The ability to generate light by nonlinear metasurfaces and shape its properties has made metasurfaces a promising platform for developing new types of nonlinear optical devices. Extending beyond the conventional two-dimensional metasurface has the potential to further enhance and control nonlinear processes in metamaterials. However, it still remains a challenge as it requires distinct and tailored phase matching schemes, different from those used in bulk nonlinear crystals. Here, we study theoretically and numerically the mechanism of second harmonic generation from a nonlinear bilayer metamaterial. We present a design approach that is based on adding nonlinear geometric phase to the layers, by a relative rotation of their meta-atoms orientation. We demonstrate how the nonlinear geometric phase affects the system through local and collective interactions, allows excitation of nontrivial solutions, and controls the second harmonic emission and directivity. It is found that the bilayer supports a bound mode with strongly enhanced dipole excitation, leading to a highly confined second harmonic field within the structure. The presented approach and results lay the foundation for designing efficient and tunable multilayer nonlinear metamaterials.

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• Geometric & topological phases
• Metamaterials
• Metasurfaces
• Nonlinear optics
• Second order nonlinear optical processes
• Dipole approximation
• Finite-element method