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Update doc/docs/Python_Tutorials/Cylindrical_Coordinates.md
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Co-authored-by: Steven G. Johnson <[email protected]>
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oskooi and stevengj authored Feb 1, 2024
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Expand Up @@ -343,7 +343,7 @@ A $y$-polarized planewave involves subtracting rather than adding the two terms

$$ \hat{E}_y = \frac{1}{2} \left[e^{i\phi}(\hat{E}_\rho + i\hat{E}_\phi) - e^{-i\phi}(\hat{E}_\rho - i\hat{E}_\phi)\right] $$

In practice, this involves performing *two* separate simulations for $m=\pm 1$. The scattered power from each simulation is then simply summed since the cross term in the total Poynting flux cancels for the different $m$ values when integrated over the $\phi$ direction. As a simplification, in the case of a material with isotropic permittivity, only one of the two simulations is necessary: the scattered power is the same for $m=\pm 1$ due to the mirror symmetry of the structure. Also, note that for an isotropic permittivity, the flux from an $x$-polarized planewave is identical to the $y$-polarized planewave.
In principle, this involves performing *two* separate simulations for $m=\pm 1$. The scattered power from each simulation is then simply summed since the cross term in the total Poynting flux cancels for the different $m$ values when integrated over the $\phi$ direction. As a simplification, in the case of a material with isotropic permittivity (and/or real permittivity), only one of the two simulations is necessary: the scattered power is the same for $m=\pm 1$ due to the mirror (and/or conjugate) symmetry of the structure.

A chiral material based on an anisotropic permittivity with principle axes not aligned with the coordinates axes breaks the mirror symmetry and thus would require two separate simulations. For a given linearly-polarized planewave, the total Poynting flux is computed by first combining the fields from the two current sources of opposite chirality in separate runs. Also, note that for an anisotropic permittivity, the flux from an $x$-polarized planewave is generally not the same as a $y$-polarized planewave.

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