1. Introduction

The development of coherent multidimensional spectroscopy [1–7] has enabled insights into the photo-induced ultrafast electronic and vibrational dynamics and couplings in a wide variety of systems by spreading the signal across multiple dimensions in frequency and/or time, correlating excitation and detection frequencies while maintaining ultrafast time resolution [8,9]. Two-dimensional electronic spectroscopy (2DES) in particular can interrogate the condensed-phase dynamics, energy transfer, and couplings in photosynthetic pigment-protein complexes [10–13] and live cells [13–15], synthetic molecular dimers [16,17] and aggregates [18,19], and semiconducting nanomaterials [20–25].

One major hurdle in realizing coherent multidimensional spectroscopy is maintaining sub-optical-cycle phase stability between femtosecond pulses defining the Fourier-transform time delays. Phase stability on the order of $\lambda /50$ is typically required for accurate Fourier transformation [26]. Short-term phase instability or errors in pulse delay timing lead to artifacts such as ghost peaks or tilted lineshapes in the resultant 2D spectrum after Fourier transformation [1,5,27,28]. This requirement becomes increasingly difficult in the visible or ultraviolet regime [29,30] because greater positional accuracy and atmospheric stability are required to maintain equivalent sub-cycle phase stability compared with infrared [31,32] or near-IR bandwidths. Fully noncollinear geometries such as the background-free ‘boxcars’ geometry provide high signal-to-noise but require an auxiliary measurement to determine the absolute signal phase, colloquially referred to as ‘phasing.’ Properly ‘phased’ data clearly distinguishes between the absorptive and dispersive contributions to the third-order nonlinear signal, allowing for accurate spectral assignment [3,33]. As such, determining the absolute signal phase in fully noncollinear geometries is often one of the largest bottlenecks in data processing but is critical to correctly interpreting the vast information content in 2D spectra. Furthermore, long-term phase drift necessitates determining the absolute signal phase for different time points independently, significantly complicating the phasing process. Although other experimental geometries such as the partially collinear pump-probe geometry [34,35] generate fully absorptive data without additional phasing procedures, they can suffer from lower sensitivity as the emitted signal overlaps with the probe pulse, meaning that detection is not background-free, and are generally unable to separate rephasing and non-rephasing contributions [36,37]. Pulse-shaping based approaches in the fully collinear [38–41], partially collinear [42–44], and fully noncollinear [45–47] geometries have been developed to circumvent the need for additional phasing protocols by phase cycling or phase modulation [48,49]. These schemes aid in scatter suppression and allow the recovery of rephasing and nonrephasing signals, but pulse-shaping based approaches are often limited by bandwidth or maximum achievable time delays of the pulse shaper. For these reasons, and due to the inherent sensitivity of background-free detection, fully noncollinear geometries based on simple translation delay stages remain widely employed despite the additional phasing and stability requirements.

Here, we demonstrate a passive post-processing phase drift correction procedure that partially alleviates the long-term experimental phase stability requirements present in fully noncollinear background-free geometries. We leverage this phase drift correction to aid in the retrieval of the absolute signal phase by a global phasing procedure. In this approach, the phase drift of scattered beam pairs is monitored via spectral interferometry. The measured phase drift of both the rephasing and coherence time domains is used to correct the signal phase in post-processing. Because this scheme corrects for experimental drift, it is not necessary to determine the absolute phase difference between beam pairs, but only record the relative phase drift collected over the course of the measurement. We show the effectiveness of this passive correction procedure in removing ∼8.5 radians of slow signal phase drift over >3 hours of experimental time, leading to a more robust determination of the absolute signal phase by employing a global phasing algorithm to generate fully absorptive 2DES spectra of wafer-scale monolayer MoS₂ [50]. Importantly, the correction procedure presented here requires no extra scans, reference beams, or optics, leading to a negligible increase in the experimental acquisition time and making it easily adaptable to existing experimental configurations where phase drift is a concern. Furthermore, this method may be used in conjunction with existing phase correction [51] or global phase-determination protocols [52–54] and should be widely applicable in multidimensional spectroscopy and general interferometry of any geometry or bandwidth.

2. Phase stability in two-dimensional spectroscopy

In background-free geometries, heterodyne detection with a local oscillator (LO) [55] measures both the complex signal field’s amplitude and phase via spectral interferometry (SI) [1,3,26,55–57]. In the common boxcars geometry, three pulses forming the corners of a box are focused onto the sample to generate a third-order signal in the phase-matched direction ${{\boldsymbol k}_{sig}} ={-} {{\boldsymbol k}_1} + {{\boldsymbol k}_2} + {{\boldsymbol k}_3}$. The fringes of the spectral interferogram arise from the temporal separation of the LO and signal and their difference in phase, in addition to the phase contributions of the individual excitation beams, $\Delta {\phi _{SI}} = {\omega _t}\Delta {T_{LO}}\; - \; {\phi _1} + {\phi _2}\; + \; {\phi _3}\; - \; {\phi _{LO}} + \; {\phi _{sig}}$ [4,7,58]. Phase information in particular allows for separating the absorptive (real) and dispersive (imaginary) components of the signal distinguishing photo-induced absorption pathway contributions to the signal from those corresponding to stimulated emission or ground state bleach [3,55]. However, in fully noncollinear geometries the absolute signal phase is initially unknown due in part to the difficulty in determining absolute timings and phase differences between multiple noncollinear beams [2,3]. Furthermore, the signal phase is sensitive to experimental phase drift or jitter during the measurement [5] because a change in time delays $\delta t$ manifest as phase changes by $\delta {\phi _i} = \omega \delta {t_i}$ for a given beam [5,58]. At 600 nm, delay times must be known to roughly 60 attoseconds to maintain a $\lambda /50$ stability typically required for accurate Fourier transformation and to avoid lineshape distortions.

Active [59–61] and passive [4,5,62] phase stabilization approaches have been developed to either track and correct or cancel phase fluctuations across common optics in the interferometer, respectively, often reporting long-term stability greater than $\lambda /100$. In passively stabilized approaches, for example, beams 1 and 3 and beam 2 and the LO are incident on common optics so that the correlated phase fluctuations $\delta {\phi _1} = \delta {\phi _3}$ and $\delta {\phi _2} = \delta {\phi _{LO}}$ act to cancel the phase fluctuations of the heterodyned signal, $\delta {\phi _{SI}} = (\delta {\phi _3} - \; \delta {\phi _1}) + (\delta {\phi _2} - \delta {\phi _{LO}}) = 0$ [5]. Regardless of the stabilization approach, however, experimental constructions remain sensitive to changing environmental conditions and mechanical instabilities. In addition to phase stability, care must be taken in proper delay line and detector wavelength calibration [63–65] and interpolation after conversion of the raw data from wavelength to frequency to avoid Fourier-transform artifacts in spectral interferometry [56,66,67] and phase-twisting of spectral features in the resulting 2D spectrum [63]. To determine the absolute signal phase, an additional phasing protocol is often required, such as fitting a projection of the 2DES data to spectrally resolved pump-probe experiments using the projection-slice theorem [3,68].

3. Passive phase drift correction using scattered light

The all-reflective two-dimensional interferometer used in this work [69] is passively phase stabilized similar to the passively stabilized interferometer using conventional optics described in Ref. [70] except at the all-reflective interferometric delay line (ARID) defining the coherence time delay where each beam is incident on separate mirrors instead of coupled delay lines. The use of separate optics on the ARID lead to possible sources of phase instability in contrast to designs of passive phase stabilization using common delay stages [70–72]. After the ARID, all four beams are incident on common optics until the sample position, after which the collinear LO and generated signal beams are spectrally dispersed onto an array detector. Beyond the short-term (∼minutes) phase-stability required to collect a single waiting time frame by scanning the coherence time, $\tau $, long-term (∼hours) phase stability is often required as well to sequentially collect a range of waiting times, T. If the interferometer is unstable on this timescale, the measured heterodyned signal phase will change over time, requiring the absolute signal phase to be determined independently for each waiting time frame.

A 2D spectrum at arbitrary phase angle of wafer-scale CVD-grown monolayer MoS₂ [50] for a waiting time of T=100 fs is shown in Fig. 1(a) after roughly removing the acquired phase from the LO time delay, ∼${\omega _t}\Delta {T_{LO}}$ [58]. The absolute phase has not been determined and so the absorptive and dispersive components are still partially mixed. Projecting the 2D spectrum onto the detection axis for all waiting times in Fig. 1(b) shows large changes between positive and negative amplitude in the waiting time dynamics across the spectrum. A waiting time trace for a single point from the lower positive cross-peak feature on the 2D spectrum in Fig. 1(c) shows similar dynamics. These oscillatory dynamics are characteristic of significant signal phase drift during the measurement, shown in Fig. 1(d) as the relative contribution of absorptive and dispersive components to the signal at a given phase angle changes over time.

 

Fig. 1. (a) Unphased 2D spectrum at arbitrary phase angle of monolayer MoS₂ at T = 100 fs. Phase drift manifests as large oscillations between positive and negative amplitude across all spectral features, evident in (b) a waterfall plot when projecting of the 2D spectra onto the detection axis for all waiting times and (c) a waiting time trace for a single point from the lower cross peak on the 2D spectrum. (d) The complex phase of the signal from Fig. 1(c) drifts >8 rad during the ∼3.2 hr measurement time.

