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\pdf_author "Marek Nečada"
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\end_header
\begin_body
\begin_layout Section
Infinite periodic systems
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\begin_inset CommandInset label
LatexCommand label
name "sec:Infinite"
\end_inset
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\begin_inset FormulaMacro
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\newcommand{\dlv}{\vect a}
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\end_inset
\begin_inset FormulaMacro
\newcommand{\rlv}{\vect b}
\end_inset
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\end_layout
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\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
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opposed to, for example, FDTD with a cubic mesh applied to a honeycomb
lattice), which makes e.g.
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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
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finite systems gives a possibility to study finite-size effects in periodic
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scatterer arrays.
\end_layout
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\begin_layout Subsection
Formulation of the problem
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\begin_inset CommandInset label
LatexCommand label
name "subsec:Quasiperiodic scattering problem"
\end_inset
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\end_layout
\begin_layout Standard
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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
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, but this time, the system shall be periodic: let there be a
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\begin_inset Formula $d$
\end_inset
-dimensional (
\begin_inset Formula $d$
\end_inset
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can be 1, 2 or 3) Bravais lattice embedded into the three-dimensional real
space, with lattice vectors
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\begin_inset Formula $\left\{ \dlv_{i}\right\} _{i=1}^{d}$
\end_inset
, and let the lattice points be labeled with an
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\begin_inset Formula $d$
\end_inset
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-dimensional integer multi-index
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\begin_inset Formula $\vect n\in\ints^{d}$
\end_inset
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, so the lattice points have cartesian coordinates
\begin_inset Formula $\vect R_{\vect n}=\sum_{i=1}^{d}n_{i}\vect a_{i}$
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\end_inset
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.
There can be several scatterers per unit cell with indices
\begin_inset Formula $\alpha$
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\end_inset
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from a set
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\begin_inset Formula $\mathcal{P}_{1}$
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\end_inset
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and (relative) positions
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\begin_inset Formula $\vect r_{\alpha}$
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\end_inset
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inside the unit cell; any particle of the periodic system can thus be labeled
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by a multi-index from
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\begin_inset Formula $\mathcal{P}=\ints^{d}\times\mathcal{P}_{1}$
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\end_inset
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.
The scatterers are located at positions
\begin_inset Formula $\vect r_{\vect n,\alpha}=\vect R_{\vect n}+\vect r_{\alpha}$
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\end_inset
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and their
\begin_inset Formula $T$
\end_inset
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-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
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LatexCommand eqref
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reference "eq:Multiple-scattering problem"
plural "false"
caps "false"
noprefix "false"
\end_inset
can be rewritten as
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\end_layout
\begin_layout Standard
\begin_inset Formula
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\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}
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\end_inset
\end_layout
\begin_layout Standard
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Due to periodicity, we can also write
\begin_inset Formula $\tropsp{\vect n,\alpha}{\vect m,\beta}=\tropsp{\alpha}{\beta}\left(\vect R_{\vect m}-\vect R_{\vect n}\right)=\tropsp{\alpha}{\beta}\left(\vect R_{\vect m-\vect n}\right)=\tropsp{\vect 0,\alpha}{\vect m-\vect n,\beta}$
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\end_inset
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.
Assuming quasi-periodic right-hand side with quasi-momentum
\begin_inset Formula $\vect k$
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\end_inset
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,
\begin_inset Formula $\rcoeffincp{\vect n,\alpha}=\rcoeffincp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
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\end_inset
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, the solutions of
\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem periodic"
plural "false"
caps "false"
noprefix "false"
\end_inset
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will be also quasi-periodic according to Bloch theorem,
\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
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\end_inset
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, and eq.
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\begin_inset space \space{}
\end_inset
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem periodic"
plural "false"
caps "false"
noprefix "false"
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\end_inset
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can be rewritten as follows
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\begin_inset Formula
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\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}
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\end_inset
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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
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\begin_inset Quotes eld
\end_inset
lattice Fourier transform
\begin_inset Quotes erd
\end_inset
of the translation operator,
\begin_inset Formula
\begin{equation}
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W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\beta}\delta_{\vect m\vect 0}\right)\tropsp{\vect 0,\alpha}{\vect m,\beta}e^{i\vect k\cdot\vect R_{\vect m}},\label{eq:W definition}
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\end{equation}
\end_inset
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evaluation of which is possible but rather non-trivial due to the infinite
lattice sum, so we cover it separately in Sect.
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "subsec:W operator evaluation"
plural "false"
caps "false"
noprefix "false"
\end_inset
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.
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\end_layout
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\begin_layout Standard
As in the case of a finite system, eq.
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\begin_inset space \space{}
\end_inset
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\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem unit cell"
plural "false"
caps "false"
noprefix "false"
\end_inset
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can be written in a shorter block-matrix form,
\begin_inset Formula
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\begin{equation}
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\left(I-TW\right)\outcoeffp{\vect 0}\left(\vect k\right)=T\rcoeffincp{\vect 0}\left(\vect k\right)\label{eq:Multiple-scattering problem unit cell block form}
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\end{equation}
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\end_inset
Eq.
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\begin_inset space \space{}
\end_inset
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\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.
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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}
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\left(I-TW\right)\outcoeffp{\vect 0}\left(\vect k\right)=0,\label{eq:lattice mode equation}
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\end{equation}
\end_inset
describes the
\emph on
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lattice modes
\emph default
, i.e.
electromagnetic excitations that can sustain themselves for prolonged time
even without external driving
\emph on
.
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\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
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\begin_inset Formula $M\left(\omega,\vect k\right)=\left(I-T\left(\omega\right)W\left(\omega,\vect k\right)\right)$
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\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.
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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.
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\end_layout
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\begin_layout Standard
Loss in the scatterers causes the solutions of
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lattice mode equation"
plural "false"
caps "false"
noprefix "false"
\end_inset
shift to complex frequencies.
If the background medium has constant real refractive index
\begin_inset Formula $n$
\end_inset
, negative (or positive) imaginary part of the frequency
\begin_inset Formula $\omega$
\end_inset
causes an artificial gain (or loss) in the medium, which manifests itself
as exponential magnification (or attenuation) of the radial parts of the
translation operators,
\begin_inset Formula $h_{l}^{\left(1\right)}\left(rn\omega/c\right)$
\end_inset
, w.r.t.
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\begin_inset space \space{}
\end_inset
the distance; the gain might then balance the losses in particles, resulting
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in sustained modes satisfying eq.
