Branch cuts illustration.
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Marek Nečada
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@ -622,7 +622,149 @@ d shapes.
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\begin_inset Formula $W\left(\omega,\vect k\right)$
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\end_inset
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, as described in the following.
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, as described in the following and illustrated in Fig.
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\begin_inset space \space{}
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:ewald branch cuts"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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\begin_inset Float figure
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placement document
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alignment document
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\align center
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\begin_inset Graphics
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filename figs/ewald_branchcuts.pdf
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width 100col%
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Left: Illustration of branch cuts in
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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\end_inset
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obtained using Ewald summation over two-dimensional square lattice in three-dim
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ensional space filled with dielectric medium with constant real refraction
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index
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\begin_inset Formula $n$
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\end_inset
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and wavenumber
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\begin_inset Formula $\kappa\left(\omega\right)=\omega n/c$
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\end_inset
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.
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The function is holomorphic in the positive imaginary half-plane.
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The points corresponding to the diffraction orders of an
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\begin_inset Quotes eld
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\end_inset
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empty
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\begin_inset Quotes erd
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\end_inset
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lattice lie on the real axis (pink), and from each of them two branch cuts
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originate: one due to the branch cut in the incomplete
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\begin_inset Formula $\Gamma$
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\end_inset
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function (orange, hyperbolic shape), and another due to the branch cut
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of
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\begin_inset Formula $\gamma(z)$
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\end_inset
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as defined in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:lilgamma"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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(blue, circular shape).
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Further non-analyticities might stem from the material model: the violet
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curve represents a branch cut originating from a complex square root in
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the refractive index
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\begin_inset Formula $n_{\mathrm{Au}}\left(\omega\right)=\sqrt{\varepsilon_{\mathrm{Au}}\left(\omega\right)}$
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\end_inset
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, where
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\begin_inset Formula $\varepsilon_{\mathrm{Au}}\left(\omega\right)$
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\end_inset
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is the Drude-Lorentz permittivity model of gold used for the scatterers.
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The other parameters used here are
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\begin_inset Formula $p_{x}=580\,\mathrm{nm}$
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\end_inset
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(lattice period),
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\begin_inset Formula $\vect k=\left(0.2\pi/p_{x},0\right)$
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\end_inset
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,
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\begin_inset Formula $n=1.52$
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\end_inset
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.
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The plot on the right shows the
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\begin_inset Quotes eld
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\end_inset
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empty
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\begin_inset Quotes erd
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\end_inset
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lattice diffraction orders on the line
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\begin_inset Formula $\vect k=\left(k_{x},0\right),x\in\left[0,\pi/p_{x}\right].$
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\end_inset
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:ewald branch cuts"
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Subsection
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@ -1002,9 +1144,9 @@ noprefix "false"
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is defined as
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\begin_inset Formula
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\[
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\gamma\left(z\right)=\left(z-1\right)^{\frac{1}{2}}\left(z+1\right)^{\frac{1}{2}},\quad-\frac{3\pi}{2}<\arg\left(z-1\right)<\frac{\pi}{2},-\frac{\pi}{2}<\arg\left(z+1\right)<\frac{3\pi}{2}.
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\]
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\begin{equation}
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\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}
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\end{equation}
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\end_inset
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@ -1203,18 +1345,33 @@ literal "false"
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\begin_inset Formula $\kappa$
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\end_inset
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away from the real axis.
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away from the real axis, as illustrated in Fig.
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\begin_inset space \space{}
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:ewald branch cuts"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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.
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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TODO Figure.
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Detailed physical interpretation of the remaining branch cuts is an open
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question.
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\end_layout
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\end_inset
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Detailed physical interpretation of the remaining branch cuts is an open
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question.
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\begin_inset Note Note
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status open
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