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Determining the absolute phase in fully noncollinear geometries is usually performed by using the projection-slice theorem to compare a projection of the real-valued 2D spectrum onto the detection axis ${\omega _t}$ to a separately acquired spectrally resolved pump-probe measurement for each waiting time, T, under identical experimental conditions [3,73]. In general, phasing is required due to both the unknown phase offsets and time delay uncertainties of the excitation pulses and LO that contribute to the measured heterodyned signal phase [2–4,74]. If there is net-zero phase and timing difference between beams 1 and 2 and between beam 3 and the LO, $\Delta {\phi _{2 - 1}} = $ $\Delta {\phi _{3 - LO}} = 0$ and $\tau = {T_{LO}} = 0,\; $then peaks in the spectrum will appear with the correct phase and sign [28]. In pump-probe measurements with a perfectly compressed pulse, there is no phase or timing ambiguity because the first two interactions between the sample and the light’s electric field occur with the same pump pulse and the signal is self-heterodyned with the probe, automatically producing fully absorptive spectra. Multiplying the 2D data by a complex phase factor determined by fitting the ${\omega _t}$-projected 2DES to the pump-probe removes the phase ambiguities of the 2DES signal to produce absorptive 2DES spectra [1,3]. Various experimental procedures employing delay time scanning [52] and theoretical post-processing [51] procedures based on spectral interferometry have been proposed to correct for experimental timing errors to aid in phasing. However, in general, 2DES measurements still require an auxiliary phasing procedure to determine the absolute phase and remain sensitive to phase drift. All-optical determination of the absolute phase using spatial [53,75] or spectral [54] fringe patterns has been demonstrated, but requires multiple additional delay scans or optical elements. Furthermore, the absolute phase determined at the start of an experimental acquisition remains susceptible to the phase drift of the interferometer. The ability to phase to background-free heterodyned transient grating (TG) has also been demonstrated [76,77] as an alternative to spectrally resolved pump-probe measurements. However, the implementation in Ref. [76] requires the comparison of two signals in separate phase-matched directions in a non-boxcars geometry to determine the absolute heterodyned TG signal phase.

When phase drift is present, the absolute signal phase generally must be determined separately for each waiting time. Furthermore, the phasing procedure often becomes more difficult for short waiting times (T<100 fs) because scatter contributions in pump-probe measurements can be difficult to separate from the signal by simple apodization, or windowing, in the Fourier domains. In addition, cross-phase modulation or coherent artifacts may also be present in the pulse overlap region, further complicating the phasing of early waiting times. If the signal phase is stable over the course of the whole data run, a global phasing algorithm can be employed to generate a single set of phase parameters for the entire dataset, significantly reducing the time required for phasing and avoiding potential difficulties in phasing single waiting times for highly scattering samples or those with weak signal.

A previous passive phase correction method by Zanni and coworkers for two-dimensional infrared spectroscopy involved scanning a full time delay to correct for long-term phase drift during a measurement [78]. Here, we demonstrate a passive scatter-based approach to correct for long-term signal phase drift with negligible increase in experimental time, requiring no additional scans or optics. The relative phase drift in the coherence and rephasing time domains is retrieved via spectral interferometry of scattered light between pulse pairs and used to correct the 2DES signal phase drift. Depending on the nature of the sample, scatter contributions can be significant, often on the same order of magnitude as the signal strength or greater, especially for samples with low optical density, such as monolayer MoS₂, where the signal strength is usually weak. Subtraction or windowing is typically used to separate the various scatter and homodyne terms from the heterodyned 2DES signal in post-processing [14]. While we leverage these scatter terms in the demonstration presented here, the same correction method should also be easily adaptable to auxiliary detection of the relative scatter phases with a scattering medium just before the sample in cases where the sample scatter contributions are inherently weaker, such as in solution-phase samples.

To perform the phase drift correction in post-processing, spectral interferograms between beams 2 and 1 and between beam 3 and the LO are collected during the 2DES measurement after each waiting time scan, imaged on an array detector and isolated from the signal and other unwanted scatter contributions using mechanical shutters. When collecting the spectral interferograms, the time delays are set so that beam 1 precedes beam 2 by 300 fs and the LO precedes beam 3 by roughly 1300 fs. The LO delay is static and identical to the LO time delay used in the data acquisition. An example set of interferograms is shown in Fig. 2(a). Fourier transformation of the frequency-domain interferograms produces sharp absolute-value peaks in the Fourier (time) domain corresponding to the time separation between the scattered beams, shown in Fig. 2(b). The 300 fs time delay between beams 1 and 2 is chosen because the largest coherence time delay scanned in generating a waiting time frame, typically $\tau $ = 90 fs, does not consistently produce well-separated peaks in the Fourier time domain from the zero time delay component. Using a 2-1 separation outside the coherence time delay requires excellent positional repeatability of the delay stages (<< 1 fs) over a wide delay range to avoid artificial measured phase drift arising from repeated positional imprecision.

