Angle resolved Mueller Polarimetry,
Applications to periodic structures
PhD Defense
Clément Fallet
Under the supervision of Antonello de Martino
Outline of the presentation
Motivations and introduction to polarization
Design and optimization of a Mueller microscope
Fourier space measurements : application to
semiconductor metrology
Real space measurements : example of characterization
of beetles
Conclusions and perspectives
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Motivations of the study
 Various applications of polarization of
light over the past decades.
 A lot of studies, but mainly driven by
classical ellipsometry  spectral
resolution (discrete angle, averaged over
the illuminated region)
 Spatial dependency of polarimetric
properties is only qualitatively assessed
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Motivations of the study
 What we propose, discrete wavelength :
 Angular resolution (averaged over the field)
 Spatial resolution (averaged over the angles)
 Possibility to use the same system for
both measurements.
 Evolution of a classical bright-field
microscope  ease of use
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LET’S TALK ABOUT POLARIZATION
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Introduction to polarization
𝑆𝑖𝑛
𝐼
𝐼𝐻 − 𝐼𝑉
=
𝐼45° − 𝐼−45°
𝐼𝐶𝐿 − 𝐼𝐶𝐷
𝑆𝑜𝑢𝑡
𝑆𝑜𝑢𝑡 = 𝑀. 𝑆𝑖𝑛
M=
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 M 11

M
 21
 M 31

 M 41
M 12
M 13
M 22
M 23
M 32
M 33
M 42
M 43
M 14 

M 24

M 34 

M 44 
6
A word about polarimeters
Mueller Polarimeter)
Polarimeter
(Stokes
B = A.M.W
 M = A-1.B.W-1
At = [S’1, S’2, S’3, S’4]
PSA Basis Stokes vectors
W = [S1, S2, S3’ S4]
PSG Basis Stokes vectors
A and W must be as close as possible to unitary Calibration : eigenvalue method
No instrument modelling
Their condition numbers must be optimized
(E.Compain 1999, S. Tyo 2000, M. Smith, 2002)
(E.Compain, Appl. Opt 38, 3490 1999)
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DESIGN & OPTIMIZATION OF A
MUELLER MICROSCOPE
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Specifications of the set-up
 Complete Mueller polarimeter at discrete λ
▪
Complete measurement of the Mueller Matrix
(4 by 4 matrix). First setup by S. Ben Hatit.
 2 imaging modes
▪ Fourier Space
 we’re not imaging the sample itself but the back
focal plane of a high-aperture microscope
objective
▪ Real space
 Design based on classical microscopy
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Epi-Illumination scheme
CCD
Aperture image : angularly resolved
Lim3
Real image : spatially resolved
Interferential
filter
5
1
2
3
4
5
–
–
–
–
–
Aperture diaphragm
Field diaphragm
PSG : Polarization State Generator
PSA : Polarization State Analyser
Aperture Mask
Lim2
retractable lens
Lim1
4
1
2
3
Source
Beamsplitter
LColl
L1
L2
Back focal plane
Strain-free
Microscope objective
Sample
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Illumination arm
Collection
lens
L2
Rays emerging
from the source
L1
Aperture
diaphragm
Field
diaphragm
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Back focal
plane
11
Detection arm
400nm pitch grating
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Choice of the objectives
 Strain-free Nikon objectives
▪
▪
Specified for quantitative polarization
No polarimetric signature in real space
 But small dichroism and birefringence
when used in Fourier space
 calibration of the objective with wellcharacterized reference samples (c-Si, SiO2 on
c-Si) (method explained in the manuscript)
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Aperture Vs Field
objective
Full Field
Maximum Aperture
5x
360µm
0-8°
20x
90µm
0-26°
50x
36µm
0-53°
100x
18µm
0-64°
with our current pinhole, the field (spot size) can be discreased
down to 10µm
 Use of a pinhole with smaller diameter to achieve 5µm
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Description of the measurements
0.2
0.2
𝑟-0.2
∝ sin 𝜃-0.2
𝑟𝑝
= tan(Ψ)
𝑟𝑠
dichroism
𝑒 𝑖Δ
c-Si wafer, 633nm
retardance
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From (x,y) to (s,p)
𝑀 𝑥, 𝑦 = 𝑅 𝜑 . 𝑀 𝑠, 𝑝 . 𝑅(−𝜑)
y
x
(x,y)
0.2
0.2
-0.2
-0.2
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(s,p)
Isotropic sample
16
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APPLICATION TO OVERLAY
CHARACTERIZATION IN THE
SEMICONDUCTOR INDUSTRY
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Motivations
 To keep increasing the power of
microprocessors, we need to decrease
the size of the transistors
 Transistor fabrication = layer by layer
 With the decrease in size (currently
22nm), better metrology is required
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Metrology requirements
 We engrave specially designed marks in the
CD
scribe lines
 We measure :
▪
The profile (critical dimension …) : ASML contract
▪
The overlay (shift between the 2 structures) :
MuellerFourier contract with Horiba Jobin Yvon and
CEA-LETI
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Overview of the metrology techniques



