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Abstract
We present the design and performance of an active stabilization system for attosecond pump-probe setups based on a Mach-Zehnder interferometer configuration. The system employs a CW laser propagating coaxially with the pump and probe beams in the interferometer. The stabilization is achieved with a standalone feedback controller that adjusts the length of one of its arms to maintain a constant relative phase between the CW beams. With this system, the time delay between the pump and probe beams is stabilized within 10 as rms over several hours. The system is easy to operate and only requires a few minutes to set up before any pump/probe measurements.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recently, the advent of extreme-ultraviolet (EUV) light pulses in the attosecond time scale has opened up new avenues for experimentalists to probe temporal aspects of electron transitions in atoms, molecules and complex systems with unprecedented precision [1–7]. In general, the temporal characterization of a given electronic process is measured by exposing the target to both an attosecond pump pulse ($\sim$100 as) and a phase-locked optical IR probe field, with a variable time delay between the two. To fully exploit the temporal resolution of attosecond pulses for time-resolved study using such a pump-probe approach, the time delay between the pump and probe pulses must be controlled with an attosecond resolution as well. This requires the ability to linearly vary the delay with time steps of the order of the pulse duration (or less), and, more importantly, to maintain it to any desired value over extended periods of time.
The major difficulty to achieve such temporal resolution comes from the fact that the spatial extent of an attosecond pulse is of the order of a few tens of nanometers ($\sim$30 nm for a 100-as pulse). As a consequence, any nanometer-scale fluctuations of the path length difference between the paths followed by the pump and probe beams inevitably lead to a timing jitter of the order of magnitude of the pulse duration itself. Such nanometer-scale fluctuations are generally inherent in any experimental setup due to beam pointing instabilities, thermal expansion of the optical table, or vibrations of the mechanical structure of the optical system resulting from the surrounding environment (air flow, vacuum pumps, chillers $\ldots$). The timing jitter can be reduced down to a few attoseconds by using an experimental setup with collinear geometry where both the attosecond pump and IR probe pulses propagate along the same path up to the target [8,9]. Nevertheless, such a geometry prevents to independently manipulate the temporal and spatial profiles of the IR fields producing the attosecond pump pulse and serving as a probe pulse, respectively. Due to the increasing number of techniques that use a temporally-shaped driving field for the attosecond pulse generation [10,11] and/or the necessity to tune the properties of the IR probe for a particular time-resolved experiment [12], Mach-Zehnder interferometers, where the two pulses propagate along different paths, emerge as a more popular scheme in the community even though they are generally less stable [13–15].
To improve their stability, a few active stabilization systems have been already proposed [15–18]. They are generally based on a CW laser propagating in both arms of the interferometer. The stabilization is achieved with an active feedback system that adjusts the length of one arm of the interferometer to maintain a constant relative phase between the two waves. The implementation of such a relatively simple approach to attosecond pump/probe setups is not straightforward, though. Indeed, a metallic foil is generally placed in the attosecond beam path to filter out the attosecond pulses from their driving IR field, which also prevents the propagation of the CW stabilizing beam. To circumvent this issue, one practical approach consists in propagating the CW laser in a spatially separate path adjacent to that followed by the attosecond and IR beams. Several designs for setting up the separate path have been recently proposed, which have demonstrated a long-term stability of the interferometer as low as 50 as [15,18]. Nevertheless, despite its versatility, such an approach only allows for a partial stabilization of the interferometer as the optics forming the separate path occasion an additional drift to that of the main path.
The coaxial propagation of the CW laser and the attosecond beam in the interferometer can be achieved by using a small-diameter metallic filter (2-3 mm) that allows a portion of the CW to pass around it [16,17]. In such configuration, the CW beams traveling along with the attosecond and the IR beams exit the interferometer non-collinearly, though, due to the inherent wedge-shape of the recombination mirror. A position sensitive detector (CCD or CMOS camera) is then required to capture the interference pattern where the two beams cross. The relative phase between the two beams are deduced from the shifts of the interference fringes using Fourier-transform interferometry [19]. This elegant approach is relatively easy to implement provided that interference fringes of good contrast are formed. However, the use of a position sensitive detector puts a limit on the maximum feedback frequency achievable (typically below 30 Hz), which is defined by the exposure time of the camera plus the computational time required to extract the relative phase from the imported 2D image. In addition, the maximum achievable stability is limited by the level of accuracy in the retrieval of the relative phase, which in turn depends on the spatial resolution of the camera and/or the number and the quality of the interference fringes, for example.
