Impact of Morphology and Scale on the Physical Properties of
Periodic/Quasiperiodic Micro- and Nano- Structures
by
Lin Jia
Bachelor of Science, Physics
Peking University (2007)
Submitted to the Department of Materials Science and Engineering
in Partial fulfillment of the Requirements for the degree of
Doctor of Philosophy in Materials Science and Engineering
at the
Massachusetts Institute of Technology
June 2012
© 2012 Massachusetts Institute of Technology. All rights reserved.
Signature of author .............................................................................................................................
Department of Materials Science and Engineering
April 13th, 2012
Certified by …....................................................................................................................................
Edwin L. Thomas
Morris Cohen Professor of Department of Materials Sciences and engineering
Thesis Supervisor
Accepted by .......................................................................................................................................
Gerbrand Ceder
Chair, Departmental Committee on Graduate Students
1
Impact of Morphology and Scale on the Physical Properties of
Periodic/Quasiperiodic Micro- and Nano- Structures
by
Lin Jia
Submitted to the Department of Materials Science and Engineering on April 13th, 2012, in
Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy in Materials
Science and Engineering
ABSTRACT
A central pillar of real-world engineering is controlled molding of different types of waves (such
as optical and acoustic waves). The impact of these wave-molding devices is directly dependent
on the level of wave control they enable. Recently, artificially structured metamaterials have
emerged, offering unprecedented flexibility in manipulating waves. The design and fabrication
of these metamaterials are keys to the next generation of real-world engineering. This thesis aims
to integrate computer science, materials science, and physics to design novel metamaterials and
functional devices for photonics and nanotechnology, and translate these advances into realworld applications. Parallel finite-difference time-domain (FDTD) and finite element analysis
(FEA) programs are developed to investigate a wide range of problems, including optical
micromanipulation of biological systems [1, 2], 2-pattern photonic crystals [3], integrated optical
circuits on an optical chip [4], photonic quasicrystals with the most premier photonic properties
to date [5], plasmonics [6], and structure-property correlation analysis [7], multiple-exposure
interference lithography [8], and the world’s first searchable database system for nanostructures
[9].
Thesis Supervisor: Edwin L. Thomas
Title: Morris Cohen Professor of Department of Materials Sciences and engineering
2
Table of Contents
List of figures………………………………………………………….……………5
List of tables………………………………………………………….…………...12
Acknowledgements……...………………………………………….…………….13
Chapter 1. Thesis overview.…………………….………………………………...15
Chapter 2. Optical micromanipulation in optical lattices…....……………….…22
2.1.Introduction…………………………………………….………..…..…22
2.2.Numerical approach to compute optical force and optical torque in
Lorentz-Mie regime.………………..............................................................24
2.3.Optical forces from Optical lattices..……………………………..…….26
2.4.Optical torques from Optical lattices...…………………………………43
Chapter 3. Two-pattern photonic crystals and the associated optical devices.........46
3.1.Introduction………………….……………….………………………...46
3.2.Two-pattern photonic crystals….……………....…………...………….47
3.3.Integrated optical devices on a optical chip……………………...…….58
3.4.Summary ……………………………..………..………………………63
Chapter 4. Photonic quasicrystals……….………………………………………...64
4.1.Introduction………………..…………………………………………64
4.2.PBG maps of quasicrystals……………………………………………65
4.3.Summary…..…………………………………………………………79
Chapter 5. Structure-photonic properties correlation via statistical approach….…81
5.1.Introduction………………..…………………………………………...81
5.2.Statistical analysis on the structural morphologies…………………….82
5.3.Summary……………………………..……………………………...…94
3
Chapter 6. A searchable database for micro- and nano-structures………………..96
6.1.Introduction………………………...…………..……...……….………96
6.2.The protocol to process structures in the database……………………97
6.3.Inverse design of experiments to fabricate target structures…………...99
6.4.Discussion……………..……………………………………………...110
6.5.Summary……………..……………………………………………….112
Chapter 7. Acoustic wave modeling………………………………...……….…..114
Chapter 8. Summary and Future plans……..…………….……………………...120
8.1.Overview of research accomplishments.……………………………120
8.2.Future research directions.……………………………………………123
Bibliography……………………………………………………………………..130
4
List of figures
Figure 2.1: optical forces for particles with (a) 120nm diameter, (b) 300nm
diameter, (c) 540nm diameter. (d) The trends for optical forces acting on particles
of different sizes superposed with the theoretical predictions.
Figure 2.2: F0 in the range of 0 < d <1.5l . The broken line is
∇σ 0
.
E02
Figure 2.3: optical forces for titanium particle of 300nm diameter.
Figure 2.4: parameter η (nm , ns ) calculated by FDTD simulation, the two component
method, and the ESA method.
Figure 2.5: (a) optical forces on PS particles arising from the 3D optical lattice.
The shapes of the optical force lines along different directions are similar. The
amplitudes of the optical forces fulfill Fmax[1,0,0] = Fmax[1,1,0]
2 = Fmax[1,1,1]
3 . (b)
Optical forces on absorbing particles arising from the 3D optical lattice.
Figure 2.6: optical torques on three kinds of cylinders. For the 140nm length
cylinder, we have not observed stable equilibrium orientations. For the 400nm
length cylinder, θ = 0 is the stable equilibrium orientation. For the 800nm length
cylinder, in addition to the stable equilibrium orientation at θ = 0 , metastable
orientation is observed at θ = ±0.5π .
Figure 3.1: (a) through (c): three sub-crystals with large TM PBG (rods of
triangular lattice, rods of square lattice and eight-fold quasicrystals). The filling
5
ratio of the sub-crystals ( f ) are also given in the figure. For quasicrystals, the
filling ratio is given by parameter r a . Here r is the radius of the rods and a is the
quasicrystal lattice parameter. The DOS calculations of the sub-crystals are shown
in (d) through (f). Two sub-crystals with large TE PBG (the honeycomb structure
and rings on a triangular lattice) are shown in (g) and (h); the corresponding DOS
calculations are shown in (i) and (j).
Figure 3.2: (a): 2-pattern crystal from the sub-crystals in Figure 3.1(b) and Figure
3.1(h). (b): The DOS plot for a(TM) a(TE) = 0.559 . (c): The DOS plot for
a(TM) a(TE) = 0.656 .
Figure 3.3: (a): 2-pattern crystal from the sub-crystals in Figure 3.1(a) and Figure
3.1(g). (b): the TE PBG and the TM PBG of the 2-pattern crystal for different
filling ratios of TE sub-crystal. (c): The DOS plot if the TE sub-crystal filling ratio
f(TE) is 0.058. (d): DOS plot if f(TE) is 0.086. (e): DOS plot if f(TE) is 0.114.
Figure 3.4: optimized 2-pattern crystals with large, complete PBG. (a): the 2pattern crystal from the sub-crystals in Figure 3.1(a) and Figure 3.1(g). Here the
filling ratio of the TE sub-crystal f(TE) = 0.086 and the filling ratio of the TM subcrystal f(TM) = 0.086 . The ratio of the periodicities of the sub-crystals
a(TM) a(TE) = 0.7157 .
The DFT of the 2-pattern crystal and the corresponding DOS
calculation are shown below the structure. (b): the 2-pattern crystal from the subcrystals in Figure 3.1(b) and Figure 3.1(g). Here f(TE) = 0.08 , f(TM) = 0.14 and
a(TM) a(TE) = 0.67 .
(c): the 2-pattern crystal from the sub-crystals in Figure 3.1(c)
and Figure 3.1(g). Here f(TE) = 0.11 and r a = 0.22 . (d): the 2-pattern crystal from the
sub-crystals in Figure 3.1(a) and Figure 3.1(h). Here f(TE) = 0.121 , f(TM) = 0.102 and
6
a(TM) a(TE) = 0.7258 .
(e): the 2-pattern crystal from the sub-crystals in Figure 3.1(b)
and Figure 3.1(h). Here f(TE) = 0.108 , f(TM) = 0.149 and a(TM) a(TE) = 0.6563 . (f): the
champion structure from the sub-crystals in Figure 3.1(c) and Figure 3.1(h). Here
f(TE) = 0.084 and r a = 0.189 .
Figure 3.5: (a): the proposed crossed waveguide based on 2-pattern crystal in
Figure 3.4(a). (b): the light intensity distribution for TE wave. The TE wave
propagates in region 2 and 4. (c): the light intensity distribution for TM wave. The
TM wave propagates in region 1 and 3.
Figure 3.6: wavelength scale polarizer based on 2-pattern crystals. (a): the
proposed polarizer based on 2-pattern crystal in Figure 3.4(a). TE and TM waves
enter in channel 1 and they are separated to channel 2 and channel 3. (b): the light
intensity distribution for TE wave. The TE wave propagates in region 1 and 3. (c):
the light intensity distribution for TM wave. The TM wave propagates in region 1
and 2.
Figure 3.7: (a): the filter structure. (b): the resonance peak for TE wave. (c): the
resonance peak for TM wave. (d): the light intensity distribution of TE wave for
on-resonance frequency: ωa 2πc = 0.438 . (e): the light intensity distribution of TE
wave for off-resonance frequency: ωa 2πc = 0.466 . (f): the light intensity
distribution of TM wave for on-resonance frequency: ωa 2πc = 0.475 . (g): the light
intensity distribution of TM wave for off-resonance frequency: ωa 2πc = 0.456 .
Figure 4.1: (a)-(e): structural evolution with fill factor for 8mm symmetric QCs
defined by level set equation (4.1) as dielectric filling ratio is varied. Here the last
7
number behind LS − 8mm / is the filling ratio of the black component. For a negative
photoresist, a black component indicates the dielectric; for a positive photoresist,
the black color indicates air features in a dielectric surrounding; (f)-(j): structural
evolution with fill factor for 10mm symmetric QCs defined by level set equation
(4.1); (k)-(o): structural evolution with fill factor for 12mm symmetric QCs
defined by level set equation (4.1).
Figure 4.2: (a)-(e): QCs of 8mm rotational symmetry from quasiperiodic tiling
patterns decorated with rods. Here the number after R − 8mm / is the filling ratio of
the black component; (f)-(j): QCs of 10mm rotational symmetry from
quasiperiodic tiling patterns decorated with rods; (k)-(o): QCs of 12mm rotational
symmetry from quasiperiodic tiling patterns decorated with rods. For R-QCs, the
black color indicates the dielectric and the white color indicates the air; whereas
for AR-QCs, the black color indicates the air and the white color indicates the
dielectric.
Figure 4.3: (a)-(f): TM photonic LDOS maps for LS-8mm, LS-10mm, and LS12mm QCs, with log (LDOS) plotted vs. fill ratio and frequency.
Figure 4.4: (a)-(d): TM photonic LDOS maps for periodic R-p6mm and the
quasiperiodic R-8mm, R-10mm, and R-12mm QCs, with log (LDOS) plotted vs.
fill ratio and frequency.
Figure 4.5: (a)-(c): TM photonic LDOS maps for AR-8mm, AR-10mm and AR12mm QCs, with log (LDOS) plotted vs. fill ratio and frequency. The AR-QCs
with a low filling ratio have a wide range of shapes and sizes of the discrete
dielectric regimes (see Figure 4.2) and they provide relatively weak TM gaps.
8
Figure 4.6: (a): the champion AR-12mm structure with the largest TE PBG
(56.5%). The filling ratio of the structure is 0.32; (b) schematic delineating the
periodic honeycomb veins (AR-p6mm): honeycomb, rhombus, and triangles; (c)
TE photonic LDOS maps for AR-12mm QCs, with log (LDOS) plotted vs. fill ratio
and frequency; (d): the LDOS plot for the champion structure ( f = 0.32 ), the TE
PBG is indicated in the figure; (e): the champion AR-12mm QC with the largest
complete PBG at f = 0.18 . The “ninja star” shape dielectric resonator particles are
indicated in the circles. This R/V (resonator-vein) structure exhibits good short
range order.
Figure 5.1: the 17 photonic structures examined. The structure index is shown
above each figure. Here R stands for rods; O stands for oval; LS stands for level set
equations and the rotational symmetry of the associated PQC is 2N. (a): rods in a
triangular lattice; (b) rods in a triangular lattice, the rods have two area values:
1.2A
and
0.6A
0.8A
; (c) rods in a triangular lattice, the rods have two area values:
1.4A
and
;
(d) rods in a square lattice; (e) rods in a parallelogram lattice; (f) through (h): QCs
from placing rods in 8mm, 10mm, 12mm QC tilings; (i) through (k): QCs from
placing ovals in 8mm, 10mm, 12mm QC tilings; (l) through (q) QCs from level set
equations. The values in the end of the structural index ( 0π ,
stand for
ϕ
there is no
ϕ
values. For N=4, all
ϕ
0.25π , 0.5π
,
0.75π , 1π )
values lead to the same structure, therefore
value at the end of the structural index.
Figure 5.2: the relationship between the overall geometric factor q and the size of
TM PBG for three permittivity contrasts.
9
Figure 6.1: the database is classified to a tree structure according to the symmetry
of the periodic structures.
Figure 6.2: the schematic of the database system incorporating structures uploaded
by researchers. A user can upload a target structure which will be compared with
the database structures. The most similar database structure is output with
structural information such as experimental fabrication method and physical
properties.
Figure 6.3: two examples of the variation of the similarity function and the visual
appearance of a particular structure and the target structure.
(a). The structures are defined by the diamond family approximation level set
equation:
F ( x, y, z ) = cos ( 2π x ) cos ( 2π y ) cos ( 2π z ) + cos ( 2π x ) sin ( 2π y ) sin ( 2π z )
+ sin ( 2π x ) cos ( 2π y ) sin ( 2π z ) + sin ( 2π x ) sin ( 2π y ) cos ( 2π z ) = t
The target structure corresponds to t = 0 . The structure similarity function ( P )
increases as the structure deviates from the target structure.
(b): the target stucture is defined by the simple cubic surface approximation:
F ( x, y , z ) = cos ( 2π Qx ) + cos ( 2π Qy ) + cos ( 2π Qz ) = 0
The target structure corresponds to Qtarget = 1 and ΔQ = Q − Qtarget .
Figure 6.4: the target photonic crystal structure with P4m2 symmetry. The
enlarged unit cell is shown on the left.
Figure 6.5: six types of phase mask are investigated. Every phase mask consists of
two layers each with the same thickness h . The layer can be a one dimensional
10
grating or air posts on a square lattice in solid dielectric background. Here h , r1 and
r2 are tunable parameters.
Figure 6.6: photonic crystal structure with P4m2 space group. Comparison of the
optimal phase mask fabricated structure (left) and the target structure (right). The
skeletal graph of the target structure is shown in blue. The key 4 fold rotoinversion,
diagonal mirror with perpendicular 2 fold rotation symmetries are captured in the
phase mask structure using circularly polarized incident radiation.
Figure 7.1: (a): a solid rectangular plate vibrates in water along the x direction. The
vibration frequency is 1.43Hz and the vibration follows the sinusoidal function:
U x = sin(2π ft ) . (b) the pressure ( p ) distribution in the water after the plate vibrates
for 3.5 seconds; (c) the displacement along y direction ( u y ); (d) the displacement
along x direction ( ux ).
Figure 7.2: LDOS for a phononic wave propagating along Z direction. The 10layer woodpile structure is periodic in X and Y directions, and we show only one
unit cell here. The green part of the structure is PMMA. Here c air is the sound
velocity in air.
11
List of tables
Table 4.1: Summary of key findings from the LDOS numerical calculations. The
champion QCs with the largest TM, the largest TE, and the largest complete PBGs
are shown by “italics”. The results are for ε 2 ε1 = 13:1 .
Table 5.1: the geometric factors and the size of PBGs.
α , β , γ are three
geometric factors. The overall factors q for three permittivity contrasts is also
shown.
12
Acknowledgements
I feel that I cannot fully capture in words the gratitude I have for all those
who have supported me throughout the past five years at MIT, but I will try.
First and foremost, I would like to express my deepest gratitude for Prof.
Edwin L. Thomas, who is the most influential mentor I’ve had in my life. His
passion for science is contagious and his support for me has been unfailing.
Throughout my Ph.D. training, he gave me important and helpful advices on
both science and life. He granted me the freedom to explore different research
areas, but always made himself available whenever I needed guidance. His
mentoring enabled me to mature scientifically and allowed me to develop my own
vision on science and technology. Thank you Ned, for your dedication, for your
insight, and for being a powerful leader.
I would like to thank Prof. Woosoo Kim for being an incredible
colleague who not only showed me amazing science but also phenomenal kindness.
Thanks for all the support Woosoo, and for telling me I need to sleep when I sent
emails at 2AM.
I would like to thank Dr. Ion Bita for being an inspiring colleague. Our long
chats on quasicrystals laid down the basis for two chapters in this Thesis.
I would like to thank Prof. Jeffrey C. Grossman and Prof. Mary C. Boyce,
my thesis committee members, for their mentorship and insights on materials
science.
13
I would like to thank Prof. John Joannopoulos, Prof. Lionel C. Kimerling,
Prof. Eli Yablonovitch, and Prof. Steven G. Johnson for their help on my photonic
crystal research. Their legendary pioneering works on photonics are truly inspiring.
I would like to thank Dr. Jae-hwang Lee for many interesting research
discussions, Dr. Martin Maldovan for his advice on numerical simulation, Dr.
Henry Koh for his help on phase mask lithography, Miss Sisi Ni for her help on
FIB, Dr. Steven Kooi for his help with various research equipments, Mr. William
DiNatale for his help on SEM and AFM, Dr. Tz-Feng Lin and Miss Yin Fan for
our interesting chats on life, and politics, and many, many others!
I would like to thank Meghan Shan, for always telling me what I need to
hear, even if it is not what I want to hear. Her unwavering faith in me keeps me
grounded and she reminds me of the goodness in this world, which has been the
most effective motivator.
