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Abstract
Stitching interferometry is an essential technique for the non-contact, high-precision measurement of large apertures or complex optical surfaces. However, the accuracy of full-aperture surface reconstruction is significantly compromised by subaperture positioning and systematic errors. To address this challenge, this study introduces a novel stitching interferometry method utilizing alternating calibration of positioning and systematic errors (SIAC). This method calibrates one type of error while maintaining the other constant, and alternates between these processes to effectively decouple the two errors, facilitating accurate phase stitching. Within this calibration framework, an iterative weighted phase stitching model employing vertical projection for estimating overlapping areas was developed to calibrate positioning errors. Additionally, the rotation measurements of a single subaperture, in conjunction with a global fitting approach, were employed to correct reference errors. Numerical simulations have confirmed the efficacy of SIAC in calibrating these errors. Moreover, experimental measurements were performed on both a plane mirror and gullwing aspheres, with the resulting stitched full-aperture phase distributions and cross-testing outcomes affirming the method's accuracy and practicality. This research provides a novel solution for stitching interferometry, enhancing the precision of optical surface measurements.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Interferometry serves as the primary method in high-precision optical surface metrology [1]. The increasing complexity and stringent precision requirements of optical components have exposed the limitations of conventional interferometers, such as large-aperture plane mirrors [2], aspheres [3], large convex mirrors [4], cylindrical surfaces [5], and X-ray mirrors [6,7], in meeting contemporary measurement needs. To overcome these challenges, researchers have explored various strategies to broaden the field-of-view of interferometry. Among these, stitching interferometry (SI) emerges as a versatile and effective solution [8,9]. SI methodologically segments the test surface into multiple subapertures (SAs), conducts scanning measurements through their displacement, and subsequently amalgamates these SAs to reconstruct the comprehensive surface topography. The use of null optics in the stitching measurement of aspheres, aimed at augmenting the dynamic range, is a noteworthy technique also encapsulated in the discussions of this paper [10–13].
Stitching interferometry can be categorized based on the configuration of divided SAs into circular SI [14], annular SI [15,16], among other variants [17,18], with circular SI being the predominant focus due to its extensive application. Independent of the SA configuration, the methodology necessitates relocating the test surface to collect data, a process complicated by the positioning errors introduced by multi-degree-of-freedom motion systems. These errors significantly impede the precision of SI, underscoring the importance of meticulous SA planning and the advancement of more reliable scanning mechanisms to mitigate inaccuracies. Furthermore, many SA stitching algorithms have been developed for the calibration of positioning errors [19–31]. Chen et al. and Okada et al. simplified the inter-SA positional relationships into polynomial expressions of piston and tilts, facilitating phase stitching through least-squares fitting [19,20]. However, this model's incomplete error characterization constrained the attainable stitching accuracy. In an advancement, Zhao et al. expanded the error model to include Zernike polynomials, broadening the scope of error calibration [21]. Sjödahl et al. employed phase point clouds to address six degrees of freedom in positional errors [22], though challenges persist in point cloud registration and global optimization efforts. Chen et al. demonstrated notable success in phase stitching by applying a Lie group representation and iterative processes akin to the Iterative Closest Point (ICP) algorithm, in conjunction with a computer-aided design (CAD) model to represent the nominal shape of the surface [23]. Hou et al. advanced cylindrical stitching interferometry through the application of dual quaternions [24]. The calibration of positioning errors has seen enhancements through the integration of fiducial markers on the test surface and the adoption of stereovision techniques [25,26].
