Polarization and electromagnetic coherence of light fields probed with nanoscatterers

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DISSERTATIONS | LASSE-PETTERI LEPPÄNEN | POLARIZATION AND ELECTROMAGNETIC COHERENCE... | No 22

LASSE-PETTERI LEPPÄNEN

POLARIZATION AND ELECTROMAGNETIC COHERENCE OF LIGHT FIELDS PROBED WITH NANOSCATTERERS PUBLICATIONS OF

THE UNIVERSITY OF EASTERN FINLAND

This thesis deals with partially coherent and partially polarized electromagnetic fields and the measurement of their statistical properties

through probing with nanoscale scatterers.

The connection of spatiotemporal and spectral coherence and their relationship to polarization

are particularly elucidated. Detection of random light beams and optical near fields with

nanoscatterers is considered theoretically and polarization probing is demonstrated with the

first proof-of-principle experiment.

LASSE-PETTERI LEPPÄNEN

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LASSE-PETTERI LEPP ¨ANEN

Polarization and

electromagnetic coherence of light fields probed with

nanoscatterers

Publications of the University of Eastern Finland Dissertations in Forestry and Natural Sciences

No 220

Academic Dissertation

To be presented by permission of the Faculty of Science and Forestry for public examination in the Auditorium C2 in Carelia Building at the University of

Eastern Finland, Joensuu, on May 20, 2016, at 12 o’clock noon.

Institute of Photonics

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Prof. Matti Vornanen, Prof. Pekka Toivanen

Distribution:

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

http://www.uef.fi/kirjasto

ISBN: 978-952-61-2100-0 (printed) ISSNL: 1798-5668

ISSN: 1798-5668 ISBN: 978-952-61-2101-7 (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: lasse-petteri.leppanen@uef.fi

Supervisors: Associate Professor Tero Set¨al¨a, D.Sc. (Tech) University of Eastern Finland

Institute of Photonics P. O. Box 111

80101 Joensuu FINLAND

email: tero.setala@uef.fi

Professor Ari T. Friberg, Ph.D., D.Sc. (Tech) University of Eastern Finland

Institute of Photonics P. O. Box 111

80101 Joensuu FINLAND

email: ari.friberg@uef.fi

Reviewers: Assistant Professor Jonathan Petruccelli, Ph.D.

University at Albany Department of Physics 1400 Washington Avenue Albany, NY 12222 USA

email: jpetruccelli@albany.edu Professor Ilkka Tittonen, D.Sc. (Tech) Aalto University

Department of Micro- and Nanosciences P. O. Box 13500

00076 Aalto FINLAND

email: ilkka.tittonen@aalto.fi

Opponent: Associate Professor Olga Korotkova, Ph.D.

University of Miami Department of Physics 1320 Campo Sano Ave.

Coral Gables, FL 33146 USA

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theory of partial polarization and electromagnetic coherence are introduced. The connection between the time-domain and frequency- domain electromagnetic degrees of coherence is illustrated by specific examples. Furthermore, the link between the temporal electromagnetic degree of coherence and the variations of the polarization Stokes parameters in Michelson’s interferometer is demonstrated.

In addition, an interferometric interpretation for the degree of polarization of optical beams is established and its validity is shown in Young’s two pinhole experiment.

In the second part of the thesis, methods to obtain polarization and coherence information about beam-like waves by probing the field with nanoparticles are analyzed. The polarization probing is demonstrated with the first-ever nanoscatterer measurements. Fur- thermore, theory for nanoprobing the spatial coherence of partially coherent beam fields is presented. Finally, possibilities to measure the polarization and coherence characteristics of arbitrary random electromagnetic fields are discussed.

Universal Decimal Classification: 531.715, 535.4, 535.5, 537.87, 620.3, 681.787

INSPEC Thesaurus: electromagnetic waves; optics; light; light coherence;

light polarization; light interference; light interferometry; Michelson in- terferometers; light scattering; nanoparticles

Yleinen suomalainen asiasanasto: sähkömagneettinen säteily; optinen sä- teily; valo; optiikka; koherenssi; polarisaatio; interferenssi; sironta; nano- hiukkaset

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Preface

I thank everyone involved in making this thesis and especially my supervisors Prof. Tero Set¨al¨a and Prof. Ari T. Friberg. I also thank my coauthors Dr. Kimmo Saastamoinen and M.Sc. Joonas Lehto- lahti for their contributions. The support and guidance I have re- ceived from the staff of the Department of Physics and Mathematics are appreciated.

I am grateful for the comments and suggestions from the reviewers of this thesis, Prof. Jonathan Petruccelli and Prof. Ilkka Tittonen. I also acknowledge the financial support from the Emil Aaltonen Foundation, the Academy of Finland, and the University of Eastern Finland.

I thank my coworkers for the discussions and activities during the coffee and lunch breaks. Finally, I express my greatest gratitude to my friends, family, and especially to Noora for all the support and help over the years.

Joensuu, April 13, 2016 Lasse-Petteri Lepp¨anen

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of polarization and electromagnetic coherence of light fields and the following selection of the author’s publications:

I L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Connection of electromagnetic degrees of coherence in space-time and space- frequency domains,”Opt. Lett. 41,1821–1824 (2016).

II L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal electromagnetic degree of coherence and Stokes-parameter modulations in Michelson’s interferometer,”Appl. Phys. B 122, 32 (2016).

III L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä,

“Interferometric interpretation for the degree of polarization of classical optical beams,”New J. Phys. 16,113059 (2014).

IV L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,”J. Opt. Soc. Am. A 31,1627–1635 (2014).

