Open Access
Add to BibSonomy
Abstract
To meet the need for rapid, high-precision, and non-contact measurement of the radius of curvature (ROC) for large quantities of spherical optics, a radius measurement method based on transverse dual differential confocal (TDDC) detection is proposed in this study. First, a template S0 with a known ROC, R 0, is axially scanned on its confocal position to obtain the fitted linear function lTDDC(z) using TDDC. Second, the template S0 is replaced by Sn, which is one of the test sample in large quantities, then the single point TDDC intensity ITDDC(Δzn) is captured without scan, which will be applied to obtain the defocus Δzn according to the linear function lTDDC(z). Finally, the ROC Rn under test is calculated using Δzn and R 0. Simulation and experiments show that the measurement accuracy can achieve 8.0 ppm, and the measurement efficiency is 60 times higher than that of the traditional differential confocal scanning measurement. Measurement based on TDDC only requires scanning once and replacing Sn N times to realize the fast, high-precision, non-contact ROC detection of N pieces of spherical optics, which enables the high-efficiency and high-precision measurement of large quantities of spherical optics.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The demand for spherical optics has been increasing with their wide applications in optical systems such as medical instruments, camera lenses and other optical lenses [1]. The radius of curvature (ROC) is one of the most important parameters which directly influences the performance of optical systems [2]. However, existing measuring methods are difficult to achieve both high-precision and high-efficiency in machining or measuring spherical optics, which significantly inhibit in the production of spherical optics.
Existing ROC measurement methods can be divided into two types: contact and non-contact types. Contact ROC measurement approaches include the template [3], spherometer [4], three-coordinate [5], and laser tracking methods [6]. The template and spherometer methods are easy to operate, and the ROC results can be obtained immediately once the measured optics are chucked, which are commonly used for measuring the ROC for large quantities optical productions. The template method is based on the principle of equal thickness interference. The absolute values of the ROC of the template and the measured part are similar, one is positive and the other is negative. If the ROCs of the test surface and the template surface are exactly same, no fringes are formed. Conversely, fringes are observed if the ROCs are different. The difference between the measured part and the template can be calculated according to the observed number of fringes. However, this method is affected by the accuracy of the template and the changes in stress during measurement; Therefore, its accuracy is low (i.e., 800 ppm) and is influenced by the operator. For the spherometer method, the ROC is calculated using the known chord length and the measured sagittal height of the sphere. This method is easy to operate, but the measurement accuracy is around 50 ppm and it will decrease with the increase of the test radius. For the three-coordinate method, the test surface is scanned to obtain a three-dimensional point cloud, and the measured ROC is obtained by sphere fitting. The accuracy of the three-coordinate method is also low (i.e., 20 ppm) [4] . The laser tracking method can accurately measure the radius of curvature of a spherical surface with a phase-measuring interferometer and a laser tracker. The laser tracker eases the alignment of the testing system, which no need to move the test piece during the measurement, its accuracy is 16 ppm. [6]. However, this measurement is usually applied for large ROC lenses. All of these methods have the risk of scratching the tested surface.
Geometric optics methods (the knife edge [7] and autocollimator [8,9]) and interferometry are the mostly applied non-contact ROC measurement methods. For the knife edge method, a knife edge is placed at the focus position of the test surface, and the distance of the measuring beam converging away from the vertex of test surface is measured, which is the ROC result. Although this method is easy to operate, it has low accuracy (i.e., 110 ppm). In the autocollimator method, laser autocollimation is used to integrate slop values with respect to distance to obtain the profile height, from which the ROC can be calculated. This method is applicable to large aperture lenses. When the ROC under test is greater than 5 m, the accuracy is 500 ppm. In contrast, the interferometry method demonstrates higher measurement accuracy and is currently widely used in high-precision measurement [10–16]. Classical interferometry uses a phase measurement interferometer to respectively fix the cat's eye position and the confocal position of the measured sphere to obtain the measured ROC, and the measurement accuracy can reach 10 ppm [11]. On this basis, Kredba et al. proposed a ROC measurement method using common-path Fizeau interferometry, which measures the cavity length using absolute wavelength tuning interferometry. When the cavity length and reference ROC are known, the ROC under test can be obtained. The ROC is measured within one shot, i.e., without the need for any mechanical movement, so its efficiency is higher than traditional interferometry method [17,18]. However, all interferometry methods have cumbersome attitude adjustment processes, and require a considerable amount of time to stabilize the interference fringes after the lens are installed. In addition, the interference fringes are easily disturbed by environmental factors such as airflow, temperature, and vibration.
