Simultaneous testing of surface figure and radius of curvature of a spherical mirror using a spatially multiplexed CGH and Zernike analysis

The surface figure accuracy and the radius of curvature (ROC) of a spherical mirror directly determine the imaging performance and stability of an optical system. Conventional computer-generated holograms (CGHs) are primarily used for detecting surface figure errors, while ROC measurement usually requires additional optical systems, thereby increasing system complexity and reducing alignment stability. This paper proposes a spatially multiplexed CGH that enables simultaneous measurement of surface figure and ROC without introducing extra optical components. The CGH consists of two functional regions: the central main hologram region, which converts a plane wave into a spherical wave to compensate for the nominal spherical mirror, and the annular reference hologram region, which converges at the spherical mirror’s vertex to produce defocus-sensitive interference fringes. A four-step phase-shifting algorithm is employed to retrieve the phase, followed by zernike fitting to extract the defocus term Z₄. The extracted Z₄ is then combined with Zemax-based inverse optimization to determine the ROC, thereby achieving simultaneous measurement of surface figure and ROC. Simulation results show high accuracy. The theoretical relative error limit of ROC measurement reaches 1.458×10^-5%, and with surface figure errors introduced, the error remains as low as 6.487×10^-5%. Further analysis confirms that the measured Z₄ term from the reference hologram is linearly correlated with ROC deviation. Using only one spatially multiplexed CGH, both surface figure and ROC of spherical mirrors can be measured simultaneously in a simple, stable setup. Therefore, the proposed method is especially suitable for compact interferometric systems requiring high-precision characterization of spherical mirrors.