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Abstract
A new alternative to calculate the null-screen for highly freeform or complex surfaces for any desired pattern to be observed on the detector is presented. To validate the proposed method, we used the Zernike polynomials to design complex surfaces with sagittas greater or equal to 40 mm, and peak to valley greater or equal to 30 mm, between the used surface and the best fit sphere. The freeform surfaces were fabricated using a 3D printer and a five-axis CNC machine. With the proposed method we can calculate the image that will be observed over the detector for any null-screen and any freeform surface that want to be analyzed. The results showed that the proposal works very well for extremely fast complex freeform surfaces (with slopes ≤ 80°), obtaining an error smaller than 0.66% in PV and 0.36% in rms in sagitta differences.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The use of freeform surfaces is becoming more common replacing spherical and aspherical surfaces, thus eliminating some optical elements or components for a given design as well as, weight and manufacturing time. Also, performance and efficiency have been improved making the freeform surfaces an attractive alternative to use in the scientific and technological areas. In astronomical instrumentation, the freeform surfaces are being used in the design of many instruments due to the advantages that they offer, for instance, increasing the degrees of freedom as compared to spherical and aspherical surfaces [1,2]; higher image quality [3,4]; flexible infrastructure [5]; among many others [6]. In the illumination field, the light can be redirected to cast the desired irradiation pattern on a target surface [7,8]; also, these complex surfaces are being implemented for uniform illumination [9,10]; or for generating the required irradiance distribution [11,12]. In aircraft design, the desired aesthetic quality of a panel layout and surface smoothness are two parameters considered in the freeform surface design [13,14]. As can be seen, the use of freeform surfaces has been growing very fast lately and these are only a few examples of all the areas that have been implementing this kind of surfaces.
The fabrication and testing of complex or freeform surfaces imply great challenges and high costs [15]. In the optical workshop it is said, “you can fabricate only what you can measure.” So, having a reliable testing method that verifies that the fabricated surface meets the design parameters becomes a great task.
So, is not surprising that currently, many research groups are working on the development of new measurement techniques for this kind of surfaces: stitching interferometry [16], point-cloud noncontact metrology [17], phase measuring deflectometry [18], among others [19,20]; all these techniques with precisions of nanometers in the analysis of surfaces with maximum sags of microns.
For the above, in this work, we propose a new alternative to design the null-screen for any freeform or complex surface that wants to be tested. The classic method to calculate the null-screen consists of calculating a set of curved lines or nearly elliptical spots on a cylinder, a plane, or a cone in such a way that by reflection on the test surface, the image consists of a perfect pattern [21–26]. Its main advantages are that the analysis of the entire surface can be obtained with a single image, this without a rotation or displacement system as is usually needed in another analysis methods; the test can be done in harsh environmental conditions; no controlled environment is needed, and additional optical components are not necessary.
We propose an exact, fast, and novelty solution to calculate the null-screen or the image that wants to be observed on the detector making the forward ray tracing of all the rays reflected to the null-screen or to the detector starting on the surface. This new option that we propose is a very simple way to obtain the positions for all the points in the null-screen or in the detector which becomes a very complex task if the ray tracing is done starting from the image that wants to be observed on the detector, as is usually done.
The design of the freeform surfaces was done by using the Zernike polynomials, and the complex surfaces were computed with sagittas greater or equal to 40 mm, and peak to valley greater or equal to 30 mm between the used surface and the best fit sphere. The complex surfaces were fabricated taking the technological advantages of 3D printers and computer numerical control machines (CNC). The results showed that the proposal works very well for extremely fast complex freeform surfaces.
The paper is organized in the next form, first we made a brief introduction about the uses and applications of the freeform surfaces. In Section 2, we show the mathematical for the null-screen calculation by using the reflection law and the equation of the straight. In Section 3, the design parameters, the fabrication method, and the experimental results for the different surfaces under test are shown. In Section 4, we show the evaluation of the freeform surfaces by using the algorithm shown in [24,25], and finally, the conclusions of this work are presented in Section 5.
2. Alternative method for the null-screen calculation
As is well known, the classical method for the null-screen design is done by making an inverse ray tracing [20–23], starting on the image that wants to be observed on the detector plane, as is seen in Fig. 1, where a is the distance from the CMOS to the pinhole; b is the distance from the pinhole to the vertex of the surface, and f is the equation of the surface under test.
While for calculating the image generated by the null-screen it is necessary to make an exact ray tracing starting on the null-screen to the surface under test (Fig. 2). For this, the equations given in [25] for calculating the positions where the rays coming from the null-screen intersect the surface by using the incenter properties are used, (Fig. 2).
Any of these two methods have an implicit difficulty which is solving a nonlinear system of equations to obtain the coordinates of the points P3(x3, y3, z3) where the rays intersect the surface (Fig. 1). The solution to this system of equations becomes more complex according to the surface that wants to be tested, it can start with something relatively simple as a conical surface to something more complex as a freeform surface (Fig. 3). These two methods have the disadvantages of high computing times and non-exact solutions.
A new alternative, that we propose for calculating both the null-screen and its image over the detector is the next: it consists of making the ray tracing starting from the coordinates of the points over the surface, P3(x3, y3, z3), to the detector and passing through the pinhole, P2(0, 0, a + b), Fig. 4(a), as x1 = x3η, y1 = y3η and z1 = z3η, where η = a/(f - b) is the parameter of the parametric equations of the straight line. Since we started from known points over the surface, we can calculate the coordinates of the points P4(x4,y4,z4) on the null-screen by using the reflection law as:
where the incident vector is defined as
Fig. 4. Plotting all the rays. a) All the rays reflected by the surface on the CMOS, and b) all the rays that intersect the null-screen, both images calculated with the proposed method.
The unit normal vector N at each point of the surface is given by
As is shown in the above equations, the coordinates over the null-screen P4(x4,y4,z4), Fig. 4(b), the detector P1(x1,y1,z1), Fig. 4(a), and the surface P3(x3,y3,z3) are all related between each.
A mesh of 3k x 3k over the entire surface was used where about 7M rays were generated and traced from the surface to the CMOS and to the cylindrical screen. Figure 4(a) shows an image of all the rays reflected by the surface over the detector, while Fig. 4(b) shows an image of all the points over the cylindrical screen.
With the proposed method we can calculate in an exact, fast, and easy way the null-screen for any complex surface that wants to be tested, e. g. freeform surfaces (Fig. 5). This can be done by selecting only the coordinates of the interest points that want to be observed and eliminating the non-interest points. The points elimination is done both, in the image to be observed on the detector (Fig. 6(a)), as in the calculated null-screen (Fig. 6(c)).
Fig. 6. Selecting only a few rays (black dots). a) Rays on the CMOS, b) theoretical image over the surface, and c) rays on the null-screen.
Due to the robustness of the proposed method, the selecting process of the interest points can be done inversely by selecting the points on the null-screen and eliminating all the non-interest points as shown in Fig. 6(c). Subsequently, we can calculate the image that will be observed on the detector since the coordinates of the points on the surface, the null-screen, and the detector are completely related in Fig. 6(a).
It is worthy to highlight that, with the proposed method we can obtain an exact solution to the calculation of the null-screen and the image that will be observed on the detector for any complex or freeform surface, in only one single step and without the needing of solving a nonlinear equations system with numerical solutions. In the next sections, we describe in detail the experimental validation of the proposed method.
3. Experimental results
Although freeform surfaces are taking a significant role in the scientific and technological areas, their fabrication is still being an enormous challenge. Nowadays, the design parameters of these kinds of surfaces only achieve values in sagitta below the 16 mm [27–29] because of the complexity of their fabrication. However, 3D printing [30–32] and computer numerical control (CNC) technologies are revolutionizing complex surface manufacturing so, taking the advantages of these technologies we decided to prove the functionality of our method by testing highly complex surfaces with sagitta greater or equal than 40mm fabricated with this technology.
For the design of the highly freeform surfaces, we used the Zernike polynomial with 15 coefficients as
In the next section, we show the parameters of the design and fabrication of the surfaces under test, the materials, and the equipment specifications used for the experimental testing.
