Optical Studies of Nano-Structures in the Beetle Cetonia Aurata

Linköping Studies in Science and Technology

Rizwana Shamim LITH-IFM-A-EX--09/2060--SE

Department of Physics, Chemistry, and Biology Linköping University, SE-581 83 Linköping, Sweden Linköping 2009

Master’s Thesis.

Linköping studies in science and technology. LITH-IFM-A-EX--09/2060--SE

Optical Studies of Nano-Structures in the Beetle Cetonia Aurata Rizwana Shamim

[email protected]

Supervisor: Hans Arwin IFM, Linkoping University

Examiner: Kenneth Jarrendahl IFM, Linkoping University

Linkoping, 20 February, 2009

Avdelning, institution Datum Division, Department Date: 20 Feb, 2009

Chemistry Department of Physics, Chemistry and Biology

Linköping University

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Svenska/Swedish Licentiatavhandling ISRN: LITH-IFM-A-EX--09/2060--SE Engelska/English Examensarbete ______C-uppsats D-uppsats Serietitel och serienummer ISSN Övrig rapport Title of series, numbering ______

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Titel

Optical Studies of Nano-Structures in the Beetle Cetonia Aurata

Författare Author

Rizwana Shamim

Sammanfattning Abstract The main objective of this thesis is to study the polarization effects of the beetle Cetonia aurata using Mueller-matrix ellipsometry. The outer shell of the beetle consists of complex microstructures which control the polarization of the reflected light. It has metallic appearance which originates from helicoidal structures. When these microstructures are exposed to polarized or unpolarized light, only left-handed circularly polarized light is reflected. Moreover, the exo-skeleton of the beetle absorbs right-handed polarized light. Multichannel Mueller-matrix ellipsometer or dual rotating compensator ellipsometer, called RC2, from J.A.Woollam is used to measure the polarization caused by different parts of beetle’s body. The 16 Mueller matrix elements are measured in the spectral range 400-800 nm at multiple angles of incidence in the range 40 0-70 0. An Optical model is developed to help us understand the nature and type of microstructure which only reflects the green colour circularly polarized light. With the help of multi- parametric modeling, we were able to find optical properties and structural parameters. The parameters are: the number of layers, the numbers of sub-layers, their thicknesses, and the orientation with respect to optical axes. This optical model describes the nanostructures which provide the reflection properties similar to the nanostructure found in the beetle Cetonia aurata. The model is also useful for analysis of the optical response data of different materials with multilayer struct ures.

Nyckelord Keyword

Cetonia aurata, Mueller-matrix ellipsometry, Helicoidal structures, Left-handed circularly polarized light

Abstract

The main objective of this thesis is to study the polarization effects of the beetle Cetonia aurata using Mueller-matrix ellipsometry. The outer shell of the beetle consists of complex microstructures which control the polarization of the reflected light. It has metallic appearance which originates from helicoidal structures. When these microstructures are exposed to polarized or unpolarized light, only left-handed circularly polarized light is reflected. Moreover, the exoskeleton of the beetle absorbs right-handed polarized light. Multichannel Mueller-matrix ellipsometer or dual rotating compensator ellipsometer, called RC2, from J.A.Woollam is used to measure the polarization caused by different parts of beetle’s body. The 16 Mueller matrix elements are measured in the spectral range 400-800 nm at multiple angles of incidence in the range 400-700. An Optical model is developed to help us understand the nature and type of microstructure which only reflects the green colour circularly polarized light. With the help of multi-parametric modeling, we were able to find optical properties and structural parameters. The parameters are: the number of layers, the numbers of sub-layers, their thicknesses, and the orientation with respect to optical axes. This optical model describes the nanostructures which provide the reflection properties similar to the nanostructure found in the beetle Cetonia aurata. The model is also useful for analysis of the optical response data of different materials with multilayer structures.

iii

Abbreviations

CCD Charged Coupled device EMA Effective Media Approximation E-vector Electric field vector FA Fast Axis MSE Mean Square error p-direction Parallel direction

PCrr SC A Polarizer, Compensator (rotating) ellipsometer, Sample, Compensator (rotating), Analyzer RC2 Dual rotator compensator SA Slow Axis s-direction Perpendicular direction TA Transmission Axis

Acknowledgments

All praise and thanks are due to the Almighty Allah, the most gracious, and the most merciful. This master’s thesis has been done at the Laboratory of Applied Optics, Department of Physics, Chemistry and Biology, Linköping University. I am thankful to my family members, teachers, students, and friends. Here I would especially want to thank the persons related to this thesis work:

• My supervisor and advisor Prof. Hans Arwin, for giving me this opportunity, for his guidance, patience and ever helping attitude. He has always encouraged me when I felt that I am cought up in some problem and showed me the wayout, with a nice smile on his face. • My examiner Prof Kenneth Järrendahl, for reading my thesis and giving good sugesstions for its improvement. • M.Sc. Roger Magnusson, Ph.D. student at the labarotory of Applied optics for helping me during the experiments and teaching me the efficient way to use the lab equipment and the computer software. • My husband Rashad.M.Ramzan, Ph.D. student at ISY Linköping University for continuous encouragement and helping me in formating this thesis. My two sons Talha and Irtiza for their ultimate cooperation during course of studies. • My mother for her unlimited sacrifices, I do not have the proper words to describe. Rizwana Shamim Linköping.

vii viii Chapter 1: Introduction

Contents

Abstract iii

Abbreviations v

Acknowledgments vii

Contents viii

Chapter 1 Introduction 1 1.1 Historic background...... 2 1.2 Scientific background ...... 3 1.3 Aim of this thesis ...... 3 Chapter 2 Theory of Polarized Light 5 2.1 The refractive index N ...... 5 2.2 Optics of anisotropic media...... 6 2.3 Polarization of light ...... 8 2.4 Linear birefringence...... 9 2.5 Optical activity...... 10 2.6 Interference colour phenomena ...... 11 2.7 Stokes parameters ...... 12 2.7.1 Examples of Stokes vectors...... 13 ix

2.8 The Jones vectors- A mathematical representation of polarized light ...... 14 2.8.1 Jones matrix of the optical system...... 16 2.8.2 Basis vectors ...... 16 2.8.3 Examples of Jones vectors...... 17 2.8.4 Mathematical representation of polarizers ...... 17 2.8.5 Jones matrix of the optical device that has been rotated ...... 20 2.9 Mueller matrix ...... 21 Chapter 3 Ellipsometry 23 3.1 Principle...... 23 3.2 Standard ellipsometry ...... 24 3.2.1 External reflection ellipsometry ...... 24 3.2.2 Internal reflection ellipsometry...... 26 3.2.3 Transmission ellipsometry...... 26 3.3 Generalized ellipsometry...... 27 3.4 Mueller-matrix ellipsometry...... 28 Chapter 4 Instrument 31 4.1 Polarization analysis ...... 31 4.2 Spectral analysis ...... 32 4.2.1 Dual rotating compensator ellipsometer...... 33 Chapter 5 Models for Cetonia aurata Optical Properties 37 5.1 The cuticle structure of the beetle...... 37 5.2 Optical model...... 39 5.2.1 The Cauchy model...... 39 5.2.2 The Liquid crystal model...... 40 Chapter 6 Results and Discussion 41 6.1 Spectroscopic ellipsometry...... 41 6.2 Data analysis...... 41 Chapter 7 Conclusions and Future Prospects 59 References 61 Appendix 65

Chapter 1

Introduction

There are many colourful creatures in our world and their fascinating colours are produced by nature. Scientists are trying to find the answers to the questions related to these colours especially those which are present in beetles and butterflies. Scientific giants like Newton, Michelson and Rayleigh have started working on these colours and their structure [1]. The optical study of the iridescent outer-shell of the beetle Cetonia Aurata has presented a microstructure which controls both the polarization and wavelength of the reflected light. The beetle’s surface exhibits left-handed circularly- polarized light, the origin of which is a helicoidal layer. Reflectivity spectra collected from the beetle are compared to the theoretical data produced using a multi-layer optical model for modeling chiral anisotropic media such as liquid crystals. There is an agreement between data obtained experimentally and from theory produced using a model that gives details about the upper isotropic layer of the beetle, the inner chiral structure, number of layers present in the endocuticle surface and the refractive indices. Cetonia Aurata beetles reflect left-handed circularly polarized light. This phenomenon was observed by Michelson in 1911. He observed the production of circularly polarized reflected light from unpolarized incident light from the scarab beetle Plusiotis resplendens [2]. This is different to the circularly polarized light reflected from a plane mirror, which undergoes a 1800 phase shift on reflection and changes its handedness. The reflected light from the beetle

1 2 Chapter 1: Introduction cuticle maintains its original handedness. This conservation of initial handedness is characteristic of the behavior of circularly polarized light interacting with a helicoidal structure which is composed of anisotropic media whose optic-axis rotates with the same handedness as that of the incident light [3].

Left Wing

Head Right Wing Scutellum

Figure 1.1: Photograph of Cetonia Aurata beetle

1.1 Historic background The Scarab species belong to two families of Scarabaeoidea: Rutelidea and Centoniidae. The family Rutelidea comprises about 200 genera and 4100 species distributed worldwide. Rutelidea are generally called leaf chafers, the most colourful species which often are known as jewel scarab beetles. Ruteline beetles have different shapes and colours. Some are metallic silver and gold. In 1911 Michelson was studying the metallic gloss of the scarab beetle Plusiotis resplendens and discovered the circular polarization of reflected light. Adult rutelines are phytophagous and feed on leaves or flowers of trees and shrubs. Adult leaf chafers emerge out with a soft and pale cuticle, but within hours, their bodies harden and the jewel scarabs show their true metallic-shiny colours [4]. 3

The family Centoniidae consists of more than 400 genera and 3200 species distributed worldwide. These beetles are commonly known as flower beetles (or fruit chafers or flower chafers). These are mostly colourful species of different sizes. Some of them are metallic green, red, blue or purple, with the majority of the iridescent species seen in the tropics. These are mostly pollen-feeders and are usually found on flowers. Adult flower-chafers prefer to eat nectar, sap and the juice and flesh of soft, ripe fruits. Their main predators are omnivorous birds and mammals [4].

1.2 Scientific background In 1911 Michelson discovered the circular polarization of reflected light from the metallic gloss of the scarab beetle Plusiotis resplendens from the family of Rutelidae. He explained metallic colouring in insects by grating-like structures and “screw structures”. Beetles have appealing bright colours and some of them have metallic appearance. Scientists have observed that some of the beetles in the zoological museum are more than 200 years old and have their colours and brightness same as they had when they were alive [12]. Their metallic look is from structures in organic materials which is both electrically and thermal insulating. The investigation of the spatial distribution and wavelength dependencies of the left-handed circular polarization of light reflected from the Scarab cuticle by using polarimetric techniques was first time done in 2005. In the red and green ranges of the spectrum while in the blue some of the ventral areas of Plusiotis resplendens are strongly left-circularly polarizing while the dorsal part is circularly unpolarized in the red and green ranges of the spectrum. It is only weakly left- or right-circularly polarized in the blue in some tiny spots [4].

1.3 Aim of this thesis In this thesis, we seek to determine the polarization effects in the beetle Cetonia aurata with Mueller-matrix ellipsometry. To describe the nanostructure which provides the reflection properties, an optical model is developed which is used to analyze the results obtained experimentally. Mueller-matrix spectroscopic ellipsometry is used to measure reflection properties of the green and shiny beetle Cetonia aurata. The reflected left-handed circularly polarized light is modeled with a chiral dielectric layer on a substrate.

Chapter 2

Theory of Polarized Light

2.1 The refractive index N When we are studying propagation of light through materials, it is convenient to discuss properties of materials in terms of the refractive index which is a complex quantity, written as: Nnik= + (2.1) where n is the usual refractive index and k is the extinction coefficient. The refractive index N is very useful as it enters directly into the expression describing propagation of a plane wave. Its real part n gives the phase velocity and the imaginary part k gives the attenuation of the electric field. From the dispersion relation of a plane wave, it is found that the dielectric function ε is related to N through ε = N 2 (2.2) The dielectric function ε is a complex number which can be written ε =+(nik)2 (2.3)

ε = εε12+ i ( 2.4)

5 6 Chapter 2: Theory of Polarized Light for isotropic materials. The relation between real and imaginary parts of N and ε are then

22 εε12 =− nkand =2 nk ( 2.5)

1 22 n =++()ε ()εε ( 2.6) 2 ( 121) ε k = 2 22 ( 2.7) 2 ε ++εε ()()121 ()

ε can be a tensor and have different values in different directions in space. For anisotropic materials, when the light interacts with materials, its optical response varies in different directions. The dielectric functionε is described as a tensor in anisotropic materials. One of the constitutive relations for anisotropic materials can be written as:

DEixyzij==εε0ij∑ ,, ( 2.8) jxyz= ,, In tensor form equation (2.8) can be written as

DExxεεεxx xy xz    DE= εε ε ε yy0yxyyyz  ( 2.9)   DEzzεεεzx zy zz 

2.2 Optics of anisotropic media A plane wave of polarized light propagates through an anisotropic medium along the z-axis of xyz Cartesian coordinate system. The properties of the wave can be uniform over any transverse plane perpendicular to the direction of propagation of the beam. The properties may be different in the beam direction [5]. We have a crystal which is oriented in a specific direction and light is falling on it. It refracts the light in one way and when oriented in the other direction, it refracts light in another way. The phenomena that a light beam falling normally on the surface of such crystal propagates straightforwardly inside the crystal only along a definite direction and for other directions, the optical beam splits into two beams. Such phenomena are called optical birefringence and this dependence of optical properties on crystalline direction is called anisotropy. 7

When a white light beam passes through this crystal and the crystal is placed perpendicular to the axis of the incident beam, two light spots of equal brightness are formed on the screen at the same point as shown in Figure 2.1.

Crystal

Figure 2.1: Light is traveling along optic axis of the crystal and no optical birefringence is seen. When rotating the crystal about the axis of the incident beam, one light spot on the screen will be at the same place while the other moves to the new position as shown in Figure 2.2.

Extra Ordinary optical wave

Crystal Ordinary optical wave

Figure 2.2: Light is falling normally on the face of the crystal and optical birefringence is seen. Figure 2.2 shows anisotropic refraction of light where the ordinary wave is perpendicular to the principal plane and extraordinary wave is in the direction of the principal plane. The phenomenon of optical birefringence can also be applied to produce polarized light [5].

8 Chapter 2: Theory of Polarized Light

Optical anisotropy is caused by the anisotropy of the medium. Anisotropic properties are only observed in crystal but not in gases and liquids (except liquid crystals), plastic materials or glasses [6 ]. Optical anisotropy is sometime seen in isotropic media because of mechanical stress or an external electric field (Kerr’s effect). Same phenomena can be seen in an alternating electric field and in the field of a powerful laser pulse [7].

2.3 Polarization of light For the study of reflection of light from surfaces, it is necessary to determine the orientation of the fields in the propagating electromagnetic waves which is its polarization state. The state of polarization of the beam is determined by dividing the beam into two components with plane of incidence as its reference. The p-direction is parallel to the plane of incidence and the s-direction is perpendicular to both the direction of propagation and the p-direction. The polarization is described by choosing the plane of propagation when there is no oblique incidence or reflection occurs. For p-component, the electric field lies in the plane and for the s-component, the electric field is perpendicular to the

plane. For light with angle of incidenceθo , the p- and s- components are shown in Figure 5.

Eip Erp E Ers H H is ip rp His Hrs

θ0 θ0 θ0 θ0 N0 N0

N1 N1 θ θ yΛ 1 1 H H E tp E ts ts zΛ xΛ tp a) b) Figure 2.3: a) The p-polarized and b) the s-polarized parts of the incident reflected and transmitted fields when the reflection is from a single surface with refractive

index No for the ambient and N1 for the substrate. The angle of incidence θo is

equal to angle of reflection θr and angle of refraction is θt The xyz-coordinates of the electric field of the propagating light wave are related with its polarization coordinates as 9

EEix==− ipcosθ00 E rx E rp cosθ

EEiy== ipsinθ00 EE ry rp sinθ ( 2.10)

EEiz== is EE rz rs The electromagnetic plane wave E(r,t) can be written as

E = Eppsˆˆ+ Es ( 2.11) The Fresnel equations for reflection and transmission of p-polarized light are

Erp NN1001cosθ − cosθ rp == ( 2.12a) ENip10cosθ + N o cosθ 1 and

Etp 2cosN00θ t p == (2.13b) ENip10cosθ + N o cosθ 1 The Fresnel equations for reflection and transmission of s-polarized light are

ENrs 0011cosθ − N cosθ rs == ( 2.13a) ENis 0011cosθ + N cosθ and

ENts 2cos00θ ts == (2.13b) ENis 0011cosθ + N cosθ

The degree of polarization is the ratio between the irradiance of the polarized

part I pol and the total irradiance Itot is given as I P = pol (2.14) Itot

2.4 Linear birefringence In cubic crystals, atoms are arranged in such a manner that the optical properties are the same in all directions. The material is optically isotropic and can be described with a scalar ε. However, crystals belonging to a hexagonal, trigonal or tetragonal system have an asymmetric structure which yields an optical anisotropy. These materials are anisotropic with a dielectric function tensor ε.

In the absence of optical activity, ε will be a symmetric tensor with εij= ε ji . If

the coordinate system coincides with the principal axes of the crystal, the non- diagonal elements are zero and we obtain in an orthogonal xyz coordinate system.

10 Chapter 2: Theory of Polarized Light

2 ε xx00N 0 0   εε==0000N 2 yy  ( 2.15)  2  00ε zz 0 0N 

where the short notations εii= ε i (ixyz= ,,) have been used and Ni is the refractive index in direction i. Nx, Ny and Nz are the refractive indices in the x-, y- and z-directions [9]. If ε xy,ε and ε z all are different, the crystal is called biaxial. If two of them e.g. ε x andε y are equal, the crystal is called uniaxial. The z-direction in a uniaxial crystal is called the optic axis and the refractive index in this direction is named the extraordinary index Nnikee= + e. The refractive indices in the x- and y-directions are the ordinary index Nnikoo= + o . For a uniaxial crystal, we thus have the following tensor

2 No 00 2 ε = 00No ( 2.16) 2 00Ne Uniaxial crystals are characterized by the equality of the two principle axes which can be taken as nnn= = xyo ( 2.17) nnnzeo= ≠ Equation (2.17) is valid for a rotation about the optic axis of the crystal [9].

A measure of birefringence is ∆nn=−eo n and can be negative or positive depending on whether no or ne is larger.

2.5 Optical activity In certain materials, the direction of linearly polarized light undergoes a continuous rotation as the light propagates in the material. Any such material that causes the E-field of an incident linear plane wave to rotate is called optically active. If the rotation is clockwise, while looking into the beam (towards the source), the material is called dextrorotatory or d-rotatory. If the rotation is counter-clockwise, it is called levorotatory or l-rotatory. In an optically active material, the dielectric tensor is nonsymmetrical. An optically active material show circular birefringence by which is meant that circularly polarized light propagates at different speed depending on if it is right- 11 handed or left-handed polarized. In other words, the material possesses two indices of refraction, nr and nl , for right-handed or left-handed polarization respectively. Like linear birefringence, we can define circular birefringence as nnlr− . If the path length in the material is d, the angle of rotation of linearly polarized light will be

π d θ =−()nn l lr ( 2.17) where θ is called specific rotatory power. d

In a similar manner, circular dichroism is defined as kklr− , where kl and kr are the extinction coefficient of the medium for left-handed and right-handed circularly polarized light, respectively. When light propagates through a linear medium, its polarization properties are usually described either by Stokes vector [10] formalism or by the polarization matrix that was introduced by Wiener [11] and Wolf [7].

2.6 Interference colour phenomena Two basic structure models have been identified to explain many of the interference colour phenomena seen: • The observed colour phenomena can be caused by a multilayer structure with alternating high and low refraction indices. These structures do not change the orientation of polarized light. • The observed colour phenomena are explained by a twisted multilayer chitin rich structure. The chitin molecules are predominantly arranged helical (Bouligand) and behave similar to liquid crystals. These structures reflect circularly polarized light. These models can explain narrowband and broadband (metallic) reflection. The `twisted` Bouligand structures are used to explain why the golden beetles can have an optically active surface, which reflect and change the plane of polarized light. Most insects with metallic appearance exhibit both specular metallic reflection and diffusive reflection. For beetles this can be a multilayer structure combined with irregular surface textures that breaks the flatness of the reflector [12]. Circularly polarized light is rare in nature. The wavelength- and species- dependent circular polarization patterns are of a rather complex nature.

12 Chapter 2: Theory of Polarized Light

The direction of rotation of the electric field vector (E-vector) of circularly polarized reflected light depends on the sense of rotation of the helix of the molecules. In living organisms, the capability to produce a given helical molecule is restricted to one sense of rotation, which has been fixed at very early stage in evolution, thus, perhaps apart from some mutants; this sense is the same for all living organisms. Since the exoskeletons of all beetles reflecting circularly polarized light consists of the same substance, the sense of rotation of the E-vector of reflected light is the same, left-handed, for all of them [13]. In the literature, only sporadic photographs taken from a few scarab species (e.g. Cetonia aurata, Plusiotis woodi) through left- and right-circular polarizer are available [4].

2.7 Stokes parameters In 1852, George Gabriel Stokes discovered that the polarization state of electromagnetic radiation (including visible light) could be represented in terms of observables and his work made foundation of the modern representation of polarized light. He defined four quantities which are functions of observables of the electromagnetic wave and are known as Stokes parameters. These values are description of incoherent and partially polarized light in terms of irradiance I, degree of polarization P and shape parameters of the polarization ellipse. Monochromatic and quasi-monochromatic light may be represented by a 4D column vector with real-valued elements with dimension of irradiance.

S0  S  S  1   0  S = =   ( 2.18) S 2   s    S3 

S1  s = S  and 3D column vector  2  S3  For unpolarized light

SSS123= ==0 ( 2.19)

For a completely polarized beam of light 2222 SSSS0123= ++ For a partially polarized light 13

22 2 2 SSSS01〉++ 2 3

The physical meanings of the Stokes vector elements are:

S0 represents the irradiance of the light wave, S1 represents the difference between the intensities of the x-and y- components i.e. the preference to either x-

or y-polarization, S2 represents the difference between the intensities of the light

waves in the +45 and − 45 directions of linear polarization and S3 represents the difference between the intensities of the right-circular state and left-circular state of polarization. A beam of light can be either neutral or partial or totally polarized. All these states are described by the Stokes vectors [10]. The monochromatic and quasi-monochromatic light may be represented by a 4D column vector which is not a real vector but a Stokes column matrix with real-valued elements having dimension of irradiance. The degree of polarization P is given by I SSS222++ s P ==pol 123 = ( 2.20) Itot SS00 where 0 ≤ P ≤ 1 Thus P = 0 indicates that the light is unpolarized, P = 1 that the light is completely polarized and 01〈〈P that the light is partially polarized.

2.7.1 Examples of Stokes vectors

Unpolarized light with irradiance I 0 has the Stokes vector

1   0 S = I 0 ( 2.21) 0   0

For normalized Stokes parameters I 0 = 1 The total polarization of elliptically polarized light in the general case will be

22 EE00xy+  EE22− S = 00xy ( 2.22) 2cosEE δ 00xy yx 2sinEE00xyδ yx

14 Chapter 2: Theory of Polarized Light

where the amplitudes E0x and E0 y and the phase difference δ yx = δ y − δ x are

2 constants. For linear polarized light with E = 0 and is normalized with E =1 0 y 0 x in the x-direction

1 1 S =   ( 2.23) 0   0

S is often written horizontally as transpose to save space

S = [110 0]t ( 2.24)

The examples of Stokes vectors are given in Table 3.1

Stokes vectors Polarization t unpolarized [1000] t Linear in the x-direction [110 0] [1100− ]t Linear in the y-direction [1010]t Linear in the+45º -direction [10− 10]t Linear in the-45º -direction t Right-handed circular [1001] t Left-handed circular [100− 1]

Table3.1: Examples of normalized Stokes vectors with their polarization where t indicates transpose

2.8 The Jones vectors- A mathematical representation of polarized light In 1941, another representation of polarized light was proposed by the American physicist R. Clark Jones. This technique was being applicable to coherent waves which are polarized. Jones vectors are vectors in an abstract mathematical space and have complex-valued elements. 15

Two waves with identical wavelength and irradiance, but different directions of their electric fields, can behave quite differently. Polarization of wave can accommodate the dependence on the direction of the electric fields. We describe the plane wave by its propagation direction and space and time dependencies of E(r,t) . If z-axis is the direction of propagation, then E(r,t) can be written as

Ezt(,)=ℜ Exˆˆ + Eyeiqz()−ω t  ( xy)  ( 2.25) or in complex form

E = Ex xˆ + E y yˆ ( 2.26)

iδ x iδ y where the complex-valued field amplitudes Exx= Ee and EEyey = represents the propagations of the fields or the complex-valued field amplitudes along the x-axis and the y-axis and are representing sinusoidal linear oscillations along two perpendicular directions [9]. The mathematical description of light propagation was presented by Clark Jones in early 1940s by introducing matrix formalism. Jones vector is written in the column vector as:

Et() E x =   Ety () ( 2.27)

where Ex ()t and E y (t) are the instantaneous scalar components of E By including z dependence we obtain

 iqz()−+ωδ t x  Eex Ezt(), =   ( 2.28)  Eeiqz()−+ωδ t y   y  which can be written as:

−+Eqztxxcos( ω δ ) Ezt(), =  ( 2.29) Eqztcos −+ω δ yy()

16 Chapter 2: Theory of Polarized Light

2.8.1 Jones matrix of the optical system When a monochromatic plane wave is passing through a non-depolarizing and frequency conserving optical system S, it will come out as a modified wave. The

relationship between incoming Ei and outgoing Eo waves is given by

EjEjEox′ = 11 ix+ 12 iy ( 2.30) EjEjEoy′ =+21 ix 22 iy

this can be written as

EEox′ jj11 12  ix =  ( 2.31) EE oy′ jj21 22  iy

where Eix and Eiy represent the electric fields in the xyz-coordinate system at the

input and Eox′ and Eoy′ represent the electric fields in the x´y´z´-coordinate system at the output and

jj11 12 J =  ( 2.32) jj21 22

is the Jones matrix of the optical system.

2.8.2 Basis vectors

For Jones vectors, basis vectors are same as the unit vectors xˆ and yˆ in coordinate systems. The Cartesian basis vectors are

ˆ 1 ˆ 0 Ex =   Ey =   ( 2.33) 0 1 so we can write ˆ ˆ Exy = Ex Ex + E y E y ( 2.34)

ˆ ˆ we can write Cartesian basis vectors as Ex′ and E y′ which are rotated with ˆ ˆ respect to the xy-system and circular basis vectors as Eq and Er defined by 17

Eq  E =   ( 2.35) Er 

The circular basis vectors in the xy-system are given by

1 ˆ i ˆ 1  1  1 ˆ i ˆ 1 1 El = Ex − E y =   and Er = Ex + E y =   ( 2.36) 2 2 2 − i 2 2 2 i

2.8.3 Examples of Jones vectors Jones vectors for linearly polarized waves for which the electric vector oscillates along a direction inclined an angel α to the x -axis is given by

cosα  E =   ( 2.37) sinα 

Jones vectors for the left- and right-circularly polarized waves are given by

1  1  1 1 Exy =   and Exy =   ( 2.38) 2 − i 2 i

The factor 1 is included for normalization. 2

2.8.4 Mathematical representation of polarizers There are four types of polarizer.

2.8.4.1 Linear polarizer The linear polarizer removes all or most of the E-vibrations in a given direction while transmitting those vibrations which are in the perpendicular direction. Unpolarized light traveling in the +z-direction passes through a plane polarizer whose transmission axis TA is vertical. The unpolarized light is represented by perpendicular (x and y) vibrations. The light transmitted include components only along the TA direction and is linear polarized in the vertical or y-direction. The horizontal component is removed by absorption. Jones vector for the linear polarizer is

18 Chapter 2: Theory of Polarized Light

0 0 J =   ( 2.39) 0 1

Y X

Linear Polarizer

Figure 2.4: Operation of a linear polarizer 2.8.4.2 Elliptical polarizer Jones vector for the elliptical polarizer is

 cos 2 α sinα cosαe −iδ  J =  +iδ 2  ( 2.40) sinα cosαe sin α 

This is the Jones matrix for any type of elliptical polarizer including a linear polarizer and a circular polarizer. For a linear horizontal polarizer α = 00

1 0 J =   ( 2.41) 0 0

For right circularly polarized light α = 450 and δ = 900

1 1 −i J =   ( 2.41) 2 i 1 

2.8.4.3 Phase Retarder The phase retarder introduces a phase difference between the E-vibrations. When light corresponding to these vibrations travels with different speeds through retarding plates, there will be a phase difference of ∆Φ between the emerging waves. Y X

FA Retardation Plate SA

Figure 2.5: Operation of a phase retarder

Figure 2.5 shows the effect of a retarding plate on unpolarized light whose vertical component travels through the plate faster than the vertical component. When the net phase difference ∆Φ = 900 , the retardation plate is called a quarter- wave plate, when ∆Φ = 1800 , it is called a half-wave plate. The Jones matrix for a retarder is given by

Φ  +i  e 2 0  J = Φ ( 2.43)  −i   0 e 2  and for a homogenous right circular retarder

 Φ Φ   cos sin  J = 2 2  Φ Φ  ( 2.44) − sin cos   2 2 

20 Chapter 2: Theory of Polarized Light

2.8.4.4 Rotator The rotator has the effect of rotating the direction of linearly polarized light incident on it by some particular angle. Vertically linear polarized light is incident on a rotator as shown in Figure 2.6. The effect of rotator element is to transmit linearly polarized light whose direction of vibration has rotated counterclockwise by an angleθ . The requirement for a rotator of angle β is that an E-vector oscillating linearly at angleθ be converted to one that oscillates linearly at angle ()θ + β . The Jones matrix for a rotator through angle + β is given by [14]

cos β − sin β  J =   ( 2.45) sin β cos β  Y X

Rotator

θ Z

Figure 2.6: Operation of a rotator

2.8.5 Jones matrix of the optical device that has been rotated In an ellipsometer, we have a cascade of optical devices which are rotated individually, so we need a fixed reference laboratory frame, called the laboratory frame. We can call this laboratory frame as uv coordinate system. Consider an optical device which is rotated an angle α where α is an angle between the x-axis of the input coordinate system of the device and u-axis of the fixed coordinate uv system. A light wave with Jones vectors Ei with reference to the uv-system is 21 incident on the optical device. The Jones vector in the input coordinate system xy is

xy uv EREii= (α ) ( 2.46) where R()α is the rotation matrix given by

cosα sinα R()α =  ( 2.47) −sinα cosα

For the optical device with Jones matrix J xy , the output Jones vector in the x´y´-system will be

xy′ ′ xy uv EJREoi= ()α ( 2.48)

This is the electric field vector after transmission through the optical device. If the xy and x´y´ coordinate systems are parallel then the output Jones vector in the uv coordinate system is

uv x′ y′ Eoo=−RE( α ) ( 2.49) and we can write

uv xy uv Eoi=−RJRE( αα) ( ) ( 2.50) Finally we can write that, in a fixed uv coordinate system, the Jones matrix J uv of a rotated optical device is given by JRuv=−( α ) JR xy (α ) ( 2.51)

2.9 Mueller matrix In 1943 Hans Mueller, a professor of physics at the Massachusetts Institute of Technology, devised a matrix method for dealing with the Stokes vectors. Stokes vectors are applicable to both unpolarized and partially polarized light. The Jones method is valid only for totally polarized light. For light which is partially polarized or unpolarized, the depolarization of light caused by an optical device can be studied by Mueller matrices. When light, either propagating through the medium or is reflected from some interface, its polarization state is changed. There can be a change in amplitude or

22 Chapter 2: Theory of Polarized Light phase or direction of the orthogonal electric field vectors [15]. The depolarizing optical system is represented by a 4x4 Mueller matrix and the state of polarization of the light waves is represented by Stokes vectors. It is analogous to Jones matrix definition to write as

SMSoi= ( 2.52) where So and Si are the Stokes vectors of the outgoing and incoming beams of light. The matrix M i is called the Mueller matrix of the optical system. The out - going Stokes vector is having a linear combination with incoming Stokes vector as

SmSmSmSmSoiiii0=+++ 11 0 12 1 13 2 14 3 SmSmSmSmS=+++ oiiii1210221232243 ( 2.53) SmSmSmSmSoiiii2=+++ 31 0 32 1 33 2 34 3

SmSmSmSmSoiiii3410421432443=+++

In matrix form equations (2.53) can be written as

SmmmmSoi0111213140  SmmmmS  oi1212223241=  ( 2.54) SmmmmSoi2313233342    SmmmmSoi3414243443  The combined effect of several optical elements in a cascade is given by

SMMoNN= −121...... MMS i ( 2.55) where M1 is the first optical device with which light encounters.

