Abstract
The physics involving the emission of coherent undulator radiation from a bunch of correlated electrons wiggling inside an undulator has been very well established in the frequency domain. With the aim of describing the process of emission of coherent undulator radiation in time domain while incorporating the effect of hyperbolic field profile of the undulator and various electron beam parameters, we present a time-domain analysis which can serve as a complimentary formulation to the frequency-domain analysis of the coherent undulator radiation. The formulation requires revisiting the equation of motion of an electron moving inside a planar undulator. An electron beam with substantial spread along the direction of the magnetic field will undergo betatron oscillations due to the hyperbolic profile of the undulator fields. We show that the betatron motion can be conveniently expressed in terms of Mathieu functions. The new set of equations of motion have been used to study the effect of betatron motion on the emitted undulator radiation. The single-particle time-domain analysis is then easily extended to multiparticle systems by correctly accounting for the phase-difference between the individual electromagnetic waves emitted by different electrons occupying different phase-space in the bunch.
Highlights
The interest in studying electromagnetic radiation emitted by energetic charged particles using particle accelerators dates back to the early 1940s
The theoretical formulation of the multiparticle coherent enhancement of the synchrotron radiation was laid down by Hirschmugl [11]. She showed that the extent of enhancement of spontaneous synchrotron radiation depends on the form-factor of the electron bunch; which gives a measure of how tightly the particles in the bunch are packed around its center of mass as compared to the wavelength of the emitted radiation
The betatron motion of an electron was expressed in time-domain in terms of the Mathieu functions which allowed in understanding its effects on the radiation temporal profile and the spectrum
Summary
The interest in studying electromagnetic radiation emitted by energetic charged particles using particle accelerators dates back to the early 1940s. The theoretical formulation of the multiparticle coherent enhancement of the synchrotron radiation was laid down by Hirschmugl [11] She showed that the extent of enhancement of spontaneous synchrotron radiation depends on the form-factor of the electron bunch; which gives a measure of how tightly the particles in the bunch are packed around its center of mass as compared to the wavelength of the emitted radiation. She derived the equation for radiated power as: 2469-9888=19=22(2)=020702(10). The dependency of the intensity of the emitted electromagnetic radiation on the electron beam bunch length has been studied
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