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Linköping Studies in Science and Technology
Doctoral Dissertation No. 1115
Theoretical studies of light
propagation in photonic and
plasmonic devices
Aliaksandr Rahachou
Department of Science and Technology
Linköping University, SE-601 74 Norrköping, Sweden
Norrköping, August 2007
To Olga.
The picture on the cover illustrates the concept of a ”photonic micropolis”.
Adopted from http://ab-initio.mit.edu/photons/micropolis.html.
Theoretical studies of light propagation in photonic and plasmonic
devices
c 2007 Aliaksandr Rahachou
Department of Science and Technology
Campus Norrköping, Linköping University
SE-601 74 Norrköping, Sweden
ISBN 978-91-85831-45-6
ISSN 0345-7524
Printed in Sweden by UniTryck, Linköping, 2007
Preface
Science is about to discover God.
I start worrying about His future.
Stanislaw Jerzy Lec, Polish poet (1909–1966)
When people ask me: ”Where do you work and what are you doing?” I
have a standard answer that I work at Linköping University and do my PhD
in physics. In most cases it is enough, people make their faces serious and say:
”O-o-o! Physics!” But sometimes, I need to explain in more detail that my
research area is actually photonics, and I study things related to propagation
of electromagnetic waves in some strange media, created by people in order to
deceive Nature. People become serious at this point and say: ”O-o-o! Photonics!” I like these moments and I like what I have been doing during these years
– photonics.
This Thesis presents the results of the four-year work that was done in the
Solid-state Electronics group at the Department of Science and Technology at
Linköping University. This is a theoretical work, which touches three main
directions in photonics, namely photonic crystals, microcavities and plasmonics. The Thesis consists of four chapters. Chapter 1 is a short introduction
where I familiarize the reader with the subject. Chapter 2 gives an introductory review of photonic structures, in Chapter 3 I present the methods that
were developed during my study, and in Chapter 4 the results are summarized
and briefly discussed. This Dissertation is based on seven papers, presented in
the Appendix.
Almost everywhere in the text I use ”we” instead of ”I”, despite my royal
roots are neither that clear nor documented anywhere. This is just to emphasize that any research is never a single person’s but a team work.
Aliaksandr Rahachou
Norrköping, Midsommar, 2007
iii
iv
Acknowledgements
Well, if you have already read the Preface, you probably know, that this Thesis
is a result of the four-year work at ITN LiU in Sweden. During this time I met
a lot of nice people, who helped me not only in my research activity, but also
supported me in everyday routine.
First of all I would like to thank Igor Zozoulenko for the brilliant supervision.
He introduced me to the fascinating area of photonics, spent really loads of time
answering my stupid questions, discussing, encouraging and sometimes pushing
me to do something ,.
I am very grateful to Olle Inganäs for the valuable discussions, initiation
of this work and the experimental input. I was also pleased to collaborate
with Kristofer Tvingstedt, whose unexpected ideas from the point of view of
an experimentalist helped me to understand the subject deeper.
Then, the guys from our group – Martin Evaldsson and Siarhei Ihnatsenka.
Despite we did a little bit different things, Martin and Siarhei not only always
understood what I was doing, but also helped me with practical things like
LATEX, elementary school-level math or other important issues that bothered
me sometimes.
Of course, all people at ITN are very kind. Thank you, people! However,
I’d like especially thank Aida Vitoria for good humor, which is, despite the
weather, season or Iraq war, remains sparkling.
Big thanks to my mother and father. Being far away from them, I feel their
love and support every day.
Thanks to my girlfriend Olga Mishchenko. She supports and helps me from
day to day, her love and kindness is just a miracle that I revealed here in
Sweden. Tack, Sverige!
I very appreciate the financial support from the Swedish Institute (SI),
Royal Swedish Academy of Sciences (KVA), National Graduate School of Scientific Computing (NGSSC), Center of Organic Electronics (COE), Centre in
Nanoscience and Technology at LiU (CeNANO) and ITN that enabled me to
start and complete this Thesis.
v
vi
Abstract
Photonics nowadays is one of the most rapidly developing areas of modern
physics. Photonic chips are considered to be promising candidates for a new
generation of high-performance systems for informational technology, as the
photonic devices provide much higher information capacity in comparison to
conventional electronics. They also offer the possibility of integration with electronic components to provide increased functionality. Photonics has also found
numerous applications in various fields including signal processing, computing,
sensing, printing, and others.
Photonics, which traditionally covers lasing cavities, waveguides, and photonic crystals, is now expanding to new research directions such as plasmonics
and nanophotonics. Plasmonic structures, namely nanoparticles, metallic and
dielectric waveguides and gratings, possess unprecedented potential to guide
and manipulate light at nanoscale.
This Thesis presents the results of theoretical studies of light propagation in
photonic and plasmonic structures, namely lasing disk microcavities, photonic
crystals, metallic gratings and nanoparticle arrays. A special emphasis has been
made on development of high-performance techniques for studies of photonic
devices.
The following papers are included:
In the first two papers (Paper I and Paper II) we developed a novel scattering matrix technique for calculation of resonant states in 2D disk microcavities
with the imperfect surface or/and inhomogeneous refraction index. The results demonstrate that the surface imperfections represent the crucial factor
determining the Q factor of the cavity.
A generalization of the scattering-matrix technique to the quantum-mechanical electron scattering has been made in Paper III. This has allowed us to treat
a realistic potential of quantum-corrals (which can be considered as nanoscale
analogues of optical cavities) and has provided a new insight and interpretation
of the experimental observations.
Papers IV and V present a novel effective Green’s function technique for
studying light propagation in photonic crystals. Using this technique we have
analyzed surface modes and proposed several novel surface-state-based devices
vii
viii
for lasing/sensing, waveguiding and light feeding applications.
In Paper VI the propagation of light in nanorod arrays has been studied.
We have demonstrated that the simple Maxwell Garnett effective-medium theory cannot properly describe the coupling and clustering effects of nanorods.
We have demonstrated the possibility of using nanorod arrays as high-quality
polarizers.
In Paper VII we modeled the plasmon-enhanced absorption in polymeric
solar cells. In order to excite a plasmon we utilized a grated aluminum substrate. The increased absorption has been verified experimentally and good
agreement with our theoretical data has been achieved.
Contributions to the papers
All the enclosed papers constitute the output of a 4-year close collaboration
between the authors, involving a permanent, almost everyday, exchange of the
ideas and discussions during the whole process. Therefore, it is hard to pick
out my own effort, but an attempt is the following:
• Paper I: A. Rahachou and I. V. Zozoulenko, Effects of boundary roughness on a Q factor of whispering-gallery-mode lasing microdisk cavities,
J. Appl. Phys., vol. 94, pp. 7929–7931, 2003
• Paper II: A. Rahachou and I. V. Zozoulenko, Scattering matrix approach
to the resonant states and Q values of microdisk lasing cavities, Appl.
Opt., vol. 43, pp. 1761–1772, 2004
In the first two papers I implemented both the serial and parallel versions
of the scattering matrix (SM) technique in Fortran 95, performed all
the calculations and summarized the results. I also derived necessary
equations for the Husimi-function analysis, developed and implemented
the ray tracing problem in the Poincaré surface-of-sections part. I also
gave an idea of the enhanced transmission of the high-Q whisperinggallery modes through a curved surface. I believe I tried to write the
papers, but... They were rewritten by Igor anyway.
• Paper III: A. Rahachou and I. V. Zozoulenko, Elastic scattering of surface electron waves in quantum corrals: Importance of the shape of the
adatom potential, Phys. Rev. B, vol. 70, pp. 233409 1–4, 2004
I adapted the SM technique to the quantum-mechanical problem and did
all the calculations. Took part in the discussions and interpretation of the
results. First several unsuccessful iterations of the paper were actually
mine...
• Paper IV: A. Rahachou and I. V. Zozoulenko, Light propagation in finite
and infinite photonic crystals: The recursive Greens function technique,
Phys. Rev. B, vol. 72, pp. 155117 1–12, 2005
ix
x
I derived some of the matrix equations (combination of the Green’s functions) and implemented the method in both serial and parallel Fortran
95 codes. I also performed all the calculations, took part in discussions.
I wrote the introduction and results/discussion parts of the paper.
• Paper V: A. Rahachou and I. V. Zozoulenko, Waveguiding properties
of surface states in photonic crystals, J. Opt. Soc. Am B, vol. 23, pp.
1679–1683, 2006
I carried out all the calculations, suggested the idea of the directional
beamer and wrote the paper.
• Paper VI: A. Rahachou and I. V. Zozoulenko, Light propagation in
nanorod arrays, J. Opt A, vol. 9, pp. 265–270, 2007
I adapted the Green’s function technique to the plasmonic applications. I
proposed some of the structures, made all the computations, summarized,
discussed and analyzed the results. Then I wrote the paper. After serious
Igor’s criticism it finally came to its present state...
• Paper VII: K. Tvingstedt, A. Rahachou, N.-K. Persson, I. V. Zozoulenko,
and O. Inganäs, Surface plasmon increased absorption in polymer photovoltaic cells, submitted to Appl. Phys. Lett., 2007
I made all the calculations, analyzed the results and wrote the theoretical
part of the paper.
Contents
Abstract
vi
Contributions to the papers
ix
Table of Contents
xi
1 INTRODUCTION
1
2 Photonic structures
2.1 Whispering-gallery-mode lasing microcavities . . . . . . . . .
2.1.1 General principle of lasing operation . . . . . . . . . .
2.1.2 Total internal reflection and whispering-gallery modes
2.2 Surface states in photonic crystals . . . . . . . . . . . . . . .
2.2.1 Photonic crystals . . . . . . . . . . . . . . . . . . . . .
2.2.2 Surface states and their applications . . . . . . . . . .
2.3 Surface plasmons . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Excitation of surface plasmons . . . . . . . . . . . . .
2.3.2 Applications of surface plasmons . . . . . . . . . . . .
2.4 Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Properties of nanoparticles and Mie’s theory . . . . .
2.4.2 Nanoparticle arrays and effective-medium theories . .
2.4.3 Applications of nanoparticles . . . . . . . . . . . . . .
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3 Computational techniques
3.1 Available techniques for studying light propagation in photonic
structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Scattering matrix method . . . . . . . . . . . . . . . . . . . . .
3.2.1 Application of the scattering matrix method to quantummechanical problems . . . . . . . . . . . . . . . . . . . .
3.3 Green’s function technique . . . . . . . . . . . . . . . . . . . . .
3.4 Dyadic Green’s function technique . . . . . . . . . . . . . . . .
xi
5
5
5
7
11
11
15
16
16
19
21
21
24
24
27
27
28
31
32
38
xii
CONTENTS
4 Results
4.1 Effect of inhomogeneities on quality factors of disk microcavities
(Papers I, II) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Quantum corrals (Paper III) . . . . . . . . . . . . . . . . . . . .
4.3 Surface-state lasers (Paper IV) . . . . . . . . . . . . . . . . . .
4.4 Surface-state waveguides (Paper V) . . . . . . . . . . . . . . . .
4.5 Nanorod arrays (Paper VI) . . . . . . . . . . . . . . . . . . . .
4.6 Surface plasmons in polymeric solar cells (Paper VII) . . . . . .
43
Bibliography
59
Appendix
I
Paper
II
Paper
III Paper
IV Paper
V
Paper
VI Paper
VII Paper
I .
II .
III
IV
V .
VI
VII
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43
45
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71
77
91
97
111
119
127
Chapter 1
INTRODUCTION
1
2
CHAPTER 1. INTRODUCTION
The idea of constructing chips that operate on light signals instead of electricity has engaged the minds of scientists during the last decade. Communicating photons instead of electrons would provide revolutionary breakthrough
not only in the performance of devices, which can distribute data at the speed
of light, but also in the capacity of transmitted data. By now, modern optical
networks can provide such a bandwidth, that even the fastest state-of-the-art
processors are unable to handle, and this trend seems to remain in nearest future. Furthermore, photons are not so strongly interacting as electrons/holes
that significantly broaden bandwidth. Speaking about present time, only photonics provides solutions for high-dense modern data storage, like CDs and
DVDs, whose capacity is constantly increasing.
Manufacturing practical photonic chips, however, brings in several challenges: first of all, lack of all-optical logic switches themselves as well as the
principles of their operation, technological difficulties in manufacturing of novel
photonic devices with the same well-developed processes for electronic chips,
and, finally, the need of novel materials. In this regard, the most promising
”building blocks” of modern photonics are photonic crystals, lasing microcavities and plasmonic devices, which, being intensively studied during the latest
decade, can provide the required functionality and microminiaturization.
Along with opportunities for integration of optical devices, photonic crystals exhibit a variety of unique physical phenomena. Photonic crystal is usually
fabricated from the same semiconductor materials as electronic chips using common chipmaking techniques like photolithography. The main reason that has
made photonic crystals so popular is their basic feature of having gaps in the energy spectrum that forbid light to travel at certain wavelengths. Such the gaps
in the spectra provide very effective confinement of the light within photonic
crystals that can be exploited as a basis for a large number of photonic devices.
Creating linear defects, for instance, will form low-loss waveguides, whereas
point defects can act as high-quality microcavities. Another unique feature
of photonic crystals with certain lattice parameters is the negative refraction
index that can be exploited for focusing and non-conventional distribution of
light on a microscopic level. In additional, real finite photonic crystals can support surface states on their boundaries, which can also be exploited for different
purposes in photonic chips.
Optical microcavities are structures that confine light and enable lasing
action on a microscopic scale. In conventional lasers, a significant portion of the
pump energy simply dissipates, and a rather high threshold power is required
to initiate the lasing effect. In contrast, microcavities can be utilized to sustain
highly efficient, almost ”thresholdless”, lasing action. Such the efficiency is
related to the existence of the natural cavity resonances. These resonances
are known as morphology-dependent resonances or whispering gallery modes.
The origin of these resonances can be addressed to ray dynamics, when the
light is trapped inside the cavity through total internal reflection against its
3
circumference. An ideal lossless cavity would trap this ”rotating” light for
infinitely long time and would have infinitely narrow lasing peaks. Combining
microcavities into arrays or coupling them to waveguides creates variety of
devices for sensing and filtering. Ultra high-quality microcavities can also be
utilized in stunning applications such as single atom detection.
Plasmonic structures is the ”State of the Art” of modern photonics. Plasmons, the electromagnetic modes localized at metal-dielectric interfaces and
metallic nanoparticles, bring in new unprecedented opportunities of guiding
and manipulating light beyond the diffraction limit. Novel plasmonic waveguides and their arrangements are able to distribute light on nanoscale, providing the missing link between highly-integrated electronic chips and larger-scale
photonic components. Enhanced field intensities of plasmonic modes are utilized in a variety of applications – from biological sensors to spectroscopy and
lasing structures.
The Thesis is organized as follows. In Chapter 2 we make a brief overview
of photonic structures under the study, namely microdisk cavities, photonic
crystals and plasmonic devices. Chapter 3 outlines the scattering matrix and
Green’s function techniques, and Chapter 4 summarizes the main results and
contains discussions.
4
CHAPTER 1. INTRODUCTION
Chapter 2
Photonic structures
2.1
2.1.1
Whispering-gallery-mode lasing microcavities
General principle of lasing operation
The word ”LASER” is an acronym for Light Amplification by Stimulated Emission of Radiation. The output of a laser is a highly-coherent monochromatic (in
a very ideal case) radiation, which can be pulsed or beamed in a visible, infrared
or ultraviolet range. The power of a laser can vary from several milliwatts to
megawatts.
The main and the most crucial component of a laser is its active medium,
which can be a solid, gas, liquid or semiconductor. In thermodynamic equilibrium nearly all atoms, ions or molecules (depending on the particular laser) of
the active medium occupy their lowest energy level or ”ground state”. To produce laser action, the majority of atoms/ions/molecules should be ”pumped”
up into the higher energy level, creating so called population inversion. Typical
three-level structure is given in Fig. 2.1(a). Pump energy here excites atoms
from the ground state to the short-lived level, which rapidly decays to the
long-lived state. At random times, some of these excited atoms/ions/molecules
will decay to the ground state on their own. Each decay is accompanied by
the emission of a single photon propagating in a random direction (spontaneous emission). However, when one of these photons encounters an excited
atom/ion/molecule, the latter will drop down to a lower energy state and emit
a new photon with exactly the same wavelength, phase, direction and polarization. This is called stimulated emission.
When a photon is emitted nearly parallel to the long side of the cavity [Fig.
2.1(b)] it will travel down to one of the mirrors and be able to get reflected back
and forth many times. Along its way, it hits excited atoms/ions/molecules and
5
6
CHAPTER 2. PHOTONIC STRUCTURES
Equilibrium
Pumping
Fast
Relaxation
Stimulated
emission
short-lived level
p
en ump
er
gy
long-lived level
hν
hν
ground level
Totally
reflecting
mirror
Active medium
(a)
Partially
reflecting
mirror
hν
(b)
Figure 2.1: (a) Three-level diagram of a lasing system. (b) Lasing cavity.
Output intensity
”stimulates” them to emit up new photons. The process acts as an avalanche
caused by a single photon which produces more and more photons via this stimulated emission process. When the energy of the photon beam becomes enough
to make the beam escape the partially reflecting mirror, a highly monochromatic and coherent ray goes out. Depending on the type of a cavity the beam
can be well collimated or appears to originate from a point/plane source.
Spontaneous
emission
Threshold PTH
Stimulated
emission
Pump power
Figure 2.2: Threshold of a laser.
One of the most important parameters of lasers is their threshold power
PT H , that can be defined as the ”critical” pumping power that corresponds
to the initiation of the stimulated emission (see Fig. 2.2). The threshold is
proportional to the threshold population difference, i.e. the minimum positive
difference in population between the long-lived and ground levels in Fig. 2.1
NT = Nll − Ng ∼
1
ω0
=
cτp
cQ
(2.1)
2.1. WHISPERING-GALLERY-MODE LASING MICROCAVITIES
7
where c is the speed of light, τp is a photon lifetime, ω0 is a resonant frequency
of a lasing mode and Q is a quality factor (Q factor hereafter) of a lasing
cavity. The main goal is obviously to minimize the threshold power, therefore
maximize the photon lifetime and cavity quality factor. The Q factor is strongly
determined by the design of a cavity. Several representative examples are given
in Fig. 2.3.
(a)
(d)
(b)
(e)
(c)
(f)
Figure 2.3: Different types of lasing cavities. (a) Confocal resonator. Employed
in a variety of gas, solid-state and chemical lasers. Two confocal mirrors (one
of them is partially reflecting) create a collimated beam parallel to the long
side of the cavity. (b) Laser diode. The cavity is created by finely polished
side walls of the structure. (c) Photonic-crystal cavity. The cavity is created
by a point inhomogeneity in a photonic-crystal lattice (see the next section for
details). Q factor can reach 105 . (d) Fabri-Perot resonator. A set of stacked
Bragg mirrors provides cavity confinement. Typical value of the Q factor is
∼ 2000. (e) Whispering-gallery disk microcavity. Light is trapped inside the
cavity, undergoing multiple ”bounces” against the side wall due to the effect
of total internal reflection. Q ∼ 104 , toroidal cavities with Q ∼ 108 have been
also reported [1]. (f) A spherical whispering-gallery droplet. Q ∼ 108 . (c-f) are
adopted from [2].
2.1.2
Total internal reflection and whispering-gallery modes
One of the most well-known mechanisms of the ray confinement in cavities is
based on the effect of total internal reflection, which is presented in Fig. 2.4.
8
CHAPTER 2. PHOTONIC STRUCTURES
The angle
θc = arcsin
n1
n2
(2.2)
is called the critical angle for total internal reflection. At larger incidence angles
θ2 the ray remains fully reflected.
For curved boundaries [see Fig. 2.4(b)] the regime of total internal reflection and the critical angle (2.2) have the same meaning. However, because of
the diffraction at the curved boundary, a leakage takes place. Transmission
coefficient for an electromagnetic wave penetrating a curved boundary in the
regime of total internal reflection reads [3]
3/2
2 nkρ
2
2
T = |TF | exp −
cos θc − cos θ
,
(2.3)
3 sin2 (θ)
where TF is a classical Fresnel transmission coefficient for an electromagnetic
wave incident on a flat surface, k is a wavevector of the incident wave, ρ is a
radius of curvature, and θ is an angle of incidence. The main goal, obviously,
is to minimize T , in order to hold the light ”trapped” inside the cavity as long
as possible.
n1
n1
θc
θc
ρ
n2
n2
(a)
(b)
Figure 2.4: (a) The regime of total internal reflection for (a) a flat surface,
(b) a curved surface. The ray falls from medium 2 to the boundary with
medium 1 (n1 < n2 ) at incidence angle θ2 and gets refracted to medium 1 at
θ1 . According to the Snell’s law, n1 sin θ1 = n2 sin θ2 . If θ2 is being increased, at
some particular incidence angle θc , angle θ1 becomes equal π/2 that corresponds
to the full internal reflection of the incident beam.
Total internal reflection is a mechanism of light localization in whisperinggallery cavities. The term whispering-gallery modes (WGMs) came after the
whispering gallery at St. Paul’s Cathedral in London, see Fig. 2.5(a), where
the quirk in its construction makes a whisper against its walls audible at the
2.1. WHISPERING-GALLERY-MODE LASING MICROCAVITIES
9
opposite side of the gallery. In whispering-gallery cavities [Fig. 2.5(b)] WGMs
occur at particular resonant wavelengths of light for a given cavity size. At these
wavelengths the light undergoes total internal reflection at the cavity surface
and remains confined inside for a rather long time. In the WGM regime the
θ>θc
(a)
(b)
Figure 2.5: (a) The dome of the St. Paul’s Cathedral in London. The white line
outlines distribution of a WG-mode. (b) Multiple reflections of a whisperinggallery mode against the circumference of the cavity.
mode is localized near the circumference of a cavity and can be assigned a
radial and angular mode numbers. The angular mode number n shows the
number of wavelengths around the circumference, and the radial mode number
l – the number of maxima in the intensity of the electromagnetic field in the
radial direction within the cavity. A typical experimental spectrum of the WG
modes is given in Fig. 2.6(a).
Each whispering-gallery lasing mode of a cavity is characterized by its quality factor Q, which, by the definition, is also related to the width of the resonant
spectral line as
Q≡
2π(stored energy per cycle)
k
=
(energy loss per cycle)
∆k
(2.4)
where ∆k is a spectral line broadening taken at the half-amplitude of the lasing
peak as it shown in Fig. 2.6(b). Q factor is also closely related to the time that
the WG mode spends trapped within a cavity, so-called ”Wigner delay time”
[4]
Q = ωτD (ωres ),
(2.5)
where ω is a resonant frequency.
The main reason of using whispering-gallery mode cavities is their high Q
values as well as excellent opportunities to be integrated into optical chips. Lasing whispering-gallery modes were first observed in spherical glass droplets. An
CHAPTER 2. PHOTONIC STRUCTURES
Intensity
10
∆ k - Broadening
k
Wavevector
(a)
(b)
Figure 2.6: (a) Experimental spectrum of a whispering-gallery lasing microcavity [5]. Angular and radial mode numbers are also given. (b) Broadening
of a lasing peak.
important step was the development of microdisk semiconductor lasers, which
exploited total internal reflection of light to achieve the perfect mirror reflectivity. These lasers – the smallest in the world at the time, were invented and
first demonstrated in 1991 by Sam McCall, Richart Slusher and colleagues at
Bell Labs. Microdisk, -cylinder or -droplet lasers form a class of lasers based on
circularly symmetric resonators, which lase in whispering-gallery modes. These
tiny lasers, however, lack for directional emission due to their circular symmetry. The experimental microlasers of Bell Labs and Yale team overcame this
limitation. They were based on a new optical resonator shaped as a deformed
cylinder (quadruple) and were highly directional. They exploited the concept
of chaotic dynamics in asymmetric resonant cavities and were introduced by
Nöckel and Stone at Yale in 1997.
By now there have been reported cavities with Q factors of order ∼ 108
[1] with characteristic diameters ∼ 100µm. The another advantages are their
relatively easy fabrication process (i.e. they can be etched on a surface [5] or
pedestal [6], highly-symmetrical spherical cavities [7] are formed through the
surface tension in silica); broad range of pumping methods (optical pump from
the outside [5] or by the build-in quantum dots [6]; use of active polymers
[8]); as well as a set of different shapes (disk, toroid, spherical, hexagonal,
quadruple) possessing unique properties.
Unfortunately, quality factors in actual fabricated microcavities are normally several orders lower than the corresponding calculated values of ideal
cavities. A degradation of the experimental Q factors may be attributed to a
variety of reasons including side wall geometrical imperfections, inhomogeneity of the refraction index of the disk, effects of coupling to the substrate or
pedestal and others. A detailed study of effects of the factors above on the characteristics and performance of the microcavity lasers appears to be of crucial
2.2. SURFACE STATES IN PHOTONIC CRYSTALS
11
importance for their optimization. Of the especial importance are the studies
of surface roughness of the cavities, as it have been demonstrated [9; 6; 10]
to be the main factor affecting the Q value. Such the studies would require a
versatile method that can deal with both the complex geometry and variable
refraction index in the cavity. In the next Chapter we develop a novel computational technique, which is capable to handle disk microcavities both with
geometrical imperfections and refraction index inhomogeneities.
2.2
2.2.1
Surface states in photonic crystals
Photonic crystals
Photonic crystals (PCs) or photonic bandgap materials are artificial structures,
which forbid propagation of light in particular ranges of frequencies, remaining
transparent for others. Photonic band gaps were first predicted in 1987 by two
physicists working independently. They were Eli Yablonovitch, at Bell Communications Research in New Jersey, and Sajeev John of the University of Toronto.
A periodic array of 1mm holes mechanically drilled in a slab of a material with
the refraction index 3.6 was found to prevent microwaves from propagating
in any direction. This structure received a name Yablonovite. Despite this
remarkable success, it took more than a decade to fabricate photonic crystals
that work in near-infrared (780-3000 nm) and visible (450-750 nm) ranges of
the spectrum and forbid light propagation in all directions. The main challenge
was to find suitable materials and technologies to fabricate structures that are
about a thousandth the size of the Yablonovite.
Let us now compare light propagation in a photonic crystal to the carrier
transport in a semiconductor. The similarity between electromagnetic waves in
PCs and de-Broglie electronic waves propagating in a crystalline solid has been
utilized to develop theories of photonic crystals. For electrons in semiconductor
materials the Schrödinger equation reads as
2 2
~ ∇
−
+ V (r) Ψ(r) = EΨ(r).
(2.6)
2m∗
In a semiconductor crystal the atoms are arranged in a periodic lattice, and
moving carriers experience a periodic atomic lattice potential
V (r + a) = V (r),
(2.7)
where a is a lattice constant. Then, there exists a wavevector k in the reciprocal
lattice such that Ψ(r) can be written as
Ψ(r) = eikr uk (r),
(2.8)
12
CHAPTER 2. PHOTONIC STRUCTURES
where uk (r + a) = uk (r) is a periodic function on the lattice. This expression is known as Bloch’s theorem. Substituting it into Eq. (2.6) one finds the
eigenfunctions uk (r) and eigenvalues Ek . The periodic potential causes formation of allowed energy bands separated by gaps. In perfect bulk semiconductor
crystals no electrons or holes can be found in these energy gaps.
The situation holds also for photons traveling through periodic structures.
Let us consider a periodic structure, e.g. a block of a transparent dielectric
√
material of the high refraction index (related to a permittivity as n = ǫ) with
”drilled” holes or, vice versa, a periodic set of high-index dielectric rods in
air background. In this case the corresponding electromagnetic wave equation
(Maxwell’s equation for the magnetic field ) reads
1
∇×
∇× H(r) = (ω 2 /c2 )H(r),
(2.9)
ǫ(r)
with the periodic dielectric function
ǫ(r + R) = ǫ(r).
(2.10)
ε1 ε2
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2.7: Examples (a-c) of 1D, 2D and 3D photonic crystals and (d-f)
corresponding band structures. (adopted from [11])
For a photon, the periodic dielectric function acts just as the lattice potential that an electron or hole experiences propagating through a semiconductor
crystal. If the contrast of the refraction indexes is large, then the most of
2.2. SURFACE STATES IN PHOTONIC CRYSTALS
13
the light will be confined either within the dielectric material or the air. This
confinement causes formation of intermingled allowed and forbidden energy regions. It is possible to adjust the positions of bandgaps by changing the size of
the air holes/rods in the material/air or by variation of the refraction index.
It is worth mentioning that the similarity between electrons in semiconductors and photons in photonic crystals is not complete. Unlike the Schrödinger’s
equation for electron waves, the Maxwell’s equations and electromagnetic waves
are vectorial that requires an additional computational effort. On the other
hand, the Schrödinger’s equation can include many-body interactions, which
are not the case for electromagnetic problems.
Another important aspect is periodicity of photonic crystals. If the periodicity in the refraction index holds only in one direction (i.e 1D photonic
crystal), only light traveling perpendicularly to the periodically arranged layers
is affected. Any 1D structure supports bandgaps. In the 2D case, light propagating in the plane perpendicular to the rods will be affected. In order to make
a complete bandgap for any direction of light propagation, a 3D structure have
to be constructed. Fig. 2.7 illustrates 1D, 2D and 3D photonic crystals along
with their band structures.
Photonic crystal devices normally operate in the frequency regions corresponding the bandgaps. The area of possible applications is constantly expanding, some representative examples are given in Fig. 2.8.
14
CHAPTER 2. PHOTONIC STRUCTURES
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 2.8: (a)Low-threshold cavity lasers. A properly designed point defect
in a photonic crystal can act as a lasing cavity. Strong confinement of the
field within the defect area enables one to achieve quality factors of order
∼ 106 [12; 13]. (b) Band-edge lasers. Photonic crystal operates at the energy
of the band edge, where the velocity of light is very low, that causes long
lifetime and high Q factor of the given state at this energy [14]. (c) Surfacestate lasers. Braking the translation symmetry of the surface of a photonic
crystal turns a surface mode into a resonant state with the high Q factor.
The unique feature of such the cavity is its location on the surface of a PC
[15; 16]. (d) Low-loss waveguides with wide curvature. In optical integrated
circuits, construction of low-loss waveguides with wide curvature is essential.
When PCs are fabricated using low-loss dielectric materials, they act as perfect
mirrors for the frequencies in the gap [17]. (e,f) Channel add/drop filters.
Enable switching and redistributing light of certain frequencies between two
or more waveguides [13; 18]. (g) Photonic bandgap microcavity in a dielectric
waveguide. Acts as a filter in dielectric waveguides, suppresses all frequency
range except for the frequencies of the resonant states of the PC-cavity [19].
(h) Optical transistor. Based on the Kerr effect. The intensity of the control
beam (transverse waveguide) affects the Kerr cell, switching the light in the
longitudinal waveguide [20].
2.2. SURFACE STATES IN PHOTONIC CRYSTALS
2.2.2
15
Surface states and their applications
0.55
0.55
0.50
0.50
0.45
0.45
ωa/2πc
ωa/2πc
Surface states or surface modes is a special type of states in a photonic crystal
that reside at the interface between a semi-infinite PC and open space, decaying
into both the crystal and air [21]. Not every PC boundary supports surface
states. For example, surface modes can be always found on the surface of
a truncated 2D hexagonal array of holes in a material. At the same time,
no surface state are found on the unmodified surface of a semi-infinite square
array of cylinders in the air background. For the latter case the surface states
appear in the bandgap of a square-lattice photonic crystal when its boundary
is modified by, e.g., truncating the surface rods, shrinking or increasing their
size, or creating more complex surface geometry [21; 22; 23; 24]. Examples of
structures supporting surface states along with their band diagrams are given
in Fig. 2.9.
0.40
0.35
0.40
0.35
0.30
0.30
0.25
0.25
0.20
0.30
0.35
0.40
ka/2π
0.20
0.30
0.45
(a)
0.35
0.40
ka/2π
0.45
(b)
Figure 2.9: Band structures for the TM modes in the ΓX direction of squarelattice photonic crystals composed of rods with diameter D = 0.4a (a is the
lattice constant) and permittivity ε = 8.9 along with the projected surface
modes. The surface rods are (a) reduced to d = 0.2a and (b) half-truncated.
The right panels show the intensity of the Ez component of the surface modes
at the energies denoted with the arrows.
So, why do the surface states in PCs attract our attention? Thanks to
their unique location, on the surface of a photonic crystal, they open up new
possibilities of coupling photonic devices to external light sources, stimulate
directional beaming [25] from the waveguide opening on the surface. It is worth
to emphasize that the surface mode residing on the infinitely long boundary
of a semi-infinite crystal represents a truly bound Bloch state with the infinite
lifetime and Q factor, and consequently does not couple or leak to air states.
We have recently shown (see Paper V) that this feature enables surface states
to be exploited as high-quality surface waveguides and directional beamers,
which, being situated on the surface of a PC, provide unique opportunities in
redistributing light in photonic chips. It has also been demonstrated (Paper
IV), [15; 16] that when the translational symmetry along the boundary of the
16
CHAPTER 2. PHOTONIC STRUCTURES
semi-infinite crystal is broken, the Bloch surface mode turns into a resonant
state with a finite lifetime. This effect can be utilized for lasing and sensing
purposes.
2.3
Surface plasmons
2.3.1
Excitation of surface plasmons
Surface plasmons (SPs) are electromagnetic surface waves that propagate along
the boundary between a metal and dielectric. They originate from collective
oscillations of the electron density in the metal near the boundary under the
external excitation. They were referred by Ritchie for the first time in 1950-th
[26], and since then have attracted increased attention due to their extraordinary ability to guide and manipulate light at nanoscale. Figure 2.10 illustrates
the p-polarized electromagnetic field (i.e. field, which has its electric component
parallel to the plane of incidence) propagating towards the boundary of two media at angle of incidence θ. Boundary conditions for the electric fields imply that
Ez
By
+++ −−− +++ −−−
ε1’>0
Ex
θ
z
δ
|E|
x
y
z
Metal (ε2’<0)
(a)
(b)
Figure 2.10: (a) Excitation of a plasmon on the metal-dielectric interface with
p-polarized light, propagating at angle of incidence θ greater than the angle
of total internal reflection. Inset illustrates the surface charges. (b) Plasmoninduced field intensity at the interface.
the Ex -component is conserved across the boundary (i.e. Ex1 = Ex2 ), whereas
the Ez -component undergoes a discontinuity, such that ε1 ε0 Ez1 = ε2 ε0 Ez2 . This
discontinuity results in polarization changes at the interface and, consequently,
additional localized surface charges [see inset to Fig. 2.10(a)]. The electromagnetic field, induced by these charges, represents a plasmonic mode, which
is localized near the interface and propagates along it. It is worth mentioning
2.3. SURFACE PLASMONS
17
that the s-polarized light (which has its magnetic components parallel to the
plane of incidence) does not generate any surface charges and, therefore, does
not excite a plasmonic mode.
The plasmonic mode is localized in the dielectric over the distance δ, which
approximately equals the half wavelength of the incoming light in the dielectric,
whereas in the metal its localization is determined by the metal skin depth
(∼ 10 nm). The propagation length of the plasmon depends on the absorbing
properties of the metal (the imaginary part of the dielectric function ε′′2 ). Thus,
for low-ε′′ metals, such as silver in the infrared, the propagation length can
reach hundreds of micrometers, but for the high-ε′′ ones (aluminum) it hardly
exceeds tens of microns [27].
The dispersion relation for a plasmonic mode reads as [28]
kx = k
ε1 ε2
ε1 + ε2
1/2
,
(2.11)
where k = 2π/λ. This relation clearly shows the condition for excitation of
a plasmonic mode: ε′2 has to be negative and |ε2 | > ε1 , which means that a
plasmonic mode can only be excited on the surface of a metal. The another
important conclusion from (2.11) is that the real part of the plasmon wavevector
is always greater than the wavevector of the exciting radiation (see Fig. 2.11).
Because of this, it is not possible to excite a plasmon on the flat surface with
a propagating light beam.
ω
ω=ck
ω
k<kx
kx
Figure 2.11: The dispersion curve of a plasmonic mode. The curve lies beyond the light cone that does not allow direct excitation of a plasmon with
propagating light.
