## Abstract

We report an experimental design for two-dimensional electronic spectroscopy (2D-ES) that avoids the need to measure notoriously weak pump-probe spectra. Retaining a fully non-collinear folded boxcar geometry, the described layout replaces pump-probe with heterodyned transient grating (het-TG). The absorptive component of the het-TG signal is measured directly, following a straightforward optimization routine. The use of het-TG achieves an improvement in signal to noise ratio by almost two orders of magnitude. As a result, 2D-ES-signals down to 0.5% can be clearly resolved.

© 2013 OSA

## 1. Introduction

Two-dimensional electronic spectroscopy (2D-ES) [1] is a modern, multifunctional technique in the ultrafast spectroscopy toolbox and is employed by a growing number of research groups. 2D-ES measures correlation between excitation- and emission-frequencies (ω_{1} and ω_{3,} respectively), to provide further insights when compared to ω_{1}-integrated techniques such as pump-probe (PP) or transient grating (TG). This is achieved by resolving so-called cross peaks (ω_{1} ≠ ω_{3}). Cross peak dynamics describe the properties of various processes, e. g. energy transfer pathways in photosynthesis [2], excitonic dynamics in semiconductors [3] and molecular J-aggregates [4], or vibronic and vibrational coherences in molecules [5].

In 2D-ES, the excitation frequency axis ω_{1} is retrieved by Fourier-transformation (FT) from the time domain along *t*_{1}, where *t*_{1} is the delay between the first two pulses in a four-wave-mixing sequence. The third pulse arrives after a so-called population time *t*_{2} and sets the system back into a state of coherence, which radiates for a time *t*_{3}. FT along *t*_{3} is achieved in the experiment by frequency-resolved detection of the emerging signal. FT along *t*_{1} is more challenging, as the pulses separated by *t*_{1} have to be stable in phase for a meaningful FT. There are several reported methods to achieve such phase stability with atto-second accuracy. The first approach used diffractive optical elements (DOEs) [6] to retrieve two identical pulse replicas and common optics after the DOE to ensure phase stability. Even though DOEs inevitably induce angular dispersion on the excitation pulses, it has been shown [7] that this effect is reversed if the sample is placed accurately in the focal plane of the collimating mirror or lens. There are several experimental designs for 2D-ES, avoiding the use of DOEs, such as active phase stabilization [8], pairwise beam manipulation [6, 9], or pulse-shaper assisted layouts, either in a fully linear design [10], in the PP-geometry [11, 12], or based on diffractive pulse shaping [13] in folded boxcars geometry. Pulse-shaping approaches to 2D-ES bear many advantages, such as fast data acquisition based on rotating frame detection or direct measurement of the desired real (absorptive) component of the 2D-signal.

One limitation imposed by the PP-phase-matching geometry is the exclusion of double-quantum 2D-signals (2Q-2D). 2Q-2D-signals are useful in determining many-body interactions in semiconductors [14] or for describing the energy levels of molecules [15] and aggregates [16]. 2Q-2D spectra require a FT along *t*_{2}, which is not possible in standard realizations of 2D-ES in PP-phase-matching geometry [11] as phase locking is not maintained along*t*_{2}.Nevertheless, the ability to retrieve absorptive spectra directly is a key advantage for PP-geometry 2D-experiments, as a process called phasing is avoided: In non-collinear phase-matching geometries, the 2D-signal is only measured as its amplitude. To separate real (absorptive) and imaginary components, the signal’s phase has to be determined by heterodyne detection with a local oscillator pulse (LO) [17]. This procedure called phasing relies on the so-called projection slice theorem [1],stating that for a common value of *t*_{2}, the purely absorptive component of a 2D-spectrum, integrated along ω_{1}, equals the absorptive component of ω_{1}-integrated techniques such as PP or TG. PP is typically used for phasing purposes as it inherently measures the absorptive component when ω_{3} is resolved. Thus, when using a non-collinear phase-matching geometry, it is necessary to record a PP-signal for all measured *t*_{2} times. In many cases the notoriously weak PP-signal creates a bottleneck for all 2D-experiments not using phase cycling schemes. Alternatively, the absorptive component of a TG-signal can be employed for phasing with the advantage that the TG-signal can be amplified by an independently tuned LO as for 2D-signals. In this article, we show how a directly measured absorptive component of a heterodyned TG-signal (het-TG) is an excellent alternative to PP for phasing purposes. The increased signal to noise ratio allows for the detection of signals down to the 0.5% level in a molecular J-aggregate (PVA/C8O3) [4]. Furthermore, the described experimental design retains a fully non-collinear folded boxcar phase-matching geometry, in which both single- and double-quantum 2D signals can be measured.

