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Fix plane wave expansion coeffs. 

Former-commit-id: ae5b827ed3b9c7646d1f98d96a784659fb460129
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Marek Nečada 2019-10-13 17:54:48 +03:00
parent 04ff84b5c9
commit f62ce5f700
1 changed files with 22 additions and 22 deletions
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lepaper/finite.lyx
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@ -135,7 +135,7 @@ Single-particle scattering
In order to define the basic concepts, let us first consider the case of In order to define the basic concepts, let us first consider the case of
electromagnetic (EM) radiation scattered by a single particle. electromagnetic (EM) radiation scattered by a single particle.
We assume that the scatterer lies inside a closed ball We assume that the scatterer lies inside a closed ball
\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ \begin_inset Formula $\closedball{R^{<}}{\vect0}$
\end_inset \end_inset
of radius of radius
@ -144,16 +144,16 @@ In order to define the basic concepts, let us first consider the case of
and center in the origin of the coordinate system (which can be chosen and center in the origin of the coordinate system (which can be chosen
that way; the natural choice of that way; the natural choice of
\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ \begin_inset Formula $\closedball{R^{<}}{\vect0}$
\end_inset \end_inset
is the circumscribed ball of the scatterer) and that there exists a larger is the circumscribed ball of the scatterer) and that there exists a larger
open cocentric ball open cocentric ball
\begin_inset Formula $\openball{R^{>}}{\vect 0}$ \begin_inset Formula $\openball{R^{>}}{\vect0}$
\end_inset \end_inset
, such that the (non-empty) spherical shell , such that the (non-empty) spherical shell
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}=\openball{R^{>}}{\vect 0}\setminus\closedball{R^{<}}{\vect 0}$ \begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}=\openball{R^{>}}{\vect0}\setminus\closedball{R^{<}}{\vect0}$
\end_inset \end_inset
is filled with a homogeneous isotropic medium with relative electric permittivi is filled with a homogeneous isotropic medium with relative electric permittivi
@ -173,7 +173,7 @@ ty
\end_inset \end_inset
in in
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$ \begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$
\end_inset \end_inset
must satisfy the homogeneous vector Helmholtz equation together with the must satisfy the homogeneous vector Helmholtz equation together with the
@ -278,8 +278,8 @@ outgoing
, respectively, defined as follows: , respectively, defined as follows:
\begin_inset Formula \begin_inset Formula
\begin{align} \begin{align}
\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(\kappa r\right)\vsh 1lm\left(\uvec r\right),\nonumber \\ \vswfrtlm1lm\left(k\vect r\right) & =j_{l}\left(\kappa r\right)\vsh1lm\left(\uvec r\right),\nonumber \\
\vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{\kappa r}\frac{\ud\left(\kappa rj_{l}\left(\kappa r\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(\kappa r\right)}{kr}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF regular} \vswfrtlm2lm\left(k\vect r\right) & =\frac{1}{\kappa r}\frac{\ud\left(\kappa rj_{l}\left(\kappa r\right)\right)}{\ud\left(kr\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(\kappa r\right)}{kr}\vsh3lm\left(\uvec r\right),\label{eq:VSWF regular}
\end{align} \end{align}
\end_inset \end_inset
@ -287,8 +287,8 @@ outgoing
\begin_inset Formula \begin_inset Formula
\begin{align} \begin{align}
\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\nonumber \\ \vswfouttlm1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh1lm\left(\uvec r\right),\nonumber \\
\vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(\kappa r\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(\kappa r\right)}{\kappa r}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF outgoing}\\ \vswfouttlm2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(\kappa r\right)\right)}{\ud\left(kr\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(\kappa r\right)}{\kappa r}\vsh3lm\left(\uvec r\right),\label{eq:VSWF outgoing}\\
& \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,\nonumber & \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,\nonumber
\end{align} \end{align}
@ -323,9 +323,9 @@ vector spherical harmonics
\begin_inset Formula \begin_inset Formula
\begin{align} \begin{align}
\vsh 1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\nonumber \\ \vsh1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\nonumber \\
\vsh 2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\nonumber \\ \vsh2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\nonumber \\
\vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition} \vsh3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition}
\end{align} \end{align}
\end_inset \end_inset
@ -473,7 +473,7 @@ noprefix "false"
\end_inset \end_inset
inside a ball inside a ball
\begin_inset Formula $\openball{R^{>}}{\vect 0}$ \begin_inset Formula $\openball{R^{>}}{\vect0}$
\end_inset \end_inset
with radius with radius
@ -483,7 +483,7 @@ noprefix "false"
and center in the origin, were it filled with homogeneous isotropic medium; and center in the origin, were it filled with homogeneous isotropic medium;
however, if the equation is not guaranteed to hold inside a smaller ball however, if the equation is not guaranteed to hold inside a smaller ball
\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ \begin_inset Formula $\closedball{R^{<}}{\vect0}$
\end_inset \end_inset
around the origin (typically due to presence of a scatterer), one has to around the origin (typically due to presence of a scatterer), one has to
@ -492,7 +492,7 @@ noprefix "false"
\end_inset \end_inset
to have a complete basis of the solutions in the volume to have a complete basis of the solutions in the volume
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}=\openball{R^{>}}{\vect 0}\setminus\closedball{R^{<}}{\vect 0}$ \begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}=\openball{R^{>}}{\vect0}\setminus\closedball{R^{<}}{\vect0}$
\end_inset \end_inset
. .
@ -514,11 +514,11 @@ The single-particle scattering problem at frequency
\end_inset \end_inset
can be posed as follows: Let a scatterer be enclosed inside the ball can be posed as follows: Let a scatterer be enclosed inside the ball
\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ \begin_inset Formula $\closedball{R^{<}}{\vect0}$
\end_inset \end_inset
and let the whole volume and let the whole volume
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$ \begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$
\end_inset \end_inset
be filled with a homogeneous isotropic medium with wave number be filled with a homogeneous isotropic medium with wave number
@ -527,7 +527,7 @@ The single-particle scattering problem at frequency
. .
Inside Inside
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$ \begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$
\end_inset \end_inset
, the electric field can be expanded as , the electric field can be expanded as
@ -549,7 +549,7 @@ doplnit frekvence a polohy
\end_inset \end_inset
If there were no scatterer and If there were no scatterer and
\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ \begin_inset Formula $\closedball{R^{<}}{\vect0}$
\end_inset \end_inset
were filled with the same homogeneous medium, the part with the outgoing were filled with the same homogeneous medium, the part with the outgoing
@ -558,7 +558,7 @@ If there were no scatterer and
\end_inset \end_inset
due to sources outside due to sources outside
\begin_inset Formula $\openball{R^{>}}{\vect 0}$ \begin_inset Formula $\openball{R^{>}}{\vect0}$
\end_inset \end_inset
would remain. would remain.
@ -1032,8 +1032,8 @@ literal "false"
with expansion coefficients with expansion coefficients
\begin_inset Formula \begin_inset Formula
\begin{eqnarray} \begin{eqnarray}
\rcoeffptlm{}1lm\left(\uvec k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right),\nonumber \\ \rcoeffptlm{}1lm\left(\uvec k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right)\cdot\vect E_{0},\nonumber \\
\rcoeffptlm{}2lm\left(\uvec k,\vect E_{0}\right) & = & -4\pi i^{l+1}\vshD 2lm\left(\uvec k\right).\label{eq:plane wave expansion} \rcoeffptlm{}2lm\left(\uvec k,\vect E_{0}\right) & = & -4\pi i^{l+1}\vshD 2lm\left(\uvec k\right)\cdot\vect E_{0}.\label{eq:plane wave expansion}
\end{eqnarray} \end{eqnarray}
\end_inset \end_inset