Zhao et al. proposed the laser differential confocal curvature radius measurement (DCRM) method [1,19], which uses the characteristic that the absolute zero point of the differential confocal curve accurately corresponds to the focus of the measurement beam to measure the surface of the measured spherical optics. The cat's eye and confocal positions are fixed to obtain the ROC to be measured. The accuracy of DCRM is 5 ppm, but it requires a great amount of time to axially scan and obtain the cat's eye and confocal positions, as well as to conduct the cumbersome attitude adjustment process. Therefore, the efficiency of this method is difficult to be improved.
In this study, radius measurement method based on transverse dual differential confocal (TDDC) detection is proposed. A template with a known ROC was axially scanned at its confocal position to obtain the linear function using TDDC. The template was replaced with the test sample to capture the single TDDC intensity without scanning, which will be applied to obtain the defocus according to the linear function. Finally, the ROC under test was calculated using the defocus and ROC of the template, thereby transforming the absolute measurement process of the ROC into a deviation measurement relative to a template. This method not only retains the advantages of the high precision of the traditional DCRM but also significantly improves the measurement efficiency. Compared with traditional confocal detection [20] and differential confocal detection [1,19], TDDC detection has bipolarity, high sensitivity, and a longer linear interval [21]. Proposed method can be used for the high-efficiency and high-precision measurement of large quantities of spherical optics.
2. TDDC principle
Traditional dual differential detection requires three detectors whose axial defocus should be precisely adjusted according to the numerical aperture of the condenser lens. Thus, the process is time-consuming [22].Herein we use a D-shaped aperture to block half of the measuring beam and detect the integrated intensity of virtual pinholes (vph1, vph2, c) on the elliptic spot with CCD substituted for traditional dual differential confocal detectors. The TDDC method prevent from precisely adjusting the detector at a certain position, thereby possible adjusting errors can be avoided.
The principle of TDDC is shown in Fig. 1. The laser source is collimated into parallel light by the lens Lc, then converged into a cone using the converging mirror Ls, and is reflected into the measuring beam by S0 at its confocal position. The measuring beam reflected on the surface under test returns through the converging lens Ls and collimating lens Lc and is reflected by PBS, after which half of the measuring beam is blocked by the D-shaped aperture and converges on the object plane of the microscopic objective lens. After passing through the microscopic objective lens M with magnification β, the light is imaged on the CCD plane. For the elliptical spot detected by CCD, an area is set where the optical axis crosses the CCD plane as the virtual confocal pinhole c and symmetrically set virtual pinholes (pre-focus vph1 and post-focus vph2) on either side of on-focus pinhole c. The integral gray value in the virtual pinholes can be regarded as their detected intensity, and dual differential processing is performed. The TDDC curve (ITDDC) is obtained from the intensity of the three virtual pinholes and the positions collected. By fitting the TDDC intensity and position in the linear interval of the curve ITDDC, the linear function lTDDC(z) and its zero crossing position are obtained, which is the confocal position. Thus, high-precision confocal detection is realized.
Then the measuring beam is reflected by the surface under test, divided by the D-shaped aperture, and imaged on the CCD, the complex amplitude response function of the image plane of the focal plane of the detection optical path is given as follows [22].
When Eq. (2) is simulated with νM = 5.18 and φ = π/2, the resulting TDDC curve is shown in Fig. 2.
In Fig. 2, as the zero point position of the curve corresponds precisely to the confocal position of the surface S0, the calibration template S0 can be accurately fixed. The part near the zero-crossing point of the curve can be approximated as a straight line. In actual measurement, the defocus is usually expressed as axial displacement z rather than u, where $u = (\pi {D^2}/2\lambda {f_S}^2) \cdot z$. Therefore, the functional relationship between the differential and the defocus amount can be approximated using the linear function lTDDC(z) as follows.
where kTDDCis the slope of the linear function and b is the intercept. The interval of the function is called the linear interval. In this interval, the axial defocus Δz can be obtained from the TDDC intensity ITDDC(Δz) and slope kTDDC.3. High-efficiency ROC measurement principle based on TDDC
A high-efficiency measurement principle, shown in Fig. 3, is proposed based on TDDC.
