Honors Thesis

Simulation of Particle Arrays for Optical Bandgap Control
By: Jed Schales
Presented to The Honors College
in partial fulfillment of the requirements for
Honors Senior Thesis
Arkansas State University
___________ _____________________________________
Date Dr. Brandon Kemp, Advisor and Thesis Chair
___________ _____________________________________
Date Dr. Ilwoo Seok
___________ _____________________________________
Date Dr. Shivan Haran
___________ _____________________________________
Date Ms. Rebecca Oliver, Director of the Honors College
April 2015

1
Simulation of Particle Arrays for Optical Bandgap Control
Jed Schales
Faculty Supervisor: Brandon Kemp, Ph.D.
(Received 27 April 2015, Accepted 30 April 2015)
The bandgap structure of a rhombohedral array of nanoparticles was
studied under various configurations and for multiple selections of
nanoparticle material. Shifting of the bandgap due to change in the
particle array configuration was studied between the 2D square lattice
and 2D triangular lattice cases in MATLAB and COMSOL. The
relationship between the MATLAB and the COMSOL results as well
as the physical meaning of both sets of data was studied and
discussed. Finally, the patterns discovered by altering the
nanoparticle material provide insight into how to realize and perfect
the control of a nano-structure’s optical bandgap.
Introduction
Photonic bandgaps are an interesting
characteristic of particular nanoscale
periodic structures. A photonic bandgap is
a property of the geometry and
electromagnetic makeup of a material that
disallows the propagation of light waves
through a structure based on the
wavelength of the incident light.
Structures that exhibit a photonic bandgap
have been studied for over one hundred
years, since as far back as 1887 when Lord
Rayleigh created a quarter-wave stack
which completely reflected an incoming
light wave[1]
. Rayleigh’s quarter-wave
stack was made up of carefully sized
alternating layers of dielectric constant
which caused incident light waves to
interfere with each other causing no light
wave to be transmitted through the
device. Based on the same concept that a
structure’s pattern can cause interference
in an electromagnetic wave depending on
its geometry and material properties,
modern day nanoscale optical structures
have taken on a wide variety of odd
shapes and used novel materials that do
not exist in nature to produce bandgaps in
multiple dimensions. Materials that
exhibit these photonic bandgaps are
becoming more and more popular within
the realm of research today due to the fact
that they can control or manipulate light
in various exciting ways.
When a light wave is not allowed to
travel through a medium, it is not halted,
but rather reflected back toward the
source. Based on the periodicity of the
nano-structure, only certain single
wavelengths or often “bands” of multiple
adjacent wavelengths are reflected creating
a gap in the transmission band, hence the
term “bandgap”. This physical
characteristic of some periodic structures
allows for certain wavelengths of light that
correspond to specific colors within the
visible spectrum to be filtered and
reflected while others are transmitted
through the structure. This would be a
novel method of displaying color on a

2
device, as the device would display a color
of light based on its physical structure and
the incident light upon the display’s
surface as opposed to utilizing a liquid
crystal display which is backlit. This
device would need to be able to display
multiple colors at the user’s command,
and thus would require a tuneable
bandgap that can shift to reflect different
wavelengths corresponding to different
colors within the visible spectrum (410nm
for violet to 670nm for red). One theory
for the control of such a device involves
using an applied electric field to
reconfigure a particle array into a new
configuration with different spatial
periodicity and thus a different bandgap.
The study of the difference in the size
and location of bandgaps between the two
extreme configurations of a rhombohedral
particle array, square lattice with the angle
between axes equal to 90° and triangular
lattice with the angle between axes equal
to 60°, are considered. This structure is
defined in both the 2D and 3D cases by
three parameters – the lattice constant, a,
which remains constant during each study
conducted, the radius of the nanoparticles,
which for this study is 0.5*a meaning that
the particles of the lattice are touching
each other, and the angle between the
axes, θ, which will only take on the values
associated with the two cases previously
mentioned. An image depicting the shape
of the rhombohedral lattice is given in
Figure 1.
Figure 1. Rhombohedral Lattice[2]
Theory
The theory and equations that follow
were taken directly from Joannopolous’
Photonic Crystals: Molding the Flow of Light[1]
.
The science behind the photonic bandgap
phenomenon can be explained by the
macroscopic Maxwell equations which are
given in Eqs. 1-4 as:
[Eq. 1]
[Eq. 2]
[Eq. 3]
[Eq. 4]
where E is the macroscopic electric field,
D is the displacement field, H is the
macroscopic magnetic field, B is the
magnetic induction field, is the free
charge density, and J is the current
density. To apply these equations to the
case being studied, we state that the wave
propagation within the dielectric material
is independent of time and that it contains
no charges or currents ( = 0 and J = 0).
We also relate the displacement field to
the electric field through Eq. 5
∑ ∑
[Eq. 5]
which can be further simplified based on
the assumptions that the field strengths
are small enough to be linear, the material
is macroscopic and isotropic, the material
dispersion is ignored, and that permittivity
is purely real and positive. From this
relationship and a similar one relating the
magnetic induction field to the magnetic
field, Eqs. 1-4 become Eqs. 6-9 below.
[Eq. 6]
[Eq. 7]

