Coherent nonlinear spectroscopy in the time domain is a powerful tool for studying the dynamics of complex systems. It is routinely used from the terahertz (THz) to the ultraviolet (UV) regime [1]. Extension of such techniques to the extreme UV (XUV) and x-ray region is on the frontier of time-resolved spectroscopy [2]. This will provide new possibilities to study energy and charge migration in systems of vast complexity with unprecedented spatial, temporal, and energetic resolution [3,4].

In order to observe coherences, it is essential to achieve interferometric stability on the order of a fraction of the laser wavelength (typically $\lambda /50$) [5], a fundamental problem with short wavelengths. Moreover, isolating notoriously weak nonlinear contributions from the overall signal requires high sensitivity. Several methods have been developed to satisfy these requirements, most of which can be classified as either phase-matching or phase-cycling techniques [1]. Phase-matching techniques rely on non-collinear coherent wave mixing, and have been recently demonstrated with XUV photons [6,7]. However, phase-matching approaches are constrained by limited phase stability, the lack of efficient scattered light suppression, and insufficient sensitivity for low-density ultra-high vacuum (UHV) samples. They also require modifications of the XUV beam paths when changing the signal-selection protocols.

In phase cycling, coherent nonlinear signals are isolated based on imprinted phase signatures [8–10]. This solves the mentioned issues in phase matching and is of great advantage to XUV applications. The phase modulation (PM) method, which can be regarded as a shot-to-shot phase-cycling technique, is one particularly sensitive variant [11–14]. Here, acousto-optical PM is combined with lock-in detection, which efficiently relaxes demands on interferometric stability and improves detection sensitivity. Its combination with highly selective “action” signal-detection in UHV environments has been demonstrated [13,15], and diverse signal-selection protocols have been established [12,16,17]. However, phase-cycling/modulation concepts have not yet been established for XUV light due to the lack of suitable optics.

An effective solution is to perform the PM on the fundamental wavelength prior to harmonic generation, which has been demonstrated with low harmonics [18,19]. High harmonics reaching down to the XUV regime can be generated in seeded free-electron lasers (FELs) using high-gain harmonic generation (HGHG) or by table-top high-harmonic generation (HHG) in noble gases. Hereby, the phase of the driving laser is imprinted on the XUV pulse [20,21]. This facilitates interferometric spectroscopy techniques using phase-locked XUV pulses generated by phase-locked driving pulses for HGHG [22] and HHG [23]. These are necessary prerequisites for implementing PM of XUV pulses through manipulation of the driving laser.

In this Letter, we introduce a compact PM setup specifically designed for driving HGHG processes to facilitate nonlinear spectroscopy at XUV wavelengths. The setup is optimized for deep UV wavelengths (250–270 nm), and its unique characteristics comprise high phase stability ($<\lambda /200$ at 268 nm), large linear intensity regime (tested up to 300 μJ pulse energy), minimized losses (65%), and high pointing stability (4.3 μrad).

A schematic of the experimental setup is shown in Fig. 1. It is designed to probe the quantum interference of electronic wave packets (WPs) [11,15], in our case for the $3s\to 6p$ transition (268 nm) in sodium (Na) (Fig. 3). A femtosecond pulse pair with controllable delay $\tau$ is generated in a Mach–Zehnder interferometer (MZI) to excite the electronic WPs, whose quantum interference is recorded as a function of $\tau$. Two acousto-optical modulators (AOMs) are placed in the MZI, each driven on a distinct continuous-wave (cw) radio frequency (RF) in phase-locked mode. This leads to a low frequency beat of the relative phase between pump and probe pulses at the AOM difference frequency ($\mathrm{\Omega }\sim \mathrm{kHz}$). This beat appears as an independent shot-to-shot modulation in the quantum interference signal. Simultaneously, phase/timing jitter of the MZI is traced with a cw reference laser (RL). Using the RL for demodulation of the $\mathrm{\Omega }$-beat note with a lock-in amplifier (LIA) leads to cancellation of the jitter and thus to an effective passive stabilization of the MZI [11]. At the same time, the WP interference signal is recorded in the rotating frame of the reference’s optical frequency, leading to a downsampling of the quantum interferences.

