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cleanup latex

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This commit is contained in:
Jan Petykiewicz 2025-01-14 22:34:52 -08:00
parent 53d5812b4a
commit 4afc6cf62e
1 changed files with 6 additions and 6 deletions
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12
meanas/fdfd/waveguide_2d.py
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@ -530,22 +530,22 @@ def exy2e(
dxes: dx_lists_t, dxes: dx_lists_t,
epsilon: vfdfield_t, epsilon: vfdfield_t,
) -> sparse.spmatrix: ) -> sparse.spmatrix:
""" r"""
Operator which transforms the vector `e_xy` containing the vectorized E_x and E_y fields, Operator which transforms the vector `e_xy` containing the vectorized E_x and E_y fields,
into a vectorized E containing all three E components into a vectorized E containing all three E components
From the operator derivation (see module docs), we have From the operator derivation (see module docs), we have
$$ $$
\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 \\
$$ $$
as well as the intermediate equations as well as the intermediate equations
$$ $$
\begin{aligned} \begin{aligned}
\gamma H_y &= \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\ \imath \beta 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 \beta H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
\end{aligned} \end{aligned}
$$ $$
@ -553,11 +553,11 @@ def exy2e(
$$ $$
\begin{aligned} \begin{aligned}
E_z &= \frac{1}{\imath \omega \gamma \epsilon_{zz}} (( E_z &= \frac{1}{- \omega \beta \epsilon_{zz}} ((
\hat{\partial}_y \hat{\partial}_x H_z \hat{\partial}_y \hat{\partial}_x H_z
-\hat{\partial}_x \hat{\partial}_y H_z) -\hat{\partial}_x \hat{\partial}_y H_z)
+ \imath \omega (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y)) + \imath \omega (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y))
&= \frac{1}{\gamma \epsilon_{zz}} (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y) &= \frac{1}{\imath \beta \epsilon_{zz}} (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y)
\end{aligned} \end{aligned}
$$ $$