  The 2$\times$2 3-dB couplers are one of the most widely used and important
components in silicon photonics. We propose an ultra-broadband and compact
2$\times$2 3-dB adiabatic coupler defined by b-splines and optimized with an
efficient supermode-injected adjoint shape optimization. By employing mode
adiabatic evolution and mode coupling at two different wavelength ranges,
respectively, we achieve an ultra-broad bandwidth of 530 nm from 1150nm
to1680nm with a power imbalance below $\pm$0.76 dB in a compact coupling length
of 30 $\mu m$ according to our simulation results. The supermode-injected
adjoint shape optimization can also be applied to the design of other photonic
devices based on supermode manipulation.
