipole plasmon modes with a broad spectral density (see discussions on polydispersity in ref . 11). A third reason might be that the actual complex refractive index of the Au nanorods differs from those used in these simulations, because of electronic confinement. For t = 300 nm, the additional propagation phase shift brought by the larger spacer thickness enables to achieve even lower reflectance values (less than 0.1) in the C spectral band (1530-1565 nm). With the metasurface structure considered, where light enters through the glass substrate, the reflectance could be further lowered by adding an antireflection coating at the air-glass interface. This shows the potential of our Au nanorod arrays to build metasurfaces acting as efficient polarizing mirrors operating over a broad range in the near infrared. Besides the high near infrared reflectance (comparable to that of an Al mirror) predicted for the ⊥ polarization, it is especially remarkable that a very low reflectance approaching 0 can be achieved in a broad spectral range in the near infrared for the // polarization. As shown in Supporting Information S6 (figure S6.2), the spectral range in which // polarized reflectance can approach 0 is much broader than for a continuous Au film/Al2O3/Al cavity with the same thicknesses. This occurs because the effective optical properties of the nanorod array under this polarization present a suitable wavelength-dependence to achieve strong interference in a broad spectral range, in contrast with an Au film that absorbs too strongly at long wavelengths. In fact, the capability of the Au nanorod array to enable strong interference for the // polarization in a broad spectral range in the near infrared is comparable with that of Cr films, one of the wisest choices to build broadband near-perfect absorbers. [19,22,23] This is seen by comparing the // polarized reflectance spectrum of the metasurface with that of a continuous Cr film/Al2O3/Cr cavity with the same thicknesses (figure S6.2): both present a similarly low reflectance for wavelengths above 1200 nm.

Fabricated metasurface and its optical properties

To test our theoretical design with experiments, we fabricated a metasurface based on the nanorod array studied above. This metasurface is schematized in figure 4a. To produce the depicted structure, an Al2O3 spacer with a 190 nm thickness was grown by pulsed laser deposition onto the nanorod array. This spacer was finally covered with a 50 nm-thick Al reflector grown by evaporation. The fabricated metasurface showed a homogeneous visual aspect over a cm 2 area.

The Rp and Rs reflectance spectra of this metasurface measured at near-normal incidence with the plane of incidence perpendicular to the nanorods are shown in figure 4b. In this configuration, Rp (resp. Rs) measurements probe the metasurface with the electric field ⊥ (resp. //) to the nanorods. The measured spectra show features similar to those predicted by simulations. In particular, in the near infrared they show a high Rp and a low Rs (near 0.3 for a broad range of wavelengths above 1000 nm). A similar strong contrast is observed in a broad range of AOIs, as seen in figure 4c that shows that Rp and Rs remain unchanged upon increasing the incidence angle to 70º. Note that this is true either when the plane of incidence is perpendicular to the nanorods or parallel to them. This confirms the suitability of the fabricated metasurfaces for acting as broadband omnidirectional polarizing mirror in the infrared.

Conclusions and outlook

In summary, we have reported self-assembled plasmonic metasurfaces tailored at the scale of few nm and grown homogeneously over cm 2 scale areas, and we have demonstrated the suitability of such metasurfaces for achieving an original thin and monolithic optical component. In particular, we considered metasurfaces with a 2D Au nanorod array/Al2O3 spacer/Al reflector cavity structure, where the nanorods present a parallel orientation, a quasimonodisperse few-nm diameter and separation, and a broad polydisperse length distribution.

These outstanding nanostructural features were achieved by growing the nanorods by pulsed laser deposition at glancing angle onto periodic nanorippled Al2O3 substrates. They endow the nanorod array with unconventional plasmonic and anisotropic effective optical properties enabling the metasurface to behave as a broadband omnidirectional polarizing mirror in the near infrared.

