Negative Hartman effect in one-dimensional photonic crystals with negative refractive materials

The Hartman effect inside the one-dimensional photonic crystals (1DPC's) composed of negative index materials (NIM's) is always negative and is reversed to the Hartman effect inside the 1DPC's composed of positive index materials (PIM's). By calculating the phases of Fourier components of a pulse accumulated inside the 1DPC's of NIM's and the evolution of the pulse inside the 1DPC's of NIM's, the origin of the negative phase time is explained. The evolution of the electromagnetic fields inside the 1DPC's of NIM's is time reversal with conjugate to that inside the 1DPC's of PIM's for real spectral pulses. An example for the practical applications to obtain the negative phase time is illustrated.