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Abstract
Freeform optics, offering high degrees of design freeform to control light propagation, have already been widely applied in various photoelectric equipment. The form quality of those optics is crucial to their opto-electronics functionalities, which requires to be measured accurately. The deflectometry is a promising technology to test the complex freeform surfaces. In general, there is a designed surface model for the monoscopic deflectometry to estimate the positions of whole measured points to solve the issue of height-slope ambiguity. However, the unknown or inaccurate surface model can induce errors into the measured normal, thereby decreasing the measurement precision. In this paper, without relying on the known surface model, the proposed method iteratively optimizes a sphere model to describe the measured surface by changing the spherical radius. In order to reduce the global error, the space-variant spheres are optimized, respectively, to estimate the whole-aperture surface coordinate. With the help of the iteration surface reconstruction process, the optimal number of the space-variant spheres is achieved to meantime obtain the final reconstructed surface. Compared to the measurements by using the plane model, the form accuracy can be improved by three times. Experiments demonstrate that the proposed method can successfully reconstruct the complex surfaces without the need of a known surface model, which can greatly improve the measuring flexibility and measurement accuracy.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Due to the rapid development of high-performance imaging systems, the optical components having aspheric and freeform surfaces have been widely applied in various areas. As a consequence, there is a high demand for the accurate and flexible measurement of their form qualities. Deflectometry is a promising measurement technology because of its higher dynamic range than the interferometric systems [1,2]. Deflectometry obtains the surface normals by displaying some encoded patterns (such as the fringe patterns [3–5]) on a screen and capturing the distorted patterns using a camera, and then reconstructs the surface by integrating the normal field [6,7]. However, there are many possible height and slope combinations to explain the rays transmitting trajectory, which is a height-slope ambiguity problem for the monoscopic deflectometry [8,9]. In order to address the ambiguity issue, a nominal surface model is utilized to specify the incident ray in the software configurable optical test system (SCOTS) [10–12], and the angular bisectors between the incident and reflected rays are iteratively assigned to the surface normals. Unfortunately, the model is not well known, such as during the grinding phase of an optic, where the surface error compared to the ideal surface changes from the millimeter-to-micron scale. Without an accurate surface model, it is nearly impossible to correctly determine the measured points’ coordinates, leading to errors in the reconstructed surface [13,14].
In recent years, when the surface model is not well known, there are three methods to accurately estimate the surface normal [8,15]. First method is to use two screens [2,16] or move a screen [17–19] to specify the incident rays, thereby calculating the surface normal. However, the measurement precision is affected by the positioning accuracy of the screen, and the measurement efficiency could be reduced, because the displayed patterns should be encoded on two screens, respectively. Second method is to iteratively calculate the measured point coordinates to specify the surface normal by using two cameras [20–22]. However, there is a contraction between the measurement precision and the reconstruction efficiency. In addition, the above two methods need the extra device to solve the issue of height-slope ambiguity. The last method is to use a flat model as the initial surface and reconstruct the final surface using the iterative operation in the SCOTS [23]. However, the flat model is not accurate enough to describe the measured surface with the complex model and with the large sagittal height. Therefore, it is important to solve the problem of unknown surface model to measure the complex freeform surfaces in monoscopic deflectometry.
In this paper, in order to solve the height-slope problem caused by the unknown or inaccurate surface model, an iterative space-variant sphere-model method is proposed to produce the precise surface to intercept deflection rays by using the space-variant sphere model and then integrate iteratively surface slopes to obtain the final surface in the monoscopic deflectometry. Section 2 presents the methodology of the proposed method. Section 3 provides experimental validation and finally, the paper is summarized in Section 4.
2. Methodology of the iterative space-variant sphere model
For monoscopic deflectometry, a known surface model is required to calculate the surface normals, which limits the scope of measurement and decreases the measurement precision of the unknown complex surface. Therefore, in this paper, an iterative data-processing technology using the space-variant sphere model is proposed to remove the need of a known surface model to reconstruct precisely the measured surface, known as iterative space-variant sphere-model deflectometry (ISSD).
