Spontaneous and Stimulated Undulator Radiation in Symmetric and Asymmetric Multi-Periodic Magnetic Fields

Konstantin Zhukovsky,Igor Fedorov

doi:10.3390/sym13010135

Abstract

In this work, the radiation from electrons in multi-periodic undulator fields with symmetric and asymmetric harmonics was analyzed using generalized Bessel functions formalism. The asymmetric, symmetric, and anti-symmetric periodic magnetic fields with harmonics were studied in order to get the enhanced radiation of the high harmonics of undulator radiation (UR). The effect on the spontaneous and stimulated UR was explored. The exact integral forms for the Bessel coefficients were obtained for undulators with general symmetric and asymmetric field harmonics. Spectral properties of the radiation from several configurations of the undulator fields with harmonics were compared with each other. The resulting spontaneous UR spectrum and harmonic intensities were obtained analytically in the form of integrals and compared with the respective results that were obtained numerically with SPECTRA program. The dimensionless scaling parameter of a free electron laser (FEL)—the Pierce parameter (ρ)—was computed and compared for the different considered undulators. We studied the differences in the behavior of the high-gain single pass FEL harmonics and the spontaneous UR harmonics in the same undulators. The undulators with variable deflection parameter (k) were considered. The effect of the k parameter (deflection parameter for a common planar undulator) on the spontaneous UR and on the high-gain FEL radiation was explored. In this context, an experiment with variable strength undulators at FLASH 2 FEL was analyzed; the shorter saturated length in high harmonic self-seeding (HHSS) regime vs. self-amplified spontaneous emission (SASE) is explained.

Highlights

The synchrotron radiation (SR) from electrons was predicted and discovered in the middle of the 20th century [1]. It was followed by the prediction [2] of undulator radiation (UR), which is the SR from relativistic charges in a spatially periodic magnetic field
An accurate description of the spontaneous UR intensity in an undulator can be done with the generalized Bessel and Airy functions; they account for the main loss factors for each UR harmonic
The phenomenological description of an free electron laser (FEL) is based on an accurate analytical description of spontaneous UR from an undulator and Bessel factors that individually account for all main losses for each harmonic

Summary

Introduction

The synchrotron radiation (SR) from electrons was predicted and discovered in the middle of the 20th century [1] It was followed by the prediction [2] of undulator radiation (UR), which is the SR from relativistic charges in a spatially periodic magnetic field. An accurate description of the spontaneous UR intensity in an undulator can be done with the generalized Bessel and Airy functions; they account for the main loss factors for each UR harmonic. UR intensity and Bessel factors were calculated analytically using the formalism of generalized Bessel and Airy functions, which accounts for the field harmonics, losses, and distortions by non-periodic magnetic components. A FLASH 2 FEL experiment with harmonic multiplication was analyzed

Numerical Approach to UR Calculations with the SPECTRA Code

Bessel Factors for UR Harmonics in Multiperiodic Magnetic Fields

Radiation from Elliptic and Planar Undulators with Field Harmonics

Harmonic Radiation in Undulators with Variable k Parameter