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@ -14,7 +14,20 @@ Submodules:
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- `waveguide_3d`: Functions for transforming `waveguide_2d` results into 3D.
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===========
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================================================================
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From the "Frequency domain" section of `meanas.fdmath`, we have
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$$
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\\begin{aligned}
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\\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
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\\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\
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\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
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-\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} &= \\imath \\Omega \\tilde{J}_{\\vec{r}} \\\\
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\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
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\\end{aligned}
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$$
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# TODO FDFD?
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# TODO PML
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@ -185,8 +185,8 @@ def e_tfsf_source(TF_region: fdfield_t,
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def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[fdfield_t, fdfield_t], fdfield_t]:
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"""
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Generates a function that takes the single-frequency `E` and `H` fields
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and calculates the cross product `E` x `H` = \\( E \\times H \\) as required
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for the Poynting vector, \\( S = E \\times H \\)
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and calculates the cross product `E` x `H` = $E \\times H$ as required
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for the Poynting vector, $S = E \\times H$
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Note:
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This function also shifts the input `E` field by one cell as required
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@ -12,82 +12,82 @@ As the z-dependence is known, all the functions in this file assume a 2D grid
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Consider Maxwell's equations in continuous space, in the frequency domain. Assuming
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a structure with some (x, y) cross-section extending uniformly into the z dimension,
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with a diagonal \\( \\epsilon \\) tensor, we have
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with a diagonal $\\epsilon$ tensor, we have
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$$
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\\begin{align*}
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\\begin{aligned}
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\\nabla \\times \\vec{E}(x, y, z) &= -\\imath \\omega \\mu \\vec{H} \\\\
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\\nabla \\times \\vec{H}(x, y, z) &= \\imath \\omega \\epsilon \\vec{E} \\\\
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\\vec{E}(x,y,z) = (\\vec{E}_t(x, y) + E_z(x, y)\\vec{z}) e^{-\\gamma z} \\\\
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\\vec{H}(x,y,z) = (\\vec{H}_t(x, y) + H_z(x, y)\\vec{z}) e^{-\\gamma z} \\\\
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\\end{align*}
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\\end{aligned}
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$$
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Expanding the first two equations into vector components, we get
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$$
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\\begin{align*}
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\\begin{aligned}
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-\\imath \\omega \\mu_{xx} H_x &= \\partial_y E_z - \\partial_z E_y \\\\
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-\\imath \\omega \\mu_{yy} H_y &= \\partial_z E_x - \\partial_x E_z \\\\
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-\\imath \\omega \\mu_{zz} H_z &= \\partial_x E_y - \\partial_y E_x \\\\
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\\imath \\omega \\epsilon_{xx} E_x &= \\partial_y H_z - \\partial_z H_y \\\\
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\\imath \\omega \\epsilon_{yy} E_y &= \\partial_z H_x - \\partial_x H_z \\\\
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\\imath \\omega \\epsilon_{zz} E_z &= \\partial_x H_y - \\partial_y H_x \\\\
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\\end{align*}
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\\end{aligned}
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$$
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Substituting in our expressions for \\( \\vec{E}, \\vec{H} \\) and discretizing:
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Substituting in our expressions for $\\vec{E}$, $\\vec{H}$ and discretizing:
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$$
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\\begin{align*}
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\\begin{aligned}
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-\\imath \\omega \\mu_{xx} H_x &= \\tilde{\\partial}_y E_z + \\gamma E_y \\\\
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-\\imath \\omega \\mu_{yy} H_y &= -\\gamma E_x - \\tilde{\\partial}_x E_z \\\\
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-\\imath \\omega \\mu_{zz} H_z &= \\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x \\\\
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\\imath \\omega \\epsilon_{xx} E_x &= \\hat{\\partial}_y H_z + \\gamma H_y \\\\
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\\imath \\omega \\epsilon_{yy} E_y &= -\\gamma H_x - \\hat{\\partial}_x H_z \\\\
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\\imath \\omega \\epsilon_{zz} E_z &= \\hat{\\partial}_x H_y - \\hat{\\partial}_y H_x \\\\
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\\end{align*}
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\\end{aligned}
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$$
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Rewrite the last three equations as
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$$
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\\begin{align*}
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\\begin{aligned}
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\\gamma H_y &= \\imath \\omega \\epsilon_{xx} E_x - \\hat{\\partial}_y H_z \\\\
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\\gamma H_x &= -\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z \\\\
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\\imath \\omega E_z &= \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x H_y - \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y H_x \\\\
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\\end{align*}
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\\end{aligned}
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$$
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Now apply \\( \\gamma \\tilde{\\partial}_x \\) to the last equation,
