Romain Rouxel

Michele Diego

Paolo Maioli

Noelle Lascoux

Fabien Vialla

Francesco Rossella

Francesco Banfi

Fabrice Vallée

Natalia Del Fatti

Aurélien Crut

Electron and Lattice Heating Contributions to the Transient Optical Response of a Single Plasmonic Nano-Object

Keywords: single nano-objects, time-resolved spectroscopy, transient optical response, gold nanoparticles, electron-phonon energy exchanges, nanoplasmonics

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

INTRODUCTION

The impulsive excitation of a metal nano-object induces a series of ultrafast relaxation processes, including internal thermalization by electron-electron and electron-phonon scattering, acoustic vibrations and thermal cooling. All these phenomena have generated a large interest, as they enable to address fundamental nanoscience questions (e.g., regarding the dependence of their associated kinetics on nano-object size) and allow the development of applications based for instance on the use of nano-objects for catalytic purposes or as generators of heat and GHz/THz acoustic waves. [START_REF] Kamat | Photochemical and Photocatalytic Aspects of Metal Nanoparticles[END_REF][START_REF] Siemens | Quasi-Ballistic Thermal Transport from Nanoscale Interfaces Observed Using Ultrafast Coherent Soft X-Ray Beams[END_REF][START_REF] Chen | Silica-Coated Gold Nanorods as Photoacoustic Signal Nanoamplifiers[END_REF][START_REF] Baffou | Thermo-Plasmonics: Using Metallic Nanostructures as Nano-Sources of Heat[END_REF][START_REF] Cahill | Nanoscale Thermal Transport. II. 2003-2012[END_REF][START_REF] Gil-Santos | High-Frequency Nano-Optomechanical Disk Resonators in Liquids[END_REF][START_REF] Crut | Acoustic Vibrations of Metal Nano-Objects: Time-Domain Investigations[END_REF] Optical pump-probe spectroscopy constitutes a powerful approach to experimentally investigate these processes. It is based on the excitation of nano-objects by a femtosecond "pump" pulse and the monitoring of their subsequent relaxation dynamics using a second time-delayed femtosecond "probe" pulse, and enables the investigation of the dynamic processes occurring on femtosecond to nanosecond timescales, provided they have an impact on the optical response of the investigated nano-objects. Many experimental investigations on ensembles of metal nano-objects (in most cases colloidal solutions or glass samples containing gold or silver nanoparticles) have been performed in the last 25 years. They have greatly clarified the kinetics of the phenomena mentioned above, and their dependences on the properties of the nanoparticles (composition, morphology, crystallinity…) and of their local environment. [START_REF] Voisin | Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles[END_REF][START_REF] Arbouet | Electron-Phonon Scattering in Metal Clusters[END_REF][START_REF] Guillet | Spectral Dependence of the Ultrafast Optical Response of Nonspherical Gold Nanoparticles[END_REF][START_REF] Hartland | Optical Studies of Dynamics in Noble Metal Nanostructures[END_REF][START_REF] Polli | Low Sensitivity of Acoustic Breathing Mode Frequency in Co Nanocrystals upon Change in Nanocrystallinity[END_REF][START_REF] Ferrera | Thermometric Calibration of the Ultrafast Relaxation Dynamics in Plasmonic Au Nanoparticles[END_REF][START_REF] Brown | Experimental and Ab Initio Ultrafast Carrier Dynamics in Plasmonic Nanoparticles[END_REF][START_REF] Maioli | Mechanical Vibrations of Atomically Defined Metal Clusters: From Nano-to Molecular-Size Oscillators[END_REF][START_REF] Ge | Thermal Transport in Au-Core Polymer-Shell Nanoparticles[END_REF] The introduction of single metal nanoparticle optical spectroscopy techniques at the beginning of the 21 th century has represented an important breakthrough in this research field. [START_REF] Sönnichsen | Drastic Reduction of Plasmon Damping in Gold Nanorods[END_REF][START_REF] Arbouet | Direct Measurement of the Single-Metal-Cluster Optical Absorption[END_REF][START_REF] Van Dijk | Detection of Acoustic Oscillations of Single Gold Nanospheres by Time-Resolved Interferometry[END_REF][START_REF] Van Dijk | Absorption and Scattering Microscopy of Single Metal Nanoparticles[END_REF][START_REF] Muskens | Femtosecond Response of a Single Metal Nanoparticle[END_REF][START_REF] Hu | Dark-Field Microscopy Studies of Single Metal Nanoparticles: Understanding the Factors That Influence the Linewidth of the Localized Surface Plasmon Resonance[END_REF] Indeed, singleparticle experiments avoid all the spurious effects associated to dispersions of nanoparticle size, shape and environment affecting measurements on ensembles of nanoparticles. In particular, this approach allows more detailed studies of damping processes, and also enables to precisely quantify the transient changes of the optical cross-sections of a nanoparticle occurring after its sudden excitation, and to correlate them with the nanoparticle's morphology and static optical properties. Up to now, singleparticle time-resolved spectroscopy techniques have been mostly exploited for the investigation of the vibrational damping of nano-objects, allowing significant progress in the understanding of this phenomenon. [START_REF] Crut | Acoustic Vibrations of Metal Nano-Objects: Time-Domain Investigations[END_REF][START_REF] Major | Damping of the Acoustic Vibrations of a Suspended Gold Nanowire in Air and Water Environments[END_REF][START_REF] Marty | Damping of the Acoustic Vibrations of Individual Gold Nanoparticles[END_REF][START_REF] Belliard | Vibrational Response of Free Standing Single Copper Nanowire through Transient Reflectivity Microscopy[END_REF][START_REF] Yu | Damping of Acoustic Vibrations of Immobilized Single Gold Nanorods in Different Environments[END_REF][START_REF] Medeghini | Controlling the Quality Factor of a Single Acoustic Nanoresonator by Tuning Its Morphology[END_REF][START_REF] Su | Acoustic Vibrations and Energy Dissipation Mechanisms for Lithographically Fabricated Plasmonic Nanostructures Revealed by Single-Particle Transient Extinction Spectroscopy[END_REF] They were also applied -although less frequently -to studies of the internal thermalization [START_REF] Muskens | Femtosecond Response of a Single Metal Nanoparticle[END_REF][START_REF] Baida | Ultrafast Nonlinear Optical Response of a Single Gold Nanorod near Its Surface Plasmon Resonance[END_REF][START_REF] Zavelani-Rossi | Transient Optical Response of a Single Gold Nanoantenna: The Role of Plasmon Detuning[END_REF] and cooling dynamics [START_REF] Rouxel | Ultrafast Thermo-Optical Dynamics of a Single Metal Nano-Object[END_REF] of metal nanoparticles.

The aim of the work described in this paper was to quantitatively investigate the contributions of electron and lattice heating to the transient optical response of a single plasmonic nanoparticle at short timescales preceding its internal thermalization, i.e., when its state cannot be defined by a single temperature (Figure 1). To do so, time-resolved experiments with pump-probe delays of up to 10 ps were performed on single gold nanodisks (NDs). They were analyzed in the framework of a theoretical model addressing the dynamics and influence on the gold dielectric function of both electron and lattice heating, and constituting a refined version of that used in a previous work, performed on single plasmonic nanoparticles with a different shape (single gold nanorods). [START_REF] Baida | Ultrafast Nonlinear Optical Response of a Single Gold Nanorod near Its Surface Plasmon Resonance[END_REF] Comparison of the measured signals with this more complete model enables to clarify the contributions of electron and lattice heating to the transient optical response of the investigated NDs at all considered timescales. We show that, for delays posterior to the excitation and the thermalization of the electron gas, time-resolved signals generally reflect a combination of electron and lattice temperature evolutions (whose typical temporal profiles are shown in Figure 1b), with probe wavelength-dependent weights. We then demonstrate that these distinct dynamics can both be selectively probed through the use of specific probe wavelengths.

EXPERIMENTAL AND THEORETICAL METHODS

Nanodisk Fabrication and Morphological Characterization. Gold NDs were nanopatterned by a combination of electron beam lithography, Au thermal evaporation, and lift-off techniques on the optically polished surface of a 480 μm thick (0001) α-Al2O3 single-crystal sapphire substrate. [START_REF] Medeghini | Controlling the Quality Factor of a Single Acoustic Nanoresonator by Tuning Its Morphology[END_REF][START_REF] Rouxel | Ultrafast Thermo-Optical Dynamics of a Single Metal Nano-Object[END_REF][START_REF] Medeghini | Signatures of Small Morphological Anisotropies in the Plasmonic and Vibrational Responses of Individual Nano-Objects[END_REF] Atomic force and scanning electron microscopy characterizations were performed to estimate their thicknesses (about 20 nm) and diameters (about 100 to 150 nm for the NDs considered in this work), respectively.

