Open Access
Add to BibSonomy
Abstract
In the past 10 years, adaptive wavefront interferometry (AWI) has been employed for measuring freeform surface profiles. However, existing AWI techniques relying on stepwise and model-free stochastic optimizations have resulted in inefficient tests. To address these issues, deterministic adaptive wavefront interferometry (DAWI) is firstly introduced in this paper based on backpropagation (BP), which employs a loss function to simultaneously reconstruct and sparsify initial incomplete interferometric fringes until they are nulled. Each iteration of BP requires two phase shifts. Through simulations, we have verified that freeform wavefront error with a peak-to-valley (PV) of up to 168 λ can be fully compensated in tens of iterations using a 1024 × 1024 pixel area of a liquid-crystal spatial light modulator. In experiments, we accomplished a null test of a freeform surface with 80% missing interference fringes in 39 iterations, resulting in a surface profile error PV of 66.22 λ and measurement error better than λ/4. The DAWI has at least 20 times fewer iterations in fringe reconstruction than the 3-step AWI methods, and nearly an order of magnitude fewer iterations in the whole process, paving the way for significantly enhanced efficiency, generality and precision in freeform surface adaptive interferometry.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As optical technology continues to advance, freeform surfaces [1] are becoming increasingly important for enhancing optical systems due to their non-rotational symmetry by providing more design flexibility, better performance [2–4], and smaller system volume [5]. The techniques for measuring freeform surfaces with high precision can generally be categorized as interferometry [6–18], profilometry [19–23] and geometric ray methods (e.g., wavefront sensing [24,25]), among which interferometry excels in high accuracy and lateral resolution. However, measuring freeform surfaces accurately in optical rough polishing stage is still a big challenge via conventional interferometric approaches due to the diverse surface profile changes and large surface profile deviations: computer-generated holograms (CGHs) [6,7] are suitable for a final test and not applicable for measuring rough polishing surfaces since the surface profile deviation normally has a peak-to-valley (PV) of tens to hundreds of wavelength and the interference fringes are incomplete; sub-aperture stitching (SAS) technique [8,9] demands careful sub-aperture alignment and calibration, leading to lower measurement efficiency; and tilt-wave interferometry (TWI) [10–12] is not suitable for measuring steep or highly curved surfaces. In recent 10 years, adaptive wavefront interferometry (AWI) [14–18] utilizing dynamic compensators such as deformable mirrors (DMs) [26] and liquid-crystal spatial light modulators (SLMs) [27–31] independently or in conjunction with null optics is considered as a superior method.
Fuerschbach et al. [13] used a DM to correct the unknown coma and higher-order aberrations when measuring a freeform surface by AWI, and a Shack-Hartmann wavefront sensor (SHS) was used to interrogate the DM-induced wavefront variations during the close-loop iteration. Chaudhuri et al. [31] performed a detailed analysis of a null-test setup consisting of an SLM and an SHS, and tested a concave freeform mirror with an aperture of 65 mm and a sag deviation of 90.6 µm. However, due to the limited number of micro-lenses for sampling, SHSs have under-sampling and aliasing errors when characterizing higher-order and large deviation aberrations, thus affecting the iteration efficiency. Huang et al. (2016) [14] proposed to use a 3-step method, namely unidentifiable fringe reconstruction (UFR), highly-dense fringe sparsification (HDFS) and decreasing fringe density to near null (DFDNN), to measure unknown freeform surfaces with larger deviations, and the stochastic parallel gradient descent (SPGD) optimizer was used in the iterative process. Meanwhile, they used fringe grayscale variance as the objective function in the first and third steps, and the indicative wavefront PV with data loss measured by a Zygo interferometer as the objective function in the second step. The measurement process was rigorous but tedious. Similarly, Xue et al. (2018, 2019) successfully employed SPGD [15] and genetic algorithm (GA) [16] optimizers, respectively, leveraging a transmissive SLM to compensate for unknown freeform wavefronts. Their process also involved 3 steps, namely UFR, HDFS and obtaining a null fringe by phase conjugation. The efficiency of GA was found to be inferior to that of SPGD. Zhang et al. (2021) [17] proposed utilizing simulated annealing (SA) and hill climbing (HC) optimizers in the