Stitching of off-axis sub-aperture null measurements of an aspheric surface

Transcription

1 Sttchng of off-axs sub-aperture null measurements of an aspherc surface Chunyu Zhao* and James H. Burge College of optcal Scences The Unversty of Arzona 1630 E. Unversty Blvd. Tucson, AZ ABSTRACT Optcal testng of a large convex aspherc surface, such as the secondary of a Rtchey-Chreten telescope, can be performed wth a Fzeau nterferometer that utlzes subaperture aspherc reference plates, each provdng a null test of a subaperture of the larger mrror. The subaperture data can be combned or sttched together to create a map of the full surface. The regon of the secondary mrror surface under test n each sub-aperture s an off-axs segment of the parent aspherc surface, therefore, the Fzeau reference requres a non-ax-symmetrc aspherc surface to match t. Msalgnment of the Fzeau reference relatve to the parent n each sub-aperture wll then result n aberratons n the measurements other than the ordnary terms of pston and tlt. When sttchng sub-aperture measurements together, the apparent aberratons due to the null lens msalgnment need to be ftted and subtracted. Ths paper presents an algorthm to perform ths partcular type of sttchng. Keywords: optcal testng, sttchng, aspherc optcs 1. INTRODUCTION It s always dffcult to nterferometrcally test a large convex aspherc surface, such as secondary mrrors for Rtchey- Chreten type of telescopes. The man reason s that a hgh qualty reference surface larger than the test surface s needed for the full aperture test of the surface. A few large convex mrrors, both sphercal and aspherc, have been tested wth full-aperture Fzeau test at the Unversty of Arzona, Steward Observatory Mrror Lab 1. Dffractve test plates were used as reference surfaces. As the telescope apertures get bgger and bgger, so do the secondary mrrors. Then the full aperture test s smply mpossble. A feasble alternatve s to test t n sub-apertures wth a small reference surface, then sttch them together to get the full surface map. Sttchng technque s not new 2-8, yet testng an aspherc convex surface wth an aspherc test plate presents new challenges. Besdes the usual pston/tp/tlt that need to be ftted over the overlapped area of the adacent sub-aperture maps, the lateral msalgnment of the aspherc test plate ntroduces other aberraton terms whch must be ftted as well and removed from the sub-aperture measurement maps. In ths paper, we present an algorthm for fttng these lateral msalgnment. The algorthm s extended from the one developed by Otsubo, et al 2. We wrote Matlab code to mplement ths algorthm. And we used ths program to smulate testng a aspherc convex mrror 1.4m n dameter wth an aspherc test plate that s the shape of an off-axs segment of the mrror under test. The smulaton results are presented. 2. ALGORITHM When usng an nterferometer to test a flat n subaperture, then sttch the measurements together to get the full aperture map, only the pston, tp and tlt for each map need to ftted to remove the dfference between the adacent maps n the overlapped area. Otsubo et al outlned the theory behnd t. As shown n Fgure 1, two adacent sub-aperture measurement over areas A and A have heght of Z A and Z A, both expressed n global coordnate (x, y). There are relatve pston, tp and tlt between these two measurements. In the overlapped area, there exsts a rght combnaton of a, b and c that makes the followng relatonshp true: czhao@optcs.arzona.edu ; phone , fax Interferometry XIV: Technques and Analyss, edted by Joanna Schmt, Katherne Creath, Catherne E. Towers, Proc. of SPIE Vol. 7063, , (2008) X/08/$18 do: / Proc. of SPIE Vol

2 z (, ) (, ) A' x y = za x y + ax+ by+ c. (1) z z (, ) (, ) A' x y = za x y + ax+ by+ c y A A z (, ) A x y x Fgure 1. Illustraton of pston/tp/tlt between the adacent subaperture measurements. Assumng there are N subaperture measurements, and the Nth one s chosen as the reference whch does not need to ft, but all other maps have to ft and compensate tp/tlt and pston to mnmze the dfference over the overlapped area between the adacent subaperture measurements. For the -th subaperture map, z', N( x, y) = z( x, y) + ax + by + c. (2) The coeffcents for tp, tlt and pston for each map, (a, b, c ) must be ftted and then compensated to get the full aperture map. Least squared ft s used: (( ) ( )) 2 Z x y ax by c Z x y ax by c. (3) mn = (, ) (, ) = 1... N = 1... N The algorthm can be generalzed when more terms need to be ftted. The fttng functons for each map may not be lmted to tp/tlt and pston. They can be any predefned functons, e.g. they can be arbtrary functons f (x,y) where =1,2,..L, and L s number of functons. Then L z' ( xy, ) z( xy, ) F f( xy, ) = +, (4), N k k k = 1 where coeffcents F k need to be ftted. Agan least squared fttng s used: L L mn = Z( xy, ) + Fk fk( xy, ) Z ( xy, ) + Fk fk( xy, ). (5) = 1... N = 1... N k= 1 k= 1 Eq. (5) can be transformed to a group of lnear equatons. For the -th sub-aperture measurement, the fttng coeffcents form an Lx1 vector R whose k-th element s F k,.e. R [ k] = F, (6) k For any two subaperture measurements, e.g. the -th and -th, we construct an Lx1 vectors P and an LxL matrx Q, where ( ) P [ k] = f ( x, y) Z ( x, y) Z ( x, y), (7) k 2 Proc. of SPIE Vol

