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Polarization Aberrations:
A Comparison of Various
Representations
Greg McIntyre,a,b Jongwook Kyeb, Harry
Levinsonb and Andrew Neureuthera
a EECS
Department, University of California- Berkeley, Berkeley, CA 94720
b Advanced Micro Devices, One AMD Place, Sunnyvale, CA 94088-3453
FLCC Seminar
31 October 2005
FLCC
Purpose & Outline
Purpose: to compare multiple representations and propose a
common ‘language’ to describe polarization aberrations for optical
lithography
Outline
• 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
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What is polarization?
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.
• Pure polarization states
Oscillating
electron
Polarization
state
Propagating EM wave
eVector
representation
in x y plane
Ey,out eiy,out
Linear Circular Elliptical
Ex,out ei x,out
• Partially polarized light =
superposition of multiple pure states
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Why is polarization important in optical
lithography?
z
x

High NA

Low NA

Z component of E-field introduced at High NA from radial
pupil component decreases image contrast
mask
TM
y
TE

Z-component
negligible
Ez = ETM sin()
= ETM NA

wafer
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Increasing NA
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Scanner vendors are beginning to engineer
polarization states in illuminator?
Purpose: To increase exposure latitude (better contrast) by
minimizing TM polarization
Choice of illumination setting depends on features to be printed.
ASML, Bernhard (Immersion symposium 2005)
TE
Polarization orientation
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Polarization and immersion work together
for improved imaging
Immersion lithography can increase depth of focus
Dry
Wet
l
a
liquid
resist
resist
NA = .95 = sin(a)
a = 71.8
Depth of focus
NA = .95 = nl sin(l)
l ~ 39.3


2n(1 - cos( ))
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Polarization and immersion work together
for improved imaging
Immersion lithography can also enable hyper-NA tools
(thus smaller features)
Dry
Wet
Last lens
element
Last lens
element
l
liquid
air
resist
resist
Total internal reflection
prevents imaging
NA = nl sin(l) > 1
Minimum feature  k 1

NA
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Polarization is needed to take full advantage
of immersion benefits
• Immersion increases DOF and/or decreases minimum feature
• Polarization increases exposure latitude (better contrast)
Dry, unpolarized
Dry, polarized
Wet, unpolarized
Wet, polarized
Wet
Dry
NA=0.95, Dipole 0.9/0.7, 60nm equal L/S (simulation)
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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.
Mask polarization
effects
Illuminator
polarization design
Polarization
aberrations
of projection
optics
Source
polarization
Wafer / Resist
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Polarization Aberrations
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Traditional scalar aberrations
Scalar diffraction theory: Each pupil location characterized by a
single number (OPD)
EWafer ( x' , y' , a ) 
1

2 1

ik  cos   x '   sin  y '  ik (  , )
E
(

,

,
a
)
e
e
dd
 Diff
0 0
Optical Path Difference
Typically defined
in Zernike’s
 n
Ein eiin
 (  , ) 
Eout eiout
 A
n , m Z n, m (  , )
n 1 m 0
defocus
astigmatism
coma
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
Ey,in ei y,in
Ey,out eiy,out
Ex,out ei x,out
Ex,in eix,in
Each pupil location no longer characterized by a single number
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Changes in polarization state
Diattenuation: attenuates
Retardance: shifts the phase of
eigenpolarizations differently
(partial polarizer)
eigenpolarizations differently
(wave plate)
Ey
Ey
E'y
Degrees of Freedom:
Degrees of Freedom:
• Magnitude
• Eigenpolarization
orientation
E'x
Ex
E'x
Ex
E'y
• Magnitude
• Eigenpolarization
orientation
•Eigenpolarization
ellipticity
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•Eigenpolarization
ellipticity
FLCC
Sample pupil (physical properties)
Total representation has 8 degrees of freedom per pupil location
Apodization
diattenuation
Scalar
aberration
retardance
However, this format is
• inconvenient for understanding the impact on imaging
• inconvenient as an input format for simulation
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Mueller-pupil
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Mueller Matrix - Pupil
Consider time averaged intensities
Stokes vector completely characterizes state of polarization
V
V
H
H
Sin
Sout
 s0   PH  PV 

