Abstract
Reconstruction of attosecond beating by interference of two-photon transitions (RABBITT) is one of the most widely used approaches to measure the time delays in photoionization. The time delay, which corresponds to a phase difference of two oscillating signals, is usually retrieved by cosine fitting or fast Fourier transform (FFT). We propose two estimators for the phase uncertainty of cosine fitting from the signal per se of an individual experiment: (i) $\sigma (\varphi _\textrm {fit}) \approx \frac {B}{A} \sqrt {\frac {2}{N}}$, where B/A is the mean-value-to-amplitude ratio, and N is the total count number, and (ii) $\sigma (\varphi _\textrm {fit}) \approx \sqrt {\frac {1-R^2}{R^2 n_\textrm {bins}}}$, where nbins is the total number of bins in the time domain, and R2 is the coefficient of determination. The former estimator is applicable for the statistical fluctuation, while the latter includes the effects from various uncertainty sources, which is mathematically proven and numerically validated. This leads to an efficient and reliable approach to determining quantitative uncertainties in RABBITT experiments and evaluating the observed discrepancy among individual measurements, as demonstrated on the basis of experimental data.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
1.1 RABBITT experiment
Our physical world evolves on many different timescales. The typical period of the electron motion in atoms or molecules is in the sub-femtosecond, i.e. attosecond (as, 1 as = $10^{-18}$ s) range, which corresponds to $T=h/\Delta E$, where the involved energy intervals are $1 \sim 100$ eV. In order to capture such ultrafast dynamics, one needs an ultrafast signal source and a corresponding detection scheme. This is achieved by attosecond extreme-ultraviolet (XUV) light or X-ray pulses generated by high-harmonic generation (HHG) [1,2] or free-electron lasers (FELs) [3,4]. The pulses can be either comb-like attosecond pulse trains (APTs) [5,6] or single attosecond pulses (SAPs) [7]. The dynamical information is extracted by pump-probe schemes that correlate the photoelectron spectra and the pump-probe time delays, where the attosecond XUV (or X-ray) pulse is the pump and a femtosecond infrared (IR) pulse serves as the probe. The schemes for APTs and SAPs are RABBITT [8–11] and attosecond streaking [12–16], respectively. In this paper, we focus on the RABBITT technique, where cosine-like beating signals are observed and the electron’s dynamics is characterized by the phase difference of two signals. We expect that the proposed methods can be readily generalized to the $\omega -2\omega$ in-situ methods [17], PROOF [16], and possibly other interferometric methods.
The origin of the beating pattern is known as the interference of different XUV-IR transition pathways [18–20]. In short, the same observed photoelectron kinetic energy corresponds to the absorption of one HHG-XUV photon and the exchange of $\pm m$ IR photons, where plus and minus correspond to absorption and emission, respectively. The transition amplitude of the $m$-pathway reads:
where $\mathcal {E}_\textrm {IR}$ and $\mathcal {E}_{q-m}$ denote the amplitudes of the dressing IR field and the $(q-m)$-order HHG, and $\mathcal {M}_{(q;m)}$ is the corresponding matrix element that contributes to the $q$-th HHG band. If several sub-pathways contribute to the $m$-pathway coherently [19,21], then $\mathcal {M}_{(q;m)}$ is the sum of those elements. The phase $\arg ( \mathcal {E}_\textrm {IR} ) = \omega \tau$ is referenced to the XUV pulse, where a greater $\tau$ indicates that the dressing IR pulse has a shorter optical path length.Let us recall the well-known electron two-slit interference experiment [22], where the electrons are detected one-by-one, and the interference of the wave function is presented by the beating probability of finding the electron as a function of position on the screen. In the RABBITT experiment, the observed signal is proportional to the probability density function (PDF) oscillating along the time axis (XUV-IR delay), and the pattern builds up while the number of detected electrons increases. For an APT containing only odd-order harmonics, the even-order sidebands (SBs) correspond to the process where $m$ is an odd number. The $q$-SB signal for a single atom or molecule can be described by:
Figure 1(a) illustrates possible pathways with $m = \pm 1, \pm 3$ for an even-order harmonic SB, which lies between two mainbands (MBs) of the XUV pulse. The high-$|m|$ pathways become important under a strong dressing field [23]; the $m = \pm 2$ pathways can be selectively enabled by a double-frequency dressing field [21,24,25]. There are also reports on conducting RABBITT with HHG frequencies separated by $1\omega$ or $3\omega$ [3,26,27]. Here we present the analysis mainly based on the interference between $m = \pm 1$ pathways, but the methods are applicable to other schemes. In our case, Eq. (2) is simplified to:1.2 Uncertainty of the measured time delay
