Study of aberrations of stepper lenses in-situ using phase shifting point diffraction interferometry

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5-1-2002

Study of aberrations of stepper lenses in-situ using

phase shifting point diffraction interferometry

P. Venkataraman

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STUDY OF ABERRAnONS OF STEPPER LENSES IN-SITU USING PHASE SHIFTING POINT DIFFRACTION INTERFEROMETRY

By

P. Venkataraman M.S. Physics University of Cincinnati

(1996)

A thesis submitted in partial fulfillment of the requirements for the degree of Masters in Science in the Chester F. Carlson Center for Imaging Science

of the College of Science Rochester Institute of technology

May 2002

Signature of the Author _

Acceptedby

/....:.';,_/~;.~Z_O_o_z...

CHESTER F. CARLSON CENTER FOR IMAGING SCIENCE

COLLEGE OF SCIENCE

ROCHESTER INSTITUTE OF TECHNOLOGY ROCHESTER, NEW YORK

CERTIFICATE OF APPROVAL

MS DEGREE THESIS

The Masters Degree Thesis of P. Venkataraman has been examined and approved by the

thesis committee as satisfactory for the thesis requirements for the

Master of Science degree

Dr. Bruce W. Smith, Thesis Advisor

Dr. Zoran Ninkov, Committee Member

THESIS RELEASE PERMISSION ROCHESTER INSTITUTE OF TECHNOLOGY

COLLEGE OF SCIENCE CHESTER F. CARLSON CENTER FOR IMAGING SCIENCE

Title of Thesis: Study of aberrations of stepper lenses in-situ using phase shifting point diffraction interferometry

I, P. Venkataraman,>-hereby grant permission to the Wallace Memorial Library ofR.I.T. to reproduce my thesis in whole or in part. Any reproduction will not be for commercial

use or profit.

Signature: _

STUDY OFABERRATIONSOF STEPPER LENSES

IN-SITU USINGPHASESHIFTING POINTDIFFRACTIONINTEROMETRY

by

P.Venkataraman

submittedto the

Chester F. Carlson

Centerfor_ImagingScience

CollegeofScience

inpartialfulfillmentoftherequirements

fortheMasterofScience Degree

atRochester Instituteof_Technology

ABSTRACT

Stepper lenses are tested _by the lens manufacturer _using various interferometric

methods like phase measurement _{interferometry (PMI),} before _they are assembled onto

the stepper or scanner. Once the system is set _{up, there} are a few methods to _study the

lens aberrations in-situ _{using interferometry. The} methods _currentlyused are directones

like directaerial imagemeasurement _(DAIM) or indirect ones in which imagesoflines

and spaces are formed in the photoresist and the aberrations inferred from scanning

electron microscope _(SEM) images of these. We propose _using phase _shifting point

diffraction _{interferometry (PSPDI)} for thepurpose of_measuring aberrations in-situ. The

method has the advantages of _{being simple,} and of _having relaxed coherence length

requirements and _{applicability} over a widewavelength range. We present results from a

prototype experimentdoneona436nm optic onanoptical bench usinga442nmHe-Cd

ACKNOWLEDGEMENTS

I would like to thank Dr. Bruce Smith for all his advice and help. I thank the

committeemembersfortheirpatience andhelp. Iwouldliketothank_{Hoyoung Kang} and

Dr. Ralph Schleif for their_discussions, Lena_Zavyalova, Tolic_{Bourov, Suraj} Bhaskaran

for their _help with mask _and, coating. I would like to thank all the staff at Center for

TABLE OF CONTENTS

ListofTables iii

ListofFigures iv

1. Introduction 1

2. Background 4

2.1 Aberrations 4

2.2 Optical_LithographyandProjection Steppers 10

2.3 Image Metrics 16

2.4 EffectofAberrations on_imaging 21

3. Experimental Procedure 29

3.1 Phase_ShiftingPoint Diffraction_{Interferometry} 29

3.2 Errorsources inexperimental_set-up 32

3.3 Five framemethod 41

3.4 Phase_Unwrapping 43

4.Experimental _Set-up 45

5. ResultsandAnalysis 48

6. Conclusions andRecommendations 57

Appendices

AppendixA 59

Appendix C 72

LISTOF TABLES

1. Table 1:Technology

roadmap forlithography 2

2. Table 2.1.1: Zernikepolynomial andtheaberrations

theyrepresent 8

3. Table 2.1.2: Microlithographic lenses forICproduction 15

LISTOF FIGURES

1. Fig.2.1.1: Aberrationwavefrontforan on-axis point object 5

2. _Fig2.1.2: Schematicof an optical_imagingsystem 5

3. Fig2.1.3: Sometypicalwavefront plots

showingdifferenttypesof aberrations. 9

4. _Fig2.2. 1a: OpticalLithography: Contactand_ProximityPrinters 11

5. _Fig2.2.1b: Optical _LithographyProjection Printers 12

6. _Fig2.2.2: Coherent andincoherent illuminationandthe_{corresponding MTF}

profiles 14

7. _Fig2.3. la: Light_Intensitydistributionof a350nmiso space atbestfocus 19

8. _Fig2.3.1b: Normalized derivativeoflight_intensitydistribution 19

9. _Fig2.3.2: Processwindowforisolated space 20

10. _Fig2.4.1: Aerial Imageofisolated_line,variation withaberration 23

11. _Fig2.4.2: PSFandMTFvariation withaberration 24

12. _Fig2.4.3: Variation inprocess window with aberration 25

13._Fig2.4.4: Variation in Strehlratiowithaberration 28

14. _Fig3.1.1: Point diffraction_{interferometryset-up} 31

15._Fig3. 1.2:AschematicofPhase ShiftingPoint DiffractionInterferometry 31

1 6. _Fig3.2.1: Errorduetoaparallelplate ₃₃

17. _Fig3.2.2: Measuredphase errordue _gratingmotion error ₃₃

1 8. _Fig3.2.3: Errordueto largerreferencepinhole ₃₆

20._Fig3.2.5: Detectormisalignment effect 39

21._Fig3.2.6: _Scatteringof referencebeamthrough thewindow 40

22. _Fig3.4. 1:Arepresentation of phase_unwrapping 43

23._Fig3.4.2: _{Unwrapping by}line_{by line,} andTakeda'smethod 44

24._Fig5. 1: Aset offive interferograms 49

25._Fig5.2: Aplot_showingcalculated wavefront coefficientsand_{repeatability} 50

26. _Fig5.3: Comparisonof algorithmsusedforfitting 52

27._Fig5.4: Zernike coefficients calculated after_{applying Park's}method 53

28. _Fig5.5: MTFplotsfrom datasheetand calculatedfrom Strehl 'sratio 55

29._FigAl: Comparisonof generated andfittedwavefronts 65

1.Introduction

TheInternational_{Technology Roadmap for Semiconductors(ITRS)(Table 1) [1]} gives

requirements for_{keylithographic capabilities such as}

resolution, total overlay, chipsize,

defect _density, mask_size,andfield size. Lithographiclensesusedin steppers_playa_very

important role in _defining the limits to which optical _lithography can be taken as _they

determine most the ITRS requirements. _{By design,} the performance ofthese lenses is

near diffraction limited. Resolution that can be obtained _by optical _lithography for a

projection stepperisgiven_by[30]:

NA

Where _ki (typical values are 0.25 to _0.8) is a constant _depending on various process

parameters, A, is the illumination wavelength (typical values _365nm(Hg arc _lamp),

248nm(KrF excimer _laser), 193nm(ArF excimer _laser), and NA is the image side

numerical aperture(typicalvaluesfrom0.28to0.8). As canbe seenfromtheequation we

can get better resolution _byreducing the wavelength or _increasing the NA or_{by doing}

both. That is exactlywhatthelens manufacturers are doing. Thedownsideofthisis that

theusabledepthoffocus_(DOF)isreduced asisevidentfromtheequation[30]:

~ ,- K-.A,

DOF =

-fe

Where _k2 is also a constant _depending on process parameters (typical values are 0.8 to

Thelens manufactureror_lithographytoolmanufacturertests thelenses, during andafter

the _assembly of the lens. This is done _by, through the lens (TTL)[2] interferometry

methods like phase measurement _{interferometry} (PMI)[3]5 direct aerial image

measurement _(DAIM)[3], point spread function measurement (PSF)[4], and

opto-electromechanical_raytrace [5].