  
P

P
s
H
V
1

S  
 s2   P45  P135 

  
 s3   PR  PL 
PH = flux of light
in H polarization
Mueller matrix defines coupling between Sin and Sout
Sout  MS in
m00
m
M   10
m20

m30
m01 m02
m11 m12
m21 m 22
m31 m32
m03 
m13 
m23 

m33 
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Mueller Matrix - Pupil
Recast polarization aberration into Mueller pupil
Mueller Pupil
m00
m
M   10
m20

m30
m01 m02
m11 m12
m21 m 22
m31 m32
m03 
m13 
m23 

m33 
m01,m10: H-V Linear diattenuation
m02,m20: 45-135 Linear diattenuation
m03,m30: Circular diattenuation
m12,m21: H-V Linear retardance
m13,m31: 45-135 Linear retardance
m23,m32: Circular retardance
16 degrees of freedom
per pupil location
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Right Circular
Mueller Matrix - Pupil
S
• Stokes vector represented as a unit vector
on the Poincare Sphere
135
45
0
• Meuller Matrix maps any input Stokes
vector (Sin) into output Stokes vector (Sout)
Sout  MS in
Linear
S’
Left Circular
• The extra 8 degrees of freedom specify depolarization, how
polarized light is coupled into unpolarized light
Represented by warping of the Poincare’s sphere
Polarization-dependent
depolarization
Uniform depolarization
Chipman, Optics
express, v.12, n.20,
p.4941, Oct 2004
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Mueller Matrix - Pupil
Advantages:
• accounts for all polarization effects
• depolarization
• non-reciprocity
• intensity formalism
• measurement with slow detectors
Disadvantages:
• 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 - Pupil
Consider instantaneous fields:
Ey,in ei y,in
Ex,in eix,in
Ey,out eiy,out
Ex,out eix,out
 E x ,out e i x ,out   J xx

i y ,out   
 E y ,out e
  J yx
Jones vector
J xy   E x ,ine i x ,in 


J yy   E y ,ine i y ,in 
Jones matrix
Elements are complex, thus 8 degrees of freedom
Vector imaging equation:
 Fxx
E x 
2 1
1

E 
(
x
'
,
y
'
,
a
,
Pol
)

F
 y
 0 0  yx
 Fzx
 E z 

Wafer
a: illumination
frequency
Fxy 
  J xx
F yy  
J yx
Fzy  
High-NA
& resist
effects
J xy   E x 
J yy   E y 
Jones
Pupil
e ik  cos  x '  sin  y ' dd
Diff
Mask
diffracted
fields
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Lens
effect
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Jones Matrix - Pupil
Jxx(mag)
Jxy(mag)
Jxx(phase) Jxy(phase)
Jyx(mag)
Jyy(mag)
Jyx(phase) Jyy(phase)
Mask coordinate
system (x,y)
y
x
i.e. Jxy = coupling between input x and output y polarization fields
Jtete(mag) Jtetm(mag) Jtete(phase) Jtetm(phase)
Pupil coordinate
system (te,tm)
TE TM
Jtmte(mag) Jtmtm(mag) Jtmte(phase) Jtmtm(phase)
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Jones Matrix - Pupil
Decomposition into Zernike
polynomials
• Lowest 16 zernikes => 128
degrees of freedom for pupil

Zernike
coefficients (An,m)
Jxx (real) Jxx (imag)
real
imaginary
Jxy (real) Jxy (imag)
n
(  , )    An ,m Z n ,m (  , )
n 1 m  0
•Annular Zernike polynomials
(or Zernikes weighted by
radial function) might be more
useful
Jyx (real) Jyx (imag)
Jyy (real) Jyy (imag)
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Similar to Totzeck, SPIE 05
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Pauli-pupil
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Pauli-spin Matrix - Pupil
Decompose Jones Matrix into Pauli-spin matrix basis
J ( H ,  , )  a0 0  a1 1  a2 2  a3 3
1 0
0  

0 1 
1 0 
1  

0

1


0 1 
2  

1 0
0  i 
3  

i 0 
a0 
a1 
a2 
a3 
J xx  J yy
mag(a0)
phase(a0)
real(a1/a0)
imag(a1/a0)
real(a2/a0)
imag(a2/a0)
real(a3/a0)
imag(a3/a0)
2
J xx  J yy
2
J xy  J yx
2
J xy  J yx
2i
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Meaning of the Pauli-Pupil
Scalar transmission
(Apodization) &
normalization constant for
diattenuation & retardance
mag(a0)
phase(a0)
Scalar phase
(Aberration)
imag(a1/a0)
real(a1/a0)
Diattenuation along
x & y axis
Retardance along
x & y axis
real(a2/a0)
imag(a2/a0)
Diattenuation along
45  & 135 axis
Retardance along
45  & 135 axis
imag(a3/a0)
real(a3/a0)
Circular
Diattenuation
Circular
Retardance
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Usefulness of Pauli-Pupil to Lithography
Pupil can be specified by only:
traditional
scalar phase
a1
Diattenuation
effects
(complex)
a2
Retardance
effects
(complex)
|a0| calculated to ensure physically realizable pupil assuming:
• no scalar attenuation
• eigenpolarizations are linear
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The advantage of Pauli-Pupils
Jones
Pauli
• 8 coupled pupil functions
• 4 independent pupil functions
(scalar effects considered separately)
(easy to create unrealizable pupil)
• 64 Zernike coefficients
• physically intuitive
• easily converted to Jones for
imaging equations
• 128 Zernike coefficients
• not very intuitive
• fits imaging equations
Jxx(mag)
Jxy(mag) Jxx(phase)Jxy(phase)
Jyx(mag)
Jyy(mag) Jyx(phase)Jyy(phase)
a1 real
a1 imag
a2 real
a2 imag
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Proposed simulation flow
(to determine polarization aberration specifications and tolerances)
Input: a1, a2, scalar aberration
Calculate a0
Simulate
Convert to Jones Pupil
J ( H ,  , )  a0 0  a1 1  a2 2  a3 3
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Simulation example
Monte Carlo simulation done with Panoramic software
and Matlab API to determine variation in image due to
polarization aberrations
Example: polarization monitor (McIntyre, SPIE 05)
Polarization
monitor
Resist
image
Intensity at center is
polarization-dependent signal
Center intensity change (%CF)
Simulate many randomly generated Pauli-pupils to determine how
polarization aberrations affect signal
0.05
Signal variation
0.04
0.03
0.02
0.01
0
-0.01 0
50
100
-0.02
-0.03
-0.04
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A word of caution…
This analysis is based on the “Instrumental Jones Matrix”
Ein
Jinstrument
Eout
J  a0 0  a1 1  a2 2  a3 3
 a0  [1 0 
J scalar
real (a3 )
imag (a3 )
real (a1 )
real (a2 )
imag (a1 )
imag (a2 )
1 
2 
 3 ]  [1 0  i
1  i
2  i
3]
a0
a0
a0
a0
a0
a0
•Apodization
•Aberration
J diattenuation
• Magnitude
• Orientation
• Ellipticity of
eignpolarization
J retardance
• Magnitude
• Orientation
• Ellipticity of
eignpolarization
“Instrumental parameters”
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Constraints of Causality & Reciprocity
JA
JC
JB
Ein
Eout
JE
JD
JF
polarization state can not depend on future
states (order dependent)
Causality:
J  J F  J E  J D  JC  J B  J A
J A  a0, A 0  a1, A 1  a2, A 2  a3, A 3 (“parameters of
element A”)
Reciprocity:
time reversed symmetry
(except in presence of magnetic fields)
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Differential Jones Matrix
J 0 , z J z ,z ' J z ', z ''
Wave Equation:
 E
 KE  0
2
z
2
z z'
Ez'  J z ,z' Ez
J  J z 1
N  lim z '
Jz
z z'
z' z
J
NJ 
z
N = differential Jones
J  e Nz General solution
Also: NE 
E
z
K   2
K  N2
N= generalized propagation vector
(homogeneous media)
EM Theory:
D  E  QE   (G)  E
symmetric

= dielectric tensor
Anti-symmetric
 xx

  yx
 zx

 xy  xz 

 yy  yz 
 zy  zz 
 N    K  Q, G
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Differential Jones Matrix
Jones (1947): N  a0 0  a1 1  a2 2  a3 3
Assumed real(ai) => dichroic property & imag(ai) => birefringent property
Barakat (1996): Jones' assumption was wrong
J  e Nz
N  e0 0  e1 1  e2 2  e3 3
real (ei )  dichroic
imag (ei )  diattenuation
Contradiction resolved for small values of polarization effects
e x  1  x  x 2 ...
 e i  a i , e0  a 0  1
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Other representations
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E-field test representation
Output electric field, given input polarization state
X
Y
45
rcp
TE
TM
Color
degree of circular polarization
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Intensity test representation
Output intensity, given input polarization state
X
Y
45
rcp
TE
TM
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Covariance & Coherency Matrix
Covariance Matrix (C)
C  kC  kC

 J xx 


kC    2 J xy 
 J

 yy 
 J xx  J yy 

1 
kt 
 J xx  J yy 
2

2
J
xy


Coherency Matrix (T)
T  kt  kt

Kt1 (mag)
Kt2 (mag)
Kt3 (mag)
Kt1 (mag)
Kt2 (mag)
Kt3 (mag)
Kt1 (phase)
Kt2 (phase)
Kt3 (phase)
Kt1 (phase)
Kt2 (phase)
Kt3 (phase)
(similar to Pauli-pupil)
(similar to Jones-pupil)
Power
• Assumes reciprocity (Jxy = Jyx)
• Convenient with partially polarized light
• Trace describes average power transmitted
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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
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