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PUBLICATIONS OF
THE UNIVERSITY OF EASTERN FINLAND
This thesis considers surface-plasmon
polaritons (SPPs), partially coherent optical
surface fields, and complementarity in
vector-light photon interference. Novel SPPs,
including a long-range higher-order metalslab mode, are predicted. Generation, partial
polarization, and electromagnetic coherence
of polychromatic SPPs and evanescent
light fields are also examined. Polarization
modulation of vectorial quantum light is
explored to uncover a new intrinsic aspect of
photon wave–particle duality.
uef.fi
PUBLICATIONS OF
THE UNIVERSITY OF EASTERN FINLAND
Dissertations in Forestry and Natural Sciences
ISBN 978-952-61-2357-8
ISSN 1798-5668
DISSERTATIONS | ANDREAS NORRMAN | ELECTROMAGNETIC COHERENCE OF OPTICAL SURFACE AND... | No 252
ANDREAS NORRMAN
Dissertations in Forestry and
Natural Sciences
ANDREAS NORRMAN
ELECTROMAGNETIC COHERENCE OF OPTICAL
SURFACE AND QUANTUM LIGHT FIELDS
ANDREAS NORRMAN
Electromagnetic Coherence
of Optical Surface and
Quantum Light Fields
Publications of the University of Eastern Finland
Dissertations in Forestry and Natural Sciences
No 252
Academic Dissertation
To be presented by permission of the Faculty of Science and Forestry for public
examination in the Auditorium AG100 in Agora Building at the University of
Eastern Finland, Joensuu, on December 16, 2016,
at 12 o’clock noon.
Institute of Photonics
Grano Oy
Jyväskylä, 2016
Editors: Prof. Pertti Pasanen, Prof. Jukka Tuomela,
Prof. Matti Vornanen, Prof. Pekka Toivanen
Distribution:
University of Eastern Finland Library / Sales of publications
[email protected]
http://www.uef.fi/kirjasto
ISBN: 978-952-61-2357-8 (printed)
ISSNL: 1798-5668
ISSN: 1798-5668
ISBN: 978-952-61-2358-5 (pdf)
ISSNL: 1798-5668
ISSN: 1798-5676
Author’s address:
University of Eastern Finland
Institute of Photonics
P. O. Box 111
80101 JOENSUU
FINLAND
email: [email protected]
Supervisors:
Professor Ari T. Friberg, Ph.D., D.Sc. (Tech)
University of Eastern Finland
Institute of Photonics
P. O. Box 111
80101 JOENSUU
FINLAND
email: [email protected]
Associate Professor Tero Setälä, D.Sc. (Tech)
University of Eastern Finland
Institute of Photonics
P. O. Box 111
80101 JOENSUU
FINLAND
email: [email protected]
Reviewers:
Professor Taco D. Visser, Ph.D.
Vrije Universiteit Amsterdam
Department of Physics and Astronomy
De Boelelaan 1081
1081 HV AMSTERDAM
THE NETHERLANDS
email: [email protected]
Professor Goëry Genty, D.Sc. (Tech)
Tampere University of Technology
Department of Physics
P. O. Box 692
33101 TAMPERE
FINLAND
email: [email protected]
Opponent:
Professor P. Scott Carney, Ph.D.
University of Illinois at Urbana–Champaign
Department of Electrical and Computer Engineering
405 North Mathews Avenue
URBANA, IL 61801
USA
email: [email protected]
ABSTRACT
This thesis encompasses fundamental theoretical research on plasmonics, electromagnetic coherence, and quantum complementarity.
Three main topics are covered: surface-plasmon polaritons (SPPs),
partial coherence of optical surface fields, and complementarity in
vectorial photon interference.
Existence of new SPP modes at planar single-interface and thinfilm geometries are predicted. It is further shown, for the first time,
how a higher-order metal-slab mode, commonly not regarded useful, may turn into a strongly confined long-range surface mode in
many situations where the fundamental long-range SPP does not
exist and the single-interface SPP propagation is negligible.
Partially coherent SPP fields are studied and a scheme to tailor
the vectorial coherence of polychromatic SPPs in the Kretschmann
setup by controlling the correlations of the source light is advanced.
Generation and coherence of purely evanescent light fields are also
examined. It is demonstrated that, for such fields, the coherence
length in air can be notably shorter than the free-space wavelength.
The analysis also reveals the possibility to excite an evanescent field
that shares the polarization properties of blackbody radiation, yet
with tunable coherence qualities.
Polarization modulation in double-pinhole photon interference
is explored to derive two general complementarity relations among
distinguishability and visibility associated with genuine vector-light
quantum fields of arbitrary state. The established framework uncovers a new intrinsic aspect of wave–particle duality of the photon,
having no correspondence within the scalar quantum treatment.
Universal Decimal Classification: 53.01, 537.8, 535-6
Keywords: theoretical physics; nanophotonics; plasmonics; optical surface
fields; surface-plasmon polaritons; light coherence; light polarization; photon interference; quantum complementarity; wave–particle duality
Asiasanat: teoreettinen fysiikka; nanofotoniikka; plasmoniikka; optiset pintakentät; koherenssi; polarisaatio; kvanttivalo; aalto-hiukkasdualismi
Preface
The research summarized in this dissertation started nearly five
years ago, at the end of 2011, when I began my doctoral studies
on electromagnetic nanophotonics in the Department of Applied
Physics at Aalto University, Espoo, Finland. However, as a result of
various factors, our theory group relocated in the beginning of 2013
to the Institute of Photonics at the University of Eastern Finland,
Joensuu, Finland, which turned this project into quite a colorful
roller-coaster journey. In particular, due to personal reasons, I have
had to carry out my research mainly from my home apartment in
Helsinki on my own. The 450 km distance to my supervisors, not to
mention the birth of my daughter seven months after the move, has
undeniably caused some real challenges and moments of despair
from time to time. Even so, now in hindsight, I believe that these
circumstances have also taught me to do and to take responsibility
of individual research in a positive manner. Of course, this would
not be the case without the constant encouragement and trust of my
supervisors, or the support of other colleagues, friends, and family
members. I would therefore now take the opportunity to thank all
these wonderful people.
First, I would like to express my sincerest gratitude to Prof. Ari
T. Friberg, my supervisor, mentor, and friend, for his invaluable
guidance and persistent support during all these years. His exceptional enthusiasm and integrity towards scientific research, his constant and untiring devotion to research projects, as well as his ability to challenge and to educate students’ thinking are simply unequalled. Our in-depth conversations, covering science, art, sport,
politics, religion, and life in general, have been very intriguing; our
numerous tough matches on the squash court have been extremely
enjoyable; our uncountable dusk-till-dawn adventures, naturally
accompanied by a certain refreshing yellowish beverage, have been
absolutely memorable. I am also indebted to my other supervisor,
Prof. Tero Setälä, for his valuable instructions, ingenious wits, and
active involvement, not only at the office level, but also through
extracurricular late-night activities. Overall, I want to thank both
supervisors for contributing to a stimulating and open academic
and social environment. It has been an honor and privilege to work
with them all these years.
I also want to extend my gratitude to my other coauthors, Prof.
Sergey A. Ponomarenko and Dr. Kasimir Blomstedt, for providing broad expertise, important insights, and essential contributions
during the course of this thesis. The perceptive comments and suggestions by the reviewers, Prof. Taco D. Visser and Prof. Goëry
Genty, are highly appreciated. In addition, I am grateful to Prof. P.
Scott Carney for accepting to be my opponent on short notice.
During my time at Aalto University and the University of Eastern Finland I have been fortunate to make acquaintance with other
talented individuals as well. Special thanks go to my former fellow colleagues, Dr. Timo Hakkarainen, Dr. Timo Voipio, and M.Sc.
Henri Kellock, with whom I have had the pleasure to share many
fruitful discussions and occasions over the years, both inside and
outside the university. My present workmates, Dr. Lasse-Petteri
Leppänen and M.Sc. Matias Koivurova, have also had a positive
influence on my well-being. For efficiency at the bureaucratic level,
Dr. Noora Heikkilä, Ms. Hannele Karppinen, Ms. Katri Mustonen,
and Ms. Marita Ratilainen deserve my sincerest gratitude.
Personal grants from the Jenny and Antti Wihuri Foundation,
the Emil Aaltonen Foundation, and the Finnish Foundation for
Technology Promotion are gratefully acknowledged. Furthermore,
I thank the Academy of Finland for funding my research through
various projects.
I am also indebted to my inspiring soulmates, Alexander, Erik,
Hannes, Janek, Jukka-Pekka, Jussi, Markku, Martin, Nikolas, Rasmus, Sampo, Simo, and Veli-Matti, of course not to disregard their
better halves, who have strongly enriched my life and provided me
with (sometimes much needed) nonscientific counterbalance, each
in their own unique way.
Last, but not least, I owe my heartfelt gratefulness to my beloved
parents for their endless empathy, kindness, and presence throughout my entire life. Above all, my deepest thankfulness and love go
to my best friend and life companion, my wonderful wife Jenni, as
well as to my most precious jewel, my sweet daughter Lily. Without their unwavering patience and support at moments of darkness
and desperation this thesis would not exist. The least I can do is to
dedicate this dissertation to them.
For my family.
Helsinki, November 24, 2016
Andreas Norrman
LIST OF PUBLICATIONS
This thesis consists of an overview of the author’s work in the field
of electromagnetic nanophotonics and the following selection of the
author’s publications:
I A. Norrman, T. Setälä, and A. T. Friberg, “Exact surface-plasmon polariton solutions at a lossy interface,” Opt. Lett. 38,
1119–1121 (2013).
II A. Norrman, T. Setälä, and A. T. Friberg, “Surface-plasmon
polariton solutions at a lossy slab in a symmetric surrounding,” Opt. Express 22, 4628–4648 (2014).
III A. Norrman, T. Setälä, and A. T. Friberg, “Long-range higherorder surface-plasmon polaritons,” Phys. Rev. A 90, 053849
(2014).
IV A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partially
coherent surface plasmon polaritons,” submitted (2016).
V A. Norrman, T. Setälä, and A. T. Friberg, “Partial coherence
and polarization of a two-mode surface-plasmon polariton
field at a metallic nanoslab,” Opt. Express 23, 20696–20714
(2015).
VI A. Norrman, T. Setälä, and A. T. Friberg, “Partial spatial coherence and partial polarization in random evanescent fields
on lossless interfaces,” J. Opt. Soc. Am. A 28, 391–400 (2011).
VII A. Norrman, T. Setälä, and A. T. Friberg, “Generation and
electromagnetic coherence of unpolarized three-component
light fields,” Opt. Lett. 40, 5216–5219 (2015).
VIII A. Norrman, K. Blomstedt, T. Setälä, and A. T. Friberg, “Complementarity and polarization modulation in photon interference,” submitted (2016).
Throughout the overview, these publications will be referred to by
Roman numerals.
The results reported in Publications I–VI and VIII have been presented in the following international conferences:
1. 5th EOS Topical Meeting on Advanced Imaging Techniques
(Engelberg, Switzerland, 2010).
2. 9th Euro-American Workshop on Information Optics (Helsinki, Finland, 2010).
3. Nordic Physics Days 2011 (Helsinki, Finland, 2011).
4. 22nd General Congress of the International Commission for
Optics (Puebla, Mexico, 2011).
5. 18th Microoptics Conference (Tokyo, Japan, 2013).
6. 1st Joensuu Conference on Coherence and Random Polarization: Electromagnetic Optics with Random Light (Joensuu,
Finland, 2014).
7. Northern Optics & Photonics 2015 (Lappeenranta, Finland,
2015).
8. Frontiers in Optics / Laser Science 2016 (Rochester, NY, USA,
2016).
9. OSA Incubator on Emerging Connections: Quantum and Classical Optics (Washington, DC, USA, 2016).
AUTHOR’S CONTRIBUTION
The author has had a central role in all aspects of the research work
reported in this thesis. The author carried out the majority of the
mathematical derivations and the numerical calculations in all the
publications. The research subjects arose from discussions with the
coauthors, with the author contributing significantly to the original ideas and to the interpretation of the results. In particular, the
notions of the long-range higher-order surface-plasmon polaritons
and the complementarity relations for quantum vector light came
from the author. The author wrote the first drafts of all publications,
which subsequently were finalized together with the coauthors.
Contents
1 INTRODUCTION
1.1 Plasmonics, electromagnetic coherence, and quantum
complementarity . . . . . . . . . . . . . . . . . . . . .
1.2 Scope of the thesis . . . . . . . . . . . . . . . . . . . .
2 ELECTROMAGNETIC SURFACE WAVES
2.1 Nomenclature . . . . . . . . . . . . . . . . . .
2.2 Field characterization . . . . . . . . . . . . . .
2.2.1 Representation . . . . . . . . . . . . .
2.2.2 Propagation . . . . . . . . . . . . . . .
2.2.3 Energy flow . . . . . . . . . . . . . . .
2.3 Existence of electromagnetic surface modes .
2.4 Fresnel coefficients and existence conditions
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4 PARTIALLY COHERENT
SURFACE-PLASMON POLARITONS
4.1 Single-interface geometry . . . . . . . . . . . . . . . .
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3 SURFACE-PLASMON POLARITONS
3.1 Single-interface modes . . . . . . . . . . . . . . . . .
3.1.1 Approximate solutions vs. exact solutions .
3.1.2 Mode types SPP I and SPP II . . . . . . . . .
3.1.3 Flow of energy . . . . . . . . . . . . . . . . .
3.2 Metal-slab modes . . . . . . . . . . . . . . . . . . . .
3.2.1 Mode class M1 . . . . . . . . . . . . . . . . .
3.2.2 Mode class M2 . . . . . . . . . . . . . . . . .
3.2.3 Mode class M3 . . . . . . . . . . . . . . . . .
3.2.4 Forward- and backward-propagating modes
3.3 Long-range modes . . . . . . . . . . . . . . . . . . .
3.3.1 Mode interchanges . . . . . . . . . . . . . . .
3.3.2 Long-range higher-order modes . . . . . . .
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CONCLUSIONS
7.1 Summary of main results . . . . . . . . . . . . . . . .
7.2 Future prospects . . . . . . . . . . . . . . . . . . . . .
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A CLASSICAL THEORY OF
ELECTROMAGNETIC COHERENCE
A.1 Nonstationary fields . . . . . . . . . . . . . . . . . . .
A.2 Stationary fields . . . . . . . . . . . . . . . . . . . . . .
A.3 Degree of polarization . . . . . . . . . . . . . . . . . .
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REFERENCES
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4.1.1 Polychromatic surface-plasmon polaritons
4.1.2 Narrowband and broadband fields . . . .
4.1.3 Plasmon coherence engineering . . . . . .
Metal-slab geometry . . . . . . . . . . . . . . . . .
4.2.1 Degree of coherence . . . . . . . . . . . . .
4.2.2 Local and global coherence length . . . . .
4.2.3 Degree of polarization . . . . . . . . . . . .
ELECTROMAGNETIC COHERENCE OF
EVANESCENT LIGHT FIELDS
5.1 Evanescent wave in total internal reflection
5.2 Random evanescent fields . . . . . . . . . .
5.2.1 Subwavelength coherence lengths .
5.2.2 Genuine 3D-polarized states . . . .
5.3 3D-unpolarized evanescent fields . . . . . .
5.3.1 Generation . . . . . . . . . . . . . . .
5.3.2 Degree of coherence . . . . . . . . .
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COMPLEMENTARITY IN PHOTON INTERFERENCE
6.1 Coherence of vectorial quantum light . . . . . . . .
6.2 Photon interference law . . . . . . . . . . . . . . . .
6.3 Visibility and distinguishability . . . . . . . . . . . .
6.4 Weak and strong complementarity . . . . . . . . . .
6.5 Wave–particle duality of the photon . . . . . . . . .
1 Introduction
The rapid progress in nano-optics and photonics during the last two
decades has opened a whole new realm of possibilities for interdisciplinary science and technology [1–4]. Groundbreaking discoveries
in such engrossing areas as superlens imaging [5,6], transformation
optics [7–9], optical cloaking [10–12], and ghost imaging [13–15],
entail extraordinary physical phenomena with fascinating potential
applications for technoscientific research and engineering. At the
same time, the advent of near-field optics [16, 17], strongly dominated by the manipulation and utilization of evanescent waves at
subwavelength dimensions [18], has allowed to surpass the traditional diffraction limit and plays a vital role in the design and manufacture of various optoelectronic nanocomponents. In particular,
tailoring novel composite materials with unforeseen precision enables fabrication of hitherto unrealizable structures to exploit light–
matter interactions of diverse nature at microscopic and nanoscopic
scales [19–23].
1.1 PLASMONICS, ELECTROMAGNETIC COHERENCE,
AND QUANTUM COMPLEMENTARITY
Modern plasmonics [24–27], constituting an exceptionally rich area
of nanophotonics [28–30], offers methods for subwavelength light
control over dimensions as small as a few nanometers [31–33], with
numerous applications in biomedical and chemical sensing [34–36],
detectors and emitters [37, 38], nanophotonic devices [39–41], lasers
[42–44], complex materials [45–47], nonlinear and quantum interactions [48, 49], as well as light shaping and holography [50–53].
Optical coherence theory, in turn, forms an important discipline in
all areas of electromagnetic light physics, by providing the basic
means to understand spectral distribution, propagation, interference, interactions, polarization, and other fundamental characteris-
Dissertations in Forestry and Natural Sciences No 252
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
Quantum Light Fields
tics of both classical and quantum wave fields [54–59].
The success of plasmonics in optical physics is mainly due to
the celebrated surface-plasmon polariton (SPP), a hybridized excitation between light and collective charge-density oscillations that
typically appear at metal–dielectric interfaces. The seemingly earliest recorded observation associated with SPPs, although not recognized at that time, is the spectral anomaly discovered by Wood
in 1902 when studying light diffraction from a metal grating [60].
Despite major efforts and contributions [61–63], Wood’s anomaly,
as it is now called, remained unexplained for nearly 40 years, until
Fano in 1941 gave it a proper description in terms of ‘superficial
waves’ [64]. Later during the 1940s, experiments performed with
fast electrons impinging on thin metal films revealed unexpected
peaks in the measured energy-loss spectrum [65–67]. In the early
1950s, Pines and Bohm, attributed for coining the phrase ‘plasmon’,
provided an explanation for these observations by realizing that the
long-range Coulomb interaction between valence electrons in metals could result in longitudinal collective plasma oscillations [68,69].
Interest towards plasma excitations started to grow in the late 1950s.
Especially, the pioneering theoretical investigations made by Ritchie
in 1957 led to the prediction of a self-sustained surface-collective excitation [70], whose existence was experimentally verified two years
later by Powell and Swan [71, 72], and subsequently by Stern and
Ferrell [73], who were the first to describe the new excitation as a
‘surface plasmon’.
The earliest publication of prism-coupled excitation of an SPP
by means of optical evanescent waves would appear to be that of
Turbadar from 1959 [74], only a year after Hopfield had invented
the term ‘polariton’ [75]. Unfortunately, Turbadar did not explicitly state that he had actually excited an SPP, whereupon the credit
for prism-coupled SPP generation has been given to Otto [76] and
to Kretschmann and Raether [77] for their experiments performed
independently in 1968. Also the important theoretical contributions of Kliewer and Fuchs [78] and of Economou [79] from the late
1960s deserve mentioning. In the following two decades SPPs were
2
Dissertations in Forestry and Natural Sciences No 252
Introduction
extensively studied by several groups, leading to the discovery of
the long-range SPP [80, 81] and eventually culminating in a seminal text by Raether in 1988 [82], which still remains an important
landmark in the field. Over the next ten years the research evolved
somewhat slowly, until an extraordinary strong light transmission
through subwavelength hole arrays due to plasmon excitations was
observed by Ebbesen and co-workers in 1998 [83], which sparked an
explosion of interest and spawned the field of modern plasmonics
as we recognize it today.
To date, plasmonics has chiefly dealt with spatially and spectrally totally coherent (and polarized) SPPs. Surface plasmons are
nonetheless known to greatly alter the spectrum, polarization, and
spatial coherence of optical near fields. For instance, some time
ago it was demonstrated that the correlation length in a fluctuating
thermal near field can extend over several tens of wavelengths if
surface plasmons are present [84]. Likewise, a thermal broadband
near field may become essentially quasimonochromatic [85], and
highly polarized [86], when surface plasmons are involved. Bodies
in thermal equilibrium may emit radiation in the form of spatially
coherent beam lobes of directionally dependent spectra if gratings
are fabricated on their surface [87]. Surface plasmons also play a
key role in modifying the coherence properties of fields transmitted through periodic hole arrays, gratings, and slits in thin metal
films [88–90]. These findings illustrate that, depending on the prevailing circumstances, the presence of surface plasmons may have
a significant effect on the coherence and polarization characteristics of electromagnetic near fields. However, a randomly fluctuating near field, with or without the influence of surface plasmons,
is generally a three-component, non-beamlike field, necessitating appropriate theoretical methods for its statistical analysis.
The rigorous, classical theory of optical coherence in the space–
time domain was established during the 1950s, primarily through
the efforts of Wolf, when the polarization matrix as well as the electromagnetic coherence matrices were introduced, together with the
equations that govern their behavior [91–95]. Two decades later,
Dissertations in Forestry and Natural Sciences No 252
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
Quantum Light Fields
the foundations of classical optical coherence theory in the space–
frequency domain were put forward by Mandel and Wolf [96, 97].
An alternative representation, first for scalar light [98–100] and only
relatively recently for vectorial light [101,102], was established. The
frequency-domain formulation may in some sense be viewed as a
more fundamental theory, as it yields insight into the inner (spectral) structure of the coherence of light. The spectral coherence
theory is also advantageous when light–matter interactions are investigated. Whereas these treatments concern stationary light fields,
i.e., light fields for which the character of the random fluctuations
does not change with time, the advent of pulsed lasers, supercontinuum light, and ultrafast detectors have in recent years awaken
interest, and highlighted the need, to study optical coherence and
polarization of nonstationary light [103–108].
The fundamental measures to characterize any partially coherent and partially polarized light field are the degree of coherence and
the degree of polarization. Conventionally, these two quantities have
been restricted, respectively, to scalar light and to two-component
(2D) fields [54]. However, recent advances in nano-optics [2] and
high-numerical-aperture imaging systems [109–111] have accentuated the importance to extend these concepts for arbitrary threecomponent (3D) light fields. Partial coherence of vectorial light may
be assessed by the electromagnetic degree of coherence [112–114],
which describes the strength of correlations that exist between all
the orthogonal components of the electric field at a pair of points.
