**1. Introduction**

250 Trends in Electromagnetism – From Fundamentals to Applications

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Recently, the idea of complex materials in which both the permittivity and the permeability possess negative real values at certain frequencies has received considerable attention. In 1967, Veselago theoretically investigated plane-wave propagation in a material whose permittivity and permeability were assumed to be simultaneously negative (Veselago, 1968). For materials with negative permittivity and permeability, several names and terminologies have been suggested, such as "left-handed" media, media with "negative refractive index" (NIR), "backward-wave" (BW) media and "double-negative"(DNG) material (Caloz & Itoh, 2005). In this book chapter, materials with negative permittivity and permeability, and hence negative index of refraction, will be referred indistinctly as left-handed metamaterials (LHMs) or metamaterials (MTMs) (Caloz & Itoh, 2005).

Metamaterials have found many applications in electromagnetic problems. For instance, numerous novel MTM-based microwave components have been proposed to control amplitudes, frequencies, and wave numbers of propagating and non-propagating electromagnetic modes (Caloz & Itoh, 2005). Advances in MTMs have also stimulated the development of new couplers with unique coupling mechanisms. Recently, coupled-line couplers (CLCs) using composite right/left-handed transmission lines (CRLH TLs), which are the special realization of transmission lines based on the metamaterial concept, with broad bandwidth and arbitrary loose/tight coupling levels have been developed. But usually these couplers occupy large length and also, because of using stubs in their structures, width of them would be large. For eliminating this drawback, we have proposed some new backward and forward coupled line couplers with high coupling levels, broad bandwidths and compact sizes, base on the CRLH TLs.

Organization of this chapter is as follows. In Section 2 theory of CRLH TLs, interdigital capacitor and their equivalent circuit models and parameters, have been explained. Section 3, at first, reviews some conventional CRLH- based CLCs and in continues presents our proposed couplers. In this section, three CLCs based on the concepts of CRLH CLCs are presented; a symmetrical backward CLC (Section 3.3.1), an asymmetrical backward CLC (Section 3.3.2) and a symmetrical forward CLC (Section 3.3.3).

Coupled-Line Couplers Based on the Composite Right/Left-Handed (CRLH) Transmission Lines 253

in this figure, an interdigital capacitor is made of some gaps. The gap meanders back and forth in a rectangular area forming two sets of fingers that are interdigital. These gaps are essentially very long and folded to use a small amount of area. By using a long gap in a small area, compact single-layer small-value series capacitors can be realized. Typically, its capacitance values range from 0.05 pF to about 0.5 pF. The capacitance can be increased by increasing the number of fingers, or by using a thin layer of high dielectric constant material

The value of series capacitance of an interdigital structure can be expressed as (Bahl, 2003):

( ) ( ) ( ) <sup>1</sup> ( ) <sup>18</sup>

As is well-known, the characteristic impedance of a CRLH TL ( *Zc* ) with equivalent circuit

*K C N l pF <sup>K</sup>*

κ

′ = − ′ (1)

(2)

= (3)

 κ

′ is effective permittivity of a strip with width *W* , *N* is the number of fingers and

2

1

1

*sh L CL R*

2

*se*

ω

 <sup>−</sup> 

<sup>=</sup> , <sup>1</sup>

ω

ω

<sup>=</sup>

ω

<sup>−</sup>

*sh*

ω

*re L s*

*c L*

*<sup>C</sup>* <sup>=</sup> , <sup>1</sup> *se L CR L* ω

(a) (b)

According to Fig. 2, the equivalent circuit model of an interdigital capacitor is similar to the equivalent circuit model of one cell of CRLH TL when *LL* → ∞. Inserting *LL* → ∞ into (2) results the characteristic impedance ( int *Zc* ) of a TL consists of cascaded interdigital

*Z Z*

*L*

*L*

*L*

*<sup>L</sup> <sup>Z</sup>*

Fig. 2. (a) Interdigital capacitor. (b) Its equivalent circuit model.

such as a ferroelectric between the conductors and the substrate (Bahl, 2003).

ε

*K k* ′ is a constant that has been presented in (Bahl, 2003).

model of Fig. 1(a) is given by (Caloz & Itoh, 2005):

where *re* ε

( ) ( ) *K k*

where

capacitors as:

π
