Abstract

In this work, Mie theory extended to the specific case of the optical second harmonic generation (SHG) from metallic nanoshells is described. Our model results from a combination of the Mie theory developed for the linear optical response of concentric nanospheres and the Mie theory developed for the SHG from nanospheres. This approach leads to a multipolar expansion of the second harmonic scattered electric fields. The total scattered intensity and the relative contribution of each multipole to the scattered wave are directly calculated within this framework. Our model is then applied to the calculation of the second harmonic cross section for nanoshells made of the most common metals used in plasmonics, namely gold and silver. Finally, the effect of the aspect ratio, i.e., the ratio between the inner and the outer radii of the metallic nanoshell, a parameter that is known to greatly impact the surface plasmon resonance properties of the system, is discussed notably in terms of the tunability of the optical SHG from metallic nanoshells.

© 2012 Optical Society of America

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2012

2011

R. Kullock, A. Hille, A. Haußmann, S. Grafström, and L. M. Eng, “SHG simulations of plasmonic nanoparticles using curved elements,” Opt. Express 19, 14426–14436 (2011).
[CrossRef]

A. Benedetti, M. Centini, M. Bertolotti, and C. Sibilia, “Second harmonic generation from 3D nanoantennas: on the surface and bulk contributions by far-field pattern analysis,” Opt. Express 19, 26752–26767 (2011).
[CrossRef]

R. Czaplicki, M. Zdanowicz, K. Koskinen, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Dipole limit in second-harmonic generation from arrays of gold nanoparticles,” Opt. Express 19, 26866–26871 (2011).
[CrossRef]

G. Gonella, and H.-L. Dai, “Determination of adsorption geometry on spherical particles from nonlinear Mie theory analysis of surface second harmonic generation,” Phys. Rev. B 84, 121402(R) (2011).
[CrossRef]

S. Wunderlich, B. Schürer, C. Sauerbeck, W. Peukert, and U. Peschel, “Molecular Mie model for second harmonic generation and sum frequency generation,” Phys. Rev. B 84, 235403 (2011).
[CrossRef]

Y. Zhang, N. K. Grady, C. Ayala-Orozco, and N. J. Halas, “Three-dimensional nanostructures as highly efficient generators of second harmonic light,” Nano Lett. 11, 5519–5523 (2011).
[CrossRef]

Z. J. Li, S. Y. Gao, and D. Han, “Tuning the mapping of second-harmonic generation in silver nanoshell,” Eur. Phys. J. Appl. Phys. 56, 10404 (2011).
[CrossRef]

S. Zaiba, F. Lerouge, A.-M. Gabudean, M. Focsan, J. Lerme, T. Gallavardin, O. Maury, C. Andraud, S. Parola, and P. Baldeck, “Transparent plasmonic nanocontainers protect organic fluorophores against photobleaching,” Nano Lett. 11, 2043–2047 (2011).
[CrossRef]

T. Utikal, T. Zentgraf, T. Paul, C. Rockstuhl, F. Lederer, M. Lippitz, and H. Giessen, “Toward the origin of the nonlinear response in hybrid plasmonic systems,” Phys. Rev. Lett. 106, 133901 (2011).
[CrossRef]

2010

N. K. Grady, M. W. Knight, R. Bardhan, and N. J. Halas, “Optically-driven collapse of a plasmonic nanogap self-monitored by optical frequency mixing,” Nano Lett. 10, 1522–1528 (2010).
[CrossRef]

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P.-F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett. 10, 1717–1721 (2010).
[CrossRef]

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interferences between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (2010).
[CrossRef]

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, 207402 (2010).
[CrossRef]

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P. F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[CrossRef]

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from Chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104, 127401 (2010).
[CrossRef]

B. Schürer, S. Wunderlich, C. Sauerbeck, U. Peschel, and W. Peukert, “Probing colloidal interfaces by angle-resolved second harmonic light scattering,” Phys. Rev. B 82, 241404 (2010).
[CrossRef]

J. Butet, G. Bachelier, J. Duboisset, F. Bertorelle, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Three-dimensionnal mapping of single gold nanaoparticles embedded in a homogeneous transparent matrix using optical second-harmonic generation,” Opt. Express 18, 22314–22323 (2010).
[CrossRef]