UFR and HDFS steps, respectively, to enhance the convergence. The approaches required thousands or tens of thousands of iterations. Wu et al. (2022) [18] improved the efficiency of the SPGD by utilizing the adaptive moment estimation (AdaM) optimizer, namely AdaM-SPGD. They also adopted the same 3-step process as Huang et al., and used three independent objective functions in each step. Recently, Zhang et al. (2023) [32] proposed a convolutional neural network (CNN) to determine the Zernike coefficients of an unknown freeform wavefront to be measured from a single incomplete interferogram, replacing the UFR step, and the AdaM-SPGD method was then utilized to accomplish the remaining steps. However, the success rate of CNNs is limited by their poor generalizability in practical scenarios, e.g., for systems with varying noise and fringes of different contrast. In summary, the existing AWI methods for measuring freeform surfaces exhibit two common features: (1) multiple steps and (2) model-free stochastic optimization, and they rely on the use of randomness in objective functions, which are typically used in systems that cannot be characterized by mathematical expressions (e.g. atmospheric disturbance), and deemed non-deterministic.
In this paper, the deterministic adaptive wavefront interferometry (DAWI) is firstly proposed based on the backpropagation (BP) theory [33]. The BP accurately characterizes the derivative relationship between the fringe gradient loss and the Zernike coefficients of the surface profile error to be compensated, thereby enabling deterministic compensation during the adaptive measurement. The DAWI transforms the current 3-step AWI from multiple problems of stochastic searches to only one problem of gradient descent in a multidimensional space. Specifically, it allows simultaneous fringe reconstruction and fringe sparsification by minimizing a loss function, and a null test is accomplished in only tens of iterations, which is nearly an order of magnitude lower than the reported number of iterations via the existing 3-step methods. The efficiency, generality and measurement precision of DAWI are demonstrated by numerical simulations and experiments.
2. Principles
2.1 Principle of DAWI
Figure 1(a) is the optical setup of DAWI employed for measurement of optical rough polishing freeform surfaces using a reflective SLM, and Fig. 1(b) is an overview of the polarization state information related to the setup. The reference beam is reflected by the PBS, as represented by the blue line; and the test beam initially passes through the PBS, carrying the wavefront modulated by the SLM and the surface profile of the FS, before being back to the camera, as indicated by the red lines. P2 is used to generate a linear polarization state parallel to the SLM transverse axis, enabling high-quality pure phase modulation [34]. For an FS in optical rough polishing, though the null optics compensates for the FS’s ideal profile, a significant uncompensated surface profile deviation remains. To enhance the image quality, S is used to filter out highly oblique rays, resulting in unidentifiable pixels) on the interferogram. The SLM produces Zernike wavefronts with high spatial resolution in order to compensate for the fringe regions with unknown phases, enabling reconstruction and sparsification of the entire fringe. By rotating P3, the intensity of the reference beam can be reduced, thus allowing adjustment of fringe contrast. The telescopic imaging system comprising lenses L1 and L2 enables the position of the camera to be conjugate to that of the SLM.
Fig. 1. Principle of DAWI: (a) optical setup; (b) polarization state information of the two interference beams. (Laser: single-longitudinal-mode laser source, P1-3: linear polarizers, BE: beam expander, M: mirror, QWP1-3: quarter-wave plates, PBS: polarization beam splitter, L1-2: lenses, S: aperture stop, PC: personal computer, NO: null optics and FS: freeform surface under test.)
During the measurement, the camera first acquires a current interferogram (${I_0}$), followed by a $\pi /2$ phase-shifting interferogram (${I_1}$) and a $3\pi /2$ phase-shifting interferogram (${I_2}$), and the SLM functions as a phase shifter [30,35]; By analyzing the three interferograms, the BP method accurately identifies areas necessitating gradient backpropagation, yielding a set of new Zernike coefficients. Then, the coefficients are sent to the SLM to generate a compensated Zernike wavefront, which facilitates subsequent reconstruction and sparsification of the fringes. The process repeats cyclically until the fringe is nulled. Finally, substituting the compensated Zernike wavefront into an optical design software, the precise surface error map of the FS can be solved by optical ray tracing.