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then substitute in for \\( \\gamma H_x \\) and \\( \\gamma H_y \\):
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Now apply $\\gamma \\tilde{\\partial}_x$ to the last equation,
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then substitute in for $\\gamma H_x$ and $\\gamma H_y$:
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$$
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\\begin{align*}
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\\begin{aligned}
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\\gamma \\tilde{\\partial}_x \\imath \\omega E_z &= \\gamma \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x H_y
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- \\gamma \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y H_x \\\\
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&= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon{xx} E_x - \\hat{\\partial}_y H_z)
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- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon{yy} E_y - \\hat{\\partial}_x H_z) \\\\
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&= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon{xx} E_x)
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- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon{yy} E_y) \\\\
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&= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon_{xx} E_x - \\hat{\\partial}_y H_z)
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- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
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&= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon_{xx} E_x)
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- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon_{yy} E_y) \\\\
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\\gamma \\tilde{\\partial}_x E_z &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
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\\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
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\\end{align*}
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\\end{aligned}
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$$
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With a similar approach (but using \\( \\gamma \\tilde{\\partial}_y \\) instead), we can get
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With a similar approach (but using $\\gamma \\tilde{\\partial}_y$ instead), we can get
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$$
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\\begin{align*}
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\\begin{aligned}
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\\gamma \\tilde{\\partial}_y E_z &= \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
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\\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
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\\end{align*}
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\\end{aligned}
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$$
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We can combine this equation for \\( \\gamma \\tilde{\\partial}_y E_z \\) with
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the unused \\( \\imath \\omega \\mu_{xx} H_z \\) and \\( \\imath \\omega \\mu_{yy} H_y \\) equations to get
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We can combine this equation for $\\gamma \\tilde{\\partial}_y E_z$ with
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the unused $\\imath \\omega \\mu_{xx} H_z$ and $\\imath \\omega \\mu_{yy} H_y$ equations to get
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$$
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\\begin{align*}
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\\begin{aligned}
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-\\imath \\omega \\mu_{xx} \\gamma H_x &= \\gamma^2 E_y + \\tilde{\\partial}_y (
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\\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
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+ \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
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@ -96,30 +96,30 @@ $$
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\\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
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+ \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
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)\\\\
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\\end{align*}
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\\end{aligned}
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$$
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However, based on our rewritten equation for \\( \\gamma H_x \\) and the so-far unused
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equation for \\( \\imath \\omega \\mu_{zz} H_z \\) we can also write
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However, based on our rewritten equation for $\\gamma H_x$ and the so-far unused
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equation for $\\imath \\omega \\mu_{zz} H_z$ we can also write
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$$
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\\begin{align*}
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\\begin{aligned}
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-\\imath \\omega \\mu_{xx} (\\gamma H_x) &= -\\imath \\omega \\mu_{xx} (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
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&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
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-\\imath \\omega \\mu_{xx} \\hat{\\partial}_x (
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\\frac{1}{-\\imath \\omega \\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x)) \\\\
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&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
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+\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
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\\end{align*}
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\\end{aligned}
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$$
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and, similarly,
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$$
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\\begin{align*}
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\\begin{aligned}
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-\\imath \\omega \\mu_{yy} (\\gamma H_y) &= -\\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
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+\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
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\\end{align*}
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\\end{aligned}
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$$
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Using these, we can construct the eigenvalue problem
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@ -137,9 +137,9 @@ $$ \\beta^2 \\begin{bmatrix} E_x \\\\
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E_y \\end{bmatrix}
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$$
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An equivalent eigenvalue problem can be formed using the \\( H_x, H_y \\) fields, if those are more convenient.
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An equivalent eigenvalue problem can be formed using the $H_x$ and $H_y$ fields, if those are more convenient.
