Weakly relativistic nonlinear orbit dynamics for intense ordinary-mode propagation near electron cyclotron resonance

The nonlinear orbit equations are investigated analytically and numerically for a constant-amplitude electromagnetic wave (ω_s, k_s) with ordinary-mode polarization propagating perpendicular to a uniform magnetic field B₀ ê_z. Relativisitic electron dynamics are essential in determining the maximum orbit excursion when the incident wave frequency ω_s is near the nth harmonic of the electron cyclotron frequency Ω_c. Coupled nonlinear equations are obtained for the slow evolution of the amplitude A(т) and phase ø(t) of the perpendicular orbit when ω_s ≈ nΩ_c. The dynamical equations are reduced to quadrature and simplified analytically. At exact resonance (ω_s = nΩ_c), if relativistic effects are (incorrectly) neglected, the maximum orbit excursion A_max approaches the unacceptably large value determined from the first (non-zero) solution to J_n(n_s Ω_s A_max) = 0, which totally invalidates the non-relativistic approximation. (Here n_s Ω_s = ck_s/Ω_c.) As a contrasting illustrative example, for n = 1, ω_s = Ω_c and moderate pump strength, weak relativistic effects limit the maximum orbit excursion to the approximate value , where є_s = n_s ω_s (V_z/c) V_q/c 1, and V_q is the axial quiver velocity in the applied oscillatory field. Physically, relativistic effects produce a nonlinear frequency shift that dynamically ‘detunes’ the particle motion from exact cyclotron resonance, thereby limiting amplitude growth.