  Circular microresonators are micron sized dielectric disks embedded in
material of lower refractive index. They possess modes of extremely high
Q-factors (low lasing thresholds) which makes them ideal candidates for the
realization of miniature laser sources. They have, however, the disadvantage of
isotropic light emission caused by the rotational symmetry of the system. In
order to obtain high directivity of the emission while retaining high
Q-factors, we consider a microdisk with a pointlike scatterer placed off-center
inside of the disk. We calculate the resulting resonant modes and show that
some of them possess both of the desired characteristics. The emission is
predominantly in the direction opposite to the scatterer. We show that
classical ray optics is a useful guide to optimizing the design parameters of
this system. We further find that exceptional points in the resonance spectrum
influence how complex resonance wavenumbers change if system parameters are
varied.
