more doc updates

.gitignore

@@ -63,3 +63,6 @@ target/
 
 
 .*.sw[op]
+
+*.svg
+*.html

make_docs.sh

@@ -1,3 +1,15 @@
 #!/bin/bash
 cd ~/projects/meanas
-pdoc3 --html --force --template-dir pdoc_templates -o doc .
+
+# Approach 1: pdf to html?
+#pdoc3 --pdf --force --template-dir pdoc_templates -o doc . | \
+#    pandoc --metadata=title:"meanas" --toc --toc-depth=4 --from=markdown+abbreviations --to=html --output=doc.html --gladtex -s -
+
+# Approach 2: pdf to html with gladtex
+pdoc3 --pdf --force --template-dir pdoc_templates -o doc . > doc.md
+pandoc --metadata=title:"meanas" --from=markdown+abbreviations --to=html --output=doc.html --gladtex -s --css pdoc_templates/pdoc.css doc.md
+gladtex -a -n -d _doc_mathimg -c white doc.html
+
+# Approach 3: html with gladtex
+#pdoc3 --html --force --template-dir pdoc_templates -o doc .
+#find doc -iname '*.html' -exec gladtex -a -n -d _mathimg -c white {} \;

meanas/fdfd/__init__.py

@@ -14,7 +14,20 @@ Submodules:
 - `waveguide_3d`: Functions for transforming `waveguide_2d` results into 3D.
 
 
-===========
+================================================================
+
+From the "Frequency domain" section of `meanas.fdmath`, we have
+
+$$
+ \\begin{aligned}
+ \\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
+ \\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\
+ \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
+    -\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} &= \\imath \\Omega \\tilde{J}_{\\vec{r}} \\\\
+ \\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
+ \\end{aligned}
+$$
+
 
 # TODO FDFD?
 # TODO PML

meanas/fdfd/functional.py

@@ -185,8 +185,8 @@ def e_tfsf_source(TF_region: fdfield_t,
 def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[fdfield_t, fdfield_t], fdfield_t]:
     """
     Generates a function that takes the single-frequency `E` and `H` fields
-    and calculates the cross product `E` x `H` = \\( E \\times H \\) as required
+    and calculates the cross product `E` x `H` = $E \\times H$ as required
-    for the Poynting vector, \\( S = E \\times H \\)
+    for the Poynting vector, $S = E \\times H$
 
     Note:
         This function also shifts the input `E` field by one cell as required

meanas/fdfd/waveguide_2d.py

@@ -12,82 +12,82 @@ As the z-dependence is known, all the functions in this file assume a 2D grid
 
 Consider Maxwell's equations in continuous space, in the frequency domain. Assuming
 a structure with some (x, y) cross-section extending uniformly into the z dimension,
-with a diagonal \\( \\epsilon \\) tensor, we have
+with a diagonal $\\epsilon$ tensor, we have
 
 $$
-\\begin{align*}
+\\begin{aligned}
 \\nabla \\times \\vec{E}(x, y, z) &= -\\imath \\omega \\mu \\vec{H} \\\\
 \\nabla \\times \\vec{H}(x, y, z) &=  \\imath \\omega \\epsilon \\vec{E} \\\\
 \\vec{E}(x,y,z) = (\\vec{E}_t(x, y) + E_z(x, y)\\vec{z}) e^{-\\gamma z} \\\\
 \\vec{H}(x,y,z) = (\\vec{H}_t(x, y) + H_z(x, y)\\vec{z}) e^{-\\gamma z} \\\\
-\\end{align*}
+\\end{aligned}
 $$
 
 Expanding the first two equations into vector components, we get
 
 $$
-\\begin{align*}
+\\begin{aligned}
 -\\imath \\omega \\mu_{xx} H_x &= \\partial_y E_z - \\partial_z E_y \\\\
 -\\imath \\omega \\mu_{yy} H_y &= \\partial_z E_x - \\partial_x E_z \\\\
 -\\imath \\omega \\mu_{zz} H_z &= \\partial_x E_y - \\partial_y E_x \\\\
 \\imath \\omega \\epsilon_{xx} E_x &= \\partial_y H_z - \\partial_z H_y \\\\
 \\imath \\omega \\epsilon_{yy} E_y &= \\partial_z H_x - \\partial_x H_z \\\\
 \\imath \\omega \\epsilon_{zz} E_z &= \\partial_x H_y - \\partial_y H_x \\\\
-\\end{align*}
+\\end{aligned}
 $$
 
-Substituting in our expressions for \\( \\vec{E}, \\vec{H} \\) and discretizing:
+Substituting in our expressions for $\\vec{E}$, $\\vec{H}$ and discretizing:
 
 $$
-\\begin{align*}
+\\begin{aligned}
 -\\imath \\omega \\mu_{xx} H_x &= \\tilde{\\partial}_y E_z + \\gamma E_y \\\\
 -\\imath \\omega \\mu_{yy} H_y &= -\\gamma E_x - \\tilde{\\partial}_x E_z \\\\
 -\\imath \\omega \\mu_{zz} H_z &= \\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x \\\\
 \\imath \\omega \\epsilon_{xx} E_x &= \\hat{\\partial}_y H_z + \\gamma H_y \\\\
 \\imath \\omega \\epsilon_{yy} E_y &= -\\gamma H_x - \\hat{\\partial}_x H_z \\\\
 \\imath \\omega \\epsilon_{zz} E_z &= \\hat{\\partial}_x H_y - \\hat{\\partial}_y H_x \\\\
-\\end{align*}
+\\end{aligned}
 $$
 
 Rewrite the last three equations as
 $$
-\\begin{align*}
+\\begin{aligned}
 \\gamma H_y &=  \\imath \\omega \\epsilon_{xx} E_x - \\hat{\\partial}_y H_z \\\\
 \\gamma H_x &= -\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z \\\\
 \\imath \\omega E_z &= \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x H_y - \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y H_x \\\\
-\\end{align*}
+\\end{aligned}
 $$
 
-Now apply \\( \\gamma \\tilde{\\partial}_x \\) to the last equation,
+Now apply $\\gamma \\tilde{\\partial}_x$ to the last equation,
-then substitute in for \\( \\gamma H_x \\) and \\( \\gamma H_y \\):
+then substitute in for $\\gamma H_x$ and $\\gamma H_y$:
 
 $$
-\\begin{align*}
+\\begin{aligned}
 \\gamma \\tilde{\\partial}_x \\imath \\omega E_z &= \\gamma \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x H_y
                                                 - \\gamma \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y H_x \\\\
-        &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon{xx} E_x - \\hat{\\partial}_y H_z)
+        &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon_{xx} E_x - \\hat{\\partial}_y H_z)
-         - \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon{yy} E_y - \\hat{\\partial}_x H_z)  \\\\
+         - \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z)  \\\\
-        &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon{xx} E_x)
+        &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon_{xx} E_x)
-         - \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon{yy} E_y)  \\\\
+         - \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon_{yy} E_y)  \\\\
 \\gamma \\tilde{\\partial}_x E_z &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
                                     \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
-\\end{align*}
+\\end{aligned}
 $$
 
