Phase cycling of extreme ultraviolet pulse sequences generated in rare gases

The development of schemes for coherent nonlinear time-domain spectroscopy in the extreme-ultraviolet regime (XUV) has so far been impeded by experimental difficulties that arise at these short wavelengths. In this work we present a novel experimental approach, which facilitates the timing control and phase cycling of XUV pulse sequences produced by harmonic generation in rare gases. The method is demonstrated for the generation and high spectral resolution characterization of narrow-bandwidth harmonics ($\approx 14\,$eV) in argon and krypton. Our technique simultaneously provides high phase stability and a pathway-selective detection scheme for nonlinear signals - both necessary prerequisites for all types of coherent nonlinear spectroscopy.


I. INTRODUCTION
Coherent nonlinear spectroscopy in the time domain is a powerful tool to study photoinduced dynamics in complex quantum systems on their natural time scale [1]. An extension to the XUV spectral regime is highly desirable, as it would in principle foster studies with attosecond temporal resolution and site or chemical selectivity [2,3]. In coherent nonlinear spectroscopy, sequences of phase-locked ultrashort laser pulses interact with a system, simultaneously exciting many quantum pathways. Subsequently, observables are measured as a function of the time delay between the pulses, which in combination with Fourier-transform allows to observe spectral signatures. Therefore these methods require generation of phaselocked pulse sequences and highly sensitive pathway-selective detection methods [1,4]. While this is readily achieved in the visible regime, the simultaneous experimental realization of both ingredients constitutes a major challenge at XUV wavelengths.
Phase-locking in the visible regime is typically achieved using a passively or actively stabilized interferometric setup, which splits, delays and recombines the pulses. At XUV wavelengths fundamental mechanical constraints limit the phase stability of most implementations and technical difficulties (for example the lack of transmissive optics and absorption of XUV light in air) impede the experimental realization. Therefore only few studies have demonstrated phase-locked XUV pulses. Among those, at free-electron lasers (FELs) phase-locking was achieved using double-pulse seeding with phase-locked pulses [5], by manipulating the trajectory of the relativistic electron bunch used for generation of the XUV radiation [6], or by elaborate interferometric setups directly in the XUV [7]. Ultimately phase-locking of several harmonics of a FEL, has led to generation of attosecond pulses at the μJ level [8]. In the case of tabletop High Harmonic Generation (HHG), phase-locked XUV pulse-sequences were generated using phase-locked double-pulse pumping [9,10], direct ma-at different angles. The nonlinear mixing signals are then radiated in specific directions and can be isolated by spatial filtering, yielding sensitive, background-free detection. Examples are the FWM beamline at the FERMI FEL [15,16] and experiments exploiting the inherent synchronization between the NIR driving pulse and the resulting XUV pulse in HHG schemes [17,18]. However, owing to the complex geometries of the FWM setup, the phase stability is limited and the overall sensitivity is constrained by straylight since FWM is restricted to detection of photons.
In a recent study we have simultaneously implemented XUV phase-locking and pathwayselective detection using double-pulse pumping of a FEL with phase-modulated pulse sequences [19]. This enabled tracking of inner-subshell valence-shell electronic coherences spanning over 28 eV and efficient background suppression. The phase-modulation technique is a highly sensitive technique to record electronic wave-packet interferences. It was originally developed in the visible regime [20], but extensions to UV wavelengths [21][22][23] and to an XUV FEL have been demonstrated [19]. In a nutshell, the relative phase of the excitation pulses is cycled on a shot-to-shot basis, leading to distinct modulation patterns in the (non)linear response of the system, which are monitored using incoherent 'action'-signals like fluorescence [20] or photoionization yields [24]. These distinct modulations are readily demodulated using a lock-in amplifier, allowing for isolation of specific excitation pathways (i.e. pathway-selective detection). Due to the lock-in detection the method is highly sensitive. For example, it enabled isolation of dipole interactions in extremely dilute atomic vapors [25]. Furthermore, a straightforward extension allows for two-dimensional electronic spectroscopy [26,27], which is a powerful technique to follow electronic dynamics in realtime.
