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Journal of the Optical Society of America B

Journal of the Optical Society of America B


  • Vol. 21, Iss. 7 — Jul. 1, 2004
  • pp: 1328–1347

Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit

Jerry I. Dadap, Jie Shan, and Tony F. Heinz  »View Author Affiliations

JOSA B, Vol. 21, Issue 7, pp. 1328-1347 (2004)

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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

OCIS Codes
(190.3970) Nonlinear optics : Microparticle nonlinear optics
(190.4350) Nonlinear optics : Nonlinear optics at surfaces
(240.6680) Optics at surfaces : Surface plasmons
(290.4020) Scattering : Mie theory
(290.5850) Scattering : Scattering, particles
(290.5870) Scattering : Scattering, Rayleigh

Jerry I. Dadap, Jie Shan, and Tony 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)

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  105. 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).
  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.
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