1. Introduction

Transient absorption spectroscopy (TAS) is a well-established technique for investigating time-dependent dynamics extending from the terahertz energy range [1–3] to the extreme ultraviolet (XUV) and X-ray energies [4–6]. During the last decade, TAS with high-order harmonic generated (HHG) probe pulses has emerged as a powerful technique for probing ultrafast electron dynamics in the XUV and soft X-ray regimes in atoms, molecules and solids on femtosecond and attosecond timescales [7–23]. The table-top nature of HHG light sources has also provided a flexible platform for utilizing tailored driving laser radiation in useful ways [24,25].

Absorbance is, by convention, defined as the logarithmic ratio of two measured intensities, $I^1$ and $I^0$. It is the attenuation (optical density) that arises with/without a sample in linear absorption spectroscopy, or in the case of transient absorption, from a second coupling laser pulse that triggers time-dependent dynamics. Multiple terms contribute to the attenuation in transmitted intensity. Contributions from reflection and scattering processes are typically minor and will be omitted here. The remaining terms are the signal of interest ${\Delta }OD_{\mathrm {signal}}$, a multiplicative noise term $F$ and an additive noise term arising from changes in the probe light source intensity $I_{\mathrm {probe}}$ in the two measurements [26,27]:

In visible/NIR transient absorption spectroscopy, it is common to use a beam splitter to record the probe spectrum before and after interaction with the sample. Knowledge of the probe light intensity allows the additive noise to be minimized and enables the detection of changes in the mean optical density ${\langle {\Delta }OD}\rangle$ as small as $\sim 10^{-6}$ [28]. In the XUV/X-ray energy region, the lack of broadband beam samplers, the limited HHG flux and the high cost of detection systems have largely prevented implementation of this approach. Instead, the probe spectrum is measured with and without the coupling laser pulse in rapid succession, in an effort to reduce any differences between $I_{\mathrm {probe}}^{\mathrm {0}}$ and $I_{\mathrm {probe}}^{\mathrm {1}}$. Averaging over many acquisitions minimizes the additive noise. The relatively low photon flux and stability of HHG sources sets a lower limit to the detectable absorbance change, typically greater than 1 mOD ($10^{-3}$ optical density) [29,30].

The main source of noise in HHG transient absorption experiments comes from the fluctuation of the light source [31], which has been shown to be correlated with both the ion yield [32] and the laser intensity [33]. Recently, Volkov et al. reported that spectral shifts of the high-order harmonics can be ascribed to intensity-induced process, which depends on both the intensity of the laser and the phase-matching in the target [31]. The dependence of the HHG fluctuation on only a few parameters results in highly structured and spectrally correlated noise. Here, we demonstrate that we can exploit the high spectral correlation of the HHG light source to model the noise based on measurements without the coupling laser. The retrieved additive noise can then be subtracted from the transient absorption spectra. Implementations of this correction have been dubbed "edge-pixel referencing" [34], meaning that the noise calibration region of the spectrum must be free from any coupling laser induced signals, so that the difference between the noise calibration and the noise underlying the measured optical density can be minimized. Application to optical TAS [26,34] and XUV attosecond TAS [27] spectroscopy have recently been reported.

In this work, we apply principal component analysis (PCA) based approaches to characterize the spectral correlations within the noise in terms of linearly independent principal components. Two techniques are implemented for subtracting the noise. In the first case, principal component regression (PCR) of the measured signal in the edge-pixel referencing region is applied to retrieve the noise floor. This procedure is similar to the one implemented by R. Géneaux et al. [27] for TAS measurements using an isolated attosecond pulse (IAP). Our results demonstrate that sufficient correlation also exists in the HHG spectrum of an attosecond pulse train (APT) to enable edge-pixel referencing for background reconstruction. In the second case, an adaptive iteratively reweighted PCR (airPCR) algorithm is introduced. In this case, we demonstrate that the background noise floor can be retrieved without making any prior assumption about the edge-pixel referencing region. We demonstrate in both cases sub-mOD sensitivity in a time-resolved study of autoionizing Rydberg states of helium. Moreover, we demonstrate that the airPCR method produces a higher mean noise power reduction compared to the standard edge-pixel referencing approach, reaching a $\sim$40x mean noise power reduction across the entire spectral range, compared to a $\sim$20x reduction outside the edge-pixel referencing region when using the edge-pixel PCR technique.