In this work, we combine the phase-modulation technique for the first time with harmonic generation in rare gases, yielding high phase stability and pathway selectivity in a single setup. Our results imply that the shot-to-shot phase cycling in the pump pulses is coherently transferred to the XUV pulses. By performing high-resolution linear interferometric cross correlations (CCs) of the XUV pulses, we find that the resulting XUV radiation is extremely narrowband. Calculations show that this is a consequence of enhanced phase matching on the high energy side of atomic resonances. Our method provides high phase stability and the necessary pathway-selective detection scheme to isolate the weak CC signals, which upon Fourier-transform, provides spectral information about the XUV light. In the present case, (DL), the phase φ i = Ω i t is controlled using acousto-optical modulators (AOMs) which are driven on distinct phase-locked radio frequencies Ω i . The interferometer is traced with a continuous wave reference laser. Its interference is recorded with a photodiode (PD) and is used as a reference for the lock-in amplifier. The UV pulses are compressed using a CaF prism compressor and focused into the gas cell. Residual UV light is reduced using an aperture. The intensity of the XUV light is monitored using a microchannel plate (MCP), whose output is fed into a boxcar averager and subsequently to the lock-in signal port. the high resolution of the Fourier-transform approach allows us to identify the contribution of closely spaced individual resonances to the overall spectrum.

A. Experimental setup
The experimental setup is shown in figure 1. The laser system (Amplitude Technologies) provides 24 fs pulses centered at 790-800 nm at a repetition rate of 1 kHz with pulse energies up to 6 mJ. For timing control and phase modulation of the intense femtosecond pulses, we employ a specialized monolithic setup that can handle high pulse energies at wavelengths of ≈ 266 nm, i.e. the third harmonic of the fundamental wavelength of the laser. The setup was previously introduced in Ref. [23]. The 266 nm pulses are generated by a commercial third-harmonic generation (THG) kit (Eksma Optics). The residual near-infrared (NIR) and second harmonic light are removed using dichroic mirrors. Wavelength tuning of the resulting ultraviolet (UV) pulses between 264 -268 nm is achieved by tuning of the fundamental NIR wavelength and simultaneously adjusting the phase-matching angles of the THG crystals.
In the phase-modulation setup (see figure 1) the UV beam is split by a 50/50 beamsplitter and the relative phase between both arms is modulated with acousto-optical modulators (AOMs). The delay τ between the two pulses is controlled by a pair of fused-silica wedges.
The position of one of the wedges is motorized, which allows to scan the delay from -1 ps to 11 ps. Unless otherwise stated, the pulses are recombined in a collinear fashion. Note that the beam path in the interferometer is not actively stabilized, in contrast to other double-pulse pumping studies [28]. The total path-length in the bulk of the optics of the interferometer (AOMs, beamsplitters, wedges) is 27 mm and therefore material dispersion temporally stretches the UV pulses. The setup is designed in a way that dispersion in both arms is balanced at τ = 0 fs. A prism-compressor is used to recompress the UV pulses to their transform-limited duration of ≈ 50 fs, which is larger than the duration of the NIR pulses due to phase-matching bandwidth limitations in the THG setup.
The compressed UV pulses are focused into a gas cell using an f = 200 mm CaF lens mounted in vacuum. Due to the onset of filamentation in the bulk material of the optics the energy per pump pulse is limited to ≈ 50 μJ. The gas cell used to generate the XUV light has a length of 6 mm and the focus is placed right at the entrance of the cell. The third harmonic (≈14 eV) of the 266 nm light is generated either in argon or krypton. The intensity of the XUV pulses is monitored with a microchannel plate (MCP) whose boxcar-averaged output iss fed into the lock-in amplifier.

B. Phase modulation scheme
The signals are detected using a phase modulation technique in combination with lock-in detection, which is described in detail in Refs. [20,22]. We outline the principle here only briefly. A detailed calculation of the signals can be found in the Supplementary Information.
