VSWF definitions
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Marek Nečada
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@ -155,7 +155,19 @@ The basic idea of MSTMM is quite simple: the driving electromagnetic field
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Repeating the same procedure with all (pairs of) scatterers yields a set
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of linear equations, solution of which gives the coefficients of the scattered
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field in the VSWF bases.
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However,
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Once these coefficients have been found, one can evaluate various quantities
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related to the scattering (such as cross sections or the scattered fields)
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quite easily.
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\end_layout
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\begin_layout Standard
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However, the expressions appearing in the re-expansions are fairly complicated,
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and the implementation of MSTMM is extremely error-prone also due to the
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various conventions used in the literature.
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Therefore although we do not re-derive from scratch the expressions that
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can be found elsewhere in literature, we always state them explicitly in
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our convention.
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\end_layout
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\begin_layout Subsection
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@ -185,12 +197,40 @@ ity
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, and that the whole system is linear, i.e.
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the material properties of neither the medium nor the scatterer depend
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on field intensities.
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Under these assumptions, the EM fields in
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Under these assumptions, the EM fields
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\begin_inset Formula $\vect{\Psi}=\vect E,\vect H$
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\end_inset
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in
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\begin_inset Formula $\medium$
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\end_inset
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must satisfy the homogeneous vector Helmholtz equation
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\begin_inset Formula $\left(\nabla^{2}+k^{2}\right)\Psi\left(\vect r,\vect{\omega}\right)=0$
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must satisfy the homogeneous vector Helmholtz equation together with the
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transversality condition
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\begin_inset Formula
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\begin{equation}
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\left(\nabla^{2}+k^{2}\right)\Psi\left(\vect r,\vect{\omega}\right)=0,\quad\nabla\cdot\vect{\Psi}\left(\vect r,\vect{\omega}\right)=0\label{eq:Helmholtz eq}
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\end{equation}
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\end_inset
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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frequency-space Maxwell's equations
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\begin_inset Formula
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\begin{align*}
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\nabla\times\vect E\left(\vect r,\omega\right)-ik\eta_{0}\eta\vect H\left(\vect r,\omega\right) & =0,\\
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\eta_{0}\eta\nabla\times\vect H\left(\vect r,\omega\right)+ik\vect E\left(\vect r,\omega\right) & =0.
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\end{align*}
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\end_inset
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\end_layout
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\end_inset
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@ -208,29 +248,165 @@ todo define
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\end_inset
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with
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\begin_inset Formula $k=TODO$
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\begin_inset Formula $k=k\left(\omega\right)=\omega\sqrt{\mubg(\omega)\epsbg(\omega)}/c_{0}$
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\end_inset
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[TODO REF Jackson?].
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Its solutions (TODO under which conditions? What vector space do the SVWFs
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actually span? Check Comment 9.2 and Appendix f.9.1 in Kristensson)
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, as can be derived from the Maxwell's equations [REF Jackson?].
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\end_layout
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\begin_layout Subsubsection
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Spherical waves
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\end_layout
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\begin_layout Standard
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Throughout this text, we will use the same normalisation conventions as
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in
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Equation
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:Helmholtz eq"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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can be solved by separation of variables in spherical coordinates to give
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the solutions – the
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\emph on
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regular
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\emph default
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and
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\emph on
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outgoing
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\emph default
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vector spherical wavefunctions (VSWFs)
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\begin_inset Formula $\vswfrtlm{\tau}lm\left(k\vect r\right)$
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\end_inset
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and
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\begin_inset Formula $\vswfouttlm{\tau}lm\left(k\vect r\right)$
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\end_inset
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, respectively, defined as follows:
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\begin_inset Formula
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\begin{align*}
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\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\
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\vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),
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\end{align*}
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\end_inset
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\begin_inset Formula
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\begin{align*}
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\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\
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\vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\\
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& \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,
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\end{align*}
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\end_inset
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where
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\begin_inset Formula $\vect r=r\uvec r$
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\end_inset
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,
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\begin_inset Formula $j_{l}\left(x\right),h_{l}^{\left(1\right)}\left(x\right)=j_{l}\left(x\right)+iy_{l}\left(x\right)$
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\end_inset
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are the regular spherical Bessel function and spherical Hankel function
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of the first kind, respectively, as in [DLMF §10.47], and
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\begin_inset Formula $\vsh{\tau}lm$
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\end_inset
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are the
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\emph on
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vector spherical harmonics
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\emph default
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\begin_inset Formula
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\begin{align*}
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\vsh 1lm & =\\
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\vsh 2lm & =\\
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\vsh 3lm & =
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\end{align*}
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\end_inset
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In our convention, the (scalar) spherical harmonics
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\begin_inset Formula $\ush lm$
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\end_inset
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are identical to those in [DLMF 14.30.1], i.e.
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\begin_inset Formula
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\[
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\ush lm=\left(\frac{\left(l-m\right)!\left(2l+1\right)}{4\pi\left(l+m\right)!}\right)^{\frac{1}{2}}e^{im\phi}\dlmfFer lm\left(\cos\theta\right)
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\]
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\end_inset
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where importantly, the Ferrers functions
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\begin_inset Formula $\dlmfFer lm$
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\end_inset
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defined as in [DLMF §14.3(i)] do already contain the Condon-Shortley phase
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\begin_inset Formula $\left(-1\right)^{m}$
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\end_inset
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.
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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TODO názornější definice.
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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The convention for VSWFs used here is the same as in [Kristensson 2014];
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over other conventions used elsewhere in literature, it has several fundamental
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advantages – most importantly, the translation operators introduced later
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in eq.
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:translation op def"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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are unitary, and it gives the simplest possible expressions for power transport
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and cross sections without additional
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\begin_inset Formula $l,m$
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\end_inset
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-dependent factors (for that reason, we also call our VSWFs as
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\emph on
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power-normalised
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\emph default
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).
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Power-normalisation and unitary translation operators are possible to achieve
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also with real spherical harmonics – such a convention is used in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "kristensson_scattering_2016"
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literal "true"
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literal "false"
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\end_inset
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.
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\end_layout
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\begin_layout Subsubsection
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Spherical waves
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\begin_layout Standard
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Its solutions (TODO under which conditions? What vector space do the SVWFs
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actually span? Check Comment 9.2 and Appendix f.9.1 in Kristensson)
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\end_layout
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\begin_layout Standard
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