2. Experiment

The femtosecond XUV transient absorption apparatus has been described in detail elsewhere [10,35]. In brief, the output of a Ti:Sapphire laser producing $13\,$mJ of $804\,$nm radiation at a repetition rate of $1\,$kHz and $35\,$fs in duration is split into two arms by a 70% / 30% beam splitter. In one arm, $4.0\,$mJ of the NIR is focused into a semi-infinite gas cell containing 70 Torr of argon to generate femtosecond XUV pulses. A differential pumping stage placed downstream from the generation point removes the gas load from the XUV propagation path. The NIR driving field is rejected by a $200\,$nm thick aluminum foil. A toroidal mirror focuses the XUV beam into a $4\,$mm long sample cell, continuously filled with helium gas, that is mounted inside an interaction chamber evacuated by a turbomolecular pump. The transmitted light is detected by an XUV spectrometer consisting of a variable line spaced grating ($1200$ lines$/$mm) and an X-ray CCD camera.

To record time-resolved data, $268\,$nm UV pulses (photon energy $E_{\mathrm {UV}} = 4.63\,$eV) with up to $300\,\mu$J of energy are produced by second harmonic generation followed by sum-frequency mixing with the NIR fundamental [35]. The UV energy is controlled with a combination of a half waveplate and a thin film polarizer to attenuate the fundamental NIR. Four high-reflective UV mirrors are used to remove the residual second harmonic and NIR fields. The UV pulses are then focused inside the sample cell with a $500\,$mm focal length lens. In the interaction region, the typical duration of the UV pulses is $100\,$fs. The XUV and UV beams are spatially overlapped inside the sample cell in a quasi-colinear geometry ($\sim 2^{\circ }$ angle) using a drilled-through mirror to transmit the XUV and reflect the UV pulses. For the data presented here, the UV beam is attenuated to energies of approximately $30\,\mu$J per pulse. The delay $t$ between the UV and XUV pulses is controlled with a motorized delay stage, with positive $t$ defined as the UV pulse preceding the XUV pulse. An optical 2-slots 101.6$\,$mm diameter chopper wheel with a $50\%$ duty cycle is placed in the path of the UV beam, whose diameter is $~$6$\,$mm. The rising and falling edges of the chopper transmission trigger CCD exposures every $125\,$ms to alternately record UV-on and UV-off spectra. The trigger delay is calibrated in order to maximize the UV transmission through the chopper during UV-on acquisitions. For each delay $t$ between the two pulses, we acquired $N_s=450$ UV-on and $N_s$ UV-off spectra. $\Delta OD_{\mathrm {raw}}$ is measured for $N_t=30$ values of $t$, corresponding to a total number of UV-off (and UV-on) spectra acquired of $p=N_sN_t=13500$, and a total acquisition time of approximately one hour.

3. Analysis of the Noise

A calibration dataset ${\Delta }OD_{\mathrm {calib}}$ is formed by pairing consecutively acquired UV-off spectra and computing the logarithm of their intensity ratios. All the $p=13500$ UV-off spectra acquired (regardless of the delay $t$), are used to form the calibration dataset. The dimensionality of ${\Delta }OD_{\mathrm {calib}}$ is therefore $n$ rows $\times p'$ columns, whereby $n$ is the number of pixels in the spectrum and $p'=p/2=6750$ is the number of paired observations. Deviations of ${\Delta }OD_{\mathrm {calib}}$ from zero encode changes in the XUV probe light source intensity between two consecutively acquired UV-off spectra throughout the experiment. We extract this information to correct the transient absorption signal ${{\Delta }OD}_{\mathrm {raw}}$ constructed from the consecutively acquired UV-on and UV-off spectra.

An exemplary UV-off XUV spectrum is shown in Fig. 1(a). We characterize the light source fluctuations by calculating the probability density function (PDF) of the absorbance noise ${\Delta }OD_{\mathrm {calib}}$ at each pixel/photon energy. The false color plot in Fig. 1(b) shows the calculated PDF, emphasizing that, while the mean absorbance is centered at zero, there is a considerable variation in the width of the distribution arising from noise. The $2\sigma _s$ standard deviation of $\Delta OD_{\mathrm {calib}}$ as a function of energy is quantified by an overlaid line, ranging from 40 to 150 mOD.

