Relating the Phase in Vibrational Sum Frequency Spectroscopy and Second Harmonic Generation with the Maximum Entropy Method

Nonlinear optical methods such as vibrational sum frequency generation (vSFG) and second harmonic generation (SHG) are powerful techniques to study the elusive structures at charged buried interfaces such as the silica/water interface. However, for an accurate evaluation of the structure formed at these buried interfaces, the complex vSFG spectra and hence the absolute phase needs to be retrieved. The maximum entropy method is a useful tool for the retrieval of complex spectra from the intensity spectra; however, one caveat is that an understanding of the error phase is required. Here we provide a physically motivated understanding of the error phase, where we show that for broadband vSFG spectra such as the silica/water, the good spectral overlap between water in the diffuse and Stern (or bonded interfacial) layers results in the absolute phase correlating with the error phase. This correlation makes the error phase sensitive to changes in Debye length from varying the ionic strength amongst other variations at the interface. Furthermore, the change in the magnitude error phase can be related to the absolute SHG phase permitting the use of an error phase model that can utilize the SHG phase to predict the error phase and hence the complex vSFG spectra. We highlight limitations of the model for narrow vSFG spectra with poor overlap between the diffuse and Stern layer spectra, such as the silica/HOD in D2O system.