Photonic crystal with left-handed components

•Photonic structures with left-handed materials possess rich spectrum of guided modes.•Fano resonances excited by incident wave might destroy the band transmission spectrum.•Excitation of eigenmodes enhanced the absorption in a broad frequency region.

We show that the periodic array of left-handed cylinders possesses a rich spectrum of guided modes when the negative permeability of cylinders equals exactly to minus value of permeability of embedding media. These resonances strongly influence propagation of electromagnetic waves through photonic structures made from left-handed materials. A series of Fano resonances excited by incident wave destroys the band frequency spectrum of square array of left-handed cylinders and increases considerably the absorption of transmitted waves.