High-quality crystals without inversion symmetry are the conventional platform to achieve optical frequency conversion via three wave-mixing. In bulk crystals, efficient wave-mixing relies on phase-matching configurations, while at the micro- and nano-scale it requires resonant mechanisms that enhance the nonlinear light-matter interaction. These strategies commonly result in wavelength-specific performances and narrowband applications. Disordered photonic materials, made up of a random assembly of optical nonlinear crystals, enable a broadband tunability in the random quasi-phase-matching (RQPM) regime and do not require high-quality materials. Here, we combine resonances and disorder by implementing RQPM in Mie-resonant spheres of a few microns realized by the bottom-up assembly of barium titanate nano-crystals. The measured second harmonic generation (SHG) reveals a combination of broadband and resonant wave mixing, in which Mie resonances drive and enhance the SHG, while the disorder keeps the phase-matching conditions relaxed. This new phase-matching regime can be described by a random walk in the SHG complex plane whose step lengths depend on the local field enhancement within the micro-sphere. Our nano-crystals assemblies provide new opportunities for tailored phase-matching at the micro-scale, beyond the coherence length of the bulk crystal. They can be adapted to achieve frequency conversion from the near-ultraviolet to the infrared ranges, they are low-cost and scalable to large surface areas.
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