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60
lepaper/T-matrix paper.bib
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@ -1,60 +0,0 @@
@article{xu_efficient_1998,
title = {Efficient {{Evaluation}} of {{Vector Translation Coefficients}} in {{Multiparticle Light}}-{{Scattering Theories}}},
volume = {139},
issn = {0021-9991},
abstract = {Vector addition theorems are a necessary ingredient in the analytical solution of electromagnetic multiparticle-scattering problems. These theorems include a large number of vector addition coefficients. There exist three basic types of analytical expressions for vector translation coefficients: Stein's (Quart. Appl. Math.19, 15 (1961)), Cruzan's (Quart. Appl. Math.20, 33 (1962)), and Xu's (J. Comput. Phys.127, 285 (1996)). Stein's formulation relates vector translation coefficients with scalar translation coefficients. Cruzan's formulas use the Wigner 3jm symbol. Xu's expressions are based on the Gaunt coefficient. Since the scalar translation coefficient can also be expressed in terms of the Gaunt coefficient, the key to the expeditious and reliable calculation of vector translation coefficients is the fast and accurate evaluation of the Wigner 3jm symbol or the Gaunt coefficient. We present highly efficient recursive approaches to accurately evaluating Wigner 3jm symbols and Gaunt coefficients. Armed with these recursive approaches, we discuss several schemes for the calculation of the vector translation coefficients, using the three general types of formulation, respectively. Our systematic test calculations show that the three types of formulas produce generally the same numerical results except that the algorithm of Stein's type is less accurate in some particular cases. These extensive test calculations also show that the scheme using the formulation based on the Gaunt coefficient is the most efficient in practical computations.},
number = {1},
journal = {Journal of Computational Physics},
doi = {10.1006/jcph.1997.5867},
author = {Xu, Yu-lin},
month = jan,
year = {1998},
pages = {137-165},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/STV5263F/Xu - 1998 - Efficient Evaluation of Vector Translation Coeffic.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/VMGZRSAA/S0021999197958678.html}
}
@book{jackson_classical_1998,
address = {{New York}},
edition = {3 edition},
title = {Classical {{Electrodynamics Third Edition}}},
isbn = {978-0-471-30932-1},
abstract = {A revision of the defining book covering the physics and classical mathematics necessary to understand electromagnetic fields in materials and at surfaces and interfaces. The third edition has been revised to address the changes in emphasis and applications that have occurred in the past twenty years.},
language = {English},
publisher = {{Wiley}},
author = {Jackson, John David},
month = aug,
year = {1998},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/3BWPD4BK/John David Jackson-Classical Electrodynamics-Wiley (1999).djvu}
}
@article{mie_beitrage_1908,
title = {Beitr{\"a}ge Zur {{Optik}} Tr{\"u}ber {{Medien}}, Speziell Kolloidaler {{Metall{\"o}sungen}}},
volume = {330},
copyright = {Copyright \textcopyright{} 1908 WILEY-VCH Verlag GmbH \& Co. KGaA, Weinheim},
issn = {1521-3889},
language = {en},
number = {3},
journal = {Ann. Phys.},
doi = {10.1002/andp.19083300302},
author = {Mie, Gustav},
month = jan,
year = {1908},
pages = {377-445},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/RM9J9RYH/Mie - 1908 - Beiträge zur Optik trüber Medien, speziell kolloid.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/F5A7EX6R/abstract.html}
}
@book{kristensson_scattering_2016,
address = {{Edison, NJ}},
title = {Scattering of {{Electromagnetic Waves}} by {{Obstacles}}},
isbn = {978-1-61353-221-8},
abstract = {This book is an introduction to some of the most important properties of electromagnetic waves and their interaction with passive materials and scatterers. The main purpose of the book is to give a theoretical treatment of these scattering phenomena, and to illustrate numerical computations of some canonical scattering problems for different geometries and materials. The scattering theory is also important in the theory of passive antennas, and this book gives several examples on this topic. Topics covered include an introduction to the basic equations used in scattering; the Green functions and dyadics; integral representation of fields; introductory scattering theory; scattering in the time domain; approximations and applications; spherical vector waves; scattering by spherical objects; the null-field approach; and propagation in stratified media. The book is organised along two tracks, which can be studied separately or together. Track 1 material is appropriate for a first reading of the textbook, while Track 2 contains more advanced material suited for the second reading and for reference. Exercises are included for each chapter.},
language = {English},
publisher = {{Scitech Publishing}},
author = {Kristensson, Gerhard},
month = jul,
year = {2016},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/3R7VYZUK/Kristensson - 2016 - Scattering of Electromagnetic Waves by Obstacles.pdf}
}

433
lepaper/arrayscat.lyx
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\begin_layout Title
Some nice title about multiple scattering approach to photonic nanoparticle
arrays (outline)
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\begin_layout Author
Marek Nečada
QPMS Photonic Multiple Scattering suite (TODO better title)
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\begin_layout Standard
@ -254,6 +478,10 @@ The purpose of SIAM Journal on Scientific Computing (SISC) is to advance
\begin_layout Quotation
SISC papers are classified into three categories:
\begin_inset Separator latexpar
\end_inset
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\begin_deeper
@ -307,12 +535,31 @@ Due to space limitations, articles are normally limited to 20 journal pages.
Category: Methods and Algorithms for Scientific Computing?
\end_layout
\begin_layout Abstract
The (somewhat underrated) T-matrix multiple scattering method (TMMSM) can
be used to solve the electromagnetic response of systems consisting of
many compact scatterers.
It largely surpasses other methods in the number of scatterers it can deal
with, while retaining very good accuracy.
\end_layout
\begin_layout Abstract
TODO REWRITE: We release a modern implementation of the method under GNU
General Public Licence, with several theoretical advancements presented
here, such as exploiting the system symmetries to further improve the efficienc
y of the method, or extending it on infinite periodic systems.
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Outline
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Intro:
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@ -343,6 +590,10 @@ my implementation
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Finite systems:
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\begin_deeper
@ -353,11 +604,19 @@ motivation (classes of problems that this can solve: response to external
\begin_layout Itemize
theory
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\end_inset
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\begin_deeper
\begin_layout Itemize
T-matrix definition, basics
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\end_inset
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@ -385,6 +644,10 @@ Example results (or maybe rather in the end)
\end_deeper
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Infinite lattices:
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\end_inset
\end_layout
\begin_deeper
@ -394,6 +657,10 @@ motivation (dispersion relations / modes, ...?)
\begin_layout Itemize
theory
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\end_inset
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\begin_deeper
@ -432,10 +699,83 @@ My implementation.
Maybe put the numerical results separately in the end.
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\begin_layout Section
TODOs
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\begin_layout Itemize
Consistent notation of balls.
How is the difference between two cocentric balls called?
\end_layout
\begin_layout Itemize
Abstract.
\end_layout
\begin_layout Itemize
Translation operators: rewrite in sph.
harm.
convention independent form.
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Truncation notation.
\end_layout
\begin_layout Itemize
Example results!
\end_layout
\begin_layout Itemize
Figures.
\end_layout
\begin_layout Itemize
Concrete comparison with other methods.
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\begin_layout Itemize
Fix and unify notation (mainly indices) in infinite lattices section.
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\begin_layout Itemize
Carefully check the transformation directions in sec.
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reference "sec:Symmetries"
plural "false"
caps "false"
noprefix "false"
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\begin_layout Itemize
The text about symmetries is pretty dense.
Make it more explanatory and human-readable.
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\begin_layout Itemize
Check whether everything written is correct also for non-symmetric space
groups.
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LatexCommand include
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@ -475,8 +858,10 @@ filename "examples.lyx"
\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "T-matrix paper"
btprint "btPrintCited"
bibfiles "tmpaper"
options "plain"
encoding "default"
\end_inset

109
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\begin_layout Subsection
The multiple-scattering problem
\begin_inset CommandInset label
LatexCommand label
name "subsec:The-multiple-scattering-problem"
\end_inset
\end_layout
\begin_layout Standard
In the
\begin_inset Formula $T$
\end_inset
-matrix approach, scattering properties of single nanoparticles in a homogeneous
medium are first computed in terms of vector sperical wavefunctions (VSWFs)—the
field incident onto the
\begin_inset Formula $n$
\end_inset
-th nanoparticle from external sources can be expanded as
\begin_inset Formula
\begin{equation}
\vect E_{n}^{\mathrm{inc}}(\vect r)=\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{t=\mathrm{E},\mathrm{M}}\coeffrip nlmt\vswfr lmt\left(\vect r_{n}\right)\label{eq:E_inc}
\end{equation}
\end_inset
where
\begin_inset Formula $\vect r_{n}=\vect r-\vect R_{n}$
\end_inset
,
\begin_inset Formula $\vect R_{n}$
\end_inset
being the position of the centre of
\begin_inset Formula $n$
\end_inset
-th nanoparticle and
\begin_inset Formula $\vswfr lmt$
\end_inset
are the regular VSWFs which can be expressed in terms of regular spherical
Bessel functions of
\begin_inset Formula $j_{k}\left(\left|\vect r_{n}\right|\right)$
\end_inset
and spherical harmonics
\begin_inset Formula $\ush km\left(\hat{\vect r}_{n}\right)$
\end_inset
; the expressions, together with a proof that the VSWFs span all the solutions
of vector Helmholtz equation around the particle, justifying the expansion,
can be found e.g.
in
\begin_inset CommandInset citation
LatexCommand cite
after "chapter 7"
key "kristensson_scattering_2016"
literal "true"
\end_inset
(care must be taken because of varying normalisation and phase conventions).
On the other hand, the field scattered by the particle can be (outside
the particle's circumscribing sphere) expanded in terms of singular VSWFs
\begin_inset Formula $\vswfs lmt$
\end_inset
which differ from the regular ones by regular spherical Bessel functions
being replaced with spherical Hankel functions
\begin_inset Formula $h_{k}^{(1)}\left(\left|\vect r_{n}\right|\right)$
\end_inset
,
\begin_inset Formula
\begin{equation}
\vect E_{n}^{\mathrm{scat}}\left(\vect r\right)=\sum_{l,m,t}\coeffsip nlmt\vswfs lmt\left(\vect r_{n}\right).\label{eq:E_scat}
\end{equation}
\end_inset
The expansion coefficients
\begin_inset Formula $\coeffsip nlmt$
\end_inset
,
\begin_inset Formula $t=\mathrm{E},\mathrm{M}$
\end_inset
are related to the electric and magnetic multipole polarization amplitudes
of the nanoparticle.