 

Fig. 2. Scatter-based phase drift retrieval. (a) Example spectral interferograms between scatter from pulse pairs 2-1 and 3-LO for a single waiting time. The data have been interpolated from wavelength to be linearly spaced in frequency. (b) Fourier transformation (F.T.) of the interferograms produces sharp absolute-value peaks in the Fourier time domain corresponding to the time delay between pulses. The negative time separation indicates that beam 1 precedes beam 2 and the LO precedes beam 3. The data are offset for clarity. (c) Retrieval of the complex phase angle at the peaks of interest in Fig. 2(b) for each waiting time tracks the relative phase drift during the measurement.

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By taking the complex phase angle of the peaks, the relative phase of the scattered beams, $\Delta {\phi _{2 - 1}}$ and $\Delta {\phi _{3 - LO}}$, is recorded for each waiting time, T. The relative phase drift of both the coherence and rephasing time domains can thus be monitored over the course of the experiment, as shown in Fig. 2(c), allowing for signal drift correction in post-processing. The phase drift correction procedure in this implementation requires ∼6 s (∼10%) of additional acquisition time for each waiting time frame, mostly due to the time required to move the delay stage to set the 300 fs delay between pulses 1 and 2. This correction therefore scales linearly with the number of population time points, and inversely with coherence time points, and sublinearly with camera exposure time.”

The phase drift of the signal $\Delta {\phi _{sig}}(T )$ is approximately the sum of the phase drifts of $\Delta {\phi _{2 - 1}}(T )$ and $\Delta {\phi _{3 - LO}}(T )$, as expected, shown in Fig. 3(a). For this particular dataset, a total phase drift of ∼8.5 radians occurred in the signal over the >3 hr time period collecting positive waiting times (T>0 fs), corresponding to an overall signal phase stability of $\lambda /2.4$. However, the phase drift is slow on the time scale of collecting a single waiting time frame (∼60 s). The 2DES signal shown in Fig. 1(d) varies an average of ∼0.047 radians per frame, or ∼$\lambda /130$, whereas the 2-1 and 3-LO phase varies on average ∼0.025 radians (∼$\lambda /250$) and ∼0.018 radians (${\sim} \lambda /350$) between frames, respectively. Although the short-term phase stability is sufficient to avoid lineshape distortions and Fourier transform artifacts, the long-term phase drift makes it difficult to retrieve fully absorptive spectra when phasing, as described above. Because the signal phase drift results from drift in both the coherence and rephasing time domains, the relative scatter phase drift during these two time delays can be used to compensate for the phase drift of the heterodyned signal, $\Delta {\phi _{SI}} ={-} \Delta {\phi _1} + \Delta {\phi _2} + \Delta {\phi _3} - \Delta {\phi _{LO}} = \Delta ( - {\phi _1} + {\phi _2}) + \Delta ({\phi _3} - {\phi _{LO}})$ [5] where the retrieved phases from the spectral interferograms $\Delta {\phi _{2 - 1}}$ and $\Delta {\phi _{3 - LO}}$ track the changes in the $- \delta {\phi _1} + \delta {\phi _2}$ and $\delta {\phi _3} - \delta {\phi _{LO}}$ contributions to the heterodyned signal phase drift, respectively. Therefore, we define the correction factor as the sum of the scatter phase terms for a given waiting time, $\Delta {\phi _{corr}}(T )= \Delta {\phi _{2 - 1}}(T )+ \; \Delta {\phi _{3 - LO}}(T )$. Multiplying the entire two-dimensional dataset by the complex factor ${e^{ - i\ast \Delta {\phi _{corr}}(T )}}$ shown in Eq. (1) as a function of waiting time, T, removes both the signal phase drift as shown by the dotted curve in Fig. 3(a) and the corresponding large amplitude phase roll oscillations of the waiting time trace, Fig. 3(b). The correction leads to an effective increase in the signal phase stability by near a factor of 30, from $\lambda /2.4$ to $\lambda /69.9$, demonstrating the possibility of performing two-dimensional spectroscopic experiments even when the long-term relative phase stability between Fourier-transform beam pairs is an order of magnitude lower than typically required. Furthermore, this phase drift correction procedure may be easily adapted to monitor and remove phase drift in other time domains or pulse orderings in other experimental implementations as well, such as in two-quantum two-dimensional spectroscopy, by recording interferograms between beam pairs defining the relevant time delays where phase drift is a concern.

