  The representation of solutions of Maxwell's equations as superpositions of
scalar wavelets with vector coefficients developed earlier is generalized to
wavelets with polarization, which are matrix-valued. The construction proceeds
in four stages: (1) A Hilbert space H of solutions is considered, based on a
conformally invariant inner product. (2) The analytic-signal transform extends
solutions from real space-time to a complex space-time domain T (double tube).
The evaluation map E_z, which sends any solution F=B+iE in H to the value F(z)
at z\in T, is bounded. The electromagnetic wavelets are defined as the adjoints
the \Psi_z=E_z^*. (3) The eight real parameters z=x+iy\in T are given a
complete physical interpretation: x\in R^4 is interpreted as a space-time point
about which \Psi_z is focussed, and the timelike vector y gives its scale and
velocity. Thus wavelets parameterized by the set of {\sl Euclidean} points
(real space, imaginary time) have stationary centers, and the others are
Doppler-shifted versions of the former. All the wavelets can be obtained from a
single "mother wavelet" by conformal transformations. (4) A resolution of unity
is established in H, giving a representation of solutions as "atomic
compositions" of wavelets parameterized by z\in E. This yields a constructive
method for generating solutions with initial data specified locally in space
and by scale. Other representations, employing wavelets with moving centers,
are obtained by applying conformal transformations to the stationary
representation. This could be useful in the analysis of electromagnetic waves
reflected or emitted by moving objects, such as radar signals.