In addition, it is invariant under unitary transformations, a natural
requirement [115], and most importantly, amounts to equivalence
between complete coherence and factorization of the coherence matrix (see below). We emphasize, though, that other coherence measures have been suggested [116–123], with different physical motivations, implications, and mathematical properties [124]. The 3D
degree of polarization, in turn, can be constructed via an expansion
of the polarization matrix in terms of the Gell-Mann matrices and
the generalized Stokes parameters [125], in analogy with the Pauli
spin matrices and the usual Stokes parameters for the customary
4
Dissertations in Forestry and Natural Sciences No 252
Introduction
2D degree of polarization [54]. In this representation, the degree
of polarization can be interpreted as the square root of the average
of the normalized correlations squared among the three orthogonal
electric-field components in a reference frame where the diagonal
elements of the polarization matrix are equal. However, also other
means to address partial polarization of 3D-light fields have been
proposed [126–132], and compared [133, 134].
Around the time of the invention of the laser, influenced by the
revolutionary experiment performed in 1956 by Hanbury Brown
and Twiss [135], Glauber formulated the quantum theory of optical
coherence in terms of nth-order correlation functions [136]. Higherorder correlations have gained major importance in quantum optics [137], since they can supply information about the nonclassical nature of light. Furthermore, such correlations have a crucial
position in the characterization of quantum polarization [138], especially in addressing the degree(s) of polarization for quantized
light, a subject which is in constant development [139]. At the heart
of Glauber’s seminal work is the equivalence between complete coherence and the factorization of the coherence matrix into a product
of two vectors. This foundation provides a more general definition for complete coherence than the original definition from 1938
of Zernike for classical scalar light [140], which connects complete
coherence with full intensity visibility in Young’s interference experiment [141]. An essential requirement for a completely coherent
field, according to the definition of Glauber, is that it must be totally
polarized. Overall, as with the part played by Wolf (and Mandel)
in the context of classical optics [142], the importance of Glauber’s
(and Mandel’s) contributions for quantum optics is hard to overestimate [143].
The principle of complementarity [144], stating that quantum objects share mutually exclusive characteristics, has had a major significance for the foundations of physics and a profound impact on
the interpretation of the fundamental nature of reality [145, 146].
The arguable most recognized manifestation of complementarity in
physics is the wave–particle duality, which places a trade-off for the
Dissertations in Forestry and Natural Sciences No 252
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
Quantum Light Fields
wave and particle qualities of quantum systems [147–150]. In the
third volume of his famous lecture series [151], Feynman declares
that this dual wave–particle behavior “... has in it the heart of quantum mechanics. In reality, it contains the only mystery.” The duality
has been given quantitative expressions in two-way interferometry
[152–154], such as the celebrated double-slit (or double-pinhole) experiment [155,156], according to which path predictability and path
distinguishability, representing different kinds of ’which-path information’, are complementary with the visibility of intensity fringes.
In the case of photons, however, interference does not necessarily
appear only as intensity fringes, but also, or exclusively, as polarization contrasts [157–159]. How complementarity is manifested, and
quantified, under such polarization modulation, has not been considered before.
1.2
SCOPE OF THE THESIS
This thesis encompasses fundamental theoretical research on several topics in electromagnetic nanophotonics. The subjects covered
can be classified into three main categories: rigorous modal studies
of SPPs (Publications I–III), partially coherent optical surface fields
(Publications IV–VII), and complementarity in vector-light photon
interference (Publication VIII). Below we give a brief overview of
the central aspects concerning these themes; in the chapters that
follow we address the motivations, results, and implications of our
research in more detail.
Publications I–III deal with novel SPP modes at planar singleinterface and metal-slab geometries. In Publication I, by utilizing a
rigorous electromagnetic treatment, we demonstrate that the standard approximate approach that is frequently used to characterize
SPPs at a single boundary can lead to false predictions even in situations where it is supposed to be valid. As a main result, we predict a
new type of backward-propagating SPP mode that does not follow
from the approximate analysis. Publication II establishes a unified
framework and classification for all possible mode solutions, in-
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Dissertations in Forestry and Natural Sciences No 252
Introduction
cluding sets of entirely new ones, at a lossy metal slab in a symmetric and lossless surrounding. While previous works have focused
merely on the region outside the slab, we investigate the propagation and energy-flow features of the modes also within the film. It
is found that the various modes may appear either as forward- or
backward-propagating waves inside the slab. In Publication III, we
show how a higher-order metal-slab mode, commonly presumed
to have little practical importance, can turn into a strongly confined
long-range surface mode in circumstances where the fundamental
long-range SPP does not exist and the propagation length of the
single-interface SPP is minuscule. This discovery, which is encountered for a broad range of materials and bandwidths, but which
seems not to have been observed or even suggested before, forms
the culmination of our modal studies.
Publications IV and V consider partially coherent SPP fields at
a planar surface and on a metal slab. In publication IV, we formulate a theory for partially coherent polychromatic SPPs excited at
a metal–air interface in the Kretschmann configuration. The formalism covers stationary as well as nonstationary SPP fields of any
spectra. The key result is establishing a generic scheme to tailor the
electromagnetic coherence of such polychromatic SPPs by controlling the coherence state of the light source. The main objective of
Publication V is to examine the fundamental ranges that the spectral degrees of coherence and polarization of a stationary two-mode
field composed of the long-range and short-range SPPs at a metallic
nanoslab can attain, regardless of the excitation method. We also
explore how the degrees are influenced when the media, frequency,
and film thickness are changed.
Publications VI and VII concern the generation, partial polarization, and spatial coherence of stationary, purely evanescent light
fields at a lossless dielectric surface. The analysis in Publication VI
shows that, for such fields, the coherence length in air can attain
values notably shorter than the free-space wavelength, in contrast
to the common view that blackbody radiation exhibits the shortest
coherence length (about half a light’s wavelength). Publication VI
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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also demonstrates that evanescent fields are generally genuine 3Dpolarized fields, highlighting the need for full 3D polarization treatment. In Publication VII, motivated by the results of Publication
VI, we explore conditions for controlled excitation of completely
3D-unpolarized evanescent light fields in specific multibeam illumination setups and investigate their electromagnetic coherence.
The results reveal the possibility to excite an evanescent field that
shares the polarization properties of blackbody radiation, yet with
tunable coherence characteristics.
Publication VIII, dealing with quantum coherence and forming
an essential part of this thesis, concerns complementarity and polarization modulation in double-pinhole photon interference. As
a main contribution, we derive two general complementarity relations for genuine vector-light quantum fields of arbitrary state. The
complementarity relations are shown to reflect two distinct, fundamental aspects of wave–particle duality of the photon, having no
correspondence in scalar quantum theory. In particular, we demonstrate that, contrary to scalar light, for pure single-photon vector
light the a priori which-path information does not attach to the intensity visibility, but to a generalized visibility, which takes into account the polarization-state modulation as well. It is also elucidated
that in the general case such complementarity is a manifestation of
complete coherence, not of quantum-state purity.
Since many topics of this thesis involve electromagnetic surface
waves at planar interfaces, we begin by introducing the basic formalism for characterizing such fields in Chap. 2. The novel SPP
modes are presented in Chap. 3 (Publications I–III), whereas in
Chap. 4 we address partially coherent SPP fields (Publications IV
and V). Chapter 5 concerns electromagnetic coherence and polarization of purely evanescent light fields (Publications VI and VII).
Complementarity and polarization modulation in vectorial dualpinhole photon interference are discussed in Chap. 6 (Publication
VIII). Finally, Chap. 7 summarizes the main conclusions and future
prospects of this work. A brief account of classical electromagnetic
coherence theory is provided in Appendix A.
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2 Electromagnetic surface
waves
Despite its broad versatility, the SPP is just a certain species in the
rich family of surface polaritons [160], manifested also via phonons,
excitons, and magnons, among others. Surface polaritons, in turn,
constitute a part of the more general class of electromagnetic surface
waves (ESWs) [161], whose attractiveness comes from their intrinsic
unique capability of strong confinement and long-range guidance
of electromagnetic energy along the supporting interface. In this
chapter, we introduce the basic formalism to characterize ESWs in
homogeneous and isotropic media. The formalism is extensively
employed in Publications I–VII.
2.1 NOMENCLATURE
In its simplest form, an ESW is a field that propagates along a planar interface with an exponentially decaying amplitude away from
the surface. Long before Fano introduced his superficial waves for
the optics community, ESWs had been studied by a completely different physics community. In 1907, while analyzing radio-wave
propagation parallel to Earth’s surface, Zenneck found solutions
of Maxwell’s equations representing ESWs at a flat interface separating two homogeneous media of different permittivities [162].
Although Zenneck showed that Maxwell’s equations allow the existence of such ESWs, he did not examine the excitation process of the
fields. Two years later Sommerfeld published an influential paper,
in which he investigated rigorously the field generated by a vertical
dipole near a conductive plane [163]. He divided the dipole field
into two parts, a ‘space wave’ and a ‘surface wave’, and concluded
that the latter dominates near the boundary and goes over into that
predicted by Zenneck as the distance is increased. However, the
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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original paper contained an erroneous calculation which was later
corrected and extended by himself [164] and others [165–169].
Sommerfeld’s original paper left its mark on the radio-engineering community, where the Zenneck wave and the Sommerfeld wave
have often been used as synonyms. Even more terms have been established over the years, which has caused misunderstandings and
confusion from time to time [170, 171]. The extensive nomenclature
for various ESWs serves as a good example: Zenneck wave, Sommerfeld wave, Norton surface wave, ground wave, improper mode,
and lateral wave [171]. In addition, contradictory results concerning Zenneck waves have been published and no consensus on their
existence seems yet to exist [172]. The problematic terminology is
not restricted to radio-wave physics only, but also concerns optical
plasmonics, where surface plasmons, surface-plasmon polaritons,
Fano waves, and even Zenneck waves are often mixed and used to
describe same or different phenomena. With the incarnation of such
exotic ESWs as Tamm waves, Dyakonov waves, and Dyakonov–Tamm
waves [161], not to disregard spoof modes [173–175], the taxonomy
has become even more involved. Nevertheless, it is important to
understand that, regardless of the complex nomenclature and the
different physical origins of these various ESWs, their essential electromagnetic character is the same.
2.2
FIELD CHARACTERIZATION
From a fundamental point of view, the complete description of an
electromagnetic field requires that both the electric field and the
magnetic field are taken into account [176]. Nonetheless, since the
interaction between light and matter takes predominantly place via
the electric field (principally through the dipole moment) [177], we
mainly focus on the electric part. As another starting point we
take all the involved media to be linear, homogeneous, isotropic,
stationary in time, spatially nondispersive, free of sources, and passive. The electromagnetic response of the medium then generally
depends on the (angular) frequency ω through a complex-valued
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Electromagnetic surface waves
(relative) electric permittivity ϵr (ω ) and a complex-valued (relative)
magnetic permeability µr (ω ), accounting to both temporal dispersion as well as absorption. Alternatively, as is common in optical
physics [177], the materials can be characterized by the refractive
√
index n(ω ) = ϵr (ω )µr (ω ). Both conventions are alternately employed throughout this thesis.
2.2.1 Representation
Let us consider a monochromatic electromagnetic plane wave in
connection of a planar interface. The wave vector is taken to be
of the form k(ω ) = k ∥ (ω )ê∥ + k ⊥ (ω )ê⊥ , where k ∥ (ω ) and k ⊥ (ω )
are complex numbers, whereas ê∥ and ê⊥ are real-valued unit vectors lying parallel and perpendicularly to the surface, respectively.
In this case k(ω ) and ê⊥ span a plane analogous to the ‘plane of
incidence’ encountered in the context of light propagation across
a boundary [177]; yet, since we are dealing with ESWs, we prefer
the more neutral term ‘propagation plane’. The electric field, at a
space–time point (r, t), can thus be decomposed into an s-polarized
part (which is perpendicular to the propagation plane) and a ppolarized part (which is parallel to it) as
E(r, t) = [ Es (ω )ŝ(ω ) + E p (ω )p̂(ω )]ei[k(ω )·r−ωt] ,
(2.1)
where Es (ω ) and E p (ω ) are the (complex-valued) amplitudes of the
s-polarized and the p-polarized field components, respectively. The
corresponding unit polarization vectors, ŝ(ω ) and p̂(ω ), obeying
k(ω ) · ŝ(ω ) = 0,
k(ω ) · p̂(ω ) = 0,
(2.2)
as required by Maxwell’s equations [176, 177], are constructed as
ŝ(ω ) = ê⊥ × ê∥ ,
p̂(ω ) = k̂(ω ) × ŝ(ω );
k̂(ω ) =
k(ω )
. (2.3)
|k(ω )|
In this way {k̂(ω ), ŝ(ω ), p̂(ω )} constitutes a right-handed and unitnormalized vector triad.
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We stress that, instead of the length |k(ω )| in Eq. (2.3), the wave
vector is customarily normalized with respect to the wave number
k(ω ) = k0 (ω )n(ω ) [2, 176–179], where k0 (ω ) is the free-space wave
number. Such a choice, however, does not yield a k̂(ω ) of unit
length in the case of a true complex-valued wave vector, for which
|k(ω )| ̸= k(ω ). The virtue of using |k(ω )| instead of k(ω ) in the
normalization is actually not merely mathematical, but rather physical, as is explained in Sec. 2.4. Another point that must be emphasized is that, regardless of the normalization, and even if ŝ(ω ) is (by
definition) perpendicular to both k̂(ω ) as well as p̂(ω ), the vectors
k̂(ω ) and p̂(ω ) are principally not mutually orthogonal when the
wave vector is complex [180], unlike sometimes stated [178]. Hence
the triad {k̂(ω ), ŝ(ω ), p̂(ω )} is generally semi-orthogonal.
2.2.2
Propagation
To elucidate the propagation characteristics of an ESW, we adopt
the notation k(ω ) = k′ (ω ) + ik′′ (ω ) for the wave vector, where the
real part k′ (ω ) = k′∥ (ω )ê∥ + k′⊥ (ω )ê⊥ describes the phase movement, while the imaginary part k′′ (ω ) = k′′∥ (ω )ê∥ + k′′⊥ (ω )ê⊥ accounts to the confinement of the field. The existence of an ESW requires, by definition, that at least k′′⊥ (ω ) ̸= 0 and k′∥ (ω ) ̸= 0. In the
special case that k′⊥ (ω ) = k′′∥ (ω ) = 0 the ESW is pure, i.e., the field
is purely evanescent perpendicularly to the boundary and strictly
propagating along it without attenuation. In general, though, both
k′⊥ (ω ) and k′′∥ (ω ) are nonzero, thereby resulting in quasibound or
pseudo ESWs [181], which are neither purely evanescent away from
the surface, nor solely propagating along it.
The condition k(ω ) · k(ω ) = k20 (ω )n2 (ω ) enables to determine
the magnitudes and the relative directions of k′ (ω ) and k′′ (ω ). In
the presence of absorption, we find that
|k′ (ω )| > |k′′ (ω )|,
′
′′
′
′′
|k (ω )| = |k (ω )|,
|k (ω )| < |k (ω )|,
12
if |n′ (ω )| > n′′ (ω ),
(2.4)
′
′′
(2.5)
′
′′
(2.6)
if |n (ω )| = n (ω ),
if |n (ω )| < n (ω ),
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Electromagnetic surface waves
where n′ (ω ) and n′′ (ω ) are, respectively, the real and imaginary
parts of the refractive index, with n′′ (ω ) > 0 [180], and
α < π/2,
if n′ (ω ) > 0,
′
(2.7)
α = π/2,
if n (ω ) = 0,
(2.8)
α > π/2,
if n′ (ω ) < 0,
(2.9)
in which α ∈ [0, π ] is the angle between k′ (ω ) and k′′ (ω ). Further,
k′⊥ (ω )k′′⊥ (ω ) < 0 ⇔ k′∥ (ω )k′′∥ (ω ) > k20 (ω )n′ (ω )n′′ (ω ),
(2.10)
k′⊥ (ω )k′′⊥ (ω ) > 0 ⇔ k′∥ (ω )k′′∥ (ω ) < k20 (ω )n′ (ω )n′′ (ω ),
(2.12)
k′⊥ (ω ) = 0 ⇔ k′∥ (ω )k′′∥ (ω ) = k20 (ω )n′ (ω )n′′ (ω ),
(2.11)
where Eq. (2.10) corresponds to the situation in which the directions
of phase movement and amplitude attenuation are antiparallel perpendicularly to the boundary, while in the case of Eq. (2.12) they
are collinear. At the transition point, Eq. (2.11), the ESW is purely
evanescent transversally to the surface.
In a lossless medium, one instead always has |k′ (ω )| > |k′′ (ω )|
and α = π/2, with the latter signifying orthogonality between the
wavefront advancement and field attenuation. Moreover,
k′⊥ (ω )k′′⊥ (ω ) = −k′∥ (ω )k′′∥ (ω ),
if n′′ (ω ) = 0,
(2.13)
stating that if the directions of phase propagation and amplitude
decay are the same parallel to the surface, then they will be opposite
perpendicularly to it (and vice versa).
Eventually, the basic physical quantities characterizing the field
propagation of an ESW are [161]
Propagation length :
Penetration depth :
Surface wavelength :
l∥ (ω ) = 1/|k′′∥ (ω )|,
l⊥ (ω ) = 1/|k′′⊥ (ω )|,
Λ(ω ) =
Surface phase velocity : vp (ω ) =
Surface group velocity :
vg ( ω ) =
2π/|k′∥ (ω )|,
ω/k′∥ (ω ),
∂ω/∂k′∥ (ω ),
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
which are frequently encountered later on.
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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2.2.3
Energy flow
To assess the (net) flux of electromagnetic energy, we make use of
the time-averaged Poynting vector for harmonic fields [2, 176],
S(r, ω ) =
1
[E(r, t) × H∗ (r, t)]′ ,
2
(2.19)
where H(r, t) is the magnetic field, the asterix denotes complex conjugation, and the prime stands for the real part. The Poynting vector itself may, with certain caution, be interpreted as describing the
direction of the energy flux, whereas the integral of S(r, ω ) taken
over a closed surface amounts to the total energy flowing through
the boundary of the considered volume. For an ESW, according to
Eqs. (2.1)–(2.3) and (2.19),
S(r, ω ) =
ϵ0 c [ Ss (ω ) + S p (ω ) + 2iSsp (ω ) ]′ −2k′′ (ω )·r
e
,
2k0 (ω )
µr∗ (ω )
(2.20)
Ss (ω ) = | Es (ω )|2 k∗ (ω ),
(2.21)
S p (ω ) = | E p (ω )|2
(2.22)
with ϵ0 and c being the vacuum permittivity and the speed of light,
respectively, and
k 2∗ ( ω )
k ( ω ),
|k(ω )|2
[k∗∥ (ω )k ⊥ (ω )]′′
∗
ŝ(ω ),
Ssp (ω ) = Es (ω ) E p (ω )
|k(ω )|
(2.23)
where the double prime in Eq. (2.23) stands for the imaginary part.
From Eqs. (2.20)–(2.23) we make three main observations.
Firstly, regardless of the polarization, the Poynting vector decays
exponentially at twice the rate of the field in the direction determined by k′′ (ω ). Secondly, in a nonabsorptive medium, when the
ESW is fully s polarized or completely p polarized, the energy-flow
direction is either parallel or antiparallel to that of phase propagation, specified by k′ (ω ). In our studies, fields whose wavefronts
move parallel to S(r, ω ) are regarded as forward-propagating waves,
while those fields for which k′ (ω ) is opposite to the energy transfer are referred to as backward-propagating waves. Thirdly, in the case
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Electromagnetic surface waves
that the ESW contains both s- and p-polarized contributions, the
Poynting vector may have a component in a direction orthogonal to
k′ (ω ). A nonzero energy flux perpendicularly to the propagation
plane is not necessarily an artifact, but has been viewed as a manifestation of the inertial effect of the photon spin [17, 18, 182], also
known as the spin-Hall effect of light [183,184]. This prominent feature is responsible for the Imbert–Fedorov effect [185,186], in which
a small transverse phase shift with respect to the propagation plane
is detected when a light beam is totally internally reflected at an interface [187]. The phenomenon is akin to the Goos–Hänchen effect
involving a longitudinal phase shift [188].
2.3 EXISTENCE OF ELECTROMAGNETIC SURFACE MODES
Up to this point, we have been dealing with a rather arbitrary and
independent ESW. According to Maxwell’s equations, however, the
ESW is unavoidably coupled to another field on the other side of
the interface via the electromagnetic boundary conditions. This fact
places constraints on the existence of the ESW.
In this section, we consider surface-bound solutions for a configuration in which there is one wave, as given by Eq. (2.1), on each
side of the interface. Such delicate, self-sustained ESW solutions
can be regarded as representing electromagnetic surface modes (ESMs)
of the system. Without loss of generality, we take the boundary to
z
ǫr2 (ω ), µr2 (ω )
E2 (r, t)
x
ǫr1 (ω ), µr1 (ω )
E1 (r, t)
Figure 2.1: Illustration of the geometry and notation for two electric plane waves at a flat
interface (z=0) between two media of relative electric permittivities ϵr1 (ω ) and ϵr2 (ω ) as
well as relative magnetic permeabilities µr1 (ω ) and µr2 (ω ).
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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lie in the Cartesian xy plane (z = 0) and the wave propagation to be
along the x axis, as illustrated in Fig. 2.1, whereby k ∥ (ω ) → k x (ω )
and k ⊥ (ω ) → k z (ω ). Under these conditions, the s-polarized and
p-polarized field components have to satisfy
s polarization :
µr2 (ω )k z1 (ω ) = µr1 (ω )k z2 (ω ),
(2.24)
p polarization :
ϵr2 (ω )k z1 (ω ) = ϵr1 (ω )k z2 (ω ),
(2.25)
where the subscripts 1 and 2 have been introduced to distinguish
the half-space regions z < 0 (medium 1) and z > 0 (medium 2), respectively. Equations (2.24) and (2.25), which are found to be symmetric with respect to the interchange of ϵr (ω ) and µr (ω ), specify the exact conditions that the two fields at the planar interface
must fulfill in order to exist. Therefore, we refer to Eqs. (2.24) and
(2.25) as the existence conditions for the two-wave system depicted in
Fig. 2.1. When Eqs. (2.24) and (2.25) are further accompanied by the
surface-bound requirements k′′z1 (ω ) < 0 and k′′z2 (ω ) > 0, we end up
with the existence conditions for ESMs.
