Most computer-aided alignment methods for optical systems are based on numerical algorithms at present, which omit aberration theory. This paper presents a novel alignment algorithm for three-mirror anastigmatic (TMA) telescopes using Nodal Aberration Theory (NAT). The aberration field decenter vectors and boresight error of misaligned TMA telescopes are derived. Two alignment models based on 3rd and 5th order NAT are established successively and compared in the same alignment example. It is found that the average and the maximum RMS wavefront errors in the whole field of view of 0.3° × 0.15° are 0.063 λ (λ = 1 μm) and 0.068 λ respectively after the 4th alignment action with the 3rd order model, and 0.011 λ and 0.025 λ (nominal values) respectively after the 3rd alignment action with the 5th order model. Monte-Carlo alignment simulations are carried out with the 5th order model. It shows that the 5th order model still has good performance even when the misalignment variables are large (−1 mm≤linear misalignment≤1 mm, −0.1°≤angular misalignment≤0.1°), and multiple iterative alignments are needed when the misalignment variables increase.

The traditional optical design process isolates the two steps of system performance optimization and tolerance allocation, making it difficult to achieve optimal design of as-built performance. To solve this problem, this paper proposes an analytical method for optimizing the as-built performance of optical systems. The method uses the nodal aberration theory to derive the wavefront aberration estimated value under the given surface decenter and tilt tolerance, and establishes the optical system as-built performance evaluation model. The as-built performance evaluation does not require a large amount of ray tracing, which can be completed only by tracking the paraxial marginal ray and the principal ray, and the calculation amount is small. The as-built performance evaluation model can be directly used as error function in optical design software for optical system optimization. A Cooke triplet system is taken as an example to compare the as-built performance optimization method, Code V and Zemax OpticStudio’s built-in optimization methods and the traditional method which optimizes only nominal performance by Monte-Carlo tolerance analysis.

We propose a design method for a three-mirror anastigmatic telescope with low misalignment sensitivity and deduce the analytic expression between the misalignment aberration and its optical parameters based on the nodal aberration theory. We establish an optical system as-built performance evaluation model. Using this model as the system’s as-built performance evaluation indicator, we can get an optical system that could have both low misalignment sensitivity and good image quality after optimization. The design results of a field bias three-mirror anastigmatic telescope show that the misalignment aberration of the system can be reduced by changing the spacing of the mirrors. When the spacing between the primary mirror and the secondary mirror increases and the spacing from the secondary mirror to the third mirror and the third mirror to the image plane decreases, the misalignment sensitivity will drop significantly. If the mirror spacing is changed by 10%, the misalignment sensitivity of the telescope optimized by our method is only about 85% of that of the traditional method.

Nodal aberration theory has been a powerful tool in studying the aberration behavior of on-axis optical systems that contain misaligned, or intentionally tilted and/or decentered components.^1,2 This theory develops from the wave aberration theory of Hopkins,³ based on the concept of shifted aberration field center attributed to Buchroeder⁴ and the vector multiplication developed by Shack and Thompson⁵ (it is later discovered that the operation had existed in the mathematical literature for about a century).^6,7 Thompson⁸ 9. 10. 11. ^{– 12} developed the concept of nodal aberration theory and further extended this theory to fifth order. This theory was extended to study optical systems with plane symmetry.¹³ It was also used to design off-axis optical systems, where the mirrors are tilted to obtain an unobscured configuration.^14,15 This theory was further extended to analytically express the effects of freeform surfaces on the net aberration fields several years ago.^16,17 Recently, the correctness of nodal aberration theory was verified through experiment with a Ritchey–Chrétien telescope.¹⁸

Active optics usually uses the computation models based on numerical methods to correct misalignments and figure errors at present. These methods can hardly lead to any insight into the aberration field dependencies that arise in the presence of the misalignments. An analytical alignment model based on third-order nodal aberration theory is presented for this problem, which can be utilized to compute the primary mirror astigmatic figure error and misalignments for two-mirror telescopes. Alignment simulations are conducted for an R-C telescope based on this analytical alignment model. It is shown that in the absence of wavefront measurement errors, wavefront measurements at only two field points are enough, and the correction process can be completed with only one alignment action. In the presence of wavefront measurement errors, increasing the number of field points for wavefront measurements can enhance the robustness of the alignment model. Monte Carlo simulation shows that, when −2 mm ≤ linear misalignment ≤ 2 mm, −0.1 deg ≤ angular misalignment ≤ 0.1 deg, and −0.2 λ ≤ astigmatism figure error (expressed as fringe Zernike coefficients C₅ / C₆, λ = 632.8 nm) ≤0.2 λ, the misaligned systems can be corrected to be close to nominal state without wavefront testing error. In addition, the root mean square deviation of RMS wavefront error of all the misaligned samples after being corrected is linearly related to wavefront testing error.