The propagation properties of evanescent waves in a multi-layered structure containing materials with negative refractive index are analyzed. The decay or magnification of the amplitude of evanescent waves in different situations is explained analytically in detail. In particular, the influence of the dissipation of the negative refractive index medium on the photon tunneling effect is discussed. The propagation and energy storage of evanescent waves are shown clearly by using the incidence of a Gauss beam.