On the interaction of light with novel artificial

On the interaction of light with novel artificial materials –

Intriguing phenomena and an extended toolbox

Der Naturwissenschaftlichen

Fakultät

der Friedrich-Alexander-Universität

Erlangen-Nürnberg

zur

Erlangung des Doktorgrades Dr. rer.

nat.

vorgelegt von

Muhammad Abdullah Tariq Butt

aus Karatschi (Pakistan)

Als Dissertation genehmigt

von der Naturwissenschaftlichen Fakultät

der Friedrich-Alexander Universität Erlangen-

Nürnberg

Abgabe bei den Berichterstattern 11. 02 2021 Tag der mündlichen Prüfung 30. 03 2021 Vorsitzender des Promotionsorgans: Prof. Dr. Wolgang

Achtziger Gutachter: Prof. Dr. Gerd Leuchs Prof. Dr. Nicolas Joly

Beginning with the name of Almighty the most gracious, I dedicate

this thesis to my parents and better half, who have supported me

throughout my doctoral journey.

Abstract

This thesis summarizes the advantage and versatility of analyzing light-mater interaction

exploiting the polarization state of light. We use customized experimental systems to develop

specialized techniques to study intriguing optical phenomena at small scales for novel artificial

materials. The quintessential parts of this report are Chapter 3 and 4, which describe the

experimental results and analysis related to projects covered in this thesis.

In the first part of this thesis, we study the case of 3D novel self-assembled carbon flakes with

orthorhombic phase of carbon intercalated with bimetallic (Au-Ag) nanoclusters, fabricated from

liquid phase solution of a supra molecular complex (SMC) using laser induced deposition process.

The fabrication of carbon flakes was done by Prof. Alina Manshina and her group at St. Petersburg

State University, Russia. The challenge however, was the small lateral dimensions of individual

carbon flakes. Therefore, we developed a scheme (microscopic Müller matrix measurement

technique) to extract the optical properties of the carbon flakes, once the incoming and outgoing

polarization states of light are known. This technique combines the benefits of polarized light

matter interaction with the back focal plane (k-space / Fourier space) microscopy and the usage

of electrically controlled liquid crystals to perform a comprehensive polarization analysis in

transmission at small scales. With the help of experimental results and theoretical modelling

(performed by collaborators at University of Ottawa, Canada), we relate the optical birefringence

in the carbon flakes to the crystalline arrangement of carbon atoms in the orthorhombic lattice.

Later, we also study the dependence of optical and geometrical properties of carbon flakes on the

fabrication parameters. To access the direct information regarding refractive index of the carbon

flake, we implement a specialized single-shot ellipsometric technique with a resolution, in the

order of the wavelength. This provided us with a preliminary estimate of complex refractive

indices of carbon flake.

In the second part of this thesis, we focus on the concept of diffraction assisted chiral scattering

in 2D metasurfaces. By designing sub-wavelength scattering structures called meta-atoms and

periodically arranging them, metasurfaces can be realized with intriguing optical properties. By

finely selecting the shape, orientation, material, and size of meta-atoms, their optical response

can be tuned. When the meta-atoms are arranged with certain periodicity, it leads to the

generation of propagating surface modes, also known as surface lattice resonances (SLR).

Resultantly, a resonant coupling of incident light beam with individual meta-atom resonance and

grazing diffracted waves can occur. We discuss the in-plane scattering of individual meta-atom

and how it could be effectively coupled to the diffraction modes of a lattice to observe asymmetric

transmission in the far-fields. We elaborate this concept for a fourfold symmetric structure

(quadrumer). The simulations and fabrication were performed in collaboration with researchers

from Tecnologico de Monterrey, Mexico and University of Ottawa, Canada (also part of Max

Planck-University of Ottawa Centre for Extreme and Quantum Photonics), respectively. For

analysis of samples, we use an experimental setup to angularly resolve transmitted light to study

asymmetric transmission in zeroth and first diffraction orders. For our case of a fourfold

symmetric quadrumer, an in-plane rotation of each quadrumer by 22.5°/-22.5° leads to

asymmetric transmission in first diffraction order. The zeroth order does not exhibit any

asymmetry. We analyze simulation and experimental results and find them in good agreement.

In the outlook chapter of this thesis, we discuss future works related to localized modification of

carbon flakes and extension of our ellipsometric scheme using polarization tailored light beams.

We also elaborate on few designs/ideas for observing asymmetric transmission in symmetric

structures as a future extension of our present work.

Zusammenfassung

Diese Arbeit beschäftigt sich mit den Vorteilen und der Vielseitigkeit der Analyse von Licht-

Materie-Wechselwirkung unter Ausnutzung des Polarisationszustands des Lichts. Wir verwenden

maßgeschneiderte experimentelle Systeme, zur Entwicklung spezieller Techniken für die

Untersuchung faszinierender optischer Phänomene auf kleinen Längenskalen in Bezug auf

neuartige künstliche Materialien. Der wesentliche Teil dieses Berichts sind die Kapitel 3 und 4, in

denen die experimentellen Ergebnisse und Analysen der in dieser Arbeit behandelten Projekte

beschrieben werden.

Im ersten Teil dieser Arbeit untersuchen wir 3D neuartige selbstorganisierter Kohlenstoffflocken

mit orthorhombischer Kohlenstoffphase, die mit bimetallischen (Au-Ag) -Nanoclustern

interkaliert sind und aus einer Flüssigphasenlösung eines supra-molekularen Komplexes (SMC)

unter Verwendung eines laserinduzierten Abscheidungsprozesseshergestellt wurden. Die

Herstellung von Kohlenstoffflocken wurde von Prof. Alina Manshina und ihrer Gruppe an der

staatlichen Universität St. Petersburg in Russland durchgeführt. Die Herausforderung sind jedoch

die geringen seitlichen Abmessungen der einzelnen Kohlenstoffflocken. Daher haben wir ein

System (mikroskopische Müller-Matrix-Messtechnik) entwickelt, um die optischen Eigenschaften

der Kohlenstoffflocken zu extrahieren, sobald die ein- und ausgehenden Polarisationszustände

des Lichts bekannt sind. Diese Technik kombiniert die Vorteile polarisierter Licht-Materie-

Wechselwirkung mit der Mikroskopie der hinteren Brennebene (k-Raum / Fourier-Raum) und der

Verwendung elektrisch gesteuerter Flüssigkristalle, um eine umfassende Polarisationsanalyse in

Transmission auf kleinen Größenskalen durchzuführen. Mit Hilfe experimenteller Ergebnisse und

theoretischer Modelle (durchgeführt von unseren Partnern an der Universität von Ottawa,

Kanada) beziehen wir die optische Doppelbrechung in den Kohlenstoffflocken auf die kristalline

Anordnung von Kohlenstoffatomen im orthorhombischen Gitter. Später untersuchen wir auch die

Abhängigkeit der optischen und geometrischen Eigenschaften von Kohlenstoffflocken von den

Herstellungsparametern. Um auf die direkten Informationen bezüglich des Brechungsindex der

Kohlenstoffflocke zugreifen zu können, wurde eine spezielle Einzelbild ellipsometrische Technik

mit einer Auflösung in der Größenordnung der Wellenlänge implementiert. Dies liefert uns eine

vorläufige Abschätzung der komplexen Brechungsindizes der Kohlenstoffflocken.

Im zweiten Teil dieser Arbeit konzentrieren wir uns auf das Verständnis der

beugungsunterstützten chiralen Streuung in 2D-Metaoberflächen. Indem wir Sub-Wellenlängen-

Streustrukturen, sogenannte Meta-atome, entwerfen und periodisch anordnen, können wir

Metaoberflächen mit verblüffenden optischen Eigenschaften realisieren. Durch genaue Auswahl

von Form, Ausrichtung, Material und Größe der Meta-atome können wir deren Reaktion auf

optische Anregung einstellen. Wenn die Meta-atome mit einer bestimmten Periodizität

angeordnet sind, führt dies zur Erzeugung von sich ausbreitenden Oberflächenmoden, die auch

als Oberflächengitterresonanzen (SLR) bezeichnet werden. Infolgedessen kann eine resonante

Kopplungdes einfallenden Lichtstrahls an eine individuelle Meta-atom-Resonanz und streifenden

gebeugten Wellen auftreten. Wir werden die Streuung einzelner Meta-atome in der Ebene

diskutieren und wie sie effektiv an die Beugungsmoden eines Gitters gekoppelt werden kann, um

eine asymmetrische Transmission im Fernfeld zu beobachten. Wir werden dieses Konzept für eine

vierfach symmetrische Struktur (Quadrumer) ausarbeiten.

Die Simulationen und die Herstellung wurden in Zusammenarbeit mit Forschern des Tecnologico

de Monterrey, Mexico, und der Universität von Ottawa, Kanada (ebenfalls Teil des Zentrums für

Extreme und Quantenphotonik der Max-Planck-Universität von Ottawa) durchgeführt. Zur

Analyse von Proben verwenden wir einen Versuchsaufbau, welcher eine Winkel-aufgelöste

Messung des transmittierten Lichts erlaubt, um die asymmetrische Transmission in nullter und

erster Beugungsordnung zu untersuchen. Für den vorliegenden Fall eines vierfach symmetrischen

Quadrumers führt eine Drehung jedes Quadrumers in der Ebene um 22,5 ° / -22,5 ° zu einer

asymmetrischen Transmission in der ersten Beugungsordnung. Die nullte Ordnung zeigt keine

Asymmetrie. Simulations- und Versuchsergebnisse weisen eine gute Übereinstimmung auf.

Im Ausblick-Kapitel dieser Arbeit diskutieren wir zukünftige Projekte zur lokalisierten Modifikation

von Kohlenstoffflocken und zur Erweiterung unseres ellipsometrischen Schemas mithilfe speziell

polarisierter Lichtstrahlen. Wir arbeiten außerdem einige Designs/ideen von Meta-atomen zur

Beobachtung der asymmetrischen Transmission in symmetrischen Strukturen als zukünftige

Erweiterung unserer gegenwärtigen Arbeit aus.

Table of Contents

1. Introduction ................................................................................................................................. 1

2. Theoretical background .............................................................................................................. 4

2.1 Electromagnetic fields, plane waves and more ..................................................................... 4

2.2 Jones and Müller formalism ................................................................................................. 14

2.3 Optical material properties .................................................................................................. 19

3. 3D self-assembled carbon-metal hybrid structure .................................................................. 30

3.1 Fabrication of novel hybrid structures ................................................................................. 31

3.2 Experimental setup .............................................................................................................. 36

3.3 Results and discussion (optical properties and fabrication effects) .................................... 50

4. Chiral 2D metasurfaces ............................................................................................................. 66

4.1 Chirality and light matter interaction .................................................................................. 66

4.2 General experimental setup and arrangements .................................................................. 76

4.3 Results and discussion (asymmetric transmission of quadrumer array) ............................. 78

5. Conclusion and Outlook ............................................................................................................ 85

A. Some mathematical relations .................................................................................................. 91

B. Additional data-ellipsometry .................................................................................................... 94

C Additional data-quadrumer array ........................................................................................... 100

References ................................................................................................................................... 102

List of publications ...................................................................................................................... 115

Acknowledgements ..................................................................................................................... 117

1. Introduction

In the last decade, miniaturization of electronic devices has pushed the field of nanotechnology

to define the new boundaries of science [1-6]. Nano-optics, a related field, deals with the

interaction of light with matter at nanoscale [7-9]. In this thesis, we demonstrate the advantage

and versatility of analyzing light-mater interaction, exploiting polarization state of light. We used

customized in-home built experimental systems [10-13] to devise ways and methods to study

intriguing optical phenomena at small scales. Most of the studies were performed in collaboration

with scientists across the globe belonging from different fields of science [14-20].

To begin our journey, initially we will consider the theoretical foundations related to the light

matter interaction in Chapter 2 of this thesis. We will discuss the polarization state of light and

how it alters when interacting with matter. This understanding will later help us to study the

inverse case. To extract the optical properties of a medium, once the incoming and outgoing

polarization states of light are known. Although being developed and used for over a century,

polarimetric analysis methods have evolved over time. Nanotechnology has provided room for

newer miniaturized devices, sophisticated analysis techniques to find their applications in

characterizing materials and structures with unprecedented accuracy and scales.

A similar case in hand, was of 3D novel self-assembled orthorhombic phase of carbon intercalated

with bimetallic (Au-Ag) nanoclusters, fabricated from liquid phase solution of a supra molecular

complex (SMC) by laser induced deposition process. The fabrication was done by Professor Alina

Manshina and her group at St. Petersburg State University, Russia [20-24]. The structure manifests

itself in the form of cuboid (called caron flake) with lateral dimensions of a few microns and

thickness of a few hundred nanometers. Due to its organo-metallic composition, it finds

application in plasmonic sensing platforms [25, 26] with future prospects of being used for light

guiding and plasmonics applications. We will discuss details about carbon flakes fabrication and

structure briefly in the first section of Chapter 3. The case, as intriguing, forced us to look and

think deeply in understanding the nature of this complex structure. Initial studies performed at

Max Planck institute for the science of light pointed towards interesting optical properties related

to carbon flake. Therefore, it was decided to perform a complete investigation into the optical

properties. The challenge however, was the small lateral dimensions of individual carbon flakes.

The commercially available polarimetric and ellipsometric setups usually provide resolution of

tens of microns which would not be helpful in our case.

Henceforth, an in-house built experimental setup [10-13, 27, 28] (previously used to study to

nanostructures) was modified to perform the polarimetric analysis of carbon flakes. The

microscopic Müller matrix measurement technique, as we call it, merges the benefits of polarized

light matter interaction with the back focal plane (k-space / Fourier space) microscopy [29] and

the usage of liquid crystal variable retarders (LCVRs) to perform a comprehensive polarization

analysis in transmission at small scales [14, 18, 30, 31].

The theoretical concepts related to this technique will already be discussed in Chapter 2, while

the experimental setup, its peculiarities and working behavior will be discussed in Chapter 3. We

1. Introduction

2

will shed light on experimental results and how they helped us in understanding the optical nature

of carbon flakes. To comprehend the contributions of bimetallic nanoclusters and orthorhombic

phase of carbon, our collaborators from University of Ottawa, Canada (also part of Max Planck-

University of Ottawa Centre for Extreme and Quantum Photonics) developed a computation

model based on experimental findings. This would be briefly discussed in the results section of

Chapter 3. The collaboration between physical chemists, material experts, experimental

physicists, and computational optics specialists from different parts of the globe, underscores the

importance of research collaboration for pushing the boundaries of science beyond geographical

borders.

With the help of experimental results and theoretical modelling, we relate the optical

birefringence in the carbon flakes to the crystalline arrangement of carbon atoms in the

orthorhombic lattice [14, 15, 18-20]. Later, we also study and examine the dependence of optical

and geometrical properties of carbon flakes on the fabrication parameters [15, 19].

The Müller matrix measurement technique adopted in transmission provides differential optical

information (for instance birefringence and dichroism). To access the direct information regarding

refractive index of the carbon flake, we implemented a specialized single-shot ellipsometric

technique with a resolution, in the order of the wavelength. The theoretical assumptions and

computational model in this regard would be elaborated in the last section of Chapter 2. The

modified experimental setup and results will be discussed in Chapter 3. This provided us with a

preliminary estimate of complex refractive indices of carbon flake. Understanding the intricate

and complex nature of the carbon flake is an ongoing work with projects planned for future

investigation. These would be discussed in Chapter 5.

In the second part of this thesis, we focused onto understanding the concept of diffraction

assisted chiral scattering in 2D metasurfaces. This work was done in collaboration with

researchers from University of Ottawa (also part of Max Planck-University of Ottawa Centre for

Extreme and Quantum Photonics), Canada and Tecnologico de Monterrey, Mexico.

Just like self-assembled structures discussed in Chapter 3 of this thesis,

metamaterials/metasurfaces also rely on the concept of structuring individual building blocks [7,

32-35]. We achieve metamaterials by fabricating sub-wavelength scattering structures called

meta-atoms and periodically arranging them. By finely selecting the shape, orientation, material,

and size of meta-atoms, we can tune the optical response of metamaterials/metasurfaces [34-

45]. When the meta-atoms are arranged with certain periodicity, it leads to generation of

propagating surface modes also known as surface lattice resonances (SLR) [46-50].These

propagating surface modes occur close to Rayleigh anomaly condition due to resonant coupling

of incident light beam with individual meta-atom resonance and grazing diffracted waves [51]. A

sharp decrease in transmission can be expected around these wavelengths (𝜆𝑅𝐴/𝑆𝐿𝑅) with very

narrow FWHM bandwidth. Typically, the spectral width of 𝜆𝑅𝐴/𝑆𝐿𝑅 is in the order of 10nm [48].

This has instigated a lot of research towards application of surface lattice resonances for sensing

devices [47, 49, 52-54]. We will review the relevant theoretical concepts related to the design of

chiral meta-atom in first section of Chapter 4. Later, we will discuss the in-plane scattering of

individual meta-atom and how it could be effectively coupled to the diffraction orders of a lattice

to observe asymmetric transmission in surface lattice resonances [48, 52-55]. We will elaborate

1. Introduction

3

this concept for a fourfold symmetric structure (quadrumer). Using, Finite difference time domain

(FDTD) simulation, we found out that for a chiral orientation ( in-plane rotation of 22.5°/-22.5°) at

certain periodicity (600nm) leads to asymmetric transmission in first diffraction order. This does

not happen for symmetric orientation (0° and 45° in-plane rotation). Besides this, due to

symmetry reasons, the zeroth order does not exhibit any asymmetric transmission. The

simulations were performed by research collaborators from Tecnologico de Monterrey, Mexico.

Later, the fabrication of requisite samples was done at University of Ottawa, Canada. For analysis

of samples, we use an experimental setup to angularly resolve transmitted light to study

asymmetric transmission in zeroth and first diffraction orders. We analyze simulation and

experimental results in last section of Chapter 4, which are found to be in good agreement.

In outlook chapter of this thesis, we elaborate on few designs of meta-atom for observing

asymmetric transmission in rotationally symmetric structures as future extension of our present

work.

2. Theoretical background

To comprehend the interaction of light with matter at nanoscale, it is crucial to know the basics

of light as a propagating electromagnetic wave. We briefly discuss the fundamentals of

propagating light and how various theoretical concepts help us to understand the case of light

matter interaction.

2.1 Electromagnetic fields, plane waves and more

Electromagnetic fields can be described in the most fundamental way by Maxwell equations [7,

56-58]. For vacuum (with absence of free charge density or current density), we can write these

equations as,

𝛁 ∙ 𝐄(𝐫, 𝑡) = 0, (2.1a)

𝛁 × 𝐄(𝐫, 𝑡) = −μ0∂𝐇(𝐫,𝑡)

∂𝑡, (2.1b)

𝛁 ∙ 𝐇(𝐫, 𝑡) = 0, (2.1c)

𝛁 × 𝐇(𝐫, 𝑡) = ε0∂𝐄(𝐫,𝑡)

∂𝑡 , (2.1d)

where μ and ε are respectively the vacuum permeability and permittivity of the medium. Here,

electric (𝐄) and magnetic (𝐇) field vectors are supposed to be a function of position and time,

represented by 𝐫 and t, respectively. By replacing the above curl fields and by taking advantage

of the divergence-free space we end up with the wave equations,

𝛁𝟐𝐮 −1

c2

∂2𝐮

∂𝑡2 = 0, (2.2)

where, 𝐮 represents electric (𝐄) and magnetic (𝐇) field vectors, c is the speed of light connected

to the permeability (μ0) and permittivity (ε0) of a medium by the relation,

c0 = 1

√μ0ε0. (2.3)

Simplification of the wave equation, by consideration of a time harmonic wave (of form 𝑒−𝑖𝜔𝑡,

where ω is the angular frequency of the wave), brings us to the vectorial Helmholtz equation, by

separation of temporal and spatial variables. For electric and magnetic fields this can be written

as,

( 𝛁𝟐 + 𝑘2 )𝐄(𝐫) = 0, (2.4a)

( 𝛁𝟐 + 𝑘2 )𝐇(𝐫) = 0, (2.4b)

with 𝑘 = ω/c0 is the wave number [58]. One of the fundamental solutions of Equation 2.4a and

2.4 b is a plane wave, described by,


5

𝐄(𝐫) = ��Eoe−i𝐤𝐫, (2.5)

where the propagation direction is along the wavevector 𝐤, 𝐄𝐨 represents the amplitude and �� is

the unit vectors defining the polarization direction. This simple solution can help in representing

different beams of light by superposition of multiple plane waves [7, 58]. One of the implications

for plane wave solutions (due to Equation 2.1a and 2.1c) is that electric and magnetic field vectors

are both perpendicular to the propagation direction 𝐤, hence limited to the transverse plane

(transverse electromagnetic (TEM) waves).

Gaussian beam solution

The paraxial approximation implies slowly varying amplitude along the propagation direction and

hence provide a beam solution diverging and converging very slowly. Gaussian beam solution,

although not a rigorous solution of Maxwell’s equations, but can be considered rather a simplified

solution of the scalar Helmholtz equation. This approximation helps us to get various solutions of

propagating paraxial beam in different coordinate systems. For example, Laguerre-Gaussian (LG)

modes in cylindrical (𝜌, 𝜃, 𝑧) and Hermite-Gaussian (HG) modes in the Cartesian (𝑥, 𝑦, 𝑧)

coordinate system [7, 56]. The fundamental solution in both cases is a TEM00 solution, which has

a Gaussian intensity profile and, as the name implies, has electric and magnetic fields in the

transverse plane, orthogonal to propagation direction. The field component along propagation

direction is ignored due to negligible value as long as paraxial approximation is valid. In the non-

paraxial regime we can expect stronger longitudinal component due to focusing of light [59],

which will not be discussed further in the purview of this thesis.

Angular spectrum representation

The electric (or magnetic) field in Cartesian coordinates for any plane of observation orthogonal

to the propagation direction (here positive 𝓏-direction) can be described as the superposition of

multiple plane waves for a spectrum of wave vectors with certain amplitudes and phases [60],

𝐄(𝐫) = ∬ ��(𝑘𝑥 , 𝑘𝑦 ; 𝑧)∞

−∞e−i (𝑘𝑥𝑥+𝑘𝑦𝑦+𝑘𝑧𝑧)d𝑘𝑥d𝑘𝑦. (2.6a)

It is possible to decompose this superposition of plane waves into longitudinal and transverse

field components,

𝐄(𝐫) = ∬ ��(𝑘𝑥 , 𝑘𝑦 ; 𝑧)∞

−∞e−i (𝑘𝑥𝑥+𝑘𝑦𝑦+𝑘𝑧𝑧)d𝑘𝑥d𝑘𝑦 = 𝐄⊥ + 𝐄∥, (2.6b)

and in Fourier space,

��(𝑘𝑥 , 𝑘𝑦 ; 𝑧) =1

4π2 ∬ 𝐄(𝑥, 𝑦, 𝑧)∞

−∞e−i (𝑘𝑥𝑥+𝑘𝑦𝑦+𝑘𝑧𝑧)d𝑥 d𝑦, (2.6c)

where 𝑘𝑥, 𝑘𝑦 are the transverse components of wavevector. Considering plane waves

propagating along positive 𝓏-direction, we define 𝑘⊥ = √𝑘𝑥2 + 𝑘𝑦

2 as the transverse wavenumber

and 𝑘∥ = 𝑘𝑧 = √𝑘2 − 𝑘⊥2 as the longitudinal wavenumber [7]. Considering the Equation 2.5a, we

can understand that 2.6b can lead to a vectorial angular spectrum depending on ��(𝑘𝑥 , 𝑘𝑦). The

angular spectrum representation is very useful in theoretically understanding laser beam


6

propagation and focusing of light waves. Moreover, considering 2.6c, the Fourier transformed

field evolves along the propagation direction in the following way,

𝐄(𝑘𝑥, 𝑘𝑦 ; 𝑧) = 𝐄(𝑘𝑥, 𝑘𝑦 ; 0)e±i𝑘𝑧𝑧, (2.7)

where the ± sign depends on the direction of propagation (+ for 𝓏 > 0 propagation direction). This

means that in reciprocal (angular spectrum) space, the field at any point along 𝓏 is equal to the

field at object plane (𝓏 = 0) times the propagator e±ikzz. Using this assumption, Equation 2.6a

gives us the angular spectrum representation (as shown in Figure 2.1) for any arbitrary value of

𝓏. Mathematically we can write it as,

𝐄(𝐫) = ∬ ��(𝑘𝑥 , 𝑘𝑦 ; 0)∞

−∞e−i (𝑘𝑥𝑥+𝑘𝑦𝑦+𝑘𝑧𝑧)d𝑘𝑥d𝑘𝑦. (2.8)

Figure 2.1 the angular spectrum representation. The incoming paraxial light field approaching

the entrance pupil (back focal plane) of the focusing lens. This in turn is projected onto a

reference sphere defined be the focal length. Each point on the reference sphere can then be

related to the coordinates (in k-space) within the entrance pupil [7]. For the parallel field

components, the incoming wavevector is rotated an angle θ, which is defined by the distance

between the optic axis and the respective wavevector times the focal length.

Using the angular spectrum representation in Equation 2.8, we can comprehend the effect of the

focusing system, by estimating the far-field representation of above plane wave spectrum. We

define a dimensionless unit vector based on a reference sphere in far field, which is defined by,

𝐬 = (𝑠𝑥 , 𝑠𝑦, 𝑠𝑧) = (𝑥

𝑟,𝑦

𝑟,𝑧

𝑟), (2.9)

where 𝑟 = √𝑥2 + 𝑦2 + 𝑧2 being the distance from origin (focal plane). As the distance r

increases, the contribution of evanescent waves 𝑘𝑥2 + 𝑘𝑦

2 > 𝑘2 (as decaying fields) is decreased

to being negligible and the integration is reduced to the circular region defined by 𝑘𝑥2 + 𝑘𝑦

2 < 𝑘2

[7]. For the electric field with radius 𝑟 → ∞, we can write,


7

𝑬∞(𝑠𝑥 , 𝑠𝑦, 𝑠𝑧) = lim𝑘𝑟→∞

∬ 𝑬(𝑘𝑥, 𝑘𝑦 ; 0)𝑒−𝑖𝑘𝑟(𝑘𝑥𝑘

𝑠𝑥+𝑘𝑦

𝑘𝑠𝑦+

𝑘𝑧𝑘

𝑠𝑧)d𝑘𝑥d𝑘𝑦𝑘⊥2 ≤ 𝑘2 . (2.10)

We solve the integral by applying the method for stationary phase [61]. This helps us to link the

far field reference sphere to initial field by relation,

𝐬 = (𝑠𝑥 , 𝑠𝑦, 𝑠𝑧) =𝑘𝑥

𝑘+

𝑘𝑦

𝑘+

𝑘𝑧

𝑘 (2.11)

This implies that, only a single incident plane wave out of plane wave spectrum defined by a wave

vector 𝐤 = (𝑘𝑥, 𝑘𝑦, 𝑘𝑧), contributes to a point on the reference sphere 𝐬 = (𝑠𝑥 , 𝑠𝑦, 𝑠𝑧). This

happens because rapidly oscillating phase terms in the integral in Equation 2.8 add up

destructively except the slowly varying phase corresponding to a certain wavevector contribute

significantly as defined in Equation 2.9. Hence, due to this elegant mathematical relation, we can

link the incident field in k-space to the far field of a focusing system.

𝑬(𝑘𝑥 , 𝑘𝑦; 0) = 𝑖𝑟

2𝜋

𝑒−𝑖𝑘𝑟

𝑘𝑧𝑬∞(𝑠𝑥 , 𝑠𝑦). (2.12)

This mathematical concept can be utilized to understand focusing of various types of light beams

and also for studying the inverse problem; to extract the response of a system in focal plane by

evaluating far field angular spectrum. This will also be of prime importance in the discussion of

experimental setups in the next chapters, where the angular spectrum of a collection microscope

objective is imaged to extract optical properties of an examined system [14, 15, 56, 62].

Polarization states of light

As shown in Equation 2.5a, a polarized light field can always be written as superposition of two

orthogonal plane waves. This could be performed in various coordinate systems [7, 63, 64]. For

two orthogonal plane waves polarized along 𝑥 and 𝑦 direction in a cartesian coordinate system

propagating along positive 𝑧-direction [65] we have

𝐄1 = 𝐄x + 𝐄y (2.13a)

𝐄x = E𝑥0��e−i (𝑤𝑡−k𝑧)eiδx, (2.13b)

𝐄y = E𝑦0��e−i (𝑤𝑡−k𝑧)eiδy and (2.13c)

We can define spatially homogenous incident light beam into three polarization types namely

linear, circular, and elliptical polarized light [7, 56, 58, 64]. We make the distinction based on the

individual amplitudes and relative phase difference (Δ δ = δx − δy) between the two constituent

orthogonal plane waves. The polarization state of light can be visualized and described in different

ways. For example, using the polarization ellipse (shown in Figure 2.2) in which, for all positions

along propagation direction (z-axis), we observe the fields in time in the 𝑥𝑦-plane. The two

defining parameters (as shown in Figure 2.2) in plotting a polarization ellipse are azimuthal angle

(θ ranging from −π

2−

π

2 ) and ellipticity angle (φ ranging from

−π

4−

π

4 ).


8

Figure 2.2 The concept of polarization ellipse with representation of common linear and circular

polarization states. The two defining parameters (as shown in Figure 2.2) in plotting a

polarization ellipse (extracted from the equations above) are azimuthal angle (θ ranging from −𝜋

2−

𝜋

2 ) and ellipticity angle (𝜑 ranging from

−𝜋

4−

𝜋

4 ), defined with the help of relative

amplitude and phase differences of orthogonal field components.

