Unify balls notations, some refs etc.
Former-commit-id: 69e5ae075b639a6aed4988b9e8801947017026a5
This commit is contained in:
Marek Nečada
parent
2b031d43da
commit
46b651d97f
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@ -260,12 +260,12 @@ status open
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\begin_inset FormulaMacro
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\newcommand{\particle}{\mathrm{\Theta}}
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\newcommand{\medium}{\Theta}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\medium}{\thespace\backslash\particle}
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\newcommand{\mezikuli}[3]{\Theta_{#1,#2}\left(#3\right)}
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\end_inset
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@ -355,7 +355,7 @@ status open
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\begin_inset FormulaMacro
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\newcommand{\closedball}[2]{B_{#1}#2}
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\newcommand{\closedball}[2]{\overline{B_{#1}\left(#2\right)}}
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\end_inset
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@ -566,7 +566,17 @@ The (somewhat underrated) T-matrix multiple scattering method (TMMSM) can
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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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(SHOULD I MENTION ALSO THE CROSS SECTION FORMULAS IN THE ABSTRACT?)
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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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Should I mention also the cross sections formulae in abstract / intro?
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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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@ -736,11 +746,6 @@ Maybe put the numerical results separately in the end.
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TODOs
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\end_layout
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\begin_layout Itemize
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Consistent notation of balls.
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How is the difference between two cocentric balls called?
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\end_layout
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\begin_layout Itemize
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It could be nice to include some illustration (example array) to the introductio
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n.
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@ -752,26 +757,12 @@ Maybe mention that in infinite systems, it can be also much faster than
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other methods.
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\end_layout
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\begin_layout Itemize
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Translation operators: rewrite in sph.
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harm.
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convention independent form.
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\end_layout
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\begin_layout Itemize
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Truncation notation.
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\end_layout
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\begin_layout Itemize
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Example results!
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\end_layout
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\begin_layout Itemize
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Figures.
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\end_layout
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\begin_layout Itemize
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Concrete comparison with other methods.
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Example results and benchmarks with BEM; figures!
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\end_layout
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\begin_layout Itemize
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@ -138,16 +138,40 @@ Single-particle scattering
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\begin_layout Standard
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In order to define the basic concepts, let us first consider the case of
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EM radiation scattered by a single particle.
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We assume that the scatterer lies inside a closed sphere
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\begin_inset Formula $\particle$
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We assume that the scatterer lies inside a closed ball
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\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
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\end_inset
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, the space outside this volume
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\begin_inset Formula $\medium$
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of radius
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\begin_inset Formula $R^{<}$
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\end_inset
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is filled with an homogeneous isotropic medium with relative electric permittiv
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ity
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and center in the origin of the coordinate system (which can be chosen
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that way; the natural choice of
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\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
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\end_inset
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is the circumscribed ball of the scatterer) and that there exists a larger
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open cocentric ball
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\begin_inset Formula $\openball{R^{>}}{\vect 0}$
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\end_inset
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, such that
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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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Is there a word for this?
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\end_layout
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\end_inset
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the (non-empty) volume
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\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}=\openball{R^{>}}{\vect 0}\setminus\closedball{R^{<}}{\vect 0}$
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\end_inset
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is filled with a homogeneous isotropic medium with relative electric permittivi
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ty
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\begin_inset Formula $\epsilon(\vect r,\omega)=\epsbg(\omega)$
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\end_inset
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@ -285,7 +309,16 @@ where
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\end_inset
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are the regular spherical Bessel function and spherical Hankel function
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of the first kind, respectively, as in [DLMF §10.47], and
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of the first kind, respectively, as in
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\begin_inset CommandInset citation
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LatexCommand cite
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after "§10.47"
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key "NIST:DLMF"
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literal "false"
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\end_inset
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, and
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\begin_inset Formula $\vsh{\tau}lm$
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\end_inset
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@ -307,7 +340,16 @@ In our convention, the (scalar) spherical harmonics
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\begin_inset Formula $\ush lm$
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\end_inset
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are identical to those in [DLMF 14.30.1], i.e.