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Note that \\( E_z \\) was never discretized, so \\( \\gamma \\) and \\( \\beta \\) will need adjustment
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Note that $E_z$ was never discretized, so $\\gamma$ and $\\beta$ will need adjustment
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to account for numerical dispersion if the result is introduced into a space with a discretized z-axis.
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@ -187,8 +187,8 @@ def operator_e(omega: complex,
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\\begin{bmatrix} \\hat{\\partial}_x \\epsilon_{xx} & \\hat{\\partial}_y \\epsilon_{yy} \\end{bmatrix}
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$$
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\\( \\tilde{\\partial}_x \\) and \\( \\hat{\\partial}_x \\) are the forward and backward derivatives along x,
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and each \\( \\epsilon_{xx}, \\mu_{yy}, \\) etc. is a diagonal matrix containing the vectorized material
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$\\tilde{\\partial}_x$ and $\\hat{\\partial}_x$ are the forward and backward derivatives along x,
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and each $\\epsilon_{xx}$, $\\mu_{yy}$, etc. is a diagonal matrix containing the vectorized material
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property distribution.
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This operator can be used to form an eigenvalue problem of the form
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@ -253,8 +253,8 @@ def operator_h(omega: complex,
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\\begin{bmatrix} \\tilde{\\partial}_x \\mu_{xx} & \\tilde{\\partial}_y \\mu_{yy} \\end{bmatrix}
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$$
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\\( \\tilde{\\partial}_x \\) and \\( \\hat{\\partial}_x \\) are the forward and backward derivatives along x,
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and each \\( \\epsilon_{xx}, \\mu_{yy}, \\) etc. is a diagonal matrix containing the vectorized material
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$\\tilde{\\partial}_x$ and $\\hat{\\partial}_x$ are the forward and backward derivatives along x,
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and each $\\epsilon_{xx}$, $\\mu_{yy}$, etc. is a diagonal matrix containing the vectorized material
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property distribution.
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This operator can be used to form an eigenvalue problem of the form
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@ -43,11 +43,11 @@ Scalar derivatives and cell shifts
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----------------------------------
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Define the discrete forward derivative as
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$$ [\\tilde{\\partial}_x f ]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$
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where \\( f \\) is a function defined at discrete locations on the x-axis (labeled using \\( m \\)).
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The value at \\( m \\) occupies a length \\( \\Delta_{x, m} \\) along the x-axis. Note that \\( m \\)
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$$ [\\tilde{\\partial}_x f]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$
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where $f$ is a function defined at discrete locations on the x-axis (labeled using $m$).
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The value at $m$ occupies a length $\\Delta_{x, m}$ along the x-axis. Note that $m$
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is an index along the x-axis, _not_ necessarily an x-coordinate, since each length
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\\( \\Delta_{x, m}, \\Delta_{x, m+1}, ...\\) is independently chosen.
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$\\Delta_{x, m}, \\Delta_{x, m+1}, ...$ is independently chosen.
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If we treat `f` as a 1D array of values, with the `i`-th value `f[i]` taking up a length `dx[i]`
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along the x-axis, the forward derivative is
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@ -62,7 +62,7 @@ Likewise, discrete reverse derivative is
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deriv_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
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The derivatives' values are shifted by a half-cell relative to the original function, and
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will have different cell widths if all the `dx[i]` ( \\( \\Delta_{x, m} \\) ) are not
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will have different cell widths if all the `dx[i]` ( $\\Delta_{x, m}$ ) are not
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identical:
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[figure: derivatives and cell sizes]
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@ -87,19 +87,20 @@ identical:
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Periodic boundaries are used here and elsewhere unless otherwise noted.
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In the above figure,
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`f0 =` \\(f_0\\), `f1 =` \\(f_1\\)
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`Df0 =` \\([\\tilde{\\partial}f]_{0 + \\frac{1}{2}}\\)
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`Df1 =` \\([\\tilde{\\partial}f]_{1 + \\frac{1}{2}}\\)
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`df0 =` \\([\\hat{\\partial}f]_{0 - \\frac{1}{2}}\\)
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`f0 =` $f_0$, `f1 =` $f_1$
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`Df0 =` $[\\tilde{\\partial}f]_{0 + \\frac{1}{2}}$
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`Df1 =` $[\\tilde{\\partial}f]_{1 + \\frac{1}{2}}$
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`df0 =` $[\\hat{\\partial}f]_{0 - \\frac{1}{2}}$
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etc.
