J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)446–448
000–000

Cyclotron-Resonance Accelerations by a Generalized EM Wave
AKIMOTO Kazuhiro and HOJO Hitoshi1
School of Science and Engineering, Teikyo University, Utsunomiya 320-8851, Japan1
Plasma Research Center, University of Tsukuba, Tsukuba 305-8577, Japan(Received: 9 December 2003 / Accepted: 26 February 2004)

Abstract
Particle accelerations by a one-dimensional, electromagnetic (EM), dispersive pulse propagating along an external
magnetic field are investigated. As the pulse amplitude increases, cyclotron-resonance accelerations (CRA) evolve tononlinear-trapping, and cause particle reflections. Both the amplitude and dispersion strongly affect those accelerations,revealing interesting phenomena of resonance bifurcations.
Keywords:
cyclotron-resonance, bifurcation, acceleration, electromagnetic wave, dispersion effect

1. Introduction

particularly valid for stationary pulses like solitons. We will
present below some numerical solutions to the equation ofmotion of electrons [1], and analyze them, increasing E0.

Accelerations of charged particles by an EM wave/pulse
have been studied extensively [1]. Here, particle accelerations
by a one-dimensional (1D), EM, dispersive wave packet in amagnetic field are studied. The pulse has a Gaussian profile
with the following generalized linearly polarized electric field(and its self-consistent magnetic field) directed perpendicular
to the z-axis along which it propagates: Ex (z, t) = E0 e–{(z – vg t)/l}2 + i (k0 z – ω0 t + θ ); here, E0 , z, vg , t, l are the amplitude, theposition, the central group-velocity, the time, and a measureof the pulse-length, respectively, and θ is the phase constant;furthermore, k0 (ω 0) is the wave number (angular frequency)of the carrier wave. If the acceleration is relatively small, thebeam-like particles with only longitudinal initial velocities v0experience the following perpendicular velocity shifts [1],∆ v⊥ =πqE0 ∆ t

2m γ0⎡ e − {ω + Ω /(αγ )} ∆t / 2 + e − {ω − Ω /(αγ )} ∆t / 2⎣202200

20

2.1 Linear cyclotron-resonanceWe start with a linear case, which may be well describedby Eq. (1). Fig. 1 depicts among others velocity shifts ofelectrons with initial velocity v0 after penetrating an EM wave
packet with En = eE0 /mcω0 = 0.001 and ln = l/(c/ω 0) = 2.0.

20+ 2 cos(2θ ) e− {ω + Ω /(αγ )} ∆t / 4 − {ω − Ω /(αγ )} ∆t / 4 ⎤⎦02. Results

22021/ 20Fig. 1(1)with α = 1 – vp /v0 , ∆t = l/(βvp), β = ⏐(v0 – vg)/(v0 – vp)⏐ andγ 0 = (1 – v02/c2)–1/2.
Other notations are standard. The phase velocity isdefined as vp = ω 0 /k0 . Among dispersion effects only vg ≠ vp ,i.e., vp = 0.1c and vg = 0.05c, will be considered, assumingthat the time-scales for other effects are sufficiently longer
than those for accelerations. The assumption should be

Perpendicular velocity shifts of electrons after
interaction with a linear pulse are plotted as a functionof v0 . The dashed curve and circles respectively show
analytical and numerical solutions for final
perpendicular velocity-shifts due to a nondispersivewave packet with vp = vg = 0.1c, respectively, which arein excellent agreement with each other. The dottedcurve shows analytical perpendicular velocity shifts (1),while diamonds depict numerical ones for a dispersive
wave packet with vp = 2vg = 0.1c.

©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: akimoto@ees.teikyo-u.ac.jp1446

Akimoto K. et al., Cyclotron-Resonance Accelerations by a Generalized EM WaveFig. 2Fig. 3

Perpendicular electron velocity-shifts after interaction
with a weakly nonlinear pulse are plotted as a functionof v0 . The perpendicular velocity shifts due to thenondispersive wave packet shown here by squares
demonstrate a bifurcation of cyclotron resonance.
Other symbols are the same as in Fig. 1.

Parallel and perpendicular electron velocity-shifts due
to a nonlinear pulse are plotted as a function of v0 .
Parallel accelerations are generally stronger thanperpendicular ones

EM pulse is expected [1] only at En > (v0 – vp)/c > – En ,while Fig. 3 shows that accelerations are realized
approximately at 2En > (v0 – vp)/c > - 2En . In addition, at0.15 > v0 /c > 0.05 net accelerations are suppressed. It turns
out that at 0.2 > v0 /c > 0 the cyclotron resonance condition
cannot be satisfied. Then, what is the acceleration mechanism
present at 0.2 > v0 /c > 0.15 and 0.05 > v0 /c > 0?

