Infinite systems WIP (how to numerical solutions)
Former-commit-id: dd3118e392b58eb9ae2260581e93028561571245
This commit is contained in:
Marek Nečada
parent
d5b6f8f5d4
commit
f167360c1e
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@ -802,6 +802,17 @@ literal "true"
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset include
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LatexCommand include
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filename "symmetries.lyx"
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literal "true"
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -290,8 +290,8 @@ outgoing
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, respectively, defined as follows:
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\begin_inset Formula
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\begin{align*}
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\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\
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\vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),
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\vswfrtlm1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh1lm\left(\uvec r\right),\\
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\vswfrtlm2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh3lm\left(\uvec r\right),
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\end{align*}
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\end_inset
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@ -299,8 +299,8 @@ outgoing
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\begin_inset Formula
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\begin{align*}
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\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\
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\vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\\
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\vswfouttlm1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh1lm\left(\uvec r\right),\\
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\vswfouttlm2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh3lm\left(\uvec r\right),\\
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& \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,
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\end{align*}
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@ -326,9 +326,9 @@ vector spherical harmonics
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\begin_inset Formula
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\begin{align*}
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\vsh 1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\\
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\vsh 2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\\
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\vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).
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\vsh1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\\
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\vsh2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\\
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\vsh3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).
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\end{align*}
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\end_inset
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@ -452,7 +452,7 @@ noprefix "false"
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\end_inset
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inside a ball
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\begin_inset Formula $\openball 0{R^{>}}$
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\begin_inset Formula $\openball0{R^{>}}$
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\end_inset
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with radius
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@ -470,7 +470,7 @@ noprefix "false"
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\end_inset
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to have a complete basis of the solutions in the volume
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\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
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\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$
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\end_inset
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.
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@ -496,7 +496,7 @@ The single-particle scattering problem at frequency
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\end_inset
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and let the whole volume
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\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
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\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$
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\end_inset
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be filled with a homogeneous isotropic medium with wave number
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@ -532,7 +532,7 @@ If there was no scatterer and
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\end_inset
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due to sources outside
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\begin_inset Formula $\openball 0R$
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\begin_inset Formula $\openball0R$
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\end_inset
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would remain.
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@ -793,7 +793,7 @@ literal "true"
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.
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Let the field in
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\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
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\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$
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\end_inset
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have expansion as in
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@ -812,7 +812,7 @@ noprefix "false"
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\end_inset
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to
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\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
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\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$
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\end_inset
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via by electromagnetic radiation is
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@ -864,8 +864,8 @@ In many scattering problems considered in practice, the driving field is
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with expansion coefficients
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\begin_inset Formula
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\begin{eqnarray}
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\rcoeffptlm{}1lm\left(\vect k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right),\nonumber \\
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\rcoeffptlm{}2lm\left(\vect k,\vect E_{0}\right) & = & -4\pi i^{l+1}\vshD 2lm\left(\uvec k\right).\label{eq:plane wave expansion}
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\rcoeffptlm{}1lm\left(\vect k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD1lm\left(\uvec k\right),\nonumber \\
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\rcoeffptlm{}2lm\left(\vect k,\vect E_{0}\right) & = & -4\pi i^{l+1}\vshD2lm\left(\uvec k\right).\label{eq:plane wave expansion}
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\end{eqnarray}
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\end_inset
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@ -1251,7 +1251,17 @@ noprefix "false"
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\end_inset
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can then be solved using standard numerical linear algebra methods.
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can then be solved using standard numerical linear algebra methods (typically,
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by LU factorisation of the
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\begin_inset Formula $\left(I-T\trops\right)$
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\end_inset
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matrix at a given frequency, and then solving with Gauss elimination for
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as many different incident
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\begin_inset Formula $\rcoeffinc$
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\end_inset
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vectors as needed).
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\end_layout
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\begin_layout Standard
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@ -324,9 +324,9 @@ noprefix "false"
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As in the case of a finite system, eq.
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can be written in a shorter block-matrix form,
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\begin_inset Formula
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\[
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\left(I-WT\right)\outcoeffp{\vect 0}\left(\vect k\right)=\rcoeffincp{\vect 0}\left(\vect k\right)
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\]
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\begin{equation}
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\left(I-WT\right)\outcoeffp{\vect 0}\left(\vect k\right)=\rcoeffincp{\vect 0}\left(\vect k\right)\label{eq:Multiple-scattering problem unit cell block form}
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\end{equation}
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\end_inset
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@ -343,8 +343,196 @@ noprefix "false"
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can be used to calculate electromagnetic response of the structure to external
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quasiperiodic driving field – most notably a plane wave.
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However, if one sets the right the right-hand side to zero, it can also
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be used to find electromagnetic lattice modes
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However, the non-trivial solutions of the equation with right hand side
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(i.e.
