Extending completeness of the eigenmodes of an open system beyond its boundary, for Green’s function and scattering-matrix calculations: data

In the related research the asymptotic completeness of eigenmodes is investigated. The asymptotic completeness of a set of the eigenmodes of an open system with increasing number of modes enables an accurate calculation of the system response in terms of these modes. Using the exact eigenmodes, such completeness is limited to the interior of the system. In paper it is shown that when the eigenmodes of a target system are obtained by the resonant-state expansion, using the modes of a basis system embedding the target system, the completeness extends beyond the boundary of the target system. This is illustrated by using the Mittag-Leffler series of the Green’s function expressed in terms of the eigenmodes, which converges to the correct solution anywhere within the basis system, including the space outside the target system. Importantly, this property allows one to treat pertubations outside the target system and to calculate the scattering crosssection using the boundary conditions for the basis system. Choosing a basis system of spherical geometry, these boundary conditions have simple analytical expressions, allowing for an efficient calculation of the response of the target system, as demonstrated for a resonator in a form of a finite dielectric cylinder.
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FIG. 1. a) RSE modes (crosses) of a dielectric sphere with ε = 9, basis radius R, and target radius 0.7R, in the complex wave number plane, along with the exact modes of the basis (dots) and target system (squares). b) Field amplitude of a mode close to a physical RS (green) and a lower and a higher order VG mode (blue, black), with arrows in a) indicating these modes for N = 39.

FIG. 2. a) Comparison of the exact analytic form of the GF (G_an) and its ML series using exact modes (G_ML) with N = 39 (k_maxR ≈ 28.5), and modes calculated via the RSE (GRSE) with N = 39 (k_maxR ≈ 20), for a source located in the gap at r′ = 0.85R (vertical dashed line), for kR = 5. b) Relative error of the ML series of the GF, with one point and both points on the surface of the basis sphere, as labelled, for kR = 5, calculated via Eq. (4) (solid) and Eq. (3) (dashed).

FIG. 3. a) Complex k-plane with modes of a target dielectric sphere (ε = 9) calculated with the RSE for l = 1, 2, both in TE and TM polarizations, with a basis k_maxR ≈ 20 (N = 158). b) Scattering cross-section of a perturbed sphere of radius R_p = 0.7R, with basis modes from a), calculated on the target (dashed lines) and basis surface (solid lines), with (teal) and without (red) VG modes, and the black line showing the exact solution.

FIG. 4. Scattering cross-section of a cylinder, with ε = 9, height and diameter of sqrt(2)R, and incoming excitation propagating along the cylinder axis (k_i), calculated with k_maxR = 13 (N = 1004). The eigenmodes in the complex plane are shown for comparison (crosses, right axis).

Research results based upon these data are published at https://doi.org/10.1103/PhysRevResearch.7.L012035