Metamaterials offer the prospect of new science and applications. They have been designed by shaping or changing the material of the individual meta-molecules to achieve properties not naturally attainable. Composite meta-molecules incorporating a magnetic component offer new opportunities. In this work we report on the interaction between a non-magnetic split ring resonator (SRR) and a thin film of yttrium iron garnet (YIG). Strong hybridized resonances are observed. While the SRR is characterized by a magnetic and electric resonance, in practice, it is found that the YIG couples strongly to this symmetric (electric) mode of the SRR. It is also demonstrated that the anti-crossing region provides fertile ground for the creation of elementary excitations such as backward volume magnetostatic waves.
© 2012 Optical Society of America
Metamaterials are man-made patterned structures which exhibit unusual electromagnetic properties. In recent years, this subject has developed into a rich research field, motivated by the promise of both ground-breaking science and device applications. It is possible to engineer metamaterials with a negative index of refraction , and to achieve ‘cloaking’ at microwave frequencies . Such desirable properties are achieved at frequencies close to the resonances of the so called meta-molecule that comprise the metamaterial.
To date, most research has concentrated on manipulating relatively simple materials such as ohmic metals and low-loss dielectrics. The most commonly employed meta-molecule is the split ring resonator (SRR), a system which has been extensively studied ever since the original proposal by Pendry et al. . A given metamaterial consists of a periodic array, in our case SRRs, providing a collective response to an electromagnetic wave. However, such systems are often restricted to a narrow frequency band due to the resonant character of the individual meta-molecule. To overcome this limitation, attention has therefore turned to tunable metamaterials.
In this paper, we report on composite structures containing active magnetic components. In particular, we discuss a hybrid structure which incorporates a high-conductivity SRR and a thin film of insulating yttrium iron garnet (YIG). In essence, there are two differing resonances to be considered: that of the SRR and that of the YIG. Since the latter is field- dependent, tunability of the composite structure can be achieved using an applied magnetic field. Consequently, the transmission can therefore be controlled, as required. Tunability of magnetic resonances in the microwave regime offers a powerful means to realize controllable, loss-compensating, multi-frequency antennas .
Kang et al.  were the first to report a magnetically tunable metamaterial, made up from an SRR array and small rods of YIG, all placed in an X-band waveguide. The resulting complex resonances were subsequently interpreted by Gollub et al. . Our research centers on the interaction between a thin (2.4 µm) YIG film and a single SRR, excited by a broadband co-planar waveguide (CPW). The results show that strong hybridization occurs between the SRR and YIG resonances. The latter is interpreted using a simple two-state model, for an SRR and YIG film respectively [7,8]. In addition, it is argued that the non-linear interaction between the SRR and YIG provides a fertile breeding ground for the excitation of spin-waves, magnetostatic surface wave (MSSW) and backwards volume magnetostatic wave (BVMSW) . In practice, Spin-waves, MSSWs and BVMSWs are easily incorporated into the two-state model.
2. Experimental arrangement
We turn now to the experimental arrangement used in this work. Martin et al.  and Falcone et al.  have shown that resonances in SRR systems can be studied and exploited in a co-planar arrangement, in close proximity to a broadband coplanar waveguide (CPW). Such an arrangement presents increased opportunities for applications as a broadband tunable device. Further, it is found that SRRs placed on the reverse side of the CPW substrate couple effectively to the CPW, resulting in lower transmission losses, while still revealing strong metamaterial resonances . Schematic diagrams, showing the physical arrangement of the CPW, SRRs, and the YIG film used in this work, can be seen in Fig. 1 . This arrangement possesses extensive advantages. The planar geometry is suited to applications and only small quantities of YIG are required.
The applied magnetic field was supplied by an electromagnet, driven by a computer- controlled DC bipolar power supply. The magnetic field strength was recorded using a Lakeshore 425 Gaussmeter. The rf field, applied perpendicular to the DC field, was delivered by a 20 GHz vector network analyzer (VNA) (HP E5071C) through low loss broadband cables. The cables were inserted through the centre of the electromagnet pole pieces and connected to the coplanar waveguide (CPW) using end launch connectors. The signal line width and gap size is set at 1.1 mm and 200 μm respectively, as shown in Fig. 1(a), which allows the CPW to be a broadband frequency device in the range 1-20 GHz with the required 50 Ω impedance. The VNA provides a frequency sweep from 300 kHz to 20 GHz while simultaneously monitoring microwave transmission over this range. In summary, the CPW provides the rf field to the overall system, exciting both the SRR and YIG film, simultaneously. Hence a transverse electromagnetic geometry is achieved. More details of the SRR can be seen in Fig. 1(b) and (c). The SRR is fabricated on the underside of the CPW, directly underneath one of the signal lines in order to maximize rf-coupling . The CPW itself was fabricated using a simple mask/photo-etching technique. A 70 μm copper layer on both sides of a 0.7 mm thick dielectric sheet was used as the substrate (FR4, εr = 4.5). The SRR, fabricated on the reverse of the CPW with dimensions specified in Fig. 1(c), is similar to the design by Shelby et al. . However, the dimensions of the SRR have been modified to produce frequencies close to the ferrimagnetic resonance (FMR) of the YIG, in the available magnetic field range 0-0.5 T.
