matlab仿真二维光子晶体最简程序

本程序为初学者使用，只考虑MT方向

下面的程序为matlab代码

只考虑MT方向

%This is a simple demo for Photonic Crystals simulation 

%This is a simple demo for Photonic Crystals simulation
%This demo is for TE wave only, so only h wave is considered.
%for TM direction only,10 points is considered.
%---------------------------------------M
%| / |
%| / |
%| / |
%| --------------------|X
%| T |
%| |
%| |
%---------------------------------------
%equation :sum_{G',k}(K+G)(K+G')f(G-G')hz(k+G')=(omega/c)^2*hz(k+G)
%G' can considerd as the index of column, and G as index of rows
%[(K+G1)(K+G1)f(G1-G1) (K+G1)(K+G2)f(G1-G2) ][hz(G1)]=(omega/c)^2[hz(G1)]
%[(K+G2)(K+G1)f(G2-G1) (K+G2)(K+G2)f(G2-G2) ][hz(G2)] [hz(G2)]
%or: THETA_TE*Hz=(omega/c)^2*Hz
%by Gao Haikuo
%date:20170411

clear; clc; epssys=1.0e-6; %设定一个最小量，避免系统截断误差或除0错误

%this is the lattice vector and the reciprocal lattice vector
a=1; a1=a*[1 0]; a2=a*[0 1];
b1=2*pi/a*[1 0];b2=2*pi/a*[0 1];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%定义晶格的参数
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
epsa = 1; %介质柱的介电常数
epsb = 13; %背景的介电常数
Pf = 0.7; %Pf = Ac/Au 填充率，可根据需要自行设定
Au =a^2; %二维格子原胞面积
Rc = (Pf *Au/pi)^(1/2); %介质柱截面半径
Ac = pi*(Rc)^2; %介质柱横截面积

%construct the G list
NrSquare = 10;
NG =(2*NrSquare+1)^2; % NG is the number of the G value
G = zeros(NG,2);
i = 1;
for l = -NrSquare:NrSquare
for m = -NrSquare:NrSquare
G(i,:)=l*b1+m*b2;
i = i+1;
end
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%生成k空间中的f(Gi-Gj)的值，i,j 从1到NG。
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
f=zeros(NG,NG);
for i=1:NG
for j=1:NG
Gij=norm(G(i,:)-G(j,:));
if (Gij < epssys)
f(i,j)=(1/epsa)*Pf+(1/epsb)*(1-Pf);
else
f(i,j)=(1/epsa-1/epsb)*Pf*2*besselj(1,Gij*Rc)/(Gij*Rc);
end;
end;
end;
T=(2*pi/a)*[epssys 0];
M=(2*pi/a)*[1/2 1/2]; %????????
X=(2*pi/a)*[1/2 0];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%对于简约布里渊区边界上的每个k，求解其特征频率
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
THETA_TE=zeros(NG,NG); %待解的TE波矩阵
Nkpoints=10; %每个方向上取的点数，
stepsize=0:1/(Nkpoints-1):1; %每个方向上的步长

TX_TE_eig = zeros(Nkpoints,NG); %沿MT方向的TE波的待解的特征频率矩阵
XM_TE_eig = zeros(Nkpoints,NG); %沿MT方向的TE波的待解的特征频率矩阵
MT_TE_eig = zeros(Nkpoints,NG); %沿MT方向的TE波的待解的特征频率矩阵

for n=1:Nkpoints %scan the 10 points along the TM direction
fprintf(['\n k-point:',int2str(n),'of',int2str(Nkpoints),'.\n']);
MT_step = stepsize(n)*(T-M)+M; % get the k
%先求非对角线上的元素
for i=1:(NG-1) % G
for j=(i+1):NG % G'
kGi = MT_step+G(i,:); %k+G
kGj = MT_step+G(j,:); %K+G'
THETA_TE(i,j)=f(i,j)*dot(kGi,kGj); %(K+G)(K+G')f(G-G')
THETA_TE(j,i)=conj(THETA_TE(i,j));
end
end
%再求对角线上的元素
for i=1:NG
kGi = MT_step+G(i,:);
THETA_TE(i,i)=f(i,i)*norm(kGi)*norm(kGi);
end
MT_TE_eig(n,:)=sort(sqrt(eig(THETA_TE))).';
end

%draw
kaxis = 0;
TXaxis = kaxis:norm(T-X)/(Nkpoints-1):(kaxis+norm(T-X));
kaxis = kaxis + norm(T-X);
XMaxis = kaxis:norm(X-M)/(Nkpoints-1):(kaxis+norm(X-M));
kaxis = kaxis + norm(X-M);
MTaxis = kaxis:norm(M-T)/(Nkpoints-1):(kaxis+norm(M-T));
kaxis = kaxis + norm(M-T);

Ntraject = 3;
figure (1)
hold on;
Nk=Nkpoints;
for k=1:NG
for i=1:Nkpoints
EigFreq_TE(i+0*Nk) = TX_TE_eig(i,k)/(2*pi/a);
EigFreq_TE(i+1*Nk) = XM_TE_eig(i,k)/(2*pi/a);
EigFreq_TE(i+2*Nk) = MT_TE_eig(i,k)/(2*pi/a);
end
plot(TXaxis(1:Nk),EigFreq_TE(1+0*Nk:1*Nk),'r',...
XMaxis(1:Nk),EigFreq_TE(1+1*Nk:2*Nk),'r',...
MTaxis(1:Nk),EigFreq_TE(1+2*Nk:3*Nk),'r');
end

grid on;
xlabel('K-Space');
yLabel('Frequency(\omegaa/2\piC)');
axis([0 MTaxis(Nkpoints) 0 1]);
set(gca,'XTick',[TXaxis(1), TXaxis(Nkpoints),...
XMaxis(Nkpoints),MTaxis(Nkpoints)]);
xtixlabel = strvcat('T','X','M','T');
set(gca,'XTickLabel',xtixlabel);