Full-vectorial finite-difference beam propagation method based on the modified alternating direction implicit method

A modified alternating direction implicit algorithm is proposed to solve the full-vectorial finite-difference beam propagation method formulation based on $H$ fields. The cross-coupling terms are neglected in the first sub-step, but evaluated and doubly used in the second sub-step. The order of two sub-steps is reversed for each transverse magnetic field component so that the cross-coupling terms are always expressed in implicit form, thus the calculation is very efficient and stable. Moreover, an improved six-point finite-difference scheme with high accuracy independent of specific structures of waveguide is also constructed to approximate the cross-coupling terms along the transverse directions. The imaginary-distance procedure is used to assess the validity and utility of the present method. The field patterns and the normalized propagation constants of the fundamental mode for a buried rectangular waveguide and a rib waveguide are presented. Solutions are in excellent agreement with the benchmark results from the modal transverse resonance method.     

Complete analysis of photonic crystal fibers by full-vectorial 2D-FDTD method

In this paper, an extended finite difference time domain (FDTD) algorithm for the full-vectorial analysis of photonic crystal fibers has been derived. For achieving a good convergence and high accuracy, a kind of modified conformal FDTD method has been applied. An anisotropic perfectly matched layer for truncation of boundary conditions has been introduced. Material and chromatic dispersions are numerically investigated for the photonic crystal fibers with different dimensions and geometrical parameters and different dispersion behaviors are exhibited.     

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 Nyström method for discretizing integral operators and it does not require analytic properties of the electromagnetic field (which are singular) at the corners.     

利用超格子模型对布拉格光纤的研究

为了研究布拉格光纤的模式特征和传输特性,提出了超格子模型,利用傅里叶级数表示光纤横向折射率分布,利用平面波展开法分析布拉格光纤的能带结构,基于厄米-主斯函数的局域正交函数展开法,从全矢量耦合波动方程出发,得到关于模式传播常量和电场展开系数的本征方程,从而分析布拉格光纤的模式特征。以高折射率芯布拉格光纤为例,实现了该算法,得到基模与二次模的横向电场分布、基模色散曲线和模式双折射。基模的模式双折射可用于衡量算法的精度,结果表明该算法精度较高。超格子模型不仅可以用于研究高折射率芯布拉格光纤,而且同样可以研究低折射率区域导光的布拉格光纤。     

基于插值的全矢量有限差分法求解阶跃光波导本征模式

给出了一种适合求解阶跃折射率分布光波导本征模式的全矢量有限差分法.该方法充分考虑了电场分量在折射率分布跃变处的不连续性, 在拉格朗日插值的基础上得到了一种精度较高的差分格式.为了验证其有效性,对一种典型结构、折射率阶跃分布的波导作了计算,并与其它方法的结果作了比较.最后对不理想刻蚀情况做了模拟计算.     

Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method

A full-vectorial mode solver in terms of the transverse magnetic field components for optical waveguides with transverse anisotropy is described by using the multidomain spectral collocation method based on Chebyshev polynomials. The waveguide cross section surrounded by the perfectly matched layers is divided into suitable number of homogeneous rectangles, and then connected with by imposing the continuities of the longitudinal field components at the dielectric interfaces shared by the adjacent rectangles, resulting in a generalized matrix eigenvalue problem. To validate the established method, results of an anisotropic square waveguide and a magnetooptic rib waveguide are presented and compared with those from the full-vectorial finite difference method, full-vectorial beam propagation method, and the experimental data.     

A Modified full-vectorial finite-difference beam propagation method based on H-fields for optical waveguides with step-index profiles

A modified full-vectorial finite-difference beam propagation method based on H-fields in solving the guided-modes for optical waveguides with step-index profiles is described. The propagation is split into two substeps. In the first substep, the field propagates in the absence of the cross-coupling terms, and then they are evaluated and double used in the second substep. An improved six-point finite-difference scheme is constructed to approximate the cross-coupling terms along the transverse directions. By using the imaginary-distance procedure, the field patterns and the normalized propagation constants of the fundamental modes for a buried rectangular waveguide and rib waveguide are presented, and the hybrid nature of the full-vectorial guided-modes is demonstrated. Solutions are in good agreement with the benchmark results from film mode matching method, which tests the validity and utility of the present method.     

