Traveling Wave Tube Eigenmode Solver for Hot Slow Wave Structure Based on Particle-In-Cell Simulations

Traveling wave tube (TWT) amplifiers are the devices of choice for several decades for radar and satellite communications applications when high power is required and also when reliability is important, like in satellite communications [1, 2]. TWTs are increasingly important to generate high power at millimeter wave and terahertz frequencies [3, 4, 5, 6, 7] where the current technology based on solid state devices struggles to generate even low power levels. An important component of the TWT is the slow wave structure (SWS), that is a guiding structure where the speed of the electromagnetic (EM) wave is reduced to match the speed of the beam electrons leading to energy transfer from the electron beam to the EM wave [8, 9]. An important mechanism for the energy transfer is the synchronization of the phase velocity $v_{ph}$ of the EM wave in the SWS with the average speed of the electrons $u_{0}$. Furthermore, the EM wave needs to have a longitudinal electric field component $E_{z}$ to interact with the electron beam to form electron bunches. Therefore the electron beam is modulated in terms of electron velocity and electron density forming a “space-charge wave” that is synchronized with the EM wave. The modes in the “hot” SWS are complex hybrid physical phenomena involving both the space-charge wave and EM field, i.e., each mode is made of these two components and may have a complex wavenumber.

The study of the “cold” eigenmodes in the SWS, i.e., the EM modes that exist without considering the interaction with the electron beam, is important to establish the onset of the synchronization condition between the electron beam’s space-charge wave and the EM wave in the SWS. Denoting with $v_{ph}$ the phase velocity of the EM mode in the cold SWS and with $u_{0}$ the average velocity of the electron beam, the initial synchronization condition is $v_{ph}\approx u_{0}$. There are various EM solvers in commercial software packages that can be used to find the dispersion diagram of the EM modes in the cold SWS. Some of the most famous commercial eigenmode solvers are provided by finite element method-based packages by Ansys HFSS and DS SIMULIA (previously known as CST Microwave Studio). Often, eigenmode solvers work under the approximation of a lossless SWS, i.e., the modes are found in a closed metallic waveguide with perfect electric conducting walls.

The interaction of an EM wave with the electron beam results in hybrid (EM+space-charge wave) modes whose phase velocity is different from the phase velocity of the cold EM eigenmode, hence the “hot” eigenmodes, i.e., the eigenmodes in interactive system, have a dispersion diagram that is different from the one of the cold EM modes, especially in the frequency region where $v_{ph}\approx u_{0}$. Although the study of the EM eigenmodes in a cold SWS is very important, the main operation of TWTs depends mainly on the eigenmodes of the hot SWS. Note that the modes of the interactive system have complex valued wavenumbers accounting for possible gain coming from the electron beam and losses in the metallic waveguide.

The modeling and design of TWTs are carried out by either theoretical models or simulations. For about the past seventy years, Pierce’s classical small signal theory has been used for the modeling and design of TWTs [10, 11]. Pierce describes the dispersion relation for hot SWS as cubic polynomial [10]. Other studies have been provided in the literature to theoretically model TWTs as in [12, 13, 14, 15, 16]. Although the theoretical models are considered as good tools for initial design of TWT, they are inaccurate and the actual performance of TWTs is assessed by performing accurate simulations.

Advances in 3D electromagnetic simulation software make it possible to accurately model and simulate complex electromagnetic structures accounting for the interaction with an electron beam. Most of the simulation and design work of TWTs is carried out using particle-in-cell (PIC) codes. A PIC solver is a self-consistent simulation method for particle tracking that calculates particles trajectories and EM fields in the time-domain [17, 18]. Some commercial computational software provides solvers based on PIC code that allows to accurately simulate driven-source problems of TWTs taking into account all physical aspects of the problem. Although PIC solvers are currently the most accurate tools to model TWTs, they require a lot of simulation time and computer memory size, especially for TWTs with large lengths, therefore, sometimes running multiple PIC simulations may not be the most practical way to start the optimization process. Several methods were proposed in literature as an intermediate step between simple qualitative theoretical models and time-consuming accurate PIC simulations. Some of thesemethods are based on merging data extracted from a 3D cold SWS electromagnetic simulator with those of a particle solver, assuming the SWS is modeled as 1D transmission line and the beam is modeled as pencil beam with infinitesimal cross-section area. These methods are used in the well known codes known as CHRISTINE [19, 20], TESLA [21]and MUSE [22]. Other methods based on merging results from 3D EM simulations and particle solver simulations are also proposed in [23, 24, 25, 26, 27].

