# Far field projections - Power integration on hemisphere

The section explains how to integrate the far fields on the default hemispherical surface. This is typically done to calculate the amount of far field power within some range of angles.

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.

## Power Integrals

In general, we want to integrate power over a given solid angle in the far field. There are 2 ways this can be done

- We integrate the fraction of total electric field intensity (|E|2) over the solid angle that we are interested in, and multiply by the normalized power transmission through the monitor in the near field.

$$far\_field\_fraction=\frac{\int_{cone}\mid\overrightarrow{E}\mid^2}{\int_{hemisphere}\mid\overrightarrow{E}\mid^2}\\Power\_norm=far\_field\_fraction*near\_field\_power\_transmission$$

- We calculate the Poynting vector in the far field and integrate the power over a given solid angle. We then normalize to the original source power. In the far field, it is easy to calculate the normal component of the Poynting vector directly from the Electric fields by using the plane wave relationships between E and H. We know the tangential components the Poynting vector are zero for a plane wave.

$$Poynting_\bot=n\sqrt{\frac{\epsilon_0}{\mu_0}}\mid\overrightarrow{E}\mid^2\\Power=\frac{1}{2}\int_{cone}real(Poynting_\bot)\\Power\_norm=\frac{Power}{SourcePower}$$

Both methods will give the same result.

The script file will calculate the normalized power in a given cone around the z-axis defined by a half-angle. It will do the calculation by both methods described above. The desired half angle can be set in the first executed line of the script file

# choose the half angle over which we will integrate

half_angle = 30; #in degrees

We expect that using a half angle of 30 degrees will account for 50% of the power. When we run with 30 degrees, we find have the following output

> solver_far_field2;

The half angle is: 30 degrees at (theta,phi)=(0,0)

The normalized transmission by Method 1 is: 45.6613 %

The normalized transmission by Method 2 is: 44.5107 %

If we try several angles we find that

Cone half angle (degrees) | Normalized transmission in cone by method 1 (%) | Normalized transmission in cone by method 2 (%) |
---|---|---|

29 |
38.4055 |
37.4377 |

30 |
45.6613 |
44.5107 |

31 |
54.7707 |
53.3905 |

As expected, we find that half the power is radiated within a half-angle cone of 30 degrees around the z-axis.

NOTE: Far field integration The function farfield3dintegrate makes integrating far field data very easy. |

NOTE: Integration with non-default far field refractive index. Additional normalization is required when using a non-default far field refractive index. See the far field refractive index page or contact Lumerical support for additional information. |