Fourier Optics treatment of Classical Relativistic Electrodynamics

TL;DR

This paper integrates Fourier optics with synchrotron radiation theory, demonstrating that SR beams can be characterized as laser-like beams using wave optics, enabling near-field reconstruction from far-field data.

Contribution

It introduces a novel approach applying Fourier optics techniques to synchrotron radiation, allowing near-field distribution reconstruction from far-field measurements.

Findings

SR beams are analogous to laser beams in wave optics.

Near-field distributions can be reconstructed from far-field patterns.

Analytical formulas for near-field and far-field relations are provided.

Abstract

In this paper we couple Synchrotron Radiation (SR) theory with a branch of physical optics, namely laser beam optics. We show that the theory of laser beams is successful in characterizing radiation fields associated with any SR source. Both radiation beam generated by an ultra-relativistic electron in a magnetic device and laser beam are solutions of the wave equation based on paraxial approximation. It follows that they are similar in all aspects. In the space-frequency domain SR beams appear as laser beams whose transverse extents are large compared with the wavelength. In practical situations (e.g. undulator, bending magnet sources), radiation beams exhibit a virtual "waist" where the wavefront is often plane. Remarkably, the field distribution of a SR beam across the waist turns out to be strictly related with the inverse Fourier transform of the far-field angle distribution. Then, we take advantage of standard Fourier Optics techniques and apply the Fresnel propagation formula to characterize the SR beam. Altogether, we show that it is possible to reconstruct the near-field distribution of the SR beam outside the magnetic setup from the knowledge of the far-field pattern. The general theory of SR in the near-zone developed in this paper is illustrated for the special cases of undulator radiation, edge radiation and transition undulator radiation. Using known analytical formulas for the far-field pattern and its inverse Fourier transform we find analytical expressions for near-field distributions in terms of far-field distributions. Finally, we compare these expressions with incorrect or incomplete literature.