1. Introduction

Since the first demonstration of two-dimensional photon-echo spectroscopy (2D-PE) in the optical domain in 1998 [1], the number of both experimental and theoretical research papers has rapidly grown. The extension of known NMR techniques first to the IR-domain and now to the study of molecular electronic resonances (MER) in the visible is, at first glance, quite straightforward. However, the requirements on the phase stability of the so-called local oscillator (LO) field needed for heterodyne detection of the third-order polarization, induced in a medium, are much higher. This is dictated by the wavelength range used in electronic 2D-PE: while the phase stability in NMR is not an issue due to the long wavelengths used, achieving phase stability of Λ/100 in the VIS demands that the phase jitter between LO and signal fields should be 5 nm or even less. The design of a multi-beam interferometer with such high phase stability in the VIS is a technical challenge that, in principle, can be solved by introducing sophisticated feedback-controlled mirror mounts etc. as was demonstrated in [2]. There are trade offs in flexibility and ease of implementation in this approach. Another possibility to measure the 2D electronic spectra involves adopting fully collinear [3] or partially collinear geometries [4]. While the first method is almost identical to the NMR approach, the second one can be considered as a modification of the traditional pump-probe experiment where two pulses, separated in time, provide the excitation. One disadvantage of both these techniques is the presence of a significant background in the measured signals which requires “phase-cycling” procedures to extract the artifact-free signals. In addition, the fully collinear geometry [3] only allows measurements with fluorescent compounds. The great advantage of the partially-collinear geometry of Grumstrup et al [4] is the automated phasing of the LO-pulse – as in any pump-probe geometry, it is always in phase with the measured signal. However, there are other trade offs in sensitivity and homodyne signal contribution (see discussion in Section 5).

An alternative solution to this problem proposed and realized in [5] is based on the non-collinear geometry leading to background-free signals. Passive stabilization of the relative phases of all 4 beams needed for generation and heterodyne detection of the PE is accomplished by introducing anti-correlations between the phases of beam pairs. The anti-correlations in the phases of ultrashort pulses were experimentally realized in [5] by means of a diffractive optic (DO) used as a beam splitter [6], and at 540 nm wavelengths, phase stabilities of Λ/40 and Λ/90 were achieved using mechanical and refractive delay stages, respectively. This technical solution was modified by Brixner et al [7] using glass-wedge based delay stages instead of rotating glass plates as demonstrated previously [5, 8], and applied to the study of the excitonic states in the FMO antenna complex [9]. However the resolution of the 2D-PE spectra [9, 10] is relatively low in comparison to NMR and often requires additional information from independent studies to assign couplings.

In multi-pulse NMR the use of pulses of different shapes, frequencies, and durations allows a significant increase in the spatial resolution of the compound’s structures, obtained from the measured multi-dimensional spectra. Such manipulation of the pulse sequence can be generally viewed as pulse shaping. A logical extension of 2D-PE spectroscopy would be to combine optical pulse shaping techniques with the existing heterodyne-detected 2D spectroscopies. This approach in general makes it possible to control the coherences and energy transfer flow in conjugated many-body quantum systems, and also helps to increase the spectral resolution of 2D spectra. For instance, an appropriate shaping of the excitation pulses increases the sharpness of cross-peaks and enhances their relative magnitudes, as was numerically demonstrated in [11] for the simplest excitonically-coupled system (a dimer). There is another important distinction from NMR. For electronic transitions, one is not dealing with two level quantum systems. The use of 2D electronic spectroscopy with shaped excitation pulses offers the unique opportunity to control the initial quantum state distribution and observe purely quantum effects associated with constructive and destructive wave function interference effects. In addition, this approach makes it possible to examine state-dependent decoherence phenomena or level-dependent dephasing. It should be noted that the implementation of programmable pulse shapers in 2D spectrometers has already been realized in both the IR and VIS wavelength domains (see, eg., [3, 4, 12, 13, 14]). However, these shapers were basically used to generate the pairs of pulses needed for four-wave mixing experiments, and acted as programmable translation stages. In principle, these optical systems could achieve arbitrary pulse shaping but so far only very simple time delays between pulse pairs have been demonstrated [12, 13]. There are technical difficulties in realizing arbitrary pulse shaping in programmable shapers that are already employed to provide delayed pulse pairs. The additional complexity of including both variable time delays and arbitrary pulse shapes is beyond the restricted phase space and finite spatial resolution of current pulse shapers. Although these difficulties can be solved in principle, 2D spectroscopy employing arbitrary pulse shapes has not been demonstrated to date, largely due to these limitations.

In this work we investigated the possibility to overcome the above-mentioned technical difficulties and to design a robust, broadband (for ~10 fs pulses in the VIS), passively phase-stabilized four-beam spectrometer in a non-collinear geometry with background-free registration of the signals by excitation with arbitrarily shaped pulses obtained using any current pulse shaper. Embedding a pulse shaper into the 2D-PE experimental setup requires a significant extension of the interferometer arms (needed for the shaper’s optics), and consequently a significant improvement of the phase stability. The schematic solution proposed by Cowan et al [5] gives good phase stability if the length of the interferometer arms is minimized typically to ~20 cm or less. This limitation is due to the coupling between translation stages - the measurement of a 2D-PE spectrum involves scanning along the “dephasing” coordinate (τ) at a fixed “waiting time” coordinate (T) which, in their scheme, requires the synchronous movement of both delay stages with sub-femtosecond precision and linearity.