The AOMs are driven at distinct radio frequencies in phase-locked mode (Ω 1 = 160 MHz, Ω 2 = Ω 1 +110 Hz). Bragg-diffraction in the AOM crystal shifts the optical frequency ω by the AOM frequency: ω → ω + Ω i . Hence, the interference of photons from both interferometer arms exhibits a low-frequency beating at a frequency of Ω 21 = Ω 2 − Ω 1 = 110 Hz (i.e. linear shot-to-shot phase cycling). To track the relative phase of the two interferometer arms a second continuous wave (CW) UV laser beam (Crylas FQCW 266) co-propagates with UV pulses through the interferometer. We record the interference after the interferometer with a photodiode and the intensity beating of the CW light severs as a reference signal: For the UV fs pulses traveling through the interferometer, the beating is sampled by the repetition rate of the fs laser (ν rep = 1 kHz Ω 21 ). Upon third-harmonic generation by the UV pulses in the gas cell, the frequency of the fs pulses is up-converted, leading to an up-conversion of the beat frequency between the XUV pulses to 3Ω 21 , respectively. We record the interfereometric CC of these modulated XUV pulse pairs: Here, W i denotes the pulse energy and A i the spectral amplitude of the XUV pulses (i = 1, 2).
This detection scheme has several advantages. The retrieved signal S oscillates at a reduced frequency of ω = ω − 3ω 0 with respect to the pulse delay τ (rotating frame detection) [29]. Accordingly, signal frequencies are reduced in our experiment by a factor of ω/ω ≤ 120, which permits sampling with much larger delay increments and thus reduces data acquisition times. Furthermore, phase noise from the interferometer appears correlated in the reference and fs laser signals. In the lock-in detection, the phase difference between the third-harmonic reference and the CC signal is computed upon which correlated phase noise cancels out to a large extent. This has the effect of efficient passive phase stabilization of the interferometer. In addition, the lock-in amplifier acts as a very steep bandpass filter combined with high amplification, extracting only signals modulated at a frequency of 3Ω 21 .
This efficiently suppresses signals e.g. from fundamental light or non-interference signals, providing a background-free output of XUV interference signals.

A. Phase modulation of XUV pulses
Third harmonic XUV pulses are generated in argon below the ionization threshold at a pressure of 89 mbar in the gas cell. The UV pulse energy is 12 μJ per pulse and the spectrum is centered at 264.8 nm. Note that under these conditions no higher-order harmonics (> 3) are observed. Linear interferometric CCs of the resulting XUV pulses, are recorded by scanning the delay between the two UV pump pulses (see figure 1). Consequently, the Fourier transform of the CC trace yields the XUV spectrum. The narrow spectrum can be explained by enhanced phase matching occurring on the high-energy side of an atomic resonance [30]. In a simple model, the phase mismatch between the generated harmonic and the laser-induced polarization is given by ∆k = ∆k at + ∆k g , where ∆k at is the phase mismatch of the neutral atomic medium and ∆k g is the Gouy-phase shift: Here n XUV and n UV are the refractive indexes of the neutral generation medium in the XUV and the UV, respectively, E is the photon energy, h is the Planck constant, c is the speed of light, q is the order of the harmonic, z the position along the propagation axis of the laser relative to the focus position and z R is the Rayleigh-range in the focus. The pressure-dependent refractive index can be calculated with Sellmeier's equation [30], using the state energies and transition dipole moments reported in Refs. [31,32]. If {E j } are the resonances of the medium, then for E > E j , ∆k at can be negative and therefore compensates the phase mismatch induced by the Gouy-phase shift. The spectrum of the generated XUV light is determined by [30] P XUV (E) where χ (3) is the third-order nonlinear susceptibility, P UV is the power of the fundamental and b = 2πE 3hc w 0 is the confocal parameter of the UV laser, determined by the focus width 2w 0 .
