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1 Opto-Electronic Engineering Article DOI:.86/oee a b k k M.5 m mm O46. A. [J]. 8 45(5): 768 Optimization of sparse subaperture arra model for stitching detection of plane wavefront Institute of Optics and Electronics, Chinese Academ of Sciences, Chengdu, Sichuan 69, China; Universit of Chinese Academ of Sciences, Beijing 49, China Abstract: The sparse subaperture stitching, the accurac of which is closel related to the arrangement, number and size of subapertures, is one of the main methods of qualit testing for large aperture optical sstems. A mathematical model was established to deduce the relation curve between the subaperture number k and fill factor M when the value of k ranges from one to infinit. As a result, the optimal arrangement laout, consisting of seven sparse subapertures, was obtained for the detection sstems below.5 m. Autocollimation interference detection of mm validated the rationalit of the arrangement. Kewords: sparse subaperture; mathematical modeling; stitching detection; wavefront reconstruction; interferometr Citation: Luo Q, Wu S B, Wang L H, et al. Optimization of sparse subaperture arra model for stitching detection of plane wave front[j]. Opto-Electronic Engineering, 8, 45(5): (6YFB5) (99-) luoqian_@6.com 768-

2 DOI:.86/oee [-] Arizona Kim [] [] [4] [5] RC [6] [7-8] [9-] [5] ) ) Sk kr M () S R S k k r S R gola6 [-] M R= r k M a b Fig. Schematic diagram of sparse subaperture detection 768-

DOI:.86/oee.8.768 ) ) ) 4) k 68 46 4 48 6n R r r 5r 7r 9r r r 5r 7r +(n-) n ) r k 8 9 5 a) r k 8 4 b) r k 9 5 4 k 9 k M k=7.77778 k=9.76. M..8.6.4 (7,.7778) (9,.76). Fig.

3 DOI:.86/oee ) ) ) 4) k n R r r 5r 7r 9r r r 5r 7r +(n-) n ) r k a) r k 8 4 b) r k k 9 k M k= k=9.76. M (7,.7778) (9,.76). Fig. Honecomb-shaped arrangement of subapertures M ) r k= R=r M= ) r k k (k ~9) Fig. The relation curve between subaperture number and filling factor (k=~9) k 4 k M n(n+) M=.7778 M=.76 M=.755 k M (k ~7) Table Subaperture number k and fill factor M (k= ~7) k M k M (k 8~9) Table Subaperture number k and fill factor M (k= 8~9) k M

DOI:.86/oee.8.768 M=.75 M=.75 M lim[ n( n )] n 4n.75 n M. (, ). (7,.77778) (7,.755) (9,.75) (69,.75).8 (9,.76) (6,.75) (7,.754) (7,.

4 The relation curve between subaperture number and filling factor (k=~ ) Zgo GPI mm 6 7 k=7 M=.77778 k= M= 5 R= (a) (c) 6 Fig.

4 DOI:.86/oee M=.75 M=.75 M lim[ n( n )] n 4n.75 n M. (, ). (7,.77778) (7,.755) (9,.75) (69,.75).8 (9,.76) (6,.75) (7,.754) (7,.758).6 (,.5) k 4 k M (k ~ ) Fig. 4 The relation curve between subaperture number and filling factor (k=~ ) Zgo GPI mm 6 7 k=7 M= k= M= 5 R= (a) (c) 6 Fig. 6 Photograph of eperimental setup (b) (d) (-, ) (, ) (, ) (-, - ) (, ) (, - ) (,- ) (e) (f) 5 Fig. 5 Optimal arrangement 7 Fig. 7 Sparse subaperture arrangement diagrams with four to nine subapertures 8 Zgo PV.8λ RMS.8λ 7 RMS 768-4

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(c) Si subapertures; (d) Seven subapertures; (e)

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direct detection with multiple subapertures.

5 DOI:.86/oee 光电工程 8 ອสШ c ໜ ԅ (a). Surface pattern after removing phase constant andtilt (b) (d) (f) (e)..4. (c) Fig ങӾࡔ ᆐ ࡅଳࠄ Љԅ௦ ࡅϏ d(a) ങ ᆐ ࡅĢ(b) ๔ ᆐ ࡅĢ(c) ঢ ᆐ ࡅĢ(d) ᆐ ࡅĢ(e) ᆐ ࡅĢ(f) ࡔ ᆐ ࡅ Fig. 9 Full aperture wavefront mapping for multiple subaperture stitching detection. (a) Four subapertures; (b) Five subapertures; (c) Si subapertures; (d) Seven subapertures; (e) Eight subapertures; (f) Nine subapertures (d) (c) (e)..6 (f) (b) (a) ങӾࡔ ᆐ ࡅଳࠄ Љ ᄐࠄ ЉϏ ԅϸе d(a) ങ ᆐ ࡅĢ(b) ๔ ᆐ ࡅĢ(c) ঢ ᆐ ࡅĢ (d) ᆐ ࡅĢ(e) ᆐ ࡅĢ(f) ࡔ ᆐ ࡅ Fig. Wavefront residuals between stitching and direct detection with multiple subapertures. (a) Four subapertures; (b) Five subapertures; (c) Si subapertures; (d) Seven subapertures; (e) Eight subapertures; (f) Nine subapertures 768-5

