quantitative mueller matrix polarimetry with diverse applications

Indian Institute of Science Education and ResearchKolkata

Integrated Dual BS − MS Dissertation

Quantitative Mueller matrix polarimetry

with diverse applications

Author:Harsh PurwarEmail: [email protected]

Supervisor:Dr. Nirmalya Ghosh

Email: [email protected]

Department of Physical Sciences,IISER-Kolkata, Mohanpur Campus, Nadia − 741 252, India

2011 − 2012

Certificate

This is to certify that the thesis entitled “Quantitative Mueller matrix polarimetry withdiverse applications” being submitted to the Indian Institute of Science Education and Research(IISER) − Kolkata in partial fulfillment of the requirements for the award of the IntegratedBS − MS degree, embodies the research work done by Harsh Purwar under my supervision atIISER − Kolkata. The work presented here is original and has not been submitted so far, inpart or full, for any degree or diploma of any other university/institute.

Place:

Date:

Dr. Nirmalya GhoshAssistant ProfessorDepartment of Physical SciencesIndian Institute of Science Education and Research, Kolkata

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Acknowledgement

It is a pleasure to thank the many people who made this thesis possible.

Foremost, I would like to express my sincere gratitude to my supervisor Dr. Nirmalya Ghoshfor his continuous support, motivation, inspiration, enthusiasm and immense knowledge. Hisguidance helped me at all times, during research and also while writing this thesis. I could nothave imagined having a better supervisor and mentor for my Masters study.

Besides my advisor, I would like to thank the rest of my thesis committee: Dr. Ayan Banerjee,Dr. Bhavtosh Bansal and Dr. Ritesh K. Singh for their encouragement, insightful commentsand for asking good questions.

I also thank Dr. Uday Kumar for important discussions on various new ideas related to theapplications of the developed strategies. I would also like to thank the Department of ChemicalSciences at IISER-Kolkata for letting me work in their labs even at odd hours.

I thank my fellow labmates in Optics and Photonics/Biophotonics Laboratory: Jalpa Soni,Sayantan Ghosh, Satish Kumar, Harshit Lakhotia, Shubham Chandel, Nandan K. Das andSubhasri Chatterjee for the stimulating discussions, for the sleepless nights we were workingtogether before deadlines, and for all the fun we have had in the last twelve months. Also Ithank my friends and family members for motivating, encouraging, inspiring and supportingme throughout.

Place: IISER-Kolkata, Mohanpur CampusDate: May 7th 2012

Harsh PurwarDepartment of Physical Sciences,Indian Institute of Science Education and Research, Kolkata

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List of Publications

Included in thesis

Jan. 2012 “Development and Eigenvalue calibration of an automatedspectral Mueller matrix system for biomedical polarimetry”Harsh Purwar, Jalpa Soni, Harshit Lakhotia, Shubham Chandel, ChitramBanerjee, Nirmalya Ghosh Proc. of SPIE, Vol. 8230 No. 823019(doi:10.1117/12.906668). Full length manuscript is under preparation.

Dec. 2011 “Quantitative polarimetry of plasmon resonant spheroidal metalnanoparticles: A Mueller matrix decomposition study” Jalpa Soni,Harsh Purwar, Nirmalya Ghosh Optics Communications, Vol. 285 Issue6, pp. 1599−1607 (doi:10.1016/j.optcom.2011.11.066).

Nov. 2011 “Enhanced polarization anisotropy of metal nano-particles andtheir spectral characteristics in the surface plasmon resonanceband” Jalpa Soni, Harsh Purwar, Nirmalya Ghosh appeared in paper ver-sion of Proc. of SPIE, Vol. 8096 No. 809624.

Other related pulications

May 2012 “A comparative study of differential matrix and polar decom-position formalisms for polarimetric characterization of complexturbid media” Satish Kumar, Harsh Purwar, Razvigor Ossikovski, I AlexVitkin and Nirmalya Ghosh communicated to Optics Communications.

Sep. 2011 “Differing self-similarity in light scattering spectra: A poten-tial tool for pre-cancer detection” Sayantan Ghosh, Jalpa Soni,Harsh Purwar, Jaidip Jagtap, Asima Pradhan, Nirmalya Ghosh, Pras-anta K. Panigrahi Optics Express, Vol. 19 No. 20, pp. 19708−16(doi:10.1364/OE.19.019717). Selected for further impact by Virtual Jour-nal for Biomedical Optics, Vol. 6 Issue 10, Nov. 2011.

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Preface

The work embodied in this thesis has been carried out in the Department of Physical Sci-ences, Indian Institute of Science Education and Research (IISER) − Kolkata during the periodAugust 2011 to April 2012. The thesis is divided into following chapters.

Chapter 1: Introduction gives a brief overview of various aspects of Mueller matrix (MM)polarimetry and its applications. It also discusses about its applicability in the field of biomed-ical optics.

Chapter 2: Basics of Polarimetry discusses about the Stokes-Mueller formalism and de-fines the basic polarization parameters. Two of the commonly used Mueller matrix measurementtechniques are also discussed very briefly highlighting their major advantages and drawbacks.

Chapter 3: Our MM Measurement Strategy describes in detail about the proposed spec-tral Mueller matrix measurement strategy and its advantages over other common approaches.It also lays emphasis on the physical realizability of the measured Mueller matrices and howcan it be achieved through optimization of the varying parameters.

Chapter 4: Calibration and Decomposition discusses two mathematical techniques insome detail namely, Eigenvalue calibration and Polar decomposition which form the backboneof the proposed measurement strategy.

Chapter 5: Results and Discussion shows some of the important calibration results, mea-sured Mueller matrices and spectral behaviour of the decomposition derived polarization pa-rameters for some common optical elements.

Chapter 6: Applications towards Tissue Characterization: This spectral MM mea-surement system was initially used for tissue characterization (to distinguish between normaland cancerous tissues). Brief introduction, experimental setup and preliminary results are pre-sented in this chapter.

Chapter 7: Applications towards Nano-plasmonics: One of the major applications ofthis work is in the field of nano-plasmonics and is discussed in this chapter. The Polar decom-posed Mueller matrices obtained using T-matrix for spheroidal metal (silver) nanoparticles andsimilar dielectric particles are investigated and some of the interesting results are shown.

Chapter 8: Conclusions and Future Directions concludes this work by describing a fewinnovative ideas which may be carried out in near future.

Bibliography contains references which have been cited in the above chapters.

Appendix A: Eigenvalue Calibration - Matlab Script shows an implementation of theEigenvalue calibration method in Matlab.

Appendix B: Polar Decomposition - Matlab Script shows an implementation of thePolar decomposition algorithm in Matlab.

Appendix C: Labview Automation Code shows front panel and block diagram for theVI designed to automate the whole experimental setup.

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Contents

Certificate ii

Acknowledgement iii

List of Publications iv

Preface v

List of Figures vii

1 Introduction 101.1 Review of related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Basics of Polarimetry 132.1 Stokes-Mueller Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Commonly used measurement approaches . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Modulation based approaches . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Direct measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Our MM Measurement Strategy 173.1 Dual rotating retarder approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Polarization State Generator (PSG) . . . . . . . . . . . . . . . . . . . . . 173.1.2 Polarization State Analyzer (PSA) . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Optimization of Generator and Aanalyzer . . . . . . . . . . . . . . . . . . . . . . 193.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Advantages of this strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Calibration and Decomposition 224.1 Eigenvalue Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1.2 Limitations and Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Results and Discussion 275.1 Calibration Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Polar Decomposition Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6 Applications towards Tissue Characterization 336.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.3 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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7 Applications towards Nanoplasmonics 377.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

8 Conclusions and Future directions 468.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Bibliography 49

A Eigenvalue Calibration - MATLAB Script 54

B Polar Decomposition - MATLAB Script 59

C Labview Automation Code 62

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List of Figures

3.1 Poincare sphere showing Stokes vectors constituting PSG (red) and PSA (blue)matrices assuming the ideal nature of both of them. x, y and z axis are the threeelements S1, S2 and S3 of the Stokes vector respectively as defined in section 2.1. 20

3.2 Schematic of the experimental setup for elastic scattering based Mueller matrixmeasurements. Key: L1: Lens, A: Aperture, P1: Fixed linear polarizers withpolarization axis oriented at any arbitrary angle (calling this polarization state asH), Q1 & Q2: Rotatable quarter waveplates (orientation angles are with respectto P1), S: Sample with unknown Mueller matrix, P2: Fixed linear polarizer(acting as analyzer), polarization axis crossed with polarizer P1. . . . . . . . . . . 20

5.1 Variation of individual elements of Polarization State Generator Matrix (W ) asa function of wavelength (λ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Variation of individual elements of Polarization State Analyzer Matrix (A) as afunction of wavelength (λ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.3 Condition number for polarization state generator matrix (W ) and polarizationstate analyzer matrix (A) calculated for the spectral region (λ = 500− 700 nm). 28

5.4 An estimate of the error in determination of the PSG and PSA matrices for thesame spectral region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.5 Variation of the individual elements of Mueller matrix for blank (no sample) overthe spectral region (λ = 500− 700 nm). Ideally Mueller matrix for blank shouldbe identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.6 Null (zero) elements of the Mueller matrix for a quarter waveplate, fast axisoriented at an angle of 23◦ for the spectral range λ = 500− 700 nm. Clearly theelemental error in the measure Mueller matrices is of the order of 0.01. . . . . . . 30

5.7 Polar decomposition derived polarization parameters namely depolarization (∆),diattenuation (d) and retardance (R) for blank Mueller matrix over the spectralregion (λ = 500 − 700 nm). Expected values of all three parameters for blankwas zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.8 Polar decomposition derived diattenuation (d) for a wide-band Glan-Thompsonlinear polarizer with polarization axis oriented at an angle of 28◦ over the spectralregion (λ = 500− 700 nm). Expected value of diattenuation (d) was one. . . . . 31

5.9 Polar decomposition derived linear retardance (δ) for a 632.8 nm quarter wave-plate with fast-axis oriented at an angle of 23◦ over the spectral region (λ =500 − 700 nm). Expected value of linear retardance was π/2 or 1.57 radians atλ = 632.8 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.1 Schematic of the experimental setup for in-elastic (fluorescence) scattering withsimultaneous imaging and spectroscopic Mueller matrix measurements. Key:L: Lens, A: Aperture, BS: Beam splitter, NDF : Neutral density filter, GT :Wide-band Glan Thompson linear polarizer, QWP : Quarter waveplate, MO:Mounting optics, F : Filter, LCV F : Liquid crystal variable filter. . . . . . . . . . 34

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6.2 First element of fluorescence Mueller matrix M11 (un-normalized) for humancervical cancer tissue biopsy slide (thickness ∼ few µm) over the spectral region(λ = 450 − 800 nm). A bandpass filter (450 − 900 nm) was used to cut off theincident wavelength 405 nm used of exciting the tissue sample. The fluorescenceemission peaks of Collagen and NADH are indicated. . . . . . . . . . . . . . . . . 35

6.3 Individual elements of fluorescence Mueller matrix for human cervical cancertissue biopsy slide (thickness ∼ few µm) over the spectral region (λ = 475− 800nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6.4 Polar decomposition derived diattenuation (d) for human cervical cancer tissuebiopsy slide (thickness ∼ few µm) over the spectral region (λ = 450−800 nm) fortwo different grades (stages) of cancer determined from histo-pathology report ofthe sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.1 The variation of the decomposition-derived net depolarization coefficient (∆) as afunction wavelength (λ = 325−525 nm) for randomly oriented silver nanoparticleswith radius r = 20 nm. The results are shown for varying aspect ratios of theoblate spheroids (ε = 1− 2) and the scattering angle was chosen to be θ = 45◦. . 42

7.2 The T-matrix computed scattering efficiency (Qsca) as a function of wavelengthfor the spheroidal silver nanoparticles of varying aspect ratios (ε = 1.25 − 2.0)and having a fixed radius of 20 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.3 The spectral variation of the decomposition-derived linear retardance δ(λ) forpreferentially oriented (see text for details on orientation) spheroidal silver nanopar-ticles with radius r = 20 nm and varying aspect ratio ε = 1.0−2.0. The scatteringangle was θ = 45◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.4 The scattering angle (θ) dependence of the decomposition-derived value of linearretardance δ for preferentially oriented spheroidal silver nanoparticle with r = 20nm and ε = 1.5. The results are shown for three different wavelengths: 365nm, 440 nm (corresponding to the peaks of the transverse and the longitudinalplasmon resonances respectively) and 400 nm (corresponding to the peak of δ(λ)and ∆(λ)). The corresponding variation of the similar dielectric particle is shownby solid circle in the figure (λ = 400 nm). . . . . . . . . . . . . . . . . . . . . . . 45

C.1 Front Panel of the automation VI. . . . . . . . . . . . . . . . . . . . . . . . . . . 62C.2 Block Diagram of the automation VI. . . . . . . . . . . . . . . . . . . . . . . . . 63

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Chapter 1

Introduction

Development of optical techniques for biomedical diagnosis is an area of considerable currentresearch interest [1, 2]. Optical methods can facilitate non-invasive and quantitative diagnosis[3]. For optical diagnosis, one mostly uses the light scattered from the tissue. Most of thelight scattered from tissue is scattered elastically i.e. without any change in frequency. Theelastically scattered light from tissue bears useful information on underlying morphological andphysiological state of tissue. The scattered light also has a very weak component which isscattered in-elastically i.e. with a change in frequency via processes like fluorescence, Ramanscattering etc. The in-elastically scattered light is characteristic of the chemical composition ofthe tissue and thus is linked with bio-chemical properties of the tissue. Therefore by carefulanalysis of elastically and in-elastically scattered light one can diagnose disease. Motivated bythis fact, various optical techniques based on scattered light (elastically or in-elastically) fromtissue are being actively perused for their diagnostic potential [2, 4, 5, 6].