For most natural materials (dielectrics, metals, semiconductors)
at optical frequencies, µr (ω ) assumes up to a very good accuracy
the value of unity [180]. In this case Eq. (2.24) states that k z1 (ω ) =
k z2 (ω ), implying that the surface-bound requirements k′′z1 (ω ) < 0
and k′′z2 (ω ) > 0 cannot both be satisfied, thereby excluding the existence of s-polarized ESMs (it should be mentioned, though, that
metamaterials as well as nonhomogeneous and anisotropic media
permit s-polarized ESMs within the optical domain [161, 189]). For
p polarization, on the other hand, there is no such proscription, and
after setting µr1 (ω ) = µr2 (ω ) = 1 as well as using the fact that the
tangential wave-vector component is continuous across the boundary, viz., k x1 (ω ) = k x2 (ω ) = k x (ω ), Eq. (2.25) yields
k x (ω ) = k0 (ω )
√
k zα (ω ) = k0 (ω ) √
16
ϵr1 (ω )ϵr2 (ω )
,
ϵr1 (ω ) + ϵr2 (ω )
ϵrα (ω )
ϵr1 (ω ) + ϵr2 (ω )
,
(2.26)
α ∈ {1, 2}.
(2.27)
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Electromagnetic surface waves
The relations above are well-established and specify the basic physical properties [Eqs. (2.14)–(2.18)] of p-polarized ESMs [161].
In some studies, the basic starting point is to neglect absorp′′ ( ω ) = ϵ′′ ( ω ) = 0, whereupon Eqs. (2.26) and (2.27)
tion, i.e., ϵr1
r2
′ ( ω ) ϵ′ ( ω ) < 0 and
lead to surface-bound solutions only when ϵr1
r2
′
′
ϵr1 (ω ) + ϵr2 (ω ) < 0. Under these conditions the ESMs are pure and
sometimes called Fano waves [161, 172, 190]. However, a more realistic scenario involves losses, at least in one of the media, rendering
the situation quite different. In particular, the inclusion of absorption results in quasibound ESMs, since the wave-vector components
generally become to include both real and imaginary parts. Yet, albeit the pureness is lost as a result of the losses, what is perhaps
less known is that the strict requirements on ϵr1 (ω ) and ϵr2 (ω ) also
vanish. In fact, when absorption is present, the interface supports
(quasibound) ESMs for almost any value of ϵr1 (ω ) and ϵr2 (ω ) [191].
2.4 FRESNEL COEFFICIENTS AND
EXISTENCE CONDITIONS
Let us next consider three monochromatic plane waves, constructed
according to Eqs. (2.1)–(2.3), at a planar boundary as illustrated in
Fig. 2.2. The setup is otherwise the same as in Fig. 2.1, but now with
two fields in medium 1 and one field in medium 2. To distinguish
the waves in medium 1, we introduce the superscripts (1) and (2)
z
E2 (r, t)
ǫr2 (ω ), µr2 (ω )
ǫr1 (ω ), µr1 (ω )
(1)
E1 (r, t)
(2)
E1 (r, t)
x
Figure 2.2: Illustration of the geometry and notation for three electric plane waves at a flat
interface (z=0) between two media of relative electric permittivities ϵr1 (ω ) and ϵr2 (ω ) as
well as relative magnetic permeabilities µr1 (ω ) and µr2 (ω ) (cf. Fig. 2.1).
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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to these fields. From the electromagnetic boundary conditions we
then find that the Fresnel reflection coefficients, with respect to the
(1)
s- and p-polarized amplitudes of the field E1 (r, t), become
(1)
rs (ω ) =
µr2 (ω )k z1 (ω ) − µr1 (ω )k z2 (ω )
r p (ω ) =
ϵr2 (ω )k z1 (ω ) − ϵr1 (ω )k z2 (ω )
(1)
µr2 (ω )k z1 (ω ) + µr1 (ω )k z2 (ω )
,
(2.28)
(1)
(1)
ϵr2 (ω )k z1 (ω ) + ϵr1 (ω )k z2 (ω )
,
(2.29)
while the transmission coefficients read as
(1)
ts (ω ) =
2µr2 (ω )k z1 (ω )
(1)
µr2 (ω )k z1 (ω ) + µr1 (ω )k z2 (ω )
,
(2.30)
(1)
t p (ω ) =
2ϵr1 (ω )k z1 (ω )
|k2 (ω )|
,
(1)
ϵr2 (ω )k (ω ) + ϵr1 (ω )k z2 (ω ) |k1 (ω )|
(2.31)
z1
(1)
(2)
where |k1 (ω )| = |k1 (ω )| = |k1 (ω )| in Eq. (2.31). We emphasize
that the derivation of Eqs. (2.28)–(2.31) assumes nothing more than
(2)
(1)
k z1 (ω ) = −k z1 (ω ) in medium 1.
As reported in Publication VI, the Fresnel transmission coefficient t p (ω ) in Eq. (2.31) differs from that given in most of the literature [1, 2, 17, 18, 25, 176, 177] if the wave vectors k1 (ω ) and/or
k2 (ω ) contain imaginary parts. The difference stems from the definition of the basis vector k̂(ω ) given by Eq. (2.3); instead of the unit
vector k̂(ω ) = k(ω )/|k(ω )|, many works use k̂(ω ) = k(ω )/k(ω )
which is not normalized to unity for a complex-valued k(ω ), as
discussed in Sec. 2.2.1. The advantage of k̂(ω ) = k(ω )/|k(ω )| is
that it always preserves the physical meaning of the transmission
coefficient as being the ratio between the complex field amplitudes
on the two opposite sides of the interface. For k̂(ω ) = k(ω )/k(ω ),
on the other hand, this is only true for purely propagating waves in
lossless media having real-valued wave vectors.
The Fresnel coefficients, Eqs. (2.28) and (2.30), and correspond-
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Electromagnetic surface waves
ingly Eqs. (2.29) and (2.31), have singularities, respectively, at
(1)
−µr2 (ω )k z1 (ω ) = µr1 (ω )k z2 (ω ),
(1)
−ϵr2 (ω )k z1 (ω )
= ϵr1 (ω )k z2 (ω ),
(2.32)
(2.33)
which we refer to as pole conditions. Under these circumstances the
(1)
(2)
field E1 (r, t) becomes vanishingly small compared to E1 (r, t) and
E2 (r, t). The zeros of the reflection coefficients in Eqs. (2.28) and
(2.29), on the other hand, are found at
(1)
µr2 (ω )k z1 (ω ) = µr1 (ω )k z2 (ω ),
(2.34)
(1)
ϵr2 (ω )k z1 (ω )
(2.35)
= ϵr1 (ω )k z2 (ω ),
(2)
respectively, corresponding to situations for which the field E1 (r, t)
vanishes. We may hence interpret Eqs. (2.34) and (2.35) as generalizations of the standard Brewster angle [177], and consequently call
them Brewster conditions.
Let us consider the p-polarized case. Physically the pole and
Brewster conditions represent a similar phenomenon in the sense
that if Eq. (2.33) or Eq. (2.35) is satisfied, the three-wave configuration in Fig. 2.2 reduces to the two-wave system in Fig. 2.1. Furthermore, we observe that Eqs. (2.25), (2.33), and (2.35) are mathematically of similar form [analogous connection is found for Eqs. (2.24),
(2.32), and (2.34) representing s polarization]. Thus the existence
condition, the pole condition, and the Brewster condition describe
the same physical two-field situation, but from slightly different
points of view.
SPPs have frequently been interpreted to correspond to the pole
condition [2,25,192,193], while Zenneck waves have been associated
with the Brewster condition [170, 194]. However, both wave types
can in fact be regarded as representing either the pole condition or
the Brewster condition. Note that we could equally well have defined the reflection and transmission coefficients with respect to the
(2)
field E1 (r, t). The expressions in Eqs. (2.28)–(2.35) would remain
the same, except for the change of the superscript (1) to (2). Since
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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(2)
(1)
k z1 (ω ) = −k z1 (ω ), the pole and Brewster conditions in Eqs. (2.32)–
(2.35) would then simply interchange their roles, i.e., the pole con(1)
dition of the field E1 (r, t) is the same as the Brewster condition
(2)
of the wave E1 (r, t), and vice versa. Hence the pole and Brewster
conditions depend on the interpretation which of the two fields in
medium 1 is considered as the ‘incident’ wave and which as the ‘reflected’ wave (a division that is not necessarily unambiguous when
dealing with complex-valued wave vectors). Figure 2.2 might sug(1)
gest that the field E1 (r, t) represents an ‘incident’ wave, whereas
(2)
the field E1 (r, t) stands for a ‘reflected’ wave. This is, however,
not the case, but the purpose of Fig. 2.2 is merely to provide visual
support for our discussion. In reality, we have not at any point spec(1)
(2)
ified the absolute directions of either k z1 (ω ) or k z1 (ω ); only their
(2)
(1)
relative directions have been determined via k z1 (ω ) = −k z1 (ω ).
20
Dissertations in Forestry and Natural Sciences No 252
3 Surface-plasmon polaritons
Ever since the foundations, many theoretical studies on SPPs have
concerned idealized, lossless metals, usually by considering the
conduction electrons as an undamped free-electron gas. Absorption is, however, an integral part of most real metals in the optical domain [195], mainly due to scattering processes and interband
transitions. This is important to take into account if one aims for
a deeper understanding as well as to improve the agreement between theory and experiments. Particular attention should also be
paid when introducing any approximations into the SPP-field analysis, since this can lead to solutions which are no longer admissible [181]. At the same time, field solutions arising from a rigorous
treatment may not appear, or even have any correspondence, within
a simplificative framework [191, 196–198].
Although the fabric for the fundamental properties of SPPs at
planar (and rough) interfaces was established nearly 30 years ago
[82], recent studies have indicated that there still remain issues that
could benefit from further critical analysis [181,196–199]. This chapter is an account of the novel single-interface SPP, metal-slab, and
long-range modes that are predicted in Publications I–III, in which
rigorous electromagnetic theory and empirical data for the material
parameters are employed.
3.1 SINGLE-INTERFACE MODES
The conceptually most fundamental SPP modes are those supported
by a single planar interface (see Fig. 2.1), whose propagation characteristics are completely specified by Eqs. (2.26) and (2.27). Whereas
for many dielectrics (and some semiconductors) ϵr (ω ) can be taken
as real-valued and positive, many metals, especially noble metals,
possess losses and a negative real part of the relative permittivity
within the visible spectrum [195]. Accordingly, we choose medium
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′ ( ω ) < 0 and ϵ′′ ( ω ) > 0,
1 to represent the absorptive metal with ϵr1
r1
while ϵr2 (ω ) of the lossless medium 2 is a positive real number.
Then again, both media are taken to be nonmagnetic, as is applicable to most natural materials in the optical regime.
3.1.1
Approximate solutions vs. exact solutions
Discussions on SPP propagation at a planar interface under the
influence of metal absorption can be found in standard textbooks
[2, 82]. Such treatments, however, employ approximate expressions
for the wave-vector components [Eqs. (A7)–(A9) in Publication I]
′′ ( ω ) ≪ | ϵ′ ( ω )|. Another conwhich rely on the assumption that ϵr1
r1
′ ( ω )| > ϵ ( ω ),
straint embedded in those approximations is that |ϵr1
r2
leading to a cut-off frequency similar to that occurring in the loss′ ( ω )| =
less free-electron model. At the cut-off frequency, where |ϵr1
ϵr2 (ω ), both the real and imaginary parts of the wave vector diverge, so that the mode becomes sort of a ‘frozen’ and infinitely
strongly localized surface ‘spot’. In contrast, such singularities are
never present in the exact expressions [Eqs. (3)–(5) in Publication I],
neither do they prohibit mode solutions in the ‘forbidden’ region
′ ( ω )| < ϵ ( ω ).
where |ϵr1
r2
The rigorous expressions can, in fact, entail rather peculiar physical features in regimes beyond the scope of the approximate frame′ ( ω )| = ϵ ( ω ) and ϵ′′ ( ω ) ≫ | ϵ′ ( ω )|,
work. For instance, when |ϵr1
r2
r1
r1
the exact formulas yield a mode solution for which
√
[
ϵr2 (ω ) ]
.
(3.1)
k x (ω ) ≈ k0 (ω ) ϵr2 (ω ) 1 + i ′′
2ϵr1 (ω )
Surprisingly, the real part of k x (ω ) in Eq. (3.1), being equal to the
wave number in the region z > 0, suggests that the mode could
be excited directly by light incident from medium 2. The imaginary
part, in turn, indicates that the damping of the mode is inversely
proportional to the losses, i.e., the larger the absorption, the longer
the propagation. Other curiosities emerging in the rigorous framework, which at first sight may appear puzzling, are the seemingly
infinite and negative group velocities due to backbent dispersion
22
Dissertations in Forestry and Natural Sciences No 252
Surface-plasmon polaritons
curves [196]. However, superluminal (including infinite) as well
as negative group velocities are encountered in situations involving
strong absorption [200], and in such cases the group velocity should
not be associated with the signal velocity [176].
3.1.2 Mode types SPP I and SPP II
In Publication I, based on an exact treatment, we demonstrate that
the approximate approach discussed above, frequently used to assess SPP propagation [29, 30, 201], may lead to inaccurate conclusions even in situations where it is supposed to hold. Yet, the main
result of Publication I is the prediction of a new type of backwardpropagating SPP mode that does not follow from the customary
approximate analysis.
To address the subject, in the following we constrain ourselves
to fields which attenuate in the positive x direction, i.e., k′′x (ω ) > 0,
but stress that the main conclusions are also valid for k′′x (ω ) < 0.
As shown rigorously in Publication I, under these conditions
k′x (ω ) > 0,
k′z2 (ω ) < 0,
(3.2)
indicating that along the x axis the phase movement is in the same
positive direction as the amplitude attenuation, while along the z
axis in the lossless medium 2 the wavefronts propagate towards the
surface. The situation is more involved for k′z1 (ω ) within the metal,
for which the exact treatment reveals a material dependency:
k′z1 (ω ) < 0,
k′z1 (ω )
k′z1 (ω )
k′z1 (ω )
′′
if ϵr2 (ω ) < ϵr1
( ω ),
< 0,
if
= 0,
if
> 0,
if
′
[ϵr1
(ω ) + ϵr2 (ω )]2
′
[ϵr1
(ω ) + ϵr2 (ω )]2
′
[ϵr1
(ω ) + ϵr2 (ω )]2
(3.3)
>
=
<
2
(ω ) −
ϵr2
2
ϵr2 (ω ) −
2
ϵr2
(ω ) −
′′2
ϵr1
( ω ),
′′2
ϵr1 (ω ),
′′2
ϵr1
( ω ).
(3.4)
(3.5)
(3.6)
We may thereby define two different types of SPPs in medium 1 as
SPP I : k′z1 (ω ) < 0,
SPP II : k′z1 (ω ) > 0,
(3.7)
which are illustrated in Fig. 3.1. The situation of k′z1 (ω ) = 0, where
the field in the metal is purely evanescent in the z direction and
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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SPP I
SPP II
Figure 3.1: Illustration of the directions of phase movement (black arrows) and field attenuation (solid-red curves) for SPP I (left) and SPP II (right) decaying to the right.
the wavefronts advance only along the surface, is a transition point
between SPP I and SPP II.
According to the approximate analysis, on the other hand, lean′′ ( ω ) ≪ | ϵ′ ( ω )| and | ϵ′ ( ω )| > ϵ ( ω ),
ing on the assumptions ϵr1
r2
r1
r1
one always has k′z1 (ω ) < 0, corresponding to SPP I. The existence of
SPP II, a prediction of the rigorous analysis, is thereby completely
excluded within the approximate framework. There is another severe issue with the approximate approach. To see this, we consider
a Ag–GaP interface at the free-space wavelength λ0 (ω ) = 632.8 nm
for which ϵr1 (ω ) = −15.85 + i1.08 [195] and ϵr2 (ω ) = 11.01 [202].
′′ ( ω ) ≪ | ϵ′ ( ω )| and | ϵ′ ( ω )| > ϵ ( ω ), as required in the
Now ϵr1
r2
r1
r1
approximate treatment, whereupon we should obtain k′z1 (ω ) < 0.
However, by looking at Eqs. (3.3)–(3.6), derived by rigorous means,
we find that these material parameter values fall under Eq. (3.6),
expressing that k′z1 (ω ) > 0. Thus the approximate treatment does
not only exclude SPP II, but it may also lead to false physical predictions in situations where it is supposed to be valid.
3.1.3
Flow of energy
To gain further insight into SPP propagation based on an exact
treatment, we consider the energy flow of the (p-polarized) SPPs
in terms of the Poynting vector given by Eq. (2.20), which yields
′′
S(r, ω ) ∝ [ϵr′ (ω )k′ (ω ) + ϵr′′ (ω )k′′ (ω )]e−2k ·r .
(3.8)
For ϵr′ (ω ) > 0 and ϵr′′ (ω ) = 0, the relation above states that S(r, ω )
is parallel to k′ (ω ) and thus, according to Eq. (3.2), the energy
24
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Surface-plasmon polaritons
transfer in the region z > 0 is tilted toward the surface in the same
direction as the wavefronts propagate. Consequently, only forwardpropagating SPPs are encountered in medium 2, as is characteristic
of electromagnetic plane waves in dielectrics. Within the metal the
situation is more involved, due to absorption, whereupon it is illustrative to consider the energy flow separately along the z and x
axes.
As concluded in Publication I, the z component of the Poynting vector is always negative within the metal, i.e., Sz1 (r, ω ) < 0,
indicating that energy is transported away from the interface (and
absorbed) in medium 1. Interestingly, this is true even in the case of
Eq. (3.5) with k′1z (ω ) = 0, for which the field is purely evanescent
perpendicularly to the surface. In view of Eq. (3.7) we then find that
SPP I is a forward-propagating wave, whereas SPP II is a backwardpropagating wave with respect to the z axis. As pointed out in the
previous section, the backward-propagating SPP II is not met in the
standard approximate framework, but is exclusively a consequence
of the rigorous treatment. Regarding the x component, the exact
analysis reveals the possibility of energy transfer in both directions:
Sx1 (r, ω ) > 0,
Sx1 (r, ω ) < 0,
′
if |ϵr1
(ω )| < ϵr2 (ω )/2,
if
′
|ϵr1
(ω )|
> ϵr2 (ω )/2,
(3.9)
(3.10)
′′ ( ω ). Hence, since k ′ ( ω ) > 0
which is completely independent of ϵr1
x
according to Eq. (3.2), a similar forward–backward behavior occurs
′ ( ω )| = ϵ ( ω ) /2,
also along the surface. At the transition point |ϵr1
r2
where Sx1 (r, ω ) = 0, no energy is transferred along the boundary.
′ ( ω )| >
The approximate treatment, on the other hand, requiring |ϵr1
ϵr2 (ω ), allows energy flow in only one direction.
It is important to understand that the physical mechanisms for
the appearance of forward–backward propagation along the z and
x axes are very different: in the former the behavior arises from the
change of direction in the phase movement, while in the latter the
feature is caused by the change in the energy-flow direction. As
we will discuss in the next chapter, also metal-slab modes possess
similar field-propagation properties.
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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3.2
METAL-SLAB MODES
What makes a thin metal slab with two boundaries fundamentally
very different from a single metal surface is that, much like a dielectric waveguide, it can support an unlimited number of modes [198].
Moreover, not only from a fundamental, but also from an application point of view, the thickness of the slab provides an important
degree of freedom, not possessed by the single-interface geometry,
which can be utilized to modify various physical properties of the
modes. How the slab thickness (among other parameters) affects
the coherence and polarization of an SPP field consisting of the two
well-known symmetric and antisymmetric modes [203], the subject
of Publication V, is discussed in the next chapter.
This section summarizes the results of Publication II, in which
a rigorous theoretical formulation based on electromagnetic plane
waves is utilized to construct a unified framework and identification for all possible mode solutions at an absorptive (nonmagnetic)
metal slab in a symmetric and lossless surrounding. The mode solutions are categorized into three main classes and divided further
into subspecies depending on their field profile and propagation
characteristics. It turns out that the various modes appear not only
as bound waves, but also as fields with an exponentially growing
amplitude away from the supporting surface. We refer to these
transversally growing fields as leaky waves, following the nomenclature used by Burke, Stegeman, and Tamir [203]. Although an
exponentially growing field amplitude is somewhat dubious, leaky
waves can be practically meaningful in a transient sense over limited regions of space [203]. Furthermore, whereas many works have
focused only on the region outside the film, we investigate the field
characteristics also inside the slab. Energy-flow considerations reveal that the various modes manifest themselves both as forwardand backward-propagating waves within the film.
To be specific, the geometry that we consider (see Fig. 3.2) encompasses an absorptive metal film of thickness d, characterized by
′ ( ω ) + iϵ′′ ( ω ) in
a complex-valued relative permittivity ϵr1 (ω ) = ϵr1
r1
26
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Surface-plasmon polaritons
z
(1)
E2 (r, t)
ǫr2 (ω )
(1)
d
E1 (r, t)
ǫr1 (ω )
x
(2)
E1 (r, t)
ǫr2 (ω )
(2)
E2 (r, t)
Figure 3.2: Illustration of the geometry and notation for electric plane waves at a lossy
metal film (|z| < d/2) of thickness d surrounded by a nonabsorptive medium (|z| ≥ d/2),
possessing relative permittivities ϵr1 (ω ) (complex) and ϵr2 (ω ) (real), respectively.
′ ( ω ) < 0 and ϵ′′ ( ω ) > 0, which is surthe region |z| < d/2, with ϵr1
r1
rounded on both sides by a lossless medium possessing a real and
positive permittivity ϵr2 (ω ). The spatial part of the (p-polarized)
electric field outside the slab reads as

 E(1) (ω )p(1) (ω )eik2(1) (ω )·r , z ≥ d/2,
2
2
(3.11)
E2 (r, ω ) =
 E(2) (ω )p(2) (ω )eik2(2) (ω )·r , z ≤ −d/2,
2
2
while the (p-polarized) field inside the film is expressed as
(1)
(1)
(1)
E1 (r, ω ) = E1 (ω )p̂1 (ω )eik1
(ω )·r
(2)
(2)
(2)
+ E1 (ω )p̂1 (ω )eik1
(ω )·r
. (3.12)
The superscripts (1) and (2) have been introduced to separate the
waves in each medium and the unit polarization vectors are constructed according to Eq. (2.3).
As is customary, we thus examine a system in which there are
two plane waves within and one plane wave on each side of the slab.
However, unlike in previous studies, we place no restrictions on the
wave-vector directions inside or outside the slab. More precisely,
whereas the tangential components of the wave vectors are continu( β)
ous across the boundaries, i.e., k xα (ω ) = k x (ω ) for all α, β ∈ {1, 2},
we make no assumptions on the transverse components, for which
(2)
(1)
one has two options in each medium, namely k zα (ω ) = ±k zα (ω ).
