Lineshape Distortions in Internal Reflection Two-Dimensional Infrared Spectroscopy: Tuning across the Critical Angle

Reflection mode two-dimensional infrared spectroscopy (R-2DIR) has recently emerged as a tool that expands the utility of ultrafast IR spectroscopy toward a broader class of materials. The impact of experimental configurations on the potential distortions of the transient reflectance (TR) spectra has not been fully explored, particularly in the vicinity of the critical angle θc and through the crossover from total internal reflection to partial reflection. Here we study the impact on the spectral lineshape of a dilute bulk solution as θc is varied across the incident angle by tuning the refractive index of the solvent. We demonstrate the significance of several distortions, including the appearance of phase twisted lineshapes and apparent changes in the spectral inhomogeneity, and show how these distortions impact the interpretation of the TR and R-2DIR spectroscopies.

All materials were obtained from Sigma-Aldrich.
For transmission measurements the samples were held in demountable liquid cell between two 1 mm thick CaF 2 windows with a 50 µm teflon spacer. Transmission spectra were measured in an FTIR spectrometer (Bruker Vertex) equipped with a DTGS detector. Linear reflectance spectra were measured in situ in the 2DIR spectrometer, where the difference in reflectance between the air-ZrO 2 and the solution-ZrO 2 interface were measured by dispersing the probe beam on a 64 element HgCdTe detector (IR Associates) using a spectrograph (Horiba Triax 190) with a 100 lines/mm grating. The spectra have been smoothed for the purpose of presentation. The baseline was removed by fitting to a 3rd order polynomial and subtracting. We chose to use the linear reflectance measured in the 2DIR spectrometer rather than measuring it in the FTIR to ensure the experimental geometry is identical between the linear spectra and the TR and R-2DIR spectra. The custom ZrO 2 prism (Supply Chain Optics) had an apex angle of 80 • and the IR beam was aligned normal to the incident face so that the angle at the sample interface is set to 50 • . The prism was positioned so that the focus of the beam was placed at the sample-prism interface.
R-2DIR and TR measurements were performed in pulse-shaper based pump probe geometry spectrometer. IR pulses were generated with a laser system composed of a Ti:Sapph regenerative amplifier (Spectra-Physics Solstice, 800 nm, 3 mJ per pulse energy, 1 kHz, 90 fs pulse duration) used to pump an optical parametric amplifier (OPA) equipped with difference frequency generation (Light Conversion TOPAS Prime). The resulting mid-IR pulse was centered at 1720 cm −1 with ∼ 200 cm −1 bandwith at full width half maximum with 100 fs duration and 15 µJ per pulse. The probe beam was split from the pump by the front face reflection off an uncoated wedged CaF 2 window. The waiting time t 2 was controlled with a delay stage in the probe line (Aerotech). The pump was passed through a 4 − f S2 pulse shaper with a Ge AOM (Phasetech QuickShape), which was used to generate a pair of pulses with controllable time delay t 1 . For 2DIR measurements, we scanned from t 1 = 0 to −3 ps in 25 fs steps using a rotating frame at 1350 cm −1 and a 2 × 2 phase cycling scheme.
The pump was compressed by maximizing the nonresonant response of the ZrO 2 prism at the prism-air interface at waiting time t 2 = 0 fs as a function of the 2nd through 6th order dispersion applied by the pulse shaper. The probe was compressed by adding AR coated Ge plates sufficient to minimize the crosscorrelation time determined from the ZrO 2 nonresonant response. The pump and probe pulses were focused onto the prism-sample interface with a 6 inch focal length 90 • off-axis parabolic mirror and the change in reflectance of the probe induced by the pump was measured with the spectrograph to obtain the detection frequency axis ω 3 . The polarizations of the pump and probe beams were set relative to the prism-sample interface using λ/2 waveplates (Alphalas) and wire grid polarizers (Thorlabs), and the detected signal was fixed to an additional analyzer in front of the spectrograph.