\end_layout
\begin_layout Standard
At a given frequency, assuming the system is linear, the relation between
the expansion coefficients in the VSWF bases is given by the so-called
\begin_inset Formula $T$
\end_inset
-matrix,
\begin_inset Formula
\begin{equation}
\coeffsip nlmt=\sum_{l',m',t'}T_{n}^{lmt;l'm't'}\coeffrip n{l'}{m'}{t'}.\label{eq:Tmatrix definition}
\end{equation}
\end_inset
The
\begin_inset Formula $T$
\end_inset
-matrix is given by the shape and composition of the particle and fully
describes its scattering properties.
In theory it is infinite-dimensional, but in practice (at least for subwaveleng
th nanoparticles) its elements drop very quickly to negligible values with
growing degree indices
\begin_inset Formula $l,l'$
\end_inset
, enabling to take into account only the elements up to some finite degree,
\begin_inset Formula $l,l'\le l_{\mathrm{max}}$
\end_inset
.
The
\begin_inset Formula $T$
\end_inset
-matrix can be calculated numerically using various methods; here we used
the scuff-tmatrix tool from the SCUFF-EM suite
\begin_inset CommandInset citation
LatexCommand cite
key "SCUFF2,reid_efficient_2015"
literal "true"
\end_inset
, which implements the boundary element method (BEM).
\end_layout
\begin_layout Standard
The singular VSWFs originating at
\begin_inset Formula $\vect R_{n}$
\end_inset
can be then re-expanded around another origin (nanoparticle location)
\begin_inset Formula $\vect R_{n'}$
\end_inset
in terms of regular VSWFs,
\begin_inset Formula
\begin{equation}
\begin{split}\svwfs lmt\left(\vect r_{n}\right)=\sum_{l',m',t'}\transop^{l'm't';lmt}\left(\vect R_{n'}-\vect R_{n}\right)\vswfr{l'}{m'}{t'}\left(\vect r_{n'}\right),\\
\left|\vect r_{n'}\right|<\left|\vect R_{n'}-\vect R_{n}\right|.
\end{split}
\label{eq:translation op def}
\end{equation}
\end_inset
Analytical expressions for the translation operator
\begin_inset Formula $\transop^{lmt;l'm't'}\left(\vect R_{n'}-\vect R_{n}\right)$
\end_inset
can be found in
\begin_inset CommandInset citation
LatexCommand cite
key "xu_efficient_1998"
literal "true"
\end_inset
.
\end_layout
\begin_layout Standard
If we write the field incident onto the
\begin_inset Formula $n$
\end_inset
-th nanoparticle as the sum of fields scattered from all the other nanoparticles
and an external field
\begin_inset Formula $\vect E_{0}$
\end_inset
(which we also expand around each nanoparticle,
\begin_inset Formula $\vect E_{0}\left(\vect r\right)=\sum_{l,m,t}\coeffripext nlmt\vswfr lmt\left(\vect r_{n}\right)$
\end_inset
),
\begin_inset Formula
\[
\vect E_{n}^{\mathrm{inc}}\left(\vect r\right)=\vect E_{0}\left(\vect r\right)+\sum_{n'\ne n}\vect E_{n'}^{\mathrm{scat}}\left(\vect r\right)
\]
\end_inset
and use eqs.
(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:E_inc"
\end_inset
)–(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:translation op def"
\end_inset
), we obtain a set of linear equations for the electromagnetic response
(multiple scattering) of the whole set of nanoparticles,
\begin_inset Formula
\begin{equation}
\begin{split}\coeffrip nlmt=\coeffripext nlmt+\sum_{n'\ne n}\sum_{l',m',t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)\\
\times\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''}.
\end{split}
\label{eq:multiplescattering element-wise}
\end{equation}
\end_inset
It is practical to get rid of the VSWF indices, rewriting (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:multiplescattering element-wise"
\end_inset
) in a per-particle matrix form
\begin_inset Formula
\begin{equation}
\coeffr_{n}=\coeffr_{\mathrm{ext}(n)}+\sum_{n'\ne n}S_{n,n'}T_{n'}p_{n'}\label{eq:multiple scattering per particle p}
\end{equation}
\end_inset
and to reformulate the problem using (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Tmatrix definition"
\end_inset
) in terms of the
\begin_inset Formula $\coeffs$
\end_inset
-coefficients which describe the multipole excitations of the particles
\begin_inset Formula
\begin{equation}
\coeffs_{n}-T_{n}\sum_{n'\ne n}S_{n,n'}\coeffs_{n'}=T_{n}\coeffr_{\mathrm{ext}(n)}.\label{eq:multiple scattering per particle a}
\end{equation}
\end_inset
Knowing
\begin_inset Formula $T_{n},S_{n,n'},\coeffr_{\mathrm{ext}(n)}$
\end_inset
, the nanoparticle excitations
\begin_inset Formula $a_{n}$
\end_inset
can be solved by standard linear algebra methods.
The total scattered field anywhere outside the particles' circumscribing
spheres is then obtained by summing the contributions (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:E_scat"
\end_inset
) from all particles.
\end_layout
\end_body
\end_document

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\lyxformat 583
\begin_document
\begin_header
\save_transient_properties true
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\end_header
\begin_body
\begin_layout Subsection
Periodic systems and mode analysis
\begin_inset CommandInset label
LatexCommand label
name "subsec:Periodic-systems"
\end_inset
\end_layout
\begin_layout Standard
In an infinite periodic array of nanoparticles, the excitations of the nanoparti
cles take the quasiperiodic Bloch-wave form
\begin_inset Formula
\[
\coeffs_{i\nu}=e^{i\vect k\cdot\vect R_{i}}\coeffs_{\nu}
\]
\end_inset
(assuming the incident external field has the same periodicity,
\begin_inset Formula $\coeffr_{\mathrm{ext}(i\nu)}=e^{i\vect k\cdot\vect R_{i}}p_{\mathrm{ext}\left(\nu\right)}$
\end_inset
) where
\begin_inset Formula $\nu$
\end_inset
is the index of a particle inside one unit cell and
\begin_inset Formula $\vect R_{i},\vect R_{i'}\in\Lambda$
\end_inset
are the lattice vectors corresponding to the sites (labeled by multiindices
\begin_inset Formula $i,i'$
\end_inset
) of a Bravais lattice
\begin_inset Formula $\Lambda$
\end_inset
.
The multiple-scattering problem (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:multiple scattering per particle a"
\end_inset
) then takes the form
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\coeffs_{i\nu}-T_{\nu}\sum_{(i',\nu')\ne\left(i,\nu\right)}S_{i\nu,i'\nu'}e^{i\vect k\cdot\left(\vect R_{i'}-\vect R_{i}\right)}\coeffs_{i\nu'}=T_{\nu}\coeffr_{\mathrm{ext}(i\nu)}
\]
\end_inset
or, labeling
\begin_inset Formula $W_{\nu\nu'}=\sum_{i';(i',\nu')\ne\left(i,\nu\right)}S_{i\nu,i'\nu'}e^{i\vect k\cdot\left(\vect R_{i'}-\vect R_{i}\right)}=\sum_{i';(i',\nu')\ne\left(0,\nu\right)}S_{0\nu,i'\nu'}e^{i\vect k\cdot\vect R_{i'}}$
\end_inset
and using the quasiperiodicity,
\begin_inset Formula
\begin{equation}
\sum_{\nu'}\left(\delta_{\nu\nu'}\mathbb{I}-T_{\nu}W_{\nu\nu'}\right)\coeffs_{\nu'}=T_{\nu}\coeffr_{\mathrm{ext}(\nu)},\label{eq:multiple scattering per particle a periodic}
\end{equation}
\end_inset
which reduces the linear problem (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:multiple scattering per particle a"
\end_inset
) to interactions between particles inside single unit cell.
A problematic part is the evaluation of the translation operator lattice
sums
\begin_inset Formula $W_{\nu\nu'}$
\end_inset
; this is performed using exponentially convergent Ewald-type representations
\begin_inset CommandInset citation
LatexCommand cite
key "linton_lattice_2010"
literal "true"
\end_inset
.
\end_layout
\begin_layout Standard
In an infinite periodic system, a nonlossy mode supports itself without
external driving, i.e.
such mode is described by excitation coefficients
\begin_inset Formula $a_{\nu}$
\end_inset
that satisfy eq.
(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:multiple scattering per particle a periodic"
\end_inset
) with zero right-hand side.
That can happen if the block matrix
\begin_inset Formula
\begin{equation}
M\left(\omega,\vect k\right)=\left\{ \delta_{\nu\nu'}\mathbb{I}-T_{\nu}\left(\omega\right)W_{\nu\nu'}\left(\omega,\vect k\right)\right\} _{\nu\nu'}\label{eq:M matrix definition}
\end{equation}
\end_inset
from the left hand side of (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:multiple scattering per particle a periodic"
\end_inset
) is singular (here we explicitly note the
\begin_inset Formula $\omega,\vect k$
\end_inset
depence).
\end_layout
\begin_layout Standard
For lossy nanoparticles, however, perfect propagating modes will not exist
and
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
will never be perfectly singular.
Therefore in practice, we get the bands by scanning over
\begin_inset Formula $\omega,\vect k$
\end_inset
to search for
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
which have an
\begin_inset Quotes erd
\end_inset
almost zero
\begin_inset Quotes erd
\end_inset
singular value.
\end_layout
\begin_layout Section
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
{
\end_layout
\end_inset
Symmetries
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "sm:symmetries"
\end_inset
\end_layout
\begin_layout Standard
A general overview of utilizing group theory to find lattice modes at high-symme
try points of the Brillouin zone can be found e.g.
in
\begin_inset CommandInset citation
LatexCommand cite
after "chapters 10–11"
key "dresselhaus_group_2008"
literal "true"
\end_inset
; here we use the same notation.
\end_layout
\begin_layout Standard
We analyse the symmetries of the system in the same VSWF representation
as used in the
\begin_inset Formula $T$
\end_inset
-matrix formalism introduced above.
We are interested in the modes at the
\begin_inset Formula $\Kp$
\end_inset
-point of the hexagonal lattice, which has the
\begin_inset Formula $D_{3h}$
\end_inset
point symmetry.
