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

Abstract

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 “ $\infty -\infty $ ” 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 $ \boldsymbol {\alpha }_{\mathrm {eff}}(\omega)=[\boldsymbol {\alpha }^{-1}-\mathbf {G}_{\mathrm {loc}}(\mathbf {r}')]^{-1}$ yields the coupled resonances from $ \det [\boldsymbol {\alpha }_{\mathrm {eff}}^{-1}(\omega)]=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 eigenfunctions are obtained numerically, providing a practical design tool for cavity–particle systems from microwave to terahertz frequencies.