Spatial Modulation Microscopy/Spectroscopy. Localization and extinction spectroscopy of individual

NDs were performed by spatial modulation spectroscopy (SMS), a far-field optical technique based on the periodic displacement of a single nano-object in the focal spot of a tightly focused light beam, which enables the extinction cross-section σext of single nano-objects to be quantitatively determined as a function of the illumination wavelength λ and light polarization angle θ. [START_REF] Arbouet | Direct Measurement of the Single-Metal-Cluster Optical Absorption[END_REF][START_REF] Crut | Optical Absorption and Scattering Spectroscopies of Single Nano-Objects[END_REF] SMS experiments were performed using a tunable Ti:sapphire oscillator combined with a visible optical parametric oscillator and a frequency-doubling BBO crystal as a light source, enabling optical studies in the 375-1040 nm wavelength range. The light beam delivered by this source was focused on the sample by a 100X microscope objective, with the direction of its linear polarization being controlled by a wire grid polarizer. The size of the laser spot at the sample surface, close to the diffraction limit (about 0.7 λ fullwidth at half-maximum), was precisely determined by two-dimensional fitting of the SMS images of small NDs obtained using an XY piezoelectric translation stage. Spatial modulation of the sample was performed at f=1.5 kHz frequency with 300 nm modulation amplitude in the Y direction, and lock-in detection at 2f. Quasicircular NDs, identified from the weak dependence of their extinction on θ, [START_REF] Medeghini | Signatures of Small Morphological Anisotropies in the Plasmonic and Vibrational Responses of Individual Nano-Objects[END_REF] were selected for the present study.

Time-Resolved Experiments. The transient optical response of individual gold NDs was measured using single-particle femtosecond pump-probe spectroscopy, [START_REF] Muskens | Femtosecond Response of a Single Metal Nanoparticle[END_REF][START_REF] Baida | Ultrafast Nonlinear Optical Response of a Single Gold Nanorod near Its Surface Plasmon Resonance[END_REF] an approach based on the illumination of the investigated nanoparticle with two spatially overlapping, tightly focused, and time-delayed femtosecond laser pulses, referred to as pump and probe. Time-resolved measurements on individual NDs were performed by combining the SMS microscope with a two-color pump-probe setup based on the femtosecond laser source described in the previous paragraph. The oscillator pulse train was split in two parts to generate the pump and probe beams. The pump beam was produced using second harmonic generation (yielding λpp in the 375-510 nm range), whereas the probe beam was either the signal from the Ti:sapphire oscillator (with λpr=750-1020 nm in this case) or the output of the optical parametric oscillator (λpr=500-730 nm, with here λpp=410 nm). Durations of approximately 1 ps and 200 fs were measured for the pump and probe pulses, respectively (the relatively long duration of pump pulses in the blue spectral region originating from their broadening through the microscope objective used in this work). A mechanical chopper operating at 30 kHz was used for pump power modulation. The pump-induced relative changes of the probe beam transmission, ΔTr/Tr, were measured using synchronous detection as a function of the time interval separating the pump and probe pulses, controlled by a mechanical delay line. This raw measured signal is directly related to the transient changes of the extinction cross-section σext(λpr) of the investigated nano-object at probe wavelength λpr via ΔTr/Tr = -Δσext (λpr)/Spr, with Spr = πdfwhm 2 /(4 ln 2) representing the area of the probe spot and dfwhm its full-width at half-maximum. [START_REF] Baida | Ultrafast Nonlinear Optical Response of a Single Gold Nanorod near Its Surface Plasmon Resonance[END_REF] As Spr can be estimated by analysis of SMS images, experiments provide access to the ultrafast variations of Δσext as a function of the pump-probe delay.

The temporal origin of the signals, defined as the instant at which the pump and probe pulses present a maximal overlap, was a posteriori determined for each signal by comparison with the complete model presented below.

Modeling of the gold dielectric function. The dielectric function ε=ε1+iε2 of bulk gold in the visible/near-infrared range can be written as the sum of two frequency-dependent components, related to interband (ib) and intraband (Drude-like) absorption mechanisms [START_REF] Crut | Optical Absorption and Scattering Spectroscopies of Single Nano-Objects[END_REF][START_REF] Kreibig | Optical Properties of Metal Clusters[END_REF][START_REF] Hövel | Width of Cluster Plasmon Resonances: Bulk Dielectric Functions and Chemical Interface Damping[END_REF] :

= + -1 = - (1)
with ωP the plasma frequency (ℏωP = 9.01 eV for bulk gold) and γ the optical scattering rate of the conduction electrons (ℏγ ≈ 50 meV for bulk gold). In this work, ε was modeled based on Johnson and Christy tables of the bulk gold dielectric function εJC. [START_REF] Johnson | Optical Constants of the Noble Metals[END_REF] The interband component of the bulk gold dielectric function at room temperature, deduced from εJC tables as = + , was theoretically reproduced using the approach developed by Rosei [START_REF] Rosei | Temperature Modulation of the Optical Transitions Involving the Fermi Surface in Ag: Theory[END_REF] and the gold band structure calculated by Rangel et al. [START_REF] Rangel | Band Structure of Gold from Many-Body Perturbation Theory[END_REF] The imaginary part of , (ω) was obtained by evaluating the total probability of a transition from an occupied state to an empty one by absorption of a photon of energy The dielectric function of gold was described using eq 1, using for and their bulk values but leaving γ as a free parameter (using ℏγ values larger than the bulk gold value is often necessary to reproduce the SPR width measured on single nano-objects, and physically corresponds to taking into account additional non-radiative plasmon damping mechanisms in metal nano-objects as compared to the bulk metal case [START_REF] Baida | Quantitative Determination of the Size Dependence of Surface Plasmon Resonance Damping in Single Ag@SiO2 Nanoparticles[END_REF][START_REF] Juvé | Size-Dependent Surface Plasmon Resonance Broadening in Nonspherical Nanoparticles: Single Gold Nanorods[END_REF][START_REF] Zijlstra | Chemical Interface Damping in Single Gold Nanorods and Its near Elimination by Tip-Specific Functionalization[END_REF][START_REF] Foerster | Plasmon Damping Depends on the Chemical Nature of the Nanoparticle Interface[END_REF] ). ND illumination by a linearly polarized plane wave was considered.

Absorption and scattering cross-sections were deduced from the computed electric field, and summed to yield the extinction one σext. The D and h values used in the modeling were adjusted so as to optimize the agreement with the investigated experimental spectrum (Figure 2a). The ∂σext/∂ε1 and ∂σext/∂ε2 partial derivatives (Figure 2b) were deduced from these calculations by computing σext with slightly modified ε1 or ε2 values and then performing discrete derivation.

Modeling of the transient optical response of gold NDs. Complete modeling of the transient optical response of gold NDs following their impulsive excitation was performed along the following steps.

-Evolution of the energy-and time-dependent distribution function f(E, t) (where E refers to the energy of the electronic state) of gold quasi-free electrons in the conduction band was computed in the framework of a bulk-like approach by numerically solving Boltzmann equation, [START_REF] Stoll | Advances in Femto-Nano-Optics: Ultrafast Nonlinearity of Metal Nanoparticles[END_REF][START_REF] Del Fatti | Nonequilibrium Electron Dynamics in Noble Metals[END_REF] including the effects of three terms, corresponding to initial pulse excitation (leading to an initially athermal electron distribution), thermalization of the electron gas by electron-electron scattering (occurring on a timescale of a few hundred fs) and electron-to-lattice energy transfer by electron-phonon scattering (complete after a few ps).

-The electron excess energy density ∆ue(t) and equivalent temperature increase ∆Te (t) are deduced from ∆f(E, t) = f(E, t) -f0 (E), with f0 the initial electronic Fermi-Dirac distribution at room temperature (T0=295 K). The lattice excess energy density ∆uL(t) and temperature increase ∆TL (t) are also computed at each time-step by applying energy conservation for the {electron+lattice} system (see below for more details).

-Knowledge of ∆f(E, t) and ∆TL (t) enables the transient change ∆ε(t) of the ND dielectric function to be determined at each time-step, by separately computing and summing its interband and Drude-like components.

