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Abstract
Freeform optics, due to the more general surface geometry that offers high degrees of design freedom to control light propagation, has already been widely used in both nonimaging optics and imaging optics. With the recent advances in design and fabrication of freeform optics, one of the remaining challenges is how to accurately measure freeform optical surfaces, especially those included in freeform refractive optics. To meet this imperative need, for the first time, we believe, present an effective simultaneous multisurface measurement method for freeform refractive optics. Instead of using a reflected optical field to reconstruct tested optical surfaces, we develop a surface reconstruction method based on a transmitted field to tackle the challenges caused by the low reflectivity and compound effect of multiple reflection of refractive surfaces. The transmitted fields from refractive elements are measured by computer-aided deflectometry in order to achieve a large measurement dynamic range and high accuracy. Using the transmitted fields, a multisurface reconstruction model based on iterative optimization is then employed to achieve the accurate multisurface measurement simultaneously. The proposed method is demonstrated to be very effective and robust in testing freeform refractive optics, with a future potential for in situ metrology.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Due to the flexible surface geometry that offers high degrees of design freedom, freeform optics has been widely applied in various optical systems (e.g., illumination [1–4], imaging [5,6], and display [7–9]) to improve system performance and decrease system size and weight. The manufacturing quality of freeform surfaces relies heavily on the metrology precision, which places high requirements on metrology tools. Therefore, a general and accurate testing method is essential to meet the high metrology requirements of freeform optics.
Various techniques have been developed for freeform surface testing. The profilometers [10–14] are capable of performing accurate measurement for arbitrary surfaces with large slope ranges; however, these methods are very time-consuming due to the point-by-point measurement process. As a time-honored and full-field technique, interferometry is a powerful method with nanometer-scale accuracy [15–17]. To determine the fabrication errors of freeform surfaces, null interferometric testing should be performed with specifically designed null optics and compensators [18], leading to high cost and inflexibility of the interferometry [19]. To improve its flexibility, non-null interferometric testing, which usually employs a stitching technique to achieve full-field measurement, has been developed. However, error accumulation in subaperture stitching cannot be avoided, and tedious system calibration [20–22] is usually required to remove retrace and alignment errors. Additionally, the slope dynamic range of interferometry is quite small, making it unsuitable for freeform optics testing. Deflectometry, with neither compensation optics nor retrace error calibration, is a fundamentally incoherent surface metrology method in the slope domain, such as the Shack–Hartmann sensor [23,24] and phase measuring deflectometry (PMD) [25,26]. Compared with interferometers, surface profiles can be measured with high flexibility and relatively large dynamic range by use of a Shack–Hartmann sensor; however, the spatial resolution is very limited due to the limited number of lens units in a Shack–Hartmann sensor. The PMD has been proved to be a powerful tool in evaluating freeform optical surfaces [27–32] with both large dynamic range and measurement accuracy comparable to interferometry. However, the PMD is only suitable for reflective surface testing, and its application in testing refractive optics is very limited due to the low reflectivity of the refractive surface.
The traditional contact-free metrology is based on the evaluation of a reflected optical field from a tested optical surface. Thus, the reflectivity of the tested surface is a key factor by which the measurement accuracy of a contact-free metrology is strongly determined. For a refractive optics with multiple optical surfaces, parasitic reflection usually takes place on the optical surfaces due to Fresnel reflection and could interfere with the obtained reflected field, introducing an additional measurement error. Consequently, some feasible remedies, including blackening the rear surfaces, immersing rear surfaces in refractive-index-adapted material, line shift, multifrequency approach, and binary pattern [33–37] are needed to address this issue. However, the multiple surfaces contained in a refractive element are usually measured sequentially by the traditional contact-free metrology, making the testing process quite time-consuming and troublesome. Thus, the measurement of freeform refractive optics has still not been well addressed and faces many unresolved challenges.
Fig. 1. Flow chart for SMM of freeform refractive optics. (a) The whole measurement procedure, procedures of (b) system geometry calibration and (c) simultaneous multisurface reconstruction.
In this paper, we develop a computer-aided deflectometric method to achieve simultaneous measurement of multiple freeform refractive surfaces with both high accuracy and large slope dynamic range. There are three main contributions of this work: (1) freeform surfaces are reconstructed from the transmitted field, by which the low reflectivity limitation is overcome and no additional parasitic reflection elimination is needed; (2) the multisurface profiles in a freeform refractive optics are measured simultaneously; and (3) the proposed method provides a general and novel testing method with high accuracy and is applicable to a broad range of various optical elements.
2. PRINCIPLE
The proposed simultaneous multisurface measurement (SMM) method for freeform refractive optics is depicted in Fig. 1(a), which can be subdivided into three distinct steps: the acquisition of transmitted wavefronts, system geometry calibration, and simultaneous multisurface reconstruction. In the first step, the transmitted wavefronts from the tested refractive element are measured by a fringe-illumination deflectometric system. In the second step, the system geometry calibration based on computer-aided reverse optimization is performed to achieve the accurate measurement of transmitted wavefronts. In the third step, the multiple optical surfaces of the freeform refractive element are simultaneously reconstructed from the measured transmitted wavefronts. More details about the proposed method are given below.
A. Acquisition of Transmitted Wavefronts
Figure 2 shows a schematic diagram of the deflectometric system for the proposed SMM method. A liquid crystal display (LCD) screen illuminates the tested refractive element with coded fringe patterns, and the fringe passing through the refractive element is distorted and recorded by a pinhole CCD camera. With the coded fringe illumination and phase-shifting method, the phase distribution of distorted fringe pattern can be obtained, from which the relationship among the illumination screen pixels, CCD camera pixels, and sampled regions on the tested surfaces is established. Then, the relationship between the incident and refracted rays can be used to calculate the surface normal vectors, from which the multisurface profiles under test can be reconstructed.
The computer-aided deflectometry is applied to acquire the transmitted wavefront from the tested refractive element. Then, an ideal ray-tracing system model with the same geometrical parameters as the actual experimental system is built in order to achieve the virtual “null” testing of the transmitted wavefronts. According to the reversibility of optical paths, the illumination screen and pinhole CCD camera in the model are set as an image plane and point source, respectively; the tested element is modeled with nominal parameters. By comparing the ideal image spot distribution obtained by tracing rays in the ideal system model and the actual spot distribution measured in the actual system, the transmitted wavefront error from the tested element can be obtained according to the transverse ray aberration model [38].
With the ideal and actual ray positions $({\varepsilon _{x, \text{ideal}}},{\varepsilon _{y, \text{ideal}}})$ and $({\varepsilon _{x,\text{actual}}},{\varepsilon _{y, \text{actual}}})$, which correspond to the image spot distributions in the ideal system model and actual system, respectively, the ray aberration $(\Delta {\varepsilon _x}\!,\Delta {\varepsilon _y})$ and the wavefront aberration ${W_{\text{ept}}}(x,\;y)$ introduced by the tested refractive element can be written as
B. System Geometry Calibration
Ray tracing of the ideal system model enables us to remove the wavefront aberrations introduced by the system geometry and structure of the tested refractive element. However, the measurement errors in the system geometrical parameters (including the tilt and decenter of each system component in various directions) due to the limited accuracy of measuring tools could introduce significant system geometrical error and deviation of wavefront aberrations. Thus, it is necessary to further optimize the geometrical parameters of the deflectometric testing system to ensure a high measurement accuracy before we reconstruct the multisurface profiles. Here, we apply the computer-aided reverse optimization model [29,32] to calibrate the system geometrical error.
