Freeform optical surfaces are nonrotationally symmetric surfaces that provide more degrees of freedom to the designer to minimize optical aberrations (i.e. astigmatism, coma, and distortion at multiple locations in the optical system) and make the system more compact by reducing size, weight, and volume , , . The freeform surfaces are typically represented by extended polynomials , ɸ-polynomial , gradient-orthogonal Q-polynomials , nonuniform rational basis spline , radial basis functions , and Zernike polynomials . These optical elements are found in nonimaging and imaging applications such as illuminations industries , , aerospace systems , biomedical engineering , , green energy , laser beam shaping , augmented reality , , and head-mounted display systems , . Along with these advantages, there are numerous challenges involved in the design, manufacturing, and metrology of freeform surfaces. The present CNC-based ultra-precision manufacturing technologies are capable to fabricate such complex surfaces , , . However, the performance of the manufacturing techniques are limited by the available metrology capabilities and a suitable feedback mechanism for an effective manufacturing process. To push the accuracy of the fabricated freeform surfaces, a measurement technique needs to be developed, which can measure the fabricated profile precisely on the machining platform, as an in situ measurement solution.
Coordinate measuring machine  can measure complex shapes, but the low-resolution data points can provide only the global shape of the freeform surface. A deflectometric technique  involves more computation to measure a freeform surface and also offers low-resolution measurement. Mechanical profiler , such as Form Talysurf, provides high-resolution measurement in a single scan. To record 3D shapes, an array of linear scans are required to be performed. Being contact in nature, such multiple linear scans destroy the surface. Optical profilers  are noncontact in nature and provide 3D shapes but have the slope limitations in single measurements. It is also a time-consuming process to scan the entire surface.
Interferometric techniques are typically used for rotationally symmetric surfaces, but by incorporating a null element in the setup it has been used for the measurement of freeform shapes also , , . These null elements are surface specific and expensive to manufacture. QED Technologies (USA) developed non-null-based interferometric systems based on stitching principles with variable optical null elements to test mild aspheres; further, it has been extended to measure the shallow freeform optics . Due to the complex setup and the requirement of vibration-insensitive platform, interferometric techniques, null or nonnull, cannot be used for in situ measurement. Tilted wave interferometry  has demonstrated the measurement capability of freeform optics. However, it is an offline measurement solution with limitations on the maximum slope that can be measured.
MarForm MFU200 (Mahr GmbH, Germany)  and LuphoScan (Taylor Hobson, UK)  are point-based sensors where the optical probe can be placed normal to the surface by means of two linear and one rotary stages. The sensor can measure rotationally symmetric surfaces with high accuracy. In principle, they can also be used for the measurement of freeform shapes, but the measurement uncertainties are larger for the measurement of high slope freeforms. Recently, a noncontact point cloud metrology technique based on swept source optical coherence tomography is developed for the measurement of freeform optics. It is coupled with an interferometric system having multiple scanning stages. It is demonstrated to measure the freeform surfaces in 1-inch diameter class and up to 10° in slope .
All the methods and the instruments discussed above measure the optical elements in offline mode. Due to the nonrotational symmetric nature of the freeform profiles, the offline measurement requires precise mounting and demounting of the surface part, failing that causes indexing errors. Shack-Hartmann sensors (SHS) have the potential to be integrated into the machining platforms due to its low vibration sensitivity, compactness, and simple principle of operation for the measurement of form errors. A typical SHS can measure large slope, of the order of 25°, depending on the diameter, focal length, and the number of lenslets in the array. We reported a scanning subaperture stitching scheme for measurement of freeform wavefronts using SHS as slope measurement sensor , , . The scheme has been used for the measurement of freeform surface in transmission mode. However, to be able to measure the profile of the freeform surface in situ and to provide the metrology feedback to the machining platform for corrective machining cycles, a measurement in reflection mode is required. We have recently described a scheme for the measurement of freeform optics in reflection mode as shown in Figure 1 . The collimated beam after reflection from a beam splitter interacts with the surface under test. The reflected wavefront from the freeform surface is diverging in nature, and it carries the information about the surface shape that is measured at the detector plane using a scanning SHS. The lateral extent of the wavefront is larger in size than the detector size of the SHS head. Hence, multiple subapertures are required to be measured by scanning the SHS along the X and Y directions to cover the entire wavefront. Finally, all the subapertures are stitched together using a stitching algorithm developed in-house , . Our first implementation of the scheme has been successfully demonstrated within the limits of the manual positioning of the SHS during scanning. Simulation studies to understand the effects of misalignments on the measurement accuracy have been presented.
