  Vertical coupling using a diffraction grating is a convenient way to couple
light into an optical waveguide. Several optimization approaches have been
suggested in order to design such a coupler; however, most of them are
implemented using algorithm based modelling. In this paper, we suggest an
intuitive method based on S-matrix formalism for analytically optimize 3-port
vertical grating coupler devices. The suggested method is general and can be
applied to any 3-port coupler device in order to achieve an optimal design
based on user constrains. The simplicity of the model allows reduction of the
optimization to two variables and the location of an absolute optimal operation
point in a 2D contour map. Accordingly, in an ideal device near 100 percent
coupling efficiency and insignificant return loss could be achieved. Our model
results show good agreement with numerical FDTD simulations and can predict the
general tendencies and sensitivities of the device behavior to changes in
design parameters. We further apply our model to a previously reported high
contrast uni-directional grating coupler device and show that additional
improvement in the coupling efficiency is achievable for that layout.