Year 1998 2001 2004 2007 2010

Technology Generation_(nm)

.25 .18 .13 .10 .07

DRAM_(bits) 256 M IG 4G 16G 64 G

Development _capability

(minfeaturesize_(nm)

.16 .11 .08 .05 .03

Minimum featuresize_(um) Isolatedlines Denselines .24 .25 .16 .18 .11 .13 .08 .10 .05 .07

Microprocessor _chip size

(mm2)

300 360 430 520 620

Fieldsize_(mm2) 484 1250 1250 1250 1250

Defect density, lithography

only@defectsize_(um)

320 .08 135 .06 60 .04 30 .04 15 .02

Masksize_(inches),quartz 6_by6 9_by9 9_by9 9_by9 9by9

Table 1:_{Technologyroadmap forlithography}[1].

Once the _lithography tool is in the _{shop floor,} indirect methods like focal surface

(FOCAL)[3], scanning electron microscope _(SEM)[3] analysis of_{images formed} in the

resist, 3-beam interference [6], In-situ interferometer _by Litel _[33] are used to evaluate

theperformanceofthestepper as a whole. PMIusesa piezo-mounted mirroron one arm

phase error is extracted fromthe interferencepatterns obtained at eachphase shiftofthe

mirror. In_DAIM, aerial _images ofisolatedspaces and dense linepatterns are magnified

and imaged on to a CCD array at multiple height positions in the neighborhood of

Gaussian_focus,andtheimages analyzedforaberrations. In_FOCAL, thefocalsurfaceof

the wavefront is measured_{by imaging} isolated lines onto a thin resist _using the optical

alignment system and_inferringthe focalplane variationoverthewholefield. InthePSF

method, thepointspread functionismeasured throughfocus andacross the fieldandthe

point spread function is used to evaluate image quality. In 3-beam interference method,

imaging condition of a fine grating _by 3-beam interference is used _{to quantify coma,}

astigmatism, and spherical aberrations. In opto-electromechanical_raytrace, the imageis

analyzed _{ray-by-ray using} a small _mechanically moved aperture to measure the _ray

aberrationandfromthesegetthewaveaberration.

Withthelensperformance_becomingmore criticalitwouldbeadvantageousifthe

lithographerhad ameans of_measuringthe aberrationsofthelens in-situ_bya simple and

accurate method, so that he/she can use the stepper _optimally to suit _every situation.

Towards this end we propose _{using Phase shifting} point diffraction _{interferometry} to

measure theaberrationsofstepperlenses in-situ. Thereasons for choosingthismethodis

because it is easyto _implement, low on hardware _{requirements,} applicable over a wide

range ofwavelengths, is accurate enoughto measuretheaberrations one would expectin

2.Background

Thischapter gives abrief introductionon_aberrations,opticallithography, image

metrics, andeffects of aberrations onimaging. 2.1 Aberrations

Aberrations or errors in image formationcan be described interms of eitherray,

or wave aberrations. Aberration is indicated _bythe failure ofall rays _emanating from a single objectpoint to intersect at a singleimage point after_passingthrough the optics.

The coordinate offset of rays in the image surface indicates the magnitude of _ray

aberration _[7], [8]. Aberrations can also be expressed as the failure ofthe path from

entrance pupil coordinates to exit pupil coordinates to be the same for rays _emanating

from the same object point. The path difference between the actual wavefront and a

perfect wavefrontatthe exit pupilplanerepresent wave aberrations.

The relation between wave and _ray aberration is that _{ray displacement} error can be

expressed as the derivative of wavefront error in terms ofpupil coordinates. In Figure

2.1.1, Po'Po" represents _{ray aberration,} QQ' represents wave aberration. Figure 2.1.1 is

foron-axis case.

(xyi)=Rlni*(dWldx,dWldy) (2.1.1)

Where _{(x;, y;)} are coordinates ofPo" with respect to _Po', and R is the distance between

exit pupil plane and the image _{plane, n,} is the refractive index on the image side, W

Exit

Punil

Q' Q (x,y, z) Generalray

0

Po"(xi,yi)

Po'

(0,0)

W

Figure 2.1.1 Aberratedwavefrontforan on-axis point object.

EntP

ExitP

AS

One _mostly encounters _rotationally symmetric optical systems like in Figure 2.1.2. The

wave aberration canbeexpressedas a power seriesintermsofobject/image(h) andpupil

(r)coordinates:

W(h{r) =

fjfjfjclpmh2l+mr2^cosm(6-eo)

1=0p=0m=0

Where degree of each term ofthe series in object and pupil coordinates is always even

(2*(l+m+n)). We can removethe explicitdependence onimage heightby _suppressingit

and also _writingpupil coordinates as _p=_{rl a}

, where 'a'

is themaximum radius ofthe

exitplane_pupil,9istheazimuthalangleinthepupil_plane,we get:

W(p,e) =

fjfjanmp"cosme

n=l m=0

Where anm = a"

JT

m)h'2l+m (2. 1.4) /=o

Theaboveis usuallywrittenas a sum of_mutuallyorthogonalpolynomials calledZernike

polynomials. Theseare givenby:

^(A^)=ZZc-[2("₊1)/(l₊3m0)]1/2C(P)cosAn6?

(2.1.5) n=0 m=0

Where cnmaretheexpansion coefficients whichdependontheimage heighth',and n and

(n-m)/2

K(p)=

Z

Ho j!(n+m/2-j)!(n-m/2-j)!

istheradial polynomial ofdegreeninp.

The orthogonalityoftheradial polynomial andtheangularfunctionare givenby:

\k(p)K(p)pJp

o 2(+l)

and

2/r

Jcosm,9cosm',^,9=_;r(l+m0)Jmm, _(2.1.8) o

Theexpansioncoefficientsare:

i in

cnm = (\//z)[2(n_+\)(\+Sm0)T2\\W(p,S)R:(p)cosm3pdpd& _(2.1.9) 0 0

Usingtheabove expressions one cancalculatethevariance oftheaberrationfunction:

*i=iicL (2.1.10)

11=1 m=0

Anexample of someZernikepolynomials withthe aberration _theyrepresentis shown in

Term

(n,m)

Fringe Zernike Polynomial Common Name

1(0,0) 1 Piston

2(1,1) R cos(ct) XTilt

3(1,1) R sin(a) YTilt

4(2,0) 2R2-1 Power

5(2,2) R2

cos(2a) 3rdOrder Astigmatism

6(2,2) R2

sin(2a) 3rdOrder45Astigmatism

7(3,1) (3R3-2R)cos(a)

3rd

Order X Coma

8(3,1) (3R3-2R)sin(a)

3rd

Order Y Coma

9(4,0) (6R4-6R2+1) 3rd

Order Spherical 10(3,3) R3 cos(3a) 3rd Order3-Point 11(3,3) R3 sin(3a) 3rd Order 45u3-Point

12(4,2) (4R4-3R2)cos(2a)

5th

Order Astigmatism

13(4,2) (4R4-3R2)sin(2a)

5th

Order45

Astigmatism

14(5,1) _{(10R5-12R3+3R)cos(a)} 5th

Order X Coma

15(5,1) (10R5-12R3+3R)sin(a)

5th

Order Y Coma

16(6,0) 20R-30R4+12R2-1 5th

Order Spherical

17(4,4) R4cos(4a)

3rd Order 4-Point 18(4,4) R4 sin(4a) 3rd Order45u 4-Point

19(5,3) (5R5-4R3)cos(3a)

5th

Order 3-Point

20(5,3) (5R5-4R3)sin(3a)

5m

Order45u

3-Point

21(6,2) (15R6-20R4+6R2)cos(2a)

7th

Order Astigmatism

22(6,2) (15R6-20R4+6R2)sin(2a) 7thOrder45u

Astigmatism

23(7,1) (35R7-60R5+30R3-4R) cos(a)

7m

Order XComa

24(7,1) (35R7-60R5+30R3-4R)sin(a)