V L.-P. Leppänen, K. Saastamoinen, J. Lehtolahti, A. T. Friberg, and T. Setälä, “Detection of partial polarization of light beams with dipolar nanocubes,”Opt. Express 24,1472–1479 (2016).

VI L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä,

“Detection of electromagnetic degree of coherence with nanoscatterers: comparison with Young’s interferometer,”Opt. Lett.

40,2898–2901 (2015).

Throughout the overview, these papers will be referred to by Ro- man numerals.

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The results reported in Publications III, IVand VIhave been presented in the following international conferences:

1. “Detecting polarization with a nanoprobe,” Electromagnetic Optics with Random Light: 1st Joensuu Conference on Co- herence and Random Polarization (Joensuu, Finland, 2014), poster.

2. “Detection of partial polarization with a nanoscatterer,” 23rd Congress, International Commission for Optics (ICO), (Santi- ago de Compostela, Spain, 2014), oral presentation.

3. “Probing of polarization and coherence in optical near fields,”

International Conference on Optics, Photonics and Photosci- ences (Havana, Cuba, 2014), invited paper.

4. “Interferometric interpretation for the degree of polarization of optical beams,” Northern Optics & Photonics 2015 (Lap- peenranta, Finland, 2015), poster.

5. “Detection of electromagnetic degree of coherence with nanoscatterers,” The Eleventh Finland-Japan Joint Symposium on Optics in Engineering (Joensuu, Finland, 2015), poster.

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of mathematical derivations and all numerical computations. In addition, he significantly contributed to the measurements reported in publications III and V. The author wrote the first drafts of all manuscripts and substantially contributed to the interpretation of the results.

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1 Introduction

The nature of light has been investigated for centuries. The first theoretical formulations were limited to the rectilinear propagation and reflection of light [1]. The polarization properties were recog- nized only later, due to the work by Bartholinus and Huygens in the 17th century. Furthermore, the wave nature of optical fields was ac- cepted after the interference experiments carried out by Young [2,3].

In addition, in the 19th century, Stokes introduced four measurable quantities to describe the polarization of light [4]. A few years later, the electromagnetic behavior of optical fields was introduced by Maxwell, paving the way to modern understanding of the nature of light. However, the statistical properties were not understood until the 20th century [5].

In fact, the early theoretical studies on the coherence properties of light, both classical [6–8] and quantum [9,10], were limited to the space-time domain. In the 1980s, some formulations were extended into the space-frequency domain [11–14]. Furthermore, the focus of the investigations was on beam fields and their directionality [15].

Later, however, tools to describe the polarization and coherence of classical light have been extended to account for the coherence of two-component and three-component and the polarization of three- component electromagnetic fields [16–19].

In recent years, the polarization and coherence properties of arbitrary three-component light, for instance random near fields and nonparaxial waves, have gained considerable interest [20–25]. The description of these various light fields requires the general polarization and electromagnetic coherence theories. Furthermore, new arrangements to acquire information on the electromagnetic fields are needed since the conventional measurement techniques applied to beams cannot be employed. In addition, the increased research on nanophotonics, such as surface plasmon polaritons [26], pho- tonic crystals [27], and nanometer-sized optical devices [28], require

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novel high-resolution characterization methods suitable for electromagnetic near fields.

The goal of this thesis is to formulate means to obtain polarization and electromagnetic coherence information from classical optical beams and general nonparaxial fields. In the analysis of nanoscale structures scanning near-field optical microscopy (SNOM) has been employed, since it provides mapping of a sample with sub- wavelength resolution [29, 30]. The SNOM technique has also been applied to obtain near-field intensity distributions [28,31,32], image formation in near fields [33], characterization of thermal near fields [34,35], and to study some polarization phenomena [23,36–40]. The operation principle of the SNOM is also suitable for our purposes.

In fact, we scan the field with nanoparticles, detect the radiation scattered by them, and obtain the polarization and coherence properties of the incident light field by knowing the parameters of the setup. The nanometer-sized metal particles can be considered as electric point dipoles, whose behavior in an external field is well known [41]. As it happens, these nanoparticles are widely used in nanophotonics [42].

With these new methods we can then characterize with high resolution, for example, various light fields ranging from a low to high degree of coherence, such as radiation from incandescent lamps, LEDs, and lasers [24, 43–45]. Further, the polarization distribution and the spatial coherence of optical near fields can be investigated, which will enable one to examine, e.g., molecular and nanoparticle clusters, microstructures, and metamaterials.

In the first part of this thesis, we introduce the fundamental concepts of the theory of partial polarization and electromagnetic coherence and it is constructed as follows. In Chap. 2, we review the coherence theory of scalar light fields and discuss the connection between the degrees of coherence in the space-time and space- frequency domains. The scalar theory is extended to cover the analysis of the polarization and coherence of electromagnetic fields in Chap. 3 and these considerations are then applied in the subse- quent chapters. In addition, the relationship between the space-

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Introduction

time and space-frequency electromagnetic degrees of coherence is investigated with a help of specific examples. It turns out that, in general, the two degrees are not equal and no simple closed- form relation exists between them (Publication I). In Chap. 4, we formulate how the temporal electromagnetic degree of coherence and the related coherence time of an optical beam can be obtained from the polarization Stokes-parameter modulations in Michelson’s interferometer (Publication II). In addition, we introduce a new interpretation for the degree of polarization of light beams. This is an interferometric explanation yielding more insight into the role of partial polarization in electromagnetic coherence and interference (Publication III).

In the second part of this thesis, we concentrate on establishing a theoretical framework for nanoparticle probing of partial polarization and electromagnetic coherence information on optical fields.

