Slow wave propagation has been of strong interest to electromagnetics because of its exotic properties that can lead to (i) small antennas, filters, and RF circuits; (ii) novel functionalities, and (iii) realize high-power RF and optical devices. Among some already demonstrated applications are (i) intense light source [1, 2], (ii) miniaturized antennas  with improved directivity, and (iii) high-power microwaves such as oscillators and amplifiers . The slow wave properties are realized using periodic layered media that exhibit effective anisotropic structure. Alternately, slow waves can be formed using coupled transmission lines (CTLs). In this case, one of the transmission lines propagates a strong forward wave, whereas the other adjacent line supports a weak backward wave .
To understand the physics of slow waves, this chapter provides an analysis of CTLs (Fig. 3.1) that are loaded with (L,C) elements in a periodic manner. As is well-known, any periodic loading supports Bloch waves  and exhibits passband and stopband behaviors. In this chapter, it is shown that the usual dispersion relation of these Bloch waves is of second order and can achieve nearly zero group velocity at the band edge. However, by changing the material properties or loading of the CTLs, as in Fig. 3.1a, the order of the dispersion curve can be altered, a process often called dispersion engineering. Specifically, higher-order dispersion curves (third or fourth) can be attained. Indeed, the use of anisotropic material in periodic stacks, instead of isotropic ones, can lead to third- or fourth-order dispersion relation  (Fig. 3.2). The third-order dispersion is achievable with magnetic photonic crystals (MPC). On the other hand, fourth-order dispersion relation leads to maximally flat curves. But this maximally flat behavior is difficult to achieve and leads to degenerate band edge (DBE) modes that are inherently narrowband phenomena. Of importance is the fact that these slow wave modes provide for large field enhancement as the group velocity nearly drops to zero. This reduction in group velocity also implies device miniaturization . Specifically, the DBE and MPC properties were applied to enhance the directivity of antennas [3, 9, 11, 12]. But as noted, DBE resonances are associated with very narrow bandwidth. Wider bandwidth is typically desired, and can be achieved by introducing magnetic anisotropy into the periodic layers. By doing so, a third-order dispersion relation was achieved , which is associated with an inflection point within the Bloch diagrams. Such a third-order relation provides for more bandwidth . The associated modes are referred to as MPCs [11, 12] and have led to highly directive, wideband, and miniature antennas. DBE and MPC modes were also realized and demonstrated by fabricating periodic volumetric stacks of dielectrics . But these material layers are bulks. Therefore, there is interest to develop printed or wire-type versions of MPC structures for more practical applications.
An important development in realizing the DBE and MPC modes came from Locker et al. , who proposed an equivalent printed form of DBE and MPC using printed microstrip lines. The microstrip lines are CTLs and depended on the substrate to emulate the behavior of volumetric DBE and MPC [14, 15]. In this chapter, a simple theoretical analysis of coupled transmission lines is presented. This is done first for the second-order dispersion curves and is then expanded to higher-order dispersion engineering. The chapter closes with the description of several applications.
Figure 3.1 (a) Top: printed coupled microstrip lines to realize the frozen mode
(finite length); middle: field strength within the coupled transmission lines showing strong field intensities in the middle due to wave velocity slowdown; bottom: wave amplitude representing field growth. (b) Slow wave propagation within a helical wire structure placed in a waveguide. (c) Slow wave propagation within a curved ring-bar structure placed in a waveguide, the latter refers to traveling wave tube applications.
Figure 3.2 Periodic material structures to realize MPC and DBE modes: (a) photonic crystals with simple dielectric stacking. Forward and backward waves are depicted by blue and green arrows. (b) Magnetic photonic crystals using volumetric stack of anisotropic materials created using metal strips printed on the dielectric layers .