-Combining Δε(t) with the ∂σext/∂ε1 and ∂σext/∂ε2 derivatives deduced from FEM simulations (see The average measured values of pump and probe pulse durations (about 1 ps and 200 fs) were considered in the modeling, for the time-dependent excitation of the electron gas and for convoluting the computed ∆σext(t), respectively. A 500 fs time constant was assumed for the electron gas thermalization by electron-electron interactions in gold, [START_REF] Del Fatti | Nonequilibrium Electron Dynamics in Noble Metals[END_REF] while the electron-to-lattice energy transfer time (≈1 ps in gold in the weak-excitation regime [START_REF] Arbouet | Electron-Phonon Scattering in Metal Clusters[END_REF][START_REF] Sun | Femtosecond-Tunable Measurement of Electron Thermalization in Gold[END_REF][START_REF] Hodak | Photophysics of Nanometer Sized Metal Particles: Electron-Phonon Coupling and Coherent Excitanon of Breathing Vibranonal Modes[END_REF] ) was left as a free parameter, as its value for nanoparticles produced by electron beam lithography may be affected by the presence of crystalline defects. [START_REF] Su | Ultrafast Electron Dynamics in Single Aluminum Nanostructures[END_REF][START_REF] Staechelin | Size-Dependent Electron-Phonon Coupling in Monocrystalline Gold Nanoparticles[END_REF] The ND thermal cooling induced by heat propagation in the substrate, investigated in detail in a recent study, [START_REF] Rouxel | Ultrafast Thermo-Optical Dynamics of a Single Metal Nano-Object[END_REF] was neglected here as it occurs on timescales of about 500 ps for 20 nm thick NDs, much longer than the <10 ps ones investigated here.

section
Electronic and lattice energy densities and temperatures. In more details, the excitation process leads to the injection in the ND of the total excess energy density uabs, which represents the integral of the instantaneous absorbed power density pabs over the pump pulse duration. uabs is computed as uabs= σabsFpp/V, with σabs and V being the ND absorption cross-section and volume, deduced from FEM optical simulations, and Fpp=Ppp/(frepSpp) the fluence of the pump beam (60 µJ/cm 2 average value in our experiments), with Ppp and Spp its average power and focal spot size, and frep the repetition rate of the laser source. An excitation temperature " # (the temperature that the electron gas would achieve at the end of its thermalization, in the absence of electron-phonon coupling) can be defined through the

relation $ % & = ' ( " )" * +,- * .
, yielding " # = / " 0 + 1 234 %

, with ce = aTe the volumetric electronic heat capacity (a≈65 J.K -2 .m -3 for gold). Similarly, an equivalent temperature (i.e., associated to a Fermi-Dirac distribution with same excess energy) can be defined from ∆ue(t), the time-dependent excess energy density in the electron gas, as " = / " 0 + ∆1 + %

. The increase of the lattice temperature (assumed to remain well-defined throughout the experiments), ∆TL(t), is calculated by considering that the energy lost by the electrons is entirely transferred to the lattice (the temperaturedependence of the lattice volumetric heat capacity cL being neglected, due to the small ∆TL≈10 K maximal lattice heating in our experiments). We note ∆Teq the equilibrium temperature rise of the ND after its full internal thermalization. [START_REF] Del Fatti | Femtosecond Optical Investigation of Electron-Lattice Interactions in an Ensemble and a Single Metal Nanoparticle[END_REF] An example of the temporal evolutions of pabs, ∆ue, ∆uL, ∆Te and ∆TL computed using pump pulse and nano-object parameters relevant for the experiments described in this paper can be found in Figure 1b.

Transient variations of the gold dielectric function. ε (eq 1; note that the spectral dependence of is not reminded in this paragraph for the sake of simplicity) is a function of both the electronic distribution f (E) and the lattice temperature TL. As ∆TL values are small in our experiments (typically less than 10 K), the dependence of ε on TL=T0+∆TL can be linearized as ε (f, TL) ≈ ε (f, T0) + ∆TL (f, T0).

Moreover, numerical investigations based on the model described in ref. [START_REF] Stoll | Advances in Femto-Nano-Optics: Ultrafast Nonlinearity of Metal Nanoparticles[END_REF] showed that the derivative does not strongly depend on the electronic distribution, and can thus be approximated by (f0, T0). Under this approximation, the modifications of ε induced by electron and lattice heating contributions can be fully separated. The transient change ∆ε(t) of the gold dielectric function can be decomposed as the sum of an electronic contribution, ∆εe(t), resulting from the modification of the electronic distribution f(t), and of a lattice one, ∆εL(t), resulting from the increase of the lattice temperature ∆TL(t):

∆ = ∆ + ∆ 8 (2)
with

9 ∆ = : , " 0 -: 0 , " 0 = ∆ + ∆ ∆ 8 = 56 5* 7 : 0 , " 0 • ∆" 8 = ∆ 8 + ∆ 8 (3)
, the equalities on the right of eq 3 corresponding to the decomposition of ∆εe,L into interband and Drude components (eq 1). Note that, in the formulation of eq. 3, ∆εL(t) is proportional to ∆TL(t) while the dependence of ∆εe(t) on the electronic temperature is more complex.

Electronic contribution to ∆ε (∆ ). The ∆ (t) term appearing in eq 3, associated to the modification of the probabilities of interband transitions to and from the conduction band caused by the evolution of the electron distribution f(E, t), was computed by reproducing the calculations discussed in the paragraph Modeling of the gold dielectric function with the f(E, t) distribution functions obtained at multiple instants t by solving the Boltzmann equation. Modification of the electronic distribution additionally generates a ∆ (t) change of the Drude component of ε (eq 3), dominated by the modification of the electron-electron scattering rate γe-e, which contributes to the overall scattering rate γ appearing in the Drude component of ε and mainly affecting its imaginary part (eq 1). The variation of the electron-electron scattering rate has been predicted to be quadratic with the electron temperature Te: [START_REF] Stoll | Advances in Femto-Nano-Optics: Ultrafast Nonlinearity of Metal Nanoparticles[END_REF][START_REF] Gurzhi | Mutual Electron Correlations in Metal Optics[END_REF] ∆ +<+

+<+ = = >? @ ħ B " -" 0 ! (4)
with γe-e ≈ 15 meV at room temperature in the visible range [START_REF] Smith | Frequency Dependence of the Optical Relaxation Time in Metals[END_REF] -a value however affected by a large uncertainty, as measurement of the individual contribution of electron-electron scattering to the overall scattering rate γ is challenging. In this work, eq 4 was used both in the thermal and athermal regimes, using the equivalent electronic temperature in the latter case.

Lattice contribution to ∆ε (∆ 8 ). Lattice heating induces a modification of both ε ib (by lattice dilation, ∆ 8 (t) term in eq 3) and ε D (by modification of the plasma frequency and scattering rate γ appearing in eq 1, respectively caused by the dilation-induced change of electron density and by the TL-dependence of electron-phonon scattering rate, ∆ 8 (t) term in eq 3). As in the small ∆TL regime considered here ∆εL can be assumed to depend linearly on ∆TL (eqs 2 and 3), the optical consequences of lattice heating can therefore be summarized by a single dε/dTL spectrum. In this work, we used the dε1,2/dTL tables derived by Stoll et al. from a theoretical model accounting for the effects mentioned above, [START_REF] Stoll | Advances in Femto-Nano-Optics: Ultrafast Nonlinearity of Metal Nanoparticles[END_REF] which were shown to allow a good reproduction of the thermo-optical dynamics of single gold NDs in the regime of large time delays, for which their electronic and lattice temperatures coincide. [START_REF] Rouxel | Ultrafast Thermo-Optical Dynamics of a Single Metal Nano-Object[END_REF] ∆ε-induced transient changes of ND extinction cross-section. ∆σext(λpr,t) was deduced from the computed gold dielectric function changes ∆ε1,2 (λpr,t) and ε1,2 derivatives of ND extinction crosssection as: was discussed in the section Modeling of the linear optical response of gold NDs, their values being shown in Figure 2b for the ND considered throughout the paper. Eq 5 implies that, similarly to ∆ε (eqs 2-3), ∆σext (λpr, t) can be decomposed as a sum of two terms (each one being the sum of interband and Drude components, as in eq 3) associated to electronic and lattice heating, respectively: (e.g., for λpr=855 nm), close to 0 (for λpr=700 nm) or slightly negative (for λpr=905 nm). Additionally, the signal measured with λpr=890 nm presents a very different shape, featuring a slow rise between 0 and 5 ps, followed by a plateau.

∆ λ , ! = 5C +,D 56 E λ ! ∆ λ , ! + 5C +,D 56 λ ! ∆ λ , ! (5)
∆ λ , ! = ∆ , λ , ! + ∆ ,8 λ , ! (6)
Time-resolved signals are plotted in Figure 3c as a function of λpr for fixed delays, corresponding to 0.7, 2.5 and 8 ps. These transient spectra present common general features, ∆σext being positive at short and large wavelengths (below ≈700 nm and above ≈900 nm), and negative in-between. However, their amplitudes and spectral shapes evolve in time (as more easily observable on the normalized spectra shown in Figure 3d), with in particular a red-shift at longer delays of the ∆σext minimum around λpr≈850 nm and of the zero crossing at λpr≈700 nm and ≈900 nm. The shape of transient extinction spectra is close to those of the ∂σext/∂ε1 and ∂σext/∂ε2 derivatives (Figure 2b) at short and large delays, respectively.

The measured transient extinction changes ∆σext(t, λpr) were quantitatively compared to the results of the numerical model described in the Methods part, which allows to compute the temporal evolutions of the electronic distribution f(E, t) and of the lattice temperature TL(t) occurring during the ND internal thermalization, as well as the induced transient changes of the gold dielectric function ∆ε =∆ε1+i∆ε2 and of the ND extinction cross-section ∆σext. This model constitutes a refined version of that used in a previous work which addressed the transient optical response of single gold nanorods, in which only the temporal variations of the ∆ term were quantitatively computed. [START_REF] Baida | Ultrafast Nonlinear Optical Response of a Single Gold Nanorod near Its Surface Plasmon Resonance[END_REF] Here, two additional ingredients were taken into account to estimate the time-dependent ∆ε changes, namely the modification of ε by lattice heating (∆ 8 and ∆ 8 terms) and that of the Drude component of ε by electron heating (∆ term) (eqs 2-3).

