Other Structures Generating Negative μ or n

To date, besides the early invented SRR, many different structures have been reported by several groups of researchers for achieving μ<0 such as the S-shaped [10], H-shaped [53], paired-rod [9] or double-wire [54], and π-shaped [55] structure.

Among these works, the S-shaped SRR is able to yield negative ε and μ at the same frequency range and thus is worth mentioning here.

SRRs exhibit a frequency band of negative ε that is higher than that of negative μ, and they do not overlap in general. A previously purposed structure of two SRRs printed on the opposite sides of the substrate has been proven to avoid bianisotropy, and has the same magnetic effect as the conventional SRR structures. The geometry of this structure is shown in Figure 2-26 (a) and its Re(ε) and Re(μ) are plotted in Figure (b) and (c). One can clearly see that negative μ happens at a lower frequency range than that of negative ε, thus there is no overlap. The structure was constructed where the branches of a single SRR are shaped like a squared S and two SRRs were printed on each side of the substrate, yielding an eight-like pattern when viewed from

Figure 2-26 (a) is the simplified ring geometry as first building block toward S structure. Metalization is applied on each side of the substrate, and the capacitances C enables current to flow around the ring under magnetic induction. (b) and (c) are the real parts of permittivity (dashed line, left axis) and permeability (solid line, right axis). Responses are the function of frequency for SRR structures and incidence, as shown in the insets. The figure is imaged from [28].

Figure 2-27 (a) is a schematic drawing of the S-shaped resonator. Note the additional capacitance C1 when compared to Figure 2-28 (a). (b) and (c) are the real parts of permittivity (dashed line, left axis) and permeability (solid line, right axis).

Responses are the function of frequency for S-shaped resonator structures and incidence, as shown in the insets. The figure is imaged from [28].

Figure 2-28 Equivalent circuit for the geometry shown in Figure 2-29 (a).

Capacitance C1 enables current to flow in each half ring when a magnetic field is applied. The figure is imaged from [28].

top (see Figure 2-27 (a)). Again, Re(ε) and Re(μ) are plotted. As can be seen from Figure 2-27 (b) and (c), the S-shaped resonator lowered the electric resonant frequency from 12 GHz to 11 GHz and increased the magnetic resonant frequency from 9 GHz to 15 GHz, and therefore made the two frequency bands overlap from 10.9 GHz to 13.5 GHz (i.e., about 2.6 GHz bandwidth). The losses (-1.75 dB/cm) are also proven to be significantly less than that of the standard split ring and rod designs (-6.53 dB/cm). An equivalent circuit is presented in Figure 2-28. Besides the two capacitances C between the top and bottom metallic strips, there is another capacitance C1 between the center metallic strips. With C1 being large and equal to C, the S-shaped resonator has a magnetic resonant frequency 3 times that of the SRR in Figure 2-26 (a), while exhibiting the same electric property. Yet the extra lengths of wire introduce an extra inductance, and the effect is to decrease the overall plasma frequency.

Chapter 3

Design of Nano-scaled SRR

LHMs were first realized at frequencies around 10 GHz (3 cm wavelength) and fabricated mostly on stacked printed circuit boards. These samples allowed a broad tunability of patterns and arrangements thus enabled investigation in several different properties. Then, microstructures with magnetic resonance frequencies at about 1 THz (300μm wavelength) were fabricated by microlithography. Using nanofabrication techniques, the LC resonance frequency could increase to about 100 THz (3 μm wavelength) [4], bringing optical frequencies into research for obtaining negative index of refraction.

Mean-field theoretical consideration indicates that the characteristic scale of SRRs should be at least tens or few hundred times smaller than the incident wavelength. In fact, previous studies have shown that by employing the SRR with a size 1/11^th of the incident wavelength, magnetic resonance can be obtained [13]. Thus, SRRs are designed with a typical size that is about one tenth the wavelength of the desired operating frequency [23]. In the microwave region, the mm-scaled SRR cell is easily fabricated, but fabrication in the optical wavelength region is difficult since the SRR pattern must be shrunk to the nanometer scale.

