Output Formulation for Symmetrically Excited One-to-$N$ Multimode Interference Coupler

In this paper, we present a closed form formulation for the output signals of one-to-N multimode interference coupler under symmetric excitation. We derive the output ports phases and show that the output phase has a quadratic dependence on the output port number. Using beam propagation simulations, we compare the analytical phase profile with the simulation results for different waveguiding structures. In the case of Si/SiO2 structures, our formulation predicts the output phase profile with errors not more than about 1°. Finally, we show that nonideal effects, such as limited number of guided modes, modal phase errors, and extension of the field profile into the cladding layers have minimal effects on the phase profile in comparison with the output amplitudes. These results can be used in variety of optoelectronic applications, where the knowledge of the phase profile is crucial, such as optical phased arrays.