General theory of beam splitting in multimode rectangular waveguides

We analyze the situation where a symmetric optical field from a square cross-section waveguide, 2a x 2a, is symmetrically injected into a multi-mode rectangular waveguide of cross-section 2a x 2b (b greater than or = a). We obtain a closed-form expression for the transverse field at points z = (r/s)(2b)(exp 2)/lambda, along the rectangular guide axis, where r and s are mutually prime. The closed-form expression provides the first direct analytical proof of the input field replication taking place in the multi-mode guide and creates a simple basis for calculating design parameters for splitters and recombiners based on such waveguides.