We have developed a new heterodyne-detected 2D-PE spectrometer with fully uncoupled translation stages so that the “dephasing” scan requires the movement of only one translation stage. This solution allows the extension of the interferometer arms by up to several meters without loss of phase stability. Even in the standard fashion (without shapers), the achieved phase stability using inexpensive mechanical translation stages is much better as compared to that reported previously in [5] (≥Λ/120 vs. Λ/40, respectively). Uncoupling the translation stages also significantly reduces the requirements for mechanical precision and linearity and as a consequence, the problem with the appearance of “ghosts” in the 2D-spectra is also diminished. We demonstrate the general application of this approach in the measurement of the electronic 2D-spectra of a model compound (Rhodamine 101 dissolved in methanol) using the same pulse shapes found to coherently control population transfer [15] and compare the 2D-spectra to the unshaped case. The effect of coherence transfer on the 2D-spectra clearly demonstrates the importance of pulse shaping protocols for 2D-spectroscopy.

2. Optical design

Figure 1 schematically shows the propagation of the beams in the new interferometer design. The signal is generated in the direction k⃗_e=k⃗₂+k⃗₃−k⃗₁ and mixed with the LO beam k⃗_LO. Uncoupling of the translation stages is achieved by an additional pass of the “mother” beam (labelled as #2) for the reading pulse k ₃ and local oscillator pulse through the roof mirror, placed on the “dephasing” translation stage DL1. The DO splits the incoming beams into two phase-coupled replica beams (k _1,2 and k _3,LO), which are collimated by L2 and further re-focused by L3 into the sample after passing through the stationary and movable roof mirrors. Figure 2(b) shows the timing diagram for the propagating pulses. Here, for comparison, a timing diagram for the previous 2D-PE setup described in [5] is also shown. As can be seen, the LO pulse now follows the PE pulse synchronously by moving the “dephasing” translation stage DL1. The four-beam interferometer has two ports: #1 for shaped excitation pulses k _1,2, and #2 for the short reading- and LO-pulses. The interferometer can be switched to the standard mono-beam fashion by blocking port #1 and inserting a removable beam splitter into beam #2. This is necessary for the calibration of the delay stages and measurement of 2D-spectra with non-shaped, short pulses.

 

Fig. 1. Schematic of the PE-interferometer with uncoupled translation stages for performing heterodyne-detected measurements. RBS - removable beam splitter (to switch from the “mono-beam” to the “dual-beam” configuration); M1-M3 - 100% mirrors; M4 - 100% roof mirror; DL1, DL2 - τ - and T- translation stages, respectively; DO - diffractive beam splitter; L1-L3 - collimating/focusing optics. The directions of all 4 beams are indicated by their wave vectors.

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Fig. 2. Timing diagrams showing the pulses’ temporal arrangement for the PE-interferometers in the previous configuration [5] (A), and present interferometers for both presented schematic solutions “variant 1” (B) and “variant 2” (C). The excitation pulses (1,2) and reading pulse (3) are sequentially applied to a medium from which an echo pulse is generated and temporally overlapped with the local oscillator pulse (LO), which is additionally shifted by a small delay time of ≈400 fs.

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As was shown in previous work [5] (timing diagram Fig. 2(a)), introducing additional delays after the DO degrades the phase stability through residual uncorrelated beam pointing instabilities. The phase fluctuations in the present interferometer (Fig. 1) cancel out independently to the presence of additional delays and/or optical elements, as can be shown by tracing the phase fluctuations (denoted by δϕ) for each beam.

The total phase of the heterodyne-detected PE-signal Δϕ_s depends on the phase difference between the pairs of beams which can be re-written as a sum of the phase differences [5]:

where the numbers denote the beams with wave vectors k⃗₁ …k⃗_LO,and ϕ_PE corresponds to the intrinsic phase of the induced polarization (PE). For the phase deviation of the measured signal caused by phase fluctuations in the excitation, reading and LO-beams we get:

Assuming that the phase fluctuations of the beams incoming through the ports #1 and #2 are uncorrelated and denoting them by δϕ _beam1,δϕ _beam2, we obtain the resulting phase fluctuations in the outgoing beams k⃗₁ …k⃗_LO:

where each term δϕ_N denotes an independent phase fluctuation introduced by an optical element N (see Fig. 1 for details). In particular, δϕ _DL1,δϕ _DL2 correspond to the phase fluctuations introduced by the movable delay stages. Substituting Eq. (3) into Eq. (2) gives:

That is, any phase fluctuations introduced not only before, but also after the DO by the inserted translation stage DL1, are fully cancelled in this interferometer. This “tracing” analysis of the phase fluctuations is fully applicable for examining any interferometer.

3. Experimental setup

We realized and tested two different versions of the interferometer with uncoupled translation stages. The optical layout of the first version was almost identical to the schematic diagram shown in Fig. 1 with collimating and focusing off-axis parabolic mirrors of 25.4 mm aperture and 152.4 mm focal length (Newport). Here the beam, passing through the delay stage DL1, is the “mother” beam for reading- and LO-pulses (k ₃,k_LO).

