We consider the hybridization of the resonance of a SRR metamaterial with the gyromagnetic material resonance of yittrium iron garnet (YIG) inclusions. The combination of an artificial structural resonance and natural material resonance generates a unique hybrid resonance that can be harnessed to make tunable metamaterials and further extend the range of achievable electromagnetic materials. A predictive analytic model is applied that accurately describes the characteristics of this SRR/YIG hybridization. We suggest that this hybridization has been observed in experimental data presented by Kang et al. [Opt. Express, 16, 8825 (2008)] and present numerical simulations to support this assertion. In addition, we investigate a design for optimizing the SRR/YIG structure that shows strong hybridization with a minimum amount of YIG material.
©2009 Optical Society of America
With the advent of engineered artificial composite structures, called metamaterials, it has become possible to construct electromagnetic materials with properties beyond those found in conventional materials. Metamaterials have been used to demonstrate novel properties such as negative refraction and reverse Cherenkov radiation, theoretically predicted nearly 40 years ago by Veselago [2, 3, 4]. Current research has focused on the ability to design complex spatially varying electromagnetic materials, such as the recent fabrication of a cloaking device at microwave frequencies . To achieve their exotic responses, metamaterials are constructed out of periodic and resonant structures and are typically dispersive. Their dispersive nature often constrains metamaterial devices to operate over a small frequency bandwidth of operation. In order to overcome this limitation there is interest in creating tunable metamaterials, which can be tuned to operate over the frequency range of interest. One recently demonstrated method of creating tunable structures is to integrate lumped elements, such as a varactor diodes, into the gap regions of SRR structures [6, 7, 8]. Due to operational limitations of integrated devices, this solution is currently limited to the low gigahertz regime. Another method for creating tunability is to situate a dielectric or magnetic material, that is itself tunable via some external bias, within the local fields of the resonant elements that compose the metamaterial structure. Tuning the natural material will in turn tune the bulk electromagnetic properties of the metamaterial. This method has been successfully demonstrated at terahertz by manipulating the Schottky barrier of a semiconductor substrate  and at microwaves using nematic liquid crystals .
As an alternative to harnessing resonant metallic circuit elements to construct metamaterials, periodic naturally resonant materials have been considered as metamaterial structures. It has been shown that a periodic yittrium iron garnet (YIG) structure can be analyzed as an effective medium and that this metamaterial structure provides a tunable permeability via a magnetic biasing field . YIG structures have also been combined with a wire mesh in order to create a negative index metamaterial structure . Recently it has been suggested that integrating resonant gyromagnetic materials [13, 1] can provide an indirect means of tuning geometrical metamaterial structures by influencing the local fields in a metamaterial unit cell. The tuning of a SRR structure has been experimentally demonstrated by Kang et al.  with a SRR/YIG rod structure. They suggest that in the weakly interacting limit, where the resonance of the YIG material is far from the resonance of the SRR structure, the interaction can be understood by considering the effective change in background permeability caused by the periodic YIG rods. But, they also note that in the strongly interacting region this simple description breaks down. In this region the combination of the resonant YIG material and the artificial magnetic resonance of the SRR structures interact to produce a unique hybrid resonance distinct from either of the individual resonances. In this paper we analyze the hybrid mode supported by a combined SRR and YIG metamaterial structure and provide an analytic understanding that is sustained by full-wave numerical simulations. We first consider a single ring SRR/YIG structure that demonstrates strong hybridization. We demonstrate that this structure has a broad range of tunability that can be harnessed to make interesting metamaterial devices. We also investigate the structure experimentally considered by Kang et al. . We extend the characterization of this hybrid structure into the strongly interacting region and show that a hybrid theory recently proposed  can be applied with reasonable agreement. Integrating gyromagnetic materials is promising because there exists a large range of these compounds which can be harnessed to provide tunability from the low to high gigahertz frequency range.
The influence of a resonant gyromagnetic material has been theoretically and numerically considered in a waveguide metamaterial structure by Gollub et al. . We develop a similar framework for the bulk SRR metamaterial structures considered here. Numerical simulations show that this hybrid theory is accurate over the the full frequency regime (including the weakly interacting regions investigated by Kang et al.).
The standard metamaterial methodology for designing electromagnetic materials is to construct a medium composed of periodically arranged resonant sub-wavelength elements. Through manipulation of the geometry of the individual metamaterial elements, the resonance of the bulk structure can be engineered to support the electromagnetic properties of interest (over a finite frequency range). To first order, the SRR structure has a resonant permeability with a characteristic Drude-Lorentz functional form . In addition to the fundamental resonance, there are also higher order electromagnetic resonances associated with the metamaterial structure. For example, the second order resonance of the SRR structure is electric in nature. Higher order resonances are not usually of practical interest for metamaterial design because they are subject to strong spatial dispersion effects that result from the incident radiation having a wavelength comparable to the unit cell size (i.e. the material acts less like a homogeneous material). We will see that the resulting hybrid resonances of the SRR/YIG material are a mix of both higher and lower order resonances in close proximity, and hence in order to obtain an accurate analytic theory, we must considered the higher order resonances in our analysis. Here, we show that considering just the first two metamaterial resonances sufficiently predicts the hybridization that results when the YIG resonance is near the first order resonance of the SRR.
We can modify the standard metamaterial Drude-Lorentz model to account for the spatial dispersion effects found in the higher order resonances (and to a lesser degree those found in the first order resonance) using a technique developed by Liu et al. . For the SRR structure, we define the characteristic Drude-Lorentz permeability of the first order magnetic resonance, Eq. (1), and similarly, the characteristic Drude-Lorentz permittivity that describes the second order electric resonance, Eq. (2),
where ω is the angular frequency, F (j) and A (j) are constants, ΓSRR(j) is the dissipation factor, and ω 0SRR(j) is the resonant angular frequency. The Drude-Lorentz resonant frequency,ω 0SRR(j), is related to the local structural SRR capacitance, C, and inductance, L, by,
where k(ω) = ω/c is the free-space wavevector and d is the unit cell size of the metamaterial. θ(ω) corresponds to the phase advance per unit cell. The effective wave impedance is then defined,
and the effective index of refraction is given by,
The effective permittivity, ε eff, and permeability, μ eff, are then determined in the usual way, μ eff = n(ω)η(ω) and ε eff = n(ω)/η(ω). Given the Drude-Lorentz parameters (F (j),A (j), ΓSRR(j), ω 0SRR(j) and the unit cell size (d), we can use Eqs. (1–6) to accurately predict the effective electromagnetic response over a wide frequency range.
The inclusion of YIG material in a SRR structure affects the local electromagnetic response of the SRR structure. Determining the nature of this influence at the unit cell level, as we propose to do, can be directly related to the bulk electromagnetic response of the metamaterial through the modified Drude-Lorentz model outlined above. We consider YIG material that is biased along the z-axis and above its saturation magnetization. Its resonant permeability is of the form,
where μ 1 and μ 2 are resonant functions of frequency given by
with γ the gyromagnetic ratio and μ 0 Ms the saturation magnetization. The resonant frequency is given by ω 0YIG = (γμ 0 H 0 - iωΓYIG) where H 0 is the DC magnetic bias field. The loss component, ΓYIG, can be further written in terms of the linewidth of the resonance, ΔH, as ΓYIG = γμ 0ΔH/ (2ω). For our analysis we ignore the off-diagonal components in Eq. (7) which only have a strong influence at resonance. We show that this is a suitable approximation if we restrict our unit cells to include thin YIG sheets.
If we consider incident TEM radiation on the SRR metamaterial structure that is incident perpendicular to the biasing field, the incident wave largely interacts with the x component of the YIG permeability tensor or μ 1(ω) in Eq. (7). The resulting inductance of the SRR structure can be calculated by abstractly conceptualized a ‘magnetic capacitor’ with parallel current sheets sandwiching a volume containing some fraction of which is filled with YIG material. Considering this simple model, the inductance of the SRR structure is found to be ,
where the parameter q is a constant that depends on the ‘effective’ volume fraction of the YIG material (q must be determined through a fitting procedure), μ 1(ω) is the relative permeability of the magnetic material, and g geom is a constant with units of length that is determined by the geometry of the SRR structure. Alternatively, the capacitance of the SRR structure is constant and can be written as, C = ε 0 h geom, where h geom is a constant with units of length that is determined by the geometry of the structure. If we plug the associated inductance and capacitance of the SRR structure into Eq. (3) we get an equation governing the new hybrid resonant frequencies of the SRR structure,
where ω 0 SRR = μ 0 ε 0 g geom h geom = 2π f 0is the resonant frequency of the SRR structure in the absence of the YIG material. One can replace ω → ω′0SRR in Eq. (11) to obtain a transcendental equation that can be solved for the new hybrid resonant frequencies .
We can further determine the effective permeability, μ eff, and permittivity, ε eff, for the hybrid structure from the modified Drude-Lorentz model by replacing ω 0SRR inEq. (1)and Eq.(2) with Eq. (11). This involves a two-step process of first performing numerical simulations of the SRR structure without the magnetic response of the YIG material, and using a least square fitting procedure of Eqs. (1–6) to determining the Drude-Lorentz parameters. Once the Drude-Lorentz parameters are determined the magnetic filling fraction constant, q, can be determined through comparison of the analytic theory to the numerical simulations of the combined SRR/YIG structure at several magnetic biasing values. Once q is known, the analytic theory provides a predictive model that is valid over the full frequency range of interaction. Further, the fres-nel equations can be solved to determine the analytic transmission and reflection of SRR/YIG metamaterial structure .
3. Numerical Simulations
Numerical simulations of combined SRR/YIG structures were carried out using HFSS (An-soft’s commercial finite-element solver) in order to investigate the resulting hybridization. HFSS supports the definition of gyromagnetic materials and calculates the full electromagnetic permeability response given by Eq. (7). The parameters of the YIG material were defined in these simulations to have a saturation magnetization μ 0 Ms = 1.7 kGs, linewidth ΔH =12 Oe, and permittivity ε = 14.7. A standard full wave analysis of a single unit cell of the SRR/YIG structure, with appropriate boundary conditions, is sufficient to fully characterize the bulk SRR/YIG metamaterial structure’s transmission and reflection coefficients.
We first considered the structure shown in Fig. 1 which consists of a single ring SRR structure with two very thin YIG strips covering the vertical sections of the split-ring. For this geometry the demagnetization fields, which can decrease the effective biasing of the material, are minimized because the YIG material is biased parallel to the strips. First, a simulation of the structure with the YIG magnetic properties set to zero was performed. From the numerically determined transmission and reflection coefficients of the unit cell structure it is possible to compute the effective permittivity and permeability through a retrieval technique . The effective constitutive parameters of the empty structure were extracted over the frequency range including the two lowest order resonances as shown in Fig. 1(b). The resonances include a first order magnetic resonance at 10.6 Ghz and a second order highly dispersive electric resonance at 23.7 Ghz. An advantage of using the single SRR geometry is the high frequency of the secondary resonance compared to that of the traditionally used double SRR structure. The hybridization effect splits both the fundamental resonance and any higher order resonances, but for the SRR resonances that are far from the resonant frequency of the YIG material, the splitting is highly damped. Hence, the single SRR structure provides hybrid resonances that are largely isolated from one another. The disadvantage of using the single SRR structure is that it has a slightly weaker response and a higher fundamental resonant frequency than its equiva-lently sized double SRR counterpart, but this can be compensate for by decreasing the unit cell size of the metamaterial and effectively increasing the filling fraction.
For the structure in Fig. 1, the Drude-Lorentz constants for the first resonance [F (1) = 0.228, ω 0SRR (1) = 2π(10.6) Ghz, ΓSRR(1) = 2π(0.00219) Ghz, A (1) = 1.13] and the second order resonance [F (2) = 0.659, ω 0SRR(2) = 2π(23.7) Ghz, ΓSRR(2) = 2π(0.0347) Ghz, A (2) = 2.99] were determined by a least square fit using Eqs. (1–6). The inclusion of Eq. (11) was then used in Eq. (1) and Eq. (2) to determine the effective hybridized permeability and permittivity of the structure as a function of the magnetic bias, H 0, and magnetic filling fraction, q. The Fresnel equations were then used to determine the theoretical transmission of the SRR/YIG metama-terial structure. Through a fitting process of the analytic to the simulated transmission, the magnetic filling fraction was determine to be q = 0.06.The comparison of the HFSS calculated transmission and the analytic solution are shown in Fig. 2 for a bias field of H 0 = 3 kG which results in a YIG resonance that is near the natural SRR resonance. We see that there is good correlation between analytic solution and simulation in both the position of the resonant frequency and magnitude. The splitting of the fundamental mode is seen, 1a and 1b, and a small contribution from the splitting of the higher order resonance, 2a, is also seen. The 1b lower resonance deviates somewhat from the magnitude of the analytic theory and this may be the result of the off-diagonal components of the YIG permeability tensor having an effect near resonance. Also plotted is the simulation of the YIG strips by themselves (without the SRR structure) and it has an almost negligible effect on transmission. This confirms that the magnetic properties of the YIG material strongly interact with the local fields generated by the SRR structure in order to create the hybridization.
We can further investigate the hybridization over a range of biasing fields as shown in Fig. 3. It can be seen that when the YIG Resonance is far from the SRR resonance we see the weak tuning suggested by Kang et al. and as the YIG resonance approaches the original SRR resonance a strong hybridization of the two resonances develops. The original resonance, 1 b, of the SRR shifts to the right and a small resonance, 1 a, appears and grows to the left. Once the YIG resonance passes the natural SRR resonance, the 1 b resonance continues to decline and eventually disappears while the 1a resonance reclaims the position of the natural SRR resonance. When the YIG and SRR resonance overlap, the hybridization is greatest with two relatively equal resonances straddling the original SRR resonance position. Also shown in Fig. 3 are the extracted permeability and permittivity from the numerical simulations. We further plot the extracted permeability of the hybrid structure over the biasing range H 0 = 1.7 – 4.5 kG on the same plot in Fig. 2(b). For metamaterial design we see that there is a wide range of tunability that can be harnessed. We can also note the slight asymmetry in the resonant peaks and this may be the result of the off diagonal components in the permeability tensor of the YIG.
Next we numerically investigated the structure experimentally considered by Kang et al.  shown in Fig. 4(a). Kang demonstrated the tunability of the SRR/YIG rod structure and provided an analysis that agreed in the weakly interacting realm. They also noted a complicated interaction occurring when the resonances were close together. We suggest that they observed the hybridization of the resonances, but for their structure, the hybridization is somewhat more intricate than the previous structure we considered. A simulation of the structure with the YIG magnetic properties set to zero was performed to determine the effective constitutive parameters of the empty structure as shown in Fig. 4(b). The two lowest order resonances are shown—a first order magnetic resonance at 10 Ghz and a second order highly dispersive electric resonance at 16.2 Ghz. The Drude-Lorentz constants for the first resonance were determined [F (1) = 0.0483, ω 0SRR(1) = 2π(9.97) Ghz, ΓSRR(1) = 0.108 Ghz, C (1) = 1.22] and the second order resonance [F (2) = 0.475, ω 0SRR(2) = 2π(16.2) Ghz, ΓSRR(2) = 0.0461 Ghz, C (2) = 7.18] by a least square fit of Eqs. (1–6). As noted above, the double SRR structure used by Kang has a second order resonances that is very close to the fundamental resonance. Also, the large size of the unit cell size, 5 mm, with respect to the SRR structure, 2.2 mm, results in strong spatial dispersion effects. In addition, the significant thickness of the YIG rod used, 0.8 mm, suggest that the off-diagonal components of the permeability tensor of the YIG material will have a significant effect. Further, the significant volume the material takes up in the unit cell suggest that it will have strong direct interaction with incident radiation. Still, the theory developed previously loosely characterizes the resulting hybridization.
In Fig. 5 the transmission as a function of several biasing fields is shown with the analytic solution overlayed. The transmission of the YIG rod in the unit cell by itself is also shown. It is interesting to note that the resulting stop band in the transmission is a combination of the hybrid resonance of the SRR and the pure YIG rod resonance. The YIG makes up a large enough fraction of the unit cell that it has a direct interaction with incident radiation. Further, the hybrid resonances are not well formed until the YIG resonance is near that of the original SRR structural resonance (H0 = 3 – 3.8 kG), possibly due to the off-diagonal permeability components having a strong influence in this structure. We can also observe that the 2a resonance, which is due to the hybridization of secondary SRR resonance, is much stronger due to the secondary SRR resonance being closer to the fundamental resonance. In addition, the 2a resonance is mixed together with the direct YIG response.
Overall we see that the tuning of the resonances is quite substantial. With a magnetic filling fraction of q = 0.08, the analytic theory roughly correlates with the three major hybrid resonances, though there is a slight offset at some frequencies. We suggest that the messy resonance noted by Kang et al. when the SRR and YIG resonances coincided was in fact the first experimental observation of the hybridization effect.
It is interesting to note that the effective magnetic filling fraction of the first structure considered (q = 0.06), that is shown Fig. 1, is approximately the same as Kang et al.’s structure, though it utilize 96% less YIG material. This shows the effectiveness of targeting the YIG material in the regions of strong local fields of the SRR structure, i.e. near the wire. The design of the optimized hybrid structure we considered in Fig. 1 could be simplified by considering a single sheet of YIG material extending across the entire unit cell instead of the double stripped structure. We consider the double stripped structure solely to demonstrate that targeting the material in the regions where the local fields of the SRR are strongest maximizes the interaction for the amount of YIG material used.
In summary, we have investigated the distinct hybrid resonance that occurs when a biased ferrite material, with a resonant magnetic response, is integrated into a SRR metamaterial. The hybrid response of these structures is tunable via adjustment of the magnetic biasing field of the ferrite material. We have provided a theory that characterizes this hybrid response over the full range of field biasing values. In designing these hybrid structures we can leverage the response of the magnetic material by targeting its position in regions of the metamaterial unit cell where the local fields are strongest. In general, hybrid metamaterials support reconfigurable electromagnetic properties and can be harnessed to extend the bandwidth of bulk metamaterials.
This research was supported by U.S. Army Research Office DOA under Grant No. W911NF-04-1-0247. We also acknowledge support from the Air Force Office of Scientific Research through a Multiply University Research Initiative under Contract No. FA9550-06-1-0279.
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