Abstract

We present a computer code for calculating near- and far-field electromagnetic properties of multilayered spheres. STRATIFY is a one-of-a-kind open-source package that allows for efficient calculation of electromagnetic near-field, energy density, total electromagnetic energy, and radiative and non-radiative decay rates of a dipole emitter located in any (non-absorbing) shell (including a host medium), and fundamental cross-sections of a multilayered sphere, all within a single program. Because of its speed and broad applicability, our package is a valuable tool for analysis of numerous light scattering problems, including but not limited to fluorescence enhancement, upconversion, downconversion, second harmonic generation, and surface enhanced Raman spectroscopy. The software is available for download from GitLab as Code 1.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (5)

X. Xu, A. Dutta, J. Khurgin, A. Wei, V. M. Shalaev, and A. Boltasseva, “TiN@TiO2 core-shell nanoparticles as plasmon-enhanced photosensitizers: The role of hot electron injection,” Laser Photonics Rev. 14(5), 1900376 (2020).
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S. Sun, I. L. Rasskazov, P. S. Carney, T. Zhang, and A. Moroz, “Critical role of shell in enhanced fluorescence of metal-dielectric core-shell nanoparticles,” J. Phys. Chem. C 124(24), 13365–13373 (2020).
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J. C. Castro-Palacio, K. Ladutenko, A. Prada, G. González-Rubio, P. Díaz-Núñez, A. Guerrero-Martínez, P. Fernández de Córdoba, J. Kohanoff, J. M. Perlado, O. Peña-Rodríguez, and A. Rivera, “Hollow Gold Nanoparticles Produced by Femtosecond Laser Irradiation,” J. Phys. Chem. Lett. 11(13), 5108–5114 (2020).
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M. Majic and E. C. Le Ru, “Numerically stable formulation of Mie theory for an emitter close to a sphere,” Appl. Opt. 59(5), 1293–1300 (2020).
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V. I. Zakomirnyi, I. L. Rasskazov, L. K. Sørensen, P. S. Carney, Z. Rinkevicius, and H. Ågren, “Plasmonic nano-shells: atomistic discrete interaction versus classic electrodynamics models,” Phys. Chem. Chem. Phys. 22(24), 13467–13473 (2020).
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2019 (5)

I. L. Rasskazov, A. Moroz, and P. S. Carney, “Electromagnetic energy in multilayered spherical particles,” J. Opt. Soc. Am. A 36(9), 1591–1601 (2019).
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Y. Eremin, A. Doicu, and T. Wriedt, “Extension of the discrete sources method to investigate the non-local effect influence on non-spherical core-shell particles,” J. Quant. Spectrosc. Radiat. Transfer 235, 300–308 (2019).
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A. Sheverdin and C. Valagiannopoulos, “Core-shell nanospheres under visible light: Optimal absorption, scattering, and cloaking,” Phys. Rev. B 99(7), 075305 (2019).
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K. L. Tsakmakidis, O. Reshef, E. Almpanis, G. P. Zouros, E. Mohammadi, D. Saadat, F. Sohrabi, N. Fahimi-Kashani, D. Etezadi, R. W. Boyd, and H. Altug, “Ultrabroadband 3D invisibility with fast-light cloaks,” Nat. Commun. 10(1), 4859 (2019).
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A. S. Kostyukov, A. E. Ershov, V. S. Gerasimov, S. A. Filimonov, I. L. Rasskazov, and S. V. Karpov, “Super-efficient laser hyperthermia of malignant cells with core-shell nanoparticles based on alternative plasmonic materials,” J. Quant. Spectrosc. Radiat. Transfer 236, 106599 (2019).
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2018 (8)

