  Efficient coupling between on-chip sources and cavities plays a key role in
silicon photonics. However, despite the importance of this basic functionality,
there are few systematic design tools to simultaneously control coupling
between multiple modes in a compact resonator and a single waveguide. Here, we
propose a large-scale adjoint optimization approach to produce wavelength-scale
waveguide--cavity couplers operating over tunable and broad frequency bands. We
numerically demonstrate couplers discovered by this method that can achieve
critical, or nearly critical, coupling between multi-ring cavities and a single
waveguide at up to six widely separated wavelengths spanning the
$560$--$1500$~nm range of interest for on-chip nonlinear optical devices.
