Dr Shuaiyuan Han

Dr Erwan Nicol

Dr Frédérick Niepceron

Dr Olivier Colombani
email: olivier.colombani@univ-lemans.fr

Dr Sandrine Pensec

Laurent Bouteiller
email: laurent.bouteiller@upmc.fr

Oligo-urea with no alkylene unit self-assembles into rod-like objects in water

Keywords: self-assembly, hydrogen bond, rod-like nanoparticle, light scattering

Long and rigid objects formed by self-assembly in water are useful as templates or for their rheological or biological properties. They are usually obtained by combining hydrogen bonding and strong hydrophobic interactions brought by an alkyl or alkylene chain.

We report a simple access to well-defined rod-like assemblies in water based on a penta-urea sticker directly connected to PEO side chains. These assemblies are characterized by an average length of several hundreds of nanometers and a monodisperse radius (4.5nm) resulting from a reduced lateral aggregation of the stickers.

Introduction

One-dimensional assemblies (e.g. actin, microtubule, collagen) are ubiquitous in Nature and show a rich variety of functions. Therefore, the preparation of rod-like objects by self-assembly of synthetic building blocks in aqueous medium has been a major objective in the past decade. [1] These anisotropic systems offer a huge potential for alignment or percolation, and have been used to form hydrogels, [2] or as templates for various materials, [3] as well as for their biological properties. [4] A stable anisotropic structure can only form if it has a thermodynamic advantage versus competing assemblies, and several concepts have been demonstrated to allow significant selectivity. In the case of block-copolymers [5] or amphiphiles that form worm-like micelles, [6] the anisotropy simply results from the favorable volume fraction of the hydrophobic part. It is however possible to go beyond this geometrical effect by introducing anisotropic interactions such as in crystallization-driven self-assembly of amphiphilic block copolymers where the assembly is guided by the crystallization of the hydrophobic blocks [7] or in rod-coil molecules [8] where the rigid hydrophobic cores tend to pack in a parallel fashion to maximize van der Waals and pi-stacking interactions. In this respect, the directionality of hydrogen bonds (Hbonds) is particularly useful, as demonstrated in the case of beta-sheet forming peptides [9] and nanotube forming cyclic peptides. [10] In another approach, Hbonds and hydrophobic interactions have been combined in peptide amphiphiles: [11] a hydrophobic alkyl chain provides a strong driving force for assembly, but the anisotropy is controlled by the Hbonds between peptides. This synergy between hydrophobic interactions and Hbonding has also been successfully applied to many non-peptidic bolaamphiphiles with benzene tricarboxamide, [12] bis-urea [13] or bis-squaramide [14] stickers associated to alkylene units. In fact, in these systems, the absence of the alkylene part has been shown to result in no assembly, because of the strong competition of water molecules for the Hbond stickers. [13a] As a result, apart from peptides, all these anisotropic systems that form in water by Hbonding also contain a hydrophobic alkyl or alkylene part. Intrigued by the counterexample provided by beta-sheet forming peptides, we wondered whether the loss of the alkylene part could be compensated by increasing the number of Hbonding units and whether this simplified design would bring some particular advantage.

Results and discussion

Synthesis

In order to test the influence of the number of Hbonds, PEO3U, PEO4U and PEO5U were synthesized with poly(ethylene oxide) chains directly connected to a sticker composed of 3, 4 or 5 urea groups, respectively (Scheme 1, Table 1 SEC in DMF (PMMA standards). c) PEO3U, PEO4U and PEO5U were dispersed directly in water respectively at 5, 20 and 0.2g/L. The values of Mw, Nagg, Rg and Rh are true values (i.e. concentration independent). d) Aggregation number: Nagg = Mw/Mw,unimer = Mw/(Mn(NMR) x Đ).

O NH 2 n NO 2 H 2 N + triphosgene, DIEA DCM NO 2 N H O N H O n H 2 , Pd/C methanol NH 2 N H O N H O n NCO OCN TEA THF N H N H O N H O n N H O N H N H O N H N H O O n N H N H O N H N H O N H O n N H O N H N H O N H N H O O n O NH 2 n + THF N H N H O N H N H O N H O N H O O n n NCO N H N H O OCN NCO OCN H 2 O ether + + PEO3U PEO4U PEO5U 1 2 3 1 3 3 1 TEA THF 2

Influence of the preparation pathway

The three polymers differing solely by the number of urea units per molecule, were initially dispersed in pure water. PEO3U and PEO4U afforded transparent solutions upon direct dispersion up to at least 10g/L, whereas PEO5U solutions were only transparent up to 1g/L.

Static and dynamic light scattering (Table 1 andSI §3) revealed that PEO3U did not selfassemble in water but remained as unimers. PEO4U solutions hardly aggregated, and PEO5U

self-assembled into very large particles.

The direct dispersion method was then compared to other preparation pathways in order to determine whether the particles formed by PEO4U and PEO5U were under thermodynamic equilibrium. For PEO5U, the extent of aggregation depended on the preparation pathway, revealing that this polymer self-assembles in water as out-of-equilibrium structures. The extent of aggregation dramatically increased for PEO5U when it was first dissolved in DMSO followed by slow addition of water down to a DMSO/water content of 1/99 (vol/vol) (Figure 1a). Considering that PEO5U was dissociated in DMSO (the scattered intensity was extremely low in this solvent), this result reveals that self-assembly of PEO5U was promoted by the addition of water, but in a kinetically controlled way leading to a better organization using the "DMSO-route" than by direct dispersion in water (see §3.6. of the SI for more details). This result infers that although no alkyl chain is present in PEO5U, hydrophobic interactions probably still play a significant role in its self-assembly.

For PEO4U, the light scattering data could not be interpreted assuming only one monodisperse population of scatterers because their dimensions would be extremely large compared to their extent of aggregation. Based on the rather low value of Mw, it was concluded that PEO4U only weakly aggregated following the DMSO-route, leading to a solution containing rather small particles (possibly unimers) mixed with a few larger particles responsible to the rather large Rg value (see SI §3.7).

To summarize, upon dispersion in water, PEO3U formed unimers, PEO4U aggregated weakly, and PEO5U self-assembled strongly. Moreover, the extent of self-assembly was promoted by the DMSO-route. The rest of the paper therefore focuses on the large supramolecular assemblies formed by PEO5U in DMSO/water 1/99 v/v. 1.5x10 7 g/mol, length L = 530nm, radius r = 4.5nm) (SI §3.4). Dashed lines: Guinier analysis (only relevant when q.Rg < 1). For the direct dispersion in water, Mw = 2.6x10 6 g/mol and Rg = 28nm, see Table 1. For the DMSO-route, Mw = 1.5x10 7 g/mol and Rg = 150nm. b) CryoTEM image of PEO5U prepared following the DMSO-route (DMSO/water = 1/99, SI §2.2).

Quantitative characterization of PEO5U in DMSO/water 1/99 v/v

CryoTEM (Figure 1b) unambiguously revealed that the DMSO-route led to the formation of long (hundreds of nanometers) and rigid rods, polydisperse in length but quite monodisperse in radius. The average radius of the cylinders was estimated to be r = 4.5nm. This value lies between the Rh of PEO3U unimers in water (Rh ~ 2nm) and a fully extended PEO chain (~DPn x 0.28nm = 14nm) [15] and is therefore compatible with the size of the PEO5U elementary brick.

It is also in perfect agreement with the radius estimated from Small Angle Neutron Scattering experiments (See SI §4).