A. D. Phan, N. B. Le, N. T. H. Lien, and K. Wakabayashi, “Multilayered plasmonic nanostructures for solar energy harvesting,” J. Phys. Chem. C 122(34), 19801–19806 (2018).
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Z. Wang, X. Quan, Z. Zhang, and P. Cheng, “Optical absorption of carbon-gold core-shell nanoparticles,” J. Quant. Spectrosc. Radiat. Transfer 205, 291–298 (2018).
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S. A. Scherbak and A. A. Lipovskii, “Understanding the second-harmonic generation enhancement and behavior in metal core-dielectric shell nanoparticles,” J. Phys. Chem. C 122(27), 15635–15645 (2018).
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P. Wang, A. V. Krasavin, F. N. Viscomi, A. M. Adawi, J.-S. G. Bouillard, L. Zhang, D. J. Roth, L. Tong, and A. V. Zayats, “Metaparticles: dressing nano-objects with a hyperbolic coating,” Laser Photonics Rev. 12(11), 1800179 (2018).
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I. L. Rasskazov, L. Wang, C. J. Murphy, R. Bhargava, and P. S. Carney, “Plasmon-enhanced upconversion: engineering enhancement and quenching at nano and macro scales,” Opt. Mater. Express 8(12), 3787–3804 (2018).
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J. Y. Lee, A. E. Miroshnichenko, and R.-K. Lee, “Simultaneously nearly zero forward and nearly zero backward scattering objects,” Opt. Express 26(23), 30393–30399 (2018).
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P. R. Wiecha, A. Arbouet, A. Cuche, V. Paillard, and C. Girard, “Decay rate of magnetic dipoles near nonmagnetic nanostructures,” Phys. Rev. B 97(8), 085411 (2018).
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U. Hohenester, “Making simulations with the MNPBEM toolbox big: Hierarchical matrices and iterative solvers,” Comput. Phys. Commun. 222, 209–228 (2018).
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2017 (9)

K. Ladutenko, U. Pal, A. Rivera, and O. Peña-Rodríguez, “Mie calculation of electromagnetic near-field for a multilayered sphere,” Comput. Phys. Commun. 214, 225–230 (2017).
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L. Meng, R. Yu, M. Qiu, and F. J. García de Abajo, “Plasmonic nano-oven by concatenation of multishell photothermal enhancement,” ACS Nano 11(8), 7915–7924 (2017).
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N. G. Khlebtsov and B. N. Khlebtsov, “Optimal design of gold nanomatryoshkas with embedded Raman reporters,” J. Quant. Spectrosc. Radiat. Transfer 190, 89–102 (2017).
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J.-F. Li, Y.-J. Zhang, S.-Y. Ding, R. Panneerselvam, and Z.-Q. Tian, “Core-shell nanoparticle-enhanced Raman spectroscopy,” Chem. Rev. 117(7), 5002–5069 (2017).
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E. I. Galanzha, R. Weingold, D. A. Nedosekin, M. Sarimollaoglu, J. Nolan, W. Harrington, A. S. Kuchyanov, R. G. Parkhomenko, F. Watanabe, Z. Nima, A. S. Biris, A. I. Plekhanov, M. I. Stockman, and V. P. Zharov, “Spaser as a biological probe,” Nat. Commun. 8(1), 15528 (2017).
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J. L. Montaño-Priede, O. Peña-Rodríguez, and U. Pal, “Near-electric-field tuned plasmonic Au@SiO2 and Ag@SiO2 nanoparticles for efficient utilization in luminescence enhancement and surface-enhanced spectroscopy,” J. Phys. Chem. C 121(41), 23062–23071 (2017).
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J. L. Montaño-Priede, J. P. Coelho, A. Guerrero-Martínez, O. Peña-Rodríguez, and U. Pal, “Fabrication of monodispersed Au@SiO2 nanoparticles with highly stable silica layers by ultrasound-assisted Stöber method,” J. Phys. Chem. C 121(17), 9543–9551 (2017).
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V. I. Zakomirnyi, I. L. Rasskazov, S. V. Karpov, and S. P. Polyutov, “New ideally absorbing Au plasmonic nanostructures for biomedical applications,” J. Quant. Spectrosc. Radiat. Transfer 187, 54–61 (2017).
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N. Sakamoto, T. Onodera, T. Dezawa, Y. Shibata, and H. Oikawa, “Highly enhanced emission of visible light from core-dual-shell-type hybridized nanoparticles,” Part. Part. Syst. Charact. 34(12), 1700258 (2017).
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2016 (3)

N. Arnold, C. Hrelescu, and T. A. Klar, “Minimal spaser threshold within electrodynamic framework: Shape, size and modes,” Ann. Phys. 528(3-4), 295–306 (2016).
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N. Passarelli, R. A. Bustos-Marún, and E. A. Coronado, “Spaser and optical amplification conditions in gold-coated active nanoparticles,” J. Phys. Chem. C 120(43), 24941–24949 (2016).
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Z. Wang, W. Gao, R. Wang, J. Shao, Q. Han, C. Wang, J. Zhang, T. Zhang, J. Dong, and H. Zheng, “Influence of SiO2 layer on the plasmon quenched upconversion luminescence emission of core-shell NaYF4:Yb,Er@SiO2@Ag nanocomposites,” Mater. Res. Bull. 83, 515–521 (2016).
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2015 (4)

