Experimental study on subaperture testing with iterative stitching algorithm

doi:10.1364/OE.16.004760

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Experimental study on subaperture testing with iterative stitching algorithm

Shanyong Chen, Shengyi Li, Yifan Dai, Lingyan Ding, and Shengyue Zeng »View Author Affiliations

Optics Express, Vol. 16, Issue 7, pp. 4760-4765 (2008)
http://dx.doi.org/10.1364/OE.16.004760

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▸ Article Outline
– Abstract
1. Introduction
2. The iterative stitching algorithm
3. Experimental verification
4. Conclusion
– References and links

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Abstract

Applying the iterative stitching algorithm, we demonstrate the power of subaperture testing through experiments. Naturally the algorithm applies to flats, spherical or aspheric surfaces. We first apply it to a silicon carbide flat mirror with larger aperture than the interferometer’s. The testing results help to obtain a high-precision mirror through five iterations of ion beam figuring. The second experiment is 37-subaperture testing of a large spherical mirror. Good consistence is observed between the stitching result and the full aperture test result using a Zygo interferometer. Finally we study the applicability of the algorithm to subaperture testing of a parabolic surface. The stitching result is consistent with the auto-collimation test result. Furthermore, the surface is tested with annular subapertures and also retrieved by our algorithm successfully.

1. Introduction

Based on the idea of “stitching”, the subaperture testing (SAT) method provides an alternative to full aperture interferometric testing of optical surfaces. It is advantageous especially for those difficult to be tested with a commercial interferometer, such as large flat mirrors, spherical surfaces with high numerical aperture, large convex surfaces, and aspheric surfaces exceeding the vertical range of the interferometer. Readers are referred to our previously published papers [1

S. Y. Chen, S. Y. Li, and Y. F. Dai, “Iterative algorithm for subaperture stitching interferometry for general surfaces,” J. Opt. Soc. Am. A 22,1929–1936 (2005). [CrossRef]

, 2

S.Y. Chen, S. Y. Li, Y. F. Dai, and Z. W. Zheng, “Iterative algorithm for subaperture stitching test with spherical interferometers,” J. Opt. Soc. Am. A 23,1219–1226 (2006). [CrossRef]

, 3

S. Y. Chen, S. Y. Li, Y. F. Dai, and Z. W. Zheng, “Testing of large optical surfaces with subaperture stitching,” Appl. Opt. 46,3504–3509 (2007). [CrossRef] [PubMed]

] for a short overview of related work involved in SAT. Recently QED technologies added asphere metrology capability to their new product, SSI_A, built upon the performance of the subaperture stitching interferometer (SSI) [4

P. Murphy, J. Fleig, and G. Forbes, et al., “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J-1–62930J-10 (2006). [CrossRef]

]. These products are competent for testing of optics up to 200 mm in diameter. For larger parts, the mechanical and optical structures need some modifications, since it is not wise to move the part. Moreover, the mechanical error introduced during alignment and nulling for subapertures is more difficult to be small enough within a larger stroke. Hence better performance is expected for the subaperture stitching algorithm.

In fact, the subaperture stitching algorithm has played a vital role all along the application of the SAT method. In our opinion, it was the algorithmic improvement that marked each stage of development of the SAT method. Obvious improvements can be observed from the Kwon-Thunen method [5

J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19–27 (1982).

]and the simultaneous fit method [6

W. W. Chow and G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. 8, 468–470 (1983). [CrossRef] [PubMed]

], to the discrete phase method [7

T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE 656, 118–127 (1986).