 

Fig. 1. Experimental setup: Ti:sapphire amplifier and subsequent third-harmonic generation (THG), 266 nm continuous wave reference laser, beam splitter (BS), acousto-optical modulators (AOMs), water cooling and vibration damping unit (DU), high-power phase-locked radio frequency (RF) driver, optical delay line (DL), piezo-controlled mirror (PM), photodiode (PD), 266 nm notch filter (NF), lock-in amplifier (LIA), ultra-high vacuum (UHV) chamber, and sodium (Na) reservoir producing an effusive atomic beam.

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As previously demonstrated [18], harmonic generation of the modulated laser pulses results in beat notes at harmonic frequencies $n\mathrm{\Omega }$, which are readily demodulated with harmonic lock-in detection. Hereby, the advantages of phase stabilization, rotating frame sampling, and high lock-in detection sensitivity are transferred onto the harmonic wavelengths, while all pulse manipulations are still performed on the fundamental wavelength. Combining this approach with HGHG would establish versatile phase-cycling schemes in the XUV range. However, it is not straight-forward to design a PM setup meeting the requirements for such applications, due mainly to the required high laser intensities, which are in a regime where most optics, in particular common phase shaping optics/devices, produce compromising nonlinear effects, e.g., two-photon absorption, self-phase modulation (SPM), and parametric fluorescence. To solve these issues, we have designed a specialized compact, monolithic MZI platform (Fig. 2).

 

Fig. 2. CAD drawing of the monolithic platform. Labels correspond to the ones in Fig. 1.

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For highly passive mechanical stability, the platform is milled out of ACP5080R aluminum alloy. All optics are UV-grade crystal or fused silica and are glued into their holders or directly to the platform body. For high beam-pointing stability, a transmissive delay line is used (pair of fused silica wedges, apex angle: 7°). Moving one wedge with a motorized linear stage provides a $\tau$-scan range of 10.5 ps at minimum increments of 20 as. A piezo-controlled mirror in one of the MZI arms enables remote fine-tuning of the pump–probe beam spatial overlap, which is important for the typically long beam paths (tens of meters) in HGHG setups. To reduce the laser intensity in the setup, exclusively optics with large apertures are used, providing an effective overall aperture of the setup of 10 mm. Moreover, focusing into the AOMs is omitted, which is in contrast to previous PM setups [11,15,19,24], but requires AOMs with large active apertures while providing high diffraction efficiency. To this end, custom-made AOMs are used (crystal quartz, length: 13 mm, active aperture: $100{\mathrm{mm}}^2$). These are driven by 20 W of RF power (home built phase-locked dual-channel driver), yielding a diffraction efficiency of 88%. Water cooling provides heat dissipation of the piezo transducers. Vibrations are damped with a flow damping system. Beam splitters (50/50 ratio) have a thickness of 6 mm, providing a compromise between surface flatness and induced dispersion. Their combination with the delay wedges leads to equal dispersion in both arms at $\tau =0\mathrm{fs}$. The total amount of glass passed by both beams is 27 mm. Resulting chirp can be compensated by grating/prism compressors or by chirped mirrors. The latter has achieved pulse durations of only 10 fs at the sample position in a PM setup using oscillator pulses at 800 nm [19].

For simplicity, linear WP interference measurements using the $3s\to 6p$ transition in Na serve as a model system to characterize the performance of the platform. However, an extension to nonlinear spectroscopy schemes is straight-forward [12,13]. The transition dipole moment for this transition is $3.54\times {10}^{-31}\mathrm{Cm}$, being a factor 60 lower than for the D line transition. An effusive atom beam is created by evaporating Na out of a heated reservoir into UHV and skimming it with apertures, generating an atom density on the order of ${10}^8{\mathrm{cm}}^{-3}$ in the interaction region. An amplified Ti:sapphire laser system operating at a variable repetition rate of $\le 5\mathrm{kHz}$ delivers 100 fs pulses with energies of 400 μJ at 804 nm. The output is frequency tripled with a set of nonlinear crystals (conversion efficiency $\sim 12\%$). After PM, the pulses are focused into the detector chamber ($f=500\mathrm{mm}$). The pulse intensity at the interaction region is $\sim 80{\mathrm{GW}/\mathrm{cm}}^2$. An ion time-of-flight mass spectrometer is used for detection of photoions, generated by absorption of a second UV photon from the probe pulse [Fig. 3(a)]. The integral photoion yield is used for lock-in detection; however, also mass gating using a boxcar integrator can be applied (not shown). For testing the performance of the platform with high-energy pulses, we used the spare seed laser system of the FERMI FEL, i.e., a frequency-tripled Ti:sapphire laser system delivering 100 fs deep UV pulses with energies up to 300 μJ at a repetition rate of 50 Hz.

