Extreme Ultraviolet Radiation With Coherence Time
Beyond 1 s
Abstract
Many atomic and molecular systems of fundamental interest possess resonance frequencies in the extreme ultraviolet where laser technology is limited and radiation sources have traditionally lacked longterm phase coherence. Recent breakthroughs in XUV frequency comb technology have demonstrated spectroscopy with unprecedented resolution at the MHzlevel, but even higher resolutions are desired for future applications in precision measurement. By characterizing heterodyne beats between two XUV comb sources, we demonstrate the capability for subHz spectral resolution. This corresponds to coherence times s at photon energies up to 20 eV, more than 6 orders of magnitude longer than previously reported. This work establishes the ability of creating highly phase stable radiation in the XUV with performance rivaling that of visible light. Further, by direct sampling of the phase of the XUV light originating from high harmonic generation, we demonstrate precise measurements of attosecond strongfield physics.

JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, CO 803090440

Stony Brook University, Departments of Chemistry and Physics, Stony Brook, NY 117943400

State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, Peopleâs Republic of China
In the infrared, visible, and nearUV spectral regions, the precision and accuracy of frequency metrology[1, 2] has vastly exceeded that of traditional spectroscopic methods. Two key enabling laser technologies driving these measurements are ultrastable lasers with long ( 1s) coherence times[3, 4] and optical frequency combs[5]. However, for wavelengths below approximately 200 nm, conventional light sources have short coherence times[6] and are not useful for frequency metrology. Future applications in precision measurement with highlycharged ions[7], helium[8], nuclear clocks[9], or hydrogenlike and heliumlike ions[10] necessitate phase stable light in the XUV. The workhorse of extreme ultraviolet (XUV) and soft xray science, synchrotron radiation[6], is both temporally and spatially incoherent. New free electron laser sources possess a high degree of spatial coherence but due to their singleshot nature, coherence times are never longer than their (femtosecond) pulse durations[11, 12, 13]. Conventional XUV lasers[14], based on shortlived population inversions in highly ionized pulsed plasmas, are spatially coherent but also temporally incoherent on scales useful for XUV frequency metrology.
In the last decade, highorder harmonic generation (HHG) using femtosecond lasers has emerged as method that can support both spatial and temporal coherence in XUV[15] and use of HHG sources for high resolution spectroscopy in the XUV have recently been demonstrated at the MHz level[16, 17, 18, 19]. In this Article, we report the observation of coherence times greater than 1 second in the XUV measured via heterodyne beating between two XUV frequency comb sources based on intracavity HHG[20, 21, 22]. Similar to heterodyne interferometry with continuouswave lasers, our frequency comb lasers in the nearinfrared (NIR) are upconverted to the XUV and then combined to form the heterodyne beatnote. The apparatus, schematically shown in Fig.(1a), is analogous to a MachZender interferometer where each arm of the interferometer contains a cavityenhanced HHG apparatus. The first beam splitter (the acoustooptic modulator) separates the NIR frequency comb and the second beam splitter (the beam combiner, Fig. (1d)) recombines the XUV frequency combs. The heterodyne signal will provide information about the phase of the XUV light and its noise properties. We use the phase information of the heterodyne signal to probe attosecond timescale, strongfield physics.
0.1 XUV comb generation and XUV interferometer
For our experiment, we pump two independent femtosecond enhancement cavities[23, 20, 21] (fsECs) with a highpower Yb:fiber frequency comb outputting 120 fs pulses at 154 MHz with an average power of 80 W centered at 1070 nm[24]. To pump two fsECs, the frequency comb is split into two by an acoustooptic modulator (AOM) such that the two resulting combs have a carrier envelope offset frequency detuning of 1 MHz. Each fsEC (XUV1 and XUV2 of Fig. (1a)) is actively stabilized using the PoundDreverHall technique with piezoelectric transducers on cavity mirrors as actuators. The carrier envelope offset is not actively stabilized, but its passive stability is sufficient for maintaining good power enhancement. Each fsEC typically operates with 4.5 kilowatts of average power and peak intensity of W cm. Xenon gas for HHG is injected at the cavity foci via quartz nozzles with aperture backed by 1 atmosphere of pressure. Harmonics are coupled out of the fsEC’s in colinear fashion using thick intracavity sapphire plates set at the Brewster angle for the fundamental radiation to limit intracavity loss. The finesse of the fsEC was intentionally kept low to mitigate the nonlinearities of the Brewster plate and the plasma[22, 25, 26], but enhancement factors of 200 were still obtained. The two XUV beams are combined (more on this below) and detected using either an electron multiplier or a photomultiplier tube (PMT) with a phosphor screen. A schematic of the optical layout is shown in Fig. (1a). Since the pump frequency combs are offset by , the resulting XUV frequency comb will have a relative detuning of and heterodyne beatnotes can be observed at these frequencies so that a series of beatnotes appears in the RF output of the detector, effectively mapping the XUV spectrum of the harmonics to RF domain (Fig. (1c,1e)).