These parameters are defined with the help of relative amplitude and phase differences of

orthogonal field components of Equations 2.13b&c. When the two orthogonal components are in

phase (φ = 0), we get linearly polarized light, with respective azimuthal direction (θ). For the

limiting cases of φ =−π

4,π

4, the light wave is said to be left and right circularly polarized,

respectively. For all other values (φ ≠ 0 &−π

4< φ >

−π

4), the light wave is said to be elliptically

polarized. For a circularly polarized light, the polarization vector propagating along a certain

direction spins with time in space about the optical axis in a helical fashion [66]. This intriguing

phenomenon has an analog in terms of optical properties of a medium, i.e., the chirality or optical

activity [67]. Together they explain interesting optical phenomenon, such as circular phase


9

retardation [68] and differential extinction [69] in a medium. These would be discussed in later

part of this thesis.

Boundary conditions and Fresnel equations

Next, we discuss boundary conditions based on polarized light interaction with an interface using

Maxwell’s equations. The incident (𝑬1), transmitted (𝑬2) and reflected (𝑬𝑟1) fields can then be

further evaluated using Fresnel equations [66, 70]. Based on the conservation of energy, the

vectors mentioned above can be written as,

|𝐄1|2 = |𝐄2|

2 + |𝐄r1|2, (2.14a)

and for Fresnel complex amplitude reflection/transmission coefficient can be related as [66, 70],

𝐄2r1⁄

TETM⁄

= c2/r1

TETM⁄

. 𝐄1

TETM⁄

, (2.14b)

where, 𝑐2/𝑟1

𝑇𝐸𝑇𝑀⁄

defines the respective complex amplitude coefficients between medium 1 & 2 [58,

65, 70]. We consider an interface between two mediums with different refractive index (n1 & n2).

The Maxwell curl and divergence equations (refer Equation 2.1a-d) leads us to the boundary

condition for the tangential electric field and normal displacement field components respectively

being continuous and mathematically shown as,

𝐄∥1 = 𝐄∥

2, 𝐃⊥1 = 𝐃⊥

2 , (2.15a)

where 𝑫 = 휀𝑟𝜖0𝑬, and 휀𝑟 being relative permittvity of the medium. In a similar fashion we can

derive the boundary condition for magnetic fields. Here, for ease of mathematical process, we

define orthogonal plane waves as parallel and perpendicular to the plane of incidence as shown

in Figure 2.3. These are mathematically defined as [7],

𝐄1 = 𝐄TM1 + 𝐄TE

1 , (2.15b)

where 𝐸𝑇𝑀1 is parallel and 𝐸𝑇𝐸

1 is perpendicular (senkrecht) to the plane of incidence as shown in

Figure 2.3. We consider the interface between two mediums as shown in Figure 2.3 with refractive

indices, permittivity and permeability referred to as n1, ε1, μ1 and n2, ε2, μ2 for medium 1 and 2

respectively. In a similar fashion, for incident and transmitted beam we can define the wavevector

considering boundary conditions as,

𝐤1 = (𝑘𝑥1, 𝑘𝑦1, 𝑘𝑧1), |𝐤1| = 𝑘1 = ω

c √ε1μ1, (2.16a)

𝐤2 = (𝑘𝑥2, 𝑘𝑦2, 𝑘𝑧2), |𝐤2| = 𝑘2 = ω

c √ε2μ2, (2.16b)

Here, the definition of transverse and longitudinal wavenumber are the same as mentioned above

for Equation 2.6c. From the boundary conditions in Equation 2.15 we can also deduce that,

𝑘𝑥1 = 𝑘𝑥2 = 𝑘𝑥 , 𝑘𝑦1 = 𝑘𝑦2 = 𝑘𝑦 and (2.17a)


10

|𝐤1| = |𝐤0|n1. (2.17b)

Implying that transverse wavenumber is conserved. In fact, this leads us also to the infamous

Snell’s law [58]. Considering the incident wavevector 𝐤1 making an angle of 𝜃 with the normal to

the interface (as shown in Figure 2.2) [7], we can define the component parallel to the interface

by,

k∥1 = |𝑘1| sin θ1 = k∥2 = |𝑘2| sin θ2 = √𝑘𝑥2 + 𝑘𝑦

2, (2.18a)

and for the longitudinal wavevector component we can write,

𝑘𝑧1 = 𝑘⊥1 = |k1| cos θ1 = √𝑘12 − 𝑘∥

2, (2.18b)

𝑘𝑧2 = 𝑘⊥2 = |k2| cos θ2 = √𝑘22 − 𝑘∥

2. (2.18c)

Considering the above-mentioned equations, we can now deduce the case for transmitted and

reflected fields at an interface by applying boundary conditions for electric and magnetic fields

and analytically solving them. For a linear homogeneous isotropic medium, the electric and

magnetic fields are related by intrinsic impedance 𝑧 = √μ

ɛ of the medium and thus we end up

with amplitude coefficients of reflection and transmission as,

𝑟12𝑇𝐸 =

𝑛1 cos𝜃𝑖−𝑛2 cos𝜃𝑡

𝑛1 cos𝜃𝑖+𝑛2 cos𝜃𝑡=

𝑘𝑧1−𝑘𝑧2

𝑘𝑧1+𝑘𝑧2, (2.19a)

𝑟12𝑇𝑀 =

𝑛2 cos𝜃𝑖−𝑛1 cos𝜃𝑡


𝑛22𝑘𝑧1−𝑛1

2𝑘𝑧2

𝑛22𝑘𝑧1+𝑛1

2𝑘𝑧2, (2.19b)

𝑡12𝑇𝐸 =

2𝑛1 cos𝜃𝑖


2𝑘𝑧1

𝑘𝑧1+𝑘𝑧2, (2.19c)

𝑡12𝑇𝑀 =

2𝑛1 cos𝜃𝑖


2𝑛1𝑛2𝑘𝑧1

𝑛22𝑘𝑧1+𝑛1

2𝑘𝑧2. (2.19d)

The amplitude coefficients are related as [64, 71],

𝑡12𝑇𝐸 = 𝑟12

𝑇𝐸 + 1, 𝑛𝑡12𝑇𝑀 = 1 − 𝑟12

𝑇𝑀. (2.19e)

The observable quantities, reflectance (R) and transmittance (T) are the power of reflected and

transmitted fields. By conservation of energy for a non-absorbing, dielectric medium,

𝑅 + 𝑇 = 1, (2.20)

where 𝑅 = 𝑟. 𝑟∗ = |𝑟2| and 𝑇 = (𝑛cos𝜃𝑡

cos𝜃𝑖) |𝑡2|.


11

Figure 2.3 The propagation of TM (a) & TE (b) light fields at an interface. The Maxwell curl and

divergence equations leads us to the boundary condition for the tangential electric field and

normal displacement field components respectively being continuous. Implying that the

transverse wavenumber is conserved, leads us to the infamous Snell’s law.

Permittivity, permeability, and refractive index of a medium

From Maxwell’s equations, we can relate the electric displacement and field by the so-called

constitutive relations. Similar case can be assumed for the magnetic flux density and field. The

constitutive relations describe the medium’s interaction with the incoming electromagnetic fields

[58, 72]. These are shown as,

𝐃 = ε0𝐄 + 𝐏, (2.21a)

B= μ0(𝐇 + 𝐌). (2.22b)


12

In Equations 2.16a and b, we introduce the mean dipole moment per unit volume expressed as

polarization and magnetization of a medium respectively (𝐏 and 𝐌). As in the previously described

case for boundary conditions (linear, homogenous, and isotropic), we can define polarization and

magnetization as,

𝐏 = ε0χe𝐄, also (1 + χe ) = εr , 𝐌 = μ0χm𝐇 , also (1 + χm ) = μr, (2.23)

where 𝜒𝑒 and 𝜒𝑚are the electric and magnetic susceptibility of the medium. 휀𝑟 and 𝜇𝑟 are the

relative permittivity and permeability of a medium (normalized with respect to free space values).

Essentially these equations serve as the starting point for understanding all sorts of light-matter

interaction. The Drude model for metals and Lorentz model for dielectrics and metals describes

the dependence of complex permittivity on angular frequency and material properties [64]. In

some instances, permittivity (휀) and permeability (𝜇) of a medium can take the form of a tensor

(anisotropic medium) or susceptibility could have higher order terms to define a nonlinear

medium response. Typically, naturally occurring materials have permittivity (휀) and permeability

(𝜇) that are dependent and change with angular frequency (𝜔). Usually, close to visible

frequencies the naturally occurring materials are non-magnetic meaning permeability close to

unity. The real part of permittivity is positive/negative, distinguishing between two naturally

occurring solids dielectrics/metals, respectively. These details will be important in later part of

this thesis regarding computation model for carbon flakes (built by our collaborator, Professor Dr.

Antonino Calà Lesina) and for considering chiroptical phenomenon.

Refractive index, which was used extensively in deriving boundary conditions, can also be related

to permittivity and permeability of a medium. Since permittivity is also a complex number and

assuming permeability being unity, a generalized form of refractive index can be expressed as a

complex number [64, 73],

�� = √𝜇𝑟휀𝑟 = 𝑛 + 𝑖𝜅, (2.24)

휀�� = 휀𝑟′ + 𝑖휀𝑟

′′, where 휀𝑟′ = 𝑛2 + 𝜅2 and 휀𝑟

′′ = 2𝑛𝜅, (2.25)

where 𝑛 / 𝜅 define the real/imaginary part of the refractive index, respectively. κ is also known as

extinction coefficient, which is related to the attenuation constant (α) by the relation, α =4𝜋𝜅

𝜆0,

where 𝜆0 is the wavelength of incident light onto the medium. 휀𝑟′ , 휀𝑟

′′ are the real and imaginary

part of complex permittivity, respectively. We also modify Equation 2.15 to accommodate

absorption in medium which now reads as,

R + T + A = 1, (2.26)

where A depicts the power of incident light beam absorbed by the interacting medium. We

discuss now two special conditions when refractive index of incident medium is lower than

transmission medium (𝑛2 > 𝑛1) , namely case for normal incidence and that of Brewster Effect

[74]. These two special conditions would later help us in defining the computational model used

for extracting refractive index.


13

Figure 2.4 Reflectance for various incident angles for TE and TM polarized input beams. The

incident medium is air, while the second medium is considered as N-BK7 glass (n=1.51). At

normal incidence, we get simplified mathematical equations for reflectance (RTE and RTM), where

TE and TM components are equal in magnitude. At Brewster angle, reflected light is completely

TE polarized. For air-glass interface this happens at an incidence angle of 56.4°.

Normal incidence: For normal incidence, first we consider the case of vanishing imaginary part of

refractive index. This leads to simplified mathematical equations for reflectance (RTE and RTM),

where TE and TM components are equal in magnitude. Considering 𝑛 = 𝑛2/𝑛1 We can write the

reflectance as

𝑅12𝑇𝐸/𝑇𝑀

= |1−𝑛

1+𝑛|2. (2.27a)

Considering the case of commonly used microscopic cover glass N-BK7 in the lab, the refractive

index at 632nm is 1.51, which leads to a reflectance of around ~ 4% as shown in Figure 2.4. For

absorbing medium (non-zero imaginary part of refractive index), the equations for reflectance

simplifies as

𝑅12𝑇𝐸/𝑇𝑀

=(𝑟𝑒𝑎𝑙(𝑛)−1)2+𝑖𝑚𝑎𝑔(𝑛)2

(𝑟𝑒𝑎𝑙(𝑛)+1)2+𝑖𝑚𝑎𝑔(𝑛)2. (2.27b)

For example, considering gold with complex refractive index of n=0.1879-3.4223i, the normally

incident light at 632 nm is reflected by 94.3%, as shown in Figure 2.5.

Brewster angle: For the dielectric loss-less case where 𝑛2 > 𝑛1, for a specific angle of incidence,

the TM reflection is completely extinguished [74]. Mathematically this is expressed as

tan 𝜃𝐵 =𝑛2

𝑛1 (2.27c)


14

Figure 2.5 Reflectance graphs for various incident angles for TE and TM reflected light. The

incident medium is air, while second medium is considered gold (n=0.1879-3.4223i). At normal

incidence, TE and TM reflectance are equal in magnitude light (94.3%,). In case of an absorbing

media, a minimum (non-zero) in TM reflectance is observed at a certain incidence angle called

pseudo-Brewster angle. As an example, for gold, this happens around 72.3°.

Where 𝜃𝐵 is called Brewster angle. This essentially means that at Brewster angle, reflected light

is completely TE polarized. For example, for cover glass N-BK7 this happens at 56.3° at 632nm, as

shown in Figure 2.4. In case of an absorbing media, a minimum (non-zero) in TM reflectance is

observed. This angle is usually also referred to as pseudo-Brewster angle [74-77]. For example, for

gold, this happens around 72.3°, as shown in Figure 2.5.

In the above discussion, till now we have concentrated on building an understanding of polarized

light field propagation, boundary conditions and conditions related to medium change (interface).

We studied the refractive index cases for normal and incidence at Brewster angle. We can already

see that material properties play an important role in polarized light matter interaction [71].

In the next section we will concentrate on understanding Jones and Müller formalism which

defines polarized light and matter interaction in terms of vectors and matrices. This helps in

defining the incident and exiting polarized light as vectors while optical properties of interacting

matter are defined in a matrix (Jones and Müller matrix). Eventually, these formalism helps in

quantifying the optical properties of a medium which can be implemented/extracted in an

experimental system [78].

2.2 Jones and Müller formalism

As briefly discussed in Equation 2.15 and later in the solution of Fresnel equations, a paraxial

polarized light field can always be written as superposition of two orthogonal plane waves [7, 56,

58, 64]. The polarization state is defined by the amplitudes and the phase difference between two

orthogonal plane waves. We will use this notion to elaborate the interaction of polarized light

with matter and how it further deepens our understanding of the optical properties of a material.


15

Jones formalism

In 1941, the American physicist R. C. Jones, introduced a formalism based on Equation 2.16 to

define polarized light and its interaction with matter in a simple equation, now known as Jones

formalism [79]. We can write the Jones matrix and input and output light beam vectors together

mathematically as

(𝐄x

out

𝐄yout) = J (

𝐄xin

𝐄yin), (2.28)

where, the fields are related to respective incident and outgoing intensities by the relation,

𝐼𝑖𝑛/𝑜𝑢𝑡 = 𝐄in/out∗ 𝐄in/out and 𝐸∗define the complex conjugate of the respective field vector [58,

79, 80]. Here, the Jones matrix 𝐽 = (𝐽11 𝐽12

𝐽21 𝐽22) is a 2x2 matrix, which defines the optical

interaction of a medium with incident light beam. The terms of the Jones matrix are usually of

complex nature (amplitude and phase terms) and hence a total of 8 independent variables are

required to completely define the interaction of light with a medium [31]. We can also

conveniently define the interaction of light with multiple optical elements by cascaded

multiplication of respective Jones matrices (𝐽1 interacts first with the incident light beam). The

incident and outgoing light waves are then related as,

(𝐸𝑥

𝑜𝑢𝑡

𝐸𝑦𝑜𝑢𝑡) = (𝐽𝑛𝐽𝑛−1 …𝐽1) (

𝐄𝑥𝑖𝑛

𝐄𝑦𝑖𝑛) = 𝐽𝑡𝑜𝑡 (

𝐄𝑥𝑖𝑛

𝐄𝑦𝑖𝑛). (2.29)

As discussed above, each element of the Jones matrix is of complex nature (assuming the form

𝐴𝑒𝑖𝜑), where 𝐴 defines the amplitude and 𝜑 defines the phase associated with the respective

field component.

Amplitude ratios of Jones matrix

As it is evident from linear matrix calculations, real numbers in Jones matrix would correspond to

a changing ratio of orthogonal components of the incoming light wave. This can be thought of as

selective polarization absorber (e.g., polarizer) with arbitrary axis position (θ) leading to light

polarized at a certain azimuthal angle (θ) without introducing any phase difference.

Mathematically this matrix can then be written as,

𝐽𝑝𝑜𝑙 = ( cos2 𝜃 cos 𝜃 sin 𝜃cos 𝜃 sin𝜃 sin2 𝜃

) (2.30)

Phase ratios of Jones matrix

By adding certain phase to one field component of the incoming light, we can induce ellipticity

(𝜑) in the outgoing light wave [81, 82]. Optical materials (crystals) which can induce such an effect

have different refractive indices along orthogonal field components of the incoming light beam.

The generalized Jones matrix for a wave retarder is hence shown as,

𝐽𝑟𝑒𝑡𝑎𝑟𝑑𝑒𝑟 = (cos2 𝜃 + 𝑒𝑖𝛿sin2 𝜃 (1 − 𝑒𝑖𝛿)𝑒−𝑖𝜌cos𝜃 sin𝜃

(1 − 𝑒𝑖𝛿)𝑒𝑖𝜌cos 𝜃 sin 𝜃 𝑒𝑖𝛿cos2 𝜃 + sin2 𝜃), (2.31)


16

where 𝛿 = 𝛿𝑥 − 𝛿𝑦 is the phase delay induced between orthognal field components of the

incoming light wave. 𝜃 is the orientation of the optic axis and 𝜌 defines the type of retardance.

For a linear retarder, 𝜌 is assumed to be zero. The phenomenon of wave retardance can be

accomplished using naturally occurring uniaxial crystals or by using sophisticated electrically

tunable retarder devices [30, 83].

For a rotation of an optical element or for redefining the coordinate system along a new

orthogonal direction, we can use coordinate transformation as,

𝐽𝑟𝑜𝑡 = 𝑅(𝛼)𝐽𝑅(−𝛼), (2.32a)

where 𝑅(𝛼) = (𝑐𝑜𝑠(2𝛼) 𝑠𝑖𝑛(2𝛼)𝑠𝑖𝑛(2𝛼) 𝑐𝑜𝑠(2𝛼)

) (2.32b)

Due to its simplicity and elegant mathematical description, Jones formalism serves as the basis

for defining polarized light matter interaction. In many cases optical properties extraction is

performed using Jones formalism even though the polarization measurements are performed

using other methods [84]. Besides the obvious advantages, Jones formalism has certain

limitations. As it can be observed from the above discussion, a completely polarized incident light

wave is necessary for the evaluation of Jones formalism. However, experimentally this is normally

not the case. This brings us to another approach for analyzing polarized light matter interaction

which can cope with partial and unpolarized light beams.

Stokes parameters and Müller formalism

In 1852, Sir G.G. Stokes introduced the idea of evaluating the polarization state of light by

measuring the intensity of light in certain projected states of light beam [85]. The vector which

describes the nature of polarized light wave is called Stokes vector and its elements are called

Stokes parameters,

𝐒 = [

𝑆0

𝑆1

𝑆2

𝑆3

] = [

𝐼𝑄𝑈𝑉

] =

[

Ex∗Ex + Ey

∗Ey

Ex∗Ex − Ey

∗Ey

Ey∗Ex + Ex

∗Ey

i(Ex∗Ey − Ey

∗Ex)]

=

[

𝐼𝑥 + 𝐼𝑦𝐼𝑥 − 𝐼𝑦

𝐼45 − 𝐼−45

𝐼𝑅𝐶𝑃 − 𝐼𝐿𝐶𝑃] . (2.33)

As observed in Equation 2.33, Stokes vector is represented by observable intensity-based

measurements in certain orthogonal set of states (linear: X/Y or H/V and 45/-45 or D/A, circular:

RCP/LCP), which shows the experimental benefit of using the Stokes vector [65, 86]. As evident

from the first Stokes parameter, it shows the total intensity of incoming light wave. The rest of

the three Stokes parameter represent the polarization state of incoming light beam. As per

definition of the polarization ellipse, the second and third parameter describes mainly the light

polarized at a certain azimuthal angle (θ) (degree of linear polarization) while fourth parameter

depicts the ellipticity (degree of circular polarization) of the light beam. Hence, for a partially or

completely polarized beam, following condition holds true, 𝑆02 ≥ 𝑆1

2 + 𝑆22 + 𝑆3

2. The Stokes vector

is related to the polarization ellipse mentioned above [65],


17

Figure 2.6 A Poincare sphere. A light beam defined by Stokes vector on the great circle of S1 & S2

is purely linear (vanishing S3). All other polarizations states have non-zero ellipticity (𝜑), with

limiting case of left/right circularly polarized light defined by 𝜑 =−𝜋

4,𝜋

4 respectively.

𝐒 = [

𝑆0

𝑆1

𝑆2

𝑆3

] = [

𝐼𝑄𝑈𝑉

] = [

𝐼𝐼𝑝 cos 2𝜃 cos 2𝜑𝐼𝑝 sin2𝜃 cos2𝜑

𝐼𝑝 sin2𝜑

] (2.34)

where 𝑝 is the degree of polarization, defined as 𝑝 =√𝑆1

2+𝑆22+𝑆3

2

𝑆0. For fully polarized light 𝑝=1, while

for partially polarized the values varies between 0 < 𝑝 < 1. Also, we can derive relations for

azimuthal orientation angle in terms of Stokes parameters as tan 2𝜃 = 𝑆2

𝑆1⁄ and for ellipticity

angle as sin2𝜑 = 𝑆3

𝑆0⁄ , with the argument of 𝑆3 defining the handedness of polarizations state.

For a completely polarized light (𝑝=1), if we normalize the Stokes vector with total intensity

(defined by 𝑆0), we get,

𝐒

𝑆0= S =

[

1𝑆1

𝑆0⁄

𝑆2𝑆0

⁄

𝑆3𝑆0

⁄ ]

= [

𝐼𝑄𝑈𝑉

] = [

1cos𝜃 cos𝜑sin𝜃 cos𝜑

sin𝜑

]. (2.35)


18

The second, third and fourth Stokes parameter can then be used to plot the polarization state on

a sphere, which was first discussed by Henry Poincare in 1892 (as shown in Figure 2.6 ). It is

defined with three axis pointing along second, third and fourth Stokes parameter. The polarization

states on the surface of the sphere would correspond to completely polarized light while any state

inside the sphere would correspond to partially polarized light. Similar to Jones formalism, we can

mathematically relate incident and output light beam after interaction with the material/medium

under study [78, 85, 87].

��𝑜𝑢𝑡 = 𝑀��𝑖𝑛 , where 𝑀 = [

𝑚00 𝑚01 𝑚02 𝑚03

𝑚10 𝑚11 𝑚12 𝑚13

𝑚20 𝑚21 𝑚22 𝑚23

𝑚30 𝑚31 𝑚32 𝑚33

] . (2.36)

Where ��𝑜𝑢𝑡 (��𝑖𝑛) are outgoing (incoming) Stokes vector of light wave while, 𝑀 is a 4x4 matrix

which defines the interaction with medium (optical system) with real valued elements. Although

the concept of Stokes vector-based polarization state evaluation was developed much earlier [87],

it was Hans Müller in 1940’s who, building up on the ideas of F. Perrin and P. Soleillit, introduced

a formalism for Müller matrices. Since Stokes vector and Müller formalism are directly related to

measuring intensities of light, henceforth, it has been the method of choice involving polarimetric

measurements [63, 78, 88].

Considering the light matter interaction, we can inversely solve the Equation 2.36. This means

that if the input and output polarized state of light is known, we can extract the optical response

of a medium, also called, Müller polarimetry. The application of Müller polarimetry can be found

in various fields including but not limited to ellipsometry [81], characterization of chemicals and

liquids [89], remote sensing [90], radar polarimetry [91] and for polarized light scattering leading

to various applications [92]. Due to the symmetries of Müller matrix, an experimentally extracted

Müller matrix should fulfill certain conditions [93-95], such as,

a. 𝑇𝑟(𝑀𝑀𝑇) ≤ 4𝑚002 ,

b. 𝑚00 ≥ 𝑚𝑖𝑗 ∈ 𝑖, 𝑗 = 0,1,2,3,

c. 𝑚002 ≥ (𝑚01

2 + 𝑚022 + 𝑚03

2 ).

A rotation of an optical element around the optical axis of a light beam can change the resultant Müller matrix. This change is dependent on the degree of rotation and can be easily calculated in a similar way as Jones formalism. For angle of rotation (𝛼), mathematically we can state that,

𝑀𝑟𝑜𝑡 = 𝑅(𝛼)𝑀𝑅(−𝛼), (2.37a)

where 𝑅(𝛼) = (

1 0 0 00 𝑐𝑜𝑠(2𝛼) 𝑠𝑖𝑛(2𝛼) 00 𝑠𝑖𝑛(2𝛼) 𝑐𝑜𝑠(2𝛼) 00 0 0 1

). (2.37b)

Since both Jones and Müller formalism deal with polarized light-matter interaction, extensive

research has been done to mathematically link both of them [84, 96]. More details can be found

in Appendix A to this thesis.


19

2.3 Optical material properties

In this section, we will elaborate on the optical properties of an examined medium and relevant

mathematical techniques to extract these properties from an experimental Müller matrix. We will

discuss some common setups for determining the Müller matrix of an examined material. Later,

we will elaborate the concept of complex refractive index retrieval and discuss the computational

model for that process.

Optical properties from Müller matrix

Here, we will briefly discuss optical properties that can be extracted from a Müller matrix. These

can be broadly categorized as, depolarization, dichroism, and birefringence of a medium. We

consider the case of Müller matrix in transmission.

Depolarization of incident light

This describes the phenomenon in which polarized light is coupled into depolarized light. It can

intrinsically happen in case of scattering or for loss of coherence in a polarized light wave. The

Müller matrix for a depolarizer can be shown as [63],

𝑀𝑑𝑒𝑝𝑜𝑙 = [

1 0 0 00 𝑝𝑥,𝑦 0 0

0 0 𝑝𝐴,𝐷 0

0 0 0 𝑐𝑝

], (2.38)

where 𝑝, 𝑝45 and 𝑝45 are the depolarization along horizontal, diagonal, and circular polarization

states, respectively. Another way to describe depolarization is also as the variation in degree of

polarization of light as discussed earlier. In an optical experiment, depolarization can lead to

unrealizable experimental results and therefore, needs to be identified and removed accordingly

[63, 71, 93, 95, 97-99].

Dichroism of a medium

Dichroism is the phenomenon whereby an incident light beam travelling through a medium

encounter differential extinction. The outgoing intensity can be maximum along one field

component while minimum along the other orthogonal field component of exiting light beam.

This optical phenomenon can also be understood as the difference in imaginary part of refractive

index, i.e., extinction coefficient along two orthogonal directions for a medium Mathematically

this can be defined as,

𝐷 =𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛

𝐼𝑚𝑎𝑥 +𝐼𝑚𝑖𝑛, (2.39a)

𝐷 =(𝜅𝑎−𝜅𝑏)𝜆

2𝜋𝑙, (2.39b)

where 𝜅 is the extinction coefficient, 𝑙 is the length for which light propagates through the

medium and a, b represents the orthogonal polarization projection. Based on the mathematical


20

definition of the Stokes vector, we can define three polarization projection sets, which are as

follows,

𝐿𝑖𝑛𝑒𝑎𝑟 𝐷𝑖𝑐ℎ𝑟𝑜𝑖𝑠𝑚 = 𝐿𝐷 =(𝜅𝑥−𝜅𝑦)𝜆

2𝜋𝑙 , (2.39c)

𝐿𝐷45 =(𝜅45−𝜅−45)𝜆

2𝜋𝑙 , (2.39d)

𝐶𝑟𝑖𝑐𝑢𝑙𝑎𝑟 𝐷𝑖𝑐ℎ𝑟𝑜𝑖𝑠𝑚 = 𝐶𝐷 =(𝜅𝑅𝐶𝑃−𝜅𝐿𝐶𝑃)𝜆

2𝜋𝑙 . (2.39e)

The information regarding dichroism of a material can be found in the first row and column of

respective Müller matrix of the medium. Linear polarizers are a good example of a material

possessing linear dichroism. The case of circular dichroism leads to chirality in a medium which

will be discussed in Chapter 4 of this thesis.

Optical birefringence of a medium

Analog to the concept of wave retardation in Jones matrix, birefringence is an intrinsic property

of a medium in which the real part of refractive index (n), introduces a phase (optical path length)

difference for the incoming light wave, depending on polarization and orientation of a medium.

