The influence of the sampling dots on th

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Optik

jou rn a l h o m e p a g e :w w w . e l s e v i e r . d e / i j l e o

The

influence

of

the

sampling

dots

on

the

analysis

of

the

wave

front

aberration

by

using

the

covariance

matrix

method

Xuelian

Yu

a,b

,

Yong

Yao

a,∗

,

Yunxu

Sun

a

,

Jiajun

Tian

a

,

Chao

Liu

a

a_Department_of_Electronic_and_Information_Engineering,_Shenzhen_Graduate_School,_Harbin_Institute_of_Technology,_Shenzhen_518055,_PR_China b_Department_of_Optics_Information_Science_and_Technology,_Harbin_University_of_Science_and_Technology,_Harbin_150080,_PR_China

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received22February2011 Accepted15June2011

Keywords: Fringeanalysis Digitalimageprocessing Wavefrontaberration Covariancematrixmethod Zerniketerm

a

b

s

t

r

a

c

t

ThecovariancematrixmethodisasimplemethodforsolvingtheZernikepolynomialwiththehigher fittingprecision.Inthispaper,itwasusedtoanalyzetheseveralopticalwavefrontsofthefinepolished aluminumdisksurfacecapturedbyaTwyman-Greeninterferometersystem.WehadfoundthatthePV (peak-to-valley)andrms(root-mean-square)valuesofthewavefrontaberrationchangeswithchanging theZerniketermandtheexpressionsfortheseveralopticalwavefrontswiththedifferentsampling dotswerewrong.Byanalyzingtherelationsamongtheconditionnumberofthecoefficientsmatrix,the Zerniketerm,andthenumberofthesamplingdots,itwasindicatedthatthenumberofthesampling dotshadonlyreducedthefluctuationthePVandthermsvaluewhiletheZerniketermincreases,but didnotchangethecasethattheexpressionsforthewavefrontaberrationwerewrongwhentheZernike termislargerthan14,especiallywhenthenumberofthesamplingdotsisless.Suchananalysiswillbe valuableinsolvingtheZernikepolynomialforthewavefrontaberrationanalysisbyusingthecovariance matrixmethodinopticaltesting.

1. Introduction

Zernikepolynomialsare widelyused todescribewave front aberrations for the interferogram analysissince themid-1970s

[1–4],becauseoftheiruniquepropertiesoveracircularpupiland relationtotheclassicalSeidel aberrationsthat provideauseful mathematicalexpressionoftheaberrationcontentinawavefront using similarterms [5–7]. The classicalGram–Schmidt method andtheleast-squaresmatrixinversionmethodhavebeenapplied to determine theZernike coefficients since 1980 [8]. Typically 37-termZernikecoefficientsareprovidedtoexpresswavefront aberrationsandthetheoreticalinterpretationoftheZernike coef-ficientsstabilityisalsogiven[9],andexperimentalinterpretation hasbeendoneonit[10].

Inthepaper,theseveralopticalwavefrontswiththedifferent samplingdotsareanalyzedbyusingthecovariancematrixmethod

[11]tosolvetheZernikepolynomials.Thispaperisorganizedas follows.Theprincipleofthecovariancematrixmethodisgivenin Section2.Athoroughprocessingprocedureofexperimentaldata fromthecircleinterferencefringeofthefinepolishedaluminum diskssurface capturedby theTwyman–Greeninterferometeris describedinSection3,inwhichthecrucialreconstruction

algo-∗_{Corresponding}_author.

E-mailaddresses:yxl-1216@sohu.com,yaoyong@hit.edu.cn(Y.Yao).

rithmisbasedonZernikepolynomialsandthecovariancematrix methodandexperimentalresultsandrelateddiscussionare pro-videdinthissection.Finally,theconclusionispresentedinSection

4.

2. Thecovariancematrixmethod

Zernikepolynomialscanbewritteninpolarcoordinatesas prod-uctsofaradialpolynomialfunctionandangularfunctions.These polynomialsaredefinedhereby

Zn(,)= Rn()n() (1)

wheretheindicesnisamodetermnumber.Theaberrationsand propertiescorrespondingtothefirstninemodetermsarelisted inTable1.TheorderingoftheZerniketermschosenistheFringe ordering[12].

Theinterferencewavefrontmaybewrittenasfollows:

wi(x,y)=q0+q1z1i(x,y)+···+qjzji(x,y)+···+qnzni(x,y) i=1,2,...,m (2)

wherewi(x,y)isthewavefrontoftheinterferencewavesurface,

zii(x,y)istheZernikepolynomialofthej-thorder,qn isthe coef-ficient,mistotalthenumberofZernikepolynomials.Usedinthe fitting,iisthei-thdatapointofm,(x,y)istherightanglecoordinate ofthei-thdatapoint.

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Table1

Theaberrationsandpropertiescorrespondingtothefirstninemodeterms.