According to this reverse optimization model, the measured wavefront aberration ${W_{\text{ept}}}$ can be expressed as
C. Simultaneous Multisurface Reconstruction
After we obtain the transmitted wavefront aberrations in the first two steps, an optimization method (e.g., the downhill simplex method [1]) is used to reconstruct the multisurface profiles of the tested refractive element simultaneously. The objective function is defined as the root mean square (RMS) of deviation between the measured transmitted wavefronts and the simulated ones with the reconstructed multisurface profiles.
Since the propagation direction of a light ray is perpendicular to the transmitted wavefront, the unit normal vectors of the refractive surfaces can be determined by the unit vectors of incident and refracted rays. According to Snell’s law, the relationship between the unit normal vector ${{\boldsymbol \Omega}_m}$ of the $m$th surface ($m = {1},\;{\ldots},\;M$, $M$ is the number of optical surfaces contained in the tested refractive element) and unit vector ${{\bf R}_m}$ of the corresponding refracted ray can be described as
Table 1. Lens Parameters for Tested Optics in Simulation
Table 2. ${XY}$ Polynomials Definition of Tested Freeform Lens
A set of sampling light rays are traced through the tested refractive element. Then, the intersection point between each incident ray and each optical surface, and that between each outgoing ray and the transmitted wavefront can be determined. For the $n$th sampled light ray, Eq. (7) can be rewritten as
Equation (8) indicates that there are $N$ equations and $N \times M$ unknowns during the measurement of each transmitted wavefront. In order to find a solution to this underdetermined problem, we need $K$($K\; \ge \;M$) transmitted wavefronts (each wavefront corresponds to a ray-transfer matrix at a different deflectometric system geometry) to convert this underdetermined problem to an overdetermined one. However, it might be very time-consuming to trace a large number of rays to reconstruct the multisurface profiles. To improve the measurement efficiency, the multisurface reconstruction could be performed based on the representation of surface profiles such as Zernike polynomials, ${XY}$ polynomials, and Legendre polynomials. Then, the surface reconstruction is converted to the process of solving the polynomial coefficients, and the iterative optimization method can be applied to solve the polynomial coefficients of multiple surfaces.
For the $K$ transmitted wavefront aberrations $\{W_{\text{ept},k}^*\}$ ($k = {1},\;{\ldots},\;K$) corresponding to different system geometries (with the optimal system geometry $\{{\bf GP}_k^*\}$ after calibration), we have actual multisurface profiles ${E_{\text{surf}}}({\Omega _{\text{ept},\{{1,2,\ldots M} \}}})$ and modeled multisurface profiles ${\tilde E_{\text{surf},k}}({{\bf \Omega} _{\text{ideal},\{{1,2,\ldots M} \}}})$ from Eq. (5),
To ensure the similar sensitivity of optimization variables in orthogonal directions, a set of additional transmitted wavefronts are recommended by rotating the tested element about the $x$ and $y$ axes (or translating the element along the $x$ and $y$ axes). It is worth noting that the proposed SMM method is applicable in the surface reconstruction based on both normal vectors and surface-fitting methods.
3. SIMULATION VERIFICATION
Two refractive lenses were tested by the ray-tracing-based simulation. One of the lenses is a traditional spherical lens with some predefined surface departures, and the other one is a freeform lens including two freeform optical surfaces that are both represented by ${ XY}$ polynomials. The lens parameters for tested optics are summarized in Tables 1 and 2.