Measured signals were quantitatively compared with complete calculations based on eqs 2 and 3 performed with ∆Teq= 6.5 K, close to the experimental values for the investigated ND (Figure 4). The global agreement between experimental and simulated signals was found to be optimal for an electron-lattice energy transfer time of τe-L≈ 800 fs at weak excitation, a value slightly smaller than the one measured on large diameter colloidal gold nanoparticles [START_REF] Arbouet | Electron-Phonon Scattering in Metal Clusters[END_REF] , possibly as a result of polycrystallinity effects [START_REF] Staechelin | Size-Dependent Electron-Phonon Coupling in Monocrystalline Gold Nanoparticles[END_REF] (the increase of this time constant for strong excitation, [START_REF] Del Fatti | Femtosecond Optical Investigation of Electron-Lattice Interactions in an Ensemble and a Single Metal Nanoparticle[END_REF][START_REF] Mongin | Ultrafast Electron-Lattice Thermalization in Copper and Other Noble Metal Nanoparticles[END_REF] illustrated in Figure S6 of the Supporting Information, is predicted by the numerical model). This initial analysis also showed that the agreement between measured and simulated time-resolved signals (in particular in terms of amplitudes) could be significantly improved in some cases by comparing experimental measurements with time-resolved signals simulated using a slightly different (within < 15 nm) λpr value. Such a fine tuning of λpr in the modeling is justified by the occasional observation of small evolutions of the linear extinction spectra of the investigated NDs during the course of time-resolved measurements.

The overall agreement between experimental (blue dots in Figure 4) and calculated (green lines) timeresolved signals is good, regarding both their absolute amplitudes and their kinetics, for all probe wavelengths investigated. Moreover, the possibility to distinguish in the computations the ∆σext components associated to electron and lattice heating (∆σext,e and ∆σext,L, eq 6, shown as dotted red and dashed black lines in Figure 4) permits to clarify the physical origins of the measured transient signals and to explain the particular temporal profiles of some of the signals shown in Figure 4, namely:

-Signals decreasing to nearly zero at the end of the ND internal thermalization (such as those shown in Figure 4a and f, corresponding to measurements with λpr=725 nm and λpr=915 nm). They correspond to the exceptional case of a very small lattice heating contribution to ∆σext. Such signals thus reflect the successive heating and cooling of the electrons in gold.

-The signal measured at λpr= 890 nm (Figure 4d), which presents a slow increase (in absolute value), in strong contrast with the rapid rise followed by a decay seen for other experimental signals. It corresponds to the opposite exceptional case of a negligible impact of electron heating on the transient extinction signal. This signal shape, here measured on the investigated gold ND only for a specific λpr value, is reminiscent of those observed in the context of measurements on aluminum NDs, for which lattice heating was predicted to have a dominant effect in the whole 500-900 nm λpr range. [START_REF] Su | Ultrafast Electron Dynamics in Single Aluminum Nanostructures[END_REF] -A third noticeable particular case is that of signals presenting a constant ∆σext background value immediately after the end of the ND excitation and electron thermalization (i.e., after about 2 ps) and prior to full ND internal thermalization, which requires the completion of electron-lattice energy transfer. This case occurs when the temporal variations of ∆σext,e and ∆σext,L induced by electronphonon energy exchanges compensate each other, so that their sum ∆σext remains constant. A timeresolved signal close to this situation was measured with λpr=855 nm (Figure 4c). The physical origin of such signal shape will be further clarified by considering the simplified model introduced below.

Other signals presenting less specific shapes are shown in Figure 4b and4e. They are representative of the majority of the time-resolved measurements performed on NDs, for which inclusion of both electron and lattice time-dependent contributions is required to account for the observed kinetics.

While previous analyses of short-timescale signals focused on the interband electronic contributions (∆ term in eq 3), leading to the quasi-vanishing of calculated signals at the end of internal thermalization, [START_REF] Baida | Ultrafast Nonlinear Optical Response of a Single Gold Nanorod near Its Surface Plasmon Resonance[END_REF] the quantitative inclusion of the lattice contribution to time-resolved signals proves crucial in the present case. The interband electronic (Drude) term (∆ term in eq 3) also added in this work (as compared to ref. [START_REF] Baida | Ultrafast Nonlinear Optical Response of a Single Gold Nanorod near Its Surface Plasmon Resonance[END_REF] ) has a minor effect for λpr in the visible range, but improves the agreement between experimental and simulated signals for λpr in the infrared range.

For a full understanding of time-dependent transient extinction changes, it is useful to relate them to the ∆Te(t) and ∆TL(t) temperature rises respectively quantifying electron and lattice heating. Whereas the lattice heating contribution to ∆ε is simply approximated as ∆εL∝∆TL in the model used (eq 3),

which is justified by the fact that ∆TL remains small in the context of our experiments (Figure 1), the electronic term ∆εe is more complex as the electronic temperature is not always defined (Te only representing an equivalent temperature in the athermal regime occurring during and shortly after excitation, in which the electron gas has to be described by its full electronic distribution) and increases much more than TL, the electronic heat capacity being much smaller than the lattice one (Figure 1).

The electronic temperature becomes well-defined only at time delays exceeding the time needed for the restoration of a Fermi-Dirac electronic distribution after excitation, i.e. a few times the characteristic duration of electron gas thermalization by electron-electron energy exchanges (about 500 fs in gold [START_REF] Del Fatti | Nonequilibrium Electron Dynamics in Noble Metals[END_REF] ). In the resulting two-temperature regime, the state of the gold ND is fully characterized by the two temperatures Te and TL, so that a quantitative modeling of the transient optical response of the metal only requires determination of the temporal evolutions of Te(t) and TL(t) (which can also be obtained in the framework of the well-known two-temperature model [START_REF] Anisimov | Electron Emission from Metal Surfaces Exposed to Ultrashort Laser Pulses[END_REF] ) and of the dependence of the gold dielectric function on these two temperatures ε(Te, TL).

In the context of our experiments, ∆Te is typically of the order of 100 K at the beginning of the twotemperature regime (Figure 1). For such electron heating, the non-linear dependence of ∆εe on ∆Te remains moderate (as shown in Figure S7 of the Supporting Information). It is therefore helpful for a qualitative understanding of the λpr-dependence of time-resolved signals to consider the lowperturbation limit of the complete model used above in the two-temperature regime. In this case, ∆εe can be written as ∆ = 56 5* +

∆" (such a proportionality between ∆ and ∆" being not assumed to establish eq. 3, only using ∆ 8 ∝ ∆" 8 ) and eq 5 can then be rewritten as

∆ λ , ! = I λ ! ∆" + I 8 λ ! ∆" 8 (7)
with

J K LI = M M" = M M M M" + M M M M" I 8 = M M" 8 = M M M M" 8 + M M M M" 8 (8)
The ∂ε1,2/∂Te and ∂ε1,2/∂TL spectra of gold (obtained by calculation of ∆εe and ∆εL spectra for small ∆Te and ∆TL values using our model, followed by numerical derivation), and the Ae and AL coefficients obtained for the illustrative ND (using the ∂σext/∂ε1,2 derivatives shown in Figure 2b) are shown in and Figure 4). Indeed, according to eq 7, time-domain signals represent a linear combination of the electron and lattice temperature evolutions (Figure 1) with probe wavelength-dependent weights, both vanishing for specific λpr values (Figure 4). The particular signals of Figure 4a and f (corresponding to λpr=725 nm and λpr=915 nm) discussed above, reflecting electronic heating and cooling, can be simply interpreted as resulting from the use of λpr values close to those for which the AL coefficient is predicted to vanish (λpr ≈710 and 930 nm, Figure 5d), in the narrow spectral range where |Ae| is not much smaller than |AL|. Similarly, the signal measured at λpr= 890 nm (Figure 4d), mainly reflecting the lattice heating dynamics, corresponds to a λpr value close to one of those where Ae=0 occurs (Figure 5b), in the narrow spectral interval where |AL|/|Ae| remains high enough to allow ∆σext,L>>∆σext,e even at small time delays when ∆Te >> ∆TL.