Lastly, I would like to thank my father and mother: Yang Qiu and Jingchun
Jia. Thank you for your unconditional support, for giving me the freedom to do
what I love even if it makes no sense to you and takes your only son half way
across the planet.
I dedicated this thesis to my family.
14
Chapter 1
Thesis overview
Controlled molding of different types of waves (such as optical and elastic
waves) is key to future applications of engineering. For example, controlling
photonic waves at the nanoscale is crucial for next generation information
technology (IT) such as quantum computing [10] and optical interconnect [11]; and
controlling the flow of both optical and elastic waves is the basis for future
cloaking devices [12-15]. The impact of these wave-molding devices is directly
dependent on the level of control they enable. Recently, artificially structured
devices have emerged (e.g., metamaterials, photonic and phononic crystals),
offering unprecedented flexibility in manipulating waves. The modeling, design
and fabrication of these devices have enabled unprecedented applications including
negative refractive index [16, 17], super lenses [18], and sub-wavelength imaging
[19].
This thesis utilizes computational tools (e.g., finite-difference time-domain)
to design novel functional devices for manipulating waves: primarily optical and
one chapter of elastic waves. We investigate six interconnected projects: optical
micromanipulation (Chapter 2), 2-pattern photonic crystals and associated optical
devices (Chapter 3), photonic quasicrystals (Chapter 4), the correlation of
15
structure-photonic properties via a statistical method (Chapter 5), a searchable
nanostructure database (Chapter 6), and elastic wave modeling (Chapter 7).
In Chapter 2, we discuss how we can use optical forces and optical torques
to control the position and orientation of micro-scale objects in optical lattices.
Maxwell pointed out that an electromagnetic wave can transfer momentum to
matter and thus exert force [20]. However, it was not until nearly one hundred
years later that Ashkin first demonstrated that this force could be useful to
manipulate objects at the micro-scale using a tightly focused laser beam [21, 22].
This breakthrough quickly lead to the widespread development of “optical
tweezers” with many important uses [23, 24], particularly in biology [25],
microfluidics [26], and particle manipulation [27]. The notion of optical traps was
also extended to atom traps [28]. An important extension of optical traps is the
optical lattice, which comes from the interference of several laser beams. The large
trapping area of an optical lattice enables it to be an effective tool for optical
sorting and optical assembly [29].
In my thesis, I developed a novel numerical method combining the finitedifference time-domain (FDTD) and Maxwell stress tensor to compute optical
forces and optical torques on Lorentz-Mie particles from optical lattices. We
16
compare our method to the two component method and the electrostatic
approximation [30]. We also discuss how the particle refractive index, shape, size
and morphology of the optical lattice influence optical forces and conditions to
form stable optical trapping wells. In addition to optical forces, optical torque from
the 1D optical lattices is discussed for particles having anisotropic shapes;
metastable and stable equilibrium orientation states are found. A detailed
understanding of the optical force and torque from optical lattices has significant
implications for optical trapping, micromanipulation and sorting of particles.
Chapter 2 shows how an optical field can control materials. Conversely,
materials can also bring strong impact on optical fields. For example, one of the
major recent revolutions in photonics, the photonic crystal, was proposed in 1987
[31]. Photonic crystals enable new approaches for controlling the flow of light
primarily because of their photonic band gaps (PBG). Chapter 3 presents a new
strategy to design two-dimensional (2D) structures with a large, complete PBG.
These structures, “2-pattern crystals”, come from the superposition of two
structures, each of which can be crystalline or quasicrystalline. One structure is
selected to have an outstanding PBG for only one polarization (transverse magnetic,
TM, for example) and it is paired with another structure that has an outstanding
transverse electric (TE) gap. Compared to the many kinds of photonic crystals
17
proposed before, our 2-pattern photonic crystals offer opportunities to adjust the
scale of each structure and the respective dielectric fill fractions to provide a
compound photonic structure with superior TE and TM properties. Importantly,
the TM and TE gaps of a 2-pattern photonic crystal are less interdependent than for
conventional photonic crystals, affording capabilities for us to tune the PBGs. By
superposing the two structures without regards to registration, we designed six new
aperiodic PBG structures having a complete photonic band gap larger than 15% for
ε 2 ε1 = 11.4 . The rod-honeycomb 2-feature photonic crystal provides the largest
complete PBG to date. The structures present more design flexibility for on-chip
photonic devices. By purposely introducing defects into the sub-photonic crystals,
on chip photonic devices for different polarizations can be integrated to bend, split
and resonate TM/TE waves simultaneously on the same plane.
In addition to 2-pattern photonic crystals, another group of photonic
structures explored in this thesis are the photonic quasicrystals (QCs). Photonic
QCs have been shown to be promising PBG materials because of their nearly
isotropic photonic band structures, defect-free localized modes [32, 33], and the
possibility to open a PBG at lower refractive index contrast than typical PC
structures [34, 35]. In Chapter 4, we provide a comprehensive analysis of the PBG
18
properties of 2D QCs, where rotational symmetry, dielectric fill factor and
structural morphology were varied systematically in order to identify correlations
between structure and PBG width at a given dielectric contrast (13:1, Si:air). The
TE and TM PBGs of 12 types of QCs are investigated (a total of 588 structures).
We discovered a 12mm QC with a 56.5% TE PBG, the largest reported TE PBG
for an aperiodic crystal to date. We also found a QC morphology comprising
“ninja star” like dielectric domains, with near circular air cores and interconnecting
veins emanating radially from the core. This interesting morphology leads to a
complete PBG of about 20%, which is the largest reported complete PBG for
aperiodic crystals.
In Chapters 3 and 4, photonic band gaps were calculated from the
geometrical structure of the photonic structures. However, the formation
mechanisms of PBGs in photonic crystals are not always clear, thus there is no
general rule to predict the size and location of PBGs. In Chapter 5, we
demonstrate a novel quantitative procedure to pursue statistical studies on the
influence of geometric properties on PBGs of photonic crystals and photonic QCs
which consist of separate dielectric particles. The geometric properties are
quantified and correlated to the size of the PBG for a wide permittivity range using
three characteristic parameters: dielectric particle shape anisotropy, size
19
distribution, and the feature-feature distance distribution. Our concept brings
statistical analysis to photonic crystal research and offers the possibility to predict
the PBG from a morphological analysis.
Presently, there is no existing database system to incorporate the emerging
number of newly designed functional micro- and nano-structures such as photonic
crystals, phononic crystals, microtrusses, plasmonic structures, and metamaterials.
Further, in micro- and nano- fabrication, the ability to achieve specific target
structures is highly desirable as the morphology of a structure greatly influences its
physical properties. In Chapter 6, we develop a database system to efficiently
classify, store, analyze and compare 2D and 3D micro- and nano- structures. This
system can also be used for high throughput experimental design for various
fabrication techniques. The database is multi-functional: it can be used to analyze
the impact of symmetry on physical properties of structures and it can also be used
to find the best structure with a particular extremized objective function among a
set of candidate structures. Further, we suggest initiation of a world-wide
cooperation to establish a public website to incorporate millions or even billions of
natural and human made structures in a comprehensive database.
20
In addition to modeling photonic waves, modeling elastic waves is important
for a wide range of applications, including phononic crystals [36], ultrasound
imaging [37, 38], seismic wave sensing [39, 40], and optomechanical interaction
[41, 42]. In Chapter 7, we discuss the application of FDTD modeling to elastic
waves. Here the partial differential equations governing elastic wave propagation
are processed using Yee’s grid convention.
In Chapter 8, we summarize the main accomplishments of this thesis and
propose several future directions.
21
Chapter 2
Optical micromanipulation in optical lattices
2.1 Introduction
An Electromagnetic field can apply forces on both static charges ( F = qE )
and moving charges ( F = qv × B ); therefore, when an EM wave propagates inside
materials, it interacts with the electrons and protons. This microscopic interaction
leads to the macroscopic effects of reflection, refraction and absorption of EM
waves. Since the momentum of the EM waves is changed during the process, the
EM waves exert forces on the materials to keep the total momentum conserved, as
was first pointed out by Maxwell in 1873 [20]. The application of optical forces
was first explored by Ashkin, using a highly focused laser beam to trap micro
objects. This breakthrough was later termed “optical tweezers” or “optical
traps”[22, 23, 25, 43]. Since then, various forms of optical fields have been
developed to achieve optical micromanipulation and optical sorting, including
optical vortex [44, 45], holographic optical tweezers [46, 47], interferometric
optical tweezers [48-50], standing wave [51-53], plasmonic resonances [54-57],
and optical lattices [26, 58, 59]. The rapid development of optical
micromanipulation provide many applications in various fields, including sorting
of cells, macromolecules, and colloids [51, 60, 61], colloidal crystals evolution [58,
22
62, 63], nanofabrication [29], quantum simulation [64], micro- and nano- motors
[65, 66], and the study of the physical properties of biology systems including
DNA at the molecular level [67, 68].
The numerical calculations of optical forces are clearly very important to
support, design, and nurture the continued fast development of the experiments and
the industrial applications of optical micromanipulation, trapping, and sorting. This
fact has led to the development of multiple numerical techniques including 1)
generalized Lorenz-Mie theory (GLMT) [69], which is suitable for calculating
optical forces on objects of particular shapes including spherical, spheroid and
cylinder shapes [69-71], 2) the gradient force intuitive approach [72], which is
suitable for a strongly focused light source, 3) the T-matrix method [73, 74], which
is suitable for highly symmetric systems, 4) the discrete dipole approximation
(DDA) [75], which needs to satisfy two validate criteria and is computational
cumbersome [75], and 5) the electrostatic approximation [30, 52, 76, 77], which
calculates the optical forces on particles with refractive index close to the
background medium.
In this chapter, we combine a previously reported 3D finite-difference timedomain (FDTD) simulation [6] and the Maxwell stress tensor ( T ) to compute both
optical forces and optical torques on particles in the Lorentz-Mie regime. Our
approach can calculate optical forces and optical torques from various forms of
23
optical fields; we specifically focus on optical lattices in this chapter. Compared to
other techniques, optical trapping and sorting using optical lattices have two
advantages: 1) the number of trapping sites is high, up to hundreds; the trapping
sites are periodic and can exhibit various optical intensity morphologies [78]; 2)
the optical sorting area is large and control can be automated to offer high
throughput sorting. These advantages enable interesting applications. For example,
optical lattices can assemble particles to make optical matters including photonic
and phononic crystals [63], or potentially even dual band gap crystals [79, 80].
Recently, optical lattices have been extended to a quasicrystalline substrate
potential, which assembles particles to the Archimedean-like tiling [58]. In
addition to micromanipulations, optical lattices can also be used to achieve passive,
parallel, and high-throughput optical sorting [26, 59]. Furthermore, the opticalmechanical coupling is observed in optical lattices: the optical binding effect [81,
82] leads collective oscillations [83] and spatially localized vibration with multiple
modes in optical matters [84].
2.2. Numerical approach to compute optical force and optical torque in
Lorentz-Mie regime
The Maxwell stress tensor is a second rank tensor expressed as:
Ti, j = Di E j + Bi H j − 0.5( E ⋅ D + B ⋅ H ) δi, j
24
(2.1)
The force on an object at a specific time can be calculated by:
⋅ ndS
f =∫ T
∂S0
(2.2)
Here ∂ S 0 is a closed surface over the object and n is the surface normal vector. The
Maxwell stress tensor is calculated by a FDTD simulation. Equation (2.2) is then
used to calculate optical forces [1]. Noting that the EM wave varies with time, the
time average force is calculated from:
1 τ
f dt
τ ∫0
F=
(2.3)
Here τ is the period of the light source. In addition to the net force, the torque can
be calculated by:
Ω=−
1
τ
dt
τ∫ ∫
0
∂S0
× rdS
n⋅T
(2.4)
Here r is the vector from the center of mass of the object to the surface element ∂S0 .
The method is versatile: it can be applied to calculate optical forces and torques on
objects of arbitrary shape and does not require system symmetry. The calculation
covers the Lorentz-Mie regime ( object size ≈ wavelength ) where most applications of
optical trapping, micromanipulation and sorting reside. The method can be applied
to both dielectric and absorbing objects.
25
2.3 Optical forces from optical lattices
In our numerical calculation, the wavelength of the light source in vacuum
( λ) is fixed at 1064nm, to relate to the wavelength of the widely-used solid state
lasers (Nd: YAG and Nd: YLF). All calculation results are applicable to other
typical wavelengths via scaling along with incorporation of the dispersion of the
refractive index. We first consider the simplest case of a one dimensional (1D)
periodic optical lattice arising from the interference of two plane waves
propagating along opposite directions in a uniform lossless medium. We assume
that the medium is water and the refractive index nm is taken as 1.3. The light
intensity is calculated by:
I=
1
τ
τ
N
N
0
i =1
i =1
∫ ( ∑ Ei ) ⋅( ∑ Ei )dt
(2.5)
N is the number of the plane waves. The electronic component of the first plane
Λ
wave is given by E1 = E0 exp(ik0 y − iωt ) z , and the electronic vector of the second
Λ
plane wave is E2 = E0 exp(−ik0 y − iωt ) z ; here k0 = 2π nm λ and the polarization of the
two plane waves is along ẑ direction. From equation (2.5), the light intensity is:
26
I = E 02 + E 02 cos(2 k 0 y )
(2.6)
From equation (2.6), the period of the optical lattice ( l ) is 409nm. We next
introduce a dielectric sphere with diameter d . The refractive index of the particle
( ns ) is taken as 1.58 ( ns nm = 1.2 ), which is typical for polystyrene spheres. We are
interested in particles which have size that is comparable to the period of the
optical lattice. Specifically, we focus on three kinds of particles: 120nm diameter
( d l = 0.293 ), 300nm diameter ( d l = 0.733 ) and 540nm diameter ( d l = 1.32 ). We
change the positions of the centers of the particles ( y ) in the optical lattices and
calculate the optical forces. From equations (2.1) though (2.4), optical forces are
directly proportional to the light source power, thus we express the force in the
form of F
E 02 .
The results are shown as the solid lines in Fig. 2.1(a), (b) and (c),
and the dashed lines are the light intensity distribution. In Fig. 2.1(a) and (b), the
optical forces push the sphere center to the high light intensity positions and trap it
there. In Fig. 2.1(c), however, optical forces push the center of the sphere to the
lowest light intensity positions. From Fig. 2.1(a), (b) and (c), we find that the
optical force can be expressed as:
F ( x, y, z ) = F0C ( x, y, z)
(2.7)
C(x, y, z) is a function of the particle position in the optical lattice and indicates how
the optical force changes with the particle position. In the 1D optical lattices,
27
C(x, y, z) is only related to y due to the system symmetry, thus it can be expressed as
C( y) . Comparing C( y) in Fig. 2.1(a), (b) and (c) demonstrates a perfect match,
which is shown in Fig. 2.1(d).
Figure 2.1: optical forces for particles with (a) 120nm diameter, (b) 300nm diameter, (c)
540nm diameter, (d) the trends for optical forces acting on particles of different sizes
superposed with the theoretical predictions.
The maximum force the particle experiences is F0 and it is a function of the
particle size and permittivity. We assume that F0 is negative when optical forces
push the particle center to the low light intensity positions. Fig. 2.2 shows how F0
changes with the diameter of the particle d . F0 increases with particle diameter until
28
the diameter is around the period of the optical lattice, and then F0 drops quickly
with increasing diameter and becomes negative in some range.
Figure 2.2: F0 in the range of 0 < d <1.5l . The broken line is
∇σ 0
.
E02
We can understand how the optical forces on particles arise from optical
lattices from examining the energy aspect. We first assume that the dielectric
particle can be modeled as a set of multiple dipoles. The interaction energy
between a single dipole p and the electric field E is −p ⋅ E . If the refractive index of
the particle is near to the background material, p and E are generally in phase, so
the particles are pushed into high light intensity positions to achieve minimum
interaction energy. When the diameter of the particle is smaller than the period of
the optical lattice, the optical forces push the particle center to the high light
intensity positions and the particle stabilizes at the light intensity peaks. However,
if the diameter of the particle is bigger than the period, the particle can span two
29
light intensity peaks within the particle when its center lies in the low light
intensity positions. As the diameter increases continually, the particle center may
return to high light intensity positions to hold three peaks in the material, so F0
oscillates as d increases. Fig. 2.2 shows the oscillation of F0 in the range of
6nm < d < 540nm ( 0.015 < d l < 1.320 ). To quantitatively explain Fig. 2.2, we define
the following parameter to represent how much of the EM field is inside the
particle:
σ = − ∫∫∫ I(x, y,z)dxdydz
(2.8)
V0
Here V0 is the region occupied by the particle and I(x,y,z) is the light intensity of the
optical lattice. The parameter σ represents the interaction energy between the
particle and the optical lattice, thus the optical force should be directly proportional
to the spatial derivative of the parameter σ :
F ∝ −∇σ = ∫∫∫ ∇I(x, y,z)dV
(2.9)
V0
For the 1D optical lattice, the maximum optical force is:
F0 = −∇σ 0 ≈ 2 E02 k0 ∫∫∫ sin [ 2k0 ( y + 0.25l )] dV
V0
30
(2.10)
In equation (2.10) the particle center is at the origin of the coordinates. We plot
∇ σ 0 E 02
as the dotted line in Fig. 2.2. The reason for the shape differences between
∇ σ 0 E 02
and F0
E 02
is that the light intensity in equation (2.8) are calculated from
light intensity of the optical lattice, the actual field inside the particle is different
from the field of the optical lattice. Despite this, ∇ σ 0
E 02
gives a very good
approximation to F0 in the range of 0 < d <l .
