Patterns of SRR

3.2.1 Unit Cell Size of SRR

There are typically two interested frequency ranges when applying for bio-sensing.

One is the visible light region and another is the middle infrared region (MIR) around 10 μm. The former is valuable due to its ease to observe. Also, the EM wave energy in this region (~2eV) is able to excite the transition between two orbital bands (e.g.

HOMO and LUMO), such that fluorescence of bio-molecules might be enhanced when applied to periodic metal patterns. This so-called metal-enhanced fluorescence (MEF) has a promising potential in detecting bio-molecules. The later, on the other hand, induces energy approximately equal to that of molecular bonding (~0.1eV).

This is helpful in “fingerprint” detection of bio-molecules. It would be rather difficult to design SRRs that operate in the visible light region using existing fabrication technology, since it requires really small line-width. Instead, by fabricating SRRs with the unit cell size about 1 μm, it is possible to obtain LH behavior at around 10 μm, which might further excite magnetism or change in bonding configuration, and thus achieve bio-sensing. Moreover, as mentioned at the beginning of this chapter, this thesis work aims at studying the scaling of the electric and magnetic resonance frequency for nano-scaled SRRs. Considering both intentions, five unit cell sizes have been chosen in our experiment: 600nm, 900nm, 1200nm, 1500nm, and 1800nm.

For a square single-ring SRR shown in Figure 3-7, ωm is proportional to 1/l while ω0 depends only on sides parallel to the electric field, following approximately the relation, ω₀ ∝a l_E . Thus ω0 shows a weaker dependence than ωm on l in Figure 3-8 (a) and is constant in Figure 3-8 (b) where only lk was tuned. The different dependence of ωm and ω0 on a/l allows the relative position of ωm and ω0 to be controlled.

Figure 3-7 Two single-ring SRRs, a square one and an orthogonal one, are shown, together with the external electric field E, propagation directions k, lattice constant a, side length of the unit cell l. The lengths, lk and lE, of the two sides of the orthogonal SRR are also shown. The figure is imaged from [20].

Figure 3-8 (a) ωm and ω0 (in units of c/a, where c is the vacuum light velocity and a is the lattice constant of the unit cell) versus a/l for a square single-ring SRR (see the left drawing of Figure 3-7). l is the SRR side length. (b) ωm and ω0 (in units of c/a) versus a/lk for an orthogonal single-ring SRR (see Figure 3-7, right panel). lk is the length of the SRR side which is perpendicular to the incident electric field, E. The figure is imaged from [20].

3.2.2 Square Single-ring SRR

For circular and square single-ring SRRs with the same geometric size, metal characteristics, and gaps, the magnetic resonance frequency of the square SRR is a little lower than that of the circular one (Figure 3-9). Nevertheless, the ω and ω

dependences on the system parameters are the same in both cases [20]. However, for simplicity of both the nanofabrication and the corresponding transmission calculations, the square form was chosen in the present work.

To make a comparison between single- and double-ring SRRs, the transmission through the double-ring SRR and that through only its outer or its inner ring SRR are presented in Figure 3-10. It can be seen that the lower magnetic resonance frequency of the double ring is essentially due to the outer ring, but with a relatively small downwards shift. This shift is caused by the additional capacitance between the rings.

The second dip of the double-ring corresponds essentially to the magnetic resonance of the inner ring with a small upwards shift. The strength of this resonance is sometimes very small, indicating that the magnetic response of the inner ring is screened by the presence of the outer one. This happens mainly in the case where the electric field is parallel to the continuous side of the SRR.

There are mainly two reasons why double-ring SRR has been the preferable form when studying LH behavior in microwave region. One advantage of the double-ring SRR compared with its outer single-ring SRR is that the magnetic resonance frequency occurs at a relatively lower frequency, thus there is a higher probability for the magnetic response to lie in the ε<0 regime in the combined system of SRRs and wires. Another advantage is that the array of double-ring SRRs possesses a stronger magnetic resonance, which might lead to a more robust LH peak. Although the desired condition is to make ωm as low as possible, one has to pay special attention to that the lowering of ωm is not associated with a weakening of the strength of the magnetic response of the SRR.

As the SRR size decreases, the actual capacitance becomes larger than estimation because Gupta’s formula is valid only when the thickness of the ring is negligible. To reduce the geometrical capacitance of the structure, employing an SRR consisting of a

single ring is advantageous [33]. It has been shown that it is not necessary to use the conventional double-ring SRR proposed by Pendry et al., a single-ring SRR with a cut also behaves as a magnetic resonator. This simplifies the fabrication, especially for small structural sizes, and potentially reduces dielectric losses, since the fields get strong only around the cuts but not between the rings anymore [34]. Therefore, for nano-scaled unit size we focus our study on single-ring SRRs

Figure 3-9 (a) Transmission (dB) versus frequency for one single-ring square (solid curve) and circular (dashed curve) SRR. The corresponding designs are shown on the left side of the panel. (b) Re(μ) as a function of frequency for the single-ring square (solid curve) and circular (dashed curve) SRR, at frequencies around the magnetic resonance frequency. The figure is imaged from [20].

Figure 3-10 Transmission (dB) versus frequency for (a) the double-ring SRR and (b) its isolated outer and inner ring SRRs. The figure is imaged from [20].

3.2.3 Geometric Parameters of SRR

Previous parametric studies showed a rather weak dependence of EM response of SRRs on the metal width and gap width. Referring to the SRRs utilized in ref. 6, 36, and 37, a square single-ring SRR is designed to have a line width that is one fifth of the side length and a gap width equal to the line width for simplicity in layout. For example, for the smallest unit cell size 600 nm, its line width and gap width is then 120 nm. In addition, corresponding CRR and 2-cut SRR are designed to demonstrate the magnetic resonance and the electric coupling effect. The layout designs of 1-cut SRR, 2-cut SRR and CRR in our experiment are drawn in Figure 3-11.

Figure 3-11 Designed unit cell of (a) 1-cut SRR, (b) CRR, and (c) 2-cut SRR.