The scheme has limitations on the maximum slope that can be measured and we could only measure a small area (~3 mm) of a freeform with shallow slope. The dynamic range of the SHS limits the maximum slope that could be measured in a subaperture. In this paper, we present an improved measurement scheme that can measure steep freeform surfaces with large aperture size. The size that can be measured is basically limited by the scanning stages used during the measurement. It can easily be incorporated in the machine head using its computer-controlled translation, tip-tilt, and rotational axes. The scheme is better suited for efficient correction of the tool path to compensate the form error. The capability of the scheme has been demonstrated by measuring the form error of a steep freeform surface. The measurements have been further used as a feedback for the correction of the tool path for form error compensation.
2 Measurement setup
Instead of placing the scanning SHS normal to the freeform wavefront, as described in Ref.  and shown in Figure 1, we chose a setup shown in Figure 2, where the SHS is placed normal to each subaperture of the wavefront. The SHS is mounted on six-axes precise computer-controlled scanning stages (tip, tilt, rotational, and three translational), which make it capable to scan the wavefront in a manner shown in Figure 2. The advantage of this configuration is that the maximum slope to be measured in each subaperture is reduced. This in turn enhances the capability to measure steep surfaces as it becomes possible to place the SHS head normal to each subaperture by following the geometry of the freeform surface. The technique offers the flexibility to measure large freeform surface/wavefront areas. Further, this configuration can be used as in situ measurement of freeform surface by integrating it into machining platform.
2.1 Alignment and calibration process
The calibration process of the setup is performed by measuring a reference optical flat before performing the measurement of freeform optics. Before the measurement, a prealignment of the SHS sensor is done to adjust its orientation to ensure the orthogonality of the SHS with respect to the incident collimated beam. Larger tilt in the incident beam needs to be avoided, which otherwise does not provide good measurement conditions and causes higher measurement uncertainty. The process uses a prealigned reference flat in the reflection mode with an alignment ring. The alignment ring is a circular plate with a central hole of diameter 0.5 mm mounted in front of the SHS. This allows only a small portion of the light beam to enter the SHS assembly and forms small spot patterns on the detector. The spot patterns are adjusted to fall inside a software-generated circular pattern to ensure the minimal tilt as shown in Figure 3. Once this prealignment is done, the reference optical flat is replaced by the freeform optics under test. As freeform surface has nonrotational symmetric nature, the alignment of both the surface and the SHS is a challenging task. Therefore, an iterative alignment process is developed for the SHS suitable for the measurement of freeform surface. In this process, SHS is aligned to the center portion of the freeform surface until the residual error is as minimum as possible compared to its geometrical surface part. Further, SHS is aligned to the two extreme corner subapertures of the surface. Thereafter, it is assumed that the SHS will remain aligned normal to the each subaperture during measurement.
2.2 Freeform wavefront reconstruction scheme
The measurement of freeform wavefront is performed by scanning the entire surface to measure all the subapertures with consistent overlapping area. Using in-house stitching software, the misalignments are corrected and subapertures are stitched to give the phase profile of the entire freeform wavefront. The size of the beam splitter is an important parameter that is required to be prudently selected based on both the divergence of the reflected beam and the slope of the freeform surface. In the case of a small beam splitter size and a steep diverging reflected wavefront, the beam splitter will itself be a limiting aperture and restricts the size of the freeform surface to be measured . A beam splitter of diameter 12.7 mm has been used in the experiment. It is assumed that any optical aberration induced due to the insertion of the beam splitter is very less and does not affect the measurement.
