The design for a rotationally symmetric four-mirror optical system with reduction magnification 5 x for deep UV (lambda = 248 nm of a KrF excimer laser) submicrometer lithography is developed. Initially by using the paraxial quantities, numerical solutions are found for the system, which is free from the four off-axial third-order aberrations-coma, astigmatism, field curvature, and distortion. Aspherization is carried out to the spherical mirror surfaces to reduce the axial and residual off-axial higher order aberrations. The numerical aperture of the final system is as large as 0.38, which gives a Rayleigh resolution of 0.4 mum and hence is useful in submicrometer lithographic applications.