A strongly nonlinear wake wave driven by an intense laser pulse can act as a partially reflecting relativistic mirror (the flying mirror) [S. V. Bulanov, et al. Bulletin of the Lebedev Physics Institute, No. 6, 9 (1991); S. V. Bulanov, et al., Phys Rev. Lett. 91, 085001 (2003)]. Upon reflection fi'om such mirror, a counter-propagating optical-fi'equency laser pulse is directly converted into high-fi'equency radiation, with a fi'cquency multiplication factor ∼ 4y2 (the double Doppler effect). We present the results of recent experiment in which the photon number in the reflected radiation was at least several thousand times larger than in our proof-of-principle experiment [M. Kando, et al., Phys Rev. Lett. 99, 135001 (2007); A. S. Pirozhkov, et al., Phys Plasmas 14, 123106 (2007)]. The flying mirror holds promise of generating intense coherent ultrashort XUV and x-ray pulses that inherit their temporal shape and polarization from the original optical-frequency (laser) pulses. Furthermore, the reflected radiation bears important information about the reflecting wake wave itself, which can be used for its diagnostics.