In the second variant of the interferometer the beam that initially passes through the delay DL1, is the “mother” beam of the two excitation pulses k ₁,k ₂ (Fig. 3). To ensure the phase cancellation conditions (Eq. (4)) this beam first hits a retroreflector RR2 in the opposite direction to the second pass of the translation stage (through the retroreflector RR3). Both retroreflectors are mounted on the same translation stage DL1. For scans of both dephasing τ and waiting T coordinates, we used identical translation stages equipped with nanopositioners having 10 nm step resolution and 100 nm repeatability (Nanomotion II, Melles Griot). Figure 2(c) illustrates the timing of the pulses which propagate through this variant of the interferometer. In this case the second excitation pulse (labelled by 2) is not moved by any translation stages, and thus serves as a starting point in the timing. We found that this version of the interferometer gives better phase stability (see below) which is due to the replacement of the roof mirrors by hollow retroreflectors. The torsional motions in a retroreflector are transferred to longitudinal fluctuations of the beam delays travelling throughout, and they are completely cancelled out (see Section 2). The retroreflectors RR1,2 with a 25.4 mm aperture (Newport) and RR3,4 with 63.5 mm aperture (Edmund Optics) have aluminum coatings, as do all other mirrors in the setup. All of the optical elements of the interferometer were mounted directly on the optical table, without any additional vibrational isolation. The interferometer is very robust and does not require any service (beam alignment, tweaking of mirrors etc.) over a period of more than 3 months. The DO beam splitter (with diffraction efficiency of ~60% in both 1st diffraction orders) has ~70 grooves/mm so that the spacing between collimated beams of 12.5×12.5 mm (boxcar geometry) prevents any overlap of the beam spots, even for 150 nm spectral bandwidth.

 

Fig. 3. Optical layout of the coherently-controlled 2D-PE setup with the variant 2 four-beam interferometer. Here NOPA - non-collinear optical parametric amplifier producing broadband (100 nm width) pulses; BS - dielectric 50/50 beam splitter (the first beam splitter in the interferometer); AOS - acousto-optical pulse shaper for coherent control of the PE; DMS - deformable mirror based shaper for compressing the NOPA light pulses to nearly transform-limited pulses with a duration of 10 fs at 565 nm; chirp compensators - for removing the intrinsic linear chirp in both the shaper’s channels; FROG - the χ⁽³⁾ frequency-resolved optical gating apparatus for the measurement and characterization of pulses; DL1, DL2 - translation stages for performing of τ - and T-scans, respectively; RR1–RR4 - hollow retroreflectors; RBS - removable 50/50 dielectric beam splitter; CP, RCP - the permanent and removable compensating glass plates, respectively; DO - diffractive beam splitter (the second beam splitter in the interferometer); PM - off-axis parabolic mirrors; M - aluminum mirrors; SC - the sample flow cell.

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The great advantage of the DO in a multi-beam interferometer is the placement of this element in the focal plane and subsequent relay imaging to the sample plane. In this configuration the spatial overlap of the wave fronts in the sample plane is optimal and does not affect the temporal resolution [16], and the interferometer itself is insensitive to any torsional distortions of the DO beam splitter since all of the beams originate from the same spot. This significantly increases the phase stability as compared to interferometers with conventional beam splitters [17] where, due to large spacing between the beams (~centimeter or more), only the longitudinal phase fluctuations cancel out. The key concept, however, with respect to the inherent passive phase stabilization for 2D spectroscopy in [17] is the explicit use of delayed pulse pairs in the interferometer design as shown previously [5]. Technically, it is irrelevant whether the anti-correlation between pairs of beams for the cancellation of phase fluctuations is introduced - by using a DO to split the beams [5] or using a “conventional” beam splitter, as was done in [17]. However, in the case of broadband applications the light intensity on the DO beam splitter is very high - for incoming pulses of 60 nJ with a duration of 10 fs (i.e. experimental conditions) focused onto a 50 µm spot (measured using calibrated pinholes), the instantaneous intensity reaches 200 GW/cm ². At this level any refractive dielectric medium, even air, will manifest a nonlinear response to the incoming fields, and in particular lead to cross-talk between the temporally overlapped pulses. Figure 4(a) shows the pump-probe scan of the DO response, measured under the above experimental conditions. Here beam #1 is the pump (modulated by a chopper), and beam #2 serves as a probe. The magnitude of cross-talk reaches ±10% in its maximum, is stretched in time over 150 fs, and shows a “wave-like” behavior. There are two solutions which circumvent this artifact - either the displacement of the pulses in time (e.g. by placing the compensating plate (CP) after the DO as done previously in 6-wave mixing experiments [18]), or the introduction of a small shift of the DO position from the focal plane of focusing mirror leading to a small displacement of beams onto the DO surface. The CP whose thickness is equal to the thickness of the beam splitter can give a ~3–4 ps temporal separation between pulses. However, by taking the PE-spectra at large waiting times there is always a possibility to get an unexpected artifact in the PE-spectra. The second method is more useful and requires only a 0.8 mm displacement of the DO from the focal plane of a mirror with f=152.4 mm in the reported setup. Such a small displacement fully eliminates the cross-talk between beams (Fig. 4(b)). Note that this cross-talk problem becomes insignificant using shaped excitation pulses where, due to temporal stretching, the instantaneous intensity is much lower.