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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As shown in Publication II, under these circumstances Maxwell’s
equations permit three different mode-solution classes, which are
(1)
(2)
M1 : k z1 (ω ) = k z1 (ω ),
(1)
(2)
(3.13)
(1)
(2)
(3.14)
k z2 (ω ) = k z2 (ω ),
(1)
(2)
k z2 (ω ) = k z2 (ω ),
(1)
(2)
k z2 (ω ) = −k z2 (ω ).
M2 : k z1 (ω ) = −k z1 (ω ),
(1)
M3 : k z1 (ω ) = −k z1 (ω ),
(1)
(2)
(2)
(1)
(3.15)
(2)
Note that the cases k z1 (ω ) = k z1 (ω ) and k z2 (ω ) = −k z2 (ω ) are
not allowed. We next review the field characteristics of each type
(1)
separately, to which end the notation k zα (ω ) = k zα (ω ) is adopted.
3.2.1
Mode class M1
The first class, M1, standing for the case where the two waves inside
the slab coincide, are characterized by exactly the same wave-vector
components as those of SPPs at a single interface,
√
ϵr1 (ω )ϵr2 (ω )
,
(3.16)
k x (ω ) = k0 (ω )
ϵr1 (ω ) + ϵr2 (ω )
k zα (ω ) = k0 (ω ) √
ϵrα (ω )
ϵr1 (ω ) + ϵr2 (ω )
,
α ∈ {1, 2}.
(1)
(3.17)
(2)
Yet, since in the present two-boundary situation k z2 (ω ) = k z2 (ω ),
the class M1 always contains one bound wave and one leaky wave
outside the film. That the wave-vector components of the M1 modes
are the same as in the single-interface case, and thereby completely
independent of the slab thickness, might at first glance be somewhat puzzling. After a second thought, however, this is quite intuitive, because as only one plane wave exists on each side of the two
interfaces, the same single-interface existence condition [Eq. (2.25)]
can be met at both boundaries. The M1 modes may thus be interpreted as corresponding to conditions for which the Fresnel reflection coefficient goes to zero outside as well as inside the film (see
Sec. 2.4).
Regarding field propagation, as we show in Publication II,
k′x (ω )k′′x (ω ) > 0,
28
k′z2 (ω )k′′z2 (ω ) < 0,
(3.18)
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Surface-plasmon polaritons
indicating that the directions of phase propagation and amplitude
attenuation are the same along the x axis, but along the z axis they
are opposite in the region |z| ≥ d/2, respectively. For k1z (ω ) in the
region |z| < d/2 we instead obtain three possibilities, which lead
us to classify three different types of M1 modes as
M1I : k′z1 (ω )k′′z1 (ω ) > 0,
(3.19)
M1II : k′z1 (ω ) = 0,
(3.20)
M1III :
k′z1 (ω )k′′z1 (ω )
< 0.
(3.21)
M1I represents modes whose phases move in the same direction as
the fields decay, while for M1III the phase advancement is opposite to that of attenuation (along the z axis). M1II stands for the
case where the field is purely evanescent in the z direction and the
wavefronts propagate only along the x axis.
In addition, it turns out that the imaginary part of the normal
wave-vector components satisfy
k′′z1 (ω )k′′z2 (ω ) < 0,
(3.22)
i.e., they have opposite signs. Equations (3.18)–(3.22) then allow altogether twelve different combinations of field propagation for the
M1 modes, of which those six corresponding to fields attenuating
in the positive x direction are illustrated in Fig. 3.3. We observe in
particular that all scenarios involve bound waves at one interface,
but leaky waves at the other.
3.2.2 Mode class M2
The M2 class, defined via Eq. (3.14), represents the general case for
which the wave vectors in the regions on the two sides of the slab
are identical and there are two transversally counter-propagating
waves within the slab (for M1 only one wave exists inside the slab).
The M2 mode solutions are thereby a generalization of the propagating plane waves that on interference at a dielectric plane-parallel
plate produce no reflected field [177].
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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M1I
M1II
M1III
M1I
M1II
M1III
Figure 3.3: Illustration of the possible directions of phase movement (black arrows) and
field attenuation (solid-red curves) for M1 modes decaying to the right. The left, middle,
and right columns represent M1I [Eq. (3.19)], M1II [Eq. (3.20)], and M1III [Eq. (3.21)],
respectively. The graphs in the bottom row are mirror images of those in the top row.
The electromagnetic boundary conditions dictate that the M2
modes come in two species with different field profiles, denoted by
the subscripts + and −, whose wave-vector components read as
√
[ m ± π ]2
k x (ω ) = k0 (ω ) ϵr1 (ω ) −
,
k0 (ω )d
(π)
,
k z1 (ω ) = m±
d√
[ m ± π ]2
k z2 (ω ) = k0 (ω ) ϵr2 (ω ) − ϵr1 (ω ) +
,
k0 (ω )d
(3.23)
(3.24)
(3.25)
where m+ (m− ) is an even (odd) integer. Qualities that make the M2
modes very different from those of the M1 class are that Eqs. (3.23)–
(3.25) stand for an infinite number of modes and depend on the
slab thickness, which Eqs. (3.16) and (3.17) instead do not. Other
prominent properties of the M2 modes are that k z1 (ω ) is purely
real [even though ϵr1 (ω ) is complex] and fully independent of the
material parameters, while k x (ω ) does not depend on ϵr2 (ω ), also
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Dissertations in Forestry and Natural Sciences No 252
Surface-plasmon polaritons
Figure 3.4: Illustration of the possible directions of phase movement (black arrows) and
field attenuation (solid-red curves) for M2 modes decaying to the right. The graph to the
right is a mirror image of that to the left.
at variance with the M1 solutions.
There are, however, similarities between the two classes. One is
that also the M2 class contains a bound wave and a leaky wave in
the surrounding, since the same complex-valued k z2 (ω ) is encountered in both regions outside the slab. In addition,
k′x (ω )k′′x (ω ) > 0,
k′z2 (ω )k′′z2 (ω ) < 0,
(3.26)
whereby the propagation of the M2 modes along the x axis and
for |z| ≥ d/2 is similar to that of the M1 modes [Eq. (3.18)]. Yet,
the situation is different for |z| < d/2 because, as Eq. (3.24) shows,
k z1 (ω ) is purely real and therefore the fields are neither decaying
nor growing in the z direction. Moreover, for the M2 class there
are two transversally counter-propagating waves within the slab
and, consequently, we cannot identify the direction of the total-field
phase movement along the z axis inside the slab.
Eventually, we end up with four different field-propagation possibilities for the M2 modes; those two representing waves decaying
in the positive x direction are illustrated in Fig. 3.4.
3.2.3 Mode class M3
The last class, M3, as defined through Eq. (3.15), is the conventionally studied scenario, involving electric fields whose normal components (or magnetic fields) are either symmetric or antisymmetric
Dissertations in Forestry and Natural Sciences No 252
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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with respect to z = 0, and which obey
Symmetric :
Antisymmetric :
[1
]
ϵr1 (ω ) k z2 (ω )
= tanh ik z1 (ω )d ,
ϵr2 (ω ) k z1 (ω )
2
[1
]
ϵr2 (ω ) k z1 (ω )
= coth ik z1 (ω )d .
ϵr1 (ω ) k z2 (ω )
2
(3.27)
(3.28)
Equations (3.27) and (3.28), both standing for an infinite number of
modes for any chosen media, frequency, and film thickness [198],
are transcendental equations for k x (ω ) and generally require numerical methods to solve. Yet, as outlined in Publication II, all of
these mode solutions can be divided (by analytical means) into two
sets depending on their behavior in the limit d → ∞.
One of the sets corresponds to the wave-vector components
√
ϵr1 (ω )ϵr2 (ω )
k x (ω ) → k0 (ω )
,
(3.29)
ϵr1 (ω ) + ϵr2 (ω )
k zα (ω ) → k0 (ω ) √
ϵrα (ω )
ϵr1 (ω ) + ϵr2 (ω )
,
α ∈ {1, 2},
(3.30)
which are exactly those obtained for SPPs at a single interface. The
modes associated with the solutions of Eqs. (3.27) and (3.28) that
approach the limits of Eqs. (3.29) and (3.30) as d → ∞ are therefore regarded (at any d) as fundamental modes (FMs) [198]. To put it
the other way around, the FMs are associated with those fields that
arise from the coupling between the SPPs supported by the individual interfaces of the slab. As concluded in Publication II, there
are only two FMs, one symmetric and one antisymmetric, for any
chosen media, frequency, and slab thickness.
The other set represents modes for which in the limit d → ∞
√
k x (ω ) → k0 (ω ) ϵr1 (ω ),
(3.31)
k z1 (ω ) → 0,
k z2 (ω ) → k0 (ω )
√
(3.32)
ϵr2 (ω ) − ϵr1 (ω ),
(3.33)
having no correspondence at a single boundary. We thereby refer to
these fields (at any d) as higher-order modes (HOMs). Unlike with the
32
Dissertations in Forestry and Natural Sciences No 252
Surface-plasmon polaritons
FMs, there are an infinite number of symmetric and antisymmetric
HOMs for any chosen media, frequency, and slab thickness [198].
Yet, since the value k z1 (ω ) = 0 represents the scenario for which
the electromagnetic field vanishes both inside and outside the film
[198], all the HOMs disappear when d → ∞.
Below we examine the general field-propagation characteristics
of the FMs and HOMs separately, but before that we want to discuss an important aspect concerning all the M3 modes. First of all,
the situation outside the slab is now different from that of M1 and
M2, because for the class M3 the waves on each side are both ei(1)
(2)
ther bound or leaky [owing to k z2 (ω ) = −k z2 (ω )]. However, there
is another fundamental difference between the classes. Unlike frequently asserted [203], we emphasize that for the M3 modes both
signs in k z2 (ω ) = ±[k22 (ω ) − k2x (ω )]1/2 , where k2 (ω ) is the wave
number in the surrounding, are not allowed for a fixed k x (ω ), since
Eqs. (3.27) and (3.28) are not invariant with respect to the change
of sign of k z2 (ω ). This implies that for a given bound (leaky) solution Eqs. (3.27) and (3.28) do not admit the corresponding leaky
(bound) mode. In other words, for any particular tangential wavevector component k x (ω ), Maxwell’s equations permit for the class
M3 only a bound or a leaky mode, but not both. This property is
at variance with the M1 and M2 classes, for which both signs are
simultaneously allowed for the transverse wave-vector component
outside the film.
Fundamental modes
The numerical analysis of Publication II suggests that only bound
FMs with k′′z2 (ω ) > 0 are allowed by Maxwell’s equations, whereas
leaky FMs possessing k′′z2 (ω ) < 0 do not exist. Furthermore, our
studies indicate that k′x (ω ) and k′′x (ω ) always have the same sign.
Consequently, according to Eq. (2.13), we then have
k′x (ω )k′′x (ω ) > 0,
k′z2 (ω ) < 0,
(3.34)
implying that the directions of wavefront propagation and amplitude attenuation of FMs are the same along the x axis (similarly to
Dissertations in Forestry and Natural Sciences No 252
33
Andreas Norrman: Electromagnetic Coherence of Optical Surface and
Quantum Light Fields
Figure 3.5: Illustration of the possible directions of phase movement (black arrows) and
field attenuation (solid-red curves) for FMs decaying to the right. The left, middle, and
right graphs correspond to the first, second, and third condition of Eq. (3.35), respectively.
the M1 and M2 modes) and the phases move towards the surfaces
outside the slab. Regarding k z1 (ω ), it turns out that all of the cases
k′z1 (ω )k′′z1 (ω ) > 0,
k′z1 (ω ) = 0,
k′z1 (ω )k′′z1 (ω ) < 0,
(3.35)
are possible [the situation with k′′z1 (ω ) = 0 does not occur], whereupon the behavior of k z1 (ω ) for |z| < d/2 of the FMs to some extent
resembles that of the M1 modes [Eqs. (3.19)–(3.21)]. Nevertheless,
the M3 class encompasses two transversally counter-propagating
waves within the slab, making the phase motion along the z axis
ambiguous (cf. M2 class).
The results above establish six separate combinations of field
propagation for the FMs, of which those three representing waves
decaying in the positive x direction are illustrated in Fig. 3.5 (note
the absence of leaky FMs).
Higher-order modes
As demonstrated in Publication II, the HOMs occur in two distinct
species. One of these, which we call HOMI s, are fields that manifest
themselves exclusively as bound waves outside the slab, regardless
of the film thickness. In other words,
HOMI : k′′z2 (ω ) > 0,
34
∀d.
(3.36)
Dissertations in Forestry and Natural Sciences No 252
Surface-plasmon polaritons
Figure 3.6: Illustration of the directions of phase movement (black arrows) and field attenuation (solid-red curves) for HOMI s decaying to the right.
The wave propagation of the HOMI s along the x axis and for |z| ≥
d/2 is similar to that of the FMs [Eq. (3.34)], i.e.,
k′x (ω )k′′x (ω ) > 0,
k′z2 (ω ) < 0,
(3.37)
whereas for k z1 (ω ) inside the film one only has [cf. Eq. (3.35)]
k′z1 (ω )k′′z1 (ω ) < 0.
(3.38)
Figure 3.6 illustrates the total wave-propagation behavior of HOMI s
attenuating in the positive x direction.
The other species, HOMII , is defined via

′′


k z2 (ω ) > 0, if d < dc ,
HOMII :
(3.39)
k′′z2 (ω ) = 0, if d = dc ,


k′′ (ω ) < 0, if d > d ,
c
z2
where dc is a critical thickness that depends on the particular mode,
media, as well as frequency. Thus the HOMII s can be either bound
(d < dc ), or leaky (d > dc ), or strictly propagating in opposite directions along the z axis outside the slab (d = dc ); the only mode type
in the M3 class with this kind of property.
The analysis in Publication II shows that the HOMII s obey
k′x (ω )k′′x (ω ) < 0,
k′z2 (ω ) > 0,
if d < dc ,
(3.40)
k′′x (ω )
k′x (ω )k′′x (ω )
k′z2 (ω )
k′z2 (ω )
> 0,
if d = dc ,
(3.41)
> 0,
if d > dc .
(3.42)
= 0,
> 0,
Dissertations in Forestry and Natural Sciences No 252
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
Quantum Light Fields
d < dc
d = dc
d > dc
Figure 3.7: Illustration of the possible directions of phase motion (black arrows) and field
attenuation (solid-red curves) for HOMII s decaying to the right, when the slab thickness
d is smaller than (left), equal to (middle), and larger than (right) the critical thickness dc .
Equations (3.40)–(3.42) especially indicate that the HOMII s possess
k′z2 (ω ) > 0, stating that the wavefronts move away from the slab for
|z| ≥ d/2, in contrast to the FMs [Eq. (3.34)] and HOMI s [Eq. (3.37)].
Regarding k x (ω ), the first case, Eq. (3.40), representing (bound)
waves for which the phase motion and amplitude attenuation are
opposite along the surfaces, and the second one, Eq. (3.41), standing for fields that are purely evanescent in the x direction (in both
media), are situations not met earlier. The last scenario, Eq. (3.42),
corresponds to (leaky) waves for which the behavior of k x (ω ) is
analogous to that of the FMs [Eq. (3.34)] and HOMI s [Eq. (3.37)].
When it comes to k z1 (ω ) of the HOMII s, our investigations in Publication II indicate that
k′z1 (ω )k′′z1 (ω ) > 0,
∀d,
(3.43)
contrary to the situation with the HOMI s [Eq. (3.38)].
In Fig. 3.7 we have summarized the three possible cases of field
propagation for HOMII s attenuating along the positive x axis. Figure 3.7 illustrates three particular qualities that make the HOMII s
very different from the other metal-slab modes. Firstly, as d < dc
(left graph), the phases advance in the negative x direction even
though the waves attenuate in the positive direction. Secondly,
when d = dc (middle graph), the fields become purely evanescent
with no phase movement at all along the x axis (in both media) and
strictly propagating along the z axis in the surrounding. Thirdly,
36
Dissertations in Forestry and Natural Sciences No 252
Surface-plasmon polaritons
for d > dc (right graph), the HOMII s outside the slab turn leaky,
with the wavefronts tilted away from the interfaces.
3.2.4 Forward- and backward-propagating modes
As a final point in Publication II, we investigate the flow of energy
of the various metal-slab modes. According to Eq. (2.20), regardless
of the mode type, the Poynting vector in the lossless surrounding
is always parallel to the real part of the wave vector and decays at
twice the rate of the field in the direction specified by the imaginary part. Hence only forward propagation (FP) occurs for the
M1–M3 modes outside the slab and their energy-flow behavior (in
that region) is illustrated by the black arrows and solid-red curves
in Figs. 3.3–3.7. The situation is significantly more involved inside
the absorptive film, where also backward propagation (BP) is possible and where the energy-flow behavior depends strongly on the
particular mode type.
In Table 3.1 we have summarized the results on FP and BP for
the M1–M3 modes within the slab. It is seen that along the x axis
both FP and BP are possible for all modes species in the M1 and
M3 classes, while for those in M2 exclusively FP occurs. Further we
observe that M1I and M1III are, respectively, the only mode types
for which FP and BP are found (and defined) in the z direction. The
M1
FP
BP
M2
I
II
III
x axis
Yes
Yes
Yes
z axis
Yes
x axis
Yes
z axis
No
M3
Symmetric
FM
Antisymmetric
HOMI HOMII
FM
HOMI HOMII
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
No
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes
Yes
Yes
Table 3.1: Summary of the possibility of FP and BP for the M1–M3 modes inside the
slab. The yellow boxes correspond to situations where the FP–BP behavior arises from the
change of direction in the energy flow, while the blue boxes represent cases in which the
behavior is caused by the change in the phase direction. The grey areas stand for scenarios
for which FP or BP is not defined.
Dissertations in Forestry and Natural Sciences No 252
37
Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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HOMII s, on the other hand, are the only mode species for which the
FP–BP behavior is caused by the change of direction in the phase
movement; for all other cases the feature stems from the change in
the energy-flow direction.
3.3
LONG-RANGE MODES
Although the single-interface SPP exhibits many useful properties,
it is also characterized by a relatively high propagation loss, which
limits its feasibility especially for waveguide purposes. The metalslab geometry, on the other hand, encompasses the salient quality
that it can support a surface mode having a much longer propagation range [191, 196, 198, 203]. This particular long-range mode, also
known as the long-range surface-plasmon polariton (LRSPP) [24, 25, 30,
32, 80, 81, 161], is a family member of the symmetric M3 modes. It
corresponds to the solution of Eq. (3.27) for which
√
(3.44)
k x (ω ) → k0 (ω ) ϵr2 (ω ),
√
(3.45)
k z1 (ω ) → k0 (ω ) ϵr1 (ω ) − ϵr2 (ω ),
k z2 (ω ) → 0,
(3.46)
in the limit d → ∞, representing a field that is vertically polarized outside the slab and of infinite extent along the x axis. If d
is small enough, the LRSPP may attain a propagation length even
several hundreds or thousands of times greater than that of the respective single-interface SPP, which makes it highly attractive for
SPP-based waveguide applications and integrated optical components [81, 204].
Unfortunately, albeit the propagation range extension of the LRSPP mitigates the limitation of the single-interface SPP, usually this
comes at the expense of reduced field confinement as the field gets
forced out of the metal and spreads progressively into the surrounding with decreasing film thickness [81]. The extended propagation length may, on the other hand, outweigh the reduced surface confinement. Depending on the situation, e.g., the operating
38
Dissertations in Forestry and Natural Sciences No 252
Surface-plasmon polaritons
frequency, some materials are more advantageous than others for
optimizing the trade-off between long propagation distance and
strong surface localization, but no single material seems to offer
superior performance for all applications [205]. The LRSPP is also
very sensitive to surface properties, and when d → 0, effects such
as surface roughness and grain size in the film start playing increased roles [82]. In addition, as the thickness decreases below a
certain threshold value, the slab becomes islandized due to the formation of voids, and the material parameters start to differ from
their bulk values [81]. Recently, however, methods which allow the
fabrication of uniform, highly smooth (surface roughness around
0.2 nm), ultra-thin (d ≈ 5 nm), low-loss films have been demonstrated [206, 207].
3.3.1 Mode interchanges
To date, the LRSPP has exclusively been associated with the (symmetric) FM, defined by Eqs. (3.29) and (3.30), which originate from
the coupling between the SPPs supported by the individual boundaries of the slab [24, 25, 30, 80–82, 161, 191, 196, 198, 203, 204]. This
might be the reason why traditionally only the FM of the M3 class
has acquired attention and physical importance, whereas the HOMs
specified via Eqs. (3.31)–(3.33), having no analogue in the singleinterface geometry and typically possessing propagation distances
of only a few nanometers, are rarely encountered in the literature
[198]. Indeed, owing to their extremely short propagation lengths,
the HOMs do not appear (at first glance) to have any other practical significance than in matching the boundary conditions upon
SPP end launching [198].
These widely held viewpoints are challenged in Publication III,
in which we demonstrate the transformation of a HOM, normally
not regarded to be useful, into a strongly confined long-range mode
in circumstances where the fundamental LRSPP does not exist and
the propagation length of the single-interface SPP is negligible. In
this process, a peculiar mode transition takes place between a HOM
Dissertations in Forestry and Natural Sciences No 252
39
Andreas Norrman: Electromagnetic Coherence of Optical Surface and
Quantum Light Fields
1.0
0.8
0.09
0.6
0.08
0.4
0.07
80
120
l x ( ω ) / λ0 ( ω )
l x ( ω ) / λ0 ( ω )
1.0
160
0.2
0.0
0
40
120
80
d [nm]
160
0.6
0.08
0.4
0.07
80
120
160
80
120
d [nm]
160
0.2
40
200
1.0
0.8
0.09
0.6
0.08
0.4
0.07
80
120
l x ( ω ) / λ0 ( ω )
l x ( ω ) / λ0 ( ω )
0.09
0.0
0
200
1.0
160
0.2
0.0
0
0.8
40
120
80
d [nm]
160
200
0.8
0.09
0.6
0.08
0.4
0.07
80
120
160
120
80
d [nm]
160
0.2
0.0
0
40
200
Figure 3.8: Propagation length l x (ω ) of the FM (solid-blue curves) and the lowest-order
HOM (dotted-red curves) at a Ag slab surrounded by SiO2 as a function of the film
thickness d for selected free-space wavelengths λ0 (ω ): 355 nm (top left), 353 nm (top
right), 352 nm (bottom left), and 350 nm (bottom right). The insets give a close-up view of
the FM–HOM interchange. The horizontal dashed-black lines corresponding to the singleinterface SPP are included for reference. The relative permittivities for Ag and SiO2 are
obtained from the empirical data of [195].
