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We describe an approach based on topology optimization that enables automaticdiscovery of wavelength-scale photonic structures for achieving high-efficiencysecond-harmonic generation (SHG). A key distinction from previous formulationand designs that seek to maximize Purcell factors at individual frequencies isthat our method not only aims to achieve frequency matching (across an entireoctave) and large radiative lifetimes, but also optimizes the equally importantnonlinear--coupling figure of merit $\bar{\beta}$, involving a complicatedspatial overlap-integral between modes. We apply this method to the particularproblem of optimizing micropost and grating-slab cavities (one-dimensionalmultilayered structures) and demonstrate that a variety of material platformscan support modes with the requisite frequencies, large lifetimes $Q > 10^4$,small modal volumes $\sim (\lambda/n)^3$, and extremely large $\bar{\beta}\gtrsim 10^{-2}$, leading to orders of magnitude enhancements in SHG efficiencycompared to state of the art photonic designs. Such giant $\bar{\beta}$alleviate the need for ultra-narrow linewidths and thus pave the way forwavelength-scale SHG devices with faster operating timescales and highertolerance to fabrication imperfections.

Cavity-enhanced second harmonic generation via nonlinear-overlap optimization