Rahul Dewan - Jacobs University

3. COMPUTATIONAL MODELING OF OPTICAL WAVE PROPAGATION
ɛ En+1 x (i, j, k) − Ex n (i, j, k)
∆t
≈
ɛ ∆t
+ σE n+ 1/2
x (i, j, k)
(
E
n+1
x (i, j, k) − Ex n (i, j, k) ) + σ (
E
n+1
x (i, j, k) − Ex n (i, j, k) )
2
= Hn+ 1/2
z (i, j + 1, k) − H n+ 1/2
z (i, j, k)
− Hn+ 1/2
y (i, j, k + 1) − H n+ 1/2
y (i, j, k)
∆y
∆z
Solving for Ex
n+1 (i, j, k), we have
E n+1
x (i, j, k) =
1
( ɛ
∆t + σ 2
)
∆y
(
H
n+1/2
z (i, j + 1, k) − H n+ 1/2
z (i, j, k) )
−
1
( ɛ
∆t + σ 2
)
∆z
(
H
n+1/2
y (i, j, k + 1) − H n+ 1/2
y (i, j, k) )
( ɛ
∆t − σ 2
)
+ ( ɛ + )E σ x n (i, j, k) (3.33)
∆t 2
Similarly, by applying the same procedure to equation (3.28) and (3.29), the explicit
finite difference equations for the other 2 components of E can be derived
E n+1
y (i, j, k) =
1
( ɛ
∆t + σ 2
)
∆z
(
H
n+1/2
x (i, j, k + 1) − H n+ 1/2
x (i, j, k) )
−
1
( ɛ
∆t + σ 2
)
∆x
(
H
n+1/2
z (i + 1, j, k) − H n+ 1/2
z (i, j, k) )
( ɛ
∆t − σ 2
)
+ ( ɛ + )E σ y n (i, j, k) (3.34)
∆t 2
E n+1
z (i, j, k) =
1
( ɛ
∆t + σ 2
)
∆x
(
H
n+1/2
y (i + 1, j, k) − H n+ 1/2
y (i, j, k) )
−
1
( ɛ
∆t + σ 2
)
∆y
(
H
n+1/2
x (i, j + 1, k) − H n+ 1/2
x (i, j, k) )
( ɛ
∆t − σ 2
)
+ ( ɛ + )E σ z n (i, j, k) (3.35)
∆t 2
The six equations (3.30)−(3.35) are the first order difference equations defining
Yee’s algorithm and the foundation of the FDTD method [79, 80]. It can be noted from
equations (3.30) till (3.35) and from Fig. 3.2, that the components of E and H are
interlaced within the unit cell and are evaluated at alternate half-time steps. Thus the
FDTD algorithm is also called leap-frog algorithm. In such a configuration, in order
to update the electric field components (E n ) the magnetic field components (H n− 1/2
)
calculated in the previous step are used, and the updated electric field components
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