1. Introduction

Sum frequency generation (SFG) is a second-order non-linear optical technique that is a powerful tool for probing interfaces by taking advantage of the symmetry properties of the complex second-order nonlinear susceptibility $\chi ^{(2)}$ [1]. The ubiquity and importance of interfaces in chemistry makes SFG a very attractive method due to its all-optical, non-invasive nature. It has found prominence with vibrational studies in particular, for determining structure and orientation of interfacial molecules. However, despite its tremendous potential, SFG spectroscopy has yet to become a widespread technique for the study of interfacial electronic structure and dynamics. Notably, experiments that require repeated spectral acquisitions over varying sample properties, such as time-resolved experiments, have proven uncommonly challenging [2–4].

In broadband electronic SFG (e-SFG), a broadband pulse $\text {E}_{\text {Vis}}(\omega _1)$ is temporally and spatially overlapped with a narrowband near-infrared pulse $\text {E}_{\text {NIR}}(\omega _2)$. The interaction of the pulses with the system susceptibility $\chi ^{(2)}$ results in the SFG signal $\text {E}_{\text {SFG}}(\omega _{1+2})$ radiated at the sum frequency in the ultraviolet region, as described by Eq. (1) [5].

Conventional SFG experiments measure the intensity I of the SFG signal, which is proportional to the square modulus of the SFG electric field, $|\text {E}_{\text {SFG}}|^2$. With proper normalization against a nonresonant reference, $|\chi ^{(2)}|^2$ can be extracted from $|\text {E}_{\text {SFG}}|^2$. Details of the normalization process can be found in reference 6 [6]. There are several disadvantages to measuring $|\chi ^{(2)}|^2$ as opposed to $\chi ^{(2)}$: (1) With the loss of phase information in $|\chi ^{(2)}|^2$, interpreting resonances becomes difficult because $\chi ^{(2)}$ is a linear combination of complex contributions, each with their own phase and amplitude. Nonresonant contributions are in the real portion of $\chi ^{(2)}$, while the imaginary part of $\chi ^{(2)}$ contains the desired spectroscopic resonances of the system [5]. Interference between multiple resonances as well as between resonant and nonresonant contributions in $|\chi ^{(2)}|^2$ means that frequencies are shifted from their true values, line shapes are distorted, and separating resonant and nonresonant signals becomes impossible [7–10]. (2) Information on the sign of the electric field is lost in intensity measurements. The sign of the electric field is related to the orientation of the oscillating dipoles that give rise to the coherent signal and can yield the "up" or "down" orientation of the interfacial molecules [6]. (3) $|\chi ^{(2)}|^2$ is proportional to the square of the density of interfacial molecules and not conducive to quantitative spectrometric experiments where intensity is expected to be proportional to sample concentration or density [11]. (4) In time-resolved experiments, spectra are acquired alternately with and without the presence of the pump pulse at each time delay. The unpumped (steady-state) spectrum is subtracted from the pumped spectrum to obtain $\Delta \text {E}_{\text {SFG}}$. To correctly interpret time-resolved data from $\Delta |\text {E}_{\text {SFG}}|^2$ requires extensive fitting analysis whereas directly measuring $\Delta \text {E}_{\text {SFG}}$ makes data analysis straightforward [2]. It has also been suggested that pump-induced phase shifts occur in the SFG signal which can only be detected with a phase-sensitive approach [12].

Phase-sensitive experiments provide a solution to these problems by detecting $\text {E}_{\text {SFG}}$ rather than $|\text {E}_{\text {SFG}}|^2$. Heterodyne detection is a phase-sensitive measurement where the interference between the SFG field and a local oscillator (LO) field is measured. The LO is generated separately from the SFG using the same incident fields $\text {E}_{\text {Vis}}$ and $\text {E}_{\text {NIR}}$ and contains the same frequencies as the SFG field. The interference with LO generates both $|\text {E}_{\text {SFG}}|^2$ and $|\text {E}_{\text {LO}}|^2$ as well as a cross term, shown in blue in Eq. (2), where $\text {E}_{\text {LO}}$ and $\text {E}_{\text {SFG}}$ are complex fields that contain frequency and phase information. The cross term is not a square modulus and can access the complex values of the electric fields.

Thus heterodyne detection enables the separation of real and imaginary parts of $\chi ^{(2)}$ and the measurement of the sign of the electric field. Since the LO is a much stronger signal than the SFG, mixing the two also increases the signal to noise ratio (SNR). However, as an interference-based measurement, obtaining correct signals in heterodyne detection hinges on phase stability between the LO and SFG fields throughout the experiment. This presents a challenging technical hurdle that is particularly highlighted in the case of electronic SFG experiments that generate visible and UV light. At these wavelengths, drifts of tens of nanometers in the relative path lengths between the SFG and LO fields lead to dramatic phase shifts of as much as $\frac {\pi }{4}$. Small errors in the optical path length caused by, for example, ambient laboratory vibrations and thermal fluctuations compromise the stability of the interferometer, making detection of heterodyned SFG signal difficult.

Most phase-sensitive SFG methods respond to the problem of stability by employing passive phase stabilization. Many vibrational and almost all electronic SFG studies use spectral interferometry of collinear LO and SFG pulses detected in frequency domain [12–19] to measure phase of the SFG signal. In a collinear geometry, the LO and SFG signal experience similar path length drift by impinging on the same optics, facilitating passive phase stabilization. However, for extended time-resolved studies of electronic spectra, passive phase stabilization is not sufficient to eliminate uncertainty in phase and amplitude between time delay measurements. Phase instability is the main barrier to performing heterodyned time-resolved electronic SFG experiments.

The need for control and stability of relative phases between pulse pairs is not unique to heterodyne detection in SFG experiments. Stable phase-locked pulse pairs are used in many nonlinear spectroscopic techniques where phase cycling is used for selective detection of quantum pathways. To address phase stability issues, many techniques have been introduced including active stabilization methods using closed-loop feedback circuits [20–24], passive stabilization using diffractive optics [25–28] or periodic measurement of a reference phase [29,30], and acousto-optic pulse shaping [31–35]. Scherer et al. used phase modulation by a piezo-electric transducer and a phase-locked-loop to achieve pulse pairs with fixed relative phase while varying inter-pulse time delay in fluorescence-detected wave-packet interferometry [22]. Phase-modulation using acousto-optic modulators (AOM) was first used by Marcus and coworkers [36] and applied to detection of 2D fluorescence [37–39] and photocurrent spectroscopy [40,41]. It has since become a popular method for dynamic rephasing in 2D electronic spectroscopy [42–46]. Phase modulation with a vibrating mirror has also been used in "pseudo-heterodyning" in scanning near-field optical microscopy (SNOM) to acquire background suppression and simultaneous measurement of phase and amplitude of near-field images [47–49].

In this paper we measure SFG spectra in the time domain with a lock-in detection scheme and introduce a technique for directly reading the relative phase between SFG and LO fields using continuous phase modulation of the LO pulse with a vibrating mirror. In the time domain the intensity of the interference of the two fields is detected as a function of varying relative time delay. A schematic of the Mach-Zehnder interferometer (MZI) set up for this purpose is shown in Fig. 1. The fundamental beams travel collinearly in each interferometer arm where they are incident on the sample or LO source. The collinear geometry of the fundamental beams simplifies alignment by ensuring that the beams are always spatially overlapped in the sample plane which is particularly important in the case of liquid interfaces, which may undergo evaporation during the experiment. The time delay between two interferometer arms is experimentally set by changing the optical path length difference between the LO and SFG fields but is beset by instabilities in the optics. In contrast to active stabilization where the phase modulation information is used in a feedback loop to correct phase in real time, we collect the relative phase data and correct for all instabilities at once in post-processing. A direct report of the relative phase between the LO and SFG fields allows us to reconstruct the time delay axis to account for unintended changes in the path length.

