Fixes suggested by Päivi up to sect. 3.2
Former-commit-id: 65d31b97bb6aa8feae9118d34ededbd3092e128a
This commit is contained in:
Marek Nečada
parent
8c559ac0b7
commit
8942753a13
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@ -243,6 +243,15 @@
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number = {6}
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}
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@book{harrington_field_1993,
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title = {Field {{Computation}} by {{Moment Methods}} ({{IEEE Press Series}} on {{Electromagnetic Wave Theory}})},
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author = {Harrington, Roger F.},
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year = {1993},
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publisher = {{Wiley-IEEE Press}},
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isbn = {978-0-7803-1014-8},
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series = {The {{IEEE PRESS Series}} in {{Electromagnetic Waves}} ({{Donald G}}. {{Dudley}}, {{Editor}})}
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}
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@article{homola_surface_1999,
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title = {Surface Plasmon Resonance Sensors: Review},
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shorttitle = {Surface Plasmon Resonance Sensors},
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@ -261,6 +270,24 @@
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number = {1}
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}
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@article{hsu_bound_2016,
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title = {Bound States in the Continuum},
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author = {Hsu, Chia Wei and Zhen, Bo and Stone, A. Douglas and Joannopoulos, John D. and Solja{\v c}i{\'c}, Marin},
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year = {2016},
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month = jul,
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volume = {1},
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pages = {1--13},
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publisher = {{Nature Publishing Group}},
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issn = {2058-8437},
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doi = {10.1038/natrevmats.2016.48},
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abstract = {Bound states in the continuum (BICs) are waves that remain localized even though they coexist with a continuous spectrum of radiating waves that can carry energy away. Their very existence defies conventional wisdom. Although BICs were first proposed in quantum mechanics, they are a general wave phenomenon and have since been identified in electromagnetic waves, acoustic waves in air, water waves and elastic waves in solids. These states have been studied in a wide range of material systems, such as piezoelectric materials, dielectric photonic crystals, optical waveguides and fibres, quantum dots, graphene and topological insulators. In this Review, we describe recent developments in this field with an emphasis on the physical mechanisms that lead to BICs across seemingly very different materials and types of waves. We also discuss experimental realizations, existing applications and directions for future work.},
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copyright = {2016 Macmillan Publishers Limited},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/WZUM4EMS/Hsu ym. - 2016 - Bound states in the continuum.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/7U67A75X/natrevmats201648.html},
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journal = {Nature Reviews Materials},
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language = {en},
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number = {9}
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}
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@book{jackson_classical_1998,
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title = {Classical {{Electrodynamics Third Edition}}},
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author = {Jackson, John David},
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@ -306,8 +306,8 @@ 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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\vswfrtlm1lm\left(\kappa\vect r\right) & =j_{l}\left(\kappa r\right)\vsh1lm\left(\uvec r\right),\nonumber \\
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\vswfrtlm2lm\left(\kappa\vect r\right) & =\frac{1}{\kappa r}\frac{\ud\left(\kappa rj_{l}\left(\kappa r\right)\right)}{\ud\left(\kappa r\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(\kappa r\right)}{\kappa r}\vsh3lm\left(\uvec r\right),\label{eq:VSWF regular}
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\vswfrtlm 1lm\left(\kappa\vect r\right) & =j_{l}\left(\kappa r\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
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\vswfrtlm 2lm\left(\kappa\vect r\right) & =\frac{1}{\kappa r}\frac{\ud\left(\kappa rj_{l}\left(\kappa r\right)\right)}{\ud\left(\kappa r\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(\kappa r\right)}{\kappa r}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF regular}
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\end{align}
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\end_inset
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@ -315,8 +315,8 @@ outgoing
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\begin_inset Formula
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\begin{align}
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\vswfouttlm1lm\left(\kappa\vect r\right) & =h_{l}^{\left(1\right)}\left(\kappa r\right)\vsh1lm\left(\uvec r\right),\nonumber \\
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\vswfouttlm2lm\left(\kappa\vect r\right) & =\frac{1}{kr}\frac{\ud\left(\kappa rh_{l}^{\left(1\right)}\left(\kappa r\right)\right)}{\ud\left(\kappa r\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(\kappa r\right)}{\kappa r}\vsh3lm\left(\uvec r\right),\label{eq:VSWF outgoing}\\
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\vswfouttlm 1lm\left(\kappa\vect r\right) & =h_{l}^{\left(1\right)}\left(\kappa r\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
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\vswfouttlm 2lm\left(\kappa\vect r\right) & =\frac{1}{kr}\frac{\ud\left(\kappa rh_{l}^{\left(1\right)}\left(\kappa r\right)\right)}{\ud\left(\kappa r\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(\kappa r\right)}{\kappa r}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF outgoing}\\
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& \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,\nonumber
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\end{align}
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@ -360,7 +360,7 @@ outgoing
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\emph on
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positive
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\emph default
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imaginary part, and gainy materials will have it negative and, for example,
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imaginary part, and gain materials will have it negative and, for example,
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Drude-Lorenz model of a lossy medium will have poles in the lower complex
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half-plane.
