Applications of Slow Waves
Slow wave applications exploit dispersion engineering and the associated ш-р diagram. Antennas are often operated near resonance frequency and near the band edge of the dispersion diagram to achieve field enhancement [3, 20, 21]. However, microwave amplifiers such as TWTs may require broader bandwidth . In that case, second-order dispersion may be appropriate. In the following, we discuss some microwave/RF applications.
Traveling Wave Tubes
The operation of TWTs is based on the Cherenkov radiation and involves the coupling of an electron beam to an RF wave .
These devices are basically amplifiers that amplify the incoming RF wave by modulating an injected electron beam from the cathode ray tube. To achieve strong coupling between the electron beam and the electromagnetic wave or mode of the waveguide, the TWT must support a slow TM01 mode  that matches the velocity of the electron beam. Also, the strength of the beam-wave coupling is dependent on the strength of the axial field component supported by the TM01 mode. Therefore, a dispersion-free TM01 mode with a strong axial electric field component facilitates efficient coupling of the electron beam. Such a dispersion-free TM01 mode can be introduced by designing slow wave structures (SWS). In this regard, SWSs play an important role in the process of electron beam to waveguide mode coupling.
Since the CTLs are made of metallic conductors and can emulate dielectric behavior in their passband, they have the potential for high-power microwave applications. Interestingly, several SWSs, e.g., helix, double-helix, ring-bar structures , have been considered and termed as slow wave circuits in the literature. To improve coupling to microwave power delivery, Fig. 3.13 shows the curved ring-bar (CRB) structure . CRB is an upgrade of the typical helical structures and can be modeled as a pair of CTLs as discussed earlier. Specifically, the top and bottom elliptically bent lines refer to the pair of transmission lines. These transmission lines are coupled through the inner rings. More bending of the elliptic feature (i.e., m > 1) of the transmission lines provides more coupling, implying a way to control the slow wave properties and interaction impedance of the TWT model (Figs. 3.1c and 3.13). In effect, the elliptic bent of the transmission line controls the effective permittivity of the propagating wave by reducing the phase velocity below c (Fig. 3.13). Indeed, the dispersion diagram in Fig. 3.13 (right) and Fig. 3.14 validates the second-order dispersion curve. The latter is, of course, the behavior caused by the inductively CTLs. The designed TWT, in this manner, can generate up to 1 MW of output power with a gain of around 30 dB, whereas typical pulsed helix TWTs can generate power up to several kilowatts . This high-power enhancement is due to the slow wave phenomena and associated propagation constants caused by the CTLs.
Figure 3.13 Curved ring-bar structure placed inside the waveguide to improve electron beam to waveguide mode coupling. The dispersion diagrams to the right show how the propagation constant and wave velocity can be controlled by the geometric property, i.e., axial ratio m.
Figure 3.14 Dispersion diagram due to the curved ring-bar structure within the waveguide. Provided computations of the dispersion diagram were carried out via full wave simulations and the CTL model using appropriate coupling parameters.