1. INTRODUCTION

Two-dimensional electronic spectroscopy (2DES) is a third-order non-linear technique which is successfully spreading in the latest years thanks to its capability of unraveling fine details of the ultrafast dynamics of the systems under investigation [1–3]. 2DES is based on the interaction of three ultrafast laser pulses with an optical sample. The experiment consists of probing the resulting third-order optical polarization as a function of the time delays between the three different interactions. A typical pulse sequence with the definition of the time variables is reported in Fig. 1(a). The capability of 2DES to scan independently the three time delays between pairs of pulses allows disentangling overlapped signals and unraveling correlations between different transitions. Suitable pulse sequence definition allows indeed the access to rephasing (R), non-rephasing (NR), and double quantum (2Q) signals, each one providing different and complementary information [4,5]. This flexibility enables investigation of molecular systems with previously unavailable detail; for example, molecular couplings and energy relaxation pathways can be directly followed and the temporal evolution of both coherent and non-coherent processes can be characterized.

 

Fig. 1. Schematic 2DES setup. Abbreviations: SM, spherical mirror; P, prism; DO, bi-dimensional diffractive optical element; DSM, donut spherical mirror; C, chopper; ND, neutral density filter; WP, wedge pair; TS, translational stage; S, sample. Panel (a) describes the pulse sequence and the time intervals definition in a 2DES rephasing experiment. Panel (b) reports a schematic representation of BOXCARS geometry, where $E_i$ are the three interacting fields, ${\mathbf{k}}_i$ are the wave vectors, and $E_{\mathrm{LO}}$ is the fourth pulse used for heterodyne detection.

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The first 2DES spectrometer [6] was based on a fully non-collinear BOXCARS geometry. Since then, other geometries have been proposed [7–11] and recently reviewed by Fuller and Ogilvie [12]. The fully non-collinear geometry remains, however, the most adopted configuration and most efforts are still being paid in the development of new technological solutions to overcome the well-known issues and to improve the quality of the data.

One of the major sources of signal degradation in fully non-collinear setups is the lack of phase stability, i.e., the capability to keep constant the phase difference between pairs of optical pulses for a given time delay [13].

Many different approaches have been used to increase the phase stability in fully non-collinear 2DES experiments, including active [14] and passive phase stabilization methods, through diffractive optics (DO) [15,16], pulse shaping [17] and, more in general, particular geometries that allow the phase-locking of specific pulse pairs [18,19]. These last methods, in particular, achieve a great phase stability working in a full optical rotating frame, but they require coupling the motion of pairs of pulses, partially losing the independent control of timings. Therefore, a better phase stability is obtained at the cost of a less versatile setup, unable to exploit the full potential of 2DES.

Here we report an experimental scheme for a fully non-collinear 2DES whose optical layout has been specifically designed to provide full independent control over all the time intervals allowing an easy retrieval of all the different phase-matching signals (R, NR, and 2Q), still maintaining excellent reliability and particularly high short- and long-term phase stability.

Experimental data recorded for a tetrahydrofuran (THF) solution of zinc phthalocyanine (ZnPc) are used to elucidate the crucial features of the proposed setup and of the corresponding data processing routines. The full potential of the described approach is exemplified through the characterization of the full 2D response of solutions of J-aggregates of the diacid form of the water-soluble tetra-($p$ sulphonato)-phenyl-porphyrin (${\mathrm{H}}_2$TPPS).

2. OPTICAL LAYOUT AND PULSE SEQUENCE

The experimental setup is illustrated in Fig. 1. The scheme is inspired to the BOXCARS geometry setup proposed by Nemeth et al. [16]. A Coherent Libra laser system generates a continuous pulse train centered at 800 nm with a repetition rate of 3 KHz. The central wavelength of laser pulses is converted in the visible range using a commercial NOPA (Light Conversion TOPAS White).

Before entering the optical setup, the pulses from the NOPA pass through a stage of pulse shaping [with an acousto-optic programmable dispersive filter (AOPDF), Fastlite Dazzler] and prisms compression to maximize the time resolution of the experiment and to guarantee full control over the phase of the different frequency components in the broadband pulse. A detailed description of the pulse shaping stage is reported in Section 3.