At the beginning of Chap. 5, we introduce a theoretical background for electromagnetic field scattering by dipolar nanoparticles. Fur- thermore, the scattered field intensities and the degree of polarization are calculated for a partially polarized Gaussian Schell-model beam (Publication IV). In addition, we demonstrate the first-ever nanoparticle probing of a light-beam polarization state (Publica- tion V). We also formulate theoretical considerations for two-probe measurements of the spatial coherence of light beams. These measurements pave the way towards nanoparticle probing of the polarization and spatial coherence information of optical near fields (Publication VI). The concluding remarks are presented in Chap. 6 with discussion on possible future prospects.

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2 Coherence of scalar fields

In this chapter, we focus on the statistical treatment of fluctuating scalar light fields [46]. The random field variations are usually too rapid to be seen by any detector, which rather measure time aver- ages. Therefore what is detected are the correlations of the field fluctuations at one or more space-time points. In this work we re- strict ourselves to second-order coherence theory, corresponding to fluctuations at one or two points. Investigation of the correlations leads to electromagnetic coherence theory, more precisely, to the description of partially coherent, partially polarized electromagnetic beams and nonparaxial waves, such as optical near fields, both in time and frequency domains. These are considered in the subse- quent chapter after a brief look into the coherence of scalar fields.

2.1 TEMPORAL AND SPATIAL COHERENCE OF SCALAR FIELDS

Consider a beam of light, which is uniformly fully polarized and therefore treatable in terms of a scalar theory. Mathematically, the field is of the formE(r,t) =E(r,t)e, whereˆ E(r,t)is a random amplitude and ˆeis a deterministic complex unit vector specifying the polarization state. We further assume that the field is stationary, i.e., no long time-scale field variations take place, and that the field is quasi-monochromatic; the bandwidth,∆ω, is small compared to the mean frequency, ω0, that is ∆ω/ω0 ≪ 1, and the time differ- ences involved are short compared to the coherence time [46].

Imagine the beam is split into two identical parts, e.g., through the use of a 50:50 beamsplitter, and introduce a time delay τ between them. If the delay is sufficiently small, that isτ ≤ ^τ^c^{, where} τ_c is the coherence time of light, and if the beams are recombined, interference fringes are formed on an observation plane. This abil- ity of light to form interference fringes is caused by temporal co-

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herence of the beams, associated with the correlations of the field fluctuations at two instants of time [46,47]. We may also investigate the spatial coherence properties of the fields, corresponding to the correlations of the field fluctuations at two spatial points. For example, insert a screen with two small openings into the beam and consider the field at a point behind the screen. If the two apertures are within the coherence area of the incident light, interference fringes are formed around the observation point [46]. The square root of the coherence area is commonly called the coherence width, while the longitudinal coherence length is the distance the light travels withinτ_c.

Mathematically the correlations between two space-time points (r₁,t) and (r2,t+τ) of a field represented by a complex analytic signal E(r,t), are quantified by a cross-correlation function called the mutual coherence function, defined as [8, 46]

Γ(r₁,r₂,τ) =�^E^∗(r₁,t)E(r₂,t+τ)�^. ^(2.1) Above the angle brackets and asterisk denote time average and complex conjugation, respectively, and owing to stationarity, the coherence function depends on time only through the time difference τ. Provided the field is ergodic the time average equals the ensemble average. Furthermore, we note two important properties of the mutual coherence function. First, it obeys a Hermiticity con- dition Γ(r₁,r₂,τ) = ^Γ^∗(r₂,r₁,−^τ), and second, it is non-negative definite [46]. When the two space-time points coincide, the mutual coherence function assumes the form

Γ(r,r, 0) =�|^E(r,t)|²�= I(r), (2.2) whereI(r)denotes the averaged intensity at pointr. For stationary beams, the intensity is constant over time. It is convenient to normalize the mutual coherence function and introduce the complex degree of coherence as [8, 46, 47]

γ(r1,r2,τ) = ^Γ(r₁,r₂,τ)

I(r₁)I(r₂)^. ^(2.3)

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Coherence of scalar fields

The degree of coherence satisfies 0 ≤ |^γ(r₁,r₂,τ)| ≤ 1, where the lower and upper limits correspond to complete incoherence (non- correlation) and coherence (correlation), respectively.

Another important coherence quantity, the cross-spectral density function, is defined as [46]

W(r₁,r2,ω) = ¹ 2π

∞

−^∞

Γ(r₁,r2,τ)e^iωτdτ, (2.4) whereωis the angular frequency. It is known that the cross-spectral density function can also be expressed as a correlation function

W(r1,r2,ω) =�^E^∗(r1,ω)E(r2,ω)�^, ^(2.5) whereE(r,ω)is a member of an ensemble of realizations that char- acterizes the field at frequencyω, and the angle brackets stand for ensemble average [13, 14]. We emphasize that E(r,ω) is not the Fourier transform ofE(r,t). The cross-spectral density function describes the spectral correlations at two points and at frequency ω and it follows a Hermiticity relation, W(r₁,r₂,ω) = W^∗(r₂,r₁,ω), and is non-negative definite [46]. In analogy to the time-domain complex degree of coherence, the normalized form of the cross- spectral density function [11, 46, 48, 49]

µ(r₁,r₂,ω) = ^W(r₁,r₂,ω)

S(r₁,ω)S(r₂,ω)^, ^(2.6) where S(rn,ω) = W(rn,rn,ω)is the spectral density (or spectrum) at point rn, n ∈ (1, 2), defines the spectral complex degree of coherence. Furthermore, the absolute value of the spectral degree of coherence is limited by 0 ≤ |^µ(r1,r2,ω)| ≤ 1, with the lower and upper limits corresponding to complete incoherence and coherence, respectively.