 

Fig. 1. Analysis of the noise of ${\Delta }OD_{\mathrm {calib}}$, formed by pairing consecutively acquired UV-off spectra. (a) Sample UV-off spectrum, corresponding to a $\sim 125\,$ms long acquisition of the 1$\,$kHz HHG source radiation. (b) False-color plot of the PDF of ${\Delta }OD_{\mathrm {calib}}$ across the XUV spectrum, demonstrating the noise associated with different harmonics. The standard deviation of the noise 2$\sigma _s$ is represented by the black line. (c) Correlation matrix of the noise in the photon energy domain, showing the PCC between any two photon energies. (d) Correlation matrix of the noise by CCD exposure, showing the PCC between any two of the first 1000 observations.

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We characterize the correlation of the noise by using the Pearson’s correlation coefficient (PCC) metric. The PCC $\rho _{x,y}$ between two variables $x$ and $y$ is defined as the covariance $\sigma _{xy}$ of the two variables divided by the product of their standard deviations $\sigma _{x}\sigma _{y}$ [36]:

A principal component analysis (PCA) is used to project the energy correlated noise onto a set of linearly independent principal components. PCA is a dimensionality-reduction technique that orthogonally transforms a large set of highly correlated data into a limited set of linearly uncorrelated principal components that accounts for as much variation in the data as possible [37]. The PCA is implemented in Python using the Scikit-learn library [38]. The first $q$ principal components returned by PCA, corresponding to the largest variance of the data, are collected in an $n\times q$ matrix ${\Delta }OD_{\mathrm {PC}}$. The absorbance noise ${\Delta }OD_{\mathrm {calib}}$ can be approximated as:

4. Methods

We present two methods for calibrating the noise floor based on linear regression of the observed spectrum on the most relevant PCs. In the edge-pixel PCR method, we restrict the fit to the edge-pixel referencing region free of UV-induced signal. In the air-PCR method, every pixel from the data is used, but weighted differently according to the signal observed.

4.1 First method: edge-pixel PCR

Figure 2 illustrates the background subtraction method using PCR on the edge-pixel referencing region. Background ${\Delta }OD$ spectra are estimated by calculating the logarithm of the intensity ratios of paired UV-off XUV spectra. These background ${\Delta }OD$ spectra form the columns of ${\Delta }OD_{\mathrm {calib}}$. Sample columns are shown in Fig. 2(a). A PCA on ${\Delta }\mathrm {OD}_{\mathrm {calib}}$ finds that $96.7\%$ of the variance of the measured $p' = 6750$ background spectra can be described with just 35 principal components, the first 8 of which are illustrated in Fig. 2(b).

 

Fig. 2. (a) First three columns of ${\Delta }OD_{\mathrm {calib}}$, obtained from two adjacent pairs of UV-off spectra ($p_1$ and $p_2$). (b) First 8 principal components ordered by decreasing relevance from top to bottom (first 8 columns of ${\Delta }OD_{\mathrm {PC}}$). (c) Recovery of additive noise across the entire spectrum (red line), compared to the mean noise (green line) for different fit regions (shaded gray). (d) Average measured spectral change $\langle \Delta OD_{\mathrm {raw}}(t)\rangle$ as a function of the UV-XUV pulse delay $t$. (e) Average spectral change $\langle \Delta OD_{\mathrm {raw}}\rangle$ over all UV-XUV pulse delays (green line) and final fit region selection (shaded gray).