7th

Order YComa

25(8,0) 70RS-₁40R6+90R4-20R2+1 TOrderSpherical

26(5,5) R5 cos(5a) 3rd Order 5-Point 27(5,5) R5 sin(5a) 3rd Order45u 5-Point

28(6,4) (6R6-5R4)cos(4a)

5W

Order4-Point

29(6,4) (6R6-5R4)sin(4a)

5m

Order45u

4-Point

30(7,3) (21R7-30R5+10R3)cos(3a) TOrder3-Point

31(7,3) (21R7-30R5+10R3)sin(3a) TOrder

45u

3-Point

32(8,2) (56R8-105R6+60R4-10R2)cos(2a)

9th

OrderAstigmatism

33(8,2) (56R8-105R6+60R4-10R2)sin(2a)

9th

Order45u

Astigmatism

34(9,1) (126R9-280R7+210R5-60R3+5R) cos(a) _} 9W

Order XComa

35(9,1) (1 26R9-280R7+2 10R5-60R3+5R)sin(a)

9th

Order YComa

36(10,0) 252R1U-630RS+560R6-210R4+30R2-1 9thOrderspherical

37(12,0) 924R,2-272Rlu+3 1 50RS-1

680R6+420R4-42R2+1

1lm

OrderSpherical

Tllt(Z2)

yaxisIn pixels o a

Astigmatism(Z5)

id';;--0) i

5-c e

I 0- ;'-'\

:'3siBrlii Hii^laW. ^*.ffr^)%%Xfiat^HrMIIil^^B

^^^^~ -5,

fe

S mmHUHi^^^^

$ -10,

150

!Er7>^

'

^*-"V~-ISO too

^s^---." X'Y

'

^^^""^vn so

^>^ ^^50

yaxisInpixels 0 0 X axisInpixels

5-10

VaxisInpixels 0 ?

yaxisInpixels

Figure 2.1.3: Sometypicalwavefrontplots

2.2 Optical_LithographyandProjectionSteppers

Lithography is the name for a sequence of_processing steps for _{structuring the}

surface of planar substrates _{[9], [10],} [1_1], [12].Wecandistinguish betweentwo types of

lithographicprocedures. Mask_lithographyand_{scanning lithography. In}mask_lithography

thepatterntobetransferredisencoded as amplitudedistributionon a_mask, andthemask

is illuminated to transfer the pattern to the wafer. _{In scanning lithography} a modulated

beam isusedto write patterns _directlyonto thewafer. One can also distinguish between

variousforms of_{lithography by}the type ofillumination_theyuse, such aselectron_beam,

x-ray, ion-beam, or optical. Electron beam and ion-beam are _scanning type, x-ray and

optical are masktype.Electronbeam_lithographyis_usuallyusedformaskmaking.

Optical _lithography,whichistheworkhorse ofthe_industry, canbeofthree_types,

contact, proximity,and projection. Alloftheseareshowninfigure 2.2. Laand2.2. Lb. In

contact_printingthe maskis _directlyplaced on thewafer and exposedto light. Themain

problem here is ofmask _defects, which arise when mask is stripped _{repeatedly from} the

wafer. Resolution down to two microns is possible. _{In proximity printing the} mask is

placed a little away from the wafer. This reduces the wear and tear in the mask but

diffraction effects reduce resolution. Inprojection_printing animageofthephotomaskis

projected _directly onto a wafer coated with photoresist through a lens. A variation of

above is_scanning or_step and repeat projectioninwhich a portion ofthemaskis imaged

onto the wafer andthe exposure repeated _by either _scanningor _stepping over the entire

Source

Mask

Wafer

Intensityprofile

ContactPrinting

Waferholder

Source

Lens

Mask

Gap

Proximity Printing

Wafer

Wafer holder

Source

Condenser

Lens

Mask

Projection

Lens

Wafer

Thebasic components of a projection stepper are ₁₎ illuminationsystem ₂₎ lens ₃₎mask

holder₄₎waferholderand₅₎alignment system.

1) Illumination system: An example ofthe two extreme _cases, coherent andincoherent

illumination is showninFigure 2.2.2. In newsteppers available now one can change

the coherence factor. _UsuallyaKohlertype illuminationis used [30]. The technique

of_focusingthe illumination source inthe entrancepupil ofthe image_forming optics

iscalledKohler illumination. Thisensuresuniformilluminationoverthefield.

2) Projection lenses: Thelensesused forprojection are some ofthebestmade andhave

near diffraction- limited performance. Projection lenses have become more

complicated withhigherdemands placedonthem. A listofsome ofthelenses_being,

and to be used in IC manufacturing is shown in Table 2.2.1. The lenses are _usually

telecentric. The exit pupil of atelecentric _lithography lens is located at _infinity, this

implies that each point onthe waferis illuminated with a cone oflightwhose axis is

perpendicularto thewafersurface.

3) Mask holder: It is _basicallya stage for_holding themask. Thestage canbemoved in

'x' and

*y'

directions, wherethe optic axis ofthe systemis _along 'z' axis. Themask

MTF

1

Coherent

v/v0

Lightsource Condenser Mask Objective

Incoherent

Partially coherent

Figure 2.2.2 Coherentandincoherentillumination,vis

the spatial_frequencyandvo isthecutoff spatial

Minimum feature

inmicrons

NA X_(nm) ki Resolution

inmicrons

Depth ofFocus

(DOF)

in microns

1 .38 436 .8 .92 3.02

.7 .4 365 .75 .7 2.28

.5 .48 365 .6 .46 1.58

.35 .6 365 .5 .35 1.01

.25 .6 248 .6 .25 .69

.18 .6 193 .5 .17 .54

Table 2.2.1: Microlithographic lenses for ICproduction[1].

4) Waferholder: This isa stage toholdthewaferinplace_usingvacuum, whileit is

exposed.This stagehasthefreedomtobemoved_alongallthe3 axes.

5) Alignmentsystem: _Usuallyuses interferometrictechniques to align themask and

the wafer. Also helpsthewafer holdertobemoved _accuratelywhile _scanning or

2.3 Image Metrics

Theperformance of a stepperlens canbemeasuredorspecifiedinthe_followingways:

1) Resolution:

W=

^-

NA

where_ki isa_constant,whichdependson processparameters, X issource wavelength

andNAtheNumerical Apertureofthelens.

Thisequation explainsthe trendfor lowerwavelengthandhigher NAlenses. The

resolutionpossibleforvariouslenses is shownin Table 2.2.1

2) Depth ofFocus (DOF):

DOF=

Mr

NA2

where_k2is a_constant,whichdependsonprocess parameters.Thedepthoffocus

values forvariouslenses isgivenin Table 2.2.1. DOFreduces whenonelowers

wavelength, orincreases NA.

3)Modulation transferfunction_(MTF) [30]. Thisistheratioofcontrastin imageto

thatinobject,asfunction ofspatial frequency. Given by:

MTF(v)=

(2/x)[Cos-\v/v0)-v/v0(l-(v/v0)2f/2] (2.3.3)

Where vo=_{2NA/A. for incoherent illumination}

and_vo=_NA/A,for

coherent

illumination.

Where

MTFP(cr,v)=

G(a,v)MTF(v) (2.3.4)

1

Gicr,v)=

4 forv<vx

1 _{sin( )}

n 2NA ^ oW V2^ \(V~Vl) 1 sin(^ ) ' ' ' ' Si' i/l

-forvx < v< v.

n 2NA _(v-v2)

(2.3.5)

1

sin(-;r

2JV4'

<_\forv2 < v

wherev, = (1

-cr) andv, =(1₊

.18cr) 1

A 2 X

aisthedegreeof coherence givenby:

CT=^

Where_NAcisNumerical Apertureof condenser_lens,NA0istheNumerical Aperture oftheobjective.

The MTF Plots forthe threekindsofilluminationis shownin Figure 2.2.2.

4) Strehlratio (SR): Thisistheratioofimage_intensityaffected_byaberrationsto

thatoftheimage_intensitywithout_anyaberrations.

