A NEW TECHNIQUE FOR TESTING LARGE OPTICAL FLAT
Qing Wang, Jinbang Chen, Rihong Zhu, Lei Chen, Yiguang Zhang
Department of Photoelectronics
Nanjing University of Science & Techno1ogyD,
Xiaolingwei 200, Nanjing, P.R.China
LABSTRACTI
This paper developed a new technique (including theory, methed and equipment etc. )for
testing larger optical flat only by means of a smaller interferometer. That is Overlapping
Subaperture Interference Testing ( OSIT ) technique. Author had established a mathematics
model for OSIT to retrieve surface of full aperture. The theoretical accuracy of retrieved sur-
face of full apertrue reached 2 / 200( p— v) . The relationship between accuracy of retrieved
wavefront of full apertrue and errors (such as system error of interferometer or position error of
subapertrues etc.) had been investigated. A computer program was established to simulate the
real procedure from testing surface data of subapertrue to retrieved wavefront of full apertrue.
The limit of expanding apertrue was discussed and it's less than 2.5—times old apreture of
interferomter with the accracy of retrieved wavefront of full apertrue better than 0.05 wave-
length (p— v) at the same time. Author successfully applied OSIT technique to a " phase shift-
ing digital flat interferometer " and expanded the testing apertrue from 250mm to 500mm with
the accuracy in expanding apertrue better than A 120 (p—v).
1. INTRODUCTION
To test an optical flat of larger aperture by a smaller reference standard flat is the general
method used in optical workshop. Simular to other two methods ( the spherical—wavefront
interferometer test and knife —edge test), the accuracy of these methods is only qualitative and it
depends on the experience of the tester, So that the method isn't appliable in moden high accu-
racy optical test. In generally, to test an optical flat , we need an interferometer with similar
dimension to the tested flat at least. But if the optical flat is larger than 3OOmm, the test is dif-
ficult, because the ordinary interferometer is too small to test such large falt. The interferomet-
er only test a smaller subaperture in the large optical flat each time. thus, we have developed a
technique to deduce the measuring interferograms of subapertures which is measured
respectively and used to retrieve the wavefront of allaperture. that is Overlapping Subaperture
Interference Testing(OSIT) Techniques. It is the quantity test and used widely.
cDOriginal name: East China Institue of Technology
O-8194-1252-X/93/$6.OO SPIE Vol. 2003 Interferometry VI (1993) / 389
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2.PRINCIPLE
The conception subaperture,introduced by W.Chow and G. lawrance is used to solve the
problem of measuring a large optical system"2. Where the reference flat is replaced by an array
of smaller optical mirrors called subapertures. they separate from each other, the processes of
measuring are finished at the same time.
Reversedly,the subapertures,used in the OSIT measurement, are the parts of the same test-
ed flat and overlap in part. They are tested by interferometer respectivly ( as shown in fig.l,
where S is the number of subaperture). The overlap is important in two reasons : One is that the
subapertures can cover the most areas of the tested flat in order to reduce the error which is
caused by the omit area. The other is that the OSIT measurement needs the part of overlap be-
tween one subaperture and another to determine and eliminate the relative piston and tilts in
order to unify the data of two subapertures.
Fig.1 Subapertures spread
Suppose that the same subaperture of a flat is measured two times by useing a Phase Digital
interferometer, so the overlapping scope full of measuring aperture. The optical path difference
data D1[X,Y] and D2[X,Y] (n = I ,2,'.,N) can be obtained by this two measurements.Where
D1(i = 1,2) is the optical path difference at point(X,Y) and N is the total sample points in each
measurement. It is obvious that D1is different from D2in two measurements,because the relative
position and tilts between the reference standard flat and tested flat are changed(as shown in
fig.2),but the surface shape of tested flat is not changed. The function W(X,Y) of optimum fit-
ting surface can be found by the lest—square algorithm:
= 1=1,2 (1)
Where the function W1(x,y) of the optimum fitting surface is formed by
390 I SPIE Vol. 2003 lnterferometry VI (1993)
tested &urfrce
overlapping rc
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w (X, Y) = P + T, • X+ T, • Y i = I ,2 (2)
Where i is the order of measurment, P is piston, This X—tilt and Tis y—tilt. By this time, a
group of parameters (P1,T1, T1) can be obtained and suited to every data of measurement.
tetlflg aperture:
I
reference 8urface 1
subapertu're i
I
reference Burface 2
I subeperture 2
;
tested surface
fitting surface ,
(a) two measurements (b) two measurements
in same subaperture in diferent subapertures
Fig.2 The change ofreference surface
When the piston and tilts are eliminated from each datum,the results of the tested
wavefront are equal:
D1[X,Y]— W1(X,Y)=D2[X,Y]— W2(X,Y) n= l,2,'••,N (3)
Eq.3 means that the relative change of the optimum fitting surface can be used to substitute
for the relative change of the tested wavefront in two measurements:
AW(X,Y)=W2(X,Y)— W1(X,Y) (4)
where the AW(X,Y) is the relative change wavefront
AW(X,Y)=AP +AT • X+AT • Y (5)
where the AP,AT4Tis the coefficients of piston and tilts.
