Complex modes and artificial magnetism in three-dimensional periodic arrays of titanium dioxide microspheres at millimeter waves

We characterize the modes with real and complex wavenumbers for both longitudinal and transverse polarization states (with respect to the mode traveling direction) in three dimensional (3D) periodic arrays of titanium dioxide (TiO₂) microspheres in the frequency range between 250 GHz and 350 GHz. Modal results are computed by using a single magnetic dipole approximation (SDA) and an SDA model with correction (SDA-WC) that assumes the array to be embedded in a host with an effective permittivity computed through Maxwell Garnett formulas. Moreover, for the transverse polarization case, modal wavenumbers are computed also by fitting the full-wave simulation magnetic field (one point per unit cell) in a finite thickness structure, and their agreement and disagreement are discussed. The longitudinal polarization is not affected by the artificial correction introduced in the SDA-WC; in the transverse polarization case, instead, the correction is needed to obtain results in better agreement with the full-wave data fit. In the observed frequency range, there are one longitudinal mode and two transverse modes, one forward and one backward, where the forward one is "dominant" (i.e., it contributes mostly to the field in the array). Therefore, in the case of transverse polarization, we describe the composite material in terms of homogenized refractive index and relative permeability, comparing results from (i) modal analysis (with and without correction), (ii) Maxwell Garnett formulas, and (iii) Nicolson-Ross-Weir retrieval method from scattering parameters of finite thickness structures. The agreement among the different methods justifies the performed homogenization procedure in the case of transverse polarization. We show that artificial magnetism is generated from a nonmagnetic composite material.