High average-power gyrotron traveling-wave tubes （gyro-TWT） have abroad applications in modern communication and radar systems^［1-3］.Conventional gyro-TWT can operate at Ka and W band with high output power， but they face significant obstacles in extension to operate at higher frequency with high average power. The reason is that these devices utilize the fundamental or low-order modes of the interaction structure to avoid mode competition； the power capability and ohmic loss in such small structures are inevitable to limit the average power at high frequency. In 2003， a single confocal waveguide gyro-TWT has been proposed by MIT. This quasioptical mode-selective interaction structure allows a stable single-mode operation in the overmoded interaction structure at high frequency and high power ^［4-6］.

The drawback of the gyro-TWT with a single confocal structure is the relatively low beam-wave interaction efficiency because the conventional annular electron beam is located in the region with different strength of the coupling to the operating mode. One of the possible solutions for mitigating this drawback is to use a double confocal structure. A double confocal waveguide consists of two identical but perpendicular single confocal waveguides in x and y direction， respectively， as shown in Fig.1. The transverse field distribution in a double confocal waveguide is more uniform than that in a single confocal waveguide， so the conventional thin annular electron beam can interact more efficiently with the RF field in a double confocal waveguide. The theoretical analysis and PIC simulation confirmed that the beam-wave interaction efficiency increases obviously in a double confocal waveguide ^［7-8］.

In order to utilize a double confocal waveguide as a beam-wave interaction structure in a gyro-TWT， it is necessary to make clear the eigenmodes and their corresponding electromagnetic （EM） characteristics in a double confocal waveguide. In this paper， the EM characteristics of a double confocal waveguide for a gyro-TWT will be investigated in details. The rest of this paper is organized as following. The investigation on eigenmodes in a double confocal waveguide is presented in section 1， in section 2， the diffraction loss is discussed. Summary and conclusion are presented in section 3.

The double confocal waveguide is an open structure formed by four identical symmetrically placed curved cylindrical mirrors， in which four mirrors with a radius of R_c and a mirror width of 2a are separated with a distance of d. EM wave is reflected among mirrors. There are two kinds of stable modes with different reflection paths in a double confocal waveguide. The first steady-state mode， namely， the superposition mode， is superposed by two perpendicular in-phase or anti-phase field， as shown in Fig. 2（a）. The second steady-state mode， namely， the ring mode， is formed by the consecutive reflection of the field among four mirrors in clockwise or counterclockwise directions， as shown in Fig. 2（b）.

All resonator dimensions are assumed large compared to a wavelength； the eigenmodes of the resonator are therefore obtainable from a self-consistent field analysis using Huygens’ Principle. The Fresnel field U_p at the point of observation is given by the surface integral ^［9］

Up=jk4π∬AUae-ikρρ1+cosθdS

where k is the propagation constant， ρ is the distance from a source point on the mirror to the point of observation and θ is the angle between ρ and the normal of the source point.

For a superposition mode， which is superposed by two perpendicular standing wave in x and y direction， respectively， and the eigensolution of the standing wave is as follows. In Fig.3， the field U₂（x₂） on mirror S₂ can be obtained in term of an integral of the known field U₁（x₁） on mirror S₁ in y direction with eq. （1）.

U2x2=jk2πd∬AU1x1e-ikρdS1

The eigenfunctions of the modes in a confocal waveguide can be obtained by making sure that the field distribution over S₁ canreproduce itself with a constant over S₂， namely， U₂（x₂） = b_mU₁（x₁）， where b_m is general complex， including both the amplitude and phase changes. In this case， eq. （2） can be rewritten as following

bmU1x1=jk2πd∬AU1x1'e-ikρ'dS1'

with

ρ'=d-x1x1'/d

For this general solution， the eigenvalues can be expressed as. The amplitude of b_m is related to the diffraction loss， and the phase of b_m determines the phase shift of the mode from S₁ to S₂， where kd is the geometric phase shift and is the additional phase shift. The phase shift of a complete field reflection in the cross section of the resonator must be an integer n times of 2π， namely. . By solving eq. （3）， the cut-off wavenumber of the superposition mode can be written as

kt=πdn+m2+14

The field between S₁ and S₂ can be calculated based on Huygens’ Principle as presented in eq. （1）. Approximately， the field distribution between S₁ and S₂ can be considered as a superposition of two Gaussian beams. The membrane function of standing wave （superposition of two Gaussian beams） can be written as ^［10-11］：

Ψmnx,y=w0wyϕm-2xwy×cosξ,n=2,4,6,...sinξ,n=3,5,7,...

，　（6）

where

ζ=ktx22Ry+kty-m+12arctan2yktw02

Ry=y1+kw022y2,wy=w01+2ykw022

ϕmτ=Hmτexp-τ22

here， H_m（τ） is the Hermite polynomial and .

Similarly， the field distribution between two mirrors in x direction can be calculated in the same way. The superposition mode can be considered as a liner superposition of two in-phase or anti-phase standing waves， and the membrane function can be expressed as following：

Ψmninx,y=Ψmnx,y+Ψmn-y,xΨmnantix,y=Ψmnx,y-Ψmn-y,x

where has been expressed in eq.（6）. With the membrane functions and of the superposition modes in a double confocal waveguide， by solving the Maxwell’s equations in a Cartesian coordinate system， the field components of TE_mn mode can be obtained.

The magnitude of the transverse electric field of the superposition mode in a double confocal waveguide is presented in Fig. 4. Fig. 4 （a）， （b）， （c） and （d） are the field of the superposition ，，and modes simulated by commercial CST software， the theoretical results of the field are presented in Fig.4 （e）， （f）， （g） and （h）. The theoretical results agree well with those obtained by the commercial CST software. It is worth noting that the theoretical calculation only includes the central area surrounded by the mirror， and the diffraction field between the mirrors is not included in the theoretical calculation.