The actual front and rear surface departures of the tested spherical lens from ideal ones are shown in Figs. 3(a) and 3(b) with the corresponding peak-to-valley (PV) values of 8.772 and 8.266 µm, respectively. To achieve accurate and full-aperture testing, the tested spherical lens was placed at four different positions, including original, rotated by 30 deg about the $x$ and $y$ axes (denoted as “Tx30” and “Ty30”), and rotated by 180 deg about the $y$ axis (denoted as “Reverse”), and consequently, four transmitted wavefronts (without missing of sampling rays on the image plane) were simulated for multisurface reconstruction, as shown in Figs. 3(c)–3(f). From these four figures, we can see that the different system geometries lead to obvious differences among the obtained transmitted wavefronts. Based on the four transmitted wavefronts, the front and rear surfaces of the tested spherical lens were reconstructed simultaneously, as shown in Figs. 3(g) and 3(h). The PV value of the reconstructed surface departure given in Fig. 3(g) equals 8.779 µm, and that equals 8.249 µm in Fig. 3(h). The residual reconstruction errors are given in Figs. 3(i) and 3(j) (PV = 0.067 µm and 0.066 µm, respectively, after 33 optimization iterations). The residual errors of reconstructed transmitted wavefronts at the four different positions (original, Tx30, Ty30, and Reverse) are shown in Figs. 3(k)–3(n). The changes in the objective function, obtained during front and rear surface departures during the iterative optimization process, are shown in Fig. 4(a), Visualization 1, and Visualization 2. From Fig. 4(a), we can observe that the objective function decreased dramatically at the early stage of the optimization process, and it took about 113.4 s (using a personal desktop computer with the configuration of Intel Core i7-7700 CPU at 3.60 GHz, 16 GB RAM, and x64-based processor) to get saturated after 14 optimization cycles, meaning the simultaneous reconstruction of freeform surfaces is fast and stable. A good agreement between the nominal and reconstructed surface profiles is achieved by the proposed SMM method, and the testing accuracy in the order of nanometers is achieved. Thus, the proposed SMM method is efficient and effective for simultaneous multisurface testing in refractive optics testing.
Fig. 3. Multisurface reconstruction results of the spherical lens in ray-tracing simulation. Nominal (a) front and (b) rear surfaces of tested spherical lens. Obtained transmitted wavefronts at the (c) original, (d) Tx30, (e) Ty30, and (f) Reverse positions. Reconstructed (g) front and (h) rear surfaces of tested element. Residual reconstruction errors of (i) front and (j) rear surfaces. Residual errors of reconstructed transmitted wavefront relative to the theoretical wavefronts at the (k) original, (l) Tx30, (m) Ty30, and (n) Reverse positions.
Fig. 4. Changes in the objective function during the optimization process. (a) Spherical lens testing; (b) freeform optics testing.
Fig. 5. Multisurface reconstruction results of the freeform optics in ray-tracing simulation. Nominal (a) front and (b) rear surfaces of tested element. Measured transmitted wavefronts at the (c) original, (d) Tx30, (e) Ty30, and (f) Reverse positions. Reconstructed (g) front and (h) rear surfaces of tested element. Residual errors of (i) front and (j) rear surface profiles relative to nominal profiles. Residual errors of reconstructed transmitted wavefront relative to the theoretical wavefronts at the (k) original, (l) Tx30, (m) Ty30, and (n) Reverse positions.
To further validate the feasibility of the proposed SMM method in a large slope dynamic range, the testing of a freeform refractive lens was simulated. The actual front and rear surface departures from their base conic surfaces are given in Figs. 5(a) and 5(b), with the PV of 302.842 and 482.501 µm, respectively. With the transmitted wavefronts [Figs. 5(c)–5(f)] obtained at the four (original, Tx30, Ty30, and Reverse) positions, the reconstructed front and rear surface departures (PV = 303.369 and 481.946 µm, respectively) from their base conic surfaces after 33 optimization cycles are shown in Figs. 5(g) and 5(h), respectively. The residual errors of reconstructed front and rear surface departures are given in Figs. 5(i) and 5(j), with the PV of 0.684 and 0.877 µm, respectively. The residual errors of reconstructed transmitted wavefronts with respect to the theoretical ones at four positions are shown in Figs. 5(k)–5(n). The changes in the objective function, front and rear surface departures from their base conic surfaces during the optimization process, are shown in Fig. 4(b), Visualization 3, and Visualization 4. From Fig. 4(b), we can see that both efficient and accurate reconstruction of multiple freeform surfaces is simultaneously achieved by the proposed SMM method. Since transmitted wavefronts are measured by the computer-aided deflectometry, the effectiveness of the proposed SMM method can be guaranteed in the testing of various optical elements with a large slope dynamic range.
