1. 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
FLCC Seminar
31 October 2005

2. Purpose & Outline Purpose 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
Outline
: to compare multiple representations and propose a common ‘language’ to describe polarization aberrations for optical lithography

3. What is polarization? Pure polarization states e- Ey,out ei
Vector representation in x y plane
Ex,out ei
Ey,out ei
y,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

4. Why is polarization important in optical lithography?
Z component of E-field introduced at High NA from radial pupil component decreases image contrast  z 
Low NA
High NA x y 
mask TM TE
= ETM NA
Ez = ETM sin() 
Z-component negligible 
wafer
Increasing NA

5. 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 orientation TE
Purpose: To increase exposure latitude (better contrast) by minimizing TM polarization

6. 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)
NA = .95 = nl sin(l)
a = 71.8
l ~ 39.3
Depth of focus

7. 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(l) > 1 l
liquid
resist
Last lens element
air
Dry
Wet
Minimum feature

8. Immersion increases DOF and/or decreases minimum feature
Polarization increases exposure latitude (better contrast)
Wet
Dry
NA=0.95, Dipole 0.9/0.7, 60nm equal L/S (simulation)
Polarization is needed to take full advantage of immersion benefits
Dry, unpolarized
Dry, polarized
Wet, unpolarized
Wet, polarized

9. Thus, polarization state is important
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

10. Polarization Aberrations

11. Traditional scalar aberrations
Scalar diffraction theory: Each pupil location characterized by a single number (OPD)
defocus
astigmatism
coma
Optical Path Difference
Ein ei in
Eout ei
out
a: illumination frequency
Typically defined in Zernike’s

12. Polarization aberrations
Subtle polarization-dependent wavefront distortions cause intricate (and often non-intuitive) coupling between complex electric field components
Ex,in ei
x,in
Ex,out ei
Ey,in ei
y,in
Ey,out ei
y,out
x,out
Each pupil location no longer characterized by a single number

13. Changes in polarization state
Diattenuation:
Retardance:
Degrees of Freedom:
Magnitude
Eigenpolarization orientation
Eigenpolarization ellipticity E x y E'
attenuates eigenpolarizations differently (partial polarizer)
shifts the phase of eigenpolarizations differently (wave plate)

14. 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

15. Mueller-pupil

16. Mueller Matrix - Pupil Consider time averaged intensities Sin SoutH V
Sin
Sout
Stokes vector completely characterizes state of polarization
PH = flux of light in H polarization
Mueller matrix defines coupling between Sin and Sout

17. Mueller Matrix - Pupil
Recast polarization aberration into Mueller pupil
Mueller Pupil
16 degrees of freedom
per pupil location
m02,m20: Linear diattenuation
m01,m10: H-V Linear diattenuation
m03,m30: Circular diattenuation
m13,m31: Linear retardance
m12,m21: H-V Linear retardance
m23,m32: Circular retardance

18. Polarization-dependent depolarization
Mueller Matrix - Pupil
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 45
135
Linear S S’
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

19. 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

20. Jones-pupil

21. Jones Matrix - Pupil Consider instantaneous fields: Ex,in ei
Ey,in ei
y,in
Ex,out ei
Ey,out ei
y,out
Elements are complex, thus 8 degrees of freedom
Jones vector
Jones matrix
Mask diffracted fields
High-NA & resist effects
Lens
effect
Jones Pupil
x,out
a: illumination frequency
Vector imaging equation:

22. Jones Matrix - Pupil Mask coordinate system (x,y)
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) x y
Jtete(mag)
Jtetm(mag)
Jtmte(mag)
Jtmtm(mag)
Jtete(phase)
Jtetm(phase)
Jtmte(phase)
Jtmtm(phase)
Pupil coordinate system (te,tm) TM TE

23. Zernike coefficients (An,m)
Jones Matrix - Pupil
Jxx (real)
Jxx (imag)
Jxy (real)
Jxy (imag)
Jyx (real)
Jyx (imag)
Jyy (real)
Jyy (imag)
Decomposition into Zernike polynomials
Annular Zernike polynomials (or Zernikes weighted by radial function) might be more useful
Lowest 16 zernikes => 128 degrees of freedom for pupil
Zernike coefficients (An,m)
real
imaginary
Similar to Totzeck, SPIE 05

24. Pauli-pupil

25. Pauli-spin Matrix - Pupil
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)

26. 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

27. traditional scalar phase Diattenuation effects
Usefulness of Pauli-Pupil to Lithography
Pupil can be specified by only: a2
(complex) a1
traditional scalar phase
Diattenuation effects
Retardance effects
|a0| calculated to ensure physically realizable pupil assuming:
no scalar attenuation
eigenpolarizations are linear

28. 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

29. Proposed simulation flow
(to determine polarization aberration specifications and tolerances)
Input: a1, a2, scalar aberration
Convert to Jones Pupil
Simulate
Calculate a0

30. Simulation example
Monte 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)
Center intensity change (%CF)
iteration
Signal variation

31. A word of caution… Jinstrument
This analysis is based on the “Instrumental Jones Matrix”
Ein
Eout
Jinstrument
Apodization
Aberration
Magnitude
Orientation
Ellipticity of eignpolarization
“Instrumental parameters”

32. Constraints of Causality & Reciprocity
Ein
Eout JA JC JD JF JB JE
Reciprocity:
time reversed symmetry
(“parameters of element A”)
Causality:
polarization state can not depend on future states (order dependent)
(except in presence of magnetic fields)

33. Differential Jones Matrix
N = differential Jones
N= generalized propagation vector (homogeneous media)
Wave Equation:
General solution
Also:
= dielectric tensor
EM Theory:
symmetric
Anti-symmetric

34. Differential Jones Matrix
Assumed real(ai) => dichroic property & imag(ai) => birefringent property
Barakat (1996):
Contradiction resolved for small values of polarization effects
Jones' assumption was wrong

35. Other representations

36. E-field test representationX Y 45
rcp TE TM
Output electric field, given input polarization state
Color degree of circular polarization

37. Intensity test representationX Y 45
rcp TE TM
Output intensity, given input polarization state

38. Covariance & Coherency Matrix
Covariance Matrix (C)
Coherency Matrix (T)
Trace describes average power transmitted
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)

39. 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

40. 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