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High-repetition-frequency HHG has recently been enabled through the use of femtosecond enhancement cavities1,2. In addition to potentially transferring frequency comb techniques3,4 to the vacuum-ultraviolet range for possible applications such as direct frequency-comb spectroscopy5,6,7, the enhancement-cavity technique allows for greater harmonic photon flux and a near-perfect Gaussian fundamental beam, which provides for exceptionally clean and high signal-to-noise studies of the HHG process itself. Here, we use these techniques and present a new set of experimental and theoretical studies of harmonics 7 to 13, generated below and near the ionization threshold in xenon gas with an intense 1,070 nm laser pulse. These harmonic orders are of great interest for the development of vacuum-ultraviolet frequency combs and it is crucial to understand how fluctuations in the driving-laser intensity influence the pulse-to-pulse coherence properties of the train of vacuum-ultraviolet pulses through the nonlinear HHG process. Production of frequency combs through HHG thus requires a systematic understanding of strong-field atomic physics8,9,10,11, but at the same time it provides a good probe for the latter subject.

Surprisingly, we find that harmonics as low as the seventh contain several contributions with different intensity-dependent phases. Our laser system permits a clean observation of interference between these phase contributions at unprecedented signal-to-noise ratios. In particular, we see experimentally a strong contribution from a component that has a large, intensity-dependent phase and we show theoretically that it originates in semi-classical laser-driven continuum dynamics. We demonstrate that this phase is significantly larger than for above-threshold harmonics owing to the influence of the atomic potential on these low-order harmonics. These measurements also provide a quantitative basis to evaluate the long-term temporal coherence of the vacuum-ultraviolet pulse train, a necessary condition for the establishment of a vacuum-ultraviolet frequency comb. Under reasonable experimental conditions, we find that the frequency-comb structure should indeed emerge. To verify this, we demonstrate for the first time coherence between successive pulses in our seventh-harmonic pulse train (λ≈153 nm), which shows a coherence time roughly five orders of magnitude larger than that shown previously12,13.

For this study, we use a passive optical cavity to enhance a high-power mode-locked femtosecond fibre laser at 1,070 nm. We are able to achieve the peak power necessary for HHG at high repetition frequencies (136 MHz), which gives a very high signal-to-noise ratio and allows the pulse-to-pulse coherence measurements shown later. Figure 1 shows the experimentally measured on-axis yields of harmonics 7 to 13 as a function of the laser intensity in the centre of the gas jet. Interestingly, all four harmonics show complicated intensity-dependent yields, with steep increases interrupted by steps. For harmonics 7 and 11, the intensity steps are more pronounced and occur at lower intensities than for harmonics 9 and 13. The insets show experimentally measured far-field spatial profiles of each harmonic at an intensity of 2×1013 W cm−2. At this intensity, harmonics 7 and 11 also show strong off-axis halos in their spatial distributions, whereas 9 and 13 do not. We also note that, in contrast to on-axis, the off-axis intensity-dependent yield shows no interference effects.

Figure 1: Harmonic yield as a function of the intensity at the centre of the xenon jet.
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a–d, Harmonics 7 to 13, respectively. The red filled circles are measured and the blue solid lines are theory (see text), both for the on-axis yield. The harmonic yields are all roughly proportional to the sixth power of the driving intensity. The horizontal dashed lines in b–d show the noise floor of the CCD camera. Theoretical harmonic yields were scaled in magnitude to offer a clear comparison. The insets show the far-field spatial profile for each harmonic. Halos on the beam profile and strong oscillations in the intensity-dependent yield are clearly visible in harmonics 7 and 11, providing clear evidence of multiple generation pathways with distinct intensity-dependent phases.

