Non-null annular subaperture stitching interferometry for steep aspheric measurement Lei Zhang, Chao Tian, Dong Liu,* Tu Shi, Yongying Yang, Hanshuo Wu, and Yibing Shen State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China *Corresponding author: [email protected] Received 19 June 2014; revised 22 July 2014; accepted 28 July 2014; posted 30 July 2014 (Doc. ID 214349); published 28 August 2014
A non-null annular subaperture stitching interferometry (NASSI), combining the subaperture stitching ideal and non-null test method, is proposed for steep aspheric testing. Compared with standard annular subaperture stitching interferometry (ASSI), a partial null lens (PNL) is employed as an alternative to the transmission sphere, to generate different aspherical wavefronts as the references. The coverage subaperture number would be reduced greatly for the better performance of aspherical wavefronts in matching the local slope of aspheric surfaces. In this way, relatively large overlapping areas can be obtained for adjustment errors correction while the error accumulation would be decreased. With the reverse optimization reconstruction (ROR) method for retrace error correction, the figure error of each subaperture can be retrieved accurately. Therefore, the testing accuracy and efficiency are thus increased. The dynamic test range is extended as well. A numerical simulation exhibits the comparison of the performance of the NASSI and standard ASSI, which demonstrates the high accuracy of the NASSI in testing steep aspheric. Experimental results of NASSI are shown to be in good agreement with those of the Zygo interferometer. © 2014 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (120.6650) Surface measurements, figure; (220.1250) Aspherics; (120.3180) Interferometry. http://dx.doi.org/10.1364/AO.53.005755
1. Introduction
Accurate interferometric testing for large and deep aspheric surfaces has always been a challenge. Traditional interferometric testing approach is limited by the spatial-frequency resolution of the interferometric system. The subaperture stitching interferometry (SSI), as an alternative, was proposed in 1980s [1–5] to overcome the limitation. According to the differences of subaperture shape, circle subaperture and annular subaperture were developed for aspheric surfaces. Circle subaperture stitching interferometry (CSSI) [1–8] is now commercially available by QED Technologies, which has the test capacity of mild aspheres [9,10]. But its complex and precise 1559-128X/14/255755-08$15.00/0 © 2014 Optical Society of America
multidimensional motion control has always limited the test accuracy. For rotational symmetric aspheric surfaces, annular subaperture stitching interferometry (ASSI) [11–15] is an effective alternative due to the requirement of only one dimension displacement. It employs a transmission sphere to generate spherical wavefronts of different radii of curvature to match the slopes in different annuli on the aspheric surface, and then combines the results of many subaperture measurements to obtain the full aperture surface map. The primary difficulty in the full aperture reconstruction is that a variable and imprecise amount of adjustment errors must be removed. Liu et al. [11] proposed the earliest reconstruction method based on Zernike circle polynomials in 1988. Subsequently, Melozzi et al. [12] presented the successive stitching methods with successive overlapping phase maps based on Zernike annular 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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polynomials. In 2004, Granados-Agustín et al. [15] improved the process by a simultaneous stitching method. Hou et al. [16] provided a more accurate reconstruction algorithm with Zernike annular polynomials and matrix method in 2006. This algorithm was experimentally demonstrated by testing a parabolic mirror of about 4 μm asphericity at 130 mm aperture the next year [17]. Meanwhile, the subaperture stitching and localization (SASL) algorithm is proposed and demonstrated by testing a parabolic mirror of about 8.7 μm asphericity at 185 mm aperture [18]. These reconstruction algorithms in ASSI are all supported by the traditional interferometer configuration, in which the spherical wavefront from transmission sphere is employed as the reference for approximate null test of each annular subaperture. However, when encountering a steep aspheric, the standard ASSI would suffer a large number and thus relatively narrow annular subaperture even if the aperture is small. It means that there are small overlapping areas which can be used to correct adjustment errors of adjacent annuli. Furthermore, the complete null test is not achievable for the varying radius of curvature of aspheric surface and thus the retrace error [19] would be induced. In addition, various errors (such as random noise, computation error) accumulation would increase with the subaperture number and the testing process would be time consuming as well. That is, the number of subapertures is one of the most important factors restraining the measurement accuracy in standard ASSI. As Melozzi et al. [12] pointed out, the measurement accuracy would be strongly affected if there were more than 10 subapertures. Many scholars pursue highprecision stitching algorithms but few attend to reduce the subaperture number. In this work, a novel non-null annular subaperture stitching interferometry (NASSI) is introduced for steep aspheric test, in which we employ a partial null lens (PNL) [20–22], as an alternative to the transmission sphere in standard ASSI, to generate different aspherical wavefronts as the references. The aspherical wavefronts would have better performance than spherical ones in matching the local slope of aspheric subapertures. This method greatly reduces the subaperture number compared to standard ASSI. Thus, relatively larger overlapping areas would be got for adjustment errors correction and error accumulation would be suppressed. Therefore, the stitching accuracy and efficiency would be increased. Moreover, a reverse iterative optimization reconstruction (ROR) method [22] based on system modeling, which is proposed in our previous work for retrace error correction in non-null test, is employed to retrieve the figure error of each subaperture. The measurement accuracy is further improved accordingly. Compared to standard ASSI, the NASSI has a clear advantage in measurement accuracy and efficiency in testing steep aspheric surfaces. In addition, the dynamic test range would be extended greatly because of 5756
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combination of the subaperture stitching ideal and part-compensation method. This paper is constructed as follows. Section 2 presents a simple description of NASSI and emphasizes the ROR method for subaperture figure error retrieve. Numerical simulation is illustrated to compare the accuracy of NASSI and standard ASSI in Section 3. Section 4 discusses some error consideration. In Section 5, experiments are carried out to verify the validity of the NASSI. A conclusion of this paper is summarized in Section 6. 2. Principle of NASSI
The NASSI system based on a modified phaseshifting Twyman–Green setup is illustrated in Fig. 1. As is shown, the collimated laser beam from the beam expander is split into two by a beam splitter. One is reflected by a reference mirror mounted on a piezoelectric ceramic transducer (PZT), serving as the reference beam; the other travels though the PNL and is then reflected by the test aspheric (ASP), forming the test beam after traveling though the PNL again. The reference and test beams interfere at the beam splitter and the resulted interferogram is imaged onto the detector by the imaging lens. When testing, the test aspheric is moved gradually away from the PNL along the optical axis with the displacement distance being monitored by a displacement measurement interferometer (DMI). The resolvable fringes of different annular subapertures can be recorded until the entire aperture is covered. Surface figure information is obtained over each subaperture and then sewed together with proper stitching algorithm. It is obvious that the crucial element in the system is the PNL, which has a simple structure and design process [9–11]. The aspheric wavefront generated by the PNL would have better performance in matching the local slope of
Fig. 1. Sketch of NASSI.