Except for the energy aspect, the optical force from optical lattices can also
be understood by the two component method [85]. The trapping abilities of optical
lattices mainly come from gradient force. The particle can be assumed as a set of
dipoles. For a single dipole, its polarization is proportional to the electric field
applied on it and can be expressed by Clausius-Mossotti relation:
P(r ,t ) = 3ε mε 0
εs − εm
E
(r ,t )
ε s + 2ε m applied
(2.11)
By employing the above equation, the gradient force on the dipole can be
expressed as:
f grad (r ,t ) = 3ε 0ε m
εs − εm
∇E2applied (r ,t )dV
2ε s + 4ε m
(2.12)
Here dV is the volume of the dipole. The total force on the particle is the integral of
equation (2.12) over the volume of particle, thus:
31
Fgrad (r) = 3ε mε 0
εs − εm
2
∇ Eapplied
(r ,t ) dV
∫∫∫
τ
2ε s + 4ε m V
(2.13)
0
In equations (2.11) through (2.13), the applied field
E applied (r ,t )
on the dipole
contains both the electric field from other dipoles and the electric field of the light
source. If the particle is small and the refractive index of the particle is near to the
background, the impact from other dipoles can be omitted, thus we have:
Fgrad ( x, y, z ) ≈ 3ε mε 0
εs − εm
∇I ( x, y, z )dV
2ε s + 4ε m ∫∫∫
V
(2.14)
0
We can also investigate the system using the electrostatic approximation [52, 76,
77]. The energy difference caused by introducing particles into optical lattices can
be expressed by:
ΔW = 0.5∫∫∫ (ED − Ei Di + HB − Hi Bi )dV
(2.15)
V
Here V is the volume of the whole system, E,D,B,Hare the fields after the particle is
introduced into the optical lattice and Ei , Di , Bi , Hi are the fields before the particle is
introduced. Equation (2.15) can be transformed into:
ΔW = −0.5ε 0ε m (
εs
− 1) ∫∫∫ EEi dV
εm
V
0
Here V0 is the volume of the particle. From equation (2.16):
32
(2.16)
Fgrad (r) = − ∇( ΔW ) τ = 0.5ε 0ε m (
εs
− 1)∫∫∫ ∇( EEi τ )dV
εm
V
(2.17)
0
The expression in equation (2.17) is not explicit because it needs information of
the optical field inside the particle.
E should
be the sum of E i and the electric field
from the material polarization. If the refractive index of the particle is near that of
the background and the particle size is small, we can omit the material polarization
and use E i to replace E ; this replacement is the so called “electrostatic
approximation” [30], which leads to:
Fgrad ( x, y, z) = 0.5ε 0ε m (
εs
− 1)∫∫∫ ∇I ( x, y, z)dV
εm
V
(2.18)
0
The two component method is a higher order approximation for calculating
optical forces compared to the ESA method because the Clausius-Mossotti relation
in the two component method gives an approximation to the electric field from the
material polarization, which has been totally omitted in ESA method. Equation
(2.14) can be deducted from equation (2.17): if the particle size is small compared
to wavelength, we can assume that
E = 3Eiε m (ε s + 2ε m )
Inserting equation (2.19) into equation (2.17) leads to equation (2.14).
33
(2.19)
No matter what method we use to understand the optical forces from optical
lattices, the forms of optical forces in equations (2.9), (2.14), and (2.18) are similar.
Thus equation (2.7) can be expressed as:
F = ε 0η (nm , ns ) ∫∫∫ ∇I ( x, y, z )dV
(2.20)
V0
In equation (2.20), the impact of the optical lattice morphology, the particle shape
and size on optical forces is represented in the volume integral ∫∫∫ ∇I ( x, y, z )dV . We
V0
calculate the volume integral to determine how the optical force changes for
particles at different positions and plot it as a solid line in Fig. 2.1(d). Comparison
of the FDTD simulation results with the predictions from physical theory shows an
excellent match. Optical force is also impacted by refractive index of the particle.
We next calculate the optical force on a titanium particle of 300nm diameter
( ns = 2.7; ns nm = 2.08 ), which is shown in Fig. 2.3.
Figure 2.3: optical forces for titanium particle of 300nm diameter.
34
The shape of optical force line in Fig. 2.3 is very similar to Fig. 2.1 and the
amplitude of the optical force increases. The impact of the particle refractive index
relative to the background medium refractive index is represented in the parameter
η (nm , ns )
in
equation
(2.20).
From
the
two
component
method:
η ( nm , ns ) = 3nm2 ( ns2 − nm2 ) ( 2 ns2 + 4 nm2 ) ; from the ESA method: η ( nm , n s ) = 0.5( n s2 − nm2 ) . We
next compare parameter η (nm , ns ) from the FDTD simulation, the two component
method, and the ESA method; the results are shown in Fig. 2.4. Compared with
the ESA method, results from the two component method match the FDTD
calculation very well in the low refractive index range because it is more accurate
than ESA. The two component method can only be applied to low relative
refractive index situations; thus it deviates from our calculation at high index
contrast ( ns nm > 1.7 ).
Figure 2.4: parameter η (nm , ns ) calculated by FDTD simulation, the two component method,
and the ESA method.
35
From Fig. 2.4, optical forces increase as the refractive index contrast
increases and then decrease. If we assume the particle as a set of dipoles, the
electric field experienced by dipole N consists of electric fields of the light source
and all the other dipoles:
Eapplied , N = Eol , N + ∑ E NM
(2.21)
M
Here
E ol , N
is the electric field of the optical lattice at the position of dipole N . E NM
is the electric field of dipole M at the position of dipole N and it is given by:
⎡⎛ k 2
ik
1
E NM = exp(ikrNM ) ⎢⎜
+ 2 − 3
⎣⎝ rNM rNM rNM
⎤
⎞
k 2 3ik
3
(
p
+
−
− 2 + 3 )rˆNM (rˆNM ⋅ p M ) ⎥
⎟ M
rNM rNM rNM
⎠
⎦
(2.22)
rNM is the distance between dipole M and dipole N . The dipole moment of dipole
N is:
p N = α (Eol , N + ∑ ENM )
(2.23)
M
Here α = 3ε mε 0
εs − εm
V and Vd is the volume of the dipole.
ε s + 2ε m d
From Equations (2.22-2.23) we have:
∑A
NM
p M = α Eol , N
M
36
(2.24)
Every dipole generates an equation like equation (2.24). In the matrix A , every
component is a polynomial function of α . If we solve the set of equations, the
final solution has the form:
p M = Z M ,1 (α )
(2.25)
Here Z M ,1 (α ) is a polynomial function of α . Inserting equation (2.25) into Equation
(2.22), we have:
ENM = Z NM (α )
(2.26)
Thus:
E applied , N = Eol , N + ∑ Z NM (α )
M
= Eol , N + ∑ Z NM ⎡⎣ 3ε 0 εmVd ( ε s − εm ) ( ε s + 2 εm ) ⎤⎦
(2.27)
M
Inserting equation (2.27) into equation (2.13), we find that optical force is a
complex polynomial function of the refractive index of the particle. Optical forces
initially increase as the refractive index of the particle increases; after refractive
index contrast exceeds some value (for 300nm spherical particles, this value is 1.8),
the optical forces fluctuate as the refractive index increases.
From Fig. 2.1, the optical force from the optical lattice is a restoring force,
and trapping potential wells form at the stable equilibrium positions. The shape of
37
optical force lines in Fig. 2.1 is very near to a sinusoidal shape, thus the depth of
the potential well is:
0.5 l
U=
∫
0.5 l
F0C(x, y,z)dy ≈
0
∫
F0sin ( 2πy l ) dy = F0l π
(2.28)
0
In real particle systems at finite temperatures, Brownian movement causes
positional fluctuation. The effect of Brownian movement on optical trapping can
be calibrated by a factor m that depends on the relative strength of the potential
versus the thermal energy:
m=
U
kT
(2.29)
It is assumed that the trapping force can overcome the effect of Brownian
movement when m > 10 [22]. From equations (2.28) and (2.29):
m=
U
Fl
F0lQ
≈ 0 =
kT πkT πkTcε0 nm E02
(2.30)
Here Q represents the power of the optical lattice ( Q = cε 0 nm E02 ). If we assume
typically that the power is 5mW μm2 and T = 300K , inserting these values into
equation (2.30) gives:
m ≈ F0 E 02 × 4.56 × 10 25
38
(2.31)
From Fig. 2.2,
F0 E 02
varies as d increases, therefore the trapping ability
changes for particles of different sizes. For example, when the diameter is around
the period of optical lattices, m ≈ 20 , the trapping force is sufficient to overcome
the Brownian movement so particles mainly reside in the high light intensity
positions. For other particle sizes such as d ≈ 1.3l , however,
F0 E 02
is nearly zero.
Here, the particles will wander among the optical lattice due to their dominant
Brownian movement, and the trapping effect can be ignored.
Next, we consider a three dimensional periodic optical lattice arising from
the interference of six plane waves. Their electric fields are depicted by:
Λ
E1 = E0 exp(ik0 y − iωt ) z
Λ
E2 = E0 exp(−ik0 y − iωt ) z
Λ
E3 = E0 exp(ik0 z − iωt ) x
Λ
E4 = E0 exp(−ik0 z − iωt ) x
Λ
E5 = E0 exp(ik0 x − iωt ) y
Λ
E6 = E0 exp(−ik0 x − iωt ) y
Here k0 =
2π nm
λ
(2.32)
(2.33)
(2.34)
(2.35)
(2.36)
(2.37)
and the light intensity is:
I ( x, y, z ) = 3E02 + E02 cos(2k0 x) + E02 cos(2k0 y ) + E02 cos(2k0 z )
39
(2.38)
We now explain the behavior of particles in this 3D optical lattice. The
highest light intensity positions are simple cubic lattice, and the period of the
optical lattice ( l ) is 409nm. Equation (2.38) is a simple approximation to the P
−
minimal surface having Pm 3 m symmetry [86, 87]. From equation (2.40), the
origin of the coordinate system is a high light intensity position. We calculate the
optical force on two kinds of spherical particles: polystyrene particle with
permittivity 2.56 and absorbing particles with permittivity 2.56 + i .The diameter of
the particles is 300nm. We change the positions of the centers of the particles
r = ( x2 + y 2 + z 2 )
0.5
along [100 ] , [110 ] , and [111] directions and calculate the optical
forces. The results are shown in Fig. 2.5.
Figure 2.5: (a) optical forces on PS particles arising from the 3D optical lattice. The shapes
of the optical force lines along different directions are similar. The amplitudes of the
optical forces fulfill Fmax[1,0,0] = Fmax[1,1,0]
2 = Fmax[1,1,1]
particles arising from the 3D optical lattice.
40
3 . (b) Optical forces on absorbing
In
Fig.
Fmax[100] = Fmax[110]
2.5(a),
optical
2 = Fmax[111]
forces
along
different
directions
fulfill
3 . This can be understood from equation (2.9):
F ∝∇I (x, y, z) . Inserting equation (2.38) into the above equation leads to:
F ∝ sin(2k0 x) xˆ + sin(2k0 y) yˆ + sin(2k0 z) zˆ
(2.39)
From equation (2.39), the restoring force pushes the particle to high light intensity
positions, where optical potential wells form. The depth of the potential well is
different along different directions and the shallowest potential is along the [100 ]
direction. Compared to the 1D optical lattice, the 3D optical lattice offers three
dimensional trapping; the depth of the potential well in 3D optical lattice is similar
to the 1D optical lattice.
In addition to dielectric particles, optical trapping has been extended to
absorbing particles [88-91]. The permittivity of absorbing particles has imaginary
part: ε = ε1 + iε 2 . For Rayleigh particles, the gradient optical force is:
F = 0.5 α ∇ E 2
In equation (2.40), α is the polarizability and can be expressed as:
α = 3Veff
ε 1 + iε 2 − ε m
ε 1 + iε 2 + 2ε m
(2.41)
41
(2.40)
In equation (2.41), the effective volume ( Veff ) is smaller than the space volume of
the particle due to the absorption [90]:
Veff = 4π
0.5 d
∫
0
⎡ 0.5d − r ⎤
r 2 exp ⎢ −
⎥⎦dr
δ
⎣
(2.42)
δ is the skin depth and can be calculated from the permittivity of the particle:
δ =
λ
(
2π 0.5 ε 12 + ε 22 − 0.5ε 1
)
(2.43)
0.5
From equation (2.40) and (2.41), the effect of absorption on optical force is
determined by the effect of absorption on polarizability:
Fε =ε1 Fε =ε1 +iε2 = α ε =ε α ε =ε +iε
1
⎡ ε −ε ⎤
= ⎢V 1 m ⎥
⎣ ε1 + 2ε m ⎦
1
2
(2.44)
⎡
ε1 + iε 2 − ε m ⎤
⎢Veff
⎥
ε1 + iε 2 + 2ε m ⎦
⎣
The permittivity absorption part ε2 increases the part ε1 + iε 2 − ε m and decreases the
ε 1 + iε 2 + 2ε m
part Veff = 4π
0.5 d
∫
0
⎡ 0.5d − r ⎤ .
r 2 exp ⎢ −
⎥⎦dr
δ
⎣
For absorbing Rayleigh particles, if the size of
particles is much smaller than skin depth, from equation (2.42):
V eff ≈ V
, therefore
absorption leads to higher gradient optical force; if the size of particles is much
bigger than skin depth, from equation (2.42): Veff
42
<< V
, therefore absorption leads
to smaller gradient optical force. For absorbing particles in Lorenz-Mie range,
however, the size of particles is usually higher than the skin depth and the electric
field inside the particle gets smaller due to the absorption. According to equation
(2.17), absorption generally leads to smaller optical force. This is confirmed by
comparing Fig. 2.5(a) and Fig. 2.5(b): absorption causes optical forces to decrease
by 25%.
2.4. Optical torques from optical lattices
Our discussions above concern spherical particles; to date, non-spherical
micro and nanostructures have been fabricated in experiments through multiple
methods such as holographic interference lithography [92], stop-flow interference
lithography [93], the modification and aggregation of spherical particles [94, 95].
In addition to optical forces, optical lattices also apply optical torques on these
non-spherical structures, offering the possibility to simultaneously control the
position and orientation of particles. We now calculate the optical torque on
particles in the 1D optical lattices discussed above. Three forms of particles are
examined: a cylinder with 140nm length and 140nm diameter (aspect ratio 1), a
cylinder with 400nm length and 80nm diameter (aspect ratio 5), and a cylinder
with 800nm length and 60nm diameter (aspect ratio 13.3); the particles have nearidentical volumes. We fix their centers at the highest light intensity positions and
rotate them along Z axis. The optical torques on them as a function of angle of the
43
cylinder axis are shown in Fig. 2.6, which shows that optical torques on particles of
various shapes are very different. Optical torques on particles can also be
understood from the aspect of energy. The interaction energy between a dielectric
particle and optical lattice is expressed in equation (2.16), thus:
Ω = − ∂ (ΔW ) ∂θ = 0.5ε 0ε m (ε s ε m − 1) ∂ ( ∫∫∫ EEi dV ) ∂θ
(2.45)
V0
Figure 2.6: optical torques on three kinds of cylinders. For the 140nm length cylinder, we
have not observed stable equilibrium orientations. For the 400nm length cylinder, θ = 0 is
the stable equilibrium orientation. For the 800nm length cylinder, in addition to the stable
equilibrium orientation at θ = 0 , metastable orientation is observed at θ = ±0.5π . From the above equation, the physical origin of optical torque is that the
particle adopts particular orientations to minimize the interaction energy. If the
refractive index of particles is near to the background (e.g. polystyrene, silica
particles in distilled water), we can assume E =
6ε mEi
; this leads to:
2ε s + 4ε m
Ω = 3ε 0ε m (ε s − ε m ) (2ε s + 4ε m ) ∂ ( ∫∫∫ I ( x, y, z )dV ) ∂θ
V0
44
(2.46)
For particles with near spherical shape or with a size much smaller than the period
of optical lattices, optical torques on them are small because the total light intensity
inside does not change much as the particles rotate with respect to the optical
lattice ( ∂ ∫∫∫ I ( x, y, z )dV ∂θ ≈ 0 ). Figure 2.6 confirms this: the 140nm length cylinder
V0
has size much smaller than the period of the optical lattice ( 409nm ) and its shape is
near to sphere compared to other cylinders calculated, thus the optical torque on it
is nearly zero compared to other cylinders. For particles with large aspect ratio and
sizes comparable to the period of optical lattices, θ = 0 is the stable equilibrium
state. For the cylinder with aspect ratio 13.3, however, the two ends of the cylinder
are both in at high intensity regimes at θ = 0.5π , so in addition to the stable
equilibrium state of θ = 0 , a metastable state of θ = 0.5π exists.
45
Chapter 3
Integrated optical devices based on 2-pattern photonic crystals
3.1. Introduction
It is generally believed that high structural symmetry is advantageous to
open a large, complete photonic band gap (PBG); therefore most PBG structures
investigated are periodic crystals or quasicrystals with high rotational symmetry
incompatible with lattice translation. Here we design a new series of twodimensional (2D) ordered aperiodic structures with large, complete PBG but
without any of the 2D symmetries: translation, rotation, mirror and glide reflection.
These structures, “2-pattern crystals”, come from the simple superposition of two
sub-crystals or quasicrystals. Our band gap optimization strategy is to first select
distinct structures that have outstanding PBG properties. We then pair one
structure that has a premier transverse magnetic (TM) gap with another structure
that has a premier transverse electric (TE) gap, adjust the scale of each structure
and their dielectric fill fractions to provide a composite structure with superior TE
and TM properties. Using this approach, we have designed six aperiodic PBG
structures having a complete photonic band gap larger than 15% for ε 2 ε1 = 11.4 .
46
Importantly, the TM and TE gaps of a 2-pattern crystal are less
interdependent than for conventional photonic crystals, affording unique
capabilities to simultaneously bend, split, couple and filter TM and TE waves. By
separately altering the respective sub-crystals, we demonstrate novel 2-pattern
crystal based polarizers, crossed waveguides, resonators and filters.