3
[Eq. 8]
[Eq. 9]
In order to separate the time
dependence from the spatial dependence
of E and H, substitutions are made such
that
[Eq. 10]
and . [Eq. 11]
This produces Eqs. 12 and 13 from
the dot equations, Eq. 6 and Eq. 8, which
simply state that no point sources or sinks
of displacement or magnetic fields are
present and also that the electromagnetic
waves must always be transverse.
[Eq. 12]
[Eq. 13]
By combining the curl equations, Eq. 7
and Eq. 9, along with Eq. 14 which is one
expression for the speed of light in a
vacuum, the “Master Equation” Eq. 15 is
obtained.
√
[Eq. 14]
( ) ( )
[Eq. 15]
This equation is very important, since
it is the one that will be used to find the
modes of the magnetic field, H(r), which
correspond to eigenfrequencies, ω, which
will be used to plot the band diagram of
the optical device. This is performed by
generalizing the eigenvalue for any
direction of incident wave vector, k, by
restating the magnetic field as
[Eq. 16]
which signifies that the magnetic field is a
plane wave that has been polarized in the
direction of H0. By applying the
transversality requirement, these plane
waves are then solutions to the master
equation and produce eigenvalues of
( )
| |
[Eq. 17]
which can be rearranged to yield the
dispersion relation
| |
√
. [Eq. 18]
Since the wave vector can differ by
multiples of 2π, the mode frequencies are
also periodic multiples of each other.
Because of this, only k values between
±π/a need to be considered. This zone of
the wave vector is commonly called the
irreducible Brillouin zone, and it takes on
a characteristic shape based on the
geometry and dimension of the
periodicity. For the case of a rhombo-
hedral structure as in this study, the
Brillouin zone is an extremely complicated
3D space defined by 12 distinct points
that can be approximately described as a
skewed trapezoidal prism as shown in
Figure 2. By simplifying the model to the
2D case, the rhombohedral structure takes
on a square or triangular lattice at the
extremes of which this study is conducted.
Therefore, the Brillouin zone is simply
defined for both cases by 3 points which
form a triangle.
Figure 2. Rhombohedral Brillouin Zone[3]

