Mass-correlated rotational Raman spectra with high resolution, broad bandwidth, and absolute frequency accuracy

Significance

Mass-correlated rotational alignment spectroscopy (mass-CRASY) is a laser spectroscopic method for the simultaneous characterization of molecular composition and rotational structure in a heterogeneous (impure) sample. The present work extends CRASY by referencing the spectroscopic data via a stable laser oscillator frequency to an external clock. Data for carbon disulfide provide mass-selected rotational spectra for multiple isotopologues with significantly improved spectroscopic accuracy compared to preceding measurements.

Abstract

We present mass-correlated rotational alignment spectroscopy, based on the optical excitation of a coherent rotational quantum wave and the observation of temporal wave interferences in a mass spectrometer. Combined electronic and opto-mechanical delays increased the observation time and energy resolution by an order of magnitude compared with preceding time-domain measurements. Rotational transition frequencies were referenced to an external clock for accurate absolute frequency measurements. Rotational Raman spectra for six naturally occurring carbon disulfide isotopologues were resolved with 3 MHz resolution over a spectral range of 500 GHz. Rotational constants were determined with single-kilohertz accuracy, competitive with state-of-the-art frequency domain measurements.

Correlated spectroscopy combines the information content of multiple spectroscopic measurements into a single multidimensional dataset. It is therefore uniquely suited for the analysis of heterogeneous signals, where a single experiment cannot resolve or assign all signal contributions. Correlated rotational alignment spectroscopy (CRASY) correlates gas-phase rotational Raman spectra with observables from short-pulse laser spectroscopy. Our initial mass-CRASY experiments resolved isotopic heterogeneities and determined isotope-specific rotational constants and fragmentation channels in carbon disulfide (CS₂) and butadiene (1, 2). Here, we describe two developments that greatly enhance the power of CRASY: (i) a method to obtain high-resolution rotational spectra from CRASY experiments and (ii) a method to reference rotational frequencies to an external clock for highly accurate spectroscopic frequency determination.

CRASY is based on time-domain rotational Raman spectroscopy and requires the excitation of a coherent superposition of rotational states and the subsequent observation of the wave packet formed by this state superposition. First related experiments were performed by Heritage et al. (3), who observed the transient birefringence caused by an evolving rotational wave packet in carbon disulfide. Felker (4) and coworkers developed a number of experimental schemes under the label rotational coherence spectroscopy (RCS). RCS was used to analyze the rotational structure of molecules and clusters with small or no dipoles (5–9). The highest resolution RCS experiments scanned opto-mechanical delays up to 2 m (10, 11) and observed rotational wave packets over a time range $\mathrm{\Delta }t\le 6$ ns. The spectroscopic resolution is limited by the Heisenberg energy uncertainty (Fourier limit), $\mathrm{\Delta }E\ge \hslash /\left(2\cdot \mathrm{\Delta }t\right)$, corresponding to less than $100$ MHz in the RCS experiments.

The spectroscopic resolution improves proportionally to the observation time range $\mathrm{\Delta }t$. Order-of-magnitude extensions of opto-mechanical delays, however, are impractical: Even the best commercial Fourier-transform spectrometers reach only a range of about 10 m [e.g., 11.7 m for the Bruker ETH-SLS 2009 FTIR spectrometer prototype at the Swiss Synchrotron Light Source (12)]. This limitation is overcome in the present work: We use discrete electronic delays to select a laser pulse from a stable laser oscillator pulse train, adding discrete nanosecond to millisecond delays with negligible timing uncertainty.

Spectroscopic accuracy does not necessarily scale with resolution but requires a reliable calibration. In the frequency domain, the inception of frequency comb spectroscopy allowed for referencing spectroscopic frequencies directly to a clock (13, 14). The resulting improvement of spectroscopic accuracy had a great impact in the fields of atomic spectroscopy, molecular spectroscopy, and metrology (15). Here, we create a time-domain equivalent to frequency comb spectroscopy by referencing rotational Raman frequencies to an external clock.

