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5.2 Theoretical Analysis 173
1
σ(i+ ,j,k)∆t ∆t
1 − 2
1
1
2ε(i+ ,j,k) ε(i+ ,j,k)
n+1 1 2 n 1 2
E z i, j, k + = E z i, j, k + +
1
2 σ(i+ ,j,k)∆t 2 σ(i+ ,j,k)∆t
1
1+ 2 1+ 2
1
2ε(i+ ,j,k) 2ε(i+ ,j,k)
1
2 2
n+ 1 1 1 n+ 1 1 1

H y 2 i + ,j,k + − H y 2 i − ,j,k +
2 2 2 2
×
∆x
n+ 1 1 1 n+ 1 1 1
H x 2 i, j − ,k + 2 − H x 2 i, j + ,k + 2
2
2
+ .
∆y
(5.18)
From (5.13)–(5.18), it is found that H at the time step n+1/2 can be
obtained by E at n and H at n − 1/2,E at n + 1 is obtained by H at n +1/2
and E at n (see Fig. 5.3). The cycle begins again with the computation of E
components based on the newly obtained H. This process continues until the
solutions remain constant. To avoid numerical instability, the time increment
must satisfy the followingCourant condition:
1
1 1 1 2
−
v max ∆t ≤ 2 + 2 + 2 . (5.19)
∆x ∆y ∆z
Here, v max is the maximum phase velocity of the electromagnetic wave.
Moreover, to prevent the reflection at the outermost space–lattice planes of
the computational domain, absorbingboundary conditions (ABCs) must be
introduced. Since first-order Mur ABCs are effective only for normally incident
plane wave, second-order Mur ABCs are often used [5.17]. On the other hand,
Berenger’s perfectly matched layer (PML) ABCs are effective for the plane
waves of arbitrary incidence, polarization, and frequency [5.18]. In the case of
a perfect conductor, the electric field equals zero on the surface.
5.2.2 Numerical Examples of Near Field Analysis
The followingare examples of the application of FDTD to problems including
the interaction between a plane wave and typical subwavelength structures.

Example 5.1. Compute the electromagnetic field around a small aperture on
the perfect conductor plane when a TM (p-polarized) plane wave is normally
incident, and show the intensity profile in two dimensions.
Solution. Figure 5.5 shows (a) a process for analysis, and (b) a 2-D Cartesian
computational domain (1500 nm × 500 nm). We consider a TM (p-polarized)
plane wave normally incident on a 100-nm-diameter aperture with the para-
meters ∆x =∆y =1 nm, ∆t =2.3586769 × 10 −19 sand n =3, 000. Figure 5.6
shows a numerical result in the domain of 75 nm×500 nm by FDTD, indicating
that the electric field is enhanced at the edge and decays rapidly [5.19].