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the external driving) set to zero,
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\begin_inset Formula
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\begin{equation}
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\left(I-WT\right)\outcoeffp{\vect 0}\left(\vect k\right)=0,\label{eq:lattice mode equation}
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\end{equation}
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\end_inset
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describes the
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\emph on
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lattice modes.
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\emph default
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Non-trivial solutions to
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:lattice mode equation"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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exist if the matrix on the left-hand side
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\begin_inset Formula $M\left(\omega,\vect k\right)=\left(I-W\left(\omega,\vect k\right)T\left(\omega\right)\right)$
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\end_inset
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is singular – this condition gives the
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\emph on
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dispersion relation
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\emph default
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for the periodic structure.
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Note that in realistic (lossy) systems, at least one of the pair
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\begin_inset Formula $\omega,\vect k$
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\end_inset
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will acquire complex values.
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\end_layout
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\begin_layout Subsection
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Numerical solution
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\end_layout
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\begin_layout Standard
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In practice, equation
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem unit cell block form"
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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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is solved in the same way as eq.
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem block form"
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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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in the multipole degree truncated form.
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\end_layout
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\begin_layout Standard
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The lattice mode problem
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:lattice mode equation"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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is (after multipole degree truncation) solved by finding
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\begin_inset Formula $\omega,\vect k$
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\end_inset
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for which the matrix
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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\end_inset
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has a zero singular value.
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A naïve approach to do that is to sample a volume with a grid in the
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\begin_inset Formula $\left(\omega,\vect k\right)$
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\end_inset
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space, performing a singular value decomposition of
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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\end_inset
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at each point and finding where the lowest singular value of
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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\end_inset
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is close enough to zero.
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However, this approach is quite expensive, for
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\begin_inset Formula $W\left(\omega,\vect k\right)$
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\end_inset
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has to be evaluated for each
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\begin_inset Formula $\omega,\vect k$
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\end_inset
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pair separately (unlike the original finite case
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem block form"
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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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translation operator
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\begin_inset Formula $\trops$
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\end_inset
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, which, for a given geometry, depends only on frequency).
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Therefore, a much more efficient approach to determine the photonic bands
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is to sample the
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\begin_inset Formula $\vect k$
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\end_inset
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-space (a whole Brillouin zone or its part) and for each fixed
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\begin_inset Formula $\vect k$
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\end_inset
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to find a corresponding frequency
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\begin_inset Formula $\omega$
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\end_inset
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with zero singular value of
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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\end_inset
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using a minimisation algorithm (two- or one-dimensional, depending on whether
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one needs the exact complex-valued
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\begin_inset Formula $\omega$
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\end_inset
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or whether the its real-valued approximation is satisfactory).
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Typically, a good initial guess for
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\begin_inset Formula $\omega\left(\vect k\right)$
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\end_inset
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is obtained from the empty lattice approximation,
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\begin_inset Formula $\left|\vect k\right|=\sqrt{\epsilon\mu}\omega/c_{0}$
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\end_inset
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(modulo lattice points; TODO write this a clean way).
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A somehow challenging step is to distinguish the different bands that can
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all be very close to the empty lattice approximation, especially if the
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particles in the systems are small.
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In high-symmetry points of the Brilloin zone, this can be solved by factorising
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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\end_inset
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into irreducible representations
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\begin_inset Formula $\Gamma_{i}$
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\end_inset
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and performing the minimisation in each irrep separately, cf.
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Section
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "sec:Symmetries"
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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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, and using the different
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\begin_inset Formula $\omega_{\Gamma_{i}}\left(\vect k\right)$
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\end_inset
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to obtain the initial guesses for the nearby points
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\begin_inset Formula $\vect k+\delta\vect k$
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\end_inset
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.
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\end_layout
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\begin_layout Subsection
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@ -617,8 +805,8 @@ reference "eq:W sum in reciprocal space"
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\begin_inset Formula
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\begin{eqnarray}
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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 \\
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W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
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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\vect0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
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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}\\
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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}
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\end{eqnarray}
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\end_inset
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@ -656,7 +844,7 @@ CHECK THE FOLLOWING EXPRESSION FOR CORRECT FUNCTION ARGUMENTS
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\begin_inset Formula
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\begin{equation}
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\sigma_{\nu}^{\mu}\left(\vect k\right)=\sum_{\vect n\in\ints^{d}\backslash\left\{ \vect0\right\} }e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\ush{\nu}{\mu}\left(\uvec{R_{n}}\right)h_{n}^{(1)}\left(R_{n}\right),\label{eq:sigma lattice sums}
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\sigma_{\nu}^{\mu}\left(\vect k\right)=\sum_{\vect n\in\ints^{d}\backslash\left\{ \vect 0\right\} }e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\ush{\nu}{\mu}\left(\uvec{R_{n}}\right)h_{n}^{(1)}\left(R_{n}\right),\label{eq:sigma lattice sums}
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\end{equation}
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\end_inset
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