Optical correlation imaging and geometric phase in electromagnetic interference

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Dissertations in Forestry and Natural Sciences

ANTTI HANNONEN

OPTICAL CORRELATION IMAGING AND GEOMETRIC PHASE IN ELECTROMAGNETIC INTERFERENCE

PUBLICATIONS OF

THE UNIVERSITY OF EASTERN FINLAND

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PUBLICATIONS OF THE UNIVERSITY OF EASTERN FINLAND DISSERTATIONS IN FORESTRY AND NATURAL SCIENCES

N:o 407

Antti Hannonen

OPTICAL CORRELATION IMAGING AND GEOMETRIC PHASE IN

ELECTROMAGNETIC INTERFERENCE

ACADEMIC DISSERTATION

To be presented by the permission of the Faculty of Science and Forestry for public examination in the Auditorium M100 in Metria Building at the University of Eastern Finland, Joensuu, on December 9th, 2020, at 12 o’clock.

University of Eastern Finland Department of Physics and Mathematics

Joensuu 2020

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Grano Oy Jyväskylä, 2020

Editors: Pertti Pasanen, Raine Kortet, Jukka Tuomela, and Matti Tedre

Distribution:

University of Eastern Finland Library / Sales of publications julkaisumyynti@uef.fi

http://www.uef.fi/kirjasto

ISBN: 978-952-61-3660-8 (print) ISSNL: 1798-5668

ISSN: 1798-5668 ISBN: 978-952-61-3661-5 (pdf)

ISSNL: 1798-5668 ISSN: 1798-5676

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Author’s address: University of Eastern Finland Institute of Photonics

P. O. Box 111 80101 JOENSUU FINLAND

email: antti.hannonen@uef.fi Supervisors: Professor Tero Setälä

University of Eastern Finland Institute of Photonics

email: tero.setala@uef.fi Professor Ari T. Friberg University of Eastern Finland Institute of Photonics

email: ari.friberg@uef.fi

Reviewers: Professor Thomas G. Brown

University of Rochester Institute of Optics 275 Hutchison Road

ROCHESTER, NY 14627-0186 USA

email: thomas.brown@rochester.edu Assistant Professor Robert Fickler Tampere University

Physics Unit P. O. Box 692 33014 TAMPERE FINLAND

email: robert.fickler@tuni.fi

Opponent: Professor Paul Urbach

Delft University of Technology Department of Imaging Physics Lorentzweg 1

2628 CJ DELFT THE NETHERLANDS email: h.p.urbach@tudelft.nl

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Antti Hannonen

Optical correlation imaging

and geometric phase in electromagnetic interference Joensuu: University of Eastern Finland, 2020

Publications of the University of Eastern Finland Dissertations in Forestry and Natural Sciences N:o 407

ABSTRACT

This thesis contains fundamental theoretical and experimental research related to classical coherence theory and electromagnetic optics. The work covers two main topics that may find important applications in polarimetric and interferometric in- vestigations of optical structures and materials.

The first topic encompasses theoretical studies of novel remote sensing techniques intended for spectral characterization of reflective and transmissive objects in ellipsometry and polarimetry. The methods are based on optical correlation imaging and make use of spatially incoherent classical light sources with Gaussian statistics and the measurement of intensity correlations. The ellipsometric schemes are designed to characterize both homogenous and inhomogeneous reflective planar samples. The ellipsometers are shown to be relatively insensitive to instrumen- tation errors and do not require source or detector calibration. Our polarimetric technique extends the use of intensity measurements to the detection of Stokes- parameter correlations and provides new information about the use of polarization state correlations in general. The design can determine the transmission matrix of any polarization sensitive object and is insensitive to inaccuracies caused by turbulent media.

The second topic covers theoretical and experimental studies related to the interference of electromagnetic waves. We demonstrate for the first time the existence of the Pancharatnam–Berry geometric phase in Young’s double-pinhole experiment and in temporal interference of light. The phase appears due to the periodic polarization state modulation caused by the superposition of two fully coherent and polarized beams. Two different formulas for the phase are established in terms of the intensities of the interfering fields together with either their polarization states or the visibility of the intensity variation. Further, we discover that the geometric and dynamical phases associated with the interference are intertwined in such a way that one cannot be changed without altering the other. The results are verified experimentally by measuring the beam Stokes parameters and the amplitude of the interference intensity modulation as well as using genuine interferometric phase measurements. Our work illustrates that two-beam interference is not yet fully understood by connecting to the phenomenon new fundamental characteristics arising from the Pancharatnam–Berry phase.

Universal Decimal Classification:535, 535.1, 535.4, 535.5

OCIS codes:030.1640, 120.2130, 120.3180, 120.5050, 260.5430, 260.3160, 350.1370 Keywords: optics, light, coherence, polarization, light interference, correlation imaging, ghost imaging, geometric phase, Pancharatnam–Berry phase, ellipsometry, polarimetry

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ACKNOWLEDGEMENTS

This thesis represents the culmination of my study at the University of Eastern Finland. However, I can by no means take all the credit for the research summarized in the thesis. The work has been made possible by the efforts of many different people and I am forever grateful that I had an opportunity to know and learn from all of them.

First and foremost, I thank my supervisors Prof. Tero Setälä and Prof. Ari T.

Friberg without whose continual encouragement and advice I would not have suc- ceeded. The discussions we have had have been thought provoking and taught me a lot about the scientific process. Next I want to acknowledge the contributions of my esteemed co-authors Prof. Bernhard Hoenders, Prof. Wolfgang Elsässer, As- soc. Prof. Tommi Hakala, Dr. Henri Partanen, Dr. Kimmo Saastamoinen, Dr. Lasse- Petteri Leppänen, Dr. Matias Koivurova, Dr. Andriy Shevchenko, Dr. Jani Tervo, M.Sc. Aleksi Leinonen, and M.Sc. Janne Heikkinen. It has been a pleasure to work with you and I look forward to new opportunities for cooperation. I am especially indebted to Dr. Henri Partanen and Dr. Kimmo Saastamoinen for their pivotal experimental work in confirming many of the theoretical results. Additionally, I extend special thanks to Prof. Wolfgang Elsässer and his research group who hosted me during my two weeks visit at the Technical University of Darmstadt in the summer 2019. As for other people at the Department of Physics and Mathematics, I wish to recognize all those of my co-workers who have assisted me in the various adminis- trative matters not directly related to my research. For financial support I thank the University of Eastern Finland’s Doctoral School and the Emil Aaltonen Foundation.