Four identical beams in BOXCARS geometry are obtained focusing the incoming beam into a 2D-DO element with spherical mirror SM3. The DO is a 2 mm thick phase mask that is etched in a 2D cross-hatched pattern where the width and the depth of the features are optimized for first-order diffraction of the incident wavelength. Since the diffraction properties of the DO are wavelength dependent, different suitably optimized DOs could be necessary to achieve maximum performances in different spectral ranges. The four first-order diffraction beams, each about 10% of the input power, are arranged in the BOXCARS configuration, as depicted in Fig. 1(b). The outgoing divergent beams are collimated and parallelized by a donut-shaped spherical mirror (DSM1), with the same focal length of the focusing mirror SM3 (500 mm).

The main advantage of using a 2D-DO is the more easy retrieval of the BOXCARS phase-matching conditions, naturally emerging from the 2D diffraction pattern, whereas the use of a 1D-DO requires an additional preliminary step of beam splitting through conventional optics and the subsequent alignment of pairs of beams to form a square. A drawback of this configuration is that the time delays between pulses can be introduced conveniently only through antiparallel wedge pairs (WP). Typically, a 1° fused silica wedge pair is employed, which is capable of providing a time delay of the order of hundreds of femtoseconds. This is enough for $t_1$ but it can be limiting for $t_2$, where one might be interested in observing phenomena with longer characteristic time scales (of the order of picoseconds). In the proposed setup this complication is overcome through the use of 4° ${\mathrm{CaF}}_2$ 25 mm long wedges, providing time delays up to 2 ps. Instead of fused silica, the choice of ${\mathrm{CaF}}_2$ is more suitable since it induces less phase distortion in broadband pulses. Such wedges require anyway a more careful calibration procedure, as described in Section 4. The insertion of wedges in the beam path could produce small deviations in the beam propagation. This is detrimental for the fulfillment of the phase-matching conditions, because any deviation, even of a small angle, on the direction of one of the beams can produce artifacts in the collected signal [20]. To correct these possible deviations, the geometry and spatial propagation of the beams is controlled with a charge-coupled device (CCD) camera, allowing the optimization of the wedges orientation.

One wedge from each pair is mounted on a computer-controlled linear delay stage (Aerotech Ant 95), whose position has to be accurately controlled to guarantee repeatability and precision of the time delays between pulses (in our setup the repeatability is of about 0.07 fs and the phase stability is higher than $\pi /20$). The fourth pulse is used as a local oscillator (LO) and its power is conveniently attenuated with a graduated neutral filter in order to maximize the heterodyne signal. Its temporal position is kept fixed during the 2DES experiment and thus it can be conveniently set as reference time for the definition of the time delays.

This configuration enables us to scan independently all the different time delays of the three exciting beams, easily obtaining the R, NR, and 2Q signals simply designing the correct pulse sequence. We found that the most suitable configuration consists of fixing the time position of the pulse generating the third interaction with the system, i.e., $E_3$ for the R and NR signals and $E_1$ for the 2Q signal. The other two pulses are scanned accordingly to construct the full 2DES spectra. This choice has the major benefit of guaranteeing a phase-locking relation between $E_3$ ($E_1$) and the LO, leading to a better phase stability. Moreover, in this way, the signal is always generated at about the same time delay from the LO and its spectral interference does not change much during the experiments, thus avoiding all the artifacts arising when the signal is moving in time. For example, when the recorded spectral interference is pushed near the limit resolution of the detection system, false decay of the signals can be possibly generated.

It must be recognized, however, that this choice can generate different issues. It indeed implies scanning the first two pulses in different ranges of their associated wedge positions, possibly resulting in different chirp in pulse 1 and 2 for each population time $t_2$. This complication can be overcome through a careful use of the pulse shaper that can be suitably tuned so to recompress the pulses at every $t_2$ delay (Section 3). Furthermore, if the phase difference between the third interaction and the LO is kept fixed, then the phase stability depends critically only on the phase difference between the first two interactions. It is thus necessary to achieve a strict control over the $t_1$ delay (and also $t_2$ for 2Q signals) and therefore particularly reliable wedge calibration procedures have been set up (Section 4).

Once the wanted time ordering is set, the four beams are focused onto the sample by DSM2 with a focal length of 200 mm. The third-order signal propagates in the LO direction and their ensuing interference is delivered to the spectrograph (Andor Shamrock 303i) equipped with a CCD camera (Andor Newton 920) for detection. The CCD is calibrated exploiting the capability of the AOPDF to produce holes in the pulse spectrum. The AOPDF is calibrated using certified absorptive filters (holmium oxide solution and didymium glass from Hellma).

During the 2DES experiment, the CCD camera collects other spurious contributions besides the signals. A suitable acquisition method and a post-processing procedure are thus necessary to remove the unwanted contributions and gain a high signal-to-noise ratio. The acquisition methodology is based on the double modulation lock-in method proposed by Augulis and Zigmantas [21] and will be discussed in detail in Section 5.

