Abstract

We propose a numerical method, based on surface integral equations (SIE), for evaluating the second harmonic (SH) scattering by metal nanoparticles (NPs) of arbitrary shape, considering both nonlocal-bulk and local-surface SH sources, induced by the electromagnetic field at the fundamental frequency. We demonstrate that the contribution of the nonlocal-bulk sources can be taken into account through equivalent surface electric and magnetic currents. We numerically solve the SIE problem by using the Galerkin method and the Rao–Wilton–Glisson basis functions in the framework of the distribution theory. The accuracy of the proposed method is verified by comparing with the SH-Mie analytical solution. As an example of a complex-shaped particle, we investigate the SH scattering by a triangular nanoprism. This method paves the way for a better understanding of the SH generation process in arbitrarily shaped NPs and can also have a high impact on the design of novel nanoplasmonic devices with enhanced SH emission.

© 2013 Optical Society of America

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    [CrossRef]
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2012

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6, 737–748 (2012).
[CrossRef]

S. Linden, F. B. P. Niesler, J. Förstner, Y. Grynko, T. Meier, and M. Wegener, “Collective effects in second-harmonic generation from split-ring-resonator arrays,” Phys. Rev. Lett. 109, 015502 (2012).
[CrossRef]

G. Gonella, W. Gan, B. Xu, and H.-L. Dai, “The effect of composition, morphology, and susceptibility on nonlinear light scattering from metallic and dielectric nanoparticles,” J. Phys. Chem. Lett. 3, 2877–2881 (2012).
[CrossRef]

C. Cirac, E. Poutrina, M. Scalora, and D. R. Smith, “Origin of second-harmonic generation enhancement in optical split-ring resonators,” Phys. Rev. B 85, 201403 (2012).
[CrossRef]

J. Xu and X. Zhang, “Second harmonic generation in three-dimensional structures based on homogeneous centrosymmetric metallic spheres,” Opt. Express 20, 1668–1684 (2012).
[CrossRef]

A. Capretti, G. F. Walsh, S. Minissale, J. Trevino, C. Forestiere, G. Miano, and L. D. Negro, “Multipolar second harmonic generation from planar arrays of au nanoparticles,” Opt. Express 20, 15797–15806 (2012).
[CrossRef]

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B 29, 2213–2221 (2012).
[CrossRef]

2011

2010

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]

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

A. Benedetti, M. Centini, C. Sibilia, and M. Bertolotti, “Engineering the second harmonic generation pattern from coupled gold nanowires,” J. Opt. Soc. Am. B 27, 408–416 (2010).
[CrossRef]

2009

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103, 257404 (2009).
[CrossRef]

M. Centini, A. Benedetti, M. Scalora, C. Sibilia, and M. Bertolotti, “Second harmonic generation from metallic 2D scatterers,” Proc. SPIE 7354, 73540F (2009).
[CrossRef]

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

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]

2008

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 7196, 757–760 (2008).

W. L. Schaich, “Second harmonic generation by periodically structured metal surfaces,” Phys. Rev. B 78, 195416 (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

B. K. Canfield, H. Husu, J. Laukkanen, B. 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]

M. Finazzi, P. Biagioni, M. Celebrano, and L. Duò, “Selection rules for second-harmonic generation in nanoparticles,” Phys. Rev. B 76, 125414 (2007).
[CrossRef]

2006

J. Nappa, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Second harmonic generation from small gold metallic particles: from the dipolar to the quadrupolar response,” J. Chem. Phys. 125, 184712 (2006).
[CrossRef]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef]

M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73, 041401 (2006).
[CrossRef]

2004

2003

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

1999

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, 4045–4048 (1999).
[CrossRef]

1997

B. Lambrecht, A. Leitner, and F. Aussenegg, “Femtosecond decay-time measurement of electron-plasma oscillation in nanolithographically designed silver particles,” Appl. Phys. B 64, 269–272 (1997).
[CrossRef]

1993

R. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag. 41, 1448–1455 (1993).
[CrossRef]

1987

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[CrossRef]

1986

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[CrossRef]

1982

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[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]

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
[CrossRef]

1968

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

1961

J. Van Bladel, “Some remarks on Green’s dyadic for infinite space,” IRE Trans. Antennas Propag. 9, 563–566 (1961).
[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]

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]

Aussenegg, F.

B. Lambrecht, A. Leitner, and F. Aussenegg, “Femtosecond decay-time measurement of electron-plasma oscillation in nanolithographically designed silver particles,” Appl. Phys. B 64, 269–272 (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, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (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]

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]

Bai, B.

B. K. Canfield, H. Husu, J. Laukkanen, B. 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]

Benedetti, A.