In this paper, we present a method to model TWTs by finding the equivalent transfer matrix of a unit-cell based on accurate PIC simulations of a SWS with a relatively small number of unit-cells, which is not very time consuming. The advantage of extracting the SWS unit-cell transfer matrix is not only to infer the characteristics of the hybrid modes of the EM-beam interactive system (the main goal of this paper) but also to predict the behavior of longer structures without need to simulate it using PIC (left to future investigations). To date, no commercial software provides an eigenmode solver for hot SWSs taking into account the interaction with the electron beam and losses and the accurate geometry of the SWS.

The method shown in this paper is based on finding the unit-cell transfer matrix through the interpretation of data extracted from PIC simulation for relatively short SWSs. We show that the method is used to approximately calculates the complex wavenumbers of the hybrid modes supported in a hot SWSs. The method also provides the contribution of the EM wave and spatial charge wave to each specific hybrid mode associated to each of the complex modal wavenumbers. The proposed solver is based on accurate PIC simulations of finite length hot SWSs and takes account of the precise SWS geometry, materials’ EM properties, electron beam cross-section area, confinement magnetic field and space charge effect. The proposed solver is based on monitoring both the EM fields and electron beam dynamics in each unit cell and then find the best transfer matrix that describe how the hybrid EM field-electron beam propagates along the TWT as shown in Fig. 1. Some simplifying assumptions are made, like linearity of EM and space charge waves excited in the TWT, and the electron beam is assumed not to lose energy while it travels along the TWT as discussed in the next section. Notably, the method provides the dispersion diagram of complex valued wavenumbers versus frequency of all the hybrid modes supported in the interactive hot SWS.

We demonstrate the performance of the proposed eigenmode to be periodically repeated solver by considering, as an illustrative example, the TWT made of a helix SWS with period $d$ consisting of $N$ unit-cells shown in Fig. 1. The input and output radio frequency (RF) signals of the structure are defined as Port 1 and Port 2. However, in the following both will be used as inputs in order to find the complex wavenumbers of the hybrid modes. It is important to point out that the following technique is general and can be applied to any kind of SWS, hence not only to helix-based SWSs. The PIC solver simulates the complex interaction between the guided EM wave and electron beam using a large number of charged particles and it follows their trajectories in self-consistent electromagnetic fields computed on a fixed mesh.

We demonstrate how the technique described in the previous section is applied considering an illustrative example made of a C-Band TWT amplifier described in [41] and shown in Fig. 3. Such TWT amplifier operates at 4 GHz and uses a solid linear electron beam with radius of 0.63 mm, with dc kinetic voltage of 3 kV and dc current of 75 mA. The axial dc magnetic field used to confine the electron beam is 0.6 T. The TWT uses a copper helix tape with inner radius of 1.26 mm and period of $d=$0.76 mm, and the metallic tape has $0.175$ mm thickness and $0.325$ mm width. The metallic circular waveguide has radius of $3$ mm and the three supports that physically hold the helix are made of Anisotropic Pyrolytic Boron Nitride (APBN) with relative dielectric constant of 5.12 and and with width of $0.7$mm. The numerical simulation consists of 15 periods of the structure. We show the numbering scheme used to define the voltages and currents representing the EM wave and the space-charge wave at the begin and end of each unit-cell in Fig. 3. While the whole structure is considered in PIC simulations, when we apply the proposed method we do not consider the first and the last state vectors (i.e., at the beginning and end of of the first and last unit cells) because the begin and the end of the SWS are close to the coaxial waveguide input and output ports, and higher order modes due to the transformation of the TEM wave in the coaxial waveguide into the SWS modes would influence the calculation of the unit-cell transfer matrix.

First, we study the system eigenmodes at single frequency by exciting Port 1 and Port 2 with 10 Watt and 5 Watt, receptively, at 4 GHz. Note that indeed, even if the actual TWT operation involves feeding from one side only, the scheme explained in the above section may need a SWS feed from both the left and right ports to be sure we excite all the modes. It is important to mention that changing the power levels at Port 1 and Port 2 shall not significantly affect the resulting eigenmode solution as long as the power levels are not too high to involve non-linear regimes. The total number of charged particles used to model the electron beam in the PIC simulation is about $10^{6}$ whereas the whole space in the SWS structure is modeled using $8\times 10^{5}$ mesh cells. Once steady state is reached, the time-domain state vector $\mathbf{\mathbf{\boldsymbol{\psi}}}_{n}(t)$ is monitored at each $n^{th}$ unit-cell. Figure 4 shows the four components (3) of the state vector at the $3^{rd}$, $8^{th}$ and $13^{th}$ unit cells in the time interval between $79T$ and $80T$, where $T=0.25$ ns is the time period at 4 GHz. The phasor-domain representation (4) of the state vector $\mathbf{\Psi}_{n}$ is calculated using (7) with $t_{ref}=79T=19.75$ ns.

The four wavenumbers of the eigenmodes of the interacting SWS system are shown in Fig. 5 based on results from Eq. (9) leading to the four red crosses, and from Eq. (13) leading to the various blue dots. The scattered blue dots represent 75 sets of four complex wavenumbers associated with 75 sets of transfer matrices obtained from Eq. (13) using 75 combinations of $q,i,j$ and $k$ to give the highest 75 determinants of the matrix $\mathbf{w}_{1,qijk}$ out of the all 330 combinations. It is important to mention that the solutions associated with the rest 255 combinations $q,i,j$ and $k$ were ignored because they result in almost singular matrices $\boldsymbol{\mathrm{w}}_{1,qijk}$ and $\boldsymbol{\mathrm{w}}_{2,qijk}$.

The results in Fig. 5 show a good agreement between the red crosses that represent the four wavenumbers obtained from the eigenvalues of $\mathbf{T}_{u,best}$ where $\mathbf{T}_{u,best}$ is the best-approximate solution of the overdetermined system in (9) and the blue dots that represent the eigenmodes of various estimates of $\mathbf{T}_{u,qijk}$ obtained from solutions of (13). It is important to point out that the small deviations between the complex wavenumbers obtained from different solutions is due to non-idealities resulting in having non-identical unit-cells along the structure. This may happen due to the electron beam non-linearity and the change of the beam kinetic voltage along the TWT, in addition to the errors due to finite mesh and finite number of charged particles used to model the TWT dynamics. The resulting eigenmodes are qualitatively in good agreement with the expected description using the theoretical transmission line-based Pierce model [11] that basically says that there exist three wavenumbers with positive real part, and one of them has positive imaginary part which describes the amplification of the EM wave along the SWS.

The wavenumber-frequency dispersion describing the eigenmodes in the hot SWS is determined by running multiple PIC simulations at different frequencies and then determining the transfer matrix of the unit-cell at each frequency using Eq. (9), i.e., the result shown by the red crosses. In other words we repeat the red-cross results shown in Fig. 5 at various frequencies. The dispersion diagram of the four modes in the hot EM-electron beam system is shown in Fig. 6 (solid lines) using 42 frequency points (42 PIC simulations). The dashed red and blue lines represent the space-charge wave (i.e., the beam line) and EM modes when they are uncoupled (i.e., in the “cold” case). The dispersion of the EM mode in the cold SWS (dashed black) is found by the finite element method-based eigenmode solver implemented in CST Studio Suite by numerically simulating only one unit-cell of the cold helix SWS.

The black solid line in Fig. 6 is the hybrid mode propagating with $\mathrm{Re}(k)$<0, i.e., in opposite direction of the beam flow and basically no beam-EM interaction occurs, and it is made mainly of EM field as also explained later on in Fig. 7. Indeed, as compared to the cold EM modes, the solid-black line of the hot SWS simulation is superposed to the dispersion line of the cold EM mode propagating in the negative $z$-direction.