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are identical to those in
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\begin_inset CommandInset citation
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LatexCommand cite
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after "14.30.1"
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key "NIST:DLMF"
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literal "false"
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\end_inset
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, i.e.
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\begin_inset Formula
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\[
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\ush lm=\left(\frac{\left(l-m\right)!\left(2l+1\right)}{4\pi\left(l+m\right)!}\right)^{\frac{1}{2}}e^{im\phi}\dlmfFer lm\left(\cos\theta\right)
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@ -319,8 +361,16 @@ where importantly, the Ferrers functions
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\begin_inset Formula $\dlmfFer lm$
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\end_inset
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defined as in [DLMF §14.3(i)] do already contain the Condon-Shortley phase
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defined as in
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\begin_inset CommandInset citation
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LatexCommand cite
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after "§14.3(i)"
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key "NIST:DLMF"
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literal "false"
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\end_inset
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do already contain the Condon-Shortley phase
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\begin_inset Formula $\left(-1\right)^{m}$
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\end_inset
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@ -418,7 +468,7 @@ The regular VSWFs
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\begin_inset Formula $\vswfrtlm{\tau}lm\left(k\vect r\right)$
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\end_inset
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constitute a basis for solutions of the Helmholtz equation
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would constitute a basis for solutions of the Helmholtz equation
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Helmholtz eq"
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@ -429,16 +479,17 @@ 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 $\openball{R^{>}}{\vect 0}$
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\end_inset
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with radius
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\begin_inset Formula $R^{>}$
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\end_inset
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and center in the origin; however, if the equation is not guaranteed to
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hold inside a smaller ball
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\begin_inset Formula $B_{0}\left(R\right)$
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and center in the origin, were it filled with homogeneous isotropic medium;
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however, if the equation is not guaranteed to hold inside a smaller ball
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\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
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\end_inset
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around the origin (typically due to presence of a scatterer), one has to
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@ -447,7 +498,7 @@ 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 $\mezikuli{R^{<}}{R^{>}}{\vect 0}=\openball{R^{>}}{\vect 0}\setminus\closedball{R^{<}}{\vect 0}$
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\end_inset
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.
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@ -469,11 +520,11 @@ The single-particle scattering problem at frequency
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\end_inset
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can be posed as follows: Let a scatterer be enclosed inside the ball
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\begin_inset Formula $B_{0}\left(R\right)$
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\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
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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 $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
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\end_inset
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be filled with a homogeneous isotropic medium with wave number
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@ -482,7 +533,7 @@ The single-particle scattering problem at frequency
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.
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Inside
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\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
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\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
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\end_inset
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, the electric field can be expanded as
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@ -504,7 +555,7 @@ doplnit frekvence a polohy
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\end_inset
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If there was no scatterer and
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\begin_inset Formula $B_{0}\left(R_{<}\right)$
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\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
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\end_inset
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was filled with the same homogeneous medium, the part with the outgoing
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@ -513,7 +564,7 @@ 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 $\openball{R^{>}}{\vect 0}$
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\end_inset
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would remain.
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@ -670,7 +721,17 @@ literal "false"
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\end_inset
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-matrix results wrong; we found and fixed the bug and from upstream version
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xxx onwards, it should behave correctly.
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xxx
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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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Not yet merged to upstream.
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\end_layout
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\end_inset
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onwards, it should behave correctly.
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\end_layout
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@ -689,8 +750,25 @@ The magnitude of the
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\begin_inset Formula $T$
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\end_inset
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-matrix of a bounded scatterer is a compact operator [REF???], so from certain
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multipole degree onwards,
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-matrix of a bounded scatterer is a compact operator
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\begin_inset CommandInset citation
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LatexCommand cite
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key "ganesh_convergence_2012"
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literal "false"
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\end_inset
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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
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\end_layout
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\end_inset
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, so from certain multipole degree onwards,
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\begin_inset Formula $l,l'>L$
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\end_inset
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@ -725,16 +803,6 @@ The magnitude of the
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\end_inset
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will also be negligible.
