Resonant-state expansion of three-dimensional open optical systems: Light scattering

A rigorous method of calculating the electromagnetic field, the scattering matrix, and scattering cross sections of an arbitrary finite three-dimensional optical system described by its permittivity distribution is presented. The method is based on the expansion of the Green's function into the resonant states of the system. These can be calculated by any means, including the popular finite-element and finite-difference time-domain methods. However, using the resonant-state expansion with a spherically symmetric analytical basis, such as that of a homogeneous sphere, allows to determine a complete set of the resonant states of the system within a given frequency range. Furthermore, it enables to take full advantage of the expansion of the field outside the system into vector spherical harmonics, resulting in simple analytic expressions. We verify and illustrate the developed approach on an example of a dielectric sphere in vacuum, which has an exact analytic solution known as Mie scattering.

Figure 1

Complex reflection coefficient $S_{lmp}^{lmp}\left(k\right)$ for TE polarization ($p=\mathrm{TE}$) and $l=3$ for a dielectric sphere of $\varepsilon =9$ in vacuum. The exact analytic result (normal line) is compared with the ML expansion using the exact RSs for $\varepsilon =9$ (thin line) and with the RSE calculation using a basis sphere of $\epsilon =4$ (thick line), both computed for a basis size of $N=65$. 3D representation with the phase given by the line color.

Figure 2

(a) Complex reflection coefficient $S_{lmp}^{lmp}\left(k\right)$ for TE polarization ($p=\mathrm{TE}$) and $l=3$ for a dielectric sphere of $\varepsilon =9$ in vacuum. $\mathrm{Re}\left(S\right)$ in black, $\mathrm{Im}\left(S\right)$ in red. The analytical result (solid line) is compared with the ML expansion using the exact RSs for $\varepsilon =9$ (dashed line) and with the RSE calculation using a basis sphere of $\epsilon =4$ (dashed-dotted line), both calculated for a RS basis size of $N=65$, and a refined RSE calculation including a first-order correction of the RS fields (dotted line). The RS wave numbers $k_n$ are given as crosses. (b) Absolute error $|S-S^{\left(\mathrm{exact}\right)}|$ of the ML (black solid line), the RSE (green solid line), and the refined RSE results (blue line). Results for basis sizes $N=65$ and 1025 as labeled. The dotted lines show asymptotic power-law dependencies as labeled.

Figure 3

As Fig. 2, but for $l=6$. The inset in (b) is a zoom around the fundamental WG RS.

Figure 4

As Fig. 2, but for TM polarization ($p=\mathrm{TM}$).

Figure 5

As Fig. 3, but for TM polarization ($p=\mathrm{TM}$).

Figure 6

(a) Partial scattering cross section ${\sigma }_l^{\mathrm{sca}}\left(k\right)$ for TE polarization and $l=3$ for a dielectric sphere of $\varepsilon =9$ in vacuum. The analytical result (black) is compared with the ML expansion for a basis size of $N=33$, using the exact RSs for $\varepsilon =9$ (red line), and the RSs determined by the RSE with refinement, using as basis a sphere with $\epsilon =4$ (blue line). (b) Absolute error $|{\sigma }_l^{\mathrm{sca}}\left(k\right)-{\sigma }_l^{\mathrm{sca}\left(\mathrm{exact}\right)}\left(k\right)|$ for ML (red line) and RSE calculation (blue line). Results for a basis size $N=33$ are given as thick lines, and for $N=1025$ as thin lines. Inset in (a): absolute error averaged over the range of $0\le kR\le 10$, as function of the basis size $N$. The dotted line gives $N^{-1}$.

Figure 7

As Fig. 6 but for TM polarization ($p=\mathrm{TM}$).

Figure 8

As Fig. 6 but for $l=6$, and basis sizes $N=32$ and 1024 as labeled.

Figure 9

As Fig. 8 but for TM polarization ($p=\mathrm{TM}$).

Figure 10

Total scattering cross section ${\sigma }^{\mathrm{sca}}\left(k\right)$ for a dielectric sphere of $\varepsilon =9$ in vacuum. The analytical result using using Mie theory [33] (black) is compared with ML expansion for a basis size of $N=64$, using the exact RSs for $\varepsilon =9$ (red line), and the RSs determined by the RSE with refinement, using as basis a sphere with $\epsilon =4$ (blue line). Inset: absolute error averaged over $0\le kR\le 10$, as function of the basis size $N$. The dotted line gives $N^{-1}$.