Chapter 3

Ellipsometry

3.1 Principle Ellipsometry is the art of measuring and analyzing the elliptical polarization of light [15]. It is used for thin films, surface and interface characterization. Alexandre Rothen in 1944 suggested the name ellipsometry and he is one of the pioneers in this field. Ellipsometry is a measurement technique which is sensitive to surface layers and is very useful for thin film metrology. It gives the full complex-valued optical response function of a sample. It has applications in different fields, from semiconductor physics to microelectronics and biology. It plays a vital role from basic research to industrial applications [16]. Ellipsometry is based on measuring phase differences and the ratios of electric field amplitudes. With ellipsometry, we are able to find the thickness of layers which are thinner than the wavelength of the incoming light and multi- layered structures with different layers can also be studied. Spectroscopic ellipsometry is such a technique which has many applications in research and industry.

23 24 Chapter 3: Ellipsometry

E p-plane p-plane E

s-plane θ o s-plane

Plane of incidence

Figure 3.1: Principle of reflection ellipsometry. The incoming light is linearly polarized and has known polarization In a reflection-based ellipsometric measurement, the change in polarization of a light beam reflected from a sample is measured. The phase and amplitude changes are different for the p-and s-polarized complex-valued electric field components. These differences are measured by an ellipsometer. The change in polarization depends on the surface and thin film properties.

3.2 Standard ellipsometry Standard ellipsometry is here described in external reflection, internal reflection and transmission configurations.

3.2.1 External reflection ellipsometry When a polarized monochromatic plane wave is incident on a surface at oblique incidence, the polarization changes on reflection. In Figure 3.1, a linearly polarized light is incident on an optically isotropic sample and is reflected. There is no coupling between parallel (p) and perpendicular (s) polarizations for an isotropic sample. After reflection, light become elliptically polarized. The incident light can be either linearly polarized or can have any state of polarization. Epi and Esi are incident electric field components. Epr and Esr are reflected electric field components. The reflectance ratio ρ is measured with an ellipsometer and is found to be the ratio between two reflection coefficients for an isotropic sample. ρ can then be written as 25

r ρ = p (3.1) rs where rp and rs are the complex-valued reflection coefficients for light polarized in the p-and s-direction, respectively. Generally ρ is defined in terms of the ratio between the states of polarization of the reflected and incident beams as χ ρ = r ( 3.2) χi where χr and χi are the complex-number representation of the states of polarization of the reflected and incident beam, respectively. χ is defined by

E χ = p ( 3.3) Es where Ep and Es are the complex-valued representation of the electric fields in

Epi Epr the p-and s-directions, respectively. With χi = and χr = ,equation Esi Esr (3.2) becomes:

E E ρ = pr si (3.4) EEsrpi

Epr Esr The reflection coefficients are rp = and rs = , so Epi Esi

ErE ρ ==prsi p (3.5) EEsrpis r In terms of ellipsometric angles ψ and ∆ , we get

r ρψ==p eei()δδps− tan i∆ (3.6) rs

rp where tanψ = is the amplitude ratio and phase difference ∆ =−δ psδ . rs The quantities ψ and ∆ are functions of the optical constants of the medium, the thin film and the substrate, the wavelength of the light, the angle of incidence and thickness of an optical film deposited on the substrate. A general observation is that ψ = 450 at both normal and glancing incidences for all isotropic materials.

26 Chapter 3: Ellipsometry

θo θo

θc θc No No N1

N1 N2

Figure 3.2: Internal reflection ellipsometry in the two-phase model (left) and three-

phase model (right). The angle of incidence θo is larger then the

critical angle θc

3.2.2 Internal reflection ellipsometry In internal reflection ellipsometry, light is incident at an interface and is reflected. Incidence is from a medium with higher refractive index than that of the reflected medium. Consider a two-phase model where a glass prism is ambient and air is the substrate. For internal reflection configuration, in the three-phase model, there will be a thin film on the prism as shown in Figure (3.2). The ambient medium must be transparent so that the light penetrates down to the interface, reflects and propagate back to the external ambient. The layer on the prism must be transparent or semi-transparent. N0 must be greater than

N1 and N2 . For total internal reflection, θo >θc

3.2.3 Transmission ellipsometry In ellipsometry, if there is transmission instead of reflection, then ellipsometric ratios for the complex transmission coefficients t p and ts are

t p i∆t ρψtt==tan e (3.7) ts

The ellipsometric parameters ρt , ψ t and ∆t are the transmission ratio, amplitude ratio and phase difference in transmission ellipsometry. 27 3.3 Generalized ellipsometry In standard ellipsometry, the sample is isotropic and has a diagonal Jones matrix. For a general anisotropic sample, we need a non-diagonal Jones matrix.

r rrpp sp  R =   (3.8) rrps ss  First index is for incident polarization and second index for reflected polarization. In generalized ellipsometry, we require at least three values of ρ measured at three different incident polarization in order to get three pairs of ()ψ ,∆ . Three complex-valued generalized ellipsometric parameters are defined as r pp i∆ pp ρψpp==tan ppe rss r ps i∆ ps ρψps==tan pse (3.9) rpp r sp i∆sp ρψsp==tan spe rss In generalized ellipsometry, we have to find six parameters

ψ pp,,,,ψψ ps sp∆∆ pp ps and ∆sp . By using equation (3.9), the reflection Jones matrix in equation (3.8) can be written as:

r  ρ ppρ sp  Rr= ss   (3.10) ρρps pp 1  The objective of generalized ellipsometry is now to determine the unknown values of ρ pp, ρ ps and ρsp . The relation between the incident and reflected light in terms of reflection Jones matrix, is given by

r EREri= (3.11) We can write it as

ErErEpr= pp pi+ sp si (3.12) ErErEsrpspisssi=+ E By using χ = p equation (3.12) can be written as: Es

28 Chapter 3: Ellipsometry

rrppχ i+ sp χr = (3.13) rrpsχ i+ ss Combining equations (3.2), (3.9) and (3.13) we have

ρ + ρχ−1 ρ = pp sp i (3.14) 1+ ρ ppρχ ps i

Hence ρ is representing a transformation of the incident polarization state χi .

For a given χi , ellipsometer gives value of ρ . In practice i =1, 2, 3 so we can find the values of ρ pp, ρ ps and ρsp .

3.4 Mueller-matrix ellipsometry From a Stokes vector of reflected light, we can describe a sample with a Mueller matrix M r . A Stokes meter is used to determine the Stokes vector of the reflected light. Mueller matrix ellipsometry is used for non-depolarizing samples. The reflection Mueller matrix elements are:

1 2222 mrrrr=+++ (3.15) 11 2 ( pp ss sp ps )

1 2222 mrrrr=−−+ (3.16) 12 2 ( pp ss sp ps )

mrrrr=ℜ ∗ + ∗  13  pp sp ss ps  (3.17)

mrrrr== ℑ ∗ + ∗  (3.18) 14  pp sp ss ps 

1 2222 mrrrr=−+− (3.19) 21 2 ( pp ss sp ps )

1 2222 mrrrr=+−− (3.20) 22 2 ( pp ss sp ps )

∗∗ mrrrr== ℜ −  (3.21) 23  pp sp ss ps 

∗∗ mrrrr== ℑ −  (3.22) 24  ppsp ss ps 

 ∗∗ (3.23) mrrrr31 =ℜ pp ps + ss sp  29

 ∗∗ (3.24) mrrrr32 =ℜ pp ps − ss sp 

 ∗∗ (3.25) mrrrr33 =ℜ pp ss + ps sp 

 ∗∗ (3.26) mrrrr34 =ℑ pp ss − ps sp 

 ∗∗ (3.27) mrrrr41 =−ℑ pp ps + ss sp 

 ∗∗ (3.28) mrrrr42 =−ℑ pp ps − ss sp 

 ∗∗ (3.29) mrrrr43 =−ℑ pp ss + ps sp 

 ∗∗ (3.30) mrrrr44 =ℜ pp ss − ps sp 

 ∗∗ (3.31) mrrrr31 =ℜ pp ps + ss sp 

For isotropic samples rrsppsppp==0, rr =and rrsss= and are described fully with equation ρψ= tan ei∆ . The Mueller matrix for isotropic sample is

112222 rr+− rr 00 22( ps) ( ps)  112222 r rr−+ rr 00 M = 22( ps) ( ps) (3.32)   ∗∗ 00ℜ−ℑrrps rr ps  00ℑℜrr∗∗ rr  ps ps r If ρψ==tan ei+ p is used, equation (3.32) is written as rs

1cos20− ψ 0 2 2 rr+ −cos 2ψ 1 0 0 M r = ps (3.33) 2 00sin2cossin2sinψψ∆ ∆  00sin2sinsin2cos− ψψ∆∆ With short notations of NC==∆=∆cos 2ψ , sin 2ψψ cos , S sin 2 sin (3.34) equation (3.33) becomes

30 Chapter 3: Ellipsometry

 100−N  2 2 rr+ −N 100 M r = ps  (3.35) 2  00CS    00−SC For a non-depolarizing sample, NS, and C are related by

NSC22+ += 21 (3.36) The complex ratio ρ is now written as CiS+ ρψ==tan ei∆ (3.37) 1+ N In terms of Mueller matrix elements, it can be expressed as 11 −++−()mm33 44 imm() 34 43 ρψ==tan ei∆ 22 1 (3.38) 1−+()mm 2 12 21 For the description of an isotropic sample three parameters N, S and C are needed in Mueller matrix ellipsometry, while for a non-depolarizing sample, from equation (3.37) we need two independent parameters ψ and ∆. For ideal instrument mmmm33== 44, 12 21 and mm34= − 43 . The most recent development in the field of ellipsometry is the Mueller matrix ellipsometer. The new generation of ellipsometers has the ability to measure the Mueller matrix along with the ellipsometer angles ∆ and ψ. We can now determine every optical property of any sample, anisotropic as well as isotropic, depolarizing as well as non-depolarizing.

Chapter 4

Instrument

In this chapter we briefly discuss the method and instrumentation used during the polarization analysis of light reflected from Cetonia aurata. Initially a simple polarizer is used to determine the polarization of the sample beetle. In a second stage, an ellipsometer system with dual rotating compensators is used to determine the Mueller matrix of the reflected polarized light.