2.2 Principle of BP
The forward propagation chain in DAWI is shown in Fig. 2, where the $N + 1$ Zernike fringe coefficients [36] are the inputs, the fringe gradient loss function ($J$) and fringe gradient loss value ($loss$) are the outputs, and BP determines the multidimensional derivative function relationship between $loss$ and ${c_n}$. Determining the functional expression of BP and finding sets of ${c_n}$ by using an appropriate gradient descent optimizer that quickly let $loss$ reach 0 are the focus of this paper. The following is the procedure for deriving the functional expression of BP:
Fig. 2. Propagation chain in DAWI. ($n$ and N represent the term number and the maximum term of the Zernike polynomials used, respectively. ${I_0}$ is the current interferogram, and ${I_1}$ and ${I_2}$ are the $\pi /2$ and $3\pi /2$ phase-shifting interferograms, respectively.)
The expressions for $\phi $ in terms of ${c_n}$ can be written as Eq. (1), and for ${I_0}$ in terms of $\phi $ and ${\phi _{test}}$ can be written as Eq. (2):
where ${Z_n}$ $({n = 0,1,2, \ldots ,N} )$ is the expression of the $n$th term of Zernike fringe polynomials [36], N is the maximum number of the Zernike terms to be used for wavefront compensation, ${\phi _{test}}$ denotes the uncompensated phase residual of the tested FS, A is the mean irradiance and $B/A$ is the fringe contrast.${I_1}$ and ${I_2}$ can be expressed as:
and$J$, defined as the central differential gradient magnitude, is written as:
where$loss$, defined as the average of the elements in J corresponding to the identifiable pixels in ${I_0}$, can be expressed as:
where M is the total number of the identifiable pixels.Obviously, the initial incomplete interferogram has a small value of $loss$; when the fringe is being reconstructed, $loss$ is increasing; and when the fringe is being sparse, $loss$ is decreasing. It is due to the non-monotonicity problem of $loss$ that the existing methods have to adopt a 3-step optimization strategy. However, the iterative process based on the BP method is independent of the size of $loss$, but the first order derivatives of $loss$. Therefore, the BP method requires only one optimization step. Since $loss$ is related to both J, ${I_0}$, $\phi $, and ${c_n}$ based on the propagation chain in Fig. 2, the derivatives of $loss$ is calculated obeying the chain rule, as:
The derivative of J with respect to ${I_0}$ is expressed as:
The derivative of ${I_0}$ with respect to $\phi $ can be derived from Eqs. (2)–(4), as:
Finally, Eq. (8) can be rewritten as:
2.3 Gradient descent optimizer
Gradient descent optimizers are a class of algorithms commonly used in machine learning to find the optimal set of parameters for a model that minimizes a given loss function. The fundamental concept in DAWI is to update the model parameters (${c_n}$, $n = 0,1,2, \ldots ,N$) iteratively in the direction of the negative gradient of $loss$ (i.e. $- dloss/d{c_n}$), which reduces the $loss$ at each step until convergence. This is a problem of gradient descent in ($N + 1$)-dimensional space, where the AdaGrad optimizer [37] with an additional exponential factor ${\rho ^t}$, named AdaGradExp, is used as the gradient descent optimizer. AdaGrad adaptively adjusts the learning rate based on the historical gradients of each parameter, resulting in better convergence. Moreover, the incorporation of ${\rho ^t}$ assists AdaGrad in achieving a steady decrease of $loss$ in the final stage of iteration, thus avoiding oscillations of the interferogram between 1-3 fringes. The optimizer is characterized by:
where t denotes the number of iterations, $t \ge 1$, G is the cumulative squared gradient, ${G^{(0 )}} = {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over 0} _{1 \times n}}$, $\eta $ is the learning rate, ${\eta ^{(0 )}}$ represents the initial learning rate, $\varepsilon $ is a smoothing term that avoids a denominator of 0 (usually in an order of ${10^{ - 8}}$), $c_n^{(0 )} = {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over 0} _{1 \times n}}$, and $\rho $ is a deceleration factor between 0.99 and 1, where its default value is 0.998.3. Results
