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PUBLICATIONS OF
THE UNIVERSITY OF EASTERN FINLAND
In this book, local-field controlled linear
and Kerr nonlinear optical properties of
subwavelength periodic nanostructures
and nanocomposites are studied. Efficient
numerical techniques and novel analytical
models have been developed to aid in these
studies. In addition, prospect for achieving low
energy optical bistability with a silicon nitride
guided mode resonance filter is examined
numerically followed by an experimental
demonstration of all-optical modulation using
such a structure.
uef.fi
PUBLICATIONS OF
THE UNIVERSITY OF EASTERN FINLAND
Dissertations in Forestry and Natural Sciences
ISBN 978-952-61-2441-4
ISSN 1798-5668
DISSERTATIONS | SUBHAJIT BEJ | LOCAL FIELD CONTROLLED LINEAR AND KERR NONLINEAR OPTICAL... | No 262
SUBHAJIT BEJ
Dissertations in Forestry and
Natural Sciences
SUBHAJIT BEJ
LOCAL FIELD CONTROLLED LINEAR AND KERR
NONLINEAR OPTICAL PROPERTIES OF
PERIODIC SUBWAVELENGTH STRUCTURES
PUBLICATIONS OF THE UNIVERSITY OF EASTERN FINLAND
DISSERTATIONS IN FORESTRY AND NATURAL SCIENCES
N:o 262
Subhajit Bej
LOCAL FIELD CONTROLLED LINEAR
AND KERR NONLINEAR OPTICAL
PROPERTIES OF PERIODIC
SUBWAVELENGTH STRUCTURES
ACADEMIC DISSERTATION
To be presented by the permission of the Faculty of Science and Forestry for public
examination in the Auditorium F100 in Futura Building at the University of Eastern
Finland, Joensuu, on March 23rd, 2017, at 12 o’clock.
University of Eastern Finland
Department of Physics and Mathematics
Joensuu 2017
Grano Oy
Jyväskylä, 2017
Editors: Pertti Pasanen, Pekka Toivanen,
Jukka Tuomela, Matti Vornanen
Distribution:
University of Eastern Finland Library / Sales of publications
[email protected]
http://www.uef.fi/kirjasto
ISBN: 978-952-61-2441-4 (print)
ISSNL: 1798-5668
ISSN: 1798-5668
ISBN: 978-952-61-2442-1 (pdf)
ISSNL: 1798-5668
ISSN: 1798-5676
ii
Author’s address:
University of Eastern Finland
Department of Physics and Mathematics
P.O. Box 111
FI-80101 JOENSUU
FINLAND
email: [email protected]
Supervisors:
Professor Jari Turunen, D. Sc.
University of Eastern Finland
Department of Physics and Mathematics
P.O. Box 111
FI-80101 JOENSUU
FINLAND
email: [email protected]
Professor Yuri P. Svirko, Ph.D.
University of Eastern Finland
Department of Physics and Mathematics
P.O. Box 111
FI-80101 JOENSUU
FINLAND
email: [email protected]
Docent Jani Tervo, Ph.D.
University of Eastern Finland
Department of Physics and Mathematics
P.O. Box 111
FI-80101 JOENSUU
FINLAND
email: [email protected]
Reviewers:
Associate Professor Hiroyuki Ichikawa, Ph.D.
Ehime University
Department of Electrical and Electronic Engineering
790-8577 Matsuyama
JAPAN
email: [email protected]
Dr. Alexey Yulin, Ph.D.
ITMO University
The Metamaterials Laboratory
Birjevaja line V.O.,14 St. Petersburg
199034 RUSSIA
email: [email protected]
Opponent:
Dr.-Ing. Bernd Kleemann
Staff Scientist
Corporate Research and Technology
Carl Zeiss AG
D-73446 Oberkochen
GERMANY
email: [email protected]
iii
Subhajit Bej
Local field controlled linear and Kerr nonlinear optical properties of periodic subwavelength structures
Joensuu: University of Eastern Finland, 2017
Publications of the University of Eastern Finland
Dissertations in Forestry and Natural Sciences
ABSTRACT
Optical properties of structured media are controlled by the local electric and magnetic fields to a great extent. Nanostructuring allows one to customize the macroscopic linear and nonlinear optical properties of these artificial media by engineering the local fields precisely. In this thesis, local electric field controlled linear and
Kerr nonlinear optical properties of subwavelength periodic nanostructures are investigated theoretically, numerically, and experimentally. Both metal-dielectric and
all-dielectric structures are examined.
In the subwavelength regime, when the feature sizes approach the wavelength of
light, approximate theoretical models may produce inaccurate results. A numerical
technique based on the Fourier-Expansion Eigenmode Method has been developed
to model optical Kerr effect in periodic structures. This numerical model serves as
an efficient and accurate tool for the design and analysis of low power all-optical
devices which rely on local field enhanced optical Kerr nonlinearity.
Light propagation in a form birefringent medium with optical Kerr medium is
considered and an analytical model is developed which accurately describes nonlinear light-matter interactions.
Possibility to achieve all-optical modulation and optical bistability with waveguide grating structures fabricated from silicon nitride thin films grown on top of
fused silica substrates are analyzed both theoretically and experimentally. The
silicon nitride films are grown by plasma enhanced chemical vapour deposition
(PECVD) and the waveguide grating structures are fabricated by electron beam
lithography and reactive ion etching techniques. Experiments are carried out using
a single-walled carbon nanotube (SWCNT) modelocked ultrafast fiber laser with an
amplifier and a tunable unit.
A full wave numerical approach based on the Fourier Modal Method is introduced to model nanocomposite optical materials. This approach can be used to
model arbitrary particle geometries and their random arrangements, finite wavelength effects, multipolar effects, and percolation. Furthermore, the proposed method
can be used to engineer large effective optical nonlinearities at the nanoscale, which
is pivotal for designing novel low power nonlinear photonic devices.
Universal Decimal Classification: 535.1, 535.3, 535.4, 535.41, 535.42, 535.421, 535.8,
535.92, 535.13, 535.18, 535.181, 535.317.2, 537.226.2, 537.226.5, 537.8
OCIS codes: 050.0050, 050.1960, 050.2065, 050.2555, 050.6624, 050.1755, 050.5745,
190.0190, 190.1450, 190.3270, 190.4360, 190.7110, 160.1245, 160.3918, 160.4330, 230.1150
Keywords: optics; micro-optics; nanophotonics; nonlinear optics; nonlinear metamaterials;
resonant nanophotonics; computational physics; computational electromagnetism; rigorous
grating theory; diffraction gratings; Kerr effect; form birefringence; resonance waveguide
grating; guided mode resonance; optical filters; optical bistability, optical switching; allv
optical device; all-optical modulation; microfabrication; nanofabrication; fiber laser; electron
beam lithography; reactive ion etching; nanostructured materials; localized surface plasmon
resonance; nanocomposites
vi
ACKNOWLEDGEMENTS
This thesis is a summary of nearly four years of work as an early stage researcher
in the Department of Physics and Mathematics at University of Eastern Finland,
Joensuu. Four years is quite a long time span and it is phenomenal how fast the
world changed during this tenure- scientists successfully cloned human stem cells,
first observation of gravitational waves were made, a historic agreement targeting
climate change was signed in Paris, Google’s artificial intelligence AlphaGo defeated
world’s top GO Player and many more to mention. In the past two years, the Department of Physics and Mathematics at UEF also encountered major changes. Like
everyone else in our unit, these had either direct or indirect impact on my research
work and turned my PhD project into a thrilling roller-coaster journey. Nevertheless, I enjoyed every part of this safari and it taught me how to stay positive and
focus on individual research during unfavourable situations. However, this journey
would be incomplete without constant encouragement and support of my academic
supervisors, my parents, my beloved wife, and close friends.
First of all, thanks to the ’Big Bang’ for creating the universe and the ’Evolution’
for creating human beings.
I would like to express my deepest gratitude to Prof. Jari Turunen for his unique
style guidance and persistent support during these years. I am thankful to my academic co-supervisor Prof. Yuri Svirko for his valuable instructions and suggestions
from time to time. I want to extend my thanks to Dr. Jani Tervo with whom I started
to work back in 2011. Without Jani’s active supervision and hands-on training during the early years of my doctoral studies, it would be impossible to complete this
thesis.
I am obliged to the past Head of Department Prof. Pasi Vahimaa, Prof. Seppo
Honkanen, and the present Head of Department Prof. Timo Jääskeläinen for providing me the opportunity to work in the encouraging atmosphere of our department.
My sincere thanks to Dr. Janne Laukkanen who trained me to work independently in our lithography laboratory. Dr. Toni Saastamoinen, Dr. Matthieu Roussey,
Dr. Viatcheslav Vanyukov, Dr. Tomi Kaplas, Dr. Hemmo Tuovinen, Prof. Tero
Setälä, and Prof. Markku Kuittinen deserve special thanks for being able to help
me and spend their valuable time whenever I needed. I also want to thank Dr.
Pertti Pääkkönen, Tommi Itkonen, and Timo Vahimaa for their kind support. For
assistance at the administrative level, Dr. Noora Heikkilä, Ms. Katri Mustonen, Ms.
Hannele Karppinen, and Ms. Marita Ratilainen deserve my sincerest gratitude.
I am indebted to Prof. Zhipei Sun for giving me an opportunity to visit and work
in his optics laboratory at Micronova, Otaniemi, Espoo. I want to acknowledge all
other collaborators especially Diao Li from Northwest University (China), and Jorge
Francés from University of Alicante (Spain) for their efforts. Special thanks to Dr.
Srikanth Sugavanam from Aston University (UK) for fruitful discussions during the
planning stage of some experimental works.
During the time at UEF, I was fortunate to meet other talented young researchers
as well. Thanks to my previous office mates Henri Partanen, and Markus Häyrinen
for the time we spent discussing academic and sometimes more refreshing nonacademic stuffs.
I really appreciate the labor made by the pre-examiners Associate Professor Hiroyuki Ichikawa and Dr. Alexey Yulin after accepting the request to review my
thesis. Their suggestions and constructive comments helped in increasing the readability of this thesis.
vii
Life is incomplete without friends. Thanks to my good friends Swarup, Avik,
Ayan, Sourav, Shovan, Apurbo, Nilabha, Hasanur for the light moments we shared
and memories we created together. Special thanks to my friend, mentor, and drinking (coffee) buddy Dr. Md. Sahidullah.
My heartfelt thanks to my beloved parents Mr. Siddheswar Bej and Mrs. Tista
Bej for their kindness, encouragement, love and enormous support throughout my
life. Thanks to my lovely sister Shampa, my brother-in-law Sanjib, and my cute
nephew Songlap for their unconditional love. Finally, I want to thank my soulmate,
my beautiful wife Lahari for completely understanding a maniac like me, accepting
me as I am and keeping faith in me even when I had started to loose faith in myself!
To humanity and global peace.
Joensuu, February 26, 2017
Subhajit Bej
viii
TABLE OF CONTENTS
1 Introduction
1.1 Background .....................................................................................
1.2 Motivation .......................................................................................
1.3 Thesis outline ..................................................................................
1
1
4
4
2 Fundamentals of electromagnetic theory
2.1 Complex representation of Electromagnetic field quantities and Fourier
decomposition ..................................................................................
2.2 Macroscopic Maxwell’s equations and their empirical basis ................
2.3 Material constitutive equations .........................................................
2.4 Electromagnetic boundary conditions ................................................
2.5 Wave equation .................................................................................
2.6 TE-TM decomposition .....................................................................
2.7 Electromagnetic energy quantities ....................................................
2.8 Electromagnetic plane wave .............................................................
2.9 Polarization of an EM plane wave .....................................................
2.10 Plane wave at planar boundary .........................................................
2.11 General field and angular spectrum representation .............................
2.12 Theory of evanescent waves .............................................................
2.13 Monochromatic plane wave in anisotropic medium ............................
2.14 Electromagnetic theory of metals ......................................................
2.14.1 Drude model .........................................................................
2.14.2 Interband transitions model ...................................................
2.15 Plasmons .........................................................................................
2.15.1 Volume plasmons ...................................................................
2.15.2 Surface plasmon polaritons ....................................................
2.15.3 Particle plasmons ...................................................................
2.16 Field encountering stack of thin films ...............................................
2.17 Recursive S-matrix algorithm ............................................................
2.18 Local field ........................................................................................
2.19 Summary .........................................................................................
7
7
8
9
11
11
13
14
14
15
16
21
22
23
25
26
27
28
29
29
32
34
36
38
41
3 Rigorous analysis of diffraction gratings
3.1 Working principle of a grating ..........................................................
3.2 Pseuodoperiodicity and grating equations .........................................
3.3 Diffraction efficiencies ......................................................................
3.4 Overview of the existing numerical modeling methods .......................
3.5 Fourier Modal method for diffraction gratings ...................................
3.5.1 Fourier factorization rules ......................................................
3.6 FMM for linear gratings with plane wave illumination .......................
3.6.1 Formulation of the eigenvalue problem ...................................
3.6.2 Solution of electromagnetic boundary conditions ....................
3.6.3 Solution for multilayered gratings ...........................................
43
43
45
47
48
48
49
50
51
55
56
ix
3.6.4 Field inside the gratings .........................................................
3.7 FMM for anisotropic crossed gratings ...............................................
3.8 Staircase approximation ...................................................................
3.9 Summary .........................................................................................
58
59
67
67
4 Light propagation in Periodic media with optical Kerr nonlinearity
4.1 Light propagation in isotropic third order nonlinear materials ............
4.2 Theory of the Optical Kerr effect in isotropic media ..........................
4.3 Modeling light-induced anisotropy with the linear FMM ....................
4.4 Symmetries in light-induced anisotropy .............................................
4.5 Numerical examples .........................................................................
4.5.1 One dimensional metallic gratings with grooves filled with
χ(3) media .............................................................................
4.5.2 1-D binary grating with TiO2 as the Kerr nonlinear material ...
4.5.3 Crossed gratings with the pillars made with Si3 N4 ..................
4.5.4 Si3 N4 resonance waveguide-grating ........................................
4.6 Summary .........................................................................................
69
69
70
73
76
81
82
82
83
85
88
5 Theory of form birefringence in Kerr-type media
89
5.1 Propagation of light in crystals ......................................................... 89
5.2 Birefringence of a uniaxial crystal ..................................................... 91
5.3 Theory of form birefringence ............................................................ 93
5.4 Form birefringence in Kerr media: analytical formulation ................... 96
5.5 Numerical examples ......................................................................... 100
5.6 Summary ......................................................................................... 102
6 All-optical modulation and optical bistability with a Silicon Nitride
waveguide grating
105
6.1 Theory of optical bistability - the Fabry-Perot resonator approach ..... 106
6.2 Working principle of a waveguide grating .......................................... 108
6.3 Silicon Nitride vs. crystalline Silicon as a nonlinear material .............. 111
6.4 Fabrication of the waveguide grating structures ................................ 113
6.4.1 Thin film deposition .............................................................. 113
6.4.2 Electron beam lithography ..................................................... 114
6.4.3 Resist technology ................................................................... 116
6.4.4 Reactive ion etching .............................................................. 116
6.5 Numerical simulation results ............................................................. 119
6.6 Experimental results ......................................................................... 126
6.7 Summary ......................................................................................... 129
7 Modeling nanocomposite optical materials with FMM
131
7.1 Nanocomposite optical materials ...................................................... 131
7.2 Quasi-static approximation and its validity ........................................ 132
7.3 Common composite geometries ........................................................ 133
7.3.1 Maxwell Garnett geometry ..................................................... 133
7.3.2 Bruggeman geometry ............................................................. 134
7.3.3 Layered composite geometry .................................................. 134
7.4 Optical properties of nanocomposites containing metal nanoparticles . 134
7.5 Rigorous modeling- methodology ...................................................... 137
7.6 Numerical examples ......................................................................... 137
x
7.6.1 Porous silicon nanostructures ................................................. 138
7.6.2 Silver nanospheres on glass substrate ..................................... 140
7.6.3 Silver nanorods embedded in a Kerr nonlinear host ................. 143
7.7 Summary ......................................................................................... 145
8 Summary, conclusions and scope of future work
147
8.1 Summary with conclusions ............................................................... 147
8.2 Scope of future work ........................................................................ 148
BIBLIOGRAPHY
151
xi
1
Introduction
This chapter includes a general background, the motivation behind this work, and
an outline of this thesis.
1.1
BACKGROUND
Nanostructured materials with exciting optical properties can be readily found in
nature. An example is opal, which contains silica nano spheres arranged in a regular
lattice, showing different colours when seen from different angles. Other examples
include the butterfly wings which are iridescent due to interference of light in treelike nanostructures [1], and a special class of beetles with chiral nanostructures
forming its exoskeleton which make it appear either green or black depending on
the handedness of the circularly polarized illuminating light [2]. In the past, strong
efforts have been made in mimicking these naturally available nanostructures to
yield unprecedented optical properties. These man-made nanostructures are widely
known as metamaterials [3–5] i.e. materials with properties beyond the conventional
materials.
Though the theory of making optical metamaterials have been understood only
recently, the first man-made metamaterials were constructed much before, without
knowing the underlying mechanisms. A particularly interesting example is the Lycurgus Cup which appears green when seen in reflected light and red in transmitted
light. It was understood later that the reason behind this phenomenon can be attributed to plasmon resonances associated with the gold and silver nanoparticles
contained in glass which forms the cup [6].
Understanding the connection between the change of bulk material properties
and structuring became possible after the construction of a diffraction grating in
1785 by D. Rittenhouse with 50 hairs positioned by the threads of two screws [7]
and later in 1821 by Joseph Von Fraunhofer who used almost the same technique
for constructing his wire diffraction grating [8]. Subsequent works include the theoretical study of a periodic stack of dielectric layers by Rayleigh [9] who realized
that a particular wavelength can be completely reflected using such a structure, and
an experiment carried out by Jagadis C. Bose with mm-wave [10]. Bose found that
even a simple book can have linearly polarizing properties which can be enhanced
by introducing metal foils in between the pages. While Bose’s experiment can be
attributed to the first systematic study of the electromagnetic properties of a composite medium, Rayleigh’s work is thought to be the first step towards the development of a new class of materials widely known as photonic crystals, where the term
’photonic crystals’ was first introduced in 1989 by Eli Yablonovitch [11]. Photonic
crystals form an important class of metamaterials having stop bands which resemble the band gap of semiconductor materials. Depending on their lattice structures
(which results from artificial structuring) these artificial media can inhibit spontaneous emission [12] or strongly confine photons [13]. Some of the very first applications of these include development of energy efficient LEDs [14] and photonic
crystal fibers [15].
1
The building block of a metamaterial is often termed as a meta-molecule which
is different from a material molecule in the sense that it displays unconventional
optical properties. Many of these meta-molecules are actually nano-size optical
resonators or resonant nanoantennas which can couple localized electromagnetic
fields as well as freely propagating radiation [16, 17]. Metal nanoparticles [18] sustaining surface plasmon-polariton and localized surface plasmon modes as well as
dielectric structures with high refractive indices [19, 20] can be employed as such
resonators. The most well known example of an optical resonator is the split-ring
resonator which was introduced by Hardy and Whitehead in 1981 [21]. In 1999, it
was realized by John Pendry et al. [22] that these split ring resonators can be used
to create artificial media with negative magnetic response. Within a year, David
Smith et al. demonstrated a split ring and wire metamaterial with simultaneously
negative permittivity and negative permeability [23]. These artificial media with
effective negative refractive index invoked the primary research interests in the field
of metamaterials and it is believed that these media can be useful in realizing optical
cloaking [5, 24–26], and super-resolution imaging [27].
Planar metamaterials are more appealing due to ease of their fabrication. Extensive research in the field of planar chiral metamaterials resulted in development
of wave plates of essentially zero thicknesses [28, 29], tunable polarization rotators,
and circular polarizers [30, 31]. Asymmetric structuring on subwavelength scale
can yield strong resonance which can trap energy on the surface of a planar metamaterial and can cause a transparent metamaterial state which is identical to electromagnetically induced transparency [32–36]. Furthermore, planar metamaterials
with optical gain media may enable lasing [37, 38]. Lastly, metasurfaces comprising
of arrays of nanoantennas with subwavelength separation between them and with
spatially varying geometric properties can be employed to shape optical wavefronts
very precisely within a distance much smaller than the wavelength of light. It is
believed that these metasurfaces will complement conventional imaging lenses in
near future [39].
Ever since the experimental demonstration of optical second harmonic generation by Franken et al. in 1961 [40], the field of nonlinear optics is continuously
blooming. Nonlinear optical processes are relatively weak in nature and strong efforts have been made to enhance these either by introducing new materials or by
enhancing light-matter interactions with the aid of nanostructuring. Increased effective nonlinear optical responses can be realized either through plasmonic resonances
in metallic nanostructures or geometrical resonances in all-dielectric subwavelength
structures.
Coupling of light to surface plasmons can produce strong local electromagnetic
fields [16,41,42] which boost the nonlinear processes. To exemplify, surface plasmon
modes at structured metal surface enhance the inherently weak Raman scattering up
to several orders of magnitude and may permit even single-molecule detection [43].
Plasmonic structures sustaining localized surface plasmons (LSP) or surface plasmon polaritons (SPP) can be used to achieve optically tunable optical properties
which rely on the mechanisms of optical Kerr nonlinearity. These SPP or LSP resonances enhance the effective refractive index change of either the metal particles or
the surrounding media by strong local field confinement and consequently lower
the light intensity required for observing nonlinearity induced changes. Examples include metal nanoparticles [44, 45], metal doped bulk media [46–48], nanorod
and nanosphere assemblies covered with nonlinear media [49, 50] etc. Due to LSP
enhanced local electric field, it has been possible also to realize photon tunnel2
ing through a nanometer size pinhole in a metal film covered with a nonlinear
polymer [51, 52]. Furthermore, enhanced nonlinear effects in plasmonic waveguides, such as metal-insulator-metal, V-groove, nano slot, or dielectric loaded waveguides [53–57], have been employed to efficiently modulate and switch the SPP signals by all-optical means. SPP or LSP based additional field enhancement effects
in these plasmonic waveguides help in achieving better figure of merit (the ratio of
modulation performance to size) as compared to the photonic waveguides [58].
Alternatively, nonlinear effects in semiconductors or dielectric materials can be
enhanced using waveguide resonances. Some of the most common examples include silicon based slot waveguides where the slots are filled with nonlinear polymer materials [59], silicon and silicon nitride ring resonators [60–62], slow-light
photonic crystal waveguides [63–66], and waveguide gratings [67–69]. Applications include electro-optic and all-optical modulation, all-optical wavelength conversion, all-optical multiplexing and demultiplexing, and supercontinuum generation [70, 71].
In recent years, the field of metamaterials has merged together with the field
of nonlinear optics to form a new class of artificial materials known as Nonlinear metamaterials [72]. These new unconventional materials have huge potential in
next-generation optical networks as one can tailor and tune their nonlinear optical
properties. The essence of constructing a nonlinear metamaterial lies in achieving
the desired nonlinear response by constructing a meta-molecule with specific linear
and nonlinear optical responses and arranging these meta-molecules in a specified
manner to yield their collective effects. Employing nonlinear metamaterials, it is
possible to enhance a particular nonlinear effect without changing its nature. One
way to achieve this is to increase the local field confinement in the region containing
the nonlinear material. In 1999, Pendry et al. [73] first suggested that introduction of
a nonlinear material into the gaps of spit ring resonators (SRR) might be beneficial to
enhance the nonlinear optical response through local electric field effects. Few years
later, it was shown [74] that such a trick produces nonlinearity in magnetic response
rather than the enhancement of electric nonlinearity. Alternatively, nonlinear process enhancement can be achieved through mutual interactions between specifically
ordered nanostructures. For example, second-harmonic generation due to surface
nonlinearity from an array of metal nanoparticles can be enhanced via better phase
matching conditions which can be accomplished by symmetry breaking with special
shaped structures and/or unit cells composed of these structures [75–80]. However,
surface defects arising from fabrication errors caused the experimental demonstrations of these effects challenging for quite a long time.
Another important class of nonlinear metamaterials employ self-action effects,
where the enhanced nonlinear properties also affect the linear properties of the
metamaterials. This may result in all-optical tuning, all-optical switching, bistability, mutistability, or modulation instability. Several theoretical and experimental
works on self-tunability of SRRs have been reported, where various nonlinear insertions into SRRs were introduced [81–84]. Intensity dependent nonlinear resistances
of these SRRs can tune a metamaterial slab between transmission, reflection, and
absorption states. Optical bistability in a resonant nanostructure occurs as a result
of the resonance peak shift with increasing power where at higher power levels
the peak becomes asymmetric. Most of the works related to optical bistability in
metamaterials to date are either theoretical or numerical [85–88].
Besides, introducing new metamaterial geometries, there has always been quest
for new materials which shows improved optical properties. One way to control the
3
material properties is by means of molecular engineering i.e. to intermix two or more
materials at the molecular level and hence form nanocomposite optical materials. In
many cases, these nanocomposites can display properties superior to those of their
constituents. Device level applications of these composite media are increasing in
number day by day [89].
Strong theoretical models and numerical techniques are indispensable to describe optical properties of micro and nanostructures in the subwavelength regime.
In many cases, effective medium theories are applied to evaluate the macroscopic
bulk properties of these structures. However, these theories can be accurate only in
the quasi-static regime i.e. when the feature sizes are much smaller than the wavelength of incoming light. Hence, if the smallest feature size approaches the wavelength, one must employ full-rigorous theories which solves James C. Maxwell’s
equations for electrodynamics [90, 91] either in the space-time/space-frequency domain or in the spatial frequency domain. In the nonlinear domain, full-wave numerical simulations are challenging as they require huge computing resources.
However, advanced numerical techniques need to be assisted by improved fabrication methodology and better experimentation techniques.
1.2
MOTIVATION
Modern nanophotonics aims at manipulating and controlling the optical properties
of bulk materials either by nanostructuring or by intermixing two or more homogeneous media at the nanoscale. The primary goal is to develop ultra-compact and
ultra-fast optical devices for fully functional photonic circuits that can be integrated
on chip with electronics [53, 92–94]. Since its development, the area of nanophotonics found a variety of applications in different spheres including optical data
storage [95, 96], super-resolution-imaging [97], bio and gas sensing [98–101], photovoltaics [102] and high performance optical computing [103].
Many areas of nanophotonics employ subwavelength resonant nanostructures.
Strong light-matter interactions can take place inside these structures due to enhanced local field [104] achieved either through the excitation of trapped electromagnetic modes or by localized resonances arising from the structure geometry [18].
These strong interactions may result in unusual macroscopic linear optical properties. Strongly confined local fields are crucial also for the intensity dependent
nonlinear optical processes, where the nonlinear response scales to the localized
field intensity, with a variety of applications ranging from frequency-conversion,
wave mixing, Raman scattering to all-optical switching. Besides the field amplitude,
structuring on the subwavelength scale allows also to control the phase and the
polarization properties of light and found useful in realization of compact polarizers [105], and wave plates [106]. However, to achieve precise control over these local
fields in optical nanostructures, one needs excellent harmony between numerical
modeling, fabrication and experimentation techniques. This invoked the motivation
for writing this thesis.
1.3
THESIS OUTLINE
This doctoral thesis is divided altogether in eight chapters. After this general introduction, Chapter 2 contains the fundamentals of electromagnetism. This chapter
covers from Macroscopic Maxwell’s equations to the concept of local fields in linear
4
and nonlinear optics. The subsequent chapters are based on the theories laid out in
Chapter 2.
In Chapter 3, we first include a general introduction to the periodically perturbed
media, working principle of a diffraction grating, and an overview of the existing
full-wave numerical simulation techniques for diffraction gratings. However, in the
subsequent sections, we emphasize the Fourier-Expansion Eigenmode Method or
simply the Fourier Modal Method (FMM) for modeling diffraction gratings. Implementations of the FMM for linear and multilayered two-dimensional (2D) gratings
and lastly for anisotropic crossed gratings (3D) are included.
In Chapter 4, we present an efficient and accurate numerical method to model
optical Kerr effect (OKE) in periodic structures. This FMM based technique accurately estimates local field enhanced optical Kerr nonlinearity in a periodic structure
and can be employed for design and analysis of low power all-optical devices.
In Chapter 5, we extend the classical theory of form birefringence to optical Kerr
nonlinear media. We develop an analytical model which can describe nonlinear
light-matter interactions in such a medium and verify its accuracy by comparing the
results obtained by this model and the full rigorous FMM based technique developed in Chapter 4.
In Chapter 6, we investigate theoretically, numerically, and experimentally the
possibility to achieve all-optical modulation and optical bistability with a waveguide
grating structure fabricated from silicon nitride thin film grown on top of fused silica substrate. FMM based numerical simulation results showing optical bistability,
fabrication methodology, and experimental results demonstrating all-optical modulation of transmitted signal are included.
In Chapter 7, we present a FMM based full wave numerical approach which can
be applied to accurately model percolation, arbitrary particle geometry and particle clustering inside a nanocomposite optical medium. The examples presented in
Chapter 7 include porous silicon nanocomposites, and metal-dielectric nanocomposites. Numerical experiments demonstrate the effect of local field on the linear
and nonlinear optical properties of these nanocomposite optical media.
Finally, we make the conclusions in Chapter 8 along with an outlook for future
research directions.
The modeling methodologies introduced in Chapter 4 and Chapter 7, the theoretical model presented in Chapter 5 and some of the theoretical as well as numerical
results included in Chapter 4, Chapter 5, and Chapter 7 are either published or
submitted for publication in peer-reviewed scientific journal articles:
• S. Bej, J. Tervo, Y. Svirko, and J. Turunen, “Modeling the optical Kerr effect in
periodic structures by the linear Fourier modal method," J. Opt. Soc. Am. B
31, 2371–2378 (2014).
• S. Bej, J. Tervo, Y. Svirko, and J. Turunen, “Form birefringence in Kerr media:
analytical formulation and rigorous theory," Opt. Lett. 40, 2913–2916 (2015) .
• S. Bej, T. Saastamoinen, Y. Svirko, and J. Turunen, “Optical properties of
nanocomposites from grating theory viewpoint," (Submitted) (2017).
Several journal articles related to the subjects covered in Chapter 6 and Chapter 7
are under preparation. Some of the results included in this thesis are also published
in the following conference proceedings:
5
• S. Bej, J. Tervo, Y. Svirko, and J. Turunen, “Fourier modal method for crossed
gratings with Kerr-type nonlinearity,” Proc. SPIE 9131, 161–169 (2014).
• S. Bej, J. Tervo, Y. Svirko, and J. Turunen, “All-optical control of form birefringence,” in 2015 European Conference on Lasers and Electro-Optics - European
Quantum Electronics Conference, (Optical Society of America, 2015), paper
CD_P_16.
• S. Bej, J. Tervo, J. Francés, Y. P. Svirko, and J. Turunen, “Analysis of all-optically
tunable functionalities in subwavelength periodic structures by the Fourier
modal method,” Proc. SPIE 9889(06), 1–9 (2016).
• S. Bej, J. Laukkanen, J. Tervo, Y. P. Svirko, and J. Turunen, “Optical bistability
in a Silicon Nitride waveguide grating,” Proc. SPIE 9894(0C), 1–10 (2016).
• J. Francés, S. Bleda, S. Bej, J. Tervo, V. Navarro-Fuster, S. Fenoll, F. J. MartínezGuardiolaa, and C. Neipp, “Efficient split field FDTD analysis of third-order
nonlinear materials in 2D periodic media,” Proc. SPIE 9889(08), 1–8 (2016).
6
2
Fundamentals of electromagnetic theory
The main subject of this thesis is the study of local field controlled linear and Kerr
nonlinear optical properties of subwavelength nanostructures and nanocomposites.
In the beginning, it is necessary to understand the basic principles of electromagnetism. This chapter deals with the basics of the electromagnetic theory for freespace optics. The more advanced topics will be covered in the following chapters.
2.1
COMPLEX REPRESENTATION OF ELECTROMAGNETIC FIELD
QUANTITIES AND FOURIER DECOMPOSITION
After James Clerk Maxwell realized that light is of electromagnetic origin, it was
essential to develop an accurate mathematical model that can describe the propagation of light and also fits well with the available experimental results. In physics,
all the field quantities that can be measured are real valued functions of space and
time. Nevertheless, in linear optics, to simplify the underlying mathematics we can
employ complex representation of the electromagnetic field quantities. Using this
complex notation, a monochromatic time harmonic stationary field can be described
as
Are (r, t) = ℜ{ A (r ) exp (−iωt)} ,
(2.1)
where A(r ) is the complex amplitude of the real valued function Are (r, t) and can be
associated with any measurable field quantity. r, t, and ω denote the position vector,
time instant, and the angular frequency of the time harmonic field respectively.
However, to describe the behavior of natural polychromatic light rigorously, we can
not use the above mentioned approach. In such a scenario, we can define a unique
complex counterpart of the real valued field Are (r, t) by introducing the concept of
temporal Fourier integral [107],
Are (r, t) =
Z ∞
−∞
Ãre (r, ω ) exp (−iωt) dω.
(2.2)
where we assume that Are (r, t) is square integrable with respect to time i.e.
Z ∞
−∞
Clearly,
Ãre (r, ω ) =
| Are (r, t)|2 dt < ∞.
1
2π
Z ∞
−∞
Are (r, t) exp (iωt) dt,
(2.3)
(2.4)
where Ãre (r, ω ) represents a space-frequency domain spectral amplitude of the real
valued field Are (r, t). Equations (2.2) and (2.4) show that any space-time domain
vector field can be expressed as a superposition of time harmonic fields with spectral
complex-amplitudes Ãre (r, ω ). Also, since Are (r, t) is real, its Fourier components
Ãre (r, ω ) satisfy the condition
∗
Ãre (r, −ω ) = Ãre (r, ω )
(2.5)
7
where the ∗ symbol is used to represent complex conjugate. The above mentioned
condition unveils that the negative frequency components do not carry additional
information which is not already contained in the positive ones. Hence, without
loss of generality we can define a new space-frequency domain function
(
0,
if ω < 0
Ã(r, ω ) =
(2.6)
2 Ãre (r, ω ) if ω ≥ 0,
where the Fourier conjugate i.e. the complex-valued space-time domain function is
defined as
Z ∞
A(r, t) =
Ã(r, ω ) exp (−iωt) dω.
(2.7)
−∞
Clearly, from Eq. (2.7), the positive part of the spectrum differs from that of its real
counterpart only by a multiplicative constant and the Fourier spectrum of the newly
defined function is written as
à (r, ω ) =
1
2π
Z ∞
−∞
A(r, t) exp (iωt) dt,
(2.8)
and can be attributed to any measurable physical quantity. Complex representation
for scalar fields can be introduced in a similar fashion.
2.2
MACROSCOPIC MAXWELL’S EQUATIONS AND THEIR EMPIRICAL BASIS
J. C. Maxwell’s theory of electricity and magnetism [90, 91] gives the fundamental
laws of electromagnetism. With the aid of these laws, which are based upon four
equations, electromagnetic field quantities can be treated classically. In principle,
electromagnetic fields in any media (which do not contain any abrupt boundary)
and in any form can be solved with these equations which are widely known as
Maxwell’s equations. In space-time domain, Maxwell’s equations can be represented
as a set of four partial differential equations written as
∂
D (r, t),
∂t
∂
∇ × E(r, t) = − B(r, t) ,
∂t
∇ · D(r, t) = ρ(r, t) ,
∇ · B(r, t) = 0 ,
∇ × H (r, t) = J (r, t) +
(2.9)
(2.10)
(2.11)
(2.12)
where E(r, t), H (r, t), D(r, t), B(r, t), J (r, t), and ρ(r, t) are the electric field, the
magnetic field, the electric displacement, the magnetic induction, the electric current
density and the electric charge density, respectively. These four equations connect
all the measurable electromagnetic field quantities. Each of these equations represents generalization of certain experimental results. Equation (2.9) is an extension
of Ampere’s law, Eq. (2.10) is Faraday’s law of electromagnetic induction in differential form, Eq. (2.11) is Gauss’s law which can be derived from Coulomb’s law,
and Eq. (2.12) signifies the fact that magnetic monopoles do not exist. Clearly, as
Maxwell’s equations are mathematical expressions of certain experimental results,
these can not be proved; however the applicability of these in any condition can be
validated.
8
Now without any loss of generality, if we make an assumption that all the
measurable electromagnetic field quantities are time-harmonic i.e. of the form of
Eq. (2.1), we can derive a new set of Maxwell’s equations in the space-frequency
domain
∇ × H (r, ω ) = J (r, ω ) − iωD(r, ω ) ,
∇ × E(r, ω ) = iωB(r, ω ) ,
∇ · D (r, ω ) = ρ(r, ω ) ,
∇ · B(r, ω ) = 0 .
(2.13)
(2.14)
(2.15)
(2.16)
Equations (2.13)-(2.16) are useful because often in optics it is more convenient to
operate in space-frequency domain than in space-time domains.
2.3
MATERIAL CONSTITUTIVE EQUATIONS
Both the space-time and the space-frequency domain Maxwell’s equations contain
more than four unknown field quantities. Hence, to solve these, we need to introduce additional equations which connect these quantities. By introducing two new
space-time domain field quantities known as the electric polarization P (r, t) and the
magnetization M (r, t), we may write these additional equations in the form
D (r, t) = ε 0 E(r, t) + P (r, t),
1
H (r, t) =
B(r, t) − M (r, t),
µ0
(2.17)
(2.18)
where ε 0 is the electric permittivity of vacuum and µ0 is the magnetic permeability of vacuum. Equations (2.17) and (2.18) connect the space-time domain electric
displacement to the electric field and the space-time domain magnetic field to magnetic induction, respectively. Both the electric polarization and the magnetization
are nonlinear functions of the electric and the magnetic fields. However, at optical
frequencies, magnetization is typically very small even if the field strength is large.
Hence it can often be neglected. However, if intense laser illumination is used,
electric polarization must generally be written in the form of a series expansion
P (r, t) = χ (1) (r ) E(r, t) + χ(2) (r ) E2 (r, t) + χ(3) (r ) E3 (r, t) + . . .
(2.19)
where the position-dependent constants χ ( j) are susceptibilities of different orders.
In the domain of linear optics, it is sufficient to retain only the first term in the
right-hand-side of Eq. (2.19). The second and third terms give rise to nonlinear
effects of second and third order, respectively. Usually at moderate field intensities,
these higher order terms are negligible and the space-time domain relation between
the electric polarization and the electric field in this context of linear optics can be
written as
Z ∞
ε
P (r, t) = 0
χ(r, t′ ) E(r, t − t′ ) dt′ ,
(2.20)
2π 0
where χ(r, t′ ) is the position dependent real-valued dielectric susceptibility tensor.
In an isotropic medium, the susceptibility assumes a scalar value (which still depends on r) and can be written as
χ(r, t′ ) = χ(r, t′ ) I,
(2.21)
9
where I is the identity matrix. The space-time domain electric current density, and
the electric displacement are also connected to the space-time domain electric field
by equations analogous to Eq. (2.20)
∞
1
σ (r, t′ ) E(r, t − t′ ) dt′ ,
2π 0
Z
ε0 ∞
D(r, t) =
ε (r, t′ ) E(r, t − t′ ) dt′ ,
2π 0
J (r, t) =
Z
(2.22)
(2.23)
and as we saw before, for isotropic media σ (r, t′ ) and ε(r, t′ ) reduce to scalar conductivity σ (r, t′ ) and scalar permittivity ε(r, t′ ) respectively. Together with the spacetime domain Maxwell’s equations (2.9)–(2.12), equations (2.20), (2.22), and (2.23)
provide the desired relations between the electromagnetic field quantities in the
space-time domain. We can derive similar expressions in the space-frequency domain for non-magnetic media by taking the Fourier transforms of equations (2.20),
(2.22), (2.23), (2.18) and using the convolution theorem [108]
P (r, ω ) = ε 0 χ(r, ω ) E(r, ω ),
J (r, ω ) = σ (r, ω ) E(r, ω ),
(2.24)
(2.25)
D(r, ω ) = ε 0 ε(r, ω ) E(r, ω ),
B(r, ω ) = µ0 H (r, ω ),
(2.26)
(2.27)
where ε(r, ω ) is defined as the relative permittivity tensor. Clearly, the spacefrequency domain equations are easier to handle as these avoid the complicacy of
performing integrations.
Equations (2.24)–(2.27) are referred as the space-frequency domain material constitutive equations. In space-time domain the electric current density and the electric
polarization are related by
∂
J (r, t) = P (r, t).
(2.28)
∂t
Using Eqs. (2.17), (2.25), (2.26), (2.28) and applying the Fourier transform, we can
derive the relationship between the relative complex permittivity (written as ε̂ (r, ω ))
and the electric conductivity in the form
ε̂ (r, ω ) = ε(r, ω ) +
i
σ (r, ω ),
ε0ω
(2.29)
which includes the effects of both the free electrons and the bound electrons. For an
isotropic linear medium the complex refractive index can be defined as
q
n̂(ω ) = n(ω ) + iκ (ω ) = ε̂(r, ω ),
(2.30)
where both n(ω ) and κ (ω ) are real functions, n(ω ) determines the phase velocity
of the propagating electromagnetic wave, and κ (ω ) determines the strength of its
damping or attenuation.
Using the complex relative permittivity as defined in Eq. (2.29) and making use
of the space-frequency domain constitutive relations Eqs. (2.24)–(2.27), Eq. (2.13) can
be written in the form
∇ × H (r, ω ) = −iωε 0 ε̂(r, ω ) E(r, ω ).
10
(2.31)
Furthermore taking divergence of Eq. (2.31) from the left and employing the vector
identity ∇ · ∇ × A = 0, we get
∇ · [ε̂(r, ω ) E(r, ω )] = 0 .
(2.32)
∇ × H (r, ω ) = −iωε 0 ε̂(r, ω ) E(r, ω ) ,
∇ × E(r, ω ) = iωB(r, ω ) ,
∇ · [ε̂(r, ω ) E(r, ω )] = 0 ,
∇ · B(r, ω ) = 0 .
(2.33)
(2.34)
Now we may rewrite the space-frequency domain Maxwell’s equations in the following form,
(2.35)
(2.36)
Much of our discussions below will be based on the set of equations (2.33)-(2.36).
2.4
ELECTROMAGNETIC BOUNDARY CONDITIONS
The validity of Maxwell’s equations in differential form at position r is based on the
assumption that the medium in the immediate vicinity of r is continuous. Clearly,
the above mentioned criteria for continuity is no longer valid at the boundary between two media. However, Maxwell’s equations in integral form still remain valid.
As throughout this thesis we will deal with Maxwell’s equations only in differential form, we need to find additional conditions that describe the field properties at
an abrupt boundary between two media. These conditions are known as electromagnetic boundary conditions [109–113]. Electromagnetic boundary conditions are
essential to describe many interesting optical phenomena that occur at boundaries
between two media such as reflection, refraction, and scattering.
Considering a sharp boundary between two media denoted by 1 and 2, and
introducing û12 as the unit surface normal vector at position r which points from
medium 1 to medium 2, we can write the space-frequency domain electromagnetic
boundary conditions in the following form
û12 (r ) · [ε̂2 (r, ω ) E2 (r, ω ) − ε̂1 (r, ω ) E1 (r, ω )] = 0 ,
û12 (r ) · [ B2 (r, ω ) − B1 (r, ω )] = 0 ,
û12 (r ) × [ E2 (r, ω ) − E1 (r, ω )] = 0 ,
1
û12 (r ) × [ B2 (r, ω ) − B1 (r, ω )] =
J (r, ω ) ,
µ0 S
(2.37)
(2.38)
(2.39)
(2.40)
where J S (r, ω ) is the surface current density at position r. It is worth mentioning
that at optical frequencies the surface current density is usually zero. However, in
case of an infinitely conducting material it must be set to non-zero value. Equations (2.37)–(2.40) signifies that across the discontinuities between two dielectrics
or finitely conducting materials, all the magnetic field components, the tangential
electric field components, and the normal components of the electric displacement
remain continuous.
2.5
WAVE EQUATION
One of the most important consequences of Maxwell’s equations is the equation
for the electromagnetic wave propagation. For a homogeneous, isotropic, sourcefree linear optical medium, the complex relative permittivity tensor ε̂(r, ω ) reduces
11
to scalar complex permittivity ε̂(r, ω ). Hence, we can apply the curl operator to
Eq. (2.34) and use Eqs. (2.33) and (2.27) to obtain
∇ × [∇ × E(r, ω )] = ω 2 ε 0 ε̂(ω ) E(r, ω ).
(2.41)
∇2 E(r, ω ) + k20 ε̂(ω ) E(r, ω ) = 0,
(2.42)
Now we use the vector identity ∇ × (∇ × A ) ≡ ∇(∇ · A ) − ∇2 A and use Eq. (2.35)
to get the wave equation for the electric field (also known as the Helmholtz wave
equation for the electric field) in the space-frequency domain
where, λ0 is the vacuum wavelength, c is the speed of light in vacuum, and k0 =
2π/λ0 = ω0 /c is the free-space wave number. Similarly, proceeding with Eq. (2.33)
we can derive the Helmholtz wave equation for the magnetic field
∇2 H (r, ω ) + k20 ε̂(ω ) H (r, ω ) = 0.
(2.43)
In the above derivations we have assumed the medium to be isotropic, linear and
source-free.
Let’s now try to derive the wave equation for nonlinear media. We begin with
the space-time domain Maxwell’s equations (2.9)–(2.12). As we are interested in the
solution in regions of space which contain no free charges and free currents, we
may put ρ = 0 and J = 0 in the space-time domain Maxwell’s equations. Also, we
assume non-magnetic media hence M in Eq. (2.18) becomes zero and we can rewrite
Eq. (2.18) in the following form
B = µ0 H.
(2.44)
We now proceed to derive the wave-equation as in the linear case. Taking curl
of Eq. (2.10), interchanging the order of the space-domain and the time-domain
derivatives and using Eqs. (2.9), (2.44), and (2.17) we get the most general form of
the wave-equation in nonlinear optics
∇×∇×E+
1 ∂2
1 ∂2 P
E
=
−
,
c2 ∂t2
ε 0 c2 ∂t2
(2.45)
where on the right hand side of Eq. (2.45) we have replaced µ0 by 1/ε 0 c2 . Under certain conditions the above written generalized form of the nonlinear wave equation
can be simplified by using the vector identity ∇ × (∇ × E) ≡ ∇(∇ · E) − ∇2 E and
putting ∇ · E = 0 as in the case of isotropic linear media. For a transverse, infinite
plane wave and for pulsed light under slowly varying amplitude approximation
∇ · E vanishes and we can write the nonlinear wave equation in the following form
∇2 E −
1 ∂2
1 ∂2 P
E
=
.
c2 ∂t2
ε 0 c2 ∂t2
(2.46)
Now, we can split P and D into linear and nonlinear counterparts such that
P = PL + PNL ,
L
D= D +D
NL
,
(2.47)
(2.48)
where DL (r, t) = ε 0 E(r, t) + PL (r, t) = ε 0 E(r, t) + χ(1) (r ) E(r, t). Hence the nonlinear
wave equation (2.46) can be rewritten as
∇2 E −
12
1 ∂2 L
1 ∂2 PNL
D
=
.
ε 0 c2 ∂t2
ε 0 c2 ∂t2
(2.49)
For an isotropic, lossless, dispersionless medium εL reduces to a scalar quantity εL
and Eq. (2.49) reduces to
∇2 E −
ε L ∂2 E
1 ∂2 PNL
=
.
ε 0 c2 ∂t2
ε 0 c2 ∂t2
(2.50)
Clearly, Eq. (2.50) has the form of an inhomogeneous driven wave equation where
the nonlinearity of the medium acts as a source term. In the absence of this source
term Eq. (2.50) reduces to the linear wave-equation (for electric field) for isotropic
medium.
2.6
TE-TM DECOMPOSITION
Let us assume that the field quantities in Maxwell’s equations (2.33)-(2.36) are yinvariant. Also, we assume that the permittivity distribution is y-invariant. Hence
all the partial derivatives with respect to y vanish in Maxwell’s equations. Now, if
we further assume that the incident field is propagating in the xz-plane, we can split
the Maxwell’s equations (2.33)–(2.36) into two sets of equations
r
i
ε0 ∂
Hx ( x, z) =
Ey ( x, z),
(2.51)
k0 µ0 ∂z
r
i
ε0 ∂
Hz ( x, z) = −
Ey ( x, z),
(2.52)
k0 µ0 ∂x
r
∂
ε0
∂
Hx ( x, z) −
Hz ( x, z) = −ik0 ε̂( x, z)
Ey ( x, z),
(2.53)
∂z
∂x
µ0
and
r
i
ε0 ∂
Hy ( x, z),
k0 ε̂ ( x, z) µ0 ∂z
r
i
ε0 ∂
Ez ( x, z) =
Hy ( x, z),
k0 ε̂ ( x, z) µ0 ∂x
r
∂
ε0
∂
Ey ( x, z) −
Ez ( x, z) = ik0
Hy ( x, z).
∂z
∂x
µ0
Ex ( x, z) = −
(2.54)
(2.55)
(2.56)
Clearly, the first set includes only the y-component of the electric field (the component of the electric field normal to the plane of incidence) and the x, z components
of the magnetic field. Hence, this set corresponds to transverse electric or TE polarization. Analogously, the second set is termed as transverse magnetic or TM
polarization. Substituting Eqs. (2.51) and (2.52) into Eq. (2.53) we can obtain a single
partial differential equation for the TE polarized set
∂2
∂2
Ey ( x, z) + 2 Ey ( x, z) + k20 ε̂( x, z) Ey ( x, z) = 0.
(2.57)
2
∂x
∂z
Similarly proceeding with the TM polarized set we can derive the single partial
differential equation for the TM polarized set
∂
∂
∂
1
∂
1
Hy ( x, z) +
Hy ( x, z)
∂x ε̂ ( x, z) ∂x
∂z ε̂( x, z) ∂z
+k20 ε̂( x, z) Hy ( x, z) = 0.
(2.58)
We shall discuss in detail the polarization properties of an electromagnetic wave in
section 2.9.
13
2.7
ELECTROMAGNETIC ENERGY QUANTITIES
Though we introduced complex notations for the electromagnetic field quantities,
measurable field quantities are always real valued. We connect the complex quantities with the real measurable field quantities by defining the concept of the energy
of the electromagnetic field. The electric energy density we (r, t), and the magnetic
energy density wm (r, t) are defined by the following relations [114]
1
Ere (r, t) · Dre (r, t),
2
1
wm (r, t) = H re (r, t) · Bre (r, t).
2
we (r, t) =
(2.59)
(2.60)
The total instantaneous energy density of the electromagnetic field is defined as the
sum of these two quantities. As the frequencies in the optical part of the spectrum is
very high ( 1015 s−1 ) and the response times of the available photo-detectors are several orders of magnitude larger than the fluctuations of the field quantities, we can
only measure the time-averaged signal [115], where we define the time-averaging of
a function f (t) by
Z T
1
h f (t)i = lim
f (t)dt.
(2.61)
T → ∞ 2T − T
The time-averaged electromagnetic energy densities which can be measured by the
available photo-detectors are given by
1
ε 0 ε(r )| E(r )|2 ,
4
1
hwm (r, t)i = µ0 | H (r )|2 .
4
hwe (r, t)i =
(2.62)
(2.63)
For plane wave illumination we define the direction of the energy flow by the direction of the Poynting vector S (r, t) = Ere (r, t) × H re (r, t). As usual, we can define the
time-averaged Poynting vector by
hS(r, t)i =
1
ℜ [ E(r ) × H ∗ (r )]
2
(2.64)
whose magnitude quantifies the intensity of the field. The superscript ∗ denotes
the complex conjugate. The time-averaged Poynting vector can be used to measure
the field intensity also for the non-plane wave illumination. Nevertheless, we must
take extra care in defining the direction of the energy flow in such scenario as there
exist cases where the z-component of the time-averaged Poynting vector might take
negative values in some small but finite regions [116].
2.8
ELECTROMAGNETIC PLANE WAVE
The simplest solution of Maxwell’s equations is the plane wave solution i.e. a field
with planar surface of constant phase. In the space-frequency domain, the plane
wave solutions for the electric field and the magnetic field take the following forms
E(r, ω ) = E0 (ω ) exp (ik · r ),
H (r, ω ) = H 0 (ω ) exp (ik · r ),
14
(2.65)
(2.66)
where E0 (ω ) and H 0 (ω ) are the vectorial complex electric and magnetic field amplitudes respectively, and k is the wave vector which defines the normal direction to
the plane of constant phase and also the propagation direction of the plane wave. In
cartesian coordinate system k is defined as k = k x x̂ + k y ŷ + k z ẑ with |k| = k = k0 n.
2.9
POLARIZATION OF AN EM PLANE WAVE
Like other vector fields, the electromagnetic wave has certain directional properties.
The directional information of an electromagnetic wave is characterized by its polarization [117]. For the most simple case of a plane wave propagating along zdirection, we can solve Eq. (2.42) i.e. the Helmholtz equation for the electric field in
the space-frequency domain and express the electric field as,
E(r, ω ) = { E0s ŝ exp [iφs (ω )] + E0p p̂ exp iφ p (ω ) } exp(ikz),
(2.67)
where ŝ and p̂ are unit vectors along two mutually orthogonal directions, E0s (alternatively E0y ) and E0p (alternatively E0x ) are the amplitudes of the electric field
components along ŝ (or ŷ) and p̂ (or x̂) respectively. φs (alternatively φy ) and φ p (alternatively φx ) are the phases of the electric field components vibrating along ŝ- and
p̂- directions respectively. The term polarization in this context is used to describe
the relations between the amplitudes and the phases of the electric field components in two mutually orthogonal directions. If, δ = φ p − φs , which is defined as
the relative phase, is a multiple of π or one of the field components is zero, we have
linearly polarized light. In the most general case for non zero field components
and arbitrary values of δ, we have elliptically polarized light. In a special situation,
when E0s = E0p , and δ = ±π/2 ± 2mπ, where m is an integer, we have circularly
polarized light. As the polarized light propagates in an optical medium, the tip of
the electric field vector traces out a specific characteristic form. In the most general
case, this trace is an ellipse as shown in Fig. 2.1.
A convenient way of representing polarized light was invented by an American
physicist R. Clark Jones in 1941 [118]. The technique prescribed by Jones is concise
and can easily be applied to coherent beams. If Ex and Ey are scalar components of
the electric field, Jones’ matrix can be written as,
E (t)
E(t) = x
.
(2.68)
Ey ( t )
Preserving the phase information of the electric field components we can write
Eq. (2.68) in the following form
E0x (t) exp (iφx )
E(t) =
.
(2.69)
E0y (t) exp iφy
Now, we can express the horizontal (TM) and the vertical (TE) polarized light using
Jones’ formalism as
E (t) exp (iφx )
E H = 0x
,
(2.70)
0
0
.
EV =
(2.71)
E0y (t) exp iφy
15
Letting E0x = E0y and φx = φy we get,
1
E = E0x exp (iφx )
.
1
(2.72)
This represents +45◦ polarized light. However, it is customary to normalize the
Jones’ vector demanding | Ex |2 + | Ey |2 = 1. After normalization we get the following
form of Jones’ vector for +45◦ polarized light
1 1
E45 = √
.
(2.73)
2 1
Similarly, the normalized Jones vectors for H-polarized, V-polarized, left-circularly
polarized (LCP) and right-circularly polarized (RCP) light can be written as
1 1
1
1
0
1
EH =
, EV =
, E LCP = √
, E RCP = √
.
(2.74)
0
1
i
−
i
2
2
Here we must mention that describing the polarization properties for a general
electromagnetic field is more demanding task and is out of scope for this thesis.
2.10
PLANE WAVE AT PLANAR BOUNDARY
Let’s now consider the simplest optical boundary-value problem of a plane wave
incident obliquely on a planar interface as shown in Fig. 2.2. The boundary or the
interface is defined by the plane z = 0. The material to the left i.e. in the half space
z < 0 has real valued permittivity ε̂ 1 whereas the permittivity of the medium to the
y′
ŝ (ŷ)
Es
δ=0
δ = π/2
x′
Ep
p̂ ( x̂)
δ=π
Figure 2.1: Polarization ellipse i.e. the trace of the tip of the electric field vector
E (r ) as a function of time. The propagation direction of the electromagnetic plane
wave is along the normal to the plane of this paper and points towards the reader.
The traces for δ = 0 and δ = π are linearly polarized.
16
right of the boundary (z > 0) ε̂ 2 might be complex. We assume that the incident
wave is partly reflected and partly transmitted at the boundary. θin is the angle of
incidence, where the subscript ’in’ stands for incidence. Similarly, the quantities
with subscripts ’ref’ and ’tra’ denote the reflected and the transmitted field quantities respectively. Furthermore, we assume that the reflected and the transmitted
fields are also plane waves. Now, omitting the explicit frequency dependence of the
field quantities at the boundary for the sake of brevity we may write the plane wave
components at the boundary in the following form
Ein (r ) = A exp [i(kin,x x + kin,z z)] ,
Bin (r ) = B0,in exp [i(kin,x x + kin,z z)] ,
(2.75)
(2.76)
Eref (r ) = R exp [i(kref · r )] ,
Bref (r ) = B0,ref exp [i(kref · r )] ,
(2.77)
(2.78)
Etra (r ) = T exp [i(ktra · r )] ,
Btra (r ) = B0,tra exp [i(ktra · r )] .
(2.79)
(2.80)
Hence, the total electric field just before the boundary and just after the boundary
can be written as
E( x, y, 0−) = A exp [i(kin,x x )] + R exp [i(k x,ref x )] ,
(2.81)
E( x, y, 0+) = T exp [i(k x,tra x )] .
(2.82)
x
kref
ktra
θref
θtra
z
θin
kin
ε̂ 1 = ℜ [ε̂ 1 ]
ε̂ 2
Figure 2.2: Geometry of a plane wave incident obliquely on a planar interface
between two media.
17
Putting equations (2.81) and (2.82) into the electromagnetic boundary conditions
(2.37) and (2.39) and using the uniqueness of the Fourier transform we get
k x,in = k x,ref = k x,tra = k x ,
(2.83)
k y,ref = k y,tra = 0,
(2.84)
A x + R x = Tx ,
Ay + Ry = Ty ,
(2.85)
(2.86)
ε̂ 1 Az + ε̂ 1 Rz = ε̂ 2 Tz .
(2.87)
The relations between the components of the magnetic induction and the electric
field components are obtained by inserting Eqs. (2.75)–(2.79) into Eq. (2.14) and can
be written as
B0,x,p = −
1
1
1
k z,p Wy , B0,y,p = (k z,p Wx − k x,pWz ), B0,z,p = k x,p Wy ,
ω
ω
ω
(2.88)
where W can be A, R or T, and p may denote the incident (in), reflected (ref) or the
transmitted (tra) field. Now we can apply Eq. (2.88) into the two other electromagnetic boundary conditions i.e. Eqs. (2.38) and (2.40) to get
−ω JS,0,x + k z,in Ay + k z,ref Ry = k z,tra Ty ,
(2.89)
k x,in Ay + k x,ref Ry = k x,tra Ty ,
(2.91)
ω JS,0,y + k z,in A x − k x,in Az + k z,ref R x − k x,ref Rz = k z,tra Tx − k x,tra Tz ,
(2.90)
where JS,0,x and JS,0,y are the components of the complex amplitude vector of the
surface current density. Eq. (2.83) and the dispersion relations for the wave vectors
give
k2x + k2z,in = k20 ε̂ 1 ,
(2.92)
k2x
k2x
(2.93)
+ k2z,ref
+ k2z,tra
=
=
k20 ε̂ 1 ,
k20 ε̂ 2 ,
(2.94)
Consequently k z,ref = ±k z,in . By examining the propagation directions in Fig. 2.2,
we may conclude that k z,ref < 0, k z,in > 0, and k z,tra > 0. Hence,
k2z,tra
− k2z,in
k z,ref = −k z,in ,
=
k20 (ε̂ 2
− ε̂ 1 ).
(2.95)
(2.96)
Also, by examining the definitions of angles in Fig. 2.2 and using Eqs. (2.83), (2.95),
and (2.96), we find that
θref = −θin ,
n̂1 sin θin = n̂2 sin θtra .
(2.97)
(2.98)
Equations (2.97) and (2.98) are the well known law of reflection and the Snell’s law,
respectively which connect the propagation angles of the reflected θref (or −θ1 ) and
transmitted waves θtra (or θ2 ) to the input angle θin (or θ1 ). We must note that if n̂2
is not real, then necessarily ℑθtra 6= 0. Now we may use Eqs. (2.95) and (2.83) to
18
rewrite the boundary value conditions i.e. Eqs. (2.85)–(2.91) in the following form
A x + R x = Tx ,
Ay + Ry = Ty ,
(2.99)
(2.100)
ε̂ 1 Az + ε̂ 1 Rz = ε̂ 2 Tz ,
−ω JS,0,x + k z,in ( Ay − Ry ) = k z,tra Ty ,
(2.101)
(2.102)
Ay + Ry = Ty .
(2.104)
ω JS,0,y + k z,in ( A x − R x ) − k x,in ( Az + Rz ) = k z,tra Tx − k x,tra Tz ,
(2.103)
Clearly, Eqs. (2.100), (2.102), (2.104) belong to the TE polarized set and Eqs. (2.99),
(2.101), (2.103) belong to the TM polarized set.
The wave vectors for the TE and the TM polarized cases are illustrated in Figs. 2.3
and 2.4 respectively. We apply Maxwell’s divergence equation to Eqs. (2.101) and
x
kref
B0,ref,TE
RTE
T TE
θ2
ktra
B0,tra,TE
z
θ1
kin
ATE
B0,in,TE
ε̂ 1 = ℜ [ε̂ 1 ]
ε̂ 2
Figure 2.3: Direction of the field vectors for TE polarized light.
(2.103) to get
k z,tra ε̂ 1 ( A x − R x ) = k z,in ε̂ 2 Tx ,
k z,in k z,tra ω JS,0,y + k z,tra ε̂ 1 ( A x − R x ) = k z,in ε̂ 2 Tx .
(2.105)
(2.106)
Hence, we conclude JS,0,y = 0 i.e. in TM polarization there is no surface current. For
TE polarization we may assume JS,0,x = 0 as long as external currents are absent.
Using Eqs. (2.99) and (2.105) we can derive the following equations which are the
19
transmission and the reflection equations for the TM polarized field
2k z,tra ε̂ 1
2n̂1 cos θ2
Ax =
Ax ,
k z,tra ε̂ 1 + k z,in ε̂ 2
n̂1 cos θ2 + n̂2 cos θ1
k z,tra ε̂ 1 − k z,in ε̂ 2
n̂ cos θ2 − n̂2 cos θ1
Rx =
Ax .
Ax = 1
n̂1 cos θ2 + n̂2 cos θ1
k z,tra ε̂ 1 + k z,in ε̂ 2
(2.107)
Tx =
(2.108)
Inspecting Fig. 2.4 we find out that ATM = sec θ1 A x , TTM = sec θ2 Tx , and RTM =
sec θ1 R x . Hence, the complex reflection and transmission amplitude coefficients
which are also known as Fresnel’s transmission and reflection coefficients are written as
TTM
2n̂1 cos θ1
=
,
ATM
n̂1 cos θ2 + n̂2 cos θ1
R
n̂ cos θ2 − n̂2 cos θ1
= TM = 1
.
ATM
n̂1 cos θ2 + n̂2 cos θ1
tTM =
(2.109)
rTM
(2.110)
Starting from Eqs. (2.100) and (2.102) and proceeding in a similar way we can derive
the Fresnel’s coefficients for the TE polarized set
TTE
2n̂1 cos θ1
,
=
n̂1 cos θ1 + n̂2 cos θ2
ATE
R
n̂ cos θ1 − n̂2 cos θ2
= TE = 1
.
n̂1 cos θ1 + n̂2 cos θ2
ATE
tTE =
(2.111)
rTE
(2.112)
The transmitted and the reflected efficiencies i.e. the normalized transmitted and
reflected energy are defined by noting that the direction of energy flow is along zx
kref
RTM
T TM
B0,ref,TM
θ2
ktra
B0,tra,TM
z
θ1
ATM
kin
B0,in,TM
ε̂ 1 = ℜ [ε̂ 1 ]
ε̂ 2
Figure 2.4: Direction of the field vectors for TM polarized light.
20
direction and hence considering the z- component of the spectral Poynting vector
which is defined as
ε ck z
Pz = 0
| E0 |2 ,
(2.113)
2k0
we can derive the expressions for the transmitted and the reflected efficiencies as
Pref,z,TM = |rTM |2 ,
RTM = (2.114)
Pin,z,TM RTE = |rTE |2 ,
k z,tra TTM 2
n̂ cos θ2
TTM =
= 2
|tTM |2 ,
n̂1 cos θ1
k z,in ATM k z,tra TTE 2
n̂ cos θ2
TTE =
= 2
| t |2 .
k z,in ATE n̂1 cos θ1 TM
(2.115)
(2.116)
(2.117)
The transmitted efficiency is meaningful only if n̂2 is real. Otherwise we have absorption and the field rapidly decays upon propagation. However, the reflected
efficiency still remains meaningful as we assumed n̂1 to be real.
2.11
GENERAL FIELD AND ANGULAR SPECTRUM REPRESENTATION
We already saw that the electric and the magnetic field vectors of a plane wave satisfies, in a homogeneous medium, the Helmholtz equations (2.42) and (2.43). We now
proceed to show that the general solutions of these equations can be represented
as superpositions of plane waves. Let us now introduce a Fourier transform pair
between the space-frequency domain and the wave vector-frequency domain (also
known as the spatial-frequency domain) electric field vectors
E(r, ω ) =
1
4π 2
ZZ ∞
−∞
Ẽ(k x , k y , z, ω ) =
Ẽ(k x , k y , z, ω ) exp i(k x x + k y y) dk x dk y ,
ZZ ∞
−∞
E(r, ω ) exp −i(k x x + k y y) dx dy.
(2.118)
(2.119)
If we now insert Eq. (2.118) into the Helmholtz equation for the electric field viz.
Eq. (2.42), we get
∂2
Ẽ(k x , k y , z, ω ) + k2z Ẽ(k x , k y , z, ω ) = 0,
(2.120)
∂z2
where
(
[k2 − k2x − k2y ]1/2 , if k2x + k2y ≤ k2 ,
(2.121)
Kz =
i[k2x + k2y − k2 ]1/2 , if k2x + k2y > k2 .
A general solution of Eq. (2.120) may be written in the following form
Ẽ(k x , k y , z, ω ) = U + (k x , k y , ω ) exp [ik z (z − z0 )] + U − (k x , k y , ω )
× exp [−ik z (z − z0 )] ,
(2.122)
where z = z0 is the reference plane. Depending on the conditions written in
Eq. (2.121) the solutions of Eq. (2.120) can be divided into two groups. The group
21
having real values of k z represents homogeneous waves. Whereas the group containing imaginary values of k z represents evanescent waves. The ’±’ signs in the subscripts represent waves propagating along +ẑ and along −ẑ respectively. Though
the evanescent waves rapidly decay exponentially with the propagating distance in
subwavelength plasmonic systems, as we will see later, they have significant roles.
Using the known value of the field at z = z0 and assuming that there is no source
in the positive half space we can write the Fourier transform pair in Eqs.(2.118) and
(2.119) in the following reformulated form
E( x, y, z, ω ) =
ZZ ∞
−∞
U + (k x , k y , ω ) exp i(k x x + k y y + k z (z − z0 ))
U + (k x , ky , ω ) =
ZZ ∞
1
4π 2
−∞
× dx dy,
E( x, y, z0, ω ) exp −i(k x x + k y y)
× dk x dk y .
(2.123)
(2.124)
The above written Fourier transform pair is known as the angular spectrum representation of the field. With this representation any general field can be written as
a superposition of plane waves. To evaluate the angular spectrum at an arbitrary
plane at z > z0 , we just need to multiply U + (k x , k y , ω ) by a constant exponential
factor.
2.12
THEORY OF EVANESCENT WAVES
As we mentioned in the earlier section, evanescent waves are extremely important
in subwavelength nanostructures and in the context of plasmonics. To understand
their behavior let us assume the same planar interface as in Fig. 2.2. By examining
the definitions of the angles and employing Eqs. (2.83), (2.95), and (2.95) we arrive
at the well known Snell’s law
n̂1 sin θin = n̂2 sin θtra ,
(2.125)
which connects the propagation angle of the transmitted wave to the angle of incidence. Now we use Eqs.(2.83) and (2.121) to obtain
[k z,tra ]2 = ε̂ 1 k20
ε̂ 2
− sin2 θ1 .
ε̂ 1
(2.126)
If we assume that the permittivity of the incident medium is higher than that of the
transmitted medium i.e. ε̂ 1 > ε̂ 2 , we get
θ1 > θc = sin−1
n̂2
,
n̂1
(2.127)
where θc is the critical angle of incidence i.e. the angle of incidence beyond which
light wave passing through a optically denser medium to the surface of a less dense
medium is no longer refracted but totally reflected. Clearly from Eq. (2.126), k z,tra
becomes imaginary which represents an evanescent field propagating parallel to the
interface but decaying exponentially along z in medium 2.
22
2.13
MONOCHROMATIC PLANE WAVE IN ANISOTROPIC MEDIUM
So far we have assumed the propagation medium to be isotropic. Let us now investigate the situation when light propagates in an electrically anisotropic medium
i.e. in crystals with only electrical anisotropy. The other assumptions regarding the
medium properties still remain valid i.e. we assume homogeneous, non-conducting,
and magnetically isotropic medium. In such a medium, the electric displacement D
and the electric field E no longer remains collinear.
The simplest relation between D and E which can account electrical anisotropy
can be written in component form as
Dj =
∑ ε jk Ek ,
(2.128)
k
where j stands for the cartesian indices (1, 2, 3), and k stands for each of 1, 2 and 3 in
turn in the summation on the right hand side of Eq. (2.128). The expressions for the
electric and the magnetic energy densities, and the Poynting vector i.e. Eqs. (2.62),
(2.63), and (2.64) still retain their validity and can be written in the following form
we = (1/2)ε 0 ∑ Ej ε jk Ek ,
(2.129a)
jk
wm = (1/2)µ0 µr H 2 ,
(2.129b)
hSi = (1/2)ℜ [E × H ∗ ] .
(2.129c)
1
∂
1 ∂ 1
Ej ε jk Ek + µ0
µr H 2 = − ∇ · ( E × H ) ,
∑
2 jk
∂t
2 ∂t
2
(2.130)
Here we have kept the relative magnetic permeability µr (which is a constant) to
preserve some symmetry in the formulae and also to include weakly magnetic crystals. However, at optical frequencies we can always replace µr by 1 in SI system of
units. Also, to write Eq. (2.129), we have omitted the explicit position (r) and time
dependence of the field quantities.
Let’s now check the consistency of Eq. (2.129) with the energy conservation principle. For this we multiply space-time domain Maxwell’s equation 1 i.e. Eq. (2.9)
with E and Maxwell’s equation 2 i.e. Eq. (2.10) with H and use the vector identity
E · (∇ × H ) − H · (∇ × E) = −∇ · ( E × H ) to write the following equation
where we have divided both sides of Eq. (2.130) by 2. Clearly, the second term on
the left hand side of Eq. (2.130) represents the rate of change of magnetic energy
∂
∂
∂
per unit volume but the first term i.e. 21 ∑ jk Ej ε jk ∂t
Ek = 14 ∑ jk ε jk Ej ∂t
Ek + Ek ∂t
Ej
represents the rate of change of the electric energy density only if
∂
∂
ε
E
E
−
E
E
(2.131)
∑ jk j ∂t k k ∂t j = 0.
jk
However, the term on the right hand side of Eq. (2.130) equals to −∇ · S. Hence,
for the validity of the energy theorem, which states that the decrease in the total
electromagnetic energy per unit time in a certain volume is equal to the net outward
flux per unit time, in differential form viz.
−∇ · S = (∂/∂t) (we + wm ),
(2.132)
23
we must have
ε jk = ε kj .
(2.133)
In words, the permittivity tensor must be symmetric with only 6 independent elements. Employing condition (2.133), Eq. (2.129a) can be written as
ε 11 E12 + ε 22 E22 + ε 33 E32 + 2ε 12 E1 E2 + 2ε 13 E1 E3 + 2ε 12 E1 E2 = 2we .
(2.134)
The right-hand side of Eq. (2.134) is a positive constant as we represents electric
energy density. Hence, Eq. (2.134) represents an ellipsoid. Any ellipsoid can be
written in a coordinate system with its axes parallel to the principal axes of the
ellipsoid. Assuming that the principal axes of the ellipsoid are along x,y, and z,
where x, y, z are also cartesian indices (the coordinate systems (x, y, z) and (1, 2, 3))
are rotated with respect to each other), we can write Eq. (2.134) in the following
simple form
Dy2
D2x
D2
+
+ z = 2we ,
(2.135)
ǫx
ǫy
ǫz
with
D x = ǫ x Ex ,
D y = ǫ y Ey ,
D z = ǫ z Ez .
(2.136)
Here ǫx , ǫy , and ǫz are the principal relative permittivities. Equations (2.135) and
(2.137) indicate that the electric displacement vector D and the electric field vector
E are in general not parallel to each other unless E directs along one of the principal axes of the crystal or all the principal relative permittivities are equal i.e. the
ellipsoid degenerates into a sphere. For a plane wave propagating in an anisotropic
medium, the direction of the electromagnetic energy flow is in general different
from the direction of the wave normal. Hence in addition to the wave-normal velocity or the phase velocity v p = c/n, n being the position dependent refractive index,
we can introduce the concept of ray (or energy) velocity vr = S/w, where S is the
amplitude of the Poynting vector and w is the total electromagnetic energy density.
Clearly from the definition, the ray velocity is equal to the energy that crosses an
area perpendicular to the flow direction in unit time divided by the energy per unit
volume. Assuming time-harmonic field of the form exp {iω [(n/c) (r · û ) − t]}, we
can write Maxwell’s equations (2.9) and (2.10) in the following forms (in regions
without currents)
(n/c)û × H = − D,
(n/c)û × E = µH,
(2.137)
where, we have used the relation B = µ0 µr H = µH. Let us now eliminate H from
Eqs. (2.137) to get the following
D=−
n2
n2
û
×
(
û
×
E
)
=
[ E − û(û · E)] = (n2 /µc2 ) E⊥ .
µc2
µc2
(2.138)
E ⊥ is the component of the electric field in a direction perpendicular to the wave
normal and lies in a plane containing E and û. Clearly from Eq. (2.137), the magnetic
field H (also the magnetic induction B) vibrates at right angles to E, D, and û.
Furthermore, D is orthogonal to û, hence H and D are transversal to the wave
propagation direction. However in an anisotropic medium, E is not transversal to
the wave normal as before. We denote the angle between D and E by α. Let us now
define a new unit vector ŝ which defines the direction of the energy flow. From the
24
definition of the Poynting’s vector, both E and H are perpendicular to ŝ. Hence, the
angle between û and ŝ is also α. Finally, we may conclude that in an anisotropic
medium, the vectors D, H, and û on one hand, and the vectors E, H, and ŝ on the
other hand form orthogonal vector triplets with the common vector H as shown in
Fig. 2.5. The theorem of equal electric and magnetic energy densities still remain
B
H
S
ŝ = |S
|
α
α
E⊥
D
û
E
Figure 2.5: Direction of the field quantities, the wave normal, and the energy flow
in an electrically anisotropic medium.
valid and by use of Eqs. (2.129) and (2.137) we can write the total electromagnetic
energy density in the following form
w=
n
S · û.
c
(2.139)
It now follows from the definitions of the phase velocity and the ray velocity and
Eq. (2.139) that
v p = vr ŝ · û = vr cos α,
(2.140)
which translates that the phase velocity in an anisotropic medium is the projection
of the ray velocity along the direction of the wave normal.
2.14
ELECTROMAGNETIC THEORY OF METALS
While studying the electromagnetic properties of metals, in many cases it is assumed for simplicity that the material media are perfectly conductive. In perfectly
conducting media, the flow of electrons is completely free and it is assumed that
the ”free electrons” can respond to the incident field infinitely quickly without being scattered. Also for an ideal metal, the electric field is ’zero’ everywhere inside
the material as in the limit ε → −∞, the electrons respond perfectly to the applied
electric field and thus cancels the external field completely. This also gives zero
electrical resistance of ideal metals. Although the assumptions made above lead to
analytical solutions of the modes of the optical field, these are inaccurate at optical
frequencies. The reason is that the free electrons inside a metal have finite mass and
25
they suffer scattering with phonons (lattice vibrations), lattice defects, surface of the
materials as well as with other electrons which impose limit on the response time of
the metals to the applied external field. In this section, we shall introduce the Drude
model and the interband transitions model for describing complex permittivities of
metals. These models are of great importance to understand the interaction between
metals and external electromagnetic fields at the atomic level. In many time-domain
solvers of Maxwell’s equations such as in FDTD [119], these can be included directly.
2.14.1
Drude model
In the Drude model (also known as the plasma model), the optical properties of
metals are portrayed by considering the optical response of a free electron gas in
an electric field. It is assumed that the gas of free electrons moves against a fixed
background of positive ion core [103]. In the plasma model, the lattice potential and
electron-electron interactions are ignored. However, it is assumed that some features
of band structure are connected to the effective mass (me ) of a single electron. Let
us denote the free electron density by Ne . Also let us assume that the electron
oscillations triggered by the external field are attenuated via collisions (scattering)
occurring with characteristic frequency γ = 1/τ, where τ is the relaxation time
of the free electron gas. The Drude model assumes that the valence electrons are
identical to the free electrons. To derive the complex relative permittivity of the
material using this model, we start from the equation of motion of a free electron
in the free electron gas which is subject to the influence of an externally applied
electric field E(r, ω ) [120]:
∂2
∂
r (t) + me γ r (t) = qe E(r ) exp(−iωt),
(2.141)
∂t
∂t2
where qe denotes the electric charge of a single electron, r (t) is the time-dependent
displacement of the electron with respect to the ion core, and γ is equivalent to the
damping parameter. Let us attempt a solution of Eq. (2.141) of the form r (t) =
r 0 exp(−iωt). Substituting this in Eq. (2.141), we get the following expression of the
electron displacement
qe
r (t) =
E (r, ω ).
(2.142)
me (ω 2 + iωγ)
The electric polarization of a single electron is defined as p(t) = qe r (t). Hence
assuming that all the electrons in the free electron gas are displaced by an equal
amount under the influence of the external field, we get the following expression
for the total electric polarization [121]:
me
P (t, ω ) = Ne p(t, ω ) = Ne qe r (t) = −
Ne q2e
E(r, ω ).
me (ω 2 + iωγ)
(2.143)
Let us now recall the material constitutive relations i.e. Eqs. (2.17) and (2.24). With
help of these and Eq. (2.143), we get the following expression for the complex permittivity [122]:
ω 2p
ω 2p τ 2
ε D (ω ) = 1 − 2
= 1− 2 2
,
(2.144)
ω + iγω
ω τ + iωτ
where ω p is termed as the volume plasma frequency and is given by
s
Ne qe
ωp =
.
(2.145)
ε 0 me
26
Equation (2.144) is known as the Drude model of the optical response of metals. The
real and the imaginary part of the complex relative permittivity ε(ω ) = ε 1 (ω ) +
iε 2 (ω ) can be calculated from
ε 1 (ω ) = 1 −
ε 2 (ω ) =
ω 2p τ 2
1 + ω2 τ2
ω 2p τ
ω (1 + ω 2 τ 2 )
,
(2.146a)
.
(2.146b)
Usually the free electron density (Ne ) in metals lies in the range 1028 − 1029 m−3 .
This leads to the plasma frequency in the ultraviolet region.
Drude model can be well used to describe optical properties of alkali metals in
the visible frequencies but for noble metals this model gives inaccurate results due
to the interband transitions.
2.14.2
Interband transitions model
The Drude model underestimates the imaginary parts of the complex permittivities
of many noble metals in the visible region. This is because the high frequency (hence
high energy) photons excite the lower band valence electrons into the conduction
band [122]. Such transitions can not be described by the Drude model. In the
interband transitions model, such type of transitions can be taken into account by
introducing oscillations of the bound electrons. In this model, the motion of the
bound electrons can be described by the following equation [123]:
∂2
∂
r (t) + me γ r (t) + mω02 r (t) = qe E(r ) exp(−iωt),
(2.147)
∂t
∂t2
where ω0 is defined as the resonance frequency of the bound electrons. Considering
a trial solution as in the previous section, we can obtain the following expression for
the complex relative permittivity
me
ε D (ω ) = 1 −
ω̃ 2p
(ω02 − ω 2 ) + iγω
where the plasma frequency is defined by
s
ω̃ p =
Ñe q2e
,
ε 0 me
,
(2.148)
(2.149)
with Ñe denoting the density of the bound electrons.
In reality, the external field can penetrate the metal surface. Atomic structure of
metals demand that the current density is the highest on the surface and it gradually
decreases to zero when we go deeper inside the metal. Hence, when an external
electromagnetic field penetrates a metal surface, it gradually decreases to zero. The
distance at which the electromagnetic field amplitude decreases to 1/e of its value on
the metal surface, is called the skindepth of the metal and is given by the following
relation
s
c
2
δ=
=
,
(2.150)
κω
σ0 ωµ0
where κ is defined in Eq. (2.30), and σ0 is the dc-conductivity of the metal as before.
From Eq. (2.150), we see that for better conductors, the skin depth is smaller.
27
2.15
PLASMONS
By plasma, we specify a medium consisting of positively and negatively charged
particles. The free electrons and ion cores inside a metal constitute a plasma with
zero net charge. Plasmons are collective oscillations of the free (conduction) electrons inside a plasma. There are three types of plasmons each with their own characteristic nature. These are outlined below.
ω = ck0
ω
ωBP
ωSPP
ωLSP
k
Figure 2.6: Dispersion diagram of bulk plasmons (black dashed line), SPP(dashdotted line) and LSP(dotted line) at metal-vacuum interface as compared to the
dispersion of light in vacuum (solid black line).
+
-
+
-
+
-
+
+
-
+ + -
+
-
+
-
-
BP
+
-
+ -
SPP
(a)
( b)
+
-
+
+
+
-
+
+ -
-
LSP
(c)
Figure 2.7: Schematic of (a) bulk plasmon, (b) a surface plasmon polariton at a
metal-dielectric interface, and (c) a localized surface plasmon at the interface of a
spherical metal nanoparticle and surrounding dielectric.
28
2.15.1
Volume plasmons
In metals, local deviations in free electron density cause restoring forces from the
fixed ionic cores which lead to simple harmonic motion of the conduction electrons.
The quanta of these oscillations are termed as bulk plasmons. The characteristic
frequencies of bulk plasmons (ωBP ) are given by Eq. (2.145). For most of the metals,
bulk plasma frequencies are higher than the frequencies of the visible light. When
light of lower frequency (than the bulk plasmon frequency) is incident on the metal,
the electric field of the incident light causes the conduction electrons to move with
respect to the positively charged ionic cores in such a way that an internal electric
field is generated which cancels out the incident electric field and creates a reflected
wave. However, for frequencies of the incident light much higher than the plasma
frequency i.e. in the limit ω ≪ ωBP , the electrons cannot keep up with light oscillations. Hence, we can make the approximation ωτ ≫ 1 and neglect the damping
term iωτ in Eq. (2.144) to get
ω 2p
ε( ω ) = 1 − 2 .
(2.151)
ω
The dispersion relation of the electromagnetic fields can be determined from the
relation k2 = |k|2 = εω 2 /c2 :
ω (k) =
q
ω 2p + (kc)2 .
(2.152)
Clearly from the dispersion curve in Fig. 2.6, for light waves with frequencies below
the bulk plasma frequency (ω < ωBP ), there is no propagation. The solid black
line represents the dispersion of light in vacuum where ω = ck0 , the dashed line
depicts the dispersion of light propagation inside the metal. For ω = ωBP , we get
from Eq. (2.152), ε(ωBP ) = 0. One can show [103] that in such a scenario collective
longitudinal excitation mode (E k k) is formed with a purely depolarizing field (E =
(−1/ε 0 )P). Since the bulk plasmons are longitudinal waves they cannot couple to
transversal electro-magnetic fields and also cannot be excited from direct irradiation.
The bulk plasmons are illustrated in Fig. 2.7(a).
2.15.2
Surface plasmon polaritons
There is another kind of plasmon which has coupled longitudinal and transverse
components along the surface of a metal. Since this type of plasmon has coupled
longitudinal and transverse parts (depicted in Fig. 2.7(b)), it is called surface plasmon polariton (SPP) [124]. In 1902, Wood observed that if a reflection grating is
illuminated with polychromatic light, narrow dark and bright bands appear in the
reflection spectrum, which are widely known as Wood’s anomalies. After almost
three decades, these anomalies were explained by Ugo Fano [125–127]. Fano concluded that these narrow bands are caused by the surface waves or the surface
plasmons polaritons, which can be excited on the surface of metallic gratings under
special conditions.
Let’s now proceed to derive the dispersion relation for a SPP by assuming a
smooth planar interface separating two half-spaces with scalar permittivities ε̂ 1 and
ε̂ 2 as shown in Fig. 2.2. This is the simplest geometry supporting SPPs. The electric
and the magnetic field components in both the half spaces can be decomposed into
two independent sets as described in section 2.6. Assuming a TE polarized incident
29
plane wave at the interface (located at z = 0), the only non-vanishing electric field
component in both half spaces can be written as
n h
io
( j)
( j)
( j)
Ey ( x, z) = Ey,0 exp i k x x + k z z ,
(2.153)
where the superscript j = 1, 2 denotes the left and the right half-spaces respectively.
Using Eqs. (2.51) and (2.52), we can find expressions for the non-vanishing magnetic
field components:
( j)
Hx ( x, z) = −
( j)
Hz ( x, z) =
( j) r
kz
k0
kx
k0
r
n h
io
ε 0 ( j)
( j)
Ey,0 exp i k x x + k z z ,
µ0
n h
io
ε 0 ( j)
( j)
Ey,0 exp i k x x + k z z .
µ0
(2.154a)
(2.154b)
As the electromagnetic boundary conditions Eq. (2.39) and (2.40) demand that the
tangential field components i.e. Ey and Hx must be continuous across the interface,
we must have
(1)
(2)
kz = kz .
(2.155)
From the definition of the wave vector we see that Eq. (2.155) is satisfied only if
ε(1) = ε(2) . Hence, we are forced to conclude that a TE polarized wave can not excite
SPPs at the interface between two different media.
If we now assume TM polarized incident field, the only non-vanishing magnetic
field component in both half-spaces can be expressed by
n h
io
( j)
( j)
( j)
(2.156)
Hy ( x, z) = Hy,0 exp i k x x + k z z .
By use of Eqs. (2.54) and (2.55) we can find the expressions for the non-vanishing
electric field components
( j)
Ex ( x, z)
( j)
( j)
k
= z ( j)
k0 ε̂
Ez ( x, z) = −
r
kx
k0 ε̂( j)
n h
io
µ0 ( j )
( j)
Hy,0 exp i k x x + k z z ,
ε0
r
n h
io
µ0 ( j )
( j)
Hy,0 exp i k x x + k z z .
ε0
(2.157a)
(2.157b)
The electromagnetic boundary conditions i.e. Eqs. (2.38)–(2.40) demand that
(1)
= ε̂(2) Ez ,
(2.158a)
(1)
(2)
(2.158b)
(2)
(2.158c)
ε̂(1) Ez
(2)
Ex = Ex ,
(1)
Hy
= Hy ,
respectively. As the tangential field components Ex and Hy must be continuous
across the interface we get using Eq. (2.157a),
ε̂(1)
(1)
kz
30
=
ε̂(2)
(2)
kz
.
(2.159)
(1)
If we now substitute for [k z ]2 = ε̂( j) k20 − k2x in Eq. (2.159), we obtain the so-called
dispersion relation of the propagating surface plasmons
s
ε̂(1) ε̂(2)
k x = kSPP = k0
.
(2.160)
ε̂(1) + ε̂(2)
The normal component of the surface plasmon wave vector is given by
s
[ε̂( j) ]2
( j)
k z = k0
.
ε̂(1) + ε̂(2)
(2.161)
As we are seeking for a wave solution that propagates along the interface and exponentially decays into both half-spaces, we must have real valued tangential wave
( j)
vector component k x = kSP . Also, the normal wave vector components k z s must
be imaginary in both half-spaces. The first condition demands that both the numerator and the denominator in the right hand side of Eq. (2.160) is either positive or
negative. The second condition demands that the denominator on the right hand
side of Eq. (2.161) is negative. These requirements are fulfilled at the interface of a
metal and a dielectric, where the absolute value of the real part of the permittivity
of the metal is larger than the permittivity of the dielectric. For real metals, below
the plasma frequency ω p , the permittivities are imaginary which leads to complex
values of kSP . The real and the imaginary parts of kSP determine the surface plasmon wavelength and the attenuation coefficient (1/e decay length) of the surface
plasmon wave along the interface respectively, i.e.
λSP =
2π
,
ℜ{kSPP }
(2.162a)
δSP =
1
.
ℑ{kSPP }
(2.162b)
and
In a similar fashion we can define the exponential decay length of the surface plasmon field perpendicular to the interface i.e. in the two half-spaces by the following
relation
1
δ( j) = ( j) .
(2.163)
ℑ k z This 1/e decay length inside the metals is equal to the skin-depth of the metals.
If we now assume that the dielectric is vacuum, in the limit of large k and assuming a Drude model for the permittivity of the metal we obtain from Eq. (2.160),
√
ωSPP = ωBP / 2.
(2.164)
The surface plasmon dispersion relation and the definition of the decay length in
Eq. (2.162b) infer that for the real metals (for example gold, silver, copper, aluminum
etc.) there is a momentum (h̄k, h̄ being the Planck’s constant) mismatch between
free-space photons and surface plasmons in the same medium and in the wavelength region where the surface plasmons are expected to propagate. This means
that the SPPs will not be excited by incident light under normal circumstances. This
is graphically illustrated in Fig. 2.6 where we see that the SPP line has a higher momentum (larger k) than the free-space light line (solid black line) for all frequencies.
31
Also, we observe that the dispersion curve of light in vacuum (solid black line) is
close to the SPP dispersion curve (black dash-dotted line) at metal-vacuum interface
only for shorter frequencies of the incident light. Hence in the lower frequency /
higher wavelength regime, the SPP is light-like and the transverse components of
the SPP field will dominate. Conversely, in lower wavelength regime, the SPP dispersion is far from the dispersion of light in vacuum and the SPP is plasmon-like
where the longitudinal components dominate.
In practical applications to excite SPP, the momenta of the incident wave vector
component along the interface and the SPP wave vector need to be matched. Some
of the common tricks which are applied to compensate for the momentum mismatch
are listed below
• Prism Coupling: The metal film in this case is illuminated through a dielectric
prism at an angle of incidence greater than the angle of total internal reflection.
The wave vector of incident light is increased in the optically dense medium.
At a certain angle of incidence θin , the component of the light wave vector
(inside the prism) parallel to the interface matches with the SPP wavevector at
metal-air interface and light is coupled to the SPP mode as shown in Fig. 2.8(a).
• Grating Coupling: A grating with grating vector parallel to the interface is introduced in this case as shown in Fig. 2.8(b). Momentum conservation can be
satisfied by the mth (m = 1, 2, 3, . . . ) evanescent diffraction order by
s
ω
ε̂(1) ε̂(2)
|k x + mK | = ℜ
,
(2.165)
c
ε̂(1) + ε̂(2)
where k x = k sin θ is the component of the incident wave vector parallel to the
interface, θ is the angle of incidence and K = 2π/|d | with d representing the
grating vector which directs parallel to the interface.
θin
θin
d
(a)
(b)
Figure 2.8: Geometry of (a) prism coupling, and (b) grating coupling for exciting
SPPs at a metal-dielectric interface.
2.15.3
Particle plasmons
There exists a third kind of plasma oscillation which is known as particle plasmon.
This arises from the scattering problem of a small subwavelength metallic nanoparticle under influence of an oscillating external electromagnetic field. Electrons are
32
pulled back to the particle surface by a restoring force which arise from the curvature of the particle. At resonance, the fields inside the particle and in its immediate
vicinity are greatly enhanced, sometimes their magnitudes can be even 100 orders
higher than that of the exciting field. As the restoring force in this case is the surface
charge (instead of the ionic cores), the particle plasmon is a transverse standing wave
as shown in Fig. 2.7(c). Due to its standing wave nature particle plasmons are also
termed as localized surface plasmons (LSPs). There are no momentum matching
conditions as the particle plasmons have a net zero momentum (standing waves).
From Fig. 2.6, we observe that the light line (solid line) will be able to cross the
dotted horizontal line (which represents the dispersion for a particle plasmon) for
any value of k.
When the particle is extremely small compared to the wavelength of the incident
field, the phase of the electric field does not change considerably over the whole
nanoparticle. Hence all the conduction electrons in the particle experience the same
driving force, causing them to act coherently. These coherently shifted electrons
under the influence of the fluctuating external electromagnetic field, resembles an
oscillator together with the restoring field. The behaviour of this oscillator is determined by the effective electron mass, surface charge density, and the geometry of
the particle. Most physical effects associated with LSPs can be explained with the
help of this simple model. Using the damped driven harmonic oscillator model we
can write the following expression
d2
d
x (t) = −ω02 x (t) − 2γ x (t) + E0 exp (iωt)
dt
dt2
(2.166)
describing the oscillations of the conduction electrons. x (t) is the displacement of
the conduction electrons under influence of the external alternating electric field
E0 exp (iωt), the restoring force is the surface polarization charge, which can be
imagined as a spring of constant k pulling the electrons back to equilibrium, the
damping term γ may arise due to the decay of the plasmon mode into radiation
mode (results in emission of a photon with the same frequency as the incident field),
phonon scattering, scattering by a lattice defect, surface scattering or absorption by
a surrounding medium. The solution of Eq. (2.166) is given by
x (t) =
E0
1
p
,
2me ω0 (ω0 − ω )2 + γ2
(2.167)
√
where ω0 = k s /m is the resonance frequency which depends on the surface charge
and hence on the geometry and the size of the particle. The LSP resonance is observed at ω = ω0 . Since the resonance frequency is proportional to the spring
constant k s , a weaker restoring force leads to a lower frequency of oscillation causing redshift of the resonance. Usually, the LSP resonances are observed in the visible
to near-infrared regime.
The geometry of the nanoparticle strongly affects the microscopic polarizability
(α) inside the particle which in turn affects the surface charge and hence the resonance frequency. For a small spherical particle, assuming a uniform static electric
field throughout its volume, if we have nonabsorbing surrounding medium, the
microscopic polarizability takes the following form [103]
α = 4πε 0 r3
εm − εd
,
ε m + 2ε d
(2.168)
33
where r is the radius of the spherical metal particle with relative permittivity ε m ,
ε d is the relative permittivity of the surrounding dielectric medium. Clearly, for
ε m = −2ε d , the denominator on the right hand side of Eq. (2.168) is zero and we
have LSP resonance. Assuming Drude model for the permittivity of the metal and
taking the dielectric to be vacuum as before, we get
√
ωLSP = ωBP / 3,
(2.169)
which is illustrated in Fig. 2.6.
2.16
FIELD ENCOUNTERING STACK OF THIN FILMS
We next examine a case of special importance in a broad range of applications in
relation to optics. This is the situation when a large number of thin homogeneous
layers are separated by plane interfaces parallel to each other as shown in Fig. 2.9.
The positions of the interfaces are given by z( j) , j = 0, 1, . . . , J − 1, J, where J is the
total number of layers. The permittivities of the layers are defined as
z ( j −1 ) < z < z j .
ε̂ (z) = ε̂( j) ,
(2.170)
Here we define region 2 as the layer between z(1) and z(2) . Let us now assume
ε̂(0) and ε̂( J +1) to be real valued i.e. we have real permittivities for the incident
and the output regions. Also, we assume that the incident field is y invariant and
propagates from region 0 towards the positive z-axis. Using the angular spectrum
representation, the input electric field can be expressed in the form
E(0+) ( x, z) =
Z ∞
−∞
h
i
( j)
U (0+) (k x ) exp ik x x + ik z (z − z(0) ) dk x ,
U (0+) (k x ) =
1
2π
Z ∞
−∞
E(0+) ( x, z(0) ) exp [−ik x x ] dx.
(2.171)
(2.172)
It can be proved easily that the y-invariance of the incident field leads to y-invariance
also in all other regions. Hence, using the angular spectrum representation the
electric field in the j-th layer can be written as
E( j,+) ( x, z) =
Z ∞
−∞
h
i
( j)
U ( j,+) (k x ) exp ik x x + ik z (z − z(0) ) dk x ,
U ( j,+) (k x ) =
E( j,−) ( x, z) =
Z ∞
−∞
1
2π
Z ∞
E( j,+) ( x, z( j)) exp [−ik x x ] dx,
h
i
( j)
U ( j,−) (k x ) exp ik x x + ik z (z − z(0) ) dk x ,
U ( j,−) (k x ) =
1
2π
−∞
Z ∞
−∞
E( j,−) ( x, z( j)) exp [−ik x x ] dx,
(2.173)
(2.174)
(2.175)
(2.176)
for j = 1, 2, . . . , J − 1, J. Here we compute the angular spectrum of the propagating
field at z( j) for the j-th layer. Now we can decompose Maxwell’s equations in j-th
34
x
ε̂(0)
ε̂(1)
ε̂(2)
ε̂(3)
ε̂( J −1)
ε̂( J )
ε̂( J +1)
z
0
z (0)
z (1)
z (2)
z (3)
z ( J −2 )
z ( J −1 )
z( J )
Figure 2.9: Schematic of light propagation in stack of thin films.
layer into two sets (TE/TM) in a similar way as described in section 2.6
∂ ( j,±)
( j,±)
Ey ( x, z) = −iωBx ( x, z),
∂z
∂ ( j,±)
( j,±)
Ey ( x, z) = iωBz ( x, z),
∂x
∂ ( j,±)
∂ ( j,±)
( j ) ( j,±)
iωµ0 ε 0 ε̂ Ey ( x, z) =
Bz ( x, z) − Bx ( x, z) ,
∂x
∂z
∂ ( j,±)
( j,±)
By ( x, z) = iωµ0 ε 0 ε̂( j) Ex ( x, z),
∂z
∂ ( j,±)
( j,±)
By ( x, z) = −iωEz ( x, z),
∂x
∂ ( j,±)
∂ ( j,±)
( j,±)
iωBy ( x, z) =
Ex ( x, z) −
Ez ( x, z) ,
∂z
∂x
(2.177)
(2.178)
(2.179)
(2.180)
(2.181)
(2.182)
Eqs. (2.177)–(2.179) contain only the y component of the electric field which is perpendicular o the xz plane i.e. the plane of propagation. Hence these form the TE
polarized set. Similarly Eqs. (2.180)–(2.182) form the TM polarized set. Following
the electromagnetic boundary conditions we can now demand the continuity of the
field quantities at the boundary z = z( j). For the TE polarized set, continuity of the
electric field gives
h
i
h
i
h
i
h
i
( j,+)
( j,−)
( j +1,+)
( j +1,−)
Ey
x, z( j) + Ey
x, z( j) = Ey
x, z( j) + Ey
x, z( j) ,
(2.183)
35
where h( j) = z( j) − z( j−1) is the thickness of the j-th layer. Uniqueness of the Fourier
representation gives
( j,+)
Uy
( j +1 )
h
( j,−)
+ Uy
( j +1,+) ( j +1)
( j +1,−) ( j +1)
f−
+ Uy
f+ ,
(2.184)
= Uy
i
( j)
where f ±
= exp ±ik z h( j) and we dropped the explicit k x dependence of Uy
for the sake of brevity. As z derivatives of the electric fields are also continuous
across the boundary, we get the following relation
( j)
( j,+)
k z Uy
( j)
( j,−)
− k z Uy
( j +1 )
= kz
( j +1,+) ( j +1)
( j +1) ( j +1,−) ( j +1)
f−
− kz
Uy
f+ .
Uy
Eqs. (2.184) and (2.185) can be conveniently written in matrix form as
#"
#
"
( j,+)
1
1
Uy
( j)
( j)
( j,−) =
−k z
kz
Uy
#
"
#"
#"
( j +1,+)
( j +1 )
1
1
Uy
f−
0
( j +1 )
( j +1 )
( j +1 )
( j +1,−) .
kz
−k z
0
f+
Uy
(2.185)
(2.186)
(2.187)
In the case of TM polarization the electromagnetic boundary conditions demand the
continuity of the y components of the magnetic induction. Proceeding similarly as
in the TE polarized case we arrive at the following matrix equation
"
#"
#
( j,+)
1
1
Vy
=
(2.188)
( j)
( j)
k z /ε̂( j) −k z /ε̂( j) Vy( j,−)
"
#"
#"
#
( j +1,+)
( j +1 )
1
1
Vy
f−
0
(2.189)
( j +1 ) ( j +1 )
( j +1 ) ( j +1 )
( j +1 )
( j +1,−) ,
kz
/ε̂
−k z
/ε̂
0
f+
Vy
( j)
where Vy denotes the angular spectrum of the magnetic induction. Here, we must
emphasize that though the y components of the magnetic induction are continuous
across the boundary their z-derivatives are discontinuous. However, the z-derivative
( j)
of Vy divided by the complex permittivity i.e. ε̂( j)s is continuous across the boundary. Owing to the similarity of the two matrix equations (2.185) and (2.186), we may
write a single matrix equation for both the TE and the TM polarized fields as
( j,+) 1
1
G
=
(2.190)
g( j) − g( j) G ( j,−)
#
" ( j +1 )
1
1
f−
0
G ( j+1,+)
,
(2.191)
( j +1 )
g ( j +1 ) − g ( j +1 )
G ( j+1,−)
0
f+
( j)
( j,±)
where g( j) = k z , G ( j,±) = Uy
( j)
for TE polarization, and g( j) = k z /ε̂( j) , G ( j,±) =
( j,±)
Vy
for TM polarization. Eq. (2.190) connects the angular spectra in regions j and
j + 1.
2.17
RECURSIVE S-MATRIX ALGORITHM
Several standard matrix algorithms exist for solving the boundary value equations
associated with the stack of thin films problem discussed in the previous section.
36
These matrix algorithms are required to connect the amplitudes of the input and
the output field quantities. However, all of these algorithms face a common difficulty which is associated with the exponential functions of the spatial variables
in the direction perpendicular to the grating plane, to be more specific with the
( j)
f ± s. Especially if the complex permittivity of the j-th layer is complex valued,
( j)
( j)
the imaginary part of k z becomes positive and f − becomes unstable. If there is
( j)
a small numerical error in computing the argument of f − , it grows exponentially
with the growing layer thickness. The well known T-matrix algorithm suffers from
this type of numerical instability. One way to overcome this is to make sure that
all the angular-spectrum components in each layer propagate only along their real,
physical propagation direction, thus eliminating the cumulative exponential error
in each step. This is done using the recursive S-matrix (also known as scattering
matrix) algorithm [128–130]. In S-matrix algorithm we take the angular spectrum
G ( j+1,−)
G ( j+1,+)
S( j )
G ( j,−)
G ( j,+)
Figure 2.10: Block diagram of S-matrix for two adjacent layers.
components G ( j+1,−) and G ( j,+) as inputs to the system and try to find the S-matrix
to derive the components G ( j+1,+) and G ( j,−) using propagation along the correct
direction only. Mathematically, this can be expressed as,
( j,+) G ( j+1,+)
G
( j )⇆( j +1)
=
S
.
G ( j,−)
G ( j+1,−)
(2.192)
Block diagram of S-matrix for two consecutive layers is shown in Fig. 2.10. We can
construct the S-matrix either for a part or for the whole system. A single S-matrix
contains all of the system’s scattering properties. The elements of the S-matrix can
be deduced from Eq. (2.190). In this context we must mention that the connection
between different layer-S-matrices are not straightforward. Lifeng Li [128–130] used
a recursive approach to solve the S-matrix problem in an efficient and stable way. In
this approach, one starts from the sub-system S-matrix S( j+1)⇆( J +1) and employs the
