%***********************************************************************
%     3-D FDTD code with PEC boundaries
%***********************************************************************
%
%     Program author: Susan C. Hagness
%                     Department of Electrical and Computer Engineering
%                     University of Wisconsin-Madison
%                     1415 Engineering Drive
%                     Madison, WI 53706-1691
%                     hagness@engr.wisc.edu
%
%     Copyright 2005
%
%     This MATLAB M-file implements the finite-difference time-domain
%     solution of Maxwell's curl equations over a three-dimensional
%     Cartesian space lattice comprised of uniform cubic grid cells.
%     Each unit cell within the Yee lattice contains electric field
%     components sampled along the edges of the cube and magnetic field
%     components normal to the faces of the cube (the dual of Fig. 3.1).
%     
%     To illustrate the algorithm, an air-filled rectangular cavity 
%     resonator is modeled.  The length, width, and height of the 
%     cavity are 10.0 cm (x-direction), 4.8 cm (y-direction), and 
%     2.0 cm (z-direction), respectively.
%
%     The computational domain is truncated using PEC boundary 
%     conditions:
%          ex(i,j,k)=0 on the j=1, j=jb, k=1, and k=kb planes
%          ey(i,j,k)=0 on the i=1, i=ib, k=1, and k=kb planes
%          ez(i,j,k)=0 on the i=1, i=ib, j=1, and j=jb planes
%     These PEC boundaries form the outer lossless walls of the cavity.
%
%     The cavity is excited by a line of current sources oriented
%     along the z-direction and located in the center of the x-y plane.
%     The source waveform is a differentiated Gaussian pulse given by 
%          J(t)=J0*(t-t0)*exp(-(t-t0)^2/tau^2), 
%     where tau=50 ps.  The FWHM spectral bandwidth of this zero-dc-
%     content pulse is approximately 7 GHz. The grid resolution 
%     (dx = 2 mm) was chosen to provide at least 10 samples per 
%     wavelength up through 15 GHz.
%
%     To execute this M-file, type "fdtd3D_pec" at the MATLAB prompt.
%
%     This code has been tested in the following Matlab environments:
%     Matlab version 6.1.0.450 Release 12.1 (May 18, 2001)
%     Matlab version 6.5.1.199709 Release 13 Service Pack 1 (August 4, 2003)
%     Matlab version 7.0.0.19920 R14 (May 6, 2004)
%     Matlab version 7.0.1.24704 R14 Service Pack 1 (September 13, 2004)
%     Matlab version 7.0.4.365 R14 Service Pack 2 (January 29, 2005)
%
%     Note: if you are using Matlab version 6.x, you may wish to make
%     one or more of the following modifications to this code: 
%       --uncomment line numbers 125 and 126
%       --comment out line numbers 180 and 188
%***********************************************************************

clear

%***********************************************************************
%     Fundamental constants
%***********************************************************************

cc=2.99792458e8;            %speed of light in free space
muz=4.0*pi*1.0e-7;          %permeability of free space
epsz=1.0/(cc*cc*muz);       %permittivity of free space

%***********************************************************************
%     Grid parameters
%***********************************************************************

ie=50;    %number of grid cells and Ex samples along x-direction
je=24;    %number of grid cells and Ey samples along y-direction
ke=10;    %number of grid cells and Ez samples along z-direction

ib=ie+1;    %grid boundary coordinates
jb=je+1;    
kb=ke+1;   

is=26;       %location of z-directed current source
js=13;       %location of z-directed current source

kobs=5;

dx=0.002;          %space increment of cubic lattice
dt=dx/(2.0*cc);    %time step

nmax=500;          %total number of time steps

%***********************************************************************
%     Differentiated Gaussian pulse excitation
%***********************************************************************

rtau=50.0e-12;
tau=rtau/dt;
ndelay=3*tau;
J0=-1.0;

%***********************************************************************
%     Material parameters
%***********************************************************************

eps=1.0;
sig=0.0;        

