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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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