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The fractional subscript \\( m + \\frac{1}{2} \\) is used to indicate values defined
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at shifted locations relative to the original \\( m \\), with corresponding lengths
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The fractional subscript $m + \\frac{1}{2}$ is used to indicate values defined
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at shifted locations relative to the original $m$, with corresponding lengths
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$$ \\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x, m + 1}) $$
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Just as \\( m \\) is not itself an x-coordinate, neither is \\( m + \\frac{1}{2} \\);
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Just as $m$ is not itself an x-coordinate, neither is $m + \\frac{1}{2}$;
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carefully note the positions of the various cells in the above figure vs their labels.
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If the positions labeled with \\( m \\) are considered the "base" or "original" grid,
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the positions labeled with \\( m + \\frac{1}{2} \\) are said to lie on a "dual" or
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If the positions labeled with $m$ are considered the "base" or "original" grid,
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the positions labeled with $m + \\frac{1}{2}$ are said to lie on a "dual" or
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"derived" grid.
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For the remainder of the `Discrete calculus` section, all figures will show
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@ -201,8 +202,8 @@ There are also two divergences,
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where `g = [gx, gy, gz]` is a fore- or back-vector field.
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Since we applied the forward divergence to the back-vector (and vice-versa), the resulting scalar value
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is defined at the back-vector's (fore-vectors) location \\( (m,n,p) \\) and not at the locations of its components
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\\( (m \\pm \\frac{1}{2},n,p) \\) etc.
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is defined at the back-vector's (fore-vector's) location $(m,n,p)$ and not at the locations of its components
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$(m \\pm \\frac{1}{2},n,p)$ etc.
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[figure: divergence]
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^^
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@ -226,23 +227,23 @@ Curls
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The two curls are then
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$$ \\begin{align*}
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$$ \\begin{aligned}
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\\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\
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[\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=
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\\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} - \\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\
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&+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} - \\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\
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&+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_y g^z_{m + \\frac{1}{2},n,p})
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\\end{align*} $$
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\\end{aligned} $$
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and
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$$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} =
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[\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$
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where \\( \\hat{g} \\) and \\( \\tilde{g} \\) are located at \\((m,n,p)\\)
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with components at \\( (m \\pm \\frac{1}{2},n,p) \\) etc.,
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while \\( \\hat{h} \\) and \\( \\tilde{h} \\) are located at \\((m \\pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})\\)
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with components at \\((m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})\\) etc.
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where $\\hat{g}$ and $\\tilde{g}$ are located at $(m,n,p)$
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with components at $(m \\pm \\frac{1}{2},n,p)$ etc.,
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while $\\hat{h}$ and $\\tilde{h}$ are located at $(m \\pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$
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with components at $(m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$ etc.
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[code: curls]
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@ -286,27 +287,27 @@ Maxwell's Equations
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If we discretize both space (m,n,p) and time (l), Maxwell's equations become
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$$ \\begin{align*}
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$$ \\begin{aligned}
|
||||
\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
|
||||
- \\hat{M}_{l, \\vec{r} + \\frac{1}{2}} \\\\
|
||||
\\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2},\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
|
||||
+ \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
|
||||
\\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\
|
||||
\\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
|
||||
\\end{align*} $$
|
||||
\\end{aligned} $$
|
||||
|
||||
with
|
||||
|
||||
$$ \\begin{align*}
|
||||
\\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{align*} $$
|
||||
$$ \\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} $$
|
||||
|
||||
where the spatial subscripts are abbreviated as \\( \\vec{r} = (m, n, p) \\) and
|
||||
\\( \\vec{r} + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}) \\),
|
||||
\\( \\tilde{E} \\) and \\( \\hat{H} \\) are the electric and magnetic fields,
|
||||
\\( \\tilde{J} \\) and \\( \\hat{M} \\) are the electric and magnetic current distributions,
|
||||
and \\( \\epsilon \\) and \\( \\mu \\) are the dielectric permittivity and magnetic permeability.
|
||||
where the spatial subscripts are abbreviated as $\\vec{r} = (m, n, p)$ and
|
||||
$\\vec{r} + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2})$,
|
||||
$\\tilde{E}$ and $\\hat{H}$ are the electric and magnetic fields,
|
||||
$\\tilde{J}$ and $\\hat{M}$ are the electric and magnetic current distributions,
|
||||
and $\\epsilon$ and $\\mu$ are the dielectric permittivity and magnetic permeability.
|
||||
|
||||
The above is Yee's algorithm, written in a form analogous to Maxwell's equations.
|
||||
The time derivatives can be expanded to form the update equations:
|
||||
|
|
@ -375,12 +376,12 @@ and combining them with charge continuity,
|
|||
Wave equation
|
||||
-------------
|
||||
|
||||
Taking the backward curl of the \\( \\tilde{\\nabla} \\times \\tilde{E} \\) equation and
|
||||
replacing the resulting \\( \\hat{\\nabla} \\times \\hat{H} \\) term using its respective equation,
|
||||
and setting \\( \\hat{M} \\) to zero, we can form the discrete wave equation:
|
||||
Taking the backward curl of the $\\tilde{\\nabla} \\times \\tilde{E}$ equation and
|
||||
replacing the resulting $\\hat{\\nabla} \\times \\hat{H}$ term using its respective equation,
|
||||
and setting $\\hat{M}$ to zero, we can form the discrete wave equation:
|
||||
|
||||
$$
|
||||
\\begin{align*}
|
||||
\\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}} \\\\
|
||||
|
|
@ -391,11 +392,11 @@ $$
|
|||
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
|
||||
-\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
|
||||
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
|
||||
-\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
|
||||
-\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
|
||||
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}})
|
||||
+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\cdot \\tilde{E}_{l, \\vec{r}}
|
||||
+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{l, \\vec{r}}
|
||||
&= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}}
|
||||
\\end{align*}
|
||||
\\end{aligned}
|
||||
$$
|
||||
|
||||
|
||||
|
|
@ -405,27 +406,27 @@ Frequency domain
|
|||
We can substitute in a time-harmonic fields
|
||||
|
||||
$$
|
||||
\\begin{align*}
|
||||
\\tilde{E}_\\vec{r} &= \\tilde{E}_\\vec{r} e^{-\\imath \\omega l \\Delta_t} \\\\
|
||||
\\tilde{J}_\\vec{r} &= \\tilde{J}_\\vec{r} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t}
|
||||
\\end{align*}
|
||||
\\begin{aligned}
|
||||
\\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
|
||||
\\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t}
|
||||
\\end{aligned}
|
||||
$$
|
||||
|
||||
resulting in
|
||||
|
||||
$$
|
||||
\\begin{align*}
|
||||
\\begin{aligned}
|
||||
\\tilde{\\partial}_t &\\Rightarrow (e^{ \\imath \\omega \\Delta_t} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{-\\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\
|
||||
\\hat{\\partial}_t &\\Rightarrow (1 - e^{-\\imath \\omega \\Delta_t}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{ \\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\
|
||||
\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
|
||||
\\end{align*}
|
||||
\\end{aligned}
|
||||
$$
|
||||
|
||||
This gives the frequency-domain wave equation,
|
||||
|
||||
$$
|
||||
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_\\vec{r})
|
||||
-\\Omega^2 \\epsilon_\\vec{r} \\cdot \\tilde{E}_\\vec{r} = \\imath \\Omega \\tilde{J}_\\vec{r}
|
||||
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
|
||||
-\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = \\imath \\Omega \\tilde{J}_{\\vec{r}}
|
||||
$$
|
||||
|
||||
|
||||
|
|
@ -435,69 +436,69 @@ Plane waves and Dispersion relation
|
|||
With uniform material distribution and no sources
|
||||
|
||||
$$
|
||||
\\begin{align*}
|
||||
\\begin{aligned}
|
||||
\\mu_{\\vec{r} + \\frac{1}{2}} &= \\mu \\\\
|
||||
\\epsilon_\\vec{r} &= \\epsilon \\\\
|
||||
\\tilde{J}_\\vec{r} &= 0 \\\\
|
||||
\\end{align*}
|
||||
\\epsilon_{\\vec{r}} &= \\epsilon \\\\
|
||||
\\tilde{J}_{\\vec{r}} &= 0 \\\\
|
||||
\\end{aligned}
|
||||
$$
|
||||
|
||||
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
|
||||
|
||||
$$
|
||||
\\begin{align*}
|
||||
\\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_\\vec{r}
|
||||
&= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_\\vec{r}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_\\vec{r} \\\\
|
||||
&= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_\\vec{r} \\\\
|
||||
&= - \\tilde{\\nabla}^2 \\tilde{E}_\\vec{r}
|
||||
\\end{align*}
|
||||
\\begin{aligned}
|
||||
\\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}}
|
||||
&= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
|
||||
&= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
|
||||
&= - \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}}
|
||||
\\end{aligned}
|
||||
$$
|
||||
|
||||
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
|
||||
|
||||
$$
|
||||
\\begin{align*}
|
||||
\\begin{aligned}
|
||||
\\tilde{\\partial}_x &\\Rightarrow (e^{ \\imath k_x \\Delta_x} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{ \\imath k_x \\Delta_x / 2} = \\imath K_x e^{ \\imath k_x \\Delta_x / 2}\\\\
|
||||
\\hat{\\partial}_x &\\Rightarrow (1 - e^{-\\imath k_x \\Delta_x}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{-\\imath k_x \\Delta_x / 2} = \\imath K_x e^{-\\imath k_x \\Delta_x / 2}\\\\
|
||||
K_x &= 2 \\sin(k_x \\Delta_x / 2) / \\Delta_x \\\\
|
||||
\\end{align*}
|
||||
\\end{aligned}
|
||||
$$
|
||||
|
||||
with similar expressions for the y and z dimnsions (and \\( K_y, K_z \\)).
|
||||
with similar expressions for the y and z dimnsions (and $K_y, K_z$).
|
||||
|
||||
This implies
|
||||
|
||||
$$
|
||||
\\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_\\vec{r} \\\\
|
||||
\\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_{\\vec{r}} \\\\
|
||||
K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2
|
||||
$$
|
||||
|
||||
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
|
||||
|
||||
$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
|
||||
|
||||
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
|
||||
|
|
@ -513,9 +514,9 @@ To get a better sense of how this works, let's start by drawing a grid with unif
|
|||
to make the illustration simpler; we need at least two cells in the x dimension to
|
||||
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
|
||||
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
|
||||
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.
|
||||
|
|
|
|||
|
|
@ -63,7 +63,7 @@ def curl_forward(dx_e: List[numpy.ndarray] = None) -> fdfield_updater_t:
|
|||
|
||||
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)
|
||||
|
||||
|
|
@ -90,7 +90,7 @@ def curl_back(dx_h: List[numpy.ndarray] = None) -> fdfield_updater_t:
|
|||
|
||||
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)
|
||||
|
||||
|
|
|
|||
|
|
@ -13,7 +13,7 @@ we have
|
|||
|
||||
$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
|
||||
|
||||
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
|
||||
|
||||
|
|
@ -27,21 +27,21 @@ Poynting Vector and Energy Conservation
|
|||
|
||||
Let
|
||||
|
||||
$$ \\begin{align*}
|
||||
$$ \\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{align*}
|
||||
\\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{align*}
|
||||
\\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}} -
|
||||
|
|
@ -49,30 +49,30 @@ $$
|
|||
&= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot
|
||||
(-\\tilde{\\partial}_t \\mu_{\\vec{r} + \\frac{1}{2}} \\hat{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} -
|
||||
\\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}}) -
|
||||
\\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t \\tilde{\\epsilon}_\\vec{r} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} +
|
||||
\\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t \\tilde{\\epsilon}_{\\vec{r}} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} +
|
||||
\\tilde{J}_{l', \\vec{r}}) \\\\
|
||||
&= \\hat{H}_{l'} \\cdot (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
|
||||
\\tilde{E}_l \\cdot (\\epsilon / \\Delta_t )(\\tilde{E}_{l'+\\frac{1}{2}} - \\tilde{E}_{l'-\\frac{1}{2}})
|
||||
- \\hat{H}_{l'} \\cdot \\hat{M}_{l-1} - \\tilde{E}_l \\cdot \\tilde{J}_{l'} \\\\
|
||||
\\end{align*}
|
||||
\\end{aligned}
|
||||
$$
|
||||
|
||||
where in the last line the spatial subscripts have been dropped to emphasize
|
||||
the time subscripts \\( l, l' \\), i.e.
|
||||
the time subscripts $l, l'$, i.e.
|
||||
|
||||
$$
|
||||
\\begin{align*}
|
||||
\\begin{aligned}
|
||||
\\tilde{E}_l &= \\tilde{E}_{l, \\vec{r}} \\\\
|
||||
\\hat{H}_l &= \\tilde{H}_{l, \\vec{r} + \\frac{1}{2}} \\\\
|
||||
\\tilde{\\epsilon} &= \\tilde{\\epsilon}_\\vec{r} \\\\
|
||||
\\end{align*}
|
||||
\\tilde{\\epsilon} &= \\tilde{\\epsilon}_{\\vec{r}} \\\\
|
||||
\\end{aligned}
|
||||
$$
|
||||
|
||||
etc.
|
||||
For \\( l' = l + \\frac{1}{2} \\) we get
|
||||
For $l' = l + \\frac{1}{2}$ we get
|
||||
|
||||
$$
|
||||
\\begin{align*}
|
||||
\\begin{aligned}
|
||||
\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}}
|
||||
&= \\hat{H}_{l + \\frac{1}{2}} \\cdot
|
||||
(-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
|
||||
|
|
@ -87,13 +87,13 @@ $$
|
|||
+\\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{align*}
|
||||
\\end{aligned}
|
||||
$$
|
||||
|
||||
and for \\( l' = l - \\frac{1}{2} \\),
|
||||
and for $l' = l - \\frac{1}{2}$,
|
||||
|
||||
$$
|
||||
\\begin{align*}
|
||||
\\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 \\ \\
|
||||
|
|
@ -101,7 +101,7 @@ $$
|
|||
+\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
|
||||
- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
|
||||
- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
|
||||
\\end{align*}
|
||||
\\end{aligned}
|
||||
$$
|
||||
|
||||
These two results form the discrete time-domain analogue to Poynting's theorem.
|
||||
|
|
@ -109,16 +109,16 @@ They hint at the expressions for the energy, which can be calculated at the same
|
|||
time-index as either the E or H field:
|
||||
|
||||
$$
|
||||
\\begin{align*}
|
||||
\\begin{aligned}
|
||||
U_l &= \\epsilon \\tilde{E}^2_l + \\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}} \\\\
|
||||
U_{l + \\frac{1}{2}} &= \\epsilon \\tilde{E}_l \\cdot \\tilde{E}_{l + 1} + \\mu \\hat{H}^2_{l + \\frac{1}{2}} \\\\
|
||||
\\end{align*}
|
||||
\\end{aligned}
|
||||
$$
|
||||
|
||||
Rewriting the Poynting theorem in terms of the energy expressions,
|
||||
|
||||
$$
|
||||
\\begin{align*}
|
||||
\\begin{aligned}
|
||||
(U_{l+\\frac{1}{2}} - U_l) / \\Delta_t
|
||||
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\
|
||||
- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
|
||||
|
|
@ -127,14 +127,14 @@ $$
|
|||
&= -\\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{align*}
|
||||
\\end{aligned}
|
||||
$$
|
||||
|
||||
This result is exact an should practically hold to within numerical precision. No time-
|
||||
or spatial-averaging is necessary.
|
||||
|
||||
Note that each value of \\( J \\) contributes to the energy twice (i.e. once per field update)
|
||||
despite only causing the value of \\( E \\) to change once (same for \\( M \\) and \\( H \\)).
|
||||
Note that each value of $J$ contributes to the energy twice (i.e. once per field update)
|
||||
despite only causing the value of $E$ to change once (same for $M$ and $H$).
|
||||
|
||||
|
||||
Sources
|
||||
|
|
@ -149,8 +149,8 @@ shape. It can be written
|
|||
|
||||
$$ 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).
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue