X-ray lasers from Inner-shell transitions pumped by the Free-electron laser.

We present a approach of generating femtosecond coherent x-ray pulses by using the self-amplified free-electron laser (SASE FEL) to pump the inner-shell x-ray lasers (ISXRL's). Theoretical simulations are performed. The gain characteristics are analyzed for the two representative schemes of inner-shell x-ray transitions, ie. the self-terminated x-ray lasing (1s)(-1)-->(2p)(-1) (lambda = 4.5nm) in carbon (Z = 6) and the quasi-stationary x-ray lasing (2p)(-1 )-->(3s)(-1) (lambda = 4.1nm) in calcium (Z = 20). When the 10fs x-ray FEL pulses are available at 284eV and 360eV with the pumping intensities of 1.2x10(15)W/cm(2) and 2x10(17)W/cm(2) for C and Ca, respectively, a net gain of 140cm(-1) can be predicted. Using a one-dimensional model, the properties of output ISXRL's are studied. By the Carbon ISXRL scheme, the multi-spiky SASE FEL x-ray pulse with chaotic temporal structure is smoothed to a temporally continuous x-ray pulse with a comparable duration but at a different wavelength. The Calcium scheme, can be used to create one single x-ray laser pulse with a duration as short as 2fs. The spectral bandwidth of the output ISXRL's is an order of 10 (-3), which is one order narrower than that of the SASE FEL's.