Flying plasma mirrors induced by intense lasers has been proposed as a promising way to generate few-cycle EUV or X-ray lasers. In addition, if such a relativistic plasma mirror can accelerate, then it would serve as an analog black hole to investigate the information loss paradox associated with the black hole Hawking evaporation. Among these applications, the reflectivity, which is usually frequency-dependent, would affect the outgoing photon spectrum and therefore impact on the analysis of the physics under investigation. In this paper, these two issues are investigated analytically and numerically with one-dimensional particle-in-cell (PIC) simulations. Based on our simulation results, we propose a new model that provides a better estimate of the reflectivity than those studied previously. Besides, we found that the peak frequency of the reflected spectrum of a gaussian incident wave deviates from the expected value, $4gamma^2omega$, due to the dependence of reflectivity on the frequency of the incident wave.
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