The consistency of the overlap area (OA) between SAs is crucial for high-precision phase stitching. However, scanning introduces variations in the reference wavefront across identical regions among different SAs, leading to inconsistencies in the OA and, consequently, diminishing the accuracy of the stitching process. To mitigate this, the calibration of systematic errors becomes imperative. One strategy involves pre-determining the reference error and subtracting it from each SA's measurement, although the emphasis is increasingly shifting towards error self-calibration during the stitching process. QED technology has emerged as a significant contributor in this domain [27], distinguishing between SA-related and unrelated error components and implementing self-calibration using free and interlocked compensators. This approach has shown to enhance data utilization and stitching accuracy. Nicolas et al. explored the conditions necessary for the calibration of additive systematic errors, proposing a method involving non-uniformly spaced SA stitching [28]. Huang et al. adopted a methodology similar to Nicolas, calibrating high-order additive systematic errors by minimizing discrepancies within the OA and subsequently stitching the reference-adjusted SAs [29]. Xue et al. introduced a self-calibration stitching method for cylindrical surfaces employing orthogonal polynomials, though the method's sensitivity to environmental and equipment variables necessitates further investigation [30]. These advancements underscore the importance of self-calibrating both positioning and systematic errors during stitching. Nonetheless, the simultaneous least-squares optimization of these errors introduces significant computational complexity and challenges in convergence stability. While reference errors are SA-unrelated, their calibration is contingent upon the precise positioning of all SAs. Moreover, the coupling between correct positioning and systematic errors complicates the achievement of high-precision phase stitching. Sequential calibration of these errors, without addressing their interdependence, impedes the attainment of optimal stitching precision.
To address these challenges, we introduce a stitching interferometry method using alternating calibration of positioning and systematic errors (SIAC). This approach is bifurcated into two principal components: first, the construction of an iteratively weighted phase stitching model that utilized projection estimation of the OA to accurately and robustly correct positioning errors; second, the employment of rotation measurements on a single SA, supplemented by global fitting techniques, to rectify systematic reference errors. Furthermore, we propose an optimization framework that alternates between these calibration processes, effectively decoupling the two types of errors. Numerical simulations were conducted to ascertain the viability of the SIAC method. Experimental validations were also carried out on a plane mirror and a gull-wing asphere, with cross-testing results corroborating the accuracy and robustness of the SIAC method.
2. Principle of SIAC
2.1 Model of stitching interferometry
To elucidate the phase-stitching problem, we present a model of stitching interferometry as depicted in Fig. 1. This model encapsulates a typical stitching system, which includes an interferometer, the surface under test (SUT), and a scanning module. The scanning module plays a pivotal role in aligning and executing scanning measurements across all SAs. Specifically, for aspherical surfaces, the scanning protocol initiates from the central SA, progressing ring-by-ring towards the periphery of SUT, as illustrated in Fig. 1(b).
Fig. 1. Optical model of stitching interferometry: (a) optical layout of the stitching system, (b) the designed lattice, (c) positioning relationship of SAs with overlap areas (OAs) indicated by red solid lines in the yoz plane.
The data collected for each SA, Si(x, y), can be characterized as:
where ${\hat{{\mathbf g}}_{i,1}} = {{\mathbf g}_{i,1}} + \Delta {{\mathbf g}_{i,1}}$ is the position of the ith SA within a global coordinate system, gi,1 and Δgi,1 are nominal value and positioning error. The global coordinate system, typically denoted as the coordinate system where the first SA is situated (e.g., CS1 in Fig. 1(c)), serves as the reference for all measurements. The OA between two SAs in the yoz plane is represented by the red solid lines. R(x, y) signifies the inherent systematic reference error of the system, while n(x, y) represents the random noise within the measurement.2.2 Iteratively weighted phase stitching with vertical projection estimation of OA
The calibration of positioning errors is a critical step in the stitching process. When discussing the calibration of these errors, this section assumes that the systematic reference error has been either corrected or is negligible. Considering two SA Si(x, y) and Sk(x, y) as examples (omitting (x, y) for convenience of description following), the transformations of their local coordinate systems CSi and CSk are
Calibration is pivotal for computing the transformations ${\hat{{\mathbf g}}_{i,k}}$ for all SAs to ensure consistency in the OA. Identifying the OA accurately is the initial step in this process. However, the challenge arises from the difficulty in pinpointing the true sets of overlapping points {Ωi,k}, due to measurement errors. Typically, the spatial normal projection of Si onto Sk is considered to approximate {Ωi,k}. This approximation necessitates an iterative optimization process to accurately calculate the spatial normal projection, which can be complex. In practice, the positioning error Δgi,k, is significantly smaller than its nominal value, gi, k. This discrepancy can be utilized to simplify the calculation of {Ωi,k}. As illustrated in Fig. 2, ${\hat{{\mathbf g}}_{i,k}}{S_i}$ the dotted line represents the coordinates of Si within the coordinate system of Sk (CSk), the black dashed line denotes the height intercept of Si in the yoz plane, and the red points indicate the projection of Si and its normal. The normal projection is estimated through the vertical projection of points pi, j (pi, j∈{Ωi,k}, j = 1, 2, …, Ni,k, Ni,k is the number of overlapping points) onto ${\hat{{\mathbf g}}_{i,k}}{S_i}$, as depicted by the blue solid line in Fig. 2. Due to the sampling step, these projection points may not align perfectly with the grid of Sk, necessitating interpolation of Sk. The nearest grid points ${\tilde{p}_{k,j}}$ in the vicinity U(pk,j) of the theoretical projection points pk,j are chosen as the actual overlapping points. These points ${\tilde{p}_{k,j}}$ can be determined by minimizing the directed distance between pi,j and the grid points within U(pk,j)
After determining {Ωi,k}, the correct position can be calculated by minimizing the inconsistency of OA
Fig. 3. Curves of the loss function and its derivative: (a) loss function, (b) derivative of the loss function, (c) weighting function applied in the calibration process.
