Objective With the continuous advancement of optical design and optical manufacturing technology, the performance of optical systems has also been significantly improved. The increase in the field of view is particularly noticeable. However, an increase in the field of view can lead to an increase in the problem of distortion. It is often difficult to correct distortion at the late stage of optical design using optical design software, so it is necessary to study the correction of distortion in optical systems with large fields of view. Freeform surfaces can be used to correct distortion in optical systems due to their large degree of freedom, but current correction methods often utilize aspherical surfaces converted to freeform surfaces, which are weak, cannot correct every field of view, and have uncontrollable surface shapes. Therefore, it is necessary to develop a freeform surface design method for correcting distortion in large-field-of-view coaxial optical systems to minimize the problems of the above design methods.

Methods A freeform lens is designed on the premise of the original system, which is inserted into the original optical system as a subsequently added field lens, and optimized as the initial structure for the next step to ultimately achieve the purpose of correcting distortion. Firstly, we analyze the indexes of the original system, and extract the pupil position and image plane information as the starting and ending points. Then the main ray of each field of view after pupil exit is deflected by ray tracing (Fig.3), and the deflection is based on the law of vector refraction (Fig.4). Then the feature data points are calculated based on the point-by-point construction method (Fig.5), and since this process is carried out in a two-dimensional coordinate system, a coordinate transformation scheme is given. Finally, based on the three-dimensional coordinate points, the data are fitted based on XY polynomial freeform surfaces, which in turn gives the freeform coefficients.

Results and Discussions Based on the above design methodology, verification and simulation were carried out using optical design software. The original system of the design was chosen as the vehicle optical system, and the results and discussions were carried out after slightly adjusting it as the original system. Freeform surfaces and aspherical surfaces were fitted as comparisons (Fig.12), and the final design architecture was given (Fig.11). Three comparative studies of the simulation results were performed. In terms of SMIA-TV distortion (Fig.13), MTF (Fig.15(a)-(b)), and Monte Carlo analysis (Tab.2), the freeform surfaces designed by this method showed different degrees of improvement compared to the original system and the aspherical surfaces, and the freeform surfaces and the aspherical surfaces showed the same degree of decrease in the relative illumination compared to the original system.

Conclusions A freeform surface design method is proposed for correcting distortion in large-field-of-view coaxial optical systems at the later stage of optical design. By analyzing the indicators of the original system, the starting point coordinates, the characteristic ray and the end point coordinates, as well as the main ray emanating from the original system are extracted. The ray tracing is utilized to deflect the system based on the point-by-point method and the law of vector refraction for the purpose of distortion correction. The coordinate conversion scheme and data point fitting method are given based on XY polynomial freeform surfaces. Examples are then designed to verify the correctness of this method, comparing the system with the freeform lens with the original system and the system with the aspherical lens in various ways. The TV distortion, relative illumination, average modulation transfer function at one-half Nyquist frequency, and 90% probability average modulation transfer function at one-quarter Nyquist frequency of the original system are −6.335 5%, 65.5%, 0.54, and 0.574, respectively; The results of the system with the addition of an aspherical lens are −0.2111%, 50.2%, 0.58, and 0.587; And the results of the system with the addition of a freeform lens are 0.000 2%, 50%, 0.64, and 0.589, respectively. After analyzing the results, it is found that this design method can effectively correct the distortion problem brought by the large field of view to the coaxial optical system. However, the correction of distortion is at the expense of the relative illuminance index. The modulation transfer function of the optical system can also be improved with redundant degrees of freedom. Compared with aspherical surfaces, freeform surfaces have greater advantages in correcting distortion than aspherical surfaces at the same location.