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