Experimental Design

The experimental scheme is illustrated in Fig. 1. Helium gas at 10 to 20 bar pressure was seeded with small quantities of CS₂ and expanded through a pulsed valve operating at 500 Hz. A 1-mm skimmer, placed at 280-mm distance from the valve, skimmed the resulting cold molecular beam to a collimation angle of $0.10^{○}$. In the spectrometer region of a time-of-flight Wiley–McLaren mass spectrometer (16), an IR laser pulse (alignment pulse) crossed the molecular beam and created a coherent wave packet by rotational Raman excitation. The wave packet was probed by resonant two-photon ionization with an UV laser pulse (ionization pulse), ionizing CS₂ via the ${}^1\mathrm{\Sigma }_u^{+}$ state. Ions were detected with a multichannel plate detector, and their time-of-flight was measured in a scaler card with 0.5 ns time resolution.

Fig. 1.

Experimental scheme. Molecules from a cold molecular beam source were excited and ionized by two laser pulses (L1 and L2) in a mass spectrometer. The time delay $\mathrm{\Delta }t$ between the laser pulses was adjusted through an opto-mechanical (OM) and an electronic pulse-selection (PS) delay: Electronic selection of laser oscillator pulses for amplification in amplifier 1 and 2 introduced discrete 12.5-ns delays. A frequency counter monitored the oscillator period and characterized the discrete delays with high accuracy. A mechanical delay stage added $\le$12.5 ns delays to allow continuous scanning from femtoseconds to microseconds.

The laser pulses originated from an amplified 796 nm femtosecond laser system consisting of a single Ti:sapphire laser oscillator (Coherent Vitara-T, 80 MHz repetition rate) and two regenerative amplifiers (Coherent, Libra USP-1K-HE-200, 1 kHz repetition rate), synchronized via electronic timing units. Amplified alignment pulses were compressed to 150–1,500 fs pulse duration and attenuated to 50–150 mJ pulse energy. Amplified ionization pulses were compressed to bandwidth-limited 45 fs pulse duration and were frequency doubled, tripled, and quadrupled with nonlinear beta barium borate crystals to give 0.1 to 0.2 $\mathrm{\mu }$J pulses at 199 nm. Alignment and ionization pulses were jointly focused with a 75 cm spherical mirror onto the molecular beam in the active spectrometer region.

The time delay between alignment and ionization pulses was controlled by coupling selected oscillator pulses into the regenerative amplifiers and by an opto-mechanical delay stage as illustrated in Fig. 1. Distinct oscillator pulses were selected for the alignment and ionization amplifiers. Because both amplifiers received their pulses from the same frequency-stable oscillator, this allowed for adding discrete delays of $\mathrm{\Delta }t=1/$(80 MHz) = 12.5 ns with negligible timing-jitter. A frequency counter (Aim-TTI TF930) monitored the oscillator repetition rate against an inexpensive GPS-stabilized clock (Leo Bodnar GPSDO). The oscillator repetition rate was stable with an Allan deviation below $10^{-10}$, and the accuracy of the frequency measurement was $\mathrm{\Delta }\nu$/$\nu \ll 10^{-8}$, limited by the counting statistics. The alignment beam was routed through a 30 cm opto-mechanical delay stage (Physik Instrumente, MD-531) with a 100 nm internal encoder. The laser beam path was folded 16-fold across the stage to obtain up to 4.8 m (16 ns) adjustable delays with $\ge 5$ fs step size, covering the delay range between oscillator pulses. Drifts in the oscillator repetition rate were corrected with the opto-mechanical delay.

For large alignment-ionization delays, a motorized mirror mount adjusted the position of the alignment laser on the molecular beam to correct for the molecular beam velocity of 1,100 m/s. A ($91.6\pm 0.4$)° angle between molecular beam and laser beams caused a small Doppler shift of $\left(1.0\pm 0.26\right)\cdot 10^{-7}$. Doppler broadening was negligible due to the small collimation angle of our skimmed molecular beam.

A chopper wheel (Thorlabs MC2000) was used to alternatingly measure alignment–ionization signal and ionization-only (reference) signal. The temporal signal modulations (ratio of signal to reference) in selected mass channels were Fourier transformed to obtain rotational Raman spectra. A detailed description of the data analysis protocol for each figure is given in SI Appendix.

