A quantum approach to realizable wigglers of free-electron lasers

Reference is made to the fact that the 'unrealizable' wiggler fields commonly used in investigating free-electron lasers do not satsify Maxwell's equations. The correct fields are known, however, and electron trajectories have been analyzed in these 'realizable' wigglers by investigating the classical relativistic equations of motion (Blewett and Chasman, 1977; Diament, 1981). The terminology of 'realizable' and 'unrealizable' wigglers used here is due to Diament. This problem is approached from a quantum mechanical point of view. The Klein-Gordon equation is used to obtain equations for the electron wave functions. After separating off some plane wave factors, cylindrical coordinates (r, phi, z) are introduced, and solutions are sought for the remaining function as a power series in r. The phi-dependence of each term in the expansion can then be isolated, and it can be shown that the remaining functions (of z alone) satisfy certain systems of ordinary differential equations with periodic coefficients. These systems comprise sets of coupled Hill equations, and because they involve sine and cosine functions, they can be thought of as generalized Mathieu equations.