Electromagnetic power is carried by the Poynting vector, so to calculate power flow through a monitor on needs to integrate the flux of this vector through the monitor. This could also be useful as an example of how to do other integrals involving monitor data. In particular, the script file shows a simple technique for integrating over an arbitrary area, such as a circle. The files in this section were created using FDTD Solutions, but a similar approach can be applied for MODE Solution’s propagator. The files in this section were created for the FDTD solver in Ansys Lumerical FDTD™, but a similar approach can be applied for the RCWA solver in FDTD, or the varFDTD solver in Ansys Lumerical MODE™.
Theory
The Poynting vector, \(\mathbf{P}\), is calculated by the cross product of the complex fields. After this calculation, integrating the vector field across a surface, such as that of a monitor, will yield the net transmitted power in Watts. Normalizing the transmitted power to the source emitted power is generally more useful than having the result in SI units of Watts.
$$
\begin{array}{c}
\mathbf{P}=\mathbf{E} \times \mathbf{H}^{*} \\
\text { Power }=\frac{1}{2} \int_{\text {Surface }} \quad \operatorname{Re}(\mathbf{P}) \cdot d \mathbf{s} \\
\text { Normalized Power }=\frac{\text { Power }}{\text { Source Power }}
\end{array}
$$
Performing the Calculation in FDTD
Frequency-domain monitors for the FDTD solver record the Poynting vector by default. Therefore, calculating power is simply a matter of integrating the Poynting vector over a surface.
The script function transmission can be used to directly calculate the total power transmitted through a power or profile monitor without explicit integration. However, to calculate the power transmitted through a portion of a monitor, you must integrate the Poynting vector over that region.
The following screenshot shows the layout of usr_integrate_poynting.fsp. A gaussian wave source emits light in the positive z direction. There is a thin gold layer with a hole in it that reflects much of the light. Monitors are placed above and below the gold layer to measure transmission and reflection.
Performing the Calculation in RCWA
The RCWA Field Monitor in the RCWA solver can be used to perform the same calculations as outlined for FDTD above, and can be used to obtain transmission inside objects in RCWA simulations. While the transmission script command is not supported in RCWA, the Poynting vector can be obtained by manually performing the cross product detailed in the Theory section of this article.
The field components obtained from the RCWA field Monitor follows the S- and P-polarizations directions. To transform these components into cartesian coordinates, you can use the Source Polarization section in the RCWA Solver Introduction article, which defines these polarizations in terms of cartesian unit vectors using incident angles \(\theta\) and \(\phi\).
Example: Integrating Over the Entire Monitor
Run the simulation, then run the script usr_integrate_poynting1.lsf. The script will first calculate the total transmission in two ways: using the built in transmission function, and manually integrating the Poynting vector. The two results should be exactly the same.
Example: Integrating Over a Portion of a Monitor
The script usr_integrate_poynting2.lsf can be used to integration the Poynting vector over a circular portion of the monitor. The script will create these two figures, showing the integration filter and the fraction of power passing through that region. Methods in this script can be applied in the same way to integrate power over a designated area for RCWA simulations.
This example script includes two other possible shapes: a rectangular region and an arbitrary polygon region, shown below.
Separating Contributions from Different Polarization
The script usr_integrate_poynting3.lsf can be used to separate the contribution from the field polarized in the Ex direction from the contribution from the Ey polarized fields. This can be roughly interpreted as measuring the fraction of power in each polarization. It is important to note that simply integrating one component of the Poynting vector can not always be interpreted in a simple way. This analysis should not be applied to your simulation without some careful thought.
In the simulation, notice that the source polarization angle is 30 degrees, which means most of the beam is polarized in the X direction, with a smaller fraction polarized in the Y direction. This is consistent with the results shown below, where approximately 2/3 of the power is X polarized and 1/3 is polarized in Y.