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.     

Analytical evaluation of chromatic dispersion in photonic crystal fibers

We present a two-dimensional modal approach for the evaluation, in an analytical manner, of chromatic dispersion in any kind of optical fiber. It combines an iterative Fourier technique to compute the propagation constant at any fixed wavelength and an analytical procedure to calculate its derivatives. The proposed formulation takes into account the effective anisotropy of the interfaces and allows us to deal with microstructured fibers, in general, and specifically with realistic photonic crystal fibers (PCFs), including arbitrary spatial refractive-index distributions of dispersive and absorbing materials. This fast and accurate numerical technique is extremely useful for both analysis and design. We show some results of analysis of PCFs with high anisotropy, and we also describe PCFs with new dispersive properties.     

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.     

Construction of Two-Dimensional Waveguides in Insulating Optical Materials by Means of Ion Beam Implantation for Photonic Applications: Fabrication Methods and Research Progress

Ion beam implantation has been used not only to modify some properties of the bulk materials but also to construct waveguide structures in many optical substrates by accurate control the refractive index in selected regions. This paper reviews the recent development of ion beam implantations on fabrication of two-dimensional (2D) optical waveguides, i.e., in cases of channel or ridge configurations, in diverse insulating optical materials by giving detailed fabrication methods and research progress obtained to date. Future prospects of practical applications in photonics are also discussed briefly. Another aim of this work is to show the challenging task of this field, that is, to create practical 2D waveguide devices applicable in any existing insulating optical materials.     

Arbitrary laser beam propagation in free space

The propagation of arbitrary laser beams in free space is examined. For this purpose, starting with an incident field of arbitrary field distribution, the intensity at the receiver plane is formulated via Huygens Fresnel diffraction integral. Arbitrary source field profile is produced by decomposing the source into incremental areas (pixels). The received field through the propagation in free space is found by superposing the contributions from all source incremental areas. The proposed method enables us to evaluate the received intensity originating from any type of source field. Using the arbitrary beam excitation, intensity of various laser beams such as cos-Gaussian, cosh-Gaussian, general type beams are checked to be consistent with the already existing results in literature, and the received intensity distributions are obtained for some original arbitrary beam field profiles. Our received intensity formulation for the arbitrary source field profiles presented in this paper can find application in optics communication links, reflection from rough surfaces, optical cryptography and optical imaging systems.     

多模干涉耦合器一般成像位置分析

自映像效应是多模波导的一个重要特性,利用多模波导自映像效应制成的多模干涉耦合器被广泛地应用于集成光学回路中。然而根据以往的文献要得出多模干涉耦合器的成像位置要经过一系列复杂的计算。根据多模干涉耦合器的自映像原理,对多模干涉耦合器的成像位置进行分析。成像位置与输入场位置和位置数密切相关,给出了成像位置的解析表达式,得出奇数、偶数的位置数的成像位置分别对应相同,并且奇、偶数的位置数所对应的成像位置之和为多模波导宽度的成像规律。用导模传输分析法验证了表达式和成像规律的正确性。根据成像位置的表达式和成像规律可以直接得出任意输入场位置和位置数所对应的成像位置。为一维和二维限制的多模干涉耦合器的设计提供了理论基础。     

(Al)GaAs/AlGaAs周期光波导中的等幅基模

本文应用周期介质光波导的模场分布函数和特征方程分析了(Al)GaAs/AlGaAs周期光波导中结构参量及介质折射率对模场分布形状的影响,研究了波导中形成等幅基模的条件,讨论了相应参量的优化选择。     

Grating-assisted coupling of optical fibers and photonic crystal waveguides

We propose and analyze a highly efficient method of coupling light from optical fibers to two-dimensional photonic crystal waveguides. Efficient coupling is achieved by positioning of a tapered fiber parallel to the linear defect, where the photonic crystal's cladding functions as a grating coupler and provides field confinement as well. Numerical simulations indicate that better than 90% transmission is possible with a full width at half-magnitude bandwidth of 12nm. It is shown that one can increase the bandwidth by increasing the field overlap between the two waveguides.     

光纤传输特性的理论计算

介绍了具有任意折射率分布的光纤传输特性参量的计算方法,利用这种方法计算并分析了三种不同折射率分布的光纤的归一化传播常量、色散和模场直径等传输特性参量.     

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.     

用科尔-霍普夫变换法求非均匀光波导导模本征方程的严格解

采用科尔-霍普夫(Cole-Hopf)变换法，将渐变折射率波导导模的本征值方程变换为里卡提(Riccati)方程，通过较简洁的数学演算导出导模的传播常量与模场分布的解析解，给出了平方律分布、对称爱泼斯坦(Epstein)分布、爱泼斯坦层的平板波导与平方律分布圆光纤4种折射率分布的计算公式。     

Circular grating coupler for creating focused azimuthally and radially polarized beams

We show a planar optical circuit design that takes light from an input waveguide and creates a focused azimuthally or radially polarized beam emanating from the surface of the substrate. It is implemented in silicon-on-insulator waveguides and does not require any external components to focus the beam. The focal spot size can be subwavelength and is potentially useful for lithography, imaging, optical data storage, optical trapping, optical excitation of molecules, or coupling to optical fibers.     