In order to enhance the wavevector of the exciting light (and thus to be
able to excite a plasmon), several techniques have been proposed. They are
18
CHAPTER 2. PHOTONIC STRUCTURES
illustrated in Fig. 2.12. The first two techniques, outlined in Fig. 2.12(a) and
(b), are based on the excitation of a plasmon with en evanescent field. If the
beam is incident at angle θ greater than critical angle of total internal reflection
θc [defined in (2.2)], it does not propagate across the interface. Instead, it
gives rise to the evanescent field with purely
p imaginary z-component ikz of the
wavevector and real x-component kx = (k 2 − (ikz )2 ) > k. This enhancement
can be used to couple the incoming radiation to the plasmonic mode. The
geometry in Fig. 2.12(a) is called Otto geometry [29] and consists of a prism
separated from a bulk metallic sample by a thin (few radiation wavelengths)
gap. The gap provides a tunnel barrier, which creates a p-polarized evanescent
mode, exciting the plasmon at the metal-air interface.
θ
d
θ
z
x
(a)
(b)
(c)
Figure 2.12: Methods of plasmon excitation. (a) Otto geometry.
Kretschmann-Raether geometry. (c) Grating coupler.
(b)
The alternative technique is a Kretschmann-Raether geometry, depicted in
Fig. 2.12(b) [30]. Here, the thin deposited metal film (< 50 nm) itself plays a
role of the evanescent tunnel barrier, and the plasmon is excited on the opposite
side of the metal.
Surface plasmons can also be excited without the coupling prisms. In order
to increase the wavevector of the propagating light, grated metallic surfaces
can be used [31]. In this case, x-component of the wavevector of the exciting
light kxinc is enhanced by the integer multiple of x-component of reciprocal unit
vector Gx of the grating
kx = kxinc + nGx = k sin θ + 2nπ/d,
(2.12)
where d is a grating period. Equation (2.12) is valid for any θ (including those
θ > θc ). Adjusting the value of d, one can alter positions of the plasmonic
resonances in the spectrum.
2.3. SURFACE PLASMONS
2.3.2
19
Applications of surface plasmons
Plasmonic nanodevices are considered to be the most promising solutions for
functional elements in photonic chips, near-field microscopy, manipulation of
atoms and others. Plasmonic devices now cover the whole range of functionality
of the traditional photonic devices, such as cavities, waveguides, apertures,
providing, however, light manipulation at a deep sub-wavelength scale. Some
of the plasmonic applications are summarized in Fig. 2.13.
(b)
(a)
(e)
(c)
(f )
(d)
(g)
(h)
Figure 2.13: Applications of surface plasmons. (a) Highly-directional plasmonic beamer [32]. Light, outgoing through the aperture in the center, couples
to surface plasmons on the grated surface that results in highly-directional
emission. (b) Plasmon-assisted extraordinary transmission through the array
of sub-wavelength holes has been demonstrated [33]. (c) Ring resonators [34],
made of grooves in a metal, can be utilized as band filters. (d) Plasmonic bandgap crystal [35]. Nano-patterned silver surface demonstrates photonic-crystallike gaps in the spectrum of plasmonic modes. (e) Nanofocusing of energy
on the tip of the adiabatic plasmonic waveguide [36]. (f) A SNOM (Scanning
Near-field Optical Microscopy) probe-based 1/4-wavelength nanoantenna [37].
Evanescent plasmonic mode from the sub-wavelength aperture couples to the
1/4-wavelength tip, resulting in the high field intensity. (g) Low-loss guiding
of light in a low-n core 2D-waveguides(n1 < n2 ) [38]. (h) V-shaped plasmonic
waveguiding grooves, splitters and Mach-Zehnder interferometers ([34] and citations therein) with a nearly zero insertion loss.
20
CHAPTER 2. PHOTONIC STRUCTURES
Special attention is now also paid to possible applications of plasmons in
photovoltaics. Figure 2.14 illustrates a typical photovoltaic device of the socalled ”third” generation. The third generation photovoltaics includes photoelectrochemical cells, polymeric and nanocrystal solar cells and is rather different from the previous semiconductor structures as it does not rely on a
traditional p-n-junction to separate photogenerated charge carriers. Instead,
the carriers are separated by the diffusion only. The device represents a multilayer stack of electrodes and active layer(s) deposited onto a transparent glass
substrate. Polymeric solar cells seem to be promising in terms of low costs and
Glass
substrate
Transparent
electrode
(ITO)
Active
layer
Electrode
(Al)
Figure 2.14: Polymeric photovoltaic solar cell.
ease of fabrication. However, the power-conversion efficiency even of the most
advanced samples does not exceed 5% [39].
Plasmons, intensively absorbing light, can create high field intensities at the
contact-active layer interfaces, facilitate electron-hole pair generation processes,
increasing, therefore, the power conversion efficiency. Recently, the plasmoncaused increased absorption has been demonstrated for light-emitting diodes
[40] with metallic nanoparticles, deposited onto the active layer of Si diodes.
The application of nanoparticles to both non-organic and organic solar cells
[41; 42] has displayed the increased short-circuit photocurrent. Rand et al.
[43] have observed the extremely-high long-range absorption enhancement in
tandem solar cells with embedded Ag nanoclusters. Nanoclusters in their paper
reported to be acting as highly-effective recombination centers.
In Chapter 4 and in Paper VII an another technique of the plasmon-induced
absorption enhancement is proposed. Instead of using nanoparticles, we use
2.4. NANOPARTICLES
21
surface plasmons, excited on metallic gratings in polymeric solar cells. It is
demonstrated that the plasmon-enhanced absorbtion leads to the increased
photocurrent in the vicinity of the plasmonic peak.
2.4
Nanoparticles
For centuries, alchemists and glassmakers have used tiny metallic particles for
creating astonishing stained-glass windows and colorful goblets. One of the
most ancient examples is the Lycurgus cup, a Roman goblet from the 4-th
century A.D., see Fig. 2.15. The gold and silver particles embedded into the
glass of the goblet absorb and scatter blue and green light.
Figure 2.15: Lycurgus cup (4-th century A.D.). When viewed in reflected light,
the goblet looks in a greenish hue, however if a light source is placed inside the
goblet, the glass appears red.
Therefore, when viewed in reflected light, the cup looks in a greenish hue,
but if a white light source is placed inside the goblet, the glass appears red
because it transmits only the longer wavelengths and absorbs the shorter ones.
Nowadays metallic nanoparticles are intensively studied due to their potential
in spectroscopy, fluorescence, biological and chemical sensing and others.
2.4.1
Properties of nanoparticles and Mie’s theory
A term nanoparticle can be applied to any object containing 3 . N . 107
atoms. Physical properties of nanoparticles are size-dependent and two different kinds of size effects can be distinguished: intrinsic and extrinsic [44].
Intrinsic effects manifest themselves for small (< 10 nm) nanoparticles and are
22
CHAPTER 2. PHOTONIC STRUCTURES
caused by a relatively small number of atoms in a nanoparticle that leads to
the quantized energy spectrum of the particle. An arrangement of the atoms
and their quantity have a strong impact on the dielectric function and optical
properties of the cluster. However, for larger nanoparticles, containing millions
of atoms, the intrinsic effects are negligible, and the dielectric function of such
a cluster is assumed to be that for the bulk material. Optical response of these
particles is fully governed by the extrinsic effects – size- and shape-dependent
responses to the external excitations, irrespective to the internal structure of
the particles.
Let us now first consider a single metallic nanoparticle, being illuminated
with electric field E of frequency ω = 2π/T (see Fig. 2.16) and the wavelength
much larger than the nanoparticle size in a quasi-static regime (i.e. in the
regime when the spatial phase of the field is assumed to be constant within
the particle). The incident electric field causes displacement of the electronic
- - +
+
+
-
+
-
+
-
+
-
- - -
+ +
+ +
+
kx
-
+
-
Ey
-
+ +
+ +
+
+
time t
time (t+T/2)
Figure 2.16: Excitation of dipole plasmonic resonance in a metallic nanoparticle.
cloud within the particle against its ion core. The displacement gives rise to
polarization charges on the opposite (for the dipole resonance) sides of the
particle and, hence, to a restoring electrostatic force, which attempts to revert
the system back to the equilibrium. After the half-period time the field changes
its direction and the charges switch their places. Therefore, the nanoparticle
acts as an oscillating system with single eigenfrequency [44]
ωp
ω1 = √ ,
3
(2.13)
where ωp is the Drude’s plasma frequency of a given metal.
The general solution of a scattering problem for an arbitrary spherical particle of radius R was given by German physicist Gustav Mie in 1908, who
calculated the absorption, scattering and extinction (absorption+scattering)
2.4. NANOPARTICLES
23
cross-sections. Start from Helmholtz equation in spherical coordinates
∇2 Ψ + k 2 Ψ = 0,
(2.14)
1 ∂ 2 ∂
∂2
1
∂
∂
1
(r
)
+
(sin
θ
)
+
.
r2 ∂r
∂r
r2 sin θ ∂θ
∂θ
r2 sin2 θ ∂φ2
(2.15)
where
∇2 =
The solutions to (2.14) can be separated in spherical coordinates as
Ψ = R(r)Θ(θ)Φ(φ)
=
∞ X
l
X
m
m
m
[Am
l cos mφPl cos θZn (kr) + Bl sin mφPl cos θZn (kr)],
l=0 m=−l
(2.16)
with Plm spherical Legendre polynomials and Zn (kr) the Bessel functions for
r < R and Hankel functions for r > R. Applying boundary conditions and
m
equating (2.16) one finds unknown coefficients Am
l and Bl . Having calculated
the coefficients one can easily obtain the extinction cross-section as
σext =
2π X∞
(2l + 1)ℜ(Al + Bl ).
k2
(2.17)
l=1
For the case R << λ, when the quasi-static limit is assumed and only the
dipole mode with l = 1 is considered, (2.17) reduces to [44]
ω
ε′′ (ω)
σext = 12π ε0 3/2 R ′
,
c
[ε (ω) + 2ε0 ]2 + ε′′ (ω)2
(2.18)
where ε0 and ε(ω) are dielectric functions of the surrounding medium and
nanoparticle respectively. It can be easily shown that the condition for the
resonance is that ε′ (ω) = −2ε0 .
For larger particles, however, the interactions of higher orders l > 1 have
stronger impact on the extinction spectra and cannot longer be neglected. The
positions of the resonances are extremely sensitive to the surrounding medium,
shape, size and symmetry of the particles and the temperature. Because of this,
nanoparticles are considered to be promising candidates for sensing applications
(see section 2.4.3 for details).
It should also be mentioned that the Mie’s theory accounts only for noninteracting spheroids, whereas for the scatterers of arbitrary shape or aggregates of particles a number of more advanced tools has been developed. Among
them are coupled-dipole approximation [45], multiple multipole technique [46],
finite-difference time-domain method [47], generalized Lorentz-Mie’s theory for
24
CHAPTER 2. PHOTONIC STRUCTURES
assemblies [48] and many others.
2.4.2
Nanoparticle arrays and effective-medium theories
Single nanoparticles are of the prime interest for the fundamental study. However, practical applications require macroscopic systems containing thousands
of particles. Moreover, many of these applications require knowledge of the
effective-medium response of such systems, i.e. knowledge of the effective dielectric function from the optical properties of the constituents.
Let us assume a set of equally-sized metallic nanoparticles with dielectric
function ε(ω) embedded into a host dielectric medium with dielectric function
εm at low filling factor f . Effective dielectric function εef f of the blend [49]
εef f (ω) − εm
ε(ω) − εm
=f
.
εef f + 2εm
ε(ω) + 2εm
(2.19)
was given by Maxwell Garnett in 1904 for non-interacting nanoparticles (low
f < 0.3) in the quasi-static limit (d << λ). His theory has been extended by
Bruggeman [50] to the case of high filling factor f & 0.5, where the effective
dielectric function is given
f
εm − εef f (ω)
ε(ω) − εef f (ω)
+ (1 − f )
= 0.
ε(ω) + 2εef f (ω)
εm + 2εef f (ω)
(2.20)
For even higher filling factors, clustering of nanoparticles and multipole
effects are expected to play a significant role in both the Maxwell Garnett and
Bruggeman theories. These factors are taken into account in the Ping Sheng
theory [51]. Further, the Maxwell Garnett theory has been extended to the
case of elliptic particles [52], to anisotropic composites [53], and others [54].
However, an effective-medium theory that accounts for non-spheroid particles
at arbitrary concentrations or touching/overlapping particles remains to be
developed.
2.4.3
Applications of nanoparticles
A number of nanoparticle applications is constantly expanding. The table below summarizes some of them and several representative illustrations are also
given in Fig. 2.17.
2.4. NANOPARTICLES
Application
Optical and photonic
Electronic
Mechanical
Thermal
Magnetic
Energy
Biomedical
Environmental
25
Description
Multi-layered structures with enhanced contrast
[55]; Anti-reflection coatings [56]; Lasing structures
[57]; Light-based detectors for cancer diagnosis [see
Fig. 2.17(b)]; Surface-enhanced Raman spectroscopy
(SERS) [see Fig. 2.17(a)].
Displays with enhanced brightness [58]; Tunableconductivity materials [59].
Improved wear resistance [60]; New anti-corrosion
coatings [61]; New structural materials and composites [62].
Enhance heat transfer from solar collectors to storage
tanks [63].
MnO particles improve detailing and contrast in MRI
scans [64].
More durable batteries [65]; Hydrogen storage applications [66]; Electrocatalysts for high efficiency fuel
cells [67]; Higher performance in solar cells [41].
Antibacterial coatings [68]; Smart sensors for proteins [69].
Clean up of soil contamination and pollution, e.g. oil
[70]; Pollution sensors [71]; More efficient and effective water filters [72].
26
CHAPTER 2. PHOTONIC STRUCTURES
(a)
(b)
(c)
(d)
Figure 2.17: Examples of nanoparticle applications (a) Spacial distribution
of nanoparticle induced SERS enhancement for two coated silver nanospheres
(adopted from [73]). (b) Gold nanoparticles stick to cancer cells and make
them shine (adopted from www.gatech.edu/news-room/release.php?id=561).
(c) Scanning electron microscope image of the nanoparticle-structured band
filter (adopted from [74]) (d) Magnetic nanoparticles produced by ”NanoPrism
Technologies, Inc” for cell labeling, magnetic separation, biosensors, hyperthermia, magnetically targeted drug-delivery and magnetic-resonance imaging
(adopted from www.nanoprism.net/ wsn/page3.html).
Chapter 3
Computational techniques
3.1
Available techniques for studying light propagation in photonic structures
By far, the most popular method for theoretical description of light propagation in photonic systems is the finite-difference time-domain method (FDTD)
introduced by Yee [75]. The method is proven to be rather flexible and has been
successfully applied to study of microcavities and photonic crystal structures.
However, despite its speed and flexibility, the FDTD technique has a serious
limitation related to the finiteness of the computational domain. As a result,
an injected pulse experiences spurious reflections from the domain boundaries
that leads to mixing between the incoming and reflected waves. In order to
overcome this bottleneck a so-called perfectly matched layer condition has been
introduced [76]. However, even using this technique, a sizable portion of the
incoming flux can still be reflected back [77]. In many cases the separation of
spurious reflected pulses is essential for the interpretation of the results, and
this separation can only be achieved by increasing the size of the computational
domain. This may enormously enlarge the computational burden, as the stability of the FDTD algorithm requires a sufficiently small time step. A severe
disadvantage of this technique in application to microcavities with tiny surface
imperfections is that the smooth geometry of the cavity has to be mapped into
a discrete grid with very small lattice constant. This makes the application
of this method to the problems, when small imperfections are studied, rather
impractical in terms of both computational power and memory.
For studying microcavities, a number of boundary-element methods has
been applied. Their essence is that they reduce the Helmholtz equation in
infinite two-dimensional space into contour integral equations defined at the
cavity boundaries. These methods include the T -matrix technique [78; 79], the
27
28
CHAPTER 3. COMPUTATIONAL TECHNIQUES
boundary integral methods [80; 81] and others [82]. In general, they are computationally effective and capable to deal with cavities of arbitrary geometry.
However, they require the refraction index to be constant within the cavity.
Numerous theoretical approaches have been developed to calculate the photonic band structure for 2D and 3D photonic crystals. The plane-wave method
[83; 84; 85], for instance, allows one to calculate the band structures of PCs
having known their Brillouin zones. Unfortunately, despite its simplicity for
the implementation and stability, the method is not suitable for dispersive
materials (for the dispersive media, a revised plane-wave technique has been
developed [86]). Moreover, for complex structures (involving e.g. waveguides,
cavities or surfaces) a large supercell has to be chosen that strongly increases
the number of plane waves in the expansion and makes the method extremely
computationally consuming.
The problem of the spurious reflections from the computational domain
boundaries does not arise in methods based on the transfer-matrix technique
[87] where the transfer matrix relates incoming and outgoing fields from one
side of the structure to those at another side. However, such the mixing leads
to divergence of the method. The scattering-matrix (SM) techniques [88; 89;
90; 91], in contrast, are free of this drawback, as the scattering matrix relates
incident and outgoing fields and their mixing is avoided. The other approaches,
free of spurious reflections, are e.g. the multiple multipole method [46; 92] and
the dyadic Green’s function method [93; 94; 95; 96] based on the analytical
expression for the Green’s function of an empty space. This method will be
described in more detail in Section 3.4.
In this Chapter we present the developed scattering matrix technique for
studying whispering-gallery mode disk microcavities with imperfect circumference and variable refraction index, the 2D recursive Green’s function technique
for a scattering problem in photonic crystals and plasmonic structures, and the
3D dyadic Green’s function technique.
3.2
Scattering matrix method
In this Section we present a method dedicated for calculation of resonant states
in dielectric disk microcavities. The motivation of the development of this
technique was that there are no theoretical tools so far, which are able to
study microcavities both with tiny surface roughness and refraction index inhomogeneities. The method is capable to handle cavities with the boundary
roughness as well as inhomogeneous refraction index. Because the majority
of experiments are performed only with the lowest transverse mode occupied,
the transverse (z-) dependence of the field is neglected and computations are
performed in 2D. The two-dimensional Helmholtz equation for z-components
3.2. SCATTERING MATRIX METHOD
29
of electromagnetic field reads as
2
1 ∂
1 ∂2
∂
+
+
Ψ(r, ϕ) + (kn)2 Ψ(r, ϕ) = 0,
∂r2
r ∂r r2 ∂ϕ2
(3.1)
where Ψ = Ez (Hz ) for TM (TE)-modes, n is a refraction index and k is a
wavevector in vacuum. Remaining components of the electromagnetic field can
be derived from Ez (Hz ) in a standard way.
A
B
∆i
R
i-th boundary
∆i
ai
ai+1
i
bi+1
ri
d
b
i-th strip
(a)
(i+1)-th strip
(b)
Figure 3.1: (a) Sketch of the geometry of a cavity with refraction index n
surrounded by air. The domain is divided in three regions. In the inner (r < d)
and in the outer regions (r > R) the refraction indexes are constant. In the
intermediate region d < r < R refraction index n is a function of both r and ϕ.
(b) The intermediate region is divided by N concentric rings of the width 2∆;
ρi is a distance to the middle of the i-th ring. Within each ring the refraction
coefficient is regarded as a function of the angle only and a constant in r. States
ai , ai+1 propagate (or decay) towards the i-th boundary, whereas states bi , bi+1
propagate (or decay) away of this boundary. The i-th boundary is defined as
the boundary between the i-th and (i + 1)-th rings.
The system is divided into three regions, the outer region, (r > R), the
inner region, (r < d), and the intermediate region, (d < r < R), see Fig.
3.1(a). We choose R and d in such a way that in the outer and the inner
regions the refraction indexes are constant whereas in the intermediate region
n is a function of both r and ϕ. In these regions the solutions to the Helmholtz
equation can be written in analytical forms
Ψin =
+∞
X
q=−∞
a0q Jq (nkr)eiqϕ ,
(3.2)
30
CHAPTER 3. COMPUTATIONAL TECHNIQUES
for the inner region, where Jq is the Bessel function of the first kind, and
Ψout =
+∞ X
q=−∞
Aq Hq(2) (kr) + Bq Hq(1) (kr) eiqϕ ,
(1)
(3.3)
(2)
for the outer region, where Hq , Hq are the Hankel functions of the first and
second kind of order q, describing incoming and outgoing waves respectively.
Scattering matrix S is defined in a standard formulation [97; 98]
B = SA,
(3.4)
where A, B are column vectors composed of expansion coefficients Aq , Bq in
Eq. (3.3). Matrix element Sq′ q = (S)q′ q gives a probability amplitude of the
scattering from incoming state q into outgoing state q ′ .
The intermediate region is divided into narrow concentric rings where the
refraction index depends only on angle ϕ [outlined in Fig. 3.1(b)]. The solutions
to the Helmholtz equation in these rings can be expressed as superpositions of
cylindrical waves. At each i-th boundary between the strips we define a local
scattering matrix, which connects states propagating (or decaying) towards the
boundary with those propagating (or decaying) outwards the boundary as
i
i
b
a
i
=
S
.
(3.5)
bi+1
ai+1
Local scattering matrices Si are derived using the requirement of the continuity
of the tangential components for the Ez - and Hz -fields at the i-th boundary.
The essence of the scattering matrix technique is the successive combination
of the scattering matrices in the neighboring regions. Thus, combining the
scattering matrices for the i-th and (i + 1)-th boundaries, Si and Si+1 , one
obtains aggregate scattering matrix S̃i,i+1 = Si ⊗Si+1 that relates the outgoing
and incoming states in rings i and i + 2 [97; 98]
i
i
b
a
i,i+1
=
S̃
,
(3.6)
bi+2
ai+2
−1 i
S̃i,i+1
= Si11 + Si12 Si+1
I − Si22 Si+1
S21 ,
11
11
11
−1 i+1
i,i+1
i+1 i
i
S̃12
= S12 I − S11 S22
S12 ,
−1
S̃i,i+1
= Si+1
I − Si22 Si+1
Si21 ,
21
21
11
−1 i i+1
i+1
S̃i,i+1
= Si+1
I − Si22 Si+1
S22 S12 ,
22
22 + S21
11
where matrices S11 , S12 , . . . define the respective matrix elements of block matrix S. Combining all the local matrices 0 ≤ i ≤ N in this manner one finally
3.2. SCATTERING MATRIX METHOD
31
obtains total matrix S̃0,N = S0 ⊗ S1 ⊗ . . . SN relating the scattering states in
the outer region (i = N ) and the states in the inner region (i = 0), which after
straightforward algebra is transformed to matrix S Eq. (3.4).
The scattering matrix provides complete information about the system under study. In order to identify resonances, one introduces the Wigner time-delay
matrix [4] averaged over incoming states as
τD (k) =
1 d
ln[det(S)],
icM dk
(3.7)
where M is a number of the incoming states. It is interesting to note that
Smith in his original paper, dealing with quantum mechanical scattering [4],
chose a letter ”Q” to define the lifetime matrix of a quantum system because of
a close analogy to the definition of the Q factor of a cavity in electromagnetic
theory. The resonant states of the cavity are manifested as peaks in the delay
time whose positions determine the resonant frequencies ωres , and the heights
are related to the Q value of the cavity according to (2.5).
3.2.1
Application of the scattering matrix method to quantum-mechanical problems
The developed scattering-matrix method was generalized to quantum-mechanical
problems. This is possible thanks to the direct similarity between the Helmholtz
and Schrödinger equations [98]:
Photons
Electrons
∇2 E = −ω 2 εE
→
∇2 Ψ = −2m/~2[E − U ]Ψ
E
→
Ψ
Polarization
→
Spin
S ∼ ℜ[−iE ∗ × (∇ × E)]
→
J ∼ ℜ[−iΨ∗ ∇Ψ]
exp(−iωt)
→
exp(−iEt/~)
The method solves a problem of quantum-mechanical (QM) scattering in
quantum corral structures [99; 100], which can be considered as QM analogues
of disk microcavivies. We calculate scattering wave function, from which one
can extract spectra and the differential conductance dI/dV of the STM tunnel
32
CHAPTER 3. COMPUTATIONAL TECHNIQUES
junction [which is proportional to the local density of states (LDOS)]
X
dI/dV ∼ LDOS(r, E) =
|ψq (r)|2 δ(E − Eq ),
(3.8)
q
where ψq (r) are scattering eigenstates of Hamiltonian Ĥ. The advance of the
method is its ability to treat a realistic smooth potential within the corral
structure.
3.3
Green’s function technique
In order to study light propagation in 2D photonic-crystal structures, we have
developed a novel recursive Green’s function technique. In contrast with the
FDTD methods, the presented Green’s function technique is free from spurious
reflections. The Green’s function of a photonic structure is calculated recursively by adding slice by slice on a basis of the Dyson’s equation that relaxes
memory requirements and makes the method easy-parallelizable. In order to
account for the infinite extension of the structure into both the air and space
occupied by the photonic crystal we make use of so-called ”surface Green’s
functions” that propagate the electromagnetic fields into (and from) infinity.
The method is widely used in quantum-mechanical calculations [101] and is
unconditionally stable.
We start from Helmholtz equation, which for the 2D case (permittivity ε(r)
is constant in the z-direction) decouples in two sets of equations for the TE
modes
∂ 1 ∂
ω2
∂ 1 ∂
Hz +
Hz + 2 Hz = 0
(3.9)
∂x εr ∂x
∂y εr ∂y
c
and for the TM modes
1
εr
∂ 2 Ez
∂ 2 Ez
+
∂x2
∂y 2
+
ω2
Ez = 0.
c2
(3.10)
Let us now rewrite equations (3.9), (3.10) in an operator form [102]
Lf =
ω 2
c
f
(3.11)
where Hermitian differential operator L and function f read
TE modes: f ≡ Hz ; LT E = −
TM modes: f =
√
εr Ez ; LT M
∂ 1 ∂
∂ 1 ∂
−
,
∂x εr ∂x ∂y εr ∂y
2
1
∂
∂2
1
= −√
+ 2 √ .
εr ∂x2
∂y
εr
(3.12)
(3.13)
3.3. GREEN’S FUNCTION TECHNIQUE
33
For the numerical solution, Eqs. (3.11)-(3.13) have to be discretized, x, y →
m∆, n∆, where ∆ is a grid step. Using the following discretization of the
differential operators in Eqs. (3.12),(3.13),
∂
∂f (x)
ξ(x)
→ ξm+ 12 (fm+1 − fm ) − ξm− 12 (fm − fm−1 ) ,
∂x
∂x
∂2
∆2 2 ξ(x)f (x) → ξm+1 fm+1 − 2ξm fm + ξm−1 fm−1
∂x
∆2
(3.14)
one arrives to finite difference equation
vm,n fm,n − um,m+1;n,n fm+1,n − um,m−1;n,n fm−1,n −
2
ω∆
fm,n ,
−um,m;n,n+1 fm,n+1 − um,m;n,n−1 fm,n−1 =
c
(3.15)
where coefficients v, u are defined for the cases of TE and TM modes as follows
TE modes: fm,n = Hz m,n ; ξm,n =
1
εr m,n
,
(3.16)
vm,n = ξm+ 12 ,n + ξm− 12 ,n + ξm,n+ 12 + ξm,n− 12 ,
um,m+1;n,n = ξm+ 12 ,n , um,m−1;n,n = ξm− 21 ,n ,
um,m;n,n+1 = ξm,n+ 12 , um,m;n,n−1 = ξm,n− 12 ;
TM modes: fm,n =
√
εr m,n Ez m,n ; ξm,n = √
1
εr m,n
(3.17)
2
vm,n = 4ξm,n
,
um,m+1;nn = ξm,n ξm+1,n , um,m−1;nn = ξm−1,n ξm,n ,
um,m;n,n+1 = ξm,n+1 ξm,n , um,m;n,n−1 = ξm,n ξm,n−1 .
A convenient and common way to describe finite-difference equations on a
discrete lattice is to introduce the corresponding tight-binding operator. For
this purpose one first introduces creation and annihilation operators, a+
m,n ,
am,n . Let the state |0i ≡ |0, . . . , 0m,n , . . . , 0i describe an empty lattice, and
state |0, . . . 0, 1m,n , 0, . . . , 0i describes an excitation at site m, n. Operators
a+
m,n , am,n act on these states according to rules [101]
a+
m,n |0i = |0, . . . 0, 1m,n , 0, . . . , 0i,
a+
m,n |0, . . . 0, 1m,n , 0, . . . , 0i
= 0,
(3.18)
34
CHAPTER 3. COMPUTATIONAL TECHNIQUES
a+m+1,n am,n
n
n+1
m m+1
Figure 3.2: Forward hopping term in Eq. (3.22).
and
am,n |0i = 0,
(3.19)
am,n |0, . . . 0, 1m,n , 0, . . . , 0i = |0i,
and they obey the following commutational relations
+
+
[am,n , a+
m,n ] = am,n am,n − am,n am,n = δm,n ,
[am,n , am,n ] =
+
[a+
m,n , am,n ]
(3.20)
= 0.
Consider an operator equation
b |f i =
L
ω∆
c
2
|f i,
where Hermitian operator
X
Lb =
(vm,n a+
m,n am,n −
(3.21)
(3.22)
m,n
+
− um,m+1;n,na+
m,n am+1,n − um+1,m;n,n am+1,n am,n −
+
− um,m;n,n+1a+
m,n am,n+1 − um,m;n+1,n am,n+1 am,n )
acts on state
|f i =
X
m,n
fm,n a+
m,n |0i.
(3.23)
The second and third terms in Eq. (3.22) correspond forward and backward
hopping between two neighboring sites of the discretized domain in the xdirection, and terms 4 and 5 denote similar hopping in the y-direction, see
Fig. 3.2. Substituting the above expressions for Lb and |f i into Eq. (3.21)
3.3. GREEN’S FUNCTION TECHNIQUE
1 … M
n
N
N-1
35
1 … M
n
N
N-1
I
I
T
R
T
R
2
1
2
1
-M+1
… -1 0 1 2 3 …
M
M+1
m
(a)
0 1 2 3 …
M
M+1
m
(b)
Figure 3.3: Schematic illustration of the system under study defined in a supercell of width N. The internal region of the structure occupies M slices. Two
representative cases are shown: (a) external regions are semi-periodic photonic
crystals with period M, (b) external regions represent a semi-infinite periodic
photonic crystal with period M to the right and air to the left. Arrows indicate
the directions on the incoming (I ), reflected (R) and transmitted (T ) waves.
and using the commutation relations and the rules Eqs. (3.18)–(3.20), it is
straightforward to demonstrate that operator equation (3.21) is equivalent to
finite difference equation (3.15).
Let us now specify structures under study. We consider light propagation
through a photonic structure defined in a supercell of width N , where one
assumes the cyclic boundary condition (i.e. row n = N + 1 coincides with row
n = 1). The photonic structure occupies a finite internal region consisting of
M slices (1 ≤ m ≤ M ).
The external regions are semi-infinite supercells extending into regions m ≤
0 and m ≥ M + 1. The supercells can represent air (or a material with the
constant refraction index) or a periodic photonic crystal. Figure 3.3 shows
two representative examples where (a) the semi-infinite waveguides represent
a periodic photonic crystal with period M, and (b) a photonic structure is
defined at the boundary between air and the semi-infinite photonic crystal.
b in a standard way
We define Green’s function of the operator L
2
b G(ω) = 1b ,
(ω∆/c) − L
(3.24)
where 1b is the unitary operator. The knowledge of the Green’s function allows
us to calculate the transmission and reflection coefficients. Indeed, let us write
down the solution of Eq. (3.21) as a sum of two terms, the incoming state |ψ i i
and the system response |ψi representing whether transmitted |ψ t i or reflected
36
CHAPTER 3. COMPUTATIONAL TECHNIQUES
Figure 3.4: Schematic illustration of the application of the Dyson’s equation
for calculation of the Green’s function for a composed structure consisting of
m+1 slices.
|ψ r i states, |f i = |ψ i i + |ψi. Substituting |f i into Eq. (3.21) and using formal
definition of the Green’s function Eq. (3.24), the solution of Eq. (3.21) can be
written in the form
b − (ω∆/c)2 |ψ i i.
|ψi = G L
(3.25)
Calculation of the whole structure starts from the internal region (i.e for
slices 1 ≤ m ≤ M in Fig. 3.3). The recursive technique based on the Dyson’s
equation is utilized, see Fig. 3.4. Our goal is to calculate the Green’s function
of the composed structure, Gm+1 , consisting of m + 1 slices. The operator
corresponding to this structure can be written down in the form
0
b m+1 = L
b 0m + b
L
l m+1 + Vb ,
(3.26)
0
b 0 and b
where operators L
l m+1 describe respectively the structure composed of
m
m slices and the stand-alone (m+1)-th slice, and Vb = Vb m,m+1 + Vb m+1,m is the
perturbation operator describing the hopping between the m-th and (m + 1)-th
slices,
Vb = Vb m+1,m + Vb m,m+1 .
(3.27)
b m+1 ,
Gm+1 = G0 + G0 VG
b 0,
Gm+1 = G0 + Gm+1 VG
(3.28)
The Green’s function of the composed structure, Gm+1 , can be calculated on
the basis of the Dyson’s equation:
3.3. GREEN’S FUNCTION TECHNIQUE
37
b0
where G0 is the unperturbed Green’s function corresponding to operators L
m
0
or b
l m+1 . Thus, starting from the Green’s function for the first slice g10 and
adding recursively slice by slice we are in the position to calculate the Green’s
function of the internal structure consisting of M slices. Explicit expressions
following from Eqs. (3.28) and used for the recursive calculations read as
0
0
Gm+1,m+1
= (I − gm+1
Um+1,m (G0m )m,m Um,m+1 )−1 gm+1
,
m+1
(3.29)
m+1,m+1
Gm+1,1
Um+1,m (G0m )m,1 ,
m+1 = Gm+1
m+1,1
0 1,1
G1,1
+ (G0m )1,m Um,m+1 Gm+1
,
m+1 = (Gm )
1,m+1
m+1,m+1
0 1,m
Gm+1 = (Gm ) Um,m+1 Gm+1
,
where the upper indexes define the matrix elements of the Green’s function.
The next step is attaching the left and right semi-infinite leads to the internal region. Starting with the left waveguide, one writes
b int+lef t = L
b int + L
b lef t + Vb ,
L
(3.30)
b int+lef t ,
Gint+lef t = G0 + G0 VG
(3.31)
b int+lef t , L
b int and L
b lef t describe respectively the system repwhere operators L
resenting the internal structure + the left waveguide, the internal structure, and
b describes the hopping between the
the left waveguide. Perturbation operator V
left waveguide and the internal structure. Applying then the Dyson’s equation
in a similar way as it has been described above,
we are in position to find Green’s function Gint+lef t of the system representing
the internal structure + the left waveguide. G0 in Eq. (3.31) is an ”unperturbed” Green’s function corresponding to the internal structure and the
semi-infinite waveguide (”surface Green’s function” Γ). The physical meaning
of the surface Green’s function Γ is that it propagates the electromagnetic fields
from the boundary slice of the semi-infinite waveguide (supercell) into infinity.
A method for calculation of the surface Green’s functions both for the case of a
semi-infinite homogeneous dielectrics, as well as for the case of a semi-infinite
photonic crystal in a waveguide geometry is given in Paper IV. Having calculated Green’s function Gint+lef t on the basis of Eq. (3.31), one proceeds in a
similar way by adding the right waveguide and calculating with the help of the
Dyson’s equation total Green’s function G of the whole system.
Having calculated matrix elements for the complete system, GM+1,0 , G0,M+1 ,
G , GM+1,M+1 , one can easily relate them to the transmission T and reflec0,0
38
CHAPTER 3. COMPUTATIONAL TECHNIQUES
tion R coefficients of the system (see Paper IV for details)
ΦM+1 T = −GM+1,0 (U0,1 Φ−M+1 Kl − Γl −1 Φ0 ),
0,0
Φ0 R = −G
(U0,1 Φ−M+1 Kl − Γl
−1
Φ0 ) − Φ0 ,
(3.32)
(3.33)
where Γl ≡ G0,0
wg is the left surface Green’s function, Kl and Φm are given by
the right-propagating Bloch eigenvectors kα+ and the corresponding eigenstates
φα
m,n in the waveguides and U0,1 is a hopping matrix between the 0-th and 1-st
slices.
3.4
Dyadic Green’s function technique
Introduced in the previous section the Green’s function technique is adapted
for the two-dimensional case. Extension of this method to the 3D-case is impractical as it would require too extensive computational resources.
