hexlaser tmatrix text; continuing
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Marek Nečada
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@article{mackowski_analysis_1991,
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title = {Analysis of {{Radiative Scattering}} for {{Multiple Sphere Configurations}}},
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volume = {433},
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issn = {1364-5021, 1471-2946},
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doi = {10.1098/rspa.1991.0066},
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abstract = {An analysis of radiative scattering for an arbitrary configuration of neighbouring spheres is presented. The analysis builds upon the previously developed superposition solution, in which the scattered field is expressed as a superposition of vector spherical harmonic expansions written about each sphere in the ensemble. The addition theorems for vector spherical harmonics, which transform harmonics from one coordinate system into another, are rederived, and simple recurrence relations for the addition coefficients are developed. The relations allow for a very efficient implementation of the ‘order of scattering' solution technique for determining the scattered field coefficients for each sphere.},
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language = {en},
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number = {1889},
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journal = {Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences},
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author = {Mackowski, Daniel W.},
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month = jun,
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year = {1991},
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pages = {599-614},
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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}
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}
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@misc{SCUFF2,
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title = {{{SCUFF}}-{{EM}}},
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author = {Reid, Homer},
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year = {2018},
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note = {http://github.com/homerreid/scuff-EM}
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}
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@article{xu_efficient_1998,
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title = {Efficient {{Evaluation}} of {{Vector Translation Coefficients}} in {{Multiparticle Light}}-{{Scattering Theories}}},
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volume = {139},
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issn = {0021-9991},
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doi = {10.1006/jcph.1997.5867},
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abstract = {Vector addition theorems are a necessary ingredient in the analytical solution of electromagnetic multiparticle-scattering problems. These theorems include a large number of vector addition coefficients. There exist three basic types of analytical expressions for vector translation coefficients: Stein's (Quart. Appl. Math.19, 15 (1961)), Cruzan's (Quart. Appl. Math.20, 33 (1962)), and Xu's (J. Comput. Phys.127, 285 (1996)). Stein's formulation relates vector translation coefficients with scalar translation coefficients. Cruzan's formulas use the Wigner 3jm symbol. Xu's expressions are based on the Gaunt coefficient. Since the scalar translation coefficient can also be expressed in terms of the Gaunt coefficient, the key to the expeditious and reliable calculation of vector translation coefficients is the fast and accurate evaluation of the Wigner 3jm symbol or the Gaunt coefficient. We present highly efficient recursive approaches to accurately evaluating Wigner 3jm symbols and Gaunt coefficients. Armed with these recursive approaches, we discuss several schemes for the calculation of the vector translation coefficients, using the three general types of formulation, respectively. Our systematic test calculations show that the three types of formulas produce generally the same numerical results except that the algorithm of Stein's type is less accurate in some particular cases. These extensive test calculations also show that the scheme using the formulation based on the Gaunt coefficient is the most efficient in practical computations.},
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number = {1},
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journal = {Journal of Computational Physics},
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author = {Xu, Yu-lin},
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month = jan,
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year = {1998},
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pages = {137-165},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/STV5263F/Xu - 1998 - Efficient Evaluation of Vector Translation Coeffic.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/VMGZRSAA/S0021999197958678.html}
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}
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@article{schulz_point-group_1999,
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title = {Point-Group Symmetries in Electromagnetic Scattering},
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volume = {16},
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issn = {1084-7529, 1520-8532},
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doi = {10.1364/JOSAA.16.000853},
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language = {en},
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number = {4},
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journal = {Journal of the Optical Society of America A},
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author = {Schulz, F. Michael and Stamnes, Knut and Stamnes, J. J.},
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month = apr,
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year = {1999},
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pages = {853},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/X9X48A6G/josaa-16-4-853.pdf}
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}
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@book{dresselhaus_group_2008,
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title = {Group {{Theory}}: {{Application}} to the {{Physics}} of {{Condensed Matter}}},
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isbn = {978-3-540-32899-5},
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abstract = {Every process in physics is governed by selection rules that are the consequence of symmetry requirements. The beauty and strength of group theory resides...},
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publisher = {{Springer, Berlin, Heidelberg}},
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author = {Dresselhaus, Mildred S. and Dresselhaus, Gene and Jorio, Ado},
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year = {2008},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/GFGPVB4A/Mildred_S._Dresselhaus,_Gene_Dresselhaus,_Ado_Jorio_Group_theory_application_to_the_physics_of_condensed_matter.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/E78682CJ/9783540328971.html}
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}
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@book{ITfC:B,
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edition = {2nd},
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title = {International {{Tables}} for {{Crystallography}}, {{Vol}}.{{B}}: {{Reciprocal Space}}},
