Multiscale Spectral Element - Boundary Integral Method for Linear and Nonlinear Nano Optical Computation

In this work, a hybrid mixed order numerical framework is proposed for multiscale linear/ nonlinear nano optical computation. Starting from the principle of the spectral element boundary integral (SEBI) method, the mixed-order SEBI solver with homogeneous Green's function is first developed for the nano-scale linear and nonlinear electromagnetic scattering analyses. The SEBI realizes the exact radiation boundary condition with a set of surface integral equations (SIE's), and discretize the whole computation domain with the fast convergent Gauss-Lobatto-Legendre (GLL) basis function. The Bloch periodic boundary condition is applied for efficient simulation of structures with periodicities in one or two directions.

For nonlinear optical simulation, full-wave solver is developed self-consistently by iteratively solving the vector Helmholtz equations at each harmonic frequency. To further address the multiscale scattering analysis in nano optics, a hybrid framework is developed by combing the SEBI solver with the dyadic periodic layered medium Green's function (PLMGF) and the domain decomposition method (DDM). Formulating the SIE's with the dyadic PLMGF, all unknowns in the background layered medium are pushed to the radiation boundaries. Thus, the whole planar layered background can be truncated from the computation domain. Considering its highly singular analytical properties, the PLMGF is carefully and systematically formulated under matrix representation. A feasible and effective technique is proposed for the on-interface PLMGF singularity extractions. By extracting the primary and secondary terms' singularities separately, all PLMGF-related SIE components can be efficiently evaluated. The DDM further reduces the memory cost for electrically large problems and enhances the framework's flexibility. Finally, a scattered field perfectly matched layer - surface integral equation (PML-SIE) radiation boundary condition is proposed to enable the non-periodic modeling. With the hybrid radiation boundary condition, the periodic and non-periodic solvers are maximumly integrated together with the minimum maintenance cost.

Benefiting from the exponential convergence and flexibility of the SEBI, computationally challenging problem can be solved with considerably reduced number of samplings. As a typical application, the multilayer defects analysis in extreme ultraviolet (EUV) lithography is studied for both 2-D and 3-D models. The light absorption engineering of graphene is also investigated around the visible spectra. Benefiting from the accuracy of the full-wave nonlinear solver, couplings between the fundamental frequency (FF) field and the higher harmonic (HH) field ignored my most previous studies can also be self-consistently analyzed in nonlinear optical simulation. With this tool, the investigation is extended to the engineering of graphene's visible spectra absorption tuning and third harmonic generation (THG) enhancement. Graphene's Kerr effects are also studied under strong surface plasmonic resonance. The hybrid higher order method's efficiency and accuracy are further validated through various multiscale nano-optical cases.