Mie Theory for Spherical Cavities

This section gives a compact overview of the spherical Green-function workflow used by the Mie driver. The implementation follows generalized Mie theory for spherical particles and cavities; the full derivation is outside the scope of this documentation. For detailed formulas and physical discussion, see [Lee2020] and standard texts on electromagnetic scattering [Bohren1983], [Chew1995].

Geometry

Mie theory is useful when the dielectric environment is radially layered rather than planar. The current workflow supports a homogeneous sphere, a simple spherical cavity, and a core-shell sphere. In the core-shell case, the regions are ordered by radius:

• exterior medium,

• shell material,

• core or cavity material.

The dyadic Green tensor still decomposes into vacuum and structure-mediated parts,

\[\mathbf{G}(\mathbf r,\mathbf r',\omega) = \mathbf{G}_0(\mathbf r,\mathbf r',\omega) + \mathbf{G}_\mathrm{Sc}(\mathbf r,\mathbf r',\omega),\]

but the scattering term is expanded in spherical rather than cylindrical waves.

Spherical-wave expansion

The electromagnetic field of a point dipole can be expanded in vector spherical wave functions. Each multipole order \(n\) and azimuthal index \(m\) has electric and magnetic components built from spherical Bessel or Hankel functions and angular functions related to associated Legendre functions. The dielectric boundaries determine Mie coefficients by enforcing continuity of the tangential electric and magnetic fields at each spherical interface.

Schematically,

\[\mathbf{G}_\mathrm{Sc} = \sum_{n=1}^{n_\mathrm{max}}\sum_{m=-n}^{n} \left( A_{nm}\,\mathbf{M}_{nm}\mathbf{M}_{nm} + B_{nm}\,\mathbf{N}_{nm}\mathbf{N}_{nm} \right),\]

where \(\mathbf{M}_{nm}\) and \(\mathbf{N}_{nm}\) are vector spherical wave functions and \(A_{nm}\), \(B_{nm}\) represent the appropriate Mie response coefficients. The truncation order \(n_\mathrm{max}\) controls the number of multipoles included.

Physical interpretation

The Mie coefficients encode localized surface-plasmon and cavity resonances of the spherical structure. Near resonance, the imaginary part of the Green tensor can vary strongly with frequency and position. In macroscopic QED this directly modifies the spectral density, resonance energy transfer, and Purcell factor. This is the mechanism discussed in [Lee2020] for frequency-dependent energy transfer coupled to localized surface plasmon polaritons.

Unlike planar Sommerfeld solvers, spherical Mie calculations do not have horizontal translational symmetry. A scan output means one fixed source and a list of explicit observer positions. A pair output means all \(\mathbf{G}(\mathbf r_i,\mathbf r_j)\) tensors for an emitter list.

Implementation notes

The implementation is in mqed.Dyadic_GF.GF_Mie and is driven by mqed.Dyadic_GF.main_mie. The tutorial Mie Green’s Function for Spherical Cavities explains the core-shell spherical-cavity example, the scan HDF5 layout, and the spectral-density workflow.

The code is intended as a practical generalized-Mie workflow for MQED studies, but it should be benchmarked for each new physical regime. In particular, the current core-shell scan path is intended for source and observer points inside the core/cavity; shell-region observation points are not yet a validated production path.

References

[Lee2020] (1,2)

M. W. Lee and L. Y. Hsu, “Controllable frequency dependence of resonance energy transfer coupled with localized surface plasmon polaritons,” J. Phys. Chem. Lett. 11, 6796–6804 (2020).

[Bohren1983]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

[Chew1995]

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).