FDTD investigation of cylindrical waveguides with azimuthal corrugations excited by axisymmetric TE modes

Abstract

In this work we investigated the scattering parameters and mode conversion of axisymmetric TE modes in smooth cylindrical waveguides with wedge-shaped azimuthal corrugations. In particular, we studied the effect of the length of the corrugated region on the scattering parameters and on the mode conversion. The field distribution was derived by the FDTD code COCHLEA and further post-processed using the COPP code. Results showed that the incident mode is converted to an ensemble of modes, the azimuthal indices of which are given by the SHM criterion. Azimuthal corrugated waveguides are usually employed in the gyrotron beam-tunnel to enhance the suppression of the parasitic modes.

Keywords

FDTD computational electromagnetics gyrotron beam tunnels

1. Introduction

In modern fusion experiments there is a need for high-power and high-frequency RF radiation for the current drive, the plasma heating [1] and for the elimination of plasma instabilities. Such radiation is usually provided by gyrotrons, in which the basic mechanism of RF generation is the Electron Cyclotron Resonance (ECR). An important part of the tube is the beam-tunnel, in which the electron beam drifts and is being compressed, to obtain the desired characteristics for the main interaction in the cavity. The beam-tunnel has been designed to operate as beam drift tube rather than resonant cavity. However, due to its complexity, undesired modes are excited by the electron beam, commonly named as parasitic oscillations, which degrade the quality of the beam and thus the main interaction efficiency [2–5]. To suppress the parasitic oscillations, several design approaches have been employed, including the stacked beam tunnel [5], the conical one with SiC coating [6] and the fully metallic one with random corrugations [7]. These designs aim to decrease the quality factor of such parasitic modes and consequently to increase their starting current, thus avoiding their excitation. The stacked beam tunnel consists of alternating copper and dielectric rings of varying radius, with azimuthal corrugations indented on the metallic parts. These indentations aim to increase the losses of the parasitic modes, in order for the latter to be further attenuated. Since these corrugations are essentially azimuthal discontinuities, they operate as mode converters. It is estimated that the mode conversion induced by such corrugations enhances the absorption of the excited modes from the dielectrics.

In the past, investigations on the parasitic oscillations in gyrotron beam tunnels were performed with semi-analytical methods without considering the presence of the azimuthal corrugations [8–10]. In [11] the quality factor and the shunt impedance in beam tunnels with azimuthal corrugations were studied using numerical tools; however, the effect of the corrugations on the mode conversion and on the propagation characteristics was not investigated. In [12] the problem of the propagation in azimuthally corrugated waveguides was studied using the Rayleigh-Fourier method for describing the corrugated boundary, being however unsuitable for structures with large corrugation depths.

The azimuthal corrugations primarily convert an incident mode to an ensemble of modes with different azimuthal indices related to the number of corrugations. For this reason, in this work, we considered the structure as a two-port mode converter and investigate the effect of the length of the corrugated region on the scattering parameters and the mode conversion. Note that, this study aims to the better understanding of the operation of the azimuthal corrugations in order to be further used in the stacked gyrotron beam tunnel for the enhancement of the suppression of the parasitic oscillations.

The study of the beam-tunnel as a whole exhibits significant challenges, both in terms of modelling, as well as regarding the computational resources needed for the latter due to its complexity and large electrical size. In this work, we are interested in the effects of the azimuthal corrugations on the mode decomposition and scattering parameters of possible parasitic modes. For this reason, we consider a simplified structure of a smooth waveguide with an azimuthally corrugated region, terminated with Complex Frequency Shifted Perfectly Matched Layers (CFS-PML) layers, which can be considered a good approximation of one ring of the structure. The structure is excited with TE modes which correspond to possible parasitic ones. CFS-PML layers are used to minimize possible reflections. In particular, we used the in-house 3D full-wave numerical code COCHLEA [13], which is based on the FDTD method in cylindrical coordinates [14], developed in C and parallelized using a hybrid OpenMP/MPI scheme. The code calculates the temporal evolution of the electromagnetic fields in a structure with a constant axis and an arbitrary cross-section, which is excited by TE or TM modes. Note that, the PML layers are separated from the ports by Δz, the axial dimension of the cell. The fields are then post-processed by the supplementary code COPP, which performs the FFT of the fields on the input and output ports of the structure and then performs the mode decomposition on a given ensemble of modes, both TE and TM. From the mode decomposition, the field expansion coefficients corresponding to the forward and backward component of each mode of the decomposition are derived. By calculating appropriate ratios of these coefficients, the scattering parameters are obtained. The remainder of this paper is organized as follows. In Section 2 the mathematical formulation of the mode decomposition is presented (which is an extension to that presented in [15]) and in Section 3 the problem under consideration and the associated structure are described. In Section 4 the geometrical properties of the structure and the corresponding numerical results are given. Finally, in Section 5 the conclusion of the study is presented.