This thesis work mainly focused on two aspects: (1) demonstrating the electric and magnetic responses of nano-scaled SRRs, which were previously done mostly for mm- or μm-scaled SRRs; (2) observing the scaling of the electric and magnetic

resonance frequency for nano-scaled SRRs. In the following sections, the design concepts of the nano-scaled SRR are presented in details.

3.1 Non-linear Scaling of SRR

Typically, SRR has a unit cell size that is 1/10^th [23] to 1/11^th [13] of the incident wavelength. However, as the operating frequency becomes higher and higher, non-linear scaling has been reported. Many approaches have been applied to study the physical reason and mechanism of this effect.

Ishikawa and T. Tanaka reported the effects of the surface resistivity and the internal reactance on the magnetic responses of the SRR by considering the delay of the current in the conductor [33]. The frequency dependence of the surface resistivity and internal reactance is depicted in Figure 3-1. As the frequency increases, both surface resistivity and internal reactance of the metals increase. The increase of the surface resistivity results in the decrease of the Q value of the SRR, and thereby, degrades the tunable range of the permeability μ. The increase of the internal reactance leads to the reduction of the SRR resonant frequency. The surface resistivity saturates at the inherent frequency of each metal about 10 THz. The saturation value of silver is remarkably smaller than those of gold and copper. The internal reactance, on the other hand, does not saturate and moves away from zero drastically as the frequency increases. As a result, in the optical frequency region, the effect of the internal reactance on the resonant frequency of the SRR was found to be more dominant than that of the surface resistivity. It can be seen in Figure 3-2 that as the geometric size of the SRR decreases, the magnetic response that is defined by the difference between maximum and minimum values of the effective μ decreases. And the resonant frequency shifts to a higher value. Note that the increase of the resonant

frequency is not linearly proportional to the sizes of the SRR due to the increase of the internal reactance.

Figure 3-1 Dispersion curves of the internal impedance of silver, gold, and copper.

In the frequency region exceeding THz, the internal reactance is more dominant than the surface resistivity, and this internal reactance decreases the resonant frequency.

The figure is imaged from [33].

Figure 3-2 Real and imaginary parts of the effective permeability of the silver SRRs as a function of the SRR pattern size. The labeling in each case indicates the unit-cell size a. The inset shows the corresponding sizes of the SRR in nanometers. The figure is imaged from [33].

Figure 3-3 Frequency dependencies of the minimum value of Re(μ) of the SRRs made of (a) silver, (b) gold, and (c) copper, with the filling factor of F = 5%, 7%, 10%, and 13%, respectively. The frequency dependencies of the imaginary part of the effective permeability lRe whose real part has the minimum value are also shown.

Only the silver SRR exhibits negative l in the visible range of 400–700 THz. The figure is imaged from [33].

The frequency dependence of the minimum value of Re(μ) of the SRR for three kinds of metals and four different filling factors were plotted in Figure 3-3. As the frequency increases, the minimum value of Re(μ) approaches unity asymptotically.

Only the silver SRR exhibits negative Re(μ) in the visible light range, while the propagation loss of silver is higher than that of gold or copper. Note that the filling factor is also important for the realization of the negative μ in visible light range.

Further studies showed that above the linear scaling regime the resonance frequency saturates due to the free electron kinetic energy, which cannot be neglected any longer in comparison with the magnetic energy [34]. Although the presence of the magnetic resonance was reported to be observed with very small unit cells, such a weak resonance was unable to conduct negative values as Re(μ) approaches unity. The maximum resonance frequency (and thus the negative Re(μ) region) increases with the number of cuts in the SRR. The 4-cut SRR retains the negative Re(μ) region for higher frequencies up to about 550 THz. As a comparison, the 2- and 1-cut SRR could only reach up to 420 THz and 280 THz (unit cell size ~60 nm), respectively. The geometries of the 1-, 2-, and 4-cut single-ring SRR are schematically presented in Figure 3-4 (a). Note that the gap widths were designed to make the capacitance of the three types of SRR approximately the same, but the experimental result revealed that the three capacitances were indeed different. This is because the formula

d C ~ wt is

not valid when d is not much smaller than w. Moreover, the periodic boundary condition along the direction of the electric field only adds asymmetric side capacitances to the SRR, as which is represented by the equivalent circuit in Figure 3-4 (b). Therefore, the three types of SRR have inequivalent capacitances and thus different saturation frequencies. The scaling of the magnetic resonance frequency as a function of the unit cell size a is shown in Figure 3-5.

