Molding the flow of light at the sub-wavelength scale has always been one of the greatest challenges of photonics, as it would allow the realization of optical circuits with a degree of integration similar to that achieved in integrated electronics. Strongly localized eigenstates of an electromagnetic (EM) field exist in small clusters of regular metallic inclusions embedded in an otherwise uniform dielectric host. The electromagnetic field can be restricted to have large amplitude in spatial regions whose linear size is much smaller than the wavelength. This ultra-localization of an EM field is achieved with the help of surface plasmons in the metallic inclusions. These plasmons screen the EM field, essentially cancelling it outside the volume of the eigenstate. This phenomenon has been proposed as the basis for a SPASER device, namely, Surface Plasmon Amplification by Stimulated Emission of Radiation. This SPASER would be a source of strong, coherent EM radiation with a size that can be much smaller than the wavelength. In this report we present results for such states which go beyond the quasi-static approximation. That is necessary in order to analyze the radiative and dissipative properties of those states, e.g., the radiative and dissipative losses.
The theory of scattering eigenstates or resonances of a monochromatic electromagnetic (EM) field is briefly
reviewed, and is then used to expand the physical field which is present in a two-constituent composite medium
when an external or incident field is applied to the system. Special attention is devoted to the case of a composite
which has the form of a finite volume of <i>p</i>-constituent (usually metallic, with electric permittivity that has a
large negative real part and a small imaginary part, both of them frequency dependent) embedded in an infinite
volume of <i>h</i>-constituent (usually a conventional dielectric, with electric permittivity that is nearly real, positive,
and frequency independent). Focusing on the case where the frequency is such that the system is close to one
of those resonances, we develop a calculation of the shape of the physical field in the composite medium, as
well as of the energy content and rate of dissipation of the physical field, and of its lifetime. Consequences for
the shape of the physical field are that it is very similar in shape to the eigenfield, and its magnitude is very
much amplified when that eigenstate is very localized in space. The size of this localization region is unrelated
to the wavelength or any other EM length such as skin depth. It is determined only by the microstructure,
whenever the micro-geometric features of the <i>p</i>-constituent are much smaller than any of the EM lengths. This
is important in any attempt to use such a resonance in order to implement a nanolens, i.e., to focus an incident
EM plane wave or other information-carrying field into a sub-wavelength-sized region.<sup>1, 2</sup> Consequences for the lifetime of the physical field are discussed-those are important for any attempt to implement a SPASER<sup>3-5</sup> (Surface Plasmon Aplification by Stimulated Emission of Radiation) device.
The strongly localized quasi-static eigenstates (also known as surface-plasmon resonances) which are found in a small nanometric cluster of spherical inclusions can form the basis for some interesting potential applications such as SPASER and nanolens. In a SPASER, a strong coherent electric field, oscillating at a frequency ω in the visible or infra-red spectral range, can be excited in a spatial region whose linear dimensions are much smaller than the wavelength appropriate to that frequency. In a nanolens an incident electromagnetic field, oscillating at such a frequency, can be focused to a spot whose size is much less than the relevant wavelength. An important property of such resonances is their finite radiative lifetime, which is infinite in the strict quasi-static limit. One needs to solve the full Maxwell's equations in order to find the radiative decay rate, and consequently the lifetime, of such an eigenstate. We develop a method for calculating such lifetimes for clusters of closely spaced spherical inclusions. We also discuss how symmetry properties of such a cluster can be exploited to ensure that certain eigenstates have especially long radiative lifetimes.