Incoherent unpolarized dipole

To simulate an incoherent unpolarized dipole source we need to perform 3 simulations. In each simulation, there must be one dipole that is orthogonal to the dipoles in the other simulations. The fields from each simulation can them be added incoherently.

Solvers: FDTD, varFDTD

See also
Sources
OLEDs methodology
FDTD and coherence
Incoherent dipole
Unpolarized beam

In practice, this means that we simulate a dipole oriented along the x, y and z axes respectively in each simulation. To obtain the unpolarized fields, we can simply sum the results incoherently. This means that

The quantity \( <|E|^2> \) refers to the time averaged electric field intensity of an unpolarized dipole source, or a large number of randomly oriented dipoles in a spatial volume much smaller than the wavelength. For a comparison with analytical results and a practical application of this approach, in both the near field and the far field, please see the OLEDs application example. For a proof of the principle, please see below.

Derivation

An unpolarized dipole source is created by a large number of incoherent dipole emitters contained in a small spatial volume that have a random orientation. It can also be created by a single dipole that is randomly re-oriented every correlation time such that all possible orientations are equally sampled on time scales typical of photodetectors.

To calculate the field distribution of an incoherent dipole, we need to average over all possible dipole orientations. The incoherent electric field intensity at position r is given by:

\[ <{|\vec{E}(\vec{r})|}^2>= \frac{1}{4\pi} \int {|\vec{E}(r,\theta,\varphi)|}^2 sin\theta \ d\theta d\varphi \]

where \(\vec{E}(r,\theta,\varphi)\) represents the electric field created at position \(r\) by a dipole (at position \(r0\)) with an orientation given by the spherical angles \(\theta\) and \(\varphi\).

A dipole of any orientation can be written as

\[\vec{p}(\theta,\varphi) = \vec{p}(\frac{\pi}{2},0)sin\theta \ cos\varphi + \vec{p}(\frac{\pi}{2},\frac{\pi}{2})sin\theta \ sin\varphi + \vec{p}(0,0) \cos \theta \]

Since Maxwell’s equations are linear, we can write the electric field from a dipole with orientation \(\theta\) and \(\varphi\) as

\[\vec{E}(\vec{r},\theta,\varphi) = \vec{E}(\vec{r},\frac{\pi}{2},0)sin\theta \ cos\varphi + \vec{E}(\vec{r},\frac{\pi}{2},\frac{\pi}{2})sin\theta \ sin\varphi + \vec{E}(\vec{r},0,0) \cos \theta \]

Substituting this into the equation for the field intensity of an incoherent dipole gives

\[ <{|\vec{E}(\vec{r})|}^2>= \frac{1}{4\pi} \int {|\vec{E}(\vec{r},\frac{\pi}{2},0)sin\theta \ cos\varphi+ \vec{E}(\vec{r},\frac{\pi}{2},\frac{\pi}{2})sin\theta \ sin\varphi + \vec{E}(\vec{r},0,0) \cos \theta|}^2 sin\theta \ d\theta d\varphi \]

It is easy (if tedious) to simply this integral with the help of the identities provided at the bottom of this page. In the end, we get

\[ <{|\vec{E}(\vec{r})|}^2>= \frac{1}{3} ( |\vec{E}(\vec{r},\frac{\pi}{2},0)|^2+ |\vec{E}(\vec{r},\frac{\pi}{2},\frac{\pi}{2})|^2 + |\vec{E}(\vec{r},0,0)|^2 )\]

Therefore, the field distribution of an incoherent, unpolarized dipole is simply the average of the field distribution of three orthogonal dipoles. The same result can easily be shown for the \(H\) field intensity or the Poynting vector.

The following identities are required to simplify the above integral:

\[\int_0^{2\pi} \sin\varphi \ d\varphi = 0 \qquad \int_0^{2\pi} \cos\varphi \ d\varphi = 0\]
\[\int_0^{2\pi} \sin^2\varphi \ d\varphi = \pi \qquad \int_0^{2\pi} \cos^2\varphi \ d\varphi = \pi\]
\[\int_0^{\pi} \sin^3\theta \ d\theta = \frac{4}{3} \qquad \int_0^{\pi} \cos^2\theta \ d\theta = \frac{4}{3}\]
\[\int_0^{2\pi} \cos\varphi \sin\varphi \ d\varphi = 0 \qquad \int_0^{\pi} \sin\theta \cos^2\theta \ d\theta = \frac{2}{3}\]
\[\int_0^{2\pi}d\varphi = 2\pi \]