Eléonore Roussel

Christophe Szwaj

Clément Evain

Bernd Steffen

Christopher Gerth

Bahram Jalali

Serge Bielawski

Phase Diversity Electro-optic Sampling: A new approach to single-shot terahertz waveform recording

THz spectroscopy is an emerging tool for detection of microorganisms and harmful compounds in the food industry, the study of proteins in biomedicine and the development of electronbeam X-ray sources for molecular imaging and lithography. Recording of THz electric field evolution in single-shot is crucially needed in terahertz spectroscopy of irreversible processes in such applications as well as for data communication in the THz portion of the spectrum where there is an abundance of untapped bandwidth. However, achieving sub-picosecond resolution over a long time window has been an open problem for electro-optic sampling -the standard technique for recording terahertz waveforms. We introduce a new conceptual framework for this open problem that is inspired by timestretch theory. The novel framework unveils a solution to this 20 year-old problem leading to a dramatic enhancement of the achievable temporal resolution. We validate this new technology in two applications. First, we present single shot recordings of long free-propagating terahertz transients with record time resolution. Second, we present recordings of ultra-short relativistic electron bunches at the European X-ray Free Electron Laser. These results show that electric signals may be now recorded with terahertz bandwidth over arbitrarily long windows, thus enabling the realization of single-shot terahertz oscilloscopes and single-shot time-domain spectroscopy systems with an arbitrary timebandwidth product.

INTRODUCTION

Recording the complete electric field of light in singleshot (including its envelope and carrier) is considered as one of the "holy grails" of terahertz science. This type of detection is largely needed for investigating and mastering novel terahertz sources, as ultrashort quantum cascade lasers [START_REF] Cappelli | Retrieval of phase relation and emission profile of quantum cascade laser frequency combs[END_REF], ultra-wide bandwidth laser-plasmabased terahertz sources [START_REF] Kress | Terahertz-pulse generation by photoionization of air with laser pulses composed of both fundamental and second-harmonic waves[END_REF], and terahertz Free-Electron-Lasers [START_REF] Hafez | Extremely efficient terahertz highharmonic generation in graphene by hot Dirac fermions[END_REF]. Such tools are also crucial for mastering novel "extreme photonic infrastuctures", such as X-ray Free-Electron Lasers [START_REF] Decking | First operation of a MHz-repetitionrate Hard X-Ray Free-Electron Laser driven by a superconducting linear accelerator[END_REF], and in the very active field of Laser-Plasma Acceleration (LPA) [START_REF] Buck | Real-time observation of laser-driven electron acceleration[END_REF], which has the ambition to replace large accelerators facilities by table-top installations. Single-shot terahertz recorders are also needed in applications of terahertz sources. This includes investigations of rapidly evolving reactions in chemistry and biology [START_REF] Hafez | Extremely efficient terahertz highharmonic generation in graphene by hot Dirac fermions[END_REF], as well as nonlinear physics (such as nonlinear spectroscopy) which use low repetition rate high power terahertz sources. From the industrial point of view, single-shot terahertz recorders would also introduce a disruptive competition with respect to latest fast scanning terahertz spectrometers, as Asynchronous Optical Sampling (ASOPS) and Electronically Controlled Optical Sampling (ECOPS) [START_REF] Bartels | Ultrafast time-domain spectroscopy based on high-speed asynchronous optical sampling[END_REF][START_REF] Kim | High-speed terahertz time-domain spectroscopy based on electronically controlled optical sampling[END_REF], in the quest for nondestructive and fast throughput inspection in, e.g., automotive and drug industries.

However, achieving electric field measurements over long durations while keeping femtosecond resolution is a largely open problem. Promising strategies consist in extending the so-called electro-optic sampling technique [START_REF] Valdmanis | Picosecond electro-optic sampling system[END_REF] to the single-shot case. The main principle consists in imprinting the electric field onto a laser pulse, and analyzing the output light. A popular scheme uses temporal encoding onto chirped laser pulses [START_REF] Jiang | Electro-optic measurement of THz field pulses with a chirped optical beam[END_REF][START_REF] Sun | Analysis of terahertz pulse measurement with a chirped probe beam[END_REF], as displayed in Figure 1a.

Although the principle seems simple, retrieving the input electric shape from the data has been the subject of active investigations during the last two decades. Deconvolution algorithms have only led to limited improvements [START_REF] Yellampalle | Algorithm for high-resolution single-shot THz measurement using in-line spectral interferometry with chirped pulses[END_REF][START_REF] Wu | Electrooptic sampling of optical pulses and electron bunches for a compact THz-FEL source[END_REF], and most research switched to alternate hardware strategies, such as encoding the information onto the transverse or angular direction [START_REF] Shan | Single-shot measurement of terahertz electromagnetic pulses by use of electro-optic sampling[END_REF][START_REF] Srinivasan-Rao | Novel single shot scheme to measure submillimeter electron bunch lengths using electro-optic technique[END_REF][START_REF] Kawada | Single-shot terahertz spectroscopy using pulse-front tilting of an ultra-short probe pulse[END_REF][START_REF] Minami | Single-shot measurement of a terahertz electric-field waveform using a reflective echelon mirror[END_REF][START_REF] Kim | Single-shot terahertz pulse characterization via two-dimensional electro-optic imaging with dual echelons[END_REF], and combining electro-optic sampling with advanced laser pulse characterization techniques [START_REF] Jamison | High-temporal-resolution, singleshot characterization of terahertz pulses[END_REF][START_REF] Walsh | The time resolved measurement of ultrashort terahertz-band electric fields without an ultrashort probe[END_REF]. However these complex designs still present important trade-offs between sensitivity, maximum recording duration and temporal resolution.

We show here that electro-optic sampling systems can be transformed into terahertz digitizers, without tradeoff on recording length and femtosecond resolution. Key to our approach is a completely new conceptual approach of electro-optic sampling, which has its roots in the theory of the photonic time-stretch analog-to-digital converter [START_REF] Bhushan | Time-stretched analogue-to-digital conversion[END_REF][START_REF] Fard | Photonic time-stretch digitizer and its extension to real-time spectroscopy and imaging[END_REF]. As we will see, this new point of view enables to obtain an extremely synthetic view of the process in Fourier domain, and to analyze electro-optic sampling in terms of frequency domain transfer functions (Fig. 1c). Using this approach, we show that it is possible to retrieve the input pulse (Fig. 1d) with unprecedented resolution, by using a retrieval algorithm based on the the concept of phase-diversity [START_REF] Kahn | Ratio squarer[END_REF][START_REF] Han | Ultrawide-Band Photonic Time-Stretch A/D Converter Employing Phase Diversity[END_REF].