4
This specific method of finding the
eigenfrequencies by directly using the
master equation is the method that is used
within COMSOL for the numerical
results, but instead of solving for H(r),
E(r) is solved which means that the results
are not strictly analytical due to the non-
Hermitian nature of using the E field
formulation of the master equation.
The method for calculating the
dispersion relation in MATLAB is very
mathematically intense, and the fields are
not solved for directly. Rather, several
mathematical methods are used so that
the dispersion relation can be directly
obtained simply based upon the dielectric
constants of the two media being
analyzed. A condensed version explaining
the theory behind the analytical
calculations follows and was taken from
the Massachusetts Institute of Technology
OpenCourseWare notes[8]
.
We define the space and spectral
domains by two three dimensional basis,
(a1, a2, a3) and (b1, b2, b3) respectively,
such that the translation vectors within
each domain can be written as
[Eq. 19]
and
[Eq. 20]
These two basis are linked since the
functions of the fields and permittivity are
periodic. This means that a relationship
between the two domains can be created
by using a Fourier expansion to state
∑ ̃ . [Eq. 21]
Because the electromagnetic (EM) fields
are also periodic, we can cast them as a
propagating function times a function
with the same periodicity as the medium
as
[Eq. 22]
where can represent either E or H, and
, indicating that the
overall function has the same periodicity
as the medium.
The master equation in H given
previously by Eq. 15 has a counterpart in
E given by
( ) . [Eq. 23]
By defining the inverse of the permittivity
function, Eqs. 15 and 23 can then be
made into a more symmetrical form.
∑ ̃
[Eq. 24]
( )
[Eq. 25]
( )
[Eq. 26]
After simplification of the lengthy process
of decomposing Eq. 22 with
representing E by using several changes of
index and variables, Eq. 23 becomes
( ) ∑ ̃ [Eq. 27]
Similarly, through decomposition of Eq.
24, Eq. 25 may be written as

5
∑ ̃
( )
[Eq. 28]
Parallel equations are derived for H, but
the transversality requirement for the
propagating wave requires that
[Eq. 29]
which allows us to define three vectors
(e1, e2, e3) such that
| | [Eq. 30a]
and , [Eq. 30b]
where these three vectors form an
orthonormal triad. The vector hG’ is then
decomposed via the triad and substituted
into Eq. 26 to eventually obtain
∑ ∑ {
}̃ ( )
[Eq. 31]
This equation is then cast in matrix form
by defining an operator Θk
(λG),(λG)’ such that
̃
| ||
| ( )
[Eq. 32]
so that
∑ ( )
[Eq. 33]
This equation is then swept for the
wave vector k across the spectral domain
G to determine the eigenfrequencies and
subsequently plot the dispersion relation.
The full MATLAB code is given in
Appendix A, and a sample snapshot of the
COMSOL work environment is included
in Appendix B.
Simulation Setup
From the results of an experiment
performed at the School of Science at
Tianjin University[5]
, to produce vivid
colors, the periodicity of the structure
should be designed so that a tight band of
wavelengths will be reflected (~50nm).
Because of this, the focus of this study
was to attempt to find a tight bandgap of
about 50nm or less that could be shifted
about the entire visible light spectrum
from 410nm to 670nm.
To first determine suitable materials
for the study, a MATLAB program
created by the faculty advisor and
modified by the author was used to find
materials that exhibited a 2D TM bandgap
within the optical region. The materials
found suitable for study were silicon,
vanadium, graphite, and poly-styrene.
These materials were chosen since they
encompassed a wide range of refractive
index values and because these materials
would be easiest to use in the manufacture
of a passive display device as mentioned
previously.
These materials were then researched
to determine their optical properties
within the visible spectrum. Values for
the refractive index and extinction
coefficient were found from an online
refractive index database[9-12]
, and these
values were then converted into real and
imaginary components of the dielectric
constant by using Eqs. 34-36.
̅ ̃ ̅̅̅
[Eq. 34]

6
[Eq. 35]
̃ [Eq. 36]
For simplification in the analysis and
because the imaginary terms of refractive
index and dielectric constant are
associated with decay over time, the
imaginary components of these quantities
were disregarded. The dielectric constant
values were used within the MATLAB
code, and the refractive index values were
used within the COMSOL simulation
studies. In order to obtain convergence
within the COMSOL simulation, the
averaged value of the refractive indices
had to be used instead of the true
frequency dependent values, which
accounts for some error.
The MATLAB analysis was
performed on a non-dimensional basis
where each parameter is normalized to the
lattice constant. The COMSOL analysis
was performed with the same geometric
parameters, but different materials. In the
COMSOL analysis, the lattice constant
was 250nm, with the radius of the
spherical particles being 125nm.
Results
The results of the MATLAB analyses
for the four different materials in both
configurations are given in Figures 3-6 on
the following pages. The square lattice
bandgap structures are given in the (a)
portions of the figures, and the triangular
structure in the (b) portions.
Following this, the results of the
COMSOL analyses for the four different
materials in both the square and triangular
configurations are given in Figures 7-14.
Due to complications in the parameter
sweep for the wave vector k within the
COMSOL analyses, only two edges of the
Brillouin zone are plotted, but previous
research conducted by the author has
shown that the fundamental band gap is
fully determined from the peaks at the
points Γ, Κ, and Μ of the square
reciprocal lattice or Γ, Χ, and Μ of the
triangular reciprocal lattice.

