When the Periodic Structure feature is used, a single axis unit vector a is defined. This vector is parallel to the normal to the excitation port boundary. Instead, when Port features are used, two axis vectors, aR and aT, are defined, one for each port boundary. The axis vectors are parallel to the normal for each port boundary. The different unit cell axis vectors are summarized in Table 2-8.

Here, C represents any of the Cartesian coordinates, x, y, or z. The reflection side is the side (boundary) where there is an excitation port. At the transmission side, there is no excitation port.

To define the propagation directions for the plane waves, two unit vector reference axes are defined, a1 and a2. When a Periodic Structure feature is used, the two reference axes are defined for the whole periodic structure. The unit cell axis vector and the two reference axes vectors are related by

Here, the subscripts 00 indicate that this is the zeroth diffraction order, k is the wave number for the material in the domain adjacent to the port boundary, α1 is the elevation angle, measured from the unit cell axis, α2 is the azimuth angle, measured from the first reference axis, kT00 is the wave vector component tangential to the port boundary, β00 is the propagation constant, and n is the port boundary normal. Note that β00 is positive and n points out from the simulation domain. So, k00 is the wave vector for the outgoing wave.

When Port features are used, the reference axes are defined for each port boundary. So,

The wave vector is defined as for the Periodic Structure case above, but the unit cell axis a and the reference axes a1 and a2 are replaced by aR, a1R, and a2R or aT, a1T, and a2T, depending on the port side (reflection or transmission). The reference axis variables are summarized in Table 2-9.

The (primitive) unit cells, the periodic structures, are arranged in an infinite two-dimensional or one-dimensional lattice. So for a point r in this lattice, the neighborhood to the point r looks the same as the neighborhood to the translated point r + T, where T is a translation vector

Here, i and j are integers and b1 and b2 are the primitive vectors that define the unit cell.

The primitive cell vectors are most often aligned with two of the sides of the unit cell. However, for hexagonal unit cells, the primitive cell is a rhomboid. So, the primitive vectors don’t align with the hexagon sides. Note that the primitive cell vectors, b1 and b2, don’t need to be orthogonal, whereas the reference axes a1 and a2 are always orthogonal.

where G1 and G2 are the reciprocal lattice vectors for the unit cell and m and n are integers. The reciprocal lattice vectors are calculated from the primitive cell vectors as

The relation above defines the propagation angles α1mn and α2mn for the diffraction order with mode numbers m and n.

The discussion about the primitive cell and reciprocal lattice vectors above describes the procedure when Periodic Structure features are used. When Port features are used, the only difference to the procedure above is that there are different sets of vectors for the reflection and transmission sides of the periodic structure. The primitive cell and reciprocal lattice vector variables are summarized in Table 2-10.

The Port (of Periodic type), Diffraction Order, Orthogonal Polarization, and Periodic Port nodes also add additional global variables that describe the properties of the plane-wave diffraction orders, used in periodic structure simulations.

In the table above, K represents the port name and M and N are diffraction order mode numbers.