In figure 3 (a) we show spectra of the generated XUV pulses acquired for different pressures in the gas cell. We observe a blue-shift and a narrowing of the spectrum with increasing pressure. This is consistent with calculations of the atomic phase-mismatch ∆k and the resulting XUV spectrum, which are shown in figure 3 (b) and (c). For the calculations, we assume a flat χ (3) between the resonances, where the generated XUV light appears. Furthermore, the Rayleigh range is used as a parameter in the calculations. It coincides with the Rayleigh range which can be determined from the experimental parameters. From the calculations one can see that phase-matching shifts to higher energies when the pressure increases. In addition, the spectral width of the phase-matching region depends on the gradient of −∆k at within the Rayleigh range. The gradient increases with increasing pressure, hence the spectral width of the XUV decreases. The experimental data and the calculations agree very well, which is remarkable considering the high spectral resolution, exceeding other studies on resonance-enhanced phase matching by a factor of > 60 [33].
The measurement was repeated using krypton as the generation medium. The UV pulse energy was set to 20 μJ per pulse and the spectrum was centered around 268.1 nm. In contrast to the argon experiment not only two resonances but several resonances of a Rydbergseries, converging to the 4p 5 ( 2 P 3/2 ) limit, occurr within the bandwidth of the UV laser.
Consequently, phase-matching is achieved at several frequencies between the neighboring resonances, resulting in a complex spectral structure [34] (see figure 4). Similar to the argon experiment a blue-shift of the peak positions and a spectral narrowing is observed when (n+2)s 3/2 4p 5 ( 2 P 3/2 ) nd 1/2 nd 3/2 69 mbar 160 mbar increasing the pressure in the gas cell. By scanning the delay from 0 ps to 9 ps a resolution of 0.5 meV FWHM is obtained by the Fourier-transform.

B. Double pump-pulse effects
So far, we analyzed the data for pulse delays larger than the pulse duration of the UV driving pulses. This was sufficient to analyze the spectral shape of the generated narrowband XUV pulses. We now turn our focus on effects occurring when the two driving UV pulses overlap temporally and spatially in the harmonic generation medium.
The perspective of generating phase-locked pulse pairs with collinear pulse sequences is appealing due to the simplicity of such setups. However, a shortcoming is the issue that the perturbation of the harmonic generation medium by the leading pulse (optical excitation and ionization of the medium), may change the amplitude and phase of the XUV radiation generated by the trailing pulse. This issue plays also an important role in HHG at high repetition rates > 10 MHz [35]. Previous studies investigated this effect by analyzing the contrast of interference fringes detected with XUV spectrometers [36,37]. Here, we study this effect in the time domain for the case of phase-matching enhanced harmonic generation near atomic resonances. As shown in the Supplementary Information, our CC measurements provide direct access to the relative phase between the XUV pulses without the need of theoretical assumptions. Yet, information about the individual amplitude of the pulses cannot be obtained due to the intrinsic symmetry of CC measurements (cf. Supplementary Information). We therefore concentrate on the relative phase shift induced by the perturbation of the harmonic generation medium.
To this end, we systematically attenuate the pulse energy of one of the two UV pumppulses (E var = 2 μ-12 μJ) while keeping the energy of the other pulse fixed (E fix = 12 μJ) and record interferometric CCs (figure 5). The power is adjusted by reducing the RF driving power of one of the AOMs. All other XUV generation parameters are the same as for the argon study at a gas cell pressure of 89 mbar. At positive delays, the pulse with fixed energy interacts with the harmonic generation medium first. Hence, for positive delays, the amount of perturbation induced by the leading pulse is the same for all measurements and a constant phase shift between the XUV pulses is expected. This is in agreement with our data, where all CC interferograms are in phase for positive delays ( figure 5(a, c). Note, that in these considerations, phase shifts introduced by the trailing pulse itself when changing its intensity can be neglected. In contrast, for negative delays, where the variable-energy pulse comes first, the perturbation of the generation medium by the leading pulse changes between the measurements. Therefore the second pulse experiences different conditions, leading to systematic phase-shifts depending on E var , as observed in our data for negative delays [ figure 5 (a, c)] These results indicate that the influence of the first pulse on the phase of the second pulse is non-negligible within our experimental conditions. Note that the different temporal position of the phase jump around zero delay stems from different discretization of the delay axis in the measurements, and that the magnitude of the phase jump is measured modulo 2π.