W. Xu, X. Min, X. Chen, Y. Zhu, P. Zhou, S. Cui, S. Xu, L. Tao, and H. Song, “Ag-SiO2-Er2O3 nanocomposites: Highly effective upconversion luminescence at high power excitation and high temperature,” Sci. Rep. 4(1), 5087 (2015).
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R. Ruppin, “Nanoshells with a gain layer: the effects of surface scattering,” J. Opt. 17(12), 125004 (2015).
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K. Ladutenko, P. Belov, O. Peña-Rodríguez, A. Mirzaei, A. E. Miroshnichenko, and I. V. Shadrivov, “Superabsorption of light by nanoparticles,” Nanoscale 7(45), 18897–18901 (2015).
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J. Waxenegger, A. Trügler, and U. Hohenester, “Plasmonics simulations with the MNPBEM toolbox: Consideration of substrates and layer structures,” Comput. Phys. Commun. 193, 138–150 (2015).
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2014 (9)

J. R. Allardice and E. C. Le Ru, “Convergence of Mie theory series: criteria for far-field and near-field properties,” Appl. Opt. 53(31), 7224–7229 (2014).
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U. Hohenester, “Simulating electron energy loss spectroscopy with the MNPBEM toolbox,” Comput. Phys. Commun. 185(3), 1177–1187 (2014).
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F. Monticone and A. Alù, “Embedded photonic eigenvalues in 3D nanostructures,” Phys. Rev. Lett. 112(21), 213903 (2014).
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R. Fleury, J. Soric, and A. Alù, “Physical bounds on absorption and scattering for cloaked sensors,” Phys. Rev. B 89(4), 045122 (2014).
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Y. Huang and L. Gao, “Superscattering of light from core-shell nonlocal plasmonic nanoparticles,” J. Phys. Chem. C 118(51), 30170–30178 (2014).
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Y. Ding, X. Zhang, H. Gao, S. Xu, C. Wei, and Y. Zhao, “Plasmonic enhanced upconversion luminescence of β-NaYF4:Yb3+/Er3+ with Ag@SiO2 core-shell nanoparticles,” J. Lumin. 147, 72–76 (2014).
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K. Ladutenko, O. Peña-Rodríguez, I. Melchakova, I. Yagupov, and P. Belov, “Reduction of scattering using thin all-dielectric shells designed by stochastic optimizer,” J. Appl. Phys. 116(18), 184508 (2014).
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C. Ayala-Orozco, J. G. Liu, M. W. Knight, Y. Wang, J. K. Day, P. Nordlander, and N. J. Halas, “Fluorescence enhancement of molecules inside a gold nanomatryoshka,” Nano Lett. 14(5), 2926–2933 (2014).
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C. Ayala-Orozco, C. Urban, M. W. Knight, A. S. Urban, O. Neumann, S. W. Bishnoi, S. Mukherjee, A. M. Goodman, H. Charron, T. Mitchell, M. Shea, R. Roy, S. Nanda, R. Schiff, N. J. Halas, and A. Joshi, “Au nanomatryoshkas as efficient near-infrared photothermal transducers for cancer treatment: benchmarking against nanoshells,” ACS Nano 8(6), 6372–6381 (2014).
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2013 (3)

F. Monticone, C. Argyropoulos, and A. Alù, “Multilayered plasmonic covers for comblike scattering response and optical tagging,” Phys. Rev. Lett. 110(11), 113901 (2013).
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P. Kannan, F. A. Rahim, X. Teng, R. Chen, H. Sun, L. Huang, and D.-H. Kim, “Enhanced emission of NaYF4:Yb,Er/Tm nanoparticles by selective growth of Au and Ag nanoshells,” RSC Adv. 3(21), 7718 (2013).
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D. G. Baranov, E. Andrianov, A. P. Vinogradov, and A. A. Lisyansky, “Exactly solvable toy model for surface plasmon amplification by stimulated emission of radiation,” Opt. Express 21(9), 10779–10791 (2013).
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2012 (7)

