Nonlinear theory and numerical simulation of slotted waveguide third harmonic cyclotron traveling wave amplifier

Figure 1 Schematic diagram of the cross section of the hollow outer slot waveguide and the electron beam. The dotted circle is the schematic diagram of the cross section of the electron beam

In zone I (0＜r＜a)

(1)
(2)
Ez=0 (3)

In zone II (a＜r＜b)

Ez=0 (4)
Er=0 (5)

(6)

in

(7)
B0=［-J′0(kcb)/Y′0(kcb)］A0 (8)

In the above formulas, E0 is the high-frequency field amplitude, Γ is the angular harmonic number, ΑΓ is the amplitude coefficient of the angular Γ subharmonic term, kc is the cutoff wave number, q is the slot number (q=1,2,…,N), and m represents the angular mode of the high-frequency field (m=0,1,2,…,N-1). The value of ΑΓ and the dispersion relation of the circuit can be determined by the boundary conditions of the electromagnetic field at r=a.

(9)

The dispersion relation is

(10)

Equation (9) shows that non-zero spatial harmonic terms exist only when the spatial harmonic order Γ=m+lN. The angular mode determines the phase difference of the high-frequency field between adjacent gaps. For each specific mode, this phase difference is m2π/N. Each angular mode is composed of countless angular harmonic terms, and its harmonic amplitude coefficient is determined by equation (9). Among all angular modes, there are two relatively important modes, namely the π mode and the 2π mode. The relative strength distribution of their angular harmonics is shown in Figure 2. As can be seen from Figure 2, the energy of the 2π mode is mainly concentrated in the zero-order harmonic term, while the energy of the π mode is mainly concentrated in the ±N/2-order harmonic terms. Therefore, the π mode is more suitable for high-order cyclotron harmonic interactions than the 2π mode. If the electron beam cyclotron harmonic order (represented by S) has been set, then the selection of the slot number N should ensure that the order Γ of the strongest non-zero angular harmonic term is equal to the cyclotron harmonic order S. For example, for the π mode, the slot number N should be equal to 2S.

Fig. 2 Schematic diagram of the relative distribution of angular harmonic amplitude to angular harmonic number (Γ). (a) π mode (m=N/2, N=6, θ0=15°), (b) 2π mode (m=0, N=6, θ0=15°)

When the angular mode m and the groove depth (i.e., the value of a/b) are determined, the cutoff wave number kc can be obtained by numerical solution method using equation (10) [6, 8, 9].

3. Self-consistent nonlinear theory
In the hot cavity, the high-frequency field is distributed slowly along the axial direction, and its distribution function for the horizontal coordinate (r, φ) is the same as that of the cold cavity. The expression of the high-frequency electric field component (TE wave) of the hot cavity in zone I is given below.

(11)
(12)
Ez=0 (13)

In the above formulas, Cmn is the normalization coefficient of the electric field, and f(z) is a complex function representing the slowly varying distribution of the high-frequency field along the Z axis. The value of Cmn is obtained by the following formula:

(14)

The following is a set of ordinary differential equations for self-consistent nonlinear beam-wave interaction. Starting
from the Lorentz formula
[8], the equations of motion of electrons under the action of high-frequency fields (E, B) and DC magnetic fields (B0) can be derived. Each electron has 6 equations of motion parameters, and only the three equations of motion parameters of velocity component and momentum space angle are given here.

(15)
(16)
(17)

In the above equations, m0 and γ are the rest mass and relativistic factor of the electron, φ is the momentum space angle, u=γv, v is the speed of the electron, as shown in Figure 1.
Starting from the active Maxwell equations, after a series of complex derivations and discretization of the current, the nonlinear beam-wave interaction field equation is obtained as follows:

(18)

In the above formula, P is the number of electron beam batches taken in one high-frequency field cycle, M is the number of circles into which the electron beam is divided considering the electron beam thickness factor, N is the number of macro electrons taken in each circle, and S is the harmonic order.〈…〉 represents the averaging of the velocity space with the initial velocity distribution function of g0(v⊥,vz). Assuming that the electron beam is a monoenergetic electron beam, the velocity dispersion mainly comes from the dispersion of the horizontal and vertical velocity ratio (V⊥/Vz). Here, the velocity dispersion is handled according to the normal distribution law, that is, the initial velocity distribution function is

Where K is the normalization constant, △vz is the average longitudinal velocity dispersion, and δ is the Dirac function.
Boundary conditions

f(z)|z=0=f(0) (19)
(20)

Where f(0) is the input high-frequency electric field amplitude.
Equations (15) to (18) are self-consistent nonlinear beam-wave interaction equations. Discretizing the electron beam into NT macro-electrons, there are a total of 6NT+2 first-order nonlinear differential equations. Combined with boundary conditions (19) and (20), the fourth-order Runge-Kutta method is used to numerically calculate the beam-wave interaction. The calculation results are given and discussed in the following part.

IV. Results and Discussion
Table 1 gives the parameters of the interaction circuit. The parameters related to each graph curve are marked in the corresponding graph. Figure 3 shows the relationship between efficiency and electron velocity ratio α when the driving power is 20W. In the figure, B0 and Bg are the DC magnetic field and the resonance point magnetic field, ω is the high-frequency field frequency, and ωc is the waveguide cutoff frequency. Since the energy of the wave in the cyclotron traveling wave tube is taken from the transverse energy of the electron, and since the transverse energy and cyclotron radius of the electron increase as the α value increases, the interaction efficiency also increases as α increases. However, when α increases to a certain value, the injection-wave interaction reaches saturation. At the same time, due to the excessively large cyclotron radius of the electron injection, the electron is intercepted on the waveguide wall, so the interaction efficiency decreases as the α value increases.

Table 1 Numerical simulation parameters and results

Figure 3 Relationship between efficiency and electron beam velocity ratio α (s=3,πmode,I=6A,V=60kV,ω/ωc=1.032,
B0/Bg=0.99)

Figure 4 shows the relationship between saturation efficiency, saturation gain and B0/Bg value, and the dotted line is the saturation gain curve. In the figure, γz is the relativistic factor of the longitudinal velocity component. The figure shows that, on the one hand, reducing the B0/Bg value is helpful to improve the saturation interaction efficiency, but the B0/Bg value cannot be too low, otherwise the detuning will be aggravated, the injection wave interaction will be difficult to achieve synchronization, and the saturation efficiency will decrease rapidly; on the other hand, increasing the B0/Bg value is conducive to improving the saturation gain. In general, the selection of the magnetic field detuning rate should be an optimal compromise between efficiency and gain.

Figure 4 Relationship between saturation efficiency and gain and B0/Bg value (s=3,π mode,I=6A,V=60kV,ω/ωc=γz,
α=1.3)

Figure 5 shows the relationship between (a) saturation efficiency and (b) saturation gain as a function of frequency when the current is 3A, 6A, and 9A. It can be seen that the saturation efficiency, saturation gain, and saturation bandwidth all increase with the increase of current. In the case of 6A and the figure, the saturation bandwidth is 7%. When the current increases from 3A to 9A, the saturation bandwidth increases from 4.6% to 8.3%.

Figure 5: (a) Saturation efficiency (b) Saturation gain versus frequency at different currents (s=3,π mode,V=60kV,α=1.3,B0/Bg=0.99)

Figure 6 shows the relationship between saturation gain and saturation efficiency at different magnetic field detuning rates and frequency. It can be seen from the figure that the magnetic field detuning rate has a great influence on the saturation gain, saturation efficiency and saturation bandwidth. The increase of B0/Bg value is conducive to the improvement of saturation gain and saturation bandwidth, but the saturation efficiency is reduced. Under the conditions shown in the figure, when the B0/Bg value increases from 0.983 to 0.998, the saturation bandwidth increases from 4.8% to 9.3%.

Figure 6 Relationship between (a) saturation gain and (b) saturation efficiency and frequency at different magnetic field detuning ratios (s=3,π mode,I=6A,V=60kV,α=1.3)

Figure 7 shows the relationship between the saturation efficiency and the harmonic order at different magnetic field detuning rates. The figure shows that the saturation efficiency decreases with the increase of the harmonic order. The lower the B0/Bg value, the greater the impact of the harmonic order on the saturation efficiency.

Figure 7 Saturation efficiency versus harmonic order (π mode, I=6A, V=60kV, α=1.3, ω/ωc=γz, rL/a=0.7)

Figure 8 shows the relationship between saturation efficiency and frequency under different harmonic orders. The figure shows that the harmonic order has a great influence on the saturation bandwidth. Under the conditions shown in the figure, when the harmonic order increases from 2 to 4, the saturation bandwidth decreases from 10.3% to 5.7%.

Figure 8 Variation of saturation efficiency with frequency at different harmonic orders (π mode, I=6A, V=60kV, α=1.3, B0/Bg=0.99, ω/ωc=γz, rL/a=0.7)

V. Conclusion
In this paper, the injection wave interaction of 95GHz 3rd harmonic cyclotron traveling wave tube is numerically calculated under the condition of single mode and without considering space charge effect and waveguide wall loss, and some important interaction laws are obtained. The research results show that slotted waveguide high-order harmonic cyclotron traveling wave tube can obtain higher interaction effect under lower magnetic field and wider frequency band. High-order harmonic interaction reduces the requirement of magnetic field of the tube, making it possible to use permanent magnets, but the increase of harmonic order will weaken the injection wave interaction efficiency and bandwidth. Since in π mode, the high-frequency field energy is mainly concentrated in In the high-order harmonic terms, while in the 2π mode, the energy is mainly concentrated in the zero-order harmonic terms. Therefore, the π mode is more conducive to high-order harmonic amplification than the 2π mode. Appropriately reducing the value of the magnetic field detuning ratio B0/Bg is conducive to improving the saturation interaction efficiency, but the saturation gain and bandwidth are reduced. Appropriately increasing the transverse and longitudinal velocity ratio (V⊥/Vz), current value and input power is conducive to improving the bandwidth, gain and interaction effect. In addition, velocity dispersion also has a great influence on the injection-wave interaction. With the increase of velocity dispersion, the injection-wave interaction efficiency and gain are reduced.