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\begin_inset space \space{}
\end_inset
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\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lattice mode equation"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
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\begin_inset Note Note
status open
\begin_layout Plain Layout
The gain in the system introduces some challenges, which we will discuss
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in Section
\begin_inset CommandInset ref
LatexCommand eqref
reference "subsec:Physical-interpretation-of"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
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\end_inset
\end_layout
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\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.
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\begin_inset space \space{}
\end_inset
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\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.
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The lattice mode problem
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\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.
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However, this approach is quite expensive, since
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\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).
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Therefore, a much more efficient but not completely robust approach to
determine the photonic bands is to sample the
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\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
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(modulo reciprocal lattice points
\begin_inset Note Note
status open
\begin_layout Plain Layout
TODO write this in a clean way
\end_layout
\end_inset
).
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A somehow challenging step is to distinguish the different bands that can
all be very close to the empty lattice approximation, especially if the
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particles in the system are small.
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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
.
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\end_layout
\begin_layout Standard
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An alternative, faster and more robust approach to generic minimisation
algorithms are eigensolvers for nonlinear eigenvalue problems based on
contour integration
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\begin_inset CommandInset citation
LatexCommand cite
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key "beyn_integral_2012,gavin_feast_2018"
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literal "false"
\end_inset
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which are able to find the roots of
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\begin_inset Formula $M\left(\omega,\vect k\right)=0$
\end_inset
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inside an area enclosed by a given complex frequency plane contour, assuming
that
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
is an analytical function of
\begin_inset Formula $\omega$
\end_inset
inside the contour.
A necessary prerequisite for this is that all the ingredients of
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
are analytical as well.
It practice, this usually means that interpolation cannot be used in a
straightforward way for material properties or
\begin_inset Formula $T$
\end_inset
-matrices.
For material response, constant permittivity or Drude-Lorentz models suit
this purpose well.
The need to evaluate the
\begin_inset Formula $T$
\end_inset
-matrices precisely (without the speedup provided by interpolation) at many
points might cause a performance bottleneck for scatterers with more complicate
d shapes.
And finally, the integration contour has to evade any branch cuts appearing
in the lattice-summed translation operator
\begin_inset Formula $W\left(\omega,\vect k\right)$
\end_inset
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, as described in the following and illustrated in Fig.
\begin_inset space \space{}
\end_inset
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:ewald branch cuts"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float figure
placement document
alignment document
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename figs/ewald_branchcuts.pdf
width 100col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Left: Illustration of branch cuts in
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
obtained using Ewald summation over two-dimensional square lattice in three-dim
ensional space filled with dielectric medium with constant real refraction
index
\begin_inset Formula $n$
\end_inset
and wavenumber
\begin_inset Formula $\kappa\left(\omega\right)=\omega n/c$
\end_inset
.
The function is holomorphic in the positive imaginary half-plane.
The points corresponding to the diffraction orders of an
\begin_inset Quotes eld
\end_inset
empty
\begin_inset Quotes erd
\end_inset
lattice lie on the real axis (pink), and from each of them two branch cuts
originate: one due to the branch cut in the incomplete
\begin_inset Formula $\Gamma$
\end_inset
function (orange, hyperbolic shape), and another due to the branch cut
of
\begin_inset Formula $\gamma(z)$
\end_inset
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if the branch is selected to be continuous for
\begin_inset Formula $-3\pi/2<\arg\left(z-1\right)<\pi/2$
\end_inset
\begin_inset Note Note
status open
\begin_layout Plain Layout
as defined in
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\begin_inset CommandInset ref
LatexCommand eqref
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reference "eq:lilgamma_old"
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plural "false"
caps "false"
noprefix "false"
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\end_inset
\end_layout
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\end_inset
(blue, circular shape).
Further non-analyticities might stem from the material model: the violet
curve represents a branch cut originating from a complex square root in
the refractive index
\begin_inset Formula $n_{\mathrm{Au}}\left(\omega\right)=\sqrt{\varepsilon_{\mathrm{Au}}\left(\omega\right)}$
\end_inset
, where
\begin_inset Formula $\varepsilon_{\mathrm{Au}}\left(\omega\right)$
\end_inset
is the Drude-Lorentz permittivity model of gold used for the scatterers.
The other parameters used here are
\begin_inset Formula $p_{x}=580\,\mathrm{nm}$
\end_inset
(lattice period),
\begin_inset Formula $\vect k=\left(0.2\pi/p_{x},0\right)$
\end_inset
,
\begin_inset Formula $n=1.52$
\end_inset
.
The plot on the right shows the
\begin_inset Quotes eld
\end_inset
empty
\begin_inset Quotes erd
\end_inset
lattice diffraction orders on the line
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\begin_inset Formula $\vect k=\left(k_{x},0\right),k_{x}\in\left[0,\pi/p_{x}\right].$
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\end_inset
\begin_inset CommandInset label
LatexCommand label
name "fig:ewald branch cuts"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
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\end_layout
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\begin_layout Subsection
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Computing the lattice sum of the translation operator
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\begin_inset CommandInset label
LatexCommand label
name "subsec:W operator evaluation"
\end_inset
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\end_layout
\begin_layout Standard
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The problem in evaluating
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\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"
\end_inset
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is the asymptotic behaviour of the translation operator at large distances,
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\begin_inset Formula $\tropsp{\vect 0,\alpha}{\vect m,\beta}\sim\left|\vect R_{\vect m}\right|^{-1}e^{i\kappa\left|\vect R_{\vect m}\right|}$
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\end_inset
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, so that its lattice sum does not in the strict sense converge for any
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\begin_inset Formula $d>1$
\end_inset
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-dimensional lattice unless
\begin_inset Formula $\Im\kappa>0$
\end_inset
.
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\begin_inset Note Note
status open
\begin_layout Plain Layout
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\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
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\end_layout
\end_inset
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The problem of poorly converging lattice sums can be solved by decomposing
the lattice-summed function into two parts: a short-range part that decays
fast and can be summed directly, and a long-range part which decays poorly
but is fairly smooth everywhere, so that its Fourier transform decays fast
enough, and to deal with the long range part by Poisson summation over
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the reciprocal lattice; these two parts put together shall give an analytical
continuation of the original sum for
\begin_inset Formula $\Im\kappa\le0$
\end_inset
.
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This idea dates back to Ewald
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\begin_inset CommandInset citation
LatexCommand cite
key "ewald_berechnung_1921"
literal "false"
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\end_inset
who solved the problem for electrostatic potentials (Green's functions
for Laplace's equation).