], the multiaperture overlap-scanning technique [8

M. Y. Chen, W. M. Cheng, and C. W. Wang, “Multiaperture overlap-scanning technique for large-aperture test.” Proc. SPIE 1553, 626–635 (1991). [CrossRef]

], and then to QED’s method with free and interlocked compensators [9

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer work-station for spherical and aspherical surfaces,” Proc. SPIE 5188, 296–307 (2003). [CrossRef]

]. Aiming to testing of large surfaces with big error of alignment and nulling, we proposed an iterative algorithm, the subaperture stitching and localization (SASL) algorithm. It has two versions, one for the planar interferometer [1

S. Y. Chen, S. Y. Li, and Y. F. Dai, “Iterative algorithm for subaperture stitching interferometry for general surfaces,” J. Opt. Soc. Am. A 22,1929–1936 (2005). [CrossRef]

], and the other for the spherical one [2

S.Y. Chen, S. Y. Li, Y. F. Dai, and Z. W. Zheng, “Iterative algorithm for subaperture stitching test with spherical interferometers,” J. Opt. Soc. Am. A 23,1219–1226 (2006). [CrossRef]

]. Advantages of the algorithm were claimed and verified through simulations, though few experiments were conducted, except the SAT of a large spherical surface [3

S. Y. Chen, S. Y. Li, Y. F. Dai, and Z. W. Zheng, “Testing of large optical surfaces with subaperture stitching,” Appl. Opt. 46,3504–3509 (2007). [CrossRef] [PubMed]

We continue to show the validity of the SASL algorithm through experiments. In Section 2, the algorithm is reviewed to show the nature of applicability to SAT of planar, spherical or aspheric surfaces. The features of the algorithm are summarized to clarify the difference from the conventional algorithm. In Section 3, we first demonstrate the power of the algorithm applied to a silicon carbide flat mirror with larger aperture than the interferometer’s. The testing results help to obtain a high-precision mirror through five iterations of ion beam figuring. The second experiment is 37-subaperture testing of a large spherical mirror. Good consistence is observed between the subaperture stitching result and the full aperture test result using a Zygo interferometer. Finally we study the applicability of the algorithm to SAT of a parabolic surface. The stitching result is consistent with the auto-collimation test result. Furthermore, the surface is tested with annular subapertures and again retrieved by our algorithm.

2. The iterative stitching algorithm

In the SASL algorithm, both the planar and the spherical versions, all phase data are transformed into the three-dimensional Cartesian coordinate frame. Denote the measurement data of interferometer by phase triplets W=(u,ν,ϕ), where ϕ is the phase difference on the pixel (u,ν). According to the test geometry with a planar Fizeau interferometer, the object coordinates are related as follows

[x, y, z] = [β u, β ν, ϕ]

(1)

where β is the scale of lateral coordinates. For SAT with a spherical Fizeau interferometer, the object coordinates are a little more complicated

[x, y, z] = [(r + ϕ) β u, (r + ϕ) β ν, r_{ts} - (r + ϕ) \sqrt{1 - β^{2} (u^{2} + ν^{2})}]

(2)

where r _ts is the radius of the transmission sphere, r is the radius of the best-fit sphere for the subaperture, and β=γ/r _ts. γ denotes the scale of lateral coordinates.

The above object coordinates are described in each subaperture (local frame) and need to be transformed into the global Cartesian frame

f_{i} w_{j, i} = g_{i}^{-1} {[x_{j, i}, y_{j, i}, z_{j, i}, 1]}^{T}

(3)

where i=1,2, …, s, and s is the number of subapertures. j=1,2, …, N_i , and N_i is the number of sampling points in the ith subaperture. g_i denotes the configuration of the global frame with regard to the local frame of the ith subaperture.

The SASL algorithm serves to compensate the uncertainty of configuration g_i , lateral scale β_i and radius r_i for the spherical version. The problem is divided into two iterative subproblems, the overlapping calculation subproblem and the configuration optimization subproblem. With the parameters g_i , β_i and r_i fixed, the computer-aided design (CAD) model is utilized to find the overlapping point pairs in the global Cartesian coordinate frame. With overlapping correspondence fixed, the configuration optimization subproblem is linearized as a linear least-squares problem and solved to obtain the optimal parameters. The two subproblems are alternatingly solved until the program converges to an acceptable tolerance.