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isbn = {978-0-7923-6592-1},
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shorttitle = {International {{Tables}} for {{Crystallography}}, {{Vol}}.{{B}}},
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publisher = {{Springer}},
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author = {Shmueli, Uri},
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year = {2001},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/3WPJZ79F/Shmueli - 2001 - International Tables for Crystallography, Vol.B R.pdf}
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}
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@book{ITfC:A,
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edition = {5th},
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series = {IUCr Series. International Tables of Crystallography},
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title = {International {{Tables}} for {{Crystallography}}, {{Vol}}.{{A}}: {{Space Group Symmetry}}},
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isbn = {978-0-7923-6590-7},
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shorttitle = {International {{Tables}} for {{Crystallography}}, {{Vol}}.{{A}}},
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publisher = {{Springer, Dordrecht}},
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author = {Hahn, Theo},
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year = {2002},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/PE7NZCZ8/Hahn - 2002 - International Tables for Crystallography, Vol.A S.pdf}
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}
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@article{dixon_computing_1970,
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title = {Computing Irreducible Representations of Groups},
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volume = {24},
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issn = {0025-5718, 1088-6842},
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doi = {10.1090/S0025-5718-1970-0280611-6},
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abstract = {How can you find a complete set of inequivalent irreducible (ordinary) representations of a finite group? The theory is classical but, except when the group was very small or had a rather special structure, the actual computations were prohibitive before the advent of high-speed computers; and there remain practical difficulties even for groups of relatively small orders . The present paper describes three techniques to help solve this problem. These are: the reduction of a reducible unitary representation into its irreducible components; the construction of a complete set of irreducible unitary representations from a single faithful representation; and the calculation of the precise values of a group character from values which have only been computed approximately.},
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language = {en-US},
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number = {111},
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journal = {Math. Comp.},
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author = {Dixon, John D.},
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year = {1970},
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keywords = {computation of characters,Computation of group representations,finite Fourier analysis,irreducible components,iterative processes,reduction of unitary representations,tensor products},
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pages = {707-712},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/I5V5XH2P/Dixon - 1970 - Computing irreducible representations of groups.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/K9A484RX/S0025-5718-1970-0280611-6.html}
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}
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@article{linton_lattice_2010,
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title = {Lattice {{Sums}} for the {{Helmholtz Equation}}},
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volume = {52},
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issn = {0036-1445},
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doi = {10.1137/09075130X},
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abstract = {A survey of different representations for lattice sums for the Helmholtz equation is made. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension \$d\_$\backslash$Lambda\$. Lattice sums are related to, and can be calculated from, the quasi-periodic Green's function and this object serves as the starting point of the analysis.},
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number = {4},
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journal = {SIAM Rev.},
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author = {Linton, C.},
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month = jan,
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year = {2010},
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pages = {630-674},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/T86ATKYB/09075130x.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/ETB8X4S9/09075130X.html}
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}
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@book{bradley_mathematical_1972,
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title = {The Mathematical Theory of Symmetry in Solids; Representation Theory for Point Groups and Space Groups},
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isbn = {978-0-19-851920-1},
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publisher = {{Clarendon Press, Oxford}},
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author = {Bradley, C. J. and Cracknell, A. P.},
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year = {1972},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/SB5ZN5WH/C.J. Bradley, A.P. Cracknell - The mathematical theory of symmetry in solids_ representation theory for point groups and space groups (1972, Clarendon Press).djvu}
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}
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@article{reid_efficient_2015,
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archivePrefix = {arXiv},
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eprinttype = {arxiv},
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eprint = {1307.2966},
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title = {Efficient {{Computation}} of {{Power}}, {{Force}}, and {{Torque}} in {{BEM Scattering Calculations}}},
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volume = {63},
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issn = {0018-926X, 1558-2221},
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doi = {10.1109/TAP.2015.2438393},
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abstract = {We present concise, computationally efficient formulas for several quantities of interest -- including absorbed and scattered power, optical force (radiation pressure), and torque -- in scattering calculations performed using the boundary-element method (BEM) [also known as the method of moments (MOM)]. Our formulas compute the quantities of interest $\backslash$textit\{directly\} from the BEM surface currents with no need ever to compute the scattered electromagnetic fields. We derive our new formulas and demonstrate their effectiveness by computing power, force, and torque in a number of example geometries. Free, open-source software implementations of our formulas are available for download online.},
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number = {8},
