1. Introduction

Coherent multidimensional optical spectroscopy is a versatile technique for studying ultrafast processes in complex systems. Several different beam geometries have been demonstrated [1]. In the conventional noncollinear phase-matching approach, the nonlinear signal is detected in form of a coherently emitted field within a four-wave mixing process [2–4]. Alternatively, population-based multidimensional spectroscopy utilizes incoherent observables like, e.g., fluorescence [5–11], photoelectrons [12,13], photocurrents [14,15], or photoions [16,17]. Both approaches have been successfully used to investigate coherent phenomena in a variety of quantum systems.

Common to all techniques is the detection of the nonlinear collective system response as a function of temporal delays between ultrashort laser pulses of an optical multipulse excitation sequence. Using two inter-pulse delays $\tau $ and t (coherence times) enables one to spectrally resolve both excitation and detection frequencies and reveal correlations between them [2]. The correlations are represented as signal peaks in a complex-valued 2D spectrum whose real and imaginary parts are interpreted as absorptive and refractive contributions, respectively [18].

Mapping of the time evolution of a 2D correlation spectrum is achieved by implementing a third delay T (population time) “between” the two coherence times $\tau ,\;t$, which enables one to investigate the sample’s relaxation dynamics (e.g., energy transfer [19–31] or exciton dynamics [32–37]) while interacting with the local environment.

In collinear excitation geometries, commonly used in population-based techniques, it is necessary to carry out the pulse delay scanning scheme with systematically and precisely tuned relative phase differences between the pulses. Phase-related pulses have been used already in 1989 to observe and analyze beatings in fluorescence signals [38]. In the multimensional context, controlling the phases individually allows one to separate various Liouville-space excitation pathways. This procedure, known as phase cycling, has become an essential tool to efficiently extract certain nonlinear signal contributions from the linear signal background [39].

The complete system response after optical excitation can be described by a density matrix that is expanded in a perturbative approach as a power series with respect to the electric-field interactions. Both population- and coherent-detection-based nonlinear spectroscopy techniques are sensitive to particular higher-order density matrix terms of this expansion. The 2D spectrum unravels line-broadening mechanisms, the homogeneous and inhomogeneous linewidths, and the frequency–frequency correlation function [40,41]. At zero population time, however, homogeneously broadened diagonal peaks may be distorted, and one should take care not to confuse this with inhomogeneous broadening [42]. Such peak distortions have negligible effect for $T\, > \,0$ and are of minor relevance for the diagonal broadening of inhomogeneous peaks. Detailed theoretical [43–45] and experimental [46] investigations of signal peak shape distortions revealed a strong dependence on the excitation beam geometry, phase-matching condition, optical density of the sample, beam overlap region, beam propagation within the sample, and population time T. Especially in noncollinear broadband 2D spectroscopy applications, minimizing cross-peak distortions becomes important and can be achieved with an appropriate geometry [47].

Apart from the “geometric” effects on distortions discussed in the previous paragraph, 2D line shapes are also influenced by the temporal profiles of the applied pulses. In a first experiment, Tian et al. employed phase cycling to isolate the 2D photon-echo contribution of electronic transitions in an atomic rubidium vapor [48]. They discussed nonidealities in their acousto-optically based pulse-sequence generation leading to “spurious” pulses, which induced an amplitude modulation with the same phase dependency as the detection signal. This study already indicated that the precise shape of the applied pulse sequence might crucially affect the obtained 2D spectrum and thus its interpretation. A systematic theoretical and experimental analysis of the effect of pulse chirp on the shape of 2D electronic spectra has been carried out by the Ogilvie group [49]. They showed, using acousto-optical pulse shaping, that peak shape asymmetries are introduced if the individual pulses in the experiment exhibit different amounts of chirp. Further, in case that all pulses are similarly chirped, distortions still affect the anti-diagonal symmetry of the signal peak and introduce negative features in the 2D spectrum. They concluded that a careful pulse characterization is absolutely necessary for correct interpretation of frequency-dependent relaxation processes via 2D spectroscopy. A recent investigation of the combined effect of high pulse intensity and chirp on population-based 2D electronic spectroscopy data in atomic rubidium vapor revealed the enhancement and suppression of distinct higher-order signal contributions [50]. We extend this further and report an automated experimental analysis procedure taking into account that the pulse shapes might additionally vary from scanning step to scanning step.

In addition to pulse shaping using an acousto-optic programmable dispersive filter [51,52], tailoring of interferometrically stable pulse sequences has been realized by a variety of other techniques and designs [53,54]. These can be categorized as time-domain [55,56] or spectral-domain approaches, with the latter mostly employing liquid-crystal-based amplitude and phase shaping [57] or acousto-optical modulators [58]. Many broadband spectral-domain techniques utilize at least one grating in 4f geometry and shape the spectral phase by modulating the refractive index for each spectral component of the femtosecond pulse in the Fourier plane. Using a dual-layer liquid-crystal-display (LCD) spatial light modulator (SLM) enables independent amplitude and phase shaping of femtosecond laser pulses [59]. Following the LCD-based approach, imperfections in the pulse-sequence generation have been reported and studied in detail like replica pulses as a result of the LCD pixelation [60], spatio-temporal coupling [61–63], pixel crosstalk or Fabry-Pérot interference effects [64]. Many of these inherent imperfections can be minimized by design, handling or using only a low-voltage working range of the pulse shaper. One additional possibility to compensate for some of the imperfections is the voltage-to-phase calibration of the individual LCD pixels (a detailed description of the calibration can be found in [54]). A significant reduction of pulse-shaper imperfections like the Fabry-Pérot amplitude modulation effect is achieved by systematically scanning the voltages of one LCD layer and correcting pixelwise deviations between expected and measured voltage-to-phase dependencies [64]. In this approach, a full correction would include an iterative optimization for every possible phase combination. A non-iterative compensation method for undesired spectral intensity modulations was also discussed [65]. Here the voltage dependency of the Fabry-Pérot effect is characterized and reduced by using a two-dimensional look-up table to correct the corresponding voltage values and ensuring that phase shaping is unaffected by this procedure. In both approaches the feedback signals used for compensation or correction have been successive measurements of the spectral intensity of the shaped pulses only, but not their spectral phase.

Here we implement a characterization method based on Fourier-transform spectral interferometry to reconstruct the full intensity and phase structure of multipulse sequences. Such a structure can be influenced by many independent factors such as power drifts, pulse-shaper imperfections or temperature fluctuations [66] throughout the scanning procedure of a multidimensional spectroscopy experiment. This may lead to both stochastic and systematic deviations of the created pulse train from the expected ideal sequence and in turn have an influence on the retrieved 2D spectra. Thus, while the general effect on 2D spectra of an overall chirp (of all excitation pulses in all measurement steps) is known, we here characterize experimentally the shape of each individual pulse sequence, for each measurement step, fully in intensity and phase. This allows us to investigate the impact of any pulse-shape deviations on the 2D spectrum and to implement 2D simulations using the exact pulse shapes instead of commonly used idealized pulse trains consisting of, e.g., infinitely short “delta” pulses or Gaussian-envelope pulses.

The paper is structured as follows: In Sec.  2, we introduce the experimental setup and describe the extraction of characteristic pulse-sequence parameters. In Sec.  3, we systematically analyze parameter deviations for phase-cycled three-pulse sequences at various time delay combinations that are characteristic for a multidimensional spectroscopy experiment. In Sec.  4, we investigate how the experimental pulse shapes including artifacts affect 2D spectra, by simulating purely absorptive 2D spectra for a molecular dimer fluorescence signal and for multiphoton photoemission from a plasmonic nanoslit. In Sec.  5, we summarize results and provide conclusions.

