## 4 Figures of Merit  
Some considerations can be made by analyzing the field components shown in Figure 5a. It is useful to define an average axial electric-field amplitude \(E_{0}=V_{0}/L\), where \(V_{0}\) is the RF voltage along the accelerating length, given by \(V_{0}=\int_{-L/2}^{L/2}E_{z}(z)dz\), with \(E_{z}(z)\) being the axial, accelerating electric field and \(L\) some characteristic length [20]. Usually, a constant gain per cell is desirable, and the cavity cells are tuned to produce a uniform field distribution along the LINAC; a parameter to evaluate this property is the so-called field _flatness_. In real scenarios, the field can tilt due to various types of distortions, and these undesirable deviations increase as the total cavity length increases. It is then of interest to evaluate the field flatness in order to quantify the field tilt. The flatness is usually calculated by examining the average electric field along the chain of LINAC cells, which is evaluated cell by cell. In our case, for the field flatness calculation, we can choose the woodpile lattice constant \(d\) in place of the cell length. In fact, as clearly seen in Figure 2b, the accelerating channel is composed of several woodpile periods (seven in the case at hand), and each period effectively resembles an accelerating cell with a local electric field. The field flatness for the presented structure results in about \(\pm 14\%\) and is shown in Figure 8. The accelerating field flatness requirements depend on the LINAC application and can be improved through several techniques, such as a fine tuning of the coupling section and by a local modification of the single cells transversal dimensions, such as the cell radius for the metallic LINACs that employ cylindrical cavities. It is useful to make some final calculations to estimate some typical LINAC parameters of merit for the case at hand. The particle energy gain, \(\Delta W\), can be expressed in terms of  
Figure 8: Electric field flatness plot vs. cell (period) number. Blue-filled dots indicate the cell-by-cell average electric field value \(E_{0}\) divided by the average of all \(E_{0}\) values for the seven cells. Black dashed lines indicate, for each cell, the length on which the average value is taken. For the presented structure, the flatness is about \(\pm 14\%\). Figure 7: Black line: phase of the longitudinal (accelerating) component, \(\angle E_{z}\), vs \(z\). Blue dashed line: light line of a relativistic particle. The operating frequency is \(f_{0}=60.074\) THz. It can be seen that the phase advancement along the structure is compatible with a particle moving at speed of light a relativistic particle “sees” a constant phase while traveling along the structure. the _accelerating gradient_, \(E_{0}T\), as \(\Delta W=qE_{0}LTcos\phi\) ([20] p. 34), where \(L=7d=14.574\)\(\upmu\)m is the accelerating channel length, \(T\) is the so-called transit-time factor, defined as  
\[T=\frac{\int_{-L/2}^{L/2}E_{z}(z)\cos(\omega t(z)-\phi(z)-\phi_{0})dz}{\int_{-L /2}^{L/2}E_{z}(z)dz} \tag{1}\]  
and \(\phi(z)=\angle E_{z}(z)\) is the accelerating field phase. It is usual to fix the phase \(\phi=0\) when the particle is at the field crest position so that maximum energy gain occurs when \(\phi=0\), which is often the choice for relativistic electrons. In our case, \(T=0.9463\approx 1\), since, as is visible from Figure 7, the phase profile is linear and well matched with the light line, although it has a small unavoidable modulation. In order to have a figure of merit independent from the input power, \(P_{0}\), it is common to define the characteristic interaction impedance, which is a measure of the effective accelerating field acting on the electrons \(E_{\rm acc}=E_{0}T\), for a given total electromagnetic power, \(P_{0}\), flowing into the system ([21] Equation (10)) ([22] p. 13):  
\[Z_{c}=\frac{(E_{0}T\lambda)^{2}}{P_{0}}\quad[\Omega] \tag{2}\]  
For the 14.574 \(\upmu\)m long structure presented above, we obtain a characteristic interaction impedance \(Z_{c}=11.4\) k\(\Omega\).