Nonstandard FDTD Theory

Beginner Course

1. Finite Difference Model
2. 1D Wave Equation
3. 1D Stability
4. 1D Boundary Condition
5. Dimensionless Form
6. 1D Example

Intermediate Course

1. 2D Wave Equation
2. 2D Maxwell's Equations
3. Scattered Field
4. 2D Stability
5. 2D Boundary Condition
6. Scatterer Representation
7. Intensity
8. 2D Example

Advanced Course

1. 3D Wave Equation
2. 3D Maxwell's Equations
3. 3D Stability
4. Conductive Media
5. Perfectly Matched Layer

4. 1D Boundary Condition

Since our computational resource is finite, boundary conditions are necessary to calculate wave propagation. We introduce three boundary conditions:

• Perfect Conductor Boundary Condition
• Periodic Boundary Condition
• Absorbing Boundary Condition

If the computational domain is surrounded by non-dispersive metals, the perfect conductor boundary condition which generates total reflection is used. If periodic structures such as photonic crystals are simulated, the periodic boundary condition is effective. In many cases, however, the unbounded uniform region such as free space given by the absorbing boundary condition is required.

Perfect Conductor Boundary Condition

Since the perfect conductor such as non-dispersive metal generates total reflection, the boundary condition is a simple fixed end. The fixed end at boundaries is given by, ψ(x,t) = 0. For example, the one-dimensional computational domain is defined by

the perfect conductor boundary condition is given by

where we write ψ_iⁿ = ψ(x,t).

Perfect Conductor Boundary Condition

Photonics has various periodic structures such as defects in photonic crystal. If we calculate scatterings for one defect, the periodic boundary condition is helpful relative to allocate many defects in very large computational domain. When the computational domain given by (1), the periodic boundary condition connects both sides,

The ideal periodic structure can be calculated in a small region by the periodic boundary condition.

1st-order Standard Mur ABC

The absorbing boundary condition (ABC) has almost no reflection at boundaries. The Mur ABC is derived from the wave equation [1]. The one-dimensional wave equation is factorized into

and we obtain one-way wave equations,

The one-way wave equation has a monochromatic wave solution which propagates only along +x or -x axis. (5) is satisfied by

whereas (6) is satisfied by

If (5) is satisfied at boundaries, it has no reflection toward +x axis. Similarly (6) has no reflection toward -x axis. Let temporal-spatial discretization is given by

Using the S-FDTD algorithm, (5) is discretized at t = (n+1/2)Δt, x = h/2,

Since we have no data on half temporal-spatial grids, they are approximated by

Substituting these approximations into (10), we obtain the "first-order standard Mur ABC,"

At another boundary x = N_xh, we similarly discretize and obtain

1st-order Nonstandard Mur ABC

The first-order Mur ABC can be optimized to monochromatic wave propagation. Substituting ψ(x,t) = e^i(kx±ωt) into (12), we find

Thus the monochromatic wave is not a solution of the difference equation (12). Replacing u_S with u'_NS, we can find

where u'_NS ≠ u_NS because (11) is additionally approximated. Hence the "first-order nonstandard Mur ABC" is given by

Bibliography:
[1] G. Mur, "Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation," Mathematics of computation, 47, 176, pp. 437-459 (1986).

Copyright (C) 2011 Naoki Okada, All Rights Reserved.