Phase-Matching and Dilution Effects in Two-Dimensional

Apr 12, 2013 - We present theoretical and experimental data for the attenuation of the cascade signal in ... Journal of Chemical Theory and Computatio...
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Phase-Matching and Dilution Effects in Two-Dimensional Femtosecond Stimulated Raman Spectroscopy Barbara Dunlap, Kristina C. Wilson, and David W. McCamant* Department of Chemistry, University of Rochester, 120 Trustee Rd., Rochester, New York 14627, United States S Supporting Information *

ABSTRACT: We present theoretical and experimental data for the attenuation of the cascade signal in two-dimensional femtosecond stimulated Raman spectroscopy (2D-FSRS). In previous studies, the cascade signal, caused by two third-order interactions, was found to overwhelm the desired fifth-order signal that would measure vibrational anharmonic coupling. Theoretically, it is found that changing the phase-matching conditions and sample concentration would attenuate the cascade signal, while only slightly decreasing the fifth-order signal. By increasing the crossing angle between the Raman pump and probe and the impulsive pump and probe, the phase-matching efficiency of the cascade signal is significantly attenuated, while the fifth-order efficiency remains constant. The dilution experiments take advantage of the difference in the concentration dependence for the fifth-order and cascade signal, in which the fifth-order signal is proportional to concentration and the cascade signal is proportional to concentration squared. Experimentally, it is difficult to see a trend in the data due to instability in signal in the phase-matching experiments and lack of signal at low concentrations in the dilution experiments.

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with previous fifth-order Raman techniques, the desired fifthorder signal is overwhelmed with signal from two consecutive third-order interactions, or cascades. Because of this, the signal does not measure anharmonic coupling or nonlinear polarizability between the modes and is instead just proportional to the product of the Raman scattering cross sections. These cascades have been a persistent problem in higher order coherent Raman spectroscopies for some time.15,16 An excellent early discussion was provided by Ivanecky and Wright.17 Recent theoretical work by Lee and co-workers describes the fifthorder and cascade processes using a time-dependent wave packet approach.18 Wave-mixing energy level (WMEL) diagrams for the “normal” third-order FSRS and for the fifth-order 2D-FSRS signals are shown in Figure 2. The WMEL diagrams are used to keep track of the amount of energy imparted to the sample by the different pulses and the vibrational state of the low- and high-frequency mode as time evolves from left to right.8,19 The energy of each vibrational level is labeled in Figure 2a and denoted by the quantum numbers (vlow, vhi) of the two vibrations with fundamental frequencies, ωlow and ωhi. The frequency of the signal for each process is labeled as ωFSRS. Figure 2a is the WMEL for a fundamental FSRS transition, where the signal appears at the Raman shift of the high frequency mode, ωFSRS = ωRpu − ωhi. The combination band is

INTRODUCTION Femtosecond stimulated Raman spectroscopy (FSRS) is an experimental technique used to measure changes in a stimulated Raman spectrum as a function of time when a photochemical reaction is initiated or vibrational modes are pumped into coherence. It has proven to be a useful tool for studying photochemistry,1−7 and its ability to impart and probe vibrational coherence has made it a promising technique for multidimensional vibrational spectroscopy. Two-dimensional (2D) spectroscopies have been used to better understand vibrational coupling in solvent and solute−solvent systems and biological structures by analysis of vibrational anharmonicity.8,9 The built-in phase-sensitive detection in FSRS has been used to observe vibrational frequency changes that are occurring faster than the vibrational dephasing time.3,10 For these reasons, it was thought that two-dimensional femtosecond stimulated Raman spectroscopy (2D-FSRS) (Figure 1) could be a Raman analogue to femtosecond two-dimensional infrared (2D-IR) spectroscopy to measure anharmonic coupling between vibrational modes. This method would measure the fifthorder nonlinear susceptibility, χ(5), which is proportional to the anharmonic coupling between two Raman-active vibrational modes. By measuring this anharmonic coupling we can learn about the mechanisms of intramolecular vibrational energy redistribution (VER or IVR). Initial work performed by Kukura et al.11,12 attributed observed signals to anharmonic coupling between the mode initially driven into coherence at frequency ωlow (Figure 1) and the mode probed by the FSRS pulses, at frequency ωhi (Figure 1). Work performed in our group by Mehlenbacher et al.13 and Wilson et al.14 later showed that, as © 2013 American Chemical Society

Special Issue: Prof. John C. Wright Festschrift Received: January 15, 2013 Revised: April 5, 2013 Published: April 12, 2013 6205

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Figure 2. WMEL diagrams for third-order FSRS and fifth-order 2DFSRS signals. Diagrams a and b describe the fundamental and combination band FSRS transitions, respectively. In each diagram, time evolves from the left to right, and dashed lines indicate dipole field coupling on the bra side of the density matrix, while solid lines represent couplings on the ket side. For both processes the Raman pump (green) and probe (red) act together to create a stimulated Raman spectrum. The frequency of the signal is generated at the Raman shift of ωFSRS, which is different for a and b. For the 2D-FSRS transitions in c and d, the impulsive pump (blue) is added to drive the low-frequency mode into coherence. Then the Raman pump and probe act on the same molecule to produce signal at either the combination frequency (c) or the difference transition frequency (d).

Figure 1. Schematic overview of the 2D-FSRS experiment. (a) Three pulses are overlapped in the sample, two short duration pulses: an initial impulsive pump (blue) and subsequent probe pulse (red) together with one long duration pulse, the Raman pump (green), which is overlapped in time with the probe. The θeffective is the crossing angle between the probe and impulsive pump, which is varied in this work. (b) The impulsive pump drives low-frequency modes into coherence. With no anharmonic coupling, the Raman pump and probe simply take a stimulated Raman spectrum of the high-frequency modes in the sample. (c) The signal from the stimulated Raman spectrum appears at ω = ωRpu − ωhi in the detected probe spectrum. When there is anharmonic coupling between the low- and high-frequency modes, sidebands may be observed in the stimulated Raman spectrum at Raman shifts of ω = ω(+) = ωRpu − (ωhi + ωlow) and ω = ω(−) = ωRpu − (ωhi − ωlow).