 

Fig. 4. Pump-probe scans of the DO beam splitter in cases of spatially-overlapped pulses (A), and after displacement of the DO from the focal position of focusing mirror (f=152.4 mm) by 0.8 mm (B).

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In both variants we used two pulse shapers, inserted in separate ports. An acousto-optical shaper in port #1 is used to generate the arbitrary pulse profiles (Dazzler, Fastlite) and perform coherently controlled 2D-PE measurements. A low-resolution pulse shaper in port #2 is used for correction of the phase spectrum of the pulses generated by the NOPA, and produces short, nearly transform-limited pulses (reading- and LO-pulses). This shaper is based on a 39-actuator electrostatic deformable mirror DM (Oko Technologies) in a typical 4f-configuration [19]. Each shaper is equipped with prism chirp compensators - two SF11-prisms spaced apart by 38 cm for compensation of intrinsic AOS-crystal chirp, and two BK7-prisms spaced by 85 cm for removal of linear chirp in the pulses generated from the NOPA, and compensation of any additional chirp generated in the refractive optical elements in the interferometer (the DO, compensation window, and the sample cell windows). The DM shaper allows the compression of NOPA-pulses centered at 565 nm down to ~10 fs FWHM, as measured using a third-order Frequency Resolved Optical Gating (FROG) setup [15] (Fig. 5(a)). It is important to note the benefits of the DM-shaper for use in the broadband 2D-PE spectrometer. The 10-fs pulse is influenced by different kinds of distortions (wave front, spectral and temporal) during its propagation through the interferometer optics. The linear chirp in the propagated pulses can be compensated by adjusting the prism chirp compensator (see Fig. 3). Figure 5(b) shows the FROG trace, measured in situ using a 1-mm thick glass slide at the sample position, after compensation of linear chirp. The resulting pulse is ~22 fs FWHM and clearly displays the phase distortions. However, after additional optimization of the DM-profile using a genetic search algorithm where the target is the magnitude of the FROG-signal (at T=0), the pulse can be compressed to approximately its initial duration. Figure 5(c) shows the FROG trace measured in situ after the DM-optimization; the pulse duration is ~13 fs FWHM and is most likely limited by the dispersion of the diffraction efficiency in the DO and the thickness of the glass slide used. The cross-FROG measurements in the case of the dual-beam operation (optimized AOS + optimized DM shapers) demonstrate a 20–22 fs FWHM of the pulses in the sample position (not shown). In this case the pulse duration is limited by the AOS (the Dazzler) whose working bandwidth allows to utilize only ~45 nm of the pulse bandwidth.

 

Fig. 5. The FROG traces of the pulse, incoming into port #2 of the interferometer (A), after passing through the interferometer and linear chirp correction (B), and following the correction of high-order phase distortions in the interferometer (C). The pulse durations (retrieved using a commercial program) are 10, 22, and 13 fs, respectively.

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Switching from the dual-beam to the mono-beam configuration requires only the insertion of a removable broadband dielectric beam splitter (Melles Griot) into beam #2, blocking of beam #1 and replacement of the compensating glass plate (see Fig. 3). This process takes ~30 sec without any additional alignments of the interferometer. The collection of the (PE+LO)- and the LO-pulse spectra were performed concurrently using a home-built 30-cm, 300 grooves/mm grating spectrometer equipped with a 512-pixel photodiode array (S3902, Hamamatsu), and a reading camera (Spectronic Devices, Ltd). The second spectrometer (Sciencetech, model 9057) serves to control the spectra of the shaped pulses. A mechanical chopper (Thorlabs, not shown) is placed before RR2 and modulates beam #1 at 1/2 of the repetition rate of the laser pulses (1 kHz). The resolution of the spectrometer (≈1nm) is sufficient to clearly resolve the interference fringes in the spectra between the PE-pulse with the LO-pulse, delayed by ΔT~390 fs and attenuated using a 0.15 mm thick glass neutral density filter (with OEM coating), placed directly into the LO-beam. We used a delayed LO-pulse in order to avoid (or minimize) the influence of this pulse on the state preparation in the sample prior the excitation/reading pulses. This consideration is important in executing coherent control protocols, whereas the LO-pulse is usually made to arrive first with unshaped pulses. The additional optical element to introduce the delay was not included into the phase stability analysis Eq. (3) since the possible phase fluctuations that it can induce, e.g. due to pointing instability of the travelling beam, are insignificantly small (a 1-mrad beam deviation leads to deviation in the propagation’s delay of 10⁻⁵ fs).

The length of the interferometer arms prior to the DO beam splitter reaches 6 meters due to the relatively large separation between the numerous optical elements in the pulse shapers. It should be noted that the whole system (laser source+2D-PE setup with shapers) is distributed over three optical tables (RS 2000, Newport) stacked together. In particular, the first beam splitter that splits the NOPA-beam to the AOS- and DM pulse shapers is located on a different table than the core of the interferometer with the translation stages, measuring spectrometers, etc. An additional source of phase instability in the present coherently-controlled 2D-PE setup is the acousto-optical pulse shaper, based on longitudinal pulse deflection. Due to inherent electronic jitter, the position of the launched acoustic wave in the AOS crystal is always slightly different from pulse to pulse which leads to an uncertainty in its temporal position due to the deflection of the travelling optical pulse under a small angle (~10 degrees). The measured RMS timing jitter of the deflected pulse is ~10 fs (~8Λ). Such large timing jitter makes the application of the Dazzler in the 2D-PE spectroscopy, based on the pump-probe geometry [4], incapable of high time resolution. However, due to the complete cancellation of the phase fluctuations (Eq. (4)) in the present non-collinear beam geometry, the coherently-controlled 2D-PE setup is robust such that even this strong phase jitter does not affect the measured 2D-PE.