Mathematically this is shown as [81],

∆n =(𝛿𝑎−𝛿𝑏)𝜆

2𝜋𝑙, (2.40a)

𝐿𝐵 = (𝑛𝑥 − 𝑛𝑦) =𝛿𝜆

2𝜋𝑙 , (2.40b)

𝐿𝐵45 = (𝑛45 − 𝑛−45) =𝛿45𝜆

2𝜋𝑙, (2.40c)

𝐶𝑟𝑖𝑐𝑢𝑙𝑎𝑟 𝐵𝑖𝑟𝑒𝑓𝑟𝑖𝑛𝑔𝑒𝑛𝑐𝑒 = 𝐶𝐵 = (𝑛𝑅𝐶𝑃 − 𝑛𝐿𝐶𝑃) =𝛿𝐶𝜆

2𝜋𝑙. (2.40d)

As it can be understood, in the case of linear birefringence case, the resultant effect is the increase

in ellipticity of the outgoing light wave. In other words, an incoming linear polarization state on

Poincare sphere moves from the great circle (S1 - S2 plane) towards north / south pole depending

on the retardance and optic axis orientation. Considering the case of uniaxial crystals as briefly

mentioned above, the incoming orthogonal field components of light wave experience different

refractive indices defined by slow axis (higher refractive index) and fast axis (lower refractive

index). Usually in crystal materials convention, the terms ordinary and extraordinary are used to

define the two principal axes: the ordinary axis, which is orthogonal and extraordinary axis, which

is parallel to optic axis of crystal, respectively. The optic axis of a crystal is defined as the direction

along which incident light beam experiences no retardation [71]. Based on above discussion, it

can be understood that for a thickness of such anisotropic crystal 𝑙, different optical path lengths

(𝑛 × 𝑙) would exist along the two principal axes leading to a phase delay in orthogonal field

components. Hence by choosing the right thickness (𝑙), the refractive index difference (∆n) and

optic axis position (𝜃), we can transform the incoming light beam to any state over a Poincare

sphere. A pure linear retarder in this case, with 𝛿 retardance and θ as optic axis can be

represented in a Müller matrix as,


21

𝑀 =

(

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

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

0 𝑠𝑖𝑛(2𝜃)𝑠𝑖𝑛(𝛿) −𝑐𝑜𝑠(2𝜃)𝑠𝑖𝑛(𝛿) 𝑐𝑜𝑠(𝛿) )

. (2.41)

Müller matrices and optical systems

As discussed above, Müller matrix can describe a set of properties which can help define optical

response of a medium. It is also pertinent to differentiate between observable quantities and

intrinsic properties of a system. For instance, extinction is an observable quantity which

depending on the length of the medium and wavelength of incoming light wave, defines

dichroism. Similar is the case for the observable transmission/retardance, which are related to

attenuation/birefringence, of a medium, respectively. For an optical system possessing

abovementioned optical properties, it can be directly correlated to a certain element of

logarithmic Müller matrix [63, 71, 81]. This can be shown as,

log(𝑀) = [

𝑇 −𝐿𝐷 −𝐿𝐷45 𝐶𝐷−𝐿𝐷 𝑝 𝐶𝑅 −𝐿𝐵45

−𝐿𝐷45 −𝐶𝑅 𝑝45 𝐿𝐵𝐶𝐷 𝐿𝐵45 −𝐿𝐵 𝑐𝑝

]. (2.42)

Figure 2.7 A non-depolarizing Müller matrix, depicting individual matrix elements and their

relation to specific optical properties in terms of refractive index of a medium. The cross-

diagonal elements are related to optical activity of a material while remaining non-diagonal

elements are related to linear anisotropies and dichroism.


22

It is important to highlight that Equation 2.42 represents element wise association of optical

properties. If we consider an isotropic, homogenous, non-depolarizing absorbing medium, we can

expect the diagonal terms to depict attenuation experienced by an incoming light beam, as shown

in Equation 2.42 and Figure 2.7.

As discussed in Equation 2.40(a-d), anisotropic medium possesses different refractive indices

along principal axis of medium. The relevant anisotropic information of a medium is present in off

diagonal elements shown as in Equation 2.42 and Figure 2.7. We can see that the differential

extinction and linear retardance effects are segregated in upper right and lower left corner of

Müller matrix, respectively. Because of the cross interaction of linear dichroism and birefringence

we can expect 𝑚12 & 𝑚21 to have residual values, although these elements also depict chiral

effects [97, 100]. For an ideal linear retarder and attenuator, we can expect 𝑚12 & 𝑚21 to be

equal in magnitude and have the same sign.

Isotropic chiral media

For definition of chiral media as discussed above, we can expect circular dichroism and retardance

from such a medium. As shown in Equation 2.42 and Figure 2.7, the cross-diagonal terms depict

the chiral properties of the medium. It should be noted that for reciprocal system 𝑚03& 𝑚30 have

same sign and magnitude, while for non-reciprocal system we can expect opposite sign of these

matrix elements. A detailed description of reciprocal/ non-reciprocal systems and their

comparison to chiral response of a medium can be found in Appendix A to this thesis. Typically,

circular retardance are small in magnitude (10-3 and lower) and hence experimentally difficult to

detect. A common experimental technique in this case is to measure circular retardance along the

optic axis of anisotropic medium (thus avoiding optical effects from linear retardance).

Polarimetric systems and Müller matrix decomposition

To experimentally record information regarding polarized light matter interaction, different

schemes are employed, which eventually are computationally evaluated to extract Müller matrix.

Some of the existing commercially available techniques/polarimetric setups are briefly mentioned

below.

Müller matrix-based polarimetry

Based on the Müller matrix approach, different polarization measurement systems are designed.

A Müller matrix is computationally extracted from the input and output Stokes vector as discussed

above. A generic structure of a Müller matrix based polarimeter is shown in Figure 2.8 [78]. The

incoming laser passes through the polarization state generator (PSG), consisting of a polarizer and

a wave retarder, to selectively generate different input polarizations states. After passing through

the sample, the light beam is projected into 6 polarizations states (H, V, A, D, RC & LC) necessary

for Stokes vector using polarizing state analyzer (PSA). Different techniques can be used to record

the respective intensities using CCD camera or photodiodes. The main difference in different

approaches is based on the three main components namely PSG, PSA and detectors. One of the

commonly used method involves using a quarter-wave retarder with a polarizer in PSA. By

rotating the QWP to certain angular positions and recording respective intensity, we can extract


23

the Stokes vector. The process is repeated for different input polarization states. Eventually, the

Müller matrix can be extracted from the recorded data by solving Equation 2.43. A detailed

description of various polarimetric techniques can be found in the literature [78]. We discuss

some of the common polarimetric techniques below.

Figure 2.8 Müller matrix-based polarimetry setup. The incoming laser passes through the

polarization state generator, consisting of a polarizer and a wave retarder, to selectively

generate different input polarizations states (𝑆𝑖𝑛) . The polarization nature (𝑆𝑜𝑢𝑡) of the

transmitted light (through the examined sample) is evaluated by polarization state analyzer.

Different techniques can be used to record the respective intensities using CCD camera or

photodiodes [78, 101].

Dual rotating retarder Müller matrix polarimeter

As the name suggests, the setup shown in Figure 2.8 is modified to have rotating retarders in both

PSG and PSA. The retardance in both cases is fixed to quarter wave retardance (λ/4). The PSG/PSA

retarders completes the 360° rotation with variable angular frequency of ratio 1:5 respectively.

The continuous rotation of retarders generate a periodic intensity signal. By performing a Fourier

analysis of the recorded signal, the elements of the Müller matrix are retrieved from the time

multiplicative constants of the periodic signal [29, 63, 78].

Müller matrix polarimeters with variable phase linear retarders

In this polarimetric setup, tunable linear retarders are used which can be electrically controlled.

The setup can consist of two linear retarders or four linear retarders [78] depending on

polarimetric configuration. In case of four variable linear retarders, two are used in both PSG and

PSA, to generate/evaluate different polarization states of light required for Stokes vector

evaluation. Liquid crystal cells are usually used for providing variable retardance by changing their

input voltages. By six different combinations of retardances, all polarization states on the Poincare

sphere can be realized [30]. In the course of this thesis, we worked on a specialized technique

(Microscopic Müller matrix measurement technique), which was used to study samples of small

lateral dimension (of a few microns), to extract the experimental Müller matrix of examined

microscopic structures [14, 15, 18]. This will be discussed in detail in experimental part of Chapter

3.


24

Decomposition of experimental Müller matrix

As briefly discussed above, if a material possesses multiple optical properties as shown in Equation

2.42, a mathematical matrix decomposition needs to be performed to extract individual optical

properties. This is performed by analytical decomposition of the Müller matrix. In the literature a

number of methods can be found, with certain constraints to extract optical properties. Some of

the known methods are polar decomposition [102], logarithmic method [63, 81], and analytical

inversion methods [31, 84]. In all cases, the core idea is to untangle the optical properties by

mathematical operations. For instance, in case of polar decomposition, we can write the equation

[63, 93, 102, 103],

𝑀 = 𝑀𝑝𝑀𝑅𝑀𝐷. (2.43)

After subsequent decomposition we can extract three separate matrices 𝑀𝑝, 𝑀𝑅 & 𝑀𝐷 for

depolarization, retardance and diattenuation, respectively [102]. The optical properties can then

be extracted from respective matrices. In case of analytical inversion, a conversion of Müller

matrix to Jones matrix (after elimination of depolarization information) is performed using the

Kronecker product relation mentioned in Appendix A to this thesis. Jones matrix calculus is then

used to extract the optical properties [84]. In the course of this thesis, we have used analytical

inversion and polar decomposition to analyze and extract optical properties from Müller matrix.

A Müller matrix of a material provides differential optical information of a medium (for instance

dichroism and birefringence as shown in Equations 2.39a-s and 2.40a-d). Many a times, it is

necessary to know the exact complex refractive indices of a medium rather than the differential

values. We will discuss the theoretical aspects regarding the direct determination of complex

refractive indices of a medium in the next section. Unlike the case of Müller matrix discussed

above in transmission, we will now define polarized light matter in reflection and will utilize

optical phenomenon explained in section 2.1.4 as starting point.

Ellipsometry

As discussed in section 2.1.4, the refractive index proves to be an important aspect of light matter

interaction and provides a greater insight into material/medium response [104]. Hence besides

other optical properties, it has been of prime importance for various fields of research, ranging

from biological samples [105, 106], semiconductors and microelectronics [107], sensing devices

[108], to various other scientific areas [71, 81]. This accentuates the importance of accurately

estimating the refractive index of a medium. With the advances in nanofabrication and

technology, more and more sophisticated techniques are required to estimate the refractive

index of structure at smaller length scales. Ellipsometry is an old field of science with a lot of

standard techniques, which work with high accuracy to extract complex refractive index of a

material [81, 107, 109, 110]. Normally, ellipsometric techniques are employed in reflection since

extraction of ellipsometric parameters is easier as compared to transmission [111]. Normally,

ellipsometers would involve setups with moving arms or tilting stages to collect ellipsometric

information at various incident angles [112]. To get rid of moving parts, which might introduce

artifacts and errors and to also enable the measurement of materials at the micron-scale, we


25

extend the capabilities of conventional ellipsometry with back focal plane imaging and structured

illumination. As discussed in Equation 2.12, focused light provides for a large angular spectrum

(multiple angles of incidence) [62] and, hence, might allow for the retrieval of ellipsometric data

in a single measurement [113]. A simple step towards this idea was reported recently using

structured light to extract the real part of the refractive index for dielectric media (negligible

absorption) [114]. The main concept is based on vanishing reflectance for TM polarized light at

certain incidence angle (Brewster effect) [74] to extract refractive index exploiting greater angular

spectrum offered by high NA microscope objective [115, 116]. Here, we will discuss computational

methods to extend the technique to the case of complex refractive index retrieval. We will use

the far-field angular spectrum representation for studying the interaction of tightly focused light

with a medium in reflection, and discuss how it can be utilized to extract complex refractive index

[117]. Later, we will define the computational fitting model, which can be used to extract the

required ellipsometric parameters. Later, in the next chapter we will discuss the experimental

setup and results for certain materials tested.

Ellipsometric equation and measurement techniques

We start with considering Equations 2.19a&b [81]. In general, it can be used as the basis of

ellipsometry leading to various methods developed to extract observables (reflectance) for

gaining knowledge about ellipsometric parameters or optical constants of a medium (complex

refractive index of a material). Using the Fresnel reflection coefficients and Jones formalism, the

reflected and incident fields are related by the mathematical expression as,

(𝐄𝑇𝑀

𝑟𝑒𝑓𝑙

𝐄𝑇𝐸𝑟𝑒𝑓𝑙) = 𝐽 (

𝐄𝑇𝑀𝑖𝑛

𝐄𝑇𝐸𝑖𝑛 ) 𝑤ℎ𝑒𝑟𝑒 𝐽 = [

𝑟𝑝𝑝 𝑟𝑝𝑠

𝑟𝑠𝑝 𝑟𝑠𝑠] , TM = 𝑝 , TE = 𝑠. (2.44)

The terms 𝑟𝑝𝑝, and 𝑟𝑠𝑠 denote the direct correlation of incoming and reflected fields, with

orthogonal field components in TM (or 𝑝, parallel) and TE ( or 𝑠, senkrecht) to plane of incidence,

respectively. The ratio of 𝑟𝑝𝑝/𝑟𝑠𝑠 can therefore, express the amplitude (diattenuation 𝜓) and

phase (retardance Δ) differences in these correlated terms. It can also be expressed

mathematically as,

𝜌 =𝑟𝑝𝑝

𝑟𝑠𝑠= tan𝜓𝑒𝑖Δ, where Δ = 𝛿𝑝

𝑟𝑒𝑓𝑙− 𝛿𝑠

𝑟𝑒𝑓𝑙 𝑎𝑛𝑑 𝜓 = |

𝑟𝑝𝑝

𝑟𝑠𝑠| = √

𝑅𝑝

𝑅𝑠. (2.45)

This is known as the ellipsometric equation and is the working principle for many commercial

ellipsometric systems [118]. Very often by polarimetric measurements, reflected light is used to

extract the retardance (Δ) and diattenuation (𝜓) of an examined system. Typically, the available

commercial ellipsometers extract retardance and diattenuation using experimental techniques

based on rotating polarizer, or phase retarders [119]. The two extracted parameters together with

a computational model (based on material properties) can then be used to extract complex

refractive index of the medium [81, 107]. As mentioned in Equation 2.45, essentially two

experimental measurements (recording 𝑅𝑝 and 𝑅𝑠) are enough at a single illumination angle to

extract complex refractive index. However, practically we are always constrained by more

unknown parameters in a system, for example sample thickness, substrate contribution, etc.


26

This is then catered by performing a sweep over a certain variable input and later using a curve

fitting model to extract refractive index [81, 107, 109, 110, 120]. Two common variable input are

wavelength and multiple angles of illumination (MAI), also known as spectroscopic ellipsometry

[118] or MAI ellipsometry [116, 120]. Methods that record experimental data over a wavelength

range and then perform material model fitting comes under spectroscopic ellipsometry. These

systems however have spatial resolution in tens of microns making them convenient for thin films

characterization.

Techniques that record data over a range of angle of incidence to extract complex refractive index

are known as MAI ellipsometry. A common example is of exploiting Brewster effect to gain access

to refractive index of a medium [75-77, 114, 117, 121, 122]. Usually, it involves a goniometer stage

to change incident angle of light beam with respect to sample. Usually a range of incidence angle

from 35-70° is used [96]. A major disadvantage in this case is that the technique involves moving

mechanical parts and becomes more complex for multilayer systems, leading to inaccuracies in

refractive index estimation. Another example of MAI ellipsometry is based on the principal angle

[96]. The incident angle 𝜃𝑝 for which TE and TM components of the reflected field has a phase

difference of π/2 is called principal angle. If a diagonal (45°) polarized light is used as incident

beam, then in reflection at principal angle we will have right circularly polarized light.

A vanishing reflectance can then be achieved at this principal angle, by a combination of QWP

followed by a linear polarizer at a specific angle called principal azimuthal angle 𝜓𝑎. The complex

refractive index [81] in this case is related by,

𝑛𝑟 = −sin𝜃𝑝 tan 𝜃𝑝 cos2𝜓𝑎, (2.46)

𝑛𝑖 = tan2𝜓𝑎. (2.47)

Similar to the Brewster effect-based MAI ellipsometry, this technique also demands illuminating

the sample with varying incidence angles sequentially to extract accurate ellipsometric

parameters, which can lead to inaccuracies due to mechanical motion.

A solution to this problem is by using single shot technique based on high NA focusing objective

[114]. As mentioned in Equation 2.12,this provides us with plane waves angular spectrum (limited

by NA of the focusing lens) which can be utilized for extracting refractive index in single

measurement. Usually, dry focusing objective of NA 0.95 are commercially available which means

an incidence angular range of 0-70° can be performed in one single measurement. To benefit from

the angular range provided by the high NA focusing objective, we would need to record angularly

resolved reflection by imagining the back focal plane of the focusing objective. In the following

we discuss important details regarding high NA focusing objective based MAI ellipsometry, as

shown in Figure 2.9.

Single shot Complex refractive index retrieval by high NA objective MAI Ellipsometry

As discussed already, the angular spectrum of incident wave vector is be defined by the focusing

objective. As an example, we consider the case of a microscope objective with NA of 0.9 (available

in our labs). Considering that the entrance pupil of such microscope objective is completely filled

with incident light beam, effectively we can access 𝜃𝑚𝑎𝑥 = sin−1 0.9 ~64.25° of incidence angle

range, making the complete range of this system from 𝜃 = −64.25 to 64.25 °and Φ= 0 − 360°


27

as shown in Figure 2.9. For incoming polarized light beam with polarization parallel to plane of

incidence (TM or P polarized light beam) we can expect to observe Brewster/pseudo-Brewster

effect in reflection at certain angle of incidence depending on refractive index of examined

medium [113].

Figure 2.9 Relation of focusing lens and equivalence to the angular spectrum. We consider the

case of a microscope objective with NA of 0.9. Considering that the entrance pupil of such

microscope objective is completely filled with incident light beam, effectively we can access

𝜃𝑚𝑎𝑥 = sin−1 0.9 ~64.25° of incidence angle. Making the complete range of this system from

𝜃 = −64.25 to 64.25 °and Φ= 0 − 360°. By recording the back focal plane (defined by 𝜃,Φ or

𝑘𝑥 , 𝑘𝑦) of focusing objective in reflection we can gather angularly resolved reflectance which can

be utilized for complex refractive index retrieval.

This can be experimentally and analytically observed in the back focal plane (defined by 𝜃,Φ or

𝑘𝑥 , 𝑘𝑦) of the microscopic objective as point of null/minimum intensity, respectively. In this case

the direct imaging of Brewster point is limited by the NA of the microscope objective and poses a

bigger constraint. Now we consider the generalized case of complex refractive index of a medium

defined by,

𝑛 =𝑛2

𝑛1⁄ = 𝑛 − 𝑖𝜅, (2.48)

where 𝑛1 is the known refractive index of the incident medium [117]. In our case we can consider

it to be unity. Then, building on our knowledge of Fresnel coefficients from 2.19a&b, we rewrite

these equations as,

𝑟12𝑁𝑇𝐸 =

cos𝜃𝑖−√𝑛2−sin2 𝜃𝑖

cos𝜃𝑖+√𝑛2−sin2 𝜃𝑖, (2.49a)

𝑟12𝑁𝑇𝑀 =

𝑛2cos𝜃𝑖−√𝑛2−sin2 𝜃𝑖

𝑛2 cos𝜃𝑖+√𝑛2−sin2 𝜃𝑖. (2.49b)

This is the generalized form of complex Fresnel amplitude coefficients. For the case |𝑛|2 ≫ sin2 𝜃𝑖

(which is true for many metals or absorbing materials).


28

Figure 2.10 A schematic of the computational and experimental process to be followed for

extraction of refractive index based on multiple angles of incidence observed in the back focal

plane of a focusing lens. This technique helps in accessing a broad angular spectrum in single

shot, hence reducing the time required for experimental measurements.

We can simplify the above equations by ignoring the sin2 𝜃𝑖 term. Mathematically [96, 117],

𝑟12𝑁𝑇𝐸 =

cos𝜃𝑖−𝑛

cos𝜃𝑖+𝑛, (2.50a)

𝑟12𝑁𝑇𝑀 =

𝑛2 cos𝜃𝑖−𝑛

𝑛2 cos𝜃𝑖+𝑛. (2.50b)

We can write Equations 2.50a&b in terms of reflectance as,

𝑅𝑇𝐸 =(𝑛−cos𝜃𝑖)

2+𝜅2

(𝑛+cos𝜃𝑖)2+𝜅2, (2.51a)

𝑅𝑇𝑀 =(𝑛−sec𝜃𝑖)

2+𝜅2

(𝑛+sec𝜃𝑖)2+𝜅2. (2.51b)

Mathematically by knowing the Brewster angle and the normalized intensity at this incident angle,

we can relate these observable quantities to the complex refractive index as,

𝑛 = (1−𝑅𝑝

1+𝑅𝑝) sec 𝜃𝐵, (2.52a)

𝜅 = (√2𝑅𝑝

1+𝑅𝑝) sec𝜃𝐵. (2.52b)


29

Where, 𝜃𝐵 is the Brewster angle and 𝑅𝑝 is the related reflected intensity. As shown in equations

2.29a&b, the presence of cos 𝜃𝑖 and sec𝜃𝑖 in TE/TM-equations describe a diverging/converging

reflectance with increasing angles, which would be a crucial parameter for fitting model of these

equations. Additionally, we normalize Equations 2.29a&b for reflectance at normal incidence,

𝑅𝑁𝑇𝐸 =

(𝑛−cos𝜃𝑖)2+𝜅2

(𝑛−cos𝜃𝑖)2+𝜅2 ×

(𝑛+1)2+𝜅2

(𝑛−1)2+𝜅2, (2.53a)

𝑅𝑁𝑇𝑀 =

(𝑛−sec𝜃𝑖)2+𝑘2

(𝑛−sec𝜃𝑖)2+𝜅2 ×

(𝑛+1)2+𝜅2

(𝑛−1)2+𝜅2. (2.53b)

In some studies, above mentioned equations have been rigorously solved to find unique solutions

for real and imaginary part of refractive index or permittivity using a goniometric (moving parts)

setup [117].

We propose a single-shot method which combines high NA focusing lens-based ellipsometry

previously used for extracting only real part of refractive index [113] with the equations 2.53a&b

to extract complex refractive index as shown in Figure 2.10. By recording the back focal plane

(defined by 𝜃,Φ or 𝑘𝑥, 𝑘𝑦) of focusing objective in reflection, we can gather angularly resolved

reflectance (𝑅𝑁𝑇𝐸 , 𝑅𝑁

𝑇𝑀). This can be done by appropriately placing a lens in the reflection arm.

Since we acquired normalized reflectance (𝑅𝑁𝑇𝐸 , 𝑅𝑁

𝑇𝑀) from experimental measurements, we

apply nonlinear least square fitting method on equations 2.49a&b and 2.53a&b to extract the

unknown in equations; complex refractive index [123, 124]. Further details on experimental and

computational setup and evaluation would be discussed in section 3.2.6 and 3.3.4. Due to high

NA focusing objective we achieve focal spot in the order of wavelength of incident light, which

provides way lower spatial resolution as compared to existing commercial ellipsometer solutions

(tens of microns).

In the next chapter we will use the theoretical consideration and experimental techniques

(discussed in this chapter briefly) about polarized light matter-interaction to investigate novel

artificial material. We will discuss, an in-house built experimental setup to perform polarimetric

analysis (with high spatial resolution) of novel orthorhombic form of carbon (carbon flakes)

intercalated with bimetal (Au-Ag) nanoclusters. Later we will apply single-shot high NA objective

MAI Ellipsometry on said structure for complex refractive index retrieval.

In Chapter 4, expanding onto the concept of optical properties of a medium, we will study

diffraction assisted chiral scattering in 2D metasurfaces. There we will study intriguing optical

phenomenon (asymmetric transmission) as a result of carefully designing the shape and

orientation of individual building blocks of metasurfaces. Later, we will perform experimental

investigation of metasurfaces based on techniques mentioned briefly here and in Chapter 3 to

compare with simulation results.

3. 3D self-assembled carbon-metal hybrid structure

In this chapter we are going to discuss novel 3D self-assembled hybrid carbon-bimetal (Au-Ag)

structures, also referred to as carbon flakes [14, 15, 22-24]. The carbon flakes were discovered

and produced at the Institute of Chemistry of St. Petersburg State University and the optical

characterization is part of this thesis. In the beginning of this chapter, we will talk about its

intriguing structure, which motivated us to understand its optical response. Later, we will discuss

the experimental schemes used to observe the optical properties of individual carbon flakes,

which are a few microns in lateral dimensions. We will analyze the optical response of these

carbon flakes and will try to merge our chemical, experimental, and computational knowledge to

build an understanding of the complex mechanism of self-assembled carbon flakes. We will also

discuss the effect of fabrication parameters on structural and optical properties. The work was

done in collaboration with specialized research groups from different parts of the world including

St. Petersburg State University, Russia and University of Ottawa, Canada.

Self-assembled metamaterials

Self-assembly, ubiquitous in nature, is an exciting and intriguing phenomenon based on the

arrangement of certain building blocks into order without any direct external guidance or control

[125-128]. Understanding the process dynamics in terms of chemical and physical transformation

is an emerging field and can help us in realizing the very fundamental question of life on earth.

Indeed, complex chemical systems and their respective self-organized growth and sustenance are

the key to this open question [129-131].

In physical chemistry, special chemical compounds also known as supramolecular complexes

(herein referred as SMC) can be transformed such that the individual constituents arrange

themselves into regular structures [132-135]. This process involves careful selection of individual

chemical building-block parameters (geometry, physical dimensions, and density to name few) to

control the self-assembly process. However, understanding the nature of morphological control

in self-assembly has remained a major question [136]. A complete morphological control of self-

assembly can obviously help in steering the overall geometry to the desired shape, whereby

getting a deeper understanding of kinetic and thermodynamic equilibrium within the structure.

Recently, many groups have looked into the possibility of organo-metallic complexes [134, 137-

140]. These nature-inspired self-assembled structures are a promising candidate for nanoparticle-

based drug delivery systems and organic electronics applications [138]. With the advancement in

technology and precise nanofabrication facilities, self-assembled metamaterials are an emerging

field combining the knowledge and expertise of multiple scientific fields [141-143]. A particularly

interesting category of such forms are centered on with intercalated metal nanoparticles [20, 21,

144-149]. Due to their hybrid organic-plasmonic properties, they are considered as promising

materials for optics, nanophotonics, electrocatalysis, organic electronics and sensing technologies

[150, 151]. In this context, exploiting different phases of carbon have naturally attracted

researchers to employ them for such cases. Covalent Organic Frameworks (COF), where

covalently bonded molecular structures of light elements (C, H, N, O), or Metal Organic

Frameworks (MOF), where metal clusters with organic ligands/linker structures assemble into

large crystalline units, address the concept of different perspectives and physical compositions


31

[138, 152-155]. An important area of research regarding these self-assembly phenomenon is the

so-called crystallinity-stability-functionality trade-off, where the structure can rearrange itself

due to external stimuli (e.g. pressure, temperature, electric fields gradient) to stay in a

thermodynamically minimum crystal state [156]. In the context of carbon, new exotic phases were

predicted under external stimuli conditions [157]. Recently, orthorhombic phase of carbon was

theoretically predicted, with intriguing optical and structural properties [154, 158-161].

In the following, we discuss a case of orthorhombic form of carbon, fabricated by self-assembly

process using laser induced deposition (LID) process [23, 24, 149]. The process is centered on the

breakdown and, later, recrystallization of a certain SMC as a result of laser irradiation at a

substrate/solution interface. LID offers a high flexibility and controllability. The LID process leads

to controllable growth of organometallic self-assembled micron-sized structures on a substrate

surface caused by the self-organization of the SMC constituents as shown in Figure 3.1a&b. The

LID of the hybrid metal-carbon structures is a straight-forward and convenient procedure that

does not require intense laser irradiation or special equipment. In the purview of this thesis, we

discuss the implications of the LID process on certain bimetallic (Au-Ag)-alkynyl ligand based SMC,

which results in the formation of [22, 23, 25] or other shapes such as flakes [14] and flowers [18]

of tunable composition, depending on the chosen SMC and the solvent [22]. One noteworthy type

of structure resulting from this process are carbon flakes (as shown in Figure 3.1b), which are the

first reported orthorhombic form of carbon (sp2 hybridized) with embedded Au-Ag nanoclusters

[14]. For the sake of brevity, we will keep our discussion in this chapter limited to carbon flakes,

which were investigated for their intriguing optical properties and may find applications in nano-

optics and spectroscopy [14, 24, 26].

3.1 Fabrication of novel hybrid structures

In purview of this thesis, we will discuss the use of alkynyl-phosphine ( PPh2(C6H4)3PPh2 ) based

ligand, which, together with heterometallic Au-Ag compound (Au13Ag12(C2Ph)20) and fluoride (PF6)

ions, helps in forming an organo-metallic SMC [23, 137, 149, 162-164]. The synthesis and chemical

characterization process will be discussed in the next section.

Chemical synthesis and characterization

The preparation of the required SMC and the fabrication of carbon flakes was done by our

collaborators at St. Petersburg State University, Russia under the supervision of Professor Alina

Manshina [20, 22, 23, 165-167]. A solution of the heterometallic-alkynyl phosphine SMC

{Au13Ag12(C2Ph)20(PPh2(C6H4)3PPh2)3}{PF6}5 was prepared in acetophenone (Aldrich, analytically

grade purity) at a specified concentration[21, 148]. The SMC solution was then placed in a cuvette

and covered with an ITO coated glass substrate (TIX 005 series from TECHINSTRO, with a thickness

of 1.1 mm), so that it is in contact with SMC solution [14, 15, 24, 168], as shown in Figure 3.1a. A

light source was focused to the substrate-solution interface from the substrate side in a fixed

position. As a light source we used a collimated He-Cd laser beam (CW, λ = 325 nm, I = 0.5 W/cm2)

with a beam waist of 𝜔0~1𝑚𝑚. The specific wavelength selected for irradiation is chosen to

correspond to the absorption spectrum of the SMC. The exposure time to irradiation can be varied


32

and usually ranges from 10-80 mins. After stopping the laser beam exposure, the substrate is

removed and washed with isopropanol (IUPAC: Propan-2-ol). The laser irradiation causes the

photoexcitation which consequently decomposes the SMC into its sub-constituents, followed by

self-organization into resulting crystalline structure (carbon flakes) as shown in Figure 3.1a&b.

The local temperature variations were found to be negligible at the substrate-solution interface,

hence confirming photoexcitation as the decomposition channel [24].

