The design of a flat-field XUV spectrograph is optimized for a high throughput, aberration-corrected spectral image in the wavelength region of 50-300Å. The XUV spectrograph consists of a toroidal mirror for beam collection and astigmatism correction, a varied-line space (VLS) concave grating for a flat field imaging, and a photographic film detector. The VLS grating theory is derived, and some problems associated with Kita and Harada's optimization routine is discussed and corrected to include a finite line source instead of a point source. Welford's finite raytracing theory is modified for VLS grating and examined for flat-focal field XUV spectrograph. Reflection matrix at the toroidal mirror and refraction matrix at VLS grating are derived and compared with those of a holographic grating. The absolute energy efficiency of a flat-focal field spectrograph is calculated considering the reflectivities of the toroidal mirror and the concave grating and the grating efficiency. The spot diagrams to check out the alignment sensitivity of the toroidal mirror and the concave grating are obtained, and the method to achieve a best imaging of XUV spectrum is discussed. The calculated resolving power of the flat-field XUV spectrograph is more than 4,000 in the aberration-corrected wavelength range. Focused spot sizes are less than 20 x $30㎛^2$. It is shown that a high throughput characteristic can also be achieved through a careful adjustment of the alignment parameters.
