Scaling of the nonlinear response of the surface plasmon polariton at a metal/dielectric interface

Plasmonic systems involve interfaces containing both a metal and a dielectric material. In an effort to investigate the scaling of the nonlinear response of the surface plasmon polariton at a metal/dielectric interface, where both the metal and the dielectric present optical nonlinearity, we introduce a figure-of-merit that quantifies the contribution of the metal and the dielectric to the nonlinear response in this specific situation. In the case of self-action of the surface plasmon polariton for the gold/dielectric interface, we predict that the dielectric nonlinear response is dominant for strongly nonlinear dielectrics such as polydiacetylenes, chalcogenide glasses or even semiconductors. The gold nonlinear response is dominant only in cases involving weakly nonlinear dielectrics such as silicon dioxide or aluminium oxide. We verify the relevance of the metric by investigating the process of optical switching via the third-order nonlinear response and discuss which gold/dielectric combinations have better switching behaviors. Nonlinear optics has enabled a variety of applications, including ultrafast lasers and amplifiers [1], optical frequency conversion [2] and nonlinear microscopy [3]. Another potential application that has garnered considerable interest in the past decades is all-optical switching [4]. Driven by the goal of making efficient nonlinear devices with low control powers or high sensitivities, efforts have focused on studying nonlinear optics at the nanoscale, notably in plasmonic systems. Surface plasmon polaritons (SPPs) are oscillations of charge density waves coupled to photons at a metal-dielectric interface [5]. SPPs stand out compared to other optical modes because their confinement is not restricted by the diffraction limit. As a result, SPPs concentrate fields at subwavelength scales to enhance the nonlinear response in metallic nanoparticles as well as on extended metal surfaces [6-8]. For almost two decades now, many theoretical proposals and experimental realizations have studied the nonlinear response of plasmonic systems, giving rise to the field of nonlinear plasmonics. Furthermore, the field of nonlinear metamaterials, in which artificial materials can be designed and fabricated that enhance nonlinear phenomena at optical frequencies, overlaps with nonlinear plasmonics [9]. Studies of nonlinear plasmonic systems or metamaterials include nonlinear plasmonic waveguides [10-15], nonlinear self-action [16], active control [17], nanofocusing [18], nonlinear switching [19,20] and four-wave mixing [21]. As was recently pointed out by several groups [22, 23], most of these studies make the assumption that the response of the metal is linear and that the nonlinearity only originates from the dielectric medium adjacent to the metal. De Leon et al. [22] and Marini et al. [23] argue that past experimental work has shown that the third-order nonlinear susceptibility χ of metals can be enormous and can exhibit ultrafast behavior at optical wavelengths; this opens up the possibility of designing nonlinear plasmonic devices based on the intrinsic nonlinear response of the metallic component of the structure. There is a great deal of uncertainty regarding the value of χ (3) for metals notably for gold, for instance due to the strong frequency dispersion of metal nonlinear susceptibilities at wavelengths below 800 nm. Past measurements of χ (3) have exhibited a variation in magnitude of more than four orders of magnitude, with a strong dependence on pulse duration [24]. In spite of this, it is important to determine the scaling of the nonlinear response from both the metal and the dielectric for a range of experimental conditions such as laser wavelength, pulse duration, metal/dielectric combination and geometry. For this reason, we introduce here an analytical figure-of-merit to assess the dominant nonlinear response of a nonlinear plasmonic system. We apply this metric to the simplest case of a plasmonic waveguide: a single metal/dielectric interface and focus on the case of gold to determine whether the dielectric or the gold nonlinear response dominates for several dielectrics. We study self-action of the SPP through the thirdorder nonlinearity and use full-wave numerical simulations to determine which nonlinear responses dominate in an optical switching scenario. Structure investigated and analytical derivation of the figure of merit. In the following, we consider solely a metal/dielectric interface as sketched in Fig. 1a. It consists of a semi-infinite metal slab that occupies the region y < 0 that is adjacent to a dielectric that occupies the region y > 0. The transverse profiles of the yand x-component of the electric field are sketched on the figure. The spatial dependence of the surface plasmon polariton (SPP) field propagating in the positive x direction is given by SPP , ) ( , ), (x, y), , ( ( ) z x y x y x y E E x y H ψ =     and may be written in the form [25]