The undulator consists of two periodic systems, each of which contains a large number of magnetic poles of alternating polarity. The strength of the transverse magnetic field of the undulator varies along its z axis according to a law close to sinusoidal with a period Î»0. A relativistic particle entering the undulator at a small angle Î± to its axis is exposed to the magnetic field of the undulator, which leads to the curvature of the initial trajectory of the particle. If the initial conditions at the entrance of the particle into the undulator are selected appropriately, then the particle at each oscillation will cross the axis of the undulator every time at the same angle Î±m. The value of this angle will be an important parameter; thus, the particle will move relative to the undulator axis along an almost sinusoidal trajectory with a period Î»0. Since the force acting from the side of the static magnetic field on the particle is always directed normally to its velocity and the radiation losses are negligible, then during motion in the undulator under consideration the absolute value of the particle velocity, like its energy, is conserved. If the velocity Î½ is constant, the lengthening of its path caused by transverse oscillations relative to the undulator axis will lead to a certain decrease in its average velocity along the undulator axis Î½IIÂ = cÎ²IIÂ in comparison with its absolute value Î½ = cÎ².
Thus, in an undulator, a particle moves along a curvilinear trajectory with variable acceleration in sign and magnitude, and, therefore, emits electromagnetic radiation. The magnitude and orientation of the electric field vector of this radiation are determined by the magnitude and direction of acceleration. With a periodic change in acceleration, the electric field of undulator radiation will also be a function that is periodic in time. It should be noted that in general this function will not necessarily be harmonic.