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 elusive structures at charged buried interfaces. However, for the separation and determination of the Stern and diffuse layer spectra at these charged interfaces, complex vSFG spectra and, hence, the absolute phase need 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, for the first time, we provide a physically motivated understanding of the error phase. Determining the error phase from simulated spectra of oscillators with a spectral overlap, we show that for broadband vSFG spectra, such as for the silica/water interface, the diffuse and Stern layers' spectral overlap within the O-H stretching window results in a correlation between the error phase and the phase shift between the responses of these layers. This correlation makes the error phase sensitive to changes in Debye length from varying the ionic strength among other variations at the interface. Furthermore, the change in the magnitude of the 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. Finally, we highlight limitations of this model for vSFG spectra with a poor overlap between the diffuse and Stern layer spectra (silica/HOD in D2O system).