In this work, we present the design of a novel stabilization system, which bears some similarity with the previously described system. It is also based on a CW laser propagating coaxially with the pump and probe beams in the interferometer. Instead of relying on the use of a position sensitive detector, though, the method employs an optical system to align coaxially the CW beams in order to produce an on-axis interference pattern that can be captured with a fast photodiode. Such a design presents several advantages over the previous method. First, it offers a larger stabilization bandwidth, which is mainly limited by the resonance frequency of the piezoelectric transducer used to adjust the interferometer path length. And, more importantly, it allows for a higher stability as the relative phase between the two CW waves is directly measured from the photodiode signal without the need of any retrieval procedure. With such a design, we report a stability of the order of 10-as rms and a feedback frequency close to 300 Hz.
2. Experimental setup
A schematic view of our experimental setup is represented in Fig. 1. It combines a femtosecond laser system and an attosecond XUV pump-IR probe beamline designed for both photoelectron and transient absorption spectroscopy measurements. The laser system is based on a commercial Ti:Sapphire regenerative amplifier operating at 10 kHz. It delivers linearly polarized 35-fs pulses with a central wavelength at 800 nm and a pulse energy of 1.5 mJ. The pulse duration can be further compressed with a gas-filled hollow-fiber compressor system which delivers 8-fs pulses with an energy around 600 $\mu$J.
Fig. 1. Schematic view of the experimental setup designed for time-resolved photoelectron and transient absorption spectroscopy measurements. It combines a femtosecond laser system, an attosecond XUV pump-IR probe Mach-Zehnder interferometer, a velocity-map imaging system (VMI), and an XUV spectrometer.
Depending on the experimental requirements, either the 35-fs or 8-fs laser beam is sent into the attosecond beamline, which is composed of an XUV-IR Mach-Zehnder interferometer, a velocity-map-imaging (VMI) system [20,21], and an XUV photon spectrometer. To minimize air flow instabilities, most of the 2m-long interferometer (roughly 2/3 of its length) is maintained under vacuum conditions. The incoming femtosecond beam enters the interferometer through the beamsplitter (BS), which reflects most of the beam in one arm (60 to 80 %), and transmitted the remaining part in the other. The reflected beam is used to produce attosecond XUV pulses via High Harmonic Generation (HHG) in a gas target [22]. The direction of the beam is steered with two flat mirrors (MX1, MX2) and focused with a 30-cm concave lens (LX) into a windowless gas cell filled with gas (typically argon or neon) at a backing pressure of tens to hundreds mbar. The gas cell consists of a movable 2-mm-diameter tube that can be precisely translated along the beam propagation direction to optimize the high harmonic generation under operational conditions. Additional optical components used to shape the temporal profile of the IR driving field for the generation of single attosecond pulse [23] or attosecond pulse trains made of odd and even harmonics [24–27] are placed in between the beamsplitter (BS) and the mirror (MX1). After the gas cell, both the XUV light and the remaining driving IR field propagate coaxially towards an ultrathin (100 to 500 nm) metallic foil (5-mm diameter) mounted on a hole-drilled window that is used to block the central part of the residual IR radiation. A 375-mm focal length, 4$^{\circ }$ grazing incidence, gold-coated toroidal mirror (TM) operating in a 2f-2f geometry, then focuses the XUV light into the velocity-map imaging system. Before entering the VMI, the attosecond XUV pulses exit the interferometer through a 2-mm hole-drilled recombination mirror (RM) located 120 mm away from the toroidal mirror.
