Frequency combs enable rapid and high-resolution multidimensional coherent spectroscopy

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Science  29 Sep 2017:
Vol. 357, Issue 6358, pp. 1389-1391
DOI: 10.1126/science.aao1090

Comb quickly through a spectral zoo

Dual-comb spectroscopy relies on a pair of laser pulses with multiple frequencies distributed like tines in a comb. It is a rapid means of characterizing atoms and molecules in fine detail, but, when applied to complex mixtures, it can produce a sea of peaks that are hard to discriminate. Lomsadze and Cundiff present a protocol to extend dual comb spectroscopy into the nonlinear regime. The cross-peaks that appear in the resulting two-dimensional spectra allow assignment of crowded features to common sources, as demonstrated for an isotopic mixture of 87Rb and 85Rb.

Science, this issue p. 1389


Dual laser frequency combs can rapidly measure high-resolution linear absorption spectra. However, one-dimensional linear techniques cannot distinguish the sources of resonances in a mixture of different analytes, nor can they separate inhomogeneous and homogeneous broadening. Here, we overcame these limitations by acquiring high-resolution multidimensional nonlinear coherent spectra with frequency combs. We experimentally differentiated and assigned the Doppler-broadened features of two naturally occurring isotopes of rubidium atoms (87Rb and 85Rb) according to the placement of their hyperfine energy states in a two-dimensional spectrum.

Optical multidimensional coherent spectroscopy (MDCS) is an extremely powerful technique developed over the past two decades for studying structure and ultrafast dynamics (15). Specifically, MDCS is a nonlinear optical technique based on concepts originating in nuclear magnetic resonance (NMR) spectroscopy (6) that enabled the determination of molecular structure. Its empowering features include the capacity to decouple homogeneous and inhomogeneous linewidths, to identify couplings between the excited states, and to track the energy redistribution (in real time) in complex systems. MDCS uses a sequence of (typically three) laser pulses to excite the sample. A multidimensional spectrum is then generated by calculating Fourier transforms of the signal with respect to the time delays between pulses and the time period during which the signal is emitted. However, current MDCS implementations have long acquisition times (when implemented with mechanical delay stages) and/or limited spectral resolution [>10 GHz, limited by spectrometer resolution or the achievable time delays (711)]. These attributes limit their applications for studying atomic systems and performing molecular fingerprint ro-vibrational spectroscopy. In addition, MDCS has not been used outside of research laboratories because of the bulky arrangements and complex phase-cycling schemes necessary to suppress background linear signals.

Here, we overcame these weaknesses of MDCS by leveraging the technique of dual-comb spectroscopy (DCS) (1214), which has emerged as a revolutionary optical method that enables the rapid acquisition of high-resolution, broad absorption spectra. DCS is similar to traditional Fourier transform spectroscopy (15), wherein the moving mechanical stage, which limits the acquisition speed, is replaced by the use of two combs with slightly different repetition rates. This repetition rate difference provides a scanning linear time delay between the signal pulses, which interrogate the sample, and the local oscillator (LO) pulses. The interference between the signal and LO combs on a photodetector produces a time-domain signal that can be Fourier-transformed to produce a radio-frequency (RF) comb spectrum that directly maps to the optical absorption spectrum of the sample. DCS has been extended into the long-wavelength region (1619) and is used for many practical applications, such as remote sensing and LIDAR (16, 2023). Substantial progress in developing microresonator combs could soon make the DCS technique compact and field-deployable (2426). However, DCS is a one-dimensional optical method, and its measured linewidths suffer from inhomogeneous broadening. In addition, DCS shares the drawback of other one-dimensional optical methods: an inability to distinguish the sources of different resonances in a sample with multiple analytes. The marriage of MDCS and DCS, which we call M-DCS2, allowed us to demonstrate rapid multidimensional coherent spectroscopy with the highest reported spectral resolution to date. We used rubidium atoms at 110°C (which have Doppler-broadened features) as a test sample and assigned the spectra of the two naturally occurring isotopes, 87Rb and 85Rb, according to the placements and couplings of their hyperfine split energy states in a two-dimensional spectrum.

The generation scheme of a four-wave-mixing (FWM) signal in the photon echo excitation sequence (27) is illustrated in Fig. 1A. The first pulse [acousto-optical modulator (AOM)–shifted pulse, shown in black] is a complex phase-conjugated pulse (E*1) that excites a coherence between the ground state and an excited state; the second pulse (delayed in time, shown in blue) converts this coherence into a population of the excited state and then converts this population into the third-order coherence that radiates a FWM signal (shown in red), which is measured. A two-dimensional coherent spectrum is produced by Fourier-transforming the signal with respect to both the delay between the pulses and the time period over which the signal is emitted.

Fig. 1 Experimental setup.

(A) Generation of a four-wave-mixing (FWM) signal in a photon echo excitation sequence. For clarity, the second and third interactions are separated in time; in the experiment, these interactions occur at the same time. |g〉 and |e〉 correspond to the ground and excited states. (B) The LO comb pulses (green) sweeping through the excitation and FWM pulses (red) at different delay times. (C) Two combs with different offset frequencies, produced from a single signal comb by means of an acousto-optical modulator (AOM), interact with Rb atoms and generate a FWM signal in a collinear geometry at different delay. PD, photodetector. The comb structure shown after the sample corresponds to the linear (black, AOM-shifted comb; blue, original comb) and FWM (red) signal comb lines in the frequency domain.

Details of the experimental setup are given in the supplementary materials and in (28). We used two home-built Kerr-lens mode-locked Ti:sapphire lasers centered at 800 nm. The repetition frequencies (Embedded Image = 93.567412 MHz and Embedded Image — 423.45 Hz) of the two lasers were phase-locked to a direct digital synthesizer. The comb offset frequencies were not locked. Phase fluctuations due to drift in the relative offset frequency, as well as optical path and residual repetition frequency fluctuations, were measured and subtracted from the FWM signal (see supplementary materials). The signal comb was split into two parts. An AOM shifted the offset frequency of one part (by 80 MHz) that was then recombined with the other part at a time delay adjusted by means of a retroreflector mounted on a mechanical stage. The combined beam was optically filtered to excite the D1 lines of both 85Rb and 87Rb atoms loaded in a cell 0.5 mm in length. The nonlinear interaction in a photon echo excitation sequence generated a FWM comb signal (Fig. 1A). The emitted FWM signal and the excitation pulses were then combined with the LO comb pulses (shown in green), which had a slightly different repetition frequency, and their interference was measured on a photodetector. The electrical signal from the detector contained both the linear and FWM signals. They were spectrally separated in the RF domain without implementing complex phase-cycling schemes (28). The delay between two excitation pulses was varied from 0 to 3.3 ns (limited by the stage length) with 10-ps steps to generate the absorption frequency axis for a two-dimensional spectrum. Figure 1B displays the evolution of the FWM signal in the time domain. Two-dimensional spectra were constructed by calculating Fourier transforms with respect to these two time axes.

Figure 2 shows the energy level diagram and the measured linear transmission spectrum of the Rb D1 lines. The natural linewidths of the hyperfine split transitions (a) to (h) were ~6 MHz; however, the transmission profile was Doppler-broadened (580 MHz at 110°C) and the hyperfine lines strongly overlapped.

Fig. 2 Rb state structure.

Top: Energy level diagram of D1 lines of 85Rb and 87Rb atoms. Bottom: Measured linear transmission spectrum. The absorption frequency is relative to an arbitrary reference, νref = 377.103258084 THz.

To demonstrate the full capability of our method, we acquired full two-dimensional energy spectra using collinearly (HHHH) and cross-linearly (HVVH) polarized excitation pulses (Fig. 3, A and B) (29). Comparison of two spectra in general helps to determine the level structure and study many-body effects (1). The negative values on the absorption axis reflect the negative phase evolution during the evolution period in the photon echo excitation sequence [in Fig. 1A, the first pulse corresponds to a complex conjugated pulse (E*1) and hence evolves with a negative frequency].

Fig. 3 Experimental and theoretical results.

(A and B) Measured two-dimensional spectra generated by collinearly and cross-linearly polarized excitation pulses. H and V, horizontal and vertical polarizations. Color scale shows normalized signal magnitude. (C and D) Diagonal slices of the spectra in (A) and (B). (E and F) Theoretical calculations showing the strength of the FWM signal generated by collinearly and cross-linearly polarized excitation pulses. νref = 377.103258084 THz; (a) to (h) refer to transitions shown in Fig. 2.

Both two-dimensional spectra provide rich information by comparison to the linear transmission spectrum. The diagonal peaks [along the (0, 0) to (10, –10) line] correspond to absorption and emission at the same [(a) to (h)] resonance frequencies. They are elongated in the diagonal direction as a result of Doppler broadening. However, along the cross-diagonal direction (for each resonance), the inhomogeneity (Doppler broadening) is removed and the line shapes reflect the homogeneous linewidth. Despite broadening of the intrinsic homogeneous linewidth along the cross-diagonal direction (due to the limitations of the scan range achievable with the mechanical stage), we are able to resolve the hyperfine structure and all possible couplings between the resonances that appear at unique locations. For example, in Fig. 3B, the peak near (1.5, –2) GHz corresponds to coupling between two excited states of 87Rb (F′ = 1 and F′ = 2) via the same ground state (F = 2), whereas the peak around (2, –9) GHz shows the coupling of two ground states of 87Rb (F = 1 and F = 2) via the same excited state (F′ = 2). The same peak analysis can be performed on 85Rb to determine all possible couplings between the excited states. It is also clear that the two-dimensional spectra do not show the coupling peaks between 87Rb and 85Rb resonances, indicating that these sources behave as independent atoms. This information is extremely valuable for chemical sensing applications, especially when probing a mixture without prior knowledge of its constituent species. The coupling information allows the decomposition of a cluttered spectrum into the individual spectra of the constituents. For Rb, this decomposition is shown in Fig. 3, A and B, by the white dashed boxes (inner and outer boxes correspond to the spectra of 85Rb and 87Rb, respectively) that can be plotted separately.