Acommon methodtoquantifytheamountofaberrationinalens isto_specifythe

root-mean square_(RMS) deviationofthewavefrontaberration. Forfringe Zernikesthisis

RMS'ik^^ikeJ (2.3.7)

m

Where _sm =k/2(n₊

1),nistheorder oftheradialpolynomial,k is 1 for

non-rotationally symmetric,and2 for_rotationallysymmetricpolynomial.Onecan getthe

Strehlratio_(SR)fromRMS using:

SR(R)= (2.3.8)

whereko=

2*n/X, Xiswavelength.

5) NILS (Normalized Image_{Log Slope) [1]}

1 r)F Givenby: NILS=_L

(2.3.9) E0 dx

where_Eois averageexposuredoseandListhelinewidth.

Fractionalchangein linewidthisrelatedtofractionalchangeinexposuredose by:

7=

2(M5)"^

This impliesthatasNILSdecreases sodoesexposurelatitude. This doesnot_uniquely

determineopticalcontributiontoprocess_latitude,because itdependson printbias

too.Anexample ofNILS isshownin Figure 2.3.1.

6) Process Window

This isa plot of usabledepthoffocusas afunctionof exposuredose.Thepresence of

aberrationsreducesthiswindow soitisauseful_waytodeterminethestepper

softwarefor_{lithographymodeling) is}shownin Figure2.3.2fora365 nmstepper

withlens_havinganNAof0.5 forcoherentillumination foraniso line500nm wide.

Intensity

248nm,NA=. 54

365 nm, NA=. 6

-300 _-ioo 100

Distance fromcenter of space_(nm)

Figure 2.3.1a: Light_intensity distributionof a350nmisolated_(iso)space atbestfocus.

.05

248nm,NA=. 54

Normalized

Derivative

365nm,NA=. 6

-200 -100 0 100 200

Distancefromright edge of space

Figure2.3.1.b: Normalized derivative ofthelight_intensity

distribution. Normalized derivative isinunits ofum [30].Asseen

Process Window

Exposure Delta

(%)

100

CD

Sidewall Angle

ResistLoss

Overlap

-1.0 -0.5 0.0 0.5 1.0

Focus

(um)

Figure 2.3.2: Process Window for 365nm,.5NAstepper withcoherent

illuminationfora500nm_{isolated(iso)}line. Theseplotsindicateusable exposure

andfocus latitudew.r.tparameterslike Critical_{Dimension(CD),} Sidewall

Angle(Profileof_{the step),}andResist Loss. Intheregionsinsidetheshowninside

thecontourstheparameters are within acceptablelimits. The overlapof allthese

showstheusableexposureandfocuslatitudewith which alltheseparameters are

2.4 Effectofaberrations onlithographic images.

Therelationbetweentheimage inthefilmatimageplane and wave aberrations canbe

written as_[13]

I(x,y,z) =

\lda0d/30J(a0,/30)

\2

(2.4.1)

Where Etheelectricfieldattheimageplaneandcanbewrittenas

E(x,y,z:_a0,/30) = FT'1

{b(a

-a0,P

-f30)P(a,[3)F(a,/3:z)eik^eik"W{a^

(2.4.2)

Where

FT"

is theinverse Fourier_Transform, J _(ao,Po)is theeffective sourcedistribution

in the projection lens entrance pupil with area S. O (ot-ao,P-Po) is the shifted Fourier

transformoftheobject. P _(a,P) istheprojectionlens transmission function. F (a,P: _{z) is}

the thinfilm contribution withinthe photoresistto adepth of'z'. e^

is the focus phase

term. elkW(a'P)

is the contribution due to the wave aberration. Here ko =

2*ti/X, X is the

wavelengthofillumination. _Using the aboveequation or _slightlymodified versions ofit

the effect ofindividual aberrationshavebeen modeled_{by inputting} individualaberration

terms in the wavefront term _W(a,p) (The individual Zernike terms are shownin Table

2.2.2).

Looking at the aberrations term by _term, the following generalizations can be

-The piston term _Zi indicates constant phase over the aperture and does not affect

imagery.

Z2 and_Z3representtiltoftheOptical Path Difference_(OPD)wavefront, andtheeffect on

imagingisa positionalshift oftheimage inx or_{y direction}ontheimageplane.This shift

canberepresented as a vector proportional to theZernike coefficients. This can be seen

in Figure 2.4.1 wherethe aerial imageof a500 nmisolated _(iso)line is shownfor a365

nm, 0.5 NAstepper,with no aberrations,andwith0.05wavesofxcomaaberration.

Z4 represents defocus. OPD surface is _quadratic, and it degrades image contrast, edge

slope, pattern _fidelity, and resolution. _Z5 and _Z6 represent astigmatism. OPD is saddle

shaped, and shifts orthogonal lines positively and negatively. For _Z$ the shift is for horizontalandvertical_lines,for_Z5theorthogonallines are noworiented at+-45 . Z7and

Zg represent coma andimage contribution from differentpupil radii shiftrelative to one

another. _Z10 and _Zn represent three clover. OPD has 3-fold symmetry and causes

undesirable_imaging artifacts._Z9representsthirdorder spherical. OPDis 3rd

order surface

andthefocus shift acrosstheimagevarieswithp.Thiseffectis shownin Figure 2.4.1.

Theeffectof aberrationscanbe inferred in manyways.

1)Aerial Image: Aerial image isaffected_byaberrations and canbesimulated_usingthe

equation2.4.1 shown_{early in}thesection. Figure 2.4.1 shows simulationdone using

Prolith fora500nmiso-lineon365 _nm, 0.5 NAstepperilluminatedwith coherent

Aerial Image

(Relative _Intensity)

-400 -200 0 200

X Position -(nm)

400

Figure 2.4.1: Aerialimageof a500nmiso lineimaged_bya365nm,0.5 NAstepper.

A-withoutaberrations, B- with0.05waves ofcomaaberration, C-with0.05 waveofspherical

aberration.These plots showtherelative _intensityatdifferentfocuspositions. Aslice parallel

tox-axisatazposition willindicate_intensityvariation_alongxatthatzlocation.

Aerial Image

(Relative _Intensity)

Aerial Image

(Relative_Intensity)

-400 -200 0 200 X Position (nm)

400 /: 1.29 1.17 1.06 0.94 0.82 0.70 0.59 0.47 0.35 0.24 0.12 0.00

Efled91.1vrev9corn, berrotai onktMgeo( ctoMed ipaee 1.5 Y r3ra A 1.2 X I 105 0S3 a3 0.15

an bJ I V-i.

~i

r-AberjoledSpacs M

' SpeoaM

I I I 0C 1.0 10 10 4.0 SO 60 7-0 8.0 10 TOO

X-flMS

MTF

Normalized_frequency

B

Figure2.4.2: Thefirst figureAshowsPSFwithout,and with.1 waves of coma aberration.

ThesecondfigureBshows variationMTFwithvarious aberrations.

In B is forno_aberration, is forsphericalaberration^ is forcoma_aberration,

2)MTForPSF: Aberrations changethe MTF characteristic of alens. Onecan alsolook

at thechange in PSF. MTF and PSF are Fourier Transformpairs of each other [32]. An

example of non-aberratedPSF and_{MTF, along}withPSF and MTF foran opticwith 0.1

waves of comaaberration are showninFigure 2.4.2. Whenaberrations arelessthanA/14

RMS, the shape ofthe central core of the PSF _{is essentially} unchanged except for a

decreaseinpeakintensity.

3)Process Window: Theeffect of aberrations canalsobeseenas areductioninprocess

window.This is shownin Figure 2.4.3 whereProcess Windowfora365 _nm,.5NA

stepperisshown with0.05 wavesof_coma,0.05 wavesofspherical,and no aberrations

forthecases of coherent and_partiallycoherentillumination (a=0.5).

Process Window

Exposure Delta_(%)

100t

50

0

-50

-100

-1.0 -0.5 0.0 0.5 1.0

Focus_(um)

y-~\

CD

SidewallAngle

ResistLoss ~

Overlap

(A)Withno aberration.

