Calculating the radiated power from incoherent, isotropic point dipole sources

This article explains how to calculate the radiated power of an incoherent, isotropic dipole source in a dielectric half space. In time domain we necessarily inject coherent sources, but in reality most sources would be considered incoherent. Often time coherent effects will have no effect on the simulation results and the distinctions irrelevant; however, in certain situations such spontaneously emitting ensembles of atoms below lasing these effects need to be considered. These type of sources cannot be reproduced in a single simulation; however, the results are easily represented as ensemble average from a number of coherent dipole point source simulations.

Here we focus on an incoherent, isotropic dipole in a dielectric half-space, since we can compare the simulation results to an analytic solutions. Advanced should users can consider adapting these techniques to their own application.

It is shown that FDTD is easily able to calculate the extraction efficiency to better than 1%, and give very accurate angular scattering in the far field up to angles of 85 degrees.

Associated Files

Introduction
Light is generated in the active layer of an LED as the injected current of electrons and holes recombines to create photons. The photons are created by a process called spontaneous emission and each photon has a random direction, phase and polarization. While in principle, the exact treatment of this process must be described quantum mechanically in terms of photons, in practice, it is possible to treat the generated light classically using electromagnetic point dipole sources. A point dipole source is simply an oscillating dipole field that has a specific orientation, phase and polarization.

The average electromagnetic field intensity of an ensemble of incoherent, isotropic dipole emitters in a small spatial volume can be calculated by

$$ <|\vec{E}|^2> = \dfrac{1}{3}[{|E_a|}^2 + {|E_b|}^2 + {|E_c|}^2] $$

where Ea, Eb and Ec are the electromagnetic fields generated by a single dipole along the x, y and z axes. Therefore we can calculate the incoherent, isotropic response with three simulations. This result is demonstrated in the appendix incoherent isotropic dipole sources.

We calculate two important results which are compared to analytic solutions. First we calculate the light extraction efficiency for a dipole located in a dielectric half-space as a function of the distance of the dipole from the interface. The results from FDTD are compared to the known analytic results from Wasey et al. Second, we calculate the angular distribution of radiation in the far field for an isotropic, incoherent ensemble of dipoles located in a dielectric half-space and compare to the straightforward theoretical calculation.

Total radiated power in dielectric half-space
The problem is a dipole emitting in a half-space of high index material, near the interface. Open the file half_space.fsp. It will look like the following screenshot.

The permittivity of the upper half-space is 2.12 and the lower half-space is 1. The dipole source radiates at 614 nm. We will calculate the extraction efficiency of the radiation into the low index (n=1) region, as a function of the distance from the dipole to the interface.

Run the script file called half_space.lsf. It will run a series of 22 simulations with the dipole place at 20 nm to 420 nm from the interface in 40 nm increments. It will run a simulation with the dipole oriented in the x-direction, then in the z-direction. It will plot the total emitted power of a set of incoherent, isotropic dipole emitters relative to the emitted power in a homogeneous material of permittivity 2.12. Finally, it will compare to the analytical results from Wasey et al. The script file wasey.lsf will calculate the analytic solution, and is required by half_space.lsf.

There total power emitted by the dipole is calculated using two different methods: the dipolepower script function, and a box of monitors.

The total radiated power is shown in the figure below. The blue (green) data was calculated using a box of monitors (dipolepower/sourcepower command). These results from FDTD compare very favorably with the analytical results of Wasey et al.

Angular distribution of power radiated in dielectric half-space
The file interface2d.fsp and interface2d.lsf can be used to compare the angular distribution of radiation from an un-polarized dipole in the XY plane in a two dimensional halfspace. This requires two simulations, where the dipole is first oriented along the x-axis, then along the y-axis.

angular_distribution_p angular_distribution_s

The theoretical results are calculated using Fresnel coefficients. Both results are plotted on the same curve as shown to the left. The angular distribution is very accurate up to angles of approximately 85 degrees. In order to remove unphysical ripple in the far field, a far field filter setting of 1 was used. Similar agreement can be achieved in 3D with three dipole simulations: for the x, y and z-axes.

Related publications
Wasey et al, “Efficiency of spontaneous emission from planar microcavities”, Journal of Modern Optics, 47, 725-741, (2000)
We gratefully acknowledge the collaboration of Horst Greiner of Philips Research in the development of this application example.

See Also
OLEDS, Spatially incoherent dipole