Collinear second harmonic generation of 20 wavelengths in a single two-dimensional decagonal nonlinear photonic quasi-crystal

We report on the observation of quasi-phase matched collinear second harmonic generation (SHG) at 20 wavelengths in a two-dimensional nonlinear photonic quasi-crystal with decagonal lattice. We show that at some wavelengths the second harmonics are generated via standard quasi-phase matching, namely a reciprocal vector exists that equals the phase-mismatch vector, while at others the SHG is as a result of the projection-based quasi-phase matching in which the momentum conservation is satisfied up to a projection of a reciprocal vector onto the direction of propagation. In spite of different generation mechanisms, the reciprocal vectors (or their projections) involved in the collinear QPM SHG can be described by a generalized equation.