2. Mode decomposition

In a cylindrical waveguide with circular cross-section, orthogonal modes are supported, and the field can be expressed as a superposition of them, each one represented by a forward and a backward component [15] $\begin{eqnarray}\mathbf{E}=\mathop{\sum }_{n}(a_{n}\mathbf{E}_{n}^{\mathit{forward}}+b_{n}\mathbf{E}_{n}^{backward}),\quad H=\mathop{\sum }_{n}(a_{n}\mathbf{H}_{n}^{\mathit{forward}}+b_{n}\mathbf{H}_{n}^{backward}),\end{eqnarray}$ (1) where a_n and b_n are the expansion terms corresponding respectively to the forward and backward component of the mode $n=∼\{m,p\}$ , where m and p are the azimuthal and radial indices of the mode n. $\mathbf{E}_{n}^{\mathit{forward}}$ , $\mathbf{E}_{n}^{backward}$ , $\mathbf{H}_{n}^{\mathit{forward}}$ , $\mathbf{H}_{n}^{backward}$ are the eigenvectors of the electric field E and magnetic field H, corresponding to the forward and backward components of mode n. It can be shown that the forward and backward components are related as follows [16] $\begin{eqnarray} \mathbf{E}_{n}^{backward}=\mathbf{E}_{t,n}^{\mathit{forward}}-\mathit{E}_{z,n}^{\mathit{forward}}\hat{z},\quad \mathbf{H}_{n}^{backward}=-\mathbf{H}_{t,n}^{\mathit{forward}}+\mathit{H}_{z,n}^{forward}\hat{z} \end{eqnarray}$ (2) where the subscript t denotes the transverse component of mode n. The orthogonality condition is expressed as $\begin{eqnarray}\int _{S}(\mathbf{E}_{n}\times \mathbf{H}_{I}^{\ast })\cdot d\mathbf{S}={\delta}_{nl}C_{n};n=\{m,p\},l=\{m^{\prime },p^{\prime }\}\end{eqnarray}$ (3) with S the normal vector to the cross-sectional area of the waveguide and δ_nl the Kronecker’s delta. C_n describes the power of the nth mode and it is given by $\begin{eqnarray}C_{n}=\int _{S}(\mathbf{E}_{n}\times \mathbf{H}_{n}^{\ast })\cdot d\mathbf{S}.\end{eqnarray}$ (4) Since we are primarily interested in the propagating mode along the z-axis, only the transverse components of the modes are involved. By substituting ((2)) in ((1)), the transverse component of the field is expressed as $\begin{eqnarray}\mathbf{E}_{t}=\mathop{\sum }_{n}(a_{n}+b_{n})\mathbf{E}_{n,t},\quad \mathbf{H}_{t}=\mathop{\sum }_{n}(a_{n}-b_{n})\mathbf{H}_{n,t}\end{eqnarray}$ (5) where, E_{n, t} and H_{n, t} are the transverse eigenvectors of the electric and magnetic field, respectively, regardless the direction (forward or backward).