Eq 7 also states that a constant σext value occurs in the two-temperature regime (as approximately achieved for the signal shown in Figure 4c and discussed above) when Ae(λpr) ∆Te(t) + AL(λpr) ∆TL(t)

remains constant, which is achieved when Ae and AL fulfill the condition Ae/AL = ce/cL (note that the electronic capacity ce can be considered constant in the framework of eq 7, which was obtained in the low-perturbation approximation). Indeed, in this case Ae ∆Te(t) + AL ∆TL(t) ∝ ∆ue(t)+ ∆uL(t) (with ∆ue and ∆uL the electronic and lattice excess energy densities introduced in the Methods part), meaning that ∆σext is then proportional to the total excess energy density in the ND, which remains approximately constant (and equal to uabs) on the <10 ps timescales considered here, much shorter than the ≈500 ps cooling characteristic times required for heat propagation from the ND to its supporting substrate. [START_REF] Rouxel | Ultrafast Thermo-Optical Dynamics of a Single Metal Nano-Object[END_REF] Noteworthy, this interpretation of the measured time-resolved signals based on eq 7 presents a large analogy with that proposed in a previous work to explain the time-resolved signals measured on longer timescales on gold nanospheres in ethanol, which were shown to exhibit a large variety of temporal shapes and to reflect a combination of two distinct dynamics (in that case, those of the temperatures of the nanospheres and of their local liquid environment, determined by the heat transfer from the former to the latter). [START_REF] Stoll | Time-Resolved Investigations of the Cooling Dynamics of Metal Nanoparticles: Impact of Environment[END_REF] Transient extinction spectra can be similarly interpreted in the framework of eq 7 as linear combinations of Ae and AL spectra with time-dependent weights (the ∆Te(t) and ∆TL(t) temperature rises), which enables to better understand the temporal variations of their measured shapes (Figure 3c-d). Indeed, the relative contribution to ∆σext of lattice heating, ∆σext,L=AL∆TL (eq 6),

is expected to be small shortly after the ND excitation (as ∆TL∼0), increasing with time as energy exchanges between the electrons and the lattice reduce the difference between ∆Te and ∆TL (Figure 1), and finally dominant once ∆Te and ∆TL are close (as in general |AL|>>|Ae|, Figure 5). The shape of transient spectra (Figure 3c,d) is thus expected to evolve from one close to that of Ae (Figure 5b) at short delays (however affected by the athermal character of the electron distribution in this time range) to one close to AL (Figure 5d) at large delays, a trend actually observed in this work.

Figure 6 presents the experimental and calculated transient extinction spectra of the investigated ND (presented using the same modeling approach and color code as in Figure 4) at three different delays, belonging to the athermal (t=0.7 ps), two-temperature (t=2.5 ps) and internally thermalized (t=8 ps)

regimes. The agreement between the measurements and the simulations is generally good, both regarding the amplitudes and shapes of the transient spectra, and improves with increasing pumpprobe delay.

Analysis of ∆σext,e and ∆σext,L components, shown as dotted red and dashed black lines in Figure 6, also fully validates the qualitative interpretation of the temporal evolution of transient spectra based on eq 7 presented above. At short delays (e.g., 0.7 ps, Figure 6a), the shape of the modeled transient spectra is dominated by the electronic contribution ∆σext,e, and close to that of the Ae coefficient shown in Figure 5b. In particular, the model correctly reproduces the sign changes experimentally observed close to the SPR at λpr = 680 and 900 nm, as well as the presence of negative and positive peaks on the blue and red sides of the SPR, respectively (the maximal extinction changes in absolute value predicted by the model being however lower than the experimental ones). Some discrepancies are however present at <700 nm wavelengths in the interband region, where the specific features predicted numerically at the different interband thresholds could not be experimentally observed.

After electronic thermalization, the transient spectra correspond to a combination of electronic and lattice contributions (t=2.5 ps, Figure 6b), as shown by the good agreement between modeled and experimental signals, which coincide on the whole investigated range, only presenting slightly different widths in the SPR domain. Conversely, the shape of the transient spectra measured for t=8 ps (Figure 6b, when the internal thermalization of the ND can be considered complete) is dominated by the lattice contribution (Figure 5d) and almost exactly corresponds to that measured at longer timescales in a previous work, in which a good agreement between the experimental and modeled spectra had already been noted. [START_REF] Rouxel | Ultrafast Thermo-Optical Dynamics of a Single Metal Nano-Object[END_REF] Measurements and analyses performed on another ND with similar thickness but smaller diameter nor their impact on the gold dielectric function, both described using a bulk-like approach in this work considering >100 nm diameter NDs.

CONCLUSIONS

In this work, we have presented time-resolved measurements of the transient optical response of an individual gold nano-object on a few ps timescales (up to its full internal thermalization), which were quantitatively modeled by taking into account both the electronic and lattice contributions to the modifications of the intraband and interband components of the gold dielectric function. The analysis of time-resolved signals shows that their sensitivities to transient electron and lattice heating strongly depend on the probe wavelength. The vanishing of these sensitivities occurring for specific probe wavelengths offers the possibility to selectively probe these phenomena, while the total energy density in the nanoparticle can also be monitored using another specific probe wavelength. This work, together with previous ones on single plasmonic nanoparticles, shows that the transient optical response induced by the optical excitation, internal thermalization and cooling of metal nanoparticles is now well understood and can be quantitatively modeled at all relevant timescales. In future, similar experiments might be performed to better understand the sensitivity of time-resolved signals to nanoparticle vibrations, a more complex process (as each vibrational mode is associated with a specific, nonuniform nanoparticle deformation), which has recently been the theme of numerical studies. [START_REF] Ahmed | Understanding How Acoustic Vibrations Modulate the Optical Response of Plasmonic Metal Nanoparticles[END_REF][START_REF] Saison-Francioso | Numerical Modeling of Acousto-Plasmonic Coupling in Metallic Nanoparticles[END_REF] Extending the quantitative investigations of the optical, mechanical and thermal properties of nano-objects produced by electron beam lithography as described in this paper and earlier ones [START_REF] Medeghini | Controlling the Quality Factor of a Single Acoustic Nanoresonator by Tuning Its Morphology[END_REF][START_REF] Rouxel | Ultrafast Thermo-Optical Dynamics of a Single Metal Nano-Object[END_REF][START_REF] Medeghini | Signatures of Small Morphological Anisotropies in the Plasmonic and Vibrational Responses of Individual Nano-Objects[END_REF] to nano-objects produced by colloidal chemistry (which typically display less crystalline defects), would also be very interesting to better understand the impact of crystallinity on the physical response of nano-objects. [START_REF] Su | Acoustic Vibrations and Energy Dissipation Mechanisms for Lithographically Fabricated Plasmonic Nanostructures Revealed by Single-Particle Transient Extinction Spectroscopy[END_REF][START_REF] Goubet | Modulating Physical Properties of Isolated and Self-Assembled Nanocrystals through Change in Nanocrystallinity[END_REF][START_REF] Gan | Crystal Structure Dependence of the Breathing Vibration of Individual Gold Nanodisks Induced by the Ultrafast Laser[END_REF] ASSOCIATED CONTENT

Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org.

Measurements and analyses on another single ND (Figures S1-S5), effect of pump fluence on timeresolved signals (Figure S6), Te-dependence of ∆εe (Figure S7). Calculations were done for a gold nanoparticle using the complete theoretical model described in the main text, with T0=295 K and Texc=750 K (the typical electron excitation temperature achieved in our experiments, corresponding to a ∆Teq=6.5 K equilibrium heating). Note that only an effective temperature (indicated by dashes) can be defined for electrons at short delays, their distribution being athermal (see main text). derivatives. These derivatives were computed using the theoretical model described in ref. [START_REF] Stoll | Advances in Femto-Nano-Optics: Ultrafast Nonlinearity of Metal Nanoparticles[END_REF] , allowing to distinguish their interband (dotted orange) and Drude (dashed gray) components.

  Modeling of the linear optical response of gold NDs) allows to quantitatively predict the transient variations ∆ λ , ! of the gold ND extinction cross-section, the quantity measured in the context of time-resolved experiments.

  Simulated time-resolved signals were finally deduced from computed transient extinction changes by performing their time-domain convolution with the intensity profile of the probe pulse, assumed Gaussian.RESULTS AND DISCUSSIONSingle gold NDs were investigated by SMS (yielding their absolute extinction cross-section spectrum σext(λ)) and time-resolved spectroscopy with different probe wavelengths λpr (yielding time-and probe wavelength-dependent transient extinction cross-section variations ∆σext(t, λpr)). These measurements and their modeling are presented in Figures2-6for an illustrative ND (a similar set of measurements on a second ND being presented in Figures S1 to S5 of the Supporting Information). The ND extinction spectrum is shown in Figure2a. It presents a resonance at 860 nm central wavelength, corresponding to the ND in plane dipolar localized surface plasmon resonance (SPR), which is well reproduced by optical FEM simulations performed with D=153 nm and h=22 nm, close to the sizes targeted during the fabrication process and those estimated by SEM and AFM imaging.[START_REF] Rouxel | Ultrafast Thermo-Optical Dynamics of a Single Metal Nano-Object[END_REF] A selection of time-resolved signals measured on this ND is shown in Figure3a. As these signals were acquired with different pump beam parameters, leading to slightly different equilibrium temperature rises, they were divided by ∆Teq, which is of the order of 10 K in our experiments. It should however be noted that this procedure does not fully suppress the ∆Teq dependence of time-resolved signals due to the fluence-dependence of the electron-lattice energy transfer time out of the low-perturbation regime[START_REF] Hodak | Photophysics of Nanometer Sized Metal Particles: Electron-Phonon Coupling and Coherent Excitanon of Breathing Vibranonal Modes[END_REF][START_REF] Link | Shape and Size Dependence of Radiative, Non-Radiative and Photothermal Properties of Gold Nanocrystals[END_REF][START_REF] Mongin | Ultrafast Electron-Lattice Thermalization in Copper and Other Noble Metal Nanoparticles[END_REF] (as shown in FigureS6of the Supporting Information), so that comparison of normalized signals is fully relevant only for low or weakly dispersed ∆Teq values (as was the case here).Large variations of amplitudes and signs of the transient extinction changes are observed as a function of the probe wavelength λpr. These signals also display very different dynamics depending on λpr, as more clearly evidenced by Figure3b, where they have been normalized by their maximal absolute amplitudes. The majority of normalized signals show a fast rise at short pump-probe delays, followed by a decay leading to an approximately constant background in the t=5-8 ps range. However, the values of the normalized signals at t=8 ps undergo large variations with λpr, being in some cases close to 1

Figure 5 .

