Manuel F Ruiz-Lopez

Josep M Anglada

Marilia T C Martins-Costa

Joseph S Francisco

Manuel F Ruiz-López

Reactivity of Undissociated Molecular Nitric Acid at the Air-Water Interface

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Rate acceleration is often observed when reactions occur at the surface of liquid water (for a recent review, see Ref. 1). In some cases, this phenomenon, known as "on-water catalysis", 2 can be explained simply by concentration effects, i.e., by the accumulation of hydrophobic or amphiphilic reactants at the air-water interface. In other cases, the catalytic effect derives from the unique properties of the water surface as a reaction medium, whose specific solvation effects can significantly alter the chemical behavior of the reagents. In principle, such effects can be quantified by interface-sensitive spectroscopic techniques or by computer simulations, but the physicochemical properties of the air-water interface remain largely unknown. [START_REF] Ruiz-Lopez | Molecular reactions at aqueous interfaces[END_REF] One of the most intriguing questions in this regard is the acid/base character of the water surface and the unexpected behavior of some acidic species at the air-water interface (see, for instance, Refs. [START_REF] Shamay | At the water's edge: Nitric acid as a weak acid[END_REF][START_REF] Wang | Depth-Dependent Dissociation of Nitric Acid at an Aqueous Surface: Car-Parrinello Molecular Dynamics[END_REF][START_REF] Baer | Investigation of Interfacial and Bulk Dissociation of HBr, HCl, and HNO3 Using Density Functional Theory-Based Molecular Dynamics Simulations[END_REF][START_REF] Mishra | Anions dramatically enhance proton transfer through aqueous interfaces[END_REF][START_REF] Mishra | Brønsted basicity of the air-water interface[END_REF][START_REF] Murdachaew | Deprotonation of formic acid in collisions with a liquid water surface studied by molecular dynamics and metadynamics simulations[END_REF] ). The present work focuses on nitric acid, HNO 3 , a strong acid that is completely dissociated at the usual concentrations in water solution (pK a =-1.3 at 300 K) but, according to several theoretical and experimental studies, 3-5, 9- 14 partially undissociated at the surfaces of various aquatic systems. These are uncommon circumstances because strong acids are expected to dissociate easily when they are hydrated by a small number of water molecules. [START_REF] Leopold | Hydrated acid clusters[END_REF] A remarkable finding of experiments and calculations is that dissociation of HNO 3 decreases by about 20% near the air-water interface compared with dissociation in bulk water solution, [START_REF] Lewis | Does nitric acid dissociate at the aqueous solution surface?[END_REF] although proton transfer can be strongly enhanced in the presence of electrolytes. [START_REF] Mishra | Anions dramatically enhance proton transfer through aqueous interfaces[END_REF][START_REF] Mishra | Quantum chemical insights into the dissociation of nitric acid on the surface of aqueous electrolytes[END_REF] Nitric acid has great environmental relevance. Its concentration in air is typically in the range 0.048-0.795 ppb at the marine boundary, 0.18-1.77 ppb in rural or semirural areas, and 4.4-8.0 ppb in polluted areas. [START_REF] Kulmala | Variations of cloud droplet concentrations and the optical properties of clouds due to changing hygroscopicity: A model study[END_REF] In the gas phase, nitric acid is essentially formed by NO 2 +OH and NO+HO 2 reactions. The aqueous phase chemistry of HNO 3 is very important too; owing to its high solubility (Henry's law constant 2.1×10 5 M•atm -1 ), increased by dissociation to nitrate (effective Henry's law constant 3.2×10 9 M•atm -1 at pH=3), [START_REF] Seinfeld | Atmospheric Chemistry and Physics: From Air Pollution to Climate Change[END_REF] the gas phase concentration in equilibrium with the aqueous medium is negligible. [START_REF] Lelieveld | The role of clouds in tropospheric photochemistry[END_REF] In the aqueous phase, HNO 3 is formed by hydrolysis of N 2 O 4 , N 2 O 5 , and other oxidized species. The main reactions of molecular HNO 3 in the gas phase are photolysis (1) both regenerating NO x /HONO species. However, these reactions have a minor impact on the chemistry of the troposphere, even in the presence of water vapor; [START_REF] Carl | OH kinetics and photochemistry of HNO3 in the presence of water vapor[END_REF][START_REF] Dulitz | Temperature-(208-318 K) and pressure-(18-696Torr) dependent rate coefficients for the reaction between OH and HNO3[END_REF][START_REF] Staikova | Ab initio investigation of water complexes of some atmospherically important acids: HONO, HNO3 and HO2 NO2[END_REF][START_REF] Gonzalez | Gas phase reaction of nitric acid with hydroxyl radical without and with water. A theoretical investigation[END_REF] the main fates of gaseous HNO 3 are thought to be dry and wet deposition, which contribute to "acid rain" and the acidification of soils and water. Accordingly, HNO 3 has traditionally been considered a sink for nitrogen oxides.

Nevertheless, in recent years, there has been increasing interest in the heterogeneous atmospheric chemistry of HNO 3 . [START_REF] Yang | Infrared spectroscopic studies of binary solutions of nitric acid and water and ternary solutions of nitric acid, sulfuric acid, and water at room temperature: Evidence for molecular nitric acid at the surface[END_REF][START_REF] Finlayson-Pitts | The heterogeneous hydrolysis of NO2 in laboratory systems and in outdoor and indoor atmospheres: An integrated mechanism[END_REF][START_REF] Dubowski | Interactions of gaseous nitric acid with surfaces of environmental interest[END_REF][START_REF] Ramazan | New experimental and theoretical approach to the heterogeneous hydrolysis of NO2: key role of molecular nitric acid and its complexes[END_REF] Chemical 27 and photochemical [START_REF] Rivera-Figueroa | Laboratory Studies of Potential Mechanisms of Renoxification of Tropospheric Nitric Acid[END_REF][START_REF] Handley | Photochemical loss of nitric acid on organic films: A possible recycling mechanism for NOx[END_REF][START_REF] Finlayson-Pitts | Reactions at surfaces in the atmosphere: integration of experiments and theory as necessary (but not necessarily sufficient) for predicting the physical chemistry of aerosols[END_REF][START_REF] Zhou | Nitric acid photolysis on forest canopy surface as a source for tropospheric nitrous acid[END_REF][START_REF] Baergen | Photochemical renoxification of nitric acid on real urban grime[END_REF][START_REF] Ye | Photolysis of nitric acid and nitrate on natural and artificial surfaces[END_REF][START_REF] Zhou | Nitric acid photolysis on surfaces in low-NOx environments: Significant atmospheric implications[END_REF][START_REF] Schuttlefield | Photochemistry of adsorbed nitrate[END_REF][START_REF] Zhu | 308 nm photolysis of nitric acid in the gas phase, on aluminum surfaces, and on ice films[END_REF][START_REF] Saliba | Reaction of gaseous nitric oxide with nitric acid on silica surfaces in the presence of water at room temperature[END_REF][START_REF] Chen | Heterogeneous photochemistry of trace atmospheric gases with components of mineral dust aerosol[END_REF] reactions have been shown to be enhanced when they occur on different types of solid surfaces. For instance, the photolysis rate constant of HNO 3 adsorbed on natural and artificial surfaces is enhanced by as much as 1-4 orders of magnitude with respect to the gas phase (see Refs. 33, 39 and references cited therein). This is a fundamental result because such surface-enhanced chemistry could be a significant contributor to renoxification mechanisms, making HNO 3 not only a sink for nitrogen oxides but also a reservoir. Possibly, catalytic effects on reactions (1) and (2) do also occur at the surface of liquid water, but despite the potential environmental importance of such processes, especially in relation to the chemistry of microdroplets in clouds, the associated renoxification routes remain unexplored.

Many theoretical studies have been devoted to the proton transfer of HNO 3 to water in a variety of aqueous environments, 3-5, 12-13, 16, 40-42 with the aim of rationalizing the available experimental data on acid dissociation. To further understand the chemistry of undissociated nitric acid "on water", in this work, we have studied the properties of the important reactions (1) and ( 2) at the air-water interface with the help of high-level ab initio calculations combined with first-principles molecular dynamics (MD) simulations. The models and methods used are briefly explained in the next section. Then, we discuss the thermodynamics of molecular nitric acid at the airwater interface, as well as the solvation effects on its main molecular properties. We briefly discuss the acid dissociation process at the interface, summarizing previous data reported in the literature and new insights obtained in our study. Finally, we report kinetic data for reactions (1) and ( 2), analyze the differences with respect to the chemistry of HNO 3 in the gas phase, and examine the atmospheric chemistry significance of our results.

