Wangtao Lu

Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations

For optical waveguides with high index-contrast and sharp corners, existing full-vectorial mode solvers including those based on boundary integral equations typically have only second or third order of accuracy. In this paper, a new full-vectorial waveguide mode solver is developed based on a new formulation of boundary integral equations and the so-called Neumann-to-Dirichlet operators for sub-domains of constant refractive index. The method uses the normal derivatives of the two transverse magnetic field components as the basic unknown functions, and it offers higher order of accuracy where the order depends on a parameter used in a graded mesh for handling the corners. The method relies on a standard Nystrom method for discretizing integral operators and it does not require analytic properties of the electromagnetic field (which are singular) at the corners.

Efficient Boundary Integral Equation Method for Photonic Crystal Fibers

Photonic crystal fibers (PCFs) with many air holes and complicated geometries can be difficult to analyze using conventional waveguide mode solvers such as the finite element method. Boundary integral equation (BIE) methods are suitable for PCFs, since they formulate eigenvalue problems only on the interfaces and are capable of computing leaky modes accurately. To improve the efficiency, it is desirable to have high-order BIE methods that calculate the minimum number of functions on the interfaces. Existing BIE methods calculate two or four functions on the interfaces, but high-order implementations are only available for those with four functions. In this paper, a new high-order BIE method is developed and it calculates two functions on the interfaces. Numerical results indicate that the new BIE method achieves exponential convergence and extremely high accuracy.

Babich's expansion and the fast Huygens sweeping method for the Helmholtz wave equation at high frequencies

In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation can be viewed as an evolution equation in one of the spatial directions. With such applications in mind, starting from Babichs expansion, we develop a new high-order asymptotic method, which we dub the fast Huygens sweeping method, for solving point-source Helmholtz equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of this method is that we develop a new Eulerian approach to compute the asymptotics, i.e. the traveltime function and amplitude coefficients that arise in Babichs expansion, yielding a locally valid solution, which is accurate close enough to the source. The second novelty is that we utilize the Huygens-Kirchhoff integral to integrate many locally valid wavefields to construct globally valid wavefields. This automatically treats caustics and yields uniformly accurate solutions both near the source and remote from it. The third novelty is that the butterfly algorithm is adapted to accelerate the Huygens-Kirchhoff summation, achieving nearly optimal complexity O ( N log ź N ) , where N is the number of mesh points; the complexity prefactor depends on the desired accuracy and is independent of the frequency. To reduce the storage of the resulting tables of asymptotics in Babichs expansion, we use the multivariable Chebyshev series expansion to compress each table by encoding the information into a small number of coefficients.The new method enjoys the following desired features. First, it precomputes the asymptotics in Babichs expansion, such as traveltime and amplitudes. Second, it takes care of caustics automatically. Third, it can compute the point-source Helmholtz solution for many different sources at many frequencies simultaneously. Fourth, for a specified number of points per wavelength, it can construct the wavefield in nearly optimal complexity in terms of the total number of mesh points, where the prefactor of the complexity only depends on the specified accuracy and is independent of frequency. Both two-dimensional and three-dimensional numerical experiments have been carried out to illustrate the performance, efficiency, and accuracy of the method.

Efficient High Order Waveguide Mode Solvers based on Boundary Integral Equations

Abstract For optical waveguides with high index contrast and sharp corners, high order full-vectorial mode solvers are difficult to develop, due to the field singularities at the corners. A recently developed method (the so-called BIE-NtD method) based on boundary integral equations (BIEs) and Neumann-to-Dirichlet (NtD) maps achieves high order of accuracy for dielectric waveguides. In this paper, we develop two new BIE mode solvers, including an improved version of the BIE-NtD method and a new BIE-DtN method based on Dirichlet-to-Neumann (DtN) maps. For homogeneous domains with sharp corners, we propose better BIEs to compute the DtN and NtD maps, and new kernel-splitting techniques to discretize hypersingular operators. Numerical results indicate that the new methods are more efficient and more accurate, and work very well for metallic waveguides and waveguides with extended mode profiles.

High order integral equation method for diffraction gratings

Conventional integral equation methods for diffraction gratings require lattice sum techniques to evaluate quasi-periodic Greens functions. The boundary integral equation Neumann-to-Dirichlet map (BIE-NtD) method in Wu and Lu [J. Opt. Soc. Am. A 26, 2444 (2009)], [J. Opt. Soc. Am. A 28, 1191 (2011)] is a recently developed integral equation method that avoids the quasi-periodic Greens functions and is relatively easy to implement. In this paper, we present a number of improvements for this method, including a revised formulation that is more stable numerically, and more accurate methods for computing tangential derivatives along material interfaces and for matching boundary conditions with the homogeneous top and bottom regions. Numerical examples indicate that the improved BIE-NtD map method achieves a high order of accuracy for in-plane and conical diffractions of dielectric gratings.

Complex modes and instability of full-vectorial beam propagation methods

Full-vectorial beam propagation methods (FVBPMs) are widely used to model light waves propagating in high-index-contrast optical waveguides. We show that the paraxial FVBPM and wide-angle FVBPMs based on diagonal Padé approximants are analytically unstable for waveguides with complex modes. The instability cannot be removed by enlarging the computational domain, increasing the numerical resolution, or using perfectly matched layers, because the complex modes are highly confined around the waveguide core.

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...

Eulerian Geometrical Optics and Fast Huygens Sweeping Methods for Three-Dimensional Time-Harmonic High-Frequency Maxwell's Equations in Inhomogeneous Media

In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that Maxwells equations may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, we propose a new Eulerian geometrical-optics method, dubbed the fast Huygens sweeping method, for computing Greens functions of Maxwells equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of the fast Huygens sweeping method is that a new dyadic-tensor-type geometrical-optics ansatz is proposed for Greens functions which is able to utilize some unique features of Maxwells equations. The second novelty is that the Huygens--Kirchhoff secondary source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics associated with the usual geometrical-optics ansatz can be treated automatically. The third novelty is that a butterfly algorithm is adapted to carry out the matrix-vector products induced by the Huygens--Kirchhoff integration in

Extending Babich's Ansatz for Point-Source Maxwell's Equations Using Hadamard's Method

O(N\log N)

Babich-Like Ansatz for Three-Dimensional Point-Source Maxwell's Equations in an Inhomogeneous Medium at High Frequencies

operations, where