  In whispering gallery mode (WGM) resonators light is guided by continuous
total internal reflection along a curved surface. Fabricating such resonators
from an optically nonlinear material one takes advantage of their exceptionally
high quality factors and small mode volumes to achieve extremely efficient
optical frequency conversion. Our analysis of the phase matching conditions for
optical parametric down conversion (PDC) in a spherical WGM resonator shows
their direct relation to the sum rules for photons' angular momenta and
predicts a very low parametric oscillations threshold. We realized such an
optical parametric oscillator (OPO) based on naturally phase-matched PDC in
Lithium Niobate. We demonstrated a single-mode, strongly non-degenerate OPO
with a threshold of 6.7 micro-W and linewidth under 10 MHz. This work
demonstrates the remarkable capabilities of WGM-based OPOs and opens the
perspectives for their applications in quantum and nonlinear optics,
particularly for the generation of squeezed light.
