  Frequency-dependent nonlinear process in microresonators is widely
acknowledged, but there is no theory available to calculate the conversion
efficiency for each resonance of the ring, except for the phase-matching one.
Similarly for azimuth-dependent nonlinear process in birefringent rings, there
is a lack of theory to analysis the conversion efficiency for each azimuth of
the ring. Consequently, it leads to old-fashioned or ill-considered coupling
position and inefficient energy conversion in birefringent microrings. Here, we
introduce spatio-temporal coupled-mode equation to describe mode spatial
properties in the cavity, compensating for the deficiency of temporal
coupled-mode equation in describing sophisticated responses. By this equation,
we find that over a wide frequency range, the extremely strong second-harmonic
generation can be achieved at different azimuth under different pumps in an
X-cut lithium niobate microring, which is important for realizing an efficient
entangled quantum light source, for example. This work will provide new ideas
and guidelines for design and applications of monolithic birefringent photonic
integrated circuits with high efficiency.
