mention no interference effects between uncorrelated dipoles in LED t…

…utorial (#2133)

* mention no interference effects between uncorrelated dipoles in LED tutorial

* Update Custom_Source.md

* Update Custom_Source.md

Co-authored-by: Steven G. Johnson <[email protected]>

doc/docs/Python_Tutorials/Custom_Source.md

@@ -17,7 +17,9 @@ This tutorial example involves computing the radiated [flux](../Introduction.md#
 ![](../images/LED_layout.png)
 </center>
 
-One can take two different approaches to computing the radiated flux based on the type of emitter: (1) random or (2) deterministic. In Method 1 (brute-force Monte Carlo), each emitter is a white-noise dipole: every timestep for every dipole is an independent random number. A single run involves all $N$ dipoles which are modeled using a `CustomSource`. The stochastic results for the radiated flux are averaged over multiple trials/iterations via [Monte Carlo sampling](https://en.wikipedia.org/wiki/Monte_Carlo_method). Method 2 exploits the property of [linear time-invariance](https://en.wikipedia.org/wiki/Linear_time-invariant_system) of the materials/geometry and involves a sequence of $N$ separate runs each with a single deterministic dipole (i.e., pulse time profile, `GaussianSource`) at different positions in the emitting layer. Because dipoles at different positions are uncorrelated, the radiated flux from the ensemble is simply the average of all the individual iterations. The two approaches converge towards identical results, but Method 1 is more computationally expensive than Method 2 due to the much larger number of trials/iterations ($\gg  N$) required to attain low noise variance.   (Even more sophisticated deterministic methods exist to reduce the number of separate simulations, especially at high resolutions; for example, replacing the point-dipole sources with a [rapidly converging set of smooth basis functions](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.81.012119) as demonstrated below, or fancier methods that exploit [trace-estimation methods](http://doi.org/10.1103/PhysRevB.92.134202) and/or transform volumetric sources to [surface sources](http://doi.org/10.1103/PhysRevB.88.054305).)
+One can take two different approaches to computing the radiated flux based on the type of emitter: (1) random or (2) deterministic. In Method 1 (brute-force Monte Carlo), each emitter is a white-noise dipole: every timestep for every dipole is an independent random number. A single run involves all $N$ dipoles which are modeled using a `CustomSource`. The stochastic results for the radiated flux are averaged over multiple trials/iterations via [Monte Carlo sampling](https://en.wikipedia.org/wiki/Monte_Carlo_method). Method 2 exploits the property of [linear time-invariance](https://en.wikipedia.org/wiki/Linear_time-invariant_system) of the materials/geometry and involves a sequence of $N$ separate runs each with a single deterministic dipole (i.e., pulse time profile, `GaussianSource`) at different positions in the emitting layer. Because dipoles at different positions are uncorrelated, the radiated flux from the ensemble is simply the average of all the individual iterations. (The interference terms between different dipoles integrate to zero when averaging over all possible phases.) The two approaches converge towards identical results, but Method 1 is more computationally expensive than Method 2 due to the much larger number of trials/iterations ($\gg  N$) required to attain low noise variance.   (Even more sophisticated deterministic methods exist to reduce the number of separate simulations, especially at high resolutions; for example, replacing the point-dipole sources with a [rapidly converging set of smooth basis functions](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.81.012119) as demonstrated below, or fancier methods that exploit [trace-estimation methods](https://arxiv.org/abs/2111.13046) and/or transform volumetric sources to [surface sources](http://doi.org/10.1103/PhysRevB.88.054305).)
+
+In principle, to compute the total power emitted upwards in *all* directions, we must also average over all possible Bloch wavevectors $k_x \in [-\pi/s_x,+\pi/s_x]$ (e.g. see [this paper](https://arxiv.org/abs/2111.13046); this is also called an ["array-scanning" method](https://doi.org/10.1109/TAP.2007.897348)).  For simplicity, however, in this tutorial we only compute the average for $k_x=0$ (i.e., the portion of the power in all diffraction orders for $k_x=0$, including normal emission).   Conversely, if one is only interested in the emitted power in a *single* direction, e.g. normal emission, then it turns out to be possible to compute the net effect using only a *single* "reciprocal" simulation as reviewed [in Yao (2022) in a general setting](https://arxiv.org/abs/2111.13046) or [in Jansen (2010) for the LED case in particular](https://doi.org/10.1364/OE.18.024522).
 
 *Note regarding normalization:* To directly compare the results for the radiated flux from the two methods, one might scale the spectrum from Method 2 in post processing to correct for the difference in spectrum between a Gaussian pulse and white noise. However, it is usually more convenient to *nondimensionalize* the results (for both methods) by dividing the flux spectrum for the textured surface with a reference spectrum computed by the *same method*, for example emission from a flat surface or in a homogeneous medium. This way, the details of the source spectra cancel automatically, and the nondimensionalized results can be compared as-is without any tricky scaling calculations.  This kind of nondimensionalized comparison is useful to determine the *emission enhancement* (or suppression) of one structure relative to another as well as the *light-extraction efficiency* (the ratio of the radiated flux to the total flux emitted by the dipoles).  In order to compute the *absolute* (not relative) light emission by a particular structure, using either Method 1 or Method 2, one would need to rescale the output ([thanks to linearity](http://doi.org/10.1103/PhysRevLett.107.114302)) to convert the input spectrum (from white noise or Gaussian) to the actual emission spectrum (e.g. determined from the gain spectrum of a light-emitting diode).