Potential for Acceleration of Simulation of Dynamic Processes in Oversized Gyrotrons by Means of Using 2.5D Particle-in-Cell Method

Abstract

Simulation of gyrotrons using the particle-in-cell method predominantly requires 3D calculations. At large oversize factors of the interaction space, it leads to lengthy calculation times. We show that under certain conditions, the dimensionality of the problem can be reduced and 2.5D PIC simulations can be applied. Using the example of a 170 GHz gyrotron with TE_28,12 operating mode, we study the influence of the external signal on the noise level in the output radiation under conditions of fluctuations of the accelerating voltage.

Appendix

The physical model used in the KARAT code is based on the Maxwell equations complemented with boundary conditions for the fields at the edges of the simulation area and material equations binding the currents with field strength values. For simulations, the simulation area is split into separate cells of the orthogonal uniform mesh parallel to coordinate axes. Charge carriers are discrete with charge values equal to integer number of the elementary charge.

Electric and magnetic fields are split into two terms:$\mathrm{E}=\widetilde{\mathbf{E}}+\overline{\mathbf{E} }$, $\mathbf{B}=\widetilde{\mathbf{B}}+\overline{\mathbf{B} }$, where $\widetilde{\mathbf{E}}$ and $\widetilde{\mathbf{B}}$ are the time-dependent fields generated by the currents and the charges present in the system while $\overline{\mathbf{E} }$ and $\overline{\mathbf{B} }$ are the quasi-static external fields. Variable fields are described by Maxwell equations:

$$\begin{array}{l}\nabla \widetilde{\mathbf{B}}=\frac{4\pi }{c}\mathbf{J}+\frac{1}{c}\frac{\partial \widetilde{\mathbf{E}}}{\partial t},\\ \nabla \widetilde{\mathbf{E}}=-\frac{1}{c}\frac{\partial \widetilde{\mathbf{B}}}{\partial t},\\ \mathbf{J}=\frac{1}{\Delta V}\sum_{s}{q}_{s}{{\varvec{v}}}_{s}\end{array}$$

(4)

where c is the speed of light, ${{\varvec{v}}}_{s}$ is the velocity of the sth particle, $\Delta V$ is the cell volume, and q_s is the part of the charge in the cell.

In turn, the motion of each particle is described by Lorentz equations:

$$\frac{d{\varvec{p}}}{dt}=q\left(\mathbf{E}+\left[\frac{\mathbf{v}}{c}\times \mathbf{B}\right]\right)$$

(5)

where $\mathbf{p}=\mathrm{m}\mathbf{v}\gamma$ is the particles’ momentum, v is its velocity, γ is the relativistic mass factor, and m and q are the rest mass and the charge of the particle.

This system of equations allows for self-consistent simulation of the particles’ dynamics in the external and induced electromagnetic fields.

In order to undertake calculations almost in any frequency band, the variables are normalized within the software by the characteristic spatial scale R determined based on the dimensions of the simulated area and the assigned number of numerical mesh points:

$$\begin{array}{c}\begin{array}{cc}{\mathbf{v}}^{*}=\mathbf{v}/c={\varvec{\upbeta}},& {t}^{*}=t\bullet c/R,\end{array}\\ \begin{array}{ccc}{\mathbf{E}}^{*}=\mathbf{E}\frac{eR}{{mc}^{2}},& {\mathbf{B}}^{*}=\mathbf{B}\frac{eR}{{mc}^{2}},& {\mathbf{J}}^{*}=\mathbf{J}\frac{{eR}^{2}}{{mc}^{3}}\end{array}\end{array}$$

(6)

The equations for variable fields and momenta take the form:

$$\begin{array}{c}\left\{\begin{array}{c}\nabla {\widetilde{\mathbf{B}}}^{\boldsymbol{*}}=4\pi {\mathbf{J}}^{\boldsymbol{*}}+\frac{\partial {\widetilde{\mathbf{E}}}^{\boldsymbol{*}}}{\partial {{\varvec{t}}}^{\boldsymbol{*}}}\\ \nabla {\widetilde{\mathbf{E}}}^{\boldsymbol{*}}=-\frac{\partial {\widetilde{\mathbf{B}}}^{\boldsymbol{*}}}{\partial {{\varvec{t}}}^{\boldsymbol{*}}}\end{array}\right.\\ \frac{{d\mathbf{p}}^{*}}{dt}={\mathbf{E}}^{*}+\left[{\varvec{\upbeta}}{\mathbf{B}}^{*}\right]\end{array}$$

(7)

Hereafter, we drop the (*) sign in the dimensionless variables.

Maxwell equations are solved using the finite-difference scheme with step-over on the hexagonal meshes with half-step shift. In cylindrical coordinates $\left(R\theta z\right)$, under the assumption of axial symmetry $\partial /\partial \theta =0$, the simulated 2D area in r-z plane is covered by a hexagonal mesh with cells of h_r and h_z dimensions. For simulations of weakly relativistic gyrotrons operating at TE-type modes, with TM modes neglected, it is sufficient to use the differential equations in the difference form only for three field components, E_θ, B_r, B_z:

$$\begin{array}{l}\frac{{\left({\widetilde{E}}_{\theta }\right)}_{i,k}^{n+1}-{\left({\widetilde{E}}_{\theta }\right)}_{i,k}^{n}}{\tau }=\frac{{\left({\widetilde{B}}_{r}\right)}_{i,k+1/2}^{n+1/2}-{\left({\widetilde{B}}_{r}\right)}_{i,k-1/2}^{n+1/2}}{{h}_{z}}-\frac{{\left({\widetilde{B}}_{r}\right)}_{i,k+1/2}^{n+1/2}-{\left({\widetilde{B}}_{r}\right)}_{i,k-1/2}^{n+1/2}}{{h}_{r}}--4\pi {\left({\mathrm{J}}_{\theta }\right)}_{i,k}^{n+1/2}\\ \frac{{\left({\widetilde{B}}_{r}\right)}_{i,k-1/2}^{n+1/2}-{\left({\widetilde{B}}_{r}\right)}_{i,k-1/2}^{n-1/2}}{\tau }=\frac{{\left({\widetilde{E}}_{\theta }\right)}_{i,k}^{n}-{\left({\widetilde{E}}_{\theta }\right)}_{i,k-1}^{n}}{{h}_{z}}\\ \frac{{\left({\widetilde{B}}_{z}\right)}_{i-1/2,j-1/2,k}^{n+1/2}-{\left({\widetilde{B}}_{z}\right)}_{i-1/2,j-1/2,k}^{n-1/2}}{\tau }=-\frac{{r}_{i}{\left({\widetilde{E}}_{\theta }\right)}_{i,k}^{n}-{r}_{i-1}{\left({\widetilde{E}}_{\theta }\right)}_{i-1,k}^{n}}{{r}_{i-1/2}{h}_{r}}\end{array}$$

(8)

Here, i and k are the numbers of mesh cells along r and z. To take into account the singularity at r = 0, we use the equality ${\left.{B}_{\theta }\right|}_{r=0}=0$ and obtain:

$$\frac{{\left({\widetilde{E}}_{z}\right)}_{i,k-1/2}^{n+1}-{\left({\widetilde{E}}_{z}\right)}_{i,k-1/2}^{n}}{\tau }=\frac{4}{{h}_{r}}{\left({\widetilde{B}}_{\theta }\right)}_{1+1/2, k-1/2,}^{n+1/2}-4\pi {\left({J}_{z}\right)}_{i,k-1/2}^{n+1/2}$$

(9)

Relativistic equations of macroparticles’ motion are integrated according to the following scheme:

$$\begin{array}{c}\frac{{\mathbf{p}}_{1}-{\mathbf{p}}^{n-1/2}}{\tau /2}={\mathbf{E}}^{n},\frac{{\mathbf{p}}_{2}-{\mathbf{p}}_{1}}{\tau }=\left[\left({\mathbf{p}}_{1}+{\mathbf{p}}_{2}\right)\frac{\mathbf{B}}{2}\right],\\ \frac{{\mathbf{p}}^{n-1/2}-{\mathbf{p}}_{2}}{\tau /2}={\mathbf{E}}^{n},{\mathrm{r}}^{n=1}={\mathrm{r}}^{\mathrm{n}}+{\mathrm{v}}^{n+1/2}\end{array}$$

(10)

where ${\mathbf{p}}_{1,2}$ are the intermediate momentum values:

$$\begin{array}{c}\begin{array}{ccc}{\mathbf{p}}_{1}={\mathbf{p}}^{n-1/2}+\frac{\tau }{2}{\mathbf{E}}^{n},& {\mathbf{p}}_{2}={\mathbf{p}}_{1}+a\left[{\mathbf{p}}_{3}{\mathbf{B}}^{^{\prime}}\right],& \begin{array}{cc}{\mathbf{p}}_{3}={\mathbf{p}}_{1}+\frac{\tau }{2}\left[{\mathbf{p}}_{1}{\mathbf{B}}^{^{\prime}}\right],& {{\mathbf{p}}_{1}}^{n+1/2}={\mathbf{p}}_{2}+\frac{\tau }{2}{\mathbf{E}}^{n},\end{array}\end{array}\\ \begin{array}{cc}{\mathbf{B}}^{^{\prime}}=\frac{{\mathbf{B}}^{n+1/2}+{\mathbf{B}}^{n-1/2}}{2}\times \frac{1}{\sqrt{1+{p}^{2}}},& a=\frac{2}{1+{\left({\mathrm{B}}^{^{\prime}}\tau /2\right)}^{2}}\end{array}\end{array}$$

(11)

To eliminate the singularity, for the particles at the system’s axis, the next-step coordinates of the particle are evaluated in the rectangular Cartesian coordinate system in the following way:

$$\begin{array}{c}\begin{array}{cc}{x}^{n+1}={x}^{n}+{\tau v}_{x}^{n+1/2};& {y}^{n}={\tau v}_{y}^{n+1/2}\end{array}\\ {v}_{x}^{n+1/2}\equiv {v}_{r}^{n+1/2};{v}_{y}^{n+1/2}\equiv {v}_{\varphi }^{n+1/2};{x}^{n}={r}^{n};{y}^{n}=0\end{array}$$

(12)

Back in cylindrical coordinates, new values of the radius vector and the velocity components are evaluated as

$$\begin{array}{c}r^{n+1}=\left[x^{n+1}+y^{n+1}\right]^{1/2},\\v_r^{n+1/2}=v_x^{n+1/2}\cos\;\alpha+v_y^{n+1/2}\sin\;\alpha;v_\varphi^{n+1/2}=-v_x^{n+1/2}\sin\;\alpha+v_y^{n+1/2}\cos\;\alpha;\end{array}$$

(13)

where $\sin\;\alpha=y^{n+1}/r^{n+1};\cos\alpha=x^{n+1}/r^{n+1}$. In the case when ${r}^{n+1}=0$, it is assumed that $\cos\;\alpha=1$ and $\sin\;\alpha=0$.