Splitting the NIR frequency comb beam at the beginning of the interferometer with the AOM is straightforward, but recombination of the XUV beams at the end of the interferometer is not. Since there are no transparent materials in the XUV, there are no standard beamsplitters available for recombination. Instead, we rely on a wavefront division scheme. A silicon wafer with a pyramidal aperture produced by KOH etching was used as the beam combiner (Fig. (1d)). The beam from XUV2 is focused through the aperture, while the reflected beam from XUV1 is much larger than the aperture. All the optics for the XUV light, including the beam combiner, were coated with boron carbide (BC) to enhance their reflectivity in the XUV[27]. In the near field, there is no overlap between the two beams, but in the far field the beams interfere with a circular, “bullseye”, fringe pattern (see Fig. (1b)). The fringe pattern can be easily observed by setting = 0 and adjusting the relative path length of the interferometer so that pulses from XUV1 and XUV2 overlap in time at detector. To observe beat signals at finite , the central portion of the fringe is selected with an aperture before detection. A key feature of this beam combination scheme is that the fringe period scales weakly as so that the interferometer can work well over a broad range of wavelengths, allowing us to simultaneously observe beats at the fundamental, the 17 harmonic, and all the harmonics in between.
Since fsEC’s XUV1 and XUV2 are both pumped by a common NIR laser, the noise in the RF beatnote is immune to the commonmode frequency noise of the Yb:fiber laser. Thus, the apparatus directly measures noise from the HHG process or the cavityplasma dynamics[25]. However, since the interferometer is not actively stabilized, there are small amounts of relative noise induced by vibrations in the optics, giving the two sources a nonzero relative linewidth that is technical in origin.
0.2 Phase noise measurement and demonstration of long coherence times
Here we show that high harmonic generation shares many common features with classical frequency multiplication of RF signals. In frequency multiplication, the power spectral density (PSD) of the phase noise increases quadratically with the harmonic order[28]. Even if the multiplication process is noiseless, any noise around the carrier will still increase with harmonic generation, thus setting a fundamental limit of how phase noise transforms under frequency multiplication and thus the achievable coherence level in the XUV (Methods). The linewidth of a carrier can be related to by the approximation[29],
(1) 
The limit of the integral is the point where approximately half of the power is in the carrier and half in the noise. Therefore, the 3 dB linewidth of the carrier will increase quadratically with harmonic order. If one is not careful to have a lownoise carrier before multiplication, the carrier will start to disappear and lead to the known phenomena of carrier collapse[30]. Fig. (2) shows the full width at half max (FWHM) of each harmonic comb tooth plotted versus harmonic order. A simple fit to FWHM = a q, where a is the scaling parameter, highlights the quadratic dependence of the linewidth. The linewidth dependence on harmonic order was also independent of intracavity power. This analysis shows that there is a fundamental scaling of phase noise from the generating laser to the resulting XUV light. For example, assuming the best available continuous wave (CW) laser with a linewidth of 30 mHz[4], and that the fundamental comb will faithfully follow the phase of the CW laser[31], the linewidth at the 17 harmonic would be 8.7 Hz. Fortunately for HHG, we do not observe any additional linewidth broadening mechanisms other than the unavoidable classical frequency multiplication.