3.2. 2-pattern photonic crystals
Many attempts have been made to achieve a complex system with
interesting properties by the assembly of geometric elements with simple
symmetries. For example, a 2D society consisting of lines, triangles, pentagons, etc
was conjured by Abbott [96]. Here we design a novel series of 2D PBG structures
by simply superposing geometric patterns without regards to preserving global
symmetries in the composite.
Photonic crystals, discovered in 1987 [31, 97], have attracted great interest
due to their ability to control the flow of light [98-100]. A large number of PBG
structures have been developed during the past twenty years. Up to now, PBG
structures can be categorized into three types: periodic structures [31, 101-103],
quasicrystals [104, 105] exhibiting higher rotational symmetries than consistent
47
with conventional periodic lattices, and even disordered structures[97, 106] with
short range order. Moreover, it is generally believed that a highly symmetric
structure, i.e. one possessing translational, rotational, mirror, and glide reflection
symmetries is most advantageous to form the largest PBG.
In our approach, we superpose a TM sub-crystal and a TE sub-crystal on the
same plane to form 2-pattern photonic crystals. The chosen TM sub-crystal has
separated dielectric features providing a large TM PBG and the chosen TE subcrystal has expanded dielectric features providing a large TE PBG. The subcrystals can be periodic or aperiodic. Interestingly, the 2-pattern crystal structures
do not belong to any of the known PBG structural categories; they are both ordered
and aperiodic. Moreover, they do not have global or even local symmetries.
A crucial property of the photonic crystal devices is a complete PBG, a
range of frequencies for which the propagation of an electromagnetic wave of
arbitrary polarization in any direction is forbidden. Designing 2D structures with a
large, complete PBG is challenging because the structural requirements are
contrary: generally, TM PBG arises from a dielectric structure with separate
features whereas TE PBG arises from a structure with a connected dielectric
48
network. In the previous literature, the complete PBG structure is mainly achieved
by drilling holes in material background [104, 105] or by increasing the local fill
fraction of a connected network at connection sites [106, 107].
Our approach is based on the rationale that the TE sub-crystal of 2-pattern
crystals can be assumed a type of geometrical perturbation to the TM sub-crystal
and vice versa. A random geometrical perturbation of a given structure shrinks the
size of the PBG [108, 109], but the PBG is generally more robust to an ordered
perturbation [110]. If the TE sub-crystal filling ratio is not high, the ordered
geometrical perturbation is not strong enough to destroy the TM PBG; therefore
the 2-pattern crystal may maintain a TM PBG, arising predominately from the TM
sub-crystal. Conversely, the TM sub-crystal can be assumed as a geometrical
perturbation to the TE PBG; therefore the 2-pattern crystal may maintain a TE
PBG especially if the TM sub-crystal filling ratio is low. Since the dominant TM
and TE PBGs mainly arise from the different sub-crystals, they should be
essentially independent of each other. Importantly, this offers the possibility to
separately tune the TM and TE PBGs by altering the respective sub-crystals and
make them overlap to generate a composite structure with a larger, complete PBG
than previously known structures.
49
In 2D, there are five factors which define the 2-pattern photonic crystal: the
morphology of the sub-crystals, the filling ratio of each of sub-crystal, the relative
scale of the sub-crystals, the relative position and the relative orientation of the
sub-crystals. Next we analyze how those factors impact the complete PBG and
demonstrate this PBG engineering strategy can successfully design a set of 2pattern crystals with large, complete PBGs. We then demonstrate several novel
features for controlling light in one of our 2-pattern crystals.
The first design factor is the morphology of the sub-crystals. Here it is
desirable to select two sub-crystals: one with either a large TE PBG and one with a
large TM PBG so as to open a large, complete PBG in the 2-pattern crystal. Here
we assume that the dielectric material is GaAs with a permittivity of 11.4. We pick
three candidates for the TM PBG sub-crystal: rods on a triangular lattice (Figure
3.1a, p6mm, the current champion structure with the largest TM PBG, 47.3% TM
PBG for 0.1 filling ratio), rods on a square lattice (Figure 3.1b, p4mm, 37.5% TM
PBG for 0.15 filling ratio), and an eight-fold quasicrystal of dielectric rods
generated from hyperspace projection [111] (Figure 3.1c, 8mm, 42.1% TM PBG
for r a = 0.185 , here r is the radius of the rods and a is the quasicrystalline length
parameter). The two candidates for the TE PBG sub-crystal both have p6mm
50
symmetry: the connected honeycomb structure (Figure 3.1g, the current champion
structure with the largest TE PBG [112], 42.5% TE PBG for 0.189 filling ratio)
and circular rings on a triangular lattice (Figure 3.1h, 23.4% TE PBG for 0.114
filling ratio).
Figure 3.1: (a) through (c): three sub-crystals with large TM PBG (rods of triangular
lattice, rods of square lattice and eight-fold quasicrystals). The filling ratio of the subcrystals ( f ) are also given in the figure. For quasicrystals, the filling ratio is given by
parameter r a . Here r is the radius of the rods and a is the quasicrystal lattice parameter.
The DOS calculations of the sub-crystals are shown in (d) through (f). Two sub-crystals
with large TE PBG (the honeycomb structure and rings on a triangular lattice) are shown
in (g) and (h); the corresponding DOS calculations are shown in (i) and (j).
51
The second factor is the relative length scale of the two sub-crystals. For the
2-pattern crystals, if we tune the scale of the TE (TM) sub-crystal, the position of
the TE (TM) PBG is shifted accordingly. The central frequency of the TE (TM)
PBG is proportional to 1 a , here a is the characteristic scale of the sub-crystal
[100]. An example is the 2-pattern crystal, shown in Figure 3.2a, made from the
superposition of the sub-crystals depicted in Figure 3.1b and Figure 3.1h. The
filling ratio of the TE sub-crystal is 0.102 and the filling ratio of the TM subcrystal is 0.149. The DOS plot for a(TM) / a(TE) = 0.559 is shown in Figure 3.2b.
Note that the TE-PBG and the TM-PBG barely overlap. To maximize the complete
PBG, we increase the periodicity of the TM sub-crystal to decrease the central
frequency of the TM-PBG. The tuning is continued until the TE-PBG and the TMPBG fully overlap, which is shown in Figure 3.2c.
Figure 3.2: (a): 2-pattern crystal from the sub-crystals in Figure 3.1(b) and Figure 3.1(h).
(b): The DOS plot for a(TM) a(TE) = 0.559 . (c): The DOS plot for a(TM) a(TE) = 0.656 .
52
The third design factor of the 2-pattern crystals is the individual sub-crystal
filling ratio. The filling ratio of each type of sub-crystal controls the respective TE
or TM gap but also modifies the strength of the ordered geometrical perturbation
on the PBG of the other sub-crystal. Therefore, altering the TE sub-crystal can in
general shrink the size of the TM PBG and the shrinkage increases with the filling
ratio of TE sub-crystal with the reverse situation holds for the TM sub-crystal. An
example 2-pattern crystal made from superposition of the sub-crystals in Figure
3.1a and Figure 3.1g is shown in Figure 3.3a. In Figure 3.3b, we fix the filling
ratio of the TM sub-crystal to 8.6% and increase the filling ratio of TE sub-crystal.
If the filling ratio of the TE sub-crystal is too high (above 14%), the TM PBG is
destroyed and the complete PBG closes. If the filling ratio is too low (below 3%),
the TE PBG of the 2-pattern crystal is low or even closes, which is also obviously
disadvantageous. Therefore, the filling ratios of the sub-crystals need to be
adjusted carefully to maximize the complete PBG. For example, for the DOS plot
shown in Figure 3.3c, the TE-PBG is smaller than TM-PBG and the complete PBG
size is determined by the TE-PBG. To maximize the complete PBG, it is necessary
to increase the TE-PBG size. Therefore we increase the filling ratio of the TE subcrystal. Although the TM-PBG shrinks because of this tuning, as shown in Figure
3.3b, the complete PBG size increases. We increase the filling ratio of the TE subcrystal until the TE-PBG and TM-PBG match with each other, which is shown in
53
Figure 3.3d. The complete PBG is maximized because if the filling ratio of the TE
sub-crystal increases further, as shown in Figure 3.3e, the TM-PBG is smaller than
the TE-PBG and the complete PBG is determined by the TM-PBG, which is
relatively smaller than the complete PBG in Figure 3.3d.
Figure 3.3: (a): 2-pattern crystal from the sub-crystals in Figure 3.1a and Figure 3.1g. (b):
the TE PBG and the TM PBG of the 2-pattern crystal for different filling ratios of TE sub-
54
crystal. (c): The DOS plot if the TE sub-crystal filling ratio f(TE) is 0.058. (d): DOS plot if
f(TE) is 0.086. (e): DOS plot if f(TE) is 0.114.
The fourth and the fifth design factors are the relative position and
orientation of the sub-crystals. Interestingly, it turns out these factors have
essentially no influence on the PBG since the changes in the relative location of the
TE sub-crystal, as an ordered geometrical perturbation to TM sub-crystal, should
not substantially vary the TM PBG of the 2-pattern crystal with the same reasoning
for the behavior of the TE PBG by the presence of the TM sub-crystal. Therefore,
the relative position and orientation of the TE/TM sub-crystals only bring minor
impact on the complete PBG of the 2-pattern crystal. The above fact is very
advantageous for the fabrication of the 2-pattern crystals because it is not
necessary to precisely adjust the relative position/orientation of the two subcrystals.
For the three selected TE sub-crystals and two TM sub-crystals shown in
Figure 3.1, six combinations can be constructed. For each combination, we
maximize the complete PBG size by tuning the filling ratios of the sub-crystals and
the relative scale of the sub-crystals. The optimized structures with their discrete
Fourier transforms (DFT) and the associated DOS calculations are shown in Figure
55
3.4. The previous best 2D PBG structure with the largest complete band gap for a
dielectric contrast of 11.4 (20.1%) is a complicated periodic dielectric network of
various diameter rods and various width veins with four-fold rotational symmetry
(p4mm) [113]. The six optimized 2-pattern crystals all have a complete PBG above
15% although as composite structures they do not possess any 2D symmetry. The
aperiodic structure depicted in Figure 3.4a has the best complete PBG (20.4%),
which is substantially the same as the current champion structure. Surprisingly, this
is the first time that an aperiodic structure without any symmetry becomes the
champion PBG structure. According to our experience, the optimized structures
generally have almost equal sub-crystal filling ratios (around 0.1). Further, because
the mid-gap frequency of the TE and TM PBGs corresponds to the wavelength
near to the periodicity of the sub-crystals, to overlap the TE and TE PBGs, the
periodicities
of
the
sub-crystals
need
( 0.1 < a(TM) a(TE) < 10 ).
56
to
be
on
the
same
order
Figure 3.4: optimized 2-pattern crystals with large, complete PBG. (a): the 2-pattern
crystal from the sub-crystals in Figure 3.1a and Figure 3.1g. Here the filling ratio of the TE
sub-crystal f(TE) = 0.086 and the filling ratio of the TM sub-crystal f(TM) = 0.086 . The
ratio of the periodicities of the sub-crystals a(TM) a(TE) = 0.7157 . The DFT of the 2pattern crystal and the corresponding DOS calculation are shown below the structure. (b):
the 2-pattern crystal from the sub-crystals in Figure 3.1b and Figure 3.1g. Here
f(TE) = 0.08 , f(TM) = 0.14 and a(TM) a(TE) = 0.67 . (c): the 2-pattern crystal from the subcrystals in Figure 3.1c and Figure 3.1g. Here f(TE) = 0.11 and r a = 0.22 . (d): the 2-pattern
crystal from the sub-crystals in Figure 3.1a and Figure 3.1h. Here f(TE) = 0.121 ,
f(TM) = 0.102 and a(TM) a(TE) = 0.7258 . (e): the 2-pattern crystal from the sub-crystals in
57
Figure 3.1b and Figure 3.1h. Here f(TE) = 0.108 , f(TM) = 0.149 and a(TM) a(TE) = 0.6563 .
(f): the champion structure from the sub-crystals in Figure 3.1c and Figure 3.1h. Here
f(TE) = 0.084 and r a = 0.189 .
3.3. Integrated optical devices on a optical chip
2D photonic crystal devices have wide applications, including components
in optical computer chips [114, 115], spontaneous emission control devices [116,
117], quantum information devices [118, 119], waveguides [120-122], lasers [123126], and optical fibers [127]. The TM/TE PBGs of the 2-pattern photonic crystals
are generally less interdependent, which is substantially different from the behavior
of the photonic crystals reported before. The above advantages make 2-pattern
crystals excellent candidates for novel optical devices because they can bend, split,
couple, and filter TM/TE waves simultaneously on the scale of wavelength. For
example, we can design waveguides and resonators which affect only the
propagation of light of a particular polarization.
A crossed waveguide is shown in Figure 3.5a. By locally removing the TM
sub-crystal, we can create a TM waveguide in regions 1 and 3. Similarly, a TE
waveguide can be created in regions 2 and 4 by simply removing the TE subcrystal. To illustrate how one can retain mode polarization through the crossing
point, we introduce TM waves into region 1 and TE waves into in region 2. The
58
light intensity distributions for TE and TM waves are shown in Figure 3.5b and
3.5c, which demonstrate that the leakage of light of one polarization to the
waveguide of the other polarization is very minor.
Figure 3.5: (a): the proposed crossed waveguide based on 2-pattern crystal in Figure 3.4a.
(b): the light intensity distribution for TE wave. The TE wave propagates in region 2 and 4.
(c): the light intensity distribution for TM wave. The TM wave propagates in region 1 and
3.
A novel type of wavelength scale polarizer device can be made from a 2pattern crystal optical circuit as shown in Figure 3.6a. Here the T-shape channel
59
consists of three regions: region 1 allows the propagation of both TE and TM
waves; region 2 allows the propagation of TM waves while blocking TE waves;
region 3 allows the propagation of TE waves while blocking TM waves. Thus, the
T-shape channel can separate TM and TE waves, which are introduced to region 1
by radiating dipoles. The time-averaged light intensity distribution for the TE wave
E2X + EY2 is shown in Figure 3.6b and the light intensity distribution for TM wave
2
E 2z is shown in Figure 3.6c. The transmittance of TM wave Ez
over 40 % and the leakage of TM wave E2z
Region 3
E2z
Region 1
Region 2
E2z
Region 1
is
is less than 1%. The
transmittance of the TE wave is nearly 90% and the leakage is again below 1%.
The degree of polarization in region 2 ( E2Z − E2X + EY2 ) ( EX2 + EY2 + E2Z ) is 0.984;
the degree of polarization in region 3 ( E2X + EY2 − E2Z ) ( EX2 + EY2 + E2Z ) is 0.985.
60
Figure 3.6: wavelength scale polarizer based on 2-pattern crystals. (a): the proposed
polarizer based on 2-pattern crystal in Figure 3.4a. TE and TM waves enter in channel 1
and they are separated to channel 2 and channel 3. (b): the light intensity distribution for
TE wave. The TE wave propagates in region 1 and 3. (c): the light intensity distribution for
TM wave. The TM wave propagates in region 1 and 2.
Creating other structures within a 2-pattern photonic crystal can lead
to interesting phenomena such as optical coupling, resonance and optical
confinement. In Figure 3.7a, we design a filter based on a ring resonator that can
be used to select TM and TE waves of particular frequencies. To demonstrate this,
we introduce light into region 1 covering the frequency range of the PBG. As the
61
optical waves propagate in the waveguide from region 1 to region 2, certain waves
of resonance frequency are coupled to the ring resonator and dropped to region 3.
Fourier analysis of the filtered waves inside region 3 is shown in Figures 3.7b and
3.7c. The resonance peaks are obvious for both TM and TE waves, which are
filtered by the ring resonator. The light intensity distributions of TE and TM waves
for on-resonance (drop) frequency and off-resonance (through) frequency are
shown in Figure 3.7e through 3.7h. The filtered TM and TE waves are output from
region 3. We can separate the TM and TE waves by connecting region 3 with the
inlet of the T-shape polarizer shown in Figure 3.6. The quality factor of the filter
device is around 100. The ring resonance frequency can be tuned by altering the
ring morphology and the local refractive index of TM and TE sub-crystals.
Figure 3.7: (a): the filter structure. (b): the resonance peak for TE wave. (c): the resonance
peak for TM wave. (d): the light intensity distribution of TE wave for on-resonance
frequency: ωa 2πc = 0.438 . (e): the light intensity distribution of TE wave for off-resonance
62
frequency: ωa 2πc = 0.466 . (f): the light intensity distribution of TM wave for on-resonance
frequency: ωa 2πc = 0.475 . (g): the light intensity distribution of TM wave for offresonance frequency: ωa 2πc = 0.456 .
3.3. Summary
In summary, a novel set of aperiodic PBG structures named “2-pattern
crystals” consisting of two sub-crystals have been investigated. Although as a
composite structure, these photonic crystals do not possess any symmetry, the six
optimized 2-pattern crystals all have complete PBGs above 15%. Further, there is
no precise translationally or orientational registration required between the two
sub-structures. The above fact makes the fabrication of the 2-pattern crystals
straightforward. Moreover, these 2-pattern crystals provide the ability to control
TM and TE wave simultaneously on the scale of the wavelength. Our designs
demonstrate strong potential for a set of novel optical circuit devices including 2pattern crystal based polarizer, crossed waveguide, and filter.
63
Chapter 4
Photonic quasicrystals
4.1. Introduction
A large photonic band gap (PBG) is highly favorable for photonic crystal
devices. One of the most important goals of PBG materials research is identifying
structural design strategies for maximizing the gap size. We provide a
comprehensive analysis of the PBG properties of two dimensional (2D)
quasicrystals (QCs), where rotational symmetry, dielectric fill factor and structural
morphology were varied systematically in order to identify correlations between
structure and PBG width at a given dielectric contrast (13:1, Si:air). The transverse
electric (TE) and transverse magnetic (TM) PBGs of 12 types of QCs are
investigated (588 structures). We discovered a 12mm QC with a 56.5% TE PBG,
the largest reported TE PBG for an aperiodic crystal to date. We also report here a
QC morphology comprising “ninja star” like dielectric domains, with near circular
air cores and interconnecting veins emanating radially around the core. This
interesting morphology leads to a complete PBG of about 20%, which is the largest
reported complete PBG for aperiodic crystals.