## 2. Results and discussion

Figure 1 shows the schematic experimental design. The incoming 570 nm/8 fs pulse is generated by a home-built non-collinear parametric amplifier (NOPA) [19] and is split by a 50:50 beamsplitter (BS). The transmitted pulse is translated by a computer controlled linear motorized stage to introduce delay *t*_{2}. Both pulses from the BS are then focused by separate spherical mirrors (SM1, radius of curvature = 750 mm) onto separate DOEs.

Two two-inch spherical mirrors (SM2, radius of curvature = 600 mm) are used to collimate the diffracted beams. Section A-A (Fig. 1) shows the employed phase-matching geometry. Pulses along *k*_{3}and LO + , LO- are 0th and 2nd order diffractions from the variable DOE, while pulses along *k*_{1} and *k*_{2} are 1st order diffractions from the static DOE (Holoeye). A pair of 2° fused silica wedges (WP2), one of which is mounted on a linear motorized stage, is used to introduce delay *t*_{1} between pulses along *k*_{1} and *k*_{2}. All other beams pass identical static wedge pairs (WP1 and WP3) to balance dispersion. The intensity of all five beams can be independently adjusted by variable neutral density filters (not shown). All beams derived from a common DOE, i.e.,*k*_{1}, and *k*_{2} and *k*_{3}, LO + and LO-,respectively, are steered by the same optics to maintain phase locking between the pulses. Over a 15 min. period, the phase stability was determined to be λ/60. This value compares well to a more compact experimental design from our group [20] with λ/90 for the same time interval.

In order to suppress unwanted low-frequency contributions, it is advantageous to record 2D-spectra at a sufficiently large signal-LO-delay, such as 540 fs in the present case. If the LO is to be used as a probe in a PP (het-TG)-experiment however, the delay should be set to zero. We achieve this by a custom made substrate (Hellma Optics), placed in the path of the excitation beams as shown in section B-B of Fig. 1. For het-TG and PP-measurements, the plate is positioned to have the same amount of glass in *k*_{1}, *k*_{2}, and *k*_{3} and the LOs, and therefore 0 fs delay between all pulses. For 2D scans, the plate is moved to a position with 400 µm less glass in *k*_{1-3}, which lets the LOs precede the excitation pulses by 540 fs.

The three excitation beams are arranged in a triangular geometry to generate 3rd order signals in the direction of the LOs. For pulses separated and ordered in time, the rephasing signal (*k*_{I} = -*k*_{1} + *k*_{2} + *k*_{3}) co-propagates with LO + , while the non-rephasing signal (*k*_{II} = *k*_{1}-*k*_{2} + *k*_{3}) emerges along LO-. This allows for the full 2D-spectrum (defined as *k*_{I} + *k*_{II}) to be measured using one of two methods: (i) delaying only *t*_{1} to negative values (via WP2 in Fig. 1) and collecting the signals along LO + (*k*_{I}) and LO- (*k*_{II}); or, similar to previous setups [21], measuring along a single phase-matching direction and scanning *t*_{1} to negative values to obtain *k*_{I }and *t*_{1} and *t*_{2} to positive values for *k*_{II}. We note that the same results cannot be achieved in a square phase-matching geometry as it requires the LOs be of different diffractive orders of the DOEs and brings upon asymmetric phase-matching between *k*_{I}, LO + and *k*_{II}, LO-. In addition to the single-quantum signals *k*_{I} and *k*_{II}, the double-quantum signal *k*_{III} = *k*_{1} + *k*_{2}-*k*_{3} can be measured along LO- for positive delays of the pulse along *k*_{1}.

The desired absorptive component of a 2D-spectrum is retrieved by employing the above mentioned projection slice theorem [1]. In its two-dimensional form, the projection slice theorem states that the projection of a function in frequency space under an angle ϕ and a slice at the same angle ϕ in the time-domain are Fourier-transform pairs. Within the framework of 2D-ES, the 2D-function in frequency-domain is the 2D-signal${S}_{\text{2D}}\left({\omega}_{1},{t}_{2},{\omega}_{3}\right)$at a given value of *t*_{2}, while the time-domain function is the 3rd-ordersignal${\widehat{S}}^{(3)}\left({t}_{1},{t}_{2},{\omega}_{3}\right)$, a complex quantity, which are related by

