  Waveform shaping and frequency synthesis based on waveform modulation is
ubiquitous in electronics, telecommunication technology, and optics. For
optical waveforms, the carrier frequency is on the order of several hundred
THz, while the modulation frequencies used in conventional devices like
electro- or acousto-optical modulators are orders of magnitude lower. As a
consequence, any new frequencies are typically very close to the fundamental.
The synthesis of new frequencies in the extreme ultraviolet (XUV), e.g. by
using relativistic oscillating mirrors, requires modulation frequencies in the
optical regime or even in the extreme ultraviolet. The latter has not been
proven possible to date. Here we demonstrate that individual strong harmonics
can indeed be generated by reflecting light off a plasma surface that
oscillates at XUV frequencies. The strong harmonics are explained by nonlinear
frequency mixing of near-infrared light and a laser-driven plasma oscillation
in the extreme ultra-violet mediated by a relativistic non-linearity.
