Parametric wave interaction in one-dimensional nonlinear photonic crystal with randomized distribution of second-order nonlinearity

We theoretically study the parametric wave interaction in nonlinear optical media with randomized distribution of the quadratic nonlinearity $chi ^{(2)}$ . In particular, we discuss the properties of second and cascaded third harmonic generation. We derive analytical formulas describing emission properties of such harmonics in the presence of $chi ^{(2)}$ disorder and show that the latter process is governed by the characteristics of the constituent processes, i.e. second harmonic generation and sum frequency mixing. We demonstrate the role of randomness on various second and third harmonic generation regimes such as Raman?CNath and ?erenkov nonlinear diffraction. We show that the randomness-induced incoherence in the wave interaction leads to deterioration of conversion efficiency and angular spreading of harmonic generated in the processes relying on transverse phase matching such as Raman?CNath interaction. On the other hand, the ?erenkov frequency generation is basically insensitive to the domain randomness.