Laguerre-Gaussian Modes in the Free Electron Laser

摘要

In a free electron laser (FEL) system knowing the optical beam characteristics is of great importance. A beam may be comprised of higher-order modes due to the interaction with the electron beam or from non-ideal operational conditions such as mirror distortions and misalignments or from imperfect injection of the electron beam. In this thesis the basic FEL theory is initially reviewed. The parabolic wave equation is then solved for the 'fundamental Gaussian mode and for higher-order modes. Working in rectangular coordinates a complete and orthogonal set of solutions involving Hermite polynomials is found. When the wave equation is solved in cylindrical coordinates we arrive at a set of solutions that contain Laguerre polynomials. The so-called Laguerre-Gaussian modes are analyzed. The evolution of these laser modes is also explored yielding quite unexpected results due to their phase structure and orbital angular momentum of light. Lastly we study a common case where higher-order optical modes appear in order to quantify the tolerances of an FEL.