In order to use a sphere to describe the measured surface, there are generally four parameters (3D (three dimension) coordinate (x0, y0, z0) of spherical center and the spherical radius r) to define a sphere in the 3D space. However, it is difficult to estimate the four unknown parameters because of insufficient conditions in the measurement process. Fortunately, in the 3D space, the normal vectors of all points on the sphere are crossing the spherical center. Therefore, 3D coordinate of the spherical center can be determined along the normal vector of an arbitrary spherical point. The distance between the spherical center and the spherical point is the spherical radius. As a result, the estimation issue of four parameters can be transformed into the optimization problem of the spherical radius r when an arbitrary normal vector is known. The initial spherical radius r can be assigned with the diameter of the measured aperture. Additionally, in monoscopic deflectometry, in order to bound the solution space, a physical point F must be measured and utilized to define the location of the measured surface model. The known normal vector to determine the spherical center’s position can be defined naturally by the physical point F, without the need of the extra information.
In the proposed ISSD technology, the normal vector of the physical point F should be calculated first. The camera pixel corresponding to the physical point F can be obtained to define the ray-pointing unit vector $\vec{N}_{\mathbf F}^R$ of the camera. The unit vector $\vec{N}_{\mathbf F}^R$ can be defined from the imaging pixel to the optical center of the camera. The screen pixel corresponding to the physical point F can be utilized to position the source location s of the incidence ray, as shown in Fig. 1(a). With the help of the position of the physical point F, the unit vector $\vec{N}_{\mathbf F}^I$ of the incidence ray can be defined from the source position on the screen to the physical point F. Based on the law of reflection, the normal vector of the physical point F can be calculated by
Fig. 1. Iterative diagram of one sphere model. (a) Iterative procedure for one sphere model. (b) Retraced deviation on the screen.
Then, by using the sphere model to intercept the reflection ray (namely the camera ray), the coordinate $({{x_{\mathbf M}},\textrm{ }{y_{\mathbf M}},\textrm{ }{z_{\mathbf M}}} )$ of arbitrary point M among the other measured points could be estimated as
After that, based on the law of reflection, the vector $\vec{N}_{\mathbf M}^I$ of the incidence ray reflected via the point M can be defined as
By calculating the intercept point between the screen and the incidence ray, the retraced point $({x_{\mathbf M}^s,\textrm{ }y_{\mathbf M}^s,\textrm{ }z_{\mathbf M}^s} )$ on the screen can be represented in
During the optimization process of the spherical radius, it is worth noting that the intercept point M(i) (i = 1:l) can be updated along the reflection ray with the change of the spherical radius, as shown in Fig. 1(a). Based on the law of reflection, the normal vector $\vec{N}_{\mathbf M}^{}$ can be also altered to deviate the retraced incidence ray from the actual ray, during the retracing process. Consequently, for all measured points, the retraced points on the screen can be deviated from the actual dots, due to the deviated retraced incidence rays, as illustrated in Fig. 1(b). As a result, the total retraced error could be changed with the change of the spherical radius r, because the radius directly affects the normal vectors of all intercept points. According to the intrinsic property of deflectometry, the retraced deviation can reflect the accuracy of the surface model. Therefore, the spherical radius r can be optimized with the goal of minimizing the retraced error. To optimize the spherical radius, the sum of the squared deviations between the retraced screen pixels and the actual screen pixels is adopted as the objective function
The objective function is a non-differentiable least-square problem, thus the derivative-based methods, such as the Gaussian-Newton method or the Levenberg-Marquardt method, cannot be applied. As a consequence, the downhill simplex algorithm can be adopted for solving this non-differentiable optimization problem [24]. Any value can be assigned as the initial spherical radius to define the initial surface U(0). The final surface U(n) can be determined by optimizing iteratively the spherical radius until minimizing the retraced error, as depicted in Fig. 1(a). During the iterative process, the retracing deviations need to be recalculated every time the radius is updated.