For linear electrodynamic problems, ruled by Helmholtz equation, the same
basic idea can be used as well, resulting in exponentially convergent summation
formulae, but the technical details are considerably more complicated than
in electrostatics.
For the scalar Helmholtz equation in three dimensions, the formulae were
developed by Ham & Segall
\begin_inset CommandInset citation
LatexCommand cite
key "ham_energy_1961"
literal "false"
\end_inset
for 3D periodicity, Kambe
\begin_inset CommandInset citation
LatexCommand cite
key "kambe_theory_1967,kambe_theory_1967-1,kambe_theory_1968"
literal "false"
\end_inset
for 2D periodicity and Moroz
\begin_inset CommandInset citation
LatexCommand cite
key "moroz_quasi-periodic_2006"
literal "false"
\end_inset
for 1D periodicity.
A review of these methods can be found in
\begin_inset CommandInset citation
LatexCommand cite
key "linton_lattice_2010"
literal "false"
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\end_inset
.
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We will not rederive the formulae here, but for reference, we restate the
results in a form independent upon the normalisation and phase conventions
for spherical harmonic bases (pointing out some errors in the aforementioned
literature) and discuss some practical aspects of the numerical evaluation.
\begin_inset Note Note
status open
\begin_layout Plain Layout
Tady ještě upřesnit, co vlastně dělám.
\end_layout
\end_inset
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\end_layout
\begin_layout Standard
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We note that the lattice sums for
\emph on
scalar
\emph default
Helmholtz equation are enough for the evaluation of the translation operator
lattice sum
\begin_inset Formula $W_{\alpha\beta}(\vect k)$
\end_inset
: in eq.
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\begin_inset space \space{}
\end_inset
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\begin_inset CommandInset ref
LatexCommand eqref
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reference "eq:translation operator singular"
plural "false"
caps "false"
noprefix "false"
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\end_inset
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we demonstratively expressed the translation operator elements as linear
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combinations of (outgoing)
\emph on
scalar
\emph default
spherical wavefunctions
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\begin_inset Formula
\begin{equation}
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\sswfoutlm lm\left(\vect r\right)=h_{l}^{\left(1\right)}\left(r\right)\ush lm\left(\uvec r\right).\label{eq:scalar spherical wavefunctions}
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\end{equation}
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\end_inset
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If we formally label
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\begin_inset Note Note
status open
\begin_layout Plain Layout
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\begin_inset Marginal
status open
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\begin_layout Plain Layout
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FP: Check signs.
\end_layout
\end_inset
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\end_layout
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\end_inset
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\begin_inset Formula
\begin{equation}
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\sigma_{l,m}\left(\vect k,\vect s\right)=\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},\vect s}\right)e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\sswfoutlm lm\left(\kappa\left(\vect s+\vect{R_{n}}\right)\right),\label{eq:sigma lattice sums}
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\end{equation}
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\end_inset
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\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Formula
\begin{equation}
\sigma_{l,m}\left(\vect k,\vect s\right)=\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},\vect s}\right)e^{i\vect{\vect k}\cdot\left(\vect R_{\vect n}-\vect s\right)}\sswfoutlm lm\left(\vect{R_{n}}-\vect s\right),\label{eq:sigma lattice sums-1}
\end{equation}
\end_inset
\end_layout
\end_inset
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we see from eqs.
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\begin_inset CommandInset ref
LatexCommand eqref
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reference "eq:translation operator singular"
plural "false"
caps "false"
noprefix "false"
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\end_inset
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,
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"
plural "false"
caps "false"
noprefix "false"
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\end_inset
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that the matrix elements of
\begin_inset Formula $W_{\alpha\beta}(\vect k)$
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\end_inset
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read
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\begin_inset Note Note
status open
\begin_layout Plain Layout
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\begin_inset Formula
\[
W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\beta}\right)\tropsp{\vect 0,\alpha}{\vect m,\beta}e^{i\vect k\cdot\vect R_{\vect m}},
\]
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\end_inset
2019-08-04 18:24:17 +03:00
\end_layout
\end_inset
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2020-06-05 10:59:41 +03:00
\begin_inset Note Note
status open
\begin_layout Plain Layout
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\begin_inset Formula
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\begin{align*}
W_{\alpha,\tau lm;\beta,\tau l'm'}(\vect k) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r_{\beta}-\vect r_{\alpha}\right),\\
W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r_{\beta}-\vect r_{\alpha}\right),\quad\tau'\ne\tau,
\end{align*}
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\end_inset
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\end_layout
\end_inset
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\begin_inset Note Note
status open
\begin_layout Plain Layout
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\begin_inset Formula
\[
W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k)=\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r_{\beta}-\vect r_{\alpha}\right),\quad\tau'\ne\tau,
\]
\end_inset
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\end_layout
\end_inset
\begin_inset Formula
\[
W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k)=\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(-\vect k,\vect r_{\alpha}-\vect r_{\beta}\right),\quad\tau'\ne\tau,
\]
\end_inset
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\begin_inset Note Note
status open
\begin_layout Plain Layout
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\begin_inset Marginal
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status open
\begin_layout Plain Layout
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Check signs
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\end_layout
\end_inset
2019-11-17 21:26:11 +02:00
\end_layout
\end_inset
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where the constant factors are exactly the same as in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:translation operator constant factors"
plural "false"
caps "false"
noprefix "false"
2019-07-01 14:50:46 +03:00
\end_inset
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.
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\end_layout
\begin_layout Standard
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The lattice sums
\begin_inset Formula $\sigma_{l,m}\left(\vect k,\vect s\right)$
\end_inset
are related to what is also called
\emph on
structural constants
\emph default
in some literature
\begin_inset CommandInset citation
LatexCommand cite
key "kambe_theory_1967,kambe_theory_1967-1,kambe_theory_1968"
literal "false"
\end_inset
, but the phase and normalisation differ.
For reader's reference, we list the Ewald-type formulae for lattice sums
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\begin_inset Formula $\sigma_{l,m}\left(\vect k,\vect s\right)$
2019-07-01 14:50:46 +03:00
\end_inset
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rewritten in a way that is independent on particular phase or normalisation
conventions of vector spherical harmonics.
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\end_layout
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2019-08-07 06:55:47 +03:00
\begin_layout Standard
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In all three lattice dimensionality cases, the lattice sums are divided
into short-range and long-range parts,
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\begin_inset Formula $\sigma_{l,m}\left(\vect k,\vect s\right)=\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)+\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)$
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\end_inset
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depending on a positive parameter
\begin_inset Formula $\eta$
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\end_inset
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.