In Table 1, we summarize features of the SASL algorithm compared with the conventional algorithm. First, it uses a unique method to determine the precise correspondence of overlapping point pairs automatically. The transformed Cartesian coordinates in the global frame are adopted to calculate the projections of all measurement points to the CAD model. Then these projections are again projected on the OXY plane and the overlapping points are recognized according to the convex hull. It is convenient to the alternating optimization. Once we get new configuration parameters after each iteration, we update the overlapping correspondence accordingly. Hence the least data preprocessing is demanded and generally it applies to subapertures of any geometrical shape. The disadvantage as a side effect is that all measurement points must be treated simultaneously. While in the conventional algorithm, the overlapping points are determined with nominal motions. It is straightforward for the flat, since only translations exist between two subapertures nominally. However, for curved surfaces tested with a spherical interferometer, the problem becomes more complicated.

Second, the alternating iterations ensure compensation of 6-dof (degree-of-freedom) uncertainties of alignment and nulling, as well as uncertainties of the radii of best-fit spheres. Consequently it is unnecessary to align and null the subapertures precisely as planned, which means a coarse platform is competent. Similar researches were published before, though they were mostly applied to flats. Tang [10

S. H. Tang, “Stitching: high-spatial-resolution microsurface measurements over large areas,” Proc. SPIE 3479, 43–49 (1998). [CrossRef]

] estimated 6-dof motion uncertainties by chi-square fitting of the deviations at overlapping points. Sjöedahl and Oreb [11

M. Sjöedahl and B. F. Oreb, “Stitching interferometric measurement data for inspection of large optical components,” Opt. Eng. 41, 403–408 (2002). [CrossRef]

] proposed an iterative method and adopted singular-value decomposition (SVD) to obtain optimal estimation of six parameters. QED technologies also suggests compensation of translation and orientation, in addition to the piston, tilts and power. The SASL algorithm differs from them in the modeling of the problem by means of Lie group, which identifies the configuration space of symmetric features. Consequently the problem is parameterized without redundancy. And the blockwise QR decomposition procedure is suggested to replace the SVD, in order to avoid the out-of-memory problem [2

S.Y. Chen, S. Y. Li, Y. F. Dai, and Z. W. Zheng, “Iterative algorithm for subaperture stitching test with spherical interferometers,” J. Opt. Soc. Am. A 23,1219–1226 (2006). [CrossRef]

3. Experimental verification

3.1. Silicon carbide flat mirror

In this experiment, the clear aperture of a SiC flat mirror is 225 mm×161 mm (elliptical). The SAT method is utilized with a 4″ interferometer and 2-dof adjustment platform (Fig. 1(a)). Totally 11 subapertures are tested one by one, as shown in Fig. 1(b). Figure 2 gives the measured results of subaperture 1 and subaperture 8, and the retrieved full aperture error with PV (peak-valley) and RMS (root-mean-square) values, respectively.

Table 1. Comparison between the SASL algorithm and the conventional algorithm

Conventional	SASL
Phase data are directly treated.	Transformed Cartesian coordinates are treated.
Only overlapping points are treated.	All measurement points are treated.
Typically piston, tilts and power are removed.	Uncertainties of 6-dof motion and radii of best-fit spheres are compensated.
Typically without iteration.	Alternating optimization with iterations.
Consumes less memory and time.	Consumes more memory and time.

Fig. 1. SAT of a SiC flat mirror.

Fig. 2. Measured subapertures and retrieved full aperture error of the SiC flat mirror.

The validity of the algorithm is supported by two experiments. First, the SAT result is provided for corrective machining of the mirror by ion beam figuring (IBF) [12

L. Zhou, Y. F. Dai, and X. H. Xie, et al.,“Model and method to determine dwell time in ion beam figuring,” Nanotechnology and Precision Engineering 5,107–112 (2007).

]. After 5 iterations of figuring and testing, the RMS error of the mirror is converged from the original 0.4λ to less than λ/50 excluding the 10 mm closest to the part edge (Fig. 3(a)). Second, the mirror is finally tested with an 800 mm aperture interferometer, which gives a consistent figure error (Fig. 3(b)).