subapertures than the spherical one in most cases. Even though each subaperture test cannot achieve null condition, the retrace error would be corrected in subsequent processes. What we need in NASSI is to vary the distance dpa to obtain resolvable interferogram of each subaperture, i.e., make the wavefront slope of each subaperture less than the Nyquist frequency (vN 0.5λ∕pixel). That is, the PNL only provides partial compensation for the longitude aberration and each subaperture would be tested in non-null configuration. Just due to the incomplete compensation, each subaperture wavefront will suffer retrace error. All these subaperture wavefront data cannot be sewed together directly before retrace error correction. The general procedure of retrieving subaperture figure error W asp m in standard ASSI can be expressed as [11,15] W asp
m
fW m − f − sRm cos θ − εm g∕2;
(1)
where W m is the subaperture wavefront, f is the nominal aspheric prescription, sRm is the spherical reference wavefront of different radius Rm, and θ is aspheric normal angle, εm is the adjustment error, respectively. However, this procedure is not effective in NASSI not only because it is inaccurate for large retrace error of steep aspheric [19], but also we are unable to predict the aspheric wavefront traveling from the PNL. Thus, an accurate and practical method is needed for retrieving the subaperture figure error. The ROR method [22] presented in our previous work for retrace error correction can be implemented on each subaperture independently. It is based on a theoretical model of experiment system, in which the test wavefront of subaperture m can be described as follows:
W m ≅ f W asp W 0m
≅
m ; dpa m ; 0 f W asp m ; d0pa m
m 1; ; M;
Min;
− f W 0asp
0 2 m ; dpa m
m
dpa
m;
(4)
we have U f W asp
m
− f W 0asp
2 m
cons Min:
(5)
When Eq. (5) is satisfied well, we have W asp
m
W 0asp
m:
(6)
In this way, each subaperture figure error can be reconstructed accurately. For ease of algorithm design and figure characterization, Zernike annular polynomials are employed to describe the wavefront and figure error in the ROR method. The Zernike coefficients of aspheric figure error in the model, selected as variables, are changed gradually until the resulting subaperture wavefront in the model is closest to the corresponding one in the experiment. After the ROR is implemented on all subapertures, each subaperture figure error can be obtained and characterized by Zernike annular polynomials as W asp
m
≅ W 0asp
m
N X
ai Zi ρ; θ;
(7)
i
where the i and N are the Zernike ordinal and term, Zi ρ; θ and ai are the Zernike polynomial and corresponding coefficient, respectively. Subsequently, all the subapertures can be sewed together with the proper stitching algorithm for relative adjustment error correction. All the subapertures data are integrated to fiducial subaperture W asp 0 , which can be expressed as W stitch W asp 0 ⊕W asp 1 ⊕ ⊕W asp
U W m − W 0m 2 cons m ; dpa m
d0pa
(2)
where M is the subaperture number, W m and W asp m are the test wavefront and subaperture figure error of subaperture m, respectively, and dpa m is the distance from the PNL to the aspheric in the experiment, while W 0m , W 0asp m , and d0pa m are the simulated counterparts in the model, respectively. In the ROR method, a closed feedback system is set up to change the parameters of the simulated surface figure error (W 0asp m ) gradually in the ray tracing program, making the simulated test wavefront (W 0m ) approach the actual test one (W m ) in the experiment constantly. This process is realized by iterative ray tracing for the test system and an optimized objective function is defined to describe this process as
f W asp
where cons is the constraint for solution space. If the model is set up accurately, i.e.,
cons (3)
M;
(8)
where the W stitch is the full aperture figure error by initial stitching and ⊕ means the operation of stitching. Then, the full aperture figure error W asp would be reconstructed with a final residual adjustment error as W asp W stitch −
4 X
bi Zi ρ; θ;
(9)
i
P where the 4i bi Zi ρ; θ is the sum of the first four Zernike terms of W stitch, which characterize the low-order misalignment aberrations of the full aperture (or fiducial subaperture). Figure 2 illustrates the whole process of aspheric figure error reconstruction in NASSI. The specific process is described as follows: • Set up the actual experimental system and model it in a ray tracing program. 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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Table 1.
Parameters of PNL
Radius (mm) Thickness (mm) Glass Diameter (mm) PNL
Fig. 2. Flow chart of the NASSI method.
• Annulus partition in the model by monitoring wavefront slope. • Locate the actual aspheric in the experiment according to the annulus partition parameters and obtain the corresponding subaperture interferograms. • The actual subaperture wavefronts W m were demodulated from corresponding interferograms and fitted with Zernike annular polynomials. Also, the Zernike annular polynomials of the subaperture wavefronts W 0m in the model are obtained. • The ROR method is implemented for each subaperture. A closed feedback system is set up to change the parameters of simulated surface figure error (W 0asp m ) gradually in the ray tracing program, until the difference between the simulated test wavefront (W 0m ) and experimental one (W m ) is less than a small threshold ε. Then, we confirm that the actual full aperture figure error can be characterized by the simulated one. • Stitch algorithm is executed for W stitch, and a final residual adjustment error is removed to obtained full aperture figure error W asp. 3. Numerical Simulation
We carried out a numerical simulation of NASSI to test a high-order aspheric with 25 mm asphericity at a clear aperture of 100 mm, which is described as z
37 X x2 y2 − 6 × 10−9 x2 y2 2 bi Zi ρ; θ; 480 i1
(10) P37 where i1 bi Zi ρ; θ is the surface figure error in forms of the first 37 Zernike annular polynomials, which is added artificially on the aspheric surface. The parameters of a matched PNL are specified in 5758
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89.148 50.678
10.6
K9
45 45
Table 1. The NASSI system was modeled in the ray trace program. The upper limit of the fringe frequency was defined as 0.125λ∕pixel to promise a large redundancy in range of the resolving capacity of the detector. In this case, the best compensation for the aspheric with the PNL induced a maximum fringe frequency 0.272λ∕pixel, which is beyond the resolving capacity of the detector. We varied the distance dpa to seek the resolvable test wavefront of the annular zone. Three subapertures were obtained and the normalized width of each overlapping area reaches 0.04 (greater than 20% of next annular). Table 2 presents the specific parameters of subaperture partition, and Fig. 3(a) provides us the corresponding interferograms. After the ROR method was performed on each subaperture, the retrace errors were removed and the figure error of the three subapertures were retrieved, which is presented in Fig. 3(b). Slight misalignments and Gaussian distribution random noise [zero-mean and noise-to-signal ratio (NSR) [16] 0.01] were added to each subaperture to simulate actual experiments condition. These additional misalignments would be removed in subsequent stitching processes based on the relatively large overlaps. As a comparison, the standard ASSI was also modeled. With an optimal transmission sphere, the full aperture is divided into at least eight subapertures with the same upper limit of the fringe frequency 0.125λ∕pixel. The partition parameters are specified in Table 3 and the corresponding interferograms are presented in Fig. 3(c). Each overlapping area is greater than 20% of the next annular. It is obvious that the annuluses close to the edge of the surface are very narrow while the overlaps are extremely small. Figure 3(d) presents the surface figure error of each subaperture obtained in the general method [11,15] in standard ASSI. Similarly, equal misalignments and Gaussian distribution random noise were added to each subaperture. The simultaneous stitching algorithm [15] was employed to reconstruct the full aperture surface figure map. The results are shown in Fig. 4, where Fig. 4(a) gives the full-aperture map of the true figure error, which is added onto the test surface by the Zernike coefficients, Figs. 4(b) and 4(c) are the reconstructed maps of figure error in NASSI and standard ASSI, Table 2.
