Coherent optical effect on time-resolved vibrational SFG spectrum of adsorbates

We present a theory to study the influence of the coherent mixing between pump-infrared and probe-visible pulse on a time-resolved sum-frequency generation (TR-SFG) spectrum for vibrations at surfaces. The general formula of the time-dependent and its Fourier transform of the SFG polarization and its Fourier transform allows us to calculate the time-resolved vibrational SFG spectrum and the transient characteristics of the SFG intensity as a function of the delay time td between the pump-infrared and probe-visible pulse. It is found the coherent optical effect manifests itself in the broadening and narrowing of the SFG spectrum with the intrinsic width of T2 at negative and positive td, respectively, being in qualitative agreement with recent experimental results. The influence of the coherent mixing on the transient behavior of the SFG intensity is also discussed in conjunction to the T2 determination.