The YIG film was grown on a single crystal yttrium aluminum garnet (100 oriented) substrate by pulsed laser deposition using a KrF excimer laser operating at 248 nm [14,15]. YIG was used as the magnetic component in this experiment, because (i) it possesses a strong narrow line-width FMR, (ii) it is characterized by negative permeability on resonance , and (iii) there are no complications from eddy current losses. The YIG film was placed directly on top of the SRR (flip-chip method) , thus allowing strong coupling to the CPW and therefore good microwave excitation. The VNA allows for collection of the S-matrix parameters, including the transmission coefficient S21. The latter describes the transmission properties of the combined SRR, YIG film and CPW.
3. Experimental results
The results presented here are for a single SRR as this avoids the complex effects reported by Kang et al.  associated with cells at the boundaries of the array. Results were also obtained using an array of 4 SRRs and a YIG film. The results proved similar to those presented here but with a frequency shift from the meta-molecule resonance. The S21 transmission was measured in a frequency vs field map (ν-Ba), in the range 1-20 GHz and 0-0.38 T. In Fig. 2 the color at a given point in the map represents the magnitude of S21. In addition, it should be noted that the following S21 map plots shown were normalized, by subtracting the S21 transmission obtained in a small (approximately zero) magnetic field (Ba = 0.001 T) from every subsequent frequency sweep. This gives a final value of |SN21|, leaving only field dependent features in the resulting plot. Such maps clearly show (i) the individual SRR and YIG resonances, and (ii) strong anti-crossing behavior, when the two resonances coincide. SRRs of various sizes were used.
The normalized SN21 map of the CPW/SRR/YIG composite system in Fig. 2 covers a reduced frequency (8.0-13 GHz) and magnetic field range (0.22-0.38 T); this then draws attention to the hybridization between the SRR and YIG resonances. The diagonal line, from the bottom left to the top right, is the expected YIG FMR absorption. For this mode, the frequency versus field follows the usual Kittel fomula . The SRR resonance appears as a horizontal line at 10.9 GHz. At that point where the YIG and SRR resonances are expected to cross, strong anti-crossing behavior occurs. In particular, at the SRR resonance at ν = 10.9 GHz, and it is seen that there is a reduction in microwave losses on resonance at a field of Ba = 0.3 T. This point is illustrated in Figs. 2 and 3 at the centre of the anti-crossing. Above and below the anti-crossing (black and blue curves respectively in Fig. 3) there are absorption troughs (negative) associated with the field-dependent YIG FMR, while exactly at the centre of the anti-crossing, there is a large reduction in the losses amounting to 1.6 dB (peak). This is indicative of an effective negative index of refraction, as obtained by He et al. , using the experimental S-parameter retrieval technique . Finally, note the presence of satellites on the high field side of the transmission peak. These features are due to the excitation of BVMSWs as discussed in [9, 21–23] and in more detail in section 4 below.
The resonances shown in Fig. 3, are similar to those described theoretically by Gollub et al. . The low frequency trough of trace Ba = 0.2777 T is the FMR resonance while the high frequency peak is associated with the symmetric electric-dipole resonance of the SRR. This has been confirmed experimentally by shorting the gaps in the ring with a short strip of conducting wire, following Ref. . This procedure eliminates the low frequency anti-symmetric magnetic resonance of the SRR at ~5 GHz (not shown), while leaving the symmetric electric-dipole resonance response at 10.9 GHz unchanged. Note that as the field is increased the YIG resonance passes through the SRR resonance, distorting the hybridization, as shown in Fig. 3 (Ba = 0.306 T) revealing the 1.6 dB reduction in losses. In general, the YIG resonance is no longer visible and the natural resonance of the SRR is observed, on either side of where the FMR would appear. On increasing the applied field still further the YIG resonance moves above that of the SRR resonance. It is also evident, on comparing the curves of Ba = 0.2777 T, and 0.3202 T in Fig. 3, that the SRR effective frequency trace is inverted in shape and shifted by 0.25 GHz to a lower frequency (Ba = 0.3202) with respect to the first trace of Ba = 0.2777. This is in agreement with the observations of Kang et al. and Gollub et al. [5,6]. A 19.76° phase reversal  was also observed overs the anti-crossing. This behavior was also found to follow a similar pattern in its reversal to that observed in the absorption ν-Ba map in Fig. 2.
Various orientations of SRR with respect to the direction of the magnetic field were also investigated, and the results of a 90° rotation can be seen in Fig. 4 . It is clear that the hybridization/interaction is much stronger for the orientation of the SRR as shown in Fig. 4(b), in particular, transmission at the centre of the anti-crossing, has increased by 1.6 to 4.0 dB. A shift in frequency of 0.25 GHz of the electric dipole resonance is also evident as the YIG FMR passes across. Clearly the orientation of the SRR with respect to the magnetization field is important, if increases in microwave transmission S21 are to be achieved. Note also concomitant increases in absorption, on both sides of the anti-crossing as shown by the arrows in Fig. 4(b). The resonance shows a similar form to that of the Fano-resonance .
