SECTION I

THE RELTRON is a high-power microwave source that velocity modulates an electron beam in its modulating cavity [1]. Velocity modulation divides the continuous beam into bunches, and the beam bunches are postaccelerated as they leave the cavity (Fig. 1). The beam that enters the modulating cavity is nonrelativistic as the anode cathode (A–K) gap is typically in the range of 100–7200 kV. In the postacceleration gap, the bunches get accelerated to relativistic velocities.

Electron overtaking occurs in the postacceleration gap as the faster electrons are accelerated a second time by the quasi-static electric fields in the postacceleration gap. Microwaves are extracted from the beam bunches as they enter the extraction cavity, causing the bunches to slow down before they are absorbed by the beam collector [2].

The reltron has attributes similar to a klystron. Both devices form beam bunches through the use of a modulating cavity and convert the beam bunches to microwaves through the use of an extraction cavity. Unlike the klystron, the reltron does not require an external magnetic field to guide the beam. The reltron's beam undergoes velocity modulation twice in the modulating cavity while the beam is velocity modulated only once in the klystron's cavity. In a klystron, the bunched beam drifts into the extraction cavity, while in the reltron, the beam bunches are postaccelerated toward the extraction cavity.

In the previous publication, Miller *et al.* [1] focused on optimizing beam bunching in the modulating cavity through the use of the $\pi/2$ mode and an optimum gap distance of $d_{\rm opt} = v/3.2f$ where $v$ is the electron velocity and $f$ is the frequency. Miller *et al.* [2] also formulated an analytic model of the extraction to determine the cavity and iris dimensions that will optimize microwave extraction from the beam bunches. By optimizing both the modulating and extraction cavities and adding the postacceleration gap, the reltron is able to achieve efficiencies of 40%–50%.

In this paper, we will focus on increasing the microwave power generated by the reltron. The general design methodology for reltrons or any electron beam microwave oscillators is to design for one mode and suppress other modes. This allows the modulating and extraction cavities to be optimized for the dominant mode and minimizes mode competition and mode hopping. When the cavities saturate, the reltron will radiate monochromatic microwaves with stable power levels. We propose an alternative design methodology which utilizes multiple modes instead of a single mode to increase the microwave power generated by the reltron. We design a side-coupled modulating cavity that resonates in the $\hbox{TM}_{010}$ and $\hbox{TM}_{110}$ modes with the desired frequency [3]. By tuning the resonant frequency of the extraction cavity, we can grow the $\hbox{TM}_{010}$ and $\hbox{TM}_{110}$ modes either separately or concurrently.

SECTION II

We examine the effects of having multiple modes by considering the beam modulation coefficient. A 1-D relativistic particle trajectory code was used to determine the modulation coefficient of the beam after it transits the reltron cavity [1]. For this paper, we will use an analytic equation that describes the beam current and examine the harmonic components within the beam.

When the velocity modulation is large [4], the beam will flow in the ballistic regime [5]. We assume that the electron enters the modulating cavity with initial velocity $v_{0}$ at time $t_{0}$. It crosses the first gap at time $t_{1}$ and crosses the second gap at time $t_{2}$. By conservation of energy, the electron will leave the modulating cavity with velocity [6] TeX Source $$v(t_{2}) = v_{0}\sqrt{1 + {MV_{\rm gap} \over V_{\rm beam}}\sin(\omega t_{2} - k_{z}z)}\eqno{\hbox{(1)}}$$ where $M = {\rm sinc}(\omega d_{\rm gap}/2v_{0})$ is the beam–wave coupling coefficient, $V_{\rm gap}$ is the voltage across the gap, $V_{\rm beam}$ is the beam voltage, $\omega$ is the radian frequency, and $k_{z}$ is the longitudinal wavenumber. The velocity allows us to express the transit phase of the electron. Differentiating the transit phase with respect to $t_{2}$, we obtain the ballistic modulating current as the beam leaves the cavity TeX Source $$I_{B} = {I_{0} \over \left[1 - X\cos(\omega t_{2} - k_{z}z)\right]}\eqno{\hbox{(2)}}$$ where $I_{0}$ is the initial beam current and the bunching parameter $X = MV_{\rm gap}\omega d_{\rm gap}/(2v_{0}V_{\rm beam}$).

