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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
(b) M to Γ Brillouin Zone Edge
Figure 8. COMSOL Silicon Triangular Bandgap Dispersion Relations
11
(a) Γ to K Brillouin Zone Edge
(b) M to Γ Brillouin Zone Edge
Figure 9. COMSOL Vanadium Square Bandgap Dispersion Relations
12
(a) Γ to X Brillouin Zone Edge
(b) M to Γ Brillouin Zone Edge
Figure 10. COMSOL Vanadium Triangular Bandgap Dispersion Relations
13
(a) Γ to K Brillouin Zone Edge
(b) M to Γ Brillouin Zone Edge
Figure 11. COMSOL Graphite Square Bandgap Dispersion Relations
14
(a) Γ to X Brillouin Zone Edge
(b) M to Γ Brillouin Zone Edge
Figure 12. COMSOL Graphite Triangular Bandgap Dispersion Relations
15
(a) Γ to K Brillouin Zone Edge
(b) M to Γ Brillouin Zone Edge
Figure 13. COMSOL Polystyrene Square Bandgap Dispersion Relations
16
(a) Γ to X Brillouin Zone Edge
(b) M to Γ Brillouin Zone Edge
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);
omega = sqrt(evalues)*a/(2*pi);
figure(2)
hold on
plot(repmat(j,size(omega)),real(omega),'*')
clear evalues; clear omega;
evalues = eig(MTM);
omega = sqrt(evalues)*a/(2*pi);
figure(3)
hold on
plot(repmat(j,size(omega)),real(omega),'*')
% Light line
evalues = eig(MatrixL);
omega = sqrt(evalues)*a/(2*pi);
if (j <= JJ/3)
figure(1)
hold on
plot(repmat(2*JJ/3+j,size(omega)),real(omega),'*','markeredgecolor','r')
figure(2)
hold on
plot(repmat(2*JJ/3+j,size(omega)),real(omega),'*','markeredgecolor','r')
A-7
figure(3)
hold on
plot(repmat(2*JJ/3+j,size(omega)),real(omega),'*','markeredgecolor','r')
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)
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');
set(ll,'fontsize',14)
axis([0 JJ 0 .5])
hold off
A-8
figure(3)
xlabel('\Gamma -> K -> M -> \Gamma','fontsize',14)
ylabel('Frequency [\omega a/(2 \pi c)]','fontsize',14)
title('Triangular: R_c/a = 0.496 (\epsilon_a = 1, \epsilon_b = 10.556)','fontsize',14)
ll = legend('TM');
set(ll,'fontsize',14)
axis([0 JJ 0 .5])
hold off
APPENDIX B COMSOL Simulation Environment
B-2
