Phase stable, shot-to-shot measurement of third- and fifth-order two-quantum correlation spectra using a pulse shaper in the pump–probe geometry

Abstract

We demonstrate the first phase stable measurement of a third-order 2Q spectrum using a pulse shaper in the pump–probe geometry. This measurement was achieved by permuting the time-ordering of the pump pulses, thus rearranging the signal pathways that are emitted in the probe direction. The third-order 2Q spectrum is self-heterodyned by the probe pulse. Using this method, one can interconvert between a 1Q experiment and a 2Q experiment by simply reprogramming a pulse shaper or delay stage. We also measure a fifth-order absorptive 2Q spectrum in the pump–probe geometry, which contains similar information as a third-order experiment but does not suffer from dispersive line shapes. To do so, we introduce methods to minimize saturation-induced artifacts of the pulse shaper, improving fifth-order signals. These techniques add new capabilities for 2D spectrometers that use pulse shapers in the pump–probe beam geometry.

I. INTRODUCTION

Linear spectroscopic methods, such as Fourier transform, infrared, and UV/Vis spectroscopy, provide information about the vibrational and electronic structure of a transition and its coupling to a bath via its peak frequency and line shape. Multidimensional optical spectroscopies provide additional observables that open new avenues of inquiry, such as studying ultrafast chemical exchange,¹ spectral diffusion,² and Fermi resonances.³ Two of the most commonly used multidimensional techniques are two dimensional infrared (2D IR) and electronic spectroscopy, which are identical except for their wavelengths. Their pulse sequences correlate the one-quantum (1Q) frequencies excited by the pump pulses to the 1Q frequencies detected after interaction with the probe pulse.

Another useful pulse sequence is the two-quantum (2Q) experiment. The 2Q spectrum correlates the doubly excited states to the fundamental transitions.^4,5 For vibrations, the 2Q spectrum reports on anharmonicities and couplings between modes.⁴ For electronic transitions, the 2Q spectrum reports on many body correlations.⁶ The third-order 2Q non-rephasing spectrum is complementary to the 1Q absorptive experiment because it contains similar information, but peaks are organized such that cross peaks appear at the combination band frequency and diagonal peaks appear at the overtone frequency along the pump axis. Therefore, 2Q spectra are informative because in some instances anharmonicities and couplings can be determined without peak fitting. Ge and co-workers showed that examining 2Q spectra in parallel with 1Q non-rephasing and rephasing spectra can help determine molecular structure.^7,8

2Q experiments measured with fifth-order pulse sequences can also be useful. Third-order 2Q spectra are entirely non-rephasing, so their peaks have phase twisted line shapes. In some cases, non-resonant responses and dipole forbidden transitions can contribute.⁹ In contrast, fifth-order 2Q measurements have both rephasing and non-rephasing pathways that add to produce absorptive line shapes. The absorptive line shapes give fifth-order 2Q spectra increased resolution over third-order 2Q spectra. In addition, fifth-order pulse sequences access the ν = 3 states, allowing determination of higher order anharmonic coupling terms⁴ and can be used to generate 3D spectra.^10,11 Knowledge of these higher order couplings can assist in the interpretation of linear spectra, where combination bands and overtones gain oscillator strength.¹²

Inspired by multi-quantum nuclear magnetic resonance experiments, Mukamel and co-workers simulated heterodyne detected, third- and fifth-order 2Q experiments.^13,14 Zanni and co-workers measured both third- and fifth-order 2Q spectra in the infrared and applied it to azide stretch modes in an ionic glass^4,15 and performed the first 3D IR experiments.¹⁶ Kim et al. performed early measurements using third-order 2Q spectra in the visible.⁵ Dostál et al. measured fifth-order 2Q spectra in the visible.¹⁷ Since their development, these methods are most often used in the visible to study exciton–exciton interactions, particularly for determining exciton binding energies.^17–25

Brossau et al. measured fifth-order 2Q visible spectra of CdSe quantum dots in the pump–probe geometry using a longitudinal pulse shaper.²⁶ Prior to the work by Brossau et al., most implementations of 2Q spectroscopy were performed in the four-wave-mixing beam geometries, where the signal field emits in a unique direction from the excitation fields. In the four-wave-mixing geometry, an external local oscillator is required for heterodyne detection and data must be phase stabilized.^27,28 Additionally, interconversion between different multidimensional experiments requires changing the optical path of the local oscillator.²⁹ Third-order 2Q measurements have only been performed in the four-wave-mixing or fully collinear beam geometry.^30,31

Herein, we introduce a modification of the pump–probe 2D pulse sequence that allows facile interconversion between third-order 1Q and 2Q experiments. We call the modified pulse sequence the permuted–pump–probe pulse sequence. Using this pulse sequence, third-order 2Q spectra can be collected in the widely used pump–probe beam geometry with a pulse shaper. In the pump–probe geometry, spectra are self-heterodyned by the probe, yielding phase stable data. Using a pulse shaper optimizes signal to noise by measuring data shot-to-shot, taking full advantage of emerging high repetition rate laser technology.³² We also measure fifth-order 2Q absorptive spectra using a transverse pulse shaper in the pump–probe geometry. To do so, we minimize artifacts caused by radio frequency amplification nonlinearities in the pulse shaper. These methods allow conversion between 1Q absorptive spectra, third-order 2Q non-rephasing spectra, and fifth-order 2Q absorptive spectra using programming and no beam alignment, expanding the capabilities of one of the most widely used 2D spectroscopic methods.

