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WG2D_SolveExEy_f.m
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WG2D_SolveExEy_f.m
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function[Ex,Ey,neff]=WG2D_SolveExEy_f(x,y,eps,lambda,nmodes,neff_guess,neff_min,neff_max)
Nx=length(x);
Ny=length(y);
Nxy=Nx*Ny;
dx=x(2)-x(1);
dy=y(2)-y(1);
k0=2*pi/lambda;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%% Building of the operators %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
AA=ones(1,Nx*Ny);
BB=ones(1,Nx*Ny-1);
BB(Ny:Ny:end)=0;
DY1=diag(BB,-1)*(-0.5) + diag(BB,1)*(0.5);
DY2=(-2)*diag(AA) + (1)*diag(BB,-1) + (1)*diag(BB,1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Axy=ones(1,(Nx-1)*Ny);
DX1 = (-0.5)*diag(Axy,-Ny) + (+0.5)*diag(Axy,Ny);
DX2 = (-2)*diag(ones(1,Ny*Nx)) + (1)*diag(Axy,-Ny) + (1)*diag(Axy,Ny);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%% Building of the Hamiltonien %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
H0 = DX2/dx^2 + DY2/dy^2 + diag(eps(:)) * k0^2;
H1 = H0 + DX1/dx * diag(1./eps(:)) * diag( DX1/dx * eps(:) ) ;
H4 = H0 + DY1/dy * diag(1./eps(:)) * diag( DY1/dy * eps(:) ) ;
H2 = DX1/dx * diag(1./eps(:)) * diag( DY1/dy * eps(:) ) ;
H3 = DY1/dy * diag(1./eps(:)) * diag( DX1/dx * eps(:) ) ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Should be the same but not working...
%H1 = H0 + DX1/dx * diag( DX1/dx * log(eps(:)) ) ;
%
%H4 = H0 + DY1/dy * diag( DY1/dy * log(eps(:)) ) ;
%
%H2 = DX1/dx * diag( DY1/dy * log(eps(:)) ) ;
%
%H3 = DY1/dy * diag( DX1/dx * log(eps(:)) ) ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Should be the same but not working...
%H1 = DY2/dy^2 + diag(eps(:)) * k0^2 + DX1/dx * diag(1./eps(:)) * DX1/dx * diag(eps(:)) ;
%
%H4 = DX2/dx^2 + diag(eps(:)) * k0^2 + DY1/dy * diag(1./eps(:)) * DY1/dy * diag(eps(:)) ;
%
%H2 = - DX1/dx * DY1/dy + DX1/dx * diag(1./eps(:)) * DY1/dy * diag(eps(:)) ;
%
%H3 = - DY1/dy * DX1/dx + DY1/dy * diag(1./eps(:)) * DX1/dx * diag(eps(:)) ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
H = [ H1 H2 ; H3 H4 ];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%% Diagonalisation of the Hamiltonien %%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Guess = neff_guess^2*k0^2;
H=sparse(H);
[psi,Beta] = eigs(H,nmodes,'LR'); %% eigen values are ordered
%[psi,Beta] = eigs(H,nmodes,Guess); %% 2 times faster but eigen values not ordered
neff=sqrt(diag(Beta))/k0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% Filtering and reshaping the Wavefunction %%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
idx1=real(neff)>neff_min;
idx2=real(neff)<neff_max;
idx=logical( idx1.*idx2);
neff=neff(idx);
psi=psi(:,idx);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% here is a small patch due to differences between Octave and Matlab
% Matlab order the eigen values while Octave reverse it
if neff(2)>neff(1)
psi=psi(:,end:-1:1);
neff=neff(end:-1:1);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
nmodes=length(neff);
Ex=psi(1:Nxy,:);
Ey=psi(Nxy+1:end,:);
Ex = reshape(Ex,[Ny,Nx,nmodes]);
Ey = reshape(Ey,[Ny,Nx,nmodes]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% First derivative X %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DX1(Ny=5,Nx=4); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% 0 0 0 0 0 |1/2 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 1/2 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 1/2 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 1/2 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 1/2| 0 0 0 0 0 | 0 0 0 0 0 %
% ----------------------------------------------------------------------------- %
% -1/2 0 0 0 0 | 0 0 0 0 0 |1/2 0 0 0 0 | 0 0 0 0 0 %
% 0 -1/2 0 0 0 | 0 0 0 0 0 | 0 1/2 0 0 0 | 0 0 0 0 0 %
% 0 0 -1/2 0 0 | 0 0 0 0 0 | 0 0 1/2 0 0 | 0 0 0 0 0 %
% 0 0 0 -1/2 0 | 0 0 0 0 0 | 0 0 0 1/2 0 | 0 0 0 0 0 %
% 0 0 0 0 -1/2| 0 0 0 0 0 | 0 0 0 0 1/2| 0 0 0 0 0 %
% ----------------------------------------------------------------------------- %
% 0 0 0 0 0 -1/2 0 0 0 0 | 0 0 0 0 0 |1/2 0 0 0 0 %
% 0 0 0 0 0 | 0 -1/2 0 0 0 | 0 0 0 0 0 | 0 1/2 0 0 0 %
% 0 0 0 0 0 | 0 0 -1/2 0 0 | 0 0 0 0 0 | 0 0 1/2 0 0 %
% 0 0 0 0 0 | 0 0 0 -1/2 0 | 0 0 0 0 0 | 0 0 0 1/2 0 %
% 0 0 0 0 0 | 0 0 0 0 -1/2| 0 0 0 0 0 | 0 0 0 0 1/2 %
% ----------------------------------------------------------------------------- %
% 0 0 0 0 0 | 0 0 0 0 0 -1/2 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 -1/2 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 -1/2 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 -1/2 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 -1/2| 0 0 0 0 0 %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Second derivative X %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DX2(Ny=5,Nx=4); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% -2 0 0 0 0 | 1 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 -2 0 0 0 | 0 1 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 -2 0 0 | 0 0 1 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 -2 0 | 0 0 0 1 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 -2 | 0 0 0 0 1 | 0 0 0 0 0 | 0 0 0 0 0 %