3 and fm( x, y) fn( x, y) Q (, ) m n =, (8) 0 = We borrow the concept of cell array from Matlab 9 and make cell arrays P, Q and R whch are defned as follows: 1. P s a (N-1)x1 cell array whose element s 2. Q s a (N-1)x(N-1)cell array whose element s N Ρ {,1} = p. (9) = 1 N Q {, } = δ Q + Q, (10) k k = 1 1 where δ = 0 =. 3. R s a (N-1)x1 cell array whose element s R {,1} = R. (11) Note that P, Q and R can be collapsed to regular vectors/matrx wth dmensons ((N-1) L)x1, ((N-1) L)x((N-1) L) and ((N-1) L)x1, respectvely. Then Eq. (5) becomes P = Q R. (12) To solve ths n Matlab s straghtforward: R = Q\ P. (13) Now we have obtaned the fttng coeffcents F k snce they are elements of R, then we can combne the subaperture measurements together to get the full aperture map. At the overlapped areas, average s taken over all the subapertures whch have vald data over ths area. Normal analyss can be performed on the sttched map, such as Zernke decomposton, subtracton of certan reference maps, etc. A further extenson of ths approach s that each subaperture map may be ftted to dfferent functons for msalgnment. Then P, Q and R are constructed dfferently, the rest of steps are the same as llustrated above. An example for ths applcaton s to test a large convex mrror wth two dfferent test plates, each used for dfferent part of the mrror along radal poston. Then the lateral msalgnment of the test plates wll have combnatons of dfferent amount of astgmatsm and coma. Assume there are N subaperture measurements, L functons denoted as f k (x,y) need to be ft for the -th subaperture measurement, and the coeffcents are R k,.e L Z ( xy, ) = Zxy (, ) + F f ( xy, ), (14) k k k = 1 where = 1, 2,, N-1 assumng the N-th measurement s used as reference wth no-msalgnment fttng needed. Defne vector P smlarly as n Eq. 7,.e. ( ) P [ k] = f ( x, y) Z ( x, y) Z ( x, y). (15) k Proc. of SPIE Vol

4 Note that n defnton descrbed by Eq. 7, P and P have the same number of elements, and the correspondng elements have the same magntude but opposte sgn. Yet, n the defnton for the more general case, there s no such relatonshp for P and P. Even the number of elements may be dfferent. Defne vector R exactly as n Eq. 6, Agan, R and R may have dfferent number of elements. R [ k] = F. (16) k Defne Q as follows: fm ( x, y) f n ( x, y) Q ( m, n) = k. (17) fm ( x, y) fn ( x, y) = k= 1... N k From the new P, and R, we can construct the cell arrays P and R exactly the same as outlned above. But Q s a lttle dfferent, Q {, } = Q. (18) Agan, P, Q and R can be collapsed to vectors and matrx, then t s straghtforward to get the best ft of the coeffcents, F k, by usng Eq SIMULATION We mplemented n Matlab the smpler verson of the algorthm where each subaperture measurement s ftted to the same group of functons. We dd a smulaton on testng a large convex aspherc mrror wth a aspherc test plate. The aspherc test plate s small such that only subaperture measurement can be made on the convex mrror (see Fgure 2). The subaperture measurements are arranged such that they cover the whole convex mrror wth suffcent overlap between them (see Fgure 3). In each subaperture, we need to ft the tp/tlt, pston and power due to the vertcal algnment change between subaperture measurements, as well as two other terms that represent the lateral algnment change. The subaperture represents a off-axs porton of the aspherc surface. When the test plate and the mrror under test are both perfect, and perfect null frnge s obtaned when they are perfectly algned. Any lateral msalgnment of the test plate n regard to the mrror under test, ether a shft along radal drecton or clockng or a combnaton of the two, wll produce apparent aberratons whch must also be ftted and taken out (see Fgure 4). Subaperture test 800 mm at 12 postons nterferometer 1400 mm Fgure 2. Testng a convex aspherc mrror wth smaller test plate. (a) Schematc of test setup. (b) Illustraton of full aperture and subaperture relaton. Proc. of SPIE Vol

