1. Introduction

The accurate knowledge of the laser pulse shape at the interaction point is crucial for many applications in ultrafast science, particularly in the presence of strong dispersion. In the case of strong-field processes, which can depend directly on the evolution of the electric field $E(t)$, the pulse characterization gains even more importance. Non-linear optical processes such as second-harmonic generation provide the key to measure the duration of femtosecond laser pulses [1] and a wide range of suitable techniques have been demonstrated over the past decades [2–4]. While a simple autocorrelation measurement only provides some information on the actual pulse extent, more detailed and accurate information including the spectral phase $\phi (\omega )$ can be obtained by techniques such as frequency-resolved optical gating (FROG) [5,6], spectral phase interferometry for direct electric-field reconstruction (SPIDER) [7] or dispersion scan [8,9]. While these traditional methods require only little pulse energy, they are limited with respect to minimal pulse duration and spectral range, as they rely on phase matching in non-linear crystals and require detectors with suitable spectral response. Moreover, full characterization of few-cycle waveforms [10] or even shorter light transients, such as synthesized light fields [11,12] requires detection of the carrier-envelope phase (CEP), or a direct measurement of $E(t)$.

The complete measurement of few-cycle pulses has been first achieved using attosecond streaking [13], and later using the petahertz optical oscilloscope [14] and attosecond resolved interferometric electric-field sampling [15]. These techniques rely on gas-phase high-harmonic generation (HHG), which implies a relatively complex setup. More recently, new pulse characterization methods have been developed which utilize non-perturbative, strong-field effects and relatively simple detection schemes. For example, tunneling ionization with a perturbation for the time-domain observation of the electric field (TIPTOE) [16] relies on the extreme non-linearity of tunneling ionization of air to create an optical gate that is sufficiently short to resolve essentially any achievable pulse duration in the visible or infrared. Very recently, the same working principle has been applied to strong-field excitation of bulk ZnO crystals [17] and similarly to multiphoton excitation in Si-based camera chips [18]. The idea of TIPTOE is to perturb the non-perturbative strong-field effect by a weak replica of the pump pulse. These effects offer essentially unlimited bandwidth and possess very high nonlinearities. In the limit of a small perturbation, however, the modulations are proportional to the perturbing electric field, which allows for a straightforward retrieval of $E(t)$ if the perturbing pulse is the pulse to characterize. While the wide range of available techniques allows one to characterize essentially any possible laser field, the characterization instrumentation is typically a secondary experiment set somewhere on the beam path. However, it is sometimes desirable to characterize pulses right at the point of interaction, for example, in the presence of strong dispersion, or if the dispersion shall be characterized [19].

Here, we demonstrate an approach that can be readily integrated into experiments on HHG from solids [20,21] and allows for the retrieval of the electric field evolution of infrared laser pulses used for HHG. We characterize MIR few-cycle laser pulses centered around 3.2 µm wavelength by means of HHG from ZnO thin-films and WS$_2$ monolayers, thus avoiding dispersion and phase matching issues. The scheme is compatible with any target and laser wavelength suitable for HHG. It requires only a few µJ of pulse energy and permits convenient detection of the signal in the visible and UV spectral ranges, i.e. where the HHG signal is routinely detected.

2. Methods

The working principle of our approach, HHG-TOE, is similar to TIPTOE [16]. The key is to introduce a perturbation to solid-state HHG, which is sufficiently weak, such that the field-dependent yield of the n-th order harmonic $S_n(E)$ can be expressed as a Taylor series,

The experiments were conducted at the Extreme Light Infrastructure-Attosecond Light Pulse Source (ELI-ALPS) located in Szeged, Hungary. The MIR laser system delivers up to 145 µJ, 45 fs CEP-stable pulses centered at 3200 nm wavelength with a repetition rate of 100 kHz [22]. A pair of wire grid polarizers are used to control the power sent into the experiment. The dispersion of the polarizers is part of the pulse compression scheme, and the pulses are routinely characterized using a commercial TIPTOE device. In front of our experiment, a pair of BaF$_2$ wedges is used for dispersion control.

The experimental setup for HHG-TOE is shown in Fig. 1. The incoming laser beam is split into a strong pump pulse ($< {5}\;\mathrm{\mu}\textrm{J}$) and a weak probe ($\sim {0.1}\;\mathrm{\mu}\textrm{J}$) by sending it off-center onto a D-shaped mirror at near-normal incidence. The D-shaped mirror is mounted on an open-loop piezo linear stage in order to control the delay between the pump and the probe pulses. The two beams are focused ($f = {200}{mm}$) into the HHG sample. We estimate that its intensity in the focus is at least two orders of magnitude below that of the pump pulse. The angle between the beams is approximately $1.5^{\circ}$, which causes a phase slip across the laser focus but does not invalidate the experimental approach, as demonstrated by our results and verified numerically below. The diameter of the focal spot of the pump beam is determined as ${70}\;{\pm}\;30\;\mathrm{\mu}\textrm{m}$ by investigating the optically-induced breakdown of sapphire, and imaging the photoluminescence from a 500 µm thick ZnO crystal placed in the focal plane. This diagnosis tool proved useful also for overlapping pump and probe beams. The harmonics generated in the HHG sample placed in the laser focus were filtered spatially from the fundamental using an iris. The transmitted harmonics are focused using a fused silica lens ($f = 100\,$mm) into a UV-NIR spectrometer (OceanFX).

 

Fig. 1. The Experimental scheme. The fundamental beam is split into 2 unequal parts using a pair of D-shaped mirrors. The concave mirror is used to focus the 2 beams into the sample. An iris is used to block the majority of the fundamental light while the high harmonic emission is largely transmitted through its opening. A fused silica lens is used to focus the transmitted light into the spectrometer. The high-harmonic spectrum is recorded as a function of the delay between the two beams.

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The samples used for HHG-TOE consist of a 0.5 mm c-cut sapphire substrate coated with roughly 50 nm thick polycrystalline ZnO or monolayers of WS$_2$ on 0.5 mm a-cut sapphire. The ZnO thin films were grown by using RF magnetron sputtering at room temperature [23]. The thickness of the ZnO layer is controlled by varying the duration of the growth process. The monolayer single crystals of WS$_2$ are grown on the sapphire substrate using chemical vapor deposition [24].

In order to validate the non-collinear geometry and explore its limitations we perform numerical simulations. The spatiotemporal field distribution $E(x,y,t)$ in the focal plane is calculated for the superposition of two non-collinear pulses similar to the ones used in the experiment, intersecting each other at an angle $\alpha$. The variation of $E(x,y,t)$ along the propagation direction and other propagation effects are neglected, as justified by the very thin samples used in the experiment. In our numerical model, we assume that HHG can be described by the rate of inter-band excitation [25]. Thus, we calculate the HHG generated in the focal region by integrating the tunnel ionization yield obtained by the Ammosov-Delone-Krainov formula [26] over the focal volume. We use the band gap of ZnO (3.2 eV) as ionization potential but have verified that the results do not significantly depend on the specific choice of the formula relating the HHG yield to the laser electric field. In particular, the results remain qualitatively the same when assuming perturbative rather than non-perturbative scaling of the harmonic emission. The calculations are carried out for various values of the pump-probe delay, in order to obtain the delay-dependent HHG yield in which the pulse shape is imprinted. Subsequently, the same pulse retrieval procedure as for the experiment is applied to the simulated HHG yield.

3. Results and discussion

Experimental results for the characterization of 45-fs pulses are presented in Fig. 2. Harmonics up to the 13th order (245 nm) have been detected, the latter being very weak and therefore ignored in the further analysis. At the temporal overlap of pump and probe pulses, varying the delay causes pronounced modulations in the HHG yield but does not notably change the shape of the spectra. The period of the modulations corresponds to the optical period of the fundamental field. At the center of the overlap, the modulation depth reaches nearly 100% and drops quickly to either side, though a small oscillating signal remains over a range of a few hundred fs, indicating the presence of a pulse pedestal. The delay-averaged yield outside the region of the overlap represents the zeroth-order in the Taylor series in Eq. (1). The modulations represent the effect of the first-order term, which we normalize to the magnitude of the zeroth-order term. For higher harmonic orders, the modulation depth tends to be larger, which is attributed to a stronger electric-field dependence of the high-order harmonics. The delay-dependent modulations for all harmonic orders are in-phase with each other.

 

Fig. 2. HHG-TOE measurement of 45-fs pulses at 3.2 µm wavelength in 50 nm ZnO films. (a) High-harmonic yield integrated over all harmonics, as a function of delay. The yield is normalized to the delay-independent signal outside the overlap. (b) Same as (a) for each harmonic. The signals are vertically displaced for visual convenience. (c) Delay-dependence of the HHG spectra with several harmonic orders indicated. The yellow dashed lines indicate the spectral ranges over which the harmonic signal is integrated for panel (b). (d) The delay-integrated HHG spectra.

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The first step in our pulse retrieval procedure is to perform a Fourier transform along the delay axis, see Fig. 3. The power spectrum shows a strong peak at zero frequency, as well as a pronounced signal at the fundamental frequency. These are associated with the zeroth and first-order terms of the Taylor series of Eq. (1). The continuous signal observed at the harmonic wavelengths for all Fourier frequencies indicates the contributions from essentially white noise. The absence of clear signals at higher frequencies, in particular at the second harmonic, indicates that the probe pulse is sufficiently weak such that the Taylor series of Eq. (1) truncates after the first order. In further measurements where a more intense probe pulse was used, the contributions of the second-order term show up as a signal oscillating at twice the fundamental frequency. However, even in this case, the pulse retrieval method yields robust results.

 

Fig. 3. Pulse retrieval from HHG-TOE measurement. (a) Frequency domain representation of the data presented in Fig. 2. The signal oscillating at the fundamental frequency is selected (the yellow squares) for further analysis. (b) Spectral power and phase extracted for the 9th harmonic, compared to those retrieved from a separate TIPTOE measurement. (c) Time-domain representation of the delay-dependent signal of the 9th harmonic (red) compared to the pulse retrieved by HHG-TOE (black), the transform-limited field (blue), and results from a TIPTOE measurement (green).

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The frequency-domain representation of the perturbation field is obtained by integrating the signal corresponding to the first-order term (inside the yellow box in Fig. 3(a) over the wavelength axis. The spectral power and phase of the probe pulse, retrieved by HHG-TOE, are shown in Fig. 3(b). The results agree well with those obtained by a commercial TIPTOE device. The presented results were obtained for the 9th harmonic but any detected harmonic order could be used equivalently. The main limitation is given by the signal-to-noise ratio, which leads to poor results for the 13th harmonic. The group delay dispersion (GDD) is obtained by fitting the phase to a second-order polynomial and weighting it with the spectral power. The extracted GDD values from the 5th, 7th, 9th, and 11th harmonic are $-{205}\;{\pm}\;{102}{fs^2}$, $-{255}\;{\pm}\;{122}{fs^2}$, $-{263}\;{\pm}\;{131}{fs^2}$, and ${-160}\;{\pm} \;{130}{fs^2}$, respectively, whereas the TIPTOE GDD value yields 27 fs². The difference between the HHG-TOE and TIPTOE results are consistent with propagation through approximately 2 mm of BaF₂, which was used for dispersion control in front of the HHG-TOE setup. However, the additional chirp may also result from spatiotemporal coupling effects, as discussed below.

The electric field in the time-domain is obtained by inverse Fourier transform and shown in Fig. 3(c). The retrieved pulse exhibits strongly reduced noise as compared to the raw signal, which indeed resembles the pulse shape rather accurately. For the harmonic orders 5 through 11 the following pulse durations are obtained, ${45.7}\;{\pm}\;{1.7}{fs}$, ${47.4}\;{\pm}\;{1.9}{fs}$, ${46.1}\;{\pm}\; {1.8}{fs}$ and ${48.3}\;{\pm}\;{1.8}{fs}$. The TIPTOE reconstruction yields ${45.3}{fs}$ and the Fourier transform limit is ${39.8}{fs}$, obtained from the independently measured spectrum of the fundamental pulse.

