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The 2$\times$2 3-dB couplers are one of the most widely used and importantcomponents in silicon photonics. We propose an ultra-broadband and compact2$\times$2 3-dB adiabatic coupler defined by b-splines and optimized with anefficient supermode-injected adjoint shape optimization. By employing modeadiabatic evolution and mode coupling at two different wavelength ranges,respectively, we achieve an ultra-broad bandwidth of 530 nm from 1150nmto1680nm with a power imbalance below $\pm$0.76 dB in a compact coupling lengthof 30 $\mu m$ according to our simulation results. The supermode-injectedadjoint shape optimization can also be applied to the design of other photonicdevices based on supermode manipulation.

Ultra-broadband and compact 2$\times$2 3-dB silicon adiabatic coupler based on supermode-injected adjoint shape optimization