Parameters of Subaperture Partition in NASSI
Subaperture dpa (mm) Normalized zone Overlapping
0
1
303.43 [0,0.8]
304.32 [0.76,0.92] 0.04
2 305.00 [0.88,1] 0.04
Fig. 3. Subaperture partition in NASSI and standard ASSI. (a) Three interferograms of subapertures in NASSI, (b) the figure error of subapertures in NASSI, (c) eight interferograms of subapertures in standard ASSI, and (d) the figure error of subapertures in standard ASSI.
Table 3.
0
1
Parameters of Subaperture in Standard ASSI
2
3
4
5
6
7
304.98 306.22 307.20 308.00 304.98 306.22 307.20 308.00 dta (mm) Normalized zone [0,0.50] [0.46,0.63] [0.60,0.72] [0.70,0.80] [0.78,0.86] [0.84,0.91] [0.90,0.96] [0.95,1] Overlapping 0.04 0.03 0.02 0.02 0.02 0.01 0.01
Fig. 4. Reconstruction results in NASSI and standard ASSI. (a) The full-aperture map of the true figure error, (b) and (c) are reconstructed subaperture maps in NASSI and standard ASSI, (d) and (e) are corresponding reconstructed errors. The aperture is normalized.
Figs. 4(d) and 4(e) are the corresponding reconstructed errors, respectively. As is shown in Figs. 4(d) and 4(e), the reconstructed error is 9.58 × 10−3 λ peak-to-vally (PV) and 9.67 × 10−4 λ root-meansquare (rms) value in NASSI and 1.169λ PV and 0.213λ rms in standard ASSI, respectively. It is obvious that standard ASSI has a poor performance in testing steep aspheric. Note that, if the upper limit of fringe frequency in standard NASSI was lower, the retrace error might be reduced but the subaperture number would increase, and thus the test accuracy would not be improved significantly. On the contrary, NASSI’s high test accuracy mainly results from few subapertures and accurate correction for retrace error.
NSRs are illustrated in Figs. 5(a) and 5(b), where five results per noise level are plotted. As is shown in Fig. 5, the PV and rms value of reconstruction error increase with NSR gradually accompanying some fluctuation. After calculation, we concluded that the relative PV and rms value would still be less than 3.9% and 2.9% even if the NSR reached 0.5. That is, the NASSI is relatively insensitive to random noise. This because the Gaussian noise we added to each subaperture is high-frequency noise, which has little influence on low-frequency figure error expressed by Zernike polynomials. Fortunately, the random noise that affects the test results in experiment generally act in forms of vibration and air fluctuations, which is just the
4. Error Consideration A.
Random Noise
Some zero-mean Gaussian distributions, acting as random noise, were added to each subaperture data to simulate experimental conditions. For convenience, the NSR is defined as σ∕rms, where σ is the standard deviation of Gaussian noise and rms is the root-mean-square value of the full aperture figure error (here 0.0683 waves). The performance of NASSI was evaluated in different noise levels. The PV and rms value of reconstruction error in different
Fig. 5. Performance of the NASSI method in different noise levels. (a) The PV value of reconstruction error in different NSR. (b) The rms value of reconstruction error in different NSR. 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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high-frequency distribution. Therefore, the NASSI is relatively accurate in actual experiment for lowfrequency figure error. B.
Subaperture Number
For the decided aspheric under test in traditional ASSI, the subaperture number is roughly fixed because each subaperture test should satisfy an approximate null condition. This issue is alleviated to some extent by NASSI because we allow a non-null condition for each subaperture test. And thus fewer subapertures can be obtained. That is, the higher the limiting frequency is allowed, the less the subaperture would be necessary but more retrace error be suffered. Thus, an overall consideration for equilibrium is necessary. It is well known that the stitching accuracy in standard ASSI is seriously limited by subaperture number due to the narrow annuluses and error accumulation [12]. We evaluated the reconstruction error of NASSI in different subaperture numbers. Figures 6(a) and 6(b) present the rms value of reconstruction error in different subaperture numbers with the same misalignments and noise level, respectively. Obviously, the misalignments and random noise would contribute to inaccuracy for error accumulation with increasing subaperture number. Small subaperture number would be desired in NASSI as well. However, the smaller subaperture configuration would suffer large retrace error as mentioned before. The ROR method has a high accuracy of retrace error correction, which is illustrated in Fig. 4(b) (no misalignments and random noise). Therefore, a small subaperture number would be allowed in the range of system resolution tolerance in NASSI, with the ROR method implemented for retrace error correction. C.
Modeling Errors
System modeling plays a crucial role in NASSI. The ROR method would be affected seriously by the modeling errors due to the iterative ray tracing upon the biased system model. The modeling errors in NASSI mainly fall into element interval errors, optical elements figure errors, refractive index errors, and misalignments.
Fig. 6. Performance of the NASSI method in different subaperture numbers. (a) The rms value of reconstruction error in case of 0.01 × tilt for each subaperture. (b) The rms value of reconstruction error in case of 0.1 NSR for each subaperture. 5760
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Slight element interval errors mainly influence the test wavefront in the form of defocus, which can be eliminated from wavefront Zernike coefficients easily. Normally, a figure error of each surface of optical elements better than PV λ∕20 is achievable, which can be ignored in most cases. The measurement accuracy for refractive index, thickness, and radius of curvature, can be up to 10−5 , 5 μm, and 0.005% [23,24], respectively. The resulted aberrations are quite small and mainly play a role in the form of defocus as well. For element misalignments in NAASI, it must be treated separately for the PNL and other common components. Misalignment aberrations of common components can be stored traditionally employing a standard reflector, and then removed from test results [25]. But the PNL that plays a crucial role in the system must be treated separately because of the absence of standard asphere used for error storing. The misalignments of the PNL are important factors affecting the wavefront aberrations. This issue is discussed in detail in [26]. Also, the misalignments of the PNL can be controlled to a small extent by some methods in [26,27]. For the experiment in Section 5, the PNL is located in a precise sleeve employing a positioning instrument, whose tilt errors and decentration errors are reduced to 30″ and micrometer dimension, respectively. The corresponding wavefront error is less than 10−3 λ. In addition, the Zernike polynomial fitting error should be treated in NASSI. In this paper, the first 37 Zernike annular polynomials [26] are employed. Residual high-frequency errors affect test results slightly because the test surface error we wanted is composed of low-frequency errors in most cases. Many experiments show that Zernike polynomial fitting errors are less than 10−6 λ. From the analyses above, we may confirm that all errors can be reduced to a small extent and the proposed NASSI can provide fairly accurate test results. 5. Experimental Results
An experiment was carried out to validate NASSI based on current equipment in the laboratory. A concave paraboloidal mirror with a vertex radius of 240.0 mm and a clear aperture of 101.0 mm was tested (wavelength λ 632.8 mm). The PNL employed is the same as the one in the numerical simulation, whose parameters are specified in Table 1. The PV values of the surface error of the PNL and reference mirror are better than λ∕20 and λ∕10, respectively. The precision linearity rail, with 1000 mm full travel, can achieve 1 um positioning accuracy. The detector employed is a 1392 × 1040 pixel CCD camera. The Renishaw DMI can precisely measure distances in range of 40 m, with an accuracy of up to 0.5 ppm. In consideration of noise and adjustment errors, the maximum slope of the subaperture wavefront vmax < vN ∕4 0.125λ∕pixel is employed to promise a large redundancy. The aspheric is translated along
Table 4.
