Abstract

The electromagnetic theory of optical second-harmonic generation from small spherical particles comprised of centrosymmetric material is presented. The interfacial region where the inversion symmetry is broken provides a source of the nonlinearity. This response is described by a general surface nonlinear susceptibility tensor for an isotropic interface. In addition, the appropriate weak bulk terms for an isotropic centrosymmetric medium are introduced. The linear optical response of the sphere and the surrounding region is assumed to be isotropic, but otherwise arbitrary. The analysis is carried out to leading order in the ratio of (a/λ), the particle radius to the wavelength of the incident light, and can be considered as the Rayleigh limit for second-harmonic generation from a sphere. Emission from the sphere arises from both induced electric dipole and electric quadrupole moments at the second-harmonic frequency. The former requires a nonlocal excitation mechanism in which the phase variation of the pump beam across the sphere is considered, while the latter is present for a local-excitation mechanism. The locally excited electric dipole term, analogous to the source for linear Rayleigh scattering, is absent for the nonlinear case because of the overall inversion symmetry of the problem. The second-harmonic field is found to scale as (a/λ)3 and to be completely determined by two effective nonlinear susceptibility coefficients formed as a prescribed combination of the surface and bulk nonlinearities. Characteristic angular and polarization selection rules resulting from the mechanism of the radiation process are presented. Various experimental aspects of the problem are examined, including the expected signal strengths and methods of determining the nonlinear susceptibilities. The spectral characteristics associated with the geometry of a small sphere are also discussed, and distinctive localized plasmon resonances are identified.

© 2004 Optical Society of America

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  106. For a sphere of arbitrary radius, an exact expression for the SH electric field for the homogeneous media can be obtained without employing the small-particle approximation. This limit (of neglecting differences in the dielectric functions) is known as the SH Rayleigh–Gans approximation, first applied by Martorell et al.78 for the case of a single nonlinear susceptibility element, χs, ⊥⊥⊥(2), and dispersionless media (K1=2k1). For the case of an isotropic surface having all three nonlinear susceptibility elements χs, ⊥⊥⊥(2), χs, ⊥ ∥ ∥(2), and χs, ∥⊥∥(2), we substitute Eq. (11) into Eq. (4) with K1=2k1. Making use of Eqs. (5a) and (5b), we obtain A(r). The resulting SH field in the radiation zone is E(2ω)(r)RG=[4πi exp(iK1r)/r]× (Ka)2(E0(ω))2 [Θ(θ, ε)θ⁁+ Φ(θ, ε)ε⁁], where Θ(θ, ε) and Φ(θ, ε) are functions given by Θ(θ, ε)=cos(θ/2) {[Γ1(θ)+ Γ2(θ)cos2(θ/2)]f(ε)+ Γ3(θ)(ε⁁0⋅ε⁁0)} and Φ(θ, ε)= −cos(θ/2)Γ1(θ)g(ε). Here, f(ε)=(ε⁁0⋅ρ⁁)2 and g(ε)=(ε⁁0⋅ρ⁁)(ε⁁0⋅ε⁁), with ρ⁁=cos εx⁁+sin εy⁁ and ε⁁= −sin εx⁁+cos εy⁁; Γ1(θ)= 2[(χs, ⊥⊥⊥(2)−χs, ⊥ ∥ ∥(2))F1(θ)− 2χs, ∥⊥∥(2)F2(θ)], Γ2(θ)= − (χs, ⊥⊥⊥(2)−χs, ⊥ ∥ ∥(2)) [F1(θ)− 2F2(θ)] −2χs, ∥⊥∥(2) [3F1(θ)− 2F2(θ)], and Γ3(θ)= −(χs, ⊥⊥⊥(2)+γ) F1(θ)−(χs, ⊥ ∥ ∥(2)+ γ)[F1(θ)−2F2(θ)]+ 2χs, ∥⊥∥(2)F1(θ), with the structure factors for δ= 2Ka sin(θ/2) of F1(θ)=(3/δ3) [(1− δ2/3)sin δ− δ cos δ] and F2(θ)=(3/δ3)[(1− δ2/2)sin δ−δ(1− δ2/6)cos δ]. (A similar treatment for SFG recently appeared in Ref. 48.) One should note that for the case of heterogeneous media, obtaining the SH field requires a full evaluation of Maxwell’s equation, as discussed and outlined in the Appendix A of this paper.
  107. C. T. Tai, “Equivalent layers of surface-charge, current sheet, and polarization in the eigenfunction-expansions of Green’s functions in electromagnetic theory,” IEEE Trans. Antennas Propag. 29, 733–739 (1981).
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    [CrossRef]
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    [CrossRef]
  111. P. D. Maker, “Spectral broadening of elastic second-harmonic light scattering in liquids,” Phys. Rev. A 1, 923–951 (1970).
    [CrossRef]
  112. V. Mizrahi and J. E. Sipe, “Phenomenological treatment of surface second-harmonic generation,” J. Opt. Soc. Am. B 5, 660–667 (1988).
    [CrossRef]
  113. K. Clays and A. Persoons, “Hyper-Rayleigh scattering in solution,” Phys. Rev. Lett. 66, 2980–2983 (1991).
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2004 (1)

J. Nappa, G. Revillod, J.-P. Abid, I. Russier-Antoine, C. Jonin, E. Benichou, H. H. Girault, and P. F. Brevet, “Hyper-Rayleigh scattering of gold nanorods and their relationship with linear assemblies of gold nanospheres,” Faraday Discuss. 125, 145–156 (2004).
[CrossRef] [PubMed]

2003 (9)

S. Roke, W. G. Roeterdink, J. E. G. J. Wijnhoven, A. V. Petukhov, A. W. Kleyn, and M. Bonn, “Vibrational sum frequency scattering from a submicron suspension,” Phys. Rev. Lett. 91, 258302 (2003).
[CrossRef]