Then we demonstrate the resulting DEOS (Diversity Electro-Optic Sampling) strategy on two different experimental applications. First we demonstrate singleshot recordings in a table-top experiment, with freepropagating terahertz pulses, thus showing the building blocks of a novel "single-shot Time-Domain Spectroscopy" system. Second, we test the DEOS approach in an accelerator context, by probing the near-field of femtosecond electron bunches at megahertz repetition rates, at the European X-ray Free-Electron Laser [START_REF] Decking | First operation of a MHz-repetitionrate Hard X-Ray Free-Electron Laser driven by a superconducting linear accelerator[END_REF] (Eu-XFEL).

RESULTS

DEOS single-shot recorder: experimental strategy and novel theoretical framework

Our experimental strategy is displayed in Figure 1a. The DEOS optical setup is extremely close to classical chirped pulse electro-optic sampling systems [START_REF] Jiang | Electro-optic measurement of THz field pulses with a chirped optical beam[END_REF][START_REF] Sun | Analysis of terahertz pulse measurement with a chirped probe beam[END_REF][START_REF] Wilke | Single-Shot Electron-Beam Bunch Length Measurements[END_REF][START_REF] Peng | Optimal chirped probe pulse length for terahertz pulse measurement[END_REF][START_REF] Fletcher | Distortion and uncertainty in chirped pulse THz spectrometers[END_REF][START_REF] Schmidhammer | Single shot linear detection of 0.01-10 THz electromagnetic fields[END_REF][START_REF] Müller | Electro-optical measurement of sub-ps structures in low charge electron bunches[END_REF][START_REF] Funkner | High throughput data streaming of individual longitudinal electron bunch profiles[END_REF], except few -though crucial -modifications. The electric field under interest E(t) is imprinted in singleshot onto a chirped laser pulse. Then we record the optical spectra in single-shot at the two polarizer outputs. If we assume a linear chirp, the optical frequency ω opt is related to the input time t by:

t = - ω opt C ( 1
)
where C is the chirp rate of the laser pulse. The optical spectra provide the input data of our retrieval algorithm. The spectra are first preprocessed in a classical way (background subtraction, amplitude scaling and wavelength to time conversion (see Fig. 1 1b.

As a key point of the DEOS method, we use a special arrangement of the crystal, waveplates, and polarizations (see the caption of Figure 1), which leads to different responses on the two output channels. We will show that this information diversity will enable a numerical retrieval of the input terahertz signal.

A careful analytical analysis first shows that the measured outputs Y 1,2 (t) are related to the input field E(t) by transfer functions H 1,2 in Fourier space. For a small input signal (see Supplementary Material, Sections IV-V) we can write:

Ỹ1 (Ω) ≈ H 1 (Ω) ∆ φ in (Ω) (2) Ỹ2 (Ω) ≈ H 2 (Ω) ∆ φ in (Ω) (3)
where the tilde denotes the Fourier transform, and Ω is the terahertz frequency at the input. ∆φ in (t) is the phase shift corresponding to the field-induced birefringence:

∆φ in (t) = βE(t) (4) with β = πd λ n 3 0 r 41 [START_REF] Buck | Real-time observation of laser-driven electron acceleration[END_REF] where n 0 is the refractive index at vanishing electric field and d the thickness of the crystal. λ is the laser wavelength in vacuum and E(t) the electric field inside the crystal.

For the specific layout described in Figure 1a, the corresponding transfer functions can be written as (see Supplementary Material, Section V):

H 1 (Ω) = √ 2 cos BΩ 2 + π 4 (6)
H 2 (Ω) = - √ 2 cos BΩ 2 - π 4 (7)
where B = 1 2C . As a first observation, the transfer functions H 1 and H 2 present nulls at specific frequencies (Fig. 1c). This explains why it is impossible to perform a deconvolution (i.e., to retrieve the input field E(t) from the recorded data) using a single channel Y 1 (t) or Y 2 (t), for broadband signals. This theoretical result on the existence of zeros is consistent with previous experimental observations [START_REF] Murakami | Dependence on chirp rate and spectral resolution of the terahertz field pulse waveform measured by electro-optic detection using a chirped optical pulse and a spectrometer and its effect on terahertz spectroscopy[END_REF][START_REF] Van Tilborg | Single-shot measurement of the spectral envelope of broad-bandwidth terahertz pulses from femtosecond electron bunches[END_REF], and ours, as we will see below. Conceptually, these nulls correspond to the dispersion penalty phenomenon which is observed in the photonic time-stretch digitizer [START_REF] Han | Ultrawide-Band Photonic Time-Stretch A/D Converter Employing Phase Diversity[END_REF], as shown in Supplementary Material, Sections IVC and VB.

Using this point of view, we also remark that it should be possible to retrieve the input signal ∆φ in (t) and E(t) by exploiting the information contained in both output signals, as long as the nulls of H 1 (Ω) and H 2 (Ω) are interleaved. In this respect, it is important to note that the relative positions of the zeros depend on the crystal orientations. For instance waveplate and polarization orientations used traditionally ([-110] axis parallel to the terahertz field) would not be acceptable, as this would lead to zeros of H 1,2 that occur at the same frequencies (see Supplementary Figure S4). In addition, the polarization arrangement considered in this article -for which the zeros are evenly interleaved -is only one of many possible choices which are compatible with the following retrieval algorithm.

Algorithm for high-resolution numerical reconstruction of the input signal

As the zeros of the transfer functions are interleaved, we can retrieve the unknown input pulse from the recorded data. The mathematical problem is well-posed, and even overdetermined, i.e., one has the freedom to chose either H 1 or H 2 for inverting the problem, except at the nulls. Several more advanced methods however exist for taking advantage of both transfer functions in the reconstruction process. We use here the so-called Maximal Ratio Combining technique (MRC) [START_REF] Kahn | Ratio squarer[END_REF][START_REF] Han | Ultrawide-Band Photonic Time-Stretch A/D Converter Employing Phase Diversity[END_REF], which is known for optimizing the signal-to-noise ratio (SNR). The input signal can be retrieved from the measurements Y 1,2 as [START_REF] Han | Ultrawide-Band Photonic Time-Stretch A/D Converter Employing Phase Diversity[END_REF]:

Ẽretr in (Ω) = β ∆φ retr in (Ω) (8)
with ∆φ retr in (Ω) =

H 1 (Ω) Ỹ1 (Ω) + H 2 (Ω) Ỹ2 (Ω) H 2 1 (Ω) + H 2 2 (Ω) (9)
where Ẽretr in (Ω) and ∆φ retr in (Ω) are the retrieved input electric field and crystal phase modulation, expressed in Fourier space. The input signal E retr in (t) (or equivalently ∆φ retr in (t)) is then obtained by performing an inverse Fourier transform

We tested this reconstruction method numerically for various parameters, and found that it is possible to retrieve the input pulse for arbitrarily long input chirped pulses (i.e., for any duration of the analysis window), down to terahertz pulse durations of the order of the Fourier-limited pulse duration. An example of retrieved input pulse is presented in Figure 1d.

