Frequency doubling by nonlinear diffraction in nonlinear photonic crystals

We investigate the process of second harmonic generation via the nonlinear diffraction in periodically poled nonlinear photonic crystals. We show that single pump beam excitation in such samples leads to the frequency doubled signal being emitted in a form of a cone (commonly referred to as Cherenkov radiation) and periodic pattern localized in the vicinity of the pump (Raman-Nath nonlinear diffraction). We show that the angle of the Cherenkov ring does not depends on poling pattern, whereas the Raman-Nath emission strongly depends on the poling structure. We study the dependence of the nonlinear diffraction pattern as a function of the incidence angle of the pump and find very good agreement between experiment and analytical formulas derived on the basis of the phase matching condition. The observed effects represent nonlinear generalization of optical diffraction in linear media and can find possible applications in second harmonic optical microscopy.