1179 lines
26 KiB
Plaintext
1179 lines
26 KiB
Plaintext
#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\pdf_title "Sähköpajan päiväkirja"
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\pdf_author "Marek Nečada"
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\begin_layout Standard
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\lang english
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\begin_inset FormulaMacro
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\newcommand{\vect}[1]{\mathbf{#1}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\ush}[2]{Y_{#1,#2}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\svwfr}[3]{\mathbf{u}_{#1,#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\svwfs}[3]{\mathbf{v}_{#1,#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffs}{a}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffsi}[3]{\coeffs_{#1,#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffsip}[4]{\coeffs_{#1}^{#2,#3,#4}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffr}{p}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffri}[3]{p_{#1,#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffrip}[4]{p_{#1}^{#2,#3,#4}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffripext}[4]{p_{\mathrm{ext}(#1)}^{#2,#3,#4}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\transop}{S}
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\end_inset
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\end_layout
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\begin_layout Section
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\lang english
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\begin_inset Formula $T$
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\end_inset
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-matrix simulations
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:T-matrix-simulations"
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\end_inset
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\end_layout
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||
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||
\begin_layout Standard
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||
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\lang english
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In order to get more detailed insight into the mode structure of the lattice
|
||
around the lasing
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||
\begin_inset Formula $\Kp$
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||
\end_inset
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||
|
||
-point – most importantly, how much do the mode frequencies at the
|
||
\begin_inset Formula $\Kp$
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||
\end_inset
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||
|
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-points differ from the empty lattice model – we performed multiple-scattering
|
||
|
||
\begin_inset Formula $T$
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||
\end_inset
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||
|
||
-matrix simulations
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
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||
key "mackowski_analysis_1991"
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||
|
||
\end_inset
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||
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for an infinite lattice based on our systems' geometry.
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We give a brief overview of this method in the subsections
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\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "sub:The-multiple-scattering-problem"
|
||
|
||
\end_inset
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||
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,
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||
\begin_inset CommandInset ref
|
||
LatexCommand ref
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||
reference "sub:Periodic-systems"
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||
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\end_inset
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||
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below.
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||
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\lang finnish
|
||
The top advantage of the multiple-scattering
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\begin_inset Formula $T$
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||
\end_inset
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||
|
||
-matrix approach is its computational efficiency for large finite systems
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||
of nanoparticles.
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||
In the lattice mode analysis in this work, however, we use it here for
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||
another reason, specifically the relative ease of describing symmetries
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
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||
key "schulz_point-group_1999"
|
||
|
||
\end_inset
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||
|
||
.
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||
A brief theoretical overview of the method is presented in subsections
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "sub:The-multiple-scattering-problem"
|
||
|
||
\end_inset
|
||
|
||
–
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||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "sub:Periodic-systems"
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||
|
||
\end_inset
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||
|
||
below.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Fig.
|
||
xxx(a) shows the dispersions around the
|
||
\begin_inset Formula $\Kp$
|
||
\end_inset
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||
|
||
-point for the cylindrical nanoparticles used in our experiment.
|
||
|
||
\lang english
|
||
The
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||
\begin_inset Formula $T$
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||
\end_inset
|
||
|
||
-matrix of a single cylindrical nanoparticle was computed using the scuff-tmatri
|
||
x application from the SCUFF-EM suite~
|
||
\lang finnish
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "SCUFF2,reid_efficient_2015"
|
||
|
||
\end_inset
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||
|
||
|
||
\lang english
|
||
and the system was solved up to the
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||
\begin_inset Formula $l_{\mathrm{max}}=3$
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||
\end_inset
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||
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||
(octupolar) degree of electric and magnetic spherical multipole.
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||
For comparison, Fig.
|
||
xxx(b) shows the dispersions for a system where the cylindrical nanoparticles
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||
were replaced with spherical ones with radius of
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||
\begin_inset Formula $40\,\mathrm{nm}$
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||
\end_inset
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||
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||
, whose
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||
\begin_inset Formula $T$
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||
\end_inset
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||
|
||
-matrix was calculated semi-analytically using the Lorenz-Mie theory.
|
||
In both cases, we used gold with interpolated tabulated values of refraction
|
||
index
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
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||
key "johnson_optical_1972"
|
||
|
||
\end_inset
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||
|
||
for the nanoparticles and constant reffraction index of 1.52 for the background
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||
medium.
|
||
In both cases, the diffracted orders do split into separate bands according
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||
to the
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||
\lang finnish
|
||
|
||
\begin_inset Formula $\Kp$
|
||
\end_inset
|
||
|
||
-point
|
||
\lang english
|
||
irreducible representations (cf.
|
||
section
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "sm:symmetries"
|
||
|
||
\end_inset
|
||
|
||
), but the splitting is extremely weak – not exceeding
|
||
\begin_inset Formula $1\,\mathrm{meV}$
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||
\end_inset
|
||
|
||
for the spherical and even less for the cylindrical nanoparticles.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
|
||
\lang english
|
||
This is most likely due to the frequencies in our experiment being far below
|
||
the resonances of the nanoparticles, with the largest elements of the
|
||
\begin_inset Formula $T$
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||
\end_inset
|
||
|
||
-matrix being of the order of
|
||
\begin_inset Formula $10^{-3}$
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||
\end_inset
|
||
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||
(for power-normalised waves).
|
||
The nanoparticles are therefore almost transparent, but still suffice to
|
||
provide enough feedback for lasing.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
The multiple-scattering problem
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "sub: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 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\svwfr 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 $\svwfr lmt$
|
||
\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 can be found e.g.
|
||
in [REF]
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
few words about different conventions?
|
||
\end_layout
|
||
|
||
\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 $\svwfs lmt$
|
||
\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\svwfs 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 polarisation 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"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The singular SVWFs 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 SVWFs,
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\svwfs lmt\left(\vect r_{n}\right)=\sum_{l',m',t'}\transop^{l'm't';lmt}\left(\vect R_{n'}-\vect R_{n}\right)\svwfr{l'}{m'}{t'}\left(\vect r_{n'}\right),\qquad\left|\vect r_{n'}\right|<\left|\vect R_{n'}-\vect R_{n}\right|.\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"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
If we write the field incident onto
|
||
\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
|
||
|
||
,
|
||
\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 eqref
|
||
reference "eq:E_inc"
|
||
|
||
\end_inset
|
||
|
||
–
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
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,
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\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
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\sum_{l,m,t}\coeffrip nlmt\svwfr lmt\left(\vect r_{n}\right)=\sum_{l,m,t}\coeffripext nlmt\svwfr lmt\left(\vect r_{n}\right)+\sum_{n'\ne n}\sum_{l,m,t}\coeffsip{n'}lmt\svwfs lmt\left(\vect r_{n'}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\sum_{l,m,t}\coeffrip nlmt\svwfr lmt\left(\vect r_{n}\right)=\sum_{l,m,t}\coeffripext nlmt\svwfr lmt\left(\vect r_{n}\right)+\sum_{n'\ne n}\sum_{l,m,t}\coeffsip{n'}lmt\sum_{l',m',t'}\transop^{l'm't';lmt}\left(\vect R_{n}-\vect R_{n'}\right)\svwfr{l'}{m'}{t'}\left(\vect r_{n}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\[
|
||
\sum_{l,m,t}\coeffrip nlmt\svwfr lmt\left(\vect r_{n}\right)=\sum_{l,m,t}\coeffripext nlmt\svwfr lmt\left(\vect r_{n}\right)+\sum_{n'\ne n}\sum_{l,m,t}\sum_{l',m',t'}\coeffsip{n'}{l'}{m'}{t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)\svwfr lmt\left(\vect r_{n}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\coeffrip nlmt=\coeffripext nlmt+\sum_{n'\ne n}\sum_{l',m',t'}\coeffsip{n'}{l'}{m'}{t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
(
|
||
\begin_inset Formula $\coeffsip{n'}{l'}{m'}{t'}=\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''}$
|
||
\end_inset
|
||
|
||
)
|
||
\begin_inset Formula
|
||
\[
|
||
\coeffrip nlmt=\coeffripext nlmt+\sum_{n'\ne n}\sum_{l',m',t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\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)\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''},\label{eq:multiplescattering element-wise}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\coeffripext nlmt$
|
||
\end_inset
|
||
|
||
are the expansion coefficients of the external field around the
|
||
\begin_inset Formula $n$
|
||
\end_inset
|
||
|
||
-th particle,
|
||
\begin_inset Formula $\vect E_{0}\left(\vect r\right)=\sum_{l,m,t}\coeffripext nlmt\svwfr lmt\left(\vect r_{n}\right).$
|
||
\end_inset
|
||
|
||
It is practical to get rid of the SVWF indices, rewriting
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
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 eqref
|
||
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 eqref
|
||
reference "eq:E_scat"
|
||
|
||
\end_inset
|
||
|
||
from all particles.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Periodic systems and mode analysis
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "sub: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
|
||
\[
|
||
\coeffs_{i\alpha}=e^{i\vect k\cdot\vect R_{i}}\coeffs_{\alpha}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
(assuming the incident external field has the same periodicity,
|
||
\begin_inset Formula $\coeffr_{\mathrm{ext}(i\alpha)}=e^{i\vect k\cdot\vect R_{i}}p_{\mathrm{ext}\left(\alpha\right)}$
|
||
\end_inset
|
||
|
||
) where
|
||
\begin_inset Formula $\alpha$
|
||
\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 eqref
|
||
reference "eq:multiple scattering per particle a"
|
||
|
||
\end_inset
|
||
|
||
then takes the form
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\coeffs_{i\alpha}=T_{\alpha}\left(\coeffr_{\mathrm{ext}(i\alpha)}+\sum_{(i',\alpha')\ne\left(i,\alpha\right)}S_{i\alpha,i'\alpha}\coeffs_{i'\alpha'}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\coeffs_{i\alpha}-T_{\alpha}\sum_{(i',\alpha')\ne\left(i,\alpha\right)}S_{i\alpha,i'\alpha'}e^{i\vect k\cdot\left(\vect R_{i'}-\vect R_{i}\right)}\coeffs_{i\alpha'}=T_{\alpha}\coeffr_{\mathrm{ext}(i\alpha)}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
or, labeling
|
||
\begin_inset Formula $W_{\alpha\alpha'}=\sum_{i';(i',\alpha')\ne\left(i,\alpha\right)}S_{i\alpha,i'\alpha'}e^{i\vect k\cdot\left(\vect R_{i'}-\vect R_{i}\right)}=\sum_{i';(i',\alpha')\ne\left(0,\alpha\right)}S_{0\alpha,i'\alpha'}e^{i\vect k\cdot\vect R_{i'}}$
|
||
\end_inset
|
||
|
||
and using the quasiperiodicity,
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\sum_{\alpha'}\left(\delta_{\alpha\alpha'}\mathbb{I}-T_{\alpha}W_{\alpha\alpha'}\right)\coeffs_{\alpha'}=T_{\alpha}\coeffr_{\mathrm{ext}(\alpha)},\label{eq:multiple scattering per particle a periodic}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\coeffs_{\alpha}-T_{\alpha}\sum_{\alpha'}W_{\alpha\alpha'}\coeffs_{\alpha'}=T_{\alpha}\coeffr_{\mathrm{ext}(\alpha)},\label{eq:multiple scattering per particle a periodic-2}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
which reduces the linear problem
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
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 $W_{\alpha\alpha'}$
|
||
\end_inset
|
||
|
||
; this is performed using exponentially convergent Ewald-type representations
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "linton_lattice_2010"
|
||
|
||
\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 $a_{\alpha}$
|
||
\end_inset
|
||
|
||
that satisfy eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:multiple scattering per particle a periodic-2"
|
||
|
||
\end_inset
|
||
|
||
with zero right-hand side.
|
||
That can happen if the block matrix
|
||
\begin_inset Formula $M\left(\omega,\vect k\right)=\left\{ \delta_{\alpha\alpha'}\mathbb{I}-T_{\alpha}\left(\vect{\omega}\right)W_{\alpha\alpha'}\left(\omega,\vect k\right)\right\} _{\alpha\alpha'}$
|
||
\end_inset
|
||
|
||
from the left hand side of
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:multiple scattering per particle a periodic"
|
||
|
||
\end_inset
|
||
|
||
is singular (here we explicitely note the
|
||
\begin_inset Formula $\omega,\vect k$
|
||
\end_inset
|
||
|
||
depence).
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
In other words, the energy bands of the lattice are given by
|
||
\begin_inset Formula
|
||
\[
|
||
\det M\left(\omega,\vect k\right)=0.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\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 sld
|
||
\end_inset
|
||
|
||
almost zero
|
||
\begin_inset Quotes srd
|
||
\end_inset
|
||
|
||
singular value.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
|
||
\lang english
|
||
Symmetries
|
||
\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"
|
||
|
||
\end_inset
|
||
|
||
; here we use the same notation.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
We analyse the symmetries of the system in the same SVWF 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.
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
The symmetry makes the
|
||
\begin_inset Formula $M\left(\omega,\vect k\right)$
|
||
\end_inset
|
||
|
||
matrix defined above invariant to the symmetry operations at the
|
||
\begin_inset Formula $\Kp$
|
||
\end_inset
|
||
|
||
-point,
|
||
\begin_inset Formula
|
||
\[
|
||
RM\left(\omega,\vect K\right)R^{-1}=M\left(\omega,\vect K\right),\qquad R\in D_{3h}.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
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
|
||
|
||
.
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
The characters of the equivalence representation
|
||
\begin_inset Formula $\Gamma^{\mathrm{equiv.}}$
|
||
\end_inset
|
||
|
||
are given by the formula
|
||
\begin_inset Formula $\chi^{\mathrm{equiv.}}=\sum_{\alpha}\delta_{R_{\alpha}\vect r_{\alpha},\vect r_{\alpha}}e^{i\vect K_{m}\cdot\vect r_{\alpha}}$
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\vect r_{\alpha}$
|
||
\end_inset
|
||
|
||
are the positions of the nanoparticles with respect
|
||
\end_layout
|
||
|
||
\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"
|
||
|
||
\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"
|
||
|
||
\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"
|
||
|
||
\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 $\svwfs lm{p,t}$
|
||
\end_inset
|
||
|
||
around the constituting nanoparticles (labeled
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
),
|
||
\begin_inset Formula
|
||
\[
|
||
\vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}=\sum_{l,m,p,t}U_{\Gamma,r,i}^{p,t,l,m}\svwfs lm{p,t},
|
||
\]
|
||
|
||
\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"
|
||
|
||
\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.
|
||
xxx.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset bibtex
|
||
LatexCommand bibtex
|
||
bibfiles "hexarray-theory"
|
||
options "plain"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|