The determination of the time delay relies on the phase retrieval of the signal, and it is important to obtain the uncertainty of the measurement. According to the Joint Committee for Guides in Metrology (JCGM), the uncertainty determination is categorized into two types: (A) evaluated by statistical methods, and (B) evaluated by other means [42–44]. In practice, the Type A uncertainty is the discrepancy among individual outcomes, and Type B uncertainty reflects the precision for each measurement. For attosecond science specifically, the outcome is sensitive to experimental conditions (e.g. the environment for XUV generation, pump-probe overlap in space and time, jitter of the delay stage, noise of the detector). Reference [31] discusses the uncertainty caused by the attochirp, and a comprehensive study of the contributing factors can be found in Ref. [35]. In general, the imperfect control of conditions can either (I) shift the expected value of the desired attosecond time delay by an amount related to specific experimental conditions, or (II) blur the oscillation pattern without changing the expected value; here the "expected value" is a conceptual limit of the average of repeating a measurement an infinite number of times under exactly the same conditions. The effect (I) may cause systematic error compared with experiments under better control of the conditions, while the effect (II) results in unequal precision of individual measurements. Qualitative or semi-quantitative estimations of single-measurement precision with corresponding weighting schemes have been applied in previous studies [14,40,41,45–51], which yields the Type A uncertainty of the studied experiments. However, to the best of our knowledge, no explicit formula has been proposed for determining the Type B uncertainty, namely, the uncertainty of a single measurement. The Type B uncertainty is meaningful when the number of individual experiments is comparatively small [42]. Furthermore, if the Type A uncertainty is much greater than the Type B uncertainty, it is an indication of the effect (I) discussed above, which provides hints for optimizing the experimental conditions.
1.3 Outline of this article
Here we propose two individual estimators based on the experimental signal per se: the $B/A$-estimator (Sec. 3) and the $R^2$-estimator (Sec. 4). We explicitly consider four effects: statistical fluctuations, smearing effect, shifting effect, and background noise, which finally fall into three categories (Sec. 2). The derived estimators are manifested by numerical simulations (Sec. 5), where the two estimators are compared. The relation between cosine fitting and FFT can be found in Sec. 6. The individual uncertainty leads to a weighting scheme (Sec. 7), where we introduce the $S$-value to quantify the consistency between the Type A and Type B uncertainties. Finally, in Sec. 8, we present a pair of examples from experimental data to demonstrate how one knows from the $S$-value whether the expected value of the time delay may be subjected to the experimental conditions on an unresolved degree of freedom.
2. Statistical fluctuations, smearing effect, shifting effect, and background noise
Experimentally, the detected electrons are binned by the step size of the delay time, and the number of counts in each bin $F_i$ obeys a Poisson distribution; when the count number is large enough ($\gtrsim 10$), the distribution can be approximated by normal distribution with variance equal to the mean value [52]
where is the PDF integrated over the time bin, which is approximately the PDF multiplied by the bin width, and $t_i$ is the central time of the $i$-th bin. $A$ is a positive real number, and we define $T = \mathrm{\pi} / \omega$ as the oscillation period of the signal. Equation (5) therefore gives the statistical fluctuation of the signal.In the derivation of Eq. (1), we have assumed that different pathways contribute coherently. In reality, the system undergoes partial decoherence, where pathways are mixed incoherently, and the total PDF is the sum of individual PDFs. The decoherence can be characterized by reconstructing the density matrix, and the origin of decoherence includes the spatial variation of the electric field, scattering of the ejected electrons, imperfect detector, and so on [53]. For RABBITT experiments, the decoherence (referred as the smearing effect) occurs both spatially and temporally.