Experimental demonstration: recording free-propagating terahertz pulses with ultra-wide time-bandwidth products

In order to test the DEOS method experimentally, we first constructed the setup displayed in Figure 2a. Terahertz pulses are produced by optical rectification of 800 nm millijoule-range laser pulses in a Zinc Telluride (ZnTe) crystal. These terahertz pulses are then analyzed by an EO sampling setup based on our design.

For test puposes, we also designed the setup so that it is possible to skip or operate the grating stetcher, without changing the setup alignment. We can thus analyze the terahertz pulses using either single-shot EO sampling (i.e., using chirped pulses), or using "traditional" EO sampling by scanning the delay between the femtosecond laser pulses and the terahertz signal. We will thus test our strategy by comparing the DEOS results to the corresponding scanned EO signals.

A typical result is presented in Figure 2e, using a time window of 20 ps. The reconstructed EO signal and the reference EO signal (i.e., obtained using scanned electrooptic sampling) are found to be extremely similar in shape. The reconstruction is even able to reproduce fine details, as the small oscillations between 5 and 20 ps which are due to the water vapor absorption in air (freeinduction decay), and multiple reflections on the Silicon plate. For comparison, the EO signal shapes Y 1,2 before reconstruction (i.e., corresponding to the previous stateof-art) are very different from the input, as can be seen in Figure 2c,d.

These results confirm that the DEOS method now enables investigations of terahertz sources, as well as Time-Domain Spectroscopy to be achieved in single-shot, with (e): Single-shot input signal retrieved from (c) and (d) using the DEOS phase-diversity-based algorithm (red). Green trace: actual input, obtained using scanned electro-optic sampling. Inset: Fourier spectra of the two terahertz signals.

simultaneously high temporal and spectral resolution. More precisely the time resolution and bandwidth limit of the DEOS method appear -as theoretically expected -similar to classical (and non-single-shot) "scanned" electro-optic sampling.

Note: determination of the reconstruction parameter

As this type of reconstruction requires the knowledge of the transfer functions H 1,2 , it is important to find a practically convenient strategy for determining the parameter B. We remarked that B can be determined in a very simple way, by analyzing the recorded data corresponding to the unknown signal. From the reconstructed signal ∆φ retr in (t), we can simulate EO sampling signals

Ỹ retr 1
(Ω) and Ỹ retr

2

(Ω):

Ỹ retr 1 (Ω) = H 1 (Ω) ∆φ retr in (Ω) (10) Ỹ retr 2 (Ω) = H 2 (Ω) ∆φ retr in (Ω) (11)
where H 1,2 , Ỹ retr 1,2 and ∆φ retr in (Ω) depend on B. Then we can perform a least-square fit of Ỹ retr 1 and Ỹ retr 2 on Ỹ1 and Ỹ2 , using B as a free parameter. Here, we perform a classical least-square fit using the following definition for the reconstruction error :

2 = +∞ -∞ dΩ Ỹ1 -Ỹ retr 1 2 + Ỹ2 -Ỹ retr 2 2 (12)
Besides its use for checking the fit quality, it is interesting to examine the reconstructed Ỹ retr 1,2 and raw Ỹ1,2 EO signals (Figure 3). Those curves clearly display the nulls that stem from the zeros of the transfer function H 1,2 expressed in Eqns. [START_REF] Bartels | Ultrafast time-domain spectroscopy based on high-speed asynchronous optical sampling[END_REF][START_REF] Kim | High-speed terahertz time-domain spectroscopy based on electronically controlled optical sampling[END_REF]. (Ω) are fitted on Ỹ1,2(Ω) using B as a free parameter. Note the presence of interleaved zeros, which play a key role in the retrieval strategy. Same data and color codes as for Figure 2.

Remarkably, this result shows that the determination of the fit parameter B does not require specific experiments to be performed. The fit can be made using any unknown input signal -including the signal to be analyzed -provided its bandwidth is sufficient.

Application to near-field measurements: relativistic electron bunch shapes at the European X-ray free-electron laser

The high resolution retrieval strategy based on phase diversity is expected to find immediate applications in single-pass Free-Electron Lasers (FELs), such as Eu-XFEL [START_REF] Decking | First operation of a MHz-repetitionrate Hard X-Ray Free-Electron Laser driven by a superconducting linear accelerator[END_REF]. FELs are based on self-amplification of stimulated radiation driven by ultra-short relativistic electron bunches with high peak current propagating in a periodic magnetic field (undulators). They provide femtosecond photon pulses with energies in the millijoule range, and wavelengths ranging from the ultraviolet to hard X-rays depending on the facilities. The emitted photon pulses depend directly on the properties of the electron bunches. Hence the diagnostics of their longitudinal shape has been the subject of an intense research the last years [START_REF] Wilke | Single-Shot Electron-Beam Bunch Length Measurements[END_REF][START_REF] Casalbuoni | Numerical studies on the electrooptic detection of femtosecond electron bunches[END_REF][START_REF] Steffen | Electro-optic time profile monitors for femtosecond electron bunches at the soft X-ray freeelectron laser FLASH[END_REF][START_REF] Wu | Comparison of the detection performance of three nonlinear crystals for the electro-optic sampling of a FEL-THz source[END_REF][START_REF] Steffen | Compact single-shot electro-optic detection system for THz pulses with femtosecond time resolution at MHz repetition rates[END_REF]. However the resolution limit of Eq. ( 14) strongly hampered the application range of electro-optic sampling. The situation is also complicated by the recent trend to megahertz-rate X-ray FELs based on superconducting technology such as FLASH [START_REF] Ackermann | Operation of a free-electron laser from the extreme ultraviolet to the water window[END_REF], Eu-XFEL [START_REF] Decking | First operation of a MHz-repetitionrate Hard X-Ray Free-Electron Laser driven by a superconducting linear accelerator[END_REF], and the LCLS-II [START_REF] Galayda | The LCLS-II: A High Power Upgrade to the LCLS[END_REF] and SHINE [START_REF] Yan | Multi-beam-energy operation for the continuous-wave x-ray free electron laser[END_REF] projects.

In order to test our DEOS reconstruction strategy in this context, we realized a proof-of-principle phasediversity setup at the EuXFEL (Figure 4a), which is the first hard X-ray FEL operating at megahertz repetition rate [START_REF] Decking | First operation of a MHz-repetitionrate Hard X-Ray Free-Electron Laser driven by a superconducting linear accelerator[END_REF] (see also Refs. [START_REF] Gisriel | Membrane protein megahertz crystallography at the European XFEL[END_REF][START_REF] Pandey | Time-resolved serial femtosecond crystallography at the European XFEL[END_REF] for examples of applica-tions). In the present case, electron bunches are generated at 1.3 MHz rate in 600 µs long bursts, every 100 ms (Figure 4b).