The six irreducible representations (irreps) of the
\begin_inset Formula $D_{3h}$
\end_inset
group are known and are available in the literature in their explicit forms.
In order to find and classify the modes, we need to find a decomposition
of the lattice mode representation
\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}=\Gamma^{\mathrm{equiv.}}\otimes\Gamma_{\mathrm{vec.}}$
\end_inset
into the irreps of
\begin_inset Formula $D_{3h}$
\end_inset
.
The equivalence representation
\begin_inset Formula $\Gamma^{\mathrm{equiv.}}$
\end_inset
is the
\begin_inset Formula $E'$
\end_inset
representation as can be deduced from
\begin_inset CommandInset citation
LatexCommand cite
after "eq. (11.19)"
key "dresselhaus_group_2008"
literal "true"
\end_inset
, eq.
(11.19) and the character table for
\begin_inset Formula $D_{3h}$
\end_inset
.
\begin_inset Formula $\Gamma_{\mathrm{vec.}}$
\end_inset
operates on a space spanned by the VSWFs around each nanoparticle in the
unit cell (the effects of point group operations on VSWFs are described
in
\begin_inset CommandInset citation
LatexCommand cite
key "schulz_point-group_1999"
literal "true"
\end_inset
).
This space can be then decomposed into invariant subspaces of the
\begin_inset Formula $D_{3h}$
\end_inset
using the projectors
\begin_inset Formula $\hat{P}_{ab}^{\left(\Gamma\right)}$
\end_inset
defined by
\begin_inset CommandInset citation
LatexCommand cite
after "eq. (4.28)"
key "dresselhaus_group_2008"
literal "true"
\end_inset
.
This way, we obtain a symmetry adapted basis
\begin_inset Formula $\left\{ \vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}\right\} $
\end_inset
as linear combinations of VSWFs
\begin_inset Formula $\vswfs lm{p,t}$
\end_inset
around the constituting nanoparticles (labeled
\begin_inset Formula $p$
\end_inset
),
\begin_inset Formula
\[
\vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}=\sum_{l,m,p,t}U_{\Gamma,r,i}^{p,t,l,m}\vswfs lm{p,t},
\]
\end_inset
where
\begin_inset Formula $\Gamma$
\end_inset
stands for one of the six different irreps of
\begin_inset Formula $D_{3h}$
\end_inset
,
\begin_inset Formula $r$
\end_inset
labels the different realisations of the same irrep, and the last index
\begin_inset Formula $i$
\end_inset
going from 1 to
\begin_inset Formula $d_{\Gamma}$
\end_inset
(the dimensionality of
\begin_inset Formula $\Gamma$
\end_inset
) labels the different partners of the same given irrep.
The number of how many times is each irrep contained in
\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}$
\end_inset
(i.e.
the range of index
\begin_inset Formula $r$
\end_inset
for given
\begin_inset Formula $\Gamma$
\end_inset
) depends on the multipole degree cutoff
\begin_inset Formula $l_{\mathrm{max}}$
\end_inset
.
\end_layout
\begin_layout Standard
Each mode at the
\begin_inset Formula $\Kp$
\end_inset
-point shall lie in the irreducible spaces of only one of the six possible
irreps and it can be shown via
\begin_inset CommandInset citation
LatexCommand cite
after "eq. (2.51)"
key "dresselhaus_group_2008"
literal "true"
\end_inset
that, at the
\begin_inset Formula $\Kp$
\end_inset
-point, the matrix
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
defined above takes a block-diagonal form in the symmetry-adapted basis,
\begin_inset Formula
\[
M\left(\omega,\vect K\right)_{\Gamma,r,i;\Gamma',r',j}^{\mathrm{s.a.b.}}=\frac{\delta_{\Gamma\Gamma'}\delta_{ij}}{d_{\Gamma}}\sum_{q}M\left(\omega,\vect K\right)_{\Gamma,r,q;\Gamma',r',q}^{\mathrm{s.a.b.}}.
\]
\end_inset
This enables us to decompose the matrix according to the irreps and to solve
the singular value problem in each irrep separately, as done in Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "smfig:dispersions"
\end_inset
(a).
\end_layout
\end_body
\end_document

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@ -81,123 +95,201 @@
\begin_layout Section
Infinite periodic systems
\begin_inset FormulaMacro
\newcommand{\dlv}{\vect a}
\end_inset
\begin_inset FormulaMacro
\newcommand{\rlv}{\vect b}
\end_inset
\end_layout
\begin_layout Standard
Although large finite systems are where MSTMM excels the most, there are
several reasons that makes its extension to infinite lattices (where periodic
boundary conditions might be applied) desirable as well.
Other methods might be already fast enough, but MSTMM will be faster in
most cases in which there is enough spacing between the neighboring particles.
MSTMM works well with any space group symmetry the system might have (as
opposed to, for example, FDTD with cubic mesh applied to a honeycomb lattice),
which makes e.g.
application of group theory in mode analysis quite easy.
\begin_inset Note Note
status open
\begin_layout Plain Layout
Topology anoyne?
\end_layout
\end_inset
And finally, having a method that handles well both infinite and large
finite system gives a possibility to study finite-size effects in periodic
scatterer arrays.
\end_layout
\begin_layout Subsection
Notation
\end_layout
\lang english
\begin_layout Standard
TODO Fourier transforms, Delta comb, lattice bases, reciprocal lattices
etc.
\end_layout
\begin_layout Subsection
Formulation of the problem
\end_layout
\begin_layout Standard
\lang english
Assume a system of compact EM scatterers in otherwise homogeneous and isotropic
medium, and assume that the system, i.e.
both the medium and the scatterers, have linear response.
A scattering problem in such system can be written as
\begin_inset Formula
\[
A_{α}=T_{α}P_{α}=T_{α}(\sum_{β}S_{α\leftarrowβ}A_{β}+P_{0α})
\]
Let us have a linear system of compact EM scatterers on a homogeneous background
as in Section
\begin_inset CommandInset ref
LatexCommand eqref
reference "subsec:Multiple-scattering"
plural "false"
caps "false"
noprefix "false"
\end_inset
where
\begin_inset Formula $T_{α}$
, but this time, the system shall be periodic: let there be a
\begin_inset Formula $d$
\end_inset
is the
\begin_inset Formula $T$
-dimensional (
\begin_inset Formula $d$
\end_inset
-matrix for scatterer α,
\begin_inset Formula $A_{α}$
can be 1, 2 or 3) lattice embedded into the three-dimensional real space,
with lattice vectors
\begin_inset Formula $\left\{ \dlv_{i}\right\} _{i=1}^{d}$
\end_inset
is its vector of the scattered wave expansion coefficient (the multipole
indices are not explicitely indicated here) and
\begin_inset Formula $P_{α}$
, and let the lattice points be labeled with an
\begin_inset Formula $d$
\end_inset
is the local expansion of the incoming sources.
\begin_inset Formula $S_{α\leftarrowβ}$
-dimensional integar multiindex
\begin_inset Formula $\vect n\in\ints^{d}$
\end_inset
is ...
and ...
is ...
\end_layout
\begin_layout Standard
\lang english
...
\end_layout
\begin_layout Standard
\lang english
\begin_inset Formula
\[
\sum_{β}(\delta_{αβ}-T_{α}S_{α\leftarrowβ})A_{β}=T_{α}P_{0α}.
\]
\end_inset
\end_layout
\begin_layout Standard
\lang english
Now suppose that the scatterers constitute an infinite lattice
\end_layout
\begin_layout Standard
\lang english
\begin_inset Formula
\[
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{\vect aα}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=T_{\vect aα}P_{0\vect aα}.
\]
\end_inset
Due to the periodicity, we can write
\begin_inset Formula $S_{\vect aα\leftarrow\vect bβ}=S_{α\leftarrowβ}(\vect b-\vect a)$
\end_inset
and
\begin_inset Formula $T_{\vect aα}=T_{\alpha}$
, so the lattice points have cartesian coordinates
\begin_inset Formula $\vect R_{\vect n}=\sum_{i=1}^{d}n_{i}\vect a_{i}$
\end_inset
.
In order to find lattice modes, we search for solutions with zero RHS
There can be several scatterers per unit cell with indices
\begin_inset Formula $\alpha$
\end_inset
from set
\begin_inset Formula $\mathcal{P}_{1}$
\end_inset
and (relative) positions inside the unit cell
\begin_inset Formula $\vect r_{\alpha}$
\end_inset
; any particle of the periodic system can thus be labeled by a multiindex
from
\begin_inset Formula $\mathcal{P}=\ints^{d}\times\mathcal{P}_{1}$
\end_inset
.
The scatterers are located at positions
\begin_inset Formula $\vect r_{\vect n,\alpha}=\vect R_{\vect n}+\vect r_{\alpha}$
\end_inset
and their
\begin_inset Formula $T$
\end_inset
-matrices are periodic,
\begin_inset Formula $T_{\vect n,\alpha}=T_{\alpha}$
\end_inset
.
In such system, the multiple-scattering problem
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem"
plural "false"
caps "false"
noprefix "false"
\end_inset
can be rewritten as
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=0
\]
\begin{equation}
\outcoeffp{\vect n,\alpha}-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect n,\alpha}{\vect m,\beta}\outcoeffp{\vect m,\beta}=T_{\alpha}\rcoeffincp{\vect n,\alpha}.\quad\left(\vect n,\alpha\right)\in\mathcal{P}\label{eq:Multiple-scattering problem periodic}
\end{equation}
\end_inset
and we assume periodic solution
\begin_inset Formula $A_{\vect b\beta}(\vect k)=A_{\vect a\beta}e^{i\vect k\cdot\vect r_{\vect b-\vect a}}$
\end_layout
\begin_layout Standard
Due to periodicity, we can also write
\begin_inset Formula $\tropsp{\vect n,\alpha}{\vect m,\beta}=\tropsp{\alpha}{\beta}\left(\vect R_{\vect m}-\vect R_{\vect n}\right)=\tropsp{\alpha}{\beta}\left(\vect R_{\vect m-\vect n}\right)=\tropsp{\vect 0,\alpha}{\vect m-\vect n,\beta}$
\end_inset
, yielding
.