N. Thantu, R. S. Schley, and B. S. Justus, “Tunable room temperature second harmonic generation in glasses doped with CuCl nanocrystalline quantum dots,” Opt. Commun. 220, 203–210 (2003).
[CrossRef]

T. S. Koffas, J. Kim, C. C. Lawrence, and G. A. Somorjai, “Detection of immobilized protein on latex microspheres by IR-visible sum frequency generation and scanning force microscopy,” Langmuir 19, 3563–3566 (2003).
[CrossRef]

Y. Zhang, M. Ma, X. Wang, D. G. Fu, H. Q. Zhang, N. Gu, J. Z. Liu, Z. H. Lu, L. Xu, and K. J. Chen, “Second-order optical nonlinearity of surface-capped CdS nanoparticles and effect of surface modification,” J. Phys. Chem. Solids 64, 927–931 (2003).
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W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. U.S.A. 100, 7075–7080 (2003).
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W. L. Mochán, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second harmonic generation in arrays of spherical particles,” Phys. Rev. B 68, 085318 (2003).
[CrossRef]

V. L. Brudny, W. L. Mochán, J. A. Maytorena, and B. S. Mendoza, “Second harmonic generation from a collection of nanoparticles,” Phys. Status Solidi B 240, 518–526 (2003).
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E. V. Makeev and S. E. Skipetrov, “Second harmonic generation in suspensions of spherical particles,” Opt. Commun. 224, 139–147 (2003).
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A. Podlipensky, J. Lange, G. Seifert, H. Graener, and I. Cravetchi, “Second-harmonic generation from ellipsoidal silver nanoparticles embedded in silica glass,” Opt. Lett. 28, 716–718 (2003).
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2002 (13)

E. C. Hao, G. C. Schatz, R. C. Johnson, and J. T. Hupp, “Hyper-Rayleigh scattering from silver nanoparticles,” J. Chem. Phys. 117, 5963–5966 (2002).
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C. Landes, M. Braun, and M. A. El-Sayed, “The effect of surface adsorption on the hyper-Rayleigh scattering of large and small CdSe nanoparticles,” Chem. Phys. Lett. 363, 465–470 (2002).
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M. R. V. Sahyun, “Hyper-Rayleigh scattering (HRS) spectroscopy applied to nanoparticulate TiO2,” Spectrochim. Acta, Part A 58, 3149–3157 (2002).
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Y. Zhang, M. Ma, X. Wang, D. Fu, N. Gu, J. Liu, Z. Lu, Y. Ma, L. Xu, and K. Chen, “First-order hyperpolarizability of ZnS nanocrystal quantum dots studied by hyper-Rayleigh scattering,” J. Phys. Chem. Solids 63, 2115–2118 (2002).
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D. V. Petrov, B. S. Santos, G. A. L. Pereira, and C. de Mello Donegá, “Size and band-gap dependences of the first hyperpolarizability of CdxZn1−xS nanocrystals,” J. Phys. Chem. B 106, 5325–5334 (2002).
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R. C. Johnson, J. T. Li, J. T. Hupp, and G. C. Schatz, “Hyper-Rayleigh scattering studies of silver, copper, and platinum nanoparticle suspensions,” Chem. Phys. Lett. 356, 534–540 (2002).
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P. J. Campagnola, A. C. Millard, M. Terasaki, P. E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogeneous structural proteins in biological tissues,” Biophys. J. 81, 493–508 (2002).
[CrossRef]

O. A. Aktsipetrov, “Nonlinear magneto-optics in magnetic nanoparticles,” Colloids Surf., A 202, 165–173 (2002).
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H. Unterhalt, G. Rupprechter, and H.-J. Freund, “Vibrational sum frequency spectroscopy on Pd(111) and supported Pd nanoparticles: CO adsorption from ultrahigh vacuum to atmospheric pressure,” J. Phys. Chem. B 106, 356–367 (2002).
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G. Ma and H. C. Allen, “Diffuse reflection broad bandwidth sum frequency generation from particle surfaces,” J. Am. Chem. Soc. 124, 9374–9375 (2002).
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H. Tuovinen, M. Kauranen, K. Jefimovs, P. Vahimaa, T. Vallius, J. Turunen, N. V. Tkachenko, and H. Lemmetyinen, “Linear and second-order nonlinear optical properties of arrays of noncentrosymmetric gold nanoparticles,” J. Nonlinear Opt. Phys. Mater. 11, 421–432 (2002).
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Y. Jiang, L. Sun, and M. C. Downer, “Second-harmonic spectroscopy of two-dimensional Si nanocrystal layers embedded in SiO2 films,” Appl. Phys. Lett. 81, 3034–3036 (2002).
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A. M. Malvezzi, M. Allione, M. Patrini, A. Stella, P. Cheyssac, and R. Kofman, “Melting-induced enhancement of the second-harmonic generation from metal nanoparticles,” Phys. Rev. Lett. 89, 087401 (2002).
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2001 (10)