64
4.2. PBG maps of quasicrystals
2D photonic quasicrystals (PQCs) have been shown to be promising PBG
materials because of their nearly isotropic photonic band structures stemming from
their high rotational symmetries [128, 129], defect-free localized modes [32, 33,
130], and the possibility to open a PBG at lower refractive index contrast than
typical PC structures [34, 35]. PQCs have attracted additional interest for a variety
of photonic applications, including lensing [131], waveguiding [132, 133],
negative refraction [134], and optical fiber devices [135].
An essential requirement for many applications of PBG materials is the gap
size. The first study of PQC structures was published in 1998 and focused on 8mm
symmetric QCs [136]. At a dielectric contrast of 10:1, Chan found a TM gap of 22%
for structures consisting of isolated dielectric rods, while a corresponding 8mm
vein-connected dielectric structure displayed a 25% TE gap and no complete PBGs
[136]. In 2000, a published report on the properties of 12-fold rotationally
symmetric QC suggested that even higher rotational symmetries could lead to
significant improvements over conventional photonic crystals, and claimed finding
a complete PBG of 30% at an astoundingly low dielectric contrast of only
4:1[104]. Unfortunately, the results from this paper were soon afterwards shown
65
to be erroneous, where structures with air rods decorating the vertices of a 12mm
square-triangular tiling exhibited only a modest complete PBG that opens at a
dielectric contrast greater than 7:1 [105]. Many theoretical investigations of QC
PBG properties ensued, including Penrose structures [137], 5mm QCs[128], 8mm
QCs [34, 138], various 12mm QCs [105, 139]. However, due to the various
differences in the focus and material assumptions in these studies, it is quite
difficult to conduct a direct comparison and detailed structure-property analysis to
identify design elements for optimizing the PBG properties of 2D QCs of different
symmetries and morphologies. For example, the impact of morphology on the size
of TM PBG has been somewhat obscured, though recently the strong influence of
the resonance of discrete dielectric features has been identified in independent
studies [137, 140-144]. While the importance of large PBG sizes for applications
of PBG material based devices has been well recognized, in the case of QC much
work remains to be done towards identifying the champion QC structures, with the
largest TM, TE and complete PBGs.
In this chapter, we conduct a comprehensive study of the impact of rotational
symmetry and morphology on the PBG of 12 different types of 588 QCs, assuming
the constitutive materials are silicon and air ( ε 2 : ε1 = 13:1 ).The QCs investigated are
66
defined by level set equations [145, 146], that can be fabricated by interference
lithography [147-151], and QCs generated by decorating quasiperiodic geometric
tilings according to a particular recipe [152, 153], which can be fabricated via Ebeam lithography or focused ion beam lithography. We examined both TE and TM
PBGs, for 8mm, 10mm, and 12mm rotational symmetries, for a very wide
dielectric fill ratio range (0.02 to 0.98), and both structural morphologies
(dielectric-filled, or air-filled). For each QC type in this extensive set of
morphologies and symmetries, we used finite-difference time-domain (FDTD)
simulations to find the optimum filling ratio that maximizes the TM or the TE PBG.
Champion QCs with the largest TM, the largest TE, and the largest complete PBG
are discussed and analyzed and the physical origin of the gaps is described.
The first family of 2D QCs investigated is generated using the following
level set equation:
N −1
f ( x, y ) = ∑ cos(2π x cos(π n N ) a + 2π y sin(π n N ) a )
(4.1)
n =0
The value of 2N in equation (4.1) sets the rotational symmetry of the associated
QCs, and a represents the characteristic unit length. QCs defined by level set
equation (4.1) can be fabricated by multiple-exposure interference lithography
67
(MEIL) [147, 154, 155]. Equation (4.1) can be used to generate two types of QC
structure morphologies, using an analogy with the photoresist tone in MEIL: (i)
positive tone, where the dielectric material of choice is found at all locations where
f ( x, y ) < t (air fills the rest of the space), and (ii) negative tone, where the dielectric
material occupies all locations where f ( x, y ) > t . Since large rotational symmetries
involve increasing fabrication difficulties due to a reduced structural contrast
associated with increasing the number of exposures, we focused our studies on
QCs with N < 7 . For N = 1,2,3 , the structures are periodic. For N = 4,5,6 , the
structures are quasiperiodic with rotational symmetries 8mm, 10mm, and 12mm
respectively. Some representative QC morphologies from this study are shown in
Figure 4.1. The rotation center of MEIL is assumed to be at the high light intensity
position of the 1D periodic fringe pattern, which generates additional mirror
symmetries. The resultant highest point group symmetries for 8, 10, and 12 fold
QCs are 8mm, 10mm, and 12mm. Here we use LS-8mm, LS-10mm, and LS-12mm
to denote 8mm, 10mm, and 12mm QCs generated from level set equation (4.1).
68
Figure 4.1: (a)-(e): structural evolution with fill factor for 8mm symmetric QCs defined by
level set equation (4.1) as dielectric filling ratio is varied. Here the last number behind
LS − 8mm / is the filling ratio of the black component. For a negative photoresist, a black
component indicates the dielectric; for a positive photoresist, the black color indicates air
features in a dielectric surrounding; (f)-(j): structural evolution with fill factor for 10mm
symmetric QCs defined by level set equation (4.1); (k)-(o): structural evolution with fill
factor for 12mm symmetric QCs defined by level set equation (4.1).
The second set of QC structures studied is generated by decorating
quasiperiodic geometric tilings obtained by 2D projection from a higher
dimensional space [152, 153]. Here, we consider the approach of placing rods
(dielectric or air) at the nodes of the chosen tilings, and filling the region between
the rods with the complementary material (dielectric rods in air, or vice-versa for
the opposite structural morphology). Representative QCs of various symmetries
69
are shown in Figure 4.2. We use R-QC (or AR-QC) to denote QCs generated from
placing dielectric (or air) rods at the nodes of the chosen tilings.
Figure 4.2: (a)-(e): QCs of 8mm rotational symmetry from quasiperiodic tiling patterns
decorated with rods. Here the number after R − 8mm / is the filling ratio of the black
component; (f)-(j): QCs of 10mm rotational symmetry from quasiperiodic tiling patterns
decorated with rods; (k)-(o): QCs of 12mm rotational symmetry from quasiperiodic tiling
patterns decorated with rods. For R-QCs, the black color indicates the dielectric and the
white color indicates the air; whereas for AR-QCs, the black color indicates the air and the
white color indicates the dielectric.
The PBG of these 2D QC structures can be determined from maps of the
local density of states (LDOS) [34, 138, 156-158] calculated using a FDTD
solution to Maxwell’s equations [1, 6]. In a LDOS map, a continuous low LDOS
region along the vertical frequency axis indicates a PBG. We first discuss the
70
results for TM polarized waves in QCs defined by the level set Equation (4.1). The
LDOS maps for the 8mm, 10mm and 12mm QCs are shown in Figure 4.3. As the
filling ratio of the structures ( f ) is varied from 0.02 to 0.98, a number of regions
with gaps are found. For the negative tone LS-8mm QC shown in Figure 4.3(a), the
largest TM PBG size is 37.0%, occurring at f = 0.18 . For the positive tone LS-8mm
QC shown in Figure 4.3(b), a similar sized TM PBG also occurs at f = 0.18 . For the
negative tone LS-10mm QC shown in Figure 4.3(c), an interesting observation is
that the lowest TM PBG width remains almost constant for a wide range of filling
fractions and this insensitivity makes it a good candidate for experimental photonic
devices. For the positive tone LS-10mm QC shown in Figure 4.3(d), the largest
TM PBG size is 37.2% at f = 0.12 . For the negative tone LS-12mm QC shown in
Figure 4.3(e), the largest TM PBG size is 40.7% for f = 0.14 . For the positive tone
LS-12mm QC shown in Figure 4.3(f), a large TM PBG of about 42% appears in
the range of 0.1 ≤ f ≤ 0.2 , implying another excellent candidate for photonic devices.
The comprehensive results of this study are collected in Table 4.1.
71
Figure 4.3: (a)-(f): TM photonic LDOS maps for LS-8mm, LS-10mm, and LS-12mm QCs,
with log (LDOS) plotted vs. fill ratio and frequency.
Quasicrystal type
Mode
f
ωlower a 2π c
ωupper a 2π c
ωmiddlea 2π c
Δω ωcenter
Negative tone LS-8mm
Positive tone LS-8mm
Negative tone LS-10mm
Positive tone LS-10mm
Negative tone LS-12mm
Positive tone LS-12mm
R-8mm
R-10mm
R-12mm
AR-8mm
AR-10mm
AR-12mm
AR-12mm
AR-12mm
R-p6mm (crystal)
R-p6mm (crystal)
R-p6mm (crystal)
Honeycomb (crystal)
Honeycomb (crystal)
AR-p6mm (crystal)
AR-p6mm (crystal)
AR-p6mm (crystal)
TM
TM
TM
TM
TM
TM
TM
TM
TM
TM
TM
TM
TE
Complete
TM
TE
Complete
TE
TM
TE
TM
Complete
0.18
0.18
0.24-0.38
0.12
0.14
0.1-0.2
0.12
0.12
0.12
0.08
0.1
0.16
0.32
0.18
0.12
0.44
any
0.22
any
0.36
0.1
0.18
0.22
0.22
~0.21
0.235
0.225
~0.285
0.292
0.280
0.294
0.225
0.26
0.417
0.26
0.41
0.297
0.292
0
0.225
0
0.252
0.483
0.413
0.32
0.32
~0.28
0.3425
0.34
~0.447
0.472
0.448
0.470
0.295
0.325
0.51
0.465
0.5
0.493
0.336
0
0.47
0
0.424
0.585
0.497
0.27
0.27
~0.245
0.289
0.2825
~0.366
0.382
0.364
0.382
0.26
0.2925
0.4635
0.362
0.455
0.395
0.314
0
0.3475
0
0.338
0.534
0.455
37.0%
37.0%
~28.5%
37.2%
40.7%
~44%
47.1%
46.1%
46.0%
26.9%
22.6%
21.2%
56.5%
19.8%
49.6%
13.77%
0
59.3%
0
50.9%
19.1%
18.5%
Table 4.1: Summary of key findings from the LDOS numerical calculations. The champion
QCs with the largest TM, the largest TE, and the largest complete PBGs are shown by
“italics”. The results are for ε 2 ε1 = 13:1 .
72
Next we analyze the TM PBGs of QCs defined by placing solid rods (R-) or
air rods (AR-) of desired diameter at the vertices of various QC geometric tilings
obtained from hyperspace projection. Interestingly, for R-8mm, R-10mm and R12mm QCs, as well as the reference case of a triangular lattice PC (R-p6mm), the
TM LDOS gaps are all found to be quite similar, see Figures 4.4(a) through 4.4(d).
The similarity of the LDOS for these structures in the region with f < 0.25 is due to
the dominant role of the Mie resonances generated by individual dielectric rods
[141, 142, 144]. Although the rotational symmetries of the QCs are different, the
identity of the individual scatterer is preserved and is found to lead to similar PBG
properties for the various quasiperiodic arrangements we investigated. At larger fill
fractions the rod-rod distance becomes small and the TM PBG closes quickly,
which can be associated with a loss of distinct Mie resonances in the case of
interconnected dielectric domains. The largest observed TM PBGs of R-p6mm, R8mm, R-10mm, and R-12mm QCs are listed in Table 4.1. The largest PBGs (above
45%) occur at f = 0.12 for the dielectric contrast of 13:1 used in this study.
Compared to TM PBGs of R-QCs, AR-QCs are found to display smaller TM PBGs,
as shown in Table 4.1 and Figure 4.5.
73
Figure 4.4: (a)-(d): TM photonic LDOS maps for periodic R-p6mm and the quasiperiodic
R-8mm, R-10mm, and R-12mm QCs, with log (LDOS) plotted vs. fill ratio and frequency.
Figure 4.5: (a)-(c): TM photonic LDOS maps for AR-8mm, AR-10mm and AR-12mm QCs,
with log (LDOS) plotted vs. fill ratio and frequency. The AR-QCs with a low filling ratio
have a wide range of shapes and sizes of the discrete dielectric regimes (see Figure 4.2) and
they provide relatively weak TM gaps.
In order to better understand why R-QCs have superior TM gap properties
compared to AR-QCs and LS-QCs, we examine their corresponding structural
characteristics. It is known that morphologies consisting of separate dielectric
domains can have a large TM PBG, thus we start with exploring the impact of the
74
dielectric domain shapes and arrangements. Based on our calculations shown in
Table 4.1 and from the data in reference [128], QCs consisting of entirely circular
shape features, e.g. rods on the nodes of quasiperiodic tilings, have the best TM
PBG properties. Further, the TM PBGs of QCs consisting of separated non-circular
features (one group of level set equation: LS-8mm, LS-10mm and LS-12mm) are
smaller than the TM PBGs of QCs consisting of near circular features (5-10 groups
of level set equation, shown in reference [128]). A non-circular element has
different scattering cross sections for waves incident from different angles, which
is disadvantageous for wave propagation suppression for all in-plane directions.
For example, the separated dielectric features of low dielectric fill AR-QCs, as
shown in Figure 4.2, are star shaped or rectangular shape with sharp corners. Such
features are strongly anisotropic scatterers; therefore their TM PBGs are small, as
can be seen in Table 4.1 and Figure 4.5. Furthermore, besides the importance of
optimizing the shape of the individual dielectric domain, it is important to
emphasize the importance of their spatial arrangement. R-p6mm PC structures
display larger TM PBG than the corresponding R-QCs, benefiting from added
coherent Bragg scattering. In R-QC structures, the larger variation of nearest
neighbor distances further limits the resonance optimization.
75
Figure 4.6: (a): the champion AR-12mm structure with the largest TE PBG (56.5%). The
filling ratio of the structure is 0.32; (b) schematic delineating the periodic honeycomb veins
(AR-p6mm): honeycomb, rhombus, and triangles; (c) TE photonic LDOS maps for AR12mm QCs, with log (LDOS) plotted vs. fill ratio and frequency; (d): the LDOS plot for the
champion structure ( f = 0.32 ), the TE PBG is indicated in the figure; (e): the champion
AR-12mm QC with the largest complete PBG at f = 0.18 . The “ninja star” shape dielectric
resonator particles are indicated in the circles. This R/V (resonator-vein) structure exhibits
good short range order.
We now proceed to examine the results for the TE LDOS calculations. In
general, TE PBGs are favored for a structure comprised of a network of connected
veins. For example, the honeycomb vein structure is the champion photonic crystal
structure with the largest TE PBG of 60% for ε1 ε 2 = 13.0 [159]. The TE PBGs of all
the QC structures examined in this study are found to be much smaller (below
76
10%), with one notable exception. In the case of the AR-12mm QC structure, we
found a very large TE PBG of 56.5% at f = 0.32 , which is the largest TE PBG
observed to date for QCs and only slightly lower than the champion honeycomb
structure [112]. The PBG gap map of AR-12mm QC and the LDOS plot for the
champion AR-12mm QC are shown in Figures 4.6(c) and 4.6(d) and the
corresponding AR-12mm morphology at f = 0.32 is shown in Figure 4.6(a). The
high rotational symmetry leads to higher density of interconnected veins. To
identify some correlations between structure and TE PBG properties, we also
examine the case of the periodic honeycomb crystal (AR-p6mm) for which we
compute the LDOS map and analyze the underlying morphology. Interestingly, we
find a very similar TE gap range and, furthermore, many common structural motifs
between the AR-12mm QC structure and the AR-p6mm PC structure, shown in
Figure 4.6(a) and 4.6(b). The primary difference between the honeycomb structure
and AR-12mm QC is that the nodes connecting the veins of the AR-12mm QC all
contain a small round central dielectric core feature. We believe that the repeating
motifs of the “ninja star” like dielectric domains, with near circular cores and
interconnecting veins emanating radially, are key for simultaneously enabling
complete gaps (overlapping TE and TM gaps), as discussed below.
77
To display a large complete PBG, the photonic structure needs to enable TM
and TE gaps that are both large and have overlapping frequency intervals. The
complete PBGs of most of the QCs examined in the present work are minimal (i.e.
below 10%). AR-12mm is the only exception with a TM PBG and a TE PBG
larger than 20%, as shown in Table 4.1. At f = 0.18 , the AR-12mm QC has the
largest complete PBG (19.8%), which is essentially the same as the largest
complete PBG found to date (20.1%) [113] and it is the largest reported value for
aperiodic crystals. A portion of the champion AR-12mm QC is shown in more
detail in Figure 4.6(e). Comparing the AR-12mm QC in Figure 4.6(a) ( f = 0.32 )
and the same structure with lower filling ratio in Figure 4.6(e) ( f = 0.18 ), we find
that as we decrease the filling ratio, the dielectric veins become thinner compared
to the dielectric core, causing the TE PBG to shrink rapidly from 56.5% to 20%
while the TM PBG increases from 0% to around 20%. The physical reason for this
behavior is that the veins contribute to the TE PBG through a destructive Bragg
interference mechanism. As the vein thickness reduces, the strength of destructive
interference decreases and the TE PBG shrinks. For thinner veins, their
corresponding connection points consist of a relatively larger dielectric portion of
the total and act as effective “particles”, as shown in Figure 4.6(e). The resonant
scattering of these “particles” enables the formation of the TM PBG. Therefore the
large, complete PBG of the AR-12mm at f = 0.18 is enabled by its morphology
78
comprising simultaneously a thin dielectric vein network, which contributes a large
TE PBG, and the set of 4-fold shape dielectric particles, which contributes a large
TM PBG. If the filling ratio is further decreased, e.g. to 0.12, the veins are broken
and the QC is no longer self-connected, leaving isolated square shape dielectric
particles – not surprisingly, in this case the corresponding TE PBG is zero while
the TM PBG is still around 20%.