Figure 3.1 Formation and structural characterization of carbon flakes. (a) A Schematic diagram

showing the typical laser-induced deposition process. The laser irradiation causes the

photoexcitation which consequently decomposes the SMC into its sub-constituents, followed by

self-organization into resulting crystalline structure (carbon flakes). (b) A Scanning electron

microscope (SEM) image of a carbon flake. (c) Energy-dispersive X-ray spectroscopy (EDX) of

flakes displaying the proportions of carbon (C), gold (Au) and silver (Ag). (d) Transmission

electron microscope (TEM) image of a single carbon flake. (e) Selected area electron diffraction

(SAED) images taken for a single flake from two different angles. The diffuse ring relating to

arbitrarily oriented nanoclusters (Au-Ag) is marked with a grey circle. (f) Estimated

orthorhombic lattice constants computed by SAED. The lattice constant ‘c’ is along thickness and

‘b’ is along the longer lateral dimension of carbon flake [14]. Sub-figures (b-f) are courtesy of

Professor Alina Manshina.

Understanding the structural properties by material characterization

The material characterization of resulting carbon flakes was performed with various instruments,

shedding light onto the structural and compositional properties. This was performed at St.

Petersburg State University, Russia. According to the elemental analysis performed with an


33

energy dispersive X-ray (EDX) spectrometer (Figure 3.1), the flake comprises of carbon, gold, and

silver in given proportions (90:5:5) as shown in Figure 3.1c. Figure 3.1(d) illustrates a transmission

electron microscopy (TEM) micrograph of an edge of a flake, in which image contrast due to

presence of nanocluster can be easily observed. Similar images were utilized to obtain

information regarding nanocluster size, which was found to have a radius of R=1.25±0.9 nm. In a

similar fashion, the distance between adjacent nanoclusters was approximated to be Dcc=7.3±1.5

nm (these parameters correspond to sample C1 as referred in Table 1). Selected area electron

diffraction (SAED) (Figure 3.1e) was performed on carbon flakes with two distinctive patterns

observed as seen in Figure 3.1e. The central region helps us to extract the lattice parameters of

monocrystalline carbon flake matrix (a=20.8±0.5Å; b=7.99±0.5Å; c=33.4±1Å; α=β=γ=90°),

corresponding to an orthorhombic crystal unit cell. Secondly, the nanoclusters in carbon flakes

result in a polycrystalline diffuse diffraction ring as shown in Figure 3.1e (marked in grey circle)

with interplanar distance (d) of about 2.35±0.25Å. Usually, for face-centered cubic lattices of

Au/Ag prominent diffraction line is observed at d111~2.36Å which resembles that of nanoclusters

in carbon flakes [169]. A comprehensive analysis of nanoclusters in carbon flakes was performed

to understand their composition [14, 24]. The analysis of the TEM contrast indicates that each of

the obtained nanoclusters within carbon flakes consist of an Au-Ag alloy, rather than being

separated into pure gold and silver nanoclusters. In general, it can be understood that the self-

assembled structure contains a crystalline (orthorhombic) carbon matrix with intercalated

bimetallic (Au-Ag) nanoclusters. This configuration by virtue of its organometallic nature is quite

intriguing and can be expected to have a wide range of applications due to crystalline anisotropy

and plasmonic properties [25, 26].

Understanding the optical response of carbon flakes

Since carbon flakes possess an orthorhombic crystalline lattice, therefore the expectation arose

that the corresponding material might also show an effect on the polarization of light interacting

with the crystal [22]. Initial studies were performed at the Max Planck Institute, Erlangen under a

polarization microscope with rotatable (in-plane) sample holder, as shown in Figure 3.2. The

measurements were done with cross polarizer configuration with horizontal (H) polarized light

used as input as shown in Figure 3.2 with black arrow. The recorded images indicated that

transmitted intensity was changed once the sample was rotated from 0° to 90° as shown in Figure

3.2. This change of intensity could be attributed to either birefringence or diattenuation

(dichroism) in the carbon flake structure. Hence, it was decided to perform a complete

polarimetric analysis of carbon flakes. A spectral polarization-resolved experimental study was

initiated and will be discussed in the next sections.


34

Fig. 3.2 Qualitative results of an individual carbon flake imaged in a polarization microscope.

Different orientations of the carbon flakes are shown [22]. The measurements are done with

cross polarization configuration where black arrows indicate the input polarization (H-polarized

light). The recorded images indicated that transmitted intensity was changed once the sample

was rotated from 0° to 90° as shown in Figure 3.2. This change of intensity could be attributed to

either birefringence or diattenuation (dichroism) in the carbon flake structure. Based on these

results, it was decided to perform a spectral polarization-resolved experimental study.

Variable fabrication parameters

In the course of this thesis, initially a single sample C1 was studied to understand the optical

effects associated with orthorhombic carbon flakes. Later, we investigated the effects of certain

fabrication parameters on the optical and structural properties of the resulting carbon flakes. For

instance, effect of electric field on carbon flakes fabrication was studied [15, 19] as shown in

Figure 3.3. The cuvette which contains the SMC solution is placed in such a way to prevent the

metal electrodes from being in contact with the SMC solution. Besides this, the influence of the

laser irradiation time and concentration of the SMC in the solution was also investigated. The

varying fabrication parameters studied during the course of this thesis are shown in Table 3.1. The

resulting carbon flakes with varying fabrication parameters are shown in Figure 3.3(b-g). The

fabrication of samples was done at St. Petersburg State University, Russia.

Table 3.1. Samples with different fabrication parameters [15]

Sample Laser irradiation

time (min)

SMC concentration

(g/l)

Electric field

(V/m)

Resulting

thickness of

carbon flakes

(nm)

C1 [14] 15 4 [170] Off 150-500

C2 40 2 Off 250-750

C3 40 2 On 250-750

C4 80 2 Off 1200-1700

C5 40 6 Off 150-750


35

Figure 3.3 Carbon flakes formed by varying fabrication parameters. For details regarding varying

fabrication parameters, see Table 3.1(a) Sketch of the laser-induced deposition setup, also

allowing for application of a DC electric field during the fabrication process. SEM micrographs

for samples mentioned in Table 3.1. (a) The sample in cuvette with particular concentration of

SMC in liquid solution is illuminated with light source for a specific amount of time. Flakes are

formed by varying laser irradiation time of (b) 40 min and (c) 80 min, varying SMC

concentrations (d) 6 g/l and (e) 2 g/l. (f) and (g) show instances of carbon flakes resulting from a

fabrication using an external electric field switched on, resulting in laterally elongated flakes.

Image taken from [15]. Sub-figures (b-g) are courtesy of Professor Alina Manshina.

The samples were shipped to Max Planck institute, Erlangen where optical examination was then

performed on selected flakes from each sample. An atomic force microscopy (AFM) measurement


36

was performed on carbon flakes to estimate the thickness of carbon flakes with mean thickness

for various samples also mentioned in Table 3.1. The parameters shown in Table 3.1, affects the

decomposition rate that all together results in the variation of the flakes morphology/dimensions

and optical properties. As mentioned above, considering the unique nature of carbon flakes, it

would be interesting to understand its optical response. One of the immediate challenges with

direct polarimetric/ellipsometric measurements using commercially available solution is their

spatial resolution (in tens of microns). As already observed from the SEM micrographs (Figure

3.3b-g), the lateral dimensions of carbon flakes is in the order of few microns.

Hence, it was deemed necessary to develop a specialized technique to experimentally study

carbon flakes. A certain customized technique in this regard was developed and later modified at

the Max Planck Institute, Erlangen [14, 18]. The initial spectral measurements were performed by

Dr. Thomas Bauer, while later complete 2D scans and detailed study for carbon flake optical

properties and further experimental evaluation was performed by author. It helped us in

performing polarimetric analysis of carbon flakes at small scales (spatial resolution in the order of

incident wavelength). This technique will be discussed in detail in this chapter. Later we will apply

high NA objective MAI Ellipsometry on carbon flakes for complex refractive index retrieval.

3.2 Experimental setup

In this section, we describe our experimental methodology for retrieving the linear optical

properties of the examined carbon flakes. We analyze and extract linear optical parameters from

Müller matrix, such as birefringence, attenuation and diattenuation.

Microscopic Müller matrix measurement technique

We describe the various experimental and computational steps followed to extract optical

properties of examined carbon flakes mentioned in Figure 3.4. The microscopic Müller matrix

measurement technique merges the benefits of polarized light matter interaction with the back

focal plane (k-space / Fourier space) microscopy [29] and the usage of liquid crystal variable

retarders (LCVRs) to perform a comprehensive polarization analysis [14, 18, 30, 31]. The technique

is a combination of various experimental and computational steps that are performed for optical

properties extraction. The technique allows for doing a spatially resolved (with nanometer

resolution) far-field polarimetric analysis of micron-sized carbon flakes by performing a raster

scan of incident light beam over the examined structure. The usage of liquid crystal variable

retarders helps to avoid any moving optical elements in the analysis part of the setup [30]. In the

following subsections we will discuss various steps involved in this process and mentioned in

Figure 3.4.

Wavelength selection and polarization state generation

For the experimental method (see Figure 3.5a (i)) we use a broadband supercontinuum white light

laser source (NKT Photonics SuperK Extreme). For wavelength-selective filtering we use an

acousto-optical tunable filter (AOTF; Gooch & Housego AOTFnC-Vis / AOTF-UV).


37

Figure 3.4 Description of the experimental procedure for the microscopic Müller matrix

measurement technique. It involves experimental and computational steps to be followed for

the retrieval of optical properties. Starting from section 3.2.1 to 3.2.5 we will explain various

stages of the measurement technique shown above.

The filter works for tuning the wavelength in a spectral range of 400-700 nm with a spectral width

of a few nanometers. The spectrally filtered light is fed into a single mode optical fiber, which acts

as a mode filter only allowing the fundamental TEM00 mode to propagate. Consequently,

spectrally selected beam with fundamental TEM00 mode is used in the experimental setup as an

input. The beam diameter is then further magnified with the help of two convex lenses as per the

requirement of the setup. With the help of four guiding mirrors, input light beam arrives at the

custom-made microscope tower, which is the heart of measurement setup to study

nanostructures (Figure 3.5a (ii)&(iii)). This setup was developed by researchers in the group over

the time and different optical elements were added/removed based on experimental

requirement [11-13, 27, 28, 171]. The polarization state of the input beam is then controlled with

a linear polarizer and a broadband quarter wave plate mounted on a flip mount. This arrangement

helps in generating four linear horizontal (H), vertical (V), diagonal (D), anti-diagonal (A) and two

circular, right circular (RC) and left circular (LC) polarization states.


38

Figure 3.5 Experimental setup used for Müller matrix measurement technique. (a) An illustration

of the experimental setup. Using a combination of Supercontinuum source and an acousto-

optical filter we selectively filter the required wavelength. The light is then guided through a

single-mode fiber to obtain fundamental TEM00 mode. After guiding the light into the

measurement tower, the beam propagates in a top-down configuration. A broadband polarizer

and a quarter-wave plate acts as the polarization state generator defines the incident

polarization states (H, V, D, A, RC and LC). The light beam is then focused onto the carbon flake

on a glass substrate, by a 0.9 NA microscope objective (effective NA approximately 0.5). The

sample is positioned precisely, and raster scanned through the beam by a 3-axis piezo stage. The

transmitted light is gathered by an oil immersion objective of 1.3 NA. For every input

polarization state and position of 3-axis piezo stage, we perform the polarization analysis by

projecting light onto six polarization states (H, V, D, A, RC & LC) by two electrically tunable liquid

crystals (LCs) and a Glan-Taylor prism as a fixed polarization analyzer. The LCs are oriented along

22.5° and 45° relative to their slow axis, to project onto six polarization states required for a

complete Stokes parameter analysis. Thereafter, the polarization projected light beam is imaged

onto a CCD camera. The imaging lens is positioned in a such a way to image the back focal plane

(Fourier space) of lower objective on camera, (b) Photo diode (PD) based scan image of the

transmitted optical intensity when scanning the carbon flake through the focal field distribution.

The sharp edges of the flake confirms the confocal arrangement of microscope objectives with

carbon flakes present in the focal plane of focusing objective. The PD based measurement was

used to precisely align the examined carbon flake in the focal spot. (c) A recorded CCD image for

transmitted light through the examined carbon flake, at a wavelength λ = 460 nm. The solid

white line depicts the highest collection angle defined by NA = 0.1 used for Müller matrix

analysis. The plateau in the central region of the flake suggests the uniformity of the material

and the appropriate selection of the input beam with respect to the flake dimensions [14].


39

Probing sample at nanoscale

After generating the required polarization state, the polarized Gaussian beam with a wavelength

λ is focused by a microscope objective as shown in Figure 3.5a (iii). In our experimental setup, we

used a microscope objective with an NA of 0.9 (Leica HCX PL FL 100x/0.9) with a back focal plane

of 3.6mm (diameter). By controlling the diameter of the input Gaussian beam, we could achieve

an effective numerical aperture, resulting in a FWHM of the focal spot of about a wavelength. The

carbon flake to be examined, is positioned in the focal region of the microscope objective.

Figure 3.6 A picture of main measurement tower of the experimental setup with marked optical

elements inside measurement tower as mentioned in Figure 3.5a (ii-iii). The light beam

propagates top-down to probe the sample in the focal plane of focusing microscope objective.

Later, the transmitted light is collected with an oil immersion objective and then through a pair

mirrors guided into the polarization state analyzer consisting of 2 LCVRs and a Glan-Taylor

polarizer (not shown above). The light beam is then guided onto either a CCD camera or a

photodetector for further evaluation [12, 13, 171].


40

The small focal spot size (on the order of the wavelength) warrants that no edge effects contribute

to the measurements results obtained from the central region of the examined carbon flake. The

higher spatial resolution is one of the major advantages of our experimental setup as compared

to commercially available polarimetric systems. To perform a 2D scan, the carbon flake is

examined across the focal spot. This is done by mounting the carbon flake sample on a 3-axis

piezo scanning system (Physik Instrument P-525CD with an E-710 controller), which provides

lateral (x/y-axis) movement resolution of ~ 2nm and along propagation direction (z-axis) of

approximately 1 nm [11-13]. We use a custom designed sample holder which can be mounted

securely onto 3-axis piezo stage and firmly holds the carbon flakes sample. To ensure the lateral

levelling of the carbon flake sample, additionally two screws mounted on the sample holder helps

in controlling the tilt of the sample holder. A set of photo diodes are used in transmission,

reflection and as reference to align the sample/experimental setup and localize the examined

carbon flake within the lateral scan range of 3-axis piezo stage (x/y/z =200µm/200µm/20µm). In

order to evaluate the homogeneity of the carbon flakes under examination, we raster scanned

the carbon flakes individually across the beam and measured the transmitted intensity for each

position, resulting in the scanning image depicted in Figure 3.5b where a typical photodiode

readout for a carbon flake is shown. For each programmed scan position of the 3-axis piezo stage,

the transmitted light is collected by an oil-immersion microscope objective with an NA of 1.3

(Leica HCX PL FL 100x/1.3 Oil) in a large angular range. This lens has an aperture size of 5.2mm.

We image the back focal plane (BFP) of collection objective (see Figure 3.5c) onto a CCD-camera

(DMK23G618, Imaging Source) with 12 bit dynamic range to acquire the angular spectrum of the

transmitted light [172], using a single achromatic lens (f=300mm). In the recorded BFP images, we

computationally evaluate only the intensity in the angular cone (defined by NA) comparable to a

near-to-normal incidence. Thus, the effective angular range for Müller matrix analysis is preferred

to be constrained by a NA = 0.1, suggesting a half angle of around 5.5° (white solid circle in Figure

3.5c). The data shown is normalized for the transmission through the bare cover glass substrate.

As can be observed, the transmission decreases significantly when the focused incoming beam is

positioned on the examined carbon flake (Figure 3,5c). Here, we find a nearly constant plateau in

the central region of the flake, suggesting that the edge effects can be ignored for a central

positioning of the beam on carbon flake. Hence, the averaged polarization parameters of the

examined flake can be computed with statistical significance. We prepared differently polarized

six input beams as mentioned above, and determine their full Stokes polarization vector after the

transmission through the flake (𝐒𝑜𝑢𝑡) [85]. The end result is then equated to the transmission

through the cover glass substrate (𝐒𝑖𝑛) next to the flake for further data evaluation. A typical

custom-made microscope tower is shown in Figure 3.6 with the marked optical elements.

Polarization resolved analysis of optical structure

We perform a polarization projection of the light beam transmitted through the flake (or the bare

glass substrate for normalization purposes) to measure the respective Stokes vector. A

combination of two electrically tunable liquid crystal variable retarders (LCVRs; Thor Labs

LCC1115-A ) followed by a fixed polarization analyzer (Glan Taylor prism; Thor labs GT10-A) were

used for this purpose [173]. Aligning the slow-axis of the LCVRs cells at 22.5° and 45° with respect

to the preferred coordinate frame allows for determining the full Stokes vector by turning


41

between zero, half-wave and quarter-wave retardance of the two LCVRs, as shown in Table 3.2.

The two LCVRS are electrically operated by a USB LC controller (ARCOptix LC controller). Since,

LCVRs are sensitive to wavelength, and temperature variations [174, 175], therefore calibration

of the LCVRs was always carried out before the experimental scan measurements [30] and will be

discussed later in this subsection.

Table 3.2 LCVR retardance and resultant polarization states [14, 15]

GENERATED POLARIZATION

STATE

RETARDANCE LCVR 1 Θ=22.5°

RETARDANCE LCVR 2 Θ=45°

HORIZONTAL λ λ

VERTICAL λ λ/2

DIAGONAL λ/2 λ

ANTIDIAGONAL λ/2 λ/2

LEFT CIRCULARLY POLARIZED

λ λ/4

RIGHT CIRCULARLY POLARIZED

λ 3λ/4

The use of LCVRs make certain that we avoid any moving optical element in the polarization

analysis section of the experimental setup. This helps in preventing rotation generated shift of the

BFP image, which can cause parasitic effects in terms of crosstalk of polarization information in

different wavevectors of the beam [14]. In the following section we will discuss the working

principle and calibration technique of liquid crystals (Figure 3.7, top) integrated into the

experimental setup to perform polarization analysis of transmitted light.

Working principle of liquid crystal variable retarders

Liquid crystals which, due to their electrically controllable optical anisotropy, find wide application

in both scientific and industrial applications, were discovered by Friedrich Reinitzer. He, together

with Otto Lehmann, did the initial investigation into liquid crystals [176]. The physical

characteristics displayed by liquid crystals are in between those of liquids and solids. They display

orientation order (direction of constituent molecules) and no positional order (ordered lattice)

which are characteristic properties of solids and liquids respectively [83, 177]. Tuning the

polarization state of light with liquid crystals can be realized by two distinct properties of LC

molecules, their birefringence and the twisted nematic (TN) phase. Due to the elongated

ellipsoidal shape with one long and two short axes, each individual liquid crystal molecule

possesses inherent birefringence. Secondly, liquid crystals demonstrate a tendency to align

themselves on application of external fields. Usually, liquid crystals can have different molecular

ordering/alignment characteristics, which is called liquid crystal phase. Commonly available liquid

crystals are thermotropic liquid crystals [83], which change their properties based on


42

temperature. At higher temperatures, liquid crystals are said to be in isotropic phase, and act as

a liquid. While at lower temperatures they have anisotropic phase where they act as solid crystals.

Some liquid crystals exhibit various phases such as nematic [178], smectic, chiral phase etc. For

our case, we will limit the discussion to the nematic phase, which is the working principle for

LCVRs used in our experimental setup.

Figure 3.7 Top: Calibration scheme for the two LCVRs used in the experimental setup. After

finding the respective optic axis position as mentioned in the text, the liquid crystal was placed

between two crossed polarizers. Bottom: An increasing DC voltage was provided sequentially,

and the output intensity was recorded. By using the relations given in Equations 3.1c & 3.2c, we

extracted the required voltages for specific retardances. The procedure is repeated for every

wavelength.

It is the most common phase in liquid crystals in which by application of external fields, LC

molecules, although without positional order, are aligned due to long range interactions, which

assists in aligning the long axis of ellipsoidal LC molecules parallel to each other, resulting in their

chains [13, 83, 179, 180]. For the operational use, the LC molecules are placed in a transparent

cell, with a fixed orientation of LC molecules at LC-layer interface defined by an alignment layer.

These alignment layers consist of photosensitive polyimides, which are patterned by polarized

light to determine the fast axis orientation of LC cell [181]. In the absence of an applied external

fields, the LC molecules orientation is determined by the alignment layer. In the case of an applied

external field, the incident light beam experiences optical anisotropy due to internal rotation of

LC molecules within LC cell. This is caused by orthogonality of alignment layer orientation at

opposite ends of LC cell [83, 173, 180, 181].

Optic axis orientation

As mentioned above, LCVR-2 and LCVR-1 have to be oriented at 45° and 22.5° respectively, to

generate the six required polarization projection states necessary for complete Stokes vector

evaluation. This is achieved by selecting different combinations of retardances mentioned in Table


43

3.2. Therefore, the optic axis alignment and retardance calibration of LCVRs are two important

steps for the calibration process. Keeping that in mind, a fast-automated calibration technique

was developed to extract the required retardance values for two LCVRs. For ease of use, both

LCVRs are placed within a mechanically rotational mount placed positioned on flip mounts

(Thorlabs FM90). In our experimental setup, initial calibration of LCVR-2 is performed with LCVR-

1 flipped out of the light beam path. Optic axis alignment is performed by placing LCVR-2 between

two crossed polarizers as shown in top part of Figure 3.7. With no applied voltage, when rotating

the LCVR-2 we observe no transmitted light for the case of slow axis of LCVR-2 parallel to the

incoming polarization state of light. This position on the rotation mount is marked as slow axis

and LCVR-2 is rotated 45° from this position.

Retardance calibration by variable voltage

The model of variable voltage application to extract the retardance profile of LCVR has been

experimentally elaborated [173, 179, 182]. A plot between increasing applied DC voltage and

transmitted intensity is shown in bottom part of Figure 3.7. This can be expressed simply in terms

of mathematical relations of Müller matrices of two linear polarizers parallel to each other, with

LCVR-2 positioned between them. Mathematically,

𝑆𝑜𝑢𝑡 = 𝑀𝑙𝑖𝑛 𝑝𝑜𝑙 0°𝑀𝑙𝑖𝑛 𝑟𝑒𝑡,45°𝑀𝑙𝑖𝑛 𝑝𝑜𝑙 0°𝑆𝑖𝑛, (3.1a)

𝐼(𝛿, 45) =1

2{S0 + 𝑆1 × (𝑐𝑜𝑠(𝛿)) + 𝑆3 × ( −𝑠𝑖𝑛(𝛿))} ] (3.1b)

Where for completely linearly 𝑦- polarized input light, 𝑆0 = 1, 𝑆1 = 1, 𝑆3 = 0, hence

𝐼(𝛿, 45) =1

2(1 + 𝑐𝑜𝑠(𝛿))] (3.1c)

The LCVR used in our experimental setup supported multiple retardance cycles. Therefore, by

applying a DC voltage ramp from 2.2-8 V with steps of ~0.007 V was applied to LCVR-2 to extract

the intensity versus applied voltage graph as shown in bottom part of Figure 3.7. The graph was

normalized w.r.t to the maximum intensity recorded and voltages associated to full, half and

quarter wave retardances are noted for further usage in the carbon flake analysis. Table 3.3

depicts the relation between normalized intensities and abovementioned retardances.

Table 3.3 Comparison of retardance and observed normalized intensity

Optical retardance

of LCVR-2

𝒄𝒐𝒔(𝜹) Normalized observable

intensity (𝑰𝒔𝒖𝒎)

λ 1 1

λ/2 -1 0

λ/4 0 ½

3λ/4 0 ½

To automate the process a software in Visual Basic was developed to gradually change the voltage

of LCVR from 2.5-8-8 V while recording the light intensity using a CCD camera. Later, LCVR-1 optic

axis orientation and retardance calibration is performed in similar fashion, while setting LCVR-2


44

at full wave retardance (see bottom part of Figure 3.7). Mathematically the intensity profile in this

case is shown as,

𝑆𝑜𝑢𝑡 = 𝑀𝑙𝑖𝑛 𝑝𝑜𝑙 0°𝑀λ,45°𝑀𝑙𝑖𝑛 𝑟𝑒𝑡,22.5°𝑀𝑙𝑖𝑛 𝑝𝑜𝑙 0°𝑆𝑖𝑛, (3.2a)

𝐼(𝛿, 22.5) = S0+ 𝑆1 × ( 0.5 + 0,5 × 𝑐𝑜𝑠(𝛿) ) + 𝑆2 × (0.5 × (1 − 𝑐𝑜𝑠(𝛿))) + 𝑆3 × (−1

√2× 𝑠𝑖𝑛(𝛿)) ] (3.2b)

Where for completely linearly y - polarized input light, 𝑆0 = 1, 𝑆1 = 1, S2 = 0 & 𝑆3 = 0, hence,

𝐼(𝛿, 22.5) = 1

2{1 + (0.5 + 0.5 × 𝑐𝑜𝑠(𝛿))} (3.2c)

For polarization projections required for Stokes analysis we are interested in full and half wave

retardances for LCVR-1. Table 3.4 depicts the relation between normalized intensities and

abovementioned retardances.

Table 3.4 Comparison of retardance and observed normalized intensity

Optical retardance

of LCVR-1

𝒄𝒐𝒔(𝜹) Normalized observable

intensity (𝑰𝒔𝒖𝒎)

λ 1 1

λ/2 -1 0.5

This set of calibrations is performed for each wavelength scan. Although the labs are equipped

with temperature stabilization, nevertheless we performed abovementioned retardance

calibration to prevent any imperfections due to temperature changes [175].

Measurement procedure and data recording

In the above subsections we have elaborated on the experimental methodology for generating

polarized gaussian beam of specific wavelength, which was later used to probe the carbon flake

sample in focal plane. Subsequently, the transmitted light that is collected and projected onto

different polarization states on a CCD camera by imaging the back focal plane of collection

microscope objective, to get access to the angular spectrum of the transmitted light. An inhouse

built custom software was used to control and automize the measurements procedure [10, 12,

13]. In the following sub-section, we will discuss some of the salient features of this software.

Image recording and data analysis

The data recording is performed using an in-house custom built solution in Visual Studio for

integrating a CCD camera, 3-axis piezo stage and photo detectors [12, 13, 171]. Certain features

were added to this code, which includes calibration of liquid crystals and polarization

measurement for complete polarimetric analysis. Detailed description can be found in section 3.5

of referred thesis [13].

Input parameters

The main user interface includes the control for 3-axis piezo stage movement as shown in Figure

3.8. The piezo control is designed to move piezo stage in an automated fashion to scan across a


45

2D grid. Besides automatically defining the scan area for experimental measurements, manual

positioning of piezo stage is also possible. For alignment and localization of the structure to be

examined, the photodetectors are also integrated, whose signal readout can be seen in color and

normalized values (a typical example of recorded data is also shown in Figure 3.5b) [10, 28].

Figure 3.8 GUI of the software used for experimental measurements [10, 28]. (a) The 2-D scan

field with lateral dimensions of scan length per line. (b) Piezo stage control with movement

options in 3D. (c) Photodetector scan displays. (d) Liquid crystal calibration interface.

LCVR calibration

To perform the retardance calibration of the two LCVRs, a part of the user interface is developed

to calibrate the LCVRs as shown in Figure 3.8. As mentioned in the previous section, to perform

retardance calibration user input for minimum and maximum applied DC voltage and number of

the measurement steps are taken as input. Liquid crystals inherently need settling time once the

applied voltage is changed. This is ensured by introducing a time delay on the order of a few

hundred milliseconds to adjust for changing LCVR retardance. After this an external trigger is sent

to the CCD camera to record an image. The recorded images were then fed into Matlab where a

routine is used to plot the total intensity versus retardance, as discussed in the previous section,

to extract the required retardance states required for Stokes analysis. The retardance calibration

process is first performed for LCVR-2 and subsequently for LCVR-1. The obtained voltages for

required retardance states for LCVR-1 and LCVR-2 were then fed into the user interface, as shown

in Figure 3.9 [10, 28] for automate polarimetric analysis of carbon flakes.

Recording the back focal plane with a CCD camera

For recording the back focal plane (BFP) of the collection microscope, the objective is imaged onto

a CCD-camera (DMK23G618, Imaging Source) with 12 bit dynamic range to get access to the

angular spectrum of the transmitted light [172], using a single achromatic lens (f=300mm). For

controlling the CCD camera, a commercially available software (IC capture version 2.4) is used.


46

The software also provides the option for external trigger, which in our case is provided by the

Visual Studio interface with certain time delay [13, 171].

Figure 3.9 Setting window of GUI. On the right side, parameters for the photodetector are

defined. While on the left side, liquid crystal retardance voltages are input for producing six

polarization states. LC delay defines the time to change voltage of liquid crystals which was set

to 15ms for nominal measurements [10, 28].

Computational evaluation of recorded data

To analyze the recorded data, a routine based on polar decomposition is implemented to extract

the Müller matrix in Matlab. For each recorded polarization projection image, the intensity of the

transmitted light within the angular spectrum cone of NA=0.1 corresponding to near normal

incident was evaluated to extract the particular Stokes vector (marked in Figure 3.5c with while

circle). The same process is performed for all input polarization states and each position step of

the 2D raster scan.