First, a template S0 with a known ROC is used, of which the ROC and the reflectivity are similar to that of the measured sample S1. The similar ROC ensures that the defocus remains in the linear interval, and the similar reflectivity ensures that it detected the same intensity at the same defocus Δz 1 for S0 and S1. Scanning is performed near the confocal position of S0 axially, the TDDC intensity and position data are fitted to obtain the linear function lTDDC(z), and the confocal position, which is the zero point of lTDDC(z), is calculated. Then, S0 is replaced with S1, the single TDDC intensity ITDDC(Δz 1) is captured, and the Δz 1 is calculated using lTDDC(z). Finally, using the geometric relationship between the fixture and the measured part, the R0 and Δz 1 are used to calculate the measured R 1.
3.1 TDDC curve calibration
The template S0 is scanned axially near its confocal position, the obtained data is shown in Fig. 4. When scanning, the template moves near the focal point of the condensing lens along the optical axis, and its position ${z_i}$ and spot images are recorded by the grating ruler and CCD.
When processing the data, the spot trajectory is fitted and the maximum point of the brightest spot is selected as the virtual pinhole c. The amount of off-axial defocus is calculated, after which virtual pinholes, denoted vph1 and vph2 on either side of pinhole c, are symmetrically set along the spot trajectory. The gray value in these areas is integrated as the intensity, after which the intensity data Ivph 1, Ivph 2, and Ic are processed with the TDDC method via Eq. (2). A set of TDDC intensity data points Im are obtained. Then, an optimum linear interval is set and a linear function lTDDC(z) is fitted to the intensity and position data in the linear interval.
The linear function is fitted to the TDDC intensity Im and the position zm of M data points captured in the interval by the least squares method to represent the functional relationship between the defocus amount and the TDDC intensity in the interval:
In Eq. (4), kTDDC is the slope of the fitted linear function and b is the intercept of the fitted linear function. Equation (3) is the expression of lTDDC(z). The zero point z0=-b/kTDDC of the linear function is the confocal position of S0. In addition, Eq. (3) also represents the functional relationship between one single TDDC intensity ITDDC(Δzn) and the defocus amount Δzn near the zero-crossing point (in the linear interval). It can be used to calculate the defocus amount Δzn of the tested object from the TDDC ITDDC(Δzn).
3.2 Obtaining Δzn without scan
The confocal position z 0 of S0 is calculated and precisely moved to z 0. Then, S0 is replaced with the test sample Sn, as shown in Fig. 5.
In Fig. 5, the points F1 and F2 remain fixed, and S0 is replaced with Sn. The sphere center of Sn locates on On. The intensity detected by the differential confocal detector also changes accordingly. Because S0 and Sn have similar reflectivity, the linear function lTDDC(z) is considered to represent the relationship between the single intensity TDDC ITDDC(Δzn) and the defocus amount Δzn of Sn. At this position, the single intensity of vph1, vph2, and c are directly captured at the confocal position of S0 without scanning, after which the single point intensities ${I_{vph1}}(\Delta {z_n})$, ${I_{vph2}}(\Delta {z_n})$, and ${I_c}(\Delta {z_n})$ are processed, similar to Eq. (2), to calculate the single TDDC intensity ITDDC(Δzn) as follows.
The TDDC intensity is substituted into the linear function lTDDC(z). When the defocus amount Δzn of Sn does not exceed the linear interval, Δzn of the test sample Sn can be calculated as follows:
3.3 Calculating Rn from Δzn
Figure 6(a) shows the geometric model for the measurement of convex surface element, as well as Fig. 6(b) shows that of the concave surface element. Because of the difference in ROCs of the two spheres, the position of the sphere center will be offset in the axial direction.
In Fig. 6, R 0 is the ROC of template S0, Rn is the ROC of Sn, and F1 and F2 are the contact points (three-dimensional edge) between the fixture and the surface. Δzn is the axial defocus between the S0 center O 0 and Sn center On, DF is the diameter of the inner ring of the fixture when measuring convex lens, and its the diameter of the outer ring of the fixture when measuring concave lens. According to the geometric relationship shown in Fig. 6, the Rn of the sample to be tested can be obtained as follows.