Process Window

Exposure Delta _(%)

100ir 50

0

-50

-100 -i i i

i--1.0 -0.5 0.0 0.5 1.0

Focus_(um)

Process Window

Exposure Delta_(%)

100

50

0

-50

-100

-1.0 -0.5 0.0 0.5 1.0

Focus_(um) nuuess vviiiuuw . u^ ^^'***\ -J"4i^^_

1 1 1 1 1 1 1

CD Sidewall Angle Resist Loss Overlap CD Sidewall Angle Resist Loss Overlap

(B)With 0.05 waves of sphericalaberration. Coherent

illiimirmtinn

(C) With 0.05 wavesof

Coma.Coherent

Exposure Delta

-100

-1.0 -0.5 0.0 0.5 1.0

Focus_(um)

CD

SidewallAngle

Resist Loss

Overlap (D)Withnoaberrations.

ProcessWindow

CD

Sidewall Angle

Resist Loss

Overlap

-1.0 -0.5 0.0 0.5

Focus_(um)

1.0 (E)With 0.05waves of spherical aberration._Partiallycoherent

illumination

Process Window

Exposure Delta_(%)

CD

SidewallAngle

Resist Loss

Overlap

-0.5 0.0 0.5

Focus_(um)

0 (F)

With 0.05waves of coma

aberration. _Partiallycoherent

illumination.

Figure 2.4.3: Alltheabove shown simulations weredoneonProlith fora365nm,0.5NAstepper_imaging

aniso line._{A, B,}Careforcohereift_{illumination,D,E,}Farefor_partiallycoherentilluminationfora=.5.As

3) Strehl'sRatio: The_followingresults shownin Figure2.4.4 arefromasimulation

doneatIBM _[31],theoptical system consideredisa257_nm, .35NA lenswith

partiallycoherentillumination (ct=0. 6). Ineachcasetheaberration valueis 0.02

waves. The_followingindicatecombinations chosen,_a) Spherical+is

Zl_{1+Z22+Z37, b)} Spherical-_isZl

1-Z22+Z37,c) Coma+isZ7+Z16+Z29, d)

Coma- _is

Z7-Z16+Z29,e)Astigmatism+isZ5+Z12+Z23,f)Astigmatism-_is

Z5-Z12+Z23, g) Combination+is Z5+Z7+Z1_{1+Z12+Z16+Z22+Z23+Z37, h)}

Combination- _{is Z5+Z7-Z11-Z12-Z16+Z22+Z23+Z37.}

Z'sindicate Zernike

coefficients(The individualZerniketermsare showninTable 2.2.2).

.9

.8

.7

Strehl's

ratio

-2 -1 0

Defocus (um)

Figure 2.3:Strehl'sratiothroughfocus for

variousaberrations.

Noaberrations

a,b,c,d,e,f

3Experimental Procedures

Inthischapterthe experimentalprocedure usedisexplained.

3.1 Phase shiftingpointdiffractioninterferometry.

In point _{diffraction interferomtry [15]} a beam diffracted from a pinhole in the

object plane falls on the test optic and the beam after _passingthrough the optic passes

through a _{partially transmitting} screen with a sub-resolution pinhole in it as shown in

Figure 3.1.1. The sub-resolution pinhole diffracts a reference beam and this interferes

with the transmitted beam to form an interference pattern. For accurate analysis ofthe

static fringe pattern, a significant number of tilt fringes need to be introduced _by

displacement ofthe reference pinhole_laterallyfromthe focusofthe testoptic. The light

intensity gettingto the sub-resolution pinhole is reduced. This with the small size ofthe

sub-resolution pinhole necessitates reduction of the _intensity of the test wave _by

attenuation.

This technique was modified _by H _Medeki, et al _[16] with the _possibility of

introducingphase shift_byusing agrating. In this _setup as shownin Figure 3.1.2 a small

pinhole atthe object plane ofthe optic to be testedproduces a_nearly spherical beamto

illuminate a_{grating kept in between}the optic and the object point. _{The grating}pitch and

thedistance between the _grating andobject side pinhole are chosen so thatmore than 2

diffractionorders pass through theoptic. Intheimageplanetwo ofthediffractionorders

are _spatially filtered with a mask which has one _window, and a sub resolution _pinhole,

The first order _beamis made to pass throughthe largepinhole _{(approximately20} to 40

times the resolution limit ofthe _optic) in a mask kept the image plane and is the test

beam. The sub resolution pinhole allows the zeroth order beam to pass _through,

generating the reference beam. The reference and the test beam interfere to produce an

interference_pattern, which can be analyzed to get the wavefront phase error due to the

testoptic. The grating allows forthe_possibilityof_varyingphase. _Bymovingthe_grating

in a directionperpendiculartoboth theoptic axis and thedirection of_gratingrulings_by

onefourthofthe _{gratingpitch, the}phase ofthe 1storderbeamchanges_by90 degrees and

that ofthezeroth orderisunaltered. Thiscan beusedto generate a set ofinterferograms

shiftedwith respectto each other_by90 _degrees,which inturn canbe analyzed_using

5-framemethod [25](The method isexplained in section3.3). The accuracy ofthemethod

depends on various _parameters, the most important _being the _quality ofthe reference

wavefront, which in turn is dependent onthe size ofthe sub-resolution pinhole. Other

factors are the 3rd order comaerror dueto the _geometryofthe setup. This effect canbe

usedtocalibratethe interferometer.We can usea maskwithtwo sub-resolution_pinholes,

and compare the _{experimentally}calculated coma to the _{theoretically} predicted value of

coma. Anotherpossible source of errorisdetectormisalignment.Alotof work onPSPDI

has been done _bythe_group atthe department ofElectrical _Engineeringat _University of

California_Berkeley, where_theyusedthemethod to characterizeEUVcameras _{[17], [18],}

Optic

Source

CCD

Pinhole Mask

Transparentmask

Withpinhole

Figure 3.1.1: Pointdiffraction_{interferometry}setup.

CCDCamera

Pinholemask

Twopinholemask

3.2 Errorsourcesintheexperimental_set-up

3.2.1 Errorduetoaparallel glass plate_beingintroducedina_convergingor

divergingbeam_oflight (Figure3.2.1).

The gratingpattern andtheimageplane pinhole mask arebothwrittenon0.09 inches

thickchrome plated glass plates.Theparallel glass plateintroduces primary

aberrations when_theyareinthepath ofa_convergingor_divergingbeamasisthecase

forthe_gratingand pinhole mask plate[8]. Theseaberrations canbeexpressedas:

W(r,0 :_h)=

as(r4

-Ahr"

cos0+4h2r2

cos2 6+

2h2r2

-4h3rcos0) (3.2.₁₎

where_asis thecoefficientofspherical aberrationandisgivenby:

(n2-\)t as - r~r~

(3.2.2)

8n3S4

Letusmake anorder ofmagnitudecalculation oftheaberrationforthe_gratingplate.

Fromthefigure it isseenthat(r/S)~NA."n"

istherefractiveindexof_glass,"S"isthe

separationbetweenglass plate andimageplane, "t"

isthethicknessof_{the plate,}"r"is

beam diameterattheplate. Thefirstterm ofequation3.2.1 willthenwillbe:

(n2-l)tNA4

1 (3.2.3)

8n3 V '

Usingthisand_{the values, t}=2286um,NA=0.028ontheobject_side,n=1.5 we getW

tobeoftheorderof6*

10"5

microns,whichisnegligible.

Ontheimagesidethepinholemaskis aligned suchthat thechrome coated part ofthe

patterngeneratedbetweenthereference andtestbeam. Wecan_onlyreduceit_by

usingtheminnest glass or quartz plate possible.

ExitP

Figure 3.2.1: Parallelglassplate ina_{converging beam. Rays}incident

onplateconvergetoP' insteadofP.

.1

.05

P-V Phase Error

(waves)

-20

0

Linearphaseshift error_(%)

20

Figure3.2.2: Measured Phase Errorvslinearphase shift error of

3.2.2 Error incalculated wavefrontdueto errorin gratingmotion.

Asdescribedinsection3.1 on experimental procedurethe_gratingneedstobe

moved_bya value equaltoonefourththatofthe_gratingpitch.Theerrorcanbereduced,

byusingalarge_gratingconstant so that therequiredmovementof_gratingiscoarse

enough. Inthepresent_set-up the_gratingneedstobemoved_by20microns,whichis

coarseenoughforthe stages_beingused withleastcount of1 micron.Also as seenin

figure 3.2.2 _[251,me use of5-bucketalgorithmimpliesthatthewavefront errordueto

errorin gratingmotionis approximatelylessthan0.002wavesforerrorsinmotion of

grating upto +-5%.

