Ma Luo

Spectral Methods and Domain Decomposition for Nanophotonic Applications

Nanophotonic applications often involve large-scale problems with excessive demand on computational resources. We develop a domain decomposition method (DDM) to reduce computer memory and central processing unit (CPU) time requirements by combining the spectral element method (SEM) and the spectral integral method (SIM) for large-scale finite periodic structures. The interior scattering subdomains within each period are modeled by the SEM while the exterior scattering problem is modeled by the SIM. The interactions between neighboring subdomains are modeled by the frequency-domain version of the Riemann solver. Numerical convergence of the Riemann solver is fast and weakly dependent on the size of the system. Two sets of examples demonstrate the typical nanophotonic applications: The first periodic system is a vertical coupling waveguide based on a photonic crystal slab which opens a way to construct and simulate optical circuits. The second periodic system is a finite-sized metamaterial with an effective negative refractive index, whose edge effects are visualized and analyzed.

Three-dimensional dispersive metallic photonic crystals with a bandgap and a high cutoff frequency.

The goal of this work is to analyze three-dimensional dispersive metallic photonic crystals (PCs) and to find a structure that can provide a bandgap and a high cutoff frequency. The determination of the band structure of a PC with dispersive materials is an expensive nonlinear eigenvalue problem; in this work we propose a rational-polynomial method to convert such a nonlinear eigenvalue problem into a linear eigenvalue problem. The spectral element method is extended to rapidly calculate the band structure of three-dimensional PCs consisting of realistic dispersive materials modeled by Drude and Drude-Lorentz models. Exponential convergence is observed in the numerical experiments. Numerical results show that, at the low frequency limit, metallic materials are similar to a perfect electric conductor, where the simulation results tend to be the same as perfect electric conductor PCs. Band structures of the scaffold structure and semi-woodpile structure metallic PCs are investigated. It is found that band structures of semi-woodpile PCs have a very high cutoff frequency as well as a bandgap between the lowest two bands and the higher bands.

Accurate determination of band structures of two-dimensional dispersive anisotropic photonic crystals by the spectral element method

The spectral element method (SEM) is used to calculate band structures of two-dimensional photonic crystals (PCs) consisting of dispersive anisotropic materials. As in the conventional finite element method, for a dispersive PC, the resulting eigenvalue problem in the SEM is nonlinear and the eigenvalues are in general complex frequencies. We develop an efficient way of incorporating the dispersion in the system matrices. The band structures of a PC with a square lattice of dispersive cylindrical rods are first analyzed. The imaginary part of the complex frequency is the time-domain decay rate of the eigenmode, which is very useful for tracing a band from discrete numerical data. Modification of the band structure of TE mode by an external static magnetic field in the out-of-plane direction is explored for this square lattice. A plasmon resonance mode is found near the plasmon frequency when the magnetic field is nonzero. The band structure of a PC with a triangular lattice is also calculated with the SEM. Other types of lattices can also be treated readily by the SEM.

Enhanced plasmonic light absorption engineering of graphene: simulation by boundary-integral spectral element method

Graphenes relatively poor absorption is an essential obstacle for designing graphene-based photonic devices with satisfying photo-responsivity. To enhance the tunable light absorption of graphene, appropriate excitation of localized surface plasmon resonance is considered as a promising approach. In this work, the strategy of incorporating periodic cuboid gold nanoparticle (NP) cluster arrays and cylindrical gold NP arrays with Bragg reflectors into graphene-based photodetectors are theoretically studied by the boundary-integral spectral element method (BI-SEM). With the BI-SEM, the models can be numerically analyzed with excellent accuracy and efficiency. Numerical simulation shows that the proposed structures can effectively engineer the light absorption in graphene by tuning plasmon resonance. In the spectra of 300 nm to 1000 nm, a maximum light absorption of 67.54% is observed for the graphene layer with optimal parameters of the photodetector model.

Enhancement of graphene’s third-harmonic generation with localized surface plasmon resonance under optical/electro-optic Kerr effects