high-frequency transitions, ωFSRS = ωRpu − (ωhi − ωlow). However, in both cases (Figure 2c and d), the strength of the signal measures the magnitude of the anharmonic force constant that mixes the two normal modes. In practice, the signal is detected as a new peak in the probe spectrum at the Raman shift of ω(±) = ωRpu − (ωhi ± ωlow). As the IRS/probe delay, Δt, is varied, the signal phase shifts, and the observed peak oscillates from a positive Lorentzian line shape, through a dispersive line shape, to a negative Lorentzian peak. These oscillations can then be Fourier transformed to produce a 2DFSRS spectrum with the oscillatory signal becoming a peak at ω2 = ωlow and ωFSRS = ωRpu − (ωhi ± ωlow). Figure 2c and d shows the possible fifth-order signals when all five driving fields act on the same molecule, but it is also possible that separate four wave mixing (FWM) processes occur on two different molecules, generating a cascade. The cascade signal travels along the same wave vector and has the same Raman shift frequency as the fifth-order signal, although it lacks the small anharmonic shift, Xhi,low. The first third-order process in the cascade, which we call four wave mixing A (FWM-A), is either a coherent anti-Stokes Raman spectroscopy (CARS) (Figure 3a) or coherent Stokes Raman spectroscopy (CSRS) (Figure 3b) interaction on molecule A, creating a new field that can interact with molecule B in a second third-order process, which we term FWM-B. There are four different combinations of the pulses that can create the cascade signals, depending on whether FWM-A is a CSRS or CARS interaction and whether the created field is the initiating (sequential cascades, Figure 3c and d) or signal-generating (parallel cascades, Figure 3a and b) field in FWM-B.

observed at ωFSRS = ωRpu − (ωhi + ωlow + Xhi,low) (Figure 2b) by a similar FSRS process. This signal frequency measures the frequency of the low- and high-frequency modes plus the anharmonic shift, Xhi,low, between them, which is usually less than 10 cm−1. The extent to which the combination band is an allowed transition depends on the extent to which anharmonic coupling, or nonlinear polarizability, mixes the two nominally independent vibrations.20 For instance, if the vibrational potential is perfectly harmonic, then no combination signal can be generated. In 2D-FSRS (Figure 2b and c), the impulsive pump is added to drive the low-frequency mode into coherence by impulsive Raman scattering (IRS). At different time delays, Δt, the Raman pump and probe act on the sample and generate a FSRS spectrum. When the impulsive pump and the first set of Raman pump and probe act on the bra side of the density matrix (Figure 2c), the signal is detected at ωFSRS = ωRpu − (ωhi + ωlow + Xhi,low); that is, the Raman shift of the combination band, including the anharmonic shift, and the strength of the signal is proportional the third-order off-diagonal force constant coupling the vibrations.13 When the impulsive pump acts on the ket side and the Raman pump and probe then act on the bra side (Figure 2d), the signal frequency does not measure the anharmonic coupling, just the difference between the low- and 6206

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length, l, but this variable is harder to reduce when using a freeflowing liquid jet, which is already just 100−200 μm thick. In this paper, we implement previous suggestions13,14 by theoretically and experimentally measuring how the fifth-order and cascade signals are affected by changing the phasematching conditions and sample concentration. In 2D-FSRS, attenuating the cascade signal is difficult because there are only three beams incident on the sample. Previous low frequency fifth-order Raman incorporated five separate beams, which allowed attenuation of the cascades with innovative and complicated focusing conditions.16,21,22

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EXPERIMENTAL SECTION Apparatus. The experimental set up has been described previously.13 Briefly, the output of an amplified Ti:sapphire regenerative amplifier (Spectra-Physics Spitfire, 1 kHz, 2.3 mJ, 100 fs, 800 nm) is split into three beams. The impulsive pump pulse at 530 nm is generated from a two-stage noncollinear phase-matched optical parametric amplifier (NOPA) and compressed with a fused silica prism pair to 27 fs as measured by second harmonic generation (SHG) autocorrelation (MiniOptic Delta). The Raman pump is produced by spectrally filtering the Ti:sapphire amplifier fundamental and frequency doubling the resultant pulse in a 3-mm beta-barium borate crystal, thereby producing a 2-ps pulse at 400 nm. A small portion of the amplifier output was frequency doubled and used to generate the probe via supercontinuum generation in a 2 mm thick CaF2 crystal. All three beams were focused into the sample by a 100-mm focal length spherical mirror. The beam diameters of the impulsive pump and Raman pump were approximately 40 μm at the sample, and the diameter of the probe is ∼20 μm. After the sample, the probe was dispersed by a spectrograph (Acton, 300 mm, 600 grooves/mm, second order diffraction) and measured by a CCD camera (Princeton Instruments Pixis 100BR). This new CCD detection system has a greatly improved signal-to-noise ratio (SNR) relative to our previous diode-array detector13,14 because of the 1 kHz readout and 500 Hz chopping of the Raman pump beam. The improved SNR makes the dilution experiments possible. Each spectrum at a particular time delay was the ratio of 500 measurements of the ratio of the probe spectrum with the impulsive pump and Raman pump on divided by the probe spectrum with the impulsive pump on, each measured in subsequent 1 ms exposures of the detector. The time delay was varied by changing the arrival time of the impulsive pump pulse relative to the temporally overlapped Raman pump and probe. Time delays were scanned from −200 to 1600 fs in 15 fs steps for the phase-matching experiments and in 10 fs steps for the dilution experiments. The time resolution was approximately Gaussian with a 40 fs full width at half-maximum (fwhm) as determined by cross correlation at the sample using the optical Kerr effect, also known as polarization gated frequency resolved optical gating, between the impulsive pump and probe.23 The sample, chloroform and/or deuterated chloroform, used as received, was illuminated in a ∼100 μm thick wire-guided liquid jet that is gravity-fed by an upper sample reservoir replenished by a peristaltic pump.6 The spectral resolution of the resultant 1D-FSRS spectra was 18 cm−1 (6 pixels) along the Raman shift (ωRpu − ω) axis. Phase-Matching Methodology. FSRS spectra were taken with different beam separations at the focusing mirror before the sample to measure the effects of beam geometry on the

Figure 3. WMEL diagrams for the cascade signals possible in 2D FSRS. A new coherent field (purple) is created by either a CARS (a and c) or CSRS (b and d) interaction on molecule A (FWM-A), which then acts on molecule B (FWM-B) to either generate the signal, as in a and b, or to initiate the second four wave mixing interaction, as in c and d. Hence, a and b are termed “parallel” cascades, and c and d are termed “sequential” cascades.

