A A Barsella

M.D M A A D Hurier

Mircea D Pichois

Habibul Vomir

Loic Hasan

Bertrand Mager

Jean Donnio

L Louis

M V V Gallani

Rastei

M Vomir

H Hasan

L Mager

B Donnio

J L Gallani

M V Rastei

Photonic Excitation of a Micromechanical Cantilever in Electrostatic Fields

Keywords: numbers: 68.37.-d, 68.37.Uv, 07.10.Cm, 68.60.-p

  L'archive ouverte pluridisciplinaire

Surface transient heating is known to generate elastic waves in mechanical systems [1,2]. Such an effect can be achieved by a modulated optical pump via photothermal actuation [3,4]. In fact, resonant coupling of different mechanical degrees of freedom with light represents a prerequisite for many optomechanical systems [5,6]. This includes the excitation of cantilever-based probes in advanced atomic force microscopy (AFM) techniques [7][8][9][10]. The photothermal effect, dominating over radiation pressure [11,12], allows for the use of micro-cantilevers as sensors in infrared [13,14] and radio frequency [6] spectral range, or for optical-induced cooling [15][16][17][18].

In addition, photoelectron excitations are known to alter the surface charge in the space-charge depletion region, via surface photovoltage effect [19,20], a phenomenon particularly relevant in semiconductors [21][22][23][24][25]. Such effects are in fact expected in any opto-mechanical system allowing light absorption and photovoltage generation. As a result, it may constitute a very efficient driving means for a mechanical eigenmode if the incident photon beam is resonantly modulated. A significant electrostatic coupling with nearby surfaces is hence expected especially in near-field configurations [26][27][28].

In the present work, we report on the interplay between photothermal and photovoltage effects upon optical excitation of a silicon AFM probe. It is first shown that the photothermal excitation is indeed a very efficient way to resonantly drive the cantilever. Then, a parabolic control of the mechanical vibration mode is demonstrated using a bipolar DC voltage, as can be expected for a capacitorlike configuration [START_REF] Sadewasser | Kelvin Probe Force Microscopy -From Single Charge Detection to Device Characterization[END_REF]. Finally, it is shown that a slightly sub-resonance modulation frequency of the laser intensity translates into a strong DC field-sign dependence of the cantilever dynamics. The numerical analyses reveal that due to their phase difference, photothermal and photovoltage effects add or subtract their action on the cantilever, depending on the DC field sign. These findings are of relevance for all AFM techniques which use an optical excitation of the cantilever sensor, and for the emergent field of photoinduced force microscopy [31][32][33][34][35] in particular, as well as for other opto-mechanical systems where intrinsic or externally controlled electrostatic fields and light absorption are involved.

The experimental setup is outlined in vacuum at pressures below of 10 -5 mbar. The sample is a thin Al layer sputtered on a glass substrate transmitting 50% of the incident light. The probe is a Si tip attached to a Si cantilever. Cantilevers with normal spring constants of the order of 0.01 N/m are used. The excitation laser is a continuous green laser. The beam intensity is modulated by a high-speed electro-optical modulator (EOM) placed between two crossed polarizers. The polarization of light did not play any other role in this work.

A power spectra density (PSD) is obtained from 16 s acquisition time of cantilever position, translating after averaging in a resolution of 0.44 Hz. The z position is controlled by the z-piezo and measured for each spectrum with respect to the tip-surface contact. Except for static deflection curves, the voltage is applied on the probe with respect to the sample. Note that when the work functions of the tip and the surface are different (inset of Fig. 1), an electrostatic field exists even without an external voltage, i.e. contact potential difference (CPD) [START_REF] Sadewasser | Kelvin Probe Force Microscopy -From Single Charge Detection to Device Characterization[END_REF].

The exciting laser beam is focused down to 5 µm on the AFM tip, which is roughly half the base of the pyramidal tip. A tip-less geometry without electric fields or photovoltage effects was used in [START_REF] Ma | [END_REF]. The EOM modulation was a sinusoidal shape, periodically changing the laser power from 0 to P , which can be up to P max = 40 mW. This is the only modulation used in this work. The modulation frequency f m is chosen close to the first normal bending mode f 0 of the cantilever. For the uncoated cantilevers used here, f 0 falls between 10 and 20 kHz, depending on the exact cantilever. An example of a PSD is shown in the inset of Fig. 1, for a modulation frequency detuned by ∆f = f m -f 0 = 6 Hz with respect to f 0 mode (thermal noise peak). The value of P was set to 1 mW in order to produce an optical excitation amplitude in the same range as the thermal noise peak.

Such an optical excitation peak in the mechanical response of the cantilever demonstrates an optomechanical coupling with the incident laser. The influence of ∆f is shown in Fig. 2(a). The diagram is constructed from 200 PSD spectra taken at different negative and positive ∆f values (step: 1 Hz). The optical sidebands, resulting from optical excitation peaks, have a linear dependence with ∆f . In addition, as |∆f | increases the optical sidebands intensity decreases, as expected for a driven harmonic oscillator presenting a finite mechanical resonance linewidth. The discontinuities of the optical sidebands from Fig. 2(a) have no real physical meaning, being just the consequence of our PSD frequency resolution, which is comparable to the width of the optical excitation peak. This effect is not relevant in PSD analyses for a fixed f m , as is the case for all data presented in the following.

A diagram built from PSD spectra acquired at different V values is shown in Fig. 2(b). The optical excitation modulation was fixed for V = 0 at ∆f = -15 Hz, i.e. below the mechanical f 0 mode frequency. It thus appears as a horizontal spectral feature in the diagram. As observed, the applied voltage induces a parabolic shift of the mechanical mode towards lower frequencies, in agreement with an attractive electrostatic force scaling with V 2 [START_REF] Sadewasser | Kelvin Probe Force Microscopy -From Single Charge Detection to Device Characterization[END_REF]31]. An evaluation of the parabola position in the diagram from Fig. 2(b) reveals a CPD value of about -0.4 V. More importantly, Fig. 2(b) gives a strong asymmetry of the cantilever dynamics for V < 0 as compared to V > 0. When the mechanical mode f 0 crosses the optical excitation frequency, the oscillation amplitude increases for V > 0, while for V < 0 the f 0 mechanical mode does not couple with the optical excitation. In addition, it can be seen that for V > 0, the oscillation amplitude at f m , i.e. intensity of optical sideband, progressively increases even before the f 0 mechanical mode reaches the resonance at around 6 V. At higher voltages, above V = 7 V, the f 0 mode also appears perturbed by the optical excitation, deviating from the parabolic downward shift. Note that the color coded bar is in log scale, highlighting in red the larger spectral amplitudes.

To gain additional insights into the optical excitation asymmetry, we performed measurements of the cantilever static deflection as a function of V. It is worth noting, that the parabolic downward shift of the f 0 mode observed in Fig. 2(b) is already a clear indication that there is an attractive electrostatic force, but the frequency shift does not provide a direct measurement of the cantilever deflection. This is because the shift of the f 0 mechanical mode depends on the gradient of the force. In contrast, the static deflection of the cantilever is modified by the force itself.

Figure 3(a) shows the static deflection as a function of the applied voltage for various lifts. To acquire the curves, the tip was first brought in contact with the surface by using the feedback loop of the microscope, The central peak corresponds to the thermal-noise peak (f0 mode). The narrow peaks correspond to the optical excitation set here at ∆f = -18 Hz. There are three spectra in each graph revealing the reproducibility.

and then retracted at specific distance with the feedback turned off. Ramping the voltage then results in a parabolic-like variation of the cantilever deflection, which is directly measured with the four-quadrant photodetector. We performed measurements every 100 nm over a range going from 200 to 2000 nm. Below 200 nm the tip is found to jump into contact at the largest V values. There are several interesting aspects that can be deduced from such measurements. First, it can be observed that all parabolae are not symmetric with respect to V = 0, as evidenced in the inset which highlights the -1.5 to +1.5 V interval. This is a measure of the CPD, which is about 0.4 V in this case. It is also interesting to see that a CPD translates in different deflections for negative and positive voltages. For instance, the curve acquired at 200 nm presents a maximum deflection asymmetry of δz ≃ 10 nm for ± 10 V. The asymmetry decreases with increasing distance, as the electrostatic force between the tip and the sample is reduced.

Figures 3 (b andc) show PSD acquired at +2.3 V and -2.3 V, respectively. At this scale the mechanical f 0 mode (central wider peak) appears unaffected by the voltage sign, except for a minor shift of about 1 Hz at +2.3 V. Despite the fact that this approaches the f 0 mode to the optical excitation peak, it cannot explain the large difference in amplification factor observed for the optical excitation peak. An optical excitation response which is almost two orders of magnitude higher at +2.3 V compared to -2.3 V is observed. Since the cantilever is basically a harmonic oscillator, there must be an additional effect that explains this large asymmetry.

Nonlinear effects giving rise to a hysteretic behavior can be ruled out, since our measurements are not performed by continuously changing the applied voltage, i.e. a PSD curve is separated from the previous one by a full approach-retract of the tip to the surface performed to ensure that the z distance remains constant. This causes the oscillator to lose information on its current state and eliminates the possibility of an amplitude hysteresis cycle. Additionally, numerical simulations using a Duffing oscillator model show that in order to have a hysteretic behavior the nonlinear coefficient (β) should be very large. This would strongly affect the shape of the resonance peak, an effect never observed.

An additional possibility is that the motion of the cantilever remains that of a harmonic oscillator, but that changing the applied voltage sign makes the optical excitation trigger a different driving. This means searching for an effect which is sensitive to voltage sign. As stated in the introduction, a laser beam illuminating a semiconductor surface can give rise to different optomechanical effects. There is the photothermal effect which is known acting as an exciting force, but not likely to be sensitive to the voltage sign. Radiation pressure also acts on the cantilever. However, the absence of an optical cavity, the low laser power, the axial illumination of the pyramidal tip, as well as the low reflectance of silicon, all speak in favor of small radiation pressure effects. Photophoretic or other radiometric forces can also be ruled out since the experiments were not conducted in air. A photovoltage generation is also expected for a semiconductor material such as silicon. A change of the cantilever's surface potential, due to mobility difference of electrons and holes, can thus be predicted. While we expected the effect to be rather small, we found that a total charge variation of a single e -is already enough to explain the occurrence of a significant electrostatic driving force. In that case, the sign of the applied voltage plays a critical role since the photovoltage has always the same sign. As a consequence, the surface charge induced by the photovoltage effect adds (or subtracts) to charges brought by the applied voltage. Moreover, the resulting electrostatic driving force, while having the same overall amplitude, is not necessarily in phase with the optical excitation.

Let us now model the cantilever as an harmonic oscillator subjected to two external driving forces: the first one caused by the photothermal effect (always in phase with the laser intensity), and the second one produced by the charge variation induced by the photovoltage (with a phase shift depending on the voltage sign). The equation of motion is written as:

m eff z = -kz- √ km eff Q ż+F PT (t)+F PV (t)+ √ 2Cη(t) (1)
with z representing the cantilever deflection having as origin the equilibrium position at V = 0, m eff the effective mass, k the elastic constant, Q the quality factor, and 1) . (c) and (d) Calculated and experimental PSD, respectively, for three applied voltages, shifting the mechanical mode f0 from supraresonance (gray curves) to sub-resonance (blue curves). The resonant case is in red. Voltages in (c): 5 V (gray), 9 V (red), 12 V (blue). Voltages in (d): 2 V (gray), 6 V (red), 11 V (blue). In Table 1 are the other parameters used for simulations.

the last term describes the effect of the thermal noise. The two time-dependent force terms represent the photothermal (F PT ) and the photovoltage (F PV ) excitations, which we write:

F PT = F PT0 [1 + cos(2πf m t)] (2)
F PV = 1 4πε 0 [q 0 + ∆q(1 + cos(2πf m t))] 2 (H + z) 2 , (3)
where H represents the distance between the surface and the equilibrium position of the cantilever tip. q 0 is the charge given by the applied voltage, which varies by ∆q as a function of time due to the photovoltage. The 1 + cos(2πf m t) term is used to describe the temporal behavior of the laser intensity and we assume for simplicity that the tip behaves as a point charge.

