Two-dimensional periodic arrays of noble metal nanospheres support a variety of optical phenomena, including bound and leaky modes of several types. The scope of this paper is the characterization of the modal dispersion diagrams of planar arrays of silver nanospheres, with the ability to follow individual modal evolutions. The metal spherical nanoparticles are described using the single dipole approximation technique by including all the retarded dynamic field terms. Polarizability of the nanospheres is provided by the Mie theory. Dispersion diagrams for both physical and nonphysical modes are shown for a square lattice of Ag nanospheres for the case of lossless and lossy metal particles, with dipole moments polarized along the x, y, and z directions. Though an array with one set of parameters has been studied, the analysis method and classification are general. The evolution of modes through different Riemann sheets and analysis of guidance and radiation are studied in detail.

This paper presents an algorithm for the acceleration of the series involved in the computation of 2-D homogeneous Green's functions with 1-D and 2-D periodicities. The algorithm is based on an original implementation of the spectral Kummer-Poisson's method, and it can be applied to the efficient computation of a wide class of infinite series. In the algorithm the number of asymptotic terms retained in Kummer's transformation is externally controlled so that any of the series that has to be accelerated is split into one series with exponential convergence and another series with algebraic convergence of arbitrarily large order. Numerical simulations have shown that there is an "optimum" number of asymptotic terms retained in Kummer's transformation for which the CPU time needed in the summation of the series is minimized. The CPU times required by Ewald's method for the evaluation of 2-D Green's functions with 1-D and 2-D periodicities have been compared with those required by the present algorithm, and the algorithm has been found to be between 1.2 and 3 times faster than Ewald's method when working in "optimum" operation conditions.

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