more doc updates

master
Jan Petykiewicz 4 years ago
parent 7d8901539c
commit 6e3cc1c3bd

3
.gitignore vendored

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

@ -1,3 +1,15 @@
#!/bin/bash #!/bin/bash
cd ~/projects/meanas 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 {} \;

@ -14,7 +14,20 @@ Submodules:
- `waveguide_3d`: Functions for transforming `waveguide_2d` results into 3D. - `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 FDFD?
# TODO PML # TODO PML

@ -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]: 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 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: Note:
This function also shifts the input `E` field by one cell as required This function also shifts the input `E` field by one cell as required

@ -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 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, 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{E}(x, y, z) &= -\\imath \\omega \\mu \\vec{H} \\\\
\\nabla \\times \\vec{H}(x, y, z) &= \\imath \\omega \\epsilon \\vec{E} \\\\ \\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{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} \\\\ \\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 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_{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_{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 \\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_{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_{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 \\\\ \\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_{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_{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 \\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_{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_{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 \\\\ \\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 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_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 \\\\ \\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 \\\\ \\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 \\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 \\\\ - \\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) \\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) \\\\ \\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) \\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) \\\\ \\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 ( -\\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}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) + \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
@ -96,30 +96,30 @@ $$
\\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x) \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) + \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
)\\\\ )\\\\
\\end{align*} \\end{aligned}
$$ $$
However, based on our rewritten equation for \\( \\gamma H_x \\) and the so-far unused However, based on our rewritten equation for $\\gamma H_x$ and the so-far unused
equation for \\( \\imath \\omega \\mu_{zz} H_z \\) we can also write equation for $\\imath \\omega \\mu_{zz} H_z$ we can also write
$$ $$
\\begin{align*} \\begin{aligned}
-\\imath \\omega \\mu_{xx} (\\gamma H_x) &= -\\imath \\omega \\mu_{xx} (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\ -\\imath \\omega \\mu_{xx} (\\gamma H_x) &= -\\imath \\omega \\mu_{xx} (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y &= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
-\\imath \\omega \\mu_{xx} \\hat{\\partial}_x ( -\\imath \\omega \\mu_{xx} \\hat{\\partial}_x (
\\frac{1}{-\\imath \\omega \\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x)) \\\\ \\frac{1}{-\\imath \\omega \\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x)) \\\\
&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y &= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
+\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\ +\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{align*} \\end{aligned}
$$ $$
and, similarly, and, similarly,
$$ $$
\\begin{align*} \\begin{aligned}
-\\imath \\omega \\mu_{yy} (\\gamma H_y) &= -\\omega^2 \\mu_{yy} \\epsilon_{xx} E_x -\\imath \\omega \\mu_{yy} (\\gamma H_y) &= -\\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
+\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\ +\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{align*} \\end{aligned}
$$ $$
Using these, we can construct the eigenvalue problem Using these, we can construct the eigenvalue problem
@ -137,9 +137,9 @@ $$ \\beta^2 \\begin{bmatrix} E_x \\\\
E_y \\end{bmatrix} 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. 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} \\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. property distribution.
This operator can be used to form an eigenvalue problem of the form 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} \\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. property distribution.
This operator can be used to form an eigenvalue problem of the form This operator can be used to form an eigenvalue problem of the form