From calculations done by Mehlenbacher et al., the cascades were predicted to have an intensity that is approximately 730 times larger than the fifth-order signal in deuterated chloroform.13 In two papers from Wilson et al.14 and Mehlenbacher et al.,13 our group proposed changes to the experiment that would attenuate the cascade signal. The first is to change the beam geometry to maximize the phase-matching of the fifth-order signal while reducing the phase-matching efficiency of the cascade signal (see phase-matching theory below). The second is to decrease the concentration of the sample, because the cascade signal is proportional to concentration squared, whereas the fifth-order signal strength is proportional to concentration.13 This was shown in our previous theoretical treatment, which established that the signal strengths of the fifth-order and cascade signals are: 2 2 |ΔI5th| ∝ lNE RPu E Ipu (α′hi )2 α′low ·

dω hi dQ low

2 2 |ΔIcascade| ∝ l 2N 2E Rpu E Ipu (α′hi )2 (α′low )2

(1a) (1b)

Here, l is the sample length, N is the number density of the sample, ERpu is the electric field of the Raman pump, EIpu is the electric field of the impulsive pump, α′ = dα/dQ is the derivative of the molecular polarizability with respect to the displacement along the vibrational mode, and dω/dQ is the anharmonicity. The fifth-order intensity (eq 1a) is directionally proportional to number density, N, and the cascade intensity (eq 1b) is proportional to number density squared, N2. (Hereafter, we will refer to N as the relative number density, that is, a 50% dilution by volume would correspond to N = 0.5.) There is a similar difference in the dependence on sample 6207

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were taken and averaged. Measurements of each concentration were repeated three times. The Raman pump energy was 7 μJ, and impulsive pump energy was 800 nJ.

sideband peak intensity (see Figure 4). The separation between the Raman pump and probe was 20 mm at a 100-mm focal

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RESULTS Phase-Matching Theory. To figure out how changing the beam geometry should affect the fifth-order and cascade signals, we calculated the phase-matching mismatch, Δk, of the signal according to the wave mixing energy level diagrams (Figures 2 and 3). For 2D-FSRS, the signal field, Es, is generated by a fifthorder polarization, P(5), but the ability of the polarization to generate a coherent field is attenuated by phase-matching terms which were ignored in our previous work.13,14 The wave equation dictates that the signal is Figure 4. Diagram of the beam focusing geometry at the sample. The diagram indicates how the change in index of refraction changes the effective separation between the probe, Raman pump, and impulsive pump. Not to scale.

⎯⇀ ⎯

Es =

⎯ (n) ⎡ Δk·l ⎤ −iΔkl /2 −4πlωs ⎯⇀ ·i· P 0 (ωs) ·sinc⎢ ·e ⎣ 2 ⎥⎦ n(ωs)

(2)

in which Δk is the phase-mismatch term that attenuates the detected signal via the sinc term and imparts a phase shift on the signal via the final exponential term.24 The phase-mismatch, Δk, was assumed to be zero in our previous work.13,14 P(n) in eq 2 corresponds to either P(5) for the fifth-order signal or to P(3) for FWM-A and FWM-B. Hence, the overall cascade phasematching efficiency is the product of the efficiencies of the FWM-A and FWM-B processes. The phase-matching equations for each signal come from conservation of momentum and are calculated by keeping track of the momentum each pulse imparts to the sample,

length mirror 100 mm from the sample. The separation between the impulsive pump and probe, xactual, in Figure 4, was varied. The separation distances were 7, 11, 15, 19, and 22 mm. A translation stage on the mirror before the focusing mirror was used to reproduce beam positions of the impulsive pump. After beam translation, overlap was adjusted to achieve maximum signal. When the beams pass into the sample, the effective separation changes because of the change in index of refraction according to Snell’s Law (Figure 4). This means that in the sample the effective separation of the beams is smaller than the actual separation of the beams at the mirror. Table 1 shows the

(+)

⃗ ‐FSRS = −kIpu1 ⃗ + kIpu2 ⃗ − kRpu ⃗ + k pr ⃗ + kRpu ⃗ k 2D

Table 1. Actual and Effective Separation of the Pump and Probe Beams Used To Vary the Phase Matching of the Cascade Signal

(−)

⃗ ‐FSRS = +kIpu1 ⃗ − kIpu2 ⃗ − kRpu ⃗ + k pr ⃗ + kRpu ⃗ k 2D (CARS)

beam impulsive pump

Raman pump

actual separation

actual angle

effective separation

effective angle

xactual (mm)

θact (deg)

xeffective (mm)

θeff (deg)

−7.0 −11.0 −15.0 −19.0 −22.5 20.0

−4.0 −6.3 −8.5 −10.8 −13.0 11.3

−4.8 −7.6 −10.3 −13.0 −15.7 13.7

−2.8 −4.3 −5.9 −7.4 −8.9 7.8

⃗ ⃗ − kIpu2 ⃗ + kRpu ⃗ = +kIpu1 kFWMA (CSRS)

⃗ ⃗ + kIpu2 ⃗ + kRpu ⃗ = − kIpu1 kFWMA (par)

⃗ ⃗ + k pr ⃗ + kFWMA ⃗ = − kRpu kFWMB (seq)

⃗ ⃗ ⃗ + kRpu ⃗ = − kFWMA + k pr kFWMB

(3)

where the vector sums follow Figures 2 and 3 in order. Figure 5 shows the vector diagrams for the 2D-FSRS(−), the CARS transition for FWM-A, and the subsequent parallel cascade (FWM-B). This figure is a visual representative of the phasematching equations in eq 3. The vector diagrams for the remaining phase-matching equations are in the Supporting Information. ⇀ In eq 3, k is defined by

comparisons between the actual and effective separation. (Note that at all crossing angles the interaction length of the three beams is much larger than the ∼100 μm sample, so no attenuation due to decreased interaction volume is expected.) Four identical scans were averaged, and each beam geometry was repeated five times. The Raman pump energy was 7 μJ, and impulsive pump energy was 600 nJ. These powers were used because this was the highest pump intensity where no nonlinear effects, such as self-focusing and self-phase modulation, were observed. Dilution Methodology. For the dilution experiments, the separation between the Raman pump and probe and impulsive pump and probe were both fixed at 20 mm, producing an effective crossing angle of 15.6° between the Raman pump and impulsive pump in the sample. The concentrations that were used are expressed as a volume ratio of chloroform to deuterated chloroform and are 100:0, 90:10, 75:25, 50:50, 25:75, 10:90, and 0:100. Twelve scans of each concentration

