  # Polarization incoherence

On this page, polarization incoherence and methods to simulate it are discussed.  FDTD
VarFDTD

### 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 an unpolarized beam we have

$\vec{E}(t) = \vec{u}(t)E_0 cos(\omega t)$

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

$\vec{p}(t) = \vec{u}(t)p_0 cos(\omega t)$

where the unit vector u(t) defines the dipole source polarization and varies on a time scale τc >> T.
In the case of a dipole source, we have

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

$\Big\langle|E|^2 \Big \rangle = \frac{1}{2}|\vec{E}_s|^2 + \frac{1}{2} |\vec{E}_p|^2$

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

$\Big\langle|E|^2 \Big \rangle = \frac{1}{3}|\vec{E}px|^2 + \frac{1}{3} |\vec{E}py|^2 + \frac{1}{3} |\vec{E}pz|^2$

### Polarization incoherence examples

For an examples on simulating polarization incoherence, see Unpolarized beam and Incoherent unpolarized dipole.