When velocity modulation is small, the beam flows in the space-charge regime [7], [8]. Substituting $n_{2} = k_{z}n_{0}v_{2}/(\omega - k_{z}v_{0})$ from the continuity equation and $v_{2} = q_{e}E_{z}/(im_{e}(\omega - k_{z}v_{0}))$ from the equation of motion, we obtain the space-charge modulating current as the beam leaves the cavity [6] TeX Source $$I_{\rm SC} = q_{e}n_{0}v_{2} + q_{e}v_{0}n_{2} = {-iq_{e}^{2}E_{z}\omega n_{0} \over m_{e}(\omega - k_{z}v_{0})^{2}}\eqno{\hbox{(3)}}$$ where $q_{e}$ is the electron charge, $E_{z}$ is the longitudinal electric field, $m_{e}$ is the electron rest mass, and $n_{0}$ is the electron density. The current is phase shifted from the electric field by $-\pi/2$. If we include the radial coordinate in an axisymmetric beam, the nonrelativistic [7], [9] or relativistic [10] space-charge modulating current is shown to be TeX Source $$I_{\rm SC} = I_{z}(\omega, \omega_{p}, v_{0})E_{z0}(rk_{r})\cos\left(\omega t_{2} - {\pi \over 2} - k_{z}z\right) \eqno{\hbox{(4)}}$$ where $\omega_{p}$ is the nonrelativistic/relativistic plasma frequency, respectively, and $k_{r}$ is the radial wavenumber.

The modulating current in the ballistic regime and in the space-charge regime is periodic in $2\pi$. As such, we can express the modulating current as a Fourier series consisting of time harmonics TeX Source $$I_{2}(t_{0}) = {\rm Re} \sum_{-\infty}^{\infty}\left[i_{n}e^{in\left(\omega t_{2} - {\pi \over 2} - k_{z}z\right)}\right]\eqno{\hbox{(5)}}$$ where TeX Source $$i_{n} = {1 \over 2\pi}\int\limits_{-\pi}^{\pi}I_{2}(t_{0})e^{-in\left(\omega t_{2} - {\pi \over 2} - k_{z}z\right)}d(\omega t_{2}).\eqno{\hbox{(6)}}$$

By conservation of charge, the charge entering the cavity is equal to the charge leaving the cavity if we neglect electron interception by the grids. We express charge conservation as $I_{2}dt_{2} = I_{0}dt_{0}$. Furthermore, we know that the electron's phase is given by $\omega t_{2} = \omega t_{0} + k_{z}2d_{\rm gap}$, where $d_{\rm gap}$ is the distance between the grids in the modulating cavity. This allows us to express the cavity transit time as a function of the initial time that the electron enters the cavity $t_{2} = \alpha t_{0}$ where $\alpha > 1$. Performing the substitutions allows us to recast (6) in terms of $t_{0}$ and implement the integration TeX Source $$\eqalignno{i_{n} = &\,{I_{0} \over 2\pi}\int\limits_{-\pi}^{\pi}e^{-in\left(\omega\alpha t_{0} - {\pi \over 2} - k_{z}z\right)}d(\omega t_{0})\cr = &\,I_{0}\sin{\rm c}(n\alpha\pi)e^{in\left({\pi \over 2} + k_{z}z\right)}.&\hbox{(7)}}$$

The modulating current is now the sum of the dc current and the time-harmonic modulating currents TeX Source $$\eqalignno{I_{2}(t_{0}) = &\,{\rm Re}\sum_{-\infty}^{\infty} \left[i_{n}e^{in\left(\omega\alpha t_{0} - {\pi \over 2} - k_{z}z\right)}\right]\cr = &\,I_{0} + 2I_{0}\sum_{n = 1}^{\infty}\left[\sin{\rm c}(n\alpha\pi)\cos(n\omega\alpha t_{0})\right].&\hbox{(8)}}$$

