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Abstract
Freeform optical components enable dramatic advances for optical systems in both performance and packaging. Surface form metrology of manufactured freeform optics remains a challenge and an active area of research. Towards addressing this challenge, we previously reported on a novel architecture, cascade optical coherence tomography (C-OCT), which was validated for its ability of high-precision sag measurement at a given point. Here, we demonstrate freeform surface measurements, enabled by the development of a custom optical-relay-based scanning mechanism and a unique high-speed rotation mechanism. Experimental results on a flat mirror demonstrate an RMS flatness of 14 nm (∼λ/44 at the He-Ne wavelength). Measurement on a freeform mirror is achieved with an RMS residual of 69 nm (∼λ/9). The system-level investigations and validation provide the groundwork for advancing C-OCT as a viable freeform metrology technique.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Freeform optical components have been shown to enable dramatic advances in optical performance and system packaging. Applications of freeform optics span both imaging and non-imaging regimes [1–3] and include dispersive spectrometers [4,5], viewfinders [6], three-mirror imagers [7], scan lenses [8], beam control optics [9], and single-photon measurement devices [10]. Surface form metrology for manufactured freeform optics remains a challenge and an active research field [11]. Desired characteristics for a freeform metrology technique include measurement uncertainty that is sufficiently lower than design tolerances, flexibility between measuring surfaces of different prescriptions, and practical measurement time. Non-contact methods are also generally preferred to minimize the risk of damage.
Existing freeform metrology techniques may be categorized by their method of illumination or interrogation on the surface under test into full-aperture, sub-aperture, and point-cloud techniques. Predominant full-aperture techniques build upon conventional Fizeau interferometry with computer-generated holograms [12,13] or active compensating elements [14,15] to null the wavefront generated by the freeform surface under test. Other full-aperture techniques include deflectometry [16,17], lateral shearing interferometry [18], and optical differentiation wavefront sensing [19]. Sub-aperture techniques include sub-aperture stitching interferometry [20], non-null zonal interferometric methods [21,22], and phase retrieval [23]. A common thread for the full-aperture and sub-aperture techniques is that they generally do not directly measure the surface sag. Significant reconstruction is necessary to back-calculate the sag from the measured data. The layout and alignment of the surface under test within the metrology setup are often coupled into the final measurement, necessitating complex calibrations and considerable downtime for different surface prescriptions.
In contrast, point-cloud techniques typically perform direct, alignment-decoupled measurements on the surface under test. Most point-cloud techniques rely on a coordinate measuring machine (CMM) metrology frame that actuates a probe and the sample platform to scan the surface under test. The probe used may be tactile with a ruby or diamond tip, quasi-tactile via scanning force microscopy [24,25], or non-tactile via optical methods. Non-contact probes may utilize low coherence interferometry [26,27], multi-wavelength interferometry [28,29], chromatic confocal sensing [30,31], and differential confocal sensing [32]. Non-contact probes may require tracing quasi-normal to the surface under test and therefore necessitate a priori knowledge of the part prescription. This requirement becomes highly challenging when the prescription is not known or when the part lacks fiducials to be oriented to the prescription. While the translation and rotation axes of a CMM frame enable large measurement volumes of hundreds of millimeters laterally, errors in these axes contribute to the measurement uncertainty. Moreover, high costs are generally associated with a CMM frame of high positioning accuracy.
Building upon optical coherence tomography (OCT), which is inherently non-contact and non-part-specific [33], we recently reported on the development of a point-cloud metrology technique referred to as cascade OCT (C-OCT) [34]. A single-point prototype setup was built to validate this technique’s working principles. Its experimental implementation with a rotating cube secondary interferometer demonstrated 26 nm precision (∼λ/24 at the He-Ne wavelength).
Here, we expand upon the architecture to include a custom optical-relay-based lateral scanning mechanism and a unique high-speed rotation mechanism for the secondary interferometer cube that varies optical path lengths for sag ranging. Optical scanning was chosen instead of the commonly done mechanical translations and rotations to eliminate motion errors and stage artifacts. A C-OCT metrology system capable of surface measurements was fully realized. In this paper, Section 2 details the requirements and instrumentation of the optical scanning sub-system and the high-speed optical cube rotation. Measurement results on a flat mirror and a freeform mirror are presented and discussed in Section 3.
