FLCC

Polarization Aberrations:A Comparison of Various

RepresentationsGreg McIntyre,a,b Jongwook Kyeb, Harry

Levinsonb and Andrew Neureuthera

a EECS Department, University of California- Berkeley, Berkeley, CA 94720b Advanced Micro Devices, One AMD Place, Sunnyvale, CA 94088-3453

FLCC Seminar31 October 2005

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Purpose

• What is polarization, why is it important• Polarization aberrations: Various representations

• Physical properties• Mueller matrix – pupil• Jones matrix – pupil• Pauli-spin matrix – pupil• Others (Ein vs. Eout, coherence- & covariance - pupil)

• Preferred representation • Proposed simulation flow & example• Causality, reciprocity, differential Jones matrices

Outline

: to compare multiple representations and propose a common ‘language’ to describe polarization aberrations for optical lithography

Purpose & Outline

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What is polarization?

Vector representation in x y plane Ex,out eiφ

Ey,out eiφy,out

x,out

• Pure polarization states

e-

• Partially polarized light = superposition of multiple pure states

Polarization is an expression of the orientation of the lines of electric flux in an electromagnetic field. It can be constant or it can change either gradually or randomly.

Linear Circular Elliptical

Oscillating electron Propagating EM wave Polarization

state

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Why is polarization important in optical lithography?

xz<

y

<<

φ

Low NA

φ

High NA

Z component of E-field introduced at High NA from radial pupil component decreases image contrast

Z-component negligible

TMTE

mask

wafer

Increasing NA

= ETM NAEz = ETM sin(φ)

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Scanner vendors are beginning to engineer polarization states in illuminator?

Choice of illumination setting depends on features to be printed.

ASML, Bernhard (Immersion symposium 2005)

Polarization orientationTE

Purpose: To increase exposure latitude (better contrast) by minimizing TM polarization

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Polarization and immersion work together for improved imaging

Immersion lithography can increase depth of focus

Dry

NA = .95 = nl sin(θl)θl ~ 39.3°

θl

liquidresist

Wet

θa

resist

NA = .95 = sin(θa)θa = 71.8°

))cos(-2n(1

θλ

=Depth of focus

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Polarization and immersion work together for improved imaging

Immersion lithography can also enable hyper-NA tools (thus smaller features)

Total internal reflection prevents imaging

NA = nl sin(θl) > 1

θl

liquidresistresist

Last lens element

Last lens element

air

WetDry

Minimum feature NAλ

1k =

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• Immersion increases DOF and/or decreases minimum feature • Polarization increases exposure latitude (better contrast)

WetDry

NA=0.95, Dipole 0.9/0.7, 60nm equal L/S (simulation)

Polarization is needed to take full advantage of immersion benefits

Dry, unpolarizedDry, polarizedWet, unpolarizedWet, polarized

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Thus, polarization state is important. But there are many things that can impact polarization state as light propagates through optical system.

Illuminator polarization design

Source polarization

Mask polarization effects

Polarization aberrations of projection optics

Wafer / Resist

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Polarization Aberrations

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Traditional scalar aberrationsScalar diffraction theory: Each pupil location characterized by Scalar diffraction theory: Each pupil location characterized by a a

single number (OPD)single number (OPD)

Typically defined in Typically defined in ZernikeZernike’’ss

( ) ( )[ ] θρρθρπ

θρθρθρ ddeeaEayxE ikyxikDiffWafer

),('sin'cos),,(),','( Φ+∏

∫ ∫=2

0

1

0

1

),(),( ,1 0

, θρθρ mnn

n

mmn ZA∑∑

∞

= =

=Φ

defocus astigmatism coma

Optical Path Difference

Ein eiφin Eout eiφout

a: illumination frequency

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Polarization aberrations

Subtle polarization-dependent wavefront distortions cause intricate (and often non-intuitive) coupling between complex electric field components

Ex,in eiφx,in Ex,out eiφ

Ey,in eiφy,in Ey,out eiφy,out

x,out

Each pupil location no longer characterized by a single number

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Changes in polarization stateDiattenuation: Retardance:

Degrees of Freedom:• Magnitude• Eigenpolarization orientation

•Eigenpolarization ellipticity

•Eigenpolarization ellipticity

Degrees of Freedom:• Magnitude• Eigenpolarization orientation

Ex

Ey E'y

E'x

attenuates eigenpolarizations differently (partial polarizer)

Ex

Ey E'y

E'x

shifts the phase of eigenpolarizations differently (wave plate)

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Sample pupil (physical properties)

However, this format is • inconvenient for understanding the impact on imaging • inconvenient as an input format for simulation

Apodization

Scalar aberration

Total representation has 8 degrees of freedom per pupil location

diattenuation

retardance

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Mueller-pupil

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Mueller Matrix - PupilConsider time averaged intensities

