Time-dependent physics of single-surface multipactor discharge with two carrier frequencies

This work investigates the time-dependent physics of multipactor discharge on a single dielectric surface with a transverse rf electric field of two carrier frequencies using a multiparticle Monte Carlo simulation model with adaptive time steps. The effects of the relative strength and phase, and the frequency separation between the two carriers are studied. Closed Lissajous curves are obtained to describe the relationship between the rf electric field parallel to the surface and the normal surface charging field in the ac saturation state. It is found that two-frequency operation can reduce the multipactor strength compared to single-frequency operation with the same total rf power, though the effect of the frequency separation is not prominent on multipactor susceptibility. Formation of beat waves is observed in the temporal profiles of the normal electric field due to surface charging with a noninteger frequency ratio between the two carrier modes. Phase space evolution of multipactor electrons is examined, revealing a periodic bunching and debunching of electrons in the surface normal direction, but a gradual debunching effect in the direction tangential to the dielectric surface. Migration of the multipactor trajectory is also demonstrated for different configurations of the two-frequency rf fields.

Figure 1

Schematic of the single-surface multipactor discharge in a normal electric field and a parallel rf field with two carrier frequencies.

Figure 2

Top row: Instantaneous rf electric field $E_y$ (solid blue lines), normal electric field $E_x$ (broken black lines), and secondary electron yield $\delta$ (dotted red lines), for (a) single-frequency rf field, $\beta =0$; (b) two carrier frequencies of the rf field with frequency ratio $n=2$, relative strength of the second carrier frequency, $\beta =1$, and initial relative phase of the second carrier frequency, $\gamma =0$; (c) two carrier modes with $n=2,\beta =1,\gamma =\pi /2$; and (d) two carrier modes with $n=2,\beta =1,\gamma =\pi$. Bottom row: (e–h) The corresponding plots of trajectories of the electric field $[E_x\left(t\right),E_y\left(t\right)]$ for (a–d), respectively. The shaded cyan region is the multipactor susceptibility region obtained by applying constant electric field, $E_{y,\mathrm{dc}}$, parallel to the surface [29, 31, 39]. In (a)–(d), the average secondary electron yield ${\delta }_{\mathrm{avg}}=1$ in the saturation regime. In all the calculations, we set $f_{\mathrm{rf}}=1\mathrm{GHz}$. For the single-frequency case, we set $E_{\mathrm{rf}}=3\mathrm{MV}/\mathrm{m},\mathrm{and}$ for the cases with two carrier frequencies, we set $E_{\mathrm{rf}}=3/\sqrt{2}\mathrm{MV}/\mathrm{m}.$

Figure 3

Top row: Instantaneous rf electric field $E_y$ (solid blue lines), normal electric field $E_x$ (broken black lines), and secondary electron yield $\delta$ (dotted red lines), for two-frequency cases with frequency ratio $n=2$, initial relative phase of the second carrier frequency, $\gamma =\pi /2$, and relative strength of the second carrier frequency, (a) $\beta =1$, (b) $\beta =0.75$, (c) $\beta =0.5$, and (d) $\beta =0.25$. Bottom row: (e–h) The corresponding plots of trajectories of the electric field $[E_x\left(t\right),E_y\left(t\right)]$ for (a–d), respectively. The shaded cyan region is the multipactor susceptibility region obtained by applying constant electric field, $E_{y,\mathrm{dc}}$, parallel to the surface [29, 31, 39]. In (a)–(d), the average secondary electron yield ${\delta }_{\mathrm{avg}}=1$ in the saturation regime. In all the calculations, we set $f_{\mathrm{rf}}=1\mathrm{GHz}$ and $E_{\mathrm{rf}}=3/\sqrt{2}\mathrm{MV}/\mathrm{m}$.

Figure 4

Multipactor susceptibility with two carrier frequencies of the rf field from MC simulation for different $\gamma$ in Eq. (1), for ${\delta }_{\max 0}=3,E_{0m}/E_{\max 0}=0.005$, relative frequency of the second carrier, $n=2,$ and relative strength of the second carrier, $\beta =1$. Top row: the relative phase is kept fixed at the initial value over time [23]; bottom row: the relative phase evolves with time.

