From 4c8a07bf200eae007a98a4c38f82c2a6935155c3 Mon Sep 17 00:00:00 2001 From: Jan Petykiewicz Date: Sun, 14 Jul 2024 22:08:30 -0700 Subject: [PATCH] Use raw strings to eliminate repeated backslashes --- meanas/fdfd/waveguide_2d.py | 250 ++++++++++++++++++------------------ 1 file changed, 125 insertions(+), 125 deletions(-) diff --git a/meanas/fdfd/waveguide_2d.py b/meanas/fdfd/waveguide_2d.py index dce3573..84b7fad 100644 --- a/meanas/fdfd/waveguide_2d.py +++ b/meanas/fdfd/waveguide_2d.py @@ -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