Treatment of the low-q data of the static light scattering (SLS) experiments with a Guinier model indicated that the particles formed by PEO5U following the DMSO-route were strongly aggregated: Rg = 150nm and Mw = 1.5x10 7 g/mol (see Figure 1 and SI §3.4). The high-q data (q > 8x10 4 cm -1 ) revealed a q -1 -dependency of the scattered light intensity, in agreement with the observation of long and rigid rods in cryoTEM (Figure 1b). [16] The SLS data were thus more quantitatively analyzed using a model of monodisperse cylinders [START_REF]The open source SASView Software[END_REF] (see SI §3.4). Since the qrange accessible with light scattering did not allow the observation of the region sensitive to the radius r of the cylinders (< 10 nm), the data were treated by fixing r = 4.5nm, according to the cryoTEM and SANS results (see SI §4). The best fit yielded a molecular weight Mw = 1.5x10 7 g/mol and a length L = 530nm, the latter value being in agreement with the cryoTEM micrographs and the value of Rg (Rg = L/Ö12 = 150nm for infinitely thin cylinders with r << L). [START_REF] Schmidt | iDynamic Light Scattering[END_REF] From the q -1 dependency of R/KC at the highest q values accessible by SLS, the mass per unit length ML of the cylinders could be deduced from (R/KC).q = ML.p (Figure S11). [START_REF] Catrouillet | [END_REF] ML = 2.6x10 4 g.mol -1 .nm -1 was found which is in excellent agreement with Mw/L = 2.8x10 4 g.mol -1 .nm -1 , confirming the consistency of the model used. It was previously determined that the distance between two Hbonded bis-ureas is 0.46nm. [20] Assuming that the distance between two H-bonded PEO5U units is the same, ML should be equal to ML(stack) = Mw, unimer/0.46 = 1.2x10 4 g.mol -1 .nm -1 if the PEO5U units are simply stacked one on top of the other with Hbonds forming between them. Considering that ML(stack) is about half of the experimental ML found, we conclude that about two PEO5U molecules can be found in the cross-section of the cylinders.

The normalized autocorrelation functions obtained from dynamic light scattering (DLS) measurements were analyzed in terms of a relaxation time (t) distribution. A single, slightly polydisperse, relaxation mode was observed for all the scattering wavevectors (Figure S8b).

However, G = <1/t>, the intensity-averaged relaxation frequency of the auto-correlation function was shown not to follow a q²-dependence along the whole q-range indicating a complex diffusion process (Figure S9). The diffusion coefficient determined by DLS, Dc, showed a q²-independence at low wavevector where the particles are probed at large observation scale. The apparent hydrodynamic radius (Rha) was thus estimated from the mean value of Dc at low wavevector according to the Stokes-Einstein equation, giving Rha = 80nm.

The hydrodynamic radius of rigid rods is related to their length L and their radius r according to equation (1) [21] .

𝑅 "# = % &'().+,( -../ 0 1+&/' .

(1

)
where L is the length of the cylinder and s=ln(L/r), with r the radius of the cylinder.

Using equation ( 1) and r = 4.5nm estimated from cryoTEM and SANS, L = 700nm was found for PEO5U. Thus, the DLS results confirm the SLS and cryoTEM data. With the exception of peptides, molecules that self-assemble into 1D structures in water through Hbonds rely on alkyl or alkylene parts to protect the Hbonds from water through strong hydrophobic interactions. [9][10][11][12][13][14] Usually, the Hbonds have been assumed to form along the rod

Hydrophobic interactions

Hydrogen bonding + hydrophobic interactions along the axis of the cylinder axis, while hydrophobic interactions caused lateral aggregation of 5 [13a] or 10 [14,22] molecules within the cross-section. In spite of the absence of alkyl/alkylene spacers, the self-assembled cylinders formed by PEO5U are very long and their characteristics are also consistent with a model where the Hbonds form along the rod-axis. Lateral aggregation is however strongly reduced (2 molecules in the cross-section) to previously reported systems (Figure 2). This reduced aggregation may be due to limited hydrophobic interactions or to the particular topology of the stickers. It is also of interest to compare the structure of the objects formed by PEO5U to a recently reported cyclic peptide-poly(ethylene glycol) conjugate. [23] Both systems form rod-like objects in water and are composed of a central Hbonded sticker (8 Hbonds versus 10 in the case of PEO5U) decorated by two PEG chains of the same size (2000g/mol). However, the aggregation number of the cyclic octapeptide is much smaller than in the case of the pentaurea (22 versus 2000).

Conclusions

We report a simple access to rod-like assemblies in water based on a penta-urea sticker decorated with PEO side chains. With the exception of peptides, this example is the first where the assembly is promoted by Hbonding aromatic stickers directly in water in the absence of alkyl chains or alkylene spacers to protect the Hbonds from the competitive water molecules.

Our cryoTEM, SLS, DLS and SANS results indicate that the formation of 1D structures is possible in water in spite of the competitive nature of the solvent and of the PEO arms. PEO3U and PEO4U hardly self-assemble in water even though they could form respectively 6 or 8

Hbonds per molecule, but PEO5U forms well-defined rods, hundreds of nanometer long and with a monodisperse radius. These pathway-dependent, out-of-equilibrium rods can conveniently be prepared using DMSO as an initial disruptive solvent. Finally, a weak lateral aggregation was noticed, with only two molecules in the cross-section compared to 5 or 10 for other systems.

Figure 1 .

 1 Figure 1. a) Normalized scattered light intensity versus wavevector for PEO5U prepared by

Figure 2 .

 2 Figure 2. Tentative model for PEO5U self-assembly.

Table 1 .

 1 and Supporting Information (SI) §1.2). Characteristics of unimers a,b and particles c obtained by direct dissolution in water (see SI).

	Scheme 1. Synthesis of PEO3U, PEO4U and PEO5U (n ~ 50)

polymer Mn (NMR) (kg/mol) a Mn (SEC) (kg/mol) b Đ SEC b Mw (kg/mol) c Nagg c,d Rg (nm) c Rh (nm) c

	PEO3U	5.1	6.1	1.2	4.7 +/-0.4	~ 1	< 20	2.0 +/-0.5
	PEO4U	5.1	6.8	1.1	85 +/-10	16 +/-2	< 20	10 +/-1
	PEO5U	5.0	7.0	1.1 2600 +/-200 450 +/-30 28 +/-4	20 +/-1
	a) 1 H NMR in

Acknowledgements

We thank Taco Nicolai for helpful discussions about the light scattering data treatment. SH acknowledges financial support from the China Scholarship Council. Jacques Jestin (LLB, Saclay) is acknowledged for assistance with the SANS experiments.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author


V Ilakovac

A Louat
email: alex.louat@u-psud.fr

A Nicolaou

J.-P Rueff

Y Joly

V Brouet

B J Kim

H Jin

S J Moon

J.-Y Kim

B.-G Park

C S Leem

J Yu

T W Noh

C Kim

S.-J Oh

J.-H Park

V Durairaj

G Cao

Phys E Rotenberg

Rev

Lett

Oxygen states in La and Rh doped Sr 2 IrO 4 probed by ARPES and O K edge RIXS

Iridates are often viewed as an equivalent of cuprates with strong spin-orbit coupling. As such, they offer a new way to study the physics of Mott insulators. The hybridization with oxygen is of course essential in both compounds and may play a crucial role in the doping process. It could be quite different in the two families, iridates being 5d transition metals with different orbital geometries. But this difference has not been studied in details so far. We present a combined ARPES and O K-edge study to document this aspect in Sr2IrO4, pure and doped with 4% La and 15% Rh. We evidence different charge transfer excitations and distinguish those associated with apical or in-plane oxygens. We observe a specific evolution of one excitation upon Rh doping, which suggests a more itinerant nature of the holes related to the apical oxygen. This gives information on the way doping proceeds in iridates.

I. INTRODUCTION

Sr 2 IrO 4 is attracting strong interest due to its large spin-orbit coupling, uncommon in the context of strongly correlated oxides. This strong SOC creates a situation with a half-filled band at the Fermi level, which is analogous to cuprates 1 . Indeed, both systems have an insulating and antiferromagnetic ground state 2 . The question on whether superconductivity could be obtained upon doping iridates has emerged 3,4 , although it has not been realized in bulk compounds up to now.

Despite the analogous situation of a half-filled band, there remains many differences, because the orbitals involved are quite different. In particular, this could give a different role to the oxygen ligand. In cuprates, the halffilled band has d x 2 -y 2 symmetry. In iridates, this role is fulfilled by the J ef f =1/2 band, with equal weight of the d xz , d yz and d xy orbitals, which do not point directly to the in-plane oxygens. On one hand, a weaker hybridization with oxygen is expected for t 2g orbitals compared to e g ; on the other hand, a larger one is expected for 5d transition metals. Moreover, contrary to the d x 2 -y 2 orbitals, the J ef f =1/2 orbitals have strong out-of-plane component, and may couple more strongly to the apical oxygen of the oxygen octahedra surrounding the Ir atom. Finally, it is well known that cuprates are charge transfer compounds, their charge transfer energy ∆ of 2-3 eV being smaller than the Mott gap (5 eV). Consequently, doped holes mainly reside in the four in-plane oxygens surrounding Cu and couple antiferromagnetically to it to form a Zhang-Rice singlet 5 . This could be different in iridates, where the Mott gap of about 0.6 eV is probably smaller than the charge transfer excitations 6 .

