Computing resonant modes of circular cylindrical resonators by vertical mode expansions

Open subwavelength cylindrical resonators of finite height are widely used in various photonics applications. Circular cylindrical resonators are particularly important in nanophotonics, since they are relatively easy to fabricate and can be designed to exhibit different resonance effects. In this paper, an efficient and robust numerical method is developed for computing resonant modes of circular cylinders which may have a few layers and may be embedded in a layered background. The resonant modes are complex-frequency outgoing solutions of the Maxwell's equations with no sources or incident waves. The method uses field expansions in one-dimensional (1D) “vertical” modes to reduce the original three-dimensional eigenvalue problem to 1D problems and uses Chebyshev pseudospectral method to compute the 1D modes and set up the discretized eigenvalue problem. In addition, a new iterative scheme is developed so that the 1D nonlinear eigenvalue problems can be reliably solved. For metallic cylinders, the resonant modes are calculated based on analytic models for the dielectric functions of metals. The method is validated by comparisons with existing numerical results, and it is also used to explore subwavelength dielectric cylinders with high-$Q$ resonances and analyze gold nanocylinders.

Figure 1

Normalized resonant frequencies of a few resonant modes on an AlGaAs cylinder of varying aspect ratio.

Figure 2

Magnitudes of the electric field $\mathbf{E}$ and scaled magnetic field $\mathbf{H}$ on the $xz$ plane for the AlGaAs cylinder at points $\mathsf{A}$ and $\mathsf{B}$ in Fig. 1, where (a) and (b) are for point $\mathsf{A}$ and (c) and (d) are for point $\mathsf{B}$.

Figure 3

Magnitudes of the electric field $\mathbf{E}$ (left) and scaled magnetic field $\mathbf{H}$ (right) on the $xz$ plane of a resonant mode for a silicon cylinder with aspect ratio $a/h=0.88211$.

Figure 4

Normalized scattering cross section of a gold circular cylinder with radius 40 nm and height 50 nm for a normal incident plane wave.

Figure 5

Comparison of the CP model and measured data of Ref. [38] for gold. Panels (a) and (b) show real and imaginary parts of $\sqrt{\varepsilon }=n+ik$.