qpms/notes/conventions.md

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] 	|   	| Yes, in the spherical harmonics definition, cf. sect. D.2.  	| \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$. |
| Reid \cite reid_electromagnetism_2016	|   |  	|   	|  \f[
	\vect E = \sum_\alpha \pr{ \wcrreg_\alpha \wfrreg_\alpha + \wcrout_\alpha wfrout_\alpha }, \\
	\vect H = \frac{1}{Z_0Z^r} \sum_\alpha \pr{ \wcrreg_\alpha \sigma_\alpha\wfrreg_\overline{\alpha} +
		 \wcrout_\alpha \sigma_\alpha\wfrout_\overline{\alpha}},
\f] where \f$ \sigma_{lmM} = +1, \sigma_{lmN}=-1, \overline{lmM}=lmM, \overline{lmN}=lmM, \f$  cf. eq. (6). The notation is not extremely consistent throughout Reid's memo.	| 	| 	|
| 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. |