There have not been many experiments specifically testing the coupling with oxygen in iridates, beyond XAS studies 7 . We propose here a coupled ARPES and RIXS study, complemented by DFT calculations, to discuss this point. There have been many ARPES studies of Sr 2 IrO 4 , undoped 1 and doped [8][9][10] , but they focused on the Ir states near the Fermi level. Similarly, previous RIXS studies, both at the Ir L edge [11][12][13] and O K edge 14,15 , focused on the low energy excitations.

With ARPES, we observe directly the oxygen valence bands and find a good agreement with the calculation, rigidly shifted by -0.55 eV, validating the use of density of states obtained from such calculations. With RIXS, we distinguish the contribution of apical and in-plane oxygens, by tuning the incoming photon energy to their different excitation thresholds. We reproduce the RIXS spectra with a simple computation based on a convolution of a product of occupied and unoccupied density of states with the resonant lorentzian. We then focus on the fluorescent or Raman nature of these excitations, which gives information on the lifetime of the intermediate state of the RIXS process 11,16 . This lifetime will typically be longer for a localized state (giving Raman behavior) and shorter for a metallic state (giving fluorescent decay). We find that at least one charge transfer excitation evolves from Raman to fluorescent behavior, as a function of the photon excitation energy.

We then study the evolution of the RIXS spectra when doping with La or Rh, which are two alternative ways to get closer to the metallic state 8,17 , inducing respectively electron 9 and hole dopings 18 . We observe a large fluorescence in the case of Rh that seems to be specific to the apical site. Our study suggests a rather strong discrepancy between the hole and the electron doping, characterized by different couplings to oxygen, which was quite unexpected. This information on the behavior of the introduced carriers could be crucial to better understand the role of oxygen in the doping process.

II. EXPERIMENTAL AND CALCULATION DETAILS

The samples were prepared using a self-flux method as reported in 8 . ARPES experiments were carried out at the CASSIOPEE beamline of SOLEIL synchrotron, with a SCIENTA R-4000 analyser and an overall resolution better than 15 meV. All data shown here were acquired at a photon energy of 100 eV and at a temperature of 50 K. O K edge (≈ 530 eV) RIXS data were acquired at the SEXTANTS beamline 20 of the synchrotron SOLEIL by means of the AERHA spectrometer 19 with an overall energy resolution of 130 meV. The scattering angle was fixed and the incoming light polarization was horizontal (π polarization). X-ray Absorption Spectroscopy (XAS) measurements were performed in the total electron yield mode with the beamline resolution set to 110 meV. RIXS measurements were performed at 25 K. XAS calculation were performed with the FDMNES code 21,22 , using the full potential, relativistic, approach, including SOC and a Hubbard correction of U = 2 eV at the iridium site, while the core-hole screening effect was minimized. In order to calculate the RIXS spectra, the site and orbital projected density of states were calculated by Wien2k program [START_REF] Blaha | WIEN2K: An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties[END_REF] using the coordinates of the I4 1 /acd space group structure measured by Crawford et al. [START_REF] Crawford | [END_REF] .

III. RESULTS AND DISCUSSION

A. ARPES in pure Sr2IrO4

In Fig. 1a, the electronic structure measured by ARPES in Sr 2 IrO 4 is displayed on a large energy window. Weak Ir t 2g states are observed between 0 and -2 eV binding energies, as already discussed in details in the literature 1,8,9 .

At higher binding energies, between -3 and -9 eV, many well defined and dispersing bands are observed. Comparison with Fig. 1b shows that they correspond well to the oxygen bands expected from an ab-initio calculation. For more clear comparison, we use here a calculation in a simplified structure, neglecting the rotation of the oxygen octahedra. This calculation gives a very similar density of states compared to that using the full structure, shown in Fig. 1c and used later for the analysis of RIXS.

In our measurement, there is a predominance of the p z orbital of the apical oxygen, which has a very strong cross section for ARPES (as it is out of plane) and is in the top cleaving plane. Indeed, we show with fat bands the weight of oxygen apical p z in Fig. 1b and this underlines nearly all dominant features in the measurement: a very strong non-dispersing band at -6 eV, a band dispersing from -4 to -5.8 eV and the top of the bands near -3 eV. As indicated by blue dotted lines, these features overlap well with our measurements, but they have to be shifted down by 0.55 eV. This value is nearly that of the gap opening in the J ef f =1/2 band. However, it is not clear whether the oxygen bands should move together with the band gap. Alternatively, a recent GW calculation predicted a 0.5 eV shift of oxygen states relative to the Ir states 25 .

The density of states presented in Fig. 1c helps to identify the origin of these bands. We can distinguish four different regions labelled A, B, C, D on the graph. The A region corresponds to the partially filled t 2g Ir orbitals, hybridizing with oxygen, B to non-bonding oxygen orbitals, C to bonding oxygen orbitals of mainly t 2g character (with the exception of the non-dispersing band at -6 eV, which results from hybridization of Ir d z 2 and oxygen apical p z ) and D to bonding oxygen orbitals of mainly d x 2 -y 2 character. The e g bands are essentially empty extending between 1 and 5 eV above E F . In the unoccupied part, the gap between t 2g and e g is 0.8 eV.

To clarify the agreement, we present in Fig. 1d integrated ARPES spectra where the dominant contribution comes from the apical oxygen (red, bottom spectra) or the in-plane oxygen (green, top spectra) site. These are tuned by the angle of incidence of light on the sample surface. We compare them with the corresponding calculation of the DOS, broadened by a convolution with a gaussian of 0.2 eV linewidth, and shifted for 0.55 eV, taking into account the gap. The qualitative shape is well described, the peaks are broader for the in-plane oxygen with a third peak D p absent at the apical site. The shift of the calculated scale brings the main features in good coincidence, especially the first edge of the valence band at -2.8 eV both for in-plane and apical sites. We note that the shift might be slightly reduced for the strong feature B a , as it occurs at 0.2 eV higher binding energy compared to the the experimental peak. The other features are too broad to discuss the possibility of such a shift.

We have obtained similar results in La and Rh doped Sr 2 IrO 4 , ruling out a strong change of the structure of the occupied oxygen bands with doping. This comparison validates the use of ab-initio calculations to analyze other measurements, such as the RIXS data we will now present to get complementary information of the role of oxygen.

B. O K-edge RIXS in pure Sr2IrO4

O K edge XAS spectra are presented in Fig. 2a. The incident light wave vector was either normal to the sample surface (NI, red dots) or in grazing incidence (GI, black dots), at 20 • relative to the sample surface. The spectra are normalized to the same value far from the edge (580 eV). Its polarization was in the horizontal plane (see inset). Consequently, mainly p x /p y oxygen orbitals are excited in NI and p z in GI, which involves in turn Ir orbitals according to their interaction. Lower part of Fig. 2a shows calculated O K edge XAS spectra and their apical/in-plane site decomposition. The unoccupied Ir 5d states are shown in Fig. 2b and their orientation relative to the O 2p orbitals is sketched in Fig. 3: Ir d xz /d yz hybridizes with the in-plane p z and the apical p x /p y , while d xy couples mainly to the in-plane p x /p y . Higher energy e g orbitals and d z 2 /d x 2 -y 2 , point to the apical p z and the in-plane p x /p y , respectively.

According to the XAS calculation, the first two peaks, at hν = 528.80 eV and hν = 529.5 eV, correspond mainly to the in-plane oxygen, but the contribution of the apical oxygen to the first peak is relevant in the NI spectra. Previous works 1,7 attribute the first peak exclusively to the apical site, on the basis of the core-level shift, due to the difference in the apical/in-plane environments. Effectively, our calculation confirms that the core-level shift of the in-plane site is 0.29 eV higher compared to the apical site, which is more evident in the GI (θ = 0 • ) calculated spectra. Despite this difference, the DFT calculation of these unoccupied states underestimate the apical character of the first XAS peak, as will be demonstrated by RIXS measurements in the following.

