  Silicon nitride ($Si_3N_4$) planar waveguide platform combined with
silicon-on-insulator (SOI) devices offer a whole new generation of
system-on-chip applications. Therefore, efficient coupling of an $Si_3N_4$
waveguide to an SOI waveguide is essential. We present a coupler to interface a
1.8 $\mu m$-wide $Si_3N_4$ waveguide to a 0.5 $\mu m$-wide SOI waveguide based
on the focusing property of the Luneburg lens. In order to match the refractive
indices of the waveguides with the edges of the lens, one side of the lens is
flattened with quasi-conformal transformation optics. The designed coupler is
implemented by graded photonic crystals. The three-dimensional numerical
simulations indicate that the 1.93 $\mu m$-long coupler has an average coupling
loss of 0.13 dB in the C-band.