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\end_layout
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@ -387,9 +387,9 @@ vector spherical harmonics
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\begin_inset Formula
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\begin{align}
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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,\nonumber \\
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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),\nonumber \\
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\vsh3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition}
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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,\nonumber \\
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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),\nonumber \\
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\vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition}
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\end{align}
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\end_inset
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@ -408,7 +408,7 @@ electric dipolar
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\end_inset
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waves
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\begin_inset Formula $\vswfrtlm21m$
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\begin_inset Formula $\vswfrtlm 21m$
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\end_inset
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, they vanish.
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@ -646,11 +646,11 @@ 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 $\closedball{R^{<}}{\vect 0}$
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\begin_inset Formula $\closedball{R^{<}}{\vect0}$
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\end_inset
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and let the whole volume
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\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
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\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$
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\end_inset
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be filled with a homogeneous isotropic medium with wave number
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@ -659,7 +659,7 @@ The single-particle scattering problem at frequency
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.
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Inside
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\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
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\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$
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\end_inset
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, the electric field can be expanded as
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@ -681,7 +681,7 @@ doplnit frekvence a polohy
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\end_inset
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If there were no scatterer and
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\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
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\begin_inset Formula $\closedball{R^{<}}{\vect0}$
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\end_inset
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were filled with the same homogeneous medium, the part with the outgoing
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@ -690,7 +690,7 @@ If there were no scatterer and
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\end_inset
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due to sources outside
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\begin_inset Formula $\openball{R^{>}}{\vect 0}$
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\begin_inset Formula $\openball{R^{>}}{\vect0}$
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\end_inset
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would remain.
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@ -836,7 +836,12 @@ literal "false"
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\end_inset
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.
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In general, simulating scattering of a regular spherical wave
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In general, elements of the
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\begin_inset Formula $T$
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\end_inset
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-matrix can be obtained by simulating scattering of a regular spherical
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wave
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\begin_inset Formula $\vswfrtlm{\tau}lm$
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\end_inset
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@ -879,24 +884,50 @@ literal "false"
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\end_inset
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.
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Note that older versions of SCUFF-EM contained a bug that rendered almost
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\begin_inset Foot
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status open
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\begin_layout Plain Layout
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Note that the upstream versions of SCUFF-EM contain a bug that renders almost
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all
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\begin_inset Formula $T$
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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
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\begin_inset Marginal
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status open
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-matrix results wrong; we found and fixed the bug in our fork available
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at
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\begin_inset CommandInset href
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LatexCommand href
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target "https://github.com/texnokrates/scuff-em"
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literal "false"
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\begin_layout Plain Layout
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Not yet merged to upstream.
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\end_inset
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in revision
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\begin_inset CommandInset href
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LatexCommand href
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name "g78689f5"
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target "https://github.com/texnokrates/scuff-em/commit/78689f5514072853aa5cad455ce15b3e024d163d"
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literal "false"
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\end_inset
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.
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However, the
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\begin_inset CommandInset href
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LatexCommand href
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name "bugfix"
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target "https://github.com/HomerReid/scuff-em/pull/197"
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literal "false"
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\end_inset
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has not been merged into upstream by the time of writing this article.
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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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\begin_layout Subsubsection
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@ -1168,7 +1199,7 @@ where
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\end_inset
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, respectively.
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Here
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\family roman
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\series medium
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\shape up
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@ -1819,8 +1850,8 @@ expansion coefficients
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they
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\emph default
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transform under translation.
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Let us assume the field can be in terms of regular waves everywhere, and
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expand it in two different origins
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We assume the field can be expressed in terms of regular waves everywhere,
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and expand it in two different origins
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\begin_inset Formula $\vect r_{p},\vect r_{q}$
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\end_inset
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@ -1980,7 +2011,7 @@ and analogously the elements of the singular operator
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\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$
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\end_inset
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) in the radial part instead of the regular bessel functions,
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) in the radial part instead of the regular Bessel functions,
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\begin_inset Note Note
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status collapsed
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@ -200,7 +200,7 @@ noprefix "false"
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\end_inset
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inside the unit cell; any particle of the periodic system can thus be labeled
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by a multiindex from
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by a multi-index from
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\begin_inset Formula $\mathcal{P}=\ints^{d}\times\mathcal{P}_{1}$
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\end_inset
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@ -271,7 +271,10 @@ noprefix "false"
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\end_inset
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, and eq.
