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Use raw strings to eliminate repeated backslashes

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Jan Petykiewicz 2024-07-14 22:08:30 -07:00
parent b47dec0317
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meanas/fdfd/waveguide_2d.py
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@ -1,4 +1,4 @@
"""
r"""
Operators and helper functions for waveguides with unchanging cross-section.
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
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
$$
\\begin{aligned}
\\nabla \\times \\vec{E}(x, y, z) &= -\\imath \\omega \\mu \\vec{H} \\\\
\\nabla \\times \\vec{H}(x, y, z) &= \\imath \\omega \\epsilon \\vec{E} \\\\
\\vec{E}(x,y,z) = (\\vec{E}_t(x, y) + E_z(x, y)\\vec{z}) e^{-\\gamma z} \\\\
\\vec{H}(x,y,z) = (\\vec{H}_t(x, y) + H_z(x, y)\\vec{z}) e^{-\\gamma z} \\\\
\\end{aligned}
\begin{aligned}
\nabla \times \vec{E}(x, y, z) &= -\imath \omega \mu \vec{H} \\
\nabla \times \vec{H}(x, y, z) &= \imath \omega \epsilon \vec{E} \\
\vec{E}(x,y,z) &= (\vec{E}_t(x, y) + E_z(x, y)\vec{z}) e^{-\gamma z} \\
\vec{H}(x,y,z) &= (\vec{H}_t(x, y) + H_z(x, y)\vec{z}) e^{-\gamma z} \\
\end{aligned}
$$
Expanding the first two equations into vector components, we get
$$
\\begin{aligned}
-\\imath \\omega \\mu_{xx} H_x &= \\partial_y E_z - \\partial_z E_y \\\\
-\\imath \\omega \\mu_{yy} H_y &= \\partial_z E_x - \\partial_x E_z \\\\
-\\imath \\omega \\mu_{zz} H_z &= \\partial_x E_y - \\partial_y E_x \\\\
\\imath \\omega \\epsilon_{xx} E_x &= \\partial_y H_z - \\partial_z H_y \\\\
\\imath \\omega \\epsilon_{yy} E_y &= \\partial_z H_x - \\partial_x H_z \\\\
\\imath \\omega \\epsilon_{zz} E_z &= \\partial_x H_y - \\partial_y H_x \\\\
\\end{aligned}
\begin{aligned}
-\imath \omega \mu_{xx} H_x &= \partial_y E_z - \partial_z E_y \\
-\imath \omega \mu_{yy} H_y &= \partial_z E_x - \partial_x E_z \\
-\imath \omega \mu_{zz} H_z &= \partial_x E_y - \partial_y E_x \\
\imath \omega \epsilon_{xx} E_x &= \partial_y H_z - \partial_z H_y \\
\imath \omega \epsilon_{yy} E_y &= \partial_z H_x - \partial_x H_z \\
\imath \omega \epsilon_{zz} E_z &= \partial_x H_y - \partial_y H_x \\
\end{aligned}
$$
Substituting in our expressions for $\\vec{E}$, $\\vec{H}$ and discretizing:
Substituting in our expressions for $\vec{E}$, $\vec{H}$ and discretizing:
$$
\\begin{aligned}
-\\imath \\omega \\mu_{xx} H_x &= \\tilde{\\partial}_y E_z + \\gamma E_y \\\\
-\\imath \\omega \\mu_{yy} H_y &= -\\gamma E_x - \\tilde{\\partial}_x E_z \\\\
-\\imath \\omega \\mu_{zz} H_z &= \\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x \\\\
\\imath \\omega \\epsilon_{xx} E_x &= \\hat{\\partial}_y H_z + \\gamma H_y \\\\
\\imath \\omega \\epsilon_{yy} E_y &= -\\gamma H_x - \\hat{\\partial}_x H_z \\\\
\\imath \\omega \\epsilon_{zz} E_z &= \\hat{\\partial}_x H_y - \\hat{\\partial}_y H_x \\\\
\\end{aligned}
\begin{aligned}
-\imath \omega \mu_{xx} H_x &= \tilde{\partial}_y E_z + \gamma E_y \\
-\imath \omega \mu_{yy} H_y &= -\gamma E_x - \tilde{\partial}_x E_z \\
-\imath \omega \mu_{zz} H_z &= \tilde{\partial}_x E_y - \tilde{\partial}_y E_x \\
\imath \omega \epsilon_{xx} E_x &= \hat{\partial}_y H_z + \gamma H_y \\
\imath \omega \epsilon_{yy} E_y &= -\gamma H_x - \hat{\partial}_x H_z \\
\imath \omega \epsilon_{zz} E_z &= \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
\end{aligned}
$$
Rewrite the last three equations as
$$
\\begin{aligned}
\\gamma H_y &= \\imath \\omega \\epsilon_{xx} E_x - \\hat{\\partial}_y H_z \\\\
\\gamma H_x &= -\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z \\\\
