1. Introduction

The time-dependent electric field of a laser pulse dictates the dynamics of strong-field laser matter interactions. Therefore, characterization of the waveform is critical for the understanding and control of these interactions. Moreover, the precise determination of these parameters is especially important for applications in the few-cycle regime, e.g. the production of extreme-ultraviolet (XUV) pulses [1, 2], which serve as the basis for much of the burgeoning field of attosecond science [3]. Assuming a Fourier-transform-limited, Gaussian pulse, the waveform can be written as E(t) = E₀ exp(−t²/τ²)cos(ωt + ϕ), where ω is the laser field frequency, ϕ is the carrier-envelope phase (CEP), E₀ is the peak electric field, and the commonly-used full-width at half of the maximum (FWHM) pulse duration in intensity is defined as $\Delta t\equiv \text{FWHM}=\tau \sqrt{2\text{ln}2}$. Thus, for a given wavelength and peak intensity, the waveform is characterized by the pulse duration, Δt, along with the carrier-envelope phase (CEP), ϕ.

Typically, the measurement of these two quantities is performed by separate devices. The CEP is most often measured and/or controlled by f–2f interferometers [4–6], but alternative methods, such as analysis of high harmonic generation [7] and quantum interference [8], have also been developed. The pulse duration is typically measured using autocorrelation, frequency-resolved optical gating (FROG) [9, 10], or spectral phase interferometry for direct electric-field reconstruction (SPIDER) [11,12]. Until now, the exceptions, which measure both pulse duration and the CEP, have been attosecond streaking [13–18] and the analysis of velocity-map images (VMI) of above-threshold ionization (ATI) from nobel gases [19, 20], which require a CEP stabilized pulse train for the lengthy measurement times and relatively complex experimental apparatuses.

We have recently proposed and implemented a technique using the ATI of Xe to determine the CEP for each individual laser shot in a kHz pulse train in real-time [21–23]. This technique moves beyond the initial work of Wittmann et al. [24] as it facilitates data tagging and real-time analysis and requires no theoretical assumptions to precisely determine the relative CEP. In this paper, we show how the same device can be used to simultaneously measure the pulse duration in real-time. This differs significantly from the recent work of Chen et al. [25] in that our analysis does not rely upon theoretical assumptions, e.g. those of quantitative rescattering theory (QRS) [26], and that we address how the measurement can be accomplished in real-time. We believe the apparatus and technique described herein is a viable alternative to existing few-cycle pulse length and/or CEP measurements as it (i) accomplishes both simultaneously and in real-time, (ii) is easy to install and operate, (iii) operates with and without CEP locking, and (iv) operates at repetition rates from approximately 1 Hz to tens of kHz, which encompasses frequencies obtainable from TW class systems to commercially available tabletop Ti:Sapphire lasers.

2. Experimental setup

Ultrashort (28 fs) laser pulses are generated by a Ti:Sapphire Femtopower™ Compact™ Pro HP / HR CEP laser system, see Fig. 1(a). The frequency spectrum is then broadened using an Ne-filled hollow-core fiber. The pulse duration is varied by changing the pressure of Ne gas within the hollow-core fiber, i.e. increasing the gas pressure within the hollow-core fiber increases the spectral broadening and, thereby, decreases the Fourier-transform limited pulse duration. Chirped mirrors (CM) are then used to compensate for the spectral phase introduced by the fiber and produce a negative chirp to accommodate the dispersion in the remaining beam paths. Next, using a thin broadband beam splitter, the laser is split into two parts — one going to the stereo-ATI measurement, Fig. 1(b), and the other going to the SPIDER, Fig. 1(c). Both beam paths have independent sets of fused silica wedges (W2 and W3) to fine tune the spectral phase and minimize the pulse length in both measurements.

 

Fig. 1 (a) Schematic of the beam path (FM: out-coupling focusing mirror, CM: chirped mirrors, PM: plane mirror, W1–W3: fused silica wedges, BS: beam splitter). (b) Schematic of the stereographic apparatus used to measure ATI [28]. (c) Schematic of the single-shot SPIDER setup (BS: beam splitter, E: etalon, WP: λ/2 waveplate, S: stretcher, Fm: focusing mirror, X: nonlinear crystal, A: aperture, OMA: optical multichannel analyzer) [11,12]. See text for details.