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journal = {IEEE Transactions on Antennas and Propagation},
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author = {Reid, M. T. Homer and Johnson, Steven G.},
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month = aug,
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year = {2015},
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keywords = {Physics - Classical Physics,Physics - Computational Physics},
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pages = {3588-3598},
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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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}
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@ -133,15 +133,158 @@
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\end_inset
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffripext}[4]{p_{\mathrm{ext}(#1)}^{#2,#3,#4}}
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\end_inset
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\begin_inset FormulaMacro
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\begin_inset FormulaMacro
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\newcommand{\transop}{S}
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\newcommand{\transop}{S}
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\end_inset
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\end_inset
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\end_layout
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\begin_layout Section
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\lang english
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\begin_inset Formula $T$
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\end_inset
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-matrix simulations
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:T-matrix-simulations"
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\end_inset
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\end_layout
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\end_layout
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\begin_layout Standard
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\begin_layout Standard
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In this approach, scattering properties of single nanoparticles are first
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\lang english
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In order to get more detailed insight into the mode structure of the lattice
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around the lasing
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\begin_inset Formula $\Kp$
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\end_inset
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-point – most importantly, how much do the mode frequencies at the
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\begin_inset Formula $\Kp$
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\end_inset
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-points differ from the empty lattice model – we performed multiple-scattering
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\begin_inset Formula $T$
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\end_inset
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-matrix simulations
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\begin_inset CommandInset citation
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LatexCommand cite
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key "mackowski_analysis_1991"
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\end_inset
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for an infinite lattice based on our systems' geometry.
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We give a brief overview of this method in the subsections
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "sub:The-multiple-scattering-problem"
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\end_inset
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,
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "sub:Periodic-systems"
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\end_inset
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below.
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\lang finnish
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The top advantage of the multiple-scattering
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\begin_inset Formula $T$
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\end_inset
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-matrix approach is its computational efficiency for large finite systems
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of nanoparticles.
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In the lattice mode analysis in this work, however, we use it here for
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another reason, specifically the relative ease of describing symmetries
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\begin_inset CommandInset citation
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LatexCommand cite
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key "schulz_point-group_1999"
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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\lang english
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The
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\begin_inset Formula $T$
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\end_inset
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-matrix of a single nanoparticle was computed using the scuff-tmatrix applicatio
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n from the SCUFF-EM suite~
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\lang finnish
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\begin_inset CommandInset citation
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LatexCommand cite
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key "SCUFF2,reid_efficient_2015"
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\end_inset
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\lang english
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and the system was solved up to the
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\begin_inset Formula $l_{\mathrm{max}}=3$
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\end_inset
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(octupolar) degree of electric and magnetic spherical multipole.
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\end_layout
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\begin_layout Standard
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\lang english
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We did not find any deviation from the empty lattice diffracted orders exceeding
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the numerical precision of the computation (about 2 meV).
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This is most likely due to the frequencies in our experiment being far
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below the resonances of the nanoparticles, with the largest elements of
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the
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\begin_inset Formula $T$
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\end_inset
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-matrix being of the order of
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\begin_inset Formula $10^{-3}$
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\end_inset
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(for power-normalised waves).
