Open Access
Add to BibSonomy
Abstract
In the simulation and tolerance analysis of mirror-based optical systems, the magnitude distribution of the surface shape error of each mirror must be established under certain values. Thus, in this study, several large-diameter meter-scale spherical mirrors were statistically processed, and a surface shape error model was constructed to control the RMS, PV, and PV/RMS of the surface shape error in Monte Carlo experiments. The findings show an improved system qualification probability from 69.2% to 99.4%. The proposed method considers the maximum evaluation of the sub-wavefront that eliminates the final co-phase adjustment for individual segments.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Owing to their larger angular resolution and large light-gathering capacity, telescopic objective lenses have been developed for large physical apertures. However, as large-aperture lenses are challenging to process and transport, several large-aperture telescopes have reflective mosaic primary mirror structures. Examples include the European Extremely Large Telescope (E-ELT) [1,2] and the Thirty Meter Telescope [3]. To eliminate the effect of the Earth's atmosphere on astronomical observations, NASA, ESA, and CSA co-developed the James Webb Space Telescope, which is currently the world's largest aperture-space-based observatory [4,5]. Moreover, NASA is developing space-based observatories, for example, the Large UV/Optical/IR Surveyor (LUVOIR), whose primary mirrors comprise more than 100 hexagonal mirrors [6].
Tolerance analysis is crucial in optical systems comprising mirrors from design to realization. A high-spatial-frequency surface shape error can generate spots larger than a low-spatial-frequency error, significantly affecting the imaging quality of the optical system. In the simulation, mirrors with a high spatial frequency surface shape error induce low-frequency component effects on the final system wavefront aberration during imaging, affecting the imaging of the optical system.
Several researchers have searched for the best-fitting spherical surface and the processing of aspherical surfaces; however, the study of face-shaped errors remains open [7–9]. In particular, a mathematical model of the surface shape error matching the actual machining situation is required for project realization. In this study, the machining surface shape errors of six-meter-scale aspherical mirrors are evaluated, and a mathematical model to control the root mean square (RMS), peak to valley (PV), and PV/RMS values of the errors is constructed. The simulation experiments demonstrate that the developed model is realistic and reduces positional errors compared to the default uniform distribution by adding different frequencies of face shape errors, enhancing the analytical validity.
For non-spliced pupil optical systems, the Monte Carlo method for system tolerance analysis is a standard and mature operation, and the internal operating procedures are integrated within the relevant optical design software. However, the traditional and well-established methods are inapplicable to the mosaic pupil optical system, making it a focal point for future development. Furthermore, the mosaic primary mirror involves segment splicing in co-phase. Thus, the traditional tolerance analysis evaluation of the entire mirror is inapplicable to this system. This study argued that the tolerance analysis for the spliced primary mirror optical system should focus on evaluating the maximum sub-wavefront error (need to refer to the same ideal spherical wavefront) of the randomized test optical system per the Monte Carlo experiment. This approach reduces the likelihood of adopting a global evaluation, resulting in a final co-phase that is challenging to correct by the individual sub-mirrors.
2. Surface shape and wavefront error description
The surface shape and wavefront error can be quantitatively described using Zernike polynomial expansion coefficients. The Zernike polynomial expression is $V_n^m = V_n^m(r,\theta )$, which is orthogonally normalized in the circular domain. Taking the separated variable method to express it as a product of the radial and angular functions, where the radial function is $R_n^m = R_n^m(r)$, and the angular function is $\Theta _n^m = \Theta _n^m(\theta )$, the Zernike polynomial can be written as
According to Fourier polynomials, the trigonometric function system is a system of orthogonal functions in the cycle, and the angular direction function $\Theta _n^m$ is
According to Zernike's literature [10], the radial function $R_n^m$ is as follows:
Due to the orthogonal normalization property of the Zernike polynomials, the expansion of the arbitrary wavefront or surface function W (r, θ) in the unit circle domain is as follows:
where $Z_n^m$ is the expansion coefficient, which is the inner product of the function to be expanded and the base function. The expansion coefficient is calculated as follows:The noncircular domain destroys the orthogonality of the Zernike polynomials. Thus, Schmidt-orthogonalization can be applied to maintain the orthogonality. However, in this study, the surface shape and wavefront were fitted, requiring a physical meaning of the aberration. Therefore, linear regression is used to perform the Zernike expansion (fitting) of the noncircular domain, and the circular domain is defined as the geometric center of the heterogeneous domain that exactly encompasses the region.