3.1 Characteristics of fringes during iteration
Based on the aforementioned principles, the evolution of fringe shape is demonstrated by four distinct states (Fig. 3), emphasizing the notable advantages offered by DAWI. Specifically, under the execution of BP, the central differential gradient magnitudes of the identifiable pixels located at the edges of the identifiable areas are back-propagated and reduced, the identifiable areas continuously expand their boundaries due to wavefront continuity (as indicated by the red arrows in Fig. 3(a)), resulting in a decrease in the ratio of unidentifiable pixels (${R_{UPs}}$). In simulations, the pixels that possess a central differential gradient magnitude exceeding π/2 are designated as unidentifiable pixels due to 2π ambiguity [38]; and the identifiable pixels (IPs) in both simulations and experiments are determined by:
where $({x,y} )$ are the coordinates of the pixels in ${I_0}$, N is a constant associated with the interferogram noise, and we took $N$=20. The symbol ${\cup} $ denotes the union operation. Therefore, ${R_{UPs}}$ can be determined by quantifying the number of remaining pixels within the effective interferometric area.Fig. 3. Four distinct states of fringe shape in DAWI: (a) initial fringe; (b) simultaneous fringe reconstruction and sparsification; (c) fringe reconstruction completed and fringes are highly sparse; and (d) null test achieved.
Meanwhile, the central differential gradient magnitudes of the identifiable pixels positioned at the central identifiable areas also experience reduction. As a result, BP method performs simultaneous fringe reconstruction and fringe sparsification, as depicted in Fig. 3(b). As the number of identifiable pixels increases, the benefits of BP become more pronounced. When ${R_{UPs}}$ approaches zero, the interferogram becomes highly sparse at the same time (Fig. 3(c)). Finally, after a few iterations, the wavefront is completely compensated, leading to null fringes (Fig. 3(d)). Hence, Fig. 3 exemplifies the intelligence of DAWI by accomplishing a null test on freeform wavefronts with large deviations through a single-step process. Notably, Fig. 3(a)-(d) were obtained by employing a small ${\eta ^{(0 )}}$ of 1 in the gradient descent optimizer, and 200 iterations were required to achieve Fig. 3(d).
3.2 Generality and efficiency test
To assess the generality and efficiency of DAWI in compensating for freeform wavefronts with varying degrees, three models were constructed. Each model consisted of 100 sets of randomly generated Zernike 37-order wavefronts, having the same image resolution of 1024 × 1024 but differing in the range of initial ${R_{UPs}}$. In these wavefronts, the coefficient for the first Zernike term, ${c_0}$ (piston), was fixed at 0 and not optimized as it did not affect the wavefront shape. The remaining coefficients (${c_1}$ to ${c_{36}}$) were randomly selected from the pre-set ranges, resulting in incomplete interferograms with different initial ${R_{UPs}}$. During the test, a large initial value of ${\eta ^{(0 )}}$ was assigned to the gradient descent optimizer process to expedite the iteration. The parameters for the three models are summarized in Table 1.
Table 1. Parameter ranges of the randomly generated Zernike freeform wavefronts
The test results for the generality and efficiency of DAWI are presented in Fig. 4. When ${R_{UPs}}$ dropped below 1% for the first time, ${\eta ^{(t )}}$ was adjusted to 1, and the BP process continued. Within a mere 16 seconds (using MATLAB software runtime, CPU model: i9-11900K), a set of 99 iterations was completed.
Fig. 4. Generality and efficiency test results of DAWI by compensating freeform wavefronts with varying degrees of deviation: (a)-(c) are the plots of $loss$ with number of iterations corresponding to the first, second and third model of test, respectively, and (d)-(f) are the variation of ${R_{UPs}}$ corresponding to (a)-(c), respectively.