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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 when it will not be negligible
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -761,7 +829,7 @@ literal "false"
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\end_inset
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by requiring that
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\begin_inset Formula $\delta\gtrsim\left(nR\right)^{L}/\left(2L+1\right)!!$
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\begin_inset Formula $\delta\gg\left(nR\right)^{L}/\left(2L+1\right)!!$
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\end_inset
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, where
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@ -886,7 +954,7 @@ 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 $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
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\end_inset
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have expansion as in
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@ -901,11 +969,11 @@ noprefix "false"
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.
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Then the net power transported from
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\begin_inset Formula $B_{0}\left(R\right)$
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\begin_inset Formula $\openball{R^{<}}{\vect 0}$
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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 $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
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\end_inset
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via by electromagnetic radiation is
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@ -917,7 +985,7 @@ P=\frac{1}{2k^{2}\eta_{0}\eta}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\ri
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\end_inset
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In realistic scattering setups, power is transferred by radiation into
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\begin_inset Formula $B_{0}\left(R\right)$
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\begin_inset Formula $\openball{R^{<}}{\vect 0}$
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\end_inset
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and absorbed by the enclosed scatterer, so
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@ -1087,25 +1155,15 @@ If the system consists of multiple scatterers, the EM fields around each
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\end_inset
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be an index set labeling the scatterers.
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We enclose each scatterer in a ball
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\begin_inset Formula $B_{\vect r_{p}}\left(R_{p}\right)$
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We enclose each scatterer in a closed ball
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\begin_inset Formula $\closedball{R_{p}}{\vect r_{p}}$
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\end_inset
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such that the balls do not touch,
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\begin_inset Formula $B_{\vect r_{p}}\left(R_{p}\right)\cap B_{\vect r_{q}}\left(R_{q}\right)=\emptyset;p,q\in\mathcal{P}$
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\begin_inset Formula $\closedball{R_{p}}{\vect r_{p}}\cap\closedball{R_{q}}{\vect r_{q}}=\emptyset;p,q\in\mathcal{P}$
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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 bacha, musejí být uzavřené!
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\end_layout
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\end_inset
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so there is a non-empty volume
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, so there is a non-empty volume
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\begin_inset Note Note
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status open
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@ -1116,12 +1174,20 @@ jaksetometuje?
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\end_inset
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\begin_inset Formula $\openball{\vect r_{p}}{R_{p}^{>}}\backslash B_{\vect r_{p}}\left(R_{p}\right)$
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\begin_inset Formula $\mezikuli{R_{p}}{R_{p}^{>}}{\vect r_{p}}$
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\end_inset
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around each one that contains only the background medium without any scatterers.
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Then the EM field inside each such volume can be expanded in a way similar
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to
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around each one that contains only the background medium without any scatterers
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(we assume that all the volume outside
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\begin_inset Formula $\bigcap_{p\in\mathcal{P}}\closedball{R_{p}}{\vect r_{p}}$
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\end_inset
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is filled with the same background medium).
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Then the EM field inside each
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\begin_inset Formula $\mezikuli{R_{p}}{R_{p}^{>}}{\vect r_{p}}$
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\end_inset
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can be expanded in a way similar to
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:E field expansion"
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@ -1135,7 +1201,7 @@ noprefix "false"
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\begin_inset Formula
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\begin{align}
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\vect E\left(\omega,\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoeffptlm p{\tau}lm\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)+\outcoeffptlm p{\tau}lm\vswfouttlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)\right),\label{eq:E field expansion multiparticle}\\
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& \vect r\in\openball{\vect r_{p}}{R_{p}^{>}}\backslash B_{\vect r_{p}}\left(R_{p}\right).\nonumber
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& \vect r\in\mezikuli{R_{p}}{R_{p}^{>}}{\vect r_{p}}.\nonumber
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\end{align}
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\end_inset
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@ -1447,10 +1513,10 @@ reference "eq:translation operator"
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below.
