From d2e9b4450fc6598084071eb80b4c86d1a75bcc6c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Mon, 8 Jul 2019 17:06:11 +0300 Subject: [PATCH] Update conventions table Former-commit-id: a6107d411fd54f0f54365e64f52ace90444d67ce --- notes/conventions.md | 12 ++++++++---- 1 file changed, 8 insertions(+), 4 deletions(-) diff --git a/notes/conventions.md b/notes/conventions.md index 6588ad6..bae6ac5 100644 --- a/notes/conventions.md +++ b/notes/conventions.md @@ -7,8 +7,8 @@ VSWF conventions | 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] | \f[ + \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$. | | Kristensson II \cite kristensson_scattering_2016 | \f[ \wfkrreg, \wfkrout= \dots \f] | | | \f[ @@ -23,8 +23,12 @@ VSWF conventions \wfet_{mn}^{(j)} = \frac{n(n+1)}{kr}\sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}P_{n}^{m}\left(\cos\theta\right)e^{im\phi}z_{n}^{j}\left(kr\right)\uvec{r} \\ +\left[\tilde{\tau}_{mn}\left(\cos\theta\right)\uvec{\theta}+i\tilde{\pi}_{mn}\left(\cos\theta\right)\uvec{\phi}\right]e^{im\phi}\frac{1}{kr}\frac{\ud\left(kr\,z_{n}^{j}\left(kr\right)\right)}{\ud(kr)}, \\ \wfmt_{mn}^{(j)} = \left[i\tilde{\pi}_{mn}\left(\cos\theta\right)\uvec{\theta}-\tilde{\tau}_{mn}\left(\cos\theta\right)\uvec{\phi}\right]e^{im\phi}z_{n}^{j}\left(kr\right) -\f] | | | \f[ +\f] | \f[ + \int_{S(kr)} \wfmt_{mn}^{(j)} \wfmt_{m'n'}^{(j)}\,\ud S = n(n+1) \abs{z_n^{(j)}}^2 \delta_{m,m'}\delta_{n,n'} ,\\ + \int_{S(kr)} \wfet_{mn}^{(j)} \wfet_{m'n'}^{(j)}\,\ud S = + \pr{\pr{n(n+1)}^2 \abs{\frac{z_n^{(j)}}{kr}}^2 + n(n+1)\abs{\frac{1}{kr}\frac{\ud}{\ud(kr)}\pr{kr z_n^{(j)}}} } \delta_{m,m'}\delta_{n,n'} , +\f] cf. \cite taylor_optical_2011, eqs. (2.40–41). I suspect that this is also wrong and \f$\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$? | +\f] | | Different sign for regular/scattered waves! Also WTF are the units of \f$ n_{ext} \f$? The whole definition seems rather inconsistent. |