We study the lasing eigenvalue problems for a periodic open optical resonator made of an infinite grating of circular dielectric cylinders standing in free space, in the E- and H-polarization modes. If possessing a "negative-absorption" refractive index, such cylinders model a chain of quantum wires made of the gain material under pumping. The initial-guess values for the lasing frequencies are provided by the plane-wave scattering problems. We demonstrate a new effect: the existence of specific grating eigenmodes that have a low threshold of lasing even if the wires are optically very thin.
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