Accuracy of three-point finite difference approximations for optical waveguides with step-wise refractive index discontinuities

A rigorous truncation error analysis of three-point finite difference approximations for optical waveguides with step-wise refractive index discontinuities is given. As the basis for the analysis we use the exact coefficients of the series that expresses the field value at a given finite difference node in terms of the field value and its derivatives at a neighbouring node. This series is applied to develop a rigorous formalism for the truncation error analysis of the three-point finite difference approximations used in the numerical modelling of light propagation in optical waveguides with step-wise discontinuities of the refractive index profile. The results show that the approximations reach O(h²) truncation error only asymptotically for sufficiently small values of the mesh size.     

Fast convergence and higher-order mode calculation of optical waveguides: perturbation method with finite difference algorithm

We present an efficient algorithm for determining mode eigenvalues as well as field distributions of optical waveguides with two-dimensional transverse refractive index profile. The algorithm is devised with the analytical perturbation correction method combined with the finite difference approximation of Helmoltz's equation. The technique is simple and does not involve solving any eigenvalue equation or matrix formalism. The algorithm reduces abruptly the computation time required for the field convergence to mode, and can calculate any higher-order modes without the need of any pre-conditioning the field w.r.t. waveguide geometry, or calculation of previous order modes and/orthogonalization. The analysis can yield precisely both scalar and polarized modes. By applying it to waveguide problems whose solutions are otherwise known, the efficacy of the method has been established.     

Analysis of arbitrary index profile planar optical waveguides and multilayer nonlinear structures: a simple finite difference algorithm

we present here a simple numerical method to obtain the mode effective indices as well as field distributions of modes of any arbitrary profile planar optical waveguide. The method is based on the solutions of scalar and semivectorial Helmoltz’s equation by finite difference algorithm and devised with a field convergence technique. This approach is quite general and can be applied straightforwardly to calculate the guided as well as quasi- or leaky modes of any arbitrary planar structure without the need to solve any eigenvalue equation or any complex matrix formalism. Besides providing the ease of application, the algorithm is particularly useful for waveguides with any graded index profile or with irregular multilayered structure and multilayered waveguides with a localised arbitrary nonlinear film. The performance of our method is verified against typical problems with analytical solutions or having results known otherwise, and is shown to yield results with good accuracy.     

Analysis of electrooptically induced waveguides in strontium barium niobate and optical modulators

An electrooptically induced optical waveguide can be formed by applying a DC voltage between electrodes fabricated on the surface of a crystal with strong electrooptic coefficients (for example, strontium barium niobate). Calculations of the refractive index profile and the optical mode size for such waveguides are presented. Approximate solutions for the fundamental mode are used to analyse the power coupling between a single-mode step-profile fibre and these waveguides. The power transmission losses in the straight and S-shaped waveguides are estimated by using a finite-difference beam propagation method (FD-BPM). The calculations predict that a new kind of modulator/switch can be realized with the induced waveguides. The important characteristics of these devices are discussed and compared with LiNbO₃ optical waveguide modulators.     

A taper coupler for integrated optics formed by ion-exchange under a non-uniform electric field

A method for the formation of gradient index tapered optical waveguides with predicted shape is proposed and described analytically. The method is applicable for structures fabricated by electric-field assisted ion-exchange in glass. The technique is based on the control of the taper shape by applying a non-uniform electrical field during the waveguide formation process. For experimental verification optical waveguides with tapers were formed by electrically induced migration of K⁺ ions. An optical coupler based on such a tapered waveguide has demonstrated 1.2° outcoupled beam divergence. The method proposed can be used to form different integrated optical elements, e.g. mode filters, horn structures etc.     

Fibre mode expansion for 2-D waveguide calculations

A numerical method is introduced which is devised to calculate the lower-order modes of optical waveguides with two-dimensional transverse index profiles embedded in a homogeneous cladding medium. We propose a method based on a series expansion using guided modes of an optical fibre. To test the method, it is applied to a three-layer optical-fibre and the results are compared with the exact analytical solutions. The error in the calculation of the modal effective indices is found to be less than 10−6. This method allows the modelling of symmetric 2 × 2 completely fused fibre couplers. Effective indices and polarization corrections are compared with results obtained by the field correction method. The adiabaticity criterion is calculated for these couplers. This revised version was published online in November 2006 with corrections to the Cover Date.     

