SFG Intensity
The output beam is collected by a detector and its intensity is measured. The intensity of the beam is given by
Here, intensity is directly proportional to the susceptibility squared and the product of the intensities of the incoming beams. The IR frequency is given as ω2 and the visible frequency is given as ω1. The constant of proportionality varies across literature, many of them including the product of the square of the output frequency, ω2 and the squared secant of the reflection angle, sec2β. Other factors include index of refractions for the three beams.
The second order susceptibility has two contributions
where χnr is the non-resonating contribution and χr 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 how to adequately correct for nonresonant interferences, it is very important to experimentally isolate the resonant contributions from any nonresonant interference, often done using the technique of nonresonant suppression.
The resonating contribution is from the vibrational modes and shows changes in resonance. It can be expressed as a sum of a series of Lorentz oscillators
where A is the strength or amplitude, ω is the frequency or energy, Γ is the damping or linewidth coefficient, and each q is a resonance mode. The amplitude is a product of μ, the induced dipole moment, and α, the polarizability. Together, this indicates that the transition must be both IR and Raman active.
The above equations can be combined to form
which used to model the SFG output over a range of wavenumbers. When the SFG system scans over a vibrational mode of the surface molecule, the output intensity is resonantly enhanced. In a graphical analysis of the output intensity versus wave number, this is represented by peaks. Depending on the system, inhomogeneous broadening and interference between peaks may occur. The Lorentz profile can be convoluted with a Gaussian intensity distribution to better fit the intensity distribution.
Read more about this topic: Sum Frequency Generation Spectroscopy, Theory
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