Abstract

We study the radiation dynamics of an electric line current source in a homogeneous, magneto-dielectric cylinder. We find that radiation by the source is most enhanced by maximizing the magnetic response of the cylinder. It is well-established that the radiation of electromagnetic sources depends on the physical environment in the vicinity of the source [1]. Until recently it was thought possible to tune only the electric properties of the materials (via the permittivity ε), however metamaterials offer the possibility of controlling both the electric and magnetic responses of a medium. This leads to markedly different radiation dynamics, in which the refractive index alone is insufficient to capture the physics of the emission of sources. We study here the emission of a source placed in a magneto-dielectric cylinder, whose magnetic properties may be controlled by changing the permeability μ, in addition to controlling the permittivity ε. We begin with the canonical situation of a 2D line source. We investigate the circumstances in which emission may be enhanced or suppressed, and outline how the inclusion of magnetic effects leads to differences in the physics of radiating sources. We denote by Γ_∞ the rate at which an (infinite) electric line current source in free space radiates energy. If we now consider the same source placed along the axis of a circular cylinder with permittivity ε and permeability μ, the emission rate changes to Γ_cyl, which is given by where R₀ is the generalized Fresnel reflection coefficient of the cylinder. For a cylinder of radius a at free-space wavenumber k₀, this is given by where Z_r = μ/ε is the relative impedance of the cylinder, and H₀ and J₀ are the Hankel and Bessel functions of the first kind respectively. In Fig. 1 we plot the emission rate, normalized to free-space, for an electric current source at the center of the cylindrical cavity versus the normalized cavity size k₀a. The refractive index of the cavity is fixed at n = 2 and the value of the pair (ε,μ) is given in the inset. As expected the emission rate oscillates as we increase the size of the cavity introducing additional resonant modes. In the case ε = μ = 2 the oscillations are quickly damped because the impedance of the cylinder matches the impedance of vacuum. However, the most interesting feature is that the emission rate is in general maximized when μ is maximized, and therefore since n is fixed, when ε is minimized. Hence the emission rate of the electric current source is in general most enhanced when the magnetic response of the cavity is maximized. A similar result has been observed previously for a (3D) spherical cavity [2]. A more surprising result emerges if we keep μ constant and allow ε to vary without imposing any restriction on the refractive index. Since μ is fixed, the increase in ε amounts to an increase in the refractive index. One would expect that as the index increases the overall emission rate would increase monotonically, however this is not the case. Figure 2 shows plots of the emission rates for μ = 3 for different values of ε between 1 and 5. We see that as ε increases, the amplitude of the oscillations of the rate decrease rapidly until the value ε = μ = 3 is attained. Beyond this, the amplitude of the oscillations increases, however the emission rate remains below that observed for a “pure” magnetic cavity (ε = 1). It is only when ε becomes equal to μ² that the maximum of emission rate becomes comparable with the pure magnetic case. This evolution can be understood in terms of the Fig. 1. Normalized √ emission rate for an electric dipole at the center of a cylindrical cavity (radius a, refractive index n = εμ = 2) in vacuum versus k₀a. The inset gives the value of (ε,μ). Fig. 2. Normalized emission rate for an electric dipole at the center of a cylindrical cavity (radius a, permeability μ = 3) in vacuum versus k₀a. The inset gives the value of ε. relative impedance (and its inverse) of the cylinder, which determines the magnitude of the generalized reflection coefficient R₀. When the source is moved off-center, we find that the influence of the magnetic response of the cavity on the dynamics of the source is more dramatic. These results provide a strong indication that the magnetic properties of metamaterial cavities will play a dominant role in the radiation dynamics of sources placed in and near these structures.

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