Finally, but definitely not the least, I want to express my gratitude to my family, friends, and relatives for their continued and outspoken support.

Joensuu, November 9th, 2020 Antti Hannonen

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LIST OF PUBLICATIONS

This thesis consists of a review of the author’s work in the field of optical correlation imaging and electromagnetic interference. The review is based on the following selection of the author’s publications:

I A. Hannonen, A. T. Friberg, and T. Setälä, "Classical spectral ghost ellipsometry," Opt. Lett. 41, 4943–4946 (2016).

II A. Hannonen, A. T. Friberg, and T. Setälä, "Classical ghost-imaging spectral ellipsometer," J. Opt. Soc. Am. A34, 1360–1368 (2017).

III A. Hannonen, B. J. Hoenders, W. Elsässer, A. T. Friberg, and T. Setälä, "Ghost polarimetry using Stokes correlations," J. Opt. Soc. Am. A37, 714–719 (2020).

IV A. Hannonen, H. Partanen, J. Tervo, T. Setälä, and A. T. Friberg,

"Pancharatnam–Berry phase in electromagnetic double-pinhole interference,"

Phys. Rev. A99, 053826 (2019).

V A. Hannonen, H. Partanen, A. Leinonen, J. Heikkinen, T. K. Hakala, A. T.

Friberg, and T. Setälä, "Measurement of the Pancharatnam–Berry phase in two- beam interference," Optica7, 1435–1439 (2020).

VI A. Hannonen, K. Saastamoinen, L.-P. Leppänen, M. Koivurova, A. Shevchenko, A. T. Friberg, and T. Setälä, "Geometric phase in beating of light waves," New J. Phys. 21, 083030 (2019).

Throughout the overview, these papers will be referred to by Roman numerals.

International conferences in which results of this research have been presented and the author’s contributions in other scientific publications are listed in Appendix A.

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AUTHOR’S CONTRIBUTION

The publications selected in this dissertation are original research papers. The research reported in them is a result of group efforts. The author contributed essen- tially to the main ideas. He had a key role in all the theoretical work that led to publications I–VI. The author performed the mathematical derivations and wrote the first drafts of publicationsI–III, V, and VI. In publicationsIV–VI the author also took part in the design of the experimental setups and the interpretation of the experimental results. All authors participated in the finishing of the research papers.

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1 Introduction

Light has an essential role in life on earth. Sunlight illuminates the world around us and provides the energy for photosynthesis that enables most plant and animal life. As such, understanding of shadows, colors, reflections, and other related phenomena has always been important to man [1, 2]. The earliest recorded thoughts on the nature of light were presented by the Greek philosophers toward the middle of the first millennium BC. Several hundred years later the work was continued by their Arab counterparts, the most famous of whom was Ibn al-Haytham (c. 965 – c. 1040). His research built upon the earlier Greek concepts and introduced a new level of thoroughness to the analysis. Together the Greek and Arabic ideas had a great influence on the birth of optical science in Renaissance Europe.

Modern optics, the scientific study of light, has its origin in the works of people like Galileo Galilei (1564 – 1642), Willebrord Snellius (1580 – 1626), and Johannes Kepler (1571 – 1630) who pioneered the use of optical instruments and introduced mathematical laws to describe light behavior. Another giant of science Sir Isaac Newton (1642 – 1727) made his contribution in 1666 through the immensely important bookOptickswhere he described, e.g., reflection, refraction, the formation of images by lenses, and the decomposition of light into its spectral components.

The latter result demonstrated that color is an intrinsic property of light and gave birth to contemporary color theory. In 1690 Christiaan Huygens (1629 – 1695) introduced the wave theory of light in his famous book Traité de la Lumière. The theory was first verified in 1801 when Thomas Young (1773 – 1829) examined the principle of interference in the momentous double-pinhole experiment now bearing his name [3, 4]. A few years after Young’s experiment, Augustine Fresnel (1788 – 1827) and François Arago (1786 – 1853) further demonstrated Huygens’ principle in diffraction, leading to the general acceptance of the wave theory of light. Fresnel and Arago were also the first to examine the interference between light of different states of polarization. They listed some of the more important properties of the phenomenon as rules which are known today as the Fresnel–Arago laws. Following these breakthroughs, discoveries in the middle of the 19th century opened the way to modern understanding of the polarization of light. First Sir George Stokes (1819 – 1903) introduced his four parameters to describe the polarization state of light [5].

Later James Clerk Maxwell (1831 – 1879) put together the famous partial differen- tial equations that describe electromagnetism [6]. Finally, in the beginning of the 20th century Max Planck (1858 – 1947) introduced the hypothesis of quantization that Albert Einstein (1879 – 1955) used to explain the photoelectric effect [7]. The results of Planck and Einstein were the spark that led to the formulation of quantum mechanics and constituted a major revolution in physics.

Modern optics encompasses many important and active subfields that have count- less applications in the current society [8, 9]. One of these is optical coherence theory which investigates the random fluctuations of light fields through statistical analysis [10, 11]. In practice, no optical field is completely coherent as all of them contain at least some randomness arising from physical processes, such as spon- taneous emission of radiation from atoms. Accordingly, coherence theory is fully

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adequate to characterize all the usual optical phenomena. Classical coherence theory was generalized in the 1950s by Emil Wolf (1922 – 2018) for scalar fields in the space-time domain [12–14]. The work stemmed from the earlier studies of coherence performed by Albert Abraham Michelson (1852 – 1931) [15–17], Pieter Hendrik van Cittert (1889 – 1959) [18], and Frits Zernike (1888 – 1966) [19]. Following Wolf, the quantum theory of coherence was pioneered in the 1960s by Roy Jay Glauber (1925 – 2018) whose work is a cornerstone of quantum optics [20, 21]. Since then coherence theory has been extended to also include electromagnetic fields [14, 22]

and for wider applicability a spectral formulation of the theory has been introduced in the space-frequency domain [14, 23–26]. Recently, the generalization of the coherence theory from scalar fields to the electromagnetic vector fields has been used to demonstrate the close relation between coherence and polarization [11, 27, 28].