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\begin_inset space \space{}
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem periodic"
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@ -327,7 +330,10 @@ noprefix "false"
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\begin_layout Standard
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As in the case of a finite system, eq.
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\begin_inset space \space{}
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem unit cell"
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@ -346,7 +352,10 @@ noprefix "false"
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\end_inset
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Eq.
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\begin_inset space \space{}
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem unit cell"
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@ -445,9 +454,15 @@ noprefix "false"
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\end_inset
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, w.r.t.
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the distance; the gain might then balance the losses in particles, resulting
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\begin_inset space \space{}
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\end_inset
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the distance; the gain might then balance the losses in particles, resulting
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in sustained modes satisfying eq.
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\begin_inset space \space{}
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:lattice mode equation"
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@ -498,7 +513,10 @@ 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 space \space{}
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\end_inset
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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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@ -509,10 +527,7 @@ 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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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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@ -994,7 +1009,10 @@ scalar
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\end_inset
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: in eq.
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\begin_inset space \space{}
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:translation operator singular"
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@ -1483,7 +1501,10 @@ If the normal component
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\end_inset
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is zero, in the last sum in eq.
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\begin_inset space \space{}
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Ewald in 3D long-range part 1D 2D"
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@ -1520,7 +1541,10 @@ noprefix "false"
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\end_inset
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can be evaluated e.g.
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using the Taylor series
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\begin_inset space \space{}
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\end_inset
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using the Taylor series
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\lang finnish
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\begin_inset Formula
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@ -1810,7 +1834,10 @@ literal "false"
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variable to the positive imaginary half-axis.
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This moves the branch cuts w.r.t.
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\begin_inset space \space{}
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\end_inset
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\begin_inset Formula $\kappa$
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\end_inset
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@ -191,10 +191,10 @@ zkontroluj reference, přidej referenci na frequency domain fem
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) include the field degrees of freedom (DoF) of the background medium (which
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can have very large volumes), whereas integral approaches such as the boundary
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element method (BEM, a.k.a the method of moments, MOM,
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element method (BEM, a.k.a the method of moments, MOM
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\begin_inset CommandInset citation
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LatexCommand cite
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key "medgyesi-mitschang_generalized_1994,reid_efficient_2015"
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key "harrington_field_1993,medgyesi-mitschang_generalized_1994,reid_efficient_2015"
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literal "false"
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\end_inset
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@ -221,8 +221,20 @@ s used in nanophotonics: there are modes in which the particles' electric
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dipole moments completely vanish due to symmetry, and regardless of how
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small the particles are, the excitations have quadrupolar or higher-degree
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multipolar character.
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These modes typically appear at the band edges where interesting phenomena
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such as lasing or Bose-Einstein condensation have been observed
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These modes, belonging to a category that is sometimes called
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\emph on
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optical bound states in the continuum (BIC)
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\emph default
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\begin_inset CommandInset citation
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LatexCommand cite
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key "hsu_bound_2016"
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literal "false"
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\end_inset
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, typically appear at the band edges where interesting phenomena such as
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lasing or Bose-Einstein condensation have been observed
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\begin_inset CommandInset citation
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LatexCommand cite
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key "guo_lasing_2019,pourjamal_lasing_2019,hakala_lasing_2017,yang_real-time_2015,hakala_boseeinstein_2018"
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@ -238,10 +250,14 @@ The natural way to overcome both limitations of CDA mentioned above is to
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take higher multipoles into account.
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Instead of a polarisability tensor, the scattering properties of an individual
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particle are then described with more general
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\emph on
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transition matrix
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\emph default
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(commonly known as
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\begin_inset Formula $T$
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\end_inset
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-matrix, and different particles' multipole excitations are coupled together
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-matrix), and different particles' multipole excitations are coupled together
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via translation operators, a generalisation of the Green's functions used
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in CDA.
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This is the idea behind the
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@ -332,12 +348,12 @@ However, the potential of MSTMM reaches far beyond its past implementations.
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This enables, among other things, to use MSTMM for fast solving of the
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lattice modes of such periodic systems, and comparing them to their finite
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counterparts with respect to electromagnetic response, which is useful
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to isolate the bulk and finite-size phenomena of photonic arrays.
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to isolate the bulk and finite-size phenomena of photonic lattices.
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Moreover, we exploit symmetries of the system to decompose the problem
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into several substantially smaller ones, which provides better understanding
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of modes, mainly in periodic systems, and substantially reduces the demands
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on computational resources, hence speeding up the computations and allowing
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for finite size simulations of systems with particle counts practically
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for finite size simulations of systems with particle numbers practically
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impossible to reliably simulate with any other method.
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\begin_inset Note Note
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status open
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Reference in New Issue