Figure 2 depicts the optimization workflow of the initial surface by using the sphere model. Depending on the help of the physical point F, the retraced error between the actual dots and the retraced dots from the camera to the screen could achieve minimization by iterating the spherical radius. In the meantime, the initial surface can be obtained by using the optimized sphere model to replace the unknown actual surface.
For the complex optical surface with small focal power, a single sphere model is not suitable to match the whole surface, due to the effects of the global errors. As a result, the surface can be divided into multiple areas, as depicted in Fig. 3. The overlapping areas are designed to provide the physical point F to the neighboring zone. Therefore, the number of overlapping areas is not important. For the case of the discontinuous overlapping areas, the depth of the overlapping point can be assigned with the average value of those different points’ depths. The optimization procedure starts at area 1 where the measured physical point F is located on. By using the above method, a sphere model can be optimized in area 1. After that, the same procedure is repeated in neighboring areas 2-19, respectively. In other areas, the new physical point should be assigned renewably with one point in the overlapping region between the neighboring area and the area after optimization, rather than the physical point F in area 1. After that, the sphere model in each neighboring area is specified until the objective function in Eq. (6) achieves the minimization. Therefore, the whole surface can be obtained by using the space-variant spheres. In order to optimize the number of the areas to minimize the global deviations, the objective function can be expressed as
where fn is the reconstructed surface by dividing into n areas, fn-1 is the reconstructed surface by dividing into n-1 areas. At the end, the full-aperture surface fn can be reconstructed from the surface slope determined by the iterative space-variant sphere model, which eliminates the impacts of unknown surface model.Figure 4 demonstrates the whole data processing flow of the ISSD technology. First, in order to obtain two different initial surface, the initial full-aperture surfaces can be divided into the t (t = 1 and 4) sub-areas. Then, the two surfaces can be reconstructed iteratively from the local slopes, which runs for k = 1:m iterations until the reconstruction process convergences, respectively. Then the comparing the deviation between those two reconstructed surfaces provided by 1 area and 4 areas, if the convergence is not met, the surface is estimated by t = (t + 1)2 sub-sphere models to reconstruct repeatedly the surface until to minimize the deviation between before and after two reconstructed surfaces. In order to reduce the iteration number, the number t can be updated in the steps of (t + 1)2. If the deviation is small enough, it is verified that the segmented area number is enough to provide an accurate initial surface. The estimation error of the initial surface can be eliminated to obtain the precision solution during the iterative surface reconstruction. In other word, the accuracy of the final reconstructed surface can be ensured by using the initial full-aperture surface integrated by the space-variant sphere models, because the estimation error of the initial surface is decreased by the space-variant sphere models. The ISSD process runs for t = 1:n iterations until to minimize the objective function as expressed by Eq. (7).
3. Experimental demonstration
To verify the performance of the proposed method, a deflectometric measuring system is established. The used camera is JAI SP-20000C-USB with resolution of 5210×3840 pixels, a frame rate of 16 fps at full resolution, and a USB3 interface. The screen is iPad mini 2 with resolution of 2048×1536 pixels and pixel size of 0.0784 mm. The measuring system is calibrated using the method described in Ref. [25]. In the experiment, the phase-shifting method [26,27] is applied to build the pixel pairs.
3.1 Experiment 1: flat mirror
A flat mirror (Ø 100 mm with flatness 63.28 nm) is measured to verify the performance of the ISSD technology without the surface model. The pose of flat mirror is positioned by assisted with the rotary stage in Ref. [25], thereby calculating the physical point F. In order to validate the measurement performance for the unknown surface model, the flat surface is reconstructed iteratively by utilizing the proposed ISSD technology to release the limitation of the known surface model. The iteration process is continued until the error between the before and after reconstructed surface is less than 1 nm or the iteration number reaches 10. The flat surface is also reconstructed by using the designed surface model, as a comparison to certify the express performance of the proposed ISSD technology. The computing time of the proposed ISSD technology is approximately 2 times to the conventional method, because the measured surface is simple enough to be estimated by one sphere model. The mirror is measured with an interferometer (Zygo GPI XP/D) as a reference. The reconstructed result by using the proposed ISSD method, as depicted in Fig. 5(a). Compared to the reference result of Zygo, the relative deviation maps of the reconstructed results with the proposed ISSD technology and that using the conventional methods are illuminated in Figs. 5(b) and 5(c). The RMS (Root-Mean-Squares) of the deviation map from the proposed ISSD method and the method using the designed surface model are 33.88 nm and 33.67 nm, respectively. The experimental results demonstrate that this ISSD method can be comparable to the conventional method when the designed surface model is unknown.
