1418 lines
33 KiB
Plaintext
1418 lines
33 KiB
Plaintext
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\pdf_title "Accelerating lattice mode calculations with T-matrix method"
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\pdf_author "Marek Nečada"
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\begin_layout Title
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Accelerating lattice mode calculations with
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\begin_inset Formula $T$
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\end_inset
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-matrix method
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\end_layout
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\begin_layout Author
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Marek Nečada
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\end_layout
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\begin_layout Abstract
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The
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\begin_inset Formula $T$
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\end_inset
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-matrix approach is the method of choice for simulating optical response
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of a reasonably small system of compact linear scatterers on isotropic
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background.
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However, its direct utilisation for problems with infinite lattices is
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problematic due to slowly converging sums over the lattice.
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Here I develop a way to compute the problematic sums in the reciprocal
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space, making the
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\begin_inset Formula $T$
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\end_inset
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-matrix method very suitable for infinite periodic systems as well.
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\end_layout
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\begin_layout Section
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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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Assume a system of compact EM scatterers in otherwise homogeneous and isotropic
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medium, and assume that the system, i.e.
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both the medium and the scatterers, have linear response.
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A scattering problem in such system can be written as
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\begin_inset Formula
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\[
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A_{α}=T_{α}P_{α}=T_{α}(\sum_{β}S_{α\leftarrowβ}A_{β}+P_{0α})
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\]
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\end_inset
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where
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\begin_inset Formula $T_{α}$
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\end_inset
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is the
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\begin_inset Formula $T$
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\end_inset
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-matrix for scatterer α,
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\begin_inset Formula $A_{α}$
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\end_inset
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is its vector of the scattered wave expansion coefficient (the multipole
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indices are not explicitely indicated here) and
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\begin_inset Formula $P_{α}$
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\end_inset
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is the local expansion of the incoming sources.
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\begin_inset Formula $S_{α\leftarrowβ}$
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\end_inset
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is ...
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and ...
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is ...
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\end_layout
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\begin_layout Standard
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...
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\sum_{β}(\delta_{αβ}-T_{α}S_{α\leftarrowβ})A_{β}=T_{α}P_{0α}.
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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Now suppose that the scatterers constitute an infinite lattice
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{\vect aα}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=T_{\vect aα}P_{0\vect aα}.
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\]
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\end_inset
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Due to the periodicity, we can write
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\begin_inset Formula $S_{\vect aα\leftarrow\vect bβ}=S_{α\leftarrowβ}(\vect b-\vect a)$
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\end_inset
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and
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\begin_inset Formula $T_{\vect aα}=T_{\alpha}$
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\end_inset
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.
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In order to find lattice modes, we search for solutions with zero RHS
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\begin_inset Formula
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\[
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\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=0
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\]
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\end_inset
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and we assume periodic solution
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\begin_inset Formula $A_{\vect b\beta}(\vect k)=A_{\vect a\beta}e^{i\vect k\cdot\vect r_{\vect b-\vect a}}$
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\end_inset
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, yielding
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\begin_inset Formula
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\begin{eqnarray*}
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\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect a\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b-\vect a}} & = & 0,\\
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\sum_{\vect bβ}(\delta_{\vect{0b}}\delta_{αβ}-T_{α}S_{\vect 0α\leftarrow\vect bβ})A_{\vect 0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
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\sum_{β}(\delta_{αβ}-T_{α}\underbrace{\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect 0\beta}\left(\vect k\right) & = & 0,\\
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A_{\vect 0\alpha}\left(\vect k\right)-T_{α}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect 0\beta}\left(\vect k\right) & = & 0.
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\end{eqnarray*}
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\end_inset
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Therefore, in order to solve the modes, we need to compute the
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\begin_inset Quotes eld
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\end_inset
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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,
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\begin_inset Formula
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\begin{equation}
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W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
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\end{equation}
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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 Section
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Computing the Fourier sum of the translation operator
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||
\end_layout
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||
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\begin_layout Standard
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The problem evaluating
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W definition"
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\end_inset
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is the asymptotic behaviour of the translation operator,
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\begin_inset Formula $S_{\vect 0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
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\end_inset
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that makes the convergence of the sum quite problematic for any
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\begin_inset Formula $d>1$
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\end_inset
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-dimensional lattice.