For every mode n taken into account in the decomposition, we consider the following cross products between the simulated fields and the mode eigenvectors, taking into account (5) $\begin{eqnarray}I_{1}=\int _{S}(\mathbf{E}_{sim}\times \mathbf{H}_{n}^{\ast })\cdot d\mathbf{S}=(a_{n}+b_{n})C_{n},\quad I_{2}=\int _{S}(\mathbf{E}_{n}^{\ast }\times \mathbf{H}_{sim})\cdot d\mathbf{S}=(a_{n}-b_{n})C_{n}^{\ast }\end{eqnarray}$ (6) where by sim the fields calculated by COCHLEA code are denoted, after their transformation in frequency domain, and by n the analytic fields are denoted, which are given by the following expressions $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}E_{r}(r,{\varphi},{\omega})=A{\displaystyle \frac{m{\omega}{\mu}_{0}}{h_{n}^{2}r}}e^{jm{\varphi}}J_{m}(h_{n}r),\quad E_{{\varphi}}(r,{\varphi},{\omega})=A{\displaystyle \frac{{\omega}{\mu}_{0}}{h_{n}}}e^{jm{\varphi}}J_{m}^{\prime }(h_{n}r)\\[10.0pt] H_{r}(r,{\varphi},{\omega})=E_{{\varphi}}(r,{\varphi},{\omega})/Z_{n},\quad H_{{\varphi}}(r,{\varphi},{\omega})=E_{r}(r,{\varphi},{\omega})/Z_{n}\end{array} & & \displaystyle\end{eqnarray}$ (7) where h_n is the transverse wavenumber of the mode n and Z_n the wave impedance of it.

For each port s, the forward (a_{n, s}) and backward (b_{n, s}) expansion terms of the mode n are calculated by $\begin{eqnarray}a_{n,s}=0.5\left(\frac{I_{1,s}}{C_{n,s}}+\frac{I_{2,s}}{C_{n,s}^{\ast }}\right),\quad b_{n,s}=0.5\left(\frac{I_{1,s}}{C_{n,s}}-\frac{I_{2,s}}{C_{n,s}^{\ast }}\right).\end{eqnarray}$ (8) The scattering parameters for each mode n are then calculated using the following relations $\begin{eqnarray}S_{11}=\frac{b_{n,in}}{a_{n,in}},\quad S_{12}=\frac{b_{n,in}}{b_{n,out}},\quad S_{21}=\frac{a_{n,out}}{a_{n,in}},\quad S_{22}=\frac{b_{n,out}}{a_{n,out}}.\end{eqnarray}$ (9) The total propagating power of the mode n is related with the forward and backward expansion terms as $\begin{eqnarray}P_{s,n}=0.5Re\{(a_{n,s}+b_{n,s})(a_{n,s}^{\ast }-b_{n,s}^{\ast })C_{n}\}.\end{eqnarray}$ (10) In addition, for each mode n we can define the power balance as $\begin{eqnarray}|S_{B,n}|^{2}=|S_{11,n}|^{2}+|S_{21,n}|^{2}-\left|\frac{b_{2,n}}{a_{1,n}}\right|^{2}\end{eqnarray}$ (11) where the subtracted term is the power corresponding to the backward component of mode n at port 2 which represents the possible reflections by the PML region, and the total power balance is defined as $\begin{eqnarray}S_{B,t}=\sqrt{\displaystyle \mathop{\sum }_{n}|S_{B,n}|^{2}}\end{eqnarray}$ (12) representing the power that exits the structure in each direction. Note that, for a lossless, non-resonant structure S_{B, t} = 1. Smaller values than unity indicate possible resonances of the structure. Of course, the presented methodology is not restricted to the specific geometry considered in this work, but it is valid to any structure where two ports are defined.

The use of analytic expressions for the modes used in the mode decomposition and scattering parameters, allows the study of the behavior of forward and backward components of each mode and at each port individually. Furthermore, faster convergence of the results is achieved with fewer computational resources compared to available commercial tools since the mode basis is analytically provided rather than numerically calculated.

