From f62ce5f7001319d381d1861887e0ff8ea075fe1d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Sun, 13 Oct 2019 17:54:48 +0300 Subject: [PATCH] Fix plane wave expansion coeffs. Former-commit-id: ae5b827ed3b9c7646d1f98d96a784659fb460129 --- lepaper/finite.lyx | 44 ++++++++++++++++++++++---------------------- 1 file changed, 22 insertions(+), 22 deletions(-) diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index 4aac440..15c97c2 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -135,7 +135,7 @@ Single-particle scattering In order to define the basic concepts, let us first consider the case of electromagnetic (EM) radiation scattered by a single particle. 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 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 that way; the natural choice of -\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ +\begin_inset Formula $\closedball{R^{<}}{\vect0}$ \end_inset is the circumscribed ball of the scatterer) and that there exists a larger open cocentric ball -\begin_inset Formula $\openball{R^{>}}{\vect 0}$ +\begin_inset Formula $\openball{R^{>}}{\vect0}$ \end_inset , 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 is filled with a homogeneous isotropic medium with relative electric permittivi @@ -173,7 +173,7 @@ ty \end_inset in -\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$ +\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$ \end_inset must satisfy the homogeneous vector Helmholtz equation together with the @@ -278,8 +278,8 @@ outgoing , respectively, defined as follows: \begin_inset Formula \begin{align} -\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(\kappa r\right)\vsh 1lm\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} +\vswfrtlm1lm\left(k\vect r\right) & =j_{l}\left(\kappa r\right)\vsh1lm\left(\uvec r\right),\nonumber \\ +\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_inset @@ -287,8 +287,8 @@ outgoing \begin_inset Formula \begin{align} -\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\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}\\ +\vswfouttlm1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh1lm\left(\uvec r\right),\nonumber \\ +\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 \end{align} @@ -323,9 +323,9 @@ vector spherical harmonics \begin_inset Formula \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 \\ -\vsh 2lm\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} +\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 \\ +\vsh2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\nonumber \\ +\vsh3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition} \end{align} \end_inset @@ -473,7 +473,7 @@ noprefix "false" \end_inset inside a ball -\begin_inset Formula $\openball{R^{>}}{\vect 0}$ +\begin_inset Formula $\openball{R^{>}}{\vect0}$ \end_inset with radius @@ -483,7 +483,7 @@ noprefix "false" 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 -\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ +\begin_inset Formula $\closedball{R^{<}}{\vect0}$ \end_inset around the origin (typically due to presence of a scatterer), one has to @@ -492,7 +492,7 @@ noprefix "false" \end_inset 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 . @@ -514,11 +514,11 @@ The single-particle scattering problem at frequency \end_inset 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 and let the whole volume -\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$ +\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$ \end_inset be filled with a homogeneous isotropic medium with wave number @@ -527,7 +527,7 @@ The single-particle scattering problem at frequency . Inside -\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$ +\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$ \end_inset , the electric field can be expanded as @@ -549,7 +549,7 @@ doplnit frekvence a polohy \end_inset If there were no scatterer and -\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ +\begin_inset Formula $\closedball{R^{<}}{\vect0}$ \end_inset were filled with the same homogeneous medium, the part with the outgoing @@ -558,7 +558,7 @@ If there were no scatterer and \end_inset due to sources outside -\begin_inset Formula $\openball{R^{>}}{\vect 0}$ +\begin_inset Formula $\openball{R^{>}}{\vect0}$ \end_inset would remain. @@ -1032,8 +1032,8 @@ literal "false" with expansion coefficients \begin_inset Formula \begin{eqnarray} -\rcoeffptlm{}1lm\left(\uvec k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right),\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{}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)\cdot\vect E_{0}.\label{eq:plane wave expansion} \end{eqnarray} \end_inset