Modeling Periodic Structures
A periodic structure is defined by the periodic unit cell. This unit cell is then repeated infinitely in one or two dimensions.
Periodic structures appear in two different modeling scenarios — driven problems and non-driven problems. For driven problems, a port is used for launching an incident wave. What is of interest here is to calculate how much of that input wave that is transmitted, reflected, absorbed, and diffracted to higher diffraction orders. For non-driven problems there is no excitation source. A typical simulation is to prescribe a certain wave vector that defines a phase relationship between the fields on the opposing parallel boundaries of the unit cell and then perform an eigenfrequency simulation to find all the frequencies and mode fields that can satisfy the prescribed periodic phase relationship. By sweeping the magnitude and direction of the wave vector, it is possible to generate band structure diagrams for the particular periodic cell.
To simplify the simulation of driven periodic problems, the Wave Optics Module has a dedicated Periodic Structure domain condition. The condition automatically adds the required ports and periodic conditions as subnodes to the Periodic Structure node. The selections for the ports and the periodic conditions are handled automatically, after the user has assigned the selection for the excitation port.
The Periodic Structure selection defines the periodic unit cell. The reference direction a1 is defined from edge and point selections in the Periodic Structure and its Reference Direction subnode. Given the first reference direction a1, the second reference direction a2 is calculated from
,
where a0 is the periodic structure axis direction that is pointing in the direction from the passive port toward the excitation port. Thus, the axis direction equals the normal direction of the excited port.
Given the reference directions, the wave vector for the incident plane wave is given by
,
where k is the material wavenumber at the excitation port and α1 and α2 are the elevation and azimuth angles, respectively. The elevation angle α1 is the angle between the wave vector kinc and the periodic structure axis (kinc⋅a0 ≤ 0, as α1 is in the range from 0 to π/2 radians). For normal incidence, this angle is zero. The azimuth angle α2 is the angle between the first reference direction a1 and the projection of the incident wave vector kinc on the plane spanned by a1 and a2 (the port planes).
The first reference direction a1 also defines the direction of the first primitive vector of the periodic unit cell. That is,
.
The length b1 is obtained from a selection of the excited port edges. In 3D, for parallelogram-like unit cells, the second primitive vector b2 is defined from the unit cell edges that are not parallel to b1. Furthermore, b2 is defined such that
(2-1).
For hexagonal unit cells, three edge vectors are defined from three consecutive hexagon edges:
•
v1 is defined from the hexagon edge selected by the user.
•
v2 is defined by the hexagon edge that starts where the edge that defines v1 ends.
•
v3 is defined by the hexagon edge that starts where the edge that defines v2 ends.
From these three vectors, the first primitive cell vector is defined by
and the second primitive cell vector is defined from
,
assuming that Equation 2-1 applies. If the cross product does not fulfill Equation 2-1, the expressions for b1 and b2 are swapped.
From the primitive vectors for the unit cell, the reciprocal lattice vectors can be calculated as
and
,
where the volume V is given by
.
In 2D, there is periodicity in only one direction. Thus, the second primitive vector has an infinite length
.
Thereby, the first reciprocal vector becomes
and, consequently,
.
The Periodic Structure and its Reference Direction subnode define the periodic structure variables in Table 2-1. The variables should be prefixed with the physics interface tag. So, to evaluate the x-component of the unit cell axis, write ewfd.axisx if the physics tag is ewfd.
Table 2-1: Periodic Structure Variables
Variable
description
available component
axisx
Unit cell axis, x-component
2D, 3D
axisy
Unit cell axis, y-component
2D, 3D
axisz
Unit cell axis, z-component
2D, 3D
aUnit1x
First reference direction, x-component
2D, 3D
aUnit1y
First reference direction, y-component
2D, 3D
aUnit1z
First reference direction, z-component
2D, 3D
aUnit2x
Second reference direction, x-component
2D, 3D
aUnit2y
Second reference direction, y-component
2D, 3D
aUnit2z
Second reference direction, z-component
2D, 3D
bx
Primitive vector, x-component
2D
by
Primitive vector, y-component
2D
bz
Primitive vector, z-component
2D
b1x
First primitive vector, x-component
3D
b1y
First primitive vector, y-component
3D
b1z
First primitive vector, z-component
3D
b2x
Second primitive vector, x-component
3D
b2y
Second primitive vector, y-component
3D
b2z
Second primitive vector, z-component
3D
Gx
Reciprocal lattice vector, x-component
2D
Gy
Reciprocal lattice vector, y-component
2D
Gz
Reciprocal lattice vector, z-component
2D
G1x
First reciprocal lattice vector, x-component
3D
G1y
First reciprocal lattice vector, y-component
3D
G1z
First reciprocal lattice vector, z-component
3D
G2x
Second reciprocal lattice vector, x-component
3D
G2y
Second reciprocal lattice vector, y-component
3D
G2z
Second reciprocal lattice vector, z-component
3D
When modeling periodic structures only using Port features, similar variables for reference directions, axes, primitive cell vectors, and reciprocal lattice vectors are defined. However, their definitions and names are slightly different from the ones discussed here. For more information, see Additional Variables for Periodic Structure Calculations.
Additional variables, related to the plane-wave modes used in periodic structure calculations, are described in
•
S-Parameter Variables
•
Jones Vector Variables
To learn how to use the Periodic Structure node to solve driven 3D periodic problems, see Hexagonal Grating: Application Library path Wave_Optics_Module/Gratings_and_Metamaterials/hexagonal_grating.
For a 2D driven periodic problem, see Plasmonic Wire Grating: Application Library path Wave_Optics_Module/Gratings_and_Metamaterials/plasmonic_wire_grating.
For a non-driven example, see Band-Gap Analysis of a Photonic Crystal: Application Library path Wave_Optics_Module/Gratings_and_Metamaterials/bandgap_photonic_crystal.