Microwave Transmission through a Single Subwavelength Annular Aperture in a Metal Plate

The resonant transmission of a small annular aperture, with a diameter much smaller than the radiation wavelength, in a thin metal plate is studied at microwave frequencies. It transpires that such an annular aperture supports several resonant guided modes, including those that are not quantized in the azimuthal direction. Such modes have resonant frequencies that are largely independent of the diameter of the annular aperture, thus being supported by annular apertures that tend to zero radius. The transmittance of such a structure at microwave frequencies is detailed and compared with the predictions of a finite element method model.

Figure 1

Experimental arrangement: Matched standard-gain source and detector horns are connected to a sweep oscillator and scalar network analyzer, respectively. Inset: Sample geometry (see text).

Figure 2

Resonant frequency squared as a function of $N^2$ for the experimental transmission data obtained at $\theta =0°$, $\psi =0°$, compared to that calculated using the FEM code. Also shown are the predictions of Eq. (1), generated using consecutive values of $N$ between 1 and 13, and a circumferential quantum number of 1. Inset: The transmission spectrum obtained at $\theta =0$, $\psi =0°$. A total of 11 resonant modes are clearly visible over the selected frequency range.

Figure 3

The time averaged magnitude [(i)] of the electric field together with the $E$- and $H$-vector magnitudes [(ii) and (iii), respectively], evaluated at the peak transmission frequency of (a) mode C (36.1 GHz) and (b) mode A (34.7 GHz). The $E$- and $H$-vectors are calculated at values of phase corresponding to maximum intensity. Plane wave radiation is incident from above with its wave vector in the $z$ direction, and incident electric vector lying parallel to the $x$ axis. Also shown (iv) is the electric vector magnitude calculated at the resonant frequency of mode A over the $xy$ plane at $z=0$ (the illuminated face of the sample).

Figure 4

Frequency dependant transmission spectra obtained for four consecutive values of $\theta$ in the range $0<\theta <60°$, in increments of 20°. Each transmission spectrum is taken at a transmission angle $\psi =60°$.

Figure 5

The field profiles calculated using the FEM code at a fixed frequency of 34.1 GHz, corresponding to the resonant frequency of mode $a^{*}$, with $\theta =60°$, $\psi =60°$. (i) The time averaged electric field magnitude. (ii) and (iii) $E$-vector magnitudes evaluated at phases separated by 180°. (iv) The $H$-vector magnitude at a phase corresponding to maximum field. (v) The $H$-vector magnitude calculated over a plane parallel to the $xy$ plane at $z=-15\mathrm{mm}$, at a phase corresponding to maximum field.

Figure 6

Resonant frequency as a function of $N$ for the experimental transmission data obtained at $\theta =60°$, $\psi =60°$, compared to the modeled response of the sample calculated using the FEM code. Also shown are the predictions of Eq. (1). Inset: Modeled resonant frequency as a function of outer radius of the annulus for the $P=0$, $N=1$ mode. In this instance, a 4.29 mm thick substrate is used, and the inner to outer radius ratio remains constant.