Negative index and mode coupling in all-dielectric metamaterials at terahertz frequencies

We report on the role of the coupling of the modes of Mie resonances in all-dielectric metamaterials to ensure negative effective index at terahertz frequencies. We study this role according to the lattice period and according to the frequency overlapping of the modes of resonance. We show that negative effective refractive index requires sufficiently strong mode coupling and that for even more strong mode coupling, the first two modes of Mie resonances are degenerated; the effective refractive index is then undeterminded. We also show that adjusting the mode coupling leads to near-zero effective index, or even null effective index. Further, we compare the mode coupling effect with hybridization in metamaterials.


Introduction
All-dielectric metamaterials (ADMs) are the promising "inflection" of metamaterials to go beyond their limits. ADMs are an alternative to metallic metamaterials. The advantages of ADMs come from their low losses and their simple geometry: they do not suffer from Ohmic losses and thus they may benefit from low energy dissipation, specially, ceramics of high dielectric permittivity and high quality factor [1,2]. In the microwave, their quality factor is greater than that of metallic metamaterials [3], and it is consequently also the case in terahertz (THz) and optical frequencies. ADMs are partly inspired by the work of Richtmyer who developed the theory of dielectric resonators, which is based on the fact that "the dielectric has the effect of causing the electromagnetic field [. . .] to be confined to the cylinder itself and the immediately surrounding region of space" [4]. Taking the matter further, O'Brien and Pendry opened the way for ADMs by considering the periodic lattice of high permittivity resonators (HPRs), thus demonstrating artificial magnetism in the microwave [2,5]. ADMs rely on the first two modes of the Mie resonances of HPR. The first mode results in resonant effective permeability that can have negative values, while the second one results in resonant effective permittivity that can also have negative values. When both are simultaneously negative, the ADM is called "double negative" (DNG) and its effective refractive index is then negative [6][7][8][9][10][11]. The unit cell of ADM thus comprises two subwavelength building blocks of simple geometry [12,13]. In analogy with chemical molecules, the unit cell is generally called a meta-dimer [14], and the two building blocks are called meta-atoms. As the two are different, the unit cell is a hetero-dimer.
The large applications of ADMs (for a review, see Ref. [15]) include perfect reflectors [16], perfect absorbers [17,18], zero-index metamaterials [19], optical magnetic mirror [20], and Fano resonances [21,22]. ADMs have been demonstrated from the microwave to the optical domain. Artificial magnetism [3,[23][24][25] and negative effective refractive index [26,27] have been, theoretically and experimentally, shown in the GHz regime. Even though artificial magnetism provided by ADMs has been experimentally demonstrated in the THz [28] and infrared ranges [29][30][31][32], DNG refractive index has not been yet demonstrated, which impedes ADMs to be the equivalent of their metallic counterpart. Besides, THz radiation is widely defined as electromagnetic radiation in the frequency range 0.3-3 THz. It permits the obtention of physical data that are not accessible using X-ray or infrared radiation and it thus finds many applications in imaging, security, quality inspection, chemical sensing, astronomy, etc. On their part, metamaterials have evolved towards the implementation of photonic components [33]. HPRs are well suited for metamaterial applications in the low THz frequency range [34].
In the following, we report on the mode coupling effect which plays a dominant role in the electromagnetic properties of metamaterials [35][36][37][38][39], notably, in the achievement of a negative effective index. Magnetic and electric mode coupling effects in ADMs have been separately studied in the microwave by Zhang et al. [36]. Herein, we report on the magneto-electric mode coupling effects in the THz domain, and we show that negative effective refractive index requires sufficiently strong mode coupling [40]. We also show that adjusting the mode coupling allows us to attain near-zero values of the refractive index, or even null effective index. Moreover, we highlight that the strongest values of the mode coupling lead to frequency mode degeneracy, for which the refractive index is undetermined.

