Nonlinear wave mixing in plasmonic structures: A transformation optics approach

Abstract

Singular structures in plasmonics, for example, touching wires, crescent cylinders, etc., are well known for enhancing the field in small volumes by several orders of magnitude. Recent studies revealed that transformation optics can provide a very powerful analytical tool to solve these class of problems. It was also pointed out that these class of structures can also be suitable candidates for energy harvesting and field enhancements were calculated to be as high as 104 close to singular points. We review some of the theoretical and the experimental results of the linear properties of these structures. Such high field enhancement would definitely invoke the nonlinear phenomenon. Our present study analytically incorporates such nonlinear phenomenon from χ(2) materials initially focusing on second-harmonic generation. We follow the route of conformal transformation optics but now extending it to nonlinear materials. To the best of our knowledge there have been no reports on such analytical techniques describing nonlinear optics at nano-scale. We identify the relations for phase and amplitude matching for the second-harmonic fields. Our approach also connects with the standard coupled-mode theory used in “macro-optics” structures such as waveguide to the Green's function approach which is extensively used in nano-optics. We identify the optimal conditions for second-harmonic generation efficiency. This approach is the starting point to understand various other nonlinear interactions such as three and four-wave mixing in singular structures.