3. Problem under consideration

The structure considered in this work is a smooth waveguide segment of length L and radius R, consisted of three segments. The first and third segments are smooth waveguides with respective lengths L₁ and L₃ and radius R, whereas in the middle segment of length L₂N wedge-shaped azimuthal corrugations are engraved on its circumference, each one with depth d and azimuthal opening W. A 2D cross section as well as a 3D representation of the structure are shown in Fig. 1, whereas the 2D r-z cross section of the structure is given in Fig. 2. Two ports are defined on the structure, the input and the output one, where on each of them the field is calculated. At the input port, the excitation with TE_{m, p} mode is applied. The latter is of soft-source type, having a temporal distribution of a Gaussian modulated sinusoidal pulse and a spatially the distribution of the TE_{m, p} mode. The fields are calculated using the COCHLEA code and then are post-processed using the supplementary code COPP for the calculation of the mode decomposition and scattering parameters by employing the methodology described in the previous section.

Fig. 1.

2D cross section (left) and 3D representation (right) of the structure considered in the study [15].

Fig. 2.

2D r-z cross section of the structure considered in the study. Note that, this figure refers to an azimuthal position where corrugations exist and the axial positions of the input and output ports are also given in the figure.

4. Numerical results

The geometrical properties of the structure under consideration are given in Table 1. We focused on the studied of the effect of the length L₂ on the mode conversion and scattering parameters and axisymmetric (m = 0) TE modes. The frequency range of the Gaussian pulse was 56 GHz–64 GHz.

Table 1
Geometrical properties of the structure under investigation

Smooth waveguide length L₁ (mm) Corrugated region length L₂ (mm) Smooth waveguide length L₃ (mm) Number of corrugations N Depth of corrugations d (mm) Width of corrugations W (degrees)

2.5 3.75 2.5 12 0.6 20

Smooth waveguide length L₁ (mm)	Corrugated region length L₂ (mm)	Smooth waveguide length L₃ (mm)	Number of corrugations N	Depth of corrugations d (mm)	Width of corrugations W (degrees)
2.5	3.75	2.5	12	0.6	20

Fig. 3.

S_{B, t} when TE_0,5 is used as the excitation.

Fig. 4.

Frequency of min(S_{B, t}) in the vicinity of 60 GHz for various excitation modes with m = 0 versus L₂.

For this study we set d = 0.6 mm, whereas L₂ = 1.25, 2.5, 3.75 and 5.0 mm. We studied the S_{B, t}, from which the resonances can be identified. In particular in Fig. 3, S_{B, t} is shown for the case where TE_0,5 was used as the excitation. It is clear that two sharp minima are present, indicating that at these frequencies the excitation mode is converted to other modes. Furthermore, it can be seen that by increasing the length, these minima shift towards smaller values. This occurs because by increasing L₂, the resonance volume increases and as a consequence, the resonance frequencies decrease. To further investigate these results, the minima of S_{B, t} in the vicinity of 60 GHz were derived, for each value of L₂ considered and for various excitation modes (Fig. 4). It is clear that the resonance frequencies decrease by increasing L₂ and they are almost identical for each of the modes used as the excitation. Therefore, the resonance frequencies when axisymmetric TE modes are used as excitation are independent of the excitation mode.

Next, we calculated the output power P_n,2 at the exit port, which correspond to each mode considered in the mode decomposition. From this quantity the modes that the excitation mode has been converted to can be found. In particular, we calculated $\begin{eqnarray}P_{n,rel}=\max (P_{n,1}+P_{n,2})/\max (P_{exc,1}+P_{exc,2})\end{eqnarray}$ (13) which denotes the maximum relative outgoing power of each mode with respect to that of the excitation mode exc. Then, the modes considered in the mode decomposition were sorted in a descending order based on their values of P_{n, rel}. The results are given in Table 2 for TE_0,5 and L₂ = 3.5 mm. Except of TE_m=0 modes, TE_m=12 modes are also excited. It is evident, that the azimuthal index of the excited modes (to which the excitation mode has been converted) is related to the azimuthal index m of the excitation mode and the number of corrugations N, according to the relation (Space Harmonic Method – SHM-criterion) $\begin{eqnarray}m_{c}=|m\pm qN|\end{eqnarray}$ (14) where q = 0,1,2, … is the harmonic index. This relation can also be found in [17] where the azimuthal mode coupling in coaxial gyrotron cavities is described, as well as in [18,19] where in general, the wavenumbers of the excited and excitation modes are related with the periodicity.