State of the art AFM (gold standard for CD metrology)
CD-SEM
Optical techniques :
•Reflectometry, classical ellipsometry (q = 70°, f =0°, 0.75 – 6.3 eV)
•Mueller matrix polarimetry (spectroscopic or angle-resolved)
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More about optical techniques

Image Based overlay (IBO) :
▪
box in box or bar in bar marks imaged with a bright-field microscope.
▪
Grating based Advanced Imaging Method (AIM) by KLA-TENCOR
▪
Limited by the aberrations and size of the marks ( 15x15 – 30x30 µm²)

Diffraction Based Overlay (DBO) : Collection of the light diffracted,
scattered and reflected by the sample and analysis as a function of either the wavelength
(spectroscopic) or the angle of incidence
▪
Empirical DBO : no modeling of the structure needed but at least 2 measurements of
calibrated targets
▪
Model-Based DBO : overlay as a parameter of the fit. Only 1 measurement needed
but model-dependent. Limited by the model and the size of the marks (30x60µm²,
ASML Yieldstar)
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THE ITRS RoadMap
2011  1.6nm
2012  1.4nm
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Properties of the Mueller matrix
The Mueller matrix elements are sensitive
to the profile structure and its asymmetry.
For a structure presenting an asymmetry,
we have :
 M ijleft  M left
ji

right
right
 M ij
i  1, 2
 M ij
 M left  M right
ij
ji

j  3, 4
where left and right stand
for the direction of the
shift in the structure.
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Simulations and RCWA

Simulation of the Mueller matrix of a superposition of 2
gratings with the same pitch but with a lateral shift

Simulation by Rigorous coupled wave analysis : All the
electromagnetic quantities (E, H and ε,μ) are expanded
in Fourier series. Simulations by T.Novikova and
M.Foldyna PhD Defense - Clément Fallet - October 18th
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Estimator
Simulations of structures of interest
0.3
0.2
0.1
0
R² = 1
0
10
20
30
Overlay (nm)




Piece-wise layer dielectric
function
Continuity of field assured by
Lalanne / Li factorization rules
Propagation of S matrices
Based on our knowledge on
Mueller matrix symmetries, we
t
compute M  M to define
possible estimators
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M  M
t
26
Description of the test samples

