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Abstract
The non-null test to detect the modulated wavefront is a widely used method in optical freeform surface measurement. In this study, the wavefront deformation in the non-null test of an optical freeform surface measurement was corrected based on the wavefront propagation model to improve measurement accuracy. A freeform surface wavefront correction (FSWC) measurement system was established to validate the proposed method. Simulation and experimental studies indicated that the proposed method can reduce the influence of freeform surface wavefront deformation in space propagation. Moreover, the freeform surface form accuracy measured by FSWC can reach a root-mean-squared value of 10 nm.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Freeform surfaces are popular in the design of optical systems because they can provide more degrees of freedom to improve performance [1,2]. With the continuous advancement of processing and inspection technology [3–5], freeform surfaces are becoming increasingly widely used in actual design and application [6,7]. Numerous methods have been developed for the application for freeform surfaces to balance the measurement accuracy and dynamic range [8]. The area measurement method is highly preferred owing to its fast measurement process and generation of a considerable amount of data.
In the null test, the freeform surface sample is measured using the pre-modulated wavefront feedback result. However, owing to the abundant degrees of freedom of the freeform surface, wavefront pre-modulation makes the realization of the null test complicated and expensive. The non-null test has more extensive applicability; the Hartmann test is a typical non-null test method. The Shack–Hartmann sensor and its variations are widely used for wavefront aberration and surface profile measurements [9,10]. With the advancement of the Shack–Hartmann sensor in centroid calculation and wavefront reconstruction [11,12], its measurement accuracy and dynamic range have been further improved. The resolution and measurement range of the Shack–Hartmann sensor can be improved by scanning [13,14]. In contrast to the null test, the non-null test does not require compensation for the shape of the measured surface. However, the wavefront loses its slope consistency after being modulated by the freeform surface, which causes the wavefront to be sensitive to the spatial position. The non-uniform wavefront modulated by the freeform surface will deform with the propagation position in space, which is more critical for freeform surfaces with a higher degree of deviation from the plane or sphere. Aspherics and quadrics can be used to correct wavefront deformation errors in the interferometric measurement of the spherical reference wavefront through the wavefront transmission equation [15,16]. Conversely, the propagation model established by the Zernike polynomial and Taylor polynomial is also used to describe changes in the free space of wavefront aberrations [17–19]. In the aforementioned methods, the propagation model is approximated by polynomials, which ignore the high-order information of freeform surfaces. In the null measurement and aberration compensation, numerical calculations use the ray-tracing method and the optical transmission matrix method to solve the free-space ray set to characterize the wavefront [20,21]. However, most of these methods adopt a paraxial approximation, and the calculation process is complicated.
By introducing the wavefront transmission model into the non-null test of the freeform surface, a simple ray-tracing method can accurately compensate for the error introduced by the wavefront deformation in free space. In this study, a freeform surface wavefront correction (FSWC) method was designed, and the FSWC measurement system was established consisting of a Shack–Hartmann wavefront sensor (SHWS) and a chromatic confocal sensor (CCS) to obtain the freeform surface modulated wavefront and optical path, respectively. The freeform surface profile was calculated using the ray-tracing method based on the wavefront propagation model and the obtained modulated wavefront and optical path. The wavefront deformation was then corrected by ray-tracing the isophase surface using the CCS data. Hence, the proposed measurement method improves the robustness of the optical freeform surface shape measurement and avoids measurement errors caused by wavefront deformation in different spatial positions. The principle of the whole measurement system is simple, reliable, and easy to trace. Other non-null test methods such as Phase-Measuring Deflectometry [22] and Phase Retrieval Based on Intensity Images [23] require complex calibration and computational procedures. Therefore, the FSWC measurement method is simpler in calibration and measurement process and easier to be implemented.
2. FSWC principle based on wavefront propagation
For a light wave developed from a unitary light source, any of its isophase surfaces satisfies
where ${W_0}$ is the incident wavefront, W is the wavefront after propagation and i indicates different mapping positions, L is the optical path. The direction of light remains unchanged before and after wavefront propagation, then the light rays satisfyBased on the wavefront propagation model, the deformation of the freeform surface wavefront in the free-space propagation process can be corrected using the ray-tracing method to obtain the 3D profile of freeform surfaces.
2.1 FSWC principle
When the incident wavefront is reflected by the surface, the wavefront is modulated by the surface shape. As shown in Fig. 1(a), the modulated wavefront changes with positions during the propagation process. In freeform surface measurements, the null test is a common measurement approach, and the reflected wavefront is corrected to a plane wave, which ignores the deformation of the wavefront propagating along the optical path. Thus, the influence of wavefront deformation leads to deviations in the final measurement results. For the isophase surface, starting from the light source,
where S is the point in the reflecting surface and i indicates different mapping positions. Based on the isophase surface relationship shown in Eq. (5), the deformation of the reflected wavefront with the spatial position can be corrected to improve the measurement accuracy of the freeform surface.
Fig. 1. (a) Deformation of freeform surface wavefront based on wavefront propagation model; (b) Schematic of FSWC principle.
The SHWS is a typical wavefront sensor that uses a microlens array to obtain the light slope for wavefront reconstruction. When using SHWS to measure freeform surfaces, the reflected wavefront changes with the shape of the target surface. Traditional measurement methods consider that the change in the reflected wavefront is twice the actual surface shape. The isophase surface method improves this calculation model, but ignores the deformation of the freeform surface wavefront in the free-space propagation process [24]. This is because the direction vector of the light changes, and the shape of the reflected wavefront changes with the transmission distance.
To reduce the influence of the wavefront deformation caused by the propagation process on the freeform surface measurement results, the proposed FSWC measurement system comprises an SHWS and a CCS. The SHWS was used to detect the reflected wavefront, whereas the CCS was used to detect the change in the optical path to calculate the surface shape through the FSWC principle, as shown in Fig. 1(b). The optical axes of the sensors are designed in parallel to maximize the dynamic range of the FSWC, which requires a high-precision displacement system in the actual measurement process.
In the SHWS design, the incident light source is a collimated beam, and its corresponding wavefront is a plane wave. The isophase surface reaches SHWS after being modulated by the reflecting surface, and Eq. (5) is transformed into
The surface shape is calculated by the ray-tracing method according to Eq. (2). The calculations of the three orthogonal directions of x, y, and z are as follows:
Calibration of installation errors is necessary during the actual measurements. Figure 2 shows the FSWC measurement process for the freeform surfaces after calibration. First, SHWS detects the freeform surface reflection wavefront; then, the displacement system moves the CCS to the same position according to the calibration result of the lateral position parameter, and detects the freeform surface displacement data. Thereafter, the displacement data are calculated by the calibration results of the angle parameters and the axial position parameters to obtain the reflection optical path of the freeform surface. Finally, the surface shape of the measured freeform surface was calculated by the ray-tracing method based on the proposed FSWC.
2.2 Simulation experiments
The principle of FSWC can be first verified through simulations, including error analysis. In the simulation experiment, a freeform surface was designed, as shown in Fig. 3(a). The reflected wavefront is then calculated. Figures 3 (b), (c), and (d) show that the wavefront of the collimated light reflected by the freeform surface is located at 0, 10, and 20 mm in the optical path, respectively.
Fig. 3. (a) Designed freeform surface; and reflected wavefront of the collimated light by the designed freeform surface located at (b) 0 mm; (c)10 mm; (d) 20 mm in the optical path.
The freeform surfaces obtained by the isophase surface method and the proposed FSWC were compared with the designed one. The deviations between the designed surface and the surface calculated by isophase surface method corresponding to the wavefront in Figs. 3 (b), (c), and (d) are shown in Figs. 4(a), (b), and (c), where the values of peak-to-valley (PV) and root-mean-square (RMS) are marked, whereas those by FSWC are shown in Figs. 4 (d), (e), and (f). After the concave wavefront propagates in free space, the area of the mapped area becomes smaller, resulting in missing boundary data. Similarly, convex wavefront will also cause missing boundary data after ray tracing (blue areas in the figure). The results in Fig. 4 indicate that the description of the reflected wavefront based on the FSWC principle is more accurate, and the calculation results of the shape of the reflecting surface are also more accurate. It is noted that there is a periodicity of the final surface deviation in the FSWC measurement results because the slope matrix information is reconstructed by the zonal method [25], which results in periodic reconstruction and interpolation errors.
Fig. 4. (a)–(c) are the residuals by the isophase surface method to calculate the surface shape of the designed freeform surfaces located at 0, 10, and 20 mm, respectively; (d)–(f) are those obtained by the FSWC method.