During the OSIT experiments, the overlapping scope is reduced, because the 2th
subaperture substiutes for the frist one in the second measurement, and only the part of
subapertures overlaped. But if the errors is ignored, two wavefront configurations which are
part of the overlapping area are also similar in two measurements. Therefore the previous meth-
od can be used in the processs of OSIT measurement, and it means that the relative change of
the optimum fitting surface in the overlapping area can be substiuted for the change of the test-
ed flat in two subapertures measurments.
Frist at all, the data of subapertures need to be processed simply by unifying in the same
plane coordinate, then by useing equation(1) , the optimum fitting surfaces can be obtained by
two group of data in their overlapping areas:
SPIE Vol. 2003 Interferometry VI (1993)! 391
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1 =(W(X,Y ) — Di[XmYm 3)2 1,2 (6)
Where Mis the total sample points of ith subaperture in its overlap area, and the function
W.(X,Y) of the optimum fitting surface are formed by equition(2). The coefficients(Pi,Txi,Tyi)
of Wi(X,Y) can be obtained by miminizing o. Because x and y are free variations , the relative
change coefficients of the reference standard flat AW(x,y) can be obtained by the following three
equations:
AP=P2 —P1
AT =T —T (7)x x2 xl
AT =T —T
y y2 yl
These parameters are suit to all the data of two subapertures, so the relative piston and tilts
in the data of the 2nd subaperture can be eliminated by these paraments:
D2{X,Y2]=D2[X,Y2]—(AP+AT •X+AT • Y) n=l,2,,N2 (8)
Where N2is the total sample points in the 2th subaperture. And the optical path difference
D;is unified with D1 in the same reference standard flat. Thus, the unified process of
subapertures is finished, and the previous process is used repeatly to unify more subapertures.
3. COMPUTER SIMULATION AND RESULTS
To test the method, the computer program OSIT is used to simulate a numerical
subaperture interferogram and retrieve the wavefront of allapertures(shown in fig.3).
the map of OPD Di[Xn1Ynl to determined P,
of subapertures sampled ATxand 1Ty by
by mnterferoineter equactIona(6), (7)
. : u:i
1 1 retrieve the unify
the map of OPD — a avefront of two
Di Xn, Yn] r-
T
subaperture by eq, 8
prodted by position errof — ________
computer or arnp1e erruc —
:zzzzzzzizzi:::± .IZIIIIIZZZ
next suhuper turd
original wavefront gauge thein(XY) and H-- accuracy — output the retrieve
location paranient8 E=out-n wavefront Vout(XY)L J L J __________ _____
Fig.3 The chart of program OSIT
To simulate the method errors, the input wavefront Win(X,Y) is the Zernike polynomials,
392 /SPIE Vol. 2003 Interferometry VI (1993)
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the coefficients are nonzero and the highest term ofZernike polynomials is Ja.
w Y) = E Z 1(X, Y) (9)
To simulate the random change of adjestment of the tested flat , we use different and arbi-
trary piston and tilt coefficients (j = I ,2,3) for the different subapertures. The remaining Zernike
cofficients are equal for all the subapertures. here we select the coefficients
A1=(—1)''A A=O.l, 0.2, 0.3, /4 (10)
where A will stand for the peak—to--valley of Win , and the different Ja will stand for Win
in different complex level of aberration (show in Table 1).
TABLE 1 the complex of input wavefront
Ja Terms Description
I
2
3
Zi
Zi-Z2
Zl-Z2+Z3
plane
4
6
6
ZI-Z2+Z3-Z4
Z1-Z2+' +Z5
ZI-Z2+ -Z6
add 2nd order
aberrations
7
S
9
10
Z1-Z2+ +Z7
ZI-Z2+" -Z8
Zl-Z2+' +Z9
Zi-Z2+" -Z1O
add 3nd order
aberrations
The another input into the program OSIT is the location for each subaperture, which in-
cludes the position of centre points and directions of subapertures in the plane coordinate. Then
a final wave—front can be obtained by OSIT program, and it is also formed by zernike
polynomials:
W01(X,Y) = EB1 • ZJ(X,Y) (11)
We gauge the accuracy of the method by looking at the relative Peak—to—Valley wavefront
error:
A— PV[E(x,y)J >< l00°/ (12)PV[W1(x,y)J 0
where PV[ ] is the peak—to—valley value of wavefront, E(x,y) is the residual error:
E(X, Y) = W Y) — W (K, Y) (13)
SPIE Vol. 2003 Interferometry VI (1 993) I 393
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It shows in calculation that the accuracy is influenced by the complex level of Win, and
inconcerned in PV{Win]. To correspond to reality, we select:
A1=(—l)''A (14)
In the absence of error, our method performs extremly well for a wide range of input
aberrations. Figure 4 is a plot of L versus A. Note that the relative accuracy is better than
1O% . With the order of aberration raising, the relative accuracy is reduced, but it is irrelevant
to A and less than 0.5% for 4 order aberration. So our method suit for the high accuracy piano
surface measuring, and if the Peak—to-—Valley of the input wavefront is 12, the theoritical accu-
racy is better than A I 200
(%)
:: %L L L T
124367S91O111213141AJ
Fig.4 The accuracy of method
To simulate the effect of errors which exist in actual measuring program. There are two im-
portant errors must be considered, one is the position errors due to the location system error,
the other is the sample errors that means the residual error of interferometer and sample ran-
dom error.
