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We report the experimentally observed abnormal refraction in metamaterials (MMs) consisting of ferrite rods and metallic wires with two kinds of configurations. Negative refraction (NR) and positive refraction (PR) are demonstrated in an MM constructed with parallel-arranged rods and wires. The frequencies of both NR and PR can be adjusted dynamically and together by an applied magnetic field and the PR occurs at frequencies slightly lower than that of the NR. The NR is attributed to simultaneously negative effective permittivity and permeability, and the PR is resulted from positive effective permittivity and permeability with the positive effective permittivity originating from electromagnetic coupling between the closest rod and wire. By making the rod cross the wire to reduce the coupling, we observed sole NR in an MM consisting of the cross-arranged rods and wires. Theoretical analysis explained qualitatively the abnormal refraction behaviors of microwave for the two kinds of MMs and it is supported by the retrieved effective parameters and field distributions.
©2011 Optical Society of America
1. Introduction
In the last several years much interest has been focused on negative refractive index material (NRIM) starting with the theoretical prediction of Veselago [1] and the pioneering experiments of Shelby et al [2]. From the original work in microwave frequency [2–4] to recent investigations on perfect lens [5–8], NRIMs have shown great potential to manipulate electromagnetic wave. Since NRIMs are not available in nature, metamaterial (MM) has been a major resort to obtain NRIMs. For example, MM consisting of metallic wires and split ring resonators (SRRs) has been used to construct NRIM in which the negative refraction (NR) is originated from simultaneously negative effective permittivity and permeability [2]. However, the structure-determined property of SRR makes the operating frequency generally inactive, which seems to be a bottleneck for its potential applications. Fortunately, by controlling equivalent capacitance C [9–15] and inductance L [16–18] of SRR, researchers have demonstrated the tunability of resonance frequency according to , but only in virtue of various additional components and within a small tunable range.
Ferrite, in comparison with SRR, can provide a more dynamically tunable negative permeability with an applied magnetic field (AMF). The simpler engineering processing, larger tunable range and faster response would make ferrite a more desirable means to achieve tunable negative permeability. In this case, the tunable NRIM based on ferrite/wire construction has been proposed theoretically [19–21] and demonstrated experimentally only with a transmitted passband in a rectangular [22–24] or a parallel plane waveguide [25]. However, it seems difficult to observe directly NR phenomena, probably because of electromagnetic coupling between the ferrite and wire. In this paper, we report a directly experimental demonstration of an abnormal refractive phenomena including frequency tunable NR and positive refraction (PR) in an MM consisting of yttrium iron garnet (YIG) rods and metallic wires with the rods paralleling to the wires. The frequencies of both NR and PR can be adjusted dynamically and together by AMF and the PR occurs at frequencies slightly lower than that of the NR. The NR is attributed to simultaneously negative effective permittivity and permeability which are originated from electric resonance of the wires and ferromagnetic resonance (FMR) of the rods respectively. The PR is attributed to simultaneously positive effective permittivity and permeability in which the positive effective permittivity would be originated from electromagnetic coupling between the closest rod and wire within the strongly resonant band for the rod. By making the rod cross the wire to reduce the coupling, we observed sole NR in an MM consisting of the cross-arranged rods and wires. Theoretical analysis explained qualitatively the abnormal refraction behaviors of microwave for the two kinds of MMs and it is supported by the retrieved effective parameters and field distributions.
2. Experimental procedure
Polycrystalline YIG rods are commercially available with a size of 0.8×0.8×10.0 mm3, saturation magnetization of 4πMs = 1700 G, dielectric constant of and resonance linewidth of ΔH=12 Oe. Copper wires are fabricated on one side of 0.25 mm thick FR-4 circuit board (ε = 4.4 and tanδ =0.014) by shadow mask/etching technique. The thickness, width and length of the wires are 0.03 mm, 0.5 mm and 10 mm respectively.
