289 lines
17 KiB
Markdown
289 lines
17 KiB
Markdown
VSWF conventions {#vswf_conventions}
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====================================
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In general, the (transversal) VSWFs can be defined using (some) vector spherical harmonics
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as follows: \f[
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\wfm\pr{k\vect r}_{lm} = \sphbes_l(kr) \vshrot_{lm} (\uvec r),\\
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\wfe\pr{k\vect r}_{lm} = \frac{\frac{\ud}{\ud(kr)}\pr{kr\sphbes_l(kr)}}{kr} \vshgrad_{lm}(\uvec r)
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+ \sqrt{l(l+1)} \frac{\sphbes_l(kr)}{kr} \vshrad_{lm}(\uvec r),
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\f]
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where at this point, we don't have much expectations regarding the
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normalisations and phases of the
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"rotational", "gradiental" and "radial" vector spherical harmonics
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\f$ \vshrot, \vshgrad, \vshrad \f$, and the waves can be of whatever "direction"
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(regular, outgoing, etc.) depending on the kind of the spherical Bessel function
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\f$ \sphbes \f$.
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We only require that the spherical harmonic degree \f$ l \f$
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is what it is supposed to be. The meaning of the order $m$ may vary depending
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on convention. Moreover, in order to \f$ \wfe \f$ be a valid "electric" multipole wave,
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there is a fixed relation between radial and gradiental vector spherical harmonics
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(more on that later).
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Let us define the "dual" vector spherical harmonics \f$ \vshD_{\tau lm} \f$ as follows:
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\f[
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\int_\Omega \vsh_{\tau lm} (\uvec r) \cdot \vshD_{\tau' l'm} (\uvec r) \, \ud \Omega
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= \delta_{\tau', \tau}\delta_{l',l} \delta_{m',m}
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\f]
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where the \f$ \cdot \f$ symbol here means the bilinear form of the vector components
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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.
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For the sake of non-ambiguity, let us first define the "canonical" associated Legendre/Ferrers polynomials
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*without* the Condon-Shortley phase.
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\f[
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\rawLeg{l}{0}(x) = \frac{1}{2^n n!} \frac{\ud^n}{\ud x^n} \pr{x^2-1}^n , \\
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\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, \\
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\rawLeg{l}{m}(x) = (-1)^\abs{m} \frac{(l-\abs{m})!}{(l+\abs{m})!} \rawLeg{l}{\abs{m}},
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\quad \abs{x} \le 1, m < 0.
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\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):
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\f[
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\dlmfFer{\nu}{} = \dlmfLeg{\nu}{} = \frac{1}{2^n n!} \frac{\ud^n}{\ud x^n} \pr{x^2-1}^n , \\
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\dlmfFer{\nu}{m}\left(x\right)=(-1)^{m}\left(1-x^2\right)^{m/2}\frac{{
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\ud}^{m}\dlmfFer{\nu}{}\left(x\right)}{{\ud x}^{m}},\\
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%\dlmfLeg{\nu}{m}\left(x\right)=\left(-1+x^2\right)^{m/2}\frac{{
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%\ud}^{m}\dlmfLeg{\nu}{}\left(x\right)}{{\ud x}^{m}},\\
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\f]
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where the connection to negative orders is
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\f[
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\dlmfFer{\nu}{m}(x) = (-1)^m \frac{\Gamma\pr{\nu-m+1}}{\Gamma\pr{\nu+m+1}}\dlmfFer{\nu}{m}(x),\\
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%\dlmfLeg{\nu}{m}(x) = \frac{\Gamma\pr{\nu-m+1}}{\Gamma\pr{\nu+m+1}}\dlmfLeg{\nu}{m}(x).\\
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\f]
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Note that there are called "Ferrers" functions in DLMF, while the "Legendre" functions have slightly
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different meaning / conventions (Ferrers functions being defined for \f$ \abs{x} \le 1 \f$, whereas
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Legendre for \f$ \abs{x} \ge 1 \f$. We will not use the DLMF "Legendre" functions here.
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One sees that \f$ \dlmfFer{l}{m} = (-1)^m \rawFer{l}{m} \f$, i.e. the Condon-Shortley phase is
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already included in the DLMF definitions of Ferrers functions.