 

Fig. 3. Downshifted WP oscillations in sodium. (a) Relevant levels and interactions with laser pulses. (b) Real part of phase-modulated pump–probe transient for a laser repetition rate of 50 Hz. For better visibility, only the first 4 ps are shown. The transient exaggeration around time zero is due to the intensity interference of overlapping pump and probe pulse. (c) Fourier-transform of the whole scan range (10.5 ps). Dashed lines indicate the transition wavenumbers.

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The RL tracing the interferometer provides up to 10 mW of 266 nm cw radiation. Its absolute frequency stability was measured to be within 100 MHz peak to peak over 12 h. The RL traces the MZI spatially overlapped with the femtosecond laser. At one exit port of the MZI, the femtosecond laser is blocked by a steep notch filter and the interference of the RL is recorded with a photodiode, serving as an electronic reference for lock-in demodulation. Due to detection in the rotating frame of this reference, the Na quantum interference frequencies are downshifted by a factor of 108, relaxing demands on phase stability and reducing acquisition times accordingly. The phase offset of the reference signal relative to the photoion quantum interference signal is calibrated as described in Ref. [11].

Figure 3 shows the data of a 10.5 ps delay scan using a delay step size of 12 fs, laser repetition rate of 50 Hz, modulation frequency of $\mathrm{\Omega }=3.7\mathrm{kHz}$, and averaging over 150 shots at each delay step. The low repetition rate is used to demonstrate the compatibility with typical seeding rates at FELs. The much higher modulation frequency is facilitated by synchronous undersampling [25] yielding high signal-to-noise ratios (SNRs).

Our data show well-resolved WP interference oscillations [Fig. 3(b)]. Note that these reflect the time evolution of electronic coherences between excited and ground states, which is in contrast to the low-frequency quantum beats between different excited (spin-orbit) states [26]. A Fourier transform yields the absorption spectrum [Fig. 3(c)]. Due to the limited scan range the fine structure splitting is not resolved. However, in a modified setup allowing for longer scan ranges (330 ps), this was well resolved (not shown). The $\mathrm{SNR}=42$ (determined by the same procedure as in Ref. [25]) is a remarkably high value considering the low sample density, small transition dipole moment, large amount of Rayleigh scattering, and high phase stability requirements at UV wavelengths. This demonstrates the high sensitivity and universal applicability of the setup to short wavelengths and low laser repetition rates.

For better comparison, Fig. 4 shows a delay scan of 200 fs length using our setup with (red) and without (blue) the PM technique. In the latter case, the $\tau$-increments are reduced to 120 as to ensure proper sampling of the 880 as period of the WP oscillations. In both runs, 2500 shots were averaged at a laser repetition rate of 5 kHz. Hence, the PM approach reduces acquisition time by two orders of magnitude due to the larger $\tau$-increment. This is of particular advantage if many adjustments and optimizations have to be carried out in between delay increments, as it is the case in HGHG experiments. At the same time, a clear advantage of PM versus conventional quantum interference spectroscopy in terms of SNR and resolution is observable in Fig. 4(b). In the PM case, a sharp spectral peak is obtained, whose width is determined by the delay range. In the conventional case, only qualitative agreement is achieved, as the resolution is limited by noise.

 

Fig. 4. Comparison of conventional (blue) with phase-modulated (red) quantum interference spectroscopy. (a) Time-domain data and (b) Fourier-transform. The dashed line indicates the $3S\to 6P$ transition in Na. Amplitudes are scaled to unity.

