In this paper, a coaxial electrostatic wiggler with corrugated inner and outer walls is investigated theoretically. The field distributions in this wiggler correspond to the special solution of a Laplace equation in a cylindrical coordinate system with two sinusoidal ripple boundaries. Through numerical analysis and comparison to the coaxial electrostatic wiggler with a corrugated outer wall, the results indicate that in a coaxial electrostatic wiggler with corrugated inner and outer walls, electric field undulation can be enhanced. The particle-in-cell simulation further demonstrates that the electrons can be modulated deeply.

## I. INTRODUCTION

A free electron laser (FEL) is a kind of device that uses a relativistic electron beam to convert electron kinetic energy into electromagnetic wave energy through a wiggler or other transduction mechanisms to generate high-power radiation ranging from microwaves, terahertz waves, and infrared waves to the visible spectrum, ultraviolet, and x rays.^{1–8} In a FEL, a relativistic electron beam is passed through a magnetostatic (or electrostatic, or electromagnetic) periodic field, known as a wiggler field. This field induces an oscillatory transverse motion to an electron beam. This transverse motion has a key role in the FEL interaction. According to the working nature of a wiggler, there are three types of wigglers: electromagnetic wave wigglers,^{8–10} magnetostatic wigglers,^{11–13} and electrostatic wigglers.^{14–16} In the electrostatic wiggler’s structure, the load ring is disposed inside the cylindrical waveguide, and either the rings are insulated from each other and connected to the static voltage of the polarity staggered reverse, or the sinusoidal ripples are corrugated with the inner surface of the circular waveguide and connected to the static voltage. Thus, the internal electrostatic field has a sinusoidal function with the distribution of the spatial variables. It also produces an alternating electric field force in the transverse plane to make the electrons swing, which is the principle of an electrostatic wiggler. Typically, an electrostatic wiggler is a configuration with a periodic electrostatic field distribution produced by the ripple corrugations on the inside surface of a metallic pipe connected to an electrostatic voltage. Other configurations, such as a two-dimensional dusty plasma crystal or a planar electrostatic system with sinusoidal ripples, have also been employed as an electrostatic wiggler for a FEL.^{17,18} Previous investigations show that, excluding the space-charge wave in the aforementioned three-wave interaction mechanism, a planar electrostatic wiggler may be favorable for a mildly relativistic electron beam to generate terahertz waves by extracting the kinetic energy of the electrons.^{19} Moreover, another study illustrated an interesting phenomenon, in which a coaxial electrostatic wiggler may pump both the kinetic energy and electrostatic potential energy of a relativistic electron beam interacting with a transverse-electric wave, resulting in wave amplification with ultrahigh gain.^{20} Thus, an electrostatic wiggler can more easily produce millimeter waves to terahertz waves with low-energy electron beams. This greatly reduces the cost of a project and facilitates engineering practice.

In Ref. 20, an electrostatic wiggle using a coaxial structure with the static voltage applied in the outer conductor was investigated. The outer shaft was periodically corrugated with sinusoidal ripples and connected to a negative voltage, and the inner shaft was smooth and grounded. In this structure, due to the periodicity of the corrugated wall of the outer shaft, the electric field distribution also changed periodically. Moreover, further study of the beam–wave interaction under this structure confirmed that the 120-GHz electromagnetic wave could achieve an ultrahigh gain of 78.6 dB, and the output power of the device reached 72 MW. The amplification of the electromagnetic wave was found by extracting both the kinetic energy and the electrostatic potential energy. Amplification better than the ordinary amplification of the electromagnetic wave was produced by extracting the kinetic energy.

In this study, we investigated a coaxial electrostatic wiggler with inner and outer walls that are both corrugated with cosine ripples with position differences. In this wiggler, more potential energy could be obtained than in the wiggler with only a corrugated outer wall. This is beneficial for obtaining higher beam–wave interaction efficiency, higher gain, and higher output power.

The rest of this paper is organized as follows. In Sec. II, the physical model and the theoretical formulas are presented. In Sec. III, the numerical calculation and nonlinear simulation of the field distribution in electrostatic wigglers are introduced. The summary and discussion are presented in Sec. IV.

## II. THEORETICAL MODEL

The profile of the coaxial electrostatic wiggler with corrugated inner and outer walls is shown in Fig. 1. The inner conductor was grounded, and the outside surface of the inner conductor was sinusoidally corrugated with a mean radius of *r*_{in}, a ripple period of *p*_{1}, and a ripple depth of *l*_{1}. The outer conductor was connected to a negative voltage −*V*_{0}, and the outside surface of the outer conductor was sinusoidally corrugated with a mean radius *r*_{out}, a ripple period *p*_{2}, and a ripple depth *l*_{2}. The electron beam was transmitted between the inner and outer conductors, with a guiding center of *r*_{L}. Adopting a cylindrical coordinate system (*r*, *θ*, *z*), the boundary functions of the inner and outer conductors were

where *k*_{1} = 2*π*/*p*_{1}, *k*_{2} = 2*π*/*p*_{2}, and *φ*_{1} and *φ*_{2} are the phases from the input port. Under the assumption that the transverse dimension is much smaller than the length in the *z* direction, the electrostatic potential *ϕ*(*r*, *z*) in the physical system leads to the boundary-value problem of the two-dimensional Laplace equation,^{18}

with the boundary conditions

Considering the periodic conditions, the general solution of Eq. (3) is shown below in square brackets,^{21}

where *ξ*_{n}, $\xi m\u2032$, *a*_{n}, $am\u2032$, *b*_{n}, $bm\u2032$, *c*_{n}, $cm\u2032$, *d*_{0}, and *d*_{1} are the integral constants, and *I*_{0} and *K*_{0} are the first and second kinds of modified Bessel functions of zero order. The last term on the right-hand side represents the effect of the ripples on the field distribution.

