A new finite-difference-based method for wide-angle beam propagation

A novel split-step finite-difference method for wide-angle beam propagation is presented. The formulation allows solution of the second-order scalar wave equation without having to make the slowly varying envelope and one-way propagation approximations. The method is highly accurate and numerically efficient requiring only simple matrix multiplication for propagation.     

The finite-difference vector beam propagation method: analysis andassessment

The newly developed finite-difference vector beam propagation method (FD-VBPM) is analyzed and assessed for application to two-dimensional waveguide structures. The general formulations for the FD-VBPM are derived from the vector wave equations for the electric fields. The stability criteria, the numerical dissipation, and the dispersion of the finite-difference schemes are analyzed by applying the von Neumann method. Important issues regarding the implementation, such as the choice of reference refractive index, the application of numerical boundary conditions, and the use of numerical solution schemes, are discussed. The FD-VBPM is assessed by calculating the attenuation coefficients and the percentage errors of the propagation constants of the TE and TM modes of a step-index slab waveguide. Several salient features of the FD-VBPM are illustrated     

Study and Design of Spiral Bent Waveguide Configuration

A new version of the scalar transverse electric（TE） wave equation in the bent waveguide is introduced. Then. TE polarized field in curved single-mode waveguides is analyzed by using the finite- difference beam propagation method（FD-BPM）. The bending loss in bent waveguides is gotten for the optical fields obtained from BPM and comparisons are made among losses of the waveguides with various curvature radiuses, refractive index differences and cross sections. Based on the results, the design of spiral bent waveguide configuration is proposed as follows： refractive index difference being of 0. 007, both width and thickness of waveguides being of 6 μm, the curvature radius in the spiral centre being of 4 mm, and the bending loss coefficient of the designed spiral bent waveguide being of 0. 302 3 dB/cm.     

用有限差分光束传输法分析过渡波导损耗

为了减小弯曲过渡波导和斜坡过渡波导的功耗,设计研究了几种过渡波导形状函数,并用有限差分光束传输法进行了模拟分析,发现过渡波导形状的选取与坡度和曲折角存在一定的条件关系,经总结给出了该条件关系表达式;并且,在分析计算不同形状过渡波导损耗的过程中,凸现出了有限差分光束传播法在光波导器件及其单元器件模拟分析中的直观性与方便性。     

Optimization design of optical waveguide in Mach-Zehnder electro-optical polymer modulator

In order to reduce transmission loss of the optical waveguide in Mach-Zehnder (M-Z) electro-optical (EO) polymer modulator,the basic iterative formula of semi-vector finite-difference beam propagation method (FD-BPM) is obtained from the scalar wave equation. The transition waveguide is combined with S-type bend branch waveguide for the M-Z EO modulator in the branch waveguide. The effects of structure parameters such as ridge width,length of the branch waveguide and interferometer spacing on the transmission loss are systematically studied by using the semi-vector FD-BPM method. The structure is optimized as an S-sine bend branch waveguide,with rib width w=7μm,length of branch waveguide L=1200μm and interferometer spacing G=22 μm. The results show that the optimized structure can reduce transmission loss to 0.083 dB,which have a certain reference value to the design of optical waveguide in M-Z polymer modulator.     

Waveguides of Arbitrary Cross Section by Solution of a Nonlinear Integral Eigenvalue Equation

The problem of electromagnetic wave propagation in hollow conducting waveguides of arbitrary cross section is formulated as an integro-differential equation in terms of fields at the waveguide boundary. Cutoff wave numbers and wall currents appear as eigenvalues and eigenfunctions of a nonlinear eigenvalue problem involving an integro-differential operator. A variational solution is effected by reducing the problem to matrix form using the method of moments. A specific solution of the problem is developed using triangle expansion functions in the method of moments. The solution is simplified by symmetry considerations and is implemented by two digital computer programs. Listings and full documentation of these programs are available. This solution yields accurate determinations of cutoff wave numbers, wall currents, and distributions of both longitudinal and transverse modal field components for the first several modes. Illustrative computations are presented for the single-ridge waveguide, which has a complicated boundary shape that does not lend itself to exact solution.     