II. EXPERIMENTAL METHODS

A. 2D IR spectrometer

The 40 W, 100 kHz output of a Yb:KGW regenerative amplifier (Carbide, Light Conversion) was used to pump an optical parametric amplifier (Orpheus Mid-IR, Light Conversion) consisting of twin OPAs that generate the signal and idler beams for difference frequency generation (DFG). The signal and idler were mixed non-collinearly in GaSe (Lyra, Light Conversion) to generate 4 microJ pulses at 5 m. The pulse bandwidth was 249 cm⁻¹ at full width at half maximum (FWHM).

The mid-IR output was used to pump a 2D IR spectrometer, the details of which are described elsewhere.³³ In brief, the spectrometer was constructed in the pump–probe geometry using a mid-IR pulse shaper. The mid-IR beam was split along a pump and probe path using a 90:10 CaF₂ beam splitter. The probe (E₃) was directed into a motorized stage delay to control the waiting time (τ₂) between the probe and pump pulses. The pump beam is directed into a horizontal mid-IR pulse shaper based on a germanium acousto-optic modulator (AOM).³⁴ The pulse shaper generates a pump–pulse pair (E₁ and E₂) separated by τ₁. A 7 s mask was used. The mask was converted from digital to analog using an arbitrary waveform generator (PXDAC4800, Signatec). The analog wave is amplified using a radio frequency amplifier (RFA1170-4-ZP(HR), Isomet Inc.). The bandwidth loss due to angular dispersion in the shaper was mitigated using active Bragg angle compensation.³⁵ Final pump bandwidth at FWHM was 198 cm⁻¹. Pump pulse duration was 142 fs.

E₁, E₂, and E₃ are focused using 90° off-axis parabolic mirrors. After overlap at the sample, the probe and signal fields were detected using a 1 × 64 element mercury cadmium telluride (MCT) array (Infrared Systems). Analog to digital conversion of the MCT signal was performed using high-repetition rate mid-IR detection electronics (Jackhammer, Phasetech Spectroscopy).³³

(Acetylacetonato)dicarbonyl iridium(I) was purchased from Sigma-Aldrich and used as received. All experiments were performed in hexanes where the antisymmetric stretch mode had an optical density (OD) of 0.16.

III. THEORY

For clarity, we introduce notation to distinguish between the different experiments in the text. R⁽ⁿ⁾_MQ,(N)R refers to the nth-order MQ (non)rephasing pathway. R⁽ⁿ⁾_MQ corresponds to an absorptive spectrum, that is, the sum of R⁽ⁿ⁾_MQ,R and R⁽ⁿ⁾_MQ,NR. The experiments discussed in this manuscript are R⁽³⁾_0Q,R, R⁽³⁾_0Q,NR, R⁽³⁾_2Q,NR, and R⁽⁵⁾_2Q.

A. The permuted–pump–probe pulse sequence

Figure 1(a) shows the pulse sequence for 2D spectroscopy in the pump–probe geometry. A pair of pump pulses, E₁ and E_2, co-propagate along the same optical path in the k₁ = k₂ direction. A probe pulse, E₃, follows a different beam path in the k₃ direction. The pump–pulse pair is separated by delay τ₁. The probe is delayed relative to E₂ by τ₂. The pulses are time-ordered such that E₁ arrives at the sample first. This pulse sequence is referred to as the pump–probe pulse sequence. Figure 1(b) shows Feynman diagrams for the ground state bleach pathways for R⁽³⁾_1Q,R, R⁽³⁾_1Q,NR, and R⁽³⁾_2Q,NR. R⁽³⁾_1Q,R and R⁽³⁾_1Q,NR emit in the −k₁ + k₂ + k₃, and k₁ − k₂ + k₃ directions. In the pump–probe geometry, they co-emit with the probe [R⁽³⁾_1Q is self-heterodyned by the probe] and add to yield absorptive line shapes. R⁽³⁾_2Q,NR emits in the k₁ + k₂ − k₃ direction and is not self-heterodyned, requiring an external local oscillator. Therefore, the measurement of R⁽³⁾_2Q,NR in this geometry is difficult when using the pump–probe pulse sequence.

FIG. 1.

Effect of time-ordering of laser pulses on the phase matching direction of R⁽³⁾_1Q,R, R⁽³⁾_1Q,NR, and R⁽³⁾_2Q,NR signals. (a) Standard pulse sequence for collecting R⁽³⁾_1Q in the pump–probe geometry. (b) Double sided Feynman diagrams for ground state bleach pathways. (c) Pulse sequence for collecting R⁽³⁾_2Q,NR. (d) Double sided Feynman diagrams for the same pathways with permuted pulse time-orderings.

Making a small modification to the pump–probe pulse sequence allows intrinsic heterodyne detection of R⁽³⁾_2Q,NR. The modified pulse sequence, shown in Fig. 1(c), is referred to as the permuted–pump–probe pulse sequence. In the permuted–pump–probe pulse sequence, E₁ is scanned in the opposite direction such that the time-ordering of the three pulses is cyclically permuted. E₂ and E₃ interact first, then E₁ interacts. Permuting the pulse ordering changes the phase matching directions of R⁽³⁾_1Q,R, R⁽³⁾_1Q,NR, and R⁽³⁾_2Q,NR. R⁽³⁾_2Q,NR now emits in the −k₁ + k₂ + k₃ direction, co-propagating with the probe. Feynman diagrams in Fig. 1(d) show that τ₁ samples different coherences in the permuted–pump–probe. The 2Q coherence of R⁽³⁾_2Q,NR now evolves over τ₁. Therefore, simply scanning E₁ in the opposite direction enables measurement of R⁽³⁾_2Q,NR spectra in the pump–probe geometry.

