  Second-order nonlinear effects, such as second-harmonic generation, can be
strongly enhanced in nanofabricated photonic materials when both fundamental
and harmonic frequencies are spatially and temporally confined. Practically
designing low-volume and doubly resonant nanoresonators in conventional
semiconductor compounds is challenging owing to their intrinsic refractive
index dispersion. In this work we review a recently developed strategy to
design doubly resonant nanocavities with low mode volume and large quality
factor by localized defects in a photonic crystal structure. We build on this
approach by applying an evolutionary optimisation algorithm in connection with
Maxwell equations solvers, showing that the proposed design recipe can be
applied to any material platform. We explicitly calculate the second-harmonic
generation efficiency for doubly resonant photonic crystal cavity designs in
typical III-V semiconductor materials, such as GaN and AlGaAs, targeting a
fundamental harmonic at telecom wavelengths, and fully accounting for the
tensor nature of the respective nonlinear susceptibilities. These results may
stimulate the realisation of small footprint photonic nanostructures in leading
semiconductor material platforms to achieve unprecedented nonlinear
efficiencies.
