The pattern place error is one of the main factors that influence the accuracy in the metrology of aspheric surfaces based on computer-generated holograms (CGHs). This error is difficult to calibrate with traditional methods. Hence, a new method for calibrating the wavefront distortion introduced by the pattern place error on the basis of the complex phase is proposed. According to the design and manufacturing process of CGHs, the main errors consist of the substrate error, pattern processing error, and imaging distortion. The substrate error includes the surface figure error, surface wedge angle error, and refractive index inhomogeneity of the substrate material. The pattern processing error includes the pattern place error, duty cycle error, and etching depth error. The imaging distortion mainly affects the mapping error of the imaging place, which should be carefully considered when the measurement data is used to process the surface. Of these errors, the substrate error can be well calibrated and compensated during data processing, and the imaging distortion can be corrected by means of image distortion processing. However, there is no way to calibrate the pattern processing error, especially the error caused by the pattern place error. The wavefront distortion induced by the pattern place error maintains a level around the sub-nanometer. Therefore, if the impact of this error can be eliminated, the measurement accuracy can be better than 0.1 nm (root mean square, RMS) theoretically. To this end, a calibration method using the complex-phase CGH which generates the auxiliary wavefront and test wavefront simultaneously is proposed in this paper.

The CGH which can generate multiple wavefronts simultaneously is designed in this paper. The wavefronts include the plane wavefront along the x direction, the plane wavefront along the y direction, the spherical wavefront, and the aspherical wavefront. The plane wavefront and spherical wavefront are auxiliary wavefronts, and the aspherical wavefront is the test wavefront which can match the aspheric surface to be tested. In this design method, the auxiliary wavefront is designed, which inversely propagates to the CGH surface first, and the test wavefront also inversely propagates to the CGH surface. The complex phase is obtained through the coherent superposition of multi-wavefronts. The auxiliary wavefront is used to calculate the pattern place error which is defined as the deviation between the actual pattern position and the design position, and then calculate the test wavefront error caused by the pattern place error. If the error can be calculated, it can be eliminated from the measurement result. The test wavefront is used to obtain the wavefront which matches the aspheric surface to be tested.

According to the aspheric surface parameters given in Table 1, a complex-phase CGH is designed to verify the proposed method in this paper, and the calibration process of the system error is described for this CGH. Using the calibration steps shown in Fig. 7, we can determine and calibrate the impact of the CGH's pattern place error-induced wavefront distortion. The calibration method contains four steps. Firstly, measuring the +1 order diffraction wavefront in the x direction. Secondly, measuring the -1 order diffraction wavefront in the x direction. Thirdly, measuring the +1 order diffraction wavefront in the y direction. Fourthly, measuring the -1 order diffraction wavefront in the y direction. Finally, the pattern place error-induced wavefront distortion is calculated through Eqs. (28)-(34).

Different kinds of errors need to be calibrated in the process of ultra-high precision testing of optical aspheric surfaces. To calibrate the pattern place error-induced wavefront distortion, this paper uses a complex-phase encoding CGH and proposes a method based on this CGH. In this method, the auxiliary wavefront (including plane wavefront in the orthogonal x and y directions) and the test wavefront (aspherical wavefront) are designed, which inversely propagate to the CGH surface, and the continuous complex phase on the CGH surface is obtained by coherent superposition of wavefronts. The complex-phase pattern is imported into the diffractive simulation software to calculate the wavefront, and the result shows that each wavefront can be obtained, and the light intensity of each wavefront can be modulated according to the amplitude. The simulation proves the feasibility of this method.