To simulate the position errors of subapertures, we add different tiny movement dx with
the subapertures location parameter, and select three difference input wavefront with
Ja=2,4,10.
Fig.SA is a plot of L versus dx / D. Where D is the diameter of subaperture(the measuring
aperture of interferometer). Shows that the effects of position errors depands on the configura-
tion of the input wavefront, means that the more the complex configuration of the tested surfer
is, the more the accuracy disturbed bigger by position errors is.
The residual error of interferometer is similar to each subaperture, because the aperture of
interferometer is the subaperture of the tested surface, but the sample random error is difference
on each sample point , so they need to be considered together. The disturbing wavefronts
G1(X,Y) are added into each subaperture, and they include two errors: one is the fixed system
error and the other is the random error which is produced randomly by computer.Their
394 /SPIE Vol. 2003 Interferometry VI (1993)
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Peak—to—Valley are equal,and it's signed as 5(ö =PV[G1(X,Y)].
I
rd o:der
d order
I irt order
•1 2 '3 4
Fig.5 (a) The influence of position errors (b) theinfluence of sample errors
It shows in calculation that the accuracy is irrelevant to the configuration of the input
wavefront, but it relates closely to the P—V of sample errors. Fig.5B is a plot of z versus ö / PV[
Win] . It shows that the relation of L and ö / PV[Win] is linear and its slope is between 2 and
2.5. Therefore, in ordinary accuracy of location (I %), the important disturbing error is the sam-
pie error of interferometer.
4. EXAMPLE OF TESTING
By the previous analysis, we can know that an interferometer with high accuracy is the es.
sential condition. A Phase Shifting Digital Flat Interferometer whose aperture is 25Omm and
accuracy is 2/ 504is choosed and shown in fig.6.
Fig. 6 Schmatic diagram of phase shifting digital flatinterferometer
The special bear table( shown in fig.7) of OSIT is built by three layers: the frist is the x— y
tilts adjuster. The second is the location machine which can move in one dimension and rotate in
perigon. The location error is less than O.5%(O.O2mm and 4'). The third is the flat bearer which
SPIE Vol. 2003 Interferometry VI (1993) /395
- t%)
3
fixed svs:ern error
sample errors
random error
1. 2 E !PVtIn (%)
____
I
[1f Cont1IcJ
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has nine bear points. The bear points can balance automatically to minimize the deformation of
large flat(eq .500mm).
Fig.7 The OSIT special tabel
There are two tested flats , one is 250mm and the other is 430 mm. The results of meas-
uring processes are showed in table 2 and fig.8.
It is successful that OSIT technique is applied into the Phase—Shifting Digital interferomet-
er and enlarge its measuring aperture from 25Omm to 5O0mm, with the accuracy of 2/20.
5%
396 ISPIE Vol. 2003 lnterferometry Vi (1993)
Fig.8 the configuement of three—dimension
fiet ba1arce bearer
rotating
p1iie .d,i&tiLg
9-point
(A) standard wavefront (B) all of retrieved wavefront
(C) centre of retrieved wa.vefront (B) residual wavefront
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Table 2 the results of testing
flat tt2bOmm (1)424mm
aubaperture 1)lbOmm t1)2öOmm
tandurd
wttvefrand
PVO 1BÔ
RM80, 051
IVO 20? ftMSO 040
(D2SOnjm Fig1 8A
retr ieved
avefrand
PVO. 1'18
RMS0, 047
PVO, l 2 RM801 200 (1)424mm Fig 8B)
PVI). 206 R1S0 042 (P2SOmm Fig SC)
re8idutil
error ve
PVO 051
R1S0 010
PV0 048 RMS0 010
1ii center of 250mm fig SD)
accuracy 27% 23%
. SUMMARY AND CONCLUSION
From the above arguments, it would seem that the technique will enlarge the aperture fur-
ther with more subapertures. But the accuracy descends along with it. The degree of overlap be-
tween two subapertures and the ratio between the allaperture to the full—covering area of
subapertures are important too. We will discuss all of them detailed in another papers.
6. REFERENCE
[1]. W.W.Chow and G.N.Lawrance "A Mathod for Subaperture Testing interferogram re-
duction" Opt. Lett. , vol. 8 , 468(1 983)
[2]. W.W.Chow and G.N.Lawrance U Influence of High Order Noise in Wavefront
Reconstruction" Proc. SPIE , vol. 440 ,31(1983)
[3]. M.Y.Chen etc. UMulti.aperture Overlap— scanning Technique for Large Aperture Test
"Proc. SPIE , vol. 1553(1991)
[4] J.B.Chen D.Z.Song etc."Large aperture—high accuracy phase shifting digital flat
interferometer" ,SPIE Vol. 2003 (to be published).
SPIE Vol. 2003 Interferometry VI (1993) /397
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