7
(a)
(b)
Figure 3. MATLAB Bandgap Structures of Silicon (n=4.59)
(a)
(b)
Figure 4. MATLAB Bandgap Structures of Vanadium (n=3.25)

8
(a)
(b)
Figure 5. MATLAB Bandgap Structures of Graphite (n=2.76)
(a)
(b)
Figure 6. MATLAB Bandgap Structures of Polystyrene (n=1.60)

9
(a) Γ to K Brillouin Zone Edge
(b) M to Γ Brillouin Zone Edge
Figure 7. COMSOL Silicon Square Bandgap Dispersion Relations

10
(a) Γ to X Brillouin Zone Edge
Figure 8. COMSOL Silicon Triangular Bandgap Dispersion Relations

11
Figure 9. COMSOL Vanadium Square Bandgap Dispersion Relations

12
Figure 10. COMSOL Vanadium Triangular Bandgap Dispersion Relations

13
Figure 11. COMSOL Graphite Square Bandgap Dispersion Relations

14
Figure 12. COMSOL Graphite Triangular Bandgap Dispersion Relations

15
Figure 13. COMSOL Polystyrene Square Bandgap Dispersion Relations

16
Figure 14. COMSOL Polystyrene Triangular Bandgap Dispersion Relations

17
Discussion
The analysis of these results consisted
first of ensuring that the band diagrams
were consistent in beginning and ending
at the same point Γ in all figures for every
band. This is indeed the case with all the
figures, taking note that some bands are
degenerate with one another meaning that
two bands may overlap leading to a
difference in the number of bands
between two configurations of the same
material (notice in Figure 7(a) that band
number 1 is actually bands number 1 and
2 from 7(b) that overlap from Γ to K).
After this check, the diagrams were
searched, with primary focus on the TM
bandgaps to find a value on the y-axis that
was not crossed by any band line. This
would indicate a frequency that is rejected
by the structure and thus reflected back to
the source. By focusing on TM bandgaps,
it is theorized by the author and faculty
supervisor that this case will correspond
to bandgaps found in the 3D case in
future work.
From Figure 3(a), silicon exhibits a 2D
TM bandgap between normalized
frequencies 0.22 to 0.34, and Figure 7(a)
confirms that there is a bandgap between
310 and 320 THz; however, it appears
that this band gap is incomplete from the
M -> Γ direction due to band 3 dipping
below band 2 into the desired gap region.
Conversely, in Figure 8(b), silicon exhibits
a 2D TM bandgap between 305 and 320
THz in the M -> Γ direction, but band 2
rises into the desired gap region as seen in
Figure 8(a).
In Figure 4(a), vanadium exhibits a 2D
TM bandgap between normalized
frequencies 0.3 to 0.4, and Figure 9(a)
confirms that there is a bandgap between
438 and 448 THz; however, it appears
that this band gap is incomplete from the
M -> Γ direction due to the same
behavior found in the silicon case where
band 3 dips below band 2 into the desired
gap region. Conversely, in Figure 10(b),
vanadium exhibits a 2D TM bandgap
between 438 and 445 THz in the M -> Γ
direction, but band 2 rises into the desired
gap region as seen in Figure 10(a). It is
interesting to note here that almost the
exact same band of frequencies is
reflected for some portion of the
dispersion relation, but the configuration
has vastly changed from the square case at
one extreme to the triangular case at the
other extreme.
In the graphite case, the same
behavior of band 3 dipping beneath band
2 is found in the square lattice analysis
from COMSOL as seen in Figures
11(a),(b). The bandgap frequencies in the
Γ -> K direction are 514 to 524 THz
according to COMSOL, and the
normalized frequencies are from 0.34 to
0.41 from MATLAB, as shown in Figure
5(a). In the triangular case, the
frequencies that constitute the gap in the
M -> Γ direction are 503 to 523 THz seen
in Figure 12(b).
Unlike the other three materials,
polystyrene had a significantly different
shaped band diagram for both lattice
configurations. In Figure 6(a) and (b), no
2D TM bandgap is found from the
MATLAB analysis, yet in both Figures
13(a) and (b), two distinct gaps are found
within two different Brillouin Zone
directions. From Figure 13(a), a bandgap
exists for the Γ -> K direction between
840 and 860 THZ, and for the M -> Γ
direction a bandgap exists between 950
and 975 THz. In Figure 14(a), a bandgap
exists between 780 and 860 THz for the Γ
-> X direction. In Figure 14(b),
COMSOL detected no eigenfrequencies
lower than 860 THz and was unable to
deliver the continued bands 1 and 2 in the
M -> Γ direction, yet there is a very large

