Coupled Resonators: Why the Simple Model Fails ✛
Context
In optics and photonics, coupled resonators form the basis of photonic molecules, lasers, and sensors. Their modes are typically described through hybridization of individual resonator modes, using an analogy with strong coupling in quantum mechanics. However, recent work has shown that resonant states of open systems grow exponentially outside the resonator, casting doubt on the applicability of such approaches. Several theories have been proposed around this problem, but all of them fail for planar systems, as demonstrated in this paper.
Methods
The author generalizes the Mittag-Leffler expansion for the dyadic Green's function so that it holds outside the resonator. To this end, a continuum of modes arising from the incompleteness of the basis outside the volume is added to the discrete set of resonant states. The resulting system of linear equations is solved numerically. To overcome the problem of exponential growth of modes with increasing distance between resonators, the theory is combined with the resonant-state expansion (RSE) method, which allows choosing an appropriate integration contour and limiting the growth.
Results
For two identical dielectric plates in contact, the new theory yields eigenvalues of wavenumbers that match the exact analytical solution. The relative error decreases as 1/$N^3$ as the number of basis modes N increases. For separations between plates up to 40 plate thicknesses, the theory with the RSE approach maintains accuracy. For coupled microresonators with Bragg mirrors, it is shown that the strong-coupling concept gives unphysical mode splitting even for high-quality states. Mode hybridization does occur, but the contribution comes not only from the fundamental modes but also from the exponentially growing tails of other states.
Implications
This work shows that the standard strong-coupling concept is inapplicable to open resonators due to the exponential growth of resonant states outside the resonator. This is a fundamental limitation that was not accounted for in previous theories. The new theory provides a rigorous way to compute hybridized modes of arbitrary three-dimensional dispersive resonators using only the modes of individual resonators and a continuum. This differs from related works, which either do not converge or require excessively large computational domains.
Future development
The theory can be extended to nonreciprocal and bianisotropic systems, as indicated in the paper. It may also be applied to resonators with more complex geometries, such as photonic crystals. The combination with RSE opens the way to efficient calculations of coupled resonators at large separations.
Impact
The results will impact the design of photonic molecules, coupled microresonators, and metasurfaces, where accurate mode hybridization is important.
Next steps
Test the theory on three-dimensional resonators such as spheres or disks, and on systems with dispersion, for example, metallic nanoparticles.
Key open problems
In the optics of coupled resonators, the question of the completeness of the basis of resonant states outside the resonator remains open. This work proposes a solution by adding a continuum, but the question of the physical interpretation of these continuum modes and their role in quantum electrodynamics remains open.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
The paper proposes a rigorous theory of coupled open resonators, generalizing the Mittag-Leffler expansion of the Green's function beyond the resonator by adding a continuum of modes. The theory is tested on two dielectric slabs and microresonators with Bragg mirrors, showing convergence with an error of 1/N^3.
Where its strength is
The strength of the work lies in the rigorous treatment of the incompleteness of the resonant-state basis outside the resonator and in demonstrating that the standard strong-coupling concept is inapplicable to open systems.
Where it could go next
- It might be interesting to compare the proposed approach with resonant-state analysis in systems with Bragg mirrors, where the coupling of discrete modes to the continuum is also important.based on: Diamond Resonator: How to Catch a Loud Echo arXiv:2603.19421
- A related work concerns a resonator interacting with a single spin: one could consider how the proposed theory describes the resonator's coupling to a quantum emitter beyond the strong-coupling approximation.based on: A Resonator That Catches a Single Spin arXiv:2603.25258
Worth trying
- Test the theory's applicability to resonators with gain, where exponential mode growth might be compensated.
- Use the mode continuum to compute spontaneous emission near open resonators.
Why this may matter
A rigorous method for calculating hybridized modes of open resonators could eliminate strong-coupling artifacts in photonic molecules and sensors, improving the accuracy of predictions.
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The archive is sparse around this work: the closest review is only 0.49, against a usual 0.64. It looks like few have drilled nearby.
The exponential growth of resonant states outside the resonator means that even very distant resonators can interact strongly through their 'tails', which contradicts the intuition of weak coupling.
Key numbers
- Error convergence rate: 1/N^3
- Maximum separation between plates: 40 plate thicknesses
- Number of basis modes for accuracy: 100
- Optimal ratio of continuum to resonant states: 1.5