The three eigenmodes with wavenumbers with positive real part (solid red, green, blue lines) are the eigenmodes affected by the interaction between the electron beam’s space-charge wave and the EM wave propagating in the same direction. The solid-red curve in the dispersion relation has $\mathrm{Im}(k)$>0 and represents the growing eigenmode that causes the amplification of the EM wave. The imaginary part of the growing mode starts to decreases with increasing frequency because the difference between the speed of the space-charge wave and that of the EM wave in the SWS (when we consider them uncoupled) increases. It is important to mention that a single PIC simulation could also be investigated to find the dispersion relation when using a moderately wide-band gaussian pulse at the input ports, however it would require excessive post processing to be able to decompose each tone behavior and it does not fully account for steady state regime resulting from the interaction with the electron beam that would otherwise includes also non-linear effects.

The hybrid modes describing the EM-charge wave interaction are further understood by looking also at their eigenvectors, besides their wavenumbers. Eigenvectors satisfy Eq. (5) and are the same used in the aforementioned discussed method to find the transfer matrix. The eigenvectors components in (4) describe the relative weight of the EM wave ($V$, $I$) versus the space-charge wave $(V_{b}$, $I_{b})$, for each of the four modes $\mathbf{\Psi}^{\mathrm{mode}m}$, with m = 1,2,3,4. The relative strength of the EM wave and space-charge wave in each eigenmode is determined in Fig. 7 by calculating the ratio between between the ac electron beam current and the EM mode equivalent transmission line current. The ratio $|I_{b}/I|$ for each of the four eigenmodes is shown using colors consistent with those in Fig. . The figure verifies that the eigenmode with EM wave propagating with $\mathrm{Re}(k)$<0 (solid-black line), in the direction opposite to the electron beam flow does not interact with the electron beam since the beam ac current component associated to this mode is vanishing. The growing eigenmode ($\mathrm{Im}(k)$>0), red line, is the one responsible of EM amplification and it shows a good EM-charge wave interaction from 3.6 GHz to 4.2 GHz which is in agreement with the bandwidth specification of the amplifier given in [41]. Indeed, the operation of TWT as an amplifier depends on having a spatially growing mode (along the SWS) and the only mechanism to have a growing mode is via the coupling to the electron beam that gives back power to the EM wave after electron bunching is formed. This plot indeed shows that the red mode is the result of synchronization and generate a very strong electron beam.

A synchronization study is performed by observing the modes’ complex wavenumber when sweeping the beam dc voltage. In Fig. 8 we show the wavenumbers for the three hot eigenmodes with positive real part (red, green, and blue curves) when changing the beam dc kinetic voltage $V_{0}$ while the dc beam current is maintained at 75 mA, at a constant frequency of 4 GHz. Solid lines represent the three eigenmodes in the interactive (i.e., hot) SWS, whereas the dashed lines represent the beam line $\beta_{0}=\omega/u_{0}$ (red dashed) and the cold EM mode with positive wavenumber (i.e., in the cold SWS). Since the frequency is fixed the cold EM mode has a fixed wavenumber $\beta_{ph}=$999.8 $\mathrm{m}^{-1}$ and it is described by the vertical black-dashed line. The phase velocity of the EM mode in the cold SWS is $v_{ph}=\omega/\beta_{ph}=0.084c$ since $\omega=2\pi(4\times 10^{9})$ rad/s, where $c=3\times 10^{8}$m/s is the speed of waves in free space. Synchronization occurs approximately in the region where the two dashed lines, beam line and EM-wave line, have the same wavenumber, i.e., the same phase velocity, which happens just above 1.8 kV. The TWT amplification factor, mainly represented by the positive imaginary part of the red eigenmode, shows a peaking around synchronization point of $V_{0}=$1.8 kV which is corresponding to a beam speed $u_{0}=\sqrt{2\eta V_{0}}=0.084c$, where $\eta=e/m$ is the charge to mass ratio of the electron.