Simulations
We consider a 2D ADM whose unit cell consists of two distinct building blocks, a magnetic block and an electric one, the former resonating in the first mode of Mie resonances and the latter resonating in the second mode [3,13,27,41]. The first mode is thus referred to as the magnetic mode and the second one as the electric mode. The ADM consists of one infinite layer made up of two sets of high permittivity square cross-section dielectric cylinders that are perpendicular to the incident wave vector (Fig. 1). The two sets of infinitely long HPRs are actually interleaved. The incident polarization is transverse electric (TE), i.e., the electric field is perpendicular to the axis of the cylinders. We studiy both spatial mode coupling and frequency mode coupling. The ADM has been numerically simulated by the means of the finite elements method software HFSS TM , which yields the Sparameters. The side lengths of the resonators are initially a m = 60 mm for the magnetic mode and a e = 90 mm for the electric one, while the lattice period is l p = 260 mm. The HPRs are equidistant and therefore, the distance between two of them is half the lattice period p 2 = l p /2 = 130 mm. The relative permittivity of the dielectric is r = 94 (titanium dioxide À TiO 2 ) and the loss tangent increases between tan d = 0.009 and 0.015 in the considered frequency range [42,43]. We are thus dealing with a high refractive index bulk material (N TiO 2 ⋍ 10).

Negative index and mode coupling
We study the mode coupling between the first two modes of Mie resonances depending on the lattice period l p (spatial mode coupling), and then depending on the frequency overlapping of the two modes (frequency mode coupling). The results of the simulation, namely the minimum n eff min of the real part of the effective index n eff as a function of the lattice period and as a function of the frequency spacing between the two modes, are reported in Figures 2 and 3, respectively. They show that the mode coupling should be strong enough to ensure negative effective index. The minima of the real part of the effective index n eff are n eff min = À2.2 and À1.9, respectively. On the one hand, increasing the mode coupling is obtained by decreasing the lattice period l p . On the other hand, it is obtained by decreasing the frequency of the second mode of Mie resonances, which stems from the increasing of the side length a e of the electric resonator. When the mode coupling is sufficient, the bandwidth of the negative effective index n eff increases with it (see insets in Figs. 2 and 3). The frequency range of negative effective index is given by the relation [44] 0 ðvÞ⋅m 00 ðvÞ þ 00 ðvÞ⋅m 0 ðvÞ 0; ð1Þ where 0 and 00 respectively denote the real and imaginary parts of the permeability m and the permittivity .

Monomode coupling
To carry out our study, we first explored monomode coupling, that is, the mode coupling due to only one mode, which arises inside a layer whose unit cell only consists of one building block, the magnetic one or the electric one. We study both cases. Consequently, we only consider the spatial mode coupling and only varied the lattice period l p which is equal to the distance p 1 between two resonators, l p = p 1 . Operating in the same frequency range, the side length of the magnetic resonator is a m = 60 mm, while that of the electric resonator is a e = 90 mm. The results of the simulation are reported in Figures 4 and 5, for both cases, and they show that the two modes behave differently. Their respective frequencies (f mr , f er ) are given by the minima of the S 12 parameter [36]. The frequency f mr of the magnetic mode increases with the lattice period l p , whereas the frequency f er of the electric one decreases. These results are consistent with that of Zhang et al [36]. Both maxima of the absorption A mr and A er (A(v) = 1 À | S 12 (v) | 2 À | S 11 (v)| 2 ) and the frequency of the resonance modes of the individual resonator are also reported. The latter provides a series of resonances whose frequency is determined by Cohn's model [1,26,45,46].
where r is the relative permittivity of the resonator, m and n the integers, a and b the side lengths of the resonator, and c is the velocity of light and its accuracy is about 5%. Equation 2 was used to design all the reported structures. For a square cross-section cylinder, a = b, and the frequencies of the first two modes of the individual resonator respectively correspond to m = n = 1 and m = 1 and n = 2. To exhibit negative refractive index, the involved resonance modes are the first mode of the magnetic resonator and the second mode of the electric resonator. Their frequencies are respectively f 11 = 0.364 THz (a = a m = 60 mm) and f 12 = 0.384 THz (a = a e = 90 mm) [27]. These are obviously constant, while the maxima of the absorption are nearly constant. It can be noticed that, as the mode coupling increases, the distance between the resonance frequencies (f mr , f er ) of the resonator inside the layer and that (f 11 , f 12 ) of the individual resonator respectively increases (cf. Figs. 4 and 5, respectively), thus demonstrating the mode coupling.