2. Experimental procedure

We utilize a technique based on analyzing the spectral interference pattern of the shaped pulse sequence and a well-known simple reference pulse, i.e., Fourier-transform spectral interferometry (FTSI). Using the principle of single-channel spectral interferometry [67], FTSI was described by Lepetit et al. as a linear technique of phase measurement [68] and enables extracting the relative spectral phase between two pulses. Its experimental implementation and possible error sources have been discussed in detail [69,70]. Note that the information about the relative spectral phase between reference pulse and shaped pulse sequence is imprinted in the interference fringe pattern in the spectrum. Since FTSI is based on a non-iterative and linear detection method [68], it is well adapted for fast and sensitive pulse characterization of multipulse sequences at low pulse energies and thus can be used simultaneously with a nonlinear spectroscopy experiment because only a small portion of the excitation light is required.

2.1 Setup

We used a noncollinear optical parametric amplifier (NOPA) of our own design to generate sub-20-fs laser pulses with a tunable central wavelength from $400 - 970\;\textrm{nm}$ at 1 MHz repetition rate [71].

Before entering the pulse-shaper setup depicted in Fig. 1(a), the laser pulses were precompressed by minimizing the second-order dispersion via a fused-silica folded two-prism compressor. Only the residual third- and higher-order dispersion was then compensated by phase shaping so that most of the device capacity was saved for tailoring the individual multipulse sequences via amplitude and phase shaping with an LCD-based pulse shaper (Jenoptik, SLM-S640d USB) [71]. The wavelength-to-pixel calibration and voltage-to-phase calibration were performed using a commercial white-light supercontinuum laser source (NKT, SuperK COMPACT).

 

Fig. 1. Experimental schematic and exemplary data. (a) Sub-20-fs input pulses (red) from the NOPA are transformed by a spatial-light-modulator-(SLM-)based pulse shaper to generate multipulse sequences for collinear multidimensional spectroscopy experiments. A reference pulse (dark gray) is separated by a thin wedge pair and can be shuttered (S) individually. Its temporal delay and intensity are controlled by a delay stage and neutral density filters (OD) relative to the shaped pulse. The unshaped reference pulse and a small fraction (∼3 %) of the experimentally used shaped pulse sequence are spatially recombined after the second thin wedge pair, guided through several additional optics (Add. optics) and spectrally interfere on a spectrometer (Spec.). For each individual pulse sequence we determine the maximum intensity of each sub-pulse ${I_{1,2,3}}$ (orange), temporal inter-pulse delays $\mathrm{\tau }$, t (light and dark green), and intra-pulse phase-offsets ${\varphi _{1,2,3}}$ (blue). (b) Temporal intensity profile (red circles) and temporal phase (dark blue circles) for a three-pulse sequence assuming ideal pulse shaping of the input pulse. The intensity maxima ${I_{1,2,3}}$ (orange circles) and temporal delays $\mathrm{\tau }$, t between these maxima where quantified by fitting the intensity profile in the gray shaded windows with a sum of three Gaussian functions (black line). The individual temporal phase offsets were obtained as the phase values at maximum intensity (light blue). (c) Same as in panel (b), but for the experimentally determined real pulse shapes.

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Further, the voltage-to-phase calibration of the first LCD layer was optimized, similar to the constant voltage correction we introduced previously [64], in the following way: We optimized the transmission for each LCD pixel by adding a pixel-dependent voltage offset ${V_{\textrm{Offset}}}(\textrm{pixel})$ to the voltage-to-phase calibration curve. For this purpose, the voltage of the first LCD layer was scanned in a range of $\Delta V \approx{\pm} 50\;\textrm{Vcounts}$ around its original voltage-to-phase calibration value ${V_{\textrm{Cal}}}(\textrm{pixel}) \in \{{100, \ldots ,2800\;\textrm{Vcounts}} \}$, while the voltage at the second LCD layer was kept constant. We monitored for each voltage scan step the transmitted spectrum of the supercontinuum laser source and performed the voltage scan twice: first, at a pulse-shaper configuration corresponding to minimum, and second, to maximum, transmission. The voltage offset ${V_{\textrm{Offset}}}(\textrm{pixel})$ for optimal minimum and maximum transmission was taken as the average from the two scans and added to the voltage-to-phase calibration of the first layer. Following this procedure, we corrected slight calibration inaccuracies and minimized intensity oscillations in the transmission spectrum due to the Fabry-Pérot effect. Note that this approach characterized the spectral intensity as feedback only, not the phase. Due to limited time and computational resources, we refrained from more sophisticated methods [65] to minimize pulse-shaper imperfections. Anyway, the purpose of the present work is to characterize the influence of any remaining pulse-shaper imperfections on 2D spectra, not to remove them all, because for other pulse-shaping approaches such correction procedures may not be applicable anyway or lead to different results, and we are interested in the accurate characterization of the real pulse shapes in all cases.

Characterization of the shaped pulse sequences in Fig. 1(a) was implemented by FTSI in the following way: We split off a small portion (${\approx} 3\;\%$ energy) of the pulse-shaper input beam with a thin fused-silica wedge pair (Hellma Optics, 1 mm thick edge, 60 arcmin, uncoated) and used it as the unshaped reference pulse. We integrated a suitable optical density filter ($\textrm{OD} = 0.3$, Thorlabs, NDUV03A) in the beam path of the reference pulse to adjust the reference pulse energy such that the spectral intensity was roughly similar to that of an unshaped pulse transmitted through the pulse shaper. The optical path of the reference pulse was matched to the propagation length of the shaped pulse taking into account the additional temporal retardation induced by the different dispersive media of the two polarizers (P), the two transmission gratings, and both liquid-crystal layers of the SLM. In this way, it was possible to set a temporal delay of around ${\tau _{\textrm{Ref}}} \approx 1\;\textrm{ps}$ between reference pulse and shaped pulse sequence within the travel range of the manual linear micrometer stage (OptoSigma, TSD-601S) featuring the retro-reflector. Temporal delays of this order ensure accurate Fourier filtering during phase reconstruction [53] with respect to the resolution of the implemented spectrometer. At the second wedge pair, a small part (∼3 % energy) of the shaped pulse sequence was recombined with the reference pulse, while the transmitted major part was directed to the spectroscopic experiment. This arrangement enables the characterization of the shaped pulse sequences used for multidimensional spectroscopy in parallel with the actually performed experiment. Thus each particular time-delay and phase-cycling correlation will be fully characterized.

The reference pulse and shaped pulse sequence were spatially superimposed on a fiber-coupled (Avantes, 250 – 2500 nm, FC-UVIR200-USB2), irradiance-corrected spectrometer (Avantes, 600 – 800 nm, AvaSpec-ULS3648) with 0.11 nm spectral resolution and the spectral interference (SI) spectrum was measured. Since the FTSI data processing requires separately measured spectra of the shaped pulse sequence and the reference pulse, we blocked either of the unwanted beam paths by automated shutters (S) after the first wedge pair when necessary.