In general, the electromagnetic coherence theory has opened new areas of research from which new information and applications are emerging continually.

The research outlined in this thesis covers two topics of classical coherence theory and electromagnetic optics. The first is related to optical correlation imaging while the second considers interference of vector light and its relation to the geometric phase.

1.1 MOTIVATION AND BACKGROUND

The motivation behind the first part of the research is to theoretically study the principle of optical correlation imaging for the purpose of introducing new improved optical devices for remote sensing. Optical correlation imaging is a novel method for obtaining an image of an object through the measurement of correlations between two separate photodetectors [29–34]. Ghost imaging was first demonstrated in the quantum regime over 20 years ago, but it has been subsequently demonstrated that thermal or pseudothermal light and classical coherence theory are adequate to describe most aspects of this unconventional phenomenon. The main benefit of the technique is its inherent insensitivity to imperfections, measurement errors, and aberrations when operating, e.g., in turbulent or low light-level conditions. The work presented in this thesis unifies classical correlation imaging with ellipsometry (PublicationsIandII) and polarimetry (PublicationIII). These two techniques are over a century-old nonperturbing and nondestructive methods used to determine the intrinsic and structural properties of various materials [35–39]. They measure the change in the polarization state of light when it interacts with the structure of a sample to obtain information on its composition and optical properties. Due to the ability of ellipsometers and polarimeters to indirectly characterize samples, they have become invaluable tools whose use has become commonplace in virtually every branch of science.

The goal of the second part of our research is to provide new insight into the role of polarization in electromagnetic interference. Interference of scalar light waves has been studied from the days of Young [3, 4] and the phenomena has been well understood in the double-pinhole arrangement since the work of Zernike [19]. However, the situation is very different when we fully take into account the polarization properties of the field. The interference of electromagnetic waves causes not just the intensity variation observed in scalar case but leads to the modulation of the polarization state of the superposition field [28, 40–43]. The resulting elegant and rich polarization features were discovered only relatively recently and are not widely

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known to all optical physicist. Our work increases the understanding by connecting the fundamentally important Pancharatnam–Berry geometric phase to the Young’s double-pinhole experiment (PublicationsIVandV) as well as to the temporal interference of vector light (Publication VI). Geometric phase describes a phase change that occurs when the polarization state of light evolves in such a manner that the initial and final polarizations are identical [44–48]. The phase is a central topic of physics and has an essential role in many applications [49]. The results of this thesis add one more new and particularly foundational aspect to the previous research.

1.2 OUTLINE

The rest of the thesis is organized as follows. In Chap. 2 we introduce the basic concepts related to random processes and scalar coherence theory. In Chap. 3 the scalar theory is extended to include two-component vector fields and, in addition, polarization properties of light are explored. The next two chapters are reserved for the review of the new research. Chapter 4 deals with optical correlation imaging and the application of the technique to ellipsomery (PublicationsIandII) and polarimetry (Publication III). Chapter 5 is in turn used to investigate the geometric phase associated with the superposition of fully coherent and polarized beams in Young’s experiment (PublicationsIVandV) and temporal interference light waves (Publica- tionVI). Finally, in Chap. 6, the main conclusions of this work are summarized, and potential future research topics are considered.

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2 Basics of scalar coherence theory

In this chapter, the relevant concepts for analyzing fluctuating, stationary scalar light fields are presented. We start by considering the elementary theory of random processes that is integral in understanding all coherence effects. Next the scalar second- order coherence functions are discussed in both space-time and space-frequency domains. These provide the means to characterize any phenomena arising from field correlations at one or two space-time points. Finally, higher-order coherence effects are discussed in terms of the fourth-order field correlations between the two detectors of an intensity interferometer. These are essential for the analysis optical correlation imaging later in the thesis. For now, we do not take into account the polarization features associated with electromagnetic fields. Electromagnetic coherence theory will be the subject of Chap. 3.

2.1 ELEMENTARY THEORY OF RANDOM PHENOMENA

All light fields, whether natural or man-made, have random fluctuations associated with them. Optical coherence theory describes physical effects arising from these fluctuations. In the optical region of the light spectrum, these fluctuations are too rapid to be observed directly using photodetectors. Accordingly, the theory of optical coherence is based on the mathematical methods developed for the analysis of random phenomena, i.e., the theory of stochastic processes.

2.1.1 Statistical ensembles

A random scalar light field at position r = (ρ,z) = (x,y,z) and time t can be expressed using a complex functionE(r,t)in the complex analytic signal representation [8, 50]. For every value ofrandt the random field E(r,t) has a probability densityp[E(r,t)]associated with it. The quantity p[E(r,t)]gives the probability that the field variable will take a valueE in a particular infinitesimal neighborhood of dE. Using the probability density, the average of the field is defined as

E(r,t)=

Ep[E(r,t)]dE, (2.1)

where the integration extends over the range of possibleEvalues. Alternatively, we can consider an ensemble (set) of all possible realizations (values) the random vari- ableE(r,t) can take. Labeling the individual realizations asEn(r,t),n ∈ {^{1, ...,}N}^, in a ensemble withNrealizations, the ensemble average is obtained through

E(r,t)= lim

N→∞

1 N

∑

N

i=1Ei(r,t). (2.2)

Equations (2.1) and (2.2) both give equivalent definitions for the ensemble average.

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2.1.2 Higher-order correlations

The probability density p[E(r,t)] does not characterize the random process completely as it contains no information about the correlation at different space-time points. For this purpose, higher-order probability densities need to be considered.

