extended nijboer-zernike analysis for vectorial diffraction … · 2006-09-18 · extended...

$: Extended Nijboer-Zernike analysis for vectorial diffraction … · 2006-09-18 · Extended Nijboer-Zernike Analysis (vectorial) IISB Fraunhofer Lithography Workshop 17-19 September$
Extended Nijboer-Zernike Analysis (vectorial)IISB Fraunhofer Lithography Workshop

17-19 September 2004Sense and Simplicity

Extended Nijboer-Zernike analysis forvectorial diffraction calculations and

aberration retrieval

1) Delft University of Technology

Optics Research Group

2) Philips Research Laboratories

Lithography group

Mathematics group

Joseph Braat1, Peter Dirksen2, Augustus J.E.M. Janssen2



OUTLINE

• What is the (extended) Nijboer-Zernike analysis?

• Axial extension of the aberrated in-focus diffraction integral to a finite focal region

• Vectorial and high-NA effects

• Future work



What is the Nijboer-Zernike analysis?

• Complete and orthogonal representation of the wavefront aberrationin the exit pupil of an optical system using circle polynomials

• Limited to a perfectly spherical support• First proposed on the unit circle by Szego in 1919• Introduced in optics by Zernike (1935) and Nijboer (thesis, 1942)

• Immensely popular thanks to Jim Wyant (interferometry) since 1980

[ ])( )exp()(

)sin()cos()(),(

maxmax,

||

,

mmmimRc

mbmaRW

mn

mn

mn

mn

mn

mn

mn

≤≤−=

+=

∑

∑ϑρ

ϑϑρϑρ



Usefulness of the Nijboer-Zernike analysis

• Optical design:The polynomials solve the aberration balancing problem in opticaldesign (Maréchal, Hopkins, Welford: 1940’s)

• Diffraction theory:Calculation of diffraction patterns for small aberration values

and modest defocusing (Nijboer, 1942, Nienhuis, 1943)• Interferometry:

Stable reconstruction of measured data;‘fringe index’ of polynomials introduced by Wyant c.s. in 1980(37 ‘first’ Zernike polynomials)

• Analytic extension of (aberrated) diffraction integrals to focal volumeand arbitrary axial position using Zernike polynomials (Janssen, 2002)



Extension through-focus N-Z analysis to high NA values

• Rigorous description of the defocus term, if necessary higher orderexpansion ( -polynomials etc.)

• Incorporation of vectorial description of focused field for arbitrary stateof polarisation

• Inclusion of radiometric effects (Abbe or other imaging conditions)

• Treatment of new (aberrated) diffraction integrals for focal volumeusing Zernike polynomials(Braat, Dirksen, Janssen & van de Nes, JOSA A, Dec. 2003)

)(04 ρR



Standard geometry of focused wave

Coordinates on pupil sphere:

)exp( ϑρνµ ii =+Coordinates in focal region:

zr ,, ϕnormalized with respect to:λ/NA (transverse direction r )

Defocus parameter f in pupil:

{ }{ }

{ }2

22

2

22

)(112

)(11)(11

)(112

NAzf

NANA

f

NAzdef

−−=→

−−−−

=

−−=Φ

λπ

ρ

ρλπ



Diffraction integral (scalar)

{ }

{ }ρ

ϑϕϑρπ

ϑρϑρρρ

πϕ

π

d

dri

iAiffrU ∫ ∫

−

×Φ=

1

0

2

02

)cos(2exp

),(exp),()exp(

1);,(

Introduction of Zernike polynomials (mapping of amplitude and phase):

{ } ∑=Φmn

mn

mn imRiA

,

|| )exp(),(exp),( ϑβϑρϑρ




∑=mn

mn

mn

m mfrVifrU,

cos),(2);,( ϕβϕ

Analytic solution for the V – functions (A.J.E.M. Janssen):

rvmnplv

vJvififfrV

l

p

jl

jlmlj

lmn π2,2/)(;

)()2()exp();(

1 0

21 =−=−= ∑ ∑∞

= =

++−

ρρπρρρ drJRiffrV mmn

mn ∫=

1

0

2 )2()()exp();(

2/)(;1

1

1

1

1)2()1( mnq

l

jlq

jp

l

l

lj

l

ljmjlmv p

lj +=

++

−

−

−−+

−

−++++−=




+= ∑ ++

mn

nmn

n mv

vJi

v

vJrU

,

111 cos)()(

2)0;,( ϕβϕ

Convergence conditions:

In-focus Nijboer-Zernike result (1942):

Extended (through-focus) result :

+= ∑

m,n

mn

mn

m00 mcos)f,r(Vi)f,r(V2)f;,r(U ϕβϕ

6

max

10error abs.,60,202,2||

:for25−≤≤≤≤≤=≤

==

qprvf

lL

ππ



Some calculated patterns

Re{V4,0}, Im{V4,0} and |V4,0|2 High-frequent pupil ‘noise’; intensity pattern image plane



Extension to high-NA case

1) Correct treatment of defocus-parameter:

−− 22

00

1exp ρsu

if

2) Inclusion of ‘radiometric’ effect (aplanatic system with magnification M=0): ( ) 4/122

01

1

ρs−

3) Finite-size point-source (accounted for by Gaussian pupil ‘transmission’): a complex defocus parameter is used!

− 2

~

exp ρf~

fiffC +=

4) Higher than quadratic effects are ‘stored’ in symmetrical polynomials of order 2n leading to the formal expression at high NA:

( )∑=

+max

0

02

2 )(expk

kkkC Rhfg ρρ

( )200 11 su −−=



Extension to vectorial treatment

The Ignatowsky / Richards & Wolf treatment of the aberrated case leads to threediffraction integrals that recall the

−+−

−+−

−++

×

−−−−=

−

−

−

∑

)exp()exp(

)2exp(2

)2exp(2

)2exp(2

)2exp(2

)exp(11

exp);,(

1,01,0

2,

20

2,

20

2,

20

2,

20

0,

,

,

20

20

ϕϕ

ϕϕ

ϕϕ

ϕβγϕ

iVisiVis

iVis

iVis

iVs

iVs

V

imis

ifsifrE

mn

mn

mn

mn

mn

mn

mn

mn

xmn

mx�

integrals and, 210 −−− III

Each aberration term creates its own ).2,1( terms)exp( and ,0, ±±=kikVV mkn

mn ϕ



New integral V to be treated

[ ]

( ) ( )

),( ofsolution

expansion series analytic theusing treatedbecan integral second The

)2()(

1111

exp11),(

:) treatment(vectorial integral-),( New

)2()( exp),(

eatment)(scalar tr integral-),( Old

||

1

0

2202

0

1||22

0||

,

,

||1

0

2

frV

drJR

ss

ifsfrV

frV

drJRiffrV

frV

mn

jmm

n

jjm

jn

mjn

mm

nm

n

mn

ρρρπρ

ρρρ

ρρρπρρ

+

+−×

−−

−−−+=

=

∫

∫



Results of analytic vectorial analysis (no aberrations)

Comparison of scalar (Airy-disc)intensity cross-section and vectorialcases:• θ = 0• θ = π / 2



Introduction of aberrations

[ ][ ]

)sin(

)cos(

RADIAL :onpolarisati of typesSpecial

ncebirefringe (phase) the and aberration wavefront W thefunction,ion transmiss theisA

),(),(2exp),(),(

),(2exp),(),(

as written isfunction pupilcomplex The

00

00

ϑϑϑϑ

ε

ϑρεϑρπϑρϑρϑρπϑρϑρ

+=

+=

+==

AA

AA

iWiAB

WiAB

y

x

yy

xx



Radiometric effect and Zernike expansion

( ) [ ] [ ]

( ) [ ]

)1(2)()(

toaccordingion normalisatwith

exp)(1

),(1

followingusually as calculated are The

. to from runs aperture; numerical theis

exp)(),(2exp1

),(),(

as written iseffect cradiometri theincludingfunction pupilcomplex The

’||||’

1

0

||1

0

2

0 4/1220

,

,

maxmax0

||

,

,4/122

0

+=

−−

+=

+−

=−

=

∫

∫ ∫

∑

ndRR

ddimRs

Bn

mmms

imRWis

BB

nnmn

mn

mn

xxm

n

xmn

mn

mn

xmn

xxR

δρρρρ

ϑρρϑρρ

ϑρπ

β

β

ϑρβϑρπρ

ϑρϑρ

π



Results of analytic vectorial analysis

Tracing the EM energy flow field through the focal region(using the analytic expression for the Poynting vector)

Linearly polarised light Right circularly polarized light





Orbital angular momentum (l=1) OAM (l=1) + right circularly polarised light





Orbital angular momentum (l=1) Orbital angular momentum (l=1) + right circular polarisation + left circular polarisation



How to obtain a parametric expression for the fielddensity in the imaging volume?