A freeform surface of phase profile shown in Figure 4A is chosen as specimen under test. It is described by a seventh-order polynomial. It is made up of polymethyl methacrylate (PMMA) with the physical dimensions of 14×14 mm and the maximum sag of 1.119 mm. It can be represented by the sag equation as
Here, the first term represents the conic section, where c is the curvature, x and y are the Cartesian coordinates, and k is the conic constant. The second term represents the Nth order polynomial with x and y sample points. Ai are the polynomial coefficients. In this study, the first term of the sag equations is taken as zero and the second term is the seventh-order polynomial. The vertex point of the phase profile is shifted by 1 mm and 1.3 mm in the X and Y directions, respectively, from the geometrical center of the wavefront. In this study, an SHS from Imagine Optic (France) is used. The dynamic range of the SHS is 0.85° (15 mrad) per lenslet. The detailed specifications of both the test specimen and the SHS are given in Table 1.
Freeform surface and SHS specifications.
|Surface under test||Scanning wavefront senor|
|Surface type||Cubic phase plate||Microlens aperture size||114×114 μm2|
|Material||PMMA||Focal length||3.8 mm|
|Profile||Seventh order polynomial||Microlens array size||4.56×3.56 mm2|
|Maximum sag||1.11 mm over 14×14 mm2||Total number of lenses in the array||40×32|
|0.376 mm over 6.27×6.27 mm2|
|Maximum slopes||Surface slope over 14×14 mm2||Pixel size of CCD||6.7×6.7 μm2|
|X slope: 0.80° (14.09 mrad)|
|Y slope: 0.68° (11.88 mrad)|
|Surface slope over 6.27×6.27 mm2||Dynamic range of SHS||0.85° (15 mrad)|
|X slope: 0.64° (11.17 mrad)|
|Y slope: 0.38° (6.63 mrad)|
The scanning SHS head needs to follow the profile of the freeform wavefront as shown in Figure 2. The position coordinates and tilts in X and Y directions for each subaperture need to be calculated first. Table 2 presents the required position coordinates and tilts of the SHS for each subaperture for the freeform surface under test. The wavefront is divided into 5×5, i.e. 25 subapertures.
SHS metrology head position coordinates and tilts required for measurement of each subaperture.
|Subaperture||Position (mm)||Angle (tilt, °)||Subaperture||Position (mm)||Angle (tilt, °)|
Simulation studies are first performed to estimate the reconstruction error of freeform wavefront from its calculated slope data before conducting the experiment. The wavefront at 34 mm distance from the vertex of the freeform is measured. It is due to the minimum possible physical distance between the freeform surface and the lenslet array of SHS head in the actual experimental setup. First, the full wavefront at 34 mm distance is calculated through ray trace. The full wavefront is of the size 11.27×11.27 mm2, which represents 6.27×6.27 mm2 actual area on the surface part. The peak-to-valley (PV) of the freeform wavefront at the propagated distance is 989.8 waves (1 wave=632.8 nm) as shown in Figure 4B. The maximum slope of the measured wavefront along X and Y directions is 0.98° (17.24 mrad) and 0.68° (12.03 mrad), respectively. It is the wavefront that needs to be measured using subaperture stitching scheme.
The reconstruction of the freeform wavefront has been performed in an ideal situation, i.e. no misalignments between adjacent subapertures using RAYTRACE software . The full measured wavefront is divided into 5×5 subapertures as shown in Figure 5A and B. The size of the individual subapertures is 2.99×2.99 mm2 and the overlapping area of 30% has been maintained between the adjacent subapertures. The scanning SHS head follows the profile of the freeform surface, as shown in Figure 2, with prior calculated position coordinates and tilts in X and Y directions for each subaperture as shown in Table 2.
The scanned subaperture slope data have been reconstructed based on a weighted cubic spline integration method  and then stitched using in-house developed subaperture stitching algorithm . A subaperture stitching algorithm minimizes the misalignments between adjacent subapertures based on the least-squares fitting method. The misalignments in the vertical plane, i.e. piston, tilt, and power, are minimized by fitting them in the overlapping zone. An iterative alignment is used to minimize the radial and rotational misalignment errors during registration in global frame before the stitching process as mentioned in Section 2.1. Once the misalignments are minimized, a data averaging method is used to connect the phase profile of the registered subapertures to obtain a full wavefront map and it has 989.79 waves of PV (Figure 5C). It is compared to its nominal profile shown in Figure 4B. Figure 5D is the residual wavefront map with PV of 0.005 waves and RMS of 0.00044 waves without any misalignment. The negligible error in the residual wavefront map shows the accuracy of the stitching process when there are no misalignments/registration error in the subapertures.
3 Experimental results
3.1 SHS head-based metrology for freeform wavefront
The freeform profile is fabricated in PMMA using a five-axes ultraprecise machining platform (Nanoform 200) in slow tool servo configuration. Figure 6 shows the experimental setup that has been developed for the measurement of freeform surface/wavefront. The measurement of the fabricated profile is performed as per the procedure described in Section 2. An He-Ne laser source (λ=632.8 nm) is used in the experiment. The area of the surface (6.27×6.27 mm2) keeping vertex point at the center has been illuminated and the divergent wavefront at a distance of 34 mm (i.e. at the detector plane) is measured. The measurements are taken as per the scheme in Section 2.2.
All the subapertures are aligned individually relative to their nominal positions (as calculated and presented in Table 2) before taking the measurements. Figure 7A shows the 25 experimentally measured subapertures. The scanned subapertures are stitched to obtain a full wavefront map. Figure 7B shows the wavefront map that exhibits a PV deviation of 986.54 waves. A point-to-point comparison has been done with the ideal freeform wavefront propagated to a distance of 34 mm (Figure 4B). The residual wavefront error is 18.4 waves of PV as shown is Figure 8.
3.2 Modeling of freeform surface profile from the measured wavefront
The experimentally stitched wavefront (Figure 7B) is back-propagated by performing the reverse raytracing as shown in Figure 9A. The following are the steps for the calculation of the surface profile under test from the experimentally measured wavefront:
- An optical surface is simulated from the measured wavefront and placed at the same position where the wavefront is measured.
- A grid of collimated rays perpendicularly emitting from the simulated surface are traced back to the curved profile of the ideal freeform element using the RAYTRACE software . The parallel rays coming from the left side are just auxiliary rays due to the software implementation and determine the starting points on the simulated freeform surface where the rays are now perpendicular to the simulated freeform surface and have all phase zero.
- The wavefront is calculated at the curved profile of the ideal freeform surface under test placed at the same distance as in the actual experiment. At this distance, the back-propagated freeform wavefront represents the surface part that has been measured.
- The simulated surface from the back-propagated freeform wavefront is then compared to the ideal geometrical surface. The residual profile error (shape errors) is obtained by taking a point-to-point difference.
The derived shape error on the freeform surface of PV deviation of 9.59 waves is shown in Figure 9B and C.
3.3 Feedback mechanism for corrective machining
The residual surface error map (Figure 9B) is used to compensate the tool path in the corrective CNC machining process. The data of residual surface error map are fed into the machine tool path using the format compatible to the machining platform, i.e. the polar coordinate system. The error map has been rearranged into the polar coordinate system to modify the tool path in a point-by-point configuration. The modified tool path is used for the corrective machining to reduce the profile error. The machined surface has been measured and the residual surface error after tool path compensation is of 3.60 waves of PV as shown in Figure 10. A significant improvement is achieved using feedback from the SHS metrology method. The above tool path compensation process proved the capability of SHS-based method for in situ metrology and feedback for form error correction. The only drawback is the limited number of data points and the resolution offered by the SHS (Table 1). However, when used in situ, it can provide fast convergence toward achieving the desired shape through error feedback mechanism.