According to the geometric relationship between S0 and the S1 in Fig. 6, the main geometric relationship of Rn comes from the triangle ΔO0OnF2 in Fig. 6(a) and 6(b). In this triangle, O0On = Δzn, O0F2 = R 0, so according to the cosine theorem, the side length OnF2, that is, the value of Rn, can be found:
Therefore, the measured Rn of both convex and concave surface can be obtained by the following equation:
4. Experiment
To verify the feasibility and accuracy of the proposed method in this study, a vertical radius measurement instrument based on TDDC was constructed, whose system diagram is shown in Fig. 7. The instrument consists of a TDDC module, motion & inspection module, and adjustment module. The TDDC module has a light source and collimator, and there is a D-shaped aperture to divide the measuring beam and collect spots. The motion& inspection module scans and locates with a motor, stud, and air bearing guide, it collects position data with a grating ruler. The adjustment module can perform two adjustments: adjusting the spatial location and orientation of Ls and S0, and an annular fixture where S0 is placed on the inner circle of the fixture without chucking such that the efficiency is significantly improved.
4.1 Experiment A: convex lens measurement
The experiment was conducted under temperature of 20 ± 0.5 °C and relative humidity of 25%±4%. A standard objective lens produced by Zygo with F = 3.3 is used as LS with aperture of 100 mm and fs = 330 mm. The focal length of the collimating lens LC is fc = 1000 mm. The wavelength of laser source is λ=632.8 nm. The model of CCD is OK_AM1160. The model of grating scale is Renishaw RGSZ20-S, T100030A linear encoder, TI0200A subdivision card. A high-precision air bearing guide rail matched with a slider are used. A Yaskawa SGM7J AC servo motor and GoogolTech GTS-400 series motion controller are used too. The diameter DF A (the subscript A stands for experiment A)of the fixture is measured to be 29.986 mm.
The template lens S0A and the test sample S1A are shown in Fig. 8. The nominal ROC of template S0A is 39.1 mm. The ROC is 39.0832 mm, as measured by DCRM [1]. S1A is another lens of the same type with S0A.
The measurement procedure is summarized in three steps:
Step 1: TDDC curve calibration. The motor drives the lead screw and the fixture is fixed on the air-floating slider with the template S0A to move along the optical axis near the confocal position, while the CCD and the grating ruler collect the image and position data when scanning. The trajectory of spots is fitted, and the maximum position of the brightest spot in all spots is taken as the pinhole c; then, the vph1 and vph2 are symmetrically located in the direction of the trajectory, where the distance from the on-focus c is νpixel.
In Fig. 9, part of the collected spot image with the TDDC curve IDDC and the linear function lTDDC(z) are shown, where the normalized off-axial defocus νM = 5.18 and CCD pixel size p = 8.3 µm. The calculated value is approximated to νpixel = 31.
The three virtual pinholes vph1, c, and vph2 are set on the colored points in the image. Then, the radius of the virtual pinholes is set as three pixels, and the integral gray value at the pinholes is collected as the intensity there. The intensity data and position data are fitted from grating ruler; then, the transverse TDDC curve ITDDCis obtained, set an optimum linear interval $z \in [{ - 0.0478\textrm{ }mm,0.0301\textrm{ }mm} ]$, and fit the intensity data and position data to the linear function lTDDC(z). Its function expression is as follows:
The zero point of Eq. (11) is the confocal position of S0A, that is, z 0A=-0.0074 mm.
Step 2: obtain Δz 1A without scanning. The template S0A is replaced with S1A, where its TDDC intensity ${I_{TDDC}}(\Delta {z_{1\textrm{A}}}) = 0.7048$, and the calculated axial defocus Δz 1A=-0.0303 mm, according to the linear function lTDDC(z). The TDDC intensity is located in the linear interval shown in Fig. 10.
Step 3: calculate R 1A from Δz 1A. The ROC (R 1A = 39.1064 mm) of S1A can be calculated via Eq. (9).
Steps 2 and 3 are then repeated for 10 measurements of S1A. R 1A values are shown in Table 1. The mean of R 1A is 39.1064 mm, the repeatability is defined as the standard deviation (STD) (i.e., 0.3 µm), and the relative repeatability is 7.7 ppm.