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The experimental results are very well reproduced by the theoretical calculations (shown in Fig. 1 for comparison) obtained by means of the coupled, non-adiabatic solutions of the time-dependent Schrödinger equation and the wave equation for a gas of xenon atoms exposed to an intense, 1,070 nm laser pulse. Our approach is described in detail in ref. 14, with the important difference that for the work described here we are directly integrating the time-dependent Schrödinger equation numerically within the single-active-electron approximation15. We are thus treating the laser electric field and the atomic potential on an equal footing, which is necessary to describe harmonics with photon energies below and close to the ionization threshold. As initial conditions for the calculation, we use the same parameters as the experiment (the calculations are carried out with 50-fs pulses to save computing time; however, we have checked that the results do not change for longer pulses). The theory results reproduce the overall increase of the yield with intensity and show prominent intensity-dependent steps in harmonics 7 and 11, and less pronounced steps in harmonics 9 and 13. Furthermore, the positions of the steps are remarkably well reproduced by theory.

Our experimental and theoretical results suggest that the observed intensity dependence is due to interference between contributions to the harmonic generation process that have different intensity-dependent phases. To explore these different contributions, we analyse the single-atom intensity-dependent dipole moment dq(I) for each harmonic q in terms of its conjugate-phase variable α, as described in detail in ref. 16. From the calculated dipole phase, we deduce the weight of each contribution, called a quantum path in the literature, over a range of intensities around I0, by means of the transform:

where W(I−I0) is a window function centred on I0. The contribution characterized by phase coefficient αj has a phase that is proportional to the intensity through φj=αjUp(I)/ℏω, where Up=I/4ω2 is the ponderomotive energy in atomic units and ω is the laser frequency. For above-threshold harmonics, the phase coefficients obtained in this way correspond well to those predicted by the semi-classical model17, including the familiar short and long trajectories with phase coefficients of α1≈0.2π and α2≈2π, respectively18. To our knowledge, this analysis has not been applied to below-threshold harmonics before.

Figure 2 shows the results of the quantum path analysis for harmonics 7–13 in xenon. All of the harmonics show multiple quantum path contributions; the two dominant ones have phase coefficients α0≈0 and α2≈2.5π–3π. To understand the origin of the α2 contribution to the below-threshold harmonics, we have studied electron trajectories in a generalized semi-classical model in which the atomic potential is present. Crucially, we have found that in the presence of the atomic potential, low-energy electrons can lose enough energy to have less than zero total energy at x=0 (the position of the ion), leading to the emission of below-threshold harmonics. In the tunnel-ionization model, this can happen only for electrons that have followed the long trajectory because they return to xi, the position of ionization, at a time when the force from the laser field is against their motion. We find that the long trajectories accumulate an intensity-dependent phase that is larger than in the no-potential case, with a phase coefficient close to 3π for the range of intensities we are interested in, rather than the 2π predicted with no potential. This is in good agreement with the result in Fig. 2. There are no negative-energy returns for short-trajectory electrons in the tunnelling model because they return to xi at a time when the laser field further accelerates them towards x=0. This leads us to designate the α0 contribution as a multiphoton process with no intensity-dependent phase and not a generalized short trajectory. We note that harmonics 3 and 5 show only an α0 contribution.

Figure 2: Quantum path distributions calculated for harmonics 7–13.
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The colour scale has been normalized for each intensity and shows only relative strengths. Harmonics 7 and 11 show large quantum path contributions at α≈0 and α≈3π that manifest themselves in the spatial profiles with halos and intensity-dependent yields with large oscillations, as shown in Fig. 1. Harmonic 9 (13) does not show prominent contributions for α>2π at intensities below ∼3×1013 W cm−2 (∼2.5×1013 W cm−2), so the features indicative of multiple quantum paths are largely absent at the driving intensities present in the experiment.