Retrieval of the Müller matrix

As discussed in previous sub section, we have acquired the Stokes vector for all incoming

polarization states for all positions in a 2D raster scan. From the transmission of the light beam

through the bare cover glass and carbon flakes, the Stokes parameters 𝑆𝑖𝑛 and 𝑆𝑜𝑢𝑡 are retrieved,

respectively. A total of 4 linear (H, V, A, D) and two circular (RCP, LCP) polarized light beams are

used for the analysis. This allows us to use the 4 x 6 matrix of the input and output Stokes vectors,

to extract the Müller matrix [29, 84, 93, 102]. Since the system of linear equations is

overdetermined, we can compute the Müller matrix 𝑀 with the help of the pseudo-inverse input

Stokes vector 𝑆𝑖𝑛+[98]. Consequently, the experimentally realized Müller matrix can be further

decomposed [31] by existing methods such as polar decomposition and analytical inversion to

extract optical properties of the examined system, as mentioned in theory chapter [78, 84, 98,

102]. Mathematically to reiterate,


47

𝐒𝑜𝑢𝑡 = 𝑀𝐒𝑖𝑛 , 𝐒𝑜𝑢𝑡 = 𝑀∆𝑀𝑅𝑀𝐷𝐒𝑖𝑛, (3.3)

where 𝑀∆,𝑀𝑅 ,𝑀𝐷 represent the polar decomposition of a Müller matrix into individual

depolarizer, retarder and diattenuator matrices. These individual matrices can then further be

used to extract the required optical properties. Besides using polar decomposition, we extracted

the optical properties using analytical inversion method [29, 31, 84, 183]. Since the thickness of

flakes is usually limited to a few hundred nanometers and in very few cases to a micron, we

effectively have low retardance approximation, which makes estimates from polar decomposition

and analytical inversion approximately comparable [31].

Ellipsometric setup

The Müller matrix measurement technique adopted in transmission provides information mainly

regarding birefringence and dichroism. To access the direct information regarding refractive index

of the medium, we implemented a technique in the reflection arm of the setup explained above.

The setup was modified to accommodate for measurements in reflection in the back focal plane

and to generate different polarization states, which will be discussed below.

Preparing polarization states of the light beam

To prepare various linear and spatially structured light beams we use a combination of linear

polarizer and an electrically tunable spiral phase plate. For the results shown in this thesis, mainly

linearly polarized light was used. Use of spatially structured light is work in progress and in future

will greatly increase the benefits of current ellipsometry scheme. A discussion regarding spatially

structured light can be found in Appendix B to this thesis. we use a set of two beam splitters

(Thorlabs; PBS101) in orthogonal orientation (before focusing of light beam) to avoid any

polarization-dependent phase and amplitude variations [184]. This arrangement allows us to

collect the light beam in reflection without any polarization errors.

Probing the sample and recording the reflected signal

After generating the required polarization state, the polarized Gaussian beam with a wavelength

λ is focused by a microscope objective (Leica HCX PL FL 100x/0.9), as shown in Figure 3.10b&c. In

this case, the beam diameter of polarized light was made to match or be greater than the diameter

of entrance pupil of focusing objective (3.6mm). This would allow us to access the complete

angular spectrum offered by the objective. The planar sample to be examined is placed in the

focal plane similar to the arrangement mentioned in Section 3.2.2. This provides us with a focal

spot comparable to wavelength of light in visible frequency range. In the present experimental

setup, we examined samples in such a way to avoid internal reflections. This is done by positioning

the examined sample top surface, in the focal region of the focusing microscope objective and

avoiding multiple reflections by using a thicker substrate. Also, we examined samples with known

relatively larger thickness (evaluated by atomic force microscopy). By ensuring the precise

positioning of the focusing microscope objective, we can expect to have a collimated beam in the

reflection path as shown in Figure 3.10c. The back focal plane (BFP) of the focusing objective is

then imaged onto a CCD-camera (DMK23G618, Imaging Source). To get access to the angular

spectrum of the reflected light [172], we use a single achromatic lens (f=100mm).


48

Figure 3.10 Sketch of the experimental setup for complex refractive index retrieval using

multiple angles of incidence exploiting angular spectrum offered by high NA focusing

microscope objectives. (a) Using a combination of Supercontinuum source and an acousto-

optical filter we selectively filter the required wavelength (632 nm used for proof of principle

measurements). (b) A broadband polarizer and a spiral phase plate is used to generate the

required polarized beam. For the results shown in this thesis, mainly linearly polarized light was

used. Use of spatially structured light is work in progress and in future will greatly increase the

benefits of current ellipsometry scheme. (c) The polarized Gaussian beam with a wavelength λ is

focused by a microscope objective. The beam diameter of polarized light beam was made to

match or be greater than the diameter of entrance pupil of focusing objective. This would allow

us to access the complete angular spectrum offered by the objective. The planar sample to be

examined is placed in the focal plane which provides us with a focal spot comparable to

wavelength of light in visible frequency range. In the present experimental setup, we examined

samples in such a way to avoid internal reflections. This is done by positioning the examined

sample top surface, in the focal region of the focusing microscope objective and avoiding

multiple reflections by using a thicker substrate.(d) The recorded reflection BFP images are then

studied in terms of angular spectrum and intensity of reflected light. The further computation

evaluation of radial profile corresponding to purely TM polarized light is performed to extract

complex refractive index by using the fitting model based on Equations 2.49a&b and 2.53a&b.


49

Image extraction and analysis

The recorded reflection BFP images (Figure 3.10d) are then studied in terms of angular spectrum

and intensity of reflected light. In order to extract the reflected intensity profile from the surface,

we normalize it w.r.t the incident light beam. This is done by placing a flat mirror just above the

focusing objective, to reflect the input beam onto the CCD camera, to record the reference input

beam Gaussian profile. Further Image processing on the recorded image is performed in Matlab.

The respective reference image is then used to normalize the BFP image of reflected light from

the examined sample’s top surface as shown in lower left corner of Figure 3.10. We define the

center of the BFP image as normal incident wavenumber (𝑘𝑥, 𝑘𝑦 = 0). The sharp edge of entrance

pupil of the focusing objective, observed in BFP image, acts as the basis for image segmentation

or defining maximum wavenumber (𝑘𝑚𝑎𝑥). As shown in bottom part of Figure 3.10, this radial

line, from the center 𝑘x/y = 0 to the outer edge, corresponds to the increasing wavenumber

𝑘𝑚𝑎𝑥. Once 𝑘x/y = 0 & 𝑘𝑚𝑎𝑥are defined, each pixel within the cropped circular BFP region can

be associated with Fourier coordinates (𝑘𝑥, 𝑘𝑦). A routine was then implemented to extract radial

(from -𝑘𝑚𝑎𝑥x to 𝑘𝑚𝑎𝑥) line segments along specified azimuthal angles (φ) of cropped circular BFP

image. These radial line profiles contain the reflected light intensity versus a transverse

wavenumber or varying incident angle data. By doing calibration measurements, the effective

𝑘𝑚𝑎𝑥=0.87 proportional to the maximum incidence angle of 60.25° was estimated for the current

measurement system and hence Fourier coordinates (𝑘𝑥 , 𝑘𝑦) were accordingly adjusted.

Computational model for ellipsometric analysis

The radial line profiles extracted are then fed as input to the fitting model developed for extracting

ellipsometric parameters. We use three different methods for extraction of complex refractive

index; firstly, direct estimation from reflected light intensity at Brewster angle using Equations

2.52a&b, approximate reflection calculations based on Equations 2.53a&b and full Fresnel

equations based on Fresnel coefficients mentioned in Equations 2.49a&b. As evident, The

nonlinear least square curve fitting function (lsqnonlin) in Matlab was used for second and third

case. The extracted radial line profiles are fitted accordingly to either TE or TM Fresnel equations

depending on incident polarization state of light. As evident, in case of first method, we use the

minimum intensity (for dielectric this is zero but for absorbing materials it is non-zero) at Brewster

angle, to approximate the complex refractive index [117]. As discussed, in our experimental setup,

the angle of incidence is limited by NA of the microscope objective. Hence, we can expect that for

materials, for which Brewster angle occurs above 60°, we will not be able to observe the

respective minima, which can lead to wrong approximations.

On the other hand, fitting models (based on Equation 2.49a&b and 2.53a&b) still can provide good

estimates, as will be discussed in the results section. This shows the two prong advantage of our

technique over previous results [113] for high AN objective MAI ellipsometry; complex refractive

index retrieval and because of fitting function we are not dependent on single point based

estimations. We perform experimental measurement on surfaces such as glass (isotropic), lithium

niobate (uniaxial) α-silicon (absorbing) for which ellipsometric data can be found in literature.

Later, we will test this scheme on the carbon flake to extract a preliminary estimate of complex

refractive index. The mapping of Fourier coordinates (𝑘𝑥 , 𝑘𝑦) to the specific pixels within circularly

cropped BFP images was kept constant for all examined samples.


50

Intensity based recording problems and assumptions

Since for each incident polarization state, we record total intensity in reflection, it is important to

discuss the assumption and constraints in regard to the computational model. In case of studying

uniaxial crystals, we ensure that the fast axis of the anisotropic sample is aligned along the

horizontal axis of experimental system. This way we can avoid cross polarization effects (off-

diagonal terms in Equation 2.44 vanishes). Hence, the information regarding cross conversion of

TM (p-) polarized light into TE (s-) polarized light in our case is not accessible. This scheme is

ensured by either keeping the incident states pure (either TE or TM polarized light) or studying

radial profiles along certain azimuthal directions (for the case of linear polarization). For incident

beams where equal contribution from TM &TE polarized fields is expected (spiral beam), a fitting

equation incorporating both TE & TM Fresnel equations is used to extract the ellipsometric

parameters.

3.3 Results and discussion (optical properties and fabrication effects)

The experimental work carried out on carbon flakes revealed some intriguing optical properties.

We start with the discussion of the optical measurements performed on sample C1. Later, we

discuss the analytical model built for carbon flakes (by Prof. Antonino Cala Lesina, University of

Ottawa) in the light of optical measurements and how it computationally verified some key

aspects of composition of the carbon flakes. Subsequently, we will study the effects of varying

fabrication parameters on optical and geometrical/structural properties of flakes. In the last part

of this chapter, we will elaborate on the result and analysis of ellipsometric measurements.

Figure 3.11 Single line scan across a carbon flake shown for comparing scans for different

polarization states. A plateau observed in the center region of the flakes, which was ultimately

used to extract optical properties of the carbon flake. An SEM image of the flake together with

the vertical scanning line are shown as inset.


51

Complete carbon flake scan results for C1

For sample C1, a spectral scan from 400-700nm was performed on a single carbon flake to

understand the underlying optical properties. The initial spectral measurements were performed

by Dr. Thomas Bauer, while later complete 2D scans and detailed study for carbon flake optical

properties and further evaluation was performed by author. The measurements were done on a

flake on sample C1 (Figure 3.12(a-d), labelled 340nm in 3.12e&f), which was selected due to its

favorable lateral dimensions (5µm x 2.2µm ). Atomic force microscopy was performed to

determine the thickness of said carbon flake (340nm). The extracted Müller matrix was

decomposed into constituent optical properties matrices. These constituent matrices were then

later used to extract the required optical properties. The alignment of recorded data was ensured

by programmed control of 3-axis piezo stage, which was set to same parameters for each input

polarization. As shown in Figure 3.11, the transmission through glass towards the end of the raster

scan shows consistent values.

Therefore, spatially averaged values of these scan points were used to calculate Stokes

parameters of glass substrate, which acted as 𝐒𝑖𝑛. To get a complete set of optical measurements

for every point in the scanned area, all scan points were taken into consideration. Therefore,

Stokes parameters of every point were calculated and used as 𝐒𝑜𝑢𝑡. An illustration of the two-

dimensional map (41 × 21 steps with a step-size of 200 nm × 270 nm) of extracted optical

properties is shown in Figure 3.12(a-d). Figure 3.12 a-d depict attenuation (calculated as 1-

normalized transmission intensity), diattenuation, optic axis orientation, and linear birefringence

across the examined carbon flake (at wavelength 480nm). As seen in Figure 3.12a, the attenuation

(A) across the carbon flake is strong and remains spatially constant around the central region. The

diattenuation (D) is found to be insignificant across the central region of the flake except at the

edges, where it can be logically assumed to be high due to edge effects (Figure 3.12b). Therefore,

the diattenuation for carbon flakes, due to its negligible effects will not be further discussed in

this thesis. The optic axis orientation (Figure 3.12c) is also used to finally calculate linear

retardance in carbon flake (Figure 3.12d). For specific wavelengths, an average optical response

was computed based on the data points from the central region of the flake, where negligible

edge effects are expected due to minimal overlap of edges and laser spot. As it can be observed

in the spectral response in Figure 3.12e, the attenuation of the carbon flake rises at shorter

wavelengths with a maximum (A=0.82) near λ = 460 nm. For longer wavelengths, the attenuation

decreases with minimum attenuation observed at longest recorded wavelength (A=0.2 at λ = 700

nm). The presence of bi-metallic nanoclusters (Au-Ag) can be expected to cause significant

contributions to the attenuation spectrum observed for carbon flakes. This will be further

discussed and verified in a later section discussing the analytical modelling of carbon flakes [14].

As shown in the 2D map of the optic axis, the fast axis orientation of the carbon flake relates very

well to its geometric shape, with the axis being parallel to the long side of the flake. As mentioned

above, the spatially averaged linear retardance for carbon flakes extracted from Müller matrix is

used to extract the linear birefringence [71]. We calculate birefringence using Equation 2.40a

where the thickness of carbon flake is determined by atomic force microscopy. Neglecting the

edge effects as shown in Figure 3.12(a-d), we can observe a spatially constant plateau in the

central region of the flakes which can be attributed to the inherent linear birefringence response

of carbon flake. As mentioned in the previous section, the structural analysis of carbon flakes

predicts an orthorhombic crystalline phase.


52

Figure 3.12 Experimental results obtained for carbon flakes in an initial run of measurements. (a-

d) Two-dimensional optical properties’ map of a flake of size 5.2 μm x 2 μm x 340 nm at λ = 460

nm, depicting attenuation, diattenuation, fast axis orientation, and birefringence. This was

performed by mounting the sample on a 3-axis piezo stage and performing a 2d raster scan as

41 x 21 lines using a 200 nm × 270 nm step size. (e & f) Spectral optical analysis of attenuation

and birefringence for various flakes of different heights and lateral dimensions (flakes chosen so

as to avoid edge effects at all wavelengths). We find that the attenuation rises at shorter

wavelengths, while the birefringence stays high and almost constant throughout the studied

spectral range. Images reproduced from Ref. [14].


53

Hence, we can relate the linear optical birefringence to the crystalline nature of carbon flake. In

order to further examine the birefringence, we perform a complete spectral analysis in the visible

spectrum (Figure 3.12f). Intriguingly, the linear birefringence is found to be almost constant (~0.09

±0.005) across the entire spectral range investigated. To verify uniformity of fabrication process

and the high birefringence, multiple flakes of varying thicknesses and lateral proportions (of at

least a few microns laterally to avoid edge effects) were studied (Figure 3.12 e&f). The evaluation

of those additional flakes was done only at some wavelengths but using the same experimental

scheme and assessment techniques. AFM scans of respective examined flakes were also

performed to approximate the height of individual carbon flake. Remarkably, all examined flakes

show similar values of birefringence (see Figure 3.12f), proving the consistency of the fabrication

process and confirming the presence of linear birefringence in crystalline carbon flakes.

Numerical modelling results and comparisons

To understand the optical response of carbon flakes, a computational model was built by

Professor Dr. Antonio Calà Lesina from the University of Ottawa, Canada. For the optical

properties present in carbon flakes two main contributions were considered: the bimetallic (Au-

Ag) nanoclusters and carbon matrix. Since the diameter of nanocluster is approximately similar to

the third dimension (c) of orthorhombic unit cell of carbon flake, it suggests that the nanoclusters

are intercalated in the carbon matrix as shown in Figure 3.13a. First part of computational study

was to model the nanoclusters present in carbon flakes. This was performed by emulating layers

of carbon sheets with intercalated nanoclusters (arranged in a face centered cubic (FCC) fashion

[14]) inspired by the first experimental study performed on carbon flake. The parameters such as

material composition of nanocluster, their effective diameter, spacing between two neighboring

nanoclusters and permittivity tensor of carbon sheets were kept variable. The initial analytical

model for nanoclusters was based on a modified Drude-Lorentz model to accommodate for

different proportions of bi-metallic particles (Au-Ag) [185] as shown in Figure 3.13ab. It was then

fitted to experimental data to find the proper approximations of parameters such as, nanocluster

inter-distance (Dcc), radius of nanoclusters (R),carbon matrix stretching factor (f), percentage

contributions of Au-Ag in nanoclusters and background refractive index (𝑛𝑏𝑎𝑐𝑘).These are briefly

discussed in Figure 3.13(b). For modelling the carbon layers, stretched graphene was first used in

the numerical model. This was accomplished by using experimental data for optical properties of

graphene obtained by spectroscopic ellipsometry [186]. This graphene model was modified by

changing its susceptibility (as discussed in Equation 2.23). The stretching factor facilitates the

modelling the carbon atoms within a layer, which is in essence equivalent to understanding

orthorhombic carbon crystalline structure. Keeping in consideration the orthorhombic crystalline

dimensions, we separate the layers of carbon by a distance of ~ 3.34 nm with background

refractive index as 𝑛𝑏𝑎𝑐𝑘 as mentioned above. The analytical model, as shown in Figure 3.13a, for

carbon flake was then simulated using the finite-difference time-domain (FDTD) approach [187].

Infinite lateral extension of modelled carbon flake is assumed based on periodic boundary

conditions in 𝑥 and 𝑦 direction, while light propagates in 𝑧 direction [188]. From the

computational model it was estimated that the average nanocluster inter-distance of 8nm

provides a solution closer to the experimental response. This is also verified by the TEM

measurements done on carbon flakes which estimates the nanocluster inter-distance Dcc=7.3±1.5

nm.


54

Figure 3.13 (a) A carbon flake model with intercalated nanoclusters of Au-Ag. Layers of carbon

with distance between interlayer of around ~3.4nm. The bimetallic nanoclusters are arranged in

the carbon matrix in face-centered cubic fashion. Inter-particle distance Dcc= 8nm between

nanoclusters is also considered. (b) Parametric study (Dcc, R, f, alloy percentage, and nback

variation) performed to fit experimentally recorded data for carbon flake with thickness 340nm.

(c) theoretical study to link orthorhombic lattice – susceptibility and birefringence vs. number of

carbon atoms [14]. Simulation work performed by Prof. Dr. Antonio Calà Lesina from the

University of Ottawa.

Similarly, nanocluster diameter is predicted to be 2.5nm, which is in good agreement with

experimental TEM results of d =2R=2.5±0.9 nm. Within clusters, a proportion of 1:1 of gold (Au)

and silver (Ag) leads to more convincing results as also predicted experimentally by XPS

measurements [14, 24]. Summarizing, the optical attenuation observed in flakes experimentally

was theoretically observed to be primarily related to nanoclusters’ composition, their Inter-

particle distance (Dcc) and radius (R). Theoretically, the birefringence in carbon flakes can be

explained by defining a orthorhombic lattice with certain number of carbon atoms distributed in

the unit cell. This is performed by redefining the susceptibility tensor, which was considered

isotropic in the above case [189]. The linear birefringence (𝑛𝑦𝑦 − 𝑛𝑥𝑥) obtained from theoretical

model is shown in Figure 3.13c, besides other material properties and is also plotted as a function

of the number of atoms per unit cell. The dashed line illustrates the experimental results for

carbon flake C1 corresponding to 𝑛𝑦𝑦 − 𝑛𝑥𝑥 = 0.9. The experimental and theoretical model

matched up for unit cell containing 300 carbon atoms per unit cell. This is in good approximation

with what we can estimate by the unit cell dimensions and considering the ratio of carbon, gold

and silver observed by polycrystalline XPS measurements [14].


55

Understanding control fabrication parameters and other optical properties

The carbon flake samples mentioned in Table 3.1, which represent different fabrication

parameters, were also experimentally investigated to understand the underlying effect of specific

fabrication parameters on the optical and structural properties of carbon flakes [15, 19]. To

extract the optical properties, we perform the experimental measurement and analysis in similar

fashion as mentioned above, for multiple flakes from samples mentioned in Table 3.1. We

examine the two optical properties of carbon flakes, i.e., attenuation and linear birefringence

across different carbon flakes. As discussed before, we evaluate the mean value for optical

property by studying the optical response in the central region of the flake where edge effects are

minimal. Besides, we perform an analysis on geometrical/structural properties of examined

carbon flakes.

Effect on the structural dimension of the carbon flakes

We used AFM to estimate the thickness of the examined flakes, while SEM was used to analyze

the lateral dimensions. SEM images of the examined flakes for variable fabrication parameters

are shown in Figure 3.3(b-g). Neglecting the slight variations in the flake’s dimensions, we can

deduce that the lateral dimensions of the examined carbon flakes are clearly correlated with

certain fabrication parameters. For instance, the SEM micrographs for DC-field switched off (see

Figure 3.3 b, e ) and for DC-field switched on (see Figure 3.3 f, g) show a visible difference in lateral

dimensions of fabricated carbon flakes. In these two scenarios (for sample C3 and C2 being E-field

‘on’ and ‘off’ respectively), other fabrication parameters were kept constant to single out the

resultant effect. It is pertinent to highlight that we observe an elongation in examined carbon

flakes along one lateral axis. The average size of carbon flakes, with electric field applied during

fabrication, reaches 40-60 µm in length and 5-10 µm in width (see Figure 3.3 f & g). These results

suggest that the lateral dimensions of flakes can be fine-tuned with the help of an applied electric

field. For the case of varying laser irradiation time, in case of C2 (C4) being 40 (80) mins

respectively, it was observed to directly impact the growth of the flakes, resulting in thicker flakes

observed by AFM measurements as shown in Table 3.1. Once more, we specifically consider the

two samples (C2 & C4) for which only laser irradiation time was changed while other fabrication

parameters were kept constant. AFM scans and SEM micrographs of the examined carbon flakes

show a direct relation. The thickness of carbon flakes was found to increase with increase in laser

irradiation time. Developing on this theory, we can assume that continued exposure to laser

irradiation helps produce requisite building block for the growth of flakes, by breakdown of SMC.

This in turn would pose a limit on the growth process which would be defined by the amount of

SMC aggregates present in the solution. It could be interesting to study this limiting phenomenon

in future studies. The attenuation of examined carbon flakes in both cases discussed above

remained unchanged, indicating uniformity in the fabrication process.

Effect of SMC concentration on attenuation

Here, we studied the effect of variable fabrication parameters on attenuation across the carbon

flakes. The attenuation across studied carbon flakes was normalized per 100nm to cater for the

varying thickness of the examined flakes (Figure 3.14).


56

Figure 3.14 Results shown for experimental measurements for carbon flakes with different

fabrication parameters. Top: The attenuation is found to vary with changing concentration of

SMC during fabrication process. We consider the case of sample C2 and C5, for which only the

SMC concentration was altered, while other fabrication parameters were kept constant. We see

a clear difference in attenuation for varying SMC concentrations. We believe that a smaller

density per unit volume of bimetallic NPs intercalated in the carbon flakes is the reason for the

noticeably lower attenuation observed for a lower SMC concentration during fabrication.

Several other factors, which could affect the attenuation of flakes include the choice of a

substrate, electro-optical effects within the flakes and the source of energy during carbon flake

fabrication process. Bottom: The linear birefringence remains unchanged for the three

fabrication parameters studied here, namely varying laser irradiation time, application of an

electric field and varying SMC density in solution. To verify uniformity of fabrication process and

the high birefringence, we examined multiple flakes of varying thicknesses and lateral

proportions. Remarkably, all examined flakes show similar values of birefringence proving the

consistency of the fabrication process and confirming the presence of linear birefringence in

crystalline carbon flakes [14, 15].


57

We observe that by varying the SMC concentration in solution, the resulting attenuation across

the individual flakes can be altered. We consider the case of sample C2 and C5, for which other

fabrication parameters were kept constant and only the SMC concentration was changed. We

observe a clear difference in attenuation for varying SMC concentrations (Figure 3.14) [15].

In the theoretical modelling reviewed in our earlier study and discussed above [14, 15], we noticed

that the attenuation varies with the change in composition, diameter, and inter-particle distance

between adjacent bimetallic nanoclusters. Consequently, we believe that a smaller density per

unit volume of bimetallic NPs intercalated in the carbon flakes is the reason for the noticeably

lower attenuation observed for a lower SMC concentration during fabrication. Several other

factors, which could affect the attenuation of flakes include the choice of a substrate, electro-

optical effects within the flakes and the source of energy during carbon flake fabrication process

etc. [15].

Independence of linear birefringence of fabrication parameters

The linear birefringence remains unchanged for the three fabrication parameters studied here,

namely varying laser irradiation time, application of an electric field and varying SMC density in

solution (bottom part of Figure 3.14). To verify uniformity of fabrication process and the high

birefringence, we examined multiple flakes of varying thicknesses and lateral proportions (Figure

3.14). A similar measurement and assessment techniques were performed with AFM scans of

respective examined flakes to approximate the height of individual carbon flake. Remarkably, all

examined flakes show similar values of birefringence (see colored data points in Figure 3.12f),

proving the consistency of the fabrication process and confirming the presence of linear

birefringence in crystalline carbon flakes. This confirms the notion that the special crystalline

carbon phase causes the carbon flakes to exhibit linear birefringence.

Table 3.5. Variable fabrication parameters and their effect on carbon flakes [15]

Varied fabrication

parameter

Effect on optical properties of

carbon flakes

Effect on geometrical properties of

carbon flakes

Electric field on or off Optical properties remain

unchanged

Lateral elongation of flakes (with

field switched on)

Laser irradiation time Optical properties remain

unchanged

Increase in thickness with increasing

irradiation time

SMC density Increase in attenuation with

increasing SMC density

No observed effect on geometrical

properties

This also conforms with the analytical model built for carbon flakes, where linear birefringence

was a consequence of the orthorhombic crystalline phase. Summarizing the results of varying

fabrication parameters, we discuss them in the Table 3.5.

Circular birefringence of flakes

We observed in a recent analysis that carbon flakes could potentially also exhibit optical activity

(circular birefringence; CB). This could be attributed to intrinsic chirality of the alkynyl phosphine

ligand, which is part of SMC and contributes as a building block for carbon flakes.


58

Figure 3.15 (a) Circular birefringence observed in carbon flakes. A 2D raster scan showing a

uniform plateau in the central region of the flake, depicting the unambiguous presence of

circular birefringence in carbon flake. Spectral representation of circular birefringence.(b)

Depolarization response for different carbon flakes on sample C1 is shown. For b (i) the effect is

more smeared out due to small lateral dimensions causing parasitic scattering contributing to

central region of flakes. Due to this, the overall optical properties’ extraction is affected;

therefore, the measurements from this structure was not considered. For The other two flakes b

(ii & iii), labelled in Figure 3.13e&f as 340nm and 300nm, depolarization is much stronger from

the edges and the central region remains unaffected. Therefore, as a Figure of merit large,

enough lateral dimensions were selected so as to avoid scattering/depolarization effects.

As shown in Figure 3.15a, we plot the CB spectrum for carbon flake. Experimentally, at the longest

wavelength (700 nm), the CB is found to be approx. 0.01, while around 480nm a maximum value

of 0.04 is calculated. It is pertinent to highlight that in naturally occurring materials CB is of fairly

lower magnitude as compared to CB found in carbon flakes. It could be interesting to further

investigate this phenomenon in future studies for potential applications related to chiral sensing.

Depolarization effects in flakes measurement

Depolarization as discussed in section 2.3.1 is an optical property which can be extracted from

Müller matrix. During the analysis of the recorded data, it was also observed that depolarization

becomes important for cases where lateral dimensions are too small to avoid edge effects. As

shown in Figure 3.15b, we can observe that the depolarization effects for smaller flake (1µm x

2.2µm) are rather spread out as compared to flakes with larger lateral dimensions. Hence, the

spread-out depolarization in smaller flakes can lead to wrong estimates of birefringence and

dichroism. In our case as discussed above, we selected flakes carefully with large enough lateral


59

dimensions to avoid such problems. Therefore, examining depolarization acts as a figure of merit

for an effective experimental analysis.

Ellipsometric measurements results

In this section we discuss the experimental results obtained with respect to the direct complex-

valued refractive index retrieval of various samples. For this study, we utilized linearly horizontal

(𝑥) and vertical (𝑦) polarized light. In future this technique can be implemented using spatially

structured light beams. More details regarding this can be found in Appendix B to this thesis.

Initially, we discuss the experimental and computational analysis for dielectric sample (glass N-

BK7). Later, we examine a non-absorbing uniaxial crystal (lithium niobate), and isotropic

absorbing substrate (α-silicon).

Figure 3.16 Different stage of ellipsometric technique applied on glass sample. A linearly (H)

polarized light is focused on the top surface of examined sample (glass, N-BK7) using a high NA

microscope objective. The reflected back focal plane image of microscope objective is recorded

on a 12-bit CCD camera. The horizontal radial profile (corresponding to TM-polarized reflected

light) is extracted and normalized with respect to normal incident wavenumber (𝑘x/y = 0). The

extracted line profile is then computationally fitted using the Equations 2.49b, 2.52a&b & 2.53b

with only unknown variables being 𝑛/𝜅 the real and imaginary part of refractive index of

examined sample. The extracted complex refractive index estimates are then shown in table as

compared to existing estimate from literature [190].


60

Lastly, we present our preliminary experimental results for carbon flakes refractive index and

analyzed them in light of optical properties extracted by microscopic Müller matrix measurement

technique. All of the experimental results mentioned below were performed at a wavelength of

632nm.