 5 Figure 5. Note that the Ae and AL coefficients of a nanoparticle introduced by eqs 7 and 8 depend on its intrinsic properties (morphology and composition), but are independent of excitation features (pump beam wavelength and fluence). While |AL| is maximal in the SPR range (where the ∂σext/∂ε1,2 derivatives are the largest, Figure 2b), |Ae| is maximal near 500 nm (where the Te-derivatives of ε1 and ε2 present large values, Figure 5a), and presents different spectral features associated to the gold interband transitions, in addition to the one in the SPR range. |Ae| presents much smaller values than |AL| for most wavelengths, except close to the interband spectral region, which implies that the transient extinction changes ∆σext =(Ae+AL)∆T≈ AL∆T of an internally thermalized ND (∆Te=∆TL=∆T) are largely dominated by the ∆σext,L component.

(

  about 100 nm) and SPR wavelength (720 nm), referred to as ND2, are shown in Figures S1 to S5 of the Supporting Information, whose content is analogous to that of Figures 2-6 of the main text. They present many common features with those performed on the ND considered throughout the main text of the paper. In particular, the time-resolved signals measured on ND2 similarly exhibit λpr-dependent shapes and amplitudes, the latter being also markedly enhanced for λpr in the ND2 SPR range (Fig S2). Simulations also allow a good quantitative reproduction of the experimental time-resolved signals and transient spectra, and demonstrate that the relative importance of electron and lattice heating contributions is highly dependent on probe delay and wavelength (Figures S3 and S5). Such similarity between the transient optical responses measured for the two distinct NDs considered in this work is consistent with the limited impact of ND morphology in the model. Indeed, the morphology of the NDs affects their simulated transient optical response only via the ∂σext/∂ε1,2 derivatives appearing in eq 5 and shown in Figures 2 and S1 but influences neither the kinetics of electron-lattice energy exchanges

Figure 1 .

 1 Figure 1. Ultrafast dynamics and induced transient optical response of a noble metal nanoparticle following its sudden excitation by a pump pulse. (a) Schematics of the energy exchanges processes relevant on timescales of a few ps and of their impact on the metal dielectric function ε and the nanoparticle extinction cross-section σext. (b) Typical example of ultrafast dynamics following the excitation of a gold nanoparticle by a pump pulse: temporal evolutions of the power density absorbed by the electrons pabs (top, corresponding to a 1 ps pump pulse duration), and of the electronic and lattice excess energy densities (∆ue and ∆uL, middle) and temperature rises (∆Te and ∆TL, bottom).Calculations were done for a gold nanoparticle using the complete theoretical model described in the main text, with T0=295 K and Texc=750 K (the typical electron excitation temperature achieved in our experiments, corresponding to a ∆Teq=6.5 K equilibrium heating). Note that only an effective temperature (indicated by dashes) can be defined for electrons at short delays, their distribution being athermal (see main text).

Figure 2 .

 2 Figure 2. Linear optical properties of gold NDs supported on a sapphire substrate. (a) SMS-measured absolute extinction spectrum of the Au single ND considered throughout the paper (black dots). It is well reproduced by FEM calculations performed with D=153 nm ND diameter and h=22 nm ND thickness (red line). (b) ∂σext/∂ε1 (purple) and ∂σext/∂ε2 (dashed pink) derivatives numerically deduced from the FEM calculations. The vertical dashed line indicates the SPR central wavelength λSPR=860 nm.

Figure 3 .

 3 Figure 3. Experimental time-resolved signals. (a) Time-resolved extinction cross-section changes measured on the investigated Au ND using different probe wavelengths (indicated in the legend), divided by the estimated equilibrium ND temperature rise ∆Teq. (b) Same signals as in (a), normalized to unitary maximal value. (c) Transient extinction spectra deduced from time-resolved signals at probe delays of 0.7, 2.5 and 8 ps. (d) Same spectra as in (c), normalized to -1 near 850 nm. The vertical dashed line in panels (c) and (d) indicates the central position of the SPR of the investigated ND (Figure 2a).

Figure 4 .

 4 Figure 4. Measured and simulated time-resolved signals at specific wavelengths. The transient extinction changes measured on the investigated single Au ND using various probe wavelengths λpr (indicated on each panel) are shown by blue dots. The transient extinction changes simulated using the complete model described in the main text are shown by green lines (λpr values of 710, 775, 850, 890, 915 and 925 nm were used for the simulations shown in panels a-f). These simulated signals were obtained by summing the contributions of electron and lattice heating, which are shown as dotted red and dashed black lines, respectively.

Figure 5 .

 5 Figure 5. Dependence of the gold dielectric function and ND extinction cross-section on electronic and lattice temperatures. (a) ∂ε1/∂Te (top, purple line) and ∂ε2/∂Te (bottom, pink line) derivatives, computed using the model described in refs 39,40 . The interband and (weak) Drude contributions to these derivatives are shown as dotted orange and dashed gray lines in the latter case (the Drude one being negligible in the former case). (b) Deduced Ae=∂σext/∂Te derivative. The contributions of ε1 and ε2 changes to Ae are shown as dashed purple and dotted pink lines, respectively. (c) ∂ε1/∂TL and ∂ε2/∂TL

  (d) Deduced AL= ∂σext/∂TL derivative for the ND considered in this work (same color codes as in (b)). The vertical dashed line in panels (b) and (d) indicates the central SPR position of the investigated ND.

Figure 6 .

 6 Figure 6. Measured and simulated transient extinction spectra at specific delays. The transient extinction spectra measured on the investigated single Au ND for pump-probe delays of (a) 0.7 ps, (b) 2.5 ps and (c) 8.0 ps are shown by blue dots. For each case, the transient extinction spectra simulated using the complete model described in the main text are shown by green lines, with the contributions of electron and lattice heating shown as dotted red and dashed black lines, respectively. The vertical dashed line indicates the central position of the SPR of the investigated ND.

Modeling of the linear optical response of gold NDs. Modeling

	ħω, considering electronic transitions around the L and X points of the first Brillouin zone. The
	procedure described above, initially used to compute the interband component of the gold dielectric
	function for a Fermi-Dirac electron distribution at room temperature, can be easily reiterated for
	different electronic temperatures and non-thermal (i.e., non Fermi-Dirac) electronic distributions, thus
	allowing to predict the temporal evolution of	in the context of time-resolved experiments, by
	computing ∆	and deducing ∆	through the Kramers-Kronig relations (see below). 40
				of the ND linear optical properties was
	performed by finite-element modeling (FEM), using the RF module of the COMSOL commercial
	software. Gold NDs were described as circular cylinders characterized by their thickness h and
	diameter D. The sapphire substrate supporting them was explicitly included in the model. Refractive
	indexes of 1.76 and 1 were used for the sapphire substrate and the medium above it, respectively. A
	perfectly matched layer was used around the simulation domain to avoid spurious finite-size effects.

ACKNOWLEDGEMENTS

This work was supported by the French National Research Agency (ANR) under the reference ANR-20-CE30-0016 (ULTRASINGLE project) and by the LABEX iMUST (ANR-10-LABX-0064) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by ANR.

AUTHOR INFORMATION

Corresponding author *E-mail: natalia.del-fatti@univ-lyon1.fr ORCID Romain Rouxel : 0000-0003-2982-5818 Paolo Maioli: 0000-0002-4199-8810 Fabien Vialla : 0000-0002-9661-3527

Francesco Rossella: 0000-0002-0601-4927

Francesco Banfi: 0000-0002-7465-8417 Natalia Del Fatti: 0000-0002-8074-256X Aurélien Crut: 0000-0003-2185-709X

Notes

The authors declare no competing financial interest.

TOC Graphic Front Cover

Following the pulsed excitation of a single nanometer-scale gold disk, the ultrafast internal relaxation dynamics are selectively studied using specific optical probe wavelengths: the electronic cloud and ionic lattice contributions are readily disentangled and extracted.


Rogelio Lozano

Sergio Salazar

Jonathan Flores

Stabilization of the PVTOL using nonlinear feedback control

Keywords: PVTOL, nonlinear systems, saturation functions, helicopter control

This paper presents a new control strategy for the well known problem of the PVTOL. The total thrust is computed using a non linear feedback compensation so that the altitude reaches the desired altitude. The horizontal position x is then controlled by choosing the orientation angle θ as a smooth saturation function of x and x˙. A proof of convergence is presented using a Lyapunov approach. The proposed control strategy is successfully tested in numerical simulations.

Introduction

The PVTOL (Planar Vertical Take-Off and Landing) problem has attracted a lot of attention during the last decades. The reason for such an interest stems from the fact that it is an important theoretical problem in the automatic control community which is clearly related to a practical application. Indeed, the PVTOL is a nonlinear dynamical system that represents the simplified longitudinal model of a helicopter or a quadrotor. The dynamic model is very simple but contains two trigonometric functions which make the stabilization problem quite challenging.