The short-range part has in all three cases the same form:
\begin_inset Note Note
status open
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2019-08-07 06:55:47 +03:00
\begin_layout Plain Layout
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FP: Check sign of s
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\end_layout
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\end_inset
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\end_layout
\begin_layout Standard
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\begin_inset Formula
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\begin{multline}
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\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{2^{l+1}i}{\kappa^{l+1}\sqrt{\pi}}\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},-\vect s}\right)\left|\vect s_{\vect n}\right|^{l}\ush lm\left(\uvec s_{\vect n}\right)e^{i\vect k\cdot\vect{R_{n}}}\\
\times\int_{\eta}^{\infty}e^{-\left|\vect s_{\vect n}\right|^{2}\xi^{2}}e^{-\kappa^{2}/4\xi^{2}}\xi^{2l}\ud\xi\\
+\delta_{\vect{R_{n}},-\vect s}\frac{\delta_{l0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{\kappa^{2}}{4\eta^{2}}\right)\ush lm\left(\uvec s_{\vect n}\right),\label{eq:Ewald in 3D short-range part}
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\end{multline}
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\end_inset
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where we labeled
\begin_inset Formula $\vect s_{\vect n}\equiv\vect s+\vect R_{\vect n}$
\end_inset
.
The formal
2020-03-15 15:04:18 +02:00
\begin_inset Formula $\left(1-\delta_{\vect{R_{n}},-\vect s}\right)$
2019-11-13 12:54:14 +02:00
\end_inset
2019-07-30 10:26:30 +03:00
2020-03-15 15:04:18 +02:00
factor here accounts for leaving out the direct excitation of a particle
by itself, corresponding to the
\begin_inset Formula $\left(1-\delta_{\alpha\beta}\delta_{\vect m\vect 0}\right)$
\end_inset
factor in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
The leaving out then causes an additional (
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\begin_inset Quotes eld
\end_inset
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2019-11-13 12:54:14 +02:00
self-interaction
\begin_inset Quotes erd
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\end_inset
2020-03-15 15:04:18 +02:00
) term on the last line of
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\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Ewald in 3D short-range part"
plural "false"
caps "false"
noprefix "false"
2019-08-07 06:55:47 +03:00
2019-07-30 10:26:30 +03:00
\end_inset
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, which appears only when the displacement vector
\begin_inset Formula $\vect s$
\end_inset
2019-08-07 06:55:47 +03:00
2020-03-15 15:04:18 +02:00
coincides with a lattice point.
Strictly speaking, this is not a
\begin_inset Quotes eld
\end_inset
short-range
\begin_inset Quotes erd
\end_inset
term, hence it is often noted separately in the literature; however, we
keep it in
\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)$
\end_inset
for formal convenience.
2019-11-13 12:54:14 +02:00
2020-03-15 15:04:18 +02:00
\begin_inset Formula $\Gamma(a,z)$
\end_inset
is the incomplete Gamma function.
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\begin_inset Note Note
status open
\begin_layout Plain Layout
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Poznámka ohledně zahození radiální části u kulových fcí?
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\end_layout
\end_inset
2020-03-15 15:04:18 +02:00
\end_layout
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\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 or continued fraction representations from
\begin_inset CommandInset citation
LatexCommand cite
key "NIST:DLMF"
literal "false"
\end_inset
.
\end_layout
2020-03-15 15:04:18 +02:00
\begin_layout Standard
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The explicit form of the long-range part of the lattice sum depends on the
lattice dimensionality.
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The long-range parts are calculated as sums over the reciprocal lattice
\begin_inset Formula $\Lambda^{*}$
\end_inset
with lattice vectors
\begin_inset Formula $\left\{ \vect b_{i}\right\} _{i=1}^{d}$
\end_inset
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lying in the same
\begin_inset Formula $d$
\end_inset
-dimensional subspace as the direct lattice vectors
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\begin_inset Formula $\left\{ \vect a_{i}\right\} _{i=1}^{d}$
\end_inset
and satisfying
\begin_inset Formula $\vect a_{i}\cdot\vect b_{j}=\delta_{ij}$
\end_inset
.
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In the following, let us label
\begin_inset Formula $\vect k_{\vect K}\equiv\vect k+\vect K$
\end_inset
, where
\begin_inset Formula $\vect K$
\end_inset
is a point in the reciprocal lattice, and let
\begin_inset Formula $\mathcal{A}$
\end_inset
be the lattice unit cell volume (or area/length in the 2D/1D cases).
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\end_layout
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\begin_layout Paragraph
Case
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\begin_inset Formula $d=3$
\end_inset
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\end_layout
\begin_layout Standard
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\begin_inset Formula
\begin{equation}
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\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=\frac{4\pi i^{l+1}}{\kappa\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{-i\vect k_{\vect K}\cdot\vect s}\frac{\left(\left|\vect k_{\vect K}\right|/\kappa\right)^{l}}{\kappa^{2}-\left|\vect k_{\vect K}\right|^{2}}e^{\left(\kappa^{2}-\left|\vect k_{\vect K}\right|^{2}\right)/4\eta^{2}}\ush lm\left(\uvec k_{\vect K}\right)\label{eq:Ewald in 3D long-range part 3D}
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\end{equation}
\end_inset
regardless of chosen coordinate axes.
Here
\begin_inset Formula $\mathcal{A}$
\end_inset
is the unit cell volume (or length/area in the following 1D/2D lattice
cases).
\end_layout
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\begin_layout Paragraph
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Cases
\begin_inset Formula $d=1,2$
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\end_inset
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\end_layout
\begin_layout Standard
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In the quasiperiodic cases, we decompose vectors into parallel and orthogonal
parts with respect to the linear subspace in which the Bravais lattice
lies (the reciprocal lattice lies in the same subspace),
\begin_inset Formula $\vect v=\vect v_{\perp}+\vect v_{\parallel}$
\end_inset
, and we label
\begin_inset Formula
\begin{equation}
\gamma_{\vect k_{\vect K}}\equiv\gamma_{\vect k_{\vect K}}\left(\kappa\right)\equiv\left(\left|\vect k_{\vect K}\right|^{2}-\kappa^{2}\right)^{\frac{1}{2}}/\kappa,\label{eq:lilgamma}
\end{equation}
\end_inset
\begin_inset Formula
\begin{equation}
\Delta_{d;j}\left(x,z\right)\equiv\int_{x}^{\infty}t^{-\frac{d_{c}}{2}-n}\exp\left(-t+\frac{z^{2}}{4t}\right)\ud t,\label{eq:Delta_j}
\end{equation}
\end_inset
where
\begin_inset Formula $d_{c}=3-d$
\end_inset
is the complementary dimension of the lattice.