This concludes our broad discussion of the experimental results. We turn now to a theoretical discussion of the anti-crossing and the finer details evident in the SN21 map of Fig. 2.
4. Anti-crossing, spin-waves, magnetostatic surface waves and backwards moving magnetostatic modes
Modeling of the anti-crossing has been performed using the COMSOL modeling package. The results replicate the observed anti-crossing, but with poor resolution due to the need to cover a wide magnetic field and frequency range. Thus more subtle details, such as the observation of BVMSWs, are precluded. However considerable insight into the problem can be gained by adopting the two-state model developed in [7,8], for a SRR in close contact with a YIG film. Following the latter, we assert that the anti-crossing between the resonant frequency of the split-ring and the uniform YIG resonance can be described by a simple 2 × 2 matrix:Fig. 2, we find δ ≈ 0.15 GHz. Further, on using this value together with (i) νSRR = 10.9 GHz, (ii) M0 = 139.26 × 103A/m  and (iii) γ = 2.949 × 1010 (Hz/T) , we find the combined response of the SRR and YIG film (solid lines) shown in Fig. 5 . From a comparison of the latter with the experimental results of Fig. 2, it is evident that the two-state model describes the anti-crossing behavior very well.
The two-state model is easily modified to take into account the generation of spin-waves and/or non-uniform Walker magnetostatic modes. For a spin-wave or MSSW the YIG resonance is shifted according to9]). Here, the energy of the spin-wave is given by E(ks) = Dks2, where D is the spin-wave stiffness factor and ks the wavenumber. From a comparison of Eqs. (2) and (4) therefore, it is evident that the generation of spin-waves with differing wave-vectors ks will give rise to a whole host of parallel lines, to the left of the uniform Walker mode. By contrast, the generation of non-uniform volume magnetostatic modes will give rise to a whole host of lines to the right of the uniform Walker mode known as BVMSWs (see Ref. , p157 and 165). Unlike spin-waves and MSSWs these are characterized by a negative group velocity νg = ∂ω/∂ks < 0: hence the description ‘backwards’. In practice, the generation of spin-waves, MSSWs or non-uniform Walker modes will be dictated by the geometry and strength of the rf field associated with the SRR. However for the thin YIG sample in question, it is evident that there are a very large number of allowed k-values, given the very large aspect ratio (thickness to width) of the thin film ~1:3 × 103 (see Refs. [21–23]). To examine this point in more detail therefore, experiments were performed using thicker bulk YIG. The results for a 5 × 5 × 0.5 mm3 thick piece (aspect ratio 1:10) can be seen in Fig. 6 . Two new features are immediately apparent. One, the YIG FMR resonance is now dominant over that of the SRR, appearing directly in the anti-crossing region. Two, the discrete nature of the BVMSWs is now very apparent on the right-hand side of the FMR. These are the magnetostatic waves first described by Dillon . Some lines are also apparent on the left-hand side of the FMR. We attribute these lines to the creation of either spin-waves or MSSWs with long wavelengths λ~1 µm. Clearly the non-linear interaction between the SRR and bulk YIG provides a fertile breeding ground for the excitation of spin-waves, surface magnons and BVMSWs. In practice, it should be possible to generate specific excitations, with well-defined ks values, using SRRs with tailored shapes, such as those shown in Fig. 1 of Ref. . However, more work will be required to determine the optimum thickness/filling factor of the YIG film, with respect to the dimensions of the SRR. Also the precise form of the rf excitation fields produced by the SRR is important. At the so-called magnetic-mode at ~5 GHz, anti-crossing exists but at a much weaker level. The interactions described above are much stronger when the SRR is excited in the symmetric-electric mode ~10.9 GHz.
In summary, it has been demonstrated that CPW-based composite SRR metamaterials, incorporating a magnetic component (YIG), can be tuned by applying a magnetic field. The coupling between the SRR and the YIG gives rise to hybridization of the two resonances, yielding a frequency response similar to that of a Fano-resonance . In particular, the anti-crossing regime allows tuning over a frequency range ~0.3 GHz, much wider than that of the SRR metamaterial taken in isolation. There is good agreement between experiment with electromagnetic simulations obtained using COMSOL, and the two-state model [7,8]. It has also been demonstrated that the anti-crossing region provides fertile ground for the creation of elementary excitations. Spin-waves, magnetostatic surface waves and backwards volume magneto-static waves have been observed. Finally, it has been shown that the orientation of the SRR with respect to the magnetization of the YIG and CPW is important. In particular, transmission, at the centre of the anti-crossing, could be increased by 1.6 to 4.0 dB, simply by rotating the SRR through 90°. Such experiments demonstrate that tunabilty is dependent not only on the strength of the magnetic field but also on the orientation of the SRR with respect to the CPW.
The authors would like to acknowledge the financial support of the Engineering and Physical Sciences Research Council (U.K.).
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