  We report on critical phenomena in the gravitational collapse of
electromagnetic waves. Generalizing earlier results that focused on dipole
electromagnetic waves, we here compare with quadrupole waves in axisymmetry. We
perform numerical simulations of dipole and quadrupole wave initial data,
fine-tuning both sets of data to the onset of black hole formation in order to
study the critical solution and related critical phenomena. We observe that
different multipole moments have different symmetries, indicating that the
critical solution for electromagnetic waves cannot be unique, at least not
globally. This is confirmed in our numerical simulations: while dipole data
lead to a single center of collapse, at the center of symmetry, quadrupole data
feature two separate centers of collapse on the symmetry axis, above and below
the center of symmetry -- reminiscent of similar findings reported for critical
collapse of vacuum gravitational waves. While the critical solution for neither
the dipole nor the quadrupole data is exactly self-similar, we find that their
approximate echoing periods appear to differ, as do the critical exponents. We
discuss whether the centers of collapse found for dipole and quadrupole data
might all have the same properties, which would suggest a "local uniqueness" of
the critical solution. Instead, we provide some evidence -- including the
differing echoing periods and critical exponents -- suggesting that the
critical solutions are distinct even locally. We speculate on the implications
of our findings for critical phenomena in the collapse of vacuum gravitational
waves, which share with electromagnetic waves the absence of a spherically
symmetric critical solution.