The control of the 2D-PE spectrometer and the data acquisition is realized using a MATLAB-based program, developed in house.

4. Characterization of phase stability

To our knowledge, there is no clear published description of phase stability in 2D-PE spectrometers; therefore a detailed explanation of the procedure used is detailed below.

The spectrometer measures the spectrum S(ω) of the incoming electrical field E(t):

where

is the sum of the induced polarization, irradiated from a medium, and the local oscillator field, delayed by ΔT, and the “coherence time” delay τ between the excitation pulses (see Fig. 2(c) for timing diagram). Here δt is the timing jitter which leads to the phase fluctuations between fields. It is caused basically by air motions and torsional fluctuations of the mirrors. The measured spectrum contains contributions of homodyne signals from the PE- and LO-pulses and mixed terms:

where the phase shift is given by δϕ=ωδt. The chopping technique removes the homodyne part corresponding to the LO-pulse, and the differential signal contains both the mixed signal (i.e. interferogram) and the spectrum of the PE-signal:

where we denote Ã(ω)=PE(ω)LO*(ω)e^iδϕ. First, we perform the inverse Fourier transform of the differential signal that gives in the time domain:

Then we split this signal into two complex-conjugate interferograms by performing the Fourier transform back to the frequency domain, but separately for the negative- and positive time domains in Eq. (9). The homodyne contribution of the PE, given by the last term, is blocked by zeroing of ΔS̃(t) around t=0. The delay of the LO-pulse should be large enough to ensure complete separation of terms in Eq. (8) in the time domain: ΔT≫τ_PE. After separation, one obtains two complex-conjugate spectra:

After removing of the oscillatory part either by multiplying by exp[iω(ΔT+τ)] or its complex-conjugate for ΔS_(ω) (assuming that the LO-delay and τ are known), we finally obtain the signal from which the actual phase can be recovered by taking the angle:

where ϕ ₀=Im{ln[PE(ω)LO*(ω)]} reflects the sum of phase spectra of the PE and LO. Assuming that ϕ ₀ does not change, the magnitude of the phase instability will correspond to the standard deviation (STD) of the monitored actual phase: |δϕ|=std(ϕ_s) (Eq. (11)). To minimize the influence of the inherent phase fluctuations in the generated laser pulses, we normally take the STD around the maximum of Ã(ω) (Λ~560 nm), where their contributions are minimal.

The measurements of the phase stability were performed at τ=0, T=0 using a glass slide as the sample. The signal was monitored during either 20–30 min or 2–3 hours (short-term or long-term phase stabilities). For the first variant of the interferometer, the short-term phase stability was Λ/70 and the long-term −Λ/35; whereas for the second variant (Fig. 3) the corresponding stabilities are Λ/125 and Λ/70. Operating with both beams (using the AOS- and DM-shapers) does not change the phase stabilities significantly: the short-term stability is slightly lower (Λ/100), but the long-term stability is slightly better (Λ/90). As an illustration, Figure 6 displays the phase fluctuations within 20 min for both mono- and dual-beam configurations. The wavelength dispersion of the measured phase stability for the dual-beam configuration is illustrated in Fig. 6(b) where for comparison the actual PE-spectrum is also shown. The STD of the phase is almost flat around the maximum of the signal where the signal to noise ratio (SNR) is maximal. Numerical modelling shows that the influence of the signal noise is proportional to its amplitude. For example, a 1% noise in a signal introduces ~0.8 mrad of noise in the measured phase fluctuations, 5%–4 mrad, etc. Under our measuring conditions, the noise limitation in the measurement of the phase stability was ≃Λ/200. The phase stability of our 2D-PE setup is basically limited by long-term drift. We performed additional experiments and found that the major factor is the long-term beam pointing stability of the incoming beam(s). For example, an angular displacement of the incoming beam by 0.3 mrad, which is comparable to the intrinsic beam divergence (0.2 mrad), leads to a phase change between the PE and LO-fields by ~Λ/10. Therefore, the day-to-day absolute phase stability (phase reproducibility), monitored during the course of a week, is only ~Λ/30. However, for really long-term experiments this issue can be readily fixed by introducing a beam pointer stabilizer.

 

Fig. 6. (A) Phase stability of the 2D-PE setup monitored within 20 min for the mono-beam (the DM-shaper arm) and the dual-beam (the DM- and AOS-shaper arms) configurations. The STD of the phase fluctuations is Λ/120 and Λ/95, respectively. (B) Dispersion of the phase fluctuations across the spectrum. The stability is maximal at ~560 nm where the measuring SNR is also maximal.

