# Divergence of Poynting vector

This section describes how to calculate the absorption per unit volume from the divergence of the Poynting vector. Divergence calculations tend to be very sensitive to numerical problems, which makes this technique inferior to the other techniques described in this section.

The loss per unit volume can be calculated from the Poynting vector by

$$\frac{dL}{dV} = \frac{1}{2} real(\nabla \cdot P)$$

Consider the example file usr_absorption_divergence.fsp. The simulation consists of a gold nanoparticle in a background material that has an index of 1.5 + i0.05. A TFSF source is used to excite the simulation and a 3D power monitor is used to measure the Poynting vector.

We want to measure the power absorbed in the gold nanoparticle. If the background was not absorbing, a box of 2D power monitors around the particle could be used to measure the total absorbed power. In this case, since the background is also absorbing, the box technique will not work. Instead, we will measure the Poynting vector as a function of (x,y,z), and calculate the loss with the above formula. The loss per volume can be integrated over the volume of the gold sphere, giving the total power absorbed by the gold.

Load the file usr_absorption_divergence.fsp, and run the simulation. When the simulation is complete, run the script usr_absorption_divergence.lsf. It will calculate the total power absorbed within the box of monitors using two techniques. First, with a box of 2D monitors measuring the transmission through each surface. Second, by calculating the loss per unit volume, then integrating over the entire volume of the box. Both techniques give 48% absorption.

Next, the loss within the gold particle is calculated by integrating the loss over the volume of the sphere. It is quite small, only 0.1% of the total power. This seems reasonable after looking at a slice of the loss per unit volume in the XY plane. There is only a small amount of absorption in the particle very close to its surface. The matching slice of the integration filter is also shown. The power absorbed in any arbitrary volume can be calculated by modifying the integration filter.

Note: 3D monitors can require large amounts of memory and computation time. Therefore, it is best to keep them as small as possible. Similarly, keep the number of frequency points small, and only record the field components that are necessary. In this example, only the Poynting vector was saved, not the E and H field components. |