Other geometry for negative permeability

Fundamentally, the basic structures of negative permeability are composite of spilt ring ant magnetic medium are just a capac

with single or double rings. The main ingredients of a reson

itance and an inductance. In fact, SRR structures with (1) a single ring and a single split [18–20] and (2) a double ring with a single split in each ring [21] have been proposed as candidates for a resonant magnetic medium. The point is that we would like to have an electrical polarizability as low as possible. With a single split ring, a large electric dipole moment would be generated across the capacitive gap (see Figure 1-6(a)), which could then dominate over the weaker magnetic dipole moment generated in the ring. When there are two

splits present, the dipole moments across opposite ends cancel each other. Therefore, one only gets a weak electric quadrupole moment (see Figure 1-6(c)) whose effects would be expected to be much weaker than those of the magnetic dipole moment. Most of the magnetic media that simultaneously generate an electric dipole moment are bi-anisotropic media, i.e. the constitutive relations are

D = εE + αH, B = βE + μH, (1.10) where α and β are the bi-anisotropy coefficients. The Ω–shaped particles first introduced as the components of a bi-anisotropic medium [22], when arranged in a periodic lattice also have a resonant ε and μ and can behave as negative magnetic materials [23]. Even the original SRR medium [3] is bi-anisotropic [24] as there is an electric dipole moment that develops across the capacitive gaps (see figure 1-6(b)), and this can also be driven by an electric field [25].

Thus the SRR is oriented such that the magnetic field is normal to the plane of the SRR and the electric field is along the SRR and can be driven by the electric field and the magnetic and electric resonances can overlap. This can be effectively resolved, of course, by rotating adjacent SRRs in the plane by 180° and the corresponding electric dipole moments would cancel. The symmetry of the single ring with two symmetrically placed capacitive gaps renders this less bi-anisotropic and electrically less active.

Figure 1-6. The SRRs develop an electric polarization too, although driven by a magnetic field. This bi-anisotropic behavior is maximal for (a) a single split ring which develops a large dipole moment. (b) The original SRR with two splits is also bi-anisotropic as it develops an electric dipole. The field lines due to the charges are also shown by thin lines.

(c) The single ring with two symmetric splits only develops a quadrupole moment and is hardly bi-anisotropic. The figure is imaged from [8].

Figure 1-7. Schematic drawings of different resonator structures: (a) single ring with one cut, (b) single ring with two cuts, (c) single ring with four cuts, (d) SRR with two cuts and (e) SRR with four cuts. Figure is imaged from [26].

People then studied how splits in ring affect resonance [26], as which was driven by the need for new resonator designs for the followi

resonant behavior at certain frequencies.

ng reasons. First of all, a conventional SRR structure is not easy to fabricate for operation at higher frequencies. As the structure is scaled down, the dimensions of the narrow split and gap regions will be very small, which may eventually lead to contact problems between the metallic regions. Conventional SRR structures were fabricated for operation at a few THz [27], but for the resonator structure working at 100 THz [28] a single ring with a single split was chosen for easier fabrication.

The second reason is that SRR structures are electrically resonant for different polarizations and propagation directions [29]. This effect would suppress the LH-behavior for 3D-constructed LHMs, where EM waves would be incident on the structure from all directions. Therefore, additional splits should be added to destroy the electrical coupling effect to the magnetic resonance.

As seen from the figure, all structures show

Simulation results are provided in Figure 1-8(b) and experimental results were compared with the simulations. The resonance frequency of one-cut ring was measured was at 4.58 GHz, whereas numerical simulations predicted it to be 4.67 GHz. For two-cut and four-cut ring resonators, measured and simulated resonance frequencies were 7.82 and 7.5 GHz; and 12.9 and 13.1 GHz, respectively. We then varied the split width of these single-ring resonators.

Figure 1-9 shows the measured resonance frequencies as a function of split width. In all cases, the magnetic resonance frequency increases with increasing split width. But the rate of changing ωm for the one-cut ring resonator (Figure 1-9(a)), two cut ring resonator (Figure 1-9(b)), and four-cut ring resonator (Figure 1-9(c)) are different. The rate of increase is larger for structures with more splits. Since the capacitance due to all splits will change, the change in total capacitance will be larger for structures having more splits. Note that the magnetic resonance frequencies increased drastically when more cuts were introduced into the system.

When the second split was placed on the ring (Figure 1-7(b)), the capacitances were connected in series. Therefore, the total capacitance was decreased approximately by a factor of 2. Because of this great amount of decrease in capacitance of individual ring resonators, the change in ωm was much larger compared to the changes owning to split widths, gap distances and metal widths.

Figure 1-8. Transmission spectra of single-ring resonator with different number of cuts are shown by (a) experiments and (b) simulations. The figure is imaged from [26].

Figure 1-9. Variation of magnetic resonance frequency with the split width of (a) the one-cut ring resonator, (b) two-cut ring resonator and (c) four-cut ring resonator. The figure is imaged from [26].

For double rings with two or four cuts (splits) in each ring, as a convention, these resonator structures are called two-cut SRR (Figure 1-7(d)) or four-cut SRR (Figure 1-7(e)), where the number of cuts presented the number of splits in each ring. The split width was initially taken with a size of d=0.2 mm. Measured and simulated transmission spectra of these structures are depicted in Figure 1-10(a) and 1-10(b), respectively. The resonance frequency of the one-cut SRR was found at 3.63 and 3.60 GHz, from both measurements and

simulations. For two-cut SRR and four-cut SRR structures, measured and simulated resonance frequencies were 6.86 and 6.45 GHz, and 12.96 and 13.2 GHz, respectively. We then varied the split width of all splits in both inner and outer rings. Figure 1-11 shows the measured resonance frequencies as a function of split width. In all cases the magnetic resonance frequency increased with increase of split width. Similar to the behavior observed in single split ring, the rate of increase in resonance frequency was also larger for structures having more splits.

Figure 1-10. Transmission spectra of SRRs with different number of cuts obtained from (a) experiments and (b) simulations. The figure is imaged from [26].

Figure 1-11. Variation of magnetic resonance frequency with the split width of (a) the one-cut SRR, (b) two-cut SRR and (c) four-cut SRR. The figure is imaged from [26].

Table 1-1 summarizes the measured resonant frequencies obtained for six different split ring resonator structures for some detailed analyses and a comparison of the results obtained in single split rings and double split rings. Columns represent the number of rings in the resonator structures, whereas the rows correspond to the number of cuts in each ring.

Increasing the number of splits increases the magnetic resonance frequency drastically, since the amount of decrease in the capacitance of the system is very large. For one-cut and two-cut resonator structures, the amount of decrease of ωm is around 1 GHz. But in the case of the four-cut structure, such a behavior is not observed. There is essentially no change in resonant frequency between both configurations. The orientation of the splits is important in this case.

Unlike the anti-symmetric orientations of splits in the one-cut and two-cut SRRs, the four-cut SRRs have symmetric orientations. So the mutual capacitance between the inner and outer rings is very small. This is due to the fact that the induced charges along both the rings have

the same sign and a similar magnitude. As a result, addition of a second ring does not affect the overall capacitance of four-cut single-ring resonator. In turn, the resonance frequency did not change appreciably.

Table 1-1. Measured magnetic resonance frequencies for six different resonator structures. The figure is imaged from [26].