Abstract

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 $\mathrm{\varphi }$. 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 $\mathrm{\varphi }$ in weakly coupled structures where the coupling coefficient between next-nearest neighboring waveguides is negligible.

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