3.2 Experiment 2: off-axis parabolic (OAP) mirror
Off-axis parabolic (OAP) mirrors are widely applied in optical systems, e.g., Cassegrain telescopes, but the interferometric measurement of OAP mirrors is troublesome and inefficient owing to the difficulty of the null measurement assisted with a standard sphere artefact. This type of surface is non-axial symmetrical, thus the conventional method specifying the designed surface model is difficult to be applied in SCOTS to specify the measured points because the position of OAP is difficult to be calibrated. An OAP mirror is measured to verify the performance of the ISSD technology without the surface model, and the ray-tracing model is shown in Fig. 6(a). The OAP mirror is designed with the parameters including a parent focal length of 45 mm, aperture of 30 mm and off-axis offset of 51.96 mm, as shown in Fig. 6(b). Additionally, the center point of the OAP, about which it rotated, was calibrated to be the physical point F, which is measured by using a height gauge with the precision of 10 um. In order to validate the measurement performance for the unknown surface model, the surface of OAP is reconstructed iteratively by utilizing the proposed ISSD technology to release the restriction of the known surface model. The surface of OAP is also reconstructed by using the plane model, as a comparison to certify the express performance of the proposed ISSD technology for the complex surface. To verify the description accuracy of the proposed ISSD technology, the OAP surface is also reconstructed by using the designed surface model. The computing time of the proposed ISSD technology is approximately 2 times to the conventional method, because the measured surface is simple enough to be estimated by one sphere model. The mirror is measured with an interferometer (Zygo GPI XP/D) as a reference.
Fig. 5. Reconstructed results of the flat mirror. (a) Reconstructed surface with the proposed ISSD technology. (b) Deviation map between ISSD and interferometry. (c) Deviation map between conventional method and interferometry.
Fig. 6. The actual deflectometric measuring system of an OAP mirror. (a) Ray-tracing of the measuring system. The screen pixels, the actual measured points, and the camera optical center are depicted in red, green, and blue, respectively. (b) Workpiece.
After reconstructing the surface, the resulting form error is compared with interferometry. These four measurement results are fitted using the first 36 terms of the Zernike polynomials [28]. The first 4 Zernike terms representing the piston, the tilt and the power, respectively are ignored, because they are relevant to the position only, rather than the underlying form of the measured surface. The surface map measured by Zygo, with the Zernike terms Z1∼Z4 removal, are depicted in Fig. 7(a). The relative departure maps of the measured results with the proposed ISSD method and that using other methods with respect to the result of Zygo are calculated separately, as shown in Fig. 7(b)–7(d). The size of the measured area is 17 × 19 mm. When the surface model is unknown, the proposed ISSD method obtains a better result closer to the result by Zygo compared to that with the plane model. Compared to the result using the designed surface model, the proposed ISSD method could have a similar result. In order to assess them quantitatively, the relative departures of the measured forms in three cases with respect to the measured result of Zygo, namely the proposed ISSD method, the method using the designed surface model and the method using the plane model, are calculated separately. The RMS of the terms Z5∼Z36 from the proposed ISSD method, the method using the designed surface model and the method using the plane model are 22.2 nm, 31.76 nm and 54.39 nm, respectively. The proposed ISSD approach has a similar result by using the traditional method based on the designed model. The proposed ISSD method's result error is decreased by approximately three times when compared to the result obtained using the plane model. As a result, the proposed ISSD technique has been shown to be capable of obtaining a trustworthy surface for a non-axial symmetrical surface with unequal principal curvature without the use of an accurate model.
