Most aspheric surfaces have been tested by interferometer with some null correctors. This approach, however, often fails when there are many aspherical terms or test surface is very steep because it is not easy to design the conventional null lens or CGH (Computer Generated Hologram). On the other hand, 3-D profilometer can measure aspheric surfaces without any null correctors; however, it takes some time to measure, which makes it unsuitable for the production line in the factory. In this paper, we apply the Hartmann test to the measurement of steep convex aspheric surfaces of which diameter is about 16 mm. In order to increase the measurement accuracy, we calibrated the test setup using a CGH that simulates the ideal test surface. We demonstrated that the significant amount of error in the test setup could be removed by this calibration process. The test results showed only 2 nm rms WFE (wave front error) difference even though the WFE of test setup was worsened by more than 0.13 μm rms. Since this method makes it possible to measure highly aspheric surface quickly and accurately, it can be used in the production line.
© 2006 Optical Society of America
In the research described below, we developed a new method to measure a small convex aspheric surface with steep slopes. The key point is to calibrate the whole testing setup using some ideal test surface and set this data as a reference to be used in the measurement of the real target. We do not need to analyze those errors; instead, if we detect any deviations from the reference, we can construct the deformation of the test surface. The Hartmann sensor would be the best to detect deviations because its wave front sensing is based on the relative displacements of beam spots with respect to their reference. In this way, this method can provide a fast and accurate measurement for the aspheric surface and be used in the production line of the factory. However, one major problem in this concept is how to procure a highly accurate test surface. We propose that a CGH can be the best ideal test surface since we can design the CGH to reflect the light to the desired direction as an ideal test surface will do. The manufacturing accuracy is comparable to that of an e-beam machine of which line accuracy is on the order of sub micrometer.
shows a schematic testing setup for a small, convex aspheric surface. This setup is quite similar to that described by Yang et al
]. The interferometer generates a collimated beam. This beam passes through the beam expander and condensing lenses and illuminates the whole surface of test surface. After reflection, the test surface is imaged on both the Hartmann sensor and the interferometer. Before measuring the test surface, a CGH that simulates the ideal target is measured by Hartmann sensor and the positions of each beam spot are stored in the computer as a reference. Then, the test surface is mounted instead of a CGH and measured again by the Hartmann sensor. The deformation of surface is determined by displacements of beam spots with respect to a reference. Even though the measurement is carried out on the Hartmann sensor, the interferometer is essential not only because it supplies the light source to the Hartmann sensor, but also because we need to align a CGH with respect to the optical setup accurately.
Fig. 1. Optical layout for the measurement of small aspheric surface.
2. Testing method for an uncoated steep convex asphere
The test surface is a convex shape with a 16 mm diameter, a 14 mm radius of curvature, 0.09 of conic constant, and it has four additional aspheric terms. Figure 2
shows the asphericity of the test surface. The maximum asphericity is about 46 λ, which makes it impossible to test the surface using a PSI (phase shifting interferometer) due to sampling theory. There were several previous researches to test the fast aspheric convex surfaces. In order to test the parabolic f/0.22 surface with a 100 mm diameter, Diaz-Uribe et al
used a set of curved lines on a cylinder in such a way that the image is a square grid [10
10. R. Diaz-Uribe and M. Campos-Garcia, “Null-screen testing of fast convex aspheric surfaces,” Appl. Opt. 39, 2670–2677 (2000). [CrossRef]
]. Campos-Garcia et al
improved a similar method to measure the large secondary convex mirror [11
11. M. Campos-Garcia, R. Diaz-Uribe, and F. Granados-Agustin, “Testing fast aspheric convex surfaces with a linear array of sources,” Appl. Opt. 43, 6255–6264 (2004). [CrossRef] [PubMed]
]. In this work, they used a linear array of sources instead of a cylindrical screen which is not suitable for the test of large optics. However, the accuracy of those methods is on the order of micrometer. It is necessary to use some null correctors to improve the measurement accuracy. In this situation, a CGH can be a good candidate for a null corrector. It is possible to design a CGH so that the incident light is normal to the test surface. However, the efficiency of a CGH can be varied, depending on the diffraction order in use. In order to test some steep surfaces, a 3rd
order diffraction beam from CGH is frequently used instead of 1st
order beam, where the efficiency is only about 6 %. This low efficiency, combining with low reflectivity of bare glass (usually less than 4 %), could be a serious problem in a Fizeau interferometric measurement due to the low fringe visibility.