Term Zernikepolynomial Meaning

0 1 Piston

1 cos x-Tilt

2 sin y-Tilt

3 22−1 Focus

4 2_cos₂ _Astigmatism_@0◦

andfocus

5 2_cos₂ _{Astigmatism@45}◦_and_focus

6 (−2+32_)cos _Coma_and_x-tilt

7 (−2+32_)sin _Coma_and_y-tilt

8 1−62₊₆4 _Spherical_and_focus

Letaij=z(xi,yi),then

⎧

⎪

⎨

⎪

⎩

a11q1+a12q2+···+a1nqn=w1 a21q1+a₂₂q₂+···+a₂_nq_n=w₂ ···

am1q1+am2q2+···+amnqn=wm

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wherem>n,andtheequationstakenfromthefittingprocedures areoftenill-conditioned.Tosolvethisproblem,inthefollowing section,thecovariancematrixmethodispresentedtodetermine theZernikecoefficients.

According to Eq. (2), the mean value of the sampling dots wi(xi,yj)isexpressedas

¯

w= 1

m

i=1

wi (4)

Therefore,Eq.(2)maybewritteninthefollowingform:

¯

w=

n

j=0

qjz¯j (5)

where ¯ziisthemeanvalueofthezji(xi,yi)oftheallsamplingdots

¯

zj= 1

m

i=1

Zji (j= 1,2,...,n) (6)

Eq.(2)subtractsEq.(4),andassumingthat

V_ki= w_i−w¯ (i= 1,2,...,m; k= n+ 1) (7)

Vji=z_ji−z¯_j(j=1,2,...,n; i=1,2,...,m) (8) Then

Vki= q1V1i+ q2V2i+ · · · + qjVji+ · · · + qnVni (j= 1,2,...n;

i= 1,2,... ,m; k= n+ 1) (9)

Aefisdefinedasthecovarianceofzeandzf,Aefmaybewritten as

Aef = 1

m

i=1

VeiVfi= 1

m

i=1

(zei−z¯e)(zfi−z¯_f)= 1

m

i=1

zeizfi−z¯ez¯e

(e,f= 1,2,...,n+ 1) (10)

According to the method mentioned above, the covariance matrixAefcanbewrittenas

⎡

⎢

⎣

A11 A21 .. .

An1

Ak1

A12

A22 .. .

An2

Ak2 · · ·

· · ·

. ._.

· · ·

A1n

A2n

.. .

Ann

Akn

A1k

A2k

.. . Ank Akk

⎤

⎥

⎦

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Fig.1. Theopticalwavefronts.(a)Thecircleinterferencefringe.(b)Thepatched imageafterthinning.

wherek=n+1,andthecovariancematrixAefEq.(11)isexpressed by A=

⎡

⎢

⎣

A11 A21 .. .

An1

A12

A22 .. .

An2 ···

· · ·

. ._.

· · ·

A1n

A2n

.. . Ann

⎤

⎥

⎦

B=

⎡

⎢

⎣

A1k

A2k

.. . Ank

⎤

⎥

⎦

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Bysolvingthefollowinglinearequations,q(q1,q2,...,qn)can beobtained

Aq= B (13)

Therefore,q0isobtainedbyEq.(5)

q0=w¯ −

n

j=1

qjz¯j (14)

Systemstabilityandthecapabilityofresistancetointerference canbeevaluatedbytheconditionnumberofthecoefficientsmatrix Aasfollows:

cond(A)=

A

A −1

(15)

where

A

and

A−1

arethevector normsofthecoefficients [image:3.595.351.522.76.396.2] [image:3.595.41.292.101.214.2]

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Fig.2. Theseveralopticalwavefrontswithdifferentsamplingdots.(a)1626;(b)2958;(c)3717;(d)4536;(e)8327.

Table2

TherelationbetweentheconditionnumbersofthecoefficientsmatrixandtheZerniketermforseveralwaveopticalfrontswithdifferentsamplingdots.