boundary value equations to find expression for the sub-system matrix S( j)⇆( J +1) in
37
terms of the elements of S ( j+1)⇆( J +1). The desired equation is of the following form
( J +1,+)
( j,+) G
G
( j )⇆( J +1)
=S
=
(2.193)
G ( j,−)
G ( J +1,−)
"
#
( j )⇆( J +1) ( j,+)
( j )⇆( J +1) ( J +1,−)
S11
G
+ S12
G
(2.194)
( j )⇆( J +1) ( j,+)
( j )⇆( J +1) ( J +1,−) ,
S21
G
+ S22
G
( j )⇆( J +1)
( j +1)⇆( J +1)
where the elements S pq
can be expressed in terms of the elements S pq
.
After some lengthy though straightforward calculations and applying the boundaryvalue condition (2.190) we can evaluate the coefficients of the S-matrix in Eq. (2.193)
( j )⇆( J +1)
S21
( j )⇆( J +1)
h
( j )⇆( J +1)
= Z21
( j )⇆( J +1)
( j )⇆( J +1) ( j +1)
i
( j )⇆( J +1) ( j )
(2.195)
( j +1) ( j +1)⇆( J +1)
(2.196)
g ,
+ Z22
,
− Z21
f + S22
h
i
( j )⇆( J +1)
( j )⇆( J +1) ( j )
( j )⇆( J +1)
( j +1)⇆( J +1) ( j +1)
,
Z11
+ Z12
g
S11
= S11
f+
h
i
( j )⇆( J +1)
( j +1)⇆( J +1) ( j +1)
( j )⇆( J +1) ( j +1)
( j )⇆( J +1)
S12
= S11
f+
Z12
g
− Z11
S22
= Z22
g
( j +1) ( j +1)⇆( J +1)
( j +1)⇆( J +1)
S22
+ S12
,
× f+
(2.197)
(2.198)
where g( j) is defined in section 2.16 and the elements of the matrix Z ( j)⇆( J +1) are
defined by the equation
Z
( j )⇆( J +1)
=
"
( j +1) ( j +1)⇆( J +1) ( j +1)
S21
f+
i
( j +1) ( j +1)⇆( J +1) ( j +1)
(
j
+
1
)
g
1 − f + S21
f+
1+
h f+
"
( j )⇆( J +1)
−1
g( j)
( j )⇆( J +1)
Z11
Z12
Z21
Z22
( j )⇆( J +1)
# −1
( j )⇆( J +1)
=
#
.
(2.199)
Eqs. (2.195)–(2.198) are the desired S-matrix recursion formulae for the film stack
problem discussed in the previous section. An S-matrix which is not operating over
a boundary between two media cannot cause any difference in the field. Hence, we
get the starting point of the recursion as S( J +1)⇆( J +1) = I, where I is the identity
matrix of appropriate size. After we derive the system S-matrix i.e. S(0)⇆( J +1), we
can use Eq. (2.193) to solve the unknown transmitted and the reflected fields.
2.18
LOCAL FIELD
All the field quantities introduced in the previous sections are the average macroscopic quantities. Here we introduce the concept of local field i.e. the field which
drives atomic transitions in a material. Local field is generally different from the
average field inside the material medium. For a medium with low atomic number
density the difference between the local field and the macroscopic ensemble average
field is not significant. However, the difference increases for materials with higher
atomic number density ≥ 1015 cm−3 [131]. In such a scenario, the influence of local field effects must be taken into consideration using proper mathematical models
which relate the local fields to the macroscopic field quantities. Using these models we can investigate the effects of the local fields on the optical properties of the
38
materials. However, the choice of the model strongly depends on the material. For
a homogeneous medium, the macroscopic electric field and the local field can be
related by the following relation
E = LEloc ,
(2.200)
where L is known as the local field correction factor. Different theoretical models
predict different expressions for this local field correction factor L. The most used
model is perhaps the Lorentz model. In the simplest version of this model, the
medium is treated as a cubic lattice of point dipoles of the same kind. To evaluate
the local field acting on a particular dipole inside the medium, we need to introduce
the concept of an imaginary spherical cavity surrounding the dipole of interest.
The radius of the imaginary cavity is assumed to be much larger than the interdipole distance though much smaller than the wavelength of the incoming light. To
deduce the expression of L, we take into consideration the exact contributions to
the local field from the dipoles situated inside the imaginary spherical cavity except
the dipole of study whereas, the effects of all the other dipoles are expressed in
terms of the average macroscopic polarization P. This approach gives the following
expression [132]
4π
Eloc = E +
P.
(2.201)
3
Now assuming the medium to be lossless, linear, and dispersionless we can derive
the well known Clausius-Mossotti relation between the macroscopic average dielectric permittivity and the microscopic polarizability α, which is defined as the relative
tendency of a dipole to be distorted from its normal shape by an external electric
field and/or by a nearby dipole. Microscopic polarizability relates the microscopic
dipole moment (p) induced in a atom of the medium to the local field acting on the
atom by the following relation
p = αEloc ,
(2.202)
where the macroscopic polarization and the microscopic dipole moment are related
by
P = N p,
(2.203)
N being the atomic number density. After some simple and straightforward calculations we arrive at the well-known Clausius-Mossotti (or Lorentz-Lorenz) relation
ε−1
4π
=
Nα.
ε+2
3
(2.204)
Also, the relation between the macroscopic average electric field and the local field
reads as
ε+2
Eloc =
E.
(2.205)
3
Comparing Eqs. (2.200) and (2.205) we get the expression for the local field correction factor
ε+2
LLor =
.
(2.206)
3
The above written expression for the local field correction factor remains valid for
homogeneous media i.e. when all the particles are of the same type.
Onsager introduced a different macroscopic model [133] for relating the local
field to the macroscopic average electric field. He treated the atom/molecule under
39
study being enclosed in a a real tiny cavity. According to his model, the field acting
on the molecule is divided into the cavity field (the field in the center of the real
cavity when the molecule is absent) and the reaction field (the cavity field correction
factor due to the presence of the molecular dipole at the center of the real cavity).
Onsager model gives the following relation between Eloc and E
Eloc =
3ε
2( ε − 1)
E+
p,
2ε + 1
(2ε + 1)r3
(2.207)
where r is the radius of the cavity. The first term on the right hand side of Eq. (2.207)
is the expression for the cavity field and the second term expresses the reaction field.
Most of the experimental works can be well described either by the Lorentz model
or by the Onsager model. Both of these models can describe the optical properties
of a guest-host system. In Lorentz model, the polarizability of the guest atom must
be the same as that of the host molecules. On the other hand, the Onsager model
is more suitable when the polarizability of the guest molecule/atom is significantly
different from that of the host molecules/atoms.
So far, our discussions were restricted to the local fields in the context of linear
optics. However, in the nonlinear optical regime, local field effects are more pronounced. To demonstrate this, let us first assume a homogeneous, centrosymmetric
medium which can be described by the Lorentz model of the local field. Also, we
assume only third-order nonlinear optical interactions at a single frequency. Hence,
the macroscopic medium susceptibility in terms of the macroscopic electric field can
be written as [134]
χ(1) + 3χ(3) | E|2 + . . . .
(2.208)
Let us now denote the microscopic counterparts i.e. the linear polarizability and the
third-order hyperpolarizability by γ(1) and γ(3) respectively. We proceed to derive
relations between γ(1) and χ(1) , and γ(3) and χ(3) respectively. Considering slowly
varying amplitudes (i.e. we assume that the amplitudes change negligibly over
distance scales equal to the wavelength of light) of the macroscopic electric field and
the macroscopic polarization, we can write the following expression for the Lorentz
local field
i
4π h (1)
Eloc = E +
P + P (3) + . . . ,
(2.209)
3
where P(1) and P(3) are the linear and the third-order nonlinear macroscopic polarizations respectively which are related to their microscopic counterparts by the
following relations
P(1) = Nγ(1) Eloc ,
(2.210a)
P(3) = Nγ(3) | Eloc |2 Eloc .
Combining Eqs. (2.210a) and (2.209) we get,
ε−1
4π (3)
(1)
P =
E+
P +... .
4π
3
(2.210b)
(2.211)
The macroscopic electric displacement vector D by definition (in the Gaussian unit)
has the following expression
D = E + 4πP(1) + 4πP(3) + . . . .
40
(2.212)
If we now substitute Eq. (2.211) into Eq. (2.212), we obtain
where
D = εE + 4πPNL ,
(2.213)
PNL = LLor P(3) + . . .
(2.214)
P = χ(1) E + PNL .
(2.215)
is termed as the macroscopic nonlinear source polarization [135]. LLor is the local
field correction factor as defined earlier. The total polarization P can be written in
the following form using Eq. (2.210a)
Furthermore, we can substitute Eq. (2.209) into Eq. (2.210b) and ignore the terms
proportional to higher than the third power of the electric field to yield
P(3) = 3Nγ(3)| LLor |2 LLor | E|2 E.
(2.216)
Finally, we substitute Eq. (2.216) into Eq. (2.214) and then substitute Eq. (2.214) into
Eq. (2.215) to write the macroscopic polarization in the following form
P = χ(1) E + 3Nγ(3)| LLor |2 LLor | E|2 E + . . . .
(2.217)
We already know that the macroscopic polarization can be written in terms of power
series expansion of the macroscopic average electric field E i.e. in the following form
P = χE = χ(1) E + 3χ(3) | E|2 E + . . . .
(2.218)
Hence, comparing Eqs. (2.217) and (2.218) we obtain
χ(1) = Nγ(1) LLor ,
(2.219a)
χ(3) = Nγ(3) | LLor |2 LLor .
(2.219b)
Equations (2.219) show that the third-order nonlinear susceptibility scales as the
third power of the local-field correction factor whereas the linear optical susceptibility scales as the first power of the local-field correction factor. Hence, the influence
of the local field on the nonlinear optical properties of a material can be significant.
In this context, we mention that significant control over the local field inside a
medium can be achieved by nanostructuring and also by intermixing more than one
homogenous media to form a composite medium. These topics will be discussed in
detail in the subsequent chapters.
2.19
SUMMARY
In this chapter we have laid out the theoretical foundation for the rest of the thesis. The concept of polarization discussed in section 2.9 plays an important part
in optics and will be used throughout the following chapters. The notion of local
field introduced in section 2.18 will be helpful for the chapters dealing with the local field effects on the linear and the nonlinear optical properties of nanostructured
materials.
41
3
Rigorous analysis of diffraction gratings
Periodic systems portray important aspects in science and technology. Moreover,
desire for order in human society gave birth to the domain of diffraction gratings
which refer to structures with periodically modulated optical properties [136–138].
Regular perturbations affect the propagation of light through these structures and
hence novel optical properties can be realized as compared to their bulk constituents.
In optics, periodic modulations are seen as changes in the medium permittivity (also
changes of permeability occurs but throughout this thesis we shall restrict ourselves
to non-magnetic materials). Since their first discovery in the 16th century by D. Rittenhouse [139], the diffraction gratings have emerged as useful optical devices and
are most commonly used in monochromators, spectrometers, wavelength division
multiplexing devices, optical pulse compressing devices, and in many other optical
instruments.
With the advancement of the domain of gratings, it was needed to develop efficient and accurate numerical tools for rigorous theoretical modeling of these. There
exists a wide variety of grating modeling methods due not only to historical reasons, but mainly to the absence of a universal approach that could efficiently solve
all diffraction problems. Though some of the methods cover broader domain of
problems, more specialized ones are usually more efficient. The main subject of this
chapter is the Fourier-Expansion Eigenmode Method or simply the Fourier Modal
Method (FMM) for modeling diffraction gratings. Here we shall discuss in detail
the implementations of the FMM for linear and multilayered gratings and lastly for
anisotropic crossed gratings. Before going into the details of the FMM, we shall
briefly discuss the working principle of a grating, and some useful terms such as
pseudoperiodic fields, grating equations, and the diffraction efficiencies.
3.1
WORKING PRINCIPLE OF A GRATING
In general, diffraction gratings can be broadly divided into two categories:
• Volume gratings: the grating structure is index-modulated, i.e, the permittivity is varying in one or two lateral directions.
• Surface relief gratings: certain spatially varying profile is fabricated into a
material having constant complex permittivity.
However, there also exist gratings, which combine these two types.
In section 2.10 we have studied the diffraction of a plane wave from a plane
interface. This results in a single reflected and a single transmitted wave. Now
we continue to investigate the situation when we replace the plane interface with
a periodically modulated interface as shown in Fig. 3.1. Periodic modulation of
the interface changes the impulsion (wavevector surface component) of the incident
wave along the surface i.e. kin,S by adding or subtracting an integer number of
grating impulses (grating vectors) D. This can be expressed by the following relation
km,S = kin,S + mD,
(3.1)
43
where D = 2(π/d)d̂ is the grating vector which directs along the unit vector d̂.
Now, if we assume that the periodicity is along x, the interface lies in the xy plane
and the incident wave lies in the xz plane as depicted in the geometry of Fig. 3.1
(i.e. non-conical mounting), we get the so-called grating equation in reflection
sin θm = sin θin + m
λin
,
d
(3.2)
where m = 0, 1, 2, . . . . Clearly, all the diffraction orders are contained in the xz
plane. In case the opto-geometric properties (refractive index and thickness) of the
interface are modulated two-dimensionally we can define two grating vectors D1
and D2 along dˆ1 and dˆ2 respectively (i.e. along the directions of periodicity) as
shown in Fig. 3.2. These create two sets of diffraction orders together with their
spatial combinations. The grating equation for the 2-D periodic case is written in
the following form
k mn,S = kin,S + mD1 + nD2 ,
(3.3)
with D1 = 2(π/d x )dˆ1 and D2 = 2(π/dy )dˆ2 .
It is worth to mention here that depending on the ratio between the grating
period d and the wavelength of the incident wave λin , diffraction gratings can be
divided into three sub-groups which are paraxial (d ≫ λin ) domain, resonance
(d ≈ λin ) domain, and quasi-static (d ≪ λin ) domain gratings [136]. In the paraxial
domain, the response of the grating becomes almost polarization independent and
the grating acts as a thin transparency [140]. In the resonance domain, the grating
response is strongly polarization dependent [141, 142]. Finally, in the quasi-static
domain the grating acts as a homogeneous anisotropic thin film with effective refractive index [143]. Both the quasi-static and the resonance domain gratings will be
discussed in detail in Chapters 4 and 5 respectively.
z
kin
y
θin
kref,0
Ein
kref,1
kref,2
d
d̂
ktra,0
ktra,1
x
ktra,2
Figure 3.1: Lamellar grating under non-conical mounting. TM polarized light is
assumed to be obliquely incident on the grating.
44
3.2
PSEUODOPERIODICITY AND GRATING EQUATIONS
Let us now consider a more general interaction geometry which is known as conical
mounting. The geometry is shown in Fig. 3.3. The plane wave with propagation
vector k forms an angle θin with z axis as before but it also forms an angle φ with
x axis. φ is known as the conical angle. Also the unit electric polarization vector
û forms an angle ψ with the plane of propagation. Considering the geometry of
y
dˆ2
dy
dˆ1
dx
x
z
Figure 3.2: Two dimensionally periodic grating geometry.
k
û
θ
φ
x
y
z
Figure 3.3: Wave propagation geometry in case of conical illumination.
45
Fig. 3.3, k can be written in the following form
k = k x x̂ + k y ŷ + k z ẑ = k0 n(sin θ cos φ x̂ + sin θ sin φŷ + cos θ ẑ),
(3.4)
where k0 is the free-space wave number and n is the refractive index of the medium.
Now we consider a laterally periodic structure with periods d x and dy along x and
y respectively as shown in Fig. 3.2. Also, we assume the periodic region to be multilayered and bounded between the planes at z = z(0) and z = z( J +1) . Complex
permittivities of the half-spaces at z < z(0) and z > z( J +1) i.e. the permittivities
of the superstrate and the substrate are ε̂(0) and ε̂( J +1) respectively. Furthermore
we assume that the complex permittivity is z-invariant inside each layer. Now, the
incident unit amplitude plane wave can be expressed as
Ein (r ) = û exp i(k x,0 x + k y,0 y + k z,0,0 z) .
(3.5)
Following the conditions of periodicity, the permittivity inside the jth layer (j =
1, . . . , J) can be written in the following form
ε̂( j) ( x, y) = ε̂( j) ( x + d x , y + dy ).
(3.6)
It then follows from the Floque-Bloch theorem [120] that the field inside the grating
region as well as in the substrate and the superstrate are pseudoperiodic i.e. laterally
periodic except for a phase factor which depends on the incident wave vector and
the transverse position. Mathematically this can be written as
E( x + dx, y + dy, z) = E( x, y, z) exp i(k x,0 d x + k y,0 dy) .
(3.7)
If we combine the pseudoperiodic field conditions in Eq. (3.7) and the angular
spectrum representation in Eqs. (2.123)–(2.124) we find that the wave vector components k x and k y can only have discrete values i.e.
k x,m = k x,0 + m2π/d x ,
(3.8)
k y,n = k y,0 + n2π/dy .
(3.9)
These equations are identical to Eq. (3.3) in component form and hold for both the
reflected and the transmitted diffraction orders. The z component of the wave vector
for (m, n)th diffraction order is given by
( j)
( j)
k z,m,n = k0 n( j) cos θm,n ,
(3.10)
where j = 0 for the incident and the reflected orders, while j = J + 1 for the transmitted orders. We can now combine Eq. (3.10) and (2.121) to deduce the threedimensional grating equation
h
h
i2
i
( j) 2
n( j) sin θn,m = n(0) sin θ (0) cos φ(0) + mλ0 /d x
h
i2
n(0) sin θ (0) sin φ(0) + nλ0 /dy .
(3.11)
The conical angles φm,n for the reflected and the transmitted orders are deduced
from the following relation
tanφm,n =
46
k y,n
n(0) sin θ (0) sin φ(0) + nλ0 /dy
= (0)
.
k x,m
n sin θ (0) cos φ(0) + mλ0 /d x
(3.12)
For non-conical incidence φ(0) = 0, if we assume 1-D periodic configuration i.e.
dy → ∞, Eq. (3.11) reduces to the conventional 1-D grating equation in transmission
( j)
n( j) sin θm = n(0) sin θ (0) + mλ0 /d,
(3.13)
where d x has been replaced by d. The 1-D periodic y invariant grating geometry for
non-conical illumination is illustrated in Fig. 3.4. Also, as the angular spectrum is
discrete which is easy to realize by combining Eqs. (3.8), (3.9) and (2.123), the angular
spectrum representations in the homogeneous regions at z < z(0) and z > z( J +1)
reduce to Rayleigh expansions [144]. Hence we arrive at the following expressions
for the reflected and the transmitted diffracted electric fields i.e. the expressions for
the fields in the homogeneous regions
(0,−)
E(0,−) (r ) = ∑ ∑ Um,n
m n
io
n
h
(0)
,
× exp i k x,m x + k y,n y − k z,m,n z − z(0)
(3.14)
( J +1,+)
E( J +1,+)(r ) = ∑ ∑ Um,n
m
(0,−)
n
io
n
h
( J +1 )
,
× exp i k x,m x + k y,n y + k z,m,n z − z( J +1)
(3.15)
( J +1,+)
where Um,n and Um,n
are the complex amplitudes of the reflected and the transmitted orders respectively.
3.3
DIFFRACTION EFFICIENCIES
Among all the experimentally measurable quantities in diffractive optics, diffraction
efficiency of a grating is one of the main quantities of interest. The diffraction
efficiency of the (m, n)th diffraction order which is written as ηm,n is defined as the
ratio of the time average of the z-component of the Poynting vector hSz,m,n (r, t)i
of the (m, n)th diffraction order and the time average of the z-component of the
Poynting vector hSz,in (r, t)i of the incident wave. Recalling Eq. (2.64) of section 2.7
we get
r
r
n ε0
µ0
1
hSz (r, t)i =
(3.16)
| Ein |2 cos θ =
| H in |2 cos θ.
2 µ0
2n ε 0
Using the definition of the diffraction efficiencies in Eq. (3.16), we can write the
efficiencies of the diffraction orders in the following form
( ( j) ) ( j) 2
( j)
h
S
(
r,
t
)i
k z,m,n Um,n ( j)
z,m,n
ηm,n =
,
(3.17)
=ℜ
hSz,in (r, t)i
k z,0,0
|Uin |2
( j)
where Um,n and Uin are the complex amplitudes of the (m, n)th diffraction order and
the incident field respectively. In Eq. (3.17) j = 0 designate the reflected orders and
j = J + 1 the transmitted orders. We can now assume the incident field to be of unit
amplitude to simplify Eq. (3.17) in the following form
( ( j) )
k z,m,n ( j) 2
hSz,m,n (r, t)i
( j)
ηm,n =
=ℜ
(3.18)
Um,n .
hSz,in (r, t)i
k z,0,0
47
In case of 1-D periodic y-invariant gratings we can define the diffraction efficiencies for TE and TM polarized light separately by decomposing Maxwell’s equations
into two sets as described in section 2.6. In non-conical illumination the diffraction
efficiencies for TE and TM polarized light can be expressed mathematically in the
following forms respectively:
( j)
ηm = ℜ
( j)
ηm = ℜ
3.4
(
( j)
k z,m
) ( j) 2
Um |Uin |2
(
) ( j) 2
( j)
n( j) k z,m Um k z,0
n(0) k z,0
|Uin |2
,
(3.19)
.
(3.20)
OVERVIEW OF THE EXISTING NUMERICAL MODELING METHODS
In all the standard full-wave numerical simulation techniques for modeling diffraction gratings, the entire three-dimensional space containing the grating is divided
into three distinct regions. The transparent semi-infinite half-space or the half space
containing the incident field located at the top i.e. at z < z(0) (the superstrate), the
periodically modulated region (which might include sub-regions) or the region containing the grating, and the bottom region at z > z( J +1) which might be opaque
to the incident field (the substrate). At first, electromagnetic fields in each region
is evaluated separately and expressed mathematically. These mathematical expressions contain unknown coefficients. Next, electromagnetic boundary conditions are
applied to the total fields at the interfaces of two subsequent regions or sub-regions
to match the electromagnetic fields at the boundaries and evaluate the unknown
coefficients by solving a set of linear algebraic equations. Values of these coefficients are used to derive quantities of interest for instance the diffraction efficiencies,
phases and polarization states of the diffracted fields, near and far field distributions
of the diffracted waves etc.
The three steps described in the above paragraph are shared by many rigorous
numerical methods for modeling diffraction gratings. Depending on the contents
of the first step (i.e. the way we solve the electromagnetic field quantities in each
region), we can classify the available methods into two groups which are spacedomain methods such as the finite-difference time-domain method (FDTD) [145] or
the finite element method [146] and spatial-frequency-domain methods, including
the differential method [147, 148] and the Fourier modal method (FMM) [149–151].
Some of the other widely used numerical methods include the integral method [152]
and the coordinate transformation method [153].
3.5
FOURIER MODAL METHOD FOR DIFFRACTION GRATINGS
In a modal method the total electromagnetic fields in each region are constructed by
means of modes. Modes can be defined as distinct, self-sustainable pattern of motion satisfying the governing law of physics and the internal boundary conditions.
Each of these grating modes satisfy Maxwell’s equations and the related internal
boundary conditions including the pseudo-periodicity conditions. The final solution of the total fields is obtained by superposing all the modes and applying the
48
external electromagnetic boundary conditions between different regions and the radiation conditions at infinity.
In the Fourier Modal Method (FMM), electromagnetic field quantities inside the
grating region are mathematically expressed in terms of Floquet-Fourier series and
the medium permittivities and permeabilities by Fourier series. Whereas, fields in
the superstrate and the substrate are written in terms of Rayleigh expansions which
we have already seen in Eqs. (3.14)–(3.15). MaxwellâĂŹs equations are first solved
separately in each z-independent layer, and then the external boundary conditions
are applied at the interfaces of two subsequent layers as discussed in the previous
paragraph to evaluate the unknown coefficients. This is efficiently done by the
recursive S-matrix propagation algorithm as discussed in section 2.17. In the FMM,
a rectangle (in general a parallelogram) is used as a building block for an arbitrary
grating profile inside a z-independent layer.
3.5.1
Fourier factorization rules
We have already mentioned in the previous section that in the FMM, complex permittivities in the modulated region are expressed in terms of Fourier series and the
field quantities in terms of Floquet-Fourier series. It was realized that to obtain better numerical convergence especially for the metallic gratings, Fourier coefficients in
the Fourier series expansions of the complex permittivities and the field quantities
should be treated in a special way. This realization led to the establishment of the
Fourier factorization rules by Lifeng Li [154]. Introduction of these rules not only
helped to improve the convergence of the FMM [155,156] but also enabled a series of
other developments in rigorous grating theory. One should notice that the complex
permittivity is discontinuous along the direction of periodicity (x direction for a 1-D
lamellar grating) and Li’s Fourier Factorization rules are needed to correctly handle
the Fourier coefficients at the boundaries of discontinuity. In section 2.4, we found
that in electromagnetic boundary conditions often we talk about the continuity of
product of two functions at the interface. For example in Eq. (2.37) we have the
continuity relation for the product of the complex permittivity and the electric field.
Correct Fourier factorization of the product of two functions should be carried out
in such cases to avoid numerical errors.
Let us now consider three periodic functions of x, f 1 ( x ), f 2 ( x ), and f 3 ( x ). The
period is d for each of these functions. Also, we assume the following relation
between these functions
f 3 ( x ) = f 1 ( x ) f 2 ( x ).
(3.21)
As the functions are periodic we can represent these in the form of Fourier series
expansion
∞
f i,n exp(i2πnx/d),
∑
fi (x) =
(3.22)
n =− ∞
for i = 1, 2, 3. The Fourier coefficients i.e. f i,n are given by
f i,n =
1
d
Z d
0
f i ( x ) exp(−i2πnx/d) dx.
(3.23)
The Fourier coefficients of f 3 ( x ) can be obtained from the Fourier coefficients of
f 1 ( x ) and f 2 ( x ) by using the Laurent’s rule:
∞
f 3,n =
∑
m =− ∞
f 1,n−m f 2,m .
(3.24)
49
Hence, the Fourier factorization of f 3 ( x ) can be expressed by
∞
f 3,n exp(i2πnx/d) =
∑
f3 (x) =
n =− ∞
∞
∞
∑
∑
n =− ∞ m =− ∞
f 1,n−m f 2,m exp(i2πnx/d).
(3.25)
Equation (3.25) contains an infinite number of Fourier coefficients of the functions f i
but to solve the diffraction problem with a computer with limited resources we can
only retain a certain number of Fourier coefficients in the above equation. However,
if we retain only up to Mth order Fourier coefficients i.e. m = − M → M in Eq. (3.24),
depending on the kind of discontinuities the functions f 1 ( x ) and f 2 ( x ) possess, the
Laurent’s rule might produce numerical errors when constructing f 3 ( x ) from the
Fourier coefficients of f 1 ( x ) and f 2 ( x ). Li examined three possible cases and made
three clear conclusions which are known as Li’s Fourier factorization rules. These
rules are listed below:
1. If f 1 ( x ) and f 2 ( x ) are piecewise smooth, bounded periodic functions and if
they do not have concurrent jump discontinuities, the product (type 1 product)
can be factorized by the truncated form of Laurent’s rule i.e.
( M)
M
f 3,n =
J f 1 Kn−m f 2,m ,
∑
(3.26)
m =− M
where J f 1 Kn−m is the symmetrically truncated Toeplitz matrix generated by the
Fourier coefficients of f 1 . The (m, n)th element of the Toeplitz matrix is f 1,n−m .
2. If f 1 ( x ) and f 2 ( x ) are piecewise smooth, bounded periodic functions and
if they have only concurrent pair-wise complementary jump discontinuities,
there product (type 2 product) cannot be factorized by the Laurent’s rule. In
such a scenario we must apply the inverse rule i.e.
( M)
f 3,n =
M
∑
1
J1/ f 1 K−
n − m f 2,m .
(3.27)
m =− M
3. If f 1 ( x ) and f 2 ( x ) are piecewise smooth, bounded periodic functions and if
they have concurrent but not pair-wise complementary jump discontinuities,
neither the Laurent’s rule nor the inverse rule holds.
3.6
FMM FOR LINEAR GRATINGS WITH PLANE WAVE ILLUMINATION
Let us now go back to the geometry in Fig. 3.4 i.e. we consider a linear y- invariant
multilayered grating under non-conical illumination. The rigorous diffraction theory for such a y- invariant but single layered structure under non-conical mounting
was first introduced by Knop [157]. Later the theory was extended for multilayered
gratings [149, 155, 156, 158–166], gratings under conical illumination [163, 167], and
finally to gratings with anisotropic materials [168, 169].
50
For the y- invariant geometry in Fig. 3.4, the complex permittivity in the j-th
layer can be expressed in the following form using Fourier series expansion
∞
ε̂( j) ( x ) =
∑
( j)
ε̂ p exp(i2πx/d),
(3.28)
p =− ∞
( j)
where ε̂ p is the pth Fourier coefficient in the expansion of ε̂( j) ( x ) and is given by
( j)
ε̂ p = (1/d)
Z d
0
ε̂( j) ( x ) exp(−i2π px/d) dx.
(3.29)
Putting p = 0 in Eq. (3.29), we can obtain the average value of the complex permittivity in the j-th layer.
3.6.1
Formulation of the eigenvalue problem
In subsection 2.6 we have already seen that if the structure is y- invariant, we can
decompose Maxwell’s equations into TE and TM polarized sets. Now as the fields in
the regions z < z(0) and z > z( J +1) can be expressed in terms of Rayleigh expansions,
for an incident field with complex amplitude Uin we have the following expressions
x
ε̂(1) ( x )
ε̂(0)
(0,−)
A1
ε̂( j) ( x )
ε̂( J ) ( x )
ε̂( J +1)
d
( J +1,+)
A1
( J +1,+)
(0,−)
A0
A0
( J +1,+)
A −1
Ain
( J +1,+)
(0,−)
A −1
A2
0
z (0) z (1)
z ( j −1 ) z ( j )
z
z ( J −1 ) z ( J )
Figure 3.4: Diffraction geometry from a multilayered y-invariant grating.
51
for the incident, the reflected and the transmitted fields
n h
io
Ey,in ( x, z) = Uin exp i k x,0 x + k z,0 z − z(0)
,
n
h
io
∞
(0,−)
(0,−)
(0)
Ey
( x, z) = ∑ Um exp i k x,m x − k z,m z − z(0)
,
(3.30)
(3.31)
m =− ∞
( J +1,+)
Ey
∞
( J +1,+)
∑
( x, z) =
Um
m =− ∞
n
h
io
( J +1 )
exp i k x,m x + k z,m
z − z ( J +1 )
,
(3.32)
( j)
where k x,m is given by Eq. (3.8) and k z,m in Eqs. (3.31) and (3.32) are given by
( j)
k z,m
(0,−)
=
( J +1,+)
q
k20 ε̂( j) − k2x,m
(3.33)
and Um , Um
are the unknown amplitudes of the reflected and the transmitted orders respectively.
Now we proceed to find the solution of the field inside the grating region. The
electric field inside the j-th layer (z( j−1) < z < z( j)) of the modulated region can be
expressed in terms of pseudo-Fourier series
∞
( j)
∑
Ey ( x, z) =
( j)
Ay,m (z) exp (ik x,m x ) ,
(3.34)
m =− ∞
( j)
where Ay,m is the amplitude of the m-th space-harmonic field which is given by
( j)
Ay,m = (1/d)
Z d
( j)
0
Ey ( x, z) exp (−ik x,m x ) dx.
(3.35)
If we substitute the expression of the complex permittivity from Eq. (3.28) and the
expression of the electric field from Eq. (3.34) into the Helmholtz equation for the
electric field for the y-invariant structure, i.e. Eq. (2.57) we get
∞
−
∑
m =− ∞
( j)
k2x,m Ay,m (z) exp (ik x,m x ) +
+k20
∞
∞
∑
∑
∂2 ( j )
A (z) exp (ik x,m x )
2 y,m
m =− ∞ ∂z
∞
∑
( j)
ε̂ p exp (i2π px/d) Ay,m (z) exp (ik x,m x ) = 0.
(3.36)
m =− ∞ p =− ∞
Let us now multiply Eq. (3.36) by the term (1/d) exp (−ik x,q ), where q is an integer
and integrate it over the grating period i.e. from x = 0 to x = d to get the following
equation
∞
∂2 ( j )
( j)
( j)
−k2x,q Ay,m (z) + 2 Ay,m (z) + k20 ∑ ε̂ q−m Ay,m (z) = 0.
(3.37)
∂z
m =− ∞
Taking a close look at the last product term of Eq. (3.37), we may conclude that Laurent’s rule i.e. Eq. (3.26) can be applied to correctly estimate the Fourier coefficients
of the product even if truncated summations are used. The reason is the following.
In TE polarization, Ey is continuous across the discontinuities along x-direction and
( j)
hence the product of ε̂ q−m and Ay,m (z) is of type 1.
52
Solution for the differential equation (3.37) has a well known form
h
i
( j)
( j)
Ay,m (z) = Am exp iγ( j) z ,
(3.38)
where γ is yet undefined constant. We can insert Eq. (3.38) into Eq. (3.37) to yield
∞
( j)
∑
m =− ∞
( j)
( j)
( j)
ε̂ q−m Am − k2x,q Aq = Aq
h
γ( j) /k0
i2
.
(3.39)
The above equation can be conveniently written in the following matrix form
h
i
h
i2
Jε̂( j) Kq−m − L x A( j) = A( j) Λ( j) /k0 ,
(3.40)
where the elements of the matrix L x are L x,q,m = k2x,q δq,m with δq,m denoting the
Kronecker delta. A( j) is the eigenvector matrix with its columns containing the
( j)
Fourier coefficients of the electric field amplitude in the j-th layer i.e. Am,p s. The
( j)
matrix Λ( j) is a diagonal matrix which contains the respective eigenvalues i.e. γ p s.
After we solve the eigenvalue problem in Eq. (3.40), we can write a single solution
of the electric field in the j-th layer. This solution is of the form
h
i
( j)
( j)
Ey,p ( x, z) = exp ±iγ p z
∞
( j)
Am,p exp(ik x,m x ).
∑
(3.41)
m =− ∞
These solutions remain invariant along z- direction and hence they closely resemble the so-called waveguide modes. However, the difference between these grating
modes and the waveguide modes lies in the fact that the grating modes do not re( j)
main confined to any specific part of the grating period. γ p s in view of Eq. (3.38)
are the propagation constants along z direction. Analogously to the plane wave
( j)
solutions of Maxwell’s equations, we may have two solutions for γ p for each eigenvalue. These two different solutions correspond to the field modes propagating
either along positive z or along negative z directions. To ensure numerical stability
while solving the diffraction problem, it is absolutely necessary to choose the sign
( j)
of γ p carefully. The rules for choosing the correct signs are as listed below
n
o
( j)
( j)
• If γ p is imaginary, we choose its sign such that ℑ γ p
> 0.
n
o
( j)
( j)
• If γ p is real, we choose its sign such that ℜ γ p
> 0.
Finally we can write the general solution of the electric field inside the j-th layer by
summing the modes propagating along +z and −z directions. The general solution
takes the following form
∞
( j)
Ey ( x, z) =
( j)
+b p exp
n
( j)
−iγ p
h
∑
p =1
z − z( j)
io
n
h
( j)
( j)
a p exp iγ p z − z( j−1)
io
!
∞
∑
( j)
Am,p exp(ik x,m x ),
(3.42)
m =− ∞
53
( j)
( j)
where a p and b p s are yet unsolved complex electric field amplitudes of the forward
propagating and the backward propagating grating modes in the j-th layer. We will
see in the next section that these complex amplitudes can be determined from the
electromagnetic boundary conditions.
In a similar way, we can proceed to write the eigenvalue equation for the TM
polarized set. Let us first write the expressions of the magnetic field outside the
grating region i.e. at z < z(0) and z > z( J +1). The incident, the reflected and the
transmitted fields can be written as
n
h
i o
Hy,in ( x, z) = Cin exp i(k x,0 x + k z,0 z − z(0) ) ,
(3.43)
n
h
io
∞
(0)
(0,−)
(0,−)
,
(3.44)
( x, z) = ∑ Cm exp i k x,m x − k z,m z − z(0)
Hy
m =− ∞
( J +1,+)
Hy
( x, z)
∞
∑
=
( J +1,+)
Cm
m =− ∞
(0,−)
io
n
h
( J +1 )
z − z ( J +1 )
,
exp i k x,m x + k z,m
(3.45)
( J +1,+)
are the unknown amplitudes of the reflected and the transwhere Cm , Cm
mitted magnetic fields respectively. To solve the field inside the grating region,
we must pay special attention to the continuity of the field components across the
boundaries of discontinuity. We can now rearrange the Helmholtz equation for the
magnetic field i.e. Eq. (2.58) such that it only contains products of type 1 or type 2.
Additionally assuming the complex permittivity inside the j-th layer of the grating
region to be z-independent we can write Eq. (2.58) in the following form
( j)
ε̂ ( x )
( j)
k20 Hy ( x, z) +
∂
∂x
∂ ( j)
1
Hy ( x, z)
(
j
)
ε̂ ( x ) ∂x
=−
∂2 ( j )
Hy ( x, z).
∂z2
(3.46)
( j)
In TM polarization, Hy ( x, z) and its z derivatives are continuous across the interfaces of discontinuity along x direction. Obviously ε̂( j) ( x ) is discontinuous across
the boundaries along x. The expression inside the curly bracket on the left hand
side of Eq. (3.46) is also discontinuous as it contains the term ε̂( j) ( x ). However, the
discontinuity jumps of the expression inside the curly bracket and that of ε̂( j) ( x ) are
concurrent and pairwise complementary. Hence their product is of type 2 and we
need to apply the inverse rule for correct Fourier factorization and to accurately estimate the Fourier coefficients of the product even if truncated summations are used.
It is easy to check that the product inside the square brackets on the left hand side of
Eq. (3.46) is also of type 2. After some lengthy though straightforward calculations
we arrive at the eigenvalue equation for TM polarized light [155]
h
i
h
i2
1
2
( j ) −1
( j)
Jζ ( j) K−
= C( j) Λ( j) ,
q− m k 0 I − L x Jε̂ Kq− m L x C
(3.47)
where ζ ( x ) = 1/ε̂( x ). The elements of the matrix L x are defined as before. C ( j)
is the eigenvector matrix with its columns containing the Fourier coefficients of the
( j)
magnetic field amplitude in the j-th layer i.e. Cm,p s. The matrix Λ( j) is a diagonal
( j)
matrix which contains the respective eigenvalues i.e. γ p s like in TE polarization. As
in TE case, we can write the general solution of the magnetic field in the following
54
form
∞
( j)
Hy ( x, z) =
( j)
+b1,p exp
( j)
n
( j)
−iγ p
h
n
h
io
( j)
( j)
a1,p exp iγ p z − z( j−1)
∑
p =1
z − z( j)
io
!
∞
∑
( j)
Cm,p exp(ik x,m x ),
(3.48)
m =− ∞
( j)
where a1,p and b1,p s are yet unsolved complex magnetic field amplitudes of the
forward propagating and the backward propagating grating modes in the j-th layer.
3.6.2
Solution of electromagnetic boundary conditions
Let us first consider a linear y- invariant binary grating i.e. a grating having only
one layer. Also, let’s assume that the grating region is located between 0 < z < h.
The continuity of the electric field across the boundaries of discontinuity along x
for TE polarized light input gives the following expressions at the boundaries z = 0
and z = h respectively
(0,−)
Uin δm,0 + Um
( J +1,+)
Um
∞
=
∑
p =1
∞
=
∑
p =1
a p + b p exp iγ p h
Am,p ,
(3.49)
a p exp iγ p h + b p Am,p .
(3.50)
From Eq. (2.40) of section 2.4, we see that the x- component of the magnetic field
over the boundaries at z = 0 and z = h is continuous. Hence we can use Eq. (2.51)
to obtain
h
i
∞ (0)
(0,−)
k z,m Uin δm,0 − Um
= ∑ a p − b p exp iγ p h Am,p ,
(3.51)
p =1
( J +1 )
( J +1,+)
k z,m Um
∞
=
∑ γp
p =1
a p exp iγ p h + b p Am,p .
(3.52)
(0,−)
Now we have four sets of equations with unknown coefficients. Solving Um
and
( J +1,+)
Um
from Eqs. (3.49) and (3.50), substituting their expressions in Eqs. (3.51) and
(3.52), and rearranging the terms we derive two sets of equations which contain the
unknown modal amplitudes in the grating region. These two sets of equations can
be written as
i
i
∞ h
∞ h
(0)
(0)
k
+
γ
A
a
+
k
−
γ
∑ z,m p m,p p ∑ z,m p exp iγ p h Am,p b p
p =1
p =1
(0)
∞
∑
p =1
h
( J +1 )
k z,m
i
− γ p exp iγ p h Am,p a p +
∞
∑
p =1
= 2k z,m Uin δm,0 ,
h
i
( J +1 )
k z,m + γ p Am,p b p = 0.
Eqs. (3.53) and (3.54) can be conveniently cast in matrix form as
i#
"h (0)
M 1 M 2 [a p ]
2k z,m Uin δm,0
=
,
M 3 M 4 [b p ]
0
(3.53)
(3.54)
(3.55)
55
with
h
i
(0)
M1,m,p = k z,m + γ p Am,p ,
h
i
(0)
M2,m,p = k z,m − γ p exp iγ p h Am,p ,
h
i
( J +1 )
M3,m,p = k z,m − γ p exp iγ p h Am,p ,
h
i
( J +1 )
M4,m,p = k z,m + γ p Am,p .
(3.56)
(3.57)
(3.58)
(3.59)
The unknown coefficients a p can be solved from the following matrix equation for
forward propagating modal amplitudes where we have replaced the coefficients b p
as a function of a p
i −1 h
i
h
(0)
a p = M 1 − M 2 M 4−1 M 3
2k z,m Uin δm,0 .
(3.60)
Once we evaluate the coefficients a p , the coefficients b p i.e. the backward propagating amplitudes can be evaluated from the relation
b p = − M 4−1 M 3 a p .
(3.61)
We can now put the known values of a p and b p in Eqs. (3.49) and (3.50) to evaluate
the reflected and the transmitted electric field amplitudes. Diffraction efficiencies
(0,−)
( J +1,+)
can be calculated using the known values of Um
and Um
and using Eq. (3.19).
Proceeding similarly we can evaluate the complex amplitudes of the magnetic field
inside and outside the grating region.
3.6.3
Solution for multilayered gratings
If the grating region is multilayered we use the S-matrix approach as described in
section 2.17 to solve the boundary-value problem. To simplify the derivation of the
( j)
S-matrix in case of multilayered gratings, let’s now introduce the quantities c p such
that
h
i
( j)
( j)
( j)
c p = a p exp iγ p h( j) ,
(3.62)
which represent the upward propagating modes at the boundary at z = z( j). The
symbol h( j) denotes the thickness of the j-th layer bounded between the boundaries
at z = z( j) and z = z( j−1). According to the electromagnetic boundary conditions,
the tangential components of the field are continuous across the boundary at z =
z( j−1). Hence, we may conveniently write the boundary-value problem in matrix
form
#
"
# "
#"
#"
( j)
c ( j −1 )
A ( j −1 )
A ( j −1 )
A( j)
A( j)
F − c( j)
=
,
(3.63)
D ( j −1 ) − D ( j −1 ) b ( j −1 )
D( j) − D( j) F (+j) b( j)
h
i
( j)
( j)
where D( j) = A( j) Λ( j) . The matrix F ± contains the elements exp ±γ p h( j) δm,p .
The above written matrix representation for the field inside the grating region is
also valid in the upper and the lower half-spaces. Hence these remain valid also
at the boundaries at z = z(0) and z = z( J ) . However, we must be careful while
56
defining the matrices in the homogeneous regions (the upper and the lower half( J +1 )
spaces). Clearly, F ±
= I as the boundary at z = z( J +1) does not exist. Also in
the homogeneous regions, γ( j) in the matrix Λ( j) corresponds to the z- component
of the wave vector. c(0) corresponds to the complex amplitudes of the incident
electric field Uin in Eq. (3.30). For unit amplitude incident field, c(0) = 1 and all the
other elements equal to zero. Additionally, b(0) and c( J +1) represent the complex
amplitudes of the reflected and the transmitted fields respectively. Furthermore, we
assume that the incident field exists only in region 0 and hence b( J +1) = 0.
As in the thin film stack problem, we need to find the S-matrix S( J +1)⇆(0) which
connects the input and the output field quantities by the following relation
"
#
"
# "
#"
#
( J +1)⇆(0)
( J +1)⇆(0)
c ( J +1 )
c (0)
c (0)
S11
S12
( J +1)⇆(0)
=S
= ( J +1)⇆(0)
,
(3.64)
( J +1)⇆(0)
b (0)
b ( J +1 )
b ( J +1 )
S21
S22
where the superscripts ( J + 1) ⇆ (0) signify that the total S-matrix must be constructed layer by layer starting from region J + 1 and proceeding towards region 0.
Starting from the sub-system S-matrix S( j+1)⇆( J +1) and employing the boundaryvalue equations we can find an expression for the sub-system matrix S ( j)⇆( J +1) in
terms of the elements of S( j+1)⇆( J +1) in a similar way as discussed in section 2.17.
Also, we can find similar expressions i.e. Eqs. (2.193)–(2.198) for the elements of
( J +1)⇆(0)
the matrix S. We note that in the homogeneous regions, the elements S12
( J +1)⇆(0)
and S22
vanish as b( J +1) = 0. We can solve for the remaining S-matrix ele-
( J +1)⇆(0)
( J +1)⇆(0)
ments S11
and S21
using Eqs. (3.63)–(3.64). After some lengthy though
straightforward calculations, we arrive at
( J +1)⇆( j −1)
S11
h
( J +1)⇆( j −1) ( j −1)
× Z 11
A
+
( J +1)⇆( j −1)
S21
( J +1)⇆( j −1) ( j −1)
= Z 21
A
( J +1)⇆( j ) ( j )
F+
i
( J +1)⇆( j −1) ( j −1)
Z 12
D
,
= S11
( J +1)⇆( j −1)
− Z 22
D ( j −1 ) ,
where the elements of the matrix Z ( J +1)⇆( j−1) are defined as
"
#
( J +1)⇆( j −1)
( J +1)⇆( j −1)
Z
Z
11
12
Z ( J +1)⇆( j−1) =
( J +1)⇆( j −1)
( J +1)⇆( j −1) =
Z 21
Z 22
"
# −1
( j ) ( J +1)⇆( j ) ( j )
A( j) [ F + S21
F + + I ] − A ( j −1 )
.
( j ) ( J +1)⇆( j ) ( j )
D ( j) [ I − F + S21
F + ] D ( j −1 )
(3.65)
(3.66)
(3.67)
Finally, the reflected and the transmitted field in the homogeneous regions are
solved using Eq. (3.64) and we get the following expressions for the reflected and
the transmitted field complex amplitudes respectively
( J +1)⇆(0)
b(0) = c(0) S21
( J +1)⇆(0)
c( J +1) = c(0) S11
= U 0,− ,
(3.68)
= U J +1,+ .
(3.69)
Once we solve the amplitudes of the electric field, we can estimate the diffraction
efficiencies using Eq. (3.19). In this context we must mention some additional properties of the S-matrix. We already know that an S-matrix which is not operating over
57
an interface does not produce any change in the field. Hence we only have the diagonal elements and these elements are equal to identity matrix I. All the off-diagonal
elements vanish for a S-matrix which is not operating over an boundary.
3.6.4
Field inside the gratings
It appears from Eq. (3.64) that the field inside the grating layers can be easily solved.
( J +1)⇆( j )
However, this is not true. The inversion of the S-matrix element S11
is unstable
and might lead to large numerical errors. Hence, we need to introduce a new Smatrix for the field calculations inside the grating layers. Let’s start from the original
field representation in Eq. (3.48) inside the j-th grating layer. We may write the
boundary value problem at z = z( j) in matrix form
"
#"
# "
#"
#
( j)
a ( j +1 )
A( j)
A( j)
A ( j +1 )
A ( j +1 )
F + a( j )
=
.
(3.70)
D( j+1) − D ( j+1) F (+j+1)b( j+1)
D( j) − D ( j)
b( j)
To avoid numerical errors, this time we start constructing the S-matrix from the 0-th
layer and proceed layer by layer to reach the region J + 1. As the field representations
as well as the way of the matrix construction are different, we define a new S-matrix
W (0)⇆( j). This W matrix connects the field in region 0 and the field inside the j-th
layer of the grating. The matrix relation reads as
"
#
"
# "
#"
#
(0)⇆( j )
(0)⇆( j )
(0)
a( j)
a(0)
W 11
W 12
(0)⇆( j ) a
=W
=
.
(3.71)
(0)⇆( j )
(0)⇆( j )
b(0)
b( j)
b( j)
W
W 22
21
As before we start from the sub-system S-matrix W (0)⇆( j) and employing the
boundary-value equations proceed to find expression for the sub-system matrix
W (0)⇆( j+1) in terms of the elements of the S-matrix W (0)⇆( j). We assume that there
is no incident field in the region J + 1, to obtain the following expression for the
amplitudes of the backward propagating field. Hence, using Eqs. (3.62) and (3.64)
we get
( J +1)⇆( j ) ( j ) ( j )
F+ a ,
b( j) = S21
(3.72)
The forward propagating field amplitudes can be deduced similarly using Eqs. (3.71)
and (3.72)
h
i
(0)⇆( j ) (0)
(0)⇆( j ) ( J +1)⇆( j ) ( j ) −1
W 11
a .
(3.73)
a( j) = I − W 12
S21
F+
(0)⇆( j )
(0)⇆( j )
The matrices W 11
and W 12
can be derived using Eqs. (3.70) and (3.71).
Again after some lengthy though straightforward calculations, we arrive at
h
i
(0)⇆( j +1)
(0)⇆( j +1) ( j )
(0)⇆( j +1) ( j )
( j ) (0)⇆( j )
W 11
= − Y 11
A + Y 12
D
F + W 11
,
(3.74)
h
i
(0)⇆( j +1)
(0)⇆( j +1) ( j +1)
(0)⇆( j +1) ( j +1)
( j +1 )
W 12
= Y 11
A
− Y 12
D
F+ ,
(3.75)
where the elements of the matrix Y (0)⇆( j+1) are defined as
"
#
(0)⇆( j +1)
(0)⇆( j +1)
Y
Y
11
12
Y (0)⇆( j+1) =
(0)⇆( j +1)
(0)⇆( j +1) =
Y 21
Y 22
"
# −1
( j ) (0)⇆( j )
− A ( j+1) A( j) [ F + W 12
+ I]
.
( j ) (0)⇆( j )
− D( j+1) D( j) [ F + W 12
− I]
58
(3.76)
From Eqs. (3.72) and (3.73) we can see that the only non-vanishing element of the
( J +1)⇆( j )
( J +1)⇆( j )
S-matrix S( J +1)⇆( j) is S21
. Hence without solving the elements S11
can solve the complex amplitudes of the transmitted field from Eq. (3.75).
3.7
, we
FMM FOR ANISOTROPIC CROSSED GRATINGS
In this section, we shall include the formulation for the most general FMM for
anisotropic media where the permittivities and the permeabilities are tensors. Also
to keep the most general form of the FMM, we assume a slanted coordinate system
as shown in Fig. 3.5. None of the coordinate axes are perpendicular to each other.
This formulations is based on Lifeng Li’s article published in 2003 i.e. Ref. [170]. Let
us first introduce the covariant and the contravariant basis vectors [150] û i and û j
such that û i · û j = δij , where δij is the Kronecker delta symbol.
In the slanted coordinate system of Fig. 3.5 these covariant and contravariant
z = x̄3
Θ
x3
y = x̄2
ζ
x2
Φ
x = x̄1 , x1
Figure 3.5: Slanted coordinate system.
tensors can be written in the form
û1 = x̂,
û2 = x̂ sin ζ + ŷ cos ζ,
(3.77a)
(3.77b)
û3 = x̂ sin Θ cos Φ + ŷ sin Θ sin Φ + ẑ cos Θ,
(3.77c)
û1 = x̂ − ŷ tan ζ + ẑ tan Θ(sin Φ tan ζ − cos Φ),
(3.78a)
2
û = ŷ sec ζ − ẑ tan Θ sin Φ sec ζ,
3
û = ẑ sec Θ,
(3.78b)
(3.78c)
59
x1 = x − y tan ζ + z tan Θ(sin Φ tan ζ − cos Φ),
2
x = y sec ζ − z tan Θ sin Φ sec ζ,
3
x = z sec Θ.
(3.79a)
(3.79b)
(3.79c)
The contravariant elements of the permittivity tensor in terms of the permittivity
tensor elements ε̂¯τχ in the Cartesian system take the form
ε̂ρσ =
∂x ρ ∂x σ ¯τχ
ε̂ ,
∂ x̄ τ ∂ x̄ χ
(3.80)
where x̄1 = x, x̄2 = y, and x̄3 = z are the Cartesian coordinates and we have used
Einstein’s summation notation, i.e. we have used summation with respect to a pair
of identical covariant and contravariant indices. Proceeding analogously for the
relative permeability tensor we find
∂x ρ ∂x σ ¯ τχ
µ̂ .
(3.81)
∂ x̄ τ ∂ x̄ χ
Here we emphasize the fact that though the relative permeability tensor µ = I at
optical frequencies, for non-orthogonal coordinate systems we also have non-zero
off-diagonal elements. Consequently, permeability tensor is no more an identity
tensor. Also, we point out that mathematically there is no difference between material anisotropy and anisotropy arising from the coordinate system. To keep the
derivations more general, we assume physically anisotropic media together with
the slanted coordinate system. For isotropic media ε̂¯τχ = ε̂δτ,χ ,µ̂¯ τχ = µδτ,χ and
Eqs. (3.80) and (3.81) reduce to
µ̂ρσ =
ε̂ρσ = ε̂gρσ ,
µ̂ρσ = µgρσ ,
(3.82a)
(3.82b)
where
∂x ρ ∂x σ
(3.83)
∂ x̄ τ ∂ x̄ χ
is the covariant metric tensor of the Cartesian system. Denoting its (gρσ ) reciprocal
by g, we get from Eq. (3.78c), g = cos2 ζ cos2 Θ. We now proceed to derive the
Eigenvalue equations. As before, we assume the optical properties of the material
to be invariant along x3 direction in a single layer j. In the following expressions we
drop the index j as well as the position dependence for the sake of brevity.
At first, we express space-frequency domain Maxwell’s equations i.e. Eqs. (2.13)–
(2.16) in covariant forms
gρσ =
∂
√
Eτ = ik0 gµρσ Nσ ,
∂x σ
∂
√
κ ρστ σ Nτ = −ik0 gε̂ρσ Eσ ,
∂x
∂ τσ
ε̂ Eσ = 0,
∂x τ
∂ τσ
µ Nσ = 0,
∂x τ
is the Levi-Civita symbol which is defined as
κ ρστ
where κ ρστ
κ 123 = κ 231 = κ 312 = 1,
κ ρστ = 0
60
κ 321 = κ 213 = κ 132 = −1,
for all other combinations,
(3.84a)
(3.84b)
(3.84c)
(3.84d)
(3.85)
p
and Nσ = µ0 /ε 0 Hσ . As the fields inside the modulated regions are pseudoperiodic, we can express them in terms of Pseudo-Fourier series as we did in case of yinvariant gratings
Eσ ( x1 , x2 , x3 ) = ∑ ∑ Eσpq ( x3 ) exp ik1,p x1 + ik2,q x2 ,
(3.86a)
p
q
Nσ ( x1 , x2 , x3 ) = ∑ ∑ Nσpq ( x3 ) exp ik1,p x1 + ik2,q x2 ,
p
(3.86b)
q
where Eσpq ( x3 ) and Nσpq ( x3 ) are the Fourier coefficients of the electric and the magnetic fields respectively. Clearly, these coefficients are x 3 dependent. From elementary tensor theory we know that the covariant components of a vector a( x1 , x2 , x3 )
i.e. av , where v is either ρ or σ, is tangential to the coordinate surface x τ = x0τ
at a fixed point (x01 , x02 , x03 ), whereas the contravariant component aτ is the normal
component at the same position. Hence recalling Eq. (2.26) of section 2.3, the contravariant components D v of the electric displacement can be written as
D1 = ε̂11 E1 + ε̂12 E2 + ε̂13 E3 ,
D2 = ε̂21 E1 + ε̂22 E2 + ε̂23 E3 ,
D3 = ε̂31 E1 + ε̂32 E2 + ε̂33 E3 .
(3.87a)
(3.87b)
(3.87c)
Clearly, the contravariant component D1 of the electric displacement is continuous
along x1 direction. Additionally, the covariant electric field components E2 and E3
are continuous along x1 .
We now write Eqs. (3.87a)–(3.87c) in the form that these includes only type 1 and
type 2 products (we introduced these two types of products while describing Li’s
Fourier factorization rules in section 3.5.1) along x1 direction.
12 12 ε̂
ε̂
D1 = ε̂11 E1 + 11 E2 + 11 E3 ,
(3.88a)
ε̂
ε̂
21 ε̂
ε̂21 ε̂12
ε̂21 ε̂13
1
22
23
D2 =
D
+
ε̂
−
E
+
ε̂
−
E3 ,
(3.88b)
2
ε̂11
ε̂11
ε̂11
31 ε̂
ε̂31 ε̂12
ε̂31 ε̂13
3
1
32
33
D =
D + ε̂ − 11
E2 + ε̂ − 11
E3 .
(3.88c)
ε̂11
ε̂
ε̂
We can conveniently cast Eqs. (3.88a)–(3.88c) in the following block matrix form
 1
 
