Phenomenological modelling of ellipsometry and Mueller matrix … · 2018-09-10 ·...

Phenomenological modelling of ellipsometry and Mueller matrix

polarimetry measurements: theory and experiment

(or how to model without a model)

Razvigor OSSIKOVSKI

LPICM, CNRS, Ecole Polytechnique, Université Paris-Saclay, 91128 Palaiseau, France

[email protected]

Collaborations and contributions

LPICM, Ecole Polytechnique: E. Garcia-Caurel, T. Novikova, A. Peinado, E. Slikboer, S. Yoo, M. Foldyna, A. Pierangelo, J. Rehbinder

University of Barcelona: O. ArteagaJohannes Kepler University: K. Hingerl, M. Miranda-Medina

University of Linköping: H. Arwin, K. Järrendahl, R. MagnussonUniversity of Zaragoza: J.J. Gil

IISER Kolkata: N. GhoshUniversity of Lille: V. Devlaminck

HORIBA Scientific: M. Stchakovsky and others…

Outline

1. What and why we model?

2. Instrumental modelling for ellipsometry and Mueller matrix polarimetry

3. Phenomenological modelling for Mueller matrix polarimetry

What and why we model?

MODEL: A schematic description or representation of something, especially a system orphenomenon, that accounts for its properties

and is used to study its characteristics (The Free Dictionary)

Polarimetric (ellipsometric) modelling

Phenomenological: decompositions & elementary

polarization properties

Electromagnetic (EM): multilayer stack, coupled

wave analysis, finiteelement

Need to relate the measured quantities (Mueller matrix elements) to the sample parameters (dielectric function, elementary properties)

Instrumental: the measurement process

Sample: Maxwell equations Sample: linear algebra

usually overlooked…

Sample-instrumentinteraction

Outline




PSA PSGdetectorlight

sourcesample

polarization stateanalyzer

polarization stategenerator

A generic polarimetric (or ellipsometric) setup

I = soutT . M . sin

N < 4 : ellipsometry (partial determination of M: usuallyM33 & M34 or M33 & M12 -> ellipsometric angles & )

N = 4: Mueller matrix polarimetry (total determination of M)

PSG & PSA: generate & analyze N ≤ 4 polarization states sin & sout

The elements of the Mueller matrix M are obtained from the knownsin & sout states and the detected intensities I

The principle of the polarimetric measurement

44434241

34333231

24232221

14131211

MMMMMMMMMMMMMMMM

Mueller matrixStokes vector Stokes vectorintensity

The parameters of the measurement process

The polarimetric measurement is an averaging process thatpotentially introduces depolarization

Depolarization depends on both the sample and the instrument

Measurement parameters p & theirinstrumental uncertainties p:

- frequency & spectral resolution - angle of incidence & its spread

- spot size A & coherence area A- time constant & measurement time t, etc.

';'' dppppwpMpM Fundamental measurement relation (convolution, averaging)

If M ≠ M then depolarization!If M = M then no depolarization:

only if M(p) = M: ‘nice’ sample or w = (Dirac delta): ‘perfect’ instrument

setupinstrumental

function

samplemeasured

sample

For more, see M. Losurdo’s poster

18-96!

sample-instrumentinteraction

Wavelength (nm)800750700650600550500450

P

0.990.9850.98

0.9750.97

0.9650.96

0.9550.95

0.9450.94

0.9350.93

0.925

Ellipsometric measurement of a thick transparent substrate, e.g. glass slide

partially coherentaddition of beams

Experimetally observeddepolarization (first case)

The finite spectral resolution (or ) of the instrument (light source or monochromator) generates depolarization (DI < 1)

d

Dep

olar

izat

ion

inde

x (D

I)

Depolarization index (DI): measures the disagreement

between M and M

courtesy of M. Stchakovsky

partially resolvedinterference fringes

DI < 1:depolarization!

: wavelength resolution

2

d

211

211

3MMtr

DIT

MM

Photon Energy (eV)0.60.550.50.450.4

M34 0

Photon Energy (eV)0.550.50.450.4

Pgen

0.95

0.9

0.85

0.8

0.75

';'' dwMM ijij

Spectral coherence: the thick substrate case

Fundamental measurement relation

Sample: strongly spectrally dependent, partially resolved interference fringesSetup: finite spectral resolution (light source or monochromator)

A special case of EM modelling: analytical expressions exist: substantial gain in simulation time!