There exists a number of techniques for 3D scattering problems. These
include the finite-difference time-domain method, [75; 47], the multiple multipole method [46; 92] and the Coupled-Dipole Approximation (CDA) method
[45], which we utilize in our work. The CDA method was further developed
by O.J.F. Martin et al. [93; 94; 95; 96] and received a name Dyadic Green’s
Function Technique. The advantage of this approach is that only the scatterer
is needed to be discretized. Moreover, this technique can be easily extended
to complex (stratified, anisotropic, etc.) backgrounds, by making use of the
corresponding free-space Green’s function.
The central and starting point of this technique is a volume integral equation formulation of the Maxwell’s equations (Lippmann-Schwinger equation).
The aim of the method is to calculate electric field E(r), scattered against
an object with volume V illuminated with incident field E0 (r). The object is
characterized by its complex dielectric function ε(r) and surrounded, in the
simplest case, by an infinite homogeneous background with permittivity ε0 .
This scatterer does not need necessary to be either homogeneous or isotropic.
Assuming exp(−iωt) time dependence of the electromagnetic waves and the
isotropic scatterer, Lippmann-Schwinger equation for optical fields reads
Z
E(r) = E0 (r) +
dr′ G0 (r, r′ ) · V (r)E(r′ ),
(3.34)
V
where V (r) is a hopping potential defined as
V (r) = k0 2 [ε(r) − ε0 ],
(3.35)
with k0 = 2π/λ0 a vacuum wavevector. G0 (r, r′ ) is the Green’s dyadic tensor
describing the background. It is obtained as a solution to the vectorial wave
3.4. DYADIC GREEN’S FUNCTION TECHNIQUE
εi
39
εB
Figure 3.5: Outline of the discretization scheme for the dyadic Green’s function
technique.
equation with a point source term [103] and analytically can be expressed as
G0 (r, r′ ) = (I +
∇∇ exp(ikB R)
)
,
4πR
kB 2
(3.36)
√
where I is the unit tensor, kB = k0 ε0 and R = |r − r′ |.
The integration in (3.34) is performed over volume V of the scatterer. Thus,
one can formally divide the calculation of the scattered field into two stages:
(1) The field is first calculated inside the scatterer and (2) having calculated
the field distribution inside the scatterer, one can easily obtain the field in any
point of space outside the scatterer making use of Eq. (3.34).
There exists a number of techniques for solving Eq. (3.34), including iterative methods [93] and the finite-element method (FEM) [104]. The first ones,
possessing the great unconditional stability and reduced storage needs, have,
however, low performance in comparison to direct linear solvers. The latter,
FEM technique, allows more fine dicretization scheme by the triagulation of
the volume. However, 3D FEM seems rather impractical even with modern
computer resources.
We have implemented a ”compromised” solver to Eq. (3.34), utilizing a
simple LU decomposition along with a ”smart” discretazation technique. We
split the scatterer into N cubic meshes with volumes Vi . The size of the mesh
is reduced towards the boundary in accordance to the procedure described in
[105] and presented in Fig. 3.5.
The discretized version of Eq. (3.34) is
X
Ei = E0 i +
G0 i,j · k0 2 (εi − ε0 )Vj Ej .
(3.37)
j
40
CHAPTER 3. COMPUTATIONAL TECHNIQUES
Equations (3.34) and (3.37) have, however, a singularity at r = r′ (i = j)
which can be avoided by removing the singularity point from the integration
volume and compensating this value by source dyad L = 1/3I. The contribution from the i-th volume can be integrated analytically by assuming this
volume to be spherical with radius Ri = [3/(4π)Vi ]1/3 . The contribution reads
as [95]
2
(3.38)
Mi = 2 [(1 − ik0 Ri ) exp(ik0 Ri ) − 1]I.
3k 0
Substituting both self-term (3.38) and the definition of the source dyad into
Eq. (3.37) we obtain
Ei = E0 i +
X
j6=i
G0 i,j ·k0 2 (εi −ε0 )Vj Ej +[Mi k0 2 (εi −ε0 )−L·
εi − ε0
]Ei . (3.39)
ε0
This equation describes the electric field inside the scattered and can be solved
with any appropriate linear solver. The resulting field distribution is then used
as an input to Eq. (3.37) for the calculation of the field anywhere outside the
scatterer.
In order to investigate a far-field response, we calculate scattering crosssection
Z
Z 2π
1 π
σ=
dθ sin θ
dϕ|E∞ (θ, ϕ)|2 ,
(3.40)
4π 0
0
where θ and ϕ are angular coordinates of a spherical coordinate system and
E∞ (θ, ϕ) is a far-field intensity on infinity
Z
E∞ (θ, ϕ) =
dr′ G∞ (θ, ϕ, r′ ) · V (r)E(r′ ).
(3.41)
V
Here G∞ (θ, ϕ, r′ ) is the Green’s propagator from a point within the scatterer
to a point located at (θ, ϕ) on infinity
G∞ (θ, ϕ, r′ ) =
1
(I − nn) exp(−ik0 n · r)
4π
where nn is a dyad product of vectors


sin θ cos ϕ
n =  sin θ sin ϕ  .
cos θ
(3.42)
(3.43)
We currently apply the dyadic Green’s function technique to study of light
scattering by 3D whispering-gallery microcavities and arrangements of nanoparticles. Some representative results are given in Fig. 3.6 that shows the scattering cross-section (a) and resonance electric field distribution (b) of a linear
3.4. DYADIC GREEN’S FUNCTION TECHNIQUE
41
1000
σ, nm 2
100
10
y
1
(a)
(b)
z
300 350 400 450 500 550 600 650 700
λ, nm
x
Figure 3.6: (a) Scattering cross-section of a 7-nanoparticle chain. The diameter
of nanoparticles is 50 nm, the inter-particle distance is 40 nm. (b) Electric field
distribution at the nanoparticle chain at 426 nm.
chain of seven 50-nm silver nanoparticles (inter-particle distance is 40 nm) in
vacuum.
The chain is illuminated from the left with the Ez -component of the field.
High field intensity between the particles characterizes the strong coupling at
the longitudinal resonance 426 nm. The position of the single-particle dipole
plasmonic resonance (∼ 360 nm) agrees very well with its analytical estimation
(2.18).
42
CHAPTER 3. COMPUTATIONAL TECHNIQUES
Chapter 4
Results
4.1
Effect of inhomogeneities on quality factors
of disk microcavities (Papers I, II)
The Q factor of a microdisk cavity is the most important parameter of the
structure. It is governed by a radiative leakage through the curved interface due
to diffraction. An estimation of the Q factor in an ideal disk cavity of a typical
diameter d ∼ 10µm for a typical WG resonance gives Q ∼ 1013 . At the same
time, reported experimentally measured values are typically in the range of
103 ∼ 104 [6] or even lower. Such the discrepancy may be attributed to different
factors such as side-wall imperfections, finite or inhomogeneous height of the
disk, non-uniform refraction index within the structure, effects of coupling to
the substrate or pedestal and others. Several experimental observations point
out the side-wall imperfections as the main factor affecting the Q value of
the cavity [9; 6; 10]. Accounting of these imperfections can be considered of
extreme importance for the design and tailoring of lasing microdisks.
Using the developed scattering-matrix method we have studied the effect of
cavity roughness and inhomogeneity of the refraction index on quality factors
of disk microcavities. A cavity with diameter d = 10µm and refraction index
hni = 1.8 is considered. Various studies indicate that a typical size of the sidewall imperfections can vary in the range of 5-300 nm (representing a variation of
the order of ∼0.05-1% of the cavity radius), but their exact experimental shape
is unfortunately not available. We thus model the interface inhomogeneities as
a superposition of random Gaussian deviations from the ideal circle of radius
R with maximal amplitude ∆r/2 and characteristic distance between the deviation maxima ∆l ∼ 2πR/50, see inset to Fig. 4.1(a). The imperfect region
is discretized into 100 concentric strips.
Figure 4.1(a) illustrates calculated Q values of the disk resonant cavity for
43
44
CHAPTER 4. RESULTS
different surface roughnesses ∆r in some representative wavelength interval
for the TM polarization. Note that we have studied a number of different
resonances and all of them showed the same trends described below.
3
-1
Ideal
20nm
50nm
100nm
200nm
TM 83,1
10
-3
10
T
Q
-7
∆l
r=
R=
5.
10
56,7~
56,7
T ch
~ T cur
0
µm
83,1
Tcur
-9
∆r
10
∆θch
-5
r
TM 56,7
r =1
.7 µ
m
r=
2.5
µm
10
10
2
-11
10
-13
10
sin (θ)
10
0.9 q=83
0.8
0.7
83,1
Tch
∆θch
q=56
0.6
0 1 2 3 4 5 6
∆θch
-15
627.5
628.0
628.5
629.0
λ (nm)
629.5
630.0
(a)
10
0.6
0.7
0.8
sin θ
0.9
(b)
Figure 4.1: (a) Dependencies Q = Q(λ) for two representative modes TM83,1
(high-Q mode) and TM56,7 (low-Q mode) for different surface roughness ∆r.
Inset sketches inhomogeneous surface geometry. (b) Dependence T = T (θ) for
several radii of curvature ρ according to Eq. (2.3). Inset shows a Poincaré SoS
for the states q = 83 and q = 56 for the cavity with ∆r = 0 (straight lines of
θ = const) and ∆r = 20nm.
The solid curve in Fig. 4.1(a) corresponds to the ideal disk cavity without
imperfections. The dependence of the averaged Q values on the surface roughness ∆r for several representative resonances is also given. A common feature
of all high-Q resonances is a drastic decrease of their maximal Q value that
occurs even for very small values of ∆r . λ/20. For example, the Q value of
resonant state TM83,1 drops from Q ≈ 1013 for an ideal disk to Q ≈ 103 for
surface roughness of only ∆r = 20 nm. However, the picture for low-Q states
is rather different. Low-Q resonances show a relatively slow decrease in their
Q values over the range of variation of ∆r. For example, for the same surface
roughness ∆r = 20 the Q value of resonant state TM56,7 decreases only by a
factor of 1.5, dropping from Q ≈ 300 to Q ≈ 200.
In order to understand this behavior we combine a Poincaré surface of section (SoS) method with analysis of ray reflection at a curved dielectric interface
[see Eq. (2.3)]. Figure 4.1(b) illustrates that transmission T , calculated by Eq.
(2.3), decreases exponentially as the difference between angle of incidence θ
and critical angle of incidence θc grows. Poincaré SoS represents dependence
of angle of incidence θ on the polar angle ϕ around the cavity, its detailed
definition and related discussion are given in Paper II. The inset to Fig. 4.1(b)
depicts the SoS for two states with q = 56 and 83 shown in Fig. 4.1(a), where
the initial angle of incidence θ0 of launched rays is related to the angular num-
4.2. QUANTUM CORRALS (PAPER III)
45
ber q. The SoS demonstrates that initially regular dynamics of an ideal cavity
(straight line) transforms into the chaotic one even for a cavity with maximum
83,1
roughness ∆r . 20nm. ∆Tch
in Fig. 4.1(b) indicates the estimated increase
in the transmission coefficient due to the broadening of the phase space, ∆θch ,
as extracted from the Poincaré SoS for the state with q = 83. This corresponds
to the decrease of ∆Q ∼ ∆T −1 ≈ 10−2 . This value is much smaller that the
actual calculated decrease of the Q factor for high-Q resonance TM83,1 .
In order to explain the rapid degradation of high-Q resonances, we focus
on another aspect of the wave dynamics. The imperfections at the surface
boundary introduce a local radius of surface curvature ρ that is smaller than
disk radius R [see inset in Fig. 4.1(a)]. One may thus expect that, with the
presence of a local surface curvature, the total transmission coefficient will be
determined by the averaged value of ρ rather than by disk radius R. Figure
4.1(b) outlines that the reduction of the local radius of curvature from 5µm
(ideal disk) to 1.7µm (∆r = 20nm) causes an increase of the transmission
coefficient by ∆Tcur ≈ 108 . This number, combined with the estimate based on
the change of ∆Tch ∼ 102 , is fully consistent with the Q factor decrease shown
in Fig. 4.1(a). We thus conclude that the main mechanism responsible for the
rapid degradation of high-Q resonances in non-ideal cavities is the enhanced
radiative decay through the curved surface because the effective local radius
(given by the surface roughness) is smaller that the disk radius.
In contrast, for the case of low-Q resonances change in the transmission coefficient due to the enhanced radiative decay ∆Tcur is of the same magnitude as
change ∆Tch due to the broadening of the phase space caused by the transition
to the chaotic dynamics (for the resonance TM56,7 in Fig. 4.1). Therefore, both
these factors play comparable roles in degradation of the low-Q WG resonances.
We have also studied the effect of the non-uniform refraction index within
the cavity and found that the decay of the Q factor in that case is of minor
importance in comparison to the case of rough surface.
4.2
Quantum corrals (Paper III)
We have adapted the scattering matrix technique developed for disk dielectric
microcavities to study of quantum-mechanical scattering in quantum corrals,
which can be considered as nanoscale analogues to dielectric cavities. Quantum
corrals consist of adatoms of noble metals or Fe, deposited by AFM on (111)
surface of Cu in a ring, triangle or other arrangements (see Fig. 4.2). Cu surface
states interact strongly with the adatoms, and the spatial variation of the
STM differential conductance reveals beautiful images of the surface standing
wave patterns in the quantum corrals. In addition, the experiments show a
series of remarkable resonant peaks in the energy spectrum of the differential
conductance dI/dV in the center of the structures. So far there have been
46
CHAPTER 4. RESULTS
Figure 4.2: Quantum corral structures. STM images, reported by IBM [106]
several reports on experimental and theoretical studies of the corral structures.
In order to describe the experimental observation [99; 100], Heller et al. [107]
have developed the multiple-scattering theory for surface electron waves in
quantum corrals. In that theory each adatom was treated as a point-like ”black
dot” δ-function potential supporting isotropic scattering of a standing wave.
The quantitative agreement with the experiment was achieved by considering
an additional inelastic channel of scattering to the bulk of the substrate. It was
also concluded that absorption is the dominant mechanism for the broadening
of the energy levels seen in the experiment. Their theory describes well the
spatial distribution of the wave function in the corrals, but overestimates the
broadening of the resonant peaks, especially at higher energies.
An alternative purely elastic scattering theory for the same quantum corral
structures was drawn by Harbury and Porod [108]. They described the adatoms
by finite-height solid potential barriers. Their findings suggest that the features
of the spectrum and the wavefunction distribution can be extremely sensitive
to the detailed shape of the scattering potential.
The advance of our scattering matrix method is that it can treat a realistic potential of the adatoms, their displacements or variety of inhomogeneities. Accounting for the realistic smooth shape of the scattering potential is known to be crucial for quantitative description of many phenomena in quantum nanostructures. The potential of each adatom is taken as
4.3. SURFACE-STATE LASERS (PAPER IV)
47
a Gaussian with half-width σ and height V0 centered at (x0 , yo ), V (x, y) =
V0 exp[−(x − x0 )2 /2σ 2 ] exp[−(y − y0 )2 /2σ 2 ].
Using the scattering matrix technique we calculate the bias voltage dependence and the spatial distribution of the local density of states (LDOS) for
60-Fe-adatom, 88.7-Å-radius circular quantum corrals reported by Heller et al.
[107]. Fig. 4.3 shows experimental and theoretical results for the voltage dependence (a) and the spatial distribution (b) of the LDOS. The Fe-adatoms are
located on the meshes of a 2.55Å triangular grid corresponding to the hexagonal Cu(111) lattice. The effective mass used in all the simulations is taken
m∗ = 0.361m0 and the electron band-edge energy E0 = 0.43 eV below the
Fermi energy of the electrons.
dI/dV (arb. units)
2.0
Experiment
Scatt. Matrix Method
Experiment
Scatt. Matrix Method
V=0.47
1.5
V=0.45
1.0
V=0.43
0.5
V=0.41
-0.4
-0.3
-0.2
-0.1
0.0
Bias potential (V)
0.1
0.2
-80 -60 -40 -20 0 20 40
Distance (Å)
(a)
60
80
(b)
Figure 4.3: (a) The experimental spectrum of differential conductance dI/dV in
the center of the 88.7-Å-radius 60-Fe-adatom circular quantum corral structure
on Cu(111) substrate (solid curve). Scattering matrix technique applied for a
smooth adatom potential with height V0 = 2.5eV and broadening σ = 1.52Å
(dashed line). (b) The experimental curves (solid lines) and results of the
scattering-matrix simulations (dashed line) for the local density of states subject to the tip position inside the circular corral for low bias voltages.
Our calculations prove the importance of the realistic potential and reproduce quantitatively all the experimental observations (see Paper III for the
details). This is in contrast to the previous theories (treating the adatoms as
point scatterers) that require additional inelastic channels of scattering into the
bulk in order to achieve an agreement with the experiment. Our findings thus
indicate that surface states are not coupled to the bulk electrons.
4.3
Surface-state lasers (Paper IV)
Surface states in photonic crystals can be exploited in a variety of lasing, sensing
and waveguiding applications. First we focus on the novel type of a lasing cavity
48
CHAPTER 4. RESULTS
that is situated on the surface of a photonic crystal and uses surface modes.
We consider a semi-infinite square-lattice photonic crystal composed of
cylinders with ε = 8.9 and diameter D = 0.4a (a is a lattice constant) in
an air background. In order to create a surface geometry sustaining surface
modes, we reduce the diameter of the rods in the outmost row to d = 0.2a.
This structure has the full fundamental bandgap for the TM-polarization in
the range of 0.33 < ωa/2πc < 0.44 and supports one surface mode, which
along with the band structure is depicted in Fig. 2.9(a). In order to create a
surface-state lasing cavity the modified surface region has to be confined along
the modified boundary that turns the surface mode into a resonant state with a
finite lifetime. For this sake a semi-infinite photonic crystal structure containing only a finite number N of the surface rods with reduced diameter d = 0.2a
was considered. These rods define a resonant cavity situated at the surface of
the photonic crystal as illustrated in the inset to Fig. 4.4(a) for the case of
N = 6. The strong confinement from three sides of the cavity is provided by
the photonic crystal operating in the bandgap.
(a)
(b)
Figure 4.4: (a) Lower panel: Spectrum of a surface state photonic bandgap
cavity. Inset illustrates a resonant cavity defined by the N = 6 surface rods
of smaller diameter d = 0.2a placed on the photonic crystal surface. Upper
panel: The dispersion relation for the surface state for the semi-infinite photonic
crystal. The dashed lines indicate the expected analytical resonant wave vectors
for the modes α = 5, 6, 7 and corresponding expected resonant frequencies. (b)
Lower panel: Calculated intensity of the Ez component for the 6-th mode shown
in Fig. 4.4(a). Upper panel: Expected field intensity at different rods is given
by the overlap of the 6-th eigenstate of the cavity with the actual positions of
the rods.
4.4. SURFACE-STATE WAVEGUIDES (PAPER V)
49
In order to calculate the quality factor of the structure at hand we apply the developed Green’s function technique. We illuminate the surface cavity with an incidence wave, which excites a resonant mode within the cavity,
compute the
R intensity of the field distribution and express the Q factor as
Q = ωΩ/(4 Sin dy) [102], where Ω characterizes the energy stored in the system and the integral over Sin is the incoming energy flux. It should be stressed
that the resonance Q value depends on the coupling of the surface state modes
with the outgoing radiation, and thus is independent on the incidence angle of
the incoming wave.
Figure 4.4(a) shows the calculated Q factor of the resonant cavity versus
the frequency of the illuminating light. In the given frequency interval there
are three lasing peaks with quality factors ∼ 105 . Note that these values might
underestimate the actual theoretical Q factors obtained within the present 2D
calculations, because even finer frequency steps in the vicinity of the resonances
are required for better resolution of the Q factors. The estimation of the
position of the peaks for the cavity at hand can be performed by making use a
formula for a Fabry-Perot resonator, whose resonant wavelengths are given by
λα = 2π/kα , with wavevector kα = πα/w. From the dispersion relation [upper
panel of 4.4(a)] it follows that only modes α = 5, 6, 7 are situated inside the
frequency interval where the surface mode exists. An estimation of the expected
positions for the resonant peaks for these modes is also given in the lower panel
of Fig. 4.4(a) where the discrepancy between the expected analytical and
calculated resonance frequencies does not exceed 0.5%.
Figure 4.4(b) depicts the intensity of the Ez component of the electromagnetic field for the resonance mode α = 6. As expected, for the TM-modes, the
field is localized in the cavity inside the rods, and the intensity decays very
rapidly both to the open space and to the crystal. The field intensity at different rods in the cavity is expected to be determined by the overlap of the α-th
eigenstate of the Fabry-Perot resonator with the actual positions of the rods in
the cavity. This overlap for the 6-th mode is also shown in Fig. 4.4(b), which
agrees perfectly with the actual calculated field intensity.
4.4
Surface-state waveguides (Paper V)
”Conventional” waveguides in photonic crystals represent line defects in periodic structures supporting guided Bloch modes whose frequency region lies in
the bandgap. These modes are strongly confined within the waveguide region
and can propagate lossless to substantial distances. Here we propose a novel
type of waveguiding structures, namely waveguides that operate on surface
modes of semi-infinite photonic crystals (surface modes, propagating along the
boundary are now waveguiding modes!) and are located on the surface of a PC.
Their ”non-traditional” location may open up new possibilities for design and
50
CHAPTER 4. RESULTS
operation of photonic structures for feeding and redistributing light in PCs.
Making use the Green’s function technique we have studied localization
properties of surface modes, their dispersion relations and an effect of inhomogeneities. The latter has been demonstrated as the one having the strongest
impact on the performance of lasing microcavities. One can expect then, that
imperfections in the shape of the rods, their displacement, or variations of the
refraction index throughout the crystal can significantly affect the waveguiding
efficiency of the surface modes.
1.0
T
0.6
0.4
0.6
Transmission
Velocity
v/c
0.8
0.8
0.4
0.2
0.2
0.34
0.36
0.38
0.40
0.42
ωa/2πc
Figure 4.5: Transmission coefficient (solid line, left axis) for the surface modes
propagating in a non-ideal surface-mode waveguide. Velocity of the surface
mode (dashed line, right axis) from Fig. 2.9. The inset shows the structure
under study, where the shaded regions denote ideal semi-infinite waveguides,
and the central region of the width of 5a represents an imperfect photonic
crystal where scattering of the Bloch surface state takes place.
We consider the semi-infinite photonic crystal from the previous Section
(ε = 8.9, D = 0.4a) with the infinitely long boundary consisting of rods of reduced diameter d = 0.2a (see the band structure in Fig. 2.9). In order to study
the effect of imperfections, the structure is split into three regions as shown
in the inset to Fig. 4.5. Two of them are left and right semi-infinite periodic
structures (perfect waveguides for surface modes), and the block of the PC in
between is an imperfect region. Utilization of the Green’s function technique
allows one to use surface Bloch modes as scattering states that propagate in
perfect waveguides from the infinity into the imperfect region where they undergo scattering. Obviously, in the case when the scattering region is absent
(perfect waveguides are attached to each other), the Bloch states propagate
freely without any losses. The imperfection is modelled by the discretization of
the scattering region differently from that in the left and right perfect waveguides.
The transmission coefficient in Fig. 4.5 for the surface mode drops quite
4.4. SURFACE-STATE WAVEGUIDES (PAPER V)
51
0.8
T
0.6
0.4
0.2
0.34 0.35 0.36 0.37 0.38
ω a/2πc
Figure 4.6: A lead-in coupler composed of a surface-state waveguide to the left
and a conventional tapered PC waveguide to the right. The size of the surface
rods gradually decreases to zero in the central region where the surface-state
waveguide transforms in a conventional PC waveguide. Intensity distribution
is shown for Ez -component of the electromagnetic field at ωa/2πc ≈ 0.365.
Arrows sketch the flow of the Poynting vector. The inset shows the transmission
coefficient subject to the energy of incoming light. Parameters of the photonic
crystal correspond to those of Section 4.3.
rapidly in the energy regions corresponding to the low velocity of the surface
state (dashed line in the Figure). This is because the backscattering probability
is greatly enhanced for the low-velocity states. Even for 5 imperfect unit cells
the transmission coefficient in energy region ωa/2πc & 0.40 is less than 1 which
makes this energy region to be hardly appropriate for waveguiding purposes.
For lower energies the structure seems to be a good candidate for a waveguide.
In order to illustrate possible applications of surface-state waveguides two
novel devices have been proposed. The first one is a light coupler that enables
feeding external light into photonic-crystal waveguides. Fig. 4.6 illustrates the
proposed structure.
In this device the external light first couples to the surface-state region,
then the surface mode enters a tapered region, where it adiabatically (the
diameter of the surface rods in the surface-state waveguide gradually decreases
to zero) is transformed into a conventional waveguiding state. The maximum
achieved transmission reaches T ≈ 0.8 (see the inset to Fig. 4.6), which is even
higher than the transmission in conventional tapered [109] or mode-matched
structures [110; 111].
The second device that we propose is a directional beamer. We demonstrate
that directional emission with the angular spread much less than in conven-
52
CHAPTER 4. RESULTS
tional waveguides can also be achieved for the case of surface-state waveguides
coupled to air. Figure 4.7 shows the Ez field intensity (a) and the directional
diagram (b) for the surface state propagating in a semi-infinite waveguide.
3.5
(a)
1.0
ωa/2πc
0.34
0.36
0.38
Sr (Θ)
T
0.6
Θ
2.5
0.2
0.34
1.5
0.38
ωa/2πc
0.42
0.5
2
|E z |, a.u.
0.0
0.5
1.0
1.5
2.0
2.5
-50
(b)
-30
-10 0 10
Θ, Degrees
30
50
Figure 4.7: (a) Intensity distribution for the Ez -component of the electromagnetic field in the surface-mode waveguide terminated to air for ωa/2πc = 0.34.
(b) Far-field radial component of the Poynting vector Sr (Θ) radiated out of a
surface-mode waveguide versus azimuthal angle Θ for different ωa/2π. Inset
shows the transmission coefficient for the surface state as a function of the
frequency.
The most of the beam intensity is localized within cone ∆Θ ∼ 20◦ . Such the
narrow beaming originates from the fact that the surface state is localized in
a wide spatial region near the surface . 10a (see discussion on the localization
in Paper V), as opposed to conventional waveguides, whose width is typically
∼ a and the corresponding diffraction cone is about 70◦ . The angular spread
λ
, is consistent with the calculated
in this case due to the diffraction, sin Θ ∼ 10a
far-field radial distribution of the Poynting vector. As the frequency of the
incoming light increases, the surface mode becomes more localized, and the
spread of the outgoing radiation increases. The effect of directional beaming
in surface-mode waveguides might find its practical application for integration
of PC-based devices with conventional fiber-optics.
4.5
Nanorod arrays (Paper VI)
In this section we will focus on light propagation in large arrays of infinitely
long nanorods. Prototypes of such the arrays have been recently fabricated
experimentally [112; 113]. These arrays represent randomly oriented or aligned
long rods (or spikes) of a material (dielectric or metal), several tens of nanometers in diameter. Nanorods can be considered as two-dimensional analogues of
nanoparticles in terms of the mode structure. This makes it possible to study
4.5. NANOROD ARRAYS (PAPER VI)
53
their properties with conventional effective-medium theories.
We consider a gelatin matrix (εmat = 2.25) with an embedded two-dimensional array of silver nanorods. The effective dielectric function of the composite can be estimated from Maxwell Garnett (MG) formula (2.19). Despite its
adequacy for small isolated circular nanoparticles, the simple MG theory, however, has certain limitations – it does not account for the shape and distribution
of the metal clusters in the dielectric medium, neglecting important polarization properties of both single non-circular particles and their arrangements.
In order to incorporate these features and study transmission characteristics of
periodic and disordered nanorod arrays we apply the recursive Green’s function
technique (see Section 3.3).
We consider 2D arrays of infinitely long silver nanorods arranged as a square
lattice in a gelatin background. Keeping the filling factor of Ag, f = 10%,
constant, we consider two cases, (a) a finite-size lattice with thickness a =
0.7µm of nanorods with diameter d = 10nm, and (b) the lattice of the same
thickness assembled from nanorods of 60 nm in diameter, see Fig. 4.8. Lattice
constants are 29 and 175 nm for cases (a) and (b) respectively.
Figure 4.8: Arrays of silver nanorods with diameter (a) 10 nm, and (b) 60 nm
embedded in an infinite gelatin background. For both cases thickness of the
layer a = 0.7µm and filling factor f =10%.
We perform numerical simulations for both the TE and TM polarizations
of light incident normally from the left to the boundary between gelatin and
the blend.
TE-modes
Let us irradiate the array of infinitely long nanorods with the TE-polarized
light. In this case the Ex and Ey components of the electromagnetic field excite coherent plasmonic oscillations on each nanorod. Figure 4.9 shows the
54
CHAPTER 4. RESULTS
calculated transmittance, reflectance and absorptance of the TE mode propagating through the arrays of nanorods.
Figure 4.9: Transmittance, reflectance and absorptance of the TE mode travelling through the square arrays of nanorods with diameter (a) 10 nm and (b)
60 nm (see Fig. 4.8 for details).
Small nanorods. Let us first concentrate on the array of nanorods with
diameter 10 nm [Fig. 4.9(a)]. In the spectra one can clearly distinguish two
regions, namely the region of high absorption (λ0 < 600 nm), containing a wide
main absorption peak at 414 nm, two minor peaks at 350 and 530 nm and the
region of high transmittance (λ0 > 600 nm). Now we will take a closer look at
these regions separately.
The position of the main extinction resonance agrees well with that obtained
from Eq. (2.19). However, in contrast to the Maxwell Garnett theory, the
spectrum contains two minor peaks at 350 and 530 nm. These peaks are
carefully studied and the results are presented in Paper VI. Rechberger [114]
has clarified the origin of these peaks in terms restoring forces between the
particles.
In order to understand the high transmittance at the wavelength region λ0 >
600 nm, we complement the transmission coefficient with the band diagram of
the nanorod array. The plasmonic band in this region is located very close
to the light line that results in a strong coupling between the incoming light
and the plasmonic Bloch states of the blend region and, consequently, high
transmittance.
Large nanorods The position of the main extinction peak for large nanorods
agrees with that one of the small particles. However, there is an essential difference in the physics behind. When the diameter of a nanoparticle increases,
higher-order dipole oscillations now contribute to the resulting extinction spectrum [44]. It has been recently shown [115] that the peak centered at ≈ 400
nm is due to the quadruple resonance of a nanorod, whereas the dipole reso-
4.5. NANOROD ARRAYS (PAPER VI)
55
nance is redshifted and overlaps with the region of the enhanced reflectance
(500 < λ0 < 700 nm). The indication in favor of this interpretation is the
narrower width of the stop-band in the transmission (60 nm against 100 nm
in the case of small rods). This is because the higher-order dipole interactions
causing the stop-band behavior for the case of large nanorods are generally
weaker.
The high-reflectance region is caused by the photonic band gap, as the
stricture has the lattice constant of the same order as the wavelength of light
and effectively represents a two-dimensional photonic crystal. The plasmonic
band extends from ωa/2πc = 0 to 0.4 (λ0 ≈ 660 nm) where it experiences
a photonic bandgap that causes the high reflectance of the structure. This
bandgap overlaps with the tail of the extinction peak near 500 nm (see Fig.
4.9).
Our numerical calculations outline the importance of geometrical factors
such as the size of the rods and their arrangement. In particular, we have
demonstrated that the interaction between adjacent nanorods brings the significant contribution to the transmission spectra, which is manifested as additional absorption peaks (that are missing in the effective-medium approach).
The Maxwell Garnett theory also disregards both the impacts of higher-order
dipole contributions and formation of photonic band gaps in the case of arrays
of larger nanorods.
TM-modes
Let us now consider the TM-polarization of the incoming light. Figure 4.10(a)
shows the transmittance, reflectance and absorptance of the TM-polarized light
for the small nanorods. Here, almost for the whole wavelength range under
study light does not penetrate the region occupied by the nanorods and gets
fully reflected back, resulting in zero transmittance. This effect can be explained by the skin-effect on the silver rods. Maxwell Garnett theory is not
able to describe the important screening properties of the rods, simply averaging
the effective dielectric constant over the structure. It is also worth mentioning,
that as we consider infinitely long nanorods, the incoming TM-mode does not
excite any plasmons on the rods and thus there is no a plasmonic contribution
in the overall transmission.
The obtained results clearly show that the resonant plasmonic oscillations in
periodic nanorod arrays represent a dominating light propagation mechanism
for the TE-polarized light, whereas for the TM modes the nanorod structure
represents a virtually perfect screen. This feature can be utilized in a nearly
100% effective polarizer.
56
CHAPTER 4. RESULTS
Figure 4.10: (a) Transmittance, reflectance and absorptance of the TM-mode
through a nanorod array of d = 10 nm. Due to the skin-effect light does not
penetrate the blend region. For λ0 < 328 nm the real part ε′ of the dielectric
function of silver (b) becomes positive and the transmission coefficient abruptly
increases.
4.6
Surface plasmons in polymeric solar cells
(Paper VII)
In this Section we utilize surface plasmons, excited at the interface of an aluminum grating and a blend of organic polymers, in order to increase efficiency
of polymeric solar cells. An increased absorption originating from surface plasmon resonances has been studied by the recursive Green’s function technique
and confirmed by experimental studies. We demonstrate that the presence of a
plasmon can lead to increased photocurrent from polymeric photovoltaic cells.
We consider a supercell (see Fig. 4.11), containing 5 periods of a sinusoidal
grating profile. In order to account for the non-ideality of the grating surface,
we have introduced a randomized surface roughness with the amplitude of ∼ 5
nm, which is a typical averaged AFM experimental value. We discretize the
structure with a uniform grid (element size is 3.4 nm) and illuminate it with
the TE-polarized light (the magnetic component of the field is parallel to the
grating lines). Having calculated transmittance T (3.32) and reflectance R
(3.33) from the recursive Green’s function technique, the absorptance can be
found as A = 100% − T − R.
We have studied separately two polymers, APFO3 [116] and APFO Green5
[117], blended with PCBM. The dielectric functions for the blends obtained by
ellipsometry is given in Paper VII and used as an input for the modelling. The
position of the plasmonic resonance for the blends was first estimated by Eq.
4.6. SURFACE PLASMONS IN POLYMERIC SOLAR CELLS (PAPER VII)57
Hz
Ex
ky
Air
Polymer Ti (optional) Al grating
Figure 4.11: The system under study. The supercell consists of 5 grating
periods (the width is 1385 nm, the grating period is 277 nm), discretized with
square grid (size of the grid element is 3.4 nm) and illuminated with the TEpolarized light.
(2.12). The absorptances of the blends on metal gratings calculated by the
recursive Greens function technique are given in Fig. 4.12.
First, we calculate the absorptance spectrum for APFO3/PCBM. The polymer blend with thickness d ≈ 150 nm is deposited directly onto the Al grating.
The simulated and measured absorptance spectra are given in Fig. 4.12(a).
The spectrum contains two resonance peaks at 450 and 625 nm, which agree
extremely well with the experimental curve. The two peaks have, however,
completely different nature. The peak at 625 nm is a first-order plasmonic
resonance and its position agrees quite well with direct analytical estimation
(598nm) from Eq. (2.12). At this resonance the electromagnetic field [inset,
Fig. 4.12(b)] is localized within the 50-nm region near the grating surface, and
its intensity is up to 7 times higher on the boundary than in the bulk of the
polymer. The resonant peak at 450 nm represents a standing wave confined
by the total internal reflection at the polymer/air interface and the reflecting
metal grating.
For APFO Green 5/PCBM we model a 90 nm thick polymer blend layer,
deposited onto the Al grating coated with a 5 nm thin Ti interfacial layer.
Fig. 4.12(b) represents the computed absorptance spectrum and the simulated
field distribution inside the cell. As the APFO Green 5/PCBM layer is thinner
than the APFO3/PCBM, the standing wave is no longer supported and there
is no the corresponding peak in the spectrum. The plasmonic resonance for
this structure is centered at 555 nm.
