Theoretical description of steady-state sum-frequency generation in molecular adsorbates

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Abstract
In this paper, we present a density-matrix formalism for treating second-order sum-frequency generation (SFG) and difference-frequency generation (DFG). The theory can treat both steady-state and time-resolved SFG and DFG. However, only the steady-state SFG will be described in this paper. As a practical application, we show how the theory can be applied to study the infrared-visible SFG. The band-shape function of SFG is derived, which consists of both real and imaginary parts. The real part of the SFG band-shape function is related to the infrared spectral band-shape function and the imaginary part is related to the real part by Kramers-Krönig relations. The temperature effect on SFG is taken into account in our expression of the SFG band-shape function. We show that for a vibrational mode to be SFG active in infrared-visible SFG measurements, it has to be both infrared and Raman active.