%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 = TX_step+G(i,:); %k+G
            kGj = TX_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 = TX_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