Hexlaser theory additions

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Marek Nečada 2018-10-04 01:46:02 +03:00
parent 77b4a5053a
commit de32e6f920
2 changed files with 346 additions and 9 deletions

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@ -15,6 +15,20 @@
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/254TXAN3/mackowski1991.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/QV6MH2N9/599.html} file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/254TXAN3/mackowski1991.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/QV6MH2N9/599.html}
} }
@article{johnson_optical_1972,
title = {Optical {{Constants}} of the {{Noble Metals}}},
volume = {6},
doi = {10.1103/PhysRevB.6.4370},
abstract = {The optical constants n and k were obtained for the noble metals (copper, silver, and gold) from reflection and transmission measurements on vacuum-evaporated thin films at room temperature, in the spectral range 0.5-6.5 eV. The film-thickness range was 185-500 \AA. Three optical measurements were inverted to obtain the film thickness d as well as n and k. The estimated error in d was $\pm$ 2 \AA, and that in n, k was less than 0.02 over most of the spectral range. The results in the film-thickness range 250-500 \AA{} were independent of thickness, and were unchanged after vacuum annealing or aging in air. The free-electron optical effective masses and relaxation times derived from the results in the near infrared agree satisfactorily with previous values. The interband contribution to the imaginary part of the dielectric constant was obtained by subtracting the free-electron contribution. Some recent theoretical calculations are compared with the results for copper and gold. In addition, some other recent experiments are critically compared with our results.},
number = {12},
journal = {Phys. Rev. B},
author = {Johnson, P. B. and Christy, R. W.},
month = dec,
year = {1972},
pages = {4370-4379},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/ANQIIJA5/PhysRevB.6.html}
}
@misc{SCUFF2, @misc{SCUFF2,
title = {{{SCUFF}}-{{EM}}}, title = {{{SCUFF}}-{{EM}}},
author = {Reid, Homer}, author = {Reid, Homer},
@ -139,7 +153,7 @@
author = {Reid, M. T. Homer and Johnson, Steven G.}, author = {Reid, M. T. Homer and Johnson, Steven G.},
month = aug, month = aug,
year = {2015}, year = {2015},
keywords = {Physics - Classical Physics,Physics - Computational Physics}, keywords = {Physics - Computational Physics,Physics - Classical Physics},
pages = {3588-3598}, pages = {3588-3598},
file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/I2DXTKUF/Reid ja Johnson - 2015 - Efficient Computation of Power, Force, and Torque .pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/LG7AVZDH/1307.html} file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/I2DXTKUF/Reid ja Johnson - 2015 - Efficient Computation of Power, Force, and Torque .pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/LG7AVZDH/1307.html}
} }

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@ -219,17 +219,39 @@ key "schulz_point-group_1999"
\end_inset \end_inset
. .
A brief theoretical overview of the method is presented in subsections
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:The-multiple-scattering-problem"
\end_inset
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Periodic-systems"
\end_inset
below.
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard
Fig.
xxx(a) shows the dispersions around the
\begin_inset Formula $\Kp$
\end_inset
-point for the cylindrical nanoparticles used in our experiment.
\lang english \lang english
The The
\begin_inset Formula $T$ \begin_inset Formula $T$
\end_inset \end_inset
-matrix of a single nanoparticle was computed using the scuff-tmatrix applicatio -matrix of a single cylindrical nanoparticle was computed using the scuff-tmatri
n from the SCUFF-EM suite~ x application from the SCUFF-EM suite~
\lang finnish \lang finnish
\begin_inset CommandInset citation \begin_inset CommandInset citation
@ -245,16 +267,56 @@ key "SCUFF2,reid_efficient_2015"
\end_inset \end_inset
(octupolar) degree of electric and magnetic spherical multipole. (octupolar) degree of electric and magnetic spherical multipole.