METHODOLOGY

MD Simulations. MD simulations were carried out to get a suitable solute-solvent sampling that was subsequently used to compute properties at different ab initio levels, as described below. For this purpose, we assumed a quantum mechanics and molecular mechanics (QM/MM) potential with electrostatic embedding [START_REF] Field | A combined quantum mechanical and molecular mechanical potential for molecular dynamics simulations[END_REF] in which HNO 3 is described at the B3LYP level [START_REF] Becke | Density-Functional thermochemistry. iii. The role of exact exchange[END_REF] using the 6-311+G(d) basis set, [START_REF] Hehre | Self-Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian-Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules[END_REF][START_REF] Hehre | Self-consistent molecular orbital methods. I Use of gaussian expansions of Slatertype atomic orbitals[END_REF][START_REF] Krishnan | Self-Consistent Molecular-Orbital Methods .20. Basis Set For Correlated Wave-Functions[END_REF] and the water solvent is described classically using a flexible TIP3P forcefield. [START_REF] Jorgensen | Comparison of simple potential functions for simulating liquid water[END_REF][START_REF] Dang | Simple intramolecular model potentials for water[END_REF] For the solute-solvent non-electrostatic interactions, we used a Lennard-Jones potential from the OPLS forcefield. [START_REF] Jorgensen | The OPLS potential functions for proteins, energy minimizations for crystals of cyclic peptides and crambin[END_REF] The choice of this combined QM/MM potential has been made based on the successful results obtained in previous studies, which showed a good balance between the different approximations for the electrostatic and non-electrostatic interaction terms. [START_REF] Martins-Costa | Reactivity of Atmospherically Relevant Small Radicals at the Air-Water Interface[END_REF][START_REF] Martins-Costa | Reactivity of Volatile Organic Compounds at the Surface of a Water Droplet[END_REF][START_REF] Anglada | Spectroscopic signatures of ozone at the air-water interface and photochemistry implications[END_REF][START_REF] Anglada | Interconnection of Reactive Oxygen Species Chemistry across the Interfaces of Atmospheric, Environmental, and Biological Processes[END_REF][START_REF] Martins-Costa | Reactivity of aldehydes at the air-water interface. Insights from molecular dynamics simulations and ab initio calculations[END_REF][START_REF] Martins-Costa | Impacts of cloud water droplets on the OH production rate from peroxide photolysis[END_REF][START_REF] Martins-Costa | Highly accurate computation of free energies in complex systems through horsetail QM/MM molecular dynamics combined with free-energy perturbation theory[END_REF][START_REF] Martins-Costa | Reaching multi-nanosecond timescales in combined QM/MM molecular dynamics simulations through parallel horsetail sampling[END_REF][START_REF] Bistafa | A Cost-Effective Method for Free-Energy Minimization in Complex Systems with Elaborated Ab Initio Potentials[END_REF] The simulations were carried out in the NVT ensemble (T=298K, time step 0.25 fs) using a Nosé-Hoover thermostat. [START_REF] Hoover | Canonical dynamics: Equilibrium phasespace distributions[END_REF][START_REF] Nosé | A unified formulation of the constant temperature molecular dynamics methods[END_REF] The simulation box contained HNO 3 and 499 H 2 O molecules. For the simulations at the interface, the size was (in Å) 24.662 x 24.662 x 130, and we imposed periodic boundary conditions along the X and Y directions. For the simulations in bulk, the size was (in Å) 24.662 x 24.662 x 24.662, and we imposed periodic boundary conditions along the X, Y, and Z directions. Our QM/MM homemade software [START_REF] Martins-Costa | Molecular dynamics of hydrogen peroxide in liquid water using a combined quantum/classical force field[END_REF] implements Gaussian 09 [START_REF] Frisch | Gaussian 09[END_REF] and Tinker 4.2 [START_REF] Ponder | Software Tools for Molecular Design 4[END_REF] code.

As we did not impose any constraints on the system, proton transfers to classical water could have occurred. To avoid this unphysical issue, the MD trajectories were stopped when the nitric acid OH bond length became larger than a pre-defined threshold. The value d OH =1.3 Å was considered the point of dissociation, based on the results of ab initio MD simulations. [START_REF] Shamay | At the water's edge: Nitric acid as a weak acid[END_REF] After appropriate system thermalization, a single MD trajectory was carried out in bulk solution. The trajectory was stopped after 83 ps owing to the OH distance increasing beyond the pre-defined threshold, indicating a proton transfer to water. Horsetail sampling [START_REF] Martins-Costa | Highly accurate computation of free energies in complex systems through horsetail QM/MM molecular dynamics combined with free-energy perturbation theory[END_REF][START_REF] Martins-Costa | Reaching multi-nanosecond timescales in combined QM/MM molecular dynamics simulations through parallel horsetail sampling[END_REF] was implemented for the simulations at the interface. In this case, there was one main (long) trajectory and many secondary (short) trajectories that complemented the sampling of the main one. The long trajectory was stopped at 267 ps to prevent unphysical proton transfer to a classical water molecule. Along this main trajectory, 75 different trajectories were launched in parallel from different starting configurations, after randomizing the atomic velocities using a Maxwell-Boltzmann distribution. Among these 75 short trajectories, nine were stopped before completion owing to fast unphysical proton transfer, while the others were carried out for ~50 ps. The total simulation time was about 3.3 ns, which was 1-2 orders of magnitude higher than in previous studies and could provide a fairly complete general overview of HNO 3 solvation and dynamics at the interface.

Electronic Spectrum. To calculate the ultraviolet-visible light spectra, we used 581 configurations distributed along the long trajectory for HNO 3 at the air-water interface. The excitation energies and oscillator strengths for the lowest five singlet electronic states were calculated using a combined multireference configuration interaction (MRCI)/MM method. The MRCI calculations were carried out with a density functional theory/MRCI program 65 linked to the Orca program. [START_REF] Neese | The ORCA program system[END_REF] The def2-TZVPPD basis set [START_REF] Weigend | Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy[END_REF][START_REF] Hellweg | Development of new auxiliary basis functions of the Karlsruhe segmented contracted basis sets including diffuse basis functions (def2-SVPD, def2-TZVPPD, and def2-QVPPD) for RI-MP2 and RI-CC calculations[END_REF] was employed in these calculations. All valence electrons were correlated, while 1s electrons of heavy atoms were frozen; the CI expansion was done over BHLYP orbitals. After some preliminary tests, we decided to include in the quantum-mechanical part (QM subsystem) the HNO 3 and the water molecules hydrogen-bonded to it (up to four water molecules); the other water molecules were treated classically as point charges. The gas phase spectrum was obtained in a similar way using 1000 configurations from a 20 ps QM simulation of isolated HNO 3 . The cross-section was obtained through a Gaussian convolution after correcting the excitation energies and oscillator strengths to fit the experimental curve in the gas phase, as explained in the Supporting Information (SI).

HNO 3 + OH Reaction Kinetics. The potential energy surface of the HNO 3 +OH reaction at the air-water interface was explored using a cluster model. Specifically, we used the 21water molecule model that we have used in other studies of similar characteristics. [START_REF] Torrent-Sucarrat | Protonation of water clusters induced by hydroperoxyl radical surface adsorption[END_REF][START_REF] Ruiz-Lopez | A New Mechanism of Acid Rain Generation from HOSO at the Air-Water Interface[END_REF][START_REF] Martins-Costa | Isoprene Reactivity on Water Surfaces from ab initio QM/MM Molecular Dynamics Simulations[END_REF] In these calculations, the stationary points were optimized and characterized, employing the B3LYP functional [START_REF] Becke | Density-Functional thermochemistry. iii. The role of exact exchange[END_REF] in conjunction with the 6-311+G(2df,2p) basis set. [START_REF] Frisch | Self-Cons¡stent Molecular Orbital Methods 25: Supplementary Functions for Gaussian Basis Sets[END_REF][START_REF] Hehre | Ab Initio Molecular Orbital Theory[END_REF] This functional has been chosen because (in contrast to other more recent functionals) it has proven to correctly describe the NO 3 radical, [START_REF] Becke | Density-Functional thermochemistry. iii. The role of exact exchange[END_REF] one of the products of the reaction, which suffers from doublet instability. [START_REF] Mclean | Symmetry breaking in molecular calculations and the reliable prediction of equilibrium geometries. The formyloxyl radical as an example[END_REF] Besides, exploratory calculations showed that the results were not significantly improved by geometry refinement at the QCISD(T) level. However, the B3LYP approach underestimates the energy barriers for hydrogen transfer processes; [START_REF] Zhang | Extending the reliability and applicability of B3LYP[END_REF] in order to obtain reliable relative energies, we carried out single-point energy calculations at all computed stationary points, using the DLPNO-CCSD(T) method [START_REF] Liakos | Comprehensive Benchmark Results for the Domain Based Local Pair Natural Orbital Coupled Cluster Method (DLPNO-CCSD (T)) for Closed-and Open-Shell Systems[END_REF] with the aug-cc-pVTZ basis set. [START_REF] Dunning | Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron through Neon and Hydrogen[END_REF][START_REF] Kendall | Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions[END_REF] For comparison, the reaction mechanism and relative energies were calculated for the naked gas phase reaction and for the reactions with one and two water molecules at the DLPNO-CCSD(T) [START_REF] Liakos | Comprehensive Benchmark Results for the Domain Based Local Pair Natural Orbital Coupled Cluster Method (DLPNO-CCSD (T)) for Closed-and Open-Shell Systems[END_REF] and CCSD(T) [START_REF] Lee | A coupled cluster approach with triple excitations[END_REF] levels with the aug-cc-pVTZ basis set; moreover, for the naked reaction, calculations with the aug-cc-pVQZ basis set and extrapolation to the CBS (complete basis set) limit [START_REF] Bak | The accurate determination of molecular equilibrium structures[END_REF] were also done. The Gaussian 09 [START_REF] Frisch | Gaussian 09[END_REF] and Orca 4.0 66 programs were employed in these calculations. The rate constants were computed using conventional transition-state theory, considering the energies obtained at the DLPNO-CCSD(T)/aug-cc-pVTZ level and the partition functions computed at the B3LYP/6-311+G(2df,2p) level. Tunneling effects were obtained with the zero-curvature approach. The Polyrate program [START_REF] Lu | POLYRATE-version 4.0.1[END_REF] was used for kinetic calculations.

