Economic tolerance is characterized by meeting the image quality requirements and minimizing processing costs, and it thus achieves looser tolerances. In nature, it is the balance of the relationship between image quality and processing costs. At present, in the research related to the establishment of the relationship function between image quality and tolerance, the theory is macro, and modulation transfer function (MTF) is mostly used as the image quality evaluation standard for detailed study, which makes the image quality single. In the analysis based on ray tracing theory, only three structural parameters, namely the radius of curvature, thickness, and refractive index are included, with fewer types. In addition, in the current method, the differential ray tracing method cannot ensure the accuracy of the image quality function in some instances. In contrast, the Youngworth method results in excessive image quality traces. Therefore, we wish to enrich the image quality evaluation methods and improve the types of structural parameters. Meanwhile, we propose methods that have high precision and can reduce the number of image quality traces.

The functional relationship between the tolerance and the image quality is usually expressed by the form of the second-order Taylor formula. In the equation, the first-order derivative and the second-order derivative are called the first-order sensitivity coefficient and the second-order sensitivity coefficient, which can be collectively referred to as the sensitivity coefficient. The sensitivity coefficient effectively measures the sensitivity of the image quality to the structural parameters. The first-order sensitivity coefficient determines the trend of the image quality. Due to the analysis of the effect of two different structural parameters on the image quality, the second-order sensitivity coefficient ensures the accuracy of the image quality function. In this paper, wavefront aberration is used as the image quality evaluation criterion. The first-order sensitivity coefficients of eccentricity and decenter structural parameters with respect to wavefront aberration are deduced and improved based on ray tracing theory, which solves the shortcoming of a few types of structural parameters. In order to address the problem of excessive image quality traces, two methods of sequential derivation and formula transfer term are proposed to establish the mathematical model of the second-order sensitivity coefficient, so as to realize high-precision, simple, and fast establishment of function.

Firstly, a doublet optical system is optimized and designed for theoretical verification (Fig. 3, Table 1). Secondly, in order to verify the accuracy of the fitting of the function between the tolerance and the image quality established by using the theory of this paper, the fitting analysis is performed for single-parameter tolerance and two-parameter tolerance and compared with the existing Youngworth method (Figs. 4-7). The verification results reveal that both the formula transfer term method and the Youngworth method have basically identical fitting accuracy for single and two parameters. The residual sum of squares is in the range of 10^-6-10^-7, but the number of traces of the formula transfer method is far less than that of the Youngworth method, which requires a smaller amount of data. However, the sequential derivation method can only be used to analyze the optical back focal length and verifies that it is linearly with the wavefront aberration. Eventually, the economic analysis of the doublet lens is carried out according to the economic tolerance theory. The set of economic tolerances for wavefront aberration of -0.5λ is listed (Table 5), as well as a graph demonstrating the relationship between wavefront aberration and cost (Fig. 8).

The application of the tolerance and image quality relationship function proposed in this paper to establish the model can enrich and improve the variety of structural parameters, effectively reduce the number of image quality traces, and ensure the high accuracy of the fitting. The research results show that the first-order sensitivity coefficients can be solved by the formula transfer term method that can avoid the integration operation with the ray tracing theory. Although image quality tracing is applied to calculate the second-order sensitivity coefficient, the number of traces is significantly reduced compared with the Youngworth method, which relieves the pressure of data storage and maintains the high fitting accuracy of the function. This method can be used to efficiently assign economic tolerances of optical systems. The sequential derivation method requires no additional tracing image quality. However, this method involves complex solutions, difficult practice, and limited application. Currently, only the back focal length that is linear with the wavefront aberration can be effectively analyzed. In addition, the proposed tolerance wavefront aberration fitting methods have significance for other image quality evaluation methods and can promote the application of economic tolerance for optical systems.