Fig. 3. Final results of the SiC flat mirror

3.2. Large spherical mirror

This experiment is designed to verify the large aperture testing capability of the SAT method. Actually it was published, where the full aperture test was conducted with a low-performance interferometer [3

S. Y. Chen, S. Y. Li, Y. F. Dai, and Z. W. Zheng, “Testing of large optical surfaces with subaperture stitching,” Appl. Opt. 46,3504–3509 (2007). [CrossRef] [PubMed]

]. It failed at local irregularities while the 37-subaperture stitching result succeeded to retrieve the full aperture error. To further verify the correctness of the SAT result, a Zygo interferometer is used for full aperture test. The results are shown in Fig. 4. It is easy to see the good consistence between them. Note that Fig. 4(a) is slightly different from the previously published result. Actually there are slight traces indicating the brims of subapertures in the latter. It is improved here by choosing more appropriate weights in the objective functions.

Fig. 4. Full aperture error of the spherical mirror.

3.3. Parabolic mirror

A parabolic mirror is tested to verify the asphere metrology capability of the SAT method. The clear aperture of the mirror is about 185 mm, and the radius of curvature at the vertex is about 640 mm. The asphericity (about 8.7µm) slightly exceeds the vertical range of an interferometer. The mirror is tested with the auto-collimation method (Fig. 5(a)) for the purpose of cross test, as shown in Fig. 8. The central obstruction is about 52 mm in diameter.

A simple 5-dof platform is built for SAT of the mirror. Three translations (X,Y, and Z) are numerically controlled, while two rotational tables (yaw and pitch) are adjusted manually (Fig. 5(b)). The error of alignment and nulling is rather big without any mechanical calibration. Totally 7 subapertures (the central one plus 6 off-axis subapertures) are tested. Figure 6 gives the measured results of subaperture 1 and subaperture 2, respectively. The full aperture error (deviations from the paraboloid) is retrieved using the spherical version of the SASL algorithm after 40 iterations (Fig. 8(b)).

Fig. 5. Experimental setup for testing a parabolic mirror.

Annular SAT of the mirror is also conducted on the same platform with three overlapping annular subapertures (Fig. 7). The SASL algorithm again succeeds to retrieve the full aperture error after 37 iterations(Fig. 8(c)). Surface errors obtained from the three methods seem to be identically distributed, though differences exist in PV and RMS values. Compared with the algorithm proposed by Hou et al. [13

X. Hou, F. Wu, L. Yang, and Q. Chen,“Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method,” Opt. Express 15,12890–12899 (2007). [CrossRef] [PubMed]

], the SASL algorithm demands no data preprocessing and the subapertures are not complementary. The only requirement is existence of appropriate overlapping region between two adjacent subapertures.

Fig. 6. Measured subapertures of the parabolic mirror.

Fig. 7. Measured annular subapertures of the parabolic mirror.

Fig. 8. Full aperture error of the parabolic mirror obtained from conventional null test as well as two different SAT methods (all with coma removed).

4. Conclusion

From the mathematical point of view, the SASL algorithm is applicable to SAT with both planar and spherical interferometers. It is verified through several experiments in applications of a flat mirror, a large spherical mirror and a parabolic mirror, respectively. The parabolic mirror is also retrieved successfully using the SASL algorithm with annular subapertures, which confirms that the algorithm applies to subapertures of any geometrical shape.

Acknowledgements

This work was supported with National Natural Science Fundation of China.

