When using finite size beams in linear systems, we typically normalize results involving power to the power spectrum of the source. When using plane waves, which in principle have infinite power, we need to consider the scattering and absorption cross sections instead.

The cross section, s, is defined such the scattered (or absorbed) power in Watts, P, is given by$$ P = \sigma I $$

where, \( I \) is the source intensity in Watts/m2 and the cross section has units of m2. In two dimensional simulations, which represent a structure which is infinite along the z axis, we generalize P to represent the power scattered per unit length, and the cross section has units of m.

## Source power vs source intensity

By default, we normalize most power results to the power of the source. For example, the script command 'transmission' will calculate the power flux through a monitor surface and normalize to the source power, returning a dimensionless quantity. To normalize instead to the source intensity, we can use the script commands

sigma = transmission(f) * sourcepower(f) / sourceintensity(f);

where f is a vector of frequency points. The result sigma will have units of m2 from a three dimensional simulation, and m from a two dimensional simulation. For plane waves and TFSF sources, the power of the source ('sourcepower' function) is simply the intensity of the source integrated over the area of the source injection plane. Normalizing to the source intensity is particularly important for non-periodic simulations (typically involving the TFSF source), since the source power depends on the (arbitrary) size of the source as setup by the user. For more information about TFSF sources specifically, please see the Power Normalization section of TFSF tips for more details.

## Angles of incidence

We have to be extremely careful about the definition of the source intensity when the source is at non normal incidence. The source intensity as returned by the script command 'sourceintensity' is calculated by integrating the power ** normal ** to the injection plane of the source. If instead, you want to normalize to the source intensity of the beam as calculated in the plane normal to the direction of propagation of the beam you need an additional factor of cos(q) where q is the nominal source angle as specified in the source properties. (Please note that this angle q should not be frequency dependent.) For example, if you download the Mie scattering example in 3D and modify the source angle to have theta of 30 degrees, you should see something like this in the yz view:

After running the simulation, we can run the usual analysis and we expect to see good agreement with the theoretical results since the scattering and absorption cross sections should not depend on angle of incidence for a sphere. Instead we will see this comparison:

The discrepancy is mainly due to the fact that our calculation defines the source intensity, I(q) with respect to the y-normal injection plane of the source even though the source angle q is 30 degrees. The theory calculates sigma using I0, which is independent of the angle of the source. Since we know that \(I(q) = I_0 \cos(q)\), we can easily modify our script by adding the lines:

theta = 30 * pi/180; # the nominal source angle is 30 degrees

Qscat = Qscat * cos(theta);

Qabs = Qabs * cos(theta);

immediately after calculating the cross section. (Please note that q is the nominal source angle and does not have to be corrected for wavelength.) We then see the results below, which are as accurate as the normal incidence results on this 5nm mesh, and converge nicely as the mesh size becomes smaller.

Note: The Mie scattering example considered here is a very particular case where the dependence of the angle of injection on wavelength does not affect the results. This is a consequence of the cross section of a sphere being independent of the injection angle. For other geometries without this symmetry, it is necessary to bear in mind that the TFSF source has the same issues as the plane wave source for Plane waves - Angled injection . This problem must be considered in addition to the normalization correction described here. |

## Periodic structures

In periodic structures, simulated with periodic or Bloch boundary conditions, the source power is integrated over one unit cell. This makes it possible to discuss quantities such as reflected power and transmitted power as fractions of the source power, as we do with finite sized beams. However, we should remember that this can be easily converted into a cross section. For example, a periodic structure with a reflection of 25% simply means that the reflection cross section is 25% of the unit cell area.

## Structures on multi-layer stacks or substrates

We must pay special attention to the meaning of the scattering and absorption cross sections when the particle is on or in a substrate or multi-layer stack. If the substrate is absorbing, then the box that measures the absorption cross section of the particle must surround only the particle. If that is not possible, then the loss per unit volume must be integrated only over the particle volume. The scattering cross section is more subtle: since the incident field is defined as the field that would exist in the multi-layer in the absence of the defect, the scattering cross section will not include any power from the specular reflection or transmission. This is ideal for detecting small scattering cross sections on surfaces that have large specular reflections. However, in the precise direction of the specular reflection it will not be possible to directly measure the simulated result since experimentally the specular reflection will dominate the small scattering from the defect. Please see the TFSF tips section for an example of setting up this type of simulation.

Note : Since BFAST assumes periodic structure, users should not use BFAST source at an angle to replace TFSF to get oblique incidence result. |