Table 1. R1A measured by TDDC instrument.
When timing the measurement, we did not include the time spent making adjustments. It takes approximately 30 s to scan the template and calibrate the curve. To perform repeated measurements on S1A, it is only necessary to repeatedly capture the single TDDC intensity ITDDC(Δz 1A) . Each capture interval is 1.5 s, and the measurement time for 10 data points is 15 s. For the same batch of samples under test, only one curve calibration is required. When the number of the sample Sn A is very large, the time used for curve calibration can be neglected; thus, it can be considered that 15 s is the time required for 10 measurements.
For comparison, the DCRM [1] was used to measure the ROC of S1A in the same environment and the same Ls with F = 3.3. In DCRM, the measured sample S1A was repeatedly measured for 10 times. The measurement time for 10 sets of data was 941 s, which includes 20 scans, 60 s per scan and 19 movements, 15 s each. This method also required some times for system self-adjustment. The ROC result and efficiency comparison for 10 measurements by the TDDC system and DCRM system are shown in Table 2.
Table 2. Comparison of TDDC and DCRM
For the same test sample, compared with the DCRM result, our ROC result had a deviation of 0.3 µm. The vital point is that the measurement speed of the high-efficiency TDDC measurement is considerably accelerated compared with that of the DCRM. The measurement speed is increased by approximately 60 times, which significantly improves the measurement efficiency for TDDC by avoiding repeated adjustment and scan. Therefore, this method leads to high-precision and high-efficiency ROC measurement.
4.2 Experiment B: ceramic balls measurement
To verify the accuracy of the proposed method, a experiment was performed. The environment and the instrument parameter was the same as section 4.1. The diameter DF B (the subscript B stands for experiment B) of the fixture is measured to be 19.998 mm. A ceramic ball is chosen as S0B whose ROC is 12.7222 mm measured by DCRM [1]. S1B is another ceramic ball of the same type with S0B.
The measurement procedure is summarized in three steps:
Step 1: TDDC curve calibration. In Fig. 11, part of the collected spot image with the TDDC curve ITDDC and the linear function lTDDC(z), where distance from the on-focus c νpixel as calculated is νpixel = 31.
The three virtual pinholes vph1, c, and vph2 are set on the colored points in the image. Then, the radius of the virtual pinholes is set as three pixels, and the integral gray value at the pinholes is collected as the intensity there. The intensity data and position data are fitted from grating ruler; Then, the transverse TDDC curve ITDDCis obtained, set an optimum linear interval $z \in [{ - 0.0341mm,0.0447\textrm{ }mm} ]$, and fit the intensity data and position data to the linear function lTDDC(z). Its function expression is as follows:
The zero point of Eq. (12) is the confocal position of S0B, that is, z 0B = 0.0038 mm.
Step 2: obtain Δz 1B without scanning. The template S0B is replaced with S1B, where its TDDC intensity ${I_{TDDC}}(\Delta {z_{1\textrm{B}}}) ={-} 0.5406$, and the calculated axial defocus Δz 1B = 0.0239 mm, according to the linear function lTDDC(z). The TDDC intensity is located in the linear interval shown in Fig. 12.
Step 3: calculate R 1B from Δz 1B. The ROC (R 1B = 12.7074 mm) of S1B can be calculated via Eq. (9).
Steps 2 and 3 are then repeated for 10 measurements of S1B. R 1B values are shown in Table 3. The mean of R 1B is 12.7064 mm, the repeatability is defined as the standard deviation (STD) (i.e., 0.14 µm), and the relative repeatability is 11.0 ppm.
Table 3. R1B measured by TDDC instrument.
The measured result of S1B is 12.7069 mm using the length measuring instrument(LMI) of an institute of legal calibration, and the measured result of S1B is 12.7072 mm using the DCRM. The comparison with the measured result of thoes methods is shown in Table 4. Besides, the result deviation is calculated based on the length measuring instrument.
Table 4. Comparison of TDDC, DCRM and MLI.
