%***********************************************************************
%     2-D FDTD TE code with PML absorbing 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 two-dimensional
%     Cartesian space lattice comprised of uniform square grid cells.
%
%     To illustrate the algorithm, a 6-cm-diameter metal cylindrical
%     scatterer in free space is modeled. The source excitation is
%     a Gaussian pulse with a carrier frequency of 5 GHz.
%
%     The grid resolution (dx = 3 mm) was chosen to provide 20 samples
%     per wavelength at the center frequency of the pulse (which in turn
%     provides approximately 10 samples per wavelength at the high end
%     of the excitation spectrum, around 10 GHz).
%
%     The computational domain is truncated using the perfectly matched
%     layer (PML) absorbing boundary conditions.  The formulation used
%     in this code is based on the original split-field Berenger PML. The
%     PML regions are labeled as shown in the following diagram:
%
%            ----------------------------------------------
%           |  |                BACK PML                |  |
%            ----------------------------------------------
%           |L |                                       /| R|
%           |E |                                (ib,jb) | I|
%           |F |                                        | G|
%           |T |                                        | H|
%           |  |                MAIN GRID               | T|
%           |P |                                        |  |
%           |M |                                        | P|
%           |L | (1,1)                                  | M|
%           |  |/                                       | L|
%            ----------------------------------------------
%           |  |                FRONT PML               |  |
%            ----------------------------------------------
%
%     To execute this M-file, type "fdtd2D" at the MATLAB prompt.
%     This M-file displays the FDTD-computed Ex, Ey, and Hz fields at
%     every 4th 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=5.0e+9;                %center frequency of source excitation
lambda=cc/freq;             %center wavelength of source excitation
omega=2.0*pi*freq;

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

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

ib=ie+1;
jb=je+1;

is=15;            %location of z-directed hard source
js=je/2;          %location of z-directed hard source

dx=3.0e-3;        %space increment of square lattice
dt=dx/(2.0*cc);   %time step

nmax=500;         %total number of time steps

iebc=8;           %thickness of left and right PML region
jebc=8;           %thickness of front and back PML region
rmax=0.00001;
orderbc=2;
ibbc=iebc+1;
jbbc=jebc+1;
iefbc=ie+2*iebc;
jefbc=je+2*jebc;
ibfbc=iefbc+1;
jbfbc=jefbc+1;

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

media=3;
%    air CU     Si
eps=[1.0 1.0     1.5];
sig=[0.0 1.0e+7  0.0];
mur=[1.0 1.0     1.0];
sim=[0.0 0.0     0.0];

%***********************************************************************
%     Wave excitation
%***********************************************************************

rtau=160.0e-12;
tau=rtau/dt;
delay=3*tau;

source=zeros(1,nmax);
for n=1:7.0*tau
    source(n)=sin(omega*(n-delay)*dt)*exp(-((n-delay)^2/tau^2));
end

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

ex=zeros(ie,jb);           %fields in main grid
ey=zeros(ib,je);
hz=zeros(ie,je);

exbcf=zeros(iefbc,jebc);   %fields in front PML region
eybcf=zeros(ibfbc,jebc);
hzxbcf=zeros(iefbc,jebc);
hzybcf=zeros(iefbc,jebc);

exbcb=zeros(iefbc,jbbc);   %fields in back PML region
eybcb=zeros(ibfbc,jebc);
hzxbcb=zeros(iefbc,jebc);
hzybcb=zeros(iefbc,jebc);

exbcl=zeros(iebc,jb);      %fields in left PML region
eybcl=zeros(iebc,je);
hzxbcl=zeros(iebc,je);
hzybcl=zeros(iebc,je);

exbcr=zeros(iebc,jb);      %fields in right PML region
eybcr=zeros(ibbc,je);
hzxbcr=zeros(iebc,je);
hzybcr=zeros(iebc,je);