-With a similar approach (but using \\( \\gamma \\tilde{\\partial}_y \\) instead), we can get
+With a similar approach (but using $\\gamma \\tilde{\\partial}_y$ instead), we can get
 
 $$
-\\begin{align*}
+\\begin{aligned}
 \\gamma \\tilde{\\partial}_y E_z &= \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
                                     \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
-\\end{align*}
+\\end{aligned}
 $$
 
-We can combine this equation for \\( \\gamma \\tilde{\\partial}_y E_z \\) with
+We can combine this equation for $\\gamma \\tilde{\\partial}_y E_z$ with
-the unused \\( \\imath \\omega \\mu_{xx} H_z \\) and \\( \\imath \\omega \\mu_{yy} H_y \\) equations to get
+the unused $\\imath \\omega \\mu_{xx} H_z$ and $\\imath \\omega \\mu_{yy} H_y$ equations to get
 
 $$
-\\begin{align*}
+\\begin{aligned}
 -\\imath \\omega \\mu_{xx} \\gamma H_x &=  \\gamma^2 E_y + \\tilde{\\partial}_y (
                                       \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
                                     + \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
@@ -96,30 +96,30 @@ $$
                                       \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
                                     + \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
                                     )\\\\
-\\end{align*}
+\\end{aligned}
 $$
 
-However, based on our rewritten equation for \\( \\gamma H_x \\) and the so-far unused
+However, based on our rewritten equation for $\\gamma H_x$ and the so-far unused
-equation for \\( \\imath \\omega \\mu_{zz} H_z \\) we can also write
+equation for $\\imath \\omega \\mu_{zz} H_z$ we can also write
 
 $$
-\\begin{align*}
+\\begin{aligned}
 -\\imath \\omega \\mu_{xx} (\\gamma H_x) &= -\\imath \\omega \\mu_{xx} (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
                                          &= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
                                             -\\imath \\omega \\mu_{xx} \\hat{\\partial}_x (
                                                 \\frac{1}{-\\imath \\omega \\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x)) \\\\
                                          &= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
                                             +\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
-\\end{align*}
+\\end{aligned}
 $$
 
 and, similarly,
 
 $$
-\\begin{align*}
+\\begin{aligned}
 -\\imath \\omega \\mu_{yy} (\\gamma H_y) &= -\\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
                                             +\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
-\\end{align*}
+\\end{aligned}
 $$
 
 Using these, we can construct the eigenvalue problem
@@ -137,9 +137,9 @@ $$ \\beta^2 \\begin{bmatrix} E_x \\\\
                      E_y \\end{bmatrix}
 $$
 
-An equivalent eigenvalue problem can be formed using the \\( H_x, H_y \\) fields, if those are more convenient.
+An equivalent eigenvalue problem can be formed using the $H_x$ and $H_y$ fields, if those are more convenient.
 
-Note that \\( E_z \\) was never discretized, so \\( \\gamma \\) and \\( \\beta \\) will need adjustment
+Note that $E_z$ was never discretized, so $\\gamma$ and $\\beta$ will need adjustment
 to account for numerical dispersion if the result is introduced into a space with a discretized z-axis.
 
 
@@ -187,8 +187,8 @@ def operator_e(omega: complex,
                  \\begin{bmatrix} \\hat{\\partial}_x \\epsilon_{xx} & \\hat{\\partial}_y \\epsilon_{yy} \\end{bmatrix}
     $$
 
-    \\( \\tilde{\\partial}_x \\) and \\( \\hat{\\partial}_x \\) are the forward and backward derivatives along x,
+    $\\tilde{\\partial}_x$ and $\\hat{\\partial}_x$ are the forward and backward derivatives along x,
-    and each \\( \\epsilon_{xx}, \\mu_{yy}, \\) etc. is a diagonal matrix containing the vectorized material
+    and each $\\epsilon_{xx}$, $\\mu_{yy}$, etc. is a diagonal matrix containing the vectorized material
     property distribution.
 
     This operator can be used to form an eigenvalue problem of the form
@@ -253,8 +253,8 @@ def operator_h(omega: complex,
                  \\begin{bmatrix} \\tilde{\\partial}_x \\mu_{xx} & \\tilde{\\partial}_y \\mu_{yy} \\end{bmatrix}
     $$
 
-    \\( \\tilde{\\partial}_x \\) and \\( \\hat{\\partial}_x \\) are the forward and backward derivatives along x,
+    $\\tilde{\\partial}_x$ and $\\hat{\\partial}_x$ are the forward and backward derivatives along x,
-    and each \\( \\epsilon_{xx}, \\mu_{yy}, \\) etc. is a diagonal matrix containing the vectorized material
+    and each $\\epsilon_{xx}$, $\\mu_{yy}$, etc. is a diagonal matrix containing the vectorized material
     property distribution.
 
     This operator can be used to form an eigenvalue problem of the form

meanas/fdmath/__init__.py

@@ -44,10 +44,10 @@ Scalar derivatives and cell shifts
 
 Define the discrete forward derivative as
  $$ [\\tilde{\\partial}_x f]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$
- where \\( f \\) is a function defined at discrete locations on the x-axis (labeled using \\( m \\)).
+ where $f$ is a function defined at discrete locations on the x-axis (labeled using $m$).
- The value at \\( m \\) occupies a length \\( \\Delta_{x, m} \\) along the x-axis. Note that \\( m \\)
+ The value at $m$ occupies a length $\\Delta_{x, m}$ along the x-axis. Note that $m$
  is an index along the x-axis, _not_ necessarily an x-coordinate, since each length
- \\( \\Delta_{x, m}, \\Delta_{x, m+1}, ...\\) is independently chosen.
+ $\\Delta_{x, m}, \\Delta_{x, m+1}, ...$ is independently chosen.
 
 If we treat `f` as a 1D array of values, with the `i`-th value `f[i]` taking up a length `dx[i]`
 along the x-axis, the forward derivative is
@@ -62,7 +62,7 @@ Likewise, discrete reverse derivative is
     deriv_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
 
 The derivatives' values are shifted by a half-cell relative to the original function, and
-will have different cell widths if all the `dx[i]` ( \\( \\Delta_{x, m} \\) ) are not
+will have different cell widths if all the `dx[i]` ( $\\Delta_{x, m}$ ) are not
 identical:
 
     [figure: derivatives and cell sizes]
@@ -87,19 +87,20 @@ identical:
     Periodic boundaries are used here and elsewhere unless otherwise noted.
 