Due to its higher pulse energy, the unshaped pulse [Fig. 1(a), upper left] can be well characterized by nonlinear methods, in our case collinear second-harmonic-generation frequency-resolved optical gating (SHG-FROG) [72]. The characterization in intensity and phase was performed in a commercial collinear SHG-FROG setup (APE, pulseCheck) in the beam path to the experiment. Since we want to characterize the pulse as accurately as possible at the position of the actual experiment, we included on the way to the SHG-FROG setup several additional optics mimicking the beam path to the experiment. In our particular setup the following additional optics were used: One ${\lambda / 2}$ wave plate (B.Halle, achromatic, 600 – 1200 nm, RAC 5.2.15 L) for final polarization adjustment, a thin vacuum chamber window (Torr Scientific, 1.5 mm thick, Fused silica), and a single fused-silica wedge (Laser Components, 2 mm thick edge, 15 arcmin, uncoated).

The measured spectrum of the unshaped pulse served as a constraint for the FROG retrieval algorithm. The retrieved spectral phase was then used to compress the pulse down to $\Delta t = 17\;\textrm{fs}$ (with a bandwidth-limited pulse duration of $\Delta {t_{\textrm{BWL}}} = 15\;\textrm{fs}$) by applying the inverted FROG phase to the SLM. Besides that, no phase or amplitude shaping was applied. The compressed pulse at the experiment was finally characterized by an additional SHG-FROG measurement and evaluation. All FROG traces and fitting results of the FROG algorithm for the compressed pulse can be found in Supplement 1.

As depicted in Fig. 1(a) we placed the same additional optics (Add. optics) used in the beam path to the SHG-FROG setup in front of the SI spectrometer (Spec.). Since the characterized compressed pulse used in the experiment passes in total two wedge pairs, a third wedge pair was introduced in front of the SI spectrometer to ensure that the pulse at the spectrometer which travels through the SLM passed the same amount of glas as the characterized compressed pulse.

The spectral phase characterization of the reference pulse has to be done once, before the spectral phases of the various shaped sequences can be determined. A direct characterization of the reference pulse via SHG-FROG would be challenging due to its low pulse energy. Since the influence of different propagation length through air on the spectral phase of the compressed pulse is negligible at the spectral range of the pulses ($600 - 800\;\textrm{nm}$), the retrieved FROG phase of the compressed pulse utilized in the experiment is an adequate description of the spectral phase of the compressed pulse at the SI spectrometer. The compressed pulse at the SI spectrometer, characterized in that way, enables a spectral-phase characterization of the reference pulse via FTSI, by analyzing the spectral-interference pattern of the spatially superimposed known compressed pulse and the unknown reference pulse (see second step in Supplement 1).

As an alternative approach, the spectral phase of the reference pulse can also be reconstructed by characterizing the uncompressed unshaped pulse instead of the compressed shaped pulse by SHG-FROG and using this spectral phase to characterize the spectral phase of the unknown reference pulse by FTSI. However, since the shaped pulse is compressed for generating multipulse sequences in any case, no evaluation step would effectively be saved by this approach.

The additional measurement time associated with the recording of the individual spectra makes up only 11 % of the 67 h overall measurement time. This could in principle be further minimized by shorter integration times down to shot-to-shot methods, or implementation of parallelized acquisition schemes of the required spectra. Note that the expenditure of measurement time for the characterization of the multipulse sequences in a 2D spectroscopy experiment strongly depends on the particular components of the FTSI setup and their arrangement.

2.2 Reconstruction of the time-domain structure

FTSI enables the reconstruction of the relative spectral phase $\Delta \varphi (\omega )$ between a reference pulse and an unknown pulse, such as a pulse sequence. If the spectral phase and intensity of the reference pulse are known, the actual spectral phase of the pulse sequence can be calculated. A rigorous description of the details concerning the individual spectral phase reconstruction steps during the FTSI data processing can be found in Supplement 1 and follows in general the literature [53,68]. While this delivers the complete electric field evolution, we here aim for a simplified description that allows analyzing deviations from the ideal case in terms of a limited number of parameters that are meaningful in multidimensional spectroscopy. For a three-pulse experiment [Fig. 1(a), top], these parameters are the two inter-pulse delays, $\tau ,\;t$ (light green, dark green), the maximum intensity of each particular pulse, ${I_{1,2,3}}$ (orange), and the phase offsets that are relevant for phase cycling, ${\varphi _{1,2,3}}$ (light blue), describing the zero-order Taylor coefficient of the individually Taylor-expanded temporal phase of the particular pulses labeled 1, 2 and 3. These parameters are retrieved in an automated fashion for both the real and ideal pulse sequences in the same manner and then quantitatively compared for each experimental measurement step.

Shown in Fig. 1(b) is an example for a pulse sequence from an ideal pulse shaper with the normalized time-domain intensity $I(t)$ (red circles) of a three-pulse sequence at time delays $\tau = 108\;\textrm{fs}$, $t = 112\;\textrm{fs}$ and phase offsets ${\varphi _1} = 0\;{{\textrm{rad}} / \mathrm{\pi }}$, ${\varphi _2} = 1.5\;{{\textrm{rad}} / \mathrm{\pi }}$, ${\varphi _3} = 1.0\;{{\textrm{rad}} / \mathrm{\pi }}$. The spectral intensity was calculated by multiplying the measured spectral intensity of the compressed pulse with the spectral intensity modulation function sent to the SLM; and the spectral phase was calculated by adding to the spectral phase, retrieved by SHG-FROG of the compressed pulse, the spectral phase modulation function sent to the SLM. After inverse Fourier transformation of the corresponding complex-valued electric field, the normalized temporal intensity and phase were obtained. As seen in Fig. 1(b), even the pulse sequence from an ideal pulse shaper reveals slight asymmetric temporal pulse shapes as expected, resulting from the non-perfect-Gaussian spectral intensity and uncompensated, residual higher-order spectral phase.

In the further discussion we refer to the pulse at smallest absolute time as the first pulse in the pulse sequence and to the pulse at largest absolute time as last pulse. By fitting the normalized time-domain intensity $I(t)$ with a sum of three Gaussian functions [Fig. 1(b), black line], we determined the inter-pulse delays $\tau ,\;t$ (green) and the maximum intensity values ${I_{1,2,3}}$ of each of the three pulses of the sequence. To optimize the stability of the fitting procedure, the data set was reduced to the gray-shaded regions in Fig. 1(b) by an intensity threshold criterion $I\, > \,0.20 \times {I_{\max }}$ with ${I _{\textrm{max}}} = \textrm{max}\{{{I_1},{I_2},{I_3}} \}$.

From the zero-order Taylor coefficients of the individual temporal phases (dark blue circles), we reconstructed the phase offsets introduced by phase cycling. Further, the uncertainty of the phase offsets, caused by linear, quadratic, or higher-order temporal phase [dark blue circles in Fig. 1(b)], was minimized by defining the phase offset ${\varphi _{1,2,3}}$ as the temporal phase value at the time of maximum pulse intensity ${I_{1,2,3}}$ (light blue circles). A detailed discussion of the phase offset extraction procedure from the temporal phase structure can be found in Supplement 1.

In Fig. 1(c) we show the experimentally characterized field for the same pulse-sequence parameters as the ideal sequence in Fig. 1(b). For the real pulse-shaper data, we evaluated the pulse-shape information in the same way as for the ideal case, only that the pulse sequence was reconstructed by FTSI from the measured SI spectrum.

Comparing the ideal and the real case reveals that the temporal delays $\tau ,\;t$ as well as the phase-offset systematic can be nicely reproduced. In the real pulse-shaper sequence, a global $10\;\textrm{fs}$ temporal shift toward positive times is observed. For multidimensional time-resolved spectroscopy, only relative time delays between individual inter-pulse pairs are relevant anyway, and a temporal offset value will not affect the retrieved 2D spectrum.