The measurement error by FSWC mainly comes from the wavefront slope error in the wavefront measurement and the displacement error in the optical path measurement. The effects of different slope errors and optical path errors on the measurement results were then studied. The shape calculation residuals of the freeform surface by the wavefront slope error and the optical path error were simulated using the Monte Carlo method, as shown in Fig. 5(a) and Fig. 5(b), respectively. In the wavefront slope error analysis, the optical path error was set as 1 µm, and the wavefront slope error was set as 1 µrad in the optical path error analysis. Each data point represents the mean of 100 Monte Carlo calculations with a fixed deviation, which is a random value that fits the error value as the standard deviation. Generally, the measurement error of the wavefront slope is in the order of microradians, and that of the optical path is in the micrometer range. The results show that the measurement accuracy of the FSWC is not sensitive to the optical path error and wavefront slope error. The effects of these errors on the measurement accuracy of the FSWC were all in the nanometer-scale levels in terms of RMS. The wavefront slope accuracy is mainly limited by the resolution of the CCD sensor and the accuracy of the spot centroid algorithm. Some of the errors in the optical path measurement originate from the limitation of the sensor's measurement accuracy, and the others originate from the measurement error of the calibration parameters after the installation.
Fig. 5. Monte Carlo method is used to simulate the influence of (a) wavefront slope error and (b) optical path error on the surface deviation of freeform surfaces.
3. Calibration of the FSWC measurement system
Installation errors exist for the FSWC measurement system, which will cause the actual parameters to be different from the design parameters. Before measurement, the calibration parameters need to be obtained to modify the measurement data through a designed calibration experiment; then, the surface shape measurement results are calculated based on the calibrated measurement data.
In the designed calibration process, the signal receiving surface and optical axis of the SHWS are used to build the reference coordinate system. The calibration process is shown in Fig. 6. Relative to the reference coordinate system, the installation error of the CCS is mainly divided into two parts: angle and position errors. The angle error refers to the pitch and yaw caused by the actual installation errors relative to the optical axis of the wavefront sensor. Position error refers to the starting point of the dispersive confocal sensor with the design parameters relative to the wavefront sensor, including two lateral and axial errors.
The calibration angle error was first measured by the wavefront sensor and confocal sensor on a standard plane. For confocal measurement data of the standard flat mirror,
From Eq. (8), because of the number of unsolved parameters, the angle calibration process needs to measure at least two planes with different inclination angles, and each plane needs to be measured at least four points as shown in Fig. 6. Subsequently, the relative angle of the confocal sensor with respect to the standard plane was obtained by performing the least square fitting on the measurement data of the confocal sensor. Two angle errors ($\alpha $, $\beta $) can be obtained by measuring the relative angle of the standard plane using the SHWS.
During the calibration process, because the displacement system is used, the actual calculation refers to the Abbe error correction between the three coordinate systems of the SHWS, CCS, and displacement system. During the angle calibration process, the calculation result of the least-squares fitting is the angle between the CCS and the displacement system. From the fitting results, the coefficients of the plane equation and the angle of the plane relative to the displacement system can be obtained. Thereafter, the relative angle obtained from the SHWS measurement plane can calculate the angular deviation of the CCS relative to the SHWS.
In the position calibration, SHWS and CCS were first used to scan a standard spherical surface after the angle calibration of the CCS data was performed. The angle calibration of the CCS data is used not only to calculate the measured data from the angle of the CCS relative to the SHWS optical axis but also needs to correct the Abbe error caused by the displacement system relative to the SHWS optical axis angle during the displacement process. By comparing the position parameters of the lowest point of the spherical surface in the two data sets after the angle calibration, the transverse position parameters of the starting point of the CCS range relative to the SHWS optical axis can be obtained. The actual displacement of the displacement system must be obtained by compensating for the Abbe error. A schematic of the calibration process is shown in Fig. 7.
Fig. 7. FSWC system position calibration diagram, ${L_H}$ is horizontal distance of CCS installation to SHWS, and ${L_V}$ is vertical distance of CCS installation to SHWS.
SHWS and CCS are used with matching position parameters to measure the standard spherical surface:
In Fig. 7, the axial position error is calculated by the least square fitting of the ray-tracing result of the wavefront sensor measurement data to the standard spherical surface.
The Monte Carlo method was used to add different errors to simulate the measurement errors of the calibration parameters under different installation conditions. The Levenberg–Marquardt-based gradient descent method was used for least squares fitting, and the initial values were all set to zero from the ideal setup. The influence of the position error of the displacement system and the reference flatness error on the angle calibration results was simulated, as shown in Figs. 8 (a) and (b). The influence of the position error of the displacement system and the standard spherical error on the position calibration results are shown in Figs. 8 (c) and (d). In the displacement system error analysis, the surface roughness was set as 10 nm, and the displacement system error was set to 100 nm in the surface roughness analysis. Each data point represents the mean of 100 Monte Carlo calculations with a fixed deviation, which is a random value that fits the error value as the standard deviation.