D
E1
 D2  = Q  E2  ,
(3.89)
E3
D3
where [169]

⌈ξ 11 ⌉−1
Q =  ⌈ε̂21 ξ 11 ⌉⌈ξ 11 ⌉−1
⌈ε̂31 ξ 11 ⌉⌈ξ 11 ⌉−1
⌈ξ 11 ⌉−1 ⌈ε̂12 ξ 11 ⌉
⌈ε̂21 ξ 11 ⌉⌈ξ 11 ⌉−1 ⌈ε̂12 ξ 11 ⌉ + ⌈ε̂22 − ε̂21 ε̂12 ξ 11 ⌉
⌈ε̂31 ξ 11 ⌉⌈ξ 11 ⌉−1 ⌈ε̂12 ξ 11 ⌉ + ⌈ε̂32 − ε̂31 ε̂12 ξ 11 ⌉

⌈ξ 11 ⌉−1 ⌈ε̂13 ξ 11 ⌉
⌈ε̂21 ξ 11 ⌉⌈ξ 11 ⌉−1 ⌈ε̂13 ξ 11 ⌉ + ⌈ε̂23 − ε̂21 ε̂13 ξ 11 ⌉ .
⌈ε̂31 ξ 11 ⌉⌈ξ 11 ⌉−1 ⌈ε̂13 ξ 11 ⌉ + ⌈ε̂33 − ε̂31 ε̂13 ξ 11 ⌉
(3.90)
61
⌈·⌉ denotes a Toeplitz matrix formed by the Fourier coefficients in x1 direction, and
ξ ij s are defined as ξ ij = 1/ε̂ij .
To simplify the notations, let us now introduce the following operator notation. We consider an arbitrary matrix O with its elements given by Oρσ , where
(ρ, σ)=(1, 2, 3). Now we define the operator lτ± in such a way that when it operates
on O we get P = lτ± (O) where
 ττ −1
(O ) ,
if ρ = τ, σ = τ



(Oττ )−1Oτσ ,
if ρ = τ, σ 6= τ
P ρσ =
(3.91)
ρτ
ττ
−
1
O ( O ) ,
if ρ 6= τ, σ = τ


 ρσ
O ± Oρτ (Oττ )−1Oτσ , otherwise.
In the above definitions we have assumed that the inverse of the matrix Oττ exists
and Oρσ are either scalars or square matrices. From the above definitions it is easy
to check that lτ+ lτ− (O) = lτ− lτ+ (O) = O. Furthermore, we define the operator Fτ
such that when it operates on an arbitrary matrix (block or ordinary) with elements
P ρσ ( x1 , x2 , x3 ), the resulting matrix R = Fτ (P ) is a block matrix with its elements
Rρσ being Toeplitz matrices generated by the Fourier coefficients of P ρσ ( x1 , x2 , x3 )
with respect to x τ . Clearly the operator Fτ changes the dimensions of the operand
matrix but the operators lτ± retain the dimensions of the operand matrix. Equation
(3.90) can be written in a compact form using the operator notations introduced
above
Q = L1 (ε̂),
(3.92)
where
Lτ = lτ+ Fτ lτ− .
(3.93)
Now, we go back to Eq. (3.88b) and rewrite this in the following form
D2 = Q21 E1 + Q22 E2 + Q23 E3 .
(3.94)
The field components D2 , E1 , and E3 are continuous along x2 direction. In the
appendix of Ref [170], Li has shown that the Fourier coefficients of the components
D2 , E1 , and E3 calculated along x1 direction are also continuous along x2 direction.
Thus, we may conclude that the vectors D2 , E1 , and E3 are also continuous along x2
direction. Multiplying the left hand side of Eq. (3.94) with (Q22 )−1 from left we get
h
i
h
i
E2 = ( Q22 )−1 D2 − ( Q22 )−1 Q21 E1 − ( Q22 )−1 Q23 E3 .
(3.95)
Eliminating the discontinuous E2 from Eqs. (3.88a) and (3.88c) we can rewrite these
in the following form
h
i
h
i
D1 = Q12 ( Q22 )−1 D2 + Q11 − Q12 ( Q22 )−1 Q21 E1
h
i
+ Q13 − Q12 ( Q22 )−1 Q23 E3 ,
(3.96)
h
i
h
i
D3 = Q32 ( Q22 )−1 D2 + Q31 − Q32 ( Q22 )−1 Q21 E1
i
h
(3.97)
+ Q33 − Q32 ( Q22 )−1 Q23 E3 ,
Examining Eqs. (3.95)–(3.97) we see that all the products are of type 1 because the
elements of the vectors are continuous along x2 direction. Hence, we can apply
Laurent’s rule for the Fourier factorization.
62
First we calculate the Fourier coefficients of all the vectors appearing in Eqs. (3.95)–
(3.97). From Eq. (3.95) we can now solve the expression of Dτ as
ρ
Dmn =
ρσ
∑ νmn,pq Eσ,pq,
(3.98)
p,q
where
ν = L2 L1 (ε̂).
(3.99)
ρσ
In the notation νmn,pq , the first index of each pair (m and p) correspond to the Fourier
coefficients along x1 direction while second index of each pair (n and q) correspond
to the Fourier coefficients along x2 direction.
Using Eqs. (3.84b), (3.89), and (3.98) we get
k y,n N3,mn + i
∂
1σ
N2,mn = −k∗0 ∑ νmn,pq
Eσ,pq ,
∂x3
p,q
(3.100a)
−k x,m N3,mn − i
∂
2σ
N1,mn = −k∗0 ∑ νmn,pq
Eσ,pq ,
∂x3
p,q
(3.100b)
2σ
k x,n N2,mn − k y,n N1,mn = −k∗0 ∑ νmn,pq
Eσ,pq ,
(3.100c)
p,q
√
where k∗0 is defined as k∗0 = k0 g. Proceeding analogously, we can derive the
relations between the magnetic induction and the magnetic field which gives
ρ
ρσ
Tmn = ∑ β mn,pq Nσ,pq ,
(3.101)
p,q
where Tσ =
p
µ0 /ε 0 Bσ and
β = L2 L1 ( µ ) .
(3.102)
Proceeding as before we now obtain
k y,n E3,mn + i
∂
E2,mn = −k∗0 ∑ β1σ
mn,pq Nσ,pq ,
∂x3
p,q
(3.103a)
−k x,m E3,mn − i
∂
E1,mn = −k∗0 ∑ β2σ
mn,pq Nσ,pq ,
∂x3
p,q
(3.103b)
k x,n E2,mn − k y,n E1,mn = −k∗0 ∑ β2σ
mn,pq Nσ,pq .
(3.103c)
p,q
From Eqs. (3.100a)–(3.100c) and Eqs. (3.103a)–(3.103c) we can now eliminate E3,mn
and N3,mn respectively to obtain first-order differential equations along x3 direction.
As in case of a y- invariant linear grating we can express the differential equations
as an eigenvalue problem which can be written as
MG = γG.
(3.104)
3
where
h γ
n corresponds
oi to the propagation constants in the x dependent term 3
3,
(
j
)
exp iγ x − x
,
 
E1
 E2 

G=
(3.105)
 N1 
N2
63
and
−µ̃23Y − X ǫ̃31

µ̃13Y − Y ǫ̃31
M=
−k∗ ǫ̃21 − (1/k∗ ) X µ̃33Y

0
−µ̃23 X − X ǫ̃32
−µ̃13 X − Y ǫ̃32
∗
−k0 ǫ̃22 + (1/k∗0 ) X µ̃33 X
k∗0 ǫ̃12 + (1/k∗0 )Y µ̃33 X
0
k∗0 ǫ̃11 − (1/k∗0 )Y µ̃33Y
k∗0 µ̃21 + (1/k∗0 ) X ǫ̃33Y
−k∗0 µ̃11 + (1/k∗0 )Y ǫ̃33Y
−ǫ̃23Y − X µ̃31
ǫ̃13Y − Y µ̃31

k∗0 µ̃22 − (1/k∗0 ) X ǫ̃33 X
−k∗0 µ̃12 − (1/k∗0 )Y ǫ̃33 X 
.

