fixes and tweaks

doc/docs/Python_Tutorials/Local_Density_of_States.md

@@ -14,11 +14,11 @@ where the $|\hat{p}(\omega)|^2$ normalization is necessary for obtaining the pow
 Planar Cavity with Lossless Metallic Walls
 ------------------------------------------
 
-The spontaneous-emission rate of a point-dipole emitter in a planar cavity bounded by a lossless metallic mirror can be tuned using the thickness of the cavity. A schematic of the cavity is shown in the figure inset below. In this example, the 3D cavity with thickness $L$ consists of a homogeneous dielectric with $n$=2.4 and the dipole wavelength is 1.0 μm with horizontal polarization. The Purcell enhancement factor, a dimensionless quantity defined relative to the bulk medium, can be computed analytically for this system using equation (7) of [IEEE J. Quantum Electronics, Vol. 34, pp. 71-76 (1998)](https://ieeexplore.ieee.org/abstract/document/655009). We will validate the simulated results using the analytic theory.
+The spontaneous-emission rate of a point-dipole emitter in a planar cavity bounded by a lossless metallic mirror can be tuned using the thickness of the cavity. A schematic of the cavity is shown in the figure inset below. In this example, the 3D cavity consists of two mirrors separated in the *z* direction by a distance $L$. The cavity consists of a homogeneous dielectric with $n$=2.4. The dipole wavelength ($\lambda$) is 1.0 μm with horizontal polarization along the *x* axis. The Purcell enhancement factor, a dimensionless quantity defined relative to the bulk medium, can be computed analytically in terms of the cavity thickness in units of the medium wavelength ($nL/\lambda$) for this system using equation (7) of [IEEE J. Quantum Electronics, Vol. 34, pp. 71-76 (1998)](https://ieeexplore.ieee.org/abstract/document/655009). We will validate the simulated results using the analytic theory.
 
-The lossless metal sidewalls of the cavity are perfectly reflective at all angles which enables the Purcell enhancement factor to be tuned over a wide range for this system. (There are other types of lossless mirrors with π phase shift based on e.g., a multilayer stack in which the index of the cavity is larger than the index of the first mirror layer but these tend to have strong angular dependence.) The cavity thickness, defined in units of the medium wavelength, is swept over the range of 0.5 to 2.5. Below a thickness of 0.5 there are no guided modes and thus the Purcell enhancement factor is zero.
+An important feature of this cavity system are the lossless metal sidewalls. These mirrors are perfectly reflective at all angles and enable the Purcell enhancement factor to be tuned over a wide range of ~0.5 to 3.0. There are other types of lossless mirrors with reflectivity of +1 (0 phase shift) rather than -1 (π phase shift) as in this example based on e.g., a multilayer stack in which the index of the cavity is *larger* than the index of the first mirror layer. These mirrors tend to have a strong angular dependence of the reflectivity and thus generally a smaller range of variation for the Purcell enhancement. In this demonstration, the cavity thickness is swept over a range of 0.5 to 2.5. Below a thickness of 0.5 there are no guided modes and thus the Purcell enhancement factor is zero.
 
-Two types of simulations are necessary for computing the Purcell enhancement factor: (1) bulk medium and (2) cavity. The `dft_ldos` featured is used to compute the LDOS in each case at a single wavelength. The Purcell enhancement factor is computed as the ratio of the LDOS measured in (2) to that from (1). Each simulation contains which three mirror symmetries that we can exploit to reduce the size of the 3D computation by a factor of eight. The cavity is infinitely extended in the *xy* plane and thus the cell is terminated using PMLs in these two directions. Because Meep uses a default boundary condition of a perfect electric conductor, there is no need to explicitly define the boundaries in the *z* direction.
+Two types of simulations are necessary for computing the Purcell enhancement factor: (1) bulk medium and (2) cavity. The `dft_ldos` featured is used to compute the LDOS in each case at a single wavelength. The Purcell enhancement factor is computed as the ratio of the LDOS measured in (2) to that from (1). Each simulation uses three mirror symmetries to reduce the size of the 3D computation by a factor of eight. The cavity is infinitely extended in the *xy* plane and thus the cell is terminated using PMLs in these two directions. Because Meep uses a default boundary condition of a perfect electric conductor, there is no need to explicitly define the boundaries in the *z* direction. The fields are timestepped until they have decayed away sufficiently due to absorption by the PMLS at the location of the pulsed source.
 
 As shown in the plot below, the results from Meep agree well with the analytic theory.
 
@@ -132,22 +132,19 @@ if __name__ == '__main__':
                     bbox_inches='tight')
 ```
 
+Square Box with a Small Opening
+-------------------------------
 
+We consider the simple example of a 2D perfect-metal $a$x$a$ cavity of finite thickness 0.1$a$, with a small notch of width $w$ on one side that allows the modes to escape. The nice thing about this example is that in the absence of the notch, the lowest-frequency $E_z$-polarized mode is known analytically to be $E_z^{(1)}=\frac{4}{a^2}\sin(\pi x/a)\sin(\pi \gamma/a)$, with a frequency $\omega^{(1)}=\sqrt{2}\pi c/a$ and modal volume $V^{(1)}=a^2/4$. The notch slightly perturbs this solution, but more importantly the opening allows the confined mode to radiate out into the surrounding air, yielding a finite $Q$. For $w \ll a$, this radiative escape occurs via an evanescent (sub-cutoff) mode of the channel waveguide formed by the notch, and it follows from inspection of the evanescent decay rate $\sqrt{(\pi/\omega)^2-(\omega^{(1)})^2}/c$ that the lifetime scales asymptotically as $Q^{(1)} \sim e^{\#/\omega}$ for some coefficient \#.
 
-
-Hollow Square Box with a Small Opening
---------------------------------------
+We will validate both this prediction and the expression for the LDOS shown above by computing the LDOS at the center of the cavity, the point of peak $|\vec{E}|$, in two ways. First, we compute the LDOS directly from the power radiated by a dipole, Fourier-transforming the result of a pulse using the `dft_ldos` command. Second, we compute the cavity mode and its lifetime $Q$ using Harminv and then compute the LDOS by the Purcell formula shown above. The latter technique is much more efficient for high Q (small $w$), since one must run the simulation for a very long time to directly accumulate the Fourier transform of a slowly-decaying mode. The two calculations, we will demonstrate, agree to within discretization error, verifying the LDOS analysis above, and $Q/V$ is asymptotically linear on a semilog scale versus $1/w$ as predicted.
 
 A lossless localized mode yields a $\delta$-function spike in the LDOS, whereas a <i>lossy</i>, arising from either small absorption or radiation, localized mode &mdash; a resonant cavity mode &mdash; leads to a Lorentzian peak. The large enhancement in the LDOS at the resonant peak is known as a [Purcell effect](https://en.wikipedia.org/wiki/Purcell_effect), named after Purcell's proposal for enhancing spontaneous emission of an atom in a cavity. This is analogous to a microwave antenna resonating in a metal box. In this case, the resonant mode's contribution to the LDOS at $\omega^{(n)}$ can be shown to be:
 
 $$\operatorname{resonant\ LDOS} \approx \frac{2}{\pi\omega^{(n)}} \frac{Q^{(n)}}{V^{(n)}}$$
 
 where $Q^{(n)}=\omega^{(n)}/2\gamma^{(n)}$ is the dimensionless [quality factor](https://en.wikipedia.org/wiki/Q_factor) and $V^{(n)}$ is the modal volume. This represents another way to compute the LDOS. In this tutorial, we will verify this expression by comparing it to the earlier one.
 
-We consider the simple example of a 2D perfect-metal $a$x$a$ cavity of finite thickness 0.1$a$, with a small notch of width $w$ on one side that allows the modes to escape. The nice thing about this example is that in the absence of the notch, the lowest-frequency $E_z$-polarized mode is known analytically to be $E_z^{(1)}=\frac{4}{a^2}\sin(\pi x/a)\sin(\pi \gamma/a)$, with a frequency $\omega^{(1)}=\sqrt{2}\pi c/a$ and modal volume $V^{(1)}=a^2/4$. The notch slightly perturbs this solution, but more importantly the opening allows the confined mode to radiate out into the surrounding air, yielding a finite $Q$. For $w \ll a$, this radiative escape occurs via an evanescent (sub-cutoff) mode of the channel waveguide formed by the notch, and it follows from inspection of the evanescent decay rate $\sqrt{(\pi/\omega)^2-(\omega^{(1)})^2}/c$ that the lifetime scales asymptotically as $Q^{(1)} \sim e^{\#/\omega}$ for some coefficient \#.
-
-We will validate both this prediction and the LDOS calculations above by computing the LDOS at the center of the cavity, the point of peak $|\vec{E}|$, in two ways. First, we compute the LDOS directly from the power radiated by a dipole, Fourier-transforming the result of a pulse using the `dft_ldos` command. Second, we compute the cavity mode and its lifetime $Q$ using `Harminv` and then compute the LDOS by the Purcell formula shown above. The latter technique is much more efficient for high Q (small $w$), since one must run the simulation for a very long time to directly accumulate the Fourier transform of a slowly-decaying mode. The two calculations, we will demonstrate, agree to within discretization error, verifying the LDOS analysis above, and $Q/V$ is asymptotically linear on a semilog scale versus $1/w$ as predicted.
-
 The simulation script is [examples/metal-cavity-ldos.py](https://github.com/NanoComp/meep/blob/master/python/examples/metal-cavity-ldos.py). The notebook is [examples/metal-cavity-ldos.ipynb](https://nbviewer.jupyter.org/github/NanoComp/meep/blob/master/python/examples/metal-cavity-ldos.ipynb).
 
 ```py