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add note describing error introduced by finite truncation of near-fie…
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…ld surface and possible workaround
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oskooi committed Sep 22, 2023
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Expand Up @@ -384,7 +384,9 @@ The total flux computed using the near and far fields is shown to be in close ag
total_flux:, 643.65058 (near), 597.72713 (far), 0.07135 (error)
```

The error decreases with increasing (1) grid resolution, (2) runtime, and (3) number of angular grid points.
The error decreases with increasing (1) grid resolution, (2) runtime, and (3) number of angular grid points. However, this only applies to a *closed* near-field surface which is not the case in this example. This is because the ground plane, which extends to infinity, contains $H_r$ and $H_\phi$ fields on its surface which are not zero (unlike the $E_r$ and $E_phi$ fields). These magnetic fields produce equivalent currents which radiate into the far field. The PML in the $r$ direction does not mitigate this effect.

Because the near-field surface actually extends to infinity in the $r$ direction, one approach to reducing the error introduced by its finite truncation would be to simply make the cell size in the $r$ direction larger (the parameter `L` in the script below). Another option which would remove this error entirely would be to simulate the same structure using a closed surface by removing the ground plane and duplicating the structure and source below the $z = 0$ plane. This is known as the method of images. See [Tutorial/Antenna above a Perfect Electric Conductor Ground Plane ](#antenna-above-a-perfect-electric-conductor-ground-plane) for a demonstration of this approach.

The simulation script is in [examples/disc_radiation_pattern.py](https://github.com/NanoComp/meep/blob/master/python/examples/disc_radiation_pattern.py).

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