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

for i=1:media 
    eaf  =dt*sig(i)/(2.0*epsz*eps(i));
    ca(i)=(1.0-eaf)/(1.0+eaf);
    cb(i)=dt/epsz/eps(i)/dx/(1.0+eaf);
    haf  =dt*sim(i)/(2.0*muz*mur(i));
    da(i)=(1.0-haf)/(1.0+haf);
    db(i)=dt/muz/mur(i)/dx/(1.0+haf);
end

%***********************************************************************
%     Geometry specification (main grid)
%***********************************************************************

%     Initialize entire main grid to free space

caex(1:ie,1:jb)=ca(1);
cbex(1:ie,1:jb)=cb(1);

caey(1:ib,1:je)=ca(1);
cbey(1:ib,1:je)=cb(1);

dahz(1:ie,1:je)=da(1);
dbhz(1:ie,1:je)=db(1);

%     Add metal cylinder

diam=20;          % diameter of cylinder: 6 cm
rad=diam/2.0;     % radius of cylinder: 3 cm
icenter=4*ie/5;   % i-coordinate of cylinder's center
jcenter=je/2;     % j-coordinate of cylinder's center

for i=1:ie
    for j=1:je
        dist2=(i+0.5-icenter)^2 + (j-jcenter)^2;
        if dist2 <= rad^2
            caex(i,j)=ca(3);
            cbex(i,j)=cb(3);
        end
        dist2=(i-icenter)^2 + (j+0.5-jcenter)^2;
        if dist2 <= rad^2
            caey(i,j)=ca(3);
            cbey(i,j)=cb(3);
        end
    end
end

%***********************************************************************
%     Fill the PML regions
%***********************************************************************

delbc=iebc*dx;
sigmam=-log(rmax/100.0)*epsz*cc*(orderbc+1)/(2*delbc);
bcfactor=eps(1)*sigmam/(dx*(delbc^orderbc)*(orderbc+1));

%     FRONT region

caexbcf(1:iefbc,1)=1.0;
cbexbcf(1:iefbc,1)=0.0;
for j=2:jebc
    y1=(jebc-j+1.5)*dx;
    y2=(jebc-j+0.5)*dx;
    sigmay=bcfactor*(y1^(orderbc+1)-y2^(orderbc+1));
    ca1=exp(-sigmay*dt/(epsz*eps(1)));
    cb1=(1.0-ca1)/(sigmay*dx);
    caexbcf(1:iefbc,j)=ca1;
    cbexbcf(1:iefbc,j)=cb1;
end
sigmay = bcfactor*(0.5*dx)^(orderbc+1);
ca1=exp(-sigmay*dt/(epsz*eps(1)));
cb1=(1-ca1)/(sigmay*dx);
caex(1:ie,1)=ca1;
cbex(1:ie,1)=cb1;
caexbcl(1:iebc,1)=ca1;
cbexbcl(1:iebc,1)=cb1;
caexbcr(1:iebc,1)=ca1;
cbexbcr(1:iebc,1)=cb1;

for j=1:jebc
    y1=(jebc-j+1)*dx;
    y2=(jebc-j)*dx;
    sigmay=bcfactor*(y1^(orderbc+1)-y2^(orderbc+1));
    sigmays=sigmay*(muz/(epsz*eps(1)));
    da1=exp(-sigmays*dt/muz);
    db1=(1-da1)/(sigmays*dx);
    dahzybcf(1:iefbc,j)=da1;
    dbhzybcf(1:iefbc,j)=db1;
    caeybcf(1:ibfbc,j)=ca(1);
    cbeybcf(1:ibfbc,j)=cb(1);
    dahzxbcf(1:iefbc,j)=da(1);
    dbhzxbcf(1:iefbc,j)=db(1);
end