The setup, based on the single EO channel detection [START_REF] Steffen | Compact single-shot electro-optic detection system for THz pulses with femtosecond time resolution at MHz repetition rates[END_REF], is depicted schematically in Figure 5. The EO setup is located just after the second bunch compressor, where the electron bunches have a beam energy of 700 MeV and a typical RMS duration in the 200 fs range at a charge of 250 pC. See Methods for details.

Results are displayed in Figure 4c,d. Figure 4c depicts the raw EO signals (after background subtraction and normalization), and Figure 4d represents the reconstructed electric field. We observe a main peak, for which the duration of 218 fs (RMS) is in very good agreement with the design value of the electron bunch duration at this location and considerably shorter than measured with conventional EOSD at the same setup [START_REF] Steffen | Compact single-shot electro-optic detection system for THz pulses with femtosecond time resolution at MHz repetition rates[END_REF]. The smaller negative peak can be attributed to a wakefield following the electron bunch, but the precise interpretation will be subject of further investigations. Typical examples are displayed in Figures 4e,f. This measurement system can now be used to perform bunch-by-bunch high resolution measurements of the bunch shape which is crucial information for the control of the bunch compression process.

As previous electro-optic sampling systems, DEOS also simultaneously measures the arrival time of the electron bunch, which is also a crucial parameter for users of the generated X-rays. The resolution of DEOS for this measurement is expected to be similar to standard EO sampling. In the present case, the arrival time jitter (Fig. 4g) is measured to be 58 fs over a bunch train, which is much lower than the bunch duration (218 fs RMS), and consistent with Ref. [START_REF] Steffen | Compact single-shot electro-optic detection system for THz pulses with femtosecond time resolution at MHz repetition rates[END_REF].

From the hardware point of view, it is important to note that only few key modifications of our initial EO system [START_REF] Steffen | Compact single-shot electro-optic detection system for THz pulses with femtosecond time resolution at MHz repetition rates[END_REF] were needed: ensuring a proper (non-standard) orientation of the GaP crystal in the vacuum chamber, and simultaneous detection of the two EO output channels. We thus think that the implementation of a double EO output channel readout will permit to relatively easily adopt the phase diversity retrieval method in existing or planned EO diagnostics at FELs or other accelerators.

DISCUSSION

It is important to note that DEOS is fundamentally different from previous techniques aiming at retrieving the input field using numerical analysis of the EO data (deconvolution [START_REF] Wu | Electrooptic sampling of optical pulses and electron bunches for a compact THz-FEL source[END_REF], and holography-inspired reconstructions [START_REF] Yellampalle | Algorithm for high-resolution single-shot THz measurement using in-line spectral interferometry with chirped pulses[END_REF]). The presence of zeros in the transfer functions limits the success of reconstruction methods that do not use the phase diversity technique to relatively narrowband signals, or short analysis windows, if there is no a priori information on properties of the solution. More precisely, deconvolutions or reconstructions from a single channel is ill-posed when the signal spectrum Ω max ex- The probe laser is reflected on a Gallium Phosphide (GaP) crystal back side. The spectrum readout is performed using a KALYPSO linear array camera operating at 1.13 MHz line rate [START_REF] Gerth | Linear array detector for online diagnostics of spectral distributions at MHz repetition rates[END_REF][START_REF] Rota | Linear array detector for highrepetition rate and real-time beam diagnostics[END_REF]. Details of laser transport and KALYPSO focusing optics are not shown -see Ref. [START_REF] Steffen | Compact single-shot electro-optic detection system for THz pulses with femtosecond time resolution at MHz repetition rates[END_REF] for further details.

ceeds the frequency Ω 0 of the first null: Ω 0 = Cπ/2 or 3Cπ/2 (depending on the channel used) for the Figure 1 setup. The limit is Ω 0 = √ Cπ for the case of the more classical situation for which the [-110] axis is parallel to the terahertz field (see Section IV of Supplementary Material for the corresponding transfer functions).

In contrast, using phase diversity-based reconstruction allows in principle arbitrarily long windows to be recorded, while preserving the same resolution. Without taking into account crystal limitations, the ultimate resolution of DEOS is expected to be limited by the compressed width of the probe laser (as observed in simulations)

τ R ≈ τ L (13)
whereas previous limitations of chirped pulse electrooptic sampling were:

τ R ≈ √ τ L × τ w (14)
where τ L is the compressed duration of the probe laser pulse. τ w is the duration of the chirped pulse, which defines the temporal window of the analysis. In this case, τ R corresponds to the shortest time scale that can be recorded without deformations. The resolution of the DEOS scheme is theoretically expected be similar to the classical "scanned" EO sampling (or Time-Domain-Spectrometers), but of course our technique is single-shot. In other words, the next ultimate barriers to be lifted should be the Pockels crystal limitations (due to the unavoidable phonon absorption lines (occuring at 5.3, 8.0 and 11 THz for ZnTe, GaAs and GaP [START_REF] Wu | Comparison of the detection performance of three nonlinear crystals for the electro-optic sampling of a FEL-THz source[END_REF]), and the laser pulse duration.

The maximum recording time window of DEOS is limited technically (but not fundamentally) by the spatial resolution of the camera used for the single-shot spectrometer. Hence we believe that next work should focus on alternate spectrum recording strategies, in particular Time Stretch Dispersive Fourier Transform (TS-DFT) [START_REF] Fard | Photonic time-stretch digitizer and its extension to real-time spectroscopy and imaging[END_REF], which may be advantageous from this point of view. This would turn the present recording system to a photonic time-stretch analog-to-digital converter [START_REF] Bhushan | Time-stretched analogue-to-digital conversion[END_REF].

It is also important to note that the MRC algorithm considered here is only one possibility, and that other more complex ones should also be considered for DEOS . In particular, the MRC algorithm relies on transfer functions and is thus valid only for small phase modulations in the crystal. In conditions of high electric fields, the formal similarity with the photonic time-stretch ADC will enable back-propagation methods to be efficient [START_REF] Stigwall | Signal reconstruction by phase retrieval and optical backpropagation in phase-diverse photonic time-stretch systems[END_REF][START_REF] Gupta | Demonstration of distortion suppression in photonic time-stretch ADC using back propagation method[END_REF].