3.4 Repeatability and reproducibility tests
Repeatability and reproducibility tests are conducted to validate the performance of the developed scheme. Repeatability is the difference between the two consecutive measurements without any change in the system. It yields the facts about the sensitivity of the experimental setup, such as vibration, temperature, and other sensor noise. Twenty such repeatability measurements at the time interval of 30 s each are performed. Figure 11A shows the average of the 20 repeatability results of subaperture measurements. The PV is 0.012 waves and the RMS is 0.0018 waves.
As the measurement scheme requires movements of SHS to scan the complete wavefront, it is important to understand the effects of positioning errors during the scanning process. It has been captured by performing reproducibility test.The measurement of the entire wavefront shown in Section 3.1 (Figure 7B) has been repeated five times. The average reproducibility of the measurements is shown in Figure 11B. The PV is 0.95 waves and the RMS is 0.19 waves. It is also observed that the reproducibility depends on the slope of the wavefront to be measured. The lesser the slope to be measured, the better is the achievable reproducibility.
4 Quantitative analysis: simulations
To understand the uncertainties involved in the measurement scheme, a quantitative analysis has been carried out. The influence of misalignment and the overlapping area of subapertures on stitching accuracy have been investigated by performing simulations.
4.1 Influence of lateral and rotational misalignments on stitching accuracy
In practical situation, the lateral (i.e. translational and rotational errors) and vertical (i.e. piston, tip-tilt, and defocus) misalignments may present due to the scanning stages of the SHS. These misalignments of the subapertures will limit the stitching accuracy. The vertical misalignments are corrected using the least-squares fitting algorithm. The lateral misalignments correction is, however, more challenging to address for achieving better stitching accuracy. To estimate the influence of the lateral misalignments, a study has been conducted by introducing various misalignments in the 5×5 subapertures array. The misaligned subapertures have been stitched and compared to the ideal wavefront. The resulted residual error represents the stitching accuracy in this investigation.
Table 3 presents the residual error when the subapertures have been misaligned by 10 μm in X and Y directions and rotated by 0.5°. The translational and rotational stages used in the experiment have the resolution of 1 μm and the 0.017° (1 arc min), respectively. The reconstructed stitching accuracy has been estimated by introducing the misalignments within resolution of the stages, and Table 4 presents the stitching error budget. It is observed that the rotational misalignments (case 4 in Table 4) are more critical than the translational misalignments. Case 6 represents a simulated situation when all the translation stages have 1 μm misalignment error and all rotational stages have 0.017°. The observed stitching error is 0.64 waves. This represents the estimated highest stitching error with the used scanning stages assuming that the positioning accuracy is within 1 μm. However, any error in the straightness of the mechanical mount/assembly may lead to larger positioning inaccuracies and needs special attention during mounting.
Stitching error budget analysis before alignment of the freeform surface.
|Case||Translations error (μm)||Angular rotational error (°)||Angular tilt error (°)||Stitching error (residual) (waves)|
Stitching error budget analysis after alignment of the freeform surface.
|Case||Translations error (μm)||Angular rotational error (°)||Angular tilt error (°)||Stitching error (residual) (waves)|
To know the effects of internal sensor error on the measurement as well as stitching process, a collimated wavefront of diameter 6.27×6.27 mm2 has been measured using the scanning stitching scheme. The total of nine subapertures of the same size (i.e. 2.99×2.99 mm2) are measured and stitched together. The overall wavefront deviation is 0.15 waves. This shows that the systematic error of the scanning SHS on the measurement is quite small.
