Ghost ellipsometry using classical light

Antti Hannonen

Master of Science ThesisApril 2016

Department of Physics and MathematicsUniversity of Eastern Finland

Antti Hannonen Ghost ellipsometry using classical light,55 pages.

University of Eastern Finland

Masters Degree Program in Photonics

Supervisors Associate Professor Tero SetalaProfessor Ari T. Friberg

Abstract

This thesis presents a new interferometric technique for the measurement ofellipsometric information by the means of classical ghost imaging. The tech-nique relies on the use of light from a classical, spatially incoherent sourcewith Gaussian statistics. The standard ghost-imaging configuration is modi-fied for reflection measurements and polarizers are added to control the stateof polarization of the beams. The results of the classical ellipsometer arealso compared to those obtained from its quantum counterpart, the so-calledentangled twin-photon ellipsometer. This ellipsometer uses polarization-entangled twin photons, generated by process of spontaneous parametricdown-conversion. In this thesis we show that entanglement is not neededfor ghost polarimetry.

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Contents

1 Introduction 1

2 Polarization and ellipsometry 42.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Jones formalism . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Reflection of polarized light . . . . . . . . . . . . . . . 72.1.3 Polarization properties of the source . . . . . . . . . . 9

2.2 Ellipsometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Definition of ellipsometric quantities . . . . . . . . . . 102.2.2 Ellipsometer configurations . . . . . . . . . . . . . . . . 12

3 Classical ghost imaging 153.1 Intensity and field correlations . . . . . . . . . . . . . . . . . . 16

3.1.1 Scalar fields . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Ghost imaging with classical scalar light . . . . . . . . . . . . 213.3 Comparison of classical and quantum regimes . . . . . . . . . 25

4 Ellipsometry using entangled photons 274.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Theoretical results . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Classical ghost ellipsometry 335.1 Basic configuration . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Using horizontally or vertically polarized light . . . . . . . . . 385.3 Adding one linear polarizer . . . . . . . . . . . . . . . . . . . . 41

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5.4 Adding a second linear polarizer . . . . . . . . . . . . . . . . . 435.4.1 Polarized light . . . . . . . . . . . . . . . . . . . . . . . 465.4.2 Unpolarized light . . . . . . . . . . . . . . . . . . . . . 47

6 Conclusions 50

References 52

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Chapter I

Introduction

Ghost diffraction and ghost imaging are techniques for obtaining the far-field diffraction pattern or the image of an object [1, 2]. These techniquesrely on the measurement of quantum coincidence rates or classical intensitycorrelations between two separate detectors, situated in two separate arms.First experimental results of quantum ghost diffraction were obtained over20 years ago [3, 4]. These experiments employed pairs of entangled photons,generated by spontaneous parametric down-conversion, consisting of the sig-nal and idler photon. In the experiments the pairs of entangled photonswere first spatially separated by sending them into two different directions,or arms. The double slit was placed in the so called test (signal) arm, whilethe reference (idle) arm was kept empty with the exception of the relevantoptical components. The separated photons then propagated through thearms and were finally detected by two faraway pointlike detectors. No in-terference pattern was observed behind the double slit, due to the lack ofspatial coherence of the used light. However, when the position on the ref-erence arm detector was scanned, the diffraction pattern could be obtainedfrom the coincidence count. This surprising phenomenon was given the nameghost diffraction, as the pattern was retrieved by only moving the detectorin the arm with no slit. Not long after the results of ghost diffraction, theobservations were expanded to the case of ghost imaging [5]. A high qualityimage of an object was obtained by measuring the coincidence counts as afunction of the reference-arm detector position.

The early results for ghost imaging and diffraction emphasised the role ofquantum entanglement. Some [6,7] even postulated that entanglement was acrucial feature needed for ghost imaging and diffraction to work. Later, aftersome debate, this was proven untrue both theoretically [8,9] and experimen-tally [10–13]. Today it is clear that classical ghost imaging can emulate mostof the features of quantum ghost imaging by using either correlated [2, 11]

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or thermal (incoherent) light [2, 12]. The main exception is the difference invisibility (contrast) of information in the coincidence regime [8, 9]. Becauseof the similarity and the advantages brought on by bright, readily-available,and simple classical light sources, classical ghost imaging keeps attractingconsiderable attention and research. This research includes imaging of tem-poral objects [14–16], imaging of pure phase objects [17], imaging troughaberrations [18] and polarimetry by classical ghost diffraction [19].

This thesis continues the research relating to classical ghost imaging bypresenting the so-called classical ghost ellipsometer. This device uses lightfrom a classical spatially incoherent source, whose statistics is described byGaussian random process. The classical ghost ellipsometer is designed toobtain ellipsometric data from a sample by the means of correlation mea-surements. For this purpose it employs a modified classical ghost-imagingconfiguration, designed for reflection measurements and with added polar-ization controlling elements. In this configuration an object with polariza-tion dependent reflectivities rs and rp is placed in the test arm. Light fromthe classical source is split by a beam splitter and directed to the two arms.The light in the reference arm propagates directly to the faraway pointlikedetector. In turn, the light in the test arm hits the object and finally thereflected light is directed to the test-arm detector. The correlations betweenintensity fluctuations at the two detectors are analysed to obtain the ellip-sometric data. The analysis is performed by using electromagnetic theory ofoptical coherence, with the help of Jones formalism and standard definitionsof ellipsometry.

The studied configuration is a classical counterpart of the previously re-ported entangled twin-photon ellipsometer [20, 21]. The twin-photon ellip-someter used pairs of polarization-entangled photons to measure ellipsometricinformation. However, the main aim of the thesis is to show the underlyingresults of the entangled twin-photon ellipsometer are obtainable with the useof classical, spatially incoherent source. This means that the results are notreliant on quantum entanglement in analogy to classical ghost diffraction andghost imaging.

This thesis is organized as follows. Chapter 2 recalls the basic formal-ism used in the presentation of polarization and reflection of light. It alsointroduces the idea behind ellipsometry, defines the measured ellipsometricquantities, and describes a few of the configurations used in classical ellipsom-etry. Next Chapt. 3 focuses on the mathematics necessary for representingthe intensity and field correlations of fields obeying Gaussian statistics. Theseresults form the foundation on which all of the theoretical work in this thesisis based. As an example, these result are used to analyse a standard classicalghost-imaging configuration. The following Chapts. 4 and 5 form the core

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of the thesis. Chapter 4 is used to describe the already briefly mentionedentangled twin-photon ellipsometer in detail. After this Chapt. 5 introducesthe classical ghost ellipsometer. its analysis, and lastly shows that the classi-cal ghost ellipsometer can obtain similar results as its quantum counterpart.Finally Chapt. 6 is reserved for summary of the main results and conclusions.

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Chapter II

Polarization and ellipsometry

Ellipsometry is defined as a non-perturbing and non-destructive techniquethat is used in the characterization of surfaces, interfaces, and thin films,by measuring the change in the state of polarization upon reflection of lightfrom them [22–24]. Ellipsometry probes the surface of a sample by usinga collimated beam of monochromatic or quasi-monochromatic light, whosepolarization states are know. After reflection from the sample, the polariza-tion state of the reflected light is analysed. From the incident and reflectedstates of polarization, the ratio of reflection coefficients absolute amplitudestogether with the relative phase difference are calculated. Using this informa-tion the structural and optical properties of the sample interface region canthen be calculated, by using electromagnetic theory and appropriate materialmodels [23]. Because the aforementioned ellipsometric analysis relies on themeasurement of polarization states, the modelling of polarized light and po-larization dependency of light sources and interfaces are of great importance.Therefore this thesis begins by focusing on the mathematical representationof the polarized light and its interactions with materials.

The first section of this chapter discusses the general polarization prop-erties of light. This section focuses on the modelling of polarized light andoptical components with the Jones formalism [25, 26]. After this reflectionand transmission of polarized light and the polarization properties of thelight source are also discussed. The second section of this chapter focuses onellipsometry. In this section the studied ellipsometric parameters are definedin detail and basic ellipsometric configurations are discussed.

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2.1 Polarization

Ellipsometry is based on the study of polarization change in the reflectedand/or transmitted light. Because of this, the modelling of polarization prop-erties is of core importance to the study of ellipsometry.

This section focuses on the mathematical methods used to describe po-larized light. Firstly, we recall the modelling of polarized light and linearoptical elements by Jones formalism. Secondly, the reflection and transmis-sion of light is modelled using electromagnetic theory. In the end of thesection we introduce the polarization matrix, used in the representation ofpolarization properties of electromagnetic fields.

2.1.1 Jones formalism

The spectral electromagnetic coherence theory of light can be employed inthe analysis of the classical situations of ghost imaging. In this case the mostgeneral way to express the complex amplitude of a beam-like polarized planeharmonic wave is as

E(r, ω) = sEs(r, ω) + pEp(r, ω), (2.1)

or in the column vector form

E(r, ω) =

[Es(r, ω)Ep(r, ω)

], (2.2)

where r denotes a position vector, ω stands for angular frequency and stogether with p are orthogonal unit vectors. The elements Es(r, ω) andEp(r, ω) denote the amplitudes of the s and p polarization components ofthe wave. Both of these components can be complex and can be expressedin an exponential form as

Es(r, ω) = |Es|exp(iξs), (2.3)

and

Ep(r, ω) = |Ep|exp(iξp), (2.4)

where we have separated the absolute amplitude and the phase ξ of thecomponents. Equations (2.3) and (2.4) imply that the column vector inEq. (2.2) can be written as

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E(r, ω) =

[|Es|exp(iξs)|Ep|exp(iξp)

]. (2.5)

The above vector is know as the Jones vector. Usually there is no need for theinformation relating to the exact amplitude and phase. Because of this, theconvention is to use the normalized form of the Jones vector, which containsall information on the polarization state. The normalization is performed bydividing both vector components by a scalar, such that the sum of the squaresof the components equals unity (|Es(r, ω)|2 + |Ep(r, ω)|2 = 1). This methodnormalizes the intensity of light to unity, as it is proportional to the squaresof the absolute values of the two components [25]. When the normalizationis performed we get

es(r, ω) =

[10

], (2.6)

for s polarized light and

ep(r, ω) =

[01

], (2.7)

for p polarized light. In the case of polarization state of 45◦ with respect tos polarization we get

e45◦(r, ω) =1√2

[11

]. (2.8)

As can be seen, the Jones vectors simplify the expression of different polar-ization states considerably.

The other benefit of the Jones formalism is in the way by which linear op-tical elements can be characterized. When a polarized beam described by itsJones vector Ein(r, ω) passes trough a linear optical element, the transmittedfield can be expressed as

Eout(r, ω) = T(r, ω)Ein(r, ω). (2.9)

Here

T(r, ω) =

[Tss(r, ω) Tsp(r, ω)Tps(r, ω) Tpp(r, ω)

], (2.10)

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is the optical element’s Jones matrix (transmission or transformation matrix).From this definition it can be seen that the Jones matrices are mathemati-cal operators that act on Jones vectors and produce new Jones vectors. Ifthe field passes trough multiple elements T1(r, ω), T2(r, ω)...TN(r, ω), theaforementioned can be expanded to

Eout(r, ω) = TN(r, ω)...T2(r, ω)T1(r, ω)Ein(r, ω). (2.11)

Optical elements that can be characterized by the Jones formalism includelinear polarizers, circular polarizers, phase retarders, isotropic phase chang-ers and isotropic absorbers [25, 26]. For example, a linear polarizer with atransmission axis of θ with respect to s polarization can expressed as

P(θ) =

[cos2θ sinθcosθ

sinθcosθ sin2θ

], (2.12)

which is a real and symmetric matrix.

2.1.2 Reflection of polarized light

The modelling of reflection and transmission at an interface of two differentmaterials, can be performed by employing electromagnetic theory. The elec-tromagnetic boundary conditions describe the interdependence of complex-valued amplitudes of electromagnetic waves at both sides of the interface [27].

The reflection and transmission coefficients, r and t, respectively, are de-fined as complex amplitude ratios

r =Er

Ein

, (2.13)

and

t =Et

Ein

, (2.14)

where Ein, Er, and Et are the incident, reflected, and transmitted electricfield amplitudes of the monochromatic plane waves, respectively. The aboveratios are dependent on the polarization state of light. However, whateveris the polarization state, the analysis of reflection and transmission can beperformed by dividing the problem into the consideration of the componentsperpendicular and parallel to the plane-of-incidence. After this, one canemploy the appropriate boundary conditions and calculate the amplituderatios. In the case where the electric field vector is perpendicular to the

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plane-of-incidence (TE polarization), the amplitude ratios can be expressedas

rs =n1cosθ1 − n2cosθ2n1cosθ1 + n2cosθ2

, (2.15)

and

ts =2n1cosθ1

n1cosθ1 + n2cosθ2, (2.16)

where θ1 is the angle of incidence, θ2 refers to the angle of transmission and n1

together with n2 represent the refractive indices in the respective mediums.Equations (2.15) and (2.16) are two of the so called Fresnel equations [25].The other two Fresnel equations are derived in the case where the electricfield vector is parallel to the plane-of-incidence (TM polarization). Whenthe proper boundary conditions are employed, the Fresnel equations take theforms

rp =n2cosθ1 − n1cosθ2n1cosθ2 + n2cosθ1

, (2.17)

and

tp =2n1cosθ1

n1cosθ2 + n2cosθ1. (2.18)

The Fresnel equations presented in Eqs. (2.15–2.18) are derived for the caseof linear, isotropic, homogeneous and dielectric media. The exact forms of theFresnel equations depends on the choice of formalism and sign convention.Here we have used the same conventions that are used in reference [25].

Often it is customary to express the above Fresnel equations in terms ofreflectance and transmittance. They are the fractions of the incident andtransmitted power, respectively. Since this is proportional to the square ofthe absolute amplitude, and the incident and reflected waves propagate inthe same medium, the reflectance is expressed as

Ri = |ri|2, i ∈ {s, p}. (2.19)

In the other hand, the transmittance takes the form of

Ti =n2cosθ2n1cosθ1

|ri|2, i ∈ {s, p}. (2.20)

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Here the mediums and directions to which the light travels are different, andbecause of this, we have additional terms in the equation.

2.1.3 Polarization properties of the source

In the case of fluctuating quasi-monochromatic electromagnetic vector field,the analysis of the polarization properties can be carried out in terms of the2 × 2 correlation matrix. This matrix is known as the polarization matrixand is expressed as

Φ(r, ω) = 〈E∗(r, ω)ET(r, ω)〉 =

[Φss(r, ω) Φsp(r, ω)Φps(r, ω) Φpp(r, ω)

], (2.21)

where T refers to the transpose of a matrix, the asterisk (*) denotes thecomplex conjugate, and angular brackets stands for an ensemble averageover monochromatic realisations. The field E(r, ω) in the above equation is asingle monochromatic realization of the fluctuating field, that is presented interms of the two orthogonal s and p polarization components, as in Eq. (2.2).The above implies that the matrix elements in Eq. (2.21) are of form

Φij(r, ω) = 〈E∗i (r, ω)Ej(r, ω)〉, i, j ∈ {s, p}, (2.22)

referring to the s and p polarization components of the field. The diagonalelements of the matrix are proportional to the intensity of the s and p com-ponent, as intensity is proportional to the square of the absolute amplitude.Hence, the off-diagonal elements can be thought analogous to the mutualintensity of the components. By normalizing the off-diagonal elements, theycan be expressed as

µij(r, ω) =Φij(r, ω)√

Φii(r, ω)Φjj(r, ω). (2.23)

From this it can be then shown that

0 ≤ |µij(r, ω)| ≤ 1, (2.24)

for i 6= j [28]. It is now clear that |µij(r, ω)| can be considered as the degreeof correlation that exist between the field components at a single point.

Let us next consider the case of unpolarized light (natural light) as anexample [28]. As can be easily understood, in this case the correlation be-tween field components is zero. According to Eq. (2.23) the off-diagonalcomponents of the polarization matrix are then

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Φps(r, ω) = Φsp(r, ω) = 0. (2.25)

Furthermore, the intensity of s and p polarized components are the same

Φss(r, ω) = Φpp(r, ω). (2.26)

These imply that for an unpolarized light beam, the polarization matrix hasthe form

Φ(r, ω) = Φss(r, ω)

[1 00 1

], (2.27)

i.e., it is proportional to the 2× 2 unit matrix [28].

2.2 Ellipsometry

Strictly mathematically, ellipsometry can be defined as a measurement of thecomplex ratios of the Fresnel reflection coefficients, rs/rp, and/or transmis-sion coefficients, ts/tp, that were defined in Sec. 2.1.2 [24]. By using thisdefinition, it can be shown that the measurement results in ellipsometry canbe expressed in terms of two ellipsometric quantities Ψ and ∆ [22–24,29].

The current part of the thesis is used to define the ellipsometric quantitiesΨ and ∆, with the help of the results of the previous section. In additionto this, we describe a general approach that is behind basic ellipsometricconfigurations.

2.2.1 Definition of ellipsometric quantities

The complex reflection coefficients presented in Eqs. (2.15) and (2.17), canbe expressed in an exponential form as

rs = |rs|exp(iφrs), (2.28)

and

rp = |rp|exp(iφrp), (2.29)

where the absolute amplitude and phase information have been separated(cf. Sec. 2.1.1). Using similar reasoning, the complex transmission coeffi-cients presented in Eqs. (2.16) and (2.18) can be expressed as

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ts = |ts|exp(iφts), (2.30)

and

tp = |tp|exp(iφtp). (2.31)

The above forms of the Fresnel reflection and transmission coefficients opena way to characterize the measurements in terms of two ellipsometric angles,also referred to as the ellipsometric parameters, and denoted by Ψ and ∆.

Because ellipsometry involves pure polarization measurements, only therelative amplitudes and phase are relevant in ellipsometric measurements.Hence, the first parameter Ψ is defined as the ratio of the amplitudes in theform

tanΨr =|rp||rs|

, 0 ≤ Ψr ≤ π/2, (2.32)

for reflected light, and

tanΨt =|tp||ts|

, 0 ≤ Ψt ≤ π/2, (2.33)

for transmitted light. As with the amplitudes, we are not interested in theindividual phases, which would be difficult to measure. Hence, we define thesecond phase parameter ∆ as

∆r = (φrp − φrs), −π ≤ ∆r ≤ π, (2.34)

for reflected light, and

∆t = (φtp − φts), −π ≤ ∆r ≤ π, (2.35)

for transmitted light. Using Eqs. (2.32–2.35) the complex ratios of Fresnelreflection and transmission coefficients can finally be expressed in terms of Ψand ∆ as

rprs

= tanΨrexp(i∆r), (2.36)

and

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tpts

= tanΨtexp(i∆t), (2.37)

respectively. Both of the ellipsometric parameters Ψ and ∆ are related tothe light’s ellipse of polarization and hence called ellipsometric angles. Theellipse of polarization is the ellipse which the locus of the electric field vectorof elliptically polarized light traces in the plane perpendicular to its directionof propagation [25]. The ellipsometric angles can be used to fully describethis polarization ellipse. This leads to another definition of ellipsometry, asthe study of the optical and structural properties of materials, based on themeasurements of the ellipse of polarization in the reflected or transmittedlight. The studied properties include thickness, refractive index and absorp-tion coefficient of the surface layer(s) [24]. From this study of polarizationellipses, the field of ellipsometry originally derived its name.

Here we have justified the ellipsometric parameters Ψ and ∆ for bothreflection and transmission ellipsometry. It is true that both configurationsare used, however, ellipsometry is mostly carried out on the reflected light.Because of this, we in this work focus on the reflection measurements, hencewhen the parameters Ψ or ∆ are mentioned later, they refer to the case ofreflected light, as defined in Eqs. (2.32) and (2.34).

2.2.2 Ellipsometer configurations

As mentioned in the beginning of this chapter, information about the po-larization states of the incident and reflected lights are needed to determinethe ellipsometric parameters Ψ and ∆. This implies that both the incidentand reflected lights polarization state must be modified or determined in thecourse of the measurements. Hence, the generic ellipsometer can be thoughtto consist of four optical components [23, 30]. These are the light source,detector, polarization state generator (PSG) and polarization state analyzer(PSA), as presented in Fig. 2.1. In reality the polarization state generatorand polarization state analyzer consist of combinations of a polarizers andretardation components and are not single elements.

As a specific example of the above generic ellipsometer, we can presentthe so called null ellipsometer. The null ellipsometer is one of the oldestand simplest of the ellipsometers. It was developed in the late 19th cen-tury by Drude and was in use for the first three quarters of the 20th cen-tury [31, 32]. It consist of a light source, polarizer, sample, compensator,analyzer and detector. These elements are arranged either in the polarizer-compensator-sample-analyzer (PCSA) or the polarizer-sample-compensator-analyzer (PSCA) arrangements. In the first arrangement the compensator

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Source

PSG Sample PSA

Detector

Figure 2.1: General approach to ellipsometry. An ellipsometerconsist of the light source, detector, polarization state gener-ator (PSG), and polarization state analyser (PSA). PSG andPSA in turn consist of a combination of polarizers and retar-dation components.

Sample ( , )

PC

A

D

φ

S

Figure 2.2: Null-ellipsometer consisting of a light source(S), linear polarizer (P), compensator (C, e.g., quarter-waveplate), linear polarization analyser (A), and an optical detec-tor (D). Angle φ represents the angle of incidence.

is before the sample and in the later after the sample, hence the acronyms.The PCSA configuration of the null ellipsometer is presented in Fig. 2.2.In the null ellipsometer the ellipsometric parameters are defined by rotating(changing the value) the polarizer and analyzer until the minimum intensityis obtained at the detector [23, 24]. This is done by first using the polarizerto transform the incoming light into a linearly polarized beam. After this thequarter-wave plate is used to generate a phase difference between the orthog-onal components. Next the reflection from the sample produces a additionalphase shift between the polarization components (parameter ∆), which canbe compensated by rotating the polarizer. The remaining light coming to thedetector is finally extinguished by rotating the analyzer. The aforementionedimplies, that in the null condition the polarizer angle gives the ellipsometricparameter ∆ while the analyzer angle yields the ellipsometric parameter Ψ.Employing this configuration all the wavelengths have to be measured indi-vidually. However, by using motors and computers the nulling process canbe automated.

The previous null ellipsometer is part of a subset of ellipsometers, inwhich the ellipsometric parameters are calculated by extinguishing the light

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intensity at the detector. The other subset of ellipsometers are the so-calledphotometric ellipsometers. Unlike the null ellipsometers, it is not intendedthat photometric ellipsometers have zero light intensity at the detector. Inthis case we are not interested in the nulling of intensity, rather the interestis in the variation of intensity. This variation is obtained by rotating oneor more of the optical components continually. For example, the previousnull ellipsometry configurations, PCSA and PSCA, can be transformed fromthe null measurement mode to the photometric mode by rotating the com-pensator continually. This will result in a sinusoidally varying intensity atthe detector, which can be Fourier analyzed to determine the ellipsometricparameters Ψ and ∆ [29, 30]. Other photometric ellipsometers include therotating analyzer ellipsometer (RAE) and the rotating polarizer ellipsometer(RPE). As the names indicate, here the rotating elements are the analyzerand polarizer, respectively. Which component is rotated depends largely onthe used light source, position of components, and what is measured (Ψ, ∆,Stokes parameters etc.) [30].

In the case of rotating analyzer and rotating polarizer ellipsometers, thecompensator is not essential and can be removed. This makes the RAE andRPE particularly suited for spectroscopic ellipsometry (SE), where measure-ments are performed at many different wavelengths [23]. In spectroscopicellipsometry a white light source along with a monochromator is employed.The monochromator is placed either before the polarizer or after the ana-lyzer [29]. In the case of spectroscopic measurements there is clearly a needto understand the wavelength dependence of optical elements. That is whythe RAE and RPE configurations used in SE do not employ compensators,which feature a distinct wavelength dependence.

Numerous other ellipsometer configurations have also been introducedthat are not presented here. These can simply use photo-elastic modulator(PEM) in place of rotating compensator or employ more elaborate rotatingPSA arrangements [23, 30]. However, they also include ellipsometers whosegeometry differs greatly from those discussed earlier [23]. These include theinterferometric ellipsometry configurations, of which two are studied later inthis thesis. The first is the non-classical entangled twin-photon ellipsometerand the other is the classical ghost ellipsometer (cf. Chapts. 4 and 5).

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Chapter III

Classical ghost imaging

Ghost imaging is a technique where an image of an object can be obtainedby measuring the photon coincidences or the correlations between intensityfluctuations at two distinct light detectors [17, 33]. The photon coincidencemeasurements are related to the case of quantum ghost imaging and themeasurements of correlations between intensity fluctuations to the case ofclassical ghost imaging [12]. The aforementioned description of the imagingsetup implies that the basic configuration consists of two distinct arms. Theobject to be imaged is placed in one arm and the light passing trough itis detected by a point-like single-pixel detector. The other detector, in theadjacent arm, is then actually used to obtain the spatial distribution of theobject by scanning another point-like detector. The latter detector never in-teracts with the object directly, hence leading to the name of ghost imaging.The ghost imaging technique can be directly extended to the case of ghostdiffraction, where the diffraction pattern of the object is obtained [8,9]. Laterin the main part of this thesis, it is also shown that the ghost imaging con-figuration can be modified for the case of ellipsometric measurements. Thisimplies that understanding ghost imaging forms the basis for the study ofthe classical ghost ellipsometer.

In this chapter the basic principle of classical ghost imaging is presentedand the differences between the classical and quantum ghost imaging are re-viewed. Before this, the intensity and field correlations of the fields obeyingGaussian statistics are studied. The understanding of the statistics of in-tensity fluctuations is needed to calculate the correlations between intensityfluctuations at the two arms. The analysis of this part focuses mainly onthe case of scalar fields. However, in the case of the intensity and field cor-relations, also the vector case is studied. The information presented in thischapter will be invaluable in understanding the classical ghost ellipsometerpresented later in this thesis.

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3.1 Intensity and field correlations

The first section of this chapter focuses on the study of intensity and field cor-relations of the fields obeying Gaussian statistics. The statistical informationrelating to these field correlations forms the basis of most of the analysis de-scribed in this thesis. This is because the classical spatially incoherent light,used as the source for the classical cases of ghost imaging and ellipsometry,is assumed to obey Gaussian statistics. This is strictly true for thermal lightand to a good approximation to a pseudo-thermal light. In addition to this,the correlation between the intensities in the two arms of the ghost imagingand ellipsometer setup is known to be responsible for the information relatedto the object. As a result, a clear understanding and a simple expression forthe correlation between the intensity fluctuations are needed.

This section first describes the intensity and field correlations of simplescalar fields obeying Gaussian statistics. After this the polarization proper-ties of the electromagnetic field are taken into account by studying vectorfields obeying Gaussian statistics. The results derived in the first part willbe mainly used in the analysis of the ghost-imaging configuration presentedlater in this chapter. The results derived in the later part form the founda-tion whereof the analysis of the classical ghost ellipsometer can be launchedin Chap. 5.

3.1.1 Scalar fields

All higher-order correlations of Gaussian variates, ∆xim , m ∈ {1, ..., n}, canbe expressed with the help of second-order correlations between pairs of vari-ates. This property is known as the Gaussian moment theorem [28] and itimplies that for any set of n indices i1, i2, ..., in

〈∆xi1∆xi2 ...∆xin〉 = 0, n odd,

〈∆xi1∆xi2 ...∆xin〉 =∑

(n−1)!!

〈∆xi1∆xi2〉...〈∆xi(n−1)∆xin〉, n even,

(3.1)

where the angular brackets denote the ensemble average and the sum is takenover all (n − 1)!! pairings. In the case of complex variates the Gaussianmoment theorem can be further represented as [28]

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〈∆z∗i1∆z∗i2...∆z∗iN ∆ziM ...∆zj1〉 = 0, N 6= M,

〈∆z∗i1∆z∗i2...∆z∗iN ∆ziM ...∆zj1〉 =

∑N !

〈∆z∗i1∆zj1〉...〈∆z∗iN

∆zjN 〉, N = M,

(3.2)

where the sum is taken over all N ! pairings. This property can now beused as a starting point from which a simple expression for the correlationbetween intensity fluctuations can be calculated for the fields with Gaussianstatistics.

In the space-frequency domain the Gaussian moment theorem implies,that when a (zero-mean) field is described by a random Gaussian process, itsspectral cross-correlation functions of order (M,N )

Υ(M,N)(r1, ..., rM+N ;ω1, ..., ωM+N) = 〈E∗(r1, ω1)...E∗(rM , ωM)

× E(rM+1, ωM+1)...E(rM+N , ωM+N)〉,(3.3)

can be expressed in terms of the lowest-order spectral cross-correlation func-tion Υ1,1 [28]. In Eq. (3.3) E(r, ω) represents the Fourier component of thefield at r. Mathematically the Gaussian moment theorem can be written as

Υ(M,N)(r1, ..., rM ; r′1, ..., r′N ;ω1, ..., ωM+N) = 0, N 6= M, (3.4)

and

Υ(M,N)(r1, ..., rM ; r′1, ..., r′N ;ω1, ..., ω2M)

=∑π

Υ(1,1)(ri1 , r′j1, ωi1 , ωj1)Υ

(1,1)(ri2 , r′j2, ωi2 , ωj2)...Υ

(1,1)(riM , r′jM, ωiM , ωjM ),

(3.5)

where π denotes summation over all the M ! possible permutations of thesubscripts. In the situation studied in this thesis N = M = 2. In this case,while replacing r′1 with r3 and r′2 with r4, Eq. (3.5) gives

Υ(2,2)(r1, r2, r3, r4;ω1, ω2, ω3, ω4) =Υ(1,1)(r1, r3;ω1, ω3)Υ(1,1)(r2, r4;ω2, ω4)

+ Υ(1,1)(r1, r4;ω1, ω4)Υ(1,1)(r2, r3;ω2, ω3),

(3.6)

If the field is next presumed stationary, the cross-correlation functions can

17

be written in terms of the cross-spectral density functions [28]. When this isdone Eq. (3.6) can be represented as

W (2,2)(r1, r2, r3, r4;ω2, ω3, ω4)δ(ω1 + ω2 − ω3 − ω4)

= W (1,1)(r1, r3;ω3)W(1,1)(r2, r4;ω4)δ(ω3 − ω1)δ(ω4 − ω2)

+W (1,1)(r1, r4;ω4)W(1,1)(r2, r3;ω3)δ(ω4 − ω1)δ(ω3 − ω2), (3.7)

where δ denotes the delta function. Furthermore assuming that r3 = r1,r4 = r2 and ω = ω1 = ω2 = ω3 = ω4, as in the cases encountered in thisthesis, the previous equation can be expressed in the form of

W (2,2)(r1, r2;ω)

= W (1,1)(r1, r1;ω)W (1,1)(r2, r2;ω) +W (1,1)(r1, r2;ω)W (1,1)(r2, r1;ω).(3.8)

Next by taking into account that the cross-spectral density can be denotedas a correlation function W (1,1)(r1, r2, ω) = 〈E∗(r1, ω)E(r2, ω)〉 over an en-semble of monochromatic realisations E(r, ω) [28], it may be seen that theabove equation can be represented as

W (2,2)(r1, r2;ω) = 〈|E(r1, ω)|2〉〈|E(r2, ω)|2〉+ |W (1,1)(r1, r2, ω)|2. (3.9)

Here the terms |E(r, ω)|2 are equal to the intensity of a realization I(r, ω).This implies that Eq. (3.9) can be written as

W (2,2)(r1, r2;ω) = 〈I(r1, ω)〉〈I(r2, ω)〉+ |W (1,1)(r1, r2, ω)|2, (3.10)

where 〈I(rj, ω)〉 represents the average intensity. Next by introducing inten-sity fluctuations in the general form of

∆Ij(rj, ω) = I(rj, ω)− 〈I(rj, ω)〉, (3.11)

it follows that

〈∆I1(r1, ω)∆I2(r2, ω)〉 = 〈I(r1, ω)I(r2, ω)〉 − 〈I(r1, ω)〉〈I(r2, ω)〉. (3.12)

The term 〈I(r1, ω)I(r2, ω)〉 can be expressed as

18

〈I(r1, ω)I(r2, ω)〉 = 〈E∗(r1, ω)E(r1, ω)E∗(r2, ω)E(r2, ω)〉 = W (2,2)(r1, r2, ω).(3.13)

Finally, it follows from Eqs. (3.10), (3.12) and (3.13), that the correlationbetween the intensity fluctuations takes on the form

〈∆I(r1, ω)∆I(r2, ω)〉 = |W (1,1)(r1, r2, ω)|2. (3.14)

The above equation implies that for light obeying Gaussian statistics thecorrelation between two intensity fluctuations can be expressed in the termsof the absolute square of the lowest-order cross-correlation function

W (1,1)(r1, r2, ω) = 〈E∗(r1, ω)E(r2, ω)〉. (3.15)

This result forms the foundation on which the analysis of classical ghostimaging is based.

3.1.2 Vector fields

When the polarization effects of an electromagnetic field are studied, both theintensity correlations of the orthogonal components need to be considered. Inthis subsection the vector field is examined and a generalized formula for thecorrelation between intensity fluctuations at two specified points is derived.

A vector field, e.g., the electric field of a plane wave, can be presented as

E(r, ω) =

[Es(r, ω)Ep(r, ω)

]. (3.16)

The amplitudes Es(r, ω) and Ep(r, ω) in the previous equation represent theorthogonal s and p polarization components, respectively, of the wave. Thetotal intensities can be expressed in terms of these components as

I(r, ω) = Is(r, ω) + Ip(r, ω), (3.17)

where Is(r, ω) = E∗s (r, ω)Es(r, ω) and Ip(r, ω) = E∗p(r, ω)Ep(r, ω) are relatedto the two field components. Following this logic further, the intensity fluc-tuation [cf. Eg. (3.11)] of the vector field can be denoted by the formula

∆I(r, ω) = ∆Is(r, ω) + ∆Ip(r, ω), (3.18)

where ∆Ii(r, ω) = Ii(r, ω) − 〈Ii(r, ω)〉, i ∈ {s, p}. Using Eq. (3.18), it can

19

be deduced that the correlation between the intensity fluctuations at twopoints may be written in terms of the sum of all the intensity fluctuationcomponents

〈∆I(r1, ω)∆I(r2, ω)〉 = 〈∆Is(r1, ω)∆Is(r2, ω)〉+ 〈∆Is(r1, ω)∆Ip(r2, ω)〉+ 〈∆Ip(r1, ω)∆Is(r2, ω)〉+ 〈∆Ip(r1, ω)∆Ip(r2, ω)〉

=∑i

∑j

〈∆Ii(r1, ω)∆Ij(r2, ω)〉, (3.19)

with the notation i, j ∈ {s, p}. The above formula holds generally for allplanar fields. However, when the field’s statistical properties are those of aGaussian random process, each correlation term in the previous summationcan be expressed in the form of Eq. (3.14). Consequently Eg. (3.19) can bewritten as

〈∆I(r1, ω)∆I(r2, ω)〉 = |Wss(r1, r2, ω)|2 + |Wsp(r1, r2, ω)|2

+ |Wps(r1, r2, ω)|2 + |Wpp(r1, r2, ω)|2

=∑i

∑j

|Wij(r1, r2, ω)|2. (3.20)

The above equation is a generalization to the electromagnetic context ofEq. (3.14) presented at the end of the previous subsection. It implies thatthe correlation between two intensity fluctuations may finally be expressedas a sum, where the summation is performed over the absolute squared ele-ments of the lowest-order cross-correlation functions. These components arepolarization dependent and can be expressed in the form of

Wij(r1, r2, ω) = 〈Ei(r1, ω)Ej(r2, ω)〉, i ∈ {s, p}. (3.21)

Equation (3.20) may also be represented in a more compact form

〈∆I(r1, ω)∆I(r2, ω)〉 = tr[W†(r1, r2, ω)W(r1, r2, ω)], (3.22)

where † refers the Hermitian transpose, tr denotes trace, and

W(r1, r2, ω) =

[Wss(r1, r2, ω) Wsp(r1, r2, ω)Wps(r1, r2, ω) Wpp(r1, r2, ω)

], (3.23)

is the cross-spectral density matrix (CSDM) in the s and p polarization basis.The CSDM is a generalization of the 2× 2 polarization matrix, discussed in

20

Sec. 2.1.3. From Eq. (3.23) it can further be deduced that the result for thecorrelation between intensity fluctuations is also independent of the choice ofthe s- and p-directions, as it should be.

3.2 Ghost imaging with classical scalar light

This section outlines the basic scalar case of classical ghost imaging that usesclassical spatially incoherent light. The geometry is presented in Fig. 3.1.The ghost-imaging configuration consists of two arms, the so-called referenceand test arms. The light originating from the source is split into the twoarms by a beam splitter (BS). An object with a space-dependent transmit-tance function T(ρ′2) is then placed in the test arm. The beam of light in thetest arm propagates a distance of zb to the sample and after passing troughit propagates another distance of zc and is detected by a point-like singlepixel detector D2. On the other hand, the light directed to the referencearm propagates a distance of za and is then detected by a scanning point-likedetector D1. After the individual intensities are measured, the correlationbetween these intensities is studied. From the correlation between intensityfluctuations information related to the object’s transmittance is obtained, asis shown in the following analysis.

As was mentioned above, the correlation between the intensity fluctu-ations is known to be responsible for the information on the object beingstudied. By applying Eq. (3.11) to the studied ghost imaging configuration,the intensity fluctuations at the detectors can be expressed as

∆I(ρα, ω) = I(ρα, ω)− 〈I(ρα, ω)〉, (3.24)

where ρα and the subscript α ∈ {1, 2} refer to one of the two arms of theghost-imaging arrangement. Next by assuming that the statistics of the fieldis that of a Gaussian random process, the correlation between two intensityfluctuations is given by Eq. (3.14). In the present case this implies

〈∆I(ρ1, ω)∆I(ρ2, ω)〉 = |W (ρ1,ρ2, ω)|2, (3.25)

where

W (ρ1,ρ2, ω) = 〈E∗1(ρ1, ω)E2(ρ2, ω)〉, (3.26)

is the cross-spectral density function between the field E1(ρ1, ω) at the ref-erence arm detector and field E2(ρ2, ω) at the test arm detector.

21

SourceBS

I1(ρ1)

I2(ρ2)

ρ1

ρ2

ρ0

< I(ρ1) I(ρ2)>

za

zb

D1

D2

zc

T(ρ'2)ρ'2

Object

Figure 3.1: Basic approach to classical ghost imaging. Spa-tially incoherent scalar light is split into two arms in the beamsplitter (BS). These arms are the so-called test arm (subscript2) and reference arms arm (subscript 1). An object with aspace-dependent transmittance function T(ρ′2) is placed inthe test arm. The resulting intensities in the arms are mea-sured by two point-like detectors (D1, D2) and the correlationbetween these intensities is measured and studied. DetectorD1 in the reference arm is scanned while D2 in the test armis kept at a fixed position.

From Eqs. (3.25) and (3.26) it is clear, that an expression for the intensityfluctuations can be formed by calculating the propagated cross-spectral den-sity function. This can be accomplished with the help of the familiar paraxialpropagation law, known as Fresnel diffraction formula [34]. When this lawis applied to the case of the studied geometry, the field at the reference armdetector can be expressed as

E1(ρ1, ω) =−ik2πza

∫E0(ρ0, ω)exp

{ik

2

[(ρ1 − ρ0)

2

za

]}d2ρ0, (3.27)

and the field at the test arm detector as

E2(ρ2, ω) =−k2

(2π)2zbzc

∫ ∫T (ρ′2)E0(ρ

′0, ω)exp

{ik

2

[(ρ′2 − ρ′0)

2

zb

]}× exp

{ik

2

[(ρ2 − ρ′2)

2

zc

]}d2ρ′0d

2ρ′2. (3.28)

In these equations, E0(ρ0, ω) represents the source field, k = ω/c0 is the wave

22

number with c0 being the vacuum speed of light, and zj, j ∈ {a, b, c} is thepropagation distance. The constant exponential phase terms exp(ikzj) asso-ciated with the previous expressions has been omitted for the sake of brevity.The range of these and all subsequent integrations extend over the wholespace. The results given for the propagated field components imply that thecross-spectral density function presented in Eq. (3.26) can be presented as

W12(ρ1,ρ2, ω) =

∫∫W0(ρ0,ρ

′0, ω)K∗1(ρ1,ρ0, ω)K2(ρ2,ρ

′0, ω)d2ρ0d

2ρ′0,

(3.29)

where W0(ρ0,ρ′0, ω) = 〈E∗0(ρ0, ω)E0(ρ

′0, ω)〉 corresponds to the cross-spectral

density function of the source and the kernels K1 together with K2, for thereference and test arm, respectively, are of the form

K1(ρ1,ρ0, ω) =−ik2πza

exp

{ik

2

[(ρ1 − ρ0)

2

za

]}, (3.30)

and

K2(ρ2,ρ′0, ω) =

−k2

(2π)2zbzc

∫T (ρ′2)exp

{ik

2

[(ρ′2 − ρ′0)

2

zb

]}× exp

{ik

2

[(ρ2 − ρ′2)

2

zc

]}d2ρ′2. (3.31)

Equation (3.29) can be developed when we first assume the source to be aclassical spatially incoherent source that is characterized by the cross-spectraldensity function of the form

W0(ρ0,ρ′0, ω) = 〈E∗0(ρ0, ω)E0(ρ

′0, ω)〉 = I0δ(ρ

′0 − ρ0), (3.32)

where δ denotes the two-dimensional Dirac delta function and I0 is a positiveconstant. The above assumption of spatial incoherence is very important inthe classical case of ghost imaging. Fundamentally, it is the spatial incoher-ence together with the Gaussian statistics of the random light that makes theclassical ghost-imaging configuration to work. As an added benefit, the as-sumption simplifies the mathematical expressions for the cross-spectral den-sity functions, as it assumes complete spatial incoherence. This means thatthe random fields at any two separate points of the source are uncorrelated.When the assumption in Eq. (3.32) is placed in Eq. (3.29) we get

23

W12(ρ1,ρ2, ω) = I0

∫K∗1(ρ1,ρ0, ω)K2(ρ2,ρ0, ω)d2ρ0, (3.33)

where it can be seen that the ρ′0 term has now been replaced by ρ0 in thekernels. Next a ghost-imaging condition for the propagation distances isdefined. This condition must be fulfilled to make the configuration to workas an imaging system. In the case of the geometry in Fig. 3.1 the conditionis of the form za = zb. When this condition is used in connection withEqs. (3.30), (3.31) and (3.33), the integrals can be calculated by rearrangingthe integration order and using the following properties of the delta function

δ(x1 − x2) =1

(2π)2

∫exp[i(x1 − x2) · k

]d2k, (3.34)

and

δ(ax) =δ(x)

|a|2, (3.35)

where x1, x2 and k are vectors and a is a scalar [35, 36]. After integrationthe solution for the cross-spectral density function is obtained as

W12(ρ1,ρ2, ω) =−ikI02πzc

T (ρ1)exp

{ik

2

[(ρ2 − ρ1)

2

zc

]}. (3.36)

Using the above result and Eq. (3.25), the correlation between the intensityfluctuations can finally be expressed as

〈∆I(ρ1, ω)∆I(ρ2, ω)〉 =

(kI02πzc

)2

|T (ρ1)|2. (3.37)

The above result indicates that the correlation between intensity fluctuationsdoes indeed contain an image of the object located in the test arm, as wasstated in the beginning of this section. This implies that the spatial profileof the object is obtained by scanning the detector in the reference arm, asthe test arm detector is kept fixed in the measurement. However, becauseof the absolute square in the transmittance term, no phase information isobtained. This means that a pure phase object cannot be imaged usingthe basic configuration of classical ghost imaging. Nonetheless there existconfigurations that can solve this problem [17, 37]. The limitations relatingto the retrieval of phase information will also be discussed in Chap. 5.

24

3.3 Comparison of classical and quantum regimes

As was shown in the previous section, the classical ghost-imaging configura-tion can be used to obtain an image of the studied object. This is not a newdiscovery, and it is found to be in a formal analogy with the case of quan-tum ghost imaging, demonstrating that quantum entanglement is not indeedneeded to obtain the desired results [8,9]. There are however differences be-tween the classical and quantum regimes. The obvious difference is in thetype of light used, the classical light wave is a spatially incoherent light withGaussian statistics, e.g., thermal light or pseudo-thermal light produced fromspatially coherent laser light by using a rotating diffuser. In turn, the quan-tum light consists of entangled photons generated by the nonlinear processof parametric down-conversion (PDC) (cf. Chap. 4). Nonetheless, the maindifference between the classical and quantum cases lies in the performance,mainly in the visibility of the information [12].

Visibility, or an image contrast, quantifies the relative difference betweenthe bright and dark areas of the picture. In the case of ghost imaging, thereare two different mathematical definitions that are commonly used for vis-ibility. In the case of the studied double-intensity correlation imaging, thefirst can be expressed as

V1 =gmax − gmin

gmax + gmin

, (3.38)

where gmax correspond to the average signal in the bright area and gmin

corresponds to the average signal in the dark area [38,39]. In the same case ofdouble-intensity correlation imaging, the second definition can be expressedas

V2 =〈∆I1∆I2〉max

〈I1I2〉max

=〈I1I2〉max − 〈I1〉〈I2〉

〈I1I2〉max

=gmax − 1

gmax

., (3.39)

where the Eq. (3.12) for the intensity fluctuations has been used as the basisof the definition [13, 38]. The last equality holds because 〈I1〉〈I2〉 representsan average background term. In both definitions the visibilities are nor-malised so that 0 ≤ V1 ≤ 1 and 0 ≤ V2 ≤ 1.

It can be shown that in the classical regime 〈∆I1∆I2〉 ≤ 〈I1〉〈I2〉 [8, 13].This implies that in the classical case visibility is never above 1/2, accordingto Eq. (3.39). Furthermore, in the quantum regime, it can be demonstratedthat the 〈∆I1∆I2〉/〈I1〉〈I2〉 is proportional to 1 + 1/〈n〉, where 〈n〉 is themean photon number per mode [8, 13]. These two facts imply that muchbetter visibility is always obtained for the quantum case, when the photon

25

number is small, i.e., in the coincidence regime. In this regime the visibilityof the quantum case approaches its maximum value of unity and the visibilityof the classical case is limited to a maximum value of 1/2. However, whenthe photon number is sufficiently large, the difference between the two casesis non-existent. Hence, better visibility is not a problem when the measure-ments are directed towards macroscopic stable objects, as there is time todo lengthy measurements with large number of data collections. However,visibility may pose a problem when time and/or sensitivity of measurementsare of great importance. This is especially true when it is taken into consid-eration that the visibility gives an estimate of the signal-to-noise ratio (SNR)of a ghost imaging configuration [13].

26

Chapter IV

Ellipsometry using entangled photons

To perform reliable and accurate measurements of ellipsometric parameters Ψand ∆, introduced in Sec. 2.2, classical ellipsometry requires well calibratedsources and detectors, or a well-characterized reference sample [20, 21, 29].These limitations are caused by the instability of classical light sources andthe imperfect nature of optical elements. To circumvent these limitations, it ispossible to use a nonclassical optical source, in an interferometric coincidence-detection scheme named as the entangled twin-photon ellipsometer. This el-lipsometer was first presented in papers [20] and [21], on which this chapteris largely based. The entangled twin-photon ellipsometer uses polarization-entangled twin photons to determine the ellipsometric parameters. The twinphotons are generated in a second-order nonlinear crystal through a processof type-II phase matched spontaneous parametric down-conversion. It canbe shown, that in contrast to the classical ellipsometers, the measurementsof entangled twin-photon ellipsometer do not rely on the calibration of thesource and detectors, or a reference sample.

This chapter is used to describe the entangled twin-photon ellipsometerand the theoretical results obtained by it. The first section focuses on thebasic configuration of the ellipsometer and after this describes the processof spontaneous parametric down-conversion. In the next section we demon-strate how the recorded coincidence rate yields the ellipsometric parameters.Finally in the end of this chapter the entangled twin-photon ellipsometer iscompared to its classical counterparts, by summarizing its advantages.

27

4.1 Configuration

Entangled twin-photon ellipsometer consists of a light source, nonlinear crys-tal (NLC), two linear polarizers and two single photon detectors, as illustratedin Fig. 4.1. These elements are arranged into an interferometric coincidence-detection scheme. In this scheme the light source is used as an optical pumpto generate successive pairs of entangled photons in the nonlinear crystal.These pairs of photons are then emitted simultaneously from the crystal intwo different directions, namely, the test and reference arms of the ellipsome-ter. The photons emitted to the test arm are first directed to the test sampleand after reflection to the linear polarizer A2 followed by the single-photondetector D2. In turn, the photons emitted to the reference arm are directedto the linear polarizer A1 directly followed by the single-photon detector D1.A coincidence circuit between the detectors is then used to register the coin-cidence rate Nc of multiple pairs of photons. From the coincidence rate theellipsometric parameters Ψ and ∆ can be calculated [see Sec. 4.2].

In this case we use the term coincidence rate, as the photon number inthe measurements is small, and therefore it would not be appropriate to usethe term correlation between intensity fluctuations. Same distinguishing fea-ture exists between classical and quantum ghost imaging, as discussed in theend of the previous chapter. Thus, due to its inherent nonclassical natureand geometry, the entangled twin-photon ellipsometer can be thought of asan extension of quantum ghost imaging. However, instead of imaging it isdesigned to perform ellipsometric measurements.

As its name suggests, the entangled twin-photon ellipsometer uses twopolarization-entangled photons to measure the ellipsometric parameters Ψand ∆. These photons are generated by the process of type-II phase matchednon-collinear spontaneous parametric down-conversion (SPDC). In the gen-eral process of spontaneous parametric down-conversion‡ part of the pump-beam photons of frequency ωpump, incident on the second-order nonlinearcrystal, are split into two photons of lower frequency and lower energy [40,41]. These photons are called the idler photon of frequency ωidler and thesignal photon of frequency ωsignal. According to energy and momentumconservation, or frequency and phase matching, the frequencies must obeyωpump=ωidler+ωsignal and the wave vectors satisfy kpump=kidler+ksignal. Thegeneral process of SPDC can be divided into two cases according to the usedphase matching condition, as depicted in Fig. 4.2. In the case of type-II phasematching [Fig. 4.2(b) and (c)], the down-converted signal and idler photonsare emitted into two cones, which have orthogonal polarization states [42].

‡Also called optical parametric generation or optical parametric fluorescence

28

Sample ( , )

D1

D2

A1

A2Source

NLC

Nc

Figure 4.1: Entangled twin-photon ellipsometer consisting ofa light source, nonlinear crystal (NLC), linear polarizers (A1,A2) and single-photon detectors (D1, D2). The type-II non-collinear crystal produces two entangled photons emitted tothe two arms of the ellipsometer. After this they pass troughthe linear polarizers and are measured at their respective de-tectors. From these measurements the coincident rate Nc isdeduced and studied. Here φ represents the angle of incidenceand Ψ together with ∆ represent the ellipsometric parametersintroduced in Sec. 2.2

The signal photon cone is ordinary polarized and the idler photon cone ex-traordinary polarized. This differs from the type-I phase matching wherethe created photons have the same polarization state and are emitted intoa single cone centered on the pump beam, as show in Fig. 4.2(a) [43]. Theopening angle of the cone(s) can differ greatly depending on the angle be-tween the crystal optic axis and the pump beam [44]. However, in the case oftype-II phase matched SPDC, there are two special situations. These are theso-called collinear and non-collinear cases, of which the non-collinear case isused in the entangled twin-photon ellipsometer. In the collinear situation thetwo orthogonally polarized cones are tangential to each other at the pumpbeam line, as show in Fig. 4.2(b). This state is not entangled, as it can bewritten as the product of the signal and idler photons states

|Ψ1〉 = |HV 〉, (4.1)

where H and V represent the horizontal (extraordinary) and vertical (ordi-nary) polarizations, respectively. In the non-collinear situation the two cones

29

intersect along two lines, as shown in Fig. 4.2(c). It can be seen that thename non-collinear originates from the fact that the two photons are emittedin two different directions, justifying the geometry in Fig. 4.1. In additionto this, the light along the two intersection lines can be described by purepolarization-entangled state

|Ψ2〉 =1√2

(|HV 〉+ exp(iβ)|V H〉), (4.2)

where the first ket (term) is for the signal photon and the second ket is forthe idler photon. As can be seen from Eq. (4.2) the state along the twodirections is indeed pure quantum state. When viewed separately the twophotons are unpolarized [45,46]. Additionally the overall phase shift betweenthe two kets is omitted, as it can be arbitrarily chosen by adjusting the NLC.The phase term β is due to the crystal birefringence that causes the signaland idler photon to emerge from the crystal with a relative time delay [44].The temporal compensation necessary to correct this time delay is done byadding an additional birefringent material in the signal and/or idler photonpath.

4.2 Theoretical results

By using a generalization of Jones matrix formalism, the coincidence rate Nc

of the entangled twin-photon ellipsometer can be shown to be [20,21]

Nc = A(tan2Ψsin2θ1cos2θ2 + cos2θ1sin2θ2

+ 2tanΨcos∆cosθ1cosθ2sinθ1sinθ2). (4.3)

Here A is a constant that includes the quantum efficiency of the detectorsand other parameters of the experimental setup, Ψ together with ∆ refer tothe ellipsometric parameters defined in Sec. 2.2.1, and θα, α ∈ {1, 2} denotesthe angle of the polarizers with respect to the horizontal direction.

From Eq. (4.3) results for the ellipsometric parameters can be obtainedusing minimum of three measurements. The simplest way to achieve this isby keeping θ2 fixed at 45◦, while adjusting θ1. For example, by first settingθ1 = 0◦ to get

Nc =1

2A, (4.4)

from which the constant A can be calculated. Second, by setting θ1 = 90◦ toobtain

30

Figure 4.2: (a) Type-I phase matched SPDC. (b) Type-IIphase matched collinear SPDC. (c) Type-II phase matchednon-collinear SPDC, used in the entangled twin-photon ellip-someter. The solid lines indicate the pump beam and thedashed lines the polarization-entangled states. The circlesdenote the cross sections of the light cones.

Nc =1

2Atan2Ψ, (4.5)

from which the first ellpsometric parameter Ψ can be calculated. Finally bysetting θ1 = 45◦ to find

Nc =1

4A(1 + tan2Ψ + 2cos∆tanΨ), (4.6)

enabling to assess the second ellipsometric parameter ∆.As can be seen from the above analysis, the result for the ellipsometric pa-

rameters can be obtained by three measurements. These measurements onlyrequire the adjustment of the reference arm polarizer A1 and do not requireany information about other system parameters (length of arms, quantumefficiency of the detectors etc.).

31

4.3 Advantages

The use of a nonclassical twin-photon light source can immediately be seento change the geometry compared to classical interferometers. The entan-glement of the source makes it possible the utilize interferometry, althoughthere are no components usually associated with such a system, such as beamsplitter(s) and wave plate(s). The absence of these components simplifies theoptical arrangement considerably, making it more reliable and at the sameavoiding the need to characterize these elements. As an additional benefitthis keeps 100% of the incoming photon flux directed to the detectors. Theentangled twin-photon nature of the source also implies that the detection ofone photon can be used to gate the arrival of its twin. This is because thedetection of one photon guarantees the existence of the other photon in theother arm. Hence, this predictability means that we essentially have beenprovided with a well calibrated optical source.

The interferometric configuration in itself offers benefits. Compared toclassical ellipsometers [cf. Sec. 2.2.2], the light beams travel independentlyin two different directions. This alleviates various problems associated withinstrumentation errors. For example, all optical elements can be placed af-ter the sample, thus removing potential beam deviation errors that changethe beams angle of incidence on the sample. This fact is important, as themeasurements of ellipsometric parameters change as a function of the angleof incidence.

Furthermore it can be shown, that the entangled twin-photon ellipsome-ter is not sensitive of the overall mismatch in the lengths of the arms. Thisincreases its robustness, as it means that the arm length difference can bechosen as desired, without changing Eq. (4.3) the coincidence rate.

Finally the process of spontaneous parametric down-conversion guaran-tees the stability of polarization along a particular direction. In the entangledtwin-photon ellipsometer only light that satisfies the type-II non-collinearphase matching condition can generate the idler and signal photons. Hence,no polarization controlling elements are necessary before the sample and stillthe polarization state can be kept stable throughout the measurements.