CONCLUSION

In conclusion, we present a novel conceptual framework for chirped pulse electro-optic sampling, that solves the "temporal resolution problem" open about 20 years ago for terahertz recorders. Technically, the key lies in the derivation of the Fourier-domain transfer function which -in turn -allows well-posed reconstruction algorithms to be possible. In practice, the resulting DEOS strategy opens the way to terahertz digitizers whose recording length and resolutions is only limited by the probe crystal speed or the laser uncompressed pulse duration. On the short term, one foreseen application of DEOS concerns a revisit of high repetition-rate Time-Domain Spectroscopy (TDS), up to megahertz rates, and consider monitoring irreversible physical and chemical processes. Another foressen application is totally opposite, and concerns the applications of very low repetition rate terahertz sources, for which scanning strategies are impractical. This concerns nonlinear TDS using acceleratorbased [START_REF] Hafez | Extremely efficient terahertz highharmonic generation in graphene by hot Dirac fermions[END_REF][START_REF] Di Mitri | Coherent THz Emission Enhanced by Coherent Synchrotron Radiation Wakefield[END_REF][START_REF] Pan | Photon diagnostics at the FLASH THz beamline[END_REF], as well as table-top-based high power terahertz sources [START_REF] Jolly | Spectral phase control of interfering chirped pulses for high-energy narrowband terahertz generation[END_REF]. Future work will include a systematic experimental study of the new limits set by the method in terms of ultimate resolution, time-window and repetition rate. In this respect, an important direction will consist in combining the present DEOS terahertz recording method with photonic time-stretch [START_REF] Bhushan | Time-stretched analogue-to-digital conversion[END_REF][START_REF] Roussel | Observing microscopic structures of a relativistic object using a time-stretch strategy[END_REF][START_REF] Evain | Direct observation of spatiotemporal dynamics of short electron bunches in storage rings[END_REF], as this should theoretically allow repetition rate to reach the hundred of megahertz range.

MATERIALS AND METHODS

Theoretical details

Proofs are given in the Supplementary Material, and we recall only main results here. The Pockels-induced phase-shift ∆φ in (t) is related to the terahertz electric field by ∆φ in (t) = βE(t) [see Eqns. [START_REF] Bartels | Ultrafast time-domain spectroscopy based on high-speed asynchronous optical sampling[END_REF][START_REF] Kim | High-speed terahertz time-domain spectroscopy based on electronically controlled optical sampling[END_REF]]. The factor β depends on the relative orientations of the crystal axes and on the polarizations of the laser and terahertz field.

For the phase-diversity scheme of Figure 1 and the setups considered in the main text: ∆φ in (t) = βE(t) (15) with β = πd λ n 3 0 r 41 [START_REF] Minami | Single-shot measurement of a terahertz electric-field waveform using a reflective echelon mirror[END_REF] where n 0 is the refractive index at vanishing electric field and d the thickness of the crystal. λ is the laser wavelength in vacuum and E(t) the electric field inside the crystal. Note that the value of β is two times smaller than for the usual layout used in classical balanced detection (see supplementary material).

Experimental recording system for the table-top experiment (Figure 2)

The laser pulses are delivered by an amplified Sapphire-Titanium laser (Coherent Astrella) with 7 mJ output, from which 3 mJ are extracted for this experiment. The emission and detection ZnTe crystals are 110-cut, with 1 mm thickness. The stretcher is a classical Treacy compressor. The Silicon filter (280 µm thick, at normal incidence) is destined to rejected the 800 nm laser light. The imaging spectrometer is composed of a reflection grating, a low-cost 1280x1024 pixels monochrome CMOS camera (UI 3240 ML NIR from IDS GmbH) equipped with a 60 mm objective (Nikkor 60 mm F2.8G ED). We also place a cylindrical lens with 100 mm focal length just before the 60 mm lens, in order to spread vertically the optical power onto the CMOS camera. The three spots are vertically binned at analysis stage, thus increasing the equivalent full-well capacity (and the signal-to-noise ratio).

At each shot n, the recorded image (see Figure 2b) provides three raw data: the spectra on the two polarization channels S 1n (λ) and S 2n (λ) (containing the information on the terahertz field), and the spectrum of the unmodulated laser S 0n (λ). The camera is trigged by the laser, and acquires 10 images per second. For this test we did not try to achieve 1 KHz repetition rate (i.e., the repetition rate of the laser), although this type of upgrade would be straightforward using a state-of-art commercial CMOS or CCD camera (as, e.g., in Ref. [START_REF] Tikan | Single-shot measurement of phase and amplitude by using a heterodyne time-lens system and ultrafast digital time-holography[END_REF]).

Data anaysis in the table-top experiment

Main data consist of single-shot camera images taken in presence of the electric fied. This image contains the three spectra S 1n (λ), S 2n (λ) and S 0n (λ). In addition we also record beforehand the same data in absence of electric field, which we will write S ref

Y 1n = S 1n σ 1 S 0n -1 (17)
Y 2n = S 2n σ 2 S 0n -1 (18)
where σ 1,2 =

This preprocessing strategy also allows us to suppress the effect of shot-to-shot fluctuations of the laser spectrum shape. In other words, Y 1,2n do not depend on the shape of S 0n . This can be shown easily, if we assume a fixed relation between the three spectra: S 0,1,2n (λ) = f 0,1,2 (λ)S Ln (λ), with S Ln (λ) the laser spectrum at shot n, and assume that f 0,1,2 (λ) are functions that do not change with time (i.e., they only are determined by the optics transmisions, and adjustment). This preprocessing should theoretically allow the measurements to reach the shot-noise limit.

The obtained Y 1,2n are then used in the MRC retrieval algorithm. We first apply a window in order to select the part of Y 1,2n (t) for which the laser spectrum intensity is well above the detection noise (in practice we select the part where the signal is larger than 10% of the maximum). Then:

• We apply a Fast Fourier Transform (FFT) to the windowed Y 1,2n (t) data.

• Apply the MRC formula displayed in Eq. ( 9).

• Finally apply and inverse FFT for obtaining the reconstructed data ∆φ retr in (t) and E retr in (t).

Experimental setup at EuXFEL (Figure 5)

The probe pulses are devivered by an amplified Ytterbium fiber laser operating at 1030 nm. The EO effect is achieved in a Gallium Phosphide (GaP) crystal, which is placed inside the vacuum chamber of the accelerator, near the electron beam [START_REF] Steffen | Compact single-shot electro-optic detection system for THz pulses with femtosecond time resolution at MHz repetition rates[END_REF]. The current setup permits to detect only one EO channel output at a time. The spectra are recorded in single-shot, using a grating spectrometer based on the KALYPSO fast linear array detector [START_REF] Gerth | Linear array detector for online diagnostics of spectral distributions at MHz repetition rates[END_REF][START_REF] Rota | Linear array detector for highrepetition rate and real-time beam diagnostics[END_REF] operated at a line rate of 1.13 MHz. A first series of electron bunch trains for one polarization is recorded and then the last half waveplate before the polarizing beam splitter is rotated by π/8 to record the complementary polarization. Hence, the two EO channel outputs can be used to reconstruct the individual electron bunch shapes within the burst using the phase-diversity technique. Note that a straightforward upgrade is also planned, with the aim to achieve simultaneous recording of both polarisations.

Data analysis in the EuXFEL experiment

With each burst, we record single-shot spectra on one of the two polarization directions S 1n (λ), S 2n (λ) and additionally unmodulated spectra with laser and no electron bunch S no bunch 1,2

, and the spectra without laser S dark 1,2 .

