The discussion starts with a single split ring resonator, since it is the simplest example for understanding the physical mechanisms that produce effective negative permittivity /permeability. For a single split ring resonator, when the applied magnetic field has a component normal to the plane of ring resonator, it will excite a current loop around the ring.
The split structure offer an effect of capacitance, then the current excited by external magnetic field will be circulated through a serials circuit of inductance and capacitance.
It is known that resonance occurs at the frequency ωm= 1
LC , in which there is a strong absorption of the external magnetic field. This effect results in an effective permeability that is smaller than zero. The primary physical mechanism is described above, being interest us news to understand how micro or nano-structures have an influence on E.M waves.
x
Below are more quantitative discussions on physical properties of SRR. Most materials have a natural tendency to be diamagnetic as a consequence of Lenz’s law. Consider the response of SRR arrays to an incident electromagnetic wave with the magnetic field normal to the plane, as shown in Figure 3-1. The SRR has a radius r, and is placed in the square lattice with a period of a. The oscillating magnetic field normal to the plane of SRR induces circumferential surface currents which tend to generate a magnetization opposing to the applied field. The axial (z-axis) magnetic field inside the cylinders is
H=H0
+J-2 2
πr
a J (3.1) where H0 is the applied magnetic field and J is the induced current per unit length of the SRR.
The third term
2 2
πr
a J is due to the depolarizing field, which is assumed to be uniform as the SRR are infinitely long along the z-axis. The electromagnetic force (emf) around the SRR can be calculated according to the Lenz’s law and is balanced by the Ohmic drop in potential：
-iωμ0πr2(H0 where the effective capacitance C = ε0επr/(3d), and ε is the relative dielectric permittivity of the material in the gap. It is assumed that the potential varies linearly with the azimuthal angle around the ring.
The system of SRRs is homogenized by adopting an averaging procedure that consists of averaging the magnetic induction, B, over the area of the unit cell while averaging the magnetic field, H, over a line along the edge of the unit cell. The averaged magnetic field is
Beff=μ0H0 (3.3) While the averaged H field outside the SRR is
Heff=H0
-2 2
πr
a J (3.4) Using above equations, it is obtained the effective relative magnetic permeability：
μeff= eff and thus it derives a resonant form of the permittivity with a resonant frequency of
0 2 3
0 0
ω = 3d
μ ε επ r (3.6) that arises from the L–C resonance of the system. The factor f = πr2/a2 is the filling fraction of the material. For frequencies larger than ω0, the response is out of phase with the driving magnetic field and μeff is negative up to the ‘magnetic plasma’ frequency of
m 2 3
0 0
ω = 3d
(1-f)μ ε επ r , (3.7) assuming the resistivity of the material is negligible, and it is seen that the filling fraction plays a fundamental role in the bandwidth over which μ < 0. It is noted that μeff =1 − πr2/a2 asymptotically at large frequencies. This is due to the assumption of a perfect conductor in above analyses. It deservers to be mentioned that μeff can attain very large values on the low frequency side of the resonance. Thus, the effective medium of SRR is going to have a very
large surface impedance Z = eff
eff
μ ε .
In the following section, the electric coupling to the magnetic resonance of split ring resonators is discussed [40]. According to [40], authors observed unexpectedly that the incident electric field coupled to the magnetic resonance of the SRR, when the E.M. waves propagated perpendicular to the SRR plane while the incident E was parallel to the gap-bearing sides of the SRR.