In summery, the entangled twin-photon ellipsometer can be seen to offermore reliable and accurate way to perform ellipsomeric measurements thanthe standard classical ellipsometers. It requires no source or detector calibra-tion and neither are its measurements dependent on the use of a referencesample.

32

Chapter V

Classical ghost ellipsometry

The classical ghost ellipsometer, introduced in this part, is an optical inter-ferometric device designed to measure the ellipsometric parameters Ψ and ∆of a sample. The measurements are carried out by measuring the intensitycorrelations between two separate arms of the interferometric configuration.In respect of its purpose, the classical ghost ellipsometer does not differ fromother classical ellipsometers, or the entangled twin-photon ellipsometer pre-sented in Chapts. 2 and 4. However, the classical ghost ellipsometer canarrive at similar results as the entangled twin-photon ellipsometer, by usinga classical light source and similar geometry. In this case the classical ghostellipsometer and entangled twin-photon ellipsometer are seen to be equiv-alent, thus proving that the underlying phenomenon is not dependent onquantum effects.

As implied above, this chapter of the thesis is focused on the derivation ofthe theoretical results of the different classical ghost ellipsometry configura-tions. In the end, it is shown that a similar geometry to that of the entangledtwin-photon ellipsometer produces analogous results when used with classicalspatially incoherent and unpolarized light. However, for the sake of complete-ness, the chapter firstly presents other more basic configurations that helpto understand and derive the final results. Because of this the chapter isdivided to four parts. In the first part, the most rudimentary configuration,similar to the classical ghost-imaging geometry, is presented. The secondpart examines the case where only horizontally or vertically polarized lightis used for the measurements. The third part adds a linear polarizer to thetest arm of the basic configuration. Finally, the last and most important partexamines the classical counterpart of the entangled twin-photon ellipsometerby adding a second linear polarizer to the test arm.

33

5.1 Basic configuration

The first section of this chapter is dedicated to the study of the simplestconfiguration that is reviewed in this thesis (shown in Fig. 5.1). This inter-ferometric geometry forms the backbone of the other configurations that arestudied later and as such is of great importance in understanding the topic.Many of the equations presented and derived in this section are also usedin the subsequent sections. This configuration also offers a simple and rela-tively easy way to start the analysis of classical ghost ellipsometry, becauseof its overall simplicity and similarity to the the configuration of the classicalghost-imaging geometry presented in Fig. 3.1.

In the studied interferometric configuration a sample with the polariza-tion dependent reflectivities rs = |rs|exp(iφs) and rp = |rp|exp(iφp) is placedin the test arm (cf. Sec. 2.2.1). After this light from a spatially incoherentsource is split into two arms in the beam splitter. Light in the reference armpropagates a distance of za directly to a point-like detector D1. Light in thetest arm in turn propagates a distance of zb, via reflection from the sample, toanother point-like detector D2. After the individual intensities are measured,the correlation between these intensities is studied in an attempt to obtaininformation about the ellipsometric parameters Ψ and ∆.

SourceBS

Sample ( , )

< I(ρ1) I(ρ2)>

I1(ρ1)

I2(ρ2)

ρ1

ρ2

ρ0 za

zb

D1

D2

Figure 5.1: The most basic attempt to measure the ellipso-metric parameters Ψ and ∆. Similarly to the classical ghost-imaging geometry of Fig. 3.1, spatially incoherent light is splitinto two arms with a beam splitter (BS) and a sample withthe polarization dependent reflectivities rs and rp is placed inthe test arm. The resulting intensities in the arms are mea-sured by two point-like detectors (D1, D2) and the correlationbetween the intensity fluctuations is measured and studied.Here φ represents the angle of incidence.

34

In analogy to classical ghost imaging, the correlation between intensity fluc-tuations ∆I(ρ, ω) = I(ρ, ω) − 〈I(ρ, ω)〉, is presumed to be responsible forthe information relating to the ellipsometric parameters Ψ and ∆. As wedemonstrated in Chap. 3, the correlation between the intensity fluctuationsof vector fields that obey Gaussian statistics can be presented as

〈∆I1(ρ1, ω)∆I2(ρ2, ω)〉 = tr[W†12(ρ1,ρ2, ω)W12(ρ1,ρ2, ω)], (5.1)

where we have used the same formalism as in Secs. 3.1 and 3.2. Additionallythe term

W12(ρ1,ρ2, ω) = 〈E∗1(ρ1, ω)ET2 (ρ2, ω)〉, (5.2)

is the cross-spectral density matrix (CSDM) between the field E1(ρ1, ω) atthe reference arm detector and the field E2(ρ2, ω) at test arm detector. Thefields E1(ρ1, ω) and E2(ρ2, ω) are themselves represented by vectors of theform

E1(ρ1, ω) =

[E1,s(ρ1, ω)E1,p(ρ1, ω)

], (5.3)

and

E2(ρ2, ω) =

[E2,s(ρ2, ω)E2,p(ρ2, ω)

], (5.4)

where the subscripts s and p now refer to horizontal and vertical polarizationcomponents, respectively. If we apply Eqs. (5.3) and (5.4) to Eq. (5.2), wecan see that the general form of CSDM can be presented as

W12(ρ1,ρ2, ω) =

[W12,ss(ρ1,ρ2, ω) W12,sp(ρ1,ρ2, ω)W12,ps(ρ1,ρ2, ω) W12,pp(ρ1,ρ2, ω)

]. (5.5)

Here we have used the fact that the terms,

W12,ji(ρ1,ρ2, ω) = 〈E∗1,j(ρ1, ω)E2,i(ρ2, ω)〉, i, j ∈ {s, p}, (5.6)

represent the familiar scalar cross-spectral density functions between differentpolarization components. Equation (5.5) together with Eq. (5.1) signify thatthe correlation between the intensity fluctuations can be written as

35

〈∆I1(ρ1, ω)∆I2(ρ2, ω)〉 = |W12,ss(ρ1ρ2, ω)|2 + |W12,ps(ρ1ρ2, ω)|2

+ |W12,sp(ρ1ρ2, ω)|2 + |W12,pp(ρ1ρ2, ω)|2. (5.7)

From this equation it is clear that an expression for the intensity fluctuationscan be formed by calculating the propagated cross-spectral density functionsof the different polarization components. This can be performed by apply-ing the familiar paraxial Fresnel propagation law to the case of the studiedgeometry. When this is done it can be seen that the propagated polarizationcomponents at the reference arm detector can be presented as

E1,j(ρ1, ω) =−ik2πza

∫E0,j(ρ0, ω)exp

{ik

2

[(ρ1 − ρ0)

2

za

]}d2ρ0, (5.8)

and the components at the test arm detector as

E2,i(ρ2, ω) =−ik2πzb|ri|exp(iφi)

∫E0,i(ρ

′0, ω)exp

{ik

2

[(ρ2 − ρ′0)

2

zb

]}d2ρ′0.

(5.9)

In the above equations the parameters are defined as in Sec. 3.2 and i, j ∈{s, p}. The results given for the propagated field components imply, that thecross-spectral density function can be expressed as

W12,ji(ρ1,ρ2, ω) =

∫∫W

(0)ji (ρ0,ρ

′0, ω)K∗1,j(ρ1,ρ0, ω)K2,i(ρ2,ρ

′0, ω)d2ρ0d

2ρ′0,

(5.10)

where W(0)ji (ρ0,ρ

′0, ω) = 〈E∗0,j(ρ0, ω)E0,i(ρ

′0, ω)〉 corresponds to the cross-

spectral density function of the source’s field components and the kernelsK1,j together with K2,i, for the reference and test arm respectively, are ofform

K1,j(ρ1,ρ0, ω) =−ik2πza

exp

{ik

2

[(ρ1 − ρ0)

2

za

]}, (5.11)

and

K2,i(ρ2,ρ′0, ω) =

−ik2πzb|ri|exp(iφi)exp

{ik

2

[(ρ2 − ρ′0)

2

zb

]}, (5.12)

with i, j ∈ {s, p}. Equation (5.10) can be further developed when we once

36

more assume the source to be classical spatially incoherent source that ischaracterized by the cross-spectral density function of form

W(0)ji (ρ0,ρ

′0, ω) = 〈E∗0,j(ρ0, ω)E0,i(ρ

′0, ω)〉 = Φjiδ(ρ

′0 − ρ0), (5.13)

where i, j ∈ {s, p}, δ denotes the two-dimensional Dirac delta function, andΦji represents one of the elements of the polarization matrix Φ of the source,as discussed in Chap. 2. When this assumption is used in Eq. (5.10), itsimplifies to

W12,ji(ρ1,ρ2, ω) = Φji

∫K∗1,j(ρ1,ρ0, ω)K2,i(ρ2,ρ0, ω)d2ρ0, (5.14)

where we see that the ρ′0 term has now been replaced by ρ0 and i, j ∈ {s, p}.The remaining integral of the preceding equation can be calculated. This canbe accomplished by using an integral identity of form∫

exp(iax2)exp(ik · x)d2x =iπ

aexp

(−ik2

4a

), (5.15)

where a is a scalar and x and k are vectors [47]. When this is done, a generalform is obtained for the cross-spectral density function of the polarizationcomponents, expressed as

W12,ji(ρ1,ρ2, ω) =ikΦji

2π(za − zb)|ri|exp(iφi)exp

{−ik

2

[ρ21za− ρ22zb

]}× exp

{−ik

2

[(ρ1zb − ρ2za)

2

zazb(za − zb)

]}, (5.16)

with i, j ∈ {s, p}. The above result can be further simplified by assumingthat the propagation distances follow the condition za = 2zb

W12,ji(ρ1,ρ2, ω) =ikΦji

2πzb|ri|exp(iφi)exp

{− ik

2

[(ρ2 − ρ1)

2

zb

]}. (5.17)

However this condition is not necessary and for the sake of generality it isbetter to use Eq. (5.16). The reason for this is that when we take the absolutesquare of W12,ji(ρ1,ρ2, ω) or calculate the product of a cross-spectral densityfunction with another’s complex conjugate, the spatial exponential termsalways vanish. Because of this, it is not necessary to restrict the lengthsof the arms in any way other than assuming that they are larger than zero(za, zb > 0) and that they are unequal (za 6= zb). Taking this into account

37

and taking the absolute square of Eq. (5.16) we get

|W12,ji(ρ1,ρ2, ω)|2 =

[kΦji

2π(za − zb)

]2|ri|2, i, j ∈ {s, p}, (5.18)

which in turn implies that the correlation between intensity fluctuations inEq. (5.7) can be expressed as

〈∆I1(ρ1, ω)∆I2(ρ1, ω)〉 =

[k

2π(za − zb)

]2[(Φ2

sp + Φ2pp)|rp|2 + (Φ2

ps + Φ2pp)|rs|2].

(5.19)

From the above equation it is clear, that the intensity fluctuations measuredin the basic configuration indeed contain some information about the sam-ple. This information is in the form of the absolute values of the amplitudereflection coefficients |rs| and |rp|. However, Eq. (5.19) does not allow toextract the ratio of reflection coefficients Ψ. In addition, the equation doesnot contain any information about the relative phase difference ∆ betweenthe reflection coefficients. This means that the basic configuration does notprovide full information about the ellipsometric parameters. However, asmentioned at the beginning of this section, this configuration forms the basison which the analysis of more complex configurations can be built.