(to be more specific, a bound and nonradiative HOMI ) and the FM,
whereby the HOM assumes the capability of long-range guidance
while the fundamental LRSPP vanishes.
As an example how the FM–HOM interchange is manifested in
the propagation length lx (ω ) = 1/k′′x (ω ) of the fields, we consider a
case involving a Ag slab surrounded by SiO2 . Figure 3.8 illustrates
the behavior of lx (ω ) for the FM and the lowest-order HOM as a
function of d in the near-ultraviolet regime. The figure shows that,
whereas for the free-space wavelength λ0 (ω ) = 355 nm (top left)
we still have the customary scenario in which the FM (solid-blue
curve) evolves into the LRSPP as d gets small, for λ0 (ω ) = 350 nm
(bottom right) the HOM (dotted-red curve) has become to stand
40
Dissertations in Forestry and Natural Sciences No 252
Surface-plasmon polaritons
for the LRSPP. This extraordinary transition happens around d ≈
110 nm between λ0 (ω ) = 353 nm (top right) and λ0 (ω ) = 352 nm
(bottom left), which is demonstrated in detail by the insets. More
precisely, the propagation length for λ0 (ω ) = 353 nm below d ≈
110 nm corresponding to the FM gets interchanged with that of the
HOM when λ0 (ω ) = 352 nm, and vice versa.
Similar flips are also possible between the FM and the secondorder, third-order, etc., HOM, implying that several HOMs may
have a larger propagation length than that of the FM. The switches
affect not only the propagation length, but also other physical properties of the modes, such as the surface wavelength, dispersion, the
penetration depths into the slab and the surrounding, and the polarization state, as demonstrated in Publication III.
3.3.2 Long-range higher-order modes
The appearance of such a FM–HOM crossover in which the lowestorder HOM evolves into a long-range mode is the main result of
Publication III. At the transition frequency, the propagation length
of the HOM may experience even a remarkable thousandfold enlargement. Phenomena of this type, where the HOMs, contrary to
common belief, acquire long-range wave guidance and thus practical significance, seem not to have been observed or even suggested
before. It is important to understand that, unlike with the fundamental LRSPP, the long-range HOM does not emerge from the coupling between the SPPs supported by the individual boundaries of
the slab. The origin of the long-range HOM is completely different: for a thick film there are no HOMs (they vanish as the slab
thickness gets large and hence have no single-boundary correspondence), but when the thickness is reduced the HOMs start to show
up and, as the thickness is small enough, one of the HOMs turns
into a long-range mode. Thus the long-range HOM does not follow from the coupling of the two single-interface SPPs on each side
of the film (their propagation lengths are generally negligible in
those situations where the long-range HOM exists), but is exclu-
Dissertations in Forestry and Natural Sciences No 252
41
Andreas Norrman: Electromagnetic Coherence of Optical Surface and
Quantum Light Fields
sively a consequence of the slab geometry and seems to come out
from ‘nothing’.
Furthermore, while the extended propagation range of the FM
is generally associated with the loss of field confinement [81], for
the HOM the situation can be different. As shown in Publication
III, the long-range HOM may have a stronger surface confinement
than the respective single-interface SPP even when the propagation
length of the former is over 100 times larger than that of the latter.
In fact, the analysis in Publication III reveals that the propagation
range extension can, remarkably, even enhance the field localization
in some cases, thus suggesting that simultaneous optimization of
strong field confinement and long-range propagation is possible for
the HOMs.
The long-range HOMs may occur for several different materials
and frequency ranges. Examples of some of these situations are
presented in Fig. 3.9 for free-space wavelengths extending from extreme ultraviolet to near infrared. Although it is not entirely clear
why these FM–HOM interchanges occur, or under what circumstances the HOM adopts long-range behavior, we have found that
a long-range HOM is supported when
′′
′
(ω ) |ϵr1
(ω )| ϵr2 (ω ).
2ϵr1
(3.47)
′ ( ω )| ϵ ( ω ); a
In particular, the condition above requires that |ϵr1
r2
regime which is commonly regarded as ‘forbidden’ in plasmonics.
Ag/GaP
Ag/SiO2
Al/vacuum
Na/vacuum
Al/SiO2
100
200
Na/SiO2
300
Cu/GaP
Ag/ZnO
Na/ZnO
400
500
λ0 (ω ) [nm]
Au/GaP
Na/GaP
600
700
800
Figure 3.9: Examples of some materials and bandwidths for which the lowest-order HOM
amounts to the LRSPP. The relative permittivities for SiO2 , Ag, Au, Cu, and Na are
from [195], and those for GaP, ZnO, and Al are from [202], [208], and [209], respectively.
42
Dissertations in Forestry and Natural Sciences No 252
Surface-plasmon polaritons
Further one observes from Eq. (3.47) that a surrounding possessing
′ ( ω ) and
a relatively small ϵr2 (ω ) demands low values for both ϵr1
′′ ( ω ), while a larger ϵ ( ω ) allows more variation for the metal.
ϵr1
r2
The ongoing and open-ended search for better and alternative plasmonic materials [210–212] provides an excellent opportunity to extend the borders for these highly localized long-range HOMs.
Dissertations in Forestry and Natural Sciences No 252
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
Quantum Light Fields
44
Dissertations in Forestry and Natural Sciences No 252
4 Partially coherent surfaceplasmon polaritons
So far, we have dealt with fully monochromatic and therefore also
completely deterministic ESWs. In reality, however, every electromagnetic field found in nature exhibits at least some degree of random fluctuations, caused by scattering processes within the involved
medium or, ultimately, by the indeterministic quantum-physical
atomic transitions in the field sources. These intrinsic, random fluctuations are manifested as partial coherence and partial polarization of
the electromagnetic field.
In this chapter, we examine partially coherent and partially polarized SPPs. Physically, the coherence and polarization states influence the interaction of SPPs with the surrounding and collections
of nanoparticles located in close proximity of the interface. For example, SPPs propagating on the surfaces of a metallic nanolayer can
form a highly sensitive interferometric biosensor [213]. Likewise, an
SPP field with varying spatial coherence may excite a random set
of molecules to radiate coherently [214]. Nanoparticle scattering is
known to depend on the polarization properties of the field, to the
extent that the field’s polarization state can be fully deduced from
measurements of the scattered radiation [215]. Studying the coherence and polarization characteristics of SPPs is hence important,
not only from a foundational, but also from an application point of
view, as coherence and polarization offer indispensable degrees of
freedom for manipulating diverse light–matter interactions.
4.1 SINGLE-INTERFACE GEOMETRY
To date, albeit SPPs have attracted a significant amount of interest
in fundamental and applied sciences, most plasmonics research has
Dissertations in Forestry and Natural Sciences No 252
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
Quantum Light Fields
involved monochromatic and, consequently, fully coherent SPPs.
Coherence-tailored polychromatic SPPs could nonetheless serve as
versatile tools, e.g., for ultrashort optical pulse manipulation in
nanostructured optoelectronic circuits [31], controlled coupling of
light-emitting elements [214,216], plasmon continuum spectroscopy
[217], and subwavelength white-light imaging [218]. At the same
time, exploring the statistical features and excitation mechanisms
of polychromatic SPP fields presents a fundamental interest.
In publication IV, we develop a theory for partially coherent
polychromatic SPPs at a metal–air interface. The formalism covers
stationary as well as nonstationary SPP fields of arbitrary spectra.
As a main result, we formulate a framework to tailor the electromagnetic coherence of such polychromatic SPPs in the Kretschmann
setup [24, 25] by controlling the correlations of the excitation light.
The connection between the coherence state of the light source and
the ensuing SPP field establishes a novel paradigm in statistical
plasmonics, which we refer to as plasmon coherence engineering.
4.1.1
Polychromatic surface-plasmon polaritons
Let us consider polychromatic SPPs generated in the Kretschmann
configuration, sketched in Fig. 4.1, with an absorptive and nonmagnetic metal film deposited on a glass prism. The film is taken to be
thick enough so that any coupling between the metal-slab modes
Air
SPP
Metal
Glass
Figure 4.1: Polychromatic SPP excitation in the Kretschmann coupling modality.
46
Dissertations in Forestry and Natural Sciences No 252
Partially coherent
surface-plasmon polaritons
can be neglected, whereupon the region near the metal–air surface
can be treated as the semi-infinite half-space geometry in Fig. 2.1.
The (p-polarized) polychromatic electric field in air then reads as
E(r, t) =
∫ ω+
ω−
E(ω )p̂(ω )ei[k(ω )·r−ωt] dω,
(4.1)
where ω−,+ specify the frequency range, E(ω ) is the spectral complex field amplitude of a monochromatic SPP at the origin (r = 0),
and p̂(ω ) is the respective polarization vector given by Eq. (2.3).
Note that we have dropped the subscript 2 (referring to medium 2)
in order to keep the notation simpler.
Now, let Eq. (4.1) represent a realization of the random electric
field. All coherence information of the polychormatic SPP field is
then encoded in the electric coherence matrix [Eq. (A.2)]
Γ ( r1 , t1 ; r2 , t2 ) =
∫ ω+∫ ω+
ω− ω−
W(r1 , ω1 ; r2 , ω2 )e−i(ω2 t2 −ω1 t1 ) dω1 dω2 , (4.2)
in which we have the spectral electric coherence matrix [Eq. (A.4)]
W(r1 , ω1 ; r2 , ω2 ) = W (ω1 , ω2 )K(ω1 , ω2 )ei[k(ω2 )·r2 −k
∗ (ω
1 )· r1 ]
,
(4.3)
including the spectral electric correlation function
W (ω1 , ω2 ) = ⟨ E∗ (ω1 ) E(ω2 )⟩,
(4.4)
and the matrix
K(ω1 , ω2 ) = [|k(ω1 )||k(ω2 )|]−1
[
]
k∗z (ω1 )k z (ω2 ) −k∗z (ω1 )k x (ω2 )
×
.
−k∗x (ω1 )k z (ω2 ) k∗x (ω1 )k x (ω2 )
(4.5)
Equation (4.2) is general as it sets no restrictions on metal dispersion, the spectrum of the field, or the spectral correlations; it covers
any partially coherent polychromatic SPP field. The spectral correlation function W (ω1 , ω2 ) of Eq. (4.4) is in this context essential,
since it determines fully the space–frequency (and hence also the
space–time) coherence characteristic of the SPPs.
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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4.1.2
Narrowband and broadband fields
As shown in Publication IV, in the case of a narrowband SPP field
for which metal dispersion can be neglected, all elements of the
electric coherence matrix in Eq. (4.2) have identical space–time dependence, i.e, the correlations among the field components propagate and attenuate in exactly the same way. Such polychromatic
narrowband SPP fields are virtually propagation invariant and also
strictly polarized, which could facilitate nearly distortion-free information transfer in plasmonic networks.
For broadband spectra, then again, dispersion in the metal can
no longer be ignored, and thus each element of Γ(r1 , t1 ; r2 , t2 ) has
to be treated separately (and numerically), whereupon the correlations between the electric-field components will in general have different space–time evolutions. Yet, also the broadband SPP fields are
highly polarized, at least for metals and optical frequency ranges
for which SPP propagation is appreciable (such as Ag and Au in the
mid and lower frequency domains of the visible spectrum), since
the polarization vectors of the SPP constituents are rather similar.
For a stationary field having an ultra-wide spectrum, e.g., thermal
radiation, the coherence length scale is only on the order of the
mean wavelength, but allowing nonzero correlations in W (ω1 , ω2 )
renders the polychromatic SPP field nonstationary and more coherent. At high levels of spectral correlations the SPPs become pulses.
4.1.3
Plasmon coherence engineering
The idea of plasmon coherence engineering is to prudently tailor
W (ω1 , ω2 ) of Eq. (4.4) into the wanted form by controlling the coherence state of the light source. In Publication IV, as a main contribution, we establish exactly how the spatio–spectral statistical properties of the stationary or pulsed excitation beam are to be tuned to
create a polychromatic SPP field with the desired coherence characteristics.
To this end, we take the illumination light incident on the prism
in the geometry of Fig. 4.1 to be a p-polarized, partially coherent,
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Dissertations in Forestry and Natural Sciences No 252
Partially coherent
surface-plasmon polaritons
z
x
Z
∆θ
θ0
ω
ω0
X
Figure 4.2: Plasmon coherence engineering with polychromatic beam illumination. The
angle θ0 between the XZ and xz frames corresponds to perfect phase matching among the
central angular spectrum mode of frequency ω0 and the respective SPP. At every frequency
within the excitation bandwidth, the angular spectrum wave of frequency ω ̸= ω0 incident
at an angle ∆θ with respect to the Z axis generates the corresponding SPP.
polychromatic beam expressed by the angular spectrum representation [2]. The electric-field amplitude of the angular spectrum mode
of frequency ω is denoted by E (k X , ω ), with k X being the tangential wave-vector component in a coordinate frame XZ, where the Z
axis makes an angle θ0 with respect to the z axis of the xz frame
(see Fig. 4.2). The second-order statistical properties of the incident
field are then specified by the spectral electric correlation function
W (k X1 , ω1 ; k X2 , ω2 ) = ⟨E ∗ (k X1 , ω1 )E (k X2 , ω2 )⟩.
(4.6)
We further choose θ0 such that in the xz frame the tangential wavevector component of the beam mode of central frequency ω0 and
k X = 0 within the angular spectrum exactly corresponds to k′x (ω0 )
of the SPP obtained from Eq. (2.26), viz.,
n ( ω0 )
ω0
sin θ0 = k′x (ω0 ).
c
(4.7)
This condition, where n(ω0 ) is the refractive index of the prism,
represents precise phase matching between the central illuminating
plane wave and the central SPP mode at the metal–air surface.
To ensure that an SPP mode is generated at every ω within the
spectral excitation bandwidth, one must impose a similar matching
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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condition for the other illumination plane waves as well. Suppose
that for an arbitrary frequency ω ̸= ω0 the angular spectrum mode
with k X = n(ω )(ω/c) sin ∆θ, where ∆θ is the angle among the wave
vector and the Z axis, couples to the respective SPP (see Fig. 4.2).
As shown in Publication IV, for beamlike illumination this implies
kX =
k′x (ω ) − k′x (ω0 )
.
cos θ0
(4.8)
At each frequency ω within the bandwidth the angular spectrum
wave satisfying Eq. (4.8) thus excites the corresponding monochromatic SPP mode. Now, since E(ω ) ∝ E (k X , ω ), with the exact coupling efficiency specified by the transmission coefficient of the slab,
we get between the SPP correlation function [Eq. (4.4)] and the correlation function of the incident light [Eq. (4.6)] the relation
W ( ω1 , ω2 ) ∝ W
[ k′ (ω ) − k′ (ω )
]
k′ (ω2 ) − k′x (ω0 )
0
1
x
x
, ω1 ; x
, ω2 . (4.9)
cos θ0
cos θ0
Equation (4.9) should be identified as an explicit (inverse) relation,
which enables one to determine exactly (e.g., numerically by iteration when the metal dispersion is known) how the spectral correlation function of the illumination source has to be tuned in order to
achieve any desired coherence properties for the ensuing polychromatic SPP field. This result is the main contribution of Publication
IV and the crux of plasmon coherence engineering.
4.2
METAL-SLAB GEOMETRY
Contrary to the single-interface geometry, the metal-slab configuration illustrated in Fig. 3.2 is capable to sustain a multitude of modes
at a given frequency. As the FMs of the M3 class are not only the most
well-known, but also the most prominent modes in a broad range
of applications, henceforth we focus on these two modes. Under
usual conditions, the symmetric FM evolves into the LRSPP when
the slab thickness decreases, while the antisymmetric FM acquires
a much smaller propagation length than that of the LRSPP and the
50
Dissertations in Forestry and Natural Sciences No 252
Partially coherent
surface-plasmon polaritons
respective single-interface SPP. Therefore, the antisymmetric FM is
commonly termed the short-range surface-plasmon polariton (SRSPP).
Owing to its unique capability of long-range guidance, the LRSPP
has over the years received considerably more attention and practical significance than the SRSPP. Nevertheless, compared to the
LRSPP, the SRSPP offers advantages as regards strong field confinement [32], and several phenomena have been reported in which
the SRSPP plays a crucial role, including plasmonic focusing [219],
stimulated-amplification supported SPP propagation [220], extraordinarily low transmission through nanopatterned films [221], and
plasmon-waveguide sensing [222]. Methods which allow efficient
excitation of the SRSPP, either simultaneously with [223] or without [224] the LRSPP, have also been presented.
Individually, the LRSPP and SRSPP are completely coherent and
polarized, but their superposition allows for the possibility of partial
coherence and partial polarization. To gain insight into the spatial
coherence and polarization properties of such a LRSPP–SRSPP superposition, it is natural to employ the degrees of coherence and
polarization, since these are the basic measures to characterize any
partially coherent and partially polarized light field. It is of particular theoretical interest to investigate how much the two degrees
may vary, as the upper and lower ranges specify to which extent
the coherence and polarization of the two-mode field can be modified (and utilized) in practice. The limits are primarily determined
by the mutual correlation between the modes, which in a practical
arrangement depends on the excitation process.
The main objective of Publication V is to examine the fundamental ranges that the spectral degrees of coherence and polarization of
such a stationary LRSPP–SRSPP field above the metallic nanoslab in
Fig. 3.2 can attain, regardless of the excitation method. In addition,
we explore how the two degrees vary within their extremal values
when the media, frequency, and film thickness are changed. Again,
as we are considering the electric field only outside the film, to keep
the notation simpler, henceforward the subscript 2 in the field amplitudes, wave vectors, etc., referring to medium 2 (surrounding) is
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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suppressed. When the constituents are mutually fully correlated,
the total field is completely coherent and polarized, thus establishing directly the upper limits for the degrees. To assess the lower
ranges, we take the modes to be mutually uncorrelated, since any
correlation among them is expected to yield a more coherent and
polarized field. For the same reason, the modes are considered to
have equal intensities at r1 = x1 ê x + (d/2)êz , which we define as
the excitation point.
4.2.1
Degree of coherence
Due to the vectorial nature of SPPs, for which the traditional scalar
degree of coherence is inadequate, we employ the generalized vector degree of coherence in Eq. (A.14) to investigate the spatial coherence of the LRSPP–SRSPP field. Under the above assumptions,
1
µ(∆r, ω ) = √
2
√
1 + κ (ω )
cos [∆k′ (ω ) · ∆r]
,
cosh [∆k′′ (ω ) · ∆r]
(4.10)
in which ∆r = r2 − r1 is the separation between the observation and
excitation points, and
�
�2
κ (ω ) = �p̂(+)∗ (ω ) · p̂(−) (ω )� ,
∆k′ (ω ) = k(+)′ (ω ) − k(−)′ (ω ),
∆k′′ (ω ) = k(+)′′ (ω ) − k(−)′′ (ω ),
(4.11)
(4.12)
(4.13)
where the superscripts (+) and (−) refer to the LRSPP and SRSPP,
respectively. The polarization term κ (ω ) in Eq. (4.11) is bounded as
1/2 ≤ κ (ω ) ≤ 1,
(4.14)
with the lower (upper) limit corresponding to d → 0 (d → ∞). We
further emphasize that generally ∆k′ (ω ) · ∆k′′ (ω ) ̸= 0.
The cosine in Eq. (4.10) indicates that, as a rule, the degree of coherence oscillates, whereupon the LRSPP-SRSPP field may show a
high (or a low) degree of coherence at certain locations even though
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Partially coherent
surface-plasmon polaritons
the two modes are mutually uncorrelated and hence do not interfere. The oscillation of µ(∆r, ω ) originates from the fact that at specific periodic distances the superposition is electromagnetically similar [225] to the total field at the excitation point (we neglect the
decay of the modes for the moment). This effect is akin to the customary oscillatory behavior of the (scalar) degree of coherence for
two uncorrelated modes in a gas laser [226].
In addition to the oscillation, the hyperbolic cosine in Eq. (4.10)
implies that µ(∆r, ω ) generally also decays, which arises from the
attenuation of the two modes owing to metal absorption. As long
as ∆r is not perpendicular to ∆k′′ (ω ), we find from Eq. (4.10) that
√
µ(∆r, ω ) → 1/ 2 when |∆r| → ∞. This value, representing partial
coherence and being independent of the material parameters, the
frequency of the field, or the thickness of the slab, is a consequence
of the different decay rates of the two modes: for a sufficiently
large |∆r|, the mode with the lower decay rate (LRSPP) dominates
the superposition and the mode with a higher decay rate (SRSPP)
can be neglected. Hence, far away from the excitation region, the
field can practically be considered as a single LRSPP which attains
an essentially constant degree of coherence.
Concerning the (global) maximum and minimum of the degree
of coherence, we get from Eq. (4.10) that
µmax (ω ) =
√
[1 + κ (ω )]/2,
µmin (ω ) =
√
[1 − κ (ω )]/2,
(4.15)
which set the fundamental limits for the domain in which µ(∆r, ω )
of the LRSPP–SRSPP field is restricted. Equation (4.15) shows that
µ2max (ω ) + µ2min (ω ) = 1, indicating that an increase of one is accompanied with a decrease of the other. From Eqs. (4.11) and (4.15) one
further finds that the extrema are bounded as
√
3/2 ≤ µmax (ω ) ≤ 1,
0 ≤ µmin (ω ) ≤ 1/2,
(4.16)
where the lower and upper limits of µmax (ω ) [µmin (ω )] correspond
to d → 0 (d → ∞) and d → ∞ (d → 0), respectively. Equation (4.16)
especially demonstrates that, irrespective of the media, frequency,
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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and slab thickness, there are always regions for which the field displays a quite high or a rather low degree of coherence.
4.2.2
Local and global coherence length
We notice from Eqs. (4.10) and (4.15) that the maximum of µ(∆r, ω )
is found at the point r1 = r2 where the two modes are generated.
Furthermore, the minimum in Eq. (4.15) is not just the global, but
also the nearest minimum with respect to the excitation point [there
exists a sequence of local minima (and maxima) due to the oscillatory behavior of µ(∆r, ω )]. Therefore, besides the actual values, it
is natural to investigate the distance between µmax (ω ) and µmin (ω )
to get a rough estimation for the domain at the excitation region
in which the field is highly electromagnetically coherent (meaning
that all field components are strongly correlated). We refer to this
particular distance as the local coherence length. As demonstrated in
Publication V, depending on the materials and frequency, the local
coherence length can be of subwavelength order for ultra-thin films,
while for larger slab thicknesses it can extend over several tens of
wavelengths.
√
Because of the property µ(∆r, ω ) → 1/ 2 when |∆r| → ∞, it
is reasonable to introduce an additional, global coherence length, as
a distance between r1 and r2 over which µ(∆r, ω ) drops from its
√
maximum value at r1 = r2 to a particular number close to 1/ 2.