We designed two kinds of unit cell and their prisms using YIG rods and copper wires: one with parallel-arranged rod and wire as shown in Fig. 1(a) and Fig. 1(b), and the other with cross-arranged rods and wires as shown in Fig. 1(c) and Fig. 1(d). In the cell with parallel-arranged configuration, a rod is pasted on the opposite side to a wire of the board with the rod and the wire being kept in parallel, while in that with the cross-arranged one, 3 parallel rods are pasted on the opposite side to 2 parallel wires of the board with the rods and the wires being kept in cross. For measurements of microwave refraction, two kinds of prisms are constructed with the unit cells shown in Fig. 1(a) and Fig. 1(c) respectively. In the prism shown in Fig. 1(b), the cells with the parallel-arranged rod and wire are arranged along y axis and x axis with a lattice of d y = d x = 5 mm. Minimum and maximum numbers of the cells arranged along y axis are 2 and 9 respectively while those along x axis are 24 and 3 respectively, which results in a wedge angle of 18.4°. In the prism shown in Fig. 1(d), the cells with cross-arranged rods and wires are arranged along y axis and x axis with lattice of d y = 6 mm and d x = 12 mm respectively. Minimum and maximum numbers of the cells arranged along y axis are 1 and 10 respectively while those along x axis are 7 and 1 respectively, which results in a wedge angle of 18.4° too.
Fig. 1 Schematics of (a) a unit cell consisting of a YIG rod and a copper wire with parallel arrangement, (b) a prism consisting of the parallel-arranged rods and wires, (c) a unit cell consisting of 3 YIG rods and 2 copper wires with cross arrangement, and (d) a prism consisting of the cross-arranged rods and wires.
Measurements of microwave refraction are performed by a setup shown in Fig. 2 . The setup and the procedure for the refraction measurement are referred to in Ref 2. The electric field E rf of the incident wave upon the prism is polarized in the direction perpendicular to the metal plates and it is parallel to the wires, while the magnetic field H rf is polarized in the direction parallel to the metal plates and it is perpendicular to the wires. Before the measurement, the setup needs to be checked aforehand with a Teflon prism [2]. The angular refraction spectra of transmitted powers with the same polarization as the incident wave are measured within frequency band between 8 and 12 GHz by a network analyzer HP8720ES connected to the setup under different AMFs H 0 with an angle step of 2° and a frequency step of 0.02 GHz. The uniform AMF provided by an electromagnet along z axis covers an area with a diameter of 170 mm, ensuring that the whole prism is biased by the AMF.
3. Experimental results and discussion
The transmitted powers of the wave with the same polarization as the incident wave through the prism shown in Fig. 1(b) are scanned within the band between 8 and 12 GHz under H 0 = 1800 Oe and 2400 Oe respectively. Figure 3 shows the power functioned as frequency and angle θ from the normal. It is evident that the transmitted beam is refracted on the negative or positive side from the normal respectively, at very close but different frequencies under a specified AMF. The frequencies in which the beams are deflected to the negative side are higher slightly than that to the positive one. As AMF increases, transmitting frequencies of both negative and positive beams increase accordingly. It is reasonable to expect that the PR and NR in the prism are different in nature with refractions of ordinary and extraordinary wave in birefringence because the PR and NR have the same polarization and they occur at different frequencies, but the ordinary and extraordinary wave in birefringence have different polarization. Both NR and PR of the prism in the transmission direction should be attributed to FMR of the rods and plasma behavior of the wires. The NR is attributed to simultaneously negative effective permittivity Re(ε eff, z) below plasma frequency of the wires and negative effective permeability Re(μ eff, x) within the antiresonant frequencies of the rods, as explained later. The PR is attributed to simultaneously positive Re(ε eff, z) and Re(μ eff, x) in which the positive Re(ε eff, z) would be originated from electromagnetic coupling between the closest rod and wire induced by the extremely large permeability Re(μ rod, x) of the rod within the resonant frequencies, as explained in detail later. Outside the transmission band, i.e., far from the FMR area of the rods away (blue area in Fig. 3), the transmitted power is nearly forbidden due to the negative Re(ε eff, z) resulted from the plasma behavior of the wires and the positive Re(μ eff, x) within the area far from the FMR.