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For further characterization of the PM setup, its phase stability was evaluated (Fig. 5). A fixed value of $\tau =1\mathrm{ps}$ was used, and the signal phase returned by the LIA was tracked for 14 h. A phase stability of 1.7° (RMS) was achieved (blue trace), corresponding to a stability of the demodulated signal better than $<\lambda /200$. Throughout a delay scan, the amount of glass in one MZI arm changes, altering the amount of SPM with respect to the other arm, possibly impairing phase stability. However, considering the delay range ($\le 750\mathrm{fs}$) and seed laser parameters ($\le 100\mathrm{\mu J}$, 70 fs) typically used with HGHG, this is negligible. Rather than from optical phase fluctuations, we assume a fair portion of the phase noise originating from other noise sources (e.g., shot noise, electronics). Thus, for the specific experiment performed in the current work, the chosen signal integration time, and corresponding signal levels, an anticipated stability of $\lambda /50$ would be achieved down to $\lambda =10\mathrm{nm}$.

 

Fig. 5. Phase stability of the setup. Blue: RL and fs laser overlap spatially in the AOMs. Red: results with spatially separated beams. Insets sketch a heat map of the AOM crystal, inferred from a thermal imager. RF transducer (heat source) is indicated on the left by a black bar; the white circle indicates the fs laser and the black one the RL.

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As an important point, phase stability crucially depends on the spatial overlap between the RL and the fs laser in the AOM crystal. Initially, we co-propagated both beams vertically displaced through the MZI to simplify beam separation afterwards. In this case, each beam exhibits a different refractive index in the crystal material due to a temperature gradient along the crystal (inset in Fig. 5), leading to a linear phase drift in the demodulated signal (red trace in Fig. 5). This is particularly significant in the case of long acquisition times.

Moreover, in the well-overlapped case, short-term fluctuations are smaller than in the spatially displaced case. This can be related to optical density fluctuations in the MZI, resulting in enhanced phase noise at larger displacement of the beams. The increase in high-frequency phase noise over time in the blue trace can be assigned to a continuous drop in laser power in the interaction region during the measurement.

Besides the minimized phase fluctuations, a small relative pointing instability between pump and probe beams is important. Without active beam stabilization, we determined pointing differences between the two MZI arms of $<4.3\mathrm{\mu rad}$ peak to peak over 20 h, measured with a beam camera placed behind one output of the MZI.

To drive highly nonlinear processes with high laser intensities, losses in the PM setup should be minimized. We determined an overall efficiency of the PM platform of 35%. Likewise, it is important to avoid nonlinear effects in the pulse manipulation setup prior to harmonic generation, as these may compromise the experiment. Most common nonlinear effects at UV wavelengths are two-photon absorptions, leading to a reduced efficiency of the setup and SPM, resulting in spectral changes in the transmitted pulses. Both parameters showed a constant behavior for a variation of the pulse energy between 10 μJ and 300 μJ (not shown). For comparison, commercial UV acousto-optical pulse shapers reach efficiencies of 20% and are limited to pulse energies $\le 10\mathrm{\mu J}$.

In summary, we presented a compact and readily implementable platform, able to perform PM of laser pulses suitable for seeding of FELs. In WP interference measurements, excellent phase stability of $<\lambda /200$ at 268 nm over 14 h was demonstrated, along with rotating frame sampling, reducing acquisition times by two orders of magnitude, and providing high detection sensitivity in ion time-of-flight measurements of dilute gas-phase samples. Our approach combines several concepts previously demonstrated [12,18,25] that allow to transfer these features to XUV light. The platform’s large linear intensity range, high damage threshold, high efficiency, and excellent pointing stability ensure stable conditions for phase-modulated double-pulse seeding at FELs. To the best of our knowledge, no other phase shaping approach/setup has been reported unifying all these features. The flexible selection protocols of the PM technique will facilitate a multitude of coherent nonlinear spectroscopy applications in the XUV, e.g., multiple quantum coherence detection [12,17] or multidimensional spectroscopy [13,16].