At lower or much higher dc voltage with respect to the synchronization point would make the beam-EM interaction is weak. This is also understood by noting that the gain starts to decay away from the interaction voltage region and the hot modes (resulting from the interaction) tend to overlap with the cold modes (without interaction) away from such region. In particular at low frequency the solid green curve tends to overlap with the cold EM mode, whereas at high frequency the solid blue line tends to overlap with the cold EM mode. The two solid red and blue curves at low dc voltage tend to overlap with the beam curve (red dashed), whereas the two solid red and green curves at high voltage tend to overlap with the beam line (red dashed): therefore these hot modes tend to overlap with the beam line (non-interactive beam) at high and low voltage despite some shifts which may be due to space charge effects. Near the synchronization point, the splitting of the solid green and blue curve (versus the red curve that does not split) represent the strong interaction between the electron beam and the EM field.

In Fig. 8 we show the complex plane mapping of the wavenumbers for the three hot eigenmodes with $\mathrm{Re}(k)>0$, shown in Fig. 8, varying the electron beam dc voltage $V_{0}$ (i.e., the speed of the electrons $u_{0}$). The black dots represent the interactive modes wavenumbers when the beam dc voltage is $V_{0}=1.8$ kV which results in an electron beam with average speed very close to the EM mode in the cold SWS, i.e., close to the synchronization point $u_{0}\approx v_{ph}$. The complex wavenumber location of the three interactive modes shown in Fig. 8 is in agreement with the three-wave theory of the Pierce model [33, 11] around the synchronization point. When $u_{0}\approx v_{ph}$, there are three modes with $\mathrm{Re}(k)$>0: two of them are waves that are slower than the electron beam average speed $u_{0}\approx v_{ph}$, and among these two, one wave is growing in the beam direction while the other one is decaying. The third mode is basically an unattenuated wave that travels faster than the beam-average speed $u_{0}\approx v_{ph}$.

The power flow for each mode of the hot (i.e. interactive) SWS is written in the form of $P_{m}(z)=P_{0,m}e^{2\mathrm{Im}(k_{m})z}$ [42], where $P_{0,m}$ is the initial amount of power carried by the same mode at $z=0$. We define the power gain resulting from the interaction between the EM and space-charge wave for as $G_{p,m}(z)=P_{m}(z)/P_{0,m}=e^{2\mathrm{Im}(k_{m})z}.$ Thus gain growth rate for each mode is $20\log(e)\mathrm{Im}(k_{m})$ dB/m. In Figure 9 we show the gain growth rate for the three interactive modes (those with $\mathrm{Re}(k_{m})>0$) versus electron-beam charges average speed $u_{0}=\sqrt{2\eta V_{0}}$ (normalized by the cold SWS phase velocity $v_{ph}=\omega/\beta_{ph}$ , where $\beta_{ph}$ is the wavenumber of the EM mode in the cold SWS). The figure shows that there is an optimal point for the interaction (where $u_{0}/v_{ph}\approx 1.18$) which is very close to synchronization point where the power transfer from the kinetic energy of the electron beam into RF power in the SWS is maximum, as discussed in [43, 10]. The figure also show that the largest power transfer, from the kinetic energy of the electron beam into RF power, occurs when the average speed of electron beam charges is roughly above the speed of the cold EM wave in the SWS.

In Appendix A we show the effect of changing the voltage definition and the SWS length on the eigenmode calculations for the same helix SWS considered in this section. There, we define the voltages representing the EM waves as the potential differences between each two successive helix loops. Results are qualitatively in agreement with what described here, and only small quantitative differences are observed.

