page_title: Second-harmonic generation
section_title: From non-planar surfaces
hierarchical_section_title: Second-harmonic generation / Optical second-harmonic generation / From non-planar surfaces
caption_attribution_description: English: A cartoon depicting second harmonic generation from ordered molecules at a small sphere surface.
context_page_description: Second-harmonic generation is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy of the initial photons, that conserves the coherence of the excitation. It is a special case of sum-frequency generation, and more generally of harmonic generation.
  The second-order nonlinear susceptibility of a medium characterizes its tendency to cause SHG. Second-harmonic generation, like other even-order nonlinear optical phenomena, is not allowed in media with inversion symmetry. However, effects such as the Bloch–Siegert shift, found when two-level systems are driven at Rabi frequencies comparable to their transition frequencies, will give rise to second harmonic generation in centro-symmetric systems.
  In some cases, almost 100% of the light energy can be converted to the second harmonic frequency. These cases typically involve intense pulsed laser beams passing through large crystals, and careful alignment to obtain phase matching.
context_section_description: Second-harmonic light can also be generated from surfaces that are ‘locally’ planar, but may have inversion symmetry (centrosymmetric) on a larger scale. Specifically, recent theory has demonstrated that SHG from small spherical particles (micro- and nanometer scale) is allowed by proper treatment of Rayleigh scattering. At the surface of a small sphere, inversion symmetry is broken, allowing for SHG and other even order harmonics to occur.
  For a colloidal system of microparticles at relatively low concentrations, the total SH signal , is given by:
  where  is the SH electric field generated by the jth particle, and n the density of particles. The SH light generated from each particle is coherent, but adds incoherently to the SH light generated by others (as long as density is low enough). Thus, SH light is only generated from the interfaces of the spheres and their environment and is independent of particle-particle interactions. It has also been shown that the second-harmonic electric field  scales with the radius of the particle cubed, a³.
  Besides spheres, other small particles like rods have been studied similarly by SHG. Both immobilized and colloidal systems of small particles can be investigated. Recent experiments using second-harmonic generation of non-planar systems include transport kinetics across living cell membranes and demonstrations of SHG in complex nanomaterials.
caption: Cartoon depicting ordered molecules at a small spherical surface. An ultrafast pump laser pumps light with frequency ω which generates light at 2ω from the locally non-centrosymmetric media.
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