Assuming quasi-periodic right-hand side with quasi-momentum
\begin_inset Formula $\vect k$
\end_inset
,
\begin_inset Formula $\rcoeffincp{\vect n,\alpha}=\rcoeffincp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
\end_inset
, the solutions of
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem periodic"
plural "false"
caps "false"
noprefix "false"
\end_inset
will be also quasi-periodic according to Bloch theorem,
\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
\end_inset
, and eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem periodic"
plural "false"
caps "false"
noprefix "false"
\end_inset
can be rewritten as follows
\begin_inset Formula
\begin{eqnarray*}
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect a\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b-\vect a}} & = & 0,\\
\sum_{\vect bβ}(\delta_{\vect{0b}}\delta_{αβ}-T_{α}S_{\vect 0α\leftarrow\vect bβ})A_{\vect 0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
\sum_{β}(\delta_{αβ}-T_{α}\underbrace{\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect 0\beta}\left(\vect k\right) & = & 0,\\
A_{\vect 0\alpha}\left(\vect k\right)-T_{α}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect 0\beta}\left(\vect k\right) & = & 0.
\end{eqnarray*}
\begin{align}
\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect n,\alpha}{\vect m,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}},\nonumber \\
\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect 0,\alpha}{\vect m-\vect n,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m-\vect n}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\nonumber \\
\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect 0,\alpha\right)\right\} }\tropsp{\vect 0,\alpha}{\vect m,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\nonumber \\
\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\beta\in\mathcal{P}}W_{\alpha\beta}\left(\vect k\right)\outcoeffp{\vect 0,\beta}\left(\vect k\right) & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\label{eq:Multiple-scattering problem unit cell}
\end{align}
\end_inset
Therefore, in order to solve the modes, we need to compute the
so we reduced the initial scattering problem to one involving only the field
expansion coefficients from a single unit cell, but we need to compute
the
\begin_inset Quotes eld
\end_inset
@ -208,23 +300,272 @@ lattice Fourier transform
of the translation operator,
\begin_inset Formula
\begin{equation}
W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\beta}\right)\tropsp{\vect 0,\alpha}{\vect m,\beta}e^{i\vect k\cdot\vect R_{\vect m}},\label{eq:W definition}
\end{equation}
\end_inset
evaluation of which is possible but quite non-trivial due to the infinite
lattice sum, so we explain it separately in Sect.
\begin_inset CommandInset ref
LatexCommand eqref
reference "subsec:W operator evaluation"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
As in the case of a finite system, eq.
can be written in a shorter block-matrix form,
\begin_inset Formula
\begin{equation}
\left(I-WT\right)\outcoeffp{\vect 0}\left(\vect k\right)=\rcoeffincp{\vect 0}\left(\vect k\right)\label{eq:Multiple-scattering problem unit cell block form}
\end{equation}
\end_inset
Eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem unit cell"
plural "false"
caps "false"
noprefix "false"
\end_inset
can be used to calculate electromagnetic response of the structure to external
quasiperiodic driving field – most notably a plane wave.
However, the non-trivial solutions of the equation with right hand side
(i.e.
the external driving) set to zero,
\begin_inset Formula
\begin{equation}
\left(I-WT\right)\outcoeffp{\vect 0}\left(\vect k\right)=0,\label{eq:lattice mode equation}
\end{equation}
\end_inset
describes the
\emph on
lattice modes.
\emph default
Non-trivial solutions to
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lattice mode equation"
plural "false"
caps "false"
noprefix "false"
\end_inset
exist if the matrix on the left-hand side
\begin_inset Formula $M\left(\omega,\vect k\right)=\left(I-W\left(\omega,\vect k\right)T\left(\omega\right)\right)$
\end_inset
is singular – this condition gives the
\emph on
dispersion relation
\emph default
for the periodic structure.
Note that in realistic (lossy) systems, at least one of the pair
\begin_inset Formula $\omega,\vect k$
\end_inset
will acquire complex values.
The solution
\begin_inset Formula $\outcoeffp{\vect 0}\left(\vect k\right)$
\end_inset
is then obtained as the right
\begin_inset Note Note
status open
\begin_layout Plain Layout
CHECK!
\end_layout
\end_inset
singular vector of
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
corresponding to the zero singular value.
\end_layout
\begin_layout Subsection
Numerical solution
\end_layout
\begin_layout Standard
In practice, equation
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem unit cell block form"
plural "false"
caps "false"
noprefix "false"
\end_inset
is solved in the same way as eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem block form"
plural "false"
caps "false"
noprefix "false"
\end_inset
in the multipole degree truncated form.
\end_layout
\begin_layout Standard
The lattice mode problem
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lattice mode equation"
plural "false"
caps "false"
noprefix "false"
\end_inset
is (after multipole degree truncation) solved by finding
\begin_inset Formula $\omega,\vect k$
\end_inset
for which the matrix
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
has a zero singular value.
A naïve approach to do that is to sample a volume with a grid in the
\begin_inset Formula $\left(\omega,\vect k\right)$
\end_inset
space, performing a singular value decomposition of
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
at each point and finding where the lowest singular value of
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
is close enough to zero.
However, this approach is quite expensive, for
\begin_inset Formula $W\left(\omega,\vect k\right)$
\end_inset
has to be evaluated for each
\begin_inset Formula $\omega,\vect k$
\end_inset
pair separately (unlike the original finite case
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem block form"
plural "false"
caps "false"
noprefix "false"
\end_inset
translation operator
\begin_inset Formula $\trops$
\end_inset
, which, for a given geometry, depends only on frequency).
Therefore, a much more efficient approach to determine the photonic bands
is to sample the
\begin_inset Formula $\vect k$
\end_inset
-space (a whole Brillouin zone or its part) and for each fixed
\begin_inset Formula $\vect k$
\end_inset
to find a corresponding frequency
\begin_inset Formula $\omega$
\end_inset
with zero singular value of
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
using a minimisation algorithm (two- or one-dimensional, depending on whether
one needs the exact complex-valued
\begin_inset Formula $\omega$
\end_inset
or whether the its real-valued approximation is satisfactory).
Typically, a good initial guess for
\begin_inset Formula $\omega\left(\vect k\right)$
\end_inset
is obtained from the empty lattice approximation,
\begin_inset Formula $\left|\vect k\right|=\sqrt{\epsilon\mu}\omega/c_{0}$
\end_inset
(modulo lattice points; TODO write this a clean way).
A somehow challenging step is to distinguish the different bands that can
all be very close to the empty lattice approximation, especially if the
particles in the systems are small.
In high-symmetry points of the Brilloin zone, this can be solved by factorising
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
into irreducible representations
\begin_inset Formula $\Gamma_{i}$
\end_inset
and performing the minimisation in each irrep separately, cf.
Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Symmetries"
plural "false"
caps "false"
noprefix "false"
\end_inset
, and using the different
\begin_inset Formula $\omega_{\Gamma_{i}}\left(\vect k\right)$
\end_inset
to obtain the initial guesses for the nearby points
\begin_inset Formula $\vect k+\delta\vect k$
\end_inset
.
\end_layout
\begin_layout Subsection
Computing the Fourier sum of the translation operator
\begin_inset CommandInset label
LatexCommand label
name "subsec:W operator evaluation"
\end_inset
\end_layout
\begin_layout Subsection
\lang english
Computing the Fourier sum of the translation operator
\end_layout
\begin_layout Standard
\lang english
The problem evaluating
\begin_inset CommandInset ref
LatexCommand eqref
@ -233,20 +574,22 @@ reference "eq:W definition"
\end_inset
is the asymptotic behaviour of the translation operator,
\begin_inset Formula $S_{\vect 0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
\begin_inset Formula $\tropsp{\vect 0,\alpha}{\vect m,\beta}\sim\left|\vect R_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect R_{\vect b}\right|}$
\end_inset
that makes the convergence of the sum quite problematic for any
that does not in the strict sense converge for any
\begin_inset Formula $d>1$
\end_inset
-dimensional lattice.
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Foot
status open
\begin_layout Plain Layout
\lang english
Note that
\begin_inset Formula $d$
\end_inset
@ -258,16 +601,20 @@ Note that
\end_inset
In electrostatics, one can solve this problem with Ewald summation.
\end_layout
\end_inset
In electrostatics, this problem can be solved with Ewald summation [TODO
REF].
Its basic idea is that if what asymptoticaly decays poorly in the direct
space, will perhaps decay fast in the Fourier space.
I use the same idea here, but everything will be somehow harder than in
electrostatics.
We use the same idea here, but the technical details are more complicated
than in electrostatics.
\end_layout
\begin_layout Standard
\lang english
Let us re-express the sum in
\begin_inset CommandInset ref
LatexCommand eqref
@ -276,11 +623,14 @@ reference "eq:W definition"
\end_inset
in terms of integral with a delta comb
\begin_inset FormulaMacro
\renewcommand{\basis}[1]{\mathfrak{#1}}
\end_inset
\end_layout
\begin_layout Standard
\lang english
\begin_inset Formula
\begin{equation}
W_{\alpha\beta}(\vect k)=\int\ud^{d}\vect r\dc{\basis u}(\vect r)S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})e^{i\vect k\cdot\vect r}.\label{eq:W integral}
@ -382,8 +732,6 @@ W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(
status open
\begin_layout Plain Layout
\lang english
Factor
\begin_inset Formula $\left(2\pi\right)^{\frac{d}{2}}$
\end_inset
@ -416,8 +764,6 @@ whole
\end_layout
\begin_layout Standard
\lang english
However, Fourier transform is linear, so we can in principle separate
\begin_inset Formula $S$
\end_inset
@ -463,7 +809,7 @@ reference "eq:W definition"
\end_inset
and legendre
and
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W sum in reciprocal space"
@ -484,7 +830,178 @@ W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\bas
\end_inset
where both sums should converge nicely.
where both sums expected to converge nicely.
We note that the elements of the translation operators
\begin_inset Formula $\tropr,\trops$
\end_inset
in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:translation operator"
plural "false"
caps "false"
noprefix "false"
\end_inset
can be rewritten as linear combinations of expressions
\begin_inset Formula $\ush{\nu}{\mu}\left(\uvec d\right)j_{n}\left(d\right),\ush{\nu}{\mu}\left(\uvec d\right)h_{n}^{(1)}\left(d\right)$
\end_inset
(TODO WRITE THEM EXPLICITLY IN THIS FORM), respectively, hence if we are
able evaluate the lattice sums sums
\begin_inset Note Note
status open
\begin_layout Plain Layout
CHECK THE FOLLOWING EXPRESSION FOR CORRECT FUNCTION ARGUMENTS
\end_layout
\end_inset
\begin_inset Formula
\begin{equation}
\sigma_{\nu}^{\mu}\left(\vect k\right)=\sum_{\vect n\in\ints^{d}\backslash\left\{ \vect 0\right\} }e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\ush{\nu}{\mu}\left(\uvec{R_{n}}\right)h_{n}^{(1)}\left(R_{n}\right),\label{eq:sigma lattice sums}
\end{equation}
\end_inset
then by linearity, we can get the
\begin_inset Formula $W_{\alpha\beta}\left(\vect k\right)$
\end_inset
operator as well.
