qpms/notes/conventions.md

VSWF conventions

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).

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. \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] 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.

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$.	\f[
\spharm[Kc]{l}{m} = (-1)^m \sqrt{\frac{(l-m)!(2l+1)}{4\pi(l+m)!}} \rawFer{l}{m}(\cos\theta) e^{im\phi},
\f] (cf. Sec. D.2), therefore it corresponds to the DLMF sph. harms.: \f[ \spharm[Kc]{l}{m} = \dlmfYc{l}{m}. \f]
Kristensson II \cite kristensson_scattering_2016	\f$ \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).

VSWF conventions

Source	VSWF definition	E/M interrelations	VSWF norm	CS Phase	Field expansion	Radiated power	Notes
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[
\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$, eq. (2.8) onwards.		Yes, in the spherical harmonics definition, cf. sect. D.2.	\f[
\vect E = k \sqrt{\eta_0\eta} \sum_n \left( \wckcreg_n \wfkcreg_n + \wckcout_n \wfkcout_n \right),
\
\vect H = \frac{k \sqrt{\eta_0\eta}}{i\eta_0\eta} \sum_n \left( \wckcreg_n \wfkcreg_n + \wckcout_n \wfkcout_n \right),
\f] but for plane wave expansion \cite kristensson_spherical_2014 sect. 2.5 K. uses a different definition (same as in Kristensson II).	\f[
P = \frac{1}{2} \sum_n \left( \abs{\wckcout_n}^2 +\Re \left(\wckcout_n\wckcreg_n^{*}\right)\right)
\f]	The \f$ \wckcreg, \wckcout \f$ coefficients have dimension \f$ \sqrt{\mathrm{W}} \f$.
Kristensson II \cite kristensson_scattering_2016	\f[ \wfkr = \dots \f] where \f$\wfkr\f$ is either of \f$ \wfkrreg, \wfkrout, \dots \f$ based on the radial (spherical Bessel) function type.	\f[
\nabla\times\wfkrreg_{\tau n} = k\wfkrreg_{\overline{\tau} n},
\f] eq. (7.7) and analogously for outgoing waves \f$ \wfkrout \f$.			\f[
\vect E = \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right),
\
\vect H = \frac{1}{i\eta_0\eta} \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right)
\f]	\f[
P = \frac{1}{2k^2\eta_0\eta} \sum_n \left( \abs{\wckrout_n}^2 +\Re \left(\wckrout_n\wckrreg_n^{*}\right)\right)
\f]	The \f$ \wckrreg, \wckrout \f$ coefficients have dimension \f$ \mathrm{V/m} \f$.
Reid \cite reid_electromagnetism_2016	By examining the code, it appears that both GetMNlmArray() and GetWaveMatrix() with argument MaxwellWaves = true (with MaxwellWaves = false it seems to calculate nonsense) return the following w.r.t. Kristensson's "complex VSWFs": \f[
\wfr_{lmM} = i\wfkc_{1lm}, \
\wfr_{lmN} = -\wfkc_{2lm}.
\f]	\f[
\nabla\times\wfr_{lmM} = -ik\wfr_{lmN}, \ \nabla\times\wfr_{lmN} = +ik\wfr_{lmM}.
\f]			\f[
\vect E = \sum_\alpha \pr{ \wcrreg_\alpha \wfrreg_\alpha + \wcrout_\alpha \wfrout_\alpha }, \
\vect H = \frac{1}{Z_0Z^r} \sum_\alpha \pr{ \wcrreg_\alpha \sigma_\alpha\wfrreg_\overline{\alpha} +
\wcrout_\alpha \sigma_\alpha\wfrout_\overline{\alpha}},
\f] where \f$ \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.
Taylor \cite taylor_optical_2011	\f[
\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} \
+\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] 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[
\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$? The whole definition seems rather inconsistent.