  It is known that for an isolated dielectric cylinder waveguide there exists
the cutoff frequency $\omega_\ast$ below which there is no guided mode. It is
shown in the paper that the infinite plane periodic array of such waveguides
possesses the guided modes in the frequency domain which is below the frequency
$\omega_\ast$. In the case of a finite array, the modes in this frequency
domain are weakly radiating ones, but their quality factor $Q$ increases with
the number of waveguides $N$ as $Q(N)\sim N^3$. This dependence is obtained
both numerically, using the multiple scattering formalism, and is justified
with a simple analytical model.