5.2 Using horizontally or vertically polarized light

As mentioned in the end of the previous section, the biggest problem of thebasic configuration was that the values of the complex amplitude reflectioncoefficients could not be extracted from the result. Because of this we nextanalyse a situation where only horizontally (s-polarization) or vertically po-larized (p-polarization) light is used in the measurement. The use of only onepolarization component can be accomplished by adding a linear polarizer A0

between the light source and the beam splitter, as shown in Fig. 5.2. Thepolarization axis is set to 0◦ or 90◦ so that it selects only the s or p polariza-tion component for the measurement. Otherwise the general geometry andthe principle of the configuration remain identical to the geometry presentedin the previous section.

Because the geometry is identical to that of the basic configuration, theanalysis of this system follows the same principles and notation as of thebasic configuration. The added polarizer does not alter the propagation ofthe fields, it only blocks the s- or p-polarization component from enteringthe system. This means that the propagated fields in Eqs. (5.8) and (5.9)

38

< I(ρ1) I(ρ2)>

SourceBS

Sample ( , )

I1(ρ1)

I2(ρ2)

ρ1

ρ2

ρ0 za

zb

D1

D2

A0

Figure 5.2: Attempt to use only horizontally or vertically po-larized light to measure the ellipsometric parameters Ψ and ∆.Spatially incoherent light from the source is first directed to alinear polarizer (A0). This polarizer selects only the horizon-tal (s-polarization) or vertical polarization (p-polarization) forthe measurement. Only after this, the light is split to the twoarms of the ellipsometer and directed to the detectors. In thisfigure φ represents the angle of incidence.

remain the same. The only difference is that there is only one polarizationcomponent and so the above equations must be expressed as

E1,i(ρ1, ω) =−ik2πza

∫E0,i(ρ0, ω)exp

{ik

2

[(ρ1 − ρ0)

2

za

]}d2ρ0, (5.20)

for the field at the reference arm detector and as

E2,i(ρ2, ω) =−ik2πzb|ri|exp(iφi)

∫E0,i(ρ

′0, ω)exp

{ik

2

[(ρ2 − ρ′0)

2

zb

]}d2ρ′0,

(5.21)

for the field at the test arm detector. In these expressions i ∈ {s, p} rep-resent one of the two possible polarization states. When we next presumethe source to be classical spatially incoherent light described by Eq. (5.13)and perform the resulting integral with Eq. (5.15), the cross-spectral densityfunction W12,ii(ρ1,ρ2, ω) = 〈E∗1,i(ρ1, ω)E2,i(ρ2, ω)〉 is realised to be of theform

39

W12,ii(ρ1,ρ2, ω) =ikΦii

2π(za − zb)|ri|exp(iφi)exp

{−ik

2

[ρ21za− ρ22zb

]}× exp

{−ik

2

[(ρ1zc − ρ2za)

2

zazb(za − zb)

]}, (5.22)

in which i ∈ {s, p} and Φii represents one of the diagonal elements of thepolarization matrix Φ describing the intensity of the studied polarizationcomponent. By applying the above expression to Eq. (5.7), describing thecorrelation between the intensity fluctuations, we finally get

〈∆I1(ρ1, ω)∆I2(ρ1, ω)〉 =

[kΦii

2π(za − zb)

]2|ri|2, i ∈ {s, p}. (5.23)

This is due to the fact that we only have one polarization component in thesystem. Equation (5.23) represents the general expression for the correlationbetween intensity fluctuations in the studied configuration. In the case of spolarized light it entails

〈∆I1(ρ1, ω)∆I2(ρ1, ω)〉 =

[kΦss

2π(za − zb)

]2|rs|2, (5.24)

with in the case of p polarized light

〈∆I1(ρ1, ω)∆I2(ρ1, ω)〉 =

[kΦpp

2π(za − zb)

]2|rp|2. (5.25)

From the last two equations it can be seen, that the addition of the linearpolarizer to the basic configuration reduces it into a scalar case. This is notsurprising as there is only one polarization component in the system at anygiven time. As the results suggest, this effectively means that the systemis almost identical to the case of scalar classical ghost imaging. Now wehave just replaced the object with a space dependent transmittance functionT(ρ′2) with a sample that has polarization dependent amplitude reflectioncoefficients rs = |rs|exp(iφs) and rp = |rp|exp(iφp). The equations also makeit clear that if we have precise information about the source (k,Φss,Φpp) andother system parameters (za, zb), it is theoretically possible to separatelymeasure the absolute values for the amplitude reflection coefficients |rs| and|rp|. With this information one could next calculate the value of the ellip-sometric parameter Ψ. However, the information about the relative phasedifference ∆ between the reflection coefficients is still missing. This is due to

40

the above-mentioned fact that we are blocking the other polarization compo-nent from entering the system. Nonetheless, the studied configuration doesillustrate that by adding a polarizer to the system, it is indeed possible toextract polarization dependent information from the sample. This fact willbe studied in more detail in the next two sections of this chapter.

5.3 Adding one linear polarizer

In the analysis of the previous section, only horizontally or vertically po-larized light was used in an attempt to extract the polarization dependentinformation. With this approach the extraction was partly accomplished,but the primary problem was that the relative phase difference between thereflection coefficients was still found to be missing in the results. In thissection we attempt to correct this by moving to a configuration that is moresimilar to the entangled-photon ellipsometer discussed in Chap. 4. This isaccomplished by adding a single linear polarizer A1 to the reference arm ofthe basic configuration, as presented in Fig. 5.3. Otherwise the general geom-etry and principle of the measurements remain identical to that of the basicconfiguration. The only difference is that the polarizer A1 now determinesthe type of polarization passed through to the test arm detector. Also inthis section we do not restrict the polarizer angle to 0◦ or 90◦ correspondingto horizontally (s-polarization) or vertically (p-polarization) polarized light,respectively.

The analysis of this configuration follows the same principles and nota-tion as the basic configuration. This means that the field components in thereference arm before the polarizer can be expressed in the familiar form of

E ′1,j(ρ1, ω) =−ik2πza

∫E0,j(ρ0, ω)exp

{ik

2

[(ρ1 − ρ0)

2

za

]}d2ρ0, (5.26)

emphasized by the prime (’) superscript. Here j ∈ {s, p} denotes one of thetwo possible polarization states. It is important to note that the aforemen-tioned equation describes the field components before the polarizer. Whencalculating the field at the detector itself, we must once again remember thatthe polarizer does not alter the propagation of the fields, it only blocks cer-tain polarization component. Because of this the field at the reference armdetector can be expressed simply as

E1(ρ1, ω) =

[cos2θ cosθsinθ

cosθsinθ sin2θ

] [E ′1,s(ρ1, ω)E ′1,p(ρ1, ω)

], (5.27)

41

< I(ρ1) I(ρ2)>

SourceBS

Sample ( , )

I1(ρ1)

I2(ρ2)

ρ1

ρ2

ρ0 za

zb

D1

D2

A1

Figure 5.3: Configuration where one linear polarizer (A1) hasbeen added to the reference arm of the basic configuration,in an attempt to measure the ellipsometric parameters Ψ and∆. Here φ represents the angle of incidence.

where angle θ is the angle of the polarization axis with respect to the hori-zontal direction. In turn the field at test arm detector is the same as in thecase of the basic configuration. Hence it can be directly expressed as

Ed2(ρ2, ω) =

[E ′2,s(ρ2, ω)E ′2,p(ρ2, ω)

], (5.28)

where the components are of the familiar form

E ′2,i(ρ2, ω) =−ik2πzb|ri|exp(iφi)

∫E0,j(ρ

′0, ω)exp

{ik

2

[(ρ2 − ρ′0)

2

zb

]}d2ρ′0,

(5.29)

with i ∈ {s, p}. By using Eq. (5.2) together with the previous equations, thecross-spectral density matrix at the end of the arms can be expressed as

W12(ρ1,ρ2, ω) =〈E∗1(ρ1, ω)ET2 (ρ2, ω)〉 = P(θ)W′(ρ1,ρ2, ω). (5.30)

Here

P(θ) =

[cos2θ cosθsinθ

cosθsinθ sin2θ

], (5.31)

is the Jones matrix of the linear polarizer and

42

W′(ρ1,ρ2, ω) =

[W ′

12,ss(ρ1,ρ2, ω) W ′12,sp(ρ1,ρ2, ω)

W ′12,ps(ρ1,ρ2, ω) W ′

12,pp(ρ1,ρ2, ω)

], (5.32)

is the cross-spectral density matrix of the basic configuration, with elementsgiven in Eq. (5.16) without the prime superscript. Using Eq. (5.30) togetherwith Eq. (5.1), it is possible to express the correlation between intensityfluctuations as

〈∆I1(ρ1, ω)∆I2(ρ2, ω)〉 = tr[W†12(ρ1,ρ2, ω)W12(ρ1,ρ2, ω)]

= tr[W′†(ρ1,ρ2, ω)P(θ)W′(ρ1,ρ2, ω)], (5.33)

where we have used the property (AB)† = B†A†. By calculating the trace,the correlation between intensity fluctuations can further be expressed as

〈∆I1(ρ1, ω)∆I2(ρ2, ω)〉 =

[k

2π(za − zb)

]2[|rs|2(cosθΦss + sinθΦps)

2

+ |rp|2(cosθΦsp + sinθΦpp)2]. (5.34)

The result given in Eq. (5.34) is somewhat similar to the result given inEq. (5.19) for the basic configuration. The main difference between the ex-pressions are the cosine and sine terms that are due to the addition of thelinear polarizer. In the general case, similarly to the case of the basic con-figuration, no information about the sample can be retrieved from the mea-surement. However, when the light being used is unpolarized, the absolutevalues of the amplitude reflection coefficients |rs| and |rp| are measurable intwo separate measurements. In the first measurement the polarizer angle isset to 0◦ and in the second to 90◦. This is possible due to the fact that thecorrelation terms Φsp and Φps of the (unpolarized) source polarization matrixΦ equal zero, as discussed in Chap. 2. However as in the case of the previousconfigurations, it is clear, that the information about the relative phase dif-ference ∆ between the reflection coefficients is still missing. A second opticalcomponent must clearly be added to the system for the measurement of thisparameter.

5.4 Adding a second linear polarizer

In the previous section, the addition of one linear polarizer to the referencearm of the basic configuration was shown to be insufficient to reveal the rel-ative phase difference between the reflection coefficients. It was made clear

43

< I(ρ1) I(ρ2)>

SourceBS

Sample ( , )

I1(ρ1)

I2(ρ2)

ρ1

ρ2

ρ0 za

zb

D1

D2

A1

A2

Figure 5.4: Configuration where a second linear polarizer (A2)has been added to the test arm of the previous configurationshown in Fig. 5.3, in a attempt to measure the ellipsometricparameters Ψ and ∆. Here φ represents the angle of incidence.

that at least another optical component is needed to acquire this information.Because of this and encouraged by the entangled twin-photon ellipsometer,this section studies the addition of another linear polarizer, A2, to the testarm of the previous system, as show in Fig. 5.4. Once more the generalgeometry and the principle of the configuration remain identical to the ge-ometry of the basic configuration. As was implied at the beginning of thischapter, it is demonstrated that this configuration leads to the results sim-ilar to the entangled twin-photon ellipsometer. Because of this, the studiedconfiguration can be considered the classical counterpart of the entangledtwin-photon ellipsometer.