The partial derivative of Eq. (5) for ${\hat{{\mathbf g}}_{i,1}}$ can be calculated as:
Perform first-order Taylor expansion on $\hat{{\mathbf g}}_{i,1}^{(t)}$
and Eq. (11) can be solved with linear least squares where ${\mathbf J} \in {{\mathbb R}^{\sum\nolimits_{i = 1}^{v - 1} {\sum\nolimits_{k = i + 1}^v {{N_{i,k}}} } \times 6v}}$, ${\mathbf b} \in {{\mathbb R}^{\sum\nolimits_{i = 1}^{v - 1} {\sum\nolimits_{k = i + 1}^v {{N_{i,k}}} } \times 1}}$. Because of the vertical projection estimation and Taylor expansion used in the calculation of the overlapping points and positioning, an iteration of the two calculations is required to reduce the approximation errors.2.3 Compensation for systematic reference error
Conventional methodologies for compensating reference errors in stitching interferometry often employ polynomial fitting, predominantly targeting the correction of low-frequency errors. The comprehensive reference error is deduced from a single SA rotation measurement combined with polynomial fitting techniques. While absolute testing of reference errors can be facilitated by the shift-rotation method, this approach introduces complexities when applied to spherical surfaces, as the shifting test alters the retrace error, complicating the accurate computation of the reference error. The rotation measurement, particularly feasible for the central SA (S1), can be expressed as:
where θnis the nth rotation angle and θ0 = 0, ${S_{1,{\theta _n}}}(x,y)$ represents the phase of S1 after rotating θn. ${\varphi _n}(x,y)$ of measured phase of the nth rotation testing. Reverse rotation of ${\varphi _n}(x,y)$ by θn and subtract from ${\varphi _0}(x,y)$ to obtain the rotational shear phase Δφ(x, y)The above equation indicates that Δφ can be approximated as the difference of Rra(x, y) in the direction of rotation. The reconstruction of Rra(x, y) can be realized through multiple rotation testing with different θ.
Owing to the rotational asymmetry of the medium- and high-frequency components, Rrs(x, y) can be assumed to be low frequency and can be written as ${R_{rs}}(x,y) = \sum\nolimits_{m = 5}^L {{A_m}{Z_m}(x,y)}$, where Z(x, y) is the Zernike polynomial, Am and L are the polynomial coefficients and total terms, respectively. Therefore, using the obtained Rra(x, y), Rrs(x, y) can be calculated by polynomial fitting using all SAs. The measured phases can be rewritten as:
The complete reference error is then subtracted from all SAs prior to the phase stitching process. It is important to highlight that the methodology delineated does not ascertain the magnitude of the reference error. In instances involving small-aperture reference mirrors, the impact of the reference error may be deemed negligible. Conversely, for larger apertures, alternative techniques such as scanning pentaprism testing must be employed to independently quantify the power of the reference error.