References and links

1.	S. Y. Chen, S. Y. Li, and Y. F. Dai, “Iterative algorithm for subaperture stitching interferometry for general surfaces,” J. Opt. Soc. Am. A 22,1929–1936 (2005). [CrossRef]
2.	S.Y. Chen, S. Y. Li, Y. F. Dai, and Z. W. Zheng, “Iterative algorithm for subaperture stitching test with spherical interferometers,” J. Opt. Soc. Am. A 23,1219–1226 (2006). [CrossRef]
3.	S. Y. Chen, S. Y. Li, Y. F. Dai, and Z. W. Zheng, “Testing of large optical surfaces with subaperture stitching,” Appl. Opt. 46,3504–3509 (2007). [CrossRef] [PubMed]
4.	P. Murphy, J. Fleig, and G. Forbes, et al., “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J-1–62930J-10 (2006). [CrossRef]
5.	J. G. Thunen and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE 351, 19–27 (1982).
6.	W. W. Chow and G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. 8, 468–470 (1983). [CrossRef] [PubMed]
7.	T. W. Stuhlinger, “Subaperture optical testing: experimental verification,” Proc. SPIE 656, 118–127 (1986).
8.	M. Y. Chen, W. M. Cheng, and C. W. Wang, “Multiaperture overlap-scanning technique for large-aperture test.” Proc. SPIE 1553, 626–635 (1991). [CrossRef]
9.	J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer work-station for spherical and aspherical surfaces,” Proc. SPIE 5188, 296–307 (2003). [CrossRef]
10.	S. H. Tang, “Stitching: high-spatial-resolution microsurface measurements over large areas,” Proc. SPIE 3479, 43–49 (1998). [CrossRef]
11.	M. Sjöedahl and B. F. Oreb, “Stitching interferometric measurement data for inspection of large optical components,” Opt. Eng. 41, 403–408 (2002). [CrossRef]
12.	L. Zhou, Y. F. Dai, and X. H. Xie, et al.,“Model and method to determine dwell time in ion beam figuring,” Nanotechnology and Precision Engineering 5,107–112 (2007).
13.	X. Hou, F. Wu, L. Yang, and Q. Chen,“Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method,” Opt. Express 15,12890–12899 (2007). [CrossRef] [PubMed]

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.4630) Instrumentation, measurement, and metrology : Optical inspection
(220.4840) Optical design and fabrication : Testing

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: February 28, 2008
Revised Manuscript: March 10, 2008
Manuscript Accepted: March 10, 2008
Published: March 24, 2008

Citation
Shanyong Chen, Shengyi Li, Yifan Dai, Lingyan Ding, and Shengyue Zeng, "Experimental study on subaperture testing with iterative stitching algorithm," Opt. Express 16, 4760-4765 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4760

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References

S. Y. Chen, S. Y. Li, and Y. F. Dai, "Iterative algorithm for subaperture stitching interferometry for general surfaces," J. Opt. Soc. Am. A 22, 1929-1936 (2005). [CrossRef]
S. Y. Chen, S. Y. Li, Y. F. Dai, and Z. W. Zheng, "Iterative algorithm for subaperture stitching test with spherical interferometers," J. Opt. Soc. Am. A 23, 1219-1226 (2006). [CrossRef]
S. Y. Chen, S. Y. Li, Y. F. Dai, and Z. W. Zheng, "Testing of large optical surfaces with subaperture stitching," Appl. Opt. 46, 3504-3509 (2007). [CrossRef] [PubMed]
P. Murphy, J. Fleig, G. Forbes, et al., "Subaperture stitching interferometry for testing mild aspheres," Proc. SPIE 6293, 62930J-1-62930J-10 (2006). [CrossRef]
J. G. Thunen and O. Y. Kwon, "Full aperture testing with subaperture test optics," Proc. SPIE 351, 19-27 (1982).
W. W. Chow and G. N. Lawrence, "Method for subaperture testing interferogram reduction," Opt. Lett. 8, 468-470 (1983). [CrossRef] [PubMed]
T. W. Stuhlinger, "Subaperture optical testing: experimental verification," Proc. SPIE 656, 118-127 (1986).
M. Y. Chen, W. M. Cheng, and C. W. Wang, "Multiaperture overlap-scanning technique for large-aperture test," Proc. SPIE 1553, 626-635 (1991). [CrossRef]
J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, "An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces," Proc. SPIE 5188, 296-307 (2003). [CrossRef]
S. H. Tang, "Stitching: high-spatial-resolution microsurface measurements over large areas," Proc. SPIE 3479, 43-49 (1998). [CrossRef]
M. Sjöedahl and B. F. Oreb, "Stitching interferometric measurement data for inspection of large optical components," Opt. Eng. 41, 403-408 (2002). [CrossRef]
L. Zhou, Y. F. Dai, X. H. Xie, et al.,"Model and method to determine dwell time in ion beam figuring," Nanotechnology and Precision Engineering 5, 107-112 (2007).
X. Hou, F. Wu, L. Yang, and Q. Chen,"Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method," Opt. Express 15, 12890-12899 (2007). [CrossRef] [PubMed]

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