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\begin_inset Foot
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status open
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\begin_layout Plain Layout
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Note that
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\begin_inset Formula $d$
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\end_inset
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here is dimensionality of the lattice, not the space it lies in, which
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I for certain reasons assume to be three.
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(TODO few notes on integration and reciprocal lattices in some appendix)
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\end_layout
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\end_inset
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In electrostatics, one can solve this problem with Ewald summation.
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Its basic idea is that if what asymptoticaly decays poorly in the direct
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space, will perhaps decay fast in the Fourier space.
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I use the same idea here, but everything will be somehow harder than in
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electrostatics.
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\end_layout
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\begin_layout Standard
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Let us re-express the sum in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W definition"
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\end_inset
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in terms of integral with a delta comb
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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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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}
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\end{equation}
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\end_inset
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The translation operator
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\begin_inset Formula $S$
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\end_inset
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is now a function defined in the whole 3d space;
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\begin_inset Formula $\vect r_{\alpha},\vect r_{\beta}$
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\end_inset
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||
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are the displacements of scatterers
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\begin_inset Formula $\alpha$
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\end_inset
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and
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\begin_inset Formula $\beta$
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\end_inset
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in a unit cell.
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The arrow notation
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\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})$
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\end_inset
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||
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means
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\begin_inset Quotes eld
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\end_inset
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||
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translation operator for spherical waves originating in
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\begin_inset Formula $\vect r+\vect r_{\beta}$
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\end_inset
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||
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evaluated in
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\begin_inset Formula $\vect r_{\alpha}$
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||
\end_inset
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||
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||
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\begin_inset Quotes erd
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\end_inset
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||
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and obviously
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\begin_inset Formula $S$
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||
\end_inset
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||
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||
is in fact a function of a single 3d argument,
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\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
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||
\end_inset
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||
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||
.
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||
Expression
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\begin_inset CommandInset ref
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||
LatexCommand eqref
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||
reference "eq:W integral"
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||
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||
\end_inset
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||
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||
can be rewritten as
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||
\begin_inset Formula
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||
\[
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||
W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
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||
\]
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||
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||
\end_inset
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||
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||
where changed the sign of
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||
\begin_inset Formula $\vect r/\vect{\bullet}$
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||
\end_inset
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||
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||
has been swapped under integration, utilising evenness of
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||
\begin_inset Formula $\dc{\basis u}$
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||
\end_inset
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||
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||
.
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||
Fourier transform of product is convolution of Fourier transforms, so (using
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||
formula
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||
\begin_inset CommandInset ref
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||
LatexCommand eqref
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||
reference "eq:Dirac comb uaFt"
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||
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||
\end_inset
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||
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||
for the Fourier transform of Dirac comb)
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||
\begin_inset Formula
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||
\begin{eqnarray}
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||
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 \\
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||
& = & \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 \\
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||
& = & \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}
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||
\end{eqnarray}
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||
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||
\end_inset
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||
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||
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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
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||
Factor
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||
\begin_inset Formula $\left(2\pi\right)^{\frac{d}{2}}$
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||
\end_inset
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||
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||
cancels out with the
|
||
\begin_inset Formula $\left(2\pi\right)^{-\frac{d}{2}}$
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||
\end_inset
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||
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||
factor appearing in the convolution/product formula in the unitary angular
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||
momentum convention.