⇀

k =

n(ω)ω r̂ c

(4)

in which n(ω) is the frequency-dependent refractive index, ω is the radial frequency of the pulse, and r̂ is the unit vector describing the direction of propagation of the beam. In eq 3, we also distinguish between the two arrows of the impulsive pump interaction, assigning kIpu1 to the upward transition and kIpu2 to the downward transition. Since the impulsive pump is a broadband pulse, the frequencies of each coupling could be anywhere within the bandwidth of the impulsive pump beam; however, their frequency difference needs to match the driven 6208

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Figure 5. Vector diagrams for the fifth-order 2D-FSRS(−), the CARS FWM-A process, and the resultant parallel cascade. Diagram a describes the downshifted fifth-order signal according to eq 3. As described in the text, the two fields from the impulsive pump (blue) are slightly different energies, so kIpu1 is longer than kIpu2. The two Raman pump (green) arrows are the same length, and the probe is red. The black dotted line is the created signal. Diagram b describes the FWM-A CARS process. The signal in this diagram is then used in the FWM-B parallel cascade (c) as the FWM-A vector (purple). Note that in a and c the resultant measured signal propagates almost parallel to the probe wavevector.

Figure 6. Simulated phase-matching efficiency of the various wave mixing processes that can generate signal at the (ν̃hi + ν̃low) = 669 + 262 cm−1 sideband. Efficiencies are shown as a function of the position of the impulsive pump (x and y axes), given that the probe (red dot) is at the origin and the Raman pump (green dot) is 7.8° to the right (positive crossing angle). (a) Fifth-order signal and (b) net parallel cascade signal. (c) Simulated phase-matching efficiency of the CSRS FWM-A signal and (d) FWM-B, which multiply together to produce the net parallel cascade efficiency shown in b. See text for details.

vibrational frequency. To account for this, ωIpu1 = ωNOPA + ωlow/2 and ωIpu2 = ωNOPA − ωlow/2, in which ωNOPA is the central frequency of the impulsive pump. This is important because, without the recognition of the frequency difference between the two dipole couplings that drive the initial coherence in ωlow, the impulsive pump would impart no momentum on the sample and each signal would be perfectly phase-matched. Each signal in eq 3 is used as the polarization wave vector, kP, to calculate the phase-mismatch between the momentum imparted to the sample, kP, and the wave vector of the resultant signal, ksig: Δk = |kP⃗ | − ksig = |kP⃗ | −

were performed using an effective crossing angle between the probe pulse (red) and Raman pump pulse (green) of +7.8°, while the effective crossing angle of the impulsive pump pulse is varied. (The effective crossing angle is the crossing angle inside the sample, which takes into account the index of refraction of the sample. As shown in Figure 4, refraction at the air/liquid interface reduces the crossing angle of the beams inside the sample so the largest possible crossing effective angle is 9.7°, rather than 14.0°, which is the largest θactual allowed using a 50 mm diameter, 100 mm focal length focusing mirror.) The position of the impulsive pump was varied to calculate phase-matching efficiency for all crossing angles from −10° to +10° in both the horizontal and vertical directions, limited by the size of the mirror before the sample. Note that, while phasematching calculations were performed for every combination of ν̃hi ± ν̃low sideband peak in the potential 2D-FSRS window, only a subset of the calculations are shown here for clarity. The contours in Figure 6 correspond to the phase-matching efficiency of the signal (eq 6) for the different transitions in the sample as the crossing angle of the impulsive pump is varied. The fifth-order signal (Figure 6a) is above 70% for all positions of the impulsive pump. The net efficiency of the parallel cascade transition (Figure 6b) is much less than the fifth-order signal especially at large impulsive pump angles. The cascade efficiency depends on two processes, (1) the four wave mixing A (FWM-A) and (2) the four wave mixing B (FWM-B) signal, shown in Figure 5c and d, respectively. The efficiency of the net cascade is the product of the efficiency of FWM-A and B, since, in order for the cascade to be generated, first the intermediate FWM-A field needs to be formed and then in a second nonlinear process, with its own efficiency, that field

n(ωsig )ωsig c

(5)

Finally, the phase-matching efficiency, F, is calculated according to ⎡ Δk·l ⎤ F = sinc⎢ ⎣ 2 ⎥⎦

(6)

−i(Δk)l/2

The phase shift, e , in eq 2 is ignored, since to-date we have ignored the phase of the 2D-FSRS signals and just calculated the magnitude of the Fourier transformed (FT) signal. Also, in this work we focus solely on the sidebands that occur at Raman shifts of ωhi ± ωlow, which can be generated by either a fifth-order mechanism or by the parallel cascade. Hence, the phase-matching of the sequential cascade, which generates a 2D signal at the Raman shifts of ωhi, will be ignored. Figure 6 shows the results of calculating the phase-matching efficiency, F, of the signal as the position of the impulsive pump beam is varied for the 262 cm−1 low-frequency upshifted from the 669 cm−1 high-frequency mode in chloroform. Early calculations (not shown) indicate that the largest range of phase-matching efficiencies could be accessed if the Raman pump and probe crossing angle was large. Hence, calculations 6209

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the four sidebands, the phase-matching efficiency of the FWMB transition varies less than the FWM-A transition as the effective crossing angle between the probe and impulsive pump increases. In particular, the FWM-B process has greater than 80% efficiency at all crossing angles for the 262 cm−1 sidebands upshifted from the 669 cm−1 (Figure 7a) and downshifted from the 3019 cm−1 (Figure 7c) high-frequency modes. For the 669 cm−1 sidebands upshifted from the 669 cm−1 (Figure 7b) and downshifted from the 3019 cm−1 (Figure 7d) modes, the efficiency of FWM-B is still usually more efficient than the FWM-A transition except at extreme angles. When the FWM-A and B efficiencies are multiplied to calculate the net cascade phase-matching efficiency, the slight phase-mismatch of FWMB is enough to cause the net cascade to be significantly attenuated. Surprisingly, the phase-matching of the sidebands downshifted from the 3019 cm−1 high-frequency mode are more efficient compared to the sideband upshifted from the 669 cm−1 high-frequency mode. Figure 8a compares the overall effect on the fifth-order and the cascade signal by showing the horizontal slice along the y =

needs to generate the detected signal in the FWM-B process. Because the sample polarization needs to collapse twice, once in FWM-A and then again in FWM-B, the overall efficiency is the product of the efficiency of each step. Hence, by multiplying the FWM-A and FWM-B phase-matching efficiencies, the efficiency of the net cascade signal can be found. As shown in Figure 6d, the FWM-B process is just as well phase-matched as the fifth-order signal. The significant attenuation of the net cascade signal actually comes from the FWM-A process being poorly phase-matched, apparent in Figure 6c. Calculations using a variety of laser wavelengths indicate that the significant attenuation of the FWM-A signal in Figure 6c is due to the dispersion in the solvent refractive index in the CARS and CSRS region around the 400 nm Raman pump wavelength. Note that, because of molecular symmetry constraints, the sideband shown in Figure 6 cannot actually have any fifth-order signal, since the 262 cm−1 mode has E symmetry and the 669 cm−1 A1 symmetry. The data are shown as an example of changes in phase-matching efficiency independent of the actual strength of the signals. A comparison of the two processes that make up the net parallel cascade transition is shown in Figure 7. The impulsive

Figure 8. Comparison of simulated phase-matching efficiency from the fifth-order process (solid black) and parallel cascade (dash dot green) as a function of impulsive pump and probe crossing angle for the (a) (ν̃hi + ν̃low) = 669 + 262 cm−1 sideband, (b) (ν̃hi + ν̃low) = 669 + 669 cm−1 sideband, (c) (ν̃hi − ν̃low) = 3019 − 262 cm−1 sideband, and (d) (ν̃hi − ν̃low) = 3019 − 669 cm−1 sideband. For each calculation, the effective Raman pump−probe crossing angle was held at +7.8°.

Figure 7. Simulated phase-matching efficiencies of the transitions that create the parallel cascade (green, dot-dashed) as a function of the crossing angle between the impulsive pump and probe. The two transitions making up the cascade are FWM-A (red, dotted) and FWM-B (blue, dashed), which multiply together to produce the net efficiency (green, dot-dashed). Efficiencies are shown for the (a) (ν̃hi + ν̃low) = 669 + 262 cm−1 sideband, (b) (ν̃hi + ν̃low) = 669 + 669 cm−1 sideband, (c) (ν̃hi − ν̃low) = 3019 − 262 cm−1 sideband, and (d) (ν̃hi − ν̃low) = 3019 − 669 cm−1 sideband. For each calculation, the effective Raman pump−probe crossing angle was held at +7.8°.

0 mm from Figure 6a and b. This demonstrates how the efficiency of the fifth-order signal and cascade signal are affected differently as the crossing angle between the probe and impulsive pump increases. The three other sidebands are shown for comparison in Figure 8b, c, and d. For all four sidebands, the cascade phase-matching efficiency decreases more than the fifth-order signal as the crossing angle between the probe and impulsive pump increases. For example, the efficiency of the fifth-order signal of the 262 cm−1 sideband upshifted from the 669 cm−1 mode (Figure 8a, solid black) decreases from a maximum of 0.926 at 0° to a minimum of 0.816 at −9.7°. The cascade phase-matching efficiency has a maximum of 0.824 at +2.7° and minimum of 0.465 at −9.7° (Figure 8a, green dash

pump could be placed anywhere on the mirror before the sample, but to simplify the experiment we decided to only change the impulsive pump position along the horizontal axis. The slices were calculated the same way as Figure 6, but only plotted along the y = 0° horizontal slice. For each sideband, the net efficiency of the parallel cascade signal (green, dot−dashed Figure 7) depends more on the FWM-A (red, dotted) phasematching efficiency than FWM-B (blue, dashed). For each of 6210

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dot). This corresponds to a 11.9% attenuation in the fifth-order signal and a 43.6% decrease in the cascade signal at large impulsive pump probe crossing angles. For the 262 cm−1 sideband downshifted from the 3019 cm−1 mode (Figure 8c), the fifth-order signal has a maximum of 0.975 at 0° and decreases to 0.900 by −9.7°. The cascade at this frequency is maximized at +3.5° with a 0.861 efficiency and decreases to 0.505 efficiency at −9.7° (Figure 8c, black). In general the dominant contribution of FMW-A to the cascade efficiency (see Figure 6c and Figure 7) means the maximum cascade signal is obtained with the impulsive pump situated nearly collinear with the Raman pump (i.e., at θeff > 0). For the 669 cm−1 sideband upshifted from the 669 cm−1 fundamental (Figure 8b), we see the fifth-order signal decrease from a maximum of 0.604 at 0° to 0.155 at −9.7°. The cascade contribution to this peak has a maximum of 0.266 at 2.7°, decreases to 0 at −5.5°, increases to 0.050 efficiency at −8.2°, and decreases to 0.034 at −9.7°. The F = 0 node in the cascade efficiency at 5.5° is caused by the zero in the sinc function in eq 5. A similar node is apparent in the 669 cm−1 sideband downshifted from the 3019 cm−1, occurring at −4.3° (Figure 8d, green dash dot). The nodes are created by the sinc term in the wave equation (eq 2) when the phase-mismatch and sample length multiply together to equal a factor of 2π, which causes the sinc term to go to zero. Unfortunately, because of the dependence on sample length, there is no simple analytical form for the angles at which these nodes occur. These nodes may be important in future experimental work because, in the ideal case, crossing the beams at this angle would completely extinguish the 669 cm−1 sidebands due to the cascade. Unfortunately, a different node angle is necessary for each impulsively driven mode frequency, so extinguishing the cascades of all pumped modes is not possible in broadband pumping schemes such as ours. Also note that the fifth-order signal is symmetric around 0° crossing angle, but the cascade signal has a maximum around +3°. Phase-Matching Experimental Results. Figure 9 shows a 2D-FSRS spectrum of CHCl3 taken with an effective crossing angle of −2.8° between the probe and impulsive pump. Figure 9a and b show the oscillations in the lineshapes of the sidebands at different time delays for the low-frequency modes that are up- and downshifted from the high-frequency modes. As can be seen in Figure 9a the sideband at 931 cm−1, which is the 262 cm −1 sideband upshifted from the 669 cm −1 fundamental, has a dispersive line shape at 100 fs (red), but the sideband has a positive Lorentzian line shape at 130 fs (blue). At 160 fs (green), it has a dispersive line shape and at 190 fs (black) a negative Lorentzian line shape. The same oscillations can be seen in the other up- and downshifted sidebands. When the scan is Fourier transformed along the Δt axis (Figure 9c) the oscillations become sideband peaks along horizontal slices at ω2/2πc = 262 cm−1, 369 cm−1, and 669 cm−1. The dotted diagonal lines are drawn along the sideband peaks that are coupled to the same high frequency mode. Figure 10 shows the comparison of the calculated efficiency of the parallel cascade and experimental signal as the separation between the impulsive pump and probe is varied. The experimental data is the FT magnitude of the sideband peaks with the right-hand vertical axis scaled so that the data at a crossing angle of −8.9° is the same intensity as the calculated efficiency. The error bars are the standard deviation of the peak heights from the five scans. Due to symmetry constraints in chloroform, the E CCl3 bend at 262 cm−1 cannot anharmoni-

Figure 9. (a) Experimental time-dependent spectra of CHCl3 showing the 262, 369, and 669 cm−1 sidebands upshifted from the 669 cm−1 fundamental and (b) up- and downshifted from the 3019 cm−1 fundamental. (c) 2D-FSRS spectrum of CHCl3 with an effective crossing angle of −2.8° between the probe and impulsive pump.

Figure 10. Comparison of theoretical efficiency and experimental intensity of sidebands. Shown are the theoretical efficiencies of the parallel cascade signals (red curve, left-hand axis) and experimental FT magnitudes (blue points, right-hand axis) of the peaks upshifted from the 669 cm−1 fundamental at the (a) 262 cm−1, (b) 369 cm−1, and (c) 669 cm−1 sidebands and downshifted from the 3019 cm−1 fundamental at (d) 262 cm−1, (e) 369 cm−1, and (f) 669 cm−1 sidebands.

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cally couple to either the A1 C−Cl stretch at 669 cm−1 or A1 C−H stretch at 3019 cm−1.13 This means that any signal that appears at the 262 cm−1 sideband peaks is only from cascade transitions and should follow the calculated efficiency of the cascade signal (Figure 10a and d). The signals at the 369 cm−1 and 669 cm−1 sideband peaks are potentially a mixture of a weak fifth-order signal and a dominant parallel cascade signal, though the fifth-order signal may be below our detection threshold. Note that, even with the attenuation of the cascade signal at large crossing angles, the cascade signal will still be the dominant signal and the experimental data should follow the general trend of the calculated phase-matching efficiency for the cascade signal for the different sidebands. This is not the case, as seen in Figure 10. The experimental signal of the sidebands does decrease with increasing crossing angle (i.e., going to larger negative angles), but not as much as is expected from the calculations. For the 262 cm−1 sideband upshifted from the 669 cm−1 fundamental, the calculated attenuation should be 32%, but the experimental attenuation is 12% as the effective crossing angle increases from −2.8° to −8.9°. The attenuation of the calculated efficiency for the 669 + 369 cm−1 sidebands is 60%, while the experimental attenuation is still on the order of 12%. Similar trends are visible for the 669 + 669, 3019 − 262, 3019 − 369, and 3019 − 669 cm−1 sidebands. Clearly, the experimental data does not exhibit the theoretically expected trends. Dilution Theory. Our previous work has highlighted the benefits of using mixed solvents to assess the contribution of cascades to the observed signal.13,14 In particular, CHCl3 and CDCl3 are useful because their C−Cl stretching vibrations, ν(H) CCl and ν(D) CCl, are close but distinct in frequency, occurring at 669 cm−1 and 650 cm−1, respectively. Hence, each mode can be driven into coherence with equivalent efficiency, and the coherent signals from the two vibrations can be easily analyzed because they occur in the same region of the 2D spectrum. Figure 11 is a diagram of the locations of the upshifted sideband peaks in a mixed sample of CHCl3 and CDCl3. Peaks HD and DH are a result of signal from intermolecular interaction when the fundamental and sideband frequency are from two different molecules. For instance, peak HD is the 650 (D) (H) cm−1 νCCl sideband upshifted from the 669 cm−1 ν CCl −1 (H) fundamental. Similarly, peak DH is the 669 cm νCCl sideband upshifted from the 650 cm−1 ν(D) CCl fundamental. Peak DD is the 650 cm−1 sideband upshifted from the 650 cm−1 fundamental in deuterated chloroform. Similarly, peak HH is the 669 cm−1 sideband upshifted from the 669 cm−1 fundamental in chloroform. Peaks DD and HH can potentially result from intramolecular coupling and contain signals from the fifth-order transition that would measure νCCl diagonal anharmonicity; however, the cascade should dominate the observed signal. We decided to examine these four peaks because experimental conditions of each peak, such as pumping efficiency and probe intensity, allow each peak to act as an internal standard for the concentration dependence of the others. Figure 12 shows, theoretically, how the signal from each sideband peak should vary with changes in the concentration of CHCl3. From eq 1, we know that the fifth-order signal is directly proportional to concentration, N, while the cascade signal is proportional to concentration squared, N2. Peaks HD and DH depend on the concentration of both solvents in the sample, because the FWM-A and B processes occur on different solvent molecules. For instance if FWM-A occurs on CHCl3, an intermediate field at ν̃Rpu − 669 cm−1 could be created by a

Figure 11. Diagram of 2D-FSRS peaks in a mixed CHCl3/CDCl3 sample. Diagram and labels show the upshifted sideband peaks for the combination of the C−Cl stretch for CHCl3 (669 cm−1) and CDCl3 (650 cm−1). Peak HH (blue) is the sideband from the ν(H) CCl stretch upshifted from the ν(H) CCl fundamental in CHCl3. Peak DD (green) is (D) (D) stretch upshifted from the νCCl the sideband from the ν CCl fundamental in CDCl3. These peaks will have fifth-order and cascade signal. The HD and DH peaks (red) are only from the cascade signal since the sideband and fundamental frequencies are from different molecules.