 

Fig. 1. The experimental schematic of the MZI. The path of the collinear fundamental colors is shown as they enter the interferometer and are incident on the LO and sample crystals. The narrowband diode laser used to measure the phase difference between the interferometer arms is also shown, travelling collinearly with the SFG and LO beams throughout the MZI.

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In addition to control over the phase stability of the interference signal, there are further advantages to interferometry in the time domain over spectral interferometry which necessarily occurs in the frequency domain. While frequency domain and time domain detection theoretically offer equivalent information, it has been shown that experimental considerations affect the measured spectra in frequency domain, particularly for vibrational spectroscopy, with significant implications for spectral lineshapes [30,50,51]. Moreover, frequency narrowing significantly reduces signal strength which decreases quadratically with pulse duration [1,7]. Combined with reduced sensitivity in frequency domain detectors, loss in signal strength makes detecting small changes in the interfacial spectrum very difficult. To our knowledge, time domain detection of heterodyne SFG has only been performed for vibrational spectra with passive phase stabilization [3,30]. The use of lock-in amplifiers with time domain detection significantly reduce noise, and eliminate large non-oscillatory signals arising from $|\text {E}_\text {LO}|^2$ term in Eq. (2). Our novel spectrometer design paves the way for phase stable time domain detection of heterodyne SFG spectra, eliminating the above mentioned concerns for vibrational spectra, while allowing electronic spectra to benefit from the advantages of single channel detectors with higher sensitivity, lower noise, and lower costs than frequency domain detectors. Achieving phase stability over extended periods will make time-resolved studies with long experiment run-times possible for investigating both electronic and vibrational interfacial states.

2. Theory

The SFG and LO fields each form one arm of a Mach-Zehnder interferometer (MZI). A time varying phase shift is applied to the LO field by reflecting the LO off a vibrating mirror, shown in Fig. 1.

Mathematical description of the underlying optics presented in this section best explains the signal processing scheme. For convenience, we denote the electric fields comprising the arms of the MZI with subscripts "1" and "2". The discussion of these fields is general and can describe any two laser light waves, pulsed or continuous, that are set up as two arms of a MZI.

The intensity of two overlapped fields is expressed as:

The vibrating mirror introduces an experimentally controlled additive phase, shown in Eq. (4), to the $\text {E}_1$ field,

We now rewrite the phase-modulated field as:

A Jacobi-Anger expansion allows for a Fourier representation of the sinusoidal frequency-modulated signal as a sum of harmonic components weighted by Bessel functions of varying order. Expanding Eq. (5) in this manner leads to the following expression for the phase-modulated field:

At each time delay, $\text {E}_1$ is modulated in real time by the vibrating mirror. This yields a modulation in the interference between the two fields that depends on their relative phase $\Delta \phi$. This modulation is illustrated in Fig. 2. The green trace is a cartoon of the unmodulated interferogram. The orange trace shows the result of the modulation at various time delays indicated by the red dots. The expansion of this signal into its Fourier series yields the series of peaks at harmonics $m$ of the vibration frequency $\nu _r$, shown in purple in Fig. 2. We denote the intensity of the peak at the $m$th harmonic of the modulation frequency as $U_m$.

 

Fig. 2. The green trace shows the interferogram, where the red dots indicate the point in time delay marked by the relative phase of the two arms, $\Delta \phi$, and the arrows indicate how the vibrating mirror modulates the interferogram intensity. The orange trace shows the interference pattern in real time due to vibrations of the mirror in one arm of the interferogram, at the phase marked by the red dots. The purple trace shows the Fourier transform spectra of the real time modulated interference. The intensities of the even and odd harmonic peaks, denoted by “$\text {U}_{\text {m}}"$, change with the value of $\Delta \phi$. For the in-phase signal where $\Delta \phi = 0$, even side bands go to zero (i.e., $U_{m=even}=0$) and for the quadrature signal where $\Delta \phi = \frac {\pi }{2}$, odd side bands go to zero.