The main difference between both pulse sequences lies in the intensities of individual pulses. The ideal sequence exhibits equal maximum intensities for all three pulses, whereas the real pulse shaper shows a 35 % reduced maximum intensity of the first pulse and a 20 % reduced maximum intensity of the middle pulse compared to the maximum intensity of the last pulse. Discrepancies such as these will be analyzed with our automated procedure for each individual scanning step, leading to a comprehensive analysis of their influence on resulting 2D spectra.

3. Systematic analysis of reconstructed pulse parameters

So far we introduced our pulse characterization procedure and exemplarily described the extraction of physical parameters defining the shaped pulse sequences. In this section, we show a systematic comparison between the extracted sequence parameters of the ideal pulse-shaper (ideal PS) and real pulse-shaper (real PS) sequences.

We scanned both inter-pulse time delays $\tau ,\;t$ from 0 to 112 fs and performed the experiment in a partially rotating frame. This enables us to sample signals at spectral frequencies within the exciting pulse spectrum centered at $\hbar {\omega _0} = 1.76\;\textrm{eV}$ $(\Delta {E_{\textrm{FWHM}}} = 0.15\;\textrm{eV})$ while using a larger delay increment [8,73], thus reducing the overall measurement time. The degree of transformation to the rotating frame is given by the $\gamma $ parameter, where $\gamma = 0$ corresponds to a fully rotating frame and $\gamma = 1$ to the laboratory frame. This leads to an additional delay-dependent temporal phase factor in the exciting pulse train, which was implemented by corresponding spectral phase shaping of the SLM and needs to be further taken into account in the phase offset extraction procedure as discussed in Supplement 1. In our case, $\gamma = 0.18$ was chosen and the delay increment set to $4\;\textrm{fs}$ to sample signals at spectral frequencies between $0.89 - 1.96\;\textrm{eV}$ with a resolution of $\delta E = 0.037\;\textrm{eV}$ [74]. To extract both rephasing and non-rephasing signal contributions necessary to construct a purely absorptive 2D spectrum, a 16-fold $1 \times 4 \times 4$ phase-cycling scheme was applied [39].

To reconstruct the time-domain structure of all pulse sequences, we acquired the intensity spectra of the reference pulse (by blocking the shaped beam), of the current pulse sequence (by blocking the reference beam), and of the superimposed SI spectrum (by opening both shutters) during each measurement step in an automated fashion. The detailed spectral phase reconstruction steps can be found in Supplement 1.

Due to the temporal overlap regime of single pulses at inter-pulse delays below 20 fs, the sequence parameters can be reliably extracted only for delays larger than 20 fs. The extracted sequence parameters represent typical three-pulse sequences with $\tau ,\;t = 20\; - \;112\;\textrm{fs}$ as commonly used in multidimensional spectroscopy experiments. This leads to a data set of 7582 real PS sequences, whose extracted parameters are systematically compared to the ideal PS sequence parameters. The reduced data set corresponds to 56 % of the $29 \times 29 \times 16 = 13456$ different pulse sequences used in the whole experiment. Such a reduction is done here for illustrating in a meaningful and systematic manner the deviations between all real and ideal pulse sequences (Fig. 2). Plotting and comparing the full electric-field characterization results for several thousand pulse shapes would not have been feasible. We note, however, that the full characterization results are available, including those for temporally overlapping pulses, and can be used to simulate 2D spectra with real pulse shapes.

 

Fig. 2. Systematic analysis of reconstructed pulse parameters measured simultaneously with a collinear multidimensional spectroscopy experiment. (a) Correlation between experimentally determined (real PS) and ideally expected (ideal PS) time delays $\mathrm{\tau }$ (light green) and t (dark green) for all temporally nonoverlapping three-pulse sequences ($\mathrm{\tau }\textrm{,}\;t \ge 24\,\textrm{fs}$). The data lining up along the diagonal (gray line) is a sign of good correlation quantified by Pearson correlation coefficients in the main text. (b) Correlation between experimentally determined (real PS) and ideally expected (ideal PS) relative phases $\Delta \varphi $ of the second ($\Delta {\varphi _{12}}$, dark blue) and third ($\Delta {\varphi _{13}}$, light blue) pulse relative to the first one of each sequence. Due to $1 \times 4 \times 4$ phase cycling, the data are clustered around $0\;{{\textrm{rad}} / \mathrm{\pi }}$, $0.5\;{{\textrm{rad}} / \mathrm{\pi }}$, $1.0\;{{\textrm{rad}} / \mathrm{\pi }}$ and $1.5\;{{\textrm{rad}} / \mathrm{\pi }}$ on the diagonal (gray line). (c) Experimentally determined delays $\mathrm{\tau }$ (light green) and t (dark green) versus measurement step, exemplarily shown for a subsection out of the full data set. The gray lines show the ideal delay steps. For the 16 measurement steps at constant $\mathrm{\tau }\,\textrm{= 84}\,\textrm{fs}$, $t = 96\,\textrm{fs}$, highlighted in orange, (d) the experimentally determined relative phases are shown within the 16-step phase-cycling (PC) scheme for the second ($\Delta {\varphi _{12}}$, dark blue) and third ($\Delta {\varphi _{13}}$, light blue) pulse, and should ideally fall onto the gray lines. (e) Squared intensity ${I^2}_{\textrm{real PS}}$ of the product of the four interacting field amplitudes $\sqrt {{I_1}} \times \sqrt {{I_2}} \times \sqrt {{I_2}} \times \sqrt {{I_3}} \propto {I^2}$ of the real pulse sequence for each $\mathrm{\tau }$, t delay combination outside of the pulse overlap region. The depicted map is averaged over the 16 phase-cycling steps $({\overline {\textrm{PC}} } )$ and normalized to the mean value of the averaged map, emphasizing deviations. (f) Squared, averaged and normalized linear photodiode signal ${I^2}_{\textrm{ref PD}}$ of an unshaped reference pulse measured simultaneously for each measurement step to monitor long-term NOPA intensity drifts.

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Figure 2(a) shows the correlation between the experimentally characterized inter-pulse delays of the real PS sequence and the ideal PS sequence. Both delay values, $\tau $ (light green) and t (dark green), accumulate along the diagonal showing very good correlation with Pearson correlation coefficients of ${|r |_\tau } = 0.99$ and ${|r |_t} = 0.99$. The four significantly deviating values are due to convergence errors in the automated fitting procedure of the three Gaussian functions and not due to an incorrect pulse sequence shape. Manual fitting of these shapes would remove the error in this plot, but we chose not do so here in order to emphasize the fully automated evaluation procedure. Anyway, for accurate simulation of 2D data, the fully characterized and not the fitted fields would be employed, such that this deviation is irrelevant.

