Theoretical analysis of optical mixing at interfaces

The art of obtaining interface-specific information via optical mixing has been continuously developed since the advent of the laser in the early sixties. For instance, generating the second harmonic on crystals with inversion symmetry necessarily leads to interface-specific signals by virtue of breaking that symmetry at an interface. In this work, two techniques of interface-specific optical mixing are analyzed theoretically. The first is an exact macroscopic treatment of general optical mixing in reflection geometry. This model is an extension of Bloembergen's model which utilized a surface layer with nonlinear and linear properties atop a semi-infinite bulk with only linear properties. This model was extended by adding a bulk median with the nonlinear properties of the bulk between the surface layer and semi-infinite bulk. This allows for the separate comparison of the surface and bulk contributions to the total signal. Solutions to the model show an overall dependence on the secant of the nonlinear 'reflection' angle in addition to the angular dependencies introduced by the layered structure. Complicated phase-matching dependencies appear in three factors to both the surface and bulk contributions. They include the usual sinc behavior of the phase-mismatch parameter, dependencies on the effective phase differences incurred in the boundary layers, and a dependence on k2t-k 2s-1 where kt and ks are the wavevectors for the mixed wave and effective source wave caused by the induced polarization. A numerical analysis of these solutions shows that for pure S-polarization optical mixing in reflection geometry, surface-specificity can be enhanced using near-grazing incident angles. If the bulk and surface dispersions are small but different, the surface signal can be isolated from the bulk signal. Signals from either S or P-polarization in a CARS experiment can be selected by average changes in angles of 0.1 degrees. In P-polarization cases, it is possible to separate signals by observing the nonlinear `reflection' at the nonlinear Brewster's angle of the bulk. Since the nonlinear Brewster's angle is different for the two regions, there may be only a surface contribution. A second technique using evanescent fields at a dielectric waveguide interface is analyzed and discussed. Three-beam and four-beam CARS experiments are compared. For non-phase matched conditions the coherence length is small, typically several microns. These are compared with the case where phasematching is achieved. Despite the shorter interaction length, the phase-matched case generally provides signals two to three times larger. Typically, there is an additional enhancement of 10 to 103 if dispersions are included. This work shows that for reflection geometry optical mixing the surface contribution can be enhanced relative to the bulk without the experimental difficulty imposed by the use of waveguides.