  We extend the rotation measure work of Burn (1966) to the cases of limited
sampling of lambda squared space and non-constant emission spectra. We
introduce the rotation measure transfer function (RMTF), which is an excellent
predictor of n-pi ambiguity problems with the lambda squared coverage. Rotation
measure synthesis can be implemented very efficiently on modern computers.
Because the analysis is easily applied to wide fields, one can conduct very
fast RM surveys of weak spatially extended sources. Difficult situations, for
example multiple sources along the line of sight, are easily detected and
transparently handled. Under certain conditions, it is even possible to recover
the emission as a function of Faraday depth within a single cloud of ionized
gas. Rotation measure synthesis has already been successful in discovering
widespread, weak, polarized emission associated with the Perseus cluster (De
Bruyn and Brentjens, 2005). In simple, high signal to noise situations it is as
good as traditional linear fits to polarization angle versus lambda squared
plots. However, when the situation is more complex or very weak polarized
emission at high rotation measures is expected, it is the only viable option.