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2.1. Stereographic ATI measurement

We utilize the facts that the ATI spectrum of Xe is dependent upon the CEP and that this dependence increases as the pulse length decreases. This is to say, the magnitude of the possible asymmetry in the electric field increases as the pulse length decreases and this asymmetry depends on the CEP and is reflected in the ATI spectrum [27]. The stereo-ATI measurement apparatus [28] is designed to take advantage of this fact and produce a real-time signal for every individual laser shot in a kHz pulse train corresponding to the CEP [21, 24] and pulse duration of that pulse.

This apparatus is depicted in Fig. 1(b) and consists of a high vacuum chamber with μ-metal shielding and two identical field-free time-of-flight (TOF) spectrometers arranged back-to-back [28]. The laser is focused into a Xe target where many atoms are ionized in each laser pulse. Those electrons emitted within a small angle around the laser polarization, ∼2°, are detected by micro-channel plate (MCP) detectors with metal anodes to the left, L, and right, R. The ATI spectra are encoded in the time-dependent electron currents for each pulse, which are then converted to voltage signals via capacitors and amplifiers.

The ATI yield is CEP dependent and the phase of this dependence varies with energy or equivalently time-of-flight (TOF) [21, 24, 27, 28]. To make this dependence more pronounced, we examine the asymmetry in the electron yield to the left (L) and right (R), i.e. A = (L – R)/(L + R). In the plateau region, the asymmetry, A, varies approximately sinusoidally with the CEP. Moreover, one can choose two regions in the spectra, which will be denoted as high and low energy, in which the sinusoidal dependence of A differs by a phase of 90°. That is to say, A_high ≈ A₀ sin(ϕ + ϕ₀) and A_low ≈ A₀ sin(ϕ + ϕ₀ + π/2), where ϕ is the CEP, ϕ₀ is an arbitrary offset, and A₀ is the amplitude of the asymmetry. In practice, the two energy regions are chosen by varying the bounds until two parameters are optimized: (i) the uniformity of the radius, i.e. the standard deviation in r is minimized, and (ii) the isotropic distribution of points in θ, i.e. the standard deviation in N(θ) is minimized. Finally, by plotting the two asymmetry values against one another in a parametric asymmetry plot (PAP), one finds that the polar angle (θ) corresponds to the CEP (ϕ), i.e. ϕ ≈ θ + ϕ₀. See Fig. 2(a) for typical data. Note that although the dependence of ϕ on θ deviates slightly from being purely linear, one can determine ϕ(θ) exactly using a pulse train in which the CEP is random and uniformly distributed [21].

 

Fig. 2 (a) Parametric asymmetry plots (PAP) for 4.02 (black), 5.70 (red), and 7.20 fs (blue). Each laser shot produces one point, 80,000 points are shown for each distribution, and the dashed circles mark the average radii for the 3 distributions. In these measurements the CEP is freely and randomly changing and uniformly distributed. (b) Radius of the PAP, r, as a function of pulse length (FWHM in intensity, Δt), where the radius is from the measured PAP and the pulse length is measured with a SPIDER (open circles). The error bars on the data points represent the standard deviation in the single-shot radial measurement and uncertainty in the pulse length measurement of the SPIDER. The data is fit with the function r = 1 – exp(−α/(Δt – β)²), where α = 9.7317 ± 1.5069, β = 1.6063 ± 0.2593, and R² = 0.9697. Additionally, the uncertainty in the pulse length measurement using a single laser shot (squares) and the ensemble of laser shots (triangles) is displayed. Points calculated using quantitative rescattering theory (QRS) [25] for a given pulse length (filled circles) are also shown. See text for details.

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In addition to revealing the CEP, the PAP is also dependent upon the pulse duration. Namely, the radial coordinate, r, in the PAP is related to the pulse duration, Δt. As the pulse length decreases the amplitude of the left-right-asymmetry (A₀) increases, i.e. as the pulse length moves from a many-cycle pulse to a few-cycle pulse, A₀ moves from 0 to 1. Moreover, as r ≈ A₀, the radial size of the PAP will increase as the pulse length decreases. This trend is illustrated in Fig. 2(a) using three different pulse lengths.