The ‘particular number’ is not unambiguous, but is chosen appropriately for each situation. The global coherence length is thereby a
rough measure for the range beyond which the degree of coherence
does not essentially change anymore, i.e., it marks the distance up
to which µ(∆r, ω ) oscillates. Physically, within the global coherence
length there are regions in which the SPP superposition at the two
points may be highly correlated and regions where it may be rather
uncorrelated. At these locations, the SPP field would interact with
nanoparticles in the vicinity of the surface in a coherent, or incoherent, manner. The investigation in Publication V reveals that, much
like the local coherence length, the global coherence length can ex-
54
Dissertations in Forestry and Natural Sciences No 252
Partially coherent
surface-plasmon polaritons
tend from subwavelength scales (thin films) to even thousands of
wavelengths (thick films).
4.2.3 Degree of polarization
Regarding the polarization degree of the LRSPP–SRSPP field, since
the constituents have only two electric-field components, the conventional (2D) treatment is formally sufficient for analyzing the polarization characteristics of the superposition. Yet, as in this case the
degree of polarization is defined in the plane parallel to the wave
vectors (xz plane, where the modes are typically elliptically polarized), the situation differs substantially from that of beamlike wave
fields for which the electric field is orthogonal to the wave vector.
As shown in Publication V, when the two modes are uncorrelated
and have equal intensities at the excitation point, the 2D degree of
polarization [Eq. (A.15)] for the SPP field above the slab becomes
P2D (r, ω ) =
√
1−
1 − κ (ω )
cosh2 [∆k′′ (ω ) · r]
,
(4.17)
where r is the position vector measured from the excitation point.
Unlike the degree of coherence in Eq. (4.10), the degree of polarization in Eq. (4.17) does not display an oscillatory term, but is
characterized only by a hyperbolic cosine. Excluding the particular direction along which P2D (r, ω ) is constant, i.e., the one that is
perpendicularly to ∆k′′ (ω ), we find from Eq. (4.17) that
P2D (r, ω ) →
{
1,
√
κ ( ω ),
|r| → ∞,
|r| → 0,
(4.18)
which are verified to be, respectively, the maximum and minimum
of P2D (r, ω ). Consequently, since 1/2 ≤ κ (ω ) ≤ 1 [Eq. (4.14)], we
conclude that the degree of polarization for the LRSPP–SRSPP superposition is always bounded as
√
1/ 2 ≤ P2D (r, ω ) ≤ 1.
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(4.19)
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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The physical origin behind the maximum, representing a fully polarized field, is quite apparent: when |r| gets large enough, the SRSPP vanishes and the field can practically be considered as a single
LRSPP for which P2D (r, ω ) = 1 regardless of the media, frequency,
film thickness, and position. By the same token, as the contribution
of the SRSPP to the degree of polarization cannot be ignored when
|r| → 0, the minimum, standing for partial polarization, depends
on the material parameters, frequency, as well as the slab thickness,
according to Eqs. (3.27), (3.28), and (4.11).
Finally, our studies in Publication V indicate that generally the
degree of polarization of the LRSPP–SRSPP field is close to unity.
Nonetheless, the analysis also suggests that for ultra-thin films,
within subwavelength distances from the excitation point, the SPP
superposition can be partially polarized and P2D (r, ω ) may fluctuate. Increasing the operating frequency reduces the polarization
degree, whereas varying the permittivity of the surrounding has a
negligible effect on it.
56
Dissertations in Forestry and Natural Sciences No 252
5 Electromagnetic coherence
of evanescent light fields
Optical evanescent waves are a special type of (pure) ESWs, formed
when a light field undergoes total internal reflection at a dielectric
boundary [18]. When interacting with matter, evanescent waves enable a phenomenon analogous to quantum mechanical tunneling
through a potential barrier [2, 18]. They also allow to study biological samples with a resolution well beyond the classical diffraction
limit [227–229] and play an important role in SPP excitation [24,25].
Evanescent waves have therefore a pivotal position in nanophotonics and for the understanding of several optical phenomena that are
confined to subwavelength dimensions.
This chapter concerns the generation, partial polarization, and
spatial coherence of statistically stationary, purely evanescent light
fields at a planar interface between two dielectric media. Contrary
to SPPs, evanescent waves generally carry three orthogonal electricfield components, whereat a rigorous 3D treatment is required to
fully describe their polarization state. Genuine 3D-polarized evanescent fields with tailored coherence properties could be utilized in
near-field probing, single-molecule detection, particle trapping, and
other polarization-sensitive surface-photonic applications.
5.1 EVANESCENT WAVE IN TOTAL INTERNAL REFLECTION
Let us consider a beam, represented as a monochromatic electromagnetic plane wave, incident onto a planar interface (z = 0) between two dielectric media (see Fig. 5.1). Both medium 1 (z < 0)
and medium 2 (z > 0), having refractive indices n1 (ω ) and n2 (ω ),
respectively, are taken lossless. The incoming wave, generally carrying both an s-polarized and a p-polarized part, hits the boundary
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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z
y
n2 ( ω )
n1 ( ω )
x
θ (ω )
ϕ(ω )
Figure 5.1: Total internal reflection at a planar interface (z = 0) between two lossless
dielectric media having refractive indices n1 (ω ) (z < 0) and n2 (ω ) (z > 0). The incident
beam impinges the surface with an azimuthal angle φ(ω ) at the angle of incidence θ (ω ).
with an azimuthal angle 0 ≤ φ(ω ) < 2π at the angle of incidence
0 < θ (ω ) < π/2. When θ (ω ) > θc (ω ), with θc (ω ) = arcsin ñ−1 (ω )
being the critical angle and ñ(ω ) = n1 (ω )/n2 (ω ) > 1, total internal reflection takes place and the transmitted field in medium 2
becomes an evanescent wave [18].
According to Eqs. (2.1)–(2.3), Maxwell’s equations, the boundary conditions, as well as Eqs. (2.30 and (2.31), the spatial part of
the electric field for the evanescent wave takes on the form
E(r, ω ) = [ts (ω ) Es (ω )ŝ(ω ) + t p (ω ) E p (ω )p̂(ω )]eik(ω )·r ,
(5.1)
where Es (ω ) and E p (ω ) are, respectively, the complex field amplitudes of the s- and p-polarized components of the incident light. In
Cartesian coordinates, the wave and polarization vectors read as
k(ω ) = k1 (ω ){sin θ (ω )[cos φ(ω )ê x + sin φ(ω )êy ] + iγ(ω )êz }, (5.2)
ŝ(ω ) = − sin φ(ω )ê x + cos φ(ω )êy ,
(5.3)
p̂(ω ) =
(5.4)
−iγ(ω )[cos φ(ω )ê x + sin φ(ω )êy ] + sin θ (ω )êz
√
,
sin2 θ (ω ) + γ2 (ω )
in which k1 (ω ) is the wave number in medium 1 and
√
γ(ω ) = ñ−1 (ω ) [ñ(ω ) sin θ (ω )]2 − 1.
58
(5.5)
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Electromagnetic coherence of
evanescent light fields
Eventually, the Fresnel transmission coefficients ts (ω ) and t p (ω ) for
the two polarizations are given by
2 cos θ (ω )
,
cos θ (ω ) + iγ(ω )
√
2ñ(ω ) cos θ (ω ) 2ñ2 (ω )γ2 (ω ) + 1
.
t p (ω ) =
cos θ (ω ) + i ñ2 (ω )γ(ω )
ts (ω ) =
(5.6)
(5.7)
We emphasize that Eqs. (5.1)–(5.7) are expressed solely in terms of
the refractive indices and the parameters associated with the incoming light. We note also that t p (ω ) in Eq. (5.7) differs from the
conventional expression [2,18] owing to our different normalization
of the wave vector (see Sec. 2.4).
The quantity γ(ω ) defined in Eq. (5.5) can be interpreted as the
decay constant of the evanescent wave. We see that the larger the
angle of incidence, the faster the wave decays with increasing distance away from the surface [Eq. (2.15)]. Another essential quantity
that characterizes the evanescent wave is its wavelength [Eq. (2.16)],
Λ(ω ) =
λ0 ( ω )
,
n1 (ω ) sin θ (ω )
(5.8)
which is readily verified to be bounded as
λ1 ( ω ) < Λ ( ω ) < λ2 ( ω ),
(5.9)
where the lower and upper limits, λ1 (ω ) and λ2 (ω ), corresponding
to θ (ω ) = π/2 and θ (ω ) = θc (ω ), are the wavelengths in medium
1 and 2, respectively. Equation (5.9) especially indicates that Λ(ω )
is always shorter than the wavelength of a propagating plane wave
above the surface, anticipating that, in random evanescent fields,
also the coherence length can be shorter than λ2 (ω ).
5.2 RANDOM EVANESCENT FIELDS
Let us now consider the case in which several, stationary incident
beams undergo total internal reflection in the modality of Fig. 5.1.
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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The beams are allowed to have different polarization states (and degrees), angles of incidence, and azimuthal angles. We let E(n) (r, ω ),
with n ∈ {1, 2, . . . , N }, be the spatial part of a monochromatic realization of the evanescent wave generated by the nth incoming beam.
The total evanescent field is then expressed by the sum
N
∑ E(n) (r, ω ),
Etot (r, ω ) =
(5.10)
n =1
in which each constituent is constructed as in Eqs. (5.1)–(5.7).
On taking an ensemble average over the realizations, the crossspectral density matrix [Eq. (A.11)] for the evanescent field becomes
N
∑
W ( r1 , r2 , ω ) =
W (mn) (ω )ei[k
(n) ( ω )· r
2 −k
(m)∗ ( ω )· r
1]
.
(5.11)
m,n=1
Here we have defined the matrix
(m)∗
Wmn (ω ) = ts
(n)
(mn)
(n)
(mn)
(n)
(mn)
(n)
(mn)
(ω )ts (ω )ϕss
(ω )ŝ(m)∗ (ω )ŝ(n)T (ω )
(m)∗
(ω )t p (ω )ϕ pp (ω )p̂(m)∗ (ω )p̂(n)T (ω )
(m)∗
(ω )t p (ω )ϕsp (ω )ŝ(m)∗ (ω )p̂(n)T (ω )
(m)∗
(ω )ts (ω )ϕ ps (ω )p̂(m)∗ (ω )ŝ(n)T (ω ),
+ tp
+ ts
+ tp
(5.12)
where the superscript T denotes matrix transpose and the factors
(mn)
(m)∗
ϕµν (ω ) = ⟨ Eµ
(n)
(ω ) Eν (ω )⟩,
(5.13)
with µ, ν ∈ {s, p} and m, n ∈ {1, 2, . . . , N }, are related to the correlations among the incident light beams. For m = n, the quantities
(mn)
ϕµν (ω ) are the elements of the 2 × 2 polarization matrix associated with the nth incident wave, while the cases m ̸= n describe the
mutual correlations between different waves.
The corresponding spectral polarization matrix [Eq. (A.12)] is
obtained by setting r1 = r2 = r in Eq. (5.11), viz.,
Φ(r, ω ) =
N
∑
W (mn) (ω )ei[k
(n) ( ω )− k(m)∗ ( ω )]· r
,
(5.14)
m,n=1
which fully describes the polarization state of the evanescent field.
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Electromagnetic coherence of
evanescent light fields
5.2.1 Subwavelength coherence lengths
The conventional wisdom in optics says that δ-correlated sources
or blackbody radiators generate the spatially most incoherent wave
fields, for which the coherence length, in a lossless medium, is
roughly half a wavelength of the light [230, 231]. Recently, these arguments have been reassessed and it was shown that, in principle,
a finite-sized source can produce a field whose coherence length
within the source may be arbitrarily short, even in the absence of
absorption [232]. In addition, it has been demonstrated that a thermal half-space source can generate an electromagnetic near field
whose longitudinal correlation length, due to absorption, may be
much shorter than the light’s wavelength [84]. In Publication VI,
we demonstrate that such subwavelength coherence lengths are also
encountered in purely evanescent fields at lossless interfaces. As a
rule, the shortest coherence lengths are observed very close to the
surface for high refractive-index contrasts.
As an example, we consider the superposition of two s-polarized
evanescent waves in the immediate vicinity of the surface (z = 0),
created by mutually uncorrelated beams sharing the same angle of
incidence θ (ω ), but having opposite azimuthal angles φ(1) (ω ) = 0
and φ(2) (ω ) = π. The amplitudes of the beams are adjusted so that
the individual evanescent waves have equal intensities at z = 0. It
follows from Eqs. (5.11) and (A.14) that for such a (fully polarized)
evanescent field the degree of coherence becomes
1
µ(∆x, ω ) = √
2
√
1 + cos[2k1 (ω ) sin θ (ω )∆x ],
(5.15)
where ∆x = x2 − x1 is the distance between the observation points
along the x axis. Owing to statistical similarity, the degree of coherence in Eq. (5.15) oscillates sinusoidally between 1 and 0, whereat
we define the coherence length for the two-wave superposition as
the distance from a maximum to a nearby minimum, viz.,
λ2 ( ω )
.
4ñ(ω ) sin θ (ω )
(5.16)
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61
lcoh (ω ) =
Andreas Norrman: Electromagnetic Coherence of Optical Surface and
Quantum Light Fields
We recognize at once that lcoh (ω ) < λ2 (ω )/4. For a high refractiveindex-contrast surface, such as GaP and air with ñ(ω ) ≈ 4 within
the optical regime [195], the coherence length of the superposition
may be as low as lcoh (ω ) ≈ λ2 (ω )/16.
5.2.2
Genuine 3D-polarized states
For 2D fields, such as beams, the 3D degree of polarization defined
in Eq. (A.16) is invariably bounded as 1/2 ≤ P3D (r, ω ) ≤ 1 [125].
The lowest value, P3D (r, ω ) = 1/2, is encountered for light that is
completely unpolarized from the traditional 2D point of view. Values within the range P3D (r, ω ) < 1/2 are thereby clear signatures of
genuine 3D fields which cannot be described with the conventional
formalism for the the degree of polarization [54].
It has been shown that light created by an optical system out of
a single, arbitrary polarized beam obeys P3D (r, ω ) ≥ 1/2 [233]. This
result covers the situation of an evanescent wave being generated
by total internal reflection at a planar interface. Nevertheless, as
demonstrated in Publication VI, a superposition of beams may produce an evanescent field with P3D (r, ω ) < 1/2. In fact, for a typical
SiO2 –air interface, already two partially polarized beams sharing
the same plane of incidence are sufficient for the excitation of a true
3D evanescent field having P3D (r, ω ) ≈ 1/4. This simple case highlights that a rigorous and full 3D treatment is generally required to
describe the polarization state of evanescent fields.
5.3
3D-UNPOLARIZED EVANESCENT FIELDS
As the investigations in Publication VI indicate the prospect to generate genuine 3D-polarized evanescent fields, one is tempted to ask:
how to generate, if even possible, a fully 3D-unpolarized evanescent
field? While a completely 3D-unpolarized state is encountered for
blackbody radiation [234, 235], situations in which totally unpolarized 3D evanescent fields can be generated by controlled means
and furnished with varying coherence properties have not previ-
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Electromagnetic coherence of
evanescent light fields
ously been reported. In Publication VII, we investigate the generation and electromagnetic coherence of unpolarized 3D evanescent
light fields in multibeam illumination at a planar dielectric interface. Our analysis reveals the feasibility to tailor evanescent fields
with polarization qualities identical to those of universal blackbody
radiation, yet with tunable spatial coherence characteristics.
5.3.1 Generation
Let us first explore how such unconventional, fully unpolarized,
genuine 3D evanescent fields could be generated above a dielectric surface under controllable circumstances. Because an unpolarized 3D light field is unambiguously represented by a polarization
matrix that is proportional to the 3 × 3 identity matrix [125], we
must search for conditions that diagonalize Φ(r, ω ) in Eq. (5.14).
To this end, we employ a specific optical multibeam configuration
in which the incoming waves are independent and have uncorrelated s- and p-polarized components. Moreover, the incident beams
are taken to share the same angle of incidence θ (ω ), with intensi(n)
(n)
ties such that ⟨| Es (ω )|2 ⟩ = Is (ω ) and ⟨| E p (ω )|2 ⟩ = I p (ω ) for
all n ∈ {1, 2, . . . , N }. In this case, the polarization matrix of the
evanescent field depends only on the height above the surface, i.e.,
Φ(r, ω ) = Φ(z, ω ), whereas the 3D degree of polarization is totally
position independent, viz., P3D (r, ω ) = P3D (ω ).
To derive the conditions under which P3D (ω ) = 0, we require
the diagonal elements of Φ(z, ω ) to be equal and the off-diagonal
elements to be zero. Unfortunately, it turns out that P3D (ω ) = 0
cannot be achieved for the simplest case N = 2. Yet, as shown in
Publication VII, if one utilizes two beams propagating in orthogonal
azimuthal directions with Is (ω )/I p (ω ) = 2ñ2 (ω ), then P3D (ω ) → 0
when θ (ω ) → θc (ω ). So, with two beams, even though it is not
possible to generate a strictly 3D-unpolarized evanescent field, one
can create a nearly unpolarized 3D evanescent field by adjusting
θ (ω ) very close to the critical angle. This result has been confirmed
to hold in the space–time domain too [236]. Things get different if
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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we consider the scenario where N ≥ 3 and the incident beams are
uniformly distributed, i.e., φ(n) (ω ) = (n − 1)2π/N. For this case,
as outlined in Publication VII, the off-diagonal elements vanish and
Φ(z, ω ) becomes a diagonal matrix when
Is (ω )
[ñ2 (ω ) − 1][ñ2 (ω ) sin2 θ (ω ) + 1]
=
.
I p (ω )
cos2 θ (ω ) + ñ2 (ω )[ñ2 (ω ) sin2 θ (ω ) − 1]
(5.17)
Equation (5.17), which is seen to be independent of N, is thus the
condition that results in P3D (ω ) = 0. In other words, for any chosen
θ (ω ) > θc (ω ) and ñ(ω ) > 1, Eq. (5.17) determines precisely how
the intensities of the incident beams must be calibrated so that the
evanescent field is strictly unpolarized in the 3D sense.
5.3.2
Degree of coherence
Besides conditions under which genuine 3D-unpolarized evanescent fields could be generated by manageable means, we also analyze the spatial coherence of the fields in Publication VII. For the optical setup discussed above, the degree of coherence in Eq. (A.14) is
independent of z1 and z2 and depends only on ∆ρ = ∆x ê x + ∆yêy ,
with ∆x = x2 − x1 and ∆y = y2 − y1 , so that µ(r1 , r2 , ω ) = µ(∆ρ, ω ).
In Fig. 5.2 we have plotted the spatial behavior of µ(∆ρ, ω ) for a
3D-unpolarized evanescent field above a SiO2 –air interface excited
by different numbers of uniformly distributed incident beams with
θ (ω ) = π/3. The penetration depth of the field is roughly λ0 (ω )/5.
Figure 5.2 reveals peculiar subwavelength patterns in the degree of
coherence, whose shapes depend on N. In particular, the structures
show periodic rotational symmetries owing to the multibeam excitation setups, but not necessarily translational symmetries. This
example illustrates the possibility to excite an evanescent field that
shares the polarization properties of blackbody radiation, yet with
radically different coherence characteristics, such as tunable subwavelength lattice-like structures in the degree of coherence.
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Electromagnetic coherence of
evanescent light fields
µ ( ∆ ρ, ω )
0.43
0.58
0.20
2
2
1
1
∆ y / λ0 ( ω )
∆ y / λ0 ( ω )
0.28
µ ( ∆ ρ, ω )
0
−1
0.39
0
−1
−2
−2
−1
0
1
∆ x / λ0 ( ω )
2
−2
−2
−1
µ ( ∆ ρ, ω )
0.20
0.39
0.58
0.18
2
2
1
1
0
−1
−2
−2
0
1
∆ x / λ0 ( ω )
2
µ ( ∆ ρ, ω )
∆ y / λ0 ( ω )
∆ y / λ0 ( ω )
0.58
0.38
0.58
0
−1
−1
0
1
∆ x / λ0 ( ω )
2
−2
−2
−1
0
1
∆ x / λ0 ( ω )
2
Figure 5.2: Spatial behavior of the degree of coherence µ(∆ρ, ω ) for a 3D-unpolarized
evanescent field above a SiO2 –air surface generated by N = 3 (top left), N = 4 (top
right), N = 5 (bottom left), and N = 6 (bottom right) uniformly distributed and uncorrelated incident beams with the angle of incidence θ (ω ) = π/3. The refractive indices are
n1 (ω ) = 1.5 and n2 (ω ) = 1, and λ0 (ω ) is the free-space wavelength.
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66
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6 Complementarity in photon
interference
The principle of complementarity is a cornerstone in physics, declaring
that quantum systems inhold mutually exclusive properties [144].
The arguable most prominent manifestation of complementarity is
the wave–particle duality, which restricts the coexistence of wave and
particle qualities of quantum objects [147–150]. In two-way interferometry, such as the double-slit experiment or a Mach–Zehnder
setup, the duality can be expressed through [152–154]
P 2 + V 2 ≤ 1,
D 2 + V 2 ≤ 1,
(6.1)
where P is the path predictability, quantifying the a priori ‘whichpath information’ (WPI), D is the path distinguishability, representing the available WPI stored in the system, and V is the intensity
visibility. For photons, however, interference does not necessarily
appear merely as intensity fringes, but also, or solely, as polarization
modulation [157–159], a feature which the two relations in Eq. (6.1)
do not account for. How complementarity is manifested under such
polarization variation is therefore of fundamental interest.
In this chapter, by exploring polarization modulation in doublepinhole photon interference, we derive two general complementarity relations which cover genuine vectorial quantum-light fields of
arbitrary state. The complementarity relations are shown to reflect
two different, intrinsic aspects of wave–particle duality of the photon, having no correspondence in scalar quantum interferometry. In
particular, it is demonstrated that, contrary to scalar light, for pure
single-photon vector light the a priori WPI does not couple to the
intensity visibility, but to a generalized visibility, which also characterizes the variation of the polarization state. We also show that
for general quantum light such complementarity is a manifestation
of complete coherence, not of quantum-state purity.
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6.1
COHERENCE OF VECTORIAL QUANTUM LIGHT
In quantum theory of optical coherence [136], all information on the
first-order statistical properties of a multicomponent and generally
nonstationary quantized light field, at two space–time points x1 and
x2 , is encoded in the electric coherence matrix
G( x1 , x2 ) = tr[ρ̂Ê(−) ( x1 )Ê(+) ( x2 )].
