1009 lines
26 KiB
Plaintext
1009 lines
26 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
|
||
\lyxformat 583
|
||
\begin_document
|
||
\begin_header
|
||
\save_transient_properties true
|
||
\origin unavailable
|
||
\textclass article
|
||
\use_default_options true
|
||
\maintain_unincluded_children false
|
||
\language english
|
||
\language_package default
|
||
\inputencoding utf8
|
||
\fontencoding auto
|
||
\font_roman "default" "TeX Gyre Pagella"
|
||
\font_sans "default" "default"
|
||
\font_typewriter "default" "default"
|
||
\font_math "auto" "auto"
|
||
\font_default_family default
|
||
\use_non_tex_fonts false
|
||
\font_sc false
|
||
\font_roman_osf true
|
||
\font_sans_osf false
|
||
\font_typewriter_osf false
|
||
\font_sf_scale 100 100
|
||
\font_tt_scale 100 100
|
||
\use_microtype false
|
||
\use_dash_ligatures true
|
||
\graphics default
|
||
\default_output_format default
|
||
\output_sync 0
|
||
\bibtex_command default
|
||
\index_command default
|
||
\float_placement class
|
||
\float_alignment class
|
||
\paperfontsize default
|
||
\spacing single
|
||
\use_hyperref true
|
||
\pdf_author "Marek Nečada"
|
||
\pdf_bookmarks true
|
||
\pdf_bookmarksnumbered false
|
||
\pdf_bookmarksopen false
|
||
\pdf_bookmarksopenlevel 1
|
||
\pdf_breaklinks false
|
||
\pdf_pdfborder false
|
||
\pdf_colorlinks false
|
||
\pdf_backref false
|
||
\pdf_pdfusetitle true
|
||
\papersize default
|
||
\use_geometry false
|
||
\use_package amsmath 1
|
||
\use_package amssymb 1
|
||
\use_package cancel 1
|
||
\use_package esint 1
|
||
\use_package mathdots 1
|
||
\use_package mathtools 1
|
||
\use_package mhchem 1
|
||
\use_package stackrel 1
|
||
\use_package stmaryrd 1
|
||
\use_package undertilde 1
|
||
\cite_engine basic
|
||
\cite_engine_type default
|
||
\biblio_style plain
|
||
\use_bibtopic false
|
||
\use_indices false
|
||
\paperorientation portrait
|
||
\suppress_date false
|
||
\justification true
|
||
\use_refstyle 1
|
||
\use_minted 0
|
||
\use_lineno 0
|
||
\index Index
|
||
\shortcut idx
|
||
\color #008000
|
||
\end_index
|
||
\secnumdepth 3
|
||
\tocdepth 3
|
||
\paragraph_separation indent
|
||
\paragraph_indentation default
|
||
\is_math_indent 0
|
||
\math_numbering_side default
|
||
\quotes_style english
|
||
\dynamic_quotes 0
|
||
\papercolumns 1
|
||
\papersides 1
|
||
\paperpagestyle default
|
||
\tablestyle default
|
||
\tracking_changes false
|
||
\output_changes false
|
||
\html_math_output 0
|
||
\html_css_as_file 0
|
||
\html_be_strict false
|
||
\end_header
|
||
|
||
\begin_body
|
||
|
||
\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
|
||
|
||
\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
|
||
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
|
||
|
||
, but this time, the system shall be periodic: let there be a
|
||
\begin_inset Formula $d$
|
||
\end_inset
|
||
|
||
-dimensional (
|
||
\begin_inset Formula $d$
|
||
\end_inset
|
||
|
||
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
|
||
|
||
, and let the lattice points be labeled with an
|
||
\begin_inset Formula $d$
|
||
\end_inset
|
||
|
||
-dimensional integar multiindex
|
||
\begin_inset Formula $\vect n\in\ints^{d}$
|
||
\end_inset
|
||
|
||
, 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
|
||
|
||
.
|
||
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
|
||
\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
|
||
|
||
|
||
\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
|
||
|
||
.
|
||
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{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
|
||
|
||
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
|
||
|
||
lattice Fourier transform
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
of the translation operator,
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
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 Standard
|
||
The problem evaluating
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:W definition"
|
||
|
||
\end_inset
|
||
|
||
is the asymptotic behaviour of the translation operator,
|
||
\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 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
|
||
Note that
|
||
\begin_inset Formula $d$
|
||
\end_inset
|
||
|
||
here is dimensionality of the lattice, not the space it lies in, which
|
||
I for certain reasons assume to be three.
|
||
(TODO few notes on integration and reciprocal lattices in some appendix)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\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.
|
||
We use the same idea here, but the technical details are more complicated
|
||
than in electrostatics.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Let us re-express the sum in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
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
|
||
\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}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
The translation operator
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
is now a function defined in the whole 3d space;
|
||
\begin_inset Formula $\vect r_{\alpha},\vect r_{\beta}$
|
||
\end_inset
|
||
|
||
are the displacements of scatterers
|
||
\begin_inset Formula $\alpha$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\beta$
|
||
\end_inset
|
||
|
||
in a unit cell.
|
||
The arrow notation
|
||
\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})$
|
||
\end_inset
|
||
|
||
means
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
translation operator for spherical waves originating in
|
||
\begin_inset Formula $\vect r+\vect r_{\beta}$
|
||
\end_inset
|
||
|
||
evaluated in
|
||
\begin_inset Formula $\vect r_{\alpha}$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
and obviously
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
is in fact a function of a single 3d argument,
|
||
\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
|
||
\end_inset
|
||
|
||
.
|
||
Expression
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:W integral"
|
||
|
||
\end_inset
|
||
|
||
can be rewritten as
|
||
\begin_inset Formula
|
||
\[
|
||
W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where changed the sign of
|
||
\begin_inset Formula $\vect r/\vect{\bullet}$
|
||
\end_inset
|
||
|
||
has been swapped under integration, utilising evenness of
|
||
\begin_inset Formula $\dc{\basis u}$
|
||
\end_inset
|
||
|
||
.
|
||
Fourier transform of product is convolution of Fourier transforms, so (using
|
||
formula
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Dirac comb uaFt"
|
||
|
||
\end_inset
|
||
|
||
for the Fourier transform of Dirac comb)
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)\nonumber \\
|
||
& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\nonumber \\
|
||
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\
|
||
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\nonumber
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Factor
|
||
\begin_inset Formula $\left(2\pi\right)^{\frac{d}{2}}$
|
||
\end_inset
|
||
|
||
cancels out with the
|
||
\begin_inset Formula $\left(2\pi\right)^{-\frac{d}{2}}$
|
||
\end_inset
|
||
|
||
factor appearing in the convolution/product formula in the unitary angular
|
||
momentum convention.
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
As such, this is not extremely helpful because the the
|
||
\emph on
|
||
whole
|
||
\emph default
|
||
translation operator
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
has singularities in origin, hence its Fourier transform
|
||
\begin_inset Formula $\uaft S$
|
||
\end_inset
|
||
|
||
will decay poorly.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
However, Fourier transform is linear, so we can in principle separate
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
in two parts,
|
||
\begin_inset Formula $S=S^{\textup{L}}+S^{\textup{S}}$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\begin_inset Formula $S^{\textup{S}}$
|
||
\end_inset
|
||
|
||
is a short-range part that decays sufficiently fast with distance so that
|
||
its direct-space lattice sum converges well;
|
||
\begin_inset Formula $S^{\textup{S}}$
|
||
\end_inset
|
||
|
||
must as well contain all the singularities of
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
in the origin.
|
||
The other part,
|
||
\begin_inset Formula $S^{\textup{L}}$
|
||
\end_inset
|
||
|
||
, will retain all the slowly decaying terms of
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
but it also has to be smooth enough in the origin, so that its Fourier
|
||
transform
|
||
\begin_inset Formula $\uaft{S^{\textup{L}}}$
|
||
\end_inset
|
||
|
||
decays fast enough.
|
||
(The same idea lies behind the Ewald summation in electrostatics.) Using
|
||
the linearity of Fourier transform and formulae
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:W definition"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:W sum in reciprocal space"
|
||
|
||
\end_inset
|
||
|
||
, the operator
|
||
\begin_inset Formula $W_{\alpha\beta}$
|
||
\end_inset
|
||
|
||
can then be re-expressed as
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\
|
||
W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect 0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
|
||
W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
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
|
||
\end_document
|