While the sign of ∆q is given by the semiconductor band distribution [20,37], the sign of q 0 depends on the polarity of the applied voltage. Thus, the maxima of F PV can either correspond to the maxima or minima of the laser intensity. Conversely, the maxima of F PT only depend on the laser intensity and is thus always in phase with it. The consequence is that the two driving forces can interfere constructively for one applied voltage sign and destructively for the opposite sign. The relation providing q 0 as a function of voltage was obtained by fitting the experimental deflection curves [Fig. 3(a)]. The values of F PT0 and ∆q are obtained by matching the height of the experimental excitation peak at V = 0 (negligible effect of F PV ).

Figures . 4 (a andb) show two calculated PSD diagrams computed for two different sets of F PT0 and ∆q values. A clear asymmetric optical excitation is seen in both cases, in good agreement with the experiments. The stronger asymmetry in Fig. 4 (b) is due to a larger ∆q value (Table 1). To further compare simulated and experimental mechanical responses, we show in Fig. 4 (c) and (d) examples of PSD curves. The broader peaks are thermal noise features of the mechanical f 0 mode, which frequency position change because of the electrostatic force, i.e. applied voltage. The f 0 mode frequency coincides with the optical excitation frequency f m at zero (red curves). The lower number of data points in the simulated time series results in a higher noise compared to the experimental case [START_REF]Keeping k, f0, and Q close to those from the experiments, significantly increases the computational effort[END_REF]. There is nevertheless a good agreement between calculated and experimental curves.

In summary, we evidenced a combined optomechanical mechanism relevant for optically driven micromechanical systems subjected to electrostatic force gradients. We found that in this configuration, the electrostatic field can be modified by the carrier dynamics within the cantilever. This gives rise to an excitation mechanism which relies on the interplay between photothermal actuation and photovoltage generation, being relevant in the presence of a modulated light absorption occurring at a frequency close to a resonant mechanical mode. As both phenomena depend on light absorption, they can represent a useful means for controlling the dynamics of micromechanical oscillators, particularly in near-field configurations. These findings can be of importance in any situation where light absorption and electric fields are applied to a micromechanical system. Specifically, our results provide a fresh insight into vibration properties of optically driven atomic force microscopy probes, and may give a new thrust to detection sensibility of various atomic force microscopy techniques.

Fig. 1 .FIG. 1 .

 11 photovoltage effects add or subtract their action on the cantilever, depending on the DC field sign. These findings are of relevance for all AFM techniques which use an optical excitation of the cantilever sensor, and for the emergent field of photoinduced force microscopy[31][32][33][34][35] in particular, as well as for other opto-mechanical systems where intrinsic or externally controlled electrostatic fields and light absorption are involved.The experimental setup is outlined in Fig.1. It consists of a modified atomic force microscope operating in

FIG. 2 .

 2 FIG. 2. (a) Mechanical response PSD diagram as a function of laser modulation detuning ∆f at V = 0, and (b) as a function of the tip voltage. In (b) the modulation fm was kept at a fixed value of 15 Hz below the mechanical mode f0 measured at V = 0 (optical sideband). The diagrams are constructed by assembling 200 thermal noise spectra in (a) and 120 in (b). Other parameters: 16 s acquisition time, z = 2000 nm. Several (28) resonant spectra near ∆f = 0 are not included in (a) as they would saturate the contrast. Note that the two diagrams do not have the same frequency interval.

FIG. 3 .

 3 FIG. 3. (a) Cantilever deflection (mean bending position) as a function of substrate voltage for tip-surface distances from 200 to 2000 nm, without laser excitation. Inset is a zoom in the interval indicated by the accolade, highlighting the CPD. The deflection asymmetry between the deflection at -10 V and +10 V is marked by ∆z for the curve measured at z = 200 nm. (b and c) PSD traces acquired at a tip voltage of +2.3 and -2.3 V respectively, with the same optical excitation.The central peak corresponds to the thermal-noise peak (f0 mode). The narrow peaks correspond to the optical excitation set here at ∆f = -18 Hz. There are three spectra in each graph revealing the reproducibility.

  FIG. 4. (a) and (b) Diagrams obtained by computing the Fourier spectrum of the cantilever position versus time for various electrostatic force values, i.e. applied voltages. The difference between (a) and (b) is a slightly larger photovoltageinduced charge variation ∆q in (b) (see Table1) . (c) and (d) Calculated and experimental PSD, respectively, for three applied voltages, shifting the mechanical mode f0 from supraresonance (gray curves) to sub-resonance (blue curves). The resonant case is in red. Voltages in (c): 5 V (gray), 9 V (red), 12 V (blue). Voltages in (d): 2 V (gray), 6 V (red), 11 V (blue). In Table1are the other parameters used for simulations.

  FPT 0 0.09 (a) 3.6 (b) pN ∆q 0.5 (a) 15 (b) e -

TABLE I .

 I Parameters used for numerical simulations of Fig 4.

We thank Ivan Favero for useful discussions, and N. Beyer for technical assistance. This work is supported by ANR METABIP (12 BS10 003 01), ANR OH-RISQUE SMARAGD (14 OHRI 0008 01), EOARD (FA8655-13-1-3001) and partially by ANR-11-LABX-0058-NIE within the Investissement d'Avenir program ANR-10-IDEX-0002-02.


V E Nefedova

S Fröhlich

F Navarrete

N Tancogne-Dejean

D Franz

A Hamdou

S Kaassamani

D Gauthier

R Nicolas

G Jargot

M Hanna

P Georges

M F Ciappina

U Thumm

W Boutu

H Merdji

Enhanced extreme ultraviolet high-harmonic generation from chromium-doped magnesium oxide

Enhanced extreme ultraviolet high-harmonic generation

from chromium-doped magnesium oxide

High-order harmonic generation (HHG) from crystals is emerging as a new ultrashort source of coherent extreme ultraviolet (XUV) light. Doping the crystal structure can offer a new way to control the source properties. Here, we present a study of HHG enhancement in the XUV spectral region from an ionic crystal, using dopant-induced vacancy defects, driven by a laser centred at a wavelength of 1.55 µm. Our numerical simulations based on solutions of the semiconductor Bloch equations and density-functional theory are supported by our experimental observations and demonstrate an increase of the XUV high harmonic yield from doped bulk magnesium oxide (MgO) compared to undoped MgO, even at low defect concentration. The anisotropy of the harmonic emission as a function of the laser polarization shows that the pristine crystal's symmetry is preserved. Our study paves the way towards the control of HHG in solids with complex defects caused by transition-metal doping.

The interaction of an electric field in the strong field regime with a semiconductor or a dielectric generates electron-hole pairs, which are subsequently accelerated by this field and emit coherent radiation via inter-and intra-band electronic transitions in the solid [START_REF] Ghimire | Generation and propagation of high-order harmonics in crystals[END_REF][START_REF] Vampa | Theoretical analysis of high-harmonic generation in solids[END_REF][START_REF] Luu | High-order harmonic generation in solids: A unifying approach[END_REF][START_REF] Tancogne-Dejean | Impact of the electronic band structure in high-harmonic generation spectra of solids[END_REF] . This coherent emission occurs as bursts of attosecond pulses that have been observed as a harmonic comb in the frequency domain. Since its first experimental observation [START_REF] Ghimire | Observation of high-order harmonic generation in a bulk crystal[END_REF] , considerable efforts were devoted to the understanding of the main mechanisms behind high-order harmonic generation in solids (HHG). This new source of extreme ultraviolet (XUV) radiation [START_REF] Vampa | High-harmonic generation in solids: bridging the gap between attosecond science and condensed matter physics[END_REF][START_REF] Garg | Multi-petahertz electronic metrology[END_REF] displays a high degree of coherence [START_REF] Kim | Generation of coherent extreme-ultraviolet radiation from bulk sapphire crystal[END_REF] , for instance, suitable for lensless diffractive imaging [START_REF] Franz | All semiconductor enhanced high-harmonic generation from a single nanostructured cone[END_REF] . It recently provided beams carrying orbital angular momentum with a selective topological charge [START_REF] Gauthier | Orbital angular momentum from semiconductor high-order harmonics[END_REF] . Recent studies also reveal that the HHG spatio-temporal properties are intricate due to the complex light-driven electronic dynamics inside the solid. Hence, as a basic application, HHG can reveal information about the crystal's electronic band structure and intrinsic crystal properties [START_REF] Tancogne-Dejean | Impact of the electronic band structure in high-harmonic generation spectra of solids[END_REF][START_REF] Lanin | Mapping the electron band structure by intraband high-harmonic generation in solids[END_REF] . Future applications will exploit the ability to manipulate the light-driven eleca) Corresponding author, email address: thumm@phys.ksu.edu b) Corresponding author, email address: hamed.merdji@cea.fr tron motion during the crystal HHG process to create optoelectronic devices operating at petahertz frequencies [START_REF] Krausz | Attosecond metrology: from electron capture to future signal processing[END_REF][START_REF] Schoetz | Perspective on petahertz electronics and attosecond nanoscopy[END_REF][START_REF] Sederberg | Attosecond optoelectronic field measurement in solids[END_REF] or to encode topological information [START_REF] Silva | Topological strong-field physics on sub-laser-cycle timescale[END_REF] . However, increasing the yield is a prerequisite for the development of a solid state HHG source and its applications. Various experimental techniques, such as plasmon- [START_REF] Han | High-harmonic generation by field enhanced femtosecond pulses in metal-sapphire nanostructure[END_REF][START_REF] Vampa | Plasmon-enhanced high-harmonic generation from silicon[END_REF] , waveguide- [START_REF] Franz | All semiconductor enhanced high-harmonic generation from a single nanostructured cone[END_REF][START_REF] Franz | Amplification of high harmonics in 3d semiconductor waveguides[END_REF][START_REF] Sivis | Tailored semiconductors for high-harmonic optoelectronics[END_REF] and antennaenchanced HHG [START_REF] Imasaka | Antenna-enhanced high harmonic generation in a wide-bandgap semiconductor ZnO[END_REF] , were already implemented to achieve this goal. Here, we propose an alternative way of boosting the HHG yield from solids based on defects. This approach exploits the modification and control of the active medium's electronic structure to tailor the high-order harmonic output signal. Since defects are a corner stone of most modern electronic devices, it appears promising to investigate their influence on the HHG yield. The addition of dopants creates electronic states in the band gap, leading to modifications of the electronic band structure [START_REF] Umebayashi | Band gap narrowing of titanium dioxide by sulfur doping[END_REF][START_REF] Kamarulzaman | Elucidation of the highest valence band and lowest conduction band shifts using XPS for ZnO and Zn 0.99 Cu 0.01 O band gap changes[END_REF] and band gaps [START_REF] Auvergne | Band-gap shrinkage of semiconductors[END_REF][START_REF] Queisser | Defects in semiconductors: some fatal, some vital[END_REF][START_REF] Oba | Design and exploration of semiconductors from first principles: A review of recent advances[END_REF][START_REF] Valentin | Spectroscopic properties of doped and defective semiconducting oxides from hybrid density functional calculations[END_REF] , allowing additional optical transitions [START_REF] Henderson | Optical spectroscopy of inorganic solids[END_REF] . Another significant effect of the doping is the appearance of point defects such as vacancies and interstitial sites [START_REF] Chadwick | Defects in solids: modern techniques[END_REF][START_REF] Mrudul | Highharmonic generation from spin-polarised defects in solids[END_REF] . According to recent theoretical investigations dopant-induced band-gap changes can substantially influence the HHG process [START_REF] Mrudul | Highharmonic generation from spin-polarised defects in solids[END_REF][START_REF] Huang | High-order-harmonic generation of a doped semiconductor[END_REF][START_REF] Yu | Enhanced high-order harmonic generation in donor-doped band-gap materials[END_REF][START_REF] Pan | Energy band splitting and high-order harmonic generation from a doped semiconductor[END_REF][START_REF] Iravani | Effects of vacancies on high-order harmonic generation in a linear chain with band gap[END_REF] . In particular, donor-doped materials were theoretically predicted to enhance the HHG yield due to electronic excitation from impurity states [START_REF] Yu | Enhanced high-order harmonic generation in donor-doped band-gap materials[END_REF][START_REF] Iravani | Effects of vacancies on high-order harmonic generation in a linear chain with band gap[END_REF] We here investigate the influence of dopant-induced vacancies on the HHG yield in chromium-doped magnesium oxide (MgO:Cr), which was extensively studied previously using experimental [START_REF] Henry | Nature of the broadband luminescence center in MgO:Cr 3+[END_REF][START_REF] Wachtman | Mass transport in oxides[END_REF][START_REF] Stavale | Cathodoluminescence of nearsurface centres in Cr-doped MgO(001) thin films probed by scanning tunnelling microscopy[END_REF][START_REF] Shablonin | Excitation of different chromium centres by synchrotron radiation in MgO:Cr single crystals[END_REF] and numerical [START_REF] Aramburu | Transition metal complexes coupled to vacancies in oxides: Origin of different properties of Cr 3+ in MgO bounded to a <100> or <110> Mg 2+ vacancy[END_REF][START_REF] Aramburu | Colour due to Cr 3+ ions in oxides: a study of the model system MgO:Cr 3+[END_REF][START_REF] Novita | Study on multiplet energies of V 2+ , Cr 3+ , and Mn 4+ in MgO host crystal based on first-principles calculations with consideration of lattice relaxation[END_REF] methods. The Cr donor-dopants were found to not only substitute atoms with Cr 3+ ions in the MgO lattice providing additional electrons, but also to cause the formation of vacancies [START_REF] Henry | Nature of the broadband luminescence center in MgO:Cr 3+[END_REF][START_REF] Wachtman | Mass transport in oxides[END_REF][START_REF] Stavale | Cathodoluminescence of nearsurface centres in Cr-doped MgO(001) thin films probed by scanning tunnelling microscopy[END_REF][START_REF] Shablonin | Excitation of different chromium centres by synchrotron radiation in MgO:Cr single crystals[END_REF] in order to maintain charge neutrality [START_REF] Shablonin | Excitation of different chromium centres by synchrotron radiation in MgO:Cr single crystals[END_REF] .

The HHG process in gases is well explained by the semiclassical "three-step model" [START_REF] Krause | High-order harmonic generation from atoms and ions in the high intensity regime[END_REF][START_REF] Corkum | Plasma perspective on strong field multiphoton ionization[END_REF][START_REF] Kulander | Super-intense laser-atom physics[END_REF] . A similar approach has been proposed in crystals, where a strong laser electric field generates electron-hole pairs, which are subsequently accel-erated by this field and emit radiation via inter-and intraband opto-electronic transitions in the solid [START_REF] Ghimire | Generation and propagation of high-order harmonics in crystals[END_REF][START_REF] Vampa | Theoretical analysis of high-harmonic generation in solids[END_REF][START_REF] Luu | High-order harmonic generation in solids: A unifying approach[END_REF][START_REF] Tancogne-Dejean | Impact of the electronic band structure in high-harmonic generation spectra of solids[END_REF] . The competition between those two mechanisms depends on the material and the driving laser parameters and, as a general feature, intraband emission dominates at lower and inter-band emission at higher HHG photon energies, the latter including the plateau and cutoff domain of the HHG spectrum. ). Although the HHG yield from MgO:Cr is larger, we do not find evidence for a HHG cutoff extension for the doped sample.

vacancies [START_REF] Henry | Nature of the broadband luminescence center in MgO:Cr 3+[END_REF][START_REF] Wachtman | Mass transport in oxides[END_REF][START_REF] Stavale | Cathodoluminescence of nearsurface centres in Cr-doped MgO(001) thin films probed by scanning tunnelling microscopy[END_REF][START_REF] Shablonin | Excitation of different chromium centres by synchrotron radiation in MgO:Cr single crystals[END_REF] to maintain charge neutrality [START_REF] Shablonin | Excitation of different chromium centres by synchrotron radiation in MgO:Cr single crystals[END_REF] . The underlying electronic excitations in pristine and Cr-doped MgO are sketched in Figs. 1 (c) and 1 (d), respectively. An electron wave packet is (1) excited from the valence band (VB) to the first conduction band (CB1) with an initial field-dressed crystal momentum at the minimum band gap (referred to as the Γ-point), where it (2) experiences intra-band acceleration. In case the laser electric field is strong enough, an electron wave packet is excited to a higher conduction band (CB2), where intra-band acceleration occurs, before (3) de-exciting to the VB upon recombination with its residual hole, emitting harmonics with photon energies equal to the instantaneous gap in the first BZ. Within the Keldysh model for strong-field ionization [START_REF] Keldysh | Ionization in the field of a strong electromagnetic wave[END_REF] and a saddle-point analysis [START_REF] Lewenstein | Theory of high-harmonic generation by low-frequency laser fields[END_REF][START_REF] Wong | Asymptotic approximations of integrals[END_REF] , a closed-form semi-quantitative approximation for the above band-gap (interband transition) yield is given by [START_REF] Navarrete | Crystal-momentumresolved contributions to high-order harmonic generation in solids[END_REF][START_REF] Navarrete | Two-color-driven enhanced high-order harmonic generation in solids[END_REF]

Y MgO (E) ∝ exp - 8 m * vb-cb1 ε 3/2 g e h E , (1)
where

m * vb-cb1 = 1 m * vb + 1 m * cb1 -1
is the reduced effective mass of the VB and CB1, given in terms of their respective effective masses, m * vb and m * cb1 , ε g the minimum band-gap energy between VB and CB1, and E the peak field strength of the driving laser pulse. Similarly, we can account for excitations to higher conduction bands, e.g., to CB2 in our calculations, as sketched in Figs. 1 (c) and 1 (d), leading to the emission of harmonics at higher frequencies upon recombination to the VB. For a fixed driving electric-field amplitude, Eq. (1) predicts that a decrease of the minimum band gap or reduced effective mass increases the HHG spectral yield above the minimum band gap. Note that the effective electron mass is a crucial parameter responsible for the electron mobility in solids. Its energy-dependent behaviour was recently investigated [START_REF] Kasmi | Effective mass effect in attosecond electron transport[END_REF][START_REF] Ambrosio | Spatiotemporal analysis of a final-state shape resonance in interferometric photoemission from Cu(111) surfaces[END_REF] . Since the Cr impurities with associated vacancies break the transitional symmetry of the MgO crystal (in the BZ of the pristine crystal), impurity levels couple with states of the valence and conduction bands along the entire BZ. Together with the dispersion-free-electron nature of MgO impurity levels [START_REF] Mrudul | Highharmonic generation from spin-polarised defects in solids[END_REF][START_REF] Medeiros | Effects of extrinsic and intrinsic perturbations on the electronic structure of graphene: Retaining an effective primitive cell band structure by band unfolding[END_REF][START_REF] Hjort | Modeling vacancies in graphite via the Hückel method[END_REF] , which implies a very large effective mass, the loss of symmetry allows us to represent impurity levels as flat bands in the BZ of the pristine sample [shown as extended horizontal lines in Figs. 1 (c,d)]. The impurity-defect band (IDB) appears close to the CB1, and the vacancy defect band (VDB) emerges close to the VB. Adapting Eq. ( 1) to transitions from the VDB in MgO:Cr, we obtain the estimate for VDB contributions to the HHG yield

Y MgO:Cr (E) ∝ exp     - 8 m * vdb-cb1 (ε cb1 -ε vdb ) 3/2 e h E     , (2)
where m * vdb-cb1 is the reduced effective mass between CB1 and VDB. Because of the large VDB effective mass, we can approximate m * vdb-cb1 ≈ m * cb1 > m * vb-cb1 . Thus, the reduced effective mass is greater than that for the pristine sample. Nevertheless, because of the reduction of the band gap induced by the vacancy, the numerator of the exponential in Eq. ( 2) is smaller than in Eq. ( 1), suggesting an enhancement of the HHG yield due to symmetry-allowed VDB -CB1 optical transitions.

For a fully quantitative description of HHG from pristine and doped MgO we numerically solve the semiconductor Bloch equations (SBEs), including the VB, IDB, CB1, and CB2 [for details see Supplementary Information (SI) SI. 3.]. Disregarding temperature effects, the Fermi energy of MgO:Cr lies approximately between VDB and CB1 [START_REF] Kittel | Quantum Theory of Solids[END_REF] , so that the VB and VDB are initially occupied.

Figure 2 (a, b) shows calculated HHG spectra from MgO and MgO:Cr, respectively. Both spectra are composed of odd harmonics (except for a dim even harmonic for the cal- in the Γ -X direction. The offset of observed highest harmonic in both spectra is about 2 eV over a broad range of laser-electric-field strengths. It approximately matches the band-gap-energy difference between doped and undoped crystals. Dashed and solid lines are linear interpolations added to guide the eye. (c) Experimental HHG spectra obtained from MgO at a vacuum laser field strength of 1.05 V/Å (14.6 TW/cm 2 ) and from MgO:Cr at a vacuum laser field strength of 0.92 V/Å (11.2 TW/cm 2 ). In MgO:Cr a weaker laser electric field results in a comparable integrated HHG yield.

(b) (a)
culated spectra from MgO:Cr) and exhibit almost the same field-strength dependence of the cutoff. The main difference between them is the increased inter-band harmonic spectral yield at above-band-gap photon energies for the material with vacancy defects. We analyze the HHG-yield enhancement by comparing the spectra in Figs. 2 (a,b) with our experimental data, estimating the intensity of the driving electric field inside the solids as detailed in the SI.7. Our laser field is sufficiently strong to excite electrons to the CB2, leading to the emission of harmonics in the XUV spectral range. Note that the consistent modeling of macroscopic optical effects, and the driving laser field strength in matter, would require merging our theoretical model with Maxwell's equations, which is beyond the scope of the current manuscript. Spectral transmission aspects are addressed in SI.2. The setup, intensity estimation, and the detailed description of the samples are given as SI. Both crystals, MgO and MgO:Cr, are 200-µm-thick with (001) oriented surfaces. We confirm the presence of Cr 3+ ions in the MgO:Cr sample by photoluminescence measurements (SI.1.). In addition, the calculated densities of states (DOS) for MgO and MgO:Cr reveal defect states above the topmost valence band in MgO:Cr (SI.4.). Our transmission measurements show a band-gap-energy reduction of 1.8 eV in MgO:Cr, compared to the pure MgO band-gap energy of 7.16 eV (SI.2.).

