cleanup latex

meanas/fdfd/waveguide_2d.py

@@ -530,22 +530,22 @@ def exy2e(
         dxes: dx_lists_t,
         epsilon: vfdfield_t,
         ) -> sparse.spmatrix:
-    """
+    r"""
     Operator which transforms the vector `e_xy` containing the vectorized E_x and E_y fields,
      into a vectorized E containing all three E components
 
     From the operator derivation (see module docs), we have
 
     $$
-    \imath \omega \epsilon_{zz} E_z &= \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
+    \imath \omega \epsilon_{zz} E_z = \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
     $$
 
     as well as the intermediate equations
 
     $$
     \begin{aligned}
-    \gamma H_y &=  \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
-    \gamma H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
+    \imath \beta H_y &=  \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
+    \imath \beta H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
     \end{aligned}
     $$
 
@@ -553,11 +553,11 @@ def exy2e(
 
     $$
     \begin{aligned}
-    E_z &= \frac{1}{\imath \omega \gamma \epsilon_{zz}} ((
+    E_z &= \frac{1}{- \omega \beta \epsilon_{zz}} ((
              \hat{\partial}_y \hat{\partial}_x H_z
             -\hat{\partial}_x \hat{\partial}_y H_z)
           + \imath \omega (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y))
-        &= \frac{1}{\gamma \epsilon_{zz}} (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y)
+        &= \frac{1}{\imath \beta \epsilon_{zz}} (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y)
     \end{aligned}
     $$