18
probability that a full TM bandgap exists
within this region if it is expected that
polystyrene’s triangular band diagram
looks similar to Figures 8(b), 10(b), and
12(b).
The triangular band diagrams shown
in Figures 3-6(b) show the proper
placement for the fundamental TM band
gap for the triangular case. Figures 8, 10,
12, and 14(b) show a vertically flipped
behavior for the fundamental TM band.
Where the MATLAB code rises sharply,
decreases to a less steep rise, then falls
while traversing the Brillouin zone, the
opposite behavior is seen in the
COMSOL results.
Conclusions
The formulation of the MATLAB
analysis and COMSOL analysis are based
upon different sciences, which explains
the difference between the two sets of
results. In the MATLAB code, the
electric field are not solved for within the
structure. Rather, the wave’s k-vector is
swept across the Brillouin zone, and a
mathematical analysis is performed by
reducing the governing equation into a
two by two permittivity matrix that solves
for H and tallies the results. In the
COMSOL code, the electric fields are
directly solved, and the modal shapes of
these fields produce eigenfrequencies
which are then used in a differential
equation analysis to determine the band
diagram. This is most likely the cause for
the contradicting results obtained between
the analyses.
The formulation of the MATLAB
code produces results that exactly match
what is expected from theory, but the
COMSOL code gave unexpected results,
and is thought to be the main culprit for
the deviation between the two results.
The trending behavior found across
the different materials during the
COMSOL analysis was unexpected, but
can be easily explained by the fact that the
only parameter that changed across trials
was the refractive index that was simply
being scaled up or down.
The two most interesting results from
the analyses are the fact that when
converting from the square to triangular
lattice structure, almost the same band of
frequencies are spanned within the gap
and that the 2D triangular polystyrene
case, with the material not expected to
exhibit a TM bandgap from the MATLAB
study actually holds the most promise to
exhibit one of all the different material
and lattice configurations.
To improve upon this work, the
author plans to improve the COMSOL
code so that a more valid parameter sweep
can be conducted across all the edges of
the Brillouin zone and eventually to
extend the COMSOL analysis to the 3D
case so that conjectures between the 2D
and 3D case can be tested. The
MATLAB portion will also be
reconfigured so that the triangular case
can show more than just the first
fundamental band, and this code will also
be extended to examine the differences
between the two program’s codes for the
3D case.
Acknowledgements
The author would like to thank his
Honors Committee for providing a wealth
of knowledge on topics related not only to
his education but also to his life and
happiness. He would also like to thank all
of the faculty within the College of
Engineering at Arkansas State University
and specifically Dr. Brandon Kemp and
Dr. Ilwoo Seok for advising him during

