ω

Figure 6.47 CRLH transmission line media dispersion relation and resonance frequencies [16].

6.2.5.2DNG applications

The DNG property, as was described previously, is characterized by other terms such as LH (Left-Handed), −k (the negative propagation constant), NRI (Negative Refractive

Index), anti-parallel nature of vp (phase velocity) and vg (group velocity), as the attribute of the properties based on the negative constitutive constants (−ε and –μ). These

properties can be implemented by combination of an array of thin wires and an array of SRRs, and periodically reactance-loaded transmission lines, in which the typical one is the CRLH transmission line. The combination of thin wires and SRR was first introduced as the metamaterial that exhibits the property of DNG, and discussions ensued about its application in various ways such as to planar lenses, filters, waveguides, and so forth, but not to radiators. Practical implementation of such metamaterials has been so far found in use of transmission lines, which are composed with lumped components periodically loaded into the structure. The typical one is the CRLH transmission line.

The CRLH transmission line constituted with a periodical connection of a series LR and CR circuit, and a shunt CL and LL circuit, as was shown in Figure 6.23, exhibits the dispersion characteristics having resonant modes as illustrated in Figure 6.24. As a conventional transmission line, a CRLH transmission line may also be open-ended or short-ended to produce a standing wave or resonance state. The resonance modes of the transmission line structure constituted with N units of the length l (= Nd; d is the unit length) are given in relation with the length l by

and the phase constant for a mode m is

The resonance modes m exist both positive and negative as shown in Figure 6.47. Each positive resonant mode (m > 0) at ω+m corresponding to β+m (RH region) has twin-negative resonant modes (m < 0) at ω−m corresponding to β−m (LH region) [53].

68Principles and techniques for making antennas small

voltage

m = ±3

β = mπ

Figure 6.48 Typical field distribution of resonance modes on the CRLH transmission line [16].

Figure 6.49 Resonance of CRLH transmission line resonator constituted of N = 4 units (balanced case) [16] (l: length of resonator, d: period).

The typical field distributions of the resonant modes on the CRLH transmission line are depicted as in Figure 6.48. It should be noted that there is a zeroth-order (ZOR) mode (m = 0), which exhibits at the transition frequency ω0 where λg = ∞ or β = 0, implying infinite wavelength or no phase variation [54, 55]. In the practical N-unit CRLH transmission line, there exist passbands as shown in Figure 6.49, where a case for N = 4 is illustrated. With N = 4, resonance modes m are 0, ±1, ±2, ±3, and when m = ±4, where lm = ± π /d, it is not resonance, but the edge of the Brillouin zone. In general, the dispersion diagram of the N-unit CRLH transmission line is limited in frequencies at the limits β = ±π /d of the Brillouin zone, and resonance occurs at m =

±(N − 1) plus m = 0. For a balanced transmission case, there exist (2N − 1) resonances, while an unbalanced case has 2N resonances. The resonance frequency ωm is calculated from

2{1 − cos(mπ/N )} = (ωL /ωm )2 + (ωm /ωR )2 − (ωsh /ωse)2/(ωR )2