The projection for ϕ = 0 leads to *t*_{1} = 0. Equation (1) becomes

The left hand side of Eq. (2) is the projection of ${S}_{\text{2D}}\left({\omega}_{1},{t}_{2},{\omega}_{3}\right)$onto the ω_{3}-axis while the signal for *t*_{1} = 0 on the right hand side is proportional to transient grating signal${\widehat{E}}_{\text{TG}}({t}_{2},{\omega}_{3}),$resolved in detection frequency ω_{3}. ${\widehat{E}}_{\text{TG}}({t}_{2},{\omega}_{3})$is complex valued an cannot be measured directly in experiment. To retrieve the desired$\Re \left[{\displaystyle {\int}_{-\infty}^{\infty}{S}_{\text{2D}}\left({\omega}_{1},{t}_{2},{\omega}_{3}\right)d{\omega}_{1}}\right],$it is common practice to replace${\widehat{E}}_{\text{TG}}({t}_{2},{\omega}_{3})$with the spectrally resolved PP-signal $\Delta {I}_{\text{PP}}\left({t}_{2},{\omega}_{3}\right)\propto {\widehat{E}}_{\text{TG}}{\left({t}_{2},{\omega}_{3}\right)}^{*}{\widehat{E}}_{\text{Pr}}\left(\omega \right)+c.c.,$ where${\widehat{E}}_{\text{Pr}}\left(\omega \right)$is the probe field in frequency domain. $\Delta {I}_{\text{PP}}\left({t}_{2},{\omega}_{3}\right)$ relates to Eq. (2) by

$\Delta {I}_{\text{PP}}\left({t}_{2},{\omega}_{3}\right)$is readily measured in the experiment depicted in Fig. 1 by blocking the pulses along *k*2 and *k*3 and using the pulses along *k*1 and LO + as a pump and probe respectively. The pulse along LO- can serve as a reference pulse if delayed by an additional glass plate (not shown). However, employing $\Delta {I}_{\text{PP}}\left({t}_{2},{\omega}_{3}\right)$for phasing as shown in Eq. (3) brings upon the above mentioned difficulties of low signal strength and can be avoided in the presented setup by measuring the absorptive component of^{${\widehat{E}}_{\text{TG}}({t}_{2},{\omega}_{3}).$This is achieved by heterodyned detection of ${\widehat{E}}_{\text{TG}}({t}_{2},{\omega}_{3})$via a local oscillator field${\widehat{E}}_{\text{LO}}.$}

The last line in Eq. (4) stems from the relations $\Delta {I}_{\text{PP}}\left({t}_{2},{\omega}_{3}\right)\propto {\widehat{E}}_{\text{TG}}{\left({t}_{2},{\omega}_{3}\right)}^{*}{\widehat{E}}_{\text{Pr}}\left(\omega \right)+c.c.$$=2\Re \left[{\widehat{E}}_{\text{Pr}}{(\omega )}^{*}{\widehat{E}}_{\text{TG}}({t}_{2},{\omega}_{3})\right]$and${\widehat{E}}_{\text{Pr}}(\omega )={\widehat{E}}_{\text{LO}}(\omega ).$In connection with Eq. (3) this leads to

This shows that a heterodyne detected transient grating signal is proportional to the spectrally resolved pump-probe signal (Eq. (4)) and can be used for purposes of phasing. For Eq. (5) to be useful, the absorptive component of the het-TG signal (right hand side of Eq. (5)) has to be determined experimentally. This procedure, described in the following, was first presented by Donaldson et al [22]. The authors measured absorptive het-TG spectra with UV excitation and IR-probing, whereas in this work we employ all-visible excitation pulses and focus on the advantages of absorptive het-TG signals with respect to phasing of electronic 2D-spectra.

In changing from pump-probe to het-TG, we are replacing an auto-heterodyned signal with a heterodyned one. In other words, the relative phase between pump and probe is irrelevant to$\Delta {I}_{\text{PP}}\left({t}_{2},{\omega}_{3}\right)$while for het-TG, the phase difference between${\widehat{E}}_{\text{TG}}({t}_{2},{\omega}_{3})$and${\widehat{E}}_{\text{LO}}(\omega )$affects their heterodyne signal${I}_{\text{het}}({t}_{2},{\omega}_{3}).$Considering all experimentally relevant phase terms, we obtain for the measured intensities along LO + and LO- for *t*_{1} = 0 fs [22, 23]

In Eq. (6), the local oscillator and signal fields along LO + and LO- are assumed to be equal. ϕ_{DOE} is changed by moving the variable DOE into and out of the drawing plane of Fig. 1. ϕ_{Win} stems from an adjustable glass window in the path of the pulse along LO-. Rotation of this window will insert more or less glass. At *t*_{1} = 0 fs, pulses along *k*_{1} and *k*_{2} become interchangeable, leading to equal signal fields ${\widehat{E}}_{\text{TG}}({t}_{2},{\omega}_{3})$ along LO + and LO-. Splitting ${\widehat{E}}_{\text{TG}}({t}_{2},{\omega}_{3})$into absorptive and dispersive components gives ${\widehat{E}}_{\text{TG}}({t}_{2},{\omega}_{3})={E}_{Disp}+i{E}_{Abs}$. After trigonometric manipulation, the difference between *I*_{LO+} and *I*_{LO-} then becomes