Some of the diagonal peaks are suppressed in the HVVH spectrum relative to the HHHH case. In Fig. 3, C and D, we plot diagonal slices of Fig. 3, A and B, respectively. The slices reveal that peaks at frequencies (g), (c), and (a) are suppressed and the peak at frequency (e) is absent for the HVVH case. To understand this behavior, we calculated the strength of each diagonal FWM signal using Clebsch-Gordan coefficients (30) and all possible double-sided Feynman diagrams (31) for each state (including the magnetic sublevels) in both cases in the circularly polarized (σ±) light bases. Our calculations show that for the HVVH case, the FWM signals for F to F′= F transitions have the opposite sign relative to the F to F′ = F ± 1 transitions. This sign flip causes partial cancellation of neighboring peaks. The calculations also show that the signal is zero for the F = 1 to F′ = 1 transition of 87Rb. For the HHHH case, all of the Feynman diagrams have the same sign that results in addition of neighboring peaks rather than cancellation. In Fig. 3, E and F, we show the results of our theoretical calculation for the strengths of the lines (normalized with natural abundance and using the transmission spectrum), which are in good agreement with the experimental results (32).

The two-dimensional plots show additional interesting behavior attributable to the complexity of the level system. The strengths of the off-diagonal peaks are not equal to the geometric mean of the corresponding diagonal peak strengths, as would be expected for a simple three-level system consisting of a single ground state coupled to two excited states. Indeed, some of the off-diagonal peaks are even weaker than their corresponding diagonal peaks. For instance, the peak at (3, –6.5) GHz in Fig. 3A is much weaker than the diagonal peaks at (3, –3) GHz and (6.5, –6.5) GHz, which correspond to the F = 3 to F′ = 3 and F = 2 to F′ = 3 transitions in 85Rb, respectively. This result can be explained in the linear polarization basis (π) by considering the fact that each hyperfine state consists of multiple (degenerate) magnetic sublevels with quantum number mF (five states for F = 2 and seven states for F = F′ = 3). All sublevels of the F = 2 and F = 3 states (in π basis) contribute to the diagonal peaks (except mF = 0 for the F = 3 to F′ = 3 transitions with zero Clebsch-Gordan coefficient); however, only (mF = –2, –1, 1, 2) substates of the F = 2 and F = 3 hyperfine states contribute to the off-diagonal peak. Our theoretical calculations show good agreement with the experimental results for this and other off-diagonal peaks.

Our results show that frequency comb–based multidimensional coherent spectroscopy offers much higher spectral resolution than traditional two-dimensional spectroscopy techniques (15, 711). The M-DCS2 technique could resolve Doppler-broadened spectral features (at 110°C) without implementing the complex laser cooling apparatus, which is applicable only for a few systems (33, 34). We emphasize that this experimental approach provides fast acquisition speed in addition to high resolution, which is crucial for remote sensing applications. In our experiment, a full two-dimensional spectrum was obtained in less than 4 min. In the future, higher acquisition speed could be achieved by using a third frequency comb with a different repetition frequency to replace the AOM and the mechanical stage in the experiment. Three combs would allow us to obtain the same two-dimensional spectrum in a few seconds (35). With the development of compact microresonator combs, this would make the system deployable to the field.

Supplementary Materials

Materials and Methods

Supplementary Text

Fig. S1

References and Notes

  1. The first three letters indicate the polarization state of the excitation pulses; the fourth letter indicates the polarization state of the detected FWM signal. H, horizontal; V, vertical.
  2. We also measured the width (FWHM) of the isolated peak (Fig. 3D, rightmost peak) to be 625 MHz, which is 7% larger than the expected value of 580 MHz at 110°C. We attribute this broadening to the limited spectral resolution in the absorption axis, which is now limited by the length of the mechanical stage (~300 MHz).
  3. This setup would also improve the resolution of the absorption frequency axis up to frep ~ 93.5 MHz (i.e., the resolution of the emission axis). The resolution for both axes can be further improved by implementing the interleaving technique well known in dual-comb spectroscopy experiments, which involves acquisition of spectra by varying either the offset or repetition frequencies.
Acknowledgments: The research is based on work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via contract 2016-16041300005. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. government. The U.S. government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation thereon. The data shown in the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. B.L. and S.T.C. are inventors on provisional patent application 62/394,771 submitted by the University of Michigan that covers “Frequency comb–based multidimensional coherent spectroscopy.”

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