Let us introduce the joint probability density function via Pn,m(r₁,...,r_n+m,t1, ...,tn+m)

=pn,m[E^∗(r₁,t1), ...,E^∗(rn,tn);E(r_n+1,tn+1), ...,E(rn+m,tn+m)], (2.3) wherenandmare positive integers and the asterisk denotes complex conjugation.

The probability density Pn,m(r₁, ...,rn+m,t₁, ...,tn+m) represents the probability that each ofE(r₁,t₁), ...,E(r_n+m,tn+m)obtains a value in an infinitesimal surrounding of dE1, ..., dEn+m. The correlation of order(n,m)can now be expressed as

Γ^(n,m)(r₁, ...,r_n+m,t1, ...,tn+m)

=E^∗(r₁,t1), ...,E^∗(r_n,tn)E(r_n+1,tn+1), ...,E(r_n+m,tn+m)^,

=

... E^∗₁...E^∗_nE_n+1...En+mPn,m(r₁, ...,r_n+m,t₁, ...,tn+m)dE₁, ..., dEn+m. (2.4)

Every higher-order probability density function contains all the information carried by the lower-order functions. In general, there thus exists an infinite hierarchy of probability densities that can be utilized in the analysis of random processes.

2.1.3 Stationarity and ergodicity

When all the probability density functions are independent of the origin of time the process is said to be statistically stationary. In optics, sunlight as well as con- tinuous laser sources are examples of statistically stationary light. Mathematically stationarity implies that

Pn,m(r₁, ...,r_n+m,t1+τ, ...,tn+m+τ) =Pn,m(r₁, ...,r_n+m,t1, ...,tn+m), (2.5) for all values ofτand positive integersnandm. Furthermore, when all realizations in an ensemble contain the same statistical information about the stationary random process it is said to be ergodic. For ergodic processes, the ensemble average equals the time average [11].

As a specific example, in the case of second-order correlations [n = m = 1 in Eq. (2.4)] withr₁ =r₂=r, the auto-correlation function of a statistically stationary field can be written as

Γ^(1,1)(r,r,t1,t2) =E^∗(r,t1)E(r,t2)=E^∗(r,t)E(r,t+τ)=Γ^(1,1)(r,r,τ), (2.6) and depends on the time arguments only through the differenceτ=t2−t1. Equa- tion (2.6) gives a measure for the statistical correlation between the field fluctuations at positionrseparated by timeτ. In this thesis, we mostly consider phenomena that can be characterized by second-order correlations. Accordingly, in these cases we suppress the explicit reference to order.

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2.2 SECOND-ORDER COHERENCE IN SPACE-TIME DOMAIN

The second-order correlations of a statistically stationary scalar field between two space-time points are described by the mutual coherence function [12, 13]

Γ(r₁,r₂,τ) =E^∗(r₁,t)E(r₂,t+τ)^. ^(2.7) The mutual coherence function is Hermitian in the sense that

Γ(r₂,r₁,−^τ) =Γ^∗(r₁,r₂,τ), (2.8) and nonnegative definite [10]. The mutual coherence function is often presented in the normalized form [12, 13]

γ(r₁,r₂,τ) = ^Γ(r₁,r2,τ)

I(r₁,t)I(r₂,t)^, ^(2.9) whereI(r_n,t) = Γ(r_n,r_n, 0) is the averaged intensity at pointr_n, n ∈ {^{1, 2}}^{. The} parameterγ(r₁,r₂,τ)is the complex spatiotemporal degree of coherence and gives a direct measure for the correlation of the fluctuating field at the two space-time points. This function satisfies 0 ≤ |^γ(r₁,r₂,τ)| ≤ 1 where the upper limit corresponds to complete correlation and the lower limit represents the absence of correlation. In the former case the field is said to be completely coherent and, in the latter, completely incoherent at points r₁, r₂ and time separation τ. The complex degree of coherence has an important physical meaning as it specifies the visibility and location of the intensity fringes in Young’s double-pinhole experiment [10, 19].

2.3 SECOND-ORDER COHERENCE IN SPACE-FREQUENCY DOMAIN In many circumstances it is more convenient to examine the spectral coherence properties rather than temporal characteristics. In the space-frequency domain the basic quantity is the cross-spectral density function, which is defined as the Fourier transform of the mutual coherence function [10]

W(r₁,r₂,ω) = ¹ 2π

_∞

−∞Γ(r₁,r₂,τ)exp(iωτ)dτ, (2.10) whereωdenotes the angular frequency. The inverse of the above function is in turn given by

Γ(r₁,r₂,τ) = _∞

0 W(r₁,r₂,ω)exp(−iωτ)dω, (2.11) where the different integration limits are due to the complex analytic signal representation. The Fourier transform pair is a generalized version of the Wiener–

Khintchine theorem that links the auto-correlation function in Eq. (2.6) to the spectral density functionW(r,r,ω) =I(r,ω). The spectral density or spectrum can be viewed as the average intensity at frequencyω. Due to the properties of the mutual coherence function and Eq. (2.10) the cross-spectral density function obeys the Hermiticity condition

W(r₂,r₁,ω) =W^∗(r₁,r₂,ω), (2.12)

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and is a nonnegative definite function [10].

Under some very general conditions, it is also possible to construct an ensemble of monochromatic sample functionsE(r,ω), in order to represent the cross-spectral density of the field as a cross-correlation function [23–25]

W(r₁,r2,ω) =E^∗(r₁,ω)E(r2,ω)^, ^(2.13) where the average is now taken over an ensemble of spectral realizations. It is important to note that regardless of the Fourier transform between W(r₁,r₂,ω) and Γ(r₁,r₂,τ), the space-frequency realizations E(r,ω) in Eq. (2.13) are not Fourier transforms of the space-time realizationsE(r,t).