\\imath \\omega E_z &= \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x H_y - \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y H_x \\\\
\\end{aligned}
\begin{aligned}
\gamma H_y &= \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
\gamma H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
\imath \omega E_z &= \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y - \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
\end{aligned}
$$
Now apply $\\gamma \\tilde{\\partial}_x$ to the last equation,
then substitute in for $\\gamma H_x$ and $\\gamma H_y$:
Now apply $\gamma \tilde{\partial}_x$ to the last equation,
then substitute in for $\gamma H_x$ and $\gamma H_y$:
$$
\\begin{aligned}
\\gamma \\tilde{\\partial}_x \\imath \\omega E_z &= \\gamma \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x H_y
- \\gamma \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y H_x \\\\
&= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon_{xx} E_x - \\hat{\\partial}_y H_z)
- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
&= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon_{xx} E_x)
- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon_{yy} E_y) \\\\
\\gamma \\tilde{\\partial}_x E_z &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
\\end{aligned}
\begin{aligned}
\gamma \tilde{\partial}_x \imath \omega E_z &= \gamma \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y
- \gamma \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
&= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z)
- \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
&= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x)
- \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y) \\
\gamma \tilde{\partial}_x E_z &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
\end{aligned}
$$
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
$$
\\begin{aligned}
\\gamma \\tilde{\\partial}_y E_z &= \\tilde{\\partial}_y \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_y \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
\\end{aligned}
\begin{aligned}
\gamma \tilde{\partial}_y E_z &= \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
\end{aligned}
$$
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
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
$$
\\begin{aligned}
-\\imath \\omega \\mu_{xx} \\gamma H_x &= \\gamma^2 E_y + \\gamma \\tilde{\\partial}_y E_z \\\\
-\\imath \\omega \\mu_{xx} \\gamma H_x &= \\gamma^2 E_y + \\tilde{\\partial}_y (
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
)\\\\
\\end{aligned}
\begin{aligned}
-\imath \omega \mu_{xx} \gamma H_x &= \gamma^2 E_y + \gamma \tilde{\partial}_y E_z \\
-\imath \omega \mu_{xx} \gamma H_x &= \gamma^2 E_y + \tilde{\partial}_y (
\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
)\\
\end{aligned}
$$
and
$$
\\begin{aligned}
-\\imath \\omega \\mu_{yy} \\gamma H_y &= -\\gamma^2 E_x - \\gamma \\tilde{\\partial}_x E_z \\\\
-\\imath \\omega \\mu_{yy} \\gamma H_y &= -\\gamma^2 E_x - \\tilde{\\partial}_x (
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
)\\\\
\\end{aligned}
\begin{aligned}
-\imath \omega \mu_{yy} \gamma H_y &= -\gamma^2 E_x - \gamma \tilde{\partial}_x E_z \\
-\imath \omega \mu_{yy} \gamma H_y &= -\gamma^2 E_x - \tilde{\partial}_x (
\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
)\\
\end{aligned}
$$
However, based on our rewritten equation for $\\gamma H_x$ and the so-far unused
equation for $\\imath \\omega \\mu_{zz} H_z$ we can also write
