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5.2 Theoretical Analysis 169
fabrication reproducibility and an artifact caused by the vertical tip displace-
ment [5.12]. Usingthe near field we can observe [5.11], read/write [5.13], and
fabricate [5.14] beyond the diffraction limit of the conventional optics.

5.2 Theoretical Analysis

The quantitative description of the near field is given by vector theory. We can
obtain analytical solutions for only limited structures, resultingin a compu-
tational solution approach to Maxwell’s equations. The finite difference time
domain (FDTD) method [5.15] is often used for such an approach because a
new structure of interest can be modeled easily by systematic mesh generation.

5.2.1 FDTD Method
FDTD is a direct solution method for Maxwell’s equations by space-grid time-
domain techniques employingno potentials. It is based on volumetric sampling
of an unknown electric field E and a magnetic field H within and surrounding
the structure of interest, and over a period of time. The samplingin space is,
typically, more than 10 per wavelength. The sampling in time is selected to
ensure numerical stability of the algorithm. At present, the best choices for
the computational algorithm and mesh remain unclear.
Since this technique requires no special knowledge, FDTD is emerging in a
variety of fields of science and engineering. Optics-related fields are concerned
with near fields, microdisk resonators, photonic band-gap devices, and col-
lidingspatial solitons. FDTD is becominga standard method in these fields.
We consider the foundation of FDTD electromagnetic field analysis using the
algorithm introduced by Yee in 1996 [5.16]. The main steps of the FDTD
algorithm are as follows:
1. Define the space domain to be computed
2. divide the domain into cells
3. rewrite Maxell’s equation (5.1) usingthe time and space derivatives
4. calculate E and H at each location under initial conditions and continue
the process cycle for the time step

∂H
∇× E = −µ ,
∂t
∂E
∇× H = σE + ε . (5.1)
∂t
We introduce the followingnotation (in one spatial dimension), denoting
a space point in a uniform, rectangular lattice as:

(x, y, z, t)=(i∆x, j∆y, k∆z, n∆t), (5.2)