Fig. 3 Transmitting power as functions of frequency and angle from the normal measured under (a) (H)0 = 1800 Oe and (b) (H)0 = 2400 Oe for the prism shown in Fig. 1(b).
To investigate the profile of the beams, the angular cross sections at 9.44 and 9.50 GHz for H 0 = 1800 Oe and at 10.80 and 10.86 GHz for H 0 = 2400 Oe are plotted in Fig. 4(a) and 4(b) respectively. It shows that the maximum powers occur at θ = 24° for 9.44 GHz and at θ = −34° for 9.50 GHz under H 0 = 1800 Oe. According to Snell’s law the index of the prism can be defined as n y = + 1.29 at 9.44 GHz and n y = −1.77 at 9.50 GHz respectively. As AMF increases from H 0 = 1800 to 2400 Oe, frequencies of the maximum powers occur at θ = 22° for 10.80 GHz and at θ = −24° for 10.86 GHz. The index of the prism under H 0 = 2400 Oe can be defined as n y = + 1.19 at 10.80 GHz and n y = −1.29 at 10.86 GHz respectively. It can be seen clearly that the frequencies of the NR and PR are very close but different under the specified AMFs. Specifically, the frequencies of PR are lower than that of NR. The similar phenomenon has ever been demonstrated in simulation reported in Ref 21. Here we just chose the frequencies with maximum powers from Fig. 3 for the discussion because the transmitting band of the beam is quite narrow and the power becomes quite weak at those frequencies with non-maximum power.
Fig. 4 Angular cross sections of the transmitting powers for the prism shown in Fig. 1(b) at (a) 9.44 and 9.50 GHz for (H)0 = 1800 Oe, and at (b) 10.80 and 10.86 GHz for (H)0 = 2400 Oe.
The transmitted powers as functions of frequency and angle from the normal through the cross-arranged prism are scanned within the band between 8 and 12 GHz under H 0 = 2800 Oe and H 0 = 3400 Oe respectively, as shown in Fig. 5 . It is evident that the transmitted beam is refracted just on the negative side from the normal without positive refractive beam observed. The disappearance of the positive refractive beam should be attributed to weakening of the electromagnetic coupling between the closest rod and wire, as explained in detail later. The sole NR can be attributed to both plasma behavior of the wires in direction of E rf and FMR of the rods in direction of H rf under AMF. As AMF increases, the transmitting frequency of the NR increases accordingly, as shown in Fig. 5 that it increases from about 9.14 GHz to 10.82 GHz as AMF increases from 2800 Oe to 3400 Oe.
Fig. 5 Transmitting powers as functions of frequency and angle from the normal measured under (a) (H)0 = 2800 Oe and (b) (H)0 = 3400 Oe respectively for the prism shown in Fig. 1(d).
The angular cross sections at 9.14 GHz for H 0 = 2800 and at 10.82 GHz for H 0 = 3400 are plotted in Fig. 6 . It shows that the maximum powers occur at θ = −32° for 9.14 GHz under H 0 = 2800 Oe and at θ = −36° for 10.82 GHz. According to Snell’s law the index of the prism shown in Fig. 1(d) can be defined as n y = −1.68 at 9.14 GHz and n y = −1.86 at 10.82 GHz respectively.
Fig. 6 Angular cross sections of the transmitting powers for the prism shown in Fig. 1(d) at (a) 9.14 GHz for (H)0 = 2800 Oe and (b) 10.82 GHz for (H)0 = 3400 Oe respectively.