The spatial smearing effect is related to the fact that the PDF is integrated over the focal volume. Mismatches in the XUV and IR focal-spot sizes, imperfect overlap, different Rayleigh lengths and the focusing of different harmonic orders at different positions along the propagation direction, among others, contribute to this effect, which can be expressed as a spatial variation of the amplitudes and phases:
The temporal smearing effect arises from the inaccuracy of defining the XUV-IR delay. Specifically, we consider the following scenarios: (i) the delay stage position is binned by the step size; (ii) the delay stage jitters randomly between the sequential electrons being detected; (iii) there is shot-to-shot XUV-IR phase jitter due to the fluctuation of generation conditions, and we assume each photoelectron originates from a different shot; (iv) if multiple scans are superimposed, there can be a long-term drift of the path length caused by thermal expansion. There can be other sources of temporal inaccuracy, but they are generalized by a redistribution function $D(\tau )$ that is normalized to 1, which describes the probability that the time delay is recorded as $t$ but the true value should be $t+\tau$. Its effect on the PDF is expressed by convolution:
The shifting effect refers to the correlated jitter, where all counts in the $i$-th bin are recorded as a delay-stage position that deviates $\Delta t_i = \eta _i / (2\omega )$ from the true value. This is relevant when multiple detected electrons are from the same shot, or at least between the typical timescale within which the jitter randomizes, which are denoted as a batch of electrons in the following discussion. The change of counts yields:
3. Cosine fitting for the binned histogram and the $B/A$-estimator
The RABBITT signal is fitted by a cosine function, which can be equivalently written as the linear decomposition into three parts:
The $B/A$ ratio is independent of $N$ or $n_\textrm {bins}$ and can easily be obtained from the signal per se ($A_\textrm {fit} \approx A$ when shifting is not very large). If the statistical fluctuation is the main uncertainty source, the phase fitting uncertainty has a concise expression:
In this case, the phase retrieval precision is proportional to $N^{-1/2}$, which agrees with the previous study [35]; it is also independent of $n_\textrm {bins}$, as shown in Fig. 3 (see Sec. 5 for details). The background noise increases the std of $\varphi _\textrm {fit}$ by a factor of ${(1 + \gamma )}^{1/2}$. The contribution of the shifting effect is $\frac {3}{2 n_\textrm {bins}} \sigma _{\eta }^2 = \frac {3}{2 n_\textrm {bins}} \sigma _\textrm {batch}, \eta ^2 / \kappa = \frac {3b}{2N} \sigma _\textrm {batch}, \eta ^2$, which ultimately depends on $N$ instead of $n_\textrm {bins}$, as numerically verified in Sec. 5.4. $R^2$-estimator
The determination of the background noise and the shifting effect needs a priori knowledge of $\gamma$ and $\sigma _{\eta }$. However, the combined uncertainty can be estimated by the coefficient of determination defined as [57]:
5. Validation by numerical simulations
5.1 $B/A$-method
Simulated RABBITT signals are generated by the accept-reject algorithm [58], and the detailed method is given in Supplement 1. In order to verify Eq. (22), the total count number $N$, the $B/A$ ratio, the number of oscillation periods $n_T$, and the bins per period $n_p$ are scanned, which perfectly agrees with the expression, as shown in Fig. 3. Note that the relation also holds for non-integer $n_p$ or $n_T$.
5.2 $R^2$-method
Figure 4 manifests that the upper limit in Eq. (26) is precise for an ideal signal that is only subject to the shifting effect with small jitter. The simulated $\sigma (\varphi _\textrm {fit})$ falls between the two limits up to $\sigma _\textrm {s} / T \sim 0.4$. Note that for white noise [59], $\sigma (\varphi _\textrm {fit}) = \mathrm{\pi} / \sqrt {3} \approx 1.81$, which corresponds to uniform distribution, while the lower limit of Eq. (26) gives $\sqrt {\mathrm{\pi} / 2} \approx 1.25$, because $R^2 n_\textrm {bins} \sim \chi ^2(k=2)$. This corresponds to the plateaus in Fig. 4(b).