%     BACK region

caexbcb(1:iefbc,jbbc)=1.0;
cbexbcb(1:iefbc,jbbc)=0.0;
for j=2:jebc
    y1=(j-0.5)*dx;
    y2=(j-1.5)*dx;
    sigmay=bcfactor*(y1^(orderbc+1)-y2^(orderbc+1));
    ca1=exp(-sigmay*dt/(epsz*eps(1)));
    cb1=(1-ca1)/(sigmay*dx);
    caexbcb(1:iefbc,j)=ca1;
    cbexbcb(1:iefbc,j)=cb1;
end
sigmay = bcfactor*(0.5*dx)^(orderbc+1);
ca1=exp(-sigmay*dt/(epsz*eps(1)));
cb1=(1-ca1)/(sigmay*dx);
caex(1:ie,jb)=ca1;
cbex(1:ie,jb)=cb1;
caexbcl(1:iebc,jb)=ca1;
cbexbcl(1:iebc,jb)=cb1;
caexbcr(1:iebc,jb)=ca1;
cbexbcr(1:iebc,jb)=cb1;

for j=1:jebc
    y1=j*dx;
    y2=(j-1)*dx;
    sigmay=bcfactor*(y1^(orderbc+1)-y2^(orderbc+1));
    sigmays=sigmay*(muz/(epsz*eps(1)));
    da1=exp(-sigmays*dt/muz);
    db1=(1-da1)/(sigmays*dx);
    dahzybcb(1:iefbc,j)=da1;
    dbhzybcb(1:iefbc,j)=db1;
    caeybcb(1:ibfbc,j)=ca(1);
    cbeybcb(1:ibfbc,j)=cb(1);
    dahzxbcb(1:iefbc,j)=da(1);
    dbhzxbcb(1:iefbc,j)=db(1);
end

%     LEFT region

caeybcl(1,1:je)=1.0;
cbeybcl(1,1:je)=0.0;
for i=2:iebc
    x1=(iebc-i+1.5)*dx;
    x2=(iebc-i+0.5)*dx;
    sigmax=bcfactor*(x1^(orderbc+1)-x2^(orderbc+1));
    ca1=exp(-sigmax*dt/(epsz*eps(1)));
    cb1=(1-ca1)/(sigmax*dx);
    caeybcl(i,1:je)=ca1;
    cbeybcl(i,1:je)=cb1;
    caeybcf(i,1:jebc)=ca1;
    cbeybcf(i,1:jebc)=cb1;
    caeybcb(i,1:jebc)=ca1;
    cbeybcb(i,1:jebc)=cb1;
end
sigmax=bcfactor*(0.5*dx)^(orderbc+1);
ca1=exp(-sigmax*dt/(epsz*eps(1)));
cb1=(1-ca1)/(sigmax*dx);
caey(1,1:je)=ca1;
cbey(1,1:je)=cb1;
caeybcf(iebc+1,1:jebc)=ca1;
cbeybcf(iebc+1,1:jebc)=cb1;
caeybcb(iebc+1,1:jebc)=ca1;
cbeybcb(iebc+1,1:jebc)=cb1;

for i=1:iebc
    x1=(iebc-i+1)*dx;
    x2=(iebc-i)*dx;
    sigmax=bcfactor*(x1^(orderbc+1)-x2^(orderbc+1));
    sigmaxs=sigmax*(muz/(epsz*eps(1)));
    da1=exp(-sigmaxs*dt/muz);
    db1=(1-da1)/(sigmaxs*dx);
    dahzxbcl(i,1:je)=da1;
    dbhzxbcl(i,1:je)=db1;
    dahzxbcf(i,1:jebc)=da1;
    dbhzxbcf(i,1:jebc)=db1;
    dahzxbcb(i,1:jebc)=da1;
    dbhzxbcb(i,1:jebc)=db1;
    caexbcl(i,2:je)=ca(1);
    cbexbcl(i,2:je)=cb(1);
    dahzybcl(i,1:je)=da(1);
    dbhzybcl(i,1:je)=db(1);
end

