quantitative mueller matrix polarimetry with diverse applications

Quantitative Mueller Matrix Polarimetrywith diverse applications

Harsh Purwar(07MS-76)

Department of Physical Sciences,Indian Institute of Science Education and Research, Kolkata

Email: [email protected]

Thesis SupervisorDr. Nirmalya Ghosh

April 26, 2012

Harsh Purwar (DPS, IISER-K) Quantitative MM Polarimetry April 26, 2012 1 / 41

Outline

1 Introduction

2 Motivation

3 MM Measurement Methods

4 Our MM Measurement Strategy

5 Experimental Setup

6 Eigenvalue Calibration Method

7 Mueller Matrix Decomposition Scheme

8 Results

9 Initial Applications on Biological Tissues

10 Applications towards nano-plasmonics

11 Conclusions


Introduction

IntroductionWhat is it?

Several calculi have been developed for analysing polarization, including those based on theJones matrix, coherency matrix, Mueller matrix, and other matrices1.

Stokes − Mueller Formalism:The Stokes parameters are a set of values that describe the polarization state of theelectromagnetic radiation first introduced by George G. Stokes in 1852.

~S =

S0

S1

S2

S3

=

IQUV

=

IH + IVIH − IVIP − IMIL − IR

Degree of polarization (DOP) is a quantity used to describe the portion of an electromagneticwave which is polarized. In Stokes formalism we have,

DOP =

√S2

1 + S22 + S2

3

S0

1Shurcliff, 1962; Gerrard and Burch, 1975; Theocaris and Gdoutos, 1979; Azzam and Bashara, 1987; Coulson,

1988; Egan, 1992.


Introduction

IntroductionMueller Calculus

Mueller matrix (M) for a polarization-altering device is defined as the matrix which transformsan incident Stokes vector S into the exiting (reflected, transmitted, or scattered) Stokes vectorS ′,

S ′4×1 = M4×4 S4×1

Mueller matrices of some common optical elements with corresponding Stokes vectors:For an ideal quarter waveplate with its fast axis oriented at an angle of 45◦:

1 0 0 00 0 0 −10 0 1 00 1 0 0

s0

s1

s2

s3

=

s0

−s3

s2

s1

For an ideal linear polarizer, transmission axis 0◦:

1 1 0 01 1 0 00 0 0 00 0 0 0

s0

s1

s2

s3

=

s0 + s1

s0 + s1

00


Motivation

MotivationWhy are we interested?

MM contains information about all the polarization properties of the sample andthus has been very useful in the following areas of (but not limiting to) scienceand technology.

Material characterization,

Measuring the thickness and refractive indices of thin films (ellipsometry),

Tissue characterization & differentiation, monitoring of Glucose levels inblood, etc.,

Remote sensing of the earth and astronomical bodies,

Applications in Metrology, Astronomy, Ophthalmology, Radar polarimetryetc.2

Stokes − Mueller calculus is applicable to randomly, fully or partially polarized

light.

2Handbook of Optics, Chapter 22, R.A. Chipman


MM Measurement Methods

MM Measurement MethodsSome commonly used approaches...

Several MM measurement methods have been proposed so far. Some of the mostadopted among them are:

1 Modulation-based methods:Uses electro-optic, magneto-optic or photoelastic modulators to rapidlychange polarization states and a lock-in based detection.

Advantages: Single measurement is sufficient to determine completeMM with very high precession and accuracy.Major Drawback: Spectral and imaging measurements are notpossible.

2 Direct Measurements, using dual rotating polarizers and retarders.

Advantages: Suits well for imaging and/or spectral measurements.Major Drawbacks: Measurements are not very accurate and areprone to huge errors.


Our MM Measurement Strategy

Our Measurement StrategyHere’s what we did...!

Dual rotating retarder approach:Consists of two rotating quarter wave plates and two fixed linear polarizers to generate sixteenelliptically polarized states.

Polarization State Generator (W) comprises of a fixed linear polarizer (P1) followed by arotatable quarter wave plate (Q1). It, in general can be used to generate any polarization state.

1 0 0 00 C2

θ1+ S2

θ1Cδ Sθ1

Cθ1(1− Cδ) −Sθ1

Sδ0 Sθ1

Cθ1(1− Cδ) S2

θ1+ C2

θ1Cδ Cθ1

Sδ0 Sθ1

Sδ −Cθ1Sδ Cδ

︸︷︷︸

MRetarder

×

1 1 0 01 1 0 00 0 0 00 0 0 0

︸︷︷︸

MPol at Horizontal

×

1000

︸︷︷︸

Unpol.