Figure 5 shows the effects of various uncertainty sources on the simulated RABBITT signals and compares the uncertainty predicted by the $B/A$- and the $R^2$-estimators based on each independent trial. It is clear that Eq. (22) underestimates the uncertainty unless the statistical fluctuation is the main uncertainty source compared with correlated jitter (Fig. 5(a) and (b)). Nonetheless, the $B/A$-estimator fails to determine the white-noise level, which is included by the $R^2$-estimator (Fig. 5(c)). The uncertainty is independent of $n_\textrm {bins}$ for small ${\sigma }_\textrm {batch}, \eta $, as shown in Fig. 5(d); the increase of the estimators for greater $n_\textrm {bins}$ is attributed to the higher-order effects. It is also notable that the uncertainty is generally closer to the lower limit of the $R^2$-estimator; therefore, we suggest simply using
to estimate the experimental uncertainty.5.3 Effect of the $4\omega$-component
When the RABBITT signal has a $4\omega$-component, as shown in Fig. 2, the expected value and variance of $\varphi _\textrm {fit}$ are not affected, according to Eqs. (15) and (19). On the other hand, the $R^2$-value for the $2\omega$-fitting cannot exceed ${A_{2\omega }^2} / {\big ( A_{2\omega }^2 + A_{4\omega }^2 \big )}$, which causes an overestimation of the uncertainty level. The $4\omega$-effect can be corrected if $R^2$ is defined based on an extension of Eq. (14):
6. Period-fitting uncertainty and spectral leakage in FFT
6.1 Propagation of the period-fitting deviation
So far, we have assumed that the oscillation period is precise. If the fitted period deviates by $\Delta T$ from the true value, the fitted phase deviates by (to the lowest order):
For cosine fitting, the period is retrieved via non-linear least-squares fitting of the reference channel. Numerical simulations (see Supplement 1 for details) indicate that6.2 FFT and spectral leakage
Although FFT uses the same principle of Eq. (14), the fitted period is restricted to an integer fraction of the time range. If a single-frequency signal contains non-integer number of periods, the oscillation is decomposed to all discrete frequencies, which is known as spectral leakage [60], as illustrated in Fig. 6. Here we only take the largest Fourier component, yet it is possible to reconstruct the signal with neighboring components, which results in an envelope, as shown in Ref. [14]. The $\Delta \varphi$ caused by spectral leakage does not affect phase differences, but reduces the $R^2$-value, and $\Delta \varphi$ does not approach zero when $N_\textrm {ref} \rightarrow \infty$. Hence, the $B/A$-estimator (Eq. (22)) is preferred, provided that the statistical fluctuation is the main uncertainty source.
7. Weighting scheme
A typical RABBITT experiment is comprised of $M$ individual experiments that are unequal precision measurements with individual uncertainties determined by Eq. (31). Here we suggest the weighting scheme used by the Particle Data Group (PDG) in particle physics [61]:
where the weight $w_j = 1/\sigma ^2_j$. The uncertainty of the weighted mean value is:8. Examples based on experimental data
Here we present two examples based on the same set of data from the laboratory-frame angular-resolved RABBITT with Ar. The electrons are detected by coincidence measurements using cold target recoil ion momentum spectroscopy (COLTRIMS) [63–65], which pairwise records the 3-dimensional momenta of the photoelectron and photoion, and $\theta$ is defined as the angle between the electron ejection and the polarization direction. The two examples show the cases where the discrepancy is dominated by the statistical fluctuation and the variation of experimental conditions, respectively, which is reflected by the $S$-value.
8.1 Angular-resolved time delays of each sideband
The angular distribution of counts is shown in Fig. 7(a). For each SB, the angular-resolved time delays are referenced to the summed signal over all angles, where negative time delays at higher angles are found (Fig. 7(b), (c)), which agrees with previous studies [32,51,66]. The mean values and uncertainties calculated without weighting or with weighting by Eqs. (34) and (36) are compared in Table 2 and in Supplement 1. $S < 1$ indicates that the uncertainty is properly addressed, which indeed explains the discrepancy between individual experiments, although in some regions the uncertainty may be overestimated. Because the electron count rate ($\sim 2$ kHz) is lower than the repetition rate (5 kHz), indicating that most electrons come from independent shots, and the interval between two counts in the same channel is $\sim 1$ s, during which jitter already randomizes, the jitter is predominantly uncorrelated and can be treated as smearing effect. Assuming that the background noise is small, the $B/A$-estimator is applicable, and the results are listed in Supplement 1, where the $S$-value is typically greater but still less or close to 1. The comparison of time delays obtained from individual experiments can be found in Supplement 1, where the discrepancies are generally well covered by the error bars. Figure 7(b) and (c) compare the results from the two estimators. Neither weighted mean value nor the uncertainty shows a substantial difference.
