General self-imaging properties in N × N multimode interference couplers including phase relations

Self-imaging properties of generalized N × N multimode interference couplers are derived. Positions, amplitudes, and phases of the self-images are directly related to the lengths and widths of the coupler by solving the eigenmode superposition equation analytically for any arbitrary length. Devices of length (M/N) 3Lc, where M is the multiple occurrence of the N self-images, are analyzed in detail. The general formalism is applied to practical N × N couplers used in integrated optics, and simple phase relations are obtained.