Parameters of Subapertures in the Experiment
Subaperture dpa (mm) Normalized zone
0
1
2
302.2155 [0,0.6]
302.0473 [0.5,0.8]
302.0241 [0.7,1]
the optical axis gradually and the displacement distance is monitored by the DMI. Three subapertures were obtained with overlap of 0.1 normalized width. The specific parameters are presented in Table 4 and corresponding interferograms are presented in Fig. 7(a). After the ROR method was implemented on each subaperture, the figure errors of the three subapertures were retrieved accurately, which are presented in Fig. 7(b). Subsequently, the simultaneous stitching algorithm was performed to obtained a full aperture figure error (here 98% of full aperture), whose 2D and 3D maps are shown in Fig. 8(a). The PV and rms values of the figure error are 0.2724λ and 0.0310λ, respectively. We also measured the paraboloidal mirror employing a ZYGO interferometer with aberration-free point method for cross test, in which the figure error of the auxiliary flat mirror achieves better than 1∕20λ. Figure 8(b) refers to the 2D and 3D maps of the figure error obtained from the ZYGO interferometer which serve as the criterion in this paper. The PV and rms values are 0.326λ and 0.037λ, respectively. Another concave paraboloidal mirror with the same nominal parameters was also measured. Figures 9(a) and 9(b) present the test results by NASSI and the Zygo interferometer, respectively. The PV and rms values of the figure error test by NASSI are 0.2660λ and 0.0350λ, respectively, and those by the Zygo interferometer are 0.277λ and 0.037λ, respectively. From Figs. 8 and 9, it is found that the test result by NASSI has a basic consistency with those of the Zygo interferometer. The PV and rms values of the test results of the two aspherics are specified in
Fig. 8. Testing results in NASSI and Zygo. (a) The reconstructed result in NASSI and (b) the surface figure error tested by Zygo.
Fig. 9. Testing results in NASSI and Zygo. (a) The reconstructed result in NASSI and (b) the surface figure error tested by Zygo.
Table 5.
PV (λ) rms (λ)
PV and RMS Values of Figure Error by NASSI and Zygo Interferometer
The First Aspheric
The Second Aspheric
NASSI
Zygo
NASSI
Zygo
0.2724 0.0310
0.326 0.037
0.2660 0.0350
0.277 0.037
Table 5, which shows the validity and high accuracy of NASSI. Of course, small variations between the results of NASSI and the Zygo interferometer may be induced by the combination of random noise, modeling errors, stitching errors, Zernike fitting errors, and so on. But these errors are small and also exist in many other mainstream ASSI methods. 6. Conclusion Fig. 7. Interferograms and figure error in NASSI. (a) The subaperture interferograms and (b) the subaperture figure error by ROR.
We proposed a NASSI for steep aspheric testing, in which the PNL is employed as an alternative to the transmission sphere in standard ASSI, and the ROR method is employed to retrieve each 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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subaperture figure error. In NASSI, the subaperture number is decreased greatly in comparison to standard ASSI, which means a sufficient overlapping area for misalignments correction and less error accumulation. Combining with the ROR method for retrace error correction, high measurement accuracy is allowed. From both numerical simulation and experiment, NASSI exhibits good performance in testing accuracy. Furthermore, NASSI combines the subaperture stitching ideal with the partcompensation method, extending the dynamic test range greatly. Also it would be time saving to test steep aspheric. The integrated NASSI workstation and more easy and accurate algorithms for full aperture reconstruction are in study in laboratory, which will be reported in our future publications. This work was partially supported by the National Natural Science Foundation of China (41305014), the State Key Laboratory of Modern Optical Instrumentation Innovation Program (MOI201208), the “985” III: First-Class Discipline Construction Program, and the the Fundamental Research Funds for the Central Universities (2013QNA5006). References 1. J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19–27 (1982). 2. W. W. Chow and G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. 8, 468–470 (1983). 3. J. E. Negro, “Subaperture optical system testing,” Appl. Opt. 23, 1921–1930 (1984). 4. S. C. Jensen, W. W. Chow, and G. N. Lawrence, “Subaperture testing approaches: a comparison,” Appl. Opt. 23, 740–745 (1984). 5. C. R. De Hainaut and A. Erteza, “Numerical processing of dynamic subaperture testing measurements,” Appl. Opt. 25, 503–509 (1986). 6. M. A. Schmucker and J. Schmit, “Selection process for sequentially combing multiple sets of overlapping surface profile interferometric data to produce a continuous composite map,” U.S. patent 5,991,461 (23 November 1999). 7. M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33, 608–613 (1994). 8. L. Yan, X. Wang, L. Zheng, X. Zeng, H. Hu, and X. Zhang, “Experimental study on subaperture testing with iterative triangulation algorithm,” Opt. Express 21, 22628–22644 (2013). 9. P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14, 38–43 (2003).