HV

Sin

inout MSS =

HV

Sout

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

33323130

23222120

13121110

03020100

mmmmmmmmmmmmmmmm

M

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−

−

−

+

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

LR

VH

VH

PPPPPPPP

ssss

13545

3

2

1

0

S

Stokes vector completely characterizes state of polarization

PH = flux of light in H polarization

Mueller matrix defines coupling between Sin and Sout

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Mueller Matrix - Pupil

Recast polarization aberration into Mueller pupil

Mueller Pupil

16 degrees of freedom per pupil location

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

33323130

23222120

13121110

03020100

mmmmmmmmmmmmmmmm

M

m02,m20: 45-135 Linear diattenuationm01,m10: H-V Linear diattenuation

m03,m30: Circular diattenuation

m13,m31: 45-135 Linear retardancem12,m21: H-V Linear retardance

m23,m32: Circular retardance

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• Stokes vector represented as a unit vector on the Poincare Sphere

• Meuller Matrix maps any input Stokes vector (Sin) into output Stokes vector (Sout)

Right Circular

Left Circular

045135Linear

S

S’

inout MSS =

• The extra 8 degrees of freedom specify depolarization, how polarized light is coupled into unpolarized light

Polarization-dependent depolarization

Represented by warping of the Poincare’s sphere

Chipman, Optics express, v.12, n.20, p.4941, Oct 2004

Uniform depolarization

Mueller Matrix - Pupil

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Mueller Matrix - Pupil

Advantages:

Disadvantages:

• accounts for all polarization effects • depolarization• non-reciprocity

• intensity formalism • measurement with slow detectors

• difficult to interpret • loss of phase information• not easily compatible with imaging equations• hard to maintain physical realizability

Generally inconvenient for partially coherent imaging

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Jones-pupil

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Jones Matrix - PupilConsider instantaneous fields:

Ex,in eiφx,in

Ey,in eiφy,in

Ex,out eiφ

Ey,out eiφy,out

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡iny

inx

outy

outx

iiny

iinx

yyyx

xyxxi

outy

ioutx

eEeE

JJJJ

eEeE

,

,

,

,

,

,

,

,ϕ

ϕ

ϕ

ϕ

Elements are complex, thus 8 degrees of freedomJones vector Jones matrix

( ) ( )[ ] θρρπ

θρθρ ddeEE

JJJJ

FFFFFF

PolayxEEE

yxik

Diffy

x

yyyx

xyxx

zyzx

yyyx

xyxx

Waferz

y

x

∫ ∫∏

+⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ 2

0

1

0

1 'sin'cos),,','(

Mask diffracted fields

High-NA & resist effects

Lenseffect

Jones Pupil

x,out

a: illumination frequency

Vector imaging equation:

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i.e. Jxy = coupling between input x and output y polarization fields

Jxx(mag) Jxy(mag)

Jyx(mag) Jyy(mag)

Jxx(phase) Jxy(phase)

Jyx(phase) Jyy(phase)

Mask coordinate system (x,y)

xy

Jtete(mag) Jtetm(mag)

Jtmte(mag) Jtmtm(mag)

Jtete(phase) Jtetm(phase)

Jtmte(phase)Jtmtm(phase)

Pupil coordinate system (te,tm)

TMTE

Jones Matrix - Pupil

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Jxx (real) Jxx (imag)

Jxy (real) Jxy (imag)

Jyx (real) Jyx (imag)

Jyy (real) Jyy (imag)

Decomposition into Zernikepolynomials

•Annular Zernike polynomials (or Zernikes weighted by radial function) might be more useful

• Lowest 16 zernikes => 128 degrees of freedom for pupil

Zernikecoefficients (An,m)

realimaginary

),(),( ,, θρθρ mnn

n

mmn ZA∑∑

∞

= =

=Φ1 0

Similar to Totzeck, SPIE 05

Jones Matrix - Pupil

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Pauli-pupil

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Pauli-spin Matrix - Pupil

33221100 σσσσθρ aaaaHJ +++=),,(

⎥⎦

⎤⎢⎣

⎡=

1001

0σ

⎥⎦

⎤⎢⎣

⎡=

0110

2σ

⎥⎦

⎤⎢⎣

⎡−

=10

011σ

⎥⎦

⎤⎢⎣

⎡ −=

00

3 ii

σ

20yyxx JJ

a+

=

21yyxx JJ

a−

=

22yxxy JJ

a+

=

iJJ

a yxxy

23

−=

Decompose Jones Matrix into Pauli-spin matrix basis

mag(a0) phase(a0)

real(a1/a0)

real(a2/a0)

real(a3/a0)

imag(a1/a0)

imag(a2/a0)

imag(a3/a0)

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Meaning of the Pauli-Pupil

mag(a0) phase(a0)

real(a1/a0)

real(a2/a0)

real(a3/a0)

imag(a1/a0)

imag(a2/a0)

imag(a3/a0)

Scalar transmission (Apodization) & normalization constant for diattenuation & retardance

Diattenuation along x & y axis

Diattenuation along 45 ° & 135° axis

Circular Diattenuation

Scalar phase (Aberration)

Retardance along x & y axis

Retardance along 45 ° & 135° axis

Circular Retardance

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Usefulness of Pauli-Pupil to LithographyPupil can be specified by only:

a2(complex)

a1(complex)

traditional scalar phase

Diattenuation effects

Retardance effects

|a0| calculated to ensure physically realizable pupil assuming:• no scalar attenuation• eigenpolarizations are linear

