Spectral Galerkin mode-matching method for applications in photonics.

Many engineered photonic devices can be decomposed into parts where the material properties are independent of one or more spatial variables. Numerical mode-matching methods are widely used to simulate such photonic devices due to the efficiency gained by treating the separated variables analytically. Existing mode-matching methods based on piecewise polynomials are more accurate than those based on the global Fourier basis or low-order finite difference, finite-element schemes, but they may exhibit numerical instability when a large number of eigenmodes are used. To overcome this difficulty, we introduce the spectral Galerkin mode matching method (SGMM) based on a global piecewise-polynomial basis and a Galerkin method to solve the eigenmodes. It is shown that the numerical eigenmodes of SGMM preserve the pseudo-orthogonality of the analytical eigenmodes. This property leads to linear systems that are typically well-conditioned. Numerical examples indicate that SGMM is more stable than other mode matching methods, and gives reliable results even when a large number of eigenmodes are used.

Non-generic bound states in the continuum in waveguides with lateral leakage channels.

For optical waveguides with a layered background which itself is a slab waveguide, a guided mode is a bound state in the continuum (BIC), if it coexists with slab modes propagating outwards in the lateral direction; i.e., there are lateral leakage channels. It is known that generic BICs in optical waveguides with lateral leakage channels are robust in the sense that they still exist if the waveguide is perturbed arbitrarily. However, the theory is not applicable to non-generic BICs which can be defined precisely. Near a BIC, the waveguide supports resonant and leaky modes with a complex frequency and a complex propagation constant, respectively. In this paper, we develop a perturbation theory to show that the resonant and leaky modes near a non-generic BIC have an ultra-high Q factor and ultra-low leakage loss, respectively. Recently, many authors studied merging-BICs in periodic structures through tuning structural parameters. It has been shown that resonant modes near a merging-BIC have an ultra-high Q factor. However, the existing studies on merging-BICs are concerned with specific examples and specific parameters. Moreover, we analyze an arbitrary structural perturbation given by δF(r) to waveguides supporting a non-generic BIC, where F(r) is the perturbation profile and δ is the amplitude, and show that the perturbed waveguide has two BICs for δ > 0 (or δ < 0) and no BIC for δ < 0 (or δ > 0). This implies that a non-generic BIC can be regarded as a merging-BIC (for almost any perturbation profile F) when δ is considered as a parameter. Our study indicates that non-generic BICs have interesting special properties that are useful in applications.

Robust and non-robust bound states in the continuum in rotationally symmetric periodic waveguides.

A fiber grating and a one-dimensional (1D) periodic array of spheres are examples of rotationally symmetric periodic (RSP) waveguides. It is well known that bound states in the continuum (BICs) may exist in lossless dielectric RSP waveguides. Any guided mode in an RSP waveguide is characterized by an azimuthal index m, the frequency ω, and Bloch wavenumber ß. A BIC is a guided mode, but for the same m, ω and ß, cylindrical waves can propagate to or from infinity in the surrounding homogeneous medium. In this paper, we investigate the robustness of nondegenerate BICs in lossless dielectric RSP waveguides. The question is whether a BIC in an RSP waveguide with a reflection symmetry along its axis z, can continue its existence when the waveguide is perturbed by small but arbitrary structural perturbations that preserve the periodicity and the reflection symmetry in z. It is shown that for m = 0 and m ≠ 0, generic BICs with only a single propagating diffraction order are robust and non-robust, respectively, and a non-robust BIC with m ≠ 0 can continue to exist if the perturbation contains one tunable parameter. The theory is established by proving the existence of a BIC in the perturbed structure mathematically, where the perturbation is small but arbitrary, and contains an extra tunable parameter for the case of m ≠ 0. The theory is validated by numerical examples for propagating BICs with m ≠ 0 and ß ≠ 0 in fiber gratings and 1D arrays of circular disks.

Cochlear implantation in a patient with congenital microtia, cochlear hypoplasia, venous anomalies of the temporal bone and laryngomalacia: Challenges and surgical considerations.

RATIONALE AND PATIENT CONCERNS: Congenital hearing loss is often caused by an inner ear malformation, in such cases, the presence of other anomalies, such as microtia, and venous anomalies of the temporal bone and laryngomalacia makes it challenging to perform cochlear implantation surgery. DIAGNOSES: This study reports the case of a 28-month-old girl with congenital profound hearing loss, laryngomalacia, and malformed inner ear, who received cochlear implantation surgery. The bony structure, vessels and nerves were first assessed through magnetic resonance imaging and computed tomography before exploring the genetic basis of the condition using trio-based whole exome sequencing. Perioperative evaluation and management of the airway was then performed by experienced anesthesiologist, with the surgical challenges as well as problems encountered fully evaluated. INTERVENTIONS: Cochlear implantation was eventually performed using a trans-mastoid approach under uneventful general anesthesia. OUTCOMES: Due to the small size of the cochlea, a short electrode FLEX24 was inserted through the cochleostomy. LESSONS: Considering the high risk of facial nerve injury and limited access to the cochlea when patients present significant bony and venous anomalies, cochlear implantation in such patients require careful preoperative evaluation and thoughtful planning. In these cases, airway assessment, magnetic resonance venography, magnetic resonance arteriography, and magnetic resonance imaging and computed tomography can be useful to minimize the risks. Intraoperative facial nerve monitoring is also recommended to assist in the safe location of facial nerve.

Computing diffraction anomalies as nonlinear eigenvalue problems.

When a plane electromagnetic wave impinges on a diffraction grating or other periodic structures, reflected and transmitted waves propagate away from the structure in different radiation channels. A diffraction anomaly occurs when the outgoing waves in one or more radiation channels vanish. Zero reflection, zero transmission, and perfect absorption are important examples of diffraction anomalies, and they are useful for manipulating electromagnetic waves and light. Since diffraction anomalies appear only at specific frequencies and/or wave vectors, and may require the tuning of structural or material parameters, they are relatively difficult to find by standard numerical methods. Iterative methods may be used, but good initial guesses are required. To determine all diffraction anomalies in a given frequency interval, it is necessary to repeatedly solve the diffraction problem for many frequencies. In this paper, an efficient numerical method is developed for computing diffraction anomalies. The method relies on nonlinear eigenvalue formulations for scattering anomalies and solves the nonlinear eigenvalue problems by a contour-integral method. Numerical examples involving periodic arrays of cylinders are presented to illustrate the new method.

Complex modes in optical fibers and silicon waveguides.

In an open optical waveguide, complex modes that are confined around the waveguide core and have a complex propagation constant may exist, even though the waveguide consists of lossless isotropic dielectric materials. However, the existing studies on complex modes are very limited. In this Letter, we consider circular fibers and silicon waveguides, study the formation mechanism of complex modes, and calculate the dispersion relations for several complex modes in each waveguide. For circular fibers, we also determine the minimum refractive-index ratio for the existence of complex modes. Our study fills a gap in optical waveguide theory and provides a basis for realizing potential applications of complex modes.

On the robustness of bound states in the continuum in waveguides with lateral leakage channels.

Bound states in the continuum (BICs) are trapped or guided modes with frequencies in radiation continua. They are associated with high-quality-factor resonances that give rise to strong local field enhancement and rapid variations in scattering spectra, and have found many valuable applications. A guided mode of an optical waveguide can also be a BIC, if there is a lateral structure supporting compatible waves propagating in the lateral direction; i.e., there is a channel for lateral leakage. A BIC is typically destroyed (becomes a resonant or a leaky mode) if the structure is slightly perturbed, but some BICs are robust with respect to a large family of perturbations. In this paper, we show (analytically and numerically) that a typical BIC in optical waveguides with a left-right mirror symmetry and a single lateral leakage channel is robust with respect to any structural perturbation that preserves the left-right mirror symmetry. Our study improves the theoretical understanding on BICs and can be useful when applications of BICs in optical waveguides are explored.

Complex modes in an open lossless periodic waveguide.

Guided modes of an open periodic waveguide, with a periodicity in the main propagation direction, are Bloch modes confined around the waveguide core with no radiation loss in the transverse directions. Some guided modes can have a complex propagation constant, i.e., a complex Bloch wavenumber, even when the periodic waveguide is lossless (no absorption loss). These so-called complex modes are physical solutions that can be excited by incident waves whenever the waveguide has discontinuities or defects. We show that the complex modes in an open dielectric periodic waveguide form bands, and the endpoints of the bands can be classified to a small number of cases, including extrema on dispersion curves of the regular guided modes, bound states in the continuum, degenerate complex modes, and special diffraction solutions with blazing properties. Our study provides an improved theoretical understanding of periodic waveguides and a useful guidance to their practical applications.

Computing resonant modes of circular cylindrical resonators by vertical mode expansions.

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.

Analyzing bowtie structures with sharp tips by a vertical mode expansion method.

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.

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.

Calculating corner singularities by boundary integral equations.

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.

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.

Vertical mode expansion method for analyzing elliptic cylindrical objects in a layered background.

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.

Nonlinear standing waves on a periodic array of circular cylinders.

A periodic array of parallel and infinitely long dielectric circular cylinders surrounded by air can be regarded as a simple two-dimensional periodic waveguide. For linear cylinders, guided modes exist continuously below the lightline in various frequency intervals, but standing waves, which are special guided modes with a zero Bloch wavenumber, could exist above the lightline at a discrete set of frequencies. In this paper, we consider a periodic array of nonlinear circular cylinders with a Kerr nonlinearity, and show numerically that nonlinear standing waves exist continuously with the frequency and their amplitudes depend on the frequency. The amplitude-frequency relations are further investigated in a perturbation analysis.

Efficient vertical mode expansion method for scattering by arbitrary layered cylindrical structures.

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.

Vertical mode expansion method for transmission of light through a single circular hole in a slab.

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.

Bilateral symmetry breaking in nonlinear circular cylinders.

Symmetry breaking is a common phenomenon in nonlinear systems, it refers to the existence of solutions that do not preserve the original symmetries of the underlying system. In nonlinear optics, symmetry breaking has been previously investigated in a number of systems, usually based on simplified model equations or temporal coupled mode theories. In this paper, we analyze the scattering of an incident plane wave by one or two circular cylinders with a Kerr nonlinearity, and show the existence of solutions that break a lateral reflection symmetry. Although symmetry breaking is a known phenomenon in nonlinear optics, it is the first time that this phenomenon was rigorously studied in simple systems with one or two circular cylinders.

Asymptotic solutions of eigenmodes in slab waveguides terminated by perfectly matched layers.

For numerical modeling of optical wave-guiding structures, perfectly matched layers (PMLs) are widely used to terminate the transverse variables of the waveguide. The PML modes are the eigenmodes of a waveguide terminated by PMLs, and they have found important applications in the mode matching method, the coupled mode theory, and so on. In this paper, we consider PML modes for two-dimensional slab waveguides. It is shown that the PML modes consist of perturbed propagating modes, perturbed leaky modes, and two infinite sequences of Berenger modes. High-order asymptotic solutions for the Berenger modes are derived using a systematic approach.

Efficient numerical method for analyzing optical bistability in photonic crystal microcavities.

Nonlinear optical effects can be enhanced by photonic crystal microcavities and be used to develop practical ultra-compact optical devices with low power requirements. The finite-difference time-domain method is the standard numerical method for simulating nonlinear optical devices, but it has limitations in terms of accuracy and efficiency. In this paper, a rigorous and efficient frequency-domain numerical method is developed for analyzing nonlinear optical devices where the nonlinear effect is concentrated in the microcavities. The method replaces the linear problem outside the microcavities by a rigorous and numerically computed boundary condition, then solves the nonlinear problem iteratively in a small region around the microcavities. Convergence of the iterative method is much easier to achieve since the size of the problem is significantly reduced. The method is presented for a specific two-dimensional photonic crystal waveguide-cavity system with a Kerr nonlinearity, using numerical methods that can take advantage of the geometric features of the structure. The method is able to calculate multiple solutions exhibiting the optical bistability phenomenon in the strongly nonlinear regime.