Phase-matched second-harmonic generation in slow-light photonic crystal waveguides

We present an analytical description of phase-matched second-harmonic generation in photonic crystal waveguides in the presence of loss. In particular, we investigate the case where the second-harmonic modes suffer from radiative losses. We use the adjoint field formalism to develop a coupled-mode theory that uses the quasinormal Bloch modes as the basis for modal expansion. To test our analytical description numerically, we propose a design in a lithium niobate photonic crystal slab waveguide, where a slow-light mode at the fundamental harmonic frequency is phase matched to a leaky mode at the second-harmonic frequency. The results of the numerical experiment agree with our analytical predictions.