4. Theoretical analysis
Based on the coordinate system shown in Fig. 1(a), the permeability of YIG rods would be a tensor with the dispersive elements μ rod, x, μ rod, y and the non-dispersive element μ rod, z = 1. It is reasonable to expect that μ rod, x = μ rod, y = μ lin due to the same dynamic shape demagnetization of H rf in x and y axis for the rod [21]. The frequency dispersive μ lin can be given by [24, 26]
where F is a proportional coefficient with F = ω m/ω r, ω m is characteristic frequency of the ferrite with ω m = 4πM s γ, γ is gyromagnetic ratio, ω r is the frequency of FMR for the rod with ω r = ω 0 + (1/2)ω m given by Kittel’s equation, ω 0 is frequency of FMR for the infinite ferrite with ω 0 = γH0, ω mp is considered as magnetic plasma frequency with and Γ(ω) is dissipation loss with . Further calculation for a YIG rod under H 0 = 2000 Oe, 4πM s = 1700 G, and α = 0.003 reveals extremely large positive Re(μ lin), i.e., Re(μ rod, x) and Re(μ rod, y), up to about 60 in the vicinity of 10 GHz, and large absolute value of negative Re(μ rod, x) and Re(μ rod, y) in the vicinity of 10.1 GHz, as shown in Fig. 7 .For the wire array, the plasma frequency is dominated generally by the lattice period and the self-inductance [27]. As AMF is parallel to the wires, it cannot produce Lorentz force to the moving electrons driven by E rf within the wires. Therefore, the effect of the AMF on the plasma frequency of the wires can be ignored in the absence of the rods. However, once the rod is introduced near the wire while it is biased by the AMF like shown in Fig. 1(a), the large positive Re(μ rod, x) and Re(μ rod, y) in the strongly resonant frequencies (near 10 GHz in Fig. 7) of the rod would result in a remarkable enhancement of magnetic field within the rod excited by a current within the wire, obeying Biot-Savart law. A significant consequence of the enhancement would be intensification of self-inductance for the wire and further enhancement of effective mass of electron within the wire. In this case, the plasma frequency of the wire grid would be reduced much more actively and the effective permittivity Re(ε eff, z) for the rod/wire unit would be positive within the frequency band with extremely large positive Re(μ rod, x) and Re(μ rod, y). Moreover, the effective permeability Re(μ eff, x) for the rod/wire unit could be calculated with effective-medium theory due to the low space-filling factor and isotropic unity permeability of the wire and air. The Re(μ eff, x) would also be positive due to the positive Re(μ rod, x) within corresponding band. The simultaneously positive Re(ε eff, z) and Re(μ eff, x) within the frequency band results in a PR index in the direction of wave propagation, as demonstrated in the above experimental observation.
In the antiresonant frequencies (near 10.1 GHz in Fig. 7), the large absolute value of negative Re(μ rod, x) would result in negative Re(μ eff, x) under consideration of isotropic unity permeability of the wire and air. Moreover, the negative Re(μ rod, x) would not affect obviously the plasma behavior of the wires because the dynamic magnetic field induced by dynamic current within the closest wire to the rod would be attenuated rapidly within the rod with the negative Re(μ rod, x). Therefore, if the antiresonant frequencies of the rods are lower than the plasma frequency of the wires, the Re(ε eff, z) and Re(μ eff, x) would be simultaneously negative which results in NR index in the direction of wave propagation, as demonstrated in the above experimental observation.
Based on the coordinate system shown in Fig. 1(c), the permeability of YIG rods would also be a tensor with the dispersive elements μ rod, x = μ lin and the non-dispersive elements μ rod, y = μ rod, z = 1. It should be noted that μ rod, x = μ lin and μ rod, y = 1 is due to the remarkably different dynamic shape demagnetization of H rf in x and y axis for the rod [21]. By arranging the rods and wires in cross, the interacting areas between the nearest rod and wire would decrease remarkably. The decreased interacting areas would result in weakening of the influence of the rods on the plasma response of the wire grid even if in the case of extremely large positive Re(μ rod, x). The PR in the prism with parallel-arranged rods and wires disappears in that with cross-arranged one due to the weakening of the electromagnetic coupling for the latter.
