Exact Local-Field Renormalization for Deep-Subwavelength Particles in Rectangular Cavities
We present a rigorous, semi-analytical framework for predicting the eigenfrequencies of a deep-subwavelength particle embedded in a perfectly conducting rectangular cavity. The formulation retains the \emph{full} cavity-mode spectrum and is therefore fully causal, in contrast to Jaynes--Cummings-type models that truncate the spectrum and fail in the strong-coupling regime. 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 ``$\infty-\infty$'' singularity of the local field. The resulting local dyadic Green function is obtained using a rapidly convergent recursive algorithm whose computational cost scales linearly with the number of spectral terms. Once the local field is known, the cavity-renormalized polarizability \[ \boldsymbolα_{\mathrm{eff}}(ω) = \left[ \boldsymbolα^{-1} - \mathbf{G}_{\mathrm{loc}}(\mathbf{r}') \right]^{-1} \] yields the coupled resonances from \[ \det\!\left[ \boldsymbolα_{\mathrm{eff}}^{-1}(ω) \right] = 0. \] Benchmark cases involving isotropic, gyrotropic, and chiral spheres confirm exponential convergence and capture both the 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 eigenmodes are obtained numerically, providing a practical design tool for cavity--particle systems spanning microwave to terahertz frequencies.