Role of a localized modulation of χ⁽²⁾ in Čerenkov second-harmonic generation in nonlinear bulk medium

We study the second-harmonic generation in quadratic nonlinear media with localized spatial modulation of χ⁽²⁾ response. We demonstrate that the emission of Čerenkov second-harmonic takes place only when the fundamental beam illuminates the region of χ⁽²⁾ variation. This proves that the sharp modulation of the χ⁽²⁾ nonlinearity constitutes a sufficient condition for the emission of Čerenkov second harmonic in bulk materials. Our calculations are in excellent agreement with simple analytical approach utilizing the concept of reciprocal vectors representing the Fourier spectrum of the modulation of χ⁽²⁾.