Fig. 6. Measurement results of a plano–convex spherical lens in experiment. Measured transmitted wavefronts at the (a) original, (b) Tx30, (c) Ty30, and (d) Reverse positions before (top) and after (bottom) system geometry calibration. Measured front spherical (top) and rear planar (bottom) surface departures by (e) the proposed SMM method after system geometry calibration and those by (f) ZYGO interferometer, and (g) measured surface differences between (e) and (f).
It should be mentioned that there are several factors that could introduce the residual wavefront error in optimization. One is the minimum slope $\Delta \theta$ that can be resolved in deflectometric testing system. When the pixel size $P = {0.265}\;\text{mm}$, $\Delta \theta = P/d{ \cdot1/50} = {35.3}\;{\unicode{x00B5}} \text{rad}$ under the 1/50 pixel uncertainty in the phase-shifting calculation. For a spatial sampling of 25 µm on exit pupil, the corresponding sag resolution is ${\sim}{0.9}\;\text{nm}$. Besides, the computational errors, including the wavefront reconstruction with the integration method and the optimization algorithms, could also introduce additional residual wavefront errors.
4. EXPERIMENTAL RESULTS
Experimental verification was also performed in this section to show the high accuracy and large slope dynamic range of the proposed SMM method. A deflectometric system based on the configuration shown in Fig. 2 was built to measure a traditional plano–convex spherical lens and a freeform lens. A coordinate measuring machine (CMM, Hexagon Global CMM, accuracy 1.7 µm) was used to measure the three-dimensional locations (system geometry) of each component in the system, which include the tested refractive lens, LCD illumination screen (Philips LCD monitor, 234E5Q, resolution: ${1920}\;\text{pixels} \times {1080}\;\text{pixels}$, pixel size: 0.265 mm) and camera (PointGrey, FL3-U3-13S2M-CS, resolution: ${1328}\;\text{pixels} \times {1048}\;\text{pixels}$, pixel size: 3.63 µm).
A. Testing of the Plano–Convex Spherical Lens
The refractive index of the plano–convex lens equals 1.5164 at the wavelength of 587.6 nm. The full diameter, thickness, and focal length of the lens are 25.4, 3, and 150 mm, respectively. The spherical lens was placed with its planar surface proximal to the LCD screen. The distance between the planar surface and the LCD screen is 228.902 mm. The tested spherical lens was placed at four different positions (i.e., the original, Tx30, Ty30, and Reverse positions) in order to achieve an accurate and full-aperture measurement. The four measured transmitted wavefronts are given in Figs. 6(a)–6(d). After the calibration of system geometry by the computer-aided reverse optimization, the surface departures of the reconstructed front spherical surface and rear planar surface are given in Fig. 6(e) (PV = 0.135 and 0.141 µm, respectively). Here, the 37-term Zernike coefficients are employed to characterize the surface departure. Figure 6(f) shows the surface departures measured by a ZYGO GPI interferometer: PV = 0.146 µm for the spherical surface and 0.162 µm for the planar surface. From Figs. 6(e) and 6(f), we can clearly see that the testing results obtained by the proposed SMM method agree well with the results by the ZYGO GPI interferometer, as shown in Fig. 6(g), indicating the effectiveness of the proposed method. The residual error of some mid-frequencies can be further decreased by employing the higher-order polynomial terms. From Figs. 6(a)–6(d), we also see that the measurement of the transmitted wavefronts is influenced by the system geometrical errors, and the computer-aided reverse optimization can significantly improve the accuracy of system geometry measurement.