Using the permuted–pump–probe pulse sequence, R⁽³⁾_1Q,R and R⁽³⁾_1Q,NR are in either a population state or interstate coherence during τ₁. Diagonal peaks will show up at zero frequency and cross peaks will show up at the interstate coherence frequency. The magnetic resonance literature refers to this as the zero quantum coherence.³⁶ Using this pulse sequence, these signals are henceforth denoted as R⁽³⁾_0Q,R and R⁽³⁾_0Q,NR. R⁽³⁾_0Q,R co-propagates with the R⁽³⁾_2Q,NR in the k₁ − k₂ + k₃ direction. R⁽³⁾_0Q,NR emits in the k₁ + k₂ − k₃ direction.

When τ₂ = 0, E₃ and E₂ arrive at the sample simultaneously. Therefore, a signal will also emit from the time-ordering in which E₃ precedes E₂ [not shown in Fig. 1(d)], contributing to the spectrum. Feynman diagrams in Fig. S1 show that R⁽³⁾_2Q,NR still emits in the −k₁ + k₂ + k₃ directions for this time-ordering. The only difference is that the phase matching for R⁽³⁾_0Q,R and R⁽³⁾_0Q,NR is switched such that R⁽³⁾_0Q,NR emits in the k₁ − k₂ + k₃ direction and R⁽³⁾_0Q,R emits in the k₁ + k₂ − k₃ direction.

We emphasize that the permuted–pump–probe pulse sequence and pump–probe pulse sequence specify experiments with different pulse time-orderings. Both pulse sequences are performed in the pump–probe beam geometry. The pump–probe pulse sequence refers to the typical case where the two pump pulses interact with the sample prior to the probe pulse. The permuted–pump–probe pulse sequence refers to the case where E₂ and E₃ arrive simultaneously, followed by E₁.

An important distinction between the pump–probe pulse sequence and permuted–pump–probe pulse sequence is the time at which the signal is emitted. When using the pump–probe pulse sequence to measure a non-rephasing pathway, the signal is emitted at τ₃ = 0, because the E₃ is the final field interaction. When using the permuted–pump–probe pulse sequence, the signal is emitted at τ₃ = τ₁ because E₁ is the final field interaction. Therefore, the phase of the R⁽³⁾_0Q,R and R⁽³⁾_2Q,NR will rotate relative to E₃ (the local oscillator) as a function of τ₁, adding a phase shift to the heterodyned signal. The signal equation for the R⁽³⁾_0Q,R and R⁽³⁾_2Q,NR experiments is given in Eqs. (1) and (2). The τ₁ dependent phase shift is shown in brackets,

$$\begin{array}{ll}\hfill E_{2Q,NR}^{(3)}{E}_3^{*}& \propto e^{-i({\varphi }_1-{\varphi }_2)}{e}^{-i({\omega }_{02}{t}_1+{\omega }_{01}{t}_2)}{e}^{-{\mathrm{\Gamma }}_2(2t_1+t2)}\hfill \\ \hfill & \times \left({\mathbf{e}}^{\mathbf{i}{\omega }_{\mathbf{3}}({\mathbf{t}}_{\mathbf{1}}+{\mathbf{t}}_{\mathbf{2}})}\right)\frac{1}{i({\omega }_3-{\omega }_{01})-{\mathrm{\Gamma }}_2},\hfill \end{array}$$ (1)

$$\begin{array}{ll}\hfill E_{0Q,R}^{(3)}{E}_3^{*}& \propto e^{-i({\varphi }_2-{\varphi }_1)}{e}^{i({\omega }_{00}{t}_1+{\omega }_{01}{t}_2)}{e}^{-(t_1{\mathrm{\Gamma }}_1+t_2{\mathrm{\Gamma }}_2)}\hfill \\ \hfill & \times \left({\mathbf{e}}^{\mathbf{i}{\omega }_{\mathbf{3}}({\mathbf{t}}_{\mathbf{1}}+{\mathbf{t}}_{\mathbf{2}})}\right)\frac{1}{i({\omega }_3-{\omega }_{01})-{\mathrm{\Gamma }}_2}.\hfill \end{array}$$ (2)

In Eqs. (1) and (2), ω₀₀, ω₀₁, and ω₀₂ are the 0Q, 1Q, and 2Q coherence frequencies, respectively. ϕ₁ and ϕ₂ are the phases applied to E₁ and E₂ using the pulse shaper. Γ₁ and Γ₂ are the population relaxation lifetime and dephasing times. The τ₁-dependent phase acts like a rotating frame, downshifting the 2Q signal and upshifting the 0Q signal by ω₃, so that they overlap at the 1Q frequencies.

An advantage of the permuted–pump–probe pulse sequence is that the R⁽³⁾_2Q,NR signal is phase stable. The phase error of the heterodyned R⁽³⁾_2Q,NR signal measured using the permuted–pump–probe geometry is shown as

$$\delta ({\varphi }_{2Q})=-\delta ({\varphi }_1)+\delta ({\varphi }_2)+\delta ({\varphi }_3)-\delta ({\varphi }_{LO}).$$ (3)

Each of the terms on the right-hand-side of Eq. (3) is the phase error in E₁, E₂, E₃, and E_LO. E₃ = E_LO because the signal is self-heterodyned. Therefore, δ(ϕ₃) = δ(ϕ_LO), causing the last two terms to cancel. When E₁ and E₂ are generated using a pulse shaper or birefringent wedges,³⁷ they follow the same optical path. Therefore, δ(ϕ₁) = δ(ϕ₂), causing the first two terms to cancel. As a result, there is no phase error in the collection of R⁽³⁾_2Q,NR.