Equation (6) can be solved by using the boundary conditions. By inserting Eq. (6) into the inner boundary condition (4), and supposing that *r*_{in} ≫ *l*_{1}, the following formula is approximately obtained:

The second term on the left-hand side of Eq. (7) can be rewritten as

To balance both sides of Eq. (7), the constant term and functional term to *z* on the left-hand side must be equal to the terms on the right-hand side,

With the series-expansion method, the left-hand side of Eq. (11) can be expressed as the form of series,

Considering that *r*_{in} ≫ *l*_{1}, the high-order harmonic terms in Eq. (12) can be neglected. Substituting the linear term −*l*_{1}/*r*_{in} · cos(*k*_{1}*z* + *φ*_{1}) into Eq. (11) and balancing both sides of Eq. (7) with the formulas of trigonometric functions, the following formulas can be obtained:

Inserting Eq. (6) into the outer boundary condition (5) and assuming *r*_{out} ≫ *l*_{2}, using the same method above, the following formulas can be obtained:

From Eqs. (9)–(21), the following integral constants are satisfied:

Finally, the special solution of the electrostatic potential can be obtained,

Substituting Eq. (30) into the potential equation $E\u2192=\u2212\u2207\varphi $ with Eqs. (22)–(29), the *r*-, *θ*-, and *z*-direction components of the electric field can be obtained as

where *I*_{1} and *K*_{1} are the first and second kinds of the modified Bessel functions of order 1.

Because phases *φ*_{1} and *φ*_{2} can be different, the position phase shift Δ*φ* = *φ*_{2} − *φ*_{1} is defined. Using Eqs. (31) and (32), the *E*_{r} and *E*_{z} field distributions not only in the corrugated inner and outer walls, but also in the corrugated inner wall with a smooth outer wall and the corrugated outer wall with a smooth inner wall can be calculated in the cases of *l*_{1} ≠ 0, *l*_{2} ≠ 0, *l*_{1} ≠ 0, *l*_{2} = 0, and *l*_{1} = 0, *l*_{2} ≠ 0.

## III. NUMERICAL CALCULATION AND COMPUTER SIMULATION

In this section, the field distributions in a coaxial electrostatic wiggler with corrugated inner and outer walls are numerically calculated and analyzed. To simplify the analysis, only the case in which the same ripple periods of the inner and outer walls *p*_{1} = *p*_{2} was investigated. The initial parameters were *r*_{in} = 2 mm, *r*_{out} = 4 mm, *l*_{1} = 0.1 mm, *l*_{2} = 0.1 mm, *p*_{1} = *p*_{2} = 2.498 mm, *r* = 2.5 mm, and *V*_{0} = 538.6 kV, which were based on the parameters in Ref. 20.

The field intensity was calculated along the *z* direction at *r* = 2.5 mm for the four cases presented in Fig. 2. The first case involved the *E*_{r} and *E*_{z} fields in a corrugated inner wall with a smooth outer wall. The second case involved the *E*_{r} and *E*_{z} fields in a corrugated outer wall with a smooth inner wall. The third case involved the *E*_{r} and *E*_{z} fields in corrugated inner and outer walls at Δ*φ* = 0. The fourth case involved the *E*_{r} and *E*_{z} fields in corrugated inner and outer walls at Δ*φ* = π.

Figure 2 shows that, in the four cases, the *E*_{r} and *E*_{z} fields periodically changed along the z direction, and the fields’ periods were the same as the ripple period. In an electrostatic wiggler, the helical electron beam is modulated by the radial and axial periodic electric forces. Thus, if the peak-to-peak value of the *E* field is higher, the modulation depth is larger. To investigate the variation of the *E* field, *E*_{r-pp} was defined as the peak-to-peak value of the *E*_{r} field, and *E*_{z-pp} was defined as the peak-to-peak value of the *E*_{z} field. The results show that, at the same ripple depth, *E*_{r-pp} and *E*_{z-pp} at a location *r* = 2.5 mm in the case of the corrugated outer wall with a smooth inner wall were much weaker than those in the other three cases. For the *E*_{r} field at a location *r* = 2.5 mm, *E*_{r-pp} in the case of the corrugated inner and outer walls at Δ*φ* = π was the largest, and *E*_{r-pp} in the case of the corrugated inner and outer walls at Δ*φ* = 0 was smaller than that in the case of the corrugated inner wall with a smooth outer wall. For the *E*_{z} field at a location *r* = 2.5 mm, *E*_{z-pp} in the case of the corrugated inner and outer walls at Δ*φ* = 0 was the largest, and the field amplitude in the case of the corrugated inner and outer walls at Δ*φ* = π was smaller than that in the case of the corrugated inner wall with a smooth outer wall.

