Analysis of input-grating couplers having finite lengths.

Planar waveguide-grating couplers are analyzed to determine the effects of grating length, the incident beam size, and the coupling length of the grating on the input-coupling efficiency. The emphasis is to examine the effects of these finite lengths on the range of angles over which input coupling occurs and the shape of the efficiency curve. A general formula for normalized input-coupling efficiency is given to relate these lengths. When any one of these characteristic lengths is much smaller than the other two, both the angular width and angular dependence of the coupling efficiency for a scan of the incident beam angle will be mainly determined by that characteristic length. This is demonstrated experimentally for two cases: one has a relatively short grating length, and the other has a short coupling length.