Wθ1=(

1 C2θ1

+ S2θ1Cδ Cθ1

Sθ1(1− Cδ) Sθ1

Sδ)T

For four different input polarizations we have,

W =

1 1 1 1

C2θ1

+ S2θ1Cδ C2

θ2+ S2

θ2Cδ C2

θ3+ S2

θ3Cδ C2

θ4+ S2

θ4Cδ

Cθ1Sθ1

(1− Cδ) Cθ2Sθ2

(1− Cδ) Cθ3Sθ3

(1− Cδ) Cθ4Sθ4

(1− Cδ)Sθ1

Sδ Sθ2Sδ Sθ3

Sδ Sθ4Sδ



Our Measurement Strategy

Polarization State Analyser (A) comprises of a rotatable quarter wave plate (Q2) followed by afixed linear polarizer (P2) crossed with P1. It is dedicated to the measurement of an unknownStokes vector.

1 −1 0 0−1 1 0 00 0 0 00 0 0 0

︸︷︷︸

MPol at Vertical

×

1 0 0 00 C2

θ1+ S2

θ1Cδ Sθ1

Cθ1(1− Cδ) −Sθ1

Sδ0 Sθ1

Cθ1(1− Cδ) S2

θ1+ C2

θ1Cδ Cθ1

Sδ0 Sθ1

Sδ −Cθ1Sδ Cδ

︸︷︷︸

MRetarder

PSA is followed by an intensity based detector, which just records total intensity or the very firstelement/row of the Stokes vector. Hence,

Aθ1=(

1 −C2θ1− S2

θ1Cδ −Sθ1

Cθ1(1− Cδ) Sθ1

Sδ)

Similarly for four different angles we have,

A =

1 −C2

θ1− S2

θ1Cδ −Sθ1

Cθ1(1− Cδ) Sθ1

Sδ1 −C2

θ2− S2

θ2Cδ −Sθ2

Cθ2(1− Cδ) Sθ2

Sδ1 −C2

θ3− S2

θ3Cδ −Sθ3

Cθ3(1− Cδ) Sθ3

Sδ1 −C2

θ4− S2

θ4Cδ −Sθ4

Cθ4(1− Cδ) Sθ4

Sδ



Criteria for Stable Mueller Matrix

From A (PSA) and W (PSG) matrices, Mueller matrix of the sample Ms is calculated asfollows. We know that,

M = A×Ms ×W

Hence if we define,

Q16×16 = A4×4 ⊗WT4×4 =⇒ (Ms)16×1 = Q−1

16×16M16×1

(Ms)16×1 is then reshaped to a 4× 4 matrix. The sample’s Mueller matrix is then decomposedusing Polar Decomposition scheme to get its basic polarization properties. Choice of orientationangles of retarders (QWP’s) is:

very crucial to get a stable and physically realizable MM.

made by maximizing the determinant of matrix Q.

For simplicity same four angles (35◦, 70◦, 105◦ & 140◦) were chosen for both the QWP’s.

Chosen angles were verified using a more rigorous approach.



Verification of Chosen Angles (θi ’s) using SVD

The Singular Value Decomposition of an m× n real or complex matrix S is a factorization of theform,

S = UDV ?

where U is a m ×m and V is a n × n unitary matrices. D is a m × n rectangular diagonalmatrix. The diagonal entries of D are known as singular values of matrix S.

The angles of the polarizers and QWPs should be chosen so thatthe corresponding Stokes vectors are distributed evenly inside thePoincare sphere.

x y

z

1

This is achieved by making sure that the singular values obtainedafter SVD of the PSG and PSA matrices are non-zero. We definethe condition number for PSG and PSA matrices as,

C# =Min{singular values}Max{singular values}

Poincare sphere showing Stokesvectors constituting PSG andPSA matrices.


Experimental Setup

Experimental SetupOur hard work!

Setup 1: Schematic of the experimental setup Setup 2: Schematic of the experimental setup

for elastic scattering. for inelastic (fluorescence) scattering.


Eigenvalue Calibration Method

Need for CalibrationHow does it help?