 

Fig. 3. Correction of the 2DES signal phase drift. (a) The 2DES signal phase drift, which is approximately the sum of the 2-1 and 3-LO drift (grey line), can be removed by the scatter-based correction, leading to high effective phase stability over many hours (dashed line). (b) The associated phase rolls in the waiting time are also removed, shown here for the lower cross peak feature.

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4. Global phasing procedure

To determine the absolute phase and produce fully absorptive spectra, we create an ${\omega _t}$ projection of the 2D data by integrating over the ${\omega _\tau }$ axis, and fit this projection to broadband spectrally resolved pump-probe (PP) data taken with the same experimental conditions using the projection-slice theorem [2,3]

After removing the long-term experimental phase drift of the signal with the scatter-based correction, we are able to employ a global fitting algorithm to produce a single set of phase parameters to phase all waiting times. In this work, we use the Nelder-Mead simplex algorithm [79], an unconstrained multidimensional nonlinear minimization procedure, via the fminsearch functionality in MATLAB [80] to minimize a cost function that is defined as the combined squared difference between the pump-probe spectrum and 2DES signal projected onto the detection axis ${\omega _t}$ for all waiting times simultaneously, Eqs. (3), (4). In the minimization, we vary an overall phase term ${\varphi _0}$, as well as linear and quadratic terms in rephasing time, ${t_c}$ and ${t_q}$ [5,68,69].

Once the phase fitting parameters ${\varphi _0},\; {t_c},{t_q}$ have been determined by the minimization procedure, they are applied to the unphased dataset by multiplication by $\exp ({i{\varphi_0} + i({{\omega_t} - {\omega_0}} ){t_c} + i{{({{\omega_t} - {\omega_0}} )}^2}{t_q}^2} )$ to generate phased, absorptive 2DES data. A global fitting algorithm is both more efficient and more robust than sequentially phasing single waiting times because it does not require assumptions of how the phase should vary over time and is less sensitive to the scatter or low signal-to-noise of a single pump-probe waiting time frame, which can be especially significant for short waiting times or in samples with weak signal. It is also possible that fitting parameters from phasing a single waiting time may be applied to the whole cube with a stable phase [86]. However, global phasing is more robust because it acts to avoid the local minima solutions of a single waiting time frame by effectively averaging noise and scatter contributions over large regions of the dataset.

A final phased, absorptive two-dimensional spectrum for MoS₂ T = 100 fs is shown in Fig. 4(a). Waterfall plots of the phased 2DES data projected onto the detection axis and spectrally resolved pump-probe are shown in Figs. 4(b) and 4(c), respectively. The two appear similar globally as confirmed by a comparison for a single T = 100 fs time slice shown in Fig. 4(d). The prominent negative-amplitude photo-induced absorption features highlight the need to generate accurately phased spectra as the spectral locations and dynamics of these features are critical in interpreting many-body phenomena in monolayer MoS₂ [87], as well as in other semiconductor systems [23].

 

Fig. 4. Global phasing of the 2DES signal after phase drift correction. (a) Real-valued, phased absorptive 2DES spectrum of monolayer MoS₂ at T = 100 fs. Waterfall plots of (b) phased 2DES data projected onto the detection axis ${\omega _t}$ and (c) spectrally resolved pump-probe. (d) Comparison of the pump-probe signal and the phased ${\omega _t}$-projected 2DES slices at T = 100 fs.

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5. Conclusions

In conclusion, we have described a passive phase drift correction procedure for two-dimensional spectroscopy based on monitoring the relative phase of excitation beam scatter during the measurement, as well as a global-fitting phasing method. This phase drift correction procedure is able to remove significant and, in principle, unlimited slow phase drift originating from phase instability in both the coherence and rephasing time delays in experiments lasting many hours. Correcting the phase drift allows for the use of more robust and efficient global phasing algorithms and greatly lowers the experimental phase stability requirements in multidimensional spectroscopy, in this demonstration by an order of magnitude. This work will make these techniques more widely accessible, while increasing the data collection and processing throughput more generally. The demonstrated procedure here should bolster and enhance the effectiveness of previously proposed phase correction or determination procedures that may suffer from phase drift. Importantly, this correction does not require additional optical components, reference beams, or scanning procedures, making it easy to implement in existing experimental apparatus.