In analogy to the analysis of the previous configurations, the analysis ofthis system follows the same principles and notation as the basic configura-tion. Because of this the field components before the polarizer in the testarm are

E ′1,j(ρ1, ω) =−ik2πza

∫E0,j(ρ0, ω)exp

{ik

2

[(ρ1 − ρ0)

2

za

]}d2ρ0, (5.35)

where j ∈ {s, p}. When calculating the field at the detector itself, it mustonce again be remembered that the polarizer does not alter the propagationof the fields, only blocks certain polarization component. Because of this thefield at the reference arm detector can be expressed as

E1(ρ1, ω) =

[cos2θ1 cosθ1sinθ1

cosθ1sinθ1 sin2θ1

] [E ′1,s(ρ1, ω)E ′1,p(ρ1, ω)

], (5.36)

44

where angle θ1 is the polarization axis of polarizer A1 with respect to thehorizontal direction. Using identical reasoning to the test arm, the fieldcomponents before the reference arm detector can be expressed as

E ′2,i(ρ2, ω) =−ik2πzb|ri|exp(iφi)

∫E0,j(ρ

′0, ω)exp

{ik

2

[(ρ2 − ρ′0)

2

zb

]}d2ρ′0,

(5.37)

with i ∈ {s, p}. And the field at the test arm detector itself as

E2(ρ2, ω) =

[cos2θ2 cosθ2sinθ2

cosθ2sinθ2 sin2θ2

] [E ′2,s(ρ2, ω)E ′2,p(ρ2, ω)

], (5.38)

where angle θ2 is the polarization axis of polarizer A2 with respect to thehorizontal direction. Now by using Eq. (5.2) together with the above expres-sions, the cross-spectral density matrix takes the form

W12(ρ1,ρ2, ω) =〈E∗1(ρ1, ω)ET2 (ρ2, ω)〉 = P(θ1)W

′(ρ1,ρ2, ω)P(θ2), (5.39)

where

P(θα) =

[cos2θα cosθαsinθα

cosθαsinθα sin2θα

], α ∈ {1, 2} (5.40)

is the Jones matrix of the linear polarizer and W′(ρ1,ρ2, ω) is the cross-spectral density matrix of the basic configuration given in Eq. (5.32), withelements provided in Eq. (5.16). Using the previous expression and Eq. (5.1),we can express the correlation between intensity fluctuations as

〈∆I1(ρ1, ω)∆I2(ρ2, ω)〉 = tr[W†12(ρ1,ρ2, ω)W12(ρ1,ρ2, ω)]

= tr[P(θ2)W′†(ρ1,ρ2, ω)P(θ1)W

′(ρ1,ρ2, ω)].(5.41)

Here we have used two distinct mathematical properties to arrive this result.Firstly the matrix multiplication property of hermitian transpose (AB)† =B†A† and secondly the cyclic property of the trace operator tr[ABCD] =tr[BCDA] = tr[CDAB] = tr[DABC]. By calculating the trace, the correla-tion between the intensity fluctuations can be expressed as

45

〈∆I1(ρ1, ω)∆I2(ρ2, ω)〉

=

[k

2π(za − zb)

]2[cos2θ2|rs|2(cosθ1Φss + sinθ1Φps)

2

+ sin2θ2|rp|2(cosθ1Φsp + sinθ1Φpp)2

+ 2cosθ2sinθ2|rs||rp|cos∆(cosθ1Φss + sinθ1Φps)(cosθ1Φsp + sinθ1Φpp)

],

(5.42)

where ∆ = φp − φs is the relative phase difference between the reflectioncoefficients.

As is evident from the above equation, the information about the relativephase difference between the reflection coefficients is now included in thecorrelation between intensity fluctuations. This seems to open a way tomeasure both ellipsometric parameters Ψ and ∆ using the configuration ofFig. 5.4. However, the result is relatively complicated and depends on thepolarization properties of the light source. Because of this, the detailedanalysis of Eq. (5.42) is divided into two parts. In the first part we considerfully polarized light, whereas the second part concerns unpolarized light.

5.4.1 Polarized light

In the case of completely polarized light (degree of polarization equals unity),which polarization state is otherwise arbitrary and described by the elementsΦji, i, j ∈ {s, p} of the polarization matrix, Eq. (5.42) can be expressed as

〈∆I1∆I2〉

=B

[cos2θ2(cosθ1Φss + sinθ1Φps)

2 + sin2θ2tan2Ψ(cosθ1Φsp + sinθ1Φpp)2

+ 2cosθ2sinθ2tanΨcos∆(cosθ1Φss + sinθ1Φps)(cosθ1Φsp + sinθ1Φpp)

],

(5.43)

where

B =

[k

2π(za − zb)

]2|rs|2, (5.44)

and tanΨ = |rp|/|rs|. The dependency on ω, ρ1 and ρ2 has been sup-pressed from the left hand side of the expression for the sake of brevity.

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From Eq. (5.43) the ellipsometric parameters can be calculated with threeseparate measurements. For example, by first setting θ1 = θ2 = 0◦ to get

〈∆I1∆I2〉 = BΦ2ss, (5.45)

specifies parameter B. Second by setting θ1 = θ2 = 90◦ to find

〈∆I1∆I2〉 = Btan2ΨΦ2pp, (5.46)

that fixes the ellipsometric parameter Ψ. Finally, the choice of θ1 = 0◦ andθ2 = 45◦ leads to

〈∆I1∆I2〉 =1

2B(Φ2

ss + tan2ΨΦ2sp + 2tanΨcos∆ΦssΦsp), (5.47)

from which the ellipsometric parameter ∆ can be determined.The above analysis shows that in the case of polarized light, it is indeed

possible to measure the ellipsometric parameters Ψ and ∆. However, it is alsoclearly evident that the result in Eq. (5.43) differs from the result presentedin Chap. 4 for the coincidence rate of the entangled twin-photon ellipsometer.Here the expression contains additional terms that need to be determined oreliminated before the ellipsometric parameters can be defined. Because ofthis, additional information about the source is needed and both polarizersmust be adjusted in the course of the measurements. In the above examplemeasurements, the source information was in the form of the intensity termsΦss and Φpp together with the correlation term Φsp of the source polarizationmatrix Φ. This is an improvement to the previous configurations in whichinformation about wave number k together with arm lengths za and zb wasalso needed. However, in the case of the entangled twin-photon ellipsometer,no information about the source or other system parameters was requiredand only the reference arm polarizer was adjusted.

5.4.2 Unpolarized light

In the case of unpolarized light the correlation terms, Φps = Φsp = 0, andthe intensity terms, Φss = Φpp = I, as discussed in Sec. 2. It follows fromthis that Eq. (5.42) assumes the form

〈∆I1∆I2〉 = C

[cos2θ2cos2θ1 + sin2θ2sin

2θ1tan2Ψ

+ 2cosθ2sinθ2cosθ1sinθ1cos∆tanΨ

], (5.48)

47

where

C =

[kI

2π(za − zb)

]2|rs|2. (5.49)

Once more the dependency on ω, ρ1 and ρ2 has been suppressed from theleft hand side of the expression for the sake of brevity. As in the case ofpolarized light, the ellipsometric parameters can be calculated with threeseparate measurements. Easiest way to achieve this is to keep θ2 fixed at 45◦

and adjust θ1. For example, by first setting θ1 = 0◦ to get

〈∆I1∆I2〉 =1

2C, (5.50)

from which the parameter C can be determined. Secondly by using θ1 = 90◦

to obtain

〈∆I1∆I2〉 =1

2Ctan2Ψ, (5.51)

from which parameter Ψ can be found. Finally, by inserting θ1 = 45◦ implies

〈∆I1∆I2〉 =1

4C(1 + tan2Ψ + 2cos∆tanΨ), (5.52)

allowing to determine parameter ∆.As when using polarized light, the above results show that it is possible

to measure the ellipsometric parameters Ψ and ∆ using classical spatially in-coherent and unpolarized light, with Gaussian statistics. The measurementsare simpler than in the case of polarized light, as there is no longer anyneed for information about the source or other system parameters. All thenecessary data (C,Ψ,∆) can be defined directly trough the measurements.Also only the reference arm polarizer needs to be adjusted in the measure-ments, simplifying the data collection considerably. When these results arecompared to the result obtained in Chap. 4, the aforementioned conclusionsimply great similarly between the studied classical ghost ellipsometer andentangled twin-photon ellipsometer. First of all, there is no difficulty in con-trolling or measuring the polarization properties of the incoming light, as thelight used is unpolarized. All the polarization manipulation can be performedafter the sample or in the reference arm of the ellipsometer. In addition, thestudied classical system is also not sensitive to an overall mismatch in thelengths of the two arms, as was discussed in the first section of this chapter.The only requirement is that the lengths are unequal za 6= zb, otherwise they

48

can be chosen as desired. The main drawback of the studied configuration,as all other classical interferometric configurations, is the use of a beam split-ter. This leads to the loss of 50 % of the light at the individual detectors.More importantly the beam splitter may induce alignment errors that leadto beam deviation, mandating the careful characterization of this element.This is important, as beam deviation may in turn lead to an error in theangle of incidence on the sample and so to errors in the measured parame-ters. Despite this, the above results compare favourably to those obtainedfor the entangled twin-photon ellipsometer in Chap. 4. In particular, theyprove that the studied case of classical ghost ellipsometer, that uses classi-cal incoherent and unpolarized light, presents many of the advantages of theentangled twin-photon ellipsometer.

49

Chapter VI

Conclusions

This thesis presented and analysed the classical ghost ellipsometer by usingelectromagnetic theory of optical coherence combined with the Jones for-malism. The studied classical ghost ellipsometer was a modified version ofthe classical ghost-imaging configuration. It was designed for ellipsometricmeasurements and the used illumination consisted of a spatially incoherent,unpolarized beam of light obeying Gaussian field statistics. The classicalghost-imaging configuration itself was modified by adding a linear polarizerto both arms of the setup. Additionally, the geometry was changed to ac-commodate reflection measurements instead of the traditional transmissionmeasurements.

The analysed classical ghost ellipsometer was also compared to its quan-tum counterpart, the entangled twin-photon ellipsometer. The comparisonfound that both ellipsometers gave rise to very similar results. Particularly, itwas seen that both ellipsometers were self-referencing. That is to say, all thestudied parameters could be defined directly trough the measurements, mak-ing source and detector calibration unnecessary. In the case of the entangledtwin-photon ellipsometer, the underlying reason for this is the two-photonquantum interference together with nonlocal polarization entanglement. Inturn, the physics behind the classical ghost ellipsometer is the presence ofcorrelations between intensity fluctuations at the detectors, or in other words,the presence of correlations between classical speckle patterns. The main dif-ference between the ellipsometers was observed to be the absence of a beamsplitter in the quantum case. This was due to the noncollinear nature ofthe quantum light source (type-II SPDC), used to produce the polarizationentangled photons. Because of this, the employment of the quantum lightsource was seen to lead to an inherently simpler optical arrangement, withfewer optical elements to characterize. However, the use of classical light isseen to offer many readily-available, cost-effective, and simple light sources.

50

These sources do not necessitate the control of phase matching conditions,or temporal compensation of the output photons.

The above summary makes it clear that both the classical and quantumellipsometers can be used to characterize a sample ellipsometrically, and bothhave their unique advantages and disadvantages. However, it is also evidentthat the fundamental physics behind the studied ellipsometers can be un-derstood in two different ways, and quantum entanglement is not needed toexplain the basic phenomena. These results compare favourably to resultsobtained between the cases of quantum and classical ghost imaging.

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