Figure 2 also shows that at different Δ*φ*, *E*_{r−pp} and *E*_{z−pp} were different, even though the ripple depths *l*_{1} and *l*_{2} were the same. Figure 3 presents the calculation results of *E*_{r−pp} and *E*_{z−pp} vs Δ*φ*. The results show that, at Δ*φ* = π, *E*_{r−pp} had a maximum value and *E*_{z−pp} had a minimum value, while, at Δ*φ* = 0, *E*_{r−pp} had a minimum value and *E*_{z−pp} had a maximum value.

Figure 4 presents the ripple-depth impacts for the *E*_{r−pp} and *E*_{z−pp} values at different beam radii *r*. It can be seen that at Δ*φ* = 0, the values of *E*_{z−pp} increased with the increase in the outer conductor ripple depth *l*_{2} and the inner conductor ripple depth *l*_{1}. The values of *E*_{r−pp} first decreased and then increased after *E*_{r−pp} dropped to 0. At Δ*φ* = π, the values of *E*_{r−pp} increased with the increase in the outer conductor ripple depth *l*_{2} and the inner conductor ripple depth *l*_{1}. The values of *E*_{z−pp} first decreased and then increased after *E*_{z−pp} dropped to 0. At both Δ*φ* = 0 and π, when the beam radius *r* was smaller, the absolute values of the slopes of *E*_{r−pp} and *E*_{z−pp} were bigger. When the beam radius *r* was larger, the absolute values of the slopes of *E*_{r−pp} and *E*_{z−pp} were smaller. This means that when the electron beam was close to the inner conductor, the *E*_{r−pp} and *E*_{z−pp} values were more sensitive to the ripple depth of the inner conductor *l*_{1}. When the electron beam was close to the outer conductor, the *E*_{r−pp} and *E*_{z−pp} values were more sensitive to the ripple depth of the outer conductor *l*_{2}.

CST three-dimensional (3D) electromagnetic simulation software was used to demonstrate the numerical calculation results from the theoretical formulas. Comparisons of the numerical calculation and CST simulation are presented in Fig. 5. The results show that, at either Δ*φ* = 0 or Δ*φ* = π, the amplitude of *E*_{r} produced with the CST simulation was slightly larger than that produced with the numerical calculation, and the amplitude of *E*_{z} as well as the values of *E*_{r−pp} and *E*_{z−pp} produced with the CST simulation were almost the same as those produced with the numerical calculation. This shows good agreement of the longitudinal component between the numerical calculation and the sCST simulation. There was a small systematic shift of the radial component calculated by the analytical formula and by the CST simulations. The reason for this may have been the neglect of the harmonic terms in Eq. (12).

To demonstrate the modulation effect of the wiggler with corrugated inner and outer walls, the particle-in-cell (PIC) simulation was performed with CST software. Figure 6(a) shows the 3D trajectory of an electron with beam energy modulation that was captured in CST. The electron trajectory in the wiggler exported from CST is presented in Fig. 6(b). The variations of the beam radial position and the beam energy in the wiggler with corrugated inner and outer walls and in the wiggler with only corrugated wall are compared in Fig. 6(c). The results show that the electron beam in the wiggler with corrugated inner and outer walls could be modulated deeply, and more potential energy could be obtained than in the wiggler with only one corrugated wall, which could be beneficial for generating a high output power in a FEL.

## IV. SUMMARY AND DISCUSSION

In this study, the theoretical model for a coaxial electrostatic wiggler with corrugated inner and outer walls, which was based on a 2D Laplace equation with coaxial sinusoidal boundaries, was presented. The characteristics of the field distribution in the coaxial electrostatic wiggler with corrugated inner and outer walls are numerically calculated and compared with those in the coaxial electrostatic wiggler with only a corrugated inner wall and those in the coaxial electrostatic wiggler with only a corrugated outer wall. The impacts of the position shifts between the inner and outer walls are also analyzed. The PIC simulation demonstrated that the electron beam could be modulated more deeply, and more potential energy could be obtained in the wiggler with corrugated inner and outer walls than in the wiggler with only a corrugated outer wall.

In engineering implementation, the assembly of a wiggler with only a corrugated inner wall and a wiggler with only a corrugated outer wall is much easier than that of a wiggler with corrugated inner and outer walls, which could also avoid impacts due to position shifts between the inner and outer walls. The results still indicate that when the beam radius was close to an inner conductor, the wiggler with only a corrugated inner wall could modulate the electron beam deeply. When the beam radius was close to the outer conductor, the wiggler with only a corrugated outer wall could modulate the electron beam deeply. The results presented in this article are useful for the development of high-power electrostatic FELs.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant No. 61971097 and the Sichuan Science and Technology, Program No. 2018HH0136. We thank LetPub (www.letpub.com) for its linguistic assistance during the preparation of this manuscript.