The inverse of Eq. (2.4) reads as Γ(r1,r2,τ) =

∞

0 W(r1,r2,ω)e⁻^iωτdω, (2.7) where the lower integration limit zero is due to the complex analytic signal representation. Equations (2.4) and (2.7) constitute the

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(generalized) Wiener-Khintchine theorem. However, while the mutual coherence function and the cross-spectral density function are related by Fourier transform, the two degrees of coherence are not Fourier transforms of each other. In fact, the following relationship between the degrees of coherence has been derived [50]

γ(r₁,r₂,τ) =

∞

0

s(r₁,ω)s(r₂,ω)µ(r₁,r₂,ω)e⁻^iωτdω, (2.8) where s(rn,ω) = S(rn,ω)/∞

0 S(rn,ω)dω, n ∈ (1, 2), are the normalized spectra atrn, with

µ(r₁,r₂,ω) = ¹ 2π

1

s(r₁,ω)s(r₂,ω)

∞

−^∞γ(r₁,r₂,τ)e^iωτdτ (2.9) being the inverse relation.

Let us recall some important features of the two degrees of coherence. Consider a quasi-monochromatic scalar field for which the normalized spectra are the same at the two observation points r₁ and r₂. Equation (2.8) then implies that the time-domain spatial coherence at τ = 0 equals the spectral degree of coherence at the central frequency ω₀ [50]. Consequently, µ(r₁,r₂,ω₀) can be obtained, e.g., from the visibility and position of the intensity fringes in Young’s interferometer near the axis where τ = 0. On the other hand, to measureγ(r₁,r₂, 0) for broadband radiation, an achromatic Fourier transformer has to be employed in the setup [51]. Furthermore, it is known that when wide-spectrum light is passed through a narrowband filter and the pass band is reduced to zero, the temporal degree of coherence approaches, not unity, but rather the spectral degree of coherence at the filter’s central frequency [50, 52–54].

To further illustrate the connection between the time-domain and frequency-domain degrees of coherence, consider, e.g., a scalar Gaussian Schell-model (GSM) beam given in Eqs. (A.1)–(A.3) of Appendix A. In addition, assume that the source obeys the scaling law [55] and thus the normalized far-zone spectrum does not depend on direction and equals the normalized spectrum at each

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source point. For a GSM source to follow the scaling law, the ex- ponent in Eq. (A.3) must satisfy ∆r/σ ∝ k∆r, where∆r = r2−^r1, σ represents the transverse coherence length, and k = ω/c is the wave number withcbeing the vacuum speed of light. The spectral degree of coherence is therefore of the form (Publication I)

µ(k∆r) = e⁻^ω²^|^∆^r^|²^/2c²^σ^µ², (2.10) where σ_µ is a constant. We also assume that Aand σ_s of the GSM beam are frequency independent. As shown in Publication I, the temporal degree of coherence assumes the form

γ(^∆r,τ) =ae⁻â²^σ²^ω^τ²^/2e⁻îa²^ω⁰^τe⁻â²^ω²⁰^|^∆^r^|²^/2c²^σ²^µ, (2.11) where a = cσµ/[c²σ_µ²+σ_ω²|^∆^r|²]^1/2 andσω represents the width of the frequency band (see Appendix A). Due to the quantity a, the temporal degree of coherence is not Gaussian in general. However, for a quasi-monochromatic GSM sourcea≈1 and henceγ(^∆r,τ)is spatially Gaussian for allτ. In addition, if alsoω =ω₀andτ=0, it follows thatγ(^∆r, 0) =µ(k∆r), as noted earlier in the general case.

2.1.1 Michelson’s interferometer

Let us next consider two frequently encountered experimental se- tups to analyze the coherence properties of a scalar light field. The first one is the Michelson interferometer, which is one of the well- known amplitude-splitting interferometers to measure temporal coherence of light beams [56].

Consider a uniform, partially temporally coherent, quasi-monochromatic scalar light beam which at time t is represented by a zero-mean complex analytic signal E(t) [46]. The light is incident on a 50:50 beamsplitter which divides the beam into two identical parts (see Fig. 2.1). The lights in the arms travel distances L1 and L2 and reflect back from the mirrors. The fields are recombined at the beamsplitter and the output field is investigated right after the recombination. One of the end mirrors can be translated leading

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BS

τ

L2

L1

τ

Figure 2.1: Illustration of Michelson’s interferometer composed of a beamsplitter (BS), two arms of lengths L₁and L₂, and mirrors at the ends. The path-length difference of the two arms induces a controlled time delayτbetween the two beams.

to a time delayτ= 2(L2−^L1)/cbetween the two fields. The total field at the output is given as [47, 57]

E(t) = ¹

2[E0(t) +E0(t+τ)eⁱ^∆^φ], (2.12) where the factor 1/2 originates from the fact that in two passings through the beamsplitter, a quarter of the light energy is retained, E0(t)is the incident field, and∆φtakes into account the phase shifts between the two paths resulting from reflections and transmissions at the beamsplitter. Inserting Eq. (2.12) into (2.2) and using Eq. (2.3), the intensity at the output takes on the form

I(τ) = ¹ 2I0+ ¹

2I0|^γ⁰(τ)|^cos{^arg[γ0(τ)] +^∆φ}^, ^(2.13) where I0 and γ0(τ) are the intensity and the complex degree of coherence of the incoming field, respectively, and arg denotes the phase of a complex number. Owing to quasi-monochromaticity

|^γ0(τ)|is slowly varying with respect toτ, and arg[γ₀(τ)] =α(τ)− ω₀τ, whereα(τ)is likewise slowly varying withτ.

According to Eq. (2.13), ifγ0(τ)�=0, the intensity at the output is sinusoidally modulated with τ exhibiting interference fringes.

The visibility of the variations is given as

V(τ) = ^max[I(τ)]−^min[I(τ)]

max[I(τ)] +min[I(τ)]^, ^(2.14)

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where max and min denote the maximum and minimum values, respectively, in the neighborhood ofτ. Inserting Eq. (2.13) into (2.14) yields

V(τ) =|^γ0(τ)|^, ^(2.15) and thus the visibility of the intensity fringes is determined by the absolute value of the degree of coherence of the input field. No- tice that V(0) = 1, while when τ increases, the visibility of the fringes decreases. The value of τ, for which V(τ) is sufficiently small, can be regarded as the coherence time τc introduced earlier.

Michelson’s interferometer thus provides a means to measure the coherence time of a light field.

2.1.2 Young’s interferometer

In his seminal work of 1938 [58], Zernike demonstrated that the visibility of the intensity fringes in Young’s interferometer is directly related to the degree of coherence at the pinholes (see Fig. 2.2). In the setup, a stationary light beam (here not necessarily quasi-monochromatic) is directed onto two small apertures on a screen A^{, the} holes diffract the light which is investigated on another screenBⁱⁿ the Fraunhofer zone [59].

A member of the ensemble of spectral field realizations (monochromatic at frequencyω) is, at pointr on the screenB^{, given by}

E(r,ω) =K1E(r₁,ω) +K2E(r2,ω), (2.16) where E(r_n,ω) is the aperture field at r_n, n ∈ (1, 2). In addition, K1 andK2are pure complex numbers specified by the shape of the pinholes [60]. Using Eq. (2.5) at a single point, the related spectral density of the field assumes the form

S(r,ω) =S⁽¹⁾(r,ω) +S⁽²⁾(r,ω) +2

S⁽¹⁾(r,ω)S⁽²⁾(r,ω)

× |^µ(r₁,r₂,ω)|^cos{^arg[µ(r₁,r₂,ω)]−^k(R2−^R1)}^, ^(2.17) whereS⁽ⁿ⁾(r,ω)is the spectral density onBwhen only the pinhole at rn is open, andRn= |^r−^rⁿ|is the distance from the opening to

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B r2

r1

A

r

Figure 2.2: Illustration of Young’s two-pinhole experiment. A beam is incident from left on screenAwith two small openings at r₁and r₂. The field diffracted from the apertures is investigated at a point r on screenBat the far zone.

the observation point, n ∈ (1, 2). Equation (2.17) can be regarded as the spectral interference law for scalar fields [46]. It indicates that the spectral density on the observation screen is not directly the sum of the spectra originating from the two pinholes, but it is modulated by the cosine term with the amplitude given by the magnitude of the spectral degree of coherence at the apertures.

In analogy to Michelson’s interferometer, the spectral visibility of interference fringes on screenB at the pointris defined as

V(r,ω) = ^max[S(r,ω)]−^min[S(r,ω)]

max[S(r,ω)] +min[S(r,ω)]^, ^(2.18) where max and min denote the maximum and minimum values aroundr. SinceS⁽¹⁾(r,ω)andS⁽²⁾(r,ω)are spatially slowly varying functions and if the spectral densities at the two openings on screen Aare equal, the visibility assumes the form [46]

V(r,ω) =|^µ(r1,r2,ω)|^. ^(2.19) Thus, a visibility measurement gives directly the absolute value of the spectral degree of coherence and the position of the interference fringes yields its phase. As noted below Eq. (2.9), for quasi- monochromatic lightV(r,ω)at the central frequency is obtained by considering the fringes close to the optical axis. For general broadband light one must insert narrowband filters of midfrequencyω₀ in front of the pinholes and consider the visibility close to the axis [46].

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3 Concepts of electromagnetic coherence

So far we have concentrated on scalar fields only. In this chapter we extend the coherence theory to two-component and three- component vectorial electric fields [1, 46, 61]. We first recall some fundamental notions of electromagnetic coherence. Following that, we consider the polarization properties of random light, i.e., the state and the degree of polarization of beam-like and general fields [1, 46, 62]. Finally, we investigate in more detail the electromagnetic spatial and temporal coherence properties of random light.

3.1 ELECTROMAGNETIC COHERENCE MATRICES

Consider an electromagnetic field with three electric-field components. A realization of the electric vector of the random, stationary electromagnetic field, at point r at time t, is written as a column vector E(r,t) = [Ex(r,t),Ey(r,t),Ez(r,t)]^T. Here the superscript T denotes transpose and Ei(r,t) is the component of the field with i ∈ (x,y,z). The coherence properties of the field are described, in analogy to the coherence function of Eq. (2.1), by the 3×^{3 electric} mutual coherence matrix defined as [6, 46, 61]

Γ(r1,r2,τ) =�Ê^∗(r1,t)E^T(r2,t+τ)�^, ^(3.1) with the elementsΓ_ij(r₁,r₂,τ) = �Êi^∗(r₁,t)Ej(r₂,t+τ)� ând(i,j) ∈ (x,y,z). The mutual coherence matrix obeys a Hermitian-like relation Γ^∗_ij(r₁,r₂,τ) = ^Γ_ji(r₂,r₁,−^τ) and a number of non-negative definiteness conditions [46].