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.     

Numerical solution of nonlocal hydrodynamic Drude model for arbitrary shaped nano-plasmonic structures using Nédélec finite elements

Nonlocal material response distinctively changes the optical properties of nano-plasmonic scatterers and waveguides. It is described by the nonlocal hydrodynamic Drude model, which – in frequency domain – is given by a coupled system of equations for the electric field and an additional polarization current of the electron gas modeled analogous to a hydrodynamic flow. Recent attempt to simulate such nonlocal model using the finite difference time domain method encountered difficulties in dealing with the grad–div operator appearing in the governing equation of the hydrodynamic current. Therefore, in these studies the model has been simplified with the curl-free hydrodynamic current approximation; but this causes spurious resonances. In this paper we present a rigorous weak formulation in the Sobolev spaces H(curl) for the electric field and H(div) for the hydrodynamic current, which directly leads to a consistent discretization based on Nédélec’s finite element spaces. Comparisons with the Mie theory results agree well. We also demonstrate the capability of the method to handle any arbitrary shaped scatterer.     

光波导等参元分析

本文提出一套用于分析任意介质光波导的等参元算法,它基于严格的E_z-H_z变分公式,采用八节点曲边四边形单元。这种类型的等参元分析法尤其适用于研究各种具有曲边边界的复杂光波导和光波导器件。作为例子,文中把这一方法用于分析阶跃型光纤,渐变型光纤、各向同性以及各向异性光纤耦合器。     

掩埋型离子交换玻璃光波导的变分分析

用变分法对离子交换法制备的掺铒光波导的传播特性进行了分析,推导出了适用于掩埋型离子交换玻璃沟道光波导中场分布传播常量的变分表达式,构建了场分布的厄米-高斯型试探解,在两种不同实验条件下,采用变分法确定了试探解中的待定参量,获得光波导中的场分布,利用传播常量的变分公式和已确定的场分布计算得到了传播常量和有效折射率。计算数据表明:导波区域的有效折射率稍稍地大于限制层的折射率,说明离子交换法制备的波导器件是弱波导;高的辅助退火电场强度和适当的退火时间下,所制备的光波导可以支持更多光模式的传输。该方法计算过程简洁、快速,计算结果与实验结果吻合。     

Optimum intermediate fibers for reducing interconnection loss: exact solution

We derive an exact analytical solution for a transmission line of N single-mode intermediate optical fibers that minimize the interconnection loss between any two dissimilar fiber modes that are well described by that paraxial scalar wave equation. Our solution shows that N optimum intermediate fibers reduce the original interconnection loss by a factor of least 1/(N+1) and that the total interconnection loss is only a function of N and the original direct interconnection loss. Our solution is not restricted to axisymmetric fibers or mode fields and therefore could be useful for reducing the interconnection loss between microstructured optical fibers, between certain slab waveguides, or between fibers and optical sources or detectors.     

Dispersion characteristics of asymmetric coupled anisotropic dielectric waveguides using FDFD

The formulation is developed in the frequency domain and the finite difference method is used for the numerical solution of the scalar wave equation, written in terms of the transverse components of the magnetic field. As a result a conventional eigenvalue problem is obtained without the presence of spurious modes due to the implicit inclusion of the divergence of the magnetic field equal to zero. The formulation is developed to include biaxial anisotropic dielectrics with an index profile varying arbitrarily in the cross section of the waveguide under analysis. This formulation is then applied to the analysis of the influence on the dispersion characteristics of the dimensions of asymmetric coupled rectangular uniaxial anisotropic dielectric waveguides. As expected, the reduction of the height or the width of one of the rectangular dielectric waveguides causes the dispersion curves to move towards higher frequencies.     

光纤模式分析的伽辽金有限元模型

提出了采用二阶吸收边界条件的全矢量平面伽辽金有限元模型,用于分析任意横截面形状和各向异性折射率分布的光纤的传导模式和泄漏模式,能精确求出各模式传输常量的实部和虚部以及模场分布,既不出现伪解,又不漏解.推导了各向异性介质全矢量耦合波动方程及其变分形式,给出了基于结点的二阶三角形单元的离散公式和单元矩阵,成功的将二阶吸收边界条件加入外边界二次线性单元的离散公式.计算表明采用该模型分析光子晶体光纤模式有效折射率与采用多极方法和基于离散函数展开的有限差分法所得结果吻合很好,采用二阶吸收边界条件计算限制损耗比一阶吸收边界条件结果精确.     

Numerical calculations of modes of optical waveguides with two-dimensional refractive index profiles by a field correction method

Abstact We describe a numerical method devised to calculate the lower-order modes of optical waveguides with two-dimensional transverse index profiles. This method is based on finite-difference approximation of the scalar wave equation and the minimization of the error in the field shape of a mode. The field correction method can be applied to waveguides with large index steps and generates the mode's scalar propagation constant and its field. To test the field correction method, it is applied to a three-layer optical fibre and the results are compared with exact solutions. The error in the calculations of the modal effective indices is found to be of the order of 10^-6. Example mode calculations are given for fused couplers and rib waveguides.