Optical multidimensional coherent spectroscopy (MDCS) has become a powerful and routine technique for studying optical properties of a wide range of materials. However, current implementations of MDCS have spectral resolution and acquisition speed limitations. In this Perspective, I describe how frequency comb technology can be used to overcome the limitations and also show the recent progress that has been made in this field.

In recent years, optical multidimensional coherent spectroscopy (MDCS)1 has become a routine technique for studying the structure, properties, and ultrafast dynamics of atomic and molecular systems and semiconductor materials.2,3 The technique is based on the concepts of Nuclear Magnetic Resonance (NMR) spectroscopy, which is widely used to study the molecular identity and structure.4 The idea of implementing it in the optical region was proposed by Tanimura and Mukamel in 1993 to probe molecular vibrations,5 and since then, multiple techniques have been developed and extended in different spectral regions.6–14 The technique is now commercially available.

Multidimensional coherent spectroscopy uses a sequence of pulses (typically three) incident on the sample that generates a four-wave mixing (FWM) signal. The emitted signal is heterodyne detected (using a local oscillator pulse) and recorded as a function of the delays between the excitation pulses. In some MDCS experiments, four pulses are used to excite a population state (instead of coherence) and the fluorescence is detected. A multidimensional coherent spectrum is then generated by Fourier transforming the signal (heterodyne detected or the fluorescence) with respect to the time delays between the pulses. Two-dimensional spectra contain rich spectroscopic information and allow the measurement of the following:

  • homogeneous linewidth of inhomogeneously broadened systems,

  • coupling and energy transfer dynamics between the states, and

  • many-body interactions (e.g., long range dipole–dipole interactions).

However, MDCS has limitations. Currently available high resolution MDCS techniques (for example, those implemented with long delay stages) require long acquisition times (minutes or hours), whereas rapid techniques provide much lower spectral resolution (limited by spectrometer resolution or time delays achievable using acousto-optic modulators). For these reasons, until recently, MDCS has been used mostly for samples with short dephasing times. In addition, currently available techniques utilize bulky apparatus that limits their applications outside the research laboratories.

To overcome these limitations, recently there has been a marriage between multidimensional coherent spectroscopy and frequency comb technology.15–17 This new approach allows the measurement of high resolution 2D spectra rapidly, which is important for rovibrational spectroscopy and also for studying the dynamics of atomic and molecular systems. Before I describe this new approach, I will give a short introduction of frequency combs18 and, particularly, dual-comb spectroscopy.19 

A frequency comb typically is a mode-locked laser that outputs a train of ultrashort pulses. The time domain representation of a comb is shown in Fig. 1 (top) where pulses are separated by 1/frep. frep is the repetition frequency of the laser. Each pulse differs from the previous one by a phase increment of Δϕce, which is referred to as the carrier–envelope phase slip and can be calculated using the following equation:

Δϕce=1vgr1vphLcωc,
(1)

where vgr and vph are the group (envelope) and phase (carrier) velocities, Lc is the laser cavity length, and ωc is the carrier frequency.

FIG. 1.

Frequency comb in the time (top figure) and frequency domains (bottom figure).

FIG. 1.

Frequency comb in the time (top figure) and frequency domains (bottom figure).

Close modal

In the frequency domain, a power spectrum of a frequency comb consists of regularly spaced frequencies, as shown in Fig. 1 (bottom). The frequency of each tooth can be written as νn = nfrep + f0, where n is an integer and f0 is the frequency of the very first tooth (n = 0), which is related to the carrier–envelope phase slip with the following expression:

2πf0=Δϕcefrep.
(2)

Since the development of the frequency comb technology, a method known as dual-comb spectroscopy (DCS) has emerged as a very powerful optical method that enables the measurement of a broad absorption spectrum rapidly and with high spectral resolution.19,20 The concept of DCS is similar to Fourier-transform infrared (FTIR) spectroscopy but contains no moving elements. In DCS, one frequency comb (typically a mode-locked laser) is used to excite the sample and the response is sampled in time with another comb (Local Oscillator, LO) that has a slightly different repetition rate. The resulting interferogram is captured by a single photodetector, as shown in Fig. 2(a). In the frequency domain, the DCS arrangement produces a Radio Frequency (RF) comb spectrum that results from these two optical combs beating against each other on the photodetector. The RF spectrum directly maps to the optical absorption spectrum of the sample, as shown in Fig. 2(b).