In order to estimate the effect of the plasmon on the photocurrent, the external quantum efficiency (EQE) has been measured under illumination of the
58
CHAPTER 4. RESULTS
(a)
(b)
Figure 4.12: (a) Calculated and measured absorptance spectrum of
APFO3/PCBM deposited directly on the Al grating. The insets show the calculated spatial distribution of the Hz -component of the electromagnetic field
in the polymer at the plasmonic resonance (625 nm) and for the standing
wave peak (450 nm). (b) Calculated absorptance spectrum of APFO Green
5/PCBM on the Al grating coated with a 5-nm thick Ti interfacial layer. The
inset demonstrates the spatial distribution of the Hz -component of the electromagnetic field in the polymer at the first-order plasmonic resonance (555 nm).
The lower dark part of the insets corresponds to the sinusoidal shaped metal
grating where no field is present.
sample with polarized light. For the TE-polarization, the APFO Green5/PCBM
solar cells demonstrate a clear ∼ 4% enhancement of the EQE in the vicinity of
the SP resonance (555 nm). However, for even stronger plasmonic resonance
in the APFO3/PCBM cell, no any indication of plasmon influence has been
observed. This discrepancy can be attributed to the mismatch in the energy
of the SP resonance which lies below the bandgap of APFO3/PCBM but but
above the gap for APFO Green5/PCBM. This mismatch leads to very weak
coupling from the SP to the excitation of APFO3/PCBM. All the measurements has been carried out by the Biomolecular and Organic Electronics group
of the Center of Organic Electronics, IFM at Linköping University.
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APPENDICES
69
I
Paper I
Effects of boundary roughness on a Q factor of
whispering-gallery-mode lasing microdisk
cavities
J. Appl. Phys., vol. 94, pp. 7929–7931, 2003
JOURNAL OF APPLIED PHYSICS
VOLUME 94, NUMBER 12
15 DECEMBER 2003
Effects of boundary roughness on a Q factor of whispering-gallery-mode
lasing microdisk cavities
A. I. Rahachou and I. V. Zozoulenkoa)
Department of Science and Technology (ITN), Linköping University, 601 74 Norrköping, Sweden
共Received 30 May 2003; accepted 23 September 2003兲
We perform numerical studies of the effect of sidewall imperfections on the resonant state
broadening of the optical microdisk cavities for lasing applications. We demonstrate that even small
edge roughness (ⱗ␭/30) causes a drastic degradation of high-Q whispering gallery 共WG兲-mode
resonances reducing their Q values by many orders of magnitude. At the same time, low-Q WG
resonances are rather insensitive to the surface roughness. The results of numerical simulation
obtained using the scattering matrix technique, are analyzed and explained in terms of wave
reflection at a curved dielectric interface combined with the examination of Poincaré surface of
sections in the classical ray picture. © 2003 American Institute of Physics.
关DOI: 10.1063/1.1625781兴
During recent years, significant experimental efforts
were put forward toward the investigation of laser emission
of dielectric and polymeric low-threshold microdisk
cavities.1–9 The high efficiency of lasing operation in such
devices is related to the existence of natural cavity resonances known as whispering-gallery 共WG兲 modes. The origin of these resonances can be envisioned in a ray optic
picture, wherein light is trapped inside the cavity through
total internal reflections on the cavity–air boundary.
One of the most important characteristics of cavity resonances is their quality factor (Q factor兲 defined as Q
⫽2 ␲ * 共stored energy兲/共energy lost per cycle兲. The Q factor
of a microdisk cavity is mostly governed by a radiative leakage through the curved interface due to diffraction. An estimation of the Q factor in an ideal disk cavity of a typical
diameter d⬃10 ␮ m for a typical WG resonance gives Q
⬃1013 关see below, Eq. 共4兲兴. At the same time, experimental
measured values reported so far are typically in the range of
103 – 104 or lower. A reduction of a Q factor may be attributed to a variety of reasons including side wall geometrical
imperfection, inhomogeneity of the height and diffraction index of the disk, effects of coupling to the substrate or pedestal, and others. Several experimental studies point out side
wall imperfections as the main factor affecting the Q factor
of the cavity.5–7 An indirect indication of the importance of
this factor in disk microcavities is provided by the observation that typical Q factors of spheroidal microcavities are
several orders of magnitude higher than those of microdisks
of comparable dimensions.1,10 This is believed to be due to
the superior quality of the microsphere surfaces where
boundary scattering may be limited by thermal fluctuations
of the surface only. Therefore, the effect of surface roughness
appears to be of crucial importance for the design, tailoring,
and optimization of Q values of lasing microdisk cavities. In
the present article, we provide a detailed numerical study of
this effect and analyze the obtained results in terms of the
wave and ray dynamics.
In order to compute the resonant states of a cavity of an
arbitrary shape, we develop an approach based on the scattering matrix technique. The scattering matrix technique is
widely used in analysis of waveguides,11 as well as in quantum mechanical simulations.12 This technique was also used
for an analysis of resonant cavities for geometries when the
analytical solution was available.13 Note that because the
problem at hand requires a fine discretization of the geometry, commonly used time-domain finite difference
methods14 would be prohibitively expensive in terms of both
computer power and memory. While a detailed description of
the calculations will be given elsewhere, we present here the
essence of the method.
We consider a two-dimensional cavity with the refraction index n surrounded by air. Because the majority of experiments are performed only with the lowest transverse
mode occupied, we neglect the transverse (z-) dependence
of the field and thus limit ourselves to the two-dimensional
Helmholtz equation. We divide our system into outer and an
inner regions. In the outer region, the refraction index n is
independent of the coordinate and the solution to the Helmholtz equation can be written in polar coordinates in the form
⫹⬁
⌿⫽
共1兲
where ⌿⫽E z (H z ) for transverse magnetic 共TM兲 关transverse
(2)
electric 共TE兲兴 modes, H (1)
q ,H q are the Hankel functions of
the first and second kind of order q describing, respectively,
incoming and outgoing waves, k⫽ ␻ /c⫽2 ␲ /␭.
We define the scattering matrix S in a standard
fashion,11–13 B⫽SA, where A,B are column vectors composed of the expansion coefficients A q ,B q for incoming and
outgoing states in Eq. 共1兲. The matrix element Sq ⬘ q gives the
probability amplitude of scattering from an incoming state q
into an outgoing state q ⬘ . In order to apply the scattering
a兲
Electronic mail: [email protected]
0021-8979/2003/94(12)/7929/3/$20.00
(1)
iq ␸
,
兺 共 A q H (2)
q 共 kr 兲 ⫹B q H q 共 kr 兲兲 e
q⫽⫺⬁
7929
© 2003 American Institute of Physics
Downloaded 10 Dec 2003 to 130.236.132.220. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
7930
J. Appl. Phys., Vol. 94, No. 12, 15 December 2003
A. I. Rahachou and I. V. Zozoulenko
FIG. 1. 共a兲 Dependence of the quality factor Q⫽Q(␭) of the circular disk
for different surface roughness ⌬r indicated in the figure; the disk radius is
R⫽5 ␮ m, the refraction index n⫽1.8. The inset illustrates a cavity where
the surface roughness ⌬r⫽200 nm and ⌬l⫽2 ␲ R/50 共the dotted line represents an ideal circular boundary, the shaded region corresponds to the cavity兲. ␳ characterizes the average radius of local curvature due to boundary
imperfections. TE modes of the cavity exhibit similar features and are not
shown here.
matrix technique, we divide the inner region into N narrow
concentric rings. At each ith boundary between the rings, we
introduce the scattering matrix Si that relates the states
propagating 共or decaying兲 toward the boundary, with those
propagating 共or decaying兲 away from the boundary. The matrices Si are derived using the requirement of continuity of
the tangential components for the E and H field at the
boundary between the two dielectric media. Successively
combining the scattering matrixes for all the boundaries,11,12
S1 丢 . . . 丢 SN, we can relate the combined matrix to the scattering matrix S.
To identify the resonant states of a resonant cavity, we
introduce the Wigner–Smith time-delay matrix Q
⫽ i/c(dS† /dk)S, 2,13,15 where the diagonal elements Qqq give
a time delay experienced by the wave incident in qth channel
and scattered into all other channels. The Q value of the
cavity is Q⫽ ␻ ␶ D (k), where ␶ D (k) is the total time delay
averaged over all M incoming channels,2,13,16
M
␶ D共 k 兲 ⫽
M
d␪␮
1
1
1 d␪
Q ⫽
⫽
,
M q⫽1 qq cM ␮ ⫽1 dk
cM dk
兺
兺
共2兲
exp(i␪␮)⫽␭␮ are the eigenvalues of the scattering matrix S,
␪ ⫽ 兺 ␮N ⫽1 ␪ ␮ is the total phase of the determinant of the matrix S, and det S⫽⌸ ␮M⫽1 ␭ ␮ ⫽exp(i␪).
Figure 1 shows calculated Q values of the disk resonant
cavity for different surface roughnesses for TM modes in
some representative wavelength intervals. Note that an exact
experimental shape of the cavity–surface interface is not
available. We thus model the interface shape as a superposition of random Gaussian deviations from an ideal circle of
radius R with a maximal amplitude ⌬r/2 and a characteristic
distance between the deviation maxima ⌬l⬃2 ␲ R/50 共see
illustration in inset to Fig. 1兲.
The solid curve in Fig. 1 corresponds to an ideal disk
cavity without imperfections. Resonant states of an ideal disk
共as well as the bound states of the corresponding closed resonator兲 are characterized by two numbers, q 关see Eq. 共1兲兴 and
FIG. 2. Dependence Q on the surface roughness ⌬r for several representative resonances. 共Each curve remains practically unchanged for different
realizations of surface roughness兲. The inset shows the dependence of local
radius of roughness curvature ␳ subject to ⌬r. Parameters of the cavity are
the same as in Fig. 1.
m. The index m is a radial wave number that is related to the
number of nodes of the field components in the radial direction r. The angular wave number q can be related to the
angle of incidence ␹ in a classical ray picture2
q⫽nkR sin ␹ .
共3兲
The dependence of the averaged Q values on the surface
roughness ⌬r is summarized in Fig. 2 for several representative resonances. A common feature of all high-Q resonances is a dramatic decrease of their maximal Q-value that
takes place for very small values of ⌬rⱗ␭/20. For example,
a Q value of at the resonant state TM83,1 drops from Q
⬇1013 for an ideal disk to Q⬇103 for a surface roughness of
only ⌬r⫽20 nm. In contrast, low-Q resonances show a
rather slow decrease in their Q values over the range of
variation of ⌬r. For example, for the same surface roughness
⌬r⫽20 nm, the Q value of the resonant state TM56,7 decreases only by a factor of 1.5, dropping to Q⬇200.
In order to understand these features, we combine a
Poincaré surface of section 共SOS兲 method with an analysis of
ray reflection at a curved dielectric interface.18 The Q value
of the cavity can be related to the transmission probability T
of an electromagnetic wave incident on a curved interface of
radius ␳ by Q⫽2nk ␳ cos ␹/T 共this expression is valid for
large angles of incidence ␹ when TⰆ1).17 In turn, for kn ␳
Ⰷ1, the transmission probability reads18
冋
T⫽ 兩 T F 兩 exp ⫺
册
2 nk ␳
共 cos2 ␹ c ⫺cos2 ␹ 兲 3/2 ,
3 sin2 共 ␹ 兲
共4兲
where T F is the classical Fresnel transmission coefficient for
an electromagnetic wave incident on a flat surface, ␹ c
⫽arcsin(1/n) is an angle of total internal reflection. Figure 3
illustrates that T decreases exponentially as the difference
␹ ⫺ ␹ c grows. It should also be noted that the roughness
⌬r⬃100– 200 nm 共i.e., of the order of ␭/5) almost completely suppress the resonances, which would diminish the
lasing action of the cavity.
The inset to Fig. 3 depicts the Poincaré SOS for two
states with q⫽56 and 83 shown in Fig. 1, where the initial
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J. Appl. Phys., Vol. 94, No. 12, 15 December 2003
FIG. 3. Dependence T⫽T( ␹ ) for several radii of curvature ␳ according to
Eq. 共4兲. Inset shows a Poincaré SOS for the states q⫽83 and q⫽56 for the
cavity with ⌬r⫽0 共straight lines of ␹ ⫽constant) and ⌬r⫽20 nm. The
number of bounces for a given angle of incidence ␹ 0 is chosen in such a
way that the total path of the ray does not exceed the one extracted from the
numerically calculated Q value for the corresponding resonance, L⫽c ␶ D
⫽Q/k.
angle of incidence ␹ 0 of launched rays is related to the angular number q by Eq. 共3兲. The SOS demonstrates that initially regular dynamics of an ideal cavity transforms into a
chaotic one even for a cavity with maximum roughness ⌬r
ⱗ20 nm. ⌬T 83,1
ch in Fig. 3 indicates the estimated increase in
the transmission coefficient due to the broadening of the
phase space, ⌬ ␹ ch , as extracted from the Poincaré SOS for
the state with q⫽83. This corresponds to the decrease of
⌬Q⬃⌬T ⫺1 ⬇10⫺2 . This value is much smaller that the actual calculated decrease of the Q factor for the high-Q resonance TM83,1 .
To explain the rapid degradation of high-Q resonances,
we concentrate on another aspect of the wave dynamics.
Namely, the imperfections at the surface boundary effectively introduce a local radius of surface curvature ␳ that is
distinct from the disk radius R 共see illustration in Fig. 1兲.
One may thus expect that with the presence of a local surface
curvature, the total transmission coefficient will be determined by the averaged value of ␳ rather than by the disk
radius R. The dependence of ␳ on surface roughness ⌬r for
the present model of surface imperfections is shown in the
inset to Fig. 2. Figure 3 demonstrates that the reduction of
the local radius of curvature from 5 ␮m 共ideal disk兲 to 1.7
␮m (⌬r⫽20 nm) causes an increase of the transmission coefficient by ⌬T cur⬇108 . This estimate, combined with the
estimate based on the change of ⌬T ch is fully consistent with
the Q-factor decrease shown in Figs. 1 and 2. We thus conclude that the main mechanism responsible for the rapid degradation of high-Q resonances in nonideal cavities is the enhanced radiative decay through the curved surface because
the effective local radius 共given by the surface roughness兲 is
smaller that the disk radius R.
For the case of low-Q resonances, the change in the
A. I. Rahachou and I. V. Zozoulenko
7931
transmission coefficient due to enhanced radiative decay
⌬T cur is of the same magnitude as the change ⌬T ch due to
the broadening of the phase space caused by the transition to
chaotic dynamics 共as illustrated in Fig. 3 for the resonance
TM56,7). Therefore, both of these factors play comparable
roles in degradation of the low-Q WG resonances.
It is worth mentioning that one often assumes that longlived high-Q resonances in idealized cavities 共e.g., in ideal
disks, hexagons, etc.兲 are not important for potential application in optical communication or laser devices.13,19 because
of their extremely narrow width. Our simulations demonstrate that it is not the case because, in real structures, the Q
values of these resonances becomes comparable to those of
intermediate-Q resonances already for small or moderate
surface roughness of ⌬r⬃10– 50 nm.
To conclude, our results pinpoint the surface roughness
as a major factor affecting the performance of microdisk
cavities and outline its role for device design and fabrication.
In particularly, we demonstrate that even small surface
roughness of the order ⌬r⬃␭/5 can cause the drastic degradation of the cavity modes, completely suppressing the lasing effect of the disk resonators.
The authors thank Olle Inganäs for stimulating discussions that initiated this work and Stanley Miklavcic and
Sayan Mukherjee for many useful discussions and conversations. One of the authors 共A.I.R.兲 acknowledges financial
support from SI and KVA.
Y. Yamamoto and R. E. Slusher, Phys. Today, 66 共1993兲; S. Arnold, Am.
Sci. 89, 414 共2001兲.
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Zee and J. P. Looney 共Academic, San Diego, 2002兲, pp. 185–226.
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B. Gayral, J. M. Gérard, A. Lemaı̂tre, C. Dupuis, L. Manin, and J. L.
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C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O.
Piquet, P. Blondy, D. Cros, and O. Marty, J. Appl. Phys. 88, 6170 共2000兲.
8
M. Theander, T. Granlund, D. M. Johanson, A. Ruseckas, V. Sundström,
M. R. Andersson, and O. Inganäs, Adv. Mater. 共Weinheim, Ger.兲 13, 323
共2001兲.
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R. C. Polson, Z. Vardeny, and D. A. Chinn, Appl. Phys. Lett. 81, 1561
共2002兲.
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V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, Opt. Lett. 26,
256 共2001兲.
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V. V. Nikolsky and T. I. Nikolskaya, Decomposition Approach to the Problems of Electrodynamics 共Nauka, Moskow, 1983兲 共in Russian兲.
12
S. Datta, Electronic Transport in Mesoscopic Systems 共Cambridge University Press, Cambridge, UK, 1995兲.
13
M. Hentschel and K. Richter, Phys. Rev. E 66, 056207 共2002兲.
14
B.-J. Li and P.-L. Liu, IEEE J. Quantum Electron. 33, 1489 共1997兲.
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II
Paper II
Scattering matrix approach to the resonant
states and Q values of microdisk lasing cavities
Appl. Opt., vol. 43, pp. 1761–1772, 2004
Scattering matrix approach to the resonant states
and Q values of microdisk lasing cavities
Aliaksandr I. Rahachou and Igor V. Zozoulenko
We develop a scattering matrix approach for the numerical calculation of resonant states and Q values
of a nonideal optical disk cavity with an arbitrary shape and with an arbitrary varying refraction index.
The developed method is applied to study the effect of surface roughness and inhomogeneity of the
refraction index on Q values of microdisk cavities for lasing applications. We demonstrate that even
small surface roughness 共⌬r ⱗ ␭兾50兲 can lead to a drastic degradation of high-Q cavity modes by many
orders of magnitude. The results of the numerical simulation are analyzed and explained in terms of
wave reflection at a curved dielectric interface, combined with an examination of Poincaré surfaces of
section and of Husimi distributions. © 2004 Optical Society of America
OCIS codes: 140.4780, 140.3410, 290.4020, 290.5880, 230.5750.
1. Introduction
Dielectric and polymeric microcavities have great potential for possible applications in lasing optoelectronic devices.1,2 In conventional lasers a significant
fraction of optical pump power is lost and a rather
high threshold power is needed to initiate the lasing
effect. In contrast, spherical and disk cavities can
be used to support highly efficient low-threshold lasing operations. The high efficiency of such devices is
related to the existence of natural cavity resonances.
These resonances are known as morphologydependent resonances or whispering-gallery modes.3
The nature of these resonances can be envisioned in
a ray optic picture in which light is trapped inside the
cavity through the total internal reflection on the
cavity–air boundary.
In dielectric cavities, optically pumped quantum
wells, wires, or dots provide an active medium that
sustains the lasing operation.4 – 8 Polymeric microcavity lasers are made with an active medium that
includes host and guest molecules.9 –11 The absorbed light is transferred from the photoexcited host
molecules in a nonradiative way by means of resonant energy transfer to the guest molecules. A stim-
The authors are with the Department of Science and Technology,
Linköping University, 601 74 Norrköping, Sweden. I. V. Zozoulenko’s e-mail address is [email protected]
Received 1 July 2003; revised manuscript received 6 October
2003; accepted 17 October 2003.
0003-6935兾04兾081761-12$15.00兾0
© 2004 Optical Society of America
ulated emission from the active medium of dielectric
and polymeric cavities is trapped in high-Q modes for
a very long time. This leads to a significant increase
of radiation intensity inside the cavity and hence to
low-threshold laser operation.
One of the most important characteristics of cavity
resonances is their quality factor 共Q factor兲 defined as
Q ⫽ 2␲*共stored energy兲兾共energy lost per cycle兲. The
high value of the Q factor results from very low radiative losses that are caused mainly by radiation
leakage due to diffraction on the curved interface.
Typical experimental values of the Q factors of dielectric or polymeric disk cavities reported so far are in
the range of ⬃103–104 or lower4 –11 共the characteristic
diameter of the lasing disk cavities is 5–20 ␮m兲. At
the same time, a theoretical estimation of the corresponding Q factor of an ideal circular disk cavity of a
representative diameter d ⬃ 10 ␮m for a typical cavity mode 共usually corresponding to the highest whispering gallery resonances for a given cavity radius兲
gives Q ⬃ 1013 关see Eqs. 共26兲 and 共27兲 below兴. A
degradation of the experimental Q factors may be
attributed to a variety of factors, including imperfections in sidewall geometry, inhomogeneity of the
disk’s diffraction index, and effects of coupling to the
substrate or pedestal to name a few. A detailed
study of the effects of the above factors on the characteristics and performance of the microcavity lasers
appears to be of crucial importance for the design,
tailoring, and optimization of Q values of lasing microdisk cavities. Such the studies would require an
effective computational method that can deal with
10 March 2004 兾 Vol. 43, No. 8 兾 APPLIED OPTICS
1761
both complex geometry and variable refraction index
in the cavity.
One of the most powerful and versatile numerical
techniques often used in photonic simulation is the
finite-difference time domain method.12–14 A severe
disadvantage of this technique in its application to
cavities with small surface imperfections is that the
smooth geometry of the cavity has to be mapped into
a discrete grid with a very small lattice constant.
This makes the application of this method to the
problem at hand rather impractical in terms of both
computational power and memory.
Another class of computational methods reduces
the Helmholtz equation in the infinite twodimensional space into contour integral equations defined at the cavity boundaries. These methods
include the T-matrix technique,15,16 the boundary integral methods,17,18 and others.19 These methods
are computationally effective and capable of dealing
with cavities with arbitrary geometries. However,
the above methods require that the refraction index
be constant inside the cavity boundary.
In the present paper we develop a new, computationally effective, and numerically stable approach
based on the scattering matrix technique that is capable of dealing with both arbitrary complex geometry and inhomogeneous refraction index inside the
cavity. Note that the scattering matrix technique is
widely used in the analysis of waveguides20 as well as
in quantum mechanical simulations.21 This technique was also used in the analysis of resonant cavities with geometries for which the analytical solution
was available.22
The main idea of the method consists of dividing
the cavity region into N narrow concentric rings. At
each ith boundary between the neighboring rings, we
calculate the scattering matrix Si that relates the
states propagating 共or decaying兲 toward the boundary to those propagating 共or decaying兲 away from the
boundary. Successively combining the scattering
matrices for all the boundaries,20,21 S1 R . . . R SN, we
eventually relate the combined matrix to the total
scattering matrix of the cavity S. To calculate the
lifetime of the cavity modes 共and therefore their Q
factor兲, we compute the Wigner–Smith lifetime matrix23 which, in turn, is expressed in terms of the total
scattering matrix S 共see Refs. 2, 23, and 24兲.
Because we combine only two scattering matrixes
at each step, it is not necessary to keep track of the
solution in the whole space. This obviously eliminates the need for storing large matrices and facilitates the computational speed. It is also well known
that the scattering matrix technique 共in contrast, for
example, with the transfer matrix technique兲 is not
plagued by numerical instability because exponentially growing and decaying evanescent waves are
separated in the course of the computation. It
should also be stressed that the scattering matrix
technique is unconditionally stable; i.e., its stability
does not depend on grid size. This is in contrast to
other techniques, such as the finite-difference time
domain method,12–14 which is stable only when cer1762
APPLIED OPTICS 兾 Vol. 43, No. 8 兾 10 March 2004
tain conditions that are imposed on steps in time and
space are satisfied. Note that the present technique
of combining S matrices is conceptually similar to the
recurrence algorithm for calculating electromagnetic
scattering from a multilayered sphere.25,26 However, in contrast to these works, the scattering matrix
technique presented here can be applied to systems
in which the refraction index varies as a function of
both radial and angular coordinates.
The paper is organized as follows. In Subsection
2.A we develop the scattering matrix technique for
disk-shaped cavities. The results of the numerical
calculations of resonant states and of Q values of
nonideal cavities are made based on the developed
technique and are presented in Section 3. We consider and compare two cases: a disk cavity of a constant refraction index n with a sidewall imperfection
共surface roughness兲 and a disk cavity of an ideal circular shape but with an inhomogeneous refraction
index n ⫽ n共r, ␸兲. The results of the numerical simulation are analyzed and explained in terms of wave
reflection at a curved dielectric interface, combined
with the examination of Poincaré surfaces of section
and of Husimi function. Finally, we present our conclusion in Section 4.
2. Scattering Matrix Approach
A.
Formalism
We consider a two-dimensional cavity with the refraction index n surrounded by air. Because the majority of experiments are performed only with the
lowest transverse mode occupied, we neglect the
transverse 共z兲 dependence of the field and thus limit
ourselves to the two-dimensional Helmholtz equation. The two-dimensional Helmholtz equation for z
components of the electromagnetic field for the case of
a slowly varying refraction index n is given by
冉
冊
⳵2
1 ⳵
1 ⳵2
⫹ 2 2 ⌿共r, ␸兲 ⫹ 共kn兲 2⌿共r, ␸兲 ⫽ 0,
2⫹
⳵r
r ⳵r r ⳵␸
(1)
where ⌿ ⫽ Ez 共Hz兲 for TM 共TE兲 modes and k is the
wave vector in vacuum. Remaining components of
the electromagnetic field can be derived from Ez 共Hz兲
in a standard way.
We divide our system into three regions: the
outer region, r ⬎ R, the inner region, r ⬍ d, and the
intermediate region, d ⬍ r ⬍ R 共see Fig. 1兲. We
choose R and d in such a way that the outer- and
inner-region refraction indices are independent of the
coordinate, whereas in the intermediate region, n is a
function of both r and ␸. In the outer region the
solution to the Helmholtz equation can be written in
the form
⫹⬁
⌿ out ⫽
兺
q⫽⫺⬁
共1兲
关A q H 共2兲
q 共kr兲 ⫹ B q H q 共kr兲兴exp共iq␸兲,
(2)
Fig. 2. Intermediate region divided by N concentric rings of width
2⌬; ␳i is the distance to the middle of the ith ring. States ai, ai⫹1
propagate 共or decay兲 toward the ith boundary, whereas states bi,
bi⫹1 propagate 共or decay兲 away from this boundary. The ith
boundary is defined as the boundary between the ith and the i ⫹
1th rings.
Fig. 1. Schematic geometry of a cavity with the refraction index
n surrounded by air. The space is divided in three regions. In
the inner 共r ⬍ d兲 and outer regions 共r ⬎ R兲, the refraction indices
are constant. In the intermediate region d ⬍ r ⬍ R, the refraction
index n is a function of both r and ␸. The intermediate region is
divided by N narrow concentric rings. In each ring the refraction
coefficient is regarded as a function only of the angle and is given
as ni ⫽ ni 共␸兲.
共2兲
where H共1兲
q , Hq are the Hankel functions of the first
and second kind of the order q that describe incoming
and outgoing waves, respectively.
We define the scattering matrix S in the standard
fashion20,21:
B ⫽ SA,
(3)
where A, B are the column vectors composed of the
expansion coefficients Aq, Bq in Eq. 共2兲. The matrix
element Sq⬘q ⫽ 共S兲q⬘q gives a probability amplitude of
the scattering from the incoming state q into the
outgoing state q⬘. Because of the requirement of the
flux conservation, the scattering matrix is unitary21:
SS ⫽ I,
†
(4)
where I is the identity matrix. The time-reversal
invariance imposes the symmetry requirement upon
the scattering matrix21
S q⬘q ⫽ S qq⬘.
(5)
These two conditions can be used to control the numerical results for the scattering matrix.
To apply the scattering matrix technique, we divide
the intermediate region into N narrow concentric
rings 共see Figs. 1 and 2兲. Within each ith ring we
write down the solution to the Helmholtz equation as
a linear superposition of the states propagating 共or
decaying兲 away from the disk center and the states
propagating 共or decaying兲 toward the disk center.
关The detailed form of these states will be given in
Subsection 2.B; see Eq. 共20兲兴. At each ith boundary
共defined as the boundary between the ith and the i ⫹
1th rings兲, we can introduce the scattering matrix Si
that relates the states propagating 共or decaying兲 toward the boundary, 兵aim其 and 兵ai⫹1
m 其, to those propagating 共or decaying兲 away from the boundary, 兵bim其
and 兵bi⫹1
m 其:
冉 冊 冉 冊
bi
ai
⫽ Si i⫹1 ,
b i⫹1
a
1 ⱕ i ⱕ N ⫺ 1,
(6)
where ai, bi are the column vectors composed of the
expansion coefficients 兵aim其, 兵bim其 关see Eq. 共20兲 below兴.
For the Nth boundary between the last Nth ring and
the outer region, the scattering matrix SN is defined
in the form
冉 冊 冉 冊
bN
aN
⫽ SN
.
B
A
(7)
In the inner region 共i ⫽ 0兲 the solution to the Helmholtz equation has the form
⫹⬁
⌿ in ⫽
兺
a 0qJ q共nkr兲exp共iq␸兲,
(8)
q⫽⫺⬁
where Jq is the Bessel functions of the order q. For
the inner boundary 共i ⫽ 0兲 between the inner region
and the first ring in the intermediate region, we define the matrix S0 according to
冉冊 冉冊
a0
a0
⫽ S0 1 .
b1
a
(9)
The brief outline of the derivation and the expressions for the scattering matrices Si are given in Subsection 2.C and in Appendix A.
The essence of the scattering matrix technique is
the successive combination of the scattering matrices
10 March 2004 兾 Vol. 43, No. 8 兾 APPLIED OPTICS
1763
in the neighboring regions. For example, combining
the scattering matrices for the ith and i ⫹ 1th boundaries, Si and Si⫹1, we obtain the combined scattering
matrix S̃i,i⫹1 ⫽ Si R Si⫹1 that relates the outgoing
and incoming states in the rings i and i ⫹ 2,20,21:
冉 冊
冉 冊
bi
ai
⫽ S̃i,i⫹1 i⫹2 ,
b i⫹2
a
i,i⫹1
11
i
i⫹1
i
i⫹1 ⫺1 i
⫽ S ⫹ S12
S11
共I ⫺ S22
S11
兲 S21,
i,i⫹1
12
i
i⫹1 i ⫺1 i⫹1
⫽ S12
共I ⫺ S11
S22兲 S12 ,
S̃
S̃
where exp共i␪␮兲 ⫽ ␭␮ are the eigenvalues of the scatN
␪␮ is the total phase of the
tering matrix S, ␪ ⫽ ¥␮⫽1
M
␭␮ ⫽
determinant of the matrix S, det S ⫽ 兿␮⫽1
exp共i␪兲.
The resonant states are manifested as peaks in the
delay time, whose positions determine the resonant
wave vectors kres, and the heights are related to the Q
value of the cavity according to
i
11
Q ⫽ ␻␶ D共k res兲.
(16)
i,i⫹1
i⫹1
i
i⫹1 ⫺1 i
S̃21
⫽ S21
共I ⫺ S22
S11
兲 S21,
i,i⫹1
i⫹1
i⫹1
i
i⫹1 ⫺1 i
i⫹1
S̃22
⫽ S22
⫹ S21
共I ⫺ S22
S11
兲 S22S12
.
(10)
Here and hereafter we use the notation S11,
S12, . . . to define the respective matrix elements of
the block matrix S. Combining step by step all the
scattering matrices for all the boundaries 0 ⱕ i ⱕ N,
we numerically obtain the total combined matrix
S̃0,N ⫽ S0 R S1 R . . . SN relating the scattering states
in the outer region 共i ⫽ N兲 and the states in the inner
region 共i ⫽ 0兲,
冉冊
冉冊
a
a
⫽ S̃0,N
.
B
A
(11)
To obtain the scattering matrix S defined by Eq. 共3兲,
we eliminate a from Eq. 共11兲 and find the relation
between S̃0,N and S,
0,N
0,N ⫺1 0,N
0,N
S ⫽ S̃21
共I ⫺ S̃11
兲 S̃12 ⫹ S̃22
.
(12)
To identify the resonant states of an open cavity,
we introduce the lifetime matrix 共often called as
Wigner–Smith time-delay matrix兲23
dS
i dS†
i
Q⫽
S ⫽ ⫺ S†
.
c dk
c
dk
(13)
The diagonal elements of this matrix give the time
delay experienced by the wave incident in the qth
channel and scattered into all other channels:
␶ qD共k兲 ⫽ Qqq ⫽
i
c
兺
q⬘
dS†qq⬘
Sq⬘q.
dk
(14)
The delay time ␶qD共k兲 experienced by a scattering
wave is totally equivalent to the lifetime ␶ ⫽ 1兾2ck⬙
of a quasibound state with complex eigenvector k ⫽
k⬘ ⫺ ik⬙2. It is interesting to note that, in his original paper dealing with quantum mechanical scattering,23 Smith chose a letter “Q” to define the
lifetime matrix of a quantum system because of a
close analogy to the definition of a Q value in electromagnetic theory. The total time delay averaged
over all M incoming channels can be expressed in
the form2,24
␶ D共k兲 ⫽
⫽
1764
1
M
M
兺␶
q⫽1
1
cM
M
兺
␮⫽1
q
D
共k兲 ⫽
冉 冊
dS†
1 i
Tr
S
Mc
dk
d␪ ␮
1 d␪
⫽
,
dk
cM dk
APPLIED OPTICS 兾 Vol. 43, No. 8 兾 10 March 2004
(15)
B. Calculation of the Wave Functions in the Intermediate
Region d ⬍ r ⬍ R
In the intermediate region the refraction index n
depends on both r and ␸. Therefore, in contrast to
the inner and outer regions, in the intermediate
region we cannot separate variables and find an
exact analytical solution to the Helmholtz equation.
We can, however, find an approximate solution to
the Helmholtz equation in each ring. To do this,
let us look for the solution in the form ⌿共r, ␸兲 ⫽
R共r兲⌽共␸兲. Substituting this solution into Eq. 共1兲,
we obtain
r ⳵R共r兲
1 ⳵ 2⌽共␸兲
r 2 ⳵ 2R共r兲
⫹
⫽⫺
2
R共r兲 ⳵r
R共r兲 ⳵r
⌽共␸兲 ⳵␸ 2
⫺ k 2n 2共r, ␸兲r 2.
(17)
Let us now assume that each ring with radius ␳i has
a vanishing width 2⌬ 3 0 共see Fig. 2兲. In this case
we can regard r as a constant within each ith ring,
r ⬇ ␳i , with the refraction index being a function
only of the angle, n共r, ␸兲 ⫽ ni 共␸兲. In this approximation the variables in Eq. 共17兲 separate such that
for ith ring we can write
⳵ 2⌽ i共␸兲
⫹ 共␨ i ⫹ k 2i 共␸兲␳ 2i 兲⌽ i共␸兲 ⫽ 0,
⳵␸ 2
(18)
⳵ 2R i共r i 兲 ⳵R i共r i 兲
⫹
⫺ ␨ iR i共r i 兲 ⫽ 0,
⳵r 2i
⳵r i
(19)
where ␨i is a constant 共which can be both positive
and negative兲 and ri ⫽ r兾␳i . The angular function
⌽i共␸兲 satisfies the cyclic boundary condition ⌽i共0兲 ⫽
⌽i共2␲兲. The solution of Eq. 共18兲 thus provides an
infinite set of eigenvalues 兵␨im其 with the corresponding eigenfunctions ⌽im共␸兲. Generally, Eq. 共18兲
must be solved numerically. For a given eigenvalue ␨im, the solution of Eq. 共19兲 for the radial wave
function can be easily written in analytical form,
and the approximate solution to the Helmholtz
equation in the ith ring 共situated to the left of ith
boundary兲 reads
⬁
⌿ i 共r i , ␸兲 ⫽
兺 兵a
i
m
Si ⫽ ⌳KA⫺1BK⌳⫺1.
exp关共⫺1兾2 ⫹ i␥ im兲r̃ i 兴
m⫽1
(20)
where r̃i ⫽ 共r ⫺ ␳i 兲兾␳i and ␥im ⫽ 共⫺1兾4 ⫺ ␨im兲1兾2. The
states in Eq. 共20兲 are grouped according to the convention adopted in Subsection 2.A. Namely, the
states propagating to the right toward the ith boundary 关exp共⫹ i␥imr̃i 兲兴 are described by the coefficients
兵am其, whereas the states propagating away from the
ith boundary 关exp共⫺i␥imr̃i 兲兴 enter with the coefficients
兵bm其. Note that if ␥im becomes imaginary, ␥ ⫽ i␬, the
state propagating toward 共away from兲 the ith boundary turns into the states decaying toward 共away from兲
this boundary.
