Beyond current 8–10-m class large astronomical telescopes, there are currently at least three extremely large telescope projects proposed world-wide. They comprise a 30-m class telescope, the California Extremely Large Telescope (CELT [1
]), a 50-m class telescope (EURO 50 [2
]), and a 100-m class telescope, the Overwhelmingly Large Telescope (OWL [3
]). These telescopes are all based on segmented primary mirrors with 1-2-m class hexagonal segments. Such telescopes pose significant technical challenges for mass production of a few hundreds—if not thousands—of 1–2 m class precision mirrors over approximately 5 years. This contrasts with the ~6-month or more delivery times that are typical for 2-m class mirrors today. If we are to prove the feasibility of constructing such massive telescopes within a reasonable time scale of approximately 15 years, the development of an efficient mass fabrication technology that comprises rapid processing and testing is essential.
There are at least four typical testing instruments for aspheric surfaces: a phase-shifting interferometer (PSI) with a null corrector, a vibration-insensitive interferometer with a null corrector, a stylus profilometer [4
4. H. S. Yang, S. W. Kim, and D. D. Walker, “A novel profilometer for nanometric form assessment for large machined surfaces,” Key Eng. Mater. 257, 225–230 (2004). [CrossRef]
], and a wave-front sensor. Among them, the interferometer with the null corrector provides the most accurate measurement. However, the Hartmann test has also been used for a long time because of its simplicity and wide dynamic range [5
5. D. Malacara, Optical Shop Testing (Wiley, New York, 1992), Chap. 10.
The Hartmann test for the fabrication of large optics has been described elsewhere. Zverev et al.
tested a 6-m mirror, using the Hartmannn pattern to find that when a maximum deformation of surface was 6 µm, the error of determining the surface deformation was approximately 0.02 µm, compared to the interferometric measurement [6
6. V. A. Zverev, S. A. Rodionov, M. N. Sokol’skii, and V. V. Usoskin, “Testing of the primary mirror of the LAT (Large Azimuthal Telescope) by the Hartmann method during its manufacture,” Sov. J. Opt. Technol. 44, 127–129 (1977).
]. In that paper, Zverev et al.
used two different testing methods; interferometric and Hartmann testing. They used the null corrector for the interferometric testing; however, when they did the Hartmann testing, the mirror was tilted about 30 min of arc such that the corrector did not intercept the Hartmann testing. This was so because they calculated the ray aberration by using prefocal and postfocal Hartmann photographs. This method took much time for processing, 3–4 h from the mounting of the mirror in the test position, which means that it is not possible to perform many iterations of polishing and testing in a day.
Servin et al.
used a CCD instead of a photographic plate to reduce the processing time and developed an algorithm with which to estimate the ray aberration from the Hartmanngram directly [7
7. M. Servin, F. J. Cuevas, D. Malacara, and J. L. Marroquin, “Direct ray aberration estimation in Hartmanngrams by use of a regularized phase-tracking system,” Appl. Opt. 38, 2862–2869 (1999). [CrossRef]
]. They extracted the spherical aberration from the Hartmanngram captured by a CCD, so the remaining aberrations had a lower dynamic range. However, this method may not be used when the CCD’s sensing area is smaller than the image area, a condition that is common for the testing of large fast mirrors.
Pfund et al.
investigated the systematic errors of nonnull Hartmann testing for aspheres that have steep wave-front slopes, such as 110 λ/mm [8
8. J. Pfund, N. Lindlein, and J. Schwider, “Misalignment effects of the Shack-Hartmann sensor,” Appl. Opt. 37, 22–27 (1998). [CrossRef]
9. J. Pfund, N. Lindlein, and J. Schwider, “Nonnull testing of rotationally symmetric aspheres: a systematic error assessment,” Appl. Opt. 40, 439–446(2001). [CrossRef]
]. They used some auxiliary lenses to illuminate the aspheric convex surface. Many systematic errors such as misalignment of surface and sensors and lateral magnification error caused by imperfect lenses were analyzed and eliminated from the measurement result to produce the correct surface form. This simulation showed maximum residual errors of less than λ/500 peak to valley (p-v). However, this method is not adequate for measurement under harsh environmental conditions, as several real-time measurements should be performed to average out the random errors.