The first peak is clearly visible in the NI spectra, meaning that it is mainly the result of d xz /d yz -p x /p y hybridization. The second peak is 2-3 times stronger for the GI compared to the NI. It thus selects d xz /d yz hybridized with the in-plane p z much more than the d xy coupled either with the in-plane or the apical p x /p y . At higher photon energy, the large feature/shoulder from about hν = 531 eV corresponds to the transition to the O 2p states hybridized with Ir e g states. The prominent peak at 531 eV is from apical oxygen 2p z coupled to 5d z 2 . O K edge RIXS spectra are presented in Fig. 4 in two ways, in the energy-loss scale (E L ), and in the emitted energy scale, see Fig. 4b,c, respectively. Features with Raman character are easily identified in the energy-loss scale, as their position does not change when the incident energy is altered, while structures with strictly fluorescent behavior do not shift in the emitted-energy scale. The Raman/fluorescent character of an excitation measures the rate of incoherent scattering of the pho- toexcited electron in the intermediate state of the RIXS process (the final state of the absorption process) 11 . If the photoexcited electron is promoted to a state where the information on its phase is conserved for a time of the order of the lifetime of the intermediate state, the absorption and emission processes will occur coherently, and the excitation will have Raman character. On the other hand, when the photoexcited electron is accommodated to a state where the information of its phase is quickly lost, the absorption and emission processes will be decoupled, and the feature in the RIXS spectra will have fluorescent behavior.

Previous studies have mainly focused on the low energy Raman structures, visible both in Ir L 3/2 edge RIXS spectra 12,13 and O K edge RIXS spectra 14,15 . These are spin-orbit exciton (feature labelled E at 0.65 eV energy loss in the Fig. 4b) and single-and bi-magnon excitations at about 100 meV, and 150 meV, respectively, visible in our data as a shoulder near the elastic peak. We focus here on the features at larger energy loss. The overall shape of the O K edge RIXS spectra strongly changes with the incoming photon energy, particularly in the hν = 528.8-530.0 eV range. At the excitation energy of the first peak, two rather sharp peaks are distinguished, which can be related to the B a and C a structures in the DOS (see Fig. 1c and Fig. 4c). When the incoming light is tuned to the second peak, corresponding mainly to the in-plane oxygen, besides two structures B p and C p , another large feature D p appears (see Fig. 1c and Fig. 4b).

From the RIXS data we can now conclude that the first peak is much more related to the excitation of the apical site than predicted by the XAS calculation.

As the RIXS spectra correspond quite well to the occupied DOS in Fig. 1, we attempt a description by a product of occupied and unoccupied site-projected density of states, convoluted by the resonant lorentzian: 26,27

F (Ω, ω) = d ρ( )ρ ( + Ω -ω) ( -ω) 2 + Γ 2 4
(1)

ρ and ρ are here densities of occupied and unoccupied oxygen 2p states, Ω and ω are the incoming and outgoing photon energies, respectively, and Γ corresponds to the core-hole lifetime broadening. The calculation procedure is illustrated in Fig. 5.

This simple approach assumes that each empty state is connected with each occupied state with the same matrix element. This approximation is not correct in the case of strong variations in the matrix elements related to the excitonic nature of the intermediate state and thus cannot describe well metal L edge RIXS of 3d compounds. It is however quite correct for the oxygen K edge even if it does not include any final-state-interference effects and thus it can not evaluate correctly the RIXS cross section. In particular, the region E L < 2 eV, where the spinorbit exciton (E in Fig. 4b), single-and bi-magnons are observed is not correctly described. Moreover, the gap of Sr 2 IrO 4 of about 0.5 eV is not included in the calculation. For the peaks at E L > 2 eV, one can expect a better agreement. Our calculation describes well fluorescencelike excitations, which shift in the E L scale. However, at the oxygen K edge, one can expect charge transfer (CT) excitation resulting of an annihilation of an oxygen 2p hole (strongly hybridized with a metal d orbital) by a photoexcited O 1s electron, followed by the decay of a different O 2p electron, filling the O 1s core hole and leaving the system in an excited state 16 . CT excitations have likely similar spectral form as the fluorescence, but are sometimes distinguished by their Raman behavior. Our calculation should therefore give good guidance to identify the origin of the peaks at E L > 2 eV. In Fig. 6, we compare experimental spectra measured under different polarization conditions for three selected photon energies with the calculation including the contribution from the oxygen dominating the XAS at this photon energy. This is the apical oxygen for hν = 528.8 eV (a) and 531 eV (c) and the in-plane oxygen for hν = 529.85 eV (b). There is a good overall correspondence and, in particular, this explains well the appearance of the D p feature in (b) and the alternation between sharp and broad features, corresponding to apical and in-plane. This comparison confirms that the contribution of the )]), while the intensity of the light emitted from the occupied pz states is decreased (enhanced). For the px/py states, the polarization dependence is opposite but less strong, as one of emitted components is perpendicular to the (kin,kout) plane and is always detected.

apical and the in-plane oxygen can be disentangled from the RIXS spectra, a conclusion already reached in 15 .

Spectra at different incident angles (θ, see Fig. 7) allow to probe the nature of the states involved in the photoabsorption and the emission process. The angular (θ) dependence at hν = 528.8 eV (Fig. 6a) is particularly strong. For ε almost parallel to the crystal axis c (θ = 20 • , GI), its intensity vanishes, while for ε almost perpendicular to c (θ = 65 • ), the intensity of all the features is strongly enhanced. This is similar to the XAS behavior and originates from the poor hybridization of Ir t 2g and apical oxygen 2p z states, strongly suppressing this absorption channel.

The angular (θ) dependence is smaller at hν = 529.85 eV (Fig. 6b) as it is connected to the in-plane oxygen, which has more equal contributions from p x /p y and p z . We have chosen this incident energy as our normalization reference because the spectra do not change considerably when θ is varied, at least in the region of the high energy loss we are interested in: the spectra at different incident energies are multiplied by a factor that yields the same integral intensity as the hν = 529.85 eV spectra of the three incident angles. Similar way of normalization was applied in the previous O K edge study of Sr 2 IrO 4 15 . The polarization dependence of the hν = 531 eV spectra (Fig. 6c), dominated again by the apical oxygen, is still strong, but completely different from that of the hν = 528.8 eV spectra. At this photon energy, there is a significant absorption for θ=20 • because of the contribution of d 2 z , which is strongly hybridized with apical p z . The intensity of the strong peak B a at E L = 5.5 eV is almost constant, but the side peaks A a and C a have opposite evolution with θ. The intensity for peak A a at E L = 4 eV is largest for θ = 20 • , while that for peak C a at E L = 8 eV is largest for θ = 65 • . This angular dependence can not be explained exclusively by the photoabsorption process but the detector position relative to the axes of the single crystal has to be taken into account. Indeed, the apical oxygen DOS (see Fig. 1) has a p x /p y related structure at low energy (A a ) and a p z feature at high energy (C a ).

For the detector position at 95 • relative to the incoming light, the emission from the p z states is enhanced for θ = 65 • , while this from the p x /p y is enhanced for θ = 20 • . The agreement of the polarization dependence with the orbital content predicted by the calculation validates this analysis.

In our calculation, all three structures are reproduced by the same procedure, which underestimates dd excitations. In a previous work, E L = 3.5 eV (our B p feature) was attributed to the equivalent of the dd (t 2g → e g ) excitation, while the two large structures in the range of E L > 6 eV are attributed to the charge-transfer (CT) excitation. 15 Nevertheless, at the Ir edge 11 , there is a peak centered at 3.2 eV, which is attributed to a CT excitation. The B p feature is the most pronounced feature in our O K edge RIXS spectra. While we cannot exclude that dd excitations contribute to this peak, we would rather expect a small dd cross section at the excitation energy corresponding to O 2p -Ir t 2g hybridized states. Moreover, there is a clear evolution of the lineshape as a function of photon energy from a sharp peak to broad to sharp again, which is well described by the change of the DOS from apical to plane and is not expected for a dd excitation. Note that a similar evolution is observed for the C p peak.

We now discuss the Raman/fluorescent behavior of the different features in our RIXS spectra. The fluorescent nature of the features observed at photon energies higher than hν = 530.5 eV is clear. For lower energies, and for E L > 2 eV, there are several components which are not resolved in the spectra, resulting in structures which are broad compared to the change of the excitation energy. Therefore their Raman/fluorescence behavior is not straightforward to decide.

At hν > 530.5 eV, sharp A a , B a , C a peaks have obviously fluorescent behavior (see Fig. 4c). The energy separation of structures B a and C a is slightly smaller (0.4 eV) than the calculated value. We noticed that the B a /C a separation is slightly (0.2 eV) reduced in ARPES, but this separation is enlarged in RIXS. This may mean that one of the peaks has not fully reached the florescence regime. The calculated structure A a is not observed in the hν = 528.80 eV spectra (see Fig. 4a). Its cross section should indeed be decreased, when the low-energy excitations, like spin-orbit exciton, single-and bi-magnons, involving O 2p states hybridized with Ir t 2g states, are prefered RIXS channels.

