  Optical parametric oscillators are widely-used pulsed and continuous-wave
tunable sources for innumerable applications, as in quantum technologies,
imaging and biophysics. A key drawback is material dispersion imposing the
phase-matching condition that generally entails a complex setup design, thus
hindering tunability and miniaturization. Here we show that the burden of
phase-matching is surprisingly absent in parametric micro-resonators adopting
monolayer transition-metal dichalcogenides as quadratic nonlinear materials. By
the exact solution of nonlinear Maxwell equations and first-principle
calculation of the semiconductor nonlinear response, we devise a novel kind of
phase-matching-free miniaturized parametric oscillator operating at
conventional pump intensities. We find that different two-dimensional
semiconductors yield degenerate and non-degenerate emission at various spectral
regions thanks to doubly-resonant mode excitation, which can be tuned through
the incidence angle of the external pump laser. In addition we show that
high-frequency electrical modulation can be achieved by doping through
electrical gating that efficiently shifts the parametric oscillation threshold.
Our results pave the way for new ultra-fast tunable micron-sized sources of
entangled photons, a key device underpinning any quantum protocol.
Highly-miniaturized optical parametric oscillators may also be employed in
lab-on-chip technologies for biophysics, environmental pollution detection and
security.

Optics