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The values of $U_{m}$ are obtained by expanding Eq. (7), isolating the cross terms, and finding their Fourier expansion coefficients. Experimentally, a lock-in amplifier locked into a harmonic frequency $m\nu _r$ multiplies the interference signal in Eq. (7) with a field $e^{-i(m\nu _{r}t)}$, effectively "picking" the term of the Fourier series at harmonic $m$. Locking into the modulations of one interferometer arm also ensures that only signal containing that modulation frequency is detected and any noise from the baseline arising from $|\text {E}_1|^2$ and $|\text {E}_2|^2$ is suppressed. Equation (8) shows the mathematical expression of the side-band intensities $U_{m}$ as detected by the lock-in amplifier. For a full derivation refer to Section (1) of Supplement 1.

As shown in Eq. (8), the intensity of each side-band has a sinusoidal dependence on the phase difference between the two fields, given by $\Delta \phi = \phi _{1}-\phi _{2}$. Successive even and odd side-bands are respectively proportional to the real ($\cos$) and imaginary ($\sin$) components of the complex cross-term of the interference signal, where $\omega _1=\omega _2$. This relationship is shown in Eq. (9).

Equations (8) and (9) show the dependence of $U_{m}$ on $\Delta \phi$ and its relationship to the interference of the $\text {E}_1$ and $\text {E}_2$ fields. The detected $U_{m}$ signals, shown in Fig. 3, are a periodic function of $\Delta \phi$, as shown in Eq. (8). Thus the $U_{m}$ signal in Fig. 3 may be treated as an interferogram whose Fourier transform is the frequency domain spectrum of the shared frequencies in $\text {E}_1$ and $\text {E}_2$ fields. This implies that the $U_m$ signal detected as a function of time delay can be used to find the frequencies present in SFG and LO fields. The lock-in detection scheme also eliminates the $|\text {E}_{\text {SFG}}|^2$ and $|\text {E}_{\text {LO}}|^2$ fields that have to be subtracted from the detected signal in most heterodyne detection techniques (see Section (1) of Supplement 1 for details).

 

Fig. 3. Time domain data collected on the three channels of the lock-in amplifier: Top two panels show the CW diode signal locked into the second and third harmonics of the mirror vibration frequency. These intensities are the $U_m$ values used for detecting relative phase between the two arms of the MZI. Bottom panel shows the SFG signal locked into the first harmonic of the mirror vibration frequency.

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Relative phase $\Delta \phi$ can be calculated from the ratio of intensity of two consecutive odd and even side bands as shown in Eq. (10) where $J_k$ and $J_l$ are $k$ (odd) and $l$ (even) order Bessel functions.

For the case of broadband light where $\text {E}_{\text {SFG}}$ and $\text {E}_{\text {LO}}$ have finite bandwidth, a simplified expression describing the fields is a sum over the number n of colors present, shown in Eq. (11).

This expression is substituted for the electric fields in Eq. (7), with each color having a different relative phase. In solving the equation for intensity, Eq. (8) becomes a sum over the number n of colors present:

Equation (10) for the broadband fields becomes:

Equation (13) is analytically intractable. Since the real source of phase instability is in path length difference, which unlike phase, is color-independent, a narrowband reference can be used for the purpose of phase detection. We copropagate a narrowband continuous wave (CW) laser diode with the $\text {E}_{\text {SFG}}$ and $\text {E}_{\text {LO}}$ fields, to be used for detecting the relative phase of the arms of the MZI. This is a valid approach given the generality of the signal processing procedure detailed above for any light analyzed by interferometry. The frequency spectrum of the CW laser is closer to a delta function than the frequency spectra of ultrashort laser pulses are, giving a more accurate value for the phase difference between the two arms that can be converted to a path length difference.

