In order to study the propagation characteristics of non-diffracting light waves in special materials and achieve better optical communication, one periodic circular dielectric structure with axial step change was proposed by combining traditional right-handed materials with bi-negative refractive index materials. Based on the generalized Huygens-Fresnel optical integral formula, the distribution characteristics of the emitted surface light intensity and the profile of the side light intensity of the Airy beam in this transmission medium were analyzed by using optical transmission matrix. The influence of negative refractive index parameters on the evolution of such light waves and its compensation mechanism were analyzed. The quantitative relationship between the negative refractive index and the length of the dielectric unit was analyzed when the output light wave was perfectly restored. The results show that, when the pore size of the medium decreases gradually, the diffraction effect of finite Airy beam is getting worse and worse. The profile of the emitted light intensity gradually transits from the Airy beam to the Gaussian beam. When the absolute value of nl is greater than nr, the bi-negative refractive index material layer is longer when the perfect light wave reduction is achieved on the exit surface. Conversely, the shorter. The study is helpful for analyzing optical wave communication in circular flat dielectrics with periodic or quasi-periodic axial step changes.