Optical waveguide coupling and the golden number

The numerical modeling of a directional coupler composed of four identical slab waveguides reveals that the ratio of the field extrema of the four propagating modes is close to the golden ratio ϕ. This is confirmed by an analytical derivation leading to the exact dispersion equation of the eigenmodes of the four-waveguide coupler and to the expression of the ratio between field extrema. A coupled wave analysis gives the analytical expression of the propagation constants of the coupler eigenmodes and of the ratio between extrema in terms of the coupling coefficients between waveguides; it shows that this ratio is strictly equal to ϕ in weakly coupled structures where the coupling coefficient between next-nearest neighboring waveguides is negligible.