  Second harmonic generation in nonlinear materials can be greatly enhanced by
realizing doubly-resonant cavities with high quality factors. However,
fulfilling such doubly resonant condition in photonic crystal (PhC) cavities is
a long-standing challenge, because of the difficulty in engineering photonic
bandgaps around both frequencies. Here, by implementing a second-harmonic bound
state in the continuum (BIC) and confining it with a heterostructure design, we
show the first doubly-resonant PhC slab cavity with $2.4\times10^{-2}$ W$^{-1}$
conversion efficiency under continuous wave excitation. We also report the
confirmation of highly normal-direction concentrated far-field emission pattern
with radial polarization at the second harmonic frequency. These results
represent a solid verification of previous theoretical predictions and a
cornerstone achievement, not only for nonlinear frequency conversion but also
for vortex beam generation and prospective nonclassical sources of radiation.