Ellipsometric parameters for a dielectric: glass

We discuss first the case of glass (N-BK7) for ellipsometric parameters extraction. By following the

scheme mentioned in Section 3.2.6, we evaluate the BFP to computationally estimate the

refractive index. The measurement on glass also served as a reference for the ellipsometric system

and to understand the constraints or limitations in this regard. We use horizontal-polarized

incident light beam to probe the sample. The reflected back focal plane image of microscope

objective is recorded on a 12-bit CCD camera as shown in top right corner of Figure 3.16. We

extracted the line profile from BFP along the horizontal axis, where a pure TM reflected light is

expected as shown in top left corner of Figure 3.16. The spurious high intensity reflections near

normal incidence can be attributed to reflections from lower interface of cover glass. We used

three different methods for extraction of complex refractive index; full Fresnel equations based

on Fresnel coefficients mentioned in Equations 2.49b, direct estimation from reflected light

intensity at Brewster angle using Equations 2.52a&b and approximate reflection calculations

based on Equations 2.53b.

From the line profile and recorded BFP images (top left corner of Figure 3.16), we can clearly

observe the vanishing intensity expected at the Brewster angle for TM reflectance from glass (N-

BK7). The extracted line profile is then computationally fitted using the equations mentioned

above with only unknown variables being 𝑛/𝜅 the real and imaginary part of refractive index of

examined sample. The nonlinear least square curve fitting function (lsqnonlin) in Matlab was used

for this purpose. The extracted complex refractive index estimates are shown in lower left corner

in Figure 3.16 in a table as compared to existing estimate from literature [190]. The estimates are

in in good agreement with the expected refractive index of glass and assures us of the workability

of this scheme. The spurious reflections near normal incidence does not cause too much deviation

to our final complex refractive index estimates. This will be discussed in detail in our next

experimental case.

Ellipsometric parameters for uniaxial non-absorbent materials: Lithium Niobate

As a second case, we perform the ellipsometric measurement on a uniaxial crystal (Lithium

Niobate). In case of a uniaxial crystal, we expect two refractive indices (ordinary 𝑛𝑜/𝜅𝑜 and

extraordinary 𝑛𝑒/𝜅𝑒) along two principal axes of the crystal. In this regard, we studied a sample

of lithium niobate and followed the similar experimental procedure as adopted for glass. We use

horizontal-polarized incident light beam to probe the sample. The sample is aligned such that the

fast axis of lithium niobate sample points in the direction of incident polarized light (H-polarized)

to avoid retardance effects as discussed in section 3.2.6. The reflected back focal plane image of

microscope objective is recorded on a 12-bit CCD camera. We extracted the line profile from BFP

along the horizontal axis, where a pure TM reflected light is expected as shown in Figure top right

corner of 3.17. Due to spurious reflections, the central part of the BFP was removed and

subsequently fitting was performed as shown in Figure top left corner of 3.17. The extracted line

profile is then computationally fitted using the Equation 2.49b, 2.52a&b & 2.53b with only


61

unknown variables being 𝑛𝑒/𝜅𝑒 the real and imaginary part of extraordinary refractive index of

examined sample, respectively. A similar procedure is adopted with linearly (V) polarized incident

light without changing the sample orientation. This meant that the slow axis of lithium niobate

sample points in the direction of incident polarized light (V-polarized). This helps us to

computationally evaluate the 𝑛𝑜/𝜅𝑜 the real and imaginary part of ordinary refractive index of

examined sample, respectively. The extracted complex refractive index estimates are shown in

left lower corner in Figure 3.17 in a table as compared to existing estimate from literature [191].

The estimates from full equations fitting model (2.49b) and approximate fitting model (2.53b) are

in good agreement with the expected refractive indices of lithium niobate and assures us of the

workability of this scheme.

Figure 3.17 Ellipsometric measurement for lithium niobate, similar to procedure adopted for

dielectric sample. A linearly (H) polarized light is focused on the top surface of examined sample

(lithium niobate) using a high NA microscope objective. The sample is aligned such that the fast

axis of lithium niobate sample points in the direction of incident polarized light to avoid

retardance effects. The reflected back focal plane image of microscope objective is recorded on

a 12-bit CCD camera. The horizontal radial profile (corresponding to TM-polarized reflected

light) is extracted and normalized with respect to normal incident wavenumber (𝑘x/y = 0). The

extracted line profile is then computationally fitted using the Equation 2.49b, 2.52a&b & 2.53b

with only unknown variables being 𝑛𝑒/𝜅𝑒 the real and imaginary part of extraordinary refractive

index of examined sample, respectively. The extracted complex refractive index estimates are

then shown in table as compared to existing estimate from literature [191]. A similar procedure

is adopted with linearly (V) polarized incident light without changing the sample orientation.

This helps us to computationally evaluate the 𝑛𝑜/𝜅𝑜 the real and imaginary part of ordinary

refractive index of examined sample, respectively.


62

The Brewster angle-intensity based estimates are way off from the values observed in literature.

This is because this method searches for the lowest intensity point and related angle of incidence,

and makes refractive index estimates based on that, as defined by Equations 2.52a&b. Since our

maximum angle of incidence is limited by the NA of the focusing objective, hence the Brewster

location-intensity based estimates do not remain accurate for the materials whose

Brewster/pseudo-Brewster angle lies outside of the NA of the focusing objective. This also

proves/validates the efficacy of our ellipsometric technique which is based on fitting of reflected

intensity profile over complete recorded angular spectrum rather than singular point (Brewster

angle-intensity estimate). Although, the central part of the BFP image was removed due to

spurious noise but the fitting model estimates are still in good agreement with literature. We

believe, the reason is related to the fact that the reflectance varies slowly for smaller

wavenumbers 𝑘x/y as compared to higher wavenumber. Thus, the effect of removal of central

part of BFP image on fitting function is negligible. This also informs us that the fitting function is

more sensitive to the gradient of the reflectance radial line profile which is an important benefit

of using our ellipsometric technique.

Figure 3.18 Ellipsometric measurement for absorbing isotropic sample (α-silicon), similar to

procedure adopted for previous cases. A linearly (H) polarized light is focused on the top surface

of examined sample (lithium niobate) using a high NA microscope objective. The reflected back

focal plane image of microscope objective is recorded on a 12-bit CCD camera. The horizontal

radial profile (corresponding to TM-polarized reflected light) is extracted and normalized with

respect to normal incident wavenumber (𝑘x/y = 0). The extracted line profile is then

computationally fitted using the Equations 2.49b, 2.52a&b & 2.53b with only unknown variables

being 𝑛/𝜅 the real and imaginary part of ordinary refractive index of examined sample,

respectively. The extracted complex refractive index estimates are then shown in table as

compared to existing estimate from literature [192].


63

Ellipsometric parameters for absorbing isotropic materials.

Next, we consider the case of absorbing material. In this regard, we study a sample of α-silicon

substrate (Figure 3.18). We use horizontal-polarized incident light beam to probe the sample. The

reflected back focal plane image of microscope objective is recorded on a 12-bit CCD camera in

similar fashion as mentioned above. We extracted the line profile from BFP along the horizontal

axis, where a pure TM reflected light is expected as shown in Figure 3.17. The extracted line profile

is then computationally fitted using the Equation 2.49b, 2.52a&b & 2.53b with only unknown

variables being 𝑛/𝜅 the real and imaginary part of ordinary refractive index of examined sample,

respectively. The extracted complex refractive index estimates are shown in left lower corner in

Figure 3.17 in a table as compared to existing estimate from literature [192]. The estimates from

full equations fitting model (2.49b) is in good agreement with the expected refractive indices of

α-silicon (from literature) and assures us of the workability of our scheme. As expected, from the

refractive index of α-silicon, the Brewster angle lies outside the NA of microscope objective.

Therefore, Brewster angle-intensity based estimates (as defined by Equations 2.52a&b) are way

off from the values observed in literature. The approximate equations-based model also show

way off estimates. This can probably be attributed to noise and approximation of equations which

leads to bad estimated of refractive index.

Ellipsometric parameters for carbon flakes

As a next case, we measure a carbon flake in a similar fashion as mentioned above. These

measurements hence provide us with first principal approximations of the refractive index of

Carbon flake. We use horizontal-polarized incident light beam to probe the sample. The sample is

aligned such that the fast axis of individual carbon flake points in the direction of incident

polarized light (H-polarized) to avoid retardance effects as discussed in section 3.2.6. The reflected

back focal plane image of microscope objective is recorded on a 12-bit CCD camera in similar

fashion as abovementioned cases. We extracted the line profile from BFP along the horizontal

axis, where a pure TM reflected light is expected as shown in top right corner of Figure 3.17.

We used three similar methods for extraction of complex refractive index as above; full Fresnel

equations based on Fresnel coefficients mentioned in Equations 2.49b, direct estimation from

reflected light intensity at Brewster angle using Equations 2.52a&b and approximate reflection

calculations based on Equations 2.53b. The extracted line profile is then computationally fitted

using the above-mentioned equations with only unknown variables being 𝑛𝑒/𝜅𝑒 the real and

imaginary part of extraordinary refractive index of examined carbon flake, respectively. A similar

procedure is then adopted with linearly (V) polarized incident light without changing the sample

orientation. This helps us to computationally evaluate the 𝑛𝑒/𝜅𝑒 the real and imaginary part of

extraordinary refractive index of examined carbon flake, respectively. The extracted complex

refractive index estimates are shown in lower left corner of Figure 3.19 in a table. The estimates

from full equations fitting model (2.49b), Brewster angle-intensity estimates (Equations 2.52a&b

) and approximate fitting model (Equation 2.53b) are all in in good agreement with each other.

The results are shown in Figure 3.19. The results seem to be compliant with the previous findings

for carbon flakes and their optical properties discussed above [14, 15, 18, 19]. From our previous

discussion of carbon flakes, we experimentally (using microscopic Müller matrix measurement

technique in transmission) found a birefringence of ~0.1, which can also be confirmed here, in

case of complex refractive index retrieval made in ellipsometric measurement.


64

Figure 3.19 Preliminary ellipsometric analysis for carbon flake, similar to procedure adopted for

previous cases. A linearly (H) polarized light is focused on the top surface of examined sample

(lithium niobate) using a high NA microscope objective. The sample is aligned such that the fast

axis of carbon flake sample points in the direction of incident polarized light to avoid retardance

effects. The reflected back focal plane image of microscope objective is recorded on a 12-bit

CCD camera. The horizontal radial profile (corresponding to TM-polarized reflected light) is

extracted and normalized with respect to normal incident wavenumber (𝑘x/y = 0). The

extracted line profile is then computationally fitted using the Equations 2.49b, 2.52a&b & 2.53b

with only unknown variables being 𝑛𝑒/𝜅𝑒 the real and imaginary part of extraordinary refractive

index of examined sample, respectively. A similar procedure is adopted with linearly (V)

polarized incident light without changing the sample orientation. This helps us to

computationally evaluate the 𝑛𝑜/𝜅𝑜 the real and imaginary part of ordinary refractive index of

examined sample, respectively. The extracted complex refractive index estimates are then

shown in table.

The extinction coefficient estimates from ellipsometric measurements are found to be on the

order of 0.08-0.18 (Figure 3.19). We can estimate the transmission of light through carbon flake

of certain thickness by applying the relation α =4𝜋𝑘

𝜆0 and use it to get an estimate of transmitted

light through carbon flake (by Beer-lamberts law). This gives us transmittance value ~ 0.44 (𝜆 =

632𝑛𝑚) for a flake thickness of 340nm which is in good agreement to results shown in Figures

3.12 & 3.14 [14, 15, 18, 19]. In this chapter we discussed novel self-assembled orthorhombic

carbon flakes intercalated with bimetallic (Au-Ag) nanoclusters which due to their organo-metallic


65

nature can find variety of applications in sensing and plasmonic applications [24-26]. we

experimentally investigated the intriguing optical properties in varying fabrication parameters.

With the help of experimental results and theoretical modelling, we relate the high optical

birefringence in the carbon flakes to the crystalline arrangement of carbon atoms in the

orthorhombic lattice [14, 15, 18-20]. We also have made the preliminary estimates of complex

refractive index of the carbon flakes using high NA objective MAI ellipsometry. Due to the high

sensitivity of carbon flakes, they are prone to optical modification which can be utilized for guiding

of light and other opto-electronics applications. The carbon flake, due to its hybrid nature and the

unique phenomenon of self-assembly, still poses many unanswered questions which can only be

answered by continued research. We will discuss some of future projects and research ideas in

Chapter 5 of this thesis.

4. Chiral 2D metasurfaces

In the previous chapter, we discussed a novel self-assembled hybrid structure consisting of an

orthorhombic phase of carbon intercalated with gold-silver nanoclusters. Just like self-assembled

structures, metamaterials/metasurfaces are constructed on the concept of structuring individual

building blocks [7, 32, 35, 42]. We achieve metamaterials by fabricating sub-wavelength scattering

structures called meta-atoms and periodically arranging them. By finely selecting the shape,

orientation, material, and size of meta-atoms, we can tune their optical response [43, 44]. The

field of metamaterials design and analysis has grown extensively since early 2000 due to

technological developments and with lithographic abilities to control surface structuring down to

nanometer scales. Although the concept of negative refractive index and non-unity permeability

was shown by V. G. Veselago [193], but it was the seminal works from J. Pendry, D. Smith and R.

Shelby who experimentally validated and fabricated such metamaterials [194]. Since then,

metamaterials–due to their light controlling and guiding phenomenon, have been used to control

the wave front, intensity, phase, and polarization state of light, both spatially and temporally, and

has instigated interest in many fields of science and industry [38, 43, 44]. Usually, this control over

fine-tuned optical response is achieved by the coupling of incident light waves to electric and

magnetic resonances of the individual meta-atom transpiring overall into control of light

propagation. In the next section, we will first briefly discuss the phenomenon of a chiral optical

response and then later relate it to the design features at the single meta-atom level.

4.1 Chirality and light matter interaction

In Chapter 2, we briefly mentioned the chiral response in the context of circularly polarized light

(in section 2.1.2) and later again while explaining optical activity in a medium (in section 2.3.1)

[63, 96, 195]. The simplest example of this intriguing phenomenon is our hands. Our left and right

hands cannot be superimposed onto each other by simple rotations in 3D-space or, in other

words, they lack mirror-symmetry. A chiral structure and its mirror image are called enantiomers

(also referred to as left and right-handed structures). Hence, a structure which can be

superimposed with its mirror image is called chiral. The most popular quote about the definition

of chirality has been coined by Lord Kelvin [196], who described it in the following way:

“I call any geometrical Figure, or group of points, chiral, and say that it has chirality if its image in

a plane mirror, ideally realized, cannot be brought to coincide with itself.”

The existence of chirality can be abundantly found in nature, ranging from microscopic to

macroscopic scales [197, 198]. Intriguing as it is, the notion of chiral behavior of natural objects,

such as proteins consisting of amino-acids being mainly left-handed and right handedness of

RNA/DNA has baffled researchers for decades [197-199]. As discussed in section 2.3.1, the

polarized light interaction with media/structures having chiral response results in optical activity.

This happens either in the form of circular extinction or circular retardance of light propagation

through such medium [63]. This has also instigated researchers to produce unique designs to

exploit optical activity for scientific and industrial applications [193, 200-205]. However, since its

first report in natural materials, optical activity is usually found to be weak [206]. Hence


67

considerable effort has been made to produce artificial chiral materials [41, 207-212]. Expanding

on our previous discussion above and in section 2.1.1 & 2.3.1 regarding Maxwell’s equations and

optically active materials, in this section we will try to understand the fundamental idea that

relates them in case of light-matter interaction. For an isotropic homogenous medium, the

respective constitutive relations were discussed earlier. One main consideration of an isotropic

medium is that an electric field (E) and electric displacement (D) are parallel to each other. This

condition holds true also for magnetization [7, 64]. In general, we can assume no magneto-electric

coupling in such media due to orthogonal electric and magnetic fields. In a bi-isotropic medium,

this condition does not remain true anymore and magneto-electric coupling can be observed.

Hence, quantities in the constitutive equations can be generalized mathematically as [213-218],

𝐃 = ε𝐄 +(χ−iγ)

c𝐇, (4.1a)

𝐁 = μ𝐇 +(χ+iγ)

c𝐄, (4.1b)

where 𝜒 & 𝛾 are nonreciprocity and chirality parameters of a medium. The presence of the factor

±𝑖 (phase factor of π/2) in front of the chiral parameter leads 𝑬 & 𝑫 being out of phase [219-221].

For an isotropic medium both 𝜒&𝛾 vanish, while for non-zero values for both factors, they define

a general bi-isotropic medium. For 𝜒 = 0 & 𝛾 ≠ 0, we define a chiral (optically active) medium,

which can be compared to the definition given by Pasteur about inner chirality of a crystal [222].

For 𝜒 ≠ 0 & 𝛾 = 0, represents a nonreciprocal magnetically active medium which produces

asymmetric optical effects in forward and backward propagation directions (also known as

Faraday’s effect) [223].

Building on the constitutive relations for chiral effects, we can derive the chiral wave equation.

The eigen-polarizations of the chiral wave equation are waves with spin angular momentum of

±1 (right and left-handed polarization). For such a wave equation, we relate corresponding wave

vectors for circularly polarized light (𝑘𝑟𝑐𝑝 𝑙𝑐𝑝⁄ ) to refractive indices as (𝑛𝑟𝑐𝑝 𝑙𝑐𝑝⁄ ) [213, 215, 216],

𝑘rcp lcp⁄ =ω

cnrcp lcp⁄ , (4.2a)

𝑛rcp lcp⁄ = n ± γ, (4.2b)

where �� = √휀𝜇 is the nominal refractive index. For a complex value 𝛾, we can relate the real part

of 𝛾 to circular birefringence and imaginary part of 𝛾 to differential circular absorption or circular

dichroism in the context of optical properties. Circularly polarized light is itself chiral, as it hold an

intrinsic sense of handedness (right-handed or left-handed) associated with the rotation of the

electric field vector. Because of this, circularly polarized light is a natural choice for probing chiral

systems. Hence it is intuitive to realize that chiroptical effects arise because of the fact that

incident circularly polarized light experiences different complex refractive indices for two circular

handedness. This can be linked directly to the definition of optical activity, discussed in section

2.3.1 and Appendix A to this thesis. In terms of a Müller matrix of a chiral structure (as shown in

Figure 2.7), optical activity corresponds to the cross-diagonal Müller matrix elements.


68

Chiral light matter interaction

The main aim in the design of chiral meta-atom is to increase chirality of individual meta-atom or

find other ways to increase the chiroptical effect. These effects can then be coupled to far field

by carefully designing a periodic array of meta-atoms. Similar to constitutive chiral relations, we

can describe chiral meta-atoms based on induced dipole moments which depend on induvial

electric, magnetic and cross polarizations [224]. At individual meta-atom level this relates to

electric and magnetic moments and resonances [225]. In this case, for a single meta-atom the

differential absorption (Δ𝑎) for incident left and right circularly polarized light is defined as [226],

Δ𝑎 =2

0ℑ(𝛼𝑒𝑚)Δ𝐶, (4.3)

where (𝛼𝑒𝑚) is magneto-electric cross-polarizability and Δ𝐶 defines the differential optical

chirality density for right and left circularly polarized light, and for monochromatic light wave the

case simplifies to,

C = − 0𝜔

2ℑ(𝐸∗. 𝐵). (4.4)

Hence the chiral effects can be considered a combination of chirality density of induvial meta-

atom and chiral incident fields. Cohen et. al. produced another mathematically elegant way to

describe the chiral effects mentioned above in terms of dissymmetry factor (g)[67, 227],

𝑔 = −ℑ(𝛼𝑒𝑚

𝛼𝑒) (

2𝐶

𝜔𝑈𝑒), (4.5)

where 𝑈𝑒 is the time average electric energy density. This equation holds true for any parity

interchangeable light waves. For a circularly polarized light they reduce to,

𝑔𝐶𝑃𝐿 = −ℑ(𝛼𝑒𝑚

𝛼𝑒) (

4

𝑐). (4.6)

Many a times, a meta-atom is achiral, but still can produce chiral optical response when probed

with circularly polarized light beams. In this chapter we will concentrate on such structures. Based

on design, composition, and orientation of such meta-atom we can induce specific electric or

magnetic resonances which leads to chiral response [16] (see Figure 4.1).

Concept of 2D and 3D chirality

In the literature, usually two variants of chirality (in terms of dimensionality) are discussed [100,

228]. A 2D meta-atom usually would have sub-wavelength thickness and sometimes are referred

to as metasurfaces (Figure 4.1). For instance, THz metasurfaces, the thickness of meta-atom

would be in the order of 20-100nm. On contrary, 3D meta-atoms have thickness greater than that

and are usually referred to as 3D-metastructures.


69

Figure 4.1 Distinguishing different forms chirality induced in 3D and 2D structures. In the

literature, usually two variants of chirality (in terms of dimensionality)are discussed [100, 228]. A

2D meta-atom usually would have sub-wavelength thickness and sometimes are referred to as

metasurfaces (Figure 4.1). For instance, THz metasurfaces, the thickness of meta-atom would be

in the order of 20-100nm. On contrary, 3D meta-atoms have thickness greater than that and are

usually referred to as 3D-metastructures. Meta-atom designs are taken from existing literature.

We can define 3D chirality (volume chirality) as when we cannot superimpose an object by

translation and rotation in three dimensions to its mirror image. Similarly, when we cannot

superimpose an object by translation and rotation in two dimensions, to its mirror image, it is said

to have 2-dimensional chirality or planar chirality [229, 230]. Based on symmetry properties, 2D

and 3D chirality have some distinct differences. For a chiral 3D structure, the sense of handedness

prevails irrespective of the direction of observation, as a consequence of Lorentz reciprocity

theorem [231]. Therefore, the chiroptical effects (optical activity) are supposed to produce same

results even when the direction of illumination is reversed. In terms of Müller matrix of a chiral

structure this means that the cross-diagonal elements would have same magnitude and sign

irrespective of illumination direction.

For a 2D chiral structures, the sense of handedness flips when observed from the opposite

direction [232-235]. Due to this reversal of handedness in 2D chiral structures, we can expect

asymmetric optical transmission (AT) depending on the direction of illumination (forward or

backward) as shown in Figure 3.2 [236-240]. As a consequence of reversal of handedness optical

activity vanishes in 2D planar structures, manifested in the Müller matrix of the chiral structure

as cross diagonal terms with same magnitude and opposite signs (𝑚03 = −𝑚30 & 𝑚12 = −𝑚21).


70

Figure 4.2 Th concept of asymmetric transmission explained for 2D metasurfaces. Adapted from

[100]. Asymmetric transmission does not violate Lorentz reciprocity theorem [241].this can be

understood once a complete polarimetric analysis is performed on transmitted light through the

2D chiral structure. For example, in case of incident right-handed circular polarization (RCP), the

transmitted intensity light in projected RCP basis remain same irrespective of wave vector

reversal (upper half of Figure 4.2). Hence asymmetric transmission is directly linked to the

conversion of polarization upon propagation through 2D chiral structure as shown. Intrinsic

chirality as the name suggests corresponds to the intrinsic handedness in the individual meta-

atom (either 2D or 3D) design. Some examples of intrinsic chirality are helix for 3D structure and

geometrical shapes of S, L or Archimedes spiral for 2D planar structures which lack in-plane

mirror symmetry.


71

Hence, optical activity can be related to 3D chirality, while 2D chirality is studied in terms of

asymmetric transmission in literature [236-240].

Asymmetric transmission does not violate Lorentz reciprocity theorem as shown in Figure 4.2

[241].this can be understood once a complete polarimetric analysis is performed on transmitted

light through the 2D chiral structure. For example, in case of incident right-handed circular

polarization (RCP), the transmitted intensity light in projected RCP basis remain same irrespective

of wave vector reversal (upper half of Figure 4.2). Hence asymmetric transmission is directly linked

to the conversion of polarization upon propagation through 2D chiral structure as shown in Figure

4.2.

Intrinsic/extrinsic chirality

In Equation 4.6 we discussed that the chiroptical response is a manifestation of either individual

meta-atom response or the polarization nature of impinging light wave. Hence, it is important to

discuss the cases of 2D and 3D chirality [234, 242] in the context of intrinsic and extrinsic chirality.

Intrinsic chirality as the name suggests corresponds to the intrinsic handedness in the individual

meta-atom (either 2d or 3D) design. Some examples of intrinsic chirality are helix for 3D structure

and geometrical shapes of S, L or Archimedes spiral for 2D planar structures which lack in-plane

mirror symmetry [243-247] as shown in Figure 4.1.

In essence, it is possible for an intrinsically achiral structure to produce chiral optical effects. This

is defined in literature as extrinsic or pseudo chirality. It can be manifested in number of ways

including, heterogeneity or arrangements of individual achiral structures, the phase, angular

spectrum, and polarization of incident light wave can be adjusted to lead to a chiroptical response

form the structure [53, 210, 234, 248-257]. Extrinsic chirality has been an intense field of study in

the last decade and has found applications in multiple fields of science [210, 234, 253-255].

In this chapter, we will discuss the special case of a 2D fourfold symmetric achiral meta-atom

(quadrumer) to induce desired asymmetric transmission by extrinsic chirality. We will start with

understanding the in-plane scattering from an achiral individual quadrumer. Later we will discuss

ways to couple the in-plane scattering of an individual quadrumer to far fields by placing them in

a periodic array.

Quadrumer design and surface lattice resonances

The simulations discussed in this section were conceptualized optimized and performed by

Professor (Assistant) Dr. N. Apurv Chaitanya. We consider the case of an achiral (four-fold

symmetric) structure consisting of four rectangular bars arranged in a square configuration, as

shown in Figure 4.3[16]. The lateral dimensions of each bar are 160nm x 80nm, while thickness is

kept to 20nm, hence can be considered as a 2D metastrucutre. We choose gold as material for

quadrumer for which the refractive index model is chosen from standard literature [258]. We use

finite difference time domain method (FDTD) [259] to simulate the said structure by

computationally solving fully vectorial Maxwell equations for light propagation and interaction.


72

Figure 4.3 Top: Schematic cross-sectional views of the studied individual quadrumer with

specific dimensions. We choose gold as material for quadrumer for which the refractive index

model is chosen from standard literature [258]. Finite difference time domain method (FDTD)

[259] was used to simulate the said structure by computationally solving fully vectorial Maxwell

equations for light propagation and interaction. We use perfectly matching layer (PML)

boundary condition along all three axes to study individual quadrumer. The light propagates in

𝑧-direction while the lateral plane of quadrumer is 𝑥𝑦-plane. Bottom: Polar plots for far-field

scattering for RCP and LCP incident light at a wavelength of 900 nm with an ambient refractive

index of 1.51. The circular differential scattering (CDS) shows an 8-lobe pattern with maxima

around multiples of 22.5° and 67.5°, while vanishing in-plane scattering around 0° and 45°. It is

also important to highlight that, for a RCP (LCP) incoming light, maximum in-plane scattering is

along multiples of 11.25° + 𝑛 × 90° (−11.25 + 𝑛 × 90°), respectively. Hence a maximum

coupling for each handedness to a lattice mode would happen for these angles. However, the

contrast (CDS) between one handedness and the other is maximized along odd multiples of

22.5. The purple/orange dashed circles corresponds to CDS maxima with opposing

positive/negative signs, respectively [16].


73

We use perfectly matching layer (PML) boundary condition along all three axes to study individual

quadrumer. To study the differential optical response of the structure, we use right and left

circularly polarized light in a spectral range from 400nm-2000nm. The incident beam is made to

propagate perpendicular to the plane of the quadrumer (normal incidence). A homogenous

surrounding refractive index of 1.51 is chosen for simulation. From the simulation results we

observe vanishing circular dichroism (by comparing quadrumer extinction cross sections for RCP

and LCP) confirming achiral nature of quadrumer, as expected from 2D meta-atom. However, as

seen in bottom part of Figure 4.3, we can observe distinct four-fold in-plane scattering patterns

for right and left circularly polarized light at a wavelength ~900nm as a function of azimuthal

angle. In the lower part of Figure 4.3, we plot a circular differential image of these two CPL states

also called as circular differential scattering (CDS). We can observe that CDS has its maxima along

odd multiples of 22.5°, while for multiples of 0° and 45°, CDS vanishes. The purple/orange dashed

circles corresponds to CDS maxima with opposing positive/negative signs, respectively. It is also

important to highlight that, for a RCP (LCP) incoming light, maximum in-plane scattering is along

multiples of 11.25° + 𝑛 × 90° (−11.25° + 𝑛 × 90°), respectively. Hence a maximum coupling for

each handedness to a lattice mode would happen for these angles. However, the contrast (CDS)

between one handedness and the other is maximized along odd multiples of 22.5°. An explanation

for CDS in four-fold quadrumer can be attributed to higher order electric resonance (octa pole)

which causes differential response for circularly polarized light [16].

Surface lattice resonances

The circular differential in-plane scattering observed in individual quadrumer is washed away

when probed in the far field experimentally. To overcome this limitation, we can periodically

arrange meta-atoms with certain periodicity. This leads to generation of propagating surface

modes also known as surface lattice resonances (SLR) [46-50]. These propagating surface modes

occur close to Rayleigh anomaly condition due to resonant coupling of incident light beam with

individual meta-atom resonance and grazing diffracted waves [51]. Mathematically this is defined

as

𝜆𝑅𝐴/𝑆𝐿𝑅 = Λ𝑛 (1

√𝑚𝑥2+𝑚𝑦

2+ sinθ) , (4.7)

where Λ is the periodicity of meta-atoms, 𝑚𝑥 & 𝑚𝑦 are the grazing diffraction orders, θ is the

incident angle (in our case normal incidence implies that this factor is considered to be zero) and

n is the refractive index of surrounding medium (considered to be same for superstrate and

substrate). The diffraction orders (𝑚𝑥 & 𝑚𝑦) are integers that define the direction of propagation

(in azimuthal angle) of grazing diffraction orders as shown in Figure 4.4a. For a certain periodicity

and surrounding refractive index, they happen at wavelengths as shown in Figure 4.4b. A sharp

decrease in transmission can be expected around these wavelengths (𝜆𝑅𝐴/𝑆𝐿𝑅) as shown in Figure

4.4c with very narrow FWHM bandwidth. Typically, the spectral width of 𝜆𝑅𝐴/𝑆𝐿𝑅 is in the order

of 10nm [48]. This has instigated a lot of research towards application of surface lattice

resonances for sensing devices [47, 49, 52-54].


