The processing precision of optical elements is increasing as current optical technology advances, as does the requirement for detection accuracy. To improve the surface shape measurement accuracy of interferometry, it is necessary to remove the influence of systematic errors including reference surface errors in the test process and obtain the absolute surface shape of the surface to be evaluated, which is the absolute testing technology of optical surface shape. The three-sided mutual detection technique, translation rotation method, and conjugate translation difference method are now the most regularly used absolute testing methods for the surface shape of planar optical components. The conjugate translation difference method is widely employed in absolute testing technology because of its short assessment cycle and no need for repeated adjustment. Translation, as a critical parameter in the conjugate translation difference approach, has a direct impact on the accuracy of surface form recognition. The traditional translation value is generally given by experience or the space size corresponding to a pixel of the detector, resulting in the measurement accuracy not reaching the optimal. In this research, the translation selection functions are the difference approximation accuracy and signal-to-noise ratio. We anticipate that the proposed approach may precisely measure the surface shape of planar optical components and enhance the scientificity and credibility of interferometry.

In this research, Gaussian white noise was added to the calculation of the absolute testing principle of conjugate translation difference surface shape to simulate the random noise introduced by environmental factors or CCD detector electronic noise in the actual measurement process. First, the error model of translation and surface reconstruction under noise was constructed by examining the accuracy and signal-to-noise ratio of differential approximation; subsequently, taking the estimated surface shape of the evaluated surface and the surface shape deviation caused by random noise as a single variable, the optimal translation interval under the allowable error of surface reconstruction was solved by utilizing the differential Zernike algorithm, and the influence of the size of random noise and the distribution of the original measured surface on the optimal translation interval was simulated and analyzed; Finally, the commercial interferometer was utilized to perform four sets of surface absolute testing experiments on the standard mirror at various translation. The measurement results were compared with the traditional three-sided mutual detection method to assess the surface measurement correctness before and after the optimization of translation.

The value of translation affects the accuracy of the differential approximation and the signal-to-noise ratio of the interferogram and then affects the final surface reconstruction result. According to the research on the surface shape measurement error induced by translation, under ideal conditions, the translation is precisely proportional to the surface shape measurement error. The larger the translation is, the larger is the surface shape measurement error (Fig. 3); in the situation of random noise interference, the ideal translation interval varies depending on the measuring environment. The larger the surface shape deviation caused by random noise, the narrower the optimal translation interval (Table 1). According to the research on the optimal translation of the original surface distribution, the broader the best translation interval is under the impact of the same noise, the greater the accuracy of the measured surface is; The surface shape distribution does not affect the value range of the optimal translation (Fig. 6). The comparison results of the conjugate translation difference method’s surface profile and the three-sided mutual inspection method’s surface profile under different translation show that only the surface shape results obtained under the optimal translation interval are consistent with the three-sided mutual inspection results, and the RMS value of the two surface shape deviation is 1.19 nm (Table 3).

Based on the inverse optimization strategy, taking the differential approximation accuracy and signal-to-noise ratio as the selected functions of the translation, we establish the translation and surface reconstruction error model under noise conditions, and study the translation and surface reconstruction error under different noise levels and initial surface error, providing theoretical guidance for selecting the appropriate translation for the tested 100 mm diameter mirror. The simulation of surface shape error under different translation shows that when the translation is small, the noise has great interference on the measurement results, that is, the signal-to-noise ratio is too low, resulting in poor surface shape reconstruction accuracy; as translation grows, the noise is efficiently suppressed, and the surface reconstruction error decreases; nevertheless, as translation increases, the surface reconstruction error increases due to the low precision of differential approximation. After the translation is optimized and determined by the random noise, initial surface shape accuracy and measurement allowable error in the actual measurement, the conjugate translation difference surface shape absolute testing technology has higher accuracy and the measurement accuracy deviation is 1.19 nm. The results of the experiments reveal that the inverse translation optimization approach improves the surface shape measurement accuracy of planar optical components. The study content fits the needs of the precision optical testing industry for fast measurement cycles and high optical element measurement accuracy.