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A linear least-squares regression fit using the first $q=35$ PCs is performed on the conventional optical density ${\Delta }OD_{\mathrm {raw}}$ (dimensions: ${n}$ pixels $\times$ ${p}$ observations), formed by pairs of UV-on and UV-off spectra. The fit is performed choosing a fitting region (edge-reference pixel region) free of UV-induced signal. Each measured transient absorption spectrum is corrected using

At a delay $\Delta t = -500\,$fs, negligible signal $\Delta {OD}_{\mathrm {signal}} \approx 0$ provides a benchmark $\Delta {OD}_{\mathrm {raw,test}}$ for testing the procedure. We perform fits within different limited regions of the spectrum. Each panel in Fig. 2(c) corresponds to a different fit region, indicated by a gray background. The fit result $\langle \Delta {OD}_{\mathrm {raw,fit}}\rangle$ (red line) is shown together with the measured noise $\langle \Delta {OD}_{\mathrm {raw,test}}\rangle$ (green line). The virtually constant red fit result for all panels in Fig. 2(c) indicates that a limited reference region is sufficient to predict the fluctuations across the entire spectral range. The performance of the method in reducing the noise can be estimated by computing the mean Noise Power Reduction ($\overline {\mathrm {NPR}}$) outside the edge-pixel referencing region. This corresponds to the weighted average:

Figure 2(d) shows a false color map of the measured mean spectral change $\langle {\Delta }OD_{\mathrm {raw}}(t)\rangle$ as a function of the UV-XUV pulse delay t. The mean spectral change $\langle {\Delta }OD_{\mathrm {raw}}\rangle$ over all UV-XUV pulse delays $t$ is shown in 2(e) (green line). No UV-induced signal is apparent in the energy regions $E < 55$eV and $E > 65.4$eV (shaded gray areas in Fig. 2(e)). This is consistent with the fact that the strongest transient features are expected at energies between 60 and 65.4 eV, corresponding to the $2s{n}p$ $^1$P$^o$ ($n\ge 2$) absorption lines of helium [39]. We used this referencing region for the final fit, for which we obtain an $\overline {\mathrm {NPR}} = 12.5\,$dB (18-fold NPR) using Eq. (6) on the test sample.

We emphasize an advantage of PCR over the correlation matrix method described in Ref. [26], that was already pointed out by R. Géneaux et al. [27]. Restricting the fit to only the first 35 principal components allows, in our implementation, to reproduce only the low frequency modulations of the noise, as shown in Fig. 2(c). Increasing the number of PCs generally leads to a higher NPR, but it introduces PCs with high frequency noise, that are not necessarily correlated between edge and signal regions. Filtering out these high frequency noise components results in a procedure that is less sensitive to weak signals (potentially present) in the edge-referencing pixel region.

4.2 Second method: airPCR

The edge-pixel PCR method requires the identification of a spectral region with no UV-induced signal, which is used as a reference for performing the fit of the noise floor. This procedure requires user intervention and is prone to some degree of variability. As an alternative, we present a different approach for calculating the PCR coefficient matrix $C$ that automatically calculates the weight matrix $W$ in Eq. (5) according to the signal observed without any user input. This technique follows an adaptive iteratively reweighted (air) algorithm similar to the one implemented in the air Penalized Least Squares (airPLS) algorithm by Z-M Zhang et al. [40]. In contrast to this procedure, we present an implementation where an air principal component regression (airPCR) is performed, allowing to take advantage of both the previous knowledge of the correlated noise and the adaptive reweighted approach.

Figure 3 compares the flowcharts of the airPCR algorithm and the edge-pixel PCR method. Both methods share the common first step illustrated in Fig. 3(a). In this step, a PCA is performed on the calibration dataset $\Delta OD_{\mathrm {calib}}$, and the $q$ most relevant PC $\Delta OD_{\mathrm {PC}}$ are selected, as described in Section 3. Then, in the edge-pixel PCR approach, a weighted PCR is performed through Eqs. (4) and 5, by setting the main diagonal entries $w_{i}$ of matrix W to 1 in the edge-referencing pixels region and 0 otherwise, as illustrated in Fig. 3(b). With the airPCR approach instead, the weight vector $\mathbf {w}$ of elements $w_i$ is computed according to the actual signal present in the data, through the iterative procedure illustrated in Fig. 3(c). We choose to use the same number of PC ($q=35$) for both approaches, in order to allow a direct comparison of the two methods.

 

Fig. 3. Flowcharts of edge-pixel PCR and airPCR methods: (a) PCA of the calibration dataset $\Delta OD_{\mathrm {calib}}$ (same for both methods), (b) edge-pixel PCR flowchart, (c) airPCR flowchart.

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In the following, we describe the iterative algorithm in more detail (an example Python code is shown in Code 1, Ref. [41]). In the first step $k=0$ of the iteration, the initial weight vector $\mathbf {w}_{k=0}$ is set everywhere to $1$, and the weight matrix $W_{k=0}$ is the identity matrix. Then, at each step of the iteration, the referenced spectra $\Delta OD_{\mathrm {ref}, k}$ are computed through the weighted PCR approach described in Eqs. (4) and 5. $\Delta OD_{\mathrm {ref}, k}$ can be used to estimate the regions of the spectra that contain UV-induced signal variations. The mean $\langle \Delta OD_{\mathrm {ref}, k}\rangle$ is expected to have small magnitude in the region of pixels free from UV-induced signal variations. Its absolute value $\mathbf {d}_k = |\langle \Delta OD_{\mathrm {ref}, k}\rangle |$ is a vector of dimension $n$ that can be used as a metric for retrieving the weight vector $\mathbf {w}_{k+1}$ of the next iteration. This is computed from:

 

Fig. 4. (a-b) Mean referenced signal $\langle \Delta OD_{\mathrm {ref}, k}\rangle$ (a) and airPCR weights $w_k$ (b) as a function of energy and for different values of the iteration index $k$ (see legend in panel a). The inset in (a) shows a y-axis magnification of $\langle \Delta OD_{\mathrm {ref}, k}\rangle$ in a selected energy region. (c) Square norm $||\mathbf {d}_k||$ of the optical density as a function of $k$. (d) Sum of all weights ${||\mathbf {w}_k||}_1$ as a function of $k$. (e) $\overline {\mathrm {NPR}}$ for the test sample (dashed line) and $\overline {\mathrm {NPR}}$ averaged over all UV-XUV delays (solid line) as a function of $k$. (c-e) The iteration index $k=46$, corresponding to the optimum result, is indicated by a vertical dashed line.

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The performance of the algorithm in reducing the noise as the iteration proceeds is tracked by calculating $\overline {\mathrm {NPR}}$ for each step in the region where signal is detected. This is done through the weighted average in Eq. (6), using the weights $\mathbf {w}_k$ and referenced spectra $\langle \Delta OD_{\mathrm {ref}, k}\rangle$ for each step. The dashed line in Fig. 4(e) shows, as a function of $k$, the $\overline {\mathrm {NPR}}$ obtained for the test dataset $\overline {\mathrm {NPR}}_{\mathrm {test}}$ (recorded at $\Delta t=-500\,$fs). For $k\le 46$, $\overline {\mathrm {NPR}}_{\mathrm {test}}$ is approximately constant, since a high effective number of pixels is used for calibrating the test noise floor (${||\mathbf {w}_0||}_1 > 420$). For $k > 46$, $\overline {\mathrm {NPR}}_{\mathrm {test}}$ starts to drop quickly, which indicates that the performance of the PCR fit on the test sample is reduced. At the same time, for $k > 46$, the square norm $||\mathbf {d}_{k}||$ is approximately constant, which indicates that no significant change of $\langle \Delta OD_{\mathrm {ref}, k}\rangle$ is observed beyond this value. The solid line in Fig. 4(e) represents $\overline {\mathrm {NPR}}_{\mathrm {avg}}$, which results from averaging over all UV-XUV delays. Its value at $k=1$ is lower than the one obtained for the test sample, due to the fact that a high weight is assigned to pixels with signals, which have a higher noise. $\overline {\mathrm {NPR}}_{\mathrm {avg}}$ reaches a maximum at $k = 46$, where its value is comparable to the one obtained for the test sample. Then it quickly drops upon further iteration. In Fig. 5, the raw $\langle \Delta \mathrm {OD}_{\mathrm {raw}}(t)\rangle$ false-color map in Fig. 5(a) is compared with the referenced $\langle \Delta \mathrm {OD}_{\mathrm {ref,k}}(t)\rangle$ false-color maps in Figs. 5(b-f), which are obtained at different values of the iteration index $k$. The corresponding standard deviations of the optical densities, averaged over all the UV-XUV delays, are displayed in the bottom panels. At $k=1$, the standard deviation is drastically reduced compared to the raw dataset. However, artifacts appear in the spectrum (e.g. around 57 eV), due to the fact that pixels with non-zero transient signal are still assigned a high weight. As soon as the iteration proceeds, the artifacts are progressively reduced and almost disappear for $k\ge 41$. On the other hand, the standard deviation stays approximately constant for $k\le 41$, while it starts to increase for larger values of $k$, which is consistent with the decrease of $\overline {\mathrm {NPR}}_{\mathrm {avg}}$ beyond $k=46$ seen in Fig. 4(e). The increase of noise for high values of $k$ is also apparent in Fig. 5(f). Artifacts can be identified in the same way as discussed by R. Géneaux et al. [27], by comparing the result of the airPCR algorithm with the referenced spectra obtained with the edge-pixel PCR approach, which are presented in the next section. Another way to identify the presence of artifacts in the referenced airPCR spectra is to track the behavior of $||\mathbf {d}_{k}||$ as the iteration proceeds. The steep change of $||\mathbf {d}_{k}||$ during the first $\approx 40$ steps is due to the progressive reduction of artifacts in the referenced spectra. When all the pixels associated with signal are assigned with low weight, $||\mathbf {d}_{k}||$ reaches a plateau, and artifacts are not visible anymore. Further iteration can only increase the noise but cannot introduce any new artifacts, due to the choice of the weight function, that progressively assigns lower weights to the region of spectra with signal.

 

Fig. 5. Top panels: raw $\langle \Delta OD_{\mathrm {raw}}(t)\rangle$ (a) and referenced $\langle \Delta OD_{\mathrm {ref}, k}(t)\rangle$ for different values of $k$ (b-f). Bottom panels: standard deviation of optical density averaged over all UV-XUV delays, for raw (a) and referenced spectra obtained for different numbers of airPCR iterations $k$ (b-f).

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Identifying the optimum result of the iterative procedure is achieved by balancing two requirements. On one hand, ${||\mathbf {w}_k||}_1$ is required to be small, so that even small signals like the one represented in the inset of Fig. 4(a) are assigned a small weight, and the fit of the noise floor is not affected by them. On the other hand, the NPR of the test sample is required to be high. This means that the effective number of pixels ${||\mathbf {w}_k||}_1$ must not decrease below a certain threshold, beyond which the PCR performance is reduced. These two requirements are satisfied best at $k = 46$, just before both $\overline {\mathrm {NPR}}_{\mathrm {avg}}$ and $\overline {\mathrm {NPR}}_{\mathrm {test}}$ start to decrease exponentially. At this $k$, $\overline {\mathrm {NPR}}_{\mathrm {test}} = 16.2$ dB and $\overline {\mathrm {NPR}}_{\mathrm {avg}} = 15.9$ dB. This value is indicated by a vertical dashed line in Figs. 4(c-e). We note that the number of iterations (and, thus, the total computation time) needed to reach the optimal value reduces as the constant $c$ in Eq. (4) is increased. At the same time, this constant has to be kept small enough to allow a sufficiently fine sampling of the parameters shown in Figs. 5(c-e), so that the optimum result can be correctly identified. The value $c = 0.2$ was chosen as a compromise between these two requirements, but we have found that, for our implementation, the effectiveness of the algorithm in reducing the noise is still preserved for any value $c \le 3$.

5. Results

Raw time-resolved transient absorption spectra of helium are shown as a false color map in Fig. 6(a). An UV-induced change in the absorption profile of the autoionizing Rydberg $2s{n}p$ $^1$P$^o$ ($n\ge 2$) states can be observed in the range of photon energies above $58\,$eV. These transient phenomena are induced by the coupling of two-electron XUV-excited states by the UV laser pulses, as previously observed by C. Ott et al. with a visible coupling pulse [14]. The $2s2p$, $2s3p$ and $2s4p$ resonances are labelled. A substantial improvement of signal-to-noise is obtained with both the edge-pixel PCR and airPCR approaches, as demonstrated by the corrected transient absorption false-color maps in Fig. 6(b) and Fig. 6(c), respectively. In Fig. 6(d), the raw mean $\langle \Delta OD_{\mathrm {raw,test}}\rangle$ in red of the test dataset (recorded at $\Delta t=-500\,$fs) is overlaid with the fit $\langle \Delta OD_{\mathrm {fit,test}}\rangle$ obtained with the edge-pixel PCR and airPCR methods, represented by green and blue lines, respectively. $\langle \Delta OD_{\mathrm {fit,test}}\rangle$ describes the baseline associated with the driving laser additive noise floor well in both cases, and a notable improvement is observed for the airPCR method. The regions of high frequency (ringing) noise around 59, 62, and 65$\,$eV correspond to minima in the HHG flux (a sample HHG spectrum is shown in Fig. 6(e) for reference); other sources of noise such as the dark-current noise of the detector are expected to dominate here.

 

Fig. 6. (a-c) False-color maps of the raw (a), edge-pixel PCR referenced (b) and airPCR referenced (c) transient absorption spectra between $57-67\,$eV. (d) Comparison of the mean noise $\langle \Delta OD_{\mathrm {raw,test}}\rangle$ (red line) of the test sample and fit $\langle \Delta OD_{\mathrm {raw,fit}}\rangle$ obtained with the edge-pixel PCR method (green line) and the airPCR method (blue line). (e) Sample XUV spectrum for reference. (f-i) Statistical analysis of the noise before (red) and after referencing with the edge-pixel PCR method (green) and the airPCR method (blue): (f-g) single pixel PDF before and after referencing for the test sample (h) standard deviations before and after referencing, averaged over all UV-XUV delays; (i) NPR averaged over all UV-XUV delays.

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The reduction in noise power through the spectrum can be estimated by performing a statistical analysis of the measured $\Delta OD_{\mathrm {raw}}$ and referenced $\Delta OD_{\mathrm {ref}}$. Figure 6(f-g) show a comparison of the PDFs of $\Delta OD_{\mathrm {raw,test}}$ and $\Delta OD_{\mathrm {ref,test}}$ for a single detector pixel corresponding to an energy $E = 58\,$eV. After referencing, the edge-pixel PCR returns a standard deviation reduced from $35\,$mOD to $6.6\,$mOD (Fig. 6(f)). With the airPCR method (Fig. 6(g)), the standard deviation reduces to $3.1\,$mOD. The standard deviations averaged over all the UV-XUV delays of the measured and referenced datasets are reported in Fig. 6(h) for all detector pixels. Figure 6(i) displays the NPR for the two methods, averaged over all the UV-XUV delays. For both the edge-pixel PCR and the airPCR approaches, the NPR inside the referencing region (shaded gray area) reaches a maximum of $\sim$23 dB (300-fold NPR). Outside of this region, it varies between a minimum of $\sim$2 dB (1.6-fold NPR) and a maximum of about $\sim$17.7 dB (60-fold NPR) for the edge-pixel PCR method. The airPCR approach leads to a stronger noise power reduction, reaching a maximum of $\sim$ 22 dB (160-fold NPR) at 57.8 eV. Thanks to the weighted approach, pixels with no signal in this region are taken into account, in addition to the pixels in the edge region, leading to a better or equally good performance throughout the entire region compared to the edge-pixel PCR approach. The results in Fig. 6(i) show that, regardless of the method used, the noise reduction at different pixels varies dramatically, as the peaks and troughs of the HHG spectrum have different stability and signal intensity. With the edge-pixel PCR approach, the mean noise reduction across all pixels on the detector is $\sim$14.8 dB (30-fold NPR), while outside the referencing region it is $\sim$12.2 dB (16.8-fold NPR). The airPCR method enables a higher mean noise reduction, equal to $\sim$16.2 dB (41.3-fold NPR) across all pixels on the detector and $\sim$16.5 dB (44.8-fold NPR) outside the edge-pixel referencing region.