The current modulation coefficient is defined as $MC_{n} = \vert I_{n} \vert/I_{0}$ where $I_{n}$ is the $n$th harmonic modulating current. From (8), we have TeX Source $$MC_{n} = 2\left\vert\sin{\rm c}(n\alpha \pi)\right\vert\eqno{\hbox{(9)}}$$ and $MC_{n}$ is dependent on $n$ and $\alpha$. The variable $\alpha$, in turn, is dependent on the transit time $t_{2}$. The sinc function has a series of local maxima/minima that decrease in magnitude with increasing $n$ and $\alpha$. $MC_{n}$ is independent of the phase angle so the modulation coefficient does not influence the position at which the current peaks. We can neglect very large values of $n$ because cavity losses increase exponentially as a function of $n$.

We plot the modulation coefficient of the first three harmonic components $MC_{1}$, $MC_{2}$, and $MC_{3}$ in Fig. 2. $MC_{1}$ reaches its first peak when $\alpha_{11} = 1.43$ and its second peak when $\alpha_{12} = 2.459$. $MC_{2}$ reaches its first peak when $\alpha_{21} = 1.23$ and its second peak when $\alpha_{22} = 1.735$. $MC_{3}$ reaches its first peak when $\alpha_{31} = 1.157$ and its second peak when $\alpha_{32} = 1.492$.

We plot the multiharmonic modulating current (8) in Fig. 3 for $n = 1$, 2, and 3 using $\alpha$ values at the peaks of $MC_{1}$, $MC_{2}$, and $MC_{3}$. We compare the multiharmonic current with the single-harmonic $(n = 1)$ modulating current. In all our plots, we set the frequency to 2.5 GHz and normalize the magnitude by $-I_{0}$.

The multimode modulating current has the highest amplitude for $\alpha_{21} = 1.23$. The current peak of the single mode is comparable to the peak of the multiple mode for $\alpha_{11} = 1.43$. We achieve larger current modulation when we extract the beam with multimodes at $\alpha$ values less than $\alpha_{11}$, which corresponds to shorter transit times. This gain in modulation reaches its apex at $\alpha_{21}$ and decreases for $\alpha$ values above or below $\alpha_{21}$. Extracting the beam at the first-order modulation coefficient maxima $\alpha_{11}$ results in the smallest current modulation when we include the higher order modes. All the multimode currents have steeper gradients $(di_{2}/dt)$ compared to the single mode. We observe the same results when we calculate the modulating current using $\alpha$ at the second peaks of $MC_{1}$, $MC_{2}$, and $MC_{3}$.

From the time-harmonic analysis, we conclude that we can increase current modulation by using multimodes and optimize mode summation to give the largest modulation. In the same manner, we can perform a spatial harmonic analysis and look for a series of modes that can increase current modulation to be greater than the modulation produced by one dominant mode.

SECTION III

We design a modulating cavity that is capable of supporting the fundamental and higher order modes. The main cavity has a radius of 4.44 cm and a gap distance of 1.27 cm. The coupling cavity has a radius of 3.17 cm and a height equal to the main cavity. We use the eigenmode solver in MAGIC [11] to find the resonant frequency of the $\pi/2$ modes in the cold modulating cavity. The $\hbox{TM}_{010}$ mode resonates at 2.49 GHz, and the $\hbox{TM}_{110}$ mode resonates at 3.50 GHz. The eigenmode solver also shows the electromagnetic field distribution inside the cavity [12]. We see the peak longitudinal electric field of the $\hbox{TM}_{010}$ mode on the center axis. The $\hbox{TM}_{110}$ mode's peak is off the center axis at a radius of 1.77 cm. This is shorter than the expected radius of 2.13 cm calculated using $x_{\max}$ where $x_{\max}$ is the variable at the maximum point of the Bessel function $J_{1}(x_{\max}) = 1.841$.