3.2.3 Referencepinhole errors

Asthepinholediameter isoffinite_size,itallowsfor finitespatial _frequency

bandwidthtopassthrough. Itisreasonabletoassumethattheseerrorsdecorrelateas a

functionof pinhole positionrelativetothepoint spreadfunctionoftheoptic.Thiseffect

canbemitigated,byan_averagingprocess wherein pinholepositionis changed_bya

fractionoftheopticpoint spreadfunction. Theothersource of erroris defectsinpinhole

shapethatistheholes_beingdifferentfromperfectcircles.Thiscanbe_reduced,by

averagingoveranensembleofequivalentsizedpinholes [24].

Forthisparticular experimentdoneontheoptic_bench, another source of erroris

micron,andthesize ofthepinhole usedinthemeasurement_{set-up is}of2microns. A

micron pinhole was_written,but didnot clear_{fullypossibly}becauseof wet_{etching done}

on athickchrome_{layer. Modeling}wasdonetoget anideaofwhaterrors anon-ideal

pinhole wouldintroduce.Theassumption wasthatidealpinhole was adeltafunction,and

thenon-idealwas a rectfunctiontwice thesize oftheideal. Theassumptionisvalid

becausetheFouriertransformof a 1 pixelimagegivestheresponseexpectedfrom ideal

deltafunction The farfield diffraction_field, whichrepresentsthewavefrontattheexit

pupil,was calculatedfortheidealand non-ideal reference pinholeand asetof37 Zernike

polynomialsfittedto theportion ofdifferencewavefront_fillinganNAof0.28. Thiswill

indicatethesystematic errorduetonon-ideal pinhole.A firstapproximationto thefar

fieldwavefrontdiffracted fromtheexperimental pinholeis thediffractionpattern ofa

coherentbeam fromasimple circular apertureinaplanar screenbasedonKirchofFs

modelofdiffraction theory.Apinhole of_{diameter, d,} diffracts aspherical wavefrontthat

fillsNA= sinG= ₁

.22A,/d.The Zernikecoefficients calculatedfornon-idealpinholefor

the436nmoptic are shownin Figure 3.2.3. Thenon-ideal 2-micronunderfillstheNAof

optic_{byapproximately} 5percent. Thecalculatederror wavefronthasnon-zero valuesfor

theaberrationterms_Z9andZ17. Thefirstoftheseis importantasthesecond one can

possibly bereducedorremoved_byPark'smethod [Appendix_B],it_beingan

3.2.4: Systematicgeometric effect_ofcoma

Becauseoftheinherent largetiltinthe_set-up thereisasystematiccoma errorin

thereconstructed wavefront_{[19, 21,}24]. _Assumingtheseparationbetweenthereference

pinholeandthewindowtolie_alongxaxis and_expressingrr,andX2(Figure_3.2.4)asa

functionof_mixingplane positioninpolarcoordinates_(p,0)andbynot_{neglecting higher}

ordertermswehave

g -0.05

-D.15

Error duetolargerreferencepinhole

-l 1

r-15 20 25

zemcoeffno

40

Figure3.2.3: Error duetolargerreferencepinhole.Theabove plot

showstheZernikecoefficients forthecalculateddifference

wavefront,betweentheidealsizedpinholeofdiameterdandthenon

Ar=rx-r2=(-^3 --(f]p)Cos0 (3.2.4)

2z z 2z

where sisthe separationbetweenpinholes,zistheseparationbetweentheimageplane

andthepinhole plane.

IntermsofZernikepolynomialstheabove equation canbewritten as:

Ar=

(C(3p3-2p)+Tp)cos0 _(3.2.5)

Where C isthecoma coefficient andTis tiltcoefficient_alongxdirection.

C= =

(sin"1

(JM)) -(NA? (3.2.6)

6 z 6 6

Thiscanbeusedtocalibratetheset-up. Ifweusetwosub-resolutionpinholesinthe

imageplane_mask, andthenmeasurethewavefrontit shouldyield_onlythe tiltand

systematic coma_error,becausenowtworeferencebeamsareinterfering._Comparingthe

calculatedsystematic comawith_{theoretical one, theset-up}canbecalibrated. This

calibrationcould notbeen done for 436nmoptic asthe2referencepinhole mask was not

written.Thevaluesin bothcases wouldbe 1 micronfor 436_nm,and.16micronsforthe

248 nm.

3.2.5: Detectormisalignmenteffect

Properalignment needsthedetectortobeperpendicularto thecentral_rayofthe

tilt yx, andyywith respecttox and y-axesrespectivelyas showninfigure3.2.5,then the

pathlengtherrordueto thisisgiven as:

Ar =

^j[yx(cos20

l)-yy

2z (3.2.7)

Thiscanbeinterpreted intermsof astigmatism anddefocuserrors and_{corresponding}

coefficients are given_respectivelyas:

and

ea=\NA\yx2+y2r2

1

(3.2.8)

ed = _(3.2.9)

whereNA=rm/z

Thevalues wouldbe 0.000154waves_(ea), 0.000054waves_(ed)for436nm perdegreeof

tilt.

2-pinhole

mask

I

Mixing

3.2.6:Errordueto_scatteringofreferencebeamthroughthespatialwindow

Inthepresent experimenttheseparationbetweenthereference pinhole andthe

spatial windowisequalto thewidth ofthewindow[23]. Inthiscasethereis a_possibility

of referencebeam_beingscatteredthrough thewindow.

Signal reachingthedetector is:

U(fx)=

A8(fx

-fe)+_{rect(^)[n(fx)}+_S(fx)]

Yy

Detectorplaneideal

alignment

Figure 3.2.5: Detectormisalignment

effect.

whereWisthewidth of_{the window,}5represents_{the pinhole,}n representsthenoise

(scatteredsignalfrompinhole_{through the window),}andSrepresentsthesignal_energy

passingthrough thewindow. This isshowninFigure 3.2.6. Thiscanbeavoided_by

changingtheseparationto 1.5times thewidthofthewindowinwhichcasethereisno

scatteringas seenin figure3.2.6.

w

k M ?

1

w

< ?

f.

Figure 3.2.6: The_toptwo figuresshowthe spatial spectrum ofthefield inthe_recording

plane, thebottomtwo figuresshow spatial spectrum of recordedirradiance. Thefigures in

theleftareforthecaseofseparationbetweenthepinhole andwindow equaltoWtheright

3.3The 5-framemethod.

Thismethod canbeused for_anyinterference _set-up, whichhasthe _possibilityof

shiftingthephase ofthe interferencepattern _{[23], [24],} [25]. Thephase _{shifting itself}can

be doneinvarious ways. Someofthemethods employed forphase_shiftingare_{a) moving}

mirror, inwhich a reference mirrorismoved inadirectionperpendicularto theopticaxis

using a piezo. _b) tilted glass plate _{c) moving diffraction grating,} in which a _grating is

moved in a direction perpendicular to the optic axis and the _{grating ruling d) Rotating}

Analyzeror Waveplate. If is the wavefront phasedifference between the test andthe

referencewavethen the_Intensityofinterferencepatternrecorded canbewritten as:

I= I0(l₊

yCosW)) (3.3.1)

This technique uses

90

phase shifts to minimize phase calibration errors. This

algorithm reduces the_possibilityofthenumerator and denominator_{tending to zero,} and

thereby reduce the _{uncertainty in} the calculation. It uses five frames of_intensity with

relative phaseshifts of a. _Ioisthezeroorder_intensity,yisthecontrastfunction.

