Resonant-state expansion applied to three-dimensional open optical systems: Complete set of static modes

We present two alternative complete sets of static modes of a homogeneous dielectric sphere, for their use in the resonant-state expansion (RSE), a rigorous perturbative method in electrodynamics. Physically, these modes are needed to correctly describe the static electric field of a charge redistribution within the optical system due to a perturbation of the permittivity. We demonstrate the convergence of the RSE toward the exact result for a perturbation describing a size reduction of the basis sphere. We then revisit the quarter-sphere perturbation treated by Doost et al. [Phys. Rev. A 90, 013834 (2014)], where only a single static mode for each angular momentum was introduced, and show that using a complete set of static modes leads to a small though non-negligible correction of the RSE result, improving the agreement with finite-element simulations. As another example of applying the RSE with a complete set of static modes, we calculate the resonant states of a dielectric cylinder, also comparing the result with a finite-element simulation.

Figure 1

TM RSs with $l=5$ and a fixed $m$ for a homogeneous perturbation Eq. (40), from the basis sphere with $\epsilon =4$ to the perturbed sphere with $\epsilon +\mathrm{\Delta }\epsilon =9$. (a) Perturbed RS wave numbers calculated using the RSE with $R{k}_{\max }=4096$ without static modes (blue crosses), as well as using the exact secular equation or the full RSE (red circles with dots). The RS wave numbers of the basis system are shown as black circles with dots. Inset: Dielectric constant profile of the basis system (black dashed line), perturbed system (blue line), and the perturbation (red line). (b) Error in the perturbed wave numbers calculated with no static modes (crosses), with VSC modes (triangles), and with VC modes (squares), for different $k_{\max }$ as labeled and color coded, as well as extrapolated using $R{k}_{\max }=4096$.

Figure 2

As Fig. 1 but for the size perturbation Eq. (41) to a sphere with the radius of $0.8R$.

Figure 3

(a) Wave numbers of the basis RSs (black circles with dots) and the perturbed RSs (blue circles) for a quarter-sphere perturbation given by Eq. (42) with $\mathrm{\Delta }\epsilon =1$, calculated by the RSE with $R{k}_{\max }=40$ using VSC static modes. A sketch of the structure is shown. (b) Enlargement of panel (a) showing the splitting due to the perturbation of $2l+1$ degenerate basis WGMs with $l=7$. The results of FEM simulations using 200 000, 100 000, 50 000, and 25 000 finite elements are shown for comparison. The insets show further enlargements around individual RSs, containing additionally data from extrapolated FEM, extrapolated RSE, RSE using only the $\lambda =0$ static modes, and no static modes, as labeled. (c) Error of the RSE wave numbers using the extrapolated RSE as reference, as function of $\mathrm{Re}\left(Rk\right)$, calculated by the RSE with different $R{k}_{\max }$ as labeled, for the RSs shown in panel (b).

Figure 4

(a) RS wave numbers of a dielectric cylinder with radius $a$ and height $2a$, with azimuthal quantum numbers $m=0,1$, and 7, and parity $+$ and $-$ as labeled, calculated using the RSE with $R{k}_{\max }=75$ ($R=a\sqrt{2}$). The leaky RS having $-\mathrm{Im}\left(Rk\right)>1.5$ are separated by a gray shadow. (b) Error of the RS wave numbers for $R{k}_{\max }=20$ and 41, using the values for $R{k}_{\max }=75$ as reference, not showing results for the leaky modes.

Figure 5

Convergence of the wave number for a selected RS of a cylinder, with $m=1$ and parity $-$. Results by the RSE with different $R{k}_{\max }$ as labeled (close to the symbol), both for the VSC (black squares) and VC (red open circles) sets of static modes. The extrapolated values for the RSE and the value calculated using comsol [9] are also shown, along with their estimated errors.