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@ -1402,132 +1402,144 @@ a structure with some (x, y) cross-section extending uniformly into the z dimens
with a diagonal <span class="arithmatex">\(\epsilon\)</span> tensor, we have</p>
<div class="arithmatex">\[
\begin{aligned}
\nabla \times \vec{E}(x, y, z) &amp;= -\imath \omega \mu \vec{H} \\
\nabla \times \vec{H}(x, y, z) &amp;= \imath \omega \epsilon \vec{E} \\
\vec{E}(x,y,z) &amp;= (\vec{E}_t(x, y) + E_z(x, y)\vec{z}) e^{-\imath \beta z} \\
\vec{H}(x,y,z) &amp;= (\vec{H}_t(x, y) + H_z(x, y)\vec{z}) e^{-\imath \beta z} \\
\nabla \times \vec{E}(x, y, z) &= -\imath \omega \mu \vec{H} \\
\nabla \times \vec{H}(x, y, z) &= \imath \omega \epsilon \vec{E} \\
\vec{E}(x,y,z) &= (\vec{E}_t(x, y) + E_z(x, y)\vec{z}) e^{-\imath \beta z} \\
\vec{H}(x,y,z) &= (\vec{H}_t(x, y) + H_z(x, y)\vec{z}) e^{-\imath \beta z} \\
\end{aligned}
\]</div>
<p>Expanding the first two equations into vector components, we get</p>
<div class="arithmatex">\[
\begin{aligned}
-\imath \omega \mu_{xx} H_x &amp;= \partial_y E_z - \partial_z E_y \\
-\imath \omega \mu_{yy} H_y &amp;= \partial_z E_x - \partial_x E_z \\
-\imath \omega \mu_{zz} H_z &amp;= \partial_x E_y - \partial_y E_x \\
\imath \omega \epsilon_{xx} E_x &amp;= \partial_y H_z - \partial_z H_y \\
\imath \omega \epsilon_{yy} E_y &amp;= \partial_z H_x - \partial_x H_z \\
\imath \omega \epsilon_{zz} E_z &amp;= \partial_x H_y - \partial_y H_x \\
-\imath \omega \mu_{xx} H_x &= \partial_y E_z - \partial_z E_y \\
-\imath \omega \mu_{yy} H_y &= \partial_z E_x - \partial_x E_z \\
-\imath \omega \mu_{zz} H_z &= \partial_x E_y - \partial_y E_x \\
\imath \omega \epsilon_{xx} E_x &= \partial_y H_z - \partial_z H_y \\
\imath \omega \epsilon_{yy} E_y &= \partial_z H_x - \partial_x H_z \\
\imath \omega \epsilon_{zz} E_z &= \partial_x H_y - \partial_y H_x \\
\end{aligned}
\]</div>
<p>Substituting in our expressions for <span class="arithmatex">\(\vec{E}\)</span>, <span class="arithmatex">\(\vec{H}\)</span> and discretizing:</p>
<div class="arithmatex">\[
\begin{aligned}
-\imath \omega \mu_{xx} H_x &amp;= \tilde{\partial}_y E_z + \imath \beta E_y \\
-\imath \omega \mu_{yy} H_y &amp;= -\imath \beta E_x - \tilde{\partial}_x E_z \\
-\imath \omega \mu_{zz} H_z &amp;= \tilde{\partial}_x E_y - \tilde{\partial}_y E_x \\
\imath \omega \epsilon_{xx} E_x &amp;= \hat{\partial}_y H_z + \imath \beta H_y \\
\imath \omega \epsilon_{yy} E_y &amp;= -\imath \beta H_x - \hat{\partial}_x H_z \\
\imath \omega \epsilon_{zz} E_z &amp;= \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
-\imath \omega \mu_{xx} H_x &= \tilde{\partial}_y E_z + \imath \beta E_y \\
-\imath \omega \mu_{yy} H_y &= -\imath \beta E_x - \tilde{\partial}_x E_z \\
-\imath \omega \mu_{zz} H_z &= \tilde{\partial}_x E_y - \tilde{\partial}_y E_x \\
\imath \omega \epsilon_{xx} E_x &= \hat{\partial}_y H_z + \imath \beta H_y \\
\imath \omega \epsilon_{yy} E_y &= -\imath \beta H_x - \hat{\partial}_x H_z \\
\imath \omega \epsilon_{zz} E_z &= \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
\end{aligned}
\]</div>
<p>Rewrite the last three equations as</p>
<div class="arithmatex">\[
\begin{aligned}
\imath \beta H_y &amp;= \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
\imath \beta H_x &amp;= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
\imath \omega E_z &amp;= \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y - \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
\imath \beta H_y &= \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
\imath \beta H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
\imath \omega E_z &= \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y - \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
\end{aligned}
\]</div>
<p>Now apply <span class="arithmatex">\(\imath \beta \tilde{\partial}_x\)</span> to the last equation,
then substitute in for <span class="arithmatex">\(\imath \beta H_x\)</span> and <span class="arithmatex">\(\imath \beta H_y\)</span>:</p>
<div class="arithmatex">\[
\begin{aligned}
\imath \beta \tilde{\partial}_x \imath \omega E_z &amp;= \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y
\imath \beta \tilde{\partial}_x \imath \omega E_z &= \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y
- \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
&amp;= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z)
&= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z)
- \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
&amp;= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x)
&= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x)
- \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y) \\
\imath \beta \tilde{\partial}_x E_z &amp;= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
\imath \beta \tilde{\partial}_x E_z &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
\end{aligned}
\]</div>
<p>With a similar approach (but using <span class="arithmatex">\(\imath \beta \tilde{\partial}_y\)</span> instead), we can get</p>
<div class="arithmatex">\[
\begin{aligned}
\imath \beta \tilde{\partial}_y E_z &amp;= \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
\imath \beta \tilde{\partial}_y E_z &= \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
\end{aligned}
\]</div>
<p>We can combine this equation for <span class="arithmatex">\(\imath \beta \tilde{\partial}_y E_z\)</span> with
the unused <span class="arithmatex">\(\imath \omega \mu_{xx} H_x\)</span> and <span class="arithmatex">\(\imath \omega \mu_{yy} H_y\)</span> equations to get</p>
<div class="arithmatex">\[
\begin{aligned}
-\imath \omega \mu_{xx} \imath \beta H_x &amp;= -\beta^2 E_y + \imath \beta \tilde{\partial}_y E_z \\
-\imath \omega \mu_{xx} \imath \beta H_x &amp;= -\beta^2 E_y + \tilde{\partial}_y (
-\imath \omega \mu_{xx} \imath \beta H_x &= -\beta^2 E_y + \imath \beta \tilde{\partial}_y E_z \\
-\imath \omega \mu_{xx} \imath \beta H_x &= -\beta^2 E_y + \tilde{\partial}_y (
\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
)\\
\end{aligned}
\]</div>
<p>and</p>
<div class="arithmatex">\[
\begin{aligned}
-\imath \omega \mu_{yy} \imath \beta H_y &amp;= \beta^2 E_x - \imath \beta \tilde{\partial}_x E_z \\
-\imath \omega \mu_{yy} \imath \beta H_y &amp;= \beta^2 E_x - \tilde{\partial}_x (
-\imath \omega \mu_{yy} \imath \beta H_y &= \beta^2 E_x - \imath \beta \tilde{\partial}_x E_z \\
-\imath \omega \mu_{yy} \imath \beta H_y &= \beta^2 E_x - \tilde{\partial}_x (
\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
)\\
\end{aligned}
\]</div>
<p>However, based on our rewritten equation for <span class="arithmatex">\(\imath \beta H_x\)</span> and the so-far unused
equation for <span class="arithmatex">\(\imath \omega \mu_{zz} H_z\)</span> we can also write</p>
<div class="arithmatex">\[
\begin{aligned}
-\imath \omega \mu_{xx} (\imath \beta H_x) &amp;= -\imath \omega \mu_{xx} (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
&amp;= -\omega^2 \mu_{xx} \epsilon_{yy} E_y + \imath \omega \mu_{xx} \hat{\partial}_x (
-\imath \omega \mu_{xx} (\imath \beta H_x) &= -\imath \omega \mu_{xx} (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
&= -\omega^2 \mu_{xx} \epsilon_{yy} E_y + \imath \omega \mu_{xx} \hat{\partial}_x (
\frac{1}{-\imath \omega \mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x)) \\
&amp;= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
&= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
-\mu_{xx} \hat{\partial}_x \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
\end{aligned}
\]</div>
<p>and, similarly,</p>
<div class="arithmatex">\[
\begin{aligned}
-\imath \omega \mu_{yy} (\imath \beta H_y) &amp;= \omega^2 \mu_{yy} \epsilon_{xx} E_x
-\imath \omega \mu_{yy} (\imath \beta H_y) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
+\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
\end{aligned}
\]</div>
<p>By combining both pairs of expressions, we get</p>
<div class="arithmatex">\[
\begin{aligned}
\beta^2 E_x - \tilde{\partial}_x (
\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
) &amp;= \omega^2 \mu_{yy} \epsilon_{xx} E_x
) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
+\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
-\beta^2 E_y + \tilde{\partial}_y (
\frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+ \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
) &amp;= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
) &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
-\mu_{xx} \hat{\partial}_x \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
\end{aligned}
\]</div>
<p>Using these, we can construct the eigenvalue problem</p>
<div class="arithmatex">\[
\beta^2 \begin{bmatrix} E_x \\
E_y \end{bmatrix} =
(\omega^2 \begin{bmatrix} \mu_{yy} \epsilon_{xx} &amp; 0 \\
0 &amp; \mu_{xx} \epsilon_{yy} \end{bmatrix} +
(\omega^2 \begin{bmatrix} \mu_{yy} \epsilon_{xx} & 0 \\
0 & \mu_{xx} \epsilon_{yy} \end{bmatrix} +
\begin{bmatrix} -\mu_{yy} \hat{\partial}_y \\
\mu_{xx} \hat{\partial}_x \end{bmatrix} \mu_{zz}^{-1}
\begin{bmatrix} -\tilde{\partial}_y &amp; \tilde{\partial}_x \end{bmatrix} +
\begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} +
\begin{bmatrix} \tilde{\partial}_x \\
\tilde{\partial}_y \end{bmatrix} \epsilon_{zz}^{-1}
\begin{bmatrix} \hat{\partial}_x \epsilon_{xx} &amp; \hat{\partial}_y \epsilon_{yy} \end{bmatrix})
\begin{bmatrix} \hat{\partial}_x \epsilon_{xx} & \hat{\partial}_y \epsilon_{yy} \end{bmatrix})
\begin{bmatrix} E_x \\
E_y \end{bmatrix}
\]</div>
<p>In the literature, <span class="arithmatex">\(\beta\)</span> is usually used to denote the lossless/real part of the propagation constant,
but in <code>meanas</code> it is allowed to be complex.</p>
<p>An equivalent eigenvalue problem can be formed using the <span class="arithmatex">\(H_x\)</span> and <span class="arithmatex">\(H_y\)</span> fields, if those are more convenient.</p>
@ -1580,15 +1592,16 @@ mu * [[-Dy], [Dx]] / mu * [-Dy, Dx] +
<p>for use with a field vector of the form <code>cat([E_x, E_y])</code>.</p>
<p>More precisely, the operator is</p>
<div class="arithmatex">\[
\omega^2 \begin{bmatrix} \mu_{yy} \epsilon_{xx} &amp; 0 \\
0 &amp; \mu_{xx} \epsilon_{yy} \end{bmatrix} +
\omega^2 \begin{bmatrix} \mu_{yy} \epsilon_{xx} & 0 \\
0 & \mu_{xx} \epsilon_{yy} \end{bmatrix} +
\begin{bmatrix} -\mu_{yy} \hat{\partial}_y \\
\mu_{xx} \hat{\partial}_x \end{bmatrix} \mu_{zz}^{-1}
\begin{bmatrix} -\tilde{\partial}_y &amp; \tilde{\partial}_x \end{bmatrix} +
\begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} +
\begin{bmatrix} \tilde{\partial}_x \\
\tilde{\partial}_y \end{bmatrix} \epsilon_{zz}^{-1}
\begin{bmatrix} \hat{\partial}_x \epsilon_{xx} &amp; \hat{\partial}_y \epsilon_{yy} \end{bmatrix}
\begin{bmatrix} \hat{\partial}_x \epsilon_{xx} & \hat{\partial}_y \epsilon_{yy} \end{bmatrix}
\]</div>
<p><span class="arithmatex">\(\tilde{\partial}_x\)</span> and <span class="arithmatex">\(\hat{\partial}_x\)</span> are the forward and backward derivatives along x,
and each <span class="arithmatex">\(\epsilon_{xx}\)</span>, <span class="arithmatex">\(\mu_{yy}\)</span>, etc. is a diagonal matrix containing the vectorized material
property distribution.</p>
@ -1735,15 +1748,16 @@ epsilon * [[-Dy], [Dx]] / epsilon * [-Dy, Dx] +
<p>for use with a field vector of the form <code>cat([H_x, H_y])</code>.</p>
<p>More precisely, the operator is</p>
<div class="arithmatex">\[
\omega^2 \begin{bmatrix} \epsilon_{yy} \mu_{xx} &amp; 0 \\
0 &amp; \epsilon_{xx} \mu_{yy} \end{bmatrix} +
\omega^2 \begin{bmatrix} \epsilon_{yy} \mu_{xx} & 0 \\
0 & \epsilon_{xx} \mu_{yy} \end{bmatrix} +
\begin{bmatrix} -\epsilon_{yy} \tilde{\partial}_y \\
\epsilon_{xx} \tilde{\partial}_x \end{bmatrix} \epsilon_{zz}^{-1}
\begin{bmatrix} -\hat{\partial}_y &amp; \hat{\partial}_x \end{bmatrix} +
\begin{bmatrix} -\hat{\partial}_y & \hat{\partial}_x \end{bmatrix} +
\begin{bmatrix} \hat{\partial}_x \\
\hat{\partial}_y \end{bmatrix} \mu_{zz}^{-1}
\begin{bmatrix} \tilde{\partial}_x \mu_{xx} &amp; \tilde{\partial}_y \mu_{yy} \end{bmatrix}
\begin{bmatrix} \tilde{\partial}_x \mu_{xx} & \tilde{\partial}_y \mu_{yy} \end{bmatrix}
\]</div>
<p><span class="arithmatex">\(\tilde{\partial}_x\)</span> and <span class="arithmatex">\(\hat{\partial}_x\)</span> are the forward and backward derivatives along x,
and each <span class="arithmatex">\(\epsilon_{xx}\)</span>, <span class="arithmatex">\(\mu_{yy}\)</span>, etc. is a diagonal matrix containing the vectorized material
property distribution.</p>
@ -2066,6 +2080,7 @@ and <code>exy2h(...)</code>, then normalizes them to unit forward power using
<div class="arithmatex">\[
\Re\left[\mathrm{inner\_product}(e, h, \mathrm{conj\_h}=True)\right] = 1,
\]</div>
<p>so the returned fields represent a unit-power forward mode under the
discrete Yee-grid Poynting inner product.</p>
</details>
@ -2721,21 +2736,23 @@ identical representatives for nearly symmetric modes.</p>
<div class="arithmatex">\[
\imath \omega \epsilon_{zz} E_z = \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
\]</div>
<p>as well as the intermediate equations</p>
<div class="arithmatex">\[
\begin{aligned}
\imath \beta H_y &amp;= \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
\imath \beta H_x &amp;= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
\imath \beta H_y &= \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
\imath \beta H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
\end{aligned}
\]</div>
<p>Combining these, we get</p>
<div class="arithmatex">\[
\begin{aligned}
E_z &amp;= \frac{1}{- \omega \beta \epsilon_{zz}} ((
E_z &= \frac{1}{- \omega \beta \epsilon_{zz}} ((
\hat{\partial}_y \hat{\partial}_x H_z
-\hat{\partial}_x \hat{\partial}_y H_z)
+ \imath \omega (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y))
&amp;= \frac{1}{\imath \beta \epsilon_{zz}} (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y)
&= \frac{1}{\imath \beta \epsilon_{zz}} (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y)
\end{aligned}
\]</div>
@ -3654,10 +3671,12 @@ E_z &amp;= \frac{1}{- \omega \beta \epsilon_{zz}} ((
<span class="arithmatex">\(\beta\)</span> to changes in the dielectric structure <span class="arithmatex">\(\epsilon\)</span>.</p>
<p>The output is a vector of the same size as <code>vec(epsilon)</code>, with each element specifying the
sensitivity of <code>wavenumber</code> to changes in the corresponding element in <code>vec(epsilon)</code>, i.e.</p>
<div class="arithmatex">\[sens_{i} = \frac{\partial\beta}{\partial\epsilon_i}\]</div>
<div class="arithmatex">\[ sens_{i} = \frac{\partial\beta}{\partial\epsilon_i} \]</div>
<p>An adjoint approach is used to calculate the sensitivity; the derivation is provided here:</p>
<p>Starting with the eigenvalue equation</p>
<div class="arithmatex">\[\beta^2 E_{xy} = A_E E_{xy}\]</div>
<div class="arithmatex">\[ \beta^2 E_{xy} = A_E E_{xy} \]</div>
<p>where <span class="arithmatex">\(A_E\)</span> is the waveguide operator from <code>operator_e()</code>, and <span class="arithmatex">\(E_{xy} = \begin{bmatrix} E_x \\
E_y \end{bmatrix}\)</span>,
we can differentiate with respect to one of the <span class="arithmatex">\(\epsilon\)</span> elements (i.e. at one Yee grid point), <span class="arithmatex">\(\epsilon_i\)</span>:</p>
@ -3665,11 +3684,13 @@ we can differentiate with respect to one of the <span class="arithmatex">\(\epsi
(2 \beta) \partial_{\epsilon_i}(\beta) E_{xy} + \beta^2 \partial_{\epsilon_i} E_{xy}
= \partial_{\epsilon_i}(A_E) E_{xy} + A_E \partial_{\epsilon_i} E_{xy}
\]</div>
<p>We then multiply by <span class="arithmatex">\(H_{yx}^\star = \begin{bmatrix}H_y^\star \\ -H_x^\star \end{bmatrix}\)</span> from the left:</p>
<div class="arithmatex">\[
(2 \beta) \partial_{\epsilon_i}(\beta) H_{yx}^\star E_{xy} + \beta^2 H_{yx}^\star \partial_{\epsilon_i} E_{xy}
= H_{yx}^\star \partial_{\epsilon_i}(A_E) E_{xy} + H_{yx}^\star A_E \partial_{\epsilon_i} E_{xy}
\]</div>
<p>However, <span class="arithmatex">\(H_{yx}^\star\)</span> is actually a left-eigenvector of <span class="arithmatex">\(A_E\)</span>. This can be verified by inspecting
the form of <code>operator_h</code> (<span class="arithmatex">\(A_H\)</span>) and comparing its conjugate transpose to <code>operator_e</code> (<span class="arithmatex">\(A_E\)</span>). Also, note
<span class="arithmatex">\(H_{yx}^\star \cdot E_{xy} = H^\star \times E\)</span> recalls the mode orthogonality relation. See doi:10.5194/ars-9-85-201
@ -3677,11 +3698,13 @@ for a similar approach. Therefore,</p>
<div class="arithmatex">\[
H_{yx}^\star A_E \partial_{\epsilon_i} E_{xy} = \beta^2 H_{yx}^\star \partial_{\epsilon_i} E_{xy}
\]</div>
<p>and we can simplify to</p>
<div class="arithmatex">\[
\partial_{\epsilon_i}(\beta)
= \frac{1}{2 \beta} \frac{H_{yx}^\star \partial_{\epsilon_i}(A_E) E_{xy} }{H_{yx}^\star E_{xy}}
\]</div>
<p>This expression can be quickly calculated for all <span class="arithmatex">\(i\)</span> by writing out the various terms of
<span class="arithmatex">\(\partial_{\epsilon_i} A_E\)</span> and recognizing that the vector-matrix-vector products (i.e. scalars)
<span class="arithmatex">\(sens_i = \vec{v}_{left} \partial_{\epsilon_i} (\epsilon_{xyz}) \vec{v}_{right}\)</span>, indexed by <span class="arithmatex">\(i\)</span>, can be expressed as
@ -4161,6 +4184,7 @@ longitudinal Poynting flux,</p>
<div class="arithmatex">\[
\frac{1}{2}\int (E_x H_y - E_y H_x) \, dx \, dy
\]</div>
<p>with the Yee-grid staggering and optional propagation-phase adjustment used
by the waveguide helpers in this module.</p>
@ -4866,6 +4890,7 @@ same sign convention used elsewhere in the package:</p>
<div class="arithmatex">\[
\sum \mathrm{overlap\_e} \; E_\mathrm{other}^*
\]</div>
<p>where the sum is over the full Yee-grid field storage.</p>
<p>The construction uses a two-cell window immediately upstream of the selected
slice:</p>
@ -4881,6 +4906,7 @@ overlap relation involving</p>
<div class="arithmatex">\[
\int ((E \times H_\mathrm{mode}) + (E_\mathrm{mode} \times H)) \cdot dn.
\]</div>
<p>E x H_mode + E_mode x H
-&gt; Ex Hmy - EyHmx + Emx Hy - Emy Hx (Z-prop)
Ex we/B Emx + Ex i/B dy Hmz - Ey (-we/B Emy) - Ey i/B dx Hmz
@ -5219,6 +5245,7 @@ along the propagation axis and applies the phase factor</p>
<div class="arithmatex">\[
e^{-i \, \mathrm{polarity} \, wavenumber \, \Delta z}
\]</div>
<p>to each copied slice.</p>
</details>
@ -5254,11 +5281,13 @@ and the fields are assumed to have the form</p>
<div class="arithmatex">\[
\vec{E}(r, y, \theta), \vec{H}(r, y, \theta) \propto e^{-\imath m \theta},
\]</div>
<p>where <code>m</code> is the angular wavenumber returned by <code>solve_mode(s)</code>. It is often
convenient to introduce the corresponding linear wavenumber</p>
<div class="arithmatex">\[
\beta = \frac{m}{r_{\min}},
\]</div>
<p>so that the cylindrical problem resembles the straight-waveguide problem with
additional metric factors.</p>
<p>Those metric factors live on the staggered radial Yee grids. If the left edge of
@ -5266,33 +5295,36 @@ the computational window is at <code>r = r_{\min}</code>, define the electric-gr
magnetic-grid radial sample locations by</p>
<div class="arithmatex">\[
\begin{aligned}
r_a(n) &amp;= r_{\min} + \sum_{j \le n} \Delta r_{e, j}, \\
r_b\!\left(n + \tfrac{1}{2}\right) &amp;= r_{\min} + \tfrac{1}{2}\Delta r_{e, n}
+ \sum_{j &lt; n} \Delta r_{h, j},
r_a(n) &= r_{\min} + \sum_{j \le n} \Delta r_{e, j}, \\
r_b\!\left(n + \tfrac{1}{2}\right) &= r_{\min} + \tfrac{1}{2}\Delta r_{e, n}
+ \sum_{j < n} \Delta r_{h, j},
\end{aligned}
\]</div>
<p>and from them the diagonal metric matrices</p>
<div class="arithmatex">\[
\begin{aligned}
T_a &amp;= \operatorname{diag}(r_a / r_{\min}), \\
T_b &amp;= \operatorname{diag}(r_b / r_{\min}).
T_a &= \operatorname{diag}(r_a / r_{\min}), \\
T_b &= \operatorname{diag}(r_b / r_{\min}).
\end{aligned}
\]</div>
<p>With the same forward/backward derivative notation used in <code>waveguide_2d</code>, the
coordinate-transformed discrete curl equations used here are</p>
<div class="arithmatex">\[
\begin{aligned}
-\imath \omega \mu_{rr} H_r &amp;= \tilde{\partial}_y E_z + \imath \beta T_a^{-1} E_y, \\
-\imath \omega \mu_{yy} H_y &amp;= -\imath \beta T_b^{-1} E_r
-\imath \omega \mu_{rr} H_r &= \tilde{\partial}_y E_z + \imath \beta T_a^{-1} E_y, \\
-\imath \omega \mu_{yy} H_y &= -\imath \beta T_b^{-1} E_r
- T_b^{-1} \tilde{\partial}_r (T_a E_z), \\
-\imath \omega \mu_{zz} H_z &amp;= \tilde{\partial}_r E_y - \tilde{\partial}_y E_r, \\
\imath \beta H_y &amp;= -\imath \omega T_b \epsilon_{rr} E_r - T_b \hat{\partial}_y H_z, \\
\imath \beta H_r &amp;= \imath \omega T_a \epsilon_{yy} E_y
-\imath \omega \mu_{zz} H_z &= \tilde{\partial}_r E_y - \tilde{\partial}_y E_r, \\
\imath \beta H_y &= -\imath \omega T_b \epsilon_{rr} E_r - T_b \hat{\partial}_y H_z, \\
\imath \beta H_r &= \imath \omega T_a \epsilon_{yy} E_y
- T_b T_a^{-1} \hat{\partial}_r (T_b H_z), \\
\imath \omega E_z &amp;= T_a \epsilon_{zz}^{-1}
\imath \omega E_z &= T_a \epsilon_{zz}^{-1}
\left(\hat{\partial}_r H_y - \hat{\partial}_y H_r\right).
\end{aligned}
\]</div>
<p>The first three equations are the cylindrical analogue of the straight-guide
relations for <code>H_r</code>, <code>H_y</code>, and <code>H_z</code>. The next two are the metric-weighted
versions of the straight-guide identities for <code>\imath \beta H_y</code> and
@ -5306,6 +5338,7 @@ and then eliminate <code>H_z</code> with</p>
H_z = \frac{1}{-\imath \omega \mu_{zz}}
\left(\tilde{\partial}_r E_y - \tilde{\partial}_y E_r\right).
\]</div>
<p>This yields the transverse electric eigenproblem implemented by
<code>cylindrical_operator(...)</code>:</p>
<div class="arithmatex">\[
@ -5315,8 +5348,8 @@ H_z = \frac{1}{-\imath \omega \mu_{zz}}
\left(
\omega^2
\begin{bmatrix}
T_b^2 \mu_{yy} \epsilon_{xx} &amp; 0 \\
0 &amp; T_a^2 \mu_{xx} \epsilon_{yy}
T_b^2 \mu_{yy} \epsilon_{xx} & 0 \\
0 & T_a^2 \mu_{xx} \epsilon_{yy}
\end{bmatrix}
+
\begin{bmatrix}
@ -5325,7 +5358,7 @@ T_b^2 \mu_{yy} \epsilon_{xx} &amp; 0 \\
\end{bmatrix}
T_b \mu_{zz}^{-1}
\begin{bmatrix}
-\tilde{\partial}_y &amp; \tilde{\partial}_x
-\tilde{\partial}_y & \tilde{\partial}_x
\end{bmatrix}
+
\begin{bmatrix}
@ -5334,12 +5367,13 @@ T_b \mu_{zz}^{-1}
\end{bmatrix}
T_a \epsilon_{zz}^{-1}
\begin{bmatrix}
\hat{\partial}_x T_b \epsilon_{xx} &amp;
\hat{\partial}_x T_b \epsilon_{xx} &
\hat{\partial}_y T_a \epsilon_{yy}
\end{bmatrix}
\right)
\begin{bmatrix} E_r \\ E_y \end{bmatrix}.
\]</div>
<p>Since <code>\beta = m / r_{\min}</code>, the solver implemented in this file returns the
angular wavenumber <code>m</code>, while the operator itself is most naturally written in
terms of the linear quantity <code>\beta</code>. The helpers below reconstruct the full
@ -5389,17 +5423,18 @@ product used by <code>waveguide_2d</code>.</p>
<p>Cylindrical coordinate waveguide operator of the form</p>
<div class="arithmatex">\[
(\omega^2 \begin{bmatrix} T_b T_b \mu_{yy} \epsilon_{xx} &amp; 0 \\
0 &amp; T_a T_a \mu_{xx} \epsilon_{yy} \end{bmatrix} +
(\omega^2 \begin{bmatrix} T_b T_b \mu_{yy} \epsilon_{xx} & 0 \\
0 & T_a T_a \mu_{xx} \epsilon_{yy} \end{bmatrix} +
\begin{bmatrix} -T_b \mu_{yy} \hat{\partial}_y \\
T_a \mu_{xx} \hat{\partial}_x \end{bmatrix} T_b \mu_{zz}^{-1}
\begin{bmatrix} -\tilde{\partial}_y &amp; \tilde{\partial}_x \end{bmatrix} +
\begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} +
\begin{bmatrix} \tilde{\partial}_x \\
\tilde{\partial}_y \end{bmatrix} T_a \epsilon_{zz}^{-1}
\begin{bmatrix} \hat{\partial}_x T_b \epsilon_{xx} &amp; \hat{\partial}_y T_a \epsilon_{yy} \end{bmatrix})
\begin{bmatrix} \hat{\partial}_x T_b \epsilon_{xx} & \hat{\partial}_y T_a \epsilon_{yy} \end{bmatrix})
\begin{bmatrix} E_r \\
E_y \end{bmatrix}
\]</div>
<p>for use with a field vector of the form <code>[E_r, E_y]</code>.</p>
<p>This operator can be used to form an eigenvalue problem of the form
A @ [E_r, E_y] = beta**2 * [E_r, E_y]</p>
@ -6288,15 +6323,15 @@ from E_r and E_y in order to then calculate E_z.</p>
<p>Returns an operator which, when applied to a vectorized E eigenfield, produces
the vectorized H eigenfield.</p>
<p>This operator is created directly from the initial coordinate-transformed equations:
$$
<p>This operator is created directly from the initial coordinate-transformed equations:</p>
<div class="arithmatex">\[
\begin{aligned}
-\imath \omega \mu_{rr} H_r &amp;= \tilde{\partial}<em yy="yy">y E_z + \imath \beta T_a^{-1} E_y, \
-\imath \omega \mu</em> E_r
- T_b^{-1} \tilde{\partial}} H_y &amp;= -\imath \beta T_b^{-1<em zz="zz">r (T_a E_z), \
-\imath \omega \mu</em>_y E_r,
-\imath \omega \mu_{rr} H_r &= \tilde{\partial}_y E_z + \imath \beta T_a^{-1} E_y, \\
-\imath \omega \mu_{yy} H_y &= -\imath \beta T_b^{-1} E_r
- T_b^{-1} \tilde{\partial}_r (T_a E_z), \\
-\imath \omega \mu_{zz} H_z &= \tilde{\partial}_r E_y - \tilde{\partial}_y E_r,
\end{aligned}
$$} H_z &amp;= \tilde{\partial}_r E_y - \tilde{\partial</p>
\]</div>
<p><span class="doc-section-title">Parameters:</span></p>
@ -6745,6 +6780,7 @@ enforces unit forward power under the discrete inner product</p>
<div class="arithmatex">\[
\frac{1}{2}\int (E_r H_y^* - E_y H_r^*) \, dr \, dy.
\]</div>
<p>The angular wavenumber <code>m</code> is first converted into the full three-component
fields, then the overall complex phase and sign are fixed so the result is
reproducible for symmetric modes.</p>