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The nanoparticles are therefore almost transparent, but still suffice to
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provide feedback for lasing.
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\end_layout
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\begin_layout Subsection
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The multiple-scattering problem
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\begin_inset CommandInset label
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name "sub:The-multiple-scattering-problem"
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\end_inset
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\end_layout
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\begin_layout Standard
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In the
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\begin_inset Formula $T$
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\end_inset
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-matrix approach, scattering properties of single nanoparticles are first
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computed in terms of vector sperical wavefunctions (VSWFs)—the field incident
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computed in terms of vector sperical wavefunctions (VSWFs)—the field incident
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onto the
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onto the
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\begin_inset Formula $n$
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\begin_inset Formula $n$
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@ -181,7 +324,7 @@ where
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\end_inset
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\end_inset
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; the expressions can be found e.g.
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; the expressions can be found e.g.
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in REF
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in [REF]
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\begin_inset Note Note
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\begin_inset Note Note
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status open
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status open
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@ -191,7 +334,7 @@ few words about different conventions?
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\end_inset
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\end_inset
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.
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(care must be taken because of varying normalisation and phase conventions).
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On the other hand, the field scattered by the particle can be (outside
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On the other hand, the field scattered by the particle can be (outside
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the particle's circumscribing sphere) expanded in terms of singular VSWFs
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the particle's circumscribing sphere) expanded in terms of singular VSWFs
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@ -206,7 +349,7 @@ few words about different conventions?
|
||||||
,
|
,
|
||||||
\begin_inset Formula
|
\begin_inset Formula
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\vect E_{n}^{\mathrm{scat}}=\sum_{l,m,t}\coeffsip nlmt\svwfs lmt\left(\vect r_{n}\right).\label{eq:E_scat}
|
\vect E_{n}^{\mathrm{scat}}\left(\vect r\right)=\sum_{l,m,t}\coeffsip nlmt\svwfs lmt\left(\vect r_{n}\right).\label{eq:E_scat}
|
||||||
\end{equation}
|
\end{equation}
|
||||||
|
|
||||||
\end_inset
|
\end_inset
|
||||||
|
|
@ -234,7 +377,7 @@ At a given frequency, assuming the system is linear, the relation between
|
||||||
-matrix,
|
-matrix,
|
||||||
\begin_inset Formula
|
\begin_inset Formula
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\coeffsip nlmt=\sum_{l',m',t'}T_{n}^{l,m,t;l',m',t'}\coeffrip n{l'}{m'}{t'}.\label{eq:Tmatrix definition}
|
\coeffsip nlmt=\sum_{l',m',t'}T_{n}^{lmt;l'm't'}\coeffrip n{l'}{m'}{t'}.\label{eq:Tmatrix definition}
|
||||||
\end{equation}
|
\end{equation}
|
||||||
|
|
||||||
\end_inset
|
\end_inset
|
||||||
|
|
@ -262,7 +405,14 @@ th nanoparticles) its elements drop very quickly to negligible values with
|
||||||
\end_inset
|
\end_inset
|
||||||
|
|
||||||
-matrix can be calculated numerically using various methods; here we used
|
-matrix can be calculated numerically using various methods; here we used
|
||||||
the scuff-tmatrix tool from the SCUFF-EM suite [REF].
|
the scuff-tmatrix tool from the SCUFF-EM suite
|
||||||
|
\begin_inset CommandInset citation
|
||||||
|
LatexCommand cite
|
||||||
|
key "SCUFF2,reid_efficient_2015"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
.