ǫ̃23 X − X µ̃32
13
32
−ǫ̃ X − Y µ̃
(3.106)
where ( X )mn,pq = k x,m δmp δnq , (Y )mn,pq = k y,n δmp δnq and
ǫ̃ = l3− (ν),
µ̃ = l3− ( β).
(3.107)
Clearly from Eqs. (3.104) and (3.106), the total number of eigenvalues depend on
the truncation order i.e. on the total number of diffraction orders along x1 and
x2 directions respectively. Let us denote the truncation orders along x1 and x2 by
P1 and P2 respectively. Hence, the size of the matrix M becomes 4P1 P2 × 4P1 P2 .
Also, the number of the grating modes inside the structure as well as the number
of eigenvalues is 4P1 P2 × 4P1 P2 . Each eigenvalue gives the propagation constant of
a mode propagating either along + x3 direction or along − x3 direction inside the
j-th layer. To keep the S-matrix recursion stable, we divide the eigenvalues into two
sets following the rules introduced in section 3.6.1. The first set corresponds to the
modes propagating along + x3 direction whereas the second set corresponds to the
modes propagating along − x3 direction (we denote these modes by the superscripts
+/−). It is worth to mention that it is better to have an equal number of elements in
the two sets to ensure improved numerical stability. Each mode has certain spatial
structure governed by its corresponding eigenvector. The expressions of the electric
and the magnetic fields inside the j-th grating layer can be written in the following
forms
n
h
io
( j)
( j ) ( j,+)
( j,+)
Eσ ( x1 , x2 , x3 ) = ∑ ∑ ∑ uq Eσ,mn,q exp iγq
x3 − x3,( j)
m n
n
q
h
x3 − x3,( j)
io
h
x3 − x3,( j)
io
exp i k1m x1 + k2n x2 ,
n
h
io
( j)
( j ) ( j,+)
( j,+)
Nσ ( x1 , x2 , x3 ) = ∑ ∑ ∑ uq Nσ,mn,q exp iγq
x3 − x3,( j)
( j ) ( j,−)
+dq Eσ,mn,q exp
( j,−)
iγq
m n
( j)
( j,−)
n
q
( j,−)
+dq Nσ,mn,q exp iγq
(3.108)
( j)
( j)
exp i k1m x1 + k2n x2 ,
(3.109)
where the indices σ = (1, 2, 3), uq and dq are the forward and the backward
propagating yet unknown modal amplitudes. As before, these modal amplitudes
can be solved using S-matrix algorithm.
After splitting the modes equally into two sets, we have 2P1 P2 number of modes
j,±
j,±
in each set and for each element Eσ,mn,q or Nσ,mn,q we have P1 P2 number of Fourier
coefficients. Hence all the eigenvectors can be arranged into four 2P1 P2 × 2P1 P2
64
matrices Rj, where j = 1, 2, 3, 4, such that
h
i
( j,+)

 E1,q
,
p = 1, 2, . . . P1 P2 ,
h
i
R1( j)
= h ( j,+) i p

pq
,
p = P1 P2 + 1, P1 P2 + 2, . . . 2P1 P2 ,
 E2,q
,
(3.110)
,
(3.111)
,
(3.112)
p
h
h
R2( j)
R3( j)
h
i
i
R4( j)
h
i
( j,−)

 E1,q
,
= h ( j,−) i p

,
 E2,q
pq
p
h
i
( j,+)

 N 1,q
,
= h ( j,+) i p

,
 N 2,q
pq
i
p
pq
p = 1, 2, . . . P1 P2,
p = P1 P2 + 1, P1 P2 + 2, . . . 2P1 P2 ,
h
i
( j,−)

,
 N 1,q
= h ( j,−) i p

,
 N 2,q
"
( j,+)
p = P1 P2 + 1, P1 P2 + 2, . . . 2P1 P2 ,
p = 1, 2, . . . P1 P2 ,
R1( j)
R3( j)
(3.113)
p = P1 P2 + 1, P1 P2 + 2, . . . 2P1 P2 .
p
In other words,
where E1
p = 1, 2, . . . P1 P2 ,

( j,+)
( j,−)
E1

E1
( j,−) 

# 
 E( j,+)
R2( j)
 2
=
 ( j,+)
R4( j)
N 1
( j,+)
N2
E2
( j,−)  ,
N1
( j,−)
N2
(3.114)


( j,+)
is a matrix of size P1 P2 × 2P1 P2 with its columns given by E1,q . The
( j,+)
size of the column vector E1,q is P1 P2 × 1.
Using the notations introduced above, we can write the continuity relation at the
interface between layers j and j + 1. Using Eqs. (3.108) and (3.109) we get
"
R1( j)
R3( j)
R2( j)
R4( j)
#"
# "
u( j)
R1( j+1)
( j) =
d
R3( j+1)
R2( j+1)
R4( j+1)
#"
j +1,+
F−
0
0
j +1,−
F+
#
#
u ( j +1 )
× ( j +1 ) ,
d
"
where
h
i
j +1,±
F−
pq
h
i
j,±
= δpq exp −iγ p h3 ,
h
j +1,±
F+
i
pq
h
i
j,±
= δpq exp iγ p h3 ,
(3.115)
(3.116)
and h3 = x3,( j+1) − x3,( j). The modal amplitudes in the j + 1th and the J + 1th layers
are connected by the S-matrix (S( j+1)⇆( J +1)) relation
#
"
#
"
( j +1 )
u ( J +1 )
( j +1)⇆( J +1) u
=
(3.117)
=S
d ( J +1 )
d ( j +1 )
#
"
( j +1)⇆( J +1) ( j +1)
( j +1)⇆( J +1) ( J +1)
Suu
u
+ Sud
d
(3.118)
( j +1)⇆( J +1) ( j +1)
( j +1)⇆( J +1) ( J +1) ,
Sdu
u
+ Sdd
d
65
Now we can write the expressions for the elements of the S-matrix as we did in the
case of thin-film stack in section 2.17.
( j )⇆( J +1)
Suu
×
( j +1)⇆( J +1) j +1,+
F+
−Suu
h
h
( j )⇆( J +1)
Z uu
R1( j) +
( j )⇆( J +1)
Sud
( j )⇆( J +1)
Z du
( j )⇆( J +1)
( j +1)⇆( J +1) j +1,+
F+
i
( j )⇆( J +1)
Z ud
R3( j) ,
(3.119)
= Suu
( j +1)⇆( J +1)
= Sud
( j )⇆( J +1)
i
R4( j+1) ,
(3.120)
( j +1)⇆( J +1)
(3.122)
R2( j+1) + Z dd
( j )⇆( J +1)
( j )⇆( J +1)
S
= Z du
R1( j) + Z dd
R3( j),
h du
i
( j )⇆( J +1)
( j )⇆( J +1)
( j )⇆( J +1)
j +1,−
Sdd
= − Z du
R2( j+1) + Z dd
R4( j+1) F +
(3.121)
,
×S dd
where
Z ( j)⇆( J +1) =
"
"
( j )⇆( J +1)
Z ud
Z du
Z dd
( j )⇆( J +1)
j +1,− ( j +1)⇆( J +1) j +1,+
Sdu
F+
(
j +1) j +1,− ( j +1)⇆( J +1) j +1,+
R4
F+
Sdu
F+
R1( j+1) + R2( j+1) F +
R3( j+1) +
( j )⇆( J +1)
Z uu
( j )⇆( J +1)
− R2( j)
− R4( j)
#
# −1
.
(3.123)
Equations (3.119)–(3.122) can be used to derive the complex amplitudes of the diffraction orders. Noting d( J +1) = 0 as before, we can find the following expressions of
the fields in regions j = 0 and j = J + 1 respectively
n
h
io
( j)
( j,+)
( j,+)
Eσ ( x1 , x2 , x3 ) = ∑ ∑ Eσ,mn exp ik3,mn x3 − x3,( j)
m n
( j,−)
+ Eσ,mn exp
n
( j,−)
+ Nσ,mn exp
n
h
io
x3 − x3,( j)
exp i k1m x1 + k2n x2 ,
n
h
io
( j)
( j,+)
( j,+)
Nσ ( x1 , x2 , x3 ) = ∑ ∑ Nσ,mn exp ik3,mn x3 − x3,( j)
where x3,( J +1) = x3,( J ).
( j,−)
ik3,mn
m n
h
io
( j,−)
ik3,mn x3 − x3,( j)
exp i k1m x1
( j,±)
k3,mn =
+ k2n x2 ,
i
1 h 3,( j)
±k mn − g13 k1m − g23 k2n ,
33
g
(3.124)
(3.125)
(3.126)
and
3,( j )
k mn
=
g
33
h
k
( j)
i2
− k21m g11
− k22n g22
− 2k1m k2n g
12
1/2
+ k1m g13 + k2n g23
(3.127)
are the third covariant and contravariant tensors components respectively of the
wave vectors of the diffraction orders.
66
3.8
STAIRCASE APPROXIMATION
In the previous sections, while formulating the eigenvalue problems inside the j-th
grating layer, we have always assumed the layer to be x3 or z independent. To solve
the multilayered grating problems i.e. to connect the fields in several z- independent
layers, we have used recursive S-matrix algorithm. If the grating profile is arbitrary
along x3 direction, we slice the grating region into many layers parallel to the grating plane and in each layer the medium boundary is locally substituted with an
x3 -invariant boundary. This type of approximation is known as the staircase approximation. As the number of slices tends to infinity and the thickness of a single slice
tends to zero, the modified structure tends the original one. Hence, the approximation seems to be reasonable.
However, several numerical experiments have demonstrated that for 1-D periodic
gratings, the staircase approximation produces accurate results only if the incident
light is TE polarized. For TM polarization and especially for highly conducting
metallic gratings, this approximation produces large numerical errors. At the sharp
edge of a wedge shaped structure, the electric field component parallel to the edge
direction is finite but the component transverse to the edge direction becomes infinite [171]. In staircase approximation, we artificially create many such edges. Hence,
for TM polarized incident field, the electric near field which should be finite becomes
infinite at the edges of the staircase boundary. This alters the total near as well as the
far fields. For one-dimensional gratings in conical mounting and crossed gratings
under any incidence condition, such type of numerical errors are unavoidable.
Also in terms of computing efficiency, staircase approximation makes the solution of the grating problem inefficient as the computation time as well as the
computing resources grow almost linearly with the total number of z-independent
slices. On the contrary, use of a larger number of slices produces more accurate
results. Thus we need a trade-off between these two. Nevertheless, the advantage
of the staircase approximation is that if it produces accurate numerical results, it is
easy to implement algorithmically.
3.9
SUMMARY
This chapter covers from the basic diffraction grating principles to the detailed rigorous mathematical formulations needed to treat these. Though several rigourous
methods exist to date for modeling diffraction gratings accurately, in this chapter
we have restricted ourselves only to the detailed mathematical formulations of the
FMM for both one-dimensionally periodic structures and two-dimensionally periodic structures with anisotropic materials (which is the most general FMM). The
one-dimensional formulation will be helpful to explain the theory of form birefringence in Chapter 5. The most general anisotropic FMM formulation will be useful
for the next chapter where we shall talk about modeling optical Kerr nonlinearity
by the linear FMM. Inclusions of the effects of periodicity along the third direction
is out of scope for this thesis.
67
4
Light propagation in Periodic media with optical Kerr
nonlinearity
After the discovery of lasers in 1960 [172], it had become possible to study the
behavior of light in optical materials at higher intensities than previously possible.
This gave birth to the field of nonlinear optics, where the term ’nonlinear’ denotes
that the response of a medium to an externally applied field is a nonlinear function
of the field. Since the discovery of second harmonic generation in 1961 [40], the field
of nonlinear optics has flourished rapidly.
Certain nonlinear processes arise when the response of a material varies with
the incident electric field to the third power (a cubic relationship). These are known
as third-order processes. The optical Kerr effect, where the refractive index of a
material depends on the intensity of light used to measure it, is one of these thirdorder nonlinear optical processes [134].
Besides the linear optical properties, nonlinear optical properties of a material
can also be tailored by nanostructuring. In the last few decades, a wide variety of
device applications of optical Kerr nonlinearity have emerged, especially in the context of integrated optics (IO) [173]. Some of these include power-dependent grating
and prism couplers [174–176], directional couplers [177], Mach-Zehnder interferometers [178], and all-optical switching devices [179,180], couplers, and gates [177,178].
Many of these nonlinear IO devices use periodically varying refractive-index profiles. Examples include optically-tunable filters, multiplexers [181], and distributedfeedback bistable optical devices [85, 182, 183]. However, free-space diffractive elements employing Kerr materials have also been proposed [184] which can be strictly
periodic (gratings) or non-periodic, such as Fresnel zone plates and other diffractive
lenses.
In this Chapter, we introduce a FMM based technique for modeling light propagation in periodic structures with optical Kerr nonlinear media and verify the accuracy of the proposed technique by comparing our results with those obtained
by other standard numerical approaches such as the differential method and the
FDTD [185, 186]. After that, we discuss about the procedure to increase the computational efficiency of our method by use of symmetries. Finally, we include several
numerical examples to demonstrate the versatility of the FMM based technique.
These examples include the roles of surface plasmon resonance and waveguide resonance on the enhancement of the optical Kerr effect in nanostructured materials.
4.1
LIGHT PROPAGATION IN ISOTROPIC THIRD ORDER NONLINEAR MATERIALS
In section 2.3, we have seen that an optical medium responds in a complicated way
if the incident field is strongly intense. Assuming the medium to be lossless and
non-dispersive (hence also instantaneous), the ith component, where i stands for the
cartesian indices 1,2, or 3 (alternatively x,y, or z), of time-dependent macroscopic
69
electric polarization can be written as a power series expansion
h
(1)
(2)
Pi (t) = ε 0 χij Ej (t) + χijk Ej (t) Ek (t)
i
(3)
+χijkl Ej (t) Ek (t) El (t) + . . .
(1)
(2)
(3)
= Pi (t) + Pi (t) + Pi (t) + . . .
(4.1)
where χ (n) (the nth order susceptibility) is a tensor of rank (n + 1) which contains
3(n+1) elements in general. In particular
(1)
(1)
(2)
(2)
(3)
(3)
Pi (t) = ε 0 χij Ej (t), Pi (t) = ε 0 χijk Ej (t) Ek (t), Pi (t) = ε 0 χijkl Ej (t) Ek (t) El (t)
are the linear, the second-order and the third-order nonlinear macroscopic polarizations respectively. Usually, the physical processes which occur as a result of the
second-order polarization are distinct from those resulting from the third-order polarization.
(3)
In general, the third-order susceptibility χijkl (r ) is a fourth-rank tensor containing 81 distinct elements. In an isotropic medium, the axes 1, 2, or 3 are equivalent.
As a result, the tensor contains only 21 non-zero elements. It is easy to show that
for an isotropic material the components of the susceptibility tensor possess the
following symmetry conditions:
χ1111 = χ2222 = χ3333,
χ1122 = χ1133 = χ2211 = χ2233 = χ3311 = χ3322,
χ1212 = χ1313 = χ2323 = χ2121 = χ3131 = χ3232,
χ1221 = χ1331 = χ2112 = χ2332 = χ3113 = χ3223.
(4.2)
Also, one can show, by requiring that the values of the nonlinear polarization are
the same when calculated in two different cartesian coordinate systems which are
rotated with respect to one another, that the tensor components χ1111 , χ1122, χ1212,
and χ1221 are related by the following equation
χ1111 = χ1122 + χ1212 + χ1221.
(4.3)
Finally, the third-order susceptibility of an isotropic third-order nonlinear material
can be conveniently written in the following compact form
χijkl = χ1122 δij δkl + χ1212 δik δjl + χ1221 δil δjk ,
(4.4)
which shows that when the field frequencies are arbitrarily chosen, the third-order
susceptibility tensor contains only three independent elements.
4.2
THEORY OF THE OPTICAL KERR EFFECT IN ISOTROPIC MEDIA
The refractive index of a third-order nonlinear material strongly depends on the
intensity of the light used to measure it. This phenomenon is termed as the optical
Kerr effect by analogy with the traditional Kerr electrooptic effect discovered in
1875 by a Scottish physicist John Kerr [187], where the refractive index change of a
material is proportional to the square of the static electric field applied across it.
70
Let us now consider that an intense wave of frequency ω is propagating in an
isotropic third-order nonlinear medium. If we assume the simple case in which the
applied field is monochromatic and is given by
E(t) = U cos ωt,
(4.5)
the time dependent third-order contribution to the nonlinear polarization can be
written as
P (3) ( t ) = ε 0 χ (3) E ( t )3 .
(4.6)
Now, using the well-known trigonometric identity cos3 ωt = (1/4) × cos 3ωt +
(3/4) cos ωt, we can write Eq. (4.6) in the form
P (3) ( t ) =
1
3
ε 0 χ(3)U 3 cos 3ωt + ε 0 χ(3)U 3 cos ωt.
4
4
The first term in Eq. (4.7) describes nonlinear response at frequency 3ω and leads
to third-harmonic generation. The second term describes a nonlinear effect at the
frequency of the incident field. Hence, this term leads to a nonlinear contribution
to the refractive index experienced by the incident wave of frequency ω, which is
known as the optical Kerr effect. The two cases mentioned above are physically
distinct from one another though might be exhibited in the same material at the
same time. The choice of the frequencies to find expressions for the respective
susceptibility tensors for these two instances can be conventionally described by
(3)
χijkl (3ω = ω + ω + ω ), for third-harmonic generation,
(3)
χijkl (ω = ω + ω − ω ), for the optical Kerr effect,
(4.7)
where for the case of third-harmonic generation, one may imagine that three photons at frequency ω are destroyed and a single photon at frequency 3ω is created.
We now go back to Eq. (4.4). For the second instance of Eq. (4.7), if we use
the intrinsic permutation symmetry of the cartesian indices, which states that the
product of Ej (ω ) and Ek (ω ) is commutable, we obtain
(3)
(3)
χ1122 (ω = ω + ω − ω ) = χ1212 (ω = ω + ω − ω ).
(4.8)
Hence, Eq. (4.4) reduces to
(3)
(3)
χijkl (ω = ω + ω − ω ) = χ1122 (ω = ω + ω − ω ) × (δij δkl + δik δjl )
(3)
+χ1221 (ω = ω + ω − ω ) × (δil δjk ).
(4.9)
For the optical Kerr effect, the nonlinear polarization can be written as
(3)
Pi (ω ) = 3ε 0 ∑ χijkl (ω = ω + ω − ω ) Ej (ω ) Ek (ω ) El (−ω ),
(4.10)
jkl
where 3 is the degeneracy factor which indicates the number of distinct permuta(3)
tions of the frequencies ω, ω, and −ω. Now, substituting χijkl from Eq. (4.9) into
Eq. (4.10) we obtain
Pi = 6ε 0 χ1122 Ei ( E · E ∗ ) + 3ε 0 χ1221 Ei∗ ( E · E),
(4.11)
71
where the superscript ∗ denotes the complex conjugate as before. Equation (4.11)
can also be written in the following vector form
P = 6ε 0 χ1122 ( E · E∗ ) E + 3ε 0 χ1221 ( E · E ) E∗ .
(4.12)
Clearly the first term on the right hand side of Eq. (4.12) has the same handedness as
the incident electric field vector E, whereas the second term has the opposite handedness due to complex conjugation. Usually, one-photon-resonant contributions to
the nonlinear coupling as depicted in Fig. 4.1 (a), leads to the first term. Whereas,
two-photon-resonant processes as shown in Fig. 4.1 (b) may lead to the first and the
second terms both.
(a)
( b)
Figure 4.1: Energy level description for (a) one-photon-resonant contributions and
(b) two-photon-resonant contributions to Kerr nonlinearity. The solid lines in the
figures represent atomic ground state whereas the dashed lines represent virtual
levels. These energy levels usually represent combined energy of one of the energy
eigenstates of the atom and of one or several photons in the radiation field [134].
Let us now combine both the linear and the Kerr nonlinear terms and write the
electric polarization in the following form
h
i
(1)
(3)
Pi (ω ) = ε 0 χij (ω ) Ej + 3χijkl (ω = ω + ω − ω ) Ej Ek El∗ .
(4.13)
(1)
In isotropic media, the linear susceptibility is a diagonal tensor, i.e. χij = (n20 −
1)δij , where n0 is the linear refractive index of the medium. This allows one to
rewrite the constitutive Eq. (4.13) in terms of the “Effective” linear susceptibility,
(eff)
Pi (ω ) = ∑ ǫ0 χij
Ej ,
(4.14)
j
where,
(eff)
χij
and
n
o
= [n20 − 1 + A| E|2 ]δij + Bℜ Ei E∗j ,
A = 6χ1122 − 3χ1221,
B = 6χ1221.
(4.15)
(4.16)
Here we introduce the coefficients A and B by following the notation introduced
by Maker and Terhune [188]. Equation (4.15) indicates that in an isotropic medium,
72
Kerr nonlinearity may result in optical anisotropy. We also note that while writing
Eq. (4.15), we have omitted the r-dependence of the susceptibility tensor for the sake
of brevity. Clearly, the first term on the right hand side of Eq. (4.15) is independent
on the polarization state of the incident field and creates the diagonal elements of
the effective susceptibility tensor. The second term is polarization-dependent and
must be retained if the intense light wave propagating in the medium is elliptically
polarized. For linearly polarized light we have a simpler expression for the nonlinear
refractive index which is written as
n = n0 + n2 I,
(4.17)
where n0 is the linear refractive index and n2 , known as the nonlinear refractive
index, gives the rate at which the refractive index changes with a change in the
incident light intensity I. One can derive the mathematical expression relating n2
and χ(3) using the approach used in Chapter 4 of Ref. [134]. In SI units, they are
related by
2
m
283 (3) m2
n2
= 2 χ
.
(4.18)
W
V2
n0
The ratio of the coefficients A and B depend on the nature of the physical process
which produces the Kerr nonlinearity. Some of the standard physical processes
which produce Kerr nonlinearity are listed below
Nonresonant electronic response: This occurs as a response of the bound electrons
to the applied intense field. This type of nonlinearity is extremely fast (response time ∼femto second) but is believed to be relatively weak and is present
in all dielectric materials. For Kerr nonlinearity arising as a result of the nonresonant electronic response, A/B = 2.
Molecular orientation: In liquids comprised of anisotropic molecules, an external
electric field tends to align the molecules along the direction of the applied
field. This alters the average microscopic polarizability per molecule. The incident intense wave then experiences a modified refractive index which results
in Kerr nonlinearity. The response time of nonlinearity in this case is usually
∼pico second and the ratio of the coefficients is A/B = −3.
Electrostriction: Electrostriction is defined as the tendency of a material to become
compressed in the presence of an external electric field. This alters the optical
density of the material and hence also the effective susceptibility. Kerr nonlinearity caused by electrostriction is believed to be rather weak and the ratio of
A and B equals to zero.
Throughout this chapter, we shall assume that the origin of the optical Kerr nonlinearity is due to nonresonant electronic response of the bound electrons.
4.3
MODELING LIGHT-INDUCED ANISOTROPY WITH THE LINEAR
FMM
We proceed to extend the existing implementations of the anisotropic FMM, as discussed in section 3.7, to treat the optical Kerr effect (OKE) in a rigorous manner.
However, we assume that the origin of the anisotropy in FMM is purely lightinduced.
73
Let us consider a two-dimensionally periodic structure as shown in Fig. 4.2, with
periods d1 and d2 along x and y directions (i.e. parallel to the cartesian coordinate
axes), respectively. In cartesian system, φ = Θ = ξ = 0 and the x3 axis is parallel
to the z axis. As we saw before, any field component Ei or Hi , i = 1, 2, 3, in any
z-independent layer can be expressed as a superposition of z-invariant modes, viz.
Ui ( x, y, z) =
∑ a p Ui,pmn exp[i(k1,m x + k2,n y + γ p z)],
(4.19)
mnp
where a p , γ p , and Uipmn represent the complex amplitude, the propagation constant,
and the transverse distribution of the mode p, respectively, and
k1,m = k1,0 + m2π/d1 ,
k2,m = k2,0 + n2π/d2 .
(4.20)
Here k1,0 and k2,0 are the x and y components of the incident-field wave vector
respectively. The matrix M which contains the material properties in the Fourierspace, along with the wave-vector components can be written in the new coordinate
system as


−s1 ε̃31
−s1 ε̃32
s1 ε̃33 s2
I − s1 ε̃33 s1
 −s2 ε̃31
−s2 ε̃32
− I + s2 ε̃33 s2 −s2 ε̃33 s1 
.
M=
(4.21)
 −ε̃21 − s1 s2 −ε̃22 + s1 s1
−ε̃23 s2
ε̃23 s1 
ε̃11 − s2 s2
ε̃12 + s2 s1
ε̃13 s2
−ε̃13 s1
Here the elements of the matrices s1 and s2 are s1,mnpq = k1,m δmp δnq /k0 , and s2,mnpq =
k2,n δmp δnq /k0 respectively, I = δmp δnq , and ε̃ ij is the ij-th element of the matrix
ε̃ = l3− l2+ F2 l2− l1+ F1 l1− ε( x, y),
(4.22)
where ε ( x, y) is the x and y dependent (complex) relative permittivity tensor. As
before Fj stands for the Toeplitz operation along the direction j, and lτ± operators
are defined by Eq. (3.91) in section 3.7. Once we solve the eigenproblem, the z
component of the electric field can be obtained from [170]
−1
E3,p = ν33
(s2 H 1,p − s1 H 2,p ),
(4.23)
where ν = l2+ F2 l2− l1+ F1 l1− ε( x, y). The matrix eigenvalue equation from which we can
solve the propagation constants and the shape of the grating modes can be written
as
γ
MF p = F p ,
(4.24)
k0
where k0 is the vacuum wave number and the eigenvector F p contains the x and y
components of the electric and magnetic fields of mode p as follows:
 
Ex
 Ey 

Fp = 
(4.25)
H x  .
Hy p
Equation (4.15) shows that in case of light-induced anisotropy, the permittivity
tensor components become field-dependent. Hence, we need to extend the existing implementation of the anisotropic linear FMM summarized above to problems
74
dealing with field-dependent permittivity tensors. This is done by following an iterative approach and assuming the medium to be isotropic and Kerr nonlinear, where
the only source of anisotropy is optical nonlinearity. The algorithm we follow is an
extended version of the algorithm introduced by Laine and Friberg for linear gratings [189], but we also consider the polarization effects in a rigorous manner. Our
approach is to some extent similar to the iterative approach with fast Fourier factorization in the differential method [185, 190]. The steps of the modeling algorithm
are as follows:
1. In each z-independent layer, we solve the eigenvalues γ and the eigenvectors
F p for modes p using the linear permittivity. Then use the S-matrix algorithm [129,130] to connect the modes in different layers and solve the complex
amplitudes a p of the grating modes in Eq. (4.19) as described in section 3.7.
2. We solve the electric-field components using Eq. (4.19) and the Fast-Fourier
Transform (FFT) algorithm in a 3D grid of n x × ny × nz points as shown in
Fig. 4.3.
3. Using the solved field components, we can evaluate the effective linear susceptibility from Eq. (4.15).
4. We go back to the eigenvalue problem in Eq. (4.24), but now using the material
parameters from the previous step.
This iterative process is continued until the computed field converges. It is worth to
mention that the method may not converge if the field change is too large between
two subsequent steps. In such a case the electric-field strength may be increased
gradually during the iteration, which ensures better convergence in most situations.
z
y
d2
x
d1
Figure 4.2: Geometry of a crossed grating. The grating structure is periodic along
x and y directions. The pillars in this figure are assumed to be made with isotropic
Kerr nonlinear material.
75
Also, like in all Fourier domain approaches, the number of Fourier coefficients
used in the computation should be high enough. A poor convergence may be as
a consequence of the Gibbs phenomenon as described in section 3.5.1. This may
strongly affect the effective linear permittivity tensor through the space-domain
field. Additionally, the sampling of the field in the real space must be dense enough,
i.e., one must use a sufficient number of layers in FMM even if the structure is invariant in the z direction.
Here we also want to emphasize the fact that since we make use of the FFT algorithm to compute the field in the (x, y, z) space, and since the number of sampling
points is retained in FFT, we must pad the Fourier-domain field by zeros before the
FFT, which in turn increases the resolution in the (x, y, z) space. This is a standard
”trick” in FFT. However, it does not affect the results, but it just allows (almost)
arbitrary sampling in Eq. (4.19). One obtains identical results with explicit summation over all included Fourier coefficients in Eq. (4.19), but FFT with zero-padding
is significantly faster.
4.4
SYMMETRIES IN LIGHT-INDUCED ANISOTROPY
In FMM, one can employ structural symmetries to reduce the computation time
and also to increase the computational efficiency by reformulating the whole eigenvalue problem into a more compact form [191–198]. The number of floating-point
operations required to solve an eigenvalue problem of size L × L is asymptotically
proportional to L3 . Hence halving the matrix by use of the symmetries reduces the
required workload to 1/8. Such type of reductions of computational efforts are of
great importance especially in the context of nonlinear optics as the size of the matrix
M in Eq. (4.21) is 2 × 2 times larger than that in the corresponding linear problem,
where one can use the isotropic FMM [150]. Also, as we saw in the previous section, the nonlinear case requires several iteration steps and each of these steps thus
requires eight times more resources than the same problem in linear optics.
Here as an example, we study the reduction of the eigenvalue problem at normal
incidence of the incoming field if the structure possess C2v symmetry (i.e. rotation
of the structure through 180◦ yields a structure which is indistinguishable from the
original one) such that ε(− x, y, z) = ε( x, y, z) and ε( x, −y, z) = ε( x, y, z). In the linear
x
y
z
Figure 4.3: Sampling of the structure in the real space. Such a sampling produces
a 3D grid of n x × n y × n z points.
76
limit i.e. when we deal with the isotropic problem, we can easily show that [191]
Ej (− x, −y, z) = Ej ( x, y, z),
Hj (− x, −y, z) = Hj ( x, y, z),
(4.26a)
(4.26b)
where j = 1 or 2. Using Eq. (4.26a) and the symmetry of the permittivity we get
the following symmetry condition for the transverse components of the electric displacement vector D( x, y, z)
D j (− x, −y, z) = D j ( x, y, z),
(4.26c)
where again j = 1 or 2, and, consequently,
∂
∂
D1 (− x, −y, z) = − D1 ( x, y, z),
∂x
∂x
∂
∂
D2 (− x, −y, z) = − D2 ( x, y, z).
∂y
∂y
(4.27a)
(4.27b)
Maxwell’s divergence equation ∇ · D( x, y, z) = 0 then implies that
D3 (− x, −y, z) = − D3 ( x, y, z).
(4.28a)
This with the help of the symmetry condition of the permittivity function, leads to
E3 (− x, −y, z) = − E3 ( x, y, z).
(4.28b)
Let us first assume that these symmetry conditions also hold for the nonlinear
case. Our assumption remains valid at least in the first step of the iteration as we
start the process from the unmodified linear permittivity tensor, which is isotropic,
for calculating the field inside the structure. It follows immediately from Eqs. (4.15),
(4.26), and (4.28b) that the effective linear relative permittivity tensor
(eff)
ε ij
(eff)
= δij + χij
/ε 0
(4.29)
obeys the symmetry relations
(eff)
ε ij
(eff)
(− x, −y, z) = ±ε ij ( x, y, z).
(4.30)
Here the − sign is chosen if either i or j (but not both) is 3, and + is chosen otherwise.
Now we proceed to apply the symmetry conditions to Eq. (4.30). Recalling the
operator relation in Eq. (3.91) of section 3.7, we can write

(Aττ )−1 ,
ρ = τ, σ = τ,



−
1
(Aττ ) Aτσ ,
ρ = τ, σ 6= τ,
lτ± Aρσ =
(4.31)
Aρτ (Aττ )−1 ,
ρ 6= τ, σ = τ,



Aρσ ± Aρτ (Aττ )−1 Aτσ , ρ 6= τ, σ 6= τ,
where Aρσ is any matrix-form element of tensor A. We find from Eqs. (4.31) and
(eff)
(4.30) that if Aij ( x, y) = l1− ǫij
( x, y), then (we drop the z-dependence for brevity)
Aij (− x, −y) = ± Aij ( x, y).
(4.32)
77
Let us now denote the Fourier coefficient of a function f ( x, y) with respect to x by
f (m) (y), i.e.,
f ( m) (y ) =
1
d1
Z d1 /2
−d1 /2
f ( x, y) exp(−i2πmx/d1 ) dx.
(4.33)
It follows from Eqs. (4.32) and (4.33) that
(−m)
Aij
( m)
(−y) = ± Aij (y).
(4.34)
( m−n)
If Bij (y) is a matrix with its elements Bij,mn (y) given by Aij
F1 Aij ( x, y). Clearly from Eq. (4.34) we get
(y), we have Bij (y) =
Bij,−m−n(−y) = ± Bij,mn (y).
(4.35)
Now denoting C(y) = l2− l1+ B(y), and making use of Eq. (4.31) and (4.35), we find
that
Cij,−m−n (−y) = ±Cij,mn (y).
(4.36)
Here we have made use of the fact that if the elements of matrix N follows the
identity N−m−n = Nmn , then also the elements of Q = N−1 obey the same identity,
i.e., Q−m−n = Qmn . In addition, we have used the identity that if N−m−n = s N Nmn
and P−m−n = s P Pmn , where s N and s P can be independently either −1 or +1, then
the product matrix R = NP obeys the relation R−m−n = s P s N Rmn . The two results
mentioned above follow at once from the definitions of the inverse matrix and the
matrix product, respectively.
In a similar fashion, one can denote the Fourier coefficients of any y- dependent
function F (y) by F [ p] , i.e.,
F[ p] =
1
d2
Z d2 /2
−d2 /2
F (y) exp(−i2π py/d2 ) dy.
(4.37)
[ p −q]
Denoting Gij,mnpq = Cij,mn , we have Gij = F2 Cij (y). It follows from Eqs. (4.36) and
(4.37) that
[− p ]
[ p]
Cij,−m−n = ±Cij,mn
(4.38)
Gij,−m−n− p−q = ± Gij,mnpq.
(4.39)
and, consequently,
One can now verify using Eqs. (4.31) and (4.39) (which is a rather straightforward
task) that ε̃ = l3− l2+ G obeys the identity
ε̃ ij,−m−n− p−q = ±ε̃ ij,mnpq.
(4.40)
From Eq. (4.21) we see that if we denote the blocks of matrix M by Mrs , r, s =
1, 2, 3, 4, their elements obey the identity
Mrs,−m−n− p−q = Mrs,mnpq.
78
(4.41)
We now focus on the symmetry conditions of the eigenvectors. First we write
Eq. (4.24) in the form
γ
Ur,mn =
k0
∑ Mrs,mnpqUs,pq,
(4.42)
pq
where U1 = E1 , U2 = E2 , U3 = H1 , and U4 = H2 . Since m, n, p, and q are dummy
indices, we may express Eq. (4.42) as
γ
Ur,−m−n =
k0
∑ Mrs,−m−n− p−qUs,− p−q
pq
= ∑ Mrs,mnpqUs,− p−q,
(4.43)
pq
where to derive the above equation, we have used Eq. (4.41). As a result, the vectors
with elements Ur,−m−n i.e. the ‘flipped’ vectors form the eigenvector of M, with the
same eigenvalues as the original one having the elements Ur,mn. To be more specific,
we are dealing with a degenerate situation in which also vectors Ur+ and Ur− with
elements
±
Ur,mn
=
1
(Ur,mn ± Ur,−m−n )
2
(4.44)
form the eigenvectors of M. These eigenvectors obey the condition Ur,±−m−n =
± . However, as we are dealing with the normal incidence, the antisymmet±Ur,mn
ric part with − sign can be omitted. Hence we may assume that all eigenvectors
obey the symmetry condition
Ur,−m−n = Ur,mn .
(4.45)
Thus we have the following symmetry conditions
E1,−m−n = E1,mn ,
H1,−m−n = H1,mn ,
E2,−m−n = E2,mn ,
H2,−m−n = H2,mn
(4.46)
which are identical to those in the linear case. We can now conclude that the field
symmetry is not changed due to the nonlinear light-matter interaction. The obtained
result may sound surprising, but it follows from the fact that all materials are intrinsically isotropic. The effective anisotropy is due to the field itself and there is
nothing in the structure that can break the existing symmetry.
Let us now consider the case in which the index m takes on values from − M
to M, and n between − N and N. Hence, the total size of the matrix M is 4(2M +
1)(2N + 1) × 4(2M + 1)(2N + 1). Making use of Eqs. (4.24) and (4.21), and the identities in Eqs. (4.41) and (4.46), we may now replace the original eigenvalue equation
(4.24) with the reduced problem in which m takes on values only from 0 to M, and


(+)
(+)
(−)
(−)
−s1 ε̃31
−s1 ε̃32
s1 ε̃33 s2
I − s1 ε̃33 s1


(+)
(−)
(−)
 −s2 ε̃(+)
−s2 ε̃32
− I + s2 ε̃33 s2 −s2 ε̃33 s1 
31

,
M =  (+)
(4.47)
(+)
(−)
(−)

−ε̃23 s2
ε̃23 s1 
−ε̃21 − s1 s2 −ε̃22 + s1 s2
(+)
(+)
(−)
(−)
ε̃11 − s2 s2
ε̃12 + s1 s2
ε̃13 s2
−ε̃13 s1
79
where
(±)
ε̃ ij,mnpq
=
ε̃ ij,mn0q,
ε̃ ij,mnpq ± ε̃ ij,mn− p−q,
p=0
.
p 6= 0
(4.48)
All matrices in blocks of M are of size ( M + 1)(2N + 1), and the total size of the
matrix M is 4( M + 1)(2N + 1) × 4( M + 1)(2N + 1). Thus the required number of
floating-point operations is asymptotically 1/23 = 1/8 of the number of operations
in the original eigenvalue problem which results in a reduction of the work-load by
8 times.
In the above calculations, we chose to reduce the dimension of the matrix in the
x direction. Analogously we can proceed in the y direction. However, we must
note that reductions in both of these directions at the same time is not possible.
This is because of the fact that the field is not symmetric in either direction, but is
symmetric only when we compare the field values at (− x, −y, z) and ( x, y, z).
Now, we shall find out that if the incident field is polarized either in the x or
y direction, the problem may be reduced further. Let us assume that we have ypolarized incident field. In the linear case, the symmetry conditions lead to [199]
E1 ( x, y, z) = − E1 (− x, y, z) = − E1 ( x, −y, z),
E2 ( x, y, z) = E2 (− x, y, z) = E2 ( x, −y, z),
H1 ( x, y, z) = H1 (− x, y, z) = H1 ( x, −y, z),
H2 ( x, y, z) = − H2 (− x, y, z) = − H2 ( x, −y, z).
(4.49)
With the help of Maxwell’s divergence equation, we find that
E3 ( x, y, z) = E3 (− x, y, z) = − E3 ( x, −y, z).
(4.50)
It then follows from Eqs. (4.15) and (4.49) that
(eff)
ε jj
(eff)
(eff)
(eff)
( x, y, z) = ε jj (− x, y, z) = ε jj ( x, −y, z),
(eff)
(eff)
ε 12 ( x, y, z) = −ε 12 (− x, y, z) = −ε 12 ( x, −y, z),
(eff)
(eff)
(eff)
ε 13 ( x, y, z) = −ε 13 (− x, y, z) = ε 13 ( x, −y, z),
(eff)
(eff)
(eff)
ε 23 ( x, y, z) = ε 23 (− x, y, z) = −ε 23 ( x, −y, z),
(eff)
(4.51)
(eff)
and we may use the identity ε ji ( x, y, z) = ε ij ( x, y, z) to derive the remaining
three elements.
Proceeding analogously to the case for the arbitrarily polarized light, we find
that
ε̃ jj,mnpq = ε̃ jj,−mn− pq = ε̃ jj,m−np−q,
ε̃ 12,mnpq = −ε̃ 12,−mn− pq = −ε̃ 12,m−np−q ,
ε̃ 13,mnpq = −ε̃ 13,−mn− pq = ε̃ 13,m−np−q,
ε̃ 23,mnpq = ε̃ 23,−mn− pq = −ε̃ 23,m−np−q
(4.52)
and hence Eq. (4.21) implies that
Mrs,mnpq = ± Mrs,−mn− pq = ± Mrs,m−np−q,
80
(4.53)
where + is chosen for rs = 11, 14, 22, 23, 32, 33, 41, 44, and − otherwise. Similarly
to the previous derivation for the symmetry of the eigenvectors, we find that the
eigenvectors obey the following conditions
E1mn = − E1−mn = − E1m−n
E2mn = E2−mn = E2m−n
H1mn = H1−mn = H1m−n
H2mn = − H2−mn = − H2m−n .
(4.54)
The results in Eq. (4.54) are in full agreement with the linear case in Eq. (4.49).
Finally, in the eigenvalue problem, Eq. (4.24), we have


(eo)
(eo)
(oo)
(ee)
I − s1 ε̃33 s1
s1 ε̃33 s2
−s1 ε̃31
−s1 ε̃32


(eo)
(ee)
(eo)
 −s2 ε̃(oo)
− I + s2 ε̃33 s2 −s2 ε̃33 s1 
−s2 ε̃32
31
,

(4.55)
M =  (oo)
(ee)
(eo)
(eo)

ε̃23 s1 
−ε̃23 s2
−ε̃21 − s1 s2 −ε̃22 + s1 s2
(oo)
(ee)
(eo)
(eo)
ε̃11 − s2 s2
ε̃12 + s1 s2
ε̃13 s2
−ε̃13 s1
where, for any matrix Ω,
(oo)
Ωmnpq = Ωmnpq − Ωmn− pq − Ωmnp−q + Ωmn− p−q ,

