Correcting method of the finite pinhole diameter influence in calculating the modulation transfer function and energy concentration of a lens based on the point spread function

The main image quality characteristics used in optical system design and fabrication for testing produced devices are the Modulation Transfer Function (MTF) and the Encircled Energy Function (EEF) which are calculated based on pinhole test objects images captured by the testing equipment. For each inspected lens, taking into account the testing equipment parameters, the pinhole diameter, which can be considered infinitely small, can be determined. Such pinholes exhibit the properties of ideal theoretical models and do not cause additional errors. However, pinholes with a diameter greater than infinitely small are most often used in practice. The paper proposes pinhole diameter influence correction methods for EEF calculation and for increasing the maximum calculable spatial frequency in MTF calculation during lens testing based on the captured images. Inverse filtering is used for the correction. The Fourier image of a pinhole with a given diameter is used as a filter. This method has limitations due to the possibility of undefined values appearing in the corrected function when dividing by values close to zero in the pinhole spectrum. To eliminate this effect, the linear interpolation of the MTF is used at frequencies where the test object spectrum function is close to zero. This approach is necessary when correcting the pinhole diameter for EEF. For this purpose, the corrected point spread function, which is the inverse Fourier transform of the MTF corrected over the entire frequency range, is calculated. The proposed method was tested on various-sized test objects generated images. As a result of the research, a divisor function threshold value that does not cause a strong increase in the corrected MTF values due to division by values close to zero was selected for MTF correction. An expression that allows the determination of the maximum test object diameter suitable for correction using the proposed method was obtained. It is shown that the corrected MTF values interpolation allows calculating the MTF and EEF using apertures 2.5 times larger in diameter than with the standard inverse filtering method, and 8 times larger than the pinhole diameter, which can be considered infinitely small. At the same time, the maximum absolute error of the corrected MTF and EEF does not exceed 0.01 relative units. The proposed method reduces the lens testing time by eliminating the need for strict pinhole diameter selection. Furthermore, the use of pinholes with a diameter 8 times larger than infinitely small ones increases the light flow during EEF and MTF testing, which is particularly important when testing lenses with low optical efficiency (small apertures).