Given previous publications focused on the nonnull Hartmann test, we considered using the null Hartmann test for a surface during fabrication. If the Hartmann test is used with null correctors, the residual dynamic range will be lower that that for the nonnull Hartmann test. This means that the dynamic range of the Hartmann test is dramatically increased when a null lens is used, which makes it possible to test a surface beginning at an early stage of polishing at which the surface error can be a few tenths of a wavelength. In this case, if a nonnull Hartmann test is carried out, the surface slopes will be large enough that the individual spots cannot be resolved at the detector. In addition, the null Hartmann test has another advantage in that a dual test is possible, particularly with an interferometric tester. As a null corrector is used, the interferometer can measure the surface that is close to the target surface. In other words, when the surface form error is far from the dynamic range of the interferometer in the early stage of polishing, the Hartmann sensor will reveal the surface shape. When the surface form is close to the target shape, the interferometer will test the surface accurately. With these dual measurements, the surface can be tested beginning with an early stage of polishing to the final smoothing of the surface.
We are developing a 0.9-m collimator that will be used to evaluate other telescopes. It is a Cassegrain telescope that consists of aspheric primary and secondary mirrors. The development includes the fabrication and testing of the mirrors, design and manufacture of a mount system, and assembly. The testing of the primary mirror is one of the most difficult problems not only because the mirror is a large aspheric mirror but also because the surface error is expected to be larger than 10 µm p-v before polishing. Fig. 1
shows the experimental setup for testing of the primary mirror. We used a modern Hartmann sensor in which the perforated plate is placed in front of the CCD instead of in front of the target surface. The interferometer generates the high-quality collimated beam that is necessary for the Hartmann sensor. This beam passes through the null correctors subsequently and illuminates the whole surface of the asphere. On its way back, the image of the asphere that is produced by the null correctors is imaged on both the Hartmann sensor and the interferometer. This means that two sensors look at the same wave front from the target surface. In this paper, technical details of the null corrector and the performance of Hartmann sensor are presented and the measurement results are shown.
Fig. 1. Experimental setup for the test of a large aspheric mirror by using dual sensors.
2. Design of the null corrector
The target mirror is a 0.9-m aspheric mirror with a 3433.41-mm radius of curvature. The asphericity of this mirror is 129.4 µm. This mirror should be polished to better than λ/20 rms wave-front error (WFE; λ=633 nm). The Hartmann sensor that we used has 45 x 45 small apertures, each of 80-µm diameter, separated by 200 µm. The distance between the apertures and the CCD is 4041.44 µm. The pixel size of the CCD is 9 µm by 9 µm, and its sensing area is 9.1 mm by 9.2 mm.
Before we designed a set of lenses as null correctors, we considered a typical nonnull Hartmann test, as shown in Fig. 2(a)
. The high-quality doublet generates a spherical wave to cover the target surface. The mirror, which is imaged onto the sensor by the doublet, includes the error of the target surface as well as the effect of the nonnull test; these errors are mostly spherical aberration. The image of the target surface should be formed at least 100 mm away from the lens for easy installation of beam splitter and the Hartmann sensor. To satisfy this requirement it is necessary to use a simple thin-lens equation to select the focal length of the lens to be greater than 97 mm. Figure 2(b)
shows the simulated wave-front slope of the target mirror at the image plane of the doublet. The doublet has diffraction-limited performance. The maximum wave-front slope is approximately 2.3 mrad. According to the geometry of the Hartmann sensor, the theoretical maximum measurable wave-front slope has the following relation:
where f is the distance between the apertures and the CCD, S is the aperture spacing, d is the size of the aperture, and λ is 0.6328 µm. Then the maximum measurable wave-front slope of the sensor is approximately 14.9 mrad, which is 7 times higher than the maximum wave-front slope of the configuration.
However, it was not easy to implement this configuration. The main reason is that the size of beams from the target surface is larger than the CCD sensing area. A ZEMAX optical design program simulation showed that the size of image is approximately 26 mm, which is nearly 3 times larger than the sensing area.
Fig. 2. (a) General configuration for the test of a target asphere by use of a Hartmann sensor. (b) The wave-front slope at the image plane.
The null correctors made it possible to accommodate all the beams from the target surface at the image plane that is located at least 100 mm away from the lenses. We confined the entrance pupil diameter of the null correctors to less than 7 mm. We also controlled the WFE of the null correctors to not more than λ/30 rms, because the target WFE of the mirror is λ/20 rms (λ=633 nm). Figure 3
shows the designed null corrector. It consists of two doublets and two singlets. The two doublets form a Keplerian telescope whose magnification is 1. By use of this telescope the distance between the null correctors and the image plane was increased to 150 mm. Two singlets were used to compensate for spherical aberration of the asphere and served as traditional null correctors. These four elements were inserted into a tube mount with some spacers and retainers.
Fig. 3. Configuration of null correctors.
After the tolerance analysis of the null correctors, the total WFE was expected to be 0.034 λ, which satisfied our requirements. However, we found that the distance between the singlets and the refractive index of glass were the most significant contributors to the total WFE. The distance tolerance between the singlets is less than 0.02 mm, which should be achievable by a mechanical spacer. Also, the tolerance of the refractive index of glass is less than 0.0005, which is measured with a minimum-deviation angle method [10
10. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1990), Chap. 4.
]. Therefore, before we use them for testing, we need to verify the properties of the null correctors in terms of the mechanical assembly and optical parameters of the lens.
A computer-generated hologram (CGH) can be used as a null corrector to test an aspheric surface [11
11. T. H. Kim, J. Burge, Y. W. Lee, and S. S. Kim, “Null test for a highly parabolic mirror,” Appl. Opt. 43, 3614–3618 (2004). [CrossRef] [PubMed]
]. In addition, it can be used as a verification surface with which to test other null correctors [12
12. J. Burge, “A null test for null correctors: error analysis,” in Quality and Reliability for Optical Systems,
J. W. Bilbro and R. E. Parks, eds., Proc. SPIE1993, 86–97 (1993).
]. Basically, the hologram was made such that it reflected the test beam from the null correctors as if it were a perfect primary mirror. In the research reported in this paper we used a single-point diamond-turned aluminum mirror instead of a CGH as a verification surface. Its diameter was 67 mm and the distance from the null corrector was approximately 184.5 mm. The shape of this mirror was an even aspheric with five additional aspheric terms. Figure 4
shows the measurement of this mirror by two methods: with a stylus profilometer and with an interferometer. The result of stylus profilometer measurement shows a W-shaped surface error with a 0.51-µm p-v error. This shape error is also seen in the interferometric measurement, with a similar p-v error. Considering that the Form Talysurf did not use the null corrector in the surface measurement, this similarity means that the null correctors were well manufactured and aligned. A more careful error analysis of the null corrector is under way.
Fig. 4. Test results for the verification mirror: (a) Form Talysurf, (b) interferometer.
3. Performance of the Hartmann sensor
To estimate the performance of the Hartmann sensor experimentally, we measured the WFE of the high-quality doublet that was described in Section 2 as the doublet was defocused from 0 to 13 mm at 1-mm intervals. The wave front was calculated by modal methods [13
13. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980). [CrossRef]
]. This result was compared with that for a ZYGO phase-shifting interferometer. As the doublet has better than λ/10 p-v performance, its manufacturing error will be close to the design tolerance and its performance will be predictable by the optical design software. The usable area is the central 4 mm in diameter, even though doublet has a 25-mm diameter. The simulation showed that the linearity of the rms WFE during 0–13 mm of defocus was maintained within 8 nm for a deviation of 1σ. Therefore the linearity of the rms WFE of the doublet is a good estimation for the measurement accuracy of the sensor.
shows the result of the ZYGO interferometer’s measurement of the WFE. It is evident that the WFE’s linearity was maintained at 3.5 nm with an error of 1 σ until 7-mm defocus; after that the rms value deviated from linearity. This was so because too many fringes occurred following much defocus that were beyond the analytical capability of the interferometer. Figure 5(b)
shows the result of the Hartmann sensor measurement. Unlike the ZYGO result, it showed good linearity during the given range of defocus. The standard deviation of linearity was approximately 10.7 nm, which is comparable to that of theoretical linearity. Furthermore, the Hartmann sensor showed its capability to measure a larger WFE than the ZYGO interferometer can analyze. The difference in slopes between the theoretical values and the experimental results is thought to come from a manufacturing error of the doublet.
