Cascading in optical third-harmonic generation by crystalline quartz

Cascading, a process whereby lower-order effects combine to contribute to a higherorder nonlinear process, in optical third-harmonic generation (THG) is reported. By using the wedge-fringing method to observe the interference between bound and free waves at the third harmonic, the presence of two bound waves is verified. Theoretical development shows one bound wave to be due to true third-order nonlinearity plus a cascaded contribution involving the bound wave of second-harmonic generation. The other bound wave arises solely due to cascading involving the free wave of second-harmonic generation. Considerations of crystal and wave-vector orientations are presented, and a method of effective fields is used to account for cascading through local fields. $\alpha$-quartz was studied, allowing verification of the cascading by symmetry as well. A treatment is presented for the THG intensity-altering effect of gases surrounding condensed-phase materials. Finally, the specific observation of interferences among the several waves allows a direct calibration of $X^{\mathrm{}\left(3\right)\mathrm{}}(-3\mathrm{}\omega ,\omega ,\omega ,\omega ,)$ relative to the product $X^{\mathrm{}\left(2\right)\mathrm{}}(-3\mathrm{}\omega ,2\mathrm{}\omega ,\omega )X^{\mathrm{}\left(2\right)\mathrm{}}(-2\mathrm{}\omega ,\omega ,\omega )$ of quartz for the 1.91-μm fundamental wavelength employed here.