qpms/Scattering and Shit.lyx

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\pdf_title "Sähköpajan päiväkirja"
\pdf_author "Marek Nečada"
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\begin_body

\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\vect}[1]{\mathbf{#1}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\ud}{\mathrm{d}}
\end_inset


\end_layout

\begin_layout Title
Electromagnetic multiple scattering, spherical waves and ****
\end_layout

\begin_layout Author
Marek Nečada
\end_layout

\begin_layout Chapter
Zillion conventions for spherical vector waves
\end_layout

\begin_layout Section
Legendre polynomials and spherical harmonics: messy from the very beginning
\end_layout

\begin_layout Subsection
Kristensson 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
P_{l}^{-m}=\left(-1\right)^{m}\frac{\left(l-m\right)!}{\left(l+m\right)!}P_{l}^{m}\left(\cos\theta\right),\quad m\ge0
\]

\end_inset

Kristensson uses the Condon-Shortley phase, so (sect.
 [K]D.2)
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Y_{lm}\left(\hat{\vect r}\right)=\left(-1\right)^{m}\sqrt{\frac{2l+1}{4\pi}\frac{\left(l-m\right)!}{\left(l+m\right)!}}P_{l}^{m}\left(\cos\theta\right)e^{im\phi}
\]

\end_inset


\begin_inset Formula 
\[
Y_{lm}^{\dagger}\left(\hat{\vect r}\right)=Y_{lm}^{*}\left(\hat{\vect r}\right)
\]

\end_inset


\begin_inset Formula 
\[
Y_{l,-m}\left(\hat{\vect r}\right)=\left(-1\right)^{m}Y_{lm}^{\dagger}\left(\hat{\vect r}\right)
\]

\end_inset


\end_layout

\begin_layout Standard
Orthonormality:
\begin_inset Formula 
\[
\int Y_{lm}\left(\hat{\vect r}\right)Y_{l'm'}^{\dagger}\left(\hat{\vect r}\right)\,\ud\Omega=\delta_{ll'}\delta_{mm'}
\]

\end_inset


\end_layout

\begin_layout Section
Pi and tau
\end_layout

\begin_layout Subsection
Taylor
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
\tilde{\pi}_{mn}\left(\cos\theta\right) & = & \sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}\frac{m}{\sin\theta}P_{n}^{m}\left(\cos\theta\right)\\
\tilde{\tau}_{mn}\left(\cos\theta\right) & = & \sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}\frac{\ud}{\ud\theta}P_{n}^{m}\left(\cos\theta\right)
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Vector spherical harmonics (?)
\end_layout

\begin_layout Subsection
Kristensson
\end_layout

\begin_layout Standard
Original formulation, sect.
 [K]D.3.3
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
\vect A_{1lm}\left(\hat{\vect r}\right) & = & \frac{1}{\sqrt{l\left(l+1\right)}}\left(\hat{\vect{\theta}}\frac{1}{\sin\theta}\frac{\partial}{\partial\phi}Y_{lm}\left(\hat{\vect r}\right)-\hat{\vect{\phi}}\frac{\partial}{\partial\theta}Y_{lm}\left(\hat{\vect r}\right)\right)\\
\vect A_{2lm}\left(\hat{\vect r}\right) & = & \frac{1}{\sqrt{l\left(l+1\right)}}\left(\hat{\vect{\theta}}\frac{\partial}{\partial\phi}Y_{lm}\left(\hat{\vect r}\right)-\hat{\vect{\phi}}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}Y_{lm}\left(\hat{\vect r}\right)\right)\\
\vect A_{3lm}\left(\hat{\vect r}\right) & = & \hat{\vect r}Y_{lm}\left(\hat{\vect r}\right)
\end{eqnarray*}

\end_inset

Normalisation:
\begin_inset Formula 
\[
\int\vect A_{n}\left(\hat{\vect r}\right)\cdot\vect A_{n'}^{\dagger}\left(\hat{\vect r}\right)\,\ud\Omega=\delta_{nn'}
\]

\end_inset

Here 
\begin_inset Formula $\mbox{ }^{\dagger}$
\end_inset

 means just complex conjugate, apparently (see footnote on p.
 89).
\end_layout

\begin_layout Section
Spherical Bessel functions
\begin_inset CommandInset label
LatexCommand label
name "sec:Spherical-Bessel-functions"

\end_inset


\end_layout

\begin_layout Standard
The radial dependence of spherical vector waves is given by the spherical
 Bessel functions and their first derivatives.
 Commonly, the following notation is adopted
\begin_inset Formula 
\begin{eqnarray*}
z_{n}^{(1)}(x) & = & j_{n}(x),\\
z_{n}^{(2)}(x) & = & y_{n}(x),\\
z_{n}^{(3)}(x) & = & h_{n}^{(1)}(x)=j_{n}(x)+iy_{n}(x),\\
z_{n}^{(4)}(x) & = & h_{n}^{(2)}(x)=j_{n}(x)-iy_{n}(x).
\end{eqnarray*}

\end_inset

Here, 
\begin_inset Formula $j_{n}$
\end_inset

 is the spherical Bessel function of first kind (regular), 
\begin_inset Formula $y_{j}$
\end_inset

 is the spherical Bessel function of second kind (singular), and 
\begin_inset Formula $h_{n}^{(1)},h_{n}^{(2)}$
\end_inset

 are the Hankel functions a.k.a.
 spherical Bessel functions of third kind.
 In spherical vector waves, 
\begin_inset Formula $j_{n}$
\end_inset

 corresponds to regular waves, 
\begin_inset Formula $h^{(1)}$
\end_inset

 corresponds (by the usual convention) to outgoing waves, and 
\begin_inset Formula $h^{(2)}$
\end_inset

 corresponds to incoming waves.
 To describe scattering, we need two sets of waves with two different types
 of spherical Bessel functions 
\begin_inset Formula $z_{n}^{(J)}$
\end_inset

.
 Most common choice is 
\begin_inset Formula $J=1,3$
\end_inset

, because if we decompose the field into spherical waves centered at 
\begin_inset Formula $\vect r_{0}$
\end_inset

, the field produced by other sources (e.g.
 spherical waves from other scatterers or a plane wave) is always regular
 at 
\begin_inset Formula $\vect r_{0}$
\end_inset

.
 Second choice which makes a bit of sense is 
\begin_inset Formula $J=3,4$
\end_inset

 as it leads to a nice expression for the energy transport.
\end_layout

\begin_layout Section
Spherical vector waves
\end_layout

\begin_layout Standard
TODO 
\begin_inset Formula $M,N,\psi,\chi,\widetilde{M},\widetilde{N},u,v,w,\dots$
\end_inset

, sine/cosine convention (B&H), ...
\end_layout

\begin_layout Standard
There are two mutually orthogonal types of divergence-free (everywhere except
 in the origin for singular waves) spherical vector waves, which I call
 electric and magnetic, given by the type of multipole source to which they
 correspond.
 This is another distinction than the regular/singular/ingoing/outgoing
 waves given by the type of the radial dependence (cf.
 section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Spherical-Bessel-functions"

\end_inset

).
 Oscillating electric current in a tiny rod parallel to its axis will generate
 electric dipole waves (net dipole moment of magnetic current is zero) moment
 , whereas oscillating electric current in a tiny circular loop will generate
 magnetic dipole waves (net dipole moment of electric current is zero).
\end_layout

\begin_layout Standard
In the usual cases we encounter, the part described by the magnetic waves
 is pretty small.
\end_layout

\begin_layout Subsection
Taylor
\end_layout

\begin_layout Standard
Definition [T](2.40); 
\begin_inset Formula $\widetilde{\vect N}_{mn}^{(j)},\widetilde{\vect M}_{mn}^{(j)}$
\end_inset

 are the electric and magnetic waves, respectively:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
\widetilde{\vect N}_{mn}^{(j)} & = & \frac{n(n+1)}{kr}\sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}P_{n}^{m}\left(\cos\theta\right)e^{im\phi}z_{n}^{j}\left(kr\right)\hat{\vect r}\\
 &  & +\left[\tilde{\tau}_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}+i\tilde{\pi}_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}z_{n}^{j}\left(kr\right)\\
\widetilde{\vect M}_{mn}^{(j)} & = & \left[i\tilde{\pi}_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}-\tilde{\tau}_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}z_{n}^{j}\left(kr\right)
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Subsection
Kristensson
\end_layout

\begin_layout Standard
Definition [K](2.4.6); 
\begin_inset Formula $\vect u_{\tau lm},\vect v_{\tau lm},\vect w_{\tau lm}$
\end_inset

 are the waves with 
\begin_inset Formula $j=3,1,4$
\end_inset

 respectively, i.e.
 outgoing, regular and incoming waves.
 The first index distinguishes between the electric (
\begin_inset Formula $\tau=2$
\end_inset

) and magnetic (
\begin_inset Formula $\tau=1$
\end_inset

).
 Kristensson uses a multiindex 
\begin_inset Formula $n\equiv(\tau,l,m)$
\end_inset

 to simlify the notation.
\begin_inset Formula 
\begin{eqnarray*}
\left(\vect{u/v/w}\right)_{2lm} & = & \frac{1}{kr}\frac{\ud\left(kr\, z_{l}^{(j)}\left(kr\right)\right)}{\ud\, kr}\vect A_{2lm}\left(\hat{\vect r}\right)+\sqrt{l\left(l+1\right)}\frac{z_{l}^{(j)}(kr)}{kr}\vect A_{3lm}\left(\hat{\vect r}\right)\\
\left(\vect{u/v/w}\right)_{1lm} & = & z_{l}^{(j)}\left(kr\right)\vect A_{1lm}\left(\hat{\vect r}\right)
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Subsection
Relation between Kristensson and Taylor
\begin_inset CommandInset label
LatexCommand label
name "sub:Kristensson-v-Taylor"

\end_inset


\end_layout

\begin_layout Standard
Kristensson's and Taylor's VSWFs seem to differ only by an 
\begin_inset Formula $l$
\end_inset

-dependent normalization factor, and notation of course (n.b.
 the inverse index order)
\begin_inset Formula 
\begin{eqnarray*}
\left(\vect{u/v/w}\right)_{2lm} & = & \frac{\widetilde{\vect N}_{ml}^{(3/1/4)}}{\sqrt{l\left(l+1\right)}}\\
\left(\vect{u/v/w}\right)_{1lm} & = & \frac{\widetilde{\vect M}_{ml}^{(3/1/4)}}{\sqrt{l\left(l+1\right)}}
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Plane wave expansion
\end_layout

\begin_layout Subsection
Taylor
\end_layout

\begin_layout Standard
\begin_inset Formula $x$
\end_inset

-polarised, 
\begin_inset Formula $z$
\end_inset

-propagating plane wave, 
\begin_inset Formula $\vect E=E_{0}\hat{\vect x}e^{i\vect k\cdot\hat{\vect z}}$
\end_inset

 (CHECK):
\begin_inset Formula 
\begin{eqnarray*}
\vect E & = & -i\left(p_{mn}\widetilde{\vect N}_{mn}^{(1)}+q_{mn}\widetilde{\vect M}_{mn}^{(1)}\right)\\
p_{mn} & = & E_{0}\frac{4\pi i^{n}}{n(n+1)}\tilde{\tau}_{mn}(1)\\
q_{mn} & = & E_{0}\frac{4\pi i^{n}}{n(n+1)}\tilde{\pi}_{mn}(1)
\end{eqnarray*}

\end_inset

while it can be shown that
\begin_inset Formula 
\begin{eqnarray*}
\tilde{\pi}_{mn}(1) & = & -\frac{1}{2}\sqrt{\frac{\left(2n+1\right)\left(n\left(n+1\right)\right)}{4\pi}}\left(\delta_{m,1}+\delta_{m,-1}\right)\\
\tilde{\tau}_{mn}(1) & = & -\frac{1}{2}\sqrt{\frac{\left(2n+1\right)\left(n\left(n+1\right)\right)}{4\pi}}\left(\delta_{m,1}-\delta_{m,-1}\right)
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Subsection
Kristensson
\end_layout

\begin_layout Standard
\begin_inset Formula $x$
\end_inset

-polarised, 
\begin_inset Formula $z$
\end_inset

-propagating plane wave, 
\begin_inset Formula $\vect E=E_{0}\hat{\vect x}e^{i\vect k\cdot\hat{\vect z}}$
\end_inset

 (CHECK, ):
\begin_inset Formula 
\[
\vect E=\sum_{n}a_{n}\vect v_{n}
\]

\end_inset


\begin_inset Formula 
\begin{eqnarray*}
a_{1lm} & = & E_{0}i^{l+1}\sqrt{\left(2l+1\right)\pi}\left(\delta_{m,1}+\delta_{m,-1}\right)\\
a_{2lm} & = & E_{0}i^{l+1}\sqrt{\left(2l+1\right)\pi}\left(\delta_{m,1}+\delta_{m,-1}\right)
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Section
Radiated energy
\end_layout

\begin_layout Standard
In this section I summarize the formulae for power 
\begin_inset Formula $P$
\end_inset

 radiated from the system.
 For an absorbing scatterer, this should be negative (n.b.
 sign conventions can be sometimes confusing).
 If the system is excited by a plane wave with intensity 
\begin_inset Formula $E_{0}$
\end_inset

, this can be used to calculate the absorption cross section,
\begin_inset Formula 
\[
\sigma_{\mathrm{abs}}=-\frac{P}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}}.
\]

\end_inset


\end_layout

\begin_layout Subsection
Kristensson
\begin_inset CommandInset label
LatexCommand label
name "sub:Radiated enenergy-Kristensson"

\end_inset


\end_layout

\begin_layout Standard
Sect.
 [K]2.6.2; here this form of expansion is assumed:
\begin_inset Formula 
\begin{equation}
\vect E\left(\vect r,\omega\right)=k\sqrt{\eta_{0}\eta}\sum_{n}\left(a_{n}\vect v_{n}\left(k\vect r\right)+f_{n}\vect u_{n}\left(k\vect r\right)\right).\label{eq:power-Kristensson-E}
\end{equation}

\end_inset

Here 
\begin_inset Formula $\eta_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}$
\end_inset

 is the wave impedance of free space and 
\begin_inset Formula $\eta=\sqrt{\mu/\varepsilon}$
\end_inset

 is the relative wave impedance of the medium.
 
\end_layout

\begin_layout Standard
The radiated power is then (2.28): 
\begin_inset Formula 
\[
P=\frac{1}{2}\sum_{n}\left(\left|f_{n}\right|^{2}+\Re\left(f_{n}a_{n}^{*}\right)\right)
\]

\end_inset


\end_layout

\begin_layout Subsection
Taylor
\end_layout

\begin_layout Standard
Here I derive the radiated power in Taylor's convention by applying the
 relations from subsection 
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Kristensson-v-Taylor"

\end_inset

 to the Kristensson's formulae (sect.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Radiated enenergy-Kristensson"

\end_inset

).
\end_layout

\begin_layout Standard
Assume the external field decomposed as (here I use tildes even for the
 expansion coefficients in order to avoid confusion with the 
\begin_inset Formula $a_{n}$
\end_inset

 in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:power-Kristensson-E"

\end_inset

) 
\begin_inset Formula 
\[
\vect E\left(\vect r,\omega\right)=\sum_{mn}\left[-i\left(\tilde{p}_{mn}\vect{\widetilde{N}}_{mn}^{(1)}+\tilde{q}_{mn}\widetilde{\vect M}_{mn}^{(1)}\right)+i\left(\tilde{a}_{mn}\widetilde{\vect N}_{mn}^{(3)}+\tilde{b}_{mn}\widetilde{\vect M}_{mn}^{(3)}\right)\right]
\]

\end_inset

(there is minus between the regular and outgoing part!).
 The coefficients are related to those from 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:power-Kristensson-E"

\end_inset

 as 
\begin_inset Formula 
\[
\tilde{p}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{-i\sqrt{n(n+1)}}a_{2nm},\quad\tilde{q}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{-i\sqrt{n(n+1)}}a_{1nm},
\]

\end_inset


\begin_inset Formula 
\[
\tilde{a}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{i\sqrt{n(n+1)}}f_{2nm},\quad\tilde{b}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{i\sqrt{n(n+1)}}f_{1nm}.
\]

\end_inset

The radiated power is then
\begin_inset Formula 
\[
P=\frac{1}{2}\sum_{m,n}\frac{n\left(n+1\right)}{k^{2}\eta_{0}}\left(\left|a_{mn}\right|^{2}+\left|b_{mn}\right|^{2}-\Re\left(a_{mn}p_{mn}^{*}\right)-\Re\left(b_{mn}q_{mn}^{*}\right)\right).
\]

\end_inset


\end_layout

\begin_layout Section
Limit solutions
\end_layout

\begin_layout Subsection
Far-field asymptotic solution
\end_layout

\begin_layout Standard
TODO start from 
\begin_inset CommandInset citation
LatexCommand cite
after "(A7)"
key "pustovit_plasmon-mediated_2010"

\end_inset


\end_layout

\begin_layout Subsection
Near field limit
\end_layout

\begin_layout Chapter
Mie Theory
\end_layout

\begin_layout Section
Full version
\end_layout

\begin_layout Section
Long wave approximation
\end_layout

\begin_layout Standard
TODO start from 
\begin_inset CommandInset citation
LatexCommand cite
after "(A11)"
key "pustovit_plasmon-mediated_2010"

\end_inset

 and around.
\end_layout

\begin_layout Chapter
Green's functions
\end_layout

\begin_layout Section
xyz pure free-space dipole waves in terms of SVWF
\end_layout

\begin_layout Section
Mie decomposition of Green's function for single nanoparticle
\end_layout

\begin_layout Chapter
Translation of spherical waves: getting insane
\end_layout

\begin_layout Chapter
Multiple scattering: nice linear algebra born from all the mess
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "dipdip"
options "plain"

\end_inset


\end_layout

\end_body
\end_document