Fourier modal method formulation for fast analysis of two-dimensional periodic arrays of graphene

Abstract

Recently, an approximate boundary condition [Opt. Lett. 38, 3009 (2013)] was proposed for fast analysis of onedimensional periodic arrays of graphene ribbons by using the Fourier modal method (FMM). Correct factorization rules are applicable to this approximate boundary condition where graphene is modeled as surface conductivity. We extend this approach to obtain the optical properties of two-dimensional periodic arrays of graphene. In this work, optical absorption of graphene squares in a checkerboard pattern and graphene nanodisks in a hexagonal lattice are calculated by the proposed formalism. The achieved results are compared with the conventional FMM, in which graphene is modeled as a finite thickness dielectric layer. We show that for the same truncation order, computation time can be reduced to one-ninth by the proposed formulation in comparison with the conventional FMM. Furthermore, the convergence rate is increased. Therefore, thanks to the improved convergence rate and reduced computational cost for a given truncation order, the computational time is saved more than 100 times for relative error of less than 1%. This is crucially important in analyzing two-dimensional periodic structures of graphene by the FMM.

Theoretical and computational analysis of second- and third-harmonic generation in periodically patterned graphene and transition-metal dichalcogenide monolayers

“…A further justification of our nonzero conductivity model is provided by the application of Ampere's law for a loop around the interface, an approach that has been recently introduced elsewhere [49,55]. This alternative model also amounts to using a modified conductivity at the interface similar to that in Eq.…”

“…However, in addition to previous work on handling 2DMs by means of Fourier series expansion methods [49,55], we also model nonlinear optical effects in 2D materials, in particular secondharmonic generation (SHG) and THG, and provide the mathematical formulation for arbitrary nonlinear optical processes. In addition, the method introduced in this work allows one to describe structured 2D materials, which themselves can be embedded in inhomogeneous 3D optical structures.…”

Analytical Circuit Model for Periodic Arrays of Graphene Disks

“…Note also that the surface current density has no component normal to the circular boundary of the disk [12]:…”

“…Equation (12) implies that resonances occur at frequencies where k p (ω) ∼ 2λ mn . Near each resonance frequency the distribution of current density is approximately given by the corresponding eigenfunction.…”

“…Take note that we neglect the effect of the perturbation on eigenfunctions by assuming that ς 1n ∼ ξ 1n . Therefore, the induced current density of a periodic array can be obtained from (9) and (12) where λ 1n is replaced by the modified eigenvalues q 1n . In this case, the resonances will be observed near the modified frequencies obtained from 2q 1n = Re(k p (ω)).…”