Ωmn00,
p = 0, q = 0




Ω
+
Ω
,
p = 0, q 6= 0
 mn0q
mn0− q
(ee)
Ωmnpq =
Ωmnp0 + Ωmn− p0,
p 6= 0, q = 0


 Ωmnpq + Ωmn− pq


+Ωmnp−q + Ωmn− p−q , p 6= 0, q 6= 0,

p = 0,
 Ωmn0q − Ωmn0−q ,
(eo)
Ωmnpq =
Ωmnpq + Ωmn− pq

−Ωmnp−q − Ωmn− p−q , p 6= 0.
(4.56)
The superscripts e and o are the abbreviations for ’even’ and ’odd’ respectively. They
represent symmetries of either even type or of odd type. In matrix M, the indices m
and n run from 1 to M and N, respectively, for rows 1 and 4, since the corresponding
field components E1 and H2 are antisymmetric. Similarly, both indices begin from
0 for rows 2 and 3 due to the symmetry of E2 and H1 . An equivalent rule applies
to the column indices p and q, which begin from 1 for columns 1 and 4 of M, and
from 0 for other two columns. Hence the total size of M is 2(2M + 1)(2N + 1) ×
2(2M + 1)(2N + 1). Finally, we may conclude that the eigenvalue problem requires
asymptotically just 1/64 of the original problem. Similar result can be obtained by
assuming the incident field to be x- polarized.
Unlike in linear problems, one cannot split the eigenproblem for an arbitrarily
polarized incident field into two smaller eigenproblems that can be solved separately
(which would mean a remarkable further improvement in computation efficiency).
This is because of the fact that the permittivity is affected by the total field, and
Eq. (4.52) does not hold unless the incident field is y polarized [199].
4.5
NUMERICAL EXAMPLES
We now apply the algorithm developed in the previous sections to periodic structures composed of isotropic nonlinear media.
81
4.5.1
One dimensional metallic gratings with grooves filled with χ(3) media
At first, we model a metallic linear grating with its grooves filled with Kerr nonlinear
material. In our numerical simulations, we use the same parameters as used by
Bonod et al. [185]. In this example, we assume that TM polarized light is normally
incident from air on the lamellar grating side. The grating pillars are made of a
metal with relative permittivity ǫr = −182.4 + i43.52. The 494 nm deep grooves
are assumed to be filled with amorphous silicon. The substrate is assumed to be
composed of the same metal as the pillars. The grating period is 1 µm.
Figure 4.4 shows the efficiency in direct reflection (reflected zeroth diffraction
order) as a function of the wavelength of the incident light. The absolute value of
the incident electric field is chosen to be 106 V/m. One may readily compare our
simulation result i.e. Fig. 4.4 with Fig. 2 in Ref. [185], and conclude that the results
are essentially identical.
0.9
0.8
0.7
η0
0.6
0.5
0.4
0.3
0.2
0.1
0
1400 1420 1440 1460 1480 1500 1520 1540 1560 1580 1600
λ [nm]
Figure 4.4: Efficiency in direct reflection as a function of the wavelength for a
metallic linear grating.
4.5.2
1-D binary grating with TiO2 as the Kerr nonlinear material
As a second example, we simulate a 1-D periodic nonlinear binary grating possessing Kerr nonlinearity in the material that fills the pillars. The substrate material
is assumed to be SiO2 with linear refractive index n0 = 1.46. The SiO2 grating
pillars are filled with amorphous (isotropic) TiO2 (linear refractive index= 2) with
χ(3) = 2.1 × 1020 m2 /V2 . The fill factor of the grating is f = 0.5 and the period of
the structure is d = 2.5 × λ0 , where λ0 = 633 nm is the wavelength of the incident
light in vacuum. The results obtained with FMM are compared with those obtained
with the SF-FDTD as implemented in Ref. [186, 200]. For the nonlinear simulations,
the intensity of the incident field was taken to be 0.5 MW/µm2 in both numerical
methods. Clearly, from Fig. 4.5, there is a high correlation between both analysis.
The small differences can be due to the difficulties in triggering the same amount
of nonlinearity which follows from the difference in defining the intensity in the
SF-FDTD cell [186, 200, 201] and the FMM approach [199].
82
η0
Linear
η±1
h/λ0
(a)
h/λ0
( b)
Figure 4.5: Diffraction efficiencies of a nonlinear binary grating computed by
means of SF-FDTD and FMM. h is the structure depth.
4.5.3
Crossed gratings with the pillars made with Si3 N4
Next example deals with a crossed diffraction grating composed of cylindrical shaped
silicon nitride (Si3 N4 ) pillars. The linear refractive index of Si3 N4 (which is amor(3)
phous in nature) is assumed to be n0 = 2, and the value of χSi3 N is taken to be
4
3.39 × 10−21 m2 /V2 as in Ref. [69]. Also, we assume that the silicon nitride pillars
are surrounded by air. The substrate material is fused silica with linear refractive
index n = 1.5. Furthermore, we presume that the surrounding media as well as
the substrate material are optically linear. This assumption remains valid as the
third-order nonlinearity of silicon nitride is two-orders of magnitude higher than
that of fused silica. Consequently, we do not get nonlinear signal from the substrate
at the used intensity level of the incident field. We first assume that Left Circularly
Polarized (LCP) light is normally incident from the substrate side, and then repeat
the simulation for 45◦ linearly polarized light. The grating period is assumed to be
d1 = d2 = 2.5λ and the height of the pillars is h = 1.25λ, where λ denotes the wavelength in vacuum. The discretization of the pillars to treat the circular boundary
was done in a similar way as in Fig. 4 of Ref. [150].
In Figure 4.6, we plot the convergence of the nonlinear FMM when the incident field intensity is fixed at 5 × 1014 W/cm2 . The sampling parameters are
n x = ny = 175 and nz = 25. Clearly, for this specific example, the method converges
with M = 8 i.e. about 17 diffraction orders are needed in this specific case.
Figures 4.7 and 4.8 show the convergence as a function of the transverse (nz )
and the longitudinal sampling (n x and ny ), respectively, where all other parameters
are kept fixed to their maximum values. The transverse sampling can be chosen
83
0.6
η00
0.5
0.4
0.3
0.2
1
2
3
4
5
6
7
8
9
10
11
M
Figure 4.6: Efficiency in direct transmission η00 as a function of the maximum
diffraction order M. The maximum order is taken to be the same both in x and y
directions.
to be large as it does not affect the computation time significantly (solution inside
a single z-independent layer). On the other hand, the computation time increases
linearly with an increase of the value of nz . Hence, it should be chosen as low as
possible but of course without sacrificing neither accuracy nor convergence. Figure
4.9 illustrates a typical convergence curve during the iteration process. Though not
absolutely necessary, in this example we increased the incident intensity from zero
to its maximum value during the iteration within the first 20 steps. Then we kept
the intensity fixed at its final value (i.e. the value at the 20-th iteration step) for a
few rounds (here five) to ensure the convergence of the method.
Figure 4.10 illustrates the effective-permittivity tensor in the middle of the
0.53
η00
0.52
0.51
0.5
0.49
60
80
100
120
140
160
180
200
n x = ny
Figure 4.7: Zero-order transmitted efficiency η00 plotted against the number of
transverse sampling points n x = n y .
structure (z = h/2) and during the final round of the iterations.
84
0.52
η00
0.49
0.46
0.43
5
10
15
20
25
30
nz
Figure 4.8: Zero-order transmitted efficiency η00 plotted against the number of
layers n z .
0.5
0.4
η00
45° LP
LCP
0.3
0.2
5
10
15
20
25
qg
Figure 4.9: Zero-order transmitted efficiency η00 plotted against the iteration steps
for LCP and 45◦ linear polarization respectively. q g is the number of iteration steps.
4.5.4
Si3 N4 resonance waveguide-grating
As a final example, we illustrate the effect of the optical Kerr nonlinearity on the
0-th order reflected signal from a waveguide-resonance structure or guided mode
resonance filter (GMRF). The structure geometry is shown in Fig. 4.11. The substrate
is made of fused silica (FS) and the grating layer as well as the waveguide layer (unetched part) is taken to be Si3 N4 . Light is normally incident on the structure from
air. The grating periods are d1 = d2 = 300 nm, Height of the pillars is 100 nm, the
thickness of the unetched part (waveguide layer) is 150 nm, and the linewidths of
the pillars are 120 nm in both transverse directions. Here, the waveguide-grating
structure is designed to act as a narrow-band reflector. We investigate the influence
of the incident field intensity and the state of polarization of the normally incident
light on the resonance peak in reflection. Figure 4.12 shows the red-shift of the res85
4
0
3
−0.2
2
−0.4
−0.6
1
4
0.1
3
0
2
−0.1
1
4
3
0.1
0
2
−0.1
1
Figure 4.10: Elements of the effective relative-permittivity-tensor ǫij(eff) ( x, y) in the
middle of the cylindrical pillars. The tensor elements in the left-hand column are 11
(top), 22 (middle), and 33 (bottom), and in the right-hand side 12 = 21 (top), 23 = 32
(middle), and 31 = 13 (bottom).
onance peak i.e. shift towards the longer wavelengths when the value of the field
intensity increases. This is in agreement with the theory of the optical Kerr effect.
As the nonlinear refractive index of Si3 N4 increases with increasing field intensity,
the resonance peak shifts in wavelength scale due to the increase (n2 of silicon nitride is positive) in mismatch between the refractive indices of the grating material
and the surrounding material.
Figure 4.13 shows how the incident state of polarization affects the resonance
peak: clearly the shift is larger with the linearly polarized light than that with the
circularly polarized light. The reason is that the nonlinear refractive index is, in
general, larger with linearly polarized states [134].
Resonance waveguide-grating structures illustrated in Fig. 4.11 can produce
strong nonlinear effects for reasonably low values of field intensities due to the
strong enhancement of local field inside the waveguide layer. This property of GMRFs can be exploited to construct novel devices especially in relation to sensing
and all-optical computing. In Chapter 6, we shall investigate these silicon nitride
waveguide-grating structures in a more detailed manner.
86
0-th order reflected field
Incident field
Grating layer
Waveguide layer
FS substrate
Figure 4.11: Guided-mode resonance filter (GMRF) structure.
1
I=1014 W/ m 2
15
I=10 W/ m 2
0.8
15
I=2 x10
W/ m 2
η00
0.6
0.4
0.2
0
496
496.4
496.8
497.2
497.6
498
λ [nm]
Figure 4.12: Red shift of resonance peak due to the change in the incident field
intensity. The incident field is assumed to be x polarized.
87
1
45 ° LP
y-pol
LCP
η00
0.8
0.6
0.4
0.2
0
496
496.4
496.8
497.2
497.6
498
λ [nm]
Figure 4.13: Resonance peak shift due to the change of polarization states of the
incident light. In these plots, the field intensity is kept fixed at 1015 W/m2 .
4.6
SUMMARY
In this Chapter, we have developed an efficient numerical tool for rigorous modeling
of optical Kerr nonlinearity in 2D periodic structures with isotropic third-order nonlinear media having instantaneous nonlinear response. For structures possessing
C2v symmetry i.e. two-fold rotational symmetry, we demonstrate that the computational workload can be reduced to 1/8-th as compared to the original nonlinear
eigenvalue problem. Furthermore, for TE or TM polarized incident field, we can
reduce the computational effort to 1/64-th as compared to that in the most general
case.
Our numerical technique produces well-converging results even if there is structural resonance and nonlinearity generated inside the structure is large as a result
of strongly confined local field. Hence we may conclude that this nonlinear FMM
can be used as a powerful numerical modeling tool for designing grating based next
generation all-optical devices as well as novel nonlinear metamaterials.
88
5
Theory of form birefringence in Kerr-type media
The term birefringence is used to denote having two different refractive indices of
a medium depending on different propagation directions and different polarization states of light [202]. Though birefringence is commonly observed in optically
anisotropic media, it can also arise in isotropic media under some special conditions. It was David Brewster who first observed that isotropic media can become
birefringent under application of mechanical stresses [203]. This type of induced
birefringence is called ’Stress birefringence’. However, there are several other ways
to make a medium optically anisotropic for example by applying an external static
electric field (D.C Kerr effect) [187], or magnetic field (Faraday effect) [113] etc. In
all the cases mentioned above, the origin of birefringence may be explained in terms
of the molecular anisotropy. However, it might be surprising to know that birefringence may also arise due to anisotropy which is larger in scale than the dimensions
of a molecule though much smaller in scale than the wavelength of light. Such type
of birefringence is termed as form birefringence.
This chapter starts with the theory of light propagation in optical crystals. After
some general discussions, we proceed to derive the wave equations governing light
propagation in a uniaxial crystal. Next we demonstrate that subwavelength gratings (SWG) with isotropic materials behave as uniaxial crystals and exhibit (form)
birefringence in the quasistatic limit. After that, we extend the theory of form birefringence in a SWG to gratings with optical Kerr nonlinear media [204]. Finally,
we include several numerical examples to elucidate our developed theory and also
to verify it by comparing with the results obtained by the nonlinear FMM [199]
developed in Chapter 4.
5.1
PROPAGATION OF LIGHT IN CRYSTALS
In section 2.13, we discussed light propagation in an electrically anisotropic medium
and found that it is possible to write the expression for the electric energy density
in a form which is equivalent to the mathematical expression for an ellipsoid. Also,
we found that we can always choose a coordinate system with axes parallel to the
principal axes of the ellipsoid to rewrite the ellipsoid equation in a more compact
form i.e.
Dy2
D2x
D2
+
+ z = constant,
(5.1)
ǫx
ǫy
ǫz
with
D x = ǫ x Ex ,
D y = ǫ y Ey ,
D z = ǫ z Ez .
(5.2)
Here the cartesian coordinate axes x, y, and z are parallel to the principal axes and
ǫ j , where j = x, y, z, are the principal dielectric constants of the crystal as before.
Let’s now substitute the expression of D from Eq. (2.128) into Eq. (2.138) to yield a
set of three homogeneous linear equations in Ex , Ey , and Ez . This set of equations
can be written as
n2 µǫ j Ej = 2 Ej − u j ( E · û) ,
(5.3)
c
89
where u j s are the cartesian components of the unit vector û. Equation (5.3) can also
be written in the following form
Ej =
(n2 /c2 )u j ( E · û)
.
(n2 /c2 ) − µǫ j
(5.4)
Multiplying Eq. (5.4) with u j , adding the resulting three component-form equations,
and finally dividing both sides by the common factor E · û, we obtain
2
uy
u2x
u2
c2
+ 2 2
+ 2 2z
= 2.
2
2
(n /c ) − µǫx (n /c ) − µǫy (n /c ) − µǫz
n
(5.5)
Some simple algebra with Eq. (5.5) yield
c2
n2
u2x
−
1
µǫx
+
u2y
c2
n2
−
1
µǫy
+
c2
n2
u2z
−
1
µǫz
= 0.
(5.6)
Let us now define the terms ’principal velocities of propagation’ by the formulae
vx = √
1
,
µǫx
vy = √
1
,
µǫy
vz = √
1
.
µǫz
(5.7)
Recalling the definition of phase velocity from section 2.13 and using Eq. (5.7), we
can rewrite Eqs. (5.4) and (5.6) in the following forms respectively
Ej =
v2j
v2j − v2p
u j (E · û) ,
( j = x, y, z),
(5.8)
2
uy
u2x
u2z
+
+
= 0.
v2p − v2x
v2p − v2y
v2p − v2z
(5.9)
Equation (5.9) is quadratic in v2p . We may now conclude that to every direction û,
there exists two phase velocities v p (we count only the positive roots as the negative
roots correspond to the negative û direction). For each of these two values of v p , Ej
and D j can be solved from Eqs. (5.8) and (5.2) respectively. Also, we see that the ratio of the electric field components as well as the electric displacement components
are real. Hence E and D are linearly polarized. Finally, we have the most important conclusion regarding light propagation in a crystalline medium with electrical
anisotropy which can be stated as: the structure of an anisotropic medium allows
two monochromatic linearly polarized plane waves (where the states of polarization
are different for these two plane waves) with two different velocities to propagate in
any given direction. By some straightforward calculus it is easy to prove [113] that
the directions of vibrations of the two electric displacement vectors D which corresponds to the two plane waves are in general orthogonal to each other as shown in
Fig. 5.1(a). We denote the two electric displacement vectors by D′ and D′′ . Clearly,
D ′ , D′′ and û form an orthogonal triplet. The ellipsoid in Fig. 5.1(a) is governed by
the equation
√
√
√
( D x / c )2 ( Dy / c )2 ( Dz / c )2
+
+
= 1,
(5.10)
ǫx
ǫy
ǫz
90
which directly follows from Eq. (5.1) and is known as the ellipsoid of wave normals.
Clearly, the semi-axes of this ellipsoid are equal to the square roots of the principal
dielectric constants ǫx , ǫy , and ǫz and coincides with in directions with the principal
dielectric axes. In the special case, when light propagates in the direction of one of
the principal axes of the ellipsoid of wave normals, say x axis, the phase velocity
becomes equal to the principal velocities of propagation vy and vz . The optic axes
of wave normals can be determined by constructing two circular sections C1 and
C2 passing through the center of the ellipsoid as shown in Fig. 5.1(b) (an ellipsoid
can have only two circular cross sections). The normals N 1 and N 2 to the surfaces
C1 and C2 are known as the optic axes of the crystal. Since the sections C1 and C2
are circular, the directions N 1 and N 2 allow only one velocity of propagation along
them and D can take any direction perpendicular to the wave normal.
5.2
BIREFRINGENCE OF A UNIAXIAL CRYSTAL
In a uniaxial crystal, two or more crystallographically-equivalent directions in one
plane exist. Usually, the plane containing these equivalent directions is perpendicular to the axis of rotational symmetry of the crystal. For a uniaxial crystal, we may
choose one of the principal axes as the distinguished direction while the other two
axes can be chosen such that they are perpendicular to the distinguished axis. If
we take z- axis as the distinguished direction, the principal dielectric constant ǫx
becomes equal to ǫy and we have the following relation
ǫ x = ǫy 6 = ǫz .
(5.11)
û
N2
D′
N1
D′′
C2
(a)
C1
( b)
Figure 5.1: The ellipsoid of wave normals. Construction of the direction of vibrations of the D vectors corresponding to the propagation direction û (a) and construction of the optic axes (b).
91
Clearly, for an uniaxial crystal the ellipsoid in Fig. 5.1(a) reduces to a spheroid with
two equal axes. Also, unlike in Fig. 5.1(b), we have only one circular cross-section
passing through the center of the spheroid. Hence, uniaxial crystals possess only
one optic axis.
Let’s us now go back to the equation of wave normals (also known as Fresnel’s
equation) i.e. Eq. (5.9). We can write this in the following form
u2x (v2p − v2y )(v2p − v2z ) + u2y (v2p − v2z )(v2p − v2x ) + u2z (v2p − v2x )(v2p − v2y ) = 0,
If we now assume the direction of the optic axis along z-axis and write v x = vy = vo ,
and vz = ve , Eq. (5.12) for an uniaxial crystal reduces to
h
i
(v2p − v2o ) (u2x + u2y )(v2p − v2e ) + u2z (v2p − v2o ) = 0,
(5.12)
where the new subscripts ’e’ and ’o’ stand for the ordinary wave and the extraordinary wave respectively which will be explained later in this section. Let us now
assume that the wave normal û forms an angle θ with the z-axis as shown in Fig. 5.3.
Thus we have
u2z = cos2 θ.
(5.13)
u2x + u2y = sin2 θ,
Using Eq. (5.13), we can write Eq. (5.12) in the following form
h
i
(v2p − v2o ) (v2p − v2e ) sin2 θ + (v2p − v2o ) cos2 θ = 0.
(5.14)
Equation (5.14) is a quadratic equation with roots
v′p2 = v2o ,
(5.15a)
v′′p 2 = v2o cos2 θ + v2e sin2 θ.
(5.15b)
Hence we see that one of the two shells of the spheroid (surface of wave normals)
is a sphere of radius v′p = vo . The other shell is an ovaloid. The wave with phase
velocity v′p which corresponds to the wave normal û propagates inside the crystal
analogous to an ordinary wave as the velocity is direction independent. This wave
is termed as ordinary wave. The other wave that corresponds to the wave normal
û propagates with a phase velocity v′′p which is direction dependent as can be seen
from Eq. (5.15b). To emphasize the contradiction with the previous case, this wave
is termed as extraordinary wave. Phase velocities of these two waves are equal only
if θ = 0 i.e. when the wave normal directs along the optic axis (here z- axis) of the
crystal. When the wave normal is perpendicular to the optic axis (here x- axis), the
difference between the phase velocities of these two waves is maximum and from
Eq. (5.15b) we have,
v′′p 2 = v2e .
(5.15c)
Due to the different phase velocities of the wave propagation inside the uniaxial
crystal, we can also define two different refractive indices by
no = c/vo ,
and
ne = c/ve ,
(5.15d)
which are known as the ordinary and the extraordinary refractive indices respectively. Now we can easily understand that light propagation inside a uniaxial crystal
gives rise to birefringence.
92
There are situations when the ordinary wave can travel faster than the extraordinary wave inside the crystal. Such type of uniaxial crystals, example include quartz,
are termed as positive uniaxial crystals. The opposite situations are observed in negative uniaxial crystals for example in Feldspar. Light propagation geometries that
correspond to these two cases are illustrated in Figs. 5.2(a) and 5.2(b) respectively.
The directions of vibrations of D′ and D′′ can be found from Fig. 5.3. The plane
containing û and the optic axis OZ is termed as the principal plane. The spheroid is
symmetrical about this plane. Clearly from the figure, the elliptical section through
O by the plane perpendicular to the wave normal is also symmetrical about the
principal plane. Hence, the principal axes are perpendicular and parallel to the
principal plane. Finally, the vector D of the ordinary wave i.e. D′ vibrates at right
angles to the principal plane, whereas the vector D of the extraordinary wave i.e.
D′′ vibrates in the principal plane as illustrated in Fig. 5.3.
5.3
THEORY OF FORM BIREFRINGENCE
Form birefringence can be exhibited by subwavelength gratings with grating periods much smaller than the wavelength of the incident light. Even if the material
which fills such a subwavelength grating is isotropic, due to anisotropy arising from
structuring one can observe strong birefringence. Hence, these gratings behave as
homogeneous effective birefringent media. This type of birefringence is usually
much stronger than the birefringence in optical crystals [205]. Also the magnitude
of form birefringence in a SWG can be tailored by varying the grating period, duty
cycle, and depth [143, 206–208].
Let’s now examine the geometry in Fig. 5.4 i.e. we consider conical illumination. This is identical to the geometry in Fig. 3.3. The name conical arises from
z
z
û
vo
û
v′′p
v′′p
vo
v′p
v′p
x
ve
(a)
x
ve
( b)
Figure 5.2: Normal surfaces of (a) positive uniaxial crystal and (b) negative uniaxial
crystal.
93
z
û
θ
D′′
o
D′
Figure 5.3: Directions of vibrations inside a uniaxial crystal.
ûin,π
(0,0)
ûr,π
ûin,σ
kin
(0,0)
θr
θin
φin
(0,0)
kr
(0,0)
ûr,σ
x
(0,0)
φr
y
z
Figure 5.4: Wave vectors of input plane wave and zero-order reflected wave as well
as directions of vibrations of TE-electric field (σ) and TM-magnetic field (π) in case
of conical illumination.
94
the fact that the wave vectors of the reflected and the transmitted orders in such
a situation lie on a conical surface. As described in section 3.6.1, assuming a yinvariant grating geometry we can derive the expressions for the grating modes in
any z- independent layer (say layer number j) starting from the space-frequency domain Maxwell’s equations in the j-th layer which are identical to Eqs. (2.33)–(2.36).
Though the structure is y- invariant, due to the light-matter interaction geometry as
shown in Fig. 5.4, everywhere
the fields are y- dependent and this dependence is of
the form exp ik y y which directly follows from Bloch’s theorem. Thus the electric
and the magnetic fields in the j-th layer can be written in the following forms
E( j) (r ) = exp ik y y E( j) ( x, z), B( j) (r ) = exp ik y y B( j) ( x, z).
(5.16)
However, it can be shown [209] that even in such a situation one can decompose
Maxwell’s equations in the j-th layer into two sets as described in section 2.6. The
( j)
( j)
first set corresponds to Ex ( x, z) = 0 and the second set corresponds to Bx ( x, z) =
0. The other field components are usually non-zero. However, we must note that
these sets are independent only inside the layers but they may be coupled at the
interfaces between the layers. Let us use the superscript ’o’ for the first set and the
superscript ’e’ for the second set from now on for reasons that will become obvious
later in this section. The eigenvalue equations in matrix form for these two sets can
be derived in a similar fashion as described in section 3.6.1. These equations take
the forms
h
i
h
i2
Jε̂( j) Kq−m − L x A( j) = A( j) Λ( j,o) /k0 ,
(5.17a)
n
h
i
o
h
i2
1
2
( j ) −1
( j)
Jζ ( j) K−
= C ( j) Λ( j,e) ,
q− m k 0 I − L x Jε̂ Kq− m L x − Y C
(5.17b)
where the elements of the matrix L x are now L x,q,m = (k2x,q + k2y )δq,m . All the other
matrices remain the same as in section 3.6.1. For the second set, now we have a new
matrix Y with elements defined as Yi,j = k2y δi,j .
Now if the grating period is much smaller in scale as compared to the wavelength
of the incoming light (usually 1/10-th or less), we reach the quasistatic limit. Under
such circumstances one may retain only the zeroth-order mode in Eqs. (5.17a) and
(5.17b) and the higher-order modes do not contribute to the final solution. Assuming plane wave illumination and the quasistatic limit one may write the eigenvalue
equations in the following forms
h
i
( j,o) 2
( j)
k2x,0 + k2y + γ0
− k20 ε̂ 0 = 0,
(5.18a)
h
i
( j)
( j)
( j,e) 2 ( j )
( j)
( j)
k2x,0 /ξ 0 + k2y ε̂ 0 + γ0
ε̂ 0 − k20 ε̂ 0 /ξ 0 = 0.
(5.18b)
h
i
( j,o) 2
k2x,0 + k2y + γ0
− k20 ε̂( j,o) = 0,
(5.19a)
Clearly, Eq. (5.18a) is of the form of the fundamental wave vector equation for a
plane wave propagating in a medium with relative complex permittivity ε̂( j,o) =
h
i
( j)
( j,o) 2
ε̂ 0 where the z- component of the wave vector is γ0
. The same holds for
Eq. (5.18b) under normal incidence. For non-normal incidence we get from Eq. (5.18)
h
i
( j,e) 2 ( j,o)
k2x,0 ε( j,e) + k2y ε̂( j,o) + γ0
ε̂
− k20 ε̂( j,o) ε( j,e) = 0,
(5.19b)
95
( j)
where we define ε( j,e) = ξ 0 . Equation (5.19b) is analogous to the wave equation for
an extraordinary wave in a uniaxial crystal i.e. Eq. (5.12). Whereas, as mentioned
before, Eq. (5.19a) is similar to the wave equation of a ordinary wave. Finally, we
may conclude that the subwavelength gratings in the quasistatic limit behave as
uniaxial crystals. This property of the subwavelength gratings can be employed to
construct wave plates, and phase retarders on demand [210–212].
5.4
FORM BIREFRINGENCE IN KERR MEDIA: ANALYTICAL FORMULATION
In the previous section we found that due to structural anisotropy, subwavelength
gratings (SWGs) behave like uniaxial optical crystals in the quasistatic regime, i.e.
when the grating period Λ ≪ λ, where λ is the wavelength of the incident light [113,
210] and one may treat these subwavelength gratings as homogeneous media with
effective optical properties. The effective medium theory (EMT) provides two different effective refractive indices for these SWGs. For the incident light with electric
field polarized along the grating period the SWG can be treated as a homogeneous
√
slab with refractive index given by the ordinary refractive index, noo = ε(oo) and
for the light with electric field polarized perpendicular to the grating period
√ the
SWG assembly can be treated as a slab with refractive index equals to nee = ε(ee)
i.e. the extraordinary refractive index.
In this section, we extend the classical form-birefringence theory (in linear optics) to the nonlinear optical domain by considering gratings with Kerr nonlinear
materials. We first proceed to derive analytical formulae for the theory of form
birefringence in Kerr media in the framework of the first-order EMT.
When a plane electromagnetic wave is normally incident on a lamellar 1D SWG
sandwiched between two homogeneous media, the form birefringence of the grating
introduces additional phase difference between the components of the transmitted
wave polarized parallel (TM) and perpendicular (TE) to the grating vector (Λ) [113].
Let us consider the lamellar grating geometry in Fig. 5.5. We assume that the incident medium is linear, homogeneous, and isotropic with refractive index ni ). The
modulated region located at 0 ≤ z < h is assumed to have a real-valued permittivity
distribution given by
ε̂( x ) =
ε̂1
ε̂2
if 0 ≤ x < l
if l ≤ x < Λ
(5.20)
The medium with index no in the half-space z > h is defined as the substrate.
Also, we assume that both lamella are made with anisotropic materials. Let us now
employ EMT to describe the fields in the modulated region.
Assuming the validity of the plane wave limit and the quasistatic limit, the electric field E and the electric displacement D can be considered constants in each
lamellae 0 ≤ x < l and l ≤ x < Λ within the grating period Λ. From the electromagnetic boundary conditions we know that the tangential electric field components, and the normal components of the electric displacement remain continuous
across the interface between two materials. Hence, inspecting Fig. 5.5, we may conclude that Dx and Ey are continuous across the interfaces at x = 0 and x = l.
Consequently, Dx and Ey have constant values within the entire structure. Hence it
96
z
ε o = ε 0 n2o
h
ε̂ 1
ε̂ 2
ε̂ 1
ε̂ 2
0
0
x
Λ
l
ε i = ε 0 n2i
Figure 5.5: Geometry of the form birefringent subwavelength-period grating. The
grating material as well as the surrounding medium are assumed to be anisotropic.
is implied from the material constitutive relation D( x ) = ε̂( x ) E( x ) that
Dx = ε xx ( x ) Ex ( x ) + ε xy ( x ) Ey ,
Dy ( x ) = ε yx ( x ) Ex ( x ) + ε yy ( x ) Ey
(5.21)
(5.22)
and, consequently, Ex and Dy become binary functions that are discontinuous at
x = 0 and x = l. We denote the fill factor of the grating by f = l/Λ, and define the
average value of a binary function across x
g1 if 0 ≤ x < l
g( x ) =
(5.23)
g2 if l ≤ x < Λ
as
h gi x =
1
Λ
Z Λ
0
g( x )dx = f g1 + (1 − f ) g2 .
(5.24)
From now on we use a similar shorthand notation for averages over products and
linear combinations of functions. Solving Ex ( x ) from Eq. (5.21) and averaging over
a grating period across x we get
1
−1
h Ex i x = h ε −
xx i x D x − h ε xy ε xx i x Ey .
(5.25)
1 −1
−1 −1
−1
D x = h ε−
xx i x h Ex i x + h ε xx i x h ε xy ε xx i x Ey ,
(5.26)
from Eq. (5.25) we obtain
which is identical to Eq. (5.21) though all the quantities in Eq. (5.26) are averaged
quantities over a grating period. Analogously, substituting the expression for Ex ( x )
from Eq. (5.21) into Eq. (5.22), we get the following expression
2 −1
1
h Dy i x = hε yx ε−
xx i x D x + h ε yy − ε xy ε xx i x Ey .
(5.27)
97
Now by substituting the expression for Dx from Eq. (5.26) into Eq. (5.27), we arrive
at
1 −1
−1
h Dy i x = h ε−
xx i x h ε yx ε xx i x h Ex i x
1 −1
−1 2
2 −1
+ h ε−
xx i x h ε xy ε xx i x + h ε yy − ε xy ε xx i x Ey .
(5.28)
Clearly, Eqs. (5.26) and (5.28), describe light-matter interactions in a SWG composed
of optically anisotropic materials. These two equations can be conveniently written
in the following matrix form
Dx
h Ex i x
= hε̂i x
,
(5.29)
h Dy i x
Ey
where hε̂i x is a 2 × 2 matrix with elements defined in Eqs. (5.26) and (5.28). For
SWG with isotropic media we have ε ij ( x ) = δij ε( x ) (i, j = x, y stand for the cartesian
indices, δij is the Kronecker delta) where
ε( x ) =
ε1
ε2
if 0 ≤ x < l
.
if l ≤ x < Λ
(5.30)
Hence the effective permittivity tensor hε̂i x becomes a diagonal tensor with the only
1
non-zero elements ε xx = hε−1 i−
x and ε yy = h εi x .
Let us now proceed to investigate the effect of Kerr nonlinearity on the form
birefringence of the structure defined in Fig. 5.5. We already saw in Chapter 4 that
in a homogeneous isotropic Kerr nonlinear medium, the electric displacement can
(eff)
be conveniently written in terms of ‘effective’ linear susceptibility χij
h
i
(eff)
Di ( x ) = ε 0 ∑ ε( x )δij + χij ( x ) Ej ( x ).
[134], i.e.,
(5.31)
j
Here i, j = x, y stand for the cartesian indices, δij is the Kronecker delta as before, and
ε( x ) = n2 ( x ) is the relative (linear) permittivity. The effective linear susceptibility,
that represents the Kerr effect in this case, can be written in the following form
h
i
(eff)
χij ( x ) = A′ ( x )|E|2 δij + B′ ( x ) ℜ Ei ( x ) E∗j ( x ) .
(5.32)
In a similar way, we can now define the corresponding effective relative permittivity
(eff)
(eff)
ε ij = ε( x )δij + χij ( x ) that combines the linear and the Kerr nonlinear contributions. For the geometry in Fig. 5.5,
(
(3)
(3)
6χ1122 − 3χ1221 if 0 ≤ x < l ,
′
A (x) =
(5.33)
0
if l ≤ x < Λ
(
(3)
6χ1122 if 0 ≤ x < l ,
B′ ( x ) =
(5.34)
0
if l ≤ x < Λ
(3)
where as usual χijkl are the elements of the general fourth-rank third-order susceptibility tensor [134]. As we discussed in section 4.2, the optical Kerr effect creates
98
off-diagonal elements in the matrix ε̂ in Eq. (5.29) and hence propagation of intense light in an isotropic medium results in light-induced optical anisotropy. Also,
Eq. (5.29) shows that the elements of the matrix ε̂ now become field-dependent.
Finally, we can employ equations (5.26), (5.28), and (5.32) to derive a mathematical expression for ε̂ which strongly depends on the intensity as well as on the
polarization state of the incident field. However, for a subwavelength grating with
finite thickness, we must account for the interference effects arising from the reflections at the interfaces located at z = 0 and z = h as these effects have non-negligible
impact on the local field inside the modulated region located at 0 < z < h.
Although in the quasistatic limit, the field is independent of x and y coordinates,
it strongly depends on the z coordinate. Consequently, ε̂ also becomes z- dependent. Hence, in the nonlinear regime, we are dealing with an effective anisotropic
structure which behaves as a uniaxial crystal but the optic axis of the crystal like
medium varies as a function of z.
The thin-film problem must be solved iteratively, for example by increasing the
intensity from a value corresponding to the linear-optics limit towards the target
value as described in section 4.2. For this, we prefer to split the modulated region into a number of z-invariant layers, and then employ the recursive S-matrix
approach as described in section 2.17, to solve the field distribution inside the
anisotropic-thin-film-stack and the interplay between the nonlinearity and the anisotropic-thin-film interference inside the component.
To get further insight to the interrelation between the elements of ε̂ and the
electric-field components, let us now proceed to derive more transparent expressions
for the mean permittivity-tensor components. Let us first combine Eqs. (5.21), (5.22),
and (5.32) to obtain
− Ex = −ε−1 Dx′ + ε−1 [( A′ + B′ )| Ex |2 Ex + A′ | Ey |2 Ex
Dy′
+ B′ ℜ( Ex Ey∗ ) Ey ],
′
′
2
(5.35)
2
′
= εEy + ( A + B )| Ey | Ey + A | Ex | Ey
+ B′ ℜ( Ex Ey∗ ) Ey ,
(5.36)
where D ′j = D j /ε 0 and we have omitted the explicit x-dependence for the sake of
brevity. We first approximate Ex ≈ ε−1 Dx′ in the right-hand sides of Eqs. (5.35) and
(5.36), and then average over the grating period to arrive at
Dx′ hε−1 i x
A′ + B′
ε4
2
A′
ε2
| Dx | Dx +
| Ey | 2 D x
x
′
B
+ h Ex i x + 2
Re( Dx Ey∗ ) Ey ,
ε x
′
A
′
′
′
2
Dy = εEy + h A + B i x | Ey | Ey +
| D x | 2 Ey
ε2 x
′
B
+
Re( Dx Ey∗ ) Ey .
ε x
=
x
(5.37)
(5.38)
Next, we combine the relation Dx′ ≈ hε−1 i x h Ex i x with Eqs. (5.37) and (5.38), to find
99
out at once that
1
(eff)
hε xx i x ≈
h ε −1 i
+
x
A′ + B′
ε4
x
|h Ex i x |2
+
hε−1 i4x
| Ey | 2
,
−1 2
x hε i x
′
Re(h Ex i x Ey∗ )
B
(eff)
(eff)
hε xy i x = hε yx i x ≈
,
ε2 x
hε−1 i2x
′
′
A
|h Ex i x |2
(eff)
′
2
hε yy i x ≈ hεi x + A + B x | Ey | +
.
ε2 x hε−1 i2x
A′
ε2
(5.39)
(5.40)
(5.41)
In the next section we will see that although the expressions given by Eqs. (5.39)–
(5.41) are approximate, they may lead to sufficiently accurate prediction of the nonlinear form birefringence. However, we note that these expressions are also z- dependent in case we have subwavelength gratings with finite thickness. From Eq. (5.40)
we find that the permittivity matrix has light-induced off-diagonal elements with
their magnitudes depending also on the polarization state inside the SWG. Since
the phase difference between Ex and Ey varies with the propagation distance inside
the structure, the direction of optic axis of the form-birefringent component varies
as a function of z.
5.5
NUMERICAL EXAMPLES
Let us first assume that the pillars i.e. the grating ridges in Fig. 5.5 are composed
of a certain class of amorphous polymer material with negligible nonlinear optical
absorption, linear refractive index n1 = 1.88 and the nonresonant third-order sus(3)
(3)
ceptibility χ1122 = χ1221 = 1.19 × 10−17 m2 /V 2 . The surrounding medium and the
medium after the component are assumed to be fused silica with n2 = no = 1.47,
and the medium of incidence is air with ni = 1. Furthermore, we assume that the intensity of the incident field is I = 5 GW/cm2 . The fill factor of the SWG is assumed
to be f = 0.7. We compare the results obtained by the analytical formulae derived
in the previous section with those obtained with the nonlinear FMM as introduced
in section 4.3. In the FMM simulations λ = 532nm and Λ = 50 nm are assumed.
Figure 5.6 elucidates the effect of Kerr nonlinearity as a function of the SWG
thickness h. With the assumed very high value of third order nonlinearity as well
as high value of the incident field intensity, the light-induced form birefringence is
seen to lead to a radical increase in the phase difference between the y and x components of the field. Also, we find that the Approximate Effective-Medium Theory
(A-EMT), given in Eqs. (5.39)–(5.41), predicts the results quite well. Since the grating
period is much smaller than the wavelength of the incident light, the general EMT
[Eqs. (5.26), (5.28), and (5.32)] is essentially identical to the results obtained by the
nonlinear FMM.
In our second example, we study the effect of the polarization state of the incident field on the light-induced form birefringence. This time we use the EMT
approach to study the same structure as in the previous example, but now with
four different incident polarization states. Clearly from the results illustrated in
Fig. 5.7 we observe that the strongest effect is obtained with an incident field that
is linearly polarized in the y direction (for this y- invariant geometry one may expect the opposite for a x- invariant geometry), and the weakest with the x-polarized
100
(Φy − Φ x )[Rad]
1.5
1.2
0.9
FMM/EMT(nl)
A-EMT(nl)
A-EMT/EMT/FMM(linear)
0.6
0.3
0
0
300
900
600
1200
1500
h [nm]
Figure 5.6: Comparison of EMT, A-EMT, and the rigorous Fourier Modal Method
(FMM) with 45◦ linearly polarized incident field.
field.
(eff)
These results can be understood by noting that in the linear limit hε xx i x ≤
(eff)
∆( Φy − Φ x )
(Φy − Φ x )[Rad]
hε yy i x , where the equality holds only if l = Λ or l = 0. Inspecting Eqs. (5.39)–(5.41),
it is easy to understand that with y- polarized field, the light-induced anisotropy is
(eff)
(eff)
larger for hε yy i x than for hε xx i x . For the x- polarized case, the light-induced form
birefringence is rather weak because the x component contributes almost equally to
(eff)
(eff)
hε xx i x and hε yy i x . The examples included so far do not uncover the information
1.5
y-Pol
x-Pol
45° LP
CP
1
0.5
0
0
300
600
900
1200
1500
900
1200
1500
h [nm]
1
0.7
y-Pol
x-Pol
45° LP
CP
0.3
0
0
300
600
h [nm]
Figure 5.7: Effect of the state of polarization of the incident field on the form
birefringence (top) and the nonlinear form birefringence viz. the difference of the
phase difference to the linear case (bottom). Here 45◦ LP means incident light that
is linearly polarized at ±45◦ with respect to the x and y axes and CP means left (L)
or right-handed (R) circularly polarized light. The oscillatory behaviours are due to
thin-film interferences which grow with film thickness.
about the polarization change upon propagation through the Kerr nonlinear form
101
∆Θ[Rad]
birefringent component. In our third example, we study the effect of Kerr nonlinearity on the polarization azimuth Θ and the ellipticity angle τ, which are defined
in the inset of Fig. 5.8. Figure 5.8 shows the light-induced change of the state of
polarization for the geometry considered in Fig. 5.5. We observe that the state of
polarization of the incident light has a significant effect on the light-induced polarization change, as can be expected based on the results discussed in the previous
example.
∆τ [Rad]
h [nm]
h [nm]
Figure 5.8: Change in the polarization azimuth (top) and the ellipticity (bottom) of
the zero-order transmitted field. In the inset we show how we define the polarization
azimuth Θ and the ellipticity angle τ. Clearly, these are identical to the definitions
introduced in section 2.9.
5.6
SUMMARY
In this Chapter, we have investigated the effects of intensity and the polarization
state of the incident field on the form-birefringence properties of subwavelength
gratings. Simple expressions are derived for the permittivity-matrix elements by
taking into account the light-induced changes. We found that in contrast to the
linear case, in which a form-birefringent SWG always acts as a negative uniaxial
crystal, the nonlinear case is much more intricate because of the role of polarization
state of light.
Although in the numerical examples, we assumed high values of the incident
intensity and the nonlinear susceptibility, which leads to extremely strong lightinduced effects, we believe that observable light-induced changes can be realized in
practice especially using ultrashort laser pulses with high peak intensity. Since the
dispersion of form birefringence is, in general, rather weak [208], we believe that it is
possible to design components for which the polarization properties stay essentially
unmodified within the spectral bandwidth of the pulses.
102
To summarize, we have developed a theory that can be used to design alloptically tunable form birefringent components which may find applications in nonlinear integrated photonics especially to construct tunable waveplates, retarders, or
phase modulators.
103
6
All-optical modulation and optical bistability with a
Silicon Nitride waveguide grating
Nonlinear optical processes can be largely enhanced using resonant optical elements. Strongly confined local fields inside these resonant structures can reach a
rather high level such that the nonlinearities become remarkable even at low input powers. To date, several structure geometries have been proposed and both
theoretical and experimental research are carried out to demonstrate all-optically
tunable functionalities by exploiting enhanced nonlinear optical responses of these
resonators. Examples include ring resonators [60–62, 213], resonant cavities induced
by defect modes in photonic crystals [63–66], resonant microspheres for whisperinggallery modes [214], metallic nanostructures with surface plasmon resonances [215],
waveguides [216], grating couplers [217], hybrid plasmonic-dielectric systems [218],
resonant waveguide gratings [67, 68] etc.
Among numerous all-optically controllable functionalities, all-optical switching
is perhaps the most fascinating one, where an incoming switching beam redirects
other beams through nonlinear light-by-light scattering [134]. In all-optical routing, it is required to control the state of an optical switch using past information
i.e. one needs to build a sequential logic circuit. One way to build such a logic
element is to employ optical bistability [219]. In an optically bistable system, the
system output can be switched between two stable states optically [134]. Usually,
bistability is observed with extremely intense light fields and is difficult to achieve
in solid state materials as the required intensity level is high enough to damage any
material. Hence, for practical applications, materials with high nonlinear optical
coefficients are needed and large interaction length is required to increase the effective nonlinearity. An alternative and more efficient way is to employ the resonant
nanostructures cited in the previous paragraph.
Another important parameter one should carefully consider while constructing
a bistable device is the response time of the nonlinear material, which may significantly affect the switching speed. Hence nonlinear materials with ultrafast response time are required as well to build fast optical switches. Since 1960’s, there
has been an extensive research focus on nonlinear optical materials. Third order
nonlinear materials (both centrosymmetric and non-centrosymmetric) such as pure
Silicon (Si) [70], Titanium di-oxide (Ti O2 ) [220], Silicon Carbide (SiC) [221], Silicon
Nitride (Si3 N4 ) [71], Chalcogenide glasses [222], and Graphene-Silicon hybrid materials [223,224] have been carefully examined in the past. Eventually, it was found out
that while some Si based devices suffer from low response time and high nonlinear
optical losses (large Two Photon Absorption (TPA), free-carrier effects etc.), Si3 N4
and chalcogenide glass based devices can offer large ultrafast third-order nonlinearities, and low TPA at the same time.
Recently, low power optical bistability in nonlinear silicon and silicon nitride
photonic crystals and ring resonators have been experimentally demonstrated [213,
225, 226]. Besides devices that can be integrated on chip, free-space bistable devices
have their own significance. These free-space devices have easy optical fan-in/out
and hence can be considered as good candidates for large-scale parallel all-optical
105
signal processing. Possibility to achieve optical bistability in diffraction gratings
with nonlinear dielectric/semiconductor materials, metallic gratings with grooves
filled with nonlinear media, multilayered gratings, parallely stacked thin films and
one dimensional slab waveguide gratings have been extensively studied theoretically [179, 183, 227–231].
Among the commonly used free-space diffractive optical components, the resonant waveguide grating structure (RWG) or guided mode resonance filter (GMRF)
is thought to be an excellent candidate for realizing all-optical sequential logic elements based on optical bistability [228, 229, 232]. A waveguide grating in its simplest form is composed of a relief grating layer, a waveguide layer and a substrate
layer [141]. Depending on the structural dimensions of these RWGs and the geometry of light-matter interaction, strong resonance phenomena can be observed which
yield reflection/transmission peaks/dips. The central frequency and linewidth of
these resonance peaks also depend on the structural parameters. At resonance,
a guided mode is excited by the incident wave which interferes with the directly
transmitted/reflected wave destructively and as a result 100 percent efficiency can
be achieved in reflection/transmission. Also, at resonance there is strong field confinement inside the structure which enhances the nonlinear effects. This enhanced