We demonstrate the utility of the proposed eigenmode solver method to find the eigenmodes in the hot serpentine SWS operating at a millimeter wave band. The interaction with the electron beam is periodic and not uniform as for the case of the helix SWS. Serpentine SWSs have recently gained a lot of interest due to the growing importance of millimeter wave and terahertz frequencies in modern applications and also due to the advancement of fabrication technologies such as LIGA (LIthographie, Galvanoformung, Abformung). As an illustrative example, we use the same geometry of serpentine SWS discussed in [44, 45], as shown in 10. The serpentine waveguide is made of copper and has rectangular cross-section of dimensions $a=1.9$ mm and $b=0.325$ mm, bending radius of $0.325$ mm (radius at half way between inner and outer radii), straight section length of 0.6 mm, beam tunneling radius of $0.175$ mm. The TWT comprises 13 unit-cells . An electron beam with dc voltage $V_{0}=20$ kV is used such that the synchronization occurs with a forward EM wave leading to amplification. In Fig 10 we show the beam line $\beta_{0}=\omega/u_{0}$ (red line) where $u_{0}=\sqrt{2\eta V_{0}}=0.28c$, and the dispersion diagram of EM modes in the cold SWS (black line). Synchronization, $v_{ph}\approx u_{0}$, between the electron beam and the forward EM wave occurs at frequencies centered at $f=88$ GHz.

Figure 10 shows the setup used for PIC simulations. We consider the electron beam to have a radius of 0.13 mm, a dc current of $0.1$ A and an axial confinement dc magnetic field of 0.6 T. Our goals is to obtaining the dispersion of the hot serpentine SWS, i.e., the complex wavenumber of the hybrid modes that account for the interaction between the electron beam and the EM wave. In our method we excite the SWS from Port 1 with 10 Watts and from Port 2 with 5 Watts. An amplifier has the input at one port and the output at the other one, but here we want to excite the supported eigenmodes sufficiently to be observed in the calculations. The serpentine supports the $\mathrm{TE_{10}}$ mode which is the only one propagating in the rectangular waveguide. Since the serpentine waveguide does not support a TEM mode, voltage and current cannot be uniquely defined. We use the equivalent representation in [28, 30, 29] that models the waveguide as a transmission line with equivalent voltage and current. Following the derivations in [28, 30, 29] for the $\mathrm{TE_{10}}$ mode in a rectangular waveguide, the transverse fields are written as

Using (18), the discrete voltages and the currents that represent the EM state at different rectangular cross-sections of the serpentine waveguide are found as

where $E_{yn}$ and $H_{xn}$ are the traverse electric and magnetic fields calculated at the center of the rectangular ($x=a/2$ and $y=b/2$) waveguide cross section as shown in the inset in Fig. 10 and they are calculated at the unit cells boundaries shown in Fig. 10.

We start by studying the eigenmode wavenumbers in the interacting SWS system at constant frequency $f=88$ GHz, which is very close to the synchronization point where $u_{0}=v_{ph}$. The four complex wavenumbers of the hybrid modes are shown in Fig. 11 based on results from Eq. (9) leading to the four red crosses, and from Eq. (13) leading to various blue dots. The scattered blue dots represent 37 sets of four complex wavenumbers associated the largest 37 determinants of the matrix $\mathbf{w}_{1,qijk}$ out of the all 126 combinations. The blue dots cluster around the four red crosses, as expected. It is important to mention that we ignored the state vectors of the first and the last two unit cells to generate our results because they may involve high order modes that affect the results. The complex wavenumbers locations of the three interactive modes (those with $\mathrm{Re}(k)$>0) shown in Fig. 11 are in good agreement with predictions of the three-wave theory of the Pierce model [33, 11].

In Fig. 11 we show the modal dispersion relation accounting for the EM-beam interaction in the serpentine SWS using 21 frequency points. The hot dispersion diagrams are obtained using the best-approximate solution $\mathbf{T}_{\mathit{u,best}}$ of the overdetermined system in (9). As in the previous section, dashed lines represent the two uncoupled systems: the beam line (dashed red) and the EM mode in the cold SWS (dashed black). The electron beam has a dc voltage of $V_{0}=20$ kV so the electron beam interacts with the forward EM wave resulting in an eigenmode with positive imaginary part leading to TWT amplification (red curve). We show only the three modes with positive real wavenumber. One wavenumber has positive imaginary part (solid red curve), which is responsible for amplification, whereas the other two modes resulting from the interaction are decaying and unattenuated modes, in agreement with the Pierce model [11]. The gain per unit-cell associated to the $m^{th}$ mode is defined as

