This page describes how to calculate fields outside of a closed surface using a box of monitors and the far field projection functions.

Lumerical provides many built in analysis groups in our object library. Please press this button to open the online library of analysis groups and select the far field category to see which analysis groups are available.

Note: The descriptions and examples of the far field projection calculation on the following pages are primarily intended for users of FDTD. For users interested in calculating far field projections with MODE, these descriptions are basically still correct, although some subtle differences do exist. |

Using the surface equivalence theorem, it is possible to show that fields radiated outside of a closed box by sources located inside the box can be determined exactly from the field components at the surface of the box. Since Maxwell's equations are linear, the fields outside of the box can be computed by calculating the far field projections for each surface of the monitor box and then summing the results.

The associated files for this section, the usr_farfield_angular.lsf script file and usr_farfield_symmetry.fsp simulation show how to use this method to obtain the electric field and intensity in the far field due to two point sources. For sake of simplicity this simulation only contains two dipoles surrounded by a box of monitors. However, this method also works when there are structures located in the monitor box. See, for example, the Mie 3D application example.

A perspective view of the simulation is shown below to the left. The yellow box in the image shows the edges of the monitors, which are grouped together in an analysis group. When the simulation has been run, the usr_farfield_angular.lsf script file can be used to run the analysis script and create the plot of the far field data shown below to the right. All of the far field projection script is contained in the analysis group, and the script file is used to run the analysis script plot the resulting data.

Note: Particle scattering with a substrate The 'box of monitor' far field projection is frequently used when studying particle scattering. This is fine, but it's important to remember the requirement that everything beyond the monitor box must be a single homogeneous material. When a substrate is present, this approach is not valid. Instead, the best way to calculate the far field scattering pattern is to use a single monitor, located above or below the particle (depending if you want scattering in the forward or backwards direction). You can then use the standard farfield3d function. When using a single monitor, it's important to make the simulation span large enough that most of the scattered light will pass through the monitor before hitting the PML absorbing boundary conditions. |

Note: Negative signs in front of far field projections when the normal to the monitor box points along the negative axis. In the case where no symmetry is used, the scat analysis group contained in the usr_farfield_symmetry.fsp simulation computes the far field projection from the sum of the far field projections from the six monitors which make up the monitor box. Notice the negative sign on some terms. E2_far = farfieldexact("x2",x,y,z) - farfieldexact("x1",x,y,z) The negative sign is required because the projection needs information about the projection surface normal. In the image below, the projection normal points outwards from the monitor box for the left monitor (x1) and right monitor (x2) are depicted with red arrows. However, the far field projection function does not know which monitor is on the left; it always assumes the monitor normal is pointing along the positive axis, as depicted with the yellow arrows. The negative sign in front of the far field projection for the x1 monitor in the equation above corrects for the fact that the yellow and red arrows point in opposite directions.
Only the farfieldexact, farfieldexact2d and farfieldexact3d commands allow projections from multiple monitors added to create a total far field projection. |

Note: Far field projections and symmetric/anti-symmetric FDTD region boundaries When symmetric or anti-symmetric boundary conditions are used in FDTD simulations, it is possible that a monitor will lie entirely outside of the simulation region. Although the monitor does not collect any data, it is possible to obtain the monitor data from other monitors in the simulation. In the usr_farfield_symmetry.fsp simulation, there are two monitors which lie outside of the simulation region, and the analysis object uses data from other monitors to obtain the contribution from the monitors which are not included in the simulation. |

Note: Far field half space analysis vs resolution and number of frequency points This calculation will take longer than the polar plot. In particular, the resolution and the number of frequency points can significantly affect the time it takes to computer the half space. Reduce either or both parameters can noticeable speed up the analysis. |

### Related publications

Allen Taflove, Computational Electromagnetics: The Finite-Difference Time-Domain Method. Boston: Artech House, (2005).