Term Samplingdots

1626 2958 3717 4536 8327

11 9.5138 10.2311 10.4614 9.9191 9.4362

12 12.3842 13.1451 13.8794 14.7010 12.3927

13 12.4358 13.4834 14.3187 14.7424 12.7156

14 12.6570 13.7405 15.1180 17.1754 13.3658

15 17.6414 19.5510 22.6623 25.2256 19.7490

16 21.0075 21.3857 26.1135 25.6053 21.3288

17 21.1219 21.4407 26.1445 26.1492 21.3696

18 21.1457 21.7963 26.6488 26.3069 21.6300

19 21.5705 25.2473 33.5285 34.0692 25.7984

20 21.6538 25.7738 34.9630 35.8958 26.6742

21 30.7620 29.8678 43.4199 46.3328 31.0206

22 30.9717 30.5599 45.3274 46.5726 31.7148

23 48.4236 39.8543 60.4320 72.4584 42.2423

24 50.0666 41.0476 63.9891 79.4444 43.9266

25 1.2322e+003 48.8052 82.5783 79.7448 47.8431

26 1.2355e+003 49.0863 82.9344 79.9127 48.1825

27 1.2844e+003 49.1205 83.2817 80.4978 48.7717

28 1.2937e+003 49.6311 83.8998 82.0042 48.9238

29 1.2982e+003 50.2438 85.1362 82.2459 49.5579

30 1.3137e+003 60.5319 110.6442 111.3772 67.8412

31 1.3251e+003 62.0512 114.3302 115.5311 69.0140

32 1.3469e+003 76.6824 150.2953 153.3583 82.0988

33 1.6905e+003 79.8453 156.3228 154.5748 83.7976

34 2.0995e+003 116.6395 245.4941 286.7746 105.5363

[image:4.595.179.409.72.412.2] [image:4.595.34.557.514.762.2]

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Fig.3. TherelativecurvesofthewavefrontaberrationwiththedifferentZernike termfrom11to35fortheseveralopticalwavefrontswithdifferentsamplingdots inFig.2.(a)ThePVofthewavefrontaberration.(b)Thermsofthewavefront aberration.

3. Experimentresultsanalysisanddiscussion

Intwo-beaminterferometrythefringepatternintensityI(x,y) asafunctionofthespatialcoordinatesisgivenby

I(x,y)= I₁(x,y)+ I₂(x,y)+ 2

I₁(x,y)I₂(x,y)cos

2W(x,y)

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whereI1(x,y)andI2(x,y)aretheintensitydistributionsofthesingle beamswhicharedeterminedseparatelybyappropriate measure-ments,andW(x,y)isthewavefrontortheirdeviationcausedby thedistancebetweenthegivenreferencesurfaceandthesample surface.Ifthegiven referencesurfaceisperfect,thecaseofthe samplesurfacecanbereflectedbyfringepattern.

The opticalwavefront wasgiveninFig.1and processedby theFCM.Fig.1(a)showsthecircleinterferencefringeoftheequal inclinationofthefinepolishedaluminumdisksurfacecaptured byaTwyman-Greeninterferometersystem,andthelaser wave-lengthis532nm.ThedeductionofZernikecoefficientsisusually alsoinfluenced bythefinitenumberofsamplingdotson inter-ferogramand their inheritedmeasurement errors, the uniform samplingtechnique[13]wasadoptedinthesamplingprocessin thepaper.Fig.1(b) shows thepatchedimage afterthinningby theautomaticprocessingfringetechnique[14].Theseveral opti-cal wave fronts with the different sampling dots were shown inFig.2.

Fortheseveralopticalwavefrontswithdifferentsamplingdots,

Fig.3showstherelativecurvesofthewavefrontaberrationwith thedifferentZerniketermfrom11to35,andTable2givethe rela-tionsbetweentheconditionnumberofthecoefficientsmatrixand theZerniketerm.Fortheseveralopticalwavefronts,thecondition

numberincreasesastheZerniketermincreasesandislessaffected bythenumberofsamplepointswhentheZerniketermislower. Notethat,astheZerniketermincreases,theconditionnumberis verylargewhenthenumberofthesamplingdotsissmall,asshown inTable2andthePVandthermsofthewavefrontaberrationis alsoverybigsimultaneously,asshowninFig.3.Fig.3alsoshows thatthenumberofthesamplingdotsonlyreducesthefluctuation thePVvalueandthermsvaluewhentheZerniketermishigher, butdotnotchangethefactthattheZernikecoefficientsis unsta-blebecausetheconditionnumberofthecoefficientsmatrixisstill bigger.

4. Conclusion

Inconclusion,theinfluenceofthesamplingdotsonthewave frontaberrationshadbeenanalyzedanddiscussed.By research-ingonthePVandthermsvaluesoftheseveralopticalwavefront aberrations withthedifferentsamplingdots,and thecondition numberofthecoefficientsmatrix,theresultsshowthatthe num-berofthesamplingdotsonlyreducesthefluctuationthePVand thermsvaluewhiletheZerniketermishigher,butwillnotchange thecasesthattheZernikecoefficientsisunstablebecausethe con-ditionnumberofthecoefficientsmatrixisstill bigger.Thisalso furthershowsthattheZernike termisonlyadecisivefactorfor keepingtrueexpressionforthewavefrontaberrationwhenusing thecovariancematrixmethodtosolvetheZernikepolynomialfor thewavefrontaberrationanalysis.

Acknowledgments

The authors acknowledge the support from the National Science Foundation of China (Grant no. 60977034) and the cooperation project in industry, education and research of Guangdong province and Ministry of Education of China (Grantno.2010B090400306).

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[image:5.595.89.246.76.366.2]