Three wave mixing of airy beams in a quadratic nonlinear photonic crystals

We study three wave mixing processes of accelerating Airy beams in quadratic nonlinear crystals. In order to frequency convert these beams, the nonlinear process should be performed with the Fourier transform of the beam, rather than with the beam itself. This was confirmed experimentally by frequency doubling of one-dimensional and two-dimensional Airy beams in a periodically poled crystal. Sum-frequency or difference frequency process between two accelerating beam enable to set the acceleration rate and direction of the generated beam.