***********************************************************************
     3-D FDTD code with PEC boundaries
***********************************************************************


     Date of this version:  February 2000

     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.

     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 an additive current source oriented
     along the z-direction.  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" at the MATLAB prompt.
     This M-file displays the FDTD-computed Ez fields at every other
     time step, and records those frames in a movie matrix, M, which
     is played at the end of the simulation using the "movie" command.

***********************************************************************

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 in x-direction
je=24;       %number of grid cells in y-direction
ke=10;       %number of grid cells in z-direction

ib=ie+1;
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;
srcconst=-dt*3.0e+11;

***********************************************************************
    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);

**********************************************************************
     Movie initialization
***********************************************************************

tview(:,:)=ez(:,:,kobs);
sview(:,:)=ez(:,js,:);

subplot('position',[0.15 0.45 0.7 0.45]),pcolor(tview');
shading flat;
caxis([-1.0 1.0]);
colorbar;
axis image;
title(['Ez(i,j,k=5), time step = 0']);
xlabel('i coordinate');
ylabel('j coordinate');

subplot('position',[0.15 0.10 0.7 0.25]),pcolor(sview');
shading flat;
caxis([-1.0 1.0]);
colorbar;
axis image;
title(['Ez(i,j=13,k), time step = 0']);
xlabel('i coordinate');
ylabel('k coordinate');

rect=get(gcf,'Position');
rect(1:2)=[0 0];

M=moviein(nmax/2,gcf,rect);

***********************************************************************
     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)+...
               srcconst*(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
***********************************************************************

if mod(n,2)==0;

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

subplot('position',[0.15 0.45 0.7 0.45]),pcolor(tview');
shading flat;
caxis([-1.0 1.0]);
colorbar;
axis image;
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]),pcolor(sview');
shading flat;
caxis([-1.0 1.0]);
colorbar;
axis image;
title(['Ez(i,j=13,k), time step = ',timestep]);
xlabel('i coordinate');
ylabel('k coordinate');

nn=n/2;
M(:,nn)=getframe(gcf,rect);

end;

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

end

movie(gcf,M,0,10,rect);