CSRS transition. Later, if FWM-B took place on CDCl3 then the signal would occur at ν̃sig = ν̃CSRS − 650 cm−1 = ν̃Rpu − (650 + 669) cm−1 and upon Fourier transformation the signal would become peak DH. Hence, the net cascade signal will be proportional to N·(1 − N), where N is the percent by volume of the CHCl3 and (1 − N) is the percent by volume of CDCl3. This concentration dependence is shown in Figure 12 for the relative FT intensity for peaks HD and DH (red), both of which should follow the same trend. For peaks DD (green) and HH (blue), the fundamental and sideband frequencies are from the same molecule, so the cascade signal is proportional to concentration squared. At 0% CHCl3, the signal from peak DD is maximum because it occurs entirely from CDCl3. As the concentration of CDCl3 decreases and concentration of CHCl3 increases, the signal from peak DD decreases according to (1 − N)2, the concentration of CDCl3 squared, and peak HH increases with the increasing concentration of CHCl3, varying like N2. Shown in Figure 12 as the dotted lines is the concentration dependence of the total signal in peaks DD and HH considering both the cascade and a small fifth-order signal. In order to highlight the concentration dependence of the total signal the fifth-order signal is arbitrarily chosen to be ∼10% of the total signal in the neat solvents. In previous work, we showed that the cascade is approximately 730 times larger that the fifth-order signal,13 so these concentration simulations would be appropriate if phase-matching had been able to attenuate the cascade by a factor of 73 relative to the fifth-order signal. Though unrealistic given our own experimental results, this magnitude of phase-matching attenuation (F = 1.3%) may be possible near the nodes in Figure 8b and d. However, we should emphasize that it is very unlikely that the fifth-order signal is as large as 10% of the cascade. In fact, it is more likely that the fifth-order signal only makes up less than one percent 6212

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Figure 12. Calculation of the concentration dependence of the sideband peaks shown in Figure 11. (a) Calculated intensity of sideband peaks as concentration is changed. Solid lines are signal from cascade transition. Dashed lines are signals from both cascade and fifth-order transition, considering a situation in which the fifth-order transition is 10% of the cascade signal in the neat solvent. (b) The ratio of cascade to total signal intensity at high and low CHCl3 concentrations.

Figure 13. Experimental 1D FSRS spectrum of the different sample concentrations. The spectrum is expanded to show (a) the νCCl stretch in CDCl3 (650 cm−1) and CHCl3 (669 cm−1) and (b) the νCD stretch at 2250 cm−1 in CDCl3 and νCH stretch at 3019 cm−1 in CHCl3.

decreases. The same is shown in Figure 13b for the νCD stretch at 2250 cm−1 in CDCl3 and νCH stretch at 3019 cm−1 in CHCl3. Unfortunately, the total magnitude of the signal for each scan does vary. For instance the 25:75 and 50:50 samples show −1 similar signal from ν(D) CCl at 650 cm , because of the unusually strong signal observed in the 50:50 sample. However, the ratio of signal from CHCl3 to CDCl3 is consistent with the nominal concentration ratios (see Supporting Information, Figure S3). The 1D-FSRS spectrum serves as a control for the 2D scans. The relative peak heights at the different concentrations in the 1D spectra confirm the expected concentration dependence of the third-order signal, which is proportional to N and (N − 1) for CHCl3 and CDCl3, respectively. Hence, we can use this as the launching point for analysis of the N2, (1 − N)2, and N(1 − N) dependent 2D-FSRS signals. Figure 14 shows how the intensity of the sideband peaks in the four-peak region varies as the concentration of CHCl3 changes. The exact location of the sideband peaks varies because of slight variations in the calibration along the probed frequency axis. The dotted diagonal lines were adjusted to match location of sideband peaks, with separation between lines kept constant. The horizontal lines are not adjusted because calibration along the pumped frequency (FT axis) is more precise. To compare the relative strengths of the signals all of the contours are plotted with the same contour scale. The white area in Figure 14e at peak HD is because the maximum peak magnitude, 0.028, is off scale. In the 50:50 sample (Figure 14a), four separate peaks can be seen, corresponding to the four

of the total signal of the neat solvent. Here, the magnitude of the fifth-order signal is chosen just for demonstration purposes. Figure 12b shows the small contribution of the cascade compared to the total signal at extreme dilutions. For example at 20% CHCl3, the cascade makes up 66% of the full signal for peak HH, whereas at 10% CHCl3 it is 50% of the signal, and at 1% CHCl3 the cascade is just 9% of the signal. Hence, under these conditions, the signal at peak HH of a 1% CHCl3 sample could be considered to be made of 91% fifth-order signal. This can be seen in Figure 12b: at very low concentrations of CHCl3 the cascade signal makes up very little of the total peak intensity for peak HH. The problem with using such a dilute concentration in the experiment is that the fifth-order signal would also be extremely small. We would be trying to measure the peak from a sideband in a 1% sample of chloroform, which is not possible given our current signal-to-noise ratio. Instead, what we hope to see is the change in percentage of cascade signal as the concentration is decreased from 100% to 10% for either chloroform or deuterated chloroform. Dilution Experimental Results. Figure 13 shows a 1DFSRS spectrum of the different concentration ratios. Figure 13a shows the νCCl stretch in CDCl3 (650 cm−1) and CHCl3 (669 cm−1). In the pure CDCl3 (red) sample, the only peak is at 650 cm−1. As the concentration of CHCl3 increases, the peak at 669 cm−1 increases by the same amount that the peak at 650 cm−1 6213

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estimate the relative magnitude of the two signals. Shown in Table 2 is a calculation of the ratio of cascade to fifth-order Table 2. Calculated Ratio of Cascade to Fifth-Order Signala sideband, ν̃hi ± ν̃low

ΔIcascade/ΔI5th

3019+669 3019+369 3019+262 3019−262 3019−369 3019−669 669+669 669+369 669+262 669−262 669−369 669−669

699 384 ∞ ∞ 397 740 749 195 ∞ ∞ 199 790

a

Calculated using eq 60 and parameters from Mehlenbacher et al., 2009, for CDCl3, assuming that the fifth-order signal is generated entirely from molecular anharmonicity.13 Since the anharmonicity and Raman cross sections are functions of electronic structure, these parameters are shared by CDCl3 and CHCl3.

signal to show the magnitude needed to attenuate the cascade signal for each of the sidebands that appear in the scan. The ratios were calculated using equations and parameters from Mehlenbacher which include experimental Raman cross sections and computed anharmonicities for chloroform.13 The table shows that if one were to use phase-matching alone to reduce the magnitude of the cascade to 0.25 samples. It is expected that a combined approach that incorporates both phase-matching and dilution to suppress the cascade signal may succeed in detecting the fifth-order 2D-FSRS spectra, which will measure the anharmonic coupling between various Raman modes of the sample.

