jan/meanas
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Fix waveguide eigenvalue derivation

Browse Source 
Thanks to Rafael Diaz Fuentes and Paolo Pintus for catching and
correcting these!
This commit is contained in:
jan 2021-07-05 15:01:08 -07:00
parent 23490694ff
commit fcca9e3ae5
1 changed files with 45 additions and 14 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

59
meanas/fdfd/waveguide_2d.py
View File

@ -69,8 +69,8 @@ $$
- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
&= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon_{xx} E_x)
- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon_{yy} E_y) \\\\
\\gamma \\tilde{\\partial}_x E_z &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
\\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
\\gamma \\tilde{\\partial}_x E_z &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
\\end{aligned}
$$
@ -78,23 +78,32 @@ With a similar approach (but using $\\gamma \\tilde{\\partial}_y$ instead), we c
$$
\\begin{aligned}
\\gamma \\tilde{\\partial}_y E_z &= \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
\\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
\\gamma \\tilde{\\partial}_y E_z &= \\tilde{\\partial}_y \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_y \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
\\end{aligned}
$$
We can combine this equation for $\\gamma \\tilde{\\partial}_y E_z$ with
the unused $\\imath \\omega \\mu_{xx} H_z$ and $\\imath \\omega \\mu_{yy} H_y$ equations to get
the unused $\\imath \\omega \\mu_{xx} H_x$ and $\\imath \\omega \\mu_{yy} H_y$ equations to get
$$
\\begin{aligned}
-\\imath \\omega \\mu_{xx} \\gamma H_x &= \\gamma^2 E_y + \\gamma \\tilde{\\partial}_y E_z \\\\
-\\imath \\omega \\mu_{xx} \\gamma H_x &= \\gamma^2 E_y + \\tilde{\\partial}_y (
\\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
) \\\\
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
)\\\\
\\end{aligned}
$$
and
$$
\\begin{aligned}
-\\imath \\omega \\mu_{yy} \\gamma H_y &= -\\gamma^2 E_x - \\gamma \\tilde{\\partial}_x E_z \\\\
-\\imath \\omega \\mu_{yy} \\gamma H_y &= -\\gamma^2 E_x - \\tilde{\\partial}_x (
\\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
)\\\\
\\end{aligned}
$$
@ -106,10 +115,10 @@ $$
\\begin{aligned}
-\\imath \\omega \\mu_{xx} (\\gamma H_x) &= -\\imath \\omega \\mu_{xx} (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
-\\imath \\omega \\mu_{xx} \\hat{\\partial}_x (
+\\imath \\omega \\mu_{xx} \\hat{\\partial}_x (
\\frac{1}{-\\imath \\omega \\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x)) \\\\
&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
+\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
-\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{aligned}
$$
@ -117,12 +126,30 @@ and, similarly,
$$
\\begin{aligned}
-\\imath \\omega \\mu_{yy} (\\gamma H_y) &= -\\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
+\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
-\\imath \\omega \\mu_{yy} (\\gamma H_y) &= \\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
+\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{aligned}
$$
By combining both pairs of expressions, we get
$$
\\begin{aligned}
-\\gamma^2 E_x - \\tilde{\\partial}_x (
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
) &= \\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
+\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\gamma^2 E_y + \\tilde{\\partial}_y (
\\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
) &= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
-\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{aligned}
$$
Using these, we can construct the eigenvalue problem
$$ \\beta^2 \\begin{bmatrix} E_x \\\\
E_y \\end{bmatrix} =
(\\omega^2 \\begin{bmatrix} \\mu_{yy} \\epsilon_{xx} & 0 \\\\
@ -137,6 +164,10 @@ $$ \\beta^2 \\begin{bmatrix} E_x \\\\
E_y \\end{bmatrix}
$$
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