PhC_2D_hex_Main.m

clear all
close all
clc

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Constants %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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c=2.99792458e8;

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%%%%%%%%%%%%%%%%%%%%%%%%%%  Plotting parameter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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band=1;
Field=1;
Epsilon=0;

AAbs=1;               %% Plot abs(E)
RReal=0;              %% Plot real(E)
IImag=0;              %% Plot imag(E)

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Parameters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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hex=1;
comb=0;

TM=0;
TE=1;

Nx=32;         % number of points on the x grid % has to be a power of 2 (32,64,128,256,512,...)
Ny=32;         % number of points on the y grid % has to be a power of 2 (32,64,128,256,512,...)
NGx=10;        % number of harmonics % has to be 2 times -1 smaller than x
NGy=11;        % number of harmonics % has to be 2 times -1 smaller than y

Nkx=10;        % number of points on the k space for the dispersion
Nky=Nkx;       % number of points on the k space for the dispersion

nmodes=5;      % number of solutions asked
Np=3;          % number of period to plot for the Field

n1 =1;         %% optical index material 1
n2 = sqrt(12); %% optical index material 2

NormUnits=1;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%% Building of the index Geometry %%%%%%%%%%%%%%%%%%%%%%%%%
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if NormUnits==1
  L=1;
elseif NormUnits==0
  L=1e-6;  
end

Lx=L;
Ly=L*sqrt(3)/2;

a1=Lx*[1 0];
a2=Lx*[  1/2  sqrt(3)/2 ];

count=1;
for jj=0:Nx-1,
  	for j=0:Ny-1,
      AAA(count,:) = jj*a1/(Nx-1) + j*a2/(Ny-1) ;
      count=count+1;
   end
end

Xhex=reshape(AAA(:,1),Ny,Nx);
Yhex=reshape(AAA(:,2),Ny,Nx);

dx=Xhex(1,2)-Xhex(1,1);
dy=Yhex(2,1)-Yhex(1,1);


if (hex==1) && (comb==1) || (hex==0) && (comb==0)
    display('Error: Select hexagonal lattice or honey-comb lattice')
    break
end

if hex==1
    a=0.3;%0.495;
    idx1 =  ( (Xhex-Lx*3/4).^2 + (Yhex-Ly/2).^2 ) < (a*L)^2;
    idx2 =  ( (Xhex-Lx*3/4+Lx/2).^2 + (Yhex-Ly/2+Ly).^2 ) < (a*L)^2;
    idx3 =  ( (Xhex-Lx*3/4-Lx/2).^2 + (Yhex-Ly/2-Ly).^2 ) < (a*L)^2;
    idx4 =  ( (Xhex-Lx*3/4-Lx).^2 + (Yhex-Ly/2).^2 ) < (a*L)^2;
    idx5 =  ( (Xhex-Lx*3/4+Lx).^2 + (Yhex-Ly/2).^2 ) < (a*L)^2;
    
    idx=idx1+idx2+idx3+idx4+idx5;

    eps = idx*n2^2 + (1-idx)*n1^2 ;
end

if comb==1
    a=0.24;
    idx1a =  ( (Xhex-Lx*3/4).^2 + (Yhex-Ly/5).^2 ) < (a*L)^2;
    idx1b =  ( (Xhex-Lx*3/4+Lx/2).^2 + (Yhex-Ly/5-Lx/sqrt(3)+Ly).^2 ) < (a*L)^2;
    idx2a =  ( (Xhex-Lx*3/4).^2 + (Yhex-Ly/5-Lx/sqrt(3)).^2 ) < (a*L)^2;
    idx2b =  ( (Xhex-Lx*3/4-Lx/2).^2 + (Yhex-Ly/5-Ly).^2 ) < (a*L)^2;
    
    idx=idx1a+idx1b+idx2a+idx2b;

    eps = idx*n2^2 + (1-idx)*n1^2 ;
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%% Reciprocal lattice vectors %%%%%%%%%%%%%%%%%%%%%%%%%%%
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NGx = 2*floor(NGx/2);           %% round to lower even number
NGy = 2*floor(NGy/2);           %% round to lower even number

b1=2*pi/Lx*[1  -sqrt(3)/3];
b2=2*pi/Lx*[0 2*sqrt(3)/3];

count=1;
GGG=[];

for jj=-NGx:NGx
for j=-NGy:NGy
    GGG(count,:)=jj*b1+j*b2;
    count=count+1;
end
end

Gxhex=reshape(GGG(:,1),2*NGy+1,2*NGx+1);
Gyhex=reshape(GGG(:,2),2*NGy+1,2*NGx+1);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%% Hexagonal Fourier Transform %%%%%%%%%%%%%%%%%%%%%%%%%
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Gamma=1./eps;
f=Gamma;

for jj=1:length(Gxhex(1,:))
for j=1:length(Gyhex(:,1))
        whex = exp( -1i*(   Gxhex(1,jj) *(Xhex-Xhex(1))*(Nx-1)/Nx + ( Gyhex(j,jj) )*(Yhex-Yhex(1))*(Ny-1)/Ny    ) );
        Ghex(j,jj) = sum(sum(f.*whex));
end
end

Gammak = Ghex*dx*dy/Lx/Ly ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% Building of the reciproque lattice vector %%%% again %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

count=1;
GGG=[];
for jj=-NGx/2:NGx/2,
    for j=-NGy/2:NGy/2,
        GGG(count,:)=jj*b1+j*b2;
        count=count+1;
    end
end

Gxhex=reshape(GGG(:,1),NGy+1,NGx+1);
Gyhex=reshape(GGG(:,2),NGy+1,NGx+1);

NGx=length(Gxhex(1,:));
NGy=length(Gyhex(:,1));
NG=NGx*NGy;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%% Building of k-space vector %%%%%%%%%%%%%%%%%%%%%%%%%%
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kx=linspace( 0 , pi/L , Nkx)*2/3;
ky=linspace( 0 , pi/Ly , Nky);

k=[
ky'*0                 ky'   
kx'                   ky(end)+kx'*0
sort(kx,'descend')'    sort(ky,'descend')'
];

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%%%%%%%%%%%%%%%%%%%%%%% NOTHING TO CHANGE ANYMORE!!! %%%%%%%%%%%%%%%%%%%%%%%%%%%
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if (TE==1 && TM==1) || (TE==0 && TM==0)
  display('Error: Select "TM" or "TE"')
  break
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Building first part of Hamitonian that is not depending on k %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

HHH=zeros(NGy,NGx,NGy,NGx);

for ix=1:NGx
for jx=1:NGx
    for iy=1:NGy
    for jy=1:NGy
        HHH(iy,ix,jy,jx) = Gammak(iy-jy+NGy,ix-jx+NGx );
    end
    end
end
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

for i=1:length(k(:,1))
  
  [psi,f0]=PhC2D_hex_PWE_f(Xhex,Yhex,Gxhex,Gyhex,k(i,:),HHH,nmodes,TE,TM);
  
  E(:,:,:,i)=psi;
  
  if NormUnits==1
    FF(:,i) = f0 * Lx / (2*pi);
  elseif NormUnits==0
    FF(:,i) = f0 * c / (2*pi) *1e-12;     % Convert in THz
    lambda(:,i)=2*pi./f0*1e6;             % Convert in wavelength (um)
  end
  
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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if AAbs==1
  EE=abs(E);
end
if RReal==1
  EE=real(E);
end
if IImag==1
  EE=imag(E);
end

if NormUnits==0
  Xhex=Xhex*1e6;
  Yhex=Yhex*1e6;
  Lx=Lx*1e6;
  Ly=Ly*1e6;
  k=k*1e-6;
end

if Field==1
    
    if TM==1 && TE==0
      figure('position',[100 50 1000 1000],'name','Ez');
    elseif TE==1 && TM==0
      figure('position',[100 50 1000 1000],'name','Exy');
    end
    colormap(jet)
    
    for ii=0:nmodes-1
        for i=1:Np
            for j=1:Np
                subplot(nmodes,3,1+3*ii)
                hold on
                pcolor( Xhex+(i-1+(j-1)/2)*Lx , Yhex+(j-1)*Ly , EE(:,:,ii+1,1) )
                contour( Xhex+(i-1+(j-1)/2)*Lx , Yhex+(j-1)*Ly ,abs(eps),1,'linewidth',2,'linecolor','w')
            end       
        end
        shading flat
        