%     RIGHT region

caeybcr(ibbc,1:je)=1.0;
cbeybcr(ibbc,1:je)=0.0;
for i=2:iebc
    x1=(i-0.5)*dx;
    x2=(i-1.5)*dx;
    sigmax=bcfactor*(x1^(orderbc+1)-x2^(orderbc+1));
    ca1=exp(-sigmax*dt/(epsz*eps(1)));
    cb1=(1-ca1)/(sigmax*dx);
    caeybcr(i,1:je)=ca1;
    cbeybcr(i,1:je)=cb1;
    caeybcf(i+iebc+ie,1:jebc)=ca1;
    cbeybcf(i+iebc+ie,1:jebc)=cb1;
    caeybcb(i+iebc+ie,1:jebc)=ca1;
    cbeybcb(i+iebc+ie,1:jebc)=cb1;
end
sigmax=bcfactor*(0.5*dx)^(orderbc+1);
ca1=exp(-sigmax*dt/(epsz*eps(1)));
cb1=(1-ca1)/(sigmax*dx);
caey(ib,1:je)=ca1;
cbey(ib,1:je)=cb1;
caeybcf(iebc+ib,1:jebc)=ca1;
cbeybcf(iebc+ib,1:jebc)=cb1;
caeybcb(iebc+ib,1:jebc)=ca1;
cbeybcb(iebc+ib,1:jebc)=cb1;

for i=1:iebc
    x1=i*dx;
    x2=(i-1)*dx;
    sigmax=bcfactor*(x1^(orderbc+1)-x2^(orderbc+1));
    sigmaxs=sigmax*(muz/(epsz*eps(1)));
    da1=exp(-sigmaxs*dt/muz);
    db1=(1-da1)/(sigmaxs*dx);
    dahzxbcr(i,1:je) = da1;
    dbhzxbcr(i,1:je) = db1;
    dahzxbcf(i+ie+iebc,1:jebc)=da1;
    dbhzxbcf(i+ie+iebc,1:jebc)=db1;
    dahzxbcb(i+ie+iebc,1:jebc)=da1;
    dbhzxbcb(i+ie+iebc,1:jebc)=db1;
    caexbcr(i,2:je)=ca(1);
    cbexbcr(i,2:je)=cb(1);
    dahzybcr(i,1:je)=da(1);
    dbhzybcr(i,1:je)=db(1);
end

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

subplot(3,1,1),pcolor(ex');
shading flat;
caxis([-80.0 80.0]);
axis([1 ie 1 jb]);
colorbar;
axis image;
axis off;
title(['Ex at time step = 0']);

subplot(3,1,2),pcolor(ey');
shading flat;
caxis([-80.0 80.0]);
axis([1 ib 1 je]);
colorbar;
axis image;
axis off;
title(['Ey at time step = 0']);

subplot(3,1,3),pcolor(hz');
shading flat;
caxis([-0.2 0.2]);
axis([1 ie 1 je]);
colorbar;
axis image;
axis off;
title(['Hz at time step = 0']);

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

M=moviein(nmax/4,gcf,rect);

%***********************************************************************
%     BEGIN TIME-STEPPING LOOP
%***********************************************************************
starttime = clock;
for n=1:nmax
    
    %***********************************************************************
    %     Update electric fields (EX and EY) in main grid
    %***********************************************************************
    
    ex(:,2:je)=caex(:,2:je).*ex(:,2:je)+...
        cbex(:,2:je).*(hz(:,2:je)-hz(:,1:je-1));
    
    ey(2:ie,:)=caey(2:ie,:).*ey(2:ie,:)-cbey(2:ie,:).*diff(hz);
    %          cbey(2:ie,:).*(hz(1:ie-1,:)-hz(2:ie,:));
    
    %***********************************************************************
    %     Update EX in PML regions
    %***********************************************************************
    
    %     FRONT
    
    exbcf(:,2:jebc)=caexbcf(:,2:jebc).*exbcf(:,2:jebc)-...
        cbexbcf(:,2:jebc).*(hzxbcf(:,1:jebc-1)+hzybcf(:,1:jebc-1)-...
        hzxbcf(:,2:jebc)-hzybcf(:,2:jebc));
    ex(1:ie,1)=caex(1:ie,1).*ex(1:ie,1)-...
        cbex(1:ie,1).*(hzxbcf(ibbc:iebc+ie,jebc)+...
        hzybcf(ibbc:iebc+ie,jebc)-hz(1:ie,1));
    