The laboratory was temperature and humidity controlled to ${\left(21\pm 0.5\right)}^{○}$C and $\left(45\pm 10\right)\%$ relative humidity Care was taken to avoid oscillatory temperature or humidity fluctuations. During high-resolution measurements, the temperature on the laser table was stable within $\pm 0.05^{○}$C, and we used the National Institute of Standards and Technology shop-floor equation to approximate the air refractive index with a relative accuracy below $10^{-7}$ (17). Delay stage encoder positions were calibrated against a temperature-insensitive optical encoder [Sony Laserscale BL57-RE, thermal expansion coefficient of $0.7\cdot 10^{-6}$ m/(m$\cdot$K)] and by measuring laser cross-correlation signals displaced by one oscillator pulse jump (12.5 ns).

Results

Fig. 2 illustrates the correlated measurement of mass and rotational Raman spectra. The photoionization mass spectrum of CS₂ showed multiple ion signals due to the presence of isotopologues, sample impurities, and molecular fragmentation processes. Mass spectra were collected after inducing a rotationally coherent wave packet with the alignment laser pulse. The rotational coherence affected the orientation of transition dipoles for the resonant two-photon ionization process and thereby created pronounced delay-dependent signal modulations, as shown for the main ¹²C³²S₂ isotopologue at mass 76 u. A Fourier analysis of the delay trace revealed the amplitudes and phases for all rotational transition frequencies that were encoded in the wave packet. The plotted power spectrum is equivalent to a Raman spectrum measured in the frequency domain.

Fig. 2.

CRASY data for carbon disulfide. (Top) The mass spectrum shows signals for multiple CS₂ isotopes (76 to 82 u) and their fragments. A gray line shows the same spectrum with 10-fold enlarged ordinate and vertical offset. Signals from sample impurities are marked with *. (Center) The mass-selected 76 u, ¹²C³²S₂ isotopologue signal shows delay-dependent signal modulation due to rotational coherence. The Inset with 100-fold enlarged abscissa shows the wave packet evolution over a 50 ps period. (Bottom) A Fourier analysis of the delay trace resolves the isotope-selective rotational Raman spectrum.

The spectroscopic bandwidth of 500 GHz was sufficient to obtain the complete rotational spectra for our cold molecular beam. The resolution for each rotational line in Fig. 2 was close to 60 MHz full width at half maximum (FWHM) and was near the Fourier limit for the scanned delay range. A greater resolution can only be achieved by scanning a longer delay range, but the collection of mass spectra for a large number of alignment–ionization delays was time-consuming and created exceedingly large datasets. We therefore used random sparse sampling to accelerate long delay scans—that is, measuring data only for a randomly selected subset of delays along an extended time axis.

Fig. 3 compares rotational Raman spectra for the main ¹²C³²S₂ isotopologue with full and sparse sampling. For the fully sampled scan, we measured mass spectra over a delay range of 15,325 ps with 1 ps delay steps, integrating signals over 3,000 laser shots for each mass spectrum (51 h data acquisition time). For the sparsely sampled scan, we measured 17,110 mass spectra over a delay range of 312.832 ns with random delay steps in multiples of 1 ps, integrating signals over 1,000 laser shots for each mass spectrum (19 h data acquisition time). Signal modulations at the main isotopologue mass 76 u were Fourier-transformed to obtain the displayed frequency spectra.

Fig. 3.

Rotational Raman spectra for mass 76 u. (Top) Continuous 1-ps sampling of a 15.3-ns delay range (total of $1\cdot 10^8$ ion counts). (Bottom) Random sparse 1-ps sampling of a 312.8-ns delay range with a similar number of samples (total of $8.5\cdot 10^6$ ion counts). Insets show data with identical ordinate but 40-fold and 1,600-fold enlarged abscissa. Please note the logarithmic ordinates.

