Exact Local-Field Renormalization for Deep-Subwavelength Particles in Rectangular Cavities

We present a rigorous, semi-analytical framework for predicting the joint eigenfrequencies of a deep-subwavelength particle embedded in a perfectly conducting rectangular cavity. The formulation retains the full cavity-mode spectrum and is therefore strictly causal, in contrast to Jaynes–Cummings–type models that truncate the spectrum and fail in strong coupling. A ladder-type Green-function renormalization is introduced: three successive subtractions—cavity minus rectangular waveguide, waveguide minus parallel- plate, and parallel-plate minus free space—remove the “∞−∞” singularity of the local field. The resulting local dyadic Green function is obtained with a rapidly convergent recursive algorithm whose cost scales linearly with the number of spectral terms. Once the local field is known, the cavity-renormalized polarizability αeff(ω) = [α−1−Gloc(r′)]−1yields the coupled resonances from det [α−1eff(ω)] = 0. Benchmark cases involving isotropic, gyrotropic, and chiral spheres confirm exponential convergence and capture both weak- and strong-coupling regimes without adjustable parameters. The method is numerically robust, applies to arbitrary material tensors, and can be extended to structured waveguides whose transverse eigen-functions are obtained numerically, providing a practical design tool for cavity–particle systems from microwave to terahertz frequencies.