We show that the previously observed linewidth originated from the differential path noise and not HHG physics so that higher levels of coherence can be observed. The differential path length noise in the interferometer can be removed by phasestabilizing the two optical paths. A small amount of pump light leaks through the XUV optics to the detection plane which is then picked off and detected on an photodetector to provide an error signal sensitive to interferometer fluctuations. The error signal is filtered and used to apply feedback correction to the AOM frequency thus removing the relative noise (Methods). With the interferometer phase locked, we can probe any noise processes intrinsic to intracavity HHG. A dramatic change in the linewidth of the XUV beatnotes is observed with a stable interferometer. The unstabilized beatnote of the pump laser and the beatnote at the 17 harmonic are show in Fig. (3a,3c) with the corresponding stabilized case in Fig. (3b,3d) with a 250 mHz resolution bandwidth. Similar resolution limited beats were observed on the 319 harmonic. The coherence of the pump lasers is faithfully transferred to the harmonics as seen by the resolutionlimited beatnote of 62.5 mHz in Fig. (3f). This is equivalent to a coherence time of 16 s or a stable phase relation maintained over consecutive pulses. This is nearly seven orders of magnitude larger coherence times than ever reported in the XUV[16, 32, 18]. This establishes that our XUV comb system is extremely phase coherent and has the capability to support subHz coherences in the XUV. Furthermore, we demonstrate that the commonmode noise rejection of this measurement scheme is not particularly sensitive to the two XUV comb systems being identical by also observing coherent beats from harmonics generated from Xenon gas in one cavity and Krypton in the other (Fig. 3e).
0.3 Application to attosecond strongfield physics
The heterodyne beatnotes in the XUV provide unique and unprecedented access to the phase of XUV radiation. We can use this to probe attosecond physics and measure the intensity dependent dipole phase[33]. This technique is a fundamentally different method for probing this phenomena than previous realizations and does not rely on extensive spatial and spectral filtering to observe interference between multiple quantum pathways[32, 34]. This technique also does not rely on referencing the short to long trajectory contributions or vice versa[35]. It represents a complementary technique to RABBITT and related methods based on photoelectron spectroscopy[36, 37, 38], which seek to determine the timedelay (or equivalently phase shift) between adjacent harmonic orders at a given intensity. In this experiment, we isolate individual harmonics and measure precisely their phase shift as a function of intensity. These phase shifts can be directly linked to temporal dynamics of the electron in the intense laser field[36]. By putting amplitude modulation (AM) on one arm of the interferometer, we can measure the amount of induced phase modulation (PM) on the XUV light (Methods). For this measurement, the peak intensity of the modulated cavity was with a 15% amplitude modulation depth. The intensity dependent dipole phase can be expressed as
(2) 
with being the laser intensity, being the pondermotive energy, being the laser frequency, and being the phase of the emitted XUV light. Due to our on axis spatial filtering needed to observe a highcontrast beatnote, we are primarily sensitive to the short trajectory. The schematic of the measurement is shown in Fig. (4a). The measurement required a twostep demodulation process to extract the amount of PM induced on the XUV light from the AM applied to the pump laser. Great care was taken to ensure that the amount of PM on the XUV light was not induced by parasitic PM on the pump laser (Methods). The phase of the XUV light depends on both the intensity of the light and the particular quantum path the electron traverses (Fig. (4b)). The result of the measurement is shown in Fig. (4c) where the of Eq. 2 is expressed in atomic units with the convention of Yost et al.[32]. The shaded purple region corresponds to values predicted by the standard semiclassical model (SCM)[33]. The grey shaded region is the range of values predicted by different approximations to below threshold harmonics[39].
Belowthreshold, short trajectory harmonics do not originate from tunnel ionization as in the abovethreshold case and are much more sensitive to the atomic potential and ionization dynamics[39]. Our measurement is able to discriminate between contributions of the standard SCM[33] and a model that includes overthebarrier (OTB) ionization[39] to confirm that the belowthreshold harmonics mostly originate from the OTB ionization. Further, our measurement for an abovethreshold harmonic (15 and 17) agrees well with the predictions of the SCM[33] and the belowthreshold with predictions of Yost et al.[32] and the theoretical framework of Hostetter et al.[39]. Further exploration of intensity dependent phases in atoms and molecules with comparison to calculations is the subject of future work. Our phase measurement technique was able to resolve phase shifts with uncertainties at the 10 rad level, which corresponds to a time uncertainty of 6 as. In contrast to typical experiments that utilize direct attosecond timing resolution[38], we measure the attosecond electron dynamics imprinted on the phase of the emitted XUV light originating from HHG[36]. With system improvements, it is feasible to extend this into the as regime, rivaling the highest achievable temporal resolution of attosecond electron dynamics[36, 40]. Thus, our apparatus provides direct, unambiguous access to the phase of XUV radiation and will prove to be a valuable tool for attosecond science and the dynamics of atoms and molecules in intense laser fields.
