  Photonic nanostructures simultaneously maximizing spectral and spatial
overlap between fundamental and second-harmonic confined modes are highly
desirable for enhancing second-order nonlinear effects in nonlinear media.
These conditions have thus far remained challenging to satisfy in photonic
crystal cavities because of the difficulty in designing a band gap at the
second-harmonic frequency. Here, we solve this issue by using instead a bound
state in the continuum at that frequency, and we design a doubly-resonant
photonic crystal slab cavity with strongly improved figures of merit for
nonlinear frequency conversion when compared to previous photonic crystal
designs. Furthermore, we show that the far-field emission at both frequencies
is highly collimated around normal incidence, which allows for simultaneously
efficient pump excitation and collection of the generated nonlinear signal. Our
results could lead to unprecedented conversion efficiencies in both parametric
down conversion and second harmonic generation in an extremely compact
architecture.