Fig. 7. The measurement results of an OAP, with the Zernike terms Z1∼Z4 removal. (a) Zygo. (b) Deviation map using ISSD. (c) Deviation map using designed surface model. (d) Deviation map using plane model.
3.3 Experiment 3: aspherical mirror
The aspheric and freeform optics can be utilized to shape beams or to compensate the aberrations, e.g. the Schmidt compensator in a telescope etc. However, the interferometry requires a null setup for measuring complex surfaces because of the limited measurement dynamic range. The deflectometry is an efficient measurement technology to measure accurately the complex surface. However, the complex surface model is difficult to be defined in the conventional deflectometry by using the designed surface model because the complex surface model will enlarge the positioning error. Those surface is complicated enough to be described difficultly by using the simple model (e.g. plane model). A germanium infrared aspheric lens is adopted to verify the proposed ISSD method when the measured surface is freeform and complex, as shown in Fig. 8(a). The nominal model of the aspheric surface is expressed as
Fig. 8. Measurement of an aspherical mirror. (a) Workpiece of germanium infrared aspheric lens. (b) Image of vertical fringe, the measured region is dashed.
To verify the reconstruction performance for the unknown complex surface model, the surface of the aspherical mirror is reconstructed iteratively by using the proposed ISSD technology. The surface is constructed by using the ISSD with single sphere to proof the necessary and expression ability to the complex surface of the multi-sphere model. The surface of the aspherical mirror is also reconstructed by using the plane model, as a comparison to certify the express performance of the proposed ISSD technology for the complex surface. As a comparison, the aspherical mirror is also reconstructed by using the designed surface model to demonstrate the description precision of the proposed ISSD technology for the complex surface. The computing time of the proposed ISSD technology is approximately 3 times to the traditional method, because the measured surface need to be estimated by 9 sphere models. For the purpose of quantitative comparison, the surface is also measured by a profilometer LUPHOScan 420HD as a reference.
These four measurement results are fitted using the first 36 terms of the Zernike polynomials. The first three Zernike terms, corresponding to the piston and the tilt, respectively, are ignored, because they are relevant to the pose of the measured surface only, but irrelevant to the form. The proposed ISSD technology is used to reconstruct iteratively the measured surface. The RMSs of the terms Z4∼Z36 of the reconstructed surface comparing to the result by LUPHOScan through iterations is visualized in Fig. 9, which decreases from 7.27 µm (1st iteration) to 4.09µm (6th iteration). Iterations shows the most significant change occurs in early iterations.
The reconstructed surface maps associated with different Zernike polynomials are shown in Fig. 10. The surface maps measured by LUPHOScan with different Zernike polynomials are shown in Fig. 10(a). The relative departure maps of the measured results with the proposed ISSD method and that using other methods with respect to the result of LUPHOScan are calculated separately, as shown in Fig. 10(b)–10(e). It is evident the proposed ISSD method obtains a similar result with the result using the designed aspherical model, and has a better result that is closer to the result by LUPHOScan compared to the result using the plane model.
Fig. 10. Reconstruction results. (a) LUPHOScan. (b) Deviation map using ISSD with multiple spheres. (c) Deviation map using ISSD with single sphere. (d) Deviation map using Aspherical model. (e) Deviation map using Plane model.
