WIP
Former-commit-id: 3ab2f0803f9399fe32c41325b9404f8e0b1ba18c
This commit is contained in:
Marek Nečada
parent
778c4b480a
commit
6092449db4
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@ -491,21 +491,12 @@ These are compatibility macros for the (...)-old files:
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\end_layout
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\begin_layout Title
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Many-particle
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\begin_inset Formula $T$
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\end_inset
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-matrix simulations in finite and infinite systems of electromagnetic scatterers
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\begin_inset Marginal
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status open
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\begin_layout Plain Layout
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(TODO better title)
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\end_layout
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\end_inset
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-matrix simulations for nanophotonics: symmetries, scattering and lattice
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modes
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\end_layout
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\begin_layout Standard
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@ -521,7 +512,7 @@ Multiple-scattering
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\end_layout
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\begin_layout Itemize
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Multiple-scattering
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Many-particle
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\begin_inset Formula $T$
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\end_inset
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@ -529,6 +520,13 @@ Multiple-scattering
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modes.
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\end_layout
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\begin_layout Itemize
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\begin_inset Formula $T$
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\end_inset
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-matrix simulations in finite and infinite systems of electromagnetic scatterers
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\end_layout
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\begin_layout Standard
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\begin_inset Note Note
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status open
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@ -617,8 +615,13 @@ The T-matrix multiple scattering method (TMMSM) can be used to solve the
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\begin_layout Abstract
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Here we extend the method to infinite periodic structures using Ewald-type
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lattice summation, and we exploit the possible symmetries of the structure
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to further improve its efficiency.
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to further improve its efficiency, so that systems containing tens of thousands
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of particles can be studied with relative ease.
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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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\begin_inset Marginal
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status open
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@ -629,6 +632,11 @@ Should I mention also the cross sections formulae in abstract / intro?
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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 Abstract
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@ -862,7 +870,7 @@ Given up
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\end_layout
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\begin_layout Itemize
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\begin_layout Standard
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\begin_inset Note Note
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status open
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@ -370,7 +370,13 @@ noprefix "false"
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describes the
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\emph on
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lattice modes.
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lattice modes
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\emph default
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, i.e.
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electromagnetic excitations that can sustain themselves for prolonged time
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even without external driving
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\emph on
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.
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\emph default
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Non-trivial solutions to
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@ -1700,7 +1706,32 @@ One pecularity of the two-dimensional case is the two-branchedness of the
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\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,z\right)$
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\end_inset
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appearing in the long-range part.
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appearing in the long-range part (in the cases
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\begin_inset Formula $d=1,3$
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\end_inset
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the function
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\begin_inset Formula $\gamma\left(z\right)$
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\end_inset
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appears with even powers, and
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\begin_inset Formula $\Gamma\left(-j,z\right)$
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\end_inset
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is meromorphic for integer
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\begin_inset Formula $j$
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\end_inset
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\begin_inset CommandInset citation
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LatexCommand cite
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after "8.2.9"
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key "NIST:DLMF"
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literal "false"
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\end_inset
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).
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As a consequence, if we now explicitly label the dependence on the wavenumber,
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\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\kappa,\vect k,\vect s\right)$
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@ -2213,8 +2244,8 @@ noprefix "false"
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translation operator:
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\begin_inset Formula
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\begin{align}
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\vswfrtlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\tropr_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),\nonumber \\
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\vswfouttlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),\label{eq:VSWFs expressed as translated dipole waves}
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\vswfrtlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\tropr_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm21{m'}\left(0\right),\nonumber \\
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\vswfouttlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm21{m'}\left(0\right),\label{eq:VSWFs expressed as translated dipole waves}
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\end{align}
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\end_inset
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@ -2241,7 +2272,7 @@ noprefix "false"
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\end_inset
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and the fact that all the other regular VSWFs except for
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\begin_inset Formula $\vswfrtlm 21{m'}$
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\begin_inset Formula $\vswfrtlm21{m'}$
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\end_inset
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vanish at origin.
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@ -2314,10 +2345,10 @@ status open
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\begin_inset Formula
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\begin{align*}
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\vect E_{\mathrm{scat}}\left(\vect r\right) & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\\
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& =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right)\\
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& =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right),\text{FIXME signs}\\
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& =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\psi_{\lambda,m-m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right)\\
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& =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right)
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& =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect0,\alpha}{\tau}lme^{i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm21{m'}\left(0\right)\\
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& =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm21{m'}\left(0\right),\text{FIXME signs}\\
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& =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\psi_{\lambda,m-m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm21{m'}\left(0\right)\\
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& =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right)
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\end{align*}
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\end_inset
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@ -2335,7 +2366,7 @@ TODO fix signs and exponential phase factors
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\begin_inset Formula
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\begin{align*}
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\vect E_{\mathrm{scat}}\left(\vect r\right) & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\\
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& =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right).
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& =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right).
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\end{align*}
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\end_inset
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