5 Fgure 3. Schematc of the arrangement of the 12 subaperture measurements, 30 degrees apart. Radal shft Nomnal poston Apparent aberraton map for 1mm radal shft: A Zernke standard coeffcents (rms nm): Z4 (power): -173 Z6 (0 astgmatsm): 122 Z7 (90 coma): 38 clockng Nomnal poston (a) Apparent aberraton map for 0.05 clockng: Zernke standard coeffcents (rms nm): Z5 (45 astgmatsm): - 48 Z8 (0 coma): 15 (b) Fgure 4. Illustraton of lateral algnment error of the test plate and assocated apparent aberraton map. (a) radal shft, (b) clockng. Sx functons are ftted for each subaperture measurement to remove the errors caused by msalgnment of the test plate relatve to the mrror under test. They are lsted n Table 1. Proc. of SPIE Vol

6 Table 1. Lst of terms ftted for each subaperture measurement. Term Expresson Correspondng test plate algnment error 1 f ( ) 1 1 pston 2 f ( x, y) x 2 = Tlt n x drecton 3 f ( 3 = Tlt n y drecton 4 f ( Focus error n z f (, ) = Z 6(, ) (, ) Radal shft n x-y plane f ( x, y ) = Z 5( x, y ) Z 8( x, y ) Clockng n x-y plane We smulated testng the mrror n 12 angular postons, 30 degrees apart, as shown n Fgure 3. The software s thoroughly tested. When there s only aberratons due to test plate msalgnment n each subaperture measurement, the sttchng returns a perfect null map of full aperture. We further smulated the nose effect by addng 3nm rms correlated nose to each subaperture map, besdes the aberratons caused by random algnment errors. A typcal subaperture map s shown n Fgure 5. And the result of a typcal sttchng run s shown n Fgure 6. nose Pston/tp/tlt/power Z5-8 = + + k Fgure 5. A typcal subaperture map s composed of three parts: correlated nose, algnment errors of pston, tp, tlt and power and algnment errors of combnatons of astgmatsm and coma. Subaperture measurements raw data: v1 Subaperture maps AFTER removng test plate algnment error: Proc. of SPIE Vol

7 Sttchng result: Sttched map: RMS: 4.1nm Ftted map: RMS 3.8nm Resdual: RMS: 1.5nm Sttched map Wth zl-z 4 removed IS Ftted map Wth zl-z4 removed Resdual map p S S ISO Fgure 6. The result of a typcal sttchng rum. 4. SUMMARY We generalzed a sttchng algorthm to deal wth cases where, besdes the usual tp, tlt and pston terms to be ftted for subaperture measurements, addtonal terms of vrtually arbtrary form need to be ftted as well. We outlned the algorthm n ths paper. We also mplemented a smpler verson n Matlab where each sub-aperture s ftted wth the same group of functons. The sttchng software we wrote n Matlab was tested extensvely and proven to be accurate. We then use t to study testng an aspherc convex mrror n subaperture measurements wth a smaller aspherc test plate. The results were presented here. The general verson of the sttchng algorthm descrbed n Secton 2 wll be mplemented n the near future. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] J. H. Burge, Fzeau nterferometry for large convex surfaces, n Optcal Manufacturng and Testng, V. J. Doherty and H. P. Stahl, Edtors, Proc. SPIE 2536, (1995). M. Otsubo, K. Okada, J Tsuuch, Measurement of large plane surface shapes by connectng small aperture nterferograms, Optcal Engneerng, 33 (2), , (1994). M. Bray, Sttchng nterferometer for large plano optcs usng a standard nterferometer, n Optcal Manufacturng and Testng II, ed. H. P. Stahl, Proc. SPIE 3134, (1997). S. Tang, Sttchng: Hgh spatal resoluton surface measurements over large areas, Proc. SPIE 3479, 43-49, S. Chen, S. L and Y. Da, Iteratve algorthm for subaperture sttchng nterferometry for general surfaces, JOSA A. 22(9), (2005). X. Hou, F. Wu, L. Yang and Q. Chen, Full-aperture wavefront reconstructon from annular sub-aperture nterferometrc data by use of Zernke annular polynomals and a matrx method for testng large aspherc surfaces, App. Opt. 45(15), (2006). P. Murphy, et al, Sttchng nterferometer for large plano optcs usng a standard nterferometer, n Imterferometry XIII: Applcatons, ed. E. L. Novak, W. Osten and C. Goreck, Proc. SPIE 6393, 62930J, (2006). C. Zhao, R.A. Sprowl, M. Bray, J.H. Burge, Fgure measurement of a large optcal flat wth a Fzeau nterferometer and sttchng technque, n Interferometry XIII, edted by E. Novak, W. Osten, C. Goreck, Proc. of SPIE 6293, 62930K, (2006). See Matlab manual or Proc. of SPIE Vol