Describing the order of Zernike polynomials requires the use of two corner symbols, m and n, which are inconvenient to describe in practice. In this study, we adopt the standard Zernike polynomial form and order used by Ansys Zemax OpticStudio because the variation in the RMS value of the function in the unit-circle domain is also normalized; thus, providing the standard Zernike polynomial coefficients is equivalent to specifying the RMS value of Zernike polynomials of that order within the unit circle domain.
The standard Zernike polynomial order describes Zi using the corner scale i, which is related to m and n, as listed in Table 1.
Table 1. Symbol i, m, n relationship
3. Error statistics and mathematical modeling
In this study, the Zernike expansion coefficients (expanded to 105 terms) of the machining errors of six large-diameter aspherical mirrors with the largest diameter of 4000 mm and the smallest diameter of 800 mm were evaluated. Owing to the deduction of the mirror centers, the Zernike polynomials no longer satisfy orthogonality. However, they can still be considered as a least-squares fit to the expansion of the original data; the fitting results and residuals are shown in Fig. 1.
Fig. 1. (a) Raw data, (b) fitted data, (c) residuals, and (d) expansion coefficients for an 1800 mm mirror.
The Zernike expansion coefficient Kij (j = 1, 2, …, 105) for each mirror face shape error Si (i = 1, 2, …, 6) is considered an absolute value. Subsequently, normalization is performed as a series of data in terms of subscript i, and the logarithm < Lj > is derived. The mathematical expression for the aforementioned description is as follows:
The results of plotting the < Lj > -j scatter diagram and the fitting line (Fig. 2) indicate that the Zernike expansion coefficient of the surface shape error of mirror machining is exponentially related to the number of terms i and floats within a certain interval.
When the function is expanded using standard Zernike polynomials, the spatial frequencies of the corresponding Zernike polynomials are of the same order when n is the same and m is different. Therefore, the random fluctuations of the coefficients of the Zernike polynomials of the face shape error during the tolerance analysis with Monte Carlo experiments should gradually decrease with an increase in n. The terms Z1 (piston), Z2 and Z3 (tilt-X and Y), and Z4 (curvature) were removed in the tolerance analysis and the actual testing because the first four terms were affected by placing the mirrors during the inspection detection of the surface shape.
The piston and tilt errors of the surface shape can be corrected as positional errors, and the fourth item is generally detected by the three-coordinate picking-point fitting spherical method that can be adjusted off-axis during mounting.
In this study, we consider a modeling exponential form, where the coefficients of the Zernike term Zi of the face shape of each mirror are generated uniformly and randomly when the Monte Carlo experiment produces a particular face shape error, and the random number distribution ranges within the interval [Zi], as given by Eq. (7).
However, the coefficients of the higher-order terms in the randomized experiment might be larger than those of the lower-order terms. Thus, Eq. (7) can be improved as follows:
The effects of k1, k2, and k3 in the mathematical model were analyzed (Fig. 3). By adjusting the values of k1, k2, and k3 to form a series of combinations, the obscuration ratio of the mirror was fixed, and 200 Monte Carlo experiments were conducted for each adjustment to record the < RMS > (value of average RMS), <PV > (value of average PV), and < PV/RMS > (value of average PV/RMS) of the mirror surface shape.
