Symmetric and anti-symmetric BCs in FDTD and MODE

Symmetry boundary conditions can be used whenever the EM fields have a plane of symmetry through the middle of the simulation region. By taking advantage of this symmetry, the simulation volume and time can be reduced by factors of 2, 4 or 8.

This topic describes the difference between symmetric and anti-symmetric boundary conditions, and how to select the appropriate boundary for your simulation. 

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Fields at the symmetry boundary

When the EM fields have a plane of symmetry, some field components must be zero at the plane of symmetry. Symmetry boundary conditions are implemented by forcing the appropriate field components to zero. The following table lists the field components that are zero for each symmetry option.

The non-zero field components are shown in the following figure. Note that the blue arrows are electric field and the green arrows are magnetic field.

Reflection symmetry rules

The electric and magnetic fields will obey certain symmetry rules with respect to reflections through the plane of symmetry. The reflection symmetry rules are shown in the figure below.

The above symmetry rules are helpful when determining the symmetry of modes found with the mode source, and resonant modes of cavities. The following figure shows the Real part of Ex and Ey of a mode that was calculated with the Mode source. The direction of propagation is +Z. Notice that Ex has the same sign on each half of the simulation region. Ey has the opposite sign on each half. Ey also goes through zero along the plane of symmetry.

Using the above symmetry rules, we can determine that there is a plane of anti-symmetry at X=0, and a plane of symmetry at Y=0. Therefore, the x min boundary should be set to anti-symmetric and the y-min boundary condition should be set to symmetric.

Symmetric Boundary Conditions for Periodic Structures

If the EM fields through a periodic structure have a plane of symmetry or anti-symmetry in the middle of a period of the structure, then set the boundary conditions as follows:

1) select the option allow symmetry on all boundary conditions

2) set the minimum and maximum boundary conditions to be either symmetric or anti-symmetric depending on the symmetry rules given above (in practice the boundary conditions are usually either both symmetric or both anti-symmetric).

Referring to the example below, changing from Setting A to Setting B will preserve periodicity while reducing the computation time needed by about 4x. Again this only applies if the structure and fields are BOTH symmetric and periodic.

Selecting the correct symmetry option with sources

This section applies for FDTD and the MODE varFDTD solver. Most sources in these solvers show their Electric or Magnetic polarization with colored arrows, where the electric field polarization is shown in blue and magnetic polarization is green. The following image shows an electric dipole on the left and a magnetic dipole on the right. The sources use a color scheme Electric field polarization is Blue

Symmetry boundary conditions use a similar color scheme:

• Symmetric BC are Blue
• Anti-Symmetric are Green.

 The following figure shows how to select the correct symmetry condition based on the source polarization.

• If the source polarization is tangential to the plane of symmetry, select the symmetry option with the same color.
• If the source polarization is normal to the plane of symmetry, select the symmetry option with the opposite color.

Examples

In the following example with the triangular structure, there is one plane of symmetry in X. The source polarization (Blue) is tangential to the X boundary. Therefore, we use Symmetric for the X min boundary condition. This simulation will run 2x faster than the equivalent simulation without any symmetry settings. Notice that the blue arrow of the source is along the edge of the region with the same color.

 This next example shows a simulation with two planes of symmetry (in X and Y) The source polarization (Blue) is normal to the X boundary, and tangential to the Y boundary. Therefore, use Anti-Symmetric for the X min boundary and Symmetric for the Y min boundary. The X and Y max boundary condition do not need to be modified. This simulation will run 4x faster than the equivalent simulation without symmetry boundary conditions. Notice that the blue arrow of the source is along the edge of the region with the same color, and sticking into the region with opposite color following the rules outlined above. 

Notes

Using the wrong symmetry option

When using symmetry boundary conditions, it is easy to select the wrong symmetry option. There will be no warning message if   the wrong symmetry option is selected, but the simulation results will be completely wrong. The easiest way to confirm that you   have the correct option is by re-running the simulation without symmetry boundary conditions. Both simulations should give the same results. If they don't, you may have selected the wrong symmetry option.

Simulation span

Do not change the size of the simulation region when using symmetry. If symmetry boundary conditions are selected, half of the   simulation region will automatically be shaded in blue or green, indicating the portion of the simulation region that will not be   directly simulated.

Unfolding data

Script commands like getdata, getelectric or getmagnetic automatically unfold the field data according to the symmetry   rules of the boundary condition. This makes it easy to create an image monitor the data over the entire simulation region, even   though only half the simulation region was actually simulated. However, there will be no data recorded by the monitor if it is   entirely in the shaded region. Some analysis group may not work when symmetry is applied but some of them contain script   commands to properly unfold the field data.