Figure 3 (a) shows the dependence of the experimental harmonic yield, integrated between 14.4 and 25.6 eV, on the driving laser electric-field strength. The total HHG yield from MgO:Cr is found to be larger than the one of pure MgO, (a) (b)

90°(Γ -X) 45°(Γ -K) 0°(Γ -X) 45°(Γ -K) 90°(Γ -X) MgO, experiment MgO:Cr, experiment 0°(Γ -X) (d) (c)
MgO, theory MgO:Cr, theory up to a laser-field strength of about 1.15 V/Å (corresponding to an intensity of 18 TW/cm 2 ). Operating at a repetition rate of 125 kHz, the damage threshold is reached at a laser electric-field strength of 1.23 V/Å (20 TW/cm 2 ) and 1.09 V/Å (15.7 TW/cm 2 ) for pure MgO and MgO:Cr, respectively. The lower damage threshold of MgO:Cr is probably associated with the increased electron density. In comparison, Fig. 3 (a) presents the numerically calculated intensities of odd harmonics for MgO and MgO:Cr, integrated over the same energy range. Our simulation results show good agreement with the experiment for pristine MgO and two discrepancies for doped MgO: (i) the experimental data show high harmonic emission starting at much lower laser-field strength, 0.71 V/Å (6.74 TW/cm 2 ), with a smaller slope than that obtained from the simulations based on solution of SBEs for doped MgO, (ii) the saturation of the HHG emission from MgO:Cr observed in the experiment is not reproduced by theory, since it involves mechanisms, such as radiation damage, which are not taken into account in the simulations. In spite of this, both, measured and calculated results exhibit a pronounced increase of the HHG spectral intensity from the MgO:Cr crystal, as the result of added transitions from the VDB.

0°(Γ -X) 45°(Γ -K) 90°(Γ -X) 0°(Γ -X) 45°(Γ -K) 90°(Γ -X) 2.
Figure 3 (b) shows the highest harmonic order we are able to detect for driving-laser-field strengths (in vacuum) between 0.82 V/Å (9 TW/cm 2 ) and 1.23 V/Å (20 TW/cm 2 ) for both samples. It also shows HHG cutoffs retrieved from our calculated spectra in Fig. 2 (a,b). Figure 3 (b) shows a linear scaling of the highest experimentally detected harmonic as a function of the peak-electric-field strength in both MgO and MgO:Cr, with almost the same slope but an energy offset of approximately 2 eV. This energy offset is close to the bandgap-energy difference between MgO and MgO:Cr of ≈ 1.8 eV that we experimentally determined independently using optical transmission measurements (SI.1). The highest harmonic order obtained from the simulations in Figs. 2 (a),(b) is identical for MgO and MgO:Cr crystals. The extension of the highest experimentally observable harmonic from MgO:Cr is not obvious. Indeed, since for the same electric-field strength the electron wave packet explores the same region of the first BZ of the MgO and MgO:Cr crystals, one might expect identical HHG cutoffs. We attribute the measured apparent extension of the HHG spectra from MgO:Cr to a higher yield in the plateau region. To further illustrate the HHG behavior, Fig. 3 (c) displays HHG spectra measured at a driving-laser peak field strength of 0.92 V/Å (11.2 TW/cm 2 ) for MgO:Cr and at 1.05 V/Å (14.6 TW/cm 2 ) for MgO. The two spectra have comparable yields even though the HHG spectrum from MgO:Cr was obtained at a lower laser field strength. This is in qualitative agreement with the simulated spectra displayed in Fig. 2. In addition, we notice in the experimental spectra that harmonics above 19 eV are less intense than those below this energy. This sharp transition between these two plateaus has been reported as contribution from higher conduction bands associated with low population transfer [START_REF] Uzan | Attosecond spectral singularities in solid-state high-harmonic generation[END_REF][START_REF] You | Laser waveform control of extreme ultraviolet high harmonics from solids[END_REF] .

Moreover, we consider the contribution of macroscopic effects. The driving laser propagation inside the crystal can play a significant role on it's initial spatial or spectral properties and the emmited harmonic radiation [START_REF] Vampa | Observation of backward high-harmonic emission from solids[END_REF] . Although the optical propagation effects related to the change of linear and nonlinear refractive indices due to doping are unlikely to play a significant role on the driving laser properties in our experiment, given the low dopant concentration of 0.5 % [START_REF] Miller | Laser sources and applications[END_REF][START_REF] Lai | Multiple pulse operation of a femtosecond Ti:sapphire laser[END_REF][START_REF] Philip | Amplification of femtosecond laser filaments in Ti:Sapphire[END_REF][START_REF] Guo | Z-scan measurement of the nonlinear refractive index of Nd 3+ , Y 3+ -codoped CaF 2 and SrF 2 crystals[END_REF] , we observe the difference of spatial and spectral properties of the harmonic beam from MgO and MgO:Cr. We detect a blueshift of the HHG spectra in MgO:Cr compared to HHG spectra from MgO at fixed laser intensity (SI.5.). We associate this wavelength shift to the increased electron density participating in HHG from MgO:Cr. Since the plasma refractive index variation acts as a diverging lens in the spatial domain, we attribute the observed blueshift to plasma-induced defocusing of the fundamental beam [START_REF] Rae | Spectral blueshifting and spatial defocusing of intense laser pulses in dense gases[END_REF][START_REF] Heins | Electron kinetic energy and plasma emission diagnosis from femtosecond laser produced air plasmas[END_REF][START_REF] Nefedova | Determination of the spectral variation origin in high-order harmonic generation in noble gases[END_REF] (and references therein). Further investigations in a loose geometry configuration or in reflection geometry would limit possible contribution of macroscopic effects [START_REF] Vampa | Observation of backward high-harmonic emission from solids[END_REF] . In a more general context, other crystals and/or higher doping rates may induce larger macroscopic aspects, that may play a significant role in enhancement or in spectral or spatial modifications of the HHG properties [START_REF] Sivis | Tailored semiconductors for high-harmonic optoelectronics[END_REF] .

Next, we study the crystal-orientation dependent HHG response of both crystals. To investigate the HHG anisotropy, we measure the HHG signal as a function of the crystal orientation with respect to the laser polarization, as sketched in Fig. 1 (a,b) and described in SI.5. Figures 4 (a) and 4 (b) present angle-dependent measured HHG spectra from MgO and MgO:Cr, recorded at their respective optimal conditions for highest HHG yields, i.e., just below the damage threshold at peak electric-field strengths of 1.23 V/Å (20 TW/cm 2 ) for MgO and 1.09 V/Å (15.7 TW/cm 2 ) for MgO:Cr. The fourfold symmetry reflects the face-centered cubic crystal structure of MgO [START_REF] You | Anisotropic high-harmonic generation in bulk crystals[END_REF] . The HHG yield is largest along the Γ -X direction while no signal is observed along the Γ -K direction. Overall, the two anisotropy dependences are comparable, which is consistent with the fact that the crystal symmetry is preserved for our low dopant concentrations. We observe that, similar to Fig. 3 (a), the harmonics emitted from MgO:Cr have a comparable yield at a lower driving-field in-tensities than harmonics from pure MgO. To analyze this behavior, we have calculated the angle-dependent HHG yield for MgO and MgO:Cr at the same peak vacuum electric-field strength of 1.23 V/Å (20 TW/cm 2 ), as shown in Figs. 4 (c) and 4 (d). These calculations are performed in reduced dimensionality, representing the crystal as a 2D square lattice using the same model electronic potential as in recent works, 2,65,66 V (x, y) = V 0 cos(ax) + V 0 cos(ay), but allowing for a modified oscillation amplitude V 0 at the defect sites in MgO:Cr [START_REF] Huang | High-order-harmonic generation of a doped semiconductor[END_REF][START_REF] Yu | Enhanced high-order harmonic generation in donor-doped band-gap materials[END_REF][START_REF] Iravani | Effects of vacancies on high-order harmonic generation in a linear chain with band gap[END_REF][START_REF] Pattanayak | Influence of vacancy defects in solid high-order harmonic generation[END_REF][START_REF] Almalki | High harmonic generation tomography of impurities in solids: Conceptual analysis[END_REF] (SI.3.). Even though the angular spread of the harmonics differs between theory and experiment, we reproduce the maximal yields measured along the Γ -X and the minimal yields along the Γ -K direction. For all crystal orientations our numerical simulation predicts a larger HHG yield for MgO:Cr. These two last characteristics are in good qualitative agreement with the present experimental findings.

In conclusion, we present the HHG yield enhancement from MgO doped with chromium atoms compared to pure a MgO crystal. We associate the extension of the highest observable harmonic in MgO:Cr to an increase of the HHG plateau intensity. The HHG yield increase from the doped crystal is interpreted as optical excitations introduced by occupied defect states in MgO:Cr that arise from Cr 3+ dopant ions and Mg-vacancy formation in the MgO lattice. This finding is in agreement with our numerical simulations and previous theoretical predictions of enhanced HHG in doped samples [START_REF] Huang | High-order-harmonic generation of a doped semiconductor[END_REF][START_REF] Yu | Enhanced high-order harmonic generation in donor-doped band-gap materials[END_REF][START_REF] Iravani | Effects of vacancies on high-order harmonic generation in a linear chain with band gap[END_REF][START_REF] Pattanayak | Influence of vacancy defects in solid high-order harmonic generation[END_REF] . The HHG anisotropy shows that the pristine MgO crystal symmetry is preserved upon doping. Our numerical results do not reproduce the experimentally observed HHG spectral blue shift and saturation, both caused by additional free electrons in the doped MgO, which we attribute to our incomplete description of the driving laser pulse in the solid. Other crystals involved in the HHG or other dopant concentrations may enchance macroscopic effects that could be exploited to boost the HH yield in specific spectral ranges. As a perspective, our study promotes the development of efficient compact XUV light sources based on doping. Moreover, it has been suggested that HHG from crystals with defects can provide a way to study dipole moments and wave functions of impurity arrangements [START_REF] Almalki | High harmonic generation tomography of impurities in solids: Conceptual analysis[END_REF] and thus potentially serve as a tomographic measurement of impurity orbitals. Other future studies will concentrate on HHG spectroscopy to extract information on the intrinsic properties of light-driven attosecond electron transport [START_REF] Liao | Distinction of electron dispersion in time-resolved photoemission spectroscopy[END_REF] . This would create attractive perspectives towards the development of all-solid-state attosecond sources and petahertz electronics [START_REF] Krausz | Attosecond metrology: from electron capture to future signal processing[END_REF][START_REF] Schoetz | Perspective on petahertz electronics and attosecond nanoscopy[END_REF][START_REF] Sederberg | Attosecond optoelectronic field measurement in solids[END_REF][START_REF] Mrudul | Highharmonic generation from spin-polarised defects in solids[END_REF][START_REF] Huang | High-order-harmonic generation of a doped semiconductor[END_REF][START_REF] Yu | Enhanced high-order harmonic generation in donor-doped band-gap materials[END_REF][START_REF] Pan | Energy band splitting and high-order harmonic generation from a doped semiconductor[END_REF][START_REF] Iravani | Effects of vacancies on high-order harmonic generation in a linear chain with band gap[END_REF][START_REF] You | Laser waveform control of extreme ultraviolet high harmonics from solids[END_REF] .

FIG. 1 :

 1 FIG. 1: Schematic of HHG in (a,c) MgO and (b,d) MgO:Cr in (a,b) coordinate and (c,d) momentum space. In (a, b) the laser propagation axis is indicated as z, around which crystals are rotated (light blue arrow) during anisotropy measurements [cf. Figs. 4 (a) and 4 (b)]. (b) A Mg vacancy replaces the closest Mg 2+ ion along the [100] (C 4v symmetry) or [110] direction (C 2v symmetry). Both C 4v and C 2v centers are observed experimentally 27,34 . (c,d) The laser electric field excites an electron wave packet from the valence band (VB) to the first conduction band (CB1) (1, blue arrow), where it experiences intra-band acceleration (2, orange arrow). At higher driver intensities, excitation to (1', blue arrow) and acceleration in (2', orange arrow) higher conductions bands (here CB2) can occur. The excited electron wave packet de-excites to the VB upon recombination with its residual hole (3, blue arrow), emitting harmonics of energy equal to the instantaneous gap in the first BZ. (d) In MgO:Cr the occupied impurity defect band (IDB) and vacancy defect band (VDB) account for additional electronic transitions that enhance the net high-harmonic yield relative to MgO.

Figure 1

 1 Figure 1 presents a scheme of the basic mechanisms involved in the HHG process in pristine and doped MgO. Here, the Cr donor-dopants not only substitute Mg atoms with Cr 3+ ions in the MgO lattice, providing additional electrons [see Figs. 1 (a) and 1 (b)], but also cause the formation of

FIG. 2 :

 2 FIG. 2: Calculated HHG spectra from (a) MgO and (b) MgO:Cr crystals for driving laser-pulse incidence and HHG emission in Γ -X direction as a function of the driver peak field strength. The laser central wavelength is 1.55 µm (corresponding to a photon energy of 0.80 eV, for details on the experimental parameters see SI.5.). Although the HHG yield from MgO:Cr is larger, we do not find evidence for a HHG cutoff extension for the doped sample.

FIG. 3 :

 3 FIG. 3: Comparison of HHG signals from MgO and MgO:Cr crystals for a driving laser central wavelength of 1.55 µm (corresponding to a photon energy of 0.80 eV), (a) Measured and calculated total harmonic yields in the Γ -X direction, integrated over photon energies between 14.4 and 25.6 eV, as a function of the peak driving laser-electric-field strength. The integrated HHG yield from MgO:Cr crystal is larger. (b) Highest observable harmonic energy versus the driving field strengthin the Γ -X direction. The offset of observed highest harmonic in both spectra is about 2 eV over a broad range of laser-electric-field strengths. It approximately matches the band-gap-energy difference between doped and undoped crystals. Dashed and solid lines are linear interpolations added to guide the eye. (c) Experimental HHG spectra obtained from MgO at a vacuum laser field strength of 1.05 V/Å (14.6 TW/cm 2 ) and from MgO:Cr at a vacuum laser field strength of 0.92 V/Å (11.2 TW/cm 2 ). In MgO:Cr a weaker laser electric field results in a comparable integrated HHG yield.

5

 5

FIG. 4 :

 4 FIG. 4: Crystal-orientation dependence of HHG in (a,c) MgO and (b,d) MgO:Cr crystals. The crystals are rotated about the laser propagation axis [z axes in Figs. 1 (a) and 1 (b)] in 3-degree steps, while the linear laser polarization is kept fixed. Measured HHG spectra for (a) MgO at a peak laser-field strength (vacuum value) of 1.23 V/Å (20 TW/cm 2 ) and (b) MgO:Cr at 1.09 V/Å (15.7 TW/cm 2 ). Due to the low doping concentration (0.5%), the four-fold symmetry of the MgO cubic crystal structure is unaltered in MgO:Cr. Calculated spectra at 1.23 V/Å for (c) MgO and (d) MgO:Cr.

ACKNOWLEDGMENTS

Authors thank M. Billon for technical support. We acknowledge support from the PETACom (Petahertz Optoelectronics Communication) FET Open H2020 grant number 829153, OPTOLogic (Optical Topologic Logic) FET Open H2020 grant number 899794, support from the French ANR through the grant PACHA (ANR-17-CE30-0008-01) and the labex PALM (ANR-10-LABX-0039-PALM), support the DGA RAPID grant "SWIM". We acknowledge the financial support from the French ASTRE program through the "NanoLight" grant, the Spanish Ministry MINECO (National Plan 15 Grant: FISICATEAMO No. FIS2016-79508-P, SEVERO OCHOA No. SEV-2015-0522, FPI), European Social Fund, Fundació Cellex, Fundació Mir-Puig, Generalitat de Catalunya (AGAUR Grant No. 2017 SGR 1341, CERCA/Program), ERC AdG NOQIA, EU FEDER, European Union Regional Development Fund -ERDF Operational Program of Catalonia 2014-2020 (Operation Code: IU16-011424), MINECO-EU QUANTERA MAQS (funded by The State Research Agency (AEI) PCI2019-111828-2/10.13039/501100011033), and the National Science Centre, Poland-Symfonia Grant No. 2016/20/W/ST4/00314. F. N. and U. T. acknowledge support from Air Force Office of Scientific Research under award number FA9550-17-1-0369 and NSF Grant No. 1802085 (theory for photoemission from surfaces). N. T.-D. thanks A. Rubio.

DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.


Anne Philippe

Marie-Claude Viano

Random sampling of long-memory stationary processes

Keywords: Aliasing, FARIMA Processes, Generalised Fractional Processes, Regularly varying covariance, Seasonal long-memory, Spectral density MS 2000 Mathematics Subject Classification: 60G10 60G12 62M10 62M15

This paper investigates the second order properties of a stationary process after random sampling. While a short memory process gives always rise to a short memory one, we prove that long-memory can disappear when the sampling law has heavy enough tails. We prove that under rather general conditions the existence of the spectral density is preserved by random sampling. We also investigate the effects of deterministic sampling on seasonal long-memory.

Introduction

The effects of random sampling on the second order characteristics and more specially on the memory of a stationary second order discrete time process are the subject of this paper.

We start from X = (X n ) n≥0 , a stationary discrete time second order process with covariance sequence σ X (h) and a random walk (T n ) n≥0 independent of X. The sampling intervals ∆ j = T j -T j-1 are independent identically distributed integer random variables with common probability law S. We fix T 0 = 0.

Throughout the paper we consider the sampled process Y defined by

Y n = X Tn n = 0, 1 . . . (1.1)
The particular case where S = δ k , the Dirac measure at point k, shall be mentioned as deterministic sampling (some authors prefer systematic or periodic sampling).

Either because it corresponds to many practical situations, or because it is a possible way to model data with missing values, there is an extensive literature on the question of sampling random processes.

Around the sixties, an important amount of publications in signal processing was devoted to the reconstruction of the spectral density of (X n ) n≥1 from a sampled version (X Tn ) n≥1 . In case of deterministic sampling, this reconstruction is prevented by the aliasing phenomenon, (which is easily understandable from formula (4.35) below). Several authors noticed that aliasing can be suppressed by introducing some randomness in the sampling procedure. See, without exhaustivity, [START_REF] Beutler | Alias-free Randomly Timed Sampling of Stochastic Processes[END_REF], [START_REF] Masry | Random Sampling and Reconstruction of Spectra[END_REF] and [START_REF] Shapiro | Alias-Free Sampling of Random Noise[END_REF] where several random schemes are introduced. The idea of sampling along a random walk was first proposed in [START_REF] Shapiro | Alias-Free Sampling of Random Noise[END_REF] where the authors proved that under some convenient hypotheses, such a sampling scheme is alias-free when the characteristic function of S, Ŝ(λ) = j≥1 S(j)e ijλ , is injective.

Later, in the domain of time series analysis, attention was particularly paid to the effect of sampling on parametric families of processes. For example the effect of deterministic sampling on ARMA or ARIMA processes is studied in [START_REF] Brewer | Some consequences of temporal aggregation and systematic sampling for ARMA or ARMAX models[END_REF], [START_REF] Niemi | The invertibility of sampled and aggregate ARMA models[END_REF] and [START_REF] Sram | Temporal aggregation in the ARIMA processes[END_REF] among others. The main result is that the ARMA structure is preserved, the order of the autoregressive part being never increased after sampling. More generally, the stability of the ARMA family by random sampling along a random walk is proved in [START_REF] Kadi | Random aggregation of uni and multivariate linear processes[END_REF] and [START_REF] Oppenheim | Échantillonnage aléatoire d'un processus ARMA[END_REF]. Precisely, if A(L)X n = B(L)ε n and A 1 (L)Y n = B 1 (L)η n are the minimal representations of X and of the sampled process Y, the roots of the polynomial A 1 (z) belong to the set { j≥1 S(j)r j k } where the r k 's are the roots of A(z). As S(1) = 1 implies | j≥1 S(j)r j k | < |r k |, a consequence is that the convergence to zero of σ Y (h) is strictly faster than that of σ X (h). So, random sampling an ARMA process along a random walk shortens its memory. In [START_REF] Oppenheim | Échantillonnage aléatoire d'un processus ARMA[END_REF] it is even pointed out that some ARMA processes could give rise to a white noise through a well chosen sampling law.

Only few papers deal with the question of the memory of the process obtained by sampling a long-memory one. The reader can find in [START_REF] Chambers | Long-memory and aggregation in macroeconomic time-series[END_REF] and [START_REF] Hwang | The Effects of Systematic Sampling and Temporal Aggregation on Discrete Time Long Memory Processes and Their Finite Sample Properties[END_REF] a detailed study of deterministic sampling and time aggregation of the FARIMA (0, d, 0) process with related statistic questions. These authors point out that deterministic sampling does not affect the value of the memory parameter d of the process. In the present paper we deal with random sampling of long memory processes.

In all the sequel, a second order stationary process X is said to have long memory if its covariance sequence is non-summable

h≥0 |σ X (h)| = ∞. (1.2)
In section 2 we present some related topics such that L p -convergence of σ Y (h) and absolute continuity of the spectrum. We show in particular that short memory is always preserved as well as absolute continuity of the spectral measure. The main results of the paper, concerning changes of memory by sampling processes with regularly varying covariances, are gathered in Section 3.

We show that the intensity of memory of such processes is preserved if E(T 1 ) = jS(j) < ∞, while this intensity decreases when E(T 1 ) = ∞. For sufficiently heavy tailed S, the sampled process has short memory, which is somehow not surprising since with a heavy tailed sampling law, the sampling intervals can be quite large. In section 4 we consider processes presenting long-memory with seasonal effects, and investigate the particular effects of deterministic sampling. We show that in some cases the seasonal effects can totally disappear after sampling.

Some features unchanged by random sampling

Taking p = 1 in Proposition 1 below confirms an intuitive claim: random sampling cannot produce long-memory from short memory. Propositions 3, 4 and 5 state that, at least in all situations investigated in this paper, random sampling preserves the existence of a spectral density.

Preservation of summability of the covariance

Proposition 1. (i) Let p ≥ 1. If |σ X | p < ∞, the same holds for σ Y ,
(ii) In the particular case p ∈ [1, 2] both processes X and Y have spectral densities linked by the relation

f Y (λ) = 1 2π +∞ -∞ π -π e -iλ Ŝ(θ) j f X (θ)dθ,
where Ŝ(θ) = E(e iθT1 ) is the characteristic function of S.

Proof. The covariance sequence of the sampled process is given by

σ Y (0) = σ X (0) σ Y (h) = E (σ X (T h )) = ∞ j=h σ X (j)S * h (j) h ≥ 1 (2.3)
where S * h , the h-times convoluted of S by itself, is the probability distribution of T h . As the sequence T h is strictly increasing, σ X (T h ) is almost surely a subsequence of σ X (h). Then, (i) follows from

h E (σ X (T h )) p ≤ E h |σ X (T h )| p ≤ h |σ X (h)| p .
The proof of (ii) is immediate, using (i)

f Y (λ) = 1 2π j∈Z e -ijλ σ Y (j) = 1 2π j∈Z e -ijλ E(σ X (T j )) = 1 2π j∈Z π -π e -ijλ E(e iTj θ )f X (θ)dθ,
where the series converges in L 2 ([-π, π]) when p = 1, the covariance being then square-summable without being summable.

Preservation of the existence of a spectral density

Concerning the existence of a spectral density some partial results are easily obtained. Firstly, it is well known that the existence of a spectral density is preserved by deterministic sampling (see (4.35) below). Second, from Proposition 1 above it follows that, for any sampling law, the spectral density of Y exists when the covariance of X is square summable. It should also be noticed that, when proving that the ARMA structure is preserved by random sampling, [START_REF] Oppenheim | Échantillonnage aléatoire d'un processus ARMA[END_REF] (see also [START_REF] Kadi | Random aggregation of uni and multivariate linear processes[END_REF] for the multivariate case) gives an explicit form of the spectral density of Y when X is an ARMA process.

The three propositions below show that preservation of the existence of a spectral density by random sampling holds for all the models considered in the present paper.

The proofs are based on the properties of Poisson kernel recalled in Appendix 5.2 .4) and the representation given in Lemma 2 of the covariance of sampled process.

P s (t) = 1 2π 1 -s 2 1 -2s cos t + s 2 s ∈ [0, 1[. ( 2
Lemma 2. For all j ≥ 0,

σ Y (j) = lim r→1 - π -π e ijθ g(r, θ)dθ (2.5)
where

g(r, θ) = 1 4π π -π f X (λ) 1 1 -re -iθ Ŝ(λ) + 1 1 -re -iθ Ŝ(-λ) dλ (2.6) = 1 4 π -π f X (λ) 1 π + P rρ (τ -θ) + P rρ (τ + θ) dλ. (2.7)
Proof. The proof of the lemma is relegated in Appendix 5.1.

Proposition 3. If f X is bounded in a neighbourhood of zero, the sampled process Y has a spectral density given by

f Y (λ) = lim r→1 1 4π π -π f X (λ) 1 1 -re -iθ Ŝ(λ) + 1 1 -re -iθ Ŝ(-λ) dλ (2.8)
Proof. In the sequel we write

Ŝ(λ) = ρ(λ)e iτ (λ) , (2.9)
often denoted ρe iτ for the sake of shortness. The proof of the proposition simply consists in exchanging the limit and integration in (2.5).

Firstly, it is easily seen that, if θ = 0, g(r, θ) has a limit as r → 1 -. Hence the proof is complete provided that conditions of Lebesgue's theorem hold.

As we can suppose that the sampling is not deterministic,

| Ŝ(λ)| < 1 ∀λ ∈]0, π]
(see [START_REF] Feller | An Introduction to Probability Theory and Applications[END_REF]). Hence, thanks to the continuity of

| Ŝ(λ)|, sup |λ|>ε | Ŝ(λ)| < 1 ∀ε > 0.
The integral (2.6) is split in two parts: choosing ε such that f is bounded on

I ε = [-ε
, ε] and using the fact that the integrand in (2.6) is positive (see (5.42)).

Iε f X (λ)Re 1 1 -re -iθ Ŝ(λ) + 1 1 -re iθ Ŝ(λ) dλ ≤ (sup Iε f X ) Iε Re 1 1 -re -iθ Ŝ(λ) + 1 1 -re iθ Ŝ(λ) dλ which leads, thanks to Lemma 13, to Iε f X (λ)Re 1 1 -re -iθ Ŝ(λ) + 1 1 -re iθ Ŝ(λ) dλ ≤ 4π(sup Iε f X )g * (r, θ) = 4π(sup Iε f X ). (2.10)
Now,

I c ε f X (λ)Re 1 1 -re -iθ Ŝ(λ) + 1 1 -re iθ Ŝ(λ) dλ = π I c ε f X (λ) 1 π + P rρ (τ -θ) + P rρ (τ + θ) dλ.
Applying (5.43) with

s = rρ(λ) < ρ(λ), t = τ (λ) ± θ, and η = 1 -sup |λ|>ε | Ŝ(λ)| yields, π I c ε f X (λ) 1 π + P rρ (τ -θ) + P rρ (τ + θ) dλ ≤ 1 + 2 η I c ε f X (λ)dλ ≤ 1 + 2 η π -π f X (λ)dλ. (2.11)
Gathering (2.10) and (2.11) leads to the result via Lebesgue's theorem.

In the next two propositions, the spectral density of X is allowed to be unbounded at zero. Their proofs are in the Appendix.

We first suppose that the sampling law has a finite expectation. It shall be proved in subsection 3.1 that in this case the intensity of memory of X is preserved.

Proposition 4. If ET 1 < ∞ and if the spectral density of X has the form f X (λ) = |λ| -2d φ(λ) (2.12)
where φ is nonnegative, integrable and bounded in a neighbourhood of zero, and where 0 ≤ d < 1/2, then the sampled process has a spectral density f Y given by (2.8).

Proof. See Appendix 5.3

The case E(T 1 ) = ∞ is treated in the next proposition, under the extra assumption (2.13) meaning that S(j) is regularly varying at infinity. We shall see in subsection 3.2 (see Proposition 7) how the parameter d is transformed when the sampling law satisfies condition (2.13). In particular we shall see that if 1 < γ < 3 -4d the covariance of the sampled process is square summable, implying the absolute continuity of the spectral measure of Y . The proposition below shows that this property holds for every γ ∈]1, 2[ Proposition 5. Assume that the spectral density of X has the form (2.12) If the distribution S satisfies the following condition

S(j) ∼ cj -γ when j → ∞ (2.13)
where 1 < γ < 2 and with c > 0, then the conclusion of Proposition 4 is still valid.

Proof. see Appendix 5.4

3 Random sampling of processes with regularly varying covariances

The propositions below are valid for the family of stationary processes whose covariance σ X (h) decays arithmetically, up to a slowly varying factor. In the first paragraph we show that if T 1 has a finite expectation, the rate of decay of the covariance is unchanged by sampling. We then see that, when E(T 1 ) = ∞, the memory of the sampled process is reduced according to the largest finite moment of T 1 .

3.1 Preservation of the memory when E(T 1 ) < ∞ Proposition 6. Assume that the covariance of X is of the form

σ X (h) = h -α L(h),
where 0 < α < 1 and where L is slowly varying at infinity and ultimately monotone (see [START_REF] Bingham | Regular variation[END_REF]). If E(T 1 ) < ∞, then, the covariance of the sampled process satisfies

σ Y (h) ∼ h -α (E(T 1 )) -α L(h) as h → ∞.
Proof. We prove that, as h → ∞

σ X (T h ) h -α L(h) = T h h -α L(T h ) L(h) a.s. --→ (E(T 1 )) -α (3.14)
and that, for h large enough,

σ X (T h ) h -α L(h) ≤ 1. (3.15)
Since T h is the sum of independent and identically distributed random variables

T h = h j=1 ∆ j , with common distribution T 1 ∈ L 1 , the law of large numbers leads to T h h a.s. --→ E(T 1 ). (3.16) Now L(T h ) L(h) a.s.
--→ 1, (3.17) indeed, T h ≥ h because the intervals ∆ j ≥ 1 for all j and (3.16) implies that for h large enough T h ≤ 2E(T 1 )h, Therefore, using the fact that L is ultimately monotone, we have

L(h) ≤ L(T h ) ≤ L(2E(T 1 )h) (3.18)
if L is ultimately increasing (and the reversed inequalities if L is ultimately decreasing). Finally (3.18) directly leads to (3.17) since L is slowly varying at infinity (see Theorem 1.2.1 in [START_REF] Bingham | Regular variation[END_REF]). Clearly, (3.16) and (3.17) imply the convergence (3.14).

In order to prove (3.15), we write

σ X (T h ) h -α L(h) = T h h -α/2 T h h -α/2 L(T h ) L(h) = T h h -α/2 T -α/2 h L(T h ) h -α/2 L(h) .
As α > 0 and

T h ≥ h, T h h -α ≤ 1.
Moreover, h -α/2 L(h) is decreasing for h large enough (see [START_REF] Bingham | Regular variation[END_REF]), so that

T -α/2 h L(T h ) h -α/2 L(h) ≤ 1.
These two last inequalities lead to (3.15) Since σ Y (h) = E (σ X (T h )) we conclude the proof by applying Lebesgue's theorem, and we get

σ Y (h) h -α L(h) → (E(T 1 )) -α .
3.2 Decrease of memory when

E(T 1 ) = ∞ If E(T 1 ) = ∞ it is known that T h /h → ∞, implying that the limit in (3.14)
is zero. In other words, in this case the convergence to zero of σ Y (h) could be faster than h -α . The aim of this section is to give a precise evaluation of this rate of convergence.

Proposition 7. Assume that the covariance of X satisfies |σ X (h)| ≤ ch -α (3.19)
where

0 < α < 1. If lim inf x→∞ x β P (T 1 > x) > 0 (3.20)
for some β ∈ (0, 1)

(implying E(T β 1 ) = ∞ ) then |σ Y (h)| ≤ Ch -α/β . (3.21)
Proof. From hypothesis (3.19),

|σ Y (h)| ≤ E(|σ X (T h )|) ≤ cE(T -α h ) Then, E(T -α h ) = ∞ j=h P (T h ≤ j) j -α -(j + 1) -α ≤ α ∞ j=0 P (T h ≤ j)j -α-1 . (3.22)
From hypothesis (3.20) on the tail of the sampling law, it follows that, for large enough h,

P (T h ≤ j) ≤ P max 1≤l≤h ∆ l ≤ j = P (T 1 ≤ j) h ≤ 1 -Cj -β h ≤ e -Ch j β . ( 3
E(T -α h ) ≤ α ∞ j=h j -α-1 e -Ch j β .
The last sum has the same asymptotic behaviour as

∞ h x -α-1 e -Ch x β dx = h -α/β h 1-β 0 u α/β-1 e -Cu du,
and the result follows since, as β < 1,

h 1-β 0 u α/β-1 e -Cu du h→∞ ----→ ∞ 0 u α/β-1 e -Cu du.
Next proposition states that the bound in Proposition 7 is sharp under some additional hypotheses.

Proposition 8. Assume that σ X (h) = h -α L(h)
where 0 < α < 1 and where L is slowly varying at infinity and ultimately monotone If

β =: sup {γ : E(T γ 1 ) < ∞} ∈]0, 1[ (3.24) then, for every ε > 0, σ Y (h) ≥ C 1 h -α β -ε . (3.25)
Proof. Let ε > 0. We have

σ X (T h ) h -α β -ε = T -α h h -α β -ε L(T h ) = T -α-βε 2 h h -α β -ε T βε 2 h L(T h ) = T h h δ -α-βε 2 T βε 2 h L(T h )
where

δ = α/β + ε α + βε 2 = 1 β α + βε α + βε/2 .
Using Proposition 1.3.6 in [START_REF] Bingham | Regular variation[END_REF],

T βε 2 h L(T h ) h→∞ ----→ +∞ a.s Moreover δ > 1 β . From (3.24), this implies E(T 1/δ 1 ) < ∞.
Then, the law of large numbers of Marcinkiewicz-Zygmund (see [START_REF] Stout | Almost sure convergence[END_REF] Theorem 3.2.3) yields

T h h δ a.s.
--→ 0 as h → ∞.

(3.26)

Therefore by applying Fatou's Lemma

σ Y (h) h -α β -ε h→∞ ----→ ∞.

The particular case of the FARIMA family

A FARIMA (p, d, q) process is defined from a white noise (ε n ) n , a parameter d ∈]0, 1/2[ and two polynomials A(z) = z p + a 1 z p-1 + . . . + a p and B(z) = z q + b 1 z q-1 + . . . + b q non vanishing on the domain |z| ≥ 1, by

X n = B(L)A -1 (L)(I -L) -d ε n (3.27)
where L is the back-shift operator LU n = U n-1 . The FARIMA (0, d, 0)

W n = (I -L) -d ε n
introduced by Granger [START_REF] Gray | On generalized fractional processes[END_REF] is specially popular.

It is well known (see [START_REF] Brockwell | Time Series: Theory and Methods[END_REF]) that the covariance of the FARIMA (p, d, q) process satisfies

σ X (k) ∼ ck 2d-1 as k → ∞, (3.28)
allowing the above Propositions 6, 7 and 8 to apply. The results can be summarised as follows.

Proposition 9. Sampling a FARIMA process (3.27), with a sampling law S such that, with some γ > 1

S(j) = P (T 1 = j) ∼ cj -γ as j → ∞, (3.29)
leads to a process (Y n ) n whose auto covariance function satisfies the following properties 1) , ∀ε > 0.

(i) if γ > 2, σ Y (h) ∼ Ch 2d-1 . (3.30) (ii) if γ ≤ 2, C 1 h (2d-1)/(γ-1)-ε ≤ σ Y (h) ≤ C 2 h (2d-1)/(γ-
(3.31)

Consequently, the sampled process (Y n ) n has

• the same memory parameter if γ ≥ 2,

• a reduced long memory parameter if 2(1 -d) ≤ γ < 2,

• short memory if 1 < γ < 2(1 -d)
Proof.

For (i), T 1 has a finite expectation. Hence Proposition 6 applies leading to

σ Y (h) ∼ ch 2d-1 .
For (ii), the conditions of Propositions 7 and 8 in section 3.2 are satisfied with β = γ -1 and α = 1 -2d, leading to (3.31) Since the intensity of memory is then reduced except to the case γ = 2.

The loss of memory is such that if 1 < γ < 2(1-d) it happens that 1-2d γ-1 > 1, implying the convergence of the series |σ Y (h)|. In this case random sampling has created short-memory.

We illustrate this last result by simulating and sampling FARIMA(0,d,0) processes.

Simulations of the trajectories are based on the moving average representation of the FARIMA (see [START_REF] Bardet | Generators of long-range dependent processes : A survey[END_REF]).

The sampling distribution is

P (S = k) = k+1 k (γ -1)t -γ dt ∼ Ck -γ , (3.32)
which is simulated using the fact that when u is uniformly distributed on [0, 1], the integer part of u 1/(1-γ) is distributed according (3.32). In Figure 1 are depicted the empirical auto covariances of a trajectory of the process X and of two sampled trajectories Y 1 and Y 2 . The memory parameter of X is d = 0.35 and the parameters of the two sampling distributions are respectively γ 1 = 2.8 and γ 2 = 1.9. According to Proposition 9, the process Y 1 has the same memory parameter as X, while the memory intensity is reduced for the process Y 2 .

Then we estimate the memory parameters of the sampled processes by using the FEXP procedure introduced by [START_REF] Moulines | Log-periodogram regression of time series with long range dependence[END_REF][START_REF] Bardet | Semi-parametric estimation of the long-range dependent processes : A survey[END_REF]. The FEXP estimator is adapted to processes having a spectral density. From Propositions 5 and 4, this is the case for our sampled processes. Figure 2 shows the estimate and a confidence interval. Two values, d = 0.1 (lower curves) and d = 0.35 (upper curves) of the memory parameter of X are considered. In abscissa, the parameter γ of the sampling distribution is allowed to vary between 1.7 and 3.3. From Proposition 9 the value of the memory parameter is d if γ ≥ 2 and d-1+γ/2 otherwise. In the case d = 0.1 short memory is obtained for γ < 1.8. In the case d = 0.35, sample distributions leading to short memory are too heavy tailed to allow tractable simulations.

Decrease of memory by heavy-tailed sampling law

See [START_REF] Gray | On generalized fractional processes[END_REF], [START_REF] Leipus | Modelling long-memory time series with finite or infinite variance: a general approach[END_REF], [START_REF] Oppenheim | long-memory with seasonal effects[END_REF] for results and references on this class of processes. Taking M = 1 and θ 1 = 0, it is clear that this family includes the FARIMA processes, which belong to the framework of section 3. It is proved in [START_REF] Leipus | Modelling long-memory time series with finite or infinite variance: a general approach[END_REF] that as h → ∞ the covariance sequence is a sum of periodic sequences damped by a regularly varying factor

σ X (h) = h 2d-1 c 1,j cos(hθ j ) + c 2,j sin(hθ j ) + o(1) (4.34)
where d = max{d j ; j = 1, . . . , M } and where the sum extends over indexes corresponding to d j = d. Hence, these models generally present seasonal longmemory, and for them the regular form σ X (h) = h 2d-1 L(h) is lost. Precisely, this regularity remains only when, in (4.33), d = max{d j ; j = 1, . . . , M } corresponds to the unique frequency θ = 0. In all other cases, from (4.34), it has to be to be replaced by σ X (h) = O(h 2d-1 ).

From the previous comments it is clear that Propositions 6 and 8 are no longer valid in the case of seasonal long-memory. That means that in this situation, it is not sure that long memory is preserved by a sampling law having a finite expectation. But it is true that long-memory can be lost. Indeed, Proposition 7 applies with α = 1 -2d where d = max{d j ; j = 1, . . . , M }, and the following result holds: Proposition 10. When sampling a process with spectral density (4.33) along a random walk such that the sampling law satisfies (3.29), the obtained process has short memory as soon as 1 < γ < 2(1 -d) where d = max{d j ; j = 1, . . . , M }.

The effect of deterministic sampling

In a first step we just suppose that X has a spectral density f X , which is unbounded in one or several frequencies (we call them singular frequencies). It is clearly the case of the generalised fractional processes. The examples shall be taken in this family. We investigate the effect of deterministic sampling on the number and the values of the singularities. Let us suppose that S = δ k , the law concentrated at k. It is well known that if X has a spectral density, the spectral density of the sampled process

Y = (X kn ) n is f Y (λ) = 1 2ℓ + 1 l j=-l f X λ -2πj 2ℓ + 1 if k = 2ℓ + 1 (4.35) f Y (λ) = 1 2ℓ   ℓ-1 j=-ℓ+1 f X λ -2πj 2ℓ + f X λ -2πℓsgn(λ) 2ℓ   if k = 2ℓ
Proposition 11. Let the spectral density of (X n ) n have N X ≥ 1 singular frequencies. Then, denoting by N Y the number of singular frequencies of the spectral density of the sampled process

(Y n = X kn ) n • 1 ≤ N Y ≤ N X .
• N Y < N X if and only if f X has at least two singular frequencies λ 0 = λ 1 such that, for some integer number j * ,

λ 0 -λ 1 = 2πj * k
Proof. We choose k = 2ℓ + 1 to simplify. The proofs for k = 2ℓ are quite similar. Now remark that for fixed j,

λ -2πj 2ℓ + 1 ∈ I j := -π -2πj 2ℓ + 1 , π -2πj 2ℓ + 1 , for λ ∈ [-π, π[.
The intervals I j are non overlapping and ∪ ℓ j=-ℓ I j = [-π, π[. Now suppose that f X is unbounded at some λ 0 . Then f Y is unbounded at the unique frequency λ * satisfying λ 0 = λ * -2πj0 2ℓ+1 for a suitable value j 0 of j. In other words, to every singular frequency of f X corresponds a unique singular frequency of f Y . As a consequence, the number of singular frequencies of f Y is at least 1, and cannot exceed the number of singularities of f X .

It is clearly possible to reduce the number of singularities by deterministic sampling. Indeed, two distinct singular frequencies of f X , λ 0 and λ 1 produce the same singular frequency λ * of f Y if and only if

λ * -2πj 0 2ℓ + 1 = λ 0 , and λ * -2πj 1 2ℓ + 1 = λ 1 ,
which happens if and only if λ 0 -λ 1 is a multiple of the basic frequency 2π 2ℓ+1 .

Proposition 11 is illustrated by the three following examples showing the various effects of the aliasing phenomenon when sampling generalised fractional processes.

We take k = 3, so that the spectral density of Y is

f Y (λ) = 1 3 f X λ -2π 3 + f X λ 3 + f X λ + 2π 3 . Example 1. Firstly consider f X (λ) = |e iλ -1| -2d
, the spectral density of the FARIMA (0, d, 0) process. Then

f Y (λ) = 1 3 |e i λ-2π 3 -1| -2d + |e i λ 3 -1| -2d + |e i λ+2π 3 -1| -2d
has, on [-π, π[, only one singularity at λ = 0, associated with the same memory parameter d as for f X (λ). The memory of Y has the same characteristics as the memory of X. This was already noticed in [START_REF] Chambers | Long-memory and aggregation in macroeconomic time-series[END_REF] and [START_REF] Hwang | The Effects of Systematic Sampling and Temporal Aggregation on Discrete Time Long Memory Processes and Their Finite Sample Properties[END_REF].

Example 2. Consider now f X (λ) = |e iλ -e 2iπ 3 | -2d |e iλ -e -2iπ 3
| -2d , the spectral density of a long-memory seasonal process (see [START_REF] Oppenheim | long-memory with seasonal effects[END_REF]). The sampled spectral density is

f Y (λ) = 1 3 1 j=-1 |e i λ-2π 3 -e 2iπ 3 | -2d |e i λ-2π 3 -e -2iπ 3
| -2d ,
and is everywhere continuous on [-π, π], except at λ = 0 where

f Y (λ) ∼ c|λ| -2d .
In other words the memory is preserved, but the seasonal effect disappears after sampling.

Example 3. Consider now a case of two seasonalities (±π/4 and ±3π/4) associated with two different memory parameters d 1 and d 2 :

f X (λ) = |e iλ -e iπ 4 | -2d1 |e iλ -e -iπ 4 | -2d1 |e iλ -e 3iπ 4 | -2d2 |e iλ -e -3iπ 4
| -2d2 .
It is easily checked that f Y has the same singular frequencies as f X , with an exchange of the memory parameters:

f Y (λ) ∼ c λ ± π 4 -2d2 near ∓ π/4 f Y (λ) ∼ c λ ± 3π 4 -2d1
near ∓ 3π/4.

