Asymmetric χ⁽²⁾-translated optical frequency combs assisted by avoided mode crossing in concentric ring resonators

χ⁽²⁾-translated microcomb generation in microresonators that possess both χ⁽²⁾ and χ⁽³⁾ nonlinear responses opens the door for ultra-broadband integrated comb sources. The interplay between the second- and third-order nonlinearities within a fixed coupling coefficient fertilizes complicated cavity dynamics which is of paramount scientific and technological potential. However, this coupling coefficient can be drastically wavelength-dependent, which is lack of consideration in previous studies. Here, we extend the range of coupling strengths to a full description and propose a new approach to delineate the spectral response of the interactions between the χ⁽²⁾ and χ⁽³⁾ nonlinearities. Critically, the underpinned physics is enabled by avoided mode crossing (AMX) in concentric double-ring microresonators. We demonstrate that the evolution of the anti-symmetric mode at fundamental wavelengths disrupts spectral symmetry, leading to asymmetric χ⁽²⁾-translated optical frequency combs at second-harmonic wavelengths. Simultaneous generation of skewed two-color optical frequency combs is numerically realized in an exemplary gallium phosphide-on-insulator platform with a coupling constant from 133.3 m^-1W^-1/2 to 7.4 m^-1W^-1/2, showing reasonable agreement with our theoretical model. Our findings provide a novel approach to shaping the optical frequency comb, which may facilitate potential applications in self-referencing and frequency metrology with desired comb spectral shapes.