Finite Difference Time Domain (FDTD) Method February 14, 2014

Yee Cell (3-D) Kane Yee (1966) suggested the following cell for evaluating discretized Maxwell Equations:

Yee Cell (2-D)

Finite Difference Equations Maxwell’s Equations are continuous. For all but the simplest geometry, closed form solutions of Maxwell’s Equations are not able to be obtained. We must convert the continuous equations (with infinitesimal differences) to discrete equations with finite differences

Discretize and Conquer Ey(i,j) = Ey(i,j) - (dt/eps_0)*(Hz(i-1,j)-Hz(i,j))*(1/dx) (Hz(i, j)-Hz(i-1, j))*(1/dx) = (eps_0)*(Ey(i,j) now - Ey(i,j) before ) / dt Hz(i,j) = Hz(i,j) – (dt/mu_0)*(Ey(i,j)-Ey(i+1,j) + Ex(i,j+1) - Ex(i,j))*(1/dx) ((Ey(i+1, j)-Ey(i, j)) – (Ex(i, j+1) – Ex(i, j)))*(1/dx) = -(mu_0)*(Hz(i, j) now - Hz(i, j) before )/dt

Convergence When we go from continuous to discrete, does the solution become “incorrect”? If you keep making the spatial step smaller and smaller, eventually the results converge to the correct result – Rule of thumb: make  x equal to 1/100 of the wavelength of the incident light The Courant Condition specifies a value for  t:

Programming I wrote examples of 1-D and 2-D FDTD in Fortran 90, C and Python. Maxim uses Fortran 90 for everything and therefore so should you. – Ask him for the Absoft Fortran 90 editor. If you try to compile and run with this editor it will give you trouble. I suggest using the cluster.

Fortran 90 Variables must be declared (but not initialized) at the very beginning Constants must be declared and initialized at the beginning After variables are declared and constants are declared and initialized, variables must be set Then you can start writing your code. End your code with the word “end”.

Code Example implicit none ! !~~~ Fundamental Constants ~~~! ! double precision, parameter :: pi= D0,c= D0 double precision, parameter :: mu0=4.0D-7*pi,eps0=1.0D0/(c*c*mu0) ! !~~~ Grid, Time Steps, Unit Cells ~~~! ! double precision, parameter :: dx=1.0d-9,dy=1.0d-9 integer, parameter :: Nt=5000 integer, parameter :: Nx=601

! !~~~ Source ~~~! ! integer, parameter :: ms=10 !m position of the incident source double precision, parameter :: E0=1.0D0 !incident pulse peak amplitude double precision pulse(Nt) double precision, parameter :: omega=2.0*pi*2.4180D14*5.00 !laser frequency in eV ! !~~~ Time Step ~~~! ! double precision, parameter :: dt=dx/(2.0D0*c) ! !~~~ Dielectric ~~~! ! double precision eps_r(Nx) double precision, parameter :: eps_rel_dielectric=1.0!4.0 double precision, parameter :: eps_rel_vacuum=1.0 integer, parameter :: xnd = 300 !Grid point where dielectric starts

! !~~~ EM field components ~~~! ! double precision Ex(Nx),Hz(Nx-1) ! !~~~ Boundary condition ~~~! ! double precision ex_low_m2 double precision ex_low_m1 double precision ex_high_m2 double precision ex_high_m1 ! !~~~ Other parameters and variables ~~~! ! integer n,i double precision t double precision, parameter :: dt_eps0=dt/eps0,dt_mu0=dt/mu0

! !~~~ Fire off the source ~~~! ! !Sinewave do n=1,Nt t=dt*n pulse(n)=E0*SIN(omega*t) enddo ! !~~~ Structure ~~~! ! do i=1,Nx if((i<xnd)) then eps_r(i)=eps_rel_vacuum else eps_r(i)=eps_rel_dielectric endif enddo

! !~~~ Initialize remaining variables ~~~! ! Ex=0.0 Hz=0.0 ex_low_m2=0.0 ex_low_m1=0.0 ex_high_m2=0.0 ex_high_m1=0.0 do n=1,Nt

!Hz Update do i=1,Nx-1 Hz(i)=Hz(i)-dt_mu0*(Ex(i+1)-Ex(i))/dx enddo !Ex Update do i=2,Nx-1 if(i==ms) then !Excitation Ex(i)=Ex(i)-(1/eps_r(i))*dt_eps0*(Hz(i)-Hz(i-1))/dx+ & pulse(n) else Ex(i)=Ex(i)-(1/eps_r(i))*dt_eps0*(Hz(i)-Hz(i-1))/dx endif enddo

! Inc Boundary Condition Ex(1) = ex_low_m2 ex_low_m2 = ex_low_m1 ex_low_m1 = Ex(2) Ex(Nx) = ex_high_m2 ex_high_m2 = ex_high_m1 ex_high_m1 = Ex(Nx-1) enddo !Nt

!Record Ex OPEN(file='Ex.dat',unit=32) do i=1,Nx write(32,*) Ex(i) enddo CLOSE(unit=32) end Import into Matlab and type Plot(Ex):

Results As one is learning FDTD, it is helpful to plot electric field over the entire grid

Results Graphs of the T and R coefficients (obtained from the Poynting vector) are measured in experiments

Next Steps Look at a single-processor 2-D code with a dielectric cylinder (ask me or Maxim for it) The absorbing boundary conditions (ABC’s) are implemented via convolutional perfectly matched layers (CPML) – It’s there and it works. In general, there’s no need to mess with it.

Later On The best metals for SPP’s are silver and gold – Metal is represented by the Drude Model The interaction of the two-level emitters with the classical EM field is handled by the Liouville-von Neumann equation Longer simulations benefit from message passing interface (MPI); the simulation is split up between several processors – Ask me or Maxim for the MPI slits code Rather than using point sources, we can define a Total Field / Scattered Field boundary to introduce a plane wave into the system