The high-order convex aspherical mirror is a crucial element in the optical system, and usually used as a secondary mirror to compensate for the off-axis aberration of optical systems, however, its inspection method has always been a major challenge. The back-to-zero detection method is adopted, and a combination of three lenses and a single refractive surface is proposed to compensate for the normal aberration of the high-order aspheric surface. Firstly, the quadratic comparison surface of the high-order aspheric surface is selected to simplify the calculation. Based on the third-order aberration theory, the initial structure of the system is solved, and the normal aberration of the high-order aspheric surface is compensated. After simulation and optimization using ZEMAX software, it is shown that the design results fully meet the requirements. And then, combined with a high-order convex aspheric reflector with an effective clear aperture of 170 mm and a vertex curvature radius of 266.8 mm, the root mean square of the mirror's surface shape accuracy is measured to be 0.019 λ (λ= 632.8 nm), which meets the actual detection requirements and verifies the feasibility of the proposed design method. This method provides a new idea for the inspection of large-diameter high-order convex aspheric surfaces.