angle resolved mueller polarimetry, applications to periodic structures
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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Angle resolved Mueller Polarimetry, Applications to periodic structures
PhD DefenseClément Fallet
Under the supervision of Antonello de Martino
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2
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
Outline of the presentation
PhD Defense - Clément Fallet - October 18th
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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
Motivations of the study
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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
Motivations of the study
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LET’S TALK ABOUT POLARIZATION
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Introduction to polarization
PhD Defense - Clément Fallet - October 18th
𝑆𝑜𝑢𝑡
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
M M M MM M M MM M M MM M M M
M=𝑆𝑜𝑢𝑡=𝑀 .𝑆𝑖𝑛
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7
A word about polarimeters
PhD Defense - Clément Fallet - October 18th
CCD
camera
PS
GP
SA
(Stokes Polarimeter)Mueller PolarimeterB = A.M.W
M = A-1.B.W-1
W = [S1, S2, S3’ S4]PSG Basis Stokes vectors
At = [S’1, S’2, S’3, S’4]PSA Basis Stokes vectors
A and W must be as close as possible to unitaryTheir condition numbers must be optimized(E.Compain 1999, S. Tyo 2000, M. Smith, 2002)
Calibration : eigenvalue method No instrument modelling(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
PhD Defense - Clément Fallet - October 18th
CCD
Back focal plane
Sample
retractable lens
Beamsplitter
34
1 – Aperture diaphragm2 – Field diaphragm3 – PSG : Polarization State Generator4 – PSA : Polarization State Analyser5 – Aperture Mask
1 2
5
Source
Aperture image : angularly resolvedReal image : spatially resolved
Interferential filter
Strain-freeMicroscope objective
LColl L1 L2
Lim1
Lim2
Lim3
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Illumination arm
Collectionlens
Aperturediaphragm
L1 L2
Back focal plane
Fielddiaphragm
Rays
em
ergi
ng
from
the
sour
ce
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Detection arm
400nm pitch grating
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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 well-characterized reference samples (c-Si, SiO2 on c-Si) (method explained in the manuscript)
Choice of the objectives
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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
𝑟 ∝ sin𝜃
c-Si wafer, 633nm
-0.2
0.2
-0.2
0.2
𝑟𝑝𝑟 𝑠
=tan (Ψ )𝑒𝑖 Δ
dichroism retardance
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From (x,y) to (s,p)
(x,y) (s,p)Isotropic sample
psx
y
-0.2
0.2
-0.2
0.2
𝑀 (𝑥 , 𝑦 )=𝑅 (𝜑 ) .𝑀 (𝑠 ,𝑝 ) .𝑅 (−𝜑 )
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APPLICATION TO OVERLAY CHARACTERIZATION IN THE SEMICONDUCTOR INDUSTRY
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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
Motivations
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We engrave specially designed marks in the scribe lines
We measure :▪ The profile (critical dimension …) : ASML contract
Metrology requirements
▪ The overlay (shift between the 2 structures) : MuellerFourier contract with Horiba Jobin Yvon and CEA-LETI
CD
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Overview of the metrology techniques
•Reflectometry, classical ellipsometry (q = 70°, f =0°, 0.75 – 6.3 eV)•Mueller matrix polarimetry (spectroscopic or angle-resolved)
State of the art AFM (gold standard for CD metrology) CD-SEM Optical techniques :
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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
4,32,1 jiMMMMMM
rightji
leftij
rightij
rightij
leftji
leftij
The Mueller matrix elements are sensitive to the profile structure and its asymmetry. For a structure presenting an asymmetry,
we have :
where left and right stand for the direction of the shift in the structure.
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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
Simulations and RCWA
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Simulations of structures of interest
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 compute to define possible estimators
tMM
0 5 10 15 20 25 300
0.10.20.3
R² = 0.999998726171532
Overlay (nm)
Estim
ator
tMM
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Description of the test samples Test samples designed and manufactured @ CEA-LETI
Nominal overlays (nm) : ±150, ± 100, ± 50, ± 40, ± 30, ± 20 ± 10, 0 Nominal CDs L1 and L2 also vary to extensively test the simulations 84 different grating combinations
50µm
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Sample 1 : CD N1 150 N2 300
Normalized Mueller matrix measurement EstimatortMME
-0.2
0.2
-0.2
0.2
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Scalar estimatorManually selected maskKept constant for all measurements of the same CD comination
Scalar estimator :
E = <E14>mask
E14
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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)
How to use our estimator?