74

It is also possible to couple the in-plane scattering of individual meta-atom to SLR. In our case

regarding CDS observed in individual quadrumer, this could lead to differential surface lattice

resonances in a quadrumer array, which has not been previously studied in literature. Hence, it is

one of the novel concepts for this study. This requires choosing the right combination of

periodicity and surrounding refractive index and understanding the geometry of individual

quadrumer to enhance the differential SLR effect.

Figure 4.4 Propagating surface modes occur close to Rayleigh anomaly condition due to

resonant coupling of incident light beam with individual meta-atom resonance and grazing

diffracted waves [51] as defined in Equation 4.7.These depend on periodicity (Λ) of meta-atoms,

the grazing diffraction orders (𝑚𝑥 & 𝑚𝑦), incident angle (θ) of impinging light beam and the

refractive index (𝑛) of surrounding medium (considered to be same for superstrate and

substrate). (a) The diffraction orders (𝑚𝑥 & 𝑚𝑦) are integers that define the direction of

propagation of grazing diffraction orders. (b) For a certain periodicity (600nm blue line,700nm

red line) and surrounding refractive index (n = 1.51), we can excite various diffraction orders at

wavelengths corresponding to Equation 4.7.(c) For a certain periodicity (600nm blue line,700nm

red line) and surrounding refractive index (n = 1.51), first diffraction order can be observed at

𝜆𝑅𝐴/𝑆𝐿𝑅 = 905, 1060𝑛𝑚 respectively. A sharp decrease in transmission can be expected around

these wavelengths (𝜆𝑅𝐴/𝑆𝐿𝑅) as shown with very narrow FWHM bandwidth. Typically, the

spectral width of 𝜆𝑅𝐴/𝑆𝐿𝑅 is in the order of 10nm. (d) Illustration of surface lattice resonances

propagating in the plane of quadrumer near first order Rayleigh anomaly. The inset correspond

to a real space image of a quadrumer array.

In this regards the next step is to define/optimize the parameters (periodicity, diffraction orders,

orientation of quadrumer etc.) to enhance the asymmetric transmission in a planar quadrumer

array. Mathematically we define asymmetric transmission as


75

AT =(𝐼𝑅𝐶𝑃−𝐼𝐿𝐶𝑃)

𝐼𝑅𝐶𝑃+𝐼𝐿𝐶𝑃. (4.8)

Where 𝐼𝑅𝐶𝑃 , 𝐼𝐿𝐶𝑃 are the transmitted intensity for case of RCP,LCP polarized incident light. We can

decompose the total transmitted intensity (for respective) into intensities correspond to certain

diffraction orders. As evident from discussion above, the differential SLR should lead to higher AT

near Rayleigh anomaly for higher diffraction orders (neglecting zeroth diffraction order). As we

move away from the Rayleigh anomaly condition the differential coupling to SLRs decreases

resulting in reduced AT. The extreme case is when the diffracted beam becomes normal to the

planar quadrumer array, where the diffracted beam has same transmitted power for RCP and LCP

incident light beam. Hence, Due to symmetry reasons no AT is observed in zeroth diffraction

order. To understand and optimize the phenomenon of asymmetric transmission in planar

quadrumer array, we perform a FDTD simulation of a quadrumer with periodic boundary

conditions in the lateral plane of quadrumer and PML boundary condition along the direction of

light propagation [16]. We choose to study asymmetric transmission in the first diffraction orders

as shown in Figure 4.4d (for illustrative purposes). The insets correspond to a real space image of

recorded surface lattice mode). An in-plane rotation of individual quadrumer leads to an

equivalent rotation of the in-plane scattering. This rotation can lead to efficient coupling of in-

plane scattering to surface lattice resonances of first diffraction order, resulting in enhanced AT.

Figure 4.5 SEM micrograph of fabricated quadrumer array. The nanofabrication was done on N-

BK7 cover glass with thickness of 170 microns without any adhesion layer (fabrication done by

and pictures courtesy of Dr. Orad Reshef, University of Ottawa, Canada).

The simulations were performed for varying orientations (in plane rotation of quadrumer ranging

from -45°to 45°). We chose four orientations which are of importance, i.e. 0°,22.5°,45°and –

22.5°(corresponds to 67.5°). These orientations were chosen based on the maxima/minima of

CDS observed in single quadrumer simulation. A sweep of periodicity was performed from 500-

1000nm. The simulation results pointed towards enhanced AT for the case of 600nm periodicity

of quadrumer. Hence the sample was fabricated with said periodicity and orientations.

200 m

1 m

22.5

22.5

45


76

Fabrication of 2D chiral metasurfaces

The fabrication of quadrumer metasurface was done at the University of Ottawa by. Dr. Orad

Reshef. For fabrication, an N-BK7 cover glass of 170 microns thickness was used. To study the

effect of quadrumer orientation (0°, 22.5°, -22.5° and 45°), four different quadrumer arrays ( each

200 x 200 microns) were fabricated with specific in-plane orientation of individual quadrumer as

shown in Figure 4.5. No adhesion layer was used between metal and glass to avoid any

asymmetric effects from surrounding metasurface environment. All four quadrumer

metasurfaces were fabricated with similar periodicity (600nm). Unfortunately, the quadrumer

array with 0° orientation had fabrication defects, hence, was not used for experimental

investigation.

4.2 General experimental setup and arrangements

For experimental measurements of fabricated quadrumer arrays, we use a modified version of

setup as described in Chapter 3. We used a broadband supercontinuum white light laser source

to spectrally probe the sample in the range of 630nm-950nm. The spectrally filtered light is fed

into a single mode optical fiber which acts as a mode filter only allowing fundamental TEM00 mode

to propagate as shown in Figure 4.6a (i). The beam diameter is then further magnified with the

help of two convex lenses as per the requirement of the setup. With the help of four guiding

mirrors, input light beam arrives to the top-down custom-made microscope tower. We use a

broadband polarizer and a quarter wave plate to generate right and left circularly polarized light

required for the experiment as shown in Figure 4.6a (ii). The circularly polarized beam was than

weakly focused (lens with f=60mm) onto the quadrumer array which has dimensions of 200µm x

200µm (Λ = 600𝑛𝑚). The sample is immersed in oil whose refractive index matches that of glass

substrate, to provide a symmetric surrounding environment. We use a 3-axis piezo stage to

precisely position the examined sample in focal spot of incident beam. An oil-immersion

microscope objective (NA 1.3) was used to collect the transmitted light as shown in Figure 4.6a

(iii). The back focal plane (BFP) of this microscope objective is imaged onto a 12 bit CCD-camera

[172], using a single achromatic lens (f=300mm). This technique helps us to angularly resolve the

transmitted light thus helping in observing specific diffraction orders. The experimental setup

used by us could be considered as an analog of a modified leakage radiation microscope [260,

261], which combines the advantages of back focal plane (k-space) microscopy [29] with

polarimetric analysis to study diffraction orders from a periodic array. We perform the

measurements in real space (Figure 4.6b) and fourier space (Figure 4.6c) to detect the propagating

surface modes and diffraction orders, respectively. To explicitly extract the optical effects (in our

case asymmetric transmission) from the quadrumer array, the focused beam must be significantly

smaller than the quadrumer array to parasitic effects from the edges. The position of diffraction

orders in the BFP image can be mathematically described as,

𝑘𝑥,𝑦 = sin 𝜃 +𝜆×√𝑚𝑥

2+𝑚𝑦2

𝑛×𝛬, since 𝜃 = 0° (for normal incidence), (4.9)


77

leads to 𝑘𝑥,𝑦 =𝜆×√𝑚𝑥

2+𝑚𝑦2

𝑛×𝛬.

Figure 4.6 An experimental sketch for the setup modified for recording diffraction orders

resolved imaging in transmission (LRM-Fourier space) and direct imaging (LRM-real space) near

Rayleigh anomaly. (a) The experimental setup used by us could be considered as an analog of a

modified leakage radiation microscope [260, 261], which combines the advantages of back focal

plane (k-space) microscopy [29] with polarimetric analysis to study diffraction orders from a

periodic array. The experimental scheme in essence is a modified form of setup shown in

Chapter 3. Slight modification are made to observe back focal plane image in transmission and

to record real space image in reflection (b). Near Rayleigh anomaly, the propagating surface

modes travel parallel to the quadrumer (in-plane), hence they can be observed by direct imaging

of quadrumer array. The asymmetric transmission effect can then be calculated as difference in

intensity of propagating surface waves generated by right and left circularly polarized incident

light. (c) The back focal plane image in transmission allows to angularly resolve the transmitted

light thus helping in observing specific diffraction orders. These diffraction orders are then

quantitively evaluated for asymmetric transmission [16].

In the recorded BFP images (Figure 4.6c), the central angular range (𝑘𝑥,𝑦 = 0) corresponds to

zeroth diffraction order, also evident from the Equation 4.9. While wavenumber for the first

diffraction order increases with wavelength. As evident, the collection angle of microscope

objective constraints the collection of first diffraction order above certain wavelength. For our

case 𝑘𝑚𝑎𝑥 = 1.3/1.51 = 0.861 which corresponds to maximum collection angle of 𝜃𝑚𝑎𝑥 =

60.25°. For the quadrumer array sample under study, this means that the first diffraction order is

collected till ~780nm, while Rayleigh anomaly happens around 905nm. This means in the spectral

region 781-900nm we are not able to observe differential surface lattice resonance manifested in

first diffraction order.

As mentioned earlier, this experimental scheme is an analog of leakage radiation microscopy in

Fourier domain, performed for normal incident weakly focused beam. Due to the limitation of


78

collection microscope objective, the asymmetric transmission effects in first diffraction order near

Rayleigh anomaly cannot be observed. Thus, while our technique to observe diffraction order

based asymmetric transmission has a certain drawback, it is still preferable for observing optical

properties at only normal incidence condition. This way, we can avoid patristic extrinsic chirality

emerging from the angular spectrum, when using focused light-based experimental setups. This

method also helps in observing the coupling of incident light beam to various diffraction orders

at normal incidence, which can be useful in the future to study higher orders differential surface

lattice resonances in rotationally symmetric achiral structures.

To observe qualitatively the asymmetric transmission at Rayleigh anomaly, we performed leakage

radiation microscopy in real space in reflection. Near Rayleigh anomaly as explained above, the

propagating surface modes travel parallel to the quadrumer (in-plane), hence they can be

observed by direct imaging of quadrumer array. This was done by placing an imaging lens in the

reflection arm and recording the real space image on a CCD camera. As shown in Figure 4.6b. The

asymmetric transmission effect can then be calculated as pixel-by-pixel difference in intensity

images of propagating surface waves generated by right and left circularly polarized light.

4.3 Results and discussion (asymmetric transmission of quadrumer array)

We experimentally evaluated the three cases of 45°, 22.5° and -22.5° in-plane rotation of the

meta-atom to understand the asymmetric transmission in zeroth and first diffraction orders. In

the following we discuss some salient aspects of the results and discuss them in light of

simulations. We will also elaborate on certain interesting outcomes, which can be interesting for

future projects.

Observing Rayleigh anomaly

As discussed in section 4.1.3, the presence of propagating surface modes near Rayleigh anomaly

results in sharp drop in total transmission, which is dependent on periodicity of meta-atoms and

their surrounding refractive index. It is pertinent to highlight that this effect is independent of

material composition of individual meta-atom. In top part of Figure 4.7, photo diode (PD) based

scan image of the transmitted optical intensity when scanning the quadrumer array through the

focal field distribution can be found. The sharp edges of the array confirms the confocal

arrangement of the focusing lens and collection objective with quadrumer array present in the

focal plane of focusing objective. Hence, the PD based measurement was used to precisely align

the examined quadrumer array in the focal spot. These measurements helped us in confirming

the presence of propagating surface modes near Rayleigh anomaly due to sudden drop in

transmission. This can be observed in bottom part of Figure 4.7. where both simulation and

experimental results are plotted. In the course of this thesis, we tried different combinations of

focusing lens as shown in bottom part of Figure 4.7. Some results for high NA focusing objective

(NA 0.9) are shown in Appendix C to this thesis. As expected, the use of higher NA focusing lens,

spectrally broadens the transmission drop near Rayleigh anomaly and can lead to patristic

extrinsic chirality emerging from the angular spectrum of focusing lens.


79

Figure 4.7 A proof of principle measurement carried out to find the optimized focusing and

collection of total transmitted light (testing performed on quadrumer array of size 100x100

microns). Top: Photo diode (PD) based scan image of the transmitted optical intensity when

scanning the quadrumer array through the focal field distribution. The sharp edges of the array

confirms the confocal arrangement of the focusing lens and collection objective with quadrumer

array present in the focal plane of focusing objective. Hence, the PD based measurement was

used to precisely align the examined quadrumer array in the focal spot. These measurements

helped us in confirming the presence of propagating surface modes near Rayleigh anomaly due

to sudden drop in transmission. Bottom: Simulated and experimental results for proof of

principle photodiode based total transmission measurements with varying NA. In both cases the

sample is illuminated with RCP light wave. the use of higher NA focusing lens, spectrally

broadens the transmission drop near Rayleigh anomaly. The sudden drop in transmission can be

observed in the experimental curve corresponding to low NA (0.02). Hence corresponding

focusing lens (f=60mm) was used later for studying asymmetric transmission in quadrumer

array.


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Therefore, for the evaluation of asymmetric transmission (results shown in following) we used

low NA focusing lens (f=60mm) corresponding to normal incidence condition. In following, we

discuss the cases of asymmetric transmission for different orientations of quadrumer array.

Vanishing asymmetric transmission for symmetric orientations.

First, we discuss the case of quadrumer array with each individual meta-atom in-plane rotated by

45° as shown in Figure 4.8a. As discussed in section 4.1.3, we estimate the transmission of light

into separate diffraction orders by performing FDTD simulation as required in our case.

Figure 4.8 Experimental results for symmetric orientation of quadrumer. (a) The in-plane 45°

orientation of individual quadrumer in the array.(b) The experimental and simulation results for

symmetric orientation of quadrumer are shown. We define three regions. The green shaded

region (630-780nm) where we could experimentally record both zeroth and first diffraction

orders. Here both simulation and experimental results are in good approximation depicting

absence of AT. In the pink shaded region (781-905nm) only zeroth order was recorded

experimentally due to microscope objective constraint discussed above. The experimental data

for zeroth order shown no AT both in simulation and experimental results. From the periodic

simulation of quadrumer (all orientations) and from Equation 4.7 we also observe that there are

no diffracted waves travelling parallel to quadrumer (in-plane) after Rayleigh anomaly. Thus, no

differential SLR are formed leading to absence of asymmetric transmission. (c) Propagating

surface modes of equal magnitude were observed by real space imaging of quadrumer array

near Rayleigh anomaly for opposite incident handedness. Hence, we observe vanishing

asymmetric transmission also in real space imaging. We can relate the phenomenon of vanishing

AT in first diffraction order to CDS observed in individual quadrumer. Since, as discussed in

section 4.1.3, the CDS of induvial quadrumer has its maxima along odd multiples of 22.5°, while

for multiples of 0° and 45°, CDS vanishes. Hence for symmetric orientation of quadrumer (45°)

the first diffraction orders align with vanishing CDS, leading to non-differential effects in SLR.

Resultantly, no asymmetric transmission is observed in first diffraction order. Since 0° and 45°

both are example of symmetric orientation of quadrumer, hence we can safely assume both of

them to behave optically in similar manner [16].


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The simulation results were then used to study asymmetric transmission in zeroth and first

diffraction order as shown in Figure 4.8b with solid black and green line, respectively. The

experimentally recorded asymmetric transmission data for zeroth and first diffraction order are

plotted with dashed black and green line, respectively. As discussed above, due to symmetric

nature of quadrumer we should not observe AT in zeroth order. Both experimental and

simulation results confirm the same conclusion.

As mentioned in previous section, due to limitation of collection objection angular range, first

diffraction order is only recorded in the range of 630-780nm (shown in Figure 4.8b as green

shaded area). In this range we observe vanishing asymmetric transmission in simulation and

experimental results as shown in Figure 4.8b. The region shaded in pink corresponds to the region

in which due to collection objective constraint first diffraction orders could not be collected.

Propagating surface modes of equal magnitude were observed by real space imaging of

quadrumer array near Rayleigh anomaly for opposite incident handedness. Hence, we observe

vanishing asymmetric transmission also in real space imaging as shown in Figure 4.8c. From the

periodic simulation of quadrumer (all orientations) and from Equation 4.7 we also observe that

there are no diffracted waves travelling parallel to quadrumer (in-plane) after Rayleigh anomaly.

Thus, no differential SLR are formed leading to absence of asymmetric transmission after Rayleigh

anomaly (region shaded white in Figure 4.8b).

We can relate the phenomenon of vanishing AT in first diffraction order to CDS observed in

individual quadrumer. Since, as discussed in section 4.1.3, the CDS of induvial quadrumer has its

maxima along odd multiples of 22.5°, while for multiples of 0° and 45°, CDS vanishes. Hence for

symmetric orientation of quadrumer (45°) the first diffraction orders align with vanishing CDS,

leading to non-differential effects in SLR. Resultantly, no asymmetric transmission is observed in

first diffraction order. Since 0° and 45° both are example of symmetric orientation of quadrumer,

hence we can safely assume both of them to behave optically in similar manner.

Asymmetric transmission for chiral orientation

Next, we consider the array of quadrumer with in-plane orientation of 22.5° and -22.5°(as shown

in Figure 4.9a&d). We perform analysis of simulation and experimental results, similar to the case

of symmetric orientation of quadrumer. As discussed in section 4.1.3, we estimate the

transmission of light into separate diffraction orders by performing FDTD simulation as required.

The simulation results were then used to study asymmetric transmission in zeroth and first

diffraction order as shown in Figure 4.9b&e with solid black and green line, respectively. The

experimentally recorded asymmetric transmission data for zeroth and first diffraction order are

plotted with dashed black and green line, respectively. Again, due to symmetric nature of

quadrumer we should not observe AT in zeroth order. Both experimental and simulation results

confirm the same conclusion.


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Figure 4.9 Experimental results for symmetric orientation of quadrumer (a,d). The in-plane 22.5°

& -22.5° orientation of individual quadrumer in the array. (b,e) The experimental and simulation

results for chiral orientation of quadrumer are shown. We define three regions. The green

shaded region (630-780nm) where we could experimentally record both zeroth and first

diffraction orders. Here both simulation and experimental results are in good approximation

depicting presence of AT in first diffraction order. In the pink shaded region (781-905nm) only

zeroth order was recorded experimentally due to microscope objective constraint discussed

above. The experimental data for zeroth order shown no AT both in simulation and

experimental results. From the periodic simulation of quadrumer (all orientations) and from

Equation 4.7 we also observe that there are no diffracted waves travelling parallel to quadrumer

(in-plane) after Rayleigh anomaly. Thus, no differential SLR are formed leading to absence of AT.

This was also experimentally confirmed (in the region shaded white, from 906-960nm). (c&e)

are differential images evaluating AT in similar fashion as described above for diffraction orders.

For the chiral orientation (22.5° and -22.5° orientation), we observe asymmetric propagating

modes with opposite signs as expected also from simulation results [16].


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The simulation results clearly show the presence of high AT, closer to wavelengths supporting

Rayleigh anomaly condition and therefore depicting the significance of SLRs in observing AT in 2D

metasurfaces. Towards shorter wavelengths, the value of AT decreases. For the case of symmetric

orientation (0°and 45°) discussed above we saw vanishing AT in first diffraction order. For the

present case of in-plane rotation of 22.5 and -22.5°, we observe AT in first diffraction order as

shown in Figure 4.9b&e. The simulation and experimental results for AT show good agreement in

the region shaded green where the first diffraction order was recorded. We observe AT with

opposite signs and equal magnitude for the case of 22.5 and -22,5°in plane orientation of

quadrumer. As mentioned, above the first diffraction order was not collected in the region shaded

pink. For propagating surface modes near Rayleigh anomaly, we perform real space measurement

for opposite incident light beam handedness to estimate AT. The images (Figure 4.9c&d) are

differential images evaluating AT in similar fashion as described above for diffraction orders. We

observe asymmetric propagating modes with opposite signs (Figure 4.9c&d) similar to

quantitative results shown in Figure 4.9b&e (green shaded region).

For the two cases (22.5 and -22,5°in plane rotation) discussed here, we can relate the

phenomenon of opposing sign, similar magnitude of AT in first diffraction order to CDS observed

in individual quadrumer. Since, as discussed in section 4.1.3, the CDS of induvial quadrumer has

its maxima along odd multiples of 22.5°, while for multiples of 0° and 45°, CDS vanishes. An in-

plane rotation 22.5° of individual quadrumer also symmetrically rotates the in-plane scattering.

This results in the first diffraction orders being aligned with maxima of CDS (positive unity, marked

with purple circle in bottom part of Figure4.3), leading to differential SLR. This leads to positive

asymmetric transmission first diffraction order. Conversely, for the case of in-plane rotation of -

22.5°, the first diffraction orders get aligned with minima of CDS (negative unity, marked with

orange circle in bottom part of Figure4.3). This leads to negative asymmetric transmission in first

diffraction order.

We can conclude that, once the meta-atom is rotated around its central point, we can expect the

CDS of individual meta-atom to also rotate by same azimuthal degree. Hence, by in-plane rotation

of the individual meta-atom, we can align the CDS maxima with the first diffraction order. As

evident, for our case of a fourfold symmetric quadrumer, this happens for an in-plane rotation of

each quadrumer by 22.5°/-22.5° leading to (positive/negative) asymmetric transmission in first

diffraction order, respectively.

The effect of asymmetric transmission in first diffraction order can be observed even in total

transmission. However, the magnitude of transmitted intensity in first diffraction orders is way

lesser than zeroth order. Therefore, our experimental setup is recommended for measuring

asymmetric transmission in higher diffraction orders at normal incidence condition.

Asymmetric transmission in higher diffraction orders

From the experimental and simulation results discussed above, we can imagine the coupling of

individual meta-atom in-plane scattered field to SLR of higher orders (diffraction orders above

first order). In this case, higher diffraction orders propagating in various azimuthal directions

(parallel to plane of quadrumer) could be chosen accordingly as shown in Figure 4.4a. By choosing


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the right periodicity, we can efficiently couple individual quadrumer scattered fields to grazing

higher diffraction orders. We can foresee, that even for an achiral orientation of a meta-atom, we

could still generate AT in certain orders propagating along the sample surface depending on CSD

of individual meta-atom.

For the case of symmetric orientation of induvial quadrumer (0° and 45°), we observe CDS maxima

along odd multiples of 22.5°(Figure 4.3), which correspond approximately to the <1,2> and <2,1>

diffraction orders (and their respective orthogonal pairs) azimuthal propagation direction (See

Figure 4.4a). By choosing right periodicity we can generate differential SLR of these orders leading

to AT even in symmetric orientation of quadrumer. The two diffraction orders <1,2> and <2,1>

work as enantiomeric pairs (equal magnitudes, opposing signs), hence in total transmission their

effect cancels out. With an experimental scheme as discussed above, we should be able to identify

AT in these higher diffraction orders. In this regard, we will briefly discuss future extensions of

present work in upcoming chapter.

Achieving asymmetric transmission from the resonant coupling of individual meta-atom CDS to

diffraction orders is a novel phenomenon and can define a new perspective for 2D metasurfaces

design.

5. Conclusion and Outlook

Novel self-assembled carbon flakes

In this thesis we have discussed light-matter interaction in context of polarized light for extracting

optical properties of examined structures. Initially, we discussed the theoretical background

regarding polarization analysis. This was later used to develop an experimental technique

(microscopic Müller matrix measurement) to understand the optical response of a 3D novel self-

assembled orthorhombic crystalline carbon phase intercalated with gold-silver nanoclusters

(carbon flakes) [14, 18]. The experimental findings and computational modelling of carbon flakes

could relate the optical birefringence in the structure to the crystalline arrangement of carbon

atoms in the orthorhombic lattice.

Figure 5.1 Observation of carbon flake modification by exposing them to higher energy electron

beams. (a) The SEM of the carbon flake under study (b-c) A variation in attenuation across

carbon flakes before and after high energy SEM imaging was observed by experimental

measurements (λ=632nm). The two optical measurements in b&c are in slightly different

orientation and position. (d) Controllable and selective modification scheme planned for future

experiments. (d) On left/right side we have bright field/dark field microscope images.


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The attenuation in carbon flakes was found to be a direct indication of bimetallic nanoclusters

present in the carbon flake. We also studied the dependence of optical and geometrical properties

of carbon flakes on fabrication parameters [15]. Due to their hybrid organometallic nature, these

self-assembled structures can find applications in the fields of sensing, electronics and plasmonics

[25, 26]. Our study provided ways that can be helpful in the future to fine-tune the optical and

geometrical properties of self-assembled structures for larger scale fabrication. The intriguing

nature of these self-assembled structures demands more investigations, which are planned in the

future. During experimental investigation, it was observed that carbon flakes are sensitive to

external stimuli and could also be modified by exposing them to electron, light, or ion beams. For

instance, a variation in attenuation across carbon flakes before and after high energy SEM imaging

(Figure 5.1a) was observed by experimental optical measurements (λ=632nm), as shown in Figure

5.1b and c. The two optical measurements in b&c are in slightly different orientation and position.

This modification can also be locally induced on flakes in a selective manner by controlling the

exposure of high energy on the flakes’ surface. As a proof of concept, a focused He-ion beam with

energies ranging up to 30keV was used to locally modify flakes, as shown in Figure 5.1d. In a next

step, a similar experimental examination of optical properties, as discussed in Chapter 3, will be

performed to understand the effects of such modifications. This selective and controllable

modification could be valuable in the future to selectively control the routing of light and for

sensing applications by applying said modifications. During the fabrication process, carbon flake

growth was also found to selectively favor regions on the substrates featuring epitaxially grown

graphene as compared to silicon carbide (SiC). This might pave the wave for selective growth of

flakes in desired regions of interest, acting as sensing pads for surface enhanced Raman

measurements [25, 26].

Furthermore, an initial study for understanding the conductivity of carbon flakes was performed

at the Institute of Applied Physics, University of Erlangen-Nuremberg. The study indicated a highly

insulating behavior also confirming a highly crystalline structure. A Kelvin probe force microscopic

measurement suggested a diode like behavior at the flake-electrode interface. The diode-like

behavior along with other factors could explain the insulation behavior found in the carbon flakes

[262].

Due to sensitive nature of carbon flakes and the large crystalline unit cell, the exact position of

carbon atoms within the unit cell is still not completely known. Currently, rigorous efforts are

being made to comprehend the underlying chemical mechanism, which helps unfold the

intriguing features of this structure.

Brewster effect based refractive index retrieval

We discussed and implemented the High NA focusing objective based refractive index retrieval

technique, which was tested on dielectric, absorbing and uniaxial crystalline structures besides

carbon flakes. The measurement system provided us with a preliminary estimate of complex

refractive indices of carbon flake. The results shown in Section 3.3.5 were mainly performed using

linearly polarized input beam. This technique could be useful in the future for extracting the local

refractive index information in smaller focal spots (in order of hundreds of nanometers) as

compared to existing commercial solutions (in order of tens of microns). One instance is the


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potential use for detecting local refractive index modification of crystalline surfaces by laser/high

energy processes (FIB, SEM or TEM).

Figure 5.2 Incident spiral beam of opposite handedness (a,b) after getting reflected (c, d) from a

surface with refractive index 0.36-0.37i (ITO refractive index near Epsilon near zero

phenomenon). In reflection arm we project the orthogonal linear polarization states after

passing through a quarter wave plate (e, f, g & h). The spatially varying polarization ellipse is also

plotted over respective angular spectrum in a, b, c & d where red/blue color depicts left-

handed/right-handed nature of the light beam, respectively.

We briefly discussed the use of polarization tailored light beams to extract the refractive index

(see Appendix B to this thesis). One of the manifestations of such beams (TM polarized-radially

polarized beam) is the observation of a null intensity ring in reflection at Brewster angle in Fourier

space, for non-absorbing medium (Figure E.2). Similarly, for absorbing medium, we can observe a

null intensity ring when using spiral beam (TM-TE polarized beam). The TE and TM components

of such beam acquire 90° phase shift in reflection at a specific incidence angle called principal

angle. Due to the rotationally symmetric nature of spiral beam, by using a quarter wave plate and

linear polarizer in reflection arm, we can observe a null intensity ring around principal angle as

shown in Figure 5.2. In the case shown, we use a surface with refractive index 0.36-0.37i

corresponding to ITO refractive index near Epsilon near zero phenomenon [263-265].

Diffraction assisted asymmetric transmission in metasurfaces

In Chapter 4 we considered the diffraction assisted asymmetric transmission in 2D metasurfaces.

This work was done in collaboration with colleagues from University of Ottawa, Canada and

Tecnologico de Monterrey, Mexico.

We discussed the in-plane scattering of individual meta-atom and how it could be effectively

coupled to the diffraction modes of a lattice to observe asymmetric transmission in the far-fields.

In our study we can conclude that, the in-plane scattering of an individual meta-atom can be chiral

(possess circular differential in-plane scattering ), regardless of the meta-atom being chiral. In our


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case of fourfold symmetric quadrumer, we observed that CDS has its maxima along odd multiples

of 22.5°, while for multiples of 0° and 45°, CDS vanishes. Hence, by in-plane rotation of the

individual meta-atom, we can align the CDS maxima with the first diffraction order, when the

quadrumer are arranged in an array (with specific periodicity). As evident, for our case of a

fourfold symmetric quadrumer, this happens for an in-plane rotation of each quadrumer by

22.5°/-22.5° leading to asymmetric transmission in first diffraction order. The zeroth order does

not exhibit any asymmetry.

Figure 5.3 Asymmetric transmission in <2,1> and <1,2> diffraction orders with symmetric

orientation of individual quadrumer. Top: an illustration of grazing diffraction orders <2,1> and

<1,2> generated at Rayleigh anomaly wavelength of 905nm, for periodicity of 1350nm. Bottom:

Asymmetric transmission only observed in <2,1> and <1,2> diffraction orders, while it vanishes

for other lower diffraction orders as expected.

We were able to confirm this through simulation and experiment, which were in good agreement.

Based on our results, we confirmed that for symmetric orientation of quadrumer (0° and 45° in-

plane rotation) there was no asymmetric transmission observed in first diffraction order. For the

chiral arrangement of individual quadrumer (22.5°/-22.5° in-plane rotation), asymmetric

transmission was observed in first diffraction order with opposing equal magnitudes. Later

towards the end of Chapter 4 we discussed the concept of asymmetric transmission in higher

order diffraction modes. For that, we need to align the maxima of the CDS along the next higher

order mode.


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Figure 5.4 Meta-atom with CDS maxima along multiples of 90° in symmetric orientation. Top:

the case of a square patch of gold with diagonal holes etched within it, with specific dimension

as shown. Bottom: FDTD simulation results for in-plane scattering for incident RCP and LCP light.

If this meta-atom is place in a periodic arrangement (600nm periodicity), we can expect

asymmetric transmission within first diffraction orders. This can lead to directional propagation

of light based on opposite handedness within first diffraction orders which can find application

in integrated nanophotonics in future.

For instance , this can happen for symmetric orientation of quadrumer where CDS has its maxima

along odd multiples of 22.5° and can be coupled to <2,1> and <1,2> diffraction orders

simultaneously, leading to asymmetric transmission in these respective orders (as shown in Figure

5.3). The top figure shows an illustration of grazing diffraction orders <2,1> and <1,2> generated

at Rayleigh anomaly wavelength of 905nm, for periodicity of 1350nm. It is important to highlight

that both <2,1> /<1,2> are generated on same wavelength and correspondingly gets coupled

majorly to in-plane scattering for RCP/LCP incident light beam, respectively (see Figure 4.3). This


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leads to AT with opposing equal magnitudes (see Figure 5.3 bottom part) in these diffraction

orders.

Therefore, unlike the case of AT in chiral orientation discussed above (and in chapter 4), for

symmetric orientation.in total transmission AT vanishes, although present in higher diffraction

orders. This argument/concept can lead in future to study some interesting phenomenon such

as AT in higher diffraction orders (<2,1> and <1,2>) for rotationally symmetric structures like

cylinders. Our initial assessment (FDTD simulations for cylinder with 40nm height and 200nm

diameter) shows promising results.

Similarly, we looked into the concept of designing meta-atom, that can generate CDS maxima

along multiples of 90° in symmetric orientation. Thus, leading to asymmetric transmission in first

diffraction orders when placed in an array (with specific periodicity). As an example, we discuss

the case of a square patch of gold with diagonal holes etched within it as shown in top part of

Figure 5.4. The lower part of Figure 5.4 shows FDTD simulation results for in-plane scattering for

incident RCP and LCP light. If this meta-atom is placed in a periodic arrangement (600nm

periodicity), we can expect asymmetric transmission within first diffraction orders. Whereby, the

first diffraction orders along vertical and horizontal direction get resonantly coupled to RCP and

LCP incident light, respectively. This can lead to directional propagation of light based on opposite

handedness within first diffraction orders which can find application in integrated nanophotonics

in future.

To conclude, the studies presented in this thesis, demonstrate the advantage and versatility of

studying of light-mater interaction exploiting polarization state of light. We used various

experimental systems to devise ways and methods to study intriguing optical phenomena at small

scales. Most of the studies were performed in collaboration with researchers across the globe

belonging from different fields of science. This helped us in understanding novel structures and

metasurfaces in a more comprehensive way, leading to ideas and concepts for future projects.

A. Some mathematical relations

Circularly polarized light and chirality

Light waves can carry angular momentum and exert torque on surfaces/media they are

interacting with in two distinct ways [266, 267]. The spin (𝜎) and orbital (𝑙) angular momentum,

which involve polarization and spatial wavefront/phase profile of light wave, respectively. We will

keep our discussion here to spin angular momentum (SAM) as it is related to polarization states

of light. Mathematically this can be written as

𝜏 =𝜎

2𝜋𝜈=

𝜎

𝜔, (A.1)

Hence, SAM carried per photon can be represented as 𝜎ћ. Right-handed circularly polarized

paraxial light beams therefore carry SAM in the direction parallel to propagation direction (𝜎 =

−1), while for left circularly polarized it is in the opposite direction (𝜎 = 1). Linear states of

polarization can be thought of as containing equal right and left circular contributions and hence

total SAM is equal zero. Circularly polarized light propagating along a certain direction spins with

time or in space about the optical axis in a helical fashion [66]. This intriguing phenomenon has

an analog in terms of optical properties of a medium, i.e., the chirality or optical activity [67].

Optical activity and reciprocity

The optical phenomenon of circular retardance together with circular dichroism are often named

as optical activity [268]. Circular dichroism results in differential extinction for incoming circularly

polarized light. While circular retardance rotates the azimuthal angle of the incoming polarization,

proportional to retardance while it does not affect the ellipticity of the light wave (incoming

circular polarization is preserved). This effect is somewhat similar to optical rotation effect

observed by the Faraday [223] in the presence of a static magnetic field but with one distinction,

i.e., reciprocity. The earliest work regarding reciprocity in optics was done by G. G. Stokes and

Helmholtz around 1850s. Lord Kelvin, Kirchhoff and Rayleigh also mentioned similar concepts of

reciprocity in fields the of thermoelectric, thermal radiation and acoustics, respectively [231].

Later H. Lorentz formulated the electromagnetic reciprocity theorem. Mathematically, by

considering two spatially separate volumes with certain current densities, we perform divergence

operation on Maxwell’s curl equations. Resultantly, we end up with Lorentz reciprocity theorem

with sources,

𝛁. (𝑬𝟏 × 𝑯𝟐 − 𝑬𝟐 × 𝑯𝟏) = 𝒋𝟏. 𝑬𝟐 − 𝒋𝟐. 𝑬𝟏. (A.2)

In terms of discussion regarding Faraday effect, it is important to remember that circular

dichroism and circular retardance are reciprocal quantities. On the other hand, Faraday effect is

non-reciprocal and depends on the direction of light propagation in relation to that of magnetic

field, hence producing vanishing chiral effects.


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Relation between Jones and Müller matrix

Since both Jones and Müller formalism deal with polarized light-matter interaction, extensive

research has been done to mathematically relate both matrices [84, 96]. Since, Jones formalism

does not deal with partially polarized light, therefore usually the first step involves removing

depolarization information from Müller matrix [95]. Later, a Müller matrix and its corresponding

Jones matrix can be related by the equation,

𝑀 = 𝐴(𝐽 ⊗ 𝐽∗)𝐴−1 also, 𝑚𝑖𝑗 =1

2𝑇𝑟(𝐽𝜎𝑖𝐽

†𝜎𝑗) ∈ 𝑖, 𝑗 = 0,1,2,3, (A.3)

𝐴 = [

1 0 0 11 0 0 −10 1 1 00 −𝑖 𝑖 0

], (A.4)

where ⊗ denotes the Kronecker product , 𝜎 is an individual Pauli matrix and A is a combination

of 3 Pauli matrices and 2x2 unitary matrix [56]. The origin of this elegant connection to Pauli

matrices lies in the concept of coherency matrices which is another way to describe polarized

light, introduced by Soleillit, Weiner and later reintroduced by E. Wolf [56, 87]. The 2x2 coherency

matrix is based on correlations between orthogonal field components of a light wave under

consideration. In fact, the field relations shown in Equation 2.33 define the elements of a

coherency matrix. The decomposition relation between coherency matrix and Stokes vector lies

in Pauli matrices, which was first Figured out by U. Fano in 1954 [269]. It is pertinent to highlight

that similar to 2x2 Pauli matrices, the linearly independent nine, 3x3 Hermitian matrices (also

known as Gell-Mann matrices [270]) can be used to define arbitrary polarized light wave in three

dimensions [80, 271, 272].

Brewster effect in uniaxial crystals

As discussed in section 2.3.1, materials which show birefringence has varying refractive indices

along their principal axis. In following we will discuss such cases and how anisotropy affects the

Brewster angle of the material.

Following from Equation 2.21a&b, we can expand constitutive relations to accommodate more

generalized solutions where the permittivity and permeability can be defined as 3 dimensional

tensors. Mathematically we can expect it to take the following form [273]

휀 = [

휀𝑥𝑥 0 00 휀𝑦𝑦 0

00 0 휀𝑧𝑧

], (A.5)

where, 휀𝑥𝑥 , 휀𝑦𝑦 & 휀𝑧𝑧 are the permittivity tensors along three principal axes of a medium [98,

273]. Intuitively, we can expect cases in which diagonal tensor elements all are identical

(isotropic), two are identical (uniaxial) or all three different (biaxial). Extending on Brewster effect

for a medium, we can discuss this effect in relation to uniaxial systems [75]. For a uniaxial crystal

in the dielectric case (no absorption in medium) two refractive indices are defined as ordinary 𝑛0

and extraordinary 𝑛𝑒. The refractive index of incident medium is denoted as 𝑛1 [75, 81]. Following

the existing mathematical notation [75], we can distinguish three cases:


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Case 1: Surface parallel to optic axis, with optic axis pointing normal to the plane of incidence. In

this case, the light wave sees only the ordinary refractive index and hence can be written in a

similar way as for an isotropic medium:

𝑏1 = tan𝜃𝐵1 =𝑛0

𝑛1. (A.6a)

Case 2: A Surface that is parallel to the optic axis, with optic axis being parallel to the plane of

incidence. In this case we expect a cross combination of ordinary and extraordinary refractive

indices and corresponding contributions to the effect at the Brewster angle

𝑏2 = sin2 𝜃𝐵2 =𝑛1

2𝑛𝑜2−𝑛𝑜

2𝑛𝑒2

𝑛14−𝑛𝑜

2𝑛𝑒2 . (A.6b)

These two cases are of more experimental importance and will be discussed later in the

experimental part.

Case 3: For a surface normal to the optic axis, we can define the Brewster angle as

𝑏3 = sin2 𝜃𝐵3 =𝑛1

2𝑛𝑒2−𝑛𝑜

2𝑛𝑒2

𝑛14−𝑛𝑜

2𝑛𝑒2 . (A.6c)

For example, taking the case of rutile (titanium dioxide in its natural crystalline form), the two

refractive indices are 𝑛0 = 1.6557 and 𝑛𝑒 = 1.4849 and we get 𝜃𝐵1 = 58.870°, 𝜃𝐵2 = 54.01°

and 𝜃𝐵3 = 60.7°.

Brewster effect in Biaxial Crystal

Following from Equation A.5, the case of a uniaxial crystal can be further extended to the biaxial

crystal case in similar manner. We can expect for a biaxial crystal a total of 12 Brewster angle

cases which, due to symmetry, are reduced to 6 unique Brewster angles [75, 121].

𝑏1 = sin2 𝜃𝛽′ =𝑛1

2𝑛𝑧2+𝑛𝑥

2𝑛𝑧2

𝑛12−𝑛𝑥

2𝑛𝑧2 , (A.7a)


2𝑛𝑧2+𝑛𝑦

2𝑛𝑧2

𝑛12−𝑛𝑦

2𝑛𝑧2 , (A.7b)


2𝑛𝑦2+𝑛𝑥

2𝑛𝑦2

𝑛12−𝑛𝑥

2𝑛𝑦2 , (A.7c)


2𝑛𝑦2+𝑛𝑧

2𝑛𝑦2

𝑛12−𝑛𝑧

2𝑛𝑦2 , (A.7d)


2𝑛𝑥2+𝑛𝑦

2𝑛𝑥2

𝑛12−𝑛𝑦

2𝑛𝑥2 , (A.7e)


2𝑛𝑥2+𝑛𝑧

2𝑛𝑥2

𝑛12−𝑛𝑧

2𝑛𝑥2 . (A.7f)

For instance, we compare the equations with experimental results for biaxial organic crystal MNA

(2-methyl-4-nitroaniline) [121].The three of the Brewster angles observed in the case of above-

mentioned experimental study, matches the estimates (𝑏2 = 65.1212°, 𝑏3 = 71.8603°, 𝑏4 =

51.7114°) observed by us, using analytical equations.

B. Additional data-ellipsometry

Use of spatially structure light beams

In Chapter 2 we discussed about Gaussian beam solutions and about various homogeneous

polarization states. Here we want to emphasize the case of spatially varying polarized light beams,

also known as vectorial optical fields. Since the inception of the concept, this has garnered a lot

of attention in various applications [274]. One particular example of such light fields are cylindrical

vector beams, where we expect an axially symmetric beam solution. We achieve this by treating

the full vector wave equation (with certain approximations [275]) gives us two solutions of

azimuthally (TE) and radially (TM) spatially polarized light beams which maintains symmetry along

propagation direction in terms of intensity and polarization [274-276]. Pertaining to the continuity

of transverse field components they exhibit singularity at the center of the beam. The symmetric

properties of these CV beams find direct applications for focusing systems, where they maintain

cylindrical symmetry in focus. It is also more convenient to describe radially and azimuthally

polarized beams as superposition of respective Hermite-Gaussian modes of orthogonal

polarization which mathematically are shown as [275],

�� 𝑟 = 𝐻𝐺10��𝑥 + 𝐻𝐺01��𝑦, (B.1a)

�� ∅ = 𝐻𝐺01��𝑥 + 𝐻𝐺10��𝑦. (B.1b)

Consequently, we can describe a generalized equation in terms of radial and azimuthal unit

vectors for general CV beam as,

�� (𝑟, 𝜑) = 𝑃[cos𝜑0 ��𝑟 + sin∅0 ��𝜑], (B.2)

where 𝜑0 is the angle by which a polarization state is rotated from radial direction. There are

multiple ways to generate these beams, while most commonly used methods involve using a

combination of spatially homogeneous polarizations (e.g. linear polarized light) together with

spatially varying polarization response of an electro-optical device (liquid crystals, spatial light

modulators)[12, 13, 171, 275]. In case of liquid crystal devices (variable spiral plate), this is

achieved by varying optic axis orientation of liquid crystals [277, 278]. In Cartesian coordinate

system (𝑥𝑦 plane), if we define the angle of optic axis 𝛼 from the x-axis, then mathematically we

can write it as

𝛼(𝑟, 𝜑) = 𝑞𝜑 + 𝛼0, (B.3)

where 𝜑 = 0 − 360, 𝛼0 defines the initial constant angle for optic axis orientation. 𝑞 is defined

as the topological charge which defines how quickly the position of optic axis changes with respect

to the azimuthal angle 𝜑, as shown in Figure B.1. A combination of linearly polarized light with

the right local optic axis orientation and retardance applied to variable spiral plate, we can achieve

different spatially structured beams.


95

Figure B.1 A schematic for converting linearly polarized beams into different cylindrical vector

beams (CV). In essence by locally varying the optic axis orientation of liquid crystals, any CV

beam can be generated. In our case the q=0.5 spiral plate is used, which together with λ/2

retardance tuning of LC and incident linear polarizations (H, V, D) can generate radial, azimuthal,

and spiral beams.

For proof of concept and for future work we evaluated the reflected light for the case of radial

and azimuthal polarized incident beams. In our case, when a horizontal/vertical polarized

Gaussian beam passes through a variable spiral plate with a topological charge of 0.5 (ARCoptix;

VSP), and tuning to retardance of λ/2, it generates radially azimuthally polarized light beam,

respectively (shown in Figure D.1) [279]. Similarly, a spiral beam can be generated by using D-

polarized light as input beam. For higher mode quality, we perform Fourier filtering of the

resultant light beam using a combination of two convex lenses with appropriately sized pinhole

placed in the common focal plane of two lenses. As mentioned before [184], we use a set of two

beam splitters (Thorlabs; PBS101) in orthogonal orientation (before focusing of light beam) to

avoid any polarization-dependent phase and amplitude variations.


96

Figure B.2 Back focal (Fourier) plane image of microscope objective imaged by 12bit CCD camera

for various incident polarized beams. Specially interesting are the case of spatially structured

radial (TM) and azimuthal (TE)-polarized light beams. Due to rotational symmetry we see

vanishing intensity ring for radial (TM)-polarized light at Brewster angle as evident from Fresnel

TM reflection coefficients [113].

This arrangement allows us to collect the light beam in reflection. Figure D.2 shows the back focal

plane CCD image of the reflected light from a glass substrate (N-BK7) using different incident

polarization state. Specially interesting are the case of spatially structured radial (TM) and

azimuthal (TE)-polarized light beams. Due to rotational symmetry we see vanishing intensity ring

for radial (TM)-polarized light at Brewster angle as evident from Fresnel TM reflection coefficients

[113]. The ellipsometric parameters extraction discussed in Chapter 3 will be extended to use

spatially structured light beams for probing samples. In the case of TM/TE polarized incident light

beam, radial profile along any azimuthal 𝜑 direction could be used for ellipsometric fitting models

defined in chapter 3, This can greatly increase the applicability and effectiveness of experimental

system. Another interesting case is related to spiral beam (TE:TM;1:1) and principal angle of a

medium. This is discussed in Chapter 5 of this thesis.


97

Constraints of the fitting model: theoretical and experimental assumptions

Expanding from our discussion in section 3.3.6, here we discuss the constraint of the fitting models

for complex refractive index retrieval. Firstly, we study the dependency of reflected polarized light

on fitting model. Here, we computationally evaluated the accuracy of the fitting model for various

combinations of TE/TM polarized light. We studied the cases for purely TM (1:0), TE (0:1) and for

equal contribution of both (1:1), as shown in Figure E.1. Since the purpose is to understand the

extent of fitting model accuracy, therefore we used the normalized reflectance as expected for

the cases of TE and TM polarized light.

Figure B.3 Analytical calculation for the retrieval of refractive index, indicating the error in

refractive index retrieval using approximate and full equations for both real and imaginary part.

Maximum incidence angle of focusing system is constant (θmax=60.25°).


98

Using Fresnel reflection coefficients, we theoretically calculated reflectance over a fixed range of

incidence angle (0°-60.25°) for varying input complex refractive indices. Then using fitting model

(based on full and approximate Equations 2.49a&b and 2.53a&b ), we computed back complex

refractive index. In Figure B.3, we plot the error (|𝑛𝑖𝑛𝑝𝑢𝑡 − 𝑛𝑟𝑒𝑡𝑟𝑖𝑒𝑣𝑒𝑑|)in retrieval of refractive

index (right side real part, left side imaginary part of refractive index in Figure B.3). Generally, the

implemented ellipsometric scheme works better for TM reflected light cases. Towards

combinations of higher real and complex part of input refractive index, we observe increasing

error. An example for such refractive index combination can be aluminum (1.44-7.5i @ 632nm),

for which ellipsometric estimates would have greater error. For approximate equations model,

we also observe small errors in retrieval for input combinations of real and imaginary part of

refractive index lesser than unity.

NA of system

Secondly, we study the dependency of fitting model on the range of incidence angle. In

experimental scheme, numerical aperture (NA) of the focusing system defines the maximum

incidence angle of incoming light beam interacting with the sample in focal plane.

Figure B.4 Analytical calculation for retrieval of refractive index. The case for varying maximum

incidence angle of focusing system is presented. We plot the absolute of deviation in refractive

index (real part) retrieval from full equations model.


99

Figure B.5 Analytical calculation for retrieval of refractive index. The case for varying maximum

incidence angle of focusing system is presented. We plot the absolute of deviation in refractive

index (imaginary part) retrieval from full equations model.

Similar to previous Figure, we use Fresnel reflection coefficients, to theoretically calculate

reflectance over a varying ranges of incidence angle for input complex refractive indices as shown

in Figure B.4 and B.5. Then using fitting model (based on Equations 2.49a&b), we computed back

complex refractive index. In Figure B.4 and B.3, we plot the error (|𝑛𝑖𝑛𝑝𝑢𝑡 − 𝑛𝑟𝑒𝑡𝑟𝑖𝑒𝑣𝑒𝑑|)in real and

imaginary part of retrieved refractive index, respectively. As intuitively imagined, increasing range

of incidence angle provides better accuracy. However, a good trade off can be found for the case

of 𝜃𝑚𝑎𝑥 = 50° which corresponds to NA~0.7.

C Additional data-quadrumer array

In the course of this thesis, we tried different combinations of focusing lens to study quadrumer

array samples. Although, for the evaluation of asymmetric transmission we used low NA focusing

lens as mentioned in section 3.4., here we will discuss the case of high NA focusing objective (NA

0.9). This would essentially correspond to a focal spot of approximately 1.5microns near Rayleigh

anomaly for first diffraction order (𝜆𝑅𝐴 = 905𝑛𝑚. We use the experimental setup as described in

section 4.2, except the interchanging of high NA focusing objective (shown in Figure C.1).

Figure C.1 Probing quadrumer array sample near Rayleigh anomaly for first diffraction order

(𝜆𝑅𝐴 = 905𝑛𝑚) with high NA (0.9) focusing objective. We use the experimental setup as

described in section 4.2, except the interchanging of high NA focusing objective (shown in Figure

F.1). We plot the BFP images recorded near Rayleigh anomaly for two chiral orientations of

quadrumer (22.5°& -22.5°). These BFP images are also superimposed with simulated in-plane

scattering of individual quadrumer for certain incident handedness of light beam (RCP for -22.5

and LCP for 22.5°). we concentrate on the first diffraction orders in the BFP which are present in

the region 1.3<NA>0.9, hence we omit the central region of the BFP image (NA<0.9).

C Additional data-quadrumer array

101

The quadrumer array was precisely positioned in the focal spot with help of 3-axis piezo stage. An

oil-immersion microscope objective (NA 1.3) was then used to collect the transmitted light as

shown in Figure C.1. The back focal plane (BFP) of this microscope objective is imaged onto a 12

bit CCD-camera [172], using a single achromatic lens (f=300mm) which helps us to angularly

resolve the transmitted light. In the right side of Figure C.1 we plot the BFP images recorded near

Rayleigh anomaly for two chiral orientations of quadrumer (22.5°& -22.5°). These BFP images are

also superimposed with simulated in-plane scattering of individual quadrumer for certain incident

handedness of light beam (RCP for -22.5 and LCP for 22.5°). We concentrate on the first diffraction

orders in the BFP which are present in the region 1.3<NA>0.9, hence we omit the central region

of the BFP image (NA<0.9). As we can observe, within each first diffraction order in BFP image, we

observe regions of high and low intensity. The regions of high intensity coincide well with the in-

plane scattering of individual quadrumer of certain handedness. The difference in high and low

intensity regions could also correspond to asymmetric transmission. However, this technique (in

present state) provided more of a qualitative analysis, rather a quantitative one, hence was not

used for further analysis.

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List of publications

This thesis is based on following works.

• Investigating the Optical Properties of a Laser Induced 3D Self-Assembled Carbon-Metal

Hybrid Structure,

Butt, M. A., Lesina, A. C., Neugebauer, M., Bauer, T., Ramunno, L., Vaccari, A., Berini, P.,

Petrov, Y., Danilov, D., Manshina, A., Banzer, P., Leuchs, G., Small 2019, 15, 1900512.

https://doi.org/10.1002/smll.201900512

• Hybrid Orthorhombic Carbon Flakes Intercalated with Bimetallic Au-Ag Nanoclusters:

Influence of Synthesis Parameters on Optical Properties,

Butt, M.A., Mamonova, D., Petrov, Y., Proklova, A., Kritchenkov, I., Manshina, A.; Banzer,

P., Leuchs, G., Nanomaterials 2020, 10, 7, https://doi.org/10.3390/nano10071376

• Diffraction assisted chiral scattering for 2D metasurfaces (in progress), Chaitanya, N. A., Butt, M. A., Reshef, O., Boyd, R. W., Banzer, P., Leon, I. D., 2021 (in preparation)

Other works, not part of this thesis

• Vectorial vortex generation and phase singularities upon Brewster reflection,

Barczyk, R., Nechayev, S., Butt, M. A., Leuchs, G., Banzer, P., Physical Review A 2019, 99,

6, https://doi.org/10.1103/PhysRevA.99.063820

• On the Application of Twisted Photonics Crystal Fiber for Quantum Communication : A polarimetric study Butt, M. A., Roth, P., Wong, G. K. L. , Russell, P. St.J., Banzer, P., Leuchs, G., 2021 (in preparation)- as part of QuNet EU project

Conference publications and poster presentations

• Investigating the Optical Properties of a Novel 3D Self-Assembled Metamaterial made of

Carbon Intercalated with Bimetal Nanoparticles,

Butt, M. A. et al., in Advanced Photonics 2018, NoTu4J.5,

https://doi.org/10.1364/NOMA.2018.NoTu4J.5

• 2D carbon allotrope with incorporated Au-Ag nanoclusters – Laser-induced synthesis and

optical characterization,

in CLEO Pacific Rim Conference 2018, Th1G.2.

https://doi.org/10.1364/CLEOPR.2018.Th1G.2

• Optical Properties of a Novel Metamaterial made of Carbon Intercalated with Au/Ag

Nanoparticles

Butt, M. A. et al., in Hole burning, single molecule and related spectroscopies 2018,

http://hbsm2018.ru/publications/

https://doi.org/10.1002/smll.201900512

https://doi.org/10.3390/nano10071376

https://doi.org/10.1103/PhysRevA.99.063820

https://doi.org/10.1364/NOMA.2018.NoTu4J.5

https://doi.org/10.1364/CLEOPR.2018.Th1G.2

http://hbsm2018.ru/publications/

List of publications

116

• Novel hybrid carbon flakes – optical investigation using microscopic Müller matrix

measurements,

Butt, M. A. et al., in Mendeleev 2019, http://mendeleev.spbu.ru/

• Microscopic Müller Matrix Analysis (Best poster award)

Butt, M. A. et al., International Conference on Advanced Optical Technologies, Erlangen,

2019, https://www.saot.fau.de/events_saot/archive/international-conference/

• Tuning the Optical and Geometrical Properties of Hybrid Carbon Flakes by Fabrication

Parameters,

Butt, M. A. et al., in OSA Advanced Photonics Congress 2020, JTu4C.14,

https://doi.org/10.1364/NOMA.2020.JTu4C.14

• Presented poster at Synthetic carbon Allotrope Symposium (2019) and Photonics Online

Meetup (2020).

http://mendeleev.spbu.ru/

https://www.saot.fau.de/events_saot/archive/international-conference/

https://doi.org/10.1364/NOMA.2020.JTu4C.14

Acknowledgements

People define success in life in terms of money, fame, power, or other material pleasures. For me

it has rather been about family, friends, nature, colleagues, and peace of mind-body. With every

passing day of my life, I realize how fortunate I have been in this regard. I bow my head humbly

to the Almighty for all these blessings.

I am thankful to my family, who kept believing in me and supported me through the difficult times.

Studying away from home meant that my parents never got the opportunity to attend my

graduation ceremonies (unfortunately this time again). I am thankful to Mr. & Mrs. Pervaiz Akhter

(uncle/aunt), who treated me as family during my bachelor studies and never made me miss

home. As an expat doctoral student in Germany, I have missed out on many important moments

with family. Especially the demise of my grandmother, who was always close to my heart. May

God bless her soul. I am thankful to my friends here in Germany and back in Pakistan, for their

unconditional support. Special shoutout to Maximillian (Maxi), Dereje (DJ), Julia, Sumair and

Shakeel (Balti) who helped me always irrespective of time or place. I was fortunate enough to

meet and make friends with a lot of international students, all of whom I cannot name here due

to space constraint but are all close to my heart. Ramlah!, this journey would have never

completed in time, without your untiring, non-stop support.

I am thankful to all colleagues, who helped me at one or another stage in this doctoral journey.

Doktorvater (Doctor Father) – the deutsch word for doctoral supervisor, amused me when I first

heard it. But over the years I have realized what it really meant. I am thankful to Gerd and Peter

for being there for me. Listening to my crazy ideas and never discouraging me. I hope that I can

build onto the wisdom and the technical knowledge that I gained from them. I am thankful to all

the research collaborators across the globe, from whom I learned a lot. A special thanks to all the

colleagues at MPL who made the research work more exciting and fun. I would thank Professor

Alina Manshina and Dr. Daria Mamonova for the interesting scientific conversations and making

the carbon flakes available to us for the purpose of my doctoral studies. A special shout out to

Mr. Mahad Hameed who helped me through procedural stuff regarding my doctoral studies back

in Pakistan.

The struggle of life continues till you die. New challenges, difficulties, and unknown

circumstances, I believe, are the beauty of life. I vow, never to stop learning and working to

fulfilling my dream of bringing scientific collaboration to Pakistan.

To all my instructors, teachers, and professors, I owe this doctorate to you all.

Documents

On the interaction of light with novel artificial