Hexlaser theory additions

Former-commit-id: 725f0dd36fef3c76f23d28367f6cb7de12264d6b

notes/hexarray-theory.bib

@@ -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}
 }
 
+@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,
   title = {{{SCUFF}}-{{EM}}},
   author = {Reid, Homer},
@@ -139,7 +153,7 @@
   author = {Reid, M. T. Homer and Johnson, Steven G.},
   month = aug,
   year = {2015},
-  keywords = {Physics - Classical Physics,Physics - Computational Physics},
+  keywords = {Physics - Computational Physics,Physics - Classical Physics},
   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}
 }

notes/hexlaser-tmatrixtext.lyx

@@ -219,17 +219,39 @@ key "schulz_point-group_1999"
 \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
 
 \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
 The 
 \begin_inset Formula $T$
 \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
 
 \begin_inset CommandInset citation
@@ -245,16 +267,56 @@ key "SCUFF2,reid_efficient_2015"
 \end_inset
 
  (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
 
 \begin_layout Standard
 
 \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 $T$
 \end_inset
 
@@ -264,7 +326,7 @@ We did not find any deviation from the empty lattice diffracted orders exceeding
 
  (for power-normalised waves).
  The nanoparticles are therefore almost transparent, but still suffice to
- provide feedback for lasing.
+ provide enough feedback for lasing.
  
 \end_layout
 
@@ -840,6 +902,267 @@ almost zero
  singular value.
 \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 10–11"
+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_inset CommandInset bibtex
 LatexCommand bibtex