RESULTS AND DISCUSSION

Thermodynamics of Molecular HNO 3 at the Air-Water Interface. The first important new result obtained in our study is the probability distribution describing the movements of molecular HNO 3 across the air-water interface. In fact, the short time-scales of previous simulations dealing with an extended interface, together with the widespread use of a biased potential fixing the position of the molecule with respect to the Gibbs-dividing surface (GDS), prevented proper characterization of such movements. The calculated density profile for the main QM/MM MD trajectory is shown in Figure 1. Remarkably, it displayed a maximum probability very close to the GDS, and the distribution extended approximately ±4Å from the GDS (see SI for the method used to calculate the GDS). In other words, the free molecular nitric acid did not migrate to bulk in the time-scale of the simulation but, on the contrary, appeared to be quite stable at the air-water interface, confirming the findings reported by experiments. Note that the thickness of the profile was comparable to that found for other small systems of atmospheric interest using a similar model. 51- 52, 58 The free-energy profile for the accommodation of HNO 3 from gas phase to the aqueous interface and the bulk solution is not available in the literature. Obtaining this quantity is important to quantify the significance of the HNO 3 interfacial chemistry, but unfortunately the computational cost is exceedingly high, especially at the ab initio level. Here, we used parallel computing of secondary trajectories to obtain an augmented probability distribution and an estimation of the interfacial free-energy well depth (Figure S1 in the SI). [START_REF] Martins-Costa | Reaching multi-nanosecond timescales in combined QM/MM molecular dynamics simulations through parallel horsetail sampling[END_REF] According to this calculation, the interface → bulk transfer process would involve not less than +1.6 kcal•mol -1 (same reference state) suggesting interface concentration excess with an estimated interface/bulk ratio equal 15. Using the experimental Henry's law constant for HNO 3 , one can deduce -8.0 kcal•mol -1 for the free energy of the gas phase → interface adsorption process. These values are subject to some uncertainties but can be considered as an approximate lower limit for the interfacial thermodynamics that were lacking in the literature.

Interfacial Solvation Effects. Table 1 summarizes the average of some molecular properties calculated for HNO 3 at the air-water interface. Compared with gas phase values, the OH bond was significantly elongated (+0.056 Å), in good agreement with the predictions of ab initio calculations in water clusters. [START_REF] Mccurdy | Nitric Acid-Water Complexes: Theoretical Calculations and Comparison to Experiment[END_REF][START_REF] Escribano | The nitric acid hydrates: Ab initio molecular study, and RAIR spectra of the solids[END_REF] Besides, the molecule underwent important electronic polarization, as indicated by the significant increase in the dipole moment (+1.59 D) and the net charge on the H atom (+0.094e). Table 1 also reports the average values of the HOMO and LUMO frontier orbitals. Both were significantly destabilized at the interface. The destabilization was greater for the HOMO than for the LUMO, a result that has consequences for the photolysis rate constant because it implies that the HOMO→LUMO electronic transition will be red-shifted at the air-water interface (see the discussion below). As shown, the interfacial solvent effects were very similar to those obtained in bulk solution, except that the dipole moment seemed to be slightly smaller at the interface, possibly owing to a lower electronic polarization in this case.

Table 1. Values of some HNO 3 parameters in gas phase (optimized geometry) and in water (averages from QM/MM MD simulations): O-H bond length (d OH , Å), O-H bond order (B), net atomic charge on H (q H , e), dipole moment (µ, D), and HOMO and LUMO energies (eV). Bond orders and atomic charges were obtained by natural bond orbital analysis. [START_REF] Reed | Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint[END_REF] To gain deeper insight into the differences between HNO 3 solvation at the interface and in bulk, it was worth comparing the solute-solvent radial distribution functions (RDFs) in both cases. The RDFs at the interface are shown in the left panel of Figure 2. Not surprisingly, the acidic proton formed a very strong H-bond with water. Indeed, the simulation data reveal that this H-bond was a permanent interaction along the whole trajectory. By contrast, the H-bonds formed by the O atoms in HNO 3 with water protons were relatively weak, especially for the O atom in the OH group of nitric acid. Integration of the first RDF peaks led to a total average of 1.5 H-bonds between nitric acid and water. Specifically, there were 1.0, 0.2, and 0.1 H-bonds for the H, O (terminal), and O (bonded to H) atoms in HNO 3 , respectively. For simplicity, the analysis of the RDFs in bulk is not detailed here, but the results (Figure S2 in the SI) were very similar, with a somewhat higher asymmetry for the RDFs of the terminal O atoms and a very slightly higher average number of H-bonds between nitric acid and water, which added up to 1.6. This suggests that the solvation modes of HNO 3 at the interface and in bulk are comparable. Previous ab initio simulations reported a similar conclusion when the HNO 3 concentration in bulk was high, [START_REF] Lewis | Does nitric acid dissociate at the aqueous solution surface?[END_REF] suggesting that an insufficient amount of water around HNO 3 is the driving force for a strong hydrogen bond interaction and its stabilization at the interface. Our longer simulations differ from this conclusion, however, in that the same (or very similar) solvation patterns were found for low HNO 3 bulk concentrations. This poses the question of the driving force for the easier dissociation in bulk. Although discussing this issue was not the aim of the present work, some comments will be made in the next section.

Figure 2 (right panel) also shows that the HNO 3 molecule had a preferential orientation at the air-water interface, with the OH moiety pointing towards the water layer. This was the expected result, considering the solvation pattern shown by the RDFs, and it parallels the conclusions reported previously. [START_REF] Lewis | Does nitric acid dissociate at the aqueous solution surface?[END_REF] Finally, it is important to emphasize that the strong H-bonddonating character of nitric acid explains the observed destabilizing solvation effect on the HOMO and LUMO energies. This is the consequence of the overall negative electrostatic potential created by the H-accepting water molecule in the vicinity of the solute (due to its dipole moment orientation) and is characteristic of H-donating systems. Acid Dissociation at the Interface. The dissociation of HNO 3 at aqueous interfaces has been addressed in detail by several authors, 3-5, 12-13, 16, 40-42 and the thermodynamics of dissociation have been shown to be strongly dependent on the location depth (i.e., the relative position with respect to the GDS). [START_REF] Wang | Depth-Dependent Dissociation of Nitric Acid at an Aqueous Surface: Car-Parrinello Molecular Dynamics[END_REF][START_REF] Baer | Investigation of Interfacial and Bulk Dissociation of HBr, HCl, and HNO3 Using Density Functional Theory-Based Molecular Dynamics Simulations[END_REF] Nevertheless, it is worth making a few comments based on the complementary results provided by our multinanosecond QM/MM MD simulations.

We have shown that the solvation shells of molecular HNO 3 at the interface and in the bulk are quite similar, despite a more favorable free energy at the interface by ~1.6 kcal•mol -1 , possibly owing to a greater loss of water-water interactions in the bulk. In the dissociation process, the first step consists of the formation of a contact ion pair (NO ! ! ••• H -OH ! ! ) that precedes the diffusion of the proton to the liquid through the Grotthuss mechanism. According to first-principles MD simulations, the ion pair lies ~1.8 kcal•mol -1 above the undissociated HNO 3 form in bulk and ~4.5 kcal•mol -1 at the interface, where formally it does not correspond to a free-energy minimum (the values have been estimated from Figure 5 in Ref. 5). The combination of these data leads to the conclusion that the ion pair is thermodynamically less stable at the interface than in bulk by ~1.1 kcal•mol -1 , and that dissociation is more favorable in bulk, as actually observed.

It is useful to estimate the lifetime of molecular HNO 3 with respect to dissociation at the interface. If one assumes the free energy required to reach the ion pair as an approximate barrier for dissociation, the lifetime is 0.32 ns (assuming a pseudofirst order process and transition state theory; similar orders of magnitude for the dissociation energy have been reported depending on depth and interface models). [START_REF] Bianco | Theoretical study of the dissociation of nitric acid at a model aqueous surface[END_REF][START_REF] Wang | Nitric Acid Dissociation at an Aqueous Surface: Occurrence and Mechanism[END_REF][START_REF] Wang | An atmospherically relevant acid: HNO3[END_REF] Interestingly, this lifetime is consistent with the number of pre-dissociation events that we observed in our QM/MM simulations (10 events within the 3.3 ns simulation time), confirming that the elongation of the OH bond beyond 1.3 Å is a suitable choice to characterize such events.