Use of the electromagnetic version of the generalized Wiener- Khintchine theorem leads to the cross-spectral density matrix [63]

W(r₁,r₂,ω) = ¹ 2π

∞

−^∞

Γ(r₁,r₂,τ)e^iωτdτ, (3.2)

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with the inverse relation as Γ(r₁,r2,τ) =

∞

0 W(r₁,r2,ω)e⁻^iωτdω. (3.3) The matrixW(r₁,r₂,ω)can also be presented as a correlation function over a suitable ensemble{^E(r,ω)}of spectral (monochromatic) realizations [64, 65]

W(r₁,r₂,ω) =�Ê^∗(r₁,ω)E^T(r₂,ω)�^, ^(3.4) with the elementsWij(r₁,r₂,ω) = �Ê^∗_i(r₁,ω)Ej(r₂,ω)� ând (i,j) ∈ (x,y,z). The cross-spectral density matrix is Hermitian in the sense thatWij(r₁,r₂,ω) =W_ji^∗(r₂,r₁,ω)and it satisfies various non-negative definiteness conditions [46].

3.2 POLARIZATION OF RANDOM LIGHT FIELDS

Regarding the degree of polarization, random light can be divided into three groups, namely unpolarized, partially polarized, and fully polarized cases. In unpolarized light, the polarization state evolves randomly with no preferred state when considered over a sufficiently long time interval [56]. If the polarization state is constant in time (though the intensity may fluctuate), light is fully polarized. The situations between the unpolarized and completely polarized limits correspond to partially polarized light. The above observations hold in the space-time domain, but below we first formulate the polarization theory in the space-frequency domain, which provides deeper insight into polarization.

3.2.1 Beam-like fields

A spectral realization, at frequencyω, of the electric field of a random, stationary electromagnetic beam propagating along thezaxis is written as a vector E(r,ω) = [Ex(r,ω),Ey(r,ω)]^T. Here Ex(r,ω) andEy(r,ω) are the two (transverse) components of the field. The polarization properties of the beam can be described by the single- point cross-spectral density matrixW(r,r,ω) =φ(r,ω). For beam

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Concepts of electromagnetic coherence

fieldsφ(r,ω)is the spectral polarization matrix of the form [46]

φ(r,ω) =

φxx(r,ω) φxy(r,ω) φ_yx(r,ω) φ_yy(r,ω)

, (3.5)

whereφ_ij(r,ω) =�^Ei^∗(r,ω)Ej(r,ω)�^,(i,j)∈ (x,y), are the elements of the matrix. Here and henceforth the lower case φ(r,ω)refers to a 2×2 polarization matrix. The polarization matrix is Hermitian, that isφ_ij(r,ω) = φ^∗_ji(r,ω), non-negative definite, and its diagonal elements are non-negative since they are spectral densities of the field components [46].

The 2×2 polarization matrix of a beam can be expressed, at any pointr, as a sum of two matrices, one corresponding to a fully polarized field and the other to unpolarized light [46]. The ratio of the intensity of the polarized part to the total intensity defines the spectral degree of polarization, expressible as [46]

P2(r,ω) =

1−^{4 det} ^φ(r,ω) tr²φ(r,ω)

1/2

, (3.6)

where the subscript 2 refers to two-component field and det and tr denote determinant and trace of a matrix, respectively. The degree of polarization is bounded by 0≤ ^P2(r,ω)≤1, with the lower and upper limits corresponding to a completely unpolarized and a fully polarized field, respectively. Even though the degrees of polarization of two fields may be the same, the beams can nonetheless be distinguished by the polarization state of the polarized part, polarization dynamics [66, 67], and irreversibility properties [68].

Besides the degree, the polarization matrix uniquely defines also the polarization state of a field. Equivalently to the four elements of the polarization matrix, the polarization information can be described in terms of the spectral polarization (one-point) Stokes pa-

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rameters defined as [1, 4, 46]

S0(r,ω) =φ_xx(r,ω) +φ_yy(r,ω), (3.7a) S1(r,ω) =φ_xx(r,ω)−^φyy(r,ω), (3.7b) S2(r,ω) =φxy(r,ω) +φyx(r,ω), (3.7c) S3(r,ω) =i

φyx(r,ω)−^φ^xy(r,ω). (3.7d) The parameter S0(r,ω) is the total spectral density of the field, whereas the parametersS1(r,ω),S2(r,ω), andS3(r,ω)describe the intensity difference of thexandylinearly polarized parts,+45^◦and

−⁴⁵^◦linearly polarized contributions, and right-hand and left-hand circularly polarized components, respectively. Thus, the Stokes parameters may also be written as [1, 69]

S0(r,ω) = Ix(r,ω) +Iy(r,ω), (3.8a) S1(r,ω) = Ix(r,ω)−Îy(r,ω), (3.8b) S2(r,ω) = Iα(r,ω)−Îβ(r,ω), (3.8c) S3(r,ω) = Ir(r,ω)−Îl(r,ω), (3.8d) where Ii(r,ω), i ∈ (x,y,α,β,r,l), denote the intensities of the x, y, +45^◦, −⁴⁵^◦, right-hand, and left-hand circularly polarized contributions. Furthermore, it is convenient to normalize the last three Stokes parameters with the intensity, resulting in

sn(r,ω) = ^Sⁿ(r,ω)

S0(r,ω)^, ⁿ∈(1, 2, 3), (3.9) for which |^sn(r,ω)| ≤ 1 as implied by the definitions. In terms of the normalized Stokes parameters the degree of polarization assumes the form [1, 46]

P2(r,ω) = 3

∑

n=1

s²_n(r,ω) _1/2

, (3.10)

consistently with Eq. (3.6).