4.3. Summary
In conclusion, we computed and analyzed in detail the LDOS of two
families of 2D photonic QCs, generated from level set equations and from
quasiperiodic tiling patterns decorated with rods, for 3 different rotational
symmetries (8, 10, and 12). We found that (1) for QCs comprising individual
dielectric features, TM PBGs are largely impacted by the feature shape and feature
inter-distance. QCs consisting of dielectric rods with uniform diameter, as
produced by decorating quasiperiodic tiling patterns, exhibit better TM PBGs than
QCs consisting of non-circular features, as typically obtained in structures
produced from level set equations. (2) R-8mm, R-10mm and R-12mm QCs are
found to exhibit large TM PBGs (above 45%), close to the TM PBG of the current
champion structure (~50% for the R-p6mm), supporting the conclusion that their
79
PBG properties are dominated by Mie resonances of the individual dielectric
domains, with only a small impact from the particular rotational symmetry. (3) the
TE PBGs of QCs investigated here are found to be very small (below or around
10%), except for the case of the AR-12mm QC where we find a very large TE
PBG (56.5% at f = 0.32 ), which is only slightly lower than the champion TE PBG
size (the honeycomb structure has a TE PBG ~ 60% for ε1 ε 2 = 13.0 at f = 0.2 ). The
AR-12mm structure has a large amount of structural motifs of honeycomb veins
and square veins, which are associated with the large TE PBG observed; (4) the
AR-12mm QC is also found to have a large, complete PBG (19.8%) at f = 0.18 . In
this case, the corresponding morphology consists of “ninja star” like dielectric
domain motifs, with an “effective” particle in the center and interconnected veins
radiating around the particle core, enabling formation of both TE and TM gaps; (5)
the DOS maps provide detailed information about PBG properties obtained from
different QC morphology design choices (symmetry, dielectric shapes, filling
ratios, and structure morphology). The optimized QCs which have the largest
PBGs are listed in Table 4.1, which enables direct comparison of the PBG
properties of this large set of QCs, and provides useful guidance for further
photonic QC device design and optimization.
80
Chapter 5
Structure-photonic properties correlation via statistical approach
5.1. Introduction
Here we demonstrate a novel quantitative procedure to pursue statistical
studies on the geometric properties of photonic crystals and photonic quasicrystals
(PQCs) which consist of separate dielectric particles. The geometric properties are
quantified and correlated to the size of the photonic band gap (PBG) for wide
permittivity range using three characteristic parameters: shape anisotropy, size
distribution, and feature-feature distribution. Our concept brings statistical analysis
to the photonic crystal research and offers the possibility to predict the PBG from a
morphological analysis.
Topology optimization procedures for photonic crystals [112] and PQCs
[128] have been developed to achieve premier PBG properties. However, the
physical meanings behind the numerical optimization procedures have not been
revealed and the detailed role of the particular geometric factors on determining
the PBG size has not yet been clearly identified.
81
In this Chapter, we demonstrate a statistical analysis on the geometric
properties of 2D photonic structures. Three geometric parameters (shape
anisotropy parameter, size distribution parameter, and feature to feature distance
distribution parameter) are identified to have strong correlation with the associated
PBGs.
5.2. Statistical analysis on the structural morphologies
Five periodic photonic crystals (Figures 5.1a through 5.1e) and 12 PQCs
(Figures 5.1f through 5.1q), all with the same filling ratio ( f
= 0.18
), are analyzed.
This particular filling ratio is selected in order for those structures to have large
transverse magnetic (TM) PBGs. The structures are chosen to cover a wide range
of morphologies and rotational symmetries (2, 4, 6, 8, 10, and 12-fold) to
effectively represent the various types of 2D PBG structures. Structures in Figure
5.1(a) through 5.1(e) are made by placing rods on a periodic lattice. The
characteristic length scale d 0 is the lattice constant. Structures in Figure 5.1(f)
through 5.1(k) are made by placing rods or ovals on the vertices of the 8-fold, 10fold, and 12-fold quasicrystal tilings generated by projection into 2D from a higher
dimensional space. The characteristic scale d 0 is the distance between the central
82
particle at rotational symmetric axis and its neighbor particles. PQCs in Figure
5.1(l) through 5.1(q) are defined by the following level set equation [145, 146]:
N-1
f(x, y) = ∑ cos ⎡⎣ 2πxcos(πn N) d0 +2πysin(πn N) d0 +φ ⎤⎦
(5.1)
n=0
Here f(x, y) is a numerical landscape in the 2D plane which assigns a value to the
general point with coordinate (x, y) . In equation (5.1), d0 is the characteristic
length scale of the PQCs. If f ( x , y ) > t , position ( x , y ) is occupied by material; if
f ( x , y ) < t , position ( x , y ) is occupied by air. Here t is a parameter which can be
tuned to change the filling ratio of the resultant structure. The PQCs have Fourier
transform patterns with 2N-fold rotational symmetry.
83
Figure 5.1: the 17 photonic structures examined. The structure index is shown above each
figure. Here R stands for rods; O stands for oval; LS stands for level set equations and the
rotational symmetry of the associated PQC is 2N. (a): rods in a triangular lattice; (b) rods
in a triangular lattice, the rods have two area values: 1.2A and 0.8A ; (c) rods in a triangular
lattice, the rods have two area values: 1.4A and 0.6A ; (d) rods in a square lattice; (e) rods in
a parallelogram lattice; (f) through (h): QCs from placing rods in 8mm, 10mm, 12mm QC
tilings; (i) through (k): QCs from placing ovals in 8mm, 10mm, 12mm QC tilings; (l)
through (q) QCs from level set equations. The values in the end of the structural index ( 0π ,
0.25π , 0.5π , 0.75π , 1π ) stand for ϕ values. For N=4, all ϕ values lead to the same structure,
therefore there is no ϕ value at the end of the structural index.
84
A large TM PBG is usually found in 2D structures consisting of separate
dielectric particles and it is recognized that the particle resonances, which
correspond to the peaks of the scattering cross section (SCS) [140], lead to the
formation of the TM PBG [137, 140-144]. If the optical wave frequency is slightly
higher than the first resonance frequency of the particle, the transmitted wave
passing through the particles is anti-phase with the incoming wave, which leads to
destructive interference and the reflectance of the incoming optical wave [140].
Resonance is believed to be the main physical reason for the formation of the TM
PBGs for frequencies falling between the first and second resonance frequencies
[140]. Such TM PBGs are rather insensitive to the positional disorder of the
particles [141, 142].
Next, we provide a statistical analysis on the features of the PBG
structures shown in Figure 5.1 and select three important geometric parameters
which strongly influence the corresponding TM PBG. The first parameter is the
shape anisotropy parameter ( α ). The particle shape brings strong impact on the
scattering properties; therefore it is highly related to the TM PBG. For a typical
non-circular particle, we can define the averaged radius: r = A π , here A is the
area of the particle. For an arbitrary point on the surface of the particle, rG denotes
G
the vector from the center of mass to the surface point. Δr = r − r indicates the
85
deviation of the particle shape from the isotropic circular shape. For circular shape
particle, Δr equals zero at all surface points. The normalized integration of Δr over
the particle i is:
Q = v∫∂S Δri2 ds Si ri
(5.2)
Here Si is the perimeter of the particle i and ∂S is the particle surface. The shape
anisotropy factor α is defined by the averaged Q for the PBG structure:
M
α = ∑ ( v∫ ∂s Δ ri2 ds Si ri ) M
(5.3)
i =1
Here
M
is the total number of particles. For the structures investigated, we choose
a fundamental domain with
M
M
~1000; ∑ is the sum for all
i=1
M
particles. The larger
the α parameter, the more the particle shape deviates from the circular shape. A
non-circular dielectric feature has different scattering cross section for waves
incident from different angles, which is disadvantageous to achieve wave
propagation suppression for all in-plane directions; therefore a larger value of α
leads to a lower TM PBG consistent with the observation that the structures
comprised of circular elements are found to open larger TM PBGs than the
structures consisting of non-circular elements [128].
86
The size of the particle determines the resonance frequency ( f ∝ 1 r ). The
second structural characterization factor β is the standard deviation of the areas of
the particles:
M
β = ∑ ( Ai − A)2 (M −1) A
i =1
(5.4)
Here A is the averaged area of the M particles. A larger β indicates a more polydisperse particle size distribution. Particles with various sizes correspond to
different resonance frequencies, which are disadvantageous for uniform and
isotropic wave propagation suppression; therefore a larger β leads to a lower TM
PBG.
Lastly, the neighbor-neighbor particle distribution is important because
optical wave confinement, which is an important property of PBG, is a local effect.
For every particle P0 , there are a group of
n neighboring particles ( P1, P2.....Pn )
with particle to particle distance d0<−>i < 1.5d0 . Here d 0<−>i is the distance
between the particle P0 and the particle Pi , and d 0 is the characteristic length scale
for the structure defined previously. 1.5d 0 is selected as the criterion to identify
whether two particles are neighbors. The normalized deviation of the particleneighbor particle distance for particle i is:
87
X=
ni
(5.5)
∑ (di <−> j − d 0 )2 (ni − 1) d 0
j =1
We define the third factor, γ , to represent the averaged
X
for the associated PBG
structure:
M
ni
i =0
j =1
γ = ∑ ∑ (di<−> j − d0 )2 (ni −1) Md0
(5.6)
Here ni is the number of neighbor particles of particle Pi . A larger γ indicates a
more poly-disperse neighbor particle distance distribution and lower local
positional order. Loss of short range order is disadvantageous for coherent
scattering and the isotropic local confinement of EM wave is hard to achieve due
to the leakage from a non-uniform neighbor particle distribution; therefore a large
γ
leads to lower TM PBG.
According to the above analysis, all three factors bring strong impact on the
TM PBG size. We define an overall morphological factor ( q ) as a linear
combination of three geometric factors:
M
M
i=1
i=1
q = cα ∑ ( v∫∂s Δri2 ds Siri ) M + cβ ∑ ( Ai− A)2 (M −1) A
M
ni
i=0
j=1
+cγ ∑ ∑ (di<−> j −d0)2 (ni−1) Md0 = cα α + cβ β + cγ γ
88
(5.7)
Here cα , cβ and cγ are positive weighting coefficients for the geometric factors,
which are functions of permittivity contrast ratio. Next, we calculate the individual
geometric factors of the 17 structures and the associated TM PBGs for three
permittivities: (1) silicon/air contrast ratio ( ε : ε
2
(ε :ε
2
1
= 11.4 : 1 ),
1
= 13 : 1 );
(2) GaAs/air contrast ratio
and (3) silicon nitride/air contrast ratio ( ε : ε
2
1
= 4 : 1 ).
The size of
PBG is determined from the local density of states calculated via finite-difference
time-domain (FDTD) method. The geometrical parameters of the 17 structures and
the associated TM PBGs are shown in Table 5.1. According to the data, we
propose a simple linear model to correlate the TM PBG and the overall geometrical
factor:
PBG size = Δω ω = b − kq
For ε : ε
2
1
= 13 : 1 ,
values of
cα = 0.97
,
c β = 1.03
(5.8)
, and
cγ = 0.5
lead to the best
correlation: the correlation coefficient ( R ) equals 0.95. The slope ( k ) in equation
(5.8) equals 0.3 and the intercept ( b ) equals 0.46. For ε
cα = 0.31 , cβ = 0.64
b = 0.45 .
For ε : ε
correlation:
2
1
, and
= 4 :1 ,
R = 0.85
,
cγ = 0.5
lead to the best correlation:
values of
k = 0.98
2
and
cα = 0.05 , c β = 0.21 ,
b = 0.246
and
: ε 1 = 11.4 : 1 ,
values of
R = 0.94 , k = 0.58
cγ = 0.5
and
lead to the best
. From inspection of the weighting
coefficients, the most important parameter at high refractive index contrast is the
89
standard deviation of the particle area distribution (
β
) while the particle
distribution factor γ is the most important parameter at low refractive index
contrast. The PBGs of the structures are plotted as the black squares in Figure 5.2
and the lines in Figure 5.2 are the linear models from equation (5.8). For ε
2
: ε 1 = 13 : 1 ,
the TM PBG sizes of the structures deviate less than 3.1% gap size from the linear
relation. For ε
2
: ε 1 = 11.4 : 1 ,
the deviation is less than 4.5% gap size. For ε
2
: ε1 = 4 : 1 ,
the
deviation is less than 6.0% gap size. Figure 5.2 indicates that the overall geometric
factor
q
and the TM PBG size are related as the trend that PBG size decreases
with increased
q
as expected.
Figure 5.2: the relationship between the overall geometric factor q and the size of TM PBG
for three permittivity contrasts.
90
Structure:
α
β
γ
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.20
0.20
0.20
0.17
0.16
0.18
0.19
0.19
0.18
0.00
0.00
0.00
0.20
0.40
0.00
0.00
0.00
0.00
0.00
0.00
0.23
0.40
0.25
0.16
0.18
0.28
0.00
0.18
0.17
0.00
0.00
0.12
0.17
0.14
0.12
0.17
0.14
0.15
0.16
0.14
0.13
0.11
0.11
PBG(%) q PBG(%)
PBG(%)
Q
45.13 0.00 49.32 0.00
36.67 0.14 40.95 0.09
37.53 0.13 42.55 0.085
37.27 0.28 41.56 0.206
27.82 0.56 30.71 0.412
R - 6mm - 0.4ΔA
44.27 0.09 44.80 0.06
R -8mm
43.0 0.13 44.38 0.085
R -10mm
42.2 0.11 45.00 0.07
R -12mm
33.3 0.48 35.50 0.254
O - 8mm
37.0 0.52 37.55 0.279
O - 10mm
36.9 0.51 38.43 0.264
O -12mm
30.1 0.77 34.80 0.4768
LS_N = 4
22.2 1.00 27.70 0.6472
LS_N = 5_φ = 0π
27.8 0.81 30.21 0.5021
LS_N = 5_φ = 0.25π
29.0 0.70 31.20 0.4141
LS_N = 5_φ = 0.5π
30.1 0.71 34.10 0.4247
LS_N = 5_φ = 0.75π
32.1 0.84 32.80 0.518
LS_N = 5_φ = π
Table 5.1: the geometric factors and the size of PBGs. α , β , γ are three geometric factors.
R - p6mm
R - p4mm
R - p2
R - 6mm - 0.2ΔA
22.06
9.88
16.43
20.26
16.52
21.47
19.93
19.38
17.64
13.33
17.77
12.93
5.57
9.86
12.98
15.46
16.05
q
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.1
0.1
The overall factors q for three permittivity contrasts is also shown.
In Figure 5.2, PCs and PQCs consisting of uniform circular rods (R-p6mm,
8mm, 10mm, and 12mm) have zero shape anisotropy parameter ( α = 0 ) and zero
standard deviation of area ( β
= 0 ).
Therefore they have low overall factors. Further,
circular particles lead to strong isotropic Mie resonance, therefore they have large
PBGs and occupy up left corner of the figure. If
R - p6mm - 0.2ΔA structure
in Figure 5.1(b) and the
A− A ≠ 0 ,
such as occurs for the
R - p6mm - 0.4ΔA structure
5.1(c), the PBG size decreases as the area deviation increases.
91
in Figure
Compared to PQCs consisting of rods, PQCs consisting of ovals (O-8mm,
O-10mm, and O-12mm) have a higher overall factor q because of the non-circular
features. The optical paths along the long axis ( nd long ) and the short axis ( nd short ) of
the oval feature are different, which leads to an anisotropic resonance condition.
Therefore they have smaller TM PBGs and occupy the middle area in Figure 5.2.
PQCs defined by level set equations, according to Figure 5.1(l) through
5.1(q), have non-circular features ( α > 0 ), non-uniform particle area distribution
(β
> 0 ),
and non-uniform neighbor particle distance distribution ( γ
> 0 ).
This leads
to high overall morphological factors and low TM PBGs. Therefore these
structures occupy the bottom right corner in Figure 5.2(a) and Figure 5.2(b). As
refractive contrast decreases, the PBG sizes of the structures also decrease. PQCs
are more robust compared to photonic crystals though they do not exhibit higher
PBGs than photonic crystals. For high permittivity contrast, a noticeable case is
LS_N = 5_φ = 0π :
the particle area distribution is highly polydisperse ( β
= 0.4 ),
so the
expected lowest TM PBG, which is supposed to be between the first and the
second resonance frequencies [140, 144], is destroyed by leaky modes and the
actual lowest TM PBG for this structure is between the second and the third
92
frequencies. Due to this shift toward higher frequency, the size of TM PBG is
relatively lower than other LS-QCs.
In the above discussion, we focused on the lowest PBG, which is usually the
largest PBG for 2D photonic crystals. Among the 17 structures investigated, the
structure of the triangular lattice ( R - p6mm ) is an interesting photonic crystal device
platform due to its large PBG. Next we use geometrical factors to predict the
impact of possible experimental errors on the size of TM PBG. For example, we
introduce a random deviation of particle area ( A = A ± δ A ) to the R - p6mm structure
with permittivity contrast
ε 2 : ε 1 = 13 : 1 .
The decrease of photonic band gap due to the
experimental fabrication error δ A can be estimated using the prediction based on
the associated geometrical factors and the linear approximation in equation (5.8).
For δ A = 0.1A , the PBG size is predicted to be 43% and the FDTD simulation
indicates that the actual PBG size is 46%. For δ A = 0.2A , the PBG size is predicted
to be 39.8% and the FDTD simulation indicates that the actual PBG size is 42%.