The peak E is the only feature showing obvious Raman behavior (see Fig. 4b) in the whole excitation energy range. We note that it disappears for photon energies corresponding to excitations above t 2g states. For the three large structures B p , C p , D p and the E-peak shoulder A p , which are quite broad and appear in a small excitation energy range of the in-plane oxygen, it is less clear. 2D map in previous O K edge RIXS measurements with the grazing incident π polarized (ε almost parallel c) in the range of hν = 529.4-529.9 eV, suggests that E L = 3.5, 6.5 and 8.8 eV have Raman character. 15 We agree that the edge of B p in Fig. 4 is better described by a Raman mode up to 529.85eV. Although it changes from apical to in-plane over this photon energy range, our ARPES spectra have shown that both DOS (apical and in-plane) have a similar edge so that this should not produce a shift. But, at 530.2 eV, there is clearly a double peak structure, that seems to originate from the apical contribution in fluorescence regime and the in-plane contribution still in Raman regime. This observation will acquire more significance as we turn to the doping dependence.

C. La-and Rh-doped Sr2IrO4

O K edge XAS spectra of pure, 15% Rh-, and 4% La-doped Sr 2 IrO 4 are shown in Fig. 8, for two incident angles θ = 90 • (NI) and 20 • (GI). Rh-doping induces holes 10,18,28 , while La dopes with electrons 8,9 . These two different ways of doping move both compounds close to a metallic state 8 . For Normal Incidence (θ = 90 • ), the double pre-peak structure, corresponding to apical and in-plane oxygens, is clearly resolved in the pure and La doped spectra, but less clearly in the Rh doped case. This might be due to a low energy shift of the in-plane p x /p y states for about 0.3 eV, induced by Ir/Rh substitution, while the apical p x /p y states are less affected. A similar effect was observed for pure Sr 2 RhO 4 7 and as a function of Rh doping 29 . This indicates a modification of the electronic environment of both oxygen sites, probably related to the weakening of the in-plane O 2p -Ir t 2g hybridization.

At the same time, for GI (θ = 20 • ), the strong feature at hν = 528.85 eV is quite similar in the pure-and the La-doped-sample spectra. Conversely, for the Rh-dopedsample, it is shifted to the lower energy, similarly to the NI peak, and enlarged on the low energy side. This might be due to the Rh-doping related accommodation of holes on apical oxygen sites and/or a distribution of values of the apical-plane splitting. O K edge RIXS spectra of the three samples are shown in Fig. 9, in the emitted energy scale. In order to compare the spectra, we adopted the same normalization scheme as for the pure system: the spectra at different excitation energies are multiplied by a factor that yields the same integral intensity as the hν = 529.85 eV spectra of the three compounds. Regarding excitations at less than E L = 1 eV, for the three systems their intensity is highest in the hν = 529.5 eV spectra, even if it seems to resonate at slightly lower energy in the Rh-doped system.

For the excitations at higher energy loss, we first remark that the spectra at high photon energies overlap well in the 3 compounds, implying there is no strong modification of the DOS itself. Regarding high energy loss structures, pure and 4% La-doped samples are quite similar. The 15% Rh-doped sample however shows differences. The fluorescent part of the B peak appears at much lower photon energy and/or with larger intensity. It also extends to lower E L in the hν = 528.80 eV spectrum.

On one hand, one would expect fluorescence to dominate more strongly when a sample becomes more metallic, which agrees with the trend observed for Rh doping. However, the difference between La and Rh is difficult to understand, as their resistivities are relatively close 8 . More specifically, the behavior in the Rh case makes it also quite clear that the fluorescent part of B is sharp, typically like the contribution from the apical site. This suggests a different behavior for electrons coupled to the apical or the in-plane site, the former ones being more strongly delocalized. From the XAS spectra, we know that the threshold for exciting apical and in-planes oxygen sites are somewhat different for Rh. From this alone, we would only expect the in-plane contribution (i.e. large B, C and D peaks) to appear at slightly smaller photon energies, which does not explain our observation. We therefore suggest that there is a real evolution of the environment of the apical site specific to Rh, which favors delocalization of the charge carriers on this site. Conversely, electrons doped by La may bound strongly with the in-plane oxygen and delocalize less efficiently.

IV. CONCLUSION

Our study gives detailed information on the oxygen states in iridates. With ARPES, we have fully mapped their dispersion, which is found in good agreement with DFT calculations, albeit with a shift of 0.55 eV to lower binding energies, that may be explained by non-local effects 25 . Using RIXS, we show how the contribution of apical and in-plane oxygens can be decoupled using different photon energies and polarizations. Their shape is well described by a convolution of a product of occupied and unoccupied DOS with the resonant lorentzian. At least one of the features in the high-energy-loss part of the RIXS spectra evolves with photon energy from a Raman (localized-like) to a fluorescent (itinerant-like) behavior. Interestingly, we find a strong difference in the case of 15% Rh substitutions (hole doping), where electrons excited to apical oxygen (unccupied) states are found to be more itinerant. This is not the case for 4% La substitu-tions (electron doping), although this doping provides a similar metallic state as the Rh doping in transport measurements. This suggests that the coupling with oxygen plays a different role in hole and electron doped case. Our O K edge RIXS results reveal an asymmetry between electron and hole doping, which is important to take into account for an accurate description of the electronic properties of doped iridates. ANR-10-LABX-0039-PALM) is acknowledged.

V.I. thanks Amélie Juhin and F. M. F. de Groot for enlightening discussions.

Figure 1 .

 1 Figure 1. (a) Energy-momentum plot of the ARPES intensity obtained in Sr2IrO4 at 100 eV photon energy along a path ΓMΓ'XΓ. (b) Band calculation for Sr2IrO4 including spin-orbit coupling but neglecting the rotation of oxygen octahedra. The weight of the apical pz oxygen orbital is represented as marker size. Blue dotted lines underline the correspondence between measured and calculated features, after a shift of 0.55 eV. (c) Density of states calculated for Sr2IrO4 including oxygen octahedra rotations. Partial weights are shown, as indicated. (d) Integrated valence bands (VB) measured by ARPES in experimental configurations favoring apical (red, top) or in-plane (green, bottom) oxygens. Calcualted DOS in (c) convoluted by a Gaussian for apical (top) and in-plane (bottom) oxygens. Indexes a/p of the structures A, B, C, D, are for apical/in-plane. The energy scales are shifted between spectra (bottom) and calculation (top) by 0.55eV.

Figure 2 .

 2 Figure 2. (a) Polarization dependence of the measured (dots) and calculated (lines) O K edge XAS spectra. Inset presents the experimental geometry. Doted lines are corresponding contributions of the apical and the in-plane oxygen sites. (b) DOS of the unoccupied Ir 5d states, hybridizing with O 2p states, for the case when the apical oxygen site is excited.

Figure 3 .

 3 Figure 3. Sketch of the relative positions of O 2p and Ir 5d orbitals. Gray lobes in upper/lower panels present Ir 5d t2g/eg orbitals. Red/blue/green lobes are for O px/py/pz orbitals hybridized with corresponding Ir states. Light gray square indicates the xy plane, including Ir and in-plane oxygens.

Figure 4 .

 4 Figure 4. O K edge XAS (a) and RIXS (b, c) spectra of Sr2IrO4. RIXS spectra, collected at photon incidence of θ = 45 • , are presented in the energy loss, EL (b) and emitted energy scale (c). Incident photon energies are indicated on the XAS spectrum by the line of the same color. Experimental geometry showing the incoming wave vector (kin), outgoing wave vector (kout), the direction of the incoming photon polarization (ε) related to the surface of the sample and its crystal axis c, is shown in the inset of (c).

Figure 5 .

 5 Figure 5. Illustraion of the RIXS calculation procedure taking the O apical DOS as example. For each energy loss EL = Ω -ω, the occupied DOS is multiplied by the unoccupied DOS shifted for (Ω -ω), and the product is then convoluted by the resonant lorentzian.

Figure 6 .

