Ya Yan Lu

Standing waves on two-dimensional periodic dielectric waveguides

Guided modes of a periodic waveguide usually exist below the light line, if the minimum period of the waveguide is used in the definition, but for some periodic waveguides, there are standing waves with the same period as the waveguide. These non-propagating waves localized around the waveguide core are special guided modes above the light line with a zero wavenumber, and they are related to transmission anomalies and other resonant phenomena. In this paper, we analyze the standing waves on two periodic waveguides: a periodic array of circular dielectric cylinders, and a dielectric slab with a periodic array of circular air-holes. Based on an efficient semi-analytic method, the frequencies of standing waves are calculated as functions of the dielectric constant and the radius of cylinders. Our work provides a basis for further studies on these waves and for realizing their potential applications.

Bound states in the continuum on periodic structures: perturbation theory and robustness

On periodic structures, a bound state in the continuum (BIC) is a standing or propagating Bloch wave with a frequency in the radiation continuum. Some BICs (e.g., antisymmetric standing waves) are symmetry protected, since they have incompatible symmetry with outgoing waves in the radiation channels. The propagating BICs do not have this symmetry mismatch, but they still crucially depend on the symmetry of the structure. In this Letter, a perturbation theory is developed for propagating BICs on two-dimensional periodic structures. The Letter shows that these BICs are robust against structural perturbations that preserve the symmetry, indicating that these BICs, in fact, are implicitly protected by symmetry.

Propagating bound states in the continuum at the surface of a photonic crystal

Bound states in the continuum (BICs) are trapped or guided modes with their frequencies within the radiation continuum. On periodic structures, BICs have interesting properties and potentially important applications. It is known that BICs can exist at the surface of a photonic crystal (PhC), and they are distinctively different from the well-known surface Bloch modes below the lightline. However, for a given structure with specific geometric and material parameters, it is difficult to predict whether BICs exist or not. In this paper, using an efficient computational method, we calculate BICs at the surface of a two-dimensional PhC consisting of dielectric rods, and determine the existence domain in the plane of the refractive index and the radius of the surface rods. The boundary of the existence domain reveals that the BICs cease to exist when the bulk PhC can no longer confine light. In addition, the frequency and wavenumber of the BIC can approach the lightline, leading to bound states on the lightline and special highly confined surface Bloch modes below the lightline.

Vertical mode expansion method for numerical modeling of biperiodic structures

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.

Robust iterative method for nonlinear Helmholtz equation

Abstract A new iterative method is developed for solving the two-dimensional nonlinear Helmholtz equation which governs polarized light in media with the optical Kerr nonlinearity. In the strongly nonlinear regime, the nonlinear Helmholtz equation could have multiple solutions related to phenomena such as optical bistability and symmetry breaking. The new method exhibits a much more robust convergence behavior than existing iterative methods, such as frozen-nonlinearity iteration, Newtons method and damped Newtons method, and it can be used to find solutions when good initial guesses are unavailable. Numerical results are presented for the scattering of light by a nonlinear circular cylinder based on the exact nonlocal boundary condition and a pseudospectral method in the polar coordinate system.

Sensitivity analysis for photonic crystal microcavities

Resonant modes in photonic crystal microcavities with large quality factors and small mode volumes are important in many applications, but they are very sensitive to geometric and physical parameters of the structure. In this paper, we develop an efficient method for computing the partial derivatives of the complex resonant frequency with respect to parameters such as radii, refractive indices, and positions of the circular cylinders, for two-dimensional photonic crystal microcavities. Like the adjoint variable method for sensitivity analysis, our method is capable of rapidly calculating the partial derivatives with respect to a large number of geometric and material parameters. The method is efficient, since it takes advantage of the many identical unit cells in photonic crystal devices and the analytic solutions for circular cylindrical structures.

New unconditionally stable ADI schemes for the wave equation

Unconditionally stable ADI (alternating direction implicit) methods are useful in time-domain numerical simulations, when the restriction on the step size in time is too severe if the standard explicit time-domain numerical methods are used. For two dimensional structures, the Maxwells equations can be reduced to a scalar wave equation for the TE (transverse electric) or TM (transverse magnetic) polarizations.

Perfectly Matched Layer Boundary Integral Equation Method for Wave Scattering in a Layered Medium

For scattering problems of time-harmonic waves, the boundary integral equation (BIE) methods are highly competitive since they are formulated on lower-dimension boundaries or interfaces and can automatically satisfy outgoing radiation conditions. For scattering problems in a layered medium, standard BIE methods based on Greens function of the background medium need to evaluate the expensive Sommerfeld integrals. Alternative BIE methods based on the free-space Greens function give rise to integral equations on unbounded interfaces which are not easy to truncate since the wave fields on these interfaces decay very slowly. We develop a BIE method based on the perfectly matched layer (PML) technique. The PMLs are widely used to suppress outgoing waves in numerical methods that directly discretize the physical space. Our PML-based BIE method uses the PML-transformed free-space Greens function to define the boundary integral operators. The method is efficient since the PML-transformed free-space Greens func...

Analyzing bowtie structures by a vertical mode expansion method

The vertical mode expansion method (VMEM) is a special numerical method for scattering problems involving cylindrical structures in a layered background. The method is implemented for structures where the cross sections have sharp corners, and used to analyze bowtie metallic nanoparticle structures with sharp corners.

Sensitivity of photonic crystal waveguide-cavity systems with respect to cylinder radii

Sensitivity analysis is important for evaluating the performance of practical devices which are affected by fabrication errors. It is also very useful in optimal designs for various structures and devices. Based on the Dirichlet-to-Neumann map method, which is an efficient numerical method for modelling photonic crystal devices, we perform a sensitivity analysis for photonic crystal waveguide-cavity systems.