The electron kinetic energy adds an electron self-inductance Le to the magnetic response which scales as 1/a while both the magnetic field inductance Lm and capacitance C of the SRR scale as a. The resonance frequency fm thus has the following a-dependence:

where c1 and c2 are independent of a. Other factors that break the linear scaling and contribute to the increase of the Ohmic losses are (1) the increased scattering of electrons at the surface of the metal and (2) the larger skin depth (scales as

a

1 ) over

metal thickness ratio. Both factors depend in a complicated way on the geometry and the surface smoothness, and thus require further experimental studies.

Figure 3-4 (a) The geometries of the 1-, 2-, and 4-cut single-ring SRR. The SRR is made of aluminum, simulated using Drude model (fp=3570 THz, fτ=19.4 THz). The parameters of the SRR are side length l=0.914a, width and thickness w=t=0.257a, and cut width d=0.2a, 0.1a, and 0.05a for the 1-, 2-, and 4-cut SRR, respectively. (b) The left panel shows the charge accumulation in a 4-cut SRR, as a result of the periodic boundary conditions in the E (and H) direction. The right panel shows the equivalent LC circuit describing this SRR. Cg is the gap capacitance and Cs the side capacitance resulting from the interaction with the neighboring SRR. The figure is imaged from [34].

Figure 3-5 Solid lines with symbols shows the scaling of the simulated magnetic resonance frequency fm as a function of the size of the unit cell a for the 1-, 2-, and 4-cut SRR, respectively. Up to the lower THz region, the scaling is linear, fm∝ 1/a.

The maximum attainable frequency is strongly enhanced with the number of cuts in the SRR ring. The hollow symbols as well as the vertical line at 1/a =17.9 μm^-1 indicate that no μ<0 is reached anymore. The nonsolid lines show the scaling of fm

calculated from Eq. 3.1 (LC circuit model). The figure is imaged from [34].

Despite the small line width that is theoretically required when scaling down the SRR, there are studies showing that the ratio of the effective sizes of the SRR to the wavelength of the incident beam is around 1, which suggests a wide scale independence [35]. This size tolerance is useful in realizing the metamaterials with negative n in the optical wavelength range.

In most experiments, the transmission is measured at a fixed incident angle for a certain range of wavelength. However, in [35], the incident angle was tuned while the wavelength was fixed by choosing different laser sources. Lasers with the output wavelength of 10.86 μm and 9.43 μm were used because the characteristic size of the SRR (10.48μm) was in between these two wavelengths. The transmission results were shown in Figure 3-6 (a) and (b). At 90° tile angle (in-plane incident) the transmission was expected to peak since the direct contribution of the incident beam to the detector

became dominant, as can be seen in the case of Al-coated planar Si (gray line).

However, when the SRRs were patterned on the substrate, absorption at in-plane incident occurred (black line). Although the absorption of parallel polarization at 9.43 μm is less significant than that at 10.86 μm, it is still observable, showing a great size tolerance. The coherence of the laser may be critical to the onset of resonance.

Moreover, an 808 nm laser was used to cross-check the rapid oscillation near the non-resonance region caused by the Fabry-Perot effect, as shown in Figure 3-6 (c).

Here, neither the rapid oscillation nor the magnetic response appeared. The transmission through Si at wavelength below 1μm is extremely low due to high loss.

And the incident wavelength is beyond size tolerance.

Figure 3-6 Normalized transmission of the SRRs. Data of the Al-coated planar Si is presented for reference. The operating wavelength is (a) 10.86 μm, (b) 9.43 μm and (c) 808 nm, and the laser power on the sample is ~100mW. The figure is imaged from [35] and edited

3.2 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.

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