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The calculations in Fig. 2 confirm our interpretation of the experimental results in Fig. 1 as manifestations of the multiple generation mechanisms. The strong halos in the far-field spatial profiles of harmonics 7 and 11 are consistent with a large α2 contribution to these harmonics at intensities below 2×1013 W cm−2. This is because the large intensity-dependent phase, coupled with the Gaussian spatial profile of the fundamental beam, produces a strongly curved harmonic wavefront19. For harmonics 9 and 13, the α2 contribution in Fig. 2 is significant only at the upper range of intensities used in this experiment. As an experimental confirmation, we observe the sudden appearance of a halo on harmonic 9 at our highest experimental intensity. This reveals that the seemingly distinct behaviour of harmonics 9 and 13 arises mainly from the higher intensity threshold necessary to observe a large α2 contribution. Although several generation pathways are familiar from HHG high above the ionization threshold, it has never been observed (or predicted) so far below threshold. Previous works have noted that harmonics close to threshold can have large intensity-dependent phases20 and that harmonics well below threshold can show a step in the intensity-dependent yield, which was interpreted as an atomic resonance effect21. Our results explain and extend these observations to harmonics well below and around threshold without invoking atomic resonances. As it turns out that the intensity dependence of the dipole phase below threshold is due primarily to continuum dynamics, we have now a general quantitative connection between the laser intensity and the phase for these harmonics.

The observation of a large intensity-dependent phase contribution to harmonics as low as the seventh in our experiment allows us to connect these measurements to several other strong-field phenomena observed in recent years. These experiments, which measured ionization dynamics rather than photon emission, all invoke motion in the combined field of the ion and laser field to explain the behaviour of electrons with energies below the ionization threshold. Ho and Eberly studied non-sequential double ionization by means of classical simulations and found that trajectories for which an ionized electron was slowed down as it approached the ion significantly enhanced the e–e interaction that leads to double ionization22. Recently, Nubbemeyer, et al. found an unexpectedly large fraction of excited neutral atoms surviving a strong laser pulse23. Their explanation, supported by classical simulations, invokes tunnel-ionization trajectories very similar to those we discuss here. Finally, Shuman et al. have studied multiphoton-assisted recombination, where an ionized electron exchanges momentum with the laser field in the presence of the Coulomb potential and reattaches to the ion24. Our observations and the high dynamic range associated with them, open the possibility to study the electron dynamics invoked in these experiments directly by means of the HHG process.

As the HHG process is extremely nonlinear, it has been a significant concern that frequency-comb coherence could not be maintained through HHG. Previous experiments have measured the coherence maintained in HHG but only over the timescale of a single pulse12,13. However, to produce a frequency comb, it is a prerequisite that coherence is maintained between successive pulses in the train as this sets the frequency-comb linewidth to be less than the repetition frequency. This comb linewidth determines the ultimate frequency resolution for experiments using the vacuum-ultraviolet frequency comb5. The α parameters discussed above are a direct measure of the intensity-to-phase noise conversion in the below-threshold generation process that is directly relevant to the linewidth of a vacuum-ultraviolet frequency-comb mode. For instance, using α parameters extracted from the experiment, we can estimate the percentage of the power that would be removed from individual 1-kHz-linewidth comb modes for a laser source with a 1% r.m.s. intensity fluctuation with a characteristic 1/f distribution for each of the generation processes25. The results are presented in Table 1. For comparison, we include α and phase values for harmonics produced high above the ionization threshold, for instance in argon at an intensity of 2×1014 W cm−2.

Table 1 Frequency-comb parameters for below-threshold harmonics () at an intensity of I=2×1013 W cm−2 and for above-threshold harmonics () at I=2×1014 W cm−2.
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Our estimates of the intensity-to-phase noise put firm limits on the intensity noise that can be tolerated to preserve the comb structure in HHG and it is clear from Table 1 that a very high degree of pulse-to-pulse coherence should be possible in principle. To put this question to rest, we demonstrate pulse-to-pulse coherence of the seventh harmonic. For this study, the harmonic radiation is collimated and sent through a cross-correlation interferometer with a delay set so that a pulse in the harmonic train will interfere with the subsequent pulse, allowing us to measure the degree of pulse-to-pulse coherence26. As the seventh-harmonic radiation is absorbed in air and most solid materials, extreme care was taken in the design and construction of this interferometer, with all necessary optical alignments being conducted under vacuum.