3.1 Additive manufacturing surfaces
The first freeform surface tested was fabricated by using a 3D printer Makerbot Replicator+ which has a print technology of fused deposition modeling with a layer resolution of 100 microns; XY positioning precision of 11 microns; Z positioning precision of 2.5 microns; 1.8° step angle with 1/16 micro-stepping. In Table 1, the Zernike coefficients are shown and the design parameters for the first surface analyzed. The departure between the calculated surface sagitta and the best fit sphere (R = 25 mm) was 36.26 mm in PV, and rms of 11.17. As can be seen, these values are extremely high which proves that the surfaces under test are extremely complex.
Table 1. Data of the Zernike surface M1-3D
Figure 7(a) shows an image of the M1-3D freeform surface fabricated with additive manufacture using polylactic acid (PLA). The 3D model was generated by using the Zemax commercial software. Note that, the PV of the surface shown in these figures was 46.619 mm and 18.377 mm in rms, which are very high values compared with reported in the literature for these kinds of surfaces. It is well known that, depending on the 3D printer model, the final sample could have some material remains and layer lines typical of the fused deposition, so, a three steps surface postprocessing was done: cleaning, polishing, and reflective film deposition [30]. In our case, the polishing was hand-made by using water sandpaper 3500 grain, and the polishing time was 15 minutes approximately. To have a reflective surface that reflected the null-screen over the detector, different kinds of paints were tested after the polishing. The best result was obtained with four thin films of bright reflective finish paint.
Fig. 7. Surface M1-3D under test. a) Printed and post-processed surface, b) surface sagitta, and c) surface contour plot.
For the experimental setup we used one XYZ mount where the cylinder with the null-screen was fixed for a better centering; the cylinder was rectified in diameter and allows to the screen having better support; one Edmund Optics CMOS color camera calibrated for distortion with a sensor resolution of 4912 × 3684 pixels, and one 8-mm fixed focal length lens; due to the robustness of the test, special illumination is not necessary so for the system illumination the ambient lighting was used. The sensor was mounted in a kinematic optical mount with a tip/tilt precise motion control; the diameter of the diaphragm was the smallest possible for matching with the theoretical calculation of the null-screen, where the pinhole size is considered as one point. The dimensions of the null-screen were calculated based on the characteristics of the printer which only prints in letter size (215.4 × 279.4 mm) or legal-size (215.4 × 355.6 mm) so the perimeter and the high of the cylinder had to be within these dimensions, besides knowing that the radius of the cylinder does not affect the projected pattern due that the null-screen test works as a dark camera; also, the dimensions of the null-screen and the distance of some spots were measured and verified. The design parameters of the cylindrical null-screen are shown in Table 2.
Table 2. Design parameters of the null-screen for surface M1-3D.
In Fig. 8(a), the calculated image to be observed over the detector is shown, while Fig. 8(b) shows the designed null-screen, both images calculated with the proposed method. The null-screen was printed on a sheet of paper and placed in a transparent acrylic cylinder rectified on an industrial lathe to avoid screen deformations, and errors during the evaluation process, subsequently, the image reflected by the surface was captured with the detector. In Fig. 9, the experimental results obtained with surface M1-3D are shown.
Fig. 8. Experimental results for Surface M1-3D. a) Calculated image to be observed on the CMOS, and b) calculated null-screen, both calculated with the proposed method.
Fig. 9. Experimental results for surface M1-3D. a) Surface M1-3D without rotation, b) surface M1-3D with a rotation of 45 degrees, c) calculated image for surface M1-3D without rotation, and d) calculated image for surface M1-3D with an angle of rotation of 45 degrees.
In Figs. 7(a) and 7(b) can be observed that the spots in both images are not completely circular as those of the theoretical images of Fig. 8(a). Also, the images look like they were out of focus or look like they have low visibility, and some color in the center of the image looks as they have some chromatic aberration but, it is important to note that these results are due to a non-well postprocessing of the printed surface, e.g. a wrong polishing, a bad reflective painted, or even a combination of both. However, a great similarity can be observed between the experimental images of Figs. 9(a) and 9(b) compared with images calculated with our method, Figs. 7(d) and 9(c), respectively. Figure 9(b) shows a 45° rotated image for the same null-screen and freeform surface.
