MMI Couplers and the Talbot Effect, Symmetries and Golden Ratio

The Talbot effect concerns the periodic self-imaging along an optical axis of a free-space optical field that is periodic in an initial transverse plane. It may be modeled by a shift-invariant linear system, fully characterized by the convolution of its impulse response. Self-imaging at integer and fractional Talbot distances of point sources on a regular grid in free space may then be represented by a transmission matrix that is circulant, symmetric, and persymmetric. The free-space Talbot effect may be mapped to the Talbot effect in a multimode waveguide by imposing the anti-symmetry of the mirror-like sidewalls created by the tight confinement of light within a high-index contrast multimode waveguide. The position of the anti-symmetry axis controls the distribution of discrete lattice points in a unit cell. For different distributions, interesting features such as conditional flexibility in the placement of access ports without altering amplitude and phase relationships, omitting ports without power penalty, closed form uneven splitting ratios, and offset access ports can be derived from the MMI coupler. As a specific example, a simple $2\times 2$ MMI coupler is shown to provide a power-splitting ratio related to the golden ratio $\phi$. The structure is amenable to planar photonic integration on any high-index contrast platform. The predictions of the theory are confirmed by simulation and verified by experimental measurements on a golden ratio MMI coupler fabricated using an SOI process.