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[waveguide_2d] Remove \gamma from docs in favor of just using \beta

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Jan Petykiewicz 2025-01-14 22:34:35 -08:00
parent 651e255704
commit 53d5812b4a
1 changed files with 36 additions and 37 deletions
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73
meanas/fdfd/waveguide_2d.py
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@ -18,8 +18,8 @@ $$
\begin{aligned}
\nabla \times \vec{E}(x, y, z) &= -\imath \omega \mu \vec{H} \\
\nabla \times \vec{H}(x, y, z) &= \imath \omega \epsilon \vec{E} \\
\vec{E}(x,y,z) &= (\vec{E}_t(x, y) + E_z(x, y)\vec{z}) e^{-\gamma z} \\
\vec{H}(x,y,z) &= (\vec{H}_t(x, y) + H_z(x, y)\vec{z}) e^{-\gamma z} \\
\vec{E}(x,y,z) &= (\vec{E}_t(x, y) + E_z(x, y)\vec{z}) e^{-\imath \beta z} \\
\vec{H}(x,y,z) &= (\vec{H}_t(x, y) + H_z(x, y)\vec{z}) e^{-\imath \beta z} \\
\end{aligned}
$$
@ -40,56 +40,57 @@ Substituting in our expressions for $\vec{E}$, $\vec{H}$ and discretizing:
$$
\begin{aligned}
-\imath \omega \mu_{xx} H_x &= \tilde{\partial}_y E_z + \gamma E_y \\
-\imath \omega \mu_{yy} H_y &= -\gamma E_x - \tilde{\partial}_x E_z \\
-\imath \omega \mu_{xx} H_x &= \tilde{\partial}_y E_z + \imath \beta E_y \\
-\imath \omega \mu_{yy} H_y &= -\imath \beta E_x - \tilde{\partial}_x E_z \\
-\imath \omega \mu_{zz} H_z &= \tilde{\partial}_x E_y - \tilde{\partial}_y E_x \\
\imath \omega \epsilon_{xx} E_x &= \hat{\partial}_y H_z + \gamma H_y \\
\imath \omega \epsilon_{yy} E_y &= -\gamma H_x - \hat{\partial}_x H_z \\
\imath \omega \epsilon_{xx} E_x &= \hat{\partial}_y H_z + \imath \beta H_y \\
\imath \omega \epsilon_{yy} E_y &= -\imath \beta H_x - \hat{\partial}_x H_z \\
\imath \omega \epsilon_{zz} E_z &= \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
\end{aligned}
$$
Rewrite the last three equations as
$$
\begin{aligned}
\gamma H_y &= \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
\gamma H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
\imath \beta H_y &= \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
\imath \beta H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
\imath \omega E_z &= \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y - \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
\end{aligned}
$$
Now apply $\gamma \tilde{\partial}_x$ to the last equation,
then substitute in for $\gamma H_x$ and $\gamma H_y$:
Now apply $\imath \beta \tilde{\partial}_x$ to the last equation,
then substitute in for $\imath \beta H_x$ and $\imath \beta H_y$:
$$
\begin{aligned}
\gamma \tilde{\partial}_x \imath \omega E_z &= \gamma \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y
- \gamma \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
\imath \beta \tilde{\partial}_x \imath \omega E_z &= \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y
- \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
&= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z)
- \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
&= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x)
- \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y) \\
\gamma \tilde{\partial}_x E_z &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
\imath \beta \tilde{\partial}_x E_z &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
\end{aligned}
$$
With a similar approach (but using $\gamma \tilde{\partial}_y$ instead), we can get
With a similar approach (but using $\imath \beta \tilde{\partial}_y$ instead), we can get
$$
\begin{aligned}
\gamma \tilde{\partial}_y E_z &= \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
\imath \beta \tilde{\partial}_y E_z &= \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
\end{aligned}
$$
We can combine this equation for $\gamma \tilde{\partial}_y E_z$ with
We can combine this equation for $\imath \beta \tilde{\partial}_y E_z$ with
the unused $\imath \omega \mu_{xx} H_x$ and $\imath \omega \mu_{yy} H_y$ equations to get
$$
\begin{aligned}
-\imath \omega \mu_{xx} \gamma H_x &= \gamma^2 E_y + \gamma \tilde{\partial}_y E_z \\
-\imath \omega \mu_{xx} \gamma H_x &= \gamma^2 E_y + \tilde{\partial}_y (
-\imath \omega \mu_{xx} \imath \beta H_x &= -\beta^2 E_y + \imath \beta \tilde{\partial}_y E_z \\
-\imath \omega \mu_{xx} \imath \beta H_x &= -\beta^2 E_y + \tilde{\partial}_y (
\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
)\\
@ -100,25 +101,24 @@ and
$$
\begin{aligned}
-\imath \omega \mu_{yy} \gamma H_y &= -\gamma^2 E_x - \gamma \tilde{\partial}_x E_z \\
-\imath \omega \mu_{yy} \gamma H_y &= -\gamma^2 E_x - \tilde{\partial}_x (
-\imath \omega \mu_{yy} \imath \beta H_y &= \beta^2 E_x - \imath \beta \tilde{\partial}_x E_z \\
-\imath \omega \mu_{yy} \imath \beta H_y &= \beta^2 E_x - \tilde{\partial}_x (
\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
)\\
\end{aligned}
$$
However, based on our rewritten equation for $\gamma H_x$ and the so-far unused
However, based on our rewritten equation for $\imath \beta H_x$ and the so-far unused
equation for $\imath \omega \mu_{zz} H_z$ we can also write
$$
\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) \\
-\imath \omega \mu_{xx} (\imath \beta 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{aligned}
$$
@ -126,7 +126,7 @@ and, similarly,
$$
\begin{aligned}
-\imath \omega \mu_{yy} (\gamma H_y) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
-\imath \omega \mu_{yy} (\imath \beta 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{aligned}
$$
@ -135,12 +135,12 @@ By combining both pairs of expressions, we get
$$
\begin{aligned}
-\gamma^2 E_x - \tilde{\partial}_x (
\beta^2 E_x - \tilde{\partial}_x (
\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_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) \\
\gamma^2 E_y + \tilde{\partial}_y (
-\beta^2 E_y + \tilde{\partial}_y (
\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
) &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
@ -165,14 +165,13 @@ $$
E_y \end{bmatrix}
$$
where $\gamma = \imath\beta$. In the literature, $\beta$ is usually used to denote
the lossless/real part of the propagation constant, but in `meanas` it is allowed to
be complex.
In the literature, $\beta$ is usually used to denote the lossless/real part of the propagation constant,
but in `meanas` it is allowed to be complex.
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
to account for numerical dispersion if the result is introduced into a space with a discretized z-axis.
Note that $E_z$ was never discretized, so $\beta$ will need adjustment to account for numerical dispersion
if the result is introduced into a space with a discretized z-axis.
"""