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GSL computes \f$ \rawFer{l}{m} \f$ unless the corresponding `csphase` argument is set to
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\f$-1\f$ (then it computes \f$ \dlmfFer{l}{m} \f$). This is not explicitly obvious from the docs
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\cite GSL,
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but can be tested by running `gsl_sf_legendre_array_e` for some specific arguments and comparing signs.
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Convention effects on symmetry operators
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----------------------------------------
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### Spherical harmonics
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Let' have two different (complex) spherical harmonic conventions connected by constant factors:
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\f[
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\spharm[a]{l}{m} = c^\mathrm{a}_{lm}\spharm{l}{m}.
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\f]
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Both sets can be used to describe an angular function \f$ f \f$
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\f[
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f = \sum_{lm} f^\mathrm{a}_{lm} \spharm[a]{l}{m}
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= \sum_{lm} f^\mathrm{a}_{lm} c^\mathrm{a}_{lm}\spharm{l}{m}
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= \sum_{lm} f_{lm} \spharm{l}{m}.
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\f]
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If we perform a (symmetry) transformation \f$ g \f$ acting on the \f$ \spharm{l}{m} \f$
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basis via matrix \f$ D(g)_{l,m;l',m'} \f$, i.e.
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\f[
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g\pr{\spharm{l}{m}} = \sum_{l'm'} D(g)_{l,m;l'm'} \spharm{l'}{m'},
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\f]
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we see
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\f[
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g(f) = \sum_{lm} f_{lm}\sum_{l'm'} D(g)_{l,m;l'm'} \spharm{l'}{m'}
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= \sum_{lm} f^\mathrm{a}_{lm} c^\mathrm{a}_{lm}\sum_{l'm'} D(g)_{l,m;l'm'} \spharm{l'}{m'}.
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\f]
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Rewriting the transformation action in the second basis
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\f[
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g\pr{\spharm[a]{l}{m}} = \sum_{l'm'} D(g)^\mathrm{a}_{l,m;l'm'} \spharm[a]{l'}{m'},\\
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g(f) = \sum_{lm} f^\mathrm{a}_{lm}\sum_{l'm'} D(g)^\mathrm{a}_{l,m;l'm'} \spharm[a]{l'}{m'},
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\f]
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and performing some substitutions,
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\f[
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g(f) = \sum_{lm} \frac{f_{lm}}{c^\mathrm{a}_{lm}}
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\sum_{l'm'} D(g)^\mathrm{a}_{l,m;l'm'} c^\mathrm{a}_{l'm'}\spharm{l'}{m'},
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\f]
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and comparing, we get
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\f[
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D(g)^\mathrm{a}_{l,m;l'm'} = \frac{c^\mathrm{a}_{lm}}{c^\mathrm{a}_{l'm'}}D(g)_{l,m;l'm'}.
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\f]
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If the difference between conventions is in particular Condon-Shortley phase,
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this means a \f$ (-1)^{m-m'} \f$ factor between the transformation matrices.
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This does not affect the matrices for the inversion and
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mirror symmetry operations with
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respect to the \a xy, \a yz and \a xz planes, because they are all diagonal
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or anti-diagonal with respect to \a m (hence \f$ m-m \f$ is either zero
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or anyways even integer).
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It does, however, affect rotations, flipping the sign of the rotations
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along the \a z axis.
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Apparently, a constant complex factor independent of \f$ l,m \f$
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does nothing to the form of the tranformation matrix.
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These conclusions about transformations of spherical harmonics
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hold also for the VSWFs built on top of them.
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Convention effect on translation operators
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------------------------------------------
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Let us declare VSWFs in Kristensson's conventions below,
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\f$ \wfkc \f$ \cite kristensson_spherical_2014,
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\f$ \wfkr \f$ \cite kristensson_scattering_2016, as the "canonical"
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spherical waves based on complex and real spherical harmonics, respectively.
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They both have the property that the translation operators \f$ \tropRrr{}{},\tropSrr{}{} \f$
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that transform
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the VSWF field expansion coefficients between different origins, e.g.
