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#LyX 2.1 created this file. For more info see http://www.lyx.org/
\lyxformat 474
\begin_document
\begin_header
\textclass article
\use_default_options true
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\language finnish
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\use_hyperref true
\pdf_title "Sähköpajan päiväkirja"
\pdf_author "Marek Nečada"
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\cite_engine basic
\cite_engine_type default
\biblio_style plain
\use_bibtopic false
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\index Index
\shortcut idx
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\end_index
\secnumdepth 3
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\quotes_language swedish
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\end_header
\begin_body
\begin_layout Standard
\lang english
\begin_inset FormulaMacro
\newcommand{\vect}[1]{\mathbf{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ush}[2]{Y_{#1,#2}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\svwfr}[3]{\mathbf{u}_{#1,#2}^{#3}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\svwfs}[3]{\mathbf{v}_{#1,#2}^{#3}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\coeffs}{a}
\end_inset
\begin_inset FormulaMacro
\newcommand{\coeffsi}[3]{\coeffs_{#1,#2}^{#3}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\coeffsip}[4]{\coeffs_{#1}^{#2,#3,#4}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\coeffr}{p}
\end_inset
\begin_inset FormulaMacro
\newcommand{\coeffri}[3]{p_{#1,#2}^{#3}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\coeffrip}[4]{p_{#1}^{#2,#3,#4}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\coeffripext}[4]{p_{\mathrm{ext}(#1)}^{#2,#3,#4}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\transop}{S}
\end_inset
\end_layout
\begin_layout Section
\lang english
\begin_inset Formula $T$
\end_inset
-matrix simulations
\begin_inset CommandInset label
LatexCommand label
name "sec:T-matrix-simulations"
\end_inset
\end_layout
\begin_layout Standard
\lang english
In order to get more detailed insight into the mode structure of the lattice
around the lasing
\begin_inset Formula $\Kp$
\end_inset
-point most importantly, how much do the mode frequencies at the
\begin_inset Formula $\Kp$
\end_inset
-points differ from the empty lattice model we performed multiple-scattering
\begin_inset Formula $T$
\end_inset
-matrix simulations
\begin_inset CommandInset citation
LatexCommand cite
key "mackowski_analysis_1991"
\end_inset
for an infinite lattice based on our systems' geometry.
We give a brief overview of this method in the subsections
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:The-multiple-scattering-problem"
\end_inset
,
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Periodic-systems"
\end_inset
below.
\lang finnish
The top advantage of the multiple-scattering
\begin_inset Formula $T$
\end_inset
-matrix approach is its computational efficiency for large finite systems
of nanoparticles.
In the lattice mode analysis in this work, however, we use it here for
another reason, specifically the relative ease of describing symmetries
\begin_inset CommandInset citation
LatexCommand cite
key "schulz_point-group_1999"
\end_inset
.
\end_layout
\begin_layout Standard
\lang english
The
\begin_inset Formula $T$
\end_inset
-matrix of a single nanoparticle was computed using the scuff-tmatrix applicatio
n from the SCUFF-EM suite~
\lang finnish
\begin_inset CommandInset citation
LatexCommand cite
key "SCUFF2,reid_efficient_2015"
\end_inset
\lang english
and the system was solved up to the
\begin_inset Formula $l_{\mathrm{max}}=3$
\end_inset
(octupolar) degree of electric and magnetic spherical multipole.
\end_layout
\begin_layout Standard
\lang english
We did not find any deviation from the empty lattice diffracted orders exceeding
the numerical precision of the computation (about 2 meV).
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 $T$
\end_inset
-matrix being of the order of
\begin_inset Formula $10^{-3}$
\end_inset
(for power-normalised waves).
The nanoparticles are therefore almost transparent, but still suffice to
provide 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 Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "hexarray-theory"
options "plain"
\end_inset
\end_layout
\end_body
\end_document