19
his research and sharing their vast
knowledge on fascinating topics. The
financial support received from Dr.
Kemp, Dr. Seok, and from the
institutional ASTATE Scholar scholarship
from Arkansas State University is greatly
appreciated and made this work possible.
References
[1] John D. Joannopoulos, et. al.
“Photonic Crystals: Molding the Flow
of Light”, Princeton University Press
(2008).
[2] "Gloss2 - Rhombahedral."
Rhombohedral, Trigonal. Princeton
University, n.d. Web. 30 Apr. 2015.
[3] "Brillouin Zone." Wikipedia.
Wikimedia Foundation, n.d. Web. 30
Apr. 2015.
[4] S. Anantha Ramakrishna and Tomasz
M. Grzegorczyk, “Physics and
Applications of Negative Refractive
Index Materials,” CRC Press (2009).
[5] Yue Liu, et. al. J. Mater. Chem., 2011,
21, 19233.
[6] HongShik Shim, et. al. Appl. Phys. Lett.,
2014, 104, 051104.
[7] Hyungyu Jin, et. al. Nature Materials,
2015, DOI: 10.1038/NMAT4247.
[8] “Study of EM waves in Periodic
Structures,” MIT Course Notes. Sept.
28 2010.
[9] Vuye et al. "Optical Constants of Si
(Silicon) N,k 0.26-0.83 µm; 20 °C."
Refractive Index of Si (Silicon). N.p.,
1993. Web. 24 Apr. 2015.
[10] Johnson and Christy. "Optical
Constants of V (Vanadium) N,k
0.188-1.937 µm." Refractive Index of
V (Vanadium). 1974. Web. 24 Apr.
2015.
[11] Djurišić and Li. "Optical Constants of
C (Carbon) - Graphite; N,k (o);
0.0310-10.332 µm." Refractive Index
of C (Carbon). 1999. Web. 24 Apr.
2015.
[12] Sultanova et al. "Optical Constants of
(C8H8)n (Polystyrene, PS) N 0.4368-
1.052 µm." Refractive Index of
(C8H8)n (Polystyrene, PS). 2009.
Web. 24 Apr. 2015.

APPENDIX A
MATLAB Code Formulation

A-2
%PBG Calculate dispersion relations for 2-D Photonic Bandgap
clear; close all;
hold off;
%%%%%%%%%%%%%%%%Constants%%%%%%%%%%%%%%%%%%%%%
a = 1; % Lattice constant
N = 4; % +/- N for beta
J = 30; % Number of k along each edge
Rc = 0.496*a; % Size of inclusions
eps_a = 1.0; % Background Dielectric Constant
eps_b = 10.556; % Particle Dielectric Constant
ka = 1/eps_a;
kb = 1/eps_b;
%fr = pi*(Rc/a)^2; % Square Lattice
fr = (2*pi/sqrt(3))*(Rc/a)^2; % Triangular Lattice
%b(1:2,1) = (2*pi/a)*[1;1]; % Square Lattice
b(1,1) = norm((2*pi/a)*[1;-1/sqrt(3)]); % Triangular Lattice
b(2,1) = norm((4*pi/(sqrt(3)*a))*[0;1]); % Triangular Lattice
z = [0;0;1];
%%%%%%%%%%%%%%%%Calculate and Sort G%%%%%%%%%%%%%%%%
j = 1;
for n = -N:1:N
for m = -N:1:N
G(1,j) = n*b(1,1);
G(2,j) = m*b(2,1);
Gn(j) = norm(G(:,j));
j = j + 1;
end;

A-3
end;
clear j;
[~,index] = sort(Gn);
G = G(:,index);
M = (N*2+1)^2; %Size of Matrix
% %%%%%%%%%%%%%%%%Calculate k vectors%%%%%%%%%%%%%%
%[SQUARE]
% k = zeros(2,J*3); %Initialize k Matrix
% kx = pi/(J*a);
% ky = pi/(J*a);
% for j = 1:1:J
% k(:,j) = [kx*j;0];
% k(:,J+j) = [kx*J;ky*j];
% k(:,2*J+j) = [kx*(J+1-j);ky*(J+1-j)];
% end;
%
% JJ = J*3; %New Index Definition
%%%%%%%%%%%%%%%%Calculate k vectors%%%%%%%%%%%%%%
%[TRIANGULAR]
k = zeros(2,J*3); %Initialize k Matrix
kx = 2*pi/(3*J*a);
ky = 2*pi/(sqrt(3)*J*a);
for j = 1:1:J
k(:,j) = [0;ky*j];
k(:,J+j) = [kx*j;ky*J];
k(:,2*J+j) = [kx*(J+1-j);ky*(J+1-j)];

