Electron orbits and stability in realizable and unrealizable wigglers of free-electron lasers

Abstract

Magnetostatic wiggler fields x coskz +y" sinkz, commonly used to model the undulators of free-electron lasers, violate Maxwell's equations and are "unrealizable. " Realizable wigglers that approximate the unrealizable ones near the axis have a radial variation and an axial field component, both of which affect electron motion. Exact helical equilibrium orbits are given for relativistic electrons in a combined uniform guide field and realizable wiggler, in cylindrical geometry. The parameter ka that measures the size of the helix also measures the imparted quiver motion, on which the gain of the laser depends. Hence, wigglers that impart substantial quiver motion necessarily have electrons far from the axis, for which the unrealizable wiggler model is not valid, A linearized stability analysis shows that the equilibrium helical orbits are either strongly unstable or else exhibit a secular growth, linear in time.The trajectory of an electron that starts from given position and velocity in combined guide field and wiggler is also found from the perturbation analysis, with corrections for realizability and for harmonics of practical wigglers, such as a bifilar winding. Although the helical orbits are either strongly or weakly unstable, a class of nonhelical bounded orbits is found when the secular behavior is suppressed.

Chaotic particle dynamics in free-electron lasers

“…Indeed, the surface-of-section plots show that the regular region of phase space diminishes in size as the wiggler amplitude is increased. In the limit where self-field effects are negligibly small, it is found that the onset of chaoticity for electron orbits with guiding center on the axis of the wiggler helix occurs whenever for the existence of helical, steady-state orbits for given electron energy Yb (Diament, 1981). Furthermore, it is shown that the onset of chaoticity for off-axis electron orbits occurs at some values of A less than Ac(0).…”

“…and yo. The motion is integrable and has been analyzed by several authors (Friedland, 1980;Diament, 1981;Freund and Drobot, 1982;Davidson and Uhm, 1982;Freund, 1983;Littlejoin, Kaufman, and Johnston, 1987;Chen and Schmidt, 1988 which determines the values of P,0 = pzo/mc in terms of the parameters aw, ao, and -Yo. Equation (3.16) is a fourth-order algebraic equation for P.0, which has at most four real roots.…”

Equilibrium electrons in free-electron lasers with a 3D helical wiggler and a guide magnetic field: Nonlinear simulations

“…In the above equations,B w is the magnetic field of a threedimensional helical wiggler with amplitude B A and wave number k w , which can be expressed as [5] …”

“…Later, it was extended to a three-dimensional (3D) helical wiggler for the electrons with on-axis guiding centers [5,6]:…”

“…Although previous perturbation analysis and Poincaré-map analysis have been devoted to this subject, fundamental approximation was presupposed in these analytical derivations; for example, either the wiggler field was restricted to have onedimensional distribution [3,4,9,11,12] or was averaged over the cyclotron angle [8], or the electron's guiding center was treated to be on axis [5][6][7]10], and the adiabatic field over the initial region of the wiggler was mostly neglected. In practice, however, these factors may substantially affect the equilibrium electrons' motion, stability, and transmission.…”