According to the comparison between the above experimental results and calibration results, it can be seen that the difference between the curvature radius measured by this method and the calibration value from calibration is 0.5 µm. Therefore, the measurement accuracy of the curvature radius of this method has been verified. At the same time, the efficiency of this method has been greatly improved compared with DCRM method, so this method can achieve high-precision and high-efficiency measurement of ROC.
5. Uncertainty analysis
When measuring the ROC of a spherical surface with the radius measurement based on TDDC, the errors primarily arise from the grating scale (u 1), the detection error (u 2) introduced by light intensity drift, the surface shape error (u 3), the error (u 4) introduced by the diameter of the fixture, and the repeated measurement error (u 5).
The length measuring system uses the RGSZ20-S grating scale produced by Renishaw. Therefore, the uncertainty component introduced by the length measurement error of the grating ruler includes the linearity of the grating scale uL, the subdivision error us and the installation error u I:
There is an intensity drift in the light spot when capturing the TDDC intensity of the measured lens that is related to the stability of the laser light source, environmental interference, and CCD dark current. The error introduced by the light intensity drift is determined as follows.
where $\varDelta {I_{vph}}$ is the relative light intensity drift, which is approximately 1% in the capture interval which is 1.5 s when measuring.The uncertainty component introduced by the TDDC intensity is determined as follows.
Because of the influence of the surface shape of the measured surface, the measurement area corresponding to the confocal position usually does not coincide with the optimal spherical surface of the measured surface,which introduced surface shape error is:
The PV value is measured by the Zygo interferometer, and the uncertainty component introduced by the surface error u3 is determined as follows.
The diameter of the fixture will introduce error when calculating the ROC. The diameter of the fixture is measured with a inside micrometer with a accuracy of±5 µm. The error of the measured value of the fixture diameter ${u_4}$ is given as follows.
The errors caused by the environment, such as temperature, vibration, and airflow, can be included in 10 repeated measurements of u 5 as follows.
Considering the above errors, the combined uncertainty uc of the ROC is as follows, where $\frac{{\partial {R_n}}}{{\partial \Delta {z_n}}}$ is the partial derivative of Rn to Δzn, and $\frac{{\partial {R_n}}}{{\partial \Delta {D_F}}}$ is the partial derivative of Rn to DF derived from Eq. (9).
To the ROC measurement result in section 4.1, the uncertainty is calculated with Eq. (20). The elative accuracy of the ROC in section 4.1 is given as follows.
As for the ROC measurement result in section 4.2, the uncertainty is calculated with Eq. (20). The elative accuracy of the ROC in section 4.2 is given as follows.
6. Conclusion
We proposed a radius measurement method based on TDDC detection with high efficiency and high precision, a vertical radius measurement device based on proposed method was constructed. A template S0 with a known ROC, R 0, was scanned at the confocal position to obtain the linear function lTDDC(z) using TDDC, then the template S0 was replaced with the test sample Sn to capture the single TDDC intensity ITDDC(Δzn) without scanning. Then, ITDDC(Δzn) was applied to get defocus Δzn with lTDDC(z). Finally, the ROC Rn under test was calculated using Δzn and R 0. The radius measurement based on TDDC is suitable for the measurement of large quantities of spherical optics with the same nominal ROC and reflectivity. It has the advantages of simple assembly and adjustment, no need for scanning, which meets the measurement requirements for huge amounts of production and enables radius measurement with high efficiency and high precision.
Funding
National Key Research and Development Program of China (2017YFA0701203); National Natural Science Foundation of China (61827826); The Civil Aerospace Technology Advance Research Project (No.D030207).
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. W. Zhao, R. Sun, L. Qiu, and D. Sha, “Laser differential confocal radius measurement,” Opt. Express 18(3), 2345–2360 (2010). [CrossRef]
2. Y. Bitou, O. Sato, and S. Telada, “Three-spherical-mirror test for radius of curvature measurement using a Fabry-Pérot cavity,” Opt. Express 27(10), 13664–13674 (2019). [CrossRef]
3. M. V. Mantravadi and D. Malacara, “Newton, Fizeau, and Haidinger Interferometers,” Optical Shop Testing (Wiley, 2020), pp. 1–45.
4. Y. Shen, P. Wang, H. Zhang, C. Hui, and K. Yang, “Comparison and analysis on the accurate measurement of spherical curvature radius,” Proc. SPIE 10839, 9th International Symposium on Advanced Optical Manufacturing and Testing Technologies: Optical Test, Measurement Technology, and Equipment, 108391 V (18 January 2019).
5. K. Yoshizumi, T. Murao, J. Masui, R. Imanaka, and Y. Okino, “Ultrahigh accuracy 3-D profilometer,” Appl. Opt. 26(9), 1647–1653 (1987). [CrossRef]
6. C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005). [CrossRef]
7. T. J. Hugo, N. v, and d. W. Lessing, “Determination of Long Radii of Curvature of Positive Lenses,” Appl. Opt. 3(4), 483–485 (1964). [CrossRef]
8. A. Thetford, “Optical shop testing: Edited by D. Malacara John Wiley,” Opt. Laser Technol. 11(1), 55 (1979). [CrossRef]
9. M. S. Virdee, “High Accuracy Form Measurement Of Specularly Reflecting Surfaces By Laser Autocollimation,” Ultraprecision Machining and Automated Fabrication of Optics. Vol. 676. International Society for Optics and Photonics, 1987.
10. T. L. Schmitz, C. J. Evans, A. Davies, and W. T. Estler, “Displacement uncertainty in interferometric radius measurements,” CIRP Ann. 51(1), 451–454 (2002). [CrossRef]
11. J. R. Lawall, “High resolution determination of radii of curvature using Fabry-Perot interferometry,” Meas. Sci. Technol. 20(4), 045302 (2009). [CrossRef]
12. U. Griesmann, J. Soons, and Q. Wang, “Measuring Form and Radius of Spheres with Interferometry,” CIRP Ann. 53(1), 451–454 (2004). [CrossRef]
13. H. Yi, R. Zhang, X. Hu, P. Yong, Z. Kai, and M. Chen, “A novel compensation method for the measurement of radius of curvature,” Opt. Laser Technol. 43(4), 911–915 (2011). [CrossRef]
14. Z. Yang, Z. Gao, Q. Yuan, J. Ye, and M. Li Yang, “Radius of curvature measurement based on wavefront difference method by the point diffraction interferometer,” Optics & Lasers in Engineering 56, 35–40 (2014). [CrossRef]
15. Z. Yang, K. Wang, J. Cheng, Z. Gao, and Q. Yuan, “Large radius of curvature measurement based on virtual quadratic Newton rings phase-shifting moiré-fringes measurement method in a nonnull interferometer,” Appl. Opt. 55(17), 4769–4775 (2016). [CrossRef]
16. M. N. Morris and E. Frey, “Short Coherence Dynamic Fizeau Interferometer with Internal Path Matching for Radius of Curvature Measurement,” in Interferometry Xix, K. Creath, J. Burke, M. B. N. Morris, and A. D. Davies, eds., Proceedings of SPIE (Spie-Int Soc Optical Engineering, 2018), 10749.
17. J. Kredba, P. Psota, M. Stašík, V. Lédl, L. Veselý, and J. Nečásek, “Absolute interferometry for fast and precise radius measurement,” Opt. Express 29(8), 12531–12542 (2021). [CrossRef]
18. P. Psota, M. Stašík, J. Kredba, V. Lédl, and J. Nečásek, “Measurement of radius of curvature directly in the interferometer confocal position,” Appl. Opt. 60(15), 4485–4490 (2021). [CrossRef]
19. W. Zhao, L. Qiu, Y. Xiao, and J. Yang, “Laser differential confocal interference multi-parameter comprehensive measurement method and its system for spherical lens,” Opt. Express 24(20), 22813–22829 (2016). [CrossRef]
20. J. Yang, L. Qiu, W. Zhao, X. Zhang, and X. Wang, “Radius measurement by laser confocal technology,” Appl. Opt. 53(13), 2860–2865 (2014). [CrossRef]
21. R. Shao, W. Zhao, L. Qiu, Y. Wang, R. Zhang, and C. Zhang, “Dual differential confocal method for surface profile measurement with a large sensing measurement range,” Appl. Opt. 59(3), 614–621 (2020). [CrossRef]
22. S. Tang, Y. Li, L. Qiu, Y. Song, W. Zhao, and S. Yang, “High-precision laser transverse differential confocal radius measurement method,” Opt. Express 29(19), 29960–29971 (2021). [CrossRef]