The polarizers and QWPs do not behave ideally (i.e. not as a perfect diattenuatorand as a perfect λ/2 retarder) over the entire spectral range. And hence the PSGand PSA matrices constructed earlier will change with wavelength.Eigenvalue Calibration,

provides exact PSG (W ) and PSA (A) matrices over the entire spectralrange.

hence automatically corrects for the non-ideal behaviour of the involvedoptical elements.

requires at least two extra measurements with samples whose form of theMM is known.



Eigenvalue CalibrationMathematical Formulation

Lets choose the two reference samples to be blank (no sample) and a diattenuating retarder andcall these measurements as b0 and b respectively. Hence,

b0 = aw , b = amw

=⇒ c = b−10 b = w−1mw , c ′ = bb−1

0 = ama−1

By definition c, m & c ′ are similar matrices and hence have same eigenvalues. We do not knowm explicitly yet, but it has the following form,

m =

1 − cos 2ψ 0 0

− cos 2ψ 1 0 00 0 sin 2ψ cos ∆ sin 2ψ sin ∆0 0 sin 2ψ sin ∆ sin 2ψ cos ∆

m has four eigenvalues (2 real & 2 imaginary),

λR1= 2τ cos2 ψ, λR2

= 2τ sin2 ψ

λC1= τ sin 2ψe−i∆, λC2

= τ sin 2ψe i∆



Eigenvalue CalibrationA little more mathematics!

Hence,

τ =λR1

+ λR2

2, ψ = tan−1

√λR1

λR2

, ∆ = log

√λC2

λC1

Mueller matrix m for the reference sample (diattenuating retarder) is constructed back fromthese eigenvalues.

Now, consider the following two matrix equations,

mX − Xc = 0, mX ′ − X ′c ′ = 0

with solutions: X = w and X ′ = a since, c = w−1mw and c ′ = ama−1.

Solving3 these two linear equations gives us our PSG (w) and PSA (a) matrices.

3Eric Compain, et. al., Applied Optics, Vol. 38, Issue 16, 1999.



Eigenvalue Calibration ResultsPSG Matrix

Figure: Polarization State Generator Matrix (W ) as a function of wavelength (λ).



Eigenvalue Calibration ResultsPSA Matrix

Figure: Polarization State Analyser Matrix (A) as a function of wavelength (λ).



Eigenvalue Calibration ResultsCondition Number for PSG & PSA Matrices

Figure: Plot of condition number for the PSG and PSA Matrices as a function of wavelength(λ). Clearly, condition number > 0.25 ∀ λ.



Eigenvalue Calibration ResultsError Estimate

Figure: Plot of an estimate of error as a function of wavelength (λ).



ResultsMueller Matrix (blank)

Figure: Mueller Matrix (M) for blank (no sample) as a function of wavelength (λ).



ResultsNull Elements from the MM for a quarter waveplate

Figure: Null (zero) elements of the Mueller matrix for a quarter waveplate plotted as afunction of wavelength (λ) showing elemental error ∼ 0.01.


Mueller Matrix Decomposition Scheme

MM DecompositionPolar Decomposition

Mueller matrix reflects lumped effects hindering their unique interpretation.

Measured MM (4 × 4) is decomposed into three 4 × 4 matrices using Polardecomposition scheme:

M = M∆MRMD ,

leading to three basic polarization properties diattenuation (D), retardance (R)and depolarization (∆).

Depolarization (∆): If an incident state is 100% polarized and the exiting statehas a degree of polarization less than unity, then the system is said to bedepolarizing.

Retardance (R) is the phase difference between the two orthogonal polarizationsof light for both linear and circular polarizations.

Diattenuation (D) is the differential attenuation of orthogonal polarizations for

both linear and circular polarization states.


Results

ResultsDiattenuation (D) for a wide-band linear polarizer

500 520 540 560 580 600 620 640 660 680 7000.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

Wavelength λ (nm)

Dia

tten

uat

ion

D

Ideal Value = 1.0

Figure: Polar decomposition derived Diattenuation (D) for a wide-band Glan-Thomson linearpolarizer as a function of wavelength (λ).


Results

ResultsLinear Retardance (δ) for a quarter waveplate

Figure: Polar decomposition derived Linear Retardance (δ) for a 633 nm quarter waveplateplotted as a function of wavelength (λ). At λ = 633 nm, measured δ = 1.54 rad.