6 DOI:.86/oee Table Comparison of parameters of stitching and direct detection No. Wavefront PV/ Wavefront RMS/ Wavefront residuals PV/ Wavefront residuals RMS/ RMS.9λ RMS.8λ % RMS.9λ 4 M k k M.5 m.5 m [] Hou X, Wu F, Yang L, et al. Status and development trend of sub-aperture stitching interferometric testing technique[j]. Optics & Optoelectronic Technolog, 5, (): 5 5.,,,. [J]., 5, (): 5 5. [] Sjödahl M, Oreb B F. Stitching interferometric measurement data for inspection of large optical components[j]. Optical Engineering,, 4(): [] Kim C J. Polnomial fit of interferograms[j]. Applied Optics, 98, (4): [4] He Y, Wang Z X, Wang Q, et al. Testing the large aperture optical components b the sub-aperture stitching interferometer [J]. Proceedings of SPIE, 7, 664: 664D. [5] Xu X D, Shen Z X, Tong G D, et al. Sparse subaperture stitching method for measuring large aperture planar optics[j]. Optical Engineering, 6, 55(): 4. [6] Yan F T, Fan B, Hou X, et al. Large-aperture mirror test using sparse sub-aperture sampling[j]. High Power Laser and Particle Beams,, (): 9 96.,,,. [J].,, (): [7] Chow W W, Lawrence G N. Method for subaperture testing interferogram reduction[j]. Optics Letters, 98, 8(9): [8] Wang Y, Fu R M, Liao Z B. Wavefront reconstruction algorithm based on sparse aperture[j]. Spacecraft Recover & Remote Sensing, 5, 6(5): 5 59.,,. [J]., 5, 6(5): [9] Chen S Y, Dai Y F, Li S Y, et al. Error reductions for stitching test of large optical flats[j]. Optics & Laser Technolog,, 44(5): [] Smith G A, Burge J H. Subaperture stitching surface errors due to noise[j]. Proceedings of SPIE, 5, 9575: 9575W. [] Hill J M, Salinari P. The large binocular telescope project[j]. Proceedings of SPIE,, 5489: [] Chung S J, Miller D W, De Weck O L. Design and implementation of sparse aperture imaging sstems[j]. Proceeding of SPIE,, 4849:

7 DOI:.86/oee Optimization of sparse subaperture arra model for stitching detection of plane wavefront Institute of Optics and Electronics, Chinese Academ of Sciences, Chengdu, Sichuan 69, China; Universit of Chinese Academ of Sciences, Beijing 49, China (-, ) (-, - ) (, ) (, ) (, - ) (, ) (, - ) Optimal arrangement Overview: The sparse subaperture stitching is one of the main methods of qualit detection for large and oversized aperture optical sstems. Tradition methods using parallel light pipe and plane mirror autocollimation have some limitations, and the are difficult to realize full-caliber high-precision wavefront aberration detection. The sparse subaperture stitching is based on the principle of interference autocollimation, and the sstem contains three parts, including interferometer, optical sstem to be measured and a plane mirror. Unlike overlapping subaperture stitching, a pluralit of small aperture plane mirror structure according to the arrangement of a certain composition sparse aperture can replace a large flat mirror. Each subaperture wavefront information was used to reconstruct the full aperture wavefront b stitching algorithm to achieve the required accurac. The precision of stitching algorithm is closel related to the arrangement, number and size of subaperture, and the coverage ratio of the subapertures in the whole region is represented b the filling factor M. Therefore the better M, the better the precision of stitching algorithm. In this paper, a mathematical model was established to deduce the relation curve between the subaperture number k and fill factor M when the value of k ranges from one to infinite. Though the relation curve between the subaperture number k and fill factor M, the maimum value of fill factor M= was obtained when k equals to 7. As a result, the coverage ratio of the subapertures in the whole region is the largest, which collects most of the information. Seven sparse subapertures, which is the optimal laout diagram for the detection sstems below.5 m was selected. Multiple subapertures ranging from 4 to 9 were stitching and detected, respectivel. After stitching, the data were processed b removing tilt The data of the stitching wavefront and the full aperture test data were subtracted, and the residual error was compared. The obtained RMS is.9λ and the direct detection RMS is.8λ when sampling the wavefront data with seven subapertures. The relative error is %. The residual wavefront RMS between stitching and direct detection is.9λ. Compared with other subaperture, the reconstructed full-aperture wavefront is the most consistent with directl detected full-aperture wavefront, which verifies the rationalit of the mathematical model. That is to sa, self-collimation interference detection of Φ mm demonstrates the rationalit of the arrangement. Citation: Luo Q, Wu S B, Wang L H, et al. Optimization of sparse subaperture arra model for stitching detection of plane wavefront[j]. Opto-Electronic Engineering, 8, 45(5): 768 Supported b Ke Project of the National Ke Research and Development Plan for Earth Observation and Navigation (6YFB5) * luoqian_@6.com 768-7

θ 1 = φ n -n 2 2 n AR n φ i = 0 1 = a t - θ θ m a t-m 3 3 m MA m 1. 2 ρ k = R k /R 0 5 Akaike ρ k 1 AIC = n ln δ 2

θ 1 = φ n -n 2 2 n AR n φ i = 0 1 = a t - θ θ m a t-m 3 3 m MA m 1. 2 ρ k = R k /R 0 5 Akaike ρ k 1 AIC = n ln δ 2 35 2 2012 2 GEOMATICS & SPATIAL INFORMATION TECHNOLOGY Vol. 35 No. 2 Feb. 2012 1 2 3 4 1. 450008 2. 450005 3. 450008 4. 572000 20 J 101 20 ARMA TU196 B 1672-5867 2012 02-0213 - 04 Application of Time Series