which is equivalent to $20\log(e)\mathrm{Im}(k_{m}d)$ dB. The imaginary part of the wavenumber of the amplification mode (solid red) is almost constant and equal to$\mathrm{Im}(kd)\approx 0.027\pi$ in the frequency range from 87 GHz to 89 GHz shown in Fig. 11, because the phase synchronization condition is almost satisfied for the considered frequency band, i.e., $\beta_{ph}\approx\beta_{0}$ over the shown band (relative band of $2.2\%$ around $88$GHz) in this example. Thus, the gain per period resulting from the amplification mode (solid red) is $20\log(e)\times 0.027\pi\approx$ 0.74 dB in this frequency range which is close to the small-signal gain 1 dB reported in [45] that was obtained by simulating the serpentine TWT amplifier at 90 GHz.

A repeatability study is performed by obtaining the dispersion relation for the serpentine hot SWS using two different numbers of unit cells (all other parameters are kept the same for the two TWT configurations as previously described). Indeed, the transfer matrix of the hot SWS unit-cell $\mathbf{T}_{u}$ should not be a function of the used number of unit-cells, therefore, the dispersion relation obtained using different numbers of unit cells should not be affected. In Fig. 12 we show the complex wavenumberscalculated from the unit-cell transfer matrix $\mathbf{T}_{u}$ estimated using data from PIC simulations of two SWSs with $N=13$ (as in the previous example) and $N=15$ unit cells. In each case, we plot 37 sets of four wavenumbers obtained from 37 estimated transfer matrices (distinct determined solutions): blue and green dots represent the cases with $N=13$ and $N=15$ unit cells, respectively. The plotted sets are the ones associated with the highest 37 determinants of the matrices $\mathbf{w}_{1,qijk}$ used to determine the unit-cell transfer matrices $\mathbf{T}_{u,qijk}$ out of the all 126 combinations for the case with $N=13$ unit cells and 330 combinations for the case with $N=15$ unit cells . The red and black crosses represent the wavenumbers obtained from the best-approximate solution of the overdetermined system for the cases with $N=13$ and $N=15$ unit cells, respectively. The clustering of wavenumbers at 88 GHz in Fig. 12 shows a good agreement between the two cases based on $N=13$ and $N=15$ unit cells. In Fig. 12 we compare the dispersion relation obtained based on the best-approximate solution of the overdetermined system (red and black crosses in Fig. 12): solid and dotted curves represent the dispersion obtained from simulating the hot SWS with $N=13$ and $N=15$ unit cells, respectively. The figure shows an almost negligible discrepancy between the dispersion diagrams obtained using two SWS lengths.

A TWT eigenmode solver to determine the complex wavenumbers of the hybrid eigenmodes in hot SWSs (i.e., accounting for the interaction of the electron beam and the EM wave) has been demonstrated using a novel approximate technique based on data obtained from PIC simulations. The technique has been able to predict the growing mode’s complex wavenumber in both TWTs studied here based on: (i) a SWS made of a circular waveguide with a helix working in the GHz range; (ii) a SWS made of a serpentine waveguide for millimeter wave amplification. The method is based on elaborating the data obtained from time domain PIC simulations, hence accounting for the precise waveguide geometry, materials, losses, and beam space charge effects. We believe that the proposed technique is a powerful tool for the understanding and the design of TWTs amplifiers.

In the present investigation we have focused on determining the complex wavenumbers of the hybrid EM-beam modes, but the same technique can also be used to estimate the performance of longer TWTs based on simulations of shorter TWTs, i.e., it can be used in the design and initial optimization of TWTs. The determination of wavenumbers and eigenvectors of all the hybrid modes supported in a TWT amplifiers discussed here can also be useful to study mode degeneracy conditions in hot SWSs as those investigated in [37, 38, 39].

This material is based upon work supported by the Air Force Office of Scientific Research award number FA9550-18-1-0355 and award number FA9550-20-1-0409. The authors are thankful to DS SIMULIA for providing CST Studio Suite that was instrumental in this study.