The energetics analysis described above is useful to explain the dissociation kinetics at the interface, [START_REF] Lewis | Does nitric acid dissociate at the aqueous solution surface?[END_REF] but understanding the whole picture would require evaluation of dissociation constants, i.e., the thermodynamics and distribution of the ions in the vicinity of the interfacial layer. Answering this question is not straightforward, even in a pure HNO 3 solution, and obtaining a reliable answer will require further experimental and theoretical work. It is worth remembering, however, that the nitrate anion has a low propensity for the air-water interface, according to simulations of neat nitrate solutions [START_REF] Minofar | Propensity for the air/water interface and ion pairing in magnesium acetate vs magnesium nitrate solutions: Molecular dynamics simulations and surface tension measurements[END_REF][START_REF] Dang | On NO3 --H2O interactions in aqueous solutions and at interfaces[END_REF][START_REF] Thomas | Molecular dynamics simulations of the solution-air interface of aqueous sodium nitrate[END_REF][START_REF] Miller | Structure of large nitrate-water clusters at ambient temperatures: simulations with effective fragment potentials and force fields with implications for atmospheric chemistry[END_REF] and several experiments, [START_REF] Otten | Observation of nitrate ions at the air/water interface by UV-second harmonic generation[END_REF][START_REF] Brown | Spatial distribution of nitrate and nitrite anions at the liquid/vapor interface of aqueous solutions[END_REF] while the interface affinity of the hydronium ion is now broadly assumed (see the discussion in Ref. 1). Besides, ion distributions are influenced by the presence of other species in real systems, and this factor must be taken into account. For example, halides in sea-salt aerosols draw NO 3 -anions closer to the surface. [START_REF] Tobias | Simulation and Theory of Ions at Atmospherically Relevant Aqueous Liquid-Air Interfaces[END_REF] Absorption Spectrum, Photolysis Rate Constant. The MRCI calculations of the electronic absorption spectrum revealed a slight red-shift for the first excitation. The average excitation energies were 4.65 eV (267 nm) in the gas phase and 4.48 eV (277 nm) at the air-water interface. The shift can be explained simply by the solvation effects on the HOMO and LUMO energies, which are mainly due to the effects of the water molecule H-bonded to the nitric acid proton. The calculated cross-sections in the spectral region of atmospheric interest at 298 K are shown in Figure 3 and compared with the recommended [START_REF] Burkholder | Chemical kinetics and photochemical data for use in atmospheric studies: evaluation number 18[END_REF] experimental values in the gas phase. The tails of the cross-sections in the 290-350 nm region, where the actinic flux rapidly increases, were relatively small and became almost negligible after 350 nm. This explains the rather low photolysis rate of gaseous nitric acid in the troposphere. [START_REF] Warneck | The Atmospheric Chemist's Companion[END_REF] However, owing to the red-shift of the first excitation at the interface, the cross-section was significantly enhanced in the actinic region, suggesting an increase in the photolysis rate constant in this medium.

The rate constant for the photolysis process ( 1) is calculated from the expression:

𝐽 ! = Φ(𝜆) ! 𝜎 𝜆 𝑞(𝜆)d𝜆, ( 3
)
where λ is the wavelength, σ is the absorption cross-section, Φ is the quantum yield, and q is the actinic flux in the troposphere (hereafter we assume the actinic flux reported in Ref. [START_REF] Finlayson-Pitts | Atmospheric Chemistry: Fundamental and Experimental Techniques[END_REF] at the Earth's surface, noontime, and no surface Albedo). Different values have been reported for the OH quantum yield in the gas phase, which in general is close to unity in the actinic region. [START_REF] Burkholder | Chemical kinetics and photochemical data for use in atmospheric studies: evaluation number 18[END_REF][START_REF] Riffault | Quantum yields for OH production in the photodissociation of HNO3 at 248 and 308 nm and H2O2 at 308 and 320 nm[END_REF] Zhu et al [START_REF] Zhu | Photolysis of Nitric Acid at 308 nm in the Absence and in the Presence of Water Vapor[END_REF] reexamined the photolysis at 308 nm and concluded that ground-state NO 2 was the main product with near-unity quantum yield. Thus, for the range of wavelengths considered here, 290 nm ≤ λ ≤ 350 nm, we assume that process (1a) is predominant and that Φ=1 is a good approximation. [START_REF] Burkholder | Chemical kinetics and photochemical data for use in atmospheric studies: evaluation number 18[END_REF][START_REF] Riffault | Quantum yields for OH production in the photodissociation of HNO3 at 248 and 308 nm and H2O2 at 308 and 320 nm[END_REF] The same quantum yield is considered for the process at the interface, assuming that solvent cage effects at the interface are not large. [START_REF] Nissenson | Enhanced photolysis in aerosols: evidence for important surface effects[END_REF] The calculation of J 1 for the photolytic process (1a) is detailed in the right part of Figure 3. As shown, the main contribution to the rate constant was from light absorption at around 310 nm, both in the gas phase and at the interface. The integrated value for J 1 in the gas phase was J 1,gas =10 -6 s -1 using the calculated cross-section, and 9.3x10 -7 s - 1 using the experimental cross-section; these values are very close to other data reported in the literature (8.2 × 10 -7 s -1 ). [START_REF] Warneck | The Atmospheric Chemist's Companion[END_REF] At the interface, our calculations showed an enhancement of J with respect to the gas phase by a factor of two, i.e., J 1,inter =2x10 -6 s -1 . This moderate increase in the photolysis rate constant of HNO 3 at the air-water interface contrasts with the strong increase reported for nitric acid adsorbed on a variety of surfaces. [START_REF] Ye | Photolysis of nitric acid and nitrate on natural and artificial surfaces[END_REF][START_REF] Ye | Matrix effect on surface-catalyzed photolysis of nitric acid[END_REF] The apparent contradiction between these data will be discussed below. Reaction of Nitric Acid with Hydroxyl Radical. We analyzed the reactivity of nitric acid with a hydroxyl radical at the air-water surface by studying the process on the surface of a cluster of 21 water molecules. [START_REF] Torrent-Sucarrat | Protonation of water clusters induced by hydroperoxyl radical surface adsorption[END_REF] The cluster employed in this study had a distorted pentagonal dodecahedron structure with a water molecule inside the cavity. It has eight possible anchoring sites for the reacting system, defined as oxygen atoms with a free lone-pair pointing out of the cluster and associated with a negative charge distribution (Figure S3). We thoroughly explored the different reaction paths on the cluster (see details in the SI), resulting in 16 elementary reactions for further consideration. Each elementary reaction begins with the formation of a pre-reactive (HO)(HNO 3 ) complex hydrogen-bonded to the water cluster. The reaction proceeds through a transition state that leads to the formation of NO 3 , interacting with the augmented water cluster (H 2 O) 22 . As in the gas phase, [START_REF] Gonzalez | Gas phase reaction of nitric acid with hydroxyl radical without and with water. A theoretical investigation[END_REF] the transition states correspond to proton-coupled electron transfer (pcet) or conventional hydrogen-atom transfer (hat) mechanisms. The two mechanisms are illustrated in Figure 4, which displays their main electronic features. In the pcet mechanism (Figure 4a), the radical of the hydroxyl group interacts with one of the terminal oxygen atoms of nitric acid, provoking the transfer of one electron from the HNO 3 moiety to the OH moiety. Simultaneously, a transfer of a proton occurs from the nitric acid to the OH moiety. The process can be described by a three-electron, two-orbital interaction between O3 and O6 (see Figure 4 for atom numbering). The hat mechanism (Figure 4b), instead, involves the simultaneous breaking and forming of two covalent bonds, namely, the OH bond of the nitric acid and HO bond of the hydroxyl radical. As shown, this is also a three electron, two-orbital process, but the electronic densities of the double and single natural orbitals involved entail O4 and O6, where the covalent bonds break and form. Further discussion on the characteristics of pcet and hat mechanisms for different systems can be found elsewhere. [START_REF] Anglada | The gas phase oxidation of HCOOH by Cl and NH2 radicals. Proton coupled electron transfer versus hydrogen atom transfer[END_REF][START_REF] Anglada | Atmospheric formation of the NO3 radical from gas-phase reaction of HNO3 acid with the NH2 radical: Proton-coupled electron-transfer versus hydrogen atom transfer mechanisms[END_REF][100][101][102][103][104][105][106] The calculated thermodynamic data for the 16 studied processes are detailed in the SI (Figures S4-S6 and Tables S1-S10). As shown, the relative energies of the pre-reactive complexes differed by up to 6.29 kcal•mol -1 (Table S3)..The activation free energies for the 16 processes are schematized in Figure 5 (Table S4). The elementary reactions are denoted by the symbol "oxtsy", where the number x identifies the anchoring site (oxygen atom) of the water cluster where the reaction takes place, and the number y identifies different elementary reactions taking place at the same site (note that x corresponds to the ordinal number of the oxygen atom in the Cartesian coordinates table provided in the SI).