We also design a dual extraction cavity that can be tuned to match the resonant frequencies of the modulating cavity. The cavity consists of a rectangular waveguide with a height of 1.7 cm and a width of 7.21 cm and a circular drift tube with a radius of 2.6 cm and a height of 5.63 cm joined to the middle of the waveguide. We tune the resonant frequency by placing two pairs of inductive irises [2] equidistant from the center axis with the same window width. The extraction cavity is a three-port cavity. The entrance of the drift tube is the first port, and the two ends of the rectangular waveguide form the second and third ports. Microwaves are extracted from the ends of the waveguide, port two and port three. We simulate the cold extraction cavity using HFSS v12 to verify the resonance frequency.

Once we have the dimensions of both cavities, we simulate the reltron using MAGIC [11]. We drive the A–K gap by a 175-kV 1-ns rise time step voltage and the postacceleration gap by a 500-kV step voltage of equal rise time. A–K gap width is 1.52 cm. Fig. 4 shows a plot of the particle position captured in a MAGIC simulation.

All the metal conductors (colored brown in Fig. 4) are perfect electrical conductors, and the dielectric insulators (colored green in Fig. 4) are lossless with a relative permittivity of three. Total number of mesh cells is 491 040 cells, and we use the explosive emission model in MAGIC to model space-charge-limited electron emission from the cathode surface.

From our HFSS simulations, we obtain the iris placement which tunes the resonance frequency of the extraction cavity to match with the $\hbox{TM}_{010}$ mode. The iris is to be placed 6.98 cm from the center axis, and the iris window is 3.6 cm. In MAGIC simulations, we use a solid beam of radius of 2.54 cm. We plot the beam current at the cathode, as it leaves the modulating cavity and prior to entering the rectangular waveguide in Fig. 5. The modulating cavity current gets amplified as it transits through the postacceleration gap via kinematic bunching [5]. As a result, the extraction cavity current has a larger peak-to-peak magnitude than the modulating cavity current.

We plot the frequency spectrum of the $E_{z}$ field absorbed by the extraction port in Fig. 6. We see the $\hbox{TM}_{010}$ mode fundamental frequency at 2.395 GHz and its second harmonic at 4.771 GHz. The $\hbox{TM}_{110}$ mode is the third highest component at 3.591 GHz.

The average microwave power absorbed by each extraction port is 40.5 MW. Total microwave power generated by the reltron is 81 MW (efficiency of 23%).

We tune the iris placement and window width in HFSS simulations to match the resonant frequency of the extraction cavity with the $\hbox{TM}_{110}$ mode. At resonance, the iris placement is 3.17 cm from the center axis, and the iris window is 3.91 cm.

In MAGIC simulations, we use an annular beam with an inner radius of 1.9 cm and an outer radius of 3.17 cm. The peak-to-peak modulating cavity current is smaller for the $\hbox{TM}_{110}$ mode (Fig. 7) compared to that of the $\hbox{TM}_{010}$ mode. As a result, the peak-to-peak extraction cavity current is also smaller for the $\hbox{TM}_{110}$ mode.

The frequency spectrum of the modulating cavity current reveals the fundamental frequency at 3.579 GHz and the second harmonic of 7.157 GHz (Fig. 8). The three current dips in Fig. 7 correspond to 3.579-GHz oscillations, and the seven dips correspond to 7.157-GHz oscillations.

We plot the frequency spectrum for the $E_{z}$ field inside the modulating cavity and the extraction port in Fig. 9. Both cavities resonate with the $\hbox{TM}_{110}$ mode fundamental frequency at 3.579 GHz and the second harmonic at 7.157 GHz. Total microwave power generated by the reltron is 74 MW (efficiency of 17%).

SECTION IV

For the reltron to generate both $\hbox{TM}_{010}$ and $\hbox{TM}_{110}$ modes, we have to tune the extraction cavity to resonate at frequencies close to the fundamental frequency of each mode and their higher order frequency harmonics. In this manner, we are capitalizing on spatial harmonics through the use of two modes and time harmonics to increase beam bunching. From HFSS simulations, we obtain the iris placement at 8.89 cm from the center axis and the iris window width of 4.16 cm. We perform a frequency sweep from 2.4 to 5.0 GHz and obtain the resonant frequencies at 2.47, 3.25, 3.42, 4.38, and 4.7 GHz (Fig. 10). This is close to the fundamental of the $\hbox{TM}_{010}$ (2.4 GHz) and $\hbox{TM}_{110}$ (3.579 GHz) modes and the second harmonic of the $\hbox{TM}_{010}$ mode (4.8 GHz) obtained from MAGIC simulations in the previous section.

In MAGIC simulations, we use a 2.54-cm-radius solid beam. The major dips in modulating cavity current correspond to the $\hbox{TM}_{010}$ mode, and the minor dips, between the major dips, correspond to the $\hbox{TM}_{110}$ mode (Fig. 11). The peak-to-peak modulating cavity current is greater than the peak-to-peak current of the single mode ($\hbox{TM}_{010}$ or $\hbox{TM}_{110}$), and it has a steeper gradient.

The peak-to-peak extraction cavity current is lower than the peak-to-peak current of the single $\hbox{TM}_{010}$ mode. The fields are higher in the dual-mode extraction cavity so the electrons experience greater deceleration. This results in a lower peak-to-peak magnitude.

The dominant frequency in the extraction port is the $\hbox{TM}_{010}$ mode at 2.395 GHz with its second harmonic at 4.791 GHz. The fundamental $\hbox{TM}_{110}$ mode resonates at 3.592 GHz and is comparable in magnitude with the fundamental $\hbox{TM}_{010}$ mode (Fig. 12). Time–frequency analyses show the three frequencies oscillating consistently after 70 ns which remain until the end of the simulation (Fig. 13).

For the single-mode simulations, microwave power saturates after 100 ns. The dual-mode reltron requires a longer time to saturate, generating consistent levels after 150 ns. Average microwave power from one extraction port is 64 MW, as shown in Fig. 14, giving a total power of 128 MW (efficiency of 35%).

SECTION V

Analysis of the harmonic components in the beam current indicates that we can achieve greater peak current through the use of three harmonic modes. MAGIC simulations show that we can increase microwave power by exciting $\hbox{TM}_{010}$ and $\hbox{TM}_{110}$ modes in the reltron concurrently. Total microwave generated by two modes is larger than the power generated by individual modes. The reltron radiates the fundamental frequency of the $\hbox{TM}_{010}$ mode, $\hbox{TM}_{110}$ mode, and the second harmonic $\hbox{TM}_{010}$ mode consistently after 70 ns into the simulation.

The dual-mode reltron is less efficient than the 3-GHz experimental reltron published by Miller *et al.* [1]. This is because the modulating cavity of the dual-mode reltron is designed to excite either the $\hbox{TM}_{010}$ mode or the $\hbox{TM}_{110}$ mode separately or concurrently. The extraction cavity is designed to be tunable to resonate at either frequency. For the 3-GHz experimental reltron, the modulating cavity's gap distance is optimized for beam bunching, and the experimental reltron uses multiple extraction cavities instead of a single extraction cavity. The experimental reltron is also driven at higher voltages than the dual-mode reltron.

S. Soh, E. Schamiloglu, and C. G. Christodoulou are with the Electrical and Computer Engineering Department, The University of New Mexico, Albuquerque, NM 87131-0001 USA (e-mail: shawnsoh@gmail.com; edl@ece.unm.edu; christos@ece.unm.edu).

R. B. Miller is with the Department of Advanced Systems, Raytheon Ktech Corporation, Albuquerque, NM 87123 USA (e-mail: bmiller@ktech.com).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

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