3. COMPRESSION AND SHAPING STAGE

The pulse properties are determined by two factors: the shape of the laser spectrum and the phases of its spectral components. The shape of the laser spectrum affects the shape of the final 2DES maps [22,23] while a distortion of the phase (like in chirped pulses) leads to longer pulse duration, reducing the resolution of the experiment [12], and to altered 2DES maps [24,25]. Therefore, it is necessary to work with transform-limited (TL) pulses, defined as the shortest possible pulses for a given optical laser spectrum, to obtain high-quality 2DES data. This condition is achieved when the phase $\varphi (\omega )$ is a linear function of the frequency $\omega$ [26]. Possible non-linear deviations of the phase, mainly caused by propagation in transmissive optics, have to be compensated to retrieve the TL (and thus temporally compressed) pulse at the sample position. In the implemented setup, this is performed by a pulse-shaping stage consisting of an AOPDF and a prism compressor [27,28], used to relieve the self-induced chirp of the Dazzler crystal.

The AOPDF employs a Taylor expansion of the phase of the electric field around the central frequency of the pulse spectrum, ${\omega }_0$. The terms of the expansion at different orders describe different properties of the temporal profile of the pulse. The linear term is related to the arrival time of the pulse, while the second order describes the linear chirp. Higher even (odd) terms correspond to phase distortions with point (axial) symmetry with respect to ${\omega }_0$ [29].

The measurement of the pulse profile is performed with the frequency-resolved optical gating (FROG) technique at the sample position [30] that reveals the frequency-resolved auto-correlation of the pulse. In our setup, we can take advantage of the BOXCARS geometry to perform FROG by replacing the sample with a non-zero third-order material, typically a solvent. The use of the same cuvette in the same position for sample and solvent guarantees identical operating conditions in FROG and 2DES measurements, keeping the same pulse properties in the two experiments. The procedure to recover a TL pulse consists of manually modifying the coefficients of the expansion via the AOPDF and checking the FROG result until the best conditions are found.

In the experiment the delay times are set modulating the thickness of the wedge medium traversed by the beams. This implies that, at different time delays, the pulses can be affected by different chirp and thus the optimal compression conditions are position dependent. Therefore we must evaluate if and when the chirp is negligible or how we can correct its impact on the final signal. This issue is particularly relevant in the $t_2$ scan during R and NR experiments, where the full thickness of the wedges is exploited. The chirp has two main effects: a decrease of the maximum signal intensity and the presence of distortions because the medium dispersion leads to frequency-dependent time delays.

Simulation of FROG time traces [29] can be used to predict the loss of heterodyne third-order signal as a consequence of the chirping of the pulses. This procedure is applied to monitor in particular how the maximum intensity of the simulated FROG decays as a function of the chirp induced during the $t_2$ scan. The pulse duration, specified as full width at half-maximum (FWHM), must be accounted for in the discussion; indeed the shorter the pulse duration, the larger the bandwidth of the laser spectrum and the bigger the effect of the dispersion. The contour plots in Fig. 2 report the ratio between the maximum of the FROG traces simulated with chirped pulses and FROG traces simulated with optimal TL pulses, as a function of the pulse duration and the imposed delay. Deviations from the ideal fully compressed condition are studied as function of wedge material [panel (a)], different pulse shaping operating parameters [panel (b)], and different pulse spectrum central frequency [panel (c)]. It is clear that the greatest aberrations are expected for short pulses at long time delays. Therefore, the capability to manage large bandwidth, and thus short pulses, sets the limit of the maximum delay that can be investigated with a negligible loss of signal. In this context the use of ${\mathrm{CaF}}_2$ wedges allows to reliably scan a time interval about 30% longer than fused silica wedges [Fig. 2(a)]. Moreover the capability of managing a specific bandwidth depends sensibly upon the central wavelength of the pulse spectrum [Fig. 2(c)]. The same simulations can also be used to find the best operating conditions of the pulse shaper at each investigated $t_2$ value. In our setup the AOPDF is indeed programmable to minimize the chirp effect at each $t_2$. Since the AOPDF is positioned before the DO, it affects in the same way all the pulses. Thus it cannot operate independently on the different pulses of the experimental sequence and the optimal TL conditions for pump pulses ($E_1$ and $E_2$) do not necessarily correspond to optimal conditions for the probe ($E_3$) pulse. The simulations in Fig. 2(b) confirm that the best choice is to set the pulse shaper apparatus so to maximize the pulse compression of the pump pulses at each $t_2$. The Sellmeier expression of ${\mathrm{CaF}}_2$ refractive index [31] is used to obtain the expansion coefficient to program the AOPDF. The choice of programming the AOPDF to keep compressed the pump pulses with respect to the probe pulse allows for another gain of about 30% in the exploitable time interval, see Fig. 2(b). The effect of the medium-induced chirp and how it can be effectively corrected through a smart programming of the AOPDF is illustrated in Fig. 3, reporting an example of FROG measurement recorded at different wedge positions.

 

Fig. 2. Contour plots showing the ratio between the maximum intensity of the FROG traces simulated with chirped pulses and with optimal TL pulses. The results are plotted as a function of the pulse duration ($x$-axis) and $t_2$ delay ($y$-axis). In the simulations the chirp reproduces the experimental conditions. The pump pulses $E_1$ and $E_2$ are considered identical and fixed in time, whereas the probe pulse $E_3$ pulse is moved in time. Comparison between the loss of FROG intensity induced by (a) the chirp in ${\mathrm{CaF}}_2$ and fused silica (FS) wedges; (b) the chirp in ${\mathrm{CaF}}_2$ wedges when the AOPDF is set to optimize the compression of the probe pulse (dotted line) or of the pump pulses (solid line); (c) considering different central wavelengths of the pulses, employing ${\mathrm{CaF}}_2$ wedges and compressing the pump pulses.

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Fig. 3. (a) Laser spectrum used in ZnPc measurements. (b) Pulse profile measured by FROG experiment at optimal TL conditions. (c) FROG experiment in the same conditions but measured at extreme positions of the wedges delay ($t_2=2\mathrm{ps}$), showing a slight chirp. (d) FROG experiment at $t_2=2\mathrm{ps}$ after the phase correction of the Dazzler that retrieves the compressed condition.

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The procedure described above is able to correct and minimize the loss of signal due to the chirp but it leaves unmodified the frequency-dependent timing distortions due to the dispersion induced by the wedges. It follows that the higher-frequency components are subjected to a longer delay than the lower-frequency components, resulting in a frequency shift after Fourier transforming from the time to the frequency domain. This leads to possible distortions of the signal along the excitation axis in 2DES maps and frequency shifts of the coherence beating detected along the $t_2$ axis. In order to estimate and possibly correct these aberrations, the time delay imposed through wedge motion can be expressed as a function of the frequency:

4. TIME DELAYS CALIBRATION

A crucial step in setting a 2DES experiment is the calibration of translation stages that regulate the wedge positions and the amount of ${\mathrm{CaF}}_2$ that the pulse passes through. A sub-femtosecond precision of the time delay within the entire scanned range is necessary to achieve a full control over the phases of the pulses and therefore obtain a meaningful 2DES map. In a non-collinear 2DES measurement the phase of the signal depends indeed on the phase of all pulses. When the time delay is controlled by a translation stage, its accuracy should be adequate to conserve the phase difference between pulses. The methodology for time delay calibration here proposed is based on the determination of the sub-cycle time delays from the local oscillator [17].

The starting point is the definition of a linear relation that associates the position $x_i$ [in millimeters (mm)] of the $i$th linear stage, and thus the position of the optical wedge, to a time delay ${\tau }_i$ [in femtoseconds (fs)] of the $i$th pulse: ${\tau }_i=c_i(x_i-z_i)$, where $c_i$ is the linear calibration coefficient (fs/mm) that has to be determined and $z_i$ (mm) are the zero time positions at which all the pulses have the same time delay from the LO. The LO is taken naturally as a reference because it is the only pulse whose timing (${\tau }_{\mathrm{LO}}$) is fixed since it is not controlled by a WP.

The procedure of time delay calibration consists of three main tasks: (i) determining time zero of the experiment and therefore the positions $z_i$ of the three linear stages that satisfy the equation ${\tau }_i(z_i)=0\mathrm{fs}$; (ii) finding the linear coefficients $c_i$ to convert a displacement of the stage in a time delay from the zero of the experiment; and (iii) finding the precise value of ${\tau }_{\mathrm{LO}}$.

The typically employed calibration procedure consists of recording the spectral interference (SI) produced by two pulses [32,33]. To this purpose, a 25 μm pinhole is placed where the four beams are focused at the sample position and the SI is recorded by the spectrometer. The Fourier transform of the SI is the simplest method to correlate the position of the stages with the time delay between the LO and the selected pulse: when a spectral interference between two pulses is Fourier transformed, a peak in the time domain is obtained, which is centered at the delay time between the interfering pulses. As this delay increases, the number of fringes in the resulting SI increases shifting the peak in the time domain at longer times. This method works only until the CCD camera and the spectrograph can spectrally separate the fringes of the interference; therefore, it could fail at long values of delay times such as 2 ps.

Moreover, the adopted pulse sequence with the third interaction fixed and phased-locked with the LO, makes particularly crucial a full control over the relative time and phase of the first two pulses. Therefore, a new procedure has been proposed to fulfill these requirements. With respect to previously proposed approaches [33,34] where each wedge is calibrated independently, our method allows a more robust determination of time-zero positions thanks to the delay correlations plots $\mathbf{H}$, that correlate the positions of couples of stages (Section 4.A). Moreover, a further step of refinement of the linear calibration coefficients has been added, ensuring a reliable phase-locking condition between the first two pulses, effectively scanned in our experimental configuration (Section 4.B).

The first step consists of estimating a set of positions $x_{i,\mathrm{in}}$ of the three stages that roughly induce the same delay from the LO. This evaluation can be performed by adjusting the stages in order to produce similar interference patterns. We subsequently scan each stage $x_i$ in a short range, about 20% of the length of the wedges, around $x_{i,\mathrm{in}}$ and obtain three matrices ${\mathbf{F}}^{(i)}$ of spectral interferograms, recorded for different positions $x_i$ and resolved in frequency $\omega$ by the spectrograph [see Fig. 4(a)]. Every column of the matrix ${\mathbf{F}}^{(i)}$ represents an interference spectrum, which evolves as the delay between the interfering beams changes due to the motion of the linear stage. At this point we have three matrices, ${\mathbf{F}}^{(1)}$, ${\mathbf{F}}^{(2)}$, and ${\mathbf{F}}^{(3)}$, that contain all the information about the evolution of the pulse delays with respect to the fixed LO as the stages move.

 

Fig. 4. Procedure for the evaluation of the zero positions of the stages exemplified for $E_2$ with $E_1$ taken as reference. (a) Graphical representation of the matrix ${\mathbf{F}}^{(2)}$; (b) matrix with elements $F_{jk}^{(2)}-F_{jl}^{(1)}$ with $F_{jl}^{(1)}$ taken at $x_{1,l}=22.65\mathrm{mm}$; (c) the solid line is the result of the integration of the absolute value of the matrix in panel (b) along the frequency dimension, the dashed line is the envelope computed with the analytic signal. The insets on the right of panels (a)–(c) show enlargements of the traces in a limited interval of $x_2$ (grey area). (d) Matrix ${\mathbf{H}}^{(2)}$ showing a clear signature only at the $(x_1,x_2)$ coordinates where the pulses are synchronous. The black line represents the linear fit performed with Eqs. (4) and (5).

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A. Evaluation of Zero Positions

The procedure is based on differences of interference spectra. One of the exciting beams, for example $E_1$, is selected as the reference pulse. One at a time, the remaining two beams are compared with the reference. Here the procedure is illustrated for beam $E_2$, but the same will be applied for beam $E_3$. The main objective of this processing is to produce, from the interference patterns ${\mathbf{F}}^{(1)}$ and ${\mathbf{F}}^{(2)}$, a delay correlation plot called ${\mathbf{H}}^{(2)}$, which correlates the positions of the two stages, showing a clear signature when the two pulses are synchronous. The elements of matrix ${\mathbf{H}}^{(2)}$ are defined as

B. Evaluation of the Calibration Coefficients

The second part of the calibration procedure consists of determining the linear coefficients $c_1$, $c_2$, and $c_3$ that allow accurately moving the pulses from the zero positions. The optimal procedure should provide conversion coefficients able to guarantee, at the maximum excursion of the linear stages, a time error much smaller than the optical cycle of the electromagnetic radiation. This accuracy assures that the final signal, recorded during experiments, is correctly phase-locked. The idea at the basis of this procedure is to mathematically compensate for the evolution of the spectral interference coupled to the movement of the wedge. In other words, we want to determine which is the exact delay to be mathematically subtracted in order to achieve a perfectly constant interference pattern. We will describe the procedure for the pulse $E_1$; the same applies for the other pulses.

First, matrix ${\mathbf{F}}^{(1)}$ is Hilbert transformed along the frequency axis in order to operate on the complex analytic signal. This allows subtracting the delay ${\tau }_1(x_1)=c_1(x_1-z_1)$ from the spectral interference multiplying by an oscillating exponential function as

The interference spectra in matrices ${\mathbf{F}}^{(i)}$ used for the previous calculation are sampled only within a limited portion of the stage movement range. This is because at large delay values the spectrometer is not able to efficiently resolve all the fringes of the SI. This is not detrimental for the accurate evaluation of the zero positions of the stages but it may generate inaccuracy at the extreme positions of the stages. This is particularly critical for pulses $E_1$ and $E_2$, whose position in time is effectively scanned during the experiment, exploiting the full length of the optical wedges. The third pulse and LO are instead fixed during R and NR measurements, and are not delayed relevantly in 2Q experiments. Therefore, we verify that the relative phase between $E_1$ and $E_2$ is conserved also at the extreme positions and possible small errors, even of fractions of an optical cycle, are corrected.

In practice, using the calibration coefficients previously determined, $x_1$ and $x_2$ are moved simultaneously so to keep ${\tau }_2-{\tau }_1$ constant. The interference pattern generated by interaction of $E_1$ and $E_2$ as a function of the position $x_1$ (or $x_2$) and $\omega$ is recorded and the matrix ${\mathbf{F}}^{(1,2)}$ is retrieved. If the spectral interference is constant for the full scanned range, then the calibration is accurate and ready for the experiments. When a small linear drift of the phase of the interference is observed, it means that the values of the linear coefficients are not fully rigorous in the whole range and they must be corrected to avoid phase drift of the third-order signal.

To this purpose, it is possible to define a corrected interference pattern as

 

Fig. 5. Distortions of rephasing 2DES maps arising when effects of phase drift at extreme positions of the delay lines are not accounted for. (a)–(c) 2DES maps obtained using the calibration coefficients before linear drift correction. (d)–(f) The same 2DES maps determined using the calibration coefficients corrected as in Eq. (10). Measures performed on a THF solution of ZnPc. (g) SI of $E_1$ and $E_2$ with $t_1=150\mathrm{fs}$ scanned over the entire wedge applying correction in Eq. (10). (h) Phase error between $E_1$ and $E_2$ after the correction.

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C. Evaluation of the Delay from the LO

The last step consists of retrieving the precise value of ${\tau }_{\mathrm{LO}}$. This task cannot be accomplished by simply looking at the SI generated by the 25 μm pinhole used for the calibration, because this configuration does not reproduce conveniently the exact working conditions of 2DES experiments. We therefore suggest to exploit the heterodyne non-resonant FROG of a solvent in a cell at the sample position. The procedure consists of collecting the interference patterns produced by the LO and the third-order non-resonant signal of the solvent generated when all the pulses are superimposed in time. During this experiment, all the stages positions $x_i$ are scanned simultaneously around their $z_i$ values, so to keep ${\tau }_{\mathrm{FROG}}={\tau }_1={\tau }_2={\tau }_3$. Therefore, we obtain a matrix of interference patterns which evolves as a function of ${\tau }_{\mathrm{LO}}-{\tau }_{\mathrm{FROG}}$, similar to matrices ${\mathbf{F}}^{(i)}$. This matrix is Fourier transformed along the frequency axis obtaining a new matrix, which has a tunnel-like shape similar to matrix ${\mathbf{H}}^{(2)}$. A minimization procedure analogous to the one reported in Eq. (5) can be applied and a more accurate value of ${\tau }_{\mathrm{LO}}$ at ${\tau }_{\mathrm{FROG}}=0\mathrm{fs}$ can be extrapolated.

The determination of the exact value of ${\tau }_{\mathrm{LO}}$ is a crucial parameter in the processing of the 2DES spectrum (see the next section). The time distance of the LO from the zero of the experiment (${\tau }_{\mathrm{LO}}$) controls indeed how dense the fringes of the interference pattern in the collected spectra are, usually is set at about 400 fs. As an example, Fig. 6 shows the effects of an error in the estimation of ${\tau }_{\mathrm{LO}}$ in the final 2DES rephasing map.

 

Fig. 6. Distortions of a rephasing 2DES map caused by an inaccurate determination of ${\tau }_{\mathrm{LO}}$. Panels (a), (b), and (c) show errors of $-15$, 0, and $+15\mathrm{fs}$ in retrieving a ${\tau }_{\mathrm{LO}}=489\mathrm{fs}$, respectively. The measure was performed on a THF solution of ZnPc at $t_2=100\mathrm{fs}$.

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5. DATA ACQUISITION AND PROCESSING

A double lock-in modulation acquisition [21] is implemented in order to suppress the spurious contributions in the recorded spectra. In this acquisition procedure, $E_2$ and $E_3$ exciting pulses are modulated by two phase-locked optical choppers at two different frequencies, ${\nu }_{E_2}$ and ${\nu }_{E_3}$, in order to isolate the signal from almost all the other contributions after a Fourier transform operation. The laser pulse train, the CCD camera, and the optical choppers are reciprocally triggered and phase-locked. To fully exploit the potential of this method, each modulation frequency must be carefully chosen. First, the signal should be extracted at a frequency ${\nu }_S$ lower than the Nyquist limit, so that ${\nu }_S<{\nu }_{\mathrm{CCD}}/2$, with ${\nu }_{\mathrm{CCD}}$ being the sampling frequency of the detector. Second, to avoid phase instability it is convenient to set the higher modulating frequency as a multiple of the lower one. However, since the choppers modulate the beams with square waves, the scattering contributions modulated at the lower frequency are distributed in all their odd harmonic frequencies. Therefore, taking the higher frequency as the odd multiple of the lower one is a convenient approach, since it guarantees that the modulation frequencies of the signal are always even multiples of the slower modulation. The signal can be eventually retrieved at both sum and difference frequencies, ${\nu }_S^{+}={\nu }_{E_2}+{\nu }_{E_3}$ and ${\nu }_S^{-}={\nu }_{E_2}-{\nu }_{E_3}$, respectively. The signal is recovered as the real part of the Fourier transform of the series of spectra acquired by the detector. There is a particular combination of relative phases between the two modulations and the CCD sampling for which the real part of the signal is maximized at both sum and difference frequencies and the maximum signal-to-noise ratio can be achieved. In our setup, ${\nu }_{\mathrm{CCD}}=500\mathrm{Hz}$, ${\nu }_{E_2}=200\mathrm{Hz}$, and ${\nu }_{E_3}=40\mathrm{Hz}$; thus the signal can be extracted at ${\nu }_S^{-}=160$ or ${\nu }_S^{+}=240\mathrm{Hz}$ or both. With this method, most of the scattering contributions are removed, since they reach the CCD camera with the frequency of modulation of a single chopper or they are not modulated at all.

However, the acquired signal still contains the unwanted homodyne third-order signal, ${|E_S|}^2$, the spurious contribution arising from the interference between the signal and the scattering of the non-modulated beam, $E_S{E}_1^{*}$, and the interference between the scattering of the modulated beams, $E_3{E}_2^{*}$, so that the raw signal acquired by the CCD can be expressed as $S^{(a)}(t_1,t_2,{\omega }_3)={|E_S|}^2+2\mathfrak{R}\{E_S{E}_{\mathrm{LO}}^{*}\}+2\mathfrak{R}\{E_S{E}_1^{*}\}+2\mathfrak{R}\{E_3{E}_2^{*}\}$. An example of rephasing data collected at a fixed population time is reported in Fig. 7(a).

 

Fig. 7. Step-by-step description of the implemented data processing procedure. (a) Raw signal $S^{(a)}$ acquired for a rephasing experiment at a fixed value of $t_2$; (b) signal $S^{(b)}$ is retrieved after applying the correction for the delay from LO; (c) signal $S^{(c)}$ is obtained after the application of the rotating frame approach and the application of a window filter in the $t_3$ domain to remove residual spurious contributions; (d) final 2DES map obtained after Fourier transform along $t_1$, shift from the reference frequency and phase correction. (e) Comparison between pump-probe spectrum (blue line) and projection onto emission axis of the total 2DES spectrum (black line). The measure was performed on a THF solution of ZnPc at $t_2=100\mathrm{fs}$.

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First, to isolate the pure signal $E_S$ from the contribution $E_S{E}_{\mathrm{LO}}^{*}$, it is necessary to enforce the time ordering between the signal and the local oscillator. The recorded interference can be expressed as $E_S{E}_{\mathrm{LO}}^{*}={\mathcal{E}}_S({\omega }_3){\mathcal{E}}_{\mathrm{LO}}({\omega }_3)e^{i{\omega }_3({\tau }_S-{\tau }_{\mathrm{LO}})}$, where ${\mathcal{E}}_S({\omega }_3)$ and ${\mathcal{E}}_{\mathrm{LO}}({\omega }_3)$ are the square root of the spectra of the corresponding fields. The local oscillator arrives at the CCD after the signal, thus we need to impose that ${\tau }_S-{\tau }_{\mathrm{LO}}$ is negative. This operation is performed (i) by inverse-Fourier transforming $S^{(a)}$ along ${\omega }_3$, (ii) by retaining only the negative part of the resulting time axis, and (iii) by Fourier transforming back. After this set of operations, the signal $S^{(a)}$ becomes a complex valued signal ${\widehat{S}}^{(a)}$: the real part is equal to the data recorded by the CCD and the imaginary part is a copy of the original data with a phase shift of $-\pi /2$. The delay ${\tau }_{\mathrm{LO}}$ can now be added and at the same time the data are divided by the square root of the LO spectrum giving

At this point a rotating frame (RF) can be applied, as previously suggested by Schlau-Cohen et al. [36]. For R and NR signals, this is done detuning the optical frequency during $t_1$ by subtracting a reference frequency ${\omega }_{\mathrm{ref}}$, which is usually taken as the central frequency of the laser bandwidth:

In order to remove the remaining spurious contributions from the signal, a time-filtering in the $t_3$ dimension is applied using a super-Gaussian function. The filter is tuned to keep only the portion of the $t_3$ axis between zero and about 100–150 fs, after which the signal is completely decayed. This operation removes most of the remaining spurious contributions in the recorded signal, including the homodyne third-order signal, because the operation in Eq. (11) shifts all these contributions by ${\tau }_{\mathrm{LO}}$ and thus outside the filter window. The initial transient of the signal can be slightly affected by $E_S{E}_1^{*}$ and $E_3{E}_2^{*}$ when the delay between signal and $E_1$ and the delay between $E_3$ and $E_2$ are inside the filter window, respectively. This condition is verified only at early values of population time and does not arise any concern for the rest of the data.

The small dispersion of the pulse spectral components affecting $t_1$ and $t_2$ delays is corrected applying Eq. (1) and interpolating the new data. This step of data processing is necessarily performed after the RF because the interpolation of the $t_1$ axis is not feasible when dealing with undersampled data.

The next step is the Fourier transform along $t_1$ (or $t_2$ in the case of 2Q maps), to obtain the final frequency-frequency maps. The resulting excitation frequency axis ${\omega }_1$ must then be shifted by ${\omega }_{\mathrm{ref}}$ to account for the rotating frame previously applied. Finally, the arbitrary phase of the signal with respect to the LO must be removed to isolate the real from the imaginary part. As suggested in [2], this last step is performed exploiting the projection-slice theorem. The same phase shift is employed for the phasing of R, NR, and 2Q signals, as long as these measurements are performed with the same calibration conditions.

At this point, the processed signal and its time evolution along $t_2$ is ready to be examined with the various data analysis procedures proposed to extract information about populations and coherent dynamics [38–48]. The versatility of this setup is exemplified in Fig. 8 that reports the pump-probe spectrum, the R and NR maps recorded at $t_2=100\mathrm{fs}$, and the 2Q map for a solution of J-aggregates of ${\mathrm{H}}_2$TPPS. This particular system represents an ideal example of how the synergic use of high-quality R, NR, and 2Q response can provide a clear picture of very subtle details of the excitonic structure and dephasing mechanisms in complex systems, not accessible by single measurements [5].

 

Fig. 8. Examples of the different responses achievable with the described setup suitably modifying the pulse sequence: (a) pump-probe spectrum; (b) rephasing and (c) non-rephasing 2DES maps at a fixed $t_2$ value (100 fs); (d) 2Q map. The measures were performed on aqueous solutions of ${\mathrm{H}}_2$TPPS J-aggregates at room temperature.

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6. CONCLUSIONS

We reported a detailed description of a versatile fully non-collinear setup for 2DES spectroscopy. Novel implementations in the optical layout and in calibration and data processing procedures have been proposed to efficiently obtain high-quality and artifacts-free 2DES spectra along different phase-matching directions. With respect to previously proposed fully non-collinear wedge-based setups, we introduced the use of 4° wedge pairs, allowing longer scanning times (up to 2 ps). The particular optical layout of the setup allows an independent control the three pulse delays, giving access to rephasing, non-rephasing, and double quantum signals simply imposing the correct pulse sequence. Possible aberrations introduced especially at long delay times by the dispersion of the wedge medium have been estimated and discussed. Correction methodologies based on a fine control of the phases of the pulses with an AODPF pulse shaper and a robust wedge calibration procedure have been proposed. Reduced acquisition times were achieved thanks to the application of an a posteriori rotating frame approach that allows a drastic reduction of the number of data points along $t_1$ to acquire a single map.