4.2 Influence of overlapping area on stitching accuracy
A consistent overlapping area is required between adjacent subapertures to minimize the effects of misalignments and provides an optimal performance of stitching process. A smaller overlapping area causes larger stitching errors, whereas a larger overlapping area results more number of subapertures and leads to longer computational/measurement time. To know the influence of the overlapping area on the stitching accuracy quantitatively, a study has been conducted by varying the overlapping area between the adjacent subapertures from 10% to 50%. The subapertures have been misaligned by the amount within the resolution of the scanning stages used as discussed in Section 4.1. The subapertures with the individual size of 2.99×2.99 mm2 (i.e. 28×28 sample points) are simulated for different overlapping areas and stitched to obtain a full wavefront map. The reconstructed stitched wavefront map is compared to the respective ideal wavefront maps. The residual errors are gradually decreasing with the increase of the overlapping area as shown in Figure 12. It is evident that the overlapping area more than 30% does not significantly increase the stitching accuracy further.
5 Validation of the SHS-based metrology using coherence correlation interferometric optical profiler
Due to the nonavailability of a standard metrology technique for freeform optics until date, the validation of the developed SHS metrology method is quite challenging. For the validation purpose, a coherence correlation interferometric optical profiler has been chosen for the metrology of freeform optics. However, the optical profilometric techniques have limitations on the maximum sag, which can be measured. The freeform surface measured in the previous section is having steep sag and hence could not be measured using the available optical profiler (Talysurf CCI 6000, Taylor Hobson make). Therefore, a freeform surface with low slopes has been fabricated and tested by the coherence correlation interferometric optical profiler and the developed scanning SHS technique. The profile of the surface has been scaled down where the slopes along X and Y directions are 0.063° (1.1 mrad) and 0.08° (1.4 mrad), respectively, and the sag of 0.111 mm over 14×14 mm2 area. The phase profile of the measured full freeform wavefront at a propagated distance of 34 mm has the maximum slopes along X and Y directions are 0.118° (2.05 mrad) and 0.069° (1.20 mrad), respectively.
The freeform wavefront has been measured at a distance of 34 mm by taking 3×3 subapertures over the area of 7.752×7.752 mm2. The actual area represented on the surface is 6.8×6.8 mm2. The size of the individual subapertures is 3.192×3.192 mm2 with the overlapping area of 28% between the adjacent subapertures. The experimentally stitched wavefront has been back-propagated at the surface and compared to the nominal profile. The residual error is 7.8 waves of PV as shown in Figure 13. The same surface has been measured using optical profiler (Talysurf CCI 6000). Figure 14 shows the surface profile map of the freeform optics measured by CCI. The PV of the surface error is 7.92 waves.
Both measurement techniques, SHS and optical profiler, are performed in reflection mode. Although both measurement schemes and measurement principles are different, the results are in good agreement. One can observe that the global error shape is quite similar in both measurements. This comparative study validates the metrology scheme developed under this research work.
In this paper, we introduce a subaperture measurement technique for freeform optics using a scanning SHS head in reflection mode. The fabrication of freeform surface is done by slow tool servo configuration of ultra-precision machining. The wavefront reflected from the freeform surface under test has been measured by capturing subapertures while moving the scanning SHS head normal to the freeform wavefront. The measured wavefront has been back-propagated to obtain the profile error. The residual profile error map is used as an input for the tool path compensation for form error correction. The measured profile error on the component after tool path correction has been reduced from the PV deviation of 9.59 to 3.60 waves. A quantitative analysis also is presented on the effects of misalignments and overlapping area of subapertures on stitching accuracy. The SHS has the potential to be incorporated into the machining platform during the freeform surface fabrication process; in addition, the feedback mechanism for the correction of the tool path demonstrated in this study can improve the surface profile accuracy efficiently. Further, the scheme is validated using coherence correlation interferometric optical profiler and the comparisons of the results are drawn.
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