jan/meanas
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Use raw strings to eliminate repeated backslashes

Browse Source
This commit is contained in:
Jan Petykiewicz 2024-07-14 22:08:30 -07:00
parent b47dec0317
commit 4c8a07bf20
1 changed files with 125 additions and 125 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

250
meanas/fdfd/waveguide_2d.py
View File

@ -1,4 +1,4 @@
""" r"""
Operators and helper functions for waveguides with unchanging cross-section. Operators and helper functions for waveguides with unchanging cross-section.
The propagation direction is chosen to be along the z axis, and all fields The propagation direction is chosen to be along the z axis, and all fields
@ -12,166 +12,166 @@ 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{aligned} \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{aligned} \end{aligned}
$$ $$
Expanding the first two equations into vector components, we get Expanding the first two equations into vector components, we get
$$ $$
\\begin{aligned} \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{aligned} \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{aligned} \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{aligned} \end{aligned}
$$ $$
Rewrite the last three equations as Rewrite the last three equations as
$$ $$
\\begin{aligned} \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{aligned} \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{aligned} \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{aligned} \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{aligned} \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{aligned} \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 $\\imath \\omega \\mu_{xx} H_x$ and $\\imath \\omega \\mu_{yy} H_y$ equations to get the unused $\imath \omega \mu_{xx} H_x$ and $\imath \omega \mu_{yy} H_y$ equations to get
$$ $$
\\begin{aligned} \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 + \gamma \tilde{\partial}_y E_z \\
-\\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 (
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x) \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y) + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
)\\\\ )\\
\\end{aligned} \end{aligned}
$$ $$
and and
$$ $$
\\begin{aligned} \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 - \gamma \tilde{\partial}_x E_z \\
-\\imath \\omega \\mu_{yy} \\gamma H_y &= -\\gamma^2 E_x - \\tilde{\\partial}_x ( -\imath \omega \mu_{yy} \gamma H_y &= -\gamma^2 E_x - \tilde{\partial}_x (
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x) \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y) + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
)\\\\ )\\
\\end{aligned} \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{aligned} \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{aligned} \end{aligned}
$$ $$
and, similarly, and, similarly,
$$ $$
\\begin{aligned} \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{aligned} \end{aligned}
$$ $$
By combining both pairs of expressions, we get By combining both pairs of expressions, we get
$$ $$
\\begin{aligned} \begin{aligned}
-\\gamma^2 E_x - \\tilde{\\partial}_x ( -\gamma^2 E_x - \tilde{\partial}_x (
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x) \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y) + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
) &= \\omega^2 \\mu_{yy} \\epsilon_{xx} E_x ) &= \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) \\
\\gamma^2 E_y + \\tilde{\\partial}_y ( \gamma^2 E_y + \tilde{\partial}_y (
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x) \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y) + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
) &= -\\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{aligned} \end{aligned}
$$ $$
Using these, we can construct the eigenvalue problem Using these, we can construct the eigenvalue problem
$$ $$
\\beta^2 \\begin{bmatrix} E_x \\\\ \beta^2 \begin{bmatrix} E_x \\
E_y \\end{bmatrix} = E_y \end{bmatrix} =
(\\omega^2 \\begin{bmatrix} \\mu_{yy} \\epsilon_{xx} & 0 \\\\ (\omega^2 \begin{bmatrix} \mu_{yy} \epsilon_{xx} & 0 \\
0 & \\mu_{xx} \\epsilon_{yy} \\end{bmatrix} + 0 & \mu_{xx} \epsilon_{yy} \end{bmatrix} +
\\begin{bmatrix} -\\mu_{yy} \\hat{\\partial}_y \\\\ \begin{bmatrix} -\mu_{yy} \hat{\partial}_y \\
\\mu_{xx} \\hat{\\partial}_x \\end{bmatrix} \\mu_{zz}^{-1} \mu_{xx} \hat{\partial}_x \end{bmatrix} \mu_{zz}^{-1}
\\begin{bmatrix} -\\tilde{\\partial}_y & \\tilde{\\partial}_x \\end{bmatrix} + \begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} +
\\begin{bmatrix} \\tilde{\\partial}_x \\\\ \begin{bmatrix} \tilde{\partial}_x \\
\\tilde{\\partial}_y \\end{bmatrix} \\epsilon_{zz}^{-1} \tilde{\partial}_y \end{bmatrix} \epsilon_{zz}^{-1}
\\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})
\\begin{bmatrix} E_x \\\\ \begin{bmatrix} E_x \\
E_y \\end{bmatrix} E_y \end{bmatrix}
$$ $$
where $\\gamma = \\imath\\beta$. In the literature, $\\beta$ is usually used to denote 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 the lossless/real part of the propagation constant, but in `meanas` it is allowed to
be complex. be complex.
An equivalent eigenvalue problem can be formed using the $H_x$ and $H_y$ 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.
@ -198,7 +198,7 @@ def operator_e(
epsilon: vfdfield_t, epsilon: vfdfield_t,
mu: vfdfield_t | None = None, mu: vfdfield_t | None = None,
) -> sparse.spmatrix: ) -> sparse.spmatrix:
""" r"""
Waveguide operator of the form Waveguide operator of the form
omega**2 * mu * epsilon + omega**2 * mu * epsilon +
@ -210,18 +210,18 @@ def operator_e(
More precisely, the operator is More precisely, the operator is
$$ $$
\\omega^2 \\begin{bmatrix} \\mu_{yy} \\epsilon_{xx} & 0 \\\\ \omega^2 \begin{bmatrix} \mu_{yy} \epsilon_{xx} & 0 \\
0 & \\mu_{xx} \\epsilon_{yy} \\end{bmatrix} + 0 & \mu_{xx} \epsilon_{yy} \end{bmatrix} +
\\begin{bmatrix} -\\mu_{yy} \\hat{\\partial}_y \\\\ \begin{bmatrix} -\mu_{yy} \hat{\partial}_y \\
\\mu_{xx} \\hat{\\partial}_x \\end{bmatrix} \\mu_{zz}^{-1} \mu_{xx} \hat{\partial}_x \end{bmatrix} \mu_{zz}^{-1}
\\begin{bmatrix} -\\tilde{\\partial}_y & \\tilde{\\partial}_x \\end{bmatrix} + \begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} +
\\begin{bmatrix} \\tilde{\\partial}_x \\\\ \begin{bmatrix} \tilde{\partial}_x \\
\\tilde{\\partial}_y \\end{bmatrix} \\epsilon_{zz}^{-1} \tilde{\partial}_y \end{bmatrix} \epsilon_{zz}^{-1}
\\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
@ -265,7 +265,7 @@ def operator_h(
epsilon: vfdfield_t, epsilon: vfdfield_t,
mu: vfdfield_t | None = None, mu: vfdfield_t | None = None,
) -> sparse.spmatrix: ) -> sparse.spmatrix:
""" r"""
Waveguide operator of the form Waveguide operator of the form
omega**2 * epsilon * mu + omega**2 * epsilon * mu +
@ -277,18 +277,18 @@ def operator_h(
More precisely, the operator is More precisely, the operator is
$$ $$
\\omega^2 \\begin{bmatrix} \\epsilon_{yy} \\mu_{xx} & 0 \\\\ \omega^2 \begin{bmatrix} \epsilon_{yy} \mu_{xx} & 0 \\
0 & \\epsilon_{xx} \\mu_{yy} \\end{bmatrix} + 0 & \epsilon_{xx} \mu_{yy} \end{bmatrix} +
\\begin{bmatrix} -\\epsilon_{yy} \\tilde{\\partial}_y \\\\ \begin{bmatrix} -\epsilon_{yy} \tilde{\partial}_y \\
\\epsilon_{xx} \\tilde{\\partial}_x \\end{bmatrix} \\epsilon_{zz}^{-1} \epsilon_{xx} \tilde{\partial}_x \end{bmatrix} \epsilon_{zz}^{-1}
\\begin{bmatrix} -\\hat{\\partial}_y & \\hat{\\partial}_x \\end{bmatrix} + \begin{bmatrix} -\hat{\partial}_y & \hat{\partial}_x \end{bmatrix} +
\\begin{bmatrix} \\hat{\\partial}_x \\\\ \begin{bmatrix} \hat{\partial}_x \\
\\hat{\\partial}_y \\end{bmatrix} \\mu_{zz}^{-1} \hat{\partial}_y \end{bmatrix} \mu_{zz}^{-1}
\\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