update figure to compute relative error from the norm before dividing…

… (not after)

doc/docs/Python_Tutorials/Custom_Source.md

@@ -304,7 +304,7 @@ if mp.am_master():
 
 There are three items to note in this script. (1) The line source spans the entire length of the cell in the $x$ direction (i.e., $L$ is `sx`). (2) The number of point dipoles in Method 2 is `sx*resolution`, one per pixel. (3) The source amplitude function in Method 3 is specified by the `amp_func` property of the [`Source`](../Python_User_Interface.md#source) object. In the case of a Fourier cosine series as conventionally written, $\cos (m\pi x)/L$ is defined over the interval $x=[0,L]$ such that $x=0$ corresponds to the *edge* of the source, not the center. Since the source region in this example is defined in $[-L/2,+L/2]$, the amplitude function must shift its $x$ coordinate argument by $+L/2$ or `0.5*sx`.
 
-Method 3 requires a convergence check in which $M$ (`nsrc` in the script) is repeatedly doubled until the change in the results are within a desired tolerance of e.g., < 1%. For this example, $M=15$ was found to be sufficient. Note that because a line source with a cosine amplitude function in *homogeneous* media is analogous to generating a planewave at a discrete angle, at each frequency $\omega$ there exists a cutoff $M$ beyond which there are *no* propagating planewaves. The cutoff $M$ can be computed analytically using the grating equation: $\sqrt{\omega^2n^2 - \left(k_x+\frac{M\pi}{L}\right)^2} = 0$, where $n$ is the refractive index of the source medium and $k_x$ is the Bloch-periodic wavevector in the $x$ direction. In this example, $\omega=2\pi$, $L=1.5$, $k_x=0$, and $n=3.45$ for which the cutoff $M$ is 11. For $M > 11$ the cosine source produces evanscent *waves in the material*, but these waves scatter into other Fourier components (including propagating waves) once they hit the grating. Thus, the source still produces propagating waves, but the amplitude of the propagating waves (and hence the contribution to the power) decreases exponentially for $M > 11$ as the coupling between the source and the grating decreases. This effect is demonstrated in the figure below which is a semilog plot of the $L_2$ norm of the error in the normalized flux of the cosine source relative to the same result computed using 75 point dipoles (Method 2) as a function of the number of terms in the cosine source. The error converges exponentially to zero until $M=11$ beyond which it is a constant value. Note that this analysis is only applicable to periodic structures where the line source extends the entire width of the cell with periodic boundary conditions.
+Method 3 requires a convergence check in which $M$ (`nsrc` in the script) is repeatedly doubled until the change in the results are within a desired tolerance of e.g., < 1%. For this example, $M=15$ was found to be sufficient. Note that because a line source with a cosine amplitude function in *homogeneous* media is analogous to generating a planewave at a discrete angle, at each frequency $\omega$ there exists a cutoff $M$ beyond which there are *no* propagating planewaves. The cutoff $M$ can be computed analytically using the grating equation: $\sqrt{\omega^2n^2 - \left(k_x+\frac{M\pi}{L}\right)^2} = 0$, where $n$ is the refractive index of the source medium and $k_x$ is the Bloch-periodic wavevector in the $x$ direction. In this example, $\omega=2\pi$ (pulse center frequency), $L=1.5$, $k_x=0$, and $n=3.45$ for which the cutoff $M$ is 11. For $M > 11$ the cosine source produces evanscent *waves in the material*, but these waves scatter into other Fourier components (including propagating waves) once they hit the grating. Thus, the source still produces propagating waves, but the amplitude of the propagating waves (and hence the contribution to the power) decreases exponentially for $M > 11$ as the coupling between the source and the grating decreases. This effect is demonstrated in the figure below which is a semilog plot of the $L_2$ norm of the error in the normalized flux of the cosine source relative to the "correct" result computed using 75 point dipoles (Method 2) as a function of the number of terms in the cosine source. The error converges exponentially to zero until $M=12$ beyond which it is a constant value. Note that this analysis is only applicable to periodic structures where the line source extends the entire width of the cell with periodic boundary conditions. A finite length source in a non-periodic cell has no such cutoff and thus will typically require a large number of cosine terms for convergence.
 
 <center>
 ![](../images/line_source_DCT_ampfunc_convergence.png)