FDTD is a fundamentally coherent simulation method. Below, we discuss three types of incoherence that are relevant for FDTD simulations, and the difficulties in direct simulation of these effects. In many physical processes, we have a combination of all types of incoherence.

The following sections provide three examples that show how to model various incoherent effects with FDTD and MODE' propagator.

## Temporal phase incoherence

The phase, j, of the light shifts randomly over time, on a time scale tc. For light near visible wavelengths, we have

$$ \vec{E}(t)=\vec{E}_{0} \cos (\omega t+\varphi(t)) $$

$$ T=\frac{2 \pi}{\omega} \approx 10^{-15} S $$

$$ \tau_{c} \approx 10^{-11} s $$

It is not practical to simulate the statistics of temporal phase incoherence because tc >> T.

## Spatial incoherence

A spatially incoherent source could be, for example, an ensemble of dipole emitters spread over a given volume. Each dipole is independent of its neighbors, and emits radiation with a random phase that varies on a time scale on the order of tc. At any given instant, the relative phases of all the dipoles are fixed, however, on a time scale of tc the relative phases of the dipoles change in a completely random fashion. The direct simulation of spatial incoherence requires simulations for very long periods of time or a large amount of ensemble averaging with one FDTD simulation per ensemble. Therefore it is not practical to simulate spatial incoherence directly in most cases.

## Polarization incoherence

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 tc >> 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 tc >> T.

In both cases, the time scale for the variation of the polarization is much larger then the optical cycle, making it unpractical to simulate the statistics of temporal polarization incoherence with FDTD.

## How to simulate incoherent phenomena with FDTD

Direct simulation of incoherent effects requires simulations that last many times the correlation time (tc) which is typically many orders of magnitude larger than the optical cycle. In some cases, ensemble averaging can be used but this requires a very large number of simulations. Therefore to simulate the physical conditions required for incoherence means simulation times that are far too long to be practical in most cases.

In general, a more efficient approach is to calculate the statistical averaging analytically rather than performing lengthy or numerous simulations. In general, this requires a small number of simulations and some post-processing of the data. In the following sections, we explain how to perform this post-processing in a number of examples that can be applied to most simulations of incoherent processes.

### See also