  Free-electron lasers (FELs) over virtually the entire electromagnetic
spectrum from microwaves through ultraviolet through hard x-rays that are
either seeded or start from noise. FELs can produce a variety of different
optical polarizations of the output radiation ranging from linear through
elliptic to circular polarization depending upon the characteristics of the
undulators used. For example, x-ray FELs use long undulator lines composed of a
series of relatively short undulators. Most such FELs use linearly polarized
undulators due to the ease of manufacture and tuning; hence the optical output
is linearly polarized. However, elliptic or circular polarizations are possible
by varying the orientation of the undulators along the line. Alternately,
APPLE-II or Delta undulator designs allow for producing undulating magnetic
fields with arbitrary polarizations. Here, we present a three-dimensional,
time-dependent formulation that self-consistently models two independent
optical polarizations and, therefore, completely treats any given sequence or
combination of undulator, undulator imperfections, and undulator degradation
There are two principal characteristics of the formulation that underpin this
capability. In the first place, the particle trajectories are integrated by
means of the full Lorentz force equations using three-dimensional analytic
models of the undulators fields. This permits an accurate model of the
interaction of the electrons with a large variety of undulator fields and
orientations. In the second place, the formulation allows the electrons to
couple simultaneously to two independent polarizations of the electromagnetic
wave and, therefore, allows the optical polarization to evolve
self-consistently along the undulator line. After first describing the
numerical model, we proceed to give some examples using different undulator
configurations that are of interest.