3. Methods

Visible and near infrared up-conversion pulses are generated by a commercial OPA system (Light Conversion Orpheus N-2H and N-3H) with two outputs at 600 nm (150 mW) and 850 nm (500 mW), with sub 30 fs pulse duration at 50 kHz repetition rate. Both fields are P-polarized. The two pulses at 600 nm and 850 nm are made collinear using an incoupling dichroic mirror (Thorlabs) before they enter a Mach-Zehnder type interferometer. A 50/50 broadband, low GDD beamsplitter (Thorlabs) is used to split the overlapped beams into two arms. Separately, a narrowband CW laser (Newport LQC660-110C) at 660 nm with the perpendicular polarization to the pulsed laser light is also incident on the beam-splitter at 90 degrees to the pulsed laser light, as shown in Fig. 1, such that it travels collinearly with the pulsed light into the interferometer.

Each arm is focused by a parabolic mirror onto a sample or local oscillator source, in reflection geometry, and is subsequently recollimated along with the resulting SFG, where one arm of the interferometer contains the sample signal and the other functions as the local oscillator field. For this proof of concept experiment, GaAs(100) crystals (MTI) were used for both sample and LO source. GaAs has a strong bulk SFG response with a broad response in the visible region which makes it an ideal LO source for future experiments. The LO arm enters a delay line with a computer controlled delay stage (PI N565), with 0.375 fs step size, corresponding to 112.5 nm, with a maximum delay of 300 fs, and the sample line is sent to a retroreflector fitted with a phase shifter driven by a piezo-electric transducer (PI S-303), vibrating the retroreflector at 800 Hz and 51 nm amplitude. The two arms are then recombined using a low GDD beam splitter (Thorlabs).

The recombined beams then go through a series of dichroic mirrors (Eksma) to separate the UV SFG signal from the visible and near IR fundamental fields as well as the CW laser. The SFG signal is directed towards the first avalanche photodiode (Thorlabs APD 130A2). The CW diode has a perpendicular polarization relative to the fundamentals and is separated from them using a Wollaston prism and is directed towards a second avalanche photodiode (Thorlabs APD 130A). The photodiode outputs are passed to a lock-in amplifier (Zurich Instruments MFLI) that also serves as the signal source for the phase shifter. The lock-in amplifier is simultaneously locked into multiple harmonics of the frequency of the vibrations.

Three channels on the lock-in amplifier are used for data collection. The CW laser signal from the second APD is the input for the first two channels, each locked into the second or third harmonics of the vibrations of the phase shifter. The third channel input is the SFG signal from the first APD and is locked into the first harmonic of the phase shifter frequency. Data is acquired by integrating this signal for 153 ms. The resulting locked-in signals are shown in Fig. 3. The CW signals are used for determining the phase difference between the two arms of the MZI.

4. Results

As previously mentioned, the CW diode beam is divided and co-propagated with both the LO and the SFG fields and is subjected to the same treatments. The phase modulation of the LO arm by the phase-shifter results in modulation of the interference signal. The lock-in amplifier detects the intensity $U_m$ of harmonic $m$ of the phase modulated signal in real-time for every time delay between the two arms of the MZI. For this experiment, two channels on the lock-in amplifier were locked into the second and third harmonics of the reference frequency, $\nu _r$, which is also used as the frequency of the phase-shifter. These channels receive signal from the APD detecting the CW diode beam and are depicted in Fig. 3(a) and (b). A third channel is locked into the first harmonic of $\nu _r$ and detects the pulsed SFG signal, shown in Fig. 3(c).

Given that the time dependent signal is described by Eq. (8), the value of the Bessel function modulates the signal intensity. The Bessel function value $J_m[\gamma ]$ depends on the wavelengths of light ($\lambda$) present in the SFG light, the value of the integer $m$, and the vibration depth ($\Delta L$) of the phase shifter. Judicious selection of the harmonic of the lock-in reference frequency and vibration depth of the phase shifter can increase the signal to noise ratio as well as eliminate unwanted fundamental wavelengths on the detector (see Supplement 1 section (2)).

