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A novel dual-band frequency selective surface (FSS) with composite element is presented, in which each periodic cell consists of a left-handed structure unit on the front side of the dielectric slab and a capacitive grid on its back side. As a result, a dual-band FSS operated over Ku-band (12 GHz) and Ka-band (35 GHz) is designed. Both simulated and measured results show that the dual-band FSS has good stability to oblique incidence waves for both TE and TM polarization and no grating lobe exists between the two passbands. The current distribution of the FSS has been investigated to explain the principle of the resonance, and the design formulas are also given.