The near to far field projection functions allow the EM fields to be calculated anywhere in the far field. The functions require the spatial near field EM data (typically from an FDTD simulation) and the desired far field spatial locations.

A simple way to understand far field projections is to view them as a decomposition of the near field data using a set of plane waves propagating at different angles as the basis for the decomposition. The end result is that the far field projections functions provide a straightforward, accurate, and numerically efficient method for calculating the EM fields anywhere in the intermediate and far field regions.

If your structure is periodic, consider using the related grating projections. If you are using DGTD, see Far field projections in DGTD overview.

## Near to far field projection simple example

The example below shows how the far field projection can be used to see the angular distribution of reflected light from a small surface feature. A beam propagating downward at an angle reflects off the surface and bump. The reflected light is recorded in the near field slightly above the surface directly from the FDTD simulation. After the simulation is complete, the far field projection can be used to project the reflected radiation to an arbitrary location. In this case, the fields are calculated on a hemispherical surface located 1m away. The final result is an image of the field intensity on the hemisphere, as viewed from above. The effects of the misalignment of the spot can clearly be seen from the far field projection results.

**Near field simulation results**

A) Screenshot of simulation, B) Reflected near field |E|^2 data on XY plane above structure

**Result from far field projection**

A) Hemispherical surface where far field is calculated (1m radius), B) View of hemisphere, looking down

Far field |E|^2 on hemisphere

## Understanding when far field projections can be used

### Requirements:

- The EM fields must be known everywhere on a plane or on a closed surface.
- The plane or closed surface must be in a single homogeneous material.
- The material must extend out to infinity. There can be no additional structures (or sources) beyond this plane or closed surface.
- The far field projection functions assume that the EM fields are zero beyond the edge of the monitor. This effectively truncates the near fields at the monitor edge. Use spatial filtering to minimize this issue.
- The material can be dispersive as long as the loss is negligible, ie k<<n. However, the loss is not taken into account for the projection to the far field.

There are two types of situations where these conditions are satisfied:

### Case 1 - fields known on a plane

When fields are known on a single surface, the projection functions can be used to calculate the fields anywhere beyond that surface. In the above screenshot, a monitor is located above the source. This monitor records all of the reflected fields. There are no additional structures or sources of light above the source. In this situation, the far field projections can be used to calculate the EM fields anywhere above the monitor plane.

### Case 2 - fields known on a closed surface

When the fields are known on a closed surface, the projection functions can be used to calculate the fields anywhere beyond that surface. In the above screenshot, a monitor surrounds the source and scattering particle. The monitor records all reflected fields. There are no additional structures or sources of light above the source. In this situation, the far field projections can be used to calculate the EM fields anywhere outside of the monitor object.

### Related publications

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

### See also

farfield3d script command, Simple example, Direction unit vector coordinates, Far field filter. Field polarization, Power integration on hemisphere, Changing the far field refractive index, Projections from a closed volume, Grating projections, Far field projections in DGTD overview