Y. Jiang, P. T. Wilson, M. C. Downer, C. W. White, and S. P. Withrow, “Second-harmonic generation from silicon nanocrystals embedded in SiO2,” Appl. Phys. Lett. 78, 766–768 (2001).
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M. J. Eilon, T. Mokari, and U. Banin, “Surface exchange effect on hyper Rayleigh scattering in CdSe nanocrystals,” J. Phys. Chem. B 105, 12726–12731 (2001).
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R. Srinivasan, Y. Tian, and I. I. Suni, “Surface plasmon effects on surface second harmonic generation during Au nanoparticle deposition onto H-Si(111),” Surf. Sci. 490, 308–314 (2001).
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N. Yang, W. E. Angerer, and A. G. Yodh, “Second-harmonic microscopy of single micrometer-size particles on a substrate,” Phys. Rev. A 64, 045801 (2001).
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L. Moreaux, O. Sandre, S. Charpak, M. Blanchard-Desce, and J. Mertz, “Coherent scattering in multi-harmonic light microscopy,” Biophys. J. 80, 1568–1574 (2001).
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T. V. Murzina, A. A. Nikulin, O. A. Aktsipetrov, J. W. Ostrander, A. A. Mamedov, N. A. Kotov, M. A. C. Devillers, and J. Roark, “Nonlinear magneto-optical Kerr effect in hyper-Rayleigh scattering from layer-by-layer assembled films of yttrium iron garnet nanoparticles,” Appl. Phys. Lett. 79, 1309–1311 (2001).
[CrossRef]

Y. Liu, E. C. Y. Yan, X.-L. Zhao, and K. B. Eisenthal, “Surface potential of charged liposomes determined by second harmonic generation,” Langmuir 17, 2063–2066 (2001).
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E. C. Y. Yan, Y. Liu, and K. B. Eisenthal, “In situ studies of molecular transfer between microparticles by second-harmonic generation,” J. Phys. Chem. B 105, 8531–8537 (2001).
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X.-M. Shang, Y. Liu, E. C. Y. Yan, and K. B. Eisenthal, “Effects of counterions on molecular transport across liposome bilayer: probed by second harmonic generation,” J. Phys. Chem. B 105, 12816–12822 (2001).
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N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87, 103902 (2001).
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2000 (7)

E. C. Y. Yan and K. B. Eisenthal, “Effect of cholesterol on molecular transport of organic cations across liposome bilayers probed by second harmonic generation,” Biophys. J. 79, 898–903 (2000).
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M. Jacobsohn and U. Banin, “Size dependence of second harmonic generation in CdSe nanocrystal quantum dots,” J. Phys. Chem. B 104, 1–5 (2000).
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H. F. Wang, T. Troxler, A. G. Yeh, and H. L. Dai, “In situ, nonlinear optical probe of surfactant adsorption on the surface of microparticles in colloids,” Langmuir 16, 2475–2481 (2000).
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B. S. Santos, G. A. L. Pereira, D. V. Petrov, C. de Mello Donegá, “First hyperpolarizability of CdS nanoparticles studied by hyper-Rayleigh scattering,” Opt. Commun. 178, 187–192 (2000).
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S. Baldelli, A. S. Eppler, E. Anderson, Y. R. Shen, and G. A. Somorjai, “Surface enhanced sum frequency generation of carbon monoxide adsorbed on platinum nanoparticle arrays,” J. Chem. Phys. 113, 5432–5438 (2000).
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E. C. Y. Yan and K. B. Eisenthal, “Rotational dynamics of anisotropic microscopic particles studied by second harmonic generation,” J. Phys. Chem. B 104, 6686–6689 (2000).
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V. L. Brudny, B. S. Mendoza, and W. L. Mochán, “Second-harmonic generation from spherical particles,” Phys. Rev. B 62, 11152–11162 (2000).
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1999 (8)

B. Lamprecht, A. Leitner, F. R. Aussenegg, “SHG studies of plasmon dephasing in nanoparticles,” Appl. Phys. B 68, 419–423 (1999).
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P. J. Campagnola, M.-D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
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A. Brysch, G. Bour, R. Neuendorf, and U. Kreibig, “Nonlinear optical spectroscopy of embedded semiconductor clusters,” Appl. Phys. B 68, 447–451 (1999).
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M. L. Sandrock, C. D. Pibel, F. M. Geiger, and C. A. Foss, Jr., “Synthesis and second-harmonic generation studies of noncentrosymmetric gold nanostructures,” J. Phys. Chem. B 103, 2668–2673 (1999).
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Y. Liu, J. I. Dadap, D. Zimdars, and K. B. Eisenthal, “Study of interfacial charge-transfer complex on TiO2 particles in aqueous suspension by second-harmonic generation,” J. Phys. Chem. B 103, 2480–2486 (1999).
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P. Galletto, P. F. Brevet, H. H. Girault, R. Antoine, and M. Broyer, “Size dependence of the surface plasmon enhanced second harmonic response of gold colloids: towards a new calibration method,” Chem. Commun. 581–582 (1999).
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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).
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P. Galletto, P. F. Brevet, H. H. Girault, R. Antoine, and M. Broyer, “Enhancement of the second harmonic response by adsorbates on gold colloids: the effect of aggregation,” J. Phys. Chem. B 103, 8706–8710 (1999).
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1998 (8)

Y. Fang, “Optical absorption of nanoscale colloidal silver: Aggregate band and adsorbate-silver surface band,” J. Chem. Phys. 108, 4315–4318 (1998).
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A. Srivastava and K. B. Eisenthal, “Kinetics of molecular transport across a liposome bilayer,” Chem. Phys. Lett. 292, 345–351 (1998).
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H. Wang, E. C. Y. Yan, Y. Liu, and K. B. Eisenthal, “Energetics and population of molecules at microscopic liquid and solid surfaces,” J. Phys. Chem. B 102, 4446–4450 (1998).
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F. W. Vance, B. I. Lemon, and J. T. Hupp, “Enormous hyper-Rayleigh scattering from nanocrystalline gold particle suspensions,” J. Phys. Chem. B 102, 10091–10093 (1998).
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E. C. Y. Yan, Y. Liu, and K. B. Eisenthal, “New method for determination of surface potential of microscopic particles by second harmonic generation,” J. Phys. Chem. B 102, 6331–6336 (1998).
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R. Antoine, M. Pellarin, B. Palapant, M. Broyer, B. Prével, P. Galletto, P. F. Brevet, and H. H. Girault, “Surface plasmon enhanced second harmonic response from gold clusters embedded in an alumina matrix,” J. Appl. Phys. 84, 4532–4536 (1998).
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J.-H. Klein-Wiele, P. Simon, and H.-G. Rubahn, “Size-dependent plasmon lifetimes and electron-phonon coupling time constants for surface bound Na clusters,” Phys. Rev. Lett. 80, 45–48 (1998).
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M. Simon, F. Träger, A. Assion, B. Lang, S. Voll, and G. Gerber, “Femtosecond time-resolved second-harmonic generation at the surface of alkali metal clusters,” Chem. Phys. Lett. 296, 579–584 (1998).
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1997 (7)

J. Martorell, R. Vilaseca, and R. Corbalán, “Second harmonic generation in a photonic crystal,” Appl. Phys. Lett. 70, 702–704 (1997).
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J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
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T. Kuroda, S. Matsushita, F. Minami, K. Inoue, A. V. Baranov, “Observation of homogeneous broadening in semiconductor nanocrystals by resonant second-harmonic scattering spectroscopy,” Phys. Rev. B 55, R16041–R16044 (1997).
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C. P. Collier, R. J. Saykaly, J. J. Shiang, S. E. Heinrichs, and J. R. Heath, “Reversible tuning of silver quantum dot monolayers through the metal-insulator transition,” Science 277, 1978–1981 (1997).
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T. Muller, P. H. Vaccaro, F. Balzer, and H.-G. Rubahn, “Size dependent optical second harmonic generation from surface bound Na clusters: comparison between experiment and theory,” Opt. Commun. 135, 103–108 (1997).
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J. M. Hartings, A. Poon, X. Pu, R. K. Chang, and T. M. Leslie, “Second harmonic generation and fluorescence images from surfactants on hanging droplets,” Chem. Phys. Lett. 281, 389–393 (1997).
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Y. R. Shen, “Wave mixing spectroscopy for surface studies,” Solid State Commun. 102, 221–229 (1997).
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1996 (3)

H. Wang, E. C. Y. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259, 15–20 (1996).
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K. B. Eisenthal, “Liquid interfaces probed by second-harmonic and sum-frequency spectroscopy,” Chem. Rev. 96, 1343–1360 (1996).
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J. P. Dewitz, W. Hübner, and K. H. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Z. Phys. D 37, 75–84 (1996).
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1995 (6)

A. Guerrero and B. S. Mendoza, “Model for great enhancement of second-harmonic generation in quantum dots,” J. Opt. Soc. Am. B 12, 559–569 (1995).
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K. Y. Lo and J. T. Lue, “Quantum-size effect on optical second-harmonic generation in small metallic particles,” Phys. Rev. B 51, 2467–2472 (1995).
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J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
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T. Götz, M. Buck, C. Dressler, F. Eisert, and F. Träger, “Optical second-harmonic generation by supported metal-clusters—size and shape effects,” Appl. Phys. A 60, 607–612 (1995).
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O. A. Aktsipetrov, P. V. Elyutin, A. A. Nikulin, and E. A. Ostrovskaya, “Size effects in optical second-harmonic generation by metallic nanocrystals and semiconductor quantum dots: the role of quantum chaotic dynamics,” Phys. Rev. B 51, 17591–17599 (1995).
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O. A. Aktsipetrov, P. V. Elyutin, A. A. Fedyanin, A. A. Nikulin, and A. N. Rubtsov, “Second-harmonic generation in metal and semiconductor low-dimensional structures,” Surf. Sci. 325, 343–355 (1995).
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1993 (2)

D. Östling, P. Stampfli, and K. H. Bennemann, “Theory of nonlinear-optical properties of small metallic spheres,” Z. Phys. D 28, 169–175 (1993).
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G. Berkovic and S. Efrima, “Second harmonic-generation from composite films of spheroidal metal particles,” Langmuir 9, 355–357 (1993).
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1992 (2)

K. Hayata and M. Koshiba, “Theory of surface-emitting second-harmonic generation from optically trapped microspheres,” Phys. Rev. A 46, 6104–6107 (1992).
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S. Sato and H. Inaba, “Observation of second harmonic-generation from optically trapped microscopic LiNbO3 particle using Nd-YAG laser,” Electron. Lett. 28, 286–287 (1992).
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1991 (3)

R. Bavli, D. Yogev, S. Efrima, and G. Berkovic, “Second harmonic-generation studies of silver metal liquid-like films,” J. Phys. Chem. 95, 7422–7426 (1991).
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C. Flytzanis, F. Hache, M. C. Klein, D. Ricard, and P. Roussignol, “Nonlinear optics in composite materials,” Prog. Opt. 29, 321–411 (1991).
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K. Clays and A. Persoons, “Hyper-Rayleigh scattering in solution,” Phys. Rev. Lett. 66, 2980–2983 (1991).
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1990 (1)

T. P. Shen and D. Rogovin, “Coherent frequency mixing in microparticle composites,” Phys. Rev. A 42, 4255–4268 (1990).
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1989 (1)

C. K. Johnson and S. A. Soper, “Nonlinear surface-enhanced spectroscopy of silver colloids and pyridine: hyper-Raman and second-harmonic scattering,” J. Phys. Chem. 93, 7281–7285 (1989).
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1988 (2)

For heterogeneous media with ε(sphere)≠ε(ambient), this term behaves like a polarization sheet at the interface, proportional to the product E (ω) Er(ω) and, hence, can be incorporated into the surface susceptibilities χs, ⊥⊥⊥(2) and χs, ∥⊥∥(2). The effect of this and any other terms that may contribute in a localized region at the interface is omitted from the analysis. These terms are incorporated in our treatment into the surface nonlinear susceptibility χ↔s(2), as has been discussed elsewhere, e.g., see Ref. 17 or P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985–7989 (1988).
[CrossRef]

V. Mizrahi and J. E. Sipe, “Phenomenological treatment of surface second-harmonic generation,” J. Opt. Soc. Am. B 5, 660–667 (1988).
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1986 (1)

X. M. Hua and J. I. Gersten, “Theory of second-harmonic generation by small metal spheres,” Phys. Rev. B 33, 3756–3764 (1986).
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1983 (1)

C. K. Chen, T. F. Heinz, D. Ricard, and Y. R. Shen, “Surface-enhanced second-harmonic generation and Raman-scattering,” Phys. Rev. B 27, 1965–1979 (1983).
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1982 (1)

G. S. Agarwal and S. S. Jha, “Theory of second harmonic-generation at a metal-surface with surface-plasmon excitation,” Solid State Commun. 41, 499–501 (1982).
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1981 (1)

C. T. Tai, “Equivalent layers of surface-charge, current sheet, and polarization in the eigenfunction-expansions of Green’s functions in electromagnetic theory,” IEEE Trans. Antennas Propag. 29, 733–739 (1981).
[CrossRef]

1970 (1)

P. D. Maker, “Spectral broadening of elastic second-harmonic light scattering in liquids,” Phys. Rev. A 1, 923–951 (1970).
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1968 (1)

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).
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1966 (1)

R. Bersohn, Y. H. Pao, and H. L. Frisch, “Double-quantum light scattering by molecules,” J. Chem. Phys. 45, 3184–3198 (1966).
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1965 (1)

R. W. Terhune, P. D. Maker, and C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684 (1965).
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1962 (1)

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
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1954 (1)

E. H. Hill, “The theory of vector spherical harmonics,” Am. J. Phys. 22, 211–214 (1954).
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1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. (Leipzig) 30, 57–136 (1909).
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1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
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1871 (1)

J. W. Strutt (Lord Rayleigh), “On the light from the sky, its polarization and colour,” Philos. Mag. 41, 107–120, 274–279 (1871).

Abid, J.-P.

J. Nappa, G. Revillod, J.-P. Abid, I. Russier-Antoine, C. Jonin, E. Benichou, H. H. Girault, and P. F. Brevet, “Hyper-Rayleigh scattering of gold nanorods and their relationship with linear assemblies of gold nanospheres,” Faraday Discuss. 125, 145–156 (2004).
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Agarwal, G. S.

G. S. Agarwal and S. S. Jha, “Theory of second harmonic-generation at a metal-surface with surface-plasmon excitation,” Solid State Commun. 41, 499–501 (1982).
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Aktsipetrov, O. A.

O. A. Aktsipetrov, “Nonlinear magneto-optics in magnetic nanoparticles,” Colloids Surf., A 202, 165–173 (2002).
[CrossRef]

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H. Unterhalt, G. Rupprechter, and H.-J. Freund, “Vibrational sum frequency spectroscopy on Pd(111) and supported Pd nanoparticles: CO adsorption from ultrahigh vacuum to atmospheric pressure,” J. Phys. Chem. B 106, 356–367 (2002).
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D. V. Petrov, B. S. Santos, G. A. L. Pereira, and C. de Mello Donegá, “Size and band-gap dependences of the first hyperpolarizability of CdxZn1−xS nanocrystals,” J. Phys. Chem. B 106, 5325–5334 (2002).
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B. S. Santos, G. A. L. Pereira, D. V. Petrov, C. de Mello Donegá, “First hyperpolarizability of CdS nanoparticles studied by hyper-Rayleigh scattering,” Opt. Commun. 178, 187–192 (2000).
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A. M. Malvezzi, M. Allione, M. Patrini, A. Stella, P. Cheyssac, and R. Kofman, “Melting-induced enhancement of the second-harmonic generation from metal nanoparticles,” Phys. Rev. Lett. 89, 087401 (2002).
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N. Thantu, R. S. Schley, and B. S. Justus, “Tunable room temperature second harmonic generation in glasses doped with CuCl nanocrystalline quantum dots,” Opt. Commun. 220, 203–210 (2003).
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X.-M. Shang, Y. Liu, E. C. Y. Yan, and K. B. Eisenthal, “Effects of counterions on molecular transport across liposome bilayer: probed by second harmonic generation,” J. Phys. Chem. B 105, 12816–12822 (2001).
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Y. Liu, E. C. Y. Yan, X.-L. Zhao, and K. B. Eisenthal, “Surface potential of charged liposomes determined by second harmonic generation,” Langmuir 17, 2063–2066 (2001).
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J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

The electric dipole, magnetic dipole, and electric quadrupole (tensor) moments are defined as p =∫x ρ(x)d x, m = (1/2c)∫x ×J (x)d x and Qij =∫[(xi xj)−r2 δij ]× ρ(x)d 3 x, respectively. By employing the relations ρ(r)=−∇⋅P (r) and J (r)=−iΩP (r), and performing integration by parts, we obtain Eqs. (7a)–(7c).

For a sphere of arbitrary radius, an exact expression for the SH electric field for the homogeneous media can be obtained without employing the small-particle approximation. This limit (of neglecting differences in the dielectric functions) is known as the SH Rayleigh–Gans approximation, first applied by Martorell et al.78 for the case of a single nonlinear susceptibility element, χs, ⊥⊥⊥(2), and dispersionless media (K1=2k1). For the case of an isotropic surface having all three nonlinear susceptibility elements χs, ⊥⊥⊥(2), χs, ⊥ ∥ ∥(2), and χs, ∥⊥∥(2), we substitute Eq. (11) into Eq. (4) with K1=2k1. Making use of Eqs. (5a) and (5b), we obtain A(r). The resulting SH field in the radiation zone is E(2ω)(r)RG=[4πi exp(iK1r)/r]× (Ka)2(E0(ω))2 [Θ(θ, ε)θ⁁+ Φ(θ, ε)ε⁁], where Θ(θ, ε) and Φ(θ, ε) are functions given by Θ(θ, ε)=cos(θ/2) {[Γ1(θ)+ Γ2(θ)cos2(θ/2)]f(ε)+ Γ3(θ)(ε⁁0⋅ε⁁0)} and Φ(θ, ε)= −cos(θ/2)Γ1(θ)g(ε). Here, f(ε)=(ε⁁0⋅ρ⁁)2 and g(ε)=(ε⁁0⋅ρ⁁)(ε⁁0⋅ε⁁), with ρ⁁=cos εx⁁+sin εy⁁ and ε⁁= −sin εx⁁+cos εy⁁; Γ1(θ)= 2[(χs, ⊥⊥⊥(2)−χs, ⊥ ∥ ∥(2))F1(θ)− 2χs, ∥⊥∥(2)F2(θ)], Γ2(θ)= − (χs, ⊥⊥⊥(2)−χs, ⊥ ∥ ∥(2)) [F1(θ)− 2F2(θ)] −2χs, ∥⊥∥(2) [3F1(θ)− 2F2(θ)], and Γ3(θ)= −(χs, ⊥⊥⊥(2)+γ) F1(θ)−(χs, ⊥ ∥ ∥(2)+ γ)[F1(θ)−2F2(θ)]+ 2χs, ∥⊥∥(2)F1(θ), with the structure factors for δ= 2Ka sin(θ/2) of F1(θ)=(3/δ3) [(1− δ2/3)sin δ− δ cos δ] and F2(θ)=(3/δ3)[(1− δ2/2)sin δ−δ(1− δ2/6)cos δ]. (A similar treatment for SFG recently appeared in Ref. 48.) One should note that for the case of heterogeneous media, obtaining the SH field requires a full evaluation of Maxwell’s equation, as discussed and outlined in the Appendix A of this paper.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H. Ponath and G. Stegeman, eds. (Elsevier, Amsterdam, 1991) pp. 353–416.

J. I. Dadap and T. F. Heinz, “Nonlinear optical spectroscopy of surfaces and interfaces,” in Encyclopedia of Chemical Physics and Physical Chemistry, J. H. Moore and N. D. Spencer, eds. (Institute of Physics, Bristol, 2001), pp. 1089–1125.

A. V. Baranov, Ya. S. Bobovich, and V. I. Petrov, “Study of surface-enhanced Raman-scattering initiated by adsorption of molecules on colloidal-silver microparticles,” Opt. Spectrosc. 58, 353–356 (1985) [ Opt. Spectrosk. 58, 578–582 (1985)].

U. Kreibig, in Handbook of Optical Properties, R. E. Hummel and P. Wißmann, eds. (CRC, Boca Raton, Fla., 1997), Vol. II, p. 145 and references therein.

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, Berlin, 1995).

R. F. Haglund, Jr., in Handbook of Optical Properties, R. E. Hummel and P. Wißmann, eds. (CRC, Boca Raton, Fla., 1997), Vol. II, p. 191 and references therein.

J. L. Cheung, J. M. Hartings, and R. K. Chang, in Handbook of Optical Properties, R. E. Hummel and P. Wißmann, eds. (CRC, Boca Raton, Fla., 1997), Vol. II, p. 233, and references therein.

S. V. Gaponenko, Optical Properties of Semiconductor Nanocrystals (Cambridge University Press, New York, 1998).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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Figures (5)

Fig. 1
Fig. 1

Scattering geometry and the linear and nonlinear response of the small sphere for the SHRS problem. kˆ denotes the direction of the incident plane wave of polarization ˆ0; Kˆ denotes the direction of the scattered light with polarization state ˆ. Inset indicates the linear and nonlinear optical responses in the three regions: ambient, interface, and the internal volume of the sphere.

Fig. 2
Fig. 2

Graphical representations of scattering patterns dP2ω/dΩ for SHRS [(a)–(c)] and for LRS [(d)–(e)]. Because of symmetry with respect to the xz plane, only half of each radiation patterns is shown. Panels (a) and (b) are plots of pure E1 (with χ10 and χ2=0) and E2 (with χ1=0 and χ20) modes, respectively, for linearly polarized excitation. Panel (c) is a plot of pure E2 mode arising from circularly polarized excitation. For comparison, the LRS radiation patterns are shown for linearly polarized (d) and circularly polarized (e) input radiation.

Fig. 3
Fig. 3

SH radiation patterns dP2ω/dΩ [(a)–(d)] and polarization anisotropy μ(θ) (e)–(h) corresponding to different values of κK1χ2/k1χ1=2, -1/2, -3, and i. The values κ=2, -1/2, and -3 correspond to a dominant nonlinear response from χs,(2), χs, (2), or χs,(2), respectively, for a homogeneous, nondispersive linear optical response. The value κ=i illustrates the absence of interference between E1 and E2 modes.

Fig. 4
Fig. 4

Spectral dependence of the total SH power, P2ω, for aluminum spheres of radii a=2.5, 5, and 10 nm relative to that from a spherical shell of fixed 2.5-nm radius. Panel (a) shows the results for linearly polarized excitation, and panel (b) shows the corresponding behavior for circularly polarized radiation. The nonlinear response is assumed to be dominated by χs,(2). Note that all four localized plasmon resonances appear in (a), but only two occur in (b).

Fig. 5
Fig. 5

Normalized SH power dP2ω/dΩψ polarized along the scattering plane at a scattering angle of θ=π/4 as a function of the polarization orientation ψ of a linearly polarized pump beam for several values of κ=K1χ2/k1χ1. The top, middle, and bottom panels indicate cases where κ is real positive (κ=1, 2), imaginary (κ=±i,±2i), and real negative (κ=-1/2,-2,-3), respectively. The case where κ=0 is indicated as a horizontal (i.e., no anisotropy) dash-dot line in all panels. A measurement of the quantity dP2ω/dΩψ permits the extraction of the complex polarization anisotropy parameter κ.

Tables (4)

Tables Icon

Table 1 Excitation-Radiation Interactions and the Corresponding Nonlinear Material Responsea

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Table 2 Directional and Polarization Selection Rulesa

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Table 3 Comparison of LRS and SHRSa

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Table 4 Coefficients bijklm for an Incident Field Polarized along x Directiona

Equations (87)

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P(2ω)(r)=Psurface(2ω)+Pbulk(2ω)=χs(2):E(ω)(r)E(ω)(r)δ(r-a)+χb(2):E(ω)(r)E(ω)(r),
χs(2)=χs,(2)rˆrˆrˆ+χs,  (2) rˆ(θˆθˆ+φˆφˆ)+χs,(2)(θˆrˆθˆ+φˆrˆφˆ+θˆθˆrˆ+φˆφˆrˆ).
Pbulk(2ω)(r)=γ(Ein(ω)·Ein(ω))+δ(Ein(ω)·)Ein(ω),
A(r)=1c  exp(iK1|r-r|)J(r)|r-r| dr,
B(r)=×A(r)=iK1rˆ×A(r),
E(r)=iε1K×B(r)=1ε1 B(r)×rˆ,
A(r)=-iK exp(iK1r)r p-(rˆ×m)-iK16 Q(rˆ),
p=P(r)dr,
m=-iK2 r×P(r)dr,
Q(rˆ)={3[(rˆ·r)P(r)+rˆ·P(r)r]-2r·P(r)rˆ}dr
E(Ω)=K12 exp(iK1r)ε1(Ω)r(Kˆ×peff)×Kˆ,
peff=p-(Kˆ×m)-iK16 Q(Kˆ).
Einc(ω)(r)=E0 exp(ik1·r)ˆ0,
P(2ω)(r)=({χs,(2)rˆ(rˆ·ˆ0)2+χs,  (2) rˆ[(θˆ·ˆ0)2+(φˆ·ˆ0)2]+2χs,(2)[θˆ(rˆ·ˆ0)(θˆ·ˆ0)+φˆ(rˆ·ˆ0)×(φˆ·ˆ0)]}δ(r-a)+iγ2k1kˆ(ˆ0·ˆ0))E02 exp(2ik1·r).
p=p0(ˆ0·ˆ0)kˆ,
m=0,
Q(Kˆ)=q0(Kˆ·ˆ0)ˆ0-13(ˆ0·ˆ0)Kˆ.
p0=8πi15k1a3E02χ1,
q0=16π5a3E02χ2,
χ1χs,(2)+4χs,  (2)-2χs,(2)+5γ,
χ2χs,(2)-χs,  (2)+3χs,(2).
peff=p0[(ˆ0·ˆ0)kˆ-κ(Kˆ·ˆ0)ˆ0].
κ=iK1q0/6p0=K1χ2/k1χ1.
E(2ω)(r)=K2 exp(iK1r)p0r[(ˆ0·ˆ0)(Kˆ×kˆ)×Kˆ-κ(Kˆ·ˆ0)(Kˆ×ˆ0)×Kˆ],
E(2ω)(r)=K2 exp(iK1r)p0r l=12m=-l(even)lclmrˆ×Xlm(θ, φ).
dP2ω(ˆ0, kˆˆ, Kˆ)dΩ
=cK142π[ε1(2ω)]3/2 |ˆ*·p|2+K162|ˆ*·Q(Kˆ)|2
-K13 Im[(ˆ*·p)(ˆ·Q(Kˆ)*)].
dP2ω(ˆ0, kˆˆ, Kˆ)dΩ=cK14|p0|22π[ε1(2ω)]3/2|(ˆ0·ˆ0)(ˆ*·kˆ)-κ(Kˆ·ˆ0)(ˆ*·ˆ0)|2.
dP2ω(ˆ0, kˆKˆ)dΩ=cK142π[ε1(2ω)]3/2 |p|2[1-(Kˆ·kˆ)2]+K162|Q(Kˆ)|2[1-|Kˆ·ˆ0|2]+K13 Im[(Kˆ·p)(Kˆ·Q(Kˆ)*)],
dP2ω(ˆ0, kˆKˆ)dΩ=cK14|p0|22π[ε1(2ω)]3/2|(ˆ0·ˆ0)×[kˆ-Kˆ(Kˆ·kˆ)]-κ(Kˆ·ˆ0)×[ˆ0-Kˆ(Kˆ·ˆ0)]|2.
dP2ωlp(ˆ0, kˆKˆ)dΩ=cK14|p0|22π[ε1(2ω)]3/2[1+|κ|2(Kˆ·ˆ0)2-|(Kˆ·kˆ)-κ(Kˆ·ˆ0)2|2],
dP2ωcp(ˆ0, kˆKˆ)dΩ=cK16|q0|22532π[ε1(2ω)]3/2×[1-(Kˆ·kˆ)4].
P2ωlp=22cK14|p0|215[ε1(2ω)]3/2(5+|κ|2),
P2ωcp=cK16|q0|290[ε1(2ω)]3/2.
dP2ωdΩx=cK14|p0|22π[ε1(2ω)]3/2 sin θ+12κ sin 2θ cos2 φ2,
dP2ωdΩx=cK16|q0|22532π[ε1(2ω)]3/2 sin2 θ sin2 2φ,
dP2ωdΩψ=cK14|p0|22π[ε1(2ω)]3/2 sin θ+12κ sin 2θ cos2 ψ2,
dP2ωdΩψ=cK16|q0|22532π[ε1(2ω)]3/2 sin2 θ sin2 2ψ.
μ(θ)dP2ω/dΩ|dP2ω/dΩ|=dP2ω/dΩ|dP2ω/dΩ|.
μ(θ)=|1+κ cos θ|2.
χ1χs,(2)[LE1(2ω)LE1(ω)LE2(ω)]+χs,  (2)32LE1(2ω)LE1(ω)LE2(ω)+52LE1(2ω)LE1(ω)LM1(ω)-χs,(2)LE1(2ω)LE1(ω)LE2(ω)-32LE1(2ω)LE2(ω)LE1(ω)+52LE1(2ω)LE1(ω)LM1(ω)+52LE1(2ω)LE1(ω)×LM1(ω)γ-δ2+LE2(ω)γ+δ2,
χ2χs,(2)LE2(2ω)LE1(ω)LE1(ω)-χs,  (2)LE2(2ω)LE1(ω)LE1(ω)+3χs,(2)LE2(2ω)LE1(ω)LE1(ω).
LEl=(2l+1)ε1lε2+(l+1)ε1,
LEl=ε2εLEl,
LMl=(2l+1)μ2lμ2+(l+1)μ1.
Re[lε2(Ω)]+(l+1)ε1(Ω)=0.
P2ωsphere=212π3153cA2 ε1(2ω)(ka)4k2As(5+|κ|2)×|χ1|2(Pω)2.
P2ωplane=25π3k2 sec2 θPε1(2ω)ε1(ω)cA|e(2ω)·χP(2):e(ω)e(ω)|(Pω)2,
η=27153(5+|κ|2)ε1(ω)ε1(2ω)cos2 θP(ka)4 AsA χ1χP2,
η(ka)4As/A.
Esc(ω)=k2 exp(ik1r)r(kˆsc×peff(ω))×kˆsc,
Ein(ω)=LE1(ω)E0-i2LM1(ω)r×(k1×E0)+i2LE2(ω)[(r·k1)E0+(r·E0)k1].
E(ω)=Er(ω)+Et(ω).
Er(ω)=ε2(ω)ε(ω)Ein,r(ω)|r=a,Et(ω)=Ein,t(ω)|r=a.
Psurface(2ω)=Ps(2ω)(θ, φ)δ(r-a),
Ps(2ω)(θ, φ)=rˆ(χs,(2)ErEr+χs,  (2)Et·Et)+2χs,(2)ErEt,
ErEr=lmblmYlm(θ, φ),
Et·Et=lmb  lmYlm(θ, φ),
2ErEt=lm[b,MlmXlm(θ, φ)+b,Elmrˆ×Xlm(θ, φ)].
blm=Ylm*ErErdΩ,
b  lm=Ylm*Et·EtdΩ,
b,Mlm=2Xlm*·ErEtdΩ,
b,Elm=2rˆ×Xlm*·ErEtdΩ.
Esurface,i(2ω)=l=12m=-ll{AM(i)(l, m)fTi(Kir)Xlm+AE(i)(l, m) 1Ki×fli(Kir)Xlm},
Bsurface,i(2ω)=1i l=12m=-ll1KAM(i)(l, m)×fli(Kir)Xlm+εi(2ω)AE(i)(l, m)fli(Kir)Xlm,
ΔDr(2ω)=Dr(2ω)(r=a+)-Dr(2ω)(r=a-)=-4πt·Ps(2ω),
ΔEt(2ω)=-4πε(2ω)tPs,r(2ω),
ΔBr(2ω)=0,
ΔHt(2ω)=4πi 2ωc rˆ×Ps(2ω).
AE(1)(l, m)=4πiK1(K1a)l+1ε1(2ω)(2l+1)!![χs,(2)b,Elm(l+1)LEl(2ω)-i(χs,(2)blm+χs,  (2)b  lm)l(l+1)LEl(2ω)],
AM(1)(l, m)=4πiK1(K1a)l+2ε1(2ω)(2l+1)!!χs,(2)b,MlmLMl(2ω).
Pbulk(2ω)=ik1LE1(ω)LM1(ω)γ-δ2+LE2(ω)γ+δ2E02zˆ.
××Ebulk,in(2ω)-ε2(2ω)K2Ebulk,in(2ω)=4πK2Pbulk(2ω).
Ebulk,in(2ω),p=-4πε2(2ω) Pbulk(2ω).
Ebulk,i(2ω),c=l=12m=-llαM(i)(l, m)fli(Kir)Xlm+αE(i)(l, m) 1Ki×fli(Kir)Xlm,
Bbulk,i(2ω),c=1i l=12m=-ll1KαM(i)(l, m)×fli(Kir)Xlm+εi(2ω)αE(i)(l, m)fli(Kir)Xlm.
αE(1)(1, 0)=8(K1a)33ε1(2ω) 2π33 Pbulk(2ω)LE1(2ω).
Xlm(θ, φ)=1l(l+1) LYlm(θ, φ),
Xlm(θ, φ)=-1l(l+1) 0,mYlm(θ, φ)sin θ, i Ylm(θ, φ)θ,
Alm*·BlmdΩ=δABδllδmm.
X1,0=i2 32π {0, 0, sin θ},
X1,1=14 3π {0, 1, i cos θ}exp(iφ),
X2,0=i4 152π {0, 0, sin 2θ},
X2,1=14 5π {0, cos θ, i cos 2θ}exp(iφ),
X2,2=-18 5π {0, 2 sin θ, i sin 2θ}exp(2iφ).
Xl,-m(θ, φ)=(-1)m+1Xlm*(θ, φ).

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