Abstract
To achieve low wavefront error and high throughput with vignetting-free target simulators, a concave-convex-concaveoff-axis spherical system is designed in this paper, which effectively eliminates astigmatism and coma of the system by fields of view (FOVs) and aperture offset and provides theoretical basis for the realization of 100% matching between the exit pupil of the field lens and entrance pupil of the three-mirror anastigmat (TMA) system. Only spherical mirror element is used in this target simulator, which not only reduces the difficulty of manufacturing and cost but also greatly reduces the difficulty of assembling and adjusting. It provides an effective scheme for the design of target simulator and has strong engineering application value.
1. Introduction
With the rapid development of remote sensing, optical systems are moving toward a large FOV and high resolution in many fields, such as space exploration, earth science, and military applications [1–3]. These optical systems are generally characterized by large aperture, high precision, and long focal length. In the detection process of the optical system, it is necessary to use small-diameter equivalent target to detect and monitor the manufacturing accuracy of the system; so, the target simulator was born. Sullivan et al. described the capabilities, design, manufacturing and integration status, and uses of OSIM, an optical simulator of the James Webb Space Telescope (JWST) Optical Telescope Element (OTE) [4]. Toshihiro proposed a design approach of a coaxis double TMA for designing a target simulator with a large field of view (FOV) [5]. TMA optical systems have strong aberration correction ability, so that it is easy to get larger FOV, larger relative aperture, and better imaging quality [6, 7]. Taking the famous Large Synoptic Survey Telescope (LSST) as an example, this coaxial TMA telescope with a FOV of 3.5° is composed of an 8.4 m primary mirror, a 3.4 m second mirror, and a 5.0 m tertiary mirror [8]. Off-axis TMA systems have great potential to provide good optical solutions to meet the requirement of high resolution while maintaining a large FOV [9–12]. Off-axis configurations usually make the aperture stop offset or the FOV biased to avoid obscuration and achieve a large FOV [13, 14].
In order to reduce the difficulty of manufacturing, cost, and installation of target simulator, a target simulator is proposed and designed, which is consisted only with spherical mirror. Also, the rotationally asymmetric aberrations which are mainly astigmatism and coma are eliminated based on FOVs and aperture offset. The engineering feasibility of the design of the target simulator has been confirmed by simulation analysis.
2. Selection of the Initial Configuration
For off-axis TMA systems with relay images, there are two types of configuration design that are widely adopted: concave-convex-concave systems and convex-concave-convex systems [7, 15].
The initial structure of the target simulator is from Offner spectrometer structure. Offner spectrometer consisted by three mirrors, and it is a finite remote imaging system. Concave-convex-concave systems have a strong ability to correct spherical, coma, astigmatism, and field curvature aberrations. In this paper, the finite remote target simulator is realized by matching the exit pupil of the field lens with the entrance pupil of the convex-concave-convex TMA system, and the optical path is shown in Figure 1(a). To simulate the finite optical path of the imaging system, the image plane and exit pupil of the imaging system are taken as the object plane and entrance pupil of the target simulator, respectively, as shown in Figure 1(b). The exit pupil of the TMA system is located behind the primary mirror (PM), and all mirrors are spherical, which reduces the difficulty of processing and assembling of the system. The marginal ray tracing of the system is shown in Figure 2.
To match the optical path of the imaging system, we focus on tracing the marginal ray behind the primary image plane, and the propagation of the marginal ray is illustrated in Figure 3. u is the paraxial ray’s incident aperture angle with respect to the optical axis, u′ is the corresponding exit aperture angle, y represents the heights of the marginal ray, and t1, t2, t3, and t4 represent the distance between the entrance pupil and PM, the distance between PM and the secondary mirror (SM), the distance between SM and the tertiary mirror (TM), and the distance between TM and the image plane, respectively.
We set transmission media refractive index as , . For PM, the optical power can be determined aswhere i (i = 1, 2, 3, 4) represents the PM, SM, TM, and the image plane, respectively, and ci and ri is the curvature and radius.
By the paraxial ray tracing, the relationship between the ray height and the ray slope on different mirrors can be expressed as follows [16]:
Similarly, the ray tracing of the chief ray can be expressed as follows:
According to the Seidel aberration theory, in a rotationally symmetric optical system, five monochromatic aberrations, including spherical aberration (SI), coma (SII), astigmatism (SIII), field curvature (SIV), and distortion (SV), can be simplified as follows:where A and represent the Snell invariant of the marginal ray and chief ray, respectively, and H is the Lagrange invariant, which can be calculated as follows:
In this paper, our main concerns are about the coma and astigmatism of the TMA. For a general optical design process, a specific Seidel coefficient is expected to be zero to meet different design requirements. An initial configuration is obtained by setting reasonable initial values.
3. Off-Axis TMA Systems Design
The parameters of the system are shown in Table 1. The FOV distribution of the object space is shown in Figure 4, and ray tracing in different FOVs is marked with different colors. The PM, SM, and TM are tilted and eccentric to obtain the initial configuration of the TMA system without central obscuration, as shown in Figure 5. The off-axis system is an effective way to avoid the system blocking and vignetting. Therefore, these two phenomena need not be considered in the following analysis.
4. Design Results and Tolerance Analysis
4.1. Design Results
The structure of the final optical system is very compact, as shown in Figure 6. The detailed parameters of the system are listed in Table 2. It can be seen in Figure 7 that the spot radius in the spectral range of 0.43∼0.8 μm is within a single pixel. Figure 8 shows the modulation transfer function (MTF) of the system and the MTF is greater than 0.5 at 34 lp/mm.
4.2. Tolerance Analysis
The tightness of the tolerance directly determines the difficulty of the processing and assembly of the system, so it is necessary to reasonably allocate the tolerance of the system, and the tolerance distribution is listed in Table 3.
We performed 100 simulations using Monte Carlo analysis to predict the performance. The diffractions MTF of the system in different FOVs under different probability distributions are listed in Table 4.
5. Conclusion
In this paper, by ray tracing of the concave-convex-concave coaxial system, the initial configuration in which the exit pupil of the TMA system is matched with the entrance pupil of the field lens is obtained, and the astigmatism and coma of the system are eliminated based on FOVs and aperture offset. By optimizing the eccentricity and tilt of the PM, SM, and TM, a low wavefront error and high throughput with vignetting-free target simulator is designed. Finally, we analyze the tolerance and engineering feasibility of the system, which confirms the validity of the target simulator that all mirrors are spherical.
Data Availability
The simulation analysis data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The research was funded by Project of Shaanxi Provincial Science and the Technology Department (Grant no. 2022GY-222) and “Belt and Road” Innovative Talent Exchange Program for Foreign Experts (Grant no. DL2022040006L).