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(8) Zhao, W.; Murdoch, K. M.; Besemann, D. M.; Condon, N. J.; Meyer, K. A.; Wright, J. C. Nonlinear Two-Dimensional Vibrational Spectroscopy. Appl. Spectrosc. 2000, 54, 1000−1004. (9) Jonas, D. Two-Dimensional Femtosecond Spectroscopy. Annu. Rev. Phys. Chem. 2003, 54, 425−463. (10) McCamant, D. W. Re-Evaluation of Rhodopsin’s Relaxation Kinetics Determined from Femtosecond Stimulated Raman Lineshapes. J. Phys. Chem. B 2011, 115, 9299−9305. (11) Kukura, P.; Frontiera, R.; Mathies, R. A. Direct Observation of Anharmonic Coupling in the Time Domain with Femtosecond Stimulated Raman Scattering. Phys. Rev. Lett. 2006, 96, 238303. (12) Frontiera, R. R.; Mathies, R. A. Polarization Dependence of Vibrational Coupling Signals in Femtosecond Stimulated Raman Spectroscopy. J. Chem. Phys. 2007, 127, 124501. (13) Mehlenbacher, R.; Lyons, B.; Wilson, K.; Du, Y.; McCamant, D. M. Two Dimensional Femtosecond Stimulated Raman Spectroscopy: Theoretical Treatment of Direct Fifth-Order and Cascading ThirdOrder Signals with Application to CdCl3. J. Chem. Phys. 2009, 131, 244512. (14) Wilson, K.; Lyons, B.; Mehlenbacher, R.; Sabatini, R.; McCamant, D. W. Two Dimensional Femtosecond Stimulated Raman Spectroscopy: Observation of Cascading Raman Signals in Acetonitrile. J. Chem. Phys. 2009, 131, 214502. (15) Blank, D. A.; Kaufman, L. J.; Fleming, G. R. Fifth-Order TwoDimensional Raman Spectra of Cs2 Are Dominated by Third-Order Cascades. J. Chem. Phys. 1999, 111, 3105−3114. (16) Golonzka, O.; Demirdoven, N.; Khalil, M.; Tokmakoff, A. Separation of Cascaded and Direct Fifth-Order Raman Signals Using Phase-Sensitive Intrinsic Heterodyne Detection. J. Chem. Phys. 2000, 113, 9893−9896. (17) Ivanecky, J. E., III; Wright, J. C. An Investigation of the Origins and Efficiencies of Higher-Order Nonlinear Spectroscopic Processes. Chem. Phys. Lett. 1993, 206, 437−444. (18) Zhao, B.; Sun, Z.; Lee, S.-Y. Quantum Theory of Time-Resolved Femtosecond Stimulated Raman Spectroscopy: Direct Versus Cascade Processes and Application to CdCl[Sub 3]. J. Chem. Phys. 2011, 134, 024307−12. (19) Lee, D.; Albrecht, A. C. In Advances in Infrared and Raman Spectroscopy; Clark, R. J. H., Hester, R. E., Eds.; Wiley: New York, 1985; Vol. 12, pp 179−213. (20) Wilson, E. B. J.; Decius, J. C.; Cross, P. C. Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra; Dover: New York, 1955. (21) Blank, D. A.; Kaufman, L. J.; Fleming, G. R. Direct Fifth-Order Electronically Nonresonant Raman Scattering from Cs2 at Room Temperature. J. Chem. Phys. 2000, 113, 771−778. (22) Kubarych, K. J.; Milne, C. J.; Lin, S.; Astinov, V.; Miller, R. J. D. Diffractive Optics-Based Six-Wave Mixing: Heterodyne Detection of the Full Chi((5)) Tensor of Liquid Cs2. J. Chem. Phys. 2002, 116, 2016−2042. (23) Trebino, R.; DeLong, K. W.; Fittinghoff, D. N.; Sweetser, J. N.; Krumbugel, M. A.; Richman, B. A.; Kane, D. J. Measuring Ultrashort Laser Pulses in the Time-Frequency Domain Using FrequencyResolved Optical Gating. Rev. Sci. Instrum. 1997, 68, 3277−3295. (24) Levenson, M. D.; Kano, S. S. Introduction to Nonlinear Laser Spectroscopy; Academic Press, Inc.: San Diego, CA, 1988. (25) Kubarych, K. J.; Milne, C. J.; Miller, R. J. D. Fifth-Order TwoDimensional Raman Spectroscopy: A New Direct Probe of the Liquid State. Int. Rev. Phys. Chem. 2003, 22, 497−532. (26) Kaufman, L. J.; Blank, D. A.; Fleming, G. R. Polarization Selectivity in Fifth-Order Electronically Nonresonant Raman Scattering from Cs2. J. Chem. Phys. 2001, 114, 2312−2331. (27) Kubarych, K. Heterodyne Detected Fifth-Order Raman Spectroscopy of Ultrafast Liquid Dynamics. Thesis, University of Toronto, Toronto, Canada, 2003.

ASSOCIATED CONTENT

S Supporting Information *

Additional phase-matching diagrams, sample dilution analysis, and 2D spectra in other spectral regions. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Telephone: (585) 276-3122. E-mail: mccamant@chem. rochester.edu. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by a National Science Foundation CAREER award, CHE-0845183. Acknowledgment is made to the donors of The American Chemical Society Petroleum Research Fund for partial support of this work.

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REFERENCES

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