        %colorbar
        if RReal==1 || IImag==1
          caxis([-1 1])
        elseif AAbs==1
          caxis([0 1])
        end
        if NormUnits==1
          title(strcat('\Gamma: w=' , num2str(FF(1+ii,1), '%.2f') ))
          xlabel('x (norm. units)')
          ylabel('y (norm. units)')
        elseif NormUnits==0 
          title(strcat('\Gamma: \lambda=' , num2str(lambda(1+ii,1), '%.2f') , 'um' ))
          xlabel('x (um)')
          ylabel('y (um)')
        end
        xlim([0 1.5*Np*Lx])
        ylim([0     Np*Ly])
    end
    for ii=0:nmodes-1
        for i=1:Np
            for j=1:Np
                subplot(nmodes,3,2+3*ii)
                hold on
                pcolor( Xhex+(i-1+(j-1)/2)*Lx , Yhex+(j-1)*Ly , EE(:,:,ii+1,1*Nkx) )
                contour( Xhex+(i-1+(j-1)/2)*Lx , Yhex+(j-1)*Ly ,abs(eps),1,'linewidth',2,'linecolor','w')
            end
        end
        shading flat
        %colorbar
        if RReal==1 || IImag==1
          caxis([-1 1])
        elseif AAbs==1
          caxis([0 1])
        end
        if NormUnits==1
          title(strcat('M: w=' , num2str(FF(1+ii,length(k)/3), '%.2f') ))
          xlabel('x (norm. units)')
          ylabel('y (norm. units)')
        elseif NormUnits==0 
          title(strcat('M: \lambda=' , num2str(lambda(1+ii,length(k)/3), '%.2f') , 'um' ))
          xlabel('x (um)')
          ylabel('y (um)')
        end
        xlim([0 1.5*Np*Lx])
        ylim([0     Np*Ly])
    end
    for ii=0:nmodes-1
        for i=1:Np
            for j=1:Np
                subplot(nmodes,3,3+3*ii)
                hold on
                pcolor( Xhex+(i-1+(j-1)/2)*Lx , Yhex+(j-1)*Ly , EE(:,:,ii+1,2*Nkx) )
                contour( Xhex+(i-1+(j-1)/2)*Lx , Yhex+(j-1)*Ly ,abs(eps),1,'linewidth',2,'linecolor','w')
            end
        end
        shading flat
        %colorbar
        if RReal==1 || IImag==1
          caxis([-1 1])
        elseif AAbs==1
          caxis([0 1])
        end
        if NormUnits==1
          title(strcat('K: w=' , num2str(FF(1+ii,length(k)*2/3), '%.2f') ))
          xlabel('x (norm. units)')
          ylabel('y (norm. units)')
        elseif NormUnits==0 
          title(strcat('K: \lambda=' , num2str(lambda(1+ii,length(k)*2/3), '%.2f') , 'um' ))
          xlabel('x (um)')
          ylabel('y (um)')
        end
        xlim([0 1.5*Np*Lx])
        ylim([0     Np*Ly])
    end
    
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

if Epsilon==1
  
  figure('position',[1100 50 500 400])
  subplot(111)
  hold on
     
  for i=1:Np
    for j=1:Np
        pcolor( Xhex+(i-1+(j-1)/2)*Lx , Yhex+(j-1)*Ly , real(eps) )
    end
  end
  shading flat
  colormap(jet)
  c=colorbar;
  title(c,'Epsilon')
  if NormUnits==1
    xlabel('x (norm. units)')
    ylabel('y (norm. units)')
  elseif NormUnits==0 
    xlabel('x (um)')
    ylabel('y (um)')
  end
  %axis equal