FIG. 2.

(a) Dual-comb spectroscopy in the time domain. The repetition rates for the signal comb and the local oscillator comb are slightly different by δ. The signal comb is used to probe the sample, and the local oscillator (LO) comb is used to read out the response. (b) Dual-comb spectroscopy in the frequency domain. The radio frequency comb spectrum directly maps to the optical absorption spectrum. f0, sig and f0,LO correspond to the offset frequencies of the signal and LO combs.

FIG. 2.

(a) Dual-comb spectroscopy in the time domain. The repetition rates for the signal comb and the local oscillator comb are slightly different by δ. The signal comb is used to probe the sample, and the local oscillator (LO) comb is used to read out the response. (b) Dual-comb spectroscopy in the frequency domain. The radio frequency comb spectrum directly maps to the optical absorption spectrum. f0, sig and f0,LO correspond to the offset frequencies of the signal and LO combs.

Close modal

To demonstrate powerful features of dual-comb spectroscopy, I will compare it to traditional spectrometers. In Fig. 3 (top), I show the energy level diagram for the rubidium (87Rb and 85Rb) D1 lines that are separated by hundreds of MHz. In Fig. 3 (bottom), the corresponding linear absorption spectrum is shown that was acquired using DCS (two 100 MHz repetition rate combs that differ only by 500 Hz). From the figure, it is clear that the energy levels are fully resolved (except the ones that are overlapped due to Doppler broadening). The spectrum was acquired in 2 ms, and the resolution of the spectrum is 100 MHz. In contrast, 100 MHz resolution is not achievable with traditional diffraction grating or prism-based spectrometers. Obviously, one can achieve 100 MHz resolution using spectrometers based on mechanical stages (1.5 m long), but obtaining the same spectrum in 2 ms is not possible as this would require the stage to move with a speed of 750 m/s.

FIG. 3.

(Top) Energy level diagrams of the D1 lines of 87Rb and 85Rb atoms. (Bottom) Measured linear absorption spectrum. νref = 377.103 258 084 THz. The figure has been adapted from Ref. 15.

FIG. 3.

(Top) Energy level diagrams of the D1 lines of 87Rb and 85Rb atoms. (Bottom) Measured linear absorption spectrum. νref = 377.103 258 084 THz. The figure has been adapted from Ref. 15.

Close modal

The example shows that DCS simultaneously provides high resolution and fast acquisition speed. Because of these qualities, DCS has now been extended to nonlinear spectroscopy as well to probe materials’ nonlinear properties with high resolution. The methods include photon echo spectroscopy,21 coherent anti-stokes Raman spectroscopy,22 and time resolved pump–probe spectroscopy.23 Recently, DCS has been applied to multidimensional coherent spectroscopy, which is discussed in Sec. III.

In this section, I will describe a marriage between dual-comb spectroscopy and multidimensional coherent spectroscopy, which was recently demonstrated.15 A simplified schematic diagram of comb-based MDCS is shown in Fig. 4(a). The experiment uses two combs that have slightly different repetition rates (100 MHz and 100–424 Hz), and they are locked to a direct-digital synthesizer. Phase fluctuations caused by drifts in the offset frequencies and optical path fluctuations are measured, and the signal is used to correct 2D spectra. The output of the signal comb is split into two parts. The offset frequency of one part is shifted using an AOM and is then recombined with the other part whose delay is adjusted by a retroreflector mounted on a mechanical stage. The combined beam is then focused to the sample of interest. The experiment uses the photon echo24 excitation scheme (A*, B, C) shown in Fig. 4(b); however, I note that the method has recently been extended to double-quantum multidimensional coherent spectroscopy as well to study the effects of thermal motion on dipole–dipole interactions in Doppler broadened atomic vapor.25,26 The emitted FWM signal is then interfered with the LO pulses on a photodetector, and the electrical signal is digitized (see Ref. 27 for details about the separation of linear and FWM signals in the RF domain). To generate the second axis, the delay between the excitation pulses varied from 0 to 3.3 ns. Two-dimensional spectra are generated by taking Fourier transforms of the emitted FWM signal with respect to these two time axes.

FIG. 4.

(a) Experimental setup for comb-based MDCS. (b) Photon echo excitation scheme. The first pulse is a complex phase-conjugated pulse (EA*) and excites a coherence between the ground state |g⟩ and the excited state |e⟩. The second pulse (EB) then converts this coherence into a population of the excited (or ground) state, and then, the same pulse (EC) converts this population into the third-order coherence that radiates the FWM signal (photon echo shown in red). (c) A measured two-dimensional spectrum generated by cross-linearly polarized excitation pulses (H stands for horizontal and V stands for vertical). The color scale shows the normalized signal magnitude. The figure has been adapted from Ref. 28.

FIG. 4.

(a) Experimental setup for comb-based MDCS. (b) Photon echo excitation scheme. The first pulse is a complex phase-conjugated pulse (EA*) and excites a coherence between the ground state |g⟩ and the excited state |e⟩. The second pulse (EB) then converts this coherence into a population of the excited (or ground) state, and then, the same pulse (EC) converts this population into the third-order coherence that radiates the FWM signal (photon echo shown in red). (c) A measured two-dimensional spectrum generated by cross-linearly polarized excitation pulses (H stands for horizontal and V stands for vertical). The color scale shows the normalized signal magnitude. The figure has been adapted from Ref. 28.

Close modal

To demonstrate improvements in resolution and acquisition speed, Doppler broadened rubidium (87Rb and 85Rb) atoms, loaded in a thin cell, were used as a sample. The temperature of the cell was set at 110 °C, and the excitation beams were filtered to excite only the D1 lines of rubidium atoms (shown in Fig. 3).

In Fig. 4(c), I show the results of the experiment. The spectrum was acquired using cross-linearly (HVVH) polarized excitation pulses. The negative values on the evolution axis reflect the negative phase evolution during the evolution period [Fig. 4(b)]. The diagonal peaks [along the (0, 0) to (10, −10) GHz line] correspond to absorption (evolution) and emission at the same [(a)–(h)] resonance frequencies, whereas off-diagonal peaks show all possible coupling of [(a)–(h)] resonances. These peaks are elongated due to Doppler broadening. Along the cross-diagonal direction, the broadening is removed (due to photon echo excitation scheme). The spectrum was acquired in under 4 min, and it is clear that the rubidium hyperfine structure is fully resolved. This high degree of resolution (350 MHz) is not achievable using 2D techniques based on diffraction grating spectrometers, and it would take hours or even days to acquire a similar spectrum using the techniques that utilize mechanical stages.

I would like to note that the experiment described above was the first demonstration of comb-based MDCS, and since then, other versions of comb-based MDCS techniques have been developed and applied to semiconductor materials and molecular samples as well.16,21,29 For example, in Refs. 15 and 16, the excitation combs were collinear and the separation of FWM and linear signals was performed in the RF domain. On the other hand, Refs. 21 and 29 used a non-collinear excitation geometry and the FWM signal was detected in the phase matched direction.

In Sec. III, I showed that frequency comb-based MDCS simultaneously provides high resolution and rapid acquisition speed; however, the experimental setup uses a mechanical stage, which is still a limiting factor. To fully leverage the advantages that are provided by frequency comb technology, recently, Lomsadze et al. proposed and demonstrated a novel approach to multidimensional coherent spectroscopy using three frequency combs.17,30 This novel approach, which is called tri-comb spectroscopy (TCS), contains no mechanical moving elements and can measure comb-resolution multidimensional coherent spectra in under half a second.

The experimental setup for tri-comb spectroscopy [shown in Fig. 5(a)] is similar to the one shown in Fig. 4(a), but the stage and the AOM are replaced with another comb. Comb 1, comb 2, and the LO comb have slightly different repetition rates that allow evolution and emission times to be scanned automatically without the use of any mechanical delay line. In the experiment, the phases of the repetition frequencies are locked to a four-channel direct-digital synthesizer (DDS). Path length and offset frequency fluctuations for each comb are measured and corrected. Pulses from comb 1 and comb 2 are used to generate a photon echo. The emitted FWM signal is then sampled after interfering with the LO comb on a photodetector. To optimize the acquisition speed of the experiment, the relative repetition frequency between comb 2 and LO comb (δ2,Lo) is set to be exactly equal to the relative repetition frequency between comb 1 and comb 2 (δ1,2). To understand why this arrangement enables a faster acquisition speed, in Fig. 5(b), I show a cartoon of the magnitude of a photon echo signal as a function of the evolution and emission times. The signal is non-zero only near the diagonal (due to photon echo). The condition δ2,Lo = δ1,2 enables the signal to be sampled along the echo (along the diagonal line) and not in the region where the signal is zero. In order to sample points off the diagonal, the phase of the DDS (serving as the reference for the LO comb) is stepped, which causes time shifts of the LO pulses, and the FWM signal is measured along the lines parallel to the diagonal [white dashed lines, (a)–(e)]. After digitizing the FWM signal, a multidimensional spectrum is generated by calculating a two-dimensional Fourier transform with respect to t′ and τ′.

FIG. 5.

(a) Experimental setup for tri-comb spectroscopy. (b) Cartoon showing the magnitude of a FWM signal as a function of emission and evolution times. (c) A measured two-dimensional spectrum generated using 365 ms data records. The color scale shows the normalized signal magnitude. The figure has been adapted from Refs. 17 and 28.

FIG. 5.

(a) Experimental setup for tri-comb spectroscopy. (b) Cartoon showing the magnitude of a FWM signal as a function of emission and evolution times. (c) A measured two-dimensional spectrum generated using 365 ms data records. The color scale shows the normalized signal magnitude. The figure has been adapted from Refs. 17 and 28.

Close modal

To demonstrate the resolution and acquisition speed improvement that can be achieved with TCS, the measurement shown in Fig. 4(c) was repeated. The results are shown in Fig. 5(c). The two-dimensional spectrum is tilted by 45° to show the spectrum in the νt and ντ coordinate system. The diagonal resolution of Fig. 5(c) is still the same as the resolution of Fig. 4(c), but the cross-diagonal resolution of Fig. 5(c) is now four times higher. The spectrum shown in Fig. 5(c) was generated by 365 ms of data, which show 600 times improvement compared to Fig. 4(c).

In this Perspective, I showed how frequency combs (particularly tri-comb spectroscopy) can revolutionize multidimensional coherent spectroscopy. They provide very high resolution, and 2D spectra can be acquired in seconds. With the help of frequency comb technology, MDCS can now be used to study fundamental processes in cold atomic and molecular systems where energy line splitting is of the order of hundreds of MHz.31,32 However, TCS has limitations and the biggest one is the fact that it requires three independent combs, which is expensive. One might question if TCS can be implemented in research laboratories and furthermore used for practical applications outside the laboratories. I would like to note that in recent years, there has been substantial progress in developing compact, high power fiber and micro-resonator laser sources.33,34 Furthermore, multiple groups have been successful at generating two, three, and even four frequency combs with different repetition rates from a single resonator.35–38 Combs are becoming commercially available, and they are already used for real time applications outside the laboratories.39,40 Combs have also been extended in different spectral regions, particularly in the mid-IR, which is important for chemical sensing applications.40,41 In the near future, TCS can be implemented with cheap and compact apparatus and can become a field deployable device.

Finally, I would like to note that TCS can be used for microscopy and chemical imaging applications to study optical materials at the single-particle level. In addition, TCS can also be extended to quad-comb spectroscopy, which will enable the measurement of the full Hamiltonian of a system of interest with unprecedented high resolution.42,43

The data that support the findings of this study are available from the corresponding author upon reasonable request.

I acknowledge the National Science Foundation (NSF) (Grant No. 1904704).

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