@ -43,11 +43,11 @@ Scalar derivatives and cell shifts
---------------------------------- ----------------------------------
Define the discrete forward derivative as Define the discrete forward derivative as
$$ [\\tilde{\\partial}_x f ]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$ $$ [\\tilde{\\partial}_x f]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$
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 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]` 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 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] 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 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: identical:
[figure: derivatives and cell sizes] [figure: derivatives and cell sizes]
@ -87,19 +87,20 @@ identical:
Periodic boundaries are used here and elsewhere unless otherwise noted. Periodic boundaries are used here and elsewhere unless otherwise noted.
In the above figure, 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. 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
$$ \\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x, m + 1}) $$ $$ \\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x, m + 1}) $$
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. 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. "derived" grid.
For the remainder of the `Discrete calculus` section, all figures will show 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. 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 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] [figure: divergence]
^^ ^^
@ -226,23 +227,23 @@ Curls
The two curls are then The two curls are then
$$ \\begin{align*} $$ \\begin{aligned}
\\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\ \\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}} &= [\\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{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{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}) &+ \\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 and
$$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} = $$ \\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}} $$ [\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$
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] [code: curls]
@ -286,27 +287,27 @@ Maxwell's Equations
If we discretize both space (m,n,p) and time (l), Maxwell's equations become 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}} \\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{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}} \\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{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
\\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\ \\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}} \\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
\\end{align*} $$ \\end{aligned} $$
with 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 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: The time derivatives can be expanded to form the update equations:
@ -375,12 +376,12 @@ and combining them with charge continuity,
Wave equation 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{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
-\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
- \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\ - \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
@ -391,11 +392,11 @@ $$
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &= \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
-\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\ -\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &= \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
-\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\ -\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}})
+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\cdot \\tilde{E}_{l, \\vec{r}} + \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{l, \\vec{r}}
&= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}} &= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}}
\\end{align*} \\end{aligned}
$$ $$
@ -405,27 +406,27 @@ Frequency domain
We can substitute in a time-harmonic fields We can substitute in a time-harmonic fields
$$ $$
\\begin{align*} \\begin{aligned}
\\tilde{E}_\\vec{r} &= \\tilde{E}_\\vec{r} e^{-\\imath \\omega l \\Delta_t} \\\\ \\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
\\tilde{J}_\\vec{r} &= \\tilde{J}_\\vec{r} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t}
\\end{align*} \\end{aligned}
$$ $$
resulting in resulting in
$$ $$
\\begin{align*} \\begin{aligned}
\\tilde{\\partial}_t &\\Rightarrow (e^{ \\imath \\omega \\Delta_t} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{-\\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\ \\tilde{\\partial}_t &\\Rightarrow (e^{ \\imath \\omega \\Delta_t} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{-\\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\
\\hat{\\partial}_t &\\Rightarrow (1 - e^{-\\imath \\omega \\Delta_t}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{ \\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\ \\hat{\\partial}_t &\\Rightarrow (1 - e^{-\\imath \\omega \\Delta_t}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{ \\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\
\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t \\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
\\end{align*} \\end{aligned}
$$ $$
This gives the frequency-domain wave equation, This gives the frequency-domain wave equation,
$$ $$
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_\\vec{r}) \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
-\\Omega^2 \\epsilon_\\vec{r} \\cdot \\tilde{E}_\\vec{r} = \\imath \\Omega \\tilde{J}_\\vec{r} -\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = \\imath \\Omega \\tilde{J}_{\\vec{r}}
$$ $$
@ -435,69 +436,69 @@ Plane waves and Dispersion relation
With uniform material distribution and no sources With uniform material distribution and no sources
$$ $$
\\begin{align*} \\begin{aligned}
\\mu_{\\vec{r} + \\frac{1}{2}} &= \\mu \\\\ \\mu_{\\vec{r} + \\frac{1}{2}} &= \\mu \\\\
\\epsilon_\\vec{r} &= \\epsilon \\\\ \\epsilon_{\\vec{r}} &= \\epsilon \\\\
\\tilde{J}_\\vec{r} &= 0 \\\\ \\tilde{J}_{\\vec{r}} &= 0 \\\\
\\end{align*} \\end{aligned}
$$ $$
the frequency domain wave equation simplifies to the frequency domain wave equation simplifies to
$$ \\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 $$
Since \\( \\hat{\\nabla} \\cdot \\tilde{E}_\\vec{r} = 0 \\), we can simplify Since $\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}} = 0$, we can simplify
$$ $$
\\begin{align*} \\begin{aligned}
\\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_\\vec{r} \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}}
&= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_\\vec{r}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_\\vec{r} \\\\ &= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
&= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_\\vec{r} \\\\ &= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
&= - \\tilde{\\nabla}^2 \\tilde{E}_\\vec{r} &= - \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}}
\\end{align*} \\end{aligned}
$$ $$
and we get and we get
$$ \\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 $$
We can convert this to three scalar-wave equations of the form We can convert this to three scalar-wave equations of the form
$$ (\\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)} $$
resulting in resulting in
$$ $$
\\begin{align*} \\begin{aligned}
\\tilde{\\partial}_x &\\Rightarrow (e^{ \\imath k_x \\Delta_x} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{ \\imath k_x \\Delta_x / 2} = \\imath K_x e^{ \\imath k_x \\Delta_x / 2}\\\\ \\tilde{\\partial}_x &\\Rightarrow (e^{ \\imath k_x \\Delta_x} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{ \\imath k_x \\Delta_x / 2} = \\imath K_x e^{ \\imath k_x \\Delta_x / 2}\\\\
\\hat{\\partial}_x &\\Rightarrow (1 - e^{-\\imath k_x \\Delta_x}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{-\\imath k_x \\Delta_x / 2} = \\imath K_x e^{-\\imath k_x \\Delta_x / 2}\\\\ \\hat{\\partial}_x &\\Rightarrow (1 - e^{-\\imath k_x \\Delta_x}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{-\\imath k_x \\Delta_x / 2} = \\imath K_x e^{-\\imath k_x \\Delta_x / 2}\\\\
K_x &= 2 \\sin(k_x \\Delta_x / 2) / \\Delta_x \\\\ K_x &= 2 \\sin(k_x \\Delta_x / 2) / \\Delta_x \\\\
\\end{align*} \\end{aligned}
$$ $$
with similar expressions for the y and z dimnsions (and \\( K_y, K_z \\)). with similar expressions for the y and z dimnsions (and $K_y, K_z$).
This implies This implies
$$ $$
\\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_\\vec{r} \\\\ \\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_{\\vec{r}} \\\\
K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2 K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2
$$ $$
where \\( c = \\sqrt{\\mu \\epsilon} \\). where $c = \\sqrt{\\mu \\epsilon}$.
Assuming real \\( (k_x, k_y, k_z), \\omega \\) will be real only if Assuming real $(k_x, k_y, k_z), \\omega$ will be real only if
$$ 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}) $$ $$ 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 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 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 to make the illustration simpler; we need at least two cells in the x dimension to
demonstrate how nonuniform `dx` affects the various components. 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. 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. Draw lines to denote the planes on which the H components and back-vectors are defined.

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

@ -13,7 +13,7 @@ we have
$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$ $$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
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 Based on this, we can set
@ -27,21 +27,21 @@ Poynting Vector and Energy Conservation
Let 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}} \\\\ \\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{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{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}}) & &+ &\\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 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{S}_{l, l', \\vec{r}}
&= \\hat{\\nabla} \\cdot (\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}}) \\\\ &= \\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}} - &= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l, \\vec{r}} -
@ -49,30 +49,30 @@ $$
&= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot &= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot
(-\\tilde{\\partial}_t \\mu_{\\vec{r} + \\frac{1}{2}} \\hat{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} - (-\\tilde{\\partial}_t \\mu_{\\vec{r} + \\frac{1}{2}} \\hat{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} -
\\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}}) - \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}}) -
\\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t \\tilde{\\epsilon}_\\vec{r} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} + \\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t \\tilde{\\epsilon}_{\\vec{r}} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} +
\\tilde{J}_{l', \\vec{r}}) \\\\ \\tilde{J}_{l', \\vec{r}}) \\\\
&= \\hat{H}_{l'} \\cdot (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) - &= \\hat{H}_{l'} \\cdot (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
\\tilde{E}_l \\cdot (\\epsilon / \\Delta_t )(\\tilde{E}_{l'+\\frac{1}{2}} - \\tilde{E}_{l'-\\frac{1}{2}}) \\tilde{E}_l \\cdot (\\epsilon / \\Delta_t )(\\tilde{E}_{l'+\\frac{1}{2}} - \\tilde{E}_{l'-\\frac{1}{2}})
- \\hat{H}_{l'} \\cdot \\hat{M}_{l-1} - \\tilde{E}_l \\cdot \\tilde{J}_{l'} \\\\ - \\hat{H}_{l'} \\cdot \\hat{M}_{l-1} - \\tilde{E}_l \\cdot \\tilde{J}_{l'} \\\\
\\end{align*} \\end{aligned}
$$ $$
where in the last line the spatial subscripts have been dropped to emphasize where in the last line the spatial subscripts have been dropped to emphasize
the time subscripts \\( l, l' \\), i.e. the time subscripts $l, l'$, i.e.
$$ $$
\\begin{align*} \\begin{aligned}
\\tilde{E}_l &= \\tilde{E}_{l, \\vec{r}} \\\\ \\tilde{E}_l &= \\tilde{E}_{l, \\vec{r}} \\\\
\\hat{H}_l &= \\tilde{H}_{l, \\vec{r} + \\frac{1}{2}} \\\\ \\hat{H}_l &= \\tilde{H}_{l, \\vec{r} + \\frac{1}{2}} \\\\
\\tilde{\\epsilon} &= \\tilde{\\epsilon}_\\vec{r} \\\\ \\tilde{\\epsilon} &= \\tilde{\\epsilon}_{\\vec{r}} \\\\
\\end{align*} \\end{aligned}
$$ $$
etc. etc.
For \\( l' = l + \\frac{1}{2} \\) we get For $l' = l + \\frac{1}{2}$ we get
$$ $$
\\begin{align*} \\begin{aligned}
\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}}
&= \\hat{H}_{l + \\frac{1}{2}} \\cdot &= \\hat{H}_{l + \\frac{1}{2}} \\cdot
(-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) - (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
@ -87,13 +87,13 @@ $$
+\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\ +\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\ - \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
- \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\ - \\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}} \\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}}
&= (\\mu \\hat{H}^2_{l - \\frac{1}{2}} &= (\\mu \\hat{H}^2_{l - \\frac{1}{2}}
+\\epsilon \\tilde{E}_{l-1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\ +\\epsilon \\tilde{E}_{l-1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\
@ -101,7 +101,7 @@ $$
+\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\ +\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\ - \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\ - \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
\\end{align*} \\end{aligned}
$$ $$
These two results form the discrete time-domain analogue to Poynting's theorem. These two results form the discrete time-domain analogue to Poynting's theorem.
@ -109,16 +109,16 @@ They hint at the expressions for the energy, which can be calculated at the same
time-index as either the E or H field: time-index as either the E or H field:
$$ $$
\\begin{align*} \\begin{aligned}
U_l &= \\epsilon \\tilde{E}^2_l + \\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}} \\\\ U_l &= \\epsilon \\tilde{E}^2_l + \\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}} \\\\
U_{l + \\frac{1}{2}} &= \\epsilon \\tilde{E}_l \\cdot \\tilde{E}_{l + 1} + \\mu \\hat{H}^2_{l + \\frac{1}{2}} \\\\ U_{l + \\frac{1}{2}} &= \\epsilon \\tilde{E}_l \\cdot \\tilde{E}_{l + 1} + \\mu \\hat{H}^2_{l + \\frac{1}{2}} \\\\
\\end{align*} \\end{aligned}
$$ $$
Rewriting the Poynting theorem in terms of the energy expressions, Rewriting the Poynting theorem in terms of the energy expressions,
$$ $$
\\begin{align*} \\begin{aligned}
(U_{l+\\frac{1}{2}} - U_l) / \\Delta_t (U_{l+\\frac{1}{2}} - U_l) / \\Delta_t
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\ &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\
- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\ - \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
@ -127,14 +127,14 @@ $$
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\ &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\
- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\ - \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\ - \\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- This result is exact an should practically hold to within numerical precision. No time-
or spatial-averaging is necessary. 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 Sources
@ -149,8 +149,8 @@ shape. It can be written
$$ f_r(t) = (1 - \\frac{1}{2} (\\omega (t - \\tau))^2) e^{-(\\frac{\\omega (t - \\tau)}{2})^2} $$ $$ f_r(t) = (1 - \\frac{1}{2} (\\omega (t - \\tau))^2) e^{-(\\frac{\\omega (t - \\tau)}{2})^2} $$
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).

@ -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://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://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://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. # A prefix to use for every HTML hyperlink in the generated documentation.
# No prefix results in all links being relative. # No prefix results in all links being relative.

@ -2,7 +2,10 @@
import os import os
import pdoc 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='{}'): def link(d, name=None, fmt='{}'):
@ -14,8 +17,33 @@
return '<a title="{}" href="{}">{}</a>'.format(d.refname, url, name) 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">${name}</span></%def> <%def name="ident(name)"><span class="ident">${name}</span></%def>
@ -377,10 +405,6 @@
</script><script async src='https://www.google-analytics.com/analytics.js'></script> </script><script async src='https://www.google-analytics.com/analytics.js'></script>
% endif % endif
% if latex_math:
<script async src='https://mpxd.net/scripts/MathJax/MathJax.js?config=TeX-AMS_CHTML'></script>
% endif
<%include file="head.mako"/> <%include file="head.mako"/>
</head> </head>
<body> <body>

@ -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>).

@ -0,0 +1,381 @@
.flex {
display: flex !important;
}
body {
line-height: 1.5em;
background: black;
color: #DDD;
max-width: 140ch;
}
#content {
padding: 20px;
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overflow: hidden;
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font-size: 130%;
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font-size: .75em;
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a {
color: #999;
text-decoration: none;
transition: color .3s ease-in-out;
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color: #18d;
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.title code {
font-weight: bold;
}
h2[id^="header-"] {
margin-top: 2em;
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.ident {
color: #7ff;
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pre code {
background: transparent;
font-size: .8em;
line-height: 1.4em;
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code {
background: #0d0d0e;
padding: 1px 4px;
overflow-wrap: break-word;
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h1 code { background: transparent }
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background: #111;
border: 0;
border-top: 1px solid #ccc;
border-bottom: 1px solid #ccc;
margin: 1em 0;
padding: 1ex;
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#http-server-module-list {
display: flex;
flex-flow: column;
}
#http-server-module-list div {
display: flex;
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min-width: 10%;
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margin-top: 0;
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.toc ul,
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list-style-type: none;
margin: 0;
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background: transparent;
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border-bottom: 1px solid #ddd;
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#index ul {
padding: 0;
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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;
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#header-classes + dl > dd {
margin-bottom: 3em;
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dd dd {
margin-left: 2em;
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margin: 10px 0;
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blockquote code {
background: #111;
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padding: 5px 10px;
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.name > span:first-child {
white-space: nowrap;
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border-left: 5px solid #eee;
padding-left: 1em;
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.inheritance em {
font-style: normal;
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/* Docstrings titles, e.g. in numpydoc format */
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font-weight: 400;
font-size: 1.25em;
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font-size: 1em;
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.desc dt code {
background: inherit; /* Don't grey-back parameters */
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.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;
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.git-link {
color: inherit;
margin-left: 1em;
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.source pre {
max-height: 500px;
overflow: auto;
margin: 0;
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list-style: none;
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img {
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font-weight: bold;
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.admonition.note,
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background: #610;
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@media screen and (min-width: 700px) {
#sidebar {
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padding: 3em 4em;
border-left: 1px solid #ddd;
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font-size: 1em;
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display: flex;
flex-direction: row-reverse;
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.toc ul ul,
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padding-left: 1.5em;
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.toc > ul > li {
margin-top: .5em;
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#sidebar h1 {
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.source {
display: none;
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}
@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;
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thead {
display: table-header-group; /* h5bp.com/t */
}
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page-break-inside: avoid;
}
img {
max-width: 100% !important;
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@page {
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p,
h2,
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orphans: 3;
widows: 3;
}
h1,
h2,
h3,
h4,
h5,
h6 {
page-break-after: avoid;
}
}
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