\end_layout
\begin_layout Standard
TODO ADD MOROZ AND OTHER REFS HERE.
\begin_inset CommandInset citation
LatexCommand cite
key "linton_one-_2009"
literal "true"
\end_inset
offers an exponentially convergent Ewald-type summation method for
\begin_inset Formula $\sigma_{\nu}^{\mu}\left(\vect k\right)=\sigma_{\nu}^{\mu(\mathrm{S})}\left(\vect k\right)+\sigma_{\nu}^{\mu(\mathrm{L})}\left(\vect k\right)$
\end_inset
.
Here we rewrite them in a form independent on the convention used for spherical
harmonics (as long as they are complex
\begin_inset Note Note
status open
\begin_layout Plain Layout
lepší formulace
\end_layout
\end_inset
).
The short-range part reads (UNIFY INDEX NOTATION)
\begin_inset Formula
\begin{multline}
\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)=-\frac{2^{n+1}i}{k^{n+1}\sqrt{\pi}}\sum_{\vect R\in\Lambda}^{'}\left|\vect R\right|^{n}e^{i\vect{\beta}\cdot\vect R}Y_{n}^{m}\left(\vect R\right)\int_{\eta}^{\infty}e^{-\left|\vect R\right|^{2}\xi^{2}}e^{-k/4\xi^{2}}\xi^{2n}\ud\xi\\
+\frac{\delta_{n0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{k}{4\eta^{2}}\right)Y_{n}^{m},\label{eq:Ewald in 3D short-range part}
\end{multline}
\end_inset
\begin_inset Note Note
status open
\begin_layout Plain Layout
NEPATŘÍ TAM NĚJAKÁ DELTA FUNKCE K PŮVODNÍMU
\begin_inset Formula $\sigma_{n}^{m(0)}$
\end_inset
?
\end_layout
\end_inset
and the long-range part (FIXME, this is the 2D version; include the 1D and
3D lattice expressions as well)
\begin_inset Formula
\begin{multline}
\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)=-\frac{i^{n+1}}{k^{2}\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\\
\times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/k\right)^{n-2j}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\gamma_{pq}\right)^{2j-1}\label{eq:Ewald in 3D long-range part}
\end{multline}
\end_inset
where
\begin_inset Formula $\xi$
\end_inset
is TODO,
\begin_inset Formula $\beta_{pq}$
\end_inset
is TODO,
\begin_inset Formula $\Gamma_{j,pq}$
\end_inset
is TODO and
\begin_inset Formula $\eta$
\end_inset
is a real parameter that determines the pace of convergence of both parts.
The larger
\begin_inset Formula $\eta$
\end_inset
is, the faster
\begin_inset Formula $\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)$
\end_inset
converges but the slower
\begin_inset Formula $\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)$
\end_inset
converges.
Therefore (based on the lattice geometry) it has to be adjusted in a way
that a reasonable amount of terms needs to be evaluated numerically from
both
\begin_inset Formula $\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)$
\end_inset
and
\begin_inset Formula $\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)$
\end_inset
.
Generally, a good choice for
\begin_inset Formula $\eta$
\end_inset
is TODO; in order to achieve accuracy TODO, one has to evaluate the terms
on TODO lattice points.
(I HAVE SOME DERIVATIONS OF THE ESTIMATES IN MY NOTES; SHOULD I INCLUDE
THEM?)
\end_layout
\begin_layout Standard
In practice, the integrals in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Ewald in 3D short-range part"
plural "false"
caps "false"
noprefix "false"
\end_inset
can be easily evaluated by numerical quadrature and the incomplete
\begin_inset Formula $\Gamma$
\end_inset
-functions using the series TODO and TODO from DLMF.
\end_layout
\end_body

197
lepaper/intro.lyx
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@ -1,29 +1,37 @@
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\begin_document
\begin_header
\save_transient_properties true
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@ -58,6 +66,8 @@
\suppress_date false
\justification true
\use_refstyle 1
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\index Index
\shortcut idx
\color #008000
@ -66,10 +76,14 @@
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\quotes_language swedish
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@ -81,6 +95,159 @@
\begin_layout Section
Introduction
\begin_inset CommandInset label
LatexCommand label
name "sec:Introduction"
\end_inset
\end_layout
\begin_layout Standard
The problem of electromagnetic response of a system consisting of many compact
scatterers in various geometries, and its numerical solution, is relevant
to many branches of nanophotonics (TODO refs).
The most commonly used general approaches used in computational electrodynamics
, such as the finite difference time domain (FDTD) method or the finite
element method (FEM), are very often unsuitable for simulating systems
with larger number of scatterers due to their computational complexity.
Therefore, a common (frequency-domain) approach to get an approximate solution
of the scattering problem for many small particles has been the coupled
dipole approximation (CDA) where individual scatterers are reduced to electric
dipoles (characterised by a polarisability tensor) and coupled to each
other through Green's functions.
\end_layout
\begin_layout Standard
CDA is easy to implement and has favorable computational complexity but
suffers from at least two fundamental drawbacks.
The obvious one is that the dipole approximation is too rough for particles
with diameter larger than a small fraction of the wavelength.
The other one, more subtle, manifests itself in photonic crystal-like structure
s used in nanophotonics: there are modes in which the particles' electric
dipole moments completely vanish due to symmetry, regardless of how small
the particles are, and the excitations have quadrupolar or higher-degree
multipolar character.
These modes typically appear at the band edges where interesting phenomena
such as lasing or Bose-Einstein condensation have been observed – and CDA
by definition fails to capture such modes.
\end_layout
\begin_layout Standard
The natural way to overcome both limitations of CDA mentioned above is to
include higher multipoles into account.
Instead of polarisability tensor, the scattering properties of an individual
particle are then described a more general
\begin_inset Formula $T$
\end_inset
-matrix, and different particles' multipole excitations are coupled together
via translation operators, a generalisation of the Green's functions in
CDA.
This is the idea behind the
\emph on
multiple-scattering
\begin_inset Formula $T$
\end_inset
-matrix method
\emph default
(MSTMM) (TODO a.k.a something??), and it has been implemented previously for
a limited subset of problems (TODO refs and list the limitations of the
available).
\begin_inset Note Note
status open
\begin_layout Plain Layout
TODO přestože blablaba, moc se to nepoužívalo, protože je težké udělat to
správně.
\end_layout
\end_inset
Due to the limitations of the existing available codes, we have been developing
our own implementation of MSTMM, which we have used in several previous
works studying various physical phenomena in plasmonic nanoarrays (TODO
examples with refs).
\end_layout
\begin_layout Standard
Hereby we release our MSTMM implementation, the
\emph on
QPMS Photonic Multiple Scattering
\emph default
suite, as an open source software under the GNU General Public License
version 3.
(TODO refs to the code repositories.) QPMS allows for linear optics simulations
of arbitrary sets of compact scatterers in isotropic media.
The features include computations of electromagnetic response to external
driving, the related cross sections, and finding resonances of finite structure
s.
Moreover, in QPMS we extensively employ group theory to exploit the physical
symmetries of the system to further reduce the demands on computational
resources, enabling to simulate even larger systems.
\begin_inset Note Note
status open
\begin_layout Plain Layout
(TODO put a specific example here of how large system we are able to simulate?)
\end_layout
\end_inset
Although systems of large
\emph on
finite
\emph default
number of scatterers are the area where MSTMM excels the most—simply because
other methods fail due to their computational complexity—we also extended
the method onto infinite periodic systems (photonic crystals); this can
be used for quickly evaluating dispersions of such structures and also
their topological invariants (TODO).
The QPMS suite contains a core C library, Python bindings and several utilities
for routine computations, such as TODO.
It includes extensive Doxygen documentation, together with description
of the API, making extending and customising the code easy.
\end_layout
\begin_layout Standard
The current paper is organised as follows: Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Finite"
\end_inset
is devoted to MSTMM theory for finite systems, in Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Infinite"
\end_inset
we develop the theory for infinite periodic structures.
Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Applications"
\end_inset
demonstrates some basic practical results that can be obtained using QPMS.
Finally, in Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Comparison"
\end_inset
we comment on the computational complexity of MSTMM in comparison to other
methods.
\end_layout
\end_body

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@ -0,0 +1,426 @@
@book{bohren_absorption_1983,
title = {Absorption and Scattering of Light by Small Particles},
abstract = {Not Available},
urldate = {2014-05-09},
url = {http://adsabs.harvard.edu/abs/1983asls.book.....B},
author = {Bohren, Craig F. and Huffman, Donald R.},
year = {1983},
keywords = {Particles,LIGHT SCATTERING,ABSORPTION,DUST,THEORY},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/HES6WJTP/(Wiley science paperback series) Craig F. Bohren, Donald R. Huffman-Absorption and scattering of light by small particles-Wiley-VCH (1998).djvu}
}
@misc{SCUFF2,
title = {{{SCUFF}}-{{EM}}},
url = {http://homerreid.dyndns.org/scuff-EM/},
author = {Reid, Homer},
year = {2018},
note = {http://github.com/homerreid/scuff-EM}
}
@article{xu_calculation_1996,
title = {Calculation of the {{Addition Coefficients}} in {{Electromagnetic Multisphere}}-{{Scattering Theory}}},
volume = {127},
issn = {0021-9991},
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\textendash{}6.},
number = {2},
urldate = {2015-11-22},
journal = {Journal of Computational Physics},
doi = {10.1006/jcph.1996.0175},
url = {http://www.sciencedirect.com/science/article/pii/S0021999196901758},
author = {Xu, Yu-lin},
month = sep,
year = {1996},
pages = {285-298},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/8B2TWTJ2/1-s2.0-S0021999197956874-main (2).pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/NCD6BBNZ/Xu - 1996 - Calculation of the Addition Coefficients in Electr.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/NDSF7KI2/S0021999196901758.html}
}
@article{xu_efficient_1998,
title = {Efficient {{Evaluation}} of {{Vector Translation Coefficients}} in {{Multiparticle Light}}-{{Scattering Theories}}},
volume = {139},
issn = {0021-9991},
abstract = {Vector addition theorems are a necessary ingredient in the analytical solution of electromagnetic multiparticle-scattering problems. These theorems include a large number of vector addition coefficients. There exist three basic types of analytical expressions for vector translation coefficients: Stein's (Quart. Appl. Math.19, 15 (1961)), Cruzan's (Quart. Appl. Math.20, 33 (1962)), and Xu's (J. Comput. Phys.127, 285 (1996)). Stein's formulation relates vector translation coefficients with scalar translation coefficients. Cruzan's formulas use the Wigner 3jm symbol. Xu's expressions are based on the Gaunt coefficient. Since the scalar translation coefficient can also be expressed in terms of the Gaunt coefficient, the key to the expeditious and reliable calculation of vector translation coefficients is the fast and accurate evaluation of the Wigner 3jm symbol or the Gaunt coefficient. We present highly efficient recursive approaches to accurately evaluating Wigner 3jm symbols and Gaunt coefficients. Armed with these recursive approaches, we discuss several schemes for the calculation of the vector translation coefficients, using the three general types of formulation, respectively. Our systematic test calculations show that the three types of formulas produce generally the same numerical results except that the algorithm of Stein's type is less accurate in some particular cases. These extensive test calculations also show that the scheme using the formulation based on the Gaunt coefficient is the most efficient in practical computations.},
number = {1},
urldate = {2015-11-18},
journal = {Journal of Computational Physics},
doi = {10.1006/jcph.1997.5867},
url = {http://www.sciencedirect.com/science/article/pii/S0021999197958678},
author = {Xu, Yu-lin},
month = jan,
year = {1998},
pages = {137-165},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/STV5263F/Xu - 1998 - Efficient Evaluation of Vector Translation Coeffic.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/VMGZRSAA/S0021999197958678.html}
}
@book{mishchenko_light_1999-1,
title = {Light {{Scattering}} by {{Nonspherical Particles}}: {{Theory}}, {{Measurements}}, and {{Applications}}},
isbn = {978-0-08-051020-0},
shorttitle = {Light {{Scattering}} by {{Nonspherical Particles}}},
abstract = {There is hardly a field of science or engineering that does not have some interest in light scattering by small particles. For example, this subject is important to climatology because the energy budget for the Earth's atmosphere is strongly affected by scattering of solar radiation by cloud and aerosol particles, and the whole discipline of remote sensing relies largely on analyzing the parameters of radiation scattered by aerosols, clouds, and precipitation. The scattering of light by spherical particles can be easily computed using the conventional Mie theory. However, most small solid particles encountered in natural and laboratory conditions have nonspherical shapes. Examples are soot and mineral aerosols, cirrus cloud particles, snow and frost crystals, ocean hydrosols, interplanetary and cometary dust grains, and microorganisms. It is now well known that scattering properties of nonspherical particles can differ dramatically from those of "equivalent" (e.g., equal-volume or equal-surface-area) spheres. Therefore, the ability to accurately compute or measure light scattering by nonspherical particles in order to clearly understand the effects of particle nonsphericity on light scattering is very important.The rapid improvement of computers and experimental techniques over the past 20 years and the development of efficient numerical approaches have resulted in major advances in this field which have not been systematically summarized. Because of the universal importance of electromagnetic scattering by nonspherical particles, papers on different aspects of this subject are scattered over dozens of diverse research and engineering journals. Often experts in one discipline (e.g., biology) are unaware of potentially useful results obtained in another discipline (e.g., antennas and propagation). This leads to an inefficient use of the accumulated knowledge and unnecessary redundancy in research activities.This book offers the first systematic and unified discussion of light scattering by nonspherical particles and its practical applications and represents the state-of-the-art of this importantresearch field. Individual chapters are written by leading experts in respective areas and cover three major disciplines: theoretical and numerical techniques, laboratory measurements, and practical applications. An overview chapter provides a concise general introduction to the subject of nonspherical scattering and should be especially useful to beginners and those interested in fast practical applications. The audience for this book will include graduate students, scientists, and engineers working on specific aspects of electromagnetic scattering by small particles and its applications in remote sensing, geophysics, astrophysics, biomedical optics, and optical engineering.* The first systematic and comprehensive treatment of electromagnetic scattering by nonspherical particles and its applications* Individual chapters are written by leading experts in respective areas* Includes a survey of all the relevant literature scattered over dozens of basic and applied research journals* Consistent use of unified definitions and notation makes the book a coherent volume* An overview chapter provides a concise general introduction to the subject of light scattering by nonspherical particles* Theoretical chapters describe specific easy-to-use computer codes publicly available on the World Wide Web* Extensively illustrated with over 200 figures, 4 in color},
language = {en},
publisher = {{Academic Press}},
author = {Mishchenko, Michael I. and Hovenier, Joachim W. and Travis, Larry D.},
month = sep,
year = {1999},
keywords = {Science / Physics / General,Science / Applied Sciences,Science / Earth Sciences / Meteorology \& Climatology,Science / Physics / Geophysics,Science / Earth Sciences / Oceanography,Science / Earth Sciences / General}
}
@article{mackowski_calculation_1996,
title = {Calculation of the {{T}} Matrix and the Scattering Matrix for Ensembles of Spheres},
volume = {13},
issn = {1084-7529, 1520-8532},
language = {en},
number = {11},
urldate = {2015-11-09},
journal = {Journal of the Optical Society of America A},
doi = {10.1364/JOSAA.13.002266},
url = {https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-13-11-2266},
author = {Mackowski, Daniel W. and Mishchenko, Michael I.},
month = nov,
year = {1996},
pages = {2266},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/V59UV9H9/josaa-13-11-2266.pdf}
}
@misc{mackowski_mstm_2013,
title = {{{MSTM}} 3.0: {{A}} Multiple Sphere {{T}} -Matrix {{FORTRAN}} Code for Use on Parallel Computer Clusters},
url = {http://www.eng.auburn.edu/~dmckwski/scatcodes/},
author = {Mackowski, Daniel W.},
year = {2013},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/RQMQMC7H/mstm-manual-2013-v3.0.pdf}
}
@book{jackson_classical_1998,
address = {{New York}},
edition = {3 edition},
title = {Classical {{Electrodynamics Third Edition}}},
isbn = {978-0-471-30932-1},
abstract = {A revision of the defining book covering the physics and classical mathematics necessary to understand electromagnetic fields in materials and at surfaces and interfaces. The third edition has been revised to address the changes in emphasis and applications that have occurred in the past twenty years.},
language = {English},
publisher = {{Wiley}},
author = {Jackson, John David},
month = aug,
year = {1998},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/3BWPD4BK/John David Jackson-Classical Electrodynamics-Wiley (1999).djvu}
}
@misc{kristensson_spherical_2014,
title = {Spherical {{Vector Waves}}},
urldate = {2014-05-20},
url = {http://www.eit.lth.se/fileadmin/eit/courses/eit080f/Literature/book.pdf},
author = {Kristensson, Gerhard},
month = jan,
year = {2014},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/7MVDRPF2/Kristensson - 2014 - Spherical Vector Waves.pdf}
}
@incollection{mishchenko_t-matrix_1999,
title = {T-Matrix {{Method}} and {{Its Applications}}},
isbn = {978-0-08-051020-0},
abstract = {There is hardly a field of science or engineering that does not have some interest in light scattering by small particles. For example, this subject is important to climatology because the energy budget for the Earth's atmosphere is strongly affected by scattering of solar radiation by cloud and aerosol particles, and the whole discipline of remote sensing relies largely on analyzing the parameters of radiation scattered by aerosols, clouds, and precipitation. The scattering of light by spherical particles can be easily computed using the conventional Mie theory. However, most small solid particles encountered in natural and laboratory conditions have nonspherical shapes. Examples are soot and mineral aerosols, cirrus cloud particles, snow and frost crystals, ocean hydrosols, interplanetary and cometary dust grains, and microorganisms. It is now well known that scattering properties of nonspherical particles can differ dramatically from those of "equivalent" (e.g., equal-volume or equal-surface-area) spheres. Therefore, the ability to accurately compute or measure light scattering by nonspherical particles in order to clearly understand the effects of particle nonsphericity on light scattering is very important.The rapid improvement of computers and experimental techniques over the past 20 years and the development of efficient numerical approaches have resulted in major advances in this field which have not been systematically summarized. Because of the universal importance of electromagnetic scattering by nonspherical particles, papers on different aspects of this subject are scattered over dozens of diverse research and engineering journals. Often experts in one discipline (e.g., biology) are unaware of potentially useful results obtained in another discipline (e.g., antennas and propagation). This leads to an inefficient use of the accumulated knowledge and unnecessary redundancy in research activities.This book offers the first systematic and unified discussion of light scattering by nonspherical particles and its practical applications and represents the state-of-the-art of this importantresearch field. Individual chapters are written by leading experts in respective areas and cover three major disciplines: theoretical and numerical techniques, laboratory measurements, and practical applications. An overview chapter provides a concise general introduction to the subject of nonspherical scattering and should be especially useful to beginners and those interested in fast practical applications. The audience for this book will include graduate students, scientists, and engineers working on specific aspects of electromagnetic scattering by small particles and its applications in remote sensing, geophysics, astrophysics, biomedical optics, and optical engineering.* The first systematic and comprehensive treatment of electromagnetic scattering by nonspherical particles and its applications* Individual chapters are written by leading experts in respective areas* Includes a survey of all the relevant literature scattered over dozens of basic and applied research journals* Consistent use of unified definitions and notation makes the book a coherent volume* An overview chapter provides a concise general introduction to the subject of light scattering by nonspherical particles* Theoretical chapters describe specific easy-to-use computer codes publicly available on the World Wide Web* Extensively illustrated with over 200 figures, 4 in color},
language = {en},
booktitle = {Light {{Scattering}} by {{Nonspherical Particles}}: {{Theory}}, {{Measurements}}, and {{Applications}}},
publisher = {{Academic Press}},
author = {Mishchenko, Michael I. and {Travis, Larry D.} and Macke, Andreas},
editor = {Mishchenko, Michael I. and Hovenier, Joachim W. and Travis, Larry D.},
month = sep,
year = {1999},
keywords = {Science / Physics / General,Science / Applied Sciences,Science / Earth Sciences / Meteorology \& Climatology,Science / Physics / Geophysics,Science / Earth Sciences / Oceanography,Science / Earth Sciences / General},
pages = {147-172}
}
@article{schulz_point-group_1999,
title = {Point-Group Symmetries in Electromagnetic Scattering},
volume = {16},
issn = {1084-7529, 1520-8532},
language = {en},
number = {4},
urldate = {2016-08-04},
journal = {Journal of the Optical Society of America A},
doi = {10.1364/JOSAA.16.000853},
url = {https://www.osapublishing.org/abstract.cfm?URI=josaa-16-4-853},
author = {Schulz, F. Michael and Stamnes, Knut and Stamnes, J. J.},
month = apr,
year = {1999},
pages = {853},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/X9X48A6G/josaa-16-4-853.pdf}
}
@article{mishchenko_t-matrix_1994,
title = {T-Matrix Computations of Light Scattering by Large Spheroidal Particles},
volume = {109},
issn = {0030-4018},
abstract = {It is well known that T-matrix computations of light scattering by nonspherical particles may suffer from the ill-conditionality of the process of matrix inversion, which has precluded calculations for particle size parameters larger than about 25. It is demonstrated that calculating the T-matrix using extended-precision instead of double-precision floating-point variables is an effective approach for suppressing the numerical instability in computations for spheroids and allows one to increase the maximum particle size parameter for which T-matrix computations converge by as significant a factor as 2\textendash{}2.7. Yet this approach requires only a negligibly small extra memory, an affordable increase in CPU time consumption, and practically no additional programming effort. As a result, the range of particle size parameters, for which rigorous T-matrix computations of spheroidal scattering can be performed, now covers a substantial fraction of the gap between the domains of applicability of the Rayleigh and geometrical optics approximations.},
number = {1\textendash{}2},
urldate = {2017-01-18},
journal = {Optics Communications},
doi = {10.1016/0030-4018(94)90731-5},
url = {http://www.sciencedirect.com/science/article/pii/0030401894907315},
author = {Mishchenko, Michael I. and Travis, Larry D.},
month = jun,
year = {1994},
pages = {16-21},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/FT8KN354/mishchenko1994.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/TB425HGN/0030401894907315.html}
}
@article{mishchenko_t-matrix_1996,
series = {Light {{Scattering}} by {{Non}}-{{Spherical Particles}}},
title = {T-Matrix Computations of Light Scattering by Nonspherical Particles: {{A}} Review},
volume = {55},
issn = {0022-4073},
shorttitle = {T-Matrix Computations of Light Scattering by Nonspherical Particles},
abstract = {We review the current status of Waterman's T-matrix approach which is one of the most powerful and widely used tools for accurately computing light scattering by nonspherical particles, both single and composite, based on directly solving Maxwell's equations. Specifically, we discuss the analytical method for computing orientationally-averaged light-scattering characteristics for ensembles of nonspherical particles, the methods for overcoming the numerical instability in calculating the T matrix for single nonspherical particles with large size parameters and/or extreme geometries, and the superposition approach for computing light scattering by composite/aggregated particles. Our discussion is accompanied by multiple numerical examples demonstrating the capabilities of the T-matrix approach and showing effects of nonsphericity of simple convex particles (spheroids) on light scattering.},
number = {5},
urldate = {2017-01-18},
journal = {Journal of Quantitative Spectroscopy and Radiative Transfer},
doi = {10.1016/0022-4073(96)00002-7},
url = {http://www.sciencedirect.com/science/article/pii/0022407396000027},
author = {Mishchenko, Michael I. and Travis, Larry D. and Mackowski, Daniel W.},
month = may,
year = {1996},
pages = {535-575},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/8EA7QMDG/Mishchenko et al. - 1996 - T-matrix computations of light scattering by nonsp.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/HNWF8F6R/0022407396000027.html}
}
@article{hakala_lasing_2017,
title = {Lasing in Dark and Bright Modes of a Finite-Sized Plasmonic Lattice},
volume = {8},
copyright = {\textcopyright{} 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.},
issn = {2041-1723},
abstract = {Plasmonic dark modes are promising candidates for lasing applications. Here, Hakalaet al. show lasing at visible wavelengths in dark and bright modes of an array of silver nanoparticles combined with optically pumped dye molecules, opening up a route to utilization of all modes of plasmonic lattices.},
language = {en},
urldate = {2017-03-28},
journal = {Nature Communications},
doi = {10.1038/ncomms13687},
url = {http://www.nature.com/ncomms/2017/170103/ncomms13687/full/ncomms13687.html},
author = {Hakala, T. K. and Rekola, H. T. and V{\"a}kev{\"a}inen, A. I. and Martikainen, J.-P. and Ne{\v c}ada, M. and Moilanen, A. J. and T{\"o}rm{\"a}, P.},
month = jan,
year = {2017},
pages = {13687},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/73KCXGAP/ncomms13687.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/J6R8MHBH/ncomms13687-s1.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/X4PNEUQN/ncomms13687.html}
}
@article{mackowski_effective_2001,
series = {Light {{Scattering}} by {{Non}}-{{Spherical Particles}}},
title = {An Effective Medium Method for Calculation of the {{T}} Matrix of Aggregated Spheres},
volume = {70},
issn = {0022-4073},
abstract = {An effective medium approach is developed for describing the radiative scattering characteristics of large-scale clusters of spheres. The formulation assumes that the waves exciting each sphere in the cluster can be described by a regular vector harmonic expansion, centered about a common origin of the cluster, and characterized by an effective propagation constant mek. By combining this description with the multiple sphere interaction equations a `homogeneous' T matrix of the cluster is derived, which is analogous to using the effective propagation constant models of the Varadans in conjunction with Waterman's EBCM. However, it is shown that the homogeneous T matrix will not automatically satisfy energy conservation because it cannot account for dependent scattering effects among the spheres. A `discrete' formulation of the T matrix is then developed which retains the effective medium description of the exciting field yet provides for energy conservation. Illustrative calculations show that the effective medium T matrix can provide accurate predictions of the cross sections and scattering matrices of clusters containing a large number of uniformly packed spheres, yet this approximation uses a fraction of the computational time required for an exact solution.},
number = {4\textendash{}6},
urldate = {2017-06-05},
journal = {Journal of Quantitative Spectroscopy and Radiative Transfer},
doi = {10.1016/S0022-4073(01)00022-X},
url = {http://www.sciencedirect.com/science/article/pii/S002240730100022X},
author = {Mackowski, Daniel W.},
month = aug,
year = {2001},
pages = {441-464},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/9E7R7IRX/Mackowski - 2001 - An effective medium method for calculation of the .pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/D75CJ78C/S002240730100022X.html}
}
@book{dresselhaus_group_2008,
title = {Group {{Theory}}: {{Application}} to the {{Physics}} of {{Condensed Matter}}},
isbn = {978-3-540-32899-5},
abstract = {Every process in physics is governed by selection rules that are the consequence of symmetry requirements. The beauty and strength of group theory resides...},
urldate = {2017-10-31},
publisher = {{Springer, Berlin, Heidelberg}},
url = {//www.springer.com/us/book/9783540328971},
author = {Dresselhaus, Mildred S. and Dresselhaus, Gene and Jorio, Ado},
year = {2008},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/GFGPVB4A/Mildred_S._Dresselhaus,_Gene_Dresselhaus,_Ado_Jorio_Group_theory_application_to_the_physics_of_condensed_matter.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/E78682CJ/9783540328971.html}
}
@article{linton_lattice_2010,
title = {Lattice {{Sums}} for the {{Helmholtz Equation}}},
volume = {52},
issn = {0036-1445},
abstract = {A survey of different representations for lattice sums for the Helmholtz equation is made. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension \$d\_\textbackslash{}Lambda\$. Lattice sums are related to, and can be calculated from, the quasi-periodic Green's function and this object serves as the starting point of the analysis.},
number = {4},
journal = {SIAM Rev.},
doi = {10.1137/09075130X},
url = {http://epubs.siam.org/doi/10.1137/09075130X},
author = {Linton, C.},
month = jan,
year = {2010},
pages = {630-674},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/T86ATKYB/09075130x.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/ETB8X4S9/09075130X.html}
}
@book{bradley_mathematical_1972,
title = {The Mathematical Theory of Symmetry in Solids; Representation Theory for Point Groups and Space Groups},
isbn = {978-0-19-851920-1},
urldate = {2018-07-25},
publisher = {{Clarendon Press, Oxford}},
url = {http://gen.lib.rus.ec/book/index.php?md5=8539E3400CF65B6CC4FAC71B9DF286C5},
author = {Bradley, C. J. and Cracknell, A. P.},
year = {1972},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/SB5ZN5WH/C.J. Bradley, A.P. Cracknell - The mathematical theory of symmetry in solids_ representation theory for point groups and space groups (1972, Clarendon Press).djvu}
}
@article{moroz_quasi-periodic_2006,
title = {Quasi-Periodic {{Green}}'s Functions of the {{Helmholtz}} and {{Laplace}} Equations},
