Nonlinear evolution of electromagnetic waves driven by the relativistic ring distribution

Relativistic ring distribution of plasma particles generates electromagnetic waves via the relativistic cyclotron resonance. The long time evolution of this so-called cyclotron maser instability at null wave number $(k=0)$ is studied in detail, by performing particle simulations using a plasma which consists of relativistic ring electrons, background positrons, and background electrons. The linear and nonlinear stages of the system evolution are discussed for both gyrotropic and nongyrotropic ring distributions. The linear theory predicts that, when the initial ring energy is strongly relativistic, there appears a critical initial ring momentum at which the system is marginally stable. Numerical simulations show, however, that the system is nonlinearly unstable even when the initial ring momentum exceeds the critical momentum. The final saturation level of the wave energy is obtained analytically.