Figure 2(b) displays a similar correlation analysis for the relative inter-pulse phase offset values that are important to validate a correct implementation of the phase-cycling scheme. For simplicity, we refer to this relative inter-pulse phase offset values in the following as relative phases. Since the phase-cycling phase values are referenced to those of the first pulse in the pulse sequence, the relative phases amount to $\Delta {\varphi _{12}} = {\varphi _2} - {\varphi _1}$ and $\Delta {\varphi _{13}} = {\varphi _3} - {\varphi _1}$. Both relative phases, $\Delta {\varphi _{12}}$ (dark blue) and $\Delta {\varphi _{13}}$ (light blue), accumulate along the diagonal at $0\;{{\textrm{rad}} / \mathrm{\pi }},\;0.5\;{{\textrm{rad}} / \mathrm{\pi }},\,\;1.0\;{{\textrm{rad}} / \mathrm{\pi }},\;1.5\;{{\textrm{rad}} / \mathrm{\pi }}$ and quantitatively demonstrate the correlation between ideal and real PS sequence with Pearson correlation coefficients of ${|r |_{\Delta {\varphi _{12}}}} = 0.99$ and ${|r |_{\Delta {\varphi _{13}}}} = 0.99$. The few uncorrelated, isolated measurement points seen in the plot result from induced phase jumps due to the phase ambiguity at zero temporal intensity in the offset phase extraction procedure. Thus, again, the automated extraction has some convergence problems for isolated cases that were not removed manually on purpose, but these are not relevant when using the fully recovered electric field for simulating 2D spectra.

Figure 2(c) depicts the extracted delays from the real PS sequences for one particular section of 175 measurement steps. These steps encode 175 different three-pulse sequences with temporal delays between $\tau ,\;t = 76 - 97\;\textrm{fs}$, out of the whole multidimensional spectroscopy experiment. For measurement steps $9522 - 9697$, the increase of the t delay (dark green) from $t = 76\;\textrm{fs}$ to $t = 96\;\textrm{fs}$ in steps of $4\;\textrm{fs}$ is reconstructed while inter-pulse delay $\tau $ stays constant at $\tau = 96\;\textrm{fs}$. The gray line shows the expected data point evolution from ideal PS sequences. After delay t reaches $96\;\textrm{fs}$, the $\tau $ delay is decreased in $4\;\textrm{fs}$ steps down to $\tau = 76\;\textrm{fs}$, while t stays constant at $t = 96\;\textrm{fs}$. This time delay scanning behavior reflects our so-called “quadratic delay scanning scheme”. The benefit of this delay scanning scheme is to recreate, in case of unexpected abortion of the experiment, a complete quadratic time map, containing all coherence-time combinations up to the current value, which can be Fourier transformed into a 2D spectrum having the same frequency resolution on both frequency axes. We prefer this “quadratic scanning” to a more conventional approach in which one coherence time is scanned completely for all time steps before proceeding by one time step in the other coherence time. From a systematic analysis of the extracted delays between ideal and real PS sequence of 7582 measurement steps, we find root-mean-square deviations between real PS and ideal PS delays of $\tau {\,_{\textrm{RMS}}}\, = 1.5\;\textrm{fs},\;t{\,_{\textrm{RMS}}}\, = 1.3\;\textrm{fs}$, which is below the $4\;\textrm{fs}$ delay increment and thus ensures temporal ordering of the constituent pulses outside of the pulse-overlap region. As seen in Fig. 2(c), the adjusted pulse-delay configuration stays unchanged for 16 measurement steps before one delay is either increased or decreased. Over the course of these 16 measurement steps the phase cycling of the pulse sequence is carried out as emphasized by the orange box in Fig. 2(c).

We apply $1 \times 4 \times 4$ phase cycling to change the phases of the second and third pulse in four steps of $0.5{{\;\textrm{rad}} / \mathrm{\pi }}$. Their experimental characterization is shown in the zoomed-in region of Fig. 2(d). Here we show the reconstructed temporal relative phases $\Delta {\varphi _{12}}$ (dark blue) and $\Delta {\varphi _{13}}$ (light blue) for all 16 phase-cycling steps (PC steps) during a certain time delay configuration $\tau = 84\;\textrm{fs}$ and $t = 96\;\textrm{fs}$. Pulse 3 is first phase-cycled from $\Delta {\varphi _{13}} = 0{{\;\textrm{rad}} / \mathrm{\pi }}$ to $1.5{{\;\textrm{rad}} / \mathrm{\pi }}$ in $0.5{{\;\textrm{rad}} / \mathrm{\pi }}$ steps, while the relative phase between pulse 1 and 2, $\Delta {\varphi _{12}}$, stays constant around $0{{\;\textrm{rad}} / \mathrm{\pi }}$. During PC steps 5 to 8, pulse 3 is again phase-cycled from $0{{\;\textrm{rad}} / \mathrm{\pi }}$ to $1.5{{\;\textrm{rad}} / \mathrm{\pi }}$, while $\Delta {\varphi _{12}}$ stays constant around $0.5{{\;\textrm{rad}} / \mathrm{\pi }}$. This procedure is continued until $\Delta {\varphi _{12}}$ and $\Delta {\varphi _{13}}$ reach $1.5{{\;\textrm{rad}} / \mathrm{\pi }}$ which corresponds to the last PC step. The reconstructed phase-cycling scheme is in qualitative agreement with the expected ideal relative phases (horizontal gray lines) of the $1 \times 4 \times 4$ phase-cycling scheme [39]. The systematic comparison of 7582 relative phases, similar to the exemplary subset from Fig. 2(d), between real PS sequence and ideal PS sequence results in root-mean-square deviations of $\Delta {\varphi _{12}}_{\;\textrm{RMS}}\, = 0.19\;{{\textrm{rad}} / \mathrm{\pi }}$ and $\Delta {\varphi _{13}}_{\;\textrm{RMS}}\, = 0.08\;{{\textrm{rad}} / \mathrm{\pi }}$. This quantifies the precision or deviation of the phase-cycling scheme for the real PS sequences.

Next, we carry out a systematic comparison of the three maximum intensity values for all reconstructed and fitted real PS sequences with respect to the corresponding ideal PS sequences outside of the pulse overlap region. This reveals that the maximum intensity of the first pulse is systematically reduced. We quantify the effect by averaging the relative deviations between real PS and ideal PS sequences for the data set, which results in $\overline {{\textstyle{{\Delta {I_1}} \over {{I_1}}}}} ={-} 38\;{\%}$. The statistic deviations are determined by the standard deviation of $\overline {{\textstyle{{\Delta {I_1}} \over {{I_1}}}}}$ resulting in ${\sigma _{\overline {{\textstyle{{\Delta {I_1}} \over {{I_1}}}}} }}\, = 11\;{\%}$. For the maximum intensity of the middle and last pulses, only small systematic deviations $\overline {{\textstyle{{\Delta {I_2}} \over {{I_2}}}}} \, ={-} 3\;{\%,}\;\overline {{\textstyle{{\Delta {I_3}} \over {{I_3}}}}} \, ={-} 4\;{\%}$ and, with respect to the first pulse comparable, statistic deviations ${\sigma _{\overline {{\textstyle{{\Delta {I_2}} \over {{I_2}}}}} \,}} = 11\;\%$ and ${\sigma _{\overline {{\textstyle{{\Delta {I_3}} \over {{I_3}}}}} }}\, = 11\;\%$ are disclosed. No further systematic variations of the individual maximum intensity values ${I_{1,2,3}}$ are observed.