The IR beam transmitted through the beamsplitter (BS) is used as a probe for attosecond experiments. It follows the path of the second arm of the interferometer defined by the mirrors MI1 to MI3, and subsequently exits the interferometer coaxially with the attosecond XUV beam by reflecting off the hole-drilled recombination mirror (RM). Before exiting the interferometer, the beam is focused with a 75-mm converging lens (LI) to the same position in the VMI spectrometer as the attosecond beam. For pump-probe experiments, the time delay between the IR and attosecond XUV pulses is controlled by varying the length of this arm. A micrometer translation stage controlling the position of the mirror (MI1) allows for a coarse temporal overlap of the two pulses, while a second movable mirror (MI2) mounted on a piezoelectric actuator provides a finer temporal adjustment of the delay down to the attosecond time scale ($\sim$2 fs/V). The temporal and spatial overlaps between the two pulses can be simultaneously optimized under operational conditions, by using the IR field driving the HHG process to mimic both the direction and the timing of the attosecond pulses in the absence of the metallic foil. A movable mirror (MM) inserted in the beam path after the recombination mirror directs both IR beams toward a CMOS camera, which captures the beam profile at focus. A fine adjustment of the spatial overlap is achieved by controlling the pointing of the mirror MI3.
After exiting the interferometer, the attosecond and IR beams interact with an effusive gas jet flowing out of a 20-$\mu$m nozzle mounted on the VMI spectrometer [20]. In the photoionization region, which is essentially defined by the waists of the beams (140$\mu$m and 80$\mu$m for the attosecond and IR beams, respectively), a maximum IR intensity of the order of 10$^{14}$ W.cm$^{-2}$ is reached (estimated from above-threshold ionization experiments [28]). Photoelectrons with energy up to 100 eV can be detected with a $4\pi$-acceptance angle and a resolution ($\Delta E/E$) as high as 1% [20]. The two beams then propagate toward a home-made XUV spectrometer located 300 mm away from the photoionization region. The spectrometer consists of a flat-field grating (FG) and a soft x-ray CCD camera. With the appropriate grating, the spectrometer covers the energy region from 10 to 250 eV.
3. Active stabilization system
A schematic view of the active stabilization system is shown in Fig. 2. It allows to continuously control and stabilize the path length difference between the two arms of the interferometer to any desired value. It is based on a 5-mW continuous wave (CW) green laser ($\lambda$=532nm) propagating in the Mach-Zehnder interferometer coaxially with both the attosecond pump and IR probe beams. The stabilization is achieved with an active feedback system that adjusts the length of one interferometer’s arm (the one followed by the IR beam) to maintain a constant relative phase between the two waves. A photodiode capturing the optical intensity fluctuations induced by any phase shift between the two interfering waves provides a feedback signal to a high-speed piezo controller, which corrects in return the position of the mirror MI2 to maintain the optical intensity at a constant value. The control of the relative path length between the two arms of the interferometer, on the other hand, is achieved with a phase shifter [29] that imparts a controlled relative phase between the two waves forcing the mirror MI2 to move to a desired position.
Fig. 2. Schematic view of the optical actuator, which allows both the stabilization and the control of the Mach-Zehnder interferometer used in the attosecond beamline. It is based on a 532-nm CW laser propagating in the interferometer coaxially with both the attosecond and IR beams. The stabilization is achieved with an active feedback system that adjusts the length of one arm of the interferometer to maintain the relative phase between the two waves constant. A phase shifter, on the other hand, imparting a controlled relative phase to the waves, is used to adjust the relative path length between the two arms of the interferometer to any desired value.
3.1 Optical system
The vertically polarized CW laser beam first combines with the femtosecond IR beam by traveling through a broadband dichroic mirror (DM1) located outside the attosecond beamline, and subsequently enters the interferometer through the beamsplitter (BS). Both the transmitted and reflected beams propagate up to the hole-drilled recombination mirror where they exit the interferometer by traveling through the hole and by reflecting off the back of the mirror, respectively. On its way toward the recombination mirror, the outer part of the reflected beam first passes around the thin-metallic filter, while its central part is blocked, and then its polarization is rotated by 90$^{\circ }$ with a hole-drilled half-wave plate. At the recombination mirror, the reflected and transmitted beams are thus orthogonally-polarized. Due to the inherent wedge-shape of the mirror, the beams are also not perfectly coaxial at the exit of the interferometer. To correct for this slight angular deviation, they are sent into a compact Michelson interferometer (5-cm arm length) where they are first spatially separated with a broadband polarizing beam-splitter (PBS), and then reflected back toward the beamsplitter to recombine. With a proper alignment of the flat mirrors (MS1, MS2), the orthogonally-polarized returning beams exit the Michelson interferometer collinearly. Finally, the beams are sent into a balanced homodyne detection system composed of a half-wave plate oriented at 22.5$^{\circ }$, a polarizing beamsplitter (PBS$_{532}$), and a balanced optical receiver (BOR) for the measurement of their relative phase. It can be readily shown that the relative phase $\varphi$ between the two waves is encoded into the signal $S$ measured by the receiver as:
where $E_{\mathrm {XUV}}$ and $E_{\mathrm {IR}}$ are the amplitudes of the waves traveling along the XUV and IR arms of the interferometer, respectively. As Eq. (1) indicates, any fluctuation of the relative phase $\varphi$ due to instabilities in the interferometer leads to a modulation of the signal $S$ measured by the balanced receiver. The stabilization is thus achieved by maintaining the signal $S$ constant (equal to 0) with a feedback loop. The receiver provides an error signal to a high-speed PI piezo controller that adjusts in return the position of the mirror MI2 in the interferometer to correct the phase shift $\varphi$ between the two waves. For the sake of driving the piezo stage safely below its resonance frequency ($\sim$ 400 Hz), the high-frequency components ($>$ 300 Hz) of the error signal are filtered out with a low-pass filter before feeding it to the controller.To ensure that the relative phase measured by the receiver is uniquely related to the instability in the Mach-Zehnder interferometer, the Michelson interferometer is also stabilized. The stabilization system is similar to the one used in the Mach-Zehnder interferometer. It is based on a 633-nm CW laser polarized at 45$^{\circ }$ with respect to the vertical direction. The laser beam’s divergence is first slightly increased with a concave lens (LS) to match the angular deviation between the two orthogonally-polarized 532-nm CW beams. The beam then combines with the 532-nm beams by reflecting off a dichroic mirror (DM2) and it is finally sent into the interferometer, where its s- and p-components propagate independently in each arm. After exiting the interferometer, the beam is separated from the 532-nm beams with a second dichroic mirror (DM3), and the relative phase between its s- and p-components is measured with a second balanced homodyne detection system and stabilized with a feedback loop in a similar way to that described above for the green CW laser.
3.2 Delay control
An Evans’ phase-shifter [29,30] placed at the exit of the Michelson interferometer is used to adjust the position of the mirror MI2 and thus vary the relative time delay between the IR and attosecond pulses. It imparts a controlled relative phase between the two 532-nm waves that forces the mirror to move to a new position so that the signal $S$ measured by the balanced receiver remains constant. It is composed of two quarter-wave plates oriented at +45$^{\circ }$ with respect to the direction of polarization of the incoming CW beams, and a rotatable half-wave plate located in between that controls the relative phase imparted to the CW beams. The cumulative effect of the three wave plates on the incoming CW beams is readily described by the Jones matrix
where $\theta$ is the angle between the horizontal direction and the half-wave plate’s fast axis. As the matrix $M$ indicates, the phase shifter imparts to the orthogonally-polarized waves a relative phase equal to $4\theta +\pi$ . By rotating the half-wave plate, the relative phase can thus be continuously tuned, so as the position of the mirror MI2 (2128 nm per revolution). Using a rotational stage with sufficient angular precision, the time delay between the IR beam and the attosecond pulse is controlled with an accuracy down to a few attoseconds ($\sim$20 as per degree).4. Results
The active stabilization system is used routinely in our laboratory to control and stabilize the delay between the attosecond and the IR pulses. It is easy to operate and only requires a few minutes to set up before any pump-probe experiments. We present below some measurements that demonstrate its capability to control the time delay between the two pulses with an attosecond temporal resolution and to maintain it to a given value over several hours.
4.1 Stability measurements
A typical long-term measurement of the relative path length drift in the Mach-Zehnder interferometer without active stabilization is shown in Fig. 3(a). The measurement was taken during day-time when the level of vibration in the laboratory is nearly at its maximum. The relative path length drift was deduced from the interference pattern produced by the 532-nm laser and captured by the CMOS camera located at the exit of the interferometer. An image of the 2D pattern was recorded every second. The integrated intensity of the central peak was normalized by the intensity of the full image to correct for both the laser intensity and the camera efficiency fluctuations. The resulting signal was then converted to the time delay between the two waves traveling in the interferometer. It can be seen that the time delay drifts considerably over the duration of the measurement. The peak-to-peak drift is about 300 as over two hours while the stability is of the order of 72 as rms. Such a temporal drift is significantly larger than the typical duration of the attosecond pulses ($\sim$100 as). The origins of the drift have not been clearly indentified. But it is worth noting that the passive stability of our interferometer is quite comparable in magnitude to those of interferometers entirely placed under vacuum conditions [15,18], which are known to be generally more stable. This indicates that possible air flows in the portion of our interferometer placed in atmospheric conditions may not be the main source of instability.