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For singular (outgoing) waves, the form of the expansion differs inside
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and outside the ball
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\begin_inset Formula $\openball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}:$
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\begin_inset Formula $\closedball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}$
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\end_inset
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:
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\begin_inset Formula
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\begin{eqnarray}
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\vswfouttlm{\tau}lm\left(k\left(\vect r-\vect r_{1}\right)\right) & = & \begin{cases}
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@ -1669,8 +1735,17 @@ and analogously the elements of the singular operator
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\end_inset
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where the constant factors in our convention read (TODO CHECK ONCE AGAIN
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CAREFULLY FOR POSSIBLE PHASE FACTORS FACTORS)
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where the constant factors in our convention read
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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 check once again carefully for possible phase factors.
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\end_layout
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\end_inset
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\begin_inset Note Note
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status collapsed
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@ -1979,8 +2054,8 @@ literal "false"
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derives only the extinction cross section formula.
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Let us re-derive it together with the many-particle scattering and absorption
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cross sections.
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First, let us take a ball circumscribing all the scatterers at once,
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\begin_inset Formula $\openball R{\vect r_{\square}}\supset\particle$
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First, let us take a ball containing all the scatterers at once,
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\begin_inset Formula $\openball R{\vect r_{\square}}\supset\bigcup_{p\in\mathcal{P}}\closedball{R_{p}}{\vect r_{p}}$
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\end_inset
|
||||
|
||||
.
|
||||
|
|
@ -2010,8 +2085,7 @@ where
|
|||
\begin_inset Formula $\outcoeffp{\square}$
|
||||
\end_inset
|
||||
|
||||
using the translation operators (REF!!!) and use the single-scatterer formulae
|
||||
|
||||
using the translation operators and use the single-scatterer formulae
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:extincion CS single"
|
||||
|
|
@ -2113,8 +2187,18 @@ noprefix "false"
|
|||
|
||||
where only the last expression is suitable for numerical evaluation with
|
||||
truncated matrices, because the previous ones contain a translation operator
|
||||
right next to an incident field coefficient vector (see Sec.
|
||||
TODO).
|
||||
right next to an incident field coefficient vector
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
(see Sec.
|
||||
TODO)
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Similarly,
|
||||
\begin_inset Formula
|
||||
\begin{align}
|
||||
|
|
|
|||
|
|
@ -175,10 +175,29 @@ superposition
|
|||
\end_inset
|
||||
|
||||
-matrix method
|
||||
\begin_inset Marginal
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
a.k.a.
|
||||
something else?
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\emph default
|
||||
(TODO a.k.a something; refs??), and it has been implemented previously for
|
||||
a limited subset of problems (TODO refs and list the limitations of the
|
||||
available).
|
||||
, and it has been implemented previously for a limited subset of problems
|
||||
\begin_inset Marginal
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
Refs; list the limitations of available codes?
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
.
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
|
@ -237,18 +256,46 @@ We hereby release our MSTMM implementation, the
|
|||
QPMS Photonic Multiple Scattering
|
||||
\emph default
|
||||
suite, as free software under the GNU General Public License version 3.
|
||||
(TODO refs to the code repositories.) QPMS allows for linear optics simulations
|
||||
of arbitrary sets of compact scatterers in isotropic media.
|
||||
|
||||
\begin_inset Marginal
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
TODO refs to the code repositories once it is published.
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
QPMS allows for linear optics simulations of arbitrary sets of compact
|
||||
scatterers in isotropic media.
|
||||
The features include computations of electromagnetic response to external
|
||||
driving, the related cross sections, and finding resonances of finite structure
|
||||
s.
|
||||
Moreover, it includes the improvements covered in this paper, enabling
|
||||
to simulate even larger systems and also infinite structures with periodicity
|
||||
in one, two or three dimensions, which can be e.g.
|
||||
used for quickly evaluating dispersions of such structures, and also their
|
||||
topological invariants (TODO).
|
||||
used for quickly evaluating dispersions of such structures
|
||||
\begin_inset Marginal
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
And also their topological invariants (TODO)?
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
.
|
||||
The QPMS suite contains a core C library, Python bindings and several utilities
|
||||
for routine computations.
|
||||
\begin_inset Marginal
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
Such as?
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
|
|
|||
Loading…
Reference in New Issue