 In the above figure,
- `f0 =` \\(f_0\\), `f1 =` \\(f_1\\)
+ `f0 =` $f_0$, `f1 =` $f_1$
- `Df0 =` \\([\\tilde{\\partial}f]_{0 + \\frac{1}{2}}\\)
+ `Df0 =` $[\\tilde{\\partial}f]_{0 + \\frac{1}{2}}$
- `Df1 =` \\([\\tilde{\\partial}f]_{1 + \\frac{1}{2}}\\)
+ `Df1 =` $[\\tilde{\\partial}f]_{1 + \\frac{1}{2}}$
- `df0 =` \\([\\hat{\\partial}f]_{0 - \\frac{1}{2}}\\)
+ `df0 =` $[\\hat{\\partial}f]_{0 - \\frac{1}{2}}$
  etc.
 
-The fractional subscript \\( m + \\frac{1}{2} \\) is used to indicate values defined
+The fractional subscript $m + \\frac{1}{2}$ is used to indicate values defined
- at shifted locations relative to the original \\( m \\), with corresponding lengths
+ at shifted locations relative to the original $m$, with corresponding lengths
  $$ \\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x, m + 1}) $$
-Just as \\( m \\) is not itself an x-coordinate, neither is \\( m + \\frac{1}{2} \\);
+
+Just as $m$ is not itself an x-coordinate, neither is $m + \\frac{1}{2}$;
 carefully note the positions of the various cells in the above figure vs their labels.
-If the positions labeled with \\( m \\) are considered the "base" or "original" grid,
+If the positions labeled with $m$ are considered the "base" or "original" grid,
-the positions labeled with \\( m + \\frac{1}{2} \\) are said to lie on a "dual" or
+the positions labeled with $m + \\frac{1}{2}$ are said to lie on a "dual" or
 "derived" grid.
 
 For the remainder of the `Discrete calculus` section, all figures will show
@@ -201,8 +202,8 @@ There are also two divergences,
 where `g = [gx, gy, gz]` is a fore- or back-vector field.
 
 Since we applied the forward divergence to the back-vector (and vice-versa), the resulting scalar value
-is defined at the back-vector's (fore-vectors) location \\( (m,n,p) \\) and not at the locations of its components
+is defined at the back-vector's (fore-vector's) location $(m,n,p)$ and not at the locations of its components
-\\( (m \\pm \\frac{1}{2},n,p) \\) etc.
+$(m \\pm \\frac{1}{2},n,p)$ etc.
 
     [figure: divergence]
                                     ^^
@@ -226,23 +227,23 @@ Curls
 
 The two curls are then
 
-  $$ \\begin{align*}
+  $$ \\begin{aligned}
      \\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\
      [\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=
         \\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} - \\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\
      &+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} - \\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\
      &+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_y g^z_{m + \\frac{1}{2},n,p})
-     \\end{align*} $$
+     \\end{aligned} $$
 
  and
 
   $$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} =
      [\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$
 
-  where \\( \\hat{g} \\) and \\( \\tilde{g} \\) are located at \\((m,n,p)\\)
+  where $\\hat{g}$ and $\\tilde{g}$ are located at $(m,n,p)$
-  with components at  \\( (m \\pm \\frac{1}{2},n,p) \\) etc.,
+  with components at $(m \\pm \\frac{1}{2},n,p)$ etc.,
-  while \\( \\hat{h} \\) and \\( \\tilde{h} \\) are located at \\((m \\pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})\\)
+  while $\\hat{h}$ and $\\tilde{h}$ are located at $(m \\pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$
-  with components at \\((m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})\\) etc.
+  with components at $(m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$ etc.
 
 
     [code: curls]
@@ -286,27 +287,27 @@ Maxwell's Equations
 
 If we discretize both space (m,n,p) and time (l), Maxwell's equations become
 
- $$ \\begin{align*}
+ $$ \\begin{aligned}
   \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
                                                                          - \\hat{M}_{l, \\vec{r} + \\frac{1}{2}}  \\\\
   \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2},\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
                                                                              + \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
   \\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\
   \\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
- \\end{align*} $$
+ \\end{aligned} $$
 
  with
 
- $$ \\begin{align*}
+ $$ \\begin{aligned}
-  \\hat{B}_\\vec{r} &= \\mu_{\\vec{r} + \\frac{1}{2}} \\cdot \\hat{H}_{\\vec{r} + \\frac{1}{2}} \\\\
+  \\hat{B}_{\\vec{r}} &= \\mu_{\\vec{r} + \\frac{1}{2}} \\cdot \\hat{H}_{\\vec{r} + \\frac{1}{2}} \\\\
-  \\tilde{D}_\\vec{r} &= \\epsilon_\\vec{r} \\cdot \\tilde{E}_\\vec{r}
+  \\tilde{D}_{\\vec{r}} &= \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}}
- \\end{align*} $$
+ \\end{aligned} $$
 
-where the spatial subscripts are abbreviated as \\( \\vec{r} = (m, n, p) \\) and
+where the spatial subscripts are abbreviated as $\\vec{r} = (m, n, p)$ and
-\\( \\vec{r} + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}) \\),
+$\\vec{r} + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2})$,
-\\( \\tilde{E} \\) and \\( \\hat{H} \\) are the electric and magnetic fields,
+$\\tilde{E}$ and $\\hat{H}$ are the electric and magnetic fields,
-\\( \\tilde{J} \\) and \\( \\hat{M} \\) are the electric and magnetic current distributions,
+$\\tilde{J}$ and $\\hat{M}$ are the electric and magnetic current distributions,
-and \\( \\epsilon \\) and \\( \\mu \\) are the dielectric permittivity and magnetic permeability.
+and $\\epsilon$ and $\\mu$ are the dielectric permittivity and magnetic permeability.
 
 The above is Yee's algorithm, written in a form analogous to Maxwell's equations.
 The time derivatives can be expanded to form the update equations:
@@ -375,12 +376,12 @@ and combining them with charge continuity,
 Wave equation
 -------------
 
-Taking the backward curl of the \\( \\tilde{\\nabla} \\times \\tilde{E} \\) equation and
+Taking the backward curl of the $\\tilde{\\nabla} \\times \\tilde{E}$ equation and
-replacing the resulting \\( \\hat{\\nabla} \\times \\hat{H} \\) term using its respective equation,
+replacing the resulting $\\hat{\\nabla} \\times \\hat{H}$ term using its respective equation,
-and setting \\( \\hat{M} \\) to zero, we can form the discrete wave equation:
+and setting $\\hat{M}$ to zero, we can form the discrete wave equation:
 
 $$
-  \\begin{align*}
+  \\begin{aligned}
   \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
       -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
                           - \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}}  \\\\
@@ -391,11 +392,11 @@ $$
   \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
     -\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}  \\\\
   \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
-    -\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
+    -\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
   \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}})
-            + \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\cdot \\tilde{E}_{l, \\vec{r}}
+            + \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{l, \\vec{r}}
             &= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}}
-  \\end{align*}
+  \\end{aligned}
 $$
 
 
@@ -405,27 +406,27 @@ Frequency domain
 We can substitute in a time-harmonic fields
 
 $$
- \\begin{align*}
+ \\begin{aligned}
- \\tilde{E}_\\vec{r} &= \\tilde{E}_\\vec{r} e^{-\\imath \\omega l \\Delta_t} \\\\
+ \\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
- \\tilde{J}_\\vec{r} &= \\tilde{J}_\\vec{r} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t}
+ \\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t}
- \\end{align*}
+ \\end{aligned}
 $$
 
 resulting in
 
 $$
- \\begin{align*}
+ \\begin{aligned}
  \\tilde{\\partial}_t &\\Rightarrow (e^{ \\imath \\omega \\Delta_t} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{-\\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\
    \\hat{\\partial}_t &\\Rightarrow (1 - e^{-\\imath \\omega \\Delta_t}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{ \\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\
  \\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
- \\end{align*}
+ \\end{aligned}
 $$
 
 This gives the frequency-domain wave equation,
 
 $$
- \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_\\vec{r})
+ \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
-    -\\Omega^2 \\epsilon_\\vec{r} \\cdot \\tilde{E}_\\vec{r} = \\imath \\Omega \\tilde{J}_\\vec{r}
+    -\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = \\imath \\Omega \\tilde{J}_{\\vec{r}}
 $$
 
 
@@ -435,69 +436,69 @@ Plane waves and Dispersion relation
 With uniform material distribution and no sources
 
 $$
- \\begin{align*}
+ \\begin{aligned}
  \\mu_{\\vec{r} + \\frac{1}{2}} &= \\mu \\\\
- \\epsilon_\\vec{r} &= \\epsilon \\\\
+ \\epsilon_{\\vec{r}} &= \\epsilon \\\\
- \\tilde{J}_\\vec{r} &= 0 \\\\
+ \\tilde{J}_{\\vec{r}} &= 0 \\\\
- \\end{align*}
+ \\end{aligned}
 $$
 
 the frequency domain wave equation simplifies to
 
-$$ \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_\\vec{r} - \\Omega^2 \\epsilon \\mu \\tilde{E}_\\vec{r} = 0 $$
+$$ \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} - \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$
 
-Since \\( \\hat{\\nabla} \\cdot \\tilde{E}_\\vec{r} = 0 \\), we can simplify
+Since $\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}} = 0$, we can simplify
 
 $$
- \\begin{align*}
+ \\begin{aligned}
- \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_\\vec{r}
+ \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}}
-   &= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_\\vec{r}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_\\vec{r} \\\\
+   &= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
-   &= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_\\vec{r} \\\\
+   &= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
-   &= - \\tilde{\\nabla}^2 \\tilde{E}_\\vec{r}
+   &= - \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}}
- \\end{align*}
+ \\end{aligned}
 $$
 
 and we get
 
-$$  \\tilde{\\nabla}^2 \\tilde{E}_\\vec{r} + \\Omega^2 \\epsilon \\mu \\tilde{E}_\\vec{r} = 0 $$
+$$  \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}} + \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$
 
 We can convert this to three scalar-wave equations of the form
 
-$$ (\\tilde{\\nabla}^2 + K^2) \\phi_\\vec{r} = 0 $$
+$$ (\\tilde{\\nabla}^2 + K^2) \\phi_{\\vec{r}} = 0 $$
 
-with \\( K^2 = \\Omega^2 \\mu \\epsilon \\). Now we let
+with $K^2 = \\Omega^2 \\mu \\epsilon$. Now we let
 
-$$  \\phi_\\vec{r} = A e^{\\imath (k_x m \\Delta_x + k_y n \\Delta_y + k_z p \\Delta_z)}  $$
+$$  \\phi_{\\vec{r}} = A e^{\\imath (k_x m \\Delta_x + k_y n \\Delta_y + k_z p \\Delta_z)}  $$
 
 resulting in
 
 $$
- \\begin{align*}
+ \\begin{aligned}
  \\tilde{\\partial}_x &\\Rightarrow (e^{ \\imath k_x \\Delta_x} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{ \\imath k_x \\Delta_x / 2} = \\imath K_x e^{ \\imath k_x \\Delta_x / 2}\\\\
    \\hat{\\partial}_x &\\Rightarrow (1 - e^{-\\imath k_x \\Delta_x}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{-\\imath k_x \\Delta_x / 2} = \\imath K_x e^{-\\imath k_x \\Delta_x / 2}\\\\
  K_x &= 2 \\sin(k_x \\Delta_x / 2) / \\Delta_x \\\\
- \\end{align*}
+ \\end{aligned}
 $$
 
-with similar expressions for the y and z dimnsions (and \\( K_y, K_z \\)).
+with similar expressions for the y and z dimnsions (and $K_y, K_z$).
 
 This implies
 
 $$
-  \\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_\\vec{r} \\\\
+  \\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_{\\vec{r}} \\\\
   K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2
 $$
 
-where \\( c = \\sqrt{\\mu \\epsilon} \\).
+where $c = \\sqrt{\\mu \\epsilon}$.
 
-Assuming real \\( (k_x, k_y, k_z), \\omega \\) will be real only if
+Assuming real $(k_x, k_y, k_z), \\omega$ will be real only if
 
 $$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
 
-If \\( \\Delta_x = \\Delta_y = \\Delta_z \\), this simplifies to \\( c \\Delta_t < \\Delta_x / \\sqrt{3} \\).
+If $\\Delta_x = \\Delta_y = \\Delta_z$, this simplifies to $c \\Delta_t < \\Delta_x / \\sqrt{3}$.
 This last form can be interpreted as enforcing causality; the distance that light
-travels in one timestep (i.e., \\( c \\Delta_t \\)) must be less than the diagonal
+travels in one timestep (i.e., $c \\Delta_t$) must be less than the diagonal
-of the smallest cell ( \\( \\Delta_x / \\sqrt{3} \\) when on a uniform cubic grid).
+of the smallest cell ( $\\Delta_x / \\sqrt{3}$ when on a uniform cubic grid).
 
