Fig. 1. Schematic description of the accelerating structure. A trigger bunch propagates in a cylindrical metallic waveguide of radius filled with an active medium. This bunch is injected into the structure with velocity larger than the Cerenkov velocity and generates an entire manifold of TM modes which propagate behind. One of the eigenmodes is amplified by the active medium and many wavelengths behind the trigger bunch the former accelerates a trailing train bunch.

Fig. 2. (a) The dynamics of the wake on the axis (solid curve), the PID, (dashed–dotted curve), and the trigger bunch profile, (dashed curve). The value of the saturated wake is shown by the dotted curve. (b) A comparison of the nonlinear wake dynamics in real units (solid curve) with the linear wake dynamics (dots). In addition, the profile of the bunch is drawn as a reference (dashed curve).

Fig. 3. The mode spectrum of the wake . Here, the single-resonance mode is .

Fig. 4. (a) A two-dimensional plot of the longitudinal wake . The green rectangle is the location of the trigger bunch. (b) The same as (a) but in the region that is marked in magenta in (a). (c) The radial dependence of the wake at .

Fig. 5. The energy conservation. Here, is the energy of the ground state and is the energy of the excited state; (d) shows the deviation from energy conservation.

Fig. 6. The wake dynamics (a) and the wake spectrum (b) for various Cerenkov parameters and bandwidths. The solid curve corresponds to and , where and are the Cerenkov and bandwidth parameters as in Figure 2. The dotted curve corresponds to and . The dashed curve corresponds to and .

Fig. 7. The dependence of the saturation value (a) and saturation length (b) on the waveguide and trigger bunch parameters. Here, the index 0 represents the parameter value as in Figure 2 or Ref. [18].

Table 1. Structure parameters of our studied example. Note that the set of parameters used here is the same as in Ref. [22].