B. Testing of the Freeform Lens
The freeform lens shown in Fig. 7(a) includes two freeform surfaces that are both represented by ${XY}$ polynomials; the parameters of these two optical surfaces are given in Tables 1 and 2. The rear surface was placed facing the illumination screen in the deflectometric testing system, and the distance between the rear surface and illumination screen was 158.192 mm. The ray mapping is given in Fig. 7(b), which clearly indicates the freeform nature of the freeform lens. From the surface departures of the front and rear surfaces from their base conic surfaces shown in Fig. 7(d), we know that the RMS departures equal 83.823 and 84.307 µm, respectively, and the corresponding PV values are 302.842 and 482.501 µm. The recorded sinusoidal fringes in the $x$ and $y$ directions are given in Fig. 7(c). The four measured transmitted wavefronts at four different positions (the original, Tx30, Ty30, and Reverse positions) are plotted in Figs. 7(g)–7(j). From these measured transmitted wavefronts, we can reconstruct both the front and rear freeform surfaces of the lens. The surface departures from their base conic surfaces are given in Fig. 7(e). From Fig. 7(e), we know the PV value of the front surface equals 305.096 µm and that of the rear surface equals 481.807 µm. Since the slope dynamic range of the tested freeform lens exceeds the measurable range of a conventional laser interferometer, the comparisons were performed between the nominal surfaces and the measured ones, as shown in Figs. 7(d) and 7(e). The residual errors of reconstructed surfaces are given in Fig. 7(f), with the PV of 2.967 and 2.389 µm, respectively. From Fig. 7(f), we can see that a good agreement between the designed and measured surfaces has been achieved, with the PV and RMS values of residual errors in the order of micrometers and submicrometers. From these results, we can see the proposed SMM method is very effective in freeform optics testing.
Fig. 7. Measurement results of a freeform refractive lens in experiment. (a) Lens model and the fabricated lens; (b) ray mapping between the incident (left) and outgoing (right) rays on image plane; (c) recorded sinusoidal fringes in $x$ (left) and $y$ (right) directions; (d) nominal front (left) and rear (right) surfaces of tested freeform lens; (e) measured front (left) and rear (right) surfaces by the proposed SMM method based on calibrated system geometry; (f) residual errors of front (left) and rear (right) surfaces between the designed and measured surfaces. Measured transmitted wavefronts at the (g) orignial, (h) Tx30, (i) Ty30, and (j) Reverse positions before (top) and after (bottom) system geometry calibration.
5. DISCUSSION
Reliability evaluation of measured transmitted wavefronts is essential for accurate SMM. The measurement uncertainty can be characterized by four major error sources: the camera distortion, uncertainty of the illumination screen pixel, optical tolerances on the tested element, and system geometrical error. The test uncertainty in deflectometry and the corresponding calibration method, including those from the camera and illumination screen, has been investigated [40]. The camera distortion due to the aberration of the camera lenses, which degrades the accuracy of phase retrieval, can be removed with the camera calibration method [41]. The illumination screen pixel uncertainty, which usually leads to a shift in the illumination screen pixel position, can be tackled by the illumination screen calibration [41]. By use of the currently available calibration methods, the errors caused by the performance of system components is negligible. The optical tolerances include those in lens thickness, the refractive index and Abbe number of the lens material. For the tested freeform lens in the simulation (Section 3), the residual RMS wavefront errors corresponding to the thickness tolerance $\pm {0.02}\;\text{mm}$, refractive index tolerance $\pm {0.0005}$, and Abbe number tolerance $\pm {0.8}\%$ are 0.009, 0.064, and 0.001 µm (those for the spherical lens are 0.06, 1.1, and 0.03 nm), respectively. Thus, the major tolerance that should be considered is that of the refractive index. It is of great interest to mention that the refractive index can be optimized by the computer-aided optimization technique introduced in Section 2.B, and the related residual measurement error would be well controlled.
The system geometrical error is related to the precision of three-dimensional measuring tools. The computer-aided system geometry calibration performed in the second step of the proposed SMM method, can effectively reduce the measurement error caused by the system geometrical error. The relationship among the system geometrical error, measurement errors of transmitted wavefronts, and surface profiles were investigated by performing ray tracing in the ideal system model with Zemax software in numerical simulation. The Zernike polynomials were employed to describe wavefront aberrations and surface profiles. Without loss of generality, the displacement and tilt errors of the tested refractive element in the $x$ direction were taken as the examples. An additional displacement error of 20 µm and a tilt error of 0.02 deg in the $x$ direction were added to the tested element. The Zernike coefficients (from ${Z_5}$ to ${Z_{10}}$) of the measured transmitted wavefront aberrations at the Tx30 position are given in Figs. 8(a) and 8(b). From Figs. 8(a) and 8(b), we can clearly see that the RMS values of residual aberrations are less than 5 nm, indicating that the high-order aberrations introduced by the system geometrical errors are significantly reduced by use of the computer-aided system geometry calibration.
Fig. 8. Deviations of wavefronts and surface profiles introduced by system geometrical error. Zernike coefficient deviations of transmitted wavefront at the Tx30 position obtained with (a) 20 µm displacement error and (b) 0.02 deg tilt error of tested element in the $x$ axis. RMS values of wavefront measurement errors at the original, Tx30, Ty30, and Reverse positions with various (c) displacement errors and (d) tilt errors of tested element in the $x$ direction, surface testing error with various (e) displacement errors and (f) tilt errors of tested element in the $x$ direction, respectively, before and after system geometry calibration.
The residual testing errors of wavefront aberrations corresponding to various displacement and tilt errors of the tested element in the $x$-axis are shown in Figs. 8(c) and 8(d), respectively, and those for surface profiles are given in Figs. 8(e) and 8(f). Figures 8(c)–8(f) clearly show that the increase in system geometrical error could lead to significant growth in the testing errors of wavefront aberrations and surface profiles, with the residual surface testing error RMS in the order of submicrometers without system geometry calibration. Thus, it is necessary to remove the aberrations introduced by system geometrical error. As mentioned above, the computer-aided system geometry calibration could effectively eliminate the system geometrical error, and the residual surface testing error after calibration is decreased to tens of nanometers. Therefore, the computer-aided system geometry calibration provides a feasible way to achieve the high measurement accuracy and strong generality of the proposed SMM method.
6. CONCLUSION
In this work, a simultaneous multisurface computer-aided deflectometry is proposed for the testing of freeform refractive optics. The proposed method relies on the transmitted wavefront rather than the reflected wavefront. This unique feature of the proposed method ensures that no additional parasitic reflection elimination is needed during testing, and the multiple surface profiles included in a freeform refractive optics can be measured simultaneously. The computer-aided techniques are used to achieve the virtual “null” acquisition of transmitted wavefront and to improve the accuracy of wavefront measurement and surface reconstruction. An accurate measurement of transmitted wavefront can still be achieved by the computer-aided system geometry calibration, even though there are obvious geometrical errors. The proposed method is highly robust and efficient and can significantly reduce the amount of human effort and system cost. The numerical and experimental verification clearly show that both high accuracy and large slope dynamic range can be achieved by the proposed method, with the testing accuracy in the order of nanometers for the traditional spherical optics and submicrometers for freeform refractive optics. This method provides a novel and general way for the testing of freeform optics and may have a significant potential for in situ metrology. Moreover, this method could be generalized to the testing of a broad range of various nonoptical elements.
Funding
National Natural Science Foundation of China (51775528); Guangxi Key Laboratory of Optoelectronic Information Processing (GD18205); Natural Science Foundation of Zhejiang Province of China (LY19F050008); China Postdoctoral Science Foundation (2017M621928).
Disclosures
The authors declare no conflicts of interest.
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