A New Angle on Second Harmonic Generation from a Small Hole in a Metal Film
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A New Angle on Second Harmonic Generation from a Small Hole in a Metal Film Esmaeil Rahimi1 · Reuven Gordon1 Received: 18 June 2020 / Accepted: 15 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Bethe’s theory treats a subwavelength aperture in a metal film as the combination of a parallel magnetic dipole and transverse electric dipole. For linear optics, this gives the usual dipole transmission; however, for nonlinear optics, it is interesting to consider how these two components interact. While many works have studied the nonlinear harmonic generation for metal nanoapertures, here we show that the Lorentz force dominates the second harmonic generation by an order of magnitude at angled incidence where the generation is maximized. The angular dependence matches that of the magnetic and electric dipoles accounting for Lambert’s cosine law. The theoretical analysis and numerical calculations agree well with past experiments. Keywords Second harmonic generation · Angle dependence · Lorentz magnetic force · Magnetic/electric dipole · Hydrodynamic theory · Nonlinear scattering theory
Introduction The relative importance of various hydrodynamic contributions to second harmonic generation (SHG) has been a topic of investigation for at least four decades [1]. While there is some debate about how to disentangle various contributions in practice [1, 2], theoretical works have suggested that the Lorentz contribution is negligible in common plasmonic structures [3]. Following this, we suggested that the underlying reason why this contribution is usually weak is because it is uncommon to have both a magnetic and electric enhancement in the same location; we then went on to design a structure having simultaneous magnetic and electric resonant contributions in the same location by having
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11468-020-01293-6) contains supplementary material, which is available to authorized users. Reuven Gordon
[email protected] Esmaeil Rahimi [email protected] 1
Department of Electrical and Computer Engineering, University of Victoria, Victoria, British Columbia, V8P 5C2, Canada
two separate resonances [4]. That structure produced an unusually large second harmonic signal dominated by the Lorentz contribution. The structure, however, was quite contrived, and so it is interesting to consider if there are more natural ways to achieve an enhanced Lorentz contribution. It is well known that subwavelength apertures in thin metal films can be treated as magnetic dipoles [5], and hence the transmission scales as the fourth power of the aperture dimension. In addition, there is an electric dipole component that arises when an out-of-plane electric field is present at the aperture [5]. This only occurs for angled incidence and so it is less commonly considered. Nevertheless, this normal contribution is important because it allows for simultaneous electric and magnetic dipoles in the same location. In this
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