Comprehensive expressions have been presented to facilitate the analysis of the surface sum-frequency generation (SFG) spectrum. The electric field components of the SFG beam for a given experimental setup have been related via appropriately defined Fresnel coefficients to the non-linear source polarization, which in turn has been related to the electric fields of exciting visible and infrared beams through the macroscopic SFG susceptibility tensor. The coefficients of transformation have been given to relate the laboratory-fixed Cartesian components of the SFG tensor to the components described in a surface-fixed axis system. The tensor components have been further related to the components of the microscopic hyperpolarizability tensor of surface species, and the explicit expressions (in terms of the Euler angles defining molecular orientation) of the transformation coefficients are presented to describe the Cartesian tensor components described in a surface-fixed axis system by the molecule-fixed components.
Chiaki Hirose, Naotoshi Akamatsu, and Kazunari Domen, "Formulas for the Analysis of the Surface SFG Spectrum and Transformation Coefficients of Cartesian SFG Tensor Components," Appl. Spectrosc. 46, 1051-1072 (1992)