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The advantage of Pauli-Pupils

Jxx(mag) Jxy(mag)

Jyx(mag) Jyy(mag)

Jxx(phase)Jxy(phase)

Jyx(phase)Jyy(phase)

Jones Pauli• 8 coupled pupil functions

(easy to create unrealizable pupil)• 128 Zernike coefficients • not very intuitive• fits imaging equations

• 4 independent pupil functions(scalar effects considered separately)

• 64 Zernike coefficients• physically intuitive• easily converted to Jones for

a1 real a1 imag

a2 real a2 imag

imaging equations

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Proposed simulation flow(to determine polarization aberration specifications and tolerances)

Input: a1, a2, scalar aberration

Convert to Jones Pupil33221100 σσσσθρ aaaaHJ +++=),,(

Simulate

Calculate a0

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Simulation exampleMonte Carlo simulation done with Panoramic software and Matlab API to determine variation in image due to polarization aberrations

Polarization monitor

Resist image

Intensity at center is polarization-dependent signal

Simulate many randomly generated Pauli-pupils to determine how polarization aberrations affect signal

Example: polarization monitor (McIntyre, SPIE 05)

-0.04-0.03-0.02-0.01

00.010.020.030.040.05

0 50 100 150

Cen

ter

inte

nsity

cha

nge

(%C

F)

iteration

Signal variation

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A word of caution…This analysis is based on the “Instrumental Jones Matrix”

Ein EoutJinstrument

scalarJ

])()()([])()()([ 30

32

0

21

0

103

0

32

0

21

0

100 11 σσσσσσσσ

aaimagi

aaimagi

aaimagi

aareal

aareal

aareala +++⋅+++⋅=

33221100 σσσσ aaaaJ +++=

iondiattenuatJ retardanceJ•Apodization•Aberration

• Magnitude• Orientation• Ellipticity of eignpolarization

“Instrumental parameters”

• Magnitude• Orientation• Ellipticity of eignpolarization

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Constraints of Causality & Reciprocity

Ein Eout

JA JC

JD JF

JB

JE

Reciprocity: time reversed symmetry

33221100 σσσσ AAAAA aaaaJ ,,,, +++= (“parameters of element A”)

Causality: polarization state can not depend on future states (order dependent)

ABCDEF JJJJJJJ ⋅⋅⋅⋅⋅=

(except in presence of magnetic fields)

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Differential Jones Matrix

zzzz EJE ',' =

1−

= −−

≡ zzz

zzJ

zzJJ

N'

lim '

'

zJNJ∂∂

=

zENE∂∂

=

', zzJ

z 'z

N = differential Jones

02

2

=−∂∂ KE

zE

2µεϖ=K

N= generalized propagation vector (homogeneous media)

Wave Equation:

2NK ∝

NzeJ = General solution

zJ ,0 '',' zzJ

Also:

EGQEED ×∇+== )(αε

ε = dielectric tensor⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

zzzyzx

yzyyyx

xzxyxx

εεεεεεεεε

GQKN ,⇔⇔⇔∴ ε

EM Theory:

symmetric Anti-symmetric

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Differential Jones Matrix

33221100 σσσσ aaaaN +++=Jones (1947):Assumed real(ai) => dichroic property & imag(ai) => birefringent property

Barakat (1996):

33221100 σσσσ eeeeN +++=NzeJ =dichroicereal i ≠)(

iondiattenuateimag i ≠)(

Contradiction resolved for small values of polarization effects

Jones' assumption was wrong

...21 xxe x ++= ii ae =∴ 100 −= ae,

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Other representations

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E-field test representation

X Y 45

rcp TE TM

Output electric field, given input polarization state

Color degree of circular polarization

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Intensity test representation

X Y 45

rcp TE TM

Output intensity, given input polarization state

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Covariance & Coherency MatrixCovariance Matrix (C)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

√=

yy

xy

xx

C

JJ

Jk 2+⋅= CC kkC

Coherency Matrix (T)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−+

√=

xy

yyxx

yyxx

t

JJJJJ

k2

21

+⋅= tt kkT

• Trace describes average power transmitted

Kt1 (mag) Kt2 (mag) Kt3 (mag)

Kt1 (phase) Kt2 (phase) Kt3 (phase)

Kt1 (mag) Kt2 (mag) Kt3 (mag)

Kt1 (phase) Kt2 (phase) Kt3 (phase)

• Assumes reciprocity (Jxy = Jyx)Power

• Convenient with partially polarized light

(similar to Jones-pupil) (similar to Pauli-pupil)

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Additional comments on polarization in lithography

• Different mathematics convenient with different aspects of imaging

• Source, mask Stokes vector• Lenses Jones vector

• Each vendor uses different terminology

• Initially, source and mask polarization effects will be most likely source of error

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Conclusion• Polarization is becoming increasingly important in lithography

• Compared various representations of polarization aberrations & proposed Pauli-pupil as ‘language’ to describe them

• Proposed simulation flow and input format

• Multiple representations of same pupil help to understand complex and non-intuitive effects of polarization aberrations