For δ A = 0.3A , the PBG size is predicted to be 36.7% and the FDTD simulation
indicates that the actual PBG size is 34.6%. Next, we introduce three kinds of
perturbations to the
R - p6mm
structure: in addition to the particle area deviation
( δ A = 0.2A ), every particle has 40% probability to have its center shifted from the
93
original coordinate
( x, y )
to
( x ± δ x, y ± δ y ) ;
here δ x, δ y = 0.14d0 . Also, we assume the
particle shape deforms to be oval with aspect ratio
d long / d short = 1.5
and random
orientation. Such a distorted structure has an overall factor ( q ) of 0.385 and the
PBG size is predicted to be 34%. The FDTD simulation indicates that the actual
PBG size is 31.9%. All 4 tests above show that the actual PBG sizes deviate from
the predictions based on geometrical factors less than 4% PBG size.
Our concept of developing a statistical parameter related to the structural
geometry can be extended to 3D although in order to be self supporting,
dielectric/air structures need to have the dielectric regions connected. For
resonance dominant TM gap formation, a "tight binding" structure is necessary,
requiring thin connecting "bonds" between the larger dielectric nodes. The q factor
in 3D will then be related to variations in the shape, sizes and distances of the
nodes as well as the overall isotropy of the structure.
5.3. Summary
In conclusion, we employed a novel statistical procedure to correlate the
geometric properties of 17 PBG structures and their TM PBG sizes. Based on our
94
analysis, to achieve a photonic crystal with a large TM PBG, one needs separate
circular particles with uniform distribution. The physical meaning of the interesting
optimization procedures proposed before [112, 128] are revealed: the PBG
optimizations, which are done numerically by tuning the coefficients of level set
equations [128] or the gradient-based algorithm known as topology optimization
[112], are essentially tuning the particle shape to the near-circular shape ( α ≈ 0 ) and
reaching a uniform particle distribution with nearly the same neighbor particle
distance ( β
≈0
and γ
≈ 0 ).
95
Chapter 6
A searchable database for micro- and nano-structures
6.1. Introduction
Periodic micro- and nano-structures are of importance in a variety of fields,
including photonic crystals [31, 97], phononic crystals [160, 161], data storage
devices [162, 163], fuel cells [164], microfluidics [165], biosensors [166], tissue
engineering [167, 168], optical devices [169], impact absorption micro-trusses
[170], plasmonic metamaterials [171], and energy applications including solar
photovoltaics [172]. Though there are extensive ongoing investigations on
functional micro and nano periodic structures, there is at present no system to
comprehensively classify, store, process, and analyze the various functional
structures that are continually emerging from these exciting research areas.
Structure database system has been established for proteins [173-175],
chemicals [176], crystal structures [122, 177, 178], minerals [119, 179],
biotechnology information [180], standard references [181], and drug designs
[182]. In this chapter, we propose a database system which incorporates various
functional micro- and nano-structures. The database connects the morphologies,
fabrication technologies, and physical properties of the functional structures. The
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advantages of the database system include: (1) enabling the design of experiments
for fabricating a given target structure; (2) finding the structure with an extremized
objective function; (3) analyzing the impact of morphology on various physical
properties.
6.2. The protocol to process structures in the database
A periodic structure can be represented by a unit cell function S ( x, y , z ) .
S ( x, y , z ) = i if the material i occupies position ( x, y , z ) within the unit cell of the
periodic structure. If S ( x, y , z ) is directly stored, a large amount of storage space is
required. For example, if the unit cell is meshed to 1000 × 1000 × 1000 grids, a single
structure requires 240MB of storage space. A database consisting of one million
structures would then need 2.4 × 105 GB of storage. To reduce the storage space, we
take advantage of the periodicity of the structures: since all periodic structures can
be described by a set of Fourier components, a small set of Fourier expansion
coefficients of S ( x, y , z ) can be used to efficiently represent the structures.
The structural unit cell function S ( x, y , z ) can be expressed as:
S ( x, y, z ) =
∑a
lmn
l ,m, n
cos(lπ x d x + mπ y d y + nπ z d z ) + ∑ blmn sin(lπ x d x + mπ y d y + nπ z d z )
l ,m,n
97
(6.1)
Here d x , d y , d z are the periodicities of the structure along the x, y, z directions. We
select one hundred Fourier coefficient groups which have the largest values of
2
2
almn
+ blmn
to represent a given structure. The choice of the number of Fourier
coefficient groups is a tradeoff between storage space, calculation time and
structural description accuracy. Using our approach, the storage of one structure
only needs 10kB and therefore a database consisting of one million structures only
occupies 10 GB, which can be stored in a personal computer (PC) or ordinary
external hard drive.
The morphologies of human-made and naturally occurring periodic
structures are unlimited. Thus any reasonably comprehensive database will be
gigantic; therefore an efficient classification scheme for the database is necessary.
We categorize the functional structures according to their symmetry using the
seven crystal systems, 32 point groups and 230 space groups. Because of
Neumann’s principle: that the macroscopic physical properties of any periodic
structure have at least the symmetry of the structure, this scheme also helps
researchers search for certain properties. The database is classified into a tree
structure which is shown in Figure 6.1.
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Figure 6.1: the database is classified to a tree structure according to the symmetry of the
periodic structures.
6.3. Inverse design of experiments to fabricate target structures
The vast applications of micro- and nano-structures have motivated the
development of multiple fabrication techniques, ranging from bottom-up
approaches including polymer phase and micro-phase separation [183, 184],
molecular self-assembly [185], and colloidal self-assembly [186] to top-down
99
approaches including phase mask lithography [187-190], interference lithography
[191-193], multiple-exposure interference lithography [194], electron-beam
lithography [195], and Helium beam lithography [196, 197]. Each technique has its
own particular group of ever growing accessible structures.
By integrating various simulation results and actual experiments, we aim to
establish a database comprised of a library of these mesoscale 3D and 2D periodic
structures and their corresponding fabrication techniques and fabrication
parameters. The structural length scale and material compositions are also included
in the database structures. The database can indicate which kinds of structures can
be fabricated using existing technologies.
One important function of the database is to assist in the experimental design
of the fabrication of a structure with particular morphology. After a database user
uploads the target structure in various formats such as CAD images, level set
equations, and Fourier coefficients, an automated program is used to both process
the target structure (extract Fourier coefficients) and to compare it with the stored
structures in the database. The most similar structures are selected and their
100
corresponding fabrication parameters are the output. Our method is schematically
shown in Figure 6.2.
Figure 6.2: the schematic of the database system incorporating structures uploaded by
researchers. A user can upload a target structure which will be compared with the
database structures. The most similar database structure is output with structural
information such as experimental fabrication method and physical properties.
In our method, we need to estimate the similarity of the target structure to
the structures in the database. The similarity of two objects is normally judged
visually and by one’s experience. To be more precise, we define the following
similarity function P to quantify the correspondence between a database structure
and the target structure:
101
P=
∑ (a
lmn
− a0 lmn ) 2 +
l ,m ,n
∑ (b
lmn
− b0 lmn ) 2
(6.2)
l ,m ,n
Here almn , blmn are Fourier coefficients of a particular database structure and a0 lmn ,
b0 lmn are Fourier coefficients of the target structure. However, the unit cell function
S ( x, y , z ) is not uniquely defined because the origin of the coordinate system is
arbitrarily chosen; thus we need to calculate P for different origin definitions and
select the smallest value of P to represent the closest similarity of the candidate
structure and the database structure.
In order to minimize additional computation we follow the convention of the
International Tables for Crystallography: the origin of the coordinate system is the
site of the highest symmetry of the space group. From equation (6.2), the similarity
function P is a positive value, the smaller the similarity function; the closer the two
structures. A value of P = 0 indicates that two structures are essentially the same.
An example of the similarity function is shown in Figure 6.3. According to our
experience (we employ one hundred Fourier coefficients to define a structures), if
P < 0.01 ,
the two structures are visually the same; if P < 0.1 , the two structures are
only slightly different; if P > 0.3 , the two structures are obviously visually different.
102
Figure 6.3: two examples of the variation of the similarity function and the visual
appearance of a particular structure and the target structure.
(a). The structures are defined by the diamond family approximation level set equation:
F ( x, y, z ) = cos ( 2π x ) cos ( 2π y ) cos ( 2π z ) + cos ( 2π x ) sin ( 2π y ) sin ( 2π z )
+ sin ( 2π x ) cos ( 2π y ) sin ( 2π z ) + sin ( 2π x ) sin ( 2π y ) cos ( 2π z ) = t
The target structure corresponds to t = 0 . The structure similarity function ( P ) increases
as the structure deviates from the target structure. (b): the target stucture is defined by the
simple cubic surface approximation:
F ( x, y , z ) = cos ( 2π Qx ) + cos ( 2π Qy ) + cos ( 2π Qz ) = 0
The target structure corresponds to Qtarget = 1 and ΔQ = Q − Qtarget .
103
For a PC with single 2.0GHz CPU, calculating the similarity function of the
target structure and one database structure requires only about one second. By
employing a 100 CPU server, the search time for a database of one million
structures reduces to around two hours. Furthermore, the tree structure of the
database avoids unnecessary computation: only the database structures which have
the same symmetry as the target structure are needed to be visited during the
search.
After comparing the target structure with the database structures, a structure
with Pmin is selected. If Pmin < 0.1 , we can assume that the structure is a close
approximation of the target structure. Sometimes we have multiple sets of
experimental parameters which all give Pmin < 0.1 . The user can then choose the
structure with the most experimentally convenient set of fabrication parameters, for
example, the highest contrast ratio ( I max I min ) in photolithography technology [191].
If Pmin > 0.1 , then additional corresponding experimental adjustments of the
experimental parameters might be needed. By examining the structures in the
database, users can observe the evolution of the structure as fabrication parameters
change and decide on the adjustments. As the database evolves and new fabrication
104
technologies are realized, the chance to find database elements which precisely
match a given target structure increases.
Next, we illustrate how this approach can work by establishing a mini
database for phase mask lithography [188, 198-200]. Here the task is to design a
2D periodic mask, located above a photoresist in the XY plane in order to fabricate
a 3D targeted structure. When a coherent light source, such as plane wave,
propagates along the Z direction and impinges on the mask, the incident light is
reflected and diffracted such that an optical field pattern forms inside the
photoresist, which can be used to fabricate 3D micro- and nano- structures.
Compared to other techniques, phase mask lithography offers the advantages of
fabrication over large areas with a very simple experimental setup [188, 190, 198].
However, the inverse design problem of phase mask lithography is challenging.
We address this problem by creating a structural database consisting of phase mask
lithography elements, to store phase mask lithography designs and their
corresponding 3D periodic structures.
We choose to explore the possibility to fabricate a layered photonic crystal
structure [201]. The structure, shown in Figure 6.4, consists of alternating rod and
105
vein layers. It can be fabricated through layer-by-layer assembly [202], which is
time-consuming and the structure size is limited. Therefore high throughput phase
mask lithography, which can fabricate large area nanostructures (above cm 2 ) is
designed to fabricate the structure. The target structure belongs to the tetragonal
crystal system. The aspect ratio is d z d x, y = 1.09 and the filling ratio is 0.3. The target
structure exhibits a large, complete photonic band gap (18% for silicon/air) and it
is an excellent candidate for waveguide device [201].
Figure 6.4: the target photonic crystal structure with P4m2 symmetry. The enlarged unit
cell is shown on the left.
In phase mask lithography, there are three experimental parameters which
strongly influence the morphology of the resulting structure: 1) the light source
power ( I ). A higher laser power employed with a negative photoresist results in a
higher filling fraction; whereas for a positive photoresist, the opposite occurs; 2)
the depth of the phase mask surface profile ( h ). As h changes, the phase
106
modulation of the propagating light changes; 3) the symmetry of phase mask
lithography system: the phase mask symmetry and the light source polarization.
Each layer of the target structure has four-fold rotational symmetry in XY plane
and the target structure has two-fold rotational symmetry; therefore we select
circularly polarized light source to provide high rotational symmetry in XY plane.
Further, we select the grating mask with two-fold rotational symmetry and the
mask with four-fold rotational symmetry (posts on a square lattice). Those two
phase mask patterns can be readily fabricated by nanoimprint lithography,
interference lithography, and E-beam lithography. Based on these patterns, we
select six phase masks with two layers [203] to create the database. The set of
phase masks are shown in Figure 6.5. A different order stacking of the layers
creates different phases between the diffracted beams from the first layer and the
diffracted beams from the second layer.
107
Figure 6.5: six types of phase mask are investigated. Every phase mask consists of two
layers each with the same thickness h . The layer can be a one dimensional grating or air
posts on a square lattice in solid dielectric background. Here h , r1 and r2 are tunable
parameters.
The correct periodicity of the phase mask can be determined from the
structural aspect ratio (see supplemental information). We choose light with
λ = 810nm , a wavelength that is effective for two-photon lithography, which
achieves a higher effective contrast ratio [199]. During the process of creating the
mini database, structures are added to the database as we vary experimental
parameters. If the variational step is too large, the resultant structures are too
sparsely distributed and do not well represent all the structures available from the
technique; if the variational step is too small, the neighboring structures are almost
108
identical and occupy unnecessary database storage space. A good selection of the
variational step is that the neighboring structures should not be visually the same
( P > 0.1 ) but should be only slightly different ( P ≤ 0.3 ). The selected experimental
parameter should be evenly distributed to cover all situations. For every phase
mask, we pick three values for r1 and r2 :
r1(2) d < 0.1 or r1(2) d > 0.9 ,
r1(2) d x , y = 0.27 , 0.54
and 0.81 because if
the fabrication of the mask is difficult and the filling ratio
of the resulting structures tends to be low. We pick three layer thickness values:
h d x , y = 0.29, 0.58, 0.86 . We choose h d x , y < 1 to avoid high phase mask aspect ratio.
The light source powers have ten levels to cover wide filling ratio range. Therefore
every binary layer phase mask corresponds to 270 experimental designs. The minidatabase we construct thus has 1620 structures.
We next calculate the similarity function between the target structure and the
database structures. It takes a personal computer around thirty minutes to locate the
minimum similarity function Pmin = 0.23 . The experimental design is: phase mask (3),
d x , y = 520nm , h = 150nm , r1 = 420nm , r2 = 420nm , Ithres I = 0.75 . To further reduce the
similarity function, we tuned the parameters and make a small database of 270
elements
based
on
phase
mask
h = 120nm, 135nm, 150nm, 165nm, 180nm, 195nm
109
,
(5).
Here
we
choose
r1(2) = 360nm, 420nm, 480nm ,
and
I thres I = 0.65, 0.7, 0.75, 0.8, 0.85 . Then we calculate the similarity function
between the target structures and database structures. The minimum similarity
function is now Pmin = 0.219 . The new experimental design is: phase mask (3),
d x , y = 520nm , h = 195nm , r1 = 420nm , r2 = 420nm , Ithres I = 0.8. The comparison between
the target structure and the structure from experimental design is shown in Figure
6.6.
Figure 6.6: photonic crystal structure with P4m2 space group. Comparison of the optimal
phase mask fabricated structure (left) and the target structure (right). The skeletal graph
of the target structure is shown in blue. The key 4 fold rotoinversion, diagonal mirror with
perpendicular 2 fold rotation symmetries are captured in the phase mask structure using
circularly polarized incident radiation.
6.4. Discussion
To design structures with extremized physical properties is quite challenging
and attracting [204]; for example, finding the photonic crystal with the largest
photonic band gap [128, 205], or a microtruss with the maximum ratio of Young’s
110
modulus to density ( E ρ ), etc. If a physical property can be calculated from the
structural morphology, it can be included as a structure element in the database. A
comprehensive database, which incorporates hundreds of thousands of structures
fabricated by various techniques, is an excellent reservoir to find a candidate
structure with an extremized physical property. Furthermore, we can analyze the
impact of changes in symmetry on the physical property by comparing the
properties of structures residing in different symmetry branches of the database.
Some functional mesostructures are non-periodic. If a functional structure
has finite size, it can be assumed periodic by employing periodic boundary
conditions. If a functional structure has unlimited size, for example, a photonic
quasicrystal [104], a representing portion of the structure can be assumed as the
unit cell and periodic boundaries are employed. Although the database is designed
for periodic structures, non-periodic structures can also be included.
For a functional mesostructure, if there is a way to link the fabrication
parameters and the physical properties with the structure, it can be incorporated in
the database. Given the fast development of nanotechnology, any comprehensive
database will be gigantic. Constructing such a large database which contains
111
functional structures fabricated from various technologies would need a worldwide cooperation. The continual update and maintenance of the database would
involve a sustained effort. If a user has a target structure in mind for a desired
function, an automated program will search the database and locate structures that
are the most similar to the target structure and output their fabrication. If a user
does not know the target structure but specifies a desired physical property, the
database can be used to find various candidates and even the optimal structure and
output the design methodology. We hope that initiating such a system will
strengthen world-wide scientific cooperation.
6.5. Summary
We have developed a database system to efficiently classify, store, analyze
and compare micro- and nano-structures. A multi-functional database incorporates
a large number of natural and human made structures uploaded by contributors.
High throughput experimental design and structural property analysis can be
achieved using the database system. After a target structure is assigned by the user,
a high throughput numerical engine compares the target structure with the
structures inside the database and outputs the most similar database structure with
associated experimental design and physical properties. The proposed database
112
incorporating millions or even billions of structures could be accessed through a
public website, which is maintained and updated via a worldwide cooperation.
113
Chapter 7
Elastic wave modeling
7.1 Finite-difference time-domain modeling of elastic waves
Controlling elastic waves have a wide range of engineering applications,
such as acoustic cloaking [15], acoustic imaging [37, 38], seismic wave sensing
[39, 40], opto-acoustic interaction [41, 42], and sub-wavelength imaging [206].
