Supplementary Information for : ” Optical transduction and routing of microwave phonons in cavity - optomechanical circuits ”

In the optomechanical cavity-waveguide coupled devices, we can change the cavity-waveguide coupling (i.e. γe) on purpose. This is achieved by the design of a lowQ mechanical cavity mode and varying the number of mirror cells (Fig. S-1a). As shown in Fig. S-1b, the blue curves are the mechanical band structure of the mirror unit cell (blue rectangle in Fig. S-1a). We design the cavity such that the mechanical cavity frequency (red dashed line in Fig. S-1b) overlaps with the band of mirror unit cell, such that the mechanical cavity mode can tunnel through the mirror cells into waveguide. Meanwhile, the optical cavity frequency (red dashed line in Fig. S-1c) lies within the optical band gap of mirror unit cell, such that the optical cavity mode keeps high-Q. By varying the number of unit cells, we find the simulated radiation mechanical coupling rate into waveguide (γe/2π) oscillates between a few MHz to as high as 30 MHz, due to the interference within the mirror unit cells.