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.
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