In this spectral region, no transverse plasmon is excited so that the array is transparent to light polarized perpendicular to the nanorods. This leads to the absence of destructive optical interference within the metasurface, which therefore reflects light with this polarization as efficiently as an Al mirror. In contrast, for light polarized parallel to the nanorods, multiple longitudinal plasmons are excited within the array at different wavelengths in the near infrared. Therefore, in this spectral region, the array absorbs light partially and the wavelength dependence of its effective optical properties lead to strong destructive interference within the metasurface over a broad range of wavelengths. This enables the metasurface to display a near-perfect absorption of near infrared light when it is polarized parallel to the nanorods, with a performance comparable to that of continuous Cr film/Al2O3 spacer/Al reflector cavity structures, one of the wisest choices to achieve broadband near-perfect absorption in the near infrared.

To our knowledge, this is the first time that a self-assembled metamaterial design harnesses nanoscale monodispersity/periodicity in one direction and nanoscale polydispersity/randomness in another direction to achieve in a broad spectral range no interference for one polarization and strong interference for the other. Indeed, in recent literature, although nanostructuration and randomness have been considered to achieve strong interference in a broad spectral range, this was done without considering or harnessing anisotropy effects. [11,18,[22][23][24][25] Note that, while in this work we grew Au nanorods with a near 10 nm diameter and separation, the nanorod dimensions could be even more finely tuned to further enhance the metamaterial optical properties. Such fine tuning can be achieved for instance by depositing the Au onto nanorippled substrates with a different period. Furthermore, the proposed metasurface concept can also be implemented using other materials than Au, for instance Ag and Al, to achieve polarizing mirrors operating in other spectral regions, such as the visible or ultraviolet.

Figure 1 .

 1 Figure 1. Metasurface based on a 2D array of parallel Au nanorods: working principle. (a) Simplified metasurface design considered as example. The nanorods are cylindrical, monodisperse (length L, diameter D), parallel and organized on a periodic array. They are separated from tip-to-tip and side-to-side by the same distance . They lie on an Al2O3 spacer coating an Al reflector. (b) Simplified representation of the key optical features of the metasurface in the near infrared, for the polarization of the incoming light ⊥ and // to the nanorods. The black arrow represents the directly reflected wave, and the red arrow represents the wave escaping after being reflected once by the Al reflector. (c) FDTD-simulated nearfield maps of a unit cell of the metasurface at normal incidence, for the ⊥ and // polarization. The wavelength was 860 nm, and we took L = 50 nm, D =  = 10 nm and thicknesses of 200 nm and 50 nm for the spacer and reflector, respectively. The map is observed in the equatorial plane of the nanorod array. The color scale ranges from 0 to 4 (resp. 0 to 15) for the ⊥ (resp. //) polarization. (d) FDTD-simulated transmittance spectra of the nanorod array (for these simulations, the Al reflector was removed). (e) FDTD-simulated reflectance spectra of the metasurface. In (d) and (e) the spectra were simulated for different values of L.

Figure 2 .

 2 Figure 2. Fabrication of 2D arrays of Au nanorods. (a) Schematic representation of the glancing incidence deposition process of Au onto a nanorippled Al2O3 film to form the nanorods. The nanorippled Al2O3 film is supported on a transparent fused silica substrate. (b) Geometry of the setup used for Au deposition, and topographic image (atomic force microscopy) of a nanorippled Al2O3 film. The black arrow represents the direction of the incident Au atoms. A corresponding height profile is shown in Supporting Information S2.1. (c) Top-view transmission electron microscopy image of a 2D array of Au nanorods grown onto nanorippled Al2O3. Inset: Macroscopic image of a nanorod array sample taken with a digital camera. (c) Distribution of the nanorod length L.

Figure 3 .

 3 Figure 3. Effective optical properties of the fabricated 2D arrays of Au nanorods and their relevance for the design of metasurfaces acting as polarizing mirror. (a) Layered model used for determining the complex effective refractive index of the nanorod array, parallel and perpendicular to the nanorods (N// and N⊥). The effective layer accounting for the nanorod array is shown in pink. Its thickness was set to 10 nm. The Al2O3 thickness was set to 40 nm. (b) Measured best fit spectra of the real part n and imaginary part k of these complex refractive indices (solid lines). The n and k spectra of Au from the Palik database are shown for comparison (dotted lines). (c) Structure of a metasurface incorporating this effective layer. It is covered by an Al2O3 spacer with thickness t and a 50 nm-thick Al reflector. Light is shined onto the metasurface through the transparent fused silica substrate. (d) Transfer matrixsimulated reflectance spectra of this metasurface at normal incidence, for the // and ⊥ polarizations, as a function of the spacer thickness. In the simulations, the complex refractive indices shown in (b) were used for the effective layer. The simulated reflectance spectrum of an Al mirror is shown for comparison.