Since the magnetic field strength will be held constant at Ωe= ω 0 , the electrons with v0 ≈ 0 are cyclotron-resonant. Overall
agreement between the theory and numerical results is rather
good. The reasons for the small discrepancies in the dispersive
case are small relative velocities between the particles andthe pulse as well as the dispersion. The numerical results
indicate the absence of θ dependence (not shown).
Perpendicular acceleration is dominant for linear pulses.
Hence, this is a transit-type acceleration, through which
particles penetrate the entire wave packet. In this velocity
range the dispersion increases accelerations and reduces theresonance width by factor of two, which may be analyticallyreduced from the factor 1/β = 2 in Eq. (1) [1].

2.4 Phase-trapping and nonresonant cyclotron
acceleration
The shift of CRA to nonlinear trapping may be explainedin terms of phase trapping [2], which is an interaction
mechanism applied to whistler waves and more recently toEM ion cyclotron waves; these waves are right-circularly
polarized EM waves with l = ∞. For particles interacting withsuch a wave there exists a second Hamiltonian H2 [2]:2.2 Weakly nonlinear cyclotron-resonance1⎛ VΩ ⎞
H2 = ⎜ γ
cosα + e ⎟k 0c ⎠2⎝ cFig. 2 shows final velocity shifts of electrons for aweakly nonlinear wave packet with En = 0.01. Here, evidently
the linear theory hardly accounts for the numerical values,
which show much broader acceleration region as well as
multi-peaks. The dispersion slightly reduces the resonancewidth, while the maximum velocity shifts remain hardly
affected. However, the most outstanding dispersion effect isthe further bifurcation of the resonance peaks. The strongest
acceleration occurs at v0 = –0.02c, while that in Fig. 1 occurs
at v0 ≈ 0. At this stage the θ dependence is still absent (notshown). Owing to the relatively small longitudinal
accelerations (not shown), the particles traverse the entire
wave packet, while being accelerated.

+γω 0 B1 Vk 0c B 0 c

2sin α cosθHere, V = ⏐v0 – vp⏐, γ is the Lorentz factor of the particle, B1
is the magnetic field amplitude of the wave, α is the pitch
angle of the particle measured in the wave frame, i.e., v⊥ /(vz– vp) = tanα. The lowest values of H2 are given when θ = π.
Therefore, H2 with v0 = 0, V = 0.1c and θ = π is plotted inFig. 4 with various values of En . The electrons which startfrom cosα = –1, i.e., cyclotron resonance [2] may travel along
the dashed line until they hit a corresponding curve, where
they are reflected. In Fig. 4 and H2 it is evident that as En is
increased the resonance center shifts from cosα = –1(v0 = 0)
toward 0(v0 = vp) and the width broadens considerably.
Namely, the overall trend of electron velocity-shifts in Fig. 3
may be at least qualitatively explained in terms of the phasetrapping.

2.3 Nonlinear cyclotron-resonanceFig. 3 presents the electron velocity-shifts due to anonlinear dispersive wave packet with En = 0.1. Diamonds
depict maximum perpendicular velocity-shifts, while theupward and downward triangles show maximum andminimum parallel ones (and significant θ dependence),
respectively. It is observed that strongly accelerated particles
are reflected by the wave packet. This case differs from theprevious ones dramatically. Its overall appearance isreminiscent of quasi-trapping (QT) [1]. However, QT by an

3. Conclusions
As the pulse amplitude increases and/or the pulse ismade more dispersive, the cyclotron resonance bifurcates,
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Akimoto K. et al., Cyclotron-Resonance Accelerations by a Generalized EM Waveshowing multi-resonance velocities, and finally transforms tophase trapping [2] characterized by large velocity shifts andbroad resonance widths. Also, as the amplitude of wave
packet increases, the transit-type CRA is transformed to thereflection-type phase-trapping, and the center of the resonancevelocity consequently shifts to vp. It is likely that thesebifurcations are caused by the finite extent of wave packet
and the phase trapping.

ReferencesFig. 4[1] K. Akimoto, Phys. Plasmas 4, 3101 (1997); ibid. 9, 3721(2002); ibid. 10, 4224 (2003).
[2] R.F. Lutomirski and R.N. Sudan, Phys. Review 147, 147(1966).

The second Hamiltonian H2 is plotted as a function ofcosα for En = 0.001, 0.003, 0.01, 0.03, and 0.1 (from topto bottom, respectively). The dashed line depicts H2 = 0.

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