1. INTRODUCTION

Multidimensional spectroscopy is one of the nonlinear spectroscopic techniques that visualizes the correlation between multiple quantum transitions of a molecular ensemble on a multidimensional time or frequency space [1,2]. 2D electronic spectroscopy (2DES) and 2D infrared spectroscopy are two representative multidimensional spectroscopic techniques for condensed-phase materials [3–15], but due to the rapid advancements in laser technologies, coherent multidimensional spectroscopy has utilized a broad range of radiations from THz [16,17] to X-ray [18]. Multidimensional spectroscopy can resolve congested and averaged spectral features often observed in condensed-phase 1D spectra and provide far richer information on molecular systems. In the case of the 2D spectroscopy, the interferometrically measured signal is determined by the transition frequencies, their correlations and fluctuations, and transition dipole moments. In particular, the correlation between the transitions at excitation and detection frequencies can be monitored by measuring the third-order signal with respect to the so-called waiting time, which is defined as the time from the impulsive photoexcitation. Moreover, the fluctuation of the transition frequency, dipole, and population exchange between the electronic or vibrational states affect the line shape of the 2D spectral peaks and their amplitudes. Therefore, 2D spectroscopy has been widely used in studying, for example, protein structure [11], hydrogen bond interaction and dynamics in liquid water [10,12], charge or energy transfer [13], solvation dynamics [10,19,20], and photochemical systems [21].

 

Fig. 1. (a) Two pulse trains from frequency-stabilized mode-locked lasers with different repetition rates. (b) The time interval $T$ between two laser pulses, which is determined by the repetition frequencies of the two lasers, increases linearly in the laboratory time $t$.

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Often, a wide observation time window is needed to monitor photoinitiated chemical, physical, or biological processes in condensed phases because typical photoexcited materials undergo multi-step parallel or sequential reactions and relaxations with different time scales before returning to the ground state. For example, the solvation dynamics, which plays a significant role in the charge transfer within a single molecule consisting of both donor and acceptor moieties or between two nearby molecules [22], has three components with different time scales, i.e., hundreds of femtoseconds, several picoseconds, and tens of picoseconds [23]. In time-resolved spectroscopy, a photoinitiated process can be investigated by observing the transition frequency and the transition dipole moment of an impulsively photoexcited chromophore by varying the waiting time $T$. Since the spectral response generated by the photoinitiated process can be measured by an optical probe pulse, $T$ can be experimentally resolved by controlling the arrival time difference between the excitation (pump) pulse and the probe pulse. Typically, femtosecond-to-picosecond time delay is generated by adjusting the optical pathlength difference between optical pulses, while electric delay generator is utilized to generate time delay longer than nanoseconds. Using a wide range of light sources and spectroscopic instruments employing different time-delay generation methods, chemical events occurring on very different time scales have been investigated.

When two pulsed lasers with different repetition frequencies are detected at a point, the arrival time difference between the two pulse trains increases progressively as multiples of the minimal time interval determined by the two repetition frequencies (Fig. 1). This automatic time-delay generation scheme is known as asynchronous optical sampling (ASOPS) [24]. The scan rate of ASOPS is proportional to the detuning frequency, and the scan range is equivalent to the repetition periods (${\gt}{{10}}\;{\rm{ns}}$) of the lasers. These characteristics of ASOPS enable rapid measurements of time-resolved multidimensional spectra with an exceptionally wide observation time window. In this tutorial paper, we introduce the theory and experiment of ASOPS-based time-resolved coherent multidimensional spectroscopy.

2. CONVENTIONAL MULTIDIMENSIONAL SPECTROSCOPY

Time-domain multidimensional spectroscopy utilizing ultrashort pulses is based on the $N$th order optical response, where $N$ is the number of light–matter interactions prior to the generation of the nonlinear signal field from the sample, necessitating up to $N + {{1}}$ femtosecond pulses to record the nonlinear response on an $n$-dimensional frequency space ($n\; \le \;N$). The optical layout of $n$-dimensional spectroscopy is schematically illustrated in Fig. 2(a). In a typical $N$th order multidimensional spectroscopy, the $N + {{1}}$ pulses are produced by a fundamental laser beam and multiple beam splitters. $N$ pulses interact with a given molecular system with nonlinear optical properties to generate an $N$ th-order signal. The signal is collinearly detected with the $(N + {{1}}){\rm{th}}$ pulse, which is called a local oscillator (LO), at a photodetector so that one can extract information on the spectral components and their phases of the signal field. The number of optical pulses can be less than $N$, depending on the demanded dimensionality. For example, two-dimensional electronic spectroscopy (2DES) typically utilizes four optical pulses (two pump pulses, probe, and LO), while pump-probe spectroscopy requires only two (pump and probe) pulses, even though both techniques belong to the same group of third-order nonlinear spectroscopy.

 

Fig. 2. (a) Optical layout of $N$th order spectroscopy with multiple time-delay generators; DG: delay generator; BS: beam splitter; PD: photodetector; LO: local oscillator. Transparent green squares indicate beam splitters. (b) Schematic representation of 2DES experiment. Each 2D spectrum is plotted with respect to the excitation and detection frequencies. The time-resolved 2D spectra correspond to the series of 2D spectra for varying waiting time $T$, which provide information on the molecular structure, chemical exchange, state-to-state energy transfer, molecular rotation, and so on, in real-time. The red line represents the time profile of the 2D ES signal at a given frequency point in the 2DES spectrum.

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In third-order spectroscopy, the first and the third light–matter interactions generate coherences or superposition states of molecular eigenstates, which are formally defined as the off-diagonal elements of the system density matrix and can be regarded as an ensemble of electronic transition dipoles oscillating with frequencies determined by the energy gap of the two eigenstates involved. In contrast, the second light–matter interaction between the second pump pulse and material generates population (diagonal elements of the density matrix), both an electron in the excited state and a hole in the ground state [1,25]. The oscillation frequencies of the first and second coherences are interrogated by varying the time delays between the two pumps (${\tau _1}$) and between the probe and LO (${\tau _2}$), respectively. The Fourier frequencies of ${\tau _1}$ and ${\tau _2}$ are defined as excitation frequency ${\omega _{\rm{exc}}}$ and detection frequency ${\omega _{\rm{det}}}$, respectively. The time evolution of the population generated by the second light–matter interaction is represented as a function of the time delay between the second pump and the probe called the waiting time $T$. The spectral information along ${\omega _{\rm{det}}}$ can be directly measured without ${\tau _2}$ scan using a dispersive spectrometer, which is a combination of dispersive optics (e.g., grating or prism) and photodetector array, e.g., charge-coupled device (CCD). A conventional data-acquisition strategy of 2D spectroscopy is to record ${\omega _{\rm{det}}}$-resolved signals at fixed ${\tau _1}$ and $T$. Finally, a 3D data set is obtained after the Fourier transformation over ${\tau _1}$ after 2D mechanical scan for ${\tau _1}$ and $T$ [Fig. 2(b)].

Femtoseconds or picoseconds of time delay between two optical pulses are typically generated by controlling the optical pathways of the two pulses generated by a single light source, e.g., mode-locked laser. One of the two pulses passes through an optomechanical delay generator to control the optical pathlength of the pulse. The delay generator consists of a retroreflector and a translational stage [DGs in Fig. 2(a)]. Because a delayed pulse travels the delayed optical pathway twice, the time delay (${T_D}$) between the two pulses is given as ${T_D} = {{2}}(x - {x_0})/v$, where ${x_0}$ is the position of the translational stage where the two pulses overlap. $v$ is either the phase velocity for optical interferometry (i.e., when ${T_D} = \;{\tau _1}$ or ${\tau _2}$) or the group velocity for waiting time scan (i.e., when ${T_D} = \;T$). Piezo translators are widely used for interferometric time-delay generation, while translational stages with DC or stepper motor are used for controlling waiting time.

Although the accuracy and precision of currently available mechanical stages for time-delay scannings are sufficiently high for femtosecond experiments, it is still challenging to study the entire photochemical and photophysical events occurring before the full recovery of photoexcited systems using a single instrument. First, the spatial overlap condition between the pump and probe or any two interfering beams depends on the optical pathlength generated by the mechanical stage because no light source can be a perfect plane wave. The changes in wavefront and beam size by a long optical-pathlength generation (e.g., ${\gt}{\rm{ns}}$) may not affect pump-probe signal significantly. However, it becomes detrimental for transient grating-type experiments [26] like 2DES because the diffraction property of the generated signal field strongly depends on the wavefront of the probe beam. Second, if a wide $T$ window has to be scanned with a small increment for high time resolution, it takes a long time when using a single mechanical delay generator. This issue is critical for vibrational wavepacket analysis of time-resolved multidimensional spectra [27] since it requires collecting extremely high signal-to-noise ratio data with a few to tens of femtosecond time resolution. To overcome such disadvantages of mechanical time-delay scanning approaches, various data acquisition methods have been proposed to develop substantially improved multidimensional spectroscopic techniques [28]. In the next section, an optical time-delay generation method that is useful for developing multidimensional spectroscopy with a wide dynamic range of $T$ and an ultrafast data acquisition speed is introduced.