4.1 Polarization analysis Light reflected from Cetonia aurata is left-handed circularly polarized and the beetle appears black if we are viewing it through right-handed circular polarizer. Figure 4.1 is showing that we have a light source, a linear polarizer and a quarter-wave retarder. These two combined together form a circular polarizer. When we will look at Cetonia Aurata with a left-circular polarizer, it will appear as having green gloss over its body surface but with a right-circular polarizer, the colour disappear and beetle look black as shown in Figure 4.2 The exoskeleton of the beetle is composed of such materials having the property of transmitting right-handed polarized light and reflecting left-handed polarized light.

31 32 Chapter 4: Instrument

Figure 4.1: A simple desktop polarization system

Figure 4.2: The color of beetle Cetonia aurata under right and left handed circular polarizer is black and green, respectively

4.2 Spectral analysis The 16 Mueller matrix elements are measured with a dual rotating compensator ellipsometer in the spectral range 300-900 nm at multiple angles of incidence in the range 40-70º with a data acquisition of 20 s/angle. 33

Spectrometer Optical Fiber Camera

Rotating Compensator Focusing Probes

Figure 4.3: Dual rotating compensator ellipsometer

4.2.1 Dual rotating compensator ellipsometer This ellipsometer system has dual rotating compensators. The setup is denoted

PCrr SC A or Polarizer, Compensator (rotating), Sample, Compensator (rotating), Analyzer. The ellipsometer is manufactured by a company named J.A.Woollam Co., Inc. The name of the instrument is RC2® and is shown in Figure 4.3. In this ellipsometer, the light is a white light source-a xenon bulb and the full spectral range is transmitted through the system. The light is polarized with a polarizer and then passes through a compensator attached to a focusing probe and then fall on the sample. After reflection from the sample it will go through the other compensator which also is attached to a focusing probe and then reaches the analyzer. After passing through the analyzer, it reaches the detector. The compensator is an optical element that changes the phase of the incident wave, delaying one of the two orthogonal light constituents by optical anisotropy

34 Chapter 4: Instrument

Figure 4.4: Compensator element is introducing a phase shift between the two electric field components of light

( nn0 ≠ e ). Each of the two orthogonal electric fields experiences a different index which produces different phase velocities in the two directions. In ellipsometry, compensators which are quarter-wave plates are used to enhance measurement accuracy. With this ellipsometer, the polarization states of the incident and outgoing lights are determined and then we are able to get the full Mueller matrix. That is the reason for the use of two compensators and hence the equipment got the name RC2. In RC2 the full spectral range is transmitted and reflected by the sample. After the analyzer the beam is diffracted at a grating and is collected by the detector.

In the ellipsometer with the configuration PC12rr(3ω ) SC (5ω ) A, the 3:5 ratios for the rotation speeds of the two compensators are chosen [17]. The ratio can be 1:5 [18]. This frequency ratio of 3:5 gives all 15 elements of normalized Mueller matrix. It gives a highest order harmonic frequency of 32ω, in the waveform of the detected irradiance. For ratio of 1:5, the frequency is 24ω.The wavelength- dependent phase retardance values for the first and second compensator areδ1 andδ2 .

In the dual compensator, the Stoke’s vector Sout of the light which is incident on the detector, can be written in terms of Mueller matrix product as

SMACM=ℜℜ−()( ) (δ )( ℜ C ). out A222 C 2 (4.1) ×ℜ−M SC()CM1111 ()()()δ ℜ C ℜ−ℜ PM Pin () PS

where Sin is the Stoke’s vector of the light incident on the polarizer.

M P ,(),(),MMCC12δ12δ M Sand M A are the Mueller matrices of the polarizer, the first 35 compensator, the second compensator, the sample and the analyzer. ℜ()x in equation 4.1 is the Mueller rotation transformation matrix for rotation by the ` ` angle x . The angles C1 andC2 are the angles of fast axes of the first and second ` ` compensator. For angles CCC11=−s 1 and CCC22=−s 2 ,Ct11= ω and Ct22= ω are the angular rotation of the elements and Cs1 and Cs2 are absolute angular phases. Cs1 and Cs2 are wavelength dependent for a multi-channel system. The expression for irradiance at the detector is proportional to first element of the Stoke’s vector and is obtained by the multiplication of the matrices in equation 4.1

22`δδ22  I =+IK01{ [cos cos(2 A ) + sin  cos() 4 C 2 − 2 AK ] 2 22 

22`δδ22  (4.2) ++−[cos sin(2ACAK ) sin  sin() 423 2 ] 22  ` −−[sinδ22 sin(2CAK 2 )] 4 } where,

22`δδ11  Kmjj=+112[cos cos(2 P ) + sin  cos() 4 C − 2 Pm ] j 22 

22`δδ11  (4.3) ++−[cos sin(2PCPm ) sin  sin() 413 2 ] j 22  ` +−[sinδ11 sin(2CPm 2 )] j 4

In equation 4.2, mjjk (1,,4;1,,4)=="" k are the elements of the sample

Mueller matrix. It is assumed that Mueller matrix is normalized, i.e. m11 =1 and ` the absolute irradiance is proportional to I0 . For the ratio of 5:3, CCC111=−5(s ) ` and CCC222=−3(s ) , the equation 4.2 become

16 I=+ I021[cos2∑ αφβφnnnn()() nC −+ 222 sin2 nC − ] (4.4) n=1

(,)α22nnβ are non-zero Fourier coefficients and φ2n are the phase-terms. From 24 non-zero Fourier coefficients of frequencies ranging from 2C to 32C, the eight coefficient of{(αβ22nn , ),n = 9,12,14,15} vanish, the remaining 16 coefficients can

36 Chapter 4: Instrument be used to determine the 15 normalized Mueller matrix elements [18]. Thus the

Mueller matrix elements m jk are the Fourier components of the modulated intensity [17]. We have a rapid-scanning multi-channel Mueller matrix ellipsometer with a capability of collecting the 15 elements of the normalized Mueller matrix from 245-1700 nm in a minimum acquisition time of 0.2 s. In this system, the rotating motors are operating at 5ω = 12.5 Hz and 37.5ω = Hz in order to get 36 detectors readings with the integration time of π which is about 5.56 ms. 36ω We have a system in which both quarter-waveplates are rotating with a constant ratio of speeds. In hardware, such system is good for a simple and fast acquisition of data [17].

Chapter 5

Models for Cetonia aurata Optical Properties

This chapter is about the optical study of the exoskeleton of beetle Cetonia aurata. The light reflected from the metallic-shiny regions of the cuticle of Cetonia aurata belonging to Scarabaeoidea family is left-handed circularly polarized. The circularly polarized gloss is all over its body and is retained after its death. Some optical models are used to study the complex nature of the cuticle structure.

5.1 The cuticle structure of the beetle Cetonia aurata has a green gloss over its body which reflects polarized or unpolarized incident light into circularly polarized light. The exoskeleton of the beetle has the property of absorbing right-handed polarized light. The helical structure of the molecules causes this property [20]. The optical study of the iridescent outer-skeleton of the beetle Cetonia aurata has shown a nanostructure which controls the polarization and wavelength distribution of the irradiance of the reflected light. The origin of this effect is the helicoidal layer. Ellipsometry spectra are collected from different areas of the

37 38 Chapter 5: Models for Cetonia aurata Optical Properties shell of the beetles like the scutellum (the triangular part behind the head), the head, the right and left wings. From the abdominal side, a spectrum is obtained which is quite noisy. These spectra are compared with those calculated from an optical model. The exoskeleton of the beetle gives stiffness and strength to the beetle and has muscles attached. The cuticle is made of a number of layers which are separated from each other by the epidermal cells as shown in Figure 5.1. With the passage of time, these layers change their chemical composition, thickness and optical properties. The cuticle is a composite structure having proteins, melanin, lipids but the most dominant component is chitin [20]. Different layers of a beetle cuticle are shown in Figure 5.1. The epicuticle is about 1-2 µm thick and is covered by thin layers of wax and cement. The exocuticle contains colour generating structures. The endocuticle contains more fluid and is a reflective part of the cuticle. The exocuticle consists of a number of layers which are placed on each other and consists of chiral structures. The chiral structure materials are twisted stacks of birefringent sublayers. The circular polarization of metallic shiny Cetonia aurata is caused by the helical structure of molecules in the chitinous cuticle [21] as shown in Figure 5.2.

Figure 5.1: Cross section of the beetle cuticle which is showing different layers [11]. 39

Figure 5.2: The exocuticle is showing layers with different directions of the refractive index [12]

5.2 Optical model The optical models involve approximations due to surface roughness; gradients in properties and depolarization which play a major role in structure determination. EMA-models are used in finding properties of composite materials. With help of the optical model, we can find optical properties and structural parameters. Typical structural parameters are number of layers and their thicknesses and material composition. For a sample with anisotropic materials, the orientation of the optic axes, parameters describing surface roughness and interface layers are such structural parameters which are modeled. This modeling is known as multi-parametric modeling. The model is useful for analysis of the optical response data of materials.

5.2.1 The Cauchy model The nature of cuticle structure is complex. Its chemical composition and optical properties along with the layer thickness can be found by using a Cauchy dispersion model. This model gives best solution and lowest mean square error (MSE). The Cauchy model was developed by Augustin Louis Cauchy-a French mathematician, who discovered that the refractive index n decreases in the material with increasing wavelength in the visible light range for a transparent material. This model is useful in photon energy ranges where the absorption is zero or very low. The equation for the Cauchy model is written as

40 Chapter 5: Models for Cetonia aurata Optical Properties

B C nA()λ =+ + λ 24λ ( 5.1)

where n is the refractive index, λ is the wavelength and A, B and C are constants specific for each material. In the ultraviolet range of the experimental data, there is an absorption tail. For this tail, the Cauchy model is improved by using an expression called Urbach’s tail and is given by

11 Bu ()− λ Cu kAe()λ = u ( 5.2)

where Au is amplitude constant and Bu is the width or broadening of the exponential tail and Cu is the absorption band edge. We can fit either Au orCu , not both of them. Equations 5.1 and 5.2 together have five parameters which are determined for modeling of the dispersion: A, B, C, Au and Bu . A, B and C are constants which are determined by fitting equation 5.1 to the experimental data. Sometimes it is sufficient to fit only A and B while ignoring C. Additional fitting parameters are Au and Bu whileCu correlates with Au .

5.2.2 The Liquid crystal model Liquid crystals have unique physical and optical properties. They are condensed matter phases that show characteristics of conventional liquids and solid crystals. They are fluids like liquids and have molecules in the liquid arranged and oriented like crystals. Some of them have orientational order and some have positional order along one spatial direction. There are different types of liquid crystals depending on their optical properties (such as birefringence). They have different texture and the contrasting area depend on the different orientation direction of its molecules but the molecules are always well ordered. In 1888, the Austrian botanical physiologist Friedrich Reinitzer described three important features of cholesteric liquid crystals (the name was given by Georges Friedel in 1922); the existence of two melting points, the reflection of circularly polarized light and the ability to rotate the polarization direction of light [22].

Chapter 6

Results and Discussion

For finding sample properties like layer thicknesses and optical constants, data analysis of spectroscopic ellipsometry is done by using an optical model. The sixteen elements of the Mueller matrix are measured for wavelengths λ in the visible range 400-800 nm.