Fig. 3. Relative path length drift in the Mach-Zehnder (a) and Michelson (b) interferometers without active stabilization and long-term stability of the Mach-Zehnder interferometer when both interferometers are actively stabilized (c). Spectral composition of the short-term stability of the interferometer without (d) and with (e) active stabilization.
As a comparison, the long-term measurement of the stability of the Michelson interferometer without active stabilization is shown in Fig. 3(b). The relative path length drift was deduced from the signal measured by the balanced homodyne detection system. As expected, the temporal drift is significantly better than that of the Mach-Zehnder interferometer. The peak-to-peak drift is of the order of 100 as over two hours while the stability is about 26 as rms. We have observed that the drift is mainly generated by temperature fluctuations in the laboratory resulting from the cooling/heating cycles of the air conditioning unit. For most of the attosecond pump-probe experiments, which are performed with a time step of the order of 100 to 200 as, such a drift would not significantly decrease the temporal resolution of the measurement. Nevertheless, to achieve the highest level of stability in the Mach-Zehnder interferometer, an active stabilization of the Michelson interferometer is necessary.
The long-term measurement of the relative path length drift of the Mach-Zehnder interferometer when both interferometers are actively stabilized is shown in Fig. 3(c). A stability of 11 as rms is achieved over a period of 2 hours. Such a level of stability is somewhat higher than those measured in similar setups [15,16,18] and is very close to those achievable with collinear attosecond setups [9].
To identify the limiting factors for the attainable stability, a short-term measurement of the interferometer stability was performed using a photodiode in place of the CMOS camera. Figures 3(e) and 3(d) show the spectral composition of the short-term stability with and without active stabilization of the interferometer, respectively. The spectrum corresponding to the unstabilized interferometer reveals several sources of vibration in our system. The region below 400 Hz is mainly dominated by vibrations in the building (0-200 Hz) and additional mechanical resonances of the experimental setup (200-400 Hz). On the other hand, the well-defined vibration peak located at 820 Hz is generated by the turbo-molecular pumps. With active stabilization, it can be seen that vibrations below 300 Hz have been almost fully canceled out, which suggests that the attainable stability is mainly limited by vibrations generated by the turbo-pumps.
4.2 Time delay control
The capability of the system to accurately control the time delay between two beams traveling in the interferometer is demonstrated in Fig. 4. The time delay was deduced from the interference pattern produced by the 532-nm CW laser using the method described previously. Figure 4(a) shows the signal measured with the camera as the angle of the Evans’ phase shifter is varied with angular steps corresponding to 20 as (1$^{\circ }$) over a range of 30 fs (1500$^{\circ }$). The signal exhibits a remarkable cosine-wave shape, which indicates that the time delay between the two waves is accurately controlled. The level of accuracy in controlling the delay is better observed in Fig. 4(b) where the time delay deduced from the modulated signal is plotted as a function of the phase shifter’s angle. As can be seen in the figure, the linear relationship between the time delay and the phase shifter’s angle is quite strong. The deviation from a perfect linear relationship mostly comes from the error in the time delay measurement around the maxima and minima of the modulated signal, where the error becomes comparable to the step size, and from the instability in the interferometer. As the statistical distribution of the delay steps shows in the subplot of Fig. 4(b), the time delay is controlled to within 10 as, which corresponds to the level of stability of the interferometer.
Fig. 4. Control of the time delay between two 532-nm waves traveling in the interferometer: Interference signal captured by the photo-detector (a) and time delay between the two waves (b) as a function of the angle of the Evans’ phase shifter. The subplot in (b) displays the statistical distribution of the delay steps deduced from the interference signal.
