Table-top optical-extreme-ultraviolet (XUV) transient absorption spectroscopy is a powerful tool to investigate photoinduced electronic and structural dynamics in atoms, molecules, and solids [1–4]. In this approach, a sample material is excited by an optical pulse and subsequently probed in reflection or transmission with a time-delayed XUV pulse produced by high-harmonic generation (HHG). Photons in the XUV spectrum can excite core electrons, and thus enable element-specific probing of the optically induced dynamics with intrinsic attosecond timing precision due to the phase-locked generation [5]. In addition, the HHG process allows for single isolated attosecond pulses. These properties enable unprecedented time resolution for the observation of sub-cycle electron dynamics [1,6,7].

Despite its wide range of applications, table-top optical-XUV transient absorption spectroscopy has been mainly utilized in the high-pump-fluence regime (>1 mJ/cm$^2$) that induce large transient signals. This is largely due to the strong nonlinearities in HHG that greatly amplify fluctuations of the driving laser pulse, leading to a typical noise floor of 0.1 - 1 mOD in state-of-the-art experiments [8–12], which presents a formidable challenge for the observation of small pump-induced XUV absorbance changes. This far-from-shot-noise-limited sensitivity hinders studies of dynamics in the low-excitation regime, as involved in carrier dynamics in photovoltaics and electronic and structural phase transitions in solids [13–15]. For example, in a solar cell, the excited carrier density by sunlight is typically on the order of $10^{14}$-$10^{17}$ cm$^{-3}$ [16,17], whereas the carrier density in a 200 nm thick silicon membrane excited by 800 nm laser irradiation with fluence of 1 mJ/cm$^{2}$ exceeds $10^{18}$ cm$^{-3}$ [18]. Substantial efforts have already been made to enhance the signal-to-noise ratio of transient absorption spectroscopy in the context of HHG sources. The most common method to trace long-term fluctuations involves a successive recording of pumped and un-pumped spectra [9,10]. Additionally, noise reduction is possible by referencing, using the laser power [19], the spectral regions unaffected by the pump-induced dynamics [8,20], or a second spectrum of the XUV pulse before the sample [21,22]. It is also possible to use an almost common path referencing by splitting up the XUV beam [23–25]. However, these approaches typically impose requirements on the sample and experimental geometry and can not be easily used in a large variety of beam lines. Although approaches with a second spectrum may also overcome noise due to both long and short term fluctuations, imperfectly matched spectra can lead to far-from-ideal noise suppression.

In this work, we utilize machine learning to find the optimal referencing given an imperfect reference spectrum. Linear and polynomial regression and a neural network model are evaluated for signal enhancement of experimental XUV transient absorption spectra. In experiments with a 60 nm thick 1T-TiSe$_2$ sample at low fluences (0.66 mJ/cm$^2$), we find that a three-layer neural network surpasses all other methods and enables studies of coherent lattice vibrations and a phase transition in this material [26]. Moreover, we exploit the capability of performing a pixel-wise referencing with sub harmonic spectral resolution using the neural network.

 

Fig. 1. (a) Experimental setup for extreme ultraviolet transient absorption spectroscopy with a reference spectrum. (b) Comparison of different referencing algorithms. (c) Scheme for referencing with a neural network, trained on the un-pumped data. The model is applied at every pump-probe time delay $\tau$. (d) Noise control scheme by comparing pumped and un-pumped signal (on/off) spectrum without additional reference spectrum.

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The experimental setup is illustrated in Fig. 1(a). We investigate a 1T-TiSe$_2$ sample with XUV pulses produced by HHG with a 2 mJ, 35-fs-duration, 800-nm-wavelength driving pulse and its second harmonic from a Ti:sapphire laser operating at 1 kHz repetition rate with 0.28 % RMS shot-to-shot stability. A second femtosecond optical pump pulse at 2 $\mu$m wavelength produced by optical parametric amplification from the same laser system is used to excite the sample. We employ discrete harmonics that are even and odd multiples of the 1.55 eV fundamental photon energy and fluctuate by 4 % RMS. The total flux exceeds $6~\text {x}~10^{9}$ photons/s/harmonic for 38 nm wavelength at the source. In contrast to conventional XUV absorption experiments [9], where the XUV beam is focused onto the sample with a toroidal mirror, here the focusing mirror before the sample is replaced by a toroidal grating (Grating 1) and the first-order diffracted XUV beam from the grating is taken as the reference spectrum. The zeroth-order beam probes the sample and is dispersed by a second grating producing the signal spectrum. The signal and the reference XUV beam illuminate the upper and lower half of a charged-coupled device (CCD, 1024 $\times$ 255 pixels), respectively. By recording the reference beam intensity $I_{ref}$, the noise of each acquisition of the signal spectrum $I_{sig}$ may be directly compensated. However, noise originating from the beam path after the reference grating such as vibrations of the sample cannot be accounted for. Transient absorption experiments measure the change of absorbance $\Delta A=-\log _{10} (I_{sig}/I_{0})$, defined as the logarithm of the pumped signal spectrum normalized by $I_{0}$ which corresponds to the transmitted probe spectrum without the pump. The time-dependent information of $I_{sig}$ is obtained by varying the delay $\tau$ between pump and probe pulse.

Most experiments use successive recording of pumped and unpumped (on/off) signal spectra ($I_{0}$ corresponds to a subsequently acquired signal spectrum without the pump) to trace fluctuations on timescales exceeding the camera acquisition time (see Fig. 1(d)). However, noise originating from fluctuations on the acquisition timescale can only be fully suppressed if $I_0$ and $I_{sig}$ are known simultaneously. In a two-spectrometer configuration, the normalization spectrum $I_0$ can be calculated by the simultaneously acquired reference spectrum $I_{0}=F(I_{ref})$. Here, $F$ is a function that relates the reference spectrum with the unpumped sample absorption.

In its simplest form, $F$ merely contains the averaged ratio between the intensity of the unpumped signal and reference spectrum (direct referencing):

In the nonlinear polynomial regression model of degree $N$, the intensity of $I_{0}$ at pixel $s$ is expressed as

The neural network model can be viewed as a further generalization, capable of fitting an arbitrary, non-polynomial function $F(I_{ref})$ to the data [28]. In the present work, we employ a three-layer feed-forward neural network (see Fig. 1(c)) constructed with libraries Keras and TensorFlow [29]. The code is available at [30]. In the model, the number of neurons from hidden layers 1-3 are 250, 100, 70, and 200, 100, 20 for correlating intensity data of each pixel and each harmonic, respectively. These numbers have been empirically tested on the presented data and equivalent data sets recorded with the same setup and slightly changed HHG parameters. Larger models were found to yield no further accuracy improvements. A rectified-linear-unit activation function [31,32] was used. The networks of 24435 and 193170 trainable parameters, corresponding to individual harmonic and pixel wise models, are optimized with stochastic gradient descent on the mean-squared error with the Adam optimizer [33] and batches of 50 data points [34]. Training of the larger model by 58240 data sets is achieved in less than three hours on a personal computer without parallelization, which presents a very moderate computational cost. In order to avoid over-fitting, the fitting is stopped when the error minimization of the testing data saturates. The presented architecture of a simple 3 layer neural network poses a good foundation for a model to describe the connection of signal and reference spectrum without any prior knowledge. In the future, more sophisticated designs may be employed to further improve the already great performance and accuracy.

Among the different models, the mean squared error (Fig. 1(b)) of the pumped spectrum before time zero and a subsequently acquired unpumped spectrum (on/off) is the largest. Marginal improvements are obtained by direct referencing (Eq. (1)) and linear regression. The polynomial regression method can reach better approximation with much smaller error, but is still surpassed by the neural network model. This indicates the importance to account for non-polynomial intensity noise imprinted by the highly nonlinear HHG process. To further analyze the performance of the different approaches, the models are applied to experimental XUV transient absorption data before time zero. Here, the pump pulse arrives after the XUV probe and we expect the transient absorption in this material system to be zero. In Fig. 2(a) the absorbance change of the 25$^\text {th}$ and 27$^\text {th}$ harmonic are compared between the on/off method, direct referencing and the neural network. The on/off approach shows the largest noise which is given by the standard deviation of the histograms. Direct referencing improves the noise level, but is highly dependent on the harmonic order. This may result from various effects including clipping of some intensity and the unequal imaging condition such as optical aberrations, imperfect simultaneous alignment of both focal planes, and spectral impurity caused by the different beam paths and optical elements. Significantly better results are achieved with the neural network yielding similar noise levels at both harmonics.

 

Fig. 2. (a) Histograms of optical density difference before time zero for harmonics 25 and 27. (b) Noise levels of the discussed referencing techniques derived by the histogram standard deviation as a function of the harmonic order. (c) Autocorrelation of the signal spectrum. (d) Cross-correlation between reference and signal spectrum. Diagonal elements (black line) are used for direct referencing.

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A comprehensive analysis of the noise level derived from the standard deviations as a function of harmonic order is presented in Fig. 2(b). While the direct referencing improves the on/off approach at almost all wavelengths, the resulting noise strongly varies between harmonics. Please note that due to geometric constraints, only harmonic 24 to 30 can be recorded in the reference spectrum, limiting the spectral applicability of the direct referencing. The machine learning approaches which includes linear and polynomial regression and the neural network show significantly better performance with less deviation over the whole spectral region. We find that the neural network outperforms the other techniques at almost all orders of harmonics for the presented data. While the linear regression shows already good results and could be applied as a simpler model, the difference to the neural network approach is still significant. The results are compared with the photon shot noise that defines the lower noise limit. At the utilized 200 msec integration time and 3 MHz readout rate, the dark current of 0.02 counts and the readout noise of 30 counts are negligible in comparison to the measured signal of $>10^{5}$ counts per pixel. The machine learning referencing reduces the noise level significantly and roughly matches the photon shot noise spectral dependence. The remaining discrepancy is likely to result form imperfect matching of signal and reference spectrum due to the different optical elements and beam paths. Furthermore, the spectral region which is not covered by the reference spectrum is prone to additional error.