3. ASYNCHRONOUS OPTICAL SAMPLING: CONCEPT AND EXAMPLES

When the repetition rates of two mode-locked lasers (MLs) are not identical (${f_{r,0}}$ and ${f_{r,0}} + \Delta {f_r}$), the pulse-to-pulse time delay between the two pulse trains linearly increases as a function of laboratory time [Fig. 3(a)]. The detuning (difference) repetition frequency $\Delta {f_r}$ between the two MLs determines the repetition rate of the automatic time-delay generation. Then, two pulses from ${{\rm{ML}}_1}$ and ${{\rm{ML}}_2}$ would overlap every ${\rm{1/}}\Delta {f_r}$, because $\Delta {f_r}$ is the beating frequency of the mixed intensity profiles of the two MLs. This automatic optical time-delay generation method has been referred to as asynchronous optical sampling (ASOPS). Here, the increment of the time interval between the two pulse trains ($\Delta T$) is determined by the repetition period difference between the two MLs as

 

Fig. 3. (a) Two pulse trains from repetition-frequency-stabilized lasers 1 and 2 with slightly different repetition rates, ${f_{r,0}}$ and ${f_{r,0}} + \Delta {f_r}$. The pulse-to-pulse delay time is ${\rm{1/}}({f_{r,0}} + \Delta {f_r})$ for Laser 1, whereas that of Laser 2 is ${\rm{1/}}{f_{r,0}}$. Therefore, the time delays between pulses from the two lasers are multiples of $\Delta T$ that is determined by ${f_{r,0}}$ and $\Delta {f_r}$ [see Eq. (1)]. (b) Schematic diagram of repetition rate stabilization processes of the two lasers.

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Equation (1) shows that $\Delta {f_r}$ is the only parameter determining the time-delay increment $\Delta T$ with fixed ${f_{r,0}}$ and with ${f_{r,0}}\; \gg \;\Delta {f_r}$. Therefore, the time-delay scanning rate ($\Delta {f_r}$) should be reduced for a high time-resolution experiment, e.g., femtosecond time-resolved measurements. $D$ ($\equiv \Delta {f_r}/{f_r}$) in Eq. (1) is the factor that down-converts the repetition period (${\rm{1/}}{f_r}$) to $\Delta T$ so that it is called the down-conversion factor of all the ASOPS-based experiments. If the phase of each optical pulse, called the carrier-envelope phase, is consistent or varies linearly over time, ASOPS can be employed for interferometry that is sensitive to the optical phase. Such ASOPS scheme requiring coherent phase is called coherent ASOPS (cASOPS), while ASOPS utilizing only time-delay generation is called incoherent ASOPS (iASOPS) [29].

The ASOPS scheme was initially used and applied for pump-probe (PP) experiments by Elzinga et al. [24,30]. They utilized two 80 MHz MLs with 10 kHz repetition-rate detuning, i.e., ${f_{r,0}} = {{80}}\;{\rm{MHz}}$ and $\Delta {f_r} = {{10}}\;{\rm{kHz}}$, to record electronic relaxation dynamics of a near-IR laser dye, which completes within the repetition period of the MLs (${\rm{1/}}{f_r}\;\sim\;{12.5}\;{\rm{ns}}$). In their experimental condition, the time-delay scan speed, which is defined as optical time delay per laboratory time, is $\Delta T/({\rm{1/}}{f_r}) = {{125}}\;{\rm{ns/ms}}$. To accomplish this time-delay scan speed with a mechanical time-delay generator, one should use a translational stage that moves as fast as a launched bullet because the speed of which should be about 100 m/s. Because of such a fast time-delay scanning speed, the ASOPS scheme has also been applied for PP microscopy [31,32].

A solid-state passive ML, which will be referred to as an ML in this tutorial for simplicity, is a representative light source with exceptionally high stability that has been used in various femtosecond time-resolved spectroscopy or imaging studies. Since an ML is based on the optical cavity, it successively generates femtosecond pulses with its intrinsic repetition period, which is the inverse of the repetition frequency. The cavity length ($L$) of the ML determines repetition frequency ${f_r}$ as ${f_r} = {v_g}/{{2}}\;L$, where ${v_g}$ is the group velocity of the femtosecond pulse inside the cavity. In the original ASOPS-based PP experiment by Elzinga et al., free-operation MLs, which are defined as MLs without repetition-rate stabilization, were employed to record electronic relaxation dynamics. Since typical electronic relaxations of chromophores in solutions are completed within several nanoseconds, $\Delta T$ for characterizing the rates of electronic energy relaxations does not have to be set below the picosecond level. Therefore, for such measurements, repetition rates do not have to be additionally stabilized unless the cavity lengths of the MLs fluctuate as much as hundreds of micrometers (e.g., ${\rm{0}}.{\rm{23}}\;{{{\rm mm}/1.8}}\;{\rm{m}}\;(\delta L/L)\;\sim\;{{10}}\;{{{\rm kHz}/80}}\;{\rm{MHz}}\;(\delta {f_r}/{f_r}))$). However, the free-operation ML is not applicable in ASOPS with femtosecond-level $\Delta T$ because the detuning frequency $\Delta {f_r}$ for femtosecond experiments should be below the fluctuation level of the repetition frequency. In this case, $\Delta T$ is not a constant for each ASOPS scan; thus, time-resolved measurements with femtosecond resolution cannot be achieved with free-operation ML-based ASOPS techniques. This problem was overcome by employing two repetition-rate stabilized MLs and optical triggers [33,34].

According to the inverse relationship between the repetition rate and the cavity length, the repetition rate of an ML can be stabilized by controlling the cavity length $L$ through a feedback loop [Fig. 3(b)]. A multichannel RF synthesizer (or multiple RF synthesizers) generates reference frequencies ($N{{\cdot}}{f_r}$ and $N{{\cdot}}({f_r} + \Delta {f_r}))$. Two servo electronics control the cavity lengths of the two MLs by comparing their repetition frequencies with the reference frequencies [Fig. 3(b)]. Reference frequency should be chosen as a high-harmonic frequency of the target frequency because the feedback loop responds faster at higher frequencies. In this tutorial, multiple repetition-rate-stabilized MLs will be called synchronized mode-locked laser (SM) in short since the lasers share the same frequency reference (e.g., atomic clock).

Figure 4(a) shows the optical layout of an SM-based pump-probe (SM-PP) experiment. The repetition rates of two MLs are stabilized by the repetition-rate stabilizing instrument depicted in Fig. 4(b). A pulse train from the ML with the repetition rate of ${f_{r,{\rm pu}}}$ is utilized as the pump, while the other with ${f_{r,{\rm pr}}}$ serves as the probe. The time delay ($T$) between the pump and the probe is automatically scanned by adopting the ASOPS scheme with ${f_{r,{\rm pu}}}$ and ${f_{r,{\rm pr}}}$ being slightly different from each other. Typically, the repetition frequency of the pump (${f_{r,{\rm pu}}} = {f_r} + \Delta {f_r}$) is set higher than that of the probe (${f_{r,{\rm pr}}} = {f_r}$) to make $T$ directly proportional to the laboratory time $t$ as $T = D{{\cdot}}(t - {t_0})$ for convenience in the subsequent data analysis, where ${t_0}$ is the time when two pulses from different MLs overlap in time and $D$ is the down-conversion factor defined in Eq. (1).

 

Fig. 4. (a) Optical layout of synchronized mode-locked laser-based pump-probe (SM-PP) spectroscopy; NLO: nonlinear optical crystal; PD: photodetector; LPF: electric low-pass filter with cutoff frequency between ${f_r}/{{2}}$ and ${f_r}$. LPF enables recording of pulse-to-pulse intensity changes. (b) Schematic representation of the SM-PP. Green areas indicate nonlinear molecular responses repetitively generated by the train of pump pulses. Blue pulses are the time profiles of sum-frequency generation (SFG) signals generated by the NLO in (a) when the pump and probe pulses overlap in time.