Figure 3 shows the interference signal as the path length of the interferometer was scanned over 600 nm in about 2 s. The pulse-to-pulse coherence is clearly visible in the figure as a cross-correlation that corresponds to the seventh-harmonic wavelength of ≈150 nm. The contrast ratio shown in the figure is only ≈6%; however, if we take into account imperfect collimation and severe power imbalance in the interferometer, we estimate that the maximum achievable contrast ratio was 10–15%. To provide a zero background for a significantly enhanced measurement contrast, we also used a lock-in detection by dithering the length of the interferometer at 392 kHz as the length of the interferometer was scanned. The demodulated signal is also shown in Fig. 3.

Figure 3: Seventh-harmonic pulse-to-pulse coherence measurement.
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a, Pulse-to-pulse measurement set-up: a vacuum-ultraviolet interferometer with the delay set for pulse n in the seventh-harmonic pulse train to interfere with pulse n+1. HR: high reflector. b, The electric field of pulse n in the seventh-harmonic pulse train that has been sent through an optical delay line so that it interferes with the n+1 pulse in the train, which is shown as the dashed red trace. c, A cross-correlation signal results as the length of the delay line is scanned. d, The blue filled circles show our experimental measurement of the cross-correlation as the path length of the interferometer is linearly scanned, showing a contrast ratio of ≈6% limited by the spatial-mode overlap and the power imbalance between the two interfering pulses. The red open circles show the signal achieved from a lock-in detection of a 392 kHz modulation of the delay line. Both sets of data are fitted with sinusoidal functions. Point 1 in c and d shows the positions of a maximum negative lock-in signal where the cross-correlation signal at d.c. has a maximum negative slope. Point 2 shows the position of a zero lock-in signal with the d.c. cross-correlation at its maximum.

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As imperfect alignment in the interferometer and pulse overlap can easily decrease the contrast ratio further, our measurement provides only a lower bound on the coherence. By measuring the pulse-to-pulse coherence, we place a lower bound on the coherence time (∼10 ns), which is roughly five orders of magnitude longer than that measured before12,13. The corresponding frequency-comb linewidth is narrower than 20 MHz.

We have found experimental and theoretical evidence for multiple contributions to the generation of below-threshold harmonics. In particular, one of these contributions is dominated by laser-driven continuum dynamics, in analogy with the semi-classical model for generation of high-order harmonics. Our measurements of the spatial and spectral profiles and of the intensity dependence of the interference pattern all confirm that the intensity-dependent phase is larger for these low-order harmonics than for high-order harmonics owing to the influence of the atomic potential. We used our measured phase coefficients to estimate the coherence one can expect in both low-order and high-order harmonic frequency combs and demonstrated that indeed, pulse-to-pulse coherence was maintained for the below-threshold harmonic generation.

Methods

The enhancement cavity.

To excite the enhancement cavity, we use a high-powered, frequency-controlled 1,070 nm fibre laser system that supplies 100-fs pulses at 136 MHz with a pulse energy of 75 nJ (refs 27, 28). When the pulse train is resonant with the cavity (obtained by actively controlling the two independent degrees of freedom of the frequency comb, the repetition frequency and the offset frequency), we achieve an enhancement of 260, which translates to an intracavity pulse energy of 19 μJ. Two 10-cm radius-of-curvature mirrors within the cavity produce a focal spot area of 960 μm2 that yields a peak intensity of 4×1013 W cm−2. To generate harmonic radiation, we inject xenon gas near the intracavity focus using a glass nozzle with a 100 μm aperture and a backing pressure of 425 torr. An extreme-ultraviolet diffraction grating is used as one element of the enhancement cavity so that a portion of the harmonic radiation generated diffracts out of the cavity and impinges on a fluorescent plate29. By imaging the fluorescence onto a CCD (charge-coupled device) camera, we are able to make measurements of the power and far-field profile for individual harmonics. To rule out possible effects from nonlinear responses of the detection scheme (the combination of sodium salicylate with a CCD camera), we retook the experimental data shown in Fig. 1 with a decreased pressure of xenon gas that resulted in an order of magnitude decrease in the harmonic flux. The essential features in Fig. 1 (the slopes and oscillations) were minimally affected, ruling out the possibility of any significant detection nonlinearity.