Fig. 8. Influence of the optical path measurement result caused by (a) position error of the displacement system and; (b) reference flatness error on the angle calibration error; and (c) position error of the displacement system and (d) standard spherical error on the position calibration error.
The simulation results show that the accuracy of the calibration parameters is of the same order of magnitude as the accuracy of the displacement system and the errors of the standard surface. In practice, the accuracy of the standard surface can often reach the order of 100 nm or higher; therefore, the accuracy of the calibration parameters is mainly limited by the accuracy of the displacement system. The Monte Carlo simulation results show that the optical path errors caused by the calibration errors are all smaller than the calibration errors. The sensitivity of the optical path error to the actual surface profile measurement reaches approximately 2×10−5, which is calculated from the maximum value of dividing all Monte Carlo calculated surface deviation RMS by the corresponding set optical path error in optical path error analysis. This shows that the FSWC is not sensitive to calibration errors in correcting the wavefront deformation of freeform surfaces.
4. Experimental studies
In the experimental studies, the designed FSWC measurement system was calibrated, and a freeform surface and a spherical surface were measured and compared for verification.
The measurement system consists of an SHWS (Imagine Optic HASO4 Broadband) and a CCS (STIL CL2-MG140), both of which have built-in light sources, as shown in Fig. 9. The FSWC system was built on a CMM displacement system (AEH Daisy 564), which can provide high-precision displacements in the X, Y, and Z directions.
Fig. 9. FSWC measurement system comprises SHWS and CCS, which is built on a high-precision displacement system.
In the calibration process, a stand plane and a sphere with high precision were used for testing to ensure the accuracy of the calibration parameters. Related specific parameters, such as the built sensor and calibration samples, are listed in Table 1.
Table 1. Parameters of the measurement experiment
The errors of the standard plane and the standard sphere in Table 1 were measured using an interferometer (Zygo GPI/XP D); the measured parameters were used to calibrate the proposed system using the aforementioned method to obtain the calibration parameters, as shown in Table 2.
Table 2. Calibration parameters calculated in the measurement experiment
In the experiment, 25 points were sampled for each of the two plane measurements in the angle calibration, and the RMS value of the residual in the optimization result was 17.7 nm. For spherical measurement in the position calibration, a total of 50 × 68 points was sampled and the residual RMS value in the optimization result was 10.1 nm.
Based on the aforementioned calibration parameters, a standard sphere with a diameter of 100 mm used in the calibration and a freeform surface with a diameter of 50 mm were measured, and the measurement results are shown in Fig. 10 and Fig. 11, respectively. The measurement results show that the FSWC method significantly improves the measurement accuracy and effectively corrects the wavefront deformation of the freeform surface during propagation. Compared with the data measured by the reference method, the surface deviation RMS measured by the FSWC method is less than 10 nm. The measurement error originates from the influence of the calibration error and is also caused by the reconstruction error in the wavefront reconstruction process.
Fig. 10. (a) Reference data of a standard spherical mirror, (b) surface shape calculation results of isophase surface methods, (c) FSWC method surface shape calculation results, (d) image of the standard spherical mirror sample, (e) surface shape calculation results deviations of isophase surface methods, (f) FSWC method surface shape calculation result deviation.
Fig. 11. (a) Reference data of freeform mirror, (b) surface shape calculation results of isophase surface methods, (c) FSWC method surface shape calculation results, (d) physical images of freeform mirror, (e) surface shape calculation results deviations of isophase surface methods, (f) FSWC method surface shape calculation result deviation.
The reference data of the standard sphere were obtained from the measurement results of the interferometer (Zygo GPI/XP D). The freeform surface used in the measurement experiment is expressed as
The position registration of the freeform surface measurement data was obtained using the least square method [26]. The RMS value of the registration residual of the freeform surface is 15.8 nm. The freeform mirror was measured using a 3D optical profiler (Taylor Hobson LuphoScan 260 HD), and the measured results were used as the reference data.
The blue area of the image boundary in Fig. 11 is due to the lack of boundary data because the mapping area is no longer regular after the freeform wavefront propagation in free space. For ease of comparison, all data points have been shown in Fig. 11. The measurement results are presented in Table 3. The measurement accuracy of the FSWC method was higher than 10 nm RMS within 5 mm × 5 mm. Compared with the isophase surface calculation method, the accuracy is significantly improved from 16.85 nm to 8.78 nm in terms of RMS values. This confirms that the proposed method using the Hartmann sensor for non-null testing can effectively improve the measurement accuracy of optical freeform surfaces.