Graphically the polarization state can be illustrated by using the unit-radius Poincar´e sphere depicted in Fig. 3.1. The polarization state and degree are uniquely specified by the Poincar´e vector

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) p

s( s3

s2

s1

s

Figure 3.1: Poincar´e-sphere illustration of the polarization state in a Stokes-parameter space. The larger sphere describing fully polarized fields is of unit radius. Partially polar- ized states are inside the sphere and the origin corresponds to an unpolarized beam. The Poincar´e vector s determines the polarization of a partially polarized beam, whereas s^(p) specifies the polarization state of the fully polarized part of that field.

s(r,ω) = [s1(r,ω), s2(r,ω), s3(r,ω)]^T, which is a vector in a space spanned by the normalized polarization Stokes parameterssn(r,ω), n ∈ (1, 2, 3). For a fully polarized beam the end point of s(r,ω) is on the surface of the sphere, while the origin corresponds to a completely unpolarized field. Other points inside the sphere correspond to spectrally partially polarized fields with the degree of polarization given by the length of the Poincar´e vector. Fields with the same degree of polarization, but different polarization states, are at equal distance from the origin. The polarization state of the polarized part is obtained by extending the Poincar´e vector to the surface of the unit sphere. Since the non-normalized Stokes param- etersS1(r,ω),S2(r,ω)andS3(r,ω)of a partially polarized field are equal to those of its fully polarized part, we may write the normalized Stokes parameters of the polarized contribution as [70]

s⁽n^p⁾(r,ω) = ^sⁿ(r,ω)

P2(r,ω)^, ⁿ∈(1, 2, 3). (3.11) The relation between the Poincar´e vectors s(r,ω) and s⁽^p⁾(r,ω) = [s⁽₁^p⁾(r,ω), s⁽₂^p⁾(r,ω),s₃⁽^p⁾(r,ω)]^T is illustrated in Fig. 3.1.

In addition to the treatment of spectral polarization, we introduce the related time-domain counterpart. The temporal polariza-

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tion matrix is defined in terms of the single-point mutual coherence matrix asΓ(r,r, 0) = J(r). According to the Wiener-Khintchine relation of Eq. (3.3), the polarization matrix assumes the form [71]

J(r) =

∞

0 φ(r,ω)dω. (3.12)

On the other hand, it can be expressed as [46, 71]

J(r) =

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

, (3.13)

where J_ij(r) = �^Ei^∗(r,t)E_j(r,t)� are the elements of the matrix and Ei(r,t) is the i component of the electric field, (i,j) ∈ (x,y). The time-domain Stokes parameters are of the form [46]

S0(r) =Jxx(r) +Jyy(r), (3.14a) S1(r) =Jxx(r)−^Jyy(r), (3.14b) S2(r) =Jxy(r) +Jyx(r), (3.14c) S3(r) =i

Jyx(r)−^J^xy(r), (3.14d) and the degree of polarization is [46]

P2(r) =

1− ^{4 det}^J(r) tr²J(r)

1/2

= 3

∑

n=1

s²_n(r) 1/2

, (3.15) where

sn(r) =Sn(r)/S0(r), n∈(1, 2, 3), (3.16) are the normalized Stokes parameters. According to Eq. (3.12) the polarization matrices in the time and frequency domains can be completely different. Consequently, the state and degree of polarization in the two domains are different as well [19, 68, 71, 72].

3.2.2 Detection of beam-field polarization

The state and the degree of polarization of a light beam can be measured by employing a waveplate-polarizer setup depicted in

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x

y

z

p) (θ ) P

θq

Q( ω) , (r E

f θq θ^p

Figure 3.2: An arrangement to measure the polarization characteristics of a beam field E(r,ω). The field propagating along the z axis is passed through a quarter-wave plate Q(θ_q) and a polarizer P(θ_p), after which the intensity is measured. The fast axis (f) of the waveplate and the polarization axis of the polarizer make the angles ofθ_qandθ_p, respectively, with the x axis in the xy plane. Positive angles are counterclockwise when looking into the field.

Fig. 3.2. A quarter-wave plate and a polarizer are placed into the path of the beam and the transmitted intensity is measured for four of their relative orientations. These four measurements lead to the four Stokes parameters introduced in Sec. 3.2.1. The formulation is presented here in the frequency domain but the principle works in the time domain as well.

Traditionally, transmission of an electromagnetic beam through an optical element, such as a waveplate and a polarizer, is described by the Jones-matrix formalism [56]. By multiplying the electric field of light incident onto a system by the related Jones matrix gives the output field. The Jones matrix of an (achromatic) polarizer is of the form [73]

P(θp) =

cos²θp cosθpsinθp

cosθpsinθp sin²θp

, (3.17)

whereθpis the angle the polarization axis makes with thexaxis (see Fig. 3.2). Positive angles are counterclockwise when the observer looks into the source of the beam. Furthermore, the Jones matrix of an (achromatic) quarter-wave plate is given as [73]

Q(θ_q) =

cos²θ_q−ⁱ^sin²^θq (1+i)cosθ_qsinθ_q (1+i)cosθqsinθq sin²θq−ⁱ^cos²^θq

, (3.18) whereθqis the angle that the fast axis of the waveplate makes with the x axis. The spectral density of an electromagnetic wave with

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the polarization matrixφ(r,ω)is after a waveplate and a polarizer expressed as (Publication IV)

I(θq,θp;r,ω) =tr[P(θp)Q^∗(θq)φ(r,ω)Q(θq)P(θp)], (3.19) where we have noted that the Jones matrices of the elements are symmetric andP(θp)is real.

