Nonlinear Optics of a Single Slightly-Relativistic Cyclotron Electron

  • A. E. Kaplan
  • Y. J. Ding
Part of the Ettore Majorana International Science Series book series (volume 49)


The interaction of microwave and optical radiation with a tiny relativistic single electron can result in strong nonlinear-optical effects1−7 based on the most fundamental mechanism of nonlinear interaction of light with matter 3 Even a slight relativistic change of mass of a single free electron may result in large nonlinear effects such as hysteresis and bistability in cyclotron resonance of the electron precessing in a dc magnetic field under the action of an EM wave1. The relativistic masse-ffect consists in the increase of the effective mass of electron, m, as its speed v, or energy W, or momentum p, increase:
$$m/{{m}_{0}} = \gamma = {{(1 - {{v}^{2}}/{{c}^{2}})}^{{ - 1/2}}} = {{(1 + {{p}^{2}}/{{m}^{2}}{{c}^{2}})}^{{1/2}}},$$
where m0 is the electron rest mass, γ = W/m0c2 is the dimensionless electron energy, and c is the speed of light. Because of very low energy losses (which are due to synchrotron radiation), the relativistic change of mass to which the hystegetic resonance is attributed may be as strikingly small as 10−6 – 10−6 Consistent with predictions1, the hysteretic (bistable) cyclotron resonance of a free electron has been subsequently observed in an experiment in which a single electron has been trapped in a Penning trap for a period of time as long as 10 months.


Cyclotron Resonance Cyclotron Frequency Cyclotron Electron Propagation Configuration Nonlinear Force 
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Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • A. E. Kaplan
    • 1
  • Y. J. Ding
    • 1
  1. 1.Department of Electrical and Computer EngineeringThe Johns Hopkins UniversityBaltimoreUSA

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