B. Processing of third-order 2Q spectra

The 2Q and 0Q signals overlap in the frequency-domain and need to be separated. Separation of signals with −ϕ₁ + ϕ₂ + ϕ₃ and ϕ₁ − ϕ₂ + ϕ₃ phase dependence can be achieved experimentally with independent phase control over the E₃ and the local oscillator (E_LO).^38,39 Because typical pump–probe 2D spectrometers do not have independent phase control over E₃ and E_LO, we demonstrate a data-processing method to separate the two signals.

The 2Q and 0Q signals can be separated by retrieving the imaginary part of the signal from the real part using a time-domain Kramers–Kronig inversion.^40,41 To perform this operation, the real valued data are inverse Fourier transformed along the ω₃ axis into the time domain. All signals where τ₃ < 0 are multiplied by 0. This dataset is Fourier transformed back into the frequency domain. The resulting data are now complex. Upon Fourier transformation along τ₁, the 2Q signal will appear in the (+ω₁, + ω₃) quadrant and the 0Q signal will appear in the (−ω₁, + ω₃) quadrant. This is still true at τ₂ = 0, where either R⁽³⁾_0Q,R or R⁽³⁾_0Q,NR will emit in the k₁ − k₂ + k₃ direction, depending on the time-ordering of the E₂ and E₃ interactions.

Finally, a phase term must be added to undo the action of the τ₁-dependent phase factor and shift the data to its correct frequency. The added phase is shown in Eq. (4). The whole operation is shown in Eq. (5),

$$\varphi ={\omega }_{01}{t}_2-({\omega }_3-{\omega }_{rf})(t_1+t_2),$$ (4)

$$\begin{array}{ll}\hfill & S(t_1,t_2,{\omega }_3)\stackrel{F{T}^{-1}}{\to }\mathrm{\Theta }(t_3)S(t_1,t_2,t_3)\stackrel{FT}{\to }\hfill \\ \hfill & \times e^{i\varphi }S(t_1,t_2,{\omega }_3)\to S_{\varphi }(t_1,t_2,{\omega }_3).\hfill \end{array}$$ (5)

In Eq. (5), S_ϕ is the processed data. When calculating ϕ, ω₃ is obtained from the calibrated probe frequency axis and ω_rf is the rotating frame frequency. The ω₀₁t₂ term is used to adjust the phase if τ₂ ≠ 0. The ω₀₁t₂ phase term was not required in experimental results reported in Sec. IV.

C. Cross peak specific 2Q spectra

We recently reported a cross-peak specific polarization scheme for R⁽³⁾_1Q spectra: ⟨0°, 0°, 60°, −60°⟩.⁴² When converting to the permuted–pump–probe pulse sequence, this polarization scheme becomes ⟨0°, 60°, 0°, −60°⟩ because the pulse ordering rearranges. Similar to ⟨0°, 0°, 60°, −60°⟩, ⟨0°, 60°, 0°, −60°⟩ is still cross peak specific. This polarization scheme interferes with the XXXX and XYXY spectra in situ, as shown in the following:

$$⟨0°,60°,0°,-60°⟩=\frac{1}{4}[⟨X_i{X}_i{X}_i{X}_i⟩-3⟨X_i{Y}_i{X}_i{Y}_i⟩].$$ (6)

In Eq. (6), the two terms on the right-hand side are the orientational correlation functions for pathways where each field interacts with the same mode, i. These functions are equal to 1 and 1/3 for XXXX and XYXY, respectively.⁴³ Therefore, diagonal overtone peaks destructively interfere and are eliminated. Thus, R⁽³⁾_2Q,NR spectra measured in ⟨0°, 60°, 0°, −60°⟩ are cross peak specific for non-parallel transition dipoles, as we demonstrate in Sec. IV. In principle, there also exists a cross peak specific polarization scheme for R⁽⁵⁾_2Q spectra.

D. Fifth-order 2Q absorptive spectra

In this section, pump–probe measurement of R⁽⁵⁾_2Q is briefly discussed and compared to R⁽³⁾_1Q. R⁽⁵⁾_2Q spectra have been previously reported in the infrared.^4,15 Figure 2(a) shows the pump–probe pulse sequence, which is used to acquire both R⁽³⁾_1Q and R⁽⁵⁾_2Q. Figure 2(b) shows Feynman diagrams for the three rephasing diagonal peak pathways in R⁽⁵⁾_2Q spectra. The pathways correspond to the ν = 0 → 1, ν = 1 → 2, and ν = 2 → 3 diagonal peaks. The difference between R⁽⁵⁾_2Q and R⁽³⁾_1Q is that E₁ and E₂ both interact twice for R⁽⁵⁾_2Q, measuring a 2Q coherence over τ₁. Similar to R⁽³⁾_1Q, R⁽⁵⁾_2Q is in a population state over τ₂ and a 1Q coherence over τ₃. R⁽⁵⁾_2Q,R and R⁽⁵⁾_2Q,NR co-propagate in the ± 2k₁ ∓ 2k₂ + k₃ direction and add constructively to produce absorptive line shapes. R⁽⁵⁾_2Q also co-propagates with R⁽³⁾_1Q, meaning that both signals are acquired simultaneously and must be isolated using phase cycling.

FIG. 2.

(a) Pulse sequence for R⁽⁵⁾_2Q and R⁽³⁾_1Q experiments. Coherences and populations excited during R⁽⁵⁾_2Q and R⁽³⁾_1Q experiments are shown in solid blue and dashed red lines, respectively. (b) Representative rephasing Feynman diagrams for the R⁽⁵⁾_2Q measurement.