Phase Correction
We implement a phase correction procedure equivalent to the one recently developed by Tek and Hamm. 1 The goal is to convert partially or fully dispersive lineshapes into the equivalent absorptive lineshape, which can be done by making use of the Kramers-Kronig transformation (KKT). Given a real valued reflectance spectrum R (ω) (either PR or ATR) converted to absorbance units we numerically calculate the KKT using the Hilbert transform as implemented in matlab to determine the imaginary part of the linear spectra. So the complex reflectance spectrum is then

S3
We then determine the frequency ω 0 of the absorption maximum from the separately measured transmission spectrum and rotate the data by the constant phase term φ 0 to make the real part of the R c (ω) maximally symmetric about ω 0 using only the single fitting parameter φ 0 . The results of each step of this procedure are illustrated in SI figure   S4.
For the CP spectra this procedure was modified to accommodate multiple overlapping absorptions. Specifically, rather than optimizing φ 0 to make the real part of the linear reflectance spectrum maximally symmetric, we instead optimize the real part of the spectrum to be maximally positive. All other aspects of the phase correction are the same.

Phenomenological Model
To illustrate how the linear reflectance, TR and R-2DIR relate to each other and depend on the relationship between θ 1 and θ c we construct a simple phenomenological model. We start by assuming the imaginary part of the refractive index for the 0 → 1 and 1 → 2 transitions can be described as complex Lorentzians with the form The linear reflectance spectra can be determined fromñ(ω) using the Fresnel equations.
For s polarized light the reflectance takes the form where the angle of transmission θ 2 is related to the angle of incidence θ 1 by Snell's law. We show the resulting reflectance spectra − log(R(ω)) for several values of θ 1 near θ c in figure 4 for both the fundamental and excited state transitions. As expected, the resulting spectrum shows an absorptive lineshape for θ 1 > θ c and a dispersive lineshape for θ 1 < θ c with varying degrees of distortion to the band as θ 1 is tuned across the region of anomalous refractive index due to the presence of the absorption band. To compute the TR spectra we simply take the ratio −∆ log R(ω) = − log (R 0→1 (ω)/R 1→2 (ω)) which we show in figure 4 E, and we can clearly see the origin of the TR lineshapes we have observed experimentally.
To calculate the 2DIR spectra we assume that k can be described via a bivariate Lorentzian where we introduce the crossterm between the excitation frequency ω 1 and the detection frequency ω 3 to allow for correlated lineshapes, with the degree of correlation determined by the parameter c (here we use c = 0.75). We consider the interaction with the pump pulses to be entirely via the absorptive component, not the dispersive component, and so when we compute n(ω 1 , ω 3 ) and R(ω 1 , ω 3 ) we perform the KKT and include ∆ only along ω 3 . Otherwise the procedure for calculating the 2DIR reflectance spectra is the same as for the linear spectra. In SI figure S8 we show the contribution to the 2DIR we calculate for the 0 → 1 transition as a function of θ 1 , as well as the difference with the 1 → 2 transition, which corresponds to the experimental spectra. These calculations illustrate how the various S5 distortions to the lineshape arise as θ 1 is varied across θ c , including the increased apparent inhomogeneity at θ c . Figure S1: IR absorption spectrum of the bare ZrO 2 prism. Inset shows details of the spectral region relevant for the present study.  Figure S3: Baseline subtraction for linear reflectance spectra.

Supplementary Figures
S7 Figure S4: Phase correction procedure for the reflectance spectra. Panel A shows the real valued reflectance spectra (R (ω)) directly acquired from the measurement and panel B shows the imaginary part (R (ω)) determined from the KKT. Panels C and D show the absolute magnitude (|R| (ω)) and phase spectra (atan2(R (ω), R (ω)) from combining the real and imaginary parts. Panels E and F show the purely absorptive and dispersive spectra obtained after performing the phase correction (R (ω) andR (ω)).    Figure S8: Model 2DIR spectra as a function of incident angle. Top row: the contribution to the signal from the 0 → 1 transition. Bottom row: the difference spectrum −∆ log R(ω 1 , ω 3 ) = − log R 0→1 (ω 1 , ω 3 ) + log R 1→2 (ω 1 , ω 3 ) which corresponds to the experimental 2DIR spectra.