How are results normalized?

Date	Category
2023-10-24 15:21:50	Sources

In many cases, Tidy3D simulations can be run, and well-normalized results can be obtained without normalizing/empty runs. This is because care is taken internally to normalize the injected power, as well as the output results, in a meaningful way. To understand this, there are two separate normalizations that happen, outlined below. Both are discussed with respect to frequency-domain results, as those are the most commonly used.

Source spectrum normalization

Every source has a spectrum associated to its particular time dependence that is imprinted on the fields injected in the simulation. Usually, this is somewhat arbitrary, and it is most convenient to take it out of the frequency-domain results. By default, after a run, Tidy3D normalizes all frequency-domain results by the spectrum of the first source in the list of sources in the simulation. This choice can be modified using the Simulation.normalize_index attribute, or normalization can be turned off by setting that to None. Results can even be renormalized after the simulation run using SimulationData.renormalize(). If multiple sources are used, but they all have the same time dependence, the default normalization is still meaningful. However, if different sources have a different time dependence, then it may not be possible to obtain well-normalized results without a normalizing run.

This type of normalization is applied directly to the frequency-domain results. The custom pulse amplitude and phase defined in SourceTime.amplitude and SourceTime.phase, respectively, are not normalized out. This gives the user control over a (complex) prefactor that can be applied to scale any source. Additionally, the power injected by each type of source may have some special normalization, as outlined below.

Source power normalization

Source power normalization is applied depending on the source type. In the cases where normalization is applied, the actual injected power may differ slightly from what is described below due to finite grid effects. The normalization should become exact with sufficiently high resolution. That said, in most cases the error is negligible even at default resolution.

The injected power values described below assume that the source spectrum normalization has also been applied.

• PointDipole: Normalization is such that the power injected by the source in a homogeneous material of refractive index \(n\) at frequency \(\omega = 2\pi f\) is given by

  \[\frac{\omega^2}{12\pi}\frac{\mu_0 n}{c}.\]

• UniformCurrentSource: No extra normalization applied.

• CustomFieldSource: No extra normalization applied.

• ModeSource, PlaneWave, GaussianBeam, AstigmaticGaussianBeam: Normalized to inject 1W power at every frequency. If supplied SourceTime.num_freqs is 1, this normalization is only exact at the central frequency of the associated SourceTime pulse but should still be very close to 1W at nearby frequencies too. Increasing num_freqs can be used to make sure the normalization works well for a broadband source. The correct usage for a PlaneWave source is to span the whole simulation domain for a simulation with periodic (or Bloch) boundaries, in which case the normalization of this technically infinite source is equivalent to 1W per unit cell. For the other sources which have a finite extent, the normalization is correct provided that the source profile decays by the boundaries of the source plane. Verifying that this is the case is always advised, as otherwise results may be spurious beyond just the normalization (numerical artifacts will be present at the source boundary).

• TFSFSource: Normalized to inject \(1W/μm^{2}\) in the direction of the source injection axis. This is convenient for computing scattering and absorption cross-sections without the need for additional normalization. Note that for angled incidence, a factor of \(1/\cos(\theta)\) needs to be applied to convert to the power carried by the plane wave in the propagation direction, which is at an angle \(\theta\) with respect to the injection axis. Note also that when the source spans the entire simulation domain with periodic or Bloch boundaries, the conversion between the normalization of a TFSFSource and a PlaneWave is just the area of the simulation domain in the plane normal to the injection axis.