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Abstract
A normal transverse laser differential confocal freeform measurement (NTDCFM) method was proposed to address the high-precision measurement difficulty of steep freeform surfaces with large variations in inclination, scattering, and reflectance. Using D-shaped diaphragm technology, the freeform surface under test (FSUT) axial variation transformed into a spot transverse movement on the detection focal plane. Meanwhile, a 2D position sensitive detector (PSD) was used to obtain the normal vector of the sampling points so that the measuring sensor’s optical axis could track the FSUT normal direction. The focus tracking method extended the sensor measurement range. Theoretical analysis and experimental results showed that the axial resolution of the NTDCFM was better than 0.5 nm, the direction resolution of the normal vector was 0.1°, the maximum surface inclination could be measured up to 90°, the sensor range was 5 mm, and the measurement repeatability of the FSUT was better than 9 nm. It provides an effective new anti-inclination, anti-scattering, and anti-reflectivity method for accurately measuring steep freeform surfaces.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Freeform surface optical elements are used in space remote sensing, extreme ultraviolet lithography, and astronomical observation because they improve the imaging quality of optical systems, enhance resolution ability, and have more degrees of freedom [1–3]. The manufacturing accuracy of a freeform surface depends on the measurement accuracy. However, the optical surface of a steep freeform surface has the characteristics of a sharp dip angle variation and an irregular normal direction, which pose challenges for measurement [4]. Additionally, the roughness of the freeform surface during processing is poor, which resulted in irregular changes in the intensity of the reflected measurement light and affected the measurement accuracy due to the scattering of the measurement light. So, the roughness of the freeform surface necessitates higher requirements for the measurement method. Recently, high-precision measurement methods for freeform surfaces have attracted increasing attention. Therefore, it is of great significance to study high-precision measurement methods for anti-inclination, anti-scattering, and anti-reflectivity of steep freeform surfaces.
Currently, the most accurate freeform surface measurement methods are non-contact methods, including interferometry, non-interferometric areal measurement, and single-point-probe measurement methods.
Interferometry includes null and non-null interferometry. Null interferometry uses a computer-generated hologram (CGH) to generate interference fringes to measure FSUT [5,6]. However, CGH have a high production cost and poor generality. The adaptive null interferometry method uses a deformable mirror, spatial light modulator, and other adaptive optical elements to compensate for wavefront distortion, making it more flexible. However, limited by the adjustable range, it is difficult to measure steep FUST [7–9]. Interferometry requires a smooth surface for the measurement object; therefore, it is difficult to measure an FSUT with poor surface roughness during machining. Non-null interferometry is used to generate interference fringes using subaperture stitching or shearing interferometry for the measurement. However, this method has poor accuracy when measuring FSUT with a large aspheric deviation [10–13].
Non-interferometric areal measurement methods include the Shaker-Hartmann wavefront measurement method and phase-measuring deflectometry, which are resistant to environmental disturbances such as vibration and air turbulence. However, these techniques are limited to FSUT with small inclination angles and high reflectances [14–17].
Single-point probe measurement methods include the 3D-coordinate measuring machine and the rotating tracking normal measurement method. The former method uses a non-contact probe, which has the advantage of a strong universality. However, owing to the limitations of the principle and structure, the sensor cannot track the normal direction of the FSUT, especially when measuring the FSUT with a wide range of inclinations, which will produce large nonlinear errors. Moreover, steep FSUT cannot be measured because of the limitation of the sensor angle range [18,19]. The latter uses the rotation axis to drive the measurement sensor and track the direction of the Y-dimensional component of the FSUT normal vector (Y-normal direction) to compensate for the limitation of sensor angle range. Typical systems include Luphoscan and Nanomefos. Luphoscan uses multiwavelength interference sensors, but it has a ± 8° inclination measurement capability [20] and cannot directly measure the local inclination angle of the FSUT. Before measurement, it is necessary to adjust the initial position deviation of the FSUT to less than 20 µm. Otherwise, excessive normal tracking error will lead to the loss of sampling points and even lead to mechanical interference between the sensor and FSUT, resulting in measurement failure. Nanomefos uses a differential confocal sensor for measurements and adopts the differential confocal method with two detectors in the front and back of the focus. The defocus deviation of the detector with two optical paths must be controlled within 10 µm, which complicates the installation and adjustment processes. Therefore, this method has the problems of difficult installation, adjustment, and high costs [21–24].
In addition, neither Luphoscan nor Nanomefos can measure the normal vector of the FSUT, resulting in the inability further to reduce the error of coordinate matching during data processing [25]. This greatly improves the accuracy requirements of the measurement system for the initial pose of the FSUT, making the steps of FSUT pose installation and adjustment especially complicated. The efficiency of FSUT measurement significantly reduced. Additionally, the machining process of many freeform surfaces requires normal vector information, and the simultaneous measurement of position and normal vector is of great significance in guiding the machining of FSUT [26,27].
To address these problems, a new method of normal transverse laser differential confocal freeform measurement is proposed in this study to solve the measurement problems of freeform surfaces, such as variations in large inclination angles, roughness, and reflectance during machining. This method can not only achieve high-precision, non-contact measurement of freeform surfaces, but can also greatly reduce the pre-adjustment requirement and coordinate matching deviation of freeform surfaces by simultaneously measuring the position and normal vector of freeform surfaces, which improves the efficiency of freeform surface measurement and has important guiding significance for further processing.
2. Principle of anti-inclination, anti-scattering, and anti-reflectance normal transverse differential confocal freeform surface measurement
Figure 1 shows the measurement principle of a normal transverse differential confocal freeform surface. Normal transverse differential confocal sensor (NTDCS) is a new sensor designed based on the transverse laser differential confocal measurement principle, normal tracking measurement principle based on 2D PSD, and differential confocal focus-tracking range extension method.
The FSUT was placed on the air-floating rotary bearing θ−axis, the NTDCS performed synchronous motion using an R-axis air bearing guide rail, Z-axis air bearing guide rail, and rotation bearing ψ−axis along the horizontal, vertical, and rotation directions, respectively, and then traced to the normal direction of the FSUT sampling points. The axial V-axis drive objective of the NTDCS performs axial high-precision scanning measurement on the FSUT. The displacements of the R-, Z-, and V-axes were obtained by using an interferometer in the metrology loop.
2.1 Principle of transverse differential confocal measurement against inclination, scattering, and reflectance
Figure 2 shows the principle of transverse differential confocal measurement based on the strict conjugate relationship between the zero crossing of the transverse differential confocal curve (TDCC) and the focus of the measurement optical path. The laser emitted from the laser device is expanded by the beam expanding mirror, reflected by the PBS, passed through the 1/4 wave plate, and irradiated to the FSUT through the objective lens. The voice coil linear stage with air bearing (VCSA) drives the objective lens Lo to scan the FSUT. The beam is reflected by the FSUT, passes through the objective lens, 1/4 wave plate, and PBS successively, and then enters the beam splitter (BS). Part of the light beam is reflected into the PSD photosensitive area to detect the spot centroid position, and the other part of the light beam is transmitted through the BS and converged by the collecting lens Lc. Half of the light beam passed through a D-shaped diaphragm, then is amplified by the microscopic objective lens M and is imaged on the two-quadrant detector. The double pinhole adjacent to the two-quadrant detector can filter out the stray light and synchronously collect the front and back confocal intensity signals. The TDCC was obtained by using normalized differential subtraction of the front and back confocal intensity signals. The absolute zero of the TDCC corresponds exactly to the height of the FSUT sampling point.
Four coordinate systems are established in the object and image spaces, where the coordinates of the point light source space are denoted as (x, y, z), as shown in Fig. 2. The coordinates of the detector, illumination optical path in the image space, and detection optical path in the object space are expressed as (xd, yd, zd), (xi, yi, zi), and (xc, yc, zc), respectively. Subsequently, the point spread functions hi (Vix, Viy, ui) and hc (Vcx, Vcy, uc, VM, β) of the illumination and detection optical paths can be expressed as
For the coordinates of the illumination light path in the image space and detection light path in the object space, the optical coordinates were normalized as
Under ideal conditions, the confocal intensity response function I (u, VM, β) is given by
The front and back confocal light intensity signals I1 (u, +VM, β) and I2 (u, -VM, β) collected by D1 and D2 are processed by differential subtraction, and the transverse differential confocal response signal ID(u, VM, β) is obtained as
Variations in the surface roughness and reflectance of the FSUT lead to a change in the light intensity of the front and back confocal axial response curves. The surface characteristic functions Re(ε,δ) is introduced to reflect the light intensity change of the detection signal, which caused by the roughness ε and reflectance δ. The front and back confocal signals were normalized, and differential subtraction was performed to obtain the normalized transverse differential confocal response signal Idiff(u, VM, β), which is expressed as
According to Eq. (6), the normalized TDCC against scattering and reflectance can be obtained, as shown in Fig. 3. The proposed method is suitable for freeform surface measurement with different roughness and reflectance.
Based on Fig. 3, the TDCC exhibits a bipolar distribution on both sides of the zero point, and the slope of the linear segment is steep. The zero point of the curve can be accurately identified using the zero-crossing trigger logic analog circuit, and the height values of the sampling points on the FSUT can be obtained accurately and quickly obtained. When β changes, the zero-point position of the transverse differential curve does not change, indicating that the method has the characteristics of an anti-inclination angle change.
2.2 Adaptive Y-normal direction tracking principle
In practice, the freeform surface has the inclinations in both the x-direction and the y-direction. Here, we analyze the inclination measurement principles with y-direction as an example. As shown in Fig. 4, the reflected beam may not be entirely collected by the objective lens during measuring the FSUT (i.e., the centroid of the returning beam deviates from the optical axis center). Therefore, a two-dimensional PSD was used as the angle feedback device of the normal tracking system in this study. The normal vector is calculated using the 2D position information of the spot obtained by the 2D PSD, and feedback control is carried out according to the value of the Y-dimensional component of the normal vector, so that the optical axis of the measurement sensor is kept in the Y-normal direction. In the absence of an inclination angle (β = 0°), the reflected beam is entirely collected by the objective lens and reaches the PSD target surface through the detection optical path, presenting a circular spot with a centroid at the center of the target surface. With the surface tilted (β ≠ 0°), the reflected beam was partially collected by the objective lens and entered the PSD target plane through the optical detection path, presenting an elliptical spot. The centroid of the spot deviated from the target surface center, assuming a uniform distribution of the light intensity.
Fig. 4. (a) Objective lens collection when measuring inclined surface. (b) Spot position and shape on 2D PSD target surface.
Based on Fig. 4, the geometric relationship between the inclination angle and centroid position is.
Based on the displacement characteristics of Eq. (7), the inclination angle of the FSUT can be obtained, and the ψ−axis is controlled based on the inclination value to drive the NTDCS to perform adaptive normal tracking.
The influence of the surface inclination angle on the change in the spot centroid offset distance $\Delta L$ is shown in Fig. 5. The offset distance of the spot centroid changed significantly as the FUST inclined angle increased, which could effectively feedback the surface inclination.
3. Analysis of the influence of key parameters of the NTDCS
3.1 Determination of radial offset VM of the physical pinhole
The slope k at the zero point of the TDCC changes as the radial offset VM of the physical double pinhole in the detection focal plane changes. In other words, the location of the physical pinhole significantly affected the measurement sensitivity of the TDCC. The slope of the TDCC at zero is expressed as
The absolute value of the slope of the TDCC at zero was equated to the measurement sensitivity S, which is given by
Figure 6(a) shows that the slope absolute value of the TDCC at zero was larger when the pinhole radial offset VM gradually increased. Figure 6(b) shows that the sensitivity S reaches a maximum value of 6 when VM is 6.2; however, the TDCC linear range is sufficiently small. Considering the integrated linear range and sensitivity, the pinhole radial offset was VM = 4.4, and the corresponding sensitivity was S = 0.58.