Figure 4 .

 4 Figure 4. Structure and optical properties of the fabricated Au nanorod array-based metasurface. (a) Schematic representation of the metasurface structure. The nanorod array is covered by a 190 nm-thick Al2O3 spacer and a 50 nm thick Al mirror. Macroscopic images of a metasurface sample taken with a digital camera, from the glass side (front) and from the reflector side (back). (b) Measured reflectance spectra of this metasurface at near normal incidence (angle of incidence AOI of 20º, plane of incidence perpendicular to the nanorods), for S (solid line, equivalent to //) and P (dotted line, equivalent to ⊥) polarizations. (c) Maps of the reflectance of this metasurface versus wavelength and AOI, for the plane of incidence perpendicular (k⊥) and parallel (k//) to the nanorods, and for S and P polarized light.

Acknowledgements

Lamellae for TEM analysis were prepared at the "Laboratorio de Microscopías Avanzadas" (Instituto de Nanociencia de Aragón-Universidad de Zaragoza). Authors acknowledge L. Casado and I. Rivas for their preparation. TEM analysis was conducted at the National Center for Electronic Microscopy (Universidad Complutense de Madrid). Dr. J. Garcia and Dr. J. de la Portilla are acknowledged for his support in the TEM analysis and the binarization of TEM images, respectively. Dr. J.I. Larruquert is acknowledged for the deposition of the Al reflector. This research was funded by the Spanish Research Agency (AEI, Ministry of Research and Innovation) and the European Regional Development Fund (ERDF) under grant RTI 2018-096498-B-I00. The work of P. Gomez was supported by the Autonomous Community of Madrid and the European Social Fund (contract E-28-2019-0767805).

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Supporting Information

Supporting Information is available from the Wiley Online Library or from the author. S1. Details about FDTD simulations; S2. Details about the metasurface fabrication; S3. Digital image analysis of the nanorod array; S4. Self-assembled 2D arrays of parallel metal nanostructures: literature data; S5. Reflectance and transmittance spectra of the nanorod array; S6. Simulation of the metasurface reflectance spectra.

Self-assembled plasmonic metasurfaces tailored down to 10 nm are reported. They combine non-conventional in-plane structural features (randomness along one direction and periodicity along the other) with a vertical resonant cavity design. This enables them to behave as broadband coherent near-perfect absorber for one polarization and high-quality mirror for the other. Therefore, they can be used as broadband omnidirectional polarizing mirror.

Self-assembled plasmonic metasurfaces

Esther Soria, Pilar Gomez, Christophe Tromas, Sophie Camelio, David Babonneau, Rosalía Serna, José Gonzalo*, and Johann Toudert* Self-assembled, 10 nm-tailored, near infrared plasmonic metasurface acting as broadband omnidirectional polarizing mirror


Caroline Matringe

Elsa Thune

Rémy Cavalotti

Ahmad Fakih

Stephan Arnaud

Nils Blanc

Nathalie Boudet

Alessandro Coati

Yves Garreau

D Babonneau

R Guinebretière
email: rene.guinebretiere@unilim.frtoc

Vasarely painting at the nanoscale on sapphire crystals

Keywords: Nanoscaled Vasarely surfaces, 2D-Self-organization, Sapphire vicinal surfaces

