## 1. Introduction

[2] Artificial composite materials (metamaterials) proved to be a feasible way to increase the degrees of freedom in the interaction of electromagnetic fields with matter from microwaves to optical frequencies.

[3] Collective resonances and wave propagation in composite materials can be characterized by modal analyses of arrays periodic in three dimensions (3D) [*Alù and Engheta*, 2007; *Benenson*, 1971; *Campione et al.*, 2011a, 2011b, 2012; *Ham and Segall*, 1961; *Shore and Yaghjian*, 2007, 2010, 2012; *Steshenko and Capolino*, 2009; *Wheeler et al.*, 2005b]. In particular, under certain circumstances of polarization and excitation, a 3D periodic array of particles with finite thickness could be described with good approximation as a homogeneous slab with effective parameters [*Campione et al.*, 2011a, 2011b, 2012; *Collin*, 1960; *Shore and Yaghjian*, 2012], such as relative permittivity (*ε*_{eff}), relative permeability (*μ*_{eff}), and refractive index (*n*_{eff}).

[4] The aim of this paper is to characterize the modes with real and complex wave number in 3D periodic arrays made of particles with spherical shape modeled through the dual (electric and magnetic) dipole approximation (DDA). The DDA is a good approximation when the two dipolar terms (or any of them) dominate the scattered-field multipole expansion. For transverse polarization, the structure is treated as a homogeneous slab with effective refractive index. Here we are interested in analyzing composite materials made of large dielectric permittivity materials (e.g., lead telluride, titanium dioxide, etc.) which in general exhibit first a magnetic then an electric resonance. The DDA is more accurate than the single dipole approximation (SDA) approach, since it accounts for electric and magnetic couplings, and the method described here could also be used to characterize the artificial magnetic (i.e.,*μ*_{eff} ≠ 1) or electric (i.e., *ε*_{eff} ≠ 1) properties of the 3D lattice. Possible routes for obtaining artificial magnetism, such as the use of split ring resonators, binary mixtures, etc., have been recently summarized in the introduction of *Campione et al.* [2012]. The present work has been indeed partly motivated by the observations made in [*Campione et al.*, 2012] about the fact that the results shown in that paper could be further improved by modeling the particle with the DDA model, instead of the two SDA models based only on magnetic dipoles shown therein.

[5] The approach described in the present paper allows for the tracking and especially for the characterization of the evolution of modes varying frequency. First, we develop the formulation with coupled electric and magnetic dipoles through DDA to model the particle behavior (as done, for example, in *Mulholland et al.* [1994] and *Shore and Yaghjian* [2010, 2012]) in order to analyze modes with real and complex wave number in 3D periodic arrays of spherical particles. The numerical procedure developed in this paper for evaluating the complex zeros of the dispersion relation uses the Ewald representation for the *dyadic* periodic Green's functions (GFs) to represent the field in 3D periodic arrays, and is partly based on previous scalar developments [*Kustepeli and Martin*, 2000; *Lovat et al.*, 2008; *Park et al.*, 1998; *Stevanoviæ and Mosig*, 2007] and dyadic developments [*Campione et al.*, 2011b, 2012]. The Ewald representation, besides providing analytic continuation to the complex wave number space, results in two series with Gaussian convergence where only a handful of terms are needed [*Ewald*, 1921; *Kustepeli and Martin*, 2000; *Lovat et al.*, 2008; *Park et al.*, 1998; *Stevanoviæ and Mosig*, 2007]. In *Shore and Yaghjian* [2010, 2012] the authors treat similar problems of arrays of spherical particles where both electric and magnetic dipoles have been considered. They evaluate the modes with complex wave number by using a method based on polylogarithmic functions, which is different from the Ewald method employed here. Furthermore, in our formulation, the direction of electric and magnetic dipoles is arbitrary.

[6] The structure of the paper is as follows. We discuss in section 2 the rigorous representation of the field in 3D periodic arrays using the dyadic GFs including the case with coupled electric and magnetic dipoles. Then, in section 3, we derive the new expressions for the dyadic GFs using the Ewald method related to both electric and magnetic dipole excitations. The convergence properties of scalar and dyadic GFs are briefly inspected as well. Last, in section 4, we analyze the modal wave numbers for transverse polarization in two composite metamaterials: one made of an array of non-magnetic lead telluride (PbTe) microspheres, and the other made of non-magnetic titanium dioxide (TiO_{2}) microspheres. Modal wave numbers are computed through various methods: DDA, SDA based on electric dipoles only (SDA-E), SDA based on magnetic dipoles only (SDA-M), and Nicolson-Ross-Weir (NRW) retrieval method from reflection and transmission of finite thickness slabs computed by full-wave simulations. The 3D lattices are then described in terms of effective refractive index, computed by using the four methods above and Maxwell Garnett (MG) formulas, and their agreement or disagreement is discussed. The procedure for the singularity regularization of the spatial series counterparts in the Ewald representations is discussed inAppendix A. Supporting mathematical expressions are reported in Appendix B.