Figures 7(a), 7(b) and 7(c) show false-color maps of the raw, edge-pixel PCR and airPCR referenced time-resolved transient absorption spectra, respectively, between $54-58.5\,$eV. They highlight weak transient features and the dramatic improvement in signal-to-noise after referencing, especially for the airPCR referenced spectra. Spectra at two UV-XUV delays before correction, after edge-pixels PCR referencing and after airPCR referencing are highlighted in Figs. 7(d), 7(e) and 7(f), respectively. Before referencing, almost no signal is distinguishable from the $\sim$2$\,$mOD noise. After the noise subtraction, however, features around $E_1 = 55.2\,$eV and $E_2 = 57.43\,$eV are revealed, with maximum absorbances of $2\,$ and $3\,$mOD, respectively. The feature at $E_1$ is attributed to coupling to the 2s4p resonance by the three photon process $E_1 + 2E_{\mathrm {UV}} = E_{2s4p} = 64.47$eV, while the feature at $E_2$ is attributed to the two photon absorption process $E_2 + E_{\mathrm {UV}} = E_{2p^2}$ that accesses the dark $2p^2$ state at an energy E$_{2p^2}$ = 62.06$\,$eV. At these two energies, the average NPR obtained with the edge-pixel PCR approach is $15.5\,$dB (35.5-fold NPR) and $15.3\,$dB (33.9-fold NPR). With the airPCR approach, the NPR for these two energies increases to $20.6\,$dB (115-fold NPR) and $17.4\,$dB (55-fold NPR).

 

Fig. 7. (a-c) Raw (a), edge-pixel PCR referenced (b) and airPCR referenced transient absorption spectra between $54-58.5\,$eV. (d-f) Spectra at $t = -200\,$fs and $t = 20\,$fs: raw (d), edge-pixel PCR referenced (e) and airPCR referenced (f). (g-j) Temporal variations of absorption features $E_1 = 55.17\,$eV (g,i) and $E_2 = 57.43\,$eV (h,j). The transients are represented before background subtraction (semi-transparent red lines), after edge-pixel PCR referencing (solid green lines), and after airPCR referencing (solid blue lines).

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Single-pixel lineouts of the time-dependent behavior of the absorption features at $E_1$ and $E_2$ after edge-pixels PCR referencing are shown in Figs. 7(g) and 7(h), respectively (green lines), while the airPCR referenced transients at these two energies are shown in Figs. 7(i) and 7(j) (blue lines). In both cases, transients before background subtraction are reported for comparison (semi-transparent red lines). Substantial oscillations in ${\Delta }OD$ are suppressed after the subtraction and clear transient features emerge. In particular at $E_1$, the airPCR approach provides a better suppression of these oscillations compared with the edge-pixel PCR approach. The shaded red, green and blue areas represent $68\%$ confidence intervals before noise subtraction, after edge-pixels PCR, and after airPCR referencing, respectively, calculated from the estimated standard deviation of the mean $\sigma _{\mathrm {mean}} = \sigma _{\mathrm {ref}} \cdot {N_s}^{1/2}$ at each delay. Before correction, the average $\sigma _{\mathrm {mean}}$ for different UV-XUV delays is $1.96\,$mOD at $E_2$ and $1.62\,$mOD at $E_1$. After the edge-pixel PCR noise subtraction, the average $\sigma _{\mathrm {mean}}$ for different UV-XUV delays is reduced to $\sim$280$\mu$OD at $E_2$ and $\sim$340$\mu$OD at $E_1$. With the airPCR approach, a further reduction of $\sigma _{\mathrm {mean}}$ is observed, reaching $\sim$220$\mu$OD at $E_2$ and just $\sim$185$\mu$OD at $E_1$.

6. Conclusions

We present two methods to minimize the noise associated with fluctuations of the HHG light source in ultrafast XUV transient absorption spectra. The methods rely on the high spectral correlation of the noise and enable the recovery of UV-induced changes in XUV-spectra with sub-mOD sensitivity.

The results obtained with the edge-pixel principal component regression (edge-pixel PCR) method complement previous attosecond transient absorption spectroscopy (ATAS) measurements using a few-cycle laser to generate isolated attosecond pulses (IAP) [27] and demonstrate that sufficient correlation also exists in the HHG spectrum of an attosecond pulse train (APT) to enable edge-pixel referencing. In this case, we demonstrate that, in contrast to an IAP, the noise reduction for an APT varies dramatically through the entire spectral range.

An adaptive iteratively reweighted principal component regression (airPCR) algorithm is presented, which allows to calibrate the noise floor without any prior assumption on the position and extent of the referencing region. With this method, a 40x mean power reduction of the noise is demonstrated across the entire spectral range. The extension of this iterative retrieval procedure to other experiments that exploit broadband highly spectrally correlated sources, such as an IAP, is straightforward.