In analogy with Eq. (2.9) in the space-time domain the cross-spectral density is normalized by setting [51–54]

µ(r₁,r₂,ω) = ^W(r₁,r₂,ω)

I(r₁,ω)I(r₂,ω)^, ^(2.14) where I(rn,ω) = W(rn,rn,ω), is the spectral density at point rn, n ∈ {^{1, 2}}^. The quantityµ(r₁,r₂,ω) is known as the complex spectral degree of coherence of the random field between the two points r₁ and r₂ at frequency ω. The normal- ized cross-spectral density plays a role analogous to the complex degree of coherence inroduced in the space-time domain and is bounded in absolute value as 0 ≤ |^µ(r₁,r₂,ω)| ≤ 1. The extreme values one and zero correspond to complete correlation and the absence of correlation, respectively. The functions µ(r₁,r₂,ω) and γ(r₁,r₂,τ) are not Fourier transforms of each other, but there is a connection between them [55].

2.4 SCALAR-FIELD INTENSITY CORRELATIONS

In Secs. 2.2 and 2.3 we discussed the formalism of second-order coherence of scalar fields in the space-time and space-frequency domains, where the key concepts are the mutual coherence and cross-spectral density functions, respectively. However, there exist some coherence effects whose treatment requires the use of higher-order correlations. The ones that are of special interest to us are the fourth-order correlation functions that describe intensity correlation between two space-time points.

These quantities are central in the Hanbury Brown–Twiss type intensity interferometer [56–59].

2.4.1 Fields obeying Gaussian statistics

We begin our analysis of intensity correlations by considering the properties of scalar fields whose statistics can be characterized by Gaussian random processes.

The fourth-order correlation functions with(n,m) = (2, 2) that describe intensity correlations are defined as

Γ^(2,2)(r₁,r₂,r₃,r₄,t1,t2,t3,t4) =E^∗(r₁,t1)E^∗(r₂,t2)E(r₃,t3)E(r₄,t4)^, ^(2.15) in the space-time domain [see Eq. (2.4)] and

W^(2,2)(r₁,r₂,r₃,r₄,ω) =E^∗(r₁,ω)E^∗(r₂,ω)E(r₃,ω)E(r₄,ω)^, ^(2.16)

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in the space-frequency domain [10]. We have written Eq. (2.16) in a simplified form analogous to Eq. (2.13), in whichE(r,ω)can be viewed to represent the field at fre- quencyω. In general, there is no simple connection between the fourth- and second- order correlation functions. However, there exists a subclass of random processes whose joint probability densities are all Gaussian distributions. These so-called Gaussian random processes satisfy the Gaussian moment theorem [10], stating that the correlations of order(n,n)can be expressed in terms of the lowest-order correlation functions. Mathematically the moment theorem implies that the correlation functions in Eqs. (2.15) and (2.16) of a zero-mean stationary field obeying Gaussian statistics can be written in terms of the mutual coherence and cross-spectral density functions as

Γ^(2,2)(r₁,r2,r3,r₄,t₁,t2,t3,t₄) =Γ(r₁,r3,t3−t₁)Γ(r2,r₄,t₄−t2)

+Γ(r₁,r₄,t4−t1)Γ(r₂,r₃,t3−t2), (2.17) W^(2,2)(r₁,r₂,r₃,r₄,ω) =W(r₁,r₃,ω)W(r₂,r₄,ω)

+W(r₁,r₄,ω)W(r₂,r₃,ω), (2.18) respectively. Thermal, pseudo-thermal, and some multimode laser radiation are examples of light fields obeying Gaussian statistics.

A situation that is of particular interest isr₃=r₁,r₄=r₂,t3=t1, andt4=t2. In this case, Eqs. (2.17) and (2.18) simplify to

Γ^(2,2)(r₁,r₂,τ) =Γ(r₁,r₁, 0)Γ(r₂,r₂, 0) +Γ(r₁,r₂,τ)Γ(r₂,r₁,−^τ), (2.19) W^(2,2)(r₁,r₂,ω) =W(r₁,r₁,ω)W(r₂,r₂,ω) +W(r₁,r₂,ω)W(r₂,r₁,ω). (2.20) Furthermore, using Eqs. (2.8) and (2.12) as well as the facts thatΓ(r,r, 0) =I(r,t) andW(r,r,ω) =I(r,ω), the above equations take the forms

Γ^(2,2)(r₁,r₂,τ) =I(r₁,t)I(r₂,t)+|^Γ(r₁,r₂,τ)|²^, ^(2.21) W^(2,2)(r₁,r₂,ω) =I(r₁,ω)I(r₂,ω)+|W(r₁,r₂,ω)|²^. ^(2.22) These are the basic formulas that govern intensity interferometry in the space-time and space-frequency domains.

2.4.2 Intensity interferometry

Let us consider the Hanbury Brown–Twiss type intensity interferometer depicted in Fig. 2.1. Two photodetectors situated at pointsr₁andr₂generate photocurrents that are proportional to the intensity of light incident on them. One signal is delayed byτafter which the signals are correlated. Analogous arrangement but withτ=0 and variable separation between the detectors corresponds to the famous Hanbury Brown–Twiss stellar interferometer, which can be used to determine the angular diameters of stars [56, 58].

The instantaneous intensities at the photodetectors are given by

I(r_n,t) =E^∗(r_n,t)E(r_n,t), n∈ {^{1, 2}}^. ^(2.23) Introducing the intensity fluctuation around their average valuesI(r_n,t)^as

∆I(r_n,t) =I(r_n,t)− I(r_n,t)^, n∈ {^{1, 2}}^, ^(2.24)

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Figure 2.1: Hanbury Brown–Twiss interferometer. Incoming light is directed to two photodetectors at positionsr1,r2, and the photocurrents they generate are correlated. A delay line is used to introduce a time delayτbetween the two signals.

the correlation of intensities in the two photodetectors can be represented as I(r₁,t)I(r₂,t+τ)=I(r₁,t)I(r₂,t)+^∆I(r₁,t)∆I(r₂,t+τ)^. ^(2.25) Above we utilized the fact that stationarity implies thatI(r,t+τ)=I(r,t)^{. The} first term on the right-hand side in the above equation is a constant while the second term denotes the correlation of intensity fluctuations.

Expressing the left-hand side of Eq. (2.25) in terms of the fluctuating field one finds that

I(r₁,t)I(r₂,t+τ)=E^∗(r₁,t)E(r₁,t)E^∗(r₂,t+τ)E(r₂,t+τ)

=Γ^(2,2)(r₁,r₂,τ). (2.26) Assuming that the field arriving at the detectors obeys Gaussian statistics, the fourth- order correlation function Γ^(2,2)(r₁,r₂,τ)in Eq. (2.26) can be expressed in terms of Eq. (2.21) as

I(r₁,t)I(r₂,t+τ)=I(r₁,t)I(r₂,t)+|^Γ(r₁,r₂,τ)|²^. ^(2.27) On substituting from Eq. (2.27) into Eq. (2.25) we find that the correlation of the intensity fluctuations is given by the expression

∆I(r₁,t)∆I(r₂,t+τ)=|^Γ(r₁,r₂,τ)|²^. ^(2.28) In the space-frequency domain, analogous analysis using Eq. (2.22) yields

∆I(r₁,ω)∆I(r₂,ω)=|W(r₁,r₂,ω)|²^. ^(2.29) Equations (2.28) and (2.29) imply that for light obeying Gaussian statistics the correlation between intensity fluctuations at detectors can be expressed in the terms of the absolute square of the second-order correlation functions. In other words, the intensity normalized correlation function of the intensity fluctuations coincides with the squared magnitude of the complex degree of coherence in the space-time and space-frequency domains. As we will see in Chap. 4, the theory of optical correlation imaging of PublicationsI–III can be understood in terms of the results of the Hanbury Brown–Twiss effect.

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3 Fundamentals of electromagnetic coherence theory

In the previous chapter we limited our discussion to scalar fields whose coherence properties can be expressed in terms of scalar functions. To fully understand optical phenomena, such as geometric phase and interference, however, it is necessary to consider the vector nature of light. We begin the chapter by extending the scalar second-order coherence theory to random electromagnetic beams by introducing suitable correlation matrices. Next the polarization properties of the field are examined in terms of the polarization matrices, degree of polarization, and the Stokes parameters. Finally, the concepts of Stokes parameters and scalar intensity correlations are generalized by introducing the two-point Stokes parameters and Stokes-parameter correlations, respectively.

3.1 SECOND-ORDER ELECTROMAGNETIC COHERENCE

We consider the electromagnetic coherence both in the space-time and space-frequency domains. We focus on the electric field only because often no significant interaction takes place between the magnetic field and matter at optical frequencies.

3.1.1 Space-time domain

An electric field realization of a random light beam at positionrand timetpropa- gating in the positivezdirection can be expressed as a vector

E(r,t) = [Ex(r,t),Ey(r,t)]^T, (3.1) where T denotes transpose. In addition,E_i(r,t),i∈ {x,y}, are the analytic signals of the Cartesian electric field components perpendicular to the propagation direction.

Assuming that the field fluctuations are stationary, the second-order correlations between two space-time points are described by the 2×2 (electric) mutual coherence matrix [10, 22, 60]

Γ(r₁,r₂,τ) =^E^∗(r₁,t)E^T(r₂,t+τ)=

Γ_xx(r₁,r₂,τ) Γ_xy(r₁,r₂,τ) Γ_yx(r₁,r₂,τ) Γ_yy(r₁,r₂,τ)

, (3.2) where Γ_ij(r₁,r₂,τ) = E_i^∗(r₁,t)Ej(r₂,t+τ)^, i,j ∈ {x,y}. The mutual coherence matrix satisfies the Hermiticity relation Γ^∗_ij(r₁,r2,τ) = Γ_ji(r2,r₁,−^τ) and is a nonnegative definite function [10].

It is convenient to represent the strength of the correlation of a vector-valued electric field using a single scalar quantity. For this purpose, we introduce the temporal electromagnetic degree of coherence as [28, 61]

γ_EM(r₁,r₂,τ) =

tr[Γ(r₁,r₂,τ)Γ(r₂,r₁,−^τ)]

tr[Γ(r₁,r₁, 0)]tr[Γ(r₂,r₂, 0] 1/2

, (3.3)

where tr denotes the trace of a matrix and tr[Γ(rn,rn, 0)]is the average intensity at pointrn,n ∈ {^{1, 2}}[62]. The electromagnetic degree of coherence is the intensity- weighted average of the magnitudes of the correlation coefficients between the field

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components. It is real valued and bounded to the interval 0 ≤ ^γEM(r₁,r₂,τ) ≤ ^1.

The values zero and unity correspond to no correlation and complete correlation among the field components, respectively. The quantityγ_EM(r₁,r₂,τ) can be seen as an extension of the scalar degree of coherence to the electromagnetic case and it reduces to|^γ(r₁,r₂,τ)| when the field is uniformly fully polarized. Physically the electromagnetic degree of coherence quantifies the modulation contrasts of the time- domain Stokes parameters (cf. Sec. 3.2.3) in Young’s double-pinhole interference experiment [28, 42].

3.1.2 Space-frequency domain

In the space-frequency domain the second-order coherence properties of an electromagnetic beam are represented by the (electric) cross-spectral density matrix, which is defined as the Fourier transform of the mutual coherence matrix [10, 60]

W(r₁,r2,ω) = ¹ 2π

_∞

−∞Γ(r₁,r2,τ)exp(iωτ)dτ, (3.4) with an inverse relation

Γ(r₁,r₂,τ) = _∞

0 W(r₁,r₂,ω)exp(−iωτ)dω. (3.5) The above Fourier transform pair is the electromagnetic form of the generalized Wiener–Khintchine theorem referred to in Sec. 2.3.

Alternatively to Eq. (3.4), the cross-spectral density matrix can be expressed as a correlation matrix of the form [63, 64]

W(r₁,r₂,ω) =^E^∗(r₁,ω)E^T(r₂,ω)=

Wxx(r₁,r2,ω) Wxy(r₁,r2,ω) Wyx(r₁,r2,ω) Wyy(r₁,r2,ω)

, (3.6) where Wij(r₁,r₂,ω) = E^∗_i(r₁,ω)Ej(r₂,ω)^, i,j ∈ {x,y}. The electric field vectors E(r,ω) = [Ex(r,ω),Ey(r,ω)]^T are members of a suitably constructed ensemble of monochromatic realizations. According to Eqs. (3.4) and (3.6) as well as the properties of the mutual coherence matrix, the cross-spectral density matrix satisfies the Hermiticity propertyW_ij(r₁,r₂,ω) =W_ji^∗(r₂,r₁,ω)and various nonnegative definite- ness conditions [10].