R⁽³⁾_1Q signal is isolated using four-frame π phase cycling and R⁽⁵⁾_2Q signal is isolated using eight-frame π/2 phase cycling,^31,44,45

$$4E_{1Q}^{(3)}{E}_3+4E_1{E}_2=S_{0,0}-S_{0,\pi }+S_{\pi ,\pi }-S_{\pi ,0},$$ (7)

$$8E_{2Q}^{(5)}{E}_3=S_{0,0}-S_{\frac{\pi }{2},0}+S_{\pi ,0}-S_{\frac{3\pi }{2},0}+S_{0,\pi }-S_{\frac{\pi }{2},\pi }+S_{\pi ,\pi }-S_{\frac{3\pi }{2},\pi }.$$ (8)

In Eqs. (7) and (8), $S_{{\varphi }_1,{\varphi }_2}$ is the signal measured with the relative phases ϕ₁ and ϕ₂. E₁E₂ is the pump scatter term. The eight-frame π/2 phase cycling scheme shown in Eq. (8) isolates 2Q signals and eliminates both transient absorption backgrounds and all scatter terms. Phase cycling was found to be accurate to 0.2%. Pulse shaping accuracy was determined by comparing the intensity of the R⁽³⁾_1Q spectrum obtained by four-frame phase cycling with the residual R⁽³⁾_1Q features appearing in the spectrum calculated using eight-frame phase cycling.

Finally, because each pump pulse interacts twice, the measured frequency of the R⁽⁵⁾_2Q free induction decay (FID) will be downshifted by two times the applied rotating frame frequency. When processing data, two-times the rotating frame frequency must be added to the pump frequency axis for spectra to appear at the correct 2Q frequencies.

E. Pulse shaper non-linearities

Nonlinearities in either the detector or the pulse shaper can introduce artifacts by creating harmonics of third-order signals that overlap with higher-order spectra, such as R⁽⁵⁾_2Q.^46,47 These artifacts are replicas of the third-order signal. In principle, they can be differentiated from 2Q spectra by their anharmonic shifts, but, in practice, their deconvolution is difficult. Detector artifacts and pulse shaper artifacts are different. Third-order detector nonlinearities can be removed from fifth-order spectra using eight-frame phase cycling. Pulse shaping nonlinearities, on the other hand, are not entirely removed by phase cycling. In this section, we describe how pulse shaping artifacts arise and introduce a method for their suppression.

A transverse pulse shaper works by modifying the spectrum of an input pulse to match the spectrum of a desired output pulse shape in the time-domain, and in this case, the spectrum of a pulse pair. Practically, this is achieved by multiplying the pulse spectrum by a complex mask function given in Eq. (9). To generate a pulse pair, a cosine amplitude modulation is applied to the spectrum of the pulse. The mask equation for a pulse pair is given in Eq. (10),

$$E_{\mathit{out}}(\omega )=M(\omega )E_{in}(\omega ),$$ (9)

$$M(\omega )=\frac{1}{2}{e}^{i[(\omega -{\omega }_{rf}){\tau }_1+{\varphi }_1]}+\frac{1}{2}{e}^{i{\varphi }_2}.$$ (10)

In Eq. (9), E_in and E_out are the input and output fields for the pulse shaper. In Eq. (10), ω_rf is the rotating frame frequency applied to the traveling pump pulse. The two complex waves correspond to the E₁ and E₂ output pulses. The mask is generated on a PC and mapped onto an RF wave. The RF wave undergoes digital to analog conversion in an arbitrary waveform generator (AWG) and is amplified by a radio frequency amplifier (RFA). The amplified waveform is converted into an acoustic wave that propagates across the aperture of the Ge AOM. Ideally, the acoustic wave on the AOM perfectly reproduces the RF wave calculated by the PC, generating a perfect pulse pair. In reality, the fidelity of the mask to the ideal waveform is limited for a variety of reasons. Distortions to the RF wave can be introduced by the AWG and RF amplification nonlinearities.^48–50

The pulse shaper nonlinearities were characterized by measuring the intensity of the pulse spectrum as a function of the RF drive amplitude.⁵¹ Figure 3(a) shows the nonlinearity at one position along the AOM aperture (blue). These data are referred to as the saturation curve. There are two non-linearities present: one at a low RF amplitude and saturation at high RF amplitude. To characterize the magnitude of the high intensity nonlinearity, the saturation curve was fitted to a 2nd-order polynomial. The quadratic coefficient of this fit is proportional to the magnitude of the nonlinearity. Figure 3(b) shows the quadratic term from the fit as a function of position along the AOM aperture. The nonlinearity varies across the AOM aperture. The nonlinearity is most intense on the side nearest the piezoelectric transducer. The nonlinearity becomes weaker with distance from the transducer and is oscillatory at the far end of the AOM aperture. This result likely varies for different RF amplifiers and AWG cards.

FIG. 3.

Effect of RF amplification non-linearity on a pump–pulse pair. (a) Saturation of pump power with increasing RF drive amplitude (blue). Linearity of pump power after pre-distorting the waveform to correct for waveform saturation (red). (b) Quadratic parameter from 2nd-order polynomial fit of the saturation curve as a function of position across the AOM aperture. (c) Amplitude mask for a 0.5 ps pump–pulse pair (black). Simulated waveform in the presence of nonlinearities (red). Pre-distorted waveform that calibrates for waveform saturation (blue). (d) Simulated electric fields of a 1 ps pump–pulse pair in the presence (orange) and absence (black) of RF amplification nonlinearities.