4.3 Application for attosecond pump/probe measurements
To test the level of performance of the stabilization system for attosecond pump/probe measurements, it was used to drive a RABITT (Reconstruction of Attosecond Beating by Interference of Two-photon Transitions) experiment [2,31], where an attosecond pulse train (APT) made of odd-order harmonics is used to ionize an atomic target in the presence of a weak IR field. In such an experiment, a spectrogram is formed by measuring the photoelectron spectra as a function of the time delay between the APT and the IR field. The photoelectron spectra exhibit peaks at photoelectron energies corresponding to one-photon absorption of the odd harmonics, and located in between, sideband peaks resulting from two-photon transitions (absorption of one XUV photon plus absorption or emission of one IR photon). Two different quantum paths involving two consecutive harmonics contribute to the same sideband quantum state, and thus interfere. As a consequence, the intensity of each sideband peak oscillates with the time delay between the APT and the IR field at twice the frequency of the IR field, as predicted by second-order perturbation theory [32]:
The experiment was performed with the setup described in section 2. Attosecond pulse trains made of odd harmonics of the fundamental 800-nm field were generated in argon via high harmonic generation. The pulse trains were then filtered to remove harmonics below the 11$^{\mathrm {th}}$ order and finally focused into an effusive gas jet of argon. The photoelectron energy spectra were measured with a velocity-map imaging system [20] and reconstructed using the DAVIS inversion procedure [21]. A typical spectrogram (measured over a period of 1.5 hours) is shown in Fig. 5(a). The delay between the APT and IR field was scanned over 30 fs with 50-as time steps. At each delay, the photoelectron energy spectrum was recorded by collecting photoelectrons produced by roughly 10$^{5}$ pulse trains. The envelope-normalized modulation of the peak intensity of sideband H18 formed by the harmonics 17$^{\mathrm {th}}$ and 19$^{\mathrm {th}}$ as a function of the delay is plotted in Fig. 5(b). Except for small deviations due to statistical fluctuations in the measurement, the modulation exhibits a clear cosine-wave shape, as described by Eq. (3). The period of the oscillation is about 1.35 fs, which corresponds to half the period of the 800-nm IR field. As displayed in Fig. 5(c), the short-time Fourier transform (STFT) of the H18-sideband intensity modulation reveals that the periodicity of the modulation is remarkably constant over the full 30-fs delay range. This is further demonstrated in Fig. 5(d) where the fluctuation of the sideband period deduced from the STFT analysis is represented as a function of time. It can be seen that the peak-to-peak fluctuation of the period is about 30 as over the duration of the measurement (1.5 hours) while the standard deviation is about 6 as, which demonstrates the capability of the stabilization system to accurately stabilize and control the time delay between the APT and IR field.
Fig. 5. RABITT experiment in argon with 50-as delay step. (a) Photoelectron energy spectra as a function of the time delay between the APT composed of odd harmonics and the IR field. (b) Envelope-normalized intensity of sideband H18 (formed by harmonics 17$^{\mathrm {th}}$ and 19$^{\mathrm {th}}$) as a function of the time delay. (c) Short-time Fourier transform of the H18-sideband intensity modulation (13.5-fs window, 50-as time step). (d) H18-sideband period deduced from the frequency measured in (c) as a function of time.
5. Conclusion
In this work, we have developed a stabilization system for attosecond pump-probe setups based on a Mach-Zehnder configuration. The system uses a CW laser propagating coaxially with the pump and probe beams in the interferometer. At the exit, the CW beams are recombined collinearly to produce a strong interference pattern allowing for a precise measurement of their relative phase. The stabilization of the interferometer is achieved with a standalone feedback controller that adjusts the length of one arm of the interferometer to maintain a constant relative phase between the CW waves. The maximum stabilization rate is set by the resonance frequency of the piezoelectric stage used. With such a system, we have shown that the relative path length of our 2m-long interferometer can be stabilized within 3 nm over several hours, which corresponds to a timing jitter between the pump and probe beams as low as 10 as rms. The system is easy to operate and only requires a few minutes to set up before any pump-probe experiments. Due to its long-term stability, the system is well-suited for low-count experiments that require data collection over an extended period of time.
Funding
Basic Energy Sciences (DE-SC0017984); Air Force Office of Scientific Research (FA9550-18-1-0333).
Acknowledgments
Authors thank the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) and the U.S. Air Force Office of Scientific Research (AFOSR) for supporting the development of the attosecond beamline and the stabilization system, respectively. M. T. and B. U. acknowledge support from the Undergraduate Research Fellowship (URF) program at Auburn University.
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