One of the first papers to deal with the stabilization of the PVTOL was [START_REF] Hauser | Nonlinear control design for slightly non-minimum phase systems: Application to V/STOL aircraft[END_REF]. This paper motivated the search for control algorithms to stabilize the PVTOL. Since the publication of (1), different techniques have been proposed to control the PVTOL. Optimal controllers were presented in [START_REF] Lin | Robust hovering control of a PVTOL aircraft[END_REF][START_REF] Jadbabaie | Control of a thrust-vectored flying wing: a receding horizon-LPV approach[END_REF][START_REF] Xin | Robust control of PVTOL aircraft with a nonlinear optimal control solution[END_REF][START_REF] Raffo | Robust nonlinear control for path tracking of a quad-rotor helicopter[END_REF] to stabilize the PVTOL. Nonlinear feedback linearization techniques to control the PVTOL have been developed in [START_REF] Olfati-Saber | Global configuration stabilization for the VTOL aircraft with strong input coupling[END_REF][START_REF] Huang | Output tracking of a non-linear non-minimum phase PVTOL aircraft based on non-linear state feedback control[END_REF][START_REF] Do | On global tracking control of a VTOL aircraft without velocity measurements[END_REF][START_REF] Shu | An augmented system approach to static output-feedback stabilization with performance for continuous-time plants[END_REF][START_REF] Consolini | A homotopy method for exact output tracking of some non-minimum phase nonlinear control systems[END_REF][START_REF] Zhu | Approximate trajectory tracking of input-disturbed PV-TOL aircraft with delayed attitude measurements[END_REF][START_REF] Marconi | High-gain output feedback for a miniature UAV[END_REF][START_REF] Aguilar-Ibanez | Robust trajectory-tracking control of a PVTOL under crosswind[END_REF]. The sliding mode controller design has also been applied to the PVTOL in [START_REF] Zhao | Nonlinear robust sliding mode control of a quadrotor unmanned aerial vehicle based on immersion and invariance method[END_REF][START_REF] Chen | Robust backstepping sliding-mode control and observer-based fault estimation for a quadrotor UAV[END_REF][START_REF] Yu-Chan | Robust stabilization of nonlinear PVTOL aircraft with parameter uncertainties[END_REF][START_REF] Aguilar-Ibanez | Stabilization of the PVTOL aircraft based on a sliding mode and a saturation function[END_REF]. Furthermore, the backstepping approach has been used in [START_REF] Abdessameud | Global trajectory tracking control of VTOL-UAVs without linear velocity measurements[END_REF][START_REF] Su | Robust output tracking control for a velocity-sensorless vertical take-off and landing aircraft with input disturbances and unmatched uncertainties[END_REF][START_REF] Aguilar-Ibáñez | A backstepping-based procedure with saturation functions to control the PVTOL system[END_REF][START_REF] Jia | Integral backstepping sliding mode control for quadrotor helicopter under external uncertain disturbances[END_REF]. Other well known techniques have been proposed to control the PVTOL as prediction based control [START_REF] Chemori | A prediction-based nonlinear controller for stabilization of a non-minimum phase PVTOL aircraft[END_REF], gain scheduling [START_REF] Wu | Gain-scheduled control of PVTOL aircraft dynamics with parameter-dependent disturbance[END_REF], singular perturbation [START_REF] Esteban | Three-time scale singular perturbation control and stability analysis for an autonomous helicopter on a platform[END_REF], fault tolerant control [START_REF] Chadli | Robust fault tolerant tracking controller design for a VTOL aircraft[END_REF], Lyapunov approach [START_REF] Brandão | PVTOL maneuvers guided by a high-level nonlinear controller applied to a rotorcraft machine[END_REF][START_REF] Kuga | Robust Adaptive Control of PVTOL System via Minimum Projection Method[END_REF], quaternions technique [START_REF] Tayebi | Attitude stabilization of a VTOL quadrotor aircraft[END_REF] and adaptive control [START_REF] Benrezki | Adaptive trajectory tracking control for VTOL-UAVs with unknown inertia, gyro-bias, and actuator LOE[END_REF][START_REF] Zou | Nonlinear robust adaptive hierarchical sliding mode control approach for quadrotors[END_REF]. [START_REF] Sanahuja | A Stabilization of n integrators in cascade with bounded input with experimental application to a VTOL laboratory system[END_REF] and [START_REF] Castillo | Real-time stabilization and tracking of a four-rotor mini rotorcraft[END_REF] proposed the use of a nested saturation control algorithm [START_REF] Teel | Global stabilization and restricted tracking for multiple integrators with bounded controls[END_REF] to stabilize the PVTOL.

An interesting control algorithm for the PVTOL was proposed in [START_REF] Tran | Adaptive control of a quadrotor aerial vehicle with input constraints and uncertain parameters[END_REF]. They used the concept of virtual or ideal forces such that the PVTOL model becomes fully actuated.

The orientation angle and the thrust required to generate such virtual forces are then computed. The first and second order derivatives of the orientation are used in the control algorithm and therefore the virtual forces have to be smooth functions.

A controller was presented in [START_REF] Aguilar-Ibañez | Stabilization of the PVTOL aircraft based on a sliding mode and a saturation function[END_REF] which is based on a combination of the traditionally PD-controller and a sliding mode controller to stabilize both the horizontal and angular variables to the desired rest position. A hierarchical control strategy was proposed in [START_REF] Naldi | Robust global trajectory tracking for underactuated VTOL aerial vehicles using inner-outer loop control paradigms[END_REF] for a miniature VTOL vehicle to track a desired trajectory globally with respect to the initial position and attitude configuration.

The contribution of paper is as follows. The PVTOL has three degree of freedom which are the displacement x, the altitude z and the angular displacement θ. The total thrust u is currently used to stabilize the altitude z which converges to a desired value. The system is then essentially reduced to the form ẍ = tan (θ) = u 1 and θ = τ. The problem is to find an expression for τ such that x converges to a desired value. One way to solve the problem is to assign a virtual input u 1 as a function of x and ẋ, to stabilize x. The desired value for θ will then be θ d = arctan(u 1 ). In order to obtain a control input τ so that θ converges to θ d we will require the first and second derivatives of θ d . This technique has been used in [START_REF] Tran | Adaptive control of a quadrotor aerial vehicle with input constraints and uncertain parameters[END_REF]. In the present paper we have been able to simplify the expression for θ d to only θ d = u 1 without the arctan function. The computation of the first and second derivatives of θ d are much simpler since arctan does no longer appear in θ d . Furthermore, the contribution can also be presented in the following conceptual alternative way. Consider the first order non linear system ẋ = tan (θ). We can use the fact that tan (θ) and θ have the same sign and propose the simple desired angle θ d = -x which does not include arctan. When θ = θ d we get ẋ =tan (x) which is Lyapunov stable. The question is whether the above technique can be extended to the second order nonlinear system ẍ = tan (θ) without recurring to the arctan function. The present paper presents a solution to the above problem and applies it to the stabilization of the PVTOL.

The paper is organized as follows : Section ?? presents the PVTOL model. Section 3 describes the altitude control law. Section 4 is devoted to the controller for the horizontal displacement x. The orientation control algorithm is given in Section 5. The numerical simulations of the proposed control strategy are shown in Section 6. Final remarks are given in the conclusions.

PVTOL mathematical model

The PVTOL model can be written as

m ẍ = u sin θ m z = u cos θ -mg θ = τ (1)
Figure 1 presents the PVTOL where f 1 and f 2 are the control forces, m is the mass, θ is the angle of the aircraft with respect to the horizontal line, g is the gravitational acceleration, x is the horizontal displacement and z is the vertical displacement. Let us define the total thrust u and the torque τ respectively as follows

u = f 1 + f 2 τ = ( f 1 -f 2 )l

Altitude control (z)

From ( 1) it follows that

m ẍ = u sin θ d + 1
(2)

m z = u cos θ d -mg + 2 (3)
where

1 = u(sin θ -sin θ d ) (4)
2 = u(cos θ -cos θ d ) (5)
where θ d is the desired orientation. The control input τ will be computed so that θθ d converges to zero exponentially

In order to control the vertical displacement z let us choose the following control input u where z d is the constant desired altitude.

u = m cos θ d [g -2 ż -(z -z d )] (6)
To avoid singularities in the above equation the desired angular position θ d will be chosen so that |θ d < π 2 | in the next section. From ( 6) into (3) the closed loop system for the z dynamics is

z + 2 ż + (z -z d ) = 2 m (7)
then z → z d , ż → 0 and u → mg cos θ d . Notice that the convergence rate is exponential. From the above it follows that for a time arbitrarily long the x dynamics reduces to

ẍ g = tan θ d (8)
4 Horizontal displacement control (x)

