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MAXWELL’s EQUATION
∇ ×E=-[∂B/∂t+M] (Faraday's law)
∇ ×B=με[ ∂E/∂t+J] (Ampere's law)
M = Msource + σ∗ H B=μ H J = Jsource+σ E
In 3D
MODES OF OPERATION
TMz mode TEz mode
Hz = 0 Ez = 0
YEE LATTICE
TEz mode
TMz mode
General representation
H is updated at ½ time steps
E is updated at full time steps
2 D YEE lattice grid
DISCRETIZATION BASED ON 2D YEE LATTICE
Collecting terms
Putting the values in Ez discretization equation
Simplifying => consider sources to be 0 and ∆x =∆y
PROBLEM SETUP
Δx = 10^-6 μm
Δy = Δx = 10^-6 μm
T = 8250 fsec (Total time)
In 1 D⇒ Δx/Δt≥c ;c is the speed of light
Therefore,Δt≤Δx/c
In 2 D⇒ Δt≤Δx/(√2 c)
S= 1/√2 , S is the stability factor
where
ε0=8.85∗10^(-12) C^2/(Nm^2 )
μ0=4π *10^(-7) N/A^2
σ=4∗10^(-4) S/m Electrical conductivity of medium
σ∗=4∗10^(-4) N/A^2 Magnetic conductivity of medium
BOUNDARY CONDITIONS
All electric field is confined within the boundary
Ez = 0 on boundaries
Ez(1,j)^n=0 (left)
Ez(i,1)^n=0 (bot)
Ez(nx,j)^n=0 (right)
Ez(i,ny)^n=0 (top)
All perpendicular magnetic field is reflected from the boundary
H⊥ = 0 on boundaries
Hx(1,j+1/2)^n=0 (left)
Hx(nx,j+1/2)^n=0 (right)
Hy(i+1/2,1)^n=0 (bot)
Hy(i+1/2,ny)^n=0 (top)
INITIAL CONDITIONS
SOURCE CONDITION
ALGORITHM
RESULTS
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