qpms/notes/cylinderT.lyx

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\begin_body

\begin_layout Standard
\begin_inset CommandInset include
LatexCommand input
filename "makrot.lyx"
literal "false"

\end_inset


\end_layout

\begin_layout Title
T-matrix of an axially symmetric particle
\end_layout

\begin_layout Standard
So we have 
\begin_inset CommandInset citation
LatexCommand cite
after "(9.12)"
key "kristensson_scattering_2016"
literal "false"

\end_inset

:
\begin_inset Formula 
\begin{align*}
R_{nn'} & =ik^{2}\iint_{S_{s}}\left(\frac{\eta}{\eta_{1}}\wfkcreg_{n}\left(k\vect r\right)\times\wfkcreg_{\overline{n'}}\left(k_{1}\vect r\right)+\wfkcreg_{\overline{n}}\left(k\vect r\right)\times\wfkcreg_{n'}\left(k_{1}\vect r\right)\right)\cdot\uvec{\nu}\,\ud S,\\
Q_{nn'} & =ik^{2}\iint_{S_{s}}\left(\frac{\eta}{\eta_{1}}\wfkcout_{n}\left(k\vect r\right)\times\wfkcreg_{\overline{n'}}\left(k_{1}\vect r\right)+\wfkcout_{\overline{n}}\left(k\vect r\right)\times\wfkcreg_{n'}\left(k_{1}\vect r\right)\right)\cdot\uvec{\nu}\,\ud S,
\end{align*}

\end_inset

where 
\begin_inset Formula $S_{s}$
\end_inset

 is the scatterer surface, 
\begin_inset Formula $\uvec{\nu}$
\end_inset

 is the outwards pointing unit normal to it, and the subscript
\begin_inset Formula $_{1}$
\end_inset

 refers to the particle inside; then
\begin_inset Formula 
\begin{equation}
T_{nn'}=-\sum_{n''}R_{nn''}Q_{n''n}^{-1}.\label{eq:T matrix from R and Q}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Let us consider the case with full rotational symmetry around the 
\begin_inset Formula $z$
\end_inset

 axis and parametrise the integral in terms of polar angle 
\begin_inset Formula $\theta$
\end_inset

.
 Let 
\begin_inset Formula $\beta$
\end_inset

 be the angle between the surface normal 
\begin_inset Formula $\uvec{\nu}$
\end_inset

 and the coordinate radial direction 
\begin_inset Formula $\uvec r$
\end_inset

.
 The infinitesimal surface area element is then
\begin_inset Formula 
\[
\ud S\left(\theta\right)=\frac{\left(r\left(\theta\right)\right)^{2}\sin\theta}{\cos\beta\left(\theta\right)}\ud\theta\,\ud\phi
\]

\end_inset

and the surface normal in local coordinates 
\begin_inset Formula 
\[
\uvec{\nu}\left(\theta\right)=\uvec r\cos\beta\left(\theta\right)+\uvec{\theta}\sin\beta\left(\theta\right),
\]

\end_inset

which also sets a convention for the sign of 
\begin_inset Formula $\beta$
\end_inset

.
\end_layout

\begin_layout Standard
For fully axially symmetric particles the integrals vanish for 
\begin_inset Formula $m\ne-m'$
\end_inset

 due to the 
\begin_inset Formula $e^{i\left(m+m'\right)}$
\end_inset

 asimuthal factor in the integrand.
 One then has 
\begin_inset Formula 
\begin{equation}
T_{nn'}=-\sum_{n''}R'_{nn''}Q'_{n''n}^{-1}\label{eq:T-matrix from reduced R and Q}
\end{equation}

\end_inset

where 
\begin_inset Formula 
\begin{align*}
R'_{nn'} & =\int_{0}^{\pi}\left(\frac{\eta}{\eta_{1}}\wfkcreg_{n}\left(k\vect r\right)\times\wfkcreg_{\overline{n'}}\left(k_{1}\vect r\right)+\wfkcreg_{\overline{n}}\left(k\vect r\right)\times\wfkcreg_{n'}\left(k_{1}\vect r\right)\right)\cdot\left(\uvec r\cos\beta\left(\theta\right)+\uvec{\theta}\sin\beta\left(\theta\right)\right)\frac{\left(r\left(\theta\right)\right)^{2}\sin\theta}{\cos\beta\left(\theta\right)}\ud\theta,\\
Q'_{nn'} & =\int_{0}^{\pi}\left(\frac{\eta}{\eta_{1}}\wfkcreg_{n}\left(k\vect r\right)\times\wfkcreg_{\overline{n'}}\left(k_{1}\vect r\right)+\wfkcreg_{\overline{n}}\left(k\vect r\right)\times\wfkcreg_{n'}\left(k_{1}\vect r\right)\right)\cdot\left(\uvec r\cos\beta\left(\theta\right)+\uvec{\theta}\sin\beta\left(\theta\right)\right)\frac{\left(r\left(\theta\right)\right)^{2}\sin\theta}{\cos\beta\left(\theta\right)}\ud\theta
\end{align*}

\end_inset

where 
\begin_inset Formula $\vect r=\vect r\left(\theta\right)=\left(r\left(\theta\right),\theta,0\right)$
\end_inset

.
 Matrices 
\begin_inset Formula $Q',R'$
\end_inset

 differ from the original 
\begin_inset Formula $R,Q$
\end_inset

 matrices in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:T matrix from R and Q"
plural "false"
caps "false"
noprefix "false"

\end_inset

 by a factor of 
\begin_inset Formula $2\pi ik^{2}$
\end_inset

, but this cancels out in the matrix product.
 
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\begin_inset Graphics
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	lyxscale 30
	width 50text%

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\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Parametrisation of cylindrical particle surface.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
For cylindrical particle of radius 
\begin_inset Formula $R$
\end_inset

 and height 
\begin_inset Formula $h$
\end_inset

, we can divide the parametrisation into three intervals 
\begin_inset Formula $\left(0,\theta_{1}\right),\left(\theta_{1},\theta_{2}\right),\left(\theta_{2},\pi\right)$
\end_inset

 where 
\begin_inset Formula $\theta_{1}=\tan^{-1}\left(2R/h\right),\theta_{2}=\pi-\tan^{-1}\left(2R/h\right)$
\end_inset

:
\end_layout

\begin_layout Enumerate
In the first section, 
\begin_inset Formula $0<\theta<\theta_{1}$
\end_inset

,
\begin_inset Formula 
\begin{align*}
r & =\frac{h}{2\cos\theta},\\
\beta & =-\theta.
\end{align*}

\end_inset


\end_layout

\begin_layout Enumerate
In the second section, 
\begin_inset Formula $\theta_{1}<\theta<\theta_{2}$
\end_inset

,
\begin_inset Formula 
\begin{align*}
r & =\frac{R}{\cos\left(\theta-\pi/2\right)}=\frac{R}{\sin\theta},\\
\beta & =-\theta+\pi/2.
\end{align*}

\end_inset


\end_layout

\begin_layout Enumerate
In the third section, 
\begin_inset Formula $\theta_{2}<\theta<\pi$
\end_inset

,
\begin_inset Formula 
\begin{align*}
r & =\frac{h}{2\cos\left(\theta-\pi\right)}=-\frac{h}{2\cos\theta},\\
\beta & =-\theta+\pi.
\end{align*}

\end_inset


\end_layout

\begin_layout Standard
Let's write VSWFs in terms of the power-normalised 
\begin_inset Formula $p,\pi,\tau$
\end_inset

 funs:
\begin_inset Formula 
\begin{align*}
\vsh_{1lm} & =\left(\uvec{\theta}\pi_{lm}-\uvec{\phi}\tau_{lm}\right)e^{im\phi}\\
\vsh_{2lm} & =\left(\uvec{\theta}\tau_{lm}+\uvec{\phi}\pi_{lm}\right)e^{im\phi}\\
\vsh_{3lm} & =\sqrt{l\left(l+1\right)}p_{lm}e^{im\theta}
\end{align*}

\end_inset


\begin_inset Formula 
\begin{align*}
\vect y_{\kappa1lm} & =\underbrace{h_{l}^{\kappa}e^{im\phi}}_{c_{\kappa lm}^{1}}\left(\uvec{\theta}\pi_{lm}-\uvec{\phi}\tau_{lm}\right)\\
\vect y_{\kappa2lm} & =\frac{1}{kr}e^{im\phi}\left(\frac{\ud\left(krh_{l}^{\kappa}\right)}{\ud\left(kr\right)}\left(\uvec{\theta}\tau_{lm}+\uvec{\phi}\pi_{lm}\right)+h_{l}^{\kappa}l\left(l+1\right)\uvec rp_{lm}\right)\\
 & =c_{\kappa lm}^{2}\left(\uvec{\theta}\tau_{lm}+\uvec{\phi}\pi_{lm}\right)+c_{\kappa lm}^{3}\uvec rp_{lm}
\end{align*}

\end_inset

The triple products than are (reminder: 
\begin_inset Formula $\uvec{\nu}\left(\theta\right)=\uvec r\cos\beta\left(\theta\right)+\uvec{\theta}\sin\beta\left(\theta\right))$
\end_inset

:
\begin_inset Formula 
\begin{align*}
\left(\vect y_{\kappa1lm}\times\vect v_{1l'm'}\right)\cdot\uvec{\nu} & =\cos\beta c_{\kappa lm}^{1}c_{\mathrm{R}l'm'}^{1}\left(-\pi_{lm}\tau_{l'm'}+\tau_{lm}\pi_{l'm'}\right)\\
\left(\vect y_{\kappa1lm}\times\vect v_{2l'm'}\right)\cdot\uvec{\nu} & =\cos\beta c_{\kappa lm}^{1}c_{\mathrm{R}l'm'}^{2}\left(\pi_{lm}\pi_{l'm'}+\tau_{lm}\tau_{l'm'}\right)\\
 & +\sin\beta c_{\kappa lm}^{1}c_{\mathrm{R}l'm'}^{3}\left(-\tau_{lm}p_{l'm'}\right)\\
\left(\vect y_{\kappa2lm}\times\vect v_{1l'm'}\right)\cdot\uvec{\nu} & =\cos\beta c_{\kappa lm}^{2}c_{\mathrm{R}l'm'}^{1}\left(-\pi_{lm}\pi_{l'm'}-\tau_{lm}\tau_{l'm'}\right)\\
 & +\sin\beta c_{\kappa lm}^{3}c_{\mathrm{R}l'm'}^{1}\left(p_{lm}\tau_{l'm'}\right)\\
\left(\vect y_{\kappa2lm}\times\vect v_{2l'm'}\right)\cdot\uvec{\nu} & =\cos\beta c_{\kappa lm}^{2}c_{\mathrm{R}l'm'}^{2}\left(\tau_{lm}\pi_{l'm'}-\pi_{lm}\tau_{l'm'}\right)\\
 & -\sin\beta c_{\kappa lm}^{3}c_{\mathrm{R}l'm'}^{2}p_{lm}\pi_{l'm'}\\
 & +\sin\beta c_{\kappa lm}^{2}c_{\mathrm{R}l'm'}^{3}\pi_{lm}p_{l'm'}
\end{align*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "Electrodynamics"
options "plain"
encoding "default"

\end_inset


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\end_body
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