    %     BACK
    
    exbcb(:,2:jebc-1)=caexbcb(:,2:jebc-1).*exbcb(:,2:jebc-1)-...
        cbexbcb(:,2:jebc-1).*(hzxbcb(:,1:jebc-2)+hzybcb(:,1:jebc-2)-...
        hzxbcb(:,2:jebc-1)-hzybcb(:,2:jebc-1));
    ex(1:ie,jb)=caex(1:ie,jb).*ex(1:ie,jb)-...
        cbex(1:ie,jb).*(hz(1:ie,jb-1)-hzxbcb(ibbc:iebc+ie,1)-...
        hzybcb(ibbc:iebc+ie,1));
    
    %     LEFT
    
    exbcl(:,2:je)=caexbcl(:,2:je).*exbcl(:,2:je)-...
        cbexbcl(:,2:je).*(hzxbcl(:,1:je-1)+hzybcl(:,1:je-1)-...
        hzxbcl(:,2:je)-hzybcl(:,2:je));
    exbcl(:,1)=caexbcl(:,1).*exbcl(:,1)-...
        cbexbcl(:,1).*(hzxbcf(1:iebc,jebc)+hzybcf(1:iebc,jebc)-...
        hzxbcl(:,1)-hzybcl(:,1));
    exbcl(:,jb)=caexbcl(:,jb).*exbcl(:,jb)-...
        cbexbcl(:,jb).*(hzxbcl(:,je)+hzybcl(:,je)-...
        hzxbcb(1:iebc,1)-hzybcb(1:iebc,1));
    
    %     RIGHT
    
    exbcr(:,2:je)=caexbcr(:,2:je).*exbcr(:,2:je)-...
        cbexbcr(:,2:je).*(hzxbcr(:,1:je-1)+hzybcr(:,1:je-1)-...
        hzxbcr(:,2:je)-hzybcr(:,2:je));
    exbcr(:,1)=caexbcr(:,1).*exbcr(:,1)-...
        cbexbcr(:,1).*(hzxbcf(1+iebc+ie:iefbc,jebc)+...
        hzybcf(1+iebc+ie:iefbc,jebc)-...
        hzxbcr(:,1)-hzybcr(:,1));
    exbcr(:,jb)=caexbcr(:,jb).*exbcr(:,jb)-...
        cbexbcr(:,jb).*(hzxbcr(:,je)+hzybcr(:,je)-...
        hzxbcb(1+iebc+ie:iefbc,1)-...
        hzybcb(1+iebc+ie:iefbc,1));
    
    %***********************************************************************
    %     Update EY in PML regions
    %***********************************************************************
    
    %     FRONT
    
    eybcf(2:iefbc,:)=caeybcf(2:iefbc,:).*eybcf(2:iefbc,:)-...
        cbeybcf(2:iefbc,:).*(hzxbcf(2:iefbc,:)+hzybcf(2:iefbc,:)-...
        hzxbcf(1:iefbc-1,:)-hzybcf(1:iefbc-1,:));
    
    %     BACK
    
    eybcb(2:iefbc,:)=caeybcb(2:iefbc,:).*eybcb(2:iefbc,:)-...
        cbeybcb(2:iefbc,:).*(hzxbcb(2:iefbc,:)+hzybcb(2:iefbc,:)-...
        hzxbcb(1:iefbc-1,:)-hzybcb(1:iefbc-1,:));
    
    %     LEFT
    
    eybcl(2:iebc,:)=caeybcl(2:iebc,:).*eybcl(2:iebc,:)-...
        cbeybcl(2:iebc,:).*(hzxbcl(2:iebc,:)+hzybcl(2:iebc,:)-...
        hzxbcl(1:iebc-1,:)-hzybcl(1:iebc-1,:));
    ey(1,:)=caey(1,:).*ey(1,:)-...
        cbey(1,:).*(hz(1,:)-hzxbcl(iebc,:)-hzybcl(iebc,:));
    