Figure 5

Top row: Horizontal (along the dielectric surface) and vertical (normal to the surface) excursions of $N=50$ multipacting macroparticles with respect to time, for (a) single-frequency rf electric field with $f_{\mathrm{rf}}=1\mathrm{GHz}$, and (b) two-frequency rf electric field with $f_{\mathrm{rf}}=1\mathrm{GHz},\theta =0$, $\beta =1,n=2,\gamma =\pi /2$. Charge contained in the macroparticles is shown in the color bar. Mean displacements of the macroparticles with respect to time are shown as projections on the horizontal and vertical planes. Bottom row: Comparison of the mean horizontal displacements for the single- and two-frequency rf electric fields with $f_{\mathrm{rf}}=1\mathrm{GHz},\theta =0,n=2$, and $\gamma =0,3\pi /16,\pi /2,\pi ,19\pi /16,\mathrm{and}3\pi /2$, for (c) $\beta =1$ and (d) $\beta =0.5$.

Figure 6

Top two rows: $v_x$ vs $x$ for the single-frequency case at times (a) $t=0.25T_{\mathrm{rf}}$, (b) $t=0.5T_{\mathrm{rf}},$ (c) $t=0.75T_{\mathrm{rf}},\left(\mathrm{d}\right)t=T_{\mathrm{rf}},\left(\mathrm{e}\right)t=1.25T_{\mathrm{rf}},\left(\mathrm{f}\right)t=1.5T_{\mathrm{rf}},\left(\mathrm{g}\right)t=1.75T_{\mathrm{rf}},\mathrm{and}\left(\mathrm{h}\right)t=2T_{\mathrm{rf}},$ where $T_{\mathrm{rf}}=1\mathrm{ns}$ is the rf period. Bottom two rows: (i–p) $v_x$ vs $x$ for the two-frequency case with $\beta =1,n=2,$ and $\gamma =\pi /2$ at the same times as (a–h), respectively.

Figure 7

Top two rows: $v_y$ vs $y$ for the single-frequency case at times (a) $t=0.25T_{\mathrm{rf}}$, (b) $t=0.5T_{\mathrm{rf}},$ (c) $t=0.75T_{\mathrm{rf}},\left(\mathrm{d}\right)t=T_{\mathrm{rf}},\left(\mathrm{e}\right)t=1.25T_{\mathrm{rf}},\left(\mathrm{f}\right)t=1.5T_{\mathrm{rf}},\left(\mathrm{g}\right)t=1.75T_{\mathrm{rf}},\mathrm{and}\left(\mathrm{h}\right)t=2T_{\mathrm{rf}},$ where $T_{\mathrm{rf}}=1\mathrm{ns}$ is the rf period. Bottom two rows: (i–p) $v_y$ vs $y$ for the two-frequency case with $\beta =1,n=2,$ and $\gamma =\pi /2$ at the same times as (a–h), respectively.

Figure 8

Top row: Multipactor susceptibility diagrams for two-frequency rf fields with frequencies (a) $f_1=1\mathrm{GHz}$ and $f_2=1.1\mathrm{GHz}$, (b) $f_1=1\mathrm{GHz}$ and $f_2=1.25\mathrm{GHz}$, and (c) $f_1=1\mathrm{GHz}$ and $f_2=1.5\mathrm{Hz}$. Bottom row: (d–(f) The instantaneous rf electric field $E_y$ (solid blue lines), normal electric field $E_x$ (broken black lines), and secondary electron yield $\delta$ (dotted red lines) for rf field configurations of plots (a–c), respectively. In (d–f), the average secondary electron yield ${\delta }_{\mathrm{avg}}=1$ in the saturation regime. In all the calculations, we set, $\gamma =0$, and $\beta =1$.

Figure 9

(a) Multipactor susceptibility diagrams for two-frequency rf fields with $\beta =1$, $\gamma =0$, and frequency ratio $1.05\le n\le 1.5.$ The shaded cyan region shows the parameter regime where the multipactor discharge develops. The upper and lower susceptibility boundaries for different frequency ratio $n$ are largely overlaid with one another. (b) Rf carrier amplitude at the lower susceptibility boundary in ac saturation state, $E_{\mathrm{rf},\mathrm{sat}}$, for two-frequency rf operation with $\beta =1$, $\gamma =0,\pi /4,\pi /2,$ and $\pi$ for time-averaged saturation values of the normal electric field $E_x=0.5\mathrm{MV}/\mathrm{m}$ (dashed lines) and $1.0\mathrm{MV}/\mathrm{m}$ (solid lines). For a fixed value of $E_x(0.5$ or $1\mathrm{MV}/\mathrm{m})$, the $E_{\mathrm{rf}}$ vs $n$ plots corresponding to different values of $\gamma$ are overlaid with one another.