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Experimental study on subaperture

testing with iterative stitching algorithm

Shanyong Chen, Shengyi Li, Yifan Dai, Lingyan Ding, Shengyue Zeng

College of Mechatronic Engineering and Automation, National University of Defense

Technology,

Changsha, Hunan, PR China, 410073

shanyongchen@tom.com

Abstract:

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.

Applying the iterative stitching algorithm, we demonstrate the

© 2008 Optical Society of America

OCIS codes: (120.3180) Interferometry; (120.4630) Optical inspection; (220.4840) Optical

testing

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).

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).

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).

4. P. Murphy, J. Fleig, G. Forbes, et al., “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE

6293, 62930J-1–62930J-10 (2006).

5. J.G. Thunen, and O. Y. Kwon, “Full aperture testing with subaperture test optics,” Proc. SPIE351, 19–27 (1982).

6. W. W. Chow, and G. N. Lawrence, “Method for subaperture testing interferogram reduction,” Opt. Lett. 8,

468–470 (1983).

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).

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).

10. S. H. Tang, “Stitching: high-spatial-resolution microsurface measurements over large areas,” Proc. SPIE 3479,

43–49 (1998).

11. M.Sj¨ oedahl, B.F.Oreb,“Stitching interferometric measurement data forinspection oflarge optical components,”

Opt. Eng. 41, 403–408 (2002).

12. L. Zhou, Y. F. Dai, X. H. Xie, et al.,“Model and method to determine dwell time in ion beam figuring,” Nan-

otechnology 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).

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1.Introduction

Based on the idea of “stitching”, the subaperture testing (SAT) method provides an alterna-

tive to full aperture interferometric testing of optical surfaces. It is advantageous especially for

those difficulttobe tested with a commercialinterferometer,suchas largeflat 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, 2, 3] for a short overview of related work involved in SAT. Recently QED technologies

added asphere metrology capability to their new product, SSIA, built upon the performance of

the subaperture stitching interferometer (SSI) [4]. 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 intro-

duced duringalignmentand nullingfor subaperturesis more difficult to be small enoughwithin

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]and the simultaneous fit method [6], to the discrete phase method [7], the

multiaperture overlap-scanning technique [8], and then to QED’s method with free and inter-

locked compensators [9]. 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,oneforthe planarinterferometer[1], andthe otherforthe spheri-

cal one [2]. Advantagesof the algorithmwere claimedand verifiedthroughsimulations, though

few experiments were conducted, except the SAT of a large spherical surface [3].

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 as-

pheric surfaces. The features of the algorithm are summarizedto 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 interfer-

ometer. 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 trans-

formed into the three-dimensional Cartesian coordinate frame. Denote the measurement data

of interferometer by phase triplets W = (u,v,φ), where φ is the phase difference on the pixel

(u,v). According to the test geometry with a planar Fizeau interferometer, the object coordi-

nates are related as follows

[x,y,z] = [βu,βv,φ]

where β is the scale of lateral coordinates. For SAT with a spherical Fizeau interferometer, the

object coordinates are a little more complicated

?

where rtsis the radius of the transmission sphere, r is the radius of the best-fit sphere for the

subaperture, and β = γ/rts. γ denotes the scale of lateral coordinates.

(1)

[x,y,z] =(r+φ)βu,(r+φ)βv,rts−(r+φ)

?

1−β2(u2+v2)

?

(2)

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Received 28 Feb 2008; revised 10 Mar 2008; accepted 10 Mar 2008; published 24 Mar 2008

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The above object coordinates are described in each subaperture (local frame) and need to be

transformed into the global Cartesian frame

fiwj,i= g−1

i

[xj,i,yj,i,zj,i,1]T

(3)

where i = 1,2,···,s, and s is the number of subapertures. j = 1,2,···,Ni, and Niis the number

of sampling points in the ith subaperture. gidenotes the configuration of the global frame with

regard to the local frame of the ith subaperture.

The SASL algorithmservesto compensatethe uncertaintyofconfigurationgi, lateral scale βi

and radius rifor the spherical version. The problem is divided into two iterative subproblems,

the overlapping calculation subproblem and the configuration optimization subproblem. With

the parametersgi, βiand rifixed,the computer-aideddesign (CAD) model is utilized to find the

overlapping point pairs in the global Cartesian coordinate frame. With overlapping correspon-

dence 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 overlap-

ping 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 sub-

apertures 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 trans-

lations 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) uncer-

tainties of alignment and nulling, as well as uncertainties of the radii of best-fit spheres. Conse-

quently 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] estimated 6-dof motion uncertainties by chi-square fitting of

the deviations at overlapping points. Sj¨ oedahl and Oreb [11] 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 prob-

lem by means of Lie group, which identifies the configuration space of symmetric features.

Consequently the problem is parameterized without redundancy. And the blockwise QR de-

composition procedure is suggested to replace the SVD, in order to avoid the out-of-memory

problem [2].

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)).

Totally11subaperturesaretestedonebyone,asshowninFig.1(b).Figure2givesthemeasured

results of subaperture 1 and subaperture 8, and the retrieved full aperture error with PV (peak-

valley) and RMS (root-mean-square)values, respectively.

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

Conventional

Phase data are directly treated.

Only overlapping points are treated.

Typically piston, tilts and power are

removed.

Typically without iteration.

Consumes less memory and time.

SASL

TransformedCartesiancoordinatesaretreated.

All measurement points are treated.

Uncertainties of 6-dof motion and radii of

best-fit spheres are compensated.

Alternating optimization with iterations.

Consumes more memory and time.

(a) Experimental setup

10

9

11

12

3

4

5

6

78

(b) Lattice scheme

Fig. 1. SAT of a SiC flat mirror.

(a) Measured subaperture 1(b) Measured subaperture 8 (c) Full aperture (PV 3.24λ, RMS 0.40λ)

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 pro-

vided for corrective machining of the mirror by ion beam figuring (IBF) [12]. 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 excludingthe 10 mm closest to the part edge (Fig. 3(a)).Second, the mirror is finally

tested with an 800 mm apertureinterferometer,which gives a consistent figure error(Fig. 3(b)).

(a) SAT (PV 0.17λ, RMS 0.012λ) (b) Full aperture test (PV 0.16λ, RMS 0.016λ)

Fig. 3. Final results of the SiC flat mirror

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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]. It failed at local irregularities while the 37-subaperture stitching result suc-

ceeded 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 previ-

ously 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.

(a) SAT (PV 2.08λ, RMS 0.275λ) (b) Full aperture test with Zygo interferometer (PV 2.19λ, RMS 0.278λ)

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

640mm.Theasphericity(about8.7μm)slightlyexceedstheverticalrangeofaninterferometer.

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)).

(a) Auto-collimation testing(b) SAT

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], the SASL algorithm demands no data preprocessing

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31 March 2008 / Vol. 16, No. 7 / OPTICS EXPRESS 4764