The wave function ⌿i⫹1共ri⫹1, ␸兲 in the i ⫹ 1th ring
共situated to the right of the ith boundary兲 is given by
a similar expression with coefficients am and bm interchanged:
⬁
兺 兵b
i⫹1
m
exp关共⫺1兾2 ⫹ i␥ i⫹1
m 兲r̃ i⫹1兴
m⫽1
⫹ a i⫹1
m exp关共⫺1兾2
i⫹1
⫺ i␥ i⫹1
m 兲r̃ i⫹1兴其⌽ m 共␸兲.
(21)
This is because in the i ⫹ 1th ring, the states
exp共⫹i␥i⫹1
m r̃i⫹1兲 propagate 共or decay兲 away from the
ith boundary, whereas the states exp共⫺␥i⫹1
m r̃i⫹1兲
propagate 共or decay兲 toward the ith boundary.
C.
Scattering Matrix Si at the ith Boundary
In this section we derive the expression for the scattering matrix Si by matching the wave functions
across the ith boundary. Using the condition of the
continuity of the tangential components of the electric and magnetic fields at the boundary between two
dielectric media, the matching conditions at the ith
boundary 共i.e., at the boundary between the ith and
the i ⫹ 1th rings兲 read
⌿ i 共r, ␸兲 ⫽ ⌿ i⫹1共r, ␸兲,
1 ⳵⌿ i 共r, ␸兲
1
⳵⌿ i⫹1共r, ␸兲
⫽ 2
,
␹ 2i 共␸兲
⳵r
␹ i⫹1共␸兲
⳵r
(22)
where ␹2i 共␸兲 ⫽ 1 for TM modes and ␹2i 共␸兲 ⫽ k2n2i 共␸兲 for
TE modes.
To derive the expression for the scattering matrix
Si in the intermediate region 共1 ⱕ i ⱕ N ⫺ 1兲, we
substitute the wave functions of Eqs. 共20兲 and 共21兲
into the boundary conditions of Eq. 共22兲. Multiplying the obtained equations by 关⌽im共␸兲兴* and integrating over the angle using the conditions of the
i
i
orthogonality 兰2␲
0 d␸关⌽m共␸兲兴*⌽m⬘共␸兲 ⫽ ␦mm⬘, we arrive
at two infinite systems of equations for the coeffii
i⫹1
cients aim, ai⫹1
After some straightm , bm, and bm .
forward algebra these systems of equations are
(23)
The scattering matrixes S , S 共for inner i ⫽ 0 and
outer i ⫽ N boundaries, respectively兲 are derived in a
similar fashion. The expression for Si given by Eq.
共23兲 holds for all the boundaries 0 ⱕ i ⱕ N. A particular form of the matrices ⌳, K, A, B is different for
three distinct cases: 共a兲 0th boundary 共the boundary
between the inner region 共i ⫽ 0兲 and the first ring i ⫽
1 in the intermediate region兲; 共b兲 ith boundary, 1 ⬍
i ⬍ N ⫺ 1 共the boundary between the ith and the i ⫹
1th rings in the intermediate region兲, and 共c兲 Nth
boundary 共the boundary between the last ring i ⫽ N
in the intermediate region and the outer region i ⫽
N ⫹ 1兲. The corresponding expressions for these
three cases are given in Appendix A, Eqs. 共A1兲–共A3兲.
0
⫹ b im exp关共⫺1兾2 ⫺ i␥ im兲r̃ i 兴其⌽ im共␸兲,
⌿ i⫹1共r i⫹1, ␸兲 ⫽
reduced to the form prescribed by Eq. 共6兲 with the
following result:
N
3. Nonideal Microdisk Cavities
In this section we apply the scattering matrix method
to the calculation of resonant states and Q values of
nonideal microdisk cavities with 共a兲 sidewall imperfections and 共b兲 circular cavities with an inhomogeneous refraction index of n ⫽ n共r, ␸兲.
The scattering matrix of the disk S defined by Eq.
共3兲 has an infinite dimension. To perform the numerical calculations, we truncate the matrix S to the
size M ⫻ M, where M ⫽ Mprop ⫹ Mevs, with Mprop and
Mevs being the number of the propagating and evanescent waves, respectively, in Eq. 共2兲. The number
of the propagating solutions Mprop equals the number
of modes in the corresponding closed circular cavity.
This number scales with nkR because it is given by
the number of real eigenvalues of the equations
Jq共nkR兲 ⫽ 0 共TM modes兲 and J⬘q共nkR兲 ⫽ 0 共TE modes兲
that define the eigenmodes of the closed circular cavity. The number of evanescent modes is chosen in
such a way that the calculated Q value does not
change with further increase of Mevs. For the dielectric cavities studied in the present paper, it is sufficient to choose Mevs ⬃ 5 ⫺ 10 to achieve the required
accuracy 共⬍0.5%兲.
The choice of the number of the rings N in the
intermediate region depends strongly on the character of the imperfections and the extent of the inhomogeneity present in the system. As a result, it is
not possible to provide a universal receipt for the
choice of N that is suitable for all systems. In our
calculations we choose N based on the requirement
that the results of the numerical calculations do not
change when N is increased 共i.e., when the ring width
⌬R is reduced兲. For example, for the case of cavities
with sidewall imperfections considered here, a sufficient accuracy 共⬍0.5%兲 was achieved for N ⫽ ⌬r兾20;
for cavities with the refractive index inhomogeneity,
N ⫽ R兾40.
To validate the present method, we perform numerical calculations for structures for which the analytical solution was available. This includes, for
example, an annular billiard consisting of a dielectric
10 March 2004 兾 Vol. 43, No. 8 兾 APPLIED OPTICS
1765
disk placed inside a larger disk with some displacement of the disk center,22 as well as an ideal circular
disk displaced from the origin of the coordinate system. In the latter case, the positions of the resonant
states and Q values are obviously independent of the
choice of the coordinate system. However, from a
computational point of view, this case is no simpler
than that of a cavity with an arbitrary shape because
the displacement from the origin lifts the radial symmetry and makes the separation of variables impossible. As an additional tool for validating the
numerical solution, we use Eqs. 共4兲 and 共5兲 to control
the unitarity and symmetry of the scattering matrix.
A.
Ideal Circular Cavity
Let us first briefly analyze the resonant states and Q
values of an ideal circular cavity with radius R and
refraction index n. In this case the scattering matrix can easily be written in analytical form. Employing the matching conditions of Eq. 共22兲 between
the wave function in the outer region r ⬎ R of Eq. 共2兲
and the wave function inside the disk given by the
Bessel function Jq共nkr兲 for r ⬍ R of Eq. 共8兲, we derive
the expression for the scattering matrix in the form22
S qq⬘ ⫽
共2兲
H 共2兲⬘
q 共kr兲 ⫺ ␰关 J⬘q共nkr兲兾J q共nkr兲兴H q 共kr兲
␦ qq⬘,
共1兲
H 共1兲⬘
共kr兲
⫺
␰关
J⬘
共nkr兲兾J
共nkr兲兴H
q
q
q
q 共kr兲
(24)
Fig. 3. Transmission coefficient T of a locally plane wave incident
on a curved surface with the radii of curvature ␳ as a function of the
incidence angle ␹ calculated from Eq. 共26兲. The angle of total
internal reflection sin ␹c ⫽ 0.56 共corresponding to n ⫽ 1.8兲. The
inset shows the dependence of the average radius of local curvature
due to boundary imperfections, ␳, subject to ⌬r for the present
model of surface roughness.
length, nk␳ ⬎⬎ 1, 共which applies to the majority of
cavities兲, the transmission probability reads27
冋
T ⫽ 兩T F兩exp ⫺
where ␰ ⫽ n共1兾n兲 for TM 共TE兲 modes. Derivatives
are taken over the full arguments in the brackets.
Resonant states of an ideal cavity can be inferred
from the scattering matrix in Eq. 共24兲 by use of Eq.
共15兲.
Each resonant state of an open disk is characterized by two wave numbers, q and m. These two
numbers are directly related to the corresponding
numbers of the closed resonator of the same radius R.
The index m is a radial wave number, and it is related
to the number of nodes of the field components in the
radial direction inside the disk. The index q is called
an angular 共or azimuthal兲 wave number because of
the analogy to quantum mechanics in which the angular momentum is given by LQM ⫽ បq. Equating
the quantum and classical angular momenta 共LClas ⫽
pR sin ␹, p ⫽ បnk兲, we find the relation between the
angular wave number and the angle of incidence ␹ in
a classical ray picture22:
q ⫽ nkR sin ␹.
(25)
Here we are mostly interested in the whisperinggallery modes with high Q values for which the angle
of incidence is larger than the angle of total internal
reflection, ␹ ⬎ ␹c 共sin ␹c ⫽ 1兾n兲. For such angles of
incidence, the transmission probability T of an electromagnetic wave incident on a curved interface of
radius ␳ is small, T ⬍⬍ 1. For the case when the
radius of curvature is much larger than the wave1766
APPLIED OPTICS 兾 Vol. 43, No. 8 兾 10 March 2004
册
2 nk␳
共cos2 ␹ c ⫺ cos2 ␹兲 3兾2 ,
3 sin2共␹兲
(26)
where TF is the classical Fresnel transmission coefficient for an electromagnetic wave incident on a flat
surface. Figure 3 illustrates that T decreases exponentially as the difference ␹ ⫺ ␹c grows. The Q
value of the whispering-gallery mode q in a cavity of
radius R is related to the transmission probability T
in Eq. 共26兲 by the relation28
Q⫽
2nkR cos ␹
,
T
(27)
where the classical incidence angle ␹ is related to
mode number q by Eq. 共25兲 and T ⬍⬍ 1.
B. Nonideal Cavities with Surface Roughness and
Inhomogeneous Refraction Index
In this section we present the results of the numerical
calculations of resonant states and of Q values of
nonideal cavities. We consider separately two cases:
共1兲 a disk cavity of a constant refraction index n but
with a sidewall imperfection 共surface roughness兲, 共2兲
a disk cavity with an ideal circular shape but with an
inhomogeneous refraction index n ⫽ n共r, ␸兲.
Various studies indicate that a typical sidewall imperfection can vary in size from 5–300 nm 共representing a variation of the order of ⬃0.05–1% of the cavity
radius兲.6 – 8,11 An exact experimental shape of the
cavity–air interface is, however, not available. We
thus model the interface shape as a superposition of
random Gaussian deviations from an ideal circle of
Fig. 4. Examples of nonideal cavities studied in the present paper in terms of 共a兲 surface roughness and 共b兲 inhomogeneous refraction
index. 共a兲 Radius of the disk R ⫽ 5 ␮m, n ⫽ 1.8, surface roughness ⌬r ⫽ 100 nm. 共b兲 R ⫽ 5 ␮m, 具n典 ⫽ 1.8, ⌬n ⫽ 5%.
radius R with a maximal amplitude ⌬r兾2 and a characteristic distance between the deviation maxima
⌬l ⬃ 2␲R兾50. In a similar fashion we model the
inhomogeneity of the diffraction index in the cavity,
where a parameter ⌬n characterizes a mean deviation of the refraction index n from its average value
具n典 ⫽ 1.8. The variation of the refraction index n can
be caused by different factors, including the presence
of quantum wells, wires, and dots that form an active
medium of the cavity; the local field intensity dependence n ⫽ n共I兲, and other factors. Examples of the
typical structures under investigation are shown in
Fig. 4.
Figure 5 shows the calculated Q values of the disk
resonant cavity for different surface roughnesses ⌬r
and the refraction index inhomogeneity ⌬n over some
representative wavelength interval. Note that we
have studied a number of different resonances, and
all of them showed the same trends described below.
Because TE modes exhibit similar features, hereafter
we concentrate only on TM modes of the cavity. The
calculated dependencies of the Q values on ⌬r and ⌬n
are summarized in the insets to Fig. 5.
Let us first concentrate on the low-Q state TM55,7
共q ⫽ 55, m ⫽ 7兲. An increase in both the surface
roughness ⌬r and the refractive index inhomogeneity
⌬n causes a graduate and rather slow decrease in the
Q value of this state, as shown in the insets to Fig. 5.
This behavior is typical for all other low-Q states. In
contrast, the high-Q resonances exhibit very different
and rather striking behavior. Namely, these resonances show a dramatic decrease of their Q-values
even for very small values of the surface roughness ⌬r
ⱗ ␭兾50. At the same time, the Q values of the cavity
decrease much more slowly when the refractive index
inhomogeneity ⌬n increases. For example, let us
choose ⌬r ⫽ 20 nm and ⌬n ⫽ 5%. For these values
of ⌬r and ⌬n, the Q value of the low-Q state TM55,7
drops by the same factor of ⬃1.3, decreasing from Q ⬇
270 to Q ⬇ 205. In contrast, for the very same surface roughness ⌬r, the Q value of a high-Q state
TM82,1 drops by the factor of ⬃1011, decreasing from
its value of Q ⬇ 4 ⫻ 1013 for an ideal disk to Q ⬇ 260.
At the same time, for the above value of ⌬n ⫽ 5%, the
Q value of this resonance decreases to the value of
Q ⬇ 1.3 ⫻ 107, which corresponds to the decrease by
Fig. 5. Dependencies Q ⫽ Q共␭兲 for two representative modes TM82,1 and TM55,7 for the cases of 共a兲 different surface roughness ⌬r 共R ⫽
5 ␮m, n ⫽ 1.8兲 and 共b兲 different refraction index inhomogeneities 关具n典 ⫽ 1.8兴. Note that in case 共b兲 the resonances shift when ⌬n varies.
For the sake of clarity, we plot all the resonances centered around their maxima of the corresponding ideal disk 共i.e., ⌬n ⫽ 0兲. The
broadening of the high-Q resonance TM82,1 is not discernible on the scale of the figure for all the values of ⌬n. Insets in 共a兲 and 共b兲 show
the dependencies Q ⫽ Q共⌬r兲 and Q ⫽ Q共⌬n兲, respectively.
10 March 2004 兾 Vol. 43, No. 8 兾 APPLIED OPTICS
1767
Fig. 6. Poincaré surfaces of section for geometrical rays corresponding to the states q ⫽ 55 共a兲–共c兲 and q ⫽ 82 共g兲–共i兲 for the cavity with
surface roughness ⌬r ⫽ 0, 20, 100 nm. The Husimi distributions for the states TM55,7 共d兲–共f 兲 and TM82,1 共j兲–共l兲 for the same values of ⌬r
used in the corresponding Poincaré SoS. ⌬␹ch indicates the broadening of the phase space due to the transition to the chaotic dynamics.
Dashed lines show the angle of total internal reflection ␹c.
a factor of ⬃104. 共Note that for the case of an ideal
cavity, the high-Q resonances are so narrow that the
numerical resolution does not allow a reliable estimation of their exact values. In this case we therefore use Eqs. 共26兲 and 共27兲 to estimate their Q values.兲
C.
Discussion
In the Subsection 2.B we found that the surface
roughness ⌬r and the refraction index inhomogeneity
⌬n that produce a similar degradation of low-Q states
have strikingly different effects on high-Q resonances. To understand these effects, we combine
Poincaré surface of section and Husimi function
methods with an analysis of ray reflection at a curved
dielectric interface. The Poincaré surfaces of section
共SoS兲 is a powerful tool for visualizing the phase
space for classical ray dynamics in cavities.2,29 We
concentrate on the surface section of the phase space
along the cavity boundary, r 僆 surf. For a given
resonant state with an angular number q, the corresponding ray is launched with the angle ␹0 ⫽
arcsin共q兾nkR兲 according to Eq. 共25兲. Each reflection
at the boundary 共characterized by the polar angle ␸
and the angle of incidence ␹兲, corresponds to a single
point in the plot. The number of bounces for a given
angle of incidence ␹0 is chosen in such a way that the
total path of the ray does not exceed that extracted
from the numerically calculated Q value for the corresponding resonance L ⫽ c␶D ⫽ Q兾kn. Figures
6共a兲– 6共c兲 and 6共g兲– 6共i兲 show a Poincaré SoS for the
geometrical rays corresponding to the states TM55,7
and TM82,1 for different values of the surface roughness ⌬r. For an ideal circular disk 共⌬r ⫽ 0兲, the
Poincaré SoS are obviously straight lines correspond1768
APPLIED OPTICS 兾 Vol. 43, No. 8 兾 10 March 2004
ing to a constant angle of incidence ␹ ⫽ ␹0. Figures
6共b兲– 6共c兲, 6共h兲 and 6共i兲 demonstrate that the dynamics of an ideal cavity transform from regular to chaotic, even for a cavity with a maximum roughness ⌬r
ⱗ 20 nm. In Figs. 6共b兲– 6共c兲, 6共h兲 and 6共i兲, ⌬␹ch approximately indicates the phase-space broadening
due to the transition to the chaotic dynamics. An
important observation is that, for a given surface
roughness ⌬r, the broadening of the phase space is
independent of the angular mode q; i.e., it is the same
for low- and high-Q states.
We complement classical Poincaré SoS with Husimi function analysis.2,29,30 The Husimi function
共often called Husimi distributions兲 H共␸, ␹兲 represents
a quantum 共wave兲 analog to a classical Poincaré SoS.
It is defined as a projection of a given cavity mode ⌿共r
僆 surf, ␸兲 共where “surf” denotes surface兲 taken at the
surface of the cavity into a Gaussian wave packet
⌽共␸⬘; ␸, ␹兲 impinging the cavity boundary with the
coordinate ␸ at the angle ␹:
H共␸, ␹兲 ⫽
兰
2␲
d␸⬘⌿共r 僆 surf, ␸⬘兲⌽共␸⬘; ␸, ␹兲,
(28)
0
where the minimum-uncertainty wave packet centered around ␸, ␹ with the dispersion in position 共␴兾
2兲1兾2 is given by
⌽共␸⬘; ␸, ␹兲 ⫽ 共␲␴兲 ⫺1兾4
兺 exp关⫺1兾2␴共␸⬘ ⫺ ␸
l
⫹ 2␲l 兲 2 ⫺ ik sin ␹共␸ ⫹ 2␲l 兲兴,
(29)
where we have chosen ␴ ⫽ 公2兾k. The Husimi distributions 关see Figs. 6共d兲– 6共f 兲 and 6共j兲– 6共l兲兴 exhibit
the same trends as the classical Poincaré SoS. In-
Fig. 7. Illustrative examples of intensity distribution Ez for the resonant state TM55,7 in cavities with ⌬r ⫽ 0 共a兲, 20 nm 共b兲, 100 nm 共c兲
and with R ⫽ 5 ␮m, n ⫽ 1.8. Dashed lines indicate boundaries of the cavity.
deed, broadening the phase space while increasing
the surface roughness ⌬r for the Husimi functions
has the same effect as the corresponding broadening
of ⌬␹ch in the Poincaré SoS. 共Illustrative examples
of the wave functions in cavities with different surface roughness ⌬n are shown in Fig. 7.兲
Figure 8 shows the Husimi distributions for a circular cavity with an inhomogeneous refraction index.
The variation of the refraction index ⌬n ⫽ 5% is
chosen in such a way that the degradation of the Q
value for the low-Q resonance TM55,7 is the same as
that for the surface roughness ⌬r ⫽ 20 nm case
shown in Fig. 6. As expected, the broadening of the
Husimi distribution due to the increase in ⌬n is of the
same order as that for the corresponding values of ⌬r
共compare Figs. 6 and 8兲.
According to Eq. 共26兲, one can expect an increase in
the transmission coefficient 共and therefore a decrease
in the Q value of the cavity兲 due to the broadening of
the phase space ⌬␹ch. This is because the incidence
angle ␹ effectively moves closer to the angle of the
total internal reflection ␹c. ⌬Tch in Fig. 3 indicates
that the estimated increase in the transmission coefficient is due to the broadening of the phase space
⌬␹ch as extracted from the Poincaré SoS for ⌬r ⫽ 20
nm and ⌬n ⫽ 5%. For the low-Q resonance TM55,7,
this corresponds to the decrease in the Q value by a
factor of ⌬Qch ⬃ ⌬T⫺1
ch ⬇ 1.5, which is consistent with
the calculated decrease in the low-Q resonances.
For the case of high-Q resonance TM82.1, the estimated decrease in the Q factor is ⌬Qch ⬃ ⌬T⫺1
ch ⬇
103–104 共see Fig. 3兲, which is consistent with the
Fig. 8. The Husimi distributions for the states TM55,7 共a兲 and
TM82,1 共b兲 for the cavity with the refraction index inhomogeneity
⌬n ⫽ 5%.
calculated decrease in this resonance for the case of
the inhomogeneous refraction index only. 共Note
that because of a rather approximate definition of
⌬␹ch we can give only a very rough estimation of the
factor ⌬Tch.兲 In contrast, for the case of high-Q resonances in the presence of surface imperfections, this
estimated value of ⌬Qch is many orders of magnitude
smaller than the actual calculated decrease in the Q
factor 共given by a factor of ⬇1011; see Fig. 5兲.
To explain the rapid degradation of high-Q resonances, we concentrate on another aspect of the wave
dynamics. Namely, the imperfections at the surface
boundary effectively introduce a local radius of a surface curvature ␳ that is distinct from the disk radius
R 关see Fig. 4共a兲兴. One may thus expect that, with the
presence of the local surface curvature, the total
transmission coefficient is determined by the averaged value of ␳ rather than by the disk radius R.
The dependence of ␳ on surface roughness ⌬r for the
present model of surface imperfections is shown in
the inset to Fig. 3. Figure 3 demonstrates that the
reduction of the local radius of curvature from 5 ␮m
共ideal disk兲 to 1.7 ␮m 共⌬r ⫽ 20 nm兲 causes an increase
in the transmission coefficient by ⌬Tcur ⬇ 108. This
estimation, combined with the estimation based on
the change of ⌬Tch is fully consistent with the actual
computed decrease of the Q factor shown in Fig. 5.
Note that the estimation of the transition coefficient
T based on Eq. 共26兲 is justified for nk␳ ⬎⬎ 1. This
condition is satisfied for a wide range of the surface
roughness ⌬r. For example, ⌬r ⫽ 100 nm 共i.e., the
surface roughness when high-Q resonances are already strongly suppressed兲 corresponds to the effective local curvature ␳ ⬇ R兾10, i.e., over the considered
wavelength interval nk␳ ⬇ 10. We thus conclude
that the main mechanism responsible for the rapid
degradation of high-Q resonances in nonideal cavities
is the enhanced radiative decay through the curved
surface because the effective local radius 共given by
the surface roughness兲 is smaller than the disk radius R.
In contrast, the degradation of low-Q resonances
共as well as the high-Q resonances with inhomogeneous refraction indices兲 is related mostly to the
broadening of the phase space caused by the transi10 March 2004 兾 Vol. 43, No. 8 兾 APPLIED OPTICS
1769
tion to chaotic dynamics. It should be noted, however, that both factors 共broadening of the phase space
and enhancement of the transmission due to a decrease in the effective radius of curvature兲 may play
comparable roles in the degradation of the low-Q
whispering-gallery resonances in the presence of surface roughness.
It is interesting to note that an analogues degradation of high-Q modes was recently found in
hexagonal-shaped microcavities, where the modes
were strongly influenced by roundings of the corners
even when the characteristic length scale 共the local
radius of curvature兲 was 1 order of magnitude
smaller than the wavelength.31 It is worth mentioning that one often assumes that long-lived high-Q
resonances in idealized cavities 共e.g., in ideal disks,
hexagons, etc.兲 are not important for potential application in optical communication or laser devices18,22
because of their extremely narrow width. Our simulations demonstrate that this is not the case because
in real structures the Q values of these resonances
become comparable with those of intermediate-Q resonances for small or moderate surface roughness of
⌬r ⬃ 10–50 nm.
4. Conclusions
In this paper we develop a new, computationally
effective, and numerically stable approach based on
the scattering matrix 共S-matrix兲 technique that is
capable of dealing with both arbitrary complex geometry and inhomogeneous refraction index inside
a two-dimensional cavity. The derivation is based
on the separation of the cavity region into N narrow
concentric rings and the calculation of the S matrix
at every boundary between the rings. The total S
matrix is obtained in a recursive way by successive
combination of the scattering matrices for all the
boundaries. To calculate the lifetime of the cavity
modes 共and therefore their Q factors兲, we compute
the Wigner–Smith time-delay matrix, which in turn
is expressed in terms of the total scattering matrix.
We apply the developed algorithm to the calculation of resonant states and Q values of nonideal microdisk cavities with sidewall imperfections and
circular cavities with an inhomogeneous refraction
index n ⫽ n共r, ␸兲. We find that the surface roughness ⌬r and the refraction index inhomogeneity ⌬n,
which produce similar degradation of low-Q states,
have strikingly different effects on high-Q resonances. In particularly, in the case of inhomogeneous refraction index, the increase in ⌬n causes a
rather gradual decrease in the Q value of high-Q
resonances. In contrast, in the presence of surface
roughness, even small imperfections 共⌬r ⱗ ␭兾50兲 can
lead to a drastic degradation of high-Q cavity modes
by many orders of magnitude.
To understand these features, we combine Poincaré SoS and Husimi function methods with an analysis of ray reflection at a curved dielectric interface.
We argue that the main mechanism responsible for
the rapid degradation of high-Q resonances in nonideal cavities with surface roughness is the enhanced
1770
APPLIED OPTICS 兾 Vol. 43, No. 8 兾 10 March 2004
radiative decay through the curved surface. This is
because the effective local radius 共given by the surface roughness兲 is smaller than the disk radius R.
In contrast, the degradation of low-Q resonances 共as
well as high-Q resonances with inhomogeneous refraction indices兲, is related mostly to the broadening
of the phase space caused by the transition to chaotic
dynamics.
Appendix A: Expressions for the Si Matrices
In this appendix we present the expressions for the
matrices ⌳, K, A, B entering Eq. 共23兲 for the scattering matrix Si relating incoming and outgoing states
at the ith boundary. We distinguish three different
cases as specified below.
Case 1: Zeroth boundary 共the boundary between
the inner region 共i ⫽ 0兲 and the first ring i ⫽ 1 in the
intermediate region兲,
⌳11 ⫽ I,
共⌳22兲 mj ⫽ exp关⫺共1兾2兲⌬ 1兴␦ mj,
⌳12 ⫽ ⌳21 ⫽ 0,
K11 ⫽ I,
共K22兲 mj ⫽ exp共i␥ m⌬ 1兲␦ mj,
K12 ⫽ K21 ⫽ 0,
A⫽
冉
冊
0
V0,1
,
⫺J⬘ U0,1P1
共J兲 mj ⫽ J m共n 0 kd兲␦ mj,
共V0,1兲 mj ⫽
兰
2␲
B⫽
冉
冊
J
⫺V0,1
,
0 ⫺U0,1Q1
共J⬘兲 mj ⫽ J⬘m共n 0 kd兲␦ mj,
exp共⫺im␸兲⌽ 1j 共␸兲d␸,
0
共U0,1兲 mj ⫽
1
n 0 k␳ 1
兰
2␲
0
␹ 20共␸兲
exp共⫺im␸兲⌽ 1j 共␸兲d␸.
␹ 21共␸兲
(A1)
Case 2: ith boundary, 1 ⬍ i ⬍ N ⫺ 1 关the boundary between the ith and the i ⫹ 1th rings in the
intermediate region兴
共⌳11兲 mj ⫽ exp关共1兾2兲⌬ i 兴␦ mj,
共⌳22兲 mj ⫽ exp关⫺共1兾2兲⌬ i⫹1兴␦ mj,
⌳12 ⫽ ⌳21 ⫽ 0,
A⫽
B⫽
共Vi,i⫹1兲 mj ⫽
冉
冉
兰
共K兲 mj ⫽ exp共i␥ m⌬ i 兲␦ mj,
冊
冊
⫺I
V
,
⫺Qi Ui,i⫹1Pi⫹1
i,i⫹1
I
⫺V
,
Pi ⫺Ui,i⫹1Qi⫹1
i,i⫹1
2␲
关⌽ im共␸兲兴*⌽ i⫹1
j 共␸兲d␸,
0
共Ui,i⫹1兲 mj ⫽
␳i
␳ i⫹1
兰
2␲
0
␹ 2i 共␸兲
关⌽ im共␸兲兴*⌽ i⫹1
j 共␸兲d␸.
2
␹ i⫹1
共␸兲
(A2)
Case 3: Nth boundary 关the boundary between the
last ring i ⫽ N in the intermediate region and the
outer region 共i ⫽ N ⫹ 1兲兴
共⌳11兲 mj ⫽ exp关共1兾2兲⌬ N兴␦ mj,
⌳ 22 ⫽ I,
6.
7.
⌳ 12 ⫽ ⌳ 21 ⫽ 0,
共K11兲 mj ⫽ exp共i␥ m⌬ N兲␦ mj,
K22 ⫽ I,
A⫽
B⫽
冉
冉
8.
K12 ⫽ K21 ⫽ 0,
冊
冊
⫺I VN,N⫹1H共1兲
,
⫺QN UN,N⫹1H共1兲⬘
9.
I ⫺VN,N⫹1H共2兲
,
PN ⫺UN,N⫹1H共2兲⬘
共H共1,2兲兲mj ⫽ H共1,2兲共kR兲␦mj,
mj,
共H共1,2兲⬘
兲 mj ⫽ H 共1,2兲⬘共kR兲␦
m
m
共VN,N⫹1兲 mj ⫽
兰
2␲
10.
关⌽ Nm共␸兲兴* exp共ij␸兲d␸,
0
共U
N,N⫹1
兲 mj ⫽ k␳N
兰
11.
2␲
0
␹ 2N共␸兲
关⌽ 1j 共␸兲兴* exp共ij␸兲d␸.
2
␹ N⫹1
共␸兲
12.
(A3)
In Eqs. 共A1兲–共A3兲 the matrices Qi, Pi are defined
according to
14.
共Pi兲 mj ⫽ 共⫺1兾2 ⫹ i␥ im兲␦ mj,
共Q 兲 mj ⫽ 共⫺1兾2 ⫺ i␥ 兲␦ mj,
i
Jm, H共1,2兲
m ,
i
m
13.
1 ⱕ i ⱕ N.
J⬘m, H共1,2兲⬘
m
and
are the Bessel and Hankel
functions and their derivatives, and ⌬i ⫽ ⌬兾␳i .
We thank Olle Inganäs for stimulating discussions
that initiated this work. We are also grateful to
Sayan Mukherjee and especially to Stanley Miklavcic
for many useful discussions and conversations. We
appreciate the correspondence with Jan Wiersig.
A. I. Rahachou acknowledges financial support from
the Swedish Institute and The Royal Swedish Academy of Sciences.
15.
16.
17.
18.
19.
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30. B. Crespi, G. Perez, and S. J. Chang, “Quantum Poincaré
sections for two-dimensional billiards,” Phys. Rev. E 47, 986 –
991 共1993兲.
31. J. Wiersig, “Hexagonal dielectric resonators and microcrystal
lasers,” Phys. Rev. A 67, 023807 共2003兲.
III
Paper III
Elastic scattering of surface electron waves in
quantum corrals: Importance of the shape of
the adatom potential
Phys. Rev. B, vol. 70, pp. 233409 1–4, 2004
PHYSICAL REVIEW B 70, 233409 (2004)
Elastic scattering of surface electron waves in quantum corrals:
of the adatom potential
Importance of the shape
A. I. Rahachou* and I. V. Zozoulenko†
Department of Science and Technology (ITN), Linköping University, S-601 74 Norrköping, Sweden
(Received 26 January 2004; revised manuscript received 9 April 2004; published 28 December 2004)
We report elastic scattering theory for surface electron waves in quantum corrals defined by adatoms on the
surface of noble metals. We develop a scattering-matrix technique that allows us to account for a realistic
smooth potential profile of the scattering centers. Our calculations reproduce quantitatively all the experimental
observations, which is in contrast to previous theories (treating the adatoms as point scatterers) that require
additional inelastic channels of scattering into the bulk in order to achieve the agreement with the experiment.
Our findings thus indicate that accounting for a realistic potential as well as using the exact numerical schemes
is important in achieving detailed agreement as well as interpretation of the experiment.
DOI: 10.1103/PhysRevB.70.233409
PACS number(s): 73.20.At, 03.65.Nk, 72.10.Fk, 68.37.Ef
Advances in modern nanotechnology made it possible to
manipulate adatoms on the surface of a metal, arranging
them into ordered structures coined as “quantum corrals.”
Using scanning tunneling microscopy (STM), Crommie et
al.1–3 studied the scattering of the surface electron waves
residing at (111) faces of Cu. These surface states interact
strongly with Fe adatoms, and the spatial variation of the
STM differential conductance revealed beautiful images of
the standing wave patterns in the quantum corrals. In addition, the experiment showed a series of pronounced peaks in
the energy spectrum of the differential conductance dI / dV at
the center of the corrals.
In order to describe the experimental observation,1,2
Heller et al.4 have developed multiple-scattering theory for
surface electron waves in quantum corrals. In their theory
each adatom was modeled as a pointlike ␦-function potential
supporting isotropic s-wave scattering. The quantitative
agreement with the experiment was achieved by assuming an
additional (inelastic) channel of scattering (presumably into
the bulk metallic states). The authors concluded that absorption is the dominant mechanism for the broadening of the
energy levels seen in the experiment, and estimated that
⬃25% of the incident amplitude is reflected, ⬃25% is transmitted, and ⬃50% is absorbed. The importance of the electron scattering to the bulk states for the level width broadening in the quantum corrals was also asserted by Crampin et
al.5 and Cramplin and Bryant.6
An alternative purely elastic scattering theory for the
same quantum corral structures was reported by Harbury and
Porod.7 They modeled the adatoms by finite-height potential
barriers, as opposed to “black dot” ␦-point absorbing scattering potential adopted in the above cited works.4–6 The elastic
theory accounts well for the spatial variation of the wave
function in the quantum corrals, but overestimates the broadening of the resonant levels, especially for higher energies.
The findings of Harbury and Porod therefore suggest that the
features of the spectrum can be sensitive to the detailed
shape of the scattering potential.
It is important to stress that accounting for a detailed
shape of a scattering potential was crucial for quantitative
description of many phenomena in quantum nanostructures.
1098-0121/2004/70(23)/233409(4)/$22.50
This, for example, includes the Hall and bend resistance
anomalies in four-terminal junctions,8 the breakdown of
quantized conductance in quantum point contacts calculated
using realistic potentials,9 and the explanation of a branched
flow in a two-dimensional electron gas.10 In the present Brief
Report we develop a scattering matrix approach that allows
us to account for a realistic smooth potential of the adatoms.
We demonstrate that for such a potential the broadening and
positions of the resonant states as well as the scattering wave
function in the quantum corrals can be quantitatively described by the inelastic theory alone, without the assumption
of any additional (inelastic) scattering channels. Our findings
thus support a conclusion of Harbury and Porod7 that the
elastic scattering model is compatible with the reported STM
data and hence our results outline the importance of accounting for a realistic potential as well as using of exact numerical techniques in detailed comparison and interpretation of
the experiment.
The differential conductance dI / dV of the STM tunnel
junction is proportional to the local density of states (LDOS)
which is given in terms of the scattering eigenstates of the
Hamiltonian Ĥ, ␺q共r兲,3
dI/dV ⬃ LDOS共r,E兲 =
兺q 兩␺q共r兲兩2␦共E − Eq兲.
共1兲
In order to calculate the scattering eigenstates ␺q共r兲 we
adopt to the problem at hand the scattering matrix
technique11 that was recently developed for the numerical
solution of the Helmholtz equation for resonant states of dielectric optical cavities with both complex geometry and
variable refraction index. This is possible because of a direct
correspondence between the Helmholtz and Schrödinger
equations.12 The advantage of the scattering-matrix technique is that it provides an efficient way to treat the smooth
realistic profile of the adatom. Note that commonly used
methods based on the discretization of the scattering domain
would be rather impractical in terms of both computation
power and memory, because the smoothly varying potential
of the adatom has to be mapped into a discrete grid with a
very small lattice constant.
233409-1
©2004 The American Physical Society
PHYSICAL REVIEW B 70, 233409 (2004)
BRIEF REPORTS
FIG. 1. The experimental spectrum of the differential conductance dI / dV at the center of the 88.7-Å-radius 60-Fe-adatom circular quantum corral structure on Cu(111) substrate (adopted from
Ref. 4) (solid curve). The calculated spectrum, dotted line: our
scattering-matrix technique applied for a smooth adatom potential
with V0 = 2.5 eV, ␴ = 1.52 Å; dashed line: multiple-scattering theory
for the ␦-barrier adatom potential with inelastic channel of scattering (adopted from Ref. 4).