It was recently shown [41-42] that the SRRs exhibited electric resonant response in addition to their magnetic resonant response. As a result of this electric response and its interaction with the electric response of the wires, the effective plasma frequency,ω , was 'p
much lower than the plasma frequency of the wires, ω . An easy criterion [41] to identify p whether an experimental transmission peak was left-handed (LH) or right-handed (RH) was proposed, i.e., If closing the gaps of the SRRs in a designed LH structure would remove only one single peak from the transmission data in the low frequency regime, it could be a strong evidence to assign that transmission peak due to the LH effect. The authors reported numerical and experimental results for the transmission coefficient of a lattice of SRRs alone for different orientations of the SRR with respect to the external electric field, E, and the direction of propagation. It was considered an obvious fact that an incident EM wave excited the magnetic resonance of the SRR only through its magnetic field; hence one could conclude that this magnetic resonance appeared only if the external magnetic field H was perpendicular to the SRR plane, which in turn implied a direction of propagation parallel to the SRR [Figures. 3-2(a), and 1(b)]. If H was parallel to the SRR [Figures. 3-2(c), and 1(d)] no coupling to the magnetic resonance was expected. However, the present work shows that this was not always the case. If the direction of propagation was perpendicular to the SRR plane and the incident E was parallel to the gap-bearing sides of the SRR [Figure 3-2(d)], an electric
coupling of the incident EM wave to the magnetic resonance of the SRR occurred. It thus means that the electric field excited the resonant oscillation of the circular current inside the SRR, influencing the behavior of either only ε(ω) [in Figure 3-2(d)] or both ε(ω) and μ(ω) [in Figure 3-2(b)].
Figure 3-2. Left–hand side: SRR geometry studied. Right–hand side: The four studied orientations of the SRR with respect to the triad k, E, H of the incident EM field. The two additional orientations, where the SRR are on the H-k plane, produce no electric or magnetic response. The figure is imaged from [40].
Four nontrivial orientations of the SRR are considered, as shown in Figure 3-2, while the measured transmission spectra, T, of the composite meta-material (CMM) are presented in Figure.3-3. The continuous line (line a) corresponds to the conventional case (Figure 3-2(a)), with H perpendicular to the SRR plane and E parallel to the symmetry axis of the SRR.
Notice that T exhibited a stop band at 8.5–10.0 GHz, due to the magnetic resonance. The dashed line (line b) shows T for the orientation of Fig. 3-2(b); here E was no longer parallel to the symmetry axis of the SRR and thus there was no longer a mirror symmetry of the combined system of SRR plus EM field. Notice that now T exhibited a much wider stop band at 8–10.5 GHz, starting at lower frequency. An interesting result was obtained by comparing T for the two cases shown in Figures. 3-2(c) and 3-2(d), for which there was no coupling to the magnetic field since H was parallel to the SRR plane. For the case of Figure 3-2(c), where
E was parallel to the symmetry axis, no feature was observed around the magnetic resonance frequency (line c) in Figure 3-3, as expected. However in the case of Figure 3-2(d), where the SRR plus EM field exhibited no mirror symmetry, a strong stop band in T around ωm was observed (line d), similar to that of the conventional case [Figure 3-2(a)]. It thereby strongly suggested that the magnetic resonance could be excited by the electric field if there was no mirror symmetry.
Figure 3-3. Measured transmission spectra of a lattice of SRRs for the four different orientations shown in Figure 3-2. The figure is imaged from [40].
At low frequencies, the SRR can basically be only represented by its outer ring. As shown in Figure 3-4, the SRR ring will experience different spatial distributions of the induced polarization, depending on the relative orientation of E with the SRR gap. If E is parallel to the no gap sides of the SRR, its polarization will be symmetric and the polarization current is only flowing up and down the sides of the SRR, as shown in Figure 3-4(a). If the SRR is turned by 90°, as shown in Fig. 3-4(b), the broken symmetry leads to a different configuration of surface charges on both sides of the SRR, connected with a compensating current flowing between the sides. This current contributes to the circulating current inside the SRR and hence couples to the magnetic resonance.
This unexpected electric coupling to the magnetic resonance of the SRR is of
fundamental importance in understanding the refraction properties of SRRs in the low frequency region of the EM spectrum.
Figure 3-4. Schematic drawing for the polarization in two different orientations of a single ring SRR. The external electric field points upward. Only in case of broken symmetry (b) a circular current will appear which excites the magnetic resonance of the SRR. The interior of the ring shows FDTD data for the polarization current component at a fixed time for normal incidence [Figures 3-2(c) and 3-2(d)] as a gray scale plot.
The figure is imaged from [40].
J&E