2009

A. G. F. de Beer, and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79, 155420 (2009).
[CrossRef]

Y. Zeng, W. Hoyer, J. J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

V. K. Valev, N. Smisdom, A. V. Silhanek, B. De Clercq, W. Gillijns, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Plasmonic ratchet wheels: Switching circular dichroism by arranging chiral nanostructures,” Nano Lett. 9, 3945–3948 (2009).
[CrossRef]

F. X. Wang, F. J. Rodriguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

2008

F. Le, D. W. Brandl, Y. A. Urzhumov, H. Wang, J. Kundu, N. J. Halas, J. Aizpurua, and P. Nordlander, “Metallic nanoparticle arrays: A common substrate for both surface-enhanced Raman scattering and surface-enhanced infrared absorption,” ACS Nano 2, 707–718 (2008).
[CrossRef]

G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B 25, 955–960 (2008).
[CrossRef]

2007

M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. 98, 026104 (2007).
[CrossRef]

B. K. Canfield, H. Husu, J. Laukkanen, B. F. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
[CrossRef]

2005

C. Loo, A. Lowery, N. J. Halas, J. West, and R. Drezek, “Immunotargeted nanoshells for integrated cancer imaging and therapy,” Nano Lett. 5, 709–711 (2005).
[CrossRef]

2004

D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett. 209, 171–176 (2004).
[CrossRef]

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, 1853–1857 (2004).
[CrossRef]

E. Prodan and P. Nordlander, “Plasmon hybridization in spherical nanoparticles,” J. Chem. Phys. 120, 5444–5454 (2004).
[CrossRef]

C. L. Nehl, N. K. Grady, G. P. Goodrich, F. Tam, N. J. Halas, and J. Hafner, “Scattering spectra of single gold nanoshells,” Nano Lett. 4, 2355–2359 (2004).
[CrossRef]

T. V. Teperik, V. V. Popov, and F. J. García de Abajo, “Radiative decay of plasmons in a metallic nanoshell,” Phys. Rev. B 69, 155402 (2004).
[CrossRef]

Y. Pavlyukh, and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70, 245434 (2004).
[CrossRef]

J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21, 1328–1347 (2004).
[CrossRef]

2003

A. Bouhelier, M. Beversluis, A. Hartschuch, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef]

E. Prodan and P. Nordlander, “Structural tunability of the plasmon resonances in metallic nanoshells,” Nano Lett. 3, 543–547 (2003).
[CrossRef]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef]

2002

C. Sönnischen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402(2002).
[CrossRef]

1999

S. J. Oldenburg, S. L. Westcott, R. D. Averitt, and N. J. Halas, “Surface enhanced Raman scattering in the near infrared using metal nanoshell substrates,” J. Chem. Phys. 111, 4729–4735 (1999).
[CrossRef]

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045 (1999).
[CrossRef]

1997

R. D. Averitt, D. Sarkar, and N. J. Halas, “Plasmon resonance of Au-coated Au2S nanoshells: Insight into multicomponent nanoparticle growth,” Phys. Rev. Lett. 78, 4217–4220 (1997).
[CrossRef]

1993

D. Östling, P. Stampfli, and K. H. Benneman, “Theory of nonlinear optical properties of small metallic spheres,” Z. Phys. D 28, 169–175 (1993).
[CrossRef]

1980

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1971

J. Rudnick and E. A. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
[CrossRef]

1951

A. L. Aden, and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Aden, A. L.

A. L. Aden, and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Ahorinta, R.

F. X. Wang, F. J. Rodriguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

Aizpurua, J.

F. Le, D. W. Brandl, Y. A. Urzhumov, H. Wang, J. Kundu, N. J. Halas, J. Aizpurua, and P. Nordlander, “Metallic nanoparticle arrays: A common substrate for both surface-enhanced Raman scattering and surface-enhanced infrared absorption,” ACS Nano 2, 707–718 (2008).
[CrossRef]

Aktsipetrov, O. A.

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from Chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104, 127401 (2010).
[CrossRef]

Albers, W. M.

F. X. Wang, F. J. Rodriguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

Ameloot, M.

V. K. Valev, N. Smisdom, A. V. Silhanek, B. De Clercq, W. Gillijns, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “Plasmonic ratchet wheels: Switching circular dichroism by arranging chiral nanostructures,” Nano Lett. 9, 3945–3948 (2009).
[CrossRef]

Andraud, C.

S. Zaiba, F. Lerouge, A.-M. Gabudean, M. Focsan, J. Lerme, T. Gallavardin, O. Maury, C. Andraud, S. Parola, and P. Baldeck, “Transparent plasmonic nanocontainers protect organic fluorophores against photobleaching,” Nano Lett. 11, 2043–2047 (2011).
[CrossRef]

Averitt, R. D.

S. J. Oldenburg, S. L. Westcott, R. D. Averitt, and N. J. Halas, “Surface enhanced Raman scattering in the near infrared using metal nanoshell substrates,” J. Chem. Phys. 111, 4729–4735 (1999).
[CrossRef]

R. D. Averitt, D. Sarkar, and N. J. Halas, “Plasmon resonance of Au-coated Au2S nanoshells: Insight into multicomponent nanoparticle growth,” Phys. Rev. Lett. 78, 4217–4220 (1997).
[CrossRef]

Ayala-Orozco, C.

Y. Zhang, N. K. Grady, C. Ayala-Orozco, and N. J. Halas, “Three-dimensional nanostructures as highly efficient generators of second harmonic light,” Nano Lett. 11, 5519–5523 (2011).
[CrossRef]

Bachelier, G.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interferences between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (2010).
[CrossRef]

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P.-F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett. 10, 1717–1721 (2010).
[CrossRef]

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J. Butet, G. Bachelier, I. Russier-Antoine, F. Bertorelle, A. Mosset, N. Lascoux, C. Jonin, E. Benichou, and P.-F. Brevet, “Nonlinear fano-profiles in the optical second-harmonic generation from silver nanoparticles,” (in press).

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[CrossRef]

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[CrossRef]

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F. Le, D. W. Brandl, Y. A. Urzhumov, H. Wang, J. Kundu, N. J. Halas, J. Aizpurua, and P. Nordlander, “Metallic nanoparticle arrays: A common substrate for both surface-enhanced Raman scattering and surface-enhanced infrared absorption,” ACS Nano 2, 707–718 (2008).
[CrossRef]

West, J.

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C. Sönnischen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402(2002).
[CrossRef]

Wilson, O.

C. Sönnischen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402(2002).
[CrossRef]

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ACS Nano

F. Le, D. W. Brandl, Y. A. Urzhumov, H. Wang, J. Kundu, N. J. Halas, J. Aizpurua, and P. Nordlander, “Metallic nanoparticle arrays: A common substrate for both surface-enhanced Raman scattering and surface-enhanced infrared absorption,” ACS Nano 2, 707–718 (2008).
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Eur. Phys. J. Appl. Phys.

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J. Opt. Soc. Am. B

Nano Lett.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Sensing with multipolar second harmonic generation from spherical metallic nanoparticle,” Nano Lett. 12, 1697–1701 (2012).
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C. Loo, A. Lowery, N. J. Halas, J. West, and R. Drezek, “Immunotargeted nanoshells for integrated cancer imaging and therapy,” Nano Lett. 5, 709–711 (2005).
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S. Zaiba, F. Lerouge, A.-M. Gabudean, M. Focsan, J. Lerme, T. Gallavardin, O. Maury, C. Andraud, S. Parola, and P. Baldeck, “Transparent plasmonic nanocontainers protect organic fluorophores against photobleaching,” Nano Lett. 11, 2043–2047 (2011).
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Y. Zhang, N. K. Grady, C. Ayala-Orozco, and N. J. Halas, “Three-dimensional nanostructures as highly efficient generators of second harmonic light,” Nano Lett. 11, 5519–5523 (2011).
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B. K. Canfield, H. Husu, J. Laukkanen, B. F. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7, 1251–1255 (2007).
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[CrossRef]

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[CrossRef]

F. X. Wang, F. J. Rodriguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
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[CrossRef]

S. Wunderlich, B. Schürer, C. Sauerbeck, W. Peukert, and U. Peschel, “Molecular Mie model for second harmonic generation and sum frequency generation,” Phys. Rev. B 84, 235403 (2011).
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Figures (5)

Fig. 1.
Fig. 1.

(a) Schema of the coordinate system. (b) Schematic representation of the nanoshell structure. The core, shell, and surrounding media are denoted by 1, 2, and 3, respectively.

Fig. 2.
Fig. 2.

(a) Extinction cross section as a function of the wavelength for a silver nanoshell (a=20nm, b=30nm). The dipolar and quadrupolar resonances are labeled 1 and 2, respectively; (b) Total SH scattering cross section calculated for the same silver nanoshell and for the dipolar (dashed line) and quadrupolar (dotted line) emission modes. Resonances are labeled from 1 to 4 (see main text for details).

Fig. 3.
Fig. 3.

Total SH scattering cross section calculated for silver nanoshells with the same shell thickness (10 nm) but for increasing core dimensions. The evolution of resonance positions is shown by the arrows.

Fig. 4.
Fig. 4.

(a) Extinction cross section as a function of the wavelength for a gold nanoshell (a=30nm, b=40nm). Only the dipolar resonance is visible because the amplitude of the quadrupolar one is too small. (b) Total SH scattering cross section calculated for the same gold nanoshell and for the dipolar (dashed line) and quadrupolar (dotted line) emission modes. Resonances are labeled from 1 to 4 (see main text for details).

Fig. 5.
Fig. 5.

Total SH scattering cross section calculated for gold nanoshells with the same shell thickness (10 nm) but for increasing core dimensions. The evolution of resonance positions is shown by the arrows.

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

Einc(ω)=lm=±1Al,mE,inc(ω)k3×jl(k3r)Xl,m+Al,mM,inc(ω)jl(k3r)Xl,m,
Al,mE,inc(ω)=12il4π(2l+1)δm,±1,
Al,mM,inc(ω)=m2il4π(2l+1)δm,±1.
Esca(ω)=lm=±1Al,mE,sca(ω)k3×hl(k3r)Xl,m+Al,mM,sca(ω)hl(k3r)Xl,m,
Ecore(ω)=lm=±1Al,mE,core(ω)k1×jl(k1r)Xl,m+Al,mM,core(ω)jl(k1r)Xl,m,
Eshell(ω)=lm=±1Al,mE,shell(ω)k2×jl(k2r)Xl,m+Al,mM,shell(ω)jl(k2r)Xl,m+Bl,mE,shell(ω)k2×yl(k2r)Xl,m+Bl,mM,shell(ω)yl(k2r)Xl,m,
(Eshell(ω)Ecore(ω))×rc-s=0,
(Hshell(ω)Hcore(ω))×rc-s=0,
(Esca(ω)+Einc(ω)Eshell(ω))×rs-sm=0,
(Hsca(ω)+Hinc(ω)Hshell(ω))×rs-sm=0,
Al,mE,sca(ω)Al,mE,inc(ω)=ε3jl(k3r)(r[rjl(k2r)]+Dlr[ryl(k2r)])ε2r[rjl(k3r)](jl(k2r)+Dlyl(k2r))ε3hl(k3r)(r[rjl(k2r)]+Dlr[ryl(k2r)])ε2r[rhl(k3r)](jl(k2r)+Dlyl(k2r))|r=b,
Al,mM,sca(ω)Al,mM,inc(ω)=jl(k3)(r[rjl(k2r)]+Elr[ryl(k2r)])r[rjl(k3r)](jl(k2r)+Elyl(k2r))hl(k3)(r[rjl(k2r)]+Elr[ryl(k2r)])r[rhl(k3r)](jl(k2r)+Elyl(k2r))|r=b,
Dl=ε2jl(k2)r[rjl(k1r)]ε1r[rjl(k2r)]jl(k1r)ε1r[ryl(k2r)]jl(k1r)ε2yl(k2r)r[rjl(k1r)]|r=a,
El=jl(k1r)r[rjl(k2r)]r[rjl(k1r)]jl(k2r)r[rjl(k1r)]jl(k2r)jl(k1r)r[ryl(k2r)]|r=a.
Al,mE,shell(ω)=(ε3hl(k3k)r[ryl(k2r)]+ε2yl(k2r)r[rhl(k3r)])n3n2(r[ryl(k2r)]jl(k2r)+yl(k2r)r[rjl(k2r)])Al,mE,sca(ω)|r=b+ε3jl(k3r)r[ryl(k2r)]ε2yl(k2r)r[rjl(k3r)]n3n2(r[ryl(k2r)]jl(k2r)+yl(k2r)r[rjl(k2r)])|r=b,
Bl,mE,shell(ω)=n2r[rjl(k3r)]n2r[rhl(k3r)]Al,mE,sca(ω)n3r[rjl(k2r)]Al,mE,shell(ω)n3r[ryl(k2r)]|r=b.
Psurf,c-s(2ω)=χc-sE(a+,ω)E(a+,ω)δ(ra),
Psurf,s-sm(2ω)=χs-smE(b,ω)E(b,ω)δ(rb+),
Psurf,c-s(2ω)=l=0m=llCl,mc-sYl,m(θ,ϕ),
Psurf,s-sm(2ω)=l=0m=llCl,ms-smYl,m(θ,ϕ),
Cl,mc-s=χl1,m1=±1l2,m2=±1l1(l1+1)l2(l2+1)Al1,mE,shelljl1(k2a)+Bl1,mE,shellyl1(k2a)k2a×Al2,mE,shelljl2(k2a)+Bl2,mE,shellyl2(k2a)k2adΩYl,m*Yl1,m1Yl2,m2,
Cl,ms-sm=χl1,m1=±1l2,m2=±1l1(l1+1)l2(l2+1)Al1,mE,shelljl1(k2b)+Bl1,mE,shellyl1(k2b)k2b×Al2,mE,shelljl2(k2b)+Bl2,mE,shellyl2(k2b)k2bdΩYl,m*Yl1,m1Yl2,m2.
dΩYl,m*Yl1,m1Yl2,m2=(2l+1)(2l1+1)(2l2+1)4π(ll1l2mm1m2)(ll1l2000).
ll1+l2,
l+l1+l2even,
m=m1+m2.
Esca(2ω)=lm=0,±2Al,mSH,sca(2ω)K3×hl(K3r)Xl,m,
Ecore(2ω)=lm=0,±2Al,mSH,core(2ω)K1×jl(K1r)Xl,m,
Eshell(2ω)=lm=0,±2Al,mSH,shell(2ω)K2×jl(K2r)Xl,m+Bl,mSH,shell(2ω)K2×yl(K2r)Xl,m,
Eshell,(2ω)Ecore,(2ω)=4πε1(2ω)Psurf,c-s(2ω),
Hshell,(2ω)Hcore,(2ω)=0,
Esca,(2ω)Eshell,(2ω)=4πε3(2ω)Psurf,s-sm(2ω),
Hsca,(2ω)Hshell,(2ω)=0,
ε1(2ω)jl(K1r)Al,mSH,core(2ω)K1r=ε2(2ω)jl(K2r)Al,mSH,shell(2ω)+yl(K2r)Bl,mSH,shell(2ω)K2r|r=a,
ε2(2ω)jl(K2r)Al,mSH,shell(2ω)+yl(K2r)Bl,mSH,shell(2ω)K2r=ε3(2ω)hl(K3r)Al,mSH,sca(2ω)K3r|r=b,
4πil(l+1)Cl,mc-sε1(2ω)=Al,mSH,core(2ω)r[rjl(K1r)]K1+Al,mSH,shell(2ω)r[rjl(K2r)]+Bl,mDH,shell(2ω)r[ryl(K2r)]K2|r=a,
4πil(l+1)Cl,ms-smε3(2ω)=Al,mSH,sca(2ω)r[rjl(K3r)]K3Al,mSH,shell(2ω)r[rjl(K2r)]+Bl,mSH,shell(2ω)r[ryl(K2r)]K2|r=b.
Al,mSH,sca(2ω)=4πil(l+1)T6T3+T5T4(T6Cl,ms-smε3(2ω)T4Cl,mc-sε1(2ω))
T1=ε2(2ω)jl(K2r)r[rjl(K1r)]ε1(2ω)jl(K1r)r[rjl(K2r)]ε1(2ω)K2(2ω)jl(K1r)|r=a,
T2=ε2(2ω)yl(K2r)r[rjl(K1r)]ε1(2ω)jl(K1r)r[ryl(K2r)]ε1(2ω)K2jl(K1r)|r=a,
T3=ε2(2ω)jl(K2r)r[rjl(K3r)]ε3(2ω)hl(K3r)r[rjl(K2r)]ε2(2ω)jl(K2r)K3|r=b,
T4=jl(K2r)r[ryl(K2r)]yl(K2r)r[rjl(K2r)]K2|r=b,
T5=n2ε3(2ω)hl(K3b)n3ε2(2ω)jl(K2b)T1,
T6=yl(K2b)T1+jl(K2b)T2.
Al,mSH,sca(2ω)=4πil(l+1)T3(Cl,ms-smε3(2ω)).
χ=aRS4[εr(ω)1]eε0mω2,
CscaSH(2ω)=c8πK32(2ω)l,m|Al,mSH,sca(2ω)|2.

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