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

if band==1
    
    figure('position',[50 570 900 450])
    
    subplot(1,2,1)
    hold on;%grid on;
    
    plot(0:length(k)-1,real(FF(1:nmodes,:))','o-')
    
    yscale=get(gca,'ylim');
    xlim([0 length(k)-1])
    
    plot( [1/3*length(k)    1/3*length(k)] , yscale , 'k')
    plot( [2/3*length(k)    2/3*length(k)] , yscale , 'k')
    plot( [3/3*length(k)    3/3*length(k)] , yscale , 'k')
    
    text(0/3*length(k) , -0.05*yscale(2) , ' \Gamma')
    text(1/3*length(k) , -0.05*yscale(2) , ' M'     )
    text(2/3*length(k) , -0.05*yscale(2) , ' K'     )
    text(3/3*length(k) , -0.05*yscale(2) , ' \Gamma')
    %xlabel('k')
    set(gca,'xticklabel',[])
        
    if NormUnits==1
      ylabel('w (2\pi/Ltot)')
    elseif NormUnits==0 
      ylabel('f (THz)')
    end
    title(strcat('R/a=',num2str(a),'; n1=',num2str(n1,'%.2f'),'; n2=',num2str(n2,'%.2f')  ))
    
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    
    subplot(1,2,2)
    hold on;grid on;
    LW=2;
    
    for i=1:nmodes
      plot3( k(:,1)*Lx/pi , k(:,2)*Ly/pi , real(FF(i,:))','o-')
    end
    set(gca,'xticklabel',[])
    set(gca,'yticklabel',[])
    xlabel('kx (pi/Lx)')
    ylabel('ky (pi/Ly)')
    if NormUnits==1
      zlabel('w (2\pi/Ltot)')
    elseif NormUnits==0 
      zlabel('f (THz)')
    end
    view(-30,15)
    
    plot3( [-1 1]*2/3 , +[1 1] , [0 0] ,'r', 'linewidth',LW )
    plot3( [-1 1]*2/3 , -[1 1] , [0 0] ,'r', 'linewidth',LW )
    plot3( +[1 2]*2/3 , +[1 0] , [0 0] ,'r', 'linewidth',LW )
    plot3( +[1 2]*2/3 , -[1 0] , [0 0] ,'r', 'linewidth',LW )
    plot3( -[2 1]*2/3 , -[0 1] , [0 0] ,'r', 'linewidth',LW )
    plot3( -[2 1]*2/3 , +[0 1] , [0 0] ,'r', 'linewidth',LW )
    
    plot3( [-1 1]*2/3 , +[1 1] , [1 1]*max(real(FF(nmodes,:))) ,'r', 'linewidth',LW )
    plot3( [-1 1]*2/3 , -[1 1] , [1 1]*max(real(FF(nmodes,:))) ,'r', 'linewidth',LW )
    plot3( +[1 2]*2/3 , +[1 0] , [1 1]*max(real(FF(nmodes,:))) ,'r', 'linewidth',LW )
    plot3( +[1 2]*2/3 , -[1 0] , [1 1]*max(real(FF(nmodes,:))) ,'r', 'linewidth',LW )
    plot3( -[2 1]*2/3 , -[0 1] , [1 1]*max(real(FF(nmodes,:))) ,'r', 'linewidth',LW )
    plot3( -[2 1]*2/3 , +[0 1] , [1 1]*max(real(FF(nmodes,:))) ,'r', 'linewidth',LW )
    
    plot3( +[1 1]*2/3 , +[1 1] , [0 1]*max(real(FF(nmodes,:))) ,'r', 'linewidth',LW )
    plot3( +[1 1]*2/3 , -[1 1] , [0 1]*max(real(FF(nmodes,:))) ,'r', 'linewidth',LW )
    plot3( -[1 1]*2/3 , +[1 1] , [0 1]*max(real(FF(nmodes,:))) ,'r', 'linewidth',LW )
    plot3( -[1 1]*2/3 , -[1 1] , [0 1]*max(real(FF(nmodes,:))) ,'r', 'linewidth',LW )
    plot3( +2*[1 1]*2/3 ,0*[1 1] , [0 1]*max(real(FF(nmodes,:))) ,'r', 'linewidth',LW )
    plot3( -2*[1 1]*2/3 ,0*[1 1] , [0 1]*max(real(FF(nmodes,:))) ,'r', 'linewidth',LW )
end

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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