/, =Io(l+_yCos(0-2a))

I2=Io(l+_yCos(0-a))

I,=Io(\+_{yCos(0)) (3.3.2)}

I4=I0(l+rCos(</>+_a))

Theseequations canbecombinedtoyield:

*\im/yS\ivi/h

(3.3.3)

I2-I4 SinaSintfi

2/3

-15

-Ix (1- 2Cos2a)Cos<t>

when a=

n/2theequation simplifiesto:

=_tan-'[ 2(/z h) ] (3.3.4)

2/3-/5-/,

V[2(/2-/4)]2+(2/3-/5-/1)2

Thephasethusinferred fromthe interferogramis_usuallywrappedintherangeof

0 to _27i, due to the nature of arctangent calculation. To get the correct phase of the

wavefrontthephase needs to beunwrapped. Thereare_manymethods of_unwrappingthe

phase. Thebasic assumption all ofthesemethodsmakeis thatit isnotpossibletohavea

3.4 Phase unwrapping:

Phaseof wavefront whendetermined_byinterferometricmethodsiswrappedin

therange of0to_2n, and needstobeunwrapped. See Figure 3.4.1 as anexample.

Thesimplest method of_unwrappinginvolvesa sequential_scanningthroughofdataline

byline as shownin Figure 3.4.2 [26]._{While scanning}thelineadjustment(aphaseof n

added or_subtracted) ismadeto thepixel valueswiththeassumptionthatphasevaries

smoothlyand_neighboringpixelscannothave a_jumpof morethann.Attheendof each

linethephasedifference betweenthecurrent pixel andthepixelontheline beloware

compared, corrected andthelinescannedintheoppositedirection. Thisimpliesthatwe

are_treatingthe2dimensional dataas wrapped one-dimensionaldata. Thisdoesnot work

always.

271

0

Unwrappedphase

Wrapped Phase

^

/

?/

/

?

V

N

Figure 3.4.1: Arepresentation of

Phasewrapping.

An improvedversionwas givenTakeda [27]. Hereeach rowis_{independently}

unwrapped.Therows nowhaveanundeterminedphase relationtoeach other. This is

Line

by

After Scan 1

After

Scan 2

Takeda'smethod

Figure 3.4.2: Schematicof_unwrapping

byline-by-linemethodandTakeda's

4Experimental_set-up

Aprototype experiment wasdoneon an opticbench_usinga436nmCarl Zeiss

optic.The_set-upused was similarto thatshowninfigure3.1.2. Thesource used was a

He-Cd laser_lasingat442nm.Theobject side pinhole was 10microns.The gratingused

hadapitch of80microns and waskeptadistanceof

'z'

awayfromtheobject_pinhole,

towards theoptic._{The grating}wasfabricatedon_{commerciallyavailable,} chromeplated

glassmasks_usinge-beam patterning.Theimagesidemaskwas alsofabricated_using

e-beampatterning. _Initiallythemask wasmadeon_{commerciallyavailable,} chrome plated

masks,butthesedidnothaveenoughattenuation. Soglass plates were coated with2000

angstroms of chrome andthenused. Theimage sidemaskhada set of pinholewritten

withthesmallpinhole sizes_{ranging from 1} to 5microns,for 2differentseparations,27.5

and50microns, correspondingto 'z'

valuesof50 and90mm.Thepinhole pair with

1-micronsize didnot clearin_{the mask; possibly due}tothe_inabilityof wet etchtoclearthe

thickerchrome. Fromthedata intheTable 4. 1 thedistancebetweenthefrontlenssurface

andobjectplaneis 440.5 mm andthedistance between last lenssurface andimageplane

is 29.4mm.

For recordingtheinterferogramanLBA 100 laser beamanalyzer_set-upwas used.

The LBA 100usesapulnix CCDcamera(120_by 120pixels). Thedatawastransferred

fromtheLBA 100to aPC usinganRS232 interface. Theinterferogram fileswere

program written_{in-house forthe}purpose[Sourcecodeis inAppendixC].Aset offive

typicalinterferograms isshownintheFigure5.1.

S-Planar 1.6750

Cat. No. 10 77 82

Reductionratio _1:10

Wavelengthrange 436+-5nm

Objectto Imagedistance 602+-_4mm

Image fielddiameter 14.5mm

Square image format 10.5_by 10.5mm

Flangefocaldistance 24.5

+-.3 mm

Free working distance Min 10mm

Lengthofbarrel 160.5 mm

Max. diameter 85.5mm

Screwthread M 76_by 1

Entrancepupil

Locationbehindtheobject

Diameter

486mm

27.2mm

Exitpupil

Locationbehindtheimage

Diameter

512mm

287mm

Effective focal length 49.1mm

DistanceofPrincipalpoints +8 mm

Numerical aperture 0.28

Diffractionlimitofresolution 1285linepair per/mm

Rayleighdepth in imagespace +-2.8microns

Dataonimageperformance

1. Full field

Distortion max.+- _0.3microns

Opticaltransferfactor for_{incoherent/partially}coherentillumination_(o=0.5)

Spatial_frequency(lpper/mm) Linewidth_(um) Optical Transferfunction_(%)

250 2 59/92

500 1 37/62

2. Restrictedfield diameter 10mm

Distortion max. 0.2microns

Optical Transferfactor for_{incoherent/partially}coherentillumination(a_=0.5)

Spatial _{frequency(lpper/mm)} Linewidth_(um) Optical Transferfunction(%)

250 2 69/96

500 1 46/67

700 0.7 30/30

5 ResultsandAnalysis

The experiment was performed on a _{Carl Zeiss g} line lens. The procedure is

explained in section 4.1. The experimental interferograms obtained for 0

orientation is

shown in figure 5.1. The Wavefront aberration of the lens was calculated _using the

algorithms described. Park's method was used to remove theunsymmetrical aberrations

due to the reference beam. _{An averaging} of 50 counts was done while _taking each

interferogramtoremove or reducerandom errors. I used2_fitting algorithms. These are

shownin appendixC. Each algorithm wasused to fitasyntheticwavefrontforwhichall

the Zernike coefficients were set to 0.1 waves. The difference or error wavefronts for

each algorithmwas calculated. The RMSforerrorwavefront after_{fitting,using}algorithm

1 was 0.00754waves, usingalgorithm2was 0.000247waves. Thevalueslooksmallbut

areimportantaswillbeseen.

A set of six interferograms each_having aphase difference of tc/2 w.r.ttheother

was taken forall the4 orientations. Wecan calculatethewavefront phase_{using 5-bucket}

algorithm. Weshouldget comparable values when phaseis calculatedwiththefirst 5 or

thelast 5 interferogramsthatiseither_{usingL, L, I3, L,} and_I5,or_{usingI2, 13, L, I5,}andI6.

This should represent repeatability. The _following plot Figure 5.2 shows the wavefront

coefficients calculated_using algorithm 1 and 2 for90

orientation of_{the optic, byusing}

Ilfff

Iff

\

i:;';

|j

; ; _; 1

"' fi

;

j

j

:;

fj

j'

j

\\

'

\ i '

^ |

'

t , j ,Bj

';

|

i ii |m i' ', ;..' .

"

* _'-: *' '

j'

I\ I : _j

'

k1 \\\\\\ _: :

i. j i _; '

|| j {

|

^ II i_j '

! ,,- !< !;

I'' i'

| ' j. jJ

'

* _;B'.| !'

!

I

' '

'

' :

<

11

illii

1 ill 1 1

1

3 1 1

ll

ii

[

>1*

1f> '

t

'

f

!:J :_{| K | I} i 1

|

;-'

j 1, ,

'

','

'

. I "'';

'

! _!

1[%i

' i ; I I ' i --.. | h ; ' ;

\) i ! i

1

Interferogram 4

Interferogram 2

Figure 5.1: Shows aset interferogramsshiftedw.r.t eachother_byn/2. These interferogramswereimagedon a120_by 120 Pulnix CCD

camera.

Interferogram 5

in ii

I !'! I

Repeatability </5 > CO CD o o N 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 ' _J

fi

[l

12 3 4 5 6 17 8 9 11011 12 13||4 1516 1718(1 9 20 21 22 23 24 25 6 27 28/28 30 313233,3435 36 37

11st five

12ndfive

zern coeffno

Repeatability 0.06 CD > 0.04 CO C "2 g " d) N

XX flJH,

Ix.

