We theoretically studied the band gap structure of one-dimensional photonic crystals constituted by alternate layers of left-handed and right-handed materials, and found that the band gap structure is different from the usual photonic crystals. With an appropriate choice of the parameters, we show that it is possible to realize an absolute band gap for either TE or TM polarization of the electromagnetic waves, which can not be found in normal photonic crystals. The wave function of electric field in the pass band is strongly localized compared with the normal photonic crystals. And we give an example in which we substitute the left-handed materials with dispersion materials, and get different energy bands in different areas.