Even though a CGH is not suitable for a null corrector in our case, we can use it as an ideal target in the test setup. We can design a CGH to reflect the light to the desired direction as an ideal target will do. By measuring a CGH with the Hartmann sensor, we can estimate all error sources in test setup like Fig.1
, including the manufacturing and alignment errors. The positions of each beam spot on the Hartmann sensor result from all kinds of test errors and can be used as a reference. This is the calibration process for the test setup. After finishing this calibration process, we measure the test surface in the same test setup. Then, we can reconstruct the deformation of the test surface by calculating the relative displacement of the spots with respect to the reference. The interferometer cannot be used for this purpose because the non-null test setup usually generates a large WFE that is beyond the dynamic range of the interferometer.
The Hartmann sensor we used in the experiment has 35×40 small apertures. Each aperture is 142 μm in diameter and is separated from its neighbor by 350 μm. The distance between the Hartmann mask and the CCD is 10.923 mm. The sensing area of CCD is 12.57 mm × 14.34 mm and its pixel size is 14 μm × 14 μm. According to the geometry of the Hartmann sensor, the theoretical maximum measurable wave front slope has the following relation [6
wherefis the distance between the apertures and the CCD, S is the aperture spacing, d is the diameter of the aperture, and λ. is 0.6328 μm. The maximum measurable wave front slope of the sensor is approximately 10.6 mrad.
Fig. 2. The asphericity of the surface. Wave means 633 nm.
To illuminate the test surface, we need to design a proper optical system which consists of beam expander and condensing lenses, which is shown in Fig. 1
. A beam from the interferometer is expanded to have a 100 mm diameter using a beam expander with magnification of 10. This beam illuminates the target through condensing lenses that consist of four single lenses. The theoretical WFE of the test setup is reduced to 16λ PV from 46λ PV of original asphericity, which is shown in Fig. 3
. In this figure, the WFE slope is a maximum 45 mrad, which is beyond the dynamic range of Hartmann sensor used in our experiment. This large residual WFE is not critical for the accurate measurement in our method, which is explained in detail in section 3.
Fig. 3. (a). Wave front error of the test setup and (b) its slope in radian.
3. Preliminary characterization
To check the feasibility of our method, we measured the WFE of high-quality doublet (λ/10 PV) of which the diameter is 8 mm. The light source is the high-quality converging beam from the ZYGO interferometer. The doublet was defocused by 20 mm and the Hartmann sensor measured the WFE. The analysis software calculated the WFE by modal methods [12
12. W. H. Southwell, “Wavefront estimation fromm wavefront slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980). [CrossRef]
]. The expected WFE was 39 μm PV, whereas the Hartmann sensor showed only about 24 μm PV. This large difference showed that the WFE of the doublet with more than 20 mm defocus exceeded the dynamic range of the Hartmann sensor used in our experiment. In this situation, this measurement was saved as a reference and successive measurements were carried out up to 20.8 mm of defocus with a 0.05 mm interval. This result was compared to that of the theoretical expectation. According to the simulation done by optical design software, the linearity of the WFE during 20.2 mm – 20.8 mm of defocus was maintained within 5 nm rms for a deviation of 1σ. Figure 4
shows the measurement result for the doublet lens and its residual error after the linear fitting. The linearity of the rms WFE is 4 nm (1σ), which is quite similar to that of the theoretical expectation. This means that even though the absolute WFE of the system exceeds the dynamic range of the Hartmann sensor, the reference made under this system can work well so that the relative WFE can be measured within a few nm rms.
Fig. 4. Measurement result of high precision doublet using a Hartmann sensor as the doublet is defocused. The reference was generated at defocus of 20 mm. (a) rms WFE measured (b) residual error after linear fitting.
As an ideal target, an on-axis and amplitude type CGH was prepared, which is shown in Fig. 5 (a)
. The CGH is a chrome-on-glass type in third-order reflection mode. According to the manufacturer, there are three kinds of error in the fabrication of a CGH; 0.01λ rms of substrate flatness error, 0.015 μm rms of origin of coordinates, and 0.04 μm rms of trajectory of rotation error. Among them, the combination of origin of coordinates and trajectory of rotation errors is about 0.043 μm rms. This error corresponds to 0.018λ rms of WFE, considering that the CGH pattern spacing is 3.03 μm rms. This means that the WFE induced by the fabrication error of a CGH is quite small compared to the measurement target accuracy (0.1λ rms) and a CGH can be used to calibrate the test setup.
It has a main CGH part to simulate the surface to be measured and an alignment CGH part for aligning itself with respect to the system. The alignment portion is essential to eliminate any alignment errors in the system. Figure 5(b)
is the test surface to be measured. This was made of Zerodur for thermal stability and manufactured using diamond turning machine (DTM). Figure 6
is a profile error of the test surface, which was measured by a 3-D profiler. Some overall form error and high peak irregular shapes in the central part can be seen. These irregular shapes come from a partial polishing process after the DTM work. The spatial period of irregular shapes is minimum 0.5 mm. Considering that the image size of test surface is 8.76 mm (magnification is 0.55) and the separation of apertures on the Hartmann sensor is 0.35 mm, these irregular shapes would not be reconstructed properly.