Mode analysis of realistic optical waveguide structures and microstructured holey fibers by modal field evolution

In this article, we describe a useful technique for calculating modes of practical optical waveguides having two-dimensional arbitrary transverse refractive index profile. The method uses a finite difference platform for evaluating Helmholtz's equation in scalar and semivectorial forms through a field evolution algorithm. The method is straightforward, easy to handle and does not involve any complex analysis or matrix formulation. We tested the accuracy of our analysis approach by applying it on a large number of realistic waveguide problems having known results or results available in the literature. The formulation has facilitated us to study the modal properties, viz., field distribution, birefringence, dispersion and mode effective area, of a variety of practical two-dimensional structures namely, planar structure, coupler, semiconductor optical waveguides, optical fibers and arbitrary profile microstructured fibers which are uniquely important in photonics and guided-wave devices. The algorithm will therefore be very useful in designing and studying any arbitrary-structure waveguides, and to explore new geometry and properties.     

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.     

WKB analysis of guided and semileaky modes in graded-index anisotropic optical waveguides

WKB analysis of graded-index anisotropic optical waveguides is briefly presented. The angular dependencies of the propagation characteristics of both guided and semileaky modes are calculated numerically for the out-diffused and in-diffused LiNbO₃ waveguides with parabolic index profile. The results for the out-diffused waveguide are compared with those obtained from the rigorous theory and the applicability of the WKG method is discussed for semileaky mode loss calculations. The angular dependence of the propagation constants of a TiO₂-diffused LiNbO₃ waveguide is demonstrated experimentally, too.     

Research on the influence of fabrication parameters on the performance of buried ion-exchanged glass waveguides using the variational method

The variational method is proposed to analyze the influence of the fabrication parameters on the performance of buried K⁺–Na⁺ ion-exchanged Er³⁺–Yb³⁺ ions co-doped glass waveguide. The unknown parameters of the Hermite–Gaussian functions as the trial field distribution are determined based on the scalar variational principle. It is demonstrated that the results calculated in this paper agree with those measured in the experiment. The mode dimensions, the effective refractive index, and the overlap factor as the functions of the fabrication parameters are investigated. These results of the variational analysis are useful for the design and optimization of Er³⁺–Yb³⁺ ions co-doped waveguides.     

Circular Fourier analysis of modes in optical waveguides under critical and supercritical conditions

An extension of the circular Fourier analysis of the mode field is presented that makes it possible to obtain a successively correctable solution to the scalar waveguide problem posed for 3D optical waveguides. The critical frequencies of rectangular and elliptic waveguides, as well as the characteristics of leaky modes in microstructural optical fibers formed by elliptical air channels, are studied.     

基于变量变换伽辽金法光波导半矢量分析

基于变量变换伽辽金法(VTGM),建立了半矢量分析光波导本征值问题的数理模型;获得了矩形光波导、脊形光波导的本征模的模场分布及归一化传播常数,计算结果和已发表的结果非常接近.半矢量VTGM采用平面映射边界条件,避免了边界截断,又能反映模场的偏振特性,是一种较为理想的光子器件分析方法.     

Direct measurement of the intersubband electro-optic coefficient in asymmetric multiple quantum wells

An extremely large electro-optic coefficient in asymmetric quantum well systems has been inferred from optical rectification measurements. To date, however, a direct measurement of the electro-optic coefficient in these materials has not been performed. We present a method to directly measure the electro-optic coefficient in an asymmetric quantum well systemconsisting of 30 Å GaAs wells, 65 Å Al(0.20)Ga(0.80)As step barriers, and 500 Å Al(0.40)Ga(0.60)As barriers. The devices consist of short waveguides with asymmetric quantum well cores. The waveguide endfaces are polished to form Fabry–Perot cavities. Electrodes are deposited running parallel to the waveguides to apply electric fields. The waveguides are analyzed using a Fourier transform infrared spectrometer. Interference fringes are observed in the reflected spectra at room temperature. The fringe spacing yields the refractive index of the material, while plots of the refractive index versus applied electric field yield the electro-optic coefficient. Absorption measurements through these waveguides showed the characteristic polarization dependent intersubband absorption. Because the electrodes appeared to be Schottky like, a large voltage was applied across the device and the voltage across the quantum wells was estimated from the device leakage current and the current-voltage characteristics of a similar device structure after scaling for length, area, and doping. There may be some uncertainty in this approximation. The maximum applied electric field is estimated to be 2.3 kV cm⁻¹. No anomalous effects were observed in the Fabry–Perot fringe pattern in the presence of this electric field, indicating the electro-optic coefficient in these waveguides is most likely less than 0.9 nm V⁻¹over a wide wavelength range. Clearly, however, measurements need to be performed at higher electric fields to accurately measure the electro-optic coefficient.     