6.1 Spectroscopic ellipsometry

6.2 Data analysis The instrument used for the measurements is RC2 which is a dual rotating compensators ellipsometer as described in Section 4.2.1. The spectral range for RC2 is 245 nm to 1690 nm. A measuring time of 10 second is used. The angles of incidence are 450 , 500 and 650 .For the measurements, the hard shell of the beetle Cetonia aurata is separated from its body and different samples are prepared for the head, the scutellum, the right wing, the left wing and the dorsal side of the beetle. Each sample was mounted on the sample stage and aligned.

41 42 Chapter 6: Results and Discussion

2 Top Layer: nt =A + B/λ Chiral Layer X Turns with Linear Rotation Φ = start direction of the helix 2 nooo = A + B λ neo = n o + ∆n Dielectric

Substrate ns =1.5

Figure 6.1: Three layers of beetle’s exocuticle surface having a single chiral layer

2 Top Layer: nt =A + B/λ Chiral Layer-2 X Turns with Linear Rotation Φ = start direction of the helix 2 n111 = A + B λ ne1 = n o + ∆ n 1 Chiral Layer-1 X Turns with Linear Rotation Φ = start direction of the helix 2 nooo = A + B λ neo = n o + ∆ n

Dielectric

Substrate ns =1.5

Figure 6.2: Four layers of beetle’s exocuticle surface with two chiral layers

After switching on the light source, measurements are taken after 10 minutes and the spectrum of the reflected light is obtained. With a small CCD-camera preview picture of the beetle is transmitted to the monitor of the computer through a USB port connected to the computer. The wavelength and polarization patterns are of complex nature. 43

1.5 Model Fi t 1.0 Exp mm12 50° Exp mm14 50° Exp mm22 50° Exp mm33 50° 0.5 Exp mm44 50°

0.0

-0.5 Muller-matrix elements

-1.0 400 500 600 700 800 Wavelength (nm)

Figure 6.3: Experimental Muller matrix elements m12, m14, m22, m33 and m44 for the scutellum of the beetle and optical model data with single chiral layer and at an angle of incidence of 500

1.5

Model Fi t 1.0 Exp mm12 50° Exp mm14 50° Exp mm22 50° Exp mm33 50° 0.5 Exp mm44 50°

0.0

-0.5 Muller-matrix elements

-1.0 400 450 500 550 600 650 Wavelength (nm)

Figure 6.4: Experimental Muller matrix elements m12, m14, m22, m33 and m44 for the scutellum of the beetle and optical model data with double chiral layer and at an angle of incidence of 650

44 Chapter 6: Results and Discussion

For the beetle scutellum, the model consists of three layers. First is a dielectric substrate with a refractive index of 1.5 as shown in Figure 6.1. On the top of this layer, there is a chiral layer containing ordinary and extraordinary refractive indices and with thickness dc. On top of this layer is a dielectric layer of certain thickness and refractive index nt. The 15 Mueller matrix elements are obtained as we have normalized Mueller matrix with first element m11=1. From these values, five elements values m12, m14, m22, m33 and m44 are shown in Figures 6.3 and 6.4 at an angle of incidence of 500. Since Cetonia aurata is green, we can see a peak in wavelength at about 525nm. Modeling is done by using the liquid crystal model where instead of the liquid crystal layer, we have a single, isotropic Cauchy layer, and then we converted the layer into an anisotropic layer. The Cauchy layer is the first layer then the next layer is “dNz” which has zero thickness and holds the refractive index difference in the ordinary and extraordinary directions. The measurement is most sensitive to this difference. The index of refraction is a function of wavelength as described in section 5.2.1, and here we are using the two constants A and B. The next layer is “Nz” which gives the ordinary optical constants with the index difference given in the “dNz” layer[23]. In the uniaxial layer, the Euler angle θ is set to 900 as the sample is placed parallel to the glass surface and angle θ gives tilt to the z-axis into the sample plane. The “Nz” layer is also of zero thickness. The uniaxial layer is coupled into the graded layer. The Φ Euler angle controls the in-plane rotation. The Φ value is varied to control the twist and Φ–zero gives the value where the twist is started. The thickness of the chiral layer appears in the graded layer and then at the top there is the low index layer as seen in Figure 6.2. From Figure 6.3, in the experimental data, firstly it appears that there are two peaks lying adjacent to one another and secondly we want to reduce the MSE value so we consider to model with two chiral layers. In Figure 6.4, after the anisotropic Cauchy layer, we model that there are two chiral layers. The values of constants A and B are given in Table 6.1. The two refractive indices in the chiral layer are no and ne and for the top layer is nt. Table 6.1 shows versus wavelength the values of constants A0, B0 for the refractive index no and A1, B1 for the refractive index ∆n and the values of refractive index ne where nneo=+∆ n and the values of constants A2, B2 for the refractive index nt of the top layer. The top layer is low refractive index layer as 1.439<1.499. Table 6.2 shows the thicknesses of the chiral layer dc as 6.87 µm and of the top layer dt as 0.298 µm. The Mean Square Error MSE is 75.9. The twist of the chiral layer starts at angle 0 Φ0 of -21 . The thickness non-uniformity of uniaxial layer is 13%. There are 22 sub-layers inside a chiral layer and there are 8 number of turns or twists.

Beetle’s scutellum with single chiral layer:

Wavelength Refractive Refractive Refractive Refractive λ(nm) index no index∆ n index ne index nt 450 1.151 0.348 1.499 1.439 500 1.207 0.284 1.491 1.375 550 1.248 0.237 1.485 1.328 600 1.279 0.200 1.480 1.292 650 1.304 0.172 1.476 1.264 700 1.324 0.149 1.474 1.241 750 1.339 0.132 1.471 1.223 800 1.352 0.117 1.469 1.209 Table 6.1: Wavelength of the incident light and the refractive indices of the Chiral layer as no with constants A0=1.446 and B0=-0.059 , ∆n with constants A1=0.009 and B1=4.538

and ne with nneo=+∆ n. The refractive index of the top layer nt with constants A2=1.102 and B2=0.068

1.55 1.50 1.45 1.40 1.35 1.30 1.25 1.20

Refractive Index Refractive neone 1.15 nono 1.10 450 500 550 600 650 700 750 800 Wavelength (nm)

Figure 6.5: Chiral layer’s refractive indices of the beetle´s scutellum

Figure 6.5 shows the graph for the refractive indices of the chiral layer where ne is the extraordinary refractive index and ne is the ordinary refractive index of the beetle´s scutellum. Figure 6.6 shows versus wavelength, the refractive index of the top layer for beetle’s scutellum.

46 Chapter 6: Results and Discussion

1.50 nt 1.45 nt 1.40 1.35 1.30 1.25 1.20 Refrective Index 1.15 1.10 450 500 550 600 650 700 750 800 Wavelength(nm)

Figure 6.6: Top layer’s refractive index of beetle´s scutellum

Thickness Angle Number of Number Thickness MSE dc (µm) Φo layers of turns dt (µm) 6.87 -21 22 8 0.298 75.9

Table 6.2: Thickness of the Cauchy layer dc and the top layer dt Beetle’s scutellum with two chiral layers:

Table 6.3 shows versus wavelength, the values of constants A0 and B0 for the refractive index no , A1 and B1 for the refractive index ∆no and the values of refractive index ne for the first chiral layer of the beetle Cetonia aurata. Figure 6.7 shows versus wavelength, the graph for the values of the extra ordinary refractive index ne and the ordinary refractive index no for the first chiral layer.Table 6.4 shows values for the second chiral layer. Figure 6.8 is for the second chiral layer of the beetle’s scutellum.

Wavelength Refractive Refractive Refractive λ index no index∆ no index ne 400 1.698 0.177 1.875 450 1.680 0.159 1.839 500 1.667 0.146 1.813 550 1.657 0.136 1.793 600 1.649 0.129 1.779 650 1.641 0.123 1.764 Table 6.3: Wavelength of the incident light and the refractive indices of the first Chiral layer as no with constants A0=1.611 and B0=-0.014 , ∆n with constants A1=0.091 and

B1=0.099 and ne with nneo=+∆ n for beetle’s scutellum 47

1.90 neone 1.85 nono

1.80

1.75

1.70

Refractive Index Refractive 1.65

1.60 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 6.7: First chiral layer refractive indices of the beetle´s scutellum

2.10 neon 2.00 e1 non1 1.90 1.80 1.70 1.60 1.50 1.40 Refractive Index 1.30 1.20 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 6.8: Second chiral layer refractive indices of the beetle´s scutellum

48 Chapter 6: Results and Discussion

Wavelength Refractive Refractive Refractive Refractive λ index n1 index ∆ n1 index ne1 index nt 400 1.588 -0.220 1.368 1.651 450 1.625 0.111 1.736 1.543 500 1.650 0.349 1.999 1.465 550 1.670 0.557 2.227 1.408 600 1.685 0.658 2.343 1.364 650 1.697 0.762 2.459 1.330 Table 6.4: Wavelength of the incident light and the refractive indices of the second Chiral layer as n1 with constants A2=1.763 and B2=-0.028 , ∆n with constants A3=1.361 and B3=-0.121 and ne1 with nne11=+∆ n 1 . and the refractive index of the top layer nt with constants A4=1.135 and B4=0.081 for beetle’s scutellum

Table 6.5 is showing thicknesses of the first chiral layer dc1 as 4.7 µm and of the second chiral layer dc2 as 3.1 µm. The thickness of top layer dt is 0.55 µm. There are 22 sub-layers inside each chiral layer and there are 8 number of turns 0 in both chiral layers. The twist of the first chiral layer starts at angle Φ0 of 6 and for the second chiral layer it is -150. The plane of incidence is perpendicular to the beetle and angle of incidence is 500. For first chiral layer, the thickness is 4.74±0.1 µm, for second chiral layer, the thickness is 3.16±0.03 µm and for top layer, thickness is 0.559 ±0.004 µm. The thickness non-uniformity of uniaxial layer is 11%. Figure 6.9 shows versus wavelength, the values of the refractive index for the top Cauchy layer.

1.70 1.65 nntt 1.60 1.55 1.50 1.45 1.40 1.35 1.30 Refractive Index 1.25 1.20 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 6.9: Top layer refractive index of beetle´s scutellum with two chiral layers

Thickness Angle Number Thickness Angle Number Thickness MSE dc1 Φ01 of turns dc2 Φo2 of turns dt µm µm µm 4.74 2 8 3.16 -17 8 0.559 43.1

Table 6.5: Thickness of the first Cauchy layer dc1, second Cauchy layer dc2 and the top layer dt,twist of chiral layerΦ0,number of turns in each layer and number of layers is 22 in each chiral layer

Table 6.5 shows the thicknesses of the first chiral layer dc1 and the second chiral layer dc2 and the top layer dt, the angle of rotation for the start of two chiral layers as Φ01and Φ02, the number of sub-layers for both chiral layer is 22 and for both chiral layers the number of twist is 8. Modelling of this data with two chiral layers have reduced the value of Mean Square Error (MSE). With single chiral layer, its value is 75.9 but with two chiral layers, it reduces to 43.1.

Beetle’s head The next experimental data is from the head of the beetle. The sample for head of the beetle is mounted on the sample stage. The light is incident on the surface with angle of incidence of 500. The measurement are taken and the spectrum of the reflected light is obtained. The monitor of the computer got the

1.5

Model Fi t 1.0 Exp mm12 50° Exp mm14 50° Exp mm22 50° 0.5 Exp mm33 50° Exp mm44 50° 0.0

-0.5

Muller-matrix elements -1.0

-1.5 400 450 500 550 600 650 Wavelength (nm)

Figure 6.10: Experimental Muller matrix elements m12, m14, m22, m33 and m44 for the head of the beetle and optical model data with double chiral layer and at an angle of incidence of 500

50 Chapter 6: Results and Discussion picture of the head of the beetle. The 15 Mueller matrix elements are obtained as we have normalized Mueller matrix with first element m11=1. From the 15 Mueller matrix elements, the five elements are chosen which are m12, m14, m22, m33 and m44. The next step is modelling of this data which is done by considering the beetle’s exocuticle surface of its head. Figure 6.10 is showing the experimental and generated data with the model fit curves. Table 6.6 shows the wavelengths and refractive indices no, ∆no and ne with constants of Cauchy layers as A0, B0 for no, A1, B1 for ∆n and A2, B2 for ne, of first chiral layer. Table 6.7 shows values for second chiral layer and the values of the dielectric top layer. Table 6.8 is showing the thicknesses of first and second chiral layer as 4.44 µm and 3.05 µm with the thickness of top layer as 0.52 µm. In each chiral layer, the number of layers are 22, the twist of chiral starts at Φ0 with value of -9 and number of turns are 8. The mean square error (MSE) is 99.2. For the first chiral layer thickness is 4.4±0.2 µm, the second chiral layer thickness is 3.05 ±0.05 µm and for top layer thickness is 0.522±0.008 µm. The thickness non- uniformity of uniaxial layer is 10%. Figure 6.11 shows versus wavelength, the graph for the values of the extra ordinary refractive index ne and the ordinary refractive index no for the first chiral layer. Figure 6.12 is for the second chiral layer of the beetle’s scutellum.

1.74 neone 1.72 nono 1.70

1.68

1.66

1.64

Refractive Index 1.62

1.60 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 6.11: First chiral layer refractive indices of the beetle´s head

Wavelength Refractive Refractive Refractive λ index no index ∆ n index ne 400 1.678 0.058 1.726 450 1.647 0.052 1.699 500 1.634 0.048 1.682 550 1.625 0.044 1.669 600 1.617 0.042 1.659 650 1.612 0.040 1.652 Table 6.6: Wavelength of the incident light and the refractive indices of the first Chiral layer as no with constants A0=1.578 and B0=0.014 , ∆n with constants A1=0.029 and

B1=0.099 and ne with nneo=+∆ n for beetle’s head

Wavelength Refractive Refractive Refractive Refractive λ index n1 index ∆ n1 index ne1 index nt 400 1.653 -0.236 1.416 1.562 450 1.675 0.122 1.797 1.534 500 1.691 0.379 2.069 1.514 550 1.702 0.569 2.271 1.499 600 1.711 0.713 2.425 1.488 650 1.718 0.826 2.544 1.480 Table 6.7: Wavelength of the incident light and the refractive indices of the second Chiral layer as n1 with constants A2=1.758, B2=-0.017, ∆n1 with constants A3=1.473,

B3=-0.121 and ne1 with nne11=+∆ n 1 and the refractive index of the top layer nt with constants A4=1.429, B4=0.021 for beetle’s head

2.50 neone1 non 2.30 1

2.10

1.90

1.70

Refractive Index Refractive 1.50

1.30 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 6.12: Second chiral layer refractive indices of the beetle´s head

52 Chapter 6: Results and Discussion

1.57 1.56 ntnt 1.55 1.54 1.53 1.52 1.51 1.50 1.49 Refractive Index Refractive 1.48 1.47 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 6.13: Top layer refractive index of the beetle´s head

Thickness Angle Number Thickness Angle Number Thickness MSE dc1 Φo of turns dc2 Φo of turns dt µm µm µm 4.44 -9 8 3.05 -9 8 0.522 99.2

Table 6.8: Thickness of the first Cauchy layer dc1, second Cauchy layer dc2, and the top layer dt, twist of each chiral layer Φ0, number of turns in each chiral layer for beetle’s head

Beetle’s left wing Figure 6.13 shows versus wavelength, the values of the refractive index for the top Cauchy layer. Figure 6.14, is showing the normalized Mueller matrix element of m12,m14,m22,m33and m44 and a curve showing the model fit values for the beetle´s left wing at an angle of incidence of 650. Table 6.9 shows versus wavelengths the refractive indices no, ∆no and ne with constants of Cauchy layers as A0, B0 and A1, B1 of first chiral layer of beetle’s left wing.

1.0 Model Fi t Exp mm12 65° 0.5 Exp mm14 65° Exp mm22 65° Exp mm33 65° Exp mm44 65° 0.0

-0.5 Muller-matrix elements

-1.0 400 450 500 550 600 650 Wavelength (nm)

Figure 6.14: Experimental Muller matrix elements m12, m14, m22, m33 and m44 for the left wing of the beetle and optical model data with double chiral layer and at an angle of incidence of 650

3.00 neone 2.80 nono 2.60 2.40 2.20 2.00 1.80 Refractive Index Refractive 1.60 1.40 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 6.15: First chiral layer refractive indices of the beetle´s left wing Figure 6.15 shows versus wavelength, the graph for the values of the extra ordinary refractive index ne and the ordinary refractive index no for the first chiral layer. Figure 6.16 is for the second chiral layer of the beetle’s left wing.

54 Chapter 6: Results and Discussion

Figure 6.17 is for the top layer of the beetle’s left wing at an angle of incidence of 650. Wavelength Refractive Refractive Refractive λ index no index ∆ n index ne 400 1.635 1.305 2.94 450 1.617 1.171 2.788 500 1.604 1.075 2.679 550 1.594 1.004 2.598 600 1.587 0.950 2.537 650 1.581 0.908 2.489 Table 6.9: Wavelength of the incident light and the refractive indices of the first Chiral layer as no with constants A0=1.548, B0=0.014, ∆n with constants A1=0.666,B1=0.099 and 0 ne with nneo=++ n.for left wing of beetle at angle of incidence of 65

Wavelength Refractive Refractive Refractive Refractive λ index n1 index ∆ n1 index ne1 index nt 400 1.253 -0.022 1.231 1.543 450 1.398 0.083 1.481 1.467 500 1.502 0.158 1.660 1.412 550 1.579 0.214 1.792 1.371 600 1.637 0.255 1.892 1.340 650 1.683 0.288 1.971 1.316 Table 6.10: Wavelength of the incident light and the refractive indices of the second Chiral layer as n1 with constants A2=1.945,B2=-0.111, ∆n1 with constants A3=0.477,

B3=-0.109 and ne1 with nnne11=++ 1 and the refractive index of the top layer nt with constants A4=1.178, B4=0.059 for beetle’s left wing. 2.00 neon 1.90 e1 non1 1.80 1.70 1.60 1.50 1.40 Refractive Index Refractive 1.30 1.20 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 6.16: Second chiral layer refractive indices of the beetle´s left wing 55

Table 6.10 is for values of the refractive indices with constants A2, B2 and A3, B3 of the second chiral layer and A4, B4 for the top layer for beetle’s left wing. Table 6.11 shows thicknesses of first and second chiral layer as 2.92 µm and 3.89 µm with the thickness of top layer as 0.59 µm. In each chiral layer, the number of layers are 22, the twist of chiral starts at Φ0 with value of 3 and number of turns are 8. The mean square error (MSE) is 46.5. For the first chiral layer thickness is 2.92±0.1 µm, second chiral layer thickness is 3.89 ±0.04 µm and for top layer thickness is 0.59±0.009µm. The thickness non-uniformity of uniaxial layer is 13%.

Thickness Angle Number Thickness Angle Number Thickness MSE dc1 Φo of turns dc2 Φo of turns dt µm µm µm 2.92 3 8 3.89 3 8 0.59 46.5

Table 6.11: Thickness of the first Cauchy layer dc1, second Cauchy layer dc2 and the top layer dt, twist of chiral layerΦ0, number of turns in each layer and number of layers in each chiral layer

1.60 ntn 1.55 t

1.50

1.45

1.40

Refractive Index Refractive 1.35

1.30 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 6.17: Top layer refractive indices of the beetle´s left wing

56 Chapter 6: Results and Discussion

Beetle’s right wing

1.5 Model Fi t 1.0 Exp mm12 65° Exp mm14 65° Exp mm22 65° 0.5 Exp mm33 65° Exp mm44 65° 0.0

-0.5

Muller-matrix elements -1.0

-1.5 400 450 500 550 600 650 Wavelength (nm)

Figure 6.18: Mueller matrix elements m12, m14, m22, m33 and m44 for beetle´s right wing with double chiral layer and at angle of incidence of 650.

Wavelength Refractive Refractive Refractive λ index n0 index ∆ n index neo 400 1.490 0.977 2.467 450 1.472 0.877 2.349 500 1.459 0.805 2.264 550 1.449 0.752 2.201 600 1.441 0.711 2.153 650 1.436 0.679 2.116 Table 6.12 Wavelength of the incident light and the refractive indices of the first chiral layer as no with constants A0=1.403, B0=0.014, ∆n with constants A1=0.498, B1=0.099 and 0 ne with nneo=+∆ n for right wing of beetle at angle of incidence of 65

Figure 6.18 shows versus wavelength, the graph for the values of normalized Mueller matrix elements of m11, m12, m22, m33 and m44 and a curve showing the model fit values. Table 6.12 shows versus wavelength the refractive indices no, ∆no and ne with constants of Cauchy layers as A0, B0 and A1, B1 of first chiral layer. Table 6.13 is for values of the refractive indices with constants A2, B2 and A3, B3 of the second chiral layer and A4, B4 for the top layer for beetle’s right 57 wing at angle of incidence of 650. Table 6.14 shows thicknesses of first and second chiral layer as 3.51 µm and 3.76 µm with the thickness of top layer as 0.82 µm. In each chiral layer, the number of layers are 22, the twist of chiral starts at Φ0 with value of 3 and number of turns are 8. The mean square error (MSE) is 46.5. For the first chiral layer thickness is 3.5±0.1 µm, second chiral layer thickness is 3.76±0.04 µm and for top layer thickness is 0.82±0.02 µm. The thickness non-uniformity of uniaxial layer is 14%.

Wavelength Refractive Refractive Refractive Refractive λ index n1 index ∆ n1 index ne1 index nt 400 1.266 0.046 1.312 1.236 450 1.407 0.157 1.565 1.202 500 1.509 0.237 1.745 1.177 550 1.584 0.296 1.879 1.159 600 1.641 0.340 1.981 1.146 650 1.685 0.375 2.061 1.136

Table 6.13: Wavelength of the incident light and the refractive indices of the second Chiral layer as n1 with constants A2=1.941, B2=-0.108 ∆n1 with constants A3=0.576,

B3=-0.096 and ne1 with nneo11=+∆ n 1 and the refractive index of the top layer nt with 0 constants A4=1.178, B4=0.059 for right wing of beetle at angle of incidence of 65

2.80 neone 2.60 no no 2.40 2.20 2.00 1.80 1.60 Refractive Index Refractive 1.40 1.20 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 6.19: First chiral layer refractive indices of the beetle´s right wing

58 Chapter 6: Results and Discussion

2.20 neone1 2.00 non1

1.80

1.60

1.40

Refractive Index Refractive 1.20

1.00 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 6.20: Second chiral layer refractive indices of the beetle´s right wing

1.26 nt 1.24 nt

1.22

1.20

1.18

1.16

Refractive Index 1.14

1.12 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 6.21: Top layer refractive indices of the beetle´s right wing

Thickness Angle Number Thickness Angle Number Thickness MSE dc1 Φo of turns dc2 Φo of turns dt µm µm µm 3.51 -28 8 3.76 -26 8 0.82 72.7

Table 6.14: Thickness of the first Cauchy layer dc1, second Cauchy layer dc2 and the top layer dt,twist of chiral layerΦ0, number of turns in each layer and number of layers in each chiral layer for right wing of beetle at angle of incidence of 650

Chapter 7

Conclusions and Future Prospects

This chapter presents the conclusions we have come to, both regarding the methods and modeling of nanostructures in the beetle Cetonia Aurota. • It is possible to find the Mueller matrix elements for different parts of the beetle like the scutellum, the head, the left wing, and the right wing in the wavelength range between 400-800 nm at the angle of incidence of 40º to 70º. • The values of the Mueller matrix elements are different for the different body part of the beetles, although their visual colour is the same metallic green with different hue. • Modeling is done with single and double chiral layers. The double layer gives lower MSE. • The thickness of the exocuticle layer of beetle is around 10 µm. • The refractive index of the top layer is generally lower then the refractive indices of the first and second chiral layer.

We also suggest some directions for the future work in this area.

59 60 Chapter 7: Conclusions and Future Prospects

• In this thesis polarization effects are observed in the wavelength of the visible light ranges from 400-800 nm. In future, work can be done in the infra-red and ultra-violet range. • The metallic green colour of the beetles helps them to camouflage through the reflection of its surroundings. The reflection of specific wavelengths from the beetle’s cuticle may be related to the spectral sensitivity of the beetles. In future, more work can be done on the wavelength sensitivity. It is needed to explore the possible link between the reflected spectra and the peak sensitivities of the photoreceptors. • Some arthropods with compound eye designs are known to have sensitivity to polarized light at ultra-violet wavelengths. In future, work can also be done on finding possible reflections at infra-red wavelengths. It is also of interest to further investigate metallic-shiny scarab beetles by circular imaging polarimetry in order to get more information about the circular polarization patterns. • Biologist can find out whether the metallic-shiny, circularly polarizing scarab beetles can perceive circular polarization and whether they can distinguish between linear and circular polarization. It means to find out the presence of quarter-wave retarder in beetle’s eyes. • More work can be done about colour perception and the implications of colour on the behavior of these beetles as we know that the scarab beetles produce a bright, brilliant, metal colour which is circularly polarized.

• The final goal might be to commercialize this research by fabrication of tunable micro-mirrors for optical applications.

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61 62 Refrences

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[18]. R.M.A. Azzam, “Photopolarimetric measurement of the Muller matrix by Fourier analysis of a single detected signal”,Opt. Lett. 2 (1978) 148-150.

[19]. K. Ichimoto, K. Shinoda, T. Yamamoto, J. Kiyohara, “Photopolarimetric Measurement System of Muller Matrix with dual rotating waveplates”, Publ. Natl. Astron. Obs. Japan, 9 (2006) 11-19.

[20]. L.D. Silva, I. Hodgkinson, P. Murray, M. Arnold, J. Leader, A. Mcnaughton, “Natural and Nanoengineered Chiral Reflectors: Structural Color of Manuka Beetles and Titania Coatings”, Taylor&Francis Group, Electromagnetics, 25 (2005) 391-408. 63

[21]. H. Arwin, J. Landin, K. Järrendahl, “Optical active cuticle structures in the beetle Cetonia aurata”, Manuscript, Linkoping University, Sweden

[22]. Wikipedia, “Liquid crystal” , http://en.wikipedia.org/wiki/Liquid_crystal

[23]. User manual, “Liquid Crystal Modeling, J.A.Woollam Co., Inc., Nebraska, U.S.A.

Appendix

2.0 Model Fi t Exp mm12 45° 1.0 Exp mm14 45° Exp mm22 45° Exp mm33 45° Exp mm44 45° 0.0

-1.0 Muller-matrix elements

-2.0 400 450 500 550 600 650 Wavelength (nm)

Figure 1: Experimental Muller matrix elements m12, m14, m22, m33 and m44 for the left wing of the beetle and optical model data with double chiral layer and at an angle of incidence of 450

65 66 Appendix

Wavelength Refractive Refractive Refractive λ index n0 index ∆ n index ne 400 1.716 0.0067 1.723 450 1.698 0.0060 1.704 500 1.685 0.0055 1.690 550 1.675 0.0052 1.680 600 1.668 0.0049 1.673 650 1.662 0.0047 1.667 Table1: Wavelength of the incident light and the refractive indices of the first Chiral layer as no with constants A0=1.629, B0=0.014, A1=0.003, B1=0.099 and ne 0 with nneo=+∆ n.for left wing of beetle at angle of incidence of 45 .

1.73 neone 1.72 nono 1.71 1.70 1.69 1.68

Refractive Index Refractive 1.67 1.66 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 2: First Chiral layer’s refractive indices of the beetle´s left wing

Wavelength Refractive Refractive Refractive Refractive λ index n1 index ∆ n1 index ne1 index nt 400 1.448 -0.137 1.311 1.592 450 1.539 0.136 1.674 1.457 500 1.604 0.331 1.934 1.360 550 1.652 0.475 2.127 1.289 600 1.689 0.584 2.273 1.234 650 1.717 0.669 2.387 1.192

Table2: Wavelength of the incident light and the refractive indices of the second Chiral layer as n1 with constants A2=1.881,B2=-0.069 ∆n1 with constants A3=1.162,B3=-0.116 and 67

ne1 with nneo11=+∆ n 1 and the refractive index of the top layer nt with constants 0 A4=0.948, B4=0.103 for left wing of beetle at angle of incidence of 45 .

2.40 neone1 2.20 non1 2.00 1.80 1.60 1.40

Refractive Index Refractive 1.20 1.00 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 3: Second Chiral layer’s refractive indices of the beetle´s left wing

1.70 ntn 1.60 t

1.50

1.40

1.30

1.20 Refractive Index Refractive 1.10 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 4: Top layer’s refractive indices of the beetle´s left wing

68 Appendix

Thickness Angle Number Thickness Angle Number Thickness MSE dc1 Φo of turns dc2 Φo of turns dt µm µm µm 4.36 -20 8 3.22 -20 8 0.61 129.8

Table 3: Thickness of the first Cauchy layer dc1, second Cauchy layer dc2 and the top layer dt,twist of chiral layerΦ0,number of turns in each layer and number of layers in each chiral layer for left wing of beetle at angle of incidence of 450. The thickness of first Cauchy layer is 4.4±0.1 µm, the second Cauchy layer is 3.22±0.08 µm and thickness of top dielectric layer is 0.61±0.02 µm. The thickness non-uniformity of the uniaxial layer is 10 %.There are 22 sub-layers in each Cauchy layer.

2.0 Model Fi t 1.5 Exp mm12 50° Exp mm14 50° 1.0 Exp mm22 50° Exp mm33 50° 0.5 Exp mm44 50°

0.0

-0.5

Muller-matrix elements -1.0

-1.5 400 450 500 550 600 650 Wavelength (nm)

Figure 5: : Experimental Muller matrix elements m12, m14, m22, m33 and m44 for the right wing of the beetle and optical model data with double chiral layer and at an angle of incidence of 500

Wavelength Refractive Refractive Refractive λ index n0 index ∆ n index ne 400 1.708 -0.0017 1.706 450 1.689 -0.0015 1.688 500 1.676 -0.0014 1.675 550 1.667 -0.0013 1.665 600 1.659 -0.0012 1.658 650 1.653 -0.0011 1.652 Table4: Wavelength of the incident light and the refractive indices of the first Chiral layer as no with constants A0=1.620, B0=0.014, A1=-0.0009, B1=0.099 and ne 0 with nneo=+∆ n.for right wing of beetle at angle of incidence of 50 .

1.71 neone 1.70 nono

1.69

1.68

1.67

1.66 Refractive Index

1.65 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 0: First Chiral layer’s refractive indices of the beetle´s right wing

Wavelength Refractive Refractive Refractive Refractive λ index n1 index ∆ n1 index ne1 index nt 400 1.509 -0.125 1.384 1.539 450 1.577 0.156 1.733 1.447 500 1.626 0.357 1.982 1.381 550 1.661 0.506 2.167 1.332 600 1.689 0.619 2.308 1.295 650 1.709 0.707 2.397 1.266

Table5: Wavelength of the incident light and the refractive indices of the second Chiral layer as n1 with constants A2=1.832,B2=-0.052 ∆n1 with constants A3=1.214,B3=-0.115 and ne1 with nneo11=+∆ n 1 and the refractive index of the top layer nt with constants 0 A4=1.099, B4=0.070 for right wing of beetle at angle of incidence of 50 .

70 Appendix

2.40 neone1 n 2.20 no1 2.00 1.80 1.60 1.40 Refractive Index Refractive 1.20 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 7: Second Chiral layer’s refractive indices of the beetle´s right wing

1.60

1.55 nntt 1.50 1.45 1.40 1.35 1.30

Refractive Index Refractive 1.25 1.20 400 425 450 475 500 525 550 575 600 625 650 Wavelength (nm)

Figure 8: Top layer’s refractive indices of the beetle´s sright wing

Thickness Angle Number Thickness Angle Number Thickness MSE dc1 Φo of turns dc2 Φo of turns dt µm µm µm 4.35 -23 8 3.20 -23 8 0.59 105.4

Table 6: Thickness of the first Cauchy layer dc1, second Cauchy layer dc2 and the top layer dt,twist of chiral layerΦ0,number of turns in each layer and number of layers in each chiral layer for right wing of beetle at angle of incidence of 500. 71

The thickness of first Cauchy layer is 4.4±0.1 µm, the second Cauchy layer is 3.2±0.1 µm and thickness of top dielectric layer is 0.59±0.008 µm. The thickness non-uniformity of the uniaxial layer is 10%. There are 22 sub-layers in each Cauchy layer.