 
 Grid description
@@ -513,9 +514,9 @@ To get a better sense of how this works, let's start by drawing a grid with unif
 to make the illustration simpler; we need at least two cells in the x dimension to
 demonstrate how nonuniform `dx` affects the various components.
 
-Place the E fore-vectors at integer indices \\( r = (m, n, p) \\) and the H back-vectors
+Place the E fore-vectors at integer indices $r = (m, n, p)$ and the H back-vectors
-at fractional indices \\( r + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2},
+at fractional indices $r + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2},
-p + \\frac{1}{2}) \\). Remember that these are indices and not coordinates; they can
+p + \\frac{1}{2})$. Remember that these are indices and not coordinates; they can
 correspond to arbitrary (monotonically increasing) coordinates depending on the cell widths.
 
 Draw lines to denote the planes on which the H components and back-vectors are defined.

meanas/fdmath/functional.py

@@ -63,7 +63,7 @@ def curl_forward(dx_e: List[numpy.ndarray] = None) -> fdfield_updater_t:
 
     Returns:
         Function `f` for taking the discrete forward curl of a field,
-        `f(E)` -> curlE \\( = \\nabla_f \\times E \\)
+        `f(E)` -> curlE $= \\nabla_f \\times E$
     """
     Dx, Dy, Dz = deriv_forward(dx_e)
 
@@ -90,7 +90,7 @@ def curl_back(dx_h: List[numpy.ndarray] = None) -> fdfield_updater_t:
 
     Returns:
         Function `f` for taking the discrete backward curl of a field,
-        `f(H)` -> curlH \\( = \\nabla_b \\times H \\)
+        `f(H)` -> curlH $= \\nabla_b \\times H$
     """
     Dx, Dy, Dz = deriv_back(dx_h)
 

meanas/fdtd/__init__.py

@@ -13,7 +13,7 @@ we have
 
 $$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
 
-or, if \\( \\Delta_x = \\Delta_y = \\Delta_z \\), then \\( c \\Delta_t < \\frac{\\Delta_x}{\\sqrt{3}} \\).
+or, if $\\Delta_x = \\Delta_y = \\Delta_z$, then $c \\Delta_t < \\frac{\\Delta_x}{\\sqrt{3}}$.
 
 Based on this, we can set
 
@@ -27,21 +27,21 @@ Poynting Vector and Energy Conservation
 
 Let
 
-$$ \\begin{align*}
+$$ \\begin{aligned}
   \\tilde{S}_{l, l', \\vec{r}} &=& &\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}}  \\\\
   &=&  &\\vec{x} (\\tilde{E}^y_{l,m+1,n,p} \\hat{H}^z_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^z_{l,m+1,n,p} \\hat{H}^y_{l', \\vec{r} + \\frac{1}{2}}) \\\\
   & &+ &\\vec{y} (\\tilde{E}^z_{l,m,n+1,p} \\hat{H}^x_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^x_{l,m,n+1,p} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}}) \\\\
   & &+ &\\vec{z} (\\tilde{E}^x_{l,m,n,p+1} \\hat{H}^y_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^y_{l,m,n,p+1} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}})
-   \\end{align*}
+   \\end{aligned}
 $$
 
-where \\( \\vec{r} = (m, n, p) \\) and \\( \\otimes \\) is a modified cross product
+where $\\vec{r} = (m, n, p)$ and $\\otimes$ is a modified cross product
-in which the \\( \\tilde{E} \\) terms are shifted as indicated.
+in which the $\\tilde{E}$ terms are shifted as indicated.
 
 By taking the divergence and rearranging terms, we can show that
 
 $$
-  \\begin{align*}
+  \\begin{aligned}
   \\hat{\\nabla} \\cdot \\tilde{S}_{l, l', \\vec{r}}
    &= \\hat{\\nabla} \\cdot (\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}})  \\\\
    &= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l, \\vec{r}} -
@@ -49,30 +49,30 @@ $$
    &= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot
             (-\\tilde{\\partial}_t \\mu_{\\vec{r} + \\frac{1}{2}} \\hat{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} -
                 \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}}) -
-      \\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t \\tilde{\\epsilon}_\\vec{r} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} +
+      \\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t \\tilde{\\epsilon}_{\\vec{r}} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} +
                 \\tilde{J}_{l', \\vec{r}}) \\\\
    &= \\hat{H}_{l'} \\cdot (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
       \\tilde{E}_l \\cdot (\\epsilon / \\Delta_t )(\\tilde{E}_{l'+\\frac{1}{2}} - \\tilde{E}_{l'-\\frac{1}{2}})
       - \\hat{H}_{l'} \\cdot \\hat{M}_{l-1} - \\tilde{E}_l \\cdot \\tilde{J}_{l'} \\\\
-  \\end{align*}
+  \\end{aligned}
 $$
 
 where in the last line the spatial subscripts have been dropped to emphasize
-the time subscripts \\( l, l' \\), i.e.
+the time subscripts $l, l'$, i.e.
 
 $$
-  \\begin{align*}
+  \\begin{aligned}
   \\tilde{E}_l &= \\tilde{E}_{l, \\vec{r}} \\\\
   \\hat{H}_l &= \\tilde{H}_{l, \\vec{r} + \\frac{1}{2}}  \\\\
-  \\tilde{\\epsilon} &= \\tilde{\\epsilon}_\\vec{r}  \\\\
+  \\tilde{\\epsilon} &= \\tilde{\\epsilon}_{\\vec{r}}  \\\\
-  \\end{align*}
+  \\end{aligned}
 $$
 
 etc.
-For \\( l' = l + \\frac{1}{2} \\) we get
+For $l' = l + \\frac{1}{2}$ we get
 
 $$
-  \\begin{align*}
+  \\begin{aligned}
   \\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}}
    &= \\hat{H}_{l + \\frac{1}{2}} \\cdot
             (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
@@ -87,13 +87,13 @@ $$
        +\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
       - \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
       - \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
-  \\end{align*}
+  \\end{aligned}
 $$
 
-and for \\( l' = l - \\frac{1}{2} \\),
+and for $l' = l - \\frac{1}{2}$,
 
 $$
-  \\begin{align*}
+  \\begin{aligned}
   \\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}}
    &=  (\\mu \\hat{H}^2_{l - \\frac{1}{2}}
        +\\epsilon \\tilde{E}_{l-1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\
@@ -101,7 +101,7 @@ $$
        +\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
       - \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
       - \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
-  \\end{align*}
+  \\end{aligned}
 $$
 
 These two results form the discrete time-domain analogue to Poynting's theorem.
@@ -109,16 +109,16 @@ They hint at the expressions for the energy, which can be calculated at the same
 time-index as either the E or H field:
 
 $$
- \\begin{align*}
+ \\begin{aligned}
  U_l &= \\epsilon \\tilde{E}^2_l + \\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}} \\\\
  U_{l + \\frac{1}{2}} &= \\epsilon \\tilde{E}_l \\cdot \\tilde{E}_{l + 1} + \\mu \\hat{H}^2_{l + \\frac{1}{2}} \\\\
- \\end{align*}
+ \\end{aligned}
 $$
 
 Rewriting the Poynting theorem in terms of the energy expressions,
 
 $$
-  \\begin{align*}
+  \\begin{aligned}
   (U_{l+\\frac{1}{2}} - U_l) / \\Delta_t
    &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\
       - \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
@@ -127,14 +127,14 @@ $$
    &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\
       - \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
       - \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
- \\end{align*}
+ \\end{aligned}
 $$
 
 This result is exact an should practically hold to within numerical precision. No time-
 or spatial-averaging is necessary.
 
-Note that each value of \\( J \\) contributes to the energy twice (i.e. once per field update)
+Note that each value of $J$ contributes to the energy twice (i.e. once per field update)
-despite only causing the value of \\( E \\) to change once (same for \\( M \\) and \\( H \\)).
+despite only causing the value of $E$ to change once (same for $M$ and $H$).
 
 
 Sources
@@ -149,8 +149,8 @@ shape. It can be written
 
 $$ f_r(t) = (1 - \\frac{1}{2} (\\omega (t - \\tau))^2) e^{-(\\frac{\\omega (t - \\tau)}{2})^2} $$
 
-with \\( \\tau > \\frac{2 * \\pi}{\\omega} \\) as a minimum delay to avoid a discontinuity at
+with $\\tau > \\frac{2 * \\pi}{\\omega}$ as a minimum delay to avoid a discontinuity at
-t=0 (assuming the source is off for t<0 this gives \\( \\sim 10^{-3} \\) error at t=0).
+t=0 (assuming the source is off for t<0 this gives $\\sim 10^{-3}$ error at t=0).
 
 
 

pdoc_templates/config.mako

@@ -18,8 +18,9 @@
     #git_link_template = 'https://github.com/USER/PROJECT/blob/{commit}/{path}#L{start_line}-L{end_line}'
     #git_link_template = 'https://gitlab.com/USER/PROJECT/blob/{commit}/{path}#L{start_line}-L{end_line}'
     #git_link_template = 'https://bitbucket.org/USER/PROJECT/src/{commit}/{path}#lines-{start_line}:{end_line}'
-    #git_link_template = 'https://CGIT_HOSTNAME/PROJECT/tree/{path}?id={commit}#n{start-line}'
+    #git_link_template = 'https://CGIT_HOSTNAME/PROJECT/tree/{path}?id={commit}#n{start_line}'
-    git_link_template = None
+    #git_link_template = None
+    git_link_template = 'https://mpxd.net/code/jan/fdfd_tools/src/commit/{commit}/{path}#L{start_line}-L{end_line}'
 
     # A prefix to use for every HTML hyperlink in the generated documentation.
     # No prefix results in all links being relative.

pdoc_templates/html.mako

@@ -2,7 +2,10 @@
   import os
 
   import pdoc
-  from pdoc.html_helpers import extract_toc, glimpse, to_html as _to_html, format_git_link
+  from pdoc.html_helpers import extract_toc, glimpse, to_html as _to_html, format_git_link, _md, to_markdown
+
+  from markdown.inlinepatterns import InlineProcessor
+  from markdown.util import AtomicString, etree
 
 
   def link(d, name=None, fmt='{}'):
@@ -14,8 +17,33 @@
     return '<a title="{}" href="{}">{}</a>'.format(d.refname, url, name)
 
 
-  def to_html(text):
+  # Altered latex delimeters (allow inline $...$, wrap in <eq></eq>)
-    return _to_html(text, module=module, link=link, latex_math=latex_math)
+  class _MathPattern(InlineProcessor):
+      NAME = 'pdoc-math'
+      PATTERN = r'(?<!\S|\\)(?:\\\((.+?)\\\)|\\\[(.+?)\\\]|\$\$(.+?)\$\$|\$(\S.*?)\$)'
+      PRIORITY = 181  # Larger than that of 'escape' pattern
+
+      def handleMatch(self, m, data):
+          for value, is_block in zip(m.groups(), (False, True, True, False)):
+              if value:
+                  break
+          wrapper = etree.Element('eq')
+          wrapper.text = AtomicString(value)
+          return wrapper, m.start(0), m.end(0)
+
+  def to_html(text: str):
+      if not latex_math and _MathPattern.NAME in _md.inlinePatterns:
+          _md.inlinePatterns.deregister(_MathPattern.NAME)
+      elif latex_math and _MathPattern.NAME not in _md.inlinePatterns:
+          _md.inlinePatterns.register(_MathPattern(_MathPattern.PATTERN),
+                                      _MathPattern.NAME,
+                                      _MathPattern.PRIORITY)
+      md = to_markdown(text, docformat='numpy,google', module=module, link=link)
+      return _md.reset().convert(md)
+
+
+#  def to_html(text):
+#    return _to_html(text, module=module, link=link, latex_math=latex_math)
 %>
 
 <%def name="ident(name)"><span class="ident">${name}</span></%def>
@@ -377,10 +405,6 @@
     </script><script async src='https://www.google-analytics.com/analytics.js'></script>
   % endif
 
-  % if latex_math:
-    <script async src='https://mpxd.net/scripts/MathJax/MathJax.js?config=TeX-AMS_CHTML'></script>
-  % endif
-
   <%include file="head.mako"/>
 </head>
 <body>

pdoc_templates/pdf.mako

@@ -0,0 +1,185 @@
+<%!
+    import re
+    import pdoc
+    from pdoc.html_helpers import to_markdown, format_git_link
+
+    def link(d, fmt='{}'):
+        name = fmt.format(d.qualname + ('()' if isinstance(d, pdoc.Function) else ''))
+        if isinstance(d, pdoc.External):
+            return name
+        return '[{}](#{})'.format(name, d.refname)
+
+    def _to_md(text, module):
+        text = to_markdown(text, module=module, link=link)
+        # Setext H2 headings to atx H2 headings
+        text = re.sub(r'\n(.+)\n-{3,}\n', r'\n## \1\n\n', text)
+        # Convert admonitions into simpler paragraphs, dedent contents
+        text = re.sub(r'^(?P<indent>( *))!!! \w+ \"([^\"]*)\"(.*(?:\n(?P=indent) +.*)*)',
+                      lambda m: '{}**{}:** {}'.format(m.group(2), m.group(3),
+                                                      re.sub('\n {,4}', '\n', m.group(4))),
+                      text, flags=re.MULTILINE)
+        return text
+
+    def subh(text, level=2):
+        # Deepen heading levels so H2 becomes H4 etc.
+        return re.sub(r'\n(#+) +(.+)\n', r'\n%s\1 \2\n' % ('#' * level), text)
+%>
+
+<%def name="title(level, string, id=None)">
+    <% id = ' {#%s}' % id if id is not None else '' %>
+${('#' * level) + ' ' + string + id}
+</%def>
+
+<%def name="funcdef(f)">
+    <%
+        returns = show_type_annotations and f.return_annotation() or ''
+        if returns:
+            returns = ' -> ' + returns
+    %>
+> `${f.funcdef()} ${f.name}(${', '.join(f.params(annotate=show_type_annotations))})${returns}`
+</%def>
+
+<%def name="classdef(c)">
+> `class ${c.name}(${', '.join(c.params(annotate=show_type_annotations))})`
+</%def>
+
+<%def name="show_source(d)">
+  % if (show_source_code or git_link_template) and d.source and d.obj is not getattr(d.inherits, 'obj', None):
+    <% git_link = format_git_link(git_link_template, d) %>
+[[view code]](${git_link})
+  %endif
+</%def>
+
+---
+description: |
+    API documentation for modules: ${', '.join(m.name for m in modules)}.
+
+lang: en
+
+classoption: oneside
+geometry: margin=1in
+papersize: a4
+
+linkcolor: blue
+links-as-notes: true
+...
+% for module in modules:
+<%
+    submodules = module.submodules()
+    variables = module.variables()
+    functions = module.functions()
+    classes = module.classes()
+
+    def to_md(text):
+        return _to_md(text, module)
+%>
+
+-------------------------------------------
+
+${title(1, ('Namespace' if module.is_namespace else 'Module') + ' `%s`' % module.name, module.refname)}
+${module.docstring | to_md}
+
+% if submodules:
+${title(2, 'Sub-modules')}
+    % for m in submodules:
+* [${m.name}](#${m.refname})
+    % endfor
+% endif
+
+% if variables:
+${title(2, 'Variables')}
+    % for v in variables:
+${title(3, 'Variable `%s`' % v.name, v.refname)}
+${show_source(v)}
+${v.docstring | to_md, subh, subh}
+    % endfor
+% endif
+
+% if functions:
+${title(2, 'Functions')}
+    % for f in functions:
+${title(3, 'Function `%s`' % f.name, f.refname)}
+${show_source(f)}
+
+${funcdef(f)}
+
+${f.docstring | to_md, subh, subh}
+    % endfor
+% endif
+
+% if classes:
+${title(2, 'Classes')}
+    % for cls in classes:
+${title(3, 'Class `%s`' % cls.name, cls.refname)}
+${show_source(cls)}
+
+${classdef(cls)}
+
+${cls.docstring | to_md, subh}
+<%
+    class_vars = cls.class_variables(show_inherited_members, sort=sort_identifiers)
+    static_methods = cls.functions(show_inherited_members, sort=sort_identifiers)
+    inst_vars = cls.instance_variables(show_inherited_members, sort=sort_identifiers)
+    methods = cls.methods(show_inherited_members, sort=sort_identifiers)
+    mro = cls.mro()
+    subclasses = cls.subclasses()
+%>
+        % if mro:
+${title(4, 'Ancestors (in MRO)')}
+            % for c in mro:
+* [${c.refname}](#${c.refname})
+            % endfor
+        % endif
+
+        % if subclasses:
+${title(4, 'Descendants')}
+            % for c in subclasses:
+* [${c.refname}](#${c.refname})
+            % endfor
+        % endif
+
+        % if class_vars:
+${title(4, 'Class variables')}
+            % for v in class_vars:
+${title(5, 'Variable `%s`' % v.name, v.refname)}
+${v.docstring | to_md, subh, subh}
+            % endfor
+        % endif
+
+        % if inst_vars:
+${title(4, 'Instance variables')}
+            % for v in inst_vars:
+${title(5, 'Variable `%s`' % v.name, v.refname)}
+${v.docstring | to_md, subh, subh}
+            % endfor
+        % endif
+
+        % if static_methods:
+${title(4, 'Static methods')}
+            % for f in static_methods:
+${title(5, '`Method %s`' % f.name, f.refname)}
+
+${funcdef(f)}
+
+${f.docstring | to_md, subh, subh}
+            % endfor
+        % endif
+
+        % if methods:
+${title(4, 'Methods')}
+            % for f in methods:
+${title(5, 'Method `%s`' % f.name, f.refname)}
+
+${funcdef(f)}
+
+${f.docstring | to_md, subh, subh}
+            % endfor
+        % endif
+    % endfor
+% endif
+
+##\## for module in modules:
+% endfor
+
+-----
+Generated by *pdoc* ${pdoc.__version__} (<https://pdoc3.github.io>).