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VALIDATION OF THE LINEARITY OF ESTIMATOR E14
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Sample 1 (N1 150 N2 300) : Linearity
-20 0 20 40 60 80 1000
2
4
6
8
10
12
f(x) = NaN x + NaNR² = 0 Estimator overlay Y
AIM overlay (nm)Gold standard
Valu
e of
the
estim
ator
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Sample 1 : comparison with simulations
-60 -40 -20 0 20 40 60 80 100
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
R² = 0.993582309213263
R² = 0.999937738839223R² = 0.999976456213567
Max(E14(mask)) simu
Linear (Max(E14(mask)) simu)
Mean(E14(mask)) simu
Overlay (nm)
valu
e of
the
estim
ator
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Sample 2 : CD N1 130 N2 300
-40.00 -20.00 0.00 20.00 40.00 60.00 80.00
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
f(x) = 0.00429680543212432 x + 0.00909874060495459R² = 0.966683795799765
mean(E14) Overlay Y
AIM overlay (nm)
estim
ator
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Sample 2 : CD N1 130 N2 300
-60 -40 -20 0 20 40 60
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
f(x) = − 0.00549447722353445 x − 0.0300912498760537R² = 0.997823073653477
mean(E14) Overlay X
AIM overlay (nm)
estim
ator
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Influence of the CD
-60 -40 -20 0 20 40 60
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
f(x) = − 0.00141753887375114 x − 0.0664328571428571
f(x) = NaN x + NaNInfluence of the CD
overlay Y 200 200Linear (overlay Y 200 200)overlay Y 200 220
nominal overlay (nm)Specified value
valu
e of
the
estim
ator
-45nm -25nm
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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.
Conclusion
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MEASUREMENTS OF THE UNCONTROLLED OVERLAY
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We distinguish the nominal overlay (specified) and real overlay
The nominal overlay is a controlled bias, intentionally introduced.
Only the uncontrolled overlay is relevant
Definitions
𝑜𝑣𝑟𝑒𝑎𝑙=𝑜𝑣𝑛𝑜𝑚+𝑜𝑣𝑢𝑛𝑐
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I
Linear fit on the measurements
Method 1
𝐸14=𝑆∗𝑜𝑣𝑛𝑜𝑚+𝑜𝑓𝑓𝑠𝑒𝑡
𝑜𝑣𝑢𝑛𝑐=− 𝑜𝑓𝑓𝑠𝑒𝑡𝑆
Given by linear regression -50 -40 -30 -20 -10 0 10 20 30 40 50
-0.16-0.14-0.12
-0.1-0.08-0.06-0.04-0.02
-2.77555756156289E-170.020.04
f(x) = − 0.0014175389 x − 0.0664328571
Overlay Y
nominal overlay (nm)
Estim
ator
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(H)
Method 2
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Verification of H
Method 1
Method 2
AIM overlay (nm)
Module 10 N1 170 N2 300, overlay Y
Method 2 is validated for high nominal overlays
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Correlation between AIM et Mueller
-100 -50 0 50 100 150
-100
-50
0
50
100
150
200
f(x) = 1.04558846108261 x + 1.52931313231716R² = 0.967448625485969
Correlation AIM - Mueller Overlay Y
AIM overlay (nm)
Mue
ller o
verla
y (n
m)
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Correlation between AIM et Mueller
-80.00 -60.00 -40.00 -20.00 0.00 20.00 40.00 60.00
-80
-60
-40
-20
0
20
40
60
f(x) = 0.94503672902141 x + 0.708960842034468R² = 0.97077398340729
Correlation AIM - Mueller Overlay X
AIM overlay (nm)
Mue
ller o
verla
y (n
m)
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Map of the overlay on a field
Map of the uncontrolled overlay (all measurement in nm)
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TMU : total measurement uncertainty
A few quality estimators
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Total measurement uncertainty (TMU) for commercial instruments
▪ AIM : TMU ~ 2nm (2008)
▪ Yieldstar : TMU = 0,2nm (2011)
▪ Nanometrics : TMU ~ 0,4nm (2010)
Comparisons with existing apparatus
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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
Conclusions
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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
Conclusions (2)
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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)
Perspectives
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MEASUREMENTSON
BEETLES
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A twisted multilayer structure : Bouligand structures
Each layer consists of a chitin structure with uniaxial anysotropy
Organization of the cuticle
L. Besseau and M.-M. Giraud-Guille, J. Mol. Biol., no. 251, pp. 197–202, 1995.
10µm
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Fit of spectroscopic Mueller ellipsometry Optical model of the cuticle (K. Järrendahl)
Spatial homogeneity is assumed; but need of a more complex model to take into account the spatial variations
Modeling of the structure
Image from K. Järrendahl.
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Compare the results obtained on same species with different characterization methods
Characterize the spatial variations of the polarimetric response to improve the model
Purpose of this study
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PhD Defense - Clément Fallet - October 18th 55
Variable Angle Spectroscopic ellipsometer RC2
Angular range 20°-70° 2θ configuration Average on the field Spectral resolution Only the specular
reflection
Angle resolved Mueller polarimeter
All incidence at a time Average on the angle
Spatial resolution All the light emitted at a
certain angle (reflection + scattering)
Comparisons of the results
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PhD Defense - Clément Fallet - October 18th 56
Cetonia aurata
Cetonia aurata 5x imageImaged area 360µm
20x imageImaged area 90µm
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PhD Defense - Clément Fallet - October 18th 57
Cetonia aurata
M14
20X
[ 1 0 0 − 10 0 0 00 0 0 0
−1 0 0 1
]
[1 0 0 00 𝑎 0 00 0 −𝑎 00 0 0 𝑏
]
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PhD Defense - Clément Fallet - October 18th 58
Chrysina argenteola
20x imageImaged area 90µm
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PhD Defense - Clément Fallet - October 18th 59
Chrysina argenteola
M14
20X
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PhD Defense - Clément Fallet - October 18th 60
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
Conclusions
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CONCLUSIONS & PERSPECTIVES
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PERSPECTIVES
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PhD Defense - Clément Fallet - October 18th 63
Understand the relationship between helicoidal structures and circular dichroism
Mimic the cuticle of beetles
Chiral structures
From G. z. Radnoczi et al. ,Physica status solidi. A. Applied
research, vol. 202, no. 7, pp. R76–R78.
Mueller Matrix @ 633nm M14
-0.2
0.2
-0.2
0.2
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PhD Defense - Clément Fallet - October 18th 64
Periodic structuresSol-gel deposited silica spheres
Real image with 100x
M12
M34
Angle resolved MM
Hexagonal symmetry visible in both the structure and the Mueller matrix
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PhD Defense - Clément Fallet - October 18th 65
CONCLUSIONS
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PhD Defense - Clément Fallet - October 18th 66
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
Conclusions
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PhD Defense - Clément Fallet - October 18th 67
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 auto-organized structures
Conclusions
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PhD Defense - Clément Fallet - October 18th 68
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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Thank you
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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. c-Si wafer), we can calibrate the objective
iobj
oobj TMTM 1 i
objo
obj TMTM 1 iobj
oobj TMTM 1
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PhD Defense - Clément Fallet - October 18th 72
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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PhD Defense - Clément Fallet - October 18th 73
Results on objective calibration
Objective calibrated with cSi @633nmDifference between calibration with cSi and SiO2Difference between calibration @532nm and 633nm
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PhD Defense - Clément Fallet - October 18th 74
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.
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PhD Defense - Clément Fallet - October 18th 75
AIM marks clockwiseanticlockwise
y marksOverlay specified along x
x marksoverlay specified along Y
10
10
10
AIM marks 30x30 µ2
20
20
5
x grating
y grating
Level 1 Level 2
Details of the mark
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Best results so far, N1 300 N2 180
-60 -40 -20 0 20 40 60
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
f(x) = − 0.00573610047238372 x − 0.00474275000000001R² = 0.989273527602751
OVY
-60 -40 -20 0 20 40 60
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
f(x) = − 0.00691868272727273 x − 0.017844R² = 0.994721736748336
OVX
Value of E14 versus nominal overlay in nm for overlay along x and y axis
Main features :- the uncontrolled overlay is
close to 0.- Highest slope in the
measured samples
Is there a correlation between the slope and the uncontrolled overlay?
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PhD Defense - Clément Fallet - October 18th 77
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)
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Intrensic properties of the MM
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PhD Defense - Clément Fallet - October 18th 79
Beetles, natural occurrence of NSD MM
The MM can be regarded as the weighted average of 3 components
1000010000100001
1000010000100001
1001000000001001
ndM
LCP Mirror HWP
From Ossikovski et al., Opt. Lett. 34, 2426-2428 (2009)
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Sum decomposition of the MM
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DOP ellipse
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Calibration of Bouligand structures