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When the pump beam is focused on an optical sample, the light–matter interaction between the pump-pulse train and the subensemble of interacting chromophores in a sample cell generates periodic absorptive and refractive responses (changes) with the same repetition frequency as ${f_{r,{\rm pu}}}$ of the pump [green areas in Fig. 4(b)]. Due to the automatically generated pump-probe time delay $T = D{{\cdot}}(t - {t_0})$, each probe pulse interacts with the pumped (excited) molecules at different waiting times [see the probe panel in Fig. 4(b)]. By recording the integrated intensities of probe pulses for every repetition period ${\rm{1/}}{f_{r,{\rm pr}}}$, the time-dependent molecular response can be measured in a time up-converted manner [data panel in Fig. 4(b)]. Here, two factors must be considered to measure the integrated intensity of each probe accurately. First, the sampling frequency of the data recorder (e.g., digitizer) must be synchronized to ${f_{r,{\rm pr}}}$. Otherwise, $T$ cannot be correctly conveyed to the data recorder. Second, the bandwidths of the data recorder and photodetector should be between ${f_{r,{\rm pr}}}/{{2}}$ and ${f_{r,{\rm pr}}}$. If one of the bandwidths is lower than ${f_{r,{\rm pr}}}/{{2}}$, the pulse-to-pulse intensity change will be blurred or filtered out. Although detection and recording bandwidths (${\gt}{f_{r,{\rm pr}}}$) do not affect the data quality, the time dependence of the probe intensity can be easily integrated out if the bandwidths are lower than ${f_{r,{\rm pr}}}$.

The time zero of a given time-resolved spectroscopy experiment is defined as the time when pump and probe pulses overlap in time at the sample position in the sample. In conventional pump-probe experiments, time zero can be set by decreasing the optical pathlength difference between the pump and the probe lines, measured from the beam splitter at which the two beams are separated to the sample at which the two beams intersect. In SM-PP, however, the pump and probe pulses overlap every ${\rm{1/}}\Delta {f_r}$ regardless of the optical pathlength difference, which means that the time zero of SM-PP should be determined in the time domain. In other words, we need a trigger synchronized to the ASOPS-based pump-probe time-delay scan with the time jitter much less than the time resolution of the instrument.

The cross-correlation signal between the pump and the probe pulses can be utilized as a trigger signal because it indicates when the two pulses overlap. Since femtosecond MLs are used for ASOPS-based experiments, the cross-correlation signal can be generated using a second-order nonlinear optical crystal producing sum-frequency generation (SFG) field, for example. As shown in Fig. 4(a), small fractions of the beams from two MLs are separated from the fundamental beams and focused at a nonlinear optical crystal [NLO in Fig. 4(a)]. By setting a proper trigger level for the SFG signal intensity, it is possible to define the time zeroaccurately for each ASOPS scan. We have shown that the time-jitter of the time zero of an SM-PP experiment with a 12.5 fs time resolution is less than 0.1 fs, which is sufficiently small enough not to affect the desired femtosecond time resolution [35].

Photothermal damage and thermal lensing effect could be problematic in an SM-PP experiment because the repetition frequencies of both the pump and probe lasers are very high, approximately 100 MHz. Non-radiative decay of a photoexcited system is governed by electron–phonon and phonon–phonon couplings. The electronic excitation generated by photoexcitation is converted into nuclear kinetic energies of intramolecular and solvent degrees of freedom through non-radiative relaxations, and this process results in the generation of heat localized around each pumped molecule. Thus, generated excessive heat or multi-photon excitation processes could induce complicated chemical reactions that eventually damage the molecular system under investigation. As a result, the measured signals would be inconsistent in time or even be irreproducible. Furthermore, the local heat at the pumped molecular ensemble in an SM-PP experiment could be accumulated during the measurements due to the high repetition rates (${\gt}{{10}}\;{\rm{MHz}}$) of the utilized MLs. This photothermal damage is particularly deleterious for solution samples.

One of the conventional methods to minimize such photothermal damages to solution samples is to refresh the solution sample before each measurement. Rotating, shaking, stirring, or using flow cells are well-known tricks. In this respect, our SM-PP has an additional capability of minimizing photothermal damage. Since SM-PP utilizes ASOPS, it is possible to block the pump and probe beams during $T$ out of the measurement window using a pair of optical shutters, which is referred to as gated sampling [35]. Although the photothermal degradation of samples could still be accumulated while the optical shutters are open, the total amount of heat accumulated in the sample can be reduced by lowering the duty cycle of the gated sampling. We showed that the degradation rate of bacteriochlorophyll $a$ in solutions could be dramatically suppressed by employing the combination of a flow cell and the gated sampling scheme [36].

Despite a few advantages of the ASOPS-based PP technique over the conventional PP methods, it should be mentioned that ASOPS may not be the ideal tool for studying every problem in photochemistry, photophysics, and photoelectric devices. Although the high repetition rate of MLs enables fast data acquisition speed, optical samples with recovery times slower than the repetition period (${\rm{= 1/}}{f_r}$) cannot be investigated by using SM-PP. According to this characteristic of SM-PP, the appropriate applications of SM-PP would be, for example, electronic relaxation of photoexcited systems without changes in spin multiplicity [35,37,38] and acoustic phonon measurement [39–41]. As briefly mentioned previously, ASOPS could be useful for developing a variety of time-resolved microscopy techniques [31] because of its fast time-delay scanning capability. Moreover, if the ASOPS is employed to record impulsive stimulated Raman scattering, it will be possible to obtain microspectroscopic images of objects, where each detector pixel can be used to measure the electronically nonresonant Raman spectrum at a specific position on the sample object [42,43].

4. DUAL FREQUENCY COMB SPECTROSCOPY

Frequency comb is a special resonator-based light source [44]. The power spectrum of a frequency comb has equally spaced narrow spectral lines, and the peak frequency of each comb line is highly stabilized. Strictly speaking, a repetition-rate-stabilized laser is not a frequency comb because its spectral lines may not be stabilized at fixed peak frequencies. In other words, the frequency comb is a repetition-frequency-stabilized laser with optical phase stability. Due to these unique optical phase-preserving characteristics, the frequency comb has been widely utilized for precision measurements and gas-phase spectroscopy [45]. When two frequency combs with slightly different repetition rates are collinearly aligned, it is possible to measure the interference fringe between them because the relative phase between the two lasers is fixed during ASOPS. High-frequency resolution spectroscopy across the broad molecular resonance spectrum with a single frequency comb requires a complicated spectrometer. However, ASOPS-based spectroscopy with two frequency combs, called dual-frequency comb (DFC) spectroscopy, can be performed with a Mach–Zehnder interferometer and a single-point detector [46]. Although two stabilized and synchronized laser systems are required, the simple instrumentation of DFC spectroscopy enables a wide range of applications compared to single frequency comb spectroscopy.

In this section, we discuss DFC spectroscopy involving a coherent ASOPS scheme with two combs having slightly different repetition frequencies. Over the past decade, DFC spectroscopy has been widely used to measure the linear spectra of a variety of gas-phase and liquid-phase samples.

If the electromagnetic pulses are described as a product of a continuous carrier wave and an envelope function, the electric field ${\textbf{E}_j}({{\textbf{r}},t})$ of a frequency comb labeled $j$ ($j = {{1}},{{2}}$ for DFC spectroscopy) can be written as a sum of monochromatic plane waves [47]

 

Fig. 5. (a) Optical layout of symmetric intrapulse DFG-based DFC spectrometer; NLO: nonlinear optical crystal; LPF: optical low-pass filter; BS: beam splitter; PD: photodetector; BPF: electric bandpass filter. Note that the two trains of pulses from combs with ${f_{\rm{ceo}}} = {{0}}$ interact with the sample in this case of the symmetric DFC spectroscopy. (b) Power spectra of two frequency combs interacting with a transition line at ${f_c}$. (c) DFC spectrum resulting from the wave-mixing between the two frequency combs in panel b. (d) Electric field representation of two frequency comb lasers. (e) Intensity profile of the total electric field (the summation of the two-frequency comb fields) in panel d. (f) Recorded DFC data at digitizer.