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Scaling down of the electronic or optoelectronic devices often requires the elaboration of twodimensional (2D) periodic structures at the nanoscale. Beside the conventional top-down fabrication technique that consists in etching macroscopic crystals by electron beam writing or advanced lithography using high energy photon beams, self-ordering offers an elegant and low cost bottom-up route [START_REF] Barth | Engineering atomic and molecular nanostructures at surfaces[END_REF]. In this field, the use of ordered vicinal surfaces of single crystals is a very efficient way to elaborate periodic templates [START_REF] Yagi | Step bunching, step wandering and faceting: selforganization at Si surfaces[END_REF][START_REF] Misbah | Crystal surfaces in and out of equilibrium: A modern view[END_REF]. In the case of stable phases, free of any solid-state phase transition or chemical decomposition in the concerned temperature-pressure range, the decrease of the surface free energy drives the out-of-equilibrium continuous selfordering process [START_REF] Misbah | Crystal surfaces in and out of equilibrium: A modern view[END_REF] that is often described as a spinodal decomposition process [START_REF] Kim | Coarsening kinetics of a spinodally decomposed vicinal Si(111) surface[END_REF]. Various vicinal surfaces of metals [START_REF] Rousset | Self-ordering of Au(111) vicinal surfaces and application to nanostructure organized growth[END_REF], semiconductors [START_REF] Men | Selforganized nanoscale pattern formation on vicinal Si(111) surfaces via two-stage faceting transition[END_REF] or insulating oxides [START_REF] Cuccureddu | Surface morphology of c-plane sapphire (-alumina) produced by hightemperature anneal[END_REF][START_REF] Sanchez | Critical effects of substrate terraces and steps morphology on the growth mode of epitaxial SrRuO3 films[END_REF] constituted by periodically spaced steps have been obtained after convenient thermal treatments, thereby resulting in the formation of one-dimensional (1D) ordered templates at the nanoscale. Sapphire, single crystal of -alumina, is playing an ever-increasingly important role as a base material for nanoelectronic systems. For example, it is commonly used as substrate for growing indium gallium nitride (InGaN) [START_REF] Nakamura | The roles of structural imperfections in InGaN-based blue light-emitting diodes and laser diodes[END_REF][START_REF] Koester | M-plane core shell InGaN/GaN multiple-quantum-wells on GaN wires for electroluminescent devices[END_REF] as well as zinc oxide (ZnO) [START_REF] Huang | Room-temperature ultraviolet nanowire nanolasers[END_REF][START_REF] Eaton | Semiconductor nanowire lasers[END_REF] epitaxial nanostructures for electroluminescent devices. More generally, sapphire surfaces are used as a substrate for the growth of inorganic [START_REF] Nakamura | The roles of structural imperfections in InGaN-based blue light-emitting diodes and laser diodes[END_REF][START_REF] Koester | M-plane core shell InGaN/GaN multiple-quantum-wells on GaN wires for electroluminescent devices[END_REF][START_REF] Huang | Room-temperature ultraviolet nanowire nanolasers[END_REF][START_REF] Eaton | Semiconductor nanowire lasers[END_REF][START_REF] Lee | Role of substrate step bunches on the growth behavior of LiNbO3 thin film on -Al2O3 substrate[END_REF][START_REF] Ago | Competition and cooperation between lattice-oriented growth and step-templated growth of aligned carbon nanotubes on sapphire[END_REF][START_REF] Bachelet | Self-patterned oxide nanostructures grown by post-deposition thermal annealing on stepped surfaces[END_REF][START_REF] Boulle | Structural and optical characterizations of high quality LiNbO3 epitaxial films grown by pulsed laser deposition[END_REF] as well as organic nanostructures [START_REF] Aoki | Immobilization of protein molecules on step-controlled sapphire surfaces[END_REF]. Recent studies have focused on the continuous evolution of the period of the steps formed by isothermal treatment of vicinal sapphire surfaces exhibiting a miscut angle m between 1° and 10° with respect to the (006) crystallographic planes [START_REF] Thune | Understanding of one dimensional ordering mechanisms at the (001) sapphire vicinal surface[END_REF]. The corresponding surfaces are constituted by a 1D periodic arrangement of prisms with a triangular saw tooth-like profile and, in the case of m = 10°, a symmetric profile can be obtained [START_REF] Matringe | Symmetric faceting of a sapphire vicinal surface revealed by Grazing Incidence Small-Angle X-ray Scattering 3D mapping[END_REF]. In this article, we show that long isothermal treatment of such a surface induces an astonishing 2D-ordering. The surface can then be described as a 2D periodic assembly of diamond-shaped facets separated from each other by parallelogram-shaped steps. As shown in fig. 1, the resulting surface pattern is very similar to the "Hexagon VII" painting realized by Vasarely and, as far as we know, such a 2D periodic topography at the nanoscale has never been elaborated through self-ordering process. This article is focused on the quantitative description, with respect to crystallographic aspects, of this "nanoscaled Vasarely surface".