    %     RIGHT
    
    eybcr(2:iebc,:)=caeybcr(2:iebc,:).*eybcr(2:iebc,:)-...
        cbeybcr(2:iebc,:).*(hzxbcr(2:iebc,:)+hzybcr(2:iebc,:)-...
        hzxbcr(1:iebc-1,:)-hzybcr(1:iebc-1,:));
    ey(ib,:)=caey(ib,:).*ey(ib,:)-...
        cbey(ib,:).*(hzxbcr(1,:)+hzybcr(1,:)- hz(ie,:));
    
    
    %***********************************************************************
    %     Update magnetic fields (HZ) in main grid
    %***********************************************************************
    
    %hz(1:ie,1:je)=dahz(1:ie,1:je).*hz(1:ie,1:je)+...
    %              dbhz(1:ie,1:je).*(ex(1:ie,2:jb)-ex(1:ie,1:je)+...
    %                                ey(1:ie,1:je)-ey(2:ib,1:je));
    hz=dahz.*hz+dbhz.*(ex(1:ie,2:jb)-ex(1:ie,1:je)-diff(ey));
    hz(is,js)=source(n);
    
    
    %***********************************************************************
    %     Update HZX in PML regions
    %***********************************************************************
    
    %     FRONT
    
    hzxbcf(1:iefbc,:)=dahzxbcf(1:iefbc,:).*hzxbcf(1:iefbc,:)-...
        dbhzxbcf(1:iefbc,:).*(eybcf(2:ibfbc,:)-eybcf(1:iefbc,:));
    
    %     BACK
    
    hzxbcb(1:iefbc,:)=dahzxbcb(1:iefbc,:).*hzxbcb(1:iefbc,:)-...
        dbhzxbcb(1:iefbc,:).*(eybcb(2:ibfbc,:)-eybcb(1:iefbc,:));
    
    %     LEFT
    
    hzxbcl(1:iebc-1,:)=dahzxbcl(1:iebc-1,:).*hzxbcl(1:iebc-1,:)-...
        dbhzxbcl(1:iebc-1,:).*(eybcl(2:iebc,:)-eybcl(1:iebc-1,:));
    hzxbcl(iebc,:)=dahzxbcl(iebc,:).*hzxbcl(iebc,:)-...
        dbhzxbcl(iebc,:).*(ey(1,:)-eybcl(iebc,:));
    
    %     RIGHT
    
    hzxbcr(2:iebc,:)=dahzxbcr(2:iebc,:).*hzxbcr(2:iebc,:)-...
        dbhzxbcr(2:iebc,:).*(eybcr(3:ibbc,:)-eybcr(2:iebc,:));
    hzxbcr(1,:)=dahzxbcr(1,:).*hzxbcr(1,:)-...
        dbhzxbcr(1,:).*(eybcr(2,:)-ey(ib,:));
    
    %***********************************************************************
    %     Update HZY in PML regions
    %***********************************************************************
    
    %     FRONT
    
    hzybcf(:,1:jebc-1)=dahzybcf(:,1:jebc-1).*hzybcf(:,1:jebc-1)-...
        dbhzybcf(:,1:jebc-1).*(exbcf(:,1:jebc-1)-exbcf(:,2:jebc));
    hzybcf(1:iebc,jebc)=dahzybcf(1:iebc,jebc).*hzybcf(1:iebc,jebc)-...
        dbhzybcf(1:iebc,jebc).*(exbcf(1:iebc,jebc)-exbcl(1:iebc,1));
    hzybcf(iebc+1:iebc+ie,jebc)=...
        dahzybcf(iebc+1:iebc+ie,jebc).*hzybcf(iebc+1:iebc+ie,jebc)-...
        dbhzybcf(iebc+1:iebc+ie,jebc).*(exbcf(iebc+1:iebc+ie,jebc)-...
        ex(1:ie,1));
    hzybcf(iebc+ie+1:iefbc,jebc)=...
        dahzybcf(iebc+ie+1:iefbc,jebc).*hzybcf(iebc+ie+1:iefbc,jebc)-...
        dbhzybcf(iebc+ie+1:iefbc,jebc).*(exbcf(iebc+ie+1:iefbc,jebc)-...
        exbcr(1:iebc,1));
    