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10. P. E. Murphy, J. Fleig, G. Forbes, and M. Tricard, “High precision metrology of demes and aspheric optics,” Proc. SPIE 5786, 112–121 (2005). 11. Y. M. Liu, G. Lawrence, and C. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. 27, 4504–4513 (1988). 12. M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993). 13. D. Malacara, M. Servin, A. Morales, and Z. Malacara, “Aspherical wavefront testing with several defocusing steps,” Proc. SPIE 2576, 190–192 (1995). 14. M. J. Tronolone, J. F. Fleig, C. Huang, and J. H. Bruning, “Method of testing aspherical optical surfaces with an interferometer,” U.S. patent 5,416,586 (16 May 1995). 15. F. Granados-Agustín, J. F. Escobar-Romero, and A. Cornejo-Rodríguez, “Testing parabolic surfaces with annular subaperture interferograms,” Opt. Rev. 11, 82–86 (2004). 16. X. Hou, F. Wu, L. Yang, S. Wu, and Q. Chen, “Full-aperture wavefront reconstruction from annular subaperture interferometric data using Zernike annular polynomials and matrix method for testing large aspheric surfaces,” Appl. Opt. 45, 3442–3455 (2006). 17. X. Hou, F. Wu, L. Yang, S. Wu, and Q. Chen, “Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method,” Opt. Express 15, 12890–12899 (2007). 18. S. Y. Chen, S. Y. Li, and Y. F. Dai, “Experimental study on subaperture testing with iterative stitching algorithm,” Opt. Express 16, 4760–4765 (2008). 19. D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in non-null aspheric testing,” Opt. Express 17, 7025–7035 (2009). 20. H. Liu, Q. Zhu, Q. Hao, and D. Sha, “Design of novel partcompensating lens used in aspheric testing,” Proc. SPIE 5253, 480–484 (2003). 21. J. J. Sullivan and J. E. Greivenkamp, “Design of partial nulls for testing of fast aspheric surfaces,” Proc. SPIE 6671, 66710W (2007). 22. D. Liu, Y. Yang, Y. Luo, C. Tian, Y. Shen, and Y. Zhuo, “Nonnull interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 74260, 74260M (2009). 23. C. Tian, Y. Yang, and Y. Zhuo, “Generalized data reduction approach for aspheric testing in a non-null interferometer,” Appl. Opt. 51, 1598–1604 (2012). 24. L. Zhang, D. Liu, T. Shi, Y. Yang, and Y. Shen, “Practical and accurate method for aspheric misalignment aberrations calibration in non-null interferometric testing,” Appl. Opt. 52, 8501–8511 (2013). 25. Y. Luo, “Research on error analysis and system optimization of aspheric partial compensatory interferometric testing method,” M.Sc. thesis (Zhejiang University, 2012). 26. Y. Luo, Y. Yang, and C. Tian, “Error analysis and preprocess of partial compensatory aspheric testing system,” J. Zhejiang Univ. 46, 636–733 (2012). 27. Zhejiang University, “Aligning device and method for partial compensating lens in aspheric non-null interference testing,” China patent 201210054455.2 (18 July 2012).
A non-null annular subaperture stitching interferometry (NASSI), combining the subaperture stitching ideal and non-null test method, is proposed for steep aspheric testing. Compared with standard annular subaperture stitching interferometry (ASSI), a partial null lens (PNL) is employed as an alternative to the transmission sphere, to generate different aspherical wavefronts as the references. The coverage subaperture number would be reduced greatly for the better performance of aspherical wavefronts in matching the local slope of aspheric surfaces. In this way, relatively large overlapping areas can be obtained for adjustment errors correction while the error accumulation would be decreased. With the reverse optimization reconstruction (ROR) method for retrace error correction, the figure error of each subaperture can be retrieved accurately. Therefore, the testing accuracy and efficiency are thus increased. The dynamic test range is extended as well. A numerical simulation exhibits the comparison of the performance of the NASSI and standard ASSI, which demonstrates the high accuracy of the NASSI in testing steep aspheric. Experimental results of NASSI are shown to be in good agreement with those of the Zygo interferometer. © 2014 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (120.6650) Surface measurements, figure; (220.1250) Aspherics; (120.3180) Interferometry. http://dx.doi.org/10.1364/AO.53.005755
1. Introduction
Accurate interferometric testing for large and deep aspheric surfaces has always been a challenge. Traditional interferometric testing approach is limited by the spatial-frequency resolution of the interferometric system. The subaperture stitching interferometry (SSI), as an alternative, was proposed in 1980s [1–5] to overcome the limitation. According to the differences of subaperture shape, circle subaperture and annular subaperture were developed for aspheric surfaces. Circle subaperture stitching interferometry (CSSI) [1–8] is now commercially available by QED Technologies, which has the test capacity of mild aspheres [9,10]. But its complex and precise 1559-128X/14/255755-08$15.00/0 © 2014 Optical Society of America
multidimensional motion control has always limited the test accuracy. For rotational symmetric aspheric surfaces, annular subaperture stitching interferometry (ASSI) [11–15] is an effective alternative due to the requirement of only one dimension displacement. It employs a transmission sphere to generate spherical wavefronts of different radii of curvature to match the slopes in different annuli on the aspheric surface, and then combines the results of many subaperture measurements to obtain the full aperture surface map. The primary difficulty in the full aperture reconstruction is that a variable and imprecise amount of adjustment errors must be removed. Liu et al. [11] proposed the earliest reconstruction method based on Zernike circle polynomials in 1988. Subsequently, Melozzi et al. [12] presented the successive stitching methods with successive overlapping phase maps based on Zernike annular 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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polynomials. In 2004, Granados-Agustín et al. [15] improved the process by a simultaneous stitching method. Hou et al. [16] provided a more accurate reconstruction algorithm with Zernike annular polynomials and matrix method in 2006. This algorithm was experimentally demonstrated by testing a parabolic mirror of about 4 μm asphericity at 130 mm aperture the next year [17]. Meanwhile, the subaperture stitching and localization (SASL) algorithm is proposed and demonstrated by testing a parabolic mirror of about 8.7 μm asphericity at 185 mm aperture [18]. These reconstruction algorithms in ASSI are all supported by the traditional interferometer configuration, in which the spherical wavefront from transmission sphere is employed as the reference for approximate null test of each annular subaperture. However, when encountering a steep aspheric, the standard ASSI would suffer a large number and thus relatively narrow annular subaperture even if the aperture is small. It means that there are small overlapping areas which can be used to correct adjustment errors of adjacent annuli. Furthermore, the complete null test is not achievable for the varying radius of curvature of aspheric surface and thus the retrace error [19] would be induced. In addition, various errors (such as random noise, computation error) accumulation would increase with the subaperture number and the testing process would be time consuming as well. That is, the number of subapertures is one of the most important factors restraining the measurement accuracy in standard ASSI. As Melozzi et al. [12] pointed out, the measurement accuracy would be strongly affected if there were more than 10 subapertures. Many scholars pursue highprecision stitching algorithms but few attend to reduce the subaperture number. In this work, a novel non-null annular subaperture stitching interferometry (NASSI) is introduced for steep aspheric test, in which we employ a partial null lens (PNL) [20–22], as an alternative to the transmission sphere in standard ASSI, to generate different aspherical wavefronts as the references. The aspherical wavefronts would have better performance than spherical ones in matching the local slope of aspheric subapertures. This method greatly reduces the subaperture number compared to standard ASSI. Thus, relatively larger overlapping areas would be got for adjustment errors correction and error accumulation would be suppressed. Therefore, the stitching accuracy and efficiency would be increased. Moreover, a reverse iterative optimization reconstruction (ROR) method [22] based on system modeling, which is proposed in our previous work for retrace error correction in non-null test, is employed to retrieve the figure error of each subaperture. The measurement accuracy is further improved accordingly. Compared to standard ASSI, the NASSI has a clear advantage in measurement accuracy and efficiency in testing steep aspheric surfaces. In addition, the dynamic test range would be extended greatly because of 5756
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combination of the subaperture stitching ideal and part-compensation method. This paper is constructed as follows. Section 2 presents a simple description of NASSI and emphasizes the ROR method for subaperture figure error retrieve. Numerical simulation is illustrated to compare the accuracy of NASSI and standard ASSI in Section 3. Section 4 discusses some error consideration. In Section 5, experiments are carried out to verify the validity of the NASSI. A conclusion of this paper is summarized in Section 6. 2. Principle of NASSI
The NASSI system based on a modified phaseshifting Twyman–Green setup is illustrated in Fig. 1. As is shown, the collimated laser beam from the beam expander is split into two by a beam splitter. One is reflected by a reference mirror mounted on a piezoelectric ceramic transducer (PZT), serving as the reference beam; the other travels though the PNL and is then reflected by the test aspheric (ASP), forming the test beam after traveling though the PNL again. The reference and test beams interfere at the beam splitter and the resulted interferogram is imaged onto the detector by the imaging lens. When testing, the test aspheric is moved gradually away from the PNL along the optical axis with the displacement distance being monitored by a displacement measurement interferometer (DMI). The resolvable fringes of different annular subapertures can be recorded until the entire aperture is covered. Surface figure information is obtained over each subaperture and then sewed together with proper stitching algorithm. It is obvious that the crucial element in the system is the PNL, which has a simple structure and design process [9–11]. The aspheric wavefront generated by the PNL would have better performance in matching the local slope of
Fig. 1. Sketch of NASSI.
