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Use raw strings to avoid double backslashes

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Jan Petykiewicz 2024-07-15 16:32:31 -07:00
commit 77715da8b4
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meanas/fdfd/__init__.py
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@ -1,4 +1,4 @@
"""
r"""
Tools for finite difference frequency-domain (FDFD) simulations and calculations.
These mostly involve picking a single frequency, then setting up and solving a
@ -19,71 +19,71 @@ Submodules:
From the "Frequency domain" section of `meanas.fdmath`, we have
$$
\\begin{aligned}
\\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
\\tilde{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= \\tilde{H}_{\\vec{r} + \\frac{1}{2}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\
\\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\
\\tilde{M}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= \\tilde{M}_{\\vec{r} + \\frac{1}{2}} e^{-\\imath \\omega l \\Delta_t} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
-\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} &= -\\imath \\Omega \\tilde{J}_{\\vec{r}} e^{\\imath \\omega \\Delta_t / 2} \\\\
\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
\\end{aligned}
\begin{aligned}
\tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\
\tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{H}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
\tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
\tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{M}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega l \Delta_t} \\
\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
-\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} &= -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\
\Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t
\end{aligned}
$$
resulting in
$$
\\begin{aligned}
\\tilde{\\partial}_t &\\Rightarrow -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\
\\hat{\\partial}_t &\\Rightarrow -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\
\\end{aligned}
\begin{aligned}
\tilde{\partial}_t &\Rightarrow -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\
\hat{\partial}_t &\Rightarrow -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\
\end{aligned}
$$
Maxwell's equations are then
$$
\\begin{aligned}
\\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} &=
\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2} \\hat{B}_{\\vec{r} + \\frac{1}{2}}
- \\hat{M}_{\\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times \\hat{H}_{\\vec{r} + \\frac{1}{2}} &=
-\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2} \\tilde{D}_{\\vec{r}}
+ \\tilde{J}_{\\vec{r}} \\\\
\\tilde{\\nabla} \\cdot \\hat{B}_{\\vec{r} + \\frac{1}{2}} &= 0 \\\\
\\hat{\\nabla} \\cdot \\tilde{D}_{\\vec{r}} &= \\rho_{\\vec{r}}
\\end{aligned}
\begin{aligned}
\tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
\imath \Omega e^{-\imath \omega \Delta_t / 2} \hat{B}_{\vec{r} + \frac{1}{2}}
- \hat{M}_{\vec{r} + \frac{1}{2}} \\
\hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
-\imath \Omega e^{ \imath \omega \Delta_t / 2} \tilde{D}_{\vec{r}}
+ \tilde{J}_{\vec{r}} \\
\tilde{\nabla} \cdot \hat{B}_{\vec{r} + \frac{1}{2}} &= 0 \\
\hat{\nabla} \cdot \tilde{D}_{\vec{r}} &= \rho_{\vec{r}}
\end{aligned}
$$
With $\\Delta_t \\to 0$, this simplifies to
With $\Delta_t \to 0$, this simplifies to
$$
\\begin{aligned}
\\tilde{E}_{l, \\vec{r}} &\\to \\tilde{E}_{\\vec{r}} \\\\
\\tilde{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &\\to \\tilde{H}_{\\vec{r} + \\frac{1}{2}} \\\\
\\tilde{J}_{l, \\vec{r}} &\\to \\tilde{J}_{\\vec{r}} \\\\
\\tilde{M}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &\\to \\tilde{M}_{\\vec{r} + \\frac{1}{2}} \\\\
\\Omega &\\to \\omega \\\\
\\tilde{\\partial}_t &\\to -\\imath \\omega \\\\
\\hat{\\partial}_t &\\to -\\imath \\omega \\\\
\\end{aligned}
\begin{aligned}
\tilde{E}_{l, \vec{r}} &\to \tilde{E}_{\vec{r}} \\
\tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{H}_{\vec{r} + \frac{1}{2}} \\
\tilde{J}_{l, \vec{r}} &\to \tilde{J}_{\vec{r}} \\
\tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{M}_{\vec{r} + \frac{1}{2}} \\
\Omega &\to \omega \\
\tilde{\partial}_t &\to -\imath \omega \\
\hat{\partial}_t &\to -\imath \omega \\
\end{aligned}
$$
and then
$$
\\begin{aligned}
\\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} &=
\\imath \\omega \\hat{B}_{\\vec{r} + \\frac{1}{2}}
- \\hat{M}_{\\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times \\hat{H}_{\\vec{r} + \\frac{1}{2}} &=
-\\imath \\omega \\tilde{D}_{\\vec{r}}
+ \\tilde{J}_{\\vec{r}} \\\\
\\end{aligned}
\begin{aligned}
\tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
\imath \omega \hat{B}_{\vec{r} + \frac{1}{2}}
- \hat{M}_{\vec{r} + \frac{1}{2}} \\
\hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
-\imath \omega \tilde{D}_{\vec{r}}
+ \tilde{J}_{\vec{r}} \\
\end{aligned}
$$
$$
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
-\\omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = -\\imath \\omega \\tilde{J}_{\\vec{r}} \\\\
\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
-\omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \omega \tilde{J}_{\vec{r}} \\
$$
# TODO FDFD?