(6.2)
Here Ê(+) ( x ) and Ê(−) ( x ) are the positive and negative frequency
parts of the total electric-field operator, ρ̂ is the density operator
characterizing the quantum state, and tr denotes the trace. Mathematically, depending on whether the light is treated as a 2D or as a
3D field, the electric coherence matrix in Eq. (6.2) is a 2 × 2 or 3 × 3
matrix which satisfies the symmetry relation G† ( x2 , x1 ) = G( x1 , x2 )
[136], with the dagger representing conjugate transpose. Physically,
the elements of G( x1 , x2 ) describe the correlations between the orthogonal field components at x1 and x2 .
A complete specification of the coherence characteristics of light
requires knowledge of all correlation orders [54]. Higher-order correlations are of major importance in quantum optics, as they can
provide information about the nonclassical properties of light [137].
For instance, second-order correlation measurements are able to
distinguish between light states with identical spectral distributions but having different photon number distributions. The first
successful demonstration of photon antibunching, offering a clear
proof of the quantum nature of light, was made by Kimble, Dagenais, and Mandel in 1977 by measuring such photon–photon correlations [237]. Yet, although higher-order correlations play a crucial
role in quantum optics, we focus exclusively on first-order correlations in this chapter, as they determine the intensity (and polarization) variation in double-pinhole interference [54].
Motivated by the classical measure in Eq. (A.6), we may define
the degree of coherence for a general vector-light quantum field as
g ( x1 , x2 ) = √
68
∥G( x1 , x2 )∥F
,
trG( x1 , x1 )trG( x2 , x2 )
(6.3)
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Complementarity in photon interference
where ∥ · ∥F is the Frobenius matrix norm. The quantity g( x1 , x2 ) includes all elements of G( x1 , x2 ) and is a measure of the correlations
that exist between all the orthogonal components of the quantized
electric field at two space–time points. As its classical counterpart,
also g( x1 , x2 ) is real, invariant under unitary transformations, and
bounded as 0 ≤ g( x1 , x2 ) ≤ 1. Most importantly, following the
steps in [113,114], one can show that g( x1 , x2 ) = 1 throughout some
domain Ω if, and only if, all the field components are fully correlated for all x1 , x2 ∈ Ω. In this case G( x1 , x2 ) factors in x1 and x2 and
the field is considered (first-order) fully coherent in Ω, consistently
with Glauber’s definition of complete coherence [136]. Likewise, for
g( x1 , x2 ) = 0 no correlations occur between any of the components,
and the photon field is incoherent. The range 0 < g( x1 , x2 ) < 1
corresponds to partial coherence.
For a 2D quantum-light field, say polarized in the xy plane, the
electric coherence matrix in Eq. (6.2) can be expressed as [158]
G ( x1 , x2 ) =
1
2
3
∑ S j ( x1 , x2 ) σ j ,
(6.4)
j =0
where σ 0 is the identity matrix, while σ 1 , σ 2 , and σ 3 are the three
Pauli spin matrices [238]. The four complex-valued quantities
S j ( x1 , x2 ) = tr[G( x1 , x2 )σ j ],
j ∈ {0, . . . , 3},
(6.5)
offering an alternative, yet an equivalent way to represent the firstorder coherence properties of the field, are quantum analogs of the
classical two-point Stokes parameters [239, 240]. For x1 = x2 = x,
they become S j ( x ) = tr[G( x, x )σ j ], which give the expectation values of the quantum Stokes operators [137], with the following interpretation: S0 ( x ) is proportional to the (average) total photon number, while S1 ( x ), S2 ( x ), and S3 ( x ) give, respectively, the (average)
differences between x- and y-polarized, +π/4- and −π/4-linearly
polarized, and right- and left-circularly polarized photons. Making
use of Eqs. (6.3) and (6.4) one readily finds that
g2 ( x1 , x2 ) =
1
2
3
∑ |s j ( x1 , x2 )|2 ,
(6.6)
j =0
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involving the normalized two-point Stokes parameters
s j ( x1 , x2 ) = √
S j ( x1 , x2 )
S0 ( x 1 ) S0 ( x 2 )
,
j ∈ {0, . . . , 3},
(6.7)
which satisfy 0 ≤ |s j ( x1 , x2 )| ≤ 1 for all j ∈ {0, . . . , 3}.
We emphasize that a representation similar to Eq. (6.4) may be
written for 3D quantum fields as well, with the expansion basis
being the Gell-Mann matrices [238], as used to introduce the Stokes
parameters for 3D classical fields [125].
6.2
PHOTON INTERFERENCE LAW
Let us now consider the double-pinhole interference experiment
with the vector nature of the photon field taken into account. The
two openings are located at r1 and r2 in an opaque screen A in
the xy plane and the emerging light (of angular frequency ω) is
observed on a screen B by a photodetector at position r and time t,
as illustrated in Fig. 6.1. The openings are assumed to be so large
that boundary effects can be neglected, but so small that in each
the field can be treated as uniform. Under these circumstances, by
r
r1
B
r2
A
Figure 6.1: Double-pinhole photon interference. Light impinges on a screen A with two
openings located at r1 and r2 . The emerging photons are investigated at a point r on the
observation screen B , where intensity fringes and polarization modulation may appear.
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Complementarity in photon interference
employing for both orthogonal field components the steps outlined
for quantized scalar light in [241], the expression for the positive
frequency part of the electric-field operator at B can be written as
Ê(+) (r, t) = K (ω )e−iωt
2
∑ (âmx êx + âmy êy )
m =1
eik(ω )rm
,
rm
(6.8)
where K (ω ) is a constant, ê x and êy are the Cartesian unit vectors,
k(ω ) is the wave number, and rm is the distance from rm to r. Moreover, the annihilation operators âmµ , associated with the µ ∈ { x, y}
polarized radial modes emanating from pinhole m ∈ {1, 2}, obey
the commutation relations
[ âmµ , ânν ] = [ â†mµ , â†nν ] = 0,
[ âmµ , â†nν ] = δmn δµν ,
(6.9)
in which δ is the Kronecker delta.
It is now straightforward to show that in the paraxial regime the
Stokes parameters in the observation plane B take on the forms
S j (r) = S′j (r) + S′′j (r) + 2[S0′ (r)S0′′ (r)]1/2
× |s j (r1 , r2 )| cos [θ j (r1 , r2 ) − k(r1 − r2 )],
j ∈ {0, . . . , 3}, (6.10)
where S′j (r) and S′′j (r) are the Stokes parameters on B when, respectively, only the pinhole at r1 or r2 is open. Furthermore, s j (r1 , r2 )
is the normalized, equal-time, two-point Stokes parameter at the
pinholes on A, given by Eq. (6.7), and θ j (r1 , r2 ) is its phase. Equation (6.10), which is formally similar to the classical electromagnetic interference law [157–159], fully describes any effect that firstorder coherence may introduce onto the intensity and the polarization state of the photons in the double-pinhole configuration. We
thereby refer to Eq. (6.10) as the photon interference law. Governing
any quantum light characterized by a density operator ρ̂, it especially signifies that double-pinhole photon interference, resulting
from spatial coherence at the two openings, does not necessarily
manifest itself solely as intensity fringes, but also, or exclusively, as
polarization-state variations on the observation screen.
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6.3
VISIBILITY AND DISTINGUISHABILITY
As the photon interference law states that the Stokes parameters on
screen B are sinusoidally modulated by the correlations that prevail
among the electric-field components at the pinholes, we may define
four separate modulation contrasts (or visibilities) via
Vj (r) =
max[S j (r)] − min[S j (r)]
,
max[S0 (r)] + min[S0 (r)]
j ∈ {0, . . . , 3},
(6.11)
where max (min) stands for the maximum (minimum). The quantity V0 (r) is the customary intensity visibility, whereas V1 (r), V2 (r),
and V3 (r) are polarization visibilities [157–159]. From Eq. (6.10) and
(6.11) one then finds that
Vj (r) = C (r)|s j (r1 , r2 )|,
j ∈ {0, . . . , 3},
(6.12)
in which we have introduced the factor
C (r) =
2[S0′ (r)S0′′ (r)]1/2
2[S0 (r1 )S0 (r2 )]1/2
,
≈
′
′′
S0 ( r ) + S0 ( r )
S0 ( r 1 ) + S0 ( r 2 )
(6.13)
with the approximation valid near the central axis where r1 ≈ r2 .
Since 0 ≤ C (r) ≤ 1, also 0 ≤ Vj (r) ≤ 1 for all j ∈ {0, . . . , 3}.
In view of Eq. (6.12), it seems natural to try to find a single quantity that characterizes the intensity and the polarization visibilities
of the photon field at the same time. Interestingly, on defining
√
1
V02 (r) + V12 (r) + V22 (r) + V32 (r),
(6.14)
V (r) = √
2
we observe from Eqs. (6.6), (6.12), and (6.14) that
V ( r ) = C ( r ) g ( r1 , r2 ).
(6.15)
Equation (6.15) is recognized to be of the same form as the standard
visibility relation met in the scalar context [241], but now V (r) and
g(r1 , r2 ) replace the intensity visibility and the traditional degree
of coherence, respectively. It is also readily verified from Eq. (6.15)
that 0 ≤ V (r) ≤ 1, with the lower limit taking place when C (r) = 0
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Complementarity in photon interference
or g(r1 , r2 ) = 0, whereas the upper limit is reached if, and only if,
C (r) = 1 and g(r1 , r2 ) = 1. We are therefore tempted to interpret
V (r) in Eq. (6.14), which includes both the intensity visibility and
the polarization-modulation contrasts, as the total visibility for the
vectorial photon field. In particular, when the average number of
photons passing through each pinhole is the same [S0 (r1 ) = S0 (r2 )]
the factor C (r) in Eq. (6.13) is unity, in which case Eq. (6.15) implies
that V (r) = g(r1 , r2 ), i.e., the total visibility on B is directly given
by the vectorial degree of coherence.
If g(r1 , r2 ) = 0, viz., the photons are completely incoherent (no
correlations exist between any of the components at the pinholes),
then V (r) = 0, stating that neither intensity nor polarization modulations are observed on screen B . And vice versa, V (r) = 0 reflects
the fact that g(r1 , r2 ) = 0. Nonetheless, when the light exhibits partial coherence at the openings, in other words g(r1 , r2 ) > 0, then
V (r) > 0 and at least one S j (r) is modulated. And conversely, any
variation in at least one of the Stokes parameters on B is a signature
of partially coherent photons at the pinholes. However, even in the
case of a fully coherent field, for which g(r1 , r2 ) = 1, Eqs. (6.12) and
(6.15) imply that one cannot have Vj (r) = 1 for all j ∈ {0, . . . , 3} at
the same time. Instead, at most two of the Stokes parameters may
exhibit maximum visibility simultaneously, in which case the other
two are zero.
Next, to distinguish the light in the pinholes, we first introduce
the intensity distinguishability through
D0 (r1 , r2 ) =
|S0 (r1 ) − S0 (r2 )|
,
S0 ( r 1 ) + S0 ( r 2 )
(6.16)
which is bounded within the interval 0 ≤ D0 (r1 , r2 ) ≤ 1. The upper
limit, D0 (r1 , r2 ) = 1, corresponds to full intensity distinguishability
and occurs if all photons pass through one pinhole only, in other
words, if S0 (r1 ) = 0 or S0 (r2 ) = 0. The lower limit, D0 (r1 , r2 ) = 0,
stands for complete intensity indistinguishability and takes place
when there is (on average) an equal number of photons in the openings, viz., S0 (r1 ) = S0 (r2 ). The range 0 < D0 (r1 , r2 ) < 1 represents
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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partial intensity distinguishability. To further differentiate the light
in the pinholes, we define also the polarization distinguishability via
Dp (r1 , r2 ) =
|S(r1 ) − S(r2 )|
,
S0 ( r 1 ) + S0 ( r 2 )
(6.17)
where S(rm ) = [S1 (rm ), S2 (rm ), S3 (rm )] is the Poincaré vector [55] in
pinhole m ∈ {1, 2}. Clearly 0 ≤ Dp (r1 , r2 ) ≤ 1, with maximal polarization distinguishability, Dp (r1 , r2 ) = 1, being possible only for orthogonally fully polarized light, whereas total absence of polarization distinguishability, Dp (r1 , r2 ) = 0, occurs when S(r1 ) = S(r2 ).
The intermediate values are signatures of partial polarization distinguishability.
6.4
WEAK AND STRONG COMPLEMENTARITY
The stage is now set for quantifying complementarity for genuine
vector-light quantum fields of arbitrary state in double-pinhole interference. As shown in Publication VIII,
D02 (r1 , r2 ) + V 2 (r) ≤ 1,
Dp2 (r1 , r2 ) + V02 (r)
≤ 1,
(6.18)
(6.19)
establishing fundamental upper limits for D0 (r1 , r2 ) and V (r), as
well as for Dp (r1 , r2 ) and V0 (r), of a vectorial photon field. Note that
in the case of scalar light (corresponding to a uniformly polarized
light field), for which V (r) → V0 (r) and Dp (r1 , r2 ) → D0 (r1 , r2 ),
the two relations above merge into D02 (r1 , r2 ) + V02 (r) ≤ 1. Most
importantly, the analysis in Publication VIII reveals that
D02 (r1 , r2 ) + V 2 (r) = 1,
if g(r1 , r2 ) = 1,
(6.20)
Dp2 (r1 , r2 ) + V02 (r)
if g(r1 , r2 ) = 1,
(6.21)
= 1,
stating that when the light at the pinholes is completely coherent in
the full vector sense, intensity distinguishability and total visibility,
as well as polarization distinguishability and intensity visibility, are
mutually exclusive quantities.
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Complementarity in photon interference
Equations (6.18)–(6.19) and (6.20)–(6.21) constitute the main results of Publication VIII. The former, making no restrictions on the
quantum state involved, are associated with weak complementarity,
because the parameters can in principle vary independently of each
other as long as the sums do not exceed unity. For the latter, on the
other hand, a variation of one parameter always changes the other
so that the sums strictly equal unity, whereupon these relations are
regarded as representing strong complementarity.
It is important to understand that strong complementarity is not
a manifestation of quantum-state purity, but of complete coherence.
To see this, we consider the pure four-photon state |ψ⟩ = |1, 1, 1, 1⟩,
where the notation |n1x , n1y , n2x , n2y ⟩ = |n1x ⟩1x |n1y ⟩1y |n2x ⟩2x |n2y ⟩2y
has been adopted, nmµ being the number of photons in the | · ⟩mµ
mode. In this case Eqs. (6.7) and (6.12) result in Vj (r) = 0 for all
j ∈ {0, . . . , 3}, whereupon neither intensity nor polarization modulations are observed. Equations (6.16) and (6.17), on the other hand,
yield D0 (r1 , r2 ) = 0 and Dp (r1 , r2 ) = 0, so there is no intensity or
polarization distinguishability either. Hence D02 (r1 , r2 ) + V 2 (r) = 0
and Dp2 (r1 , r2 ) + V02 (r) = 0, contradicting strong complementarity.
A profound quality that concerns all the complementarity relations (6.18)–(6.21) is that they govern, not only quantum light, but
also classical light, either scalar of vectorial in nature, since the definitions for the visibilities and distinguishabilities do not care about
whether the electromagnetic field is quantized or classical.
6.5 WAVE–PARTICLE DUALITY OF THE PHOTON
Although the complementarity relations (6.18)–(6.21) cover light of
any state, it is of particular (and fundamental) interest to investigate how complementarity is manifested for a single photon in the
presence of polarization modulation. Let us therefore examine in
detail the case of an arbitrary, pure single-photon state
|ψ⟩ = c1x |1, 0, 0, 0⟩ + c1y |0, 1, 0, 0⟩
+ c2x |0, 0, 1, 0⟩ + c2y |0, 0, 0, 1⟩ ,
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(6.22)
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Andreas Norrman: Electromagnetic Coherence of Optical Surface and
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in which the coefficients, with |cmµ |2 giving the probability to find
the photon µ ∈ { x, y} polarized in pinhole m ∈ {1, 2}, are normalized such that |c1x |2 + |c1y |2 + |c2x |2 + |c2y |2 = 1.
From Eqs. (6.12) and (6.22) we first obtain (for clarity, we adopt
lower-case symbols for this single-photon case)
∗
∗
c2y |,
c2x + c1y
v0 (r) = 2|c1x
(6.23)
∗
∗
v2 (r) = 2|c1x
c2y + c1y
c2x |,
(6.25)
∗
∗
v1 (r) = 2|c1x
c2x − c1y
c2y |,
v3 ( r ) =
∗
2|c1x
c2y
∗
− c1y
c2x |,
(6.24)
(6.26)
revealing that for the (pure) single-photon state (6.22) all four visibilities may be nonzero. Equations (6.6), (6.7), and (6.22) indicate
further that g(r1 , r2 ) = 1, signifying that not only in the scalar context [241], but also within the general vector framework the (pure)
one-photon field is always completely coherent. Eventually, owing
to full coherence, the strong complementarity relations (6.20) and
(6.21) hold for any single-photon state possessing the form (6.22).
Indeed, concerning total visibility and intensity distinguishability,
from Eqs. (6.13)–(6.16) and (6.22) we find that
√
v ( r ) = 2 p1 p2 ,
d0 ( r1 , r2 ) = | p1 − p2 |,
(6.27)
where pm = |cmx |2 + |cmy |2 is the probability for the photon to pass
through pinhole m ∈ {1, 2}, and clearly
d20 (r1 , r2 ) + v2 (r) = 1.
(6.28)
For the polarization distinguishability, Eqs. (6.17) and (6.22) yield
√
∗ c + c ∗ c |2 ,
dp (r1 , r2 ) = 1 − 4|c1x
(6.29)
2x
1y 2y
implying together with Eq. (6.23) that
d2p (r1 , r2 ) + v20 (r) = 1,
(6.30)
also in full agreement with Eq. (6.21).
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Complementarity in photon interference
The complementarity relations (6.28) and (6.30) may be regarded
as reflecting two distinct aspects of wave–particle duality of the photon. In particular, we observe that d0 (r1 , r2 ) in Eq. (6.27) coincides
with the path predictability P of Eq. (6.1). For scalar light, this a priori WPI of any pure one-photon state satisfies, using our notation,
d20 (r1 , r2 ) + v20 (r) = 1 [152, 153]. Nevertheless, such a relationship is
principally no longer valid for vectorial light (see below), because
Eq. (6.1) does not take into account the polarization modulation,
which Eq. (6.28) instead does. In other words, within the general
vector framework the initial WPI becomes coupled with total visibility, not just intensity visibility, revealing a novel fundamental
aspect of photon wave–particle duality. This finding, which has no
correspondence in scalar quantum interferometry, is another major
result of Publication VIII.
As an example, let us consider the scenario with c1y = c2x = 0
√
and c1x = c2y = 1/ 2, which means that
√
|ψ⟩ = (|1, 0, 0, 0⟩ + |0, 0, 0, 1⟩)/ 2.
(6.31)
The photon is now in a superposition of being x polarized at pinhole 1 with probability p1 = 1/2 and y polarized at pinhole 2 with
probability p2 = 1/2. Making use of Eqs. (6.23), (6.27), and (6.31)
one readily verifies that v0 (r) = 0 and d0 (r1 , r2 ) = 0, whereupon
d20 (r1 , r2 ) + v20 (r) = 0 although the state (6.31) is pure, contradicting the first relation of Eq. (6.1), where the equality sign holds for
pure states [152, 153]. Yet, albeit intensity distinguishability in this
situation is zero, from Eqs. (6.27) and (6.31) we find that total visibility is maximal, viz., v(r) = 1, and hence the complementarity
relation (6.28) holds. Moreover, Eqs. (6.29) and (6.31) yield maximal
polarization distinguishability, dp (r1 , r2 ) = 1, whereat the complementarity relation (6.30) is also satisfied.
The fact that v0 (r) = 0 and v(r) = 1 signals that the state (6.31)
exhibits exclusively polarization modulation, and Eqs. (6.24)–(6.26)
reveal that v1 (r) = 0, v2 (r) = 1, and v3 (r) = 1. As S′j (r) and S′′j (r) in
the photon interference law (6.10) are spatially slowly varying functions near the central axis on B , the respective normalized Stokes
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parameters, s j (r) = S j (r)/S0 (r) with j ∈ {0, . . . , 3}, are found to be
s1 (r) = 0,
s2 (r) = cos k∆r,
s3 (r) = sin k∆r,
(6.32)
where ∆r = r1 − r2 and Eq. (6.7) has been used. Recalling the physical meanings of the Stokes parameters, here s1 (r) = 0 indicates
that at every point on B the photon is equally likely x polarized or
y polarized. More precisely, in repeated single-photon experiments
one would get an equal distribution of x- and y-polarized light at
the observation screen. The two oscillatory terms, s2 (r) and s3 (r),
are more intriguing. In particular, at k∆r = 2lπ with l ∈ Z we find
that s2 (r) = 1, whereas for k∆r = (2l + 1)π one has s2 (r) = −1 [in
both situations s3 (r) = 0]. The former (latter) condition signifies
that if a −π/4-linear (+π/4-linear) polarizer is placed in front of
the detector, a count signal is never obtained from those locations.
On the other hand, for k∆r = (2l + 12 )π and k∆r = (2l − 12 )π we see
that s3 (r) = 1 and s3 (r) = −1 [now s2 (r) = 0], respectively, with
the former (latter) indicating that at those places a left-circularly
(right-circularly) polarized photon is never detected.
Finally, because the state (6.31) is orthogonally polarized in the
openings, the photon is also completely which-path marked; for instance, if we were to measure y-polarized (x-polarized) light, then
whenever a count signal is obtained from the detector we know that
the photon has passed the second (first) pinhole. The same concerns
all orthogonal polarization states of the photon, and in such situations dp (r1 , r2 ) = 1. By contrast, when the polarization is uniform in
the openings, so that dp (r1 , r2 ) = 0, there is no chance to obtain any
path information of the photon from a polarization measurement.
Following these reasonings, in scenarios where 0 < dp (r1 , r2 ) < 1
one could gain partial knowledge about the photon’s path. We may
therefore identify dp (r1 , r2 ) in Eq. (6.29) as quantifying the a posteriori WPI of the photon, i.e., the available WPI that can be extracted
retrodictively by polarization measurements. Hence, for the pure
single-photon state (6.22), the polarization distinguishability shares
similarity with the path distinguishability D of Eq. (6.1) [154].
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7 Conclusions
In this thesis we have presented results relating to theoretical research in three topics of electromagnetic nanophotonics: novel SPP
modes (Publications I–III), partial coherence of optical surface fields
(Publications IV–VII), and complementarity in genuine vector-light
photon interference (Publication VIII). Below we summarize the
main conclusions of our work and discuss potential future research.