subapertures than the spherical one in most cases. Even though each subaperture test cannot achieve null condition, the retrace error would be corrected in subsequent processes. What we need in NASSI is to vary the distance dpa to obtain resolvable interferogram of each subaperture, i.e., make the wavefront slope of each subaperture less than the Nyquist frequency (vN 0.5λ∕pixel). That is, the PNL only provides partial compensation for the longitude aberration and each subaperture would be tested in non-null configuration. Just due to the incomplete compensation, each subaperture wavefront will suffer retrace error. All these subaperture wavefront data cannot be sewed together directly before retrace error correction. The general procedure of retrieving subaperture figure error W asp m in standard ASSI can be expressed as [11,15] W asp
m
fW m − f − sRm cos θ − εm g∕2;
(1)
where W m is the subaperture wavefront, f is the nominal aspheric prescription, sRm is the spherical reference wavefront of different radius Rm, and θ is aspheric normal angle, εm is the adjustment error, respectively. However, this procedure is not effective in NASSI not only because it is inaccurate for large retrace error of steep aspheric [19], but also we are unable to predict the aspheric wavefront traveling from the PNL. Thus, an accurate and practical method is needed for retrieving the subaperture figure error. The ROR method [22] presented in our previous work for retrace error correction can be implemented on each subaperture independently. It is based on a theoretical model of experiment system, in which the test wavefront of subaperture m can be described as follows:
W m ≅ f W asp W 0m
≅
m ; dpa m ; 0 f W asp m ; d0pa m
m 1; ; M;
Min;
− f W 0asp
0 2 m ; dpa m
m
dpa
m;
(4)
we have U f W asp
m
− f W 0asp
2 m
cons Min:
(5)
When Eq. (5) is satisfied well, we have W asp
m
W 0asp
m:
(6)
In this way, each subaperture figure error can be reconstructed accurately. For ease of algorithm design and figure characterization, Zernike annular polynomials are employed to describe the wavefront and figure error in the ROR method. The Zernike coefficients of aspheric figure error in the model, selected as variables, are changed gradually until the resulting subaperture wavefront in the model is closest to the corresponding one in the experiment. After the ROR is implemented on all subapertures, each subaperture figure error can be obtained and characterized by Zernike annular polynomials as W asp
m
≅ W 0asp
m
N X
ai Zi ρ; θ;
(7)
i
where the i and N are the Zernike ordinal and term, Zi ρ; θ and ai are the Zernike polynomial and corresponding coefficient, respectively. Subsequently, all the subapertures can be sewed together with the proper stitching algorithm for relative adjustment error correction. All the subapertures data are integrated to fiducial subaperture W asp 0 , which can be expressed as W stitch W asp 0 ⊕W asp 1 ⊕ ⊕W asp
U W m − W 0m 2 cons m ; dpa m
d0pa
(2)
where M is the subaperture number, W m and W asp m are the test wavefront and subaperture figure error of subaperture m, respectively, and dpa m is the distance from the PNL to the aspheric in the experiment, while W 0m , W 0asp m , and d0pa m are the simulated counterparts in the model, respectively. In the ROR method, a closed feedback system is set up to change the parameters of the simulated surface figure error (W 0asp m ) gradually in the ray tracing program, making the simulated test wavefront (W 0m ) approach the actual test one (W m ) in the experiment constantly. This process is realized by iterative ray tracing for the test system and an optimized objective function is defined to describe this process as
f W asp
where cons is the constraint for solution space. If the model is set up accurately, i.e.,
cons (3)
M;
(8)
where the W stitch is the full aperture figure error by initial stitching and ⊕ means the operation of stitching. Then, the full aperture figure error W asp would be reconstructed with a final residual adjustment error as W asp W stitch −
4 X
bi Zi ρ; θ;
(9)
i
P where the 4i bi Zi ρ; θ is the sum of the first four Zernike terms of W stitch, which characterize the low-order misalignment aberrations of the full aperture (or fiducial subaperture). Figure 2 illustrates the whole process of aspheric figure error reconstruction in NASSI. The specific process is described as follows: • Set up the actual experimental system and model it in a ray tracing program. 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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Table 1.
Parameters of PNL
Radius (mm) Thickness (mm) Glass Diameter (mm) PNL
Fig. 2. Flow chart of the NASSI method.
• Annulus partition in the model by monitoring wavefront slope. • Locate the actual aspheric in the experiment according to the annulus partition parameters and obtain the corresponding subaperture interferograms. • The actual subaperture wavefronts W m were demodulated from corresponding interferograms and fitted with Zernike annular polynomials. Also, the Zernike annular polynomials of the subaperture wavefronts W 0m in the model are obtained. • The ROR method is implemented for each subaperture. A closed feedback system is set up to change the parameters of simulated surface figure error (W 0asp m ) gradually in the ray tracing program, until the difference between the simulated test wavefront (W 0m ) and experimental one (W m ) is less than a small threshold ε. Then, we confirm that the actual full aperture figure error can be characterized by the simulated one. • Stitch algorithm is executed for W stitch, and a final residual adjustment error is removed to obtained full aperture figure error W asp. 3. Numerical Simulation
We carried out a numerical simulation of NASSI to test a high-order aspheric with 25 mm asphericity at a clear aperture of 100 mm, which is described as z
37 X x2 y2 − 6 × 10−9 x2 y2 2 bi Zi ρ; θ; 480 i1
(10) P37 where i1 bi Zi ρ; θ is the surface figure error in forms of the first 37 Zernike annular polynomials, which is added artificially on the aspheric surface. The parameters of a matched PNL are specified in 5758
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89.148 50.678
10.6
K9
45 45
Table 1. The NASSI system was modeled in the ray trace program. The upper limit of the fringe frequency was defined as 0.125λ∕pixel to promise a large redundancy in range of the resolving capacity of the detector. In this case, the best compensation for the aspheric with the PNL induced a maximum fringe frequency 0.272λ∕pixel, which is beyond the resolving capacity of the detector. We varied the distance dpa to seek the resolvable test wavefront of the annular zone. Three subapertures were obtained and the normalized width of each overlapping area reaches 0.04 (greater than 20% of next annular). Table 2 presents the specific parameters of subaperture partition, and Fig. 3(a) provides us the corresponding interferograms. After the ROR method was performed on each subaperture, the retrace errors were removed and the figure error of the three subapertures were retrieved, which is presented in Fig. 3(b). Slight misalignments and Gaussian distribution random noise [zero-mean and noise-to-signal ratio (NSR) [16] 0.01] were added to each subaperture to simulate actual experiments condition. These additional misalignments would be removed in subsequent stitching processes based on the relatively large overlaps. As a comparison, the standard ASSI was also modeled. With an optimal transmission sphere, the full aperture is divided into at least eight subapertures with the same upper limit of the fringe frequency 0.125λ∕pixel. The partition parameters are specified in Table 3 and the corresponding interferograms are presented in Fig. 3(c). Each overlapping area is greater than 20% of the next annular. It is obvious that the annuluses close to the edge of the surface are very narrow while the overlaps are extremely small. Figure 3(d) presents the surface figure error of each subaperture obtained in the general method [11,15] in standard ASSI. Similarly, equal misalignments and Gaussian distribution random noise were added to each subaperture. The simultaneous stitching algorithm [15] was employed to reconstruct the full aperture surface figure map. The results are shown in Fig. 4, where Fig. 4(a) gives the full-aperture map of the true figure error, which is added onto the test surface by the Zernike coefficients, Figs. 4(b) and 4(c) are the reconstructed maps of figure error in NASSI and standard ASSI, Table 2.