Intensity calibration.

The peak power within the enhancement cavity was determined accurately by measuring the average power within the enhancement cavity and the pulse duration of the driving field. The average power within the cavity was determined by monitoring the transmitted light through one cavity mirror for which the transmission was well calibrated and the pulse duration was determined by autocorrelation of the transmitted light. As the enhancement cavity defines the resonant spatial mode, the mode volume at the focus can be determined accurately by a simple calculation. Together with the determination of the peak power, this calculation provides a dependable calibration of the peak intensity within the enhancement cavity.

The use of an optical cavity ensures that the spatial profile of the optical cavity field is well defined, power independent and can be precisely determined. The beam profile of the transmitted fundamental radiation was measured and found to be a near-perfect TEM00 as a result of the transverse-mode filtering provided by the enhancement cavity. This mode structure is maintained at the intracavity focus so that we can be confident that the harmonics were being generated by a high-quality TEM00 beam. The exact location of the experimental laser focus relative to the gas jet, which is known only to within 200 μm, is inferred by aligning the experimental and theoretical result at the lowest intensity step for harmonic 7. This results in all of the intensity-step positions being remarkably consistent between theory and experiment.

Experimental determination of phase-coefficient values.

To extract values for the phase coefficients α0 and α2 shown in Table 1, we made careful measurements of the inner and outer parts of the spatial (or spectral) profile of harmonic 7 at a well-known intensity30. If we assume that the harmonic has a Gaussian spatial (temporal) profile with an extra intensity-dependent phase and a focal spot (duration) that is one half of the infrared spot (pulse), we find α2≈2.7π and α0≤0.2π. Likewise, using the intensity separation between the two prominent steps in Fig. 1 as the ‘period’ of the interference between the two contributions, we extract a value of α2−α0≈3π (ref. 31). All three of these measurements are thus in excellent agreement with the theoretical predictions.

Coherence measurements.

For the pulse-to-pulse coherence measurements, the harmonic radiation outcoupled with the intracavity diffraction grating was sent through a vacuum-ultraviolet interferometer (see Fig. 4). A MgF2 beam splitter, calcium fluoride lenses and aluminium mirrors optimized for operation in the vacuum-ultraviolet were used for the construction of the interferometer. As the reflectivity of the MgF2 beam splitter was only due to the Fresnel reflection (0.09 for 152 nm light and S polarization), the ratio of power in the two arms was ≈0.12. One end-mirror of the interferometer was mounted on a piezoelectric transducer (PZT) so that the optical delay of the interferometer could be scanned and quickly modulated. To detect radiation at the output port, we used a conventional photomultiplier tube (PMT) with a thin layer of sodium salicylate on the front surface. The sodium salicylate fluoresces at a wavelength of about 420 nm when struck by the seventh-harmonic radiation, which matches very well to the peak detection efficiency of the PMT. To observe the interference with a better contrast, we applied a 392 kHz modulation to the PZT. The PZT and mounted mirror had a natural resonance close to this frequency, allowing enhanced mirror travel. The demodulated signal is basically the first-order derivative of the d.c.-based interference measurement.

Figure 4: Experimental set-up for the demonstration of vacuum-ultraviolet pulse-to-pulse coherence.
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A cavity-enhanced Yb-fibre frequency comb is used to generate harmonics in xenon gas. Photodiodes (PD) are used to monitor the reflected and transmitted fundamental light and maintain resonance between the frequency comb and the enhancement cavity1,2. An intracavity vacuum-ultraviolet diffraction grating (VUV-dg) is used to outcouple the seventh-harmonic radiation29, which is then sent through a cross-correlation interferometer. One mirror in the delay arm of the interferometer is mounted on a PZT, allowing scans of the differential path length. The seventh-harmonic radiation at the output of the interferometer is detected by monitoring the fluorescence from a sodium-salicylate-coated plate with a PMT.

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