Variational schemes and geometric simulations for a hydrodynamic-electrodynamic model of surface plasmon polaritons

Abstract

A class of variational schemes for the hydrodynamic-electrodynamic model of lossless free electron gas in a quasi-neutral background is developed for high-quality simulations of surface plasmon polaritons. The Lagrangian density of lossless free electron gas with a self-consistent electromagnetic field is established, and the dynamical equations with the associated constraints are obtained via a variational principle. Based on discrete exterior calculus, the action functional of this system is discretized and minimized to obtain the discrete dynamics. Newton-Raphson iteration and the biconjugate gradient stabilized method are equipped as a hybrid nonlinear-linear algebraic solver.Instead of discretizing the partial differential equations, the variational schemes have better numerical properties in secular simulations, as they preserve the discrete Lagrangian symplectic structure, gauge symmetry, and general energy-momentum density. Two numerical experiments were performed. The numerical results reproduce characteristic dispersion relations of bulk plasmons and surface plasmon polaritons, and the numerical errors of conserved quantities in all experiments are bounded by a small value after long term simulations.In the past two decades, there have been impressive developments and significant advancement in applications of Surface Plasmon Polaritons (SPPs), bringing many new ideas into traditional electromagnetics and optics, such as the lithography beyond the diffraction limit, chip-scale photonic circuits, plasmonic metasurfaces, bio-photonics, etc. [1][2][3][4][5][6][7][8][9][10]. In the field of metal optics, plasmonics focuses on the collective motions of free electron gas in metal with self-consistent and external electromagnetic fields whose first-principle model is the classical particle-field theory [1][2][3]. Direct applications of the first-principle model in macroscopic simulations face many obstacles, such as the nonlinearity, the multi-scale, and the huge degrees-of-freedom. As a simplification, linearized phenomenological models, e.g., the Drude-Lorentz (DL) model, are widely used to describe macroscopic plasmonic phenomena [2,3]. In a mesoscopic context, kinetic and hydrodynamic descriptions are basic physical models of plasmonics, which involve both the dynamics of free electron gas and an electromagnetic field. Therefore, high-quality numerical schemes and simulations based on the hydrodynamic model are necessary in plasmonic research.The physics of SPPs can be described by the free electron gas model whose dynamical equations are hydrodynamic and Maxwell's equations [1]. For Maxwell's equations, many numerical schemes, such as the Finite-Difference Time-Domain (FDTD) method, the Finite Element (FE) method, and the Method of Moments (MoM) have been developed, that are widely used in modern electromagnetic engineering, Radio Frequency (RF) and microwave engineering, terahertz engineering, optical engineering, metamaterial design, accelerator design, fusion engineering, radio astrophysics, geophysics...

Symmetries and Local Conservation Laws of Variational Schemes for the Surface Plasmon Polaritons

“…In this work we prove that the variational schemes constructed in Ref. 12 admit discrete local conservation laws. In Sec.II, the discrete local charge conservation law is proven via the gauge symmetry of the diacrete action functional, and the discrete Noether's theorem is obtained.…”

“…In order to preserve the Lagrangian symplectic structure of the model, we constructed a class of variational schemes in Ref. 12 , which are known as the structure-preserving geometric algorithms. The discrete Hamilton's principle based variational integrator developed by J. Marsden and M. West for Lagrangian systems provide us with an alternative numerical approach to achieve the preservation of discrete symplectic 2-form which was first realized by K. Feng in canonical Hamiltonian systems 9,10,[17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] .…”

“…The preservation of discrete local conservation laws and mathematical structures that are discrete analogs of the continuous ones admitted by the physical system, can be recognized as a standard 10,11 . In this work, we give a detailed discussion about the previously constructed variational schemes for the hydrodynamicelectrodynamic model of lossless free-electron gas in a quasi-neutral background which are used for simulating surface plasmon polaritons (SPPs) 12 . The discrete local conservation laws admitted by the variational schemes are derived, in addition to the preservation of Lagrangian symplectic structure, both of which ensure the good properties of the schemes in secular simulations.…”

“…The Lin's constraint factor µ recognized as a general Lagrangian coodinate is used to complete the vorticity structure 14,15 . By taking the total variation of the action functional S = T Ω Ldx 3 dt with fixed boundary and leading the variational derivatives of S with respect to the fields q = (n, v, A, φ, α, λ, µ) equal to 0, we obtain the Euler-Lagrange equations of lossless freeelectron gas as 12 ,…”

“…These desirable features and the conservation of Lagrangian symplectic structure proven in Ref. 12 make the algorithm a powerful tool in the study of SPPs using the hydrodynamic-electrodynamic model.…”