2.2. Few-cycle SPIDER

For spectral phase sensitive pulse characterization, we used a commercial SPIDER system (FC SPIDER by APE GmbH) that is specifically designed to measure few-cycle pulses [11]. Fig. 1(c) shows the schematic setup of this device, which splits the laser into a spectral amplitude characterization part with a fundamental spectrometer (OMA1) and a spectral phase characterization part with a second spectrometer (OMA2) for recording the up-converted pulses.

The introduction of the spectral shear between the two test pulses is realized by their non-collinear type-II interaction with the third, stretched pulse. The resulting sum-frequency generation (SFG) signal is detected by an UV-sensitive spectrometer. The incidence angle at the etalon is balanced between a higher Fresnel reflectivity and reduced intensity modulation of the transmitted beam for robust phase detection [12]. To support the robustness of the measurement, especially in case of fluctuating laser sources, the spectral power density is measured independently, yet synchronously with the spectral phase.

3. Results and discussion

3.1. pulse duration

Using the stereo-ATI and SPIDER measurements in tandem, we are able to determine the dependence of the radial dimension in the PAP, r, on the pulse duration, Δt. This relationship is shown in Fig. 2(b) and closely fits a function of the form r = 1 – exp(−α/(Δt – β)²). In the measured range, the slope of this curve increases as the pulse length decreases, so the measurement is more precise as the pulse length decreases. For few-cycle pulses roughly in the range of 4 to 7 fs, the radial dependence is especially strong and allows one to determine pulse length differences of a few 100 attoseconds. Additionally, this calibration does not depend upon any theoretical assumptions and is in rough agreement with the limited number of data points predicted using quantitative rescattering theory (QRS) [25] (see filled circles in Fig. 2(b)).

The aforementioned fitting function is based on a simple model. The peak value of a cosine pulse is E(t=0) = E₀ and the absolute value at the maximum of the next half cycle is E(t=π/ω) = E₀ exp(−(π/ωτ)²). Thus, if the half-cycle yields depend on the peak values of the electric field to some power, we expect the L–R asymmetry in the ATI yield to follow the form 1 – exp(−α/Δt²). Additionally, the offset, β, is added to better fit the data and can be thought of as the minimum pulse length required to produce rescattered electrons. Although this model is quite crude, it fits the data extremely well, with α = 9.7317 ± 1.5069, β = 1.6063 ± 0.2593, and R² = 0.9697.

Inverting this fit yields the pulse length as a function of the PAP radius, $\Delta t=\beta +\sqrt{-\alpha /\text{ln}\left(1-r\right)}$. Using this formula and propagating the uncertainties in the fit and radial measurement, one can determine the uncertainty in the stereo-ATI pulse length measurement. As one would expect and shown in Fig. 2(b), the error when using one laser shot (squares) is larger than when using a large ensemble of laser shots (triangles). However, unlike the CEP, which can vary over the full 2π range from shot-to-shot, the pulse duration is typically much more stable. Therefore, using the average radial value along with the single-shot CEP value still allows one to determine the waveform of individual laser pulses in a train. Furthermore, the number of values being averaged can be easily varied from one to many depending on the desired application.

3.2. Spectral phase

As compared to methods that retrieve the spectral phase of the pulse, e.g. SIPDER [11, 12] or frequency-resolved optical gating (FROG) [9, 10], our technique has a distinct disadvantage — we must assume a temporal Gaussian pulse shape. In many cases this is an appropriate assumption as spectral phase introduced by frequency broadening, e.g. hollow core fiber compression, is well compensated for using the appropriate chirped-mirrors. However, when one pushes the limits towards shorter pulses, the time-bandwidth product (TBP) typically increases to a point where assuming a Gaussian pulse shape is erroneous. This is illustrated in Figs. 3(a–d), which show the intensity envelopes and the spectral representation of a relatively Gaussian 6.3 fs pulse and a more complex 4.1 fs pulse. Nevertheless, as our technique is insensitive to structures below ∼30% of the peak intensity, the stereo-ATI measurement provides an excellent measurement of the pulse duration of the primary intensity envelope. Moreover, as many strong-field processes show a highly exponential dependence on the electric field, determination of primary pulse structure is often adequate.

