[waveguide_2d] Remove \gamma from docs in favor of just using \beta
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@ -18,8 +18,8 @@ $$
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\begin{aligned}
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\begin{aligned}
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\nabla \times \vec{E}(x, y, z) &= -\imath \omega \mu \vec{H} \\
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\nabla \times \vec{E}(x, y, z) &= -\imath \omega \mu \vec{H} \\
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\nabla \times \vec{H}(x, y, z) &= \imath \omega \epsilon \vec{E} \\
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\nabla \times \vec{H}(x, y, z) &= \imath \omega \epsilon \vec{E} \\
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\vec{E}(x,y,z) &= (\vec{E}_t(x, y) + E_z(x, y)\vec{z}) e^{-\gamma z} \\
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\vec{E}(x,y,z) &= (\vec{E}_t(x, y) + E_z(x, y)\vec{z}) e^{-\imath \beta z} \\
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\vec{H}(x,y,z) &= (\vec{H}_t(x, y) + H_z(x, y)\vec{z}) e^{-\gamma z} \\
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\vec{H}(x,y,z) &= (\vec{H}_t(x, y) + H_z(x, y)\vec{z}) e^{-\imath \beta z} \\
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\end{aligned}
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\end{aligned}
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$$
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$$
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@ -40,56 +40,57 @@ Substituting in our expressions for $\vec{E}$, $\vec{H}$ and discretizing:
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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-\imath \omega \mu_{xx} H_x &= \tilde{\partial}_y E_z + \gamma E_y \\
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-\imath \omega \mu_{xx} H_x &= \tilde{\partial}_y E_z + \imath \beta E_y \\
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-\imath \omega \mu_{yy} H_y &= -\gamma E_x - \tilde{\partial}_x E_z \\
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-\imath \omega \mu_{yy} H_y &= -\imath \beta E_x - \tilde{\partial}_x E_z \\
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-\imath \omega \mu_{zz} H_z &= \tilde{\partial}_x E_y - \tilde{\partial}_y E_x \\
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-\imath \omega \mu_{zz} H_z &= \tilde{\partial}_x E_y - \tilde{\partial}_y E_x \\
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\imath \omega \epsilon_{xx} E_x &= \hat{\partial}_y H_z + \gamma H_y \\
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\imath \omega \epsilon_{xx} E_x &= \hat{\partial}_y H_z + \imath \beta H_y \\
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\imath \omega \epsilon_{yy} E_y &= -\gamma H_x - \hat{\partial}_x H_z \\
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\imath \omega \epsilon_{yy} E_y &= -\imath \beta H_x - \hat{\partial}_x H_z \\
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\imath \omega \epsilon_{zz} E_z &= \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
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\imath \omega \epsilon_{zz} E_z &= \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
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\end{aligned}
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\end{aligned}
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$$
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$$
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Rewrite the last three equations as
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Rewrite the last three equations as
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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\gamma H_y &= \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
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\imath \beta H_y &= \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
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\gamma H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
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\imath \beta H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
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\imath \omega E_z &= \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y - \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
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\imath \omega E_z &= \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y - \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
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\end{aligned}
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\end{aligned}
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$$
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$$
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Now apply $\gamma \tilde{\partial}_x$ to the last equation,
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Now apply $\imath \beta \tilde{\partial}_x$ to the last equation,
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then substitute in for $\gamma H_x$ and $\gamma H_y$:
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then substitute in for $\imath \beta H_x$ and $\imath \beta H_y$:
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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\gamma \tilde{\partial}_x \imath \omega E_z &= \gamma \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y
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\imath \beta \tilde{\partial}_x \imath \omega E_z &= \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y
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- \gamma \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
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- \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
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&= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z)
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&= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z)
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- \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
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- \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
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&= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x)
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&= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x)
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- \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y) \\
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- \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y) \\
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\gamma \tilde{\partial}_x E_z &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
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\imath \beta \tilde{\partial}_x E_z &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
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+ \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
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+ \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
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\end{aligned}
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\end{aligned}
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$$
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$$
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With a similar approach (but using $\gamma \tilde{\partial}_y$ instead), we can get
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With a similar approach (but using $\imath \beta \tilde{\partial}_y$ instead), we can get
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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\gamma \tilde{\partial}_y E_z &= \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
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\imath \beta \tilde{\partial}_y E_z &= \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
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+ \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
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+ \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
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\end{aligned}
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\end{aligned}
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$$
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$$
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We can combine this equation for $\gamma \tilde{\partial}_y E_z$ with
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We can combine this equation for $\imath \beta \tilde{\partial}_y E_z$ with
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the unused $\imath \omega \mu_{xx} H_x$ and $\imath \omega \mu_{yy} H_y$ equations to get
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the unused $\imath \omega \mu_{xx} H_x$ and $\imath \omega \mu_{yy} H_y$ equations to get
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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-\imath \omega \mu_{xx} \gamma H_x &= \gamma^2 E_y + \gamma \tilde{\partial}_y E_z \\
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-\imath \omega \mu_{xx} \imath \beta H_x &= -\beta^2 E_y + \imath \beta \tilde{\partial}_y E_z \\
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-\imath \omega \mu_{xx} \gamma H_x &= \gamma^2 E_y + \tilde{\partial}_y (
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-\imath \omega \mu_{xx} \imath \beta H_x &= -\beta^2 E_y + \tilde{\partial}_y (
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\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
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\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
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+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
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+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
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)\\
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)\\
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@ -100,22 +101,21 @@ and
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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-\imath \omega \mu_{yy} \gamma H_y &= -\gamma^2 E_x - \gamma \tilde{\partial}_x E_z \\
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-\imath \omega \mu_{yy} \imath \beta H_y &= \beta^2 E_x - \imath \beta \tilde{\partial}_x E_z \\
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-\imath \omega \mu_{yy} \gamma H_y &= -\gamma^2 E_x - \tilde{\partial}_x (
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-\imath \omega \mu_{yy} \imath \beta H_y &= \beta^2 E_x - \tilde{\partial}_x (
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\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
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\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
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+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
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+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
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)\\
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)\\
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\end{aligned}
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\end{aligned}
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$$
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$$
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However, based on our rewritten equation for $\gamma H_x$ and the so-far unused
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However, based on our rewritten equation for $\imath \beta H_x$ and the so-far unused
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equation for $\imath \omega \mu_{zz} H_z$ we can also write
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equation for $\imath \omega \mu_{zz} H_z$ we can also write
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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-\imath \omega \mu_{xx} (\gamma H_x) &= -\imath \omega \mu_{xx} (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
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-\imath \omega \mu_{xx} (\imath \beta H_x) &= -\imath \omega \mu_{xx} (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
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&= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
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&= -\omega^2 \mu_{xx} \epsilon_{yy} E_y + \imath \omega \mu_{xx} \hat{\partial}_x (
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+\imath \omega \mu_{xx} \hat{\partial}_x (
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\frac{1}{-\imath \omega \mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x)) \\
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\frac{1}{-\imath \omega \mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x)) \\
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&= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
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&= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
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-\mu_{xx} \hat{\partial}_x \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
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-\mu_{xx} \hat{\partial}_x \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
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@ -126,7 +126,7 @@ and, similarly,
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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-\imath \omega \mu_{yy} (\gamma H_y) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
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-\imath \omega \mu_{yy} (\imath \beta H_y) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
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+\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
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+\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
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\end{aligned}
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\end{aligned}
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$$
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$$
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@ -135,12 +135,12 @@ By combining both pairs of expressions, we get
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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-\gamma^2 E_x - \tilde{\partial}_x (
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\beta^2 E_x - \tilde{\partial}_x (
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\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
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\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
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+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
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+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
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) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
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) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
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+\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
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+\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
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\gamma^2 E_y + \tilde{\partial}_y (
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-\beta^2 E_y + \tilde{\partial}_y (
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\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
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\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
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+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
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+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
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) &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
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) &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
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@ -165,14 +165,13 @@ $$
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E_y \end{bmatrix}
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E_y \end{bmatrix}
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$$
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$$
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where $\gamma = \imath\beta$. In the literature, $\beta$ is usually used to denote
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In the literature, $\beta$ is usually used to denote the lossless/real part of the propagation constant,
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the lossless/real part of the propagation constant, but in `meanas` it is allowed to
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but in `meanas` it is allowed to be complex.
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be complex.
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An equivalent eigenvalue problem can be formed using the $H_x$ and $H_y$ fields, if those are more convenient.
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An equivalent eigenvalue problem can be formed using the $H_x$ and $H_y$ fields, if those are more convenient.
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Note that $E_z$ was never discretized, so $\gamma$ and $\beta$ will need adjustment
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Note that $E_z$ was never discretized, so $\beta$ will need adjustment to account for numerical dispersion
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to account for numerical dispersion if the result is introduced into a space with a discretized z-axis.
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if the result is introduced into a space with a discretized z-axis.
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"""
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"""
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