Almost all of the elementary reactions found took place through the pcet mechanism (in red in Figure 5), but two of them occurred via the hat mechanism (in blue in Figure 5). Attempts to find other elementary reactions via hat mechanisms converged to pcet processes. As shown in Figure 5 and Table S4, the computed free-energy barriers were lower in the pcet mechanisms: they ranged between 6.65 and 9.18 kcal•mol -1 (o25ts1 and o7ts3, respectively) for the elementary reactions going through the pcet mechanism, and between 9.89 and 10.97 kcal•mol -1 for the reactions via the hat mechanism (o7ts4 and o22ts3, respectively). These differences in stability may be attributed to two main factors. The first is the effect of hydrogen bond interactions. In all transition states, there is a hydrogen bond between one oxygen atom of the water cluster and the hydrogen atom of the hydroxyl radical moiety, and a second (weaker) hydrogen bond between a dangling hydrogen atom of the cluster and one of the oxygen atoms of the HNO 3 moiety. In each case, there is a small charge transfer in the direction reverse to the H donation, facilitating or hampering the electron and proton transfer processes, and therefore producing a stabilizing or destabilizing effect. [START_REF] Gonzalez | Gas phase reaction of nitric acid with hydroxyl radical without and with water. A theoretical investigation[END_REF]103 The second factor is the site of the cluster where the reaction takes place. Each site is surrounded by a distinct micro-environment whose electrostatic potential affects the stability of the stationary point in a different way (see SI for full details). The structures involved in the most favorable process predicted in this study (o25ts1) are drawn in Figure 6.

The experimental kinetic constant for reaction (2) in the gas phase reported by Brown et al 107 amounts 𝑘 ! !"# = 1.2 × 10 -13 cm 3 •molecule -1 •s -1 ; other similar data are available in the literature. [START_REF] Dulitz | Temperature-(208-318 K) and pressure-(18-696Torr) dependent rate coefficients for the reaction between OH and HNO3[END_REF] Using the DLPNO-CCSD(T) thermodynamic calculations (see SI), we deduce a huge increase of this constant on the surface of the water cluster relative to the gas phase, with 𝑘 ! !"#$ /𝑘 ! !"# ~10 4 . Further analysis showed that the main effect was due to the hydrogen bonds formed with water molecules in the first solvation shell, although remaining water molecules in the cluster provided a significant long-range effect, which would conceivably be even larger if an extended interface were considered. We conclude, therefore, that 𝑘 ! !"#$% ~ 1.2 × 10 -9 cm 3 •molecule -1 •s -1 represents a lower limit for the kinetic constant of the HNO 3 +OH reaction at the air-water interface. Atmospheric Significance. To discuss the atmospheric significance of our results, it is worth obtaining some orders of magnitude for the rates of reaction ( 1) and ( 2), R 1 and R 2 , for representative concentrations of nitric acid. In the gas phase, we will take [HNO 3 ] = 5ppb (1.2x10 11 molecules•cm -3 ) characteristic of polluted areas. [START_REF] Kulmala | Variations of cloud droplet concentrations and the optical properties of clouds due to changing hygroscopicity: A model study[END_REF] In the water droplets, we will assume pH=4, which is representative of acid rain pH (typically lying between 4.0 and 4.6), and a nitrate concentration equal 100 µM, which is also in the range of acid rain measurements. 108 The aqueous phase to gas phase concentration ratio taken here is comparable to that predicted in photochemical box model simulations. 109 It is worth noting that inside a cloud, all the nitric acid can be considered to be dissolved in the aqueous phase and that the equilibrium gas-phase concentration is negligible. [START_REF] Lelieveld | The role of clouds in tropospheric photochemistry[END_REF] Assuming the conditions above, and using the pK a of nitric acid, together with the free energy profile obtained in our work, one can estimate the interface concentration of undissociated nitric acid as 4.4x10 12 molecule•cm -3 , which is about one order of magnitude larger than [HNO 3 ] in polluted air.

The calculated HNO 3 photolysis rate constant J 1 at the surface of liquid water is only slightly higher than in gas phase and therefore the predicted rate R 1 = J 1 [HNO 3 ] grows by one order of magnitude due to the concentration increase estimated above. Such small effect is not expected to influence the overall atmospheric burden of NO 2 since the total interfacial volume is small compared to gas phase. Interestingly, this finding differs from results reported on different artificial and natural solid surfaces. [START_REF] Rivera-Figueroa | Laboratory Studies of Potential Mechanisms of Renoxification of Tropospheric Nitric Acid[END_REF][START_REF] Handley | Photochemical loss of nitric acid on organic films: A possible recycling mechanism for NOx[END_REF][START_REF] Finlayson-Pitts | Reactions at surfaces in the atmosphere: integration of experiments and theory as necessary (but not necessarily sufficient) for predicting the physical chemistry of aerosols[END_REF][START_REF] Zhou | Nitric acid photolysis on forest canopy surface as a source for tropospheric nitrous acid[END_REF][START_REF] Baergen | Photochemical renoxification of nitric acid on real urban grime[END_REF][START_REF] Ye | Photolysis of nitric acid and nitrate on natural and artificial surfaces[END_REF][START_REF] Zhou | Nitric acid photolysis on surfaces in low-NOx environments: Significant atmospheric implications[END_REF][START_REF] Schuttlefield | Photochemistry of adsorbed nitrate[END_REF][START_REF] Zhu | 308 nm photolysis of nitric acid in the gas phase, on aluminum surfaces, and on ice films[END_REF][START_REF] Saliba | Reaction of gaseous nitric oxide with nitric acid on silica surfaces in the presence of water at room temperature[END_REF][START_REF] Chen | Heterogeneous photochemistry of trace atmospheric gases with components of mineral dust aerosol[END_REF][START_REF] Ye | Matrix effect on surface-catalyzed photolysis of nitric acid[END_REF] The reason for this discrepancy is not clear. It has generally been assumed that the strong increase of the photolytic rate constant on surfaces is due to the red-shift provoked by the distorted structure of nitric acid. However, at the air-water interface, important structural changes have also been predicted, while only a moderate concomitant red-shift has been obtained. It therefore appears that interaction with water is not enough per se to produce a strong increase in the photolysis rate constant. As a corollary, one may deduce that photolysis on surfaces should involve effects beyond hydration, which may include the interaction of HNO 3 with organic species, the formation of dimers or complexes with hydrated nitrate anions, etc.

The case of the reaction with OH leads to quite different conclusions. Using the recommended [START_REF] Warneck | The Atmospheric Chemist's Companion[END_REF] annual average concentration of OH radicals in air, [OH] = 1.16 x 10 6 molecule•cm -3 , and the experimental gas phase kinetic constant, we obtain a gas phase reaction rate R 2 = k 2 [HNO 3 ] [OH] = 1.7 x 10 4 molecule•cm -3 •s -1 , which indicates a small contribution to renoxification of the HNO 3 gas phase chemistry, as outlined in the introduction. In aqueous droplets, the concentration of the OH radicals is not in equilibrium with the gas phase because due to their high reactivity, the radicals are consumed as they are formed. Indeed, [OH] in aqueous droplets is in general much greater than in gas phase owing to specific OH sources. 110 Uptake from the gas phase is one of such sources but production in situ from ozone and peroxides photolysis may be up to 5 orders of magnitude larger. [START_REF] Anglada | Spectroscopic signatures of ozone at the air-water interface and photochemistry implications[END_REF][START_REF] Martins-Costa | Impacts of cloud water droplets on the OH production rate from peroxide photolysis[END_REF]111 We use here, as a lower limit, the equilibrium concentration derived from thermodynamic simulations, 112 which leads to 1.0 x 10 10 molecule•cm -3 . Consequently, the reaction rate at the interface is estimated to be R 2 = 5.3 x 10 13 molecule•cm -3 •s -1 , which is ~9 orders of magnitude larger than the same process in air, and ~6 orders of magnitude larger than the production rate from the O 3 +NO 2 reaction (the main NO 3 source in air) in polluted urban environments. [113][114] The photochemically unstable NO 3 radicals may diffuse and heterogeneously react with compounds adsorbed on the surface, such as isoprene or other terpenes. But due to the slightly hydrophobic character of NO 3 , 115 a fraction of the formed radicals should be released to the gas phase and contribute to renoxification mechanisms. In principle, reactions (1) and (2) can also occur in bulk solution, but their contribution to renoxification is expected to be less important than for interfacial chemistry. The concentrations of undissociated nitric acid and OH 112 are much lower in that case, and the small amounts of the formed nitrate radicals, not likely to escape to the air, should react quickly in the aqueous phase 116 before wet deposition.

CONCLUSIONS

In water droplets, most of nitric acid is in the form of nitrate, but a fraction of undissociated HNO 3 exists at the surface. Previous studies have reported that surficial dissociation decreases by about 20% compared with bulk in neat water, [START_REF] Lewis | Does nitric acid dissociate at the aqueous solution surface?[END_REF] but the atmospheric significance of the resulting concentration is still under debate. The water content of clouds, the size of droplets, their chemical composition, and the thermodynamic conditions are certainly important parameters that need to be considered.

In this work, we have studied the photolysis and the reaction with OH of undissociated nitric acid at the air-water interface with the help of high-level theoretical methods. The main results are as follows. 1) HNO 3 appears to be quite stable at the interface; we predict a free energy of adsorption of ~-8.0 kcal•mol -1 (gas phase → interface process) and an interface preference with respect to bulk of ~1.6 kcal•mol -1 . 2) The solvation pattern and the solvation effect of HNO 3 properties at the interface are not very different compared with solvation in bulk. 3) The reduced acid dissociation at the interface is most probably due to the lower stability of the ion pair formed in the early steps of the process. 4) The photolysis rate constant at the interface in the actinic region is larger than that in the gas phase by a factor of 2, which is smaller than the increase reported on several solid surfaces, suggesting the existence of effects other than hydration in the latter case. 5) The redox reaction with OH is hugely enhanced at the interface with respect to gas phase due to significant reaction rate increase and large reactant accumulation, therefore this process may be a significant contributor to renoxification processes in the troposphere.

