We describe the electromagnetic theory of second-harmonic generation from the surface of a sphere that is small compared to the wavelength of light. In a leading-order expansion, the second-harmonic radiation is emitted in nonlocally excited electric dipole and locally excited electric quadrupole modes. We analyze the problem with particular emphasis on experimental aspects, such as the radiation pattern, polarization selection rules, determination of nonlinear susceptibilities, radiation efficiency, and spectral characteristics.

We report on the evolution of the thickness-dependent electronic band structure of the two-dimensional layered-dichalcogenide molybdenum disulfide (MoS2). Micrometer-scale angle-resolved photoemission spectroscopy of mechanically exfoliated and chemical-vapor-deposition-grown crystals provides direct evidence for the shifting of the valence band maximum from Γ to K, for the case of MoS2 having more than one layer, to the case of single-layer MoS2, as predicted by density functional theory. This evolution of the electronic structure from bulk to few-layer to monolayer MoS2 had earlier been predicted to arise from quantum confinement. Furthermore, one of the consequences of this progression in the electronic structure is the dramatic increase in the hole effective mass, in going from bulk to monolayer MoS2 at its Brillouin zone center, which is known as the cause for the decreased carrier mobility of the monolayer form compared to that of bulk MoS2.

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/), the particle radius to the wavelength of the incident light, and can be considered as the Rayleigh limit for secondharmonic 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/) 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.

We report strong third-harmonic generation in monolayer graphene grown by chemical vapor deposition and transferred to an amorphous silica (glass) substrate; the photon energy is in threephoton resonance with the exciton-shifted van Hove singularity at the M point of graphene. The polarization selection rules are derived and experimentally verified. In addition, our polarization-and azimuthal-rotation-dependent third-harmonic-generation measurements reveal in-plane isotropy as well as anisotropy between the in-plane and out-of-plane nonlinear optical responses of graphene. Since the third-harmonic signal exceeds that from bulk glass by more than 2 orders of magnitude, the signal contrast permits background-free scanning of graphene and provides insight into the structural properties of graphene.