We consider a two-dimensional ring-shaped corral structure. Experimental observations suggest that adatoms
strongly disturb the local charge density at the finite distance
⬃7 Å.3 An exact experimental shape of the adatom potential
is not available. We thus model this potential as a Gaussian
with the half-width ␴ and the height V0 centered at the
location 共x0 , y o兲, V共x , y兲 = V0 exp关−共x − x0兲2 / 2␴2兴exp关−共y
− y 0兲2 / 2␴2兴 (see below, inset to Fig. 3).
In order to calculate the scattering eigenstates we divide
the quantum corral into inner, outer, and intermediate regions. In the inner region (inside the corral) and in the outer
region (outside the corral) the adatom potential is negligible,
V共x , y兲 = 0. Therefore, in these two regions the solution to the
Schrödinger equation can be written in analytical form. Introducing the polar coordinates we can write for the wave
function outside the corral,
+⬁
⌿out =
共1兲
iq␸
,
兺 关AqH共2兲
q 共kr兲 + BqHq 共kr兲兴e
q=−⬁
共2兲
共2兲
where H共1兲
q and Hq are the Hankel functions of the first and
second kind of the order q describing, respectively, incoming
and outgoing waves; k = 冑2m*E / ប, with m* being the effective electron mass. The expression for the wave function
inside the corral ⌿in can be written in a similar fashion as an
expansion over Bessel functions Jq.
We introduce the scattering matrix S in a standard
fashion,11,12 B = SA, where A and B are column vectors composed of the expansion coefficients Aq and Bq for incoming
and outgoing states in Eq. (2). The matrix element Sq⬘q gives
the probability amplitude of scattering from an incoming
state q into an outgoing state q⬘. In order to apply the scattering matrix technique we divide the intermediate region
[i.e., the region where the adatom potential V共x , y兲 is distinct
from zero] into N narrow concentric rings. At each ith
boundary between the rings we introduce the scattering matrix Si that relates the states propagating (or decaying) towards the boundary, with those propagating (or decaying)
FIG. 2. The experimental curves (solid lines, adapted from Ref.
4) for the LDOS subject to the tip position inside a circular corral
for low bias voltages. The calculated LDOS, dashed line: our scattering matrix technique applied for a smooth adatom potential with
V0 = 2.5 eV, ␴ = 1.52 Å. Parameters of the structure and are the
same as those in Fig. 1. Voltages are given in volts and measured
relatively to the bottom of the surface-state band. All the theory
voltages are shifted by ⫺0.01 V relative to the experiment.
away from the boundary. The matrices Si are derived using
the requirement of continuity of the wave function and its
first derivative at the boundary between the two neighboring
rings. (Note that in our calculations a typical ring width was
chosen ⬃0.1 Å, and for the purpose of the wave function
matching the ring was divided on ⬃400 sectors in the angular direction.) Successively combining the scattering matrices for all the boundaries,11,12 S1 丢 ¯ 丢 SN, we can relate the
combined matrix to the scattering matrix S. With the help of
the scattering matrix S we determine the wave function in
the outer region for every incoming state q in Eq. (2). Using
the expression for the matrices Si we then recover the corresponding wave functions in the intermediate region as well
as the wave function ⌿in in the inner region.
Note that in the scattering-matrix technique one combines
only two scattering matrices at each step. Hence, it is not
necessary to keep track of the solution for the wave function
in the whole space. This obviously eliminates the need for
storing large matrices and facilitates the computational
speed.
Using our scattering matrix technique we calculate the
bias voltage dependence and the spatial distribution of the
LDOS for 60-Fe-adatom, 88.7-Å-radius circular quantum
corrals reported by Heller et al.4 (Figs. 1 and 2). The Fe
adatoms are placed on the meshes of a 2.55-Å triangular grid
corresponding to the hexagonal Cu(111) lattice. The effective
mass used in all the simulations was taken as m* = 0.361m0
and the electron band-edge energy E0 = 0.43 eV below the
Fermi energy of the electrons.1,7 In the absence of applied
voltage V these parameters correspond to the wavelength of
electrons ␭ = 30 Å. For the parameters of the adatom potential we use7 V0 = 2.5 eV, ␴ = 1.52 Å, which correspond to
those used by Harbury and Porod,7 who modeled the adatoms as hard wall finite potential barriers of 1.52-Å diameter
of the height of 2.5 eV.
Figure 1 also shows corresponding results of the multiplescattering theory of Heller et al.4 Both theories show a simi-
233409-2
PHYSICAL REVIEW B 70, 233409 (2004)
BRIEF REPORTS
FIG. 3. The calculated spectrum of the differential conductance
dI / dV at the center of the circular quantum corral structure for
various widths ␴ and heights V0 of the adatom potential (the scattering efficiency of the potential is kept constant). Parameters of the
structure are the same as those in Fig. 1. Inset shows the schematic
shape of the potential.
lar level of agreement with the experimental data in both the
peak positions and level broadenings for the differential conductance as well as in the number and the peak positions for
the spatial LDOS distribution throughout the cross section of
the quantum corral. But in contrast to the case of ␦ scatterers
used in Ref. 4 our model agrees quantitatively well with the
experimental data without introduction of additional inelastic
scattering channels.
It should be pointed out that the spectrum calculated on
the basis of the scattering-matrix technique shows relatively
broad peaks with decreasing amplitude at lower energy. It is
interesting to note that this feature in our calculated spectrum
is in fact closer to the experimental data than predictions of
other theories giving narrow peaks with high
amplitude.3–7,13,14 (Note also that a simple convolution with a
Gaussian function broadens the narrow peaks at the bottom
of the spectra and brings theory closer to the experiment.5)
We do not have a full explanation for the difference for lowenergy peaks between different theoretical approaches. One
of possible reasons is that all other theories calculate LDOS
from the imaginary part of the scattering Green’s function,
whereas we calculate LDOS directly from the scattering
eigenstates of the Hamiltonian, Eq. (1).
Figure 3 represents the differential conductance spectra
dI / dV at the center of the quantum corral structure where the
width ␴ and the height V0 of the adatom potential are varied,
but its scattering efficiency is kept constant [i.e, 兰V共r兲dr
= const]. The best agreement with the experiment is achieved
for ␴ ⬇ 1.5 Å and V0 = 2.5 eV, which is in agrement with the
results reported by Harbury and Porod.7 It is often assumed
that because the spatial extent of the scattering potential
共⬃7 Å兲 is small compared to the wavelength of the incoming
particles 共␭ ⬃ 30 Å兲, the adatom potential can be treated as a
point scatterer or even as a continuous boundary V共r兲 ⬃ ␦共r
− r0兲.3,4,13–15 The results presented in Fig. 3 clearly show that
even though ␴ Ⰶ ␭, the finite width of the scattering potential
affects strongly the observed characteristic of the systems.
Our calculations thus signify the importance of the shape of
the scattering potential for achieving the quantitative agree-
FIG. 4. Effect of the nonideal positioning of the adatoms on the
differential conductance of the quantum corral. Parameters of the
structure and the adatom potential are the same as those in Figs. 1
and 2. The inset illustrates the displacement of the scatterers from
their ideal positions (white circles) in a circular geometry for twolattice shift.
ment with the experiment of the voltage dependence of
dI / dV.
The experiment2 suggests that for high energies of incoming electrons (i.e., for large tip voltages) the significant fraction of Fe adatoms can move from their original positions.
We therefore study the effect of this displacement on the
shape and broadening of the resonant states of the quantum
corral. Figure 4 shows the calculated differential conductance for a quantum corral for the case when the adatoms of
an ideal circular corral are randomly shifted from their positions by one and two lattice constant (as illustrated in the
insets to Fig. 4). A one-lattice shift does not seem to have a
significant effect on the broadening of the peaks in the differential conductance. The deviation from an ideal circular
case become rather noticeable for the shift in adatom positions by two lattice sites. As expected, these deviations are
more pronounced for larger energies of incoming electrons.
Note that we performed simulations for different realization
of the ensembles of scatterers (keeping the average shift
fixed to one or two lattice sites), and all of them are almost
indistinguishable. This is because of the self-averaging character of scattering in a quantum corral due to a large number
of scatterers. Our calculations thus pinpoint the deviation
from a regular arrangements of scatterers as an additional
factor that should be taken into account for achieving the
quantitative agreement with the experiment.
Let us now discuss the coupling between the surface and
the bulk metallic states. In the assumption of purely inelastic
scattering the widths of the resonances in the differential
conductance spectrum given by the multiple scattering
theory of Heller et al. are far too narrow compared to
experiment.3 In order to reproduce the experiment one thus
assumes that each adatom acts as an absorbing “black dot”
providing an inelastic channel of scattering from the surface
to presumably bulk states. This is done by introducing the
complex phase shift4 and phenomenological self
energy.3–6,13,14 By fitting these parameters an excellent agreement with the experiment can be achieved. Our results, how-
233409-3
PHYSICAL REVIEW B 70, 233409 (2004)
BRIEF REPORTS
ever, demonstrate that the experimental data are fully consistent with the model of elastic scattering with a smooth
realistic potential even without invoking inelastic channels of
scattering. Following Harbury and Porod7 we conclude that
inability of the multiple-scattering theory to reproduce the
experiment without invoking the inelastic channel is most
probably related to the s-wave approximation. In contrast,
our method is numerically exact in the sense that it is not
limited to the s-wave scattering and all partial waves as well
as mode mixing between them are taken into account by the
exactly calculated scattering matrices.
It should be stressed, however, that our scattering model
is strictly two-dimensional (2D). Fully three-dimensional
scattering calculations were reported by Crampin and
Bryant,6 where they used self-consistent potentials and included partial waves up to l = 3. Unfortunately, our method
cannot be easily extended into three dimensions and we are
thus not in the position to comment on the importance of the
coupling between surface and bulk states within our model.
However, it is important to stress that a vast majority of the
Financial support from Vetenskapsrådet (I.V.Z) and the
National Graduate School of Scientific Computing (A.I.R.) is
gratefully acknowledged. We appreciate a discussion with G.
Hansson.
9 M.
*Electronic address: [email protected]
†Electronic
papers use 2D models of scattering for the interpretation of
the experiment.3,4,13,14 In view of the excellent agreement
between our 2D calculations and the experiment we conclude that taking into account a realistic potential as well as
using the exact numerical schemes not limited to the s-wave
approximation is important in achieving a detailed agreement
as well as interpretation of the experiment.
To conclude, we studied the scattering of electron waves
by quantum corral structure for the case of realistic smooth
potential of adatom scatterers. We achieved a detailed agreement with the experiment without introducing an additional
inelastic channels of scattering. Our findings also suggest
that accounting for a realistic potential shape may be of particular importance for the quantitative description and interpretation of quantum mirage experiments.
address: [email protected]
1 M. F. Crommie, C. P. Lutz, and D. M. Eigler, Nature (London)
363, 524 (1993).
2
M. F. Crommie, C. P. Lutz, and D. M. Eigler, Science 262, 218
(1993).
3 For a review see, G. A. Fiete and E. J. Heller, Rev. Mod. Phys.
75, 933 (2003).
4 E. J. Heller, M. F. Crommie, C. P. Lutz, and D. M. Eigler, Nature
(London) 369, 464 (1994).
5 S. Crampin, M. H. Boon, and J. E. Inglesfield, Phys. Rev. Lett.
73, 1051 (1994).
6
S. Crampin and O. R. Bryant, Phys. Rev. B 54, R17 367 (1996).
7 H. K. Harbury and W. Porod, Phys. Rev. B 53, 15 455 (1996).
8 H. U. Baranger, D. P. DiVincenzo, R. A. Jalabert, and A. D.
Stone, Phys. Rev. B 44, 10 637 (1991).
Laughton, J. A. Nixon, J. H. Davies, and H. U. Baranger,
Phys. Rev. B 43, 12 638 (1991); M. Laughton, J. R. Barker, J.
A. Nixon, and J. H. Davies, ibid. 44, 1150 (1991).
10
M. A. Topinka, B. J. LeRoy, R. M. Westervelt, S. E. J. Shaw, T.
Fleischmann, E. J. Heller, K. D. Maranowski, and A. C. Gossard, Nature (London) 410, 183 (2001).
11 A. I. Rahachou and I. V. Zozoulenko, J. Appl. Phys. 94, 7929
(2003); A. I. Rahachou and I. V. Zozoulenko, Appl. Opt. 43,
1761 (2004).
12 S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge
University Press, Cambridge, 1995).
13
J. Kliewer, R. Berndt, and S. Crampin, New J. Phys. 3, 22
(2001).
14 K.-F. Braun and K.-H. Rieder, Phys. Rev. Lett. 88, 096801
(2002).
15 A. Lobos and A. A. Aligia, Phys. Rev. B 68, 035411 (2003).
233409-4
IV
Paper IV
Light propagation in finite and infinite photonic
crystals: The recursive Greens function
technique
Phys. Rev. B, vol. 72, pp. 155117 1–12, 2005
PHYSICAL REVIEW B 72, 155117 共2005兲
Light propagation in finite and infinite photonic crystals:
The recursive Green’s function technique
A. I. Rahachou and I. V. Zozoulenko
Department of Science and Technology, Linköping University 601 74, Norrköping, Sweden
共Received 19 April 2005; revised manuscript received 7 June 2005; published 24 October 2005兲
We report a computational method based on the recursive Green’s function technique for calculation of light
propagation in photonic crystal structures. The advantage of this method in comparison to the conventional
finite-difference time domain 共FDTD兲 technique is that it computes Green’s function of the photonic structure
recursively by adding slice by slice on the basis of Dyson’s equation. This eliminates the need for storage of the
wave function in the whole structure, which obviously strongly relaxes the memory requirements and enhances
the computational speed. The second advantage of this method is that it can easily account for the infinite
extension of the structure both into air and into the space occupied by the photonic crystal by making use of
the so-called “surface Green’s functions.” This eliminates the spurious solutions 共often present in the conventional FDTD methods兲 related to, e.g., waves reflected from the boundaries defining the computational domain.
The developed method has been applied to study scattering and propagation of the electromagnetic waves in
the photonic band-gap structures including cavities and waveguides. Particular attention has been paid to
surface modes residing on a termination of a semi-infinite photonic crystal. We demonstrate that coupling of
the surface states with incoming radiation may result in enhanced intensity of an electromagnetic field on the
surface and very high Q factor of the surface state. This effect can be employed as an operational principle for
surface-mode lasers and sensors.
DOI: 10.1103/PhysRevB.72.155117
PACS number共s兲: 42.70.Qs, 41.20.Jb, 78.67.⫺n
I. INTRODUCTION
Optical microcavities and photonic crystals 共PC兲 have received increased attention in recent years because of the
promising prospects of applications in a future generation of
optical communication networks.1,2 Examples of successfully demonstrated devices include lasers, light emitting diodes, waveguides, add-drop filters, delay lines, and many
others.3
By far the most popular method for the theoretical description of light propagation in these systems is the finitedifference time-domain method 共FDTD兲 introduced by Yee.4
The success of the FDTD method is due to its speed, flexibility, and ease of computational storage requirements. The
limitation of the FDTD technique is related to the fact that
the computational domain is finite. As a result, an injected
pulse experiences spurious reflections from the domain
boundaries, which leads to mixing between the incoming and
reflected waves. In order to overcome this limitation the socalled perfectly matched layer condition has been introduced.5 However, even with this technique, a sizable part of
the incoming flux can still be reflected back.6 In many cases
the separation of spurious pulses is essential for the interpretation of the results, and this separation can only be achieved
by increase of a size of the computational domain.7 This may
lead to a prohibitive amount of computational work, because
the stability of the FDTD algorithm requires a sufficiently
small time step.
The problem of the spurious reflections from the computational domain boundaries does not arise in the methods
based on the scattering matrix technique, where the incident
and outgoing fields are related with the help of the scattering
matrix.8–12 Other approaches where the spurious reflections
1098-0121/2005/72共15兲/155117共12兲/$23.00
are avoided include, e.g., a multiple multipole method,13 and
a Green’s function method14 based on the analytical expression for the Green’s function for an empty space. The main
objective of the present paper is to present a computational
approach based on the recursive Green’s function technique
that can account for an infinite extension of a photonic crystal. In this technique the Green’s function of the photonic
structure is calculated recursively by adding slice by slice on
the basis of Dyson’s equation. In order to account for the
infinite extension of the structure both into air and into the
space occupied by the photonic crystal we make use of the
so-called “surface Green’s functions” that propagate the electromagnetic fields into infinity. In this paper we present a
method for calculation of the surface Green’s functions both
for the case of a semi-infinite homogeneous dielectrics, as
well as for the case of a semi-infinite periodic structure 共photonic crystal兲. This makes it possible to apply the Green’s
function technique for investigation of a variety of important
structures including waveguides and cavities in infinite or
semi-infinite photonic crystals, as well as to study the effect
of the surface states and the coupling of waveguide Bloch
modes to the external radiation. Note that the recursive
Green’s function technique is widely used for quantum mechanical transport calculations15–18 and is proven to be unconditionally numerically stable for various discretization
schemes.
The article is organized as follows. In Sec. II we present a
general formulation of the problem. A description of the recursive Green’s function technique is given in Sec. III. This
section also provides a recipe for the calculation of Bloch
states in a periodic structure as well as the surface Green’s
function. Technical details of the calculations are given in
Appendixes A–C. Several examples of the application of the
155117-1
©2005 The American Physical Society
PHYSICAL REVIEW B 72, 155117 共2005兲
A. I. RAHACHOU AND I. V. ZOZOULENKO
developed method are given in Sec. IV. The conclusions are
presented in Sec. V.
following discretization of the differential operators in Eqs.
共5兲 and 共6兲,19
⌬2
II. GENERAL FORMULATION OF THE PROBLEM
We start with Maxwell’s equations in two dimensions
⌬2
1
␻
⵱ ⫻ 兵⵱ ⫻ E共r兲其 = 2 E共r兲,
␧r共r兲
c
2
⵱⫻
再
冎
共1兲
where r = xi + yj, ⵱ = 共⳵ / ⳵x兲i + 共⳵ / ⳵y兲j, ␧r共r兲 is the relative dielectric constant, and the electric and magnetic field vectors
E共r , t兲 = E共r兲exp共−i␻t兲 and H共r , t兲 = H共r兲exp共−i␻t兲. If the
dielectric constant ␧r共r兲 is independent on z, the Maxwell’s
equations decouple in two sets of equations for the TE modes
共Hz , Ex , Ey兲,
− um,m;n,n−1 f m,n−1 =
TE modes:
冊
− i ⳵Ez
,
Hx =
␻␮0 ⳵ y
i ⳵Ez
.
␻␮0 ⳵x
共3兲
Let us rewrite the equations for Hz , Ez 共2兲 and 共3兲 in an
operator form2
TM modes:
f ⬅ H z,
f = 冑␧rEz,
␰m,n =
1
,
␧rm,n
um,m−1;n,n = ␰m−1/2,n ,
um,m;n,n+1 = ␰m,n+1/2,
um,m;n,n−1 = ␰m,n−1/2 ,
f m,n = 冑␧rm,nEzm,n,
␰m,n =
共9兲
1
冑␧rm,n ,
冑␧ r
冉
⳵2
⳵2
2 +
⳵x
⳵y2
冊冑
1
␧r
um,m−1;nn = ␰m−1,n␰m,n ,
um,m;n,n+1 = ␰m,n+1␰m,n,
um,m;n,n−1 = ␰m,n␰m,n−1 .
+
兩0, . . . 0,1m,n,0, . . . ,0典 = 0,
am,n
共10兲
共11兲
and
am,n兩0典 = 0,
⳵ 1 ⳵
⳵ 1 ⳵
−
, 共5兲
LTE = −
⳵x ␧r ⳵x ⳵ y ␧r ⳵ y
1
um,m+1;nn = ␰m,n␰m+1,n,
+
am,n
兩0典 = 兩0, . . . 0,1m,n,0, . . . ,0典,
共4兲
f,
LTM = −
f m,n = Hzm,n,
A convenient and common way to describe finitedifference equations on a numerical grid 共lattice兲 is to introduce the corresponding tight-binding operator. For this purpose we first introduce creation and annihilation operators,
+
, am,n. Let the state 兩0典 ⬅ 兩0 , . . . , 0m,n , . . . , 0典 describe an
am,n
empty lattice, and the state 兩0 , . . . 0 , 1m,n , 0 , . . . , 0典 describe an
+
, am,n act on
excitation at the site m , n. The operators am,n
these states according to the rules16
2
where the Hermitian differential operator L and the function
f reads
TE modes:
共8兲
f m,n ,
2
,
vm,n = 4␰m,n
1 ⳵ 2E z ⳵ 2E z
␻2
+ 2 Ez = 0,
2 +
2
␧r ⳵x
⳵y
c
␻
c
2
um,m+1;n,n = ␰m+1/2,n,
TM modes:
共2兲
and for the TM modes 共Ez , Hx , Hy兲,
冉冊
␻⌬
c
vm,n = ␰m+1/2,n + ␰m−1/2,n + ␰m,n+1/2 + ␰m,n−1/2 ,
− i ⳵Hz
,
␻ ␧ 0␧ r ⳵ x
Lf =
冉 冊
where the coefficients v , u are defined for the cases of TE
and TM modes as follows:
i ⳵Hz
,
Ex =
␻ ␧ 0␧ r ⳵ y
Hy =
共7兲
vm,n f m,n − um,m+1;n,n f m+1,n − um,m−1;n,n f m−1,n − um,m;n,n+1 f m,n+1
⳵ 1 ⳵
⳵ 1 ⳵
␻2
Hz +
Hz + 2 Hz = 0,
⳵x ␧r ⳵x
⳵ y ␧r ⳵ y
c
冉
⳵2
␰共x兲f共x兲 → ␰m+1 f m+1 − 2␰m f m + ␰m−1 f m−1 ,
⳵x2
we arrive at the finite difference equation
1
␻2
⵱ ⫻ H共r兲 = 2 H共r兲,
␧r共r兲
c
Ey =
⳵
⳵ f共x兲
→ ␰m+1/2共f m+1 − f m兲 − ␰m−1/2共f m − f m−1兲,
␰共x兲
⳵x
⳵x
am,n兩0, . . . 0,1m,n,0, . . . ,0典 = 兩0典
共12兲
and they obey the following commutational relations:
.
+
+
+
兴 = am,nam,n
− am,n
am,n = ␦m,n ,
关am,n,am,n
共6兲
For the numerical solution, Eqs. 共4兲–共6兲 have to be discretized, x , y → m⌬ , n⌬, where ⌬ is the grid step. Using the
+
+
,am,n
兴 = 0.
关am,n,am,n兴 = 关am,n
Consider an operator equation
155117-2
共13兲
PHYSICAL REVIEW B 72, 155117 共2005兲
LIGHT PROPAGATION IN FINITE AND INFINITE…
L̂兩f典 =
冉 冊
␻⌬
c
2
兩f典,
共14兲
where the Hermitian operator
+
+
am,n − um,m+1;n,nam,n
am+1,n
L̂ = 兺 共vm,nam,n
m,n
+
+
− um+1,m;n,nam+1,n
am,n − um,m;n,n+1am,n
am,n+1
+
− um,m;n+1,nam,n+1
am,n兲
共15兲
acts on the state
+
兩0典.
兩f典 = 兺 f m,nam,n
共16兲
m,n
Substituting the above expressions for L̂ and 兩f典 in Eq. 共14兲,
and using the commutation relations and the rules Eqs.
共11兲–共13兲, it is straightforward to demonstrate that the operator equation 共14兲 is equivalent to the finite difference equation 共8兲. Note an apparent physical meaning of the last four
terms in Eq. 共15兲: terms 2 and 3 describe forward and backward hopping between two neighboring sites in the x direction, and terms 4 and 5 denote similar hopping in the y
direction. In the next section we outline the Green’s function
formalism for solution of Eq. 共14兲.
III. THE RECURSIVE GREEN’S FUNCTION TECHNIQUE
A. Basics
Let us first specify structures under investigation. We consider light propagation through a photonic structure defined
in a waveguide 共supercell兲 of the width N, where we assume
the cyclic boundary condition 共i.e., the row n = N + 1 coincides with the row n = 1兲. The photonic structure occupies a
finite internal region consisting of M slices 共1 艋 m 艋 M兲.
The external regions are semi-infinite waveguides 共supercells兲 extending into regions m 艋 0 and m 艌 M + 1. The
waveguides can represent air 共or a material with a constant
refractive index兲, or a periodic photonic crystal. Figure 1
shows two representative examples where 共a兲 the semiinfinite waveguides represent a periodic photonic crystal
with the period M, and 共b兲 a photonic structure is defined at
the boundary between air and the semi-infinite photonic
crystal.
Let us first define the scattering states for the structures
under consideration. The translation invariance along the supercell dictates the Bloch form for the ␣th incoming state
兩␺␣i 典,
N
兩␺ ␣i 典 =
+
␣
am,n
兩0典,
兺 ␾m,n
兺 eik mn=1
m⬉0
+
␣
共17兲
where k␣+ 共k␣− 兲 is the Bloch wave vector of the right␣
propagating 共left-propagating兲 state ␣, and ␾m,n
is the corresponding Bloch transverse eigenfunction satisfying the
Bloch condition
␣
␣
␾m,n
= ␾m+M,n
.
共18兲
The transmitted and reflected states, 兩␺␣t 典 and 兩␺␣r 典, can be
written in a similar form,
FIG. 1. 共Color online兲 Schematic illustration of the system under study defined in a waveguide 共supercell兲 of the width N. An
internal region of the structure occupies M slices. Two representative cases are shown: 共a兲 external regions are semiperiodic photonic
crystals with the period M; 共b兲 external regions represent a semiinfinite periodic photonic crystal with the period M to the right and
air to the left. Arrows indicate the directions on the incoming 共I兲,
reflected 共R兲, and transmitted 共T兲 waves.
兩␺␣t 典 =
兺
N
+
␤
am,n
兩0典,
兺 t␤␣eik␤关m−共M+1兲兴 兺 ␾m,n
+
m⭌M+1 ␤
共19兲
n=1
N
兩␺␣r 典 =
+
␤
am,n
兩0典,
兺 ␾m,n
兺 兺 r␤␣eik mn=1
m⬉0 ␤
−
␤
共20兲
where t␤␣共r␤␣兲 stands for the transmission 共reflection兲 amplitude from the incoming Bloch state ␣ to the transmitted 共reflected兲 Bloch state ␤. Note that in the general case the wave
␣
can be different in the
vectors k␣± and the Bloch states ␾m,n
left and right waveguides 关see, e.g., Fig. 1共a兲, when the photonic structure is defined at the boundary air–photonic crystal兴. The method of calculation of the Bloch states for an
arbitrary periodic structure is described below in Sec. III C.
We define Green’s function of the operator L̂ in a standard
way,
„共␻⌬/c兲2 − L̂…G共␻兲 = 1̂,
20
共21兲
where 1̂ is the unitary operator. The knowledge of the
Green’s function allows one to calculate the transmission and
reflection coefficients. Indeed, let us write down the solution
of Eq. 共14兲 as a sum of two terms, the incoming state 兩␺i典 and
the system response 兩␺典 representing whether the transmitted
155117-3
PHYSICAL REVIEW B 72, 155117 共2005兲
A. I. RAHACHOU AND I. V. ZOZOULENKO
or reflected states, 兩␺t典 or 兩␺r典, 兩f典 = 兩␺i典 + 兩␺典. Substituting 兩f典
into Eq. 共14兲 and using the formal definition of the Green’s
function Eq. 共21兲, the solution of Eq. 共14兲 can be written in
the form
兩␺典 = G„L̂ − 共␻⌬/c兲2…兩␺i典.
共22兲
Calculating
the
matrix
elements
具M + 1 , n兩␺典
⬅ 具0 兩 aM+1,n␺典 and 具0 , n 兩 ␺典 ⬅ 具0 兩 a0,n␺典, of the right and left
hand side of Eq. 共22兲, we arrive at the N ⫻ N system of linear
equations for the transmission and reflection amplitudes 共see
for details Appendix A兲,
⌽M+1T = − GM+1,0共U0,1⌽−M+1Kl − ⌫l−1⌽0兲,
⌽0R = − G0,0共U0,1⌽−M+1Kl − ⌫l−1⌽0兲 − ⌽0 ,
共23兲
共24兲
where the matrix elements 共T兲␤␣ = t␤␣, 共R兲␤␣ = r␤␣; G
and G0,0 are the Green’s function matrixes with the elements
M+1,0
共Gm,l兲n,p = 具0兩am,nGa+l,p兩0典.
共25兲
0,0
⌫l ⬅ Gwg
is the left “surface Green’s function” corresponding
only to part of the whole structure, namely, to the semiinfinite waveguide 共supercell兲 that extends to the left, −⬁
⬍ m 艋 0. The physical meaning of the surface Green’s function ⌫ is that it propagates the electromagnetic fields from
the boundary slice of the semi-infinite waveguide 共supercell兲
into infinity. A method for calculation of the surface Green’s
functions both for the case of a semi-infinite homogeneous
dielectrics, as well as for the case of a semi-infinite photonic
crystal in a waveguide geometry, is described below in Sec.
III D. The matrices Kl and ⌽m are given by the rightpropagating Bloch eigenvectors k␣+ and the corresponding
␣
in the waveguides,
eigenstates ␾m,n
共Kl兲␣␤ = exp共ik␣+ 兲␦␣␤,
␣
共⌽m兲n␣ = ␾m,n
,
共26兲
FIG. 2. Schematic illustration of the application of Dyson’s
equation for calculation of Green’s function for a composed structure consisting of m + 1 slices 共see text for details兲. The operators
0
L̂0m and l̂m+1
describe respectively the structure composed of m
0
+ V̂ corslices, and the 共m + 1兲th slice. The operator L̂m+1 = L̂0m + l̂m+1
responds to the composed structure of m + 1 slices, where V̂ is the
perturbation operator describing the hopping between the mth and
共m + 1兲th slices.
0
of
nearest neighbors. Suppose we know Green’s function Gm
0
0
the operator L̂m, as well as Green’s function gm+1 of the
0
, correspond to a single 共m + 1兲th slice,
operator l̂m+1
0
+
+
= 兺 共vm+1,m+1am+1,n
am+1,n − um+1,m+1;n,n+1am+1,n
am+1,n+1
l̂m+1
n
+
− um+1,m+1;n+1,nam+1,n+1
am+1,n兲.
共The method of calculation of Green’s function for a single
slice is outlined in Appendix C.兲 Our aim is to calculate
Green’s function of the composed structure, Gm+1, consisting
of m + 1 slices. The operator corresponding to this structure
can be written down in the form
0
0
+ l̂m+1
+ V̂,
L̂m+1 = L̂m
and the diagonal “hopping matrix” U0,1 is defined as
共U0,1兲n,n⬘ = u0,1;n,n⬘␦n,n⬘ .
共27兲
关Note that the matrix Kl in Eqs. 共23兲 and 共24兲 refers to the
right-propagating states in the left waveguide.兴 In the following sections we describe the recursive Green’s function technique based on the successive use of the Dyson’s equation,
introduce the method for the calculation of Bloch states in a
periodic structure, and outline the way to calculate the surface Green’s function ⌫.
In order to calculate Green’s function of the internal structure 共i.e., for the slices 1 艋 m 艋 M兲 we utilize the recursive
technique based on Dyson’s equation; see Fig. 2.
In order to illustrate this technique let us consider a struc0
describing this
ture consisting of m slices. The operator L̂m
structure can be written down in the form
r
0
L̂m
共30兲
0
l̂m+1
and
are given by the expresswhere the operators
ions Eqs. 共30兲 and 共29兲, and V̂ = V̂m,m+1 + V̂m+1,m is the perturbation operator describing the hopping between the mth and
共m + 1兲th slices,
+
am+1,n
V̂ = V̂m+1,m + V̂m,m+1 = − 兺 共um,m+1;n,nam,n
n
+
+ um+1,m;n,nam+1,n
am,n兲.
共31兲
The Green’s function of the composed structure, Gm+1, can
be calculated on the basis of Dyson’s equation20
B. Recursive technique based on Dyson’s equations
0
= 兺 vra+r ar − 兺 ur,r+⌬a+r ar+⌬ ,
L̂m
共29兲
共28兲
r,⌬
where r = m⬘ , n⬘共1 艋 m⬘ 艋 m ; 1 艋 n⬘ 艋 N兲, and the summation
over ⌬ in the second term is performed over all available
Gm+1 = G0 + G0V̂Gm+1 ,
Gm+1 = G0 + Gm+1V̂G0 ,
0
共32兲
where G is the “unperturbed” Green’s function correspond0
0
or l̂m+1
. For the sake of completeness,
ing to the operators L̂m
a brief derivation of Dyson’s equation is given in Appendix
B. Thus, starting from Green’s function for the first slice g01
and adding recursively slice by slice, we are in the position
to calculate Green’s function of the internal structure consist-
155117-4
PHYSICAL REVIEW B 72, 155117 共2005兲
LIGHT PROPAGATION IN FINITE AND INFINITE…
ing of M slices. Explicit expressions following from Eqs.
共32兲 and used for the recursive calculations are given below,
m+1,m+1
0
0 m,m
0
= 关I − gm+1
Um+1,m共Gm
兲 Um,m+1兴−1gm+1
,
Gm+1
m+1,1
m+1,m+1
0 m,1
= Gm+1
Um+1,m共Gm
兲 ,
Gm+1
1,1
0 1,1
0 1,m
m+1,1
= 共Gm
兲 + 共Gm
兲 Um,m+1Gm+1
,
Gm+1
1,m+1
0 1,m
m+1,m+1
= 共Gm
兲 Um,m+1Gm+1
,
Gm+1
共33兲
where the upper indexes define the matrix elements of the
+
Green’s function Gm,m⬘ = 具0兩am,nGam
兩0典. This recursive
⬘,n⬘
technique is proven to be unconditionally numerically
stable.15–17 The performance of the method is determined
by the size of the system of linear equations 共33兲 which
we solve when we add each consecutive slice. This system
is solved M times, where M is the number of slices of the
internal structure 共in the x direction兲. The size of Eqs. 共33兲 is
N ⫻ N, where N is a number of discretization points in the
y direction. Typical dimensions of the equations used for
computations of the structures reported in Sec. IV are
⬃200⫻ 200.
In order to calculate the Green’s function of the whole
system, we have to connect the internal structure with the left
and right semi-infinite waveguides. Starting with the left
waveguide, we write
L̂int+left = L̂int + L̂left + V̂,
共34兲
where the operators L̂int+left, L̂int, and L̂left describe respectively the system representing the internal structure ⫹ the
left waveguide, the internal structure, and the left waveguide.
The perturbation operator V̂ describes the hopping between
the left waveguide and the internal structure. Applying then
the Dyson equation in a similar way as we described above,
Gint+left = G0 + G0V̂Gint+left ,
共35兲
we are in the position to find the Green’s function Gint+left of
the system representing the internal structure ⫹the left waveguide. G0 in Eq. 共35兲 in an “unperturbed” Green’s function
corresponding to the internal structure and the semi-infinite
waveguide 共the “surface Green’s function” ⌫兲. Having calculated the Green’s function Gint+left on the basis of Eq. 共35兲,
we proceed in a similar way by adding the right waveguide
and calculating with the help of the Dyson’s equation the
total Green’s function G of the whole system.
FIG. 3. 共Color online兲 Schematic illustration of the calculation
of Bloch states in an infinite periodic structure 共see text for details兲.
The operator L̂cell describes a unit cell under consideration, 1 艋 m
艋 M, and L̂out describes the rest of the structure. The hopping
between the cell and the rest of the structure is described by the
operator V̂.
