On surfaces with prescribed shape operator

The problem of immersing a simply connected surface with a prescribed shape operator is discussed. I show that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface with one family of principal curves being geodesic, the space of such realizations is a convex set in an affine space of dimension at most 3. The cases where this maximum dimension of realizability is achieved are analyzed and it is found that there are two such families of shape operators, one depending essentially on three arbitrary functions of one variable and another depending essentially on two arbitrary functions of one variable. The space of realizations is discussed in each case, along with some of their remarkable geometric properties. Several explicit examples are constructed.

Gold nanoparticles on polarizable surfaces as Raman scattering antennas.

Surface plasmons supported by metal nanoparticles are perturbed by coupling to a surface that is polarizable. Coupling results in enhancement of near fields and may increase the scattering efficiency of radiative modes. In this study, we investigate the Rayleigh and Raman scattering properties of gold nanoparticles functionalized with cyanine deposited on silicon and quartz wafers and on gold thin films. Dark-field scattering images display red shifting of the gold nanoparticle plasmon resonance and doughnut-shaped scattering patterns when particles are deposited on silicon or on a gold film. The imaged radiation patterns and individual particle spectra reveal that the polarizable substrates control both the orientation and brightness of the radiative modes. Comparison with simulation indicates that, in a particle-surface system with a fixed junction width, plasmon band shifts are controlled quantitatively by the permittivity of the wafer or the film. Surface-enhanced resonance Raman scattering (SERRS) spectra and images are collected from cyanine on particles on gold films. SERRS images of the particles on gold films are doughnut-shaped as are their Rayleigh images, indicating that the SERRS is controlled by the polarization of plasmons in the antenna nanostructures. Near-field enhancement and radiative efficiency of the antenna are sufficient to enable Raman scattering cyanines to function as gap field probes. Through collective interpretation of individual particle Rayleigh spectra and spectral simulations, the geometric basis for small observed variations in the wavelength and intensity of plasmon resonant scattering from individual antenna on the three surfaces is explained.