%***********************************************************************
%     Updating coefficients
%***********************************************************************

ca=(1.0-(dt*sig)/(2.0*epsz*eps))/(1.0+(dt*sig)/(2.0*epsz*eps));
cb=(dt/epsz/eps/dx)/(1.0+(dt*sig)/(2.0*epsz*eps));
da=1.0;
db=dt/muz/dx;

%***********************************************************************
%     Field arrays
%***********************************************************************

ex=zeros(ie,jb,kb);
ey=zeros(ib,je,kb);
ez=zeros(ib,jb,ke);
hx=zeros(ib,je,ke);
hy=zeros(ie,jb,ke);
hz=zeros(ie,je,kb);

%figure
%set(gcf,'DoubleBuffer','on')

%***********************************************************************
%     BEGIN TIME-STEPPING LOOP
%***********************************************************************

for n=1:nmax
   
%***********************************************************************
%     Update electric fields
%***********************************************************************

ex(1:ie,2:je,2:ke)=ca*ex(1:ie,2:je,2:ke)+...
                   cb*(hz(1:ie,2:je,2:ke)-hz(1:ie,1:je-1,2:ke)+...
                       hy(1:ie,2:je,1:ke-1)-hy(1:ie,2:je,2:ke));

ey(2:ie,1:je,2:ke)=ca*ey(2:ie,1:je,2:ke)+...
                   cb*(hx(2:ie,1:je,2:ke)-hx(2:ie,1:je,1:ke-1)+...
                       hz(1:ie-1,1:je,2:ke)-hz(2:ie,1:je,2:ke));
                    
ez(2:ie,2:je,1:ke)=ca*ez(2:ie,2:je,1:ke)+...
                   cb*(hx(2:ie,1:je-1,1:ke)-hx(2:ie,2:je,1:ke)+...
                       hy(2:ie,2:je,1:ke)-hy(1:ie-1,2:je,1:ke));
                    
ez(is,js,1:ke)=ez(is,js,1:ke)+...
               J0*(n-ndelay)*exp(-((n-ndelay)^2/tau^2));

%***********************************************************************
%     Update magnetic fields
%***********************************************************************

hx(2:ie,1:je,1:ke)=hx(2:ie,1:je,1:ke)+...
                   db*(ey(2:ie,1:je,2:kb)-ey(2:ie,1:je,1:ke)+...
                       ez(2:ie,1:je,1:ke)-ez(2:ie,2:jb,1:ke));
                
hy(1:ie,2:je,1:ke)=hy(1:ie,2:je,1:ke)+...
                   db*(ex(1:ie,2:je,1:ke)-ex(1:ie,2:je,2:kb)+...
                       ez(2:ib,2:je,1:ke)-ez(1:ie,2:je,1:ke));
                
hz(1:ie,1:je,2:ke)=hz(1:ie,1:je,2:ke)+...
                   db*(ex(1:ie,2:jb,2:ke)-ex(1:ie,1:je,2:ke)+...
                       ey(1:ie,1:je,2:ke)-ey(2:ib,1:je,2:ke));
                    
%***********************************************************************
%     Visualize fields
%***********************************************************************

timestep=int2str(n);
tview(:,:)=ez(:,:,kobs);
sview(:,:)=ez(:,js,:);

subplot('position',[0.15 0.45 0.7 0.45]),imagesc(tview');
shading flat;
caxis([-1.0 1.0]); 
colorbar;
axis image; axis xy; 
title(['Ez(i,j,k=5), time step = ',timestep]);
xlabel('i coordinate'); ylabel('j coordinate');

subplot('position',[0.15 0.10 0.7 0.25]),imagesc(sview');
shading flat;
caxis([-1.0 1.0]); 
colorbar;
axis image; axis xy;
title(['Ez(i,j=13,k), time step = ',timestep]);
xlabel('i coordinate'); ylabel('k coordinate');

pause(0.05)

%***********************************************************************
%     END TIME-STEPPING LOOP
%***********************************************************************

end