1 2 3 4 51 6 7 8 9 10 1112 13 14 1516 17 181

i4ji

P'2?l

LoJil

I

2nd five

zern coeff no B

Figure5.2: AshowsthewavefrontZernike coefficients calculated

usingalgorithm1. B shows wavefrontZernikecoefficients

calculated_usingalgorithm2.Foralgorithm 1 theRMS oferror

Theabove plotsindicatethat_{repeatability is}good andisindependentof algorithm.The

RMSvaluesforthewavefrontsmatchtowithin0.0056wavesforalgorithm1 andwithin

0.0009wavesforalgorithm2.Algorithm 1 hasmore errorinthe_estimatingthe_following

coefficients_{9,10,15,16,17,28,36,37}andthisiswhatleadsto thelargervaluesforthese

coefficientsinthecalculated wavefront.Thecomparison ofthe twoalgorithmsisshown

in Figure 5.3.

Tilt:_{The experimentally}observedtiltinthe_{interferogram,} inlinepairsper mm

(lp/mm)is: observedtiltis 9.8666waves or 19.73 lp/mmover8.5mmof_CCD,thatis

2.32 lp/mm. Thecalculatedvaluefrom geometry usingyoung's doubleslit equation:tilt

in lp/mm isgiven_bysl(X*Z). Where's' isthedistance betweenthepinholeandwindow

(27.5microns),

'A,'

isthewavelength(442_nm), and 'Z'

is thedistancebetweenthe

pinhole maskandCCDplane(2.75cm). _Puttingin_{the values, the tilt}is foundtobe 2.26

lp/mm.

Thevalues gotfrom usingalgorithm2were usedto calculatethewavefront

coefficients foreachofthe4orientationoftheoptic. Park'smethod(Appendix_B)was

usedtocalculatethewavefront with reference errorsremoved.Figure 5.4showsthe

aberration coefficients ofthewavefrontafter_{applying Park's}method. Strehl'sRatio

Algorithmcomparision

0.102

> TO

5

03 >

8=

d) o o

o>

N 0.096

Dnewalg

old_alg

12 3 4 5 6 7 8 9 10 1112 13 14 15 16 17 18 19 20 2122232425 26 27 28 29 30 3132 33 34 35 36 37

Zerncoefficient no

Figure 5.3: This showsthecomparison oftwoalgorithms usedtofit

thewavefrontandestimateZernikecoefficients.Herefitis donetoa

syntheticwavefront with each coefficient_havinga value of0.1. The

WavefrontcoeffafterPark'smetod

0.07 0.06 0.05 004

a>

ii 0.03

CO >

8 01

E o

CD N

-0.01

-0.02

-0.03

-0.04

12 3 4 5 6 7 S 9 10 11 12 13 14. 15 16 17 18 19 20 21,22 23 24 25 26 27 28,29 30 31 32 33 34 35 36 37

niJ

TV -crrrcr n

~

-n , _ n

\

1

'irir '

lii~'

11 '

f,r

ij-j

zern coeff no

Figure 5.4: ThisshowstheZernike coefficients after

MTF

The MTF of an optical system withknownaberrations canbecalculated_usingan

equationfrom chapter4of'TheArt andScienceofOptical_Design',byR.R. Shannon

[32].

Theequations are:

DTF(Q)= _[arccos(fi)

-QVl-^2]

whereQ. =

71

V

V _(5-1)

ATF(Cl) = 1

-(

^=-

J

(l

-0.5)2

)

MTF(0)=_{DTF(Ci)ATF(Q)}

Intheabove equationsDTFistheaberrationfree MTFof_lens,ATFistheaberration

transferlens_dependingontheWavefront aberrationsofthelens(Wrms)- Figure5.5

showsthe calculatedMTF usingthecalculated_Wrmsvalueof.0505waves,andtheMTF

MTF

-calculated -fromdatasheet

0.4 0.6 0.8

normalized spatial_freq

Figure 5.5: TheaboveshowsMTFplotsof436-nm

opticfromthedatasheet andfromcalculation_using

Total errorinthemeasurement:

1₎ Errordueto_gratingglass plate: RMSof errorwavefrontis7.92*1

0"5

waves

2) Errorduetomeasurement(couldincludeerrordueto_gratingmotion,but I have

consideredthatseparately):RMS of error wavefrontis 0.0009waves.

3) Errorduetomotion of_{grating assuming I}moveto within 1 micron: RMSof error

wavefrontis 0.001587waves.

4) Errorduetodetectortilt_(assumingdetectortiltiswithin2degreestoaplane

normaltooptic axis): RMS of error wavefrontis 0.000188waves

Assumingtheseerrorstobe independentthe totalerrorcalculatedforthemeasured

6 Conclusions andRecommendations

The results obtained for the Zeiss lens show that PSPDI can be used for _testingof

stepperlenses. The measured values of aberrations andStrehl's ratio are in the expected

order of magnitude. The analysis end has been verified and works fine. The things

learned are that one ofthe most crucial components is the sub-resolution pinhole. The

smallerthepinhole_is, thebetterthe _accuracyofthemeasured aberrations. Butonehasto keep also in mind that the smaller pinhole willreduce the contrast inthe interferogram.

The set-up canbecalibrated_byusinga2-sub-resolutionpinholemask and_calculatingthe

coma got from interferogram _analysis, and _comparingthe resultwith expected coma for

the_{set-up (refer}section3.2.4). The gratingmotion needstobecalibrated and micrometer

stages with _accuracy of 0.1 micron need to be used as mentioned in section 3.2.2. As

mentioned in section 3.2.3 for future measurements one needs to take averages of

measurements_bymovingthepinhole mask_byafractionoftheopticPSF and also_taking

averages over a number of pinhole pairs. One can get abetterestimate ofthe total error

by doingthefollowing:

a) Measurementerror: Take 10interferogramsshifted w.r.t each other_bya phase of

ti/2, thisallowsforcalculation ofwavefrontphase 5times.This impliesI can

generate 10differencewavefronts. LetthestandarddeviationoftheRMS ofthese

b) Makeanother set ofmeasurementswiththeposition ofmicrometerscrew_moving

the_grating,moved_by.2micronfrom idealpositionforeachofthefive

successive positions. Calculatethewavefront phaseforeach_case; calculatethe

differencewavefront w.r.tidealcase.Let_a2bethestandarddeviationoftheRMS

ofthesedifferencewavefronts.

c) Let0-3 representtheRMSoftheerrorwavefrontduetoglass_{plate(grating)}

d) Let _cj4representtheRMSoftheerror wavefrontduetodetectortilt_(assuming

detectorplaneisnormaltooptic axis within 1 degree).

Thetotal errorintheRMS ofmeasuredwavefrontthenis:

ct=(^₊

^

ct32+ct42)-5 (6.1)

10 5

Appendix A

Zernike_fitting

WenowdiscussthevectorformulationforZernikepolynomial surface _fittingand

extraction ofZernikecoefficients.Awavefront canberepresented as alinear

combination ofZernikepolynomials.

W(x,y)=

YJctjZj(x,y)

In discreteform:

Wr(xr,yr)=

YjajZj(xr,yr)

Where _{Wr (xr, yr) is} the value of wavefront at point (xr, _yr) and Zj's are Zernike

polynomials and a,-'s are the coefficients. Let there be n points (n>m). _Usually m = _37.

Writingtheaboveinmatrixform

^iCWi)

W2(x2,y2)

^,(W).

^iCwi). -^(xp^i)

zx(x2,y2), zm(x2,>>2)

Writing

Wx

W.

using(3),equation₍₂₎ canbewritten as:

W=[ZbZg

ZyCWl)

~ax

zM2,y2)

=h> a2

zMn>y)_

=_a

-. Zm]*a

(4)

(5)

Letthemeasuredvalue of actual wavefront W begiven_by Wl.Thensumof squaresis

givenby:

S= [Wl-_[Z,,

-,Zm]*a]

thatis:

n m

r=\ j=\

S=

Y,(mr-ZajZAXr>yr)r

n

Now minimizing Sw.r.t

a,-weget:

(6)

(7)

YwKzA*r,yr)

YLajzMr,yr)zk{xr,yr)

r=\ j=\

r=\

(8)

WIZ,

w\z

ZXZX,

~lZlZm

ZmZ\1 'ZmZn

(9)

ChoosingZ/stobeorthonormalonehas:

WIZ,

W1Z

ZxZx,0,-

0,Z2Z2,-

0,0,-o

,0

7 7

(10)

Thecoefficients canthenbewritten as:

Becausewe