一种简单精准的渐变折射率分布光波导分析方法

渐变折射率分布的光波导分析对光波导器件的设计和研究至关重要, 近年来已提出了多种分析方法, 然而在简便性或准确性上都存在着不足. 为此, 提出了一种分析渐变折射率分布光波导的方法, 能够结合现有的Wentzel-Kramers-Brillouin近似法和离散化的波动方程, 构建模场分布, 再结合变分运算方程和修正的模式本征方程, 计算出较为精确的有效折射率. 与其他分析方法相比, 该方法较为简单, 而且有一定的精度.     

Bragg光纤包层制作误差对导模特性的影响

对Bragg光纤中包层制造误差对导模特性的影响进行了深入的分析和研究,给出了一般N层波导中模式场标量波动方程求解的传输矩阵方法,并将其应用在Bragg光纤模式场的分析当中。分别研究了Bragg包层存在折射率误差和厚度误差时对光纤基模模式场的有效折射率、模场半径和芯区功率占比的影响,并利用统计的方法分析了不同制作误差时导模特性的相对变化。折射率误差在1‰的水平时,基模有效折射率、模场半径和芯区功率占比的方差水平为1.55×10~(-4)、1.04×10~(-1)和1.79×10~(-2);厚度误差在2%的水平时模式场参量的方差水平分别为3.61×10~(-5)、1.84×10~(-2)和2.35×10~(-3)。     

离子交换KTP晶体光波导及其各向异性波导性能研究

采用离子交换法制备了Ｒｂ：ＫＴＰ晶体光波导，在不同波长的耦合光下测量了ＴＥ和ＴＭ偏振波导模式的有效折射率，拟合了波导层折射率的余误差函数分布，证明了波导折射率增量的各向异性，给出了其色散关系；最后计算并讨论了Ｒｂ：ＫＴＰ波导的扩散特性。     

Determination of refractive index profiles in diffused optical waveguides

An improved method for the reconstruction of diffused waveguide index profiles from experimental mode spectra is proposed. The method uses a parabolic approximation between successive points. It is tested on the example of an exponential profile, and applied to the reconstruction of index profiles in Ag⁺ ion-exchanged waveguides from measured mode spectra. This method appears to work well also in the cases of very few mode wave-guides.     

Optical and microwave analysis of mushroom-type waveguides for traveling wave electroabsorption modulators based on asymmetric intra-step-barrier coupled double strained quantum wells by full-vectorial method

The finite difference method is exploited for a full-vectorial analysis of mushroom-type waveguides for traveling wave electroabsorption modulators (TWEAM) based on asymmetric intra-step-barrier coupled double strained quantum wells (AICD-SQW). In this analysis, the discontinuities of the normal components of the electric field across abrupt dielectric interfaces which are known as the limitations of scalar and semivectorial approximation methods are considered. The optical field distributions in mushroom-type TWEAM based on AICD-SQW and conventional ridge-type TWEAM of the same active region for 1.55 μm operation are presented. The important parameters in the high-frequency TWEAM design such as optical effective index which defines optical velocity and transverse mode confinement factor are calculated. Then, the transmission line microwave properties (microwave index, microwave loss, and characteristic impedance) of TWEAMs are obtained. The modulation response of mushroom-type TWEAM is calculated using circuit model by considering interaction between microwave and optical fields in waveguide and compared with conventional ridge-type TWEAM. It is found that increasing the width of p-cladding layer with the same i-layer to reduce the resistance in p-i-n mushroom-type waveguide of TWEAM based on AICD-SQW can improve the microwave propagation loss and thus the high-speed electro-optical response.     

Fabrication and characterization of polyethylene terephatate optical waveguides

In this paper, the fabrication and characterization of polyethylene terephatate (PET) thin film integrated optical waveguide is presented. Dip-coating technique has been used to fabricate the optical waveguide using PET. Various waveguide properties, such as mode effective refractive index, optical loss, dispersion and waveguide thickness as a function of temperature of waveguides have been reported. The optical loss are found to be <1 dB cm^–1.