In Eq. (7)it can be seen if ϕ_{Win} is chosen to be zero, the dispersive signal components vanish. The desired *E*_{Abs} can be further enhanced by optimizing sin(ϕ_{DOE}-ϕ_{TG}). This means that the difference signal in Eq. (7) yields the absorptive components of a het-TG signal in the same fully non-collinear phase-matching geometry used for electronic 2D-measurements. On a daily basis, the absorptive het-TG signal is optimized via the following steps:

- 1.
*t*_{1}= 0 fs and calibration of WP2 are determined by interference on a pinhole. This procedure was described in detail previously [5] and is crucial to the present optimization, as the signals along LO + and LO- are only equal for*t*_{1}= 0 fs, turning into rephasing (*k*_{I}) or non-rephasing signals (*k*_{II}) for*t*_{1}≠ 0 fs. - 2. The homodyne TG-signals, i.e.,${\left|{\widehat{E}}_{\text{TG}}({t}_{2},{\omega}_{3})\right|}^{2}$ along LO + and LO- are set equal by choosing appropriate detection areas on the CCD-chip. This step ensures that the het-TG signals along LO + and LO- are equal, which is a stated prerequisite for the validity of Eqs. (6) and (7).
- 3. The measured LO-intensities ${\left|{\widehat{E}}_{\text{LO+,LO-}}(\omega )\right|}^{2}$ are set equal to ${\left|{\widehat{E}}_{\text{TG}}({t}_{2},{\omega}_{3})\right|}^{2}$ by variable neutral density filters to ensure maximal fringe visibility in heterodyned detection.
- 4. For
*t*_{1}<< 0 fs, possible differences between the LO-intensities along LO + and LO- are subtracted as a background measurement. - 5. The global maximum of the difference signal
*I*_{LO+}-*I*_{LO-}is found iteratively by systematically varying both the position of the variable DOE (ϕ_{DOE}) and the rotation angle of the glass window (ϕ_{Win}). We note, for the desired zero phase differences, the signals along LO- and LO + are mirror images of each other and therefore out of phase. For strongly modulated signals, it is often easier to optimize this property rather than the overall amplitude of signal difference, though experimentally challenging.

An example of the data quality achieved using this procedure is given in Fig. 2.

Figure 2(a) shows a comparison between a PP-signal of PVA/C8O3 at *t*_{2} = 100 fs and an absorptive het-TG signal recorded at the same excitation pulse energy of 0.7 nJ per pulse (1.4 × 10^{13} Photons/cm^{2}).The PP-signal was still discernible from noise at a maximal signal of 6 ΔmOD. At the same energy level, the absorptive het-TG trace shows the same spectral features, but an improvement in signal to noise ratio by almost two orders of magnitude (86). Accordingly, the energy per pulse can be reduced to 0.2 nJ (3.9 × 10^{12} Photons/cm^{2}), while maintaining a relatively noise free absorptive het-TG signal. Comparison of the 2D-signal’s projected absorptive component and the corresponding absorptive het-TG-signal shows that the projection slice theorem holds (Fig. 2(b) and Eq. (5)). Phasing against absorptive het-TG instead of PP enables the phase of the 2D-signal to be determined with significantly higher accuracy. As an example, we plot the absorptive component of the 2D-signal from PVA/C8O3 at *t*_{2} = 100 fs in Fig. 3.

The strongest features in the 2D-spectrum in Fig. 3 are the positive ground state bleach and stimulated emission peaks near the diagonal line shown in black. Details of the underlying dynamics can be found elsewhere [24]. The ability to accurately determine the 2D-signal’s phase makes it possible to analyze low-intensity-features. The cross peak between the first (16750 cm^{−1})and the third peak (17400 cm^{−1}) in the absorption spectrum can be resolved clearly down to the 0.5% signal level. At such low signal levels, pulse propagation and beam geometry effects start to take effect and can be investigated quantitatively in the presented experiment [25].

## 3. Summary

Het-TG is an excellent alternative to notoriously weak PP-signals with respect to phasing. The experimental layout presented offers reduced data acquisition time by the simultaneous detection of *k*_{I} and *k*_{II}, along with the possibility to measure *k*_{III}-signals. The last point is an advantage over 2D-experiments in PP-geometry. The low excitation pulse energies possible in the presented setup are especially beneficial for molecular aggregates, as high-intensity effects such as exciton-exciton annihilation are avoided.

## Acknowledgment

The authors acknowledge helpful discussion with Paul M. Donaldson and funding by the Austrian Science Fund (FWF): START project Y 631-N27.

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