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||
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||
\end_layout
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||
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||
\end_inset
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||
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||
As such, this is not extremely helpful because the the
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||
\emph on
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||
whole
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||
\emph default
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||
translation operator
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||
\begin_inset Formula $S$
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||
\end_inset
|
||
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||
has singularities in origin, hence its Fourier transform
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||
\begin_inset Formula $\uaft S$
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||
\end_inset
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||
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||
will decay poorly.
|
||
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||
\end_layout
|
||
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||
\begin_layout Standard
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||
However, Fourier transform is linear, so we can in principle separate
|
||
\begin_inset Formula $S$
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||
\end_inset
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||
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||
in two parts,
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||
\begin_inset Formula $S=S^{\textup{L}}+S^{\textup{S}}$
|
||
\end_inset
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||
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||
.
|
||
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||
\begin_inset Formula $S^{\textup{S}}$
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||
\end_inset
|
||
|
||
is a short-range part that decays sufficiently fast with distance so that
|
||
its direct-space lattice sum converges well;
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||
\begin_inset Formula $S^{\textup{S}}$
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||
\end_inset
|
||
|
||
must as well contain all the singularities of
|
||
\begin_inset Formula $S$
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||
\end_inset
|
||
|
||
in the origin.
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||
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 should converge nicely.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Finding a good decomposition
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The remaining challenge is therefore finding a suitable decomposition
|
||
\begin_inset Formula $S^{\textup{L}}+S^{\textup{S}}$
|
||
\end_inset
|
||
|
||
such that both
|
||
\begin_inset Formula $S^{\textup{S}}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\uaft{S^{\textup{L}}}$
|
||
\end_inset
|
||
|
||
decay fast enough with distance and are expressable analytically.
|
||
With these requirements, I do not expect to find gaussian asymptotics as
|
||
in the electrostatic Ewald formula—having
|
||
\begin_inset Formula $\sim x^{-t}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $t>d$
|
||
\end_inset
|
||
|
||
asymptotics would be nice, making the sums in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:W Short definition"
|
||
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:W Long definition"
|
||
|
||
\end_inset
|
||
|
||
absolutely convergent.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The translation operator
|
||
\begin_inset Formula $S$
|
||
\end_inset
|
||
|
||
for compact scatterers in 3d can be expressed as
|
||
\begin_inset Formula
|
||
\[
|
||
S_{l',m',t'\leftarrow l,m,t}\left(\vect r\leftarrow\vect 0\right)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect r},\phi_{\vect r}\right)z_{p}^{(J)}\left(k_{0}\left|\vect r\right|\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $Y_{l,m}\left(\theta,\phi\right)$
|
||
\end_inset
|
||
|
||
are the spherical harmonics,
|
||
\begin_inset Formula $z_{p}^{(J)}\left(r\right)$
|
||
\end_inset
|
||
|
||
some of the Bessel or Hankel functions (probably
|
||
\begin_inset Formula $h_{p}^{(1)}$
|
||
\end_inset
|
||
|
||
in the meaningful cases; TODO) and
|
||
\begin_inset Formula $c_{p}^{l,m,t\leftarrow l',m',t'}$
|
||
\end_inset
|
||
|
||
are some ugly but known coefficients (REF Xu 1996, eqs.
|
||
76,77).
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The spherical Hankel functions can be expressed analytically as (REF DLMF
|
||
10.49.6, 10.49.1)
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\right)!}{2^{k}k!\left(n-k\right)!},\label{eq:spherical Hankel function series}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
so if we find a way to deal with the radial functions