Test samples designed and manufactured @ CEA-LETI
Nominal overlays (nm) : ±150, ± 100, ± 50, ± 40, ±50µm
30, ± 20 ± 10, 0
Nominal CDs L1 and L2 also vary to extensively test the simulations
 84 different grating combinations
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Sample 1 : CD N1 150 N2 300
0.2
0.2
Normalized Mueller matrix measurement
-0.2
Estimator E  M  M
-0.2
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t
Scalar estimator
Manually selected mask
Kept constant for all
measurements of the same CD
comination
Scalar estimator :
E = <E14>mask
E14
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How to use our estimator?
 2 possibilities
▪
▪
1 – Check the linearity of the estimator based
on the overlay actually present on the wafer.
Gold standard established by Advanced
Imaging Method (AIM)
2 – Measurement of the uncontrolled overlay
(overlay in addition of the nominal overlay)
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VALIDATION OF THE LINEARITY OF
ESTIMATOR E14
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Sample 1 (N1 150 N2 300) : Linearity
Estimator overlay Y
0.14
Value of the estimator
0.12
y = 0.0016x - 0.003
R² = 0.9936
0.1
0.08
0.06
0.04
0.02
0
-10
0
-0.02
-0.04
10
20
30
40
50
60
70
80
90
AIM overlay (nm)
Gold standard
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Sample 1 : comparison with simulations
0.4
R² = 1
0.3
value of the estimator
R² = 0.9999
-60
0.2
0.1
R² = 0.9936
0
-40
-20
0
20
40
60
80
-0.1
Max(E14(mask)) simu
-0.2
Mean(E14(mask)) simu
-0.3
Mean(E14(mask)) exp
100
-0.4
Overlay (nm)
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Sample 2 : CD N1 130 N2 300
mean(E14) Overlay Y
y = 0.0043x + 0.0091
R² = 0.9667
0.35
0.3
0.25
0.2
estimator
0.15
-40.00
0.1
0.05
-20.00
0
0.00
-0.05
20.00
40.00
60.00
80.00
-0.1
-0.15
-0.2
AIM overlay (nm)
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Sample 2 : CD N1 130 N2 300
mean(E14) Overlay X
0.3
0.2
y = -0.0055x - 0.0301
R² = 0.9978
estimator
0.1
0
-60
-40
-20
0
20
40
60
-0.1
-0.2
-0.3
-0.4
AIM overlay (nm)
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Influence of the CD
Influence of the CD
-45nm
-25nm
0.15
overlay Y 200 200
value of the estimator
0.1
-60
overlay Y 200 220
0.05
0
-40
-20
0
20
40
60
-0.05
-0.1
-0.15
y = -0.0014x - 0.0664
-0.2
-0.25
-0.3
y = -0.0047x - 0.1185
-0.35
nominal overlay (nm)
Specified value
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Conclusion
 Estimator OK linear with overlay
measured by AIM, which is considered as
gold standard.
 Consistency between X and Y overlays.
 The slope highly depends on the CD of
the gratings.
 Value of the experimental estimator
smaller than predicted by simulations.
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MEASUREMENTS OF THE
UNCONTROLLED OVERLAY
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Definitions
 We distinguish the nominal overlay
(specified) and real overlay
𝑜𝑣𝑟𝑒𝑎𝑙 = 𝑜𝑣𝑛𝑜𝑚 +
𝑜𝑣𝑢𝑛𝑐
 The nominal overlay is a controlled bias,
intentionally introduced.
 Only the uncontrolled overlay is relevant
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Method 1
 If 𝑜𝑣 = 𝑜𝑣𝑛𝑜𝑚 + 𝑜𝑣𝑢𝑛𝑐 = 0  𝐸14 = 0
 Linear fit on the measurements
Overlay Y
𝐸14 = 𝑆 ∗ 𝑜𝑣𝑛𝑜𝑚 + 𝑜𝑓𝑓𝑠𝑒𝑡
0.04

