2.9 KiB
2.9 KiB
VSWF conventions
Source | VSWF definition | VSWF norm | CS Phase | Field expansion | Radiated power | Notes |
---|---|---|---|---|---|---|
Kristensson I \cite kristensson_spherical_2014 | \f[ \wfkcreg, \wfkcout= \dots \f] | \f[ | ||||
\vect E = k \sqrt{\eta_0\eta} \sum_n \left( \wckcreg_n \wfkcreg_n + \wckcout_n \wfkcout_n \right), | ||||||
\ | ||||||
\vect H = \frac{k \sqrt{\eta_0\eta}}{i\eta_0\eta} \sum_n \left( \wckcreg_n \wfkcreg_n + \wckcout_n \wfkcout_n \right), | ||||||
\f] but for plane wave expansion \cite kristensson_spherical_2014 sect. 2.5 K. uses a different definition (same as in Kristensson II). | \f[ | |||||
P = \frac{1}{2} \sum_n \left( \abs{\wckcout_n}^2 +\Re \left(\wckcout_n\wckcreg_n^{*}\right)\right) | ||||||
\f] | The \f$ \wckcreg, \wckcout \f$ coefficients have dimension \f$ \sqrt{\mathrm{W}} \f$. | |||||
Kristensson II \cite kristensson_scattering_2016 | \f[ \wfkrreg, \wfkrout= \dots \f] | \f[ | ||||
\vect E = \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right), | ||||||
\ | ||||||
\vect H = \frac{1}{i\eta_0\eta} \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right) | ||||||
\f] | \f[ | |||||
P = \frac{1}{2k^2\eta_0\eta} \sum_n \left( \abs{\wckrout_n}^2 +\Re \left(\wckrout_n\wckrreg_n^{*}\right)\right) | ||||||
\f] | The \f$ \wckrreg, \wckrout \f$ coefficients have dimension \f$ \mathrm{V/m} \f$. | |||||
Reid \cite reid_electromagnetism_2016 | ||||||
Taylor \cite taylor_optical_2011 | \f[ | |||||
\wfet_{mn}^{(j)} = \frac{n(n+1)}{kr}\sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}P_{n}^{m}\left(\cos\theta\right)e^{im\phi}z_{n}^{j}\left(kr\right)\uvec{r} \ | ||||||
+\left[\tilde{\tau}{mn}\left(\cos\theta\right)\uvec{\theta}+i\tilde{\pi}{mn}\left(\cos\theta\right)\uvec{\phi}\right]e^{im\phi}\frac{1}{kr}\frac{\ud\left(kr,z_{n}^{j}\left(kr\right)\right)}{\ud(kr)}, \ | ||||||
\wfmt_{mn}^{(j)} = \left[i\tilde{\pi}{mn}\left(\cos\theta\right)\uvec{\theta}-\tilde{\tau}{mn}\left(\cos\theta\right)\uvec{\phi}\right]e^{im\phi}z_{n}^{j}\left(kr\right) | ||||||
\f] | \f[ | |||||
\int_{S(kr)} \wfmt_{mn}^{(j)} \wfmt_{m'n'}^{(j)},\ud S = n(n+1) \abs{z_n^{(j)}}^2 \delta_{m,m'}\delta_{n,n'} ,\ | ||||||
\int_{S(kr)} \wfet_{mn}^{(j)} \wfet_{m'n'}^{(j)},\ud S = | ||||||
\pr{\pr{n(n+1)}^2 \abs{\frac{z_n^{(j)}}{kr}}^2 + n(n+1)\abs{\frac{1}{kr}\frac{\ud}{\ud(kr)}\pr{kr z_n^{(j)}}} } \delta_{m,m'}\delta_{n,n'} , | ||||||
\f] cf. \cite taylor_optical_2011, eqs. (2.40–41). I suspect that this is also wrong and \f$\delta_{m,m'}\f$ should be replaced with \f$\delta_{m,-m'}\f$. | \f[ | |||||
\vect E = \sum_{mn} \pr{-i \pr{\wcetreg_{mn}\wfetreg_{mn} + \wcmtreg_{mn}\wfmtreg{mn}} +i \pr{\wcetout_{mn}\wfetout_{mn} + \wcmtout_{mn}\wfmtout_{mn}}}, \ | ||||||
\vect H = n_{ext}\sum_{mn} \pr{- \pr{\wcmtreg_{mn}\wfetreg_{mn} + \wcetreg_{mn}\wfmtreg{mn}} + \pr{\wcmtout_{mn}\wfetout_{mn} + \wcetout_{mn}\wfmtout_{mn}}}, | ||||||
\f] | Different sign for regular/scattered waves! Also WTF are the units of \f$ n_{ext} \f$? The whole definition seems rather inconsistent. |