The unitarity of split-operator finite difference andfinite-element methods: Applications to longitudinally varyingsemiconductor rib waveguides

Split-operator finite-difference and finite-element techniques are applied to the calculation of losses in semiconductor rib waveguide Y junctions. It is shown that, unlike the finite-difference procedure, which is unitary for both uniform and nonuniform grid-point spacings, considerable care must be exercised in formulating the split-operator finite-element method in order to preserve the power in the propagating electric field. The calculations are performed in the context of the Fresnel approximation to the scalar Helmholtz equation, which yields accurate solutions for transverse electric (TE)-polarized electric fields in rib waveguides far from cutoff     

The semivectorial beam propagation method

Details of an efficient semivectorial beam propagation method are presented. The propagation is determined by solving the finite-difference equation in alternating directions. Unlike the traditional split-operator beam propagation method, in the semivectorial method the index term is not separated from the spatial differentiations. Since boundary conditions at dielectric interfaces can be satisfied by two orthogonal polarizations individually, the method is semivectorial. The performance can be enhanced by using a nonuniform grid. By modifying the routine, it can also be applied to periodic structures. Numerical results for rib waveguides are presented     

Application of modified finite-difference formulas to the analysisof z-variant rib waveguides

Modified finite-difference formulas for a general position of an interface are applied to the propagating beam analysis of z-variant rib waveguides. The modified formula based on the semivectorial H-field is found to be more insensitive to variation in an interface position than that on the E-field. A discretization error is satisfactorily reduced in tilted and tapered rib waveguides     

Improved full-vectorial beam propagation method with high accuracy for arbitrary optical waveguides

An improved finite-difference (FD) full-vectorial beam propagation method is introduced for arbitrary optical waveguides with a dramatic improvement in accuracy. This method is developed based on the generalized Douglas scheme and a novel FD formula for the cross-coupling terms which, expressed in explicit form, is independent of specific types of waveguides. The present method is demonstrated for a strongly guiding rib waveguide.     

Spectral index method for polarized modes in semiconductor ribwaveguides

A fast way of modeling polarized modes in semiconductor rib waveguides is presented and shown to be in excellent agreement with a semivectorial finite-difference computer program. The method uses little storage and has been implemented on a personal computer     

The design of triple rib waveguide couplers by the discretespectral index method

The operations of three-guide couplers are usually controlled by the width and refractive index of the guides. However, when rib waveguides are used in three-guide couplers and power dividers, we propose to use the dimensions of the ribs as the governing parameters instead. Formulating a new transcendental equation using the discrete spectral index (DSI) method, triple rib waveguide couplers are accurately analyzed for the first time. The new accurate and fast analysis of triple rib waveguides shows that the propagation constants of the various modes are dominantly controlled by the height of the middle rib. A systematic and simple procedure is consequently introduced to design quasi-transverse electric (TE) polarized triple rib waveguide couplers, which ensures maximum power transfer from one outermost guide to the other outermost guide. For triple rib power dividers, the height of the middle rib is found to control the amount of power that can be transferred from the excited middle rib waveguide to the two outer rib waveguides. Maximum power division is achieved when the three ribs of the waveguide are identical     

High-accuracy finite-difference equations for dielectric waveguide analysis II: dielectric corners

For part I see ibid., p. 1210, 2002. We present a discussion of the behavior of the electric and magnetic fields satisfying the two-dimensional Helmholtz equation for waveguides in the vicinity of a dielectric corner. Although certain components of the electric field have long been known to be infinite at the corner, it is shown that all components of the magnetic field are finite, and that finite-difference equations may be derived for these fields that satisfy correct boundary conditions at the corner. These finite-difference equations have been combined with those derived in the previous paper to form a full-vector waveguide solution algorithm of unprecedented accuracy. This algorithm is employed to provide highly accurate solutions for the fundamental modes of a previously studied standard rib waveguide structure.     

A Compact 2-D Finite-Difference Frequency-Domain Method for Dispersion Characteristics Analysis of Trapezoidal-Ridge Waveguides

A compact two-dimension (2-D) finite-difference frequency-domain (FDFD) method is used to analyze the dispersion characteristics of single and double trapezoidal-ridge waveguides. The general 2-D FDFD formulation under orthogonal curvilinear coordinate system is derived from the difference form of Maxwell’s equations, and modified difference formulas at the trapezoidal-ridge section are built. After implementing the boundary conditions, the 2-D FDFD formulation is concluded as an eigen equation and then constructed by a highly sparse matrix. By solving the matrix-eigen equation, the dispersion characteristics of the ridged waveguides can be obtained. Computed results are in good agreements with the previously published and simulated ones, which prove the correctness of the method.     

Semivectorial Helmholtz beam propagation by Lanczos reduction

By using the finite-difference method to perform Lanczos reduction, a semivectorial Helmholtz beam propagation algorithm is demonstrated. The applicability of this algorithm is no longer limited to paraxial beams and scalar fields. Mode indices of rib waveguides are calculated and compared to previously published data. Losses of Y -branch for two orthogonal polarizations are also presented. This algorithm is more efficient than the conventional beam propagation method in determining the mode index. To calculate the radiation loss however, it requires much more computational effort. More than 30 Krylov vectors are needed to avoid numerical dissipation     

An Extension of the Effective Index Method to Analyze Leakage Losses in Rib Type Waveguides

An extension of the effective index method is proposed as a tool to investigate leakage losses in two-dimensional waveguides. The above method, referred to as the Extended Effective Index Method (EEIM), utilizes the Transfer Matrix Technique (TMT) for the calculation of the complex propagation constants. The results show that the EEIM can be applied to conventional rib leaky waveguides as well as to rib ARROW leaky waveguides. The results for rib ARROW waveguides show excellent agreement when compared to those obtained with Finite Element Method.     