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Absorption lines of materials can be resolved using two frequency combs with different repetition rates through the frequency mixing or beating between the two frequency combs. The down-converted RF frequency ${f_{\rm{RF}}}$ by the frequency mixing can be represented as

The ${f_{\rm{ceo}}}$ values of spectral lines in a repetition-frequency-stabilized laser are identical. Thus, the differential frequency between two spectral lines does not have a ${f_{\rm{ceo}}}$-dependent term, where the differential frequency (${f_{m - n}}$) of two spectral lines with fluctuating ${f_{\rm{ceo}}}$ can be expressed as

As a matter of fact, the generation of a mid-IR frequency comb from a femtosecond laser was proposed in 2004 [49]. Since then, it has been reported that ultrabroadband mid-IR frequency comb can be generated from an ultrabroadband Ti:Sapphire laser [50] and supercontinuum generated at photonic crystal fiber [51] as well. It is also possible to develop a mid-IR source based on combining a fundamental beam source and a frequency-shifted source by photonic crystal fiber [52].

The optical layout of the DFG-based DFC spectroscopy is shown in Fig. 5(a). Two repetition-frequency-stabilized MLs are focused at nonlinear optical crystals [NLOs in Fig. 5(a)], which satisfy the phase-matching condition for the DFG, to generate mid-IR frequency combs with different repetition frequencies. The fundamental beams are reflected or absorbed by an optical low-pass filter [LPF in Fig. 5(a)]. Two mid-IR frequency combs are combined at a beam splitter [BS in Fig. 5(a)] and interfere with each other. After the sample, the power spectra of the two frequency combs contain the spectral information (absorptivity or transmissivity) of the sample, as represented in Fig. 5(b). Since photodetector cannot measure optical frequency, only the down-converted RF frequency would remain after the photodetection [Fig. 5(c)]. The down-conversion factor $D$ in Eq. (1) is applicable for the down-conversion in DFG-based DFC spectroscopy as well. An electric bandpass filter [BPF in Fig. 5(a)] can be utilized to remove unwanted frequency components for better data quality.

From the time-domain viewpoint, the zero ${f_{\rm{ceo}}}$ in the DFG-based DFC results in the fixed carrier-envelope phase of each pulse for a mid-IR frequency comb laser. Due to the repetition-rate detuning $\Delta {f_r}$, the time delay between the two pulse envelopes of the two frequency combs linearly increases as a function of time [Fig. 5(d)]. A photodetector measures the intensity of the total electric field that depends critically on the interference between the two frequency comb fields, in either one or both of which are the signal field attenuated and shifted compared to those of the incident field by the sample according to its susceptibility [Fig. 5(e)]. Finally, the interference fringe between the two frequency combs is recorded in a time-upconverted manner, where the time increment for an optical interference ($n\Delta T$) is converted to the repetition period, ${T_{r,0}} = {\rm{1/}}{f_{r,0}}$ [Fig. 5(f)]. This time up-conversion is equivalent to the frequency down-conversion represented in Figs. 5(b) and 5(c). Because the relative phases between the optical fields from the two lasers should be fixed and stable throughout the measurements, the DFC linear spectroscopy is one of the coherent ASOPS (cASOPS) techniques.

When the optical sample is allowed to interact with only one ML, one can achieve asymmetric DFC spectroscopy, similar to the asymmetric placement of an optical sample in conventional Fourier-transform spectroscopy with both movable and fixed mirrors. The experimental setup of the asymmetric DFC spectroscopy and the power spectra of the two lasers after the sample are shown in Figs. 6(a) and 6(b), respectively. Here, the field–matter interaction of the sample with a train of pulses from ${{\rm{ML}}_1}$ at a repetition frequency of ${f_{r,0}} + \Delta {f_r}$ induces changes in the phase and amplitude of the incident beam. Due to the non-zero frequency-dependent refractive index and extinction coefficient of the sample, the transmitted electric field of ${{\rm{ML}}_1}$ is both phase-shifted and attenuated, respectively, from the original form in Eq. (2) as follows:

 

Fig. 6. (a) Optical layout of asymmetric intrapulse DFG-based DFC spectrometer; NLO: nonlinear optical crystal; LPF: optical low-pass filter; BS: beam splitter; PD: photodetector; BPF: electric bandpass filter. Note that only the train of pulses from Comb 1 with ${f_{\rm{ceo}}} = {{0}}$ interacts with the sample, and the pulses from Comb 2 with ${f_{\rm{ceo}}} = {{0}}$ act as time-upconverting local oscillator pulses in this case of the asymmetric DFC spectroscopy. (b) Power spectra of ${{\rm{ML}}_1}$ (red) and ${{\rm{ML}}_2}$ (purple) after the BS combines the two beams.

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Considering that ${f_{m,j}}$ is given by Eq. (4), we can see that the first and second terms in Eq. (8) have frequency components at integer multiples of ${f_{r,0}} + \Delta {f_r}$ and ${f_{r,0}}$, respectively. On the other hand, the interference term between the transmitted ${{\rm{ML}}_1}$ and the fundamental (reference) ${{\rm{ML}}_2}$ beams contains unique spectral components with frequencies less than one half the free spectral range ${f_r}$ (${=} {f_{r,0}}$) of the optical sampling, as can be seen from Eq. (5) using $n\sim{\rm{1/}}({{2}}D)$, where the inverse down-conversion factor ${\rm{1/}}D$ corresponds to the minimum mode number after which two comb lines from ${{\rm{ML}}_1}$ and ${{\rm{ML}}_2}$ are brought together again, i.e., $({\rm{1/}}D)({f_r} + \Delta {f_r}) = ({\rm{1/}}D + {{1}}){f_r}$, and the factor of 1/2 comes from the need to avoid aliasing effects (Nyquist condition). Therefore, the optical spectral response of the sample in the frequency range from $\Delta {f_{\rm{ceo}}}/D$ to ${f_r}/({{2}}D)$ can be uniquely mapped to the RF region between $\Delta {f_{\rm{ceo}}}$ and ${f_r}/{{2}}$. Furthermore, the FT spectrum of the time-domain interferogram would show a series of bands that repeatedly appear every repetition frequency ${f_r}$. In practice, the oscillating voltage signal from the photodetector passes through an RF bandpass filter. It is amplified and sampled at ${f_r}$ with a high-speed digitizer that is synchronized to ${f_r}$. Only the interferometric signal below ${f_r}/{{2}}$ from the last term in Eq. (8) can be selectively measured through this procedure. In short, to map the entire optical bandwidth of interest $\Delta {f_{\rm{opt}}}$ into radio frequencies, ${f_r}$ should be chosen to satisfy the inequality $D\Delta {f_{\rm{opt}}} \le {f_r}/{{2}}$.

Based on the fast measurement time and molecular fingerprinting characteristics, mid-IR DFC spectroscopy can be applied in various research fields such as atmosphere analysis [53] and biochemical [54] and chemical [55] reaction dynamics.

In the following two sections, we discuss several nonlinear spectroscopy techniques employing multiple repetition-rate-stabilized MLs or synchronized MLs, including pump-probe spectroscopy, 2D electronic spectroscopy, and time-resolved Raman spectroscopy. Most of these methods are based on the third-order optical response of molecular samples to incident radiations. In general, if the synchronized MLs also have comb stability, the third-order polarization produced by the field–matter interactions of chromophores with three applied frequency comb fields can be written as the summation of the polarization components ${\textbf{P}}_{\textit{mpq}}^{(3)}({{\textbf{r}},t})$ with integers $m$, $p$, and $q$ as follows:

Here, ${{\textbf{R}}^{(3)}}({{t_3},{t_2},{t_1}})$ is the third-order response function of the sample, ${{\textbf{S}}^{(3)}}$ is the corresponding susceptibility in the frequency domain, and the eight terms, including the “seven other terms” not shown, explicitly arise from the triple product of ${\cal E}({{\textbf{r}},t}) = [{{\textbf{E}}({{\textbf{r}},t}) + {\rm{c}.\rm{c}}.}]/2$ in Eq. (9). Also counting the triple summation over three comb fields ($j$, $k$, $l = {{1}}$, 2, 3), there exist ${{{3}}^3} \times {{8}}\;{ = }\;{{216}}$ terms contributing to the third-order polarization ${{\textbf{P}}^{(3)}}({{\textbf{r}},t})$. However, imposing a specific phase-matching condition, invoking the rotating wave approximation, using a phase-cycling technique, or putting an appropriate bandpass filter before a photodetector, one can selectively measure a set of particular four-wave-mixing signals. Each component ${\textbf{P}}_{\textit{mpq}}^{(3)}({{\textbf{r}},t})$ contributing to the third-order polarization acts as the source for generating a signal electric field ${\textbf{E}}_{\textit{mpq}}^{(3)}({{\textbf{r}},t})$. One can measure both the phase and amplitude of ${\textbf{E}}_{\textit{mpq}}^{(3)}({{\textbf{r}},t})$ using an interferometric detection scheme. In Section 6, we outline an experimental realization and a theoretical description [56] of the 2D electronic spectroscopy employing synchronized MLs based on the third-order polarization introduced previously.

5. PUMP-PROBE SPECTROSCOPY: TWOFOLD TIME-MULTIPLEXING

The power spectrum of a femtosecond laser that we used in PP experiments is broader than ${{2000}}\;{\rm{c}}{{\rm{m}}^{- 1}}$ (60 THz). It is comparable to or even broader than electronic transition lines of nanoparticles, semiconductors, and molecules in condensed phases. Thus, the spectral evolution reflecting the relaxation dynamics of materials is imprinted in the PP signal. Therefore, the PP signal spectrally resolved is useful for distinguishing and monitoring the time evolutions of distinct transition lines associated with different chemical species or states. In conventional PP spectroscopy, a dispersive spectrometer, which is the combination of a dispersive optical element (prism or grating) and a photodetector array, is widely utilized to record spectral information of a PP signal at fixed $T$. The readout rate of the photodetector array is less than 1 kHz (for charge-coupled devices). It is too slow to be utilized in ASOPS-based experiments because the data recording rate should be synchronized to the repetition rate of the probe ML (e.g., 80 MHz).

Interferometer is also a type of spectrometer that is used to measure the spectroscopic response of optical samples by allowing the signal to interfere with a reference field. Similarly, PP signals can be resolved using an interferometer. The interferometric PP spectroscopy (PPS) can be achieved by using an LO that is split from the probe beam before the sample and is collinearly detected with the probe after the sample. PP spectrum at a fixed PP-time delay ($T$) can be constructed by Fourier transforming the interference fringe generated by the LO and probe fields [57]. Because interferometric spectroscopy requires single-point detection with one data recorder, it costs substantially less than the dispersive spectrometer. Nevertheless, the use of an interferometric spectrometer is not preferred due to its slow data acquisition time. Although each interferometric data point can be recorded while the translational stage is moving, the stage velocity must be sufficiently slow, considering the repetition frequency of the light source, the bandwidth of the detector, and the maximum sampling rate.

In general, the mechanical time-delay scan rate is even lower than the kHz level of readout rate of dispersive spectrometers, and this relatively slow scan speed can be exploited for the interferometric SM-PP spectroscopy (SM-PPS). Figure 7(a) schematically represents the optical layout of our SM-PPS. As in conventional interferometric PPS [57], a fraction of the probe beam with the repetition frequency of ${f_{r,{\rm pr}}} = {f_{r,0}}$ is separated before the sample, which is used as LO. The LO beam is combined with the probe beam after the sample. If the scan speed for the time delay (${\tau _2}$) between the probe and LO pulses is sufficiently slow, the time-dependent PP response can be assumed to be measured at a fixed ${\tau _2}$ owing to the rapid ASOPS-based $T$ scan. In other words, 2D data acquisition with respect to ${\tau _2}$ and $T$ can be achieved during only one ${\tau _2}$ scan. Such a fast 2D time-delay scan is called twofold time multiplexing (TTM). The recorded data show an interference fringe containing periodic $T$-dependent PP signals [Fig. 7(b)], where each signal separated by ${\rm{1/}}\Delta {f_r}$ corresponds to different ${\tau _2}$ values. Finally, 2D PPS data can be constructed after reshaping the raw 1D data into 2D array data and then Fourier transforming it with respect to ${\tau _2}$ [Fig. 7(c)] [58].

 

Fig. 7. (a) Optical layout of interferometrically detected SM-PPS; LO: local oscillator; PD: photodetector. (b) Schematic representation of typical raw SM-PPS data (solid black line). Linear response (red dashed line) is overlaid for comparison. (c) Final SM-PPS data of a laser-dye (IR125) solution measured with SM-PPS [58] on femtosecond (bottom) and nanosecond (top) time windows.

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Figure 7(c) shows the PP spectra of a laser-dye (IR125) solution measured with SM-PPS [58]. The PP spectrum in the nanosecond window (top panel) is measured with $\Delta T = {{200}}\;{\rm{fs}}$ ($\Delta {f_r} = {1.28}\;{\rm{kHz}}$ and ${f_r} = {{80}}\;{\rm{MHz}}$). The disappearance of the spectral response represents the depopulation of the electronically excited state and the recovery of the ground state. The PP spectrum in the femtosecond window (bottom panel) is measured with $\Delta T = {{4}}\;{\rm{fs}}$ ($\Delta {f_r} = {25.6}\;{\rm{Hz}}$). The dynamic redshift represents the solvation dynamics of the photoexcited state, while the oscillatory feature can be attributed to the vibronic coupling associated with the photoexcitation [59].

Time-delay scanning speed of ASOPS, ${v_{\rm{ASOPS}}}$, is defined as one repetition period multiplied by repetition-rate detuning, ${v_{\rm{ASOPS}}} = \Delta {f_{r/}}{f_{r,0}}$. In general, ${f_{r,0}}$ is fixed, and thus $\Delta {f_r}$ is the only variable determining this speed, ${v_{\rm{ASOPS}}}$. Since $\Delta {f_r}$ is proportional to $\Delta T$ according to Eq. (1), $\Delta T$ should be carefully chosen by considering (i) the maximum time resolution of the experiment and (ii) the fastest molecular dynamics of interest. Condition (i) is determined by the characteristics of the light source, while condition (ii) is set by the sample.

For practical twofold time-multiplexing applications, ASOPS and the interferometric scan must be independent of each other. The mechanical scan speed for the ${\tau _2}$ scan should be set by considering ${v_{\rm{ASOPS}}}$ and the observation time window ${T_{\rm{obs}}}$. However, the two time-delay scannings cannot be orthogonal, so the linear interferometric signal would influence SM-PPS data as shown in Fig. 7(b). To make the effect of linear response negligible, the observation window (desired scan range) ${T_{\rm{obs}}}$ and stage velocity ${v_{\rm{stage}}}$ should be set as follows to make $T$ and ${\tau _2}$ almost independent of each other:

The left-hand side of Eq. (11) represents the travel range of the mechanical stage while ASOPS generates a pump-probe time delay of ${T_{\rm{obs}}}$. The right-hand side of Eq. (11) is a quarter of the lowest detection wavelength. ${v_{\rm{stage}}}$ is the only adjustable parameter in Eq. (11) because the other parameters are predetermined by the sample properties.

As illustrated in Fig. 7(a), the probe beam needs to be focused and then collimated to make the signal field interfere with the LO in SM-PPS. This focusing-and-collimation process makes it difficult to match the wavefront of the probe beam to that of the LO beam due partly to the imperfection of optics, uncompensated astigmatism, and unavoidable scattering from the sample cell. Therefore, it is practically difficult to obtain high visibility for the probe-LO interferogram, so that the presence of an intense DC component, which is proportional to the total intensity of the detected electric field, restricts the use of a preamplifier with limited amplified voltage. The DC component and slowly fluctuating ${\tau _2}$-independent components can be eliminated by AC coupling. It should be noted is that every AC filter has a finite cut-on frequency, ${f_{\rm cut - on}}$, which must be set lower than the optical frequency down-converted by the mechanical ${\tau _2}$ scan. The latter can be calculated by multiplying ${v_{\rm{stage}}}$ and the lowest optical frequency (${\nu _L}$) of the system,

Absorptive PPS, also called transient absorption spectroscopy, records the absorbance difference before and after a pump pulse interacts with the sample. Similarly, the PPS interferogram at $T \lt{{0}}$ provides the real and imaginary parts of the reference spectrum. The accuracy of the measured relative phase between the probe and the LO fields during a $T$ scan is important in interferometric PPS. Since the phase accuracy is undermined mostly by the mechanical instability (e.g., vibrations of opto-mechanical devices for the probe and LO beams), their adverse effects could be minimized by a rapid ASOPS scan.

The fast data acquisition speed of SM-PPS enables the measurement of the PP spectra with an extremely high signal-to-noise ratio due to a vast number of averaging. Such a high signal-to-noise ratio may not be necessary for the general purpose of time-resolved spectroscopy, where just the peak intensity is needed. However, coherent vibrational wavepacket analyses of SM-PPS and multidimensional spectroscopy data require signals with very high signal-to-noise ratios. Coherent vibrational wavepacket represents intramolecular nuclear motions of a subensemble of molecules under observation [27]. The photophysical or photochemical process perturbs the propagation of the photoexcitation-induced vibrational wavepacket. The coherence of vibrational modes coupled to the photoinitiated process would be redistributed [60] or decoherenced [61]. Thus, coherent vibrational wavepacket analysis has been proven to be helpful in studying the reaction dynamics of various photochemical and photophysical processes in condensed phases. The amplitude of oscillating coherent wavepackets is typically 10 times smaller than electronic PP signal amplitude, which is why the PP signals with extremely high signal-to-noise ratio are required to identify those wavepacket oscillations. Although increasing the pump pulse energy would enhance the signal-to-noise ratio, it often causes unwanted signals that could be attributed to higher-order responses, coherent artifacts, solvent non-resonant Raman scattering, or photodamage [62,63]. ML as the light source of SM-PPS provides moderate pulse energy (${\sim}{\rm{nJ}}$) and a high repetition rate. Therefore, SM-PPS can be used to study coherent vibrational wavepackets efficiently due to its fast data acquisition speed and moderate pulse energy.

Table 1. Characteristics of Conventional (Step Scan-Based) 2DES Techniques (Boxcars and Pulse-Shaping) and SM-2DES

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In addition to the fast data acquisition speed, SM-PPS has other advantages over conventional PPS techniques with a dispersive spectrometer. Interferometric measurement of SM-PPS in the time domain provides both real and imaginary parts of the pump-probe signal field, which are respectively attributed to the absorption and refraction properties. The transient refractive index cannot be measured using a dispersive spectrometer unless the relative phase of LO with respect to the probe is precisely controlled. Although the real part of the complex linear susceptibility can be deduced from its imaginary (absorptive) part using the Kramers–Kronig relation, both the transient refractive index and absorption in the time domain could be measured simultaneously by employing an interferometric detection method as demonstrated by us. Furthermore, it should be mentioned that, because the refractive index is related to the density of materials, the photothermal dynamics such as heat expansion and dissipation can be studied using the interferometric SM-PPS in the future.

6. MULTIDIMENSIONAL SPECTROSCOPY

In this section, two state-of-the-art nonlinear spectroscopy techniques utilizing synchronized MLs (SM) are introduced: SM-two-dimensional electronic spectroscopy (SM-2DES) and SM-time-resolved Raman spectroscopy.

2DES requires three-dimensional data acquisition that results in ${N_1}{N_2}{N_T}$ data points as briefly explained in Section 2, where ${N_1}$ and ${N_2}$ are the numbers of data points for the ${\tau _1}$- and ${\tau _2}$ scans, respectively, and ${N_T}$ is the number of $T$ data points. This implies that the data acquisition time should be the product of the single-point data acquisition time (${T_{\rm{DAQ}}}$) and ${N_1}{N_2}{N_T}$. Conventional 2DES techniques often utilize the combination of a dispersive spectrometer and a photodetector array for rapid and effective data acquisition since that combination enables recording one-dimensional spectral information (e.g., detection frequency) without an interferometric ${\tau _2}$-delay scan. As a result, the ${N_2}$ dependence is eliminated by using a dispersive spectrometer so that the data acquisition speed would be ${T_{\rm{DAQ}}}{N_1}{N_T}$, where ${T_{\rm{DAQ}}}$ is the readout time of the photodetector array in this case (Table 1).

In SM-2DES, a complete $T$ scan is done in ${\rm{1/}}\Delta {f_r}$, so that the total 2DES data acquisition time (${T_{\rm{DAQ}}}{N_1}{N_2}{N_T}$) should be replaced by ${N_1}{N_2}/\Delta {f_r}$ (Table 1). In a typical 2DES experiment for condensed-phase materials, the time windows of ${\tau _1}$ and ${\tau _2}$ scans do not have to be long due to short electronic dephasing times (${\lt}\;{{20}}\;{\rm{fs}}$) [64], where ${N_1}$ and ${N_2}$ values of about 50 are sufficiently large for interferometrically resolving excitation and detection frequencies. In contrast, large ${N_T}$ (e.g., ${N_T} \gt {{500}}$) is often necessary for $T$ scan to observe the vibrational coherence due to vibronic coupling-induced excitations of vibrations in an electronically excited state or the time evolution (lifetime) of a photoexcited system with high detection sensitivity. Therefore, considering these aspects, we showed that the data acquisition time of SM-2DES (${N_1}{N_2}/\Delta {f_r}$) is shorter than that of conventional 2DES techniques (${T_{\rm{DAQ}}}{N_2}{N_T}$) [36]. Here, ${T_{\rm{DAQ}}}$ of conventional 2DES techniques is usually comparable to or longer than ${\rm{1/}}\Delta {f_r}$ (${\sim}$ ${{26}}\;{\rm{ms}}$) because ${T_{\rm{DAQ}}}$ includes the communication time between the instruments and computer and/or the travel time for the mechanical time delay. Similarly, as will be shown later in this section, the data acquisition speed of the time-resolved Raman spectroscopy can be dramatically enhanced using multiple repetition-rate-stabilized MLs.

A. 2D Electronic Spectroscopy

2DES is a four-wave-mixing technique measuring the third-order response of materials [3]. From the data interpretation point of view, 2DES can be considered as an excitation-frequency-resolved PPS. Excitation frequency resolvability is necessitated for the systems where the experimentally excited states are spectrally indistinguishable (e.g., weakly coupled molecular excitons or nearly degenerate excitons).

Cundiff and coworkers experimentally demonstrated a frequency-comb 2D electronic spectroscopy (FC-2DES). They employed two [65,66] or three [67] frequency combs to record the 2D spectral response of Rb atomic vapor and quantum-well materials, respectively. In FC-2DES, all the interacting fields (two pumps and probe) propagate collinearly through the sample so that the four-wave-mixing signal is generated along the same direction of the incident beams. The 2DES signal was separated in the down-converted frequency domain using a frequency modulation technique [65]. On the other hand, conventional multidimensional spectroscopy, including our ASOPS-based nonlinear spectroscopic techniques, used spatial phase-matching conditions, e.g., ${{\textbf{k}}_{\rm s}} = - \;{{\textbf{k}}_{{\rm p}1}} + {{\textbf{k}}_{{\rm p}2}} + {{\textbf{k}}_{\rm{pr}}}$ in Fig. 8(a), to measure the signal selectively. In SM-2DES, the scanning of two electronic coherence times and subsequent Fourier transformation yield the 2D electronic spectrum that displays the correlation between electronic excitation and emissive transitions.

 

Fig. 8. (a) Optical layout of SM-2DES; LO: local oscillator; PD: photodetector. The wavevector matching equation in this figure is for the measurement of rephasing signal. The delay time between the two pump pulses from the pump laser is scanned using a translational stage. Likewise, the delay time between the probe pulse and LO pulse is also scanned using another translational stage. However, the waiting time, which is the time delay between the second pump pulse and the probe pulse, is automatically scanned by employing the ASOPS scheme. (b) 2DES spectra of bacteriochlorophyll $a$ in propanol at waiting times of 5, 50, 500, and 5000 fs [36]. The $x$ and $y$ axes are the excitation and detection wavelengths in nm.

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In the cases of the photochemical and photophysical processes of highly disordered condensed-phase materials, the electronic transition linewidth often exceeds the spectral bandwidth of light sources because of ultrashort electronic dephasing times (${\lt}{{100}}\;{\rm{fs}}$). Then, the high optical frequency resolution of FC-2DES is not advantageous not only because of its low duty cycle (coherence time/ASOPS ${\rm{scan}}\;{\rm{range}} = {{100}}\;{\rm{fs}}/10 \;{\rm{ns}}\;{ = }\;{{10}^{- 5}}$) but also because of an under-sampling issue when one wants to enhance the data acquisition speed. Therefore, for molecular systems with broad spectral linewidths, the iASOPS-based 2DES technique could be of a better use for measuring the full 3D spectral responses.

The optical layout of our SM-2DES is illustrated in Fig. 8(a). Two interferometric time delays for excitation and detection frequencies [${\tau _1}$ and ${\tau _2}$ in Fig. 8(a)] are generated by two optomechanical delay generators, while the pump-probe time delay ($T$) is scanned employing the ASOPS scheme. $T$ and ${\tau _2}$ are simultaneously scanned using the same TTM technique as in the case of the interferometric SM-PPS. A single $T - {\tau _2}$ scan is executed at a fixed ${\tau _1}$, and then ${\tau _1}$ is varied discretely for each $T - {\tau _2}$ scan, i.e., step-scan mode. Therefore, the data acquisition time, ${t_{\rm{DAQ}}}$, for measuring a complete three-dimensional 2DES dataset is given as the product of the number of ${\tau _1}$-points, ${N_{\tau 1}}$, and the $T - {\tau _2}$ scanning time ${t_{\rm{TTM}}}$. Practical ${t_{\rm{DAQ}}}$ with 800 nm centered 7-fs optical pulses is approximately 8 min (${N_{\tau 1}} = {{70}}$ and ${t_{\rm{TTM}}} = {{7}}\;{\rm{s}}$). It should be noted that ${t_{\rm{DAQ}}}$ is independent of the number of recorded $T$ points. Thus, it is several times faster than even the most recently demonstrated fast 2DES-measurement technique [28] for long waiting time measurements (e.g., ${N_{T}} \gt {{1000}}$).

There are two critical factors that affect the quality of SM-2DES signals. First, the fluctuation of ${\tau _1}$ that results from the instability of pathlength difference between the two pump pulses can be detrimental to the data quality. This is mainly determined by the mechanical stabilities of the optomechanics in a given SM-2DES setup. In order to minimize the effect of the optomechanical fluctuations largely attributable to vibrations, normal reflection angle of mirror on tilted translational stage [68] or wedge pair [69] have been employed for the fine controlling of time-delay generation with improved ${\tau _1}$-scanning stability. The second factor is the low accuracy of ${\tau _2}$ scanning. However, this issue may not be critical if the stage controller can record the absolute position of the ${\tau _2}$-dependent interferogram with the sampling rate synchronized to $\Delta {f_r}$. However, the absolute position of ${\tau _2}$ is usually uncertain due to the bidirectional repeatability and non-uniform acceleration of the stage. For the relative phase correction, we employed a reference Mach–Zehnder interferometer. The reference interferometer consists of the beams from the probe laser separated before and after the ${\tau _2}$ stage. The reference interferogram is recorded with the same trigger signal used for the ${\tau _2}$ scan and with the sampling rate of $\Delta {f_r}$. The relative positions of the reference interferogram and a $T$-multiplexed ${\tau _2}$ scan dataset are fixed during the entire measurement so that the relative phase among different ${\tau _2}$ scan datasets can be accurately measured.

As an application of SM-2DES, we studied the excited state dynamics of bacteriochlorophyll $a$ (BChl $a$) in solutions. The fluorescence lifetimes of BChl $a$, which is a photosynthetic pigment of bacterial photosynthetic complexes, and bacterial light-harvesting complex II (LH2) are about 1.5 ns [36] and 3 ns [70], respectively. Since the fluorescence lifetimes of these molecular systems are sufficiently shorter than the repetition period of typical MLs (12.5 ns), they are appropriate systems to be studied using SM-2DES. Although the photooxidative degradation is an important issue for both BChl $a$ and LH2, it has been shown that the stability against the photodegradation can be dramatically improved by employing the combination of gated sampling and sample cooling [36]. Consequently, the excited-state dynamics of BChl $a$ solution [Fig. 8(b)] [36] and LH2 [71,72], where we could study coherent vibrational wavepacket oscillations, could be successfully investigated with SM-2DES.

B. Theory of 2DES with Synchronized MLs

Here, we sketch a brief theoretical description of the 2DES spectroscopy using synchronized MLs whose optical configuration is shown in Fig. 8(a) [56]. We first specify the three incident ML fields and the LO field for heterodyne detection as follows:

The third-order polarization induced by the incident field ${\cal E}({{\textbf{r}},t}) = [{{E_{p1}}({{\textbf{r}},t}) + {E_{p2}}({{\textbf{r}},t}) + {E_{\rm{pr}}}({{\textbf{r}},t}) + {\rm{c}.\rm{c}}.}]/2$ is obtained using Eq. (9), of which only six terms have the matching phase factor ${e^{i{{\textbf{k}}_s} \cdot {\textbf{r}}}}$. Furthermore, considering the time ordering of the three pulses within a time span of $0 \le t \lt1/({2{{\Delta}}{f_r}})$, which well encompasses typical lifetimes of relevant electronically excited states, only one of the six terms is expected to contribute predominantly to the observed signal for waiting times longer than the incident pulse duration [56]. The polarization from this term is given by

If the signal field ${E^{(3)}}$ is much weaker than ${E_{{\rm{LO}}}}$, the last term in Eq. (15), representing the interference between the two fields, can be separated by ignoring the first term and subtracting the second term that is a known quantity. If this is not the case, the first two terms can be removed by Fourier transform and high-pass filtering of the measured intensity with respect to ${{{\tau}}_1}$ and ${{{\tau}}_2}$ because only the last term depends on both time variables [36]. The interference signal can then be written as

This signal oscillates along the measurement time $t$ with the frequency given by

If the experimental setup is designed to detect only the slowly oscillating interference terms in the RF domain with a low-pass filter or slow-response detector, the terms with non-zero coefficients of ${{{\omega}}_{r,0}}$ are discarded due to the relation ${{\Delta}}{{{\omega}}_r} \ll {{{\omega}}_{r,0}}$ and only the terms in Eq. (16) with the following values of $n$ and associated frequency factors are detected:

With this, the heterodyne-detected signal $I({{\tau _2},t,{\tau _1}})$ can be written as follows:

This equation confirms that we obtained a rephasing signal because the first and third frequency arguments of ${S^{(3)}}$ representing coherence oscillation frequencies have opposite signs (note that, because ${a_{\textit{nj}}}$ in Eq. (20) is sharply peaked at $n = {{0}}$, only terms with ${-}L{{{\omega}}_{r1}} + \omega _2^{L + N} \approx {\omega _{c2}}$ and ${-}\omega _1^M \approx - {\omega _{c1}}$ contribute dominantly to the signal).

To obtain a 2D spectrum with explicit dependence on the waiting time $T$, we rewrite the triple summation in Eq. (19), approximating the summation over $L$ by the integral over a frequency variable ${{\omega}}$, which corresponds to $L{{\Delta}}{{{\omega}}_{\rm{r}}}$ and performing Fourier transforms over ${\tau _1}$ and ${\tau _2}$. After a lengthy derivation [56], the 2D spectrum ${S_{2{\rm{D}}}}({{{{\omega}}_{{{{\tau}}_2}}},t,{{{\omega}}_{{\tau _1}}}})$ is obtained as

C. Time-Resolved Impulsive Stimulated Raman Spectroscopy

Time-resolved impulsive stimulated Raman spectroscopy (TR-ISRS) is a fifth-order spectroscopy technique that can be used to monitor the structural evolutions of photoexcited molecules [73–75]. Although TR-ISRS has not been considered to be two-dimensional spectroscopy, it should be noted that, among various time-resolved Raman spectroscopy techniques, TR-ISRS is a technique requiring a two-dimensional time-delay scan [76]. TR-ISRS utilizes pump–dump–probe geometry, where the three beams spatially overlap at the sample [Fig. 9(a)]. In TR-ISRS, the pump, dump, and probe pulses are referred to as the actinic pump, Raman pump, and Raman probe, respectively. An actinic pump initiates a photochemical reaction of interest, and a Raman pump generates an impulsive Raman response after a certain time delay (${T_1}$) from the actinic pump pulse. The impulsive Raman response generated at ${T_1}$ is monitored by the Raman probe with varying the time delay between the Raman pump and probe (${T_2}$).

 

Fig. 9. (a) Simplified optical layout of SM-TR-ISRS; PD: photodetector. The actinic pump, Raman pump, and Raman probe pulses were generated from three synchronized repetition frequency-stabilized MLs with different repetition frequencies. Therefore, the SM-TR-ISRS involves two iASOPS schemes with employing TTM scheme. (b) Schematic representation of raw SM-TR-ISRS data. (c) TR-ISRS signal of IR144 methanol solution [77]. (d) Fourier transform of (c) with respect to ${T_2}$.

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The two time delays (${T_1}$ and ${T_2}$) in TR-ISRS can be scanned by adopting the TTM scheme with one ASOPS and one mechanical stage similar to interferometric SM-PPS. The scan type (ASOPS with a mechanical time-delay generator versus dual ASOPS) for each time delay should be determined based on the expected scan length of ${T_1}$. Dual ASOPS should be chosen for a time delay with a longer scan length, while the other is scanned by a translational stage. For example, if the structural evolution of a photoexcited chromophore in solution is of interest, the scan length of ${T_1}$ should be about three times the excited-state lifetime, and that of ${T_2}$ is about three times the vibrational coherence time. For molecules in the condensed phase, vibrational coherence time is less than 10 ps, while electronic relaxation completes in the nanosecond time scale. Because the scan length of ${T_1}$ is much longer than ${T_2}$, ${T_1}$ should be scanned by employing the iASOPS scheme while ${T_2}$ is scanned mechanically.

It is also possible to use a dual-iASOPS scheme [Fig. 9(a)] for TR-ISRS with three SMs instead of using the TTM method with one mechanical time delay. For the application of the dual-iASOPS approach to TR-ISRS, the repetition rates of the actinic pump, Raman pump, and probe lasers should be ${f_{r,0}} + \Delta {f_r} + \delta {f_r}$, ${f_{r,0}} + \Delta {f_r}$, and ${f_{r,0}}$, respectively [77]. ${T_2}$ is scanned by the detuning frequency of $\Delta {f_r}$, where the corresponding time increment is given by Eq. (1) as $\Delta {T_2} = \Delta {f_r}/f_r^2$. ${T_1}$ is assumed to be fixed during a single ${T_2}$ scan, so that the increment for each repetition period ($\delta {T_1}$) of ${T_1}$ should be small enough compared to $\Delta {T_2}$. Then, the ${T_1}$ increment during a single ${T_2}$ scan is given by

The TR-ISRS signal then shows oscillatory features associated with vibrational wavepackets of Raman-active modes of the electronically excited state over ${T_2}$ at each ${T_1}$ position, as schematically illustrated in Fig. 9(b). For 2D analysis, the one-dimensional data should be reshaped into a 2D space consisting of ${T_1}$ and ${T_2}$ after removing the non-oscillating trajectories and ${T_1}$-independent (data at ${T_1} \lt{{0}}$) third-order oscillating signals [Fig. 9(c)]. Time-resolved Raman spectrum is finally obtained by Fourier transforming the oscillating time profiles with respect to ${T_2}$ for varying ${T_1}$ [Fig. 9(d)]. Figures 9(c) and 9(d) are the TR-ISRS data of IR144 dissolved in methanol and measured with three repetition-rate-stabilized lasers [77].

Here, the two detuning frequencies $\Delta {f_r}$ and $\delta {f_r}$ should be chosen by considering the time resolutions and Nyquist frequency, much like interferometric SM-PP. While the time jitter for ${T_2}$ scan is small enough not to affect the time resolution of ${T_2}$ [35], the time jitter for ${T_1}$ scan should be estimated before the measurement because it critically depends on the performance of electric instruments (e.g., piezo translators and servo electronics for feedback loops) and the intrinsic stability of the MLs. The uncertainty of ${T_1}$ can be measured by accumulating the histogram of the time difference between cross-correlation functions between the actinic pump and Raman pump and between the actinic pump and probe with $\delta {f_r} = {{0}}$ (fixed ${T_1}$). In our previous study demonstrating the first TR-ISRS with dual-iASOPS TTM, the ${T_1}$ uncertainty was about 10 ps, so we set $\delta {f_r} = {{77}}\;{\rm{mHz}}$ to obtain $\Delta {T_1} = {25.07}\;{\rm{ps}}$ with ${f_r} = {{80}}\;{\rm{MHz}}$ and $\Delta {f_r} = {38.4}\;{\rm{Hz}}$ [77]. According to $\delta {f_r}$, the data acquisition time for a single 2D (${T_1}$ and ${T_2}$) dataset was about 13 s. This fast data acquisition of TR-ISRS cannot be achieved with other mechanical stage-based TR-ISRS techniques.

An optical trigger generated by the pulse trains from the Raman pump laser and probe laser provides the absolute ${T_2}$ reference (e.g., ${T_2} = {{0}}$) for each ${T_2}$ scan. Here, another optical trigger, generated by the SFG of two pulse trains from the actinic pump and Raman pump lasers, is needed to define ${T_1} = {{0}}$ and to trigger the data acquisition with respect to the ${T_1}$ scan. Note that the sampling rates and detection bandwidths of the two optical triggers are different. The ${T_2}$-trigger is synchronized to the data recording system such that the sampling rate is ${f_r}$ (e.g., 80 MHz), and the detection bandwidth should be higher than ${f_r}/{{2}}$. An additional discriminator is necessary for the ${T_1}$-trigger. Because the uncertainty in triggering times should be less than ${\rm{1/}}\Delta {f_r}$ (e.g., 26 ms), slow-response discriminators (e.g., software) can be utilized.

For a proof-of-principle experiment, we measured the time-resolved ISRS spectrum of a laser dye (IR144) solution, where the data acquisition speed was very high (1 h for recording 280 datasets). We found fast (${\lt}{{50}}\;{\rm{ps}}$) and slow (${\gt}{{300}}\;{\rm{ps}}$) components that are associated with structural evolutions of the excited IR144 after the solvent reorganization dynamics of the system. The dual-iASOPS-based TR-ISRS could also be applied in various photochemical systems (e.g., isomerization [73,78], proton transfer [79], bond-formation [80], and protein structural evolution [81]) widely studied by using the conventional femtosecond time-resolved Raman spectroscopy techniques such as femtosecond stimulated Raman spectroscopy and TR-ISRS.

7. CONCLUDING REMARKS AND PERSPECTIVE

Photochemical reactions and photoinitiated biological functions involve multiple parallel or sequential processes taking place in a wide time window from femtoseconds to microseconds and beyond. Time-resolved multidimensional spectroscopy can monitor the time evolution of the correlations between quantum transitions and thereby enables the identification of chemical species or electronic states involved in a photoinitiated process and the investigation of the reaction kinetics. The time-delay generation method determines the dynamic range, precision, and accuracy of the delay time in most of the time-resolved spectroscopy experiments. Mechanical time-delay generation by varying optical pathlength difference is advantageous for scanning short time-delay ranges (e.g., 10 fs to 1 ns) with high precision. In contrast, an electric time-delay generator is advantageous for scanning a longer time delay (${\gt}{{100}}\;{\rm{ps}}$). However, it isn’t easy to expect consistency between the data sets with different time windows recorded by employing two independent methods. In this regard, ASOPS is considered to be a powerful alternative by enabling time-delay scans over a wide dynamic range with a single instrument and rapid scan rate.

ASOPS-based multidimensional techniques resolve optical frequency interferometrically using single-point detectors rather than array-type spectrometers. The use of single-point detectors not only makes instrumentation simple, but also is advantageous for experiments necessitating multichannel data acquisition. For example, the 45-degree-polarized probe beam in PPS and the transient grating signal in 2DES can be separated using a polarizing beam splitter. The separated beams that are parallel and perpendicular to the polarization direction of the pump can be simultaneously detected by using two identical single-point detectors. Recently, we demonstrated that the polarization-controlled SM-2DES spectroscopy of photosynthetic light-harvesting complex II reveals that the wavepacket dynamics of two different exciton states can be selectively measured and promotes the electronic excitation transfer between the two states [72].

We have demonstrated synchronized ML-based multidimensional spectroscopic techniques and investigated ultrafast excited-state dynamics of molecules [36] and photosynthetic protein complexes [71] in solutions. A key experimental aspect of these studies is that the coherent vibrational wavepackets created by vibronic couplings could be measured with a high signal-to-noise ratio owing to the exceptional data acquisition speed of ASOPS. Similarly, the detailed mechanisms of ultrafast photochemical or photophysical processes can be unveiled by synchronized ML-based multidimensional spectroscopic techniques through coherent wavepacket analysis.

Many molecular systems and functional materials of interest could have excited-state lifetimes that are comparable to or even longer than the repetition period (${\sim}{{10}}\;{\rm{ns}}$) of MLs used in our recent ASOPS time-resolved spectroscopic studies. Therefore, it would be desirable to use relative low-repetition-rate lasers, e.g., long cavity laser system [82] and cavity-dumped laser [83], for those systems with slow relaxation dynamics. Although a pair of low-repetition-rate lasers might not be of immediate use for studying very slow electronic or vibrational coherence due to the limited bandwidth of the currently available feedback method used to control for repetition rate, such frequency-stabilized low-${f_r}$ lasers will be of critical use for the scanning of waiting time in various multidimensional spectroscopic techniques.

Synchronized ML-based experiments have the potential to be applied in the development of various linear and nonlinear spectromicroscopy techniques. DFC-based spectromicroscopy has already been demonstrated by employing frequency-to-space conversion [84,85]. We anticipate that all-optically-controlled nonlinear spectromicroscopy would also become possible using multiple synchronized MLs and combs, and employing appropriate time delay to space conversion schemes.