Benichou, E.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B 29, 2213–2221 (2012).
[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]

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

J. Nappa, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Second harmonic generation from small gold metallic particles: from the dipolar to the quadrupolar response,” J. Chem. Phys. 125, 184712 (2006).
[CrossRef]

Bertolotti, M.

Beversluis, M.

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

Biagioni, P.

M. Finazzi, P. Biagioni, M. Celebrano, and L. Duò, “Selection rules for second-harmonic generation in nanoparticles,” Phys. Rev. B 76, 125414 (2007).
[CrossRef]

Bloembergen, N.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

Bouhelier, A.

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

Bratschitsch, R.

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103, 257404 (2009).
[CrossRef]

Brevet, P. F.

J. Nappa, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Second harmonic generation from small gold metallic particles: from the dipolar to the quadrupolar response,” J. Chem. Phys. 125, 184712 (2006).
[CrossRef]

Brevet, P.-F.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B 29, 2213–2221 (2012).
[CrossRef]

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

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]

Bunton, P. H.

M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73, 041401 (2006).
[CrossRef]

Butet, J.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B 29, 2213–2221 (2012).
[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]

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

Canfield, B.

Canfield, B. K.

B. K. Canfield, H. Husu, J. Laukkanen, B. 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]

Capretti, A.

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Figures (5)

Fig. 1.
Fig. 1.

Close up of the triangular mesh near the pth edge. The union of the two triangles T1=Tp+, T2=Tp is the support of the basis function fp. For the triangle Tp+=T1 we show the curves lj1, their tangent vectors tj1 and binormal vectors mj1 with j=2, 3, 4.

Fig. 2.
Fig. 2.

SH radiated power per unit solid angle (W/sr) of a gold sphere with diameter (a)–(d) 20 nm, (e)–(h) 100 nm, and (i)–(l) 200 nm. Different combinations of SHG polarization sources have been considered; namely (γ,χ(2),χ(2))=(1,0,0) in panels (a), (e), (i); (0,1,0) in panels (b), (f), (j); (0,0,1) in panels (c), (g), (k). In panels (d), (h), (l) the sources have been weighted using Eq. (33) with (a,b,d)=(1,1,1). The sphere is excited by an x-polarized plane wave of unitary intensity propagating along the positive z axis, with wavelength 520 nm. The diagrams are in linear scale.

Fig. 3.
Fig. 3.

Relative error of dP(2ω)/dΩ calculated with the SH-SIE with respect to the SH-Mie solution for a sphere of diameter D=100nm as a function of the inclination angle θ for (a) φ=0° and (b) φ=90°. Different combinations of SHG polarization sources have been considered; namely, (γ,χ(2),χ(2))=(1,0,0) (black line); (0,1,0) (red line); (0,0,1) (blue line); (0.82+0.44j,3.271.76j,1.64+0.88j)·1020m2/V (green line).

Fig. 4.
Fig. 4.

Radiation diagram of dP(2ω)/dΩ (W/sr) for a gold sphere with diameter: (a) 20 nm, (b) 100 nm, and (c) 200 nm. The sphere is excited by an x-polarized plane wave of unitary intensity propagating along the positive direction of the z axis, with wavelength 520 nm. The radiated power is evaluated at 260 nm. The diagrams are in linear scale. The sources have been weighted using Eq. (33) with (a,b,d)=(1,1,1).

Fig. 5.
Fig. 5.

(a) and (c) Magnitude of the field Ee(ω) at the fundamental frequency on the surface of the triangular nanoprism excited by a monochromatic (λ=690nm) plane wave of unitary intensity propagating along the positive z axis and linearly polarized along (a) x and (c) y, and the corresponding radiated power per unit solid angle dP(ω)/dΩ (inset). (b) and (d) Radiation diagram of dP(2ω)/dΩ (W/sr) for both the (b) x-polarized and (d) y-polarized pump.

Equations (66)

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{Ee(ω,s)=Ee(ω)E0(ω)He(ω,s)=He(ω)H0(ω)ine{Ei(ω,s)=Ei(ω)Hi(ω,s)=Hi(ω)ini.
{×El(ω,s)=jωμ0Hl(ω,s)×Hl(ω,s)=jωεlEl(ω,s)inl,withl=i,e,{n×(Ee(ω,s)Ei(ω,s))=n×E0(ω)n×(He(ω,s)Hi(ω,s))=n×H0(ω)onS,
Pb(2ω)=ε0χb(2):Ei(ω)(Ei(ω)),
Pb(2ω)=ε0γ(Ei(ω)·Ei(ω))+ε0δ(Ei(ω)·)Ei(ω),
Pb(2ω)=Pγ(2ω)+Pδ(2ω),
PS(2ω)=ε0χS(2):Ei(ω)Ei(ω)|S,
PS(2ω)=ϵ0[χ(2)nnn+χ(2)(nt1t1+nt2t2)+χ(2)(t1nt1+t2nt2)]:Ei(ω)Ei(ω)|S,
πS(e)=j2ωPtS,
πS(m)=1ε0n×SPnS,
{×Ee(2ω)=j2ωμ0He(2ω)×He(2ω)=j2ωεeEe(2ω)ine,
{×Ei(2ω)=j2ωμ0Hi(2ω)×Hi(2ω)=j2ωεi{2ω}Ei(2ω)+j2ω(Pγ+Pδ)ini,
{n×(Ee(2ω)Ei(2ω))=πS(m)n×(He(2ω)Hi(2ω))=πS(e)onS,
{Epart=Eγ(2ω)+Eδ(2ω)Hpart=Hδ(2ω).
Eγ(2ω)=ε0γεi{2ω}(Ei(ω)·Ei(ω)).
Eδ(2ω)=j2ωμ0ViJδ(r)gi(2ω)(rr)dV1εi{2ω}[Viρδ(r)gi(2ω)(rr)dV+Sηδ(r)gi(2ω)(rr)dS],
Hδ(2ω)=×ViJδ(r)gi(2ω)(rr)dV,
Jδ=j2ωPδ,ρδ=·Pδ,ηδ=n·Pδ|S
{Ei(2ω)=E˜i(2ω)+EpartHi(2ω)=H˜i(2ω)+Hpart{Ee(2ω)=E˜e(2ω)He(2ω)=H˜e(2ω),
{×E˜l(2ω)=j2ωμ0H˜l(2ω)×H˜l(2ω)=j2ωεl{2ω}E˜l(2ω)inlwithl=i,e,
{n×(E˜e(2ω)E˜i(2ω))=πS(m)πγ(m)πδ(m)n×(H˜e(2ω)H˜i(2ω))=πS(e)+πδ(e)onS,
πγ(m)=1ε0n×SPnγ,πδ(m)=n×Eδ(2ω)|S,πδ(e)=n×Hδ(2ω)|S,
Pnγ=ε02γεi{2ω}(Ei(ω)·Ei(ω))|S.
{×El(Ω,s)=jΩμ0Hl(Ω,s)×Hl(Ω,s)=jΩεl{Ω}El(Ω,s)inlwithl=i,e,{n×(Ee(Ω,s)Ei(Ω,s))=π0(Ω,m)n×(He(Ω,s)Hi(Ω,s))=+π0(Ω,e)onS,
{π0(ω,e)=n×H0(ω),π0(ω,m)=n×E0(ω).
{π0(2ω,e)=πS(e)+πδ(e),π0(2ω,m)=πS(m)+πδ(m)+πγ(m).
Ee(Ω){je(Ω,e),je(Ω,m)}(r)={0ifri,Ee(Ω,s)(r)ifre,He(Ω){je(Ω,e),je(Ω,m)}(r)={0ifri,He(Ω,s)(r)ifre,
{je(Ω,e)=+n×He(Ω,s)|Se,je(Ω,m)=n×Ee(Ω,s)|Se,
Ei(Ω){ji(Ω,e),ji(Ω,m)}(r)={Ei(Ω,s)(r)ifri0ifre,Hi(Ω){ji(Ω,e),ji(Ω,m)}(r)={Hi(Ω,s)(r)ifri0ifre,
{ji(Ω,e)=n×Hi(Ω,s)|Siji(Ω,m)=+n×Ei(Ω,s)|Si.
{ji(Ω,e)+je(Ω,e)=π0(Ω,e),ji(Ω,m)+je(Ω,m)=π0(Ω,m).
Ee(Ω,t){je(Ω,e),je(Ω,m)}n×je(Ω,m)=0,
He(Ω,t){je(Ω,e),je(Ω,m)}+n×je(Ω,e)=0,
Ei(Ω,t){ji(Ω,e),ji(Ω,m)}+n×ji(Ω,m)=0,
Hi(Ω,t){ji(Ω,e),ji(Ω,m)}n×ji(Ω,e)=0,
El(Ω,t){j(e),j(m)}=n×n×El(Ω){j(e),j(m)}|Sl,Hl(Ω,t){j(e),j(m)}=n×n×Hl(Ω){j(e),j(m)}|Sl.
C(Ω)x(Ω)=y(Ω),
C(Ω)=|ζeTe(Ω,t)+Ke(Ω,t)ζiTi(Ω,t)Ki(Ω,t)Ke(Ω,t)ζe1Te(Ω,t)+Ki(Ω,t)ζi1Ti(Ω,t)I0I00I0I|,
x(Ω)=|je(Ω,e)je(Ω,m)ji(Ω,e)ji(Ω,m)|,y(Ω)=|12n×π0(Ω,m)12n×π0(Ω,e)π0(Ω,e)π0(Ω,m)|,
fp(r)={fp+(r)=+lp2Ap+(rvp+)rTp+fp(r)=lp2Ap(rvp)rTp0otherwise,
Ei(ω)(r)|Si=Ei,t(ω)(r)|Si+Ei,n(ω)(r)|Sin,
Ei,t(ω)(r)|Si=n×ji(ω,m)(r),Ei,n(ω)(r)|Si=jS·ji(ω,e)(r)ωεi{ω}.
SPnS={j=2,3,4(PnS|Tj(r)PnS|T1(r))mj1δj1+j=5,6(PnS|Tj(r)PnS|T2(r))mj2δj2ifrT1T2,SPnS|T1(r)ifr1,SPnS|T2(r)ifr2,
dP(2ω)dΩ(k)=limr+[r22ζeEe(2ω)(k)2].
[χ(2),χ(2),γ]=[a4,b2,d8][ϵi(ω)ϵ01]emω2,
ξ=|dP(2ω)dΩdPMIE(2ω)dΩ|/|dP(2ω)dΩ|,
E(Ω)(r)=El(Ω){j(Ω,e),j(Ω,m)}(r),
H(Ω)(r)=Hl(Ω){j(Ω,e),j(Ω,m)}(r),
El(Ω){j(Ω,e),j(Ω,m)}=ζl{Ω}Tl(Ω){j(Ω,e)}+Kl(Ω){j(Ω,m)}+{0ifrS,+ζl{Ω}2jkl{Ω}[S·j(Ω,e)]n12n×j(Ω,m)ifrSi,ζl{Ω}2jkl{Ω}[S·j(Ω,e)]n+12n×j(Ω,m)ifrSe,Hl(Ω){j(Ω,e),j(Ω,m)}=Kl(Ω){j(Ω,e)}+Tl(Ω){j(Ω,m)}ζl{Ω}+{0ifrS,+12n×j(Ω,e)(r)+[S·j(Ω,m)(r)]2jζl{Ω}kl{Ω}nifrSi,12n×j(Ω,e)(r)[S·j(Ω,m)(r)]2jζl{Ω}kl{Ω}nifrSe.
Kl(Ω){w}(r)=Sw(r)×gl(Ω)(rr)dS,
Tl(Ω){w}(r)=jkl{Ω}Sgl(Ω)(rr)w(r)dS1jkl{Ω}Sgl(Ω)(rr)S·w(r)dS,
Kl(Ω,t){·}=n×n×Kl(Ω){·}|Sl,
Tl(Ω,t){·}=n×n×Tl(Ω){·}|Sl.
fp,πS(m)=Σfp·πS(m)dS.
fp,πS(m)=I1+I2,
I1=1ε0T1n×SPnS|T1·fp+dS+1ε0T2n×SPnS|T2·fpdS,
I2=1ε0j=2,3,4Σ(PnS|TjPnS|T1)fp·n×mj1δj1dS+1ε0j=5,6Σ(PnS|TjPnS|T2)fp·n×mj2δj2dS.
I1=1ε0j=2,3,4lj1PnS|T1fp+·tj1dl+1ε0j=1,5,6lj2PnS|T2fp·tj2dl.
Fδlji=F|Tj+F|Ti2δlji.
fp·n×mj1δj1={12fp+·n×mj1δj1j=3,4,12(fp++fp)·n×mj1δj1j=2,fp·n×mj2δj2=12fp·n×mj2δj2j=5,6,
I2=12ε0j=2,3,4lj1(PnS|TjPnS|T1)fp+·tj1dl+12ε0j=1,5,6lj2(PnS|TjPnS|T2)fp·tj2dl.
fp,πS(m)=12ε0j=2,3,4lj1(PnS|Tj+PnS|T1)fp+·tj1dl+12ε0j=1,5,6lj2(PnS|Tj+PnS|T2)fp·tj2dl.
fp,n×πS(m)=lpε0[1A1T1PnS|T1dS1A2T2PnS|T2dS].
fp,πδ(m)=r=±{j2ωμ0ViJδ(r)·Tprn×fpr(r)gi(2ω)(rr)dSdV+1εi{2ω}Viρδ(r)Tprgi(2ω)(rr)fpr(r)·dldV+1εi{2ω}Sηδ(r)Tprgi(2ω)(rr)fpr(r)·dldS},
fp,n×πδ(m)=r=±{j2ωμ0ViJδ(r)·Tprfpr(r)gi(2ω)(rr)dSdV+lpεi{2ω}rAprViρδ(r)Tprgi(2ω)(rr)dSdV+lpεi{2ω}rAprSηδ(r)Tprgi(2ω)(rr)dSdS}.
fp,πδ(e)=r=±ViJδ(r)·Tpr(n×fpr)×gi(2ω)(rr)dSdV,
fp,n×πδ(e)=r=±ViJδ(r)·Tprfpr×gi(2ω)(rr)dSdV.

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