Given by linear regression
-50
𝑜𝑣𝑢𝑛𝑐
Estimator
0.02
𝑜𝑓𝑓𝑠𝑒𝑡
=−
𝑆
-30
0
-10
-0.02
y = -0.0014x - 0.0664
10
30
50
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
 𝑜𝑣𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 = 𝑜𝑣𝑛𝑜𝑚 + 𝑜𝑣𝑢𝑛𝑐
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-0.16
nominal overlay (nm)
40
Method 2
 𝐸14 𝑑 ∝ 𝑑 + 𝑜𝑣𝑢𝑛𝑐
 𝐸14 −𝑑 ∝ −𝑑 + 𝑜𝑣𝑢𝑛𝑐
 𝑜𝑣𝑢𝑛𝑐 =
𝐸14 𝑑 +𝐸14(−𝑑)
𝑑
𝐸14 𝑑 −𝐸14(−𝑑)
 𝑜𝑣𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 = 𝑜𝑣𝑛𝑜𝑚 + 𝑜𝑣𝑢𝑛𝑐
hypothesis ∶ 𝑜𝑣𝑢𝑛𝑐 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (H)
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Verification of H
Module 10 N1 170 N2 300, overlay Y
Method 2 is validated for high nominal overlays
Method 1
Method 2
AIM overlay (nm)
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Correlation between AIM et Mueller
Correlation AIM - Mueller Overlay Y
200
Mueller overlay (nm)
150
-100
y = 1.0456x + 1.5293
R² = 0.9674
100
50
0
-50
0
50
100
150
-50
-100
AIM overlay (nm)
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Correlation between AIM et Mueller
Correlation AIM - Mueller Overlay X
Mueller overlay (nm)
y = 0.945x + 0.709
R² = 0.9708
60
40
20
-80.00
-60.00
-40.00
-20.00
0
0.00
20.00
40.00
60.00
-20
-40
-60
-80
AIM overlay (nm)
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Map of the overlay on a field
 Map of the uncontrolled overlay 𝑜𝑣𝑢𝑛𝑐
(all measurement in nm)
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A few quality estimators
 𝑇𝐼𝑆 ∶ 𝑡𝑜𝑜𝑙 𝑖𝑛𝑑𝑢𝑐𝑒𝑑 𝑠ℎ𝑖𝑓𝑡 = 1,12𝑛𝑚
 𝜎𝑇𝐼𝑆 = 1,1nm
 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = 0,88 𝑛𝑚
 TMU : total measurement uncertainty
𝑇𝑀𝑈 =
𝑇𝐼𝑆² + 𝜎𝑇𝐼𝑆 ² + 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛²
𝑇𝑀𝑈 = 1,80𝑛𝑚
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Comparisons with existing apparatus
 Total measurement uncertainty (TMU) for
commercial instruments
▪
AIM : TMU ~ 2nm (2008)
▪
Yieldstar : TMU = 0,2nm (2011)
▪
Nanometrics : TMU ~ 0,4nm (2010)
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Conclusions
 Characterization of the overlay with a
(fast), non-destructive technique. No
modelling required but 2 very-well
characterized structures for calibration
 Uncertainty relatively small ~ 2nm
 Measurements in 20 x 20µm² boxes
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Conclusions (2)
 Very good linearity of the scalar estimator
respect to the overlay defect (R² between
0,94 and 0,99)
 However, experimental values of the
estimators are lower than what simulation
predicted.
 Estimators are very sensitive to the
chosen mask
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Perspectives
 Possibility to go down to 5 x 5µm² boxes
with the correct pinhole
 Automatic selection of the mask
 Increase the repeatability of the
measurements to decrease Tool Induced
Shift and its variability to decrease total
uncertainty
 Integrate CD measurement through fitting of
the Mueller matrix to approach Ausschnitt’s
MOXIE (Metrology Of eXtremely Irrational
Exuberance)
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MEASUREMENTS
ON
BEETLES
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Organization of the cuticle
 A twisted multilayer structure : Bouligand
structures
 Each layer consists of a chitin structure
with uniaxial anysotropy
L. Besseau and M.-M. Giraud-Guille, J. Mol. Biol., no. 251, pp.
197–202, 1995.
10µm
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Modeling of the structure
 Fit of spectroscopic Mueller ellipsometry
 Optical model of the cuticle (K. Järrendahl)
Image from K. Järrendahl.
 Spatial homogeneity is assumed; but need of a
more complex model to take into account the
spatial variations
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Purpose of this study
 Compare the results obtained on same
species with different characterization
methods
 Characterize the spatial variations of the
polarimetric response to improve the
model
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Comparisons of the results
Variable Angle Spectroscopic
ellipsometer RC2
Angle resolved Mueller
polarimeter









