Shichang She and Ya Yan Lu, "Improved Dirichlet-to-Neumann map method for scattering by circular cylinders on a lattice," J. Opt. Soc. Am. A 29, 1999-2004 (2012)
A simple model for two-dimensional photonic crystal devices consists of a finite number of possibly different circular cylinders centered on lattice points of a square or triangular lattice and surrounded by a homogeneous or layered background medium. The Dirichlet-to-Neumann (DtN) map method is a special method for analyzing the scattering of an incident wave by such a structure. It is more efficient than existing numerical or semianalytic methods, such as the finite element method and the multipole method, since it takes advantage of the underlying lattice structure and the simple geometry of the unit cells. The DtN map of a unit cell is a relation between a wave field component and its normal derivative on the cell boundary, and it can be used to avoid further computation inside the unit cell. In this paper, an improved DtN map method is developed by constructing special DtN maps for boundary and corner unit cells using the method of fictitious sources.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
The relative errors (R.E.) are calculated based on the reference solution .
Table 4.
Numerical Solutions for the Dielectric Slab with 10 Holesa
Method
at point
R.E.
DtN(i),
0.00038
DtN(i),
0.00039
DtN(i),
0.00049
DtN(i),
0.00028
DtN(i),
0.00036
DtN(i),
0.00043
DtN(i),
0.00042
FEM,
—
FEM,
—
FEM,
—
The DtN results are obtained with . The FEM results are obtained on a triangular mesh with nodes. Relative errors are calculated based on the most accurate FEM solution.
Tables (4)
Table 1.
Maximum Relative Errors of Normal Derivatives of Test Functions on the Edges of a Boundary Unit Cella
Horizontal Line
U curve
V curve
5
0.052689
0.0310944
0.051555
5
0.064214
0.0106649
0.009561
5
0.007494
0.0002250
0.000284
5
0.000385
0.0002906
0.000153
0
0.000011
0.0000031
0.000003
0
0.000282
0.0000003
0.000002
and are the order and the center of the multipole testing function given in Eq. (8), respectively.
Table 2.
Example 1: at Point Computed by the Improved and Original DtN Map Methods, the Multipole Method, and the FEM
The relative errors (R.E.) are calculated based on the reference solution .
Table 4.
Numerical Solutions for the Dielectric Slab with 10 Holesa
Method
at point
R.E.
DtN(i),
0.00038
DtN(i),
0.00039
DtN(i),
0.00049
DtN(i),
0.00028
DtN(i),
0.00036
DtN(i),
0.00043
DtN(i),
0.00042
FEM,
—
FEM,
—
FEM,
—
The DtN results are obtained with . The FEM results are obtained on a triangular mesh with nodes. Relative errors are calculated based on the most accurate FEM solution.