  We study the optical transmission of a waveguide that is side-coupled to a
high-$Q$ circular microresonator. The coupling is critical if the intrinsic
resonator losses equate the coupling losses to the waveguide. When this
happens, the transmittance of the waveguide displays resonance dips with
maximal depth as the frequency is swept through the resonators resonances. We
show that multiple configurations, parameterised by the minimal distance
between the resonator and the waveguide, can lead to critical coupling. Indeed,
for a sufficiently large resonator radius, the flow of power between the
waveguide and the resonator can change sign several times within a single pass.
This leads to an oscillatory coupling parameter as a function of the separation
distance. As a result, multiple geometrical configurations can lead to critical
coupling, even if the waveguide lies in the equatorial plane of the resonator.
These results are explained using coupled-mode theory and full wave numerical
simulations. In the vicinity of secondary or higher-order critical coupling,
the depth of the transmittance dip is very sensitive to the environment. We
discuss how this effect can be exploited for sensing purpose. Alternatively, by
actively controlling the environment in the secondary critical configuration,
the waveguide/resonator system can be driven as an optical switch.