In comparison to the TIPTOE result, the HHG-TOE signals show stronger pre- and post-pulses. We attribute this difference to the usage of the D-shaped mirror to create an intense and a weak replica of the incoming pulse. The weak perturbation field, which is characterized by HHG-TOE, is sampled from the edge of the beam profile in our experiment. Due to spatiotemporal coupling effects, the pulse shape at the edges of the beam profile may be less clean as compared to the center [27]. The TIPTOE device is less affected by such beam imperfections as the probe pulse is obtained from the center of the pulse by using a holey mirror to split the beam into two parts. This design would be advantageous also for HHG-TOE. In our current setup, one may sample the cleaner region of the beam profile by cropping the beam using an iris aperture in front of the D-shape mirror. This was done in the measurements presented below.

To test the flexibility of HHG-TOE, we have performed measurements on WS$_2$ monolayers, which are presented in Fig. 4. In addition to the odd harmonics, which are generated in centrosymmetric materials such as c-cut ZnO, also even-order harmonics can be generated in WS$_2$ [28]. In our experiment, these harmonics exhibit the same type of modulation as the odd-order harmonics. We select the 10th harmonic for pulse retrieval and conduct the same procedure as for the signal produced in ZnO thin-films above. The retrieved pulse has a duration of ${40.7}\;{\pm} \;{1.4}{fs}$, in very good agreement with the separately obtained TIPTOE result and the Fourier transform limit of the measured spectrum of 40.6 fs. Our results demonstrate that the HHG-TOE scheme is suitable for on-target pulse characterization even in the case of a single atomic layer.

 

Fig. 4. HHG-TOE measurement for 45-fs pulses at 3.2 µm wavelength in monolayer WS$_2$ grown on 0.5 mm thick, a-cut sapphire. (a) Delay-dependent HHG yield integrated over the entire spectral range of the measurement. The yield is normalized to the delay-averaged yield. (b) Same as (a) but for harmonic orders 10 through 5, from top to bottom. (c) Delay-dependent high-harmonic spectrum measured with an exposure time of 300 ms per time step. Harmonic orders are indicated. The inset shows a microscope image of the sample, with the monolayer WS$_2$ flake used for the measurements marked. The hole was created by the laser to mark the spot. (d) High-harmonic spectrum integrated over the delay. (e) Spectral power and phase of the probe pulse, obtained from the delay-dependent signal recorded for the 10th harmonic, compared to results obtained from TIPTOE. (f) Time-domain representation of the delay-dependent signal of the 10th harmonic (red) compared to the pulse retrieved by HHG-TOE (black), the transform-limited field (blue) and results from TIPTOE measurement (green).

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In Fig. 5 we present numerical results, which further validate our experimental results. The simulations were obtained for laser pulses similar to the ones used in the experiments and with an intensity ratio of 100:1 in the focus. Our simulations confirm that the retrieved pulse possesses a CEP value equal to the CEP difference between probe and pump pulses. The pulse duration of the retrieved pulse is 46.4 fs, somewhat longer than that of the probe pulse used for simulations 42.5 fs.

 

Fig. 5. Numerical simulations of HHG-TOE for a few-cycle pulse centered at 3.2 µm. (a) The 42-fs pump and probe pulses in the time domain with different CEP values. The insets show the beam profiles of the pump (left) and probe (right) pulses normalized to their peaks. (b) Simulated HHG yield for different harmonic orders, offset vertically for visual convenience. The yield is normalized to the delay-averaged yield. The inset shows the beam profile of the superposition of pump and probe at zero delay. (c) The retrieved pulse has a pulse duration of 46.4 fs, a CEP of $\pi /2$ and is plotted along with the orignal pulse and the raw signal before Fourier filtering. (d) Fourier domain representation of the simulated HHG signal (black line and symbols) and the 42.5-fs probe pulse (green lines and symbols).

