RESUMO
Open subwavelength cylindrical resonators of finite height are widely used in various photonics applications. Circular cylindrical resonators are particularly important in nanophotonics, since they are relatively easy to fabricate and can be designed to exhibit different resonance effects. In this paper, an efficient and robust numerical method is developed for computing resonant modes of circular cylinders which may have a few layers and may be embedded in a layered background. The resonant modes are complex-frequency outgoing solutions of the Maxwell's equations with no sources or incident waves. The method uses field expansions in one-dimensional (1D) "vertical" modes to reduce the original three-dimensional eigenvalue problem to 1D problems and uses Chebyshev pseudospectral method to compute the 1D modes and set up the discretized eigenvalue problem. In addition, a new iterative scheme is developed so that the 1D nonlinear eigenvalue problems can be reliably solved. For metallic cylinders, the resonant modes are calculated based on analytic models for the dielectric functions of metals. The method is validated by comparisons with existing numerical results, and it is also used to explore subwavelength dielectric cylinders with high-Q resonances and analyze gold nanocylinders.
RESUMO
Bowtie structures of metallic nanoparticles are very effective in producing strong local fields needed in many applications. Existing numerical studies on bowtie structures are limited to those with rounded tips. Due to the field singularities at sharp edges and corners, accurate numerical solutions for bowtie structures with mathematically sharp tips are difficult to obtain. Based on an improved vertical mode expansion method (VMEM) that incorporates boundary integral equation techniques for domains with corners, we analyze bowtie structures with truly sharp tips. Numerical results are presented to reveal the effects of a few key factors including the distance between the tips, the apex angle and the substrate.
RESUMO
Accurate numerical solutions for electromagnetic fields near sharp corners and edges are important for nanophotonics applications that rely on strong near fields to enhance light-matter interactions. For cylindrical structures, the singularity exponents of electromagnetic fields near sharp edges can be solved analytically, but in general the actual fields can only be calculated numerically. In this paper, we use a boundary integral equation method to compute electromagnetic fields near sharp edges, and construct the leading terms in asymptotic expansions based on numerical solutions. Our integral equations are formulated for rescaled unknown functions to avoid unbounded field components, and are discretized with a graded mesh and properly chosen quadrature schemes. The numerically found singularity exponents agree well with the exact values in all the test cases presented here, indicating that the numerical solutions are accurate.
RESUMO
Due to the existing nanofabrication techniques, many periodic photonic structures consist of different parts where the material properties depend only on one spatial variable. The vertical mode expansion method (VMEM) is a special computational method for analyzing the scattering of light by structures with this geometric feature. It provides two-dimensional (2D) formulations for the original three-dimensional problem. In this paper, two VMEM variants are presented for biperiodic structures with cylindrical objects of circular or general cross sections. Cylindrical wave expansions and boundary integral equations are used to handle the 2D Helmholtz equations that appear in the vertical mode expansion process. A number of techniques are introduced to overcome some difficulties associated with the periodicity. The method is relatively simple to implement and highly competitive in terms of efficiency and accuracy.
RESUMO
The vertical mode expansion method (VMEM) [J. Opt. Soc. Am. A31, 293 (2014)] is a frequency-domain numerical method for solving Maxwell's equations in structures that are layered separately in a cylindrical region and its exterior. Based on expanding the electromagnetic field in one-dimensional vertical modes, the VMEM reduces the original three-dimensional problem to a two-dimensional (2D) problem on the vertical boundary of the cylindrical region. However, the VMEM has so far only been implemented for structures with circular cylindrical regions. In this paper, we develop a VMEM for structures with an elliptic cylindrical region, based on the separation of variables in the elliptic coordinates. A key step in the VMEM is to calculate the so-called Dirichlet-to-Neumann (DtN) maps for 2D Helmholtz equations inside or outside the ellipse. For numerical stability reasons, we avoid the analytic solutions of the Helmholtz equations in terms of the angular and radial Mathieu functions, and construct the DtN maps by a fully numerical method. To illustrate the new VMEM, we analyze the transmission of light through an elliptic aperture in a metallic film, and the scattering of light by elliptic gold cylinders on a substrate.
RESUMO
A relatively simple and efficient numerical method is developed for analyzing the scattering of light by a layered cylindrical structure of arbitrary cross section surrounded by a layered background. The method significantly extends an existing vertical mode expansion method (VMEM) for circular or elliptic cylindrical structures. The original VMEM and its extension give rise to effective two-dimensional formulations for the three-dimensional scattering problems of layered cylindrical structures. The extended VMEM developed in this paper uses boundary integral equations to handle the two-dimensional Helmholtz equations that appear in the vertical mode expansion process. The method is applied to analyze the transmission of light through subwavelength apertures in metallic films and the scattering of light by metallic nanoparticles.
RESUMO
An efficient method is developed for rigorously analyzing the scattering of light by a layered circular cylindrical object in a layered background, and it is applied to the study of the transmission of light through a subwavelength hole in a metallic film, where the hole may be filled by a dielectric material. The method relies on expanding the electromagnetic field (subtracted by one-dimensional solutions of the layered media) in one-dimensional modes, where the expansion "coefficients" are functions satisfying two-dimensional Helmholtz equations. A system of equations is established on the boundary of the circular cylinder to solve the expansion "coefficients." The method effectively reduces the original three-dimensional scattering problem to a two-dimensional problem on the boundary of the cylinder.