Parameters of Subaperture Partition in NASSI
Subaperture dpa (mm) Normalized zone Overlapping
0
1
303.43 [0,0.8]
304.32 [0.76,0.92] 0.04
2 305.00 [0.88,1] 0.04
Fig. 3. Subaperture partition in NASSI and standard ASSI. (a) Three interferograms of subapertures in NASSI, (b) the figure error of subapertures in NASSI, (c) eight interferograms of subapertures in standard ASSI, and (d) the figure error of subapertures in standard ASSI.
Table 3.
0
1
Parameters of Subaperture in Standard ASSI
2
3
4
5
6
7
304.98 306.22 307.20 308.00 304.98 306.22 307.20 308.00 dta (mm) Normalized zone [0,0.50] [0.46,0.63] [0.60,0.72] [0.70,0.80] [0.78,0.86] [0.84,0.91] [0.90,0.96] [0.95,1] Overlapping 0.04 0.03 0.02 0.02 0.02 0.01 0.01
Fig. 4. Reconstruction results in NASSI and standard ASSI. (a) The full-aperture map of the true figure error, (b) and (c) are reconstructed subaperture maps in NASSI and standard ASSI, (d) and (e) are corresponding reconstructed errors. The aperture is normalized.
Figs. 4(d) and 4(e) are the corresponding reconstructed errors, respectively. As is shown in Figs. 4(d) and 4(e), the reconstructed error is 9.58 × 10−3 λ peak-to-vally (PV) and 9.67 × 10−4 λ root-meansquare (rms) value in NASSI and 1.169λ PV and 0.213λ rms in standard ASSI, respectively. It is obvious that standard ASSI has a poor performance in testing steep aspheric. Note that, if the upper limit of fringe frequency in standard NASSI was lower, the retrace error might be reduced but the subaperture number would increase, and thus the test accuracy would not be improved significantly. On the contrary, NASSI’s high test accuracy mainly results from few subapertures and accurate correction for retrace error.
NSRs are illustrated in Figs. 5(a) and 5(b), where five results per noise level are plotted. As is shown in Fig. 5, the PV and rms value of reconstruction error increase with NSR gradually accompanying some fluctuation. After calculation, we concluded that the relative PV and rms value would still be less than 3.9% and 2.9% even if the NSR reached 0.5. That is, the NASSI is relatively insensitive to random noise. This because the Gaussian noise we added to each subaperture is high-frequency noise, which has little influence on low-frequency figure error expressed by Zernike polynomials. Fortunately, the random noise that affects the test results in experiment generally act in forms of vibration and air fluctuations, which is just the
4. Error Consideration A.
Random Noise
Some zero-mean Gaussian distributions, acting as random noise, were added to each subaperture data to simulate experimental conditions. For convenience, the NSR is defined as σ∕rms, where σ is the standard deviation of Gaussian noise and rms is the root-mean-square value of the full aperture figure error (here 0.0683 waves). The performance of NASSI was evaluated in different noise levels. The PV and rms value of reconstruction error in different
Fig. 5. Performance of the NASSI method in different noise levels. (a) The PV value of reconstruction error in different NSR. (b) The rms value of reconstruction error in different NSR. 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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high-frequency distribution. Therefore, the NASSI is relatively accurate in actual experiment for lowfrequency figure error. B.
Subaperture Number
For the decided aspheric under test in traditional ASSI, the subaperture number is roughly fixed because each subaperture test should satisfy an approximate null condition. This issue is alleviated to some extent by NASSI because we allow a non-null condition for each subaperture test. And thus fewer subapertures can be obtained. That is, the higher the limiting frequency is allowed, the less the subaperture would be necessary but more retrace error be suffered. Thus, an overall consideration for equilibrium is necessary. It is well known that the stitching accuracy in standard ASSI is seriously limited by subaperture number due to the narrow annuluses and error accumulation [12]. We evaluated the reconstruction error of NASSI in different subaperture numbers. Figures 6(a) and 6(b) present the rms value of reconstruction error in different subaperture numbers with the same misalignments and noise level, respectively. Obviously, the misalignments and random noise would contribute to inaccuracy for error accumulation with increasing subaperture number. Small subaperture number would be desired in NASSI as well. However, the smaller subaperture configuration would suffer large retrace error as mentioned before. The ROR method has a high accuracy of retrace error correction, which is illustrated in Fig. 4(b) (no misalignments and random noise). Therefore, a small subaperture number would be allowed in the range of system resolution tolerance in NASSI, with the ROR method implemented for retrace error correction. C.
Modeling Errors
System modeling plays a crucial role in NASSI. The ROR method would be affected seriously by the modeling errors due to the iterative ray tracing upon the biased system model. The modeling errors in NASSI mainly fall into element interval errors, optical elements figure errors, refractive index errors, and misalignments.
Fig. 6. Performance of the NASSI method in different subaperture numbers. (a) The rms value of reconstruction error in case of 0.01 × tilt for each subaperture. (b) The rms value of reconstruction error in case of 0.1 NSR for each subaperture. 5760
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Slight element interval errors mainly influence the test wavefront in the form of defocus, which can be eliminated from wavefront Zernike coefficients easily. Normally, a figure error of each surface of optical elements better than PV λ∕20 is achievable, which can be ignored in most cases. The measurement accuracy for refractive index, thickness, and radius of curvature, can be up to 10−5 , 5 μm, and 0.005% [23,24], respectively. The resulted aberrations are quite small and mainly play a role in the form of defocus as well. For element misalignments in NAASI, it must be treated separately for the PNL and other common components. Misalignment aberrations of common components can be stored traditionally employing a standard reflector, and then removed from test results [25]. But the PNL that plays a crucial role in the system must be treated separately because of the absence of standard asphere used for error storing. The misalignments of the PNL are important factors affecting the wavefront aberrations. This issue is discussed in detail in [26]. Also, the misalignments of the PNL can be controlled to a small extent by some methods in [26,27]. For the experiment in Section 5, the PNL is located in a precise sleeve employing a positioning instrument, whose tilt errors and decentration errors are reduced to 30″ and micrometer dimension, respectively. The corresponding wavefront error is less than 10−3 λ. In addition, the Zernike polynomial fitting error should be treated in NASSI. In this paper, the first 37 Zernike annular polynomials [26] are employed. Residual high-frequency errors affect test results slightly because the test surface error we wanted is composed of low-frequency errors in most cases. Many experiments show that Zernike polynomial fitting errors are less than 10−6 λ. From the analyses above, we may confirm that all errors can be reduced to a small extent and the proposed NASSI can provide fairly accurate test results. 5. Experimental Results
An experiment was carried out to validate NASSI based on current equipment in the laboratory. A concave paraboloidal mirror with a vertex radius of 240.0 mm and a clear aperture of 101.0 mm was tested (wavelength λ 632.8 mm). The PNL employed is the same as the one in the numerical simulation, whose parameters are specified in Table 1. The PV values of the surface error of the PNL and reference mirror are better than λ∕20 and λ∕10, respectively. The precision linearity rail, with 1000 mm full travel, can achieve 1 um positioning accuracy. The detector employed is a 1392 × 1040 pixel CCD camera. The Renishaw DMI can precisely measure distances in range of 40 m, with an accuracy of up to 0.5 ppm. In consideration of noise and adjustment errors, the maximum slope of the subaperture wavefront vmax < vN ∕4 0.125λ∕pixel is employed to promise a large redundancy. The aspheric is translated along
Table 4.
Parameters of Subapertures in the Experiment
Subaperture dpa (mm) Normalized zone
0
1
2
302.2155 [0,0.6]
302.0473 [0.5,0.8]
302.0241 [0.7,1]
the optical axis gradually and the displacement distance is monitored by the DMI. Three subapertures were obtained with overlap of 0.1 normalized width. The specific parameters are presented in Table 4 and corresponding interferograms are presented in Fig. 7(a). After the ROR method was implemented on each subaperture, the figure errors of the three subapertures were retrieved accurately, which are presented in Fig. 7(b). Subsequently, the simultaneous stitching algorithm was performed to obtained a full aperture figure error (here 98% of full aperture), whose 2D and 3D maps are shown in Fig. 8(a). The PV and rms values of the figure error are 0.2724λ and 0.0310λ, respectively. We also measured the paraboloidal mirror employing a ZYGO interferometer with aberration-free point method for cross test, in which the figure error of the auxiliary flat mirror achieves better than 1∕20λ. Figure 8(b) refers to the 2D and 3D maps of the figure error obtained from the ZYGO interferometer which serve as the criterion in this paper. The PV and rms values are 0.326λ and 0.037λ, respectively. Another concave paraboloidal mirror with the same nominal parameters was also measured. Figures 9(a) and 9(b) present the test results by NASSI and the Zygo interferometer, respectively. The PV and rms values of the figure error test by NASSI are 0.2660λ and 0.0350λ, respectively, and those by the Zygo interferometer are 0.277λ and 0.037λ, respectively. From Figs. 8 and 9, it is found that the test result by NASSI has a basic consistency with those of the Zygo interferometer. The PV and rms values of the test results of the two aspherics are specified in
Fig. 8. Testing results in NASSI and Zygo. (a) The reconstructed result in NASSI and (b) the surface figure error tested by Zygo.
Fig. 9. Testing results in NASSI and Zygo. (a) The reconstructed result in NASSI and (b) the surface figure error tested by Zygo.
Table 5.
PV (λ) rms (λ)
PV and RMS Values of Figure Error by NASSI and Zygo Interferometer
The First Aspheric
The Second Aspheric
NASSI
Zygo
NASSI
Zygo
0.2724 0.0310
0.326 0.037
0.2660 0.0350
0.277 0.037
Table 5, which shows the validity and high accuracy of NASSI. Of course, small variations between the results of NASSI and the Zygo interferometer may be induced by the combination of random noise, modeling errors, stitching errors, Zernike fitting errors, and so on. But these errors are small and also exist in many other mainstream ASSI methods. 6. Conclusion Fig. 7. Interferograms and figure error in NASSI. (a) The subaperture interferograms and (b) the subaperture figure error by ROR.
We proposed a NASSI for steep aspheric testing, in which the PNL is employed as an alternative to the transmission sphere in standard ASSI, and the ROR method is employed to retrieve each 1 September 2014 / Vol. 53, No. 25 / APPLIED OPTICS
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subaperture figure error. In NASSI, the subaperture number is decreased greatly in comparison to standard ASSI, which means a sufficient overlapping area for misalignments correction and less error accumulation. Combining with the ROR method for retrace error correction, high measurement accuracy is allowed. From both numerical simulation and experiment, NASSI exhibits good performance in testing accuracy. Furthermore, NASSI combines the subaperture stitching ideal with the partcompensation method, extending the dynamic test range greatly. Also it would be time saving to test steep aspheric. The integrated NASSI workstation and more easy and accurate algorithms for full aperture reconstruction are in study in laboratory, which will be reported in our future publications. This work was partially supported by the National Natural Science Foundation of China (41305014), the State Key Laboratory of Modern Optical Instrumentation Innovation Program (MOI201208), the “985” III: First-Class Discipline Construction Program, and the the Fundamental Research Funds for the Central Universities (2013QNA5006). References 1. J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19–27 (1982). 2. W. W. Chow and G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. 8, 468–470 (1983). 3. J. E. Negro, “Subaperture optical system testing,” Appl. Opt. 23, 1921–1930 (1984). 4. S. C. Jensen, W. W. Chow, and G. N. Lawrence, “Subaperture testing approaches: a comparison,” Appl. Opt. 23, 740–745 (1984). 5. C. R. De Hainaut and A. Erteza, “Numerical processing of dynamic subaperture testing measurements,” Appl. Opt. 25, 503–509 (1986). 6. M. A. Schmucker and J. Schmit, “Selection process for sequentially combing multiple sets of overlapping surface profile interferometric data to produce a continuous composite map,” U.S. patent 5,991,461 (23 November 1999). 7. M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33, 608–613 (1994). 8. L. Yan, X. Wang, L. Zheng, X. Zeng, H. Hu, and X. Zhang, “Experimental study on subaperture testing with iterative triangulation algorithm,” Opt. Express 21, 22628–22644 (2013). 9. P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Opt. Photon. News 14, 38–43 (2003).
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