A-4
end;
JJ = J*3; %New Index Definition
%%%%%%%%%%%%%%%%%Solution%%%%%%%%%%%%%%%
for j=1:1:JJ % for each k
tic
% Build Matrix
clear e1; clear e2; clear e3;
clear e1p; clear e2p; clear e3p;
clear Mat2x2;
for m=1:1:M
% Unit Vectors
e3 = (k(:,j)+G(:,m)); e3 = e3/norm(e3); e3 = [e3;0];
e1 = cross(e3,z); e1 = e1/norm(e1);
e2 = cross(e3,e1); e2 = e2/norm(e2);
for mp=1:1:M
%%%%%%PC Light Line Points%%%%%
Gp = norm(G(:,mp)-G(:,m));
if (Gp==0)
kappa = ka*fr+kb*(1-fr);
kappaL = 1;
else
kappa = (ka-kb)*2*fr*besselj(1,Gp*Rc)/(Gp*Rc);

A-5
kappaL = 0;
end;
% Prime Unit Vectors
e3p = (k(:,j)+G(:,mp)); e3p = e3p/norm(e3p); e3p = [e3p;0];
e1p = cross(e3p,z); e1p = e1p/norm(e1p);
e2p = cross(e3p,e1p); e2p = e2p/norm(e2p);
Mat2x2 = [dot(e2,e2p),-dot(e2,e1p);-dot(e1,e2p),dot(e1,e1p)];
Matrix(2*m-1:2*m,2*mp-1:2*mp) =
kappa*norm(k(:,j)+G(:,m))*norm(k(:,j)+G(:,mp))*Mat2x2;
MTM(m,mp) = kappa*norm(k(:,j)+G(:,m))*norm(k(:,j)+G(:,mp))*Mat2x2(1,1);
MTE(m,mp) = kappa*norm(k(:,j)+G(:,m))*norm(k(:,j)+G(:,mp))*Mat2x2(2,2);
% Light line
MatrixL(2*m-1:2*m,2*mp-1:2*mp) =
kappaL*norm(k(:,j)+G(:,m))*norm(k(:,j)+G(:,mp))*Mat2x2;
end; %mp
end; %m
evalues = eig(Matrix);
omega = sqrt(evalues)*a/(2*pi);
figure(1)
hold on
plot(repmat(j,size(omega)),real(omega),'*')

A-6
clear evalues; clear omega;
evalues = eig(MTE);
figure(2)
hold on
clear evalues; clear omega;
evalues = eig(MTM);
figure(3)
hold on
% Light line
evalues = eig(MatrixL);
if (j <= JJ/3)
figure(1)
hold on
plot(repmat(2*JJ/3+j,size(omega)),real(omega),'*','markeredgecolor','r')
figure(2)
hold on

A-7
figure(3)
hold on
end; % if j
j
JJ
toc
end; %j Loop for k
figure(1)
xlabel('Gamma -> K -> M -> Gamma','fontsize',14)
ylabel('Frequency [omega a/(2 pi c)]','fontsize',14)
title('Triangular Lattice: R_c/a = 0.496 (epsilon_a = 1, epsilon_b = 10.556)','fontsize',14)
ll = legend('TE & TM');
set(ll,'fontsize',14)
axis([0 JJ 0 .5])
hold off
figure(2)
title('Triangular Lattice: R_c/a = 0.496 (epsilon_a = 1, epsilon_b = 10.556)','fontsize',14)
ll = legend('TE');
axis([0 JJ 0 .5])
hold off

A-8
figure(3)
title('Triangular: R_c/a = 0.496 (epsilon_a = 1, epsilon_b = 10.556)','fontsize',14)
ll = legend('TM');
axis([0 JJ 0 .5])
hold off

APPENDIX B
COMSOL Simulation Environment

Honors Thesis

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