Learning Forum

[Shin-Sung Kim]

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 be added incoherently.nIn 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 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 can be expressed as follows:nFor proof of the principle, please see below.nDerivationnAn 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.nTo calculate the field distribution of an incoherent dipole, we need to average overall possible dipole orientations. The incoherent electric field intensity at position r is given by:nwhere E(r,θ,φ)  represents the electric field created at position r  by a dipole (at position r0) with an orientation given by the spherical angles θ and φ.nA dipole of any orientation can be written asnSince Maxwell’s equations are linear, we can write the electric field from a dipole with orientation θ and φ asnSubstituting this into the equation for the field intensity of an incoherent dipole givesnIt is easy (if tedious) to simplify this integral with the help of the identities provided at the bottom of this page. In the end, we getnTherefore, 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.nThe following identities are required to simplify the above integral:nn