Angular range 20°-70°
2θ configuration
Average on the field
Spectral resolution
Only the specular
reflection
All incidence at a time
Average on the angle
Spatial resolution
All the light emitted at a
certain angle (reflection
+ scattering)
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Cetonia aurata
Cetonia aurata
5x image
Imaged area 360µm
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20x image
Imaged area 90µm
56
Cetonia aurata
1 0
0 𝑎
0 0
0 0
0
0
−𝑎
0
20X
1
0
0
−1
0
0
0
0
0
0
0
0
−1
0
0
1
M14
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0
0
0
𝑏
Chrysina argenteola
20x image
Imaged area 90µm
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Chrysina argenteola
20X
M14
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Conclusions
 Difficult to accurately compare the results
obtained with different techniques.
 But still, common features arise
 Only a preliminary work, a lot remains to
be done.
 To our knowledge, nobody has ever
published spatially resolved Mueller
matrices for beetles
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CONCLUSIONS & PERSPECTIVES
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PERSPECTIVES
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Chiral structures
 Understand the
relationship
between helicoidal
structures and
circular dichroism
 Mimic the cuticle of
beetles
From G. z. Radnoczi et al. ,Physica status solidi. A. Applied
research, vol. 202, no. 7, pp. R76–R78.
Mueller Matrix @ 633nm
0.2
0.2
-0.2
-0.2
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M14
63
Periodic structures
Sol-gel deposited silica spheres
M12
Real image with 100x
Angle resolved MM
 Hexagonal symmetry visible in both the structure
and the Mueller matrix
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M34
64
CONCLUSIONS
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Conclusions
 Optimization of a Mueller microscope
▪
▪
Better illumation scheme  Modified Köhler
Good calibration of the objective without any
prior modelling but only a (Ψ,Δ) matrix
assumption
 Measurements in both real and reciprocal
space, different kind of applications
presented
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Conclusions
 In Fourier space
▪
▪
Characterization of the overlay with a (fast),
non-destructive technique. No modelling
required but 2 very-well characterized
structures
Uncertainty relatively small ~ 2nm
 In real space
▪
▪
Accurate spatial characterization of
entomological structures
Major step for the study of the autoorganized structures
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Acknowledgements
 Financial support of the French National
Research Agency (ANR) through the joint
project MuellerFourier with CEA-LETI and
Horiba Jobin-Yvon.
 Hans Arwin, Kenneth Järrendahl and Roger
Magnusson at LiU.
 Special thanks to Tatiana Novikova and Bicher
Haj Ibrahim for their help and support.
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M 1  T obj  M  T obj
o
i
M 1  T obj  M  T obj
o
i
M 1  T obj  M  T obj
o
i
Calibration of the objective
 Assumptions :
▪
▪
Objective can be described by a (Ψ,Δ)
matrix.
The MM in forward and backward directions
are equal = Mobj
 By measuring an isotropic sample (eg. cSi wafer), we can calibrate the objective
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Calibration of the objective
 Mmeas = Mobj * McSi * Mobj
(Ψ,Δ) matrices commute
 Mmeas = Mobj² * McSi

Δmeas = 2 Δobj + ΔcSi
tanΨmeas = tanΨobj² * tanΨcSi
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Results on objective calibration
Difference
between
calibration
Difference
between
calibration
with@633nm
cSi and 633nm
SiO2
Objective
calibrated
with@532nm
cSi
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Calibration of reflectivity
 M11 is not calibrated in the ECM.
 B = τ A’.M.W’.Isource with τ, total
transmission of the device
 M = 1/c .A’-1.B.W’-1 with c= τ.Isource
 By measuring well-known samples, we
can calibrate the factor c.
PhD Defense - Clément Fallet - October 18th
74
Details of the mark
AIM marks
30x30 µ2
10
x marks
overlay specified
along Y
y marks
Overlay specified
along x
20
5
10
20
10
x grating
Level 1
Level 2
y grating
AIM marks
clockwise
anticlockwise
PhD Defense - Clément Fallet - October 18th
75
Best results so far, N1 300 N2 180
OVY
0.4
Value of E14 versus nominal
overlay in nm for overlay along
x and y axis
0.3
y = -0,0057x - 0,0047
R² = 0,9893
0.2
0.1
0
-60
-40
-20
0
20
40
60
Main features :
- the uncontrolled overlay is
close to 0.
- Highest slope in the
measured samples
-0.1
-0.2
-0.3
-0.4
OVX
0.4
0.3
Is there a correlation between
the slope and the uncontrolled
overlay?
y = -0.0069x - 0.0178
R² = 0.9947
0.2
0.1
0
-60
-40
-20
0
20
40
60
-0.1
-0.2
-0.3
-0.4
PhD Defense - Clément Fallet - October 18th
76
Intrensinc properties of the MM
A Stokes non-diagonalizable Mueller matrix (NSD MM) : theory
Image and equation from Ossikovski et al, Opt. Lett. 34, 974-976 (2009)
77
PhD Defense - Clément Fallet - October 18th
Intrensic properties of the MM
PhD Defense - Clément Fallet - October 18th
78
Beetles, natural occurrence of NSD MM
 The MM can be regarded as the weighted
average of 3 components
M  nd
1

0


0

1
0
0
0
0
0
0
0
0
LCP
 1
1


0
0
 
0
0 


 1
0
0
0
1
0
0
1
0
0
0

0

0

1
1

0

0

0
Mirror
0
0
1
0
0
1
0
0
0

0

0

1
HWP
From Ossikovski et al., Opt. Lett. 34, 2426-2428 (2009)
PhD Defense - Clément Fallet - October 18th
79
Sum decomposition of the MM
PhD Defense - Clément Fallet - October 18th
80
DOP ellipse
PhD Defense - Clément Fallet - October 18th
81
Calibration of Bouligand structures
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82