8.2 Time delays between sidebands
The phase difference between SBs, as given in Eq. (4), has contributions from both the attochirp ($\tau _\textrm {XUV}$) and the atomic time delay ($\tau _\textrm {A}$). Here we focus on the precision of retrieving this total time delay from the experimental data, while the assignment to each component and comparison to theoretical calculations can be found in Ref. [31]. The relative time delays of SBs (with signal summed over all angles and referenced to SB-12) are plotted in Fig. 8; the uncertainties obtained by statistical approach and the $R^2$-estimator are compared in Table 3 . We have intentionally used all data measured in a period of about one week, including some individual experiments with inappropriate spatial or temporal overlap (e.g. the inset panel of Fig. 8(a)) that yield barely visible oscillations. The estimated uncertainty of these signals are large, and therefore they contribute little to the final result. One can immediately notice that there are outliers with relatively small statistical uncertainties, as compared in Fig. 8(b) and (c), where the relative phases between SB-12 and SB-14 show noticeable differences. The $S$-values are significantly greater than 1, which means that the discrepancy among individual experiments cannot be fully attributed to the statistical fluctuations and the above-considered effects; this indicates that the attochirp of each individual experiment may not stay constant. This is not surprising, since the gas pressure for HHG was tuned every day to maximize the XUV flux, which is related with the phase matching conditions and has an effect on the pulse structure [67–71]. It agrees with a previous observation that the attochirp varies on different days but remains stable during a few hours [31]. Besides, the atomic phase can also be affected by the intensity of the dressing field [72] which varies when the pump-probe overlap changes. Nevertheless, if channels are referenced within the same sideband, as shown in Sec. 8.1, the effect is cancelled, which results in smaller $S$-values.
9. Conclusion
We have shown that the uncertainty of phase retrieval can be extracted along with the fitted phase from the signal per se of a single experiment. We have proven that the statistical fluctuation caused by the Poisson distribution of each bin gives $\sigma (\varphi _\textrm {fit}) \approx \frac {B}{A} \sqrt {\frac {2}{N}}$ (the $B/A$-estimator). This expression is particularly concise, as the binning effect, uncorrelated jitter, and long-term drift are included in the decrease of $A$ (the smearing effect). For a given $B/A$ ratio, the uncertainty is inversely proportional to $\sqrt {N}$.
In order to include the correlated jitter (the shifting effect) and the background noise, the $R^2$-estimator: $\sigma (\varphi _\textrm {fit}) \approx \sqrt {\frac {1-R^2}{R^2 n_\textrm {bins}}}$ based on the coefficient of determination was proposed. Although it formally depends on $n_\textrm {bins}$, $R^2$ is also a function of $n_\textrm {bins}$, and the overall ratio is in accordance with the $B/A$-estimator for the statistical fluctuation. The $R^2$-estimator, however, is sensitive to the $4\omega$-component of the signal and the spectral leakage when FFT is applied, while the $B/A$-estimator is more robust.
Under the assumption that the phase jitter affects the electrons by constant batches (Eq. (13)), we showed that the shifting effect is independent of $n_\textrm {bins}$. If the background noise obeys $\sigma _\textrm {noise}^2 \propto B$, then its contribution to the uncertainty is also independent of $n_\textrm {bins}$. The main effect of the bin width is the binning effect, which reduces the $A$-value if the bins are too wide; otherwise finer bins do not provide improvement of precision.
A weighting scheme based on the individual uncertainty was proposed, which allows one to combine data of unequal qualities, as illustrated with real experimental data. The $S$-value allows one to check whether the considered uncertainty sources explain the discrepancy among different measurements, as demonstrated by the two examples. When the experimental conditions may alter the expected value of the measured time delay (Sec. 8.2), if one conducts the experiment with better monitoring, the expected outcome will fall on one of those individual experiments, which may deviate from the average of individual experiments under varying conditions. It is therefore suggested to search for the corresponding condition (e.g. HHG gas pressure, dressing field intensity); it not only improves the repeatability and reproducibility, but also potentially leads to new physical insight and discoveries.
Our quantitative uncertainty determination provides a framework of data analysis for future work, which can be extended for the uncertainty analysis of TURTLE fitting and for the two-dimensional energy–time delay RABBITT signal with overlapping or congested bands in the energy domain [36–41]. The related works are in progress.
Funding
ETH Zürich (ETH grant 41-20-2).
Acknowledgments
The authors thank A. Schneider and M. Seiler for technical support of the experimental setup.
Disclosures
The authors declare no conflicts of interest.
Data availability
The experimental data presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. The python codes for numerical simulations can be found in Ref. [73] .
Supplemental document
See Supplement 1 for supporting content.
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