N. Calander, D. Jin, and E. M. Goldys, “Taking plasmonic core-shell nanoparticles toward laser threshold,” J. Phys. Chem. C 116(13), 7546–7551 (2012).
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A. Priyam, N. M. Idris, and Y. Zhang, “Gold nanoshell coated NaYF4 nanoparticles for simultaneously enhanced upconversion fluorescence and darkfield imaging,” J. Mater. Chem. 22(3), 960–965 (2012).
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P. Yuan, Y. H. Lee, M. K. Gnanasammandhan, Z. Guan, Y. Zhang, and Q.-H. Xu, “Plasmon enhanced upconversion luminescence of NaYF4:Yb,Er@SiO2@Ag core-shell nanocomposites for cell imaging,” Nanoscale 4(16), 5132–5137 (2012).
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B. Jankiewicz, D. Jamiola, J. Choma, and M. Jaroniec, “Silica-metal core-shell nanostructures,” Adv. Colloid Interface Sci. 170(1-2), 28–47 (2012).
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D. V. Guzatov and V. V. Klimov, “The influence of chiral spherical particles on the radiation of optically active molecules,” New J. Phys. 14(12), 123009 (2012).
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U. Hohenester and A. Trügler, “MNPBEM - A Matlab toolbox for the simulation of plasmonic nanoparticles,” Comput. Phys. Commun. 183(2), 370–381 (2012).
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T. J. Arruda, F. A. Pinheiro, and A. S. Martinez, “Electromagnetic energy within coated spheres containing dispersive metamaterials,” J. Opt. 14(6), 065101 (2012).
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2011 (3)

J. J. Wang, G. Gouesbet, G. Gréhan, Y. P. Han, and S. Saengkaew, “Morphology-dependent resonances in an eccentrically layered sphere illuminated by a tightly focused off-axis Gaussian beam: parallel and perpendicular beam incidence,” J. Opt. Soc. Am. A 28(9), 1849–1859 (2011).
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Z. Ruan and S. Fan, “Design of subwavelength superscattering nanospheres,” Appl. Phys. Lett. 98(4), 043101 (2011).
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D.-K. Lim, K.-S. Jeon, J.-H. Hwang, H. Kim, S. Kwon, Y. D. Suh, and J.-M. Nam, “Highly uniform and reproducible surface-enhanced Raman scattering from DNA-tailorable nanoparticles with 1-nm interior gap,” Nat. Nanotechnol. 6(7), 452–460 (2011).
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2010 (8)

F. Zhang, G. B. Braun, Y. Shi, Y. Zhang, X. Sun, N. O. Reich, D. Zhao, and G. Stucky, “Fabrication of Ag@SiO2@Y2O3:Er nanostructures for bioimaging: Tuning of the upconversion fluorescence with silver nanoparticles,” J. Am. Chem. Soc. 132(9), 2850–2851 (2010).
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Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
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Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105(1), 013901 (2010).
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A. E. Miroshnichenko, “Off-resonance field enhancement by spherical nanoshells,” Phys. Rev. A 81(5), 053818 (2010).
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T. J. Arruda and A. S. Martinez, “Electromagnetic energy within magnetic spheres,” J. Opt. Soc. Am. A 27(5), 992–1001 (2010).
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A. Moroz, “Non-radiative decay of a dipole emitter close to a metallic nanoparticle: Importance of higher-order multipole contributions,” Opt. Commun. 283(10), 2277–2287 (2010).
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A. K. Kodali, M. V. Schulmerich, R. Palekar, X. Llora, and R. Bhargava, “Optimized nanospherical layered alternating metal-dielectric probes for optical sensing,” Opt. Express 18(22), 23302 (2010).
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A. K. Kodali, X. Llora, and R. Bhargava, “Optimally designed nanolayered metal-dielectric particles as probes for massively multiplexed and ultrasensitive molecular assays,” Proc. Natl. Acad. Sci. 107(31), 13620–13625 (2010).
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2009 (3)

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: the effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem. C 113(8), 3041–3045 (2009).
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M. A. Ochsenkühn, P. R. T. Jess, H. Stoquert, K. Dholakia, and C. J. Campbell, “Nanoshells for surface-enhanced Raman spectroscopy in Eukaryotic cells: Cellular response and sensor development,” ACS Nano 3(11), 3613–3621 (2009).
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M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
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2008 (2)

A. Alù and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett. 100(11), 113901 (2008).
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A. Moroz, “Electron mean free path in a spherical shell geometry,” J. Phys. Chem. C 112(29), 10641–10652 (2008).
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2007 (3)

2006 (4)

K. Hasegawa, C. Rohde, and M. Deutsch, “Enhanced surface-plasmon resonance absorption in metal-dielectric-metal layered microspheres,” Opt. Lett. 31(8), 1136–1138 (2006).
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O. G. Tovmachenko, C. Graf, D. J. van den Heuvel, A. van Blaaderen, and H. C. Gerritsen, “Fluorescence enhancement by metal-core/silica-shell nanoparticles,” Adv. Mater. 18(1), 91–95 (2006).
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J. Zhang, I. Gryczynski, Z. Gryczynski, and J. R. Lakowicz, “Dye-labeled silver nanoshell-bright particle,” J. Phys. Chem. B 110(18), 8986–8991 (2006).
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L. R. Hirsch, A. M. Gobin, A. R. Lowery, F. Tam, R. A. Drezek, N. J. Halas, and J. L. West, “Metal nanoshells,” Ann. Biomed. Eng. 34(1), 15–22 (2006).
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2005 (3)

A. Moroz, “A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere,” Ann. Phys. 315(2), 352–418 (2005).
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A. Moroz, “Spectroscopic properties of a two-level atom interacting with a complex spherical nanoshell,” Chem. Phys. 317(1), 1–15 (2005).
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S. Schelm and G. B. Smith, “Internal electric field densities of metal nanoshells,” J. Phys. Chem. B 109(5), 1689–1694 (2005).
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2004 (1)

G. Raschke, S. Brogl, A. S. Susha, A. L. Rogach, T. A. Klar, J. Feldmann, B. Fieres, N. Petkov, T. Bein, A. Nichtl, and K. Kürzinger, “Gold nanoshells improve single nanoparticle molecular sensors,” Nano Lett. 4(10), 1853–1857 (2004).
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2003 (2)

L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. 100(23), 13549–13554 (2003).
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W. Yang, “Improved recursive algorithm for light scattering by a multilayered sphere,” Appl. Opt. 42(9), 1710–1720 (2003).
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2002 (4)

F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65(11), 115418 (2002).
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K. P. Velikov, A. Moroz, and A. Van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
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A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66(11), 115109 (2002).
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C. Graf and A. van Blaaderen, “Metallodielectric colloidal core-shell particles for photonic applications,” Langmuir 18(2), 524–534 (2002).
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2001 (2)

M. J. A. de Dood, L. H. Slooff, A. Polman, A. Moroz, and A. van Blaaderen, “Modified spontaneous emission in erbium-doped SiO2 spherical colloids,” Appl. Phys. Lett. 79(22), 3585–3587 (2001).
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M. J. A. de Dood, L. H. Slooff, A. Polman, A. Moroz, and A. van Blaaderen, “Local optical density of states in SiO2 spherical microcavities: Theory and experiment,” Phys. Rev. A 64(3), 033807 (2001).
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2000 (2)

W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. 84(13), 2853–2856 (2000).
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A. Moroz, “Photonic crystals of coated metallic spheres,” Europhys. Lett. 50(4), 466–472 (2000).
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1999 (2)

A. Moroz and C. Sommers, “Photonic band gaps of three-dimensional face-centred cubic lattices,” J. Phys.: Condens. Matter 11(4), 997–1008 (1999).
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R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16(10), 1824–1832 (1999).
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1998 (1)

S. Oldenburg, R. Averitt, S. Westcott, and N. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288(2-4), 243–247 (1998).
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1997 (2)

R. D. Averitt, D. Sarkar, and N. J. Halas, “Plasmon resonance shifts of Au-coated Au2S nanoshells: Insight into multicomponent nanoparticle growth,” Phys. Rev. Lett. 78(22), 4217–4220 (1997).
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Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36(21), 5188–5198 (1997).
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1995 (1)

1994 (4)

T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a coated sphere: investigation of the volume-averaged source function and resonance positions,” Appl. Opt. 33(33), 7789 (1994).
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J. A. Lock, J. M. Jamison, and C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33(21), 4677 (1994).
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J. Sinzig and M. Quinten, “Scattering and absorption by spherical multilayer particles,” Appl. Phys. A: Solids Surf. 58(2), 157–162 (1994).
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H. S. Zhou, I. Honma, H. Komiyama, and J. W. Haus, “Controlled synthesis and quantum-size effect in gold-coated nanoparticles,” Phys. Rev. B 50(16), 12052–12056 (1994).
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1991 (1)

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26(6), 1393–1401 (1991).
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1990 (2)

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Supplementary Material (1)

NameDescription
» Code 1       the computer program

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Figures (2)

Fig. 1.
Fig. 1. A sketch of the multilayered sphere in a homogeneous isotropic host medium with permittivity $\varepsilon _h=\varepsilon _{N+1}$ and permeability $\mu _h=\mu _{N+1}$ . Center of the sphere is located at the origin of the reference system.
Fig. 2.
Fig. 2. Comparison between our code (top, STRATIFY) and freely available BEM-based package (bottom, MNPBEM). (a),(d) Normalized fundamental cross-section ( $Q=\sigma /\pi r_2^{2}$ ) of SiO ${}_2$ @Au core-shell sphere with $\{r_1,r_2\}=\{50,55\}$ nm, in air host; (b),(e) intensity of the electric field for SiO ${}_2$ @Au@SiO ${}_2$ @Au matryoshka with $\{r_1,r_2,r_3,r_4\}=\{10,13,36,48\}$ nm, in water host, at $\lambda =690$ nm [111, cf. Fig. 3(b)]; (c),(f) normalized spontaneous decay rates ( $\tilde {\Gamma } = \Gamma / \Gamma _\textrm {rad;0}$ ) for the dipole emitter located at $r_d$ distance from the center of Au@SiO ${}_2$ core-shell sphere with $\{r_1,r_2\}=\{50,70\}$ nm, in water host, at $\lambda =614$ nm. Run times of codes on a laptop with 2.6 GHz 6-Core Intel Core i7 processor are shown on top of each plot. For a fair comparison, the same number of points for $\lambda$ (801) in (a),(c), same $700\times 700$ mesh in (b),(e), and same number of points for $r_d$ (29) in (c),(f) are used. Cut-offs $\ell _\textrm {max}=4$ and $\ell _\textrm {max}=10$ are used in (a) and (b), according to Eq. (35) and Eq. (36), respectively, while decay rates in (c) are calculated with $0.01$ accuracy. Note typically more than $50$ times faster speed of our package compared to the freely available BEM-based package.

Equations (40)

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E = i c ω ε ( × H )   , H = i c ω μ ( × E )   .
× [ × F p L ( k , r ) ] = k 2 F p L ( k , r )
F M L ( k n , r ) = f M L ( k n r ) Y L ( m ) ( r )   , F E L ( k n , r ) = 1 k n r { ( + 1 ) f E L ( k n r ) Y L ( o ) ( r ) + d d r [ r f E L ( k n r ) ] Y L ( e ) ( r ) }   ,
F ~ E L ( k n , r ) = 1 k n × F M L ( k n , r )   , F ~ M L ( k n , r ) = 1 k n × F E L ( k n , r )   ,
Y L ( m ) = i ( m ) ! ( + m ) ! 2 + 1 4 π ( + 1 ) [ e ^ ϑ i m P m ( cos ϑ ) sin ϑ e ^ φ d P m ( cos ϑ ) d ϑ ] exp ( i m φ )   , Y L ( o ) = i ( m ) ! ( + m ) ! 2 + 1 4 π P m ( cos ϑ ) exp ( i m φ ) e ^ r   , Y L ( e ) = i ( m ) ! ( + m ) ! 2 + 1 4 π ( + 1 ) [ e ^ ϑ d P m ( cos ϑ ) d ϑ + e ^ φ i m P m ( cos ϑ ) sin ϑ ] exp ( i m φ )   ,
P m ( x ) = ( 1 ) m 2 ! ( 1 x 2 ) m / 2 d + m d x + m ( x 2 1 )   .
E p ( r ) = L F p L ( k n , r ) = L [ A p L ( n ) J p L ( k n , r ) + B p L ( n ) H p L ( k n , r ) ]   .
H E ( r ) = i ε n μ n L [ A E L ( n ) J M L ( k n , r ) + B E L ( n ) H M L ( k n , r ) ]   , H M ( r ) = i ε n μ n L [ A M L ( n ) J E L ( k n , r ) + B M L ( n ) H E L ( k n , r ) ]   .
( A p L ( n + 1 ) B p L ( n + 1 ) ) = T p + ( n ) ( A p L ( n ) B p L ( n ) )   , ( A p L ( n ) B p L ( n ) ) = T p ( n ) ( A p L ( n + 1 ) B p L ( n + 1 ) )   ,
[ T p + ( n ) ] 1 = T p ( n )   , [ T p ( n ) ] 1 = T p + ( n )   ,
T M ( n ) = i ( η ~ ζ ( x ) ψ ( x ~ ) μ ~ ζ ( x ) ψ ( x ~ ) η ~ ζ ( x ) ζ ( x ~ ) μ ~ ζ ( x ) ζ ( x ~ ) η ~ ψ ( x ) ψ ( x ~ ) + μ ~ ψ ( x ) ψ ( x ~ ) η ~ ψ ( x ) ζ ( x ~ ) + μ ~ ψ ( x ) ζ ( x ~ ) )   ,
T E ( n ) = i ( μ ~ ζ ( x ) ψ ( x ~ ) η ~ ζ ( x ) ψ ( x ~ ) μ ~ ζ ( x ) ζ ( x ~ ) η ~ ζ ( x ) ζ ( x ~ ) μ ~ ψ ( x ) ψ ( x ~ ) + η ~ ψ ( x ) ψ ( x ~ ) μ ~ ψ ( x ) ζ ( x ~ ) + η ~ ψ ( x ) ζ ( x ~ ) )   ,
T M + ( n ) = i ( ζ ( x ~ ) ψ ( x ) / η ~ ζ ( x ~ ) ψ ( x ) / μ ~ ζ ( x ~ ) ζ ( x ) / η ~ ζ ( x ~ ) ζ ( x ) / μ ~ ψ ( x ~ ) ψ ( x ) / η ~ + ψ ( x ~ ) ψ ( x ) / μ ~ ψ ( x ~ ) ζ ( x ) / η ~ + ψ ( x ~ ) ζ ( x ) / μ ~ )   ,
T E + ( n ) = i ( ζ ( x ~ ) ψ ( x ) / μ ~ ζ ( x ~ ) ψ ( x ) / η ~ ζ ( x ~ ) ζ ( x ) / μ ~ ζ ( x ~ ) ζ ( x ) / η ~ ψ ( x ~ ) ψ ( x ) / μ ~ + ψ ( x ~ ) ψ ( x ) / η ~ ψ ( x ~ ) ζ ( x ) / μ ~ + ψ ( x ~ ) ζ ( x ) / η ~ )   ,
T p ( n ) = j = n 1 1 T p + ( j )   , M p ( n ) = j = n N T p ( j )   .
( A p L ( N + 1 ) B p L ( N + 1 ) ) = T p ( N + 1 ) ( A p L ( 1 ) B p L ( 1 ) )   .
[ T p ( N + 1 ) ] 1 = M p ( 1 )   , [ M p ( 1 ) ] 1 = T p ( N + 1 )   .
B E L ( 1 ) = B M L ( 1 ) 0   .
T 21 ; p ( N + 1 ) / T 11 ; p ( N + 1 ) = T p ,
σ sca = p , σ sca ; p = π k h 2 p , ( 2 + 1 ) | T p | 2   , σ abs = p , σ abs ; p = π 2 k h 2 p , ( 2 + 1 ) ( 1 | 1 + 2 T p | 2 )   , σ ext = p , σ ext ; p = 2 π k h 2 p , ( 2 + 1 ) ( T p )   ,
S ( ϑ ) = 2 + 1 ( + 1 ) [ T E d P 1 ( cos ϑ ) d ϑ + T M P 1 ( cos ϑ ) sin ϑ ]   , S ( ϑ ) = 2 + 1 ( + 1 ) [ T M d P 1 ( cos ϑ ) d ϑ + T E P 1 ( cos ϑ ) sin ϑ ]   ,
W = n = 1 N W n = n = 1 N r n 1 r n w n ( r ) r 2 d r   , w n ( r ) = 1 4 [ G e ( ε n ) | E ( r ) | 2 + G m ( μ n ) | H ( r ) | 2 ] d Ω   ,
G e ( ε n ) = ( ε n )   , G m ( μ n ) = ( μ n )   ,
G e ( ε n , ω ) = ( ε n ) + 2 ω γ D ( ε n )   ,
| E | 2 d Ω = 2 π | E 0 | 2 = 1 [ ( 2 + 1 ) | f ¯ M | 2 + ( + 1 ) | f ¯ E , 1 | 2 + | f ¯ E , + 1 | 2 ]   , | H | 2 d Ω = 2 π | E 0 | 2 | ε n | | μ n | = 1 [ ( 2 + 1 ) | f ¯ E | 2 + ( + 1 ) | f ¯ M , 1 | 2 + | f ¯ M , + 1 | 2 ]   .
A ¯ p ( n ) = { 1 / T 11 ; p ( N + 1 )   , n = 1   , M 11 ; p ( n ) + M 12 ; p ( n ) T 21 ; p ( N + 1 ) T 11 ; p ( N + 1 )   , 1 < n < N + 1   , 1   , n = N + 1   ,
B ¯ p ( n ) = { 0   , n = 1   , M 21 ; p ( n ) + M 22 ; p ( n ) T 21 ; p ( N + 1 ) T 11 ; p ( N + 1 )   , 1 < n < N + 1   , T 21 ; p ( N + 1 ) T 11 ; p ( N + 1 )   , n = N + 1   .
r n 1 r n r 2 d r | E | 2 d Ω = 2 π | E 0 | 2 r 3 x 2 x 2 = 1 [ ( 2 + 1 ) F ¯ M + ( + 1 ) F ¯ E , 1 + F ¯ E , + 1 ] | r = r n 1 r = r n   , r n 1 r n r 2 d r | H | 2 d Ω = 2 π | E 0 | 2 r 3 x 2 x 2 | ε n | | μ n | = 1 [ ( 2 + 1 ) F ¯ E + ( + 1 ) F ¯ M , 1 + F ¯ M , + 1 ] | r = r n 1 r = r n   .
x 2 x 2 2 x , F ¯ p   x ( | f ¯ p ( x ) | 2 + | f ¯ p + 1 ( x ) | 2 ) ( 2 + 1 ) ( f ¯ p ( x ) f ¯ p + 1 ( x ) ) .
Γ rad Γ rad;0 = 3 2 x d 4 N rad = 1 ( + 1 ) ( 2 + 1 ) | F E ( x d ) | 2   , Γ rad Γ rad;0 = 3 4 x d 2 N rad = 1 ( 2 + 1 ) [ | F M ( x d ) | 2 + | F E ( x d ) | 2 ]   , Γ nrad Γ rad;0 = 3 k d 3 2 x d 4 N nrad ( ε a ) > 0 ( ε a ) = 1 ( + 1 ) ( 2 + 1 ) I E ; a | D E ; a ( x d ) | 2   , Γ nrad Γ rad;0 = 3 k d 3 4 x d 2 N nrad ( ε a ) > 0 ( ε a ) = 1 ( 2 + 1 ) [ I M ; a | D M ; a ( x d ) | 2 + I E ; a | D E ; a ( x d ) | 2 ]   ,
N rad host = η d 3 ε d ε h η h 3   , N rad shell = ( η d η h ) 6 ( ε h ε d ) 2   , N nrad host = η d 3 η h 3 ε h ε d 2   , N nrad shell = 1 ε d
F p ( x d ) = { ψ ( x d ) M 21 ; p ( 1 )   , n d = 1   , T 11 ; p ( n d ) ψ ( x d ) + T 21 ; p ( n d ) ζ ( x d ) T 11 ; p ( n d ) M 22 ; p ( n d ) T 21 ; p ( n d ) M 12 ; p ( n d )   , 1 < n d N   , ψ ( x d ) + T 21 ; p ( N + 1 ) T 11 ; p ( N + 1 ) ζ ( x d )   , n d = N + 1   ,
D p ; a ( x d ) = { ψ ( x d ) M 21 ; p ( 1 )   , n d = 1   , T 11 ; p ( n d ) ψ ( x d ) + T 21 ; p ( n d ) ζ ( x d ) T 11 ; p ( n d ) M 22 ; p ( n d ) T 21 ; p ( n d ) M 12 ; p ( n d )   , 1 < n d < n a   , M 22 ; p ( n d ) ψ ( x d ) + M 12 ; p ( n d ) ζ ( x d ) T 11 ; p ( n d ) M 22 ; p ( n d ) T 21 ; p ( n d ) M 12 ; p ( n d )   , n a < n d N   , ζ ( x d )   , n d = N + 1   .
I M ; a = 1 | k a | 2 a | A M ; a ψ ( k a r ) + B M ; a ζ ( k a r ) | 2 d r   , I E ; a = ( + 1 ) | k a | 4 a | A E ; a ψ ( k a r ) + B E ; a ζ ( k a r ) | 2 d r r 2 + 1 | k a | 2 a | A E ; a ψ ( k a r ) + B E ; a ζ ( k a r ) | 2 d r   .
A p ; a = { 1   , n a = 1 and n d < N + 1   , T 11 ; p ( n a )   , 1 < n a < n d < N + 1   , M 12 ; p ( n a )   , n d < n a   , M 11 ; p ( n a ) + M 12 ; p ( n a ) T 21 ; p ( N + 1 ) T 11 ; p ( N + 1 )   , n d = N + 1   ,
B p ; a = { 0   , n a = 1   , T 21 ; p ( n a )   , 1 < n a < n d < N + 1   , M 22 ; p ( n a )   , n d < n a   , M 21 ; p ( n a ) + M 22 ; p ( n a ) T 21 ; p ( N + 1 ) T 11 ; p ( N + 1 )   , n a 1 and n d = N + 1   .
max = { x N + 4 x N 1 / 3 + 1   , 0.02 x N 8   , x N + 4.05 x N 1 / 3 + 2   , 8 < x N   ,
max = x N + 11 x N 1 / 3 + 1   .
ε n ε n , bulk + ω p 2 ω 2 + i γ D ω ω p 2 ω 2 + i γ ω   , γ = γ D + υ F L eff   , L eff = 4 3 r n 3 r n 1 3 r n 2 + r n 1 2   ,
F = γ exc γ exc;0 q ems q ems;0   , q ems = Γ rad / Γ rad ; 0 Γ rad / Γ rad ; 0 + Γ nrad / Γ rad ; 0 + ( 1 q ems;0 ) / q ems;0

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