the ability to perform time-resolved measurements with interfacial sensitivity. Nonlinear susceptibility For a given nonlinear optical process, the polarization which generates the output is given by where is the th order nonlinear susceptibility, for . It is worth noting that all the even order susceptibilities become zero in centrosymmetric media. A proof of this is as follows. Let be the inversion operator, defined by for some arbitrary vector . Then applying to the left and right hand side of the polarization equation above gives Adding together this equation with the original polarization equation then gives which implies for in centrosymmetric media. Q.E.D. [Note 1: The final equality can be proven by mathematical induction, by considering two cases in the inductive step; where is odd and is even.] [Note 2: This proof holds for the case where is even. Setting gives the odd case and the remainder of the proof is the same.] As a second-order nonlinear process, SFG is dependent on the 2nd order susceptibility , which is a third rank tensor. This limits what samples are accessible for SFG. Centrosymmetric media include the bulk of gases, liquids, and most solids under the assumption of the electric-dipole approximation, which neglects the signal generated by multipoles and magnetic moments. At an interface between two different materials or two centrosymmetric media, the inversion symmetry is broken and an SFG signal can be generated. This suggests that the resulting spectra represent a thin layer of molecules. A signal is found when there is a net polar orientation. SFG intensity The output beam is collected by a detector and its intensity is calculated using where is the visible frequency, is the IR frequency and is the SFG frequency. The constant of proportionality varies across literature, many of them including the product of the square of the output frequency, and the squared secant of the reflection angle, . Other factors include index of refractions for the three beams. The second order susceptibility has two contributions where is the non-resonating contribution and is the resonating contribution. The non-resonating contribution is assumed to be from electronic responses. Although this contribution has often been considered to be constant over the spectrum, because it is generated simultaneously with the resonant response, the two responses must compete for intensity. This competition shapes the nonresonant contribution in the presence of resonant features by resonant attenuation. Because it is not currently known