0.4 Summary and future outlook
We demonstrated that HHG is extremely phase coherent. We have identified the primary noise requirements on the pump laser and shown that it is possible to support coherence times greater than 1 s in the XUV. We have also developed an interferometer capable of operating from 1070 nm  56 nm, but with different optics and a new beam combiner[41], extension to even shorter wavelengths is possible. Such an apparatus will be a vital tool for future work in dualcomb spectroscopy, Fourier transform spectroscopy[42], high resolution molecular spectroscopy[43], attosecond electron dynamics in intense laser fields[44], and HHG spectroscopy[45]. We have successfully probed physics at attosecond timescales using the tools of frequency metrology to measure the intensitydependent dipole phase. Future work will require improved output coupling[46, 47, 48] and powerscaling[17, 49, 50] schemes to extend the high level of phase coherence to shorter wavelengths possibly enabling spectroscopy of a nuclear isomer transition where highly phasestable light will be needed for excitation.
Methods
0.5 Acoustooptic Modulator and Beat Detection.
In order to create a small frequency offset between the two XUV sources, we rely on an AOM to frequency shift the pump laser such that the relative detuning is 1 MHz. Since there are no available AOMs at this frequency, we drive the AOM such that 1 MHz =  . Therefore, we will observe two beats at frequencies less than , one at and the conjugate at . The conjugate beatnote is sensitive to any noise in . To remove this, we derive by phaselocking a voltagecontrolled oscillator to detuned by 1 MHz such that =  1 MHz. This removes the dependence of from the conjugate beatnote and puts it in . Therefore, the low frequency beatnote can be detected noise free where we have detectors of adequate bandwidth.
0.6 Phase Noise.
A signal oscillating at a frequency with phase modulation can be expressed as
(3) 
where is the modulation depth and is the modulation frequency. When is small, we can express the power in the first order modulation sideband relative to the carrier by , where are Bessel functions. We can extend this to the case of general phase modulation and not at a particular discrete tone by . We can define the phase noise power spectral density as the noise around a carrier within a bandwidth as
(4) 
By integrating , one arrives at Eq. 1 and gets the approximate linewidth of a signal. The point is the cutoff of the integral where there is approximately half the power in the carrier and half in the noise.
0.7 PhaseStable Interferometer.
The XUV light is very sensitive to path length fluctuations and any phase fluctuations in the driving laser. Therefore, it is imperative to keep the interferometer stable in order to generate the highest levels of coherence in the XUV. The passive stability is sufficient for subkHz levels of coherence. To stabilize the interferometer, we use the small amount of pump laser light that copropagates with the XUV light. Since the pump light diverges more than the XUV light, it is simple to separate it out before the detection plane. A small amount of light is picked off and sent to a photodetector where a 1 MHz beatnote is used to measure the phase fluctuations in the interferometer cause by mechanical noise on the mirrors. The beatnote is mixed with an RF synthesizer to generate an error signal. The error signal is filtered and the correction signal is applied to the 1 MHz reference frequency for the AOM. Since the fluctuations of the interferometer are small, the phaselock loop that sets the AOM frequency can easily compensate for the small, necessary corrections.
0.8 Measurement of Intensity Dependent Dipole Phase.
The intensity dependent dipole phase is expressed in Eq. 2. By trying to measure the intensity dependent phase that results from HHG, we are effectively measuring the AMPM coupling with the AM being on the pump laser and the PM being on the XUV light. We can mathematically describe a beatnote signal as
(5) 
is the frequency of the beatnote and is the frequency of the applied modulation and is its phase. A is the amplitude modulation depth and P is the phase modulation depth. Each beatnote is characterized by its own values for A and P. To avoid confusion, the subscripts will refer to which signal it represents. For example, A is for the amplitude modulation of pump laser and P is the phase modulation of the q harmonic. Our task is to determine the values for A and P at the pump and the harmonics. By taking the ratio of P to A and using proper units, we can extract the intensity dependent phase coefficient .