The results of the Monte Carlo experiments (Fig. 4) indicate that the value of < PV/RMS > varies exponentially with respect to k2, independently of k1 and k3; therefore, a limiting minimum of < PV/RMS > exists. The effect of k1 on the < RMS > and < PV > values is linear. Thus, the generation of random numbers of Zernike polynomial coefficients describing the mirror machining error can be performed by first determining the value of the machined < RMS/PV>, from which the value of the k2 parameter can be determined, and then the values of < RMS > can be adjusted by k1 and k3.
Therefore, this model can be used in Monte Carlo experiments for tolerance analysis to control the RMS, PV, and PV/RMS values of the mirror shape error according to the machining capability. The high-spatial-frequency Zernike terms are generally lower in magnitude than the low-spatial-frequency Zernike terms in each experiment according to the probability.
4. Tolerance analysis of mosaic aperture telescopes
The mosaic primary mirror telescope is shown in Fig. 5, and the design parameters of the optical system are listed in Table 2. This telescope aperture is approximately 10 m, and the optical structure is a coaxial four-reflector system with four aspherical mirrors. The primary mirror M1 was spliced into 18 segments, denoted as S1, S2,…, S18. The secondary, tertiary, and quaternary mirrors were labeled M2, M3, and M4, respectively. Two types of coordinates for this optical system were considered: a global coordinate system (GC), and each segment had its local coordinate system C1, C2, … C18. The local coordinate system was used for the positional description of the segment, and the global coordinate system was used for the positional descriptions of M2, M3, and M4, as illustrated in Fig. 6
Table 2. Optical system specifications
The telescope segments are deformable mirrors with adaptive optical correction. In addition to the adaptive optics system, each segment incorporates a large stroke corrector to mitigate the defocus aberration. This design eliminates the tolerance analysis for piston, tilt, and defocus in the wavefront. Moreover, separate statistics on the number of defocus guides the corrector designs. The remaining aberrations can only be corrected using deformable mirrors; thus, the remaining aberrations must be limited.
The rigid body has six degrees of freedom in space: Decenter-X, Decenter-Y, Decenter-Z (piston), and Tilt-X, Tilt-Y, and Tilt-Z (clocking). Owing to the rotational symmetry of the optical system, Tilt-Z can be represented as Decenter-Y or Decenter-X. Therefore, the positional error for tolerance analysis has five degrees of freedom.
Figure 7(a) illustrates the histogram of the distribution and cumulative probability of counting the RMS of the wavefront aberration of each segment and the RMS value of the maximum segment wavefront aberration of each experiment (Fig. 7(b)) under the same tolerance assignment (note that the number of statistical samples in Fig. 7(a) is 18 times larger than that in Fig. 7(b) because 18 segments are considered). As illustrated in Fig. 7(a), all segments of the wave aberration RMS are considered. In particular, the 99.8% probability is less than 0.64 wavelengths, whereas Fig. 7(b) presents that the maximum sub-wave aberration RMS statistics for each experiment with a probability of less than 0.64 wavelengths is 96.4%, and a probability of approximately 3.5% indicating that the segment wave aberration cannot be corrected. Mosaic aperture optical systems require sub-wavefront co-phasing; therefore, they cannot accept individual segments that are not correctable by adaptive optics.
Fig. 7. Comparison of tolerance analysis evaluations. (a) RMS histogram of all segments wavefront (WF) and (b) RMS histogram of maximum segments WF.
Therefore, the tolerance analysis of the mosaic-aperture optical system must be constrained to the probability distributions of the maximum sub-wavefront aberration RMS and PV for each Monte Carlo test after removing the piston, tilt, and defocus.
In addition, the segments are off-axis; however, measurements of their curvature are fitted as full mirrors. Therefore, the tolerance analysis must remove the amount of Decenter-Z due to the curvature change (Fig. 8).
Based on the machining and adaptive optics correction capability, the final tolerance requirements are sub-wavefront RMS with a 99.8% probability of less than 2 µm and PV with a 99.8% probability of less than 10 µm. The final tolerance assignments are listed in Table 3.