Results

Advantages of Our Measurement Strategy

Independent of source and detector (spectrometer) polarization responses.

Simultaneous spectroscopic and spatial mapping (imaging). Can beincorporated with microscopic arrangement.

Exact PSG and PSA matrices and their wavelength dependence need not beknown.

Is fully automated. Takes just a few minutes for measurement of completespectral (400 − 800 nm) MM.

Mueller matrix elemental error < 0.01.

Capable of detecting small polarization signal even in presence of largebackground depolarization noise.


Initial Applications on Biological Tissues

Applications towards Tissue Characterization



Fluorescence MM PolarimetryChallenges & Prospects

1 Significant change in tissue fluorescence from normal to malignant stage.

2 Collagen and NADH −→ identified as biomarkers for cancer.

3 Tissue fluorescence spectra −→ convolution of spectra of various fluorophores.

4 Diagnostic parameters: Spectra, Yield & Decay Kinetics.

5 Only spectral signature can not be used to distinguish between the normal and cancerous

tissues for the following few reasons.

Spectral signatures of these biomolecules vary from patient to patient and alsodepend on the region we are probing.Are hugely influenced by the absorption of light from absorbers like blood. Hencenot suited for in-vitro examination.

Also the emission and absorption bands of NADH and Collagen overlap to an extent

that it is not practically possible to study them separately.

Fluorescence MM on the other hand can bring out the structural differences by targetingspecific molecules. The fibrous structure of collagen gives large diattenuation as opposedto NADH, which is isotropic.

Hence this can be used as a tool for Cancer Diagnosis.



Preliminary ResultsM11 Element of Fluorescence MM for Human Cervical Cancer Biopsy Slides

Collagen Em. Spectra

NADH Em. Spectra

Figure: M11 element of fluorescence Mueller matrix (un-normalized) measured from the Humancervical tissue biopsy slides (thickness ∼ few µm) shown as a function of wavelength (λ).



Preliminary ResultsFluorescence MM for Human Cervical Cancer Biopsy Slides

Figure: Individual elements of fluorescence Mueller matrix measured from the Human cervicaltissue biopsy slides (thickness ∼ few µm) shown as a function of wavelength (λ).



Preliminary ResultsDiattenuation for Human Cervical Cancer Biopsy Slides

Figure: Diattenuation measured from the Human cervical tissue biopsy slides (thickness ∼ fewµm) with different stages of cancer shown as a function of wavelength (λ).


Applications towards nano-plasmonics




Introduction to SPRSurface Plasmon Resonance

Plasmon resonance: Collective oscillations of free electrons at the metal-dielectric interface inresonance with the incident electromagnetic (EM) field.

Can be propagating as on the planar metal-dielectric interfaces as well as localized as on metalnanoparticles.

Numerous practical applications:

Contrast enhancement in optical imaging.

Ultra-high sensitive chemical and biomedical sensing, bio-molecular manipulation,labelling, detection.

Surface enhanced spectroscopy (Raman and fluorescence).

Development of new optical devices: Plasmonic wave guiding nano-devices, opticalinformation processing and data storage.



Methodology

The scattering matrices for Ag (silver)nanoparticles and similar dielectricparticles were generated in its surfaceplasmon spectral region(λ = 380− 600 nm).

Reported values of n and k (real &imaginary part of refractive index)were used in the calculation.

Decomposition analysis was performedon S(θ) (T-matrix computed) forrandomly oriented spheroidalnano-particles (metal & dielectric) tostudy the depolarization behaviour.

Similar decomposition analysisperformed on S(θ) for preferentiallyoriented spheroidal nano-particles tounderstand the underlying mechanismof depolarization.



ResultsDepolarization for randomly oriented metal nanoparticles

Depolarization characteristics show distinct spectral features with a peak at aroundλ ∼ 380− 410 nm.

Figure: For r = 20 nm and varying ε.



ResultsLongitudinal & Transverse plasmon resonances in spheroids

The magnitude of depolarization peaks around the overlap spectral region of the twodipolar plasmon bands.

Figure: For r = 20 nm and varying ε.



ResultsEnhanced Retardance and Diattenuation in preferentially oriented metal nanoparticles

Parameters

Estimated values for the polarization parametersat λ = 400 nm, θ = 45◦ and ε = 1.5.