volume = {39},
issn = {0305-4470},
abstract = {A classical problem of free-space Green's function G 0{$\Lambda$} representations of the Helmholtz equation is studied in various quasi-periodic cases, i.e., when an underlying periodicity is imposed in less dimensions than is the dimension of an embedding space. Exponentially convergent series for the free-space quasi-periodic G 0{$\Lambda$} and for the expansion coefficients D L of G 0{$\Lambda$} in the basis of regular (cylindrical in two dimensions and spherical in three dimension (3D)) waves, or lattice sums, are reviewed and new results for the case of a one-dimensional (1D) periodicity in 3D are derived. From a mathematical point of view, a derivation of exponentially convergent representations for Schl{\"o}milch series of cylindrical and spherical Hankel functions of any integer order is accomplished. Exponentially convergent series for G 0{$\Lambda$} and lattice sums D L hold for any value of the Bloch momentum and allow G 0{$\Lambda$} to be efficiently evaluated also in the periodicity plane. The quasi-periodic Green's functions of the Laplace equation are obtained from the corresponding representations of G 0{$\Lambda$} of the Helmholtz equation by taking the limit of the wave vector magnitude going to zero. The derivation of relevant results in the case of a 1D periodicity in 3D highlights the common part which is universally applicable to any of remaining quasi-periodic cases. The results obtained can be useful for the numerical solution of boundary integral equations for potential flows in fluid mechanics, remote sensing of periodic surfaces, periodic gratings, and infinite arrays of resonators coupled to a waveguide, in many contexts of simulating systems of charged particles, in molecular dynamics, for the description of quasi-periodic arrays of point interactions in quantum mechanics, and in various ab initio first-principle multiple-scattering theories for the analysis of diffraction of classical and quantum waves.},
language = {en},
number = {36},
urldate = {2018-08-14},
journal = {J. Phys. A: Math. Gen.},
doi = {10.1088/0305-4470/39/36/009},
url = {http://stacks.iop.org/0305-4470/39/i=36/a=009},
author = {Moroz, Alexander},
year = {2006},
pages = {11247},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/268RXLJ4/Moroz - 2006 - Quasi-periodic Green's functions of the Helmholtz .pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/MGA5XR44/dlserr.pdf}
}
@article{linton_one-_2009,
title = {One- and Two-Dimensional Lattice Sums for the Three-Dimensional {{Helmholtz}} Equation},
volume = {228},
issn = {0021-9991},
abstract = {The accurate and efficient computation of lattice sums for the three-dimensional Helmholtz equation is considered for the cases where the underlying lattice is one- or two-dimensional. We demonstrate, using careful numerical computations, that the reduction method, in which the sums for a two-dimensional lattice are expressed as a sum of one-dimensional lattice sums leads to an order-of-magnitude improvement in performance over the well-known Ewald method. In the process we clarify and improve on a number of results originally formulated by Twersky in the 1970s.},
number = {6},
urldate = {2018-08-14},
journal = {Journal of Computational Physics},
doi = {10.1016/j.jcp.2008.11.013},
url = {http://www.sciencedirect.com/science/article/pii/S0021999108005962},
author = {Linton, C. M. and Thompson, I.},
month = apr,
year = {2009},
keywords = {Helmholtz equation,Ewald summation,Clausen function,Lattice reduction,Lattice sum,Schlömilch series},
pages = {1815-1829},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/YMRZHBY4/Linton ja Thompson - 2009 - One- and two-dimensional lattice sums for the thre.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/Z8CFQ6S9/S0021999108005962.html}
}
@book{olver_nist_2010,
edition = {1 Pap/Cdr},
title = {{{NIST}} Handbook of Mathematical Functions},
isbn = {978-0-521-14063-8},
urldate = {2018-08-20},
publisher = {{Cambridge University Press}},
url = {http://gen.lib.rus.ec/book/index.php?md5=7750A842DAAE07EBE30D597EB1352408},
author = {Olver, Frank W. J. and Lozier, Daniel W. and Boisvert, Ronald F. and Clark, Charles W.},
year = {2010},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/ZJ5LBQ8W/Olver ym. - 2010 - NIST handbook of mathematical functions.pdf}
}
@article{NIST:DLMF,
title = {{{NIST Digital Library}} of {{Mathematical Functions}}},
url = {http://dlmf.nist.gov/},
key = {DLMF},
note = {F.~W.~J. Olver, A.~B. Olde Daalhuis, D.~W. Lozier, B.~I. Schneider, R.~F. Boisvert, C.~W. Clark, B.~R. Miller and B.~V. Saunders, eds.}
}
@article{reid_efficient_2015,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1307.2966},
title = {Efficient {{Computation}} of {{Power}}, {{Force}}, and {{Torque}} in {{BEM Scattering Calculations}}},
volume = {63},
issn = {0018-926X, 1558-2221},
abstract = {We present concise, computationally efficient formulas for several quantities of interest -- including absorbed and scattered power, optical force (radiation pressure), and torque -- in scattering calculations performed using the boundary-element method (BEM) [also known as the method of moments (MOM)]. Our formulas compute the quantities of interest \textbackslash{}textit\{directly\} from the BEM surface currents with no need ever to compute the scattered electromagnetic fields. We derive our new formulas and demonstrate their effectiveness by computing power, force, and torque in a number of example geometries. Free, open-source software implementations of our formulas are available for download online.},
number = {8},
urldate = {2018-09-23},
journal = {IEEE Transactions on Antennas and Propagation},
doi = {10.1109/TAP.2015.2438393},
url = {http://arxiv.org/abs/1307.2966},
author = {Reid, M. T. Homer and Johnson, Steven G.},
month = aug,
year = {2015},
keywords = {Physics - Classical Physics,Physics - Computational Physics},
pages = {3588-3598},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/I2DXTKUF/Reid ja Johnson - 2015 - Efficient Computation of Power, Force, and Torque .pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/LG7AVZDH/1307.html}
}
@article{guo_lasing_2019,
title = {Lasing at \${{K}}\$ {{Points}} of a {{Honeycomb Plasmonic Lattice}}},
volume = {122},
abstract = {We study lasing at the high-symmetry points of the Brillouin zone in a honeycomb plasmonic lattice. We use symmetry arguments to define singlet and doublet modes at the K points of the reciprocal space. We experimentally demonstrate lasing at the K points that is based on plasmonic lattice modes and two-dimensional feedback. By comparing polarization properties to T-matrix simulations, we identify the lasing mode as one of the singlets with an energy minimum at the K point enabling feedback. Our results offer prospects for studies of topological lasing in radiatively coupled systems.},
number = {1},
urldate = {2019-01-10},
journal = {Phys. Rev. Lett.},
doi = {10.1103/PhysRevLett.122.013901},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.013901},
author = {Guo, R. and Ne{\v c}ada, M. and Hakala, T. K. and V{\"a}kev{\"a}inen, A. I. and T{\"o}rm{\"a}, P.},
month = jan,
year = {2019},
pages = {013901},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/TDGW4CZ5/Guo ym. - 2019 - Lasing at $K$ Points of a Honeycomb Plasmonic Latt.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/8BW4R9F6/PhysRevLett.122.html}
}
@article{mie_beitrage_1908,
title = {Beitr{\"a}ge Zur {{Optik}} Tr{\"u}ber {{Medien}}, Speziell Kolloidaler {{Metall{\"o}sungen}}},
volume = {330},
copyright = {Copyright \textcopyright{} 1908 WILEY-VCH Verlag GmbH \& Co. KGaA, Weinheim},
issn = {1521-3889},
language = {en},
number = {3},
urldate = {2014-11-30},
journal = {Ann. Phys.},
doi = {10.1002/andp.19083300302},
url = {http://onlinelibrary.wiley.com/doi/10.1002/andp.19083300302/abstract},
author = {Mie, Gustav},
month = jan,
year = {1908},
pages = {377-445},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/RM9J9RYH/Mie - 1908 - Beiträge zur Optik trüber Medien, speziell kolloid.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/F5A7EX6R/abstract.html}
}
@book{kristensson_scattering_2016,
address = {{Edison, NJ}},
title = {Scattering of {{Electromagnetic Waves}} by {{Obstacles}}},
isbn = {978-1-61353-221-8},
abstract = {This book is an introduction to some of the most important properties of electromagnetic waves and their interaction with passive materials and scatterers. The main purpose of the book is to give a theoretical treatment of these scattering phenomena, and to illustrate numerical computations of some canonical scattering problems for different geometries and materials. The scattering theory is also important in the theory of passive antennas, and this book gives several examples on this topic. Topics covered include an introduction to the basic equations used in scattering; the Green functions and dyadics; integral representation of fields; introductory scattering theory; scattering in the time domain; approximations and applications; spherical vector waves; scattering by spherical objects; the null-field approach; and propagation in stratified media. The book is organised along two tracks, which can be studied separately or together. Track 1 material is appropriate for a first reading of the textbook, while Track 2 contains more advanced material suited for the second reading and for reference. Exercises are included for each chapter.},
language = {English},
publisher = {{Scitech Publishing}},
url = {http://gen.lib.rus.ec/book/index.php?md5=00CCB3E221E741ADDB2E236FD4A9F002},
author = {Kristensson, Gerhard},
month = jul,
year = {2016},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/ZRYZ4KLK/Kristensson - 2016 - Scattering of Electromagnetic Waves by Obstacles.pdf}
}
@article{ganesh_convergence_2012,
title = {Convergence Analysis with Parameter Estimates for a Reduced Basis Acoustic Scattering {{T}}-Matrix Method},
volume = {32},
issn = {0272-4979},
abstract = {Abstract. The celebrated truncated T-matrix method for wave propagation models belongs to a class of the reduced basis methods (RBMs), with the parameters bein},
language = {en},
number = {4},
urldate = {2019-07-03},
journal = {IMA J Numer Anal},
doi = {10.1093/imanum/drr041},
url = {https://academic.oup.com/imajna/article/32/4/1348/654510},
author = {Ganesh, M. and Hawkins, S. C. and Hiptmair, R.},
month = oct,
year = {2012},
pages = {1348-1374},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/2CRM9IEU/ganesh2012.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/KLKJBTZU/Ganesh ym. - 2012 - Convergence analysis with parameter estimates for .pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/N5H8B7SF/654510.html}
}
@book{chew_fast_2000,
series = {Artech {{House Antennas}} and {{Propagation Library}}},
title = {Fast and {{Efficient Algorithms}} in {{Computational Electromagnetics}}},
isbn = {978-1-58053-152-8},
urldate = {2019-07-31},
publisher = {{Artech House Publishers}},
url = {http://gen.lib.rus.ec/book/index.php?md5=2A7D2CE03DB8CFC14E7189E9A441F759},
author = {Chew, Weng Cho and Jin, Jian-Ming and Michielssen, Eric and Song, Jiming},
year = {2000}
}
@article{pourjamal_lasing_2019,
title = {Lasing in {{Ni Nanodisk Arrays}}},
abstract = {Lasing in Ni Nanodisk Arrays},
language = {en},
urldate = {2019-07-31},
journal = {ACS Nano},
doi = {10.1021/acsnano.9b01006},
url = {https://pubs.acs.org/doi/suppl/10.1021/acsnano.9b01006},
author = {Pourjamal, Sara and Hakala, Tommi K. and Ne{\v c}ada, Marek and {Freire-Fern{\'a}ndez}, Francisco and Kataja, Mikko and Rekola, Heikki and Martikainen, Jani-Petri and T{\"o}rm{\"a}, P{\"a}ivi and van Dijken, Sebastiaan},
month = apr,
year = {2019},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/C4SN68I6/Pourjamal ym. - 2019 - Lasing in Ni Nanodisk Arrays.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/S6AU6FV9/acsnano.html}
}