Figure 3(c) plots the simulated effect of the measured pulse shaper nonlinearity on the mask. The black curve is the ideal amplitude mask function used to shape a pulse pair with a 0.5 ps delay. The red curve is the simulated amplitude mask function subject to measured pulse shaping nonlinearities. Nonlinearities cause the actual mask to deviate significantly from the ideal mask. Figure 3(d) shows the output pulses for a mask that generates a 1 ps pulse pair. A 1 ps delay was chosen to clearly show each field generated by the pulse shaper nonlinearity. The saturation nonlinearity is intrinsic to the pulse shaper and does not depend on the applied mask. The black fields are the simulated output in the absence of pulse shaping nonlinearities. The red fields are the simulated shaper output including nonlinearities. Harmonics added to the waveform by nonlinearities introduce additional pulses at multiples of the pump–pulse delay.

To understand how these pulses arise and how they affect the 2D spectrum, the saturated mask is expressed by a power series expansion,⁴⁷

$$M_{\mathit{actual}}(\omega )=M_{\mathit{ideal}}(\omega )+a(\omega )M_{\mathit{ideal}}{(\omega )}^2+b(\omega )M_{\mathit{ideal}}{(\omega )}^3+\cdots .$$ (11)

The first order term generates the desired two pump fields, henceforth referred to as $E_1^{(1)}$ and $E_2^{(1)}$, where the superscript indicates the order of expansion. To obtain fields generated by the second order term, the M(ω) in Eq. (9) is replaced with the second order term in Eq. (11),

$$\begin{array}{ll}\hfill E_{\mathit{out}}^{(2)}& ={\tilde{E}}_1^{(2)}{e}^{i[\omega (t+2{\tau }_1)-2{\omega }_{rf}{\tau }_1+2{\varphi }_1]}+2{\tilde{E}}_2^{(2)}\hfill \\ \hfill & \times e^{i[\omega (t+{\tau }_1)-{\omega }_{rf}{\tau }_1+{\varphi }_1+{\varphi }_2]}+{\tilde{E}}_3^{(2)}{e}^{i[\omega t+2{\varphi }_2]}.\hfill \end{array}$$ (12)

Equation (12) shows that the second order pulse shaping nonlinearity introduces three new fields: $E_1^{(2)}$, $E_2^{(2)}$, and $E_3^{(2)}$. $E_1^{(2)}$ appears at t = 2τ₁, $E_2^{(2)}$ appears at t = τ₁, and $E_3^{(2)}$ appears at t = 0. The relative phases of $E_1^{(2)}$, $E_2^{(2)}$, $E_3^{(2)}$ in terms of the phases applied to $E_1^{(1)}$ and $E_2^{(1)}$, are 2ϕ₁, ϕ₁ + ϕ₂, and 2ϕ₂, respectively. These fields are plotted in Fig. S2.

These new fields can interact with the sample to produce third-order polarizations that emit signals. Most of these signals will be removed by the phase matching geometry, eight-frame phase cycling, or appear at 1Q frequencies. Those signals will not overlap with the fifth-order 2Q spectrum. However, the third-order signal pumped by $E_1^{(2)}$ and $E_3^{(2)}$ and probed by E₃ is an artifact that spectrally overlaps with R⁽⁵⁾_2Q. The phase difference between $E_1^{(2)}$ and $E_3^{(2)}$ is the same as the phase dependence of R⁽⁵⁾_2Q. Thus, this artifact survives eight-frame phase cycling and phase matching. The 1Q free induction decay (FID) for this signal is sampled by artifact pulses at a timestep 2Δt₁, whereas R⁽⁵⁾_2Q is sampled by ideal pulses at Δt₁. Therefore, when Fourier transforming the data with respect to Δt₁, the artifact appears at 2Q frequencies with elongated pump line shapes (see the supplementary material).

Artifacts are suppressed by pre-distorting the RF waveform such that RF amplification nonlinearities result in generation of the ideal waveform.^48,49 This is achieved by determining a calibration function that linearizes the pump power with respect to the RF drive amplitude. Pulse shapers can be accurately calibrated using feedback algorithms that optimize the shaped pulse phase and amplitude.⁵⁰ Here, we performed a simpler amplitude calibration method that does not require nonlinear pulse characterization methods.

The normalized saturation curve is a transfer function that represents the effect of the pulse shaper amplitude nonlinearity. Therefore, the inverse of the normalized saturation curve is the calibration function that pre-distorts the RF waveform,

$$R{F}_{\mathit{cal}}(\omega )=T^{-1}[R{F}_{\mathit{ideal}},\omega ]R{F}_{\mathit{ideal}}(\omega ).$$ (13)

In Eq. (13), T⁻¹ is the inverse of the normalized saturation curve and the calibration function. RF_cal is the pre-distorted waveform.

In this case, artifacts were optimally suppressed by exclusively correcting for the saturation nonlinearity. The saturation nonlinearity was linearized by calculating the calibration function from a 2nd-order polynomial fit of the saturation curve (see the supplementary material for details). The red curve in Fig. 3(a) shows the shaper nonlinearity after calibration. The shaper output is linear at high intensity, while the low-intensity nonlinearity is left uncorrected.

R⁽⁵⁾_2Q spectra can be further improved by subtracting off a calculated artifact spectrum. This spectrum is calculated from the R⁽³⁾_1Q spectrum that was obtained simultaneously using four-frame phase cycling. We refer to this method as “calculated artifact subtraction.” To calculate the artifact, time domain R⁽³⁾_1Q data at t₁ = 0, 2Δt₁, 4Δt₁, 6Δt₁… are retained and data at t₁ = Δt₁, 3Δt₁, 5Δt₁, 7Δt₁… are discarded. The under-sampled dataset is then zero-padded after the FID to recover the original data dimensions. This operation mimics the under-sampling of the FID by $E_1^{(2)}$ and $E_3^{(2)}$, generating a 2D spectrum that appears at 2Q frequencies with elongated pump line shapes. The calculated artifact spectrum can then be scaled and subtracted from the 2Q spectrum measured with eight-frame phase cycling. When doing this, it is important that the final τ₁ delay is long enough to measure the entire R⁽³⁾_1Q FID. Also note that this method only corrects for artifacts introduced by the second order term in Eq. (11). This method is described graphically in Fig. S3.