5. Simulations
In order to verify the mentioned above theoretical analysis, we simulate sole metallic wire and our two kinds of MMs including parallel-arranged configuration and cross-arranged one using CST Microwave Studio, a Maxwell’s equations solver. The geometry and dimensions of the two kinds of unit cell for the simulation are chosen to be consistent with the experimental studies. The permeability of the rod is assigned with μ rod,x = μ rod,y = μ lin and μ rod,z = 1 for the parallel-arranged configuration, while μ rod,x = μ lin and μ rod,z = μ rod,y = 1 for the cross-arranged one, where μ lin is calculated under H 0 = 2000 Oe. The μ lin is extremely large in the frequency band for the simulation, specifically, from 29.8 + i3.0 to 59.9 + i15.1. Dimension of the unit cells is 5.0×5.0×5.0 mm3. A thin metallic wire, a pair of parallel-arranged wire and rod spaced with a 0.5 mm thick FR-4 board, and a pair of cross-arranged one are centered within the cell respectively. The simulations use electric and magnetic boundary conditions on the boundaries of z and x directions respectively, and two open ports on the boundaries of y direction, simulating the field distribution and the scattering parameter response of a single infinite layer medium to a normally incident plane wave.
The simulated distributions of magnetic energy density of the sole wire, the parallel-arranged rod/wire and the cross-arranged rod/wire are shown in Fig. 8 with the same color scale. It can be seen that density of the induced magnetic field around the sole wire is relatively weak and the field decreases as the distance to the wire increases [Fig. 8(a)], obeying Biot-Savart law. Figure 8(b) and 8(c) show that the introduced rod parallel to the wire enhances dramatically the magnetic field around the wire and the cross one influences the field distribution very little. The results of the field distributions agree well with the theoretical analysis mentioned above.
Fig. 8 Distributions of magnetic energy density of (a) the sole wire at 10.08 GHz, (b) the parallel-arranged MM at 10.08 GHz and (c) the cross-arranged MM at 10.08 GHz.
By solving the inverse problem we can retrieve the effective material properties from simulated scattering parameters using the retrieval procedure [28]. Figure 9 shows the retrieved effective permeability μ eff,x, permittivity ε eff,z and index n eff,y for the parallel-arranged and cross-arranged configurations. It can be found that dispersions of μ eff,x and ε eff,z for the parallel-arranged configuration exhibit antiresonant type and resonant type respectively. The Re(μ eff,x) and Re(ε eff,z) are simultaneously positive within the frequency band between 10.05 and 10.08 GHz that results in positive Re(n eff,y) in the band. For the cross-arranged configuration, the Re(μ eff,x) is positive and the Re(ε eff,z) is negative within whole simulating frequency band that results in very small Re(n eff,y) and very large Im(n eff,y), implying the electromagnetic wave cannot transmit in the structure within the corresponding band. The results of the retrieved effective parameters agree well with the theoretical analysis mentioned above.
Fig. 9 Retrieved effective permeability, permittivity and index. (a), (b) and (c) refer to the parallel-arranged rod/wire, while (d), (e) and (f) refer to the cross-arranged one.
6. Conclusion
In conclusion, we demonstrated experimentally abnormal refractions in MMs consisting of YIG rods and metallic wires. An NR and a PR are demonstrated in an MM with the rods paralleling to the wires. The frequencies of both NR and PR can be adjusted dynamically and together by AMF and the PR occurs at frequencies slightly lower than that of the NR. The NR is attributed to simultaneously negative effective permittivity and permeability. The PR is resulted from positive effective permittivity and permeability with the positive permittivity originating from electromagnetic coupling between the closest rod and wire. The sole NR is observed in another MM by making the rod cross the wire to reduce the coupling. The retrieved field distribution and effective parameters agree well with the theoretical analysis. The experimental realization of the tunable NR in the presented MMs would offer a robust support for the realization of many desirable devices such as tunable perfect lens.
Acknowledgements
This work is supported by the National Science Foundation of China under Grant Nos. 90922025, 51032003, 61077029, 50921061, 50902081 and 60978053 and by the China Postdoctoral Science Foundation under Grant No. 20100480013.
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