Table 3. Tolerance Allocation (3σ)
If each order of Zernike face shape error is loaded with an equal magnitude under the same tolerance assignment, then the probability of the RMS being less than 2 µm is 69.2%. The probability of PV being less than 10 µm is 74.2% (Fig. 9). In contrast, using the surface shape error model, the probability of the wavefront aberration RMS being less than 2 µm is 99.4%. The probability of PV being less than 10 µm is 99.8% (Fig. 10). A more realistic loading of the surface shape error can relax the other tolerance values and render the tolerance assignment more reasonable.
Fig. 10. Mathematical model proposed in this study: Distribution of Zernike terms of surface sag error RMS and PV.
5. Conclusion
Tolerance analysis is crucial to project implementation and provides a basis for subsequent component processing, equipment selection, and adaptive optics corrector layouts. The high-spatial-frequency surface shape error in the optical system produces greater dispersion spots and other effects on imaging compared with the low-frequency error. Therefore, establishing an accurate mathematical model for the face shape error is worth studying.
This study evaluated the surface shape errors in the optical machining of six-meter-scale aspherical mirrors. A mathematical model of the face shape sag error was constructed and applied to the tolerance analysis of a 10-meter scale aperture mosaic primary mirror telescope with a meter-scale segment aperture. The developed model was more realistic and improved the positional tolerance probability from 69.2% to 99.4% compared to the default equal-magnitude loading of the face shape for each spatial frequency error, enhancing the feasibility of the project.
Furthermore, for a mosaic primary mirror optical system, the traditional evaluation criterion of the unblocked aperture tolerance analysis is no longer applicable. This study proposed a method for considering the maximum evaluation of the sub-wavefront to guarantee that the final co-phase adjustment will not occur in the case of individual segments that are challenging to correct. Future studies can improve the mathematical model of the face shape error by considering the actual mirror machining process.
Funding
National Natural Science Foundation of China (62375266).
Acknowledgments
We would like to thank the editors and the reviewers for their useful feedback that improved this paper. The authors thank the financial support of the National Natural Science Foundation of China.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. S. Gladysz, B. F. Castella, R. Rebolo, et al., “Imaging extrasolar planets with the European Extremely Large Telescope,” EPJ Web of Conf. 16, 07003 (2011). [CrossRef]
2. M. Dimmler, J. Marrero, S. Leveque, et al., “E-ELT M1 test facility,” Proc. SPIE 8444, Ground-based and Airborne Telescopes IV, 84441Y (2012).
3. G. Chanan and M. Troy, “Chromatic effects in narrowband phasing of the Keck Telescope segments: theory and numerical simulations,” Proc. SPIE 10700, Ground-based and Airborne Telescopes VII, 107001E (2018).
4. D. S. Acton, S. Knight, A. Contos, et al., “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE 8442, Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, 84422 H (2012).
5. M. D. Perrin, D. S. Acton, C-P. Lajoie, et al., “Preparing for JWST wavefront sensing and control operations,” Proc. SPIE 9904, Space Telescopes and Instrumentation 2016: Optical, Infrared, and Millimeter Wave, 99040F (2016).
6. I. Laginja, R. Soummer, L. Mugnier, et al., “Analytical tolerancing of segmented telescope co-phasing for exo-Earth high-contrast imaging,” J. Astron. Telesc. Instrum. Syst. 7(01), 015004 (2021). [CrossRef]
7. Q-Y. Meng, Zi-C. Qin, C-Ming Ren, et al., “Review of optical systems desensitization design methods,” Chinese Opt. 15(5), 863–877 (2022). [CrossRef]
8. U. Fuchs and S. Kiontke, “Breathing life into aspheric dreams,” Adv. Opt. Technol. 5(3), 229–235 (2016). [CrossRef]
9. G. W. Forbes, “Manufacturability estimates for optical aspheres,” Opt. Express 19(10), 9923–9942 (2011). [CrossRef]
10. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934). [CrossRef]