2.4 Alternating calibration
This analysis presupposes that the reference error has been adequately compensated during the calibration of positioning errors, with the precise positioning utilized for the calculation of the reference error. For stitching interferometry, the intertwining of positioning ${\hat{{\mathbf g}}_{i,1}}$ (Eq. (18)) and reference errors within the phase data implies that calibrating one type of error invariably influences the determination of the other. Consequently, these errors should not be calibrated in isolation. Yet, attempting to concurrently optimize all errors, akin to the QED self-calibration algorithm, renders the problem ill-posed, escalating computational complexity and adversely affecting convergence stability. To navigate these challenges, this study advocates for an SIAC model. This model iteratively calibrates one category of error—positioning or systematic—before transitioning to the other, with this alternation persisting until both errors converge. The procedural flow of SIAC is illustrated in Fig. 4 and unfolds as follows:
Input: Phase dataset {Si} of all SAs; nominal positioning {gi,1}; Ending conditions σ for alternating calibration; initiate alternating count e = 0 and initial reference error R(x, y) = 0.
Step A: Calibration of positioning error
- A1. Apply a global coordinate transformation to the dataset {Si−R}, using {gi,1} from the previous alternating calculation, and perform partial interpolation on the dataset; initiate iteration count t = 0.
- A2. Estimate the overlapping points {Ωi,k} using vertical projection and normal distance.
- A3. Filter the overlapping points through compatibility analysis to remove outliers.
- A4. Update the weight factor based on the outcomes of the preceding iteration to refine the positioning error calibration.
- A5. Recalculate the positioning for all SAs and iterate through steps A2 to A5 until optimal positioning is achieved, as indicated by minimal variation across iterations; set e = e + 1.
Step B: Calibration of systematic error
- B1. Determine the rotationally asymmetric component Rra(x, y) using the phase set {φn} from multiple rotations of the central SA.
- B2. Perform polynomial fitting on all SAs to obtain the rotationally symmetric component Rrs(x, y) utilizing the updated $\{{{{\hat{{\mathbf g}}}_{i,1}}} \}$.
- B3. Update the systematic reference error, R(x, y), with the obtained Rra(x, y) and Rrs(x, y) values.
- B4. Subtract the recalculated R(x, y) from all SAs to refresh the dataset {Si−R}.
Alternating calibration
Alternate between performing Step A and Step B until the predefined ending condition (σ) is met. Following this process, a stitched map is generated through the weighted fusion of all SAs.
Output: Stitched map.
The ending condition for the SIAC process is defined as
3. Numerical simulation and analysis
The verification of the SIAC method was conducted through numerical simulations focusing on a plane mirror with an 800 mm aperture, as depicted in Fig. 5(a). The full-aperture phase metrics, peak to valley (PV) and root mean square (RMS), were determined to be 0.368λ and 0.056λ, respectively. To mitigate computational demands, the sampling interval was set at 1 pixel/mm. The systematic reference error is illustrated in Fig. 5(b), featuring PV and RMS values of 0.246λ and 0.035λ, respectively. The structured lattice for the simulation, indicating the acquisition sequences, is presented in Fig. 5(c), with the number of SAs totaling nine and an aperture size of 450 mm. The overlap rate was established at 40% between the central SA and the outer ring, and approximately 37% within the ring itself. Data acquisition for the SAs was achieved by fixing the interferometer and completing the full-aperture measurement through translation and rotation of the test mirror. To simulate actual conditions, random that the positioning error for the first SA (1st SA) was zero. The introduction of rigid body errors, denoted as Δgi,1, simulated the positioning error during the test positioning errors were introduced to the SAs based on their motion mode. The specific values of these added positioning errors, including rotation errors (α, β and γ around the x, y, and z axes, respectively) and translation errors (Tx, Ty, and Tz along the x, y, and z axes, respectively), along with the defocusing error (D), are summarized in Table 1. These errors were considered achievable, reflecting the accuracy attainable by contemporary scanning systems. The postures of all SAs, relative to these positioning errors, are detailed in Fig. 5(d).
Fig. 5. Setup for the numerical simulation: (a) test plane mirror, (b) systematic reference error, (c) lattice design for SA acquisition, (d) spatial postures of all SAs.
Table 1. Random positioning errors introduced for each SA in the numerical simulation
The phases obtained from the numerical simulations, which aimed to validate the SIAC method, are showcased in Fig. 6. In this setup, all SAs were aligned to their nominal positions within the coordinate system established by the 1st SA. The measurement outcomes for the 1st SA at rotation angles of 0°, 54°, 90°, and 144° are depicted in Figs. 6(a) to 6(d), respectively. Utilizing the SIAC method, the test surface was reconstructed, with initial iteration results displayed in Fig. 7. Figure 7(a) reveals noticeable inconsistencies among the OAs, while Fig. 7(b) illustrates the discrepancy between the calculated phase and the ground truth, exhibiting PV and RMS values of 0.354λ and 0.053λ, respectively. The presence of similar phase errors across each SA serves as evidence that the reference error has yet to be fully calibrated. The correction of the reference error is further illustrated in Figs. 7(c) and 7(d), where the PV and RMS of the residual are reduced to 0.021λ and 0.009λ, respectively. The process of calibrating the positioning error demonstrates how residual reference errors can introduce inaccuracies into the calculation of the positioning$\{{{{\hat{{\mathbf g}}}_{i,1}}} \}$. Such inaccuracies, in turn, can negatively impact the global fitting of the rotationally symmetric component of the reference error, as evidenced in Fig. 7(d). The findings from Fig. 7 underscore the inherent challenge in achieving accurate phase stitching through a single calibration due to the intricate interplay and coupling between positioning and systematic errors. This complexity highlights the necessity for the SIAC, which iteratively alternates between calibrating positioning and systematic errors to progressively refine the phase stitching accuracy.
Fig. 6. Interferometric measurements of SAs with induced positioning errors: (a)-(d) results for the 1st SA at rotations of 0°, 54°, 90°, and 144°, respectively; (e)-(l) phases of SAs 2-9.
Fig. 7. Initial iteration results from the simulation: (a) stitched full-aperture phase, (b) discrepancy between stitched phase and ground truth, (c) extracted reference error, (d) residual between calculated reference phase and systematic reference error from Fig. 5(b).
Following the implementation of the SIAC method, the ending condition was met after 33 iterations, demonstrating the method's iterative refinement capability. The convergence of the calibration process is illustrated in Fig. 8, indicating the effectiveness of the alternating calibration strategy in achieving precise phase stitching. The reconstructed full-aperture result, depicted in Fig. 9(a), shows the absence of noticeable inconsistencies within the phase, signifying a close approximation to the ground truth as shown in Fig. 5(a). The residual phase, presented in Fig. 9(b), with PV and RMS values of 0.022λ and 0.003λ, respectively, underscores the high level of accuracy attained. This residual primarily stems from the calibration residual of the reference error, characterized in Fig. 9(d) as high-order spherical aberration. In the simulation, the fourth, ninth, and sixteenth terms of the Zernike polynomials were employed to fit the rotationally symmetric components, suggesting that incorporating higher-order terms could further refine the fitting accuracy. Table 2 provides a comparative analysis of the calculated positioning errors and the mean relative errors across each degree of freedom, revealing that the calibration precision for γ, Tx, and Ty is marginally lower in comparison to α, β, and Tz. This discrepancy is attributed to the plane phase's gradual gradient change, which results in a reduced sensitivity of the OA inconsistency to the errors associated with γ, Tx, and Ty relative to the other motion axes. The calibration residual of the defocusing error (D) is primarily due to challenges in completely correcting the positioning errors. The outcomes of the SIAC method, as demonstrated through the phase distribution and numerical values, affirm its capability to achieve high-precision calibration and stitching, especially when confronted with the complex interplay of large coupling errors.
Fig. 9. Final simulation outcomes: (a) reconstructed full-aperture phase, (b) residual between the stitched result and ground truth, (c) recalibrated reference error, (d) residual of the recalibrated reference error and the systematic reference error depicted in Fig. 5(b).
Table 2. Calculated positioning errors for all SAs in the simulation.
