diff --git a/make_docs.sh b/make_docs.sh index 7ce9e39..6d99b5f 100755 --- a/make_docs.sh +++ b/make_docs.sh @@ -12,7 +12,7 @@ cd ~/projects/meanas rm -rf _doc_mathimg pdoc --pdf --force --template-dir pdoc_templates -o doc . > doc.md pandoc --metadata=title:"meanas" --from=markdown+abbreviations --to=html --output=doc.htex --gladtex -s --css pdoc_templates/pdoc.css doc.md -gladtex -a -n -d _doc_mathimg -c white doc.htex +gladtex -a -n -d _doc_mathimg -c white -b black doc.htex # Approach 3: html with gladtex #pdoc3 --html --force --template-dir pdoc_templates -o doc . diff --git a/meanas/fdfd/__init__.py b/meanas/fdfd/__init__.py index 624d576..1829cf9 100644 --- a/meanas/fdfd/__init__.py +++ b/meanas/fdfd/__init__.py @@ -1,4 +1,4 @@ -""" +r""" Tools for finite difference frequency-domain (FDFD) simulations and calculations. These mostly involve picking a single frequency, then setting up and solving a @@ -19,71 +19,71 @@ Submodules: From the "Frequency domain" section of `meanas.fdmath`, we have $$ - \\begin{aligned} - \\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\ - \\tilde{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= \\tilde{H}_{\\vec{r} + \\frac{1}{2}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\ - \\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\ - \\tilde{M}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= \\tilde{M}_{\\vec{r} + \\frac{1}{2}} e^{-\\imath \\omega l \\Delta_t} \\\\ - \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}}) - -\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} &= -\\imath \\Omega \\tilde{J}_{\\vec{r}} e^{\\imath \\omega \\Delta_t / 2} \\\\ - \\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t - \\end{aligned} + \begin{aligned} + \tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\ + \tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{H}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\ + \tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\ + \tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{M}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega l \Delta_t} \\ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}}) + -\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} &= -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\ + \Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t + \end{aligned} $$ resulting in $$ - \\begin{aligned} - \\tilde{\\partial}_t &\\Rightarrow -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\ - \\hat{\\partial}_t &\\Rightarrow -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\ - \\end{aligned} + \begin{aligned} + \tilde{\partial}_t &\Rightarrow -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\ + \hat{\partial}_t &\Rightarrow -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\ + \end{aligned} $$ Maxwell's equations are then $$ - \\begin{aligned} - \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} &= - \\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2} \\hat{B}_{\\vec{r} + \\frac{1}{2}} - - \\hat{M}_{\\vec{r} + \\frac{1}{2}} \\\\ - \\hat{\\nabla} \\times \\hat{H}_{\\vec{r} + \\frac{1}{2}} &= - -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2} \\tilde{D}_{\\vec{r}} - + \\tilde{J}_{\\vec{r}} \\\\ - \\tilde{\\nabla} \\cdot \\hat{B}_{\\vec{r} + \\frac{1}{2}} &= 0 \\\\ - \\hat{\\nabla} \\cdot \\tilde{D}_{\\vec{r}} &= \\rho_{\\vec{r}} - \\end{aligned} + \begin{aligned} + \tilde{\nabla} \times \tilde{E}_{\vec{r}} &= + \imath \Omega e^{-\imath \omega \Delta_t / 2} \hat{B}_{\vec{r} + \frac{1}{2}} + - \hat{M}_{\vec{r} + \frac{1}{2}} \\ + \hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &= + -\imath \Omega e^{ \imath \omega \Delta_t / 2} \tilde{D}_{\vec{r}} + + \tilde{J}_{\vec{r}} \\ + \tilde{\nabla} \cdot \hat{B}_{\vec{r} + \frac{1}{2}} &= 0 \\ + \hat{\nabla} \cdot \tilde{D}_{\vec{r}} &= \rho_{\vec{r}} + \end{aligned} $$ -With $\\Delta_t \\to 0$, this simplifies to +With $\Delta_t \to 0$, this simplifies to $$ - \\begin{aligned} - \\tilde{E}_{l, \\vec{r}} &\\to \\tilde{E}_{\\vec{r}} \\\\ - \\tilde{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &\\to \\tilde{H}_{\\vec{r} + \\frac{1}{2}} \\\\ - \\tilde{J}_{l, \\vec{r}} &\\to \\tilde{J}_{\\vec{r}} \\\\ - \\tilde{M}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &\\to \\tilde{M}_{\\vec{r} + \\frac{1}{2}} \\\\ - \\Omega &\\to \\omega \\\\ - \\tilde{\\partial}_t &\\to -\\imath \\omega \\\\ - \\hat{\\partial}_t &\\to -\\imath \\omega \\\\ - \\end{aligned} + \begin{aligned} + \tilde{E}_{l, \vec{r}} &\to \tilde{E}_{\vec{r}} \\ + \tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{H}_{\vec{r} + \frac{1}{2}} \\ + \tilde{J}_{l, \vec{r}} &\to \tilde{J}_{\vec{r}} \\ + \tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{M}_{\vec{r} + \frac{1}{2}} \\ + \Omega &\to \omega \\ + \tilde{\partial}_t &\to -\imath \omega \\ + \hat{\partial}_t &\to -\imath \omega \\ + \end{aligned} $$ and then $$ - \\begin{aligned} - \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} &= - \\imath \\omega \\hat{B}_{\\vec{r} + \\frac{1}{2}} - - \\hat{M}_{\\vec{r} + \\frac{1}{2}} \\\\ - \\hat{\\nabla} \\times \\hat{H}_{\\vec{r} + \\frac{1}{2}} &= - -\\imath \\omega \\tilde{D}_{\\vec{r}} - + \\tilde{J}_{\\vec{r}} \\\\ - \\end{aligned} + \begin{aligned} + \tilde{\nabla} \times \tilde{E}_{\vec{r}} &= + \imath \omega \hat{B}_{\vec{r} + \frac{1}{2}} + - \hat{M}_{\vec{r} + \frac{1}{2}} \\ + \hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &= + -\imath \omega \tilde{D}_{\vec{r}} + + \tilde{J}_{\vec{r}} \\ + \end{aligned} $$ $$ - \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}}) - -\\omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = -\\imath \\omega \\tilde{J}_{\\vec{r}} \\\\ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}}) + -\omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \omega \tilde{J}_{\vec{r}} \\ $$ # TODO FDFD? diff --git a/meanas/fdfd/functional.py b/meanas/fdfd/functional.py index 74b4263..ba2bd70 100644 --- a/meanas/fdfd/functional.py +++ b/meanas/fdfd/functional.py @@ -189,10 +189,10 @@ def e_tfsf_source( def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[cfdfield_t, cfdfield_t], cfdfield_t]: - """ + r""" Generates a function that takes the single-frequency `E` and `H` fields - and calculates the cross product `E` x `H` = $E \\times H$ as required - for the Poynting vector, $S = E \\times H$ + and calculates the cross product `E` x `H` = $E \times H$ as required + for the Poynting vector, $S = E \times H$ Note: This function also shifts the input `E` field by one cell as required diff --git a/meanas/fdfd/operators.py b/meanas/fdfd/operators.py index ff0a95c..3a489a7 100644 --- a/meanas/fdfd/operators.py +++ b/meanas/fdfd/operators.py @@ -45,14 +45,14 @@ def e_full( pec: vfdfield_t | None = None, pmc: vfdfield_t | None = None, ) -> sparse.spmatrix: - """ + r""" Wave operator - $$ \\nabla \\times (\\frac{1}{\\mu} \\nabla \\times) - \\Omega^2 \\epsilon $$ + $$ \nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon $$ del x (1/mu * del x) - omega**2 * epsilon for use with the E-field, with wave equation - $$ (\\nabla \\times (\\frac{1}{\\mu} \\nabla \\times) - \\Omega^2 \\epsilon) E = -\\imath \\omega J $$ + $$ (\nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon) E = -\imath \omega J $$ (del x (1/mu * del x) - omega**2 * epsilon) E = -i * omega * J @@ -131,14 +131,14 @@ def h_full( pec: vfdfield_t | None = None, pmc: vfdfield_t | None = None, ) -> sparse.spmatrix: - """ + r""" Wave operator - $$ \\nabla \\times (\\frac{1}{\\epsilon} \\nabla \\times) - \\omega^2 \\mu $$ + $$ \nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu $$ del x (1/epsilon * del x) - omega**2 * mu for use with the H-field, with wave equation - $$ (\\nabla \\times (\\frac{1}{\\epsilon} \\nabla \\times) - \\omega^2 \\mu) E = \\imath \\omega M $$ + $$ (\nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu) E = \imath \omega M $$ (del x (1/epsilon * del x) - omega**2 * mu) E = i * omega * M @@ -188,28 +188,28 @@ def eh_full( pec: vfdfield_t | None = None, pmc: vfdfield_t | None = None, ) -> sparse.spmatrix: - """ + r""" Wave operator for `[E, H]` field representation. This operator implements Maxwell's equations without cancelling out either E or H. The operator is - $$ \\begin{bmatrix} - -\\imath \\omega \\epsilon & \\nabla \\times \\\\ - \\nabla \\times & \\imath \\omega \\mu - \\end{bmatrix} $$ + $$ \begin{bmatrix} + -\imath \omega \epsilon & \nabla \times \\ + \nabla \times & \imath \omega \mu + \end{bmatrix} $$ [[-i * omega * epsilon, del x ], [del x, i * omega * mu]] for use with a field vector of the form `cat(vec(E), vec(H))`: - $$ \\begin{bmatrix} - -\\imath \\omega \\epsilon & \\nabla \\times \\\\ - \\nabla \\times & \\imath \\omega \\mu - \\end{bmatrix} - \\begin{bmatrix} E \\\\ - H - \\end{bmatrix} - = \\begin{bmatrix} J \\\\ - -M - \\end{bmatrix} $$ + $$ \begin{bmatrix} + -\imath \omega \epsilon & \nabla \times \\ + \nabla \times & \imath \omega \mu + \end{bmatrix} + \begin{bmatrix} E \\ + H + \end{bmatrix} + = \begin{bmatrix} J \\ + -M + \end{bmatrix} $$ Args: omega: Angular frequency of the simulation diff --git a/meanas/fdfd/waveguide_2d.py b/meanas/fdfd/waveguide_2d.py index dce3573..eaae21c 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 @@ -253,9 +253,10 @@ def operator_e( mu_yx = sparse.diags(numpy.hstack((mu_parts[1], mu_parts[0]))) mu_z_inv = sparse.diags(1 / mu_parts[2]) - op = omega * omega * mu_yx @ eps_xy + \ - mu_yx @ sparse.vstack((-Dby, Dbx)) @ mu_z_inv @ sparse.hstack((-Dfy, Dfx)) + \ - sparse.vstack((Dfx, Dfy)) @ eps_z_inv @ sparse.hstack((Dbx, Dby)) @ eps_xy + op = (omega * omega * mu_yx @ eps_xy + + mu_yx @ sparse.vstack((-Dby, Dbx)) @ mu_z_inv @ sparse.hstack((-Dfy, Dfx)) + + sparse.vstack((Dfx, Dfy)) @ eps_z_inv @ sparse.hstack((Dbx, Dby)) @ eps_xy + ) return op @@ -265,7 +266,7 @@ 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 +278,18 @@ 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 @@ -320,10 +321,10 @@ def operator_h( mu_xy = sparse.diags(numpy.hstack((mu_parts[0], mu_parts[1]))) mu_z_inv = sparse.diags(1 / mu_parts[2]) - op = omega * omega * eps_yx @ mu_xy + \ - eps_yx @ sparse.vstack((-Dfy, Dfx)) @ eps_z_inv @ sparse.hstack((-Dby, Dbx)) + \ - sparse.vstack((Dbx, Dby)) @ mu_z_inv @ sparse.hstack((Dfx, Dfy)) @ mu_xy - + op = (omega * omega * eps_yx @ mu_xy + + eps_yx @ sparse.vstack((-Dfy, Dfx)) @ eps_z_inv @ sparse.hstack((-Dby, Dbx)) + + sparse.vstack((Dbx, Dby)) @ mu_z_inv @ sparse.hstack((Dfx, Dfy)) @ mu_xy + ) return op diff --git a/meanas/fdfd/waveguide_cyl.py b/meanas/fdfd/waveguide_cyl.py index 6b3a160..0d3be0b 100644 --- a/meanas/fdfd/waveguide_cyl.py +++ b/meanas/fdfd/waveguide_cyl.py @@ -73,8 +73,10 @@ def cylindrical_operator( omega2 = omega * omega - op = (omega2 * diag((Tx, Ty)) + pa) @ diag((a0, a1)) + \ + op = ( + (omega2 * diag((Tx, Ty)) + pa) @ diag((a0, a1)) - (sparse.bmat(((None, Ty), (Tx, None))) + pb / omega2) @ diag((b0, b1)) + ) return op diff --git a/meanas/fdmath/__init__.py b/meanas/fdmath/__init__.py index eb8b6de..8a6b784 100644 --- a/meanas/fdmath/__init__.py +++ b/meanas/fdmath/__init__.py @@ -1,4 +1,4 @@ -""" +r""" Basic discrete calculus for finite difference (fd) simulations. @@ -43,11 +43,11 @@ Scalar derivatives and cell shifts ---------------------------------- Define the discrete forward derivative as - $$ [\\tilde{\\partial}_x f]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$ + $$ [\tilde{\partial}_x f]_{m + \frac{1}{2}} = \frac{1}{\Delta_{x, m}} (f_{m + 1} - f_m) $$ where $f$ is a function defined at discrete locations on the x-axis (labeled using $m$). - The value at $m$ occupies a length $\\Delta_{x, m}$ along the x-axis. Note that $m$ + The value at $m$ occupies a length $\Delta_{x, m}$ along the x-axis. Note that $m$ is an index along the x-axis, _not_ necessarily an x-coordinate, since each length - $\\Delta_{x, m}, \\Delta_{x, m+1}, ...$ is independently chosen. + $\Delta_{x, m}, \Delta_{x, m+1}, ...$ is independently chosen. If we treat `f` as a 1D array of values, with the `i`-th value `f[i]` taking up a length `dx[i]` along the x-axis, the forward derivative is @@ -56,13 +56,13 @@ along the x-axis, the forward derivative is Likewise, discrete reverse derivative is - $$ [\\hat{\\partial}_x f ]_{m - \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m} - f_{m - 1}) $$ + $$ [\hat{\partial}_x f ]_{m - \frac{1}{2}} = \frac{1}{\Delta_{x, m}} (f_{m} - f_{m - 1}) $$ or deriv_back(f)[i] = (f[i] - f[i - 1]) / dx[i] The derivatives' values are shifted by a half-cell relative to the original function, and -will have different cell widths if all the `dx[i]` ( $\\Delta_{x, m}$ ) are not +will have different cell widths if all the `dx[i]` ( $\Delta_{x, m}$ ) are not identical: [figure: derivatives and cell sizes] @@ -88,19 +88,19 @@ identical: In the above figure, `f0 =` $f_0$, `f1 =` $f_1$ - `Df0 =` $[\\tilde{\\partial}f]_{0 + \\frac{1}{2}}$ - `Df1 =` $[\\tilde{\\partial}f]_{1 + \\frac{1}{2}}$ - `df0 =` $[\\hat{\\partial}f]_{0 - \\frac{1}{2}}$ + `Df0 =` $[\tilde{\partial}f]_{0 + \frac{1}{2}}$ + `Df1 =` $[\tilde{\partial}f]_{1 + \frac{1}{2}}$ + `df0 =` $[\hat{\partial}f]_{0 - \frac{1}{2}}$ etc. -The fractional subscript $m + \\frac{1}{2}$ is used to indicate values defined +The fractional subscript $m + \frac{1}{2}$ is used to indicate values defined at shifted locations relative to the original $m$, with corresponding lengths - $$ \\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x, m + 1}) $$ + $$ \Delta_{x, m + \frac{1}{2}} = \frac{1}{2} * (\Delta_{x, m} + \Delta_{x, m + 1}) $$ -Just as $m$ is not itself an x-coordinate, neither is $m + \\frac{1}{2}$; +Just as $m$ is not itself an x-coordinate, neither is $m + \frac{1}{2}$; carefully note the positions of the various cells in the above figure vs their labels. If the positions labeled with $m$ are considered the "base" or "original" grid, -the positions labeled with $m + \\frac{1}{2}$ are said to lie on a "dual" or +the positions labeled with $m + \frac{1}{2}$ are said to lie on a "dual" or "derived" grid. For the remainder of the `Discrete calculus` section, all figures will show @@ -113,12 +113,12 @@ Gradients and fore-vectors -------------------------- Expanding to three dimensions, we can define two gradients - $$ [\\tilde{\\nabla} f]_{m,n,p} = \\vec{x} [\\tilde{\\partial}_x f]_{m + \\frac{1}{2},n,p} + - \\vec{y} [\\tilde{\\partial}_y f]_{m,n + \\frac{1}{2},p} + - \\vec{z} [\\tilde{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$ - $$ [\\hat{\\nabla} f]_{m,n,p} = \\vec{x} [\\hat{\\partial}_x f]_{m + \\frac{1}{2},n,p} + - \\vec{y} [\\hat{\\partial}_y f]_{m,n + \\frac{1}{2},p} + - \\vec{z} [\\hat{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$ + $$ [\tilde{\nabla} f]_{m,n,p} = \vec{x} [\tilde{\partial}_x f]_{m + \frac{1}{2},n,p} + + \vec{y} [\tilde{\partial}_y f]_{m,n + \frac{1}{2},p} + + \vec{z} [\tilde{\partial}_z f]_{m,n,p + \frac{1}{2}} $$ + $$ [\hat{\nabla} f]_{m,n,p} = \vec{x} [\hat{\partial}_x f]_{m + \frac{1}{2},n,p} + + \vec{y} [\hat{\partial}_y f]_{m,n + \frac{1}{2},p} + + \vec{z} [\hat{\partial}_z f]_{m,n,p + \frac{1}{2}} $$ or @@ -144,12 +144,12 @@ y in y, and z in z. We call the resulting object a "fore-vector" or "back-vector", depending on the direction of the shift. We write it as - $$ \\tilde{g}_{m,n,p} = \\vec{x} g^x_{m + \\frac{1}{2},n,p} + - \\vec{y} g^y_{m,n + \\frac{1}{2},p} + - \\vec{z} g^z_{m,n,p + \\frac{1}{2}} $$ - $$ \\hat{g}_{m,n,p} = \\vec{x} g^x_{m - \\frac{1}{2},n,p} + - \\vec{y} g^y_{m,n - \\frac{1}{2},p} + - \\vec{z} g^z_{m,n,p - \\frac{1}{2}} $$ + $$ \tilde{g}_{m,n,p} = \vec{x} g^x_{m + \frac{1}{2},n,p} + + \vec{y} g^y_{m,n + \frac{1}{2},p} + + \vec{z} g^z_{m,n,p + \frac{1}{2}} $$ + $$ \hat{g}_{m,n,p} = \vec{x} g^x_{m - \frac{1}{2},n,p} + + \vec{y} g^y_{m,n - \frac{1}{2},p} + + \vec{z} g^z_{m,n,p - \frac{1}{2}} $$ [figure: gradient / fore-vector] @@ -172,15 +172,15 @@ Divergences There are also two divergences, - $$ d_{n,m,p} = [\\tilde{\\nabla} \\cdot \\hat{g}]_{n,m,p} - = [\\tilde{\\partial}_x g^x]_{m,n,p} + - [\\tilde{\\partial}_y g^y]_{m,n,p} + - [\\tilde{\\partial}_z g^z]_{m,n,p} $$ + $$ d_{n,m,p} = [\tilde{\nabla} \cdot \hat{g}]_{n,m,p} + = [\tilde{\partial}_x g^x]_{m,n,p} + + [\tilde{\partial}_y g^y]_{m,n,p} + + [\tilde{\partial}_z g^z]_{m,n,p} $$ - $$ d_{n,m,p} = [\\hat{\\nabla} \\cdot \\tilde{g}]_{n,m,p} - = [\\hat{\\partial}_x g^x]_{m,n,p} + - [\\hat{\\partial}_y g^y]_{m,n,p} + - [\\hat{\\partial}_z g^z]_{m,n,p} $$ + $$ d_{n,m,p} = [\hat{\nabla} \cdot \tilde{g}]_{n,m,p} + = [\hat{\partial}_x g^x]_{m,n,p} + + [\hat{\partial}_y g^y]_{m,n,p} + + [\hat{\partial}_z g^z]_{m,n,p} $$ or @@ -203,7 +203,7 @@ where `g = [gx, gy, gz]` is a fore- or back-vector field. Since we applied the forward divergence to the back-vector (and vice-versa), the resulting scalar value is defined at the back-vector's (fore-vector's) location $(m,n,p)$ and not at the locations of its components -$(m \\pm \\frac{1}{2},n,p)$ etc. +$(m \pm \frac{1}{2},n,p)$ etc. [figure: divergence] ^^ @@ -227,23 +227,23 @@ Curls The two curls are then - $$ \\begin{aligned} - \\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\ - [\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= - \\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} - \\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\ - &+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} - \\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\ - &+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_y g^z_{m + \\frac{1}{2},n,p}) - \\end{aligned} $$ + $$ \begin{aligned} + \hat{h}_{m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2}} &= \\ + [\tilde{\nabla} \times \tilde{g}]_{m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2}} &= + \vec{x} (\tilde{\partial}_y g^z_{m,n,p + \frac{1}{2}} - \tilde{\partial}_z g^y_{m,n + \frac{1}{2},p}) \\ + &+ \vec{y} (\tilde{\partial}_z g^x_{m + \frac{1}{2},n,p} - \tilde{\partial}_x g^z_{m,n,p + \frac{1}{2}}) \\ + &+ \vec{z} (\tilde{\partial}_x g^y_{m,n + \frac{1}{2},p} - \tilde{\partial}_y g^z_{m + \frac{1}{2},n,p}) + \end{aligned} $$ and - $$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} = - [\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$ + $$ \tilde{h}_{m - \frac{1}{2}, n - \frac{1}{2}, p - \frac{1}{2}} = + [\hat{\nabla} \times \hat{g}]_{m - \frac{1}{2}, n - \frac{1}{2}, p - \frac{1}{2}} $$ - where $\\hat{g}$ and $\\tilde{g}$ are located at $(m,n,p)$ - with components at $(m \\pm \\frac{1}{2},n,p)$ etc., - while $\\hat{h}$ and $\\tilde{h}$ are located at $(m \\pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$ - with components at $(m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$ etc. + where $\hat{g}$ and $\tilde{g}$ are located at $(m,n,p)$ + with components at $(m \pm \frac{1}{2},n,p)$ etc., + while $\hat{h}$ and $\tilde{h}$ are located at $(m \pm \frac{1}{2}, n \pm \frac{1}{2}, p \pm \frac{1}{2})$ + with components at $(m, n \pm \frac{1}{2}, p \pm \frac{1}{2})$ etc. [code: curls] @@ -287,27 +287,27 @@ Maxwell's Equations If we discretize both space (m,n,p) and time (l), Maxwell's equations become - $$ \\begin{aligned} - \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} - - \\hat{M}_{l, \\vec{r} + \\frac{1}{2}} \\\\ - \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2},\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}} - + \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\ - \\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\ - \\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}} - \\end{aligned} $$ + $$ \begin{aligned} + \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &= -\tilde{\partial}_t \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} + - \hat{M}_{l, \vec{r} + \frac{1}{2}} \\ + \hat{\nabla} \times \hat{H}_{l-\frac{1}{2},\vec{r} + \frac{1}{2}} &= \hat{\partial}_t \tilde{D}_{l, \vec{r}} + + \tilde{J}_{l-\frac{1}{2},\vec{r}} \\ + \tilde{\nabla} \cdot \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} &= 0 \\ + \hat{\nabla} \cdot \tilde{D}_{l,\vec{r}} &= \rho_{l,\vec{r}} + \end{aligned} $$ with - $$ \\begin{aligned} - \\hat{B}_{\\vec{r}} &= \\mu_{\\vec{r} + \\frac{1}{2}} \\cdot \\hat{H}_{\\vec{r} + \\frac{1}{2}} \\\\ - \\tilde{D}_{\\vec{r}} &= \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} - \\end{aligned} $$ + $$ \begin{aligned} + \hat{B}_{\vec{r}} &= \mu_{\vec{r} + \frac{1}{2}} \cdot \hat{H}_{\vec{r} + \frac{1}{2}} \\ + \tilde{D}_{\vec{r}} &= \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} + \end{aligned} $$ -where the spatial subscripts are abbreviated as $\\vec{r} = (m, n, p)$ and -$\\vec{r} + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2})$, -$\\tilde{E}$ and $\\hat{H}$ are the electric and magnetic fields, -$\\tilde{J}$ and $\\hat{M}$ are the electric and magnetic current distributions, -and $\\epsilon$ and $\\mu$ are the dielectric permittivity and magnetic permeability. +where the spatial subscripts are abbreviated as $\vec{r} = (m, n, p)$ and +$\vec{r} + \frac{1}{2} = (m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2})$, +$\tilde{E}$ and $\hat{H}$ are the electric and magnetic fields, +$\tilde{J}$ and $\hat{M}$ are the electric and magnetic current distributions, +and $\epsilon$ and $\mu$ are the dielectric permittivity and magnetic permeability. The above is Yee's algorithm, written in a form analogous to Maxwell's equations. The time derivatives can be expanded to form the update equations: @@ -369,34 +369,34 @@ Each component forms its own grid, offset from the others: The divergence equations can be derived by taking the divergence of the curl equations and combining them with charge continuity, - $$ \\hat{\\nabla} \\cdot \\tilde{J} + \\hat{\\partial}_t \\rho = 0 $$ + $$ \hat{\nabla} \cdot \tilde{J} + \hat{\partial}_t \rho = 0 $$ implying that the discrete Maxwell's equations do not produce spurious charges. Wave equation ------------- -Taking the backward curl of the $\\tilde{\\nabla} \\times \\tilde{E}$ equation and -replacing the resulting $\\hat{\\nabla} \\times \\hat{H}$ term using its respective equation, -and setting $\\hat{M}$ to zero, we can form the discrete wave equation: +Taking the backward curl of the $\tilde{\nabla} \times \tilde{E}$ equation and +replacing the resulting $\hat{\nabla} \times \hat{H}$ term using its respective equation, +and setting $\hat{M}$ to zero, we can form the discrete wave equation: $$ - \\begin{aligned} - \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= - -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} - - \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\ - \\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= - -\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\ - \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &= - \\hat{\\nabla} \\times (-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}) \\\\ - \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &= - -\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\ - \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &= - -\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\ - \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) - + \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{l, \\vec{r}} - &= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}} - \\end{aligned} + \begin{aligned} + \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &= + -\tilde{\partial}_t \hat{B}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} + - \hat{M}_{l-1, \vec{r} + \frac{1}{2}} \\ + \mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &= + -\tilde{\partial}_t \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} \\ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &= + \hat{\nabla} \times (-\tilde{\partial}_t \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}}) \\ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &= + -\tilde{\partial}_t \hat{\nabla} \times \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} \\ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &= + -\tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \tilde{E}_{l, \vec{r}} + \hat{\partial}_t \tilde{J}_{l-\frac{1}{2},\vec{r}} \\ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) + + \tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \cdot \tilde{E}_{l, \vec{r}} + &= \tilde{\partial}_t \tilde{J}_{l - \frac{1}{2}, \vec{r}} + \end{aligned} $$ @@ -406,27 +406,27 @@ Frequency domain We can substitute in a time-harmonic fields $$ - \\begin{aligned} - \\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\ - \\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} - \\end{aligned} + \begin{aligned} + \tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\ + \tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} + \end{aligned} $$ resulting in $$ - \\begin{aligned} - \\tilde{\\partial}_t &\\Rightarrow (e^{ \\imath \\omega \\Delta_t} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{-\\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\ - \\hat{\\partial}_t &\\Rightarrow (1 - e^{-\\imath \\omega \\Delta_t}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{ \\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\ - \\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t - \\end{aligned} + \begin{aligned} + \tilde{\partial}_t &\Rightarrow (e^{ \imath \omega \Delta_t} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{-\imath \omega \Delta_t / 2} = -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\ + \hat{\partial}_t &\Rightarrow (1 - e^{-\imath \omega \Delta_t}) / \Delta_t = \frac{-2 \imath}{\Delta_t} \sin(\omega \Delta_t / 2) e^{ \imath \omega \Delta_t / 2} = -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\ + \Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t + \end{aligned} $$ This gives the frequency-domain wave equation, $$ - \\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}}) - -\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = -\\imath \\Omega \\tilde{J}_{\\vec{r}} e^{\\imath \\omega \\Delta_t / 2} \\\\ + \hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}}) + -\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\ $$ @@ -436,48 +436,48 @@ Plane waves and Dispersion relation With uniform material distribution and no sources $$ - \\begin{aligned} - \\mu_{\\vec{r} + \\frac{1}{2}} &= \\mu \\\\ - \\epsilon_{\\vec{r}} &= \\epsilon \\\\ - \\tilde{J}_{\\vec{r}} &= 0 \\\\ - \\end{aligned} + \begin{aligned} + \mu_{\vec{r} + \frac{1}{2}} &= \mu \\ + \epsilon_{\vec{r}} &= \epsilon \\ + \tilde{J}_{\vec{r}} &= 0 \\ + \end{aligned} $$ the frequency domain wave equation simplifies to -$$ \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} - \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$ +$$ \hat{\nabla} \times \tilde{\nabla} \times \tilde{E}_{\vec{r}} - \Omega^2 \epsilon \mu \tilde{E}_{\vec{r}} = 0 $$ -Since $\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}} = 0$, we can simplify +Since $\hat{\nabla} \cdot \tilde{E}_{\vec{r}} = 0$, we can simplify $$ - \\begin{aligned} - \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} - &= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\ - &= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\ - &= - \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}} - \\end{aligned} + \begin{aligned} + \hat{\nabla} \times \tilde{\nabla} \times \tilde{E}_{\vec{r}} + &= \tilde{\nabla}(\hat{\nabla} \cdot \tilde{E}_{\vec{r}}) - \hat{\nabla} \cdot \tilde{\nabla} \tilde{E}_{\vec{r}} \\ + &= - \hat{\nabla} \cdot \tilde{\nabla} \tilde{E}_{\vec{r}} \\ + &= - \tilde{\nabla}^2 \tilde{E}_{\vec{r}} + \end{aligned} $$ and we get -$$ \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}} + \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$ +$$ \tilde{\nabla}^2 \tilde{E}_{\vec{r}} + \Omega^2 \epsilon \mu \tilde{E}_{\vec{r}} = 0 $$ We can convert this to three scalar-wave equations of the form -$$ (\\tilde{\\nabla}^2 + K^2) \\phi_{\\vec{r}} = 0 $$ +$$ (\tilde{\nabla}^2 + K^2) \phi_{\vec{r}} = 0 $$ -with $K^2 = \\Omega^2 \\mu \\epsilon$. Now we let +with $K^2 = \Omega^2 \mu \epsilon$. Now we let -$$ \\phi_{\\vec{r}} = A e^{\\imath (k_x m \\Delta_x + k_y n \\Delta_y + k_z p \\Delta_z)} $$ +$$ \phi_{\vec{r}} = A e^{\imath (k_x m \Delta_x + k_y n \Delta_y + k_z p \Delta_z)} $$ resulting in $$ - \\begin{aligned} - \\tilde{\\partial}_x &\\Rightarrow (e^{ \\imath k_x \\Delta_x} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{ \\imath k_x \\Delta_x / 2} = \\imath K_x e^{ \\imath k_x \\Delta_x / 2}\\\\ - \\hat{\\partial}_x &\\Rightarrow (1 - e^{-\\imath k_x \\Delta_x}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{-\\imath k_x \\Delta_x / 2} = \\imath K_x e^{-\\imath k_x \\Delta_x / 2}\\\\ - K_x &= 2 \\sin(k_x \\Delta_x / 2) / \\Delta_x \\\\ - \\end{aligned} + \begin{aligned} + \tilde{\partial}_x &\Rightarrow (e^{ \imath k_x \Delta_x} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{ \imath k_x \Delta_x / 2} = \imath K_x e^{ \imath k_x \Delta_x / 2}\\ + \hat{\partial}_x &\Rightarrow (1 - e^{-\imath k_x \Delta_x}) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{-\imath k_x \Delta_x / 2} = \imath K_x e^{-\imath k_x \Delta_x / 2}\\ + K_x &= 2 \sin(k_x \Delta_x / 2) / \Delta_x \\ + \end{aligned} $$ with similar expressions for the y and z dimnsions (and $K_y, K_z$). @@ -485,20 +485,20 @@ with similar expressions for the y and z dimnsions (and $K_y, K_z$). This implies $$ - \\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_{\\vec{r}} \\\\ - K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2 + \tilde{\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \phi_{\vec{r}} \\ + K_x^2 + K_y^2 + K_z^2 = \Omega^2 \mu \epsilon = \Omega^2 / c^2 $$ -where $c = \\sqrt{\\mu \\epsilon}$. +where $c = \sqrt{\mu \epsilon}$. -Assuming real $(k_x, k_y, k_z), \\omega$ will be real only if +Assuming real $(k_x, k_y, k_z), \omega$ will be real only if -$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$ +$$ c^2 \Delta_t^2 = \frac{\Delta_t^2}{\mu \epsilon} < 1/(\frac{1}{\Delta_x^2} + \frac{1}{\Delta_y^2} + \frac{1}{\Delta_z^2}) $$ -If $\\Delta_x = \\Delta_y = \\Delta_z$, this simplifies to $c \\Delta_t < \\Delta_x / \\sqrt{3}$. +If $\Delta_x = \Delta_y = \Delta_z$, this simplifies to $c \Delta_t < \Delta_x / \sqrt{3}$. This last form can be interpreted as enforcing causality; the distance that light -travels in one timestep (i.e., $c \\Delta_t$) must be less than the diagonal -of the smallest cell ( $\\Delta_x / \\sqrt{3}$ when on a uniform cubic grid). +travels in one timestep (i.e., $c \Delta_t$) must be less than the diagonal +of the smallest cell ( $\Delta_x / \sqrt{3}$ when on a uniform cubic grid). Grid description @@ -515,8 +515,8 @@ to make the illustration simpler; we need at least two cells in the x dimension demonstrate how nonuniform `dx` affects the various components. Place the E fore-vectors at integer indices $r = (m, n, p)$ and the H back-vectors -at fractional indices $r + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, -p + \\frac{1}{2})$. Remember that these are indices and not coordinates; they can +at fractional indices $r + \frac{1}{2} = (m + \frac{1}{2}, n + \frac{1}{2}, +p + \frac{1}{2})$. Remember that these are indices and not coordinates; they can correspond to arbitrary (monotonically increasing) coordinates depending on the cell widths. Draw lines to denote the planes on which the H components and back-vectors are defined. @@ -718,14 +718,14 @@ composed of the three diagonal tensor components: or $$ - \\epsilon = \\begin{bmatrix} \\epsilon_{xx} & 0 & 0 \\\\ - 0 & \\epsilon_{yy} & 0 \\\\ - 0 & 0 & \\epsilon_{zz} \\end{bmatrix} + \epsilon = \begin{bmatrix} \epsilon_{xx} & 0 & 0 \\ + 0 & \epsilon_{yy} & 0 \\ + 0 & 0 & \epsilon_{zz} \end{bmatrix} $$ $$ - \\mu = \\begin{bmatrix} \\mu_{xx} & 0 & 0 \\\\ - 0 & \\mu_{yy} & 0 \\\\ - 0 & 0 & \\mu_{zz} \\end{bmatrix} + \mu = \begin{bmatrix} \mu_{xx} & 0 & 0 \\ + 0 & \mu_{yy} & 0 \\ + 0 & 0 & \mu_{zz} \end{bmatrix} $$ where the off-diagonal terms (e.g. `epsilon_xy`) are assumed to be zero. diff --git a/meanas/fdmath/functional.py b/meanas/fdmath/functional.py index e33aa93..3a10a00 100644 --- a/meanas/fdmath/functional.py +++ b/meanas/fdmath/functional.py @@ -62,7 +62,7 @@ def deriv_back( def curl_forward( dx_e: Sequence[NDArray[numpy.float_]] | None = None, ) -> fdfield_updater_t: - """ + r""" Curl operator for use with the E field. Args: @@ -71,7 +71,7 @@ def curl_forward( Returns: Function `f` for taking the discrete forward curl of a field, - `f(E)` -> curlE $= \\nabla_f \\times E$ + `f(E)` -> curlE $= \nabla_f \times E$ """ Dx, Dy, Dz = deriv_forward(dx_e) @@ -91,7 +91,7 @@ def curl_forward( def curl_back( dx_h: Sequence[NDArray[numpy.float_]] | None = None, ) -> fdfield_updater_t: - """ + r""" Create a function which takes the backward curl of a field. Args: @@ -100,7 +100,7 @@ def curl_back( Returns: Function `f` for taking the discrete backward curl of a field, - `f(H)` -> curlH $= \\nabla_b \\times H$ + `f(H)` -> curlH $= \nabla_b \times H$ """ Dx, Dy, Dz = deriv_back(dx_h) diff --git a/meanas/fdtd/__init__.py b/meanas/fdtd/__init__.py index 63be295..171c4f4 100644 --- a/meanas/fdtd/__init__.py +++ b/meanas/fdtd/__init__.py @@ -1,4 +1,4 @@ -""" +r""" Utilities for running finite-difference time-domain (FDTD) simulations See the discussion of `Maxwell's Equations` in `meanas.fdmath` for basic @@ -11,9 +11,9 @@ Timestep From the discussion of "Plane waves and the Dispersion relation" in `meanas.fdmath`, we have -$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$ +$$ c^2 \Delta_t^2 = \frac{\Delta_t^2}{\mu \epsilon} < 1/(\frac{1}{\Delta_x^2} + \frac{1}{\Delta_y^2} + \frac{1}{\Delta_z^2}) $$ -or, if $\\Delta_x = \\Delta_y = \\Delta_z$, then $c \\Delta_t < \\frac{\\Delta_x}{\\sqrt{3}}$. +or, if $\Delta_x = \Delta_y = \Delta_z$, then $c \Delta_t < \frac{\Delta_x}{\sqrt{3}}$. Based on this, we can set @@ -27,81 +27,81 @@ Poynting Vector and Energy Conservation Let -$$ \\begin{aligned} - \\tilde{S}_{l, l', \\vec{r}} &=& &\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\\\ - &=& &\\vec{x} (\\tilde{E}^y_{l,m+1,n,p} \\hat{H}^z_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^z_{l,m+1,n,p} \\hat{H}^y_{l', \\vec{r} + \\frac{1}{2}}) \\\\ - & &+ &\\vec{y} (\\tilde{E}^z_{l,m,n+1,p} \\hat{H}^x_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^x_{l,m,n+1,p} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}}) \\\\ - & &+ &\\vec{z} (\\tilde{E}^x_{l,m,n,p+1} \\hat{H}^y_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^y_{l,m,n,p+1} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}}) - \\end{aligned} +$$ \begin{aligned} + \tilde{S}_{l, l', \vec{r}} &=& &\tilde{E}_{l, \vec{r}} \otimes \hat{H}_{l', \vec{r} + \frac{1}{2}} \\ + &=& &\vec{x} (\tilde{E}^y_{l,m+1,n,p} \hat{H}^z_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^z_{l,m+1,n,p} \hat{H}^y_{l', \vec{r} + \frac{1}{2}}) \\ + & &+ &\vec{y} (\tilde{E}^z_{l,m,n+1,p} \hat{H}^x_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^x_{l,m,n+1,p} \hat{H}^z_{l', \vec{r} + \frac{1}{2}}) \\ + & &+ &\vec{z} (\tilde{E}^x_{l,m,n,p+1} \hat{H}^y_{l',\vec{r} + \frac{1}{2}} - \tilde{E}^y_{l,m,n,p+1} \hat{H}^z_{l', \vec{r} + \frac{1}{2}}) + \end{aligned} $$ -where $\\vec{r} = (m, n, p)$ and $\\otimes$ is a modified cross product -in which the $\\tilde{E}$ terms are shifted as indicated. +where $\vec{r} = (m, n, p)$ and $\otimes$ is a modified cross product +in which the $\tilde{E}$ terms are shifted as indicated. By taking the divergence and rearranging terms, we can show that $$ - \\begin{aligned} - \\hat{\\nabla} \\cdot \\tilde{S}_{l, l', \\vec{r}} - &= \\hat{\\nabla} \\cdot (\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}}) \\\\ - &= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l, \\vec{r}} - - \\tilde{E}_{l, \\vec{r}} \\cdot \\hat{\\nabla} \\times \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\\\ - &= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot - (-\\tilde{\\partial}_t \\mu_{\\vec{r} + \\frac{1}{2}} \\hat{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} - - \\hat{M}_{l, \\vec{r} + \\frac{1}{2}}) - - \\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t \\tilde{\\epsilon}_{\\vec{r}} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} + - \\tilde{J}_{l', \\vec{r}}) \\\\ - &= \\hat{H}_{l'} \\cdot (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) - - \\tilde{E}_l \\cdot (\\epsilon / \\Delta_t )(\\tilde{E}_{l'+\\frac{1}{2}} - \\tilde{E}_{l'-\\frac{1}{2}}) - - \\hat{H}_{l'} \\cdot \\hat{M}_{l} - \\tilde{E}_l \\cdot \\tilde{J}_{l'} \\\\ - \\end{aligned} + \begin{aligned} + \hat{\nabla} \cdot \tilde{S}_{l, l', \vec{r}} + &= \hat{\nabla} \cdot (\tilde{E}_{l, \vec{r}} \otimes \hat{H}_{l', \vec{r} + \frac{1}{2}}) \\ + &= \hat{H}_{l', \vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l, \vec{r}} - + \tilde{E}_{l, \vec{r}} \cdot \hat{\nabla} \times \hat{H}_{l', \vec{r} + \frac{1}{2}} \\ + &= \hat{H}_{l', \vec{r} + \frac{1}{2}} \cdot + (-\tilde{\partial}_t \mu_{\vec{r} + \frac{1}{2}} \hat{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} - + \hat{M}_{l, \vec{r} + \frac{1}{2}}) - + \tilde{E}_{l, \vec{r}} \cdot (\hat{\partial}_t \tilde{\epsilon}_{\vec{r}} \tilde{E}_{l'+\frac{1}{2}, \vec{r}} + + \tilde{J}_{l', \vec{r}}) \\ + &= \hat{H}_{l'} \cdot (-\mu / \Delta_t)(\hat{H}_{l + \frac{1}{2}} - \hat{H}_{l - \frac{1}{2}}) - + \tilde{E}_l \cdot (\epsilon / \Delta_t )(\tilde{E}_{l'+\frac{1}{2}} - \tilde{E}_{l'-\frac{1}{2}}) + - \hat{H}_{l'} \cdot \hat{M}_{l} - \tilde{E}_l \cdot \tilde{J}_{l'} \\ + \end{aligned} $$ where in the last line the spatial subscripts have been dropped to emphasize the time subscripts $l, l'$, i.e. $$ - \\begin{aligned} - \\tilde{E}_l &= \\tilde{E}_{l, \\vec{r}} \\\\ - \\hat{H}_l &= \\tilde{H}_{l, \\vec{r} + \\frac{1}{2}} \\\\ - \\tilde{\\epsilon} &= \\tilde{\\epsilon}_{\\vec{r}} \\\\ - \\end{aligned} + \begin{aligned} + \tilde{E}_l &= \tilde{E}_{l, \vec{r}} \\ + \hat{H}_l &= \tilde{H}_{l, \vec{r} + \frac{1}{2}} \\ + \tilde{\epsilon} &= \tilde{\epsilon}_{\vec{r}} \\ + \end{aligned} $$ etc. -For $l' = l + \\frac{1}{2}$ we get +For $l' = l + \frac{1}{2}$ we get $$ - \\begin{aligned} - \\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} - &= \\hat{H}_{l + \\frac{1}{2}} \\cdot - (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) - - \\tilde{E}_l \\cdot (\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} - \\tilde{E}_l) - - \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l + \\frac{1}{2}} \\\\ - &= (-\\mu / \\Delta_t)(\\hat{H}^2_{l + \\frac{1}{2}} - \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}) - - (\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} \\cdot \\tilde{E}_l - \\tilde{E}^2_l) - - \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l + \\frac{1}{2}} \\\\ - &= -(\\mu \\hat{H}^2_{l + \\frac{1}{2}} - +\\epsilon \\tilde{E}_{l+1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\ - +(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}} - +\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\ - - \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\ - - \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\ - \\end{aligned} + \begin{aligned} + \hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}} + &= \hat{H}_{l + \frac{1}{2}} \cdot + (-\mu / \Delta_t)(\hat{H}_{l + \frac{1}{2}} - \hat{H}_{l - \frac{1}{2}}) - + \tilde{E}_l \cdot (\epsilon / \Delta_t)(\tilde{E}_{l+1} - \tilde{E}_l) + - \hat{H}_{l'} \cdot \hat{M}_l - \tilde{E}_l \cdot \tilde{J}_{l + \frac{1}{2}} \\ + &= (-\mu / \Delta_t)(\hat{H}^2_{l + \frac{1}{2}} - \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}}) - + (\epsilon / \Delta_t)(\tilde{E}_{l+1} \cdot \tilde{E}_l - \tilde{E}^2_l) + - \hat{H}_{l'} \cdot \hat{M}_l - \tilde{E}_l \cdot \tilde{J}_{l + \frac{1}{2}} \\ + &= -(\mu \hat{H}^2_{l + \frac{1}{2}} + +\epsilon \tilde{E}_{l+1} \cdot \tilde{E}_l) / \Delta_t \\ + +(\mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}} + +\epsilon \tilde{E}^2_l) / \Delta_t \\ + - \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\ + - \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\ + \end{aligned} $$ -and for $l' = l - \\frac{1}{2}$, +and for $l' = l - \frac{1}{2}$, $$ - \\begin{aligned} - \\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} - &= (\\mu \\hat{H}^2_{l - \\frac{1}{2}} - +\\epsilon \\tilde{E}_{l-1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\ - -(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}} - +\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\ - - \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\ - - \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\ - \\end{aligned} + \begin{aligned} + \hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}} + &= (\mu \hat{H}^2_{l - \frac{1}{2}} + +\epsilon \tilde{E}_{l-1} \cdot \tilde{E}_l) / \Delta_t \\ + -(\mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}} + +\epsilon \tilde{E}^2_l) / \Delta_t \\ + - \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\ + - \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\ + \end{aligned} $$ These two results form the discrete time-domain analogue to Poynting's theorem. @@ -109,25 +109,25 @@ They hint at the expressions for the energy, which can be calculated at the same time-index as either the E or H field: $$ - \\begin{aligned} - U_l &= \\epsilon \\tilde{E}^2_l + \\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}} \\\\ - U_{l + \\frac{1}{2}} &= \\epsilon \\tilde{E}_l \\cdot \\tilde{E}_{l + 1} + \\mu \\hat{H}^2_{l + \\frac{1}{2}} \\\\ - \\end{aligned} + \begin{aligned} + U_l &= \epsilon \tilde{E}^2_l + \mu \hat{H}_{l + \frac{1}{2}} \cdot \hat{H}_{l - \frac{1}{2}} \\ + U_{l + \frac{1}{2}} &= \epsilon \tilde{E}_l \cdot \tilde{E}_{l + 1} + \mu \hat{H}^2_{l + \frac{1}{2}} \\ + \end{aligned} $$ Rewriting the Poynting theorem in terms of the energy expressions, $$ - \\begin{aligned} - (U_{l+\\frac{1}{2}} - U_l) / \\Delta_t - &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\ - - \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\ - - \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\ - (U_l - U_{l-\\frac{1}{2}}) / \\Delta_t - &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\ - - \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\ - - \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\ - \\end{aligned} + \begin{aligned} + (U_{l+\frac{1}{2}} - U_l) / \Delta_t + &= -\hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}} \\ + - \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\ + - \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\ + (U_l - U_{l-\frac{1}{2}}) / \Delta_t + &= -\hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}} \\ + - \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\ + - \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\ + \end{aligned} $$ This result is exact and should practically hold to within numerical precision. No time- @@ -147,10 +147,10 @@ of the time-domain fields. The Ricker wavelet (normalized second derivative of a Gaussian) is commonly used for the pulse shape. It can be written -$$ f_r(t) = (1 - \\frac{1}{2} (\\omega (t - \\tau))^2) e^{-(\\frac{\\omega (t - \\tau)}{2})^2} $$ +$$ f_r(t) = (1 - \frac{1}{2} (\omega (t - \tau))^2) e^{-(\frac{\omega (t - \tau)}{2})^2} $$ -with $\\tau > \\frac{2 * \\pi}{\\omega}$ as a minimum delay to avoid a discontinuity at -t=0 (assuming the source is off for t<0 this gives $\\sim 10^{-3}$ error at t=0). +with $\tau > \frac{2 * \pi}{\omega}$ as a minimum delay to avoid a discontinuity at +t=0 (assuming the source is off for t<0 this gives $\sim 10^{-3}$ error at t=0). diff --git a/meanas/fdtd/energy.py b/meanas/fdtd/energy.py index 75938f3..43ea3a1 100644 --- a/meanas/fdtd/energy.py +++ b/meanas/fdtd/energy.py @@ -12,7 +12,7 @@ def poynting( h: fdfield_t, dxes: dx_lists_t | None = None, ) -> fdfield_t: - """ + r""" Calculate the poynting vector `S` ($S$). This is the energy transfer rate (amount of energy `U` per `dt` transferred @@ -44,16 +44,16 @@ def poynting( The full relationship is $$ - \\begin{aligned} - (U_{l+\\frac{1}{2}} - U_l) / \\Delta_t - &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\ - - \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\ - - \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\ - (U_l - U_{l-\\frac{1}{2}}) / \\Delta_t - &= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\ - - \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\ - - \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\ - \\end{aligned} + \begin{aligned} + (U_{l+\frac{1}{2}} - U_l) / \Delta_t + &= -\hat{\nabla} \cdot \tilde{S}_{l, l + \frac{1}{2}} \\ + - \hat{H}_{l+\frac{1}{2}} \cdot \hat{M}_l \\ + - \tilde{E}_l \cdot \tilde{J}_{l+\frac{1}{2}} \\ + (U_l - U_{l-\frac{1}{2}}) / \Delta_t + &= -\hat{\nabla} \cdot \tilde{S}_{l, l - \frac{1}{2}} \\ + - \hat{H}_{l-\frac{1}{2}} \cdot \hat{M}_l \\ + - \tilde{E}_l \cdot \tilde{J}_{l-\frac{1}{2}} \\ + \end{aligned} $$ These equalities are exact and should practically hold to within numerical precision. diff --git a/pdoc_templates/html_helpers.py b/pdoc_templates/html_helpers.py index a0be764..5e58405 100644 --- a/pdoc_templates/html_helpers.py +++ b/pdoc_templates/html_helpers.py @@ -321,7 +321,6 @@ class _ToMarkdown: """Wrap URLs in Python-Markdown-compatible .""" return re.sub(r'(?)\s]+)(\s*)', r'\1<\2>\3', text) -import subprocess class _MathPattern(InlineProcessor): NAME = 'pdoc-math'