## 2 Design Approach  
The targeted design specification of the proposed multiple beam klystrons is given in Table 1. Targeted nominal bandwidth has been taken as 500 MHz because similar bandwidths are being considered for 5G systems envisaged in the Ka-band frequency range. The total beam current (therefore individual current of beamlets) has been kept in a range such that the required focusing magnetic field can be obtained by using permanent magnets, therefore avoiding complexity and requirement for magnet power supply. RF performance of a klystron, such as its gain, bandwidth, efficiency, etc., depends on the RF section's design. Because the RF section is an ensemble of RF cavities joined together by the drift tubes at particular distances, the design of the RF section involves the extraction of RF cavity parameters and the drift tube dimensions. The approach followed for design of the proposed device is given in Figure 1.  
The energy exchange between the accelerated electron beam and the RF signal takes place at the cavity gap. There are conflicting requirements in deciding the gap diameter and spacing. For efficient beam-wave interaction, both parameters should be small compared with electron transit time through the gap. However, very small gap diameter imposes severe complications in electron gun and magnetic focusing system design. Moreover, the small gap spacing increases the gap capacitance resulting in a reduced gain-bandwidth product. For small gap spacings, there are possibilities of voltage breakdown and resultant beam loss. Therefore, both parameters are optimized to achieve a compromised value of the same, keeping the set specification in mind. Moreover, proper coupling irises (or loops) are required for input and output cavities so that the device will have good matching at the input to the RF driver to the load at the output port. The analytical approach [24; 25; 27; 34; 35] used for designing for expression required to design the RF section is presented in the following section. Ladder-shaped resonant cavities have been considered for larger shunt impedances (with extended interaction configuration), and multiple numbers of beams have been considered for low operating voltage (and individual beamlet current) and microfabrication and 3D fabrication compatibility. Initial cavity dimensions (height and width) have been determined using the conventional analytical formula [25; 27; 35; 36; 37; 38]. \[f_{mn}=\frac{c}{2\sqrt{\mu_{r}\epsilon_{r}}}\sqrt{\left(\frac{m}{a}\right)^{2 }+\left(\frac{n}{b}\right)^{2}} \tag{1}\]  
where resonant frequency is denoted by \(f_{mn}\) of \(\mathrm{TE_{mn}}\) or \(\mathrm{TM_{mn}}\) mode of a resonator with sides \(a\) and \(b\). Light velocity in free space is represented by \(c\). The relative permeability and \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Specifications** \\ \hline Operating Frequency & 28.5 GHz \\ Bandwidth & 500 MHz (min) \\ Beam Voltage & 4 kV \\ Beam Current & 0.196 A \\ No. of beams & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design specification of proposed multiple beam klystron. Figure 1: Flowchart of RF section design procedure. permittivity of the medium are represented by \(\mu_{r}\) and \(\varepsilon_{r}\). It may be noted that TM mode has been considered where the electric field inside the gaps does not vary much in the beam propagation direction. For efficient RF amplification in the extended interaction cavity, different configurations of the operating mode are possible. Among these modes, the 2\(\pi\) mode of operation has been focused on because this mode generally provides maximum coupling between the RF signal and beam. The pitch length (\(p\)) and gap length (\(G\)) of the EIK cavities can be obtained using the Floquet theorem of periodicity. Phase velocity \(v_{p}\) of the space harmonics is presented by  
\[v_{p}=\frac{\omega}{\beta_{e}} \tag{2}\]  
where \(\omega\) = angular frequency, \(\beta_{e}\) = Beam propagation constant.