However as we do not record the unmodulated laser spectra S 0n (λ) of the same laser pulses in the XFEL experiment, we cannot compensate for the shot-to-shot fluctuations of the laser faster than about 2kHz. The data analysis is then similar to the 800 nm experiment case. We thus define the EO signals (before applying the MRC reconstruction) as:

Y 1n = S 1n -S dark 1 S no bunch 1 -S dark 1 -1 (19)
Then we apply the MRC algorithm in exactly the same way than for the previously described table-top (800 nm) experiment.

analysis has been performed by ER and SB (table-top experiments), and ER, BS, CS, and SB (FEL experiments). All authors participated to the manuscript redaction.

FIG. 1 .

 1 FIG. 1. (a)Principle of the DEOS single-shot recording method. The electric field pulse shape under interest is imprinted onto a chirped laser pulse in single-shot, using a Pockels electro-optic crystal, quarter and half-wave plates (QWP and HWP) and a polarizing beam-splitter. The first part of the optical setup (spectral encoding) is classical, except for the polarization directions and crystal axis. Here we show that from the spectra recorded on the two outputs, it is possible to retrieve without ambiguity the input electric field evolution with high resolution and abritrarily long durations, using a novel point of view, based on phase-diversity (see text). (b,c,d) Main steps of our reconstruction method (numerical simulation). (b) Raw electro-optic sampling (EO) signal corresponding to the optical spectra after background subtraction. (c) Transfer function corresponding to the two polarization outputs. (d) Comparison of the retrieved (red) and actual (black squares) input signals. The compressed laser pulse is Gaussian with 40 fs FWHM and is chirped to 10 ps. The input terahertz pulse is a Gaussian with duration 500 fs FWHM. The laser polarization, QWP, and HWP axes are oriented at 45, 0 and 22.5 degrees respectively with respect to the vertical axis.

  FIG. 2. Single-shot recording of free-propagating terahertz pulses over a window of the order of 20 ps. (a): DEOS experimental setup. ZnTe: 1 mm-thick, 110-cut Zinc Telluride crystal, HWP: Half-wave plates, QWP: Quarter-wave plate, PBS: Wollaston polarizing cube beam-splitter. The beams emerging from the PBS are in the plane perpendicular to the figure. (b) Raw camera image containing the single-shot spectra of the two polarization outputs S1,2n, and the unmodulated laser spectrum S0n. (c) and (d): EO signals on two polarizations channels (after background subtraction and normalization, see Methods).(e): Single-shot input signal retrieved from (c) and (d) using the DEOS phase-diversity-based algorithm (red). Green trace: actual input, obtained using scanned electro-optic sampling. Inset: Fourier spectra of the two terahertz signals.

FIG. 3 .

 3 FIG. 3. Fit providing the reconstruction parameter B from a single-shot recording. Dots: Fourier spectra | Ỹ1,2(Ω)| of experimental data before reconstruction. Lines: spectra | Ỹ retr 1,2 (Ω)| computed from the retrieved input. Ỹ retr 1,2(Ω) are fitted on Ỹ1,2(Ω) using B as a free parameter. Note the presence of interleaved zeros, which play a key role in the retrieval strategy. Same data and color codes as for Figure2.

FIG. 4 .

 4 FIG. 4. Electron bunch shapes recorded at the European Xray Free-Electron Laser (EuXFEL). (a) Picture from inside the 3km-long accelerator tunnel. The electro-optic sampling setup (see Fig. 5) is placed just upstream of the picture, after the first bunch compressor. (b) timing of the electron bunches in the conditions of the experiment. (c) Electrooptic signals Y1,2 of a single bunch before reconstruction. (d) Reconstructed electric field. Shaded areas: superposition of single-shot curves, color curves: average over 255 bursts. (e) Electro-optic signal of two bursts (i.e., 800 electron bunches in total). (f ) Shape of one bunch (with bunch number 200 within the burst) versus burst number. (g) Arrival time versus bunch number. Shaded areas: RMS arrival time fluctuations, color curve: average over 255 bursts. The EO data are low-pass filtered to 2.5 THz.

FIG. 5 .

 5 photodiode array detector (1.13 MHz repetition rate)

1 and S ref 2 and S ref 0 .

 20 We obtain the EO sampling signals Y 1n and Y 2n at each shot n (displayed in Fig.1b) in the following way:

Y 2n = S 2n -S dark 2 S no bunch 2 -S dark 2 - 1

 2221

ACKNOWLEDGEMENTS

This work has been supported by the following funds: Ministry of Higher Education and Research, Nord-Pas de Calais Regional Council and European Regional Development Fund (ERDF) through the Contrat de Plan État-Région (CPER photonics for society), LABEX CEMPI project (ANR-11-LABX-0 007), ME-TEOR CNRS MOMENTUM grant, ANR-DFG ULTRA-SYNC project (ANR-19-CE30-0031). This work also used the PhLAM femtosecond laser facility in 2016 and 2017. The a uthors would like to thank Marc Le Parquier and Nunzia Savoia (PhLAM) for their work on the operation of the Titanium-Sapphire laser.

CONFLICT OF INTERESTS

The authors declare that they have no conflict of interest

CONTRIBUTIONS

The PhLAM team realized the theoretical and numerical investigations: establishment of transfer function versus crystal arrangement, analysis algorithm based on MRC, and simulation codes. These investigations have been based on the knowledge established by UCLA (BJ) on transfer functions, phase diversity and MRC, in the framework of photonic time-stretch. Theoretical and numerical calculations have been performed by SB and ER. accompanies the manuscript on the Light: Sciences and Applications website (http://nature.com/lsa).


F Pantellini

G Le Chat

S Belheouane

N Meyer-Vernet

A Zaslavsky

On the detection of nano dust using spacecraft based boom antennas

Keywords: Interplanetary dust, plasma interaction with antennas, photoemission PACS: 96.50.Dj, 52.40.Fd, 79.60.Ht, 79.60.Ht

High rate sampling detectors measuring the potential difference between the main body and the boom antennas of interplanetary spacecraft have been shown to be an efficient means to detect impacts of small dust grains in the nanometer size range (see Solar Physics, 256,

INTRODUCTION

Following a nano dust grain impact on the spacecraft body a dense plasma cloud with a free charge of the order Q ∼ 3 10 -12 C [1,2] expands around the impact point (see Fig. 2a,b). For dust grains in the nanometer size domain the cloud's electrostatic field has decreased below the spacecraft's own field E SC when reaching a cm order radius. The cloud's electrons are then rapidly recollected by the spacecraft leaving a pure ion cloud pursue expansion (Fig. 2c). The recollected electrons decrease the spacecraft's body charge by -Q leading to a reduction of its potential with respect to other parts of the satellite electrically disconnected from the main body (e.g. the boom antennas). The instrumentally measured potential pulse δV SC is related to the free charge Q and the spacecraft body capacitance C SC via

δV SC = Γ Q C SC ( 1
)
where Γ is an order unity gain factor. Such impulsive variations of the spacecraft's body potential are indeed measured by the high time resolution TDS (Time Domain Sampler) instrument on STEREO which samples the potential between three mutually orthogonal boom antennas and the spacecraft body. In most cases [3], the detected voltage pulses are not of equal amplitude on the three antennas with one antenna presenting a much stronger signal as illustrated by the characteristic example of Fig. 1.