Method readily extendible to handle incidence angle spread

K. Hingerl and R. Ossikovski, Opt. Lett. 41, 219 (2016)

35-µm-thick polymer layer on c-Si substrate measured on an IR polarimeter

M34=

sin

2si

n

1DIdepolarization!

courtesy of A. Peinado and E. Garcia-CaurelDep

olar

izat

ion

inde

x

Ellipsometric measurement of a patterned sample:

98 nm SiO2 on c-Si

Wavelength (nm)800750700650600550500450

P

1.02001.00000.98000.96000.94000.92000.90000.88000.86000.84000.82000.80000.7800

Wavelength (nm)800750700650600550500450

¶ (ß)

80.075.070.065.060.055.050.045.040.035.030.025.020.015.010.0

£ (ß)

220.0

210.0

200.0

190.0

180.0

170.0

160.0

150.0

140.0

130.0

120.0

110.0

100.0

90.0

80.0

Experimentally observeddepolarization (second case)

The finite spot size of the probing light generates depolarization

Dep

olar

izat

ion

inde

x (D

I)

spot

Ellipsometric angles

SiO2

c-Si

courtesy of M. Stchakovsky

DI < 1:depolarization!

Spatial coherence: the finite spot size problem

The overall response is a weighted average of the individual responses

Msub Mgr

Mmix

??

measured

weighted average:Mmix = Mgr + (1 – ) Msub

Totally incoherentsuperposition of gratingand substrate: a specialcase of the fundamentalrelation when A << AA: coherence area

Substrate: block-diagonal Grating: full matrix

Mixture: full matrix

A 700-nm-period grating

on c-Si substrate

rrrrrr ddAwMM ijij''' ;,

Fundamental measurement relation

R. Ossikovski and K. Hingerl, Opt. Lett. 41, 4044 (2016)

Substrate-grating mixture: final results

Retrieval of Mgr from the « mixture » possible on « real-life » data

Mmix ____ Mgr ------Mgr calc.------

R. Ossikovski et al., Appl. Opt. 53, 6030 (2014)M. Foldyna et al., Opt. Express 17, 12794 (2009)

Very good agreement between

measured and retrieved grating

Mueller matrix Mgr

Mgr retrieved from Mmixwithout knowing (the block-diagonal) Msubfrom the weighted

average

Conclusions on instrumental modelling

The results of a polarimetric experiment depend not only on the

sample itself, but also on the measurement instrument

Whenever necessary, build not only an optical model of the

sample, but also an instrumental model (describing the sample -

instrument interaction)

Determine the properties of your instrument: wavelength

resolution, incidence angle spread, spot size, measurement time,

coherence areas (or lengths) of the light source and detector

Outline




Constraints on measuredMueller matrices: filtering

Experimental Mueller matrices must be tested and if needed, filteredfor realizability and apparent depolarization, prior to processing!

1. The measured Mueller matrix M must be realizable, i.e. its associatedcovariance matrix H(M) must be positive semi-definite, H(M) 0

(all H eigenvalues i 0: Cloude criterion)

jiji

ijM σσH

4

1,41

(2.) If there is no depolarization, the measured Mueller matrix M must bealso non-depolarizing, i.e. equivalent to J, the (complex) Jones matrix.

M is then called Mueller-Jones matrix.

1* TJJTM

Before proceeding any further: measured Mueller matrices must satisfy certain conditions!

e.g., kick out all negative i

kick out all but one i (the largest one)

S.R. Cloude, Optik 75, 26 (1986); R. Ossikovski, Opt. Lett. 37, 578 (2012)

The challenges of experimental polarimetry

Fast, reliable and easy-to-use Mueller matrix polarimeters has made possible the characterization of complex structures:

anisotropic materials (crystals, gratings, etc.) biological and organic samples (beetles, tissues, etc.) turbid media, complex solutions (phantoms), etc.

So what to do when a model is not readily available?How do we retrieve information from the polarimetric measurement without any preliminary knowledge?

ExperimentalMueller matrix M

(generally depolarizing)

Electromagnetic modelrelating the physical

properties of the structure to M

The model complexity is proportional to that of the structure!

classic physical approach

Decompositions: why decompose?

To reduce an arbitrary complex structure to a set of simpler and familiar components (to simplify)

To get a better physical insight without any preliminaryinformation (to understand / to get an impression of)

To obtain a standard « equivalent circuit » parameterization of any Mueller matrix (to compare)

Matrix decomposition is a universal phenomenologicalapproach (whether an EM model exists or not)

Experimental Mueller matrix M

(generally depolarizing)

M1

M3

M2

…

phenomenological approach

What to do with the measured M without an EM model: decompose it!

The elementary building blocks

Any non-depolarizing M is a sequence of these two blocks (in either order)

Diattenuatorchange in E amplitudesa symmetric matrix MDdiattenuation vector Dpartial polarizer

Retarderchange in E phasesan orthogonal matrix MRretardance vector Rwaveplate

name

actiondescriptorpropertyexample

Example: plane surface (, ) : M M = M M

cossin00sincos00

00100001

2sin002sin00

0012cos002cos1

cos2sinsin2sin00sin2sincos2sin00

0012cos002cos1

00

J.J. Gil and E. Bernabeu, Optik 76, 67 (1987)

Generalizations of two familiar poalrization devices: the partial polarizer and the retardation waveplate

diattenuator retarder

The third block

Any depolarizing M is a sequence containing a depolarizer (together with the other building blocks: diattenuator, retarder)

name

actiondescriptor

propertyexample

Depolarizerrandom change in E amplitudes / phasesa diagonal Md or a special non-diagonal Mnddepolarization index DI (depolarization)suspension of scattering particles

cb

ad

0000000000001

M

aand

000000000001

21

21

M

two canonical forms: diagonal non-diagonal

R. Ossikovski, J. Opt. Soc. Am. A 27, 123 (2010)

1,, cba

Sum (parallel)decompositions

The sum decompositions represent any depolarizing M as a parallel combination of non-depolarizing components Mk

Cloude / Gil M = 1M1 + 2M2 + 3M3 + 4M4 , k > 0

Note: the sum decompositions require no depolarizers (depolarization through incoherent addition)

Mk : non-depolarizingcomponents whosecontributions add

incoherently

( i : eigenvalues of the covariance matrix H of M )

S.R. Cloude, Optik 75, 26 (1986)J.J. Gil and I. San José, J. Opt. Soc. Am. A 30, 1078 (2013)

1 M1

M2

M3

M4

2

3

4

« equivalent circuit » :

eigenvalue / arbitrary

The incoherent mixture of a perfect (linear or circular) polarizerand a plane mirror produces a non-diagonal depolarizer

Cuticle area of Cetonia aurata beetle (measured in reflection)M can be represented as a linear combination of two basic matrices

21;0,1 2.1.

R. Ossikovski et al., Opt. Lett. 34, 974 (2009); 34, 2426 (2009)H. Arwin et al., Opt. Express 23, 1951 (2015)

Sum decomposition of a complex biological reflector

Mueller matrix reflection image Mueller matrix parameterization

1000010000100001

1001000000001001

ndM

circular polarizer(diattenuator)

plane mirror(retarder)

Two distinct areas in the image

Poincaré sphere mapping by depolarizing Mueller matrices

The Poincaré sphere mapping allows for geometricalclassification and interpretation of depolarizing Mueller matrices

Sout = M Sin : Poincaré sphere -> ellipsoidM ~ Md (diagonal) : zero or two contact points (centered ell.)M ~ Mnd (non-diagonal): one contact point (eccentic ell.)

Md Mnd

R. Ossikovski et al., J. Opt. Soc. Am. A 30, 2291 (2013)

diagonal depolarizer non-diagonal depolarizer

e.g. turbid

medium

e.g. Cetoniaauratabeetle

Product (series)decompositions

The product decompositions represent any depolarizing M as a series combination of the three basic blocks

M1 M2 Mk…

Lu-Chipman (forward): Mf MR MD

Reverse: MD MR MrSymmetric: MD2 MR2 M MR1 MD1

Note: unlike M, Mf and Mr are not elementary depolarizers (i.e., they can be further decomposed down to M Md or Mnd)

S.-Y. Lu and R.A. Chipman, J. Opt. Soc. Am. A 13, 1106 (1996)R. Ossikovski et al., Opt. Lett. 32, 689 (2007)R. Ossikovski, J. Opt. Soc. Am. A 26, 1109 (2009)


Given a general (experimental) depolarizing Mueller matrix M…

Different decompositions generate images with different contrasts:contrast enhancement capability

Product decompositions in polarimetric imaging

Biological (meat slice) sample: diattenuation – depolarization images from two decompositions

(a) (b)

(c) (d)

Observation:(a) has sharper contrast than (b)

(a) – (b) Reverse (MD Mr)(c) – (d) Lu-Chipman (M MD)

Explanation:meat slice behaves roughly as MD Md

R. Ossikovski et al., phys. stat. sol. (a) 205, 720 (2008)

Some samples are not discrete systems, but rather continuous media

From discrete to continuous polarimetric description

MmM

dzd propagation eq. for M along z

m: differential Mueller matrix

For a uniform non-depolarizing M:i.e. no z-dependence of m

)exp( dmM LMm lnd (d: optical path-length)

R.M.A. Azzam, J. Opt. Soc. Am. 68, 1756 (1978)R.C. Jones, J. Opt. Soc. Am. 38, 671 (1948)

Most appropriate description of the polarimetric response of a continuous medium in a transmission polarimetric experiment

Differential Mueller matrix formalism

R. Ossikovski, Opt. Lett. 36, 2330 (2011)

« equivalent circuit » :a special case of product decomposition

M1 M2 M3 … Mn: infinitesimal slabs

In practice, to determine m, one evaluates L, the Mueller matrix logarithm

Continuous media: elementary polarization properties

The elementary polarization properties fully describe the polarimetric response of a continuous medium (depolarizing or not)

Six real elementary properties grouped in three complex pairs: L, L’ and C

Defined from the differential matrix m of a non-depolarizing Mueller matrix M:

Dichroic / diattenuation (D) propertiesBirefringent / retardance (B) properties

L: linear, 0° - 90° L = LD + i LBL’: linear, 45° - 135° L’ = LD’ + i LB’C: circular, R – L C = CD + i CB

0'0'

'0'0

LBLBCDLBCBLDLBCBLD

CDLDLD

mL

Phenomenological definition throughpairs of complex refractive indices:

example on the C property

LR

LR

kklCD

nnlCB

2

2

R.M.A. Azzam, J. Opt. Soc. Am. 68, 1756 (1978); R.C. Jones, J. Opt. Soc. Am. 38, 671 (1948)

Elementary properties from the Mueller matrix logarithm

Unlike with « classic » product decompositions, polarization and depolarization effects occur simultaneously, not sequentially.The Mueller matrix logarithm represents any depolarizing M as

continuously depolarizing along the optical path

L = ln M L = Lm + Lu

: depolarizing infinitesimal slabs

Lm: non-depolarizing part Lu: depolarizing part

R. Ossikovski, Opt. Lett. 37, 220 (2012)R. Ossikovski and V. Devlaminck, Opt. Lett. 39, 1216 (2014)


M1 M2 M3 … Mn

Lm = ½ (L – G LT G) contains the mean values of the elementary propertiesLu = ½ (L + G LT G) contains their variances-covariances

0'0'

'0'0

LBLBCDLBCBLDLBCBLD

CDLDLD

m mL

Depolarizing Mueller matrix M:

Fluctuations are responsible for the depolarizing polarimetric behaviour of the continuous medium

22***

*22**

**22*

***

2

L'LCL'ReLCReL'LIm

CL'ReLCL'LReLCIm

LCReL'LReCL'CL'Im

L'LImLCImCL'Im

A

A

A

A

u mL

Statistical interpretation of depolarization

The elementary properties Pi are assumed to be fluctuating, Pi = Pi + Pi

The depolarization results from the non-zero variances-covariances of the fluctuating elementary properties, PiPj

* and |Pi|2

If no fluctuations, i.e. Pi = 0, then no depolarization

R. Ossikovski and O. Arteaga, Opt. Lett. 39, 4470 (2014)J. J. Gil and R. Ossikovski, Polarized Light: the Mueller Matrix Approach (CRC Press, 2016)

Polarimetric response of a depolarizing continuous medium of thickness d:M = exp(Lm + Lu) = exp(md + ½m2d2)

Experimental validation of the fluctuating medium model

µm-size TiO2 spheres in a plastic host matrix

Experimental results: depolarization from scattering

N. Agarwal et al., Opt. Lett. 40, 5634 (2015)

2

22

22

2

L2000

0CL00

00CL0000

A

A

AA

u mL

‘pure’ scattering: all elementary properties equal zero, Lm = 0so that M = exp(Lu) = exp(½m2d2)

Experimental results:

Parabolic behaviour of the threedepolarizations with the thickness d

(in accordance with the model)

Rotational invariance: diagonal depolarizer

1, 2 and 3: anisotropicdepolarizations (L, L’ and C)

thickness d0 1 2 3 4 5 6

-5

-4

-3

-2

-1

0

1.5 mg/ml

3 mg/ml Exp. 1 Exp. 3

Fit 1 Fit 3

1

and

3

number of layers n

6 mg/ml

Basic features of phenomenological modelling

Even when you think you do not know anything, you still know something: transmission/ reflection configuration, continuous/discrete

medium, depolarizing/nondepolarizing system, etc., so why not use this knowledge!

The phenomenological modelling approachappears as a powerful tool for the analysis ofexperimental Mueller matrices without anyelectromagnetic model or preliminary information

Phenomenological modelling is not just plain data-reduction method, but rather allows for a deeperphysical interpretation in terms of familiar devicesor elementary polarization properties

more info here, if not ‘allergic’ to algebra…

Conclusion

In French/English languages:Sire, on peut tout faire avec des baïonettes sauf s’asseoir dessus !(Sir, one can do anything with bayonets but sit on them!)

minister Talleyrand to emperor Napoleon, around 1800

In polarimetric-ellipsometric language:One can obtain anything from a model but new physics!

New physics is obtained by processing the model resultswhile using your own brain and experience

THANK YOU !

Phenomenological modelling of ellipsometry and Mueller matrix … · 2018-09-10 ·...

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