Experimental

Sapphire samples, cut with a miscut angle m = 10° with respect to the (006) planes and in such a way that the step edges were parallel to the [110] direction, were purchased from the MTI corporation Richmond, CA, USA. The 2D-ordering was promoted by annealing through isothermal treatment at 1250 °C under a PO2 = 1 bar atmosphere.

The morphology of sapphire vicinal surfaces was observed using Atomic Force Microscopy (AFM) with a MultiMode SPM Microscope in tapping mode under ambient atmosphere at room temperature. AFM images were taken with a silicon cantilever with an integrated pyramidal tip of typical radius of curvature smaller than 10 nm (manufacturer's data). The force constant was 80 N/m, and the resonant frequency range was from 200 kHz to 400 kHz. The topographic images were drawn with the WSxM software [START_REF] Horcas | A software for scanning probe microscopy and a tool for nanotechnology[END_REF].

Grazing Incidence X-ray Diffraction (GIXD) experiments were done on the MED instrument of the SixS beamline at the Synchrotron SOLEIL, in Gif-sur-Yvette, France. The samples were irradiated by a 12 keV X-ray beam under a grazing incidence angle equal to 0.30°. The diffracted intensity around several reciprocal space nodes was recorded under in-plane geometry using a point detector.

Grazing Incidence Small-Angle X-ray Scattering (GISAXS) experiments were performed on the BM02 French CRG beamline [START_REF] Chahine | Advanced non-destructive in situ characterization of metals at the French CRG D2AM / BM02 beamline at the ESRF[END_REF] at the European Synchrotron Radiation Facility (ESRF)

in Grenoble, France. All the GISAXS maps were recorded using a monochromatic X-ray beam at 9.659 keV and under a grazing incidence slightly higher than 0.23° that is the critical angle for total reflection of sapphire at this energy. The GISAXS maps were recorded from samples mounted onto a diffractometer allowing careful sample orientation with respect to the crystallographic axis of the crystal. We used a XPAD solid state detector located 5235 mm away from the sample and we recorded (qy,qz) maps as a function of the azimuthal angle  with a step of 0.5°. Measurement of 721 maps in the full 360° range allows complete reconstruction of the 3D GISAXS maps. 2D maps were analyzed according to the distorted-wave Born approximation using the FitGISAXS package [START_REF] Babonneau | FitGISAXS: software package for modelling and analysis of GISAXS data using IGOR Pro[END_REF]. According to the classical convention, the q x ⃗⃗⃗⃗ component of the scattering vector q ⃗ corresponds to the projection of the X-ray primary beam onto the sample surface, the q z ⃗⃗⃗⃗ component is orthogonal to the scattering surface and the q y ⃗⃗⃗⃗ component is orthogonal to the two others.

Results and discussion

-alumina crystallizes under the R3 ̅ c space group. The crystallographic structure of this rhombohedral phase is usually described using a hexagonal multiple cell and, all along the article, this hexagonal indexing will be used. Sapphire exhibits a 3-fold symmetry around the c-axis in the hexagonal setting and the [110], [1 ̅ 00] and [01 ̅ 0] directions lying in the (006)