The more general conclusion of this study is that the chemistry of undissociated strong acids in aqueous environments, which represents an unusual situation in chemistry, has important specificities that should be investigated in greater detail. Indeed, fundamental processes as important as acidbase reactions at the surface of water are still poorly understood, [START_REF] Ruiz-Lopez | Molecular reactions at aqueous interfaces[END_REF] and unraveling the associated mechanisms is probably one of the most appealing chemical problems to address in the field of aqueous interfaces.

ASSOCIATED CONTENT Supporting Information

Details on calculations, HNO 3 free energy profile, radial distribution functions in bulk water, Tables S1-S6, Figures S1-S6 and atomic coordinates for the HNO 3 +OH reaction on the 21-water molecules cluster are provided as supporting information (file type PDF). The Supporting Information is available free of charge on the ACS Publications website.

4 Figure 1 .

 41 Figure 1. Density profiles for HNO 3 and water (Z=0 corresponds to the center of the simulation box; the GDS lies at Z=11.7 Å) and illustrative snapshot from the QM/MM simulation at the air-water interface. The density units are g•cm -3 for water; we use arbitrary units for HNO 3 .

Figure 2 .

 2 Figure 2. Left: radial distribution functions obtained for HNO 3 at the air-water interface. The curves correspond to solute-solvent hydrogen bonds in which nitric acid (NA) acts as a donor (H NA •••O w ) or as an acceptor (O NA •••H w ). In the latter case, the dashed line corresponds to the oxygen atom of nitric acid bonded to H, while the plain lines correspond to the other two oxygen atoms. Right: probability distribution of the OH vector orientation.

Figure 3 .

 3 Figure 3. Calculated and experimental absorption cross-sections of HNO 3 (left) and partial photolysis rates (right). Plain and dashed lines correspond to calculations, while circles correspond to experimental values.

Figure 4 .

 4 Figure 4. Structure of two transition states illustrating the electronic features in the (a) proton-coupled electron transfer (pcet) and (b) conventional hydrogen atom transfer (hat) reaction mechanisms. The orbital occupation is indicated.

Figure 5 .

 5 Figure 5. Computed free energy barrier (at 298K) for the 16 elementary reactions investigated.

Figure 6 .

 6 Figure 6. Structures for the most favorable HNO 3 +OH reaction path predicted on the 21-water molecules cluster.

  and reaction with OH (2),

	HNO 3 + hν → HONO + O( 3 P) (λ≤393nm)	(1c)
	HNO 3 + OH → NO 3 +H 2 O	(2)
		HNO 3 + hν → NO 2 + OH (λ≤604nm)	(1a)
		HNO 3 + hν → NO 2 * + OH (λ≤381nm)	(1b)

ACKNOWLEDGMENT

JMA thanks the Catalan CSUC for providing computational resources. MMC and MFRL are grateful to the French CINES (project lct2550) for providing computational resources. The authors thank Prof. Christel M. Marian and Prof. Martin Kleinschmidt for providing them with the DFT/MRCI program.

Funding Sources

JMA thanks the Generalitat de Catalunya (Grant No. 2017SGR348) and Spanish Ministerio de Economía y Competitividad, Project No. PID2019-109518GB-I00 for financial support

AUTHOR INFORMATION

Corresponding Author anglada@iqac.csic.es (JMA) frjoseph@sas.upenn.edu (JSF) manuel.ruiz@univ-lorraine.fr (MFRL)

Author Contributions

The manuscript was written through contributions of all authors.


Alexandre Lebel

Alexey Tikan

Stephane Randoux

Pierre Suret

AND François Copie
email: francois.copie@univ-lille.fr

Single-shot observation of breathers from noise-induced modulation instability using heterodyne temporal imaging

We report phase and amplitude measurements of large coherent structures originating from the noise-induced modulation instability in optical fibers. By using a specifically-designed time-lens system (SEAHORSE) in which aberrations are compensated, the complex field is recorded in single-shot over long durations of 200 ps with sub-picosecond resolution. Signatures of Akhmediev breather-like patterns are identified in the ultrafast temporal dynamics in very good agreement with numerical predictions based on the nonlinear Schrödinger equation. !

Modulation instability (MI) is unarguably one of the most ubiquitous nonlinear phenomenon since it appears in fields as different as hydrodynamics [1], plasma physics [2], Bose-Einstein condensates [3] or optics [4], to name a few. MI is a long-wave instability at the origin of the amplification of arbitrary small perturbations of a homogeneous field either in space or time [5]. In the context of fiber optic experiments, a very common scenario is the so-called noise induced MI in which the instability of a continuous wave (cw) field (referred to as the pump) is triggered by the random spectral components that fall within the MI gain bandwidth [6]. Following its first observation more than 3 decades ago [4], noise induced MI in optical fibers has been the subject of numerous experimental studies and has often played a key role in the implementation of single-shot detection methods due to its intrinsic stochastic nature and fast time scale. For instance techniques like the time stretch-dispersive Fourier transformation method (TS-DFT) have revealed shot-to-shot fluctuations of the Fourier power spectrum of MI [7][8][9][10] while the so-called time lens systems have enabled the first single-shot intensity recordings of the breakup of optical fields with a picosecond timescale [11,12]. Single-shot spatiotemporal diagrams of MI dynamics have been recently captured using a recirculating fiber loop operating at low cw pumping power [13,14]. Also, the time measurement of the amplitude and phase fluctuations of random optical wavefields has been reported in Refs. [15][16][17]. In particular, an heterodyne time-lens, has been used to observe the dynamics of partially coherent waves in optical fiber [17]. In this work, we use the heterodyne time-lens technique in its digital time-holography or SEAHORSE (spatial encoding arrangement with hologram observation for recording in single shot the electric field) configuration [17] to study the local emergence of breathers structures in the dynamics of the MI process in optical fiber experiments. The simultaneous measurement of the intensity and the phase of the field in single-shot over a temporal window of 200 ps allows us to identify local emergence of quasi-periodic pulse trains accompanied with a characteristic structuration of the phase reminiscent of the Akhmediev breather.

The experimental setup exploites the SEAHORSE technique described for the first time in Ref. [17] and is schematically described in Fig. 1(a). The cw emission from single-mode fiber laser at 1550 nm (signal beam) is chopped down by an acousto-optic modulator (AOM) into 50 ns-long pulses to mitigate Brillouin scattering, and amplified by an Erbium-doped fiber amplifier (EDFA). It is then launched in a 500 m-long single-mode fiber (SMF-28) where it experiences MI before entering the detection stage described hereafter. For the ultra-fast analysis of the signal beam, we used the SEAHORSE technique introduced in Ref. [17] which consists in a spectral encoding system coupled to a heterodyne measurement that allows simultaneous measurement of the intensity and phase of the field. To perform the time-lens effect, the signal beam is nonlinearly mixed with ∼ 200 ps-long chirped pulse (800 nm, pump beam derived from a Coherent Astrella Ti:Sa amplifier) in a BBO crystal. From this interaction, a beam at 528 nm is generated via sum-frequency generation (SFG). Through this process, the ultra-fast fluctuations of the signal are encoded on the spectrum of the 528 nm beam. A diffraction grating (1600 lines/mm) is then used to map the frequency domain onto the pixels of a high-speed camera along the horizontal axis (over a 512 pixels window). Overall, this setup performs a time-to-frequency-to-space mapping that allows acquisitions over a large temporal window of ∼ 200 ps with a resolution shorter than 1 ps. For comparison, in Ref. [17], this setup was used in a different configuration yielding a more restricted field of view of ∼ 40 ps and a resolution of ∼ 80 fs.

In addition to the spectral encoding system, a reference beam provided by a cw laser at 1550 nm is used to perform a heterodyne measurement of the phase of the signal. To do so, the beams are elliptically shaped before they overlap in the BBO crystal (see Fig. 1(b) for an example of camera recording). The reference beam makes a small angle relative to the signal and pump beams resulting in horizontal fringes of interference that are recorded along the vertical axis of the camera. The relative phase between the signal beam and the reference beam is then easily retrieved by tracking the vertical position of the Example of a single-shot recording in false colors. (c) Mapping of the time on the horizontal axis of the camera taking account (solid blue line) or not (dashed blue line) of the cubic contribution to the phase of the pump pulses (red line is the derivative of the solide blue line). cw: continuous wave, AOM: acoustooptic modulator, EDFA: Erbium-doped fiber amplifier, CPL: chirped pulse laser fringes along the horizontal axis. The angle between the beams is ajusted so that around 5 horizontal fringes are observed (Fig. 1(b)). Note that the coherence timescale of the reference laser (∼ 3 µs) is much longer than the temporal acquisition window such that the phase of the reference beam is effectively constant over each acquisition window. Frames are recorded at a 1 kHz repetition rate which is fixed by the pump laser. Numerical processing of the data is performed afterwards to retrieve both the intensity and the phase of the field of each frame. As a consequence of the large temporal field of view in the present configuration (5 times longer than in precedently reported experiments [11,12]), significant aberrations on the borders of the window are observed. Indeed, in analogy with the regular thin lens that affects a field by imposing a transverse quadratic phase at leading order, the time-lens effect is obtained by imposing a longitudinal quadratic phase to the signal. This is done experimentally at the stage of SFG where the chirp of the pump pulse is transfered to the 528 nm beam. However, it is well known that the chirp applied to pulses by the stretcher in a regenerative amplifier is not perfectly linear because of higher order dispersion effects [18,19]. As a consequence of this imperfect linear chirp, there is a significant cubic contribution to the phase resulting in the temporal aberrations.

