1474 lines
34 KiB
Plaintext
1474 lines
34 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\pdf_author "Marek Nečada"
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\shortcut idx
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\end_header
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\begin_body
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\begin_layout Section
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Infinite periodic systems
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Infinite"
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\dlv}{\vect a}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\rlv}{\vect b}
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\end_inset
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\end_layout
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\begin_layout Standard
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Although large finite systems are where MSTMM excels the most, there are
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several reasons that makes its extension to infinite lattices (where periodic
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boundary conditions might be applied) desirable as well.
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Other methods might be already fast enough, but MSTMM will be faster in
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most cases in which there is enough spacing between the neighboring particles.
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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
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lattice), which makes e.g.
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application of group theory in mode analysis quite easy.
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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Topology anoyne?
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\end_layout
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\end_inset
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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.
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\end_layout
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\begin_layout Subsection
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Formulation of the problem
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\end_layout
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\begin_layout Standard
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Let us have a linear system of compact EM scatterers on a homogeneous background
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as in Section
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "subsec:Multiple-scattering"
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plural "false"
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caps "false"
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noprefix "false"
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||
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||
\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 $d$
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\end_inset
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-dimensional (
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\begin_inset Formula $d$
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\end_inset
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||
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can be 1, 2 or 3) Bravais lattice embedded into the three-dimensional real
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space, with lattice vectors
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\begin_inset Formula $\left\{ \dlv_{i}\right\} _{i=1}^{d}$
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\end_inset
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, and let the lattice points be labeled with an
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\begin_inset Formula $d$
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\end_inset
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-dimensional integer multi-index
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\begin_inset Formula $\vect n\in\ints^{d}$
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\end_inset
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, so the lattice points have cartesian coordinates
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\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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.
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There can be several scatterers per unit cell with indices
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\begin_inset Formula $\alpha$
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\end_inset
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||
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from set
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\begin_inset Formula $\mathcal{P}_{1}$
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||
\end_inset
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||
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and (relative) positions inside the unit cell
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\begin_inset Formula $\vect r_{\alpha}$
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||
\end_inset
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; any particle of the periodic system can thus be labeled by a multiindex
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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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.
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The scatterers are located at positions
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\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
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\begin_inset Formula $T$
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\end_inset
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-matrices are periodic,
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\begin_inset Formula $T_{\vect n,\alpha}=T_{\alpha}$
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\end_inset
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.
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In such system, the multiple-scattering problem
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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can be rewritten as
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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\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}
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\end{equation}
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\end_inset
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\end_layout
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||
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||
\begin_layout Standard
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||
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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.
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||
Assuming quasi-periodic right-hand side with quasi-momentum
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||
\begin_inset Formula $\vect k$
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||
\end_inset
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,
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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
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem periodic"
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plural "false"
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||
caps "false"
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||
noprefix "false"
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||
\end_inset
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||
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will be also quasi-periodic according to Bloch theorem,
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\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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||
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||
\begin_inset CommandInset ref
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||
LatexCommand eqref
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||
reference "eq:Multiple-scattering problem periodic"
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||
plural "false"
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||
caps "false"
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||
noprefix "false"
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||
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||
\end_inset
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||
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can be rewritten as follows
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||
\begin_inset Formula
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||
\begin{align}
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||
\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 \\
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\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 \\
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\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 \\
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\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}
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\end{align}
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||
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||
\end_inset
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||
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||
so we reduced the initial scattering problem to one involving only the field
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||
expansion coefficients from a single unit cell, but we need to compute
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the
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\begin_inset Quotes eld
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||
\end_inset
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||
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||
lattice Fourier transform
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||
\begin_inset Quotes erd
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||
\end_inset
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||
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||
of the translation operator,
|
||
\begin_inset Formula
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||
\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}
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||
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||
\end_inset
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||
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evaluation of which is possible but quite non-trivial due to the infinite
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lattice sum, so we explain 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"
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||
plural "false"
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||
caps "false"
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||
noprefix "false"
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||
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||
\end_inset
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||
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||
.
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||
\end_layout
|
||
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||
\begin_layout Standard
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||
As in the case of a finite system, eq.
|
||
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||
\begin_inset CommandInset ref
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||
LatexCommand eqref
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||
reference "eq:Multiple-scattering problem unit cell"
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||
plural "false"
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||
caps "false"
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||
noprefix "false"
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||
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||
\end_inset
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||
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||
can be written in a shorter block-matrix form,
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||
\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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||
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||
\end_inset
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||
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||
Eq.
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||
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||
\begin_inset CommandInset ref
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||
LatexCommand eqref
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||
reference "eq:Multiple-scattering problem unit cell"
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||
plural "false"
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||
caps "false"
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||
noprefix "false"
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||
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||
\end_inset
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||
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||
can be used to calculate electromagnetic response of the structure to external
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||
quasiperiodic driving field – most notably a plane wave.
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||
However, the non-trivial solutions of the equation with right hand side
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||
(i.e.
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||
the external driving) set to zero,
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||
\begin_inset Formula
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||
\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}
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||
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||
\end_inset
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||
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describes the
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\emph on
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||
lattice modes.
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||
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||
\emph default
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||
Non-trivial solutions to
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||
\begin_inset CommandInset ref
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||
LatexCommand eqref
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||
reference "eq:lattice mode equation"
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||
plural "false"
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||
caps "false"
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||
noprefix "false"
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||
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||
\end_inset
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||
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||
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
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||
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||
is singular – this condition gives the
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||
\emph on
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||
dispersion relation
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||
\emph default
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||
for the periodic structure.
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||
Note that in realistic (lossy) systems, at least one of the pair
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||
\begin_inset Formula $\omega,\vect k$
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||
\end_inset
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||
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||
will acquire complex values.
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||
The solution
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||
\begin_inset Formula $\outcoeffp{\vect 0}\left(\vect k\right)$
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||
\end_inset
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||
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||
is then obtained as the right
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||
\begin_inset Note Note
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||
status open
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||
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||
\begin_layout Plain Layout
|
||
CHECK!
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||
\end_layout
|
||
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||
\end_inset
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||
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||
singular vector of
|
||
\begin_inset Formula $M\left(\omega,\vect k\right)$
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||
\end_inset
|
||
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||
corresponding to the zero singular value.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
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||
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, since
|
||
\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 but not completely robust 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 reciprocal lattice points
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
TODO write this in a clean way
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
).
|
||
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 system 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 Standard
|
||
An alternative, more robust approach to generic minimisation algorithms
|
||
is Beyn's contour integral method
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "beyn_integral_2012"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
which finds the roots of
|
||
\begin_inset Formula $M\left(\omega,\vect k\right)=0$
|
||
\end_inset
|
||
|
||
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
|
||
|
||
, 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
|
||
|
||
as defined in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:lilgamma"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\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
|
||
\begin_inset Formula $\vect k=\left(k_{x},0\right),k_{x}\in\left[0,\pi/p_{x}\right].$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:ewald branch cuts"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Computing the lattice 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 in evaluating
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:W definition"
|
||
|
||
\end_inset
|
||
|
||
is the asymptotic behaviour of the translation operator,
|
||
\begin_inset Formula $\tropsp{\vect 0,\alpha}{\vect m,\beta}\sim\left|\vect R_{\vect m}\right|^{-1}e^{i\kappa\left|\vect R_{\vect m}\right|}$
|
||
\end_inset
|
||
|
||
, so that its lattice sum does not in the strict sense converge for any
|
||
|
||
\begin_inset Formula $d>1$
|
||
\end_inset
|
||
|
||
-dimensional lattice unless
|
||
\begin_inset Formula $\Im\kappa>0$
|
||
\end_inset
|
||
|
||
.
|
||
\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
|
||
|
||
The problem of poorly converging lattice sums has been originally solved
|
||
for electrostatic potentials with Ewald summation
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "ewald_berechnung_1921"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Its basic idea is to decompose the lattice-summed function in two parts:
|
||
a short-range part that decays fast and can be summed directly, and a long-rang
|
||
e 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 the reciprocal lattice.
|
||
The same idea can be used also in the case of linear electrodynamic problems,
|
||
just the technical details are more complicated than in electrostatics.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:translation operator singular"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
we demonstratively expressed the translation operator elements as linear
|
||
combinations of (outgoing) scalar spherical wavefunctions
|
||
\begin_inset Formula $\sswfoutlm lm\left(\vect r\right)=h_{l}^{\left(1\right)}\left(r\right)\ush lm\left(\uvec r\right)$
|
||
\end_inset
|
||
|
||
, because for them, fortunately, exponentially convergent Ewald-type summation
|
||
techniques have been developed
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
add refs
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "moroz_quasi-periodic_2006,linton_one-_2009,linton_lattice_2010,kambe_theory_1967,kambe_theory_1967-1,kambe_theory_1968"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
and can be applied to our case.
|
||
If we formally label
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Marginal
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
FP: Check signs.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\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(\vect{R_{n}}-\vect s\right),\label{eq:sigma lattice sums}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
we see from 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:W definition"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
that the matrix elements of
|
||
\begin_inset Formula $W_{\alpha\beta}(\vect k)$
|
||
\end_inset
|
||
|
||
read
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\beta}\right)\tropsp{\vect 0,\alpha}{\vect m,\beta}e^{i\vect k\cdot\vect R_{\vect m}},
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\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*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Marginal
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Check signs
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
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"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For reader's reference, we list the Ewald-type formulae for lattice sums
|
||
|
||
\begin_inset Formula $\sigma_{l,m}\left(\vect k,\vect s\right)$
|
||
\end_inset
|
||
|
||
from
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "linton_lattice_2010"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
rewritten in a way that is independent on particular phase or normalisation
|
||
conventions of vector spherical harmonics.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In all three lattice dimensionality cases, the lattice sums are divided
|
||
into short-range and long-range parts,
|
||
\begin_inset Formula $\sigma_{l,m}\left(\vect k,\vect s\right)=\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)+\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)$
|
||
\end_inset
|
||
|
||
depending on a positive parameter
|
||
\begin_inset Formula $\eta$
|
||
\end_inset
|
||
|
||
.
|
||
The short-range part has in all three cases the same form:
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
FP: Check sign of s
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{multline}
|
||
\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{R_{n}+\vect s}\right|^{l}\ush lm\left(\vect{R_{n}+\vect s}\right)e^{i\vect k\cdot\left(\vect{R_{n}+\vect s}\right)}\\
|
||
\times\int_{\eta}^{\infty}e^{-\left|\vect{R_{n}+\vect s}\right|^{2}\xi^{2}}e^{-\kappa/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(\vect{R_{n}+\vect s}\right).\label{eq:Ewald in 3D short-range part}
|
||
\end{multline}
|
||
|
||
\end_inset
|
||
|
||
The formal
|
||
\begin_inset Formula $\left(1-\delta_{\vect{R_{n}},-\vect s}\right)$
|
||
\end_inset
|
||
|
||
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 (
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
self-interaction
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
) term on the last line of
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Ewald in 3D short-range part"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, which appears only when the displacement vector
|
||
\begin_inset Formula $\vect s$
|
||
\end_inset
|
||
|
||
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.
|
||
|
||
\begin_inset Formula $\Gamma(a,z)$
|
||
\end_inset
|
||
|
||
is the incomplete Gamma function.
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Poznámka ohledně zahození radiální části u kulových fcí?
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The long-range part for cases
|
||
\begin_inset Formula $d=1,2$
|
||
\end_inset
|
||
|
||
reads
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
FP: check sign of
|
||
\begin_inset Formula $\vect k$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\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\\
|
||
\times\sum_{\vect K\in\Lambda^{*}}\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)}{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\\
|
||
\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}
|
||
\end{multline}
|
||
|
||
\end_inset
|
||
|
||
and for
|
||
\begin_inset Formula $d=3$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\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^{*}}\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(\vect k+\vect K\right).\label{eq:Ewald in 3D long-range part 3D}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Here
|
||
\begin_inset Formula $\mathcal{A}$
|
||
\end_inset
|
||
|
||
is the unit cell volume (or length/area in the 1D/2D lattice cases).
|
||
The sums are taken 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
|
||
|
||
satisfying
|
||
\begin_inset Formula $\vect a_{i}\cdot\vect b_{j}=\delta_{ij}$
|
||
\end_inset
|
||
|
||
.
|
||
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"
|
||
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}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
The Ewald parameter
|
||
\begin_inset Formula $\eta$
|
||
\end_inset
|
||
|
||
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 $\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)$
|
||
\end_inset
|
||
|
||
.
|
||
For one-dimensional, square, and cubic lattices, the optimal choice is
|
||
|
||
\begin_inset Formula $\eta=\sqrt{\pi}/p$
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
is the direct lattice period
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "linton_lattice_2010"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Marginal
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Whatabout different geometries?
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Marginal
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
FP: I have some error estimates derived in my notes.
|
||
Should I include them?
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For a two-dimensional lattice, the incomplete
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
-function
|
||
\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,z\right)$
|
||
\end_inset
|
||
|
||
in the long-range part has a branch point at
|
||
\begin_inset Formula $z=0$
|
||
\end_inset
|
||
|
||
and special care has to be taken when choosing the appropriate branch.
|
||
If the wavenumber of the medium has a positive imaginary part,
|
||
\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
|
||
sum with the principal value of the incomplete
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
function being used in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Ewald in 3D long-range part 1D 2D"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
For other values of
|
||
\begin_inset Formula $\kappa$
|
||
\end_inset
|
||
|
||
, the branch choice is made in such way that
|
||
\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.
|
||
|
||
\begin_inset Formula $\kappa$
|
||
\end_inset
|
||
|
||
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
|
||
|
||
.
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Detailed physical interpretation of the remaining branch cuts is an open
|
||
question.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
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.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
(I HAVE SOME DERIVATIONS OF THE ESTIMATES IN MY NOTES; SHOULD I INCLUDE
|
||
THEM?)
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\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 8.7.3 from
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "NIST:DLMF"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|