Ultrafast Plasmonic Control of Second Harmonic Generation

Efficient frequency conversion techniques are crucial to the development of plasmonic metasurfaces for information processing and signal modulation. In principle, nanoscale electric-field confinement in nonlinear materials enables higher harmonic conversion efficiencies per unit volume than those attainable in bulk materials. Here we demonstrate efficient second-harmonic generation (SHG) in a serrated nanogap plasmonic geometry that generates steep electric field gradients on a dielectric metasurface. An ultrafast pump is used to control plasmon-induced electric fields in a thin-film material with inversion symmetry that, without plasmonic enhancement, does not exhibit an an even-order nonlinear optical response. The temporal evolution of the plasmonic near-field is characterized with ~100as resolution using a novel nonlinear interferometric technique. The ability to manipulate nonlinear signals in a metamaterial geometry as demonstrated here is indispensable both to understanding the ultrafast nonlinear response of nanoscale materials, and to producing active, optically reconfigurable plasmonic device

The Delta Conjecture

International audienceWe conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author