A vector beam propagation method for guided-wave optics

A vector beam propagation method (BPM) for the modeling and simulation of electromagnetic wave propagation in optical guided-wave devices is proposed. The vector BPM is based on a finite-difference scheme which takes into account the boundary conditions of the transverse electric fields over the waveguide cross section. By incorporating the vector boundary conditions in the finite-difference scheme, the polarization property of the propagation electromagnetic waves is included in the analysis. The method is applied to a directional coupler made of parallel slabs, and is shown to be in excellent agreement with the exact solutions.<>     

A study and optimization of eigenmode calculations using theimaginary-distance beam-propagation method

Recently, it has been shown that if the paraxial wave equation is modified such that fields travel imaginary distances, the field resulting from an imaginary-distance propagation is the fundamental mode of an optical waveguide. For the finite-difference beam-propagation method, we derive the factor by which each eigenmode is amplified during one propagation step. This amplification factor places limits on the inputs-step size, input field, effective index guess, and implicitness parameter-and gives clues on how to optimize the inputs. In particular, we identify and study two optimal sets of inputs, which can reduce computational time significantly. We can obtain the fundamental mode and its propagation constant within a few propagation steps     

A novel beam propagation method for large refractive index steps and large propagation distance

Within the accuracy of the scalar equation, a novel beam propagation method (BPM) is proposed for the calculation of integrated optical waveguide structures with large refractive index steps Delta n. Propagation increments in excess of a wavelength are possible for Delta n approximately=2.5. In this way, high stability and accuracy can be maintained over a large total propagation length. The calculation time is kept low by using a finite-difference scheme.<>     

带有增益介质包层的两个平行圆柱形纳米金属棒构成的表面等离子体光波导的数值模拟

设计了一种带有增益介质包层的两个平行圆柱形纳米金属棒构成的表面等离子体光波导,基于频域有限差分法,对这种波导所支持的基模的能流密度分布、有效折射率、传播长度和模式面积随几何结构参数和电磁参数的依赖关系进行了分析。结果表明,沿纵向的能流主要分布在两个圆柱形金属棒所形成的中间区域。通过调节这种波导的几何参数及电磁参数,可以调节模式的传播特性。在增益介质的辅助下,传播距离明显增大。这种表面等离子体光波导可以用于光子器件集成领域和传感器领域。     

A joint finite-difference and FFT full vectorial beam propagationscheme

A full vectorial beam propagation scheme is developed and it is applied on 3-dimensional waveguide structures. The formulation is based on the coupled wave equations for the transverse electric field. Each propagation step is performed by utilizing both the FFT and a finite-difference implementation. Under this perspective the offered advantages of FFT and finite-differences are exploited within a single propagation step resulting in a joint propagation scheme. The scheme is applied on a step-index circular fiber where analytical solutions are readily available for cross-checking. Moreover, the dependence of the phase constant on the reference refractive index is discussed. The polarized modes and the effective mode indices are derived in the case of rib waveguides by performing propagation along imaginary distance. Further, the rib waveguide coupler is examined and the energy transfer is simulated     

A compact 2-D full-wave finite-difference frequency-domain methodfor general guided wave structures

A compact two-dimensional (2-D) full-wave finite-difference frequency-domain method is proposed for the analysis of dispersion characteristics of a general guided wave structure. Because the longitudinal field components are eliminated in the proposed method, only four transverse field components are involved in the final resulting eigen equation. This feature considerably reduces the required CPU time as compared to the existing approaches by which six field components are comprised. Additionally, unlike other 2-D finite-difference schemes that determine the eigenfrequency for a given propagation constant, the new method finds the propagation constant β for a given k_o (frequency). The new method has been verified by examining the computed results of a number of typical guided wave structures with the published results. Very good agreement is achieved     

Propagation of soliton waves in nonlinear dielectric slab waveguides of parabolic index of refraction

The pulse propagation in a non-linear slab waveguide of parabolic index of refraction is treated by using differential equation techniques. A graded index dielectric slab waveguide free of material dispersion with a cubic order non-linearity is considered. The electromagnetic wave inside the waveguide is described in terms of a non-linear equation. Slowly varying envelope function representation is employed to develop a non-linear partial differential equation for the unknown envelope function of the electric field. An averaging method over the transverse direction is applied to reduce the unknown envelope function non-linear differential equation into a form resembling the well known non-linear Schrödinger differential equation. This equation is solved by applying the Inverse Scattering Method. The N-soliton solution is developed and presented explicitly for the practical case of the single mode dielectric slab waveguide. Numerical results presenting single and double soliton propagation are also given.     