local field inside a RWG, might also be helpful to realize low energy optical bistability in practice.
Here, using the nonlinear FMM developed in Chapter 4, we perform numerical experiments with GMRFs having different structural parameters. The results of
these experiments demonstrate that low energy optical bistability in reflection/transmission for normally incident field can be observed by strong nonlinear light-matter
interactions in Silicon Nitride waveguide-gratings having 2-D periodicity. We fabricate the RWGs from PECVD synthesized Silicon Nitride thin films on top of quartz
substrates with standard electron beam lithography and reactive ion etching techniques. Finally, we perform the experiments with a single wall Carbon nanotube
(SWCNT) modelocked ultrafast fiber laser and demonstrate all-optical modulation
of the transmission spectra of these SiN RWGs.
This Chapter starts with the working principle of a RWG. Then we define the
term optical bistability and briefly discuss about the theory of optical bistability of a
Fabry-Perot resonator. After that, discussions are made regarding the merits of using Silicon Nitride as a material for constructing these RWGs. Next, we present the
fabrication methodology for these nanogratings and then the numerical simulation
results. Finally, the experimental procedure is described and experimental results
are included.
6.1
THEORY OF OPTICAL BISTABILITY - THE FABRY-PEROT RESONATOR APPROACH
Certain nonlinear optical systems can possess more than one output states for a
given value of input. This phenomenon is termed as ‘Optical multistability’. Bistability refers to a case where there are two output states for a specific input state.
Let us now recall the very first experimental demonstration of optical bistability
[233]. In that experiment, a Fabry-Perot resonator with nonlinear optical media
embedded inside its resonant cavity was considered. Geometry of such a structure
is shown in Fig. 6.1. In the sketch, M1 and M2 are two identical loss-less dielectric
mirrors with reflection coefficient ρ, reflectance R, transmission coefficient τ, and
106
transmittance T. A, Ar , and At are the incident, reflected, and the transmitted
field amplitudes respectively. B and C are the amplitudes of the forward and the
backward propagating fields inside the resonator, where the field amplitudes are
measured at the inner surface of M1 . Now, we can represent the internal fields of
the resonant cavity in terms of the incident field using the following relations
C = ρB exp (2inωL/c − 2αL) ,
(6.1)
B = τ A + ρC.
(6.2)
Here n is the refractive index and α is the absorption coefficient of the cavity which
are functions of the incident field intensity (I), and L is the length of the cavity.
Here, we have assumed that n and α are spatially invariant. We eliminate C from
Equations (6.1) and (6.2) to yield
B=
τA
.
1 − ρ2 exp[2inωL/c − αL]
(6.3)
Equation (6.3) can predict optical bistability if n and/or α have strong dependence
on I. In this context, we must mention that considering the strength of the intensity dependence of n or α, optical bistability may be classified into two categories
which are pure absorptive and pure refractive types. Here, we shall consider only
pure refractive bistability i.e. we consider nonlinear materials with negligible nonlinear absorption/two-photon absorption (TPA) is placed inside the resonator cavity.
Furthermore, we assume that the linear loss of the cavity is negligible. Hence, in
Eq. (6.3), we put α = 0 to obtain
B=
τA
1 − ρ2 exp[2inωL/c]
.
(6.4)
Again, making use of the relation ρ2 = R exp (iφ), in Eq. (6.4) and assuming TE
polarized light incidence we get,
B=
τA
τA
=
.
1 − R exp[i(φ + 2n0 ωL/c + 2n2 IωL/c)]
1 − R exp[iδ]
(6.5)
Here, n0 and n2 are the linear and the nonlinear refractive indices of the cavity
material respectively, δ is the total phase shift acquired in a cavity round trip and
I = IB + IC = | B|2 + |C |2 ≃ 2IB (ignoring the standing wave nature of the field
inside the cavity). Next, we use the relation Iin = 2n1 ǫ0 c | A|2 for the incident field
intensity, where n1 is the linear refractive index of the medium to the left of the
mirror M1 , to readily obtain the following expression from Eq. (6.5)
IB
1/T
=
.
Iin
1 + (4R/T 2 ) sin2 (δ/2)
(6.6)
Equation (6.6) shows that for a particular value of the incident field intensity (Iin ),
there might exist more than one distinct values of the cavity intensity (IB ). Under a
special circumstance, when IB has three distinct solutions for the range of the input
intensities Iin , It = | At |2 vs. Iin plot looks like the curves in Fig. 6.2. Clearly from the
figure, any value of Iin between I1 and I2 yields two distinct values of It depending
on the previous state (in terms of the intensity) of the system.
107
M1
M2
A
At
B
Ar
C
L
Figure 6.1: Geometry of a Fabry-Perot resonator with optical Kerr nonlinear
medium filling its cavity.
It
I1
Im
I2
Iin
Figure 6.2: Typical input-output relationship of an optically bistable system. Bistability can be observed for a range of incident field intensities lying between I1 and
I2 .
6.2
WORKING PRINCIPLE OF A WAVEGUIDE GRATING
In this section, we present a simple ray picture model for describing the nature of
the interaction of the grating-waveguide structure with an incident plane wave. A
resonance waveguide grating (RWG) or a guided mode resonance filter (GMRF) in
its simplest form is shown in Fig. 6.3. It consists of three layers- a thin grating
layer of thickness δ and linear refractive index n2 , a waveguide layer of thickness t
and having the same refractive index (n2 ) as the grating layer, and a substrate layer
108
A
θ
a
a
d
O1
n2
O2
δ
F
ξ
C
B
t
c
E1
c
D
E
Figure 6.3: Geometric interpretation of a resonant waveguide grating [141] in its
simplest form consisting of a grating layer of thickness δ, a waveguide layer of
thickness t and a substrate layer. θ is the angle of incidence of the incoming plane
wave and ξ is the angle of diffraction. The higher order diffracted rays are not
shown in the figure.
(refractive index n3 ). When the structure is illuminated with a plane wave from the
grating side, part of the incident wave is directly transmitted through the structure
(D) and part is diffracted by the grating layer and then might get trapped inside
the waveguide layer (B) [141]. Again, this trapped wave B might face total internal
reflection at the interface between the waveguide layer and the substrate and emerge
as C as shown in Fig. 6.3. Part of C gets diffracted out of the waveguide layer and
emerges as E which is clearly collinear with D. On special conditions, which depend
on the grating period (d), δ, t, the angle of incidence (θ) of the incoming plane wave,
and the refractive indices n1 , n2 ; D and E might interfere destructively resulting
in complete loss of transmission. Again, part of F gets reflected from the interface
between the waveguide layer and the substrate and after getting diffracted again by
the grating layer as depicted in Fig. 6.3, might emerge as E1 which is also collinear
with D and E. The cycle is repeated and hence the system is analogous to a Fabryperot resonator described in section 6.1.
We can describe the grating diffraction in B by the following relation
n1
2π
2π
2π
sin θ + m
= n2
cos(ξ ).
λin
d
λin
(6.7)
To understand the resonance mechanisms qualitatively, let us consider for now that
D and E are the only interfering transmitted waves. Here we emphasize that one
must retain all the higher order diffracted waves (Ei , where i = 1, 2, 3, . . . ) and
should use multiple interference model as implemented in Ref. [141] for an accurate
description of resonance phenomenon and while calculating the spectral bandwidth
of the resonance. Alternatively, one can use FMM (as described in Chapter 3) to
109
model the optical properties of a GMRF.
The complete destructive interference of D and E is governed by certain phase difference conditions between these two waves. We find that the total phase difference
between D and E is (under the assumption that the grating layer is infinitely thin)
ΦTotal = Φ p + Φr + 2Φd
(6.8)
where, Φ p is the phase difference gained due to the optical path length difference
between the incident wave and the wave traveling in the waveguide, Φr is the phase
difference acquired by the total internal reflection at the interface, and 2Φd denotes
the phase difference attained through the phase shift associated with each of the
diffractions. The leading edge of the incident wavefront a strikes the grating at
point O1 , where it is diffracted towards the bottom surface and then reflected to
the top surface as wave front c, whose leading edge strikes the grating at point O2 ,
which corresponds to point O1 , located an integer number of grating periods from
O1 . By some geometrical constructions, using Eqs. (6.8) and (6.7) and assuming TE
polarized incident field we get,
ΦTotal = 2n3
2π sin(ξ )
2πl
t+m
+ 2Φ f + 2Φn − π.
λin
d
(6.9)
Here, l is the distance (l = q × d, where q is an integer) between the points O1 and
O2 as illustrated in Fig. 6.3, φ f denotes the Fresnel phase shift at the interface, Φn
is the Fresnel phase shift acquired due to the refractive index mismatch between
the grating material (n2 ) and the surrounding medium (n1 ), and m can take any
integer value. At resonance, the structure supports guided modes inside the waveguide layer (which eventually leaks into the superstrate). The following condition is
required to support a guided mode
2n3
2π sin(ξ )
t + mq2π + 2Φ f + 2Φn = m2π.
λin
(6.10)
From Eqs. (6.9) and (6.10), we see that for m = 1, i.e. for the first diffraction orders, the phase difference between the waves D and E is π. Thus, total destructive
interference takes place between these two waves for a particular range of values
(depends on the structure geometry) of λin (wavelength of the incident wave) and
one can achieve 100 percent efficiency in direct reflection. Thus the structure acts as
a narrow band reflector. With increasing value of n2 , this resonance peak usually
gets shifted towards the longer wavelengths. Now, if the grating is made with Kerrnonlinear materials, assuming TE polarized light input, its refractive index can be
described by the following equation
n2t = n2 + nnl I,
(6.11)
where, I is the incident field intensity, and nnl is the nonlinear refractive index of the
grating material as defined in Chapter 4. A moderately intense field can change the
effective refractive index of the grating layer as well as the waveguide layer due to
the local field enhanced Kerr nonlinearity inside these layers and hence can tune the
structure out of resonance. As a result, 100 percent to 0 percent efficiency in zeroth
order reflected field can be obtained.
110
6.3
SILICON NITRIDE VS. CRYSTALLINE SILICON AS A NONLINEAR
MATERIAL
Silicon-on-insulator (SOI) has already been established as a platform for linear photonics and initiated the development of silicon photonic chip industry [234]. Some
of the advantages of SOI are as listed below
• SOI platform can reap the benefits of exploiting already well established complementary metal-oxide-semiconductor (CMOS) infrastructure.
• It is possible to combine both electronics and photonics on the same chip.
• High refractive index of silicon allows tight light confinement.
• Silicon has high Kerr nonlinear coefficient (n2 ) and hence when combined with
its high linear index, can be effective for low power nonlinear photonics.
However, in crystalline bulk silicon, the optical Kerr nonlinearity competes with
parasitic nonlinear absorption effects which include two-photon absorption (TPA)
effects, free-carrier effects, and defect-induced effects. TPA and TPA generated freecarrier effects are significant in all telecommunication bands with wavelengths (λ)
shorter than 2000 nm. Though the free-carrier effects can be compensated by using
a p-i-n junction which sweeps out the carriers [235], there is no way to overcome the
TPA in the telecommunication window not even by engineering waveguide dimensions. Hence, the band structure of silicon poses a fundamental limit for using it as
a material for nonlinear optics in the near infrared region.
Numerous all-optical switching devices have been demonstrated lately with silicon based resonant micro and nanostructures. It has been realized and also proved
experimentally that the switching operations in silicon occur mainly due to freecarrier induced nonlinearity [60,236]. The response time of these devices are limited
by the lifetime of these free carriers (∼ 100 ps). Clearly, this also limits the realization of ultrahigh speed all-optical operations. Hence, there have been a quest for
finding a material with either shorter carrier lifetime [237, 238] or negligible TPA.
Historically, Aluminium Gallium Arsenide (AlGaAs) is the first material proposed for nonlinear optics in the telecommunication band [239]. AlGaAs is a semiconductor material with larger bandgap and it is possible to tune its nonlinearity by
changing the alloy composition. Another important platform is based on Chalcogenide glasses [240] which has very high Kerr nonlinearity. However, there are
fabrication challenges associated with both of these platforms.
Very recently, a new platform based on Silicon Nitride (SiN) has been established
for nonlinear integrated photonics [71]. SiN is a CMOS compatible material and in
the past, it has been used as a platform for linear integrated optics. Historically, it
has been very challenging to achieve a low-loss, thick (> 250 nm) SiN layers due to
the associated tensile film stress which results in cracking of the films. However, in
recent years, it has been possible to grow thicker films using both plasma-enhanced
chemical vapour deposition [PECVD] [241] and low-pressure chemical vapour deposition [LPCVD] [242] techniques. Perhaps, the first nonlinear effect in a SiN waveguide was demonstrated in 2008 [241], where nonlinearity induced resonance peak
shift of a ring resonator was experimentally demonstrated. Here we list some of the
merits of SiN as a material for nonlinear optics
111
• SiN has a larger refractive index (≃1.8-2) as compared to SiO2 which allows
small waveguide dimensions.
• Large bandgap of SiN makes it optically transparent already in the visible
spectral regime (from λ = 400 nm).
• SiN has negligible TPA coefficient in the near infrared (communication wavelengths) and the free carrier effects are believed to be minimal even if input
power ∼100 mW is used.
• As SiN is grown by deposition process, multilayered structures can be grown
at ease (layer stack flexibility).
• It has moderately high n2 .
• Linear absorption losses in amorphous SiN can be extremely low.
• Thermo-optic coefficient of SiN is very low and hence thermal nonlinearities
are minimal.
• Integration with CMOS electronics is possible.
Though numerous works based on SiN waveguides, where the SiN layers were
grown by LPCVD, have been reported [243–246], there are not much references
available where the SiN films are grown by PECVD. PECVD as compared to LPCVD
(process temperature about 800◦ C) is a lower temperature (400◦ C) process which is
advantageous for hybrid integration. Table 6.1 shows a comparison between the SiN
films grown by these two techniques. In this context, we must mention that there is
Table 6.1: Comparison between PECVD and LPCVD grown SiN films.
Deposition temp.
PECVD
80-400◦C
LPCVD
800◦ C
r.i. (λ =800 nm)
1.89
2.03
Film stress
Manageable
High
Absorption
Water peaks at 1520 nm
Low
Uniformity
Medium
High
Etch selectivity
No
Good vs. SiO2
Content
Trace of H
Stoichiometric SiN
a possibility to control the ratio of the silicon and the nitride contents [247] during
the deposition process, which is crucial in optimizing the nonlinear properties of the
deposited films for case specific applications. Recently, strong third harmonic generation in a silicon-rich nitride (PECVD) waveguide grating [69] has been reported.
Also, a micro ring resonator based Kerr switch has been demonstrated experimentally where the silicon-rich nitride thin film is synthesized by PECVD [62].
112
6.4
FABRICATION OF THE WAVEGUIDE GRATING STRUCTURES
Before going into the detailed fabrication process, we shall first briefly describe the
principles associated with some of the most important steps involved in lithographic
fabrication of these SiN RWG samples.
6.4.1
Thin film deposition
In this subsection, we briefly describe the thin film coating techniques we used in
this work.
Evaporation
In thermal evaporation, both the material source and the substrate are placed inside
a vacuum chamber to avoid reactions between vapour and atmosphere. There are
two ways to heat the source material. These are resistive heating or heating by high
energy electron beam. As evaporation is a line-of sight technique, there is usually a
thickness gradient on the substrate [248]. One way to overcome this is to rotate the
sample during deposition which in turn averages out the thickness variation. In our
work, Kurt J. Lesker lab-18 evaporation unit was used.
Chemical vapour deposition
Chemical vapour deposition (CVD) is a well known and widely used materialprocessing technique [249]. Though the primary application of CVD is thin-film
coating of solid state materials, it can also be used to produce bulk and composite
materials. In a CVD process, one or more precursor gases are flown into the reactor
chamber which contains heated objects to be coated. Chemical reactions near and
on these heated surfaces result in thin-film deposition. The chemical by-products
generated from the reactions and unused precursor gases are exhausted out of the
chamber. Depending on the process parameters and reactor configurations, there
are many variants of CVD. It can be either hot-wall or cold-wall reactor type. The
process pressure can range from sub-torr level to above-atmospheric level. Depending on the type of the CVD, process temperature can also vary in between 200-1600
◦ C. There also exists a variety of enhanced CVD processes which may involve the
use of plasmas, ions, lasers, hot filaments etc. to increase deposition rate and/or
decrease process temperature (economic process). Hot wall reactor CVDs are run at
very high temperatures and at low pressures (∼10 Torr). Example include LPCVD.
Some of the advantages of CVD are as listed below
• CVD films are conformal in nature (film thickness on the substrate side walls
equals to the film thickness on the top of the substrate).
• A wide range of material can be deposited with high purity.
• Deposition rate is high.
• High vacuum is not needed as compared to physical vapour deposition.
However, there are also some disadvantages which include
• CVD precursors need to be volatile at near-room temperature which is nontrivial for a large number of elements in the periodic table.
113
• Precursors can be highly toxic.
• Precursors can be costly.
• Chemical byproducts can be hazardous.
• High temperature deposition might not be suitable for many substrates.
• The stresses in the films (resulting from high temperature deposition) may
cause mechanical instabilities.
An example for a relatively lower temperature CVD is plasma enhanced chemical
vapour deposition [250]. In PECVD, electrical energy is used to generate plasma
(glow discharge). Due to the electrical energy transferred into the precursor gas
mixture, reactive ions and radicals, neutral atoms and molecules etc. are formed
which are in highly excited state. These reactive and energetic species, which are
formed by collision in the gas phase, interact with the substrate and thin films are
deposited. The process temperature may lie in between 300-400◦C. Some of the
properties of PECVD films include good adhesion, and uniformity.
6.4.2
Electron beam lithography
Electron beam lithography (EBL) is a technique for creating small patterns in a highenergy sensitive chemical called resist. It was originally developed for the electronics
industry [251] but later was adopted also by the optics community [252, 253]. The
resolution of EBL is far better as compared to optical lithography [254, 255] and it
can be used to generate more complex patterns. Another advantage is that a wide
variety of substrates can be used. However, EBL system is relatively slow and hugely
expensive. Hence, it is not suitable for industrial applications but more appropriate
for research purposes.
The first EBL was constructed in late 1960s by adding a beam blanking unit, a
pattern generator, and an interferometric stage controlling unit to a scanning electron microscope. Schematic of the column in the Vistec EBPG5000+ES HR EBL tool
(the system used in the present work) is presented in Fig. 6.4. It includes an electron source, an alignment system, a blanker, magnetic lenses, a beam deflector, a
stigmator, apertures and a detector. The source generates an electron beam using
a thermal field emission gun, or by thermionic emission. The emitted electrons are
accelerated through the column by application of voltage. The alignment system is
used to center the beam in the column and the lenses are used for focusing purposes. Apertures are used to limit the beam and also to block stray electrons. The
blanker is responsible to turn on and turn off the beam. By use of the deflector, the
beam is scanned on the sample surface. Any imperfection in beam alignment may
cause beam astigmatism which is corrected by the stigmator. Electron detectors are
used to help in focusing of the beam and to find the alignment marks. The sample is
placed on a high-precision movable stage (x-y) in a chamber placed underneath the
column. Clearly, a vacuum system is needed inside the column and the chamber.
The area which can be patterned without moving the stage is called the main field.
Structure with larger size (than the main field) is patterned by moving the x-y stage
to the center of the next main field. This movement can be done with high precision
using interferometric stage measurement system. Several EBL systems are available
in market. The Vistec EBPG5000+ES HR is a Gaussian beam system which provides
better resolution as compared to other beam systems though the system throughput
114
Emitter
Suppressor
Extractor
Lens
Focus
Anode
Gun
alignment
Lens
Blanking
Blanking
Deflection
Final lens
Substrate
Figure 6.4: Schematic of the column of the Vistec EBPG5000+HR electron beam
patterning tool. (Courtesy of Vistec Lithography Ltd.)
is not good enough for very-large-scale integration (VLSI) manufacturing industries.
This system can operate with 20 kV, 50 kV or 100 kV voltage with up to 50 MHz
pattern generator. With 50 kV and 100 kV operating voltages, the maximum main
field sizes are 409.6 × 409.6 µm and 256 × 256 µm respectively. The beam exposure
115
time (t) on a surface area S is determined from the following relation [256]
t = σs S/I,
(6.12)
where σs is the surface charge density widely known as the dose and I is the beam
current.
6.4.3
Resist technology
Resists are sensitive to the energy of the electron beam. Due to absorption of energy
from the beam, the molecular weight of the resist material in the exposed regions
is changed which enables dissolution of either the exposed or the unexposed area
by a chemical, called the developer [257]. Some of the sought properties of the electron beam (e-beam) resists include high resolution, high sensitivity, and proper etch
resistance (important while transferring the resist pattern to the substrate). E-beam
resists are dissolved in a liquid solvent and can be sprayed on the substrate using
several techniques such as spray coating, roll coating, dip coating or spin coating.
In most cases, spin coating technique is used as this provides better layer uniformity. After the coating, the solvent is evaporated by pre-heating or soft-baking. The
baking time and the temperature affect the exposure and the development [258].
After the e-beam exposure, the pattern created in the resist is developed by immersion, spray, or puddle method. The later two are automated processes with better
reproducibility.
Depending on the response to the energy of the electrons, resists can be divided
into two categories• Positive resist- The exposed areas become more soluble in the developer.
• Negative resist- The exposed areas become less soluble in the developer.
Generally, positive resists are more suitable to create binary profiles [259] and the
negative resists for making continuous profiles [260]. However, there exist some
exceptions [261, 262]. To fabricate the RWG samples, we use AR-P 6200 which is a
positive tone resist.
6.4.4
Reactive ion etching
After the pattern is generated in the resist, it is required to transfer it to the substrate.
This is done by etching. In this process, the material is removed selectively in the areas defined by the etching mask. There are both dry and wet etching processes. Wet
etching is a chemical process and usually isotropic in nature which means that the
material is removed from all directions. On the contrary, dry etching is anisotropic
in nature.
Reactive ion etching (RIE) [263] is a dry etching method, which uses plasma
containing charged particles (initiated by a strong radio frequency (RF) field) to remove material. The material removal can be due to processes which are physical or
chemical in nature. These two processes can occur simultaneously. In physical etching, also known as sputtering, the ions remove material by collisions. This process
is highly directional and anisotropic. In chemical process, material is removed by
chemical reactions with the etchant and the target material which generates volatile
products [255]. This process is isotropic in nature as the chemical reactions occur
116
along the horizontal direction. In order to make the etching process highly selective,
one needs to control both the physical and the chemical parts of process carefully.
In all dry etching processes, both the material to be etched and the mask material
are removed. The ratio of their etching depths is termed as selectivity. For higher
values of selectivity, deeper structures can be fabricated.
Chromium etching
Chromium (Cr) etching is performed in chlorine atmosphere. One can add suitable
amount of oxygen to increase the Cr etch rate without losing selectivity over the
resist mask [264]. In Cr etching process, resists such as ZEP, ARP-6200, or HSQ can
be used as etching masks. However, SiO2 or TiO2 can also be used.
SiN etching
Nitrides can be etched using gases containing fluorine such as CHF3 , CF4 , or SF6 .
The physical part of the etching can be done by adding O2 , He, or Ar. In our work,
we used a gas mixture of CHF3 and O2 . Addition of O2 suppresses formation of
fluorocarbon on nitrile layer which slows down the etch rate. It also increases the
etch rate [265] of the resist hence decreases the selectivity. However, in our work,
we needed to etch only a shallow part of Si3 N4 . Hence, decrease of etch selectivity
was not a crucial issue.
The fabrication process flow is shown in Fig. 6.5. The fabrication steps are listed
as belowSilicon Nitride
Quartz
Cr Deposition
Cr
Silicon Nitride
Quartz
E beam exposure
Resist Coating
Cr Etching
ARP 6200 1:1
Cr
Silicon Nitride
Cr
Silicon Nitride
Quartz
Quartz
Cr Removal
Silicon Nitride Etching
Silicon Nitride
Quartz
Resist removal
Silicon Nitride
Quartz
Quartz
Figure 6.5: Fabrication process flow of the Silicon Nitride waveguide gratings.
117
• Silicon Nitride thin film of thickness 400 nm (±10%) was deposited on top of a
quartz substrate with 1 inch diameter and thickness 0.5 mm using PEVCD. A
gas mixture of (2% SiH4 /N2 ): NH3 =100:3 was used. The process temperature
and pressure were 300◦ C and 1000 mTorr, respectively.
• Cr layer of 50 nm layer thickness was deposited on top of Si3 N4 film using
lab-18 evaporation unit.
• The Cr layer was coated with positive resist AR-P 6200 of layer thickness≃220
nm and the sample was pre-baked at 150◦ for 3 minutes.
• Grooves of the grating were patterned using electron beam lithography tool.
• The exposed resist layer was developed with ethyl 3-ethoxypropionate (EEP).
• Cr was etched in Plasmalab100 with a gas mixture of Cl2 (54 sccm) and O2 (4
sccm) for 3 minutes and 30 seconds. In this dry etching process, we used AR-P
6200 as the etch mask.
• Residual resist layer was removed by means of O2 cleaning (30 sccm O2 ) for
30 seconds.
• Si3 N4 etching was performed in Plasmalab80 with a gas mixture of CHF3 /O2 =
45/10 sccm. Suitable etching time for the desired grating depths were found
out during the process.
• Finally, chromium was removed by means of Cr wet etch solution and the
samples were cleaned in acetone and isopropanol.
Fig. 6.6 shows a side cut view (a) and a top view SEM (b) of a fabricated two
dimensionally periodic structure (periodic square pillars).
(a)
(b)
Figure 6.6: (a) side cut and (b) top view SEM images of a 2-D periodic waveguide
grating structure.
118
6.5
NUMERICAL SIMULATION RESULTS
Most of the numerical experiments are performed with waveguide gratings which
are periodic along two mutually orthogonal directions (in this case x and y) with
equal grating periods along x and y i.e. with square shaped pillars as shown in
Fig. 6.6.
The possibility to construct optical bistable devices based on all-dielectric waveguide gratings having only one dimensional periodicity have been numerically studied before by Magnusson et. al. [232]. However, our numerical analysis reveal that
the 2-D periodic gratings have better angular tolerance as compared to 1-D periodic
waveguide grating structures (in its simplest form) with similar spectral response.
Also, to achieve high quality factor (a quantity used to measure the sharpness of
resonances) with a 1-D periodic GMRF, tight fabrication tolerances are needed.
To exemplify, let us compare the diffraction efficiencies in the 0-th order reflected
light for two waveguide grating geometries as illustrated in Fig. 6.7. The grating pad
c
δ
t
SiN
SiN
SiO2
(a)
( b)
Figure 6.7: (a) 1-D periodic and (b) 2-D periodic waveguide grating geometries.
rameters for the geometries in Fig. 6.7(a) and Fig. 6.7(b) are listed in Table 6.8. The
refractive indices of the grating material (i.e. silicon nitride) and the substrate are
assumed to be n g = 1.934 and ns = 1.4442 respectively. Figure 6.8 (a) shows that the
quality factors (Q) of the resonances, which are defined as the ratios of the resonance
full widths at half of maxima (∆λmax ) to the resonance peak wavelengths (λmax ), are
almost the same. Clearly, to attain same Q-resonance with 1-D periodic geometries,
shallower grating layers are needed, which is very challenging in fabrication point
of view. Figure 6.8 (b) shows the 0-th order reflected efficiencies of the 1-D and the
Table 6.2: Grating parameters for the 1-D and 2-D periodic geometries in Fig. 6.7.
Grating period (d)
1-D
942 nm
2-D
1035 nm
Grating layer thickness (δ)
20 nm
40 nm
Waveguide layer thickness (t)
369.3 nm
349.3 nm
Linewidth (c)
0.6 × d
0.6 × d
2-D periodic structures (with grating dimensions given in Table 6.8) as a function of
the angle of incidence (θ) of the incoming TE polarized plane wave of wavelength
119
λmax = 1548.9 nm. From the plots, it is clear that the 2-D periodic geometry offers
better angular tolerance. Hence, 1-D periodic waveguide gratings with very high Qs are more prone to get tuned out of resonance. Figures 6.9(a) and (b) show electric
field intensity distributions inside and outside the 1-D periodic waveguide grating
structure for θ = 0◦ and θ = 0.3◦ respectively at λmax = 1548.9 nm. Clearly, the
structure gets tuned out of resonance for angle of incidence θ = 0.3◦ and the localized electric field intensity is much weaker as compared to the case with normal
incidence of light i.e. for θ = 0◦ .
In Figs. 6.10 (a) and (b), the reflected efficiencies in 0-th diffraction orders corresponding to the 1-D and the 2-D periodic geometries respectively are plotted as
functions of both θ and λ. Clearly from the plots, it is challenging to design and
realize a high Q 1-D periodic waveguide grating structure as a slight deviation from
the normal incidence can tune the structure out of resonance. Also, a modest change
in the grating dimensions of the fabricated samples can largely shift the resonance
across the wavelength. On the contrary, 2-D periodic geometries have better angular tolerance and any shift in resonance resulting from the fabrication errors can be
compensated by varying the angle of incidence of the incoming light.
After initial design, the 2-D periodic grating structures are fabricated by the
1
1D RWG
2D RWG
ηr00
0.8
0.6
0.4
0.2
0
1548
1548.4
1548.8
1549.2
1549.6
1550
λ [nm]
(a)
1
2D RWG
1D RWG
ηr00
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
θ [Degrees]
( b)
Figure 6.8: (a) Comparison between spectral responses in direct reflection of a 1-D
periodic and a 2-D periodic waveguide grating. The grating parameters are listed
in Table 6.8. (b) Efficiency in direct reflection plotted as a function of the angle of
incidence of the incoming TE polarized plane wave with wavelength λmax = 1548.9
nm. With almost the same Q resonances, the 2-D periodic geometry offers better
angular tolerance.
120
−1500
2000
1500
−500
0
1000
500
1000
x [nm]
x [nm]
−1000
500
1500
−200
0
200
−1500
7
−1000
6
−500
5
0
4
500
3
1000
2
1500
1
600
400
z [nm]
(a)
−200
0
200
400
z [nm]
( b)
600
Figure 6.9: Localized electric field intensity distributions inside and outside the 1-D
periodic waveguide grating structures at λmax = 1548.9 nm. (a) and (b) correspond
to the cases with θ = 0◦ and θ = 0.3◦ respectively. The grating geometries are
highlighted by the white lines. TE polarized light is assumed to be incident from air
on the grating side.
ηr00
λ [nm]
1548
0.8
1548.5
0.6
1549
0.4
1549.5
1550
0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
θ [Degrees]
(a)
ηr00
λ [nm]
1548
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1548.5
1549
1549.5
1550
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
θ [Degrees]
( b)
Figure 6.10: Efficiency in direct reflection as functions of θ and λ for (a) a 1-D and
(b) a 2-D waveguide grating having about the same Q factors.
121
lithographic techniques as described in the previous section. Numerical simulations are performed with the values of the grating dimensions (grating periods d x
and dy , thickness of grating layer δ, thickness of waveguide layer t, fill factors f x
and f y ) obtained from SEM and optical profilometer measurements. The refractive
indices of the SiN thin films and the substrates are measured with ellipsometeric
technique [266].
In Table 6.3, we present the measured values of the refractive indices, and the
grating dimensions for samples indicated by S2 and S3 . Grating dimensions and
refractive indices for sample Ss presented in Table 6.3 are chosen arbitrarily. FigTable 6.3: Specifications of the gratings, and the refractive indices of SiN films and
substrates evaluated at λ = 1550 nm. Values corresponding to samples S2 and S3
are evaluated by SEM and optical profilometric measurements, whereas the values
corresponding to Ss are chosen arbitrarily.
d x = dy = d [nm]
S2
1027
S3
1030
Ss
1030
n g (λ =1550 nm)
1.95
1.95
1.95
ns (λ =1550 nm)
1.4442
1.4442
1.4442
δ [nm]
41
55
20
t [nm]
345
341
380
fx = fy = f
0.6
0.6
0.6
ures 6.11 (a)–(c) show the shifts of the resonance peaks in reflection towards the
longer wavelengths with growing intensity of the incident electric field (I0 ) for samples S2 , S3 , and Ss respectively. These curves are plotted with the FMM based technique [199] introduced in Chapter 4. Here, we have assumed normal incidence of
light from air on the grating sides. Also, we have used the experimentally measured
value of χ(3) = 3 × 10(−20) m2 /V2 for the SiN thin films as reported in ref. [69]. The
SiN films used in ref. [69] and the films employed in our work were grown under
similar PECVD conditions. The nonlinear absorption of SiN is neglected around
λ = 1550 nm, the Kerr nonlinear response of SiN is assumed to be instantaneous
and the substrate as well as the surrounding media are modeled as linear materials.
It is clear from the plots that the gratings with lower δ/t value, yield resonances
with higher Q-factors as with increasing value of t, the waveguide modes get less
leaky. As the field enhancements inside the gratings are directly proportional to the
Q-factors, Kerr-nonlinearity induced resonance peak shift can be achieved at lower
field intensities for gratings with higher Q-s. From the plots, we notice that at higher
field intensities, the resonance curves become asymmetric, which might result in optical bistability.
To investigate further, for each sample we choose a specific wavelength from
Figs. 6.11 (a)–(c), at which the reflection efficiency is about 10-20% of maximum efficiency on the falling edge of the resonance curve (for the linear case). Intensities
of 0-th order reflected fields at those specific wavelengths are plotted with respect
to the incident electric field intensities (Iin ). Figures 6.12(a) and (b) show such plots
for S2 and S3 , where the two branches in each plot (solid black and dotted red
lines) correspond to the cases when the incident field intensities are increased and
122
1
0.8
Linear
I0=10 MW/cm2
S2
I0=20 MW/cm2
ηr00
0.6
0.4
0.2
0
1540
1540.5
1541
1541.5
1542
1542.5
1543
λ [nm]
(a)
1
0.8
Linear
2
I 0 =50 MW/cm
S3
ηr00
0.6
0.4
0.2
0
1543
1543.5
1544
1544.5
1545
1545.5
1546
1546.5
1547
λ [nm]
( b)
1
ηr00
Ss
0.8
Linear
2
I 0=300 KW/cm
0.6
I 0=500 KW/cm2
I 0=700 KW/cm2
0.4
0.2
0
1563.2
1563.3
1563.4
1563.5
1563.6
1563.7
1563.8
λ [nm]
(c)
Figure 6.11: Efficiency in direct reflection plotted as function of λ for RWG samples
(a) S2 , (b) S3 , and (c) Ss respectively with their specifications given in Table 6.3. For
each sample, several plots corresponding to different incident field intensities are
included.
123
decreased respectively using staircase functions. Step-sizes for these staircase functions are chosen carefully to assure numerical convergence. Clearly, the plots for
S2 and S3 show optical bistability at the chosen wavelengths λS2 = 1541.56 nm and
λS3 = 1545.7 nm respectively for a range of values of Iin . From the figures, we see
that the cut-off intensities for observing bistable switching at λS2 and λS3 are 15
MW/cm2 and 40 MW/cm2 for the samples S2 and S3 respectively.
Figures 6.13(a) and (b) show the reflected and the transmitted field intensities
2.5
x 10
11
Increasing branch
Decreasing branch
r [Watts/m2 ]
Iout
2
λS2 = 1541.56 nm
1.5
1
0.5
0
1.2
1.3
1.4
1.5
1.6
Iin
1.7
1.8
1.9
2
2.1
2.2
x 10
[Watts/m2 ]
11
(a)
x 10
11
6
r [Watts/m2 ]
Iout
5
4
Increasing branch
Decreasing branch
λS3 = 1545.7 nm
3
2
1
0
2.5
3
3.5
4
Iin
4.5
[Watts/m2 ]
5
5.5
6
11
x 10
( b)
Figure 6.12: 0-th order reflected field intensities plotted as a function of the incident electric field intensities for (a) S2 and (b) S3 . The solid black lines and the
dotted red lines are the plots corresponding to the increasing and the decreasing
field intensities respectively and are marked by arrows.
respectively plotted against the incident field intensity for the sample Ss . The cut-off
intensity required for observing bistable switching at λSs in this case is about 750
124
KW/cm2 . Optical bistability at wavelengths which are slightly different than those
used in our calculations, can also be observed for all the samples. Nevertheless,
the cut-off intensity as well as the nature of the hysteresis loop will be different if
we choose a different working wavelength. It is worth mentioning that in all the
numerical simulations presented in this section, we assumed monochromatic plane
wave incidence.
r [Watts/m2 ]
Iout
10
x 10
9
Increasing branch
Decreasing branch
8
6
λ = 1563.62 nm
4
2
0
5
5.5
6
6.5
7
Iin
7.5
8
8.5
9
9.5
[Watts/m2 ]
10
9
x 10
(a)
5
x 10
9
t
Iout
[Watts/m2 ]
4
3
Increasing branch
Decreasing branch
2
λ = 1563.62 nm
1
0
5
5.5
6
6.5
7
7.5
Iin [Watts/m2 ]
( b)
8
8.5
9
9.5
10
9
x 10
Figure 6.13: (a) 0-th order reflected field intensity and (b) 0-th order transmitted
field intensity plotted as a function of the incident electric field intensity for the
sample Ss . The solid black lines and the dotted red lines are the plots corresponding
to the increasing and the decreasing field intensities respectively and are marked by
arrows.
125
6.6
EXPERIMENTAL RESULTS
Schematic of the experimental set-up is illustrated in Fig. 6.14. We use a fiber laser
module providing transform-limited 495 fs optical pulses with pulse repetition rate
17.7 MHz. The central wavelength (λ p ) of the pulses is 1560.5 nm and their 3 dB
(decibel unit defined as 10 × log10 η, where η is the power efficiency) spectral bandwidth is 6.7 nm. Mode locking is performed using single wall carbon nanotube
(SWNT) saturable absorber. The intracavity polarization controller (PC) is used to
adjust polarization for mode-locking optimization. In this section, we do not go
into the details of constructing the fiber laser module. The interested reader is encouraged to read ref. [267] for a detailed description. The fiber-integrated tunable
Fiber laser module
Cavity PC
OSA
PC
BP filter
Amplifier
Chopper
U-bench collimator
Sample
Attenuator
Figure 6.14: Schematic of the experimental set-up.
band pass (BP) filter TB-TWF-1550 (PriTel, Inc USA) is used to precisely tune the
central wavelength of the pulses inside the optical communication C-band and the
KOA 3000-Keopsys Erbium doped fiber amplifier (EDFA) unit is used to amplify
the optical pulses. The pulses after the tunable unit and the amplifier unit are measured using a second-harmonic generation autocorrelator APE Pulse-check 50 and
the pulse train is measured by an oscilloscope connected with a photodetector (not
shown in Fig. 6.14). The measured pulse duration of the amplified pulses is 0.84 ps
at λ p = 1547.3 nm, the fundamental pulse repetition rate is 17.7 MHz, and the 3 dB
spectral bandwidth of the pulses after amplification is > 50 nm. The polarization
controller after the amplifier unit is used to study the polarization dependence of
the sample response (transmission). To ensure collimated beam, we use an optical U-bench collimator. The tunable attenuator, and the sample mounted inside a
holder that can be tilted across the horizontal and the vertical axes (to ensure normal
incidence of light) are placed inside the U-bench collimator. Inside the collimator
bench, we also use an electronic chopper to reduce the average input power and
hence to minimize the thermo optical effects. The output of the U-bench collimator
is connected to an optical spectrum analyzer (OSA) which measures the transmitted
signal from the sample.
Figures 6.15 (a)–(b) show the comparison between the numerical simulation re126
sults and the experimental results showing diffraction efficiencies (in dB units defined as 10 × log10 ηt00 , where ηt00 is the transmission efficiency) in direct transmission for the samples S2 and S3 respectively with TE polarized light at the input. The
reference signals used to plot the experimental curves, are the transmitted signals
from the respective samples when they are out of resonance.
Clearly, the experimental results show 97-99% maximum efficiency (−16-20 dB)
in transmission inside a narrow spectral width. Broadening of the resonance widths
in the experimental results can be attributed to the fabrication defects, SEM measurement errors (for measuring the grating dimensions) and to various loss mechanisms
(include material loss, waveguide loss etc.) in the linear optical regime. Resonance
peak position mismatch between the numerical simulation and the experimental results can be ascribed to the uncertainties associated with the SEM, the profilometer,
and the ellipsometer measurements of the grating dimensions and the refractive indices of the SiN films.
In Figs. 6.16 (a)–(b), we plot the transmission efficiency spectra for the samples
10 × log10 [η00t ] [dB]
0
−10
Experimental
Simulation
−20
−30
−40
−50
−60
1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546
λ [nm]
(a)
10 × log10 [η00t ] [dB]
0
Experimental
Simulation
−10
−20
−30
−40
1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550
λ [nm]
( b)
Figure 6.15: Numerical simulation results vs. experimental results showing efficiency spectra in direct transmission for samples (a) S2 and (b) S3 respectively with
incident electric field intensity I = 100 W/cm2 .
127
S2 and S3 respectively for several values of the incident electric field intensity. The
spot diameter Dbeam of the collimated beam is evaluated using the Knife-Edge measurement technique [268]. The measured value of the beam diameter Dbeam = 700
µm meets the requirement for the minimum illumination area on the incident plane
as described in ref. [232]. The tunable attenuator placed in front of the sample is
used to tune the peak power of the pulses and hence to control the intensity level
of the incident field. Both Figs. 6.16 (a) and (b) show modulation of transmitted sig-
10 × log10 [η00t ] [dB]
0
−5
−10
I=0.11 MW/cm 2
I=0.41 MW/cm 2
I=0.72 MW/cm2
0
−5
−10
−15
−15
−20
1539.5
1540
1540.5
1541
1541.5
1542
1542.5
1543
−20
1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546
λ [nm]
(a)
10 × log10 [η00t ] [dB]
0
I=0.118 MW/cm2
−5
2
I=0.725 MW/cm
I=1.012 MW/cm2
−10
−15
1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550
λ [nm]
( b)
Figure 6.16: Experimental results showing efficiency spectra in direct transmission
for samples (a) S2 and (b) S3 respectively at different intensity levels of the incident
field. Inset of (a) shows blue shift of the resonance peak by 0.16 nm at the incident
field intensity of 0.41 MW/cm2 .
nals from the samples with increasing field intensities. Shifts of the resonance peaks
across the wavelength are understood as results of the refractive index changes of
SiN. Using the chopper, we found out that these effects only depend on the peak
power levels of the pulses and reducing the average power does not have any effect
on the resonance peak. Hence we may conclude that the refractive index changes
are not associated with the thermo-optic effects. Also, we notice that at higher intensities, the resonance becomes wider and shallower. This can be understood by
128
noting that the SiN material used in our work contains silicon nanocrystals of size
1-2 nm [269]. At higher intensities, free-carriers are generated inside these SiN films
containing silicon nanocrystals which are responsible for the observed strong nonlinear absorption. However, due to the quantum-size effects associated with the
silicon nanocrystals [62], the electronic Kerr effect in SiN is also enhanced. Besides,
nonlinear absorption, free-carriers also cause blue-shift of resonance as the refractive
index of SiN decreases with increased concentration of free carriers. On the contrary,
the electronic Kerr effect causes red-shift of the resonances as the refractive index
of SiN increases with increasing light intensity. Usually, the electronic Kerr effect is
instantaneous having response time ∼fs, whereas the response time of free carrier
effects in SiN can be ∼ps or sub-ps. At higher intensities, these two effects compete
with each other. The blue-shifts of the resonance curves in Fig. 6.16 (a) is due to
the dominance of the free-carrier Kerr effects in sample S2 which has narrower resonance (and hence greater field enhancement) as compared to S3 . The red-shift of
the resonance curves in Fig. 6.16 (b) is due to the dominance of the electronic Kerr
effect. Clearly from Fig. 6.16 (a), for sample S2 , free-carrier induced resonance shifts
are ≈ 0.16 nm (shown in the inset of Fig. 6.16 (a)) and ≈ 0.35 nm at the incident