In order to assess them quantitatively, the relative departures of the measured forms in four cases with respect to the measured result of LUPHOScan, namely the proposed ISSD method, the proposed ISSD with single sphere, the conventional method using the designed aspherical model and the method using the plane model, are calculated separately. The RMSs of the terms Z4∼Z36 are analyzed, as presented in Fig. 10(b)–10(e). The RMS of the total departures in the terms Z4∼Z36 decreases from 6.20 µm for the method using the proposed ISSD with single sphere to 4.09 µm for the proposed ISSD method with multiple spheres. Especially, the RMS of the total departures in the terms Z4∼Z11 decreases by about 2.5 µm, demonstrating that the multi-sphere model can decrease the low-order errors. The RMS of the total departures in the terms Z4∼Z36 decreases from 12.36 µm for the method using a plane model to 4.09 µm for the proposed ISSD method. The result demonstrates that without the known surface model, the proposed ISSD technology can improve the reconstruction precision compared to the method using a plane model, which is especially distinct for the complex surface. For the comparison between the proposed ISSD method and the conventional method using the designed aspherical model, there is an interesting result. On one hand, the RMSs of the total departures in the terms Z12∼Z36 are 0.85 µm and 0.74 µm for the proposed ISSD method and the traditional method, respectively. The departure in the high order Zernike terms can be ignored, demonstrating that the proposed ISSD method can express the complex high-order surface not to induce the extra errors. On the other hand, comparing to the result with the proposed ISSD method, the RMS of the total departures in the terms Z4∼Z11 increase by about 1 µm for the conventional method. The main error focuses on the low and medium order Zernike terms. Interestingly, the proposed ISSD method behaves better than the conventional method using the designed aspherical model. The main reason is that the location calibration including the position and the pose is important and difficult for the conventional method. And the surface with the large sagittal height and the complex model could enlarge the location error during the surface description in the deflectometry. Therefore, there is an inaccurate surface model defined in the deflectometric system to reduce the reconstruction precision for the conventional method. The surface deviation caused by the positioning error behaves mainly in the low spatial frequencies. However, for the proposed ISSD method, the spherical radius optimization can reduce the location error, particularly the pose error to describe more accurately the measured surface. In the deflectometric system, the surface slope is sensitive to the error of the tilt and the displacement. Therefore, compared to the conventional method using the designed surface model, the proposed ISSD method is not sensitive to the location error and releases the calibration difficulty.
4. Summary
In monoscopic deflectometry, without the known surface model, it is difficult for the complex surfaces to solve the height-slope ambiguity problem, thereby decreasing the surface accuracy. The proposed ISSD technique can effectively estimate the complex surfaces by utilizing the iterative space-variant sphere model to provide more accurate surface reconstruction results across all spatial frequencies for the deflectometry. Compared to the conventional methods, the proposed ISSD method has the following advantages:
- 1. High flexibility and high stability. There is only a physical point to be calibrated in advance to bound the solution space of the reconstructed surfaces and simplify the optimization complexity of the spherical model in the proposed ISSD technology. Benefiting from the simple optimization process of only one parameter, the proposed ISSD method can provide a more precise surface model to reduce the positioning errors of the traditional method, and releases the calibration challenge.
- 2. Enhancement of measurement scope. The height-slope ambiguity issue is a vital limiting factor to the measurement accuracy of the phase measuring deflectometry, especially for unknown complex surfaces. Through the proposed ISSD method, the surface slope error caused by the unknown surface model problem can be solved to reconstruct accurately the surfaces. In addition, the measured surface model can be removed to enhance the measurement scope of the deflectometry for the cases including the unknown surface model and the inaccurate surface model.
The proposed ISSD method aims to reduce the reconstruction error caused by the inaccurate and unknown surface model of the measured complex surfaces. The form accuracy can be enhanced by approximately three times when compared to the measurement results utilizing the plane model. When the surface model is unknown or inaccurate, the enhancement of the measurement accuracy is achieved solely by using the new data processing method, without demanding extra equipment or calibration techniques. The proposed ISSD method can produce a higher accurate result by using purely the intrinsic property of deflectometry that is hidden in measurement data. This highlights the importance of the ISSD approach taking greater advantage of the information in the measured deflectometric data. There is only a physical point to be measured to bound the solution space of the surface, thereby releasing the difficult of the calibration. As a result, this method is convenient to be applied and promising to the measurement of imprecise and unknown complex freeform surfaces.
Funding
Shanghai Sailing Program (20YF1454800, 22YF1454800); Natural Science Foundation of Shanghai (21ZR1472000, 51875107).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
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