Consider the following function θ d which depends on x and ẋ

θ d = -k 1 tanh ẋ g -k 2 tanh (x + ẋ) g (9)
where 0 < k i < π/4 for i = 1, 2. The control input τ will be chosen so that θ converges exponentially to θ d . The expression for τ is given in section 6. Notice from the above that |θ d < π 2 | and this avoids the singularity in [START_REF] Olfati-Saber | Global configuration stabilization for the VTOL aircraft with strong input coupling[END_REF]. Therefore we will assume that after some transient θ = θ d . The RHS of (8) could then be rewritten as

tan θ d = -tan (A + B) (10)
where

A k 1 tanh ẋ g ; B k 2 tanh (x + ẋ) g (11)
Consider the following trigonometric identity

tan (A + B) = tan A + tan B D (12)
where

D 1 -tan A tan B (13)
Notice that D = 0 since 0 < k i < π 4 . We define

y x g (14)
(9) can also be rewritten as

tan θ d = - tan A + tan B D = - σ 1 ( ẏ) + σ 2 (y + ẏ) D (15)
where

σ 1 ( ẏ) = tan(k 1 tanh ẏ) σ 2 (y + ẏ) = tan(k 2 tanh (y + ẏ)) (16)
Then from ( 8), ( 14) and ( 15) it follows

ÿ = - 1 D [σ 1 ( ẏ) + σ 2 (y + ẏ)] (17)
Next it will be proved that the closed loop system ( 17) is asymptotically stable using a Lyapunov approach. We propose the following Lyapunov function candidate

V = y+ ẏ 0 σ 2 (ξ)dξ + 1 2 ẏ2 (18)
Notice that V is positive definite since x i σ(x i ) > 0. Differentiating we obtain:

V = σ 2 (y + ẏ)[ ẏ + ÿ] + ẏ ÿ = σ 2 (y + ẏ) ẏ + [σ 2 (y + ẏ) + ẏ](- 1 D [σ 1 ( ẏ) + σ 2 (y + ẏ)]) = σ 2 (y + ẏ) ẏ - 1 D [σ 2 (y + ẏ)σ 1 ( ẏ) + σ 2 2 (y + ẏ) + σ 1 ( ẏ) ẏ + σ 2 (y + ẏ) ẏ] = 1 D σ 2 (y + ẏ) ẏ[1 -σ 1 ( ẏ)σ 2 (y + ẏ)] - 1 D [σ 2 (y + ẏ)σ 1 ( ẏ) + σ 2 2 (y + ẏ) + σ 1 ( ẏ) ẏ + σ 2 (y + ẏ) ẏ] = - 1 D [σ 2 2 (y + ẏ)σ 1 ( ẏ) ẏ + σ 1 ( ẏ)σ 2 (y + ẏ) + σ 2 2 (y + ẏ) + σ 1 ( ẏ) ẏ] ≤ - 1 D [σ 2 2 (y + ẏ)σ 1 ( ẏ) ẏ - 1 2 σ 2 1 ( ẏ) - 1 2 σ 2 2 (y + ẏ) + σ 2 2 (y + ẏ) + σ 1 ( ẏ) ẏ] ≤ - 1 D [σ 2 2 (y + ẏ)σ 1 ( ẏ) ẏ + 1 2 σ 2 2 (y + ẏ) + 1 2 σ 1 ( ẏ) ẏ] < 0 ( 19
)
where we have used the facts that

-σ 1 σ 2 ≤ 1 2 σ 2 1 + 1 2 σ 2 2 ( 20
)
and

σ 2 1 ( ẏ) ≤ ẏσ 1 ( ẏ) (21)
or

|σ 1 ( ẏ)| ≤ | ẏ| which is equivalent to: ẏ -σ 1 ( ẏ) > 0 for ẏ > 0 ẏ -σ 1 ( ẏ) < 0 for ẏ < 0 ( 22
)
Since σ 1 ( ẏ) is a saturation function (see ( 16)) it is clear that the above inequalities hold for large values of ẏ orẏ. Figure 2 shows that the inequalities also holds for values around the origin for the upper-bound of k 1 = π 4 . We conclude V is positive definite and V is negative definite. Therefore y and ẏ converge to zero, i.e. x and ẋ converge to zero.

Attitude control (θ)

The torque control input τ is given by

τ = θd -2( θ -θd ) -(θ -θ d ) (23)
where

θ d = -k 1 tanh ẋ g -k 2 tanh x + ẋ g θd = -k 1 ẍ g sech 2 ẋ g -k 2 ẍ + ẋ g sech 2 ẋ + x g θd = 2k 1 ẍ g 2 tanh ẋ g sech 2 ẋ g + 2k 2 ẍ + ẋ g 2 tanh ẋ + x g sech 2 ẋ + x g - -k 1 ... x g sech 2 ẋ g -k 2 ... x + ẍ g sech 2 ẋ + x g (24)
where ẍ and ...

x are not required to be measured since they are given by the expressions (see ( 2) and ( 6))

ẍ = u sin θ d ... x = u sin θ d + u θd cos θ d u = m cos θ d [g -2 ż -(z + z d )] u = m cos θ d [-2 z -ż] - m[g -2 ż -(z + z d )](-θd sin θ d ) cos 2 θ d (25)
where z can be computed from [START_REF] Huang | Output tracking of a non-linear non-minimum phase PVTOL aircraft based on non-linear state feedback control[END_REF] in terms of z and ż. Substituting ( 23) into (1) it follows

θ -θd + 2( θ -θd ) + (θ -θ d ) = 0 (26)
Therefore θθ d and θθd converge to zero. The control input ( 6) is basically composed of a nonlinear compensation of cos θ d , and a linear state feedback. If the control input u has to satisfy a lowerbound (u > 0) and an upperbound, we can replace the state feedback part on z and ż by saturation functions as in [START_REF] Yu-Chan | Robust stabilization of nonlinear PVTOL aircraft with parameter uncertainties[END_REF] such that the closed loop for the z dynamics will have the form given in [START_REF] Aguilar-Ibanez | Stabilization of the PVTOL aircraft based on a sliding mode and a saturation function[END_REF] with D = 1. The corresponding values of k 1 and k 2 could then be used to reduce the size of the saturation functions in [START_REF] Yu-Chan | Robust stabilization of nonlinear PVTOL aircraft with parameter uncertainties[END_REF] as well as to reduce the upperbound of |θ d | in (9).

Simulation results

This section presents the numerical simulation results obtained with the proposed controller. Furthermore we will compare this algorithm with respect to another control algorithm. The second control algorithm which we will call State Feedback is obtained as follows. The altitude z will be controlled as before by using the control input u in [START_REF] Olfati-Saber | Global configuration stabilization for the VTOL aircraft with strong input coupling[END_REF]. The torque control input τ will be given also as before, i.e. as in [START_REF] Wu | Gain-scheduled control of PVTOL aircraft dynamics with parameter-dependent disturbance[END_REF] but the desired angular position θ d will be simplified as follows

θ d = -k 1 ẋ g -k 2 x + ẋ g (27)
Therefore the State Feedback control algorithm is similar to the proposed algorithm but with a desired angle θ d which is linear with respect to x and ẋ. The stability of this State Feedback control strategy is obtained from [START_REF] Huang | Output tracking of a non-linear non-minimum phase PVTOL aircraft based on non-linear state feedback control[END_REF] for the altitude dynamics z, from [START_REF] Wu | Gain-scheduled control of PVTOL aircraft dynamics with parameter-dependent disturbance[END_REF] for the convergence of θ to θ d , and from ( 8) and ( 27) for the local stability of the x dynamics assuming that θ is so small that tan(θ) ≈ θ. We may expect a similar behavior of the two control algorithms when x and ẋ are close to the origin.

In the simulations we have used the following controller parameters for both algorithms: m = 1, k a = 20, k b = 10, k 1 = 5 and k 2 = 3. We have used the following initial conditions for z and θ: z(0) = 0.5 and θ(0) = 0. The desired altitude is z d = 1. We have run 2 simulations with 2 different initial conditions for x. Figures 3 and4

Conclusions

This paper has presented a new control strategy for the well known problem of the PVTOL. The altitude control has been obtained by non linear compensation which guarantees that the altitude will reach the desired altitude in a short period of time. A desired orientation has been proposed as a function of smooth saturation function of x and ẋ. It has been proved that when the orientation reaches the desired orientation then the closed loop for the horizontal displacement is Lyapunov stable. The proposed control strategy has been tested in numerical simulations.

Figure 1 :

 1 Figure 1: The PVTOL diagram.

Figure 2 :

 2 Figure 2: Plot of ẏσ 1 ( ẏ) showing that inequalities (22) hold.

Figure 3 :

 3 Figure 3: The vertical displacement z, the angular displacement θ and the horizontal displacement x with initial condition x(0) = 0.5

Figure 4 : 5 (Figure 5 :Figure 6 :

 4556 Figure 4: Control inputs u and τ with initial condition x(0) = 0.5

Acknowledgments

This project was funded by UTC, CNRS, UMR 7253 Heudiasyc and UMI-LAFMIA 3175 CNRS, CINVESTAV. The authors are grateful to CONACYT (National Council of Science and Technology) for its support.

Data Availability Statement

The data that support the findings of this study are openly available in https://figshare.com at http://doi.org/10.6084/m9.figshare.16599044.