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5. Phasing of the PE-spectra in a coherently-controlled experiment

The creation of the 2D-spectra from the measured 2D-PE is quite straightforward; however, to set the absolute phases of the Re- and Im-parts of the 2D-spectrum it is necessary to know the absolute temporal delay of the LO-pulse with interferometric precision. This delay is an inherent property of the 150-µm thick neutral density filter which is placed in the LO-beam. Therefore, it can be determined once using e.g. short pulses in the mono-beam configuration of the 2D-PE setup and then used for the analysis of data taken with the shaped excitation pulses. It is important to note that in the present coherently-controlled 2D-PE setup only unshaped pulses pass through the LO-filter.

The known methods for “phasing” of PE-spectra (i.e., determination of the LO-delay) can be conditionally divided into interferometric [20, 21] and comparative based approaches [22], where the pump-probe spectrum is compared to the integrated 2D-PE spectrum. The interferometric methods need the introduction of additional elements into the interferometer (a submicron sized pinhole in the measuring cell [20] or an additional beam splitter placed into the focused beams [21]) which potentially can lead to artifacts when determining the LO-delay, especially when working with broadband pulses (>2000 cm ⁻¹) in the VIS. The comparative method, based on a “projection slice theorem” [22], does not perturb the setup; however, the procedure is quite involved and requires a preliminary treatment of the measured 2D-PE data and the creation of a corresponding 2D-spectrum (often by including the pulse’s phase distortions) with it following integration over the ωt coordinate (see, e.g. [23, 5, 24]). Even after an extensive exploration of parameter space, the coincidence of the spectra, recovered from PE-data and pump-probe measurements, is never complete. We examined this phasing method and found some uncertainties and potential inaccuracies in its applications. For example, a correction pre-factor ω_t/n(ω_t) should not be included into the treatment (see below). In addition, the phasing should be performed by shifting the spectra’s phase not with a constant phase (which is subject to the fit procedure), but with the frequency-dependent phase factor δϕ=ω_tδ_t where δ_t is a constant. The expression which links pump-probe and 2D-PE spectra in [22] is general and allows to determine the LO-phase at any delay time T; however, it is enough to determine it for the particular case T=0 and use it for the phasing of 2D-spectra taken at different T-delays. Therefore, we developed a simplified method for the phasing of the 2D-PE spectra which is based on a direct comparison of the pump-probe spectrum measured at zero delay, and the PE-spectrum, recovered from heterodyne-detected PE-signal measured at τ=0,T=0.

The physical origin of this method is simple - as it follows from conventional four-wave mixing theory, the third-order induced polarization P ⁽³⁾ driven by pulses having equal shapes and zero delays between them, is invariant with respect to the permutation of the various pulses (see Eqs. (11.2) and (11.3) on p. 323 in [25]). Therefore, at these conditions the induced polarization irradiated from a medium E_out, is identical for pump-probe, photon echo and transient grating measuring configurations. Moreover, the induced polarization is equally affected by the medium for all cases as well (in the weak-field regime E_out (ω)∝[ω/n(ω)]P ⁽³⁾(ω)), and thus the above mentioned correction pre-factor should not be included into the treatment of spectra. Since pump-probe spectroscopy is inherently a heterodyne-detected technique where the LO-pulse exactly coincides in time with the induced polarization, for differential pump-probe spectrum we can write, using Eqs. (7)–(8):

where Ã_pp(ω)=PP(ω)LO*(ω) is the spectral product of the radiated P ⁽³⁾ and the LO pulses, and the last term corresponds to the homodyne part of the pump-probe signal S_pp(ω)=|PP(ω)|². In contrast to the heterodyne-detected PE, this contribution to the pump-probe signal is always present and cannot be removed from the signal (chopping only removes the contribution of the homodyne LO-signal). This term is ignored in the projection slice theorem [22, 23] and in its applications, as well as in the treatment of the measured 2D spectra using the pump-probe geometry [4]. However, it can not be ignored if the pump-probe signal is relatively strong (induced changes in absorbance ≥2%), and should be included rigorously. On the other hand, the measured differential PE-signal at τ=0 is (from Eq. (8)):

where Ã_pe(ω)=PE(ω)LO*(ω), and S_pe(ω)=|PE(ω)|² is the pure homodyne PE-spectrum. The delay in the LO-filter ΔT can be measured with fs-precision directly, for example by changing the 2D-PE interferometer to a FROG configuration. Assuming that the LO-delay is ΔT=ΔT ₀+δ t and ΔT ₀ is known (already measured), we can remove fast oscillations in the measured interferograms by multiplying the heterodyne terms by factors of exp(±iωΔT ₀). This procedure requires splitting the measured PE-spectrum into two complex-conjugate interferograms ΔS ₊(ω)=Ã_pe(ω)e ^iωΔT, ΔS_(ω)=(ΔS ₊(ω))* as described above. The homodyne PE-signal theoretically can also be recovered from the measured PE-spectrum using a low-frequency window in the Fourier-transformation; but in reality the measured signal is always affected by stray light, light scattering from the sample, its fluorescence, etc. However, this homodyne signal can be simply synthesized using the complex-conjugate interferograms and the known LO-spectrum S_LO(ω)=|LO(ω)|²:

After removing the fast oscillations and constructing the homodyne PE part of the signal we have:

Due to the invariance of P ⁽³⁾ at T=0 all terms in Eqs. (12) and (15) are identical since PP(ω)≡PE(ω), and a direct comparison of both spectra is possible:

Thus, by varying δ t one can find its value for which both pump-probe and PE-spectra coincide completely.