pdoc_templates/pdoc.css

@@ -0,0 +1,381 @@
+  .flex {
+    display: flex !important;
+  }
+
+  body {
+    line-height: 1.5em;
+    background: black;
+    color: #DDD;
+    max-width: 140ch;
+  }
+
+  #content {
+    padding: 20px;
+  }
+
+  #sidebar {
+    padding: 30px;
+    overflow: hidden;
+  }
+
+  .http-server-breadcrumbs {
+    font-size: 130%;
+    margin: 0 0 15px 0;
+  }
+
+  #footer {
+    font-size: .75em;
+    padding: 5px 30px;
+    border-top: 1px solid #ddd;
+    text-align: right;
+  }
+    #footer p {
+      margin: 0 0 0 1em;
+      display: inline-block;
+    }
+    #footer p:last-child {
+      margin-right: 30px;
+    }
+
+  h1, h2, h3, h4, h5 {
+    font-weight: 300;
+  }
+  h1 {
+    font-size: 2.5em;
+    line-height: 1.1em;
+    border-top: 20px white;
+  }
+  h2 {
+    font-size: 1.75em;
+    margin: 1em 0 .50em 0;
+  }
+  h3 {
+    font-size: 1.4em;
+    margin: 25px 0 10px 0;
+  }
+  h4 {
+    margin: 0;
+    font-size: 105%;
+  }
+
+  a {
+    color: #999;
+    text-decoration: none;
+    transition: color .3s ease-in-out;
+  }
+  a:hover {
+    color: #18d;
+  }
+
+  .title code {
+    font-weight: bold;
+  }
+  h2[id^="header-"] {
+    margin-top: 2em;
+  }
+  .ident {
+    color: #7ff;
+  }
+
+  pre code {
+    background: transparent;
+    font-size: .8em;
+    line-height: 1.4em;
+  }
+  code {
+    background: #0d0d0e;
+    padding: 1px 4px;
+    overflow-wrap: break-word;
+  }
+  h1 code { background: transparent }
+
+  pre {
+    background: #111;
+    border: 0;
+    border-top: 1px solid #ccc;
+    border-bottom: 1px solid #ccc;
+    margin: 1em 0;
+    padding: 1ex;
+  }
+
+  #http-server-module-list {
+    display: flex;
+    flex-flow: column;
+  }
+    #http-server-module-list div {
+      display: flex;
+    }
+    #http-server-module-list dt {
+      min-width: 10%;
+    }
+    #http-server-module-list p {
+      margin-top: 0;
+    }
+
+  .toc ul,
+  #index {
+    list-style-type: none;
+    margin: 0;
+    padding: 0;
+  }
+    #index code {
+      background: transparent;
+    }
+    #index h3 {
+      border-bottom: 1px solid #ddd;
+    }
+    #index ul {
+      padding: 0;
+    }
+    #index h4 {
+      font-weight: bold;
+    }
+    #index h4 + ul {
+      margin-bottom:.6em;
+    }
+    /* Make TOC lists have 2+ columns when viewport is wide enough.
+       Assuming ~20-character identifiers and ~30% wide sidebar. */
+    @media (min-width: 200ex) { #index .two-column { column-count: 2 } }
+    @media (min-width: 300ex) { #index .two-column { column-count: 3 } }
+
+  dl {
+    margin-bottom: 2em;
+  }
+    dl dl:last-child {
+      margin-bottom: 4em;
+    }
+  dd {
+    margin: 0 0 1em 3em;
+  }
+    #header-classes + dl > dd {
+      margin-bottom: 3em;
+    }
+    dd dd {
+      margin-left: 2em;
+    }
+    dd p {
+      margin: 10px 0;
+    }
+    blockquote code {
+      background: #111;
+      font-weight: bold;
+      font-size: .85em;
+      padding: 5px 10px;
+      display: inline-block;
+      min-width: 40%;
+    }
+      blockquote code:hover {
+        background: #101010;
+      }
+      .name > span:first-child {
+        white-space: nowrap;
+      }
+      .name.class > span:nth-child(2) {
+        margin-left: .4em;
+      }
+    .inherited {
+      color: #777;
+      border-left: 5px solid #eee;
+      padding-left: 1em;
+    }
+    .inheritance em {
+      font-style: normal;
+      font-weight: bold;
+    }
+
+    /* Docstrings titles, e.g. in numpydoc format */
+    .desc h2 {
+      font-weight: 400;
+      font-size: 1.25em;
+    }
+    .desc h3 {
+      font-size: 1em;
+    }
+    .desc dt code {
+      background: inherit;  /* Don't grey-back parameters */
+    }
+
+    .source summary,
+    .git-link-div {
+      color: #aaa;
+      text-align: right;
+      font-weight: 400;
+      font-size: .8em;
+      text-transform: uppercase;
+    }
+      .source summary > * {
+        white-space: nowrap;
+        cursor: pointer;
+      }
+      .git-link {
+        color: inherit;
+        margin-left: 1em;
+      }
+    .source pre {
+      max-height: 500px;
+      overflow: auto;
+      margin: 0;
+    }
+    .source pre code {
+      font-size: 12px;
+      overflow: visible;
+    }
+  .hlist {
+    list-style: none;
+  }
+    .hlist li {
+      display: inline;
+    }
+    .hlist li:after {
+      content: ',\2002';
+    }
+    .hlist li:last-child:after {
+      content: none;
+    }
+    .hlist .hlist {
+      display: inline;
+      padding-left: 1em;
+    }
+
+  img {
+    max-width: 100%;
+  }
+
+  .admonition {
+    padding: .1em .5em;
+    margin-bottom: 1em;
+  }
+    .admonition-title {
+      font-weight: bold;
+    }
+    .admonition.note,
+    .admonition.info,
+    .admonition.important {
+      background: #610;
+    }
+    .admonition.todo,
+    .admonition.versionadded,
+    .admonition.tip,
+    .admonition.hint {
+      background: #202;
+    }
+    .admonition.warning,
+    .admonition.versionchanged,
+    .admonition.deprecated {
+      background: #02b;
+    }
+    .admonition.error,
+    .admonition.danger,
+    .admonition.caution {
+      background: darkpink;
+    }
+
+  @media screen and (min-width: 700px) {
+    #sidebar {
+      width: 30%;
+    }
+    #content {
+      width: 70%;
+      max-width: 100ch;
+      padding: 3em 4em;
+      border-left: 1px solid #ddd;
+    }
+    pre code {
+      font-size: 1em;
+    }
+    .item .name {
+      font-size: 1em;
+    }
+    main {
+      display: flex;
+      flex-direction: row-reverse;
+      justify-content: flex-end;
+    }
+    .toc ul ul,
+    #index ul {
+      padding-left: 1.5em;
+    }
+    .toc > ul > li {
+      margin-top: .5em;
+    }
+  }
+
+@media print {
+  #sidebar h1 {
+    page-break-before: always;
+  }
+  .source {
+    display: none;
+  }
+}
+@media print {
+    * {
+        background: transparent !important;
+        color: #000 !important; /* Black prints faster: h5bp.com/s */
+        box-shadow: none !important;
+        text-shadow: none !important;
+    }
+
+    a[href]:after {
+        content: " (" attr(href) ")";
+        font-size: 90%;
+    }
+    /* Internal, documentation links, recognized by having a title,
+       don't need the URL explicity stated. */
+    a[href][title]:after {
+        content: none;
+    }
+
+    abbr[title]:after {
+        content: " (" attr(title) ")";
+    }
+
+    /*
+     * Don't show links for images, or javascript/internal links
+     */
+
+    .ir a:after,
+    a[href^="javascript:"]:after,
+    a[href^="#"]:after {
+        content: "";
+    }
+
+    pre,
+    blockquote {
+        border: 1px solid #999;
+        page-break-inside: avoid;
+    }
+
+    thead {
+        display: table-header-group; /* h5bp.com/t */
+    }
+
+    tr,
+    img {
+        page-break-inside: avoid;
+    }
+
+    img {
+        max-width: 100% !important;
+    }
+
+    @page {
+        margin: 0.5cm;
+    }
+
+    p,
+    h2,
+    h3 {
+        orphans: 3;
+        widows: 3;
+    }
+
+    h1,
+    h2,
+    h3,
+    h4,
+    h5,
+    h6 {
+        page-break-after: avoid;
+    }
+}