|
||
\begin_inset Formula $s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $q=1,2$
|
||
\end_inset
|
||
|
||
in 2d case or
|
||
\begin_inset Formula $q=1,2,3$
|
||
\end_inset
|
||
|
||
in 3d case, we get absolutely convergent summations in the direct space.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
2d
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Assume that all scatterers are placed in the plane
|
||
\begin_inset Formula $\vect z=0$
|
||
\end_inset
|
||
|
||
, so that the 2d Fourier transform of the long-range part of the translation
|
||
operator in terms of Hankel transforms, according to
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Fourier v. Hankel tf 2d"
|
||
|
||
\end_inset
|
||
|
||
, reads
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{multline*}
|
||
\uaft{S_{l',m',t'\leftarrow l,m,t}^{\textup{L}}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\
|
||
\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\frac{\pi}{2},0\right)e^{i(m'-m)\phi}i^{m'-m}\pht{m'-m}{h_{p}^{(1)\textup{L}}\left(k_{0}\vect{\bullet}\right)}\left(\left|\vect k\right|\right)
|
||
\end{multline*}
|
||
|
||
\end_inset
|
||
|
||
Here
|
||
\begin_inset Formula $h_{p}^{(1)\textup{L}}=h_{p}^{(1)}-h_{p}^{(1)\textup{S}}$
|
||
\end_inset
|
||
|
||
is a long range part of a given spherical Hankel function which has to
|
||
be found and which contains all the terms with far-field (
|
||
\begin_inset Formula $r\to\infty$
|
||
\end_inset
|
||
|
||
) asymptotics proportional to
|
||
\begin_inset Formula $\sim e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $q\le Q$
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $Q$
|
||
\end_inset
|
||
|
||
is at least two in order to achieve absolute convergence of the direct-space
|
||
sum, but might be higher in order to speed the convergence up.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Obviously, all the terms
|
||
\begin_inset Formula $\propto s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $q>Q$
|
||
\end_inset
|
||
|
||
of the spherical Hankel function
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:spherical Hankel function series"
|
||
|
||
\end_inset
|
||
|
||
can be kept untouched as part of
|
||
\begin_inset Formula $h_{p}^{(1)\textup{S}}$
|
||
\end_inset
|
||
|
||
, as they decay fast enough.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The remaining task is therefore to find a suitable decomposition of
|
||
\begin_inset Formula $s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $q\le Q$
|
||
\end_inset
|
||
|
||
into short-range and long-range parts,
|
||
\begin_inset Formula $s_{q}(r)=s_{q}^{\textup{S}}(r)+s_{q}^{\textup{L}}(r)$
|
||
\end_inset
|
||
|
||
, such that
|
||
\begin_inset Formula $s_{q}^{\textup{L}}(r)$
|
||
\end_inset
|
||
|
||
contains all the slowly decaying asymptotics and its Hankel transforms
|
||
decay desirably fast as well,
|
||
\begin_inset Formula $\pht n{s_{q}^{\textup{L}}}\left(k\right)=o(z^{-Q})$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $z\to\infty$
|
||
\end_inset
|
||
|
||
.
|
||
The latter requirement calls for suitable regularisation functions—
|
||
\begin_inset Formula $s_{q}^{\textup{L}}$
|
||
\end_inset
|
||
|
||
must be sufficiently smooth in the origin, so that
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\pht n{s_{q}^{\textup{L}}}\left(k\right)=\int_{0}^{\infty}s_{q}^{\textup{L}}\left(r\right)rJ_{n}\left(kr\right)\ud r=\int_{0}^{\infty}s_{q}\left(r\right)\rho\left(r\right)rJ_{n}\left(kr\right)\ud r\label{eq:2d long range regularisation problem statement}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
exists and decays fast enough.
|
||
|
||
\begin_inset Formula $J_{\nu}(r)\sim\left(r/2\right)^{\nu}/\Gamma\left(\nu+1\right)$
|
||
\end_inset
|
||
|
||
(REF DLMF 10.7.3) near the origin, so the regularisation function should
|
||
be
|
||
\begin_inset Formula $\rho(r)=o(r^{q-n-1})$
|
||
\end_inset
|
||
|
||
only to make
|
||
\begin_inset Formula $\pht n{s_{q}^{\textup{L}}}$
|
||
\end_inset
|
||
|
||
converge.
|
||
The additional decay speed requirement calls for at least
|
||
\begin_inset Formula $\rho(r)=o(r^{q-n+Q-1})$
|
||
\end_inset
|
||
|
||
, I guess.
|
||
At the same time,
|
||
\begin_inset Formula $\rho(r)$
|
||
\end_inset
|
||
|
||
must converge fast enough to one for
|
||
\begin_inset Formula $r\to\infty$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The electrostatic Ewald summation uses regularisation with
|
||
\begin_inset Formula $1-e^{-cr^{2}}$
|
||
\end_inset
|
||
|
||
.
|
||
However, such choice does not seem to lead to an analytical solution for
|
||
the current problem
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:2d long range regularisation problem statement"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
But it turns out that the family of functions
|
||
\begin_inset Formula
|
||
\[
|
||
\rho_{\kappa,c}(r)\equiv\left(1-e^{-cr}\right)^{\text{\kappa}},\quad c>0,\kappa\in\nats
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
leads to satisfactory results, as will be shown below.