In the Cartesian coordinate system， the coupling factor （Form factor） characterizing the Lorentz force on electrons can be expressed as^［2］

Ls=12π∫02π1kt∂∂X+i∂∂YΨmnX,Ydθ2

X=rbcos(θ),Y=rbsin(θ)

Fig. 5 presents the normalized beam-wave coupling factor of the double confocal mode and the single confocal modeTE₀₆ versus the guiding center radius R_b， and the beam-wave coupling factor in a double confocal waveguide is greater than that in a single confocal waveguide.

The other kind of mode in a double confocal waveguide is the ring mode. The ring mode is formed by the consecutive reflection among four mirrors in clockwise or counterclockwise directions.

Based on the scalar formulation of Huygens’ Principle in eq. （1）， if we know the field distribution U₁ （y₁） on mirror M₁， we can calculate the field U₂ （y₂） on mirror M₂. Then， the field U₂ （y₂） is reflected to M₃ in the same way to form U₃ （y₃）. It is conceivable that after multiple reflections the field will eventually reach a steady state. U₃ （y₃） and U₁ （y₁） satisfy the following relationship： U₃（y₂） = b_mU₁（y₁）. The resulting integral equation is presented as follows

bmU1y1=∬M2dS2∬M1dS1K12K23U1y1'

In eq. （13）， K_mn are defined as

Kmn=jkcosθ2πe-jkρmnρmn

where ρ_mnrepresents the distance between any two points on mirror M_m and mirror M_n ^［12］.

ρmn=l+ymyncos2θ/l+ym-ynsinθ-ym2+yn2cos2θl-Rccosθ/2lRccosθ

For this general solution， the eigenvalues can be expressed as. By solving eq. （13）， the cutoff wavenumber of the ring mode TE_0n can be written as

kt=π2ln+32π

where n is an integer for the mode index which represents the number of maxima from M₁ to M₃.The electric field at arbitrary point between M₁ and M₂ can be obtained by calculating the integral of eq.（1） with U₁（y₁） presented in eq.（13）.

The field distribution between two adjacent mirrors also can be considered as a standing-wave field formed by the superposition of Gaussian beams propagating clockwise and counterclockwise. The expression of the membrane function of the ring mode can be written as：

Ψ0nringx,y=Ψ0nx-Rc22,y+Ψ0nx+Rc22,y+Ψ0n-y+Rc22,x+Ψ0n-y-Rc22,x

where has been expressed in （6）.

The transverse field distribution of the ring modes is shown in Fig. 7. Fig. 7 （a） and （b） is the simulation result of the field distribution from CST， Fig. 7 （c） and （d） is the theoretical calculation result. The theoretical results agree well with those obtained by the commercial CST software.

Table I gives the cutoff frequency of the modes whose cutoff frequency is around 263GHz in a double confocal waveguide with R_c=3.6mm and a=1.2mm. Clearly， the cutoff frequencies of the superposition mode and the ring mode calculated by eqs. （5） and （16） agree well with the simulation results from CST.

The mode densities in a double confocal waveguide， in a single confocal waveguide and in a cylindrical waveguide are presented in Fig. 8， it is obvious that the mode density in a double confocal waveguide is higher than that in a single confocal waveguide， but less than that in a cylindrical waveguide. This is because there are more stable paths for the modes in a double confocal waveguide than in a single confocal waveguide. In addition， the in-phase superposition mode and the anti-phase superposition mode are the degenerate mode， so only one curve is used to represent these two modes in Figure 8（c）.

For a double confocal waveguide， it is possible to diffract part of power out of the waveguide by means of the gaps among the mirrors； it is helpful against the parasitic oscillation in a gyro-TWT. Table II gives the diffractive loss of different modes in a double confocal waveguide with R_c=3.6mm and a=1.2mm. It is found that the diffractive loss of the anti-phase superposition mode TE₀₆ mode is far less than that of the potential parasitic modes.

Fig. 9 shows the effect of the gap size on the diffraction losses of the anti-phase superposition mode TE₀₆ mode， which is expected to be the operating mode of the Gyro-TWT.

The electromagnetic characteristics of the double confocal waveguide for a gyro-TWT is investigated in details. There are two kinds of steady-state modes in a double confocal waveguide， namely， the superposition mode and the ring mode. The superposition mode includes in-phase and anti-phase superposition mode. With the scalar formulation of Huygens’ Principle， the eigenvalue and the field distribution of these two kinds of modes have been calculated， and the theoretical calculation results agree well with those from the commercial CST software. Meanwhile， the diffractive loss has also been calculated with CST software. If the anti-phase superposition mode TE₀₆ mode is chosen as an operating mode in a gyro-TWT， the diffractive loss of the potential parasitic modes， such as the ring mode TE₀₈ and TE₀₉ modes， the superposition mode TE₀₆ （in-phase） TE₁₅ （in-phase and anti-phase） modes is far greater than that of the operating mode. It means that the potential parasitic modes can be suppressed effectively by means of its own diffractive loss in a double confocal waveguide. The mode density in a double confocal waveguide is higher than that in a single confocal waveguide， but far lower than that in a cylindrical waveguide. Compared to the single confocal waveguide， a higher beam-wave interaction efficiency can be obtained in a double confocal waveguide for a gyro-TWT， it is an appropriate choice to choose a double confocal waveguide as the beam-wave interaction structure in a gyro-TWT.