For comparison, Fig.
xxx(b) shows the dispersions for a system where the cylindrical nanoparticles
were replaced with spherical ones with radius of
\begin_inset Formula $40\,\mathrm{nm}$
\end_inset
, whose
\begin_inset Formula $T$
\end_inset
-matrix was calculated semi-analytically using the Lorenz-Mie theory.
In both cases, we used gold with interpolated tabulated values of refraction
index
\begin_inset CommandInset citation
LatexCommand cite
key "johnson_optical_1972"
\end_inset
for the nanoparticles and constant reffraction index of 1.52 for the background
medium.
In both cases, the diffracted orders do split into separate bands according
to the
\lang finnish
\begin_inset Formula $\Kp$
\end_inset
-point
\lang english
irreducible representations (cf.
section
\begin_inset CommandInset ref
LatexCommand ref
reference "sm:symmetries"
\end_inset
), but the splitting is extremely weak not exceeding
\begin_inset Formula $1\,\mathrm{meV}$
\end_inset
for the spherical and even less for the cylindrical nanoparticles.
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard
\lang english \lang english
We did not find any deviation from the empty lattice diffracted orders exceeding This is most likely due to the frequencies in our experiment being far below
the numerical precision of the computation (about 2 meV). the resonances of the nanoparticles, with the largest elements of the
This is most likely due to the frequencies in our experiment being far
below the resonances of the nanoparticles, with the largest elements of
the
\begin_inset Formula $T$ \begin_inset Formula $T$
\end_inset \end_inset
@ -264,7 +326,7 @@ We did not find any deviation from the empty lattice diffracted orders exceeding
(for power-normalised waves). (for power-normalised waves).
The nanoparticles are therefore almost transparent, but still suffice to The nanoparticles are therefore almost transparent, but still suffice to
provide feedback for lasing. provide enough feedback for lasing.
\end_layout \end_layout
@ -840,6 +902,267 @@ almost zero
singular value. singular value.
\end_layout \end_layout
\begin_layout Section
\lang english
Symmetries
\begin_inset CommandInset label
LatexCommand label
name "sm:symmetries"
\end_inset
\end_layout
\begin_layout Standard
A general overview of utilizing group theory to find lattice modes at high-symme
try points of the Brillouin zone can be found e.g.
in
\begin_inset CommandInset citation
LatexCommand cite
after "chapters 1011"
key "dresselhaus_group_2008"
\end_inset
; here we use the same notation.
\end_layout
\begin_layout Standard
We analyse the symmetries of the system in the same SVWF representation
as used in the
\begin_inset Formula $T$
\end_inset
-matrix formalism introduced above.
We are interested in the modes at the
\begin_inset Formula $\Kp$
\end_inset
-point of the hexagonal lattice, which has the
\begin_inset Formula $D_{3h}$
\end_inset
point symmetry.
\begin_inset Note Note
status open
\begin_layout Plain Layout
The symmetry makes the
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
matrix defined above invariant to the symmetry operations at the
\begin_inset Formula $\Kp$
\end_inset
-point,
\begin_inset Formula
\[
RM\left(\omega,\vect K\right)R^{-1}=M\left(\omega,\vect K\right),\qquad R\in D_{3h}.
\]
\end_inset
\end_layout
\end_inset
The six irreducible representations (irreps) of the
\begin_inset Formula $D_{3h}$
\end_inset
group are known and are available in the literature in their explicit forms.
In order to find and classify the modes, we need to find a decomposition
of the lattice mode representation
\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}=\Gamma^{\mathrm{equiv.}}\otimes\Gamma_{\mathrm{vec.}}$
\end_inset
into the irreps of
\begin_inset Formula $D_{3h}$
\end_inset
.
