  The gravitational waves emitted by a perturbed black hole ringing down are
well described by damped sinusoids, whose frequencies are those of quasinormal
modes. Typically, first-order black hole perturbation theory is used to
calculate these frequencies. Recently, it was shown that second-order effects
are necessary in binary black hole merger simulations to model the
gravitational-wave signal observed by a distant observer. Here, we show that
the horizon of a newly formed black hole after the head-on collision of two
black holes also shows evidence of non-linear modes. Specifically, we identify
one quadratic mode for the $l=2$ shear data, and two quadratic ones for the
$l=4,6$ data in simulations with varying mass ratio and boost parameter. The
quadratic mode amplitudes display a quadratic relationship with the amplitudes
of the linear modes that generate them.