Fig. 5. Comparison of the results of measurement of the high-quality doublet with (a) ZYGO phase-shifting interferometer and (b) the Hartmann sensor. OPD is optical path difference.
Assuming that this difference comes only from the manufacturing error of the doublet, the comparison of differences should give a good estimation of measurement accuracy, particularly of the Hartmann sensor relative to the ZYGO interferometer. Figure 6
shows the difference between the theoretical values and the measurement results of the rms WFE of a doublet as the defocus error varies from 0 to 6 mm. The maximum difference between two sensors is less than 5 nm, except for the zero defocus, which implies the measurement accuracy of Hartmann sensor relative to the ZYGO interferometer. The 30-nm of difference at zero defocus is thought to be due to CCD noise of the Hartmann sensor and served as the minimum measurement accuracy of the sensor.
Fig. 6. Differences between the rms WFE estimation and measurement results.
shows the experimental setup for testing of the primary mirror, including the test setup shown in Fig. 1
. The interferometer and the Hartmann sensor were mounted on stages with 5 degrees of freedom and assembled on the same plate that slides along the optical axis of the mirror.
is the test result at the beginning stage of the polishing process. The top pictures show captured fringes from the interferometer, and the bottom ones are taken from the Hartmann test. Because of the large deviation of the surface form, some parts generate too many fringes or even cause the ray to go outside the null optics, which results in the absence of fringe. Therefore the interferometer cannot tell us about the surface error. The Hartmann test, however, shows a clear image of the surface error over the full aperture; here the p-v WFE is approximately 38.6 µm, which corresponds to a 19.3-µm p-v figure error.
shows the test result after 1 month of polishing, where the polishing was guided by feedback from the null Hartmann test. It shows some fringe pattern that is now much straightened and easier to understand. However, the vibration of the floor and air fluctuation prevented the interferometer from calculating the WFE from the fringes. The Hartmann sensor showed WFEs of approximately 1.06 µm p-v and 0.12 µm rms, which were generated by averaging of 30 repeated measurements to reduce random error. We moved the mirror to an environmentally controlled room to carry out the interferometric measurement. This measurement showed only 0.06 µm p-v or 0.03 µm rms WFE difference, which is comparable to the measurement error of Hartmann sensor described in Section 3. This result confirms that a Hartmann sensor can give a reliable test result under noisy laboratory conditions. However, another source of this difference is thought to be the high-frequency wiggles shown in the fringe pattern. Considering that the number of apertures used in the Hartmann test is 34×34, these high-frequency wiggles might be beyond the spatial resolution of the Hartmann sensor.
Fig. 7. Experimental setup for the test of the 0.9-m aspheric mirror.
5. Concluding remarks
We have proposed using a null Hartmann test with a phase-shifting interferometer to measure the surface error of a 0.9-m hyperbolic mirror. When the surface has a large surface error, a null Hartmann test is carried out; when the surface is close to the target, a null interferometric test is performed. With this system it was possible to test and polish a mirror with wave-front errors from 38.6 to 1.06 µm peak to valley. Furthermore, this method has proved to be useful for measuring surfaces in noisy laboratory conditions. The measurement difference between the null Hartmann test under much vibration and air turbulence and the null interferometric test carried out in an environmentally controlled room was only 0.06 µm p-v or 0.03 µm rms WFE. This mirror is being polished further to reach λ/20 rms WFE.
We plan to use a computer numerical control machine in conjunction with the test setup described in this paper. If a large WFE is measured early in the polishing process, a CNC machine can do the lapping for rapid removal of material instead requiring polishing. After correction, the surface is polished rapidly and tested again to check for any remaining surface error. We anticipate that this method will accelerate the fabrication of large surfaces and bring us one step closer to the mass production of mirror segments to be used in extremely large telescopes.
Fig. 8. Test results (a) at the start of polishing and (b) after 1 month of polishing.