Furthermore, the phase scale in figure 5 (c) has an offset-calibration uncertainty due to the phase-transfer function of the electronics (for example photo-diodes, amplifiers) used in the signal and reference pathways. We did not perform a phase calibration in this study, however with a straightforward calibration also the absolute phase scale could be retrieved [25].
The influence of the leading pulse on the harmonic generation from the trailing pulse can be omitted when the harmonic generation takes place in two spatially separated volumes of the generation medium, as previously demonstrated [10,[38][39][40]. However, this leads to a reduced spatial overlap of both beams in the far field, making it more challenging to achieve good signal-to-noise performance. Here, the phase modulation scheme is of advantage in maintaining the signal interference contrast as it suppresses non-interfering signal contribu-tions. In figure 5 (b) results are shown that were obtained under conditions where the foci are separated by ≈ 100 μm and both pulses have equal energy. The separation of the two foci is realized by slightly misaligning the interferometer. In this case no phase shifts occur in the interferogram [see figure 5 (c)], when going from negative to positive delays. The lower signal quality in this data originates mainly from the reduced signal amplitude due to the reduced spatial overlap of the beams.

IV. DISCUSSION AND CONCLUSION
We successfully generated phase-locked XUV pulse-pairs and gained independent control over the delay and relative phase of the XUV pulses through manipulation of the pump pulses. This novel level of control was reached by implementing a phase-cycling scheme for the XUV pulses, which allowed for pathway-selective detection of individual signal contributions and rotating frame detection, efficiently reducing data acquisition times. Here, the pathway-selective detection selects a single harmonic and isolates the interference signals from non-modulated background signals. This is especially beneficial for photoion/-electron spectroscopy in the XUV-regime, where many background events due to one-photon XUV ionization are present.
In addition, due to the phase modulation technique the majority of the phase jitter originating from the interferometric pump-pulse setup is efficiently removed in the lock-in demodulation process, yielding a phase-stability of 4.5 • over 30 min. Note that this stabilization method is passive and therefore requires no active stabilization of the interferometer optics.
Due to the high resolution of the Fourier-transform approach (0.5 meV FWHM) we were able to measure gas-pressure dependent spectral widths and shifts of narrow-band XUV light generated in argon and krypton. This is a much higher resolution than achieved in typical XUV Fourier transform spectroscopy studies. Our data is in good agreement with a theoretical model, showing that the narrow-bandwidth generation is a consequence of enhanced phase-matching at the high-energy side of atomic resonances in the generation medium. The high spectral resolution, resolving peak positions, widths and shapes may foster and challenge detailed theoretical studies including quantum models. In the Fouriertransform approach, the resolution limit is given by the inter-pulse delay range. With XUV Ramsey-comb spectroscopy even higher resolution in the MHz range may be achieved [41].
By looking at pump-intensity dependent phase-shifts in the CC measurements it was found that in our experimental conditions the XUV generation from the individual UV pulses was not independent and leads to phase-shifts in the XUV pulse generated by the trailing UV pulse. These intensity dependent phase-shifts were directly measured by our all-optical approach. Furthermore, we could confirm that perturbations in the harmonic generation medium by leading pulses can be omitted using spatially well-separated foci for both driving pulses [37].
Our results significantly extend and improve existing XUV time-domain spectroscopy schemes [9,42] by uniquely combining pathway-selective detection with high phase-stability and high spectral resolution. As such, this work may pave the way for many applications of (non)linear coherent time-domain spectroscopy in the XUV spectral range. A straightforward extension to higher harmonics will provide access to higher photon energies and shorter pulse durations, and in combination with additional probe pulses, dynamics may be probed with high spectro-temporal resolution.

ACKNOWLEDGMENTS
We gratefully acknowledge the support of Ahmet Akin Uenal and Evgenii Ikonnikov in preparing the laser system and setting up the XUV-beamline.