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mention that the number of independent dipole simulations is the same…
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… in cylindrical and 3D coordinates
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oskooi committed Apr 3, 2024
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[Tutorial/Radiation Pattern of a Disc in Cylindrical Coordinates](Near_to_Far_Field_Spectra.md#radiation-pattern-of-a-disc-in-cylindrical-coordinates) demonstrated the procedure for computing the radiation pattern of a *single* dipole (actually a "ring" current source with angular dependence $e^{im\phi}$). [Tutorial/Nonaxisymmetric Dipole Sources](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) described the method for modeling a point dipole at $r > 0$ in cylindrical coordinates using a Fourier-series expansion of the fields in $\phi$. [Tutorial/Extraction Efficiency of a Light-Emitting Diode](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) described the procedure for computing the extraction efficiency of a dipole at $r = 0$. These three demonstrations can be combined to compute the extraction efficiency for a point dipole *anywhere* in the cylindrical cell. Computing the extraction efficiency of an actual light-emitting diode (LED), however, involves a collection of spatially incoherent dipole emitters. [Tutorial/Stochastic Dipole Emission in Light Emitting Diodes](Custom_Source.md#stochastic-dipole-emission-in-light-emitting-diodes) described a method for computing the emission of a collection of dipoles using a series of single-dipole calculations and then averaging the emission profiles in post processing. The example used a 2D simulation involving a 1D binary grating (or photonic crystal). This tutorial demonstrates how this approach for modeling spatially incoherent dipoles can be extended to cylindrical coordinates for structures with rotational symmetry.

Note: an important performance benefit of computing the extraction efficiency of a disc in cylindrical rather than 3D Cartesian coordinates is that the set of dipoles within a quantum well (QW) only needs to span a line rather than a 2D surface. This means that the number of single-dipole simulations necessary for convergence can be significantly reduced.
Note: in the case of a disc, the set of dipoles within the quantum well (QW) which spans a 2D surface only needs to be computed along a line. This means that the number of single-dipole simulations necessary for convergence is the same in cylindrical and 3D Cartesian coordinates.

Note: for randomly polarized emission from the QW, each dipole requires computing the emission from the two orthogonal "in-plane" polarization states of $E_r$ and $E_\phi$ separately and averaging the results in post processing. In this example, only the $E_r$ polarization state is used.

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