Harmonic Control in a Free Electron Laser Using an N-Frequency Undulator

We propose a theoretical framework for a multi-frequency undulator, a generalized innovative undulator design that incorporates multiple magnetic field components with distinct frequencies to enhance spectral control over harmonics in a free-electron laser. By formulating a generalized analytical expression for the radiation intensity and small-signal gain, we demonstrate how the superposition of several frequencies enables controlled excitement of multiple harmonics with improved intensity distribution. The proposed scheme naturally reduces to conventional well-known cases of planar, two-frequency, and three-frequency undulators, thereby validating the proposed novel scheme. The presence of multiple field components provides an additional degree of freedom for tailoring the output radiation spectrum, allowing for the selective enhancement of both even and odd harmonics without increasing the electron beam energy. We investigate multiple planar undulator designs, offering valuable insights into the modifications of undulator geometry for free-electron lasers. The analysis is based on the Liénard–Wiechert formalism, generalized Bessel functions, and the free electron laser pendulum equation, capturing the coupled electron dynamics and radiation processes in multi-frequency magnetic fields and offering practical guidance for undulator configuration in advanced free electron laser applications.