Table 2

Excited modes and relative power for various modes used as an excitation and L₂ = 3.5 mm

Excitation mode	Excited modes (relative power)
TE_0,1	TE_0,1 (1), TE_0,5 (0.22), TE_12,2 (0.16), TE_0,6 (0.15), TE_12,1 (0.08)
TE_0,2	TE_0,2 (1), TE0_,5 (0.23), TE_12,2 (0.17), TE_0,6 (0.16), TE_12,1 (0.08)
TE_0,3	TE_0,3 (1), TE_0,5 (0.23), TE_12,2 (0.17), TE_0,6 (0.15), TE_12,1 (0.08)
TE_0,4	TE_0,4 (1), TE_0,5 (0.25), TE_12,2 (0.18), TE_0,6 (0.16), TE_12,1 (0.08)
TE_0,5	TE_0,5 (1), TE_12,2 (0.22), TE_0,6 (0.2), TE_12,1 (0.10), TE_0,4 (0.09)

Fig. 5.

Relative outgoing power for L₂ = 3.5 mm and various excitation modes.

The P_{n, rel} of some indicative excited modes are given in Fig. 5. It can be seen that, by increasing the radial index of the incident mode p, P_{n, rel} for n = TE_0,6 and TE_12,2 increases. This result is attributed to the fact that by increasing p for a specific mode, the transverse wavenumber increases. Therefore, the field penetration in the azimuthal corrugations is facilitated and the modes with similar transverse wavenumbers will be excited. Of course, the azimuthal indices of the excited modes follow the criterion presented above, in this case for q = 1 and q = 0.

To demonstrate the validity of the results derived by COCHLEA, the latter were compared to those derived by CST Studio Suite [20]. The |S₂₁| of TE_0,5 versus L₂ was calculated for d = 0.6 mm and depicted in Fig. 6 for COCHLEA and in Fig. 7 for CST Studio Suite. It is clear that, despite the inherent differences between the two codes, the results of COCHLEA are in a very good agreement with those derived by CST Studio Suite. Furthermore, the frequency of the minimum of the S_{b, t} in the vicinity of 60 GHz was also calculated by CST Studio Suite (Fig. 8). The corresponding results present a similar behavior with those obtained by COCHLEA, whereas the small discrepancy observed is attributed to the inherent differences between the two codes’ solvers.

Fig. 6.

|S₂₁| for excitation with TE₀₅ for various values of L₂ and results derived by COCHLEA.

Fig. 7.

|S₂₁| for excitation with TE₀₅ for various values of L₂ and results derived by CST Studio Suite.

Fig. 8.

Frequency of min(S_{B, t}) in the vicinity of 60 GHz for various excitation modes with m = 0 versus L₂ from results derived by CST Studio Suite.

5. Conclusion

In this work, an azimuthally corrugated waveguide was studied using the FDTD code COCHLEA. In particular, the mathematical formulation of the mode conversion and scattering parameters was derived followed by the study of the effect of the length of the corrugated region on the mode conversion. It was found that, by increasing the aforementioned length the mode conversion is facilitated and other modes than the incident one are also excited, the azimuthal indices of which follow the SHM criterion. The results were also verified against those derived by CST Studio Suite and found that are in a very good agreement.

Footnotes

Acknowledgement

This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 – EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

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FDTD investigation of cylindrical waveguides with azimuthal corrugations excited by axisymmetric TE modes

Abstract

Keywords

1. Introduction

2. Mode decomposition

Table 1 Geometrical properties of the structure under investigation Smooth waveguide length L1 (mm) Corrugated region length L2 (mm) Smooth waveguide length L3 (mm) Number of corrugations N Depth of corrugations d (mm) Width of corrugations W (degrees) 2.5 3.75 2.5 12 0.6 20

Footnotes

Acknowledgement

References

Table 1
Geometrical properties of the structure under investigation

Smooth waveguide length L₁ (mm) Corrugated region length L₂ (mm) Smooth waveguide length L₃ (mm) Number of corrugations N Depth of corrugations d (mm) Width of corrugations W (degrees)

2.5 3.75 2.5 12 0.6 20