In this work, we compensate the aberrations during the numerical processing of our experimental recordings for the first time to our knowledge. The principle is summarized in Fig. 1(c). In the aberration-free case, there is a linear mapping between time and camera pixel coordinate (dashed blue line). However, the cubic contribution to the phase of the pump leads to a nonlinear (quadratic) relation between those two quantities (solid blue curve). In practice, aberrations of the time-lens system are evaluated experimentally with a double-pulse experiment: the spacing in pixels between two pulses separated by a known delay is measured over the full acquisition window. This allows the determination of the time interval between adjacent pixels along the window (red line) and consequently, to retrieve the nonlinear mapping between temporal coordinate and horizontal axis of the camera (See Supplement for details on the calibration procedure and signal retrieval algorithm).

Before entering the description of the experiments and to facilitate the comprehension of our results, we describe in Fig. 2 a typical numerical simulation of the nonlinear Schrödinger equation (NLSE):

i ∂Ψ ∂z = β 2 2 ∂ 2 Ψ ∂t 2 -γ|Ψ| 2 Ψ, ( 1
)
where Ψ is the envelope of the optical wavefield, z is the longitudinal variable describing propagation distance, t the time defined in the frame traveling at group velocity of the carrier wave, β 2 the group velocity dispersion coefficient at the carrier frequency, and γ is the Kerr coefficient. Figure 2(a) shows as a 2D color plot the longitudinal evolution of a cw field initialy perturbed by realistic noise over 1 km on a temporal window of 150 ps featuring MI. Parameters are chosen according to our experiments and are given in the caption of Fig. 2. Figure 2(a) illustrates how the initial noisy condensate evolves towards a quasi-regular pattern (between 400 m and 600 m) before exhibiting a more complex spatio-temporal dynamics. The right panels highlight the temporal dynamics at two particular stages by showing both the intensity and the phase of the field at the initial state (Fig. 2(c)) and close to the so-called maximum compression point (MCP) of (Fig. 2(b)). The MCP can be defined in a statistical manner as the point where the Fourier spectrum is the broadest and appears as a saturation of the initial linear stage of MI (exponential growth of the spectral components) [6]. The initial state features small random fluctuations of intensity (blue curve) and uncorrelated fluctuations of the phase with very small excursion (red curve). On the other side, at the MCP, the field exhibits the formation of high intensity peaks separated by a duration close to the MI period (T MI = 2π |β 2 |/(2γP 0 ) ∼ 7.1 ps). Remarkably, the phase profile is now also highly structured and clear phase jumps of ∼ π are observed surrounding each of the high intensity peaks like, for instance, between -50 ps and 0 ps). This is reminiscent of the local appearance of structures that are close to solitons on finite background (SFB) also called breathers [20,21]. In the present case, the field observed in this window resembles the so-called Akhmediev breather (AB) solution close to its MCP. The AB, whose intensity spatio-temporal dynamics is illustrated in Fig. 2(d), is a particular solution of Eq. ( 1) that is localised in z but periodic in t [22]. At its MCP (see Fig. 2(e)), it takes the form of regularly spaced peaks. In particular, consecutive zero intensity points are associated to a succession of ±π phase jumps giving rise to a periodic (square) phase profile (solid red curve) [23]. Recent experiments realised using fiber optics and photorefractive crystals have revealed traces of such ABs dynamics in the context of the seeded MI process [24][25][26][27]. We identify local reminiscent traces of this in our numerical simulation. Lastly, a common feature of the fundamental SFBs is that the sign of the phase flips when crossing the MCP (dashed red line is the phase just after the MCP). The stochastic nature of the noise-induced MI hinders the observation of the exact AB solution.

We have specifically designed an experiment to confirm the noteworthy features described in Fig. 2. Using the setup described in Fig. 1(a) the ultra-fast temporal dynamics of the field (d,e) Same layout as previously but for an exact AB solution. The dashed red line in (e) is the phase of the AB just after the maximum compression. Parameter for the AB is a = 0.25 according to notation in Ref. [20]. can be accessed in single-shot at different stages of the MI process by varying the length of the fiber on the signal path (dashed box in Fig. 1(a)). In this letter we focus on the dynamics in the vicinity of the MCP which, for an input power of 6 W, occurs after ∼ 500 m of propagation. This distance depends on the initial noise level within the MI gain bandwidth [28]. A convenient indicator to assess the randomness of the initial field is the normalized variance of the power P = |Ψ| 2 which is defined as σ 2 = ( P 2 -P 2 )/ P 2 where • denotes average over time. We have evaluated a posteriori that in our experiments σ 0 = σ(z = 0) ∼ 0.07. Note that we did not control the amount of noise in our system which mostly originates from the amplified spontaneous emission of the EDFA. The initial stage is conveniently accessed by removing the fiber spool and connecting the output of the EDFA directly to the detection apparatus. In each case, we have acquired 25 000 frames which corresponds to a total of 5 µs effective recording duration.

Figure 3 shows selected, fully processed, experimental acquisitions at the initial stage (c) and after propagation in the 500 m-long fiber (b). The temporal traces are interpolated in the spectral domain via the zero-padding technique to increase the sampling frequency by a factor 20. Intensity is normalised such that the average over all 25 000 frames is 1. Overall, traces in Fig. 3 share strikingly resembling features with Figs. 2(b) and(c).

After propagation, we observed strongly modulated fields due to MI. The intensity profile exhibits, as expected, an overall irregular pattern where quasi-periodic structures can still be identified. This is in particular the case between 0 ps and 50 ps where an almost regular train of pulses is observed. The period of this train is ∼ 7.4 ps which in good agreement with the period associated to the frequency of maximum MI gain estimated from our experimental parameters (∼ 7.1 ps). Quite noteworthy are also the small "bumps" between each pulses that are well captured by our recording system and which are also reminiscent of the AB solution (see Fig. 2(e)). Regarding the phase of the field, its structure is now strongly correlated to the intensity with large oscillations associated to high intensity peaks and rather flat profiles in regions where MI is not fully developed yet. In the region where the field is almost periodic, phase jumps close to π are observed which is indicative of local proximity to the MCP. Sign flipping of the phase jumps between peaks are also observed in a similar fashion as the one observed in the numerical simulation. It is important to mention that, contrary to Fig. 2(b,c), the field in Fig. 3(c) is not related to the one in Fig. 3(b) as they come from different input pulses. Also, while the field presented in Fig. 3(b) has been selected among 25 000 frames, the features that it exhibits are found on most of the frames, though the size of regular patterns typically vary between 2 and 10 structures.

Taking advantage of the simultaneous measurement of the phase and the amplitude reported in Fig. 3(b) we have performed the local nonlinear spectral analysis of some of the nonlinear structures observed in our experiment (see Fig. 3(d)). Following the methodology described in Ref. [29], we have isolated and analyzed the structures observed between 0 ps and 50 ps and we show the result for 3 of them highlighted in the left panel. Remarkably each individual structure has a nonlinear spectrum (in the sense of the periodic inverse scattering transform (IST) method) composed of three bands (right panels), which means that it can be locally identified as representing a breather solution of the NLSE, see e.g. [27,29].

Finally, Fig. 3(a) illustrates qualitatively the impact of the aberrations of the pump chirp on the reconstruction of the field. The superposition of the intensity of the field with and without (solid blue and dashed green lines resp.) correction of the aberrations shows that the cubic contribution to the phase of the pump is responsible for a stretching (squeezing resp.) of the time on the left (right resp.) side of the frame in accordance with the results shown in Fig. 1(c).

The direct measurement of the full complex field in the temporal domain over a 200 ps-long window allows for the computation of the corresponding single-shot spectrum in the Fourier space with a comfortable 5 GHz resolution. The span of the calculated spectrum is limited to 1 THz by the 1 ps resolution of the SEAHORSE setup. Figure 4 shows the single-shot spectrum associated to the field of Fig. 3(b) (grey line) along with the average spectrum computed from the 25 000 frames (blue line).

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Fig. 4. Single-shot Fourier spectrum computed from the complex field of Fig. 3(b) along with the average spectrum computed over 25 000 single-shot acquisitions and a trace from an optical spectrum analyzer (OSA) recorded simultaneously.

While the single-shot spectrum is inherently very noisy, the smoothing brought by the averaging allows clear observation of the standard MI sidebands at frequency detunings of ∼ ±125 GHz (i.e. ∼ 8 ps timescale). For comparison, the spectrum acquired using a standard optical spectrum analyzer (OSA) with 2 GHz resolution directly at the output of the fiber is superimposed as a dashed red line and shows a very good agreement. The most noticeable discrepancy is found in the wings of the spectrum where the OSA trace presents higher power density than the spectrum computed from the SEAHORSE acquisitions. This can be ascribed to the fact that an optical signal is measured unnecessarily by the OSA during a large dead time lying between consecutive pulses. On the other hand, the time lens acquisitions are intrinsically performed over a temporal window fully comprised in the signal pulses, which results in a more accurate representation of the MI spectrum.