Estimates of the field intensity around the cloud using Coulomb's law for a Q point charge indicate that the field is far too small to boost the antenna's potential to the observed high levels (like the level of the x antenna in Fig. 1). On the other hand the field turns out to be large enough to substantially increase the angular momentum of a majority of photoelectrons, effectively reducing the photoelectron return current. This occurs because the antenna radius r 0 (= 1cm on STEREO) is much smaller than the typical radius of the photoelectron cloud surrounding the antenna, i.e. much smaller that the photoelectron Debye length λ ph ∼ 1m at 1 AU [2]. The smallness of the ratio r 0 /λ ph implies that most photoelectrons move on high excentricity orbits which can be easily disconnected from the antenna (Fig. 2c). The transitional interruption of the photoelectron return current induces a positive charging of the antenna and a measurable variation of the electrostatic potential difference δV a between the spacecraft and the one antenna exposed to the cloud's field.

In [3,2] it has been argued that the "critical" radius R C (see equation 2), which is a rough estimate of the maximum extension of the cloud's field has to do with the its maximum expansion before dilution in the ambient plasma. In this work we point out that R C does more pertinently represent the distance beyond which the cloud's field is efficiently screened by the ambient electrons.

STEREO-TDS OBSERVATIONS

Typical temporal voltage profiles measured by TDS on the three boom antennas after a nano dust impact is shown in Fig. 1. Dust impact profiles are characterized by a rapid rise phase on a time scale of the order of a few tens of µs [3] followed by a slow recovery phase on ms time scale. The recovery phase has been shown to be given by the discharge time RC a , where C a ≈ 60pF is the antenna capacitance and R = (dI ph /dφ a ) -1 is the effective resistance of the antenna at a potential φ a subject to a photoelectron current I ph [4]. Tens of thousands pulses of this type have been detected on STEREO. While for large grains in the µm size domain the amplitudes are approximately equal on the three antennas, smaller, nm size grains are generally characterized by a strong pulse on one antenna and two, roughly equal but much smaller pulses, on the other antennas. Example of voltage pulses detected by TDS on each of the three boom antennas on the STEREO spacecraft following the impact of a nano meter dust particle on the main body of the spacecraft. Note that the pulse is substantially stronger on the x-antenna compared to both the y and z antennas.

In the simplest imaginable scenario a dust particle impacting on the spacecraft body (Fig. 2a) generates an expanding plasma cloud as shown in Fig. 2b. Electrons from the cloud are rapidly recollected by the spacecraft's electrostatic field E SC adding a negative charge -Q to the spacecraft body (Fig. 2c). In small clouds generated by nano dust impacts, the electric field decreases below the spacecraft's field early during the expansion when the cloud's radius is smaller than 10 cm implying complete recollection of the cloud's electrons on µs time scale [2]. The accumulation of charges on the spacecraft body modifies its potential with respect to the conductively disconnected antenna (Fig. 2c). The electrostatic potential between the spacecraft body and the boom antenna suddenly increases by δV SC (cf Eq. ( 1)). In this simple scenario the potential profile should be similar on all antennas and provide a direct measure of the free charge produced by the impact. This is indeed the case for a small but significant fraction of the impacts detected on STEREO identified as µm grains impacts. However, most of the detected events are characterized by a strong peak on one antenna and two substantially smaller peaks on the other two antennas as in the example of Fig. 1. These are identified as nm grains impacts.

In appendix A of Zaslavsky et al [3] it is suggested that the strong peak occurs on the antenna which has a portion of length l within the expanding plasma cloud at the time of its largest expansion, i.e. at the time when its density has decreased to the level of the ambient density n a and its radius grown to R C :

R 3 C = 3Q 4πn a ∝ δV SC . (2)
As we shall in the next section R C should rather be viewed as the maximum extension of the electrostatic potential of the cloud due to screening rather than its maximum physical extension in terms of e.g. ion density.

For a given ambient plasma density n a , equation ( 1) provides a direct relation between the cloud radius R C and δV SC . Zaslavsky et al then make the ansatz that the peak amplitude of the potential on the antenna with the strongest signal is

δV a = Γ k B T e l L (3)
where e is the elementary charge, k B the Boltzmann constant, L the total length of the antenna and T a free parameter with the dimension of temperature. For an impact at a distance r from the base of the antenna, the length l in equation ( 3) is just l = (R 2 Cr 2 ) 1/2 (see Fig. 2). Assuming dust grains are distributed uniformly in space, the impact probability must increase linearly with r up to a maximum distance R max (of the order of the linear size of the spacecraft body) beyond which no impacts can occur. The expectation for δV a is then given by [3]

δV a =      αT R 3 C , R C ≤ R max αT R 3 C 1 -1 - R 2 max R 2 C 3/2 , R C > R max (4
) where α ≡ 2Γ/(3LR 2 max ). Equations ( 4) and ( 2) do imply the linear relation δV a ∝ T δV SC for clouds with R C ≤ R max . For larger clouds R C > R max the relation between δV a and δV SC becomes sub linear allowing for an experimental determination of R max . Direct measurements of δV a and δV SC as reported on Figure A.10 of [3] do indeed show that equation (4) does a remarkable work in fitting the measurements setting Γ = 0.5, n a = 5cm -3 (a typical value at 1 AU), R max ≈ 0.85m, and a temperature T of the order 2.2 to 2.5 eV. Quite remarkably, an excellent fit of the measurements could be obtained with only two free parameters R max and T . A physical interpretation of the empirical law (4) is presented in the next section.

PHYSICAL MECHANISM

The statistical analysis of the dust impact associated voltage pulses from STEREO-TDS shows that the pulse amplitudes do only depend on R C , or, equivalently on the free charge Q, and an ad hoc temperature T in the range 2.2 to 2.5 eV. Incidentally, at 1AU, photoelectrons around spacecraft have temperatures of this order [4] suggesting that photoelectrons are a key ingredient in any viable scenario describing the origin of the strong voltage pulse on the x-antenna on Fig. 1. The very fact that the voltage pulses are intimately related to photoelectrons is one reason for ruling out the possibility that the pulses are a direct measure of the electrostatic potential structure of the cloud itself. The other reason is that the electrostatic potential of a cloud generated by the impact of a nano dust is orders of magnitude too weak to rise the antenna's potential to the observed values [2].

Having established that the voltage pulse is due to a perturbation of the photoelectron cloud surrounding the antenna as illustrated in Fig. 2c we make a quantitative estimate of the amplitude of the voltage pulse which is expected if a portion of the antenna of length l does become disconnected from its photoelectrons (which no longer fall back onto the antenna's surface) during the pulse rise time τ. Measurements from TDS on STEREO indicate that τ is typically of the order of 10 to 80µs with a marked peak around 40 µs.