With altered physiological state, the concentration, size, shape and refractive index of thedifferent extra cellular or intra-cellular organelles of tissue show prominent changes. These al-terations have characteristic and varied impact on the measured elastic scattering signal fromtissue. The parameters of elastically scattered light that can be used to quantify these mor-phological features in tissue are the wavelength dependence of scattered light intensity (or thewavelength variation of the scattering coefficient), the angular variation of scattering or thepolarization properties of scattered light. While considerable work has been carried out on theuse of the first two approaches [2], there appear a few reports only on the use of polarimetriclight scattering for tissue diagnosis [7, 8]. Measurement of the polarization properties of scat-tered light from tissue might provide additional diagnostic information on tissue as comparedto unpolarized measurements. For example, since the rate of depolarization of incident linearlyand circularly polarized light depends on the morphological parameters like the density, size(and its distribution), shape and refractive index of scatterers present in the medium [9], thisinformation may be used for quantitative tissue diagnosis. In addition to isotropic depolariza-tion property, many constituents of tissue also show intrinsic polarization properties such asretardance (linear retardance arising due to a difference in phase between two orthogonal linearpolarization, and circular retardance arising due to a difference in phase between right and leftcircularly polarized light) and dichroism or diattenuation (linear diattenuation arising due to thedifferential attenuation of two orthogonal linear polarization, and circular diattenuation arisingdue to the differential attenuation of right and left circularly polarized light). For example, col-lagen is an important structural protein in the human body that has intrinsic linear retardanceproperty. The structural aggregates of collagen (micro-fibrils, fibrils and fibers that causes theretardance property of collagen) play an important role in maintaining the mechanical propertyof tissue and the orientation and arrangement of collagen is closely related to the structuraland functional properties of tissues and organs. In various kinds of tissue abnormalities suchas, erythema, psoriasis, actinic keratosis, neurofibroma and the numerous types of carcinomas

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(like basal cell, squamous cell, melanomas etc.), the structural and functional properties of col-lagen aggregates show distinct changes. Such changes are expected to lead to variation in theirretardance property. Measurement and quantification of linear retardance of collagen in tissuemay thus provide a useful tool for diagnosis of various kinds of tissue abnormalities. Glucose isanother important tissue constituent that possesses intrinsic circular retardance property dueto its asymmetric chiral structure. Its presence in tissue leads to rotation of the plane of linearlypolarized light about the axis of propagation (known as optical rotation). Measurement of cir-cular retardance or optical rotation of scattered light from tissue might facilitate non-invasivemonitoring of glucose level in human tissue. Thus it appears that measurement of the differentpolarization properties (depolarization, retardance and diattenuation) of elastically scatteredlight from tissue might come out to be useful for various diagnostics applications. Muellermatrix measurement technique is an experimental approach that can facilitate simultaneousquantification of all these polarization parameters of a medium. We therefore propose to carryout detailed studies on measurement of Mueller matrix of elastically scattered light from variousbiological samples (including tissue samples) to explore possibility of quantitative biomedicaldiagnosis using the polarization properties of scattered light. In contrast to elastic scatteringmeasurements, fluorescence spectroscopy probes biochemical changes that take place with theonset and progression of disease. Fluorescence spectroscopy has particularly been investigatedfor diagnosis of cancer. The parameters of autofluorescence (fluorescence from endogenous tis-sue fluorophores) that can be used for diagnosis include differences in the static spectra, decaykinetics or polarization properties of fluorescence. While considerable work has been carried outon the first two aspects [3, 10, 11], there appear only a few reports on exploiting the polariza-tion properties of fluorescence from tissue for cancer diagnosis. Limited literature available onthis aspect has demonstrated that polarized fluorescence measurements have several importantadvantages over unpolarized fluorescence measurements for tissue diagnosis [7, 8]. As comparedto the two orthogonal polarization state measurements used in the previous polarimetric studies[8, 12], measurements of Mueller matrix of fluorescence emitted from tissue would allow oneto obtain a more complete information on polarization properties of fluorescence. This there-fore might turn out to be an efficient approach to fully exploit the advantages of polarizedfluorescence measurements and design the optimal polarization scheme for quantitative tissuediagnosis. Motivated by the promise of this approach, we also plan to investigate this aspect indetails.

1.1 Review of related literature

As noted earlier, while the inelastically scattered light (fluorescence and Raman) bears usefulbiochemical information on tissue, the elastically scattered light also contains useful informationon underlying morphological and physiological state of tissue. Elastic light scattering measure-ments thus provides a tool to assess tissue morphometry by recovering typical scatterers sizesand their refractive indices. Attempts have therefore been made by several researchers to useelastic scattering measurements for quantitative tissue diagnosis [2, 6]. The general strategyfollowed in most of these studies to quantify the morphological parameters (concentration,size, shape and refractive index of cellular, sub-cellular or extra cellular organelles) has beento analyze the wavelength dependence of elastic scattering signal [2] or the angular variationof scattering [2]. Calculations based on Mie theory of light scattering have widely been usedfor this purpose [13, 14]. Most of these studies have been performed using unpolarized elas-tic scattering measurements and there appear a few reports only on the use of polarized elasticscattering measurements for biomedical diagnosis [4]. Limited literature available on this aspecthas revealed that measurement of polarization properties of scattered light like depolarization,retardance and diattenuation can also facilitate quantification of useful physiological and mor-phological parameters of tissue thus, providing a useful tool for diagnosis of various kinds of

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tissue abnormalities. Since Mueller matrix contains detailed information about all the polar-ization properties of the medium in its different elements, this can be used to quantify theseparameters very efficiently. This would however, require suitable theoretical approaches (math-ematical models) to extract the different polarization parameters from the measured Muellermatrix and to interpret and analyze these parameters in terms of the physiological and morpho-logical features of tissue. This aspect has not been investigated in sufficient details and thereappear only a few reports on this aspect [4, 15]. It has recently shown that the ambiguity of thechange in the orientation angle of the polarization vector due to scattering arises due to the com-bined effect of linear diattenuation and linear retardance of light scattered at large angles andcan be decoupled from the pure optical rotation (arising due the presence of chiral molecules)component using polar decomposition of Mueller matrix [16]. For this purpose, the methoddeveloped earlier [15] for polar decomposition of Mueller matrix was extended to incorporateoptical rotation in the medium. The results of these preliminary studies were encouraging anddemonstrated that polar decomposition of Mueller matrix can facilitate accurate quantificationof optical rotation introduced by the presence of chiral molecules in a turbid medium, like forexample glucose in human tissue. Apart from its use in glucose sensing in tissue, the developedpolar decomposition of Mueller matrix based approach might find wide range of applications inquantitative biomedical diagnosis. We thus plan to investigate this aspect in more details.

1.2 Objective

The main objective of the proposed work is to develop techniques to measure the various polar-ization properties of elastically and in-elastically (fluorescence) scattered light from biologicalsamples, to develop methodologies to interpret and analyze the measured polarization prop-erties in terms of the underlying morphological, biochemical features of the sample and touse this information for quantitative biomedical diagnosis. Since, Mueller matrix provides acomplete description of the polarization characteristics of the medium, focus is on the measure-ment of Mueller matrix and quantification of the different polarization parameters of scatteredlight (elastically and in-elastically (fluorescence) both) from the measured Mueller matrix forbiological samples.

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Chapter 2

Basics of Polarimetry

Polarimeters are optical instruments used for determining the polarization properties of lightbeams and samples [17]. Polarimetry, the science of measuring polarization, is most simply char-acterized as radiometry with polarization elements. Typical applications of polarimeters includethe following: remote sensing of the earth and astronomical bodies, calibration of polarizationelements, measuring the thickness and refractive indices of thin films (ellipsometry), spectro-scopic studies of materials, and alignment of polarization-critical optical systems [17, 18, 19, 20].

2.1 Stokes-Mueller Formalism

Since 1852, when George G. Stokes gave his formalism to describe the polarization state of lightor in general of an electromagnetic radiation, several calculus have been developed, includingthose based on Jones matrix, coherency matrix, Mueller matrix and other matrices1. Of thesemethods, the Mueller calculus is most generally suited for describing intensity-measuring instru-ments, including most polarimeters, radiometers, and spectrometers, and is used exclusively inthis thesis.

In the Mueller calculus, developed by Hans Mueller in 1943, the Stokes vector S is usedto describe the polarization state of a light beam, and the Mueller matrix M to describe thepolarization-altering characteristics of a sample. This sample may be a surface, a polarizationelement, an optical system, or some other light-matter interaction which produces a reflected,refracted, diffracted, or scattered light beam. The Stokes vector is defined relative to the fol-lowing six intensity measurements I performed with ideal polarizers in front of a photo-diode,spectrometer or a CCD [17]:

IH −→ Intensity of horizontally polarized component (0◦).IV −→ Intensity of vertically polarized component (90◦).IP −→ Intensity of +45◦ polarized component.IM −→ Intensity of −45◦ polarized component.IL −→ Intensity of left circularly polarized component.IR −→ Intensity of right circularly polarized component.

The stokes vector is defined as [17, 13, 21],

S =

s0

s1

s2

s3

=

IH + IVIH − IVIP − IMIL − IR

(2.1)

1Shurcliff, 1962; Gerrard and Burch, 1975; Theocaris and Gdoutos, 1979; Azzam and Bashara, 1987; Coulson,1988; Egan, 1992

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where s0, s1, s2 and s3 are Stokes vector elements. s0 gives the total intensity. The Stokesvector does not need to be measured by these six ideal measurements; what is required is thatother methods reproduce the Stokes vector defined in this manner. Ideal polarizers are notrequired. Further, the Stokes vector is a function of wavelength, position on the object, and thelights direction of emission or scatter.

From the Stokes vector, the following polarization parameters are determined [17],

• Degree of Polarization: describes the portion of an electromagnetic wave which ispolarized be it linear, cicular or elliptical.

DOP =

√s2

1 + s22 + s2

3

s0(2.2)

• Degree of Linear Polarization: describes the portion of an electromagnetic wave whichis linearly polarized.

DOLP =

√s2

1 + s22

s0(2.3)

• Degree of Circular Polarization: describes portion of an electromagnetic wave whichis circularly polarized.

DOCP =s3

s0(2.4)

Note: Degree of polarization (DOP) for a unpolarized light is zero whereas for partially polar-ized light is less than unity.

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

S′ =

s′0s′1s′2s′3

= MS =

m11 m12 m13 m14

m21 m22 m23 m24

m31 m32 m33 m34

m41 m42 m43 m44

s0

s1

s2

s3

(2.5)

The Mueller matrix is a 4×4 matrix with real valued elements. The Mueller matrix M(k, λ) fora device is always a function of the direction of propagation k and wavelength λ. The Muellermatrix is an appropriate formalism for characterizing polarization measurements because itcontains within its elements all of the polarization properties: diattenuation, retardance, de-polarization, and their form, either linear, circular, or elliptical. When the Mueller matrix isknown, then the exiting polarization state is known for an arbitrary incident polarization state.Table 2.1 is a compilation of Mueller matrices for common polarization elements, together withthe corresponding transmitted Stokes vector.

2.2 Commonly used measurement approaches

There are a number of Mueller matrix measurement approaches that researchers have proposedin years. As technology improves more of such approaches and strategies are expected to come.In this section two of such approaches that have been widely accepted by scientists and are in usenamely modulation based and direct measurement approaches are briefly described highlightingtheir advantages and major drawbacks.

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Table 2.1: Mueller matrices for some of the common optical elements along with the corre-sponding transmitted Stokes vector.

Nonpolarizing element1 0 0 00 1 0 00 0 1 00 0 0 1

s1

s2

s3

s4

=

s1

s2

s3

s4

An absorber

a 0 0 00 a 0 00 0 a 00 0 0 a

s1

s2

s3

s4

=

as1

as2

as3

as4

A linear polarizer, transmission axis 0◦

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

s1

s2

s3

s4

=

s1 + s2

s1 + s2

00

A linear diattenuator, axis θ◦, intensity transmittances, q, r

q + r (q − r) cos 2θ (q − r) sin 2θ 0(q − r) cos 2θ (q + r) cos2 2θ + 2

√qr sin2 2θ (q + r − 2

√qr) sin 2θ cos 2θ 0

(q − r) sin 2θ (q + r − 2√qr) sin 2θ cos 2θ (q + r) cos2 2θ + 2

√qr sin2 2θ 0

0 0 0 2√qr

Linear retarder, fast axis θ, retardance δ

1 0 0 00 cos2 2θ + sin2 2θ cos δ sin 2θ cos 2θ(1− cos δ) − sin 2θ sin δ0 sin 2θ cos 2θ(1− cos δ) cos2 2θ + sin2 2θ cos δ cos 2θ sin δ0 sin 2θ sin δ − cos 2θ sin δ cos δ

2.2.1 Modulation based approaches

Modulation based approaches make use of photo-elastic [22], electro-optic or magneto-opticsmodulators to rapidly change the polarization states and a lock-in based point detection with aphoto-diode to measure the intensity of these rapidly changing polarization states. One of themain advantages of these strategies is that due to the use of modulated polarization and lock-inbased detection the measurements are very accurate and precise. Even a single measurementmade using this approach is sufficient to determine all sixteen elements of the Mueller matrix.Such experimental setups are capable of extracting even a weak polarized component of lightfrom the strong depolarization background noise.

On the other hand one of the major drawback of such strategies is that they can not be usedfor neither imaging nor spectral measurements. This is essentially because of the limitationimposed by the use of lock-in with a photo-diode which is capable of only a point measurement.And secondly the functioning of the photo-elastic modulators also depend on the wavelengthand can not be used for spectral measurements.

15

2.2.2 Direct measurements

Direct measurement strategies refers to the calculation of the Mueller matrix directly fromthe intensity measurements corresponding to different incident and analyzed polarization statesgenerated by rotating polarizers or wave retarders [23] or both. This measurement strategy isbest suited for imaging and spectral Mueller matrix measurements and since does not reqiureany extra measurements, if automated could give very quick results. Although is not veryaccurate and is prone to huge elemental errors. In order to get a more acurate result one hasto increase the number of measurements which might not always be possible. Moreover taking36 or 49 measurements in order to find the sixteen unknown elements of a 4× 4 Mueller matrixis a shear wastage of resources and time.

