characterizing illumination angular uniformity with phase...
TRANSCRIPT
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