Due to the good results obtained with surface M1-3D, where we noted a great similarity between the experimental and the theoretical images proving the functionality of the proposed method qualitatively, the next step was to prove the method quantitatively. For having images with a better quality to analyze, we decided to fabricate the freeform surfaces with subtractive manufacturing using a CNC machine. Next, we describe the analysis done with these surfaces.
3.2 Subtractive manufacturing surfaces
The CNC machine used for the fabrication of the freeform surfaces was a 5 axis machine with linear tolerances of ±270µm in the x axis, ±130µm in the y axis, and ±50µm in the z axis; angular tolerance of ± 0.5°; radial tolerance of ±0.25 mm; and a final superficial finishing of Ra< 3.2 µm in aluminum 6061-T6. After the CNC machine finishes the fabrication process, a postprocessing to remove the cutter marks of the surface is needed; in our case, the polishing was done by hand until a mirror finishing is achieved. Next, we describe the design parameters of the two surfaces fabricated with the CNC machine.
3.2.1 Surface M1-CNC
The first freeform surface generated with the CNC machine was designed using the same parameters of surface M1-3D shown in the above section, (Table 1). The Zernike coefficients used for this surface were Z5 y Z7. Figure 10 shows an image of this CNC generated surface.
After the freeform surfaces were fabricated with the CNC machine and to verify that the surfaces were well-generated, a qualitative analysis was first done. Taking advantage that with the proposed method we can calculate any image that wants to be observed on the detector for any highly freeform surface, we made the qualitative analysis calculating the null-screen and the image to be observed on the detector for the logo of the Universidad Nacional Autónoma de México (UNAM), Fig. 11(a).
Fig. 11. Qualitative evaluation of surface M1-CNC. a) Theoretical image to be observed over the detector, and b) null-screen, both images calculated with the proposed method.
For this test, we calculated the theoretical images to be observed over the detector for different decentering of the cylinder by using the null-screen of Fig. 11(b). The displacements were +2 mm (Fig. 12(a)) and -2 mm (Fig. 12(b)), both in the z axis. The decentering were realized starting on the positions (x0=y0=z0=0) for the freeform surface, and (x'0=y'0=z'0=0) for the cylinder, both centered, (Fig. 12(c)).
Fig. 12. Different cylinder decentering of surface M1-CNC in the z axis. Experimental image, a) decentering of +2 mm; b) decentering of -2 mm; c) cylinder without decentering; theoretical image, d) decentering of +2 mm; e) cylinder with decentering of -2 mm; f) cylinder without decentering.
As can be observed in the experimental images of Fig. 12, we have shown that with our method we are able of calculating the images that will be observed on the detector no matter the alignment conditions are, even the cylinder, the surface under test, or both are decentered; this can be done for any image that wants to be observed on the detector for any highly freeform surface under test. In this same Fig. 12, we can observe a dark zone that corresponds to the reflex of the camera image where there are no points for analyzing; the area of this zone depends on the cylinder size (the higher is the cylinder, the lower is the dark zone), the surface form (the greater is the change in the slope, the smaller is the dark zone), and the image that wants to be observed on the CMOS. However, theoretically, more sampling points can be had over the center of the CMOS but the theoretical images were simulated with this dark zone for having images more similar to the experimental ones.
The experimental and theoretical images obtained for the null-screen of Fig. 8(b) are shown in Fig. 13 where different decentering for both, the null-screen and the surface under test can be observed. The decentering were the next: displacement of 5.7 mm in the z axis (Fig. 13(a)); 30° counterclockwise rotation (Fig. 13(b)); three cylinder decentering at the same time: 120° clockwise rotation, 0.5 mm displacement in the z axis, and 1 mm displacement in the y axis (Fig. 13(c)). All the decentering were realized starting on the positions where both the cylinder and the surface are centered, (Fig. 13(d)).
Fig. 13. Theoretical and experimental images for surface M1-CNC. a) Displacement of 5.7 mm in the z axis, b) 30° counterclockwise rotation, c) 120° clockwise rotation; 0.5 mm displacement in the z axis; and 1 mm displacement in the y axis, and d) centered surface.
Due to the complexity of the freeform surfaces, the alignment and centering of the experimental setup turn in a difficult task even more if this process is done by observing the experimental image over the detector, this for matching both images the experimental with the theoretical one. These small errors in the alignment can be observed in the small differences that exist between the experimental and theoretical images of Fig. 13, however, the great similarity between them results undeniable proving the functionality of our method. It is important to note that, our method can retrieve the decentering of any experimental image always the centroids of the spots can be calculated. This analysis was also done with a second freeform surface fabricated with subtractive manufacturing, the design and fabrication parameters for this surface are shown next.
3.2.2 Surface M2-CNC
To give a bigger challenge to our method and for proving its scopes and functionality, a second more complex freeform surface was designed, fabricated, and tested, (Fig. 14). The Zernike coefficients for this surface were Z5, Z8, Z9, Z10, Z11, Z13 Y Z14, their values are shown in Table 3. The design parameters for this surface were: diameter 50 mm; PV 50.767 mm; and 19.048 rms.
Table 3. Data of the Zernike coefficients of surface M2-CNC
In Fig. 15, the null-screen calculated for surface M2-CNC are shown. The first one with dots (Fig. 8(a)), and the second one with the UNAM logo, (Fig. 11(a)).
It is worth noting that two greatest challenges had to be considered in the calculation of the null-screen: one due to the complexity of the freeform surface and the second one the complexity in the calculation of the null-screen or image to be observed on the detector but due to the robustness of the proposed method, the results shown its functionality no matter what complex be the image or the freeform surface that wants to be tested, see Fig. 16.
Fig. 16. Theoretical and experimental images of surface M2-CNC. a) Grid pattern, experimental image; b) UNAM logo, experimental image; c) grid pattern, theoretical image; and d) UNAM logo, theoretical image.
The small differences between the experimental and theoretical images of Fig. 16 could be due to small errors in the alignment in the experimental setup or differences between the designed and the fabricated surface. In the next section is described in detail the quantitative analysis done to obtain these differences.
4. Surface evaluation
Having the proper analysis software (e.g. correlation, image processing, pattern recognition, etc.), the evaluation of the experimental images could be done for any image that wants to be observed on the detector for example those of the UNAM logo, Fig. 10. In our case, the evaluation is done by using the coordinates of the centroids in the experimental image, is for this reason that, the quantitative analysis was done with null-screens generated with spots, Fig. 15(a), and the algorithm we used was a variant of the algorithm described in [24]; the only image processing was a binarization of the captured image with the main aim of finding the centroids of each spot, where our method finds the centroid as the center of mass of the spot compared with other methods where the centroid is calculated taking into account the intensity of the image. Next, we show the results for each surface.
4.1 Surface M1-CNC
To simplify the evaluation of the centroids in the experimental images, a binarization of the experimental images must be done. The results obtained after the binarization and the evaluation process of the experimental image of Fig. 13(d) are shown in Table 4 where a comparison between the design coefficients and the coefficients recovered with our algorithm can be seen.
Table 4. Recovered values of the Zernike surface M1-CNC.
Table 5 shows the decentering coefficients (x0, y0, z0) and (x'0, y'0, z'0) of the surface M1-CNC and the null screen, respectively, both recovered with our algorithm.
Table 5. Decentering coefficients recovered with our algorithm (mm) for surface M1-CNC.
As can be seen, our algorithm can distinguish between the decentering coefficients due that, as is well known, the coefficients of tilt-tip Z2 and Z3 in Table 4 are the surface rotations over the x and y axis; while the decentering coefficients xo and yo in Table 5 are the surface translations over the x and y axis. On the other hand, the coefficients Z1 and z0 are the translations over the z axis which can mix between them but, these coefficients are associated with mechanical errors that do not affect the analysis for recovering the surface shape.
Figure 17(a) shows the contour plots of the theoretical surface while Fig. 17(b) shows the contour plots of the surface recovered with our algorithm. The differences in sagitta between the theoretical and the recovered surface are shown in Fig. 17(c). These differences were 0.3089 mm in PV, and 0.1675 mm in rms, corresponding to 0.66% and 0.36%, respectively.
