This section describes the "Assume structure is periodic" options of the far field projections tab.

As discussed in previous sections, far field projections are typically used for isolated structures rather than periodic arrays. However, it is possible to use the projection functions to calculate the approximate far field distribution of finite sized periodic arrays. For infinitely periodic structures, the grating functions should be used.

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. |

## Non-periodic projection

By default, the far field functions assume that the near fields are zero beyond the monitor boundary (as discussed in the Far field filtering section). This occurs even when using periodic boundary conditions. Physically, this acts like an aperture placed over the structure, only allowing radiation from one period of the structure to propagate to the far field. The aperture causes strong diffraction, which tends to make these projections not very useful.

See the image below. To simulate an infinite plane wave source (red) incident on a periodic array of structures (blue), we model a single period (orange) by using periodic boundary conditions. The default far field projection gives the far field distribution (yellow) from a single period. It is as if an aperture (black) only allows light from one period to propagate to the far field.

The final result is simply the projection from a single period:

$$\overrightarrow{E}^{far}=\overrightarrow{E_0}^{far}$$

The following figure shows the reflected far field |E|^2 distribution from the FDTD Blaze grating application example when using the default far field projection settings. To reproduce this figure, run blaze_grating.fsp which you can download from the Blazed Grating Application Gallery example. Using the Analysis window to plot the far field projection of the reflection monitor. Don't select the 'assume periodic' option.

For structures with a finite number of periods, there are two cases:

- The device is uniformly illuminated by the source. In other words, the source beam is much larger than the finite sized device. 'Top hat' illumination is appropriate for this case.
- The device is much larger than the source beam. In this case, the source is only illuminating a portion of the device and 'Gaussian' illumination should be used.

## Top hat illumination

You will notice that as the number of periods is increased, the far field distributions will become sharper. Diffraction effects are lessened when more periods are included.

The final result is the phase correct sum of m periods, where kbloch=0 for normal incidence,

$$\overrightarrow{E}^{far}=\overrightarrow{E_0}^{far}\sum_{m-2}^2e^{i(k_{inplane}-k_{bloch})ma}$$

If we calculate the far field projection of the blaze structure assuming 10 periods with top hat illumination, notice how the features become sharper than the single period illumination. More periods will make the features even sharper.

## Gaussian illumination

The difference between Top hat and Gaussian illumination tends to be small. Projections using Gaussian illumination are smoother because the Gaussian weighting function is smoother than the top hat function.

The final result is the phase correct sum of m periods with a Gaussian weighting, where kbloch=0 for normal incidence,

$$\overrightarrow{E}^{far}=\overrightarrow{E_0}^{far}\sum_{m=-\infty}^{\infty}e^{i(k_{inplane}-k_{bloch})ma}e^{-\left(\frac{m}{5/2}\right)^2}$$

The Gaussian illumination projection looks very similar to the top hat illumination.

## Infinite periodic structures

As the number of periods is increased, the sharp features in the above figures become delta functions. When studying true gratings, you should consider using the Grating projections (GP) functions , rather than the far field projections.

Note: Periodic structures - far field filter The far field filter option should not be used for periodic structures. Set it to zero when using the 'assume periodic' option. |