Two voices separated by a wall can still hear each other, if the wall is thin and the voices are strong. That is roughly how coupled optical resonators work: light trapped in one microresonator senses the presence of its neighbor through its resonant states, the eigenmodes of an open system with energy leakage. But these modes have a treacherous property: their "tails" grow exponentially outside the resonator. The wall between two resonators does not dampen the whisper; it amplifies it into a shout.

For a long time, physicists described such systems using strong coupling, an approximation that considers only nearest neighbors. The modes of individual resonators were considered a good basis, and their hybridization a way to predict the pair's behavior. But for open systems, this approximation fails: even two glass plates touching each other cannot be described by such a model. The reason is that resonant states outside the resonator do not form a complete set; discrete modes are insufficient, and one also needs a continuum of modes, a continuous set of states outside the resonator.

The new theory generalizes the Mittag-Leffler expansion, a representation of a function through its poles, for the Green's function, the system's response to a point source. Previously, this expansion worked only inside the resonator; now it has been extended beyond the boundary by adding a continuum to the discrete modes. It is as if an invisible background of countless whispers were added to a choir of soloists. For two identical dielectric plates, the theory yields eigenwavenumbers that match the exact analytical solution, and the error decreases as 1/N³ as the number of basis modes N increases. Accuracy is maintained for plate separations up to 40 plate thicknesses.

The practical benefit is clear: photonic molecules, coupled microresonators, and metasurfaces require accurate calculations of mode hybridization. The new theory provides a rigorous way to compute modes of arbitrary three-dimensional dispersive resonators using only the modes of individual resonators and the continuum. This opens the door to designing optical devices with predictable properties. Perhaps in a few years, engineers will be able to design lasers and sensors without relying on approximations that break down on the simplest structures.