Infinite systems WIP (how to numerical solutions)
Former-commit-id: dd3118e392b58eb9ae2260581e93028561571245
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Marek Nečada
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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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@ -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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