pulse peak intensities of 0.41 MW/cm2 and 0.72 MW/cm2 respectively. For S2 at
λ = 1540.7 nm, about 36% change in efficiency in direct transmission is observed
due to the free-carrier dominated optical Kerr effect.
6.7
SUMMARY
This Chapter covers numerical simulation and experimental results showing the
possibilities to achieve optical bistability and all-optical modulation of optical signals inside the optical communication C band with silicon nitride resonance waveguide grating structures. While the numerical simulation results show prospects for
achieving optical bistability, experimental results demonstrate all-optical modulation of signal by the optical Kerr effect.
129
7
Modeling nanocomposite optical materials with FMM
To date, there exists a large variety of optical materials which are suitable for specific applications. Nevertheless, still there is need for new materials with properties
superior to the existing ones. Hence, controlling the properties of optical materials
is a subject of eminent importance. One can tailor the properties of a material in several ways. Nanostructuring at subwavelength scale can yield unprecedented optical
properties which gave birth to the field of optical metamaterials [3]. Another way to
control optical properties is by means of molecular engineering i.e. by intermixing
two or more materials at the molecular level [270]. Best results can be produced by
merging the two aforementioned methods.
In this chapter, we shall first briefly discuss the theory of intermixing two or
more materials to form new media, which are widely known as nanocomposites.
However, we shall restrict our discussions mainly to the optical properties of the
nanocomposites. After that, we present analytical models for three common types
of nanocomposite geometries. Although accurate and efficient under certain conditions, these models have limitations. We propose an alternative technique for
accurate modeling of the nanocomposites which is based on FMM. Finally, we
present several numerical examples produced with our FMM based model for various nanocomposite geometries containing both linear and Kerr nonlinear materials.
These include porous silicon and glass-metal nanocomposites. In the quasi-static
limit, some of these examples are validated using the standard effective medium
models.
7.1
NANOCOMPOSITE OPTICAL MATERIALS
Nanocomposite media are formed by mixing two or more homogeneous materials
in nanoscale. Though the particles forming the nanocomposites are usually much
smaller than the wavelength of light, each of the constituents in the mixture retain
their individualities and can be characterized by their respective permittivities. Optical properties of the nanocomposite materials can be controlled by tailoring the ratio of the constituents, changing the mixing morphology, and controlling the shapes
and sizes of the individual nanoparticles [271]. Upon proper tailoring, nanocomposites can portray the best properties of the individual constituents. In some cases
they can show properties which even exceed those of their constituents.
In recent years, these new classes of artificial materials have emerged as efficient
alternatives to conventional media in optical applications. For example, these materials gave birth to the photonic crystals which enabled dispersion control to achieve
phase-matched nonlinear optical processes [272, 273]. Composite media where the
quantum dots are embedded in dielectric matrix evolved as efficient media for
mode-locking of solid state lasers [274,275]. Also, nanoscale ceramic composite gain
media for lasing [276,277] which possess improved optical properties as well as thermal properties [278, 279] have been reported. Besides the lasing properties, one can
also exploit the local field effects in nanocomposites to obtain improved nonlinear
optical properties as in the nonlinear optical regime, the material response scales
131
as several powers of the local-field correction factor. Several experiments related to
these showed promising results [280–283].
With advancement of nanofabrication techniques, nanocomposite materials are
becoming more and more important. Among several types of composite media,
glass-metal nanocomposites (GMNs) have remained at the focus of nanocomposite
research for a long time. GMNs are glasses containing silver, copper, gold, nickel or
other metal nanoparticles. Their optical properties are determined by surface plasmon resonance (SPR) of the metal nanoparticles contained in these. In the vicinity
of the metal nanoparticles, field is strongly enhanced at the SPR frequencies, which
in turn leads to enhanced optical nonlinearities of GMNs. This giant enhancement
of optical nonlinearity can be used in optoelectronics applications [284]. Also, the
enhancement of the photoluminescence [285, 286] and SERS [287, 288] signals in the
immediate neighbourhood of the individual metal nanoparticles, make GMNs appealing for development of novel active media for solid state lasers and biophotonics
respectively. GMNs can be useful also for data storage applications [289].
Another important class of artificial material is porous silicon (por-Si). Por-Si is a
composite medium containing nanometer-size silicon crystals separated by air pores
of similar size [290] and can be formed by electrochemical etching of crystalline silicon [291, 292]. Porous silicon (Por-Si) is promising material for several practical
applications including sensing, photovoltaic technology, and lasing [293–300]. Also,
due to the air pores, there is strong local field confinement inside por-Si structures
which may also be beneficial for nonlinear optical applications [301–303].
7.2
QUASI-STATIC APPROXIMATION AND ITS VALIDITY
Effective medium theories can be applied to describe optical properties of nanocomposite materials. However, several characteristic length scales are associated with
the nanocomposites. The size (d) of the individual grains which form the nanocomposites is the limit at which the individual constituents can still be characterized by
their permittivities. Usually, the lower limit is 1-2 nm [304]. However, below d ∼ 1
nm, quantum mechanical effects may dominate. The upper-limit is determined by
the validity of the quasi-static approximation when the wavelength of the incident
field is infinite (as compared to the individual grain sizes) and the time-variation
of the optical field can be neglected. Hence, in the framework of the quasi-static
approximation, the individual particles forming the composite medium behave as if
they were placed in a static electric field and they exhibit screening surface charge.
Due to this, local field inside the medium varies. Under specific conditions, electric field can get concentrated inside the grains with lower optical densities (lower
refractive indices). Beyond the quasi-static limit, we already reach the region of
finite-wavelength and waveguiding effects start to take place. In the guided wave
regime, electric field tends to be confined within the regions with higher optical
densities.
Aspens and Egan performed extensive theoretical and experimental studies [304–
306] on the validity of the quasi-static limit in the context of nanocomposite media
formed by pressed spherical Al2 O3 particles. They made conclusions that if the
sizes of the individual grains do not exceed 0.25λ (λ is the wavelength of the incident light), effective medium theories can still be applied. Between d ∼ 0.25λ-0.5λ,
quasi-static approximation starts to break and beyond d ∼ 0.5λ strong waveguiding
effects become dominant. However, in this context we should emphasize the fact
132
that the threshold for the validity of the quasi-static limit is different for different
composite media. For silver nanoparticles in glass, the quasi static approximation
already starts to break if the size of the metal grains d ≅ 100 nm or if the volume
concentration of the silver and glass in the composite becomes comparable.
In effective medium theories, one needs to perform spatial averaging to derive
the average material properties. This introduces another length scale. The length
over which spatial averaging can be done should be much larger than the individual grain size though much smaller than λ. In a work by LeBihan et al. [307], this
length was determined to be λ/4 but once again this might not be the same for all
nanocomposites.
7.3
COMMON COMPOSITE GEOMETRIES
Mainly three different types of composite geometries have been extensively investigated in the past which are Maxwell-Garnett composites [308, 309], Bruggeman
composites [310], and layered composites [282, 311]. These three composite geometries are illustrated in Fig. 7.1.
εe
εg
εh
ε2
ε1
εi
(a)
( b)
(c)
Figure 7.1: Three types of common composite structures (a) Maxwell-Garnett type,
(b) Bruggeman type, and (c) layered composites.
7.3.1
Maxwell Garnett geometry
In the Maxwell-Garnett geometry (shown in Fig. 7.1(a)), the composite medium is
assumed to be formed by a collection of small spherical/ellipsoidal nanoparticles
distributed in a host medium. The dimensions of these nanoparticles are assumed
to be much smaller than the wavelength. Also, the separation distances between the
inclusion particles are assumed to be much larger as compared to their characteristic size though much smaller than the wavelength. In this model, it is assumed that
the host medium completely surrounds the inclusions. The effective dielectric constant (ε eff ) of the composite can be derived using Maxwell-Garnett effective medium
theory which gives the following relation
ε eff − ε h
ε − εh
= fi i
,
ε eff + 2ε h
ε i + 2ε h
(7.1)
where, ε h and ε i are the dielectric constants of the host and the inclusion materials
respectively, and f i is the volume fraction of the inclusion in the composite. The
Maxwell-Garnett model can predict the existence of plasmon resonances i.e. the
case when ε i + 2ε h = 0. However, due to asymmetrical treatment of the host and the
inclusion, different results are obtained if we interchange ε i and ε h in Eq. (7.1). Also,
133
this model is applicable only if the volume fraction of the inclusions is extremely
small i.e. f i ≪ 1 and hence gives incorrect results when the host and the inclusion
have comparable volume fill fraction.
7.3.2
Bruggeman geometry
In the Bruggeman model, the host and the inclusion are treated symmetrically. Each
particle of the constituents is assumed to be embedded in an effective medium with
relative permittivity ε eff . The equation which defines ε eff is given by
f1
ε 1 − ε eff
ε − ε eff
+ f2 2
= 0,
ε 1 + 2ε eff
ε 2 + 2ε eff
(7.2)
where ε 1 and ε 2 are the relative permittivities of the constituent materials 1 and
2 respectively, f 1 and f 2 are their volume fractions. Clearly, in the limit f 1 ≪ f 2 ,
if we replace f 1 by f i , f 2 by f h , ε 1 by ε i , and ε 2 by ε h , Eq. (7.2) reduces to the
Maxwell-Garnett relation i.e. to Eq. (7.1). Due to the symmetrical treatment of
the host and inclusion, Bruggeman model can describe percolation (drastic increase
of the conductivity of a composite material at certain volume concentrations of its
constituents). However, this model cannot predict surface plasmon resonance. Also,
when f i and f h are comparable, this model starts to break.
7.3.3
Layered composite geometry
There exists a third kind of composite geometry which is shown in Fig. 7.1(c). It
consists of alternating layers of two homogeneous materials e and g with different
optical properties. Thicknesses of these layers are much smaller than the wavelength. This type of composites are artificially anisotropic.
If the electric field of the incident light is polarized parallel to the layer surfaces,
the effective relative permittivity is given by the following expression
ε eff = f e ε e + f g ε g ,
(7.3)
which is a simple volume average of the relative permittivities of materials e and
g. Electric field in this case stays uniform inside the composite as electromagnetic
boundary conditions require that the tangential electric field components should be
continuous across the interface of materials e and g. However, for incident light with
electric field polarized perpendicular to the layers, the effective relative permittivity
can be evaluated from the following expression
fg
1
fe
= + .
ε eff
εe
εg
(7.4)
Clearly in this case, electric field is distributed inside the layers in a nonuniform
manner resulting in inhomogeneous local fields.
7.4
OPTICAL PROPERTIES OF NANOCOMPOSITES CONTAINING
METAL NANOPARTICLES
In subsection 2.15.3, we saw that if a metal nanoparticle is exposed to an electric
field, the shift of the conduction electrons with respect to the metal’s ionic core
134
induces surface charge on the other side of the metal particle. This surface charge
in turn causes restoring force. Hence, in an alternating electric field, the shifted
conduction electrons together with the restoring force behaves as an oscillator. At a
specific frequency, strong resonance is observed, which is also known as localized
surface plasmon resonance (LSPR). This resonance causes giant enhancement of the
near field.
We also saw that the resonance frequency depends on the free electron density
of the metal and also on the geometry of the particle. Mie proposed a solution
of Maxwell’s equations for spherical metal particles which can explain the origin
of LSP. According to Mie [312], different eigenmodes of the spherical particles are
dipolar or multipolar in nature. In the quasi-static limit, i.e. when the size of the
metal particles are extremely small as compared to the wavelength, the external
electric field can be assumed to be static. Let us now recall Mie’s solution for the
microscopic polarizability from subsection 2.15.3
α = 4πr3
ε i (ω ) − ε h
,
ε i (ω ) + 2ε h
(7.5)
where r is the radius of the particle as before, ε i (ω ) is the frequency dependent
dielectric constant of metal nanoparticles and ε h is the dielectric function of the surrounding medium. Thus the microscopic polarization (p) of the metal nanoparticle
embedded in a transparent dielectric matrix is given by
p(ω ) = αε 0 E0 (ω ) = 4πε 0 r3
ε i (ω ) − ε h
E .
ε i (ω ) + 2ε h 0
(7.6)
The absorption cross-section (σ) can be calculated by following the procedure described in Chapter 4 of Ref. [313]. The detailed derivation is out of scope for this
thesis. σ is evalauated by the following mathematical expression
σ (ω ) = 12πr3
ω (3/2)
ε i1 (ω )
ε
.
c h
[ε i1 (ω ) + 2ε h ]2 + ε i2 (ω )2
(7.7)
Here E0 is the external electric field, ε i1 (ω ) and ε i2 (ω ) are the real and the imaginary
parts of the frequency-dependent dielectric constant of the metal particle which
according to Drude-Sommerfeld theory is given by
ε i (ω ) = ε b + 1 −
2
ωBP
,
ω 2 + iγω
(7.8)
where ωBP is the bulk plasmon frequency, ε b is the complex permittivity associated with the interband transitions of the core electrons inside the atom, and γ is
the damping constant as before. While deducing Eq. (7.7), we have assumed that
the permittivity of the surrounding medium is real. Clearly, from Eqs. (7.5) and
(7.7), resonance occurs when the denominator on the right hand side of Eq. (7.7) is
minimum. This gives,
ε i1 (ωLSP ) = −2ε h .
(7.9)
On fulfillment of the condition in Eq. (7.9), the local field and the dipole moment
in the immediate neighbourhood of the nanosphere grow resonantly and can have
magnitudes enhanced by several orders. Furthermore, Eq. (7.9) requires the real
part of the dielectric constant of the metal to be negative which indeed is the case
135
for noble metals in the visible frequency regime. Combining Eqs. (7.9) and (7.8), we
can describe the position of the LSP resonance using the following expression
2
ωBP
− γ2 .
ℜ[ε b ] + 1 + 2ε h
(7.10)
4π
ε i (ω ) − ε h
abc
,
3
ε h + [ε i (ω ) + ε h ] L j
(7.11)
2
ωLSP
=
Clearly from Eq. (7.10), the interband transitions of the core electrons have significant influence on the position of LSP resonance. Usually for silver (Ag) nanoparticles embedded in glass, LSPR is observed around λ = 415 nm. For gold (Au), and
copper (Cu) nanoparticles, LSPR is observed at slightly higher wavelengths (528 nm
and 570 nm respectively). Resonance for interband transitions for Ag is observed
far away from the LSPR at about 310 nm. Equation (7.10) also describes the effect of
the surrounding medium on the position of the resonance. Clearly, an increase of ε h
shifts the resonance peak towards the longer wavelengths.
Besides these, the size of the metal nanoparticle also strongly influences the LSPR
position. The position remains almost constant for nanoparticles with radii r < 1015 nm. However, for particles with larger radii, LSPR starts to shift towards longer
wavelengths. These effects are known as extrinsic size effects [314–318]. For smaller
particles (< 1 nm), spill-out of electrons from the particle surface may result in an
inhomogeneous dielectric function. As a consequence, very broad plasmon bands
can be observed for extremely small nanoparticles.
Lastly, the shapes of the metal nanoparticles have strong impact on plasmon
resonances. For ellipsoidal nanoparticles with semi major axes a, b, and c, three different LSP modes are observed which are associated with the three principal axes.
The microscopic polarizability in this case takes the following form
α j (ω ) =
where L j is known as the geometrical depolarization factor with ∑ L j = 1. For a
spherical particle L a = Lb = Lc = 1/3 and hence we have only one LSP mode. In
the most general case, when the propagation direction of the incoming light does
not coincide with any of the axes of the ellipsoid, we obtain three separate LSP
bands in the absorption spectra. These three bands correspond to the oscillations
of the free electrons along three axes [314]. In case, light is polarized along one of
the principal axes of the ellipsoid, we observe only one LSP band. These dichroic
properties of elongated metal nanoparticles have been largely employed in the past
to construct broad-band high-contrast polarizers [319].
Let us now consider the situation when we have many metal nanoparticles embedded in a dielectric host medium. In the quasi-static limit, the effective dielectric
constant of such a medium is given by Eq. (7.1) i.e. by the Maxwell-Garnett theory.
By rearranging Eq. (7.1), we can write it in the form
ε eff (ω ) = ε h
[ε i (ω ) + 2ε h ] + 2 f i [ε i (ω ) − ε h ]
,
[ ε i (ω ) + 2ε h ] − f i [ε i (ω ) − ε h ]
(7.12)
where all the parameters are defined as in subsection 7.3.1. However, the MG
(Maxwell-Garnett) model cannot be used to explain the extrinsic size effects i.e.
the multipolar effects. Also, when the metal concentration is high, i.e. when we
have small separation distance between two nanoparticles, strong collective dipolar
interactions between the nanoparticles may occur which results in strong enhancement of the linear and the nonlinear optical properties in the vicinity of the metal
136
particles. These collective dipolar interactions also cause broadening and red-shift
of absorption spectra. MG theory fails to explain these effects also and one must employ rigorous theories. In the next section, we shall present a method for modeling
nanocomposite optical materials accurately which is based on the FMM.
7.5
RIGOROUS MODELING- METHODOLOGY
To model the nanocomposites with FMM, we create a window that serves as an artificial period or a unit cell representing the whole composite medium. Figure 7.2
shows such a unit cell which is repeated along the orthogonal cartesian coordinate
axes x and y. However, in the most general case, the coordinate axes can be nonorthogonal to each other. We divide the artificial period of size D × D, where size
of D can be ∼ λ, into sub-cells (in this case square boxes) each of size d × d as illustrated in Fig. 7.2. Material inside each small square box is assigned a distinct value
of permittivity. In the limit d ≪ D, boundaries of a particle with arbitrary geometry
can be modeled accurately. Also, effects arising from random arrangements of the
nanoparticles and clustering of nanoparticles inside the unit cell can be taken into
account properly. In Fig. 7.2 we illustrate the way we model particles with arbitrary
geometries (marked as type ‘1’), with size approaching the wavelength of the incoming field (marked as type ‘2’), overlapping of two nanoparticles (marked as type ‘3’),
and infinitesimal interspacing between two particles (marked as type ‘4’). Along the
propagation direction, we slice the structure into several thin layers as described in
Chapter 3. This way, any arbitrary profile along the propagation direction can also
be well fitted. However, it is worth mentioning that although fineness of meshing
inside the artificial period does not significantly affect the computation efficiency
(sampling in xy plane does not increase the number of linear algebraic equations to
be solved by S-matrix approach), the computation workload increases linearly with
an increase of the number of layers along the propagation direction. By averaging
over ensembles of such structural realizations, we can determine optical properties
of the nanocomposite; with a sufficiently large unit cell, results obtained for a single
realization are good approximations of the ensemble average.
This FMM based model can correctly describe the collective dipolar oscillations
in case two or more nanoparticles form a cluster with small interspacing. Also, this
model does not impose any restriction on the size of the individual nanoparticles
(unless they are so small that quantum effects take place). Hence, it can also predict
multipolar effects. Furthermore, if the volume fill fraction of the host material and
the inclusion are comparable to each other, most of the effective medium models
fail and we need to employ full-rigorous approaches. In case, the host (or the inclusion) is amorphous in nature and possess optical Kerr nonlinearity with real valued
χ(3) , we employ the technique described in Chapter 4 to derive the effective Kerr
nonlinearity of the composite medium.
7.6
NUMERICAL EXAMPLES
The method described in the previous section is used to model the linear optical properties of all-dielectric as well as metal-dielectric nanocomposites. Furthermore, we investigate plasmon enhanced Kerr nonlinear optical properties of a metaldielectric composite medium.
137
1
(0, D )
y
d
d
2
3
(D, D )
x
4
Figure 7.2: Unit cell representing a nanocomposite structure. Figure shows the
way we model particles with arbitrary geometries (marked as type ‘1’), with size
approaching the wavelength of the incoming field (marked as type ‘2’), overlapping
of two nanoparticles (marked as type ‘3’), and infinitesimal interspacing between
two particles (marked as type ‘4’).
7.6.1
Porous silicon nanostructures
In por-Si composite media, which are prepared by electrochemical etching of crystalline silicon, the air pores are usually oriented along a particular crystallographic
axis and depending on their sizes (a), por-Si can be classified into three categoriesmicroporous (a < 5 nm), mesoporous (a ≈ 5 − 100 nm), and macroporous (a > 100
nm). Top view of a monolayer mesoporous silicon nanocomposite is shown in
Fig. 7.3. In the quasi-static limit i.e. when the pore sizes are much smaller than the
wavelength of light, one can apply effective medium theories to derive the optical
properties of por-Si. However, for the mesoporous and the macroporous samples,
finite wavelength effects may come into play and for samples with porosity in the
range 10-90 % effective medium theory fails as shown in Ref. [303]. Furthermore, for
pore sizes < 1 nm, strong quantum mechanical effects are observed. In this thesis,
we will treat porous-Si media from a classical perspective i.e. we assume that the
pore sizes are at least few nanometers.
To model the porous-Si media using the full-wave numerical simulation technique described in the previous section, we use the structural unit cells of size 200
nm × 200 nm as shown in Figures 7.4 (a)–(c). These unit cells are used to mimic
the actual structures. Three different configurations of the unit cell are used to investigate the effects of random pore arrangements on the optical properties. Here,
we emphasize on the birefringence properties of these nanocomposites. First, we
model microporous monolayer structures grown on top of crystalline silicon substrates where the largest pore dimension is 18 nm. Thickness of the porous layer is
30 nm and refractive index of silicon is assumed to be 3.5. Volume fill fraction of
138
Figure 7.3: Scanning electron microscopy (SEM) image showing top-view of a
sponge-like mesoporous silicon nanocomposite medium.
the air pores in the composite media, corresponding to the three configurations in
Figs. 7.4(a)–(c) are the same.
Numerical simulation results reveal that these microporous composite media can
be described by the effective medium theory as the feature sizes are much smaller
than the wavelength of the incoming plane wave (λ = 1550 nm) and light does not
get diffracted into higher orders. Results obtained with the FMM based rigorous
analysis agree well with those obtained with the effective medium theory. The effective indices of these composites for y- and x- polarized incident fields are listed in
Table 7.1 which are calculated by treating the porous layers as optically anisotropic
homogeneous effective media. The tabulated effective index data show that the degree of birefringence (difference between Ny and Nx ) for different configurations
is non-identical although the effective volumes of the pores are almost the same.
Hence, rigorous modeling of the randomness in pore arrangements is necessary to
accurately describe any experimental result. Next, we study model mesoporous
(a)
(b)
(c)
Figure 7.4: Unit cells of microporous model structures. (a), (b), (c) show three
distinct pore arrangements inside the unit cell of size 200 nm × 200 nm. Volume fill
fractions of the pores for these three configurations are the same.
silicon structures where all feature sizes are 10 times of those shown in Figs. 7.4 (a)–
(c) i.e. the largest pore dimension is now 180 nm. However, we use configurations
with exactly the same pore arrangements inside the unit cell as in the microporous
139
Table 7.1: Effective indices of the microporous silicon layers of thickness h = 30
nm on top of silicon substrates. Wavelength of the incident light is λ = 1550 nm. Ny
and Nx represent effective indices for y- and x- polarized incident fields respectively.
(a), (b), and (c) represent three distinct pore arrangements inside the unit cell of size
200 nm × 200 nm as shown in Figs. 7.4 (a)-(c) respectively.
Configuration type
Ny
Nx
(a)
2.31
2.25
(b)
2.31
2.29
(c)
2.31
2.26
model structures for the sake of comparison. We find out that for λ = 1550 nm,
these mesoporous structures can not be correctly described in terms of effective indices as the porous layers already start to diffract light into higher orders. Hence,
rigorous numerical treatment is essential. Simulation results show that the sum of
the diffraction efficiencies in direct reflection and direct transmission for these composites range between 92-96% and the rest 4-8% goes into higher diffraction orders.
In Table 7.2, we list degree of birefringence i.e. φx00 − φy00 (measured in terms of
the phase differences introduced by these porous slabs between the x and y polarized transmitted electric field components) of these mesoporous composites corresponding to configurations (a), (b), and (c). In each case, thickness of porous layer
was assumed to be 300 nm. Clearly, the degree of birefringence strongly depends on
the pore arrangements. As silicon is a high index material, for structures with larger
Table 7.2: Degree of birefringence (φx00 − φy00 ) of the mesoporous silicon layers of
thickness h = 300 nm on top of silicon substrates. Wavelength of the incident light
is λ = 1550 nm. (a), (b), and (c) represent three distinct pore arrangements inside
the unit cell of size 2000 nm × 2000 nm as shown in Figs. 7.4 (a)-(c) respectively.
Configuration type
(φx00 − φy00) in radians
(a)
0.4287
(b)
0.2943
(c)
0.3530
pore sizes, the effects of local electric fields, which govern the optical properties of
these nanocomposites, become stronger. Hence, to understand the characteristics of
these media, it would be helpful to locate these regions with strongly confined local
electrical fields. The strength of our FMM based technique lies in the fact that it can
be employed to locate the hot spots i.e. the regions with maximum field concentrations. Figures 7.5 (a)–(c) show the transmitted near field (electric) intensity maps for
the pore arrangements corresponding to Figs. 7.4 (a)–(c) respectively.
7.6.2
Silver nanospheres on glass substrate
One of the most common metal-dielectric nanocomposite media is silver nanoparticles arranged on top of a glass substrate and surrounded by a dielectric medium.
Plasmon resonances of such composite media are observed around λ = 400 nm and
140
the absorption spectra (at resonance) of these nanocomposites strongly depend on
the surrounding dielectric media, the shape and the size of the silver nanoparticles
and also on their volume fill fractions inside the composites.
We start with a test object in the form of silver spheres arranged on top of a glass
substrate (monolayer) and arrayed into an infinite rectangular lattice. This is the
same object used to plot Fig. 2(a) in Ref [320]. The radius of the spheres is taken to
be a = 24 nm, the refractive indices of the surrounding medium and the substrate
are nd = 1 and ns = 1.52 respectively, and the refractive index data of silver is taken
from Ref [321]. Separation between two consecutive nanoparticles define the volume fill fraction of silver in the monolayer composite medium. We first assume that
d = 120 nm i.e. the size of the unit cell in Fig. 7.2 is 120 nm ×120 nm and it contains
only one silver particle. The thickness of the composite monolayer is 48 nm. Figure 7.6(a) shows the transmission spectra of the monolayer structure for normally
incident TE polarized light obtained with both the FMM based technique and the
Maxwell-Garnett formula in Eq. (7.1).
Clearly, these two results are in close agreement. We also see that the result
obtained with FMM is in very good agreement with the FDTD result in Fig. (2 a)
in Ref [320]. With d = 120 nm, volume fill fraction (Vf ) of silver in the composite medium is Vf = 0.084 which is ≪ 1. Hence, the Maxwell-Garnett theory can
correctly estimate the location of the plasmon peak though it slightly overestimates
the peak amplitude. In Figure 7.6(b), we plot the absorption cross-section spectra
(σabs) of the composite medium, which is defined as σabs = −(1/a) log[ηt00 ] (ηt00 is
the diffraction efficiency in 0-th order transmitted field), for different values of Vf .
To change Vf , we simply change the value of d i.e. the size of the unit cell in the
FMM simulations. From the plots, we see that the mismatch between the results
obtained with FMM and the Maxwell-Garnett theory increases as Vf increases. The
FMM result showing broadening of the absorption spectra at Vf = 0.48 is due to
collective plasmon oscillations of the silver particles. Finally, we may conclude that
the quasi-static approximation breaks at higher values of Vf and one must apply
rigorous techniques to accurately estimate the nature of plasmon resonance. We
now proceed to investigate the effects of particle clustering on the absorption crosssection spectra of such a monolayer composite medium. For this, we consider a unit
cell containing nine silver nanoparticles. Four different arrangements as shown in
Figs. 7.7(a)-(d) are considered. These figures show the cross-sections of the particles
0.5 0.4 0.3 0.2 0.1
0.5 0.4 0.3 0.2 0.1
(a)
(b)
0.7
0.5
0.3
0.1
(c)
Figure 7.5: Transmitted near field intensity maps for mesoporous silicon structures. (a), (b), and (c) correspond to the configurations shown in Figs. 7.4 (a)-(c)
respectively.
141
1
η t00
0.8
MG theory
FMM
0.6
0.4
0.2
0
300
320
340
360
380
400
420
440
400
420
440
λ [nm]
(a)
−(1/a) log[η t00]
0.1
0.08
0.06
0.04
Vf =0.084 FMM
Vf =0.084 MG
Vf =0.3 FMM
Vf =0.3 MG
Vf =0.48 FMM
Vf =0.48 MG
0.02
0
300 320
340
360
380
λ [nm]
(b)
Figure 7.6: (a) Efficiency (ηt00 ) in direct transmission plotted as a function of the
wavelength (λ) of the incident field where the volume fill fraction of silver nanoparticles inside the composite medium is Vf = 0.084 (corresponding to layer thickness
of 48 nm) and normal incidence of light is assumed, (b) Absorption cross-section
spectra of monolayer composite media for different Vf -s plotted both with FMM
and the Maxwell-Garnett theory.
Config. 0
Config. 1
Config. 2
Config. 3
Figure 7.7: Cross-sections of the unit cells across the mid planes of the particles
representing Configuration 0, Configuration 1, Configuration 2, and Configuration
3. Vf = 0.3 for all of these four configurations.
142
− (1/a ) log[η t00]
across their mid plane. Vf = 0.3 for all of these i.e. we only change the locations
of the particles inside the unit cell without changing the volume fill fraction of silver particles inside the composite media. The configuration in Fig. 7.7(a) is already
simulated in Fig. 7.6 where all the particles are equally spaced. In all other configurations, the size of the period is the size of the unit cell containing nine particles.
Figure 7.8 shows the absorption cross-section spectra for these four different configurations. The difference is notable for Configuration 1 where collective plasmon
oscillations take place because of extremely small gaps (1.4 nm) between the clustered particles. For Configuration 2, the gap size is increased to a limit (5.2 nm)
such that collective plasmon resonances (which are observed at relatively higher
wavelengths) vanish. Collective plasmon oscillations disappear also for Configuration 3, where the clustered particles just touch each other along x and y directions.
To illustrate the outset of collective plasmon oscillations, we plot the local electric
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
300
Config. 0
Config. 1
Config. 2
Config. 3
320
340
360
380
400
420
440
λ [nm]
Figure 7.8: Absorption cross-section spectra of monolayer composite media for
four different configurations as shown in Figs. 7.7 (a)–(d) with Vf = 0.3008.
field intensity distributions across the mid-planes of the particles for Configurations
0 and 1 at two different wavelengths λ1 = 363 nm and λ2 = 435 nm respectively.
These two wavelengths are marked by arrows in Fig. 7.8. The field intensity plots
in Figs. 7.9 (a)–(d) show that the local electric field concentrations in the small gaps
between the clustered particles for Configuration 1 are extremely high because of
the collective plasmon resonance. For λ2 = 435 nm, field concentration in the gaps
between the closely-packed particles in Configuration 1 is even higher which gives
rise to additional absorption peak around λ2 = 435 nm as illustrated in Fig. 7.8.
7.6.3
Silver nanorods embedded in a Kerr nonlinear host
As a final example, we simulate a monolayer nanocomposite medium consisting
of cylindrical shaped silver nanorods embedded inside a nonlinear polymer host
medium with instantaneous Kerr nonlinearity. These nanorods are assumed to be
arrayed in an infinite rectangular lattice with their axes along the z- direction. The
linear refractive index and the third-order susceptibility of the host matrix are assumed to be nd = 1.7, and χ(3) = 10(−17) m2 /V2 respectively. Radii and heights of
these silver nano cylinders are taken to be a = 26 nm and h = 50 nm respectively.
Separations between two consecutive silver particles in the array are assumed to
be d = 65 nm both along x and y directions. Furthermore, the substrate medium
is glass with refractive index ns = 1.47. Plots in Fig. 7.10 show the absorption
cross-section spectra of the composite layer with TE polarized (y-polarized) incident
143
Config . 0, λ = 363 nm
Config . 0, λ = 435 nm
30
30
20
20
10
10
0
(a )
Config . 1, λ = 363 nm
Config . 1, λ = 435 nm
30
30
20
20
10
10
0
(c)
0
(b )
(d )
0
Figure 7.9: Electric field intensity distributions inside the cross-sections of the unit
cells across the mid planes of the silver spheres for (a) Configuration 0 at λ = 363
nm, (b) Configuration 0 at λ = 435 nm, (c) Configuration 1 at λ = 363 nm, and (d)
Configuration 1 at λ = 435 nm. Vf = 0.3.
− (1/h ) log[η t00 ]
wave. These two plots correspond to the linear case and the nonlinear case with
the intensity of the incoming plane wave I = 100 MW/mm2 respectively. Clearly,
due to plasmon enhanced optical Kerr effect, the absorption peak gets enhanced
and red-shifted. In this example, we have modeled only the host medium as a Kerr
nonlinear medium with the FMM based technique introduced in Chapter 4. These
results can be understood by noting the fact that with increasing field intensity, the
refractive index of the surrounding polymer material also increases.
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
350
I=10 GW/cm2
Linear
380
410
440
470
500
530 550
λ [nm]
Figure 7.10: Light induced shift of absorption cross-section spectra of monolayer
composite media consisting of silver nano cylinders embedded in a nonlinear polymer host material with instantaneous Kerr nonlinearity. The linear refractive index
of the polymer is n d = 1.7 and its third-order susceptibility is χ(3) = 10(−17) m2 /V2 .
Radii and heights of the nano cylinders are a = 26 nm and h = 50 nm respectively.
144
7.7
SUMMARY
In this chapter, using the well-known Fourier Modal Method commonly used to
model diffraction gratings or periodic structures, we have developed a technique
which can be used to mimic nanocomposite optical materials and describe their optical properties. Arbitrary particle geometries, random particle arrangements inside
the host medium as well as the effects of particle clustering can be accurately modeled using this method. Furthermore, finite wavelength effects, multipolar effects,
and local field controlled effective Kerr nonlinear properties of the nanocomposites
can be accurately described by this scheme. Hence, this serves as a unified approach
for modeling the optical properties of nanocomposite media and fills the gap where
the most commonly used effective medium theories fail.
145
8
Summary, conclusions and scope of future work
Within the scope of this thesis, we have investigated the role of local electric fields
in controlling the linear and the Kerr nonlinear optical properties of subwavelength
periodic structures. Most of the numerical studies included in this thesis are based
on the Fourier modal method (FMM) which has been discussed in detail in Chapter
3. This Chapter includes the summary along with the main conclusions made on
the basis of the theoretical, numerical, and experimental results presented in this
doctoral thesis and lastly a brief discussion regarding the possible future research
directions.
8.1
SUMMARY WITH CONCLUSIONS
In the subwavelength regime, as the feature sizes approach the wavelength of light,
quasi-static approximations start to break due to the dominance of waveguiding effects and one needs to employ full-rigorous theories such as the FMM. In Chapter
4, we have shown that the FMM can be used to model local field controlled optical
Kerr nonlinearity in periodic structures containing isotropic nonlinear materials, if
an iterative approach is taken. The accuracy of this FMM based approach is verified. Furthermore, we demonstrate the possibility of increasing the computational
efficiency of the proposed approach by use of symmetries in light-matter interaction
geometry. Numerical results included in Chapter 4 show the convergence of the
method, and its suitability for the design and analysis of local-field enhanced low
power all-optical devices.
• Conclusion: The FMM can be applied to model optical Kerr nonlinearity in
periodic structures using an iterative approach. Symmetries in light-matter
interaction geometry can be employed to greatly reduce the computational
efforts of this FMM based approach. The developed numerical method can
be used to accurately estimate the local field controlled optical Kerr effect in
subwavelength periodic structures.
In Chapter 5, we have developed an analytical model which can describe nonlinear
light-matter interactions inside a form birefringent Kerr nonlinear optical medium
and verified its accuracy by comparing the results obtained with this theoretical
model with those obtained by the full rigorous FMM based technique introduced in
Chapter 4. Theoretical results suggest that in the quasi-static regime, both the linear
and the Kerr nonlinear optical properties of structured media can be accurately
described using approximative theories as these theories can estimate the local fields
precisely. The results also show that in contrast to the linear case, in which a formbirefringent subwavelength grating always acts as a negative uniaxial crystal, the
nonlinear case is much more intricate because of the role of the polarization state
of light. The developed analytical model may aid in design of all-optically tunable
form birefringent wave plates and retarders.
• Conclusion: It is possible to develop an analytical model which can describe
the theory of form birefringence in optical Kerr nonlinear media. Results based
147
on this analytical model show that a form birefringent medium with optical
Kerr nonlinearity still behaves like a uniaxial crystal. The developed analytical
theory can be employed to design all optically tunable wave plates.
In Chapter 6, we discussed about the possibilities to achieve optical bistability and
all-optical modulation of signals with resonance waveguide grating structures. We
have chosen silicon nitride as a material to construct the resonant gratings. Numerical simulation results show optical bistable switching inside the optical communication C band both in reflected and transmitted signals from these gratings under
normal incidence of light. The waveguide grating structures are fabricated by electron beam lithography and reactive ion etching techniques from silicon nitride thin
films grown on top of fused silica substrates. Experiments have been carried out
using single wall carbon nanotube modelocked ultrafast fiber laser together with an
amplifier and a tunable unit. Experimental results show all-optical modulation of
transmitted signals by the local field enhanced optical Kerr effect in silicon nitride
resonance waveguide grating structures.
• Conclusion: Nonlinear light-matter interactions inside a silicon nitride resonant waveguide grating is greatly enhanced due to strongly confined local
field. These structures can be exploited to construct low energy optical bistable
devices as well as all-optical modulators which rely on the mechanisms of the
optical Kerr effect.
In Chapter 7, we have shown the influence of the local fields on the linear and the
Kerr nonlinear optical properties of nanocomposites. In this chapter, we demonstrate how the rigorous FMM can be employed to model nanocomposite optical materials. Numerical results display that in the quasi-static regime, effective medium
theories can be applied to accurately model the nanocomposites. Also, in case of
metal-dielectric nanocomposites, Maxwell-Garnett effective medium theory may accurately predict the plasmon resonances if the volume fill fraction of the spherical
shaped inclusions inside the host medium is ≪ 1. However, to accurately model arbitrary particle geometries, random particle arrangements inside the host medium,
effects of particle clustering, multipolar effects and finite wavelength effects one
must employ full rigorous approaches because the optical properties in the above
mentioned cases strongly depend on the localized fields inside the composite media.
• Conclusion: Local fields can strongly influence the optical properties of nanocomposites. The FMM based full rigorous analysis introduced in Chapter 7
serves as a unified and accurate approach for modeling the optical properties of nanocomposite media and fills the gap where the most commonly used
effective medium theories fail.
8.2
SCOPE OF FUTURE WORK
The results presented in this thesis suggest several possible extensions. For example, the FMM based approach presented in Chapter 4 which assumes the incident
wave to be continuous in nature and having infinite extent, can be extended to timedomain optical pulses and also to light beams having finite extent. This extension
would be useful for studying optical solitons.
Furthermore, to develop the theory for light propagation in a form birefringent
Kerr medium in Chapter 5, we assumed normal incidence of light which simplified
148
our mathematical model substantially. Nevertheless, it might be possible to generalize our analytical approach for oblique incidence of light. Also, in future it might
be possible to experimentally demonstrate all-optical control of form birefringence.
Resonant waveguide gratings in their simplest form were analyzed in Chapter
6. It might be possible to reduce further the threshold intensity required to observe
light-induced changes using multilayered structures or by means of coupled resonant gratings. Another possibility would be to use materials having superior nonlinear optical properties, viz. materials having higher electronic Kerr nonlinearity
and ultrafast response time, to construct these gratings. The experiment presented
in this thesis was carried out using only one light beam. To demonstrate all-optical
switching mechanisms in future, it will need to build a pump-probe system with
two beams.
In Chapter 7, we have seen that remarkable optical properties can be attained
using properly tailored nanocomposite optical materials. In future, periodically patterned nanocomposites having tailored unit cells can be designed and fabricated to
attain striking optical properties with vast scope of applications in optical data storage, sensing, and photovoltaics.
In this thesis, most of our discussions remained confined to nonlinear modulation of amplitude and phase of the incident wave. It would be interesting also to
design and analyze structures that can give rise to light-induced polarization rotation i.e. nonlinear optical activity. Lastly, all examples included in this thesis were
based on free-space propagation of light. For certain applications, it would be necessary to build systems where all light beams are maintained within the integrated
circuitry. Hence, it would be interesting to extend our study for on-chip light propagation.
149
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PUBLICATIONS OF
THE UNIVERSITY OF EASTERN FINLAND
In this book, local-field controlled linear
and Kerr nonlinear optical properties of
subwavelength periodic nanostructures
and nanocomposites are studied. Efficient
numerical techniques and novel analytical
models have been developed to aid in these
studies. In addition, prospect for achieving low
energy optical bistability with a silicon nitride
guided mode resonance filter is examined
numerically followed by an experimental
demonstration of all-optical modulation using
such a structure.
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Dissertations in Forestry and Natural Sciences
ISBN 978-952-61-2441-4
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DISSERTATIONS | SUBHAJIT BEJ | LOCAL FIELD CONTROLLED LINEAR AND KERR NONLINEAR OPTICAL... | No 262
SUBHAJIT BEJ
Dissertations in Forestry and
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SUBHAJIT BEJ
LOCAL FIELD CONTROLLED LINEAR AND KERR
NONLINEAR OPTICAL PROPERTIES OF
PERIODIC SUBWAVELENGTH STRUCTURES