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Eigenvalue problems for electromagnetic resonant states on open dielectricstructures are non-Hermitian and may have exceptional points (EPs) at which twoor more eigenfrequencies and the corresponding eigenfunctions coalesce. EPs ofresonant states for photonic structures give rise to a number of unusual wavephenomena and have potentially important applications. It is relatively easy tofind a few EPs for a structure with parameters, but isolated EPs provide noinformation about their formation and variation in parameter space, and it isalways difficult to ensure that all EPs in a domain of the parameter space arefound. In this paper, we analyze EPs for a dielectric slab containing aperiodic array of circular cylinders. By tuning the periodic structure towardsa uniform slab and following the EPs continuously, we are able to obtain aprecise condition about the limiting uniform slab, and thereby order andclassify EPs as tracks with their endpoints determined analytically. It isfound that along each track, a second order EP of resonant states (with acomplex frequency) is transformed to a special kind of third order EP with areal frequency via a special fourth order EP. Our study provides a clear andcomplete picture for EPs in parameter space, and gives useful guidance to theirpractical applications.

Exceptional points of Bloch eigenmodes on a dielectric slab with a periodic array of cylinders