16

Chapter 3

Our MM Measurement Strategy

3.1 Dual rotating retarder approach

The dual rotating retarder approach refers to the two rotating retarders (here quarter wave-plates) used along with two fixed linear polarizers to generate sixteen elliptically polarized statesso as to determine the sixteen unknown elements of the 4 × 4 Mueller matrix. A schematic ofthe experimental setup is shown in Fig. 3.2.

3.1.1 Polarization State Generator (PSG)

Polarization state generator (PSG) consists of a fixed linear polarizer (P1) with the choice oforienting its polarization axis at any arbitrary orientation angle followed by a rotatable retarder(Q1) (quarter waveplate here), orientation of the fast axis is with respect to the polarizer P1.If both, polarizer and retarder are allowed to change their orientations, PSG in principal canbe used to generate any polarization state of light. As mentioned in the section 2.1 Muellermatrix of an optical element essentially describes how the polarization of the incident light istransformed by that optical element. The un-polarized light or in terms of Stokes formalismthe Stokes vector S0 = (1 0 0 0)T for un-polarized light is incident on the polarization stategenerator which is then transformed to some other polarization state or Stokes vector dependingupon the configuration of the elements in PSG. For some orientation angle lets say θ1 of theretarder the output stokes vector Wθ1 can be calculated from the Mueller matrices for theretarder and polarizer as follows,

Wθ1 =

1 0 0 00 C2

θ1+ S2

θ1Cδ Sθ1Cθ1(1− Cδ) −Sθ1Sδ

0 Sθ1Cθ1(1− Cδ) S2θ1

+ C2θ1Cδ Cθ1Sδ

0 Sθ1Sδ −Cθ1Sδ Cδ

︸︷︷︸

MRetarder

×

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

︸︷︷︸MPol at Horizontal

×

1000

︸︷︷︸

Unpol.

(3.1)

Wθ1 =

1

C2θ1

+ S2θ1Cδ

Cθ1Sθ1(1− Cδ)Sθ1Sδ

(3.2)

where, Cδ = cos δ, Sδ = sin δ, Cθi = cos 2θi and Sθi = sin 2θi, θi’s being orientation anglesof the fast axis of retarder (Q1) with respect to the polarizer P1 and δ being its retardance(δ = π/2 radians for a quarter waveplate). For measuring complete Mueller matrix i.e. allsixteen elements we need to generate and analyze four different polarization states. In thisstrategy four input polarization states are generated by rotating the retarder in PSG to fourdifferent orientation angles (θ1, θ2, θ3 & θ4). The corresponding output stokes vectors for these

17

four orientations of retarder are clubbed together to get the 4× 4 W or PSG matrix such thateach column of W represents an incident Stokes vector or incident polarization state (incidentat the sample) as shown below.

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)Sθ1Sδ Sθ2Sδ Sθ3Sδ Sθ4Sδ

(3.3)

3.1.2 Polarization State Analyzer (PSA)

The construction of polarization state analyser (PSA) is very similar to that of the generatorand comprises of a rotatable quarter waveplate (Q2), fast axis again oriented with respect topolarizer P1, followed by a fixed linear polarizer (P2) crossed with P1. Again if both retarderand polarizer are allowed to change PSA can be used to measure any unknown incident Stokesvector or polarization state. Here since the order with respect to the incident light of retarderand polarizer is reversed the corresponding Mueller matrices also get reversed in the construc-tion of the PSA matrix. For some orientation of the quarter wave plate Q2 lets say θ1, thetransformation matrix would be given by,

Aθ1 =

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 Sθ1Cθ1(1− Cδ) −Sθ1Sδ

0 Sθ1Cθ1(1− Cδ) S2θ1

+ C2θ1Cδ Cθ1Sδ

0 Sθ1Sδ −Cθ1Sδ Cδ

︸︷︷︸

MRetarder

(3.4)

(symbols have the same meaning as defined earlier). As in most polarimeters as well as herepolarization state analyzer is followed by an intensity based detector, which just records thetotal intensity falling on it, given by the first element or row of the output Stokes vector, andsince,

Sout = Aθ1Sin

only first row of Aθ1 is required for determination of the first element of Sout, given by,

Aθ1 =(

1 −C2θ1− S2

θ1Cδ −Sθ1Cθ1(1− Cδ) Sθ1Sδ

)(3.5)

Just as in PSG we generated four input polarization states here we need to analyze four polar-ization states resulting in 16 measurements of each generated state (total four in number) withall configurations of the analyzer. Clubbing the four configurations of the polarization stateanalyzer for four orientation angles (lets say θ1, θ2, θ3 & θ4) of the retarder (Q2) we have,

A =

1 −C2

θ1− S2


1 −C2θ2− S2


1 −C2θ3− S2


1 −C2θ4− S2


(3.6)

The construction of the polarization state generator (W ) and analyzer (A) matrices is nowcomplete. The exact configurations of both, though needs optimization so as to yield bestpossible outcomes. From W (PSG) and A (PSA) matrices, Mueller matrix for the unknownsample Ms is calculated as follows. We know that,

M = A×Ms ×W

where, M is a matrix formed by the intensity measurements for the sample S with Muellermatrix Ms. Hence, if we define,

Q16×16 = A4×4 ⊗W T4×4 (3.7)

18

=⇒ (Ms)16×1 = Q−116×16M16×1 (3.8)

(Ms)16×1 is then reshaped back to the 4× 4 matrix.

3.2 Optimization of Generator and Aanalyzer

As has been shown in the above sections the constructions of the polarization state generator andanalyzer is very crucial in this measurement strategy. To determine their exact configurationswe need to choose four polarization states for PSG and four for PSA by fixing the orientationangles of the two rotatable retarders (quarter waveplates here) (Q1 andQ2). Choosing the anglesof the retarder Q1 belonging to PSG would essentially fix the four Stokes vectors, ellipticallypolarized in this case, incident on the sample. Similarly for four angles of Q2 the ellipticallypolarized states that would be detected (only total intensity is recorded which correspondsto the first element S0 of the Stokes vector) get fixed. One basic criteria for choosing thesepolarization states could be to maximize the determinant of the 16 × 16 matrix Q defined byequation 3.7 and to make certain that it is strictly positive, since matrix Q has to be invertibleotherwise the samples Mueller matrix Ms could not be calculated using 3.8. And since, here themeasurements are done over a spectral range, λ = 400 − 800 nm, the optics constituting PSGand PSA is very likely to change its polarization characteristics with the wavelength. The 633nm quarter waveplate does not behave like a quarter waveplate at other wavelengths as is shownin Fig. 5.9. To be on the safer side the determinant of the matrix Q is maximized because Qmay change with wavelength for a particular configuration of the PSG and PSA matrices suchthat its determinant approaches zero at certain other wavelength and this dependency of thematrix Q or of PSG and PSA matrices is not known a-priori.

The orientation angles for the two quarter waveplates chosen by maximizing the determinantof the matrix Q were 35◦, 70◦, 105◦ and 140◦. For simplicity these angles for the two quarterwaveplates were kept same and close to the round figure.

The above chosen angles were also verified using a more rigorous approach of singular valuedecomposition and a quantity named condition number was defined as follows,

Condition No. =min {singular values}max {singular values}

(3.9)

These singular values must be positive otherwise the resulting Mueller matrix from these cal-culations will not be physically realisable [24, 17, 25]. The variation of condition number withwavelength for the chosen configuration of PSG and PSA matrices is shown in Fig. 5.3. Thefreedom that the group of four Stokes vectors constituting PSG and PSA can be rotated so asto cover or span all the points on the Poincare sphere because of the fact that any polarizationstate or configuration of the polarizer P1 can be arbitrarily chosen and called Horizontal andorientations of rest of the optics (P2, Q1 and Q2) is with respect to this polarizer P1. Thisenable the rotation of these stokes vectors together on the Poincare sphere. For illustrationthese Stokes vectors are shown in Fig. 3.1 on the Poincare sphere.

19

x y

z

1

Figure 3.1: Poincare sphere showing Stokes vectors constituting PSG (red) and PSA (blue)matrices assuming the ideal nature of both of them. x, y and z axis are the three elements S1,S2 and S3 of the Stokes vector respectively as defined in section 2.1.

3.3 Experimental Setup

A schematic of the experimental setup is as shown in Fig. 3.2. It consists of a 50 W OceanOptics Xenon lamp, two wide-band linear polarizers P1 and P2; P1’s polarization axis is fixedat any arbitrary orientation angle and P2 is crossed with P1, two 633 nm Thorlabs quarterwaveplates (Q1 & Q2) mounted on computer controlled rotational stages and an Ocean OpticsUSB4000 fibre-optic spectrometer used as a detector for the spectral measurements. The wholesetup rests on a goniometer which can be used for studies based on angular dependence ofscattering [2].

Figure 3.2: Schematic of the experimental setup for elastic scattering based Mueller matrixmeasurements. Key: L1: Lens, A: Aperture, P1: Fixed linear polarizers with polarizationaxis oriented at any arbitrary angle (calling this polarization state as H), Q1 & Q2: Rotatablequarter waveplates (orientation angles are with respect to P1), S: Sample with unknown Muellermatrix, P2: Fixed linear polarizer (acting as analyzer), polarization axis crossed with polarizerP1.

The sample S is mounted in between the polarization state generator and analyzer arrange-ment and for calibration purposes a wide-band Glan Thompson linear polarizer and a singlewavelength quarter wave retarder was used. The setup was completely automated using Lab-view with the aid of the VI’s and SDK’s provided by the manufacturers of these instruments.The front panel and block diagram for the automation VI are shown in appendix C. Obtainedresults are presented in the following chapters.

20

3.4 Advantages of this strategy

Following are some of the main advantages of this measurement strategy over other commonlyused approaches listed in section 2.2:

1. Independent of source and detector polarization responses: Even if the source does not emitcompletely un-polarized light and detectors like spectrometer uses a grating to separatevarious wavelengths of the light falling on it, it may have some intrinsic polarizationresponse which need not be corrected separately in this measurement strategy, since thetwo linear polarizers P1 and P2 are always fixed at horizontal and vertical polarizationstates respectively. The polarizer P1 will pass the horizontally polarized component oflight which will always have the same intensity even if the source is not ideal.

2. Simultaneous spectral and imaging measurements: It is capable of recording spectral andspatial mapping or imaging simultaneously by using liquid crystal variable filters togetherwith a charge coupled device (CCD). The ability to generate PSG and PSA matricesthrough Eigenvalue calibration method described in section 4.1 for the entire spectral rangemakes it very efficient in terms of accuracy for spectral Mueller matrix measurements. Theelemental error in the measured Mueller matrices is ∼ 0.01 over the entire spectral rangeas shown in Fig. 5.6.

3. Is completely automated: Since the measurement system is completely automated, it isvery precise and take a very little time (about a few minutes) for recording all sixteenspectral measurements. This also reduces chances of human errors.

21

Chapter 4

Calibration and Decomposition

4.1 Eigenvalue Calibration

The Eigenvalue Calibration Method is the first which uses the matrix formalism for the globalexperimental setup to be calibrated [26]. It is capable of determining the polarization stategenerator (PSG) and analyzer (PSA) matrices over the entire spectral range thus automaticallycorrecting for the non-ideal behaviour and wavelength dependence of the optics comprisinggenerator and analyzer [27, 28, 29]. The only requirement of this method is to take a few extrameasurements for the chosen reference or calibration samples. The form of the Mueller matricesfor these reference samples must be known a-priori. Its wavelength dependence may or may notbe known. A brief description [26] of this method follows.

4.1.1 Mathematical Formulation

Let us choose our calibration (or reference) sample to be a diattenuating retarder just for thepurpose of illustration, in a more general case any optical element whose form of the Muellermatrix is known can be used for calibration purpose, and call the set of measurements (refers tothe matrix formed by the sixteen intensity-based measurements for different polarization states)as B. The eigenvalue calibration method requires another set of measurements for blank (nospecific sample, hence its Mueller matrix is a 4× 4 identity matrix), and let us denote it by B0.These measurements can be expressed in terms of polarization state generator (PSG) matrix(W ), polarization state analyzer (PSA) matrix (A) (both unknown at this point) and Muellermatrix of the sample (M), whose form is only known (its spectral variation need not be known)as follows.

B0 = AW, B = AMW (4.1)

since Mueller matrix for blank is identity. Another set of matrices C and C ′ is constructed suchthat one of them is independent of the PSA matrix A and other is independent of PSG matrixW .

C = B−10 B = W−1MW, C ′ = BB−1

0 = AMA−1 (4.2)

Clearly, matrices C, M and C ′ form a group of similar matrices and thus have same eigenvalues.The eigenvalues of the matrix C (calculated from measurements B and B0) are computed andare equated to the eigenvalues of the Mueller matrix of the calibration sample M . This canalways be done since the form of the Mueller matrix for the calibration sample is known (oneof the requirements of this method). Here, a diattenuating retarder is chosen as one of thereference samples whose Mueller matrix is known to have 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 ∆

(4.3)

22

where, ψ and ∆ are the ellipsometric parameters for the diattenuating retarder and τ is itstransmittance. Matrix M has four eigenvalues, two of which are real (λR1 , λR2) and twocomplex (λC1 , λC2).

λR1 = 2τ cos2 ψ

λR2 = 2τ sin2 ψ

λC1 = τ sin 2ψe−i∆

λC2 = τ sin 2ψei∆

(4.4)

Solving for the parameters, ψ, ∆, and τ , we have,

τ =λR1 + λR2

2, ψ = tan−1

√λR1

λR2

, ∆ = log

√λC2

λC1

(4.5)

From these measured values of the ellipsometric parameters Mueller matrix M for the referencesample (diattenuating retarder) is constructed using (4.3). If more than one reference sam-ples are taken for calibration their individual Mueller matrices for various wavelengths can beconstructed like this.

Consider the following matrix equations,

MX −XC = 0, MX ′ −X ′C ′ = 0 (4.6)

After calculating C, C ′ and M from the measurements as described above, (4.6) with solutions:X = W and X ′ = A since, (4.2) can be solved using the least square method[26] as describedbelow. The PSG matrix W4×4 is written as a vector W16×1 = {wn}n=1...16 in a basis set{Gn}n=1...16 such that,

W4×4 =16∑n=1

wnGn (4.7)

W4×4 =

w1 w2 w3 w4

w5 w6 w7 w8

w9 w10 w11 w12

w13 w14 w15 w16

=16∑n=1

wnGn (4.8)

where,

G =

1 0 0 00 0 0 00 0 0 00 0 0 0

,

0 1 0 00 0 0 00 0 0 00 0 0 0

,

0 0 1 00 0 0 00 0 0 00 0 0 0

, · · · · · · ,

0 0 0 00 0 0 00 0 0 00 0 0 1

T

(4.9)In this basis the mapping H(X) = MX−XC, hence H(W ) = 0 acts on vectors with 16 elements,and is represented by a 16× 16 real matrix H, so that the following applies:

H(W16×1) = 0 (4.10)

The matrix H is calculated in the following way:

(H16×16)ij ={

(H(Gi))F}j

(4.11)

where F denotes the flattening operation,1 2 3 45 6 7 89 10 11 1213 14 15 16

F

=(

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

(4.12)

23

The least-squares solution of equation 4.10 is given by the relation:

KW16×1 = 0

where,K = HTH

The 16× 16 matrix K is diagonizable, and because W16×1 is the only solution of HW16×1 = 0,W16×1 is the unique eigenvector associated with the null eigenvalue. In practice H(W ) is notexactly zero because of experimental errors, but still the following applies to the eigenvalues (λ)of K:

0 ≈ λ1 << λ2 < λ3 < λ4 < . . . < λ16

It should be noted here that an unknown misalignment of the various reference samples causesan increase of λ1/λ16. The minimum value of this ratio corresponds to the case that the appliedMueller matrices of the reference samples are exactly the Mueller matrices of the used referencesamples. It follows now that W16×1 is the eigenvector associated with the smallest eigenvalueof K. The vector W16×1 is written back as a 4× 4 matrix W in accordance with equation 4.12.From W , A is calculated as A = B0W

−1 where, B0 is the measurement matrix for blank.The results for PSG (W ) and PSA (A) matrices for the entire concerned spectral range

(λ = 500− 700 nm) are shown in Figs. 5.1 and 5.2 respectively.

4.1.2 Limitations and Advantages

The Eigenvalue calibration method described in above few sections plays a major role in theproposed measurement strategy. For most calibration methods the choice of a proper calibrationsample is very crucial since most of the results depend on a few extra measurements recordedfor these calibration samples. However, for eigenvalue calibration method this choice is veryeasy to make. The only requirement of this procedure is that the form of the Mueller matrixfor the chosen reference sample(s) must be know a-priori. Its wavelength dependence may ormay not be known. Through this calibration technique one can automatically correct for theerrors [30, 31, 32] due to non-ideal behaviour of the optics over the spectral range constitutingpolarization state generator (PSG) and analyzer (PSA) arrangements. Small deviations ormisalignments in the orientations or configurations of the chosen reference samples are alsotaken care off by the Eigenvalue calibration method since these values are calculated from themeasurments using the eigenvalues of the matrix C or C ′ before reconstructing the Muellermatrix for these reference samples and are not assumed to be those set by hand.

4.2 Polar Decomposition

In polar decomposition method, a given 4× 4 Mueller matrix M for a medium is decomposedor written as a product of three 4× 4 matrices [33],

M = M∆MRMD (4.13)

where the depolarizer matrix M∆ denotes the depolarization effects both linear and circular ofthe medium/sample, retardance matrix MR takes care of linear and circular (or optical rotation)retardance effects and MD called the diattenuation matrix includes the diattenuation effects.For optically clear media, the validity of this method was first given by Lu and Chipman[33]. Some basic polarization properties of the medium/sample, namely, diattenuation (d)(differential attenuation of orthogonal polarizations, both linear and circular), depolarizationcoefficients (∆, linear and circular), linear retardance (δ) (difference in phase between the twoorthogonal linear polarizations), and circular retardance (ψ) (difference in phase between rightand left circularly polarized light) are then derived from these decomposed matrices [4] asdescribed in the following section.

24

4.2.1 Mathematical Formulation

The diattenuation matrix MD is defined as [33, 4],

MD =

[1 ~dT

~d mD

](4.14)

where mD is a 3× 3 sub-matrix with the standard form,

mD =√

(a− d2)I + (1−√

1− d2)ddT

here, I is a 3× 3 identity matrix, ~d is the diattenuation vector with d its unit vector defined as,

~d =1

M11

[M12 M13 M14

]Tand d =

~d

|~d|

The magnitude of the diattenuation vector is given by,

|~d| = 1

M11

√M2

12 +M213 +M2

14

The coefficients M12 and M13 represents the linear diattenuation along horizontal (vertical) and+45 (−45) linear polarizations respectively and M14 represents circular diattenuation. Now,from (4.13) we have,

M∆MR = MM−1D = M ′ (4.15)

The matrices MR, M∆ and M ′ have the following form,

M∆ =

[1 ~0T

P∆ m∆

], MR =

[1 ~0T

~0 mR

]and M ′ =

[1 ~0T

P∆ m′

](4.16)

where,

P∆ =~P −m~d

1− d2

and polarization vector,

~P =1

M11

[M21 M31 M41

]Tand m′ is a 3× 3 submatrix of M ′, which can be written as,

m′ = m∆mR (4.17)

The depolarization ∆ can be calculated from the M∆ matrix as [33, 4],

∆ =1

3|Tr (M∆)− 1| (4.18)

And, now from (4.17) we have,mR = m−1

∆ m′

From (4.16) retardance matrix MR can be calculated and the value of total retardance is givenby the following,

R = cos−1

{Tr(MR)

2− 1

}(4.19)

The values for optical rotation ψ and linear retardance δ can also be determined from the matrixMR as follows,

δ = cos−1

(√[(MR)22 + (MR)33]2 + [(MR)32 − (MR)23]2 − 1

)(4.20)

25

ψ = tan−1

((MR)32 − (MR)23

(MR)22 + (MR)33

)(4.21)

Due to the non-commuting nature of matrix multiplication, six different combinations for de-composition of the Mueller matrix are possible. It has been shown that among these six decom-position permutations, the one given by (4.13) or its reverse order (M = MDMRM∆) alwaysproduces physically realizable matrices [34].

26

Chapter 5

Results and Discussion

Some of the important results including those of eigenvalue calibration method and Muellermatrix measurements are shown in the following sections.

5.1 Calibration Results

As mentioned earlier in section 4.1 that the optical elements comprising the polarization stategenerator and analyzer do not behave ideally and also depend on the wavelength of light. Toincorporate this behaviour of polarizers and quarter waveplates the setup was calibrated usingthe Eigenvalue calibration method. From a few extra measurements using this calibration

Figure 5.1: Variation of individual elements of Polarization State Generator Matrix (W ) as afunction of wavelength (λ).

procedure the actual PSA and PSG matrices could be determined for the entire spectral range(λ = 500 − 700 nm). Fig 5.1 shows the variation of individual elements of the polarizationstate generator (or W matrix) with wavelength. Each column of this matrix represents a Stokesvector corresponding to the optimized orientation angles of the quarter waveplate Q1 which are

27

35◦, 70◦, 105◦ and 140◦. For example first column represents the normalized Stokes vector asconstructed in subsection 3.1.1 for θ1 = 35◦ and so on.

In Fig. 5.2 individual elements of polarization state analyzer (A) matrix are shown as afunction of wavelength (λ) from 500 − 700 nm. Here as constructed in subsection 3.1.2 eachrow represent the top row of the combined Mueller matrix of the optical elements forming PSA.Please note that unlike PSG rows or columns of PSA can not be called the Stokes vectorsdirectly. Rows when multiplied by the incident Stokes vector on the PSA setup gives the totalintensity i.e. first element of the output Stokes vector.

Figure 5.2: Variation of individual elements of Polarization State Analyzer Matrix (A) as afunction of wavelength (λ).

Figure 5.3: Condition number for polarization state generator matrix (W ) and polarizationstate analyzer matrix (A) calculated for the spectral region (λ = 500− 700 nm).

Condition number defined by equation 3.9 is plotted in Fig. 5.3 for both polarization stategenerator and analyzer matrices as a function of wavelength (λ = 500− 700 nm). It should be

28

noted that the value of condition number for both PSA and PSG over the entire spectral rangeis strictly positive (greater than 0.25). This ensures that the chosen PSA and PSG matricesfor the entire spectral range would yield physically realizable Mueller matrices for the chosensample.

Figure 5.4: An estimate of the error in determination of the PSG and PSA matrices for thesame spectral region.

Fig. 5.4 shows an estimate of the error in determination of the PSG and PSA matrices usingthe Eigenvalue calibration method which may have incurred due to a possible misalignment ofvarious reference samples. As stated earlier in subsection 4.1.1 a small value of the ratio ofthe smallest to the largest eigenvalue of the matrix K16×16 corresponds to the case when theapplied (assumed for calculation) Mueller matrices of the reference samples are exactly theMueller matrices of the used reference samples. The observed value of this ratio for the entirespectral range is plotted in Fig. 5.4 and is of the order of 10−3. It is very well known that the

Figure 5.5: Variation of the individual elements of Mueller matrix for blank (no sample) overthe spectral region (λ = 500− 700 nm). Ideally Mueller matrix for blank should be identity.

29

Mueller matrix for a nonpolarizing element is a 4× 4 identity matrix. So for blank, where thesample is absent the Mueller matrix should ideally be an identity. Fig. 5.5 shows individualelements of the Mueller matrix for blank obtained using the proposed strategy for the spectralrange 500− 700 nm. Diagonal elements are close to unity and rest are close to zero.

Figure 5.6: Null (zero) elements of the Mueller matrix for a quarter waveplate, fast axis orientedat an angle of 23◦ for the spectral range λ = 500− 700 nm. Clearly the elemental error in themeasure Mueller matrices is of the order of 0.01.

One of the best ways to see how accurate these measurements are in determining the Muellermatrices for various optical elements is to plot the null (or zero) elements for that particularelement. Fig. 5.6 shows the null elements of the Mueller matrix for a quarter wave retarder,fast axis oriented at an angle of 23◦ from the horizontal of polarizer P1 over the spectral range.From Table 2.1 it can be seen that M12, M13, M14, M21, M31 and M41 are the null elementsfor all retarders in general and are shown.

30

5.2 Polar Decomposition Results

Figure 5.7: Polar decomposition derived polarization parameters namely depolarization (∆),diattenuation (d) and retardance (R) for blank Mueller matrix over the spectral region (λ =500− 700 nm). Expected values of all three parameters for blank was zero.

The measured Mueller matrices for different samples were decomposed using Polar decom-position method described in section 4.2 to get the basic polarization parameters namely depo-larization ∆, diattenuation d, and retardance R along with their linear and circular components.Fig. 5.7 shows the values of these parameters for the blank Mueller matrix (Fig. 5.5). Ideallyfor blank all these parameters should be zero. Polar decomposition derived diattenuation for

Figure 5.8: Polar decomposition derived diattenuation (d) for a wide-band Glan-Thompsonlinear polarizer with polarization axis oriented at an angle of 28◦ over the spectral region(λ = 500− 700 nm). Expected value of diattenuation (d) was one.

a wide-band Glan Thompson linear polarizer with polarization axis aligned at an angle of 28◦

with respect to horizontal of polarizer P1 in the spectral range λ = 500 − 700 nm is shownin Fig. 5.8. Since an ideal linear polarizer acts like a perfect diattenuator its diattenuation is

31

expected to be close to unity. Fig. 5.9 shows polar decomposition derived linear retardance (δ)

Figure 5.9: Polar decomposition derived linear retardance (δ) for a 632.8 nm quarter waveplatewith fast-axis oriented at an angle of 23◦ over the spectral region (λ = 500−700 nm). Expectedvalue of linear retardance was π/2 or 1.57 radians at λ = 632.8 nm.

for a single wavelength quarter wave retarder with fast axis oriented at an angle of 23◦ withrespect to horizontal of polarizer P1 in the spectral range λ = 500− 700 nm. Since this is a 633nm quarter waveplate we would expect that at λ = 633 nm the value of the linear retardanceshould be close to π/2 radians. The observed value at λ = 632.8 nm is 1.54 radians. Also theneed for calibration specially in spectral measurements is quite clear from this figure.

32

Chapter 6

Applications towards TissueCharacterization

6.1 Introduction

Several studies conducted in the past decade have established the potential of laser inducedfluorescence technique for diagnosis of cancer [12]. However, the use of polarized fluorescencespectroscopy for cancer diagnosis has only been explored recently [35]. These studies demon-strated that polarized fluorescence spectroscopic measurements have important advantages overunpolarized fluorescence measurement for cancer diagnosis [36, 35]. For example, a complica-tion often encountered in using static autofluorescence spectra from tissue for cancer diagnosisis the large site to site variability in both intensity and line-shape of the fluorescence spec-tra of sites belonging to the same class (normal or cancerous). Moreover, for epithelial tissuethe contrast in autofluorescence from cancerous and non-cancerous sites is known to stronglydepend on the differences in depth distribution of endogenous fluorophores in the superficialepithelial and the underlying connective tissue layer. Studies have demonstrated that that theuse of polarized fluorescence instead of conventional unpolarized fluorescence can address boththese issues. It was observed that the effect of wavelength dependent absorption in tissue (forexample by blood) is reduced in polarized fluorescence spectra thereby leading to reduced site-to-site variability in intensity and line shape as compared to the unpolarized fluorescence. Thisfollows because the polarized fraction of the total fluorescence comes only from a few transportscattering lengths and therefore bears minimal signature of absorption and scattering propertiesof tissue. For the same reason, fluorescence emitted from deeper layers of tissue is expectedto get more depolarized due to multiple scattering as compared to that emitted from superfi-cial layers. Measurement of fluorescence polarized at varying angles with respect to excitationlinear polarization was therefore used to accomplish depth resolved fluorescence measurementsin tissue. As compared to the polarimetric technique used in these studies, measurement ofMueller matrix of fluorescence emitted from tissue would allow one to obtain more completeinformation on polarization properties of fluorescence. This therefore might turn out to be anefficient approach to fully exploit the advantages of polarized fluorescence measurements anddesign the optimal polarization scheme for quantitative tissue diagnosis.

6.2 Experimental Setup

A schematic of the experimental setup for in-elastic (fluorescence) scattering with simultane-ous imaging and spectroscipic Mueller matrix measurements designed especially for biomedicalapplications is as shown in Fig. 6.1. The three light sources, 405 nm & 473 nm lasers for ex-citing the fluorophores in tissues and other samples and a Xenon lamp required for Eigenvalue

33

Figure 6.1: Schematic of the experimental setup for in-elastic (fluorescence) scattering withsimultaneous imaging and spectroscopic Mueller matrix measurements. Key: L: Lens, A:Aperture, BS: Beam splitter, NDF : Neutral density filter, GT : Wide-band Glan Thompsonlinear polarizer, QWP : Quarter waveplate, MO: Mounting optics, F : Filter, LCV F : Liquidcrystal variable filter.

calibration of the setup. The laser soucres can be changed according to the requirements of theexperiment. A 450 nm filter F is used with 405 nm laser after the collecting lens L4 to cutoff the incident 405 nm radiations reflected from the tissue or glass slide surface and only thefluorescence signal was recorded using an Andor CCD spectrograph. The PSG and PSA havesimilar arrangement as was for the elastic scattering Mueller matrix measurement setup Fig.3.2.

6.3 Preliminary Results

As mentioned earlier we are mainly interested in probing one of the endogenous fluorophores intissues namely, Collagen which has been found to undergo structural and biochemical changesduring the progression of a disease like Cancer. But since the absoption and emission spectraof Collagen overlaps to a huge extent with that of NADH it was not always possible distinguishthe cancerous tissues from healthy one by just looking at the fluorescence spectra or yield. Inthe fluorescence spectra thus we expect to see two close emission peaks corresponding to thesefluorophores.

In Fig. 6.2 first element (M11) of the fluorescence Mueller matrix for human cervical cancertissue biopsy slide (tissue thickness ∼ few µm) over the entire spectral range (λ = 450 − 800nm) is shown. Clearly two broad convoluted emission peaks can be seen which correspond tothe emission spectrum of Collagen and NADH as is indicated in the figure.

Individual elements of the fluorescence Mueller matrix (normalized with respect to M11)

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Figure 6.2: First element of fluorescence Mueller matrix M11 (un-normalized) for human cervicalcancer tissue biopsy slide (thickness ∼ few µm) over the spectral region (λ = 450 − 800 nm).A bandpass filter (450 − 900 nm) was used to cut off the incident wavelength 405 nm usedof exciting the tissue sample. The fluorescence emission peaks of Collagen and NADH areindicated.

for the same human cervical tissue biopsy slide are shown in Fig. 6.3 for the spectral region(λ = 475− 800 nm).

Fig. 6.4 shows the behaviour of diattenuation obtained from the measured fluorescenceMueller matrix for the human cervical cancer biopsy slides a few microns thick over the indicatedspectral range for two different grades (stages) of cancer. This gradation was done on the basisof the histo-pathology report. Grade 1 indicates the onset of cancer while Grade 3 refers toa more severe stage. It should be noted here that diattenuation peaks at a wavelength whichcoincides with the peak of the emission spectra of Collagen indicated in Fig. 6.2. Collagendue to its fibrous structure is expected to give higher diattenuation as compared to that due toNADH with an isotropic structure.

It should be noted that the results shown in this section are based on just initial studiesand shows that such polarization based fluorescence Mueller matrix polarimetry has potentialto be used for tissue characterization and diagnostic applications.

35

Figure 6.3: Individual elements of fluorescence Mueller matrix for human cervical cancer tissuebiopsy slide (thickness ∼ few µm) over the spectral region (λ = 475− 800 nm).

Figure 6.4: Polar decomposition derived diattenuation (d) for human cervical cancer tissuebiopsy slide (thickness ∼ few µm) over the spectral region (λ = 450− 800 nm) for two differentgrades (stages) of cancer determined from histo-pathology report of the sample.

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Chapter 7

Applications towardsNanoplasmonics

7.1 Introduction

Studies on optical properties of noble metal nanoparticles and nanostructures are of considerablecurrent interest both from the point of fundamental understanding and potential applications[13, 37, 38]. These attribute to their unique optical properties which are governed by the col-lective oscillations of the free (conduction) electrons, the so-called surface plasmon polaritons,in resonance with the incident electromagnetic (EM) field, known as surface plasmon resonance[13]. The surface plasmon can either be propagating, for instance at planar metal-dielectric in-terfaces, or localized as in the case of metal nanoparticles. In both cases, this leads to stronglyenhanced and highly localized electromagnetic fields. The surface plasmon resonance in metalnanoparticles and nanostructures are being pursued for numerous practical applications in var-ious branches of science and technology. Ultra-high sensitive chemical and biomedical sensing,bio-molecular manipulation, labeling, detection, contrast enhancement in biomedical imaging,surface enhanced spectroscopy (Raman and fluorescence), development of new generation opti-cal devices including plasmonic wave guiding nano-devices, optical information processing anddata storage, are just a smattering of its diversified uses [37, 39, 40, 41, 42]. In addition to thepotential applications, basic studies on various optical properties of the metal nanoparticles arealso important in context to fundamental understanding of light-matter interaction.

Although extinction, absorption and scattering are still the primary optical properties of in-terest, various other approaches are also brought to bear on these metal nanoparticles, includingSurface Enhanced Raman Scattering (SERS), a variety of non-linear scattering measurements(hyper Rayleigh, hyper Raman and second harmonic generation etc.) and time resolved mea-surements [39, 40, 43, 44, 45]. In context to the investigation and applications of the above-mentioned optical properties, synthesis and characterization of metal nanoparticles having dif-ferent controllable size and shapes, have drawn considerable attention [46, 47]. This followsbecause control of the size and shapes of these nanoparticles facilitates tuning of the plasmonresonance bands to desirable wavelengths for specific applications [46]. Moreover, such controlover size and shapes can also enhance and optimize other potentially applicable optical effectssuch as the local field enhancement, giant non-linear responses, wave guiding properties etc. [47].The measurement on the resulting optical properties of the nanoparticles (or nano-structures)are conventionally done using a variety of experimental tools, Rayleigh scattering measurementsin dark field microscope, confocal microscopic imaging (and spectroscopy), near field scanningoptical microscopy (NSOM), are to name a few [48, 49]. These techniques have been usedto conduct both spectroscopic and imaging studies on nano-structures, single nanoparticles toensemble of particles. Analogously, the EM interaction of light with such nanoparticles andstructures have been modeled using various numerical methods, such as the discrete dipole ap-

37

proximation (DDA), the T-matrix method, finite difference time domain (FDTD) simulations,finite element calculations, the multiple multipole method etc. [50, 51].

The polarization properties of light play an important role in the interaction of light withsuch non-spherical metal nanoparticles [52]. Thus, knowledge on the polarization properties ofthe incident and scattered light and their spectral (wavelength dependent) characteristics arecrucial for both fundamental understanding of the interactions and for optimizing experimentalparameters for many practical applications. Indeed, recent experimental studies have suggestedthat recording the changes in the polarization emission pattern (spectral and angular) are usefulfor instance, in in-situ monitoring and controlling of size and shape of nanoparticles duringsynthesis, improving schemes for plasmonic sensing, single molecule detection and importantlyfor enhancing contrast in biomedical imaging [47, 52, 53, 54].

As noted previously, specially engineered non-spherical metal nanoparticles are actively be-ing pursued as contrast agents in in-vitro and in-vivo biomedical imaging (diagnosis and therapy)employing optical imaging techniques such as optical coherence tomography (OCT), photo-acoustic tomography, non-linear microscopy (two photon and second harmonic generation) anddark-field microscopy etc. [55, 56]. A major difficulty encountered in such nanoparticle-basedbiomedical imaging is the fact that the strong scattering from the biological (tissue or cell) dielec-tric structures often swamps the scattering from the nanoparticles, thus limiting detectability ofnanoparticles in such media. Polarization control may play an important role in this regard forisolating scattering of plasmonic nanoparticles from the background Rayleigh / Mie scatteringof biological dielectric structures and thereby enhancing contrast in nanoparticle-based biomed-ical imaging. In previous studies exploring this possibility, the polarization information hasusually been measured or modelled in terms of the conventionally defined degree of polarizationof light (or polarization anisotropy) [52, 53, 54]. The polarization information is obtained eitherby performing measurements of co-polarized and crossed-polarized light intensity with respectto a given input polarization state of incident light, or by recording selected Stokes vector ele-ments (4 × 1 intensity vector describing the polarization state of light [13, 21]. Note however,that for scattering medium (or particles) exhibiting simultaneous several polarization effects(as should be the case for complex nano-bio composites where the most common polarimetryevents are depolarization, retardance and diattenuation, arising from both the metal nanopar-ticles and the biological background; these are defined afterwards), the constituent scatteringand the polarization effects contribute in a complex interrelated way to the measured degree ofpolarization or the selected Stokes vector elements of light [4, 5, 8]. These therefore representseveral lumped effects. Scattering from even optically inactive scatterers can exhibit complexscattering-induced diattenuation and retardance effects, in addition to depolarization [5, 8]. Ex-traction, quantification and unique interpretation of the individual, intrinsic polarimetry effectsof the non-spherical metal nanoparticles (or nano-structures) are thus expected to provide bet-ter insight into the polarizing interaction of light with such nanoparticles, which in turn can beexploited for contrast enhancement in nanoparticle-based biomedical imaging. Polar decompo-sition of Mueller matrix (a 4×4 matrix that describes the transfer function of any medium in itsinteraction with polarized light, and contains complete information about all the polarizationproperties of a medium [13, 21] is one such approach that facilitates extraction and quantifi-cation of the individual intrinsic polarimetry characteristics from any complex system [33, 5].To the best of our knowledge, use of such inverse polarimetry analysis methods on scatteringMueller matrices from plasmon resonant metal nanoparticles, have not been investigated before.We have thus theoretically investigated possibility (and plausible advantages) of such quanti-tative plasmon polarimetry by employing polar decomposition method on scattering Muellermatrices from non-spherical (as representative of non-sphericity, we have chosen the spheroidalshape) metal (silver) nanoparticles. The scattering Mueller matrices for the spheroidal silvernanoparticles in their surface plasmon resonance bands were generated using T-matrix methodfor light scattering [14, 57] and these were subjected to the polar decomposition approach for

38

quantitative polarimetry analysis. The analysis indeed revealed interesting spectral (wavelengthdependent) and angular behaviour of the constituent polarimetry characteristics of the metalnanoparticles, which were distinctly different from similar dielectric particles. The details ofthese results are presented in the following sections and their implications for contrast enhance-ment in nanoparticle-based biomedical imaging are discussed.

7.2 Theory

Mie theory provides the exact solution of scattering of electromagnetic plane wave by a sphericalscatterer [13]. In the framework of Mie theory, the Stokes vector of the incident (Si) and thescattered (So) light can be related through the so-called scattering matrix (M(θ)) as,

So = M(θ)Si (7.1)

where θ is the scattering angle, the Stokes vectors (Si, So) and M(θ) are defined in the scatteringplane (plane containing the incident and the scattered light). For spherically symmetric particle,the scattering matrix has a very simple block diagonal structure with only four independentelements [13, 14],

M(θ) =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 −M34 M33

(7.2)

These elements for known spherical scatterers (with known radius r, and refractive index ns)can be computed using Mie theory embedded inside a surrounding medium (refractive indexnm) [13].

For non-spherical scatterers having arbitrary shapes, the form of scattering Mueller matrixM(θ) is however, far more complex, essentially having non-zero values for all the matrix el-ements [13, 57, 14]. As noted previously, there do exist several methods for computing lightscattering from such non-spherical particles [14]. Among these, the T-matrix approach is one ofthe most powerful and widely used tools [57, 58, 59]. This approach, based on directly solvingMaxwell’s equations, provides accurate expressions of the scattering parameters for both singleand composite non-spherical particles. Although the method is potentially applicable to anyparticle shapes, most practical implementations of this technique pertain to particles havingrotationally symmetric shapes [57, 58]. Thus T-matrix code developed by Mishchenko et al.[58] were used to compute the scattering Mueller matrices M(θ) for spheroidal nanoparticles(both preferentially and randomly oriented) investigated in this study. Note that for any axi-ally symmetric non-spherical particle, the co-ordinates of the particle reference frame and thelaboratory reference frame may not coincide in general. However, for conceptual and practicalreasons (discussed later), the spheroids were preferentially oriented such that the particle refer-ence frame coincides with the laboratory reference frame. In this case, the computed scatteringmatrix (for any scattering angle θ) relates the Stokes vector of scattered light to that of theincident light defined with respect to the scattering plane (as in (7.1)). For computing thescattering matrix for randomly oriented spheroidal particles, orientation averaging was carriedout as described in reference [58].

The scattering matrices for both randomly and preferentially oriented spheroidal nanoparti-cles were generated for varying sizes and shapes. The sizes and the shapes of the spheroids werespecified by the radius of equal surface area sphere (r) and aspect ratio (ε) respectively [57, 58].Note that values of ε > 1 and ε < 1 correspond to oblate and prolate spheroids respectively.Computation of the scattering matrices [M(θ, λ), λ is the wavelength] were performed for slivernanoparticles in their surface plasmon resonance spectral region (λ = 325 − 525 nm). Theoptical properties of silver (real and imaginary part of refractive index; ns and ks) are avail-able in literature [60]. The refractive index of the surrounding medium was kept nm = 1.33.

39

The scattering matrices over the same spectral region were generated for spheroidal dielectricnanoparticles also, for varying sizes and shapes. The refractive indices of the dielectric particleswere assumed to be constant over the spectral region (ns = 1.59 and ks = 0).