The first and second models, focusing on small and medium-sized initial ${R_{UPs}}$, respectively, demonstrated that all randomly generated freeform wavefronts were completely compensated, resulting in null fringes, within 40 and 60 adaptive compensations, respectively. The initial wavefront PV values in the second model ranged from 47 λ to 128 λ, where λ represents the test wavelength. Notably, the slowest test in the second model had an initial wavefront PV of 93 λ and an initial ${R_{UPs}}$ of 47.39% (389,532 unidentifiable pixels), which required 64 iterations to reconstruct the initial fringes (as indicated by the blue line in Fig. 4(e)).
The third model examined large initial ${R_{UPs}}$, with the initial wavefront PV in this model ranging from 92 λ to 168 λ. 98% of the tests achieved null fringes within 99 iterations, with the fastest compensation occurring in just 28 iterations. Two tests failed to converge to a loss value less than 2 within 99 iterations due to an inappropriate ${\eta ^{(0 )}}$. Slight modifications to ${\eta ^{(0 )}}$ (e.g., changing 10 to 11) could accelerate the test speed. Therefore, it is recommended to explore different values of ${\eta ^{(0 )}}$ when compensating for freeform wavefronts with a large initial ${R_{UPs}}$. However, the average decline lines of $loss$ (represented by the black dashed lines in Fig. 4(a-c)) indicate that DAWI is expected to achieve null testing in tens of iterations for freeform wavefronts with an initial ${R_{UPs}} \le 90\%$.
In summary, the test results exhibit the remarkable generality and efficiency of DAWI in adaptive null test, surpassing the performance of the 3-step AWI methods [14–18,32].
3.3 Experimental results
The experimental setup is depicted in Fig. 5. The Laser was a single-longitudinal-mode laser operating at 635 nm. The laser beam was emitted through a single-mode fiber. The SLM utilized in the experiment was the Holoeye Pluto-1-VIS-016, featuring a pixel resolution of 1920 × 1080 and a pixel size of 8 µm [34]. The PYTHON2000 camera sensor, with a pixel resolution of 1920 × 1200 and a pixel size of 4.8 µm, was utilized. L1 and L2 represented doublets with focal lengths of 250 mm and 150 mm, respectively. This configuration ensured that the image possessed a lateral resolution of 8 µm/pixel, matching that of the SLM (as illustrated in Fig. 5(b)). If the lateral resolution of the image differed from that of the SLM, image resizing should be performed to align the number of pixels in the test area with that of the SLM. To minimize the impact of bull’s-eye rings, resulting from dust and particle diffraction, as well as black matrix diffraction caused by the SLM’s pixel fill rate not reaching 100%, an extended source generator (ESG) [39] was incorporated into the Laser. The ESG consisted of a single-mode fiber collimator, a rotating ground glass, and a multimode fiber (MMF), and the extended source was emitted from the fiber port (FP). The aperture of the FS under test was 25 mm. The first 16 terms of Zernike fringe polynomials were utilized to create compensated wavefronts. Notably, the phase modulation nonlinearity and spatial nonuniformity of the SLM were corrected using the method in Ref. [40], and SLM substrate unevenness error were considered and corrected beforehand.
Fig. 5. (a) The experimental setup of DAWI (M: mirror, MTS: motorized translation stage, 2D AF: 2-dimensional adjustment frame and 4D AF: 4-dimensional adjustment frame); (b) the initial interferogram.
Figure 5(b) is the initial interferogram of the FS with an initial ${R_{UPs}}$ of 80% (∼650,000 unidentifiable pixels). The test process of DAWI is shown in Fig. 6(a). A starting value of 10 was assigned to ${\eta ^{(0 )}}$, and after 39 iterations, null testing was achieved. Notably, during the test, we manually reduced the value of ${\eta ^{(t )}}$ to ensure a faster and more stable gradient descent, as evident in Visualization 1. Figure 6(b) is the wavefront error map of the final compensated wavefront generated by the SLM after doing ray tracing. Furthermore, Fig. 6(c) presents the surface profile error map of the FS after removing the tilt error in the result of Fig. 6(b). The map exhibits a wavefront PV value of 66.22 λ and an RMS value of 15.53 λ (λ = 635 nm). Figure 6(d) gives the measurement difference between the null test measurement obtained using a Zygo Verifire HDX interferometer (with a pixel resolution of 3.4K × 3.4 K) and the DAWI, which reveals that the measurement precision of the surface profile error PV via the DAWI is better than λ/4 (only 0.17 λ).
Fig. 6. Experimental results of DAWI: (a) plot of $loss$ with the number of iterations; (b) the wavefront error map of the final compensated wavefront generated by the SLM after doing ray tracing; (c) surface profile error map of the FS; and (d) measurement differences between DAWI and high-resolution interferometric null test. (Visualization 1)
Moreover, to emphasize the advantages of DAWI, a comparative experiment was performed utilizing the 3-step AWI method with SPGD and Adam-SPGD gradient descent optimizers, as shown in Fig. 7. Evidently, the 1st step (UFR) of the 3-step method displayed poor robustness and low measurement efficiency. The SPGD gradient descent optimizer exhibited pronounced oscillations during the initial stage and converged to local optima in the later stage, completing a total of 343 iterations. On the other hand, the Adam-SPGD gradient descent optimizer demonstrated a slightly more stable descent, still requiring 235 iterations. Nonetheless, it is worth noting that the efficiency of the 3-step method considerably lags behind that of the BP method, regardless of the optimizer used (i.e. SPGD or Adam-SPGD). The BP method achieved fringe reconstruction in only 11 iterations, showcasing a disparity of over twenty-fold between the two methods.
Fig. 7. Experimental results of the 1st step (UFR) of the 3-step AWI with SPGD and Adam-SPGD gradient descent optimizers.
4. Conclusion
In this study, we propose the BP method for adaptive measurement of freeform surfaces with large deviations, comparing it to existing 3-step methods. Each iteration of BP requires two phase shifts. The distinct derivability of the method leads to its designation as DAWI. Demonstrating its generality and efficiency, simulations and experiments were conducted. Simulation results indicate that DAWI has a high probability of achieving null testing within tens of iterations, consuming only 16 seconds, even when compensating for unknown freeform wavefronts with an initial ${R_{UPs}}$ of 90% and the first 37 Zernike fringe coefficients ranging from-15 to 15. In experiments, we successfully measured a freeform surface with a surface error PV of 66.22 λ using a pixel area of approximately 1K × 1 K on the SLM, even with an initial ${R_{UPs}}$ as high as 80%. It took 39 iterations to make the initial incomplete fringes nulled. Notably, the number of iterations required for fringe reconstruction is more than twenty times less than whether using the SPGD-based or Adam-SPGD-based 3-step method. Experimental results confirm that the BP method achieves a measurement precision of surface profile error PV better than λ/4.
The advantages of DAWI can be summarized as follows:
- (1) Robustness: DAWI can handle highly dense fringes, as long as 90% of the initial interferogram areas is incomplete and the first 37 Zernike fringe coefficients range from -15 to 15;
- (2) Convergence: With deterministic functional relations governing the entire compensation process, only tens of iterations are required in the multidimensional spatial gradient descent procedure;
- (3) Computational efficiency: DAWI relies on basic matrix operations, ensuring computational efficiency;
- (4) Practicality: DAWI can measure various complex freeform surfaces, meeting the general requirements for testing freeform surfaces with significant deviations.
In conclusion, the BP method enables adaptive measurement of complex freeform surfaces with high efficiency, high generality and high precision. These advantages present challenges to existing AWI techniques.
Funding
National Key Research and Development Program of China (2020YFA0714500, 2022YFE0204800); International Partnership Program of Chinese Academy of Sciences (181231KYSB20200040); Chinese Academy of Sciences President's International Fellowship Initiative (2023VMB0008).
Acknowledgments
We thank X.Z., M.X. and J.S. provided theoretical knowledge consultation. Wei W. and S.L. reviewed the manuscript and provided suggestions. S.L. and J.S. provided the financial support.
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. J. P. Rolland, M. A. Davies, T. J. Suleski, et al., “Freeform optics for imaging,” Optica 8(2), 161–176 (2021). [CrossRef]
2. W. Jahn, M. Ferrari, and E. Hugot, “Innovative focal plane design for large space telescope using freeform mirrors,” Optica 4(10), 1188–1195 (2017). [CrossRef]
3. A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018). [CrossRef]
4. T. Yang, G.-F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light: Sci. Appl. 6(10), e17081 (2017). [CrossRef]
5. J. Reimers, A. Bauer, K. P. Thompson, et al., “Freeform spectrometer enabling increased compactness,” Light: Sci. Appl. 6(7), e17026 (2017). [CrossRef]
6. J. C. Wyant and V. P. Bennett, “Using computer generated holograms to test aspheric wavefronts,” Appl. Opt. 11(12), 2833–2839 (1972). [CrossRef]
7. H. Shen, R. Zhu, Z. Gao, et al., “Design and fabrication of computer-generated holograms for testing optical freeformsurfaces,” Chin. Opt. Lett. 11, 032201 (2013). [CrossRef]
8. X. Zhang, H. Hu, X. Wang, et al., “Challenges and strategies in high-accuracy manufacturing of the world’s largest SiC aspheric mirror,” Light: Sci. Appl. 11(1), 310 (2022). [CrossRef]
9. X. Hou, F. Wu, L. Yang, et al., “Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method,” Opt. Express 15(20), 12890–12899 (2007). [CrossRef]
10. S. Christian, B. Rolf, G. Antonia, et al., “Tilted Wave Fizeau Interferometer for flexible and robust asphere and freeform testing,” Light: Advanced Manufacturing 3(4), 1–12 (2022). [CrossRef]
11. I. Fortmeier, M. Stavridis, A. Wiegmann, et al., “Evaluation of absolute form measurements using a tilted-wave interferometer,” Opt. Express 24(4), 3393–3404 (2016). [CrossRef]
12. S. Y. Chen, S. Xue, D. D. Zhai, et al., “Measurement of freeform optical surfaces: trade-off between accuracy and dynamic range,” Laser Photonics Rev. 14(5), 1900365 (2020). [CrossRef]
13. K. Fuerschbach, K. P. Thompson, and J. P. Rolland, “Interferometric measurement of a concave, φ-polynomial, Zernike mirror,” Opt. Lett. 39(1), 18–21 (2014). [CrossRef]
14. L. Huang, H. Choi, W. Zhao, et al., “Adaptive interferometric null testing for unknown freeform optics metrology,” Opt. Lett. 41(23), 5539–5542 (2016). [CrossRef]
15. S. Xue, S. Y. Chen, Z. B. Fan, et al., “Adaptive wavefront interferometry for unknown free-form surfaces,” Opt. Express 26(17), 21910–21928 (2018). [CrossRef]
16. S. Xue, W. X. Deng, and S. Y. Chen, “Intelligence enhancement of the adaptive wavefront interferometer,” Opt. Express 27(8), 11084–11102 (2019). [CrossRef]
17. L. Zhang, R. Liu, J. Wu, et al., “Freeform surface adaptive interferometry assisted with simulated annealing-hill climbing algorithm,” Measurement 181, 109597 (2021). [CrossRef]
18. J. Wu, C. Hu, R. Liu, et al., “Adam SPGD algorithm in freeform surface in-process interferometry,” Opt. Express 30(18), 32528–32539 (2022). [CrossRef]
19. D. F. Elkott, H. A. Elmaraghy, and W. H. Elmaraghy, “Automatic sampling for CMM inspection planning of free-form surfaces,” Int. J. Prod. Res. 40(11), 2653–2676 (2002). [CrossRef]
20. G. Y. He, Y. C. Sang, K. R. Pang, et al., “An improved adaptive sampling strategy for freeform surface inspection on CMM,” Int. J. Adv. Manuf. Technol. 96(1-4), 1521–1535 (2018). [CrossRef]
21. Y. J. Shen, J. J. Ren, N. D. Huang, et al., “Surface form inspection with contact coordinate measurement: a review,” Int. J. Extrem. Manuf. 5(2), 022006 (2023). [CrossRef]
22. Taylor Hobson Company, “LUPHOScan 850 HD,” https://www.taylor-hobson.com/products/non-contact-3d-optical-profilers/luphos/luphoscan-850-hd.