At 1AU, the charge of photoelectrons emitted during a time τ by a boom of length l, radius r 0 and oriented perpendicularly to the Sun-satellite direction is of the order Q a (l) ∼ j ph,1AU 2r 0 lτ

where j ph,1AU = 5 10 -5 A/m 2 . Thus, if as a result of the presence of the dust impact generated cloud the emitted photoelectrons are temporarily unable to fall back over a portion of the antenna a positive charge Q a (l) must accumulate on its surface. The measured potential of the antenna then rises by an amount δV a proportional to the emitted charge times the inverse of the antenna capacitance C a (≈ 60pF on STEREO)

δV a = Γ Q a (l) C a . ( 6
)
Given a typical charging time τ = 40µs we obtain Q/l = 4 10 -11 C/m from ( 5) and with Γ = 0.5 a voltage pulse of the order δV a /l = 0.32V/m. Typical clouds for nano meter sized dust grain impacts extend to R C 1m. Assuming l = 0.5m the voltage pulse expected based on ( 6) is δV a = 0.16V a value compatible with the strongest events reported in [3]. Not surprisingly, equation ( 6) shows that the ansatz (3) rests on the assumption of a rough equipartition between the average potential energy of the photoelectrons bound to a boom antenna of length l (and total charge Q) and their average thermal energy, viz. Concerning question (1). In order to appreciate how small the cloud's internal field can be, let us assume the extreme case of a cloud with all electrons stripped off by the spacecraft's field (cf Fig. 2c). The total free charge in a cloud due to a dust grain of 10 -20 kg hitting the spacecraft at 400km/s is of the order of Q = 3 10 -12 C. For a spherical cloud we can then use the Coulomb expression Q/4πε 0 R C to compute the potential difference between R C and infinity. Now, for R C = 1m the potential difference is down to a ridiculous 0.027 V, clearly insufficient to substantially alter the energy of a typical ∼ 2eV photoelectron. As we shall see, the field is nevertheless strong enough to prevent photoelectrons emitted by the antenna to be recollected by the same antenna after a ballistic flight, the standard scenario under stationary conditions (Fig. 1). The reason for most photoelectrons being emitted and recollected by the same antenna is that this is the only way to compensate for the strong current of photoelectrons emitted by the sunlit antenna, the other sources of electrons, and particularly the solar wind being unable to provide the necessary flux at 1AU. The key point in our answer to question (1) is that the cloud of photoelectrons around the boom antenna extends to meter order distances from the antenna [2] implying high excentricity orbits for most photoelectrons. Under such circumstances conservation of the angular momentum M = mrv θ (v θ being the azimuthal velocity component in cylindrical coordinates ) implies that the transverse velocity component v θ (r) = v θ (r 0 )r 0 /r is smaller by a factor 10 to 100 over most of its trajectory which spans from the antenna's surface at r 0 = 1cm out to a distance of the order of the photoelectron Debye length λ D ∼ 1m [2]. The average v θ of a photoelectron along its trajectory being a factor r 0 /λ D 1 smaller than at r = r 0 and smaller by a similar factor with respect to the average radial velocity v r , a minor energy kick from the cloud's electrostatic field may suffice to change v θ by a factor two or more, possibly enough to disconnect the trajectory from the antenna surface. The required energy kick to make a photoelectron become disconnected from the antenna can be evaluated to be of the order of the typical energy of a photoelectron (∼ 2eV) multiplied by the small ratio (r 0 /λ D ) 2 . From the above discussion we obtain (r 0 /λ D ) 2 in the range 10 -4 to 10 -2 and a minimum energy kick in the range 2 10 -4 to 2 10 -2 eV which is much less than the above estimate of 0.027 V for the cloud's field at R C .

Let us now come to question (2) and the interpretation of the critical radius R C . As already stated above, a few µs after the impact essentially all cloud electrons have been recollected by the spacecraft. At that time its radius is of the order of a couple of cm and the ambient electrons do immediately start moving towards the positive remnant. The number of ambient electrons in a sphere of radius 1.26R C < λ D is sufficient to completely screen the charge on a short time scale R C /v e , where v e = (k B T /m e ) 1/2 is the electron thermal velocity. For sub-Debye values of R C the oscillations associated with the screening are strongly Landau damped and the electric field beyond R C immediately averaged to zero. The screening process is a fast process involving the spatial scale R C and the electron thermal velocity. It is sufficiently fast to account for the short time scale of the growth phase in the potential profiles (see Fig. 1). On the other hand, the cloud expansion invoked in [3,2] is a slow process involving R C and the cloud expansion velocity which is of the order the ion thermal velocity [5]. The associated time scale may be short enough to account for the rapid growth of the potential profiles of Fig. 1 but is definitely slower than screening. In addition, if no screening is invoked there is no reason for the electric field of the cloud to be confined within R C arising the question of why only the portion of antenna within R C appears to be affected by the cloud.

Let us now address question (3) on the characteristic charging time τ in equation (5). According to the above discussion, the charging starts as soon as the cloud's electrons have been recollected by the spacecraft body, i.e. after a few µs. Charging stops as soon as the photoelectrons have reestablished a static field around the antenna. The typical time scale for this process is of the order of the inverse photoelectron plasma frequency 2π/ω ph ≈ 10µs (for a reasonable average density n ph = 100cm -3 for the photoelectron population around the antenna [2]).

CONCLUSIONS

We have presented a revised model of the dust impact associated voltage pulses detected by the TDS instrument on STEREO. In the new scenario the critical radius R C (cf equation 2) is no longer interpreted as the maximum expansion of the impact generated plasma cloud as first proposed in [3,2] but rather as the radius at which the cloud's positive charge is efficiently screened by the ambient electrons.

FIGURE 1 .

 1 FIGURE 1.Example of voltage pulses detected by TDS on each of the three boom antennas on the STEREO spacecraft following the impact of a nano meter dust particle on the main body of the spacecraft. Note that the pulse is substantially stronger on the x-antenna compared to both the y and z antennas.

  three questions remain open: (1) how do the photoelectrons become disconnected from the antenna despite the smallness of the cloud's internal electrostatic field. (2) What is the meaning of R C ? (3) What determines the charging time τ.

FIGURE 2 .

 2 FIGURE 2. Following a dust impact on the main body of a spacecraft (a) a plasma cloud made of ions and electrons expands radially from the impact point (b) at some distance r from a boom antenna. Early during the expansion, the cloud electrons are recollected by the spacecraft under the effect of the spacecraft's own electrostatic field E SC (c). The cloud's electric field is strong enough to modify the photoelectrons moving on high excentricity orbits so that they are no longer collected by the antenna. During this time the antenna increases its positive charge inducing rising its voltage by δV a with respect to the spacecraft's body.