The ratio of the intensities $U_2$ and $U_3$, shown in Fig. 3(a) and (b) respectively, are obtained at each time delay. From this ratio, the phase difference at each point in the time delay axis is obtained according to Eq. (10) and is shown in Fig. 4(a). This corresponds to the conceptual drawing in Fig. 2 where $\Delta \phi$ in each row can be obtained from the ratio of $U_2$ and $U_3$ intensities shown on the purple trace.

 

Fig. 4. Calculated $\Delta \phi$ plotted against (a) the original time delay axis and (c) the new reconstructed delay axis. Red circles show areas where phase drifts lead to non-smooth traces with the original axis. Plotted against the new axis, these traces look much smoother. Red lines in (a) show error boundaries for $\Delta \phi$ values. (Note the error is too small at some points to distinguish the upper and lower bounds in the figure.) (b) shows a zoomed in portion of (a) to more clearly show the error bars. (d) shows the original time axis subtracted from the new time axis, plotted against the new time axis, highlighting the difference between the two axes and the amount of path length drift throughout a scan. The orange shaded region is an upper bound on the estimated error.

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Figure 4(a) shows the measured value of $\Delta \phi$ plotted against the original time delay axis. Given a high sampling rate and stable conditions, this plot should be a perfect saw-tooth shape, as the phase changes from $-\pi$ to $\pi$ for each period of the wavelength. Instabilities in phase can be seen in the form of "kinks" in the plot in Fig. 4(a), circled in red. These kinks disappear from the plot when phase is plotted against a time delay axis that corresponds to the true phase difference. To achieve this, we recreate the time delay axis using the measured $\Delta \phi$ values.

The direct proportionality relationship linking time delay between two pulses and their path length difference is well-known. We replace the original time delay axis which was based on the reported displacement of the mechanical stage in the interferometer with a new computed time delay axis based on the detected $\Delta \phi$. The relative phase of two pulses is a direct reporter of their temporal delay with respect to each other and therefore a reporter of their path length difference. However, where phase is a periodic function of the wavelength, the path length difference between the two pulses is a monotonic function. Phase difference $\Delta \phi$ can be used to calculate path length difference $\Delta x$ as a fraction of the wavelength for the first period, with subsequent periods added in series, as shown in Eq. (14). The value of n, denoting the climbing number of periods, increases by 1 every time $\Delta \phi$ completes a $2\pi$ cycle. The new delay axis is thus constructed based on the computed phase difference between the two interferometer arms.

Figure 4(b) shows $\Delta \phi$ plotted against the reconstructed delay axis. The points on the scan where phase instabilities had occurred, circled in red, now present a smooth trace. The original axis only accounts for path length differences due to scanning the stage and is informed only by the stage controller. However, the new time delay axis accounts for path length differences due to scanning the stage as well as those that are due to instabilities caused by slowly drifting optics. It also accounts for errors in reported stage positions. This is reflected in Fig. 4(d) which shows the difference between the original axis and the new axis, plotted against the new axis. In this plot, it is apparent that the two axes are different by slight and varying amounts at each time delay, highlighting where instabilities have occurred in the optical paths. This drift occurs over a range of 2 fs (equivalent to 600 nm) which is a significant amount for UV/Vis wavelengths (200 nm to 700 nm). The orange trace shows the propagated error for calculated values of path length difference $\Delta x$ demonstrating that the path length drift lies outside the error boundaries. (For a discussion of error analysis, please refer to Supplement 1.) Figure 4 proves the phase detection scheme works well to detect instabilities in the path lengths of the arms of the MZI.

The SFG signal collected by the third channel on the lock-in amplifier plotted against the recalculated time delay axis is Fourier transformed to show the SFG spectra in Fig. 5. Fast Fourier transform algorithms require evenly spaced sample points for constructing the frequency axis. The original time delay axis was informed only by the steps of the mechanical stage controller, performing a regularly spaced scan, and was therefore uniform. The new axis, accounting for changes to the programmed path length change due to drifts and instabilities, is not uniform. For this irregularly spaced time delay axis, the MATLAB algorithm for Non-Uniform Fast Fourier Transform is used to compute the spectra [52,53]. Traces plotted with the new time delay axis show a much higher signal to noise ratio, narrower linewidths, and more accurate position of the peaks in the frequency domain, as shown in Fig. 5. The SFG peaks at 3.54 eV were fitted with a Lorentzian and are shown to have linewidths of 0.18(4)eV and 0.08(1)eV for the uncorrected and corrected spectra respectively (see Table S1 for details). The frequency resolution of the spectra is determined by the inverse of the product of the number of time delay points and scan step size. For this scan with steps size of 0.375 fs and 800 points, the resolution is approximately 0.0138 eV. These spectra show the tremendous improvement that is achieved by detecting and correcting for phase drifts with this spectrometer. The same spectra taken in the frequency domain with a spectrometer (Ocean Insight Maya2000 Pro) are shown in Fig. S5 in Supplement 1 for comparison. They show the same lineshapes and frequencies as the spectra achieved from the time domain spectrometer and the SFG peak has a linewidth of 0.081(4)eV, which is in good agreement with the corrected spectra. It is worth noting that the frequency domain spectrometer could not detect the bluest SHG color at 4.09 eV (300 nm) likely due to the quantum efficiency of the spectrometer.

 

Fig. 5. Fourier transform, averaged over ten scans, of (a) the raw diode data showing the frequency of the diode at 1.88 eV. (b) Frequency domain GaAs spectra showing three peaks at 2.92 eV (second harmonic of 850 nm fundamental), 3.54 eV (SFG of 850 nm and 600 nm), and 4.09 eV (second harmonic of 600 nm). (c) and (d) Non-uniform Fourier transform of the phase corrected data from (a) and (b) showing narrower peaks with less noise and higher intensity.

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5. Summary and outlook

We have developed a new design for phase-sensitive detection of electronic sum frequency generation spectra. By using continuous phase modulation of the local oscillator in the time domain we are able to measure the relative phase between the local oscillator and the SFG signal. This enables us to account for any phase drifts arising from instabilities in the optical set-up, resulting in much higher SNR, correct lineshapes, and spectral accuracy in SFG spectra.

Phase stability in the spectra facilitates heterodyning to achieve simultaneous detection of phase and amplitude. Heterodyne detection decouples real and imaginary parts of the detected SFG electric field, which will afford accurate spectra, eliminate non-resonant contributions, and elucidate the orientation of molecules at the interface. Slow phase drifts that distort phase-sensitive data over long scans are corrected through directly measuring the relative phase difference between LO and SFG signals and translating it into a new, more accurate time delay axis. Time-domain detection allows for more accurate SFG spectra with high frequency resolution while taking advantage of more sensitive and cost-effective single channel detectors. Collinear geometry of the two fundamentals $\text {E}_{\text {Vis}}$ and $\text {E}_{\text {NIR}}$ leads to a smaller room for error in the phase of reference with respect to the sample, such that normalizing the sample spectrum by the reference spectrum will yield accurate phase and amplitude for $\chi ^{(2)}$.

This spectrometer design will pave the way for time-resolved phase-sensitive experiments without risking compromising the data due to phase drifts over time. Future iterations of this spectrometer will take advantage of a broadband white light source along with a NIR $\text {E}_2$ pulse to probe multiple resonances at once, providing a wealth of data with each scan. The methods used in this paper and the relevant signal processing can be more broadly applied to any interferometric measurements that require high phase stability, breaking ground for further innovative spectroscopic techniques in the visible and ultraviolet regions.