A novel Fourier based 3D full-vectorial beam propagation method

In this paper a Fourier based full-vectorial beam propagation method (BPM) with perfectly matched layer (PML) absorbing boundary conditions is presented. The method starts with the slowly varying envelope coupled wave equations, including coordinate-stretching type PMLs, which are discretised using the Galerkin method with Fourier basis functions in the transversal direction. The proposed method is then assessed in a wide variety of situations, including 2D propagation of TE and TM waves and 3D full-vectorial situations. The obtained results show that good accuracy can be obtained even for complex strongly-guiding waveguides with a small number of basis functions.     

A search-and-track algorithm for controlling the number of guided modes of planar optical waveguides with arbitrary refractive index profiles

A search-and-track algorithm is proposed for controlling the number of guided modes of planar optical waveguides with arbitrary refractive index profiles. The algorithm starts with an initial guess point in the parameter space that supports a specific number of guided modes. Then, it searches for, and tracks, the boundaries of this space or another space supporting different number of modes. It does so by monitoring the sign of a unified cutoff dispersion function. The algorithm is applied to both symmetric and asymmetric silicon-based parabolic-index waveguides. It shows that unlike asymmetric waveguides, the single-mode condition of symmetric waveguides is controlled by TM-, as opposed to TE-, polarization. This abnormal polarization control is strongest for high index contrast waveguides of sub-micrometer core sizes. The results are verified by full-vectorial beam propagation method.     

用基于磁场的改进型有限差分法分析光波导的全矢量本征模

提出一种改进的有限差分法,用以求解全矢量磁场波方程,分析光波导承载的全矢量本征模.离散交叉项时,采用与波导结构无关的六点差分格式,考虑磁场分量的导数在芯包分界处的不连续性,比传统四点差分格式有更高的计算精度.分析阶跃型光纤、矩形和脊形光波导的全矢量本征模问题,给出全矢量基模的磁场分布及其归一化传播常数,揭示全矢量本征模的混合特性,所得结果与解析法、高精度模横向谐振法的结果吻合,验证了方法的收敛性与计算精度.     

High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces

This work overcomes the difficulty of the previous matched interface and boundary (MIB) method in dealing with interfaces with non-constant curvatures for optical waveguide analysis. This difficulty is essentially bypassed by avoiding the use of local cylindrical coordinates in the improved MIB method. Instead, novel jump conditions are derived along global Cartesian directions for the transverse magnetic field components. Effective interface treatments are proposed to rigorously impose jump conditions across arbitrarily curved interfaces based on a simple Cartesian grid. Even though each field component satisfies the scalar Helmholtz equation, the enforcement of jump conditions couples two transverse magnetic field components, so that the resulting MIB method is a full-vectorial approach for the modal analysis of optical waveguides. The numerical performance of the proposed MIB method is investigated by considering interface problems with both constant and general curvatures. The MIB method is shown to be able to deliver a fourth order of accuracy in all cases, even when a high frequency solution is involved.     

Transmission loss and dispersion in plastic terahertz photonic band-gap fibers

We present an investigation into the transmission loss and dispersion of terahertz waves in plastic photonic band-gap fibers having a cladding with a finite number of air hole rings. The leakage loss and absorption loss caused by background material are analyzed by a full-vectorial two-dimensional finite difference frequency domain method and the lowest power transmission loss of 6.126 dB/m at 1.75 THz is realized. Numerical results show that a larger diameter-to-pitch ratio is suitable for lower transmission loss and lower group-velocity dispersion in plastic terahertz photonic band-gap fibers. PACS 41.20.Jb; 41.20.Cv; 42.81.Dp     

Analysis of air-guiding photonic bandgap fibers

We present what is to our knowledge the first theoretical analysis of air-guiding photonic bandgap fibers. The fibers are characterized by a large hollow core and a microstructured cladding exhibiting photonic bandgap effects. Using an efficient, full-vectorial numerical method, we explain the operational principle of the fibers and obtain detailed information about the properties of the air-guided modes. This information includes accurate determination of the modes' spectral extent, cutoff properties, and mode-field distributions.     