2. System design
The C-OCT metrology system developed is shown schematically in Fig. 1. Requirements for advancing from single-point to surface measurement are mainly two-fold, one concerning the optical scanning and the other the data acquisition speed.
2.1 Optical scanning requirements and instrumentation
First, the optical scanning sub-system must be capable of scanning an area while maintaining near-diffraction-limited performance at the focus and high telecentricity over a flat image plane. In-depth discussions of these performance metrics and their effects on the measurement uncertainty have been detailed in [35], where a custom telecentric objective lens was developed for this specific application. However, to achieve the full telecentricity capability of the objective lens, the scanner mirror responsible for both directions of the orthogonal scan must coincide with the objective lens’s entrance pupil plane. The large entrance pupil diameter and the accepted object angle range of the objective lens determined that galvanometer mirrors or polygon scanners should be used, both of which typically offer one-dimensional scans only. For this freeform metrology application, galvanometer mirrors were chosen for their superior angular positioning capability, which is essential for correlating to the beam’s location on the sample plane in length units with high repeatability, the process of which is described in Section 3. Moreover, the angular positioning capability enables each position to be stably held while the rotating cube in the secondary interferometer completes the sag measurement.
Given that two 1D galvanometer mirrors are required, a custom afocal pupil relay was developed to image the objective lens entrance pupil to an additional accessible location. This relay is based on an Offer geometry used in an afocal configuration. Its all-reflective nature is highly compatible with the broad spectral bandwidth of an OCT system. The prescription is given in Table 1, and the design layout is shown in Fig. 2. Off-the-shelf components were substituted after optimization for the primary (32-837, Edmund Optics) and the secondary, which was made by coating the convex surface of an off-the-shelf plano-convex lens (KPX232, Newport). For a 10 mm entrance pupil diameter and ±7.4° of optical scan angle, as determined by the objective lens specifications, this design achieved a nominal root-mean-square (RMS) wavefront error of 0.09 waves at the C-OCT system’s central wavelength of 900 nm. As shown in Fig. 1, the primary interferometer is designed with a common-path configuration where the reference and the sample arm experience the same optical components. As such, the residual aberrations of this pupil relay are not expected to affect the measurement.
Fig. 2. (a) To-scale layout drawing of the pupil relay design in the YZ plane. (b) Perspective view of the pupil relay with the two galvanometer mirrors (S1 and S6) labeled and ray-paths showing the range and direction of the angular scan.
Table 1. Prescription of the pupil relay, shown sequentially to the ray-path.
2.2 Acquisition speed requirements and instrumentation
The second requirement for surface measurements is sufficiently fast data acquisition speed for each position within a point-cloud measurement. Note that millions of data points are recorded to trace the fringe bursts for each position on the sample, the peaks of which locate the sag measurement. Intuitively, the faster the cube rotation speed in the secondary interferometer, the higher the data acquisition speed. However, several system-level considerations need to be balanced simultaneously. These considerations include the laser source repetition rate, the cube’s optical path difference (OPD) scan, the digitization speed of the analog-to-digital converter (ADC), and the desired number of samples per cycle for the fringe bursts.
The process for calculating the target rotation speed of the cube is as follows. The sinusoidal oscillation in the detected fringe burst is proportional to $\cos [k \cdot OPD(\theta )]$, where k is the wavenumber and $OPD(\theta )$ is the OPD generated by the cube versus rotation angle. It has been shown that there are four regions of non-zero OPD scan per revolution of the cube due to its symmetry, and within each OPD scan region $OPD(\theta )$ is wavelength-dependent and highly linear [34]. Therefore, the fringe oscillation may be approximated as $\cos (k \cdot {C_1} \cdot \theta )$, where C1 is a wavelength-dependent constant corresponding to the linear fit slope of $OPD(\theta )$. As shown in Fig. 1, a frequency stabilized He-Ne laser is co-located with the broadband beam (referred to as the NIR beam and centered around 900 nm) within the secondary interferometer. It was found that ${C_1}(\lambda = 633\textrm{ nm}) \approx 0.4713\textrm{ mm/}^\circ $ and ${C_1}(\lambda = 900\textrm{ }nm) \approx 0.4686\textrm{ mm/}^\circ $. The process described here may be used to calculate the expected signal frequency for both the He-Ne and NIR beams.