{ } : with44

*33

*22

*11

20220 EEEEEEnEnw rrE ++== εε

!in time eunit volumper exposureresist thedetermineshat quantity t theis

)exp()exp(

)2exp(2

)2exp(2

)2exp(2

)2exp(2

)exp(11

exp);,(

exp

1,01,0

2,

20

2,

20

2,

20

2,

20

0,

,

,

20

20

ttt

wdW

iVisiVis

iVis

iVis

iVs

iVs

V

imis

ifsifrE

E

mn

mn

mn

mn

mn

mn

mn

mn

xmn

mx

∆∆∂

∂=

−+−

−+−

−++

×

−−−−=

−

−

−

∑

ϕϕ

ϕϕ

ϕϕ

ϕβγϕ�



Basic term in the expression for the exposure

{ } { }

∑ ∑

∑

∑ ∑

=

+

−=++

−−−

−

−

+=

−+−−

=−−×

=

max max

max0

)(

)(*0

,0,*0

02/

*,

0,0

*00

2/

)(

00

’,’

’*,’,

’*’

’*,’

’’

’

, ’,’,

),(),()1(

),(),();,(

t)coefficien- g(dominatinion approximat- Small

),(),()’(exp2/)’(exp

)exp()’exp()exp()exp(

);,(

ν

ν

νµ

νµµµ

νµ

νµϕµπ

νµ

µν

µν

µϕµπϕ

βαε

βαϕ

ββ

βαϕπ

ϕϕβϕϕαϕ

frVfrVee

frVfrVeeefrG

frVfrVlkmmimmi

ilimViikimVi

frG

lkii

lkii

lkikl

mn

mln

mkn

mn

mn

mln

mn

m

mn mn

mkn

mn

m

kl



Representation of light intensity (fully polarised case)

[ ] { } { }{ }[ ] { } { }{ }[ ] [ ]{ }[ ] [ ]{ }

[ ] { } { }{ }

+−−

+++−

+++−

++−

+−−

+

=

=+

=

=

−+−+

−−

−−

−−

),(Im)Re(2),(Re||||2

),()Im(21),()Im(21

),()Im(21),()Im(212

),(Im)Re(2),(Re||||

),(Im)Re(2),(Re||||

),(

4);,(

purposes)ion normalisatfor 1|||(|

1,1*

1,1222

0

1,1*

1,1*2

0

2,2*

2,2*

40

2,0*

2,0222

0

2,0*

2,0222

0

00

20

20

22

,

,

ββββ

ββββ

ββββ

ββββ

ββββββ

εϕ

ββββ

GabGbas

GabGabs

GabGabs

GabGbas

GabGbas

G

snfrw

ba

b

a

rE

mn

ymn

mn

xmn



Strehl intensity

case! vectorialin the also trueremainsrelation the

expansion, Zernikein the includedeffect cradiometri With the

ion)approximat(scalar

)exp()(1

)exp()(1

),(1

A

expansion type- usingintensity (Srehl) axisOn

200

00

2

0

1

0,

2

0

1

0,

2

0

1

0S

β

βϑρρϑρπ

β

ϑρρϑρβπ

ϑρρϑρπ

β

π

π

π

=→

=

=

==

−

∫ ∫∑

∫ ∫ ∑

∫ ∫

S

mn

mn

mn

mn

mn

mn

I

ddimR

ddimR

ddB



Best focus

- focus

+ focus

Wavefront

Complex pupil function reconstruction remainspossible in the vectorial case?

Intuitive picture of aberration retrieval



Further developments

• Complex amplitude retrieval of the pupil function at large NA shouldremain possible thanks to the parametric description of the intensitydistribution in the focal volume using the Zernike ß-expansion and theextended Nijboer-Zernike theory applied to the vectorial diffraction.

• Much of the numerical work in solving the system of equations for theß-coefficients has to be done only once and the results can betabulated and stored for further use.

• Simultaneous retrieval of aberration and birefringence possible?

extended nijboer-zernike analysis for vectorial diffraction … · 2006-09-18 · extended...

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