Resonances and stability in arrays of pseudodipoles

Abstract

The resonance properties of certain arrays consisting of the simplest possible Maxwell scatterer, the pseudodipole, are studied. T. T. Wu's analysis of the uniformly spaced infinite linear array of pseudodipoles is generalized to allow for certain classes of pseudodipole rotations; in all cases, it is shown that these arrays can admit zero-width resonances. It is then shown that the uniformly spaced circular array can admit resonances whose widths tend to zero exponentially as the number of pseudodipoles tends to infinity while the arc-length spacing remains constant-a property which has been studied by A. Grossmann and T. T. Wu for the quantum mechanical analogue of the pseudodipole. From the asymptotic analysis of the circular array of pseudodipoles, it is observed that instability can sometimes arise. It is conjectured that this stability problem can be eliminated by modifying the original pseudodipole definition to allow for a frequency-dependent pseudodipole parameter. With this generalization, the precise nature of the instability is investigated. Based on a recent result showing that the pseudodipole can be derived as the limit of a resonant cylindrical dipole, a formula for the frequency dependence of the pseudodipole parameter is proposed.