Although graphene’s particularly strong third-order susceptibility has drawn intensive attention in theoretical and experimental studies, its low bulk nonlinear response heavily emphasizes the nanostructure’s design for a sufficient magnitude of third-harmonic generation (THG). Meanwhile, currently few tools are available for accurate theoretical analyses of graphene’s nonlinear performance within a relatively complex structure, which renders the design of graphene-based nonlinear optoelectronic devices even more challenging. In this work, a high-accuracy self-consistent numerical solver based on the boundary-integral spectral element method is first proposed for the THG problem. Starting from the coupled vector wave equations, the proposed solver solves for the fundamental frequency field and third-harmonic field together iteratively, and it covers the optical/electro-optic Kerr effects ignored by most previous THG studies. After validating the proposed method with the comparison between numerical results and experimental data, we extend our study to the THG enhancement strategy with ultrastrong localized surface plasmon resonances (LSPRs) and Kerr effects. For both optical and electro-optic Kerr effects, the systematic simulation is performed for graphene’s THG within the incident spectra of 400–1000 nm. Compared with the THG of floating single-atom-layer graphene, numerical results show that under specific LSPR engineering, graphene’s THG backward emission is enhanced by 4.4×105 times. Simultaneously applying the electro-optic Kerr process can further boost the THG emission. However, its contribution is only secondary compared with LSPR. This study is also extended to bilayer and trilayer graphene models under strong LSPR.

Boundary integral spectral element method analyses of extreme ultraviolet multilayer defects

Extreme ultraviolet (EUV) lithography is an emerging technology for high-density semiconductor patterning. Multilayer distortion caused by mask defects is regarded as one of the critical challenges of EUV lithography. To simulate the influence of the defected nanoscale structures with high accuracy and efficiency, we have developed a boundary integral spectral element method (BI-SEM) that combines the SEM with a set of surface integral equations. The SEM is used to solve the interior computational domain, while the open boundaries are truncated by the surface integral equations. Both two-dimensional (2D) and three-dimensional (3D) EUV cases are simulated. Through comparing the performance of this method with the conventional finite element method (FEM), it is shown that the proposed BI-SEM can greatly decrease both the memory cost and the computation time. For typical 2D problems, we show that the BI-SEM is 11 and 1.25 times more efficient than the FEM in terms of memory and CPU time, respectively, while for 3D problems, these factors are over 14 and 2, respectively, for smaller problems; realistic 3D problems that cannot be solved by the conventional FEM can be accurately simulated by the BI-SEM.

A spectral element method calculation of extraordinary light transmission through periodic subwavelength slits

A spectral element method together with a surface integral equation as the radiation boundary condition is used to simulate the scattering properties of periodic subwavelength slits. The surface integral equation utilizes the periodic Greens function in the wave number space and is solved by the method of moments, while the interior inhomogeneous medium is modeled by the spectral element method. The solution convergence is found to be exponential; i.e., the error decreases exponentially with the order of basis functions. To our knowledge, such a fast solver with spectral accuracy is new in the scattering problem of periodic structures. Scattering properties of a gold slit grid within the whole wavelength-incidence angle parameter space are investigated, with the confirmation that strong transmission of light through subwavelength slits is achievable.

Enhancement of second-harmonic generation in an air-bridge photonic crystal slab: simulation by spectral element method

The enhancement of second-harmonic generation (SHG) across a 2D photonic crystal (PC) slab consisting of GaP is investigated in this study with the three-dimensional spectral element method (SEM). The in-plane band structure is calculated, and it is compared with the peaks of the SHG to reveal the mechanisms behind the enhancement. The numerical result from the SEM shows that, under normal incidence, the scattered power of the SHG is enhanced for the eigenstates with large decay rates, while the stored energy of the SHG is enhanced for the eigenstates with a zero decay rate. The SHG is enhanced under two conditions: (i) phase matching between the fundamental and second-harmonic (SH) fields and (ii) symmetry matching between the field pattern of the resonant eigenstate and the generated SH polarization field. Compared with a homogeneous dielectric slab, the air-bridge PC slab can enhance the SHG by 4 orders of magnitude.

The Auxiliary Differential Equations Perfectly Matched Layers Based on the Hybrid SETD and PSTD Algorithms for Acoustic Waves

The perfectly matched layer (PML) absorbing boundary condition has been proven to absorb body waves and surface waves very efficiently at non-grazing incidence. However, the traditional PML would generate large spurious reflections at grazing incidence, for example, when the sources are located near the truncating boundary and the receivers are at a large offset. In this paper, a new PML implementation is presented for the boundary truncation in three-dimensional spectral element time domain (SETD) for solving acoustic wave equations. This method utilizes pseudospectral time-domain (PSTD) method to solve first-order auxiliary differential equations (ADEs), which is more straightforward than that in the classical FEM framework.

Self-consistent analyses of third harmonic generation enhancement with compact plasmonics

The nonlinear plasmonic process is investigated for the enhancement of third harmonic generation (THG) with a self-consistent numerical solver. Quantitive analyses show that, the nonlinear plasmonics behaves as an effective strategy for THG. In addition, the proposed numerical approach can effectively analyze the nonlinear plasmonic process, where the conventional method is not robust in general.