    %     BACK
    
    hzybcb(1:iefbc,2:jebc)=dahzybcb(1:iefbc,2:jebc).*hzybcb(1:iefbc,2:jebc)-...
        dbhzybcb(1:iefbc,2:jebc).*(exbcb(1:iefbc,2:jebc)-exbcb(1:iefbc,3:jbbc));
    hzybcb(1:iebc,1)=dahzybcb(1:iebc,1).*hzybcb(1:iebc,1)-...
        dbhzybcb(1:iebc,1).*(exbcl(1:iebc,jb)-exbcb(1:iebc,2));
    hzybcb(iebc+1:iebc+ie,1)=...
        dahzybcb(iebc+1:iebc+ie,1).*hzybcb(iebc+1:iebc+ie,1)-...
        dbhzybcb(iebc+1:iebc+ie,1).*(ex(1:ie,jb)-exbcb(iebc+1:iebc+ie,2));
    hzybcb(iebc+ie+1:iefbc,1)=...
        dahzybcb(iebc+ie+1:iefbc,1).*hzybcb(iebc+ie+1:iefbc,1)-...
        dbhzybcb(iebc+ie+1:iefbc,1).*(exbcr(1:iebc,jb)-...
        exbcb(iebc+ie+1:iefbc,2));
    
    %     LEFT
    
    hzybcl(:,1:je)=dahzybcl(:,1:je).*hzybcl(:,1:je)-...
        dbhzybcl(:,1:je).*(exbcl(:,1:je)-exbcl(:,2:jb));
    
    %     RIGHT
    
    hzybcr(:,1:je)=dahzybcr(:,1:je).*hzybcr(:,1:je)-...
        dbhzybcr(:,1:je).*(exbcr(:,1:je)-exbcr(:,2:jb));
    
    %***********************************************************************
    %     Visualize fields
    %***********************************************************************
    n_rec = 20100;
    if mod(n,n_rec)==0;
        endtime = clock;
        eltime = etime(endtime,starttime);
        fprintf('time per step %8.4f  per point %9.2e \n',eltime/n_rec,eltime/n_rec/ie/je)
        starttime = endtime;
        timestep=int2str(n);
        pcolor(ex');
         shading flat;
         caxis([-80.0 80.0]);
         axis([1 ie 1 jb]);
        
%         subplot(3,1,1),pcolor(ex');
%         shading flat;
%         caxis([-80.0 80.0]);
%         axis([1 ie 1 jb]);
%         colorbar;
%         axis image;
%         axis off;
%         title(['Ex at time step = ',timestep]);
%         
%         subplot(3,1,2),pcolor(ey');
%         shading flat;
%         caxis([-80.0 80.0]);
%         axis([1 ib 1 je]);
%         colorbar;
%         axis image;
%         axis off;
%         title(['Ey at time step = ',timestep]);
%         
%         subplot(3,1,3),pcolor(hz');
%         shading flat;
%         caxis([-0.2 0.2]);
%         axis([1 ie 1 je]);
%         colorbar;
%         axis image;
%         axis off;
%         title(['Hz at time step = ',timestep]);
        
        nn=n/n_rec;
        M(:,nn)=getframe(gcf,rect);
        
    end;
    
    %***********************************************************************
    %     END TIME-STEPPING LOOP
    %***********************************************************************
    
end

movie(gcf,M,0,10,rect);
movie(gcf,M,1,10,rect);