Modeling of graded index waveguide fabricated by ion exchange on Er3+ doped glass

A general numerical tool,based on thermal diffusion equation and full-vectorial eigen-mode equation,has been presented for the systematic analysis of graded index channel waveguide fabricated by ion exchange on Er3+ doped glass.Finite difference method with full-vectorial formulation(FV-FDM) is applied to solving the full-vectorial modes of graded index channel waveguide for the first time.The coupled difference equations based on magnetic fields in FV-FDM are derived from the Taylor series expansion and accurate formulation of boundary conditions.Hybrid nature of vectorial guided modes for both pump(980 nm) and signal light(1550 nm) are demonstrated by the simulation.Results show that the fabrication parameters of ion exchange,such as channel opening width and time ratio of second step to first step in ion exchange,have large influence on the properties of waveguide.By optimizing the fabrication parameters,maintenance of monomode for signal light and improvement of the gain dynamics can be achieved in Er3+ doped waveguide amplifier(EDWA) fabricated by ion exchange technique.This theoretical model is significant for the design and fabrication of EDWA with ion exchange technique.Furthermore,a single polarization EDWA,which operates at wavelength from 1528 nm to 1541 nm for HE polarization,is numerically designed.     

Beam propagation analysis using higher-order full-vectorial finite-difference method

We develop a beam propagation method based on full-vectorial higher-order finite-difference formulation to model optical waveguides. Simulation of a photonic crystal fiber coupler is used as assessment of different schemes and convergence behaviors. Results including error convergence and field calculation show that they are improved by using higher-order scheme.     

Full-vectorial analysis of optical waveguides by the finite difference method based on polynomial interpolation

Based on the polynomial interpolation, a new finite difference (FD) method in solving the full-vectorial guided-modes for step-index optical waveguides is proposed. The discontinuities of the normal components of the electric field across abrupt dielectric interfaces are considered in the absence of the limitations of scalar and semivectorial approximation, and the present FD scheme can be applied to both uniform and non-uniform mesh grids. The modal propagation constants and field distributions for buried rectangular waveguides and optical rib waveguides are presented. The hybrid nature of the vectorial modes is demonstrated and the singular behaviours of the minor field components in the corners are observed. Moreover, solutions are in good agreement with those published early, which tests the validity of the present approach.     

Near fields and far fields generated by sources in the presence of dielectric structures with cylindrical symmetry

A full-vectorial integral equation method is presented for calculating near fields and far fields generated by sources in the presence of general finite-sized dielectric structures with cylindrical symmetry. The method is relevant for modeling of a class of antenna designs and some optical components with cylindrical symmetry, e.g., vertical-cavity surface-emitting lasers, microdisk lasers, and light-emitting diodes.     

改进的全矢量有效折射率方法分析光子晶体光纤的色散特性

得到了全矢量有效折射率方法中归一化等效纤芯半径(p/Λ)随归一化空气孔直径(d/Λ)和归一化频率(Λ/λ)的变化关系，修正了p/Λ为固定值的不合理取法，得到了改进的全矢量有效折射率方法(IFVEIM).用IFVEIM计算的基模有效折射率和总色散等模式特性与多极法的结果和试验值相比较都具有很好的一致性，将基模有效折射率的误差减小了2—3个数量级.用IFVEIM详细地分析了光子晶体光纤的色散特性，得到了色散值、波长、孔间距、d/Λ之间的相互变化关系. 关键词： 光子晶体光纤 有效折射率方法 等效纤芯半径 色散     

Determination of temperature distribution and control parameter in a two-dimensional parabolic inverse problem with overspecified data

In this paper, we present a new algorithm to solve a two-dimensional parabolic inverse problem with a source parameter, which appears in many physical phenomena. A linearized compact difference scheme for this problem is constructed using the finite difference method. The discretization accuracy is second-order in time and fourth-order in space. We obtain the unique solvability and present an alternating direction implicit algorithm to solve this difference scheme. The results of numerical experiments are presented to demonstrate the accuracy of this algorithm.     

Characteristic analysis of a directional coupler in horizontal multiple-slotted silicon wires with slanted sidewalls

The characteristics of the directional couplers based on the horizontal multiple-slotted waveguides with slanted sidewalls are analyzed in detail by using a mode solver based on the full-vectorial finite element method with perfectly matched layers absorbing boundary conditions. The coupling length of the directional coupler as functions of the structural parameters including waveguide spacing, sidewalls angle, the thickness and refractive index of the slot regions, both in quasi-TE and quasi-TM modes, is obtained. The effective indexes and the field distributions of the even and odd modes also are presented. The numerical results show that polarization-independent directional couplers can be realized by properly choosing structural and material parameters.