Fig. 6. (a) Transverse differential confocal response curves under different pinhole radial offset VM. (b) Sensitivity under different pinhole radial offset VM.
In addition, the actual optimal radial offset of the physical pinhole is related to the numerical aperture (NA) value of the collecting lens Lc and wavelength of the light source, and the actual optimal radial offset yd can be calculated as
where $\sin ({\alpha _c})$ denotes the NA value of the Lc.3.2 Influence of the FSUT's inclination in the X-axis direction on axial response characteristics of the TDCC
The FSUT has a local inclination along the X-axis direction, that is, the equivalent model of the variation in the optical pass region when the FSUT has a local inclination along the edge of the D-shaped diaphragm, as shown in Fig. 7. The blue solid line circle represents the illuminating light profile of the full pupil and the red dotted line circle represents the reflected detection light profile. The intersection of the two circles represents the area where the detection light passes through the objective lens. The shaded area represents the light blocking area of the D-shaped diaphragm. The normalized distance between the detecting light center and the illumination light center is T, which can be obtained using geometric optical relations. Distance T satisfies Eq. (11).
where f is the focal length of the objective lens, R is the actual pupil radius, and β is the inclination angle of the FSUT.Fig. 7. Equivalent model of variation in the light pass region when the FSUT has inclination. (a) Inclination in the positive direction of the X-axis. (b) Inclination in the negative direction of the X-axis.
The pupil function of the detection optical path is defined by Eq. (12) when the FSUT has inclination along the positive direction of the X-axis.
According to Eqs. (6), (12), and (13), the variation of the normalized TDCC and sensitivity at different inclinations of the FSUT along the X-axis direction under the conditions of laser wavelength of 632.8 nm, measurement objective NA = 0.8, and pinhole normalized radial offset VM = 4.4 are shown in Fig. 8.
Fig. 8. (a) TDCC at different inclinations of FSUT along the X-axis direction. (b) Variation trends of sensitivity at different inclinations.
The maximum measurement sensitivity is 0.58 when the inclination angle is 0°, as shown in Fig. 8(b). The sensitivity gradually decreases as the inclination angle increases. The influence of the change in the positive and negative inclination angles on the sensitivity is symmetric and consistent and has high sensitivity within the range of ±20°, both of which are better than 0.3.
3.3 Influence of the FSUT's inclination in the Y-axis direction on axial response characteristics of the TDCC
The equivalent model of the variation in the optical pass region when the FSUT has a local inclination along the Y-axis (i.e., an inclination perpendicular to the edge of the D-shaped diaphragm) is shown in Fig. 9.
Fig. 9. Equivalent model of variation in the light pass region when the FSUT has inclination. (a) Inclination in the positive direction of the Y-axis. (b) Inclination in the negative direction of the Y-axis.
The pupil function of the detection optical path is defined by Eq. (14) when the FSUT has inclination along the positive Y-axis.
- (1) Distance T = $|{f\tan (2\beta )/R} |$ ≤ 1
Pupil function of the detection optical path is expressed as
$${P_c}({x_{cp}},{y_{cp}},\beta ) = \left\{ \begin{array}{l} 1,{\kern 5pt}\left\{ {\begin{array}{c} {{x_{cp}} \in \left[ { - \sqrt {1 - {y^2}} ,\sqrt {1 - {y^2}} } \right]}\\ {{y_{cp}} \in [{ - 1, f\tan (2\beta )/R/2} ]} \end{array}} \right.\textrm{ } \cup \\ \left\{ {\begin{array}{c} {{x_{cp}} \in \left[ { - \sqrt {1 - {{[{y - f\tan (2\beta )/R} ]}^2}} ,\sqrt {1 - {{[{y - f\tan (2\beta )/R} ]}^2}} } \right]}\\ {{y_{cp}} \in [{f\tan (2\beta )/R/2, 0} ]} \end{array}} \right.\\ \\ 0,{\kern 5pt}\textrm{ else} \end{array} \right.$$ - (2) Distance T = $|{f\tan (2\beta )/R} |$ > 1
Pupil function of the detection optical path is given by
$${P_c}({x_{cp}},{y_{cp}},\beta ) = \left\{ \begin{array}{ll} 1,&\textrm{ }\left\{ {\begin{array}{c} {{x_{cp}} \in \left[ { - \sqrt {1 - {{[f\tan (2\beta )/R/2]}^2}} ,\sqrt {1 - {{[f\tan (2\beta )/R/2]}^2}} } \right]}\\ {{y_{cp}} \in \left[ { - \sqrt {1 - {x^2}} ,\sqrt {1 - {x^2}} + f\tan (2\beta )/R} \right]} \end{array}} \right.\textrm{ }\\ 0,&\textrm{ else} \end{array} \right.$$
Under the condition that the system parameters remain unchanged, variations in the normalized TDCC and sensitivity at different inclinations of the FSUT along the Y-axis direction can be shown in Fig. 10 using Eqs. (6), (14), (15) and (16).
Fig. 10. (a) TDCC at different inclinations of FSUT along the X-axis direction. (b) Variation trends of sensitivity at different inclinations.
The measurement sensitivity decreased as the inclination angle increased and the maximum sensitivity was 0.58 when the inclination angle was 0°, as shown in Fig. 10. However, the influence of the positive and negative angles of the Y-axis on sensitivity was asymmetrical. The sensitivity was still high in the wide range of 0° to 20° when the Y-axis tilted in the negative direction. This sensitivity was better than 0.3, and the sensitivity gradually changed (almost unchanged) in the range of 0° to −5°. When the angle of the Y-axis was positive, the sensitivity significantly decreased as the angle increased, and the sensitivity was better than 0.22 in the range of 0° to 5°.
According to the analysis in Sections 3.2 and 3.3, the measurement sensitivity of the TDCC decreased when the FSUT had a local inclination angle along the X- and Y-axes. The axial response curve of the TDCC exhibits the following characteristics.
- 1) The zero point of the TDCC under different inclination angles of the FSUT coincided (i.e., the measurement surface inclination angle variation did not influence the zero-point position of the TDCC). This method had high anti-inclination measurement ability.
- 2) The influence of the FSUT inclination along the X and Y directions on the sensitivity was inconsistent, and the measurement angle range was large in the X direction (up to ±20°) and small in the Y direction (−20° to 5°) because of the noncircular symmetry of the D-aperture.
In this study, the Y-direction was coincident with the normal rotation tracking direction of the sensor when designing the optical path of the sensor by changing the direction of the D-aperture, which effectively avoided the problem of a small measurable inclination range in the Y-direction.
3.4 Analysis of resolution characteristics
The slope at the zero point can be obtained by taking the partial derivative of u in Eq. (6), the resolution of the axial response can be obtained using
The NTDCS has a theoretical axial resolution up to 0.5 nm, which can meet the requirements of FSUT measurement.
4. System construction and experimental verification
4.1 Measurement system
The high-precision FSUT test system shown in Fig. 11 was constructed according to the principle of normal transverse differential confocal freeform surface measurement shown in Fig. 1 and 2.
The FSUT measurement system includes a computer and controller system A, a normal transverse differential confocal sensor B, an adaptive normal tracking system C, a precision rotary, and a guide rail system D.
NTDCS uses a He-Ne laser with a wavelength of λ = 632.8 nm. An objective lens with numerical aperture NA = 0.80, focal length f = 1.8 mm, working distance W.D. = 3.4 mm, magnification N = 100× was selected. In this study, a voice coil linear stage with air bearings developed by our research group was used as the objective lens driver. The objective lens driver had a 5 mm coaxial driving stroke range and 0.5 nm motion resolution. Combined with the focus-tracking strategy of the TDCC, the measurement range of the NTDCS was effectively extended. The motor had a round-trip motion frequency of 150 Hz in the range of 100 µm. The grating ruler for position feedback was HEIDENHAIN's LIP281 with a grating period of 2.048 µm and an interpolation error of 0.4 nm. The interferometer in the metrology loop is an IDS3010 from the Attocube Company, with a resolution of 1 pm.
In NTDCS, the posterior pupil diameter of the objective lens is 2.88 mm, indicating that the length and width of the target surface of 2D PSD should be greater than 2.88 mm. Considering that the spot size was enlarged in the defocusing process, the target surface size was tentatively 7 × 7 mm. The angle-measurement resolution of the normal tracking system was designed to be 0.1°. Based on Eq. (7), the $\Delta L$ value is 3.1 µm when the angular resolution is 0.1°. In other words, the PSD position resolution should be better than 3.1 µm.
Following the aforementioned analysis, Hamamatsu's S5991 was selected for the 2D PSD, with a target surface size of 9 × 9 mm and a position resolution of 1.5 µm, and the corresponding angle resolution was 0.047°; that is, the theoretical angle measurement resolution of the 2D PSD inclination-measuring module built with S5991 was 0.047°. Based on Fig. 5, $\Delta L$ is only 0.65 mm when the FSUT inclination β value is 20°, which is much smaller than the range of S5991.
The rotation angle of the ψ-axis was fed back by the HEIDENHAIN circular grating ERA4202C, which had 20000 signal cycles. Equipped with the ERA4280 analog reading head, the controller was used to perform a 200-fold subdivision with a resolution of up to 0.324". The normal tracking system based on the inclination angle feedback from the 2D PSD inclination-measuring module controls the high-precision rotation axis to drive the NTDCS to perform the rotation, and realizes the adaptive normal tracking measurement of the FSUT in the range of 90° for the inclination angle.
4.2 Performance experiment of NTDCS
4.2.1 Optical resolution test
The measurement resolution of the NTDCS system was tested, and the results are shown in Fig. 12. The motion resolution of the VCSA is 0.5 nm. The optical measurement resolution of the sensor is 0.5 nm when β = 0°. The optical resolution of the sensor is 2 nm at β = ±5°. It can satisfy the measurement requirements of a steep FSUT.
Fig. 12. Resolution test diagram of NTDCS. (a) 0° resolution test diagram, (b) −5° resolution test diagram, and (c) 5° resolution test diagram.
4.2.2 Single-point focusing experiment
NTDCS was used to conduct a single-point focusing experiment on a non-inclined surface, and the light intensity response of the output of the double pinhole was synchronously collected to obtain the signal of the front and back confocal intensities. Normalized differential subtraction was performed on the front and back confocal signals, and a transverse differential confocal axial response intensity curve was obtained, as shown in Fig. 13.
The width of the curve fitting linear segment is 0.72 µm, and a bipolar characteristic exists on both sides of the zero point, which makes it easy to achieve accurate zero-crossing trigger detection. A high-precision focusing measurement of the FSUT can be realized through the exact correspondence relationship between the curve zero point of TDCC and height value of the FSUT sampling point.
4.2.3 Inclination focusing experiment in X and Y directions
Single-point focusing experiments were performed on surfaces with different inclination angles in the X- and Y-directions to verify the inclination measurement capability of the NTDCS system. Transverse differential confocal test curves were obtained at different inclination angles, as shown in Fig. 14.
Fig. 14. Transverse differential confocal curves at different inclination angles in X and Y directions. (a) Curves at different angles in the X direction. (b) Curves at different angles in the Y direction.
As shown in Fig. 14(a), the slope of the TDCC at the zero point becomes decreases (i.e., the measurement sensitivity decreases) as the inclination angle along the X-direction increases, and the zero-point position of the curve remains unchanged, which is consistent with the simulation results. This result proves that NTDCS has an inclination measurement capability of ±20° in the X-direction.
As shown in Fig. 14 (b), the slope at the zero point of the TDCC exhibited an asymmetric downward trend in the positive and negative directions, and the zero-point position remained unchanged as the inclination angle along the Y-direction increased, which is consistent with the simulation results. This result proves that NTDCS has an inclination measurement capability of 20° and 10° in the negative and positive Y directions, respectively.
4.2.4 Test of the response characteristics of the PSD inclination-measuring module
The ψ-axis was used to drive the NTDCS system to rotate from -15° to 15° in 1° steps. Figure 15(a) illustrates the output voltage value of the corresponding PSD inclination-measuring module, and Fig. 15(b) shows the residual values of the five repeated test results.
The ψ-axis was used to drive the NTDCS to rotate 0.5° with steps of 0.1°, and the output voltage of the PSD inclination-measuring module was simultaneously measured. Figure 16 shows the test results, which indicate that the resolution of the PSD inclination-measuring module can reach 0.1°.
4.3 Measurement experiment of the freeform surface
A large-inclination aspheric generatrix with diameter D = 280 mm was tested to verify the measuring capability of the proposed method in the range of large inclination angles. The design model of the aspheric surface is as follows.
Fig. 17. Measurement results of the aspheric surface with a large inclination angle. (a) The residuals of the measurement and calibration data. (b) Difference between the residuals of the measurement and calibration data.
The calibration data were obtained using a ZYGO interferometer. The PV and RMS values of the measured results were 120 nm and 14 nm, respectively, which are similar to the calibration results. Although the generatrix position was marked during the measurement, we could not guarantee that the measurement results of any direction generatrix by our method and ZYGO were similar, and the maximum deviation between the measurement and calibration results was approximately 40 nm, as shown in Fig. 17(b).
A freeform surface (off-axis aspheric surface with an effective aperture of Φ150 mm) was verified the measurement accuracy of the proposed method. The sampling-point interval was less than 1 mm, the number of sampling points was about 36000, the movement speed of R- and Z-axes was 0.1 mm/s, the scanning frequency of V-axis was 50 Hz, and the speed of θ-axis was 6 r/min. The measurement results of the null interferometry and proposed methods are shown in Figs. 18(a) and (b), respectively. PV and RMS values measured by the null interferometry method were 97.5 nm and 10.1 nm, respectively. While, the PV and RMS values measured using our method were 105 nm and 12 nm, respectively. Figure 18(c) shows the PV and RMS values of the residuals of 10 repeated measurements. The experimental results demonstrated that the proposed method could achieve high-precision measurement of a Φ150 mm FSUT, and the repeated measurement accuracy was approximately 8.3 nm, which had good measurement reproducibility.
Fig. 18. Measurement results of FSUT. (a) Measurement result of the null interferometry method. (b) Residual diagram of the proposed method. (c) The PV and RMS values of the residuals of 10 repeated measurements.
5. Conclusion
In this study, a new method of normal transverse differential confocal freeform surface measurement was proposed, which not only greatly simplifies the optical path of the system, effectively suppresses external environment interference, and improves the stability of the system. Moreover, the position and normal vector can be measured simultaneously, which can significantly reduce the difficulty in FSUT installation and adjustment and has guiding significance for further optical machining. In addition, the linear section of the TDCC can be used for direct measurement when the NTDCS maintains the normal direction of the sampling point, which can further improve the measurement efficiency.
The experimental results showed that the repeated measurement accuracy of the proposed method was better than 9 nm. Moreover, the proposed method has significant advantages such as high measurement accuracy, a large measurement range, and resistance to large variations in inclination, scattering, and reflectance. This study provides a new measurement technique for steep freeform surfaces.
Funding
National Key Research and Development Program of China (2017YFA0701203); National Science Fund for Distinguished Young Scholars (51825501); Civil Aerospace Technology Advance Research Project (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.
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