7.3 Results and Discussion

The scattering Mueller matrix computed for randomly oriented spheroidal nanoparticles usingT-matrix method (as discussed in section 7.2) for both silver nanoparticles, in their plasmonresonance spectral region (λ = 325 − 525 nm) and for dielectric particles having identical sizeand shape were decomposed using Polar decomposition method described in section 4.2 to yieldindividual polarimetry characteristics. The matrices were computed for varying radius (r) andaspect ratios (ε) of the spheroidal nanoparticles at varying scattering angles (θ = 0◦ − 180◦).Table 7.1 gives an example of the decomposition of the scattering matrix (M) for randomlyoriented spheroidal silver nanoparticles having radius (of equal surface area sphere) of r = 20 nmand aspect ratio of ε = 1.5 (ε > 1, oblate spheroid) to get the individual polarization parameters;diattenuation (d), net depolarization coefficient (∆) and linear retardance (δ) are listed in thebottom part of the table. The wavelength was chosen to be 400 nm and the scattering angle wasθ = 45◦. The reason for choosing this wavelength is discussed subsequently. Individual values forthe linear and circular depolarization coefficients (∆L and ∆C) are also listed. For comparisionthe decomposition-derived polarization parameters for spheroidal dielectric nanoparticles havingidentical size and shape are also included in the table. As is apparent from the table, thescattering matrix for the silver nanoparticles resembles a diattenuating depolarizer Muellermatrix [33]. A perfect depolarizing Mueller matrix has diagonal form, with its diagonal elementsrepresenting linear and circular depolarization. Additionally, the non-zero value for the M12

element indicates presence of horizontal (vertical) diattenuation effect. Most importantly, forthis scattering angle (θ = 45◦), the net depolarization coefficient ∆ for the silver nanoparticlesis significantly higher than the corresponding value for the similar dielectric particles (for whichdepolarization is negligible). Similar trend was observed for all other scattering angles (althoughat around θ = 90◦, the value of ∆ for the dielectric particles was also considerably high), someof which are presented subsequently. It can also be noted that for the metal nanoparticles,the circular depolarization coefficient is higher compared to the linear depolarization coefficient(∆C > ∆L). This is in agreement with the fact that for Rayleigh scatterers (having sizes muchsmaller compared to the wavelength, r << λ), depolarization of incident circular polarization isstronger compared to the depolarization of incident linear polarization [61, 62]. This differencein relative rate of depolarization of linearly and circularly polarized light can be attributed tothe different mechanism of depolarization of the two. While randomization of the incident fieldvector’s direction (here as a consequence of scattering from randomly oriented non-sphericalparticles) is solely responsible for the depolarization of incident linear polarization state, thedepolarization of circular polarization state is additionally influenced by the randomization ofthe helicity (handed-ness) [61, 62]. Since for Rayleigh scatterers, the latter effect is predominant,depolarization of circularly polarized light is more pronounced for such scatterers [61, 62].

In Fig. 7.1, the variation of the decomposition-derived net depolarization coefficient (∆)as a function wavelength (λ = 325 − 525 nm) for the randomly oriented silver nanoparticleswith radius r = 20 nm is shown for varying aspect ratios of the oblate spheroids (ε = 1 − 2)and the scattering angle was once again chosen to be θ = 45◦. As expected, due to sphericalsymmetry, there is no depolarization (∆ ∼ 0) for shperical particles and the value of ∆ increaseswith increasing aspect ratio of the spheroidal nanoparticles. Interestingly, the depolarizationcharacteristics show distinct spectral features with a peak around λ ∼ 380− 410 nm. The peakof the depolarization curve is also observed to show slight shift towards shorter wavelengths withincreasing ε. Although the magnitude of the decomposition-derived depolarization coefficientswere different at other scattering angles, the observed trends were qualitatively similar to that

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Table 7.1: Top: The T-matrix computed scattering matrix (M) for randomly oriented spheroidalsilver nanoparticles having radius of r = 20 nm and aspect ratio of ε = 1.5. The wavelengthwas 400 nm and the scattering angle was θ = 45◦. The constituent basis matrices obtainedvia decomposition (section 4.2) are shown in the 2nd row. Bottom: The values for the polar-ization parameters (diattenuation (d), net depolarization coefficient (∆) and linear retardance(δ)) extracted from the decomposed matrices. Individual values for the linear and circular de-polarization coefficients (∆L and ∆C) are also listed. The decomposition-derived polarizationparameters of randomly oriented spheroidal dielectric nanoparticles having identical size andshape (3rd column).

1.00 −0.16 0 0−0.16 0.48 0 0

0 0 0.45 −0.010 0 0.01 0.08

1.00 0 0 0−0.08 0.47 0 0

0 0 0.46 00 0 0 0.09

M∆

1.00 0 0 0

0 1.00 0 00 0 1.00 −0.020 0 0.02 1.00

MR

1.00 −0.16 0 0−0.16 1.0 0 0

0 0 0.99 00 0 0 0.99

MD

ParametersEstimated values for the polarization parameters

Spheroidal metal Spheroidal dielectricnanoparticle nanoparticle

d 0.161 0.3315∆ 0.662 0.01∆L 0.536 0.01∆C 0.914 0.01δ (rad.) 0.021 0

for θ = 45◦.In order to understand the origin of the observed spectral characteristics of the depolariza-

tion coefficients, in Fig. 7.2, we show the computed scattering efficiency (Qsca) as a functionof wavelength for the spheroidal silver nanoparticles of varying aspect ratios (ε = 1.25 − 2.0)and having a fixed radius of 20 nm. The observed two distinct peaks in the scattering spec-tra are known to be due to the plasmon resonances corresponding to the longitudinal and thetransverse dipolar polarizabilities [37, 46]. The weaker peak at around 365 nm and the strongerpeak at around 440 nm (e.g., for ε = 1.5) can be identified due to the surface plasmon reso-nance along the short axis (transverse) and the long axis (longitudinal) of the oblate spheroidsrespectively. The relative intensities of the two bands and their spectral positions are controlledby the relative strength of the two orthogonal dipolar plasmon polarizabilities. Since these aredetermined by the aspect ratio, with increasing ε, the two resonance peaks start moving apart(the transverse and the longitudinal one gets blue and red shifted respectively) [37, 13, 46]. Thetwo peaks however, coalesce to a single peak (at ∼ 410 nm) for a sphere. Comparing the spec-tral characteristics of the depolarization coefficient (Fig. 7.1) to that of the plasmon resonancebands (Fig. 7.2), it is evident that the magnitude of ∆ peaks around the overlap spectral regionof the two dipolar plasmon bands. In what follows, the reason for such intriguing depolarizationcharacteristics is investigated by performing polar scattering matrix decomposition analysis onpreferentially oriented spheroidal nanoparticles.

In Table 7.2, the results of the polar decomposition of the scattering matrix for preferen-tially oriented spheroidal silver nanoparticles having r = 20 nm and ε = 1.5 are shown. Thewavelength was chosen to be 400 nm and the scattering angle was θ = 45◦. Several interest-ing features can be noted from the elements of the scattering matrix M . Even at the forward

41

Figure 7.1: The variation of the decomposition-derived net depolarization coefficient (∆) as afunction wavelength (λ = 325− 525 nm) for randomly oriented silver nanoparticles with radiusr = 20 nm. The results are shown for varying aspect ratios of the oblate spheroids (ε = 1− 2)and the scattering angle was chosen to be θ = 45◦.

scattering angle (θ = 45◦), the matrix elements M33 and M44 show negative values which corre-spond to phase reversal. Moreover, the matrix elements M34 and M43 show considerably highvalues corresponding to strong retardation effects [13, 62, 63]. It is well known that for dielectricRayleigh particles, the phase reversal (the so-called flipping of helicity of circularly polarizedlight) occurs beyond a scattering angle of θ > 90◦ [13, 63, 61, 62]. Further, for such Rayleighparticles, the elements M34 and M43 are always ∼ zero [63, 61, 62]. However, for these metalnanoparticles, strong phase retardation effects are evident in the forward scattering angles aswell. Indeed the decomposition analysis yields a value of linear retardance of δ = 1.99 radiansfor this scatterer (δ > π/2 radian corresponds to phase reversal or helicity flipping). As is ap-parent from the decomposition-derived polarization parameters listed in the table (bottom partof Table 7.2), no such scattering-induced retardation effects (δ ∼ 0) are observed for similardielectric particles or for spherical silver nanoparticles. Analogous to the diattenuation effect,the scattering-induced linear retardance arises due to the difference in phase between the scat-tered light polarized parallel and perpendicular to the scattering plane [5, 8]. This effect mayoccur (even in the forward scattering angle) for larger sized Mie scatterers (sizes comparableor larger than wavelength) due to the contribution of higher order multi-poles in the scattering[37, 13, 46, 47]. However, for these metal nanoparticles, the contribution of the higher ordermulti-poles (including the quadrupole) is not expected to be significant. This can also be con-firmed from the scattering spectra shown in Fig. 7.2; the contribution of quadrupole plasmonpolarizability is usually associated with appearance of additional resonance peak at shorterwavelengths [46, 47], which is clearly absent here. The additional scattering-induced linearretardance δ thus appears to originate due to the inherent phase retardation between the twoorthogonal dipolar plasmon polarizabilities (the longitudinal and the transverse dipolar plasmonpolarizabilities oscillating with a phase difference between them). In addition to the observedlinear retardance, the decomposition process also yields significant values for scattering-induceddiattenuation (d) for all the nanoparticles (spheroidal metal and dielectric nanoparticles andspherical metal nanoparticles). In contrast to the results observed for the randomly orientedspheroidal nanoparticles, for these preferentially oriented nanoparticles, the magnitude of d ishowever, higher for the metal nanoparticles compared to its dielectric counter parts. Further, as

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Figure 7.2: The T-matrix computed scattering efficiency (Qsca) as a function of wavelength forthe spheroidal silver nanoparticles of varying aspect ratios (ε = 1.25 − 2.0) and having a fixedradius of 20 nm.

expected, none of these preferentially oriented spheroidal nanoparticles (metal and the dielec-tric nanoparticles) and the spherical metal nanoparticles are observed to exhibit depolarizationeffects (∆ = 0).

Figure 7.3: The spectral variation of the decomposition-derived linear retardance δ(λ) for pref-erentially oriented (see text for details on orientation) spheroidal silver nanoparticles with radiusr = 20 nm and varying aspect ratio ε = 1.0− 2.0. The scattering angle was θ = 45◦.

The decomposition analysis was performed at various wavelengths covering the surface plas-mon band (λ = 325 − 525 nm) of the preferentially oriented spheroidal nanoparticles. Thespectral variation of the decomposition-derived δ(λ) for spheroidal silver nanoparticles withr = 20 nm and varying ε = 1.0 − 2.0 (for θ = 45◦), are shown in Fig. 7.3. The magnitudeof δ(λ) is observed to increase with increasing aspect ratio of the spheroids. Importantly, thespectral variations of δ(λ) for these preferentially oriented spheroidal metal nanoparticles areobserved to be qualitatively similar to the corresponding variations of the depolarization coef-ficient ∆(λ) for the randomly oriented nanoparticles (Fig. 7.1). As for the case of ∆(λ), themagnitude of δ peaks around the overlap spectral region of the two dipolar plasmon bands. The

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Table 7.2: Top: the T-matrix computed scattering matrix (M) for preferentially orientedspheroidal silver nanoparticle having radius of r = 20 nm and aspect ratio of ε = 1.5. Thewavelength was 400 nm and the scattering angle θ = 45◦. The constituent basis matricesobtained via polar decomposition are shown in the 2nd row. Bottom: the values for the polar-ization parameters (d, ∆ and δ) extracted from the decomposed matrices (2nd column). Thedecomposition-derived polarization parameters of preferentially oriented spheroidal dielectricnanoparticle having identical size and shape (3rd column) and spherical silver nanoparticlehaving radius r = 20 nm (4th column).

1.00 −0.43 0 0−0.43 1.00 0 0

0 0 −0.37 −0.820 0 0.82 −0.37

1.00 0 0 00 1.00 0 00 0 1.00 00 0 0 1.00

M∆

1.00 0 0 0

0 1.00 0 00 0 −0.42 −0.910 0 0.82 −0.42

MR

1.00 −0.43 0 0−0.43 1.00 0 0

0 0 0.90 00 0 0 0.90

MD

ParametersEstimated values for the polarization parameters

Spheroidal metal Spheroidal dielectric Spherical metalnanoparticle nanoparticle nanoparticle

d 0.435 0.388 0.333∆ 0 0 0δ (rad.) 1.99 0 0.011

spectral variations of δ exhibit long tails spanning the entire longitudinal and the transverseplasmon resonance bands (as was also observed for ∆(λ)).

The above analysis performed on preferentially oriented spheroidal nanoparticles revealsthat the observed enhanced depolarization (and its intriguing spectral characteristics) for ran-domly oriented spheroidal metal nanoparticles originates from the presence of strong linearretardance effect in the individual oriented nanoparticles (due to the inherent phase differencesbetween the longitudinal and the transverse plasmon polarizabilities). This follows because thescattering Mueller matrix from randomly oriented particles can be thought of being due to theaveraging (over all possible orientation) of the individual Mueller matrices from preferentiallyoriented particles. Since incoherent addition of pure retarder Mueller matrices (from preferen-tially oriented nanoparticles) having random orientation of retardation axes leads to the Muellermatrix corresponding to a pure depolarizer [4, 33], this manifests as strong depolarization (∆)of light for randomly oriented spheroidal metal nanoparticles. The spectral characteristics of∆(λ) for the randomly oriented particles are thus determined by the corresponding spectralcharacteristics of δ(λ) for the preferentially oriented ones.

In order to investigate this intriguing polarization characteristics further, in Fig. 7.4, weshow the scattering angle (θ) dependence of the decomposition-derived value of linear retardanceδ for preferentially oriented spheroidal silver nanoparticles with r = 20 nm and ε = 1.5. Theresults are shown for three different wavelengths; 365 nm, 440 nm (corresponding to the peaks ofthe transverse and the longitudinal plasmon resonances respectively) and 400 nm (correspondingto the peak of δ(λ) and ∆(λ)). The corresponding variation of the similar dielectric particleis shown by solid line in the figure (λ = 400 nm). As discussed previously, the variation ofδ(θ) for the dielectric particle resemble the usual behaviour for Rayleigh particle [63, 61, 62],where one observes a phase reversal (δ > π/2 rad.) beyond a scattering angle of θ > 90◦, andthe magnitude of δ (for any value of θ) is either close to zero or π. In contrast, the trend isreversed for the metal nanoparticles at 400 nm, where δ is greater than π/2 rad. (indicating

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Figure 7.4: The scattering angle (θ) dependence of the decomposition-derived value of linearretardance δ for preferentially oriented spheroidal silver nanoparticle with r = 20 nm andε = 1.5. The results are shown for three different wavelengths: 365 nm, 440 nm (correspondingto the peaks of the transverse and the longitudinal plasmon resonances respectively) and 400nm (corresponding to the peak of δ(λ) and ∆(λ)). The corresponding variation of the similardielectric particle is shown by solid circle in the figure (λ = 400 nm).

phase reversal) in the forward scattering angles (θ < 90◦) and no phase reversal is observed inthe backscattering (δ < π/2 rad. for θ > 90◦). Moreover, there is appreciable retardance in theentire angular region (δ is significantly different from zero or π). The corresponding variationsof δ(θ) for 365 nm and 440 nm are different from that of 400 nm in that, here the phase reversalis not observed in the forward scattering angles. Never-the-less at either of these wavelengthsappreciable value of linear retardance δ is apparent.

The results presented above demonstrate that the polarization behaviour of scattered lightfrom spheroidal metal nanoparticles in their surface plasmon resonance spectral region is sig-nificantly different from similar dielectric nanoparticles. It might be useful to relate theseresults to the experimentally observed enhanced depolarization of light from metal nanoparti-cles [54, 53]. Indeed recent experimental studies have shown that the depolarization of lightfrom colloidal metal nanoparticles (silver and gold) in their surface plasmon resonance bands aresignificantly higher as compared to that observed from colloidal solutions of similar dielectricparticles [54, 53]. The combined T-matrix and polar matrix decomposition analysis presentedabove reveals that the presence of two dipolar plasmon resonances having a difference in phasebetween them manifests as additional linear retardance for preferentially oriented individualspheroidal metal nanoparticles and that for randomly oriented nanoparticles, orientation aver-aging of such linear retardance lead to the observed strong depolarization effects in the lattercase. It is also pertinent to note here that the results of the combined T-matrix and polarmatrix decomposition analysis presented above are for oblate spheroids (aspect ratio ε > 1).

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Chapter 8

Conclusions and Future directions

8.1 Conclusions

A highly sensitive and fully automated spectral Mueller matrix polarimeter for both elastic andinelastic scattering (fluorescence) has been developed. It is capable of recording Mueller matriceswith an elemental error of the order of 0.01 over the whole spectral range (λ = 500− 700 nm)within a few minutes time. It uses two rotating wave retarders and two fixed linear polarizersto generate sixteen elliptically polarized states which are then recorded using a spectrometer.It has been shown that these wave retarders and polarizers may or may not function ideallyfor all wavelengths never-the-less their actual values can be calculated using the Eigenvaluecalibration (EVC) procedure proposed by E. Compain et. al. [26] for the entire spectral range.The method of calculating the unknown 4 × 4 Mueller matrix for any sample in general fromthese sixteen spectral measurements has been developed and discussed in detail in section 3.1.The correctness of this method has also been verified by the checking for the desired resultsof the estimated polarization parameters and Mueller matrics for the known standard opticalelements like blank, polarizer, quarter waveplate etc.

Similar strategy was applied for measurement of the complete spectral Mueller matrix forinelastic scattering (fluorescence basically) and had initially been used to characterize humancervical cancer tissues on the basis of subtle morphological and physiological changes and showspromising aspects. Since Mueller matrix stores all the polarization characteristics of the samplesuch complete spectral Mueller matrix measurements strategies may have important biomedicalapplications especially for cancer diagnosis or for detecting any abnormalities in tissues ingeneral.

This fluorescence based spectral Mueller matrix measurement technique also showed promis-ing aspects in the field of material characterization (mainly for fluorescent materials). Thefluorescence Mueller matrices for some of the fluorescent dyes like Coumarin 102 and Coumarin152 were recorded with the aim to study their polarization characteristics and some interestingresults were obtained, though not reported in this thesis since the study could not be completeddue to the constraint of time.

Derivation, quantification and understanding of the intrinsic plasmon polarization charac-teristics aided by the Mueller matrix decomposition method may have important implicationsin metal nanoparticles-based biomedical imaging. The polarization characteristics of scatteredlight from spheroidal metal (silver) nanoparticles in their surface plasmon resonance spectralregion were investigated using polar decomposition of scattering Mueller matrices. The de-composition analysis on the scattering matrices (computed using T-matrix approach) frompreferentially oriented nanoparticles revealed the presence of strong linear retardance effectsin the surface plasmon spectral band of the spheroidal metal nanoparticles. Moreover, thederived linear retardance δ(λ) for these preferentially oriented spheroidal metal nanoparticlesshowed distinct spectral characteristics, the magnitude of δ peaking around the spectral overlap

46

region of the transverse and the longitudinal dipolar plasmon resonance bands. Interestingly,the helicity flipping of circularly polarized (or phase reversal) shows anomalous behavior forthe spheroidal metal nanoparticles at wavelengths corresponding to the overlap region of thetransverse and the longitudinal dipolar plasmon resonance bands. In contrast to usual Rayleighparticles, phase reversal is exhibited in the forward scattering angles (δ > π/2 at θ < 90◦),whereas no phase reversal is observed in the backscattering (δ < π/2 radian for θ > 90◦).The observed linear retardance effects were attributed to the inherent differences in phases,of the two orthogonal (longitudinal and the transverse) dipolar plasmon polarizabilities of thespheroidal metal nanoparticles. The analysis also revealed that when averaged over all possi-ble orientations of the particles (for randomly oriented spheroids), addition of the retardancematrices having random orientation of axes manifests as stronger depolarization (∆) of lightin spheroidal metal nanoparticles as compared to their dielectric counterparts (as observed inrecent experimental studies [53, 54]). Consequently, the spectral variations of ∆(λ) for therandomly oriented nanoparticles resemble that for the variation of δ(λ) for the preferentiallyoriented ones.

Since the spectral behaviour of the constituent polarimetry characteristics of the non-spherical metal nanoparticles are distinctly different from dielectric particles, these can beexploited to develop polarization-controlled novel schemes for enhancing contrast in plasmonresonant nanoparticle-based biomedical imaging [55, 56].

8.2 Future Directions

As noted above that the spectral behaviour of the constituent polarimetry characteristics ofthe non-spherical metal nanoparticles are distinctly different from dielectric particles and thuscan be exploited to develop novel schemes for contrast enhancement in biomedical imaging.For example, co- and crossed circular polarization imaging scheme (with proper choice of wave-length of light) can be adopted to discriminate (thus enhance contrast) plasmonic scatteringfrom the background Rayleigh/Mie scattering of biological (tissue or cell) dielectric structuresin single particle microscopic imaging. This follows because the scattered light from spheroidalmetal nanoparticles will have opposite helicity of circular polarization as compared to theRayleigh/Mie scattered light from background dielectric structures. Similarly, co- and crossedlinear polarization detection scheme can be used (exploiting the much stronger depolarizationeffect and its unique spectral characteristics) to distinguish plasmonic scattering from back-ground dielectric medium for imaging ensemble of particles [54]. One should however, note thatthe nanoparticle-based biomedical imaging applications would additionally be confounded bystrong multiple scattering effects (specifically in biological tissues, multiple scattering causesextensive depolarization) and by other simultaneously occurring complex tissue polarimetryevents (these include linear birefringence due to anisotropic tissue structures and optical rota-tion due to optically active (chiral) molecules and structures) [64]. Moreover, in applicationsinvolving ensemble of nanoparticles, one would expect additional depolarization effects causedby multiple scattering within the ensemble of nanoparticles also. With regard to the first issue,it is pertinent to note that the intrinsic polarization characteristics (linear retardance and de-polarization) of the non-spherical (spheroidal, in this study) metal nanoparticles exhibit verydistinct spectral characteristics (which is known a-priori and can be controlled by varying theaspect ratio of the spheroids) and is expected to be different from the spectral polarimetry ef-fects of background biological (tissue) dielectric structures. This therefore offers the possibilityof exploiting multi-spectral polarimetric imaging (whereby polarimetric imaging is performedat multiple selected wavelengths) for discriminating against background polarization character-istics of biological tissues/cells. For example, polarization images (either full Mueller matriximage or selected combination of linear and circular polarization state images) captured atwavelength corresponding to the peak of the depolarization or linear retardance of the metal

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nanoparticles and similar images captured at slightly different wavelength may be combinedto minimize background polarization effects from tissue dielectric structures. In general, fullMueller matrix imaging with appropriate combination of wavelengths and the matrix elements(either directly from the measured Mueller matrix or the constituent basis matrices obtainedvia the decomposition analysis) or the derived polarization parameters, may thus facilitate arobust approach for contrast enhancement in nanoparticle-based imaging. The effects of multi-ple scattering on quantitative polarimetry characteristics of ensemble of spherical/non-sphericalmetal nanoparticles also need to be studied.

In addition to biomedical diagnostics and imaging, the derived polarization parameters (andtheir unique spectral characteristics) may also be exploited for monitoring and controlling of sizeand shape of metal nanoparticles during synthesis. Moreover, such quantitative polarimetry mayalso have useful applications in optimizing/enhancing the sensitivity of the metal nanoparticles(or nanostructures)-based plasmonic sensors, as well as in other important areas related tonano-plasmonics.

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Appendix A

Eigenvalue Calibration - MATLABScript

The Eigenvalue calibration was done for each wavelength (w) so as to get PSG (W) and PSA(A) matrices at all wavelengths. bo and b are measurement vectors for blank and referencesamples respectively. Following is the matlab script that was used for implementing eigenvaluecalibration method. The credits for developing this script goes to Harshit Lakhotia, Jalpa Soniand myself.

1 function[Evals, W, A]=programcalibration(w,b,bo)2 Bo = reshape(bo(:,w,2),4,4);3 B1 = reshape(b(1,:,w,2),4,4);4 B2 = reshape(b(2,:,w,2),4,4);5 B3 = reshape(b(3,:,w,2),4,4);6 B4 = reshape(b(4,:,w,2),4,4);7 C1 = (inv(Bo))*B1;8 C2 = (inv(Bo))*B2;9 C3 = (inv(Bo))*B3;

10 C4 = (inv(Bo))*B4;11 lambda1=eig(C1);12 lambda2=eig(C2);13 lambda3=eig(C3);14 lambda4=eig(C4);15 taup = real(lambda2(1));16 taur = (lambda3(3) + lambda3(4))/2;17 r = (angle(lambda3(2)/lambda3(1)))/2;18 if r>019 r = pi - r;20 else21 r = -r;22 end23 psih = pi/4;24 M1 = (taup/2)*[1,0.56,0.83,0;25 0.56,0.31,0.46,0;26 0.83,0.46,0.69,0;27 0,0,0,0];28 M2 = (taup/2)*[1, -0.83, 0.56, 0;29 -0.83, 0.69, -0.46, 0;30 0.56, -0.46, 0.31, 0;

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31 0, 0, 0, 0];32 cq = cos(2*psih);33 sq = sin(2*psih);34 cd = cos(r);35 sd = sin(r);36 M3 = taur*[1,-0.69*cq,-0.72*cq,0;37 -0.69*cq,(0.48+(0.52*cd*sq)),(0.50-(0.50*cd*sq)),-0.72*sd*sq;38 -0.72*cq,(0.50-(0.50*cd*sq)),(0.52+(0.48*cd*sq)),0.69*sd*sq;39 0,0.72*sd*sq,-0.69*sd*sq,cd*sq];40 M4 = taur*[1,0.72*cq,-0.69*cq,0;41 0.72*cq,(0.52+(0.48*cd*sq)),(-0.5+(0.5*cd*sq)),-0.69*sd*sq;42 -0.69*cq,(-0.5+(0.5*cd*sq)),(0.48+(0.52*cd*sq)),-0.72*sd*sq;43 0,0.69*sd*sq,0.72*sd*sq,cd*sq];44 U1 = [1,0,0,0;0,0,0,0;0,0,0,0;0,0,0,0];45 U2 = [0,1,0,0;0,0,0,0;0,0,0,0;0,0,0,0];46 U3 = [0,0,1,0;0,0,0,0;0,0,0,0;0,0,0,0];47 U4 = [0,0,0,1;0,0,0,0;0,0,0,0;0,0,0,0];48 U5 = [0,0,0,0;1,0,0,0;0,0,0,0;0,0,0,0];49 U6 = [0,0,0,0;0,1,0,0;0,0,0,0;0,0,0,0];50 U7 = [0,0,0,0;0,0,1,0;0,0,0,0;0,0,0,0];51 U8 = [0,0,0,0;0,0,0,1;0,0,0,0;0,0,0,0];52 U9 = [0,0,0,0;0,0,0,0;1,0,0,0;0,0,0,0];53 U10 = [0,0,0,0;0,0,0,0;0,1,0,0;0,0,0,0];54 U11 = [0,0,0,0;0,0,0,0;0,0,1,0;0,0,0,0];55 U12 = [0,0,0,0;0,0,0,0;0,0,0,1;0,0,0,0];56 U13 = [0,0,0,0;0,0,0,0;0,0,0,0;1,0,0,0];57 U14 = [0,0,0,0;0,0,0,0;0,0,0,0;0,1,0,0];58 U15 = [0,0,0,0;0,0,0,0;0,0,0,0;0,0,1,0];59 U16 = [0,0,0,0;0,0,0,0;0,0,0,0;0,0,0,1];60 G1G1 = M1*U1 - U1*C1;61 G1G2 = M1*U2 - U2*C1;62 G1G3 = M1*U3 - U3*C1;63 G1G4 = M1*U4 - U4*C1;64 G1G5 = M1*U5 - U5*C1;65 G1G6 = M1*U6 - U6*C1;66 G1G7 = M1*U7 - U7*C1;67 G1G8 = M1*U8 - U8*C1;68 G1G9 = M1*U9 - U9*C1;69 G1G10 = M1*U10 - U10*C1;70 G1G11 = M1*U11 - U11*C1;71 G1G12 = M1*U12 - U12*C1;72 G1G13 = M1*U13 - U13*C1;73 G1G14 = M1*U14 - U14*C1;74 G1G15 = M1*U15 - U15*C1;75 G1G16 = M1*U16 - U16*C1;76 G2G1 = M2*U1 - U1*C2;77 G2G2 = M2*U2 - U2*C2;78 G2G3 = M2*U3 - U3*C2;79 G2G4 = M2*U4 - U4*C2;80 G2G5 = M2*U5 - U5*C2;81 G2G6 = M2*U6 - U6*C2;

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82 G2G7 = M2*U7 - U7*C2;83 G2G8 = M2*U8 - U8*C2;84 G2G9 = M2*U9 - U9*C2;85 G2G10 = M2*U10 - U10*C2;86 G2G11 = M2*U11 - U11*C2;87 G2G12 = M2*U12 - U12*C2;88 G2G13 = M2*U13 - U13*C2;89 G2G14 = M2*U14 - U14*C2;90 G2G15 = M2*U15 - U15*C2;91 G2G16 = M2*U16 - U16*C2;92 G3G1 = M3*U1 - U1*C3;93 G3G2 = M3*U2 - U2*C3;94 G3G3 = M3*U3 - U3*C3;95 G3G4 = M3*U4 - U4*C3;96 G3G5 = M3*U5 - U5*C3;97 G3G6 = M3*U6 - U6*C3;98 G3G7 = M3*U7 - U7*C3;99 G3G8 = M3*U8 - U8*C3;

100 G3G9 = M3*U9 - U9*C3;101 G3G10 = M3*U10 - U10*C3;102 G3G11 = M3*U11 - U11*C3;103 G3G12 = M3*U12 - U12*C3;104 G3G13 = M3*U13 - U13*C3;105 G3G14 = M3*U14 - U14*C3;106 G3G15 = M3*U15 - U15*C3;107 G3G16 = M3*U16 - U16*C3;108 G4G1 = M4*U1 - U1*C4;109 G4G2 = M4*U2 - U2*C4;110 G4G3 = M4*U3 - U3*C4;111 G4G4 = M4*U4 - U4*C4;112 G4G5 = M4*U5 - U5*C4;113 G4G6 = M4*U6 - U6*C4;114 G4G7 = M4*U7 - U7*C4;115 G4G8 = M4*U8 - U8*C4;116 G4G9 = M4*U9 - U9*C4;117 G4G10 = M4*U10 - U10*C4;118 G4G11 = M4*U11 - U11*C4;119 G4G12 = M4*U12 - U12*C4;120 G4G13 = M4*U13 - U13*C4;121 G4G14 = M4*U14 - U14*C4;122 G4G15 = M4*U15 - U15*C4;123 G4G16 = M4*U16 - U16*C4;124 g1=reshape(G1G1’,16,1);125 g2=reshape(G1G2’,16,1);126 g3=reshape(G1G3’,16,1);127 g4=reshape(G1G4’,16,1);128 g5=reshape(G1G5’,16,1);129 g6=reshape(G1G6’,16,1);130 g7=reshape(G1G7’,16,1);131 g8=reshape(G1G8’,16,1);132 g9=reshape(G1G9’,16,1);

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133 g10=reshape(G1G10’,16,1);134 g11=reshape(G1G11’,16,1);135 g12=reshape(G1G12’,16,1);136 g13=reshape(G1G13’,16,1);137 g14=reshape(G1G14’,16,1);138 g15=reshape(G1G15’,16,1);139 g16=reshape(G1G16’,16,1);140 H1 = [g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,g14,g15,g16];141 g1=reshape(G2G1’,16,1);142 g2=reshape(G2G2’,16,1);143 g3=reshape(G2G3’,16,1);144 g4=reshape(G2G4’,16,1);145 g5=reshape(G2G5’,16,1);146 g6=reshape(G2G6’,16,1);147 g7=reshape(G2G7’,16,1);148 g8=reshape(G2G8’,16,1);149 g9=reshape(G2G9’,16,1);150 g10=reshape(G2G10’,16,1);151 g11=reshape(G2G11’,16,1);152 g12=reshape(G2G12’,16,1);153 g13=reshape(G2G13’,16,1);154 g14=reshape(G2G14’,16,1);155 g15=reshape(G2G15’,16,1);156 g16=reshape(G2G16’,16,1);157 H2 = [g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,g14,g15,g16];158 g1=reshape(G3G1’,16,1);159 g2=reshape(G3G2’,16,1);160 g3=reshape(G3G3’,16,1);161 g4=reshape(G3G4’,16,1);162 g5=reshape(G3G5’,16,1);163 g6=reshape(G3G6’,16,1);164 g7=reshape(G3G7’,16,1);165 g8=reshape(G3G8’,16,1);166 g9=reshape(G3G9’,16,1);167 g10=reshape(G3G10’,16,1);168 g11=reshape(G3G11’,16,1);169 g12=reshape(G3G12’,16,1);170 g13=reshape(G3G13’,16,1);171 g14=reshape(G3G14’,16,1);172 g15=reshape(G3G15’,16,1);173 g16=reshape(G3G16’,16,1);174 H3 = [g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,g14,g15,g16];175 g1=reshape(G4G1’,16,1);176 g2=reshape(G4G2’,16,1);177 g3=reshape(G4G3’,16,1);178 g4=reshape(G4G4’,16,1);179 g5=reshape(G4G5’,16,1);180 g6=reshape(G4G6’,16,1);181 g7=reshape(G4G7’,16,1);182 g8=reshape(G4G8’,16,1);183 g9=reshape(G4G9’,16,1);

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184 g10=reshape(G4G10’,16,1);185 g11=reshape(G4G11’,16,1);186 g12=reshape(G4G12’,16,1);187 g13=reshape(G4G13’,16,1);188 g14=reshape(G4G14’,16,1);189 g15=reshape(G4G15’,16,1);190 g16=reshape(G4G16’,16,1);191 H4 = [g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,g14,g15,g16];192 K = (H1’*H1 + H2’*H2+ H3’*H3 +H4’*H4);193 [V,LAMBDAK] = eig(K);194 W16 = V(:,1);195 Evals = LAMBDAK;196 for pp=1:16197 temp(pp)=Evals(pp,pp);198 end199 clear Evals;200 Evals=temp;201 clear temp;202 W=reshape(W16,4,4)’;203 for i = 1:4204 W(:,i) = W(:,i)/W(1,i);205 end206 a = Bo*inv(W);207 A=reshape(a,4,4);208 for i = 1:4209 A(i,:) = A(i,:)/A(i,1);210 end

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Appendix B

Polar Decomposition - MATLABScript

The obtained Mueller matrices (muel) were decomposed using Polar decomposition to get thebasic polarization properties, net depolatization (depol), diattenuation (D), total retardance(R), optical rotation (rotation) and linear retardance (lin reta) for all wavelengths. Followingis the matlab script that was used for implementing this method. The credits for developingthis script goes to Dr. Nirmalya Ghosh and Jalpa Soni.

1 function[rvec,dvec,R,D,depol,rotation,lin reta]=polardecomposition paper(muel)2 format long3 I=[1 0 0;4 0 1 0;5 0 0 1];6 pvec=[muel(2,1),muel(3,1),muel(4,1)]*(1/muel(1,1));7 dvec=[muel(1,2),muel(1,3),muel(1,4)]*(1/muel(1,1));8 D=((muel(1,2)^2+muel(1,3)^2+muel(1,4)^2)^0.5)*(1/muel(1,1));9 m=(1/muel(1,1))*[muel(2,2),muel(2,3),muel(2,4);

10 muel(3,2),muel(3,3),muel(3,4);11 muel(4,2),muel(4,3),muel(4,4)];12 D1=(1-D^2)^0.5;13 if D==014 muel 0=muel/muel(1,1);15 else16 mD=D1*I+(1-D1)*dvec’*dvec/D^2;17 MD=muel(1,1)*[1,dvec;18 dvec’,mD];19 diattenuation = ((MD(1,2)^2+MD(1,3)^2+MD(1,4)^2)^0.5)*(1/MD(1,1));20 muel 0=muel*inv(MD);21 end22 m 1=[muel 0(2,2) muel 0(2,3) muel 0(2,4);23 muel 0(3,2) muel 0(3,3) muel 0(3,4);24 muel 0(4,2) muel 0(4,3) muel 0(4,4)];25 l 0=eig(m 1*m 1’);26 m 0=inv(m 1*m 1’+((l 0(1)*l 0(2))^0.5+(l 0(2)*l 0(3))^0.5+(l 0(3)*l 0(1))^0.5)*I);27 m 00=(l 0(1)^0.5+l 0(2)^0.5+l 0(3)^0.5)*m 1*m 1’+I*(l 0(1)*l 0(2)*l 0(3))^0.5;28 if det(m 1)>=029 mdelta=m 0*m 00;30 else

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31 mdelta=-m 0*m 00;32 end33 [v,mdeltaf] = eig(mdelta);34 depol=1-(abs(mdelta(1,1))+abs(mdelta(2,2))+abs(mdelta(3,3)))/3;35 depol1 =1-(abs(mdeltaf(1,1))+abs(mdeltaf(2,2))+abs(mdeltaf(3,3)))/3;36 nul=(pvec’-m*dvec’)/D1^2;37 Mdelta=[1 0 0 0;38 nul mdelta];39 Mdeltaf =[1 0 0 0;40 nul mdeltaf];41 Mdinv=inv(Mdelta);42 MR=Mdinv*muel 0;43 trmR=(MR(2,2)+MR(3,3)+MR(4,4))/2;44 argu=trmR-1/2;45 if abs(argu)>146 if argu>047 R=acos(1);48 else49 R=acos(-1);50 end51 else52 R=acos(argu);53 end54 cssq 10=(MR(2,2)+MR(3,3))^2+(MR(3,2)-MR(2,3))^2;55 tan rot=(MR(3,2)-MR(2,3))/((MR(2,2))+(MR(3,3)));56 de=cssq 10^0.5-1;57 if de>0.99999999999958 de=1;59 end60 if de<-0.9999999999961 de=-1;62 end63 lin reta=acos(de);64 rotation=atan(tan rot);65 if tan rot<0.00000000166 rotation=rotation+pi;67 end68 rotation=rotation/2;69 if (MR(3,2)-MR(2,3))<0.070 if (MR(2,2)+MR(3,3))<0.071 rotation=rotation+pi/2;72 end73 end74 if (MR(3,2)-MR(2,3))<0.075 if (MR(2,2)+MR(3,3))>0.076 rotation=rotation+pi/2;77 end78 end79 if abs(MR(3,2)-MR(2,3))<=0.000000001 && abs(MR(2,2)+MR(3,3))>0.000000000180 rotation=0;81 end

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82 if abs(sin(R))<=0.00000000183 a3=((1+cos(lin reta))/2)^0.5;84 a1=(MR(3,4)+MR(4,3))/(4*a3);85 a2=(MR(4,2)+MR(2,4))/(4*a3);86 else87 D2=1/(2*sin(R));88 a1=D2*(MR(3,4)-MR(4,3));89 a2=D2*(MR(4,2)-MR(2,4));90 a3=D2*(MR(2,3)-MR(3,2));91 end92 rvec=[1,a1,a2,a3]’;93 if abs(cos(R))>=0.999999999994 C1=MR(2,2)+MR(3,3);95 C2=MR(2,3)-MR(3,2);96 if abs(C1)<0.000000000197 MR=MR*[1 0 0 0; 0 1 0 0; 0 0 -1 0; 0 0 0 -1];98 rotation=0.5*acos((MR(2,2)+MR(3,3))/2);99 lin reta=pi;

100 end101 if C1<1.999999999102 if abs(C2)<0.0000000001103 MR=MR*[1 0 0 0; 0 -1 0 0; 0 0 -1 0; 0 0 0 1];104 dum=MR(2,2)+MR(3,3)-1;105 lin reta=acos(dum);106 if dum>=1107 lin reta=0;108 end109 if dum<=-1110 lin reta=pi;111 end112 rotation=pi/2;113 end114 end115 end116 orientation = 0.5*atan(rvec(3)/rvec(2));117 mr1=MR*inv(rota(rotation));118 a1a=(mr1(3,4)-mr1(4,3));119 a2a=(mr1(4,2)-mr1(2,4));120 a3a=(mr1(2,3)-mr1(3,2));121 rveca=[1,a1a,a2a,a3a]’;122 orientationa = 0.5*atan(rveca(3)/rveca(2));123 mr2=inv(rota(rotation))*MR;124 a1b=(mr2(3,4)-mr2(4,3));125 a2b=(mr2(4,2)-mr2(2,4));126 a3b=(mr2(2,3)-mr2(3,2));127 rvecb=[1,a1b,a2b,a3b]’;128 orientationb = 0.5*atan(rvecb(3)/rvecb(2));129 orientationc = 0.5*acos(MR(3,4)/sin(lin reta));130 return

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Appendix C

Labview Automation Code

Figure C.1: Front Panel of the automation VI.

The automation of the spectral Mueller matrix polarimeter was done using Labview. Fig.C.1 is a snapshot of the front panel of the Labview code used for automation. The code wasdesigned by myself using the virtual instruments (VIs), software development kits (SDKs) andother drivers provided along with each instrument.

The block diagram behind the above control is shown in Fig. C.2. Please note that this VIconsists of several other sub VIs which are not shown here. These were either supplied by themanufacturers or were designed by me.

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Figure C.2: Block Diagram of the automation VI.

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quantitative mueller matrix polarimetry with diverse applications

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