To validate the predictive power when the reference and signal spectra do not share the exact same wavelength ranges, we study the intensity correlation of different harmonic orders between the unpumped signal and reference spectra. Figures 2(c) and 2(d) show the correlation coefficients among harmonics of the signal spectrum and between signal and reference spectrum, respectively. Most off-diagonal elements in both correlation maps have $> 0.4$ magnitude, indicating that the intensities of different harmonics are strongly correlated by the HHG process. The black line in Fig. 2(d) shows components that are used in direct referencing. As the intensities between different regions of the harmonic spectrum are highly correlated, it is feasible to predict the signal spectrum using a reference spectrum with partially overlapped spectral region. In addition, the off-diagonal components provide additional information, which are used by the linear regression model. Nonlinear correlations fitted by the polynomial regression and the neural network model are not captured by the correlation coefficients. In principle, any other correlated quantity, like laser power, HHG source variables or total XUV intensity could be used for de-noising with machine learning.

Finally, we demonstrate the signal enhancement capability by showing a pump-probe trace of the 25$^{\text {th}}$ harmonic for on/off, direct and neural network referencing in Fig. 3(a). The $9.4 \times 10^{-4} \, \text {OD}$ standard deviation of the spectrum with on/off referencing aligns with the typical noise level $\approx 1$ mOD found in most transient absorption studies [8] and can be improved by a factor of two with direct referencing. An improvement by more than one order of magnitude in sensitivity to $9 \times 10^{-5}$ OD is achieved with the neural network. In addition, the neural network model can be applied to obtain a pixel-wise correlation between the signal and reference spectra. As most attosecond sources offer a broadband supercontinuum covering large portions of the XUV region, pixel-wise correlation offers noise-reduction without the loss of spectral resolution due to binning. Transient absorption spectra as a function of pump-probe delays and XUV photon energies are compared in Fig. 3(b) and Fig. 3(c) between the established on/off approach and the neural network referencing, respectively. For clarity, the spectral regions with low XUV intensities (owing to the discrete nature of harmonics) are omitted. While the pump induced response remains at a similar level, the neural network referencing drastically improves the visibility of the pump probe data. Again one order of magnitude improvement down to roughly $10^{-4}$ OD is visible. Note that while polynomial regression can also handle nonlinearities in the spectral correlation, it is computationally unfeasible to fit single pixel spectra due to the large number of fit parameters. For example, in the case of a 1024-pixel detector, $10^{14}$ fit parameters would be needed to model a polynomial of 5$^{\text {th}}$ degree in polynomial regression.

 

Fig. 3. (a) Transient absorption of the 25$^{\text {th}}$ harmonic evaluated with different referencing techniques. The error bars are identical for every delay and defined by the standard deviation at the particular harmonic. (b) Pixel-wise spectrum obtained with the conventional on/off approach. (c) Same as (b) but with the neural network referencing.

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The feasibility of single pixel referencing renders the neural network algorithm ideally suited for normalizing the broadband spectra from a wide range of attosecond beam lines. The reference spectrum can be acquired without wavelength calibration or complete coverage of the signal spectrum. Furthermore, instead of implementing a second spectrum, any measured quantity which is correlated to the harmonic intensity such as laser power, total XUV intensity, pulse duration, pulse shape, gas pressure or beam position may be use as input in the machine learning algorithm. As a side note, the neural network should always be trained during the data acquisition on the unpumped data such that the model can accurately capture fluctuations of the HHG source during the measurement. Comparing with the widely used on/off referencing, the neural network method does not require additional experiment time since pumped data are recorded by default for the on/off referencing in quick succession to un-pumped spectra. Note that when using a pre-trained model for noise-reduction of datasets in a different experiment, the performance of the model may degrade due to differences in HHG parameters.

In summary, we developed an algorithm for optimal noise correction in transient absorption spectroscopy with a reference spectrum. Thereby, we extend the applications of data driven algorithms in optics and photonics [35] which already showed great success in X-ray diffraction and spectroscopy for noise suppression and signal extraction [36]. Among the tested algorithms, the three-layer neural network approach shows the best result and improves the noise level by an order of magnitude compared to the established on/off referencing method. This sensitivity increase opens the field of femtosecond to attosecond XUV transient absorption spectroscopy in the weak excitation regime such as photoinduced phase transitions in photovoltaics in operando or strongly correlated materials like 1T-TiSe$_2$. As the neural network approach utilizes the nonlinear intensity correlation between different pixels, it does not require wavelength-calibrated reference spectra, nor does it rely on signal and reference beams covering the same spectral region. This largely relaxes the conditions for experimental instrumentation and highlights the wide application of our method to both narrowband and broadband sources in the field of transient spectroscopy. On a broader scale, the presented combination of referencing with machine learning may be applied to a large variety of experiments with fluctuating probes to enhance the sensitivity.