The cube speed in rotations per minute (RPM) is correlated to the angular speed ω, as $\omega = 6 \cdot RPM$. Since $\theta = \omega t$, where t is time, the fringe oscillation may be further re-written as $\cos (k \cdot {C_2} \cdot t)$, where ${C_2} = \omega {C_1}$. Therefore, the period T of the fringe oscillation is
where λ is the wavelength. The desired number of samples per cycle SPC, which should be at least four, and the ADC digitization speed ${v_{ADC}}$ are related via Therefore, Eq. (1) and Eq. (2) are combined to obtainSince a sufficient number of pulses is required within the integration time of each data point [34], the photodetector’s frequency bandwidth must be faster than the fringe oscillation frequency determined by Eq. (1) and Eq. (2), while at least twice slower than the laser repetition rate. This statement assumes that the photodetector performs the necessary analog integration, as ADCs generally do not integrate without additional circuitry. For the ADC digitization speed, a practical consideration to balance is the available memory of the system’s controlling computer (PC). A higher digitization speed results in more data points and a larger variable size to process and store, which may be upward of hundreds of gigabytes for the entire point-cloud.
It was found that the optical cube should be driven at around 1,750 RPM in the current C-OCT metrology system implementation. This rotation speed enables the use of an off-the-shelf silicon photodetector with an 11 MHz bandwidth (PDA100A2, ThorLabs), which is compatible with the 80 MHz repetition rate of the broadband laser used (SuperK Versa, Koheras). A minimum of 44 MSamples/s ADC speed ensures 8 samples per cycle, which is readily achieved with the ADC used (ATS9350, AlazarTech). The theoretical maximum data acquisition speed with this cube rotation speed is 8.6 milliseconds per position on the sample.
As this rotation speed is not achievable via typical motorized rotation stages, a custom actuating assembly was developed, as shown in Fig. 3. An air-bearing spindle (SS-55, New Way Air Bearings) was used to stably rotate the cube at high speeds while minimizing vibration transfer from the actuating motor to the rest of the interferometer system. Several custom components were designed and manufactured in-house, including a cube holder, a motor mounting plate, a cylindrical height adapter, and a motor mounting fork.
Fig. 3. Perspective view of (a) the cube holder plate, (b) the air bearing spindle, and (c) the motor mounting plate. Three troughs at 120° angles were manufactured into the motor plate to allow for spindle airflow exhausted via the spindle’s center through-hole. (d) Assembly for the high-speed rotation of the optical cube in the secondary interferometer, shown here with an aluminum cube mock-up of the same dimension as that of the optical cube.
The optical cube was epoxied to the cube holder mounted to the spindle rotor top surface, enabling unobstructed access to all four facets. The epoxy used (Light Weld 429-GEL, Dymax) was chosen to satisfy the tensile strength requirement during rotation. An additional advantage of using a UV-curable epoxy is the ample time before curing for fine alignment of the cube’s perpendicularity with respect to its holder, which was performed using an alignment telescope.
The motor plate attaches the shaft of a brushless DC motor (F40 PRO III with custom 200KV, T-Motor) to the spindle rotor bottom surface. This motor choice was determined by rotation speed stability tests carried out across different motor types, including DC motors, a die grinder, and Dremel tools. The DC motors consistently showed speed variations less than 0.05%, which was at least an order of magnitude better than the other motors tested.
The spindle stator was mounted on a custom cylindrical height adapter, which positions the cube at a suitable height for the secondary interferometer beam path. The adapter’s opening enables the DC motor stator to be attached to the mounting fork that connects to a separate structure away from the optical table, reducing vibration transfer. This opening also allows for the air exhaust flow from the spindle. As such, its size and direction are controlled.
The target rotation speed of 1,750 RPM was experimentally achieved. The assembly shown in Fig. 3(d) was mounted on two orthogonal goniometer stages and an XY translation stage to enable fine alignment, all of which have locking mechanisms. With rigid mounting, it was found that the vibration propagated downward and was absorbed rapidly by each component within the assembly stack until no vibration was observed at the optical table level. The acoustics from the rotation was minimal. Additionally, it was found that this assembly is capable of stable rotation up to 15,000 RPM, providing ample room for future speed upgrades.
