Research was conducted on the axially symmetric four-mirror system (magnification + 1/5) for use in optical lithography using an ArF excimer laser beam. The initial design is derived from an extensive numerical calculation that makes the sum of the third-order aberration coefficients very small (SIGMA(j=i)V\S(j)\ < 10(-6). Using an optimization method (damped least-squares method) the finite aberrations are reduced; then, to obtain the diffraction-limited performance, three surfaces are aspherized. The final system has NA = 0.38 for the ArF excimer laser line (lambda = 0.193 mum) and depth of focus of 1.2 mum over a 2.6 x 2.6 cm2 object field. Nearly all rays fall within the Airy disk in the image plane. The resolution is 540 cycles mm-1 at MTF = 0.5 level for axial object point. We consider that the present four-mirror system may be further refined for use in soft x-ray lithography.