Observation of Proximity Resonances in a Parallel-Plate Waveguide

Experiments with dielectric scatterers in a parallel-plate waveguide have verified for the first time the existence of proximity resonances in two dimensions. A numerical solution to the scattering problem supports the analysis of the experimental data. It has recently been shown that two resonant s-wave scatterers placed close together produce two resonances in the spectrum of the combined system [1]. The first, which remains s-wave in character, is shifted down in energy and broadened with respect to the original single scatterer resonance. The second resonance, which is p-wave in character, is shifted up an equal amount in energy and can have a very narrow width. In fact, the width of the p-wave resonance vanishes as the scatterers approach each other. This second resonance has been dubbed the proximity resonance. Proximity resonances are important in a number of physical contexts, including scattering of sound from small identical bubbles in liquids [2,3], and scattering and emission of light from nearby dipole scatterers [4,5] where a proximity resonance effect has long been known under the name of Dicke super-radiance and sub-radiance. In Ref. [1], the effect was discussed for particle scattering from two identical atoms (or other identical scatterers) for the first time. Here we discuss yet another context, the classical scattering of electromagnetic waves from dielectric discs. At the same time (however see the caveat below) the system we describe mimics quantum scattering from two adjacent potential wells in two dimensions [6,7]. For the purposes of modeling the experiment, we developed a method of solving the scattering problem involving cylindrical basis functions centered on each disc. It turned out that the point scatterer model [8,9], which was used in the original discussion of proximity resonances [1], was not sufficient to accurately model the experiment. In order for the point scatterer model to be applicable, at least two conditions must be met: r λ, and r d, where r is the physical radius of each scatterer, λ the wavelength, and d the distance between the scatterers. In our experiments, the first condition was always met, but the second was not. Other work [10] indicates that there may be a similar effect present in the bound state spectrum of two nearby dielectric discs in a parallel-plate waveguide. Szmytkowski et al. [11] have found theoretically a similar resonance with fixed scattering length point interactions. The picture to keep in mind when thinking about the proximity resonance is the following: imagine two nearby point sources of unit amplitude, situated much closer together than a wavelength. When these sources are in phase, amplitude will add up nearly in phase everywhere in space, and the amplitude far from the sources will be appreciable. The far field intensity clearly will be s-wave in character. When the sources are out of phase, amplitude will interfere destructively everywhere, and the far field intensity will be much reduced compared to the in-phase case. Now, for a scattering resonance, the width of the resonance is proportional to the rate at which amplitude escapes from the neighborhood of the scattering system. This rate is proportional to the ratio |ψfar/ψnear|, where ψfar is the far field amplitude and ψnear is the amplitude in the near field. This ratio will remain finite for the in-phase pair of scatterers, and vanish for the out of phase pair, as their separation goes to zero. This narrows the proximity resonance as the scatterers are brought closer together. The waveguide, shown in Fig. 1, consisted of two parallel copper plates, 1 m square, separated by a 1 cm gap. To minimize the effect of waves reflected off the edges of the waveguide, the perimeter was lined with a 11.5 cm thick layer of microwave absorber (C-RAM LF-79, Cuming Microwave Corp.), designed to provide 20 dB of attenuation in the reflected wave intensity at frequencies above 600 MHz. Without the absorber, there would be substantial reflections of both the incident and scattered wave off the edges of the waveguide, which would produce strong cavity modes and unnecessarily complicate the analysis. The important effect of the absorber was to allow the waveguide to behave as if it were infinite in extent in the directions parallel to the plates, and thus support oscillations at all frequencies. The scatterers were cylindrical in shape (radius: 2 mm, height: 1 cm) and had a measured dielectric constant of = 77± 1. Each disc had an individual s-wave scattering resonance at 2.3 GHz with a 1.1 GHz width. They were illuminated with microwaves from a point source located 25 cm away from the midpoint of the two scatterers. The field in the waveguide could be measured at eight points located on a circle of 25 cm radius centered at the midpoint between the scatterers. Antennae were inserted perpendicular to the plates to launch the incident wave and measure the field. Such antennae couple to an electric field perpendicular to the plates. For a plate separation of 1 cm and frequencies below