|
||||||
\end_layout
|
\end_layout
|
||||||
|
|
||||||
\begin_layout Standard
|
\begin_layout Standard
|
||||||
|
|
@ -276,12 +426,229 @@ The singular SVWFs originating at
|
||||||
|
|
||||||
in terms of regular SVWFs,
|
in terms of regular SVWFs,
|
||||||
\begin_inset Formula
|
\begin_inset Formula
|
||||||
|
\begin{equation}
|
||||||
|
\svwfs lmt\left(\vect r_{n}\right)=\sum_{l',m',t'}\transop^{l'm't';lmt}\left(\vect R_{n'}-\vect R_{n}\right)\svwfr{l'}{m'}{t'}\left(\vect r_{n'}\right),\qquad\left|\vect r_{n'}\right|<\left|\vect R_{n'}-\vect R_{n}\right|.\label{eq:translation op def}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
Analytical expressions for the translation operator
|
||||||
|
\begin_inset Formula $\transop^{lmt;l'm't'}\left(\vect R_{n'}-\vect R_{n}\right)$
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
can be found in
|
||||||
|
\begin_inset CommandInset citation
|
||||||
|
LatexCommand cite
|
||||||
|
key "xu_efficient_1998"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
.
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\begin_layout Standard
|
||||||
|
If we write the field incident onto
|
||||||
|
\begin_inset Formula $n$
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
-th nanoparticle as the sum of fields scattered from all the other nanoparticles
|
||||||
|
and an external field
|
||||||
|
\begin_inset Formula $\vect E_{0}$
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
,
|
||||||
|
\begin_inset Formula
|
||||||
\[
|
\[
|
||||||
\svwfs lmt\left(\vect r_{n}\right)=\sum_{l',m',t'}
|
\vect E_{n}^{\mathrm{inc}}\left(\vect r\right)=\vect E_{0}\left(\vect r\right)+\sum_{n'\ne n}\vect E_{n'}^{\mathrm{scat}}\left(\vect r\right)
|
||||||
\]
|
\]
|
||||||
|
|
||||||
\end_inset
|
\end_inset
|
||||||
|
|
||||||
|
and use eqs.
|
||||||
|
|
||||||
|
\begin_inset CommandInset ref
|
||||||
|
LatexCommand eqref
|
||||||
|
reference "eq:E_inc"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
–
|
||||||
|
\begin_inset CommandInset ref
|
||||||
|
LatexCommand eqref
|
||||||
|
reference "eq:translation op def"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
, we obtain a set of linear equations for the electromagnetic response (multiple
|
||||||
|
scattering) of the whole set of nanoparticles,
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\begin_layout Standard
|
||||||
|
\begin_inset Note Note
|
||||||
|
status open
|
||||||
|
|
||||||
|
\begin_layout Plain Layout
|
||||||
|
\begin_inset Formula
|
||||||
|
\[
|
||||||
|
\vect E_{n}^{\mathrm{inc}}\left(\vect r\right)=\vect E_{0}\left(\vect r\right)+\sum_{n'\ne n}\vect E_{n'}^{\mathrm{scat}}\left(\vect r\right)
|
||||||
|
\]
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\begin_layout Plain Layout
|
||||||
|
\begin_inset Formula
|
||||||
|
\[
|
||||||
|
\sum_{l,m,t}\coeffrip nlmt\svwfr lmt\left(\vect r_{n}\right)=\sum_{l,m,t}\coeffripext nlmt\svwfr lmt\left(\vect r_{n}\right)+\sum_{n'\ne n}\sum_{l,m,t}\coeffsip{n'}lmt\svwfs lmt\left(\vect r_{n'}\right)
|
||||||
|
\]
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\begin_layout Plain Layout
|
||||||
|
\begin_inset Formula
|
||||||
|
\[
|
||||||
|
\sum_{l,m,t}\coeffrip nlmt\svwfr lmt\left(\vect r_{n}\right)=\sum_{l,m,t}\coeffripext nlmt\svwfr lmt\left(\vect r_{n}\right)+\sum_{n'\ne n}\sum_{l,m,t}\coeffsip{n'}lmt\sum_{l',m',t'}\transop^{l'm't';lmt}\left(\vect R_{n}-\vect R_{n'}\right)\svwfr{l'}{m'}{t'}\left(\vect r_{n}\right)
|
||||||
|
\]
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
|
||||||
|
\begin_inset Formula
|
||||||
|
\[
|
||||||
|
\sum_{l,m,t}\coeffrip nlmt\svwfr lmt\left(\vect r_{n}\right)=\sum_{l,m,t}\coeffripext nlmt\svwfr lmt\left(\vect r_{n}\right)+\sum_{n'\ne n}\sum_{l,m,t}\sum_{l',m',t'}\coeffsip{n'}{l'}{m'}{t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)\svwfr lmt\left(\vect r_{n}\right)
|
||||||
|
\]