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Hankel transforms of the long-range parts
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
3d
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{multline*}
|
||
\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\
|
||
\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect k},\phi_{\vect k}\right)\left(-i\right)^{p}\usht p{z_{p}^{(J)}}\left(\left|\vect k\right|\right)
|
||
\end{multline*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
(Appendix) Fourier vs.
|
||
Hankel transform
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Three dimensions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Given a nice enough function
|
||
\begin_inset Formula $f$
|
||
\end_inset
|
||
|
||
of a real 3d variable, assume its factorisation into radial and angular
|
||
parts
|
||
\begin_inset Formula
|
||
\[
|
||
f(\vect r)=\sum_{l,m}f_{l,m}(\left|\vect r\right|)\ush lm\left(\theta_{\vect r},\phi_{\vect r}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
Acording to (REF Baddour 2010, eqs.
|
||
13, 16), its Fourier transform can then be expressed in terms of Hankel
|
||
transforms (CHECK normalisation of
|
||
\begin_inset Formula $j_{n}$
|
||
\end_inset
|
||
|
||
, REF Baddour (1))
|
||
\begin_inset Formula
|
||
\[
|
||
\uaft f(\vect k)=\frac{4\pi}{\left(2\pi\right)^{\frac{3}{2}}}\sum_{l,m}\left(-i\right)^{l}\left(\bsht{f_{l,m}}{}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where the spherical Hankel transform
|
||
\begin_inset Formula $\bsht l{}$
|
||
\end_inset
|
||
|
||
of degree
|
||
\begin_inset Formula $l$
|
||
\end_inset
|
||
|
||
is defined as (REF Baddour eq.
|
||
2)
|
||
\begin_inset Formula
|
||
\[
|
||
\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
Using this convention, the inverse spherical Hankel transform is given by
|
||
(REF Baddour eq.
|
||
3)
|
||
\begin_inset Formula
|
||
\[
|
||
g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\,k^{2}\bsht lg(k)j_{l}(k),
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
so it is not unitary.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
An unitary convention would look like this:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Then
|
||
\begin_inset Formula $\usht l{}^{-1}=\usht l{}$
|
||
\end_inset
|
||
|
||
and the unitary, angular-momentum Fourier transform reads
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
\uaft f(\vect k) & = & \frac{4\pi}{\left(2\pi\right)^{\frac{3}{2}}}\sqrt{\frac{\pi}{2}}\sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right)\nonumber \\
|
||
& = & \sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right).\label{eq:Fourier v. Hankel tf 3d}
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
Cool.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Two dimensions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Similarly in 2d, let the expansion of
|
||
\begin_inset Formula $f$
|
||
\end_inset
|
||
|
||
be
|
||
\begin_inset Formula
|
||
\[
|
||
f\left(\vect r\right)=\sum_{m}f_{m}\left(\left|\vect r\right|\right)e^{im\phi_{\vect r}},
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
its Fourier transform is then (CHECK this, it is taken from the Wikipedia
|
||
article on Hankel transform)
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\uaft f\left(\vect k\right)=\sum_{m}i^{m}e^{im\phi_{\vect k}}\pht mf_{m}\left(\left|\vect k\right|\right)\label{eq:Fourier v. Hankel tf 2d}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where the Hankel transform of order
|
||
\begin_inset Formula $m$
|
||
\end_inset
|
||
|
||
is defined as
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\,g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
which is already self-inverse,
|
||
\begin_inset Formula $\pht m{}^{-1}=\pht m{}$
|
||
\end_inset
|
||
|
||
(hence also unitary).
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
(Appendix) Multidimensional Dirac comb
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
1D
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
This is all from Wikipedia
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Definitions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
Ш(t) & \equiv & \sum_{k=-\infty}^{\infty}\delta(t-k)\\
|
||
Ш_{T}(t) & \equiv & \sum_{k=-\infty}^{\infty}\delta(t-kT)=\frac{1}{T}Ш\left(\frac{t}{T}\right)
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Fourier series representation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
Ш_{T}(t)=\sum_{n=-\infty}^{\infty}e^{2\pi int/T}\label{eq:1D Dirac comb Fourier series}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Fourier transform
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
With unitary ordinary frequency Ft., i.e.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\uoft f(\vect{\xi})\equiv\int_{\mathbb{R}^{n}}f(\vect x)e^{-2\pi i\vect x\cdot\vect{\xi}}\ud^{n}\vect x
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
we have
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\uoft{Ш_{T}}(f)=\frac{1}{T}Ш_{\frac{1}{T}}(f)=\sum_{n=-\infty}^{\infty}e^{-i2\pi fnT}\label{eq:1D Dirac comb Ft ordinary freq}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and with unitary angular frequency Ft., i.e.
|
||
\begin_inset Formula
|
||
\[
|
||
\uaft f(\vect k)\equiv\frac{1}{\left(2\pi\right)^{n/2}}\int_{\mathbb{R}^{n}}f(\vect x)e^{-i\vect x\cdot\vect k}\ud^{n}\vect x
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
we have (CHECK)
|
||
\begin_inset Formula
|
||
\[
|
||
\uaft{Ш_{T}}(\omega)=\frac{\sqrt{2\pi}}{T}Ш_{\frac{2\pi}{T}}(\omega)=\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-i\omega nT}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Dirac comb for multidimensional lattices
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Definitions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Let
|
||
\begin_inset Formula $d$
|
||
\end_inset
|
||
|
||
be the dimensionality of the real vector space in question, and let
|
||
\begin_inset Formula $\basis u\equiv\left\{ \vect u_{i}\right\} _{i=1}^{d}$
|
||
\end_inset
|
||
|
||
denote a basis for some lattice in that space.
|
||
Let the corresponding lattice delta comb be
|
||
\begin_inset Formula
|
||
\[
|
||
\dc{\basis u}\left(\vect x\right)\equiv\sum_{n_{1}=-\infty}^{\infty}\ldots\sum_{n_{d}=-\infty}^{\infty}\delta\left(\vect x-\sum_{i=1}^{d}n_{i}\vect u_{i}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Furthemore, let
|
||
\begin_inset Formula $\rec{\basis u}\equiv\left\{ \rec{\vect u}_{i}\right\} _{i=1}^{d}$
|
||
\end_inset
|
||
|
||
be the reciprocal lattice basis, that is the basis satisfying
|
||
\begin_inset Formula $\vect u_{i}\cdot\rec{\vect u_{j}}=\delta_{ij}$
|
||
\end_inset
|
||
|
||
.
|
||
This slightly differs from the usual definition of a reciprocal basis,
|
||
here denoted
|
||
\begin_inset Formula $\recb{\basis u}\equiv\left\{ \recb{\vect u_{i}}\right\} _{i=1}^{d}$
|
||
\end_inset
|
||
|
||
, which satisfies
|
||
\begin_inset Formula $\vect u_{i}\cdot\recb{\vect u_{j}}=2\pi\delta_{ij}$
|
||
\end_inset
|
||
|
||
instead.
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Factorisation of a multidimensional lattice delta comb
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
By simple drawing, it can be seen that
|
||
\begin_inset Formula
|
||
\[
|
||
\dc{\basis u}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $c_{\basis u}$
|
||
\end_inset
|
||
|
||
is some numerical volume factor.
|
||
In order to determine
|
||
\begin_inset Formula $c_{\basis u}$
|
||
\end_inset
|
||
|
||
, let us consider only the
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
zero tooth
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
of the comb, leading to
|
||
\begin_inset Formula
|
||
\[
|
||
\delta^{d}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\delta\left(\vect x\cdot\rec{\vect u_{i}}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
From the scaling property of delta function,
|
||
\begin_inset Formula $\delta(ax)=\left|a\right|^{-1}\delta(x)$
|
||
\end_inset
|
||
|
||
, we get
|
||
\begin_inset Formula
|
||
\[
|
||
\delta^{d}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert ^{-1}\delta\left(\vect x\cdot\frac{\rec{\vect u_{i}}}{\left\Vert \rec{\vect u_{i}}\right\Vert }\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
From the Osgood's book (p.
|
||
375):
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\dc A(\vect x)=\frac{1}{\left|\det A\right|}\dc{}^{(d)}\left(A^{-1}\vect x\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Applying both sides to a test function that is one at the origin, we get
|
||
|
||
\begin_inset Formula $c_{\basis u}=\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert $
|
||
\end_inset
|
||
|
||
SRSLY?, and hence
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\dc{\basis u}(\vect x)=\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert \dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right).\label{eq:Dirac comb factorisation}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Fourier series representation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Utilising the Fourier series for 1D Dirac comb
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:1D Dirac comb Fourier series"
|
||
|
||
\end_inset
|
||
|
||
and the factorisation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Dirac comb factorisation"
|
||
|
||
\end_inset
|
||
|
||
, we get
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\dc{\basis u}(\vect x) & = & \prod_{j=1}^{d}\left\Vert \rec{\vect u_{j}}\right\Vert \sum_{n_{j}=-\infty}^{\infty}e^{2\pi in_{i}\vect x\cdot\rec{\vect u_{i}}}\\
|
||
& = & \left(\prod_{j=1}^{d}\left\Vert \rec{\vect u_{j}}\right\Vert \right)\sum_{\vect n\in\mathbb{Z}^{d}}e^{2\pi i\vect x\cdot\sum_{k=1}^{d}n_{k}\rec{\vect u_{k}}}.
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Fourier transform (OK)
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
From the Osgood's book https://see.stanford.edu/materials/lsoftaee261/chap8.pdf,
|
||
p.
|
||
379
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\uoft{\dc{\basis u}}\left(\vect{\xi}\right)=\left|\det\rec{\basis u}\right|\dc{\rec{\basis u}}^{(d)}\left(\vect{\xi}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
And consequently, for unitary/angular frequency it is
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
\uaft{\dc{\basis u}}\left(\vect k\right) & = & \frac{1}{\left(2\pi\right)^{\frac{d}{2}}}\uoft{\dc{\basis u}}\left(\frac{\vect k}{2\pi}\right)\nonumber \\
|
||
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\dc{\rec{\basis u}}^{(d)}\left(\frac{\vect k}{2\pi}\right)\nonumber \\
|
||
& = & \left(2\pi\right)^{\frac{d}{2}}\left|\det\rec{\basis u}\right|\dc{\recb{\basis u}}\left(\vect k\right)\nonumber \\
|
||
& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\dc{\recb{\basis u}}\left(\vect k\right).\label{eq:Dirac comb uaFt}
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
On the third line, we used the stretch theorem, getting
|
||
\begin_inset Formula
|
||
\[
|
||
\dc{\recb{\basis u}}\left(\vect k\right)=\dc{2\pi\rec{\basis u}}\left(\vect k\right)=\left(2\pi\right)^{-d}\dc{\rec{\basis u}}\left(\frac{\vect k}{2\pi}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Convolution
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\left(f\ast\dc{\basis u}\right)(\vect x)=\sum_{\vect t\in\basis u\ints^{d}}f(\vect x-\vect t)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
So, from the stretch theorem
|
||
\begin_inset Formula $\uoft{(f(A\vect x))}=\frac{1}{\left|\det A\right|}\uoft{f\left(A^{-T}\vect{\xi}\right)}=\left|\det A^{-T}\right|\uoft{f\left(A^{-T}\vect{\xi}\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
From
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Dirac comb factorisation"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:1D Dirac comb Ft ordinary freq"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\[
|
||
\uoft{\dc{\basis u}}(\vect{\xi})=\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert \dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|