qpms/lepaper/infinite.lyx

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\pdf_author "Marek Nečada"
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\begin_body

\begin_layout Section
Infinite periodic systems
\begin_inset CommandInset label
LatexCommand label
name "sec:Infinite"

\end_inset


\begin_inset FormulaMacro
\newcommand{\dlv}{\vect a}
\end_inset


\begin_inset FormulaMacro
\newcommand{\rlv}{\vect b}
\end_inset


\end_layout

\begin_layout Standard
Although large finite systems are where MSTMM excels the most, there are
 several reasons that makes its extension to infinite lattices (where periodic
 boundary conditions might be applied) desirable as well.
 Other methods might be already fast enough, but MSTMM will be faster in
 most cases in which there is enough spacing between the neighboring particles.
 MSTMM works well with any space group symmetry the system might have (as
 opposed to, for example, FDTD with cubic mesh applied to a honeycomb lattice),
 which makes e.g.
 application of group theory in mode analysis quite easy.
\begin_inset Note Note
status open

\begin_layout Plain Layout
Topology anoyne?
\end_layout

\end_inset

 And finally, having a method that handles well both infinite and large
 finite system gives a possibility to study finite-size effects in periodic
 scatterer arrays.
\end_layout

\begin_layout Subsection
Notation
\end_layout

\begin_layout Standard
TODO Fourier transforms, Delta comb, lattice bases, reciprocal lattices
 etc.
\end_layout

\begin_layout Subsection
Formulation of the problem
\end_layout

\begin_layout Standard
Let us have a linear system of compact EM scatterers on a homogeneous background
 as in Section 
\begin_inset CommandInset ref
LatexCommand eqref
reference "subsec:Multiple-scattering"
plural "false"
caps "false"
noprefix "false"

\end_inset

, but this time, the system shall be periodic: let there be a 
\begin_inset Formula $d$
\end_inset

-dimensional (
\begin_inset Formula $d$
\end_inset

 can be 1, 2 or 3) lattice embedded into the three-dimensional real space,
 with lattice vectors 
\begin_inset Formula $\left\{ \dlv_{i}\right\} _{i=1}^{d}$
\end_inset

, and let the lattice points be labeled with an 
\begin_inset Formula $d$
\end_inset

-dimensional integar multiindex 
\begin_inset Formula $\vect n\in\ints^{d}$
\end_inset

, so the lattice points have cartesian coordinates 
\begin_inset Formula $\vect R_{\vect n}=\sum_{i=1}^{d}n_{i}\vect a_{i}$
\end_inset

.
 There can be several scatterers per unit cell with indices 
\begin_inset Formula $\alpha$
\end_inset

 from set 
\begin_inset Formula $\mathcal{P}_{1}$
\end_inset

 and (relative) positions inside the unit cell 
\begin_inset Formula $\vect r_{\alpha}$
\end_inset

; any particle of the periodic system can thus be labeled by a multiindex
 from 
\begin_inset Formula $\mathcal{P}=\ints^{d}\times\mathcal{P}_{1}$
\end_inset

.
 The scatterers are located at positions 
\begin_inset Formula $\vect r_{\vect n,\alpha}=\vect R_{\vect n}+\vect r_{\alpha}$
\end_inset

 and their 
\begin_inset Formula $T$
\end_inset

-matrices are periodic, 
\begin_inset Formula $T_{\vect n,\alpha}=T_{\alpha}$
\end_inset

.
 In such system, the multiple-scattering problem 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem"
plural "false"
caps "false"
noprefix "false"

\end_inset

 can be rewritten as
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\outcoeffp{\vect n,\alpha}-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect n,\alpha}{\vect m,\beta}\outcoeffp{\vect m,\beta}=T_{\alpha}\rcoeffincp{\vect n,\alpha}.\quad\left(\vect n,\alpha\right)\in\mathcal{P}\label{eq:Multiple-scattering problem periodic}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Due to periodicity, we can also write 
\begin_inset Formula $\tropsp{\vect n,\alpha}{\vect m,\beta}=\tropsp{\alpha}{\beta}\left(\vect R_{\vect m}-\vect R_{\vect n}\right)=\tropsp{\alpha}{\beta}\left(\vect R_{\vect m-\vect n}\right)=\tropsp{\vect 0,\alpha}{\vect m-\vect n,\beta}$
\end_inset

.
 Assuming quasi-periodic right-hand side with quasi-momentum 
\begin_inset Formula $\vect k$
\end_inset

, 
\begin_inset Formula $\rcoeffincp{\vect n,\alpha}=\rcoeffincp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
\end_inset

, the solutions of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem periodic"
plural "false"
caps "false"
noprefix "false"

\end_inset

 will be also quasi-periodic according to Bloch theorem, 
\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
\end_inset

, and eq.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem periodic"
plural "false"
caps "false"
noprefix "false"

\end_inset

 can be rewritten as follows
\begin_inset Formula 
\begin{align}
\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect n,\alpha}{\vect m,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}},\nonumber \\
\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect 0,\alpha}{\vect m-\vect n,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m-\vect n}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\nonumber \\
\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect 0,\alpha\right)\right\} }\tropsp{\vect 0,\alpha}{\vect m,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\nonumber \\
\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\beta\in\mathcal{P}}W_{\alpha\beta}\left(\vect k\right)\outcoeffp{\vect 0,\beta}\left(\vect k\right) & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\label{eq:Multiple-scattering problem unit cell}
\end{align}

\end_inset

so we reduced the initial scattering problem to one involving only the field
 expansion coefficients from a single unit cell, but we need to compute
 the 
\begin_inset Quotes eld
\end_inset

lattice Fourier transform
\begin_inset Quotes erd
\end_inset

 of the translation operator,
\begin_inset Formula 
\begin{equation}
W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\beta}\right)\tropsp{\vect 0,\alpha}{\vect m,\beta}e^{i\vect k\cdot\vect R_{\vect m}},\label{eq:W definition}
\end{equation}

\end_inset

evaluation of which is possible but quite non-trivial due to the infinite
 lattice sum, so we explain it separately in Sect.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "subsec:W operator evaluation"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
As in the case of a finite system, eq.
 can be written in a shorter block-matrix form,
\begin_inset Formula 
\begin{equation}
\left(I-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}
\end{equation}

\end_inset

 Eq.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem unit cell"
plural "false"
caps "false"
noprefix "false"

\end_inset

 can be used to calculate electromagnetic response of the structure to external
 quasiperiodic driving field – most notably a plane wave.
 However, the non-trivial solutions of the equation with right hand side
 (i.e.
 the external driving) set to zero, 
\begin_inset Formula 
\begin{equation}
\left(I-TW\right)\outcoeffp{\vect 0}\left(\vect k\right)=0,\label{eq:lattice mode equation}
\end{equation}

\end_inset

describes the 
\emph on
lattice modes.

\emph default
 Non-trivial solutions to 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lattice mode equation"
plural "false"
caps "false"
noprefix "false"

\end_inset

 exist if the matrix on the left-hand side 
\begin_inset Formula $M\left(\omega,\vect k\right)=\left(I-T\left(\omega\right)W\left(\omega,\vect k\right)\right)$
\end_inset

 is singular – this condition gives the 
\emph on
dispersion relation
\emph default
 for the periodic structure.
 Note that in realistic (lossy) systems, at least one of the pair 
\begin_inset Formula $\omega,\vect k$
\end_inset

 will acquire complex values.
 The solution 
\begin_inset Formula $\outcoeffp{\vect 0}\left(\vect k\right)$
\end_inset

 is then obtained as the right 
\begin_inset Note Note
status open

\begin_layout Plain Layout
CHECK!
\end_layout

\end_inset

 singular vector of 
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset

 corresponding to the zero singular value.
\end_layout

\begin_layout Subsection
Numerical solution
\end_layout

\begin_layout Standard
In practice, equation 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem unit cell block form"
plural "false"
caps "false"
noprefix "false"

\end_inset

 is solved in the same way as eq.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem block form"
plural "false"
caps "false"
noprefix "false"

\end_inset

 in the multipole degree truncated form.
\end_layout

\begin_layout Standard
The lattice mode problem 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lattice mode equation"
plural "false"
caps "false"
noprefix "false"

\end_inset

 is (after multipole degree truncation) solved by finding 
\begin_inset Formula $\omega,\vect k$
\end_inset

 for which the matrix 
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset

 has a zero singular value.
 A naïve approach to do that is to sample a volume with a grid in the 
\begin_inset Formula $\left(\omega,\vect k\right)$
\end_inset

 space, performing a singular value decomposition of 
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset

 at each point and finding where the lowest singular value of 
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset

 is close enough to zero.
 However, this approach is quite expensive, for 
\begin_inset Formula $W\left(\omega,\vect k\right)$
\end_inset

 has to be evaluated for each 
\begin_inset Formula $\omega,\vect k$
\end_inset

 pair separately (unlike the original finite case 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem block form"
plural "false"
caps "false"
noprefix "false"

\end_inset

 translation operator 
\begin_inset Formula $\trops$
\end_inset

, which, for a given geometry, depends only on frequency).
 Therefore, a much more efficient approach to determine the photonic bands
 is to sample the 
\begin_inset Formula $\vect k$
\end_inset

-space (a whole Brillouin zone or its part) and for each fixed 
\begin_inset Formula $\vect k$
\end_inset

 to find a corresponding frequency 
\begin_inset Formula $\omega$
\end_inset

 with zero singular value of 
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset

 using a minimisation algorithm (two- or one-dimensional, depending on whether
 one needs the exact complex-valued 
\begin_inset Formula $\omega$
\end_inset

 or whether the its real-valued approximation is satisfactory).
 Typically, a good initial guess for 
\begin_inset Formula $\omega\left(\vect k\right)$
\end_inset

 is obtained from the empty lattice approximation, 
\begin_inset Formula $\left|\vect k\right|=\sqrt{\epsilon\mu}\omega/c_{0}$
\end_inset

 (modulo lattice points; TODO write this a clean way).
 A somehow challenging step is to distinguish the different bands that can
 all be very close to the empty lattice approximation, especially if the
 particles in the systems are small.
 In high-symmetry points of the Brilloin zone, this can be solved by factorising
 
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset

 into irreducible representations 
\begin_inset Formula $\Gamma_{i}$
\end_inset

 and performing the minimisation in each irrep separately, cf.
 Section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Symmetries"
plural "false"
caps "false"
noprefix "false"

\end_inset

, and using the different 
\begin_inset Formula $\omega_{\Gamma_{i}}\left(\vect k\right)$
\end_inset

 to obtain the initial guesses for the nearby points 
\begin_inset Formula $\vect k+\delta\vect k$
\end_inset

.
\end_layout

\begin_layout Subsection
Computing the Fourier sum of the translation operator
\begin_inset CommandInset label
LatexCommand label
name "subsec:W operator evaluation"

\end_inset


\end_layout

\begin_layout Standard
The problem evaluating 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"

\end_inset

 is the asymptotic behaviour of the translation operator, 
\begin_inset Formula $\tropsp{\vect 0,\alpha}{\vect m,\beta}\sim\left|\vect R_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect R_{\vect b}\right|}$
\end_inset

 that does not in the strict sense converge for any 
\begin_inset Formula $d>1$
\end_inset

-dimensional lattice.
\begin_inset Note Note
status open

\begin_layout Plain Layout
\begin_inset Foot
status open

\begin_layout Plain Layout
Note that 
\begin_inset Formula $d$
\end_inset

 here is dimensionality of the lattice, not the space it lies in, which
 I for certain reasons assume to be three.
 (TODO few notes on integration and reciprocal lattices in some appendix)
\end_layout

\end_inset


\end_layout

\end_inset

 In electrostatics, this problem can be solved with Ewald summation [TODO
 REF].
 Its basic idea is that if what asymptoticaly decays poorly in the direct
 space, will perhaps decay fast in the Fourier space.
 We use the same idea here, but the technical details are more complicated
 than in electrostatics.
\end_layout

\begin_layout Standard
Let us re-express the sum in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"

\end_inset

 in terms of integral with a delta comb
\begin_inset FormulaMacro
\renewcommand{\basis}[1]{\mathfrak{#1}}
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
W_{\alpha\beta}(\vect k)=\int\ud^{d}\vect r\dc{\basis u}(\vect r)S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})e^{i\vect k\cdot\vect r}.\label{eq:W integral}
\end{equation}

\end_inset

The translation operator 
\begin_inset Formula $S$
\end_inset

 is now a function defined in the whole 3d space; 
\begin_inset Formula $\vect r_{\alpha},\vect r_{\beta}$
\end_inset

 are the displacements of scatterers 
\begin_inset Formula $\alpha$
\end_inset

 and 
\begin_inset Formula $\beta$
\end_inset

 in a unit cell.
 The arrow notation 
\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})$
\end_inset

 means 
\begin_inset Quotes eld
\end_inset

translation operator for spherical waves originating in 
\begin_inset Formula $\vect r+\vect r_{\beta}$
\end_inset

 evaluated in 
\begin_inset Formula $\vect r_{\alpha}$
\end_inset


\begin_inset Quotes erd
\end_inset

 and obviously 
\begin_inset Formula $S$
\end_inset

 is in fact a function of a single 3d argument, 
\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
\end_inset

.
 Expression 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W integral"

\end_inset

 can be rewritten as
\begin_inset Formula 
\[
W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
\]

\end_inset

where changed the sign of 
\begin_inset Formula $\vect r/\vect{\bullet}$
\end_inset

 has been swapped under integration, utilising evenness of 
\begin_inset Formula $\dc{\basis u}$
\end_inset

.
 Fourier transform of product is convolution of Fourier transforms, so (using
 formula 
\begin_inset Note Note
status open

\begin_layout Plain Layout
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Dirac comb uaFt"

\end_inset


\end_layout

\end_inset

(REF?) for the Fourier transform of Dirac comb)
\begin_inset Formula 
\begin{eqnarray}
W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)\nonumber \\
 & = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\nonumber \\
 & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\
 & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\nonumber 
\end{eqnarray}

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
Factor 
\begin_inset Formula $\left(2\pi\right)^{\frac{d}{2}}$
\end_inset

 cancels out with the 
\begin_inset Formula $\left(2\pi\right)^{-\frac{d}{2}}$
\end_inset

 factor appearing in the convolution/product formula in the unitary angular
 momentum convention.
 
\end_layout

\end_inset

As such, this is not extremely helpful because the the 
\emph on
whole
\emph default
 translation operator 
\begin_inset Formula $S$
\end_inset

 has singularities in origin, hence its Fourier transform 
\begin_inset Formula $\uaft S$
\end_inset

 will decay poorly.
 
\end_layout

\begin_layout Standard
However, Fourier transform is linear, so we can in principle separate 
\begin_inset Formula $S$
\end_inset

 in two parts, 
\begin_inset Formula $S=S^{\textup{L}}+S^{\textup{S}}$
\end_inset

.
 
\begin_inset Formula $S^{\textup{S}}$
\end_inset

 is a short-range part that decays sufficiently fast with distance so that
 its direct-space lattice sum converges well; 
\begin_inset Formula $S^{\textup{S}}$
\end_inset

 must as well contain all the singularities of 
\begin_inset Formula $S$
\end_inset

 in the origin.
 The other part, 
\begin_inset Formula $S^{\textup{L}}$
\end_inset

, will retain all the slowly decaying terms of 
\begin_inset Formula $S$
\end_inset

 but it also has to be smooth enough in the origin, so that its Fourier
 transform 
\begin_inset Formula $\uaft{S^{\textup{L}}}$
\end_inset

 decays fast enough.
 (The same idea lies behind the Ewald summation in electrostatics.) Using
 the linearity of Fourier transform and formulae 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W sum in reciprocal space"

\end_inset

, the operator 
\begin_inset Formula $W_{\alpha\beta}$
\end_inset

 can then be re-expressed as
\begin_inset Formula 
\begin{eqnarray}
W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\
W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect 0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
\end{eqnarray}

\end_inset

where both sums expected to converge nicely.
 We note that the elements of the translation operators 
\begin_inset Formula $\tropr,\trops$
\end_inset

 in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:translation operator"
plural "false"
caps "false"
noprefix "false"

\end_inset

 can be rewritten as linear combinations of expressions 
\begin_inset Formula $\ush{\nu}{\mu}\left(\uvec d\right)j_{n}\left(d\right),\ush{\nu}{\mu}\left(\uvec d\right)h_{n}^{(1)}\left(d\right)$
\end_inset

 (TODO WRITE THEM EXPLICITLY IN THIS FORM), respectively, hence if we are
 able evaluate the lattice sums sums
\begin_inset Note Note
status open

\begin_layout Plain Layout
CHECK THE FOLLOWING EXPRESSION FOR CORRECT FUNCTION ARGUMENTS
\end_layout

\end_inset


\begin_inset Formula 
\begin{equation}
\sigma_{\nu}^{\mu}\left(\vect k\right)=\sum_{\vect n\in\ints^{d}\backslash\left\{ \vect 0\right\} }e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\ush{\nu}{\mu}\left(\uvec{R_{n}}\right)h_{n}^{(1)}\left(R_{n}\right),\label{eq:sigma lattice sums}
\end{equation}

\end_inset

then by linearity, we can get the 
\begin_inset Formula $W_{\alpha\beta}\left(\vect k\right)$
\end_inset

 operator as well.
\end_layout

\begin_layout Standard
TODO ADD MOROZ AND OTHER REFS HERE.
 
\begin_inset CommandInset citation
LatexCommand cite
key "linton_one-_2009"
literal "true"

\end_inset

 offers an exponentially convergent Ewald-type summation method for 
\begin_inset Formula $\sigma_{\nu}^{\mu}\left(\vect k\right)=\sigma_{\nu}^{\mu(\mathrm{S})}\left(\vect k\right)+\sigma_{\nu}^{\mu(\mathrm{L})}\left(\vect k\right)$
\end_inset

.
 Here we rewrite them in a form independent on the convention used for spherical
 harmonics (as long as they are complex
\begin_inset Note Note
status open

\begin_layout Plain Layout
lepší formulace
\end_layout

\end_inset

).
 The short-range part reads (UNIFY INDEX NOTATION)
\begin_inset Formula 
\begin{multline}
\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)=-\frac{2^{n+1}i}{k^{n+1}\sqrt{\pi}}\sum_{\vect R\in\Lambda}^{'}\left|\vect R\right|^{n}e^{i\vect{\beta}\cdot\vect R}Y_{n}^{m}\left(\vect R\right)\int_{\eta}^{\infty}e^{-\left|\vect R\right|^{2}\xi^{2}}e^{-k/4\xi^{2}}\xi^{2n}\ud\xi\\
+\frac{\delta_{n0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{k}{4\eta^{2}}\right)Y_{n}^{m},\label{eq:Ewald in 3D short-range part}
\end{multline}

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
NEPATŘÍ TAM NĚJAKÁ DELTA FUNKCE K PŮVODNÍMU 
\begin_inset Formula $\sigma_{n}^{m(0)}$
\end_inset

?
\end_layout

\end_inset

and the long-range part (FIXME, this is the 2D version; include the 1D and
 3D lattice expressions as well)
\begin_inset Formula 
\begin{multline}
\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)=-\frac{i^{n+1}}{k^{2}\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\\
\times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/k\right)^{n-2j}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\gamma_{pq}\right)^{2j-1}\label{eq:Ewald in 3D long-range part}
\end{multline}

\end_inset

where 
\begin_inset Formula $\xi$
\end_inset

 is TODO, 
\begin_inset Formula $\beta_{pq}$
\end_inset

 is TODO, 
\begin_inset Formula $\Gamma_{j,pq}$
\end_inset

 is TODO and 
\begin_inset Formula $\eta$
\end_inset

 is a real parameter that determines the pace of convergence of both parts.
 The larger 
\begin_inset Formula $\eta$
\end_inset

 is, the faster 
\begin_inset Formula $\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)$
\end_inset

 converges but the slower 
\begin_inset Formula $\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)$
\end_inset

 converges.
 Therefore (based on the lattice geometry) it has to be adjusted in a way
 that a reasonable amount of terms needs to be evaluated numerically from
 both 
\begin_inset Formula $\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)$
\end_inset

 and 
\begin_inset Formula $\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)$
\end_inset

 .
 Generally, a good choice for 
\begin_inset Formula $\eta$
\end_inset

 is TODO; in order to achieve accuracy TODO, one has to evaluate the terms
 on TODO lattice points.
 (I HAVE SOME DERIVATIONS OF THE ESTIMATES IN MY NOTES; SHOULD I INCLUDE
 THEM?)
\end_layout

\begin_layout Standard
In practice, the integrals in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Ewald in 3D short-range part"
plural "false"
caps "false"
noprefix "false"

\end_inset

 can be easily evaluated by numerical quadrature and the incomplete 
\begin_inset Formula $\Gamma$
\end_inset

-functions using the series TODO and TODO from DLMF.
\end_layout

\end_body
\end_document