Spheroidal metal Spheroidal dielectric Spherical metalnanoparticle nanoparticle nanoparticle

d 0.435 0.388 0.333∆ 0 0 0

δ (rad.) 1.99 0 0.011

Magnitude of linear retardance δ(λ) peaksaround the overlap spectral region of the twodipolar plasmon bands ∆(λ).

Stronger diattenuation (Dmetal > Ddielectric ).

No depolarization ∆ ∼ 0.

For r = 20 nm, θ = 45◦ and varying ε.



ResultsRetardance of preferentially oriented metal nanoparticles

Dielectric nanoparticle: Usual behaviour

Phase reversal (δ > π2

rad.) inbackscattering θ > 90◦.

δ(for any value of θ) is either close tozero or π.

Metal nanoparticle at 400 nm: Phase reversal orhelicity flipping even in forward scattering angles(δ > π

2rad. θ < 90◦)

Additional scattering-induced δ due the inherentphase retardation between the two orthogonaldipolar plasmon polarizabilities.

Longitudinal and the transverse dipolar plasmonpolarizabilities oscillating with a phase difference.

For r = 20 nm, ε = 1.5 at specific λ’s.


Prospective Applications & Outlook

Quantitative MM Plasmon Polarimetry: Prospectiveapplications & Outlook

Quantitative differences in intrinsic polarization parameters of non-sphericalmetal nanoparticles & background tissue (cell) dielectric structures.

Polarization can be used as additional contrast mechanism to discriminateagainst background Rayleigh / Mie scattering (with optimal choice of λ).

Can be exploited in combination with other spectroscopic approaches(Fluorescence & Raman).

In-situ monitoring and controlling of size & shape of nanoparticles duringsynthesis.

May have implications in bio-sensing exploiting polarization to enhancesensitivity.


Conclusions

Conclusions

A highly sensitive automated spectral Mueller matrix polarimeter for bothelastic and inelastic scattering (fluorescence) has been developed andcalibrated using Eigenvalue Calibration method.

It had initially been used to characterize human cervical cancer tissues.

It had also been used to study the characteristics of some of the fluorescentdyes like Coumarin 102 and Coumarin 152.

Nano-plasmonics

Enhanced depolarization (and its spectral characteristics ∆(λ)) for randomlyoriented spheroidal metal nanoparticles originates from the presence ofstrong linear retardance δ(λ) effect in the individual oriented nanoparticles.

Enhanced diattenuation d & linear retardance δ for preferentially orientedspheroidal metal nanoparticles.


Acknowledgement

Acknowledgement

Dr. Nirmalya Ghosh, IISER-Kolkata.

Prof. Asima Pradhan, IIT-Kanpur.

Dr. Ayan Banerjee, IISER-Kolkata.

Dr. Uday Kumar, IISER-Kolkata.

I am also very thankful to my friends Jalpa Soni, Sayantan Ghosh, Satish Kumar,

Harshit Lakhotia, Shubham Chandel, Nandan K. Das and Subhasri Chatterjee for

numerous healthy discussions and for their constant help, motivation and support

throughout the project.


Publications

Publications...in last one year!

Included in Thesis“Development and Eigenvalue calibration of an automated spectral Mueller matrix systemfor biomedical polarimetry” H. Purwar, et. al., Proc. of SPIE, Vol. 8230 No. 823019,2012. (Full length manuscript is under preparation).

“Quantitative polarimetry of plasmon resonant spheroidal metal nanoparticles: A Muellermatrix decomposition study” J. Soni, et. al., Optics Communications, Vol. 285 Issue 6,2011.

“Enhanced polarization anisotropy of metal nano-particles and their spectralcharacteristics in the surface plasmon resonance band” J. Soni, et. al., Proc. of SPIE,Vol. 8096 No. 809624, 2011.

Other related Publications“Differing self-similarity in light scattering spectra: A potential tool for pre-cancerdetection” S. Ghosh, et. al. Optics Express, Vol. 19 No. 20, 2011. Selected for furtherimpact by Virtual Journal for Biomedical Optics, Vol. 6 Issue 10, 2011.

“A comparative study of differential matrix and polar decomposition formalisms forpolarimetric characterization of complex turbid media” S. Kumar, et. al. to be submittedto Optics Communications.


Thank you for your time.

Harsh Purwar07MS-76

Department of Physical Sciences,Indian Institute of Science Education and Research, Kolkata

Email: [email protected]


quantitative mueller matrix polarimetry with diverse applications

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