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\begin_layout Plain Layout
|
||||||
|
\begin_inset Formula
|
||||||
|
\[
|
||||||
|
\coeffrip nlmt=\coeffripext nlmt+\sum_{n'\ne n}\sum_{l',m',t'}\coeffsip{n'}{l'}{m'}{t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)
|
||||||
|
\]
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
(
|
||||||
|
\begin_inset Formula $\coeffsip{n'}{l'}{m'}{t'}=\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''}$
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
)
|
||||||
|
\begin_inset Formula
|
||||||
|
\[
|
||||||
|
\coeffrip nlmt=\coeffripext nlmt+\sum_{n'\ne n}\sum_{l',m',t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''}
|
||||||
|
\]
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\begin_layout Standard
|
||||||
|
\begin_inset Formula
|
||||||
|
\begin{equation}
|
||||||
|
\coeffrip nlmt=\coeffripext nlmt+\sum_{n'\ne n}\sum_{l',m',t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''},\label{eq:multiplescattering element-wise}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
where
|
||||||
|
\begin_inset Formula $\coeffripext nlmt$
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
are the expansion coefficients of the external field around the
|
||||||
|
\begin_inset Formula $n$
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
-th particle,
|
||||||
|
\begin_inset Formula $\vect E_{0}\left(\vect r\right)=\sum_{l,m,t}\coeffripext nlmt\svwfr lmt\left(\vect r_{n}\right).$
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
It is practical to get rid of the SVWF indices, rewriting
|
||||||
|
\begin_inset CommandInset ref
|
||||||
|
LatexCommand eqref
|
||||||
|
reference "eq:multiplescattering element-wise"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
in a per-particle matrix form
|
||||||
|
\begin_inset Formula
|
||||||
|
\begin{equation}
|
||||||
|
\coeffr_{n}=\coeffr_{\mathrm{ext}(n)}+\sum_{n'\ne n}S_{n,n'}T_{n'}p_{n'}\label{eq:multiple scattering per particle p}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
and to reformulate the problem using
|
||||||
|
\begin_inset CommandInset ref
|
||||||
|
LatexCommand eqref
|
||||||
|
reference "eq:Tmatrix definition"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
in terms of the
|
||||||
|
\begin_inset Formula $\coeffs$
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
-coefficients which describe the multipole excitations of the particles
|
||||||
|
\begin_inset Formula
|
||||||
|
\begin{equation}
|
||||||
|
\coeffs_{n}=T_{n}\left(\coeffr_{\mathrm{ext}(n)}+\sum_{n'\ne n}S_{n,n'}\coeffs_{n'}\right).\label{eq:multiple scattering per particle a}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
Knowing
|
||||||
|
\begin_inset Formula $T_{n},S_{n,n'},\coeffr_{\mathrm{ext}(n)}$
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
, the nanoparticle excitations
|
||||||
|
\begin_inset Formula $a_{n}$
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
can be solved by standard linear algebra methods.
|
||||||
|
The total scattered field anywhere outside the particles' circumscribing
|
||||||
|
spheres is then obtained by summing the contributions
|
||||||
|
\begin_inset CommandInset ref
|
||||||
|
LatexCommand eqref
|
||||||
|
reference "eq:E_scat"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
from all particles.
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\begin_layout Subsection
|
||||||
|
Periodic systems and mode analysis
|
||||||
|
\begin_inset CommandInset label
|
||||||
|
LatexCommand label
|
||||||
|
name "sub:Periodic-systems"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\begin_layout Standard
|
||||||
|
\begin_inset CommandInset bibtex
|
||||||
|
LatexCommand bibtex
|
||||||
|
bibfiles "hexarray-theory"
|
||||||
|
options "plain"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
|
||||||
\end_layout
|
\end_layout
|
||||||
|
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue