## Abstract

We demonstrate a magnetically tunable and two-dimensional (2D) left-handed material (LHM) consisting of an array of ferrite rods and metallic wires by experiments and simulations. It shows that the ferrite rod has a 2D isotropic negative permeability. By combining the ferrite rods with metallic wires, we observe experimentally a 2D LH passband that can be tuned dynamically, continuously and reversibly by an external magnetic field within in a wide frequency range with a response of 3.5 GHz/kOe. Retrieved effective parameters based on simulated scattering parameters show that operating frequency and value of negative refraction index can be conveniently tuned by changing the external magnetic field.

©2009 Optical Society of America

## 1. Introduction

Materials with simultaneously negative permittivity and negative permeability, i.e., left-handed materials (LHMs), have aroused great interest of researchers for their unique electromagnetic (EM) properties [1–8]. From the original work of negative refraction in microwave frequency [4] based on simultaneously negative effective permittivity and permeability to recent investigations on perfect lens [5], LHMs have shown great potential to manipulate EM wave. Since LHMs are not available in nature, much attention has been paid to artificial structures, such as metallic split ring resonators (SRRs) and wires structures [2–4], etc. Among them, SRR as an artificially-designed magnetic component possesses negative permeability (NP) derived from its strong magnetic response to EM wave. However, an SRR exhibits only one-dimensional (1D) NP because of its anisotropic structure, which restricts applications of the LHMs. And in order to obtain 2D or 3D NP, more complex structures are required [6]. Besides, SRR structure itself is largely an inactive one, although considerable investigations have been carried out on tuning resonance frequency *ω*
_{res} of SRR by adjusting its effective capacitance *C* or inductance *L* according to ${\omega}_{\mathrm{res}}=1\u2044\sqrt{\mathrm{LC}}$. In this case, the tunable components such as varactors [7], capacitors [8], semiconductors [9], liquid crystal [10] or ferrite [11,12] have to be introduced into the system of the resonator. Therefore, non-structural materials which can provide intrinsically tunable NP properties would be more desirable.

Recently, insulated ferrites [13–17] such as magnetized yttrium iron garnet (YIG) slab [15,16] and Ba-M ferrite slab [17] have been studied as natural NP media for constructing tunable LHMs where the NP is derived from ferromagnetic resonance (FMR). Theoretically, the magnetized infinite ferromagnetic medium is 2D isotropic in the plane perpendicular to the magnetizing direction [18]. However, permeability of the thin slab shows only 1D negative in such plane due to its dynamic demagnetization of the magnetic field of EM wave [17]. In this paper, we aim to apply the ferrite materials to the construction of tunable 2D LHM within X-band. We experimentally demonstrate that the rodlike ferrite shows 2D NP and the combination of such rods with metallic wires exhibits 2D LH transmission property which can be conveniently tuned over X-band by an applied magnetic field **H**
_{0}. Based on the simulated scattering parameters and the retrieval procedures for the combination, we demonstrate negative refraction index whose operating frequency and value can be tuned conveniently by the field **H**
_{0}.

## 2. Experiments

The commercially available YIG rods with a size of 0.8×0.8×10.0 mm^{3}, saturation magnetization of 4*πM*
_{s}=1700 Gs and a resonance linewidth of Δ*H*=12 Oe were used in our experiment, as shown in Fig. 1(a). Using a shadow mask/etching technique, we fabricated 0.6 mm thick FR-4 dielectric substrates (*ε*
_{r}=4.4 and tan*δ*=0.014) with copper wires with a cross section of 0.5×0.03 mm^{2}, length of 10 mm and spacing of 5 mm on one side. We pasted the rods back-to-back with the wires on the other side of the substrates and then assembled the rod-wire units into an array, as shown in Fig. 1(b). The microwave transmission (S_{21}) through the array was measured by a vector network analyzer HP8720ES using an X-band rectangular waveguide WR90 under different applied magnetic fields **H**
_{0}, which is provided by an electromagnet along the *z* axis. An EM wave with TE_{10} mode propagates along the *y* axis with an electric field polarized along the *z* axis.

First, S_{21} of sole YIG rods array is measured under different **H**
_{0} as shown in Fig. 2(a). It can be seen that, when H_{0}=0 Oe, S_{21} of the YIG rods array approaches to 0 dB, which corresponds to the simultaneously positive permeability and permittivity with low loss. As **H**
_{0} increases, a sharp dip emerges and its frequency can be dynamically and continuously tuned by changing **H**
_{0}, as shown in Fig. 2(b) where the dip frequency increases from 8.14 to 11.8 GHz as **H**
_{0} rises from 1600 to 2650 Oe. Compared with the method of SRR’s resonance frequency modulating, the present approach for ferrite would be distinctly advantageous for its more strong tuning capacity with less engineering processing.

This transmission behavior of the magnetized YIG rods is produced by FMR. We consider the transversely propagating EM wave along the *y* axis with the **H**
_{0} along the *z* axis in an infinite ferrite medium. For generality, we assume that the ferrite has an isotropic relative permittivity *ε*
_{r}, no matter **H**
_{0} is applied or not. In this case, the EM wave is divided into an ordinary wave and an extraordinary wave. The wave number of extraordinary wave can be derived from Maxwell’s equations and expressed as [19]

where

and

In Eqs. (2) and (3), *ω*
_{m}=4*πM*
_{s}
*γ* is characteristic frequency of the ferrite, *γ* is gyromagnetic ratio, *ω*
_{0}=*γ*H_{0} is frequency of FMR and *α* is damping coefficient of ferromagnetic precession. In Eq. (1), we can treat the term of (*µ*
^{2}-*κ*
^{2})/*µ* as extraordinary permeability *µ*
_{inf}(*ω*). At the ignorance of higher-order infinitesimal of *α*, we can obtain *µ*
_{inf}(*ω*) as

where *F* is a proportional coefficient of *ω*
_{m} to *ω*
_{0} with *F*=*ω*
_{m}/*ω*
_{0}, *ω*
_{mp} is considered as magnetic plasma frequency with ${\omega}_{\mathrm{mp}}=\sqrt{{\omega}_{0}\left({\omega}_{0}+{\omega}_{m}\right)}$ and Γ(*ω*) is a frequency dependent dissipation loss with Γ(*ω*)=[*ω*
^{2}/(*ω*
_{0}+*ωm*)+*ω*
_{o}+*ω*
_{m}]*α*.

We then consider a 2D square array with a lattice constant of *d* which is composed of thin and homogenous ferrite cylinders with a radius of *r*. In this case, each component cylinder has the same internal dynamic magnetic field in its radial direction due to the rotational symmetry. Therefore, the array has magnetically 2D isotropy in radial plane of the cylinder as a whole. If **H**
_{0} is imposed on the cylinder lengthwise, the frequency of FMR is given by Kittel’s equation as (*ω*
_{r}=*ω*
_{0}+(1/2)*ω*
_{m}. Assuming *d*≪*λ*, we can view the array as a homogenized medium. According to effective medium theory, effective extraordinary permeability of the medium can be expressed approximately as

where *η*=2*r*/*d* is fractional volume occupied by the cylinders. Substituting *ω*
_{r} for *ω*
_{0} in the expressions of *F*, *ω*
_{mp} and Γ(*ω*), and choosing 4*πM*
_{s}=1700 Gs, *α*=0.003, *d*=5 mm and *r*=0.6 mm, we calculate *µ*
_{eff}(*ω*) under different fields of H_{0}=1600, 2000 and 2400 Oe, as shown in Fig. 3. It can be seen that the array of the cylinders exhibits a NP in the vicinity of *ω*
_{mp}, which can be tuned by applied magnetic field. Within the NP frequency range, the EM wave cannot propagate and the forbidden dip of the transmission appears correspondingly [Fig. 2(a)]. Note that for technological reasons, square cross sections are used in the experiments instead of circular ones, without influencing the qualitative results.

By combining the YIG rods with metallic wires, we observe a passband within the stop bands of the rods alone and wires alone under H_{0}=2000 Oe as shown in Fig. 4(a). The wires array has negative permittivity within a wide region below the electric plasma frequency, as explained in detail by Pendry et al [2]. The rods array has NP in the vicinity of *ω*
_{mp} under the specified **H**
_{0}. For the combination, the passband emerges just in the region where both permittivity and permeability are negative. It is considered that this passband would be LH as in the LHMs consisting of SRRs and metallic wires [20,21]. Furthermore, the passband measured under different **H**
_{0} reveals its magnetically tunable property, as shown in Fig. 4(b) where the passband frequency increases from 8.2 to 11.7 GHz as **H**
_{0} rises from 1600 to 2600 Oe. Figure 4(c) shows an approximately linear dependence of the passband frequency on **H**
_{0} with a sensitively response rate of about 3.5 GHz/kOe. As **H**
_{0} decreases, the passband turns to low frequency direction, and after removing **H**
_{0}, the transmission spectrum returns to its original zero-field state, i.e., the passband disappears (data are not shown here), indicating a continuity and reversibility of the tuning. The disappearance of the passband after removing **H**
_{0} is resulted from the negligible remanence of YIG. Moreover, a similar tunable passband can also be observed in rotated combination on *z* axis for *π*/2 [Fig. 4(d)], which reveals a 2D isotropy in *xoy* plane of the passband. The 2D isotropy is attributed to 2D isotropic NP of the rods.

## 3. Simulations

We simulate ferrite rod alone and our combination structure using CST Microwave Studio, a Maxwell’s equations solver. The geometry, dimensions and materials’ parameters for the simulation are chosen to be roughly consistent with the experimental studies. The unit cell of a ferrite rod alone with the cell dimension of 4.4×4.4×10 mm^{3} is shown in Fig. 5(a). A square-sectioned rod with a size of 0.8×0.8×10 mm^{3}, 4*πM*
_{s} of 1700 G, resonance linewidth of 12 Oe and *ε*
_{r} of 14.3 is centered within the cell. For the combination simulation, two copper cylinders with a diameter of 0.2 mm and a length of 10 mm are positioned symmetrically at the laterals of the ferrite rod, as shown in Fig. 5(b). Symmetrical arrangement of the cylinders ensures a reciprocal transmission between port 1 and port 2, i.e., S_{21}=S_{12} and S_{11}=S_{22}. The simulations use electric and magnetic boundary conditions on the transverse boundaries and two open ports to simulate the scattering parameter response of a single infinite layer medium to a normally incident plane wave.

Solving the inverse problem we can retrieve the effective material properties from scattering parameters using the retrieval procedure [22]. Nevertheless, the effective material properties could not be retrieved from experimental data because it proves difficult to obtain precise measurements of the transmission and reflection by applying the electromagnet and repeatedly assembling the waveguides and matching loads. Instead of experimental data, the simulated scattering parameters are used to the retrieve of the material parameters.

Figure 6 shows retrieved effective permeability *µ*
_{eff} for the ferrite rod alone. It can be found that *µ*
_{eff} has a resonant dispersion and Re(*µ*
_{eff}) appears negative in the vicinity of *ω*
_{mp}. Moreover, *ω*
_{mp} increases as **H**
_{0} increases, indicating a magnetically tunable property of the NP frequency range. The retrieved results agree well with the calculated one according to Eq. (5).

Figure 7 shows simulated magnitude in dB and phase in deg of S_{11} and S_{21} for the combination of the rod with the wires under H_{0}=2200 Oe. It can be seen that in the vicinity of 10.22 GHz the magnitude of S_{21} exhibits a narrow passband and the phase shows a jump correspondingly.

The effective refraction index *n*
_{eff} is retrieved from the simulated scattering parameters for the combination, as shown in Fig. 8. It can be seen that the real part of the index Re(*n*
_{eff}) is negative [Fig. 8(a)] in the frequency range where the combination functions well due to the simultaneously negative Re(*ε*
_{eff}) and Re(*µ*
_{eff}) [Fig. 8(b) and (c)], indicating that the combination should be a LHM. The frequency range of negative Re(*n*
_{eff}) is rather sensitive to the applied magnetic fields **H**
_{0}. On the one hand, frequency related to the minimum of Re(*n*
_{eff}) shifts remarkably when **H**
_{0} is “rough” tuned, as shown in Fig. 8(a) where the frequency increases from 9.0 to 10.68 GHz as **H**
_{0} increases from 1800 to 2400 Oe. On the other hand, value of Re(*n*
_{eff}) at a certain frequency is modulated obviously when **H**
_{0} is “fine” tuned, as shown in Fig. 7(d) where Re(*n*
_{eff}) increases from -2.53 to -0.30 at 10.06 GHz and decreases from -0.66 to -2.51 at 10.20 GHz, respectively, as **H**
_{0} increases from 2180 to 2220 Oe. With all dips of Re(*n*
_{eff}) in Fig. 8 included, Fig. 9 briefly summarizes the magnetically tuning effect of the frequency and the value of Re(*n*
_{eff}).

## 4. Conclusion

In conclusion, we demonstrate the magnetically tunable 2D isotropic LHM composed of YIG rods and metallic wires by experiments and simulations. We observe experimentally a LH passband within the stop bands of the ferrite rods alone and metallic wires alone. The passband can be tuned dynamically, continuously and reversibly by an applied magnetic field within whole X-band with a sensitive rate of 3.5 GHz/kOe. Numerical simulation confirms the negative refractive index whose operating frequency and value can be adjusted conveniently by altering the applied magnetic field. The experimental demonstration of tunable negative refraction can be carried out using a prism composed of copper/ferrite metamaterial and this work is in progress. It is desirable that the operating frequencies of such LHMs can extend from gigahertz up to terahertz band by selecting different ferrites and adaptive fields. Therefore, the realization of such kind of LHMs will bring about more potential applications in the construction of novel devices such as broadband perfect lens.

## Acknowledgement

This work is supported by the National Science Foundation of China under Grant Nos. 50425204, 50632030, 50621201 and 10774087.

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