planes are thus equivalent (see fig. 2a). Our samples were all cut with a miscut angle m equal to 10° with respect to the (006) planes and in such a way that the step edges were parallel to the The topography of the sapphire vicinal surface heat-treated for 32 hours can be described as an assembly of three-dimensional (3D) objects characterized by a certain shape and periodically spaced from one another. Following the crystallographic general approach, we can describe this ordered surface by a 2D lattice and plane group, with each node of this plane group being decorated by the same 3D motif. Both the plane group and the motif were determined through X-ray scattering experiments, which are able to provide statistical information averaged over macroscopic dimensions. In order to achieve high surface sensitivity, all the experiments were realized under grazing incidence. In the reciprocal space, each node (RSN) associated to the X-ray diffraction of such a surface exhibits an intensity distribution that corresponds to the 3D-Fourier transform of the motif shape. Thus, grazing incidence X-ray diffraction experiments were conducted with the aim of getting full 3D maps of several RSNs. 3a and ref. [START_REF] Matringe | Symmetric faceting of a sapphire vicinal surface revealed by Grazing Incidence Small-Angle X-ray Scattering 3D mapping[END_REF]). A longer thermal treatment induces the modification of the steps that present a zigzag pattern made of an alternation of [210] and [120] segments, respecting the surface symmetry (see fig. 3b). The complete description of the 1D to 2D transition process that could be related to meandering [START_REF] Pimpinelli | Equilibrium step dynamics on vicinal surfaces[END_REF][START_REF] Jeong | Steps on surfaces: experiment and theory[END_REF] of the steps is out of the scope of this article. Nevertheless, during this facet reorganization process, the overall surface step density, fixed by the miscut angle m, is kept constant (i.e. new steps cannot be created or destroyed) and the average step direction is conserved to maintain the global macroscopic surface orientation. Moreover, the steps cannot cross each other during the process. Therefore, the steps collectively organize in new zig-zag directions creating new facets along the [110] direction. According to this scenario, The centered rectangular cell associated to the cm plane group [START_REF]International Tables for Crystallography Volume A: Space-group symmetry[END_REF] is superimposed over the drawing displayed in fig. 3d (in green lines and red nodes). Of course, as shown hereafter, depending on the duration of the thermal treatment, the area of the facets evolves and thus the dimensions of this cell, which correspond to the x and y periods of the tiling, change accordingly. These values were determined through grazing incidence small-angle X-ray scattering (GISAXS) 3D mapping of the reciprocal space [START_REF] Babonneau | Formation of nanoripples on amorphous alumina thin films during low-energy ion-beam sputtering: Experiments and simulations[END_REF] by collecting series of (qy,qz)

GISAXS maps for different azimuthal angles  by increments of 0.5°. In this way, we were able to reconstruct the full 3D GISAXS map of the sapphire vicinal surface heat-treated for 32 hours (see fig. 4a). We observed that the scattered intensity is concentrated in almost vertical According to the schematic representation reported in fig. 3d, the period x is equal to the longest diagonal of the diamond-shaped (006) facet. Moreover, given the geometrical constraints, the ratio between x and y is

Λ x Λ y = √3 [ tan(𝛾 𝑚 -𝛼 𝑚 ) cos(𝛼 𝑚 ) tan(𝛾 𝑚 -𝛼 𝑚 )+sin(𝛼 𝑚 ) ] = 0.752 (1)
In accordance with Eq. 1, the nanoscaled Vasarely surface, which has been obtained through self-organization process after 32 hours of thermal treatment, exhibits two periods equal respectively to x = 84.5 nm and y = 112 nm. One of the key parameters of these surfaces is of course the control of the x and y values and accordingly the mean surface area of the (006) facets. In fact, it turns out that these values can be tuned by changing the duration of the isothermal treatment. The (qx,qy) GISAXS map obtained for a sample heat-treated for 96 hours is reported in fig. 4c. The overall distribution of the scattered intensity is very similar to the one reported in fig. 4b corresponding to a treatment duration of 32 hours. This is attested, in particular, by the fact that the centered rectangular reciprocal lattice drawn in fig. 4b can also be superimposed to the (qx,qy) map obtained after 96 hours of thermal treatment (see fig. 4c). It means that increasing the duration of the thermal treatment causes a homothetic growth of the facets through a coalescence process. From a quantitative point of view, we found that the y value increases from 112 nm after 32 hours of thermal treatment to 188 nm after 96 hours. The area of the diamond-shaped (006) facets is given by √3 6

Λ 𝑥 2 and thus according to the equation (1), it increases from about 2075 nm 2 after 32 h of thermal treatment to 5847 nm 2 after 96 h, i.e., by a factor of approximately 2.8 close to the duration ratio of 3. Therefore, this finding indicates that the 2D-coalescence process at 1250 °C under oxygen results in a quasi-linear isothermal growth in terms of (006) facet mean area.

Conclusions

To conclude, we have shown that a long duration heat-treatment realized under atmospheric pressure of oxygen of a specific vicinal sapphire surface can induce the formation of nanoscaled Vasarely surfaces made of (006) facets with a tunable size in the range of a few tens of nanometers. Such surfaces might offer considerable possibilities as, for example, substrates for the growth of active devices based on 2D-ordered nanostructures.

Figure 1

 1 Figure 1 Nanoscaled Vasarely surface on a sapphire crystal. (a) Modelling of the surface from analysis of X-ray scattering measurements. (b) Reproduction of the "Hexagon VII" Vasarely painting realized in 1969.

[

  fig.2b) as previously fully discussed[START_REF] Matringe | Symmetric faceting of a sapphire vicinal surface revealed by Grazing Incidence Small-Angle X-ray Scattering 3D mapping[END_REF]. The surface presents a periodic faceting with (006)

  fig.2f and 2gshow hk reciprocal space maps (RSMs) recorded for the 113 RSN at different l

Figure 2

 2 Figure 2 Topography of a nanoscaled Vasarely surface after isothermal treatment at 1250 °C under oxygen. (a) Sapphire crystallographic directions belonging to the (006) planes. Appearance of the 2D ordering evaluated by AFM, after 8 h (b); 9 h (c); 10 h (d) and 32 h (e) (insert: the corresponding 2D-Fourier transform); (f) and (g) 2Dreciprocal space maps orthogonal to the c* axis of the 113 RSN at 0.5 reciprocal lattice unit respectively below and above the center of this node for the sample heat treated for 32 h; (h) Geometrical description of the respective arrangement of the corresponding diffracting facets of the same sample.

Figure 3

 3 Figure 3 From 1D to 2D self-organization. (a) Modelling of the 1D ordered surface obtained after a thermal treatment of 8 hours; (b) formation of (2𝟏15) and (1𝟐15) facets alternated with (006) facets without specific order; (c) closepacked ordering of the {2𝟏15} facets; (d) modelling of the 2D surface made of diamond-shaped (006) facets separated by (2𝟏15) and (𝟏215) parallelogram-shaped steps; (e) AFM image of the surface recorded after a thermal treatment of 32 hours under oxygen.

  fig.4b). It is worth to note that two of the scattering streaks observed in this figure are strictly

Figure 4

 4 Figure 4 Nanoscaled Vasarely surfaces probed by GISAXS 3D mapping. (a) 3D GISAXS map and (b) projection of this map along the qz axis for a vicinal sapphire surface heat-treated for 32 hours at 1250 °C under oxygen. (c) Projection of the 3D GISAXS intensity along the qz axis for a treatment duration of 96 hours. The centered rectangular reciprocal cells are superimposed to the maps reported in the (b) and (c) 2D projections.

Acknowledgments

This work has been carried out partially within the QMAX Project No. ANR-09-NANO-031 funded by the French National Agency (ANR) in the frame of its program in Nanosciences, Nanotechnologies and Nanosystems (P3N2009). We acknowledge the synchrotron SOLEIL and the ESRF for provision of beamtime at the synchrotron radiation facilities. The authors express their gratitude towards the Limousin Region for financial support of the PhD salaries of AF and CM. and R. Guinebretière with an active contribution of S. Arnaud, N. Blanc and N. Boudet. GIXD measurements were done at the SixS beamline at SOLEIL by C. Matringe, D. Babonneau, R. Guinebretière, A. Coati and Y. Garreau. Quantitative analysis of both GISAXS and GIXD were done by C. Matringe, D. Babonneau, R. Guinebretière, A. Coati and Y. Garreau. R.

Guinebretière wrote the manuscript with contributions from all the co-authors.

Competing interests

The authors declare no competing financial interests.

Author contributions