Finally, an optical sensor (8518T96, McMaster-Carr) connected to a tachometer (8518T632, McMaster-Carr) was used to synchronize the cube rotation with the galvanometer mirror positioning, as shown in Fig. 1. In the current implementation, the tachometer monitors a reflective target on the DC motor and outputs a square-wave pulse per revolution, which triggers the ADC and advances the galvanometer mirrors. This configuration obtains up to four consecutive measurements per galvanometer mirror position, depending on the relative position between the reflective target and the cube.
The overall layout of the C-OCT metrology system is shown in Fig. 4. The current free-space implementation of C-OCT is a standalone metrology system occupying an area that is approximately 1 m by 2 m. There has been growing interest in integrating metrology within manufacturing machines, which has its associated advantages and trade-offs [11]. It is envisioned that a fiber-based implementation may enable the integration of C-OCT where an in-situ configuration is preferred.
Fig. 4. Layout and beam paths of the C-OCT metrology system. In the current implementation, the sample platform is mounted horizontally beneath the beamsplitter in the primary interferometer to accommodate non-standard size freeform optical components.
3. Surface measurement results and discussions
For each sag measurement within the C-OCT measurement point cloud, the same general processing steps are applied as detailed in [34]. Over the point cloud as a whole, a two-dimensional calibration is necessary to map the known voltage inputs to the galvanometer mirrors, denoted as ${X_V}$ and ${Y_V}$, to the unknown coordinates on the sample plane in length units, denoted as ${X_{mm}}$ and ${Y_{mm}}$. This mapping is expressed as
In the theoretical scenario where the two galvanometer mirrors are aligned exactly orthogonal, the cross-terms B and C reduce to zero. Here, a distortion grid target (58-536, Edmund Optics) is measured, and the dots are detected to obtain their centroid locations, as shown in Figs. 5(a)–5(b). Under the assumption of an ideal distortion grid, the $({X_{mm}},{Y_{mm}})$ positions are known. While a minimum of two coordinate sets would be sufficient to solve for the mapping matrix analytically, the entire measured grid is used in a fitting process to mitigate inevitable experimental and numerical noise. Since the expansion of Eq. (4) is
both equations may be viewed as planar surface definitions. Therefore, the A and B coefficients may be found by fitting a plane to the known ${X_{mm}}$ coordinates versus the detected voltage coordinates. The same procedure is repeated for ${Y_{mm}}$ to find the C and D coefficients. This fitting is shown in Figs. 5(c)–5(d), and the R-squared values for both are higher than 99.999%.Fig. 5. (a) C-OCT measurement of a distortion grid. The region within the yellow box is enlarged in (b) to show the detected outlines of the dots overlayed in blue. The detected centroids are used to find the ABCD mapping matrix. The surface fitting to obtain the coefficients A and B are shown in (c), and that for C and D are shown in (d).
The distortion grid was measured three times consecutively, over which the mean and standard deviation of the mapping matrix was found to be
First, a flat mirror (PF10-03-P01, ThorLabs) was measured against a flat mirror reference (20Z40ER.2, Newport) with the C-OCT system over a central 12 mm diameter region. The measurement was performed with a point cloud of 101 by 101 points. It was found that the data transfer of the ADC used was the speed bottleneck, resulting in a total point-cloud acquisition time of ∼56 minutes. Both mirrors were also measured with a commercial Fizeau interferometer to establish a baseline, where the reference mirror showed 6.7 nm RMS and the sample mirror 5.2 nm RMS after removing piston, tip, and tilt via Zernike fitting. Note that the Fizeau measurements were over a broader region of 24 mm in diameter to accommodate the lack of concentration fiducials to precisely register between the C-OCT and Fizeau measurements. The vertical mounting of the reference mirror with a nylon-tip screw within the C-OCT setup was maintained in the Fizeau measurement, which was why its RMS flatness is on par with that of the sample mirror. Both surface maps are shown in Figs. 6(a)–6(b). Three consecutive surface measurements were taken with the C-OCT system. These measurements were fitted to Zernike polynomials up to 100 terms in the j ordering [36] to perform low-pass filtering while sufficiently covering the form regime. Piston, tip, and tilt were removed. The mean of the measurements is shown in Fig. 6(c), which shows a peak-to-valley (PV) of 110.6 nm and an RMS of 14.4 nm (∼λ/44 at the He-Ne wavelength). The standard deviation (SD) of the measurements is shown in Fig. 6(d) with a mean of 5.3 nm.
Fig. 6. A flat mirror was measured with the C-OCT system. The (a) reference and (b) sample mirror measured with a commercial Fizeau interferometer. (c) The mean of three consecutive C-OCT measurements, showing an RMS of 14.4 nm (∼λ/44). (d) The standard deviation of the three measurements with a mean of 5.3 nm.
Next, a freeform mirror was measured to demonstrate proof-of-concept of C-OCT for freeform metrology. Three consecutive measurements were made, and the mean of the measurements is registered rigidly in all six degrees of freedom to the nominal prescription via an optimization routine that minimizes their RMS difference. The nominal prescription and the C-OCT measurement after registration are shown in Fig. 7(a) and Fig. 7(b), respectively. Their Fringe Zernike [37] coefficients comparison are plotted in Fig. 7(c), showing excellent agreement. Similar to the processing of the flat mirror measurements, the residual difference was fitted to Zernike polynomials up to 100 terms in the j ordering and shown in Fig. 7(d). The PV of the residual was found to be 345.8 nm, and the RMS residual was found to be 69.1 nm (∼λ/9 at the He-Ne wavelength), which meets the mirror’s manufacturing tolerance for form.
Fig. 7. A freeform mirror was measured with the C-OCT system. (a) Nominal prescription of the freeform mirror. (b) Mean of three consecutive C-OCT measurements. (c) Fringe Zernike coefficients comparison between the nominal and the measurement, showing excellent agreement. (d) Measurement residual after subtracting the nominal, the RMS of which is 69.1 nm (∼λ/9).
Comparing the residual of the flat mirror measurement versus that of the freeform mirror, it is theorized that the dominant residual error is in the scanning, as the line-like artifact observable in Fig. 6(c) and Fig. 7(d) are both along the measurement’s fast axis. Investigations are planned to quantify the scanning artifacts and their repeatability, following which the artifacts may be calibrated from the measurements to further reduce the residual error. Nevertheless, both the flat mirror and the freeform mirror measurements showed excellent residual RMS values. These results validate the optical scanning sub-system and the high-speed cube rotation mechanism described in Section 2 and demonstrate the C-OCT system’s viability for measuring freeform optical surfaces.
For the next-generation development of the C-OCT metrology system, several areas are anticipated to be impactful for residual error mitigation. The first area is the fine-tuning of the optical scanning sub-system. The current common-path configuration of the primary interferometer mitigates errors due to optical system aberrations, as the sample and the reference arms share the same optical components. However, this configuration may obscure errors due to slight misalignments in the positions of the two galvanometer mirrors with respect to each other across the pupil relay and the positions of both mirrors with respect to the telecentric objective lens. Having an additional, separate reference arm in the primary interferometer is expected to aid the fine-tuning of these positions.
The second area is the quantification and mitigation of relevant environmental factors’ effect on the measurement results. A quantified understanding and control of the environment are anticipated to aid the de-coupling of inherent system errors from environment-induced errors, enabling the further reduction of measurement uncertainty.
Moreover, with the C-OCT technique verified and the metrology system demonstrated, an important future study is the quantification and expansion of the measurement volume as bound by desired levels of measurement uncertainty. A simulation model was developed in prior work to quantify metrology uncertainty within a conventional OCT architecture [27]. Revisiting this model in the context of C-OCT in parallel with experimental investigations is anticipated to provide additional insights for uncertainty reduction.