Appendix

Proof of Lemma 2

Let us consider the two z-transforms of the bounded sequence σ Y (j):

σ- Y (z) = ∞ j=0 z j σ Y (j) |z| < 1 and σ+ Y (z) = ∞ j=0 z -j σ Y (j) |z| > 1.
On the first hand, from the representation

σ Y (j) = E(σ X (T j )) = π -π f X (λ) Ŝ(λ) j d λ , we have σ- Y (z) = ∞ j=0 z j π -π f (λ) Ŝ(λ) j dλ = π -π f (λ) 1 -z Ŝ(λ) dλ, |z| < 1(5.36) σ+ Y (z) = ∞ j=0 z -j π -π f (λ) (S(λ)) j dλ = π -π f (λ) 1 -Ŝ(λ) z dλ, |z| > 1(5.37)
On the second hand, let C r be the circle |z| = r. If 0 < r < 1, for all j ≥ 0

1 2iπ Cr σ- Y (z)z -j-1 dz = 1 2π π -π (re iθ ) -j σ- Y (re iθ )dθ = ∞ l=0 r l-j σ Y (l) 2π π -π
e i(l-j)θ dθ = σ Y (j), (5.38) and, similarly, if r > 1 where g(r, θ) is defined in (2.6). As the first member does not depend on r, (2.5) is proved.

1 2iπ Cr σ+ Y (z)z j-1 dz = 1 2π π -π (re iθ ) j σ+ Y (re iθ )dθ = ∞ l=0 r j-l σ Y (l) 2π π -π e i(j-l)θ dθ = σ Y (j), ( 5
Let us now prove (2.7). Firstly,

Im 1 1 -re -iθ Ŝ(λ) + 1 1 -re -iθ Ŝ(-λ) = 1 2 1 1 -re -iθ Ŝ(λ) + 1 1 -re -iθ Ŝ(-λ) - 1 1 -re iθ Ŝ(-λ) - 1 1 -re iθ Ŝ(λ)
is an odd function of λ. Hence, the imaginary part of the integrand in (2.6) disappears after integration. Secondly,

Re 1 1 -re -iθ Ŝ(λ) + 1 1 -re -iθ Ŝ(-λ) = 1 2 1 -r 2 | Ŝ(λ)| 2 |1 -re -iθ Ŝ(λ)| 2 + 1 -r 2 | Ŝ(λ)| 2 |1 -re -iθ Ŝ(-λ)| 2 + 2 , (5.42)
and the proof is over.

A few technical results

Hereafter we recall some properties used in the paper.

Lemma 12. The Poisson kernel (2.4) satisfies

2πP s (t) ≤ 1 -s 2 (1 -s) 2 = 1 + s 1 -s ≤ 2 η ∀s < 1 -η, with s < 1. ( 5
(t) = | sin t| -1 .
The following result comes from spectral considerations. Lemma 13. With ρ = ρ(λ) and τ = τ (λ) defined in (2.9), for all r ∈ [0, 1[ and

θ ∈] -π, π[, g * (r, θ) := 1 4 π -π 1 π + P rρ (τ -θ) + P rρ (τ + θ) dλ = 1 (5.46)
Proof. Notice first that if f X ≡ 1, the process X is a white noise with Var(X 1 ) = 2π. Hence, the sampled process is also a white noise with variance 2π. Applying (2.6) to a white noise, we get 2πδ 0 (j) = r -j π -π e ijθ g * (r, θ)dθ.

(5.47) for all r ∈ [0, 1[. Since r j δ 0 (j) = δ 0 (j), we can rewrite (5.47) as

2πδ 0 (j) = π -π e ijθ g * (r, θ)dθ.
This means that g * (r, θ) is the Fourier transform of (2πδ 0 (j)) j . Consequently g * (r, θ) ≡ 1.

Proof of Proposition 4

As for Proposition 3, the proof consists in finding an integrable function g(θ) .48) For that purpose, we need the following estimation of Ŝ(λ) near zero. Since u 0 < π/2, this implies arg(1 -u) < πu 0 /2. From this and inequality (5.49) we obtain

such that |g(r, θ)| ≤ g(θ) ∀r ∈]0, 1[, θ ∈] -π, π[. ( 5
|1 -Ŝ(λ)| = (1 -e iλ ) j≥1 S(j) 1 -e ijλ 1 -e iλ = |1 -e iλ | j≥1 S (j)e
|λ| ≤ 1 C =⇒ |τ (λ)| ≤ πC|λ| 2 (5.50) For a fixed θ > 0 let λ 0 be such that φ is bounded on [-λ 0 , λ 0 ]. Denoting b(θ) = min λ 0 , 1 C , θ πC , we separate [-π, π] into four intervals: [-π, -b(θ)[, [-b(θ), 0[, ]0, b(θ)], ]b(θ), π].
In the sequel we only deal with the two last intervals, and concerning the integrand in (2.7) we only treat the part P rρ (τ -θ):

I 1 (θ) + I 2 (θ) = b(θ) 0 f (λ)P rρ(λ) (τ (λ) -θ)dλ + + π b(θ)
f (λ)P rρ(λ) (τ (λ) -θ)dλ.

• Bounding I 1 : From (5.50), since b(θ) ≤ θ/(πC), we have |τ (λ)| ≤ θ/2 which implies

θ 2 ≤ |τ (λ) -θ|.
Via (5.44) and (5.45), this leads to

P rρ (τ -θ) ≤ P rρ (θ/2) ≤ 1 2θ . Consequently I 1 ≤ sup [0,b(θ)] (φ(λ)) 2θ b(θ) 0 λ -2d dλ = sup [0,b(θ)] (φ(λ)) 2θ (b(θ)) -2d+1 ≤ C 1 θ -2d
since b(θ) ≤ θ/(πC) and -2d + 1 > 0.

• Bounding I 2 : When λ > b(θ), we have λ -2d ≤ C 2 max{θ -2d , 1} for some constant C 2 . Hence

I 2 ≤ C 2 max{θ -2d , 1} π -π φ(λ)P rρ (θ -τ )dλ.
(5.51)

Since φ is bounded in a neighbourhood of zero, the arguments used to prove Proposition 3 show that the integral in (5.51) is bounded by a constant. Finally I 1 + I 2 is bounded by an integrable function g(θ) and the proposition is proved.

Proof of Proposition 5

The following lemma gives the local behaviour of Ŝ(λ) under assumption (2.13).

Lemma 14. Since S(j) ∼ cj -γ with 1 < γ < 2 and c > 0,

|λ| 1-γ (1 -Ŝ(λ)) → Z, if λ → 0.
(5.52)

where Re(Z) > 0 and Im(Z) < 0.

Proof. From the assumption on S(j), It is clear that c 1 > 0, and c 2 < 0 follows from the fact that Γ(x) < 0 for

x ∈] -1, 0[ and cos (πγ/2) < 0.

In the sequel we take λ > 0. From this lemma, if λ is small enough (say 0 ≤ λ ≤ λ 0 ), We only consider the integral on the three last domains and the part P rρ (τθ) of the integrand in (2.7).

• When λ ∈]0, c(θ)/2], inequality (5.55) and properties (5.44) and (5.45) of the Poisson kernel lead to P rρ (θ -τ ) ≤ P rρ (θ -c ′ 3 λ γ-1 ) ≤ C θ -c ′ 3 λ γ-1 , whence

I 1 = c(θ)/2 0 f (λ)P rρ (θ -τ )dλ ≤ C ′ 1 c(θ)/2 0 λ -2d θ -c ′ 3 λ γ-1 dλ ≤ C ′ 1 1 2 ( θ c ′ 3 ) 1/(γ-1) 0 λ -2d θ -c ′ 3 λ γ-1 dλ = C ′ 2 θ -2d+1 γ-1 -1 2 -1 γ-1 0 u -2d+1 γ-1 -1 1 -u du.
Since -2d+1 γ-1 -1 > -1, the last integral is finite, implying

I 1 ≤ C ′ 3 θ -2d+1 γ-1 -1 ,
which is an integrable function of θ.

• Thanks to (5.43) and to the r.h.s. of (5.54), we have on the interval ]c(θ)/2, where the function between brackets is integrable because

-2d + 2 -γ γ -1 = -2d + 1 γ -1 -1 > -1.
• Finally,

. 23 )

 23 Gathering (3.22) and (3.23) then gives

9 Figure 1 :

 91 Figure 1: Auto covariance functions of X [left] and of two sub sampled processes corresponding to γ = 2.8 [middle] and γ = 1.9 [right]. The number of observed values of the sub sampled processes is equal to 5000

Figure 2 :

 2 Figure 2: Estimation of the long memory parameter for d = 0.1 and d = 0.35 as function of parameter γ. The confidence regions are obtained using 500 independent replications. The circles represent the theoretical values of the parameter obtained in Proposition 9. For each γ, the estimation of d is evaluated on 5000 observations

  iθ Ŝ(λ) dλ dθ, if r < 1 in (5.40) into its conjugate when r < 1, and r for 1/r when r > 1 leads to gives σ Y (j) = r -j π -π e ijθ g(r, θ)dθ, , ∀r ∈ [0,1[ (5.41)

  the fact that |u| ≤ u 0 < 1 =⇒ |1 -u| > 1 -u 0 and sin(arg(1 -u)) < u 0 .

j≥1S(j)( 1 -

 1 cos(jλ)) ∼ λ→0 c j≥1 j -γ (1 -cos(jλ)) and j≥1 S(j) sin(jλ) ∼ λ→0 c j≥1 j -γ sin(jλ).Then, using well known results of Zygmund([23]; pages 186 and 189)j≥1 j -γ (1 -cos(jλ)) ∼ λ→0 c γ -1 |λ| γ-1 =: c 1 |λ| γ-1 . and j≥1 j -γ sin(jλ) ∼ λ→0 cΓ(1 -γ) cos πγ 2 |λ| γ-1 =: c 2 |λ| γ-1

c 3 λ′ 3 1/(γ- 1 ).

 331 γ-1 ≤ τ (λ) ≤ c ′ 3 λ γ-1(5.53) and1 -c 4 λ γ-1 ≤ ρ(λ) ≤ 1 -c ′ 4 λ γ-1 (5.54)where the constants are positive. For a fixed θ > 0, we deduce from (5.53)λ < min λ 0 , θ c implies 0 < θ -c ′ 3 λ γ-1 ≤ θ -τ (λ) (5.55) After choosing λ 1 ≤ λ 0 such that φ is bounded on [0, λ 1 ] define c(θ) = min λ 1 , Then we split [-π, π] into six intervals [-π, -λ 1 [, [-λ 1 , -c(θ)/2[, [-c(θ)/2, 0[, ]0, c(θ)/2], ]c(θ)/2, λ 1 ], ]λ 1 , π].

λ 1 ] 3 λ γ- 1 . 3 1 - 1 +′ 4  λ -2d+2-γ 1 + min λ 1

 131311411 P rρ (θ -τ ) ≤ C ′ Since φ is bounded on this domain, sup 0<λ≤λ1 φ(λ) -2d + 2 -γ λ -2d+2-γ (c(θ)/2) -2d+2-γ ≤ C ′ 3 sup 0<λ≤λ1 φ(λ) -2d + 2 -γ λ -2d+2-γ (c(θ)/2) -2d+2-γ = C

f

  (λ)P rρ (θ -λ)dλ ≤ λ -2d )P rρ (θ -λ)dλ

-2d γ -1 > 1 -2d, if 1 < γ ≤

2,

which has already been treated since φ is bounded near zero.

Gathering the above results on I 1 , I 2 and I 3 completes the proof.