We need to extract the relevant parameters of Eq. 5 at both the harmonic of interest and the IR simultaneously. To do this, we use a two step demodulation process. By taking Eq. 5 and mixing it with a stable RF signal (LO1) at a frequency of and relative phase offset , we get
(6)  
(7) 
We have ignored terms at . We can further low pass the signals at and obtain a “DC” signal
(8) 
Eq. 8 will be one of our primary signals. Note that the phase is set by the phase of LO1. This also assumes that the phase of the XUV beatnote is stable. This is true when we phase stabilize our interferometer.
The signal (t) contains terms that oscillate at the applied modulation frequency . We can demodulate our signal once more using a lockin amplifier at the correct phase and ignore terms at 2 to obtain
(9) 
is our second signal. With Eq. 8, Eq. 9 and some independently measured parameters (for example, the AMAM coupling), we can extract our parameters of interest. To measure the AA coupling, we can use our beatnote signals. Our XUV beatnotes are directly proportional to the amount of XUV power in each beam. The amount of beatnote power can also be easily measured on an RF spectrum analyzer. By changing the amount of power in one of the enhancement cavities and observing how the beatnote power changes, we can determine how much the XUV power must have changed for a given laser intensity change. A can be determined by
(10)  
(11) 
with dB being the measured beatnote power. By applying amplitude modulation to the pump laser on one arm of the interferometer, we can control A very well. It is also easily measured with a photodetector. By varying the phase of LO1, we can measure and (Eq. 8 and Eq. 9) simultaneously. With the modulation (A,P) turned off, tells us the phase of the beat. With the modulation on, the relative phase between and can tell us the ratio of A/P. Since A can be measured independently, we can extract the amount of phase modulation, P. This procedure needs to be done with the IR signal and the XUV signal simultaneously to prevent any systematic errors.
0.9 Signal Correction
The signal correction of P P  P is justified by experimental observations. First, the quadratic scaling of the linewidth shows that the phase of the XUV light is directly linked to the phase of the NIR light. Any phase shift on the NIR is at the harmonic. For the beatnote signals, this also implies that the phase modulation depth P is related to the modulation depth in the NIR by PP. This relation is also justified by observing the beatnote signals in the presence of phase modulation on the NIR laser. By measuring the amount of phase modulation on the NIR laser and the amount induced on the XUV beatnotes via the relation of the carrier to modulation sideband heights, we have verified the relation PP. Our lockin detection methods also produce the same relation when addition phase modulation is placed on the NIR laser.
0.10 Modulation Effects
Modulating the intensity has notable effects on the neutral/plasma density ratio inside the enhancement cavity. However, the modulation is slow at 2 kHz and the cavity feedback loop can easily follow it to maintain resonance. The effect has been previously systematically studied[25]. Due to the neutral/plasma density changes, there is a small amount of phase modulation induced by amplitude modulation on the NIR laser. This was verified by measuring the effect with gas present and absent. This can easily corrupt the intensity dependent phase measurement and necessitates the correction described previously. Further, any cavity oscillation due to bistability[25] can render the signals too noisy. Therefore, by careful measurement of the phase modulation on the NIR laser and the XUV beatnotes, we can extract the intensity dependent phase originating from HHG and not other macroscopic effects.
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We acknowledge technical contributions and collaborations from Axel Ruehl, Ingmar Hartl, and Martin Ferrman. This work is supported by NIST, AFOSR, and the NSF Physics Frontier Center at JILA.

The authors declare that they have no competing financial interests.

Correspondence and requests for materials should be addressed to J.Y. (email: ye@jila.colorado.edu) or C.B. (email: craig.benko@colorado.edu).
* Current Address: AOSense, Sunnyvale, CA 940852909
** Current Address: MaxPlanckInstitut für Quantenoptik, HansKopfermannStr. 1, 85748 Garching, Germany 
All of the authors contributed to the design and performed the experiment. All of the authors discussed the results and contributed to the final manuscript.
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