Then
\begin_inset Formula
\begin{multline}
\sigma_{l}^{m}\left(\vect k,\vect s\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa}\frac{\left(2l+1\right)!!}{\kappa^{l}}\sum_{\vect K\in\Lambda^{*}}e^{-i\vect k_{\vect K}\cdot\vect s}\times\\
\times\sum_{j=0}^{l}\frac{\left(-1\right)^{j}}{j!}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\Delta_{d;j}\left(\frac{\kappa^{2}\gamma_{\vect k_{\vect K}}^{2}}{4\eta^{2}},-i\kappa\gamma_{\vect k_{\vect K}}\left|\vect s_{\perp}\right|\right)\times\\
\times\sum_{l'=\max\left(0,l-2j\right)}^{l-j}4\pi i^{l'}\left(2\left|\vect s_{\bot}\right|\right)^{2j-l+l'}\frac{\left|\vect k_{\vect K}\right|^{l'}}{\left(2l'+1\right)!!}\sum_{m'=-l'}^{l'}\ush{l'}{m'}\left(\uvec k_{\vect K}\right)\times\\
\times\int\ud\Omega_{\vect r}\,\ush lm\left(\uvec r\right)\ushD{l'}{m'}\left(\uvec r\right)\left(\frac{\left|\vect r_{\perp}\right|}{\left|\vect r\right|}\right)^{l-l}\left(\frac{-\vect r_{\perp}\cdot\vect s_{\perp}}{\left|\vect r_{\perp}\right|\left|\vect s_{\perp}\right|}\right)^{2j-l+l'}.\label{eq:Ewald in 3D long-range part 1D 2D}
\end{multline}
\end_inset
The angular integral on the last line of
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Ewald in 3D long-range part 1D 2D"
plural "false"
caps "false"
noprefix "false"
\end_inset
gives a set of constant coefficients characteristic to a chosen convention
for spherical harmonics and coordinate axes; relatively simple closed-form
expressions are obtained for 2D periodicity if we choose the lattice to
lie in the
\begin_inset Formula $xy$
\end_inset
plane, so that both
\begin_inset Formula $\vect r_{\perp},\vect s_{\perp}$
\end_inset
are parallel to the
\begin_inset Formula $z$
\end_inset
axis, as done in
\begin_inset CommandInset citation
LatexCommand cite
key "kambe_theory_1968"
literal "false"
\end_inset
, see also Supplementary Material.
In the special case
\begin_inset Formula $\vect s_{\perp}=0$
\end_inset
the expressions can be considerably simplified as most of the terms vanish
and
\begin_inset Formula $\Delta_{d;j}\left(x,0\right)=\Gamma\left(1-d_{c}/2-j,x\right)$
\end_inset
, but the general case is needed for evaluating the fields in space (see
Section
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Periodic scattering and fields"
plural "false"
caps "false"
noprefix "false"
\end_inset
) or if there is an offset between two particles in a unitcell that is not
parallel to the lattice subspace.
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
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Reasonable explicit forms assume that the lattice lies inside the
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\begin_inset Formula $xy$
\end_inset
-plane
\begin_inset Formula $\left(\theta=\pi/2\right)$
\end_inset
.
\begin_inset Foot
status open
\begin_layout Plain Layout
If a different coordinate system for
\begin_inset Formula $\sigma_{l,m}\left(\vect k,\vect s\right)$
\end_inset
is needed, one can always perform the lattice summation in the coordinate
system described here, and rotate the result a posteriori using Wigner
matrices, according to
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Wigner matrices"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\end_inset
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The component of
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\begin_inset Formula $\vect s$
\end_inset
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normal to the lattice is then parallel to the
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\begin_inset Formula $z$
\end_inset
axis,
\begin_inset Formula $\vect s=\vect s_{\parallel}+\vect s_{\perp}=\vect s_{\parallel}+s_{\perp}\uvec z$
\end_inset
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.
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With these assumptions
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\begin_inset Note Note
status open
\begin_layout Plain Layout
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FP: check sign of
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\begin_inset Formula $\vect k$
\end_inset
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\end_layout
\end_inset
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\begin_inset Formula
\begin{multline}
\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{\kappa^{2}\mathcal{A}}\pi^{3/2}2\left(\left(l-m\right)/2\right)!\left(\left(l+m\right)/2\right)!\times\\
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\times\sum_{\vect K\in\Lambda^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot s}?}}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{l-\left|m\right|}\left(-1\right)^{j}\left(\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)\right)^{2j+1}\Delta_{j}\left(\frac{\kappa^{2}\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)^{2}}{4\eta^{2}},-i\kappa\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)s_{\perp}\right)\times\\
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\times\sum_{\substack{s\\
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j\le s\le\min\left(2j,l-\left|m\right|\right)\\
l-n+\left|m\right|\,\mathrm{even}
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}
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}\frac{1}{\left(2j-s\right)!\left(s-j\right)!}\frac{\left(-\kappa s_{\perp}\right)^{2j-s}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-s}}{\left(\frac{1}{2}\left(l-m-s\right)\right)!\left(\frac{1}{2}\left(l+m-s\right)\right)!}\label{eq:Ewald in 3D long-range part 1D 2D-1}
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\end{multline}
\end_inset
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\end_layout
\end_inset
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\begin_inset Note Note
status open
\begin_layout Plain Layout
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\begin_inset Formula
\begin{multline}
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\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{\kappa^{d}\mathcal{A}}\pi^{2+\left(3-d\right)/2}2\left(\left(l-m\right)/2\right)!\left(\left(l+m\right)/2\right)!\times\\
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\times\sum_{\vect K\in\Lambda^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot s}?}}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{\left[\left(l-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-2j}\Gamma\left(\frac{d-1}{2}-j,\frac{\kappa^{2}\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)^{2}}{4\eta^{2}}\right)}{j!\left(\frac{1}{2}\left(l-m\right)-j\right)!\left(\frac{1}{2}\left(l+m\right)-j\right)!}\times\\
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\times\left(\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)\right)^{2j+3-d}\label{eq:Ewald in 3D long-range part 1D 2D-1}
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\end{multline}
\end_inset
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\end_layout
\end_inset
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\begin_inset Note Note
status open
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\begin_layout Plain Layout
where If the normal component
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\begin_inset Formula $s_{\bot}$
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\end_inset
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is zero, in the last sum in eq.
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\begin_inset space \space{}
\end_inset
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\begin_inset CommandInset ref
LatexCommand eqref
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reference "eq:Ewald in 3D long-range part 1D 2D"
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plural "false"
caps "false"
noprefix "false"
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\end_inset
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only one term (
\begin_inset Formula $s=2j$
\end_inset
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) will remain if
\begin_inset Formula $l-\left|m\right|$
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\end_inset
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is even; for
\begin_inset Formula $l-\left|m\right|$
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\end_inset
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odd, the sum will vanish completely.
Moreover,
\begin_inset Formula $\Delta_{j}\left(x,0\right)=\Gamma\left(1/2-j,x\right)$
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\end_inset
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.
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\end_layout
\end_inset
\end_layout
\begin_layout Standard
If
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\begin_inset Formula $s_{\bot}\ne0$
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\end_inset
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, the integral
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\begin_inset Formula $\Delta_{d;j}\left(x,0\right)$
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\end_inset
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can be evaluated e.g.
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\begin_inset space \space{}
\end_inset
using the Taylor series
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\lang finnish
\begin_inset Formula
\[
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\Delta_{d;j}\left(x,z\right)=\sum_{k=0}^{\infty}\Gamma\left(1-\frac{d_{c}}{2}-j-k,x\right)\frac{\left(z/2\right)^{2k}}{k!}
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\]
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\end_inset
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which has infinite radius of convergence and is the first choice for small
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\begin_inset Formula $z$
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\end_inset
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\lang english
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.
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Kambe
\begin_inset CommandInset citation
LatexCommand cite
key "kambe_theory_1968"
literal "false"
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\end_inset
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mentions a recurrence formula that can be obtained integrating
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Delta_j"
plural "false"
caps "false"
noprefix "false"
2019-11-13 12:54:14 +02:00
\end_inset
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by parts (with signs corrected here):
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\begin_inset Formula
\begin{equation}
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\Delta_{d;j+1}\left(x,z\right)=\frac{4}{z^{2}}\left(\left(\frac{1}{2}-j\right)\Delta_{d;j}\left(x,z\right)-\Delta_{d;j-1}\left(x,z\right)+x^{\frac{d_{c}}{2}-j}e^{-x+\frac{z^{2}}{4x}}\right)\label{eq:Delta_j recurrent}
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\end{equation}
\end_inset
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with the first two terms for 2D periodicity
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\begin_inset Formula
\begin{align*}
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\Delta_{2;0}\left(x,z\right) & =\frac{\sqrt{\pi}}{2}e^{-x^{2}+\frac{z^{2}}{4x}}\left(w\left(-\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)+w\left(\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)\right),\\
\Delta_{2;1}\left(x,z\right) & =i\frac{\sqrt{\pi}}{z}e^{-x^{2}+\frac{z^{2}}{4x}}\left(w\left(-\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)-w\left(\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)\right),
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\end{align*}
\end_inset
where
\begin_inset Formula $w\left(z\right)=e^{-z^{2}}\left(1+2i\pi^{-1/2}\int_{0}^{z}e^{t^{2}}\ud t\right)$
\end_inset
is the Faddeeva function.
However, the recurrence formula
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Delta_j recurrent"
plural "false"
caps "false"
noprefix "false"
\end_inset
is unsuitable for numerical evaluation if
\begin_inset Formula $z$
\end_inset
is small or
\begin_inset Formula $j$
\end_inset
is large due to its numerical instability.
\end_layout
\begin_layout Standard
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\begin_inset Note Note
status open
\begin_layout Plain Layout
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and if the normal component
\begin_inset Formula $s_{\perp}$
\end_inset
is zero,
\begin_inset Formula
\begin{multline}
\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{\kappa^{d}\mathcal{A}}\pi^{2+\left(3-d\right)/2}2\left(\left(l-m\right)/2\right)!\left(\left(l+m\right)/2\right)!\times\\
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\times\sum_{\vect K\in\Lambda^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot s}?}}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{\left[\left(l-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-2j}\Gamma\left(\frac{d-1}{2}-j,\frac{\kappa^{2}\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)^{2}}{4\eta^{2}}\right)}{j!\left(\frac{1}{2}\left(l-m\right)-j\right)!\left(\frac{1}{2}\left(l+m\right)-j\right)!}\times\\
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\times\left(\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)\right)^{2j+3-d}\label{eq:Ewald in 3D long-range part 1D 2D z = 0}
\end{multline}
\end_inset
\end_layout
\end_inset
\begin_inset Note Note
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status open
\begin_layout Plain Layout
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The function
\begin_inset Formula $\gamma\left(z\right)$
\end_inset
used in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Ewald in 3D long-range part 1D 2D z = 0"
plural "false"
caps "false"
noprefix "false"
\end_inset
is defined as
\begin_inset Formula
\begin{equation}
\gamma\left(z\right)=\left(z-1\right)^{\frac{1}{2}}\left(z+1\right)^{\frac{1}{2}},\quad-\frac{3\pi}{2}<\arg\left(z-1\right)<\frac{\pi}{2},-\frac{\pi}{2}<\arg\left(z+1\right)<\frac{3\pi}{2}.\label{eq:lilgamma_old}
\end{equation}
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\end_inset
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\end_layout
\end_inset
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\begin_inset Note Note
status open
\begin_layout Plain Layout
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\begin_inset Marginal
status open
\begin_layout Plain Layout
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FP: I have some error estimates derived in my notes.
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Should I include them?
\end_layout
\end_inset
2019-11-13 12:54:14 +02:00
\end_layout
\end_inset
\end_layout
\begin_layout Standard
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One pecularity of the two-dimensional case is the two-branchedness of
\begin_inset Formula $\gamma_{\vect k_{\vect K}}\left(\kappa\right)$
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\end_inset
and the incomplete
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\begin_inset Formula $\Gamma$
\end_inset
-function
\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,z\right)$
\end_inset
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appearing in the long-range part (in the cases
\begin_inset Formula $d=1,3$
\end_inset
the function
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\begin_inset Formula $\gamma_{\vect k_{\vect K}}\left(\kappa\right)$
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\end_inset
appears with even powers, and
\begin_inset Formula $\Gamma\left(-j,z\right)$
\end_inset
is meromorphic for integer
\begin_inset Formula $j$
\end_inset
\begin_inset CommandInset citation
LatexCommand cite
after "8.2.9"
key "NIST:DLMF"
literal "false"
\end_inset
).
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As a consequence, if we now explicitly label the dependence on the wavenumber,
\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\kappa,\vect k,\vect s\right)$
\end_inset
has branch points at
\begin_inset Formula $\kappa=\left|\vect k+\vect K\right|$
\end_inset
for every reciprocal lattice vector
\begin_inset Formula $\vect K$
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\end_inset
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.
If the wavenumber
\begin_inset Formula $\kappa$
\end_inset
of the medium has a positive imaginary part,
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\begin_inset Formula $\Im\kappa>0$
\end_inset
, then the translation operator elements
\begin_inset Formula $\trops_{\tau lm;\tau'l'm}\left(\kappa\vect r\right)$
\end_inset
decay exponentially as
\begin_inset Formula $\left|\vect r\right|\to\infty$
\end_inset
and the lattice sum in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"
plural "false"
caps "false"
noprefix "false"
\end_inset
converges absolutely even in the direct space, and it is equal to the Ewald
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sum with the principal branches used both in
\begin_inset Formula $\gamma\left(z\right)$
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\end_inset
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and
\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,z\right)$
\end_inset
\begin_inset CommandInset citation
LatexCommand cite
key "linton_lattice_2010"
literal "false"
2019-11-13 12:54:14 +02:00
\end_inset
.
For other values of
\begin_inset Formula $\kappa$
\end_inset
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, we typically choose the branch in such way that
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\begin_inset Formula $W_{\alpha\beta}\left(\vect k\right)$
\end_inset
is analytically continued even when the wavenumber's imaginary part crosses
the real axis.
The principal value of
\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,z\right)$
\end_inset
has a branch cut at the negative real half-axis, which, considering the
lattice sum as a function of
\begin_inset Formula $\kappa$
\end_inset
, translates into branch cuts starting at
\begin_inset Formula $\kappa=\left|\vect k+\vect K\right|$
\end_inset
and continuing in straight lines towards
\begin_inset Formula $+\infty$
\end_inset
.
Therefore, in the quadrant
\begin_inset Formula $\Re z<0,\Im z\ge0$
\end_inset
we use the continuation of the principal value from
\begin_inset Formula $\Re z<0,\Im z<0$
\end_inset
instead of the principal branch
\begin_inset CommandInset citation
LatexCommand cite
after "8.2.9"
key "NIST:DLMF"
literal "false"
\end_inset
, moving the branch cut in the
\begin_inset Formula $z$
\end_inset
variable to the positive imaginary half-axis.
This moves the branch cuts w.r.t.
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\begin_inset space \space{}
\end_inset
2019-11-13 12:54:14 +02:00
\begin_inset Formula $\kappa$
\end_inset
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away from the real axis, as illustrated in Fig.
\begin_inset space \space{}
\end_inset
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:ewald branch cuts"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
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\begin_inset Note Note
status open
\begin_layout Plain Layout
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Detailed physical interpretation of the remaining branch cuts is an open
question.
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\end_layout
\end_inset
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\begin_inset Note Note
status open
\begin_layout Plain Layout
(I HAVE SOME DERIVATIONS OF THE ESTIMATES IN MY NOTES; SHOULD I INCLUDE
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THEM?)
\end_layout
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\end_inset
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\end_layout
\begin_layout Subsubsection
Choice of Ewald parameter and high-frequency breakdown
\end_layout
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\begin_layout Standard
The Ewald parameter
\begin_inset Formula $\eta$
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\end_inset
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determines the pace of convergence of both parts.
The larger
\begin_inset Formula $\eta$
\end_inset
is, the faster
\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)$
\end_inset
converges but the slower
\begin_inset Formula $\sigma_{l,m}^{\left(L,\eta\right)}\left(\vect k,\vect s\right)$
\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 $\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)$
\end_inset
and
\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)$
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\end_inset
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.
For one-dimensional, square, and cubic lattices, the optimal choice for
small frequencies (wavenumbers) is
\begin_inset Formula $\eta=\sqrt{\pi}/p$
\end_inset
where
\begin_inset Formula $p$
\end_inset
is the direct lattice period
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\begin_inset CommandInset citation
LatexCommand cite
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key "linton_lattice_2010"
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literal "false"
\end_inset
.
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\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Marginal
status open
\begin_layout Plain Layout
Whatabout different geometries?
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\end_layout
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\end_inset
\end_layout
\end_inset
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However, in floating point arithmetics, the magnitude of the summands must
be taken into account as well in order to maintain accuracy.
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\end_layout
\begin_layout Standard
There is a particular problem with the
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\begin_inset Quotes eld
\end_inset
central
\begin_inset Quotes erd
\end_inset
reciprocal lattice points in the long-range sums for which the real part
of
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\begin_inset Formula $\left|\vect k_{\vect K}\right|^{2}-\kappa^{2}$
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\end_inset
is negative: the incomplete
\begin_inset Formula $\Gamma$
\end_inset
function present in the sum (either explicitly or in the expansions of
\family roman
\series medium
\shape up
\size normal
\emph off
\nospellcheck off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $\Delta_{j}$
\end_inset
)
\family default
\series default
\shape default
\size default
\emph default
\nospellcheck default
\bar default
\strikeout default
\xout default
\uuline default
\uwave default
\noun default
\color inherit
grows exponentially with respect to the negative second argument, with
asymptotic behaviour
\begin_inset Formula $\Gamma\left(a,z\right)\sim e^{-z}z^{a-1}$
\end_inset
.
Therefore for higher frequencies, the parameter
\begin_inset Formula $\eta$
\end_inset
needs to be adjusted in a way that keeps the value of
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\begin_inset Formula $\Gamma\left(a,\left(\left|\vect k_{\vect K}\right|^{2}-\kappa^{2}\right)/4\eta^{2}\right)$
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\end_inset
within reasonable bounds.
\begin_inset Note Note
status open
\begin_layout Plain Layout
Setting a target maximum magnitude for
\begin_inset Formula $M$
\end_inset
, so that
\begin_inset Formula $\left|\Gamma\left(a,-\left|z\right|\right)\right|\lesssim M$
\end_inset
, using the asymptotic estimate
\begin_inset Formula $\Gamma\left(a,-\left|z\right|\right)\sim e^{-\left|z\right|}\left|z\right|^{a-1}$
\end_inset
, we get
\begin_inset Formula
\begin{align*}
e^{-\left|z\right|}\left|z\right|^{a-1} & \lesssim M,\\
-\left|z\right|\left(a-1\right)\log\left|z\right| & \lesssim\log M.
\end{align*}
\end_inset
\end_layout
\end_inset
If we assume that
\begin_inset Formula $\vect k$
\end_inset
lies in the first Brillouin zone, the minimum real part of the second argument
of the
\begin_inset Formula $\Gamma$
\end_inset
function will be
\begin_inset Formula $\left(\left|\vect k\right|^{2}-\kappa^{2}\right)/4\eta^{2}$
\end_inset
, so setting
\begin_inset Formula $\eta\ge\sqrt{\left|\kappa\right|^{2}-\left|\vect k\right|^{2}}/2\log M$
\end_inset
eliminates the exponential growth in the incomplete
\begin_inset Formula $\Gamma$
\end_inset
function, where the constant
\begin_inset Formula $M$
\end_inset
is chosen to represent the (rough) maximum tolerated magnitude of the summand
with regard to target accuracy.
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This adjustment means that, in the worst-case scenario, with growing wavenumber
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one has to include an increasing number of terms in the long-range sum
in order to achieve a given accuracy, the number of terms being proportional
to
\begin_inset Formula $\left|\kappa\right|^{d}$
\end_inset
where
\begin_inset Formula $d$
\end_inset
is the dimension of the lattice.
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\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Formula
\[
-\frac{\left|\left|\vect k\right|^{2}-\kappa^{2}\right|}{4\eta^{2}}\left(a-1\right)2\log\frac{\left|\left|\vect k\right|^{2}-\kappa^{2}\right|^{\frac{1}{2}}}{2\eta}\lesssim\log M
\]
\end_inset
\end_layout
\end_inset
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\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
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\begin_layout Subsection
Physical interpretation of wavenumber with negative imaginary part; screening
\begin_inset CommandInset label
LatexCommand label
name "subsec:Physical-interpretation-of"
\end_inset
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\end_layout
\begin_layout Plain Layout
left out for the time being
\end_layout
\end_inset
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\end_layout
\begin_layout Subsection
Scattering cross sections and field intensities in periodic system
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\begin_inset CommandInset label
LatexCommand label
name "subsec:Periodic scattering and fields"
\end_inset
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\end_layout
\begin_layout Standard
Once the scattering
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem unit cell block form"
plural "false"
caps "false"
noprefix "false"
\end_inset
or mode problem
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lattice mode equation"
plural "false"
caps "false"
noprefix "false"
\end_inset
is solved, one can evaluate some useful related quantities, such as scattering
cross sections (coefficients) or field intensities.
\end_layout
\begin_layout Standard
For plane wave scattering on 2D lattices, one can directly use the formulae
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:extincion CS multi"
plural "false"
caps "false"
noprefix "false"
\end_inset
,
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:absorption CS multi alternative"
plural "false"
caps "false"
noprefix "false"
\end_inset
, taking the sums over scatterers inside one unit cell, to get the extinction
and absorption cross sections per unit cell.
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From these, quantities such as absorption, extinction and scattering coefficien
ts are obtained using suitable normalisation by unit cell size, depending
on lattice dimensionality.
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\end_layout
\begin_layout Standard
Ewald summation can be used for evaluating scattered field intensities outside
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scatterers' circumscribing spheres: this requires expressing VSWF cartesian
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components in terms of scalar spherical wavefunctions defined in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:scalar spherical wavefunctions"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
Fortunately, these can be obtained easily from the expressions for the
translation operator:
\begin_inset Formula
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\begin{align}
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\vswfrtlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\tropr_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),\nonumber \\
\vswfouttlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),\label{eq:VSWFs expressed as translated dipole waves}
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\end{align}
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\end_inset
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which follows from eqs.
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\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:regular vswf translation"
plural "false"
caps "false"
noprefix "false"
\end_inset
,
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:singular vswf translation"
plural "false"
caps "false"
noprefix "false"
\end_inset
and the fact that all the other regular VSWFs except for
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\begin_inset Formula $\vswfrtlm 21{m'}$
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\end_inset
vanish at origin.
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For the quasiperiodic scattering problem formulated in section
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Quasiperiodic scattering problem"
plural "false"
caps "false"
noprefix "false"
\end_inset
, the total electric field scattered from all the particles at point
\begin_inset Formula $\vect r$
\end_inset
located outside all the particles' circumscribing sphere reads, using eqs.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:translation operator singular"
plural "false"
caps "false"
noprefix "false"
\end_inset
,
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:sigma lattice sums"
plural "false"
caps "false"
noprefix "false"
\end_inset
,
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:scalar spherical wavefunctions"
plural "false"
caps "false"
noprefix "false"
\end_inset
,
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Formula
\begin{align}
\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\psi_{\lambda,m-m'}\left(\vect d\right),\label{eq:translation operator singular-1}
\end{align}
\end_inset
\begin_inset Formula
\begin{equation}
\sigma_{l,m}\left(\vect k,\vect s\right)=\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},\vect s}\right)e^{i\vect{\vect k}\cdot\left(\vect R_{\vect n}-\vect s\right)}\sswfoutlm lm\left(\kappa\left(\vect{R_{n}}-\vect s\right)\right),\label{eq:sigma lattice sums}
\end{equation}
\end_inset
\begin_inset Formula
\begin{align*}
\vect E_{\mathrm{scat}}\left(\vect r\right) & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\\
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& =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right)\\
& =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right),\text{FIXME signs}\\
& =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\psi_{\lambda,m-m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right)\\
& =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right)
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\end{align*}
\end_inset
\end_layout
\begin_layout Plain Layout
TODO fix signs and exponential phase factors
\end_layout
\end_inset
\begin_inset Formula
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\begin{multline}
\vect E_{\mathrm{scat}}\left(\vect r\right)=\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)=\\
=\sum_{\alpha\in\mathcal{P}_{1}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\vswfrtlm 21{m'}\left(0\right)\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(-\vect k,\vect r-\vect r_{\alpha}\right).\label{eq:Scattered fields in periodic systems}
\end{multline}
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\end_inset
In the scattering problem, the total field intensity is obtained by adding
the incident field to
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Scattered fields in periodic systems"
plural "false"
caps "false"
noprefix "false"
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\end_inset
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; whereas in the lattice mode problem the total field is directly given
by
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Scattered fields in periodic systems"
plural "false"
caps "false"
noprefix "false"
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\end_inset
.
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\end_layout
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\end_body
\end_document