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\f[
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\wfkcreg(\vect{r}) = \tropRrr{\vect r}{\vect r'} \wfkcreg(\vect{r'}),
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\f]
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actually consist of two different submatrices $A,B$ for the same-type and different-type
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(in the sense of "electric" versus "magnetic" waves) that repeat themselves once:
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\f[
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\begin{bmatrix} \wfkcreg_1(\vect{r}) \\ \wfkcreg_2(\vect{r}) \end{bmatrix}
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= \begin{bmatrix} A & B \\ B & A \end{bmatrix}(\vect{r} \leftarrow \vect{r'})
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\begin{bmatrix} \wfkcreg_1(\vect{r'}) \\ \wfkcreg_2(\vect{r'}) \end{bmatrix}.
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\f]
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(This symmetry holds also for singular translation operators \f$ \tropSrr{}{} \f$
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and real spherical harmonics based VSWFs \f$ \wfkr \f$.)
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However, the symmetry above will not hold like this in some stupider convention.
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Let's suppose that one uses a different convention with some additional coefficients
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compared to the canonical one,
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\f[
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\wfm_{lm} = \alpha_{\wfm lm} \wfkc_{1lm},\\
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\wfe_{lm} = \alpha_{\wfe lm} \wfkc_{2lm}.\\
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\f]
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and with field expansion (WLOG assume regular fields only)
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\f[ \vect E = c_{\wfe l m} \wfe_{lm} + c_{\wfm l m } \wfm_{lm}. \f]
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Under translations, the coefficients then transform like
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\f[
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\begin{bmatrix} \alpha_\wfe(\vect{r}) \\ \alpha_\wfm(\vect{r}) \end{bmatrix}
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= \begin{bmatrix} R_{\wfe\wfe} & R_{\wfe\wfm} \\
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R_{\wfm\wfe} & R_{\wfm\wfm}
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\end{bmatrix}(\vect{r} \leftarrow \vect{r'})
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\begin{bmatrix} \alpha_\wfe(\vect{r'}) \\ \alpha_\wfm(\vect{r'}) \end{bmatrix},
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\f]
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and by substituting and comparing the expressions for canonical waves above, one gets
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\f[
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R_{\wfe,lm;\wfe,l'm'} = \alpha_{\wfe lm}^{-1} A_{lm,l'm'} \alpha_{\wfe l'm'},\\
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R_{\wfe,lm;\wfm,l'm'} = \alpha_{\wfe lm}^{-1} B_{lm,l'm'} \alpha_{\wfm l'm'},\\
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R_{\wfm,lm;\wfe,l'm'} = \alpha_{\wfm lm}^{-1} B_{lm,l'm'} \alpha_{\wfe l'm'},\\
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R_{\wfm,lm;\wfm,l'm'} = \alpha_{\wfm lm}^{-1} A_{lm,l'm'} \alpha_{\wfm l'm'}.
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\f]
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If the coefficients for magnetic and electric waves are the same,
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\f$ \alpha_{\wfm lm} = \alpha_{\wfe lm} \f$, the translation operator
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can be written in the same symmetric form as with the canonical convention,
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just the matrices \f$ A, B\f$ will be different inside.
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If the coefficients differ (as in SCUFF-EM convention, where there
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is a relative \a i -factor between electric and magnetic waves),
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the functions such as qpms_trans_calculator_get_AB_arrays() will
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compute \f$ R_{\wfe\wfe}, R_{\wfe\wfm} \f$ for \f$ A, B \f$ arrays.
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The remaining matrices' elements must then be obtained as
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\f[
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R_{\wfm,lm;\wfe,l'm'} = \alpha_{\wfm lm}^{-1} \alpha_{\wfe lm}
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R_{\wfe,lm;\wfm,l'm'} \alpha_{\wfm l'm'}^{-1} \alpha_{\wfe l'm'}
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= g_{lm}R_{\wfe,lm;\wfm,l'm'}g_{l'm'}, \\
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R_{\wfm,lm;\wfm,l'm'} = \alpha_{\wfm lm}^{-1} \alpha_{\wfe lm}
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R_{\wfe,lm;\wfe,l'm'} \alpha_{\wfe l'm'}^{-1} \alpha_{\wfm l'm'}
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= g_{lm}R_{\wfe,lm;\wfe,l'm'}g_{l'm'}^{-1},
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\f]
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where the coefficients \f$ g_{lm} \f$ can be obtained by
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qpms_normalisation_factor_N_M().
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Literature convention tables
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----------------------------
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### Legendre functions and spherical harmonics
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| Source | Ferrers function | Negative \f$m\f$ | Spherical harmonics |
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|------------------------|-----------------------|--------------------|---------------------|
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| DLMF \cite NIST:DLMF | \f[
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\dlmfFer{\nu}{m}\left(x\right)=(-1)^{m}\left(1-x^2\right)^{m/2}\frac{{
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\ud}^{m}\dlmfFer{\nu}{}\left(x\right)}{{\ud x}^{m}}
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\f] | \f[
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\dlmfFer{\nu}{m}(x) = (-1)^m \frac{\Gamma\pr{\nu-m+1}}{\Gamma\pr{\nu+m+1}}\dlmfFer{\nu}{m}(x)
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\f] | Complex (14.30.1): \f[
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\dlmfYc{l}{m} = \sqrt{\frac{(l-m)!(2l+1)}{4\pi(l+m)!}} e^{im\phi} \dlmfFer{l}{m}(\cos\theta).
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\f] Real, unnormalized (14.30.2): \f$
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\dlmfYrUnnorm{l}{m}\pr{\theta,\phi} = \cos\pr{m\phi} \dlmfFer{l}{m}\pr{\cos\theta}
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\f$ or \f$
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\dlmfYrUnnorm{l}{m}\pr{\theta,\phi} = \sin\pr{m\phi} \dlmfFer{l}{m}\pr{\cos\theta}
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\f$. |
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| GSL \cite GSL | \f[
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\Fer[GSL]{l}{m} = \csphase^m N \rawFer{l}{m}
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\f] for non-negative \f$m\f$. \f$
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\csphase\f$ is one by default and can be set to \f$
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-1\f$ using the functions ending with \_e with argument `csphase = -1`. \f$
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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$
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N= \sqrt{\frac{(l-m)!(2l+1)}{4\pi(l+m)!}} \f$. |
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| Kristensson I \cite kristensson_spherical_2014 | \f$ \rawFer{l}{m} \f$ | As in \f$ \rawFer{l}{m} \f$. | \f[
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\spharm[Kc]{l}{m} = (-1)^m \sqrt{\frac{(l-m)!(2l+1)}{4\pi(l+m)!}} \rawFer{l}{m}(\cos\theta) e^{im\phi},
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\f] (cf. Sec. D.2), therefore it corresponds to the DLMF sph. harms.: \f[ \spharm[Kc]{l}{m} = \dlmfYc{l}{m}. \f] |
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| Kristensson II \cite kristensson_scattering_2016 | \f$ \rawFer{l}{m} \f$ | As in \f$ \rawFer{l}{m} \f$. | \f[
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\spharm[Kr]{\begin{Bmatrix}e \\ o\end{Bmatrix}}{l}{m} =
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\sqrt{2-\delta_{m0}}\sqrt{\frac{(l-m)!(2l+1)}{4\pi(l+m)!}}
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\rawFer{l}{m}(\cos\theta)
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\begin{Bmatrix}\cos\phi \\ \sin\phi\end{Bmatrix},
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\f] \f$ m \ge 0 \f$. Cf. Appendix C.3. |
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| Reid \cite reid_electromagnetism_2016 | Not described in the memos. Superficial look into the code suggests that the `GetPlm` function *does* include the Condon-Shortley phase and spherical harmonic normalisation, so \f[
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\Fer[GetPlm]{l}{m} = (-1)^m \sqrt{\frac{(l-m)!(2l+1)}{4\pi(l+m)!}} \rawFer{l}{m}
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\f] for non-negative \f$ m \f$. | N/A. Must be calculated manually. | \f[
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\spharm[GetYlm]{l}{m}(\theta,\phi) = \Fer[GetPlm]{l}{m}(\cos\theta) e^{im\phi},\quad m\le 0, \\
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\spharm[GetYlm]{l}{m}(\theta,\phi) = (-1)^m\Fer[GetPlm]{l}{\abs{m}}(\cos\theta) e^{-im\phi},\quad m<0,
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\f] and the negative sign in the second line's exponent is quite concerning, because that would mean the asimuthal part is actually \f$ 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$s. However, it seems that `GetYlmDerivArray` has it consistent, with \f[
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\spharm[GetYlmDerivArray]{l}{m} = \dlmfYc{l}{m}
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\f] for all \f$m\f$, and this is what is actually used in `GetMNlmArray` (used by both `SphericalWave` in `libIncField` (via `GetMNlm`) and `GetSphericalMoments` in `libscuff` (via `GetWaveMatrix`)) and `GetAngularFunctionArray` (not used). |
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### VSWF conventions
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| Source | VSWF definition | E/M interrelations | VSWF norm | CS Phase | Field expansion | Radiated power | Notes |
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|--- |--- |--- |--- |--- |--- |--- |--- |
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| Kristensson I \cite kristensson_spherical_2014 | \f[ \wfkc = \dots \f] where \f$\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}, \\
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\wfkcreg_{2lm} = \frac{1}{k}\nabla\times\wfkcreg_{1lm},
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\f] and analogously for outgoing waves \f$ \wfkcout \f$, eq. (2.8) onwards. | | Yes, in the spherical harmonics definition, cf. sect. D.2. | \f[
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\vect E = k \sqrt{\eta_0\eta} \sum_n \left( \wckcreg_n \wfkcreg_n + \wckcout_n \wfkcout_n \right),
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\\
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\vect H = \frac{k \sqrt{\eta_0\eta}}{i\eta_0\eta} \sum_n \left( \wckcreg_n \wfkcreg_n + \wckcout_n \wfkcout_n \right),
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\f] but for plane wave expansion \cite kristensson_spherical_2014 sect. 2.5 K. uses a different definition (same as in Kristensson II). | \f[
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P = \frac{1}{2} \sum_n \left( \abs{\wckcout_n}^2 +\Re \left(\wckcout_n\wckcreg_n^{*}\right)\right)
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\f] | The \f$ \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},
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\f] eq. (7.7) and analogously for outgoing waves \f$ \wfkrout \f$. | | | \f[
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\vect E = \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right),
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\\
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\vect H = \frac{1}{i\eta_0\eta} \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right)
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\f] | \f[
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P = \frac{1}{2k^2\eta_0\eta} \sum_n \left( \abs{\wckrout_n}^2 +\Re \left(\wckrout_n\wckrreg_n^{*}\right)\right)
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\f] | The \f$ \wckrreg, \wckrout \f$ coefficients have dimension \f$ \mathrm{V/m} \f$. |
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| Reid \cite reid_electromagnetism_2016 | By examining the code, it appears that both `GetMNlmArray()` and `GetWaveMatrix()` with argument `MaxwellWaves = true` (with `MaxwellWaves = false` it seems to calculate nonsense) return the following w.r.t. Kristensson's "complex VSWFs": \f[
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\wfr_{lmM} = i\wfkc_{1lm}, \\
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\wfr_{lmN} = -\wfkc_{2lm}.
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\f] | \f[
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\nabla\times\wfr_{lmM} = -ik\wfr_{lmN}, \\ \nabla\times\wfr_{lmN} = +ik\wfr_{lmM}.
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\f] | | | \f[
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\vect E = \sum_\alpha \pr{ \wcrreg_\alpha \wfrreg_\alpha + \wcrout_\alpha \wfrout_\alpha }, \\
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\vect H = \frac{1}{Z_0Z^r} \sum_\alpha \pr{ \wcrreg_\alpha \sigma_\alpha\wfrreg_\overline{\alpha} +
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\wcrout_\alpha \sigma_\alpha\wfrout_\overline{\alpha}},
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\f] where \f$ \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).
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\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[
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\wfmt_{mn} = \sqrt{n(n+1)} \wfkc_{1nm}, \\ \wfet_{mn} = \sqrt{n(n+1)} \wfkc_{2nm}.
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\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'} ,\\
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\int_{S(kr)} \wfet_{mn}^{(j)} \wfet_{m'n'}^{(j)}\,\ud S =
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\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'} ,
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\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[
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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}}}, \\
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\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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