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VSWF conventions {#vswf_conventions}
====================================
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In general, the (transversal) VSWFs can be defined using (some) vector spherical harmonics
as follows: \f[
\wfm\pr{k\vect r}_{lm} = \sphbes_l(kr) \vshrot_{lm} (\uvec r),\\
\wfe\pr{k\vect r}_{lm} = \frac{\frac{\ud}{\ud(kr)}\pr{kr\sphbes_l(kr)}}{kr} \vshgrad_{lm}(\uvec r)
+ \sqrt{l(l+1)} \frac{\sphbes_l(kr)}{kr} \vshrad_{lm}(\uvec r),
\f]
where at this point, we don't have much expectations regarding the
normalisations and phases of the
"rotational", "gradiental" and "radial" vector spherical harmonics
\f$ \vshrot, \vshgrad, \vshrad \f$, and the waves can be of whatever "direction"
(regular, outgoing, etc.) depending on the kind of the spherical Bessel function
\f$ \sphbes \f$.
We only require that the spherical harmonic degree \f$ l \f$
is what it is supposed to be. The meaning of the order $m$ may vary depending
on convention. Moreover, in order to \f$ \wfe \f$ be a valid "electric" multipole wave,
there is a fixed relation between radial and gradiental vector spherical harmonics
(more on that later).
Let us define the "dual" vector spherical harmonics \f$ \vshD_{\tau lm} \f$ as follows:
\f[
\int_\Omega \vsh_{\tau lm} (\uvec r) \cdot \vshD_{\tau' l'm} (\uvec r) \, \ud \Omega
= \delta_{\tau', \tau}\delta_{l',l} \delta_{m',m}
\f]
where the \f$ \cdot \f$ symbol here means the bilinear form of the vector components
without complex conjugation (which is included in the "duality" mapping).
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The problem with conventions starts with the very definition of associated Legendre / Ferrers functions.
For the sake of non-ambiguity, let us first define the "canonical" associated Legendre/Ferrers polynomials
*without* the Condon-Shortley phase.
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\f[
\rawLeg{l}{0}(x) = \frac{1}{2^n n!} \frac{\ud^n}{\ud x^n} \pr{x^2-1}^n , \\
\rawLeg{l}{m}(x) = \pr{1-x^2}^{m/2} \frac{\ud^m}{\ud x^m} \rawLeg{l}{0},\quad\abs{x}\le 1, m \ge 0, \\
\rawLeg{l}{m}(x) = (-1)^\abs{m} \frac{(l-\abs{m})!}{(l+\abs{m})!} \rawLeg{l}{\abs{m}},
\quad \abs{x} \le 1, m < 0.
\f]
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DLMF \cite NIST:DLMF has for non-negative integer \f$m\f$ (18.5.5), (14.6.1), (14.9.3):
\f[
\dlmfFer{\nu}{} = \dlmfLeg{\nu}{} = \frac{1}{2^n n!} \frac{\ud^n}{\ud x^n} \pr{x^2-1}^n , \\
\dlmfFer{\nu}{m}\left(x\right)=(-1)^{m}\left(1-x^2\right)^{m/2}\frac{{
\ud}^{m}\dlmfFer{\nu}{}\left(x\right)}{{\ud x}^{m}},\\
%\dlmfLeg{\nu}{m}\left(x\right)=\left(-1+x^2\right)^{m/2}\frac{{
%\ud}^{m}\dlmfLeg{\nu}{}\left(x\right)}{{\ud x}^{m}},\\
\f]
where the connection to negative orders is
\f[
\dlmfFer{\nu}{m}(x) = (-1)^m \frac{\Gamma\pr{\nu-m+1}}{\Gamma\pr{\nu+m+1}}\dlmfFer{\nu}{m}(x),\\
%\dlmfLeg{\nu}{m}(x) = \frac{\Gamma\pr{\nu-m+1}}{\Gamma\pr{\nu+m+1}}\dlmfLeg{\nu}{m}(x).\\
\f]
Note that there are called "Ferrers" functions in DLMF, while the "Legendre" functions have slightly
different meaning / conventions (Ferrers functions being defined for \f$ \abs{x} \le 1 \f$, whereas
Legendre for \f$ \abs{x} \ge 1 \f$. We will not use the DLMF "Legendre" functions here.
One sees that \f$ \dlmfFer{l}{m} = (-1)^m \rawFer{l}{m} \f$, i.e. the Condon-Shortley phase is
already included in the DLMF definitions of Ferrers functions.
GSL computes \f$ \rawFer{l}{m} \f$ unless the corresponding `csphase` argument is set to
\f$-1\f$ (then it computes \f$ \dlmfFer{l}{m} \f$). This is not explicitly obvious from the docs
\cite GSL,
but can be tested by running `gsl_sf_legendre_array_e` for some specific arguments and comparing signs.
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Convention effect on translation operators
------------------------------------------
Let us declare VSWFs in Kristensson's conventions below,
\f$ \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.
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Literature convention tables
----------------------------
### Legendre functions and spherical harmonics
| Source | Ferrers function | Negative \f$m\f$ | Spherical harmonics |
|------------------------|-----------------------|--------------------|---------------------|
| DLMF \cite NIST:DLMF | \f[
\dlmfFer{\nu}{m}\left(x\right)=(-1)^{m}\left(1-x^2\right)^{m/2}\frac{{
\ud}^{m}\dlmfFer{\nu}{}\left(x\right)}{{\ud x}^{m}}
\f] | \f[
\dlmfFer{\nu}{m}(x) = (-1)^m \frac{\Gamma\pr{\nu-m+1}}{\Gamma\pr{\nu+m+1}}\dlmfFer{\nu}{m}(x)
\f] | Complex (14.30.1): \f[
\dlmfYc{l}{m} = \sqrt{\frac{(l-m)!(2l+1)}{4\pi(l+m)!}} e^{im\phi} \dlmfFer{l}{m}(\cos\theta).
\f] Real, unnormalized (14.30.2): \f$
\dlmfYrUnnorm{l}{m}\pr{\theta,\phi} = \cos\pr{m\phi} \dlmfFer{l}{m}\pr{\cos\theta}
\f$ or \f$
\dlmfYrUnnorm{l}{m}\pr{\theta,\phi} = \sin\pr{m\phi} \dlmfFer{l}{m}\pr{\cos\theta}
\f$. |
| GSL \cite GSL | \f[
\Fer[GSL]{l}{m} = \csphase^m N \rawFer{l}{m}
\f] for non-negative \f$m\f$. \f$
\csphase\f$ is one by default and can be set to \f$
-1\f$ using the functions ending with \_e with argument `csphase = -1` . \f$
N\f$ is a positive normalisation factor from from `gsl_sf_legendre_t` . | N/A. Must be calculated manually. | The asimuthal part must be calculated manually. Use `norm = GSL_SF_LEGENDRE_SPHARM` to get the usual normalisation factor \f$
N= \sqrt{\frac{(l-m)!(2l+1)}{4\pi(l+m)!}} \f$. |
| Kristensson I \cite kristensson_spherical_2014 | \f$ \rawFer{l}{m} \f$ | As in \f$ \rawFer{l}{m} \f$. | \f[
\spharm[Kc]{l}{m} = (-1)^m \sqrt{\frac{(l-m)!(2l+1)}{4\pi(l+m)!}} \rawFer{l}{m}(\cos\theta) e^{im\phi},
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\f] (cf. Sec. D.2), therefore it corresponds to the DLMF sph. harms.: \f[ \spharm[Kc]{l}{m} = \dlmfYc{l}{m}. \f] |
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| Kristensson II \cite kristensson_scattering_2016 | \f$ \rawFer{l}{m} \f$ | As in \f$ \rawFer{l}{m} \f$. | \f[
\spharm[Kr]{\begin{Bmatrix}e \\ o\end{Bmatrix}}{l}{m} =
\sqrt{2-\delta_{m0}}\sqrt{\frac{(l-m)!(2l+1)}{4\pi(l+m)!}}
\rawFer{l}{m}(\cos\theta)
\begin{Bmatrix}\cos\phi \\ \sin\phi\end{Bmatrix},
\f] \f$ m \ge 0 \f$. Cf. Appendix C.3. |
| Reid \cite reid_electromagnetism_2016 | Not described in the memos. Superficial look into the code suggests that the `GetPlm` function *does* include the Condon-Shortley phase and spherical harmonic normalisation, so \f[
\Fer[GetPlm]{l}{m} = (-1)^m \sqrt{\frac{(l-m)!(2l+1)}{4\pi(l+m)!}} \rawFer{l}{m}
\f] for non-negative \f$ m \f$. | N/A. Must be calculated manually. | \f[
\spharm[GetYlm]{l}{m}(\theta,\phi) = \Fer[GetPlm]{l}{m}(\cos\theta) e^{im\phi},\quad m\le 0, \\
\spharm[GetYlm]{l}{m}(\theta,\phi) = (-1)^m\Fer[GetPlm]{l}{\abs{m}}(\cos\theta) e^{-im\phi},\quad m< 0 ,
\f] and the negative sign in the second line's exponent is quite concerning, because that would mean the asimuthal part is actually \f$ e^{i\abs{m}\phi} \f$. _Is this a bug in scuff-em_ ? Without it, it would be probably equivalent to DLMF's \f$ \dlmfYc{l}{m} \f$s for both positive and negative \f$ m\f$s. However, it seems that `GetYlmDerivArray` has it consistent, with \f[
\spharm[GetYlmDerivArray]{l}{m} = \dlmfYc{l}{m}
\f] for all \f$m\f$, and this is what is actually used in `GetMNlmArray` (used by both `SphericalWave` in `libIncField` (via `GetMNlm` ) and `GetSphericalMoments` in `libscuff` (via `GetWaveMatrix` )) and `GetAngularFunctionArray` (not used). |
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### VSWF conventions
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| Source | VSWF definition | E/M interrelations | VSWF norm | CS Phase | Field expansion | Radiated power | Notes |
|--- |--- |--- |--- |--- |--- |--- |--- |
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| Kristensson I \cite kristensson_spherical_2014 | \f[ \wfkc = \dots \f] where \f$\wfkc\f$ is either of \f$ \wfkcreg, \wfkcout, \dots \f$ based on the radial (spherical Bessel) function type. | \f[
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\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$, eq. (2.8) onwards. | | Yes, in the spherical harmonics definition, cf. sect. D.2. | \f[
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\vect E = k \sqrt{\eta_0\eta} \sum_n \left( \wckcreg_n \wfkcreg_n + \wckcout_n \wfkcout_n \right),
\\
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\vect H = \frac{k \sqrt{\eta_0\eta}}{i\eta_0\eta} \sum_n \left( \wckcreg_n \wfkcreg_n + \wckcout_n \wfkcout_n \right),
\f] but for plane wave expansion \cite kristensson_spherical_2014 sect. 2.5 K. uses a different definition (same as in Kristensson II). | \f[
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P = \frac{1}{2} \sum_n \left( \abs{\wckcout_n}^2 +\Re \left(\wckcout_n\wckcreg_n^{*}\right)\right)
\f] | The \f$ \wckcreg, \wckcout \f$ coefficients have dimension \f$ \sqrt{\mathrm{W}} \f$. |
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| Kristensson II \cite kristensson_scattering_2016 | \f[ \wfkr = \dots \f] where \f$\wfkr\f$ is either of \f$ \wfkrreg, \wfkrout, \dots \f$ based on the radial (spherical Bessel) function type. | \f[
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\nabla\times\wfkrreg_{\tau n} = k\wfkrreg_{\overline{\tau} n},
\f] eq. (7.7) and analogously for outgoing waves \f$ \wfkrout \f$. | | | \f[
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\vect E = \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right),
\\
\vect H = \frac{1}{i\eta_0\eta} \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right)
\f] | \f[
P = \frac{1}{2k^2\eta_0\eta} \sum_n \left( \abs{\wckrout_n}^2 +\Re \left(\wckrout_n\wckrreg_n^{*}\right)\right)
\f] | The \f$ \wckrreg, \wckrout \f$ coefficients have dimension \f$ \mathrm{V/m} \f$. |
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| Reid \cite reid_electromagnetism_2016 | By examining the code, it appears that both `GetMNlmArray()` and `GetWaveMatrix()` with argument `MaxwellWaves = true` (with `MaxwellWaves = false` it seems to calculate nonsense) return the following w.r.t. Kristensson's "complex VSWFs": \f[
\wfr_{lmM} = i\wfkc_{1lm}, \\
\wfr_{lmN} = -\wfkc_{2lm}.
\f] | \f[
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\nabla\times\wfr_{lmM} = -ik\wfr_{lmN}, \\ \nabla\times\wfr_{lmN} = +ik\wfr_{lmM}.
\f] | | | \f[
\vect E = \sum_\alpha \pr{ \wcrreg_\alpha \wfrreg_\alpha + \wcrout_\alpha \wfrout_\alpha }, \\
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\vect H = \frac{1}{Z_0Z^r} \sum_\alpha \pr{ \wcrreg_\alpha \sigma_\alpha\wfrreg_\overline{\alpha} +
\wcrout_\alpha \sigma_\alpha\wfrout_\overline{\alpha}},
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\f] where \f$ \sigma_{lmM} = +1, \sigma_{lmN}=-1, \overline{lmM}=lmN, \overline{lmN}=lmM, \f$ cf. eq. (6). The notation is not extremely consistent throughout Reid's memo. | | |
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| Taylor \cite taylor_optical_2011 | \f[
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\wfet_{mn}^{(j)} = \frac{n(n+1)}{kr}\sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}\Fer[Taylor]{n}{m}\left(\cos\theta\right)e^{im\phi}z_{n}^{j}\left(kr\right)\uvec{r} \\
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+\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)}, \\
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\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] Assuming the Legendre functions \f$ \Fer[Taylor]{n}{m} \f$ here do contain the Condon-Shortley phase (AFAIK not explicitly stated in the book), i.e. \f$\Fer[Taylor]{l}{m} = \dlmfFer{l}{m} \f$, then the relation to Kristensson's waves is \f[
\wfmt_{mn} = \sqrt{n(n+1)} \wfkc_{1nm}, \\ \wfet_{mn} = \sqrt{n(n+1)} \wfkc_{2nm}.
\f] | | \f[
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\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–
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\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}}},
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\f] | | Different sign for regular/scattered waves! Also WTF are the units of \f$ n_{ext} \f$? The whole definition seems rather inconsistent. |
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