Finally, we analyze the impact of the systematic variations of ${I_{1,2,3}}$ on the scaling of the (rephasing) signal contribution in a 2D spectrum. At fourth order in perturbation theory (in the case of population-detected methods), the signal results from four electric-field interactions and thus scales with ${S_{\textrm{reph}}} \propto \sqrt {{I_1}} \times \sqrt {{I_2}} \times \sqrt {{I_2}} \times \sqrt {{I_3}} \propto {I^2}$. In our current three-pulse and phase-cycling scheme, the second pulse interacts twice [39]. The ${I^2}$ scaling of ${S_{\textrm{reph}}}$ is analyzed as a function of both delays $\tau $ and t resulting in a 2D delay map. We average ${I^2}$ for each delay combination over all 16 phase-cycling steps and normalize the whole averaged delay map with respect to its global mean value. Thus, in the ideal case, assuming no ${I_{1,2,3}}$ variations, one would expect a structureless ${I^2}$ delay map of value 1.0. In Fig. 2(e) we depict the experimental results using the maximum intensity values ${I_{1,2,3}}$ extracted from the real PS sequences. Since the extraction of maximum intensity crucially depends on the uniqueness of the three Gaussian fits, we restricted our analysis further to pulse delays $\tau ,\;t \ge 36\;\textrm{fs}$ to avoid interference effects of temporally overlapping pulses. The deviation from the ideal case scenario is seen in Fig. 2(e) and quantified by an overall root-mean-square fluctuation of ${I_{\textrm{real PS}}}{^2_{\,\textrm{RMS}}}\, = 5\;\%$. Note that the graphical representation emphasizes deviations because the color range extends only from 0.9 to 1.1. The fluctuation exhibits a delay-related systematic, especially at delays $\tau ,\;t = 64\;\textrm{fs}$ and $\tau ,\;t = 92\;\textrm{fs}$ (decreased ${I_{\textrm{real PS}}}^2$) as well as $\tau ,\;t = 48\;\textrm{fs}$ and $\tau ,\;t = 100\;\textrm{fs}$ (increased ${I_{\textrm{real PS}}}^2$).

To investigate the origin of the systematic deviations of Fig. 2(e), we monitored the pulse energy of the unshaped pulse simultaneously with the SI measurements and the multidimensional spectroscopy experiment, using a linear photodiode (PD) as an integrating detector. Due to the long integration time of our photoemission electron microscope (Elmitec, AC-PEEM) [71], the overall measurement time of this particular experiment was ${\approx} 67\;\textrm{h}$ and thus prone to long-term pulse energy drifts of the NOPA system. This raises the question if the deviations of Fig. 2(e) result from NOPA drifts of the input beam rather than systematic pulse-shaping variations of ${I_{1,2,3}}$. For comparison, we thus plot the reference measurement in Fig. 2(f). To generate this map, we plot the squared pulse energy of the unshaped reference pulse for the same delay combinations and phase-cycling steps as in Fig. 2(e), again averaged over the 16 phase-cycling steps, normalized as described above, and plotted in the same color bar range. Figure 2(f) reveals analogous scaling fluctuations around the ideal case value of ${I_{\textrm{ref}\;\textrm{PD}}}^2\, = 1.0$. Comparing Fig. 2(e) and (f) shows that the overall scale of the fluctuation of the reference pulse, ${I_{\textrm{ref PD}}}{^2_{\,\textrm{RMS}}}\, = 2\;\%$, is smaller than that observed in the SI-extracted data, ${I_{\textrm{real PS}}}{^2_{\,\textrm{RMS}}}\, = 5\;\%$, and that the deviation from 1.0 slowly decreases for larger delays. However, the delay-related systematics are very similar in both maps. We thus conclude from this comparison that the deviations observed in Fig. 2(e) are mainly caused by long-term pulse-energy drifts of the NOPA system and not by systematic pulse-shaper imperfections.

Summarizing Sec.  3, we analyzed and quantified the pulse-shaping precision of three-pulse sequences with respect to specific parameters that are relevant in 2D spectroscopy and compared real and ideal pulse-shaping behavior systematically for a full data set of time delay and relative phase variations. We now investigate the impact of deviations from the ideal case on 2D spectra.

4. Simulation of 2D spectra with real and idealized pulse sequences

To investigate if deviations in time delays, relative phases and maximum intensities of the real PS sequence (Fig. 2) influence the 2D spectrum in population-based collinear multidimensional spectroscopy, we simulated purely absorptive 2D spectra while using the experimentally characterized electric fields for each scanning step as simulation input.

The collective system response after optical excitation was described by a density matrix approach whose full time evolution was given by the Lindblad quantum master equation, taking into account system–bath interactions like decoherence and dissipation [75,76]. Solving this equation enables the determination of the off-diagonal elements of the density matrix $\rho $, known as system coherences, as well as of diagonal elements corresponding to the populations of the system.

For illustration, we chose a fictitious system but employed typical simulation parameters for molecular pure dephasing times [77] and coupling strength [78] to model, in a first example, the fluorescence emission of a homogeneous molecular dimer. In Fig. 3(a) we show the simulated level scheme of the dimer in the exciton basis [79]. The energy difference between the ground-state level $|0 \rangle $ and the two-exciton state $|3 \rangle $ (with both molecules simultaneously excited), was chosen in this example as $2\hbar {\omega _0}$, with $\hbar {\omega _0} = 1.755\;\textrm{eV}$ and central frequency ${\omega _0} = 2.67\;\textrm{f}{\textrm{s}^{ - 1}}$ of the exciting three-pulse sequence (red). The one-exciton states $|1 \rangle $ and $|2 \rangle $ describe superpositions of one molecule in the ground state and the other molecule in the excited state. We modeled the symmetric Davydov energy splitting around $\hbar {\omega _0}$ by $- {\textstyle{{\Delta E} \over 2}}$ and $+ {\textstyle{{\Delta E} \over 2}}$, where $\Delta E$ is related to the coupling strength between the individual molecules. Note that we defined the coupling strength such that the resulting Davydov splitting $\Delta E$ is within the spectral width $(\Delta {E_{\textrm{FWHM}}} = 150\;\textrm{meV})$ of our unshaped pulse spectrum. The individual transition dipole moments, pure dephasing and dissipation times defining the system are given in Supplement 1. Since the fluorescence signal decays on a nanosecond time scale, which is much larger than the summed maximum delays $({\approx} 250\;\textrm{fs)}$ of the exciting pulse sequence, an infinite dissipation time for the $|1 \rangle \to |0 \rangle $ transition was chosen.

 

Fig. 3. 2D spectra with real and ideal pulse sequences. (a) Three-pulse excitation sequence ${E_{\textrm{inc}}}$ (red) and quantum four-level system, modelling fluorescence from a molecular dimer. The energy distance between the ground-state level $|0 \rangle $ and the doubly excited level $|3 \rangle $ is $2\hbar {\omega _0}$ with ${\omega _0}$ describing the center frequency of the incoming pulse sequence ${E_{\textrm{inc}}}$. The two singly excited levels $|1 \rangle $ and $|2 \rangle $ exhibit a Davydov splitting of $\Delta E$ (detailed description see main text). (b) Exemplary Feynman diagram illustrating a rephasing pathway. To simulate 2D spectra, we calculate the fluorescence yield ${Y_{\textrm{Fl}}}$ by summing over excited-state populations. (c) Simulated absorptive 2D spectra (real parts) using experimentally determined fields for each scanning step (Real PS) or idealized pulse sequences (Ideal PS). (d) The same exciting three-pulse sequences ${E_{\textrm{inc}}}$ were used to simulate nonlinear photoemission (green) 2D spectra for a plasmonic gold nanoslit sample. (e) The local electric field ${E_{\textrm{loc}}}$, generated by ${E_{\textrm{inc}}}$, depends on the actual time delays and phases and thus alters the nonlinear photoemission yield ${Y_{\textrm{PE}}}$. We set the nonlinearity to $N = 4.2$ corresponding to a dominating four-photon photoemission process. (f) Simulated absorptive 2D spectra (real parts) obtained from ${Y_{\textrm{PE}}}$ using experimentally determined fields for each scanning step (Real PS) or idealized pulse sequences (Ideal PS).