Light propagation characteristics through the annular coupled-cavity waveguides based on the two-dimensional square-lattice photonic crystal

The light propagation characteristics through the annular coupled-resonator cavity waveguides are systematically analyzed by the finite-difference time-domain (FDTD) method. It is found that this kind of waveguide has more minbands owing to the increasing of the cavity’s size, compared with the traditional line-typed coupled-resonator waveguide. The group velocity of light propagation can be reduced for a further degree when the adjacent annular cavities are interlaced in the perpendicular direction, and a group velocity about 0.00067c (c is the light speed in vacuum) can be obtained.     

Investigation of Terahertz Wave Propagation Along Shielded Dielectric Multiple-Slot Waveguide

In this paper, a shielded dielectric multiple-slot waveguide is presented and demonstrated by theoretical calculation at terahertz frequencies. Some electromagnetic analysis of waveguide currently employed simplified intrinsic frequency dispersion models for the bulk conductivity of normal metals used in terahertz wave structures. This paper has compared various conductivity models for gold between 0.1-3 THz. The dielectric loss, conductor loss, total propagation loss with different conductivity models and E-field distribution has been obtained by numerically solving the complex eigenvalue equation for the propagation constant. The analysis results are in agreement with the assumption of R. Sun et al. and the experimental results obtained by H. Sun et al. The propagation loss deviation between various models is less than 3% at 0.2-0.5 THz. Comparisons with other slot waveguides are also given. The analysis results show that the proposed line has lower propagation attenuation than other slot waveguides.     

Higher order formulation of absorbing boundary conditions for finite-difference time-domain method

A simple formulation of absorbing boundary conditions with higher order approximation is proposed for the finite-difference time-domain (FD-TD) method. Although this formulation is based on the third order approximation of the one-way wave equations the authors have succeeded in reducing it to an equation in a form quite similar to the second order approximation.<>     

Propagation and coupling properties of integrated opticalwaveguides-an integral equation formulation

The propagation and coupling properties of integrated optical waveguides are analyzed by means of the electric field integral equation approach. The kernel of the integral equation is the Green's function of a two-layered medium. The Galerkin's method is then employed to solve the integral equation numerically. The set of basis and test functions consists of entire domain plane wave functions. Fast convergence and superior accuracy are the advantages of the chosen set of basis and test functions. The method is used to compute the propagation and coupling properties of several structures. Very good agreement is observed with previously published results. Field distributions of several coupled mode structures, such as the symmetrical and asymmetrical coupler are also investigated and presented. Finally, the same method is used to produce the field distribution of waveguides having more complex cross section like the trapezoidal waveguide     

The study of the transmission of GHz waves.in the oceanic low altitude waveguide in the PEM-based gridded method

A parabolic equation method （PEM）-based discrete algorithm is proposed and is used to obtain the field distribution in the evaporation duct space. This method not only improves the computing speed, but also provides the flexibility to adjust the simulation accuracy. Numerical simulation of the wave propagation in the oceanic waveguide structure is done. In addition, the initial field distribution and progressive steps are determined. The loss model in the waveguide is solved through the numerical solution. By comparing the characteristics of the radio wave propagation in the duct and in the normal atmospheric structure, we analyses the radio transmission over the horizon detection in the oceanic waveguide.     

Finite-difference approach to the solution of time-domain integralequations for layered structures

A numerical algorithm for the analysis of transient electromagnetic fields in planar structures is proposed based on the time-domain magnetic-field integral equation (MFIE), electric-field integral equation (EFIE), and the marching-on-in-time approach. The field vectors are represented in terms of vector potential functions which are calculated either by integration or by the three-dimensional (3-D) wave equation according to the geometry of the structure. Thus, the algorithm combines the advantages of integral equation techniques and finite-difference schemes. While this approach is applicable to any geometries, it is especially suitable for multilayered planar structures and is competitive to the finite-difference time-domain (FDTD) method in the case of open and radiating problems. Theoretical results are verified by the analysis of a pulse propagation in a homogeneous open-end microstrip line     

Numerical analysis of dielectric-rod waveguides with deepcorrugation

Dielectric-rod waveguides with deep surface corrugation is analyzed by using the finite-difference time-domain method. In order to mitigate the load in numerical calculation and to analyze a wide region, the method is formulated in two-dimensional form. It is shown that disks located in a line function as a waveguide of low loss in a frequency region. The procedure of deciding the propagation constant from the analyzed field is presented. The complex propagation constant is obtained and the dependence of the amount of radiation on the corrugation amplitude is discussed