diff --git a/notes/conventions.md b/notes/conventions.md index feb0f00..578deb4 100644 --- a/notes/conventions.md +++ b/notes/conventions.md @@ -63,6 +63,65 @@ GSL computes \f$ \rawFer{l}{m} \f$ unless the corresponding `csphase` argument i but can be tested by running `gsl_sf_legendre_array_e` for some specific arguments and comparing signs. +Convention effect on translation operators +------------------------------------------ + +Let us declare VSWFs in Kristensson's conventions below, +\f$ \wfkc \f$ \cite kristensson_spherical_2014, +\f$ \wfkr \f$ \cite kristensson_scattering_2016, as the "canonical" +spherical waves based on complex and real spherical harmonics, respectively. +They both have the property that the translation operators \f$ \tropRrr{}{},\tropSrr{}{} \f$ +that transform +the VSWF field expansion coefficients between different origins, e.g. +\f[ + \wfkcreg(\vect{r}) = \tropRrr{\vect r}{\vect r'} \wfkcreg(\vect{r'}), +\f] +actually consist of two different submatrices $A,B$ for the same-type and different-type +(in the sense of "electric" versus "magnetic" waves) that repeat themselves once: +\f[ + \begin{bmatrix} \wfkcreg_1(\vect{r}) \\ \wfkcreg_2(\vect{r}) \end{bmatrix} + = \begin{bmatrix} A & B \\ B & A \end{bmatrix}(\vect{r} \leftarrow \vect{r'}) + \begin{bmatrix} \wfkcreg_1(\vect{r'}) \\ \wfkcreg_2(\vect{r'}) \end{bmatrix}. +\f] +(This symmetry holds also for singular translation operators \f$ \tropSrr{}{} \f$ +and real spherical harmonics based VSWFs \f$ \wfkr \f$.) + +However, the symmetry above will not hold like this in some stupider convention. +Let's suppose that one uses a different convention with some additional coefficients +compared to the canonical one, +\f[ + \wfm_{lm} = \alpha_{\wfm lm} \wfkc_{1lm},\\ + \wfe_{lm} = \alpha_{\wfe lm} \wfkc_{2lm}.\\ +\f] +and with field expansion (WLOG assume regular fields only) +\f[ \vect E = c_{\wfe l m} \wfe_{lm} + c_{\wfm l m } \wfm_{lm}. \f] +Under translations, the coefficients then transform like +\f[ + \begin{bmatrix} \alpha_\wfe(\vect{r}) \\ \alpha_\wfm(\vect{r}) \end{bmatrix} + = \begin{bmatrix} R_{\wfe\wfe} & R_{\wfe\wfm} \\ + R_{\wfm\wfe} & R_{\wfm\wfm} + \end{bmatrix}(\vect{r} \leftarrow \vect{r'}) + \begin{bmatrix} \alpha_\wfe(\vect{r'}) \\ \alpha_\wfm(\vect{r'}) \end{bmatrix}, +\f] +and by substituting and comparing the expressions for canonical waves above, one gets +\f[ + R_{\wfe,lm;\wfe,l'm'} = \alpha_{\wfe lm}^{-1} A \alpha_{\wfe l'm'},\\ + R_{\wfe,lm;\wfm,l'm'} = \alpha_{\wfe lm}^{-1} B \alpha_{\wfm l'm'},\\ + R_{\wfm,lm;\wfe,l'm'} = \alpha_{\wfm lm}^{-1} B \alpha_{\wfe l'm'},\\ + R_{\wfm,lm;\wfm,l'm'} = \alpha_{\wfm lm}^{-1} A \alpha_{\wfm l'm'}. +\f] + +If the coefficients for magnetic and electric waves are the same, +\f$ \alpha_{\wfm lm} = \alpha_{\wfe lm} \f$, the translation operator +can be written in the same symmetric form as with the canonical convention, +just the matrices \f$ A, B\f$ will be different inside. + +If the coefficients differ (as in SCUFF-EM convention, where there +is a relative \a i -factor between electric and magnetic waves), +the functions such as qpms_trans_calculator_get_AB_arrays() will +compute \f$ R_{\wfe\wfe}, R_{\wfe\wfm} \f$ for A, B arrays. + + Literature convention tables ---------------------------- diff --git a/notes/mathjax_newcommands.js b/notes/mathjax_newcommands.js index f1aa3a7..06e0a81 100644 --- a/notes/mathjax_newcommands.js +++ b/notes/mathjax_newcommands.js @@ -31,6 +31,8 @@ MathJax.Hub.Config({ spharm: ["{{Y_{\\mathrm{#1}}}_{#2}^{#3}}", 3, ""], // Spherical harmonics spharmR: ["{{Y_{\\mathrm{#1}}}_{\\mathrm{#1}{#2}{#3}}", 4, ""], // Spherical harmonics csphase: "\\mathsf{C_{CS}}", // Condon-Shortley phase + tropSrr: ["{{S^\\mathrm{#1}}\\pr{{#2} \\leftarrow {#3}}}", 3, ""], // Translation operator singular + tropRrr: ["{{R^\\mathrm{#1}}\\pr{{#2} \\leftarrow {#3}}}", 3, ""], // Translation operator regular // Kristensson's VSWFs, complex version (2014 notes) wfkc: "{\\vect{y}}", // any wave