Bulk magnetic terahertz metamaterials based on dielectric microspheres

Rigid metamaterials were prepared by embedding TiO2 microspheres into polyethylene. These structures exhibit a series of Mie resonances where the lowest one is associated with a strong dispersion in the effective permeability. Using THz spectroscopy, we confirmed the magnetic nature of the observed resonance.


I. INTRODUCTION AND BACKGROUND
W E investigated THz metamaterials based on titanium dioxide (TiO 2 ) microsphere resonators. TiO 2 is a suitable dielectric material due to its high permittivity and low dielectric losses. In our previous work, the microspheres were dispersed in air, forming nearly a single-layer sample enclosed between two sapphire wafers [1]. Here we embedded the TiO 2 microspheres into a polyethylene (PE) matrix which enabled us to prepare a rigid bulk metamaterial with a controllable concentration of microresonators. We performed measurements using time-domain THz spectroscopy to characterize the response of such structures and to confirm its magnetic character. The retrieved spectra of effective dielectric permittivity and magnetic permeability are discussed within Mie theory and Maxwell-Garnett effective medium model.

II. SAMPLES
TiO 2 microspheres with a diameter of a few tens of micrometers were prepared by the bottom up approach described in [1]. They were mixed with PE powder and a pressure of 14 MPa was used to prepare rigid pellets with random spatial distribution of the TiO 2 microspheres. Pellets with thicknesses of 1 and 3 mm were prepared.

III. SIMULATIONS
Dispersed dielectric microspheres with high permittivity represent a Mie resonance-based metamaterial, where the effective response relies on the resonance of individual elements while the coupling between isolated microparticles has only minor influence on the resonance properties. This allows us to simplify the theoretical investigations: we consider a periodic array of the microspheres and neglect the influence of the disorder. Note that this metamaterial behavior fundamentally differs from that of photonic crystals where the optical response is controlled mainly by the coupling between the constituents [2]. We employed a finite-difference time-domain (FDTD) simulation software package MEEP [3] to calculate the transmission and reflection spectra of TiO 2 microspheres with diameter d = 35 μm arranged in a square lattice a×a and embedded in the centre of a slab with dielectric permittivity ε H and thickness a. The filling fraction of TiO 2 in such a sample is s = πd 3 /(6a 3 ). In order to account for the losses, the permittivity of rutile was considered as 92 + 2if , where f is a frequency in THz [1]. The Nicolson-Ross-Weir (NRW) method [4]- [6] was used to retrieve the effective dielectric permittivity ε and effective magnetic permeability μ from the simulated spectra of the metamaterial.
We investigated how the filling fraction and the ratio between the permittivity of the microspheres and the host matrix affect the position and the strength of the magnetic response associated with the first Mie mode (Fig. 1). This resonance is located around 0.9 THz and its position changes only weakly in the studied range of the parameters. In Fig. 1(a) we observe that for ε H = 2 the permeability becomes negative for filling fractions s > 5%. However, the strength of the resonance and the minimum negative value of the effective permeability decrease with increasing ε H , as shown in Fig. 1(b). For s = 18%, the range of the negative effective permeability completely disappears for the host material permittivity exceeding ∼ 5.

IV. RESULTS
Time-domain THz experiments were performed using a custom-made setup described in [7]. For a 3-mm thick pellet with a very low filling fraction of TiO 2 microspheres (0.15%) we extracted the complex dielectric permittivity and magnetic permeability from the transmittances T 0 and T 1 corresponding to the direct pass and to the first echo in the sample, respectively. A magnetic resonance around 0.9 THz is clearly resolved (see Fig. 2). This confirms the effective magnetic behavior of the metamaterial associated with the first Mie mode.
For higher filling fractions, since T 1 dramatically decreases, we carried out our investigations based exclusively on the transmittance amplitude obtained from long time-domain scans which involved the sum of all measurable Fabry-Perot reflections. In order to gain a further insight into the properties of the metamaterial, we employed a dynamic Maxwell-Garnett theory [8], in which we accounted for the distribution of the sizes of the microspheres. We found a good match between the calculated and measured spectra when two mean particle radii (r 1 = 17 μm and r 2 = 13.5 μm) were considered with a distribution width of 1 μm (see Fig. 3). These parameters are supported by an optical microscope analysis, which revealed a comparable ellipticity of the microparticles. Furthermore, the filling fractions obtained from the fit by the Maxwell-Garnett theory are in good agreement with those determined from the weights of the components (nominal values).

V. CONCLUSION
Using a bottom up approach, we fabricated rigid metamaterials made of TiO 2 microresonators embedded in polyethylene. We experimentally confirmed the magnetic effective response in the vicinity of the first Mie resonance near 0.9 THz. Using FDTD calculations of the effective response we found out that a range of negative effective magnetic permeability can be achieved for sufficiently high filling fractions and contrasts between the permittivities of the resonators and the embedding medium [9]. The developed structures are prototypes of cheap mechanically stable terahertz metamaterials.

VI. ACKNOWLEDGMENT
This work was supported by the LabEx AMADEus (ANR-10-LABX-42) and by the Czech Science Foundation under Grant No. 14-25639S.