L̂ = L̂cell + L̂out + V̂,
where the operators L̂cell and L̂out describe respectively
the cell under consideration 共1 艋 m 艋 M兲, and the outside
region including all other slices −⬁ ⬍ m 艋 0 and M + 1 艋 m
⬍ ⬁, and V̂ is the hopping operator between the cell and
slices m = 0 and m = M + 1. Write the total wave function
+
兩0典 in the form
兩␺典 = 兺m,n␺m,nam,n
兩␺典 = 兩␺cell典 + 兩␺out典,
共37兲
where 兩␺cell典 and 兩␺out典 are respectively wave functions in the
cell and in the outside region. Substituting Eqs. 共36兲 and 共37兲
into Eq. 共14兲, we obtain 兩␺cell典 = GcellV̂兩␺out典, where Gcell is the
Green’s function of the operator L̂cell. Calculating the matrix
elements 具1 , n 兩 ␺典 and 具M , n 兩 ␺典, this equation can be written
in the matrix form,
1,M
␺1 = G1,1
cellU1,0␺0 + Gcell U1,0␺M+1
共38a兲
M,M
␺M = GM,1
cell U1,0␺0 + Gcell U1,0␺M+1 ,
共38b兲
where the vector column ␺m = 共␺m,1 . . . ␺m,N兲 , and where we
used UM,M+1 = U0,1 共because of the periodicity兲 and U0,1
= U1,0 关according to the definition of U, Eq. 共27兲兴. It is convenient to rewrite Eq. 共38a兲 in a compact form
T
C. Bloch states of the periodic structure
In this section we describe the method for calculation of
the Bloch states in periodic waveguides 共supercells兲 using
the Green’s function technique. A similar method was used
for calculation of Bloch states in quantum-mechanical
structures.18
Consider a unit cell of a periodic waveguide occupying
M slices, 1 艋 m 艋 M; see Fig. 3.
Rewrite the operator corresponding to the whole structure
in the form
共36兲
T1
where T1 =
冉
冉 冊 冉 冊
␺M+1
␺1
= T2
,
␺M
␺0
− G1,M
cell U1,0 0
GM,M
cell U1,0
I
冊
,
T2 =
冉
− I − G1,1
cellU1,0
0
GM,1
cell U1,0
冊
,
共39兲
with I being the unitary matrix. The wave function of the
periodic structure has Bloch form,
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PHYSICAL REVIEW B 72, 155117 共2005兲
A. I. RAHACHOU AND I. V. ZOZOULENKO
兩␺典 = Gwg兩s典,
共44兲
where 兩␺典 is the wave function that has to satisfy Bloch
conditions 共40兲. Applying Dyson’s equation between the
slices 0 and 1 we obtain
1,−M
0,−M
Gwg
= ⌫rU1,0Gwg
,
FIG. 4. 共Color online兲 A schematic diagram illustrating calculation of the surface Green’s function ⌫ of a periodic structure 共see
text for details兲.
␺M+m = eikxMI␺m .
共40兲
Combining Eqs. 共39兲 and 共40兲, we arrive at the eigenequation
for Bloch wave vectors and Bloch states,
T −1
1 T2
冉 冊
冉 冊
␺1
␺1
= eikxM
,
␺0
␺0
共41兲
determining the set of Bloch eigenvectors kx␣ and eigenfunctions ␺␣, 1 艋 ␣ 艋 N.
To improve numerical stability of Eq. 共41兲, it may be
rewritten in the form11
共T1 + T2兲−1T1
冉 冊
冉 冊
␺1
␺1
= 共eikxM + 1兲−1
.
␺0
␺0
共42兲
This technique allows one to avoid overflows and underflows in the eigensolver routine when eigenvalues with
兩eikxM兩 Ⰷ 1 and 兩eikxM兩 Ⰶ 1 are calculated.
In order to separate the left- and right-propagating states
we compute the Poynting vector integrated over transverse
direction, whose sign determines the direction of propagation. Bloch state propagating in a waveguide 共supercell兲 defined in a photonic crystal is illustrated below in Fig. 5共c兲.
Poynting vector can be expressed as follows2
S␣共y兲 =
1
Re关E␣共y兲 ⫻ H␣* 共y兲兴.
2
共43兲
Note that for the case of the waveguide defined in air,
M = 1, and Green’s functions Gcell in Eq. 共39兲 is simply
given by Green’s function of a single slice g0 共see Appendix
C for details of calculation of g0兲.
D. The surface Green’s function ⌫
Consider a semi-infinite Bloch waveguide 共supercell兲 of
the periodicity M extending in the region −M 艋 m ⬍ ⬁ as
depicted in Fig. 4.
Suppose that an excitation 兩s典 is applied to its first slice
m = −M. Introducing the Green function Gwg corresponding
to the operator L̂wg describing the waveguide, one can write
down the response to the excitation 兩s典 in the form
共45兲
1,1
is the right surface Green’s function. 共Note
where ⌫r ⬅ Gwg
that because the waveguide is infinitely long and periodic,
M+1,M+1
2M+1,2M+1
1,1
= Gwg
= Gwg
= . . . etc.兲 Taking the matrix
Gwg
elements 具1 , n 兩 ␺典 of Eq. 共44兲 and making use of Eq. 共45兲, we
obtain for each Bloch state ␣, ␺1␣ = ⌫rU1,0␺0␣. The latter equation can be used for determination of ⌫r,
⌫rU1,0 = ⌿1⌿−1
0 ,
共46兲
where ⌿1 and ⌿0 are the square matrixes composed of matrix columns ␺1␣ and ␺0␣, Eq. 共46兲. If the waveguide is open to
the left, its surface Green’s function is the same as the surface Green’s function of the corresponding waveguide open
to the right, ⌫l = ⌫r. Note that for the case of the waveguide
defined in air the surface Green’s function 共46兲 simplifies to
⌫rU1,0 = K, where K is defined according to Eq. 共26兲.
IV. APPLICATIONS OF THE METHOD
To reveal the power of the method we study three model
systems defined in 2D square-lattice photonic crystal. First,
we calculate a transmission coefficient and quality factor 共Q
factor兲 of several representative types of microcavities in infinite PCs. Then we focus on semi-infinite crystals where we
investigate the effect of surface states, and, finally, we consider a semi-infinite PC with a waveguide opening to the
surface. For the bulk crystal we choose a structure composed
of cylindrical rods with the permittivity ␧r = 8.9 and the diameter of a rod d = 0.4a in a vacuum background, where a is
the size of the unit cell. Each unit cell is discretized into 25
points in both x and y directions.
Most photonic crystal devices operate in a band gap. The
structure at hand has a complete band gap for TM modes in
the frequency range 0.32ⱗ ␻a / 2␲c ⱗ 0.44,1 and does not
have a complete band gap for the TE polarization. Because
of this, we will hereafter consider the TM modes only.
The developed method allows one to treat structures unlimited in x direction, whereas in y direction the structure of
interest is confined within a supercell with imposed cyclic
boundary conditions. This leads to the finite size effects in a
photonic band structure. If the supercell consists of more
than one elementary cell, additional bands appear along with
the bands for infinite PC 关Figs. 5共a兲 and 5共b兲兴, as the result of
the imposed boundary conditions in the transverse direction.
A similar finite size effect emerges when air waveguides
共supercells兲 are attached to the system of interest. Even
though we send a wave from an open space, we use a finite
number of propagating modes. Solution of the eigenvalue
problem 共4兲 for the air supercell gives a discrete set of right2
冑 2 2
propagating eigenstates km
x = ␻ / c − 共2␲m / w兲 , where w is
the width of the supercell, and m is integer such that
max兩m兩 ⬍ ␻w / 2␲c. Thus, a wave incident from air effec-
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A. Microcavity
FIG. 5. 共Color online兲 共a兲 Band diagram for the rightpropagating TM mode of an infinite 2D photonic crystal 共␧r = 8.9,
d = 0.4a兲 in ⌫X direction. PC has a fundamental bandgap in the
frequency range 0.28ⱗ ␻a / 2␲C ⱗ 0.44 共filled with gray in the figure兲. The line in the fundamental bandgap corresponds to a guided
mode in a waveguide created by removing a central row of rods
from the PC as shown in the inset. 共b兲 Additional bands 共encircled兲
originated from the finite size effect. The waveguide 共supercell兲
contains three unit cells in the transverse direction as illustrated in
the inset. 共c兲 Bloch state propagating in the PC waveguide at
␻a / 2␲c = 0.38.
In this section we consider a microcavity defined in a
waveguide in an infinite PC. The waveguide is created by
removing a single central row of cylinders, such that in the
energy range corresponding the fundamental bandgap only
one waveguide mode can propagate. Band diagram of the
waveguide mode is shown in Fig. 5共a兲.
Three different cavities are introduced in order to show
the effect of geometry and demonstrate the importance of
proper design of a cavity. The first cavity is defined by two
rods placed on the lattice sites, see insets in Fig. 7. In the
second structure the diameter of the rods is doubled, and for
the third cavity we place two rods from each side of the
cavity to achieve better confinement. A dependence of the
transmission coefficient on the incoming wave frequency is
depicted in Fig. 7共a兲. We would like to stress that in the
calculation of the transmission coefficient, the incoming,
transmitted and reflected states are the Bloch states of a
waveguide 关shown in Fig. 5共c兲兴, such that all spurious reflections from PC interfaces or computational domain boundaries are avoided.
The fundamental parameter of cavity resonances is their
Q factor defined as Q = 2␲␻ⴱ 共stored energy兲/共energy lost
tively propagates only at certain incidence angles, determined by the ratio of the longitudinal and transverse wave
vectors tan ␣ = kmy / km
x , as illustrated in Fig. 6. Note that this
finite size effect 共caused by the cyclic boundary conditions in
the y direction兲 might in some cases represent a drawback of
the method.
FIG. 6. Dispersion relation for the air supercell of the width of
9a. Effective angles of incidence are determined by the angular
wave number m. Inset shows the effective angles of incidence for
m = −2 , −1 , 0 , 1 , 2.
FIG. 7. 共Color online兲 共a兲 Transmission coefficient of three cavity structures versus frequency. 共b兲 Intensity of the Ez component of
the electromagnetic field in the double-wall cavity at the resonance
共␻a / 2␲c = 0.3952兲.
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A. I. RAHACHOU AND I. V. ZOZOULENKO
per cycle兲, which can be rewritten in the following form:
⍀
Q=␻
4
冕
,
共47兲
Sindy
where ⍀TM = 兰关␧␧0兩Ez兩2 + ␮0共兩Hx兩2 + 兩Hy兩2兲兴dx dy and ⍀TE
= 兰关␮0兩Hz兩2 + ␧␧0共兩Ex兩2 + 兩Ey兩2兲兴dx dy characterizes the energy
stored in the system respectively for TM and TE polarizations and the integral over Sin is the incoming energy flux.
Equation 共47兲 can be also expressed as a well-known relation
Q = ␻ / ⌬␻ where ␻ is the resonant frequency and ⌬␻ is the
width of the resonant peak at half-maximum.
The resonance peak for the single-wall cavity is centered
at ␻a / 2␲c = 0.3952 and has Q factor 35.5. As expected, the
highest Q factor 共327.7兲 is achieved for the case of doublerod walls. Resonance peak in the case of larger rods is
shifted to the higher energy values 共␻a / 2␲c = 0.4281兲 because of the decrease of the effective size of the cavity. The
lower Q factor in this case 共25.07兲 is because the larger rods
disrupt destructive interference in a band gap of the PC.
Note that the width of the supercell used in the computations has to be large enough to ensure that the intensity of the
field decays to zero at the domain boundaries. At the same
time, it is desirable to have the size of the computational
domain as small as possible. For the present computations,
keeping this tradeoff in mind, we have chosen a supercell
consisting of seven unit cells in the y direction. This choice
seems to be sufficient, as the field intensity decreases by five
orders of magnitude within the length of two lattice constants
from the waveguide towards the supercell boundaries.
Finally, to confirm our results and to verify the developed
method, we performed calculations for the cavities and
waveguides in PC studied by Li et al.11 and found a full
agreement with their results.
B. Surface states
In the previous section we considered wave propagation
in an infinite photonic crystal. Another aspect of interest is
the effect of the surface in semi-infinite photonic crystals that
can accommodate a localized state 共surface mode兲 decaying
both into air and into a space occupied by the photonic
crystal.1,21 In the present section we study the coupling between an incident radiation and the surface states. Note that a
surface mode residing on the surface of an infinite 共in the y
direction兲 photonic crystal represents a truly bound state with
the infinite lifetime. However, because of the used cyclic
boundary conditions, our system is effectively confined in
the transverse direction. As a result, the translation symmetry
is broken, and the surface mode turns into a resonant state
with a finite lifetime. Using the developed method, we calculate the Q factor of the surface modes. Our findings indicate that the surface modes, thanks to their high Q factors,
can be used for lasing and sensing applications.
We study two semi-infinite photonic crystal structures that
support localized surface modes. In the first case a surface
row of cylinders is composed of half-truncated rods1 共structure 1兲, and in the second case the cylindrical and half-
truncated rods in the surface row are interchanged as shown
in Fig. 8 共structure 2兲. In order to calculate the Q factor of
the structures at hand, we illuminate the semi-infinite photonic crystal by an incidence wave 共that excites the surface
modes兲 and compute the intensity of the field distribution.
Note that the calculated field distribution includes the contributions from both the surface mode exited by the incident
light, as well as the incident and reflected waves. This leads
to a nearly constant off-resonance background in the dependence Q = Q共␻兲 that is caused by the contribution of the incident and reflected waves in the total field intensity in Eq.
共47兲. To remove this background we calculate the Q factor of
a structure without surface states. We choose this structure as
a semi-infinite photonic crystal with all identical cylindrical
rods, which is known not to support surface modes.1 Then
the obtained value is subtracted from the calculated value of
the Q factor of the system under study. Note that in the
calculation of the Q factor, the surface integration in Eq. 共47兲
is performed over the area depicted in Fig. 8.
Figure 9 shows a Q factor of structures 1 and 2 as a
function of the frequency of the illuminating light. For both
structures the Q factor reaches ⬃104. Figures 8共a兲 and 8共c兲
show Ez-field distribution for structures 1 and 2 at the resonance. For a comparison, a field distribution for a structure
that does not support a surface mode 共a semi-infinite photonic crystal with all identical cylindrical rods兲 is shown in Fig.
8共e兲. In the latter case the field intensity rapidly decays into
the bulk of the photonic crystal, whereas for the structures
supporting the surface modes, the intensity is strongly localized at the boundary row of rods. It is also worth mentioning
that for the latter case the intensity of the field in the surface
mode exceeds the incoming light intensity by four orders of
magnitude, such that the light intensity in the air region is
not visible in the figures 关compare Figs. 8共a兲 and 8共c兲 with
8共e兲兴.
One can easily estimate the position of the resonant
frequency for the surface modes. Indeed, the outermost row
of the cylinders 共where the surface state resides兲 can be
considered as a resonator with the characteristic resonant
wavelengths following from the cyclic boundary conditions and given by ␭␣ = 2␲ / k␣, where
k␣ =
2␲␣
,
w
共48兲
␣ is the mode number and w is the width of the waveguide.
The surface state for structure 1 exists only in a limited frequency interval, 0.33ⱗ ␻a / 2␲c ⱗ 0.37 共the dispersion relation of the surface mode of this structure is given in Ref. 1兲.
It follows from this dispersion relation that all the modes
given by Eq. 共48兲, except ␣ = 4, are situated outside this interval, whereas the mode ␣ = 4 corresponds to the frequency
␻a / 2␲c = 0.365. This estimated frequency agrees very well
with the actual calculated resonant frequency ␻a / 2␲c
⬇ 0.359; see Fig. 9.
Figures 8共b兲 and 8共d兲 show Poynting vector distribution
for both structures at the resonance. For the structure 1 the
Poynting vector is “curling” along the boundary, showing a
low speed of the surface state. In contrast, for the structure 2,
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LIGHT PROPAGATION IN FINITE AND INFINITE…
FIG. 8. 共Color online兲 Ez field and Poynting vector distributions for the structure 1 关共a兲,共b兲兴 and for the structure 2 关共c兲,共d兲兴 at the resonant
frequencies 共marked by arrows in Fig. 9兲. 共e兲 Ez field distributions for the structure that does not support surface modes 共a semi-infinite
photonic crystal with all identical cylindrical rods兲. In all cases the structures are illuminated by the incident wave propagating in the second
mode m = 2 共see Fig. 6兲.
the Poynting vector exhibits a rapid flow of energy along the
boundary. Another difference between these structures is a
very broad and rather strong “background” peak in the structure 2 in the region 0.34ⱗ ␻a / 2␲c ⱗ 0.35 共with Q factor up
to ⬃100兲. The presence of such a peak indicates that the
corresponding surface state can be rather robust to various
kinds of imperfections that are always present in real struc-
FIG. 9. Dependencies Q = Q共␻兲 for structures 1 and 2 共solid and
dashed lines, respectively兲. Arrows indicate the resonances for
which the field intensities and Poynting vectors are visualized in
Fig. 8.
tures and which are known to broaden the resonances and
lead to decrease of the Q factor.12 These two examples of
photonic crystals illustrate that with proper structure design
one can engineer and tailor properties of the surface states
into the required needs.
High values of the Q factors of the surface modes residing
at the interface of the photonic crystal structures indicate that
these systems can be used for lasing and sensing applications. The lasing effect has been demonstrated for different
photonic crystal structures including band-gap defect mode
lasers,22 distributed feedback lasers,23 and bandedge lasers.24
Utilization of the high-Q factor of the surface modes represent another way to sustain lasing emission. To achieve lasing effect careful design of the surface and surface mode
engineering should be performed and the developed method
seems to be a suitable tool for this purpose.
Note that, for the structures considered so far in this section, the transverse confinement giving rise to the surface
mode resonances is achieved rather artificially by imposing
the cyclic boundary conditions. Let us now outline the design of a realistic device that can be used for the experimental implementation of a surface state cavity. Consider a structure composed of a semi-infinite photonic crystal containing
only a finite number N of the surface rods of the reduced
diameter d = 0.5D defining a resonant cavity as illustrated in
Fig. 10. In this design the resonant modes of the cavity are
not affected by the choice of transverse boundary conditions,
as the electromagnetic field in the photonic crystal decays
very rapidly outside the cavity area.
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A. I. RAHACHOU AND I. V. ZOZOULENKO
FIG. 10. 共Color online兲 共a兲 Frequency dependence of the Q factor of a surface state photonic band gap cavity. Inset illustrates the
resonant cavity defined by the N = 6 surface rods of the smaller
diameter d = 0.5D placed on the photonic crystal surface. 共b兲 The
dispersion relation for the surface state for the semi-infinite photonic crystal. The dashed lines indicate the expected resonant wave
vectors for the modes ␣ = 5 , 6 , 7 and corresponding expected resonant frequencies.
Figure 10共a兲 shows a Q factor of the resonant cavity as a
function of the frequency of the illuminating light. In the
given frequency interval there are three cavity modes with
the Q factors of the order of ⬃105 − 106. It is expected however that in actual photonic structures realized typically in a
slab geometry, the Q factor will be reduced due to the radiative decay in the direction perpendicular to the plane of the
photonic crystal 共which has not been accounted for in the
present 2D calculations兲.
The estimation of the position of the resonant peaks for
the cavity at hand can be performed in the same way used for
the structures with truncated cylinders 共see Fig. 9 and related
discussion兲. The structure at hand can be considered as a
conventional Fabry-Pérot resonator whose resonant wavelengths are given by ␭␣ = 2␲ / k␣, with the wave vector k␣
= ␲␣ / w 关note the absence of factor 2 in contrast with Eq.
共48兲兴. From dispersion relation Fig. 10共b兲 it follows that only
modes ␣ = 5 , 6 , 7 are situated inside the frequency interval
where the surface mode exists. An estimation of the expected
positions for the resonant peaks for these modes is also
shown in Fig. 10 where the discrepancy between the expected and calculated resonance frequencies does not exceed
0.5%.
Figure 11 illustrates the intensity of the Ez component of
the electromagnetic field for the resonance mode ␣ = 6. As
expected, the field is localized in the cavity inside the rods,
and the intensity dies off very quickly both to the open space
and to the photonic crystal. The field intensity at different
rods in the cavity is expected to be determined by the overlap
of the ␣th eigenstate of the Fabry-Pérot resonator with
the actual positions of the rods in the cavity. This overlap
for the 6th mode is shown in Fig. 11, which agrees very well
with the actual calculated intensity distribution pattern.
We also performed calculations for different numbers of rods
FIG. 11. 共Color online兲 Lower panel: Calculated intensity of
the Ez component for the 6th mode of a resonant cavity shown in
Fig. 10. Upper panel: Expected field intensity at different rods is
given by the overlap of the 6th eigenstate of the cavity with the
actual positions of the rods.
N = 5 – 11 and we always find an excellent agreement between the calculated and expected resonant frequencies as
well as between the intensity distributions.
C. Waveguide coupled to the open space
The last example of application of the method presented
here is a semi-infinite photonic crystal with a waveguide
coupled to the surface; see Fig. 12. It has been recently demonstrated that a surface of a photonic crystal can serve as a
kind of antenna to beam the light emitted from the waveguide in a single direction.25,26 These findings outline the
importance of investigation of the surface modes in the photonic band-gap structures that can eventually open up the
possibilities to integrate such devices with conventional fiber
optic devices.
FIG. 12. 共Color online兲 Ez field distributions at the surface of a
truncated photonic crystal with a waveguide. 共a兲 The surface is
composed of cylinders with parameters identical to those in the bulk
of the crystal, and 共b兲 the surface cylinders are two times smaller
than the cylinders in the bulk.
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LIGHT PROPAGATION IN FINITE AND INFINITE…
In the present section we consider two different crystal
terminations to illustrate the effect of the surface on propagation of the light emitted from the waveguide. In the first
case the surface is composed of cylinders with parameters
identical to those in the bulk of the crystal, and in the second
case the surface cylinders are two times smaller than the
cylinders in the bulk.
The Bloch state propagating in a waveguide in the photonic crystal couples with the states in air and the resulting
field distributions is shown in Fig. 12. The first structure does
not support the surface mode, and hence the light intensity
distribution in the air region exhibits a typical diffraction
pattern. However, for the case of the second structure the
field distribution in the air region is drastically different. In
this case the Bloch state in the waveguide couples with the
surface state localized at the crystal termination, such that the
whole surface acts as a source of radiation.
waveguide 共supercell兲 geometry. The ␣th Bloch state in the
lattice can be written in the form
+
␣
兩␺␣典 = 兺 eik␣m␾m,n
am,n
兩0典,
+
共A1兲
m,n
where summation is performed over all lattice sites and the
␣
satisfies the conditions 共18兲. Substituting Eq.
function ␾m,n
共A1兲 into Eq. 共14兲, we arrive at the finite difference equation
valid for all sites m , n
+
+
␣
␣
␣
− um,m+1;n,neik␣␾m+1,n
− um,m−1;n,ne−ik␣␾m−1,n
vm,n␾m,n
␣
␣
− um,m;n,n+1␾m,n+1
− um,m;n,n−1␾m,n−1
=
冉 冊
␻⌬
c
2
␣
␾m,n
.
共A2兲
Consider now the incoming state 兩␺␣i 典, Eq. 共17兲. Substituting
Eq. 共17兲 into Eq. 共14兲 and using Eq. 共A2兲 we obtain
V. CONCLUSIONS
We have developed a method based on the recursive
Green’s function technique for the numerical study of photonic crystal structures. The method is proven to be an effective and numerically stable tool for design and simulation of
both infinite photonic crystals and photonic crystals with
boundaries. In the present method the Green’s function of the
photonic structure is calculated recursively by adding slice
by slice on the basis of Dyson’s equation. In order to account
for the infinite extension of the structure both into air and
into the space occupied by the photonic crystal we make use
of the so-called “surface Green’s functions” that propagate
the electromagnetic fields into infinity. This eliminates the
spurious solutions 共often present in the conventional FDTD
methods兲 related to, e.g., waves reflected from the boundaries defining the computational domain. The developed
method has been applied to scattering and propagation of
electromagnetic waves in photonic band-gap structures including cavities and waveguides. In particular, we have
shown that coupling of the surface states with incoming radiation may result in enhanced intensity of the electromagnetic field on the termination of the photonic crystal and a
very high Q factor of the surface modes localized at this
termination. This effect can be employed as an operational
principle for surface-mode lasers and sensors.
Note added. We note that since the submission of this
article, Xiao and Qiu have also reported similar results concerning a possibility to use the surface states as high-Q resonant cavities.27
冋 冉 冊册
L̂ −
␻⌬
c
2
␣ +
兩␺␣典 = eik␣ 兺 u0,1;n,n␾1,n
a0,n兩0典
+
n
␣ +
a1,n兩0典.
− 兺 u1,0;n,n␾0,n
共A3兲
n
Substituting this equation into Eq. 共22兲, calculating the matrix elements 具M + 1 , n 兩 ␺典 and 具0 , n 兩 ␺典, and using the relations
GM+1,0 = − GM+1,1U1,0⌫l ,
共A4兲
G0,0 = ⌫l − G0,1U1,0⌫l ,
共A5兲
that follow from Dyson’s equation, we arrive at Eqs. 共23兲
and 共24兲 determinig the transmission and reflection amplitudes.
APPENDIX B: DERIVATION OF THE DYSON’S
EQUATION
Let L̂0 be the operator describing an unperturbed system
and V̂ be a perturbation. In our case the unperturbed system
consists of several subsystems, e.g., m slices of the internal
structure and 共m + 1兲th slice, and the perturbation corresponds to the coupling 共hopping兲 between them 共see Fig. 2兲.
The operator of the total 共perturbed兲 system reads
L̂ = L̂0 + V̂.
ACKNOWLEDGMENTS
Partial financial support from the National Graduate
School in Scientific Computing 共A.I.R.兲 is acknowledged.
G−1 = 共␻⌬/c兲2 − L̂ = 共␻⌬/c兲2 − L̂0 − V̂ = 共G0兲−1 − V̂.
APPENDIX A: CALCULATION OF THE TRANSMISSION
COEFFICIENT
In this appendix we provide a detailed derivation of Eqs.
共23兲 and 共24兲. Consider first an infinite periodic structure in a
共B1兲
Let G0 and G be the Green’s functions of the unperturbed
and the total 共perturbed兲 systems, respectively. Starting with
the definition of the Green’s function 共21兲, we obtain
共B2兲
Multiplying this expression from the left with G and from
the right with G0 we arrive at Dyson’s equations
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A. I. RAHACHOU AND I. V. ZOZOULENKO
G0G−1G = G0共G0兲−1G − G0V̂G ⇒ G = G0 + G0V̂G.
共B3兲
Similarly one can also show that
G = G0 + GV̂G0 .
and calculating the matrix elements 共¯兲m,m;n,n⬘
+
⬅ 具0 兩 am,n . . . am,n
兩 0典, we arrive at the N ⫻ N system of linear
⬘
equations for the matrix elements of the Green’s function of
a single slice gm,
N
共B4兲
兺
n⬙=1
APPENDIX C: THE GREEN’S FUNCTION FOR A SINGLE
SLICE
2
册
␦n,n⬙ − lm,m;n,n⬙ gm,m;n⬙,n⬘ = ␦n,n⬘ ,
共C2兲
lm,m;n,n⬙ = vm,n␦n,n⬙
共C3兲
N
− um,m;n⬙−1,n⬙␦n,n⬙−1 − um,m;n⬙+1,n⬙␦n,n⬙+1 .
共C4兲
n=1
Note that because of the cyclic boundary conditions in the n
direction, the matrix elements um,m;1,N and um,m;N,1 are distinct from zero and defined according to um,m;N,1 = um,m;0,1
and um,m;1,N = um,m;N+1,N.
+
+
am,n − um,m;n,n+1am,n
am,n+1
lˆm = 兺 共vm,nam,n
+
− um,m;n+1,nam,n+1
am,n兲.
共C1兲
Using this operator in the definition of Green’s function 共21兲,
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where the matrix element of the operator l̂m reads
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155117-12
V
Paper V
Waveguiding properties of surface states in
photonic crystals
J. Opt. Soc. Am. B, vol. 23, pp. 1679–1683, 2006
A. I. Rahachou and I. V. Zozoulenko
Vol. 23, No. 8 / August 2006 / J. Opt. Soc. Am. B
1679
Waveguiding properties of surface states
in photonic crystals
A. I. Rahachou and I. V. Zozoulenko
Department of Science and Technology, Linköping University, 601 74, Norrköping, Sweden
Received January 3, 2006; revised March 17, 2006; accepted March 24, 2006; posted March 30, 2006 (Doc. ID 67038)
We propose and analyze novel surface-state-based waveguides in bandgap photonic crystals. We discuss the
surface-mode band structure, the field localization, and the effect of imperfections on the waveguiding properties of the surface modes. We demonstrate that surface-state-based waveguides can be used to achieve directional emission out of the waveguide. We also discuss the application of the surface-state waveguides as efficient light couplers for conventional photonic crystal waveguides. © 2006 Optical Society of America
OCIS codes: 130.2790, 160.3130, 230.7370, 240.6690.
1. INTRODUCTION
Photonic crystals (PCs) have attracted increasing attention in the past decade due to their unique properties and
possible applications in integrated optical and photonic
devices such as light-emitting diodes, delay lines,
waveguides, and lasers.1,2 Among the variety of PC-based
devices, waveguides play a crucial role not only as optical
interconnections but also as active elements in wide-angle
branches,3 channel add–drop filters,4,5 tapered couplers,6
optical switches,7 etc. Waveguides represent line defects
in periodic crystal structures supporting guided Bloch
modes whose frequency is located in the bandgap. These
modes are strongly confined within the waveguide region
and can propagate without loss to substantial distances.
In this paper we propose a novel type of waveguiding
structures, namely, waveguides that operate on surface
states of semi-infinite PCs and are located on the surface
of a PC. Employing surfaces of PCs as waveguides may
open up new possibilities for design and operation of photonic structures for feeding and redistributing light in
PCs.
Surface states reside at the interface between a PC and
open space, decaying into both media1 and propagating
along the boundary. In a square-lattice PC the surface
states appear in the bandgap when a boundary of a PC is
modified in some way by, e.g., truncating the surface rods,
shrinking or increasing their size, or creating more complex surface geometry.1,8–12 The surface modes in a semiinfinite PC represent truly Bloch states with the infinite
lifetime and Q factor, and consequently do not couple to
incoming or outgoing radiation. At the same time, it has
been demonstrated that when the translational symmetry
along the boundary of the semi-infinite crystal is broken,
the surface mode turns into a resonant state with a finite
lifetime, which can be utilized for lasing and sensing
applications.12,13 It has also been recently shown that
with the help of surface modes it is possible to achieve directional beaming from the waveguide opening on the
modified surface of a PC,14,15 where surface states,
coupled with outgoing waveguide radiation, suppress diffraction and focus the outgoing beam. At the same time,
0740-3224/06/081679-5/$15.00
to our knowledge there have been no studies of guiding
properties of PC surfaces.
To study surface states in PCs, we apply a novel computational method based on the recursive Green’s function technique.12 The advantage of this method is that it
allows us to calculate and use surface Bloch modes as
scattering states of the system, which makes it possible to
compute the transmission coefficients for surface modes
and corresponding field distributions.
2. SURFACE BAND STRUCTURE
We consider a semi-infinite square-lattice PC composed of
cylinders with ⑀ = 8.9 and diameter D = 0.4a (a is a lattice
constant) in an air background. We study two different
surface geometries, shown in Figs. 1(a) and 1(b), supporting the surface states where the outermost rods have reduced diameters d = 0.2a and enlarged d = 0.68a, respectively. This PC has a fundamental bandgap for TM
polarization in the range of 0.33⬍ ␻a / 2␲c ⬍ 0.44 and supports one surface mode for the case of Fig. 1(a) and two
modes for the case of Fig. 1(b). The surface modes for
these two structures show different patterns of field localization. For the structure of Fig. 1(a) the field intensity
has one maximum within each rod and extends into the
air, quickly decaying into the crystal. For low energies a
significant part of the field intensity extends into a wide
⬃5 – 10a air layer near the surface of the PC. This can be
attributed to the proximity between the dispersion curve
of the surface state and the light line, where the group velocity of the surface state, ␷ = ⳵E / ⳵k, is close to c [see Figs.
1(a) and 4(a)]. As the energy increases, the dispersion
curve moves away from the light line, and the field becomes mainly concentrated on the surface rods. For the
case of the structure of Fig. 1(b) with enlarged surface
rods, the field is mostly located within each cylinder and
has a node oriented either horizontally (mode I) or vertically (mode II). In contrast with the case of Fig. 1(a), the
intensity of both surface modes is mainly localized within
the surface rods and its extent to the air is small for the
whole energy range.
© 2006 Optical Society of America
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J. Opt. Soc. Am. B / Vol. 23, No. 8 / August 2006
A. I. Rahachou and I. V. Zozoulenko
3. EFFECT OF INHOMOGENEITIES
Fig. 1. (Color online) Band structures for TM modes in the ⌫X
direction of square-lattice PCs composed of rod diameters D
= 0.4a and permittivity ␧ = 8.9 along with the projected surface
modes. The diameters of the surface rods are (a) d = 0.2a and (b)
d = 0.68a. The thick black line denotes the light line. The lower
panels show the normalized intensity of the Ez component in different points of surface-mode dispersion curves.
Fig. 2. (Color online) Fragments of the band structures for TM
modes in the ⌫X direction of infinite square-lattice test PCs composed of rods with ␧ = 8.9 and diameters (a) d = 0.2a, (b) d
= 0.68a along with the projected surface modes of the semiinfinite PC of Fig. 1. Field distributions (Ez components) for the
corresponding bands are given in the insets.
Let us concentrate now on the surface-mode dispersion
and the intensity distribution in structures with reduced
and enlarged surface cylinders. To this end, we construct
two test PCs entirely consisting of corresponding surface
rods (i.e., with diameters d = 0.2a and d = 0.68). Their band
structures along with the projected surface states for the
structures shown in Figs. 1(a) and 1(b) are represented in
Fig. 2. The dispersion curve of the surface state for the
structure in Fig. 1(a) begins at the light line and remains
nearly linear up to ␻a / 2␲c ⲏ 0.40, where its slope slowly
decreases and finally reaches zero. Figure 2(a) demonstrates that the shape of the surface state closely follows
the valence band of the test crystal in the ⌫X direction.
The same situation also holds for the structure of Fig.
1(b), where both surface bands mimic the bulk levels in
the conduction band of the corresponding test PC [given
in Fig. 2(b)]. Field distributions for the corresponding
bands are given in the insets of Fig. 2 and outline the relations between the surface-state bands and the corresponding bands of the test PCs. It is worth mentioning
that both surface bands for the structure of Fig. 1(b) have
a lower velocity in comparison with the structure in Fig.
1(a). Fast surface states are known as the most suitable
for waveguiding applications, whereas slow modes can attract interest in structures for “slowing light”16 or in
surface-state cavities.12,13
Let us focus on the effect of inhomogeneities of the PC
(imperfections in the shape of the rods, their displacement, or variation of the refractive index throughout the
crystal, etc.) on the waveguiding properties of surface
states. It has been demonstrated previously that such imperfections strongly affect the performance of lasing
microcavities.17,18 We will show here that such imperfections can cause a profound effect on the waveguiding efficiency of the surface modes.
To study the effects of imperfections, we divide the system under study into three regions as shown in the upper
panel of Fig. 3. Two of the regions are left and right semiinfinite periodic structures (perfect waveguides for surface modes), and the block of the PC in between is an imperfect region. Utilization of Green’s function technique
allows us to use surface Bloch modes as scattering states
that propagate in perfect waveguides from infinity into
the imperfect region where they undergo scattering. Obviously, in the case when the scattering region is absent
(perfect waveguides are attached to each other), the Bloch
states propagate freely without any scattering.
Because the model is numerical, the discretization of
the circular rods of the PCs using a square grid lattice obviously leads to deviations from an ideal circular geometry, as illustrated in Fig. 3. These deviations can be
treated as inhomogeneities or roughness of the structure.
We stress that discretization of the rods in the periodic
waveguides is deliberately chosen to be different from
that for the central region (see Fig. 3). The central scattering region represents a PC of a width of 5 unit cells,
each of them discretized into 25 meshes 共⬃␭ / 50兲 in both
the x and y directions. Figure 4 shows the velocities of the
surface states in both structures and corresponding transmission coefficients. (We note that when the discretization of each cell in the central region is the same as for the
unit cells in the left and right waveguides, the transmission coefficient through the structure is unity.)
The transmission coefficients for each surface mode
drop quite rapidly in the energy regions corresponding to
Fig. 3. Discretization details of the (a) semi-infinite periodic
waveguides and (b) the central region. The upper panel shows
the structure under study, where the shaded regions denote ideal
semi-infinite waveguides, and the central region of the width of
5a represents an imperfect PC where scattering of the Bloch surface states takes place.
