On this page, polarization incoherence and methods to simulate it are discussed.
Solvers
FDTD
VarFDTD
See also
 Incoherent unpolarized dipole
 Understanding coherence in FDTD simulations
 Understanding temporal incoherence in FDTD
Problems with direct simulation
In a similar manner to temporal phase incoherence, the polarization of a beam or the orientation of a dipole source depends on time. In the case of a polarized beam we have
\begin{equation}
\vec{E}(t)=\vec{u}(t) E_{0} \cos (\omega t)
\end{equation}
where the unit vector u(t) defines the beam polarization and varies on a time scale τc >> T.
In the case of a dipole source, we have
\begin{equation}
\vec{p}(t)=\vec{u}(t) p_{0} \cos (\omega t)
\end{equation}
where the unit vector u(t) defines the dipole source polarization and varies on a time scale τc >> T.
In both cases, the time scale for the variation of the polarization is much larger than the optical cycle, making it unpractical to simulate the statistics of temporal polarization incoherence with FDTD.
Recommended simulation method
FDTD simulations have well defined polarization. For a beam, unpolarized results are obtained by adding the results of 2 orthogonal polarization simulations incoherently using the equation
\begin{equation}
\left\langleE^{2}\right\rangle=\frac{1}{2}\left\vec{E}_{s}\right^{2}+\frac{1}{2}\left\vec{E}_{p}\right^{2}
\end{equation}
The derivation of this equation can be found on the Unpolarized beam page.
In the case of a dipole, the results of the three orthogonal polarizations can be added incoherently using the equation
\begin{equation}
\left\langleE^{2}\right\rangle=\frac{1}{3}\vec{E} p x^{2}+\frac{1}{3}\vec{E} p y^{2}+\frac{1}{3}\vec{E} p z^{2}
\end{equation}
Polarization incoherence examples
To simulate an unpolarized beam or plane wave source, we need to perform 2 simulations with orthogonally polarized beams. The fields from each simulation can then be added incoherently.
Solvers
FDTD
VarFDTD
Associated files
usr_unpolarized_beam.lsf (1.4 KB)
usr_unpolarized_beam.fsp (16.9 KB)
See also
Introduction
To simulate an unpolarized beam or plane wave source, two simulations with orthogonal polarizations will be required. Results for an unpolarized source can then be calculated by incoherently summing results from the two polarized simulations according to the formula below. See the Derivation section at the bottom of this page for details.
$$\left\langleE^{2}\right\rangle=\frac{1}{2}\left\vec{E}_{s}\right^{2}+\frac{1}{2}\left\vec{E}_{p}\right^{2}$$
In practice, this means that we simulate a beam with a “polarization angle” of 0 and then a beam with a “polarization angle of 90”, as shown below. The quantity /( \left\langleE^{2}\right\rangle /) refers to the time averaged electric field intensity of an unpolarized beam source.
$$ \left\langleE^{2}\right\rangle=\frac{1}{2}\vec{E}_ 0^{2}+\frac{1}{2}\vec{E} _{90}^{2} $$
Example
The files usr_unpolarized_beam.fsp and usr_unpolarized_beam.lsf can be used to reproduce the following results.
Screenshot of the simulation
A focused beam propagating at an angle of 60 degrees. The beam is incident upon a surface with an index of 2.
 The angle of incidence is very close to Brewster’s angle for this structure. Therefore, we expect quite different behavior for the two polarizations.

P polarized light will have higher transmission (low reflection)

S polarized light will have lower transmission (high reflection)

Field profile from a P polarized beam
Notice that the transmitted fields have an amplitude of about 0.3 (highlighted with a green circle). 
Field profile from a S polarized beam
Notice that the transmitted fields have an amplitude of about 0.2 (highlighted with a green circle). 
Field profile from an unpolarized beam
The field profile is simply the average of the P and S polarized beams (highlighted with a green circle). 
Fraction of power transmitted into the substrate
As expected, the P polarized beam has a higher transmission. Once again, the unpolarized transmission is simply the average of the transmissions from the P and S simulations.Transmission (P polarization): 98%
Transmission (S polarization): 72%
Transmission (unpolarized): 85%
Derivation
To calculate the response of a system to an unpolarized beam, we need to average over all possible input polarizations:
Due to the linearity of Maxwell’s equations, we can represent the electric field of an arbitrarily polarized incoming beam as a sum of two orthogonal polarizations:
Therefore, the integral can be computed as follows:
The following identities are required to simplify the above integral: