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 a 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 systems gives a possibility to study finite-size effects in periodic
 scatterer arrays.
\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) Bravais 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 integer multi-index 
\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}\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}
\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.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem unit cell"
plural "false"
caps "false"
noprefix "false"

\end_inset

 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, 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

 rewritten in a way that is independent on particular phase or normalisation
 conventions of vector spherical harmonics.
 The derivations of these formulae can otherwise be
\begin_inset Note Note
status open

\begin_layout Plain Layout
except for eq.
 xxx todo
\end_layout

\end_inset

 found in 
\begin_inset CommandInset citation
LatexCommand cite
key "moroz_quasi-periodic_2006,linton_lattice_2010"
literal "false"

\end_inset

.
 
\begin_inset Note Note
status open

\begin_layout Plain Layout
\begin_inset CommandInset citation
LatexCommand cite
key "linton_lattice_2010"
literal "false"

\end_inset

 (case 
\begin_inset Formula $\vect s=0$
\end_inset

, all lattice dimensionalities), 
\begin_inset CommandInset citation
LatexCommand cite
key "kambe_theory_1968"
literal "false"

\end_inset

 (
\begin_inset Formula $d=2$
\end_inset

) , 
\begin_inset CommandInset citation
LatexCommand cite
key "moroz_quasi-periodic_2006"
literal "false"

\end_inset

 (
\begin_inset Formula $d=1$
\end_inset

)
\end_layout

\end_inset


\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 explicit form of the long-range part of Ewald sum depends on the lattice
 dimensionality.
 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

 lying in the same subspace as the direct lattice vectors 
\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

.
 
\end_layout

\begin_layout Standard
For case 
\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^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot 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(\vect k+\vect K\right)\label{eq:Ewald in 3D long-range part 3D}
\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

\begin_layout Standard
For cases 
\begin_inset Formula $d=1,2$
\end_inset

, reasonable explicit forms assume that the lattice lies inside the 
\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

 Moreover, we require that the component of 
\begin_inset Formula $\vect s$
\end_inset

 normal to the lattice is parallel to the 
\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

 (this holds automatically for 
\begin_inset Formula $d=2$
\end_inset

).
 With these assumptions
\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^{*}}\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)}{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 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\\
\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)}{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 z = 0}
\end{multline}

\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 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}
\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 z = 0"
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