 6 Figure 6. Incident angle dependent O K edge RIXS spectra of Sr2IrO4 for hν = 528.80 eV (a), 529.85 eV (b), 531.00 eV (c). The (grazing) incidence angle was set to θ = 20 • , 45 • and 60 • . The vertical solid line indicates the position of the spin-orbit exciton. The calculated spectra should be compared to the experiment performed at θ = 45 • . The Ba/Bp is aligned with the most prominent peak in the experimental spectra.

Figure 7 .

 7 Figure 7. For the detector position at a scattering angle which is almost equal to 90 • , the absorption to the unoccupied pz states is enhanced [θ = 20 • ] (suppressed [θ = 65• )]), while the intensity of the light emitted from the occupied pz states is decreased (enhanced). For the px/py states, the polarization dependence is opposite but less strong, as one of emitted components is perpendicular to the (kin,kout) plane and is always detected.

Figure 8 .

 8 Figure 8. XAS spectra of pure, 4% La-doped and 15% Rhdoped Sr2IrO4, for two incident angles, NI (θ = 90 • ) and GI (θ = 20 • ).

Figure 9 .

 9 Figure 9. Selected O K edge RIXS spectra of pure, 4% La-and 15% Rh-doped Sr2IrO4 presented in the energy loss (a) and emitted energy (b) scale, collected at θ = 45 • . Vertical solid lines in the energy loss scale indicate features with Raman behavior, while dotted lines in the emitted energy scale points to features which behave as fluorescence and are related to the DOS structures.

Financial

  support from the French National Agency ANR-15-CE30-0009-01 SOCRATE and the Investisse-ment d'AvenirLabEx PALM (Grant No.


Vincent Bacot

Guillaume Durey

Mathias Fink

Emmanuel Fort
email: emmanuel.fort@espci.fr

Phase-conjugate mirror for water waves driven by the Faraday instability

Significance Statement (120-word-maximum statement about the significance of their research) Non-linear and parametric processes are central in mastering wave propagation. They are at the core of numerous applications in optics and acoustics. In the case of water waves, the Faraday instability is the archetype of a parametric excitation. A periodic vertical excitation results in the destabilization of a liquid interface. However, it has always been interpreted as an instability, completely overshadowing its potential to control water wave propagation. In this paper, we unveil the ability of this temporal modulation to act as a phase-conjugate mirror. We show that waves refocus to any initial emitting source position, whatever the shape of the mirror. This paper thus revisits this historical instability and opens up new exciting perspectives in water wave manipulation.

Introduction

In 1831, Faraday discovered that submitting a bath to a periodic vertical acceleration resulted in the destabilization of its surface above an acceleration threshold [START_REF] Faraday | On a Peculiar Class of Acoustical Figures; and on Certain Forms Assumed by Groups of Particles upon Vibrating Elastic Surfaces[END_REF] . This parametric instability, known as the Faraday instability, is driven by the modulation of the effective gravity. Surface waves appear as standing waves, modulated at half the excitation frequency. This instability has been extensively studied, both experimentally and theoretically, in the steady-state regime [START_REF] Benjamin | The stability of the plane free surface of a liquid in vertical periodic motion[END_REF][START_REF] Douady | Experimental-Study of the Faraday Instability[END_REF]4 . In this regime, the wave pattern is dominated by the geometry of the bath, the boundaries and menisci acting as wave sources. The observed wave field is thus a combination of some modes of the cavity defined by the shape of the bath [START_REF] Benjamin | The stability of the plane free surface of a liquid in vertical periodic motion[END_REF] . The extended literature on the Faraday instability has focused on aspects like the non-linear hydrodynamic regimes [START_REF] Westra | Patterns of Faraday waves[END_REF][START_REF] Domino | Faraday wave lattice as an elastic metamaterial[END_REF] , the influence of the depth or of the viscosity of the liquid [START_REF] Kumar | Mechanism for the Faraday instability in viscous liquids[END_REF][START_REF] Cerda | Faraday's instability in viscous fluid[END_REF] , the excitation under several frequencies [START_REF] Kumar | Mechanism for the Faraday instability in viscous liquids[END_REF][START_REF] Douady | Pattern selection in faraday instability[END_REF] or the ability to generate droplets [START_REF] Tsai | Linear theory on temporal instability of megahertz faraday waves for monodisperse microdroplet ejection[END_REF][START_REF] Tsai | Ejection of Uniform Micrometer-Sized Droplets from Faraday Waves on a Millimeter-Sized Water Drop[END_REF] . However, all these studies only consider the steady-state regime with standing wave patterns, after the transient growth of the instability, and no external wave input. This is in stark contrast with non-linear (NL) optics, in which parametric forcing has been very fruitful to master optical beams. In particular, this has enabled the design of phase-conjugate mirrors [START_REF] Hellwarth | Generation of time-reversed wave fronts by nonlinear refraction[END_REF][START_REF] Yariv | Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing[END_REF][START_REF] Born | Principles of Optics[END_REF] (PCM). Typically, a monochromatic wave field with an arbitrary shape enters the non-linear optical crystal and generates a counter-propagating, time-reversed wave field. In the four-wave mixing (FWM) implementation, the parametric modulation is provided by two external pump beams [START_REF] Yariv | Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing[END_REF] .

On the contrary, in fluid mechanics, the Faraday instability has always been analyzed within the framework of parametric instabilities, thus ignoring its additional ability to control and manipulate propagating waves. In this paper, we revisit it as a means to perform phase conjugation on water waves, through a periodic time modulation of the medium properties. We show that it does act as a PCM for water waves at the Faraday frequency.

The paper is organized as follows. In the first section, we show the formal analogy between the Faraday instability and an optical PCM obtained by a FWM configuration in a NL optical material. Section 2 is devoted to the description of the experimental implementation, in which the water-wave PCM is confined to a specific area in the bath, taking advantage of the Faraday threshold dependence on the liquid depth. Section 3 presents and discusses the experimental results for standard point and plane source geometries. In addition, using arbitrary mirror shapes, we show that PCM refocusing is independent of the geometry of the mirror. In this section, we provide evidence for the analogy between the Faraday instability and an optical PCM produced by NL optics. In optics, one of the most common ways to achieve phase conjugation is through the backward degenerate four-wave mixing configuration, first proposed by Hellwarth in 1977 [START_REF] Hellwarth | Generation of time-reversed wave fronts by nonlinear refraction[END_REF] . A schematic of the configuration is shown in Figure 1a). When a monochromatic point source emits a wave into the NL optical crystal, the PCM generates a counter-propagating wave which refocuses at the source position. This clearly shows that a PCM acts as a monochromatic timereversal mirror. Note also that contrary to standard mirrors, this is independent of the shape of the crystal. It is based on the third-order susceptibility of a non-linear (NL) optical medium. Two counter-propagating, high-intensity plane waves, called pump waves, are sent on the NL medium.

Theoretical developments

Neglecting the vector nature of the fields, their associated electric fields are given by and , where is the time angular frequency, the wave vector, and the wave amplitude. When a signal wave with the same angular frequency and a wave vector is sent into the NL medium, it generates a counter-propagating idler wave with the same angular frequency and a wave vector equal to . Under the usual assumptions that the high intensity pump waves are not depleted by the (weak) NL interaction and remain constant [START_REF] Born | Principles of Optics[END_REF] , the signal and the idler modes are coupled, and can be expressed as a set of coupled equations in the presence of the non-linear polarization:

(

where . The two modes and are the phase-conjugates of each other: they are coupled by sources proportional to their complex conjugate (right hand-side of the equation). This is equivalent to time-reversal for monochromatic waves. The real electric field of the idler mode at time satisfies:

(

where is the real electric field associated with the signal mode at time . Note that the effect of the pump waves is equivalent to a temporal modulation of the refractive index with the doubled frequency 14 :

(

Let us now show that these equations are formally analogous to the ones which describe the propagation of water waves on a vibrated bath with a time angular frequency . In the deep water regime (where the liquid depth verifies ), neglecting viscosity, the equation for the free surface elevation field ζ can be written in the spatial Fourier space in the form of a Mathieu equation [START_REF] Benjamin | The stability of the plane free surface of a liquid in vertical periodic motion[END_REF][START_REF] Pietschmann | Faraday waves under time-reversed excitation[END_REF] :

(4) with the norm of the wave vector, and the time angular frequency satisfying the dispersion relation of linear gravito-capillary waves , being the gravity acceleration, the surface tension and the density of the liquid. is the amplitude of the acceleration of the bath vibrated at the angular frequency . In the general case, Floquet analysis shows that the solutions of equation (4) are of the form , where is a complex number and is a periodic function at the angular frequency . Only the unstable modes corresponding to are amplified 4 . In the weak excitation limit ( ) valid in our experiments, it can be shown that these unstable modes must satisfy the dispersion relation of free surface waves, with frequencies selected by multiples of half the bath vibration frequency:

, where is an integer. Let us assume that in such a modulated bath, a signal plane wave of frequency and of wave vector is introduced. Its spatial Fourier transform is Due to the modulation of the bath, other frequency components of the wave field may emerge. However, in the weak excitation limit, only the components of frequency are generated 2 , so that we can assume that the Fourier transform of the total wave field can be written:

, where . and are slowlyvarying envelopes compared to in the weak approximation limit. Equation (4) can therefore be decomposed into a set of coupled equations for the components at angular frequency and :

(5)

These coupled equations are identical to the coupled equations ( 1) describing the PCM in optics, the change of sign of the angular frequency being equivalent to a change of sign of the wave vector.

Thus, the Faraday instability can be interpreted as a PCM for surface waves. Note that the formal analogy with optical PCM predicts that the initial and the phase-conjugated waves converge to the same amplitude at large times [START_REF] Born | Principles of Optics[END_REF] . This is why the modulation of the medium and thus the Faraday instability are associated with standing waves in the long run. In the next section, we give experimental evidence of this new interpretation of the Faraday instability. is the time taken by the emitted wave to reach the wall after crossing the deep end of the bath and is the emission duration.

Experimental setup and procedures

Figure 1b) shows a schematic of the water-wave analogous configuration using the Faraday instability. The role of the counter-propagating vertical optical pumps is played by the vertical vibration of the bath. Since the Faraday threshold strongly depends on the depth of the liquid [START_REF] Eddi | Unpredictable Tunneling of a Classical Wave-Particle Association[END_REF] , we use the bathymetry (underwater topography) to split the bath into distinct areas mimicking the wave propagation in free space or in the optical NL crystal. In the shallow region, the excitation acceleration is far from the threshold, hence the vertical vibration has only negligible effects on the wave propagation: it appears similar to that of an unperturbed bath. In the deep water region, the vertical acceleration is set slightly above the Faraday threshold. In this region, any propagating wave triggers a counter-propagating wave.

The experimental setup is shown in Figure 2. A 25x25 cm 2 square container (made of polylactic acid with a 3D printer) is set on a vertically vibrating shaker (B&K Vibration Exciter Type 4808, controlled by means of a waveform generator). The bath is divided in two adjacent rectangular areas with different depths: the shallow and deep ends are respectively mm and mm deep.

The sinusoidal vertical acceleration satisfies with the frequency set to 35 Hz, and the Heaviside step function. The walls of the bath are tilted 30° from the vertical direction, in order to limit the excitation of waves at the excitation frequency by the menisci at the boundaries. Care is taken that the water wets the slope on each side.

The wave source is obtained using a loudspeaker operated by a second waveform generator. The loudspeaker produces variations of pressure in a hermetic cavity equipped with plastic tubing.

Depending on the type of geometry needed for the wave source, the tubing can be plugged on various systems attached to the water container: a straw, whose tip is placed above the shallow end of the pool, at a distance of 2 cm ahead of the boundary with the deep end, is used as a point source;

a flared hollow solid, with a series of one-millimeter holes arranged in a line with a three-millimeter spacing, is used as a planar source. An intense LED light (Constellation 60) is placed approximately one meter above the bath. A sheet of diffusing paper, on which a radial Gaussian filter is printed, is positioned under the source to create a gradient in the illumination light. A semi-reflecting mirror is used to redirect the light reflected by the waves towards the camera, which records movies at 500 frames per second.

As shown in Figure 2b), for a given acceleration above the Faraday instability threshold in the deep end, the instability grows exponentially with a typical time , which depends on the distance to the threshold. This time is evaluated through the monitoring of , the variance of the intensity of the camera pixels imaging the deep end of the pool. Initially, the surface of the water is flat, is close to zero. At time the vertical vibration is turned on. The menisci at the boundaries of the bath act as a source of propagating ripples at the excitation frequency, hence increases to a small value. In the absence of a source signal, this noise triggers a pattern of standing waves at the Faraday frequency which starts building up in the deep end of the pool (see inset of Figure 2b). This results in a sharp increase of . This instant defines which is typically of the order of ten seconds Once the instability is fully established, the waves reach their maximum amplitude and reaches a plateau.

To produce a signal wave, the loudspeaker is sent ten periods of a sinusoid at the Faraday frequency Hz, over a duration . Figure 2c) shows a top view of the bath during the emission of a signal wave from a point source. Figure 2d) shows the space-time plot of the wave emission by the point source, taken along the x-axis (red line in Fig. 2c). The signature of the outward-propagating nature of the waves is readily observed in the wave front orientation. It takes a time for the waves to reach the deep end and cross it. Both Figures 2c) and 2d) were produced in the absence of a vertical excitation of the bath.

A typical experiment combines the vertical vibration with the production of a signal wave. It is crucial in these experiments that the Faraday instability be triggered in the deep end by the signal wave and not by the fluctuations of the bath or by the boundary-induced ripples. Thus, the time sequences are set so that . This is possible since is typically of the order of ten seconds, while is less than one second.

Despite the precautions taken to physically attenuate them, the boundaries of the bath emit small ripples throughout our experiments. Although they do not interfere with, nor prevent the observation of the waves generated by the PCM, these ripples were removed numerically to improve the aesthetic appearance of the movies. Since all the experimental devices were fully synchronized, enabling a very high level of reproducibility, each experiment was run twice under the exact same conditions. A first movie was recorded with the vertical vibration of the bath alone, then a second one followed, featuring both the vertical vibration and the signal wave. To obtain a better image of the signal wave and of its associated counter-propagating wave, the first movie was subtracted from the second one. This operation removes the boundary-induced ripples, which superpose exactly on the two movies. All the following experimental movies were produced using this method.

Movie S1). c) Space-time plot associated to the image sequence along the solid red line in a) and b). d)

Normalized spectral response of the water-wave PCM for various acceleration amplitudes .

Experimental results and discussions

We first study the response of the water wave PCM to the signal wave from a point source in the configuration presented in Fig. 2 (see Supplementary Movie S1). Figure 3a) shows a snapshot of the wave field during the emission at s. The signal wave is propagating outward and enters the deep end of the bath. This image appears similar to that of Fig. 2c), which is taken in the absence of a vertical excitation, because the PCM does not yet generate a phase-conjugated wave. In Fig. 3b), at s, the source has stopped emitting and the PCM is emitting a phase-conjugated wave in the shallow end, that refocuses at the position of the source and diverges again on the other side of it.

The signal wave has triggered the Faraday instability in the deep end and has imposed its shape as a boundary condition. In the absence of this perturbation, the surface of the deep end of the bath would remain flat since . In the deep end, the Faraday instability is visible with standing waves at half the excitation frequency, with a shape that is given by the signal wave field and not by the boundaries of the basin. However, the boundary-induced ripples eventually take over, imposing the wave field pattern in the deep end, as in standard stationary Faraday experiments. The shape of the propagating wave produced in the shallow end is thus lost. Figure 3c) shows the space-time plot along the horizontal x-axis of Fig. 3a) and 3b). It clearly shows that after the initial step, in which the wave source produces an outward propagating field, a counter-propagating wave is produced in the deep end, resulting in the presence of symmetric wave fronts. These phase-conjugated waves are sustained even after the source has stopped emitting.

We now focus on the response of the PCM to various excitation frequencies. The wave profiles of the source are identical to the one described above, except for their frequency. The amplitude of the phase-conjugated wave is measured for each frequency through the light reflected on the surface reaching the camera, at the position of the source. Unfortunately, this measurement is not quantitative: the amplitude of the refocusing phase-conjugated waves can only be inferred qualitatively from the change in the reflection amplitude. The spectral response of the PCM is thus normalized by its maximum amplitude value obtained at the Faraday frequency. Figure 3d) shows the normalized spectral response of the PCM for three different bath accelerations, , and

. The shapes of the spectra are very similar with a sharp peak at the Faraday frequency, and the full width of the peak is approximately Hz. This spectral response is independent of the excitation amplitude of the vertical excitation of the bath. The sharp decrease of the PCM response when the source frequency deviates from the Faraday frequency is due to the phase mismatch between the incident wave and the phase-conjugated wave, due to a large dispersion in water. For Hz, the coherence length is equal to approximately 5 Faraday wavelength which is already smaller than the width of the deep end [START_REF]Nonlinear Optics[END_REF] . Although the phase-conjugated wave amplitude is not obtained qualitatively, the reflected light variations increase significantly with the acceleration amplitude: by a factor 1.5 and 2.75 when the acceleration amplitude increases by 1.35 and 1.6, respectively. This is the signature of a strong increase in the efficiency of the PCM with increasing pumping. The theoretical dependence given by the model in eq. ( 5) gives a linear dependence with the excitation amplitude. Another classical configuration which exemplifies the difference between a standard mirror and a PCM is the case of an incident plane wave. Contrary to a standard mirror, for which the incident wave is reflected symmetrically to the normal of the mirror plane, a PCM produces waves which are phase-conjugated and counter-propagating. Figures 4a) and4b) show two snapshots of a plane wave impinging towards a water-wave PCM and the resulting reflected phase-conjugated emission from the PCM (see Supplementary Movie S2). The time profile of the source is the same as that of Figure 2d. Figure 3a shows the wave field during the emission at time s and Figure 3b when the source has stopped emitting, at s. The triggered Faraday instability is clearly visible in the deep end as well as the counter-propagating phase-conjugated wave emitted in the shallow end.

The ability of PCMs to generate phase-conjugated waves is independent of their shape because there are no phase-matching conditions involved. Unlike standard mirrors, they can assume any shape. We have tested this unique property in an experiment in which the mirror is shaped like France, as

shown in Figures 4c) to 4f). The shallow end of the bath is located within the borders of the country.

It is surrounded by a deeper bath acting as a PCM for water waves when excited parametrically. experiment the wave speed modulation is substantial. This is in sharp contrast with the typical values for optical PCMs, for which the wave modulation is very small, even for very large amplitudes of the pump beams (pulsed beams must be used to increase the efficiency of these types of PCMs). Indeed, typical values of the third-order susceptibility are in the range m 2 /V 2 .

The phase-conjugated mode is generated immediately as the source wave enters the deep end of the bath. Each spatial wave front generates periodically at the Faraday frequency a co-propagating wave, which adds up coherently with the incident wave, and a counter-propagating wave, which refocuses on the source. As the wave propagates into the deep end, the produced waves add up constructively from each propagating wave front, increasing the forward-propagating mode and the phaseconjugated mode. Following eq. ( 5), the amplitude of the phase-conjugated mode increases with the acceleration modulation. The right-hand side of the equation represents the sources which produce this mode. There is no discontinuity in the phenomenon due to the Faraday instability threshold.

However, above the Faraday threshold, ripples produced by the boundaries of the deep end of the bath will eventually get amplified. The characteristic time involved is long since the sources which produce it are initially null (right hand side of eq. ( 5)). The growth rate is the result of the competition between the modulation of the vertical acceleration and the damping rate (which in turn is fixed by viscosity and boundary conditions). Self-oscillations can also be reached in optical FWM configurations but usually for crystal thicknesses much larger than the light wavelength [START_REF] Yariv | Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing[END_REF][START_REF] Bloom | Observation of amplified reflection by degenerate four-wave mixing in atomic sodium vapor[END_REF] .

The efficiency of Faraday-based PCMs opens up the possibility of creating thin sub-wavelength PCMs, that could be used for time-reversed flat lenses [START_REF] Pendry | Time Reversal and Negative Refraction[END_REF] , whose implementation is still very challenging with other types of waves [START_REF] Palomba | Optical negative refraction by four-wave mixing in thin metallic nanostructures[END_REF][START_REF] Harutyunyan | Controllable optical negative refraction and phase conjugation in graphite thin films[END_REF] .

This paper revisits the Faraday instability as a new way to control the propagation of water waves.

This parametric instability is based on a modulation of the effective gravity, which induces a wave velocity modulation. Equivalently, several others parametric instabilities using modulated electric or magnetic fields could be performed on fluids to induce similar modulations of the wave velocity [23][START_REF] Robinson | The electric field at a water surface stressed by an AC voltage[END_REF][START_REF] Briskman | PARAMETRIC EXCITATION OF INSTABILITY OF A FLUID IN MAGNETIC AND ELECTRIC FIELDS[END_REF] .

They could thus be revisited as well, being interpreted as PCMs to control wave propagation. These parametric controls appear much more versatile than the gravity modulation of the Faraday instability, in particular regarding the ability to perform differentiated control in different areas of the medium. In addition, the bathymetry technique used in this paper to confine the PCM to a particular area of the bath has the drawback of increasing the damping in the shallow region. This limits the wave propagation length, and can be avoided with these other parametric controls, enabling new types of exciting experiments with a longer propagation length, like the control of focusing and propagation in complex media [START_REF] Lerosey | Focusing beyond the diffraction limit with far-field time reversal[END_REF] .

Figure 1 .

 1 Figure 1. Schematics of phase-conjugate mirrors for optics and water waves. a) Optical

Figure 2 .

 2 Figure 2. Setup and experimental procedures. a) Schematics of the experimental setup. b) Temporal

Figure 3 .

 3 Figure 3. Reflection of the signal wave emitted by a point source by a water-wave PCM based on the

Figure 4 .

 4 Figure 4. Snapshots of various water-wave PCM configurations. a) and b) Snapshots of a plane wave

Figures

  Figures 4c) and 4e) show the circular outward-propagating waves emitted by point sources, placed

Acknowledgements

The authors are grateful to Yves Couder, Matthieu Labousse and Sander Wildeman for insightful discussions. The authors acknowledge the support of the AXA research fund and the French National Research Agency LABEX WIFI (Laboratory of Excellence ANR-10-LABX-24).

Supplementary Information for

Phase-conjugate mirror for water waves driven by the Faraday instability Vincent Bacot, Guillaume Durey, Antonin Eddi, Mathias Fink and Emmanuel Fort Corresponding author: Emmanuel Fort Email: Emmanuel.fort@espci.fr

This PDF file includes:

Captions for movies S1 to S4

Other supplementary materials for this manuscript include the following: The source and the bathymetry configurations are the ones presented in Figure 2. The vertical line indicates the separation between the shallow end on the left and the deep end on the right, which acts as a PCM. The point pressure source on the left first emits 10 periods of a pulse at 17.5 Hz. These signal waves propagate in the shallow end before entering the deep end. There, they trigger a phaseconjugate wave which refocuses on the source. The bath is excited vertically at the frequency of 35 Hz, above the Faraday threshold, when the source starts emitting. The movie is acquired at 500 frame/s and played 20 times slower.

Supplementary Information Text

Movie S2. Reflection of the signal wave emitted by a plane source by a water-wave PCM based on the Faraday instability.

The bathymetry configuration is the one presented in Figure 2. The vertical line indicates the separation between the shallow end on the left and the deep end on the right, which acts as a PCM. The plane pressure source on the left first emits 10 periods of a pulse at 17.5 Hz. These signal waves propagate in the shallow end before entering the deep end. There, they trigger a plane phase-conjugate wave which propagates back on the source. The bath is excited vertically at the frequency of 35 Hz, above the Faraday threshold, when the source starts emitting. The movie is acquired at 500 frame/s and played 20 times slower.

Movie S3. Water-wave PCM having the shape of France excited at the location of Paris.

The shallow end of the bath is located within the boundaries of the country. It is surrounded by a deeper liquid bath acting as a PCM for water waves. A point source positioned at the location of Paris emits 10 periods of a pulse at 17.5 Hz. The circular signal waves propagate in the shallow end before entering the deep end. There, they trigger a phase-conjugate wave which refocuses on the source. The bath is excited vertically at the frequency of 35 Hz, above the Faraday threshold, when the source starts emitting. The movie is acquired at 500 frame/s and played 20 times slower.

Movie S3. Water-wave PCM having the shape of France excited at the location of Lyon.

The shallow end of the bath is located within the boundaries of the country. It is surrounded by a deeper liquid bath acting as a PCM for water waves. A point source positioned at the location of Lyon emits 10 periods of a pulse at 17.5 Hz. The circular signal waves propagate in the shallow end before entering the deep end. There, they trigger a phase-conjugate wave which refocuses on the source. The bath is excited vertically at the frequency of 35 Hz, above the Faraday threshold, when the source starts emitting. The movie is acquired at 500 frame/s and played 20 times slower.