Modulation and demodulation of particle beams at optical frequencies

Abstract

This paper presents a convenient formulation for the discussion of experiments that are designed to modulate particle beams, such as an electron beam, and to demodulate this beam at optical frequencies. Such experiments have recently received considerable interest, and controversy, since the initial paper by Schwarz and Hora in 1969 which reported to have achieved such effects with light at 4880 A. The formalism is fully relativistic and yields conclusions that both differ from and are more general than other attempts at relativistic formulations. The formulation utilizes the concept of a density matrix to describe the particle beam, and so one may determine the effects of poor beam collimation and of nonmonoenergetic distributions without resorting to the artifice of assuming some particular wave packet. The formalism displays the beneficial effects of collimation and energy selection and, in particular, the optimum design of a « polychromator » with regard to both energy and transverse-momentum selection. The formalism for the demodulation is used to calculate directly the radiation intensity rather than the mean amplitude of the modes of the radiation field. Thus one obtains expressions for the incoherent background that is also radiated during the demodulation process and may lead experimentally to poor signal-to-noise ratios. It is shown that this background is slightly different for boson and for fermion beams. It is found, in a model-independent way, that the maximum possible signal-to-noise ratio is much poorer than that indicated by Schwarz and Hora. The recent suggestion by Schmieder of detection of modulation by the presence of a second modulator is also considered in this paper. Additional complications relating to this type of experiment are found.