Compare Reid's (SCUFF-EM) and Taylor's to Kristensson VSWFs.
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Marek Nečada
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@ -113,7 +113,7 @@ Literature convention tables
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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[ \wfkcreg, \wfkcout= \dots \f] | \f[
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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$, eq. (2.8) onwards. | | Yes, in the spherical harmonics definition, cf. sect. D.2. | \f[
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@ -123,7 +123,7 @@ Literature convention tables
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\f] but for plane wave expansion \cite kristensson_spherical_2014 sect. 2.5 K. uses a different definition (same as in Kristensson II). | \f[
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P = \frac{1}{2} \sum_n \left( \abs{\wckcout_n}^2 +\Re \left(\wckcout_n\wckcreg_n^{*}\right)\right)
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\f] | The \f$ \wckcreg, \wckcout \f$ coefficients have dimension \f$ \sqrt{\mathrm{W}} \f$. |
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| Kristensson II \cite kristensson_scattering_2016 | \f[ \wfkrreg, \wfkrout= \dots \f] | \f[
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| Kristensson II \cite kristensson_scattering_2016 | \f[ \wfkr = \dots \f] where \f$\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$. | | | \f[
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\vect E = \sum_n \left( \wckrreg_n \wfkrreg_n + \wckrout_n \wfkrout_n \right),
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@ -132,7 +132,10 @@ Literature convention tables
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\f] | \f[
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P = \frac{1}{2k^2\eta_0\eta} \sum_n \left( \abs{\wckrout_n}^2 +\Re \left(\wckrout_n\wckrreg_n^{*}\right)\right)
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\f] | The \f$ \wckrreg, \wckrout \f$ coefficients have dimension \f$ \mathrm{V/m} \f$. |
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| Reid \cite reid_electromagnetism_2016 | | \f[
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| Reid \cite reid_electromagnetism_2016 | By examining the code, it appears that both `GetMNlmArray()` and `GetWaveMatrix()` with argument `MaxwellWaves = true` (with `MaxwellWaves = false` it seems to calculate nonsense) return the following w.r.t. Kristensson's "complex VSWFs": \f[
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\wfr_{lmM} = i\wfkc_{1lm}, \\
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\wfr_{lmN} = -\wfkc_{2lm}.
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\f] | \f[
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\nabla\times\wfr_{lmM} = -ik\wfr_{lmN}, \\ \nabla\times\wfr_{lmN} = +ik\wfr_{lmM}.
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\f] | | | \f[
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\vect E = \sum_\alpha \pr{ \wcrreg_\alpha \wfrreg_\alpha + \wcrout_\alpha \wfrout_\alpha }, \\
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@ -140,10 +143,12 @@ Literature convention tables
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\wcrout_\alpha \sigma_\alpha\wfrout_\overline{\alpha}},
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\f] where \f$ \sigma_{lmM} = +1, \sigma_{lmN}=-1, \overline{lmM}=lmM, \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)!}}P_{n}^{m}\left(\cos\theta\right)e^{im\phi}z_{n}^{j}\left(kr\right)\uvec{r} \\
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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] | | \f[
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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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@ -33,14 +33,16 @@ MathJax.Hub.Config({
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csphase: "\\mathsf{C_{CS}}", // Condon-Shortley phase
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// Kristensson's VSWFs, complex version (2014 notes)
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wfkc: "{\\vect{y}}", // any wave
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wfkcreg: "{\\vect{v}}", // regular wave
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wfkcout: "{\\vect{u}}", // outgoing wave
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wckcreg: "{a}", // regular wave coeff
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wckcout: "{f}", // outgoing wave coeff
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// Kristensson's VSWFs, real version (2014 book)
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wfkrreg: "{\\vect{v}}", // regular wave
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wfkrout: "{\\vect{u}}", // outgoing wave
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wfkr: "{\\vect{y}_{\\mathrm{r}}}", // any wave
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wfkrreg: "{\\vect{v}_{\\mathrm{r}}}", // regular wave
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wfkrout: "{\\vect{u}_{\\mathrm{r}}}", // outgoing wave
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wckrreg: "{a}", // regular wave coeff
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wckrout: "{f}", // outgoing wave coeff
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