\begin_inset Note Note
status open
\begin_layout Plain Layout
The characters of the equivalence representation
\begin_inset Formula $\Gamma^{\mathrm{equiv.}}$
\end_inset
are given by the formula
\begin_inset Formula $\chi^{\mathrm{equiv.}}=\sum_{\alpha}\delta_{R_{\alpha}\vect r_{\alpha},\vect r_{\alpha}}e^{i\vect K_{m}\cdot\vect r_{\alpha}}$
\end_inset
where
\begin_inset Formula $\vect r_{\alpha}$
\end_inset
are the positions of the nanoparticles with respect
\end_layout
\end_inset
The equivalence representation
\begin_inset Formula $\Gamma^{\mathrm{equiv.}}$
\end_inset
is the
\begin_inset Formula $E'$
\end_inset
representation as can be deduced from
\begin_inset CommandInset citation
LatexCommand cite
after "eq. (11.19)"
key "dresselhaus_group_2008"
\end_inset
, eq.
(11.19) and the character table for
\begin_inset Formula $D_{3h}$
\end_inset
.
\begin_inset Formula $\Gamma_{\mathrm{vec.}}$
\end_inset
operates on a space spanned by the VSWFs around each nanoparticle in the
unit cell (the effects of point group operations on VSWFs are described
in
\begin_inset CommandInset citation
LatexCommand cite
key "schulz_point-group_1999"
\end_inset
).
This space can be then decomposed into invariant subspaces of the
\begin_inset Formula $D_{3h}$
\end_inset
using the projectors
\begin_inset Formula $\hat{P}_{ab}^{\left(\Gamma\right)}$
\end_inset
defined by
\begin_inset CommandInset citation
LatexCommand cite
after "eq. (4.28)"
key "dresselhaus_group_2008"
\end_inset
.
This way, we obtain a symmetry adapted basis
\begin_inset Formula $\left\{ \vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}\right\} $
\end_inset
as linear combinations of VSWFs
\begin_inset Formula $\svwfs lm{p,t}$
\end_inset
around the constituting nanoparticles (labeled
\begin_inset Formula $p$
\end_inset
),
\begin_inset Formula
\[
\vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}=\sum_{l,m,p,t}U_{\Gamma,r,i}^{p,t,l,m}\svwfs lm{p,t},
\]
\end_inset
where
\begin_inset Formula $\Gamma$
\end_inset
stands for one of the six different irreps of
\begin_inset Formula $D_{3h}$
\end_inset
,
\begin_inset Formula $r$
\end_inset
labels the different realisations of the same irrep, and the last index
\begin_inset Formula $i$
\end_inset
going from 1 to
\begin_inset Formula $d_{\Gamma}$
\end_inset
(the dimensionality of
\begin_inset Formula $\Gamma$
\end_inset
) labels the different partners of the same given irrep.
The number of how many times is each irrep contained in
\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}$
\end_inset
(i.e.
the range of index
\begin_inset Formula $r$
\end_inset
for given
\begin_inset Formula $\Gamma$
\end_inset
) depends on the multipole degree cutoff
\begin_inset Formula $l_{\mathrm{max}}$
\end_inset
.
\end_layout
\begin_layout Standard
Each mode at the
\begin_inset Formula $\Kp$
\end_inset
-point shall lie in the irreducible spaces of only one of the six possible
irreps and it can be shown via
\begin_inset CommandInset citation
LatexCommand cite
after "eq. (2.51)"
key "dresselhaus_group_2008"
\end_inset
that, at the
\begin_inset Formula $\Kp$
\end_inset
-point, the matrix
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
defined above takes a block-diagonal form in the symmetry-adapted basis,
\begin_inset Formula
\[
M\left(\omega,\vect K\right)_{\Gamma,r,i;\Gamma',r',j}^{\mathrm{s.a.b.}}=\frac{\delta_{\Gamma\Gamma'}\delta_{ij}}{d_{\Gamma}}\sum_{q}M\left(\omega,\vect K\right)_{\Gamma,r,q;\Gamma',r',q}^{\mathrm{s.a.b.}}.
\]
\end_inset
This enables us to decompose the matrix according to the irreps and to
solve the singular value problem in each irrep separately, as done in Fig.
xxx.
\end_layout
\begin_layout Standard \begin_layout Standard
\begin_inset CommandInset bibtex \begin_inset CommandInset bibtex
LatexCommand bibtex LatexCommand bibtex