Fig. 5 (a) CGH as an ideal surface, (b) Test surface to be measured.
Fig. 6. Surface error measured by 3-D profiler, UA3P from Panasonic Co.
Actually, in order to have enough margins for the accurate measurement, it is important to locate the spot away from the boundary of its sub-aperture so that the conventional reconstruction algorithm works properly. For this, we adjusted the positions of sub-aperture cells by changing the offset of position and number of cells in the software. Figure 7
shows the reference generated by measuring the CGH in the testing setup. In this figure, the minimum distance between the center of spot and boundary of sub-aperture is only 0.04 mm (at circle A in Fig. 7
), which is too close to the boundary. This short distance may give the incorrect reconstruction if there is some further deformation at the test surface. To ensure an accurate reconstruction, we confined the aperture size to 7.7 mm (88% of the clear aperture). This region is shown in Fig. 7
as circle B. This will give the capability of measuring few μm deviation of test surface from the ideal.
shows the measured WFE of the beam expander by the interferometer. The four elements in the condensing lenses would add additional WFE to the error in the beam expander, 0.08 μm rms. Figure 8(b)
is the measurement result using reference generated by measuring CGH. The coarse WFE from 35 × 40 cells was interpolated using a cubic spline fitting. This picture shows the main feature of test surface profile error. Even though the detail of the central part in Fig. 6
is not revealed, the concave shape is similar to the averaged central portion. The PV error is 1.627 μm and rms error is 0.163 μm, which is only 16 nm rms higher than that of 3-D profiler. This slight difference may come from the average of detail structure of the central part in the Hartmann sensor measurement and different measurement area.
Fig. 7. Reference generated by measuring CGH. Spot in circle A is the closest to the boundary of sub-aperture. Circle B is the measurement area of test surface.
In Fig. 8
, it was not clear how much the WFE in the test setup was removed in the measurement of the test surface. To check this, the beam expander was intentionally defocused to generate 0.21 μm rms error, which is shown in Fig. 9(a)
. The CGH was aligned using an interferometer again and the new reference was generated on the Hartmann sensor. The test surface was measured again with respect to this new reference. Figure 9(b)
shows the measurement result of the test surface. Even though beam expander showed significant defocus aberration compared to Fig. 8(a)
, the test surface error was quite close to Fig. 8(b)
and the difference in rms WFE was only 2 nm. This means that the error in test setup can be significantly removed by the calibration with a CGH.
Fig. 8. Measurement result of test surface (a) WFE of the beam expander, and (b) surface profile of the test surface (z-unit is micrometer).
Fig. 9. Measurement result after intentional misalignment of beam expander (a) WFE of the beam expander, and (b) surface profile of the test surface (z-unit is micrometer).
5. Concluding remarks
We proposed a new method to test the steep aspheric convex surfaces, using a combination of a Hartmann test with a CGH. The CGH is served as a reference surface and the test setup is calibrated using this CGH. Then, the measurement accuracy is significantly increased as the error in the test setup is considerably removed. Experimentally, the test surface error was changed by only 2 nm rms even though the error in the beam expander in the test setup was increased 0.13 μm rms. Also, this measurement result was within 16 nm rms compared to that of 3-D profilometer. Since this method makes it possible to measure highly aspheric surfaces quickly and accurately, we expect to use it in the production line environments.
Currently, we are modifying the reconstruction software of the Hartmann sensor to make it possible to control the size and location of each sub-aperture separately. By the manual control in the software, we can make each beam spot positioned near the center of corresponding sub-aperture wherever they are located on the sensor. This can increase the dynamic range of the test setup, which makes it possible to measure steeper aspheric surface or some free form surfaces.
We thank to Dr. Gun-Hee Kim at KBSI for generating the test surface and Dr. Sang-Suk Kim at KOPTI for the surface measurement using the UA3P. The discussions with Dell Olsen at Spiricon, Inc. for the generation of reference file in the Hartmann sensor were valuable, as well.
References and links
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(090.1760) Holography : Computer holography
(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation
(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure
Instrumentation, Measurement, and Metrology
Original Manuscript: February 23, 2006
Revised Manuscript: March 31, 2006
Manuscript Accepted: April 11, 2006
Published: April 17, 2006
Vol. 1, Iss. 5 Virtual Journal for Biomedical Optics
Ho-Soon Yang, Jae-Bong Song, In-Won Lee, and Yun-Woo Lee, "Testing of steep convex aspheric surface with a Hartmann sensor by using a CGH," Opt. Express 14, 3247-3254 (2006)