% ----------------------------------------------------------------------------- %
% 1 0 0 0 0 |-2 0 0 0 0 | 1 0 0 0 0 | 0 0 0 0 0 %
% 0 1 0 0 0 | 0 -2 0 0 0 | 0 1 0 0 0 | 0 0 0 0 0 %
% 0 0 1 0 0 | 0 0 -2 0 0 | 0 0 1 0 0 | 0 0 0 0 0 %
% 0 0 0 1 0 | 0 0 0 -2 0 | 0 0 0 1 0 | 0 0 0 0 0 %
% 0 0 0 0 1 | 0 0 0 0 -2 | 0 0 0 0 1 | 0 0 0 0 0 %
% ----------------------------------------------------------------------------- %
% 0 0 0 0 0 | 1 0 0 0 0 |-2 0 0 0 0 | 1 0 0 0 0 %
% 0 0 0 0 0 | 0 1 0 0 0 | 0 -2 0 0 0 | 0 1 0 0 0 %
% 0 0 0 0 0 | 0 0 1 0 0 | 0 0 -2 0 0 | 0 0 1 0 0 %
% 0 0 0 0 0 | 0 0 0 1 0 | 0 0 0 -2 0 | 0 0 0 1 0 %
% 0 0 0 0 0 | 0 0 0 0 1 | 0 0 0 0 -2 | 0 0 0 0 1 %
% ----------------------------------------------------------------------------- %
% 0 0 0 0 0 | 0 0 0 0 0 | 1 0 0 0 0 |-2 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 1 0 0 0 | 0 -2 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 1 0 0 | 0 0 -2 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 1 0 | 0 0 0 -2 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 1 | 0 0 0 0 -2 %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% First derivative Y %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DY1(Ny=5,Nx=4); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% 0 1/2 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% -1/2 0 1/2 0 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 -1/2 0 1/2 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 -1/2 0 1/2| 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 -1/2 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% ----------------------------------------------------------------------------- %
% 0 0 0 0 0 | 0 1/2 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 -1/2 0 1/2 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 -1/2 0 1/2 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 -1/2 0 1/2| 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 -1/2 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% ----------------------------------------------------------------------------- %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 1/2 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 -1/2 0 1/2 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 -1/2 0 1/2 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 -1/2 0 1/2| 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 -1/2 0 | 0 0 0 0 0 %
% ----------------------------------------------------------------------------- %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 1/2 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 -1/2 0 1/2 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 -1/2 0 1/2 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 -1/2 0 1/2 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 -1/2 0 %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%% Second derivative Y %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DY2(Ny=5,Nx=4); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% -2 1 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 1 -2 1 0 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 1 -2 1 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 1 -2 1 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 1 -2 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% ----------------------------------------------------------------------------- %
% 0 0 0 0 0 |-2 1 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 1 -2 1 0 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 1 -2 1 0 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 1 -2 1 | 0 0 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 1 -2 | 0 0 0 0 0 | 0 0 0 0 0 %
% ----------------------------------------------------------------------------- %
% 0 0 0 0 0 | 0 0 0 0 0 |-2 1 0 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 1 -2 1 0 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 1 -2 1 0 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 1 -2 1 | 0 0 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 1 -2 | 0 0 0 0 0 %
% ----------------------------------------------------------------------------- %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 |-2 1 0 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 | 1 -2 1 0 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 1 -2 1 0 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 1 -2 1 %
% 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 1 -2 %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%