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More to the convention table 

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Marek Nečada 2019-07-08 18:18:20 +03:00
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notes/conventions.md
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@ -2,24 +2,31 @@ VSWF conventions
================ ================
| Source | VSWF definition | VSWF norm | CS Phase | Field expansion | Radiated power | Notes | | Source | VSWF definition | E/M interrelations | 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[ | Kristensson I \cite kristensson_spherical_2014 | \f[ \wfkcreg, \wfkcout= \dots \f] | \f[
\wfkcreg_{1lm} = \frac{1}{k}\nabla\times\wfkcreg_{2lm}, \\
\wfkcreg_{2lm} = \frac{1}{k}\nabla\times\wfkcreg_{1lm},
\f] and analogously for outgoing waves \f$ \wfkcout \f$, eq. (2.8) onwards. | | Yes, in the spherical harmonics definition, cf. sect. D.2. | \f[
\vect E = k \sqrt{\eta_0\eta} \sum_n \left( \wckcreg_n \wfkcreg_n + \wckcout_n \wfkcout_n \right), \vect E = k \sqrt{\eta_0\eta} \sum_n \left( \wckcreg_n \wfkcreg_n + \wckcout_n \wfkcout_n \right),
\\ \\
\vect H = \frac{k \sqrt{\eta_0\eta}}{i\eta_0\eta} \sum_n \left( \wckcreg_n \wfkcreg_n + \wckcout_n \wfkcout_n \right), \vect H = \frac{k \sqrt{\eta_0\eta}}{i\eta_0\eta} \sum_n \left( \wckcreg_n \wfkcreg_n + \wckcout_n \wfkcout_n \right),
\f] but for plane wave expansion \cite kristensson_spherical_2014 sect. 2.5 K. uses a different definition (same as in Kristensson II). | \f[ \f] but for plane wave expansion \cite kristensson_spherical_2014 sect. 2.5 K. uses a different definition (same as in Kristensson II). | \f[
P = \frac{1}{2} \sum_n \left( \abs{\wckcout_n}^2 +\Re \left(\wckcout_n\wckcreg_n^{*}\right)\right) P = \frac{1}{2} \sum_n \left( \abs{\wckcout_n}^2 +\Re \left(\wckcout_n\wckcreg_n^{*}\right)\right)
\f] | The \f$ \wckcreg, \wckcout \f$ coefficients have dimension \f$ \sqrt{\mathrm{W}} \f$. | \f] | The \f$ \wckcreg, \wckcout \f$ coefficients have dimension \f$ \sqrt{\mathrm{W}} \f$. |
| Kristensson II \cite kristensson_scattering_2016 | \f[ \wfkrreg, \wfkrout= \dots \f] | | | \f[ | Kristensson II \cite kristensson_scattering_2016 | \f[ \wfkrreg, \wfkrout= \dots \f] | \f[
\nabla\times\wfkrreg_{\tau n} = k\wfkrreg_{\overline{\tau} n},
\f] eq. (7.7) and analogously for outgoing waves \f$ \wfkrout \f$. | | | \f[
\vect E = \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right), \vect E = \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right),
\\ \\
\vect H = \frac{1}{i\eta_0\eta} \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right) \vect H = \frac{1}{i\eta_0\eta} \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right)
\f] | \f[ \f] | \f[
P = \frac{1}{2k^2\eta_0\eta} \sum_n \left( \abs{\wckrout_n}^2 +\Re \left(\wckrout_n\wckrreg_n^{*}\right)\right) P = \frac{1}{2k^2\eta_0\eta} \sum_n \left( \abs{\wckrout_n}^2 +\Re \left(\wckrout_n\wckrreg_n^{*}\right)\right)
\f] | The \f$ \wckrreg, \wckrout \f$ coefficients have dimension \f$ \mathrm{V/m} \f$. | \f] | The \f$ \wckrreg, \wckrout \f$ coefficients have dimension \f$ \mathrm{V/m} \f$. |
| Reid \cite reid_electromagnetism_2016 | | | | \f[ | Reid \cite reid_electromagnetism_2016 | | \f[
\vect E = \sum_\alpha \pr{ \wcrreg_\alpha \wfrreg_\alpha + \wcrout_\alpha wfrout_\alpha }, \\ \nabla\times\wfr_{lmM} = -ik\wfr_{lmN}, \\ \nabla\times\wfr_{lmN} = +ik\wfr_{lmM}.
\f] | | | \f[
\vect E = \sum_\alpha \pr{ \wcrreg_\alpha \wfrreg_\alpha + \wcrout_\alpha \wfrout_\alpha }, \\
\vect H = \frac{1}{Z_0Z^r} \sum_\alpha \pr{ \wcrreg_\alpha \sigma_\alpha\wfrreg_\overline{\alpha} + \vect H = \frac{1}{Z_0Z^r} \sum_\alpha \pr{ \wcrreg_\alpha \sigma_\alpha\wfrreg_\overline{\alpha} +
\wcrout_\alpha \sigma_\alpha\wfrout_\overline{\alpha}}, \wcrout_\alpha \sigma_\alpha\wfrout_\overline{\alpha}},
\f] where \f$ \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. | | | \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. | | |
@ -27,7 +34,7 @@ VSWF conventions
\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} \\ \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)}, \\ +\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) \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[ \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)} \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 = \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'} , \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'} ,