The FWHM resolution in the sparse data were 2.9 MHz, a factor 20 better than in the fully sampled scan and close to the Fourier limit. The increased resolution came at the cost of noise: The noise floor in the sparse data were 30-fold higher than in the fully sampled dataset. Part of the noise difference was due to a lower ion count rate and the shorter signal integration period in the sparse scan.

Fig. 4 shows the rotational band assignment for several CS₂ isotopologues in the sparse dataset. Line positions were fitted (cf. SI Appendix) to obtain rotational constants, which are summarized in Table 1. For the main isotopologue, we assigned nine lines to the vibrational ground state ${}^g\mathrm{\Sigma }_1$ and determined a rotational constant of $B=3.2715170\left(\pm 7\right)$ GHz. [Errors are given as 1$\sigma$ standard deviation in the last digit(s).] The distortion constant of $D=355\left(\pm 3\right)$ Hz was determined with rather low precision because only low rotational states were observed.

Fig. 4.

Band assignment for CS₂ isotopologue spectra. (Top) The spectrum for the main isotopologue ¹²C³²S₂ (76 u) showed one rotational progression for the vibrational ground state (${\mathrm{\Sigma }}_g$) and two progressions for the first vibrationally excited bending states (${\mathrm{\Pi }}_u^e$ and ${\mathrm{\Pi }}_u^f$). (Middle) The spectrum for mass 77 u showed progressions for the ¹³C³²S₂ (¹³C) and the ³³S¹²C³²S (³³S) isotopologues. (Bottom) The spectrum for mass 78 u showed one progression for the ³⁴S¹²C³²S (³⁴S) isotopologue. Insets show data with enlarged abscissa.

Table 1.

Observed CS₂ isotopologues, isotopologue masses (u), rotational constants (kHz), and centrifugal distortion constants (Hz)

			Rotational constant $B_{eff}$*			Centrifugal distortion $D_{\mathit{\text{eff}}}$
Isotopologue	Mass	State	This work	Literature	Ref.	This work	Literature	Ref.
32S12C32S	76	${\mathrm{\Sigma }}_g$	3,271,517.0 (0.7)	3,271,516.5 (1.5)	20	355 (3)	352.79 (9)	20
	76	${\mathrm{\Pi }}_u^e$	3,276,738 (9)	3,276,759 (12)	18	347 (58)	359.1 (6)	18
	76	${\mathrm{\Pi }}_u^f$	3,279,064 (7)	3,279,077 (11)	18	386 (38)	360.6 (6)	18
32S13C32S	77	${\mathrm{\Sigma }}_g$	3,271,634.6 (1.3)	3,271,637.8 (0.3)	21	346 (7)	350.76 (2)	21
32S12C33S	77	${\mathrm{\Sigma }}_g$	3,221,849.4 (2.6)	3,221,843 (11)	18	329 (13)	341.3 (1.1)	18
32S12C34S	78	${\mathrm{\Sigma }}_g$	3,175,020.4 (1.2)	3,175,024 (8)	18	318 (6)	332.6 (5)	18
32S13C34S	79	${\mathrm{\Sigma }}_g$	3,175,120 (18)	3,175,115.1 (1.3)	21	318 (95)	332.94 (15)	21
34S12C34S	80	${\mathrm{\Sigma }}_g$	3,079,422 (9)	3,079,377 (27)	18	339 (48)	346 (16)	22

The most precise literature values are given for comparison. Values in parentheses denote the 1σ standard deviation for the corresponding last digits.

^*For the vibrationally excited Π_u states, B_eff differs from B due to the l-type splitting.

We also resolved 11 weak lines for the first vibrationally excited state of the doubly degenerate bending mode ${\mathrm{\Pi }}_u$. The latter showed a rovibrational splitting ($l$-type splitting) by $q\left(J\left(J+1\right)\right)$ into levels with $e$ and $f$ symmetry (18). The rotational constant for this state was $B=3.277901\left(\pm 10\right)$ GHz with a distortion constant of $D=367\left(\pm 68\right)$ Hz and an $l$-type splitting constant of $q_v=\mathrm{2,327}\left(\pm 10\right)$ kHz.

In the mass channel 77 u, we assigned 5 and 7 lines for the vibrational ground states of the ¹³C³²S₂ and ³³S¹²C³²S isotopologues. The fitted rotational constants (cf. Table 1) showed a significantly higher uncertainty due to the low isotope abundance of ¹³C (1.1%) and ³³S (0.75%) (19). In the mass channel 78 u, we assigned 15 lines for the vibrational ground state of the ³⁴S¹²C³²S isotopologue (8.5% abundance). In mass channel 79 u, six lines for the ³⁴S¹³C³²S isotopologue (0.09% abundance) could be assigned based on their close proximity to the lines of the ³⁴S¹²C³²S isotopologue. Six lines emerged above the sampling noise in mass channel 80 u and could be assigned to the ³⁴S¹²C³⁴S isotopologue (0.18% abundance).

Discussion and Outlook

We should discuss two fundamental claims of this paper in some detail: (i) Presented broadband rotational Raman spectra offer order-of-magnitude improved resolution compared with preceding time-domain measurements. (ii) The measurements represent a time-domain equivalent to frequency comb spectroscopy and yield absolute-frequency spectra.

The spectroscopic resolution of previous time-domain rotational Raman spectra was limited by the available opto-mechanical delay stages. The longest available delays were in the range of a few nanoseconds, based on $\le 2$ m mechanical translation stages (10, 11). We overcame this limit with the electronic selection of oscillator pulses from a stable laser oscillator, thereby adding discrete 12.5-ns delay increments to an opto-mechanical delay. The resulting delays are only limited by the laser repetition rate but can still be sampled with femtosecond step sizes. To acquire spectroscopic data, we scanned delays exceeding 300 ns, thereby obtaining a spectroscopic resolution below 3 MHz FWHM, representing a 50-fold improvement over preceding RCS experiments.

The spectroscopic range in Fourier-transform measurements is limited according to the Shannon–Nyquist theorem (23) and cannot exceed $1/\left(2\cdot t_{\mathit{\text{step size}}}\right)$. We used 1-ps steps, and the resulting spectroscopic range of 500 GHz was sufficient to resolve the complete rotational spectra in our cold molecular beam. The acquisition of correlated broadband, high-resolution data imposed a serious burden for the data collection and analysis: Already the lower resolution data shown in Fig. 2 required the measurement of $>\mathrm{15,000}$ mass spectra for signal and reference over the course of more than 2 d. Each mass spectrum contained 150,000 mass signals, creating signal and reference datasets with approx. $4.6\cdot 10^9$ data points.

With random sparse sampling, we greatly reduced the measurement time and data quantity. As discussed in the field of multidimensional NMR spectroscopy (24, 25), sparse sampling led to a modest degradation of the signal-to-noise ratio. Our data acquisition speeds were still severely limited by the used single-ion counting electronics, precluding higher resolution measurements. We expect that future CRASY experiments, based on the detection of larger signals, can collect similar or better quality data in a fraction of the measurement time. It is worthwhile to point out that our experiments are not affected by Doppler broadening, which remains below $\mathrm{\Delta }{\nu }_{\mathit{\text{Db}}}/\nu =6.6\cdot 10^{-9}$ due to the small collimation angle of the molecular beam.

The accuracy of CS₂ isotopologue rotational constants derived from our measurements was $<1$ kHz for the main isotopologue and a few kilohertz for naturally occurring heavier isotopologues. Through a simple measurement of the laser oscillator frequency, our measured spectroscopic frequencies were referenced with an accuracy $\mathrm{\Delta }\nu$/$\nu \ll 10^{-8}$ to an external clock. Uncertainties due to temperature, air pressure, or humidity affected only the opto-mechanical delay accuracy and created a relative uncertainty of $<1\cdot 10^{-6}$ for the opto-mechanical ($<12.5$ ns) delay. Upon every oscillator pulse jump, this uncertainty was reset to the much smaller uncertainty of the oscillator frequency measurement. The opto-mechanical delay uncertainty therefore scaled inversely to the number $N$ of oscillator jumps and became negligible for large $N$. Measured frequencies were therefore absolute frequency values within the measurement precision. Our rotational constants agreed well with all literature values from high-resolution rovibrational spectroscopy (see Table 1). Kummli and coworkers (7, 9) reported a significantly different rotational constant of B = 3.271 549 2 ($\pm 18$) GHz for the main CS₂ isotopologue ground state, based on RCS spectroscopy. (The striking 18 sigma deviation of this value from ours and literature values may be due to a typographic error: The tabulated rotational constant in ref. 7 indicates a 10-fold higher standard deviation than the values claimed in the paper abstract and text.)

Modern chirped-pulse Fourier-transform microwave (FTMW) experiments achieve an instrument resolution and accuracy down to a few kilohertz, with a spectroscopic range exceeding 10 GHz (26). Doppler broadening (typically in the 100 kHz range) reduces the effective resolution, which is nevertheless far beyond the resolution achieved here. The accuracy of rotational constants determined here, however, was comparable to the 4 kHz frequency precision claimed for FTMW spectroscopy (26). We expect that CRASY experiments will be complementary to FTMW spectroscopy: CRASY explores rotational Raman transitions instead of dipole transitions, covers a larger spectral range ($\ge 10$ GHz for chirped-pulse FTMW experiments and 500 GHz for our data), and allows correlation to ion masses or other spectroscopic observables.

There are illustrative parallels between the CRASY experiment reported here and frequency comb measurements (13, 14, 27). Both experiments rely on the discrete emission properties of mode-locked laser oscillators. Such oscillators emit a discrete pulse train in the time domain and, as expected from the Fourier theorem, a discrete line spectrum in the frequency domain. Frequency comb spectroscopy exploits the discrete frequency property and compares an interferometric beating pattern to an external clock to determine and lock the line frequencies in an active feedback loop. Spectroscopy with the comb lines measures absolute transition frequencies with an accuracy approaching that of the clock.

High-resolution CRASY exploits the discrete time domain property of the oscillator pulse train to observe the beating pattern between quantum mechanical states. The measurement accuracy is tied to an external clock through a measurement of the oscillator repetition rate. CRASY does not require a frequency- or phase-locked oscillator but requires merely a stable oscillator repetition rate on the time scale of the oscillator frequency measurement. Instead of active frequency stabilization, oscillator drifts are corrected with an adjustment of time delays in the opto-mechanical delay line. The resulting lock between clock and measured spectroscopic frequencies is equivalent to that in frequency comb spectroscopy; hence, our measurement is a time-domain equivalent to frequency comb spectroscopy.

The inherent value of correlated spectroscopic data are directly related to the type and quantity of the collected data: Signals that cannot be resolved along one spectroscopic axis may be distinguished along another and are therefore easily separated. The correlation of molecular mass (analyzing molecular composition) and rotational spectra (analyzing molecular structure) allowed us to separate rotational spectra for 6 (this work) or 10 (1) CS₂ isotopologues in a single measurement. Signals for small isotopologues were observed in separate mass channels and were therefore not obscured by the baseline noise of larger signals.

The correlation of mass and rotational structure, as presented here, might become a valuable tool for molecular structure determination and for the analysis of isotope effects. The potential for CRASY spectroscopy, however, goes beyond the data discussed in this manuscript. Future CRASY experiments may correlate rotational spectra with any other spectroscopic observable that can be interrogated with ultrafast laser pulses: A coherent rotational wave packet will modulate any spectroscopic signal that is tied to a molecular transition dipole with well-defined orientation in the molecular frame. This should be relevant for the analysis of any heterogeneous sample, where independent spectroscopic measurements cannot separate signal contributions from individual sample components. The majority of molecular samples are inherently heterogeneous: Natural or synthesized samples are rarely of pure structural, isomeric, or isotopic composition. Traditional spectroscopic experiments excel in the analysis of abundant species and purified compounds. CRASY will facilitate the characterization of inherently impure samples (e.g., reactive species, tautomers, or weakly bound molecular clusters).

SI Appendix

SI Appendix describes the data analysis protocol for each figure and explains how rotational constants were fitted to the data.