4. Experimental verification
4.1 Plane stitching interferometry results
The proposed SIAC method's effectiveness and precision were further validated through an experiment involving plane-stitching interferometry. This experiment utilized a test mirror with a 90 mm aperture and designed the SA apertures to be 50 mm each, achieving an overlap rate of 39.1%. The SAs were arranged in a square (3 × 3) array, with the scanning accomplished via two-dimensional translation. For data collection, a Zygo GPI interferometer with a 4-inch testing aperture was employed, capable of capturing full-aperture measurements of the test mirror. The interferograms gathered during this experiment are illustrated in Fig. 10. In this practical application, the SIAC method was applied to reconstruct the test phase. Given that the scanning was limited to two-dimensional translation, the model for the positioning error was simplified to encompass two-dimensional translation and rotation only. The results obtained using a conventional stitching method, which employs global least squares for direct stitching, are displayed in Fig. 11(a). Here, noticeable inconsistencies within the OAs, highlighted by black elliptical dashed lines, indicate the presence of significant uncalibrated positioning and reference errors. Contrastingly, Fig. 11(b) presents the final stitched phase achieved using the SIAC method, where the phase distribution exhibits a seamless transition between the SAs, with no apparent inconsistencies. The PV and RMS values of the stitched phase are reported as 0.025λ and 0.005λ, respectively.
Fig. 11. Comparison of stitched phases for a plane mirror: (a) phase stitched using global least squares, (b) phase stitched with the SIAC method, (c) full-aperture phase as measured by a Zygo GPI interferometer, (d) reference error determined through a three-flat absolute test, (e) full-aperture phase after subtracting the reference error, (f) residual between the SIAC stitched phase (b) and the full-aperture phase (e).
The accuracy of the SIAC method was quantitatively evaluated against a full-aperture measurement of the test mirror, obtained using the Zygo GPI interferometer. The full-aperture results, depicted in Fig. 11(c), recorded PV and RMS values of 0.031λ and 0.005λ, respectively. To further refine the accuracy, the phase of the reference mirror, determined through a three-flat absolute test, was utilized to calibrate the full-aperture results [32]. The outcome of this absolute test is shown in Fig. 11(d), revealing PV and RMS values of 0.030λ and 0.006λ, respectively. With the reference error subtracted, the corrected full-aperture phase, presented in Fig. 11(e), exhibited PV and RMS values of 0.028λ and 0.004λ, respectively. This comparison underscores the necessity of compensating for reference errors, especially when the test surface possesses high quality, as the full-aperture measurement alone may not accurately reflect the surface's true shape. The residual between the SIAC result and the full-aperture phase, after reference error calibration, is displayed in Fig. 11(f), showing PV and RMS values of 0.013λ and 0.001λ, respectively. The results illustrated across Fig. 11 affirm that the SIAC methodology is capable of achieving high-precision stitching interferometry for a plane mirror in case of coupling of positioning and reference error.
4.2 Gull-wing asphere stitching interferometry results
To further validate the performance of the SIAC method, it was applied to a previous stitching test of a gull-wing asphere (GWA) [12]. This test was particularly challenging due to the asphere's curvature sign reversal at the inflection point, leading to changes in concavity. The GWA had an aperture of 20 mm, defined by the equation with
Fig. 12. A Surface analysis of the Gull Wing Asphere (GWA): (a) gradient curve, (b) second-order differential curve indicating the inflection point, (c) deviation from the best-fit sphere illustrating asphericity.
A variable-sign curvature compensation stitching interferometric system was built to measure GWA, as shown in Fig. 13. The lattice designed for the GWA test showed 52 SAs with four measurements of the middle SA and two SA rings. The four angles of rotation tested for the middle SA were 0°, 54°, 111°, and 166°, along with two concentric SA rings rotated at 30° and 10°, comprising 12 and 36 SAs respectively. The experimental setup and the collected interferograms are detailed in Fig. 14, showing the fringe patterns for both the middle SA and the first SAs of rings 1 and 2 for illustrative purposes. Prior to phase stitching, corrections for retrace error and mapping distortions were applied to the collected phases, with the detailed correction process available in referenced literature [12]. The corrected phases are presented in Fig. 15. After correction of the inherent error, all the SA are stitched in the coordinate system.
Fig. 14. Experimental interferograms from GWA stitching test: (a)-(c) display the fringe patterns of the middle SA and the first SAs of rings 1 and 2, respectively, for visual reference; (d)-(f) show rotation measurements of the middle SA at angles of 54°, 111°, and 166°.
Fig. 15. Corrected phases of subapertures (SAs) for GWA: (a) phase of the central SA, (b) and (c) phases of the first SAs of rings 1 and 2, with retrace error and mapping distortions corrected.
Both the SIAC and conventional global least-squares phase stitching methods were employed to reconstruct the full-aperture phase. The results are shown in Fig. 16. The conventional method, illustrated in Fig. 16(a), yielded a phase distribution with PV and RMS values of 2.990λ and 0.414λ, respectively. This approach exhibited notable deficiencies in stitching performance, particularly in addressing the coupling of multiple positional errors and system reference errors, as evidenced by apparent inconsistencies within the OAs due to calibration residual errors. The ending condition of SIAC is set as τ=10% and E = 100. Terms 11, 22, 37, 56, and 79 of the Zernike polynomials are used to calculate the rotationally symmetric component of the reference error is calculated. After 27 alternating calibrations, the error in the OAs stabilized, indicating that the final condition had been achieved. The resultant stitched phase, illustrated in Fig. 16(b), exhibited PV and RMS values of 2.030λ and 0.273λ, respectively. Further elucidation is provided in Figs. 16(d) to 16(f), which display the rotationally asymmetric and symmetric components of the reference error, alongside the synthesized reference error. The comparison of phase distributions in Fig. 16 reveals a marked improvement in the consistency of OAs and the invisibility of stitching faults in the phase reconstructed by SIAC, outperforming the conventional method. This advancement underscores the efficacy of SIAC in addressing and compensating for the intricate interplay of positioning and systematic errors.
Fig. 16. Results from GWA stitching tests: (a)-(c) phases reconstructed using the conventional method, the SIAC method, and LuphoScan 260, respectively; (d)-(f) depict the rotationally asymmetric and symmetric components of the reference error, alongside the combined reference error.
To further validate the accuracy and consistency of the SIAC method, a cross-test was performed using LuphoScan 260, with the findings presented in Fig. 16(c). The cross-test results yielded PV and RMS values of 1.689λ and 0.254λ, respectively, affirming the SIAC method's compatibility with the LuphoScan point scanning outcomes. Notably, the RMS deviation surpasses the λ/50 benchmark, evidencing the method's high degree of precision. The smooth phase distribution and numerical values gleaned from this comparative analysis validate the SIAC method's superior performance in achieving high-precision stitching interferometry under the challenge of coupling errors.
5. Conclusion
In this study, we demonstrated the effectiveness of stitching interferometry through the SIAC. By dissecting the phase information from subapertures, we developed a novel alternating calibration model that successfully decoupled positioning from systematic errors. Within this framework, an iteratively weighted phase stitching method, which utilized vertical projection for the precise estimation of overlapping areas, was constructed to enhance the calibration of positioning errors. Moreover, rotation measurements of the central subaperture, combined with global fitting, were employed to correct for systematic reference errors. Our investigations, bolstered by simulations and experimental validations, confirmed the feasibility and accuracy of the proposed SIAC method. Cross-validation tests further substantiated its robustness and reliability for the interferometric analysis of optical surfaces. Future efforts were directed towards elaborating a more comprehensive model of positioning errors to encompass some nonrigid errors. Additionally, the calibration of nonadditive systematic errors, such as retrace errors and mapping distortions, will also be added into alternating calculation to augment the precision of the SIAC method. This progression aimed to broaden the applicability and enhance the accuracy of stitching interferometry, ensuring superior measurement and reconstruction fidelity for complex optical surfaces.
Funding
National Natural Science Foundation of China (62305161, 62305169, 62375128, 62171225); China Postdoctoral Science Foundation (2023M731684, 2023M731752).
Acknowledgments
This research is funded by Jiangsu Funding Program for Excellent Postdoctoral Talent. We thank Nikolaus Berlakovich from Vienna University of Technology for email discussion on registration algorithm.
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.
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