Make the choice for stupid-convention A,B parts of trans. op. in the docs.

Former-commit-id: 2cd1ab77b9c456701d1ec5135f88c7e603f3b22b
This commit is contained in:
Marek Nečada 2019-07-12 08:42:23 +03:00
parent f239ae0afe
commit fda3c620f4
2 changed files with 61 additions and 0 deletions

View File

@ -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. 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 Literature convention tables
---------------------------- ----------------------------

View File

@ -31,6 +31,8 @@ MathJax.Hub.Config({
spharm: ["{{Y_{\\mathrm{#1}}}_{#2}^{#3}}", 3, ""], // Spherical harmonics spharm: ["{{Y_{\\mathrm{#1}}}_{#2}^{#3}}", 3, ""], // Spherical harmonics
spharmR: ["{{Y_{\\mathrm{#1}}}_{\\mathrm{#1}{#2}{#3}}", 4, ""], // Spherical harmonics spharmR: ["{{Y_{\\mathrm{#1}}}_{\\mathrm{#1}{#2}{#3}}", 4, ""], // Spherical harmonics
csphase: "\\mathsf{C_{CS}}", // Condon-Shortley phase 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) // Kristensson's VSWFs, complex version (2014 notes)
wfkc: "{\\vect{y}}", // any wave wfkc: "{\\vect{y}}", // any wave