23. B. R. Swain, C. Dorrer, and J. Qiao, “Telephoto-lens-based Optical Differentiation Wavefront Sensor for freeform metrology,” Opt. Express 29(23), 38395–38403 (2021). [CrossRef]
24. D. R. Burada, K. K. Pant, M. Bichra, et al., “Experimental investigations on characterization of freeform wavefront using Shack-Hartmann sensor,” Opt. Eng. 56(08), 084107 (2017). [CrossRef]
25. J. Yu, F. Z. Fang, and Z. J. Qiu, “Aberrations measurement of freeform spectacle lenses based on Hartmann wavefront technology,” Appl. Opt. 54(5), 986–994 (2015). [CrossRef]
26. M. J. Booth, “Adaptive optical microscopy: the ongoing quest for a perfect image,” Light: Sci. Appl. 3(4), e165 (2014). [CrossRef]
27. N. Savage, “Digital spatial light modulators,” Nat. Photonics 3(3), 170–172 (2009). [CrossRef]
28. C. Maurer, A. Jesacher, S. Bernet, et al., “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5(1), 81–101 (2011). [CrossRef]
29. Z. Zhang, Z. You, and D. Chu, “Fundamentals of phase-only liquid crystal on silicon (LCOS) devices,” Light: Sci. Appl. 3(10), e213 (2014). [CrossRef]
30. L. Hu, L. Xuan, Y. Liu, et al., “Phase-only liquid-crystal spatial light modulator for wave-front correction with high precision,” Opt. Express 12(26), 6403–6409 (2004). [CrossRef]
31. R. Chaudhuri, A. Wansha, R. Porras-Aguilar, et al., “Implementation of a null test for freeform optics using a high-definition spatial light modulator,” Opt. Express 30(24), 43938–43960 (2022). [CrossRef]
32. R. Liu, J. Wu, S. Zhou, et al., “Fast recovery of sparse fringes in unknown freeform surface interferometry,” Opt. Express 31(5), 7144–7158 (2023). [CrossRef]
33. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” Nature 323(6088), 533–536 (1986). [CrossRef]
34. HOLOEYE Photonics AG, “LCOS microdisplay technology,” https://holoeye.com/lcos-microdisplays/.
35. B. Villalobos-Mendoza, F. S. Granados-Agustín, D. Aguirre-Aguirre, et al., “Phase shifting interferometry using a spatial light modulator to measure optical thin films,” Appl. Opt. 54(26), 7997–8003 (2015). [CrossRef]
36. R. R. Shannon and J. C. Wyant, Applied Optics and Optical Engineering, Volume 11 (Academic Press, 1992).
37. J. Duchi, E. Hazan, and Y. Singer, “Adaptive Subgradient Methods for Online Learning and Stochastic Optimization,” Journal of Machine Learning Research 12, 2121–2159 (2011). [CrossRef]
38. J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express 19(6), 5126–5133 (2011). [CrossRef]
39. M. N. Morris, M. Naradikian, and J. Millerd, “Noise reduction in dynamic interferometry measurements,” in Conference on Interferometry XV - Techniques and Analysis (San Diego, CA, 2010), p. 77900O.
40. Z. Zeng, Z. Li, F. Fang, et al., “Phase Compensation of the Non-Uniformity of the Liquid Crystal on Silicon Spatial Light Modulator at Pixel Level,” Sensors 21(3), 967 (2021). [CrossRef]
