Unitary transformation of general nonoverlapping-image multimode interference couplers with any input and output ports

An explanation of optical unitary transformation is presented for general nonoverlapping-image multimode interference (MMI) couplers with any number of input and output ports. The light transformation in the MMI coupler can be considered as an optical field matrix acting on an input light column vector. We investigate the general phase principle of output light image. The complete proof of nonoverlapping-image MMI coupler's optical unitarity along with the phase analysis of matrix element is provided. Based on a two-dimensional finite-difference time-domain (2D-FDTD) simulation, the unitary transformation is obtained for a 4 × 4 nonoverlapping-image MMI coupler within a deviation of 4 × 10⁻² for orthogonal invariance and 8 × 10⁻² for transvection invariance in the C-band spectral range. A compact 1 × 4 splitter based on cascaded MMI coupler is proposed, showing a phase deviation less than 5.4° while maintaining a low-loss performance in C-band spectra. (paper)