Objective  With the flourishing development of the applied optics, there are higher and higher requirements for the imaging quality of the optical system. With multiple design degrees of freedom, the freeform surface has excellent aberration compensation capability and is widely applied in the imaging system. Thus, it is very important to select a suitable design method and successfully design a freeform surface which can effectively compensate the aberrations. Considering the design requirement of compensating aberrations, it is very appropriate to choose the design methods guided by aberration theory. Nevertheless, the optimization and design method based on classical scalar aberration theory (SAT) may not give a good result because it is mainly applicable to rotational symmetric systems and there is aberration characterization error when the SAT method is used in a non-rotational symmetric system. What's different is that the nodal aberration theory (NAT) can accurately provide the relationship between wave aberration and various terms of Zernike type freeform surfaces. So, if adopting the optimization method guided by NAT, a freeform surface with better aberration compensation ability may be attained.

  Methods  Firstly, on the basis of the wavefront aberration distribution from Zernike type freeform given by NAT, combined with self-developed iterative solution algorithm, a freeform surface optimization and design method guided by NAT is introduced in this paper. Secondly, in order to investigate the NAT optimization method's effect, a decentered telescope system is built as an example and the proposed method is utilized to optimize the freeform surface to compensate the aberrations of the decentered telescope system. Moreover, the aberration compensation experiment for the decentered telescope system is conducted by SLM loading freeform surface phase maps. Finally, the simulation and experimental results demonstrate that the freeform surface optimized by NAT method has better aberration compensation ability compared with that optimized by SAT method.

  Results and Discussion   Firstly, after clarifying the aberration characteristics of the decentered telescope system (Fig.4), the aberration compensation freeform surfaces are optimized and designed by NAT method and SAT method respectively. The optimized result shows that the wavefront aberrations of this system reduce sharply (Fig.5). Compared to SAT optimization method's result, the residual aberration is significantly smaller by NAT optimization (Fig.5). Secondly, according to the simulation spot shape and RMS radius at the image surface, it is also found that the spot size of the system has smaller RMS radius optimized by NAT method (Fig.6). Then, the MTF curve indicates the optimized decentered telescope system by NAT method has better imaging quality after the aberration compensation freeform surface is introduced (Fig.7). Finally, by means of the SLM loading optimized freeform surfaces phase maps, the experiment on compensating the aberration of the decentered telescope system is carried out (Fig.8). The experimental results also demonstrate that the aberration of the system could be effectively compensated by freeform surface (Fig.10), and the surface optimized by NAT method has stronger aberration compensation ability. Thus, combined with simulation and experimental results, it is concluded that the NAT optimization method has better performance in optimizing freeform surfaces for aberration compensation and image quality improvement.

  Conclusions  Aiming at the optimization and design of the aberration compensation freeform surface for the imaging system, a NAT optimization method is investigated in this paper. In order to explore this method's effect and compare with traditional SAT optimization method, these two methods are used to optimize the freeform surfaces for compensating the aberration of the decentered telescope system. More than that, an aberration compensation experiment for the decentered telescope is also carried out through SLM loading freeform surfaces phase maps, which could realize the same wavefront modulation effect as the freeform surfaces. Both simulation and experimental results show that better aberration compensation and imaging quality improvement effect can be achieved by using the NAT optimization method. Moreover, the proposed NAT optimization method also has great potentials in many applications, such as building individual optical model of human eye, evaluating visual quality of refractive surgery and optimizing the corneal removals for refractive surgeries, which are typical research issues in optometry.