  To optimize the coupling efficiency of grating couplers, we extend an
analytical model for the design of apodized grating couplers, taking into
account the constraints on the upper and lower bounds of the scattering
strength as determined by fabrication technology. We prove that our model gives
the global optimum distribution of the scattering strength with the presence of
such constraints. We apply our model to the standard problem of coupling from a
silicon chip to a single-mode fiber, as well as more complex problems including
the coupling to a vortex beam, and the design of a focusing grating. In the
problem of coupling to a vortex beam, we validate our model with full-wave
numerical simulations. For this problem, our design obtains efficiency that is
significantly higher compared with existing designs. Our theoretical study
provides a solid foundation for apodized grating coupler design in different
material platforms, and our deterministic algorithm can function as a
near-optimum starting point for advanced optimizations.