7.1 SUMMARY OF MAIN RESULTS
In Publication I, we investigated by rigorous means single-interface
SPP propagation in the presence of metal absorption. Such SPPs are
conceptually the most fundamental ones in plasmonics and, since
absorption is an inherent element of metals in the optical regime,
incorporation of losses is necessary. It was shown that the conventional approximate analysis that is frequently utilized to estimate
SPP propagation may yield inaccurate conclusions even in cases
where it is presumed to hold. Most importantly, our exact approach
predicted the existence of a new type of backward-propagating SPP
mode, sharing similarity with fields encountered in metamaterials,
which is totally excluded within the approximate treatment. These
results do not only underline the essential differences between the
rigorous and approximate frameworks, but also convey an important message: just accounting for losses is not sufficient to get reliable results, attention is also to be paid when introducing simplifications into the SPP-field analysis.
Publication II dealt with mode solutions at an absorptive metal
film in a symmetric and lossless surrounding. The specific aim was
to identify all possible plane-wave mode solutions in such a geometry that follow from Maxwell’s equations and to classify them according to their field-propagation characteristics. In addition to the
ones reported in literature, sets of entirely new mode types were
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found, manifested both as bound waves and leaky waves. Likewise, we showed that some commonly accepted mode solutions are
not actually admitted by Maxwell’s equations. Moreover, whereas
previous studies have concentrated only on the region outside the
film, we analyzed the properties of the various modes also within
the slab. It was found that in the film both forward- and backwardpropagating waves may occur, depending on the mode type and the
material parameters. The mode classes can be interpreted as representing resonance conditions generalized from ones encountered
in conventional optics.
In Publication III, we demonstrated for the first time the conversion of a higher-order mode (HOM) at a thin metal slab, normally
not regarded useful, into a long-range surface mode in situations
where the long-range fundamental mode (FM) does not exist and
the propagation distance of the single-interface SPP is negligible.
The finding of this novel electromagnetic near-field phenomenon
constitutes the pinnacle of our modal studies. In this process, an
unexpected mode interchange occurs between a HOM and the FM,
at which point the propagation length of the HOM may experience
even a thousandfold enlargement. The discovery of such a mode
crossover, which may take place for many different material combinations and frequency bandwidths, is anything but trivial, since
it cannot be predicted from the mode dispersion relations. In addition, unlike with the FM, the HOM’s long-range behavior does not
originate from the coupling of the SPPs on the two slab boundaries
and, remarkably, in some cases it comes with even increased surface confinement. Our results, which follow from rigorous electromagnetic theory, thus provide deeper insights into the foundations
of subwavelength thin-film plasmonics and may find use in future
nanophotonics and optoelectronics.
A framework to customize the vectorial coherence of polychromatic SPPs in the Kretschmann setup by controlling the correlations
of the excitation light was advanced in Publication IV. To this end,
the general space–time coherence matrix, valid for stationary and
nonstationary SPP fields of arbitrary spectra and spectral correla-
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Conclusions
tions, was analytically determined. As a key result, we derived the
relation between the correlation functions of the light source and
the SPP field, enabling one to ascertain the illumination coherence
to achieve the desired coherence state for the polychromatic SPP
field. We also showed that narrowband SPPs are virtually propagation invariant and fully polarized. Quite surprisingly, even broadband SPPs of widely variable coherence were revealed to possess
a high degree of polarization, at least for metals and optical frequencies for which SPP propagation is appreciable. Publication IV
establishes a novel paradigm in statistical plasmonics, referred to
as plasmon coherence engineering, which could be instrumental
for sensor applications, interferometry, spectroscopy, and controlled
nanoparticle excitation.
Publication V concerned the spatial coherence and polarization
of a stationary two-mode SPP field consisting of the long-range and
short-range SPPs at a metallic nanofilm. These two SPP modes are
prominent in thin-film plasmonics, and in our study the short-range
SPP played a pivotal role facilitating coherence and polarization
modifications. The main objective was to examine the fundamental limits that the spectral degrees of coherence and polarization
of such a two-mode field can assume, irrespective of the excitation
process, and how the degrees vary within their extremal values
when the media, frequency, and film thickness are altered. As full
correlation naturally yields complete coherence and polarization,
we took the modes uncorrelated to assess the lower ranges. It was
shown that, due to electromagnetic similarity, such an SPP field
always exhibits regions of quite a high (low) degree of coherence.
At these locations, the two-mode field would interact with nearby
nanoparticles in a coherent (incoherent) manner. We also demonstrated that the coherence lengths may extend from subwavelength
scales to tens or hundreds of wavelengths. Finally, Publication V
indicated that with ultra-thin films the generally highly polarized
SPP field can be partially polarized within subwavelength distances
from the excitation region.
In Publication VI, we analyzed the electromagnetic coherence
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and partial polarization of stationary, purely evanescent light fields
generated in total internal reflection at a lossless dielectric interface.
Employing the spectral degree of coherence, we showed that for
such fields the coherence length in air can be significantly shorter
than the free-space wavelength, especially for high refractive-index
contrasts. The coherence length is typically smallest in the immediate vicinity of the surface, but may get very large already within a
wavelength from it. We also adopted the 3D degree of polarization
to demonstrate that already two beams are sufficient to generate
a true 3D evanescent field which cannot be described by the conventional polarization theory. Publication VI revealed that, in general, the coherence and polarization properties of electromagnetic
surface fields at subwavelength scales may, even in the absence of
absorption, differ notably from those a wavelength or more away
from the supporting interface.
Motivated by these results, in Publication VII we explored conditions under which stationary, fully 3D-unpolarized evanescent
light fields could be generated by manageable means at a lossless
dielectric surface in a configuration involving multiple illumination
beams. It was shown that by using two incident beams it is possible
to excite a nearly unpolarized 3D evanescent field, but to achieve
a strictly 3D-unpolarized state requires at least a three-beam setup.
We also investigated the spectral electromagnetic coherence of such
fields and demonstrated that their degrees of coherence may vary
considerably, exhibiting diverse subwavelength lattice-like patterns,
depending on the applied modality. These findings suggest the feasibility to customize evanescent light fields with polarization characteristics identical to those of universal blackbody radiation, yet
with adjustable coherence properties.
Finally, in Publication VIII, by examining polarization modulation in double-pinhole photon interference, we formulated two
general complementarity relations for genuine vectorial quantumlight fields of arbitrary state. To this end, we derived the photon
interference law and a generalized visibility relation, which characterize any effect that spatial coherence at the openings may have
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Conclusions
on the number and polarization variations of the photons in the
observation plane. In addition, we introduced an intensity distinguishability and a polarization distinguishability to differentiate the
light in the pinholes. The complementarity relations, establishing
links between these quantities, were identified to reflect two separate features of wave–particle duality of the photon. In particular,
we demonstrated that the a priori which-path information of singlephoton vector light does not couple to the intensity visibility, but to
the total visibility, which accounts also for polarization modulation.
This discovery reveals an intrinsic aspect of photon wave–particle
duality, not reported earlier to our knowledge. It was also shown
that for a general quantum-light field such complementarity is a
manifestation of complete coherence, not of quantum-state purity.
Photon-polarization modulation thereby entails novel, fundamental
physical facets of quantum complementarity.
7.2 FUTURE PROSPECTS
Altogether, in this thesis, we have addressed only a limited number
of research topics within the realms of plasmonics, electromagnetic
coherence, and quantum complementarity. There are naturally still
a rich diversity of physics to explore and important open questions
that deserve further investigation in all of these areas.
The novel metal-slab mode solutions derived in this work concerned a symmetric environment. What about an asymmetric metalslab or a multi-surface geometry? Can a rigorous analysis adapted
for such configurations also reveal new mode solutions, or even
whole mode-solution classes, with features similar as or drastically
different from the ones met in the symmetric case? Are mode interchanges, analogous to those discovered in Publication III, encountered in the asymmetric geometry too? There are also unanswered
issues related to the symmetric setup. For instance, the reason why
just one of the modes, either the higher-order or the fundamental
mode, evolves into a long-range mode is still unclear. This naturally
raises the question: could two, or even more, long-range modes ex-
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ist simultaneously for given media, frequency, and slab thickness?
The feasibility of multiple long-range modes would undoubtedly
be of interest in thin-film plasmonics.
Exploring the coherence and polarization characteristics of genuine three-component SPP fields, and their classical, semi-classical,
and quantum interactions with nanostructures, constitutes a broad
research area of fundamental importance. In particular, extending
the scheme of plasmon coherence engineering presented in Publication IV from one-dimensional SPP propagation to planar dimensions would lead to new, exotic, and highly versatile SPP fields, necessitating rigorous 3D electromagnetic coherence and polarization
theory. Naturally, designing and constructing the required excitation light sources of customized spatio–spectral coherence present
a whole problem area on its own. Using nanoscatterers or scanning
near-field optical microscopy could be a promising step towards
probing such novel electromagnetic surface fields.
Although quantum optics and quantum information have a long
history, studies on the role of polarization in quantum coherence
and quantum interference phenomena appear ominously absent.
Will the photon interference law established in Publication VIII predict new discoveries? For instance, is there a quantum analog to the
degree of polarization being a measure of polarization modulation
in beam self-interference, a consequence of the classical electromagnetic interference law? In multi-pinhole photon interference involving polarization modulation entirely new complementary features
would in all likelihood appear. What these entities are, how they are
quantified via complementarity relations, and what novel physics
they reveal, remain a challenge. Overall, quantum complementarity
with polarization, and the emerging connections between classical
and quantum light, constitute rich and promising research areas.
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A Classical theory of electromagnetic coherence
This Appendix provides a brief overview of the basic concepts to
characterize second-order electromagnetic coherence in classical optical fields. The formalism, which is used in Chaps. 4 and 5 to assess electromagnetic coherence of optical surface fields, covers nonstationary and stationary light, both in the space–time and space–
frequency domains. We note that the terminology in classical coherence theory differs by a factor of two from that in quantum theory
of optical coherence; the correlation order in classical theory follows the power of field amplitude, while in the quantum context it
follows the power of intensity (photons).
The second-order statistical properties of a classical electromagnetic field are specified by the electric, magnetic, and two mixedfield coherence matrices [54], which constitute the foundation for
the modern treatment of partial coherence (and polarization). Although a full description of the electromagnetic coherence of light
requires that all four matrices are taken into account, it is often sufficient to consider only electric-field correlations, as optical processes
are primarily manifested via the electric field [177]. Consequently,
henceforth we focus solely on the electric field, which we represent,
at a space–time point (r, t), as a complex analytic signal [54],
E(r, t) =
∫ ∞
0
E(r, ω )e−iωt dω,
(A.1)
where the column vector E(r, ω ) is the frequency-domain Fourier
transform of the actual real-valued field. The real and imaginary
parts of the complex analytic signal representation form a Hilbert
transform pair, which is equivalent to the fact that E(r, t) is an analytic (as the name implies) and regular function in the lower half
of the complex plane with respect to t [54]. Moreover, it is a natural
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generalization of the complex representation that is frequently used
in the context of real monochromatic fields and plays an important
role in quantum optics.
A.1
NONSTATIONARY FIELDS
All information about the second-order coherence properties of a
nonstationary classical light field is in the space–time domain included in the electric coherence matrix [107]
Γ(r1 , t1 ; r2 , t2 ) = ⟨E∗ (r1 , t1 )ET (r2 , t2 )⟩,
(A.2)
where the asterix and superscript T stand for complex conjugation
and matrix transpose, respectively, and the angle brackets denote
ensemble averaging. Physically, the elements of Γ(r1 , t1 ; r2 , t2 ) encompass the correlations among the orthogonal field components at
two space–time points (r1 , t1 ) and (r2 , t2 ). The polarization features
of the nonstationary light are encoded in the polarization matrix
J(r, t) = Γ(r, t; r, t),
(A.3)
where the diagonal elements give the average intensity of each orthogonal component, and where the off-diagonal elements characterize the correlations prevailing between the orthogonal field components at a single space–time point.
In the space–frequency domain, the spectral electric coherence
matrix of a nonstationary field is expressed directly via the Fourier
transform E(r, ω ) of E(r, t) as [107]
W(r1 , ω1 ; r2 , ω2 ) = ⟨E∗ (r1 , ω1 )ET (r2 , ω2 )⟩,
(A.4)
whose elements describe the spatial correlations of the components
at two frequencies ω1 and ω2 . The single-point, single-frequency
correlations are characterized by the spectral polarization matrix
Φ(r, ω ) = W(r, ω; r, ω ),
86
(A.5)
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Classical theory of
electromagnetic coherence
with the diagonal elements yielding the intensity of each orthogonal field component, and with the off-diagonal elements specifying
the correlations among the orthogonal components at (r, ω ).
The partial coherence of a vector-light field is conveniently quantified in the space–time and space–frequency domains by the electromagnetic temporal [112] and spectral [113] degrees of coherence.
The physical basis for these measures, which amount to the totality
of correlations existing between all the orthogonal field components
for a pair of points, are the intensity visibility and the polarization
modulation in Young’s interference experiment [158]. For nonstationary light, they are defined, respectively, through
∥Γ(r1 , t1 ; r2 , t2 )∥F
,
γ ( r1 , t1 ; r2 , t2 ) = √
trJ(r1 , t1 )trJ(r2 , t2 )
∥W(r1 , ω1 ; r2 , ω2 )∥F
,
µ ( r 1 , ω1 ; r 2 , ω2 ) = √
trΦ(r1 , ω1 )trΦ(r2 , ω2 )
(A.6)
(A.7)
where ∥ · ∥F is the Frobenius matrix norm [238] and tr stands for the
trace. Both quantities are bounded between zero and unity, with the
latter (former) limit representing complete coherence (incoherence)
and taking place if, and only if, all orthogonal field components between (r1 , t1 ) and (r2 , t2 ) in the space–time domain and (r1 , ω1 ) and
(r2 , ω2 ) in the space–frequency domain are fully correlated (uncorrelated). In particular, γ(r1 , t1 ; r2 , t2 ) = 1 and µ(r1 , ω1 ; r2 , ω2 ) = 1
in a domain are equivalent with the factorization of Γ(r1 , t1 ; r2 , t2 )
and W(r1 , ω1 ; r2 , ω2 ), respectively, which is considered to be a fundamental property of a completely coherent field [136].
We stress that other measures for characterizing partial coherence in vectorial light fields have been put forward [118–123], with
different mathematical properties and physical implications [124].
A.2 STATIONARY FIELDS
The second-order statistical properties of a stationary vector-light
field, i.e., a field for which the character of the random fluctuations
does not change with time, but depend only on the time separation
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τ = t2 − t1 , are in the space–time domain completely specified by
the electric coherence matrix [54]
Γ(r1 , r2 , τ ) = ⟨E∗ (r1 , t)ET (r2 , t + τ )⟩,
(A.8)
with the angle brackets denoting ensemble or time averaging. The
space–time polarization matrix of the stationary light is given by
J(r) = Γ(r, r, 0),
(A.9)
which, contrary to the nonstationary case, is time independent.
As a stationary field is not square integrable over time, whereby
it does not have a Fourier transform, the transition from space–time
domain to space–frequency domain is not as straightforward as for
nonstationary light. Nevertheless, in most relevant physical situations the elements of Γ(r1 , r2 , τ ) can be taken as square integrable,
whereupon the spectral coherence matrix is defined as the Fourier
transform of Γ(r1 , r2 , τ ) [54], viz.,
W ( r1 , r2 , ω ) =
1
2π
∫ ∞
−∞
Γ(r1 , r2 , τ )eiωτ dτ.
(A.10)
A distinguishing feature of the space–frequency domain representation of a stationary field is that different frequency components
are fully uncorrelated; spectral correlations induce nonstationary
light. Equation (A.10) and the fact that the frequencies are uncorrelated constitute the generalized Wiener–Khintchine theorem [54].
Moreover, albeit W(r1 , r2 , ω ) is not directly defined as a correlation
matrix of the two fields, it has an important and useful property,
namely, it can be expressed as a correlation matrix over an ensemble {E(r, ω )e−iωt } of monochromatic field realizations [102], i.e.,
W(r1 , r2 , ω ) = ⟨E∗ (r1 , ω )ET (r2 , ω )⟩.
(A.11)
We point out that here E(r, ω ) is not the Fourier transform of E(r, t).
The representation in Eq. (A.11) is a fundamental result that enables
the analysis of stationary light frequency by frequency.
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The spectral polarization properties of the stationary light field
are encoded in the spectral polarization matrix
Φ(r, ω ) = W(r, r, ω ).
(A.12)
While the space–time domain polarization matrix J(r) is independent on time, the space–frequency polarization matrix Φ(r, ω ) depends on the frequency.
Similarly to nonstationary light, it is favorable to assess the partial coherence of a stationary vector-light field by employing the corresponding temporal and spectral degrees of electromagnetic coherence. For stationary light, they are given, respectively, by [112, 113]
∥Γ(r1 , r2 , τ )∥F
,
γ ( r1 , r2 , τ ) = √
trJ(r1 )trJ(r2 )
∥W(r1 , r2 , ω )∥F
.
µ ( r1 , r2 , ω ) = √
trΦ(r1 , ω )trΦ(r2 , ω )
(A.13)
(A.14)
As for nonstationary light, the quantities γ(r1 , r2 , τ ) and µ(r1 , r2 , ω )
are nonnegative, invariant under unitary transformations, and attain their maximum value (unity) when the fields at points r1 and
r2 , and at time separation τ or frequency ω, are completely coherent. If these conditions hold in a domain, the matrices Γ(r1 , r2 , τ )
and W(r1 , r2 , ω ) factorize [59, 113, 114].
A.3 DEGREE OF POLARIZATION
The degree of polarization is a measure that reflects the amount of
correlations prevailing between the orthogonal electric-field components of the fluctuating light field at a single point. The traditional
formulation of this quantity has been restricted to two-component
(2D) light fields, such as beams and far fields, whose electric-field
vector fluctuates approximately in a plane transverse to the propagation direction [54, 55]. Nevertheless, as highlighted by novel advancements in nano-optics [2] and high-numerical-aperture imaging systems [109–111], there are situations in which this approximation is no longer valid. Therefore, the concept of the degree of polarization must be extended to also include genuine three-component
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(3D) light fields with wave fronts of arbitrary form. In the following
we limit our discussion to stationary light in the space–frequency
domain, but emphasize that analogous considerations apply in the
space–time domain and for nonstationary light.
For a 2D-light field, the spectral polarization matrix Φ(r, ω ) in
Eq. (A.12) can be uniquely expressed as a sum of two matrices, one
of which corresponds to fully unpolarized light, being proportional
to the 2 × 2 identity matrix, and another one which represents completely polarized light. In this case, the 2D degree of polarization,
P2D (r, ω ), is defined as the ratio of the spectral density of the polarized part to that of the total field, which can be written as [54, 55]
P2D (r, ω ) = 2
√
trΦ2 (r, ω ) 1
− .
tr2 Φ(r, ω ) 2
(A.15)
The 2D degree of polarization satisfies 0 ≤ P2D (r, ω ) ≤ 1, with the
lower and upper limits corresponding to totally unpolarized and
completely polarized light, respectively, while the intermediate values stand for partial polarization. In a coordinate frame where the
intensities of the two orthogonal components are equal P2D (r, ω ) coincides with the absolute value of the correlation coefficient among
the components. Moreover, P2D (r, ω ) can be given an interferometric interpretation as an ability of a light beam to exhibit polarization
modulation when it is allowed to interfere with itself [159].
Whereas for 2D light the polarization matrix can be expressed
unambiguously as a sum of two matrices, one representing unpolarized light and the other fully polarized light, for 3D-light fields
such a decomposition does generally not exist [55,57]. Accordingly,
another approach must be taken in order to define the degree of
polarization for genuine 3D-light fields. The generalization of the
degree of polarization for 3D light can be obtained by considering the expansion of Φ(r, ω ) in terms of 3 × 3 Gell-Mann matrices
and generalized Stokes parameters [125], although other methods
for assessing partial polarization of 3D-light fields have been proposed [126–134]. In this approach, the resulting expression for the
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Classical theory of
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3D degree of polarization, P3D (r, ω ), is
√
3 trΦ2 (r, ω ) 1
− .
P3D (r, ω ) =
2 tr2 Φ(r, ω ) 3
(A.16)
As its 2D counterpart, the 3D degree of polarization is bounded
between zero and unity. The lower limit P3D (r, ω ) = 0 represents
3D-unpolarized light, encountered when the spectral densities of
the components are the same and no correlation exists between any
of them. The maximum value, P3D (r, ω ) = 1, corresponds to fully
polarized light and takes place if, and only if, all the components
are mutually fully correlated. Any other value of P3D (r, ω ) stands
for a partially polarized light field.
The physical meaning of P3D (r, ω ) becomes apparent by writing
it in a coordinate frame oriented in such a way that the diagonal
elements of Φ(r, ω ) are equal (an orientation which can always be
found) [125]. In such a coordinate system the 3D degree of polarization turns into a direct measure for the average correlations between
the orthogonal electric-field components, analogously to P2D (r, ω ).
In addition, if the 3 × 3 polarization matrix can be represented as
a sum of two matrices corresponding to a fully unpolarized and a
completely polarized part, then P3D (r, ω ) is the ratio of the intensity
of the polarized part to the total field intensity [215].
Finally, we note that the 2D and 3D degrees of polarization are
connected to the spectral degree of coherence in Eq. (A.14) via
√
1
1 + P2D (r, ω ),
2
√
1
1 + 2P3D (r, ω ),
µ(r, r, ω ) =
3
µ(r, r, ω ) =
(A.17)
(A.18)
both stating that, for vectorial light, the equal-point degree of coherence is unity if, and only if, the light field is completely polarized.
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PUBLICATIONS OF
THE UNIVERSITY OF EASTERN FINLAND
This thesis considers surface-plasmon
polaritons (SPPs), partially coherent optical
surface fields, and complementarity in
vector-light photon interference. Novel SPPs,
including a long-range higher-order metalslab mode, are predicted. Generation, partial
polarization, and electromagnetic coherence
of polychromatic SPPs and evanescent
light fields are also examined. Polarization
modulation of vectorial quantum light is
explored to uncover a new intrinsic aspect of
photon wave–particle duality.
uef.fi
PUBLICATIONS OF
THE UNIVERSITY OF EASTERN FINLAND
Dissertations in Forestry and Natural Sciences
ISBN 978-952-61-2357-8
ISSN 1798-5668
DISSERTATIONS | ANDREAS NORRMAN | ELECTROMAGNETIC COHERENCE OF OPTICAL SURFACE AND... | No 252
ANDREAS NORRMAN
Dissertations in Forestry and
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ANDREAS NORRMAN
ELECTROMAGNETIC COHERENCE OF OPTICAL
SURFACE AND QUANTUM LIGHT FIELDS