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We calculated 2D spectra using the experimentally characterized electric fields ${E_{\textrm{inc}}}$ (red) for each scanning step at corresponding time delays $\tau ,\,\;t$ and relative phases (blue dashed) corresponding to the $1 \times 4 \times 4$ phase-cycling scheme as displayed in Fig. 3(a). Extraction of, e.g., the rephasing signal contribution (or others) via phase cycling allows us to monitor the temporal evolution of the system via different pathways through the various states of the density matrix. These different pathways can be visualized by double-sided Feynman-diagrams, of which an exemplary one is depicted in Fig. 3(b) for one particular rephasing pathway. Double-sided Feynman diagrams can be analogously derived for non-rephasing and other signal contributions.

To calculate the fluorescence yield ${Y_{\textrm{Fl}}}$ (red) of the molecular dimer system, we used the time-domain electric fields from either the reconstructed real PS pulse sequence or the ideal PS pulse sequence as exciting fields, ${E_{\textrm{inc}}}$, and propagated the density matrix $\rho (t)$ by the Lindblad equation [10]. The detection signal ${Y_{\textrm{Fl}}}$ was defined by summing over the excited-state populations of the density matrix $\rho (t^{\prime})$ after time $t^{\prime}\, > > \,\tau + T + t$ as shown in Fig. 3(b). Since we use a three-pulse basis, the population time is $T = 0\;\textrm{fs}$. We obtained the absorptive spectrum by adding the real part of the rephasing and non-rephasing spectra after inverting the sign of the ${\omega _\tau }$ axis for the rephasing spectrum [74].

In previous work, we have taken into account pulse-shaper imperfections in fluorescence-detected two-dimensional micro-spectroscopy and corrected time-domain artifacts by subtracting low-excitation-power time-domain maps from high-excitation-power maps before phase cycling [64]. In the present work, we go beyond that and analyze the impact of pulse-shaper imperfections by explicit measurement of all involved electric fields. Note that we observe a significant signal contribution along the diagonal of the simulated rephasing time-domain map, similar to the time-domain artifact discussed in experimental data [64], using either the ideal PS reconstructed or real PS reconstructed pulse sequences. We find that the amplitude of this artifact is significantly increased in case of the real PS sequences. Thus we corrected our simulation results following the earlier approach [64]. The rephasing and non-rephasing contributions were then extracted by the corresponding phase-cycling scheme, followed by 2D Fourier transformation. A detailed description of the procedure is given in Supplement 1.

The result is depicted in Fig. 3(c). Here we plot the corrected purely absorptive spectrum using the reconstructed time-domain structure of the real PS (left) and the ideal PS pulse sequences (right). The corresponding rephasing and non-rephasing 2D spectra and corrected time-domain maps can be found in Supplement 1, respectively. Both absorptive spectra show the same characteristic features like diagonal and off-diagonal peak positions, line shapes and relative signal amplitudes. The position of the diagonal peaks correlate with the eigenenergies of the one-exciton states $|1 \rangle $ and $|2 \rangle $ (see Supplement 1) and the energy difference between the diagonal peaks of the here simulated homogeneous dimer system corresponds to the Davydov splitting $\Delta E$. The off-diagonal peaks appear due to the nonzero exitonic coupling $J = {\textstyle{{\Delta E} \over 2}}$ between the one-exciton states $|1 \rangle $ and $|2 \rangle $ [79]. The small signal distortion along the diagonal in the Real PS spectrum originates from a small leftover of the time-domain artifact (see Supplement 1) resulting from the small pulse-shaper imperfections quantified in Fig. 2. The dominating negative sign of the absorptive signal within the spectral region covered by the laser pulse spectrum ($\Delta E = 1.63 - 1.86\;\textrm{eV}$ foot-to-foot) reflects the dominating fourth-order field interaction with the corresponding sign in the perturbation expansion.

Note that the small deviations between the real and ideal PS sequences do not significantly change the features in the obtained corrected 2D spectra. We further conclude from this investigation that the small deviations of the real PS sequences discussed in Sec.  3 lead to increased time-domain artifacts which influence the resulting 2D spectrum. These distortional effects can be sufficiently minimized by performing a high-minus-low-excitation-power correction that in turn facilitates the correct interpretation of the resulting 2D spectrum. Nonetheless, if one desires to simulate experimental data as accurately as possible, one can use the fully characterized fields as an input to 2D simulations, potentially along with corrections for “geometric” signal distortion [43–46].

As a second example, we now analyze the impact of experimentally characterized pulse sequences on electron photoemission experiments [Fig. 3(d)]. We chose the sample-dependent work function to be significantly larger than the energy of the interacting photons $\hbar {\omega _0} = 1.775\;\textrm{eV}$, thus making multiphoton photoemission the detection signal. The efficiency of multiphoton photoemission is in general lower compared to single-photon photoemission. The latter can be realized using a separate ionization pulse at sufficiently high photon energy or work-function reduction via Cs evaporation [80]. We wanted to investigate multiphoton photoemission because the relevant interaction order is higher in nonlinearity than in the case of the fluorescence example, providing a qualitatively different response. In particular, we simulated nonlinear photoelectron emission [Fig. 3(d, green)] from a plasmonic gold nanoslit sample (Au), which we previously investigated [81], after optical excitation with the same experimentally determined fields ${E_{\textrm{inc}}}$ as in the first example. The work function of gold varies between $(4.6 - 5.6)\;\textrm{eV}$ depending on the amount of carbon coverage [82]. Due to strong plasmonic field enhancement [83,84] of ${E_{\textrm{loc}}}$, the possibility of multiphoton photoemission at high nonlinear order $N\, > \,3$ is significantly increased [81] meaning that at photon energies of $\hbar {\omega _0} = 1.775\;\textrm{eV}$ six or more field interactions take place to generate photoemitted electrons via a three-photon process.

The spatially resolved response function of the plasmonic nanoslit, which was retrieved from finite-difference time-domain (FDTD) simulations (FDTD Solutions, Lumerical Inc.), enables the calculation of the local electric field ${E_{\textrm{loc}}}$ [81]. In Fig. 3(e), we show exemplarily the resulting momentary local electric field ${E_{\textrm{loc}}}$ for two consecutive phase-cycling steps (light and dark blue) at one particular time-delay combination $\tau ,\;t$ of the exciting pulse sequence. We define the nonlinear local photoemission yield as the time-integrated local intensity to the power of N. Motivated by a measured, spatially resolved, power-law dependence of the local photoemission that was fitted with an $N = 4.2$ dependence (not shown), we used the same order of nonlinearity in our simulation to model photoemission dominated by a four-photon process. The nonlinear photoemission detection signal ${Y_{\textrm{PE}}}$ was calculated for each pulse sequence separately. ${Y_{\textrm{PE}}}$ corresponds to the time-integrated nonlinear local photoemission yield, which in a last evaluation step was spatially integrated along the long slit axis covering one local electric field spot [shaded red in Fig. 3(d)] within the nanoslit.