A. I. Rahachou and I. V. Zozoulenko
Vol. 23, No. 8 / August 2006 / J. Opt. Soc. Am. B
1681
rods in the surface-state waveguide gradually decreases
to zero in the region of the conventional PC waveguide as
shown in Fig. 5. In this device an incoming state in the
surface-mode waveguide region enters a tapered region
where it is adiabatically transformed into a conventional
waveguiding state.
The maximum achieved transmission reaches T ⬇ 0.8
(see inset to Fig. 5). We should also mention that careful
optimization of the surface geometry may further improve
the performance of surface-state waveguide couplers, but
such work is out of the scope of the present paper. We also
note that our two-dimensional calculations do not account
for the radiative decay in the direction perpendicular to
the plane of the PC.
Fig. 4. (Color online) (a) Velocity of different surface modes from
Fig. 1. (b) Transmission coefficient for surface modes propagating
in a nonideal surface-mode waveguide.
the low velocity of the surface state. This is because the
backscattering probability is greatly enhanced for the
low-speed states. Slow states for a structure with enlarged surface rods are the most strongly affected. Even
for five imperfect unit cells, the transmission coefficients
for both modes approaches 1 only in a very narrow energy
range, which makes these states hardly appropriate for
waveguiding purposes. At the same time, the transmission coefficient for the fast surface state in the structure
of Fig. 1(a) with the reduced boundary rods remains at 1
in a wide energy region up to ␻a / 2␲c ⬃ 0.40, which makes
it a better candidate for waveguiding applications.
B. Directional Emitter
The width of a conventional waveguide in a PC is of the
order of the wavelength of light ␭. Because of this, the
beam launched from a semi-infinite PC into open space is
diffracted at the waveguide opening in a strong angular
spread ⬃2␲. It has been recently shown that it is possible
to achieve directional emission out of PC waveguides with
corrugated terminations supporting leaky or evanescent
surface states.14,15 We demonstrate here that directional
emission with the angular spread much less than in conventional waveguides can also be achieved for the case of
surface-state waveguides coupled to air. Figure 6(a)
shows the Ez field intensity and a directional diagram for
the surface state propagating in a semi-infinite waveguide corresponding to the structure with the surface rods
of a reduced diameter. Most of the beam intensity is localized within the range of ⌬⍜ ⬃ 20°. It should also be noted
that the coupling of the surface state to air is rather high.
The inset to Fig. 6(b) shows the transmission coefficient
for the surface state in the semi-infinite surface
4. APPLICATIONS OF SURFACE-STATE
WAVEGUIDES
A. Light Coupler
Because of the unique location on the surface of the PC,
surface-state waveguides can be exploited in a variety of
novel applications. In this paper we focus on two of them,
introducing a novel light lead-in structure and sketching
the possibility to use a surface-state waveguide as a directional emitter.
Feeding light into waveguides in PCs composed of dielectric rods in an air background is a complicated challenge, as normally it requires extremely accurate positioning of a dielectric waveguide and precise mode
matching.19,20 Even in this case, diffraction at the waveguide termination usually hampers coupling, and the efficiency of such lead-in systems hardly exceeds 60%.
Other coupling techniques, such as utilization of adiabatic dielectric tapers,21 can improve the device performance but exhibits high sensitivity to parameters of the
tapers.
In this paper we propose a novel coupler based on
waveguiding properties of surface states. Figure 5 illustrates such a lead-in structure composed of a surfacestate waveguide on the left and a conventional tapered
PC waveguide on the right. The diameter of the surface
Fig. 5. (Color online) A lead-in coupler structure composed of a
surface-state waveguide on the left and a conventional tapered
PC waveguide on the right. The size of the surface rods gradually
decreases to zero in the central region where the surface-state
waveguide transforms into a conventional PC waveguide. The intensity distribution is shown for the Ez component of the electromagnetic field at ␻a / 2␲c ⬇ 0.365. Arrows depict the flow of the
Poynting vector. The transmission coefficient subject to the energy of incoming light is given in the inset. Parameters of the PC
correspond to the structure in Fig. 1(a).
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J. Opt. Soc. Am. B / Vol. 23, No. 8 / August 2006
A. I. Rahachou and I. V. Zozoulenko
5. CONCLUSION
We put forward a novel concept for waveguiding structures based on surface modes in bandgap photonic crystal
structures. We analyze the surface-mode band structure,
field localization, and the effect of imperfections on the
waveguiding properties of the surface modes. To illustrate
applications of the surface-state waveguides, we suggest a
new principle for feeding light into a photonic crystal
waveguide and demonstrate that a semi-infinite surfacestate waveguide can be used as a directional emitter.
A.
Rahachou,
[email protected]
[email protected];
I.
Zozoulenko,
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Fig. 6. (Color online) (a) Intensity distribution for the Ez component of the electromagnetic field in the surface-mode waveguide terminated to air for ␻a / 2␲c = 0.34. (b) Far-field radial
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␻a / 2␲. The inset shows the transmission coefficient for the surface state as a function of the frequency. Parameters of the waveguide correspond to the structure in Fig. 1(a).
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VI
Paper VI
Light propagation in nanorod arrays
J. Opt. A: Pure Appl. Opt., vol. 9 pp. 265–270, 2007
IOP PUBLISHING
JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS
J. Opt. A: Pure Appl. Opt. 9 (2007) 265–270
doi:10.1088/1464-4258/9/3/010
Light propagation in nanorod arrays
A I Rahachou and I V Zozoulenko
Solid State Electronics, Department of Science and Technology, Linköping University,
Norrköping, SE 601 74, Sweden
E-mail: [email protected] and [email protected]
Received 1 November 2006, accepted for publication 25 January 2007
Published 14 February 2007
Online at stacks.iop.org/JOptA/9/265
Abstract
We study the propagation of TM- and TE-polarized light in two-dimensional
arrays of silver nanorods of various diameters in a gelatin background. We
calculate the transmittance, reflectance and absorption of arranged and
disordered nanorod arrays and compare the exact numerical results with the
predictions of the Maxwell–Garnett effective-medium theory. We show that
interactions between nanorods, multipole contributions and formations of
photonic gaps affect strongly the transmittance spectra that cannot be
accounted for in terms of the conventional effective-medium theory. We also
demonstrate and explain the degradation of the transmittance in arrays with
randomly located rods as well as the weak influence of their fluctuating
diameter. For TM modes we outline the importance of the skin effect, which
causes the full reflection of the incoming light. We then illustrate the
possibility of using periodic arrays of nanorods as high-quality polarizers.
Keywords: nanorods, particle plasmons, Maxwell–Garnett theory, Green’s
function technique
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Resonance properties of nanoparticles have been observed for
centuries thanks to the beautiful colours of gold- and silverpatterned stained glasses. Over the last decade nanopatterned
materials have attracted even more increased attention due to
their unique electronic and optical characteristics. Nowadays,
they are considered as promising candidates for a wide variety
of applications in subwavelength waveguiding [1, 2], enhanced
Raman scattering spectroscopy [3], nonlinear optics [4],
photovoltaics [5], biological/medical sensing [6] and many
others.
A characteristic size of metallic nanoparticles d is about an
order of magnitude smaller than the wavelength of incoming
light λ, which can excite collective oscillations of electron
density inside the particle—plasmons. The plasmon excitation
results in an enhanced extinction (extinction = absorption
+ scattering) as well as an increased intensity of the
electromagnetic field near the particle [7].
The important issue that makes nanoparticles so attractive
for sensing applications is the effect of the geometry and
size of nanoparticles and the surrounding environment on the
position of the plasmonic resonance [7–10]. For example,
1464-4258/07/030265+06$30.00
the presence of antibodies in cells affected by cancer modifies
the environment for gold nanoparticles placed on a tissue and
results in a shift of extinction peak that can be easily imaged
by conventional microscopy [11].
Recently it has also been demonstrated [12, 13] that
embedding metallic nanoparticles into a polymeric matrix
provides a larger contrast in the effective refractive index of
the blend material, being much lower or higher than that of a
pure polymer. Developing such materials can facilitate creating
high-contrast-index photonic polymer crystals.
Nanoparticles assembled in nanochains can also be
applied as subwavelength waveguides [2, 14, 15]. In the case
of closely spaced particles the coupling (and light propagation)
arises from the evanescent dipole field from each particle,
which excites a plasmon on its neighbour. This excitation
travels along the chain, making the electron density within all
the particles oscillate in resonance.
In the present paper we will focus on light propagation
in large arrays of infinitely long nanorods. Prototypes of such
arrays have been recently fabricated experimentally [16, 17].
These arrays represent randomly oriented or aligned long rods
(or spikes) of a material (dielectric or metal), several tens
of nanometres in diameter. Despite significant progress in
© 2007 IOP Publishing Ltd Printed in the UK
265
A I Rahachou and I V Zozoulenko
Blend
Gelatin
I
R
Gelatin
ε′
ε ′′
ε ′ , ε ′′
nanofabrication technologies, to our knowledge, however, the
theoretical description of light propagation in nanorod arrays
is still missing.
This paper is organized as follows. In section 2 we
outline transmittance properties of nanorod arrays within
the framework of the Maxwell–Garnett effective-medium
theory. In section 3 we present numerical modelling of light
propagation through periodic arrays of nanorods and compare
the results with the predictions of the Maxwell–Garnett theory.
In section 4 the effect of various types of disorder is studied.
T
A=1-T-R
2. Effective-medium theory
We consider a gelatin matrix with an embedded twodimensional array of silver nanorods. The effective dielectric
function εeff (ω) of that composite can be estimated from
Maxwell–Garnett theory, developed more than 100 years
ago [7]:
εrod (ω) − εmat
εeff (ω) − εmat
= f
,
εeff (ω) + 2εmat
εrod (ω) + 2εmat
(1)
where f = S2 /S1 is the filling factor of the nanorods
embedded into the matrix, S1 is the active area of the matrix
and S2 is the total cross-sectional area of the nanorods. The
dielectric function of the gelatin matrix is εmat = 2.25. The
dielectric function εrod (ω) of the nanorods is taken from the
SOPRA database1 for the bulk material. The Maxwell–Garnett
theory is valid for relatively small nanoparticles (nanorods) (up
to several tens of nanometres) at low concentrations (less than
30%). The dielectric function (here and hereafter all the spectra
are given with respect to the wavelength of light in a vacuum
λ0 ) of the Ag(10%)–gelatin blend is presented in figure 1(a).
The dielectric function in figure 1(a) characterizes the
blend as a highly dispersive lossy material with an absorption
peak centred around 414 nm. According to Mie’s theory this
peak corresponds to the plasmon resonance of a single Ag
spherical nanoparticle in gelatin. The position of the peak
obeys the well-known relation εrod = −2εmat [7]. In order
to study light propagation through a layer of the blend we
consider a 2D ‘sandwich-like’ structure consisting of semiinfinite gelatin ‘waveguides’ connected to the blend region (see
inset to figure 1(b)). The structure is assumed to be infinite
in the z direction; thus the solution to Maxwell’s equations
decouples into TE (where the vector of a magnetic field is
parallel to z ) and TM (where the vector of an electric field
is parallel to z ). The transmission, reflection and absorption
for both polarizations are given in figures 1(b) and (c),
respectively.
It is easy to see that, for both TE and TM polarizations,
there exists a gap (or a stop-band) in the transmission caused
by the enhanced absorption near the extinction resonance peak.
However, the reflectance and absorption within the stop-band
possess distinct behaviour for different polarizations. When
the real part of the dielectric constant of the blend becomes
negative (400 < λ0 < 425 nm) the reflectance of the TE
mode increases due to increased contrast against the dielectric
function of the gelatin matrix (which causes a dip in the
absorption). At the same time, for TM-polarized light the
1
URL: http://www.sopra-sa.com/more/database.asp.
266
λ
Figure 1. (a) Dielectric function of a blend of silver nanorods
(nanoparticles) with a concentration 10% embedded into a gelatin
background. Transmittance, reflectance and absorption of the TE (b)
and TM (c) modes propagating through a 0.7 μm thick layer of
Ag(10%)–gelatin blend are shown. Inset in (b) outlines the system
under study.
reflectance sharply increases up to 1 because of the metallic
character of the blend in this region and the enhanced skin
effect. For both polarizations Bragg’s reflections from the
boundaries of the blend region, manifesting themselves as
minima and maxima, are clearly seen for λ0 > 500 nm.
Despite its adequacy for small isolated circular nanoparticles, a simple Maxwell–Garnett theory, however, has certain
limitations. Namely, it does not account for the shape and distribution of metal clusters in the dielectric medium, neglecting
important polarization properties of both single non-circular
particles and their arrangements [18, 19]. In order to incorporate these features and study transmission characteristics of
periodic and disordered nanorod arrays we apply the recursive
Green’s function technique [20].
3. Periodic nanorod arrays
We now focus on 2D arrays of infinitely long silver nanorods
arranged as a square lattice in a gelatin background. Keeping
the filling factor of Ag, f = 10%, constant, we consider two
cases: (a) a finite-size lattice with thickness a = 0.7 μm of
nanorods with the diameter d = 10 nm, and (b) the lattice of
the same width assembled from nanorods of 60 nm in diameter,
see figure 2. Lattice constants are 29 and 175 nm for cases (a)
and (b), respectively.
Such a choice of nanorod sizes is motivated by the
essential difference in polarization properties of small and
large nanoparticles [10]. If the nanoparticle is small enough
(d λ0 ), according to Mie’s theory, only the dipole
plasmonic oscillations contribute to the extinction spectra,
whereas for larger particles higher-order resonances contribute
Light propagation in nanorod arrays
(a)
(b)
y
a
z
a
x
Figure 2. Arrays of silver nanorods with diameter (a) 10 nm, and
(b) 60 nm embedded in an infinite gelatin background. For both
cases the thickness of the layer a = 0.7 μm and the filling factor
f = 10%.
λ
λ
Figure 4. Transmittance, reflectance and absorption of a single (a),
a pair of horizontally (b) and vertically (c) aligned, and (d) four
coupled nanorods for the TE-polarized light. The inter-rod distances
are taken as 29 nm, equal to the lattice constant for the array
(figure 2(a)). The size of the computational domain is also the same
as that for the nanorod arrays in figure 2.
λ
λ
Figure 3. Transmittance, reflectance and absorption of a TE mode
travelling through the square arrays of nanorods with diameter
(a) 10 nm and (b) 60 nm (see figure 2 for details).
to the spectra as well [10]. Using the recursive Green’s
function technique we perform numerical simulations for both
TE and TM polarizations of light falling normally from the left
to the boundary between gelatin and the blend.
3.1. TE-modes
When a nanosized metallic nanoparticle is illuminated by
light, the electric components of an electromagnetic field
excite collective oscillations of electronic plasma inside the
particles—plasmons. If the particles are arranged into chains,
these plasmonic oscillations possess a resonant character that
facilitates the propagation of light along the chain. Such chains
have been intensively studied in the literature [2] as promising
candidates for subwavelength waveguiding.
Let us irradiate the array of infinitely long nanorods with
TE-polarized light. In this case E x and E y components of the
electromagnetic field excite coherent plasmonic oscillations on
each nanorod. Figure 3 shows the calculated transmittance,
reflectance and absorption of a TE mode propagating through
the arrays of nanorods.
Small nanorods. Let us first concentrate on an array of
nanorods with the diameter 10 nm (figure 3(a)). In the spectra
one can clearly distinguish two regions, namely the region
of high absorption (λ0 < 600 nm), containing a wide main
absorption peak at 414 nm, two minor peaks at 350 and 530 nm
and the region of high transmittance (λ0 > 600 nm). Now we
will take a closer look at these regions separately.
The position of the main extinction resonance agrees well
with that obtained from equation (1). However, in contrast to
the Maxwell–Garnett theory, the spectrum contains two minor
peaks near 350 and 530 nm. In order to explain them one needs
to account for the effect of coupling between several nanorods.
For this reason we compare light propagation through (a) a
single isolated nanorod (diameter 10 nm), (b) two coupled
nanorods aligned parallel to the light propagation direction, (c)
those aligned perpendicularly and (d) four coupled nanorods.
These four cases are presented in figure 4.
For the case of a single isolated rod (figure 4 (a)) only
one peak near 410 nm emerges, which is in good agreement
with the analytical value of 414 nm. For the cases (b) and
(c) of twin coupled nanorods the additional peaks, centred at
355 and 620 nm respectively, appear. Their origin has been
thoroughly studied in [18] and clarified in terms of enhanced
[case (b)] and weakened [case (c)] restoring forces between
the particles. However, for the system of four particles (d)
these forces partially compensate for each other, and the minor
resonances move closer towards the main peak.
Let us now focus on the wavelength region λ0 > 600 nm,
where TE-polarized light propagates at high transmittance.
In order to understand this behaviour, we complement the
transmission coefficient with the band diagram of the nanorod
array. It should be mentioned that, in general, a band
diagram represents propagating Bloch states (states with real
eigenvalues). However, as the metallic rods (or nanoparticles)
are absorbing, all the states in the blend will be decaying and
eventually die off at infinity. Yet all the Bloch eigenvalues in
such systems have imaginary components. In figure 5(a) we
represent a band structure (real parts of eigenvalues) in the X
direction for the states with the smallest imaginary parts.
267
A I Rahachou and I V Zozoulenko
ω
ω
π
π
λ
λ
π
π
Figure 5. Band diagrams of the nanorod arrays from figures 2(a) and (b), respectively. The dashed line outlines the light cone.
Large nanorods. For nanorods with the diameter 60 nm, the
position of the main extinction peak agrees with that of the
small particles. However, there is an essential difference in
the physics behind this. When the diameter of a nanoparticle
increases, higher-order dipole oscillations now contribute to
the resulting extinction spectrum [7]. It has been recently
shown [10] that the peak centred at ≈400 nm is due to
the quadruple resonance of a nanorod, whereas the dipole
resonance is redshifted and overlaps with the region of the
enhanced reflectance (500 < λ0 < 700 nm). The indication
in favour of this interpretation is a narrower width of the
stop-band in the transmission (60 nm against 100 nm in the
case of small rods). This is because the higher-order dipole
interactions causing the stop-band behaviour for the case of
large nanorods are generally weaker.
Now let us clarify the origin of the high-reflectance region.
The lattice constant for this structure is 175 nm. This is of the
same order as the wavelength of light, such that the structure
effectively represents a two-dimensional photonic crystal. The
plasmonic band in figure 5(b) extends from ωa/2πc = 0 to
0.4 (λ0 ≈ 660 nm) where it experiences a photonic bandgap
that causes the high reflectance of the structure. This bandgap
overlaps with the tail of the extinction peak near 500 nm (see
figure 3).
ε ′, ε ′′
The dispersion curve in figure 5(a) has a small bump
around 550 nm, which is caused by the minor extinction
resonance. The band is located very close to the light line that
results in a rather strong coupling between the incoming light
and the plasmonic Bloch states of the blend region. Such strong
coupling explains the high transmittance in the red wavelength
region.
ε′
ε ′′
λ
Figure 6. (a) Transmission, reflection and absorption coefficients of
the TM mode through a nanorod array of d = 10 nm. Due to the skin
effect light does not penetrate the blend region. For λ0 < 328 nm the
real part ε of the dielectric function of silver (b) becomes positive
and the transmission coefficient abruptly increases.
rods and thus there is no plasmonic contribution in overall
transmission.
However, for very short wavelengths (λ0 < 328 nm) the
real part of the dielectric function of silver becomes positive
(see figure 6(b)) and the blend behaves like a lossy dielectric
rather than a metal. This results in non-zero transmission in
this region.
The obtained results clearly show that resonant plasmonic
oscillations in periodic nanorod arrays represent a dominating
light propagation mechanism for the TE-polarized light,
whereas for the TM modes the nanorod structure represents
practically a perfect screen. This features can be utilized in a
nearly 100% effective polarizer.
3.2. TM modes
Let us now consider the TM polarization of the incoming light.
Figure 6(a) shows the transmittance, reflectance and absorption
of the TM-polarized light for the small nanorods. In contrast to
the Maxwell–Garnett picture (figure 1), almost for the whole
wavelength range under study light does not penetrate the
region occupied by nanorods and gets fully reflected back,
resulting in zero transmittance. This discrepancy can be
explained by the skin effect on the silver rods. At the same
time, the Maxwell–Garnett theory (1) fully disregards the
important screening properties of the rods, simply averaging
the effective dielectric constant over the structure. It is also
worth mentioning that, as we consider infinitely long nanorods,
the incoming TM mode does not excite any plasmons on the
268
4. Disordered nanorod arrays
As we have demonstrated in the preceding section, TEpolarized incident light in the off-resonance wavelength region
propagates through periodic arrays of small nanorods at very
high transmission. Now we introduce some disorder in this
array and consider two separate cases, namely when nanorods
are arranged in a square lattice but have randomly varying
diameter, and rods of equal diameter, randomly distributed
within the layer. For both cases the filling factor f = 10%
is kept constant and the distribution is taken as uniform.
Figures 7(a), (c) and (b), (d) demonstrate the transmittance,
reflectance and absorbtion for both cases.
Light propagation in nanorod arrays
dielectric function is derived by inserting the total aggregate
polarizabilities instead of that of a single isolated particle into
the Maxwell–Garnett theory.
5. Conclusions
λ
λ
Figure 7. Transmission, reflection and absorption coefficients of the
TE-polarized light propagating through disordered nanorod arrays.
(a), (c) Two different configurations of nanorods arranged in a square
lattice, their d diameter randomly varying from of 5 to 20 nm. (b),
(d) Nanorods with fixed d = 10 nm are randomly distributed within
the layer. Insets show the actual geometries of the structures.
The transmission characteristics for the structure with the
random diameter of nanorods (figures 7(a) and (c)) closely
resemble those for the array of fixed-sized nanoparticles
(figure 3 (a)). The main difference is that the weaker dipole
interactions between adjacent particles of different diameters
cause a minor narrowing of the stop-band (60 nm versus
100 nm in the ideal case) and a slight degradation of the minor
extinction peak. It should be emphasized that the transmission
properties of arrays with different distributions of nanorod
diameter (figures 7(a) and (c)) are virtually the same.
For the case of the randomly distributed equal-sized
nanorods (figures 7(b) and (d)) the situation changes. In
contrast to the previous case of the ordered nanorod array
with random diameter, the absorption spectra in the region
λ0 > 600 nm are extremely sensitive to the geometry of the
structure. It is interesting to note how clustering of nanorods
manifests itself. The overall absorption in the region λ0 >
600 nm is much higher (and the transmission is lower) in
comparison to the periodic lattice, as it consists of the averaged
multiple absorbtion peaks of closely situated, touching or
overlapping nanorods. Since the inter-rod distances are not
constant any longer, each single rod is now affected by many
dipole interactions of different strengths from neighbouring
rods. Reflectances, however, are practically identical and
not significantly higher than for the periodic case. It can be
explained that due to its non-periodicity this structure absorbs
better than it reflects. The clustering and more complex
interactions of nanorods influence the region λ0 < 600 as well.
The main and minor absorption peaks for the structure, see
figure 7(b), almost overlap, whereas for figure 7(d) they are
still well separated.
We should specially mention that, in order to incorporate
the effect of nanoparticle aggregates into Maxwell–Garnett
theory, several approaches were suggested [7] (see also
references therein).
In that case the effective-medium
We have studied the propagation of TE- and TM-polarized
light in two-dimensional arrays of silver nanorods in a gelatin
background. In order to calculate transmittance, reflectance
and absorption in arrays of ordered and disordered nanorods
we applied the recursive Green’s function technique and
compared the obtained numerical results with predictions
of the Maxwell–Garnett effective-medium theory. We have
demonstrated that this theory describes adequately only the
case of the TE-polarized light propagating in ordered arrays
of small (∼ 10 nm), well-separated nanorods and only in the
frequency interval outside the main plasmonic resonance.
Our numerical calculations outline the importance of
geometrical factors such as the size of the rods and their
distribution.
In particular, we have demonstrated that
interaction between adjacent nanorods brings a significant
contribution to the transmission spectra, which is manifested as
additional absorption peaks (that are missing in the effectivemedium approach).
The Maxwell–Garnett theory also
disregards both the impact of higher-order dipole contributions
and the formation of photonic bandgaps in the case of arrays of
large nanorods.
We have also studied the effect of disorder on the
transmittance of the nanorod arrays. We have introduced two
types of disorder: (a) ordered array with randomly varying
nanorod diameters, and (b) a random distribution of nanorods
of the same size within the blend. The disorder in rod
placement leads to a strong suppression of the transmission
(and the enhanced absorption) due to plasmonic resonances
related to the clustering of the rods. We have demonstrated
that clustering effects are sensitive to the actual geometry of
the structure. In contrast, the impact of randomly varying
diameters of the rods is much less profound.
Despite its partial adequacy for the TE-polarized light,
the Maxwell–Garnett effective-medium theory is shown to be
invalid for the case of TM polarization. It simply averages
the effective dielectric function inside the blend, missing the
important screening properties of the metallic nanorods and
characterizing the blend as a (partially) transparent medium.
In contrast, the numerical modelling shows the strong skin
effect that fully prohibits the propagation of the TM modes
through the structure. The region of high transmittance for
the TE modes and the strong skin effect for the TM modes
makes the nanorod arrays promising candidates for highquality polarizers.
Acknowledgments
We would like to thank Olle Inganäs for stimulating and fruitful
discussions. We acknowledge partial financial support from the
Center for Organic Electronics at Linköping University. Useful
conversations with Nils-Christer Persson are also appreciated.
269
A I Rahachou and I V Zozoulenko
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VII
Paper VII
Surface plasmon increased absorption in
polymer photovoltaic cells
submitted to Appl. Phys. Lett., 2007
Surface plasmon increased absorption in polymer photovoltaic cells
Kristofer Tvingstedt, Nils-Krister Persson, and Olle Inganäs
Biomolecular and Organic Electronics, Center of Organic Electronics,
IFM, Linköping University, SE-581 83 Linköping, Sweden∗
Aliaksandr Rahachou and Igor V. Zozoulenko
Solid State Electronics, Center of Organic Electronics,
ITN, Linköping University, SE-601 74 Norrköping, Sweden
We demonstrate the triggering of surface plasmons at the interface of a metal grating and a photovoltaic bulk heterojunction blend of alternating polyfluorenes and a C60 derivative. An increased
absorption originating from surface plasmon resonances is confirmed by experimental reflection
studies and theoretical modeling. Plasmonic resonances are further confirmed to influence the extracted photocurrent from devices. More current is generated at the wavelength position of the
plasmon resonance peak. High conductivity forms of the polymer PEDOT:PSS are used to build
inverted sandwich structures with top anode and bottom metal grating, facilitating for triggering
and characterization of the surface plasmon effects.
Conjugated polymers have emerged as promising materials for inexpensive and flexible photovoltaic cells, as
liquid based printing techniques enables production on
large scale at a lower price than for inorganic based solar
cell [1 ]. A new route to increase the photon absorption in
thin film photovoltaics is to exploit surface plasmon resonances at the polymer/electrode interface. Present day
polymer photovoltaic cells are comprised by blends of a
conjugated polymer and a soluble C60 derivative. These
bulk heterojunction solar cells now show power conversion efficiencies up to 4–52,3 The quantum efficiency of
thin film organic solar cells is however still limited, and
one cause is the comparatively low carrier mobility. A
thinner film between electrodes will lower the probability for charge recombination and increase the carrier drift
velocity by the higher electric field. However, a minimum
film thickness is always required for sufficient photon absorption. The appeal of surface plasmons (SP) for thin
film photovoltaics is due to the presence of an intense optical near field enhancement at the metal/dielectric interface. The strong field associated with a SP is evanescent
into dielectric materials up to half the involved wavelength. In the conjugated polymers exploited for photovoltaic applications the evanescent plasmon field extends
more than 100 nm whereas it only extends a few nanometers into the metal. Hence the SP seems suited for absorption and exciton generation enhancement in thin film
photovoltaic devices.
Surface plasmons are best described as confined light
waves that propagate along the conducting surface of a
metal4 . They are trapped on the metal surface due to
strong interaction with the free charge carriers of the conductor. Recent studies5,6 exploiting metal nanoparticles
on top of inorganic Si diodes have displayed increased
absorption and increased photocurrent response. Previously, organic small molecules have been coevaporated
together with metal clusters to excite localized plasmons
that couple to the active material, demonstrating both
increased absorption and photocurrent6–8 . Though no
reports are known on organic photovoltaic devices using
periodic gratings for triggering propagating SPs, diffraction gratings have been used to study the effects of SPs
in organic light emitting diodes by Hobson et al9,10 .
The dispersion relation of a surface plasmon, propagating in the x-direction can be written as:
µ
¶1/2
ε1 ε2
kx = k0
,
(1)
ε1 + ε2
where k = 2π/λ is a free-space wave vector, ε1 and ε2 are
the permittivities of the dielectric and metal respectively.
The position of the plasmonic resonance λ = 2π/k0 for
the normal incidence of light upon a grating can roughly
be estimated by the following relation:
µ
¶1/2
2π
ε1 ε2
= k0
,
(2)
d
ε1 + ε2
where d is the period of the grating. To trigger a surface plasmon at a metal/dielectric interface with periodic nanostructures it is further essential that the aspect ratio of the metal structures is sufficient to enable
coupling. We have used soft lithographical replication
techniques11,12 to generate metal gratings with high optical quality. Previous grating printing methods13,14 exploits soft embossing which is not fully capable of replicating deep nanostructures in thin films. Therefore we
have preferred to deposit the thin polymer blend films
on a prestructured Al metal nanostructure with sufficient aspect ratio. Figure 1(a) displays an AFM scan
of such a grating with period of 277 nm and height of
50 nm. To enable charge collection of generated carriers a thin layer of highly conducting diethylene-glycol
doped PEDOT:PSS15 is deposited on top of the polymer/PCBM blend. The device is hence best described as
an inverted polymer solar cell with a periodically nano
structured Al bottom cathode [Fig. 1(b)]. The oxide
formed on Al blocks charge transport; therefore to extract current from such an inverted cell it is necessary to
use a thin layer of Ti/TiOx on top, which has recently
been demonstrated16 to enable charge collection.
2
(a)
(a)
(b)
(b)
FIG. 1: (a) AFM scan of the surface of the metal grating
prior to polymer deposition. (b) Schematic drawing of the
inverted solar cell with nanostructured bottom cathode and
transparent PEDOT:PSS top anode.
FIG. 2: (a) Refractive index (N) and extinction coefficient
(k) for 1:4 blends of APFO3 and PCBM. (b) N and k for
1:3 blends of APFO Green5 and PCBM. Insets display the
molecular structure of the exploited materials.
Two polymers, APFO317 and APFO Green518 , have
been blended with PCBM (Fig. 2). The dielectric function for the blends, the refractive index (N) and extinction coefficients (k) obtained by ellipsometry19 is presented in Fig. 2 and used as input for the simulation
of the SPs. The triggering of SPs at the metal/organic
interface is demonstrated by measuring the reflection of
monochromatic polarized light from samples placed at
the backside of an integrating sphere, thereby collecting
reflected light at all angles. As the samples are illuminated with light polarized parallel to the grating lines, no
major difference from the reflection from a planar sample
can be identified, since this polarization is not capable
of launching SPs. When samples are illuminated with
light polarized perpendicular to the grating lines, a new
feature is observed. A strong resonance reflection dip
occurs for the APFO3/PCBM blend at a wavelength of
620 nm where the blend itself has little absorption. For
the APFO Green5/PCBM blend the plasmon resonance
is located at a wavelength of 550 nm. This resonance
fills up the dip in the absorption spectra of this material
and the sample accordingly appears dark and colorless
in reflection. The absorptance A=100-R for the blends
on metal gratings under polarized illumination vs. the
absorptance from planar cells is given in Fig. 3.
In the simulations we consider a supercell containing 5
periods of sinusoidal grating profile, slightly distorted to
make it more realistic. Adjacent supercells are connected
through cyclic boundary conditions. In order to account
for the non-ideality of the grating surface, we have introduced a randomized surface roughness with an amplitude
of 5 nm, which is a typical averaged AFM experimental value. We discretize the structure with uniform grid
(element size is 3.4 nm) and illuminate it with light polarized perpendicularly with respect to the grating lines.
Having calculated the transmittance T and reflectance
R from the recursive Greens function technique, the absorptance can be found as A = 100-T-R.
First, we calculate the absorptance spectrum for
APFO3/PCBM. A polymer blend with thickness d150
nm is deposited directly onto the Al grating. The simulated and measured absorptance spectra are given in Fig.
4(a). The spectrum contains two resonance peaks at 450
and 625 nm, which agree extremely well with the experimental curve. The two peaks have however completely
different nature. The peak at 625 nm is a first-order plasmonic resonance and its position agrees quite well with
a direct analytical estimation (598nm) from Eqn. (2).
At this resonance the electromagnetic field (inset, Fig.
4b) is localized within the 50 nm region near the grat-
3
(a)
(a)
(b)
(b)
(a) Measured absorptance (A=100-R) for
FIG. 3:
APFO3/PCBM on Al grating illuminated with different polarizations compared to a planar absorptance spectra. (b) Absorptance spectra for APFO Green5/PCBM on Ti/Al grating
illuminated with different polarizations compared to a planar
absorptance spectra.
ing surface, and its intensity is up to 7 times higher on
the boundary than in the bulk of the polymer. The resonant peak at 450 nm represents a standing wave confined
by total internal reflection at the polymer/air interface
and the metal grating reflection. For the APFO Green
5/PCBM we model a 90 nm thick polymer blend layer,
deposited onto the Al grating coated with a 5 nm thin
Ti interfacial layer. Fig. 4b represents the computed absorptance spectrum and the simulated field distribution
inside the cell. As the APFO Green 5/PCBM layer is
thinner than the APFO3/PCBM, the standing wave is
no longer supported and there is no corresponding peak
in the spectrum. The plasmonic resonance for this structure is calculated to be centered at 555 nm.
The surface plasmon resonance for the two polymers
substantially alters the absorption profile, compared to
planar samples. We measure the external quantum efficiency (EQE) under illumination of the sample with polarized light. The low bandgap APFO Green5 displays
a clear influence around the SP resonance wavelength of
555 nm when illuminated with perpendicularly polarized
light (Fig. 5). For the APFO3/PCBM cell no influence
could be observed at the SP resonance wavelength of 620
FIG. 4: (a) Calculated and measured absorptance spectrum
of APFO3/PCBM deposited directly on the Al grating. The
insets show the calculated spatial distribution of the Hy component of the electromagnetic field in the polymer at the
plasmonic resonance (625 nm) and for a standing wave peak
(450 nm). (b) Calculated absorptance spectrum of APFO
Green 5/PCBM on the Al grating coated with a 5 nm thick
Ti interfacial layer. The inset demonstrates the spatial distribution of the Hy -component of the electromagnetic field in
the polymer at the plasmonic resonance (555 nm). The lower
dark part of the insets corresponds to the sinusoidal shaped
metal grating where no field is present.
nm. The different behavior is attributed to mismatch of
the energy of the SP resonance. This energy is lower than
the bandgap of the APFO3/PCBM but higher than the
bandgap of APFO Green5/PCBM. We therefore suggest
that no significant coupling from the SP to excitation of
APFO3/PCBM can occur.
In conclusion, this study demonstrates the possibility of influencing the generated photocurrent from thin
film organic photovoltaic cells with propagating surface
plasmons launched on nanostructured electrodes. An increase of photocurrent is observed at the resonant position of the SP, when the SP resonance has higher energy than the bandgap of the absorbing polymer. The
good match between the measured and the simulated reflectance demonstrates the strong influence surface plasmons have on absorption in these thin film devices. At
present gratings are far from ideal, due to inhomogeneity
and the presence of the required Ti interfacial layers. We
consider these effects on the plasmon intensity to be a
4
drawback that requires a remedy that will be carefully
studied elsewhere. We suggest that the strong optical
field intensity associated with a propagating SP wave is
responsible for enhanced exciton generation inside the
photoactive bulk heterojunction blend. The condition
deduced from the comparative study is that the SP energy must be higher than the bandgap of the material.
Acknowledgments
FIG. 5: EQE for APFO Green5/PCBM when illuminated
with light polarized perpendicular or parallel to the grating
lines of the sample. At the resonance position of 555 nm we
see an increase of photocurrent when exciting a plasmon.
∗
1
2
3
4
5
6
7
8
9
10
11
12
Electronic address: [email protected]
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