4. Summary
In this work, we developed a C-OCT metrology system towards surface form metrology of freeform optical components. The requirements and instrumentation to achieve surface measurements were detailed. A custom pupil relay was developed to optically scan the sample surface with two galvanometer mirrors, enabling the optically relayed mirrors to be located at the entrance pupil of a custom telecentric objective lens. A high-speed rotation mechanism was developed for the secondary interferometer optical cube. Stable rotation was achieved for the current target speed of 1,760 RPM and up to 15,000 RPM to accommodate future speed upgrades. Surface measurement on a flat mirror showed an RMS residual of 14.4 nm (∼λ/44 at the He-Ne wavelength). Measurement on a freeform mirror was achieved with an RMS residual of 69.1 nm (∼λ/9). The system-level investigations and validation provide the groundwork for advancing C-OCT as a viable freeform metrology technique.
Funding
Corning Incorporated Office of STEM Graduate Research Fellowship; National Science Foundation (IIP-1338877, IIP-1338898, IIP-1822026, IIP-1822049); II-VI Foundation.
Acknowledgments
We thank Synopsys, Inc. for the education license of CODE V and LightTools. Gratitude is expressed to John Uchal, Ivan Suminski, Connor Kasper, and Zihao Chen for investigating an early prototype of the cube rotation assembly. We thank Gustavo A. Gandara-Montano and Jonathan C. Papa for stimulating discussions.
Disclosures
The authors declare no conflicts of interest.
References
1. K. P. Thompson and J. P. Rolland, “Freeform optical surfaces: a revolution in imaging optical design,” Opt. Photonics News 23(6), 30–35 (2012). [CrossRef]
2. S. Wills, “Freeform Optics: Notes from the Revolution,” Opt. Photonics News 28(7), 34–41 (2017). [CrossRef]
3. R. Wu, Z. Feng, Z. Zheng, R. Liang, P. Benítez, J. C. Miñano, and F. Duerr, “Design of Freeform Illumination Optics,” Laser Photonics Rev. 12(7), 1700310 (2018). [CrossRef]
4. J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light: Sci. Appl. 6(7), e17026 (2017). [CrossRef]
5. C. Yoon, A. Bauer, D. Xu, C. Dorrer, and J. P. Rolland, “Absolute linear-in-k spectrometer designs enabled by freeform optics,” Opt. Express 27(24), 34593–34602 (2019). [CrossRef]
6. A. Bauer, M. Pesch, J. Muschaweck, F. Leupelt, and J. P. Rolland, “All-reflective electronic viewfinder enabled by freeform optics,” Opt. Express 27(21), 30597–30605 (2019). [CrossRef]
7. E. M. Schiesser, A. Bauer, and J. P. Rolland, “Effect of freeform surfaces on the volume and performance of unobscured three mirror imagers in comparison with off-axis rotationally symmetric polynomials,” Opt. Express 27(15), 21750–21765 (2019). [CrossRef]
8. Y. Zhong, Z. Tang, and H. Gross, “Correction of 2D-telecentric scan systems with freeform surfaces,” Opt. Express 28(3), 3041–3056 (2020). [CrossRef]
9. S. Sorgato, R. Mohedano, J. Chaves, M. Hernández, J. Blen, D. Grabovičkić, P. Benítez, J. C. Miñano, H. Thienpont, and F. Duerr, “Compact illumination optic with three freeform surfaces for improved beam control,” Opt. Express 25(24), 29627–29641 (2017). [CrossRef]
10. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20(3), 2110–2115 (2012). [CrossRef]
11. J. P. Rolland, M. A. Davies, T. J. Suleski, C. Evans, A. Bauer, J. C. Lambropoulos, and K. Falaggis, “Freeform optics for imaging,” Optica 8(2), 161–176 (2021). [CrossRef]
12. K. Creath and J. C. Wyant, “Use of Computer-Generated Holograms in Optical Testing,” in Handbook of Optics, Volume II, 2nd ed. (McGraw-Hill Inc., 1995).