Bimodal coupling
We next explore the mode coupling inside the ADM, namely, the unit cell now consists of the two building blocks. The variation of the resonance frequencies (f mr , f er ) of both modes is reported in Figures 6 and 7 corresponding to the spatial mode coupling and the frequency mode coupling, respectively. These frequencies are still given by the minima of the S 12 parameter [36]. Decreasing the lattice period l p , i.e., the distance p 2 between the resonators, increases the mode coupling. Varying the overlapping of the two modes stems from the decreasing of the frequency of the electric mode, which also increases the mode coupling [35]. The curves are shaped as "tuning forks" and show that the frequencies (f mr , f er ) of the two modes of resonance are moving closer together as the mode coupling increases. To highlight this effect, the frequencies (f 11 , f 12 ) of the resonance modes of the individual resonator are again shown in these figures. For both the magnetic mode and the electric one, the distance between the frequency (f mr ,f er ) of the resonator inside the ADM and the frequency (f 11 , f 12 ) of the individual one respectively increases as the mode coupling increases. This anew evidences the mode coupling inside the ADM. Theses curves also show that further increasing the mode coupling gives rise to a frequency degeneracy in both cases of mode coupling, that is, the two resonance frequencies of the two modes become equal, f mr = f er . The S 12 parameter and the absorption A are reported in Figure 8 as a function of frequency for several values of lattice period l p , which is relative to the spatial mode coupling. Two ranges of lattice period l p are considered, one is out of the frequency degeneracy regime (250 l p 400 mm) and the second one in the frequency degeneracy regime (200 l p 248 mm). It can be noticed that the frequency of both maxima of absorption are practically constant, whereas the frequency of both minima of the S 12 parameter vary with the lattice period l p . The latter move closer together as the lattice period l p increases, until they merged, which corresponds to the frequency degeneracy. The crossing point is reached when the lattice period is equal to l pc = 250 mm. The merged minima of the S 12 parameter are very weak (≲ À 51dB) for l p = 248 mm, which corresponds to the minimum of the real part of the effective index n eff min (l p = 248 mm) = À2.2. (See effective index curves inserted in Figs. 2 and 3 which are continuous, but at the limit of continuity.) For greater values of the mode coupling, that is, for lattice period l p smaller than 248 mm, the effective parameters could not be extracted, being not continuous through all frequencies [47]; the effective refractive index n eff is then undetermined. We observed the same frequency degeneracy behavior when studying the frequency mode coupling (results not shown).
The mode coupling effect we report here is different from hybridization [48] which is observed with plasmonic metameterial [14,49], split-ring resonators metametarials [50], inductor-capacitor resonators [51], cut wires [52], nano-wires [53], nano-rings [54], nano-particle dimers [55], or silicon nanoparticles [56]. In these cases, the coupling between the two identical meta-atoms which constitute the dimer leads from a trapped mode to the formation of new hybridized modes because it lifts the degeneracy of the mode of the individual meta-atoms. Hybridization may be used to yield negative refractive index [50,51,54]. The unit cell is then a homo-dimer and the negative effective index is achieved by playing with the mode coupling so as to overlap two hybridized modes which are of different kind: magnetic orelectric. In the case we report here, the mechanism is different since it takes the reverse way,   Figure 6. The side length of the magnetic resonator is a m = 60 mm and the lattice period l p is 260 mm.
because increasing the mode coupling leads from two separate modes to a trapped one. The unit cell of our ADM is a hetero-dimer because the two meta-atoms are not identical, and moreover, each one resonates in a different kind of mode: the magnetic one and the electric one. Hence, increasing the mode coupling leads to the trapped mode putting together the two separate modes. Nevertheless, the mode coupling has to be strong enough to ensure a negative effective index, the unit cell being either a homo-dimer or a hetero-dimer.

Effective parameters
In our ADM, a magnetic moment ensues from the magnetic mode giving rise to resonant effective permeability m eff . Similarly, an electric moment ensues from the electric mode giving rise to resonant effective permittivity eff . The two are perpendicular to each other (see e.g., Fig. 1 in reference [41]). The mode coupling arises from the interaction between these electromagnetic moments and it changes with the separating distance. The side lengths of the resonators are afresh a m = 60 mm and a e = 90 mm. Both resonances consequently modify the effective refractive index n eff (v) of the ADM. The effective electromagnetic parameters m eff , eff and n eff are reported in Figure 9 relative to two values of the lattice period l p = 360 mm and l p = 260 mm. They are extracted from the S-parameters using the common retrieval method described in references [47,[57][58][59][60][61][62][63]. The antiresonance behavior of the effective permittivity e eff around the magnetic mode frequency, which is inherent in metamaterials, can be observed [62][63][64][65]. In the former case (low mode coupling), the real part of the effective index n eff does not reach negative values, whereas it does in the latter case (strong mode coupling), then satisfying equation 1. The minimum value of the real part of the effective index is then n eff min (l p = 260 mm) = À1.5. In the former case, the real part of the effective index n eff is below unity and close to zero. Its minimal value is n eff min (l p = 360 mm) ≲ 0.04, demonstrating that adjusting the mode coupling makes ADMs suitable for epsilon-near-zero (ENZ) metamaterials [66,67] (see also Fig. 2), or even null effective index [19]. It can also be noticed that the mode coupling strongly enhances the amplitude of both resonances, notably the electric one, and that it brings closer together the two modes. In addition, as we are dealing with a high refractive index bulk material (N TiO 2 ≃ 10), the wavelength inside it is about one-tenth of that in vacuum. Consequently, we calculated the static effective permittivity, i.e., beyond the resonances, from the Maxwell-Garnett model [68]. It is equal to eff = 2.1 and eff = 2.9 corresponding to both values of the lattice period l p = 360 mm and l p = 260 mm, respectively, which is in good agreement with the results of the simulation (see Fig. 9a and b-middle).
To engineer the electromagnetic properties of an ADM, one can consequently play with either mode couplings: the spatial mode coupling or the frequency mode coupling. Figure 10 gathers the role of both mode couplings: the stronger the two mode couplings, the more negative the effective index and the larger the bandwidth of negative effective index (cf. Figs. 2 and 3). Combining the two mode couplings, negative effective index as low as n eff = À2.8 is obtained.

Dielectric function, phonons and strontium titanate (SrTiO 3 )
Other high permittivity materials, having low losses, can be investigated to study the mode coupling inside ADMs at THz frequencies, e.g., SrTiO 3 [69][70][71][72]. However, the dielectric function r (v) of these high permittivity materials is dispersive, because of lattice vibrations, namely, due to optical phonons [73]. Their frequency is in the THz range, and we are concerned with the transverse optical phonon of lowest frequency (TO 1 ). The TO 1 phonon frequency of SrTiO 3 is 2.70 THz [69], while that of TiO 2 is 5.6 THz [69,73]. The dielectric function r (v) is described by the classical oscillator model or the Four-Parameter Semi Quantum (FPSQ) model [69,71,73]. Measurements in the THz frequency, reported in the literature, are in good agreement with these models for TiO 2 [42,43] and SrTiO 3 [71,72,74,75]. The two yield the same dielectric function r (v) at the operating frequency.   10. Effect of both mode couplings (spatial mode coupling (SMC) and frequency mode coupling (FMC)): minimal value of the effective refractive index n eff as a function of the distance p 2 between two resonators for several values of the side length a e of the electric resonator. The side length of the magnetic resonator is a m = 60 mm. The two arrows denote increasing mode coupling. The minimum of the real part of the effective index is n eff min = À2.8.
The latter also reflect that the imaginary part of the dielectric function r (v) strongly increases around the frequency of the TO 1 phonon. We also simulated a similar ADM but consisting of HPRs made of SrTiO 3 and found that it exhibits the same mode coupling effects (results not shown here). The real part of the effective index reaches values as low as n eff min = À7, meaning that the effect of the mode coupling is stronger when a higher permittivity material is used ( r ≃ 300 [72] as compared with r = 94), because this heightens the resonances.
However, losses (imaginary part of r (v)) resulting from the TO 1 phonon greatly increase and therefore limit the operating range at THz frequencies. In addition, the real part of the dielectric function r (v) falls down at higher frequency [72,74], which drastically modifies the Mie resonances. The lower permittivity of TiO 2 leads to greater side lengths a m and a e of each resonator (see Eq. (2)) and therefore, it facilitates their fabrication. Consequently, TiO 2 is more suitable for ADM applications at THz frequencies.

Conclusion
We have studied mode coupling effects in ADMs at THz frequencies, and we show that the mode coupling has to be sufficiently strong to ensure negative effective index of refraction. Tuning the first two modes of Mie resonances of an ADM by adjusting the mode coupling allows the setting of the effective index from a near-zero value to a negative value. We studied both spatial mode coupling and frequency mode coupling. Increasing both brings the modes closer together until they are merged. Thus, we highlight the frequency degeneracy of the first two resonance modes, namely, the two frequencies are equal, and the effective index is then undetermined. At the crossing point, the effective index reaches its lowest value.
Author contribution statement E.A. supervised the study, analyzed the results and wrote the paper. S.M. made the simulation and analyzed the results.