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Two‐dimensional materials such as graphene have become crucial components of most state‐of‐the‐art plasmonic devices. The possibility of not only generating plasmons in the terahertz regime, but also tuning them in real time via chemical doping or electrical gating make them compelling materials for engineers seeking to build accurate sensors. Thus, the faithful modeling of the propagation of linear waves in a layered, periodic structure with such materials at the interfaces is of paramount importance in many branches of the applied sciences. In this paper, we present a novel formulation of the problem featuring surface currents to model the two‐dimensional materials which not only is free of the artificial singularities present in related approaches, but also can be used to deliver a proof of existence, uniqueness, and analytic dependence of solutions. We advocate for a surface integral formulation which is phrased in terms of well‐chosen Impedance–Impedance Operators that are immune to the Dirichlet eigenvalues which plague the Dirichlet–Neumann Operators that appear in classical formulations. With a High‐Order Perturbation of Surfaces approach we are able to give a straightforward demonstration of this new well‐posedness result which only requires the verification that a finite collection of explicitly stated transcendental expressions be nonzero. We further illustrate the utility of this formulation by displaying results of a High‐Order Spectral numerical implementation which is flexible, rapid, and robust.

We present an efficient numerical method for simulating the scattering of electromagnetic fields by a multilayered medium with random interfaces. The elements of this algorithm, the Monte Carlo–transformed field expansion method, are (i) an interfacial problem formulation in terms of impedance-impedance operators, (ii) simulation by a high-order perturbation of surfaces approach (the transformed field expansions method), and (iii) efficient computation of the wave field for each random sample by forward and backward substitutions. Our perturbative formulation permits us to solve a sequence of linear problems featuring an operator that isdeterministic, and its LU decomposition matrices can be reused, leading to significant savings in computational effort. With an extensive set of numerical examples, we demonstrate not only the robust and high-order accuracy of our scheme for small to moderate interface deformations, but also how Padé summation can be used to address large deviations.

Graphene has transformed the fields of plasmonics and photonics, and become an indispensable component for devices operating in the terahertz to mid-infrared range. Here, for instance, graphene surface plasmons can be excited, and their extreme interfacial confinement makes them vastly effective for sensing and detection. The rapid, robust, and accurate numerical simulation of optical devices featuring graphene is of paramount importance and many groups appeal to Black-Box Finite Element solvers. While accurate, these are quite computationally expensive for problems with simplifying geometrical features such as multiple homogeneous layers, which can be recast in terms of interfacial (rather than volumetric) unknowns. In either case, an important modeling consideration is whether to treat the graphene as a material of small (but non-zero) thickness with an effective permittivity, or as a vanishingly thin sheet of current with an effective conductivity. In this contribution we ponder the correct relationship between the effective conductivity and permittivity of graphene, and propose a new relation which is based upon a concrete mathematical calculation that appears to be missing in the literature. We then test our new model both in the case in which the interface deformation is non-trivial, and when there are two layers of graphene with non-flat interfacial deformation.

Graphene is now a crucial component of many device designs in electronics and optics. Just like the noble metals, this single layer of carbon atoms in a honeycomb lattice can support surface plasmons, which are central to several sensing technologies in the mid-infrared regime. As with classical metal plasmons, periodic corrugations in the graphene sheet itself can be used to launch these surface waves; however, as graphene plasmons are tightly confined, the role of unwanted surface roughness, even at a nanometer scale, cannot be ignored. In this work, we revisit our previous numerical experiments on metal plasmons launched by vanishingly small grating structures, with the addition of graphene to the structure. These simulations are conducted with a recently devised, rapid, and robust high-order spectral scheme of the authors, and with it we carefully demonstrate how the plasmonic response of a perfectly flat sheet of graphene can be significantly altered with even a tiny corrugation (on the order of merely 5 nm). With these results, we demonstrate the primary importance of fabrication techniques that produce interfaces whose deviations from flat are on the order of angstroms.