  We numerically investigate the threshold of black-hole formation in the
gravitational collapse of electromagnetic waves in axisymmetry. We find
approximate power-law scaling $\rho_{\rm max}\sim (\eta_*-\eta)^{-2\gamma}$ of
the maximum density in the time evolution of near-subcritical data with
$\gamma\simeq 0.145$, where $\eta$ is the amplitude of the initial data. We
directly observe approximate discrete self-similarity in near-critical time
evolutions with a log-scale echoing period of $\Delta\simeq 0.55$. The critical
solution is approximately the same for two families of initial data, providing
some evidence of universality. Neither the discrete self-similarity nor the
universality, however, are exact. We speculate that the absence of an exactly
discrete self-similarity might be caused by the interplay of electromagnetic
and gravitational wave degrees of freedom, or by the presence of higher-order
angular multipoles, or both, and discuss implications of our findings for the
critical collapse of vacuum gravitational waves.