Figure 3(f) shows the simulated absorptive 2D spectra for the experimentally determined (left) and the ideal sequence (right). The absorptive spectra were obtained, just as described for the fluorescence example, from the corresponding rephasing and non-rephasing spectra. The positive sign of the signal in the absorptive spectra can be explained by a dominating signal contribution resulting from higher perturbation order, compared to the case of the fluorescence signal simulations, e.g., eight electric-field interactions. Note that, in contrast to the fluorescence 2D spectra simulations, it is not necessary to perform the “high-power-minus-low-power” correction procedure, because no signal contribution along the diagonal of the rephasing time-domain map is seen (see Supplement 1).

Both spectra depicted in Fig. 3(f) display the same triangular-shaped signal with a maximum around the central frequency of the pulse spectrum corresponding to $\hbar {\omega _0} = 1.755\;\textrm{eV}$. The small deviations between the real and the ideal pulse sequences, quantified in Sec.  3 and Fig. 2, do not significantly influence the resulting absorptive spectra in this example. Minor variations are visible between the rephasing and non-rephasing spectra as reported in Supplement 1. Thus, the impact of signal artifacts due to non-perfect pulse shaping is decreased as compared to the fluorescence case (for which the described power correction procedure was carried out). The improved robustness against pulse-shaper imperfections results if the detection signal is dominated by signal contributions from higher-order interactions. In this case, the enhanced nonlinear scaling of the sample signal contribution with respect to the optical exciting fields dominates more easily the systematics of the phase-cycled time-domain maps, whereas the diagonal signal artifact, due to pulse shaper imperfections, is strongly suppressed (see Supplement 1). This decreased sensitivity to imperfect pulse shaping is especially beneficial for multiphoton photoemission spectroscopy [71] or multiphoton coherent two-dimensional electronic mass spectrometry [16].

In the present particular example and implementation, ideal and real pulse sequences are quite similar which is a sign of the stability of the experiment and validity of the pulse-shaper design and calibration procedure. However, depending on the particular setup, wavelength regime, investigated quantum system, scanning range, data acquisition procedure and many other parameters, the outcome of such a pulse-shape analysis might be different and the deviations might be more severe. Examples for origins of imperfections are: pointing fluctuations of the laser that might lead to intensity fluctuations at the sample and time–frequency shifts after wavelength conversion; fluctuations of the spectral intensity and phase of the laser source; unstable frequency conversion in NOPA, hollow-core fiber or other filamentation processes; temperature variations at the pulse-shaper element leading to drifts in the generated pulse shapes; unintended pulse-shape modifications when approaching the available “maximum time window” of the pulse shaper; effects of space–time coupling on the pulse structure depending on the optical resolution versus pixel size of the pulse-shaper element; and other effects. In general, the magnitude and relevance of such deviations for 2D spectra and particular signal contributions is not known beforehand. The presented method characterizes all deviations experimentally, thus taking into account all sources of deviation within one procedure.

In both investigated cases of fluorescence or photoelectron emission, and also in other measurement modalities using either population-based or coherent-field detection, it is possible to use the fully characterized electric fields of each data acquisition scanning step to calculate 2D spectra with the precise experimental pulse shapes, and thus to facilitate a direct comparison between theory and experiment.

5. Conclusion

We implemented a measurement procedure for fully characterizing the electric field, in intensity and phase, of femtosecond multipulse sequences in each individual scanning step of pulse-shaper-assisted collinear multidimensional spectroscopy. Characterization was carried out simultaneously with the actual spectroscopy experiment employing Fourier-transform spectral interferometry. This means that the real shape of the pulse sequence is reconstructed for each measurement step and thus can be used in simulations. This is more accurate than using idealized pulses and more accurate than using one experimentally determined pulse shape that is, however, incorrectly assumed to be the same for all pulses of the excitation sequence and all measurement steps. The precise electric-field reconstruction, carried out in a fully automated fashion, is beneficial to verify and improve pulse-shaper calibration and correction methods.

In our specific case, the analysis provided information on the accuracy of relevant multipulse parameters achieved with a specific pulse-shaper implementation. Deviations between real and ideal pulse-shaper sequences were quantified outside of the temporal pulse overlap regime. The accuracies of inter-pulse delays were determined to be $\tau {\,_{\textrm{RMS}}}\, = 1.5\;\textrm{fs},\;t{\,_{\textrm{RMS}}}\, = 1.3\;\textrm{fs}$ for the coherence times, and relative phases between individual pulses were found to have accuracies of $\Delta {\varphi _{12}}{\,_{\textrm{RMS}}}\, = 0.19\;{{\textrm{rad}} / \mathrm{\pi }}$ and $\Delta {\varphi _{13}}{\,_{\textrm{RMS}}}\, = 0.08\;{{\textrm{rad}} / \mathrm{\pi }}$. The extracted maximum intensities of the pulses constituting a real PS sequence exhibited a systematically decreased maximum intensity of the first pulse of $\overline {{\textstyle{{\Delta {I_1}} \over {{I_1}}}}} \, ={-} 38\;{\%}$, small systematic deviations of the second and third pulse of $\overline {{\textstyle{{\Delta {I_2}} \over {{I_2}}}}} \, ={-} 3\;{\%,}\;\overline {{\textstyle{{\Delta {I_3}} \over {{I_3}}}}} \, ={-} 4\;{\%}$, and statistical deviations of about ${\sigma _{\overline {{\textstyle{{\Delta {I_1}} \over {{I_1}}}}} ,\;\overline {{\textstyle{{\Delta {I_2}} \over {{I_2}}}}} ,\overline {\;{\textstyle{{\Delta {I_3}} \over {{I_3}}}}} }}\, = 11\;{\%}$ compared to the constituent ideal PS pulses. Note that a systematically smaller intensity of any of the pulses in relation to the others is irrelevant at a given order of perturbation theory because the absolute intensities of the participating pulses factor out of the response-function convolution integral and thus only affect the overall magnitude of the signal, not the shape of resulting 2D spectra. Systematic variations in the scaling of the nonlinear rephasing signal ${S_{\textrm{reph}}}$ were related to intensity variations and explained by long-term pulse-energy drifts of the NOPA system, and not due to pulse-shaping imperfections.

We simulated the impact of the complete experimental pulse shapes on signal features in 2D spectra for two exemplary systems: first, the fluorescence signal of a molecular dimer system and second, the nonlinear photoemission signal from a plasmonic nanoslit sample. In both cases, we found that even with the quantified deviations between real and ideal pulse sequences, very similar characteristic 2D spectral features resulted, such as peak positions and line shapes in purely absorptive 2D spectra. In the case of fluorescence detection, additional corrections by a low- and high-field-amplitude simulation had to be applied. We further found that this correction procedure is not necessary when highly nonlinear signals, as in multiphoton photoemission, are detected.

The precise magnitude of deviations between real and ideal pulse-shaper sequence will vary from setup to setup depending on implementation details. Independent of that, the general method demonstrated here, i.e., automated Fourier-transform spectral interferometry on each individual pulse sequence, can be used for the precise simulation of obtained experimental spectra with the real applied pulse shapes. This approach can be applied in a straightforward manner also to other pulse-shaping methods, such as those based on acousto-optic modulators within 4f setups or acousto-optic programmable dispersive filters. While pulse-shaping artifacts will likely be different in those cases, the automated characterization, in parallel with the 2D spectroscopic experiment, will allow taking into account the correct electric fields in all simulation and interpretation efforts, and improve the reliability when comparing theoretical and experimental spectra. This procedure eliminates a possible source of disagreement and thus improves the confidence levels for comparison and interpretation.