Figure 4(a) shows the R⁽⁵⁾_2Q spectrum of the symmetric stretch mode of the (acetylacetonato)dicarbonyl iridium(I) (IDC) model compound without the use of any artifact suppression methods. Third-order artifacts appear along the pump axis at 2ω₀₁ = 4004 cm⁻¹. Along the probe axis, artifact spectra only contain features at the ν = 0 → 1 and ν = 1 → 2 transition frequencies of 2002 and 1987 cm⁻¹. R⁽⁵⁾_2Q spectra appear along the pump axis at the first overtone frequency 2ω₀₁ − Δ₁₂ = 3989 cm⁻¹. In addition to the fundamental and first overtone peaks along the probe axis, R⁽⁵⁾_2Q spectra also have a ν = 2 → 3 peak at 1971 cm⁻¹. Artifact features obscure R⁽⁵⁾_2Q spectra.

FIG. 4.

Performance of artifact removal from R⁽⁵⁾_2Q spectra. (a) Spectrum of the symmetric stretch diagonal peak without any artifact removal techniques. (b) Reduction of artifacts by pre-distorting waveform to calibrate for non-linearity. (c) Combination of waveform calibration and calculated artifact subtraction.

Figure 4(b) shows the R⁽⁵⁾_2Q spectrum after calibrating for pulse shaper nonlinearities. Artifacts are significantly reduced. Figure 5(c) shows the R⁽⁵⁾_2Q spectrum using waveform calibration and calculated artifact subtraction. The R⁽⁵⁾_2Q features are now unobscured by artifacts, clearly revealing the Δ₁₂ and Δ₂₃ anharmonic shifts. The signal to artifact ratio in Fig. 4(c) is 7:1. Artifacts are suppressed 19-fold when using both methods. The degree of attenuation required to obtain clean R⁽⁵⁾_2Q spectra depends on the relative intensity between the R⁽⁵⁾_2Q features and third--order artifact features. Assuming that all other things are equal, the degree of attenuation required to obtain clean R⁽⁵⁾_2Q spectra of a vibrational mode increases as its transition dipole decreases.

FIG. 5.

Measurement and processing of R⁽³⁾_2Q,NR and R⁽³⁾_0Q,R spectra using the permuted–pump–probe pulse sequence. Both (+ω₁, + ω₃) and (−ω₁, + ω₃) quadrants of the spectra are shown. (a) Spectrum acquired in the pump–probe geometry without any data processing. (b) Spectrum after separating R⁽³⁾_2Q,NR from R⁽³⁾_0Q,R. (c) Fully processed R⁽³⁾_2Q,NR and R⁽³⁾_0Q,R spectra after applying phase in Eq. (4).

IV. RESULTS

In this section, we first show how R⁽³⁾_2Q,NR and R⁽³⁾_0Q,R spectra are processed in the permuted–pump–probe geometry. We then proceed to report combination band specific R⁽³⁾_2Q,NR spectra and fully absorptive R⁽⁵⁾_2Q spectra.

Figure 5(a) shows the (+ω₁, + ω₃) and (−ω₁, + ω₃) quadrants of data collected using the permuted–pump–probe pulse sequence prior to any data processing. Features in each quadrant are mirror images of each other. The spectra are the sum of the R⁽³⁾_2Q,NR and R⁽³⁾_0Q,R appearing at 1Q frequencies due to the τ₁-dependent phase shift discussed in Sec. III B. The anharmonic shift for R⁽³⁾_2Q,NR spectra leads to imperfect cancellation of the R⁽³⁾_2Q,NR and R⁽³⁾_0Q,R dispersive features, yielding phase twisted line shapes. In the upper quadrant, anti-diagonal features (black oval) correspond to R⁽³⁾_2Q,NR spectra, whereas diagonal features (blue oval) correspond to R⁽³⁾_0Q,R spectra.

Figure 5(b) shows the spectra after separation of R⁽³⁾_2Q,NR and R⁽³⁾_0Q,R pathways. The R⁽³⁾_2Q,NR and R⁽³⁾_0Q,R spectra now appear in the (+ω₁, + ω₃) and (−ω₁, + ω₃) quadrants, respectively. Figure 5(c) shows the spectra in Fig. 5(b) after applying the phase in Eq. (4). R⁽³⁾_0Q,R spectra are flipped along the pump axis for consistency with processing methods for 2D rephasing spectra.⁵² R⁽³⁾_2Q,NR now appear at 2Q frequencies and R⁽³⁾_0Q,R now appear at 0Q frequencies. τ₂ = 0 for this measurement, so R⁽³⁾_0Q,NR pathways will also contribute to the spectra. The R⁽³⁾_0Q,NR spectra overlap with the R⁽³⁾_0Q,R spectra at 0Q frequencies.