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We have extensively tested the limitations of HHG-TOE using the numerical model. Since our model treats the HHG process like tunneling ionization, the following conclusions are applicable to TIPTOE as well. With regard to the non-collinear geometry used in our experiment, we find that the pulse retrieval is quite robust with respect to the non-collinear angle between the beams. However, the modulation depth of the HHG yield depends on the angle between the beams. This is not surprising, as for increasing angles, regions in the focus have opposite temporal phase between the two pulses (see inset of Fig. 5(b). Hence, the modulations caused by these regions cancel out each other and thereby affect the signal-to-noise ratio of the measurement, which can have an impact on the accuracy of the pulse retrieval. For the experimental value of $1.5^\circ$, the modulation depth is sufficient to allow for robust pulse retrieval, as in the experiment.

The impact of noise on the pulse retrieval has been investigated by considering both intensity fluctuations and phase fluctuations when calculating the HHG yield. In the simulations, whose results are presented in Fig. 5, the root-mean-square (rms) of the intensity fluctuations is 2% of the peak intensity, and the phase fluctuations are 200 mrad rms. The retrieval of the main pulse is reliable, as long as the noise induced by intensity fluctuations is small compared to the modulations induced by the probe pulse. For example, with the intensity ratio of 100:1, we found that intensity fluctuations as high as 5%, and phase fluctuations as high as 1 rad are tolerable and the retrieved pulse duration is within 15% of the true pulse duration. However, it is important to note that the Fourier filter applied in the retrieval tends to turn white noise into oscillations at the central frequency of the filter window. This creates an artificial pedestal in the retrieval.

For the impact of the intensity ratio between pump and probe, we confirm the above conclusion that the second-order oscillations associated with the quadratic term in the Taylor expansion (Eq. (1)) does not impact the pulse retrieval. However, for higher probe intensities where third or higher-order oscillations become significant, the retrieved pulse may substantially deviate from the true pulse shape observed. The oscillations at the fundamental frequency will be a mixture of higher order terms with the first order, and the Fourier filtering would not differentiate between the different orders. A similar issue may occur for extremely broadband probe pulses whose spectrum exceeds a full octave. Especially in this case, the probe pulse intensity should be chosen sufficiently low such that the quadratic term in Eq. (1) is negligible as for the data shown in Fig. 3(a).

Finally, we address the mismatch between the retrieved and true pulse durations. This is related to the slight deviations in the leading and trailing edges of the retrieved pulse shape from the true pulse shape, as is visible in Fig. 5. A small deviation in these regions persists for any choice of non-collinear angle or intensity ratio. Only when a pump pulse as short as a half optical cycle is chosen, the pulse duration of the retrieved pulse perfectly matches that of the probe pulse used in the simulations. This indicates that the slight error is a consequence of the finite duration of the temporal gate created in the nonlinear interaction driven by the pump pulse. Hence, choosing higher harmonic orders for HHG-TOE may be advantageous due to their higher nonlinearities. Our simulations confirm this consideration, i.e. for higher nonlinearity the pulse duration of the retrieved pulse matches more closely that of the original pulse. However, even though the experimental results of the higher harmonics exhibit stronger modulations (cf. Figure 2), indicating a stronger nonlinearity for these orders, we do not clearly observe a reduction of pulse duration with higher harmonic order. Presumably, this is due to the decreased signal-to-noise ratio for high harmonic orders.

4. Conclusions and outlook

Our experimental results demonstrate that HHG from solids is suitable to accurately characterize ultrashort laser pulses. The scheme presented here appears particularly useful to measure the electric field evolution of laser pulses used for HHG experiments, as the laser pulse is characterized at the very point of interaction. This property makes HHG-TOE suitable to characterize linear and non-linear propagation at the nanoscale. Moreover, information on the HHG process is included in the form of a precise measurement of the field-dependence of various high-harmonic orders, i.e. $\frac {\partial S_n}{E}$. We envision that HHG-TOE may become a standard tool for pulse characterization in solid-state HHG experiments, as it is simple to implement and there are essentially no limitations on the type of material used for the experiment.