任意截面波导特性的非正交频域有限差分研究

本文提出了一种求解波导本征问题的非正交频域有限差分(FD-FD)算法。该算法中的数值离散网格可与任意复杂边界形状吻合一致。文中通过所建立的本征方程对波导中各模式的截止频率进行了求解,数值结果与已有结果符合良好。     

Efficient beam propagation techniques

It is shown that suitable modifications of the free-space-propagation operator in the standard fast-Fourier-transform beam propagation procedure yield a far more rapid code in the context of semiconductor rib waveguide calculations. The procedure curtails the diffraction experienced by the high-order Fourier components of the electric field and mirrors the structure of the recently introduced finite-difference beam propagation algorithm. It is used to investigate the propagation losses of a Y-junction composed of single-mode rib waveguides and illuminated by its normalized guided mode     

A wavelet formulation of the finite-difference method: full-vectoranalysis of optical waveguide junctions

We have developed an efficient, large-stencil finite-difference scheme of the time-dependent Maxwell's curl equations based on the wavelet-collocation formulation in the time-domain. The proposed scheme enables, for the first time within a limited computational resource, full-vector analysis of three-dimensional rib waveguides that are typically used in integrated planar optical devices. The formulation takes advantage of compactly-supported interpolating bases to expand and represent the electric and magnetic fields. Moreover, unlike the well-known beam propagation methods, the numerical scheme is based on the first-principle algorithm with no explicit approximation, and thus rigorous and versatile for various types of boundary conditions. We demonstrate the efficiency of the method by first analyzing a straight rib-waveguide and examining the convergence of the results. Then we investigate a Y-shaped junction structure that is electrically too large to analyze with the conventional finite-difference time-domain scheme     

半矢量模式求解在光波导弯曲损耗计算中的应用

以柱坐标下的半矢量波动方程为基础,采用基于完美匹配层(PML)边界条件的有限差分方法,对弯曲波导进行模式求解,进而得到波导弯曲引起的辐射损耗.基于计算得到的直弯波导的模场分布,采用二维重叠积分法计算了两者连接时的过渡损耗.计算结果与已有实验结果符合较好.采用该方法,研究了SOI脊型波导的弯曲损耗与波导结构参数之间的关系,并对直弯波导的连接进行了优化.     

A hybrid implicit-explicit FDTD scheme for nonlinear opticalwaveguide modeling

A hybrid implicit-explicit finite-difference time-domain (FDTD) method for solving the wave equation in nonlinear optical waveguiding structures is proposed. The new scheme combines the computational simplicity of the explicit scheme in linear medium regions with the superior stability property of the partially implicit scheme in regions of nonlinear materials, thus eliminating potential problems of instability associated with nonlinearity. Simulation results for Kerr-type nonlinear slab waveguides and corrugated waveguides are presented and compared with those obtained using the conventional noniterative FDTD scheme     

Characteristic equation and properties of electron waves in periodic structures

The finite-difference theory of excitation of periodic waveguides is used to derive the characteristic equation for electron waves formed during interaction of an electron beam with forward and counter-propagating electromagnetic waves of periodic waveguides (slow-wave structures). The derived characteristic equation describes interaction of electrons and waves in passbands and stopbands of periodic waveguides and contains known solutions for ??smooth?? slow-wave structures and resonator slow-wave structures near cutoff frequencies as particular cases. Several analytical solutions allowing comparison of amplification and propagation properties of electron waves inside and at the edges of passbands and stopbands of periodic waveguides are found.     

Radiation losses of tapered dielectric waveguides: a finitedifference analysis with ridge waveguide applications

The finite-difference third-order simplified wave equation method is used to study the radiation losses of optical waveguides with step (abrupt) and parabolic tapers, including the radiation losses of the tapered ridge waveguides used as electro-optic phase modulators for tuning the frequencies of infrared lasers. The radiation losses of the parabolic taper given by Burns et al. is found to be under 5%. The results of a step taper are compared with Marcuse's; and found to be in good agreement. The numerical results are also compared with those from the coupled wave equations used by Sporloder and Unger     

Analysis of metallic waveguides with rectangular boundaries byusing the finite-difference method and the simultaneous iteration withthe Chebyshev acceleration

A numerical procedure based on the finite-difference method and simultaneous iteration of the power method in conjunction with the Chebyshev acceleration technique is utilized to analyze the metallic waveguides. Due to the efficiency of the present sparse matrix eigenproblem solver, lots of unknowns can be used in the domains of the waveguide cross-sections. Therefore, accurate cutoff wavenumbers or frequencies can be obtained by using the simple finite-difference method for the commonly used metallic waveguides such as the L-shaped, single-ridged, double-ridged, and rectangular coaxial waveguides. Some discrepancies with the numerical results in the recent literature are found and detailed discussions are provided to verify the correctness of the present results     

A new semirigorous analysis of rib waveguides

A semirigorous analysis of rib waveguides embedded in air is proposed. The analysis is based upon a scalar analysis and two different vectorial correction schemes, through either a perturbation formula or a scalar calculation with modified effective dimensions. The scalar field is computed with a very efficient spectral technique. The method converges rapidly, and it compares favorably with recent work on rib waveguides. It can be extended to other planar structures, such as ridge waveguides     

离散谱折射率法优化设计深刻蚀、单模GaAs/GaAlAs脊形光波导

采用离散谱折射率法对深刻蚀GaAs/GaAlAs多层脊形光波导的特性作了详细的理论分析,并对所获得的较大截面、低损耗的单模脊形光波导的制作容差性作了进一步的分析．计算表明,用离散谱折射率法获得的单模脊形光波导具有较大的制作容差性．