 

Fig. 3 (a)–(d) Temporal and frequency spectra for 6.27 (top two panels) and 4.08 fs (bottom two panels) pulses derived from a SPIDER measurement. (e) Typical data for the PAP radius, r, as a function of fused silica introduced in the beam path for a few-cycle pulse with poorly balanced higher-order dispersion. (f) Same as (e) transformed to pulse length using the aforementioned r → Δt conversion. See text for details.

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Although our technique does not retrieve the spectral phase, some general information about the spectral phase can be determined. The more the spectral phase deviates from being flat, the smaller the maximum achievable radius of the PAP, r_max, will be with respect to the radius corresponding to the Fourier transform limited (FTL) temporal pulse duration, r_FTL. In other words, the closer the measured pulse duration is to the FTL and r_max → r_FTL, the flatter the spectral phase.

In practice, one will optimize the pulse duration by varying the amount of dispersive material in the beam path, e.g. by inserting fused silica wedges (W1 in Fig. 1(a)). A pulse carrying higher-order dispersion will typically lengthen and shorten in a non-monotonic way as dispersion is added or removed. This is because the phase for different parts of the spectrum becomes linear for different amounts of fused silica dispersion. For pulses with poorly balanced higher-order dispersion, we observe several local maxima for the radius, r, of the PAP as a function of the glass thickness (see Figs. 3(e) and (f)), while spectra with well balanced dispersion exhibit a single maximum. In general, the closer the radius of the PAP is to the FTL and the fewer and closer together the minima are in the dispersion scan, the flatter the spectral phase. Thus, one can use the stereo-ATI measurement alone to optimize the spectral phase.

3.3. Waveform characterization

The most important advantage of our technique is that the stereo-ATI measurement also yields the CEP of each laser pulse. Knowing the laser power and frequency and assuming a Gaussian pulse, this information allows one to reconstruct the time-dependent electric field of each pulse. To our knowledge, this is the only device capable of retrieving the approximate temporal waveform for single few-cycle laser pulses.

In addition to the stereo-ATI apparatus, we have developed and employ electronics, which allow for the calculation of the asymmetry parameters (A_high and A_low) along with r and θ within ∼20 μs of the laser interaction [21, 23]. This combination enables real-time characterization of the waveform for each laser pulse in a kHz pulse train. That is to say: (i) There is no dead time between laser shots or groups of laser shots. (ii) The parameters are determined for each laser shot completely independent of adjacent pulses, i.e. although values from multiple pulses can be subsequently averaged to decrease error if so desired, there is no inherent averaging. These properties make this technique ideal for data tagging [21–23] and feedback loops.

Moreover, these capabilities make for a highly effective tool for tuning and monitoring ultra-short pulse characteristics. By simply displaying the asymmetry parameters on an oscilloscope in parametric mode, one has a continuous real-time monitor of the pulse characteristics while optimizing the pulse to the desired specifications. Furthermore, as the stereo-ATI requires the optical alignment of just two elements — the focusing mirror and the interaction chamber, the apparatus is trivial to align. Additionally, only ∼30 μJ of pulse energy is needed. Thus, when a small portion of the laser energy can be sacrificed, this measurement/monitoring can be done in parallel to a primary measurement by splitting off a portion of the main laser beam.

4. Conclusion

In conclusion, we have demonstrated a novel technique to determine the approximate waveform of few-cycle Ti:Sapphire laser pulses using the ATI spectrum of Xe. In addition to allowing for the single-shot CEP determining and tagging [21–24], this technique allows for the simultaneous retrieval of the pulse duration without the need for CEP locking. This technique relies upon no theoretical assumptions and has been calibrated using a few-cycle SPIDER [11, 12]. The measurement is applicable to sub-10 fs pulses and is especially sensitive to changes in pulse duration for sub-6 fs pulses, where we estimate the accuracy to be better than ±0.4 fs. Moreover, the real-time response of the system is ideal for the feedback and monitoring necessary to achieve pulse durations pushing the limits of current Ti:Sapphire laser sources. Because of these properties along with the ease of installation and operation, we believe that this technique is a viable alternative to existing few-cycle pulse length and/or CEP measurements.