Modulation instability of an electromagnetic wave interacting with a counterpropagating electron beam under condition of cyclotron resonance absorption

In this paper, transmission of a monochromatic wave through a counterpropagating electron beam under the condition of cyclotron resonance absorption is studied by theoretical analysis and numerical simulation. Conditions of the modulation instability (MI) are analyzed. The MI strongly affects the regimes of transmission. We also derive explicit periodic stationary solutions expressed in terms of elliptic Jacobi functions, as well as bright- and dark-soliton solutions. Analysis of these solutions allows obtaining threshold values of the driving power and frequency for the different regimes of transmission, such as cyclotron absorption, multifrequency self-modulation oscillations, and stationary single-frequency propagation. The theoretical predictions are verified by numerical simulation. In this way, we obtain the conditions at which a continuous-wave driving signal disintegrates into a close-to-periodic train of microwave soliton pulses.

Figure 1

Scheme of the resonance cyclotron interaction of an initially rectilinear electron beam with a backward electromagnetic wave.

Figure 2

Dispersion diagram $\omega \left(k\right)$. The asymptote $\omega =-k$ is shown with the dashed line.

Figure 3

Qualitative picture of the potential well profiles $W\left(I\right)$ for different $I_0$: (a) $\delta >-2$, (b) $\delta <-2$.

Figure 4

Phase trajectories in the $\left(I-I_0,dI/\phantom{dIdZ}dZ\right)$ plane for $\delta =0$.

Figure 5

Bright solitons (34) at different values of $\delta$.

Figure 6

Homoclinic loops in the phase plane for $\delta =-1.0$ and $\delta =-3.0$.

Figure 7

Dark solitons for different values of $\delta$.

Figure 8

Domains of nontransmission (1), self-modulation (2) (including generation of soliton pulse trains), and stationary signal transmission (3,4) on the $\left(\delta ,a_0\right)$ parameter plane. The results of numerical simulation are shown with circles. The theoretical boundaries (35) and (38) are shown with solid lines. The cutoff frequencies in the case of boundless medium (21) are shown with dashed lines. The boundary between domains 2 and 3 is plotted at different lengths of the system $L$. The boundaries of nontransmission band (red and blue dots) are the same for any lengths $L$.

Figure 9

Profiles of the field amplitude along the system in the nontransmission band for different values of $\delta$ ($a_0=1.0$, $L=13$). The numerical data and the analytical solutions in the form of an immobile bright soliton (34) are shown by the solid and dashed lines, respectively.

Figure 10

Generation of a close-to-periodic train of SIT solitons at the upper boundary of the nontransmission band at $\delta =1.48$. (a) Spatiotemporal pattern of the field amplitude. (b) Waveform of the output signal. Other parameters are the same as in Fig. 9.

Figure 11

Waveform $\left|a\left(\tau \right)\right|$ at $Z=4.0$. Other parameters are the same as in Fig. 9.

Figure 12

Generation of an irregular SIT soliton train at $\delta =1.53$. (a) Spatiotemporal pattern of the field amplitude. (b) Waveform of the output signal. Other parameters are the same as in Fig. 9.

Figure 13

Self-modulation regime without generation of soliton train at $\delta =2.10$. Spatiotemporal pattern of the field amplitude. (b) Waveform of the output signal. Other parameters are the same as in Fig. 9.

Figure 14

Stationary transmission in the domain of convective MI at $a_0=1$, $\delta =3.50$. (a) Spatiotemporal pattern of the field amplitude. (b) Waveform of the output signal. (c) Stationary profiles of $\left|a\right|$ and $\left|p\right|$.

Figure 15

Stationary profiles of the field amplitude at $a_0=0.5$ and different values of $\delta$ (a) and at $\delta =-2.22$ and different values of $a_0$ (b). At $a_0=0.5$ and $\delta =-2.22$, the field profile exactly coincides with the dark immobile soliton (39) shown with the dashed line.

Figure 16

Spatiotemporal pattern of the field amplitude at the nonstationary absorption regime at $\delta =-2.22$, $a_0=1.9$, and $L=13$.

Figure 17

Nonstationary regime with traveling dark solitons at $\delta =-2.22$, $a_0=0.3$, and $L=20$. (a) Spatiotemporal pattern of the field amplitude. (b) Waveform $\left|a\left(\tau \right)\right|$ at $Z=10$.