It is very important to note that the described phasing of the LO based on the invariance properties of the induced polarization (and thus can be thought of as an invariance theorem) does not require any perfect, transform-limited pulses nor any correction to their imperfections. The phase distortions in the pulses do not appear in the expressions; only one condition must be satisfied - all the pulses must be identical (and of course, centered at zero delays).

6. Experimental results

In order to demonstrate the capabilities of the coherently-controlled 2D-PE spectrometer, we conducted measurements of the 2D-spectra in Rhodamine 101/MeOH (Rh101) using non-shaped, ~10 fs FWHM duration transform-limited pulses, and shaped pulses with amplitude-and phase profiles corresponding to the optimal pulse found in the coherent control experiments of the population transfer, reported in [15]. This optimal pulse increases the efficiency of population transfer in the weak-field regime from the ground S ₀ to first electronic state S ₁ by approximately 20% as compared to the transform-limited pulse, and for us it was interesting to examine the coherences in the Rh101, created with such an excitation pulse. The amplitude-and phase shaping masks, applied to the AOS for these excitation pulses, are shown in Figure 8 (c,d). The shaped pulse consists of a series of sub-pulses, spaced apart by ~150 fs (the pulse’s temporal and spectral characterization is given in [15]) that correspond to driving a low frequency intramolecular mode strongly coupled to the electronic transition.

The experiments were conducted at room temperature using P-polarized fields in all beams with typical energies of 10 nJ per pulse at the sample position (the LO-pulse was attenuated by 1 order in magnitude). The sample with an OD=0.35 at the absorbance maximum was circulated through a 0.4-mm path length flow cell with 0.15 mm glass windows (the influence of the sample’s absorptivity on the generated PE can be eliminated using the absorption correction procedure described in [26, 27], and the intrinsic PE-spectrum can be recovered from the measured spectra, even in case of high absorbance of the sample). By using non-shaped pulses, for each delay step of Δτ=2 fs 100 differential spectra were averaged in order to get good SNR. For shaped excitation pulses we averaged 800 spectra at each delay step (Δτ=4 fs) since their intensities are much lower (due to temporal stretching of the excitation pulses), and the magnitude of the signal was lowered proportionally. To set the absolute phases in the measured Re- and Im-parts of the 2D-spectra, the absolute delay shift in the LO-filter was defined from the comparison of the measured zero-delay pump-probe spectrum and the PE-spectrum, measured at τ=0,T=0, as described above. Figure 7 displays a coincidence of the pump-probe and PE-spectra at the optimal LO-delay (388.973 fs); the STD of the residual is 5% which is comparable to the STD of the measurement noise which is relatively high due to the displacement of the PE-pulse maximum position forward in time by ~15 fs (see in Fig. 8(a)); thus its magnitude at τ=0 is small. The 2D-spectra were created in a conventional way, but with some modifications. Briefly, the procedure is as follows.

1) From the PE-data for each delay τ we first extract ΔS ₊(ω_t,τ), Eq. (10), and their complex conjugate values using the known absolute value of the LO-delay ΔT:

2) The PE-spectrum can be written using its modulus and phase as $\mathrm{PE}({\omega }_t,\tau )=\mid \mathrm{PE}({\omega }_t,\tau )\mid e^{i{\varphi }_{\mathrm{pe}}({\omega }_t,\tau )}$; therefore the modulus can be recovered immediately by multiplication of ΔS ₊(ω_t,τ) and ΔS_(ω_t,τ):

where for normalization we are using the actual LO-spectra recorded at each delay point, and an auxiliary program that does not lead to an increase in noise by division of the spectra at their wings. This is especially important for the electronic 2D-spectroscopy of molecules, where the PE- and excitation spectra have comparable widths.

3) The phase angle is calculated from the interferograms in a straightforward manner:

This phase contains the contributions not only of the intrinsic PE-phase, but also the phase spectrum of the LO-pulse. For now we are not performing any corrections to the LO-phase assuming that this pulse is almost transform-limited (see in Fig. 5(c) the FROG-trace, measured in situ), which is a good approximation.

4) The measured spectra are equally-spaced in the wavelength domain (and thus unequally-spaced in the frequency domain). In order to get an equally-spaced 2D-spectrum in both ω_τ,ω_t coordinates, after synthesis of the PE-spectrum using its modulus and phase (Eqs. (18) and (19)) we perform first the inverse Fourier transform of the spectra from the wavelength- to t-space using an auxiliary program that works with unequally-spaced frequency scales. If the PE-spectra have “slowly-varying” amplitudes (which is the common case for measurements with non-shaped pulses), a simple linear interpolation of scales can be applied; however, this is no longer applicable in case of highly-structured spectra.

5) The 2D-PE thus obtained in the time domain, PE(t,τ), is then Fourier-transformed to the frequency domain using a standard 2D-FT routine.

 

Fig. 7. Illustration of the phasing of the local oscillator pulse using the pump-probe and the PE spectra, recorded at zero delays. The residuals plotted in the bottom panel show a good coincidence of the spectra (both panels have identical scaling of Y-axes).

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Fig. 8. Comparison of the homodyne-detected PE from Rhodamine 101 in methanol measured at T=0 with short (~10 fs FWHM) excitation pulse (A) and shaped excitation pulses (B). Note difference in the τ scan ranges. (C,D) Amplitude and phase masks applied to the AOS for generating shaped pulses, respectively.

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Figure 8 compares a homodyne-detected PE from the Rh101, measured with unshaped- and shaped pulses. A significant difference in signals can readily be seen by eye - shaping of the excitation pulses leads to the appearance of fine structure across the spectrum. While the PE-signal generated with unshaped pulses consists of a single pulse with ~15–20 fs width, the PE-signal created with the shaped excitation pulses displays a series of well-separated PE-pulses. They arise due to temporal overlap of different sub-pulses in the shaped excitation pulse along the τ -scan. In control measurements performed in both cross-FROG and PE-configurations with the shaped pulses and a 1-mm glass plate instead of the Rh101 sample, we also observed these sub-pulses in the PE-signal, but without any structure in the spectra (not shown). These results are fully supported by numerical simulations, performed in the approximation of an instantaneously-responding non-linear medium (glass). In order to ensure the the absence of any artifacts that can be caused by the AOS (the Dazzler), we also checked for the absence of spectral spatial chirp in the focal plane by scanning the focal spot with a small pinhole and recording the spectra (for both amplitude and phase, and amplitude only shaping).

The 2D-spectra corresponding to the PE, measured using shaped excitation pulses, show even more unusual features. Excitation with short, unshaped pulses, yields a 2D-spectrum that has a classical shape showing inhomogeneous broadening, and some structure in ω_t space (Fig. 9).

 

Fig. 9. 2D-PE spectrum of Rhodamine 101 recorded at T=0 using a 10-fs excitation pulses. (A) - real part, (B) - imaginary part, and (C) - the power spectrum.

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Fig. 10. 2D-PE spectrum of Rhodamine 101 recorded at T=0 using shaped excitation pulses and a 10-fs reading pulse. (A) - real part, (B) - imaginary part, and (C) - the power spectrum.

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The latter feature can be better viewed in the 2D-power spectrum |PE(ω_τ,ω_t)|². With shaped excitation pulses, the 2D-spectrum changes dramatically - fine structure appears in both ω_t and ω_τ domains (Fig. 10). The ω_t -section in Fig. 11(b) shows a periodical structure and a central peak with a FWHM of ~30cm ⁻¹ that is in the limit of the apparatus resolution. In this regard, it is interesting to compare this slice with the pump-probe spectra, recorded in [15] using the same shaped pulses. From comparison of the 2D-spectra, measured with both amplitude- and phase shaping (Fig. 10), and with the amplitude shaping only (Fig. 11) one can conclude that the amplitude modulation in the spectrum of the excitation pulse leads to the appearance of structure in the 2D-PE spectrum in ω_τ space, whereas the phase shaping adds the structure along the ω_t coordinate. It is not yet clear what is the physical mechanism leading to these distinctive peculiarities in the 2D-spectra. One can speculate that such a shaped pulse prepares an excited state wavepacket/superposition state that has weaker couplings to the intramolecular/intermolecular bath and thus longer coherence times. Alternatively, interference effects could lead to an effective decrease in the system-bath coupling resulting in the appearance of very narrow resonant lines in the spectrum (fine structure). Irrespectively, we are observing decoherence effects that depend explicitly on the state preparation. This first experiment demonstrates how important the application of coherently-controlled 2D-spectroscopy can be in studying the electronic coherence in quantum systems. Detailed analysis of these observations will be published separately.

 

Fig. 11. The 2D-PE power spectrum of Rhodamine 101 measured with excitation pulses having only amplitude shaped profiles and a constant phase across their spectra, and a 10-fs reading pulse (left). The cross-sections at ω_τ=17850 cm ⁻¹ (568.8 nm) from the 2D-spectra obtained with the amplitude only and with the both amplitude and phase shaping are shown for comparison (right).

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7. Summary

We have developed and realized a robust coherently-controlled 2D-PE spectrometer that can be applied to the study of phase control of coherence, flow of energy transfer in conjugated many-body systems, and can increase the resolution of 2D-spectra by using appropriately designed excitation pulses. The first coherently-controlled electronic 2D-spectra are clearly demonstrated. The spectrometer is a passively-stabilized four-beam interferometer based on a DO beam splitter, and allows the execution of PE-measurements with a phase stability better than Λ/100. A simple phase tracing method for examining of the phase fluctuations in any spectrometer has also been introduced and applied to the current design. The upgrade of the phase stability is achieved through an improvement to the interferometer design, namely, the uncoupling of the translation stages, which permits the use of relatively low precision mechanical translation stages for both τ - and T-scans. In this paper we also developed a rigorous method to measure the phase stability, and improved the procedure for the phasing of the LO with respect to the PE. Looking forward, this development is particularly important for the use of complex pulse shapes. The procedure is based on the invariance properties of the third-order induced polarization under certain conditions, and was successfully applied in the demonstrated measurements. We also experimentally demonstrated that diffractive optics can be successfully used in broadband ultrafast 2D-PE spectroscopies – the present spectrometer has a spectral bandwidth >5000 cm ⁻¹ (500–690 nm; measured with unshaped pulses).