Table 3. Measurement results of the FSWC method compared with isophase surface methods
The robustness of the FSWC method is further studied by uncertainty analysis of the actual experimental parameters through Monte Carlo simulations. Firstly, the uncertainty of the optical path is calculated by calibrating the experimental parameters. The uncertainty of each parameter is obtained by 100 times of Monte Carlo simulation, and the results are shown in Table 4.
Table 4. The uncertainty analysis of the optical path
The Monte Carlo analysis is carried out by integrating the surface shape calculation process, and the uncertainty analysis of the final measurement result is shown in Table 5.
Table 5. The uncertainty analysis of the FSWC method measurement result
Through the above uncertainty analysis, the FSWC method is not sensitive to the calibration error, and the main error comes from the wavefront reconstruction error of SHWS.
5. Discussion
The Southwell model [25] was used to reconstruct the wavefront of the zonal method in the measurement experiment. Compared with the Zernike polynomial model method [27], the wavefront reconstructed by the zonal method causes the slope error to accumulate during the reconstruction process. Although the zonal method introduces slope errors and leads to surface deviations in the reconstruction process, it is not affected by the loss of higher-order terms in the polynomial fitting process. The SHWS is used to measure and reconstruct the wavefront of the point diffraction standard spherical wave. The reconstruction error in terms of the RMS of the zonal method was less than 6 nm. In the experiment, the residual RMS values of the measurement results are all less than 10 nm, and the wavefront reconstruction error of SHWS itself has become the main error source.
This study designs and builds an FSWC based on the SHWS. As a freeform surface measurement method for non-null tests, FSWC does not require additional hardware for different freeform surfaces to make the wave surface fit the surface shape. As shown by the simulation experiment, the final calculation result is not sensitive to the optical path and slope errors; the calibration method is also easy to implement. This will significantly reduce the difficulty of realization when using the FSWC method to improve the accuracy of freeform surface measurements.
Simultaneously, owing to the sampling of the wavefront by the sub-aperture, the lateral resolution is lost [28]. Moreover, the wavefront propagation process conforms to the Huygens Fresnel principle. Wavefront reconstruction is based on wavefront continuity, which also leads to the loss of high-frequency information. The dynamic range is limited by the focal length and number of SHWS microlens arrays [29]. However, the required optical system is simple, and all calculation results can be traced. The FSWC method does not require additional filtering, adjustment, or other special processing of the data. The optical system can also be added to a beam expanding system or a focusing system according to specific requirements. The FSWC method can be widely applied to other wavefront measurement sensors, such as shear interferometers. In the FSWC measurement system, the CCS can provide optical path data, so that the SHWS can detect the wavefront at a certain distance. Due to the wavefront transmission characteristics, the spatial transmission process will reduce the wavefront slope, which will effectively improve the dynamic range of the FSWC measurement system. On the premise of ensuring the incident light flux, increasing the free space transmission distance can further improve the dynamic range of the FSWC measurement system.
6. Conclusions
This study presents a freeform measurement method based on FSWC. The proposed FSWC measurement system is designed based on SHWS and CCS, which can use the ray-tracing method to apply the FSWC method to significantly improve the measurement accuracy for freeform surfaces. A calibration method that only requires a high-precision displacement system, and standard sphere was designed for FSWC, and the calibration accuracy of this calibration method was verified by simulation. Finally, by measuring spherical surface, and freeform surface, it was verified that the measurement accuracy based on the FSWC method reached an RMS value of 10 nm compared with the measurement data from commercial instruments.
In addition, the principle of the entire measurement system is based on the ray-tracing method, which is simple, reliable, and traceable. Because the FSWC measurement system integrates SHWS and CCS, the measured results also include the absolute position of the measured freeform surface relative to the SHWS coordinate system. This will provide a foundation for further inspection requirements for freeform surfaces. The calibration method involved is easy to implement, and the measurement accuracy is not sensitive to the actual calibration error, which reduces the difficulty of implementing the method.
Funding
National Key Research and Development Program of China (2017YFA0701200); National Natural Science Foundation of China (52075100); Science and Technology Commission of Shanghai Municipality (19ZR1404600).
Acknowledgments
The authors would like to express sincere thanks for the technical support from Dr. Junhua Wang of Fudan University.
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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