In analogy to the temporal degree of coherence, it is possible to introduce a scalar measure for the correlations among the field components in the space-frequency domain. Such a quantity is the spectral electromagnetic degree of coherence, defined as [28, 65]

µ_EM(r₁,r₂,ω) =

tr[W(r₁,r₂,ω)W(r₂,r₁,ω)]

tr[W(r₁,r₁,ω)]tr[W(r2,r2,ω] 1/2

, (3.7)

where tr[W(r_n,r_n,ω)] is the spectral density at point r_n, n ∈ {^{1, 2}}. The spectral degree of coherence is bounded within 0≤ ^µEM(r₁,r₂,ω) ≤ 1 with the upper and lower limits corresponding to complete correlation and total lack of correlation among the field components, respectively. As in the space-time domain, parameter µ_EM(r₁,r₂,ω) is a vector-field generalization of the scalar degree of coherence

|^µ(r₁,r₂,ω)|. Equation (3.7) can be interpreted as a measure of the modulation contrasts of the four spectral Stokes parameters (cf. Sec. 3.2.3) in Young’s interference experiment [40, 41]. In contrast to the scalar case, no straightforward connection exists betweenγ_EM(r₁,r₂,τ)and µ_EM(r₁,r₂,ω). However, in some specific circumstances a relationship between the two degrees has been established [66].

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3.2 POLARIZATION

The polarization of an electromagnetic field describes the behavior of the electric field vector [67, 68]. Light whose polarization state evolves with maximal randomness is said to be unpolarized while light whose polarization state remains unaltered is completely polarized. Partially polarized light refers to the situations between these two cases. The mathematical analysis of the polarization of random electromagnetic beams can be carried out using the 2×2 correlation matrices that are single-point versions of the mutual coherence and cross-spectral density matrices.

Visually the polarization state of light is described by using the Stokes parameters and the Poincaré sphere.

3.2.1 Polarization matrices

The time-domain polarization properties of the two-component electromagnetic beams are described using the temporal polarization matrix [8, 22, 69]

J(r) =^E^∗(r,t)E^T(r,t)=

Jxx(r) Jxy(r) Jyx(r) Jyy(r)

, (3.8)

where Jij(r) = E_i^∗(r,t)Ej(r,t)^, i,j ∈ {x,y}. The polarization matrix is the equal- time mutual coherence matrix evaluated at a single point, i.e.,J(r) = Γ(r,r, 0). The diagonal elements of the matrix denote the average intensities of the orthogonal x andy components while the off-diagonal elements give the equal-time correlation between them.

The spectral polarization matrix corresponds to the single point cross-spectral density matrix,W(r,r,ω) = Φ(r,ω). Utilizing the Wiener–Khintchine theorem of Eqs. (3.4) and (3.5), we find the relations [70]

Φ(r,ω) = ¹ 2π

_∞

−∞Γ(r,r,τ)exp(iωτ)dτ, (3.9) J(r) =

_∞

0 Φ(r,ω)dω. (3.10)

The former equation indicates that partial spectral polarization is determined by the electromagnetic temporal coherence. Furthermore, both expressions demonstrate that the states of partial polarization can be different in the two domains [70, 71].

Besides Eq. (3.9), the polarization matrix can be expressed using Eq. (3.6) in the form

Φ(r,ω) =^E^∗(r,ω)E^T(r,ω)=

Φ_xx(r,ω) Φ_xy(r,ω) Φyx(r,ω) Φyy(r,ω)

, (3.11)

where Φ_ij(r,ω) = E_i^∗(r,ω)Ej(r,ω)^, i,j ∈ {x,y}. The diagonal elements of the matrix are proportional to the spectral densities of the components while the off- diagonal elements are measures of the magnitude of their correlation at frequency ω.

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3.2.2 Degree of polarization

The strength of the correlation between the orthogonal field components can be expressed in terms of the correlation coefficients [10, 11]

jxy(r) = ^J^xy(r) Jxx(r)Jyy(r)

, (3.12)

φxy(r,ω) = ^Φ^xy(r,ω)

Φxx(r,ω)Φyy(r,ω)^, ^(3.13)

corresponding to the normalized off-diagonal elements of the polarization matrices J(r) and Φ(r,ω), respectively. The coefficients satisfy 0 ≤ |jxy(r)| ≤ ^{1 and} 0 ≤ |^φxy(r,ω)| ≤ 1, where unity denotes complete correlation and zero represents absence of correlation between the orthogonal components. Due to the Hermiticity of the polarization matrices,jxy(r) =j^∗_yx(r)andφxy(r,ω) =φ_yx^∗ (r,ω)hold.

Next, we consider the cases of full correlation and complete noncorrelation more closely. When |jxy(r)| = |^φxy(r,ω)| = 0 the off-diagonal elements of the polarization matrices satisfy Jxy(r) = Jyx(r) = 0 and Φ_xy(r,ω) = Φ_yx(r,ω) = 0.

Additionally, setting the diagonal elements to be equal, i.e., Jxx(r) = Jyy(r) and Φ_xx(r,ω) = Φ_yy(r,ω), leads to temporally and spectrally unpolarized beams, respectively. Accordingly, the polarization matrices have the forms

J_u(r) =Jxx(r) 1 0

0 1

, (3.14)

Φ_u(r,ω) =Φ_xx(r,ω) 1 0

0 1

, (3.15)

and are thus proportional to the 2×2 unit matrices. When|jxy(r)|=|^φxy(r,ω)|=1 the light is fully polarized and the polarization matrices can be expressed as

J_p=

Jxx JxxJyyexp(iα) JxxJyyexp(−iα) Jyy

, (3.16)

Φ_p=

Φxx ΦxxΦyyexp(iα) Φ_xxΦ_yyexp(iα) Φ_yy

, (3.17)

where α and α are real. In the above equations the dependency on position and frequency was dropped for brevity. Equations (3.14) – (3.17) hold irrespective of the particular choice of thexandydirections.

In general, the polarization matrixJ(r) [Φ(r,ω)] of any beam can be uniquely expressed as a sum of a completely unpolarized,J_u(r)[Φ_u(r,ω)], and a fully polarized,J_p(r)[Φ_p(r,ω)], constituents [10, 11, 22]. The degree of polarization is defined as the ratio of the intensity (spectral density) of the polarized part to that of the total field

P(r) = ^tr[J_p(r)]

tr[J(r)] =

1−^4det[J(r)]

tr²[J(r)]

1/2

, (3.18)

P(r,ω) = ^tr[Φ_p(r,ω)]

tr[Φ(r,ω)] =

1−^4det[Φ(r,ω)]

tr²[Φ(r,ω)]

1/2

, (3.19)

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where det denotes the determinant of a matrix. Using Eqs. (3.14) and (3.16) in Eq. (3.18) we observe that the valuesP(r) = 0 andP(r) =1 correspond to temporally unpolarized and polarized beams, respectively. When the degree of polarization obeys 0<P(r)<1 the light is partially polarized in the time domain. Similar reasoning can be applied to the spectral degree of polarization,P(r,ω). In view of Eqs. (3.18), (3.19), and (3.10) the temporal and spectral degrees of polarization can have different values [70, 71].

3.2.3 Stokes parameters

Besides the polarization matrix representation, an alternative and equivalent way to characterize the polarization properties of a random light beam is to use the Stokes parameters. They are defined in terms of the polarization matrix elements as [5, 67–69]

S0(r) =Jxx(r) +Jyy(r), (3.20a) S1(r) =Jxx(r)−Jyy(r), (3.20b) S2(r) =Jxy(r) +Jyx(r), (3.20c) S3(r) =i[Jyx(r)−Jxy(r)], (3.20d) in the space-time domain and

S0(r,ω) =Φxx(r,ω) +Φyy(r,ω), (3.21a) S1(r,ω) =Φxx(r,ω)−^Φyy(r,ω), (3.21b) S2(r,ω) =Φ_xy(r,ω) +Φ_yx(r,ω), (3.21c) S3(r,ω) =i[Φ_yx(r,ω)−^Φxy(r,ω)], (3.21d) in the space-frequency domain. The first quantity S0 denotes the total intensity (spectral density) of the field while the polarization Stokes parametersS1,S2, and S3 give the intensity difference between thex and y linearly polarized parts,+45^◦ and−⁴⁵^◦linearly polarized parts, and right-hand and left-hand circularly polarized parts, respectively.

The temporal and spectral degrees of polarization in Eqs. (3.18) and (3.19) can be rewritten in terms of the polarization Stokes parameters as

P(r) = ₃

n=1

∑

s²_n(r) 1/2

, (3.22)

P(^r,^ω) = ₃

n=1

∑

s²_n(r,ω) 1/2

, (3.23)

where

sn(r) = ^Sⁿ(r)

S0(r)^, ⁿ∈ {^{1, 2, 3}}^, ^(3.24) sn(r,ω) = ^Sⁿ(r,ω)

S0(r,ω)^, ⁿ∈ {^{1, 2, 3}}^, ^(3.25) are the intensity (spectral density) normalized versions of the Stokes parameters in Eqs. (3.20b) – (3.20d) and (3.21b) – (3.21d). The degree of polarization thus describes

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Figure 3.1: Unit radius Poincaré sphere in the polarization space defined by the orthonormal unit vectors ˆs₁, ˆs₂, and ˆs₃. The length of the Poincaré vector ˆP is equal to the degree of polarization while the position of the vector specifies the polarization state of the polarized part. The surface of the sphere denotes fully polarized states while the partially polarized states are inside the sphere. The origin (O) represents an unpolarized field.

the length of a vector whose components are the normalized Stokes parameters.

This vector is the normalized Poincaré vector which in the temporal and spectral domain, respectively, is given by [67, 68]

Pˆ(r) =sˆ₁s₁(r) +sˆ2s2(r) +sˆ3s3(r), (3.26) Pˆ(r,ω) =sˆ₁s1(r,ω) +sˆ₂s2(r,ω) +sˆ₃s3(r,ω), (3.27) where ˆs_n, n ∈ {^{1, 2, 3}}, are the orthogonal unit vectors. Every polarization state has a unique Poincaré vector and can be graphically visualized in terms of the unit Poincaré sphere depicted in Fig. 3.1. The points on the sphere’s surface denote a light beam that is fully polarized (P = 1) with linear and circular polarization states located on the equator and the poles, respectively. The points inside the sphere denote partially polarized states (P < 0) with the origin (O) corresponding to the case of completely unpolarized light (P = 0). The analysis of the geometric phase that arises in electromagnetic interference, discussed in PublicationsIV–VI and summarized in Chap. 5, is based on the Poincaré sphere representation.

3.3 TWO-POINT STOKES PARAMETERS

The Stokes parameters introduced in Sec. 3.2.3 are defined in terms of the elements of the two polarization matricesJ(r)andΦ(r,ω). Accordingly, they describe the correlation between the orthogonal components of the electric field vectors at a single point only. However, when we consider how the polarization state of a field changes on propagation [11, 72] or in diffraction [40] (PublicationsIVandV), it is necessary

Optical correlation imaging and geometric phase in electromagnetic interference

Dissertations in Forestry and Natural Sciences

ANTTI HANNONEN

OPTICAL CORRELATION IMAGING AND GEOMETRIC PHASE IN ELECTROMAGNETIC INTERFERENCE

Antti Hannonen

OPTICAL CORRELATION IMAGING AND GEOMETRIC PHASE IN

ELECTROMAGNETIC INTERFERENCE

TABLE OF CONTENTS

1 Introduction

2 Basics of scalar coherence theory

∑

3 Fundamentals of electromagnetic coherence theory

∑

∑