13. P. Zhou and J. H. Burge, “Fabrication error analysis and experimental demonstration for computer-generated holograms,” Appl. Opt. 46(5), 657–663 (2007). [CrossRef]
14. 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]
15. R. Chaudhuri, J. C. Papa, and J. P. Rolland, “System design of a single-shot reconfigurable null test using a spatial light modulator for freeform metrology,” Opt. Lett. 44(8), 2000–2003 (2019). [CrossRef]
16. M. C. Knauer, J. Kaminski, and G. Hausler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004). [CrossRef]
17. P. Su, M. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. Butel, Y. Wang, R. E. Parks, and J. H. Burge, “Aspheric and freeform surfaces metrology with software configurable optical test system: a computerized reverse Hartmann test,” Opt. Eng. 53(3), 031305 (2013). [CrossRef]
18. Y. B. Seo, H. B. Jeong, H. G. Rhee, Y. S. Ghim, and K. N. Joo, “Single-shot freeform surface profiler,” Opt. Express 28(3), 3401–3409 (2020). [CrossRef]
19. B. R. Swain, C. Dorrer, and J. Qiao, “High-performance optical differentiation wavefront sensing towards freeform metrology,” Opt. Express 27(25), 36297–36310 (2019). [CrossRef]
20. P. E. Murphy and C. Supranowitz, “Freeform testability considerations for subaperture stitching interferometry,” Proc. SPIE 11175, 35 (2019). [CrossRef]
21. E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33(24), 2973–2975 (2008). [CrossRef]
22. I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, W. Osten, and C. Elster, “Evaluation of absolute form measurements using a tilted-wave interferometer,” Opt. Express 24(4), 3393–3404 (2016). [CrossRef]
23. A. M. Michalko and J. R. Fienup, “Development of a concave freeform surface measurement using transverse translation-diverse phase retrieval,” Opt. Eng. 59(06), 1 (2020). [CrossRef]
24. H.-U. Danzebrink, L. Koenders, G. Wilkening, A. Yacoot, and H. Kunzmann, “Advances in scanning force microscopy for dimensional metrology,” CIRP Ann. 55(2), 841–878 (2006). [CrossRef]
25. K. Kubo, “Laser based asphere and freeform measurement technology by UA3P,” in Proc. of 11th Laser Metrology for Precision Measurement and Inspection in Industry (2014), pp. 1–6.
26. V. G. Badami and T. Blalock, “Uncertainty evaluation of a fiber-based interferometer for the measurement of absolute dimensions,” Proc. SPIE 5879, 587903 (2005). [CrossRef]
27. J. Yao, A. Anderson, and J. P. Rolland, “Point-cloud noncontact metrology of freeform optical surfaces,” Opt. Express 26(8), 10242–10265 (2018). [CrossRef]
28. O. Jusko, M. Neugebauer, H. Reimann, P. Drabarek, M. Fleischer, and T. Gnausch, “Form Measurements by Optical and Tactile Scanning,” Proc. SPIE 7133, 71333A (2008). [CrossRef]
29. M. Wendel, “Measurement of freeforms and complex geometries by use of tactile profilometry and multi-wavelength interferometry,” Proc. SPIE 11171, 20 (2019). [CrossRef]
30. J. Cohen-Sabban, J. Gaillard-Groleas, and P.-J. Crepin, “Quasi-confocal extended field surface sensing,” Proc. SPIE 4449, 178–183 (2001). [CrossRef]
31. S. DeFisher, “Advancements in non-contact metrology of asphere and diffractive optics,” Proc. SPIE 10448, 50 (2017). [CrossRef]
32. R. Henselmans, “Non-contact measurement machine for freeform optics,” Ph.D. dissertation, Technische Universiteit Eindhoven (2009).
33. W. Drexler and J. G. Fujimoto, Optical Coherence Tomography Technology and Applications (Springer, 2008). [CrossRef]
34. D. Xu, A. G. Coleto, B. Moon, J. C. Papa, M. Pomerantz, and J. P. Rolland, “Cascade optical coherence tomography (C-OCT),” Opt. Express 28(14), 19937–19953 (2020). [CrossRef]
35. D. Xu, R. Chaudhuri, and J. P. Rolland, “Telecentric broadband objective lenses for optical coherence tomography (OCT) in the context of low uncertainty metrology of freeform optical components: from design to testing for wavefront and telecentricity,” Opt. Express 27(5), 6184–6200 (2019). [CrossRef]
36. Y. Niu, “Robust numerical evaluation of Zernike polynomials and the generation of 3D freeform surface meshes,” Master’s thesis, University of Rochester (2018).
37. CODE V, “Appendix A. Zernike polynomials,” in CODE V Lens System Setup Reference Manual Version 11.0 (2017).