Figure 6(a) shows the R⁽³⁾_2Q,NR spectrum measured using the XYXY polarization scheme after applying the data processing scheme shown above, where the first and third polarizations are E₂ and E₁. Diagonal peaks appear at the first overtone frequencies, 2ω₀₁ − Δ₁₂, which are (ω_pump, ω_probe) = (3987, 2000) and (4141, 2078) cm⁻¹, respectively, for IDC. The diagonal anharmonicities measured from this spectrum were 13 and 15 cm⁻¹ for the symmetric mode and antisymmetric mode, respectively. The cross peaks appear at (4051, 2000) and (4051, 2078) cm⁻¹, which appear at the combination band frequency along the pump axis. IDC consists of two strongly coupled oscillators, so the combination band frequency is ω₁ + ω₂ − β₁₂, where β₁₂ is the coupling strength between the two carbonyl local modes. The coupling strength was determined from the spectrum to be 27 cm⁻¹. Both the coupling and anharmonicity values agree with the values obtained from 2Q spectra measured in the four-wave-mixing geometry.⁴ Figure 6(b) shows the R⁽³⁾_2Q,NR spectrum collected using the ⟨0°, 60°, 0°, −60°⟩ polarization scheme. Overtone peaks on the diagonal are suppressed, making the cross peaks the largest signals.

FIG. 6.

2Q spectra measured in the pump–probe geometry. (a) R⁽³⁾_2Q,NR spectra measured in the XYXY and (b) ⟨0°, 60°, 0°, −60°⟩ polarization schemes. (c) R⁽⁵⁾_2Q measured at τ₂ = 500 fs in XXXXXX using both waveform calibration and calculated artifact subtraction.

Figure 6(c) shows the R⁽⁵⁾_2Q spectrum collected in XXXXXX. This spectrum is fully absorptive, increasing the spectral resolution over R⁽³⁾_2Q,NR, which has phase twisted line shapes.⁴ New peaks appear, which arise from the ν = 2 → 3 transitions. A complete description of peak assignments is given by Ding et al.⁴ In addition to diagonal peaks at 2ω₀₁ − Δ₁₂ and cross peaks at ω₁ + ω₂ − β₁₂, a new set of cross peaks are present at 2ω₀₁ − Δ₁₂. These peaks correlate the overtone transition of one mode to the fundamental transition of a mode that it is coupled to.

V. DISCUSSION AND CONCLUSION

2Q experiments were more common when 2D optical spectroscopy was first invented because the first implementations of 2D spectroscopy were in the four-wave-mixing geometry, where phase error correction and use of an external local oscillator were required.^27,28 2Q spectra were often analyzed in parallel with 1Q spectra to interpret vibrational and electronic eigenstructure.^7,8 For example, Ge and co-workers modeled third-order 2Q spectra to resolve conformers of N-acetyl-l-prolinamide in CDCl₃ and help determine the peptide length at which 3₁₀ helical structures developed in Aib oligopeptides.^7,8 Those publications are good examples of how 2Q spectral analysis is a useful tool for disentangling complicated spectra.

Since the invention of 2D spectroscopy, the pump–probe geometry has emerged as a relatively straightforward way to collect 1Q absorptive spectra: the most common 2D experiment.⁵³ One of the few remaining drawbacks of the pump–probe geometry is that some signals, such as the third-order 2Q signal, emit in a different phase matching direction. Therefore, as the pulse-shaping pump–probe geometry grew more popular, 2Q experiments were performed less frequently.

The development of the permuted–pump–probe pulse sequence reported here eliminates this drawback and enables 2Q experiments in the pump–probe geometry. By re-programming the pulse shaper and postprocessing the data to extract the 2Q signals, anyone with a pulse shaping 2D spectrometer can now measure a 2Q spectrum. The permuted–pump–probe pulse sequence retains the phase stability and shot-to-shot readout typical of 1Q experiments performed with a pulse shaper. We also demonstrate the recently reported ⟨0°, 0°, 60°, −60°⟩ polarization scheme,⁴² adapted to the 2Q pulse sequence, which eliminates diagonal peaks from the spectra to better resolve the cross peaks.

We also demonstrate the measurement of fifth-order 2Q absorptive spectra in the infrared using a transverse pulse shaper. Demonstration of fifth-order 2Q absorptive spectra in the pump–probe with a transverse pulse shaper is important because RF amplification nonlinearities manifest themselves differently in pulse trains generated by transverse and longitudinal pulse shapers.²⁶ Many pulse-shaping 2D spectrometers use transverse pulse shapers. We acquired fifth-order 2Q absorptive spectra with high signal to noise. The spectra contain third-order artifacts that arise from RF amplification nonlinearities in the pulse shaper. We present two methods that efficiently suppress artifacts when used in tandem. The spectra may be further improved by using more advanced RF waveform calibration protocols. The electrical engineering community has placed significant effort into correcting for RF amplification nonlinearities.⁵⁴ Therefore, we suggest incorporating their methods into pulse shaping 2D spectroscopy to further improve fifth-order 2Q spectra.

The third- and fifth-order 2Q experiments were inspired by multiquantum NMR pulse sequences. In NMR spectroscopy, the pulse trains are entirely collinear and computer programmed. Indeed, the samples are too small compared to the wavelength of the radio frequency pulses to use phase matching. The pump–probe geometry utilizes phase matching to eliminate many background signals and when implemented with a pulse shaper, can switch between pulse sequences solely by computer control.³⁹ Selecting pathways partially by phase matching and partially by computer programming is a useful combination.

SUPPLEMENTARY MATERIAL

See the supplementary material for a description of alternate time orderings in the permuted–pump–probe geometry, simulated electric fields of artifact pulses, graphical description of calculated artifact subtraction, and experimental details regarding empirical linearization of saturation nonlinearity.

AUTHOR DECLARATIONS

Conflict of Interest

M.T.Z. is a co-owner of PhaseTech Spectroscopy, Inc., which sells 2D IR/visible spectrometers, pulse shapers, and high-repetition rate MCT detection electronics.

DATA AVAILABILITY

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