  Eigenvalue problems for electromagnetic resonant states on open dielectric
structures are non-Hermitian and may have exceptional points (EPs) at which two
or more eigenfrequencies and the corresponding eigenfunctions coalesce. EPs of
resonant states for photonic structures give rise to a number of unusual wave
phenomena and have potentially important applications. It is relatively easy to
find a few EPs for a structure with parameters, but isolated EPs provide no
information about their formation and variation in parameter space, and it is
always difficult to ensure that all EPs in a domain of the parameter space are
found. In this paper, we analyze EPs for a dielectric slab containing a
periodic array of circular cylinders. By tuning the periodic structure towards
a uniform slab and following the EPs continuously, we are able to obtain a
precise condition about the limiting uniform slab, and thereby order and
classify EPs as tracks with their endpoints determined analytically. It is
found that along each track, a second order EP of resonant states (with a
complex frequency) is transformed to a special kind of third order EP with a
real frequency via a special fourth order EP. Our study provides a clear and
complete picture for EPs in parameter space, and gives useful guidance to their
practical applications.
