%***********************************************************************
%     1-D FDTD code with simple radiation boundary conditions
%***********************************************************************
%
%     Program author: Susan C. Hagness
%                     Department of Electrical and Computer Engineering
%                     University of Wisconsin-Madison
%                     1415 Engineering Drive
%                     Madison, WI 53706-1691
%                     608-265-5739
%                     hagness@engr.wisc.edu
%
%     Date of this version:  February 2000
%
%     This MATLAB M-file implements the finite-difference time-domain
%     solution of Maxwell's curl equations over a one-dimensional space
%     lattice comprised of uniform grid cells.
%
%     To illustrate the algorithm, a sinusoidal wave (1GHz) propagating 
%     in a nonpermeable lossy medium (epsr=1.0, sigma=5.0e-3 S/m) is 
%     modeled.  The simplified finite difference system for nonpermeable
%     media (discussed in Section 3.6.6 of the text) is implemented.
%
%     The grid resolution (dx = 1.5 cm) is chosen to provide 20
%     samples per wavelength.  The Courant factor S=c*dt/dx is set to
%     the stability limit: S=1.  In 1-D, this is the "magic time step."
%
%     The computational domain is truncated using the simplest radiation
%     boundary condition for wave propagation in free space: 
%
%                      Ez(imax,n+1) = Ez(imax-1,n)
%
%     To execute this M-file, type "fdtd1D" at the MATLAB prompt.
%     This M-file displays the FDTD-computed Ez and Hy fields at every 
%     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

freq=1.0e+9;                %frequency of source excitation
lambda=cc/freq;             %wavelength of source excitation
omega=2.0*pi*freq;

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

ie=200;                     %number of grid cells in x-direction

ib=ie+1;

dx=lambda/20.0;             %space increment of 1-D lattice
dt=dx/cc;                   %time step
omegadt=omega*dt;

nmax=round(12.0e-9/dt);     %total number of time steps

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

eps=1.0;
sig=5.0e-3;

%***********************************************************************
%     Updating coefficients for space region with nonpermeable media
%***********************************************************************

scfact=dt/muz/dx;

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

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

ez(1:ib)=0.0;
hy(1:ie)=0.0;

%***********************************************************************
%     Movie initialization
%***********************************************************************

x=linspace(dx,ie*dx,ie);

subplot(2,1,1),plot(x,ez(1:ie)/scfact,'r'),axis([0 3 -1 1]);
ylabel('EZ');

subplot(2,1,2),plot(x,hy,'b'),axis([0 3 -3.0e-3 3.0e-3]);
xlabel('x (meters)');ylabel('HY');

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
%***********************************************************************

ez(1)=scfact*sin(omegadt*n);

rbc=ez(ie);
ez(2:ie)=ca*ez(2:ie)+cb*(hy(2:ie)-hy(1:ie-1));
ez(ib)=rbc;

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

hy(1:ie)=hy(1:ie)+ez(2:ib)-ez(1:ie);

%***********************************************************************
%     Visualize fields
%***********************************************************************

if mod(n,2)==0;

rtime=num2str(round(n*dt/1.0e-9));

subplot(2,1,1),plot(x,ez(1:ie)/scfact,'r'),axis([0 3 -1 1]);
title(['time = ',rtime,' ns']);
ylabel('EZ');

subplot(2,1,2),plot(x,hy,'b'),axis([0 3 -3.0e-3 3.0e-3]);
title(['time = ',rtime,' ns']);
xlabel('x (meters)');ylabel('HY');

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

end

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

end

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