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
$$
\\begin{aligned}
-\\imath \\omega \\mu_{xx} (\\gamma H_x) &= -\\imath \\omega \\mu_{xx} (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
+\\imath \\omega \\mu_{xx} \\hat{\\partial}_x (
\\frac{1}{-\\imath \\omega \\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x)) \\\\
&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
-\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{aligned}
\begin{aligned}
-\imath \omega \mu_{xx} (\gamma H_x) &= -\imath \omega \mu_{xx} (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
&= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
+\imath \omega \mu_{xx} \hat{\partial}_x (
\frac{1}{-\imath \omega \mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x)) \\
&= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
-\mu_{xx} \hat{\partial}_x \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
\end{aligned}
$$
and, similarly,
$$
\\begin{aligned}
-\\imath \\omega \\mu_{yy} (\\gamma H_y) &= \\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
+\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{aligned}
\begin{aligned}
-\imath \omega \mu_{yy} (\gamma H_y) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
+\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
\end{aligned}
$$
By combining both pairs of expressions, we get
$$
\\begin{aligned}
-\\gamma^2 E_x - \\tilde{\\partial}_x (
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
) &= \\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
+\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\gamma^2 E_y + \\tilde{\\partial}_y (
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
) &= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
-\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{aligned}
\begin{aligned}
-\gamma^2 E_x - \tilde{\partial}_x (
\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
+\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
\gamma^2 E_y + \tilde{\partial}_y (
\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
) &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
-\mu_{xx} \hat{\partial}_x \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
\end{aligned}
$$
Using these, we can construct the eigenvalue problem
$$
\\beta^2 \\begin{bmatrix} E_x \\\\
E_y \\end{bmatrix} =
(\\omega^2 \\begin{bmatrix} \\mu_{yy} \\epsilon_{xx} & 0 \\\\
0 & \\mu_{xx} \\epsilon_{yy} \\end{bmatrix} +
\\begin{bmatrix} -\\mu_{yy} \\hat{\\partial}_y \\\\
\\mu_{xx} \\hat{\\partial}_x \\end{bmatrix} \\mu_{zz}^{-1}
\\begin{bmatrix} -\\tilde{\\partial}_y & \\tilde{\\partial}_x \\end{bmatrix} +
\\begin{bmatrix} \\tilde{\\partial}_x \\\\
\\tilde{\\partial}_y \\end{bmatrix} \\epsilon_{zz}^{-1}
\\begin{bmatrix} \\hat{\\partial}_x \\epsilon_{xx} & \\hat{\\partial}_y \\epsilon_{yy} \\end{bmatrix})
\\begin{bmatrix} E_x \\\\
E_y \\end{bmatrix}
\beta^2 \begin{bmatrix} E_x \\
E_y \end{bmatrix} =
(\omega^2 \begin{bmatrix} \mu_{yy} \epsilon_{xx} & 0 \\
0 & \mu_{xx} \epsilon_{yy} \end{bmatrix} +
\begin{bmatrix} -\mu_{yy} \hat{\partial}_y \\
\mu_{xx} \hat{\partial}_x \end{bmatrix} \mu_{zz}^{-1}
\begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} +
\begin{bmatrix} \tilde{\partial}_x \\
\tilde{\partial}_y \end{bmatrix} \epsilon_{zz}^{-1}
\begin{bmatrix} \hat{\partial}_x \epsilon_{xx} & \hat{\partial}_y \epsilon_{yy} \end{bmatrix})
\begin{bmatrix} E_x \\
E_y \end{bmatrix}
$$
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
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
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.
@ -198,7 +198,7 @@ def operator_e(
epsilon: vfdfield_t,
mu: vfdfield_t | None = None,
) -> sparse.spmatrix:
"""
r"""
Waveguide operator of the form
omega**2 * mu * epsilon +
@ -210,18 +210,18 @@ def operator_e(
More precisely, the operator is
$$
\\omega^2 \\begin{bmatrix} \\mu_{yy} \\epsilon_{xx} & 0 \\\\
0 & \\mu_{xx} \\epsilon_{yy} \\end{bmatrix} +
\\begin{bmatrix} -\\mu_{yy} \\hat{\\partial}_y \\\\
\\mu_{xx} \\hat{\\partial}_x \\end{bmatrix} \\mu_{zz}^{-1}
\\begin{bmatrix} -\\tilde{\\partial}_y & \\tilde{\\partial}_x \\end{bmatrix} +
\\begin{bmatrix} \\tilde{\\partial}_x \\\\
\\tilde{\\partial}_y \\end{bmatrix} \\epsilon_{zz}^{-1}
\\begin{bmatrix} \\hat{\\partial}_x \\epsilon_{xx} & \\hat{\\partial}_y \\epsilon_{yy} \\end{bmatrix}
\omega^2 \begin{bmatrix} \mu_{yy} \epsilon_{xx} & 0 \\
0 & \mu_{xx} \epsilon_{yy} \end{bmatrix} +
\begin{bmatrix} -\mu_{yy} \hat{\partial}_y \\
\mu_{xx} \hat{\partial}_x \end{bmatrix} \mu_{zz}^{-1}
\begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} +
\begin{bmatrix} \tilde{\partial}_x \\
\tilde{\partial}_y \end{bmatrix} \epsilon_{zz}^{-1}
\begin{bmatrix} \hat{\partial}_x \epsilon_{xx} & \hat{\partial}_y \epsilon_{yy} \end{bmatrix}
$$
$\\tilde{\\partial}_x$ and $\\hat{\\partial}_x$ are the forward and backward derivatives along x,
and each $\\epsilon_{xx}$, $\\mu_{yy}$, etc. is a diagonal matrix containing the vectorized material
$\tilde{\partial}_x$ 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
property distribution.
This operator can be used to form an eigenvalue problem of the form
@ -265,7 +265,7 @@ def operator_h(
epsilon: vfdfield_t,
mu: vfdfield_t | None = None,
) -> sparse.spmatrix:
"""
r"""
Waveguide operator of the form
omega**2 * epsilon * mu +
@ -277,18 +277,18 @@ def operator_h(
More precisely, the operator is
$$
\\omega^2 \\begin{bmatrix} \\epsilon_{yy} \\mu_{xx} & 0 \\\\
0 & \\epsilon_{xx} \\mu_{yy} \\end{bmatrix} +
\\begin{bmatrix} -\\epsilon_{yy} \\tilde{\\partial}_y \\\\
\\epsilon_{xx} \\tilde{\\partial}_x \\end{bmatrix} \\epsilon_{zz}^{-1}
\\begin{bmatrix} -\\hat{\\partial}_y & \\hat{\\partial}_x \\end{bmatrix} +
\\begin{bmatrix} \\hat{\\partial}_x \\\\
\\hat{\\partial}_y \\end{bmatrix} \\mu_{zz}^{-1}
\\begin{bmatrix} \\tilde{\\partial}_x \\mu_{xx} & \\tilde{\\partial}_y \\mu_{yy} \\end{bmatrix}
\omega^2 \begin{bmatrix} \epsilon_{yy} \mu_{xx} & 0 \\
0 & \epsilon_{xx} \mu_{yy} \end{bmatrix} +
\begin{bmatrix} -\epsilon_{yy} \tilde{\partial}_y \\
\epsilon_{xx} \tilde{\partial}_x \end{bmatrix} \epsilon_{zz}^{-1}
\begin{bmatrix} -\hat{\partial}_y & \hat{\partial}_x \end{bmatrix} +
\begin{bmatrix} \hat{\partial}_x \\
\hat{\partial}_y \end{bmatrix} \mu_{zz}^{-1}
\begin{bmatrix} \tilde{\partial}_x \mu_{xx} & \tilde{\partial}_y \mu_{yy} \end{bmatrix}
$$
$\\tilde{\\partial}_x$ and $\\hat{\\partial}_x$ are the forward and backward derivatives along x,
and each $\\epsilon_{xx}$, $\\mu_{yy}$, etc. is a diagonal matrix containing the vectorized material
$\tilde{\partial}_x$ 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
property distribution.
This operator can be used to form an eigenvalue problem of the form