## 3 Numerical Results  
The structure is simulated and tuned by using the commercial software Ansys HFSS [18] and CST Studio Suite [19] with the objective to minimize _IL_ and enhance the overall wave transmission from inputs 1 and 2 to outputs 3 and 4, at the same time optimizing the quality of the TE\({}_{10}\)-to-TM\({}_{01}\) mode conversion. The obtained results are equivalent between the two software packages; therefore, only the HFSS results are reported. Figure 4 shows the S-parameters: the operational bandwidth at \(IL=0.5\) dB is 60.06-60.08 THz, and the mode conversion is optimal at \(f_{0}=60.074\) THz. Figure 5 shows the electric field components along the accelerating waveguide axis at the frequency of 60.074 THz, for a length of \(7d=14.574\) um and input power of 1 kW: it can be seen that the field presents a strong longitudinal component \(|E_{z}|\), while the transversal components, \(|E_{x}|\) and \(|E_{y}|\), are almost equal to zero, as in the case of a TM\({}_{01}\)-like mode. Nevertheless, the accelerating field reaches a maximum of \(\simeq\)1.2 GV/m with an input power of 1 kW, thus confirming that the presented solution meets the required performance needed for future DLAs setups.  
Figure 4: \(|S_{ba}|\) and \(|S_{aa}|\) for the presented EBG woodpile coupler, where \(a=1+2\) and \(b=3+4\). See Figure 2b for I/O ports numbering.  
Figure 3: Projected band diagram of the isolated accelerating waveguide along the axis (\(z\) axis). The diagram shows a band-gap (white-colored) extending from 50 to 60 THz, depending on the value of the propagation constant. The lines running through the band-gap are guided modes introduced by the longitudinal defect. In particular, the red line is a TM\({}_{01}\)-like mode, with an axial component suitable for particle acceleration. The light line (black dash-dotted curve) intercepts the guided TM\({}_{01}\)-like mode at the frequency of 60.057 THz: this synchronous point is chosen as a working point to obtain the acceleration of relativistic electrons.  
Figure 6a shows the electric field vector plot of the accelerating TM\({}_{01}\)-like mode along a \(xy\) structure slice at an arbitrary \(z\) position along the accelerating waveguide axis. For the same \(z\) position, Figure 6b shows the electric field components along the \(x\) axis. As expected, on axis (\(x=0\)), the transversal components are almost equal to zero while the longitudinal component has high amplitude. Finally, Figure 7 shows the phase advancement of the TM\({}_{01}\)-like mode along the accelerating hollow-core waveguide at the operating frequency. It is possible to conclude that the phase is suitable to accelerate a speed-of-light particle since its advancement follows the light line.  
Figure 5: Electric field components along the accelerating waveguide axis at \(f_{0}=60.074\) THz, for a length of \(7d=14.574\) μm. The average field, obtained with an input power \(P_{0}=1\) kW, results in \(E_{0}=0.726\) GV/m.  
Figure 6: (**a**) Electric field vector plot of the accelerating TM\({}_{01}\)-like mode along a \(xy\) structure slice for an arbitrarily chosen \(z\) position. (**b**) Electric field components along the \(x\) axis (blue dashed horizontal line of **a**) for the same \(z\) position, at \(f_{0}=60.074\) THz and with \(P_{0}=1\) kW. Green dashed vertical lines indicate the defect width, \(w_{d}\).