In summary, we report the single-shot observation of extended coherent structures, which are locally reminiscent of Akhmediev breathers, emerging in the nonlinear development of MI in fiber optics experiments. While intensity measurements covering 40 ps have been reported in the pioneering work of Ref. [11], the use of a heterodyne time-lens system enables here the simultaneous recording of both intensity and phase of the field over a very large temporal window of 200 ps allowing for observations with unprecedentedly great completeness. This enables reconstruction of the associated single-shot Fourier spectrum while also providing the opportunity to perform nonlinear spectral analysis of the observed structures.

It is important to emphasize that in this algorithm, the influence of B 3 on the spectral encoding of time t(ω) = B 2 ω + B 3 2 ω 2 is compensated. On the contrary, we neglect here the influence of B 3 on the phases in the integral. The study of these aberrations in the case of signal with broader spectrum is beyond the scope of this Letter and this point deserves further investigations.

NUMERICAL PROCESSING AND ABERRATIONS COMPENSATION : A. Calibration of time

The SEAHORSE technique is based on the spectral encoding of the signal. The spectral profile of the SFG signal is observed on a sCMOS camera where the frequency components are equally spaced on the x axis by means of a simple grating. A given pixel p x of the horizontal axis corresponds to a given frequency ω and thus to a given time through Eq. (S9).

The first step of the analysis is to find the quantitative relationship between time and the index px of the pixel:

t = a(px -px 0 ) + b 2 (px -px 0 ) 2 (S12)
where a ∝ B As an example, two different snapshots of the double pulses recorded by using the SEAHORSE are plotted in Fig. S1. By using numerous recorded frames, we measure the distance in pixel between the two pulses as a function of their mean position px on the camera (see the blue curve in Fig. S2).

By using an optimization procedure, we evaluate px 0 , a and b of Eq. (S12) for which the measured interval ∆t remains constant over the whole window of observation (see the red curve in Fig. S2):

px 0 = 232 (S15)
Note that, as expected, the evaluated value of px 0 is close to the centroid of the signal. Note finally that the optimization procedure is an iterative process taking into account the digital focus (see below).

B. Signal retrieval

Above, the correspondence between the time t and the index px of the horizontal pixel of the camera has been established.

In a second step, the intensity and phase of Y(t) [see Eq. (S11)] are retrieved from the raw data recorded onto the camera. A typical single-shot pattern recorded by the camera is plotted in Fig. S3.a). Roughly speaking, for a given time, the intensity |Y(t)| 2 corresponds to the sum of the pattern over a vertical line divided by p(ω) [see Fig. S3.b)]. The phase is encoded in the fringes positions and is retrieved by means of Fourier transform [see Fig. S3.d)]. See [1] for a detailed description of the full procedure.

In a last step, we perform the Final data analysis described in Sec. 1. Conceptually, this step corresponds to a digital focus analogous in the time domain to the spatial digital holography. The intensity and phase of the retrieved signal is plotted in Fig. S3.(e,f).

It is important to note that the influence of the third order dispersion in the spectral encoding of time is rigorously compensated in the data analysis. On the contrary, for the signals under investigation in this Letter, the influence of the third order dispersion in the digital focus is neglected. The compensation of aberrations for signals with broader spectrum deserves further investigations.

Fig. 1 .

 1 Fig. 1. (a) Schematic description of the experimental setup. (b) Example of a single-shot recording in false colors. (c) Mapping of the time on the horizontal axis of the camera taking account (solid blue line) or not (dashed blue line) of the cubic contribution to the phase of the pump pulses (red line is the derivative of the solide blue line). cw: continuous wave, AOM: acoustooptic modulator, EDFA: Erbium-doped fiber amplifier, CPL: chirped pulse laser

Fig. 2 .

 2 Fig. 2. Typical numerical simulation of the NLSE exhibiting noise-driven MI. (a) Spatio-temporal diagram of the intensity for the first 1000 m of propagation. Complex field at (c) the initial state and (b) after 420 m of propagation. P 0 = |Ψ(z = 0)| 2 = 6 W, β 2 = -20 ps 2 /km, γ = 1.3 /W/km.(d, e) Same layout as previously but for an exact AB solution. The dashed red line in (e) is the phase of the AB just after the maximum compression. Parameter for the AB is a = 0.25 according to notation in Ref.[20].

Fig. 3 .

 3 Fig. 3. Experimental complex fields recorded and reconstructed (c) before and (b) after propagation in 500 m of singlemode fiber. (a) shows the reconstructed intensity of the same field with (solid blue line) and without (dashed green line) compensation of the aberrations. (d) (left panel) Focus on the quasi-periodic pattern of (b) and (right panels) periodic IST profiles of isolated structures. Parameters are the same as in the caption of Fig. 2.

  2 and b ∝ B 3 . a and b are measured by using a double pulse experiment. Two pulses separated by 81.6 ps are generated by launching a picosecond mode-locked laser (Pritel) beam into a Michelson interferometer. The two pulses are recorded by using the SEAHORSE device. Even though the time interval between the two pulses is fixed at the output of the Michelson interferometer, the pixel interval between the two recorded pulses depends on their positions in the window of measurement because b = 0.

Fig. S1 .

 S1 Fig. S1. Double pulse recorded by using the SEAHORSE. The separation ∆px between the two pulses recorded on the camera depends on their mean position px within the measurement window. Blue and green curve are two different snapshots.

Fig. S2 .

 S2 Fig. S2.Blue dot : number of pixels ∆px between the centroids of the two pulses as a function of their mean position < px > in the window. Red dot : time interval between the two pulses ∆t computed from the nonlinear pixel to time conversion Eq. (S12). Blue and red solid lines are linear fits related to the same color.

Fig. S3 .

 S3 Fig. S3. a) Typical single-shot pattern recorded by the camera. b) Green line : intensity profile extracted from a). Orange : intensity profile divided by the averaged envelop p(ω). c) Intensity |Y(t)| 2 d) Phase of Y(t). In c) and d) the nonlinear pixel to time conversion of Eq. (S12) has been used. e) Intensity of the studied signal |S(t)| 2 f) Phase of S(t) g) Principle of the digital focus enabling to retrieve (e,f) from (c,d).

Funding. European Regional Development Fund ( CPER Photonics for Society P4S); Ministry of Higher Education and Research, Hauts de France council; Agence Nationale de la Recherche (CEMPI ANR-11-LABX-0007, DYDICO ANR-16-IDEX-0004).

Single-shot observation of breathers from noise-induced modulation instability using heterodyne temporal imaging: supplemental document

The purpose of this Supplemental Material is to provide some details about the methods used to extract the amplitude and the phase of the optical signal from the raw data recorded in the experiments. In the first demonstration of the SEAHORSE (spatial encoding arrangement with hologram observation for recording in single shot the electric field), the field of view was a few tens of picoseconds and aberrations were neglected [1]. In this Letter, we extend the field of view up to 200 ps and we demonstrate that aberrations induced by the third order dispersion can be significantly compensated in the data analysis. The theoretical background needed to describe the operation of the SEAHORSE is briefly given in Sec. I. Details about numerical data processing and aberration compensation are provided in Sec. II.

SIGNAL RETRIEVAL IN THE SEAHORSE (THEORY)

The notations used throughout this Supplementary document for the field and the Fourier transform are listed below.

Direct Fourier Transform

S(t) is the slowly-varying amplitude of the electric field e(t) = S(t) e i(ω 0 t) + c.c. under study:

The slowly-varying amplitude of the pump pulse is P(t). The Fourier components of the chirped pump pulse read :

B 2 is the group velocity dispersion (GVD) experienced by the pump pulse in the regenerative amplifier (Astrella, Coherent) B 3 is the third order dispersion (TOD) experienced by the pump pulse.

In the BBO crystal, the pump intensity is much higher than the signal intensity. In the approximation of weak conversion, the field generated by sum frequency generation (SFG) is proportional to P(t) and S(t) [2] SFG(t) ∝ S(t) P(t) (S4)

In the SEAHORSE, a monochromatic reference beam is also launched into the BBO crystal and the SFG between the pump and the reference provides a field that is simply proportional to P(t): SFG R (t) ∝ P(t).

The second part of the SEAHORSE is made of a single-shot spectrum analyser. We thus measure the modulus of SFG(ω) and the relative phase between SFG(ω) and SFG R (ω).

Finally, the recorded data X(ω) thus reads :

Parameters of the experiments

We neglect the last two terms in the integral. Moreover, as ω max ≫ ω ′ max one can approximate p(ωω ′ ) ≈ p(ω). Finally, one gets :

Assuming stationarity of the statistics of the random signal S(t), the Fourier components are δ-correlated : S(ω ′ ) S * (ω ′′ ) = N(ω ′ )δ(ω ′ω ′′ ). By averaging the data over many realizations, we measure p(ω):

Eq. (S6) finally reads:

Final data analysis

Starting from Eq.(S11), the signal S(t) is retrieved by using the simple algorithm:

1/ Compute Ỹ(ω), the Fourier transform of Y(t) 2/ Apply the operator of propagation e +iB 2 ω 2 /2 3/ Perform the inverse Fourier transform