Design and optimization of the grating monochromator for soft X ray self seeding FELs The emergence of Free Electron Lasers (FEL) as a fourth generation of light sources is a breakthrough. FELs operating in the X ray range (XFEL) allow one to carry out completely new experiments that probably most of the natural sciences would benefit. Self amplified spontaneous emission (SASE) is the baseline FEL operation mode: the radiation pulse starts as a spontaneous emission from the electron bunch and is being amplified during an FEL process until it reaches saturation. The SASE FEL radiation usually has poor properties in terms of a spectral bandwidth or on the other side longitudinal coherence. Self seeding is a promising approach to narrow the SASE bandwidth of XFELs significantly in order to produce nearly transformlimited pulses. It is achieved by the radiation pulse monochromatization in the middle of an FEL amplification process. Following the successful demonstration of the self seeding setup in the hard X ray range at the LCLS there is a need for a self seeding extension into the soft X ray range. Here a numerical method to simulate the soft X ray self seeding (SXRSS) monochromator performance is presented. It allows one to perform start to end self seeded FEL simulations along with (in our case) GENESIS simulation code. Based on this method the performance of the LCLS self seeded operation was simulated showing a good agreement with an experiment. Also the SXRSS monochromator design developed in SLAC was adapted for the SASE3 type undulator beamline at the European XFEL. The optical system was studied using Gaussian beam optics wave optics propagation method and ray tracing to evaluate the performance of the monochromator itself. Wave optics analysis takes into account the actual beam wavefront of the radiation from the coherent FEL source third order aberrations and height errors from each optical element. The monochromator design is based on a toroidal VLS grating working at a fixed incidence angle mounting without both entrance and exit slits. It is optimized for the spectral range of 300 − 1200 eV providing resolving power above 7000. The proposed monochromator is composed of three mirrors and the grating. Start to end simulation as a case study of the selfseeded European XFEL performance with a proposed SXRSS monochromator is presented. It shows that the laser pulse power reaches a TW level with its spectral density about eighty times higher than that of the conventional SASE pulse at saturation.