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Poznámky z Xu 

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68
dipdip.bib
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@ -924,6 +924,40 @@ and 3.4 (p, 389+) too.},
file = {mishchenko2003.pdf:/home/necadam1/.zotero/zotero/9uf64zmd.default/zotero/storage/T27R8CQ6/mishchenko2003.pdf:application/pdf;Snapshot:/home/necadam1/.zotero/zotero/9uf64zmd.default/zotero/storage/W6XKVZJQ/10.html:text/html}
}
@article{xu_radiative_2003,
title = {Radiative scattering properties of an ensemble of variously shaped small particles},
volume = {67},
url = {http://link.aps.org/doi/10.1103/PhysRevE.67.046620},
doi = {10.1103/PhysRevE.67.046620},
abstract = {This paper presents a rigorous solution to the scattering of a monochromatic plane wave by an arbitrary configuration of wavelength-sized small particles that can be of different shape, structure, size, and composition. A T-matrix formulation is developed for the calculation of optical cross sections and the asymmetry parameter of such an ensemble of scatterers in both fixed and random orientations. The solution is based on the T matrix Tjl, that is, the inverse of the coefficient matrix of boundary condition equations. A linear system containing Tjl is derived to efficiently solve the T matrix, which is required in the practical implementation of the solution.},
number = {4},
urldate = {2015-11-22},
journal = {Physical Review E},
author = {Xu, Yu-lin},
month = apr,
year = {2003},
pages = {046620},
file = {APS Snapshot:/home/necadam1/.zotero/zotero/9uf64zmd.default/zotero/storage/8ABZEH74/PhysRevE.67.html:text/html;PhysRevE.67.046620.pdf:/home/necadam1/.zotero/zotero/9uf64zmd.default/zotero/storage/XP3JQJKU/PhysRevE.67.046620.pdf:application/pdf}
}
@article{xu_calculation_1996,
title = {Calculation of the {Addition} {Coefficients} in {Electromagnetic} {Multisphere}-{Scattering} {Theory}},
volume = {127},
issn = {0021-9991},
url = {http://www.sciencedirect.com/science/article/pii/S0021999196901758},
doi = {10.1006/jcph.1996.0175},
abstract = {One of the most intractable problems in electromagnetic multisphere-scattering theory is the formulation and evaluation of vector addition coefficients introduced by the addition theorems for vector spherical harmonics. This paper presents an efficient approach for the calculation of both scalar and vector translational addition coefficients, which is based on fast evaluation of the Gaunt coefficients. The paper also rederives the analytical expressions for the vector translational addition coefficients and discusses the strengths and limitations of other formulations and numerical techniques found in the literature. Numerical results from the formulation derived in this paper agree with those of a previously published recursion scheme that completely avoids the use of the Gaunt coefficients, but the method of direct calculation proposed here reduces the computing time by a factor of 4–6.},
number = {2},
urldate = {2015-11-22},
journal = {Journal of Computational Physics},
author = {Xu, Yu-lin},
month = sep,
year = {1996},
pages = {285--298},
annote = {N.B. Erratum  J. Comput. Phys. 134, 200 (1997). },
file = {ScienceDirect Full Text PDF:/home/necadam1/.zotero/zotero/9uf64zmd.default/zotero/storage/NCD6BBNZ/Xu - 1996 - Calculation of the Addition Coefficients in Electr.pdf:application/pdf;ScienceDirect Snapshot:/home/necadam1/.zotero/zotero/9uf64zmd.default/zotero/storage/NDSF7KI2/S0021999196901758.html:text/html}
}
@article{epton_multipole_1995,
title = {Multipole {Translation} {Theory} for the {Three}-{Dimensional} {Laplace} and {Helmholtz} {Equations}},
volume = {16},
@ -1311,6 +1345,40 @@ http://www.sciencedirect.com/science/article/pii/S0021999197956874},
year = {2015}
}
@article{xu_scattering_2014,
title = {Scattering of electromagnetic radiation by three-dimensional periodic arrays of identical particles},
volume = {31},
issn = {1084-7529, 1520-8532},
url = {https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-31-2-322},
doi = {10.1364/JOSAA.31.000322},
language = {en},
number = {2},
urldate = {2015-11-22},
journal = {Journal of the Optical Society of America A},
author = {Xu, Yu-Lin},
month = feb,
year = {2014},
pages = {322},
file = {josaa-31-2-322.pdf:/home/necadam1/.zotero/zotero/9uf64zmd.default/zotero/storage/ZEG474H8/josaa-31-2-322.pdf:application/pdf}
}
@article{xu_scattering_2013,
title = {Scattering of electromagnetic waves by periodic particle arrays},
volume = {30},
issn = {1084-7529, 1520-8532},
url = {https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-30-6-1053},
doi = {10.1364/JOSAA.30.001053},
language = {en},
number = {6},
urldate = {2015-11-22},
journal = {Journal of the Optical Society of America A},
author = {Xu, Yu-Lin},
month = jun,
year = {2013},
pages = {1053},
file = {josaa-30-6-1053.pdf:/home/necadam1/.zotero/zotero/9uf64zmd.default/zotero/storage/BMACCB6D/josaa-30-6-1053.pdf:application/pdf}
}
@article{blake_surface_2015,
title = {Surface plasmon-polaritons in periodic arrays of {V}-grooves strongly coupled to quantum emitters},
url = {http://arxiv.org/abs/1504.00938},

218
worknotes.lyx
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@ -82,6 +82,17 @@
\end_inset
\begin_inset FormulaMacro
\newcommand{\uvec}[1]{\mathbf{\boldsymbol{\hat{#1}}}}
{\boldsymbol{\hat{\mathbf{#1}}}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ud}{\mathrm{d}}
\end_inset
\end_layout
\begin_layout Title
@ -167,17 +178,218 @@ Computing classical Green's functions
\end_layout
\begin_layout Subsection
Boundary element method
T-Matrix method
\end_layout
\begin_layout Subsection
T-Matrix method
\begin_layout Subsubsection
VSWF decomposition
\end_layout
\begin_layout Standard
Expressions for VSWF in Xu
\begin_inset CommandInset citation
LatexCommand cite
after "(2)"
key "xu_calculation_1996"
\end_inset
:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{eqnarray*}
\vect M_{mn}^{(J)} & = & \left(i\uvec{\theta}\pi_{mn}(\cos\theta)-\uvec{\phi}\tau_{mn}(\cos\theta)\right)z_{n}^{(J)}(kr)e^{im\phi},\\
\vect N_{mn}^{(J)} & = & \uvec rn(n+1)P_{n}^{m}(\cos\theta)\frac{z_{n}^{(J)}(kr)}{kr}e^{im\phi}\\
& & +\left(\uvec{\theta}\tau_{mn}(\cos\theta)+i\uvec{\phi}\pi_{mn}(\cos\theta)\right)\\
& & \phantom{+}\times\frac{1}{kr}\frac{\ud\left(rz_{n}^{(J)}(kr)\right)}{\ud r}e^{im\phi},\\
& = & ...
\end{eqnarray*}
\end_inset
where
\begin_inset Formula $z_{n}^{(J)}$
\end_inset
denotes
\begin_inset Formula $j_{n},y_{n},h_{n}^{+},h_{n}^{-}$
\end_inset
for
\begin_inset Formula $J=1,2,3,4$
\end_inset
, respectively, and
\begin_inset Formula
\begin{eqnarray*}
\pi_{mn}(\cos\theta) & = & \frac{m}{\sin\theta}P_{n}^{m}(\cos\theta),\\
\tau_{mn}(\cos\theta) & = & \frac{\ud P_{n}^{m}(\cos\theta)}{\ud\theta}=-\sin\theta\frac{\ud P_{n}^{m}(\cos\theta)}{\ud\cos\theta}.
\end{eqnarray*}
\end_inset
The expressions for
\begin_inset Formula $\vect M_{mn}^{(J)},\vect N_{mn}^{(J)}$
\end_inset
are dimensionless.
\end_layout
\begin_layout Standard
Expansions for the scattered fields are
\begin_inset CommandInset citation
LatexCommand cite
after "(4)"
key "xu_calculation_1996"
\end_inset
:
\begin_inset Formula
\begin{eqnarray*}
\vect E_{s}(j) & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}iE_{mn}\left[a_{mn}^{j}\vect N_{mn}^{(3)}+b_{mn}^{j}\vect M_{mn}^{(3)}\right],\\
\vect H_{s}(j) & = & \frac{k}{\omega\mu}\sum_{n=1}^{\infty}\sum_{m=-n}^{n}E_{mn}\left[b_{mn}^{j}\vect N_{mn}^{(3)}+a_{mn}^{j}\vect M_{mn}^{(3)}\right].
\end{eqnarray*}
\end_inset
These expansions should be OK in SI units (take the Fourier transform of
\begin_inset Formula $\nabla\times\vect E=-\partial\vect B/\partial t$
\end_inset
and
\begin_inset Formula $\vect B=\mu\vect H$
\end_inset
).
For internal field of a sphere, the (regular-wave) expansion reads
\begin_inset Formula
\begin{eqnarray*}
\vect E_{I}(j) & = & -\sum_{n=1}^{\infty}\sum_{m=-n}^{n}iE_{mn}\left[d_{mn}^{j}\vect N_{mn}^{(1)}+c_{mn}^{j}\vect M_{mn}^{(1)}\right],\\
\vect H_{I}(j) & = & -\frac{k}{\omega\mu}\sum_{n=1}^{\infty}\sum_{m=-n}^{n}E_{mn}\left[c_{mn}^{j}\vect N_{mn}^{(1)}+d_{mn}^{j}\vect M_{mn}^{(1)}\right]
\end{eqnarray*}
\end_inset
(note the minus sign; I am not sure about its role) and the incident field
(incl.
field from the other scatterers) is assumed to have the same regular-wave
form
\begin_inset Formula
\begin{eqnarray*}
\vect E_{i}(j) & = & -\sum_{n=1}^{\infty}\sum_{m=-n}^{n}iE_{mn}\left[p_{mn}^{j}\vect N_{mn}^{(1)}+q_{mn}^{j}\vect M_{mn}^{(1)}\right],\\
\vect H_{i}(j) & = & -\frac{k}{\omega\mu}\sum_{n=1}^{\infty}\sum_{m=-n}^{n}E_{mn}\left[q_{mn}^{j}\vect N_{mn}^{(1)}+p_{mn}^{j}\vect M_{mn}^{(1)}\right].
\end{eqnarray*}
\end_inset
The common multipole-dependent factor is given by
\begin_inset Formula
\[
E_{mn}=\left|E_{0}\right|i^{n}(2n+1)\frac{\left(n-m\right)!}{\left(n+m\right)!}
\]
\end_inset
which
\begin_inset Quotes eld
\end_inset
is desired for keeping the formulation of the multisphere scattering theory
consistent with that of the Mie theory.
It ensures that all the expressions in the multisphere theory turn out
to be identical to those in the Mie theory when one is dealing with a cluster
containing only one sphere and illuminated by a single plane wave
\begin_inset Quotes erd
\end_inset
.
(According to Bohren&Huffman
\begin_inset CommandInset citation
LatexCommand cite
after "(4.37)"
key "bohren_absorption_1983"
\end_inset
, the decomposition of a plane wave reads
\begin_inset Formula
\[
\vect E=E_{0}\sum_{n=1}^{\infty}i^{n}\frac{2n+1}{n(n+1)}\left(\vect M_{o1n}^{(1)}-i\vect N_{e1n}^{(1)}\right),
\]
\end_inset
where the even/odd VSWF and
\begin_inset Formula $m\ge0$
\end_inset
convention is used.)
\end_layout
\begin_layout Standard
\emph on
It should be possible to just take it away and the abovementioned expansions
are still consistent, are they not?
\end_layout
\begin_layout Subsubsection
Mie scattering
\end_layout
\begin_layout Standard
For the exact form of the coefficients following from the boundary conditions
on the spherical surface, cf.
\begin_inset CommandInset citation
LatexCommand cite
after "(12–13)"
key "xu_calculation_1996"
\end_inset
.
For the particular case of spherical nanoparticle, it is important that
they can be written as
\begin_inset CommandInset citation
LatexCommand cite
after "(14–15)"
key "xu_calculation_1996"
\end_inset
\begin_inset Formula
\begin{alignat*}{1}
a_{mn}^{j} & =a_{n}^{j}p_{mn}^{j},\quad b_{mn}^{j}=b_{n}^{j}q_{mn}^{j},\\
c_{mn}^{j} & =c_{n}^{j}q_{mn}^{j},\quad d_{mn}^{j}=d_{n}^{j}p_{mn}^{j},
\end{alignat*}
\end_inset
in other words, the Mie coefficients do not depend on
\begin_inset Formula $m$
\end_inset
but solely on
\begin_inset Formula $n$
\end_inset
(which is not surprising and probably follows from the Wigner-Eckart theorem).
\end_layout
\begin_layout Subsection
T-Matrix resummation (multiple scatterers)
\end_layout
\begin_layout Subsection
Boundary element method
\end_layout
\begin_layout Subsection
BEM→TM
\end_layout