Using Eq. (3.19) with four pairs of orientations of the elements we can extract the complete polarization matrix φ(r,ω). A possible set of orientations is (θq,θp) = (0, 0), (0, π/4), (π/4, π/4), (π/2, π/2), which results in

φxx(r,ω) =I(0, 0;r,ω), (3.20) φyy(r,ω) =I(π/2,π/2;r,ω), (3.21) φxy(r,ω) =φ_yx^∗ (r,ω) = I(π/4,π/4;r,ω)−¹

2

I(0, 0;r,ω) +I(π/2,π/2;r,ω)+i

I(0,π/4;r,ω)

−¹₂^I(0, 0;r,ω) +I(π/2,π/2;r,ω). (3.22) Other possible orientation combinations exist, but those given here are convenient in practice. Indeed, one first places the quarter-wave plate’s fast axis and the polarizer’s polarization axis parallel, then rotates the polarization axis byπ/4, next turns the waveplate’s fast axis byπ/4, and finally one rotates both elements by an additional π/4 so that they are both at an angle of π/2 from the original position.

3.2.3 Nonparaxial fields

Investigation of polarization in the previous sections was limited to beam-like fields with two electric-field components. In general, however, an electric field may possess three components expressed by the column vector E(r,ω) = [Ex(r,ω), Ey(r,ω), Ez(r,ω)]^T. In analogy to the two-component case, we may write the polarization matrix of the three-component fields as

Φ(r,ω) =�^E^∗(r,ω)E^T(r,ω)�^, ^(3.23)

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with the elements Φ_ij(r,ω) = �^E_i^∗(r,ω)Ej(r,ω)�^, (i,j) ∈ (x,y,z). Further, as for beam fields the polarization properties can be described in terms of the (generalized) polarization Stokes parameters. However, since the related polarization matrix has, in general, nine independent parameters, in total nine Stokes parameters are required. These are explicitly written as [16, 25]

Λ₀ =^Φxx+^Φyy+^Φzz, Λ₅= ³₂i(^Φxz−^Φ^zx), Λ₁ = ³₂(^Φxy+^Φyx), Λ₆= ³₂(^Φyz+^Φzy), Λ₂ = ³₂i(^Φ_xy−^Φyx), Λ₇= ³₂i(^Φ_yz−^Φzy),

Λ₃ = ³₂(^Φxx−^Φ^yy), Λ₈= ^√₂³(^Φxx+^Φyy−²^Φ^zz), Λ₄ = ³₂(^Φ_xz+^Φ_zx),

(3.24)

where we have dropped the position and frequency dependency for brevity. The first Stokes parameter Λ₀ is the total spectral density of the field, whereas the other eight define the polarization state.

The pairs (Λ₁,Λ₂),(^Λ₄,Λ₅), and(^Λ6,Λ₇)are analogous to(S2,S3) of the two-component case but now in the xy, xz, and yz planes, respectively. Furthermore,Λ₃ is analogous to S1 andΛ₈ describes the excess spectral density of the x and theycomponents over the zcomponent [16].

In contrast to beam fields, the 3×3 polarization matrix cannot in general be expressed as a sum of two matrices, one corresponding to a fully unpolarized three-component field and the other to a fully polarized field [16]. Therefore, we define the degree of polarization, in analogy to Eq. (3.10), as [16]

P3(r,ω) = 8

∑

n=1

λ²_n(r,ω) 1/2

, (3.25)

whereλn(r,ω) = ^Λn(r,ω)/√

3Λ₀(r,ω),n ∈(1, . . . , 8), are the normalized Stokes parameters. We note that also other definitions for the degree of polarization have been proposed in the litera- ture [21, 62, 74–76]. However, the one in Eq. (3.25) has the property

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that when the intensities of the Cartesian field components are the same, the degree of polarization reflects the average correlation that exists between the components [16]. This feature is shared by the degree of polarization of beam fields [60]. Further, when the two smallest eigenvalues of the (Hermitian) polarization matrix are the same, then the polarization matrix is expressible as a sum of two matrices corresponding to a fully polarized and completely unpolarized light, and P3(r,ω)is the ratio of the intensity in the polarized part to the total field (Publication IV). In addition, by inserting Eq. (3.24) into (3.25), the degree of polarization becomes

P₃²(r,ω) = ³ 2

trΦ²(r,ω) tr²Φ(r,ω)−¹₃

, (3.26)

which is bounded by 0≤^P³(r,ω)≤^1.

It is possible to present the polarization properties of three- component fields in terms of the Stokes vector in an eight-dimen- sional Stokes-parameter space in a manner similar to the Poincar´e- sphere construction of beam fields (Publication IV). However, due to high number of dimensions such a treatment would not introduce any additional value to visualization.

The two degrees of polarization in Eqs. (3.6) and (3.26) do not, in general, imply the same numerical value for a light beam. More precisely, by considering the field E(r,ω) = [Ex(r,ω),Ey(r,ω), 0]^T one can show that

P₃²(r,ω) =1−^{3 det}^φ(r,ω)

tr²φ(r,ω) ^, ^(3.27) whereφ(r,ω) is the 2×2 polarization matrix related to the transverse components. The difference to Eq. (3.6) is the factor 3 instead of 4 in the numerator. In addition, for beam fields 1/2≤^P3(r,ω)≤ 1, with the lower limit corresponding to a plane wave which is unpolarized in the two-component treatment [16]. Hence, such a field is not unpolarized in the three-component picture, which is intu- itively clear as the electric field is confined to a plane.

Polarization and electromagnetic coherence of light fields probed with nanoscatterers

uef.fi

Dissertations in Forestry and Natural Sciences

LASSE-PETTERI LEPPÄNEN

LASSE-PETTERI LEPPÄNEN

Polarization and

electromagnetic coherence of light fields probed with

nanoscatterers

Preface

Contents

1 Introduction

2 Coherence of scalar fields

3 Concepts of electromagnetic coherence

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∑

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