The theoretical modeling is indispensable in the development of phononic
crystals and acoustic metamaterials. Here we discuss a finite-difference timedomain (FDTD) approach to model the propagation of acoustic waves. In the
elastic range, the equation of motion is:
∂ 2ui
ρ 2 = τ ij , j + f i
∂t
(7.1)
Here ρ is the density of the material, u is the displacement of the material, τ is the
stress, and f is the body force applied to the material. We assume f = 0 and adopt
the Einstein notation:
τ ij , j =
∂τ i1 ∂τ i 2 ∂τ i 3
+
+
∂x1 ∂x2 ∂x3 114
(7.2)
The sub-indices i and j indicate the directions of the displacement, stress,
and strain. Here we use a Cartesian coordinate system, and 1, 2, 3 ( x1, x2 , x3 ) refer to
the X, Y, and Z directions, respectively. For fi = 0 , equation (7.1) can be expressed
as:
∂ 2u1 ∂τ11 ∂τ12 ∂τ13
=
+
+
∂t 2
∂x1 ∂x2 ∂x3
(7.3)
∂ 2u2 ∂τ 21 ∂τ 22 ∂τ 23
ρ 2 =
+
+
∂t
∂x1 ∂x2 ∂x3
(7.4)
∂ 2u3 ∂τ 31 ∂τ 32 ∂τ 33
=
+
+
∂t 2
∂x1 ∂x2 ∂x3
(7.5)
ρ
ρ
Equations (7.3) through (7.5) can be further rewritten as:
∂v1 ∂τ 11 ∂τ 12 ∂τ 13
=
+
+
∂t
∂x1 ∂x2
∂x3
∂v
∂τ
∂τ
∂τ
ρ 2 = 21 + 22 + 23
∂t
∂x1
∂x2
∂x3
∂v
∂τ
∂τ
∂τ
ρ 3 = 31 + 32 + 33
∂t
∂x1
∂x 2
∂x 3
ρ
(7.6)
(7.7)
(7.8)
Here vi is the velocity along the direction i . Next we discuss how to express the
stress τ ij in terms of the displacement. The constitutive law for a general material is
τ ij = Cijkl ε kl
C ijkl
(7.9)
is a fourth rank elastic tensor and has 81 components. Due to the symmetry of
this system, τ ij = τ ji and ε kl = ε lk . The above facts lead to a simplified expression
( 6 × 6 matrix CIJ ) for the fourth rank elastic constant tensor C ijkl :
115
Cijkl
⎛ C11 C12
⎜C
⎜ 21 C22
⎜C
C32
= C IJ = ⎜ 31
0
⎜ 0
⎜ 0
0
⎜
0
⎝ 0
C13
C23
0
0
0
0
C33 0
0 C44
0
0
0
0
0
0
C55
0
0 ⎞
0 ⎟⎟
0 ⎟
⎟
0 ⎟
0 ⎟
⎟
C66 ⎠
(7.10)
For isotropic structures (cubic crystal system), there are only 3 independent
parameters. For transverse isotropic structures (tetragonal crystal system), there are
5 independent parameters. For anisotropic structures (orthotropic crystal system),
there are 9 independent parameters.
Here
the
sub-indices
I , J = 1, 2, 3, 4, 5, 6
represent
xx , yy , zz , yz , xz , xy
respectively. Inserting equation (7.10) into equation (7.9) leads to:
∂τ 11
∂t
∂τ 22
∂t
∂τ 33
∂t
∂τ 23
∂t
∂τ 13
∂t
∂τ 12
∂t
∂v1
∂v
∂v
+ C12 2 + C13 3
∂x1
∂x2
∂x3
∂v
∂v
∂v
= C21 1 + C22 2 + C23 3
∂x1
∂x2
∂ x3
∂v
∂v
∂v
= C31 1 + C32 2 + C33 3
∂x1
∂x 2
∂ x3
∂v ∂v
= 0.5 × C44 × ( 2 + 3 )
∂x3 ∂x2
∂v ∂v
= 0.5 × C55 × ( 1 + 3 )
∂x3 ∂x1
∂v ∂v
= 0.5 × C66 × ( 1 + 2 )
∂x2 ∂x1
= C11
(7.11)
(7.12)
(7.13)
(7.14)
(7.15)
(7.16)
For homogeneous isotropic solids:
C44 = C55 = C66 = 2G =
116
E
(1 +ν )
(7.17)
C12 = C21 = C23 = C32 = C13 = C31 =
C11 = C22 = C33 =
Here E is the Young’s modulus,
G
νE
(1 +ν )(1 − 2ν )
(1 −ν ) E
(1 +ν )(1 − 2ν )
(7.18)
(7.19)
is shear modulus, and ν is Poisson’s ratio. For
fluidic materials:
C44 = C55 = C66 = 0
C11 = C12 = C13 = C21 = C22 = C23 = C31 = C32 = C33 = K
τ 11 = τ 22 = τ 33 = − p
Here
K
is the bulk modulus of the fluid and
(7.20)
(7.21)
(7.22)
p is the pressure. Equations (7.6)
through (7.8) and equations (7.11) through (7.16) are then greatly simplified:
ρ
∂v1 ∂p
=
∂t ∂x1
(7.23)
ρ
∂v2 ∂p
=
∂t ∂x 2
(7.24)
ρ
∂v3 ∂p
=
∂t ∂x3
(7.25)
∂v ∂v ∂v
∂p
= −K ( 1 + 2 + 3 )
∂t
∂x1 ∂x2 ∂x3
(7.26)
After the above mathematical transformation, the governing equations for elastic
waves in fluids (acoustic waves) become linear partial differential equations which
are readily solved. Here we give an exemplary simulation result:
117
Figure 7.1: (a): a solid rectangular plate vibrates in water along the x direction. The
vibration frequency is 1.43Hz and the vibration follows the sinusoidal function:
U x = sin(2π ft ) . (b) the pressure ( p ) distribution in the water after the plate vibrates for 3.5
seconds; (c) the displacement along y direction ( u y ); (d) the displacement along x direction
( ux ).
Next, we test the computation method by comparing our results with the
results from previous reference [207]. We calculate the local density of states
(LDOS) for an elastic wave propagating in a 10-layer woodpile phononic crystal
consisting of PMMA and air. The unit cell of the woodpile structure is shown in
Figure 7.2. It is periodic along X and Y directions with periodicity a0 . According to
LDOS, a phononic band gap along the Z direction between 0.6ωa 0 2πcair and
118
1ωa 0 2πcair exist, which matches very well with the result calculated from the finite
element method [207].
Figure 7.2: LDOS for a phononic wave propagating along Z direction. The 10-layer
woodpile structure is periodic in X and Y directions, and we show only one unit cell here.
The green part of the structure is PMMA. Here c air is the sound velocity in air.
119
Chapter 8
Summary and future directions
The last section reviews the main accomplishments of this thesis, and lists future
directions of our research.
8.1 Overview of research accomplishments
1). Optical force and optical torque computation (Chapter 2)
Using the three dimensional finite-difference time-domain (FDTD) and the
Maxwell stress tensor, we calculated the radiation force and optical torque from
optical lattices on various types of particles. We find that the optical force from
optical lattices is a function of particle refractive index, size, shape, orientation,
and the detailed morphology of optical lattice (equation 2.20). Absorption
generally leads to smaller optical forces. The optical force reaches the maximum
when the size of particle is around the period of the optical lattice. To form stable
optical traps for polystyrene particles ( n = 1.58 ), the energy density of the optical
lattice should be at or above
5mW μ m 2 .
For particles near to a spherical shape or
with a size much smaller than the period of the optical lattice, the optical torque is
small compared to particles with size comparable to the period of the optical lattice
and large aspect ratio. Both metastable and stable equilibrium orientation states
exist for cylinder particles in the one dimensional optical lattice.
120
2). Optical devices on a chip (Chapter 3)
Photonic crystal devices have demonstrated unique properties that enable
wavelength-scale optical devices with ultra-high energy efficiency [31, 98, 208]. In
Chapter 3, we developed a novel 2-pattern photonic crystal that offers the unique
ability to integrate optical devices for different polarizations (TM, TE or both) on
the same plane. Using these 2-pattern photonic crystals, we have further designed
devices that can bend, split, couple, and filter TM/TE waves simultaneously [4].
These devices include a wavelength-scale polarizer, a low crosstalk circuit and a
high-Q resonator for both TM and TE waves. Our work provides an important step
towards absolute on-chip optical wave control by offering a versatile platform that
can realize a wide range of important optical devices.
3). Photonic quasicrystals (Chapter 4)
The ability of photonic crystals to mold the flow of optical waves is
primarily due to the presence of a photonic band gap (PBG). Photonic quasicrystals
demonstrate premier PBG properties at low refractive index contrast and therefore
greatly expand the selection of suitable materials for photonic devices. Chapter 4
provides a comprehensive analysis of PBG properties of 558 two-dimensional (2D)
quasicrystals (QCs). Importantly, we discovered two novel QCs, which have the
best PBG properties to date [5]: a 12mm QC with a 56.5% TE PBG, the largest
121
reported TE PBG for an aperiodic crystal, and a QC comprising “throwing star”like dielectric domains, with near-circular air cores and interconnecting veins
emanating radially around the core. This interesting morphology leads to a
complete PBG of ‫׽‬20%.
4). Statistical analysis on structural morphology (Chapter 5)
The size of the PBG in a photonic crystal device determines the quality of
the device and the frequency range of waves that such a device can control. The
fundamental formation mechanisms of PBGs in photonic crystals are not always
clear, thus there is no general rule to design a given PBG. In Chapter 5, we
develop a statistical approach for predicting band gaps of 2D photonic crystals and
quasicrystals based on their morphology. The approach integrated knowledge from
mathematics, scientific computing and physics to identify three geometric factors
that correlate strongly with the size of PBG. This work helped elucidate the TM
PBG formation mechanisms relevant to a given dielectric structure consisting of
separated particles.
5). Searchable database of nanostructures (Chapter 6)
Periodic nanostructures have a wide range of applications including fuel
cells, photonic/phononic crystals, data storage devices and tissue engineering. As
122
new structural forms emerge, there is a need for a unified system to classify and
store, as well as allow researchers to process and analyze these structures. In
Chapter 6, we report the development of a database system to connect the
morphologies, fabrication techniques and some physical properties of a library of
two- and three-dimensional periodic structures. The system is empowered by an
automated algorithm to efficiently store and compare structures (a database of one
million structures requires only 10GB of storage and 2 hrs of searching based on
2011 IT technology). Users of this system input target structures, and the algorithm
analyzes the target and compares it with structures in the database to output the
fabrication parameters and physical properties of the most similar structure.
8.2 Future research directions
Future research direction (1): 2-pattern photonic crystal fabrication
In Chapter 3, we proposed a new set of photonic crystal devices based on 2pattern photonic crystals. Two advantages of 2-pattern crystals are the following:
1) the 2-pattern photonic crystals are reasonably tolerant of possible
experimental errors, including variations of air gaps between dielectric regions. For
example, we introduced a 40% random variation in the diameter of the rods (
r = r0 ++r0 , here + r0 is a random value between [ −0.4r0 , 0.4r0 ] ) and 40% random
123
variation in the thickness of the honeycomb walls ( h = h0 ++h0 , here +h0 is a random
value between [ −0.4h0 , 0.4h0 ] ) to the rod-honeycomb structure, and the complete
PBG is still large (16% for ε 2 : ε1 = 11.4 :1 , compared to the 20.4% PBG of the 2pattern photonic crystal without defects). Furthermore, we also introduced 30%
G
G
G
G
random variation in the positions of rods ( r = r0 ++r0 , here +r0 is a random value
between [0.3a(TM ), −0.3a(TM )] ) and the complete PBG only drops to 15%.
2) different from casual photonic crystal devices, 2-pattern photonic crystal
devices mold the flow of both TM and TE waves on the same plane. In our
previous work, we have demonstrated the design of various 2-pattern photonic
crystal devices including wavelength scale polarizer, crossed waveguide, and a
high-Q polarizer for both TM and TE waves [4].
One future research direction stemming from this work can be the
experimental fabrication of 2-pattern photonic crystals and the optical
characterization of the associated devices. In our previous work, we have
demonstrated the morphology of six 2-pattern photonic crystals [3]. For the visible
and near-IR frequency regimes ( 300nm < λ < 2 μ m ), experimental techniques that
can be used to fabricate 2-pattern photonic crystals include nano-imprint
lithography [209], electron beam lithography [162, 195], and focused ion beam
124
(FIB). The materials of photonic crystals for visible and near IR regime can be
silicon or GaAs. The optical characterization can be performed using NSOM
equipment. The TM and TE wave intensity can be measured to test the wave
control abilities of 2-pattern photonic crystal devices, such as the 2-pattern
photonic crystal polarizer and resonators shown in reference [4].
One important feature of photonic crystal devices is that photonic properties
are not limited by frequency and length scale. Fabricating structures on a
millimeter scale is much easier than on nanometer/micrometer scales. 2-pattern
photonic crystals with millimeter scale features can be fabricated to control
terahertz (THz) waves. One possible material for 2-pattern photonic crystal in THz
regime is Aluminum ( ε Al : ε 0 ≈ 7 :1 ). Even though the permittivity contrast is lower
than Silicon: air contrast ( ε Si : ε 0 ≈ 16 :1 ), the PBG is still quite large (10%).
Future research direction (2): photonic quasicrystal devices
In Chapter 4, we examined photonic quasicrystals based on level set
equations and also created from a projection algorithm. Photonic QCs have
rotational symmetries forbidden in periodic crystals; therefore they offer unique
optical properties including isotropic photonic bands [128, 129], defect-free
localized modes [32, 33, 130], polychromatic band gaps [210], and the possibility
125
to open a PBG at quite low refractive index contrast [34, 35]. Therefore photonic
QCs are excellent candidates for nonlinear interaction [210, 211], waveguiding
[132, 133, 212, 213], plasmonics [214], negative refraction [134], and optical
communication [135].
Using the photonic QCs with large TE and TM PBGs as platforms [5, 8], we
can create photonic devices (cavities, resonators, waveguides) within a medium
with high local rotational symmetries (8-, 10-, and 12-fold). A negative tone LS10mm QC with filling ratio around 0.24-0.38 is an excellent candidate for photonic
devices [5]. It can be readily fabricated using multiple exposure interference
lithography [5, 147]. Its lowest TM PBG width remains almost constant for a wide
range of filling fractions and this insensitivity makes it a good candidate for
photonic devices. Photonic quasicrystal devices can be fabricated by removing or
adding rows of features to the QC. Since the LS-10mm QC structures can be
fabricated by MEIL, direct laser writing could be combined in the fabrication
process to write devices directly.
Several possible device designs can be envisioned:
126
1) A slow light waveguide via removing structural features. The photonic
bands near the PBG edges are ultra-flat for quasicrystals, which offer slow light
effects for all wave vectors.
2) Polychromatic resonators by removing/adding local features. A 2D
structure with polychromatic PBGs offers fundamentally new abilities to
manipulate photons on chips by allowing cavities with multiple resonance
frequencies of integer ratios. Such cavities can be used to significantly enhance
nonlinear interactions [215]. However, only one QC platform was shown and the
cavity quality factors were not fully optimized [215]. We observed more than ten
polychromatic PBGs in the PBG maps we plotted in Chapter 4 [5, 8]. One future
direction for this work is to pursue a more comprehensive study to maximize
cavity quality factors via Fourier analysis and gradient-based optimization, and to
propose nonlinear devices based on QC cavities.
3) Phononic (acoustic/elastic wave) quasicrystals. In our previous work [5,
7], we created quasicrystals using level set equations and a projection algorithm
and examined their photonic properties. Their phononic properties can be
interesting from high symmetry. COMSOL software can be useful for phononic
band gap simulation.
127
Future research direction (3): structural morphology-PBG analysis for three
dimensional photonic crystals.
In Chapter 5, we established a structural morphology-PBG analysis for 2D
photonic crystals. Our concept of developing a statistical parameter (overall factor q)
arising from various structural factors [7] can be extended to three dimensions (3D).
In order to be self supporting, dielectric/air structures need to have connected
dielectric regions. The PBG in 3D photonic crystals can arise from 2 aspects: 1)
Bragg diffraction. The scattered fields from different structural portions interfere
destructively with each other and form a photonic band gap; 2) resonance based
PBG, a structure consisting of dielectric nodes is needed. The dielectric nodes act
as strong electromagnetic (EM) field reflector to stop the propagation of EM waves.
The correlation between the PBG and the structural morphology of a 3D
photonic crystal could be more complex than an 2D case we investigated earlier [7].
The q factor in 3D will then be related to variations in the shape, sizes and distances
of the nodes as well as the overall degree of isotropy of the structure. The q factor
should incorporate the following parameters:
128
(1): Spatial isotropy parameter.
The spatial distribution of materials should demonstrate some similarity for
different directions to form a complete PBG. The isotropy parameter can be
quantified from several aspects: (a) the Fourier coefficient signals along different
directions. The strength, frequency distribution and signal peaks can be compared
between different directions to judge the isotropy of the structure; (b) interface
distribution. We can measure the interface distribution along different directions to
judge isotropy. For example, the mean length of direction vector in different
materials can be compared for different directions.
(2): dielectric node analysis.
In our previous work [7], the three parameters were used to analyze the
geometric properties of dielectric nodes and to correlate with the size of PBG: (a)
M
the shape anisotropy factor α = ∑ ( v∫∂s Δri2 ds Si ri ) M ; (b) the standard deviation
i =1
of the areas of the particles: β =
of dielectric nodes
M
∑ ( Ai − A)2 ( M − 1) A
i =1
; and (c) the spatial distribution
n
i
.
∑ ( di<−> j − d 0 )2 ( ni − 1) Md 0
i =0 j =1
M
γ= ∑
also be used to analyze dielectric nodes in 3D.
129
Those three parameters can
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