A X-ray source, comprising a laser, of a pulse duration of at most 40 fs, instantaneous power of at least about 80 TW, a pulse repetition rate of at least 1 Hz; an optical compressor spectrally shaping the laser beam; focusing optics in the range between f#10 and f#15; and a gas target of electron density after ionization by the laser beam in a range between 1018 cm3 and 1019 cm−3; wherein the focusing optics focuses the laser beam in the gas target, and interaction of the focused laser beam with the gas target generates an X-ray beam, with a focused laser amplitude a0, given by a0=0.855 [IL (1018W/cm2)λL,2 (μm)]1/2, where IL is the on-target laser intensity and λL is the laser wavelength, of at least 2 and a P/Pc ratio value of at least 20, with P being the beam power and Pc a critical power given by Pc=17 (nc/n) GW where n is the electron density and nc is a critical electron density at which the plasma acts as a mirror reflecting the laser beam.