Fig. 17. Contour plots of surface M1-CNC. a) Theoretical surface; b) recovered surface; and c) differences between the theoretical and the recovered surface.
As can be seen in Fig. 17, the contour plots of the theoretical and the recovered surface are very similar between them with a rms of 0.1675 mm proving the functionality of the proposed method, we associate the rms value with the error of the mechanical fabrication of the CNC machine which has an error in the range of hundreds of microns.
4.2 Surface M2-CNC
For evaluating surface M2-CNC we also did a binarization process, the image used for the analysis was the image shown in Fig. 16(a). In Table 6 a comparison between the design coefficients and the coefficients recovered by our algorithm is shown.
Table 6. Recovered values of the Zernike surface M2-CNC.
Table 7 shows the decentering coefficients (x0, y0, z0) and (x'0, y'0, z'0) of the surface M2-CNC and the null screen, respectively, both recovered with our algorithm.
Table 7. Decentering coefficients recovered by our algorithm (mm) for surface M2-CNC.
Figure 18(a) shows the contour plots of the theoretical surface while Fig. 18(b) shows the contour plots of the surface recovered with our algorithm. The differences in sagitta between the theoretical and the recovered surface are shown in Fig. 18(c). These differences were 0.5705 mm in PV, and 0.1819mm in rms, corresponding to 1.12% and 0.04%, respectively.
Fig. 18. Contour plots of surface M2-CNC. a) Theoretical surface; b) recovered surface, and c) differences between the theoretical and the recovered surface.
With these results, we have demonstrated qualitatively and quantitatively that we can calculate the null-screen for any image that wants to be observed on the detector no matter its complexity or the complexity of the freeform surface that is being tested. The proposed method is an exact, elegant, and novelty solution to design the null-screen or the image that wants to be observed on the detector for any convex/concave highly freeform surface that with traditional methods cannot be tested. We observed that the method is very sensitive to misalignments and positioning which increase when highly freeform surfaces are being tested, however, these decentering can be easily retrieved with our algorithm so, a robust system for alignment can be had. Our experimental investigations were limited to the case of Zernikés surfaces but, the equations shown here have the versatility to be used with any polynomial function that describes convex/concave complex or highly freeform surfaces. However, an extensive and detailed analysis would be interesting for analyzing point to point how drastic changes in the slope affect the method because, for these kinds of surfaces, the null-screen could have overlapping spots that complicate the analysis [33,34], and one point over the null-screen could be reflected by different zones of the surface and to produce two or more images over the CMOS. Besides all these, the data recovered with our algorithm which can be seen as a global fit of the surface under test can be used for making the integration process with the traditional method for reducing errors and enhancing the precision of it.
5. Conclusions
In this paper, we presented an exact, fast, and novelty alternative to design the null-screen of any image that wants to be observed on the CMOS for testing highly freeform surfaces with great and very fast changes in the slope. We showed that, with this technique, is possible to evaluate, in a simple manner, the shape of highly freeform surfaces with large deformations, with PV values greater than 46.619 mm, and values greater than 18.377 mm in rms. It is important to note that the PV and rms values were obtained by comparing our results against the design parameters, so these errors could be lower if we compare our results against another analysis method but, at this moment this kind of surfaces with sagitta greater than 6 mm have not been reported in the literature. Another advantage of this work is that with this alternative method we can obtain the null-screen and the image that will be observed on the detector without the needing of solving nonlinear equations systems with numerical solutions so, the computing time is greatly reduced.
We recovered with a simple process the Zernike coefficients to make a direct analysis of highly freeform surfaces, this without the needing of polynomial fits and reducing the errors added with these issues. With our recovering algorithm